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+[    {        "title": "2303.15085v1.Temperature_dependent_study_of_the_spin_dynamics_of_coupled_Y__3_Fe__5_O___12___Gd__3_Fe__5_O___12___Pt_trilayers.pdf",        "content": "Temperature dependent study of the spin dynamics of coupled\nY3Fe5O12/Gd 3Fe5O12/Pt trilayers\nFelix Fuhrmann,\u0003Sven Becker, Akashdeep Akashdeep, and Gerhard Jakob\nInstitute of Physics, University of Mainz,\nStaudingerweg 7, Mainz 55128, Germany\nZengyao Ren\nGraduate School of Excellence \\Materials Science in Mainz\" (MAINZ),\nStaudingerweg 9, Mainz 55128, Germany and\nInstitute of Physics, University of Mainz,\nStaudingerweg 7, Mainz 55128, Germany\nMathias Weiler\nFachbereich Physik and Landesforschungszentrum OPTIMAS,\nRheinland-Pf alzische Technische Universit at\nKaiserslautern-Landau, 67663 Kaiserslautern, Germany\nMathias Kl aui\nInstitute of Physics, University of Mainz,\nStaudingerweg 7, Mainz 55128, Germany\nGraduate School of Excellence \\Materials Science in Mainz\" (MAINZ),\nStaudingerweg 9, Mainz 55128, Germany and\nCenter for Quantum Spintronics, Norwegian University\nof Science and Technology, Trondheim 7491, Norway\n(Dated: March 28, 2023)\n1arXiv:2303.15085v1  [cond-mat.mtrl-sci]  27 Mar 2023Abstract\nIn this study, we investigate the dynamic response of a Y3Fe5O12(YIG)/ Gd3Fe5O12(GdIG)/ Pt\ntrilayer system by measurements of the ferromagnetic resonance (FMR) and the resulting pumped\nspin current detected by the inverse spin Hall e\u000bect. This trilayer system o\u000bers the unique op-\nportunity to investigate the spin dynamics of the ferrimagnetic GdIG, close to its compensation\ntemperature. We show that our trilayer acts as a highly tunable spin current source. Our exper-\nimental results are corroborated by micro-magnetic simulations. As the detected spin current in\nthe top Pt layer is distinctly dominated by the GdIG layer, this gives the unique opportunity to\ninvestigate the excitation and dynamic properties of GdIG while comparing it to the broadband\nFMR absorption spectrum of the heterostructure.\nINTRODUCTION\nIn recent years, the \feld of magnonics has been growing. Encoding the information by\nspin angular momentum instead of moving charge carriers can potentially decrease energy\nconsumption [1, 2]. This fuels interest in developing magnonic devices, which can be used for\nmagnon logic operations and o\u000ber potentially increased speed and lower power consumption\n[1, 3]. Rare earth garnets like Y 3Fe5O12(YIG) o\u000ber a unique platform with long-distance\nmagnon propagation, enabled by its low Gilbert damping constant of down to \u000b\u001910\u00005[4{7].\nGd3Fe5O12(GdIG) is a compensated ferrimagnetic rare earth garnet with a temperature-\ndependent net magnetization that vanishes at the magnetic moment compensation temper-\natureTComp\u0019295 K [8]. Heterostructures of these materials provide an interesting static\nmagnetic system and allow to study the spin dynamics of the coupled heterostructure [9{\n11]. Currently, the study of antiferromagnets is also an active research area, as it promises\nmaterials with resilience against external magnetic \felds, long-distance spin transport and\nnaturally high resonance frequencies [12, 13]. Antiferromagnet-ferromagnet heterostructures\n[14] and ferrimagnetic systems, especially close to their compensation temperature, provide\nan interesting platform to study antiferromagnetic (and antiferromagnetically coupled) spin\ndynamics with more accessible magnetic properties [15{17]. Until now, the investigation of\nthe dynamics of ferrimagnets close to their compensation temperature has been challenging\n[18, 19]. We show that the coupling to a second layer can be used to facilitate such studies.\nIn this study, we experimentally investigate a YIG/GdIG/Pt thin-\flm heterostructure\n2schematically depicted in Fig. 1 a). We observe a strong impact of the magnetic con\fgu-\nration of our heterostructure on spin dynamics, spin pumping and spin Seebeck e\u000bect. We\nshow that the generated spin current originates in the GdIG layer, which gives us the unique\nopportunity to investigate the GdIG spin dynamics, aided by the coupling to the YIG layer,\nclose to the compensation temperature. This temperature range is usually more di\u000ecult to\nstudy because of the diverging linewidth of single GdIG layers at TComp [18]. Furthermore,\nwe study the spin current generation in our system, which is tunable by temperature, exter-\nnal \feld and relative layer thickness [9]. We drive the ferromagnetic resonance (FMR) modes\nof our heterostructure and measure the spin current which is pumped across the GdIG/Pt\ninterface when the resonance condition is satis\fed [20, 21]. This spin current, resulting from\nthe spin pumping (SP), is detected by means of the inverse spin Hall e\u000bect (iSHE) in the\nPt top layer [22] in the experimental con\fguration sketched in Fig 1 b). We gain more\ninformation about the switching behavior of our GdIG layer by observing the spin Seebeck\ne\u000bect (SSE) [23]. The SSE measurements are performed by applying an out-of-plane thermal\ngradient as depicted in Fig. 1 c). This geometry is referred to in literature as the longitudi-\nnal spin Seebeck e\u000bect (LSSE) [24]. We compare these results with SSE measurements, in\nwhich the gradient is generated by microwave heating during the SP measurements [25]. The\nmicrowave-induced SSE is helpful as a measure to determine the switching of the top GdIG\nlayer during the SP measurement itself, and is less susceptible to temperature mismatch\ncompared to remounting the sample in another setup.\nSAMPLE CHARACTERISTICS\nThe investigated sample is a trilayer of YIG, GdIG, and Pt, grown on a GGG (001)\nsubstrate. The thicknesses of YIG and GdIG are chosen to be 36 nm and 30 nm respectively.\nThe growth of the sample via pulsed laser deposition was optimized and the coupling between\nYIG and GdIG moments was investigated in Ref. [9].\nThe relative alignment of the YIG and GdIG layer magnetizations was investigated pre-\nviously by SQUID magnetometry and spin Hall magnetoresistance [9, 10]. With this in-\nformation and the (microwave-induced) SSE measurements, we can determine the relative\nalignment in our sample. Figure 1 d) shows an illustration of the switching \feld (and thus\nrelative alignment) versus temperature for the YIG 36 nm/GdIG 30 nm/Pt 4 nm sample.\n3Figure 1. a) Illustration of the YIG / GdIG / Pt trilayer without the GGG substrate. The spin\ncurrent JSwhich is generated potentially by both, the YIG and GdIG layers, is converted into\na charge current JCin the Pt layer via the iSHE. b) Illustration of the sample placement on a\nCPW. The sample is connected with two thin wires, left and right. The voltage ViSHE, generated\nin the Pt layer of the sample is measured by a Lock-In technique. c) Illustration of the SSE sample\nstack with a constant temperature gradient. The temperature gradient is established between the\nPt-heater on top of the sample and the copper piece (in contact with the VTI of the cryostat).\nThe gradient is estimated by comparing the resistance of the Pt-heater and Pt-sensor below the\nsample. The zoomed section illustrates the iSHE in the Pt after a spin current is channeled into\nthe Pt top layer. d) Illustration of the YIG / GdIG / Pt trilayer alignment and switching \feld\ndependent on temperature. Orange arrows depict the Gd sublattice, and the red (purple) arrows\nillustrate the direction of the combined Fe-sublattices of YIG (GdIG). The HZeeline indicates the\nrough temperature dependence of the magnetic \feld necessary to switch the respective layer.\nFor temperatures below the bilayer compensation temperature TC;B, the net moment of the\nGdIG layer Mnet;GdIG is larger than the one of the YIG layer Mnet;YIG. AtTC;B, the two\nlayers have the same net moment, which are antiferromagnetically coupled via the Fe-Fe\nsublattices [9, 10] and thus compensate each other. For temperatures above TC;Band below\nTC;GdIG (the compensation temperature of single layer GdIG), the net moment of the YIG\n4layerMnet;YIGis larger than that of the GdIG layer Mnet;GdIG. Under small external ap-\nplied in-plane magnetic \felds, the net magnetization is parallel to the applied \feld while the\nindividual layer magnetizations are antiparallel. For lager external magnetic \felds, the Zee-\nman energy exceeds the exchange coupling, and the layer with the smaller net moment also\nswitches so that both magnetizations are parallel with the external magnetic \feld. While\nthe net moments Mnet;GdIG andMnet;YIGare antiferromagnetically coupled below TC;GdIG,\nthe two layer magnetizations Mnet;GdIG andMnet;YIGare always parallel above TC;GdIG. The\nresults of this trilayer structure are compared to the FMR data of a single GdIG layer of a\ncomparable thickness of 30 nm and a YIG 36 nm/Pt 4 nm bilayer \flm.\nRESULTS AND DISCUSSION\nFerromagnetic resonance absorption\nTo obtain the colormaps in Fig. 2 a) we use a Vector Network Analyzer (VNA) to measure\nthe microwave absorption of the sample. In this case, the frequency is swept by the VNA\nfor each external magnetic \feld step \u00160H, resulting in a broadband FMR measurement\n(bbFMR) [26]. The obtained raw data is then processed by the derivative Divide (dD)\nalgorithm following Maier-Flaig et al. [27].\ndDS21=S21(!;H 0+ \u0001H\u0006)\u0000S21(!;H 0\u0000\u0001H\u0006)\nS21(!;H 0)\u0001H\u0006(1)\n\u0019\u0000i!A0d\u001f\nd!(2)\nHere,dDS21is thus e\u000bectively the normalized derivative of the S21-parameter (transmission\nparameter) with respect to magnetic \feld [27].\nIn our FMR measurements, we compare VNA-FMR colormaps recorded at di\u000berent tem-\nperatures. We measured our single-layer YIG and GdIG \flms \frst, to compare them later\nto the features of our heterostructure (Fig. 2 a), b)).\nFor the heterostructure, the signal di\u000bers strongly in its shape and temperature depen-\ndence from those of the single layers. For a temperature of 50 K we observe two distinct\nmodes, one at lower and one at higher frequencies (Figure 2 a). Both modes do not be-\nhave as one would expect from the Kittel equation for in-plane applied external magnetic\n\felds for single layers. However, the linewidth and signal strength of the mode at higher\n5Figure 2. a) Broadband VNA-FMR measurement at 50 K. The raw data is processed by deriva-\ntive divide and an \u000bt-\flter. The resonance linewidth is extracted by \ftting to the derivative of\nthe susceptibility for N resonances [28]. b) The plot shows the linewidth at 50 K from the YIG\n36 nm/ Pt 4 nm and GdIG 30 nm samples (red and orange respectively) and the trilayer (blue) in\ncomparison.\nfrequencies (HF mode) is more compatible with the GdIG single layer (Fig. 2 b)). The\nincrease of the slope of the mode towards lower temperatures was also previously observed\nfor GdIG [29]. The lower mode (LF mode) resembles the behavior of the single YIG layer.\nHowever, for temperatures in which the two modes are close to each other, it becomes clear\nthat the behavior is increasingly complex for the bilayer system. To identify the features\nand investigate their origin in more detail, we complement the FMR data with the SP and\nSSE measurements.\n6Figure 3. a) Broadband VNA-FMR measurement at 200 K. The raw data is processed by\nderivative divide and an \u000bt-\flter, which \flters signals by frequency apart from an allowed window.\nThe resonance linewidth is extracted by \ftting to the derivative of susceptibility for N resonances\n[14, 28]. b) The plot shows the SSE measurement by microwave heating at 200 K of the trilayer. c)\nExtracted linewidth at 200 K from the trilayer and the YIG sample in comparison. d) Measurement\nofViSHE from SP at resonance with di\u000berent excitation frequency at 200 K. The background colors\nof the plot red (yellow and blue) refer to the ranges in which the GdIG layer is antiparallel (switching\nand parallel) to the magnetic \feld.\nSpin pumping at ferromagnetic resonance\nFurther inspection of the FMR spectrum and generation of spin currents was performed\nby the observation of the inverse spin Hall e\u000bect (iSHE) in the Pt top layer [21, 22]. At\nthe resonance condition, a spin current is pumped from the YIG/GdIG bilayer into the\nadjacent Pt layer. This generated spin current is dominated by the spin dynamics in the\nGdIG layer, which o\u000bers a chance to investigate the GdIG layer in further detail. While the\nFMR measurements are sensitive to both the GdIG and YIG spin dynamics, the SP and\n7SSE measurements are complementary, as they only re\rect the GdIG spin dynamics. The\ninvestigation of magnetic \feld sweeps, especially at the switching \felds of the respective\nlayers o\u000bers the opportunity to study the spin current origin. The sample is placed on top\nof the CPW with a thin insulating tape, such that there is no electrical contact from the\nCPW to the Pt \flm. The contacts are made by a thin copper wire and silver paste to \fx\nthe wire on the Pt layer.\nFigure 4. a) FMR spectrum of the micromagnetic simulation. The magnetization is estimated\nfrom literature to match a temperature of 250 K. The raw data is processed by derivative divide.\nb) DC-Spin Seebeck e\u000bect measurement, indicating the switching of the top GdIG layer above\n0:4 T. c) Extracted resonance frequency fresat 250 K for the trilayer and a single YIG layer in\ncomparison. d) Measurement of ViSHE from SP at resonance with di\u000berent excitation frequency at\n200 K. A constant o\u000bset of 2 µV is used to separate the signal at di\u000berent frequencies visibly in\nthe plot.\nThe alternating current is generated by an MW-sweeper, which is ampli\fed and fed into\nthe CPW, generating an alternating magnetic \feld at the sample. The alternating magnetic\n\feld is modulated in its amplitude by a frequency of 30 Hz (DC). The resulting voltage\n8ViSHE is measured in the Pt layer by a Lock-In technique with a Stanford Research (SR)\n830 with a SR-530 pre-ampli\fer, improving the signal-to-noise ratio and removing artifacts\nin the signal due to small temperature \ructuations inside the cryostat. The measurements\nare performed as \feld sweeps with a constant frequency of the alternating \feld, at di\u000berent\ntemperatures. At a temperature of 200 K we can observe a sign-change of the generated\nsignal for a sweep from low to high magnetic \felds (lower and higher resonance frequencies)\n(Fig. 3 d). In this temperature range ( TC;B< T < T C;GdIG), the net moment of the YIG\nlayer is larger than the GdIG net magnetic moment (see Fig. 1 d)). Thus, at low \felds,\nthe YIG layer magnetization is parallel to the magnetic \feld, while the GdIG magnetization\nis oriented antiparallel. By comparing the sign change to the indicated ranges in Figure\n3 d), one can observe the voltage sign at resonance following the GdIG orientation. It is\nnoteworthy, that the FMR-spectrum shows a clear resonance line during the GdIG switching\n(compare Fig. 3 a) and b)). This suggests, that the YIG can still be excited during the\nGdIG switching, and the FMR spectrum is dominated by the YIG layer in this \feld range.\nThis is supported by the narrowing of the linewidth during the GdIG switching ( \u00190:2 T to\n0:5 T) (see 3 c)) From the SSE measurements it is clear that the switching is not abrupt,\nbut occurs over an extended \feld range. In this range, the linewidth is compatible with the\nlinewidth measured in the single YIG layer. It appears that we are not sensitive to a spin\ncurrent originating from YIG in this range, as there is no signal of the expected linewidth\nand sign observed in the ViSHEsignal. From SSE measurements we can estimate the 50 %\nswitching \feld value. For GdIG switched by 50 % (at \u00190:3 T) we do not see any distinct\npeak, supporting the assumption that there is no signi\fcant transmitted spin current from\nthe YIG. The sign change of the voltage signal can be explained easily by a switching of the\nGdIG layer magnetization. As the magnetization direction of the spin current source (the\nGdIG layer) inverts, the spin current polarization changes, leading to an inverted voltage\nViSHE(Fig. 3 d)).\nFor a temperature sweep across TC;GdIG one can also expect a sign change. The GdIG net\nmoment is inverted across the compensation temperature, as the combined moment of the\nFe sublattices exceeds the Gd sublattice moment. Thus, the net magnetization direction of\nGdIG is inverted, leading to an inversion of the iSHE voltage (see Fig. 5 a)). This again\nsignals, that the spin current is generated in the GdIG layer, as there is no sign-change of a\npossible YIG-spin current expected. This sign change from the SP-generated spin current is\n9compatible with the net magnetization orientation of the GdIG layer, which is inverted at\nthe compensation temperature. In contrast, the spin current generated from SSE depends\non the orientation of the Fe sublattices [23, 30, 31]. The orientation of the sublattices does\nnot change across the compensation temperature due to the coupling to the YIG layer (see\nFig. 1 d)), which explains the absence of this sign-change in the SSE measurement of the\nbilayer compared to previous studies [23, 31].\nFor GdIG single layers, at TC;GdIG one can observe a divergence of the linewidth of the\nFMR signal. We can extract a linewidth from Fig. 5 a), which clearly shows no such\ndivergence of the linewidth in our case. This divergence was studied previously and linked\nto the relation between net angular momentum and total angular moment of the material\n[18, 32]. However, the experimental investigation is di\u000ecult, due to the decrease in signal\nintensity and increasing linewidth. In our system, however, the relation between the net\nangular momentum and total angular momentum is shifted by the coupling to the YIG\nlayer. Thus, close to the compensation temperature, we still observe a signal originating\nfrom the GdIG layer (Fig. 5 a)). While the linewidth does change slightly across TC;GdIG, it\nis compatible with the damping estimated in Ref. [18], supporting the approach of extracting\nthe damping for ferrimagnetic materials close to their compensation temperature di\u000berently\ncompared to ferromagnets.\nSpin Seebeck e\u000bect measurement\nThe SSE measurements performed in the course of this study give important insight into\nthe static magnetic state of the system for the respective applied external magnetic \feld as\nit follows the magnetization of our GdIG top layer [9, 31].\nWe compare the SSE signal caused by microwave heating with the SSE signal from a\nsimple continuous temperature gradient, to verify the \feld dependence of the SSE signal\nwhich is measured during the application of the FMR. For the latter, the sample is clamped\nin between a cold sink, on which a Pt-stripe is attached, and a resistive Pt-heater [31]. When\napplying a current to the top Pt-heater, a temperature gradient is established perpendicular\nto the sample plane. The voltage which builds up on the sample is measured with an HP\n34420A nanovoltmeter. For the SSE signal during the microwave application, we measure the\nViSHEduring the \feld sweep. By applying a strong enough microwave power, we incidentally\n10Figure 5. a) ViSHE at \fxed resonance frequency of 6 :5 GHz for di\u000berent temperatures. The temper-\nature range reaches across the GdIG compensation temperature. The sign of VISHE switches at\nT\u0019270 K. b) The low-\feld step of ViSHE has it's origin in the spin Seebeck e\u000bect. The observed\nSSE signal shows no sign change in contrast to the SP signal over the same temperature range as\nin a). c) ViSHE of the SSE by mircowave heating for low temperatures [23, 30, 31]. d) Microwave\nabsorption (red curve) and ViSHE (blue curve) in comparison at 295 K and at 6 :5 GHz.\nheat the \flm, additionally inducing the spin Seebeck e\u000bect [33, 34]. This background signal\nis dependent on the magnetization direction of the GdIG layer, which gives us a direct\ncomparison between the switching state of the top layer and the FMR signal. This signal\nincludes the distinct peak of the SP at resonance (at \u00160H\u00190:15 T), see Figure 4 b). When\nwe compare it to the DC-SSE, we can see a good agreement in the \feld dependence (apart\nfrom the aforementioned SP-signal peak at FMR).\nAs observed in Ref. [9], the SSE signal is dominated by the GdIG top layer. This means,\nthat for the appropriate temperature range, we can see the switching during the \feld sweep\nof the GdIG layer. We do not observe a clear contribution of the YIG bottom layer, even if\na superposition of spin currents could be apparent in any of our SSE measurements [3, 35].\n11It appears, that the spin current generated in the YIG layer cannot penetrate through the\nGdIG layer due to its larger damping and the interface between YIG and GdIG.\nAfter comparison with the microwave-generated SSE, we can see good agreement with\nthe DC-SSE signal. For further investigations, we mainly use the microwave-generated SSE.\nThis enables a more meaningful comparison, as no remounting of the sample is needed.\nA good indication, that the observed background signal during the microwave application\nindeed stems from the SSE originating from the GdIG, is the sign change at low temper-\natures. The signal vanishes at low temperature, which agrees with the sign change of the\nSSE in GdIG due to a change of occupied magnon modes (Fig. 5 c))[23]. As the microwave\nheating power is dependent on the microwave frequency, for the applied constant microwave\npower, the frequency is kept constant at f\u00196:5 GHz while sweeping the magnetic \feld.\nSimulation\nThe data from our experiments are compared to micromagnetic simulations (Fig.4 a)).\nThese were conducted using OOMMF via the Ubermag meta-package [36] written for python.\nThe initial parameters are set by literature values. We assume the damping parameter \u000bYIG\nto be 3\u000210\u00003which we extracted from single layer YIG measurements and compatible with\nPLD-grown YIG samples [37]. The GdIG damping is estimated to be 5 \u000210\u00002from [18].\nThe magnetization of each layer is estimated from literature values for a set temperature\n[38]. An interfacial coupling between YIG and GdIG is assumed and estimated from the\nexchange constant, determined in [9]. The result of such a simulation is shown in Figure 4 a).\nThe overall signal shape is comparable to our FMR spectrum. However, in the simulation,\nwe can see a higher order mode, which we could not resolve in our experimental data. The\nexchange-driven mode [39, 40] with negative frequency to \feld dispersion seems also to be\nonly a weak signal. However, the e\u000bect of its anti-crossing with the main mode is captured\nby the measurement (Fig. 4 c)).\nCONCLUSION\nIn this study, we investigated the spin dynamics and spin currents in a trilayer of YIG\n/ GdIG / Pt. With complementary measurements of the spin current from SSE and SP at\n12FMR, and comparison to our micromagnetic simulations, we can explain the features of the\nFMR spectrum.\nWe even observed the GdIG FMR of this system close to the compensation temperature\nTC;GdIG. 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Chris Hammel* and Fengyuan Yang* \nDepartment of Physics, The Ohio State University, Columbus, OH, 43210, USA  \n†These authors made equal contributions to this work  \n*Emails: hammel@physics.osu.edu; fyyang@physics.osu.edu  \n \nWe observe  highly efficient dynamic spin injection from Y3Fe5O12 (YIG) into NiO, an \nantiferromagnetic ( AF) insulator, via strong coupling , and robust spin propagation in NiO up to \n100-nm thickness  mediated by its AF spin correlations .  Strikingl y, the insertion of a thin NiO \nlayer  between YIG and Pt significantly enhances the spin currents  driven into Pt, suggesting \nexceptionally high spin  transfer  efficiency at both YIG/ NiO and NiO /Pt interfaces .  This offers a \npowerful platform for studying AF spin pumping and AF dynamics as well as for exploration of \nspin manipulation in tailored structures comprising metallic and insulating ferromagnets, \nantiferromagnets and nonmagnetic materials.   \n \nPACS:  75.50.E e, 75.70.Cn , 76.50.+g,  81.15.Cd   \n  2 \n Spin transport in ferromagnetic (FM) and nonmagnetic materials (NM) has been \nextensively studied [1 -5].  Pure spin currents driven from FMs to metals or semiconductors by \nferromagnetic resonance (FMR) or thermal spin pumping have attracted especially intense \ninterest [6 -16].  Another important class of magnetic materials, antiferromagnets,  are not \nexpected to enable spin transport; thus, the possibility of spin transport by AF excitations  \nremains largely unexplored.  FMR spin pumping in FM/NM bilayers  relies on transfer of  angular \nmomentum from  the precessing FM magnetization to the conduction electrons in the NM to \ngenerate spin currents [ 6-12, 14, 15].  Insulating FMs are known [7] to support spin transport \nthrough magnon currents .  Simultaneous spin and magnon accumulation at a NM/FM -insulator \ninterface accompanied by the interconversion of spin current Js to magnon current Jm has been \npredicted [17, 18] .  AFs, both metallic and insulat ing, can also sustain propagating spin \nexcitations , potentially allow ing transport of spin current .  Our recent ly demonstrat ed growth of \nhigh-quality YIG thin films which enable  mV-level inverse spin Hall effect (ISHE) spin pum ping \nsignals [14, 15, 19] provides an effective  platform  for observation  of spin transport in AFs.  \nWe grow  20-nm YIG films on Gd 3Ga5O12 (111) substrates , followed by  deposition of \nNiO and Pt layers  using  off-axis sputtering  [14, 15, 19-24].  X-ray d iffraction (XRD) scan of one \nof our  YIG film s in Fig. 1(a) shows clear Laue oscillations, demonstrating its high crystalline \nquality.   Fig. 1 (b) show s an FMR derivative absorption spectrum for one (YIG -1) of the 20-nm \nYIG film s studied in this letter  taken at radio -frequency (rf) f = 9.65 GHz and power Prf = 0.2 \nmW with an in -plane magnetic field  (H), from which we obtain a  peak -to-peak linewidth ( H) of \n8.5 Oe [25].  Atomic force microscopy (AFM)  measurements  of a bare YIG film and a \nYIG/ NiO(20 nm) bilayer shown in Figs. 1 (c) and 1 (d) reveal  root-mean -square (rms) roughness \nof 0.197 and 0.100  nm, respectively, demonstrating  the smooth surfaces of both YIG and NiO.  3 \n Our s pin pumping measurements are performed at room temperature  on ~1 mm  5 mm \nsamples in an FMR cavity with a DC field applied along the short edge , as illustrated in Fig. \n1(e).  For YIG (20 nm) /Pt(5 nm)  bilayers a t YIG resonance, the  dynamical coupling between the  \nprecessing  YIG magnetization and the conduction electrons in Pt produces  a pure spin current, \nJs, into Pt, which is converted to a net charge current  via the ISHE  [8-10, 26], resulting in an \nISHE voltage (VISHE) along the  length of the samples .  Figure 1(f) shows VISHE vs. H - Hres \nspectra , where Hres is the FMR resonance field, at H = 90 and 270 (two in -plane fields), which \nexhibits an ISHE voltage of 3.04 mV , the highest value we have observed [14 , 15].   \nIn this letter , we focus on three series of YIG/Pt bilayers and YIG/NiO/Pt trilayers \nprepared from three 20-nm YIG films labeled YIG -1, YIG -2 and YIG -3 with FMR linewidth s \n(bare YIG)  of 8.5, 1 0.4 and 2 2.6 Oe, respectively.   The -2 XRD scan of a 100 -nm NiO film \ndeposited on GGG  (111) substrate shown in Figure 1(g) indicates that the NiO films are \npolycrystalline with a preferred orientation along <111>.  The top panels in Figs. 2(a)-2(c) show \nVISHE vs. H - Hres spectra for the three  YIG/Pt bilayer s at Prf = 200 mW , which give  VISHE = 3.04 \nmV , 6 04 V, and 14 6 V, respectively.  The three YIG/Pt bilayers are selected to have a wide \nrange of ISHE voltages due to t he difference in YIG  film/interface  quality .  \nTo characterize  spin transport  in AF insulator s, we insert a layer of NiO, an AF with a \nbulk Néel temperature over 500 K,  between YIG (20 nm)  and Pt (5 nm)  in all three YIG series.   \nThe insulating nature  of the NiO films is confirmed by electrical measurements.   The middle \npanels in Figs. 2a-2c shows VISHE vs H spectr a for the three series of  YIG/NiO(1  nm)/Pt trilayers .  \nStrikingly, we observe a significant enhancement , relative to Pt directly on YIG,  of the spin \npumping signal s in all three samples : VISHE = 4.71 mV (from 3.04 mV), 1.20 mV (from 604 V), \nand 1.03 mV (from 146 V), relative increases of 1.55, 1.99, and 7.05 for t he YIG -1, YIG -2, and 4 \n YIG-3 sample s, respectively.   Since the blocking temperatures ( Tb) of 1 - or 2-nm NiO films \nshould be below room temperature [ 27, 28 ] (Tb is expected to exceed 300 K at ~5 nm NiO \nthickness [2 9]), this indicates that the root of this enhancement of spin pumping efficiency in \nYIG/NiO/Pt trilayers lie s in the AF fluctuations of NiO [ 30]. \nThe dependence of the spin current injected into Pt on t he NiO thickness provides clues \nas to length scale, and hence the mechanism  underlying spin pumping  observed here .  Figure s \n3(a)-3(c) show semi -log plots of  the dependencies of VISHE on tNiO for the three series of trilayers .  \nWe observe three important features .  First, for thin NiO interlayers,  tNiO = 1 or 2 nm, the ISHE \nvoltages increase  with increasing  tNiO.  After peaking, t he spin pumping signals of the trilayers \nremain higher than the values of corresponding YIG/Pt bilayers for tNiO up to  5 nm for the YIG -1 \nand YIG -2 series and  up to  tNiO > 10 nm  for the YIG -3 series.   This is in notable contrast to the \nsuppression of VISHE by more than two orders of magnitude when a 1 -nm nonmagnetic insulator \nSrTiO 3 (STO) is inserted between YIG and Pt, as shown in Fig. 3(d) [15].   The enhanced ISHE \nvoltages suggest  that the overall  spin conver sion efficiency [5, 10, 12] of the entire  YIG/NiO/Pt  \ntrilayer  is higher than the YIG/Pt bilayer with direct contact  [14, 19], indicating that the YIG/NiO \nand NiO/Pt interfaces are exceptionally efficient in trans ferring spins.   At the YIG/NiO interface, \na strong, short -range exchange interaction [31] couples the FM  magnetization in YIG with the \nAF moments in NiO [2 9].  At YIG resonance, the precessing  YIG magnetization excites the AF \nmoments at the YIG/NiO interface.  The enhancement at tNiO  2 nm suggests that a prominent \nrole for AF spin fluctuations in the spin transfer process.  \nSecond, following this initial enhancement, VISHE decays exponentially in all three series \nof YIG/NiO( tNiO)/Pt trilayers , implying diffusive spin transport  in the AF  insulator NiO .  This  \npresumably proceeds by means of either magnons  (excitations of ordered AF spins when Tb is 5 \n above measurement temperature  Tm) or AF fluctuations (excitations of dynamic but AF \ncorrelated spins when Tb is below Tm).  Least -squares fits to 𝑉ISHE =𝑉ISHE(𝑡NiO=1 nm)e−𝑡NiO/𝜆 \nin the range 1 nm  tNiO  50 nm indicate diffusion lengths  = 8.8, 9.4, and 11 nm for the YIG-1, \nYIG-2, and YIG -3 series, respectively , as compared to  = 0.19 nm for the YIG/STO/Pt trilayers .  \nThe AF magnons or fluctuations in NiO carry the angular momentum across the NiO thickness \nto the NiO/Pt interface, where the angular momentum is transferred across the NiO/Pt interface \nto the conduction electrons of Pt, generating a spin current in Pt.   \nLastly, the decay of VISHE slow s at tNiO > 50 nm relative to thinner NiO.   The bottom \npanels in Figs. 2 (a), 2(b) and 2(c) show  the VISHE vs. H - Hres spectra for the three YIG/NiO(100 \nnm)/Pt trilayers which give VISHE = 1.85, 0.61 and 0.51 V, respectively.   The insulating nature  \nof YIG and NiO rule s out anisotropic magnetoresistance or anomalous Hall effect  [32].  \nMagnetic proximity effect in Pt is not expected given that Pt is on top of antiferromagnetic NiO  \n[33].  The slow decay in thick NiO suggests longer decay length in ordered AF.  \nMeanwhile, the YIG/NiO exchange coupling also induces extra damping in YIG which \nbroadens the FMR linewidth.  Figures 3(e) -3(g) show the NiO thickness dependence of H for \nthe three s eries of YIG/NiO/Pt trilayers, all of which exhibit an initial decrease in H, followed \nby an increases and eventual saturation at large NiO thickness.  This behavior can be understood \nas follows.  For very thin NiO (e.g., 1 or 2 nm), Tb is well below room temperature and the AF \nfluctuation induced extra damping on YIG is small.  However, an insulator as thin as 1 nm can \neffectively decouple YIG and Pt [see Fig. 3(g)] and greatly  reduce the spin pumping induced \nextra damping by Pt.  As a result, H decreases first in very thin NiO regime.  As NiO thickness \nincreases, AF correlation  becomes more robust and YIG/NiO exchange coupling grows stronger, \nwhich leads to an increase in damping and H.  This is in clear contrast with YIG/ SrTiO 3/Pt 6 \n trilayers in Fig. 3(g) in which H monotonically decreases before reaching saturation.    \nExchange coupling between a FM and an AF can potentially lead to exchange bias and \nenhanced coercivity ( Hc) [31].  Figure 4(a) shows the  room temperature in-plane magnetic \nhysteresis loops of a 20-nm YIG film and YIG/NiO (tNiO) bilayers with  tNiO = 2, 5, 10, 20, and 50 \nnm.  The bare YIG film exhibits a square hysteresis loop with a very small Hc of 0.36 Oe and a \nvery sharp magnetic switching with most of the reversal completed within 0.1 Oe, implying \nexceptionally high magnetic uniformity.  As tNiO increases, Hc continuously rises and reaches \n1.53 Oe for YIG/NiO(50 nm), as shown in the inset to Fig. 4(a).  We do not observe clear \nexchange bias in YIG/NiO bilayer (no exchange bias has been reported in YIG -based structures).  \nTo verify that  the observed spin transport  across NiO in YIG/NiO/Pt trilayers does not \narise from  spurious effects, we grow four different heterostructures on YIG -1 and measure the ir \nspin pumping signal s as shown in Fig. 4 (b).  The first sample , YIG/NiO(5  nm)/Cu(10  nm)/NiO(5  \nnm)/Pt(5  nm), in which we insert a 10 -nm Cu spacer in between two 5 -nm NiO layers, exhibits  \nVISHE = 1.25 V.  This value is three orders of magnitude smaller than the  value of  1.42 mV for \nYIG-1/NiO(10 nm)/Pt [Fig. 3(a)] , indicating that spin current can still propagate from YIG to Pt \nacross the three spacers , but the combined spin conductance of the three -layer/ four-interface \nsystem is much smaller than for the YIG/ NiO/Pt trilayer s.  Replac ing the 10 -nm Cu with a 5-nm \nSiO x layer  in control sample (2)  eliminates any  detectable spin pumping signal , demonstrating \nthat Cu can conduct spin current while Si Ox blocks spin flow.   The third control sample,  \nYIG/NiO(5  nm)/Cu(10  nm), shows no ISHE signal, confirming that the observed spin pumping \nsignal  for YIG/NiO/Cu/NiO/Pt  indeed come s from the ISHE  in Pt .  Lastly, the YIG/Cu(10 \nnm)/NiO(10 nm)/Pt  structure shows a small but clear ISHE signal of 0.20  V, indicating that \nwhile YIG/Cu(10 nm)/NiO(10 nm)/Pt can still propagate spins, the spin transfer efficiency is not 7 \n as high as that in YIG/NiO(5 nm)/Cu(10 nm)/NiO(5 nm)/Pt(5 nm) .  \nAltogether, t his suggests  the following multiple -stage spin conversion in YIG/NiO(5 \nnm)/Cu(10 nm)/NiO(5 nm)/Pt(5 nm) : 1) at the YIG/NiO interface, the precessing YIG \nmagnetization injects angular momentum into the first NiO , producing  AF excitations ; 2) the  AF \nexcitations  carry the angular  momentum to the first NiO/Cu interface, where they are  converted \nto a spin curr ent in Cu carried by the conduction electrons; 3) the spin current  in Cu propagates \nto the second Cu/NiO interface where it  is convert ed back to AF excitations  in the second NiO \nlayer ; 4) the  AF excitations in the second NiO layer transfer the angular momentum to the \ninterface with Pt, where they are  converted to  a spin current in Pt, resulting in an ISHE voltage.  \nFigure 4 (c) shows the FMR derivative absorption spectra of a bare YIG -1, a YIG-\n1/NiO(20 nm) and  a YIG-1/SiO x(20 n ym) bilayer  measure d at f = 9.65 GHz , which reveal  that a \n20-nm NiO significantly broadens  the linewidth while SiO x has essentially no effect on the YIG \nlinewidth .  This suggests that the AF ordering  or AF fluctuations  in NiO plays an important role \nin the damping of YIG.   Figure 4 (d) gives  the frequency dependenc ies of H for the three \nsamples shown in Fig. 4 (c), all of which exhibit  a linear relationship with frequency .  From least-\nsquares fit s to the data in Fig. 4 (d), we obtain  the Gilbert damping constant   = 5.9  10-4, 5.9  \n10-4, and 2.5   10-3 for YIG -1, YIG -1/SiOx, and YIG -1/NiO, respectively  [34].  The 20-nm NiO \nclearly enhances the damping in YIG while the damping in YIG/ SiO x is almost the same as in \nbare YIG.   This indicates  that the AF moments in NiO exchange couple  to the YIG \nmagnetization in a way similar to the exchange bias in FM/AF bilayers  [31], which causes \nadditional damping in the FM . \nIn summary, we report observation of spin transport  in AF insulator NiO  and significant  \nenhancement of spin pumping signals  with insertion of a thin NiO spacer between YIG and Pt .  8 \n The enhanced spin pumping  indicat es excellent  spin conver sion efficiency  at the YIG/NiO and \nNiO/Pt interfaces as well as robust  spin transport in  NiO mediated by AF magnons or AF \nfluctuations .  The magnitude of spin currents in NiO decreases exponentially with decay lengths \nof ~10  nm within 1 nm  tNiO  50 nm.   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Rev. B  84, 054423 (2011).  \n33. A. Hoffmann, M. R. Fitzsimmons, J. A. Dura, and C. F. Majkrzak , Phys. Rev. B  65, 024428 \n(2001 ).   \n34. S. S. Kalarickal, P. Krivosik, M. Z. Wu, C. E. Patton, M. L. Schneider, P. Kabos, T. J. Silva, \nand J. P. Nibarger, J. Appl. Phys.  99, 093909 (2006).   12 \n Figure captions:  \nFigure 1.   (a) A -2 XRD  scan of a 20 -nm YIG epitaxial film on G d3Ga5O12 (111) substrate  \nnear the YIG  (444) peak.  (b) A room -temperature FMR derivative absorption spectrum of a 20 -\nnm YIG film  (YIG -1) with an in -plane  DC magnetic  field and microwave power  Prf = 0.2 mW ; \nthe linewidth is 8.5 Oe.  AFM images of (c) a 20-nm bare YIG film and (d) a YIG/NiO(20 nm) \nbilayer over an area of 10 m  10 m, which exhibit  rms roughness of 0.197 and 0.100  nm, \nrespectively.  (c) Schematic of ISHE measurement  on YIG/Pt bilayer and YIG/Ni O/Pt trilayers . \n(d) VISHE vs. H – Hres spectra at θH = 90 and 270 (two opposite  in-plane fields) at  Prf = 200 mW \nfor a YIG-1/Pt(5 nm)  bilayer . (e) -2 XRD scan of a 100 -nm NiO film on GGG(111) substrate.  \nFigure 2.   VISHE vs. H - Hres spectra of YIG(20 nm)/NiO( tNiO)/Pt(5 nm) heterostructures  at Prf = \n200 mW using (a) YIG-1, (b) YIG-2, and (c) YIG-3 with differing characteristics .  The top, \nmiddle, and bottom panels are for YIG/Pt bilayers, YIG/NiO(1 nm)/Pt, and YIG/NiO(100 nm)/Pt \ntrilayers, respectively.  \nFigure 3.   Semi -log plots of the NiO thickness dependencies of the ISHE voltages for YIG(20 \nnm)/NiO( tNiO)/Pt(5 nm) trilayers  using (a) YIG-1, (b) YIG-2, and (c) YIG-3. Inset: VISHE as a \nfunction of NiO thickness from 0 to 10 0 nm for the three series of samples , where the horizontal \ndashed lines mark the values of VISHE for the YIG/Pt bilayers . (d) Semi -log plot of VISHE as a \nfunction of  the SrTiO 3 barrier thickness for YIG /SrTiO 3/Pt trilayers . FMR linewidth s as a \nfunction of NiO thickness for (e) YIG -1, (f) YIG -2, and (g) YIG -3 based trilayers, and (h) as a \nfunction of SrTiO 3 thickness in YIG /SrTiO 3/Pt trilayers.  \nFigure 4.  (a) Room temperature magnetic hysteresis loops of a single YIG(20 nm) film and \nYIG/NiO bilayers with NiO thicknesses of 2, 5, 10, 20, and 50 nm, which give coerci vities of \n0.36, 0.42, 0.67, 0.93 , 1.31, and 1.53 Oe, respectively.  The inset shows the NiO thickness 13 \n dependence of coercivity.   (b) VISHE vs. H - Hres spectra of four heterostructures, including: (1) \nYIG(20 nm) /NiO (5 nm) /Cu(10 nm) /NiO (5 nm) /Pt(5 nm)  (black) , (2) YIG/NiO(5 nm)/SiO x(5 \nnm)/NiO(5 nm)/Pt(5 nm) (purple), (3) YIG/NiO(5 nm)/Cu(10 nm) (blue) , and (4)  YIG/Cu(10 \nnm)/NiO(10 nm)/Pt(5 nm)  (red) , taken at Prf = 200 mW.  The spectra are offset for clarity.   (c) \nFMR derivative absorption spectra  taken at f = 9.65 GHz  and (d) frequency dependenc ies of \nFMR linewidth of  a bare YIG -1 film, a YIG-1/NiO(20  nm) and a YIG-1/SiO x(20 nm) bilayer .  \n  14 \n  \n \nFigure 1 . \n  \n100103106\n50 51 52Intensity (c/s)\n2 (deg)(a) YIG(20 nm)\n-505\n2550 2600 2650 2700dIFMR /dH (a.u.)\n H (Oe)H e (b)\nYIG-1\n(d) (c)\n103104105106\n20 40 60 80Intensity (counts)\n2 (deg)NiO(111)\nNiO(222)GGG(444)GGG(222)(g)\n(e)\n-3-2-10123\n-100 0 100VISHE (mV)\nH - Hres (Oe)\nH = 270o(f)\nYIG-1/Pt15 \n  \n \nFigure 2 . \n  \n0100020003000YIG-1/Pt\n(a)\n0200040006000VISHE (V)YIG-1/NiO(1 nm)/Pt\n012\n-120 -60 060120\nH - Hres (Oe)YIG-1/NiO(100 nm)/Pt0200400600YIG-2/Pt\n(b)\n050010001500YIG-2/NiO(1 nm)/Pt\n00.20.40.6\n-120 -60 060120\nH - Hres (Oe)YIG-2/NiO(100 nm)/Pt050100150YIG-3/Pt\n(c)\n05001000YIG-3/NiO(1 nm)/Pt\n00.20.40.6\n-120 -60 060120\nH - Hres (Oe)YIG-3/NiO(100 nm)/Pt16 \n  \nFigure 3.   \n100101102103104VISHE (V)(a)\n nm\nYIG-1/NiO( tNiO)/Pt020004000\n0246810\ntNiO (nm)VISHE (V)\n100101102103VISHE (V)\n9.4 nm(b)\nYIG-2/NiO( tNiO)/Pt050010001500\n0246810\ntNiO (nm)VISHE (V)\n100101102103\n0 20 40 60 80 100VISHE (V)\ntNiO (nm)11 nm(c)\nYIG-3/NiO( tNiO)/Pt05001000\n0246810VISHE (V)\ntNiO (nm)\n101102103\n0 0.2 0.4 0.6 0.8 1VISHE (V)\n tSTO (nm)YIG/STO( tSTO)/Pt = 0.19 nm(d)1020H (Oe)\nYIG-1/NiO( tNiO)/Pt(e)\n102030\nYIG-2/NiO( tNiO)/Pt(f)H (Oe)\n0 20 40 60 80 100102030\ntNiO (nm)YIG-3/NiO( tNiO)/Pt(g)H (Oe)\n0510152025\n00.2 0.4 0.6 0.8 1H (Oe)\ntSTO (nm)YIG/STO( tSTO)/Pt(h)17 \n  \nFigure 4. \n \n-1.5-1-0.500.511.5\n-100 -50 0 50 100VISHE (V)\n H - Hres (Oe)(1) YIG/NiO/Cu/NiO/Pt\n(4) YIG/Cu/NiO/Pt(3) YIG/NiO/Cu(2) YIG/NiO/SiOx/NiO/Pt(b)\n-101\n-20-10 01020dIFMR /dH (a.u.)\nH - Hres (Oe)YIG-1\nYIG-1/NiO\nYIG-1/SiOx\n(c)\n0102030\n0 5 10 15 20H (Oe)\nf (GHz)YIG-1/NiO\nYIG-1/SiOx\nYIG-1(d)-101\n-4 -2 0 2 4\nH (Oe)M/Ms\nsingle YIG \nYIG/NiO(2 nm)\nYIG/NiO(5 nm)\nYIG/NiO(10 nm)\nYIG/NiO(20 nm)\nYIG/NiO(50 nm)(a)\n00.511.5\n0 20 40Hc (Oe)\ntNiO (nm)"    },    {        "title": "1412.7712v1.Surface_sensitivity_of_the_spin_Seebeck_effect.pdf",        "content": "Surface sensitivity of the spin Seebeck e\u000bect\nA. Aqeel,1I. J. Vera-Marun,1B. J. van Wees,1and T. T. M. Palstra1,a)\nZernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, 9747 AG Groningen,\nThe Netherlands\n(Dated: 12 October 2021)\nWe have investigated the in\ruence of the interface quality on the spin Seebeck e\u000bect (SSE) of the bilayer\nsystem yttrium iron garnet (YIG) \u0000platinum (Pt). The magnitude and shape of the SSE is strongly in\ruenced\nby mechanical treatment of the YIG single crystal surface. We observe that the saturation magnetic \feld\n(HSSE\nsat) for the SSE signal increases from 55.3 mT to 72.8 mT with mechanical treatment. The change in\nthe magnitude of HSSE\nsatcan be attributed to the presence of a perpendicular magnetic anisotropy due to the\ntreatment induced surface strain or shape anisotropy in the Pt/YIG system. Our results show that the SSE\nis a powerful tool to investigate magnetic anisotropy at the interface.\nI. INTRODUCTION\nThe discovery of the spin Seebeck e\u000bect (SSE)1in\ninsulators triggered the modern era of the \feld of\nspin caloritronics2. In insulators, instead of moving\ncharges, only spin excitations (magnons) drive the non-\nequilibrium spin currents. In the spin Seebeck e\u000bect, spin\ncurrents are thermally excited in a ferromagnet FM and\ndetected in a normal metal NM deposited on the FM. The\nbilayer NM/FM system in the SSE provides the oppor-\ntunity to separately tune the properties of both layers to\noptimize the magnitude and magnetic \feld dependence\nof the SSE e\u000bect. The platinum (Pt) and yttrium iron\ngarnet (YIG) bilayer system has attracted considerable\nattention for studying the spin Seebeck e\u000bect1,3{5and\nfor other spin dependent transport experiments1,6{13.\nPlatinum (Pt) has a large inverse spin Hall response14\nwhereas YIG is an ideal ferromagnetic insulator due to\nlow magnetic damping2,6,15and a large band gap16at\nroom temperature.\nThe origin of the spin Seebeck e\u000bect is commonly ex-\nplained by the di\u000berence in the magnon temperature\nin the FM and the phonon temperature in the NM,\n\u0001Tmp17,18. When the temperature gradient rTis ap-\nplied across the NM/FM system, it creates a \u0001 Tmpbased\non the thermal conductivities of the magnon and the\nphonon subsystems17. This \u0001Tmpinduces a spin cur-\nrent density at the interface which is detected in the nor-\nmal metal Pt by the inverse spin Hall e\u000bect (ISHE). The\nISHE signal depends on a scaling parameter, the interfa-\ncial SSE coe\u000ecient Ls, related to how e\u000eciently the spin\ncurrent density can be created across the interface under\na certain \u0001 Tmp. The resulting spin Seebeck signal scales\nlinearly with the length of the NM ( lPt), therefore for the\nPt/YIG system\nVISHE/lPt: Ls:rT (1)\nThe scaling parameter Lsis proportional to the real\npart of the spin mixing conductance g\"#\nrat the inter-\na)e-mail: t.t.m.palstra@rug.nlface. The spin mixing conductance g\"#\nrand therefore the\nSSE are very sensitive to the interface quality19. In re-\ncent years substantial e\u000bort has been made to improve\nthe spin mixing conductance on thin \flms of YIG19,20\nand bulk crystals16,21. Unlike thin \flms, bulk crystals\nneed an extra surface polishing step for the device fab-\nrication, due to the initial surface roughness. The pol-\nishing of the crystal surface can in\ruence the spin mix-\ning conductance in several ways. Apart from changing\nthe surface roughness, mechanical polishing can change\nthe magnetic structures at the interface by inducing a\nsmall perpendicular anisotropy at the surface layer of the\nYIG crystal22{24. However, the e\u000bect of polishing on the\nspin Seebeck e\u000bect (SSE) has not yet been systematically\nstudied. In this paper, we report the e\u000bect of mechan-\nical surface treatment of the YIG single crystals on the\nSSE e\u000bect. This systematic study reveals the surface\nsensitivity of the SSE and indicates new ways of surface\nmodi\fcation for improved thermoelectric e\u000eciency.\nII. EXPERIMENTAL TECHNIQUE\nIn this study, we use the longitudinal con\fguration3\nfor the spin Seebeck e\u000bect where the temperature gradi-\nent is applied across a NM/FM interface and parallel to\nthe spin current direction Js. In Fig. 1(a), we illustrate\nschematically the device con\fguration for measuring the\nSSE used in this study. The sample consists of a sin-\ngle crystal YIG slab and a Pt \flm sputtered on a (111)\nsurface of the YIG crystal. When an out-of-plane (along\nz-axis) temperature gradient is applied to the Pt/YIG\nstack, spin waves are thermally excited. The spin waves\ninject a spin current along the z-axis and polarize the\nspins in the Pt \flm close to the interface parallel to\nthe magnetization of the YIG crystal, as illustrated in\nFig. 1(b). Due to the strong spin-orbit coupling in the Pt-\n\flm, the spin polarization \u001bis converted to an electrical\nvoltageVISHE. The single crystals of YIG with the same\npurity were used in all measurements. The YIG crystals\nwere grown by the \roating zone method along the (111)\ndirection and commercially available from Crystal Sys-\ntems Corporation company, Hokuto, Yamanashi Japan .arXiv:1412.7712v1  [cond-mat.mtrl-sci]  22 Dec 20142\n(a)\n(c) (d)\n2 nm\n1.5 \n1.0 \n0.0 0.5 (b)\nM\nEISHE\nPt \nYIGJsMσ\ny\nxz\nθ\nHPt VISHE\nYIG\n    ∆ T\nMHeater (T+∆T) \nHeat sink (T) \nFIG. 1. (a) Device con\fguration of the longitudinal SSE\nwhererTrepresents the temperature gradient across the\nPt/YIG system. (b) Detection of spin current by the ISHE.\nThe orange arrows indicate the spin polarization \u001bat the in-\nterface of the Pt/YIG system. M, JSandEISHE represent the\nmagnetization of YIG, spatial direction of the thermally gen-\nerated spin current, and electric \feld induced by the ISHE,\nrespectively. \u0012represents the angle between the external mag-\nnetic \feld H in the x-y plane and the x axis. (c) AFM height\nimage of a single crystal YIG surface (20 x 20 \u0016m2) for sample\nS1. (d) a comparison between the magnetic \feld dependence\nofVISHE at \u0001T= 3.6 K for sample S1 and the magnetization\nM of the YIG crystal.\nA diamond saw was used to cut the crystals. The YIG\ncrystals were cleaned ultrasonically \frst in acetone and\nthen ethanol baths.\nThree di\u000berent types of surfaces were prepared for sam-\nples S1 - S3 by the following treatments:\n\u000fFor S1: the YIG crystals were grinded with abrasive\ngrinding papers (SiC P1200 - SiC P4000) at 150 rpm\nfor 1h. After grinding, diamond particles were used\nwith a sequence of 9 \u0016m, 3\u0016mand 1\u0016mat 300 rpm for\n30 mins, respectively. To remove the surface strain or\nsurface damage due to diamond particles22{24, colloidal\nsilica OPS (oxide polishing suspension) with a particle\nsize of 40 nm was used, which can give mechanical as\nwell as chemical polishing. To remove the residuals of\npolishing particles, samples were heated at 200\u000eC for\n1h at ambient conditions. Then crystals were cleaned\nby acetone and ethanol in an ultrasonics bath before\ndepositing the Pt layer on top.\n\u000fFor S2: grinding, polishing and cleaning of the samples\nwere done in the same way as described for S1. How-\never, the colloidal silica OPS was not used for sample\nS2. Thus, the strained or damaged surface layer due\nto diamond polishing was retained.\n\u000fFor S3: no mechanical polishing was done to obtain \rat\nsurfaces as done for samples S1 and S2. After cleaningin the same way as done for samples S1 and S2, Pt was\ndeposited on the unpolished YIG crystal surface.\nTABLE I. Surface treatment, surface roughness, and orienta-\ntion of the YIG crystals for di\u000berent samples.\nSamples Polishing Roughness Orientation\nS1 Silica <3 nm (111)\nS2 diamond \u001512 nm (111)\nS3 no >300 nm (111)\nThe surface treatments are summarized in Table I. The\nmeasurements of the spin Seebeck e\u000bect (SSE) were per-\nformed in the following way. The samples were magne-\ntized in the xy plane of the YIG crystal by an external\nmagnetic \feld H, as shown in Fig. 1. To excite the spin\nwaves an external heater generates a temperature gradi-\nentrTacross the Pt/YIG stack where the temperature\nof heat sink is denoted as T. The thickness of the YIG\n(Pt bar) is 3 mm (15 nm) for all samples. Regarding\nthe lateral dimension of the Pt bar, the length (width)\nvaries from 5 mm-3 mm (2.5-1.5) with all samples having\nratios 2:1. The surface of the YIG crystals was analyzed\nby atomic force microscopy (AFM) before deposition of\nthe Pt \flm on top. The observed spin Seebeck signals\nshow a small o\u000bset which we removed. The \feld at which\n95% of the SSE signal saturates is de\fned as HSSE\nsat. The\nmagnetization M of the YIG crystal with a dimension of\n2 mm x 1 mm was measured with a SQUID magnetome-\nter.\nIII. RESULTS AND DISCUSSION\nFig. 1(c) shows the AFM height image of sample S1\nwith a surface roughness smaller than 3 nm. A distinct\nVISHE signal appears and saturates around \u001855.3 mT,\nwhich is close to the \feld required to saturate the mag-\nnetization of the YIG crystal, as illustrated in Fig. 1(d).\nSimilarly, the YIG surface of sample S2 was analyzed\nby AFM. Fig. 2(a) shows that sample S2 has a surface\nroughness around \u001812 nm with strip-like trenches at\nthe surface. A clear spin Seebeck response has been ob-\nserved for sample S2 by changing the applied magnetic\n\feld H. The signal saturates at relatively higher values\nof H (\u001866.1 mT) compared to the magnetization of YIG\nas shown in Fig. 2b. In addition, we checked the mag-\nnetic \feld dependence of the spin Seebeck response at\nlow-temperatures for sample S2, the temperature depen-\ndence of the HSSE\nsatis given in Fig. 2(c). As the YIG\ncrystal is a 3D isotropic ferrimagnet, the temperature\ndependence of the magnetic order parameter obeys a T2\nuniversality scaling25. To understand the temperature\ndependence of HSSE\nsat, we \ftted HSSE\nsatat low tempera-\ntures by assuming Tc= 553 K as shown in the inset of\nFig. 2(c). The temperature dependence of HSSE\nsatclosely3\nc) \n0512 nm(a) (b) \n(d) (c) \nθ (o)\nFIG. 2. (a) AFM height image of a single crystal YIG surface\nfor sample S2 (20 x 20 \u0016m2). (b) Comparison between the\nH dependence of VISHE at \u0001T= 3.6 K in sample S2 and\nthe magnetization M of the YIG crystal. (c) Temperature\ndependence of HSSE\nsat. The inset shows HSSE\nsatas a function\nofT\"where\"= 2. (d)VISHE as a function of the external\nmagnetic \feld direction \u0012in the Pt/YIG system at a \fxed\nmagnetic \feld 80 mT.\nobeys theT2universality behavior of the order parame-\nter of the YIG crystal with exponent \"= 2. It suggests\nthat theHSSE\nsatdirectly depends on the order parameter\nof the YIG crystal. To con\frm further the origin of the\nobserved signal, H is rotated in the x-y plane. The VISHE\nsignal follows the expected sinusoidal dependence for a\nspin Seebeck signal, as shown in Fig. 2(d).\nUnlike the samples S1 and S2, sample S3 has a very\nlarge surface roughness ( >300 nm) as shown in Fig. 3(a).\nNevertheless, a clear spin Seebeck signal was observed as\nshown in Fig 3(b).\n(a) (b) \n350 nm\n200 \n100 \n0\nFIG. 3. (a) AFM height image of the YIG surface for sam-\nple S3 (20 x 20 \u0016m2) and (b) a comparison between the H\ndependence of VISHE at \u0001T = 7.5 K in sample S3 and the\nmagnetization M of the YIG crystal.\nFrom equation 1, it follows that the inverse spin Hall\nvoltageVISHE is proportional to the applied temperature\ngradientrTand the length of the Pt bar lPt.VISHE\nincreases by reducing the thickness of the Pt \flm tPt,\nfor both the spin pumping26and the SSE27experiments.\nTherefore to compare samples with di\u000berent Pt thicknesswe can de\fne a parameter C as follows26{28:\nC=1\ntanh[tPt\n2\u0015Pt]\u001aPtlPt\ntPtVISHE\nrT(2)\nHere,rTis de\fned as the temperature di\u000berence\nacross the Pt/YIG stack normalized with the thickness\nof the YIG crystal, \u001aPtis the resistivity of Pt and \u0015Pt\nis the spin di\u000busion length of Pt. In these experiments,\nunlike\u001aPt,\u0015Ptcannot be measured directly therefore\nwe assumed that it remains constant for di\u000berent sam-\nples. Note that for all samples discussed here tPt>2\u0015Pt\n(where\u0015Pt= 1:5nm12,13) so the tanh[tPt\n2\u0015Pt] term is ap-\nproximately equal to 1 leading to VISHE/1=tPt. More-\nover, the C parameter is independent of the YIG thick-\nness when the thickness is larger than the magnon mean\nfree path and therefore it can be used as an indicator\nof changes in other parameters related to the interfacial\nmechanisms of the SSE.\nTABLE II. Comparison of the resistance R of the Pt \flm,\nthe C parameter and the HSSE\nsatfor the SSE response in bulk\nsingle crystals and thin \flms\nBulk crystals Thin \flms\nS1 S2 S3 Ref.3Ref.13\nR (\n) 33.8 52.2 119 - -\nC (10\u00008V \n\u00001K\u00001) 0.917 1.369 0.043 0.554 1.105\nHSSE\nsat(mT) 55.3 66.1 72.8 40 2.5\nThe resistance of the Pt \flm varies for the samples\nS1-S3, nevertheless all samples have similar resistance\nwithin an order of magnitude as shown in Table II. The\nobserved change in the resistance is correlated with the\nroughness of the crystals, although we do not observe\nthe same scaling for the SSE response. For example, the\nresistance of sample S2 is 50% higher than sample S1\nwhereas the SSE signal for sample S2 is only 30% higher\nthan sample S1. Furthermore the resistance of sample\nS3 is almost four times bigger than sample S1, however\nthe SSE response actually follows the opposite trend, it\nis actually more than an order of magnitude lower than\nthe response of the samples S1 and S2. Therefore, we\nestablish that the dominant mechanism relevant for the\nobserved di\u000berences in the SSE signal is not the resistiv-\nity of the NM \flms but the quality of the NM/FM inter-\nface. Sample S1 gives a C parameter that is comparable\nto the value reported for thin \flms and bulk crystals as\nshown in Table II. However, sample S2 shows 30% bigger\nand sample S3 shows more than an order of magnitude\nsmaller value of the C parameter than sample S1. The\nobserved variation in the value of the C parameter in-\ndicates the importance of mechanical treatment induced\nsurface e\u000bects that we will discuss below.\nBased on the experimental conditions listed in Table I\nand the results summarised in Table II, we propose a4\npossible mechanism for our observations. Fig. 4(a-c)\nschematically illustrates possible interface morphologies\nand the surface magnetization for the NM/FM system,\nfor di\u000berent interface conditions between the NM \flm\nand the FM crystal. Fig. 4(a) represents the case for\na NM \flm deposited on an atomically \rat FM crystal.\nHere, the case for sample S1 corresponds to Fig. 4(a).\nFig. 4(b) depicts a situation for a NM deposited on a \rat\nFM crystal but having a small perpendicular anisotropy\nat the surface. The situation represented in Fig. 4(b)\ncorresponds to the case for sample S2. The surface of\nthe YIG crystal for sample S2 contains trenches due to\npolishing of the YIG crystal with coarse diamond parti-\ncles as shown in Fig. 2(a). The trenches at the interface\ncan induce strain or shape anisotropy resulting in a per-\npendicular anisotropy at the interface. The presence of\na small perpendicular anisotropy at the interface would\nincrease the HSSE\nsatcompared to the bulk magnetization\nof the YIG crystal, which has been clearly observed for\nsample S2 (see Fig. 2(b)).\nIn addition, the magnitude of the SSE signal can also\nchange if the mechanical polishing changes the atomic\ntermination for the density of Fe atoms that are in direct\ncontact with the Pt metal. If the density of Fe atoms at\nthe surface is larger than the bulk of the YIG, the ob-\nserved SSE signal would be larger16,21. The increase in\nthe SSE signal for sample S2 compared to sample S1 can\nbe attributed to di\u000berent chemical termination due to\npolishing with coarse diamond particles. Fig. 4(c) shows\nthe case for a rough interface between the NM and the\nFM crystal which corresponds to the situation for sample\nS3. In case of sample S3, the lack of further mechanical\ntreatment after cutting with a diamond saw leaves a very\nrough surface of the YIG crystal. The HSSE\nsatis around\n72.8 mT for sample S3 as shown in Fig. 3(b). The in-\ncrease in the value of HSSE\nsatfor sample S3 compared to\nthe magnetization of YIG can be due to a non-uniform\nmagnetization at the interface resulting from high surface\nroughness of the YIG crystal.\nFig. 4(d) gives a comparison of the magnitude of the\nSSE signal in terms of the C parameter (as de\fned in\nequation 2) for samples with di\u000berent mechanical treat-\nments. The observed signal for sample S3 is smallest\ncompared to other samples. This can be explained due\nto the increase of surface roughness7,16resulting in the\nsmall spin mixing conductance at the interface. The sam-\nple S1 has the lowest surface roughness, however the SSE\nsignal observed for sample S2 is the largest compared to\nthe samples S1 and S3 as shown in Fig. 4(d). Therefore,\nfor the largest roughness of sample S3 we see a relation\nbetween roughness and the SSE signal, but not for the\nsamples S1 and S2. Hence, the roughness is not the only\nparameter and this might be related to the more abra-\nsive nature of the diamond particles leaving a di\u000berent\nchemical termination at the interface.\nTo compare the line pro\fle of the VISHE signal, in\nFig. 4(e) the signals are normalized by their value at H\n= 90 mT, where they reach saturation. Fig. 4(e) shows\n(d) (a)\n(b) \nFMNM\nMFMNM\nM\n(e)\n(c)\nFMMNMFIG. 4. (a-c) A schematic illustration of the interface mor-\nphologies of the NM/FM system for di\u000berent surface treat-\nments of the FM where orange arrows represent rT: (a) An\natomically \rat interface, (b) an interface with a perpendic-\nular anisotropy and (c) a rough interface. (d) Comparison\nbetween the magnitude of the C parameter and (e) compari-\nson between the line pro\fle of the SSE signal as a function of\nH for all samples.\nthat the line pro\fle of the SSE signal changes with mov-\ning from soft silica to coarse diamond particle polishing.\nFor the samples S1 and S2 the VISHE is very small at zero\napplied \feld compared to the value measured at 90 mT.\nHowever, for sample S3 the VISHE is almost 64% of the\nvalue measured at H = 90 mT. The value of HSSE\nsatis high-\nest for sample S3 with the largest surface roughness and\nlowest for sample S1 with the smallest surface roughness.\nTherefore, the HSSE\nsatdirectly correlates with the rough-\nness of sample. The large deviation in the magnitude of\nSSE signal and the HSSE\nsatin the YIG crystals with di\u000ber-\nent surface treatments emphasizes the surface sensitivity\nof the spin Seebeck e\u000bect. Our results indicate that not\nonly the surface roughness but actual atomic structures\nand chemical termination at the interface also play an\nimportant role in the SSE.\nIV. CONCLUSIONS\nIn conclusion, we have shown a strong dependence of\nthe spin Seebeck signal on the interface condition of the\nPt/YIG bilayer system. We observed a change of 18 mT\nin the saturation \feld of the SSE signal by changing the\ntype of polishing. Furthermore we observe the change\nin the magnitude of the SSE signal for di\u000berent samples.\nNo de\fnite relation has been found between the SSE re-\nsponse and the sample roughness. However, we observe\na direct correlation between the HSSE\nsatand the roughness\nof sample, as the former increases by moving from soft\ntoward coarse particle polishing. To understand the ori-\ngin of the magnitude and change in the saturation \feld\nHSSE\nsatfor the observed SSE signal, due to di\u000berent types\nof surface treatments, the crystal surfaces need to be in-5\nvestigated further in detail.\nACKNOWLEDGMENTS\nWe would like to acknowledge J. Baas, H. Bonder, G.H.\nten Brink, M. de Roosz and J. G. Holstein for technical\nassistance. This work is supported by the Foundation\nfor Fundamental Research on Matter (FOM), the Nether-\nlands Organisation for Scienti\fc Research (NWO), Marie\nCurie ITN Spinicur NanoLab NL, and the Zernike Insti-\ntute for Advanced Materials National Research Combi-\nnation.\n1K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi, J. Ieda,\nT. Ota, Y. Kajiwara, H. Umezawa, H. Kawai, G. E. W. Bauer,\nS. Maekawa, and E. Saitoh, Nature Mater. 9, 894 (2010).\n2G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Nature Mater.\n11, 391 (2012).\n3K.-i. Uchida, H. Adachi, T. Ota, H. Nakayama, S. Maekawa, and\nE. Saitoh, Appl. Phys. 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Uchida, Y. Fujikawa, and E. Saitoh, Phys. Rev.\nB85, 144408 (2012)."    },    {        "title": "2403.07254v1.Negative_orbital_Hall_effect_in_Germanium.pdf",        "content": " \n \n \n \nNegative orbital Hall effect in Germanium  \nE. Santos,1 J. E. Abrão ,1 J. L. Costa ,1 J. G. S. Santos,1 J. B. S. Mendes,2 and A. Azevedo1 \n1Departamento de Física, Universidade Federal de Pernambuco, 50670 -901, Recife, Pernambuco, \nBrazil.  \n2Departamento de Física, Universidade Federal de Viçosa, 36570 -900, Viçosa, Minas Gerais, Brazil.  \n \n \nOur investigation reveals a groundbreaking discovery of a negative inverse orbital Hall effect (IOHE) in Ge \nthin films. We employed the innovative orbital pumping technique where spin -orbital coupled current is injected \ninto Ge films using YIG/Pt(2)/Ge (𝑡𝐺𝑒) and YIG/W(2)/Ge (𝑡𝐺𝑒) heterostructures. Through comprehensive \nanalysis, we observe significant reductions in the signals generated by coherent (RF -driven) and incoherent \n(thermal -driven) spin -orbital pumping techniques. These reductions are attributed to the presence of a \nrema rkable strong negative IOHE in Ge, showing its magnitude comparable to the spin -to-charge signal in Pt. \nOur findings reveal that although the spin -to-charge conversion in Ge is negligible, the orbital -to-charge \nconversion exhibits large magnitude. Our res ults are innovative and pioneering in the investigation of negative \nIOHE by the injection of spin -orbital currents . \n \n \n \nThe orbital Hall effect (OHE) occurs when there is \na transverse flow of orbital angular momentum (OAM) \ninduced by an external electric field [1 -4], like the spin Hall \neffect (SHE) [5 -7]. Notably, the OHE operates \nindependently of the existence of spin -orbit coupling \n(SOC). Recent studies [4, 8] have revealed substantial \nnegative values f or orbital Hall conductivity  (𝜎𝑂𝐻) across \ndifferent materials. However, these materials often exhibit \nsignificant spin Hall conductivity  (𝜎𝑆𝐻) presenting \nchallenges in isolating distinct spin and orbital \ncontributions. Addressing this complexity, our \nunderstanding of OHE is evolving, with persistent \ninvestigations using various strategies aimed at unraveling \nthe intrinsic and extrinsic mechanisms  underlying this \nphenomenon. Exploration of the intrinsic and extrinsic \nmechanisms governing the OHE has encompassed different \nclasses of materials, from transition metals and \nsemiconductors to two -dimensional materials and \ntopological insulators [9 -19]. Despite this extensive \ninvestigation, there has been a notable absence of \nexperimental studies investigating semiconductors \nmaterials. This gap in knowledge is particular ly intriguing \ngiven the potential of group IV semiconductors, such as Ge \nand Si, to serve as exceptional platforms for spintronics \nphenomena [20 -23]. Ge has a much higher carrier mobility \nthan Si, which can be used to improve the performance of \ntransistors based on this material. Currently, Ge has \napplications in optical fibers and optical  tweezers, while Si -\nGe alloys play a role in microchip manufacturing, with \nfeatur e sizes on the chips reaching 7 nm [24 -27]. This \nunique combination of properties establishes group IV \nsemiconductors as attractive for fundamental research in \nOHE and practical advances in spin -orbitronics \napplications.  \nIn turn, a recent theoretical work [28] has discovered \nthat Ge has 𝜎𝑂𝐻𝐺𝑒~−1270 (ℏ/𝑒)(Ω∙cm)−1  and 𝜎𝑆𝐻𝐺𝑒~1.6×\n10−1 (ℏ/𝑒)(Ω∙cm)−1, making it a prime material for \nstudying orbital effects. Materials with negative 𝜎𝑂𝐻, like \nGe, play a crucial role in distinguishing orbital from spin \neffects and offer insights for the development of new OAM -\nbased devices. Nonetheless, to our knowledge, the OHE in \nGe has not been investigated to date. In this work, we \nexperimentally inve stigate orbital -charge conversion in Ge thin films using the inverse orbital Hall effect. The \nmeasurements were carried out at room temperature using the \nspin pumping driven by ferromagnetic resonance (SP -FMR) \nand longitudinal spin Seebeck effect (LSSE) te chniques. We \nfabricated samples of YIG/Pt, YIG/Ge, YIG/Pt/Ge, and \nYIG/W/Ge, where YIG refers to yttrium iron garnet \n(Y3Fe5O12) grown on (111) -oriented Gadolinium Gallium \nGarnet (Gd 3Ga5O12, GGG), by liquid phase epitaxy (LPE) \nthrough the traditional PbO/B 2O3 flux method. The quality of \nthe YIG samples is attested by the small FMR linewidth, \nwhich is less than 1 Oe (see figure 1(b)). All other t hin films \nwere deposited using magnetron sputtering at room \ntemperature,  with a working pressure of 2.8 mTorr and a base \npressure of 1.5×10-7 Torr. All investigated samples have  \nlateral  dimensions of 3.0 x 1.5 mm.  \nFigure 1(a) illustrates the SP -FMR process in a \nYIG/Pt bilayer. The precessing magnetization injects a spin \ncurrent den sity into the Pt layer, given by  \n𝐽⃗𝑆=(ℏ𝑔𝑒𝑓𝑓↑↓/4𝜋𝑀2)(𝑀⃗⃗⃗×𝑀⃗⃗⃗̇) [29]. T his spin current \nmanifests itself as two components: an AC represented by the \norange vectors and a DC represented by the fixed red arrow \nparallel to the applied field. The DC component induces a spin \naccumulation that diffuses into the Pt layer, characteri zed by \nspin polarization ( 𝜎̂𝑆) oriented along the z -axis with \ncharacteristic diffusion length, typically spanning a few \nnanometers in materials with large SOC [10] . Figures 1 (b -d) \nshows the derivatives of the FMR absorption curves with the \nexternal field applied in -plane for bare YIG(400), \nYIG(400)/Ge(8) and YIG(400)/Pt(8). The values in \nparentheses denote the film thicknesses in  nanometers. To \nexcite the FMR condition, the samples  were placed at the \nbottom of a rectangular microwave resonant cavity operating \nat 9.41 GHz, with an inci dent power of 15 mW. For more \ndetails on the SP -FMR technique, refer to [10, 30]. The \nexperimental data were fitted using the derivative of a \nLorentzian curve. The numerical fit yields FMR linewidths of \n∆𝐻=0.89 𝑂𝑒, ∆𝐻=0.90 𝑂𝑒 and ∆𝐻=2.00 𝑂𝑒, for bare \nYIG, YIG/Ge(8) and YIG/Pt(8), respectively. The spin \npumping effect in ferromagnetic(FM)/normal -metal(NM) \nintroduces additional damping in the FMR process, meaning \nthat an extra term is added to the Landau -Lifshitz -Gilbert  \nFIG. 1. (a) Illustration of the SP -FMR process where the red arrows \nrepresent the DC component of the spin current 𝐽⃗𝑆∝ (𝑀⃗⃗⃗×𝑀⃗⃗⃗̇) that \ndiffuses into the Pt layer, where 𝐽⃗𝑆 transform into 𝐽⃗𝐿𝑆 due to SOC. \n(b-d) FMR absorption curves (black symbols) for: bare YIG(400), \nYIG(400)/Ge(8), and YIG(400)/Pt(8), respectively. The solid red \nlines were obtained by fitting the data with derivative of a \nLorentzian curve.  The weak absorption peaks that appear below \nand above the resonance field of the uniform mode are due to \nsurface and volume magnetostatic modes. (e) shows the solid \ncurves obtained by the numerical fits, where the large increase in \n∆𝐻 of YIG/Pt(8) is highlighted.   \n \nequation [ 29].  The increase in FMR linewidth is described \nby ∆𝐻𝑆𝑃=ℏ𝜔0𝑔𝑒𝑓𝑓↑↓4⁄𝜋𝑡𝐹𝑀𝑀, where 𝑔𝑒𝑓𝑓↑↓ is the real part \nof the effective spin mixing conductance, 𝜔0 is the angular \nfrequency, 𝑡𝐹𝑀 and 𝑀 are the thickness and magnetization \nof the FM, respectively. Comparing figures 1(b) and 1(c), \nwe observe practically no change in linewidth. As expected, \ndue to the large SOC of Pt, the FMR linewidth of YIG/Pt(8) \nin figure 1(d) exhibited a large increase to 2 Oe, compared \nwith bare YIG. Figure 1(e) shows the numerical fits of the \nthree bilayers, where the increase of ∆H in the YIG/Pt(8) \nbilayer due to the SP process is highlighted.  \nSubsequently, SP -FMR measurements were \nconducted by attaching electrodes at the edges of the metal \nlayers using silver paint. All subsequent SP -FMR \nmeasurements were obtained at the same frequency of 9.41 \nGHz and using an incident power of 43 mW. Figures 2  (a) \nand 2(b) illustrate the electrical current generated by the SP -\nFMR process as function of the external applied field, for \nYIG/Pt(8) and YIG/Ge(8), respectively. The electrical \ncurrent is defined as 𝐼𝑁𝑀=𝑉⁄𝑅𝑆, where 𝑉 is the electrical \nvoltage that is directly measured via a nanovoltmeter and 𝑅𝑆 \nthe electrical sheet resistance of the film. As illustrated in \nfigure 1(a), the DC spin current, represented by the red \narrows and possessing spin polarization 𝜎̂𝑆 along z -axis, \ndiffuses into the Pt layer. The large SOC in Pt leads to the \ncoupling of spin and  orbital states [9 -11,31]. Consequently, \nthe spin -orbital current along the 𝑦̂ direction is expressed as \n𝐽⃗𝑆(𝑦)=𝐴sinh [(𝑡𝑁𝑀−𝑦)/𝜆1]\nsinh (𝑡𝑁𝑀/𝜆1)𝑦̂+𝐵sinh [(𝑡𝑁𝑀−𝑦)/𝜆2]\nsinh (𝑡𝑁𝑀/𝜆2)𝑦̂. Here, \nthe constants A and B are to be determined via boundary \nconditions, 𝜆1 and 𝜆2 are diffusion lengths, that depend on \nboth the spin diffusion length 𝜆𝑆, orbital diffusion length 𝜆𝐿, \nand diffusion coupling parameter 𝜆𝐿𝑆 [11, 31]. Conversion \nof spin current to charge current occurs due to the inverse \nspin Hall effect (ISH E) [32,33]. The relationship between \nspin current ( 𝐽⃗𝑆) and charge current ( 𝐽⃗𝐶), within the NM,  is \ngiven by 𝐽⃗𝐶=𝜃𝑆𝐻(𝜎̂𝑆×𝐽⃗𝑆), where 𝜃𝑆𝐻=(2𝑒/ℏ)𝜎𝑆𝐻/𝜎𝑒 is \nthe spin Hall angle, representing the spin -charge conversion \nefficiency and 𝜎𝑒 is the electric conductivity. The orbital \ncurrent generated by the coupling 𝐿⃗⃗∙𝑆⃗ is 𝐽⃗𝐿(𝑦)=\n𝐶𝛿𝐿𝑆𝐽⃗𝑆(𝑦), where the dimensionless constant C represents \nthe strength of this relationship, and 𝛿𝐿𝑆=±1 indicates the \nSOC signal. The conversion of orbital current to charge \ncurrent occurs due to the IOHE [1,10,34,35]. Analogously \nto ISHE, we can write the mathematical relationship  \nFIG. 2. SP -FMR signals for (a) YIG/Pt(8), and (b) YIG/Ge(8). Both \nsignals obey the ISHE equation, 𝐽⃗𝐶=𝜃𝑆𝐻(𝜎̂𝑆×𝐽⃗𝑆). Due to the weak \nstrength of SOC in Ge, the ISHE signal is significantly decreased \ncompared to the Pt signal, while both materials exhibit 𝜎𝑆𝐻>0. \nInset of figure (a) defines the angle 𝜙.  \n  \nbetween the orbital current and the charge current as 𝐽⃗𝐶=\n𝜃𝑂𝐻(𝜎̂𝐿×𝐽⃗𝐿), where 𝜃𝑂𝐻=(2𝑒/ℏ)𝜎𝑂𝐻/𝜎𝑒 is the orbital Hall \nangle and 𝜎̂𝐿 is the orbital polarization,  that couples parallel  \n(positive SOC) or antiparallel (negative SOC) to 𝜎̂𝑆. The \nresultant charge current is the cumulative effect of currents \ngenerated through both the ISHE and IOHE, denoted as \n𝐽⃗𝐶𝑒𝑓𝑓=𝐽⃗𝐶𝐼𝑆𝐻𝐸+𝐽⃗𝐶𝐼𝑂𝐻𝐸. It is noteworthy that despite these \neffects describing similar phenomena, their physical origins \nare distinct [1,6,8,10]. Furthermore, the polarity of the SP -\nFMR signal is determined by the strength of spin orbit \ncoupling in the material. Figure 2 shows th e results of SP -\nFMR measurements conducted on two distinct samples: \nYIG/Pt(8) and YIG/Ge(8) at 𝜙=0° (blue symbols), 𝜙=\n180°  (red symbols), and 𝜙=90° (black symbols), where 𝜙 \nis defined in the inset of figure 2(a). As Pt has strong SOC, \nwe expect large spin and orbital currents represented by 𝐽⃗𝑆 and \n𝐽⃗𝐿. The spin Hall orbital Hall conductivities for Pt are denoted \nas 𝜎𝑆𝐻𝑃𝑡~2012 (ℏ/𝑒)(Ω∙cm)−1and 𝜎𝑂𝐻𝑃𝑡~144 (ℏ/𝑒)(Ω∙\ncm)−1[8], respectively. Consequently, the dominant \ncontribution to the charge current produced via SP -FMR in \nYIG/Pt(8) is primarily due to ISHE. Note that the SP -FMR \nsignals in figure 2 obey the ISHE equation. A peak with \npositive polarity is observed for 𝜙=0°, while the signal \nchanges its polarity for 𝜙=180°  and at 𝜙=90° the \nmeasured signal is null. The substantial SP -FMR signal \nobserved in Pt (figure 2(a)) contrasts with the weak signal  in \nGe (figure 2(b)), attributed to its weak SOC. The ratio \nbetween the SP-FMR signals generated in YIG/Ge(8) and the \none generated in YIG/Pt(8) is calculated as, 𝐼𝑌𝐼𝐺 /𝐺𝑒(8)𝑃𝑒𝑎𝑘/\n𝐼𝑌𝐼𝐺 /𝑃𝑡(8)𝑃𝑒𝑎𝑘~2.5×10−4. Furthermore, the negligible  SOC in \nGe results in scarce orbital current generation during spin \ncurrent propagation within the material.  \nIn the next step of our work, we studied YIG/Pt(2)/Ge \nheterostructures, where the YIG/Pt(2) bilayer is used to inject \nan orbital current into Ge films. Notably, in the previous \nYIG/Ge configuration, the SP process injects only pure spin \ncurrent in Ge. The strong SOC of Pt couples 𝐿⃗⃗ and 𝑆⃗, resulting \nin an entangled current 𝐽⃗𝐿𝑆. The current 𝐽⃗𝐿𝑆 reaches Ge layer, \nwhere exclusively the orbital current is converted into charge \ncurrent 𝐽𝐶𝐺𝑒 through the IOHE. In YIG/Pt(2)/Ge, the effective \ncharge current is 𝐽⃗𝐶𝑒𝑓𝑓=(2𝑒/ℏ)𝜃𝑆𝐻𝑃𝑡 (𝜎̂𝑆×𝐽⃗𝑆𝑃𝑡)+(2𝑒/\nℏ)𝜃𝑂𝐻𝐺𝑒(𝜎̂𝐿×𝐽⃗𝐿𝐺𝑒), where 𝜎̂𝐿=𝜎̂𝑆=𝑧̂, 𝐽⃗𝑆=𝐽𝑆𝑦̂, 𝐽⃗𝐿=𝐽𝐿𝑦̂, \n𝜃𝑆𝐻𝑃𝑡>0 and 𝜃𝑂𝐻𝐺𝑒<0. Analyzing the equation for 𝐽⃗𝐶𝑒𝑓𝑓, it \nbecomes clear that the first term generates a current along +𝑥̂ \ndirection, while the second term is along the −𝑥̂ direction . \nThis indicates a reduction in the measured voltage value. The \neffective charge current generated in YIG/Pt/Ge 𝐽⃗𝐶𝐺𝑒 is \nobtained by the subtraction between the 𝐽⃗𝐶𝑒𝑓𝑓 and 𝐽⃗𝐶𝑃𝑡(2). To \n \nFIG. 3. (a) Ge thickness dependence of the SP -FMR peak signal of \nYIG/Pt(2)/Ge( 𝑡𝐺𝑒). Blue data corresponds to 𝜙=0° and red data \ncorresponds to 𝜙=180° . Due to the negative IOHE of Ge, a \ngradual reduction in the signal is observed with increasing the \nthickness of the Ge layer. (b) IOHE signal for Ge as a function of \nthickness, with theoretical fit using 𝐼𝐼𝑂𝐻𝐸𝐺𝑒=𝐷𝑡𝑎𝑛 ℎ(𝑡𝐺𝑒/2𝜆𝐿𝐺𝑒), \nwhere we found 𝜆𝐿𝐺𝑒=(4.0±0.6) nm, and ∆𝐼𝑆𝑃−𝐹𝑀𝑅 =𝐼𝑃𝑡(2)−\n𝐼𝑃𝑡(2)/𝐺𝑒. We observed that the greater the thickness of Ge, the \nmore intense the IOHE signal, which in magnitude is \napproximately equal to the ISHE of Pt(2) for 𝑡𝐺𝑒>30 nm. The \ninset in (a) illustrates the physical scheme of the effective charge \ncurrent in YIG/Pt/Ge, measured by SP -FMR for  𝜙=0° and 𝜙=\n180° . \n  \nbetter understand this behavior, we fabricated  a series of \nYIG/Pt(2)/Ge( 𝑡𝐺𝑒) samples, varying the thickness of the Ge \nlayer from 2 nm to 50 nm, where we analyze 𝐼𝑌𝐼𝐺 /𝑃𝑡(2)/𝐺𝑒𝑃𝑒𝑎𝑘. \nIn figure 3(a), when 𝑡𝐺𝑒=0 nm, 𝐼𝑌𝐼𝐺 /𝑃𝑡(2)𝑃𝑒𝑎𝑘 reaches a \nmaximum of approximately 600  nA, signifying no \ncontribution from the Ge layer. With 𝑡𝐺𝑒=2 nm, \n𝐼𝑌𝐼𝐺 /𝑃𝑡(2)/𝐺𝑒(2)𝑃𝑒𝑎𝑘 is around 370  nA, revealing that only 2 nm \nof Ge is sufficient to induce a negative orbital -charge \nconversion, equivalent to roughly 60% of the ISHE in \nYIG/Pt(2). As the thickness of the Ge layer is progressively \nincreased, a consistent decline in the signal becomes \nevide nt. For example, at 𝑡𝐺𝑒=10 nm, 𝐼𝑌𝐼𝐺 /𝑃𝑡(2)/𝐺𝑒(10)𝑃𝑒𝑎𝑘 is \napproximately 140  nA indicating a reduction of 76% of the \nISHE in YIG/Pt(2). The experimental data saturates for \n𝑡𝐺𝑒>30 nm, where 𝐼𝑌𝐼𝐺 /𝑃𝑡(2)/𝐺𝑒(30)𝑃𝑒𝑎𝑘~0. Therefore, at the \nsaturation, |𝐼𝐼𝑂𝐻𝐸𝐺𝑒|~𝐼𝐼𝑆𝐻𝐸𝑃𝑡. In 𝜙=0°, as shown by the blue \nsymbols, we have 𝜎̂𝑆‖𝜎̂𝐿∥𝐻⃗⃗⃗. Upon inverting the external \nmagnetic field 𝐻⃗⃗⃗ for 𝜙=180° , illustrated in figure 3(a) by \nthe red symbols , 𝜎̂𝑆 and 𝜎̂𝐿 invert direction while remaining \nparallel, owing to the strong SOC of Pt. By inverting 𝜎̂𝑆 we \nexpect a negative ISHE in Pt and a positive IOHE signal in \nGe, according to the respective ISHE and IOHE equations. \nConsequently, at 𝜙=180° , the SP -FMR signal tends \ntowards zero in a similar way. The inset in figure 3(a) \nillustrates the physical scheme of the effective charge \ncurrent in YIG/Pt/Ge, measured by SP -FMR for  𝜙=0° and \n𝜙=180° . \nFigure 3(b) shows the IOHE signals for Ge films \nranging in thickness from 0 to 50 nm, for  𝜙=0° (blue \nsymbols) and 𝜙=180°  (red symbols) . These signals were \nobtained by calculating the difference between \n𝐼𝑌𝐼𝐺 /𝑃𝑡(2)/𝐺𝑒−𝐼𝑌𝐼𝐺 /𝑃𝑡(2). Given that Ge exhibits negligible \nSOC, we can distinctly analyze the spin and orbital  \nFIG. 4. (a) Schematically shows the LSSE configuration.  LSSE \nmeasurement for (b) YIG/Pt8), (c) YIG/Ge(8), and (d) IOHE for Ge \nfilms, where the orbital current was injected from YIG/Pt(2), where \n∆𝐼𝐿𝑆𝑆𝐸 =𝐼𝑃𝑡(2)−𝐼𝑃𝑡(2)/𝐺𝑒. Theoretical fit using 𝐼𝐼𝑂𝐻𝐸𝐺𝑒=\n𝐴𝑡𝑎𝑛 ℎ(𝑡𝐺𝑒/2𝜆𝐿𝐺𝑒), where we found 𝜆𝐿𝐺𝑒=(7.5±0.5) nm.  \n \ncontributions of the 𝐽⃗𝐿𝑆 current that reaches the Ge layer. As \nconfirmed in our previous result, the spin component is nearly \nnull, highlighting the importance of the orbital component in \nthis context. The orbital flow within the Ge layer has a well -\ndefined orbital diffusion leng th 𝜆𝐿. Using 𝐼𝐼𝑂𝐻𝐸𝐺𝑒=\n𝐷𝑡𝑎𝑛 ℎ(𝑡𝐺𝑒/2𝜆𝐿𝐺𝑒) [9-11] it is possible to estimate 𝜆𝐿𝐺𝑒 in the \nYIG/Pt(2)/Ge heterostructures, where D is constant. Fitting \nthe experimental data in figure 3(b), we found 𝜆𝐿𝐺𝑒=\n(4.0±0.6) nm. To contextualize our result concerning 𝜆𝐿𝐺𝑒, \nwe will discuss some details of the spin and orbital diffusion \nlength s. A fundamental distinction exists between spin and \norbital transport. Contrary to intuition, the crystal field does \nnot quench nonequilibrium OAM as effectively as it \nsuppresses equilibrium OAM [31]. This is attributed to the \npresence of degenerate orbital states that play a crucial role in \nlong-range orbital transport [36]. Orbital degeneracy is \ngene rally protected against crystal field splitting, allowing \norbital momentum to traverse longer distances compared its \nspin counterpart. However, in materials with weak SOC, long \nspin diffusion lengths are expected [37 -40]. For example, in \nGe is expected spin diffusion len gths of orde r of micrometers \n[37-40]. In our experiment, a simultaneous injection of both \nspin current and orbital current occurs, represented by the \ncoupled current 𝐽⃗𝐿𝑆. The distinction between spin and orbital \ncontributions relies on the knowledge of 𝜎𝑆𝐻 and 𝜎𝑂𝐻. \nNotably, in Ge, the IOHE signals are expected be much larger \nthan the ISHE signals [28]. Thus, we can conclude that the \ndiffusion length found is not predominantly associated with \nspin. On the contrary, the Orbital diffusion lengths in Ge \ncould potential ly exceed what our experiments reveal, as we \ndo not directly inject an orbital current into the Ge layer. \nInitially, the orbital current is generated within the Pt layer. \nPrior to reaching the Ge layer, this orbital current must flow \nthrough the Pt film. A long this pathway, the strong SOC of Pt \nimposes constraints on both spin and orbital diffusion lengths, \nwhere 𝜆𝑆𝑃𝑡~1.5 nm or even shorter [10 ]. Consequently, a \nsignificant reduction in the orbital current occurs within the \ninitial 2 nm before reaching the Ge layer. Additionally, the \nresistivity mismatch at the Pt(2)/Ge interface further reduces \nits magnitude. This phenomenon can explain the orbital \ndiffusion length observed in our experiments.  \nAnother notable result is presented in figure s 3(c) and \n3(d). We successfully replicate results similar to those in \nfigure 3 (a), employing W instead of Pt. We fabricated the \nfollowing samples: YIG/W(2), YIG/W(2)/Ge( 𝑡𝐺𝑒), and \nYIG/W(2)/Ti(8). In figure 3(c), the results for \nYIG/W(2)/Ge( 𝑡𝐺𝑒) are shown for 𝜙=0° and 𝜙=180° . \nFigure 3(d) shows the signal for YIG/W(2), with \n𝐼𝑆𝑃−𝐹𝑀𝑅𝑃𝑒𝑎𝑘~−170  nA. Upon the addition of Ge(50) to the \nYIG/W(2) sample, 𝐼𝑆𝑃−𝐹𝑀𝑅𝑃𝑒𝑎𝑘 was reduced to around −25 nA. \nAlthough  a reduction in signal magnitude was observed, it \nwas considerably less significant than in samples utilizing \nPt. On the other hand, figure 3(d) also shows the signal for \nYIG/W(2)/Ti(8)  in comparison to YIG/W(2). An increase in \nthe signal by nearly a factor of  2 was observed.  \nIt is known that W has 𝜎𝑆𝐻𝑊=−768  (ℏ/𝑒)(Ω∙\ncm)−1 and 𝜎𝑂𝐻𝑊=4664  (ℏ/𝑒)(Ω∙cm )−1 [8]. \nConsequently, an SP -FMR signal is expected to originate \nfrom spin -orbital to charge conversion via both ISHE and \nIOHE, with the latter having a significantly greater \nmagnitude. The effective charge current in YIG/W/Ge is \ngiven by 𝐽⃗𝐶𝑒𝑓𝑓=(2𝑒/ℏ)[𝜃𝑆𝐻𝑊 (𝜎̂𝑆×𝐽⃗𝑆𝑊)+𝜃𝑂𝐻𝑊 (𝜎̂𝐿×\n𝐽⃗𝐿𝑊)+𝜃𝑂𝐻𝐺𝑒(𝜎̂𝐿×𝐽⃗𝐿𝐺𝑒)], where 𝜃𝑆𝐻𝑊<0, 𝜃𝑂𝐻𝑊>0, 𝜃𝑂𝐻𝐺𝑒<0, \n𝜎̂𝐿=−𝑧̂, 𝜎̂𝑆=𝑧̂, 𝐽⃗𝑆=𝐽𝑆𝑦̂ and 𝐽⃗𝐿=𝐽𝐿𝑦̂. Upon analyzing \nthe equation for 𝐽⃗𝐶𝑒𝑓𝑓 we observed that first and second terms \ncontribute −𝑥̂ direction, while the third term contributes in \nthe +𝑥̂, resulting in a reduction in  the signal. However, we \nobserved a less substantial reduction in the signal in the \nsample with W compared to the one with Pt. This \ndiscrepancy can be attributed to intrinsic characteristics of \nW and Pt. In a preliminary analysis, we can state the \nfollowing: (i) W has a smaller SOC than Pt and a large  \nelectrical resistivity [41, 42], resulting in a lower 𝐽⃗𝐿𝑆 current. \n(ii) The large  value of 𝜎𝑂𝐻𝑊 leads to a more pronounced \norbital -charge conversion in W compared Pt, causing the \nresidual orbital current reaching the Ge layer to be \ndecreased.  \nThe fluctuations observed in the data of figure 3(c) \nare linked to variations in the spin conductivity  [42] arising \nfrom the coexistence of  α and β phases of W. The \nsimultaneous presence of α and β phases in W thin films is \na relatively common occurrence in those produced through \nsputtering  [43]. In the β phase ( 𝑡𝑊<10 nm) the resistivity \nof the films is very high, while in the α phase it decreases \nconsiderably [ 43]. Despite this challenge, a consistent \nreduction in the signal is evident in figure 3(c), with the \ndashed line serving as a visual guide. It is worth noting that \nthe reduction in signal measured in W/Ge films follows a \nsimilar trend to that IOHE of the Ge  layer. Figure 3(d) \ncorroborates our theoretical hypotheses: (i) spin current \ninjection into the YIG/NM bilayers results in the  \naccumulation of spins and, due  to the strong SOC of the \nNM, generates a collinear orbital current. (ii) In materials \nwith negative SOC, the orbital polarization is antiparallel to \nthe spin polarization.  Note that the introduction of a Ti film \non top of the YIG/W bilayer resulted in a gain of the signal. \nThis increase can be exclusively attributed to orbital \ncurrents, given that Ti exhibits negligible SOC [11]. The \nreversal of orbital polarity within W, of the 𝐽⃗𝐿𝑆 current that \nreaches the Ti film, generates an orbital -charge conversion \nin the same direction (negative) as the YIG/W bulk signal. \nThis effect significantly enhances the SP -FMR signal. \nInterestingly, this increase was not observed in Ge, where \nthe negati ve 𝜎𝑂𝐻, along with the inversion of orbital \npolarity, leads to a positive orbital -charge conversion due to \nIOHE in Ge films. This conversion occurs in the opposite \ndirection to the bulk signal of W. Finally, the shorter orbital diffusion length can be attributed to the large resistivity of the \nbeta-phase W films, which quickly dissipates the 𝐽⃗𝐿𝑆 current. \nHence, for a heterostructure works as an effective orbital -\ncurrent injector, it is essential to employ a NM with strong \nSOC, like Pt, in the fabrication of YIG/NM structures. \nNonetheless, the presence of highly resistive phases (like β -\nW) may imp ede the successful injection of orbital currents \ninto adjacent films.  \nWe also employed the LSSE technique to excite spin \ncurrents and induce orbital currents in YIG/Pt(2)/Ge( 𝑡𝐺𝑒). \nLSSE consists of applying a thermal gradient to generate spin \ncurrents from the magnon flow generated in the YIG bulk \n[44], as illustrated in figure 4(a) . To create the temperature \ngradient, we utilized a Peltier module, and the resulting \ntemperature difference ( 𝛥𝑇) between the bottom and top of \nthe sample was measured using a differential thermocouple. \nThe IOHE voltage due to LSSE was detected between the two \nsilver -painted electrodes positioned at the edges of the Pt film. \nThe underlying physical mechanism is simi lar to SP -FMR, \nwith 𝐼𝐿𝑆𝑆𝐸 =𝛿𝑆𝐿𝑆𝑆𝐸 ∇𝑇,  and 𝛿𝑆𝐿𝑆𝑆𝐸 is the Seebeck -like spin \ncoefficient, including contributions from spin and/or orbital \neffects.  \nWe investigated the behavior of the DC electric \ncurrent ( 𝐼𝐿𝑆𝑆𝐸 =𝑉𝐿𝑆𝑆𝐸 /𝑅𝑆) arising from the IOHE in response \nto the sweep of the external magnetic field. Figure 4(b) shows \nthe LSSE signals for YIG/Pt(8) and figure 4(c) shows the \nLSSE signals for YIG/Ge(8) under temperature differences of \n∆𝑇=0𝐾 (black), ∆𝑇=5𝐾 (red) and ∆𝑇=10𝐾 (blue) . We \nobserved a very small YIG/Ge(8) signal, when compared to \nthe YIG/Pt(8) signal, similar to the trend observed in SP -\nFMR. Figure 4(d) shows the IOHE for Ge, and through fitting \nthe IOHE signal, we obtained 𝜆𝐿𝐺𝑒=(7.5±0.5) nm. \nFurthermore, there is a difference in the diffusion length \nmeasured by SP -FMR and LSSE, which has already been \ndiscussed in [9 -11]. Although both processes (SP -FMR and \nLSSE) are of spin current injection, there is a basic difference \nbetween the two proce sses. While in SP -FMR the spin \ninjection is interfacial, in LSSE the spin injection is due to the \nmagnon current generated in the YIG bulk.  \nIn conclusion, our work investigates the IOHE in Ge \nfilms. Although Ge has negligible SOC, it manifests a \nsubstantial and negative IOHE value, comparable in \nmagnitude to the ISHE signal observed in Pt. Efficient \ninjection of spin -orbital currents has been achieved through of \nheterostructures, specifically YIG(400)/HM(2), where the \nheavy metal (HM) layer could be Pt or W. Through a careful \ncombination of different stack layers and considering the spin \nand orbital conductivities, we elucidate successfully the  \nexciting results obtained through the SP -FMR and LSSE \ntechniques. Our study highlights the critical role of orbital \npolarization in influencing IOHE, a factor that can be \ncontrolled through the heavy metal SOC signal. By exploring \nmaterials with a promine nt IOHE and a negligible ISHE, we \neffectively isolate and distinguish spin effects from orbital \neffects. These discoveries not only contribute valuable \ninsights to the field of orbitronics, but also have potential \napplications in the development of electro nic devices based \non orbital angular momentum flow and spin -orbital coupling . \n \nThis research is supported by Conselho Nacional de \nDesenvolvimento Científico e Tecnológico (CNPq), \nCoordenação de Aperfeiçoamento de Pessoal de Nível \nSuperior (CAPES) (Grant No. 0041/2022), Financiadora de \nEstudos e Projetos (FINEP), Fundação de Amparo à Ciência \ne Tecnologia do Estado de Pernambuco (FACEPE), \nUniversidade Federal de Pernambuco, Multiuser Laboratory Facilities of DF -UFPE, Fundação de Amparo à Pesquisa do \nEstado de Minas Gerais (FAPEMIG) - Rede de Pesquisa em \nMateriais 2D and Rede de Nanomagnetismo, and INCT of \nSpintronics and Advanced Magnetic Nanostructures (INCT -\nSpinNanoMag), CNPq 406836/2022 -1. \n \n \n \n[1] D. Go, D. Jo, C. Kim, and H. -W. Lee. 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Saitoh1, 2, 4, 5\n1)Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan\n2)WPI Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577,\nJapan\n3)PRESTO, Japan Science and Technology Agency, Saitama 332-0012, Japan\n4)CREST, Japan Science and Technology Agency, Tokyo 102-0076, Japan\n5)Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195,\nJapan\n(Dated: 1 June 2022)\nSpin current injection from sputtered yttrium iron garnet (YIG) films into an adjacent platinum layer has been\ninvestigated by means of the spin pumping and the spin Seebeck effects. Films with a thickness of 83 and\n96 nanometers were fabricated by on-axis magnetron rf sputtering at room temperature and subsequent post-\nannealing. From the frequency dependence of the ferromagnetic resonance linewidth, the damping constant has\nbeen estimated to be (7:0\u00061:0)\u000210\u00004. Magnitudes of the spin current generated by the spin pumping and the\nspin Seebeck effect are of the same order as values for YIG films prepared by liquid phase epitaxy. The efficient\nspin current injection can be ascribed to a good YIG jPt interface, which is confirmed by the large spin-mixing\nconductance (2:0\u00060:2)\u00021018m\u00002.\nI. INTRODUCTION\nSpintronics is an aspiring field of electronics which in-\ncorporates the spin degree of freedom into charge-based\ndevices. Among the main interests in spintronics are gen-\neration, manipulation and detection of spin current, the\nflow of spin angular momentum. Pure spin current unac-\ncompanied by charge current has high potential to open\na path to new information technology free from the Joule\nheating.\nFor spin current generation in thin-film systems, two dy-\nnamical methods are the spin pumping1–8and the spin See-\nbeck effect.9–14In spin pumping [SP, Fig. 1(a)], spin cur-\nrent is generated by magnetization dynamics in the ferro-\nmagnet. The magnetization vector of a ferromagnet irra-\ndiated by a microwave precesses when the ferromagnetic\nresonance condition is fulfilled. This precession motion\nrelaxes not only by damping processes inside the ferro-\nmagnet (F), but also by emission of spin current into the\nadjacent non-magnetic conductor (N) by exchange inter-\naction at the FjN interface.1–3In the spin Seebeck effect\n(SSE), spin current is generated in the presence of a tem-\nperature gradient across the ferromagnet. The simplest\nsetup for SSE is the so-called longitudinal configuration\n[Fig. 1(b)], where the temperature difference is applied\nparallel to the direction of spin injection. Given that the\nferromagnet is attached to a non-magnetic conductor, spin\ncurrent is emitted from the ferromagnet into the neighbour-\ning non-magnetic metal by thermal spin pumping.15,16\nNotably, these two mechanisms of spin current gener-\nation do not require that the ferromagnet be a conductor.\nThe use of an insulator enables generation of pure spin cur-\nrents and limits transport mediated by conduction electrons\nto the adjacent non-magnetic metal. The ferrimagnet yt-\ntrium iron garnet (Y 3Fe5O12, YIG) is a material of choice\nas a spin current injector due to its highly insulating prop-\nerties and high Curie temperature (550 K).17In addition,\nits low magnetic loss properties at microwave frequencies\na)Electronic mail: lustikova@imr.tohoku.ac.jpmake it ideal for efficient spin injection. The magnetiza-\ntion damping in YIG is two orders of magnitude lower than\nthat in ferromagnetic metals.18\nAmong the various fabrication methods of YIG, liquid-\nphase epitaxy (LPE) is known for its ability to produce\nhigh-quality single-crystal films17,19which have been used\nextensively in spintronics experiments.7,10,20However, it\nis difficult to produce films thinner than a few hundred\nnanometers by the LPE method.21Since it has been shown\nthat the interface damping due to SP increases with de-\ncreasing thickness of the ferromagnetic film,2,22synthesis\nof YIG films with thickness below 100 nm is desirable for\nthe study of interface effects. Conversely, the increase and\nsaturation of the spin Seebeck signal with increasing YIG\nthickness has been interpreted as evidence that SSE orig-\ninates in bulk magnonic spin currents.23Therefore, thin\nYIG films are also useful for probing the physics of the\nspin Seebeck effect.\nIn quest of controlling YIG thickness at nanometer\nFIG. 1. Schematic illustrations of the experimental setup for\nspin pumping (SP) and the spin Seebeck effect (SSE). (a) SP and\nthe inverse spin Hall effect (ISHE). H,hac,M(t),jsandsdenote\nthe static magnetic field, the microwave magnetic field, the mag-\nnetization vector, the direction of spin current generated by SP\nand the spin-polarization vector of the spin current, respectively.\nThe bent arrows in the Pt layer denote the motion of the elec-\ntrons under the influence of the spin-orbit coupling which leads\nto the appearance of a transverse electromotive force (ISHE). (b)\nLongitudinal SSE in a YIG jPt bilayer film. ÑTdenotes the tem-\nperature gradient. Spin current is generated along ÑTdue to SSE\nand the electromotive force by ISHE in Pt appears in a direction\nperpendicular both to the sample magnetization and to the tem-\nperature gradient.arXiv:1410.5987v1  [cond-mat.mes-hall]  22 Oct 20142\nscale, the growth of thin films by pulsed-laser deposition\n(PLD)24–28and sputtering29–43has attracted interest. In\nthe sputtering method, the growth of crystals can be real-\nized either by direct epitaxial growth via sputtering at high\ntemperatures29,32,36,38or by sputtering at room tempera-\nture and subsequent post-annealing.30–35,38Direct epitax-\nial growth at high temperature can provide crystals of ex-\ncellent quality.36However, the sputtering rates are usually\nvery low30,34and the sample quality sensitive to the condi-\ntions during deposition. In contrast, sputtering at ambient\ntemperature is technologically more accessible as it does\nnot require a high process temperature and enables faster\ndeposition.30\nAlthough there are various industrial advantages to the\nsputtering method, such as high compatibility with the\nsemiconductors technology, suitability for coating of large\nareas, and dryness of the preparation process, there are\nonly a limited number of reports on the use of sput-\ntered YIG films in spintronics experiments.36,37,39–43In\nthis work, we grow thin YIG films by sputtering and sub-\nsequent post-annealing and confirm epitaxial growth by\ntransmission electron microscopy (TEM). By measuring\nSP and SSE, we demonstrate that the obtained YIG films\nare an efficient spin current generator comparable to LPE\nfilms.\nII. METHODS\nYIG films were deposited by on-axis magnetron rf sput-\ntering on gadolinium gallium garnet (111) (Gd 3Ga5O12,\nGGG) substrates with a thickness of 500 mm. The choice\nof substrate was due to the close match of the lattice con-\nstants and of the thermal expansion coefficients of GGG\nand YIG.33The sputtering target had a nominal compo-\nsition of Y 3Fe5O12. The base pressure was 2 :3\u000210\u00005\nPa. The substrate remained at ambient temperature dur-\ning sputtering. The pressure of the pure argon atmo-\nsphere was 1 :3 Pa. The deposition rate was fairly high\nat 2:7 nm/min with a sputtering power of 100 W. The as-\ndeposited films were non-magnetic; according to Refs. 29–\n35 such films are amorphous. Crystalization was realized\nby post-annealing in air at 850\u000eC for 24 hours. In this\nstudy, we focus on films with a thickness of 83 and 96\nnanometers. The thickness was determined by X-ray re-\nflection (XRR) and TEM. The structure of the samples was\ncharacterized by high-resolution TEM. X-ray photoelec-\ntron spectroscopy confirmed a Y:Fe stoichiometry 3:4.4.\nMicrowave properties were analyzed using a 9.45-GHz\nTE011cylindrical microwave cavity and a coplanar trans-\nmission waveguide in the 3-10 GHz range. The waveguide\nhad a 2-mm-wide signal line and was designed to a 50-\nWimpedance. The width and length of the samples were\nw=1 mm and l=3 mm, respectively.\nFor the spin injection experiments, the annealed YIG\nsamples were coated by a platinum film by rf sputter-\ning. Spin current injected into the platinum layer was de-\ntected electrically using the inverse spin Hall effect [ISHE,\nFig. 1(a)]. ISHE originates in the spin-orbit interaction\nwhich bends the trajectories of electrons with opposite\nspins and opposite velocities in the same direction and pro-\nduces an electric field transverse to the direction of the spin\ncurrent.4–6,44Platinum was chosen for its high conversionefficiency from spin current to charge current.8\nSpin pumping was performed at room temperature in a\ncylindrical 9.45-GHz TE 011cavity at a microwave power\nPMW=1 mW (corresponding to a microwave field m0hac=\n0:01 mT) in a setup illustrated in Fig. 1(a). The sample was\nplaced in the centre of the cavity where the electric field\ncomponent of the microwave is minimized while the mag-\nnetic field component is maximized and lies in the plane\nof the sample surface. A static magnetic field was applied\nperpendicular to the direction of the microwave field and\nto the direction in which the voltage was measured.8Mea-\nsurements were performed on a set of three samples. The\nthickness of the Pt layer was dN=14 nm, the thickness of\nthe YIG layer dF=96 nm.\nThe SSE experiment was performed in a longitudi-\nnal setup identical to that of Ref. 14 on three YIG(83\nnm)jPt(10 nm) samples. The length, the width, and the\nthickness of the samples were LV=6 mm, w=1 mm, and\nLT=0:5 mm, respectively. The sample was sandwiched\nbetween two insulating AlN plates with high thermal con-\nductivity. The upper AlN plate (on top of the Pt layer) was\nthermally connected to a Cu block held at room tempera-\nture. The bottom AlN plate (under the GGG substrate) was\nplaced on a Peltier module. The width of the upper AlN\nplate (5 mm) was slightly shorter than the sample length\n(6 mm) in order to take electrical contacts with tungsten\nneedles. The samples were placed in a 10\u00002Pa vacuum in\norder to prevent heat exchange with the surrounding air. A\nstatic magnetic field was applied in the plane of the sample\nsurface perpendicular to the direction in which the voltage\nwas measured.\nIII. RESULTS AND DISCUSSION\nA. Structural and microwave properties\nFigures 2(a)-(e) present the structural properties of the\n96-nm-thick films observed by TEM. A magnified view of\nthe GGGjYIG interface and the diffraction pattern at this\ninterface are shown in Figs. 2(a) and 2(b), respectively.\nThe YIG grows epitaxially on the GGG substrate. Nei-\nther defects nor misalignment in the lattice planes were\nobserved in the TEM images [Fig. 2(a)]. As shown in Fig.\n2(b), the diffraction pattern consists of a single reciprocal\nlattice confirming perfect alignment of the GGG and YIG\nstructures.\nAn image of the whole cross section of a GGG jYIGjPt\nsample is shown in Fig. 2(c). The YIG film contains spher-\nical defects with a diameter of roughly 10 nm. However,\nthese are suppressed in the vicinity of the YIG jPt inter-\nface. TEM imaging of as-deposited films revealed a uni-\nform amorphous Y-Fe-O layer indicating that the spher-\nical structures emerge during post-annealing. A magni-\nfied view of these objects is given in Fig. 2(d). They do\nnot possess crystalline structure. This can be also inferred\nfrom the fact that only a single reciprocal lattice, corre-\nsponding to epitaxial growth, was observed in the diffrac-\ntion pattern. We speculate that these defects are voids\nwhich appear due to the volume change in the transition\nfrom amorphous to crystalline phase. There is a possibility\nthat these structures contain residual amorphous material3\nFIG. 2. Structural and microwave\nproperties of the YIG films. (a)\nTEM image of the GGG jYIG in-\nterface. (b) Selected area diffrac-\ntion at the GGGjYIG interface with\nthe electron beam along the [0 11]\naxis. (c) The cross section of a\nGGGjYIGjPt sample. (d) Magni-\nfied view of the spherical defects in\nthe YIG structure. (e) Magnified\nview of the YIGjPt interface. (f)\nIn-plane FMR derivative absorption\nspectrum measured in a microwave\ncavity. The fit is a derivative of\nthe Lorentzian function. (d) Fre-\nquency scan of the FMR peak-to-\npeak linewidth Wmeasured using\na coplanar waveguide (circles) and\nvalues obtained in a microwave cav-\nity (squares). The fitting function is\ngiven by Eq. (1).\nleft over in the crystallization. We expect that these struc-\ntures, due to their location inside the film, do not affect\nspin injection efficiency because spin injection originates\nin the spin transport at the F jN interface.2,22\nThe YIGjPt interface is magnified in Fig. 2(e). One\ncan see that the YIG maintains its crystal structure up to\nthe top of the layer. The surface of the YIG film is flat\nwith a roughness less than 1 nm. This clean interface is of\nadvantage for efficient spin injection.45\nXRR measurement on a bare YIG film yielded a YIG\nsurface roughness of (0:008\u00060:002)nm. In contrast,\nthe GGGjYIG interface roughness was (0:6\u00060:1)nm.\nThe fact that the roughness at the GGG jYIG interface was\nmany times larger than that at the YIG surface can be as-\ncribed to substrate damage caused by on-axis sputtering.\nFigure 2(f) shows the ferromagnetic resonance (FMR)\nderivative absorption spectrum dI=dHof a 96-nm-thick\nYIG film measured in a microwave cavity at PMW=1 mW.\nIt consists of a single Lorentzian peak derivative with a\npeak-to-peak linewidth W=0:38 mT. This corresponds\nto a single FMR mode with damping proportional to the\nlinewidth. The linewidth in a broad set of samples var-\nied in the range of 0 :4\u00000:6 mT. These values are among\nthe lowest reported on sputtered YIG films.35,36,38The ef-\nfective saturation magnetization Meffwas determined from\nthe dependence of the FMR field on the direction of the\nstatic magnetic field with respect to the sample plane.8\nThe obtained value Meff= (103\u00064)kA/m is lower than\nthe saturation magnetization value for bulk YIG crystal\n(140 kA/m).46The decrease in the saturation magnetiza-\ntion might be a result of a deficiency in Fe atoms indicated\nby the off-stoichiometry.\nFigure 2(g) shows the frequency fdependence of the\npeak-to-peak linewidth Wmeasured on a 83-nm-thick YIGfilm using a coplanar waveguide. Using a linear fit46\nW=W0+4pp\n3a\ngf (1)\nwith gyromagnetic ratio g=1:78\u00021011T\u00001s\u00001deter-\nmined from the frequency dependence of the FMR field,47\nwe obtain a damping constant a= (7:0\u00061:0)\u000210\u00004. This\nvalue is more than ten times larger than the value for bulk\nsingle crystals (3\u000210\u00005),18but is slightly smaller than\nother values reported on films prepared by sputtering38,40\nand only three times higher than values reported on LPE\nfilms.21,48The increase in the damping constant is proba-\nbly due to two-magnon scattering on defects in the film.49\nB. Spin pumping and spin Seebeck e\u000bect\nThe results of the SP experiment are shown in Fig. 3.\nFigure 3(a) compares the integrated FMR spectra of the\nplain YIG film and the YIG jPt bilayer measured on one\nYIG sample prior to and after Pt coating. The linewidth in\nthe YIGjPt bilayer increases on average by 30% as com-\npared to the linewidth in the bare YIG layer. This corre-\nsponds to enhanced damping of the magnetization preces-\nsion in the YIGjPt sample. This enhancement is caused by\nthe transfer of spin angular momentum to conduction elec-\ntrons in Pt near the YIG jPt interface, indicating successful\nspin injection.\nSimultaneously with the FMR peak of the ferromagnet,\na voltage signal appears across the Pt layer, as shown in\nFig. 3(b). The spectral shape of the voltage signal is\na Lorentzian with the same centre and full-width-at-half-\nmaximum as the FMR spectrum of the YIG [Fig. 3(a)].\nThis is an expected behaviour in ISHE, where the gener-\nated voltage at field His proportional to the microwave4\nFIG. 3. Results of the spin pumping measurement on a YIG(96\nnm)jPt(14 nm) bilayer at PMW=1 mW ( m0hac=0:01 mT). (a)\nThe integrated FMR spectrum of a YIG sample before and af-\nter Pt coating (YIG and YIG jPt, respectively). (b) ISHE voltage\nsignal measured on the Pt layer at qH=\u000090\u000eoverlaid with a\nLorentzian fit. (c), (d) FMR derivative spectra [(c)] and spec-\ntral shapes of the voltage signals [(d)] at selected qHvalues.\n(e) Angle dependence of the peak value of the ISHE voltage\n(“data”) overlaid with the calculated dependence (“calc”). The\ninset shows the definition of qH.absorption intensity I(H).8\nThe fact that the detected voltage signal is due to ISHE\nis confirmed by the qHdependence, where qHis the an-\ngle between the surface normal and the magnetic field [see\ninset of Fig. 3(e)]. The FMR derivative spectra and the\nspectral shapes of the voltage signals at selected values\nofqHare shown in Figs. 3(c) and (d), respectively. The\nspectral shape of the voltage copies the shape of I(H)even\nwhen the magnetic field is tilted out of the sample plane\n(qH=\u000645\u000e). The sign of the voltage reverses by reversing\nthe direction of the magnetic field and the signal vanishes\nwhen the magnetic field is perpendicular to the sample sur-\nface ( qH=0\u000e). This is a signature of ISHE, where the elec-\ntromotive force is generated along the vector product of the\nspin polarization and the spin current, EISHEµjs\u0002s.5,8\nFigure 3(e) shows the qHdependence of the peak value\nof the voltage signal. The black curve is the expected qH\ndependence of the ISHE voltage calculated following the\nprocedure in Ref. 8. The magnitude of the ISHE voltage is\nproportional to the injected spin current, VISHEµjssinqM,\nwhere the spin current magnitude jsis given by Eq. (12)\nin Ref. 8\njs=g\"#\nrg2(m0hac)2¯h\u0014\nm0Meffgsin2qM+q\n(m0Meff)2g2sin4qM+4w2\u0015\n8pa2\u0002\n(m0Meff)2g2sin4qM+4w2\u0003 : (2)\nHere qMis the angle between the magnetization vec-\ntor and the surface normal, and g\"#\nrthe real part of\nthe spin mixing conductance. The relation between\nqHand qMis determined by the resonance condi-\ntion(w=g)2= [m0HRcos(qH\u0000qM)\u0000m0Meffcos2qM]\u0002\u0002\nm0HRcos(qH\u0000qM)\u0000m0Meffcos2qM\u0003\n[Eq. (9) in Ref. 8]\nand the static equilibrium condition 2 m0Hsin(qH\u0000qM)+\nm0Meffsin2qM=0 [Eq. (6) in Ref. 8]. To numerically\ncalculate Eq. (2), we used w=5:94\u00021010s\u00001,g=\n1:78\u00021011s\u00001T\u00001andMeff=103 kA/m. The result of the\ncalculation is in very good agreement with the data, pro-\nviding another piece of evidence that the observed voltage\nis due to ISHE.\nFigure 4 shows the results of the longitudinal SSE mea-\nsurement. Figure 4(a) gives the magnetic field m0Hdepen-\ndence of the voltage signal Vmeasured on the Pt layer for\nselected values of temperature difference DTbetween the\nbottom and the top of the sample. We observed a voltage\nsignal whose sign is reversed by reversing the direction of\nthe magnetic field. Upon increasing the magnetic field,\nthe magnitude of the signal increases monotonically until\nreaching a saturation value. This m0Hdependence of V\nreflects the magnetization curve of YIG.14The saturation\nvalue of the voltage increases with increasing temperature\ngradient. No signal was observed for DT=0 K. As shown\nin Fig. 4(b), the magnitude of the voltage at m0H=30 mT\nis linear in DT. This behaviour is consistent with ISHE in-\nduced by SSE, where the spin current generated across the\nYIGjPt interface is proportional to the temperature gradi-\nentÑT.12C. Spin injection e\u000eciency\nThe normalized value of the ISHE electromotive force\nobserved in the SP experiment is EISHE=(m0hac) = ( 150\u0006\n30)mV/(mm\u0001mT). This is a few times higher than the value\nreported on a 4.5 mm-thick LPE film, EISHE=(m0hac) =39\nmV/(mm\u0001mT), measured at the same equipment,45and\ncomparable with values reported for LPE films of 1.2- mm\nthickness [160 mV/(mm\u0001mT) in Ref. 7].\nAs for the ISHE voltage in the SSE measurement, using\nthe experimental value V= (5:6\u00061:2)mV atDT=10 K,\nwe obtain a normalized voltage V\u0002LT=LV= (0:47\u00060:10)\nFIG. 4. Results of the SSE measurement on a YIG(83 nm) jPt(10\nnm) bilayer. (a) Magnetic field m0Hdependence of the voltage\nsignal Von the Pt layer measured for different values of temper-\nature difference DTacross the GGGjYIGjPt sample in the lon-\ngitudinal SSE configuration. (b) DTdependence of the voltage\nmagnitude at m0H=30 mT.5\nmV . This is also of the same order as the value for YIG\nprepared by LPE (1 mV in Ref. 14 for a 4.5- mm-thick film).\nFinally, we estimate the spin mixing conductance at the\nYIGjPt interface. The efficiency of the transfer of spin an-\ngular momentum at the F jN interface is described by the\nreal part of the spin-mixing conductance g\"#\nr,7,50–52which\nis given by52–54\ng\"#\nr=4pMsdF\ngmBp\n3g\n2w\u0000\nWF=N\u0000WF\u0001\n: (3)\nHere, gis the g-factor, mB=e¯h=(2me) =9:27\u000210\u000024\nJ\u0001T\u00001the Bohr magneton, Msthe saturation magnetiza-\ntion; and WFandWF=Nare the peak-to-peak linewidth\nof the FMR spectrum in the bare ferromagnetic film and\nin the FjN bilayer, respectively. Using g=2:12,Ms\u0019\nMeff=103 kA/m, w=g=0:334 T, dF=96 nm, WF=\n(0:40\u00060:03)mT and WF=N= (0:52\u00060:02)mT, we ob-\ntain a spin-mixing conductance g\"#\nr= (2:0\u00060:2)\u00021018\nm\u00002. This value is of the same order as those in other re-\nports on the YIGjPt interface, e.g. g\"#\nr=1:3\u00021018m\u00002\nin Ref. 45. Thus, both in SP and in SSE, we have obtained\nspin injection efficiencies comparable to those reported at\nLPE-made-YIGjPt bilayer samples. This result suggests\nthat defects inside the YIG film do not significantly affect\nthe transfer of spin angular momentum at the interface with\nPt. The high spin injection efficiency is promoted by the\npresence of a regular garnet structure in the vicinity of the\ninterface with Pt as well as the clean interface, as observed\nby TEM imaging.\nIt is worth noting that based on Eqs. (2) and (3), a\ndecrease in magnetization from 140 kA/m to 103 kA/m\nshould lead to a 28 % decrease in the injected spin current.\nA corresponding suppression of the inverse spin Hall volt-\nage should be observed. However, the errors in the spin\npumping and the SSE voltage measurements were 20 %\nand 21 %, respectively. This degree of error does not al-\nlow to discuss the effect of the decreased magnetization on\nspin pumping.\nTo conclude, we estimate the spin Hall angle of Pt from\nthe obtained data. The injected spin current determined\nfrom Eq. (2) is js=5:3\u000210\u000010J/m2, where we have used\ng\"#\nr=2:0\u00021018m\u00002,m0hac=0:01 mT, a=7:0\u000210\u00004,\ng=1:78\u00021011T\u00001s\u00001,w=5:94\u00021010s\u00001,Meff=103\nkA/m, and qH=qM=\u000090\u000e. The peak value of the in-\nverse spin Hall voltage is given by8\nVISHE=dEqSHElNtanh(dN=2lN)\ndNsN\u00122e\n¯h\u0013\njs; (4)\nwhere dEis the distance of the electrodes, qSHEthe spin\nHall angle, lNthe spin diffusion length in Pt, dNthe thick-\nness of the Pt layer, and sNthe conductivity of Pt. Using\nVISHE=3:9mV ,dE=2:6 mm, dN=14 nm, sN=3:1\u0002106\nW\u00001m\u00001, and js=5:3\u000210\u000010J/m2, we obtain qSHE=\n0:029 or 0 :007 for Pt spin diffusion length lN=1:4 nm,55\nor 10 nm,56respectively. Both values are within the range\nof spin Hall angles reported for Pt.IV. CONCLUSIONS\nIn summary, we have prepared YIG films by the sput-\ntering method and investigated their structural and mi-\ncrowave properties as well as spin current generation from\nthese films. The results show that the presented fabrica-\ntion method, consisting of sputtering at room temperature\nand post-annealing in air, provides epitaxial YIG films\nwith thickness below 100 nm which have excellent mi-\ncrowave properties in spite of defects in the structure. The\nspin injection efficiency observed in spin pumping and in\nthe spin Seebeck effect is comparable with that for high-\nquality films prepared by liquid phase epitaxy. The above\npreparation of garnet films is relatively straightforward and\nthe technological requirements modest. Moreover, this\nmethod offers a possibility to control the YIG thickness\nat the nanometer scale. These results are of potential use\nin spintronics research.\nACKNOWLEDGMENTS\nWe thank S. Ito from Analytical Research Core for Ad-\nvanced Materials, Institute for Materials Research, To-\nhoku University, for performing transmission electron mi-\ncroscopy on our samples. 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Lett. 99, 226604 (2007)."    },    {        "title": "1905.01522v2.Identification_and_time_resolved_study_of_YIG_spin_wave_modes_in_a_MW_cavity_in_strong_coupling_regime.pdf",        "content": "1 \n  \n \nIdentification and time-resolved study of YIG spin wave \nmodes in a MW cavity in strong coupling regime \nAngelo Leo1,2,*, Anna Grazia Monteduro1,2, Silvia Rizzato2, Luigi Martina1,3, Giuseppe \nMaruccio1,2,* \n1Department of Mathematics and Physics, University of Salento, Via per Arnesano, 73100, Lecce, Italy \n2CNR NANOTEC - Istituto di Nanotecnologia, Via per Arnesano, 73100 Lecce, Italy, \n3 INFN – Sezione di Lecce, Via per Arnesano, 73100 Lecce, Italy \n \n \nABSTRACT  \nRecently, the hybridization of microwave-frequency cavity modes with collective spin excitations \nattracted large interest for the implementation of quantum computation protocols, which exploit the \ntransfer of information among these two physical systems. Here, we investigate the interaction among \nthe magnetization precession modes of a small YIG sphere and the MW electromagnetic modes, \nresonating in a tridimensional aluminum cavity. In the strong coupling regime, anti-crossing features \nwere observed in correspondence of various magnetostatic modes, which were excited in a \nmagnetically saturated sample. \nTime-resolved studies show evidence of Rabi oscillations, demonstrating coherent exchange of \nenergy among photons and magnons modes. To facilitate the analysis of the standing spin-wave \npatterns, we propose here a new procedure, based on the introduction of a novel functional variable. \nThe resulting easier identification of magnetostatic modes can be exploited to investigate, control and \ncompare many-levels hybrid systems in cavity- and opto-magnonics research. \n \nKeywords:  YIG – Cavity electrodynamics – Ferromagnetic resonance – Magnetostatic modes – \nStrong coupling \n \n* corresponding authors: angelo.leo@unisalento.it , giuseppe.maruccio@unisalento.it   2 \n INTRODUCTION  \nCombining different fundamental excitations is a recent route for quantum computation applications, \nwith the promise to stimulate the development of new hybrid quantum technologies and protocols. \nIndeed, it was suggested that encoding information in different physical systems can provide \nadvantages in overcoming the strict requirements in terms of decoherence timescales and capacity to \nprocess the information, which can be difficult to match together. In this respect, a crucial requirement \nis the achievement of a strong coupling regime between the respective fundamental excitations in two \nphysical systems. Recent findings demonstrated the capability to obtain a robust hybridization among \nlight quanta and different excitations at low temperature, by employing trapped atoms [1], nitrogen \nvacancy (NV) centers in diamonds [2], superconducting (SC) qubits [3] and spin impurities in Si [4]. \nIn this frame, magnons exhibited strong stability in coupling with photons, when they are excited in \nferro/ferri-magnetic (FM) materials, especially if Yttrium Iron Garnet (YIG) single crystals are used \n[5]. In contrast with paramagnetic spin ensembles, which at room temperature (RT) are weakly \ncoupled to the photons, YIG presents at least a three orders greater net spin density, which permits to \nget the strong coupling. To couple spin waves (SW) with electromagnetic (EM) signals, a convenient \nway is to confine the YIG in a three-dimensional (3D) microwave (MW) cavity [6]. It was reported \nthat this yields the stable formation of two-level systems and magnon-cavity polaritons, as a \nconsequence of the hybridization among MW photons and the fundamental magnetostatic mode (also \nknown as Kittel mode and corresponding to ferromagnetic resonance (FMR)) [7-13]. Even non-\nuniform magnetostatic modes (MSMs) can be sustained by the material depending on its shape and \nthey can be also coupled to cavity modes [14, 15]. The resulting pattern of spectral features is more \ncomplex and its association with specific modes can be not straightforward.  \nIn the present work, we investigate the interaction among the magnetization precession modes, in a \nsmall magnetically-saturated YIG sphere, and the MW electromagnetic modes, resonating in a \ntridimensional aluminum cavity, at room temperature. A rich spectrum characterized by several anti-\ncrossing features is observed, because of the strong coupling regime in correspondence of various \nmagnetostatic modes. Time-resolved studies show evidence of Rabi oscillations, demonstrating (for \nthe first time at room temperature) coherent exchanges of energy among photons and the involved \nmagnons modes. For facilitating the analysis of the stationary spin-wave patterns, here we propose a \nnew procedure, based on the introduction of a novel functional variable, related to the magnetic \ncharacteristics of the FM material and to the applied external electromagnetic field. Notably, plotting \nthe data with respect to this variable, we obtain a direct identification of the involved MSMs.  \n \n 3 \n EXPERIMENTAL SETUP AND CAVITY MODES  \nThe investigated system is composed by a single crystal YIG sphere, with 1 mm of diameter [16], \nlocated into an aluminum cavity with inner dimensions of 44 × 22 × 9 mm3. The Yttrium Iron Garnet \n(YIG, Y 3Fe5O12) is chosen for its peculiar characteristics. In particular, its magnetic moment comes \nfrom Fe+3 ions in the 6S 5/2 ground state and YIG behaves as a ferrimagnetic insulator with a  550 K \nCurie temperature, a typical saturation magnetization MS = 0.178 T, spin density of 4.22·1027 m-3, an \nexchange constant  = 3×10-12 cm2 [17]. A crucial characteristics for the applications is the low \nmagnetic damping and a correlated narrow linewidth of 2.3·10-3 mT [17]. This property has favored \nthe use of the YIG crystals in optical and radiofrequency devices, such as microwave oscillators, \ncirculators and optical isolators, since many decades.  \nFirst, the YIG sphere is placed in a central position at the bottom of one semi cavity of the ground \nwall as shown in Fig. 1.a. Such a position corresponds to the magnetic antinode for the fundamental \n(TE101) mode [8], in order to maximize the interaction of magnonic modes with the MW field ( Fig. \n1.b). Then, the cavity resonator is placed between the poles of a GMW electromagnet ( Fig. 1.e) \ngenerating a magnetostatic field, whose intensity is swept from 250 mT up to 330 mT (with steps of \n0.2 mT). Similarly, for further studies on the second (TE 102) mode, the YIG sphere is placed on one \nof the three TE 102 magnetic antinodes, located on the junction plane of the two semi cavities (this \nconfiguration is reported in Fig. 1.c). More precisely, the sphere is put close to the rounded wall of \nthe resonator, on one of the lateral antinodes as shown in Fig 1.d. In this case, the magnetostatic field \nintensity is swept from 360 mT to 440 mT, in order to get the strong coupling regime. The resonator \nis excited by an Agilent MXG N5183A signal generator, while transmission measurements are \nperformed by an Agilent MXA N9010 spectrum analyser. Both devices are controlled by a homemade \nLABVIEW software. Specifically, the frequency is swept in a range of 320 MHz, centered around \nthe first (or the second) cavity eigenfrequency, at fixed magnetic field, and spectroscopic \nmeasurements were performed applying a 0 dBm input power.  \nAt room temperature, zero DC magnetic field and loaded with the YIG sphere, experimentally the \ncavity exhibits the TE 101 mode at 𝜔/2𝜋 = 8.401 GHz, the TE 102 mode at 𝜔/2𝜋 = 10.361 GHz. The \nloaded quality factor 𝑄 of the mere cavity with YIG at TE 101 is 4000, with insertion loss 𝐼𝐿 = -33.1 \ndB. This leads to an estimated intrinsic Q-factor 𝑄=𝑄/(1−10ூ/ଶ)  ≃ 4100. The second mode \nexhibits a 𝑄 of 4300, with 𝐼𝐿 = -30.16 dB and 𝑄 = 4450. In these conditions, the system remains \nlossy coupled to the measurement setup. At both the TE 101 and TE 102 magnetic antinodes, the insertion \nof the YIG sphere does not perturb significantly the EM signal, in absence of a drive magnetostatic \nfield.  Indeed a resonant frequency shift of less than 0.1% is observed. Furthermore, the ratio between \nthe crystal volume VYIG and the magnetic modal volume 𝑉 is 𝑉ூீ / 𝑉 ≈ 2·10-4 , which also justifies \nthe observed negligible variation of the quality factor. 4 \n  \nFigure 1. (a) 3D aluminium semi-cavity loaded by a YIG sphere and (b) FEM-simulated MW magnetic field distribution \nfor the loaded cavity at TE 101 in a perpendicular static magnetic field H. (c) semi-cavity loaded by YIG sphere on lateral \nside and (d) corresponding MW magnetic field for TE 102 (black dots on FEM simulations refer to position of sphere, in \ncorrespondence of the magnetic antinodes in the two configurations). (e) Experimental setup showing the loaded cavity \nbetween the electromagnet poles for application of the static magnetic field. \nSTRONG COUPLING REGIME    \nThe loaded-cavity spectrum as a function of the perpendicular magnetic field is reported in Fig. 2. \nWhile measuring at frequencies around the first photonic mode, sweeping of the magnetostatic field \nintensity gives a rich magnonic spectrum, which is characterized by the presence of anti- crossings \npoints around TE 101, as shown in the 2D map in Fig. 2.b. The ferromagnetic resonance (FMR, known \nas Kittel mode [7]) lies at 294.1 mT, as indicated by a yellow arrow. Additionally, a series of MSMs \nare exhibited, which are pointed out by dotted yellow lines, whose separation gradually reduces \ntowards lower magnetostatic fields. Fig. 2.a illustrates how the signal amplitude at TE 101 varies as a \nfunction of the magnetic field: the transmitted electromagnetic signal takes the form of Fano \nresonances [18, 19] and it is significantly reduced in correspondence of the avoided crossings, with a \nsplitting of resonance peak (in Fig. 2.b). On the other hand, the 2D map at TE 102 shows only two \nclearly visible avoided crossings ( Fig 2.c), in addition to the main uniform precession resonance \n(FMR) at 391.0 mT, indicated by the yellow arrow. The relative signal amplitude from the cavity at \nTE102 as a function of static magnetic field is reported in Fig. 2.d. \n \n          \n5 \n Figure 2. (a) Transmission amplitude at TE 101 = 8.401 GHz for magnetic field ranging from 250 mT to 330 mT.  (b) Cavity \nresponse as a function of bias magnetic field and frequency near the fundamental TE 101 mode. Dashed yellow lines in the \n2D maps of amplitude refers to identified MSMs.  (c) Cavity response near TE 102 for magnetostatic field ranging from 360 \nmT to 440 mT. (d) Transmission amplitude at TE 102 = 10.361 GHz for magnetic field ranging from 360 mT to 440 mT (in \nthe second configuration with sphere on cavity lateral side).  \n \n \na \n \nc \n b \n \nd \n 6 \n The avoided level crossings in the map of transmitted signal amplitude 𝑇 can be described by means \nof an input-output formalism [9, 13, 20, 21]: \n𝑇(𝜔) = \n(ఠିఠ)ିభ\nమ(ଶା)ା∑||మ\nషభ\nమംశೕ(ഘషഘ)   ,    (1)   \nwhere  𝜅 /2𝜋=(2𝜅+𝜅௧)=𝜔 /2𝜋𝑄 is the photonic damping for each mode resonating at \n𝜔 /2𝜋, which takes into account the damping 𝜅 through the single connectors and the internal \ndamping int associated to the mere aluminium cavity, 𝑗 is the imaginary unit number, the index i \nidentifies a specific magnetostatic mode, whose frequency linewidth is 𝛾 and the interaction strength \nof the hybrid mode for the whole magnonic system is \n\nଶగ=బ,√ே\nଶగ=ቆఎට(ഋబℏഘ)\nೇቇ\nଶగ√𝑁,          (2)   \nwhich is related to the coupling strength  𝑔,/2𝜋 for a single interacting spin through the number N \nof net spins in the examined sample [22]. Notably, \nଶగ refers to coupling strength between the two \n(magnonic and photonic) states and is evaluated as the frequency mismatch among the resonant peaks \nof the hybrid modes ( Fig. 3, on the right). The quantity Γ =𝑔𝜇/ℏ is the gyromagnetic ratio of spin \nensemble, where 𝑔 is Landé g-factor, 𝜇 is Bohr magneton, ℏ is reduced Planck constant and 𝜇 is \nthe vacuum permeability. The spatial overlap coefficient 𝜂=∫ுሬሬ⃗∙ெሬሬ⃗\nுೌೣெೌೣೊಸ\n௦𝑑𝑉 between the \ntwo subsystems (i.e. cavity and sphere modes) is calculated taking into account the driving MW \nmagnetic field 𝐻ሬሬ⃗ and the complex time-dependent off- z axis sphere magnetization 𝑀ሬሬ⃗ for the \nconsidered mode, while 𝐻௫ and 𝑀௫ correspond to their maximum values in the sphere volume \nVYIG. As a further figure, the cooperativity of the two levels system is defined as 𝐶=𝑔ଶ/𝛾𝜅. \nIn Fig. 3, details of the spectra of cavity toward hybridization, near the fundamental magnestostatic \nmode (FMR), are reported for cavity modes TE 101 and TE 102 at fields indicated by the vertical lines \nof corresponding colour. When the subsystems are fully coupled, 𝑔/2𝜋 corresponds to the frequency \nmismatch between the resonance peaks, and broadenings 𝛾/2𝜋 and 𝜅/2𝜋 of both waves modes are \ncomparable. In these conditions, obtained when the magnetostatic field intensity is 294.1 mT for the \nTE101 and 391.0 mT for the TE 102 mode, cavity dissipation rates 𝜅/2𝜋  are 3.2 MHz and 3.0 MHz, \nrespectively. Finally, the magnonic damping 𝛾/2𝜋  is 2.5 MHz. For a FM sphere, the uniform \nprecession frequency 𝜔ிெோ2𝜋⁄ is related to the external field 𝐻 by ఠಷಾೃ\nଶగ=𝛤𝐻 [23]. Since the \ncavity mode frequencies and 𝜔ிெோ2𝜋⁄ must match when the two subsystems are strongly coupled, \nit is then possible to estimate the gyromagnetic ratio 𝛤 by posing 𝜔ிெோ=𝜔= 2𝜋 𝛤𝐻 and by \nsubstituting the corresponding 𝐻 values then we obtain 𝛤≈ 28.76 GHz/T for the YIG sphere with 7 \n respect to the value 28 GHz/T reported in [24]. In Table 1, the magnetostatic fields as well as the \nevaluated coupling parameters, photonic/magnonic dampings and cooperativity C are summarized \nfor all the observed anticrossing points (and will be discussed in more details in the next section with \nreference to the MSM identification). \n \nFigure 3. Strong coupling between the fundamental magnetostatic mode in the YIG sphere and the TE 101 and TE 102 cavity \nmodes. (a) 2D maps around TE 101 when the magnetostatic field ranges between 289 mT and 299 mT. The spectra \ncorresponding to coloured lines are reported on the left, where formation of magnons-polariton systems is shown. (b) 2D \nmaps around TE 102 when the magnetostatic field ranges between 386 mT and 396 mT. The spectra corresponding to \ncoloured lines are reported on the right. When the systems are fully coupled, g/2π is the frequency mismatch between the \nresonance peaks, and broadenings γ/2π and κ/2π of both standing waves modes are similar. \nIDENTIFICATION OF MSM  \nIn order to identify and analyse the magnetization precession phenomena corresponding to the \nobserved anticrossing features, a discussion in terms of magnetostatic theory is useful [24-27]. The \nMSMs resonant frequencies 𝑓 (and dispersion relation of spin wave modes, generally) in spheroids \ninserted in MW cavity working at frequency 𝜔/2𝜋 can be derived from characteristic equation in \nterms of associated Legendre functions 𝑃(𝑓,𝐻): \n    𝑛+1+𝜉 ᇲ(,ுబ)\n(,ுబ)±𝑚ெೄ\nమுమିమ= 0,   (3)  \nwhere 𝐻= 𝐻−𝑀ௌ/3,   𝜉=ቀ1+ு\nெೄ−మ\nమெೄுቁଵ/ଶ\n and  𝑃ᇱ(𝑓,𝐻)=ௗ(,ுబ)\nௗకబ.  \nThe index 𝑛 ∈ ℕ labels localization of MSM on the surface with respect to the external magnetostatic \nfield 𝐻ሬሬ⃗, while 𝑚 (|𝑚|≤ 𝑛) refers to the angular momentum.  \na \n \nb \n 8 \n The MSMs relative to indexes 𝑛 and 𝑚 are then labelled as ( 𝑛,𝑚) and are grouped in families as a \nfunction of the value 𝑛−|𝑚|. Generally the resonant frequencies of MSM are dispersive, except the \nmode families 𝑛−|𝑚|= 0 and 1. In these cases, the Eq. 3 assumes the simplified forms: \n  \nெೄ−ுబ,\nெೄ+ଵ\nଷ=\nଶାଵ (𝑛=𝑚),     (4)  \n\nெೄ−ுబ,(శభ)\nெೄ+ଵ\nଷ=\nଶାଷ (𝑛=𝑚+1).     (5)  \nThe Eq. 4 with fixed 𝑛=𝑚= 1 gives 𝑓=ఠಷಾೃ\nଶగ=𝛤𝐻,ଵଵ, corresponding to uniform magnetization \nprecession. \nIf the FM sphere is immersed in a confined magnetic field oscillating at frequency 𝜔/2𝜋 and a strong \ncoupling regime is reached at 𝐻, , the resonance frequencies of the two subsystems must match.  \nBy imposing this condition it is possible to determine the indexes 𝑚 and 𝑛 of the MSMs associated \nto each anticrossing. Thus, after extrapolating the 𝐻,  values associated to the various \nanticrossings observed at  𝑓≈𝜔ଵଵ/2𝜋  = 8.405 GHz for the map at TE 101 reported in Fig. 2.a [28], \na preliminary identification of ( 𝑚,𝑚) MSM can be carried out. Accordingly to Eq. 4, by using an \napproximate values for Γ (also estimated from 𝐻,ଵଵ=ఠಷಾೃ\nଶగ  ) and of 𝑀ௌ (0.178 T in literature) [24],  \nwe exploit the discrete nature of the indexes to facilitate association. Moreover, analysing the ( 𝑚,𝑚) \nMSM, the spacing of nine consecutive splittings as a function of the external field is observed to \nreduce,  moving far from FMR condition in Fig. 2. Subsequently, the trend of the magnetostatic field \n𝐻, corresponding to ( 𝑚,𝑚) MSM anticrossing conditions ( Fig. 4.a) is exploited to evaluate  the \nsaturation magnetization 𝑀ௌ, as a fitting parameter for Eq. 4 [29].  \nHowever, the identification of the involved MSM following this procedure is not immediate and \nhinder further analysis from Fig. 2. Starting from this observation and the previously discussed \ntheory, for a more straightforward identification of the different MSM, here we propose as an ansatz \nto rearrange data in order to plot the resonator signal as a function of cavity frequency and \n[𝐻,−(ఠ\nଶగ௰−𝑀ௌ/6)]ିଵ (Fig. 4.b). The reason is that (according to mathematical manipulations \nof Eq. 4 and 5 ), this is expected to bring to equally spaced features for each family when 𝑛−|𝑚|=\n0 or 1, as it is demonstrated from results reported in Fig. 4.b. In this frame, the identification of MSM \nof (𝑚,𝑚) and (𝑚+1,𝑚) families at 𝜔ଵଵ/2𝜋 is more direct. As a further improvement, in Fig. 4.c \nthe signal is plotted as a function of −12ൗ+𝑀ௌ/4ቂ𝐻,−(ఠ\nଶగ௰−𝑀ௌ/6)ቃିଵ\n, which is obtained by \na rearrangement of Eq. 4. It results that the local minima in oscillations of cavity amplitude are shown \non an x-axis now indicative of the mode (𝑚,𝑚) index, clearly visible until the 9th excitation. Notably, 9 \n in this procedure, even if for 𝑀ௌ a value from literature is employed, anyhow the discrete nature of \nthe indexes would allow a simple association of the mode.  \nFor (𝑚+1,𝑚) MSM identification, the magnetic field axis should be modified following Eq. 5. In \nthis case, the main differences with the previous calculation are the presence of -3/2 as coefficient \nand a scaling of 3/4 instead of -1/2 and 1/4, respectively. Apparently, several features corresponding \nto modes (4+3k, 3+3k), k ∈ ℕ, are detected up to 𝑚 equal to 27.  However, the most pronounced \nabsorptions are limited to a few recognizable modes, which are degenerate with respect to other \n(𝑚,𝑚) ones,  thus these features could be also ascribed to the latters. Only smaller (𝑚+1,𝑚) MSM \nsignatures can be distinguished separately in Fig. 2.a (see red arrows), as the (5, 4) and (6, 5) modes. \nAs a result, we conclude that the (𝑚+1,𝑚) MSM family is less coupled to the cavity.  \nIn Table 1, the identified (m, m) MSMs observed for the YIG sphere within the cavity for the first \ntransverse electric mode are reported together with their corresponding magnetostatic fields, coupling \nparameters and photonic/magnonic dampings. Cooperativity C among FMR and TE 101 is at least one \norder greater than values at other frequency splittings. Moreover, the photonic and magnonic losses \nwere estimated to be comparable for all the different MSM modes. \nH0 (T) MSM - \nTE101 /2 \n(MHZ) /2 \n(MHZ) g/2 \n(MHZ) C \n0.2718 (9, 9) 3.9 4.0 4.1 1.1 \n0.2722 (8, 8) 2.9 4.2 4.7 1.8 \n0.2728 (7, 7) 3.1 3.3 5.8 3.3 \n0.2734 (6, 6) 3.1 3.1 7.1 5.2 \n0.2746 (5, 5) 3 3.1 8.7 8.1 \n0.2760 (4, 4) 3.1 2.8 10.9 13.6 \n0.2780 (3, 3) 2.5 2.5 14.4 33.2 \n0.2826 (2, 2) 1.6 2.3 25.1 171.2 \n0.2882 (5, 4) 2.0 4.0 8.0 8.0 \n0.2941 (1, 1)FMR 2.5 3.2 53.5 1431.1 \n \nTable 1. Summary of obtained parameters for TE 101 mode. \n 10 \n  \nFigure 4. (a) Trend of the magnetic fields at resonance conditions as a function of (m, m) MSM order at TE 101. (b) 2D \nmap as a function of frequency near TE 101 and the introduced functional variable. In this frame, MSMs appear equally \nspaced. (c) At 8.405 GHz, (m, m) modes are recognized up to m = 9. On the other hand, (m+1, m) excitations are not all \nvisible (only the one corresponding to indexes reported in red and in large part being degenerate with (m, m) modes). \n \n \n \nb \nc 11 \n TIME-RESOLVED MEASUREMENTS  \nFor further insight in the strong coupling regime, we investigate also the time evolution of the strongly \ncoupled system, applying a pulsed 3 s signal, modulated at the cavity resonance frequency.  The \nsignal is collected by an ultrafast digital sampling oscilloscope (Tektronix DSA8300). The input \namplitude was set at 10 dBm. The digital sampling oscilloscope is set in order to acquire the envelope \nof the MW signal by sampling every 125 ps. The spin ensemble dynamics heavily influences the \nsignal amplitude, and both rise and fall times of cavity signal. In Fig. 5a, the cavity response at TE 101 \nis shown: for lower values of the magnetostatic field, the signal is weakly modified, then it \nsignificantly grows for all the pulse duration, with a sensitive increase of cavity relaxation timescale. \nSuccessively, by a further increasing of the field, the signal drastically reduces and then increases \nagain and this phenomenon is repeated many times in correspondence of the anticrossing fields. The \nproposed ansatz was employed to facilitate analysis also of this set of (time-resolved) measurements \nleading to Fig. 5.b, where MSMs up to (9, 9) are visible as increased absorptions [30]. The time scan \nrelative to the coupling among the MW cavity TE 101 and the (1,1) magnetostatic mode is shown in \nFig. 5.c. Periodic fluctuations of cavity signal amplitude during charging and relaxation of the system \nare clearly visible. These oscillations are a further indication of on-resonance hybridization among \nmagnons and photons, which is reached when 𝜔/2𝜋=𝑓, resulting in a Rabi splitting in two \npeaks at energies ℏ2𝜋ൗ(𝜔±𝑔). The energy stored inside the cavity decays in an exponential \nmanner, but with periodic oscillations among the two states, demonstrating coherent exchange of \nenergy among photon and magnon modes.  \n 12 \n                                                    \nFigure 5 . (a) Transmission of a rectangular 3 s microwave pulse at TE 101 through the cavity versus time and magnetic \nfield. (b) time-resolved 2D map with x axis rescaled as a function of MSM index. (c) Dynamics at resonance corresponding \nto strong coupling between TE 101 cavity mode and (1, 1) MSM. \n \nCONCLUSION    \nThe strong interaction regime between MSM modes excited in an YIG sphere and photonic modes in \na 3D cavity resonator was investigated at RT. A rich spectrum was observed with several anti-\ncrossing features due to coupling to the fundamental FMR mode as well as additional magnetostatic \nmodes. As an ansatz for enabling a simple identification of the various MSMs, we proposed a \nrescaling procedure in order to plot the resonator signal as a function of cavity frequency and a novel \nfunctional variable, just from a mathematical rearrangement of the eigenvalue equations for the \nMSMs in a spheroids (for both the (𝑚,𝑚) and (𝑚+1,𝑚)  families)). This procedure was applied to \nboth the frequency-dependent and time-resolved scans in magnetic field and allowed us to recognize \nthe  (𝑚,𝑚) modes as the ones more coupled to the cavity. Magnetostatic fields, coupling parameters, \nphotonic/magnonic dampings and cooperativity C were evaluated for all modes up to the (9,9) one. \na \nc b \n 13 \n Rabi oscillations were clearly visible in time scans, demonstrating (for the first time at room \ntemperature) coherent exchange of energy among photons and the involved magnons modes. The \neasier identification of magnetostatic modes can be exploited to investigate, control and compare \nmany-levels hybrid systems in cavity- and opto-magnonics research [31-33].  \n \nACKNOWLEDGEMENTS \nThis work was financially supported by the MIUR-PRIN Project (prot.2012EFSHK4) and Regione \nPuglia NABIDIT – NANOBIOTECNOLOGIE e SVILUPPO PER TERAPIE INNOVATIVE Project \n(F31D08000050007), L.M. was partially supported by the INFN-IS MMNLP. \n \n \nREFERENCES       \n \n1. Dudin, Y. and A. Kuzmich, Strongly interacting Rydberg excitations of a cold atomic gas.  Science, 2012. \n336(6083): p. 887-889. \n2. 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Slightly shifted from 8.401 GHz due to proximity to strong coupling condition.  \n29. The estimated value of gyromagnetic ratio is 28.76 GHz/T, while the experimental value of saturatioin \nmagnetization is 0.176 T. \n30. Values estimated from time-resolved set of data: gyromagnetic ratio 29.24 GHz/T, saturation \nmagnetization 149.49 mT. \n31. Liu, Z.-X., et al., Phase-mediated magnon chaos-order transition in cavity optomagnonics.  Optics letters, \n2019. 44(3): p. 507-510. \n32. Yang, Y., et al., Control of the magnon-photon level attraction in a planar cavity.  Physical Review \nApplied, 2019. 11(5): p. 054023. \n33. Bhoi, B., et al., Abnormal anticrossing effect in photon-magnon coupling.  Physical Review B, 2019. \n99(13): p. 134426. \n "    },    {        "title": "1506.05935v1.All_electrical_coherent_control_of_the_magnetization_in_thin_Yittrium_Iron_Garnet_film.pdf",        "content": "arXiv:1506.05935v1  [cond-mat.mes-hall]  19 Jun 2015All electrical coherent control of the magnetization in thi n Yittrium Iron Garnet film\nO. Wid1, M. Wahler1, N. Homonnay1, T. Richter1, and G. Schmidt*1,2\n1Institut f¨ ur Physik, Martin-Luther-Universit¨ at Halle- Wittenberg,\nD-06099, Germany*\n2IZM, Martin-Luther-Universit¨ at Halle-Wittenberg,\nD-06099, Germany*\nWe demonstrate coherent control of time domain ferromagnet ic resonance by all electrical excita-\ntion and detection. Using two ultrashort magnetic field step s with variable time delay we control the\ninduction decay in yttrium iron garnet (YIG). By setting sui table delay times between the two steps\nthe precession of the magnetization can either be enhanced o r completely stopped. The method\nallows for a determination of the precession frequency with in a few precession periods and with an\naccuracy much higher than can be achieved using fast fourier transformation. Moreover it holds the\npromise to massively increase precession amplitudes in pul sed inductive microwave magnetometry\n(PIMM) using low amplitude finite pulse trains. Our experime nts are supported by micromagnetic\nsimulations which nicely confirm the experimental results.\nCoherent control of spin precession is well known from\nsemiconductor spintronics. There for example ultrashort\npulses of circularly polarized light are used to induce a\nspin polarized carrierpopulation in a non-magneticsemi-\nconductor. The spin polarized carriers precess in an ex-\nternal magnetic field and the precession is detected by\ntime resolved Faraday rotation [1] or Kerr rotation [2].\nWhen the excitation pulses are applied in sequence syn-\nchronized with the precession frequency the polarization\nincreases or decreases depending on the relative phase of\nprecession and excitation [3], [4], [5]. If a train consist-\ning of a large number of pulses is used to increase the\nsignal far beyond the single excitation case the effect is\nalso named resonant spin amplification [6], [7]. While\nthese methods employ optical excitation and/or detec-\ntion a similar scheme can be envisaged for time domain\nferromagnetic resonance using electrical excitation by ul-\ntrashort magnetic field steps and electrical detection by\ninduction. As will be shown a main advantage of this\nresonant excitation especially in PIMM is the massively\nenhanced resolution. In addition using a synchronized\npulsetrainmayreducetheproblemswhichnormallyarise\nin PIMM from the unfavorable ratio of the very large ex-\ncitation and the very small inductive response which are\ndifficult to separate.\nThebasicprincipleofall-electricalcoherentmagnetiza-\ntion control is illustrated in figure 1. We consider a mag-\nnetic sample with the magnetization /vectorMwhich is aligned\nalong a constant magnetic field /vectorH0along the x-axis (Fig.\n1, (a)). To excite the magnetization out of its equilib-\nriumstate weapplyasmallmagnetic fieldstep /vectorHy1(with\n|/vectorHy1| ≪ |/vectorH0|) at time t= 0 with ultra-short rise time\nalong the y-axis, which tilts the magnetic field towards\n/vectorH1. The angle between /vectorH1and/vectorMis dubbed Θ and at\ntimet= 0 is identical to the angle between the magnetic\nfield vectors /vectorH0and/vectorH1which we name Θ 0. The magne-\ntization now startsto precess around /vectorH1accordingto the\nLLG equation (Fig. 1, (b)) with a precession frequency\nwhich is in good approximation ω=|/vectorH0|γ. The preces-x\nyzx\nz\nyH0\nM MH1Hy1(a) (b)\nFIG. 1: Basic concept of Pulsed Inductive Microwave Mag-\nnetometry (PIMM). As the initial state the magnetization is\naligned along the magnetic field /vectorH0(a). The first field-step\n/vectorHy1tilts the direction of the effective magnetic field, which\nleads to damped precession of the magnetization around the\nnew direction /vectorH1.\nsion amplitude depends on the magnitude of /vectorHy1ormore\nprecise on the angle Θ. This first part corresponds to the\nbasic principle of the PIMM experiment [8], [9], [10]. To\ndescribe the precession in an analytical way we use the\napproximation of a small precession angle assuming that\nthe change of the x-component of /vectorM(Mx) is negligible\nor at least small compared to the maximum change of\nthe components MyandMz. We then have\nMy=M0[1−e−t/τcosωt], (1)\nMz=−M0[e−t/τsinωt] (2)\nwith\nM0=|/vectorM|tan(Θ0), (3)\ntan(Θ0)≈ |/vectorHy1|/|/vectorH0| (4)\nand a phase angle φ=ωtwhereωis the precession fre-\nquency.\nFor coherent control a second field step /vectorHy2is used\natt=t2which can be applied either in +y or -y di-2\n/s45/s49/s48/s48 /s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48 /s53/s48/s48 /s54/s48/s48 /s55/s48/s48 /s56/s48/s48/s45/s52/s48/s48/s45/s51/s48/s48/s45/s50/s48/s48/s45/s49/s48/s48/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48\n/s77\n/s48/s77\n/s122/s32/s91/s97/s114/s98/s46/s32/s117/s46/s93\n/s77\n/s121/s32/s91/s97/s114/s98/s46/s32/s117/s46/s93/s77\n/s48/s50\nFIG. 2: Time evolution of MyandMzin adampedsystem ex-\ncited by two subsequent pulses in +y-direction. Black curve :\nPrecession from t= 0 tot=t2, Red curve: Precession after\nt=t2.\nrection and with variable time delay. Depending on the\ndirection of the second step this either tilts the magnetic\nfield further away from /vectorH0towards a new vector /vectorH2or\ncompensates /vectorHy1and restores the external field /vectorH0. De-\npending on the time delay and the sign of /vectorHy2this af-\nfects the precession of /vectorMin different ways. At the time\nt2the magnetization starts to precess around a new axis\nwith a new precession amplitude and a new phase angle.\nFor a detailed description we first consider the case of\n/vectorHy2=/vectorHy1. The center of the precession is then located\naty= 2|/vectorHy2|andz= 0. As we can see in figure 2 the\nnew precession amplitude at t=t2which we name M02\nis given as\nM02=/radicalBig\n(M0[e−t2/τsinωt2])2+(M0[1+e−t2/τcosωt2])2\n(5)\n=M0(e−t2/τ)/radicalBig\n(sinωt2)2+(1/e−t2/τ+cosωt2)2\n(6)\nThe precession angle φ2at which the new precession\nstarts can be calculated to\nφ2= arctan/parenleftbiggsinωt2\n1/e−t2/τ+cos(ωt2)/parenrightbigg\n(7)\nThe time development of MyandMzcan then be writ-\nten as\nMy= 2M0−M02e−(t−t2)/τcos[ω(t−t2)+φ2] (8)\nMz=−M02e−(t−t2)/τsin[ω(t−t2)+φ2]. (9)\nFor a second field step of opposite direction the cen-\nter of precession moves back to the original equilibriumdirection y=z= 0 and the previous equations are trans-\nformed to:\nM02=M0(e−t2/τ)/radicalBig\n(sinωt2)2+(1/e−t2/τ−cosωt2)2\n(10)\nφ2= arctan/parenleftbigg−sinωt2\n1/e−t2/τ−cos(ωt2)/parenrightbigg\n. (11)\nThe time development of MyandMzcan then be writ-\nten as\nMy=M02e−(t−t2)/τcos[ω(t−t2)+φ2] (12)\nand M z=M02e−(t−t2)/τsin[ω(t−t2)+φ2].(13)\nIn order to better visualize the consequences we first\nconsider the case of no damping or τ=∞and/vectorHy1in\n+y direction. Given a precession period 1 /ωandt≤t2\nwe have Mz= 0 for all times t=nπ/ωwhenever n\nis an integer number corresponding to phase angles of\ninteger multiples of π. For even n, we also have My= 0.\nFor uneven n we have My= 2M0. For even n a field\nstep in +y direction results in the maximum of the new\nprecession amplitude M02= 2M0. The phase angle is\nφ2= 0 so there is no phase jump. For uneven n a +y\nstep has the opposite effect. The magnetization at that\ntime is at y= 0 and the field H2points along the x-axis.\nWe thus have /vectorM×/vectorH≈0 whicheffectively terminates the\nprecession. For -y pulses the situation is again reversed.\nApplying the second pulse at even n stops the precession\nwhileforunevenn theprecessionamplitude ismaximized\ntoM02= 2M0.\nFor intermediate phase angles φ(t2) we also obtain in-\ntermediate values for the precession amplitude. In gen-\neral we can state that a +y step reduces the precession\namplitude for\n(2nπ+2π/3)< φ(t2)<(2nπ+4π/3) (14)\nwhile a -y step reduces the amplitude for\n(2nπ−π/3)< φ(t2)<(2nπ+π/3).(15)\nIn the other cases the step in the respective direction\nleads to an increase of the amplitude. In all those cases a\nphasejumpoccurs( φ2/negationslash= 0). Theamplitudeonlyremains\nunchanged for\nφ(t2) = 2nπ±2π/3 (16)\nin case of a +y step and for\nφ(t2) = 2nπ±π/3 (17)\nin case of a -y step. In those cases only the center of\nprecession changes and a phase jump takes place.3\nWe now include finite damping into our picture. For\nlow damping the description given above is still a good\napproximation for small numbers of n meaning for the\nfirst (few) precession periods. For higher damping the\npicture becomes more complex, however, we can still\nevaluate a few special cases. For φ2=nπwith even\nn and a +y step as well as for uneven n and a -y step the\namplitudeisstillmaximized, however,nowthemaximum\namplitude is\nM02=M0(1+e−t2/τ) (18)\nFor even n and a -y step or for uneven n and a +y step\nthe amplitude is minimized to\nM02=M0(1−e−t2/τ) (19)\nbut no longer reduced to zero. Here a phase jump of\nπoccurs at t=t2.\nObviously it is still possible to stop the precession\ncompletely if the height of the second field step /vectorHy2is\nmatched to /vectorHy2= (e−t2/τ−1)/vectorHy1.\nIn the experimental setup the sample is placed face\ndown on the signal line of a coplanar waveguide. Volt-\nage steps are applied to the waveguide and induce the\nfield steps /vectorHy1and/vectorHy2. The resulting precession of the\nmagnetization in the sample is detected by the induced\nvoltage. A Sampling Oscilloscope (DSA 8300) equipped\nwith Time Domain Reflectometry (TDR) sampling mod-\nules (80E08 and 80 E10) is used to produce the voltage\nsteps and to detect the induced signal. The voltage steps\nhave amplitudes of 250mV, a repetition rate of 200kHz,\nand a respective rise time of 12 −20ps which depends\non the used module. The polarity of the step and a de-\nlay time can be selected independently for each channel.\nThe system can display the reflected and the transmitted\nsignalasafunctionoftime. Foralltimedomainmeasure-\nments a digital filter is used to suppress high frequency\nnoise. Because the voltage induced by the precession is\nextremely small compared to the applied step a reference\nmeasurement without precession must be subtracted as\ndescribed by Silva et al. [8]. A second set of coils is\nused to apply an external field in y-direction [8]. In this\nconfiguration a field pulse in y direction does not induce\nany precession and yields the reference data which can\nbe subtracted from the original measurement.\nWe investigate thin films of different ferromagnetic\nmaterials namely YIG, Permalloy and La0.7Sr0.3MnO3\n(LSMO). Here only the results for YIG are presented and\ndiscussed. However, it should be noted that for all inves-\ntigated materials the results are in good agreement with\nmicromagnetic simulations and frequency domain FMR\nexperiments.\nFor the present experiment we use a 23 nmthin YIG\nfilm fabricated by pulsed laser deposition (PLD) on a\n(111) oriented Gd3Ga5O12(GGG) substrate, using an\noxygenpressureof0 .033mbar,alaserfluencyof2 J/cm2,/s53/s53 /s54/s48 /s54/s53 /s55/s48 /s55/s53 /s56/s48 /s56/s53 /s57/s48 /s57/s53 /s49/s48/s48/s48/s44/s48/s48/s44/s49/s48/s44/s50/s48/s44/s51\n/s51/s53/s53/s32/s79/s101\n/s49/s56/s50/s32/s79/s101\n/s50/s53/s32/s79/s101/s83/s105/s103/s110/s97/s108/s32/s91/s109/s86/s93\n/s116/s105/s109/s101/s32/s91/s110/s115/s93\nFIG. 3: Field dependent measurement of time-domain FMR.\nlaser repetition rate of 5 Hz, a substrate temperature of\n900◦C and a growth rate of 0 .5 nm/min. In frequency\ndomainFMR the film showsa linewidt of8 Oe at 10GHz.\nFigure 3 shows conventional time domain measurements\nat different fields /vectorH0using only one voltage step to in-\nduce the precession of the magnetization. The high qual-\nity of the YIG film allows us to observe the oscillation\nover a time of 50 ns. At low magnetic fields a beat-\ning is observed (Fig. 3, 25Oe) which is caused by two\nresonances with slightly different frequencies (0 .68GHz\nand 0.72GHz) which can be determined by Fast Fourier\nTransformation (FFT). These multiple resonances are\nalsoconfirmedbyfrequencydomainFMRmeasurements.\nIn the following we apply two voltage steps with different\ntime delays in order to investigate different cofigurations\nfor coherent control. The two voltage steps are of oppo-\nsite sign but similar magnitude corresponding to /vectorHy2as\na -y step which restores the original /vectorH0after the second\nstep.\n/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52/s48/s44/s48/s48/s48/s44/s48/s53/s48/s44/s49/s48/s48/s44/s49/s53/s48/s44/s50/s48/s48/s44/s50/s53\n/s40/s51/s41/s32/s100/s101/s108/s97/s121/s58/s32/s51/s57/s50/s32/s112/s115/s40/s50/s41/s32/s100/s101/s108/s97/s121/s58/s32/s49/s57/s54/s32/s112/s115/s40/s49/s41/s32/s114/s101/s102/s101/s114/s101/s110/s99/s101/s32/s109/s101/s97/s115/s117/s114/s101/s109/s101/s110/s116/s32/s119/s105/s116/s104/s111/s117/s116/s32\n/s32/s32/s32/s32/s32/s116/s104/s101/s32/s115/s101/s99/s111/s110/s100/s32/s115/s116/s101/s112/s32\n/s32/s32/s32/s32/s32/s40/s112/s101/s114/s105/s111/s100/s32/s111/s102/s32/s111/s115/s122/s105/s108/s108/s97/s116/s105/s111/s110/s58/s32/s51/s57/s50/s112/s115/s41/s32/s115/s105/s103/s110/s97/s108/s32/s91/s109/s86/s93\n/s116/s105/s109/s101/s32/s91/s110/s115/s93\nFIG. 4: Time-domain FMR, using one voltage step (1). With\nthe second voltage step, applied with suitable delay times t he\nprecession can be maximized (2) or completely stopped (3).\nFigure 4 shows three examples, which are also theoret-4\nically discussed at the beginning of the paper. In the first\ncase the precession is induced by one voltage step only\nto establish the ordinary precession pattern. In case (2)\nthe second voltage step is applied in the maximum of the\noscillation. According to our theory this doubles the pre-\ncession amplitude. Indeed we observe a signal which is\napprox. twice as large as for a single pulse. When the\n-y step is applied after a full precession corresponding\ntoφ(t2) = 2πthe precession is stopped (Fig. 4,(3)) as\nexpected. Besides the purely analytical approach to the-\noryourexperimentsarealsosupportedbymicromagnetic\nsimulations using OOMMF ([11]).\nBesides the two cases of φ(t2) =πandφ(t2) = 2πalso\nmeasurements for other delay times and angles are per-\nformed. In fig5 the amplitude ofthe precessionisplotted\nover the delay time. The diagram shows that by evalu-\nating the time between the minima of the curve we can\ndetermine the procession period with very high accuracy,\nin fact much better than by using FFT or simply mea-\nsuring the period of the oscillation in the measurement.\nCoherent control thus greatly enhances the precision of\nthe measurement.\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48 /s53/s48/s48 /s54/s48/s48 /s55/s48/s48/s48/s44/s48/s48/s44/s53/s49/s44/s48/s49/s44/s53/s50/s44/s48/s50/s44/s53/s51/s44/s48\n/s32/s97/s110/s97/s108/s121/s116/s105/s99/s97/s108/s32/s100/s101/s115/s99/s114/s105/s112/s116/s105/s111/s110\n/s32/s109/s101/s97/s115/s117/s114/s101/s109/s101/s110/s116\n/s32/s115/s105/s109/s117/s108/s97/s116/s105/s111/s110/s32/s40/s110/s111/s114/s109/s97/s108/s105/s122/s101/s100/s41/s97/s109/s112/s108/s105/s116/s117/s100/s101/s32/s91/s97/s114/s98/s46/s32/s117/s46/s93\n/s116/s105/s109/s101/s32/s100/s101/s108/s97/s121/s32/s91/s112/s115/s93\nFIG. 5: Amplitudes of the precession versus the time delay.\nThe dotted line shows the amplitude of the precession in-\nduced by only one voltage step and corresponds to the refer-\nence measurement in figure 4. The simulated data have been\nnormalized, assuming that the amplitudes of the oscillatio ns\ninduced by one voltage step are equal for the simulation and\nthe experiment.It should finally be noted that this method holds sev-\neral advantages with respect to standard PIMM. By us-\ning a sequence of multiple alternating steps (in other\nwords a sequence of pulses) we can massively enhance\nthe precession amplitude for samples with low damping.\nAs we know the precession amplitude after one period is\nreduced by a factor of e−t/τ. It is obvious that multiple\nexcitationsofidenticalmagnitudewhich areapplied after\neach period can increase the amplitude up to a magni-\ntudeMmaxwhich satisfies the condition:\nMmax=M0\n1−e−t/τ(20)\nThis means that if the decay of the precession is only\n1% a maximum gain of 100 is possible. In the presence\nof several resonances with only small difference in fre-\nquency using multiple steps or pulses can also help to\nfacilitate the analysis. In PIMM two resonances close to\neachotherresult inabeating patternin the time domain.\nExcitation with multiple steps which are spaced by just\none period do little to suppress one of the two lines. If,\nhowever,the spacing is a suitable multiple of a precession\nperiod one of the resonances can be enhanced while the\nother one is suppressed. It may thus be possible to selec-\ntively excite a single resonance even if when its frequency\nis close to another resonance.\nWe have shown that by using two short magnetic field\nsteps with varying time delay it is possible to coherently\ncontrol ferromagnetic resonance in a PIMM geometry.\nThe method can be used to increase the sensitivity of\nthe method to enhance the temporal resolution and even\nto selectively excite single resonance modes even if other\nmodes with different frequencies exist.\nThis work was funded by the DFG in the SFB 762 and\nby the EC in the project IFOX.\n[1] S. A. Crooker, J. J. Baumberg, F. Flack, N. Samarth,\nand D. D. Awschalom, Phys. Rev. Lett. 77, 2814 (1996).\n[2] J. M. Kikkawa, I. P. Smorchkova, N. Samarth, and D. D.\nAwschalom, Science 277, 1284 (1997).\n[3] K. Yamaguchi, M. Nakajima, and T. Suemoto, Phys.\nRev. Lett. 105, 237201 (2010).\n[4] F. Hansteen, A. Kimel, A.Kirilyuk, and T. Rasing, Phys.\nRev. B73, 014421 (2006).\n[5] G. Wolf, Diploma thesis (2007).\n[6] J. M. Kikkawa and D. D. Awschalom, Phys. Rev. Lett.\n80, 4313 (1998).[7] I. Malajovich, J. M. Kikkawa, D. D. Awschalom, J. J.\nBerry, and N. Samarth, Journal of Applied Physics 87,\n5073 (2000).\n[8] T. J. Silva, C. S. Lee, T. M. Crawford, and C. T. Rogers,\nJournal of Applied Physics 85, 7849 (1999).\n[9] A. B. Kos, T. J. Silva, and P. Kabos, Review of Scientific\nInstruments 73, 3563 (2002).\n[10] I. Neudecker, G. Woltersdorf, B. Heinrich, T. Okuno,\nG. Gubbiotti, and C. Back, Journal of Magnetism and\nMagnetic Materials 307, 148 (2006).\n[11] M. J. Donahue and D. G. Porter, OOMMF User Guide,5\nVersion 1.0 , NISTIR 6376, National Institute of Stan-\ndards and Technology, Gaithersburg, MD (1999)."    },    {        "title": "1903.01718v2.Optimal_mode_matching_in_cavity_optomagnonics.pdf",        "content": "Optimal mode matching in cavity optomagnonics\nSanchar Sharma,1Babak Zare Rameshti,2Yaroslav M. Blanter,1and Gerrit E. W. Bauer3, 1\n1Kavli Institute of NanoScience, Delft University of Technology, 2628 CJ Delft, The Netherlands\n2Department of Physics, Iran University of Science and Technology, Narmak, Tehran 16844, Iran\n3Institute for Materials Research & WPI-AIMR & CSRN, Tohoku University, Sendai 980-8577, Japan\nInelastic scattering of photons is a promising technique to manipulate magnons but it su\u000bers\nfrom weak intrinsic coupling. We theoretically discuss an idea to increase optomagnonic coupling in\noptical whispering gallery mode cavities, by generalizing previous analysis to include the exchange\ninteraction. We predict that the optomagnonic coupling constant to surface magnons in yttrium iron\ngarnet (YIG) spheres with radius 300 \u0016m can be up to 40 times larger than that to the macrospin\nKittel mode. Whereas this enhancement falls short of the requirements for magnon manipulation in\nYIG, nanostructuring and/or materials with larger magneto-optical constants can bridge this gap.\nMagnetic insulators such as yttrium iron garnet (YIG)\nare promising for future spintronic applications such as\nlow power logic devices [1], long-range information trans-\nfer [2], and quantum information [3]. Their excellent\nmagnetic quality [4, 5] implies spin waves or magnons,\nthe excitations of the magnetic order, are long-lived. Mi-\ncrowaves in high quality cavities and striplines couple\nstrongly to magnons with long (mm) wavelengths [6{\n12], i.e. the rate of energy exchange between the two\nsystems is higher than their individual dissipation rates,\nbut not to short wavelengths (except under special ge-\nometries [13]). Magnons can be injected electrically by\nmetallic contacts [14, 15], but only in rather small num-\nbers. Here, we focus on the coherent coupling of magnetic\norder and infrared laser light with sub- \u0016m wavelengths,\nthat is enhanced by using the magnet as an optical cavity\n[16{18].\nBy the high dielectric constant and almost perfect\ntransparency in the infrared [19, 20], sub-mm YIG\nspheres support long-living whispering gallery modes\n(WGMs) [16, 21]. The photons, with energy deep within\nthe band gap, scatter inelastically by absorbing or cre-\nating magnons [22, 23]. This is known as Brillouin light\nscattering (BLS) [24], which is enhanced in an optical\ncavity [16{18, 21, 25{29]. These results led to predictions\nof the Purcell e\u000bect [30] (optically induced enhancement\nof magnon linewidth), magnon lasing [31] and magnon\ncooling [32]. However, the models addressed only the\nmagnetostatic magnon modes, i.e. ignored retardation\nand the exchange interaction, with only small overlap\nwith the WGMs [16{18, 25, 29, 33, 34]. Thus, the ob-\nserved and predicted coupling rates were too low to be\nable to optically manipulate magnons [31, 32]. Higher\noptomagnonic coupling can be achieved by reducing the\nsize of the magnets down to optical wavelengths [35],\nbut this requires nanostructuring of the magnet [36{38].\nCoupling to magnons in a non-uniform magnetization\ntexture is large [39]. Here, we suggest and analyze a\nmethod to increase coupling in a conventional set-up of\na uniformly magnetized sub-mm YIG sphere by coupling\nto exchange-dipolar modes with wavelengths comparable\nto the WGMs.\nBulk magnons in \flms with both exchange and dipo-lar interactions have been extensively studied [40{42].\nIn thick \flms, exchange reduces the life time of surface\nmagnons by mixing with bulk states [43{45], while in\nthinner \flms exchange leads to modes with partial bulk\nand surface character [45, 46]. Here, we address mag-\nnetic spheres with radii that are large enough to support\nsurface exchange-dipolar magnons.\nOur system is sketched in Fig. 1. A ferromagnetic\nsphere acts as a WGM resonator in which photons in-\nteract with the magnetic order via standard proximity\ncoupling to an optical prism or \fber. The frequency\nof photons is 4 to 5 orders of magnitude larger than\nmagnons at similar wavelengths, thus the incident and\nscattered photons have nearly the same frequency and\nwavelength. Forward scattering of photons occurs via\nmagnons of large wavelength \u0018100\u0016m, which is a pro-\ncess that is well described by a purely dipolar theory [33].\nHere we discuss back scattering of photons by magnons\nwith sub\u0000\u0016m wavelengths that are a\u000bected signi\fcantly\nby exchange. We show that the exchange generates mag-\nnetic modes that have a near ideal overlap with the opti-\ncal WGMs, with an optomagnonic coupling limited only\nby the bulk magneto-optical constants.\nWe \frst brie\ry review the basics of cavity optomagnon-\nics and derive an upper bound for the optomagnonic cou-\npling constant in resonators in Sec. I. We model the\nmagnetization dynamics by the Landau-Lifshitz equation\nintroduced in Sec. II. The spatial amplitude of surface\nexchange-dipolar magnons is discussed in Sec. III, with\ndetails of the derivation in App. A. The optomagnonic\ncoupling constants found in Sec. IV are compared with\nthe upper bound found in Sec. I. We conclude with dis-\ncussion and outlook in Sec. V.\nI. CAVITY OPTOMAGNONICS\nHere we summarize the basic theory of magnon-photon\ncoupling in spherical optical resonators [33]. The electric\nand magnetic \felds of the optical modes in a spherical\nresonator are labeled by orbital indices fl;m;\u0017gand a\npolarization \u001b2fTM;TEg. They become optical whis-\npering gallery modes (WGMs) at extremal cross sectionsarXiv:1903.01718v2  [cond-mat.mes-hall]  24 May 20192\nHapp\nR\nAinWPMA\nWQ\nMAWQ\nArefˆrˆφ\nˆθ\nFIG. 1. A sphere made of a ferromagnetic dielectric in prox-\nimity to an optical \fber or prism. A magnetic \feld satu-\nrates the magnetization. The input photons in the \fber, Ain,\nleak into the whispering gallery modes (WGMs) fWPg. The\nlatter can be re\rected by magnons fMAgof twice the angu-\nlar momentum into the blue, via WP+MA!WQ, or red,\nWP!WQ+MA, sideband. The photons fWQgcan leak back\ninto the \fber and be observed in the re\rection spectrum.\nwhenl;m\u001df1;jl\u0000mjg. WGMs are traveling waves in\nthe\u0006\u001e-direction with dimensionless wavelength 2 \u0019=m:\n\u0017\u00001 andl\u0000mare the number of nodes in the optical\n\felds in the rand\u0012direction. The electric \feld of these\nmodes is ETM=E(r)^\u0012andETE=E(r)^rwhere [47],\nE(r) =EYm\nl(\u0012;\u001e)Jl(kr): (1)\nHereJlis the Bessel function of order l[Eq. (A10)] and\nYm\nlis a scalar spherical harmonic [Eq. (A3)]. The wave\nnumberk, forl\u001d1 [47]\nkR\u0019l+\f\u0017\u0012l\n2\u00131=3\n\u0000P\u001b; (2)\nwhereRis the radius of the sphere, \f\u00172\nf2:3;4:1;5:5;:::gare the negative of the zeros of Airy's\nfunction Ai ( x),PTM =ns=p\nn2s\u00001, andP\u00001\nTE =\nnsp\nn2s\u00001.Eis a normalization constant chosen such\nthat the integral over the system volume\nZ\u0014\u000fs\n2jEj2+1\n2\u00160jBj2\u0015\ndV=~!\n2; (3)\nwherei!B=r\u0002E,\u000fs=\u000f0n2\ns, and!=kc=nswithns\nbeing the refractive index of the sphere. Then\nE=s\n~!\n2\u000fsR3Nl(kR); (4)where\nNl(x)4=Z1\n0~r2d~rJ2\nl(x~r)\n\u0019J2\nl(x)\u0000Jl+1(x)Jl\u00001(x)\n2; (5)\nand the approximation holds again for l\u001d1. The angu-\nlar dependence for l=mwithl\u001d1, [47]\nYl\nl(\u0012;\u001e)\u0019\u0012l\n\u0019\u00131=4\nexp\u0014\n\u0000l\n2\u0010\u0019\n2\u0000\u0012\u00112\u0015eil\u001e\np\n2\u0019;(6)\nis a narrow Gaussian around \u0012=\u0019=2 with a widthp\n2=l\nand a traveling wave along the circle with wave number\nl=R. The radial dependence for l\u001d1 [48]\nJl(kr)\u0019\u00122\nl\u00131=3\nAi (x\u0000\f\u0017); (7)\nwhere the radial coordinate is scaled to\nx=l\n(l=2)1=3\u0010\n1\u0000r\nR\u0011\n: (8)\nThe leading interaction between magnons and WGMs\nis 2-photon 1-magnon scattering. Consider a TM po-\nlarized WGM P\u0011fp;\u0000p0;\u0016gthat scatters into a TE-\npolarized WGM Q\u0011fq;q0;\u0017gby absorbing a magnon\nA(to be generalized below). We take in the following\np0>0 and thus, back(forward) scattering corresponds to\nq0>0(q0<0). The coupling constant depends on the\nmodes as [22, 23],\nGPQA =ns\u000f0\u00150\n\u0019MsZ\nEPE\u0003\nQ(\u0002CMA;\u001a\u0000i\u0002FMA;\u001e)dV;\n(9)\nwhere the integral is over the sphere's volume, \u00150is the\nvacuum wavelength of the incident light, Msis the sat-\nuration magnetization, \u0002 Fis the Faraday rotation per\nunit length, \u0002 Cis the Cotton-Mouton ellipticity per unit\nlength, and MA;\u001e(MA;\u001a) is the\u001e(\u001a)-component of A-\nmagnons.\nFor the uniform precession of the magnetization, i.e.\nthe Kittel mode K, [49]\nMK;\u001e=iMK;\u001a=s\n~\rMs\n2Vsph; (10)\nwhereVsphis the volume of the sphere, and \ris the\nmodulus of the gyromagnetic ratio. We normalized the\nmagnetization as\nZ\nRe\u0002\niM\u0003\n\u001eM\u001a\u0003\ndV=~\rMs\n2; (11)\nequivalent to Eq. (B14). The coupling constant is \fnite\nonly when q0+p0= 1,p\u0000jp0j=q\u0000jq0j, and\u0016=\u00173\n[27, 33]. The coupling constant, independent of optical\nmodes,\njGPQKj=GK=c(\u0002F+ \u0002C)\nnsp\n2sVsph; (12)\nwheres=Ms=\r~is the spin density. For the parameters\nin Table I, GK= 2\u0019\u00029:1 Hz.\nAn upper bound on GPQA for a given set of WGMs can\nbe found by maximizing it over all normalized functions\nfMA;\u001a(r);MA;\u001e(r)g. The solution Moptgives the mag-\nnetization pro\fle with highest optomagnonic coupling.\nLater, we show that there exists eigenstates that are\nclose to Mopt. We consider circularly polarized magnons\nMA;\u001e=iMA;\u001aand discuss the e\u000bect of \fnite ellipticity\nbelow. By the method of Lagrange multipliers,\nL=Z\nEPE\u0003\nQM\u001edV\u0000\u0015\u0012Z\nM\u0003\n\u001eM\u001edV\u0000~\rMs\n2\u0013\n(13)\nis stationary at M\u001e=Mopt\n\u001e. We \fnd\nMopt\n\u001e=E\u0003\nPEQ\n\u0015/Jp(kPr)Jq(kQr)Yp0\npYq0\nq; (14)\nwith\n\u0015=s\n2\n\r~MsZ\njEPEQj2dV: (15)\nTherefore\nGPQ4=jGPQ;optj=c(\u0002F+ \u0002C)\nnsp\n2sVPQ; (16)\nde\fning the e\u000bective overlap volume\nVPQ=\u0010R\njEPj2dV\u0011\u0010R\njEQj2dV\u0011\nR\njEPj2jEQj2dV: (17)\nThe WGMs which are most concentrated to the sur-\nface have mode numbers p=p0andq=q0. Since the\nmagnon frequency \u00181\u000010 GHz, is much smaller than\nthat of the photons, \u0018200 THz, the incident and scat-\ntered photons have nearly the same frequency, implying\np\u0019q[see Eq. (2)]. The Bessel function Jpapproaches\nthe Airy function Ai( x) forp;q\u001d1 [see Eq. (7)],\nMopt\n\u001e/Ai (x\u0000\f\u0016) Ai (x\u0000\f\u0017)e\u0000p(\u0019\n2\u0000\u0012)2\nei(p+q)\u001e;(18)\nwhere the coordinate xis given by Eq. (8) after the sub-\nstitutionl!p. This is a traveling wave in \u001e-direction\nand a Gaussian in \u0012-direction. Its radial dependence for\nthe lowestf\u0016;\u0017gis plotted in Fig. 2, showing signi\fcant\nvalues only very close to the surface. The overlap volume\n(17) now reads\nVPQ\u0019\u00122\np\u00137=6R3\u00193=2\f\fAi0(\u0000\f\u0016) Ai0(\u0000\f\u0017)\f\f\nR1\n0Ai2(x\u0000\f\u0016) Ai2(x\u0000\f\u0017)dx;(19)\n0.980 0.985 0.990 0.995 1.000\nr/R−0.200.20.40.6EPEQ(arb.)µ= 1,ν= 1\nµ= 1,ν= 2\nµ= 2,ν= 2FIG. 2. The r-dependence of the product of the electric \feld\nof WGMs, in arbitrary units, for p=p0=q=q0= 3000 and\nradial mode numbers \u0016;\u00172f1;2g. For the parameters of our\nsystem in Table I, this corresponds to photons with free space\nwavelength\u00191:3\u0016m. The magnons that match these pro\fles\nhave the largest optomagnonic coupling, cf. Eq. (14).\nForp= 3000 and \u0016=\u0017= 1,Vsph=VPQ\u00191600, re\recting\nthe localized nature of the WGMs.\nFor light with \u00150= 1:3\u0016m,p= 3190 for a YIG\nsphere with parameters in Table I. For the \frst modes\nf\u0016;\u0017;GPQ=(2\u0019)g=f1;1;364 Hzg,f1;2;224 Hzg, and\nf2;2;304 Hzg;soGPQ\u001dGK. For a \fxed \u00150,p/R,\nandGPQ/R\u000011=12can be further enhanced by reduc-\ning the diameter.\nMagnetic anisotropies and dipolar interaction can de-\nform the circular precession of the magnons into an el-\nlipse. Solving the above problem for a hypothetical lin-\nearly polarized magnetization precession, e.g. by letting\nM\u001e!1 andM\u001a!0 while maintaining Eq. (11), leads\nto a divergingGPQ!1 . But such strong linear polar-\nization are di\u000ecult to achieve in practice and ellipticity is\ntypically limited to \u001810%, also valid in the calculations\nbelow.\nA similar analysis for PandQbeing TE and TM\npolarized, respectively, reveals the same results with\n\u0002F+\u0002C!\u0002F\u0000\u0002Cand thus reduced couplings by a fac-\ntor 0:45. It is therefore advantageous to input TM pho-\ntons over TE for larger blue sideband (magnon absorp-\ntion) [22, 50]. The coupling constant concerning magnon\nemission processes follows a very similar discussion since\nGblue\nPQA =G\u0003\nQPA.\nII. LANDAU-LIFSHITZ EQUATION\nHere we derive the equations for the magnetic eigen-\nmodes which will later be shown to approximate the opti-\nmal pro\fle derived above. The parameters for a standard\nYIG sphere are given in table I. The Gilbert damping\ndoes not a\u000bect the magnon mode shapes to leading order4\n\u0015ex ns Ms \r=(2\u0019)\n109nm 2 :2 140 kA/m 28 GHz/T\n\u0002F \u0002C Happ\u0000Ms=3R\n400 rad/m 150 rad/m 200 mT =\u00160 300\u0016m\nTABLE I. Parameters for a standard YIG sphere: exchange\nconstantAex[40, 51], refractive index ns[40], saturation\nmagnetization Ms[40], gyromagnetic ratio \r[40], Faraday\nrotation angle \u0002 F[52, 53], Cotton-Mouton ellipticity \u0002 C\n[21, 54, 55]. We assume the applied dc \feld Happand the\nradiusRbased on typical experimental setup [16{18].\nand is disregarded. The magnetization dynamics then\nobeys the Landau-Lifshitz equation\ndM\ndt=\u0000\r\u00160M\u0002He\u000b; (20)\nwhere Mis the magnetization, \u00160is the free space per-\nmeability, and the e\u000bective magnetic \feld\nHe\u000b=Happ^z+2Aex\n\u00160M2sr2M+Hdip; (21)\nwhereHappis the applied \feld that saturates the magne-\ntization toMsin the ^z-direction,Aexis the exchange con-\nstant, and Hdipis the dipolar \feld that solves Maxwell's\nequations in the magnetostatic approximation:\nr\u0002Hdip= 0;r\u0001Hdip=\u0000r\u0001M; (22)\nwhich is valid for magnons with wavelengths su\u000eciently\nsmaller than c=!\u00181 cm [56]. The amplitudes m=\nM\u0000Ms^zare taken to be small. The dipolar \feld has a\nlarge dc and a small ac component, Hdip=Hdemag +hdip;\nwhere the demagnetization \feld Hdemag =\u0000Ms^z=3 for\na sphere. We disregard the small magneto-crystalline\nanisotropies in YIG.\nThe scalar potential hdip=\u0000r satis\fes\nr2 =r\u0001m: (23)\nAfter substitution into Eq. (20), linearizing in m, and in\nthe frequency domain @=@t!\u0000i!,\n\u0014\n\u0006!+!a\u0000!s\nk2exr2\u0015\nm\u0006=\u0000!s@\u0006 ; (24)\nwhere we used the circular coordinates m\u0006=mx\u0006imy\nand@\u0006=@x\u0006i@y. Here!a=\r\u00160(Happ\u0000Ms=3),!s=\r\u00160Ms, and the inverse exchange length\n2\u0019\n\u0015ex=kex=s\n\u00160M2s\n2Aex: (25)\nWe callm\u0000(m+) the Larmor(anti-Larmor) component\nsincem+= 0 for a pure Larmor precession. Outside the\nmagnet\nr2 o= 0: (26)\nThe coupled set of di\u000berential equations (23)-(26) are\nclosed by boundary conditions derived from Maxwell's\nequations at the interface,\n (R) = o(R) ;\u0000@r (R) +mr(R) =\u0000@r o(R):(27)\nThe \frst condition is required for a \fnite hdipat the\nsurface, while the second one enforces continuity of the\nnormal component of the magnetic \feld hdip+m. At\nlarge distances, the magnetic \feld vanishes implying a\nconstant potential which can be chosen to be zero,\n o(r!1 ) = 0: (28)\nThe boundary conditions for the magnetization de-\npends on the surface morphology and is complicated\nby the long range nature of the dipolar interaction\n[46, 57, 58]. Here, we present calculations for pinned\nboundary conditions, mx;y(R) = 0, valid when the sur-\nface anisotropy is high [44, 57, 58] . This is not very\nrealistic for samples with high surface quality but su\u000e-\nciently accurate for our purposes, as justi\fed in Sec. III.\nIII. EXCHANGE-DIPOLE MAGNONS\nHere we discuss the amplitude of the magnons in di-\nelectric magnetic spheres which resemble the ideal mag-\nnetization distribution derived in Sec. I. These are the\nsurface exchange-dipolar magnons localized at the equa-\ntor derived in App. A. Similar problems have been ad-\ndressed in Refs. [42, 46] for di\u000berent geometries.\nAnalogous to the photons discussed above, magnons\nin spheres are characterized by three mode numbers\nfl;m;\u0017g. Their amplitudes are a linear combination of\nthree terms given in Eq. (29) [cf. Eqs. (A22)-(A23)] with\n`dispersion' relations in Eq. (30) [cf. Eq. (A7)]. The par-\ntial waves appear with coe\u000ecients \u0010de\fned below.\nm\u0006(r) =m0Ym\u00061\nl\u00061(\u0012;\u001e)\u0014\n\u0010dip;\u0006\u0010r\nR\u0011l\u00061\n+\u0010ex;\u0006Jl\u00061(kr)\nJl\u00001(kR)+\u0010s;\u0006Il\u00061(\u0014r)\nIl\u00001(\u0014R)\u0015\n: (29)\nk2\nk2ex=!sq\u0000!DE\n!s;\u00142\nk2ex=!sq+!DE\n!s; ! sq=r\n!2+!2s\n4; ! DE=!a+!s\n2: (30)5\nHerekex;!s;!aare de\fned below Eq. (24), !DEis the\nfrequency of the surface magnons in a purely dipolar the-\nory [59, 60], and the normalization constant m0is deter-\nmined below.f`dip',`ex',`s'grefers tofdipolar, exchange,\nsurfacegrespectively.\nThe ratios of anti-Larmor ( m+) and Larmor ( m\u0000)\ncomponents is a measure of the ellipticity [see Eq. (A24)]:\n\u0010dip+= 0;\u0010ex+\n\u0010ex\u0000=!sq\u0000!\n!s=2;\u0010s+\n\u0010s;\u0000=!sq+!\n!s=2:(31)\nThe coe\u000ecients \u0010read for pinned boundary conditions\nm(R) = 0 [see Eqs. (A25)-(A26)],\n\u0010dip;\u0000=!sq\n!s=2; \u0010ex;\u0000=\u0000\u00142\nk2ex; \u0010s;\u0000=\u0000k2\nk2ex: (32)\nClose to the boundary, the `dip' and `s' terms dominate,\nbut the `ex' term in m\u0006takes over for r=R< 1\u00001=l.\nThe dipolar (subscript `dip') term in Eq. (29) de-\ncays exponentially with distance from the surface with\na length scale R=l. This solution is not a\u000bected by ex-\nchange [49, 60] because r2\u0010\nYm\nl(\u0012;\u001e)\u0000r\nR\u0001l\u0011\n= 0. For\nl\u001d1 the surface term (subscript `s') simpli\fes by the\nasymptotics of the Bessel function to\nIl\u00001(\u0014r)\nIl\u00001(\u0014R)\u0019 p\nl2+\u00142R2\u0000lp\nl2+\u00142R2+l!\nIl+1(\u0014r)\nIl\u00001(\u0014R)\n\u0019exp\u0014\n\u0000p\nl2+\u00142R2R\u0000r\nR\u0015\n: (33)\nThis is again an exponential decay, but on an even shorter\nscaleR=p\nl2+\u00142R2than the dipolar term. At \frst\nglance, it appears to have a large negative exchange en-\nergy,/\u0000\u00142, but its total contribution to the energy is\nsmall due to its very small mode volume. Both `dip'\nand `s' terms are important to satisfy the boundary con-\nditions, but they do not contribute signi\fcantly to the\noptomagnonic coupling because the optical WGMs pen-\netrate much deeper into the magnet [see Fig. 2]. The\nexchange `ex' function in Eq. (29), on the other hand,\nresembles a photon WGM when kR\u0019l[see Sec. I]. We\nshow below that this condition is satis\fed by magnons\nwith\u0017 >0.\nWe now turn to the magnon eigenfrequencies and\nmodes for \fxed landmwith\u0017\u00150 [using App. A]. For\n\u0017= 0,!2\n0\u0019!2\na+!a!sand mode amplitudes Eq. (29)\napproach\nm\u001e\u0019l3=2r\n\r~Ms\n2R3Ym\nl(\u0012;\u001e)\u0010r\nR\u0011l\u00001\u0012\n1\u0000r2\nR2\u0013\n(34)\nandm\u001a=\u0000im\u001ewhenkexR\u001dp\nl, which is the case for\ntypical experimental conditions discussed below. We nor-\nmalizedm\u001eaccording to Eq. (B14). Note that (only) the\nresults for\u0017= 0 depend strongly on the surface pinning.For non-zero \u0017\u0018O(1), analogous to Eq. (2) for the\nphotons,\nk\u0017R=l+\f\u0017\u0012l\n2\u00131=3\n; (35)\nwhere\f\u00172f2:3;4:1;5:5;:::gare again the negative of\nthe zeros of Airy's function. We compute coe\u000ecients\nf\u0010dip;\u0000;\u0010ex;\u0000;\u0010s;\u0000;\u0010dip;+;\u0010ex;+g\u0019f 3:5;3:4;0:1;0:5;1:0g.\nAlthough\u0010ex\u0018\u0010dip, the energy of the `dip' term is much\nsmaller than that of the `ex' term because the former\nis localized to a small skin depth \u0018R=land therefore\ndoes not contribute much when integrated over the mode\nvolume. We disregard `dip' and `s' terms at the cost of\nan error scaling as /l\u00001=3. The magnetization\nm\u001e(r)\u0019s\n\r~Ms\n2R3Nl(kR)Ym\nl(\u0012;\u001e)Jl(k\u0017r) tan\u0012e(36)\nm\u001a(r)\u0019\u0000im\u001e(r) cot2\u0012e; (37)\nforr=R< 1\u00001=l, whereNis given by Eq. (5). Since the\nmagnetic \feld generated by magnetic dipoles is ellipti-\ncally polarized, the magnetization precesses on an ellipse\nwith major and minor axes along \u001a\u001a\u001aand\u001e\u001e\u001e, respectively.\nThe ellipticity is parametrized by the angle \u0012e, given by\ntan\u0012e=s\n\u0010ex;\u0000\u0000\u0010ex;+\n\u0010ex;\u0000+\u0010ex;+=s\n!s=2\u0000!sq+!\n!s=2 +!sq\u0000!:(38)\nThe amplitudes (36) are normalized according to\nEq. (B14).\nForR= 300\u0016m andl= 6000 [see Sec. IV], 2 \u0019R=l\u0019\n300 nm is the magnon wavelength for a typical experi-\nment. The \u001e-component of the magnetization m\u001efor\n\u0017\u00143 is plotted in Fig. 3, while m\u001alooks similar to m\u001e\nafter scaling (not shown for brevity). \u0017 >0 modes con-\ntribute signi\fcantly to the coupling with large overlap\nfactors [see Sec. IV for explicit expressions].\nFor the parameters in Table I, we \fnd !a= 2\u0019\u0002\n5:6 GHz, and !s= 2\u0019\u00024:9 GHz. Putting kR=lin\nEq. (30), we get the frequency !N= 2\u0019\u00028:4 GHz.\n!0= 2\u0019\u00027:7 GHz, while frequencies for \u0017=f1;2;3g\nare!\u0017=!N+ 2\u0019\u0002f7:5;13:2;17:9gMHz respectively.\nWe estimate the linewidth of the magnons \u0018\u000bG!\u0017, in\nterms of the (geometry-independent) bulk Gilbert con-\nstant\u000bG= 10\u00004[5, 37]. The frequency splittings are an\norder of magnitude larger than the typical line width, so\nthe magnon resonances are well de\fned. The exchange\nmode has a small ellipticity tan \u0012e= 0:8.\nAt these frequencies the `surface' term in Eq. (29)\nhas wavelengths 2 \u0019=\u0014\u0017\u001960 nm. It decays much faster\ninto the sphere than the wavelength of infrared light,\n>500 nm in YIG, which validates our statements above.\nWe assumed perfect pinning at the boundary,\nm\u0006(R) = 0;which is realistic only when surface\nanisotropies are strong [46, 57, 58]. While Eqs. (29)-(31)\ndo not depend on the boundary conditions, the relative\nweights of three waves, f\u0010dip;\u0000;\u0010ex;\u0000;\u0010s;\u0000gdo. However,6\n0.980 0.985 0.990 0.995 1.000\nr/R−15−10−505101520mφ(arb.)ν= 0\nν= 1\nν= 2\nν= 3\nFIG. 3. Radial dependence of m\u001e= (m+e\u0000i\u001e\u0000m\u0000ei\u001e)=2\nfor\u0017\u00143 andl= 6000 with parameters from Table I. \u0017= 0\nresembles a purely dipolar wave and is localized to 1 >r=R>\n1\u00002=l. For\u0017 > 0 the magnetization is dominated by the\nBessel function except for the region occupied by the \u0017= 0\nmode.\nthe validity of Eq. (36) depends only on the fact that the\nenergy is dominated by the Bessel function which still\nholds for imperfect pinning and \u0017 >0. We estimate the\ncontributions of surface exchange waves to the magnon\nmode energy by the parameter\n\u0011=j\u0010dip;\u0000j2\nj\u0010ex;\u0000j2J2\nl(kR)R\n(r=R)2ldrR\nJ2\nl(kr)dr: (39)\nFor a \flm, the squared ratio of the \u0010coe\u000ecients is\u00181\n[46], which should be the case also for a sphere with cur-\nvatureRmuch larger than the magnon wavelength R=l.\nThe second fraction is of O(l\u00001=3). Therefore \u0011\u001c1;\nimplying that the energy is indeed dominated by the\nBessel function as assumed in Eq. (36). Reduced pin-\nning changes the magnetization pro\fle near the surface,\nr=R> 1\u00001=l, but not the coupling of states with \u0017 >0\nto the WGMs.\nIV. OPTOMAGNONIC COUPLING\nWe calculate the coupling constant GPQA given by\nEq. (9). Consider an incident TM-polarized optical\nWGMP\u0011fp;\u0000p0;\u0016gthat re\rects into a TE-polarized\nWGMQ\u0011 fq;q0;\u0017gby absorbing a magnon A\u0011\nf\u000b;\u000b0;\u0018g. Their frequencies are, respectively, !P,!Q,\nand!A\u001c!P,!Q. By energy conservation, !P\u0019!Q\nand thus,p\u0019q[see Eq. (2)]. For the modes localized\nnear the equator, \u0012=\u0019=2, the indices x\u0019x0where\nx2fp;q;\u000bg. The conservation of angular momentum in\nthez-direction [33], cf. Eq. (43), implies p0+q0=\u000b0.\nFor\u00150\u00191:3\u0016m, Eq. (2) and Table I give p\u00193000 for\n\u0017P\u0018O(1). Summarizing, p\u0019p0\u0019q\u0019q0\u0019\u000b=2\u0019\n\u000b0=2\u00193000.From Figs. 2 and 3, we observe that the radial magnon\namplitude can be close to the optimal pro\fle. This is\nalso the case in the azimuthal \u0012-direction close to the\nequator (not shown). Here, we con\frm this observation\nby explicitly calculating the mode overlap integrals.\nThe coupling constant Eq. (9) can be written\nGPQA =c(\u0002F+ \u0002C)\nnsp\n2sR3APQARPQA; (40)\nin terms of the dimensionless angular and radial overlap\nintegrals,APQA andRPQA.\nThe angular part,\nAPQA =Z\nY\u0000p0\npY\u000b0\n\u000b\u0010\nYq0\nq\u0011\u0003\nsin\u0012d\u0012d\u001e: (41)\nis a standard integral that can be written in terms\nof Clebsch-Gordan coe\u000ecients hl1m1;l2m2jl3m3i. For\np;q;\u000b\u001d1,\nAPQA\u0019rpq\n2\u0019\u000bhpp0;qq0j\u000b\u000b0ihp0;q0j\u000b0i: (42)\nWithx=x0wherex2fp;q;\u000bg, the Gaussian approxi-\nmation [Eq. (6)] leads to\nAPQA\u0019\u000e\u000b;p+q(pq\u000b)1=4\n\u00193=4pp+q+\u000b\u0019\u000e\u000b;p+qp1=4\n3:97;(43)\nwhere in the second step, we used p\u0019q\u0019\u000b=2.APQA\nvanishes when \u000b6=p+q, re\recting the conservation of\nangular momentum in the z-direction. The angular over-\nlap is optimal because Y\u000b\n\u000b/Yp\npYq\nqforp\u0019q\u0019\u000b=2;\nwhich equals the angular part in Eq. (18). For p= 3000,\nAPQA = 1:9.\nWe discuss the radial overlap \frst for the magnon \u0018= 0\nwith magnetization given by Eq. (34). Then\nR(0)\nPQA =ZR\n0\u000b3=2Jp(kPr)Jq(kQr)p\nNp(kPR)Nq(kQR)r\u000b+1(R2\u0000r2)\nR\u000b+4dr\n(44)\nwherefkP;kQgare the photon wave numbers, Eq. (2).\nSince the magnetic amplitude is signi\fcant only near the\nsurface, we may linearize the optical \felds (the Bessel\nfunctions) close to R. Using Eq. (2) and the Airy's func-\ntion approximation [48] , cf. Eq. (7)\nJp(kPr)\u001922=3Ai0(\u0000\f\u0016)\np2=3h\nPTM+p\u0010\n1\u0000r\nR\u0011i\n;(45)\nand\nNp(kPR)\u0019\u00122\np\u00134=3Ai02(\u0000\f\u0016)\n2: (46)\nSimilar results hold for fp;P;\u0016;P TMg!fq;Q;\u0017;P TEg.\nForp\u0019q\u0019\u000b=2,\nR(0)\nPQA =r2\np\u0014\nPTMPTE+PTM+PTE+3\n2\u0015\n:(47)7\nForp= 3000 and ns= 2:2,R(0)\nPQA = 0:08 and the cou-\nplingG(0)\nPQA = 2\u0019\u00022:8 Hz is of the same order as that\nto the Kittel mode, GK= 2\u0019\u00029:1 Hz [see Sec. I] [33].\nWe emphasize that this result depends strongly on the\nmagnetic boundary condition (taken to be fully pinned\nhere) and only indicates the smallness of the coupling.\nThe magnetization Eq. (36) for \u0018\u00151 gives\nRPQA\nMe\u0019ZR\n0dr\nRJp(kPr)Jq(kQr)J\u000b(kAr)p\nNp(kPR)Nq(kQR)N\u000b(kAR);(48)\nto leading order in \u000b, where\nMe=tan\u0012e\u0002F+ cot\u0012e\u0002C\n\u0002F+ \u0002C: (49)\nFor a YIG sphere with parameters in table I, the ellip-\nticity of the magnons tan \u0012e= 0:8 andMe\u00190:95. The\nparameterMetakes into account that m\u001aandm\u001econ-\ntribute di\u000berently to the coupling being proportional to\nthe magneto-optical constants \u0002 Cand \u0002F, respectively\n[see Eq. (9)]. In YIG \u0002 F>\u0002Cin the infrared [see Ta-\nble I], so the coupling is reduced because jm\u001ej<jm\u001aj\n[see Eqs. (36) and (37)].\nThe Bessel functions asymptotically become Airy's\nfunctions, Eq. (7),\njRPQAj\nMe\u0019p\n2p1=3Z1\n0A\u0016(x) A\u0017(x) A\u0018\u0010\n22=3x\u0011\ndx;\n(50)\nwhere the scaled radial coordinate x\nx=l\n(l=2)1=3\u0010\n1\u0000r\nR\u0011\n; (51)\nand the normalized Airy's function,\nAo(x) =Ai (x\u0000\fo)\f\fAi0(\u0000\fo)\f\f: (52)\nRPQA mainly depends on the radial structure of the\nmode amplitudes with a weak scaling factor of p1=3. We\nsummarize results as f\u0016;\u0017;\u0018;RPQAg;where\u0018is chosen\nto maximizeRPQA for givenf\u0016;\u0017g. Forp= 3000, we\n\fndf1;1;1;8:02g,f1;2;1;3:64g, andf2;2;3;5:63g, much\nlarger than the dipolar mode R(0)\nPQA = 0:08.\nFor a given pair ( P;Q), we de\fne GPQas the maxi-\nmum over all GPQA. Withx=x0wherex2fp;q;\u000bg,\nthe angular momentum of the magnon is \fxed by the\nWGMs, see Eq. (43). The radial index can be found by\nmaximizing the integral appearing in Eq. (50) by enumer-\nating it for each \u0018. The maximum appears at \u0018\u0018O(1)\nfor\u0016;\u0017\u0018O(1), so we do not need to go beyond \u0018= 10.\nWe present the \fnal results in the table II, where\nGPQ\u00182\u0019\u0002200 Hz. This can be compared with the\nmaximum coupling possible for WGMs, GPQdiscussed\nin Sec. I. We \fnd GPQ=GPQ=MeMrwhereMeis given\nin Eq. (49) and the radial `mismatch'\nMr=21=3R1\n0A\u0016(x) A\u0017(x) A\u0018\u0000\n22=3x\u0001\ndxqR1\n0A2\u0016(x) A2\u0017(x)dx: (53)\u0016 \u0017 \u0018 GPQ=(2\u0019)Mr\n1 1 1 304 0 :88\n1 2 1 138 0 :65\n2 2 3 213 0 :74\n1 3 2 144 0 :82\n2 3 4 130 0 :66\n3 3 5 180 0 :70\nTABLE II. The calculated optomagnonic coupling for a given\nf\u0016;\u0017gand\u0018chosen to maximize GPQA.Mris the radial\noverlap de\fned in the text, such that Mr= 1 for the ideal\nmagnetization distribution. Mr\u00181 indicates high overlap.\nTable II indeed shows Mr\u0018O(1) implying a near ideal\nmode matching. Furthermore, GPQ\u001dGK, the coupling\nto the Kittel mode. By doping with bismuth, the cou-\npling can be increased tenfold [61] to GPQ\u00182\u0019\u00022 kHz.\nWe see that GPQ=GPQdoes not depend on Rand hence\nboth scale GPQ;GPQ/R\u00000:9. For a microsphere with\nR= 10\u0016m (p\u0019100),GPQ\u00182\u0019\u00024 kHz is possible in\nYIG, but fabrication is challenging. A very similar the-\nory as outlined here can be applied to YIG disks when\ntheir aspect ratio is close to unity and the demagneti-\nzation \felds are approximately uniform. Scaling those\ndown by nanofabrication of thin \flms may be the most\nstraightforward option to enhance the coupling in other-\nwise monolithic optical wave guide structures.\nThe above analysis for magnon cooling via TM !TE\nscattering can be generalized, similar to the discussion\nat the end of Sec. I. The coupling constant Gcool\nTE!TMis\nsmaller by a factor \u0002 F\u0000\u0002C=(\u0002F+ \u0002C) = 0:45. Also,\nby Hermiticity,jGpump\n\u001b!\u001b0j=\f\fGcool\n\u001b0!\u001b\f\fif the directions of\nmotion are reversed as well.\nA-magnons are e\u000eciently cooled by the process P+\nA!Qwhen the magnon annihilation rate exceeds that\nof the magnon equilibration. For the internal optical dis-\nsipation\u0014intand the leakage rate of photons into the \fber\n\u0014ext, the cooperativity should satisfy [32]\nC=4G2\nPQAnP\n(\u0014int+\u0014ext)\u0014A>1 (54)\nwherenPis the number of photons in P-mode,\u0014A\u0018\n2\u0019\u00020:5MHz is the magnon's linewidth in YIG, and \u0014int\u0018\n2\u0019\u00020:1\u00000:5 GHz [16{18]. We assumed !P+!M=!Q\nfor simplicity. In terms of input power Pin, [32]\nnP=4\u0014ext\n(\u0014int+\u0014ext)2Pin\n~!P: (55)\nThe cooperativity Cis maximized at \u0014ext=\u0014int=2 for a\ngiven input power.\nForGPQA\u00182\u0019\u0002200 Hz,CPQA = 1 fornP\u0018\n109\u00001010requiring large powers Pin\u001850\u00001000mW\nfor!P= 2\u0019\u0002200THz. However, required Pincan be\nsigni\fcantly reduced by scaling or doping as discussed\nabove: a tenfold increase in Gcauses a hundredfold de-\ncrease in required input power. Similar arguments hold8\nfor magnon pumping processes P!A+Q0. The steady\nstate number of magnons is governed by a balance of all\ncooling and pumping processes, whose analysis we defer\nto a future work.\nThe strong coupling regime is reached under the con-\nditionGPQApnP>(\u0014int+\u0014ext);\u0014Awhich again re-\nquires an unrealistically large nP>1012forGPQA\u0018\n2\u0019\u0002200 Hz and powers exceeding kilowatts, because\nof the large optical linewidths observed in typical YIG\nspheres [16{18]. The optical lifetime is limited by mate-\nrial absorption [16] and thus, can be improved only at the\ncost of reduced magneto-optical coupling. 2-3 orders of\nmagnitude improvement in coupling constant is required\nto bridge this gap.\nV. DISCUSSION\nWe modeled the magnetization dynamics in spher-\nical cavities in order to \fnd its optimal coupling to\nWGM photons. We \fnd that selected exchange-dipolar\nmagnons localized close to the equator (but not the\nDamon-Eshbach modes) are almost ideally suited to play\nthat role. We predict an up to 40-fold increase in the\ncoupling constant, implying a 1000-fold larger signal in\nBrillouin light scattering, as compared to that of the\n(unexcited) Kittel mode. Further improvement requires\nsmaller optical volumes or higher magneto-optical con-\nstants.\nThe option to shrink the cavity and optical volume is\nlimited by the wavelength \u00150=ns. For\u00150= 1:3\u0016m and\nns= 2:2, a cavity with an optical volume of \u00153\n0=n3\nsgives\nan upper limit\u00182\u0019\u000250 kHz for pure YIG. In a Bi:YIG\nsphere of radius\u0018\u00150=ns, the optical \frst Mie resonance\nmay strongly couple with the Kittel mode [35].\nThe coupling can be enhanced by the ellipticity an-\ngle\u0012eof the magnetization, which is controlled by crys-\ntalline anisotropy, saturation magnetization, and geome-\ntry. Linear polarization \u0012e!0 or\u0012e!\u0019=2 would lead\nto a diverging coupling, but in practice magnons are close\nto circularly polarized, \u0012e\u0019\u0019=4. For YIG spheres the\nweak ellipticity even suppresses the coupling, Me<1 in\nEq. (49).\nIn purely dipolar theory, the surface magnons are chi-\nral, i.e. only modes with m> 0 exist. Then, from Fig. 1,\nmagnon creation is not allowed leading to improved cool-\ning of magnons [32]. When the exchange interaction kicks\nin, propagation is not unidirectional [62], but we still ex-\npect suppression of the red sideband (magnon creation).\nWe leave an analysis of the chirality of exchange-dipolar\nmagnons to a future article.\nWe \fnd that light may e\u000eciently pump or cool cer-\ntain surface (low wavelength) magnons that do not cou-\nple easily to microwaves. This could be used to manipu-\nlate macroscopically coherent magnons, raising hopes of\naccessing interesting non-classical dynamics in the fore-\nseeable future.ACKNOWLEDGMENTS\nWe thank T. Yu, S. Streib, M. Elyasi, and K. Sato\nfor helpful input and discussions. This work is \fnan-\ncially supported by the Nederlandse Organisatie voor\nWetenschappelijk Onderzoek (NWO) as well as Grant-\nin-Aid for Scienti\fc Research (Grant No. 26103006) of\nthe Japan Society for the Promotion of Science (JSPS).\nAppendix A: Exchange-dipolar magnons\nHere, we solve Eqs. (23)-(26) with Maxwell boundary\nconditions, Eq. (27), and pinned surface magnetization\nm\u0006(R) = 0. The magnetization in the linearized LL\nequation, Eq. (24), can be eliminated in favor of the\nscalar potential  , Eq. (23) [46],\n\u0014\u0000\nO2\u0000!2\u0001\nr2+!sO\u0012\nr2\u0000@2\n@z2\u0013\u0015\n = 0;(A1)\nwhereO=!a\u0000Dexr2withDex=!s=k2\nex. The general\nsolution for a sphere is complicated because the magne-\ntization breaks the rotational symmetry, but it can be\nsimpli\fed for the surface magnons near the equator. The\nansatz\n (r) =Ym\nl(\u0012;\u001e)\t(r); (A2)\nwhere\nYm\nl(\u0012;\u001e) = (\u00001)ms\n2l+ 1\n4\u0019(l\u0000m)!\n(l+m)!Pm\nl(cos\u0012)eim\u001e\n(A3)\nare spherical harmonic functions with associated Legen-\ndre polynomials\nPm\nl(x) =(\u00001)m\n2ll!\u0000\n1\u0000x2\u0001m=2dl+m\ndxl+m\u0000\nx2\u00001\u0001l;(A4)\nleads tor2 =Ym\nl^Ol\t where\n^Ol=1\nr2@\n@r\u0012\nr2@\n@r\u0013\n\u0000l(l+ 1)\nr2(A5)\nhave spherical Bessel functions of order las eigenfunc-\ntions. The surface magnons with large angular momen-\ntumlare localized near the equator and have a large\n\\kinetic energy\" along the equator. The con\fnement\nalong the\u0012-direction is not so strong, however, so the\nmagnon amplitude looks like a \rat tire. A posteriori,\nwe \fndk\u0012/p\nl;whilek\u001e/l. For large l, the terms\n@2\nz\u0019R\u00002@2\n\u0012near the equator, may therefore be disre-\ngarded in Eq. (A1). This gives a cubic in ^Ol, similar to\na magnetic cylinder [42],\n^Ol\u0010\n^Ol+k2\u0011\u0010\n^Ol\u0000\u00142\u0011\n\t = 0; (A6)9\nwhere\nDexk2=!sq\u0000!a\u0000!s\n2; D ex\u00142=!sq+!a+!s\n2;(A7)\nwhere\n!sq=r\n!2+!2s\n4: (A8)\n\u0014is real and kis real as well when ! >p\n!2a+!a!s,\nwhich is the case for k\u0019l=R, i.e. waves propagating\nalong the equator [see Sec. IV].\nConsider the eigenvalue equation ^Ol\t\u0016=\u0000\u00162\t\u0016with\nreciprocal \\length scales\" \u00162f0;k;i\u0014g. Its two linearly\nindependent solutions are spherical Bessel functions of\n\frst and second kind, which in the limit l\u001d1 are pro-\nportional to Bessel functions of \frst [ Jl(\u0016r)] and second\n[Yl(\u0016r);not to be confused with the spherical harmonic\nYm\nl] kind, respectively. Yl(\u0016r) diverges at r= 0, so inside\nthe sphere \t \u0016=Jl(\u0016r). Thus, Eq. (A6) has three lin-\nearly independent solutions, f\t0;\tk;\ti\u0014gand the gen-\neral solution is\n\t =3X\ni=1\u000biJl(\u0016ir)\n\u0016iJl\u00001(\u0016iR); (A9)\nwhere\u00161!0,\u00162=k,\u00163=i\u0014,\u000biare integration\nconstants, and the Bessel functions\nJl(z) =1X\nr=0(\u00001)r\nr!(r+l)!\u0010z\n2\u00112r+l\n: (A10)\nThe spatial distribution of the three components are dis-\ncussed in more detail in the main text [see Sec. III].\nBringing back the angular dependence,  =Ym\nl\t [see\nEq. (A2)], the derivative @\u0006=@x\u0006i@y(introduced in\nSec. II)\n@\u0006 =Ym\nle\u0006i\u001e3X\ni=1\u000bi\nJl\u00001(\u0016iR)\u0012\nJ0\nl(\u0016ir)\u0007mJl(\u0016ir)\n\u0016i\u001a\u0013\n;\n(A11)\nwhere@\u0006=@x\u0006i@y. Close to the equator, \u001a\u0019rand\nusingl\u001djl\u0000mj,\n@\u0006 \u0019\u0007Ym\u00061\nl\u000613X\ni=1Jl\u00061(\u0016ir)\nJl\u00001(\u0016iR); (A12)where we used the recursion relations [48]\nJ\u000b\u00061(x) =\u000b\nxJ\u000b(x)\u0007J0\n\u000b(x) (A13)\nandYm\u00061\nl\u00061\u0019e\u0006i\u001eYm\nlthat holds for l\u001d1;jl\u0000mj. Solv-\ning Eq. (24) for magnetization,\nm\u0006(r) =Ym\u00061\nl\u000613X\ni=1\u0010i;\u0006Jl\u00061(\u0016ir)\nJl\u00001(\u0016iR); (A14)\nwith coe\u000ecients\n\u0010i;\u0006=!s\u000bi\n!\u0006~!i; (A15)\nand ~!i=!a+Dex\u00162\ni.\nOutside the magnet,  osatis\fes a Laplace equation\nEq. (26). Using the continuity of magnetic potential and\n o!0 atr!1 ,\n o=Ym\nl(\u0012;\u001e)\u0012R\nr\u0013l+1 3X\ni=1\u000biJl(\u0016iR)\n\u0016iJl\u00001(\u0016iR):(A16)\nThe integration constants \u000biare governed by the\nboundary conditions: Maxwell boundary conditions,\nEq. (27), and pinned magnetization boundary condition\nfor the LL equation m\u0006= 0;which we justi\fed a pos-\nteriori in Sec. III. Demanding m\u0000(r=R) = 0 and\n@r( \u0000 o)jr=R= 0 gives\n3X\ni=1!s\u000bi\n!\u0000~!i= 0 =3X\ni=1\u000bi; (A17)\nwhich is solved by\n\u000b1=m0(!\u0000~!1)(~!2\u0000~!3)\n!s; (A18)\n\u000b2=m0(!\u0000~!2)(~!3\u0000~!1)\n!s; (A19)\n\u000b3=m0(!\u0000~!3)(~!1\u0000~!2)\n!s; (A20)\nwherem0is a normalization constant.\nWe now arrive at the solution discussed in the main\ntext, Sec. III. With f\u00161;\u00162;\u00163g=f0;k;i\u0014g\nlim\n\u00161!0Jl(\u00161r)\u00191\nl!\u0010\u00161r\n2\u0011l\n;Jl(i\u0014r) =ilIl(\u0014r);(A21)\nwhereIis the modi\fed Bessel function. The above holds\nalso forl!l\u00061. Substituting into Eq. (A14),10\nm\u0000=Ym\u00001\nl\u00001\u0014\n\u00101;\u0000\u0010r\nR\u0011l\u00001\n+\u00102;\u0000Jl\u00001(kr)\nJl\u00001(kR)+\u00103;\u0000Il\u00001(\u0014r)\nIl\u00001(\u0014R)\u0015\n(A22)\nm+=Ym+1\nl+1\u0014\n0 + \u00102;+Jl+1(kr)\nJl\u00001(kR)\u0000\u00103;+Il+1(\u0014r)\nIl\u00001(\u0014R)\u0015\n: (A23)\n0 1 2 3 4 5 6 7\n(ω−ωN)/ωs[×10−3]−4−2024R1(ω)\nR2(ω)\nFIG. 4. The resonance condition R1=R2gives the allowed\nmagnon frequencies when the magnetization is pinned at the\nsurface.!Nis the frequency at which kR=l.\nIn spite ofJl\u00001(\u00161r)!0, the \frst term of m\u0000is \fnite\nwhile that of m+vanishes. The Bessel function ratios in\nthe third terms are real even though Jl(i\u0014r) need not be.\nAccording to Eq. (A15) the polarization does not\ndepend on the coe\u000ecients \u000bi. Withf~!1;~!2;~!3g=\nf!a;!sq\u0000!s=2;\u0000!sq\u0000!s=2g,!2\nsq=!2+!2\ns=4\n\u00102;+\n\u00102;\u0000=!+!s=2\u0000!sq\n!\u0000!s=2 +!sq: (A24)\nA similar result holds by substituting \u00102\u0006!\u00103\u0006and\n!sq!\u0000!sq. Multiplying the numerator and denomina-\ntor in the above equation by !\u0000!s=2\u0000!sq, we arrive at\nthe form Eq. (31) in the main text.\nSubstituting \u000bifor the pinned boundary conditions,\nEqs. (A18-A20), into Eq. (A15)\n\u00101;\u0000=m02!sq\n!s(A25)\n\u00102;\u0000=\u0000m0!a+!sq+!s=2\n!s; (A26)\n\u00103;\u0000=m0!a\u0000!sq+!s=2\n!s: (A27)\nThe above solutions satisfy Maxwell's boundary condi-\ntions, Eq. (27), and m\u0000(R) = 0 by design [see Eq. (A17)].\nThe last condition m+(R) = 0 gives the resonance con-ditionR1(!) =R2(!), where\nR1(!) =\u0000Jl+1(kR)\nJl\u00001(kR);R2(!) =k2\n\u00142!sq+!\n!sq\u0000!Il+1(\u0014R)\nIl\u00001(\u0014R):\n(A28)\nThe roots of the above equation are counted by \u0017\u00150.\nFork > 0, the lowest root \u0017= 0 occurs near k\u00190 at\nfrequency!\u0019p\n!2a+!a!s. The next and higher roots\noccurs only around kR&las plotted in Fig. 4 [the root\n\u0017= 0 is to the far left of the origin]. R1is a rapidly\nvarying function, while R2\u00191:2 is nearly constant. Suf-\n\fciently far from the zeroes of Jl\u00001(kR),R1<0 and\nat the crossing with R2;R1\u00191:2. This implies that at\nmagnon resonances, Jl\u00001(kR)\u00190 orkR\u0019l+\f\u0017(l=2)1=3,\nwhile!(k) is given by Eq. (A7). Their explicit values are\ndiscussed in Sec. III\nAppendix B: Normalization\nThe classical Hamiltonian for a sphere that leads to\nthe LL equation, Eq. (20), reads [40]\nH=\u0000\u00160Z\u0014\u0012\nHapp\u0000Ms\n3\u0013\nMz+m\u0001he\u000b\n2\u0015\ndV; (B1)\nwhere\nhe\u000b=2Aex\n\u00160M2sr2m+hdip: (B2)\nand the integral is over all space. The solution of the\nlinearized LL equation of motion gives a complete set of\nmodes with spatiotemporal distribution mp(r)e\u0000i!ptand\nfrequencies !p. We may expand the \felds\nA(r) =X\np;!p>0\u0002\nAp(r)\u000bp+A\u0003\np(r)\u000b\u0003\np\u0003\n; (B3)\nwhereApis the amplitude of any of fmx;my;hx;hygof\nthep-th mode. Here and below the sum is restricted to\npositive frequencies. We have !a=\r\u00160(Happ\u0000Ms=3),\n!s=\r\u00160Ms, and\nMz\u0019Ms\u0000m2\nx+m2\ny\n2Ms: (B4)\nEq. (20) relates mpandhp,\n!shx;p=!amx;p+i!pmy;p (B5)\n!shy;p=!amy;p\u0000i!pmx;p (B6)11\nInserting these into the Hamiltonian ,\nH=\u00160\n2X\npq\u0002\nXpq\u000bp\u000bq+X\u0003\npq\u000b\u0003\np\u000b\u0003\nq+Ypq\u000bp\u000b\u0003\nq+Y\u0003\npq\u000b\u0003\np\u000bq\u0003\n;\n(B7)\nwhere\nXpq=i!q\n!sZ\n(my;pmx;q\u0000mx;pmy;q)dV (B8)\nYpq=i!q\n!sZ\u0000\nmx;pm\u0003\ny;q\u0000my;pm\u0003\nx;q\u0001\ndV: (B9)\nFollowing Ref. [49], we \fnd orthogonality relations\nbetween magnons. For bp=hp+mp;r\u0001bp= 0 from\nMaxwell's equations and\nZ\n \u0003\nqr\u0001bpdV= 0; (B10)\nwhere the scalar potential  qobeysr2 q=r\u0001mq.\nIntegrating by parts and using h\u0003\nq=\u0000r \u0003\nq,\nZ\n(hp+mp)\u0001h\u0003\nqdV= 0: (B11)\nUsing the same relation with p$qand subtracting,\nZ\u0000\nmp\u0001h\u0003\nq\u0000m\u0003\nq\u0001hp\u0001\ndV= 0: (B12)Substituting the mode-dependent \felds hp(q)from\nEqs. (B5)-(B6), we \fnd that ( !p\u0000!q)Ypq= 0. 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Giles1, Zihao Yang2, John S. Jamison1, and Roberto C. Myers1-3 \n \n1Department of Materials Science and Engineeri ng, The Ohio State University, Columbus, OH, \n43210, USA Email: myers.1079@osu.edu  , Web site: http://myersgroup.e ngineering.osu.edu  \n2Department of Electrical and Computer Engine ering, The Ohio State University, Columbus, \nOH, 43210, USA \n3 Department of Physics, The Ohio State University, Columbus, OH, USA \n \nThe spin diffusion length for thermally exc ited magnon spins is measured by utilizing a \nnon-local spin-Seebeck effect meas urement. In a bulk single crysta l of yttrium iron garnet (YIG) \na focused laser thermally excites magnon spins. The spins diffuse laterally and are sampled using \na Pt inverse spin Hall effect detector. Ther mal transport modeling and temperature dependent \nmeasurements demonstrate the absence of spurious  temperature gradients be neath the Pt detector \nand confirm the non-local nature of the experi mental geometry. Remarkably, we find that \nthermally excited magnon spins in YIG travel over 120 µm at 23 K, indicating that they are robust \nagainst inelastic scatteri ng. The spin diffusion length is found to  be at least 47 µm and as high as \n73 µm at 23 K in YIG, while at ro om temperature it drops to less than 10 µm. Based on this long \nspin diffusion length, we envision the development of thermally pow ered spintronic devices based \non electrically insulating, but  spin conducting materials.  \n  2\nI. INTRODUCTION \n \nSpin currents consist of a flux of angular momentum [1]. In ma gnetically ordered \ninsulators, spin currents are carried by magnons. These spin currents are typically generated via \nthe spin Hall effect [2] or through spin pumpi ng, in which a microwave magnetic field excites \nspin-waves of approximately 10 GHz [3] . Such  microwave frequency spin-waves have been \nproposed for use in logic devices with low power  consumption [4], and recently an all-magnon \nbased transistor was demonstrated [5]. The de- phasing length of such coherent spin-waves has \nbeen measured to be 31 m at room temperature in YIG [6].  \nRecently, the spin Seebeck effect (SSE)  [7–10]  has been shown to provide an alternative \nmethod for spin excitation in ma gnetic insulators. An  induced temperatur e gradient produces \nthermally-excited incoherent magnons that have  much higher energies  (500-6000 GHz) than the \ncoherent microwave frequency magnons  previously discussed (1-10 GHz )  [11]. It is expected that \nthermally-generated magnons will couple more strongl y to the lattice, and have shorter lifetimes \nand diffusion lengths than the microwave generate d spin-waves. Nevertheless, these spin currents \nhave stimulated a great interest within the field of spin caloritronics [ 12]. Additionally, they may \nresult in the discovery of f undamentally new physics, such as  the Bose-Einstein condensation \n(BEC) of superfluid magnons at the Pt/YIG interface [13]. Howeve r, the realization of such \nadvances is hampered by the lack of quantitative measurements of the length scale over which these thermally-excited magnons exist.  \nIn a recently reported thickness dependent measurement of SSE, Kehlberger et al. suggest \na thermal magnon spin diffusion length of  approxi mately 100 nm at room temperature [14]. The \nobserved saturation of the SSE signal for films thic ker than 100 nm was inte rpreted to indicate the \nlength scale over which thermal magnons could contri bute to the signal.  Th eoretically, this length 3\nscale was predicted to be 70 nm at 300 K, based on modeling of the SSE in YIG/Pt [15], which is \nconsistent with this observation. Kehlberger et al. also find that the thermally excited magnon \npropagation length reaches ~ 7 µm at 50 K. However, the strong in terface sensitivity of the SSE \nleads to measurement uncertainty due to sample variation i.e each Pt/YIG interface is different \nbetween different samples [16]. Additionally, in the longitudinal SSE geometry, the applied heat \nflows directly into the spin detector, allowing for the possibility of contam ination of the measured \ninverse spin hall signal from other charge-based thermo-electric eff ects [17] such as the anomalous \nNernst effect (ANE). This motivates us to design a new experiment capable of probing the \ndiffusion length of thermally-excited magnon spins out side of the vicinity of any thermal gradients \nwhile simultaneously controlling for interface se nsitivity, hence making a measurement that is \nimmune to either parasitic thermoelec tric effects or in terface sensitivity.  \nHere we present systematic measurements of thermally-excited magnon spin diffusion in \nYIG, by utilizing a non-local detectio n geometry to sample spin currents induced by SSE. In metals \nand semiconductors, a similar non-local geometry wa s successful in measuri ng pure electron spin \ncurrents diffusing in the absence of parasitic effects caused by an  applied electric field [18]. \nBecause spin diffusion in the non-local measurements is caused solely by the gradient in chemical potential and not by driving forces , pure spin currents are directly isolated in  such measurements, \nthereby providing a clean measurement of the diffusi onal spin flow in solids . By utilizing this non-\nlocal spin injection/de tection geometry, pure magnonic spin cu rrents are similarly isolated from \nthe parasitic effects associated w ith an applied thermal gradient. Th is allows for an experimental \ndetermination of the magnon spin diffusion length in YIG. A similar non-local magnon spin measurement in YIG was very recently reported by Cornelissen et al. [19], where they utilized the \nspin Hall effect to inject a spin current into YIG and measured a spin diffusion length of 9 m at 4\nroom temperature. In the current experiment, we extend those measurements to 23 K and find a \nremarkable increase in the spin diffusion length up to 47 m upon cooling the sample. The pure \nmagnon spin current signal is observed even beyond 120 m. Our measurements clearly show that, \njust as in the case of elect ron spin diffusion, magnon spins ar e preserved over many inelastic \nscattering events. \nII. NON-LOCAL SPIN-SEEBECK  \nA. Experimental Setup \nThermal spin injection and detection is pe rformed utilizing the opto-thermal method [20], \nin which a focused laser is absorbed by a Pt f ilm that has been e-beam evaporated onto a bulk \nsingle crystal of YIG. A laser is modulated with  an optical chopper and th e resulting transverse \nvoltage is sampled using a lock-in amplifier. Instead of using a continuous Pt film to detect the spin current, we pattern the film into a central Pt  spin detector and an array of electrically and \nspatially isolated Pt absorption pa ds, as shown schematically in Fig. 1(a). When the laser is focused \non an absorption pad, a local hot spot is  produced, generating a thermal gradient (\nT) whose \nmagnitude is varied by adjusting the average power of the beam. T excites magnons in the YIG \nbeneath the absorption pad, producing a spin curren t. Magnon spins that have laterally diffused \nunderneath the spin detector can th en transfer their spin  via spin pumping into  the spin detector, \nwhere they are converted into  an electron spin current ( Jsz), which is then converted into a \ntransverse voltage al ong the y-direction ( Vy) by the inverse spin Hall effect (ISHE) [7] . The \nmagnitude of  Vy is proportional to the  magnitude of the spin current and its sign tracks the sign of \nthe spin polarization injected into the detector , thereby following the si gn of the in-plane (xy-\nplane) magnetization ( M) of YIG. \nB. Thermal Transport in Non-Local Configuration 5\nTo ensure that the de tected spins were generated non-lo cally, no significant heat flow can \noccur across the interface between  the Pt spin detector and the YIG layer beneath. This is \nconfirmed by using a finite-element method (FEM ) to simulate heat flow and steady-state \ntemperature gradient profiles caused by the heati ng laser. When the laser is focused on the Pt \nabsorption pad, there are thermal gr adients in both the vertical ( Tz) and lateral ( Tx) directions. \nFEM simulations of Tz and  Tx directly below the heating laser are plotted in Fig. 1(b) and Fig. \n1(c). The average laser power is 29.9 mW and the spot diameter is 7.5 m. The simulations are for \n23 K. Clearly, in the non-local c onfiguration, heat flows from the absorption pad into the YIG. If \nthe absorption pad is close enough to  the detection pad, lateral heat  flow through the YIG may lead \nto Tx beneath the detector. However, because the det ector is a thermal open circuit, the laterally \ndiffusing heat cannot then diffuse vertically in to the detector (see Appendix A). FEM modeling \nverifies this fundamental point, showing that Tz is not present at the spin-detector. This verifies \nthe non-local detection principle that the sampled spins are not generated at the detector/YIG \ninterface as in longitudinal SSE [9]. FEM modeling further confirms that lateral heat flow, as \nindicated by Tx, decays to 1% of its peak value within 22.9 ± 1.08 m. Therefore, when the laser \nis positioned at a di stance greater than 23 m from the spin detector neither substantial vertical, \nnor lateral, heat flow exists  beneath the Pt detector. Any Vy is attributable only to the diffusion of \na pure magnonic spin current thermally  produced at the remote Pt ab sorber. As will be shown in \nthe following section and in Appendix B, even when the laser is closer than 23 m the effects of \nTx are not present. \nIII. RESULTS AND DISCUSSION \nIn Fig. 2, the raw transverse vo ltage across the detection pad, Vy, is shown as a function of \nan in-plane magnetic field ( Bx) for measurements carried out in both the local and non-local 6\nconfigurations. In the lo cal configuration, the lase r is positioned directly on the Pt detection strip. \nAs these spins are both generated an d detected in the presence of Tz, Vy may contain contributions \nfrom both the longitudinal SSE [9] and the anomal ous Nernst effect (ANE ) [21,22] . A schematic \nof the local configuration is show n in Fig. 2(a), and the corresponding Vy, measured across the Pt \ndetection strip as a function of Bx, is presented in Fig. 2(b) . Vy saturates to 746  3 nV at >|50| mT, \nwhich matches the saturation fi eld of YIG, and the sharp Vy hysteresis is consistent with the small \ncoercive field of YIG. The M-H curve for this sa mple is included in Fig. 2(b) for comparison. We \ndefine the magnitude of the local voltage VL as the magnitude of the hysteresis loop. The inset plots \nVL over a range of laser powers, re vealing a linear relationship consistent with both the SSE and \nANE dependence on Tz. Note that in the absence of addition al measurements it is not possible in \nthe local configuration to determine the relati ve contributions of the SSE and the ANE to VL  [23]. \nIn the non-local configuration, th e laser is positioned on one of the electri cally isolated Pt \nabsorption pads, as depict ed in Fig. 2(c). Since Tz = 0, ANE cannot contribute to VNL. The non-\nlocal configuration detects only the thermally-induced magnon spin s that have di ffused beneath \nthe detector. Fig. 2(d) shows the corresponding Vy in the Pt detection strip as a function of Bx, if \nthe laser is on an absorption pad that is 54 m from the detection pad. Again, Vy tracks the magnetic \nhysteresis of YIG. VNL is defined similarly to VL, as the magnitude of the measured hysteresis loop \nin the non-local configuration, but no longer cont ains any possible ANE contribution. It scales \nlinearly with laser power, however its overall intensity is an or der of magnitude smaller than VL. \n\tImportantly, VNL also exhibits the same sign as VL. This experimentally confirms that there is no \naccidental Tz across the detector. If VNL were due to accidental heat flow from the YIG back into \nthe detector, then Tz, and therefore VNL would change sign when the laser was moved from the 7\ndetector to the absorber pad.  This possibility is ruled out  on both fundamental grounds, as \ndescribed in Section II, and on experimental  grounds (lack of obser ved sign change).  \nWe also consider the possibility that para sitic heat flow could occur through the spin \ndetector by thermal conduction through the detect or wires leading to local magnon generation at \nthe detector/YIG interface and contamination of VNL. To generate the observed VNL  would require \nuse of unphysical thermal conditions (see Appe ndix A) and 0.4-mm thick wires. Since our \nmeasurement wires are 25 m in diameter and the sample is held in high vacuum (1×10-7 torr), \nsuch parasitic thermal leakage can be ruled out. \nTo spatially map the thermally-g enerated magnon spin current, VNL is extracted from raw \nhysteresis traces (as in Fig. 2) with the laser positioned above absorption pads at increasing \ndistances from the spin detector ( Δx). Fig. 3(a) plots VNL extracted from such measurements as a \nfunction of Δx at 23 K. The same measurement is carri ed out at 280 K and plotted in Fig. 3(b). \nAdditionally, the modeled Tx at both temperatures are plotte d in Fig. 3 to compare the spin \ntransport versus the thermal transport. Since non-negligible  Tx is predicted near the edge of the \nspin detector, then in this region magnon spin transport may be driven by Tx as in transverse \nSSE [7,8]. However, this is ruled out by compar ing the measurements at 23 K to those at 280 K. \nAt low temperature, Tx decays much more rapidly than VNL, whereas at 280 K, VNL is negligible \nat all values of Δx. Conversely, Tx at high temperature exhibits a much larger magnitude than at \nlow temperature due to the decrease in thermal conductivity of Pt and YI G at high temperature, \nhowever no VNL is observed at these conditions. The lack of VNL at high temperature is attributable \nto the reduction of the magnon spin  diffusion length, which agrees w ith a recent re port of room \ntemperature measurement of magnon  spin diffusion in YIG of 9 m [19], which is smaller than \nthe spatial resolution of our current experiment.  8\nThe VNL data are fit to a single decaying exponentia l over various ranges. The spin current \ndecays exponentially with distance, consistent wi th the solution to the spin-diffusion equation at \nsteady-state. From the exponential fits to ܸே\tሺ∆ݔሻൌ\tܸே௦݁ି∆௫/ఒೄ∗ we obtain the effective magnon \nspin diffusion length ߣௌ∗. Fig. 4 plots the results of non-local measurements  carried out at three \ndifferent laser powers, togeth er with the exponential fits. ߣௌ∗ exhibits an average value of 47 m \nindependent of laser power. The lack of a change in ߣௌ∗ with laser power imp lies that the magnon \nspin scattering and lifetime are insensitive to  the applied temperature gradient. There is no \nsignificant variation for fits to data sets that include data points gathered with the laser focused on \ndiffering sets of absorption pads (see Appendix B). Additionally, Tx decays much more rapidly \nthan VNL (see Appendix A). This analysis indicates that  lateral heat flow has no detectable impact \non the ߣௌ∗ measurements under the current conditions. \nIn addition to analyzing the im pact of lateral heat flow on ߣௌ∗, we also consider the effect of \nthe Pt absorption pads that are along the path between the absorber under illum ination and the spin \ndetector. Since Pt is a well-known spin absorber [24], these unused pads will act as magnon sinks, \nthereby reducing the number of spins that  reach the spin detector, and reducing ߣௌ∗ from its intrinsic \nvalue ߣௌ. A quantitative 2D FEM analysis of the magnon spin diffusion process in the presence \nof spin sinking indicates that 47 ൏\tߣௌ\t൏ 73\t݉ߤ(  Appendix D).  \nWe now compare the diffusion of magnon spins in YIG with the diffu sion of magnon heat. \nThe mean free path of magnons in bulk YIG was recen tly calculated to be 4 µm at 20 K [25]. Note \nthat this mean free path reflects that of the magnons re sponsible for thermal transport, i.e. those \nmagnons that contribute to the thermal conductivity of YIG. Therefore the m ean free path reflects \nthe distance between inelastic sc attering events. However, our meas urements demonstrate that at 9\nthe same temperature, magnon spins diffuse more than one order of magn itude larger distances \n(47 ൏\tߣௌ\t൏ 73\t݉ߤ  .) \nTwo explanations are offered for this observa tion. First, magnons reta in their spin even \nafter multiple inelastic scattering events. In th at view, the long range magnons being detected are \ninelastically scattered ma gnons that still carry spin. This si mple picture mirrors the case for \nelectron spins in solids, where the spin-flip rate of electrons is typically far slower than the charge \nscattering rate. Thus, like electrons, magnons reta in their spin polarization between successive \nlinear momentum scattering events . In the second view, during ther mal excitation of YIG, the \nthermal magnons that carry the majo rity of the heat scat ter at short distances  due to their short \nwavelengths and stronger interac tion with the lattice. However, a small subset of the magnons, \nwhich are referred to as sub-ther mal magnons and that are at much lower energies and have longer \nwavelengths, do not couple effectiv ely to the lattice and therefore exhibit much longer mean free \npaths. As recently suggested [25], this can expl ain the ~100 nm length s cale of longitudinal SSE \nat room temperature [14], the high magnetic field suppression of spin-Seebeck at room \ntemperature [22,26] and is also c onsistent with our current observ ations at low temperature.  \nThe long range diffusion of magnon spins at 23 K ( 47 ൏\tߣௌ\t൏ 73\t݉ߤ ,) is larger than what \nwas recently reported at room  temperature by Cornelissen et al. (9 m) [19], which indicates a \ntemperature dependent inelastic scattering process for magnons in agreement with Boona and \nHeremans [25]. It is interestin g to note the surprising similarity between the low temperature \ndiffusion length of thermally-excited magnon spins in YIG compared to the room temperature \ncoherence length of microw ave spin-waves (Pirro et al. ) [6]. Clearly additional measurements \nfocusing on the temperature varia tion of magnon spin diffusion ar e required to un derstand the \nscattering processes further and connect the fi elds of microwave and thermal spintronics.  10\nSince thermally-induced magnon spin s in YIG are preserved over 100 m, they may be \nused in laterally-patterned spin-current based devices powered by waste-heat. The low temperature \nspin diffusion length of YIG is far longer than that of n-GaAs (~6 m at 4.2 K) [27]. It is \nworthwhile to consider using magnon spin conduc tors as the lateral spin channels (spin \ninterconnects) in existing spin -based device proposals [28].  \nAPPENDIX A: THERMAL MODELING \nI. Thermal Transport Modeling \nThermal modeling is carried out using the commercially available 3D finite element \nmodeling (FEM) software COMSOL [29]. Thr ee dimensional steady-st ate heat diffusion \nequations are solved in Pt/YIG bi-layer structures using FEM. In the model, the YIG is 5 mm ൈ 5 \nmm ൈ  0.5 mm. The Pt spin detector is 265 μm ൈ  265 μm ൈ  10 nm (center region) and the \nabsorption pads are 10 μm ൈ  10 μm ൈ  10 nm. All geometrical parameters match those of the \ndevice used in the non-local measurement. The bottom surface of the YIG is fixed at ambient \ntemperature, mimicking the coppe r heat sink used in the measur ement, while all other surface \nboundaries are thermally insulating, representing the vacuum in the cryostat. The laser is modeled \nas a Gaussian beam with a wavelengt h of 715 nm, and spot size of 1/e2 radius r = 3.75 μm. The \nspot size of the laser was acquired by a knife edge measurement [30]. \nThe characteristics of the modeled laser match those of the laser employed in the \nmeasurement (width of 150 fs, 80 MHz repetition ra te, modulated at a frequency of ~2 kHz). As \nultrafast modeling (not shown here ) indicates that a quasi-steady state is reached between each \nultrafast pulse, we modeled the laser beam to consis t of a square-wave (in r eality a train of pulses) \nmodulated at 2 kHz. The laser is absorbed in the center of the detector and is incident on the Pt \nbetween 0.614 and 0.886 ms. A smoothi ng zone of 0.025 ms is used for the rising and falling of 11\nthe laser power corresponding to a rise time of 12.5 μs. The dark blue circles and light blue \nrectangles are the temporal re sponse of the temperature along th e optical axis at the Pt/YIG \ninterface and 10 μm beneath the interface, respectively. In  both cases, the temperature rapidly \nsaturates to the steady-state solution with a rise time (10%~90%) of ~12.6 μs and ~23.5 μs, \nrespectively. The resulting time profile of the varia tion of temperature of the Pt/YIG bilayer system \nunder laser excitation is shown in Fig. 5. Given th at the system rapidly approaches steady-state \nand the negligible residual excess temperature,  we solve the time-inde pendent heat diffusion \nequations under steady-state conditions for all of the modeling presented. Separate simulations are \ncarried out for various average la ser powers, including 10.5, 21.3 and 29.9 mW. \nThe reflectivity (R) and absorption coefficient ( α) of Pt and YIG are calculated from the \ncomplex dielectric constant ε = ε1 + iε2 at 23 K (300 K), where, for Pt [31], ε1 = -16.26 (-14.36) \nand ε2 = 16.86 (23.708) and for YIG [32], ε1 ==4.28 (5.55) and ε2 ==0 (0) at a wavelength of 715 \nnm. The values of ܴ and ߙ are found to be 73.2%  (69.5% ) and 78.3ൈ10\t݉ିଵ (80.1ൈ10\t݉ିଵ) \nfor Pt, and 12.1%  (16.4%) and 0 (0) for YIG, respectively. We are only aware of low temperature \noptical data for Pt acquired at  77 K, and 87 K for YIG. As ܴ and ߙ for Pt and YIG do not vary \nsignificantly at low temperature (77 K and 87 K) [33,34] and room temperature (300 K) [35] we \nexpect that ܴ and ߙ for Pt and YIG at 23 K are similar to th eir values at 77 K. Due to the thin \nnature of the Pt layer, we also considered the reflected power from the Pt/YIG interface that is \nbeing re-absorbed in the Pt layer. The reflectivity  of the interface is calc ulated using the Fresnel \nequation [35] yielding ܴ௧ൌ 56%  (46% ). The thermal properties of Pt and YIG are \nacquired at 23 K and 300 K while the mass density  data is measured at 300 K. All physical \nparameters used in the simula tions are listed in Table I. \nII. Lateral heat flow 12\nThe simulated ܶ௫ resulting from a heating laser wi th an average power of 10.5 mW, at \nthe Pt/YIG interface, for 23 K and 300 K, are comp ared in Fig. 6. In this simulation, the laser is \nincident on 10 m wide Pt absorption pad at a position of 60 m from the edge of the Pt detector. \nܶ௫ is greater at 300 K due to the decreased therma l conductivity of Pt and YIG. Although the \nmagnitudes are different, the decay length of ܶ௫ is relatively unchanged at the two temperatures \nmodeled, which is clearly illu strated by comparing the two dimensional contour map of ܶ௫ shown \nin Fig. 6(b) to that  shown in Fig. 1(c). \nIII. Thermal loss analysis \nHeat loss due to convection cooling ( ܳ௩௦௦) and black body radiation ( ܳௗ௦௦) at the \nboundary are estimated using the Stefan-B oltzmann law and Newton’s law  [36],  \nܳ௩௦௦ൌ݄∗ܣ∗ሺܶ௦௨െܶሻ (1) \nܳௗ௦௦ൌߝ∗ߪ∗ ܣ∗ሺܶ௦௨ସെܶସሻ (2) \nwhere ݄ is the heat transfer coefficient, ܣ is the sample surface area, ߝ is the emissivity, ߪ is the \nStefan-Boltzmann constant, and ܶ௦௨ and ܶ are the temperature of  the sample surface and \nambient, respectively. For the heat conduction loss, as all the measurements  are conducted within \na cryostat with a pressure below 4ൈ10ି\tݎݎݐ , the heat transfer coefficient in this case is ݄ൌ\n0\t݉/ܹଶܭ( for vacuum) [37] for which the convection loss ܳ௩௦௦ൌ0. For the radiation loss, using \nߪ ൌ 5.67ൈ10ି଼\t݉/ܹଶܭସ, ߝ௧ൌ7 . 3ൈ1 0ିସ,  ܣ ൌ 10ൈ10\t݉ߤ  and ܶ௦௨ൌ 30\tܭ(  higher than \nthe actual surface temperature on sample to estimate the upper bound of ܳௗ௦௦, actual ܶ௦௨ is ~28 \nK) [37,38]  we find ܳௗ௦௦ ~10ିଵଶ\tܹ݉ , which is negligible in comparison to the laser power \n(30\tܹ݉ .) Heat loss due to convection cooling and black body radiation is  therefore safe to neglect \nin the simulations. Besides convection and radiati on losses, an additional heat loss mechanism is 13\nthermal conduction along the two voltage leads (ann ealed Au wire) from the sample surface and \nis estimated by Fourier’s law of thermal conduction  [36], \nܳௗ௦௦ൌߢ∗ܣ∗ܶ(  3) \nwhere ߢ is the thermal conductivity and ܶ is the temperature gradient between two ends of the \nwire. Using ߢ௨ൌ 11.4\tܭ݉ܿ/ܹ , ܣൌߨ∗ቀଶହ.ସ\tఓ\nଶቁଶ\nൌ 506.7\t݉ߤଶ, and ܶ ൌଶଷ.ଵହିଶଷ\nଵ\tൌ\n0.015\t݉ܿ/ܭ , we find ܳௗ௦௦ൌ1 . 7ൈ1 0ିଷ\tܹ݉  which is ~ 0.005\t%  of the absorbed laser power \nat 30 mW [39]. Here, 23.015K is the maximu m temperature under the wire that is ~\t300\t݉ߤ  away \nfrom the laser spot. Note that ܳௗ௦௦ is overestimated si nce the wire is 25.4 ݉ߤ in diameter and the \ntemperature rise in the other contact region (between Au wire and Pt surface) is smaller than \n0.015\tܭ  making the average ܶ smaller than 0.015\nൌ1 . 5ൈ1 0ି\n. However, even with the \nexaggerated value, ܳௗ௦௦ is still a small value compared to the total input power, therefore the heat \nloss due to the heat cond uction through the wires is also neglec ted in the simulati on, and we employ \nthermally insulating boundary conditions at the sample surface. \nIV. Estimation of Longitudinal SSE due to Tz \nWhen the laser is incident dire ctly on the Pt detector, in th e local configuration, an electric \nfield is produced in the Pt beneath the laser due  to the longitudinal spin-Seebeck effect (LSSE). \nSince the laser power follows a Gaussian distribution, the Tz beneath the laser is not uniform as \nit is in the conventional LSSE measurement. Therefore, it is not appropriate to use a single value \nof Tz to predict a measureable transver se voltage in the detection pad ( ܸ௬ሻ.  \nIn the local measurement configuration, this non-uniformity of the laser induced Tz leads \nto a spatially-dependent electric field in the Pt, EISHE, which decreases with increasing distance \nfrom the laser spot. The electric field is terminat ed in the region where the  measurement wire is 14\nattached, due to the equi-potential nature of  the wire. To account for the non-uniformity of Tz \nbeneath the laser, we integrat e the FEM predicted values of Tz at Pt/YIG interface, along a line \nthrough the center of the regi on illuminated by the laser ( ܶߘ௭݈݀) .We find that ܶߘ௭݈݀ exhibits \nthe same power dependence as ܸ௬, as shown in Fig. 7. The slope is different onl y in magnitude, \nimplying that this integral, ܶߘ௭݈݀ is proportional to the measured transverse voltage across the \ndetection pad, ܸ௬. \nThis allows us to determine a linear relation between the thermal gradients beneath the \ndetection pad and the measured voltages. We plot ܸ௬ as a function of ߘܶ௭݈݀  and thereby \ndetermine the effective longitudinal spin-Seebeck effect coefficient ( ߙௌௌா) of our local \nmeasurement configuration. This is found to be ~102 nV/K, as shown in Fig. 8. Table II provides \na summary of the values previously desc ribed, including the FEM modelled values of ܶ௭_௫\t(the \nmaximum thermal gradient at the Pt/YIG interface beneath the heating lase r) and the calculated \nvalues of ܶ௭݈݀ for lines through the center of for regions  beneath the heati ng laser) for three \ndifferent laser powers in the local configuration. ܸ௬ at each power is also included.  \nWe can now use these results to determine how large a spurious ܶ௭ beneath the detection \npad would have to be to cause the observed ܸ௬\tin our non-local measurements. To do this, we \nconsider a specific non-local measurement in whic h the laser is focused on an absorption pad 54 \nμm from the detection pad. The measured ܸ௬ is 25 nV. From linear extrapolation of the ܸ௬\tvs. \nܶ௭݈݀ graph, it is found that in order to produce a ܸ௬\tvalue of 25 nV a value of ܶ௭݈݀ =  0.34K \nis needed beneath the detection pad (See Fig. 8) . As previously discussed, radiative losses and \nconductive losses through the measurement wi res attached to th e detection pad (12.7 m radius) \nresult in at most a ܶ௭\t1.5ൈ10ି\n\t\t to flow through the detection pad. This would yield a V y of 15\nat most, 0.5 pV. It is illu strative to calculate the diameter of detection wires that would be needed \nto account for the measured non-local voltage. As previously stated, this would require ܶ௭݈݀  \n =0.34 K. We find that wires of 200 m radii (0.4 mm diameter) are necessary and that a \ntemperature difference across the length of the wires of Twire = T cryostat  – T Pt = 168 K would be \nrequired to allow for the required heat flow through the Pt detecti on strip. Given that the actual \nwires utilized are more than one order of magn itude smaller in diameter (>100× smaller cross-\nsectional area and thermal conductance), and that an unphysical plat inum temperature, T Pt= -188 \nK, would be required to obtain Twire = 168 K (T cryostat  = 20 K), then Tz cannot contribute to the \nnon-local signal.  \nAPPENDIX B: IMPACT OF LATERAL HEAT FLOW ON MAGNON SPIN DIFFUSION \nA single decaying exponential is fit to multiple VNL data sets containing a varying number \nof data points. Fits were constructed from data sets that include ∆x values > 23 m, where FEM \nmodeling predicts that Tx is negligible (defined as zone II),  and also from data sets with 0 < ∆x \n< 23 m, where Tx may not be negligible (def ined as zone I). As can be seen from Tables III and \nIV, there is no significa nt variation in the R2 value between fits includi ng data points from both \nzones I and II and those containing data points exclusively from z one II. In addition, Fig. 9 and \nFig. 10 show that the fits lie within the error ba rs for single exponential decaying functions fit to \nVNL values from both zones I a nd II. This indicates that ߘTx shows negligible effect on magnon \nspin diffusion under the experi mental conditions tested. \nAPPENDIX C: MATERIALS, METHODS,  AND MEASUREMENT SYSTEMATICS  \nI. Sample processing and characterization \nSingle-side-polished single-crystal <100>  YIG samples with dimensions of 5 ൈ 5 ൈ 0.5 \nmm3 are obtained commercially from the Princeton Scientific Corporation. The polished surface 16\nof the YIG is atomically flat and epi-rea dy with a surface roughness of 0.27 nm measured by \natomic force microscopy (AFM) using a Bruker Di mension Icon AFM tool. The 10 nm Pt is e-\nbeam evaporated on the YIG at a rate of 0.15\tܣሶ/ݏ using a Kurt Lesker LAB 18 deposition system \nwith a base pressure < 2.5ൈ10ି torr. Prior to the Pt depos ition, the YIG samples are cleaned \nusing solvent and DI water, followed by a five-minute dehydration bake at 150 Ԩ in atmosphere \nand a second one-hour in-situ bake at 150 Ԩ. There is no vacuum break between the in-situ baking \nand the actual Pt deposition in order to ensure a high quality Pt/YIG interf ace. The Pt is patterned \ninto a 265\tൈ265\t݉ߤ(  center region) detection pad and 10\tൈ10\t݉ߤ  absorbtion pads with 3\t݉ߤ  \nspacing using standard photo-lithography and ݈ܥଶ/ܨܥସ based reactive-ion etching (RIE). The \nmagnetic properties of the YIG sample are m easured by superconducting quantum interference \ndevice (SQUID) magnetometry using a Quantum Design MPMS XL. The saturation magnetization \nis 220\t݉ܿ/ݑ݉݁ଷ and the coercivity is 5 ܱ݁ with an in-plane magnetic field (same orientation as \nSSE measurement) at 20 K. \nII. Opto-thermal spin-Seebeck measurement \nThe Pt detector is wired with two 25.4 \t݉ߤ diameter gold wires and the YIG is attached to \na copper block heat sink using silv er paste. All measurements ar e conducted in high vacuum in a \ncustomized closed cycle He cryostat coupled to  an electromagnet capab le of producing magnetic \nfields of  0.25 T at the sample. A Chameleon Ultra II Ti:Sapphire laser is focused through a \nreflective microscope objective, producing a  7.5 m diameter laser spot on the sample surface. \nMechanically chopping the laser at 2 kHz to serv e as a reference frequency allows the induced \nvoltage to be measured using standard lock-in technique. Error bars are calculated based on the \nstandard deviation of the value V L and V NL extracted from the ma gnetic field dependent \nmeasurements as defined in the text. 17\nIII. Measurement systematics \nAdditional non-local measurements are carried ou t for the laser positioned to the left and \nthe right side of the Pt spin detector and the results are shown in Fig. 9 and Fig. 10. With only one \noutlying point (59 m), these systematic measurements re veal almost identical values of ߣௌ∗ (~47 \nm) showing the lack of a laser power dependen ce on left/right asymmetry, as well as confirming \nthe reproducibility of the experiment. \nNon-local measurements are also carried out  on a different Pt/YIG sample (20141004). As \nshown in Fig. 11, both V L and V NL track the magnetization. When the laser is focused on the \ndetection pad, V L is ~270 nV, while when the laser is 47 μm away, V NL is approximately 9 nV. As \nthe laser is moved away from the detection pad, V NL falls off exponentially, as expected. A single \ndecaying exponential fits the data with V NL from both zones I and II with an adjusted R2 value of \n.998 (Table V). ߣௌ∗ is constant regardless of the relative lase r position (left side versus right side of \nthe spin detector) shown in Fig. 12.  \nAPPENDIX D: EFFECT OF SPIN SINKING AND THE UPPER BOUND OF MAGNON SPIN \nDIFFUSION LENGTH \nAs previously mentioned, it is well known that  Pt acts as a strong spin absorber. However, \nthe measured value of ߣௌ∗ൌ 47\t݉ߤ  does not take this spin sinking into account. Therefore it serves \nas a lower bound for ߣௌ. It is important to also establish an upper bound for ߣௌ by taking into \naccount the loss of magnons en route to the detector  via spin sinking from the unused Pt absorption \npads. To do this we perform 2D FEM to solv e the diffusion equation for nonequilibrium magnons \nin the presence of Pt spin sinks. Th e magnon diffusion is specified by [40] \nܦଶ݉ߜെ݉ߜ\n߬௧ൌ0 (4)18\nwhere ܦ is the magnon diffusivity in YIG, ݉ߜ is the magnetic moment density (magnon density \ntimes 2µ B), and ߬௧ is the magnon lifetime. The magnon flux from YIG into a Pt spin sink is \ndescribed by \nെ݊ො\t∙\t\tሺെ݉ߜܦ ሻൌെ ܩ݉ߜ (5)\nwhere ݊ො is a unit vector normal to the YIG surface and ܩ is the spin convertance. ܩ is \nestimated as [41] \nܩ\t~ܽܵߨூହܬ௦ௗଶ݃ሺߝிሻܶ\nܽெܶி(6)\nwhere ܽூ and ܽெ are the lattice constants of the YIG and the Pt respectively,  ܬ௦ௗ is the exchange \ninteraction, ݃ሺߝிሻ is the Pt density of states at the Fermi energy and ܶி is the Fermi temperature \nof Pt. Using ܽூൌ 12.376\t Å, ܽெൌ 3.900\t Å, ܬ௦ௗൌ 1.000\tܸ݁݉ , ݃ሺߝிሻൌ 1.164\t∙\t10ଶଷ\t/ݏ݁ݐܽݐݏ\nሺܸ݁ ∙\t݉ܿଷሻ [42], and ܶிൌ 9.812∙\t10ସ  [42] ܩ is calculated to be  31.598 m/s. Note that \nalthough these parameters are repo rted at 300K, they are expected  to be relatively temperature \nindependent. This estimate of ܩ serves as a theoretical upper bound on the degree of spin sinking \nfrom YIG into Pt. From reference [41], we utilize ߬௧ൌ\t10ି\t.ݏ \nFollowing the calculation of ܩ, eqns. (3) and (4) are simulta neously solved in order to \ncalculate the magnon dens ity profile during the non-local meas urements. The geometry of the \nFEM is identical to the experiment al setup and consists of a 265 µm spin de tector centered on a 5 \nmm wide YIG crystal that is 500 µm thick. The right most edge of the detector is at  x = 0. The 10 \nµm wide Pt absorption pads are spaced 3 µm apar t, with each pad further from the edge of the \ndetector. The boundary conditions force ݉ߜ = 0 at the edges of the YIG crystal, a few mm from \nthe point of magnon excitation.  \nResults of the FEM modeling are shown in Fi g. 13a, where a 2D map of the steady state \n݉ߜ distribution is plotted. In this example, the laser is incident in the center  of the Pt absorber at 19\nx = 47 µm. The interfacial magnon density profile is  plotted by slicing the full 2D profile along \nx at z = 0, i.e. at the Pt/YIG in terface, and shown in Fig. 13b. Thr ee cases are shown to illustrate \nthe effect of spin sinking on the magnon density profile. As expected, spin sinking reduces the \noverall magnon density. \nThe derivative of the magnon density profile, ݉ߜ, along the z-direction evaluated at z = \n0, ௭݉ߜ, is plotted in Fig. 13c along x, which shows the magnon concentration gradient \nresponsible for the vertical diffusion of magnons into  the absorbers and spin detector. To determine \nthe total magnon spin current flowing into the detector, ܬ௦௭, we calculate the average of ௭݉ߜ at \nz = 0 over the entire detector  width along x and multiply by –ܦ ,i.e. \tܬ௦௭ൌ\nെ\nଶହ௭݉ߜ\n୶ୀିଶହ݀x. The above procedure is carried out with the magnon injection point \nbeing varied along each of the various detector pads (just as in the experiment) and ܬ௦௭ is determined \nunder each of these conditions. The results are plotted in Fig. 13d. \nThe value of ܦ is treated as a fit parameter and adjusted to achieve ߣௌ∗ = 47 µm, in order to \nmatch the experimental measurements to the FE M model (dashed orange line in Fig. 13d). From \nthis fit we determine a magnon diffusivity of ܦൌ 0.0053 m2/s and therefore since ߣ௦ൌ\tඥ߬ܦ௧ \nwe find ߣ௦ൌ 73\t݉ߤ . Because the theoretical maximum value of  ܩ and ߬௧ were used in this \nmodel, it defines the upper bound for λ௦. Therefore, 47 ൏\tߣௌ\t൏ 73\t݉ߤ . \nܩ, which parameterizes the spin sinking ability of the Pt absorbers, was estimated using \nthe largest reasonable parameters from the literatur e. As such, it is the theoretical maximum. The \nother parameter that will have the largest impact  on the numerical solution to eqns (3) and (4) is \n߬௧. To determine how sensitive our model is to this parameter, a value of ߬௧ൌ1 0ି଼\t\tݏwas also \nchosen as an input parameter. Usi ng this value, a fit parameter of ܦൌ 0.41  m2/s is found, yielding 20\nλ௦ = 64 ݉ߤ .Even by varying ߬௧ two orders of magnitude, the calculated value for λ௦ remains \nrelatively constant. \nACKNOWLEDGEMENTS \nThe authors thank J. P. Heremans and S. R. Boona  for useful discussions. This work was primarily \nsupported by the Army Research Office MURI W911NF-14-1-0016. J.J. acknowledges the Center \nfor Emergent Materials at The Ohio State Un iversity, an NSF MRSEC (Award Number DMR-\n1420451), for providing partial f unding for this research. \n  21\n[1] I. Žuti ć, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). \n[2] D. D. Awschalom and M. E. Flatté, Nat. Phys. 3, 153 (2007). \n[3] C. W. Sandweg, Y. Kajiwara, A. V. Chuma k, A. A. Serga, V. I. Vasyuchka, M. B. \nJungfleisch, E. Saitoh, and B. Hillebrands, Phys. Rev. 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Two-\ndimensional color maps of ܶߘ ,directly below the absorption pad and in the x-z plane, are \npresented. ܶߘ in the vertical direction, ( ܶߘ௭ሻ\t is shown in (b) and ܶߘ  in the lateral direction ( ܶߘ௫ሻ \nis shown in (c). Temperature grad ients are plotted as a function of  distance from the edge of the \nspin detector ( ∆ݔ )and as a function of depth from the surface of YIG (z). The edges of the \nabsorption pad are indicated by dash ed lines in revealing that both ܶߘ௫ and ܶߘ௭ are well isolated \nfrom the edge of the spin detector (at ∆ݔ ൌ 0  m). The color scale is logarithmic.  \n\nFIG. 2. (a) Schematic of the local detection geom etry. (b) The transverse voltage across the spin \ndetector ( Vy) as a function of the applied in-plane magnetic field ( Bx) with the laser positioned on \nthe spin detector as shown in (a). The measured magnetization ( M) is included for comparison. \nThe magnitude of the loca l voltage is defined as VL, as shown in (b) and may contain components \nnot purely due to spin currents. The inset plots VL as a function of laser power. (c) Schematic of \nthe non-local spin-Seebeck geometry. (d) Vy as a function of Bx with the laser positioned on a \nremote absorption pad as shown in (c). M is also plotted for comp arison. The magnitude of the \nnon-local signal, VNL, is proportional to the magn on spin current diffusing fr om the absorber to the \ndetector, which is plotted in the in set as a function of laser power. \n\nFIG. 3. (a) Measurement of V NL as the laser is focused on abso rption pads of increasing distance \nat 23 K (black) compared to FEM in-plane spuri ous temperature gradients at the edge of the 24\ndetection pad (blue) at 23K. (b) Comparison of same measurement/predict ion as (a) but at 280K. \nAs can be seen, even though FEM predicts highe r spurious temperature gradients at 280K V NL \nbecomes negligible, proving that spurious temper ature gradients cannot be the cause of the signal \nmeasured at 23K.  FIG. 4. (a) The magnitude of the non-local signal V\nNL  as a function of laser position from the edge \nof the spin detector ( ∆ݔ )measured with three different laser powers. The signal reveals exponential \ndecay of the magnon spin current produced at the absorber and diffusing to the spin detector. ܶߘ௫ \nis included for comparison (dashed lines are guides to the eye). The VNL data are well fit to a single \ndecaying exponential (solid lines) to obtain the pure magnon spin diffusion length ( ߣௌ∗) plotted in \n(b) as a function of laser power. \n\nFIG. 5. The temporal response of the temperature along the optical axis after having been heated \nby the laser pulse train modulated at 2 kHz, with a fluence of 3.2ൈ10ଵௐ\nయ. Dark blue circles \nindicate the response at the Pt/Y IG interface, while light blue re ctangles indicate the response 10 \nµm below the interface. The red dashed line indicates  the power of the laser at the center of the \noptic axis. The blue dashed lines represent calculated steady state values of the temperature rise.  \n\nFIG. 6. (a) A comparison of the one dimensional ܶ௫ profile at the Pt/YIG  interface at 23 K and \n300 K. The laser is focused 60 \t݉ߤaway from the detector. (b) Two dimensional temperature \ngradient contour map of ܶ௫ (cross-section view) with  the laser absorbed 60 \t݉ߤaway from the \ndetector at 300 K. The average laser pow er used for the simulation was 10.5 mW.  \n 25\nFIG. 7. A comparison of ߘܶ௭݈݀ and ܸ௬ as a function of laser power shows the slopes are nearly \nequal, implying that ܸ௬∝ܶ௭݈݀ .ܶ௭ is integrated in the y-dire ction at the Pt/YIG interface, \nacross the center of the heating laser. \n \nFIG. 8. The linear fit (red dotted line) between ܸ௬ (black dots) and ܶ௭݈݀ .An LSSE coefficient \nof 102\tܭ/ܸ݊  is extracted from the slope. The blue spot corresponds to a ܸ௬ of 25 nV. From the \nlinear relationshi p, this requires a corresponding ܶ௭݈݀ of 0.34K. This implies ܶ௭ is negligible \nbeneath the detection pad. \n \nFIG. 9. An analysis of V NL for multiple different sets of absorption pads located on either side of \nthe spin detector.  The la ser powers are 10.5 mW in (a), (b),  2 1.3 mW in (c), (d), and 29.9 mW in \n(e), (f). Green and red dotted lines correspond to the best fit single decaying exponential when the laser is to the left (x < 0) or right (x > 0) of the spin detector, respectivel y. Extracted spin diffusion \nlengths (in μm) are shown in the corresponding figure pane ls.  All measurements were performed \nat 23 K. As expected, the extracted spin diffusi on length does not depend on the set of absorption \npads chosen. \n \nFIG. 10. Data from Fig. 9 plotted on logarithmic scal e in order to emphasize the validity of a single \nexponential fit. The fits lie within error bars for all data points.   \n \nFIG. 11. (a) V L is measured with the laser focused di rectly on the spin detector on sample \n20141004. The signal tracks the magnetization, following the same trends as seen on sample 26\n20140804. (b) V NL is measured on sample 20141004, with the laser on an absorption pad 47 μm \naway. As seen in sample 20140804, V NL is an order of magnitude smaller than V y, and is attributed \nto diffusion of the magnon mediated spin cu rrent. Comparison of to Fig. 2 (sample 20140804) \nconfirms that the non-local measur ement is not sample dependent. \n FIG. 12. Laser power is 10.5 mW. Green and red dotted lines correspond to the best fit single \ndecaying exponential when the laser is to the left or  right side of the spin detector, respectively. \nExtracted spin diffusion lengths (in μm) are shown in the corresponding figure panels.  All \nmeasurements were performed at 24 K.   \nFIG. 13. (a) A 2D map of the steady state ݉ߜ\n distribution, depicting the experimental setup \nwhere the laser is focused on the Pt absorber at x = 47. (b) The interfacial magnon density profile \nalong x at z = 0 (Pt/YIG interface). Three scenarios are depicted, showing the result with no spin \nsinking, when only the detector serves as a spin sink and when unused Pt absorbers act as spin \nsinks. (c) The derivative of the magnon density profile, ݉ߜ, along the z-direction evaluated at z \n= 0, ௭݉ߜ. The value of ݉ߜ under the detector is integrated to determin e the total magnon spin \ncurrent flowing into the detector, ܬ௦௭. (d) The results of the integrat ion described in (c) when the \nlaser is focused on varying absorption pads. By varyng ߬௧ over two orders of magnitude the value \nof λ௦ remains relatively constant. \n \n  27\n \n   \nFIG.1(SingleColumn;Coloronline) z \n10 20 30 40 50 60 70 80 90 100 110-50-40-30-20-100Tz\nxm)z (m)23K\n29.9 mW\n10 20 30 40 50 60 70 80 90 100 110-50-40-30-20-100\n (K/m)Tx\nxm)z (m)\n10-410-310-210-110023K\n29.9 mWa) \nPtabsorbers\nYIG\nDetector  \nܸூௌுா \nܬ௦(diffusion)  laser \nߘܶ௭\nߘܶ௫\nܤ௫\nx \ny \nb) \nc) \n265 x 265 µm Detector, 10 x 10 µm Absorber28\n   \nFIG.2(SingleColumn;Coloronline) -200 -150 -100 -50 0 50 100 150 200-800-600-400-2000200400600800VLVy (nV)\nBx (mT)x = 0 m-240-160-80080160240M (emu/cm3)10 15 20 25 30200400600800VL (nV)\nPower (mW)\nPtabsorbers  \nDetector  \nVy \nYIG \n x \nz \ny \nJS \nBx \nPtabsorbers  \nVy \nYIG \n JS \nBx \nDetector  \n-200 -150 -100 -50 0 50 100 150 200-40-2002040\nVNL  Vy (nV)\nBx (mT)x = 54 m-240-160-80080160240M (emu/cm3)10 15 20 25 30102030VNL (nV)\nPower (mW)a) \nc) \nd) \nb) \n∆x \n265 x 265 µm Detector, 10 x 10 µm Absorber29\n  a) \nb) 0 30 60 90 120 150048121620\nx (m)VNL (nV)23K\n0.000.050.100.150.200.250.30Tx(K/m)\n0 30 60 90 120 150048121620\nx (m)VNL (nV)280K\n0.000.050.100.150.200.250.30Tx(K/m)\nFIG.3(SingleColumn;Coloronline) 30\n \n  \n0 20 40 60 80 100 12001020304050607029.9 mW\n21.3 mW\n10.5 mWTx (K/m)\nx (m)23K\n0.000.010.020.030.040.050.06VNL (nV)\n8 1 21 62 02 42 83 240455055\n s (m)\nPower (mW)b) a) \nFIG.4(SingleColumn;Coloronline) 31\n  \n0.5 0.6 0.7 0.8 0.9 1.0012345678\nSS valueT rise (K)\nt (ms)SS value23K\n29.9mW\n0.00.51.01.52.02.53.03.5 P (x1016W/m3)\nFIG.5(SingleColumn;Coloronline) 32\n \n  \n-100 0 100 200 300-20-15-10-505101520\nDetectorTx (K/m)300K\n23KTx (K/m)\nx (m)-1.5-1.0-0.50.00.51.01.5a) \n10 20 30 40 50 60 70 80 90 100 110-50-40-30-20-100\n10-410-110210-310-2101(K/m)Tx\nxm)z (m)\n100\nFIG.6(SingleColumn;Coloronline) b) 33\n  \n10 20 302345678\nslope=26 nV/mW\nP (mW)slope=0.25 K/mW\n200300400500600700800Tz dl (K)Vy (nV)23 K\nFIG.7(SingleColumn;Coloronline) 34\n \n  \n0123456780100200300400500600700800\nTz dl (K)\n  Vy (nV)\nLSSE coefficient=102 nV/K23K\nFIG.8(SingleColumn;Coloronline) 35\n \n   \n                      \n \n      0 25 50 75 100 1250510152025VNL (nV)\nx (m)s = 59.44 ± 1.50\n0 25 50 75 100 12501020304050s = 59.44 ± 1.50VNL (nV)\nx (m)\n0 25 50 75 100 125010203040506070s = 49.17 ± 1.16VNL (nV)\nx (m)0 25 50 75 100 1250510152025\ns = 44.30 ± 1.27VNL (nV)\nx (m)\n0 25 50 75 100 1251020304050s = 46.57 ± 1.66VNL (nV)\nx (m)\n0 25 50 75 100 12510203040506070s = 46.37 ± 1.47VNL (nV)\nx (m)a) b)\nc) d)\ne) f)\nFIG.9(DoubleColumn;Coloronline) 36\n \n  \n0 25 50 75 100 125110100VNL (nV)\nx (m)10.3 mW\nx < 0\n0 25 50 75 100 125110100\n21.3 mW\nx < 0VNL (nV)\nx (m)\n0 25 50 75 100 125110100VNL (nV)\nx (m)29.9 mW\nx < 00 25 50 75 100 1250.1110100\n10.3 mW\nx > 0VNL (nV)\nx (m)\n0 25 50 75 100 125110100\n21.3 mW\nx > 0VNL (nV)\nx (m)\n0 25 50 75 100 125110100\n29.9 mW\nx > 0VNL (nV)\nx (m)a) b)\nc) d)\ne) f)\nFIG.10(DoubleColumn;Coloronline) 37\n \n  \n-200 -150 -100 -50 0 50 100 150 200-270-180-90090180270Vy (nV)\nHx (mT)24K\n10.4 mW\nx = 0 mVL\n-240-160-80080160240\nM (emu/cm3)\n-200 -150 -100 -50 0 50 100 150 200-15-10-5051015Vy (nV)\nHx (mT)24K\n10.4 mW\nx = 47 mVISHE\n-240-160-80080160240\nM (emu/cm3)a) \nb) \nFIG.11(SingleColumn;Coloronline) 38\n \n  \n0 25 50 75 100 125051015202530VNL (nV)\nx(m)s = 40.72 ± 1.60\n0 25 50 75 100 125051015202530s = 39.21 ± .96VNL (nV)\nx(m)a) b)\nFIG.12(DoubleColumn;Coloronline) 39\n \n  a) \nb) \nc) \nd) 0 1 53 04 56 07 59 0 1 0 5-50-40-30-20-100\nmm\n(B/m3)x(m)z (m)\n2x10-24x10-27x10-21x10-12x10-1\n-10 0 10 20 30 400.00.10.2detector spin sinkno spin sink mm (B/m3)\nx (m) detector + absorber spin sink\n-10 0 10 20 30 40-1.0-0.8-0.6-0.4-0.20.0101520\ninjector absorber absorber absorber detectorJSz (10-5 B/m2s) no spin sink\n detector spin sink\n detector + absorber spin sinkJSz (B/m2s)zmm (103 B/m4)\nx (m)\n25 50 75 100 125051015202530\nth=10-8 s, s=64m \nx (m)th=10-6 s, s=73m\n0246810\nFIG.13(SingleColumn;Coloronline) 40\nTABLE I. Refractive index ( ݊ ,)reflectivity (R), absorption coefficient ሺߙሻ, density ( ߩ ,)thermal \nconductivity ( ߢ )and heat capacity ( ܥ )at 23 K and 300K used in the simulation. \n  n R (to \nair) α (1/m) ρ (kg·m3) κ (W·m-\n1·K-1) C (J·kg-\n1·K-1) \nPt 23K \n300K 1.89 + 4.54ia \n2.58 + 4.59ib 73.2%\n69.5%78.3 x 106 a \n80.1 x 106 b21450c \n21450c 350d \n72c 11.7e \n130c \nYIG 23K \n300K 2.07 + 0if \n2.36+0if 12.1%\n16.4%0 \n0 5245c \n5245c 109.5g \n6c 5.6g \n570c \na Reference [29]. \nb Reference [33]. \nc Reference [35]. \nd Reference [39]. \ne Reference [43]. \nf Reference [32]. \ng Reference [30]. \n \nTABLE I.  (Double Column)  \n  41\nTABLE II. ߙௌௌா, ܶ௭݈݀ ,ܶ௭_௫  and ܸ௬ are shown as a function of laser power for local \nmeasurement. \n \nTABLE II.  (Double Column)  \n  P (mW) ܶ௭_௫  (K/݉ߤ )ܶ ௭݈݀( K) ܸ௬ (nV) \n10.5 0.78 2.60 238.1 \n21.3 1.58 5.27 537.1 \n29.9 2.21 7.39 745.9 42\nTABLE III. Multiple data sets containing V NL values from both zones I and II and from exclusively \nfrom zone II (Sample 20140804). Values indicate no significant differenc e, indicating that ߘTx can \nbe considered negligible.  \n Left Side \n(x < 0)     Right Side \n(x > 0)  \nPower \n(mW)  Absolute value \nof ∆x (μm) Adj R2  Power \n(mW)  Absolute value \nof ∆x (μm) Adj R2 \n10.5 79 – 118 0.9323  10.5 80 – 119 0.99136 \n 66 – 118  0.96958   67 - 119 0.99618 \n 53 – 118  0.98175   54 – 119  0.97702 \n 40 – 118  0.98757   41 – 119  0.98384 \n 27 – 118  0.99469   28 – 119  0.99112 \n 14 – 131  0.99596   15 – 119  0.99451 \n21.3 79 – 118 0.97598  21.3 80 – 119 0.97851 \n 66 – 118  0.98487   67 - 119 0.99017 \n 53 – 118  0.99246   54 – 119  0.99556 \n 40 – 118  0.99696   41 – 119  0.99661 \n 27 – 118  0.99863   28 – 119  0.99801 \n 14 – 131  0.99622   15 – 119  0.99375 \n29.9 79 – 118 0.99882  29.9 80 – 119 0.99001 \n 66 – 118  0.99963   67 - 119 0.98997 \n 53 – 118  0.99981   54 – 119  0.99546 \n 40 – 118  0.9948   41 – 119  0.99657 \n 27 – 118  0.99629   28 – 119  0.99679 \n 14 – 131  0.99654   15 – 119  0.99581 \n* Yellow regions refer to data sets with V NL values from zones I and II. Blue regions refer to data \nsets containing V NL values from zone II only.  \n  TABLE III.  (Double Column; Color online)  \n  43\nTABLE IV. Multiple data sets containing V NL values from both zones I and II and from \nexclusively from zone II (Sample 20141004). Va lues indicate no significant difference, \nindicating that ߘTx can be considered negligible. \n Left Side \n(x < 0)     Right Side \n(x > 0)  \nPower \n(mW)  Absolute value \nof ∆x (μm) Adj R2  Power \n(mW)  Absolute value \nof ∆x (μm) Adj R2 \n10.5 47 – 125 0.99724  10.5 47 – 125  0.99709 \n 34 – 125  0.99749   34 – 125  0.99391 \n 21 – 125  0.99876   21 – 125  0.99695 \n 8 – 125  0.99545   8 – 125  0.99843 \n* Yellow regions refer to data sets with V NL values from zones I and I. Blue regions refer to data \nsets containing V NL values from zone II only. \n \n \nTABLE IV.  (Double Column; Color online)  \n  44\nTABLE V. Decaying single exponen tial fit parameters as a function of laser power and laser \nposition relative to the spin detector.  \n  Left Side \n (x < 0)   Right Side  \n(x > 0)  \nPower \n(mW)  V0 (nV)  λs (μm) Adj R2 V0 (nV)  λs (μm) Adj R2 \n10.5 22.2 ± .41 59.44 ± 1.50 .996 28.5 ± .70 44.30 ± 1.27 .996 \n21.3 57.1 ± 1.20 49.30 ± 1.30 .996 61.2 ± 1.69 46.57 ± 1.66 .994 \n29.9 78.9 ± 1.57 49.17 ± 1.16 .997 86.7 ± 2.34 46.35 ± 1.47 .995 \n*Sample 20140804 \n  Left Side  \n(x < 0)   Right Side  \n(x > 0)  \nPower \n(mW)  V0 (nV)  λs (μm) Adj R2 V0 (nV)  λs (μm) Adj R2 \n10.5 29.4 ± .48 39.21 ± .96 .998 29.1 ± .80 40.72 ± 1.60 .995 \n*Sample 20141004 \n \nTABLE V.  (Double Column)  \n \n"    },    {        "title": "2005.06429v1.Waveguide_cavity_optomagnonics_for_broadband_multimode_microwave_to_optics_conversion.pdf",        "content": "Research Article Vol. X, No. X / April 2016 / Optica 1\nWaveguide cavity optomagnonics for broadband\nmultimode microwave-to-optics conversion\nNA ZHU1, XUFENG ZHANG1, XU HAN1, CHANG-LING ZOU1,\nCHANGCHUN ZHONG2, CHIAO-HSUAN WANG2, LIANG JIANG2,AND\nHONG X. TANG1,*\n1Department of Electrical Engineering, Y ale University, New Haven, Connecticut 06511, USA\n2Department of Applied Physics, Y ale University, New Haven, Connecticut 06511, USA\n*Corresponding author: hong.tang@yale.edu\nCompiled May 14, 2020\nCavity optomagnonics has emerged as a promising platform for studying coherent photon-spin\ninteractions as well as tunable microwave-to-optical conversion. However, current implemen-\ntation of cavity optomagnonics in ferrimagnetic crystals remains orders of magnitude larger in\nvolume than state-of-the-art cavity optomechanical devices, resulting in very limited magneto-\noptical interaction strength. Here, we demonstrate a cavity optomagnonic device based on in-\ntegrated waveguides and its application for microwave-to-optical conversion. By designing a\nferrimagnetic rib waveguide to support multiple magnon modes with maximal mode overlap to\nthe optical field, we realize a high magneto-optical cooperativity which is three orders of magni-\ntude higher compared to previous records obtained on polished YIG spheres. Furthermore, we\nachieve tunable conversion of microwave photons at around 8.45 GHz to 1550 nm light with a\nbroad conversion bandwidth as large as 16.1 MHz. The unique features of the system point to\nnovel applications at the crossroad between quantum optics and magnonics.\n© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement\nhttp://dx.doi.org/10.1364/optica.XX.XXXXXX\n1. INTRODUCTION\nExploitation of hybrid platforms to combine microwave and pho-\ntonic circuits is of great interest for its importance to realize both\nclassical and quantum hybrid signal transduction, storage, and\nprocessing [ 1–5]. Coherent microwave-to-optical conversion has\nbeen realized at various experimental platforms, for example, by\ncoupling with auxiliary excitations such as phonons in optome-\nchanical systems [ 6–13], and direct microwave-light interaction\nvia the electro-optic approach [ 14–16]. While benefiting from the\nsmall mode volume to enhance the interactions, most systems\nintrinsically lack tunability and have relatively narrow operating\nbandwidths ( <1 MHz), limiting their practical applications. In\nrecent years, the coherent, cavity-enhanced interaction between\noptical photons and solid state magnons has attracted intensive\nattentions in both fundamental research and device applications,\nbecause of the appealing properties of magnons such as long\nspin lifetime and large-bandwidth tunability. In particular, sin-gle crystalline ferrimagnetic insulator yttrium iron garnet (YIG,\nY3Fe5O12) has emerged as a promising candidate for integrating\nmagnons in hybrid quantum systems to bridge different types\nof excitations. It has been widely adopted to investigate inter-\nactions among spin waves, microwaves, acoustic waves, and\noptical excitations [ 17–29]. YIG exhibits very low dissipation\nfor all these information carriers and in particular, as an optical\nmaterial, it shows very low optical loss in the telecom c-band\n(0.13 dB/cm) [ 30]. Because of these advantages, the feasibility of\nusing single crystalline YIG for realizing microwave-to-optical\nconversion is of significant interest [ 22,31–36]. However, pre-\nvious studies in this area mainly focused on bulk crystals with\nthe uniform magnon mode (Kittel mode) [ 37], resulting in the\nlow magneto-optical interaction strength and limited conversion\nbandwidth [18–21, 26, 38].\nIn this paper, we experimentally demonstrate the multimode-\nmagnon assisted, broadband, and tunable conversion betweenarXiv:2005.06429v1  [cond-mat.mes-hall]  13 May 2020Research Article Vol. X, No. X / April 2016 / Optica 2\n(c) \nYIG \n5 µm  GGG (a) \n(b) \n𝜔𝑎 𝜔𝑚  𝜔𝑏 𝜔𝑚 TM  TE  Magnon  \n0 1 \n0 1 \nMicrostrip  \nGGG  YIG  Sapphire  (d) \n100 m \n|E| |E| \n0 1 \n|M| \nFig. 1. (a) Schematic of the experimental assembly of the integrated optomagnonic waveguide device. The optical light is coupled\nin and out of the YIG rib waveguide via cleaved fibers. A rfmicrostrip cavity is aligned beneath the waveguide for magnon exci-\ntations. (b) A frequency domain representation of the triple-resonance-enhanced frequency conversion process. A strong optical\npump light is applied to the TE optical mode (pump mode). Photons can be converted between the magnon modes and TM optical\nmode (signal mode). The electric field distributions in the cross-section view for optical TE, TM, and the magnetic field distribu-\ntion of the magnon modes are shown in the upper panel, respectively. (c) False-color scanning electron microscope image of the\nwaveguide cross-section view. (d) The rfmagnetic field distribution across the YIG waveguide at the cross-section view.\nmicrowave and optical light in a single crystalline ferrimagnetic\nYIG thin film waveguide configuration. The waveguide simulta-\nneously supports a series of Fabry-Pérot (FP) optical resonances\nand multiple magnon resonances formed by forward volume\nmagnetostatic standing waves (FVMSWs) [ 39]. With the care-\nfully designed YIG rib-waveguide geometry, the magnon modes\nand optical modes are both confined in a small mode volume\nwith large mode overlap, which results in significantly enhanced\nmagneto-optical interactions compared with previous studies in\nYIG spheres [ 18,19,26]. At the same time, a second optical cav-\nity mode is engineered to resonantly boost the intra-cavity pump\nphoton number. Taking advantage of the triple-resonance inte-\ngration by realizing the energy and phase conservation among\nmagnon and optical photons, we demonstrate a microwave-\nto-optical photon conversion efficiency that is three orders of\nmagnitude higher compared with current state-of-the-art results\non YIG-based platforms [ 18,19,26,40]. Furthermore, with the\nexistence of multiple magnon resonances, a broad frequency con-\nversion bandwidth exceeding 16.1 MHz has been realized. Our\ndemonstration sheds light on the potential of the patterned YIG\nrib waveguide as a new platform for magnon-based coherent\ninformation processing.\n2. FRAMEWORK OF WAVEGUIDE CAVITY OPTO-\nMAGNONICS MEDIATED CONVERSION\nThe architecture of the optomagnonic device is illustrated in Fig.\n1(a). The system consists of multiple coupled modes in three dif-\nferent domains: a microwave cavity mode supported by a half- l\nresonator, magnon modes formed by forward volume magneto-\nstatic standing waves in an etched YIG waveguide, and opticalmodes engineered to have the desired dispersion relation for\nsupporting the triple-resonance enhanced frequency conversion,\nwhich means that the magnon, input and output optical photons\nare simultaneously on resonance. The magnons are coupled\nto the microwave cavity which is driven by the input itinerant\nmicrowave photons through an inductively coupled microwave\nfeedline. Due to the coexistence of the magnonic and the optical\ncavity modes and the strong magneto-optical interaction, input\noptical photons can be inelastically scattered by the magnons\ninto a single sideband mode with orthogonal polarization via\ntheFaraday effect [18,21,26,41], with the frequency difference\nmatching the magnon mode frequency. This can be described\nby an intuitive picture that, under the microwave drive, the\npolarization of the pump light oscillates at the frequency of the\nmagnons, and thus produce the desired optical sideband under\nphase matching conditions.\nThe linear interaction system Hamiltonian of coupled modes\nunder the rotating-wave approximation can be written as\nHint=å\ni¯hgo,i\u0010\nab†+a†b\u0011 \u0010\nmi+m†\ni\u0011\n+å\ni¯hge,i\u0010\ncm†\ni+c†mi\u0011\n(1)\nwhere a\u0000\na†\u0001\n,b\u0000\nb†\u0001\n,c\u0000\nc†\u0001\n,mi\u0000\nm†\ni\u0001\nare the annihilation (cre-\nation) operators for optical pump mode, optical signal mode, mi-\ncrowave mode, and i-th order magnon mode, respectively. ge,i,\nandgo,iare the electro-magnonic and photon-number-enhanced\nmagneto-optical coupling strength for the i-th order magnon\nmode, respectively [22, 34, 42].\nThe triple-resonance enhanced conversion process has been\nproposed recently [ 19,21,26], showing a dramatic enhancement\nof the magneto-optical coupling strength by several orders ofResearch Article Vol. X, No. X / April 2016 / Optica 3\n0.00.30.60.9\n  Transmission (%)\n1546.2 1546.5 1546.8 1547.10.00.30.6\n  Trans. (%)\nl (nm)\n1542.630 1542.6450.00.20.40.60.8\nWavelength (nm)\n  Trans. (a.u.)\n1557.83 1557.84 1557.850.30.71.0\nWavelength (nm)\n  Trans. (a.u.)𝑄TE ~ 220  k \n𝑄TM ~ 194  k \n1540 1545 1550 15550.00.30.60.9\n  Transmission (%)\nWavelength (nm)\n1546.5 1547.08.38.48.58.68.7Df (GHz)\nWavelength  (nm)\nDf (c) (a) \nTM  TE  \n(e) (d) (b) \n(f) \nMagnons  \nFig. 2. (a) & (c) are the optical transmission spectra of both TE (red) and TM (green) polarizations. (b) and (d) are the zoom-in single\nresonance spectra for the two polarizations with the fitted Qfactor, respectively. (e) is the zoom-in spectra representing the fre-\nquency difference between adjacent resonances. (f) Dispersion engineering of the frequency difference between two neighboring\nmodes, plotted against the wavelength. The grey area depicts the target frequency band where magnons are excited.\nmagnitude under the realization of energy, spin, and orbit an-\ngular momentum conservation relations. In our system, the\ndevice is engineered to satisfy the triple-resonance condition,\nas depicted in the Fig. 1(b). The dispersion relations of optical\ntransverse-electric (TE) and transverse-magnetic (TM) modes\nare carefully engineered to make their effective refractive indices\ndiffer slightly from each other, thus, leading to the smooth tun-\ning of the frequency difference between two adjacent modes\nover a large optical scanning range. At the same time, the fre-\nquencies of magnon modes can be tuned linearly by varying the\nstatic bias magnetic field ~Bo, according to the dispersion relation\nwmµg\f\f\f~Bo\f\f\f, with g=2p\u00022.8MHz/Oe being the gyromag-\nnetic ratio. Thus, with the magnon tunability and optical mode\ndispersion engineering, the frequency difference between adja-\ncent TE and TM optical modes can be conveniently chosen to\nmatch the magnon resonant frequencies, as well as the resonant\nfrequency of the microwave cavity. The implementation of the\nmicrowave resonator, instead of using broadband microstrip\nantenna, also boosts up the magnon read-out efficiency via the\ncavity enhancement. Besides the engineered dispersions, the\nmagneto-optical coupling strength is greatly enhanced by the\nlarge overlap between the optical and the magnonic modes, as\nshown in the mode profiles in the Fig. 1(b), thanks to the high\nmode confinement from the rib waveguide geometry.\nDuring the experiment, a strong coherent drive tone is ap-\nplied to the TE-polarized optical pump mode (a), leading to\na pump enhanced magneto-optical coupling strength go,i=pnaGo,i, where nais the intracavity photon number and Go,i\nis the single photon magneto-optical coupling strength. The\nmagnons are excited by the microwave photons, due to themagnon and microwave coupling in the hybrid system. At the\nsame time, thanks to the magneto-optical coupling, input optical\nphotons can be inelastically scattered into a single sideband op-\ntical signal mode (b)by the magnons. The on-chip conversion\nefficiency for the i-th order magnon mode hi, under the triple\nresonance enhanced condition, is written as (See Supplementary\nMaterial Section 4)\nhi=4Com,iCem,izezo\f\f1+Com,i+Cem,i\f\f2, (2)\nwhere Cem,i\u00114g2\ne,i\nkekm,iandCom,i\u00114g2\no,i\nkokm,iare the electro-magnonic\nand magneto-optical cooperativities, respectively, ze=ke,e/ke\nand zo=ko,e/koare the extraction ratios. And ke,e,ke,ko,e\nandkoare the external coupling and total dissipation rates for\nmicrowave and optical signal modes, respectively. km,iis the i-th\nmagnon dissipation rate. It is worth noting that the total on-chip\nconversion efficiency hhas multiple contributions from adjacent\nmagnon modes ( h\u0015hi), and h\u0019hiis only valid when keand\nge,iare much smaller than the FSR of the magnon modes.\n3. DEVICE DESIGN AND TRIPLE-RESONANCE INTE-\nGRATION\nA rib waveguide is fabricated in a 5- mm-thick single crystalline\nYIG thin film (<111> oriented) on the 500- mm-thick Gadolium\nGallium Garnet (GGG) substrate that simultaneously supports\noptical and magnon resonances. The top layer of the rib waveg-\nuide has a width of 5 µm and a height of 1.2 µm, as shown in the\nFig. 1(c), which is superimposed on a 200- µm-wide bottom ribResearch Article Vol. X, No. X / April 2016 / Optica 4\n(b) \nFrequency (GHz)  \n|Soe|2 (a.u.) \nFrequency (GHz)  -3 dBm  0 dBm  2.5 dBm  16.1 MHz  (a) (d) \nTLD  DUT  PD \nVNA  \nFPC \nTE pump  \n1 2 3 40.02.65.2\n   \nOptical pump power (mW)  int \n(c) \n-30-20-100|See|2 (dB)\n8.35 8.40 8.45 8.50 8.55-40-30-20|Soe|2 (dB)Filter  \nFig. 3. (a) Illustration of the experimental setup. The polarization of the light from a tunable laser diode (TLD) is adjusted by a fiber\npolarization controller (FPC) and passes through a polarizer aligned at TE polarization. The microwave input is sent into the device-\nunder-test (DUT) via a vector network analyzer (VNA). The beat signal between the optical pump and the converted sideband field\nis measured using a polarizer and a fast photodetector (PD), and fed into VNA after the signal amplification. (b) The measured\n(scatter) and fitted (line) microwave reflection spectrum jSeej2, and the corresponding microwave-to-optical conversion spectrum\njSoej2. (c) The optical power dependence of the microwave-to-optical conversion magnitude jSoej2. The conversion bandwidth\nat full width of half maximum (FWHM) is around 16.1 MHz with 2.5 dBm pump power. (d) Extracted (scatter) and fitted (line)\ninternal conversion efficiency as a function of the optical pump power. The orange, green, purple, and navy squares correspond to\nthe colored curves in (b) & (c) respectively.\nlayer along its center line. The bottom layer of the YIG rib waveg-\nuide, serving as a magnonic waveguide, is etched by 5 µm into\nthe GGG layer for the spin wave mode confinement. The waveg-\nuide is 4.2-mm long, confining spin waves in both longitudinal\nand transverse directions. The magnetic field distribution of the\nmagnon modes at the cross-section view is plotted in the Fig.\n1(b). The surface of the device is covered by a 6- µm-thick silicon\ndioxide layer for mechanical protection and minimizing the op-\ntical scattering loss. The optical light is sent into the waveguide\nvia a cleaved optical fiber. A copper coplanar half- lmicrowave\nresonator is placed beneath the YIG waveguide, which can be\nused to excite magnons efficiently. With high-precision fabri-\ncation and metallic reflection coating on the highly polished\nwaveguide facets, the YIG waveguide Fabry-Pérot cavity, which\nis inherently an excellent magnonic cavity, exhibits high opti-\ncal quality ( Q) factors for both TE and TM polarizations with\nengineered dispersion relations to support the triple resonance\ncondition.\nThe optical waveguide Fabry-Pérot cavity supports both TE\nand TM fundamental optical modes with the slight difference in\ntheir effective refractive indices, resulting in two sets of modes\nwith very close free spectral range (FSR). Here, the silicon dioxide\ncladding layer, YIG layer, and GGG substrate have the refractive\nindices of 1.44, 2.20, and 1.94, respectively. The field distributions\nof the fundamental optical modes (TE & TM) are confined in the\nwaveguide center, where the 5- mm-wide etched step locates. As\nwe can see from the optical mode profiles in Fig. 1(b), the field\ndistributions are very similar for both TE and TM, with an aspect\nratio around 1 and a mode size \u00185\u00025mm2. At the same time,\nboth the height and width of the rib waveguide are larger thanthe optical wavelength ( \u00181.55mm), resulting in relatively small\ngeometry introduced dispersion. The effective refractive indices\nof TE and TM modes are simulated via COMSOL Multiphysics\nwith the values to be 2.1937 and 2.1934, respectively, close to the\nbulk YIG refractive index. With a small dispersion difference,\nthe frequency difference between adjacent TE and TM optical\nmodes slowly varies within the measured wavelength range,\nwhich can conveniently match input microwave frequencies\nwithin a single device.\nFigures 2(a) & (c) show the optical transmission spectra for\nboth TE and TM input lights. The optical Qfactors for both polar-\nizations are fitted in Figs. 2(b) & (d), with a value achieving near\n200,000 for both polarizations. As shown in Fig. 2(e), the TE and\nTM modes have very similar dispersion relations, and thus, the\nfrequency difference between two adjacent modes can smoothly\nvary as a function of wavelength. The relation between Dfand\nthe wavelength is extracted in Fig. 2(f). The grey area denoted\nthe frequency range where the magnon modes locate. The fre-\nquency difference between two adjacent modes smoothly varies\nfrom 8.7 GHz to 8.4 GHz in a measurement range from 1546\nto 1547.3 nm, triggering triple-resonance enhanced frequency\nconversion when Dfmatches the magnon frequency.\n4. MICROWAVE TO OPTICAL CONVERSION\nThe experimental setup for measuring the microwave-to-optical\nconversion is depicted in Fig. 3(a). The optical pump is sent to\nthe device from a tunable laser, with the polarization controlled\nby a fiber polarization controller and a polarizer. The device is\nbiased perpendicularly by a static magnetic field with the fieldResearch Article Vol. X, No. X / April 2016 / Optica 5\naround 4800 Oe. The microwave input is sent to the microstrip\nfeedline via a VNA to excite the magnons. The beat signal\nbetween the optical pump field and the converted sideband field\nis measured using a polarizer and a fast photodiode, then fed\nback into the VNA.\nWe first characterize the magnon modes by measuring the\nmicrowave reflection spectrum. The magnon modes are con-\nfined in the slab layer of the YIG rib waveguide centered at the\nrib structure. Since the film is perpendicularly magnetized, the\nFVMSWs are excited which form standing waves between the\ntwo waveguide facets along the length direction [ 43–45], when-\never the wavevectors kequal ip/l, where l=4.2mm is the\nlength of the waveguide and iis the mode number ( i= 1, 3, 5,\n7,. . . ). The magnonic waveguide is aligned at the center of the\nmicrowave coplanar cavity, and driven by a nearly uniform rf\nfield, as plotted in Fig. 1(d). Here, only these modes with odd\nmode number can be excited efficiently, because the even modes\nhave cancelled coupling strength with the uniform microwave\ncavity field.\nThe microwave reflection spectrum is shown in Fig. 3(b),\nillustrating the coupling between the microwave mode and mul-\ntiple magnon modes. The microwave mode has the resonance\naround 8.444 GHz with the intrinsic and external dissipation\nrates ke,i/2p= 85 MHz and ke,e/2p= 165 MHz, respectively\n(See Supplementary Material Section 3). The multiple magnon\nmodes are under-coupled with the microwave cavity with an\nFSR around 7 MHz and resonant linewidth km,i/2paround 3.6\nMHz for each magnon resonance. The electro-magnonic cou-\npling strength between the fundamental magnon mode and\nmicrowave mode ge,1/2phas been extracted to be 13.3 MHz,\ncorresponding to an electro-magnonic cooperativity Cem,1 =\n4g2\ne,1/kekm,1to be 0.80. Up to four magnon modes are clearly ob-\nserved in the reflection spectrum. When the optical pump mode\nis at 1547.021 nm, according to the transmission spectra, the\nfrequency difference between optical pump and signal modes is\naround 8.445 GHz, fulfilling the triple resonance condition for\nthe lowest order magnon mode.\nThe microwave-to-optical conversion spectrum jSoej2is mea-\nsured by injecting a microwave field ( \u00003dBm) and monitoring\nthe optical beating signal, which is simultaneously taken with\nthe reflection spectrum of jSeej2in the Fig. 3(b) upper panel. The\nresults show the broad conversion band peaked near the lowest\norder magnon mode frequency with the detectable frequency\nspan broader than 50 MHz, thanks to the multi-mode assisted\nfrequency conversion process, which is two to three orders of\nmagnitude broader than other conversion devices [7].\nThe system conversion efficiency is calibrated (Supplemen-\ntary Material Section 4), and the highest on-chip efficiency is\nachieved at the third order ( i=3) magnon mode, estimated\nto be 1.08\u000210\u00008at the resonant frequency of 8.451 GHz when\nthe optical pump power is 4.8 dBm. The internal conversion\nefficiency at the this magnon resonant frequency hint=h/zezo\nis calibrated to be 5.19\u000210\u00007. From the measurement results,\nthe single photon magneto-optical coupling strength is fitted to\nbeGo,3/2p= 17.2 Hz (Supplementary Material Section 4). Such\ncoupling strength is four orders of magnitude higher than the\nfrequency conversion realized at YIG bulk crystal [ 18], and 50\ntimes higher than the result demonstrated at the YIG sphere\nutilizing WGM-resonance-enhancement [ 19,26,40]. With 4.8\ndBm pump light, yielding 1.77 \u0002106intra-cavity photon number,\nthe photon-number-enhanced coupling strength go,3/2p= 22.59\nkHz, corresponding to an enhanced magneto-optical coopera-\ntivity Com,3 = 4g2\no,3/kokm,1=4.06\u000210\u00007. The parameters for\n0204060\n  Converted optical power (pW)\nDf (GHz)(a) \n(c) \n0.25 0.50 0.7501 \nl/2 waveplate angle ( p)Normalized pump light (a.u.)01\nNormalized converted light (a.u.)\nTE  TM  TLD  DUT  \nSG \nFPC \nPolarizer  \nTE pump  \nOSA  \nl\n2 WP \n(b) \n(TM)  \n(TE)  \n8.45 8.30 8.50 𝜋/4 \n𝜋/2 Fig. 4. (a) The experimental setup to measure the polarization\nrelation of the optical pump and converted lights. The sig-\nnal generator (SG) is used for the microwave input. The l/2\nwaveplate (WP) is used to adjust the polarizations of the out-\nput lights, then sent through a polarizer to a optical spectrum\nanalyzer (OSA). (b) The orthogonal optical polarizations be-\ntween the pump light and the converted light. The solid lines\nare the sinusoidal fitting. (c) Power of the converted sideband\nwhen the waveplate is rotated at different angles.\nthe first four magnon modes are fitted and listed in the Table 1.\nThejSoej2spectrum is also fitted by considering the frequency\nconversion assisted by the first four magnon modes ( i=1, 3, 5,\n7), as shown in the solid line of the Fig. 3(b) bottom panel. The\nfitting results matches very well with the measurement up to the\nresonant frequency of 7-th magnon mode ( i=7,\u00188.47 GHz).\nHere, the discrepancy between the fitting and measurement at\nhigh frequency tail end is due to the conversion process assisted\nby the higher order magnon modes along both longitudinal and\ntransverse directions, which are not resolvable in the reflection\njSeej2spectrum, and thus, not included in the fitting.\nThejSoej2spectra yield the broadening effect instead of dis-\ncrete magnon features as shown in the reflection spectrum. This\nis because the microwave-to-optical conversion is non-zero for\nthe other adjacent magnon modes with the detuned frequencies.\nWhen the system satisfies the triple-resonance condition at one\nmagnon mode, for example, the lowest-order magnon mode ( i=\n1), the conversion process is dominated by this magnon mode\ndue to the cavity enhancement, meanwhile, the adjacent magnon\nmodes ( i= 3, 5) also contribute to the conversion. This is because\nthe FSR of the magnon modes are relatively small ( \u00187 MHz)\nand comparable to the magnon linewidth ( \u00183.6 MHz). At the\nsame time, the optical linewidth ( \u00181.6 GHz) is much larger than\nthe frequency band where magnons exist. Considering those\nfactors, the adjacent modes, although are slightly detuned from\nthe triple resonance condition, still have non-negligible contribu-\ntion to the conversion process. Thus, the conversion at a specific\nfrequency is the collective addition assisted by multiple magnon\nmodes, enabling the broadband conversion. The detailed numer-\nical fitting for each magnon mode and the collective conversion\nprocess is further illustrated in the Supplementary Material.Research Article Vol. X, No. X / April 2016 / Optica 6\nTable 1. List of parameters\ni-th magnon modewm,i\n2p(GHz)km,i\n2p(MHz)ge,i\n2p(MHz) Cem,igo,i\n2p(kHz) Com,i(\u000210\u00007)Go,i\n2p(Hz) hi(\u000210\u00007)\ni=1 8.445 3.55 13.3 0.80 22.59 3.72 17.2 3.68\ni=3 8.451 3.25 13.2 0.87 22.59 4.06 17.2 4.04\ni=5 8.460 3.50 8.5 0.33 14.68 1.59 11.18 1.17\ni=7 8.468 3.22 5.0 0.12 12.42 1.21 9.46 0.45\nFigure 3(c) plots the microwave-to-optical conversion spectra\njSoej2when the optical pump power is increased. The conver-\nsion efficiency increases linearly with the pump power. The 3-dB\nconversion bandwidth is measured to be around 16.1 MHz. In\nFig. 3(d), the internal conversion efficiency at different optical\npump power is extracted when the microwave input frequency\nis fixed at the lowest magnon mode with a \u00003 dBm input power,\nwhich clearly shows a linear power dependence and hence a\nlinear magnon-to-photon conversion process.\nAnother distinct property of magnon-mediated microwave-\nto-optical conversion is the orthogonal polarizations between\nthe pump light and converted signal light, as required by spin\nmomentum conservation [ 18,26]. This feature is experimentally\nconfirmed by tuning the polarization of both the transmitted\noptical pump and the converted signal light by using a l/2\nwaveplate, as illustrated in Fig. 4(a). The input optical pump is\naligned at TE polarization by using the fiber polarization con-\ntroller and a polarizer before the device. The l/2waveplate\nrotates the polarization axis of both transmitted pump and the\nconverted light together, before passing through the second po-\nlarizer for detection. The polarizer at the output side has the\npolarization axis aligned at TE as well. The amplitude change\nwill behave out-of-phase, if the lights have orthogonal polar-\nizations, which is measured by an OSA. For a l/2waveplate,\nif the waveplate is rotated by y, it is equivalent to rotate the\npolarization axis of a linear polarized light passing through it\nby2y[46]. As depicted in Fig. 4(b), the transmitted power of\nthe pump light and signal light has p/4phase difference by\nrotating the angle of the waveplate, corresponding to a p/2\nangle between the polarization axis, clearly demonstrating the\northogonal polarizations between those two lights.\nWe further demonstrate such polarization dependence by\nalternating the rotating angle yof the l/2waveplate before\nthe spectrometer. The converted optical sideband is measured\ndirectly via a tunable Fabry-Pérot spectrometer. The linewidths\nof the measured sideband signals measured in the Fig. 4(c)\nare not the physical linewidths of the light; but instead they\nonly represent the finite resolution (67 MHz) of the filter in the\nspectrometer. In this measurement, the polarizer at the output\nend is aligned at TE, thus, the TE-polarized light can transmit\nwith minimal loss ( \u00181 dB), but\u001832 dB rejection ratio for the\nTM-polarized light. When the waveplate is rotated by p/4, the\npolarization axes of both pump and converted lights are rotated\nbyp/2(TE –> TM, TM –> TE). The magnitude of the converted\nlight is maximized at this rotating angle, while minimized when\ny=p/2(maintaining the original polarizations), as shown\nin the Fig. 4(c). The results verify that the converted optical\nsideband is TM-polarized, orthogonal to that of the pump light.5. DISCUSSION AND OUTLOOK\nThe system conversion efficiency can be further improved by a\nvariety of efforts in both magnonic and photonic engineering.\nFirst, the optical metallic reflective coatings can be further im-\nproved by utilizing the distributed Bragg reflector (DBR) coating\non the facets to achieve higher reflectivity, leading to higher in-\ntrinsic optical Qfactors. Second, by further decreasing the device\nvolume, such as fabricating micro-disk or ring resonators [ 47]\nusing sputtered crystallized films [ 47], the magneto-optical cou-\npling strength will be dramatically improved because of further\nreduction of the mode volume. Third, other magnetic materials\nwith larger Faraday constant such as ion doped YIG will offer\nstronger magnon-photon interaction [ 38], leading to enhanced\ncoupling strength. Lastly, measurement at cryogenic tempera-\nture will be beneficial to the improvement of the performance of\nmicrowave cavity made of superconducting materials [13, 16].\nIn conclusion, we have developed a waveguide cavity opto-\nmagnonic system that achieves multi-mode assisted broadband\nmicrowave-to-optical frequency conversion. By carefully engi-\nneering the optical dispersions and optimizing the mode over-\nlaps between optical and magnon modes, the vacuum magneto-\noptical coupling strength has been increased by three order of\nmagnitude compared with previous studies. In particular, a\nbroad conversion bandwidth up to 16.1 MHz centered at 8.45\nGHz has been achieved, thanks to the collective conversion\nprocess assisted by multiple magnon modes. Lastly, this opto-\nmagnonic device demonstrates high tunability for both optical\nand magnon modes to accommodate different microwave fre-\nquencies within one integrated device.\nFUNDING\nWe acknowledge funding from National Science Foundation\n(EFMA-1741666). H.X.T acknolwedge support from a DARPA\nMTO/MESO grant ( N66001-11-1-4114), an ARO grant (W911NF-\n18-1-0020) and Packard Foundation.\nACKNOWLEDGMENTS\nThe authors acknowledge fruitful discussions with the team\nmembers of the EFRI Newlaw program led by Ohio State Uni-\nversity: Andrew Franson, Seth Kurfman, Denis R. Candido,\nKatherine E. Nygren, Yueguang Shi, Kwangyul Hu, Kristen S.\nBuchanan, Michael E. Flatté, and Ezekiel Johnston-Halperin.\nThe authors thank M. Power and M. Rooks for the assistance in\ndevice fabrications.\nSUPPLEMENTAL DOCUMENTS\nSee Supplement 1 for supporting content.Research Article Vol. X, No. X / April 2016 / Optica 7\nREFERENCES\n1. H. J. 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Our results represent an e\u000ecient way of analyzing the Rashba-Edelstein\ne\u000bects.\nTo ArXiv\n1. Introduction\n2D spintronics has gained an important meaning in data storage technologies, in\nmany those cases, the 2D materials are non-magnetic, however, magnetism can be\ninduced at the interface of those materials. Among some methods, two of them are\nbroadly applied to induce magnetism on 2D material. The \frst method is to introduce\nvacancies or adding atoms producing spin polarization [1, 2, 3]. The other one is\nto induce magnetism of the adjacent magnetic materials via the magnetic proximity\ne\u000bect [4, 5, 6, 7]. Recently was discovered that 2D magnetic van der Waals crystals\nhave intrinsic magnetic ground states at the atomic scale, providing new opportunities\nin the \feld of 2D spintronics [8, 9]. Furthermore, it was discovered that several\nmaterials as single-layer graphene (SLG) and molybdenum disul\fde (MoS 2) [10, 11, 12]\ncan also be used for spin-charge current conversion [13, 14, 15, 16, 17, 18]. Due to\ntheir layered structures, MoS 2and SLG can be easily prepared with one or several\natomic layers to explore the transport properties. SLG and semiconducting MoS 2\nhave 2D electronic states that are expected to exhibit remarkable pseudospin and spin-\nmomentum locking, respectively [10, 19, 20, 21, 22]. These are essential ingredientsarXiv:2210.03854v1  [cond-mat.mtrl-sci]  8 Oct 2022Wibson W. G. Silva and Jos\u0013 e Holanda................................................................... 2\nfor the charge-to-spin current conversion by the direct Rashba-Edelstein e\u000bect (REE)\nor for spin-to-charge current conversion by the inverse Rashba Edelstein e\u000bect (IREE).\nAnother fundamental ingredient is the broken inversion symmetry at material surfaces\nand interfaces [1, 2, 3, 23, 24, 25, 26, 27].\nAlthough the change of electrical resistance of ferromagnets has been studied for\na long time, providing a fundamental understanding of spin-dependent transport in\ndi\u000berent structures [24, 25, 26], the transport properties of 2D materials still present\nthemselves as a challenge. One of the most important e\u000bects in spin-dependent transport\nis the spin Hall magnetoresistance (SMR) [27, 28, 29]. In 3D materials, the SMR is\nexplained by the spin-current re\rection and reciprocal spin-charge conversion caused\nby the simultaneous action of the spin Hall e\u000bect (SHE) [30, 31, 32] and inverse spin\nHall e\u000bect (ISHE) [33]. The challenge is to explore the magnetoresistance induced in\n2D materials [34, 35]. In this paper, we present a study based on direct and inverse\nRashba-Edelstein e\u000bects that describes the magnetoresistance in 2D materials, which is\ncalled of Rashba-Edelstein magnetoresistance (REMR).\n2. 2D materiais in contact with a magnetic insulator\nThe REMR is induced by the simultaneous action of direct and inverse Rashba-Edelstein\ne\u000bects and therefore a nonequilibrium proximity phenomenon. The magnetoresistance\nstudy was carried out with arrangement as illustrated in Fig. 1 below. The e\u000bects\nx\ny\n0\nDl2\nFM2DM\nz\nFigure 1. (online color) Illustration of the sample structure used to study the Rashba-\nEdelstein magnetoresistance (REMR).\nof the spin current in 2D materials are very important for phenomena of transport.\nConsidering the Ohm's law for 2D materials with direct and inverse Rashba-Edelstein\ne\u000bects and therefore a nonequilibrium proximity phenomenon can be understood by\nthe relation between thermodynamic driving force and currents that re\rects Onsager'sWibson W. G. Silva and Jos\u0013 e Holanda................................................................... 3\nreciprocity by the symmetry of the response matrix:\n0\nBBBBB@~JC\n~JSx\n~JSy\n~JSz1\nCCCCCA=1\nR2D0\nBBBBB@1 ^x\u0002 ^y\u0002 ^z\u0002\n1\n\u0015REE^x\u00021\n\u0015REE0 0\n1\n\u0015REE^y\u0002 01\n\u0015REE0\n1\n\u0015REE^z\u0002 0 01\n\u0015REE1\nCCCCCA0\nBBBB@\u0000r\u0016C=e\n\u0000r\u0016Sx=2e\n\u0000r\u0016Sy=2e\n\u0000r\u0016Sz=2e1\nCCCCA;(1)\nwhere e=jejis the electron charge, R 2Dis the resistance of 2D material, \u0016Cis the\ncharge chemical potential, ~ \u0016Sis the spin accumulation, ~JCis the charge current density\nand~JSis the spin current density. The direct Rashba-Edelstein is represented by the\nlower diagonal elements that generate the spin currents in the presence of an applied\ncurrent density, which generates an electric \feld, in the following chosen to be in the\n^xdirection~E=Ex^x=\u0000^x(@x\u0016C=e). On the other hand, the inverse Rashba-Edelstein\ne\u000bect is governed by element above the diagonal that connect the gradients of the spin\naccumulations to the charge current density. The spin accumulation ~ \u0016Sis obtained from\nthe spin-di\u000busion equation in the 2D materials\nr2~ \u0016S=~ \u0016S\n\u00152\nSD; (2)\nwhere\u0015SDis the spin-di\u000busion length. Spin accumulation is always due to spin di\u000busion,\nwhich even for a 2D material such as graphene has spin di\u000busion in the z-direction. For\n2D materials with thickness l2Din the ^xdirection the solution of equation (2) is\n~ \u0016S(z) =~ pe\u0000z=\u00152D+~ qez=\u00152D; (3)\nwhere the constant column vectors ~ pand~ qare determined by the boundary conditions\nat the interfaces. According to Eq. (2), the spin current in 2D materials consists of spin\ndi\u000busion process. For a system homogeneous in the x-yplane, the spin current density\n\rowing in the ^ zdirection is\n~Jz\nS(z) =\u0000\u00121\n2eR2D\u0015REE\u0013\n@z~ \u0016Sz\u0000JREE\nSO^y; (4)\nwhereJREE\nSO =Ex=R2D\u0015REEis the bare Rashba-Edelstein current, i. e., the spin current\ngenerated directly by the REE and \u0015REEis the REE length. At the interfaces z=l2D\nandz= 0 the boundary conditions demand that ~Jz\nS(z) is continuous. The spin current at\nz=l2Dinterface vanishes, ~Jz\nS(z=l2D) =~J2D\nS= 0. On the other hand, in general at the\nmagnetic interface the spin current density ~JFM\nSis governed by the spin accumulation\nand spin-mixing conductance [36], such that:\n~JFM\nS( ^m) =gr^m\u0002 \n^m\u0002~ \u0016S\ne!\n+gi \n^m\u0002~ \u0016S\ne!\n; (5)\nwhere ^m= (mx;my;mz)Trepresents a unit vector along the magnetization and\ng\"#=gr+igithe complex spin-mixing interface conductance per unit length and\nresistance. It is agreed that grcharacterizes the e\u000eciency of the interfacial spin transport\nand the imaginary part gican be interpreted as an e\u000bective exchange \feld acting on theWibson W. G. Silva and Jos\u0013 e Holanda................................................................... 4\nspin accumulation. According with Eq. (5) a positive current corresponds to up spins\nmoving from FM towards 2D. In particular for FM insulator, this spin current density\nis proportional to the spin transfer torque acting on the ferromagnet\n~ \u001cSTT=\u0000\u0016h\n2e^m\u0002\u0010\n^m\u0002~JFM\nS\u0011\n=\u0016h\n2e~JFM\nS( ^m): (6)\nUsing the boundary conditions discussed before, it is possible to determine the\ncoe\u000ecients ~ pand~ q, which leads to the spin accumulation for structures 2DM/FM\n~ \u0016S(z) =\u0000\u0016SO2\n4sinh\u0010\n2z\u0000l2D\n2\u00152D\u0011\nsinh\u0010\nl2D\n2\u00152D\u00113\n5^y+\n2e\u00152D\u0015REER2D2\n4cosh\u0010\nz\u0000l2D\n2\u00152D\u0011\nsinh\u0010\nl2D\n2\u00152D\u00113\n5~JFM\nS( ^m) (7)\nwhere\u0016SO\u0011j~ \u0016S(0)j= 2e\u00152D\u0015REER2DJREE\nSOtanh(l2D=2\u00152D) is the spin accumulation at\nthe interface in the absence of spin transfer, i. e., when g\"#= 0. Furthermore, according\nwith Eq. (5), the spin accumulation at z= 0 becomes\n~ \u0016S(0) =\u0016SO^y+ 2\u00152D\u0015REER2Dcoth l2D\n\u00152D!\n\u0002\n[grf^m( ^m\u0001~ \u0016S(0))\u0000~ \u0016S(0)g+gi^m\u0002~ \u0016S(0)] (8)\nwhere ^m\u0001~ \u0016S(0) and ^m\u0002~ \u0016S(0) here are\n^m\u0001~ \u0016S(0) =my\u0016SO; (9)\n^m\u0002~ \u0016S(0) =\u0016SO2\n4\u0010\n^m\u0002^y\nR2D\u0015REE\u0011\n+\u0010\n2my\u00152Dgicoth\u0010\nl2D\n\u00152D\u0011\u0011\n^m\n1\nR2D\u0015REE+ 2\u00152Dgrcoth\u0010\nl2D\n\u00152D\u00113\n5\n\u00002\n42\u00152Dgicoth\u0010\nl2D\n\u00152D\u0011\n1\nR2D\u0015REE+ 2\u00152Dgrcoth\u0010\nl2D\n\u00152D\u00113\n5~ \u0016S(0); (10)\nand\n~ \u0016S(0) =\u0016SO\"(A(1 +A) +B2) ^m+B( ^m\u0002^y) + (1 +A)^y\nA2+B2#\n; (11)\nwhere\nA= 2\u00152D\u0015REER2Dgrcoth l2D\n\u00152D!\n(12)\nand\nB= 2\u00152D\u0015REER2Dgicoth l2D\n\u00152D!\n: (13)\nThe spin current through the FM/2DM interfaces reads\n~JFM\nS=\u0012\u0016SO\neR2D\u0015REE\u00132\n4Im8\n<\n:g\"#\n1\nR2D\u0015REE+ 2\u00152Dg\"#coth\u0010\nl2D\n\u00152D\u00119\n=\n;3\n5( ^m\u0002^y)Wibson W. G. Silva and Jos\u0013 e Holanda................................................................... 5\n+\u0012\u0016SO\neR2D\u0015REE\u00132\n4Re8\n<\n:g\"#\n1\nR2D\u0015REE+ 2\u00152Dg\"#coth\u0010\nl2D\n\u00152D\u00119\n=\n;3\n5^m\u0002( ^m\u0002^y):(14)\nIn this way, the spin accumulation is,\n~ \u0016S(z)\n\u0016SO=Im2\n48\n<\n:2\u00152Dg\"#\n1\nR2D\u0015REE+ 2\u00152Dg\"#coth\u0010\nl2D\n\u00152D\u00119\n=\n;8\n<\n:cosh\u0010\nz\u0000l2D\n\u00152D\u0011\nsinh\u0010\nl2D\n\u00152D\u00119\n=\n;3\n5( ^m\u0002\n^y) +Re2\n48\n<\n:2\u00152Dg\"#\n1\nR2D\u0015REE+ 2\u00152Dg\"#coth\u0010\nl2D\n\u00152D\u00119\n=\n;8\n<\n:cosh\u0010\nz\u0000l2D\n\u00152D\u0011\nsinh\u0010\nl2D\n\u00152D\u00119\n=\n;3\n5^m\u0002( ^m\u0002\n^y)\u00002\n4sinh\u0010\n2z\u0000l2D\n2\u00152D\u0011\nsinh\u0010\n2z\u0000l2D\n2\u00152D\u00113\n5^y; (15)\nthen leads to the distributed spin current in 2DM\n~Jz\nS(z)\nJREE\nSO=\u0000Im2\n48\n<\n:2\u00152Dg\"#\n1\nR2D\u0015REE+ 2\u00152Dg\"#coth\u0010\nl2D\n\u00152D\u00119\n=\n;8\n<\n:sinh\u0010\nz\u0000l2D\n\u00152D\u0011\nsinh\u0010\nl2D\n\u00152D\u00119\n=\n;3\n5( ^m\u0002\n^y)\u0000Re2\n48\n<\n:2\u00152Dg\"#\n1\nR2D\u0015REE+ 2\u00152Dg\"#coth\u0010\nl2D\n\u00152D\u00119\n=\n;8\n<\n:sinh\u0010\nz\u0000l2D\n\u00152D\u0011\nsinh\u0010\nl2D\n\u00152D\u00119\n=\n;3\n5^m\u0002( ^m\u0002\n^y) +2\n4cosh\u0010\n2z\u0000l2D\n2\u00152D\u0011\n\u0000cosh\u0010\nl2D\n2\u00152D\u0011\ncosh\u0010\nl2D\n2\u00152D\u00113\n5^y: (16)\nThe inverse Rashba-Edelstein e\u000bect drives a charge current in the x-yplane by the\ndi\u000busion spin current component \rowing along the zdirection. The total longitudinal\n(along ^x) component is\nJC;long (z)\nJCO= 1 + 4 \u0015REE\nl2D!22\n4cosh\u0010\nz\u0000l2D\n\u00152D\u0011\ncosh\u0010\nl2D\n\u00152D\u0011+\u0010\n1\u0000m2\ny\u00113\n5\u0002\nRe2\n48\n<\n:2\u00152Dg\"#tanh\u0010\nl2D\n2\u00152D\u0011\n1\nR2D\u0015REE+ 2\u00152Dg\"#coth\u0010\nl2D\n\u00152D\u00119\n=\n;8\n<\n:sinh\u0010\nz\u0000l2D\n\u00152D\u0011\nsinh\u0010\nl2D\n\u00152D\u00119\n=\n;3\n5 (17)\naand transverse or Rashba-Edelstein component is\nJC;trans (z)\nJCO= 4 \u0015REE\nl2D!2\n[mxmyRe\u0000myIm]\u0002\nRe2\n48\n<\n:2\u00152Dg\"#tanh\u0010\nl2D\n2\u00152D\u0011\n1\nR2D\u0015REE+ 2\u00152Dg\"#coth\u0010\nl2D\n\u00152D\u00119\n=\n;8\n<\n:sinh\u0010\nz\u0000l2D\n\u00152D\u0011\nsinh\u0010\nl2D\n\u00152D\u00119\n=\n;3\n5 (18)Wibson W. G. Silva and Jos\u0013 e Holanda................................................................... 6\nwhereJCO=Ex=(R2D\u0015REE) it is the charge current driven by the external electric\ncurrent. Expanding the longitudinal resistance governed by the current in the x-\ndirection of the applied \feld to leading order in \u00152\nREEand averaging the electric currents\nover the 2DM thickness, it is \fnded\n(R2D)long= \u0015REEEx\nJC;long!\n\u0019R2D+ \u0001R(0)\n2D+ (1\u0000m2\ny)\u0001R(1)\n2D; (19)\nand\n(R2D)trans\u0019\u0000 JC;trans\nEx!\u00121\nR2D\u0015REE\u00132\n=mxmy\u0001R(1)\n2D+mz\u0001R(2)\n2D;(20)\nwhere\n\u0001R(0)\n2D\nR2D=\u0000 2\u0015REE\nl2D!2 2\u00152D\nl2D!\ntanh l2D\n2\u00152D!\n; (21)\n\u0001R(1)\n2D\nR2D= 2\u0015REE\nl2D!2 \u00152D\nl2D!\nRe2\n42\u00152Dg\"#tanh2\u0010\nl2D\n2\u00152D\u0011\n1\nR2D\u0015REE+ 2\u00152Dg\"#coth\u0010\nl2D\n\u00152D\u00113\n5; (22)\n\u0001R(2)\n2D\nR2D= 2\u0015REE\nl2D!2 \u00152D\nl2D!\nIm2\n42\u00152Dg\"#tanh2\u0010\nl2D\n2\u00152D\u0011\n1\nR2D\u0015REE+ 2\u00152Dg\"#coth\u0010\nl2D\n\u00152D\u00113\n5 (23)\nand \u0001R(0)\n2D<0, this suggests that the resistance is reduced by the Rashba interaction.\n3. Discussion and applications\n3.1. Di\u000berent 2D magnetoresistances\nFor small thickness (2D surface) l2D\u001c\u00152Dthe equations (21), (22) and (23) are written\nas\n\u0001R(0)\n2D\nR2D=\u0000 2\u0015REE\nl2D!\n; (24)\n\u0001R(1)\n2D\nR2D= 2 \u0015REE\nl2D!2\"grR2D\u0015REEl2D\nl2D+ 2grR2D\u0015REE\u00152\n2D#\n(25)\nand\n\u0001R(2)\n2D\nR2D=\u00002 \u0015REE\nl2D!2\"giR2D\u0015REEl2D\nl2D+ 2giR2D\u0015REE\u00152\n2D#\n: (26)\nIn Fig. 2, it is shown the di\u000berent 2D magnetoresistances \u0001 Ri\n2D=R2Das a function\nof thickness l2Dwith i= 0, 1, 2, gr= 2:4\u0002107m\u00001\n\u00001,gi= 2:4\u0002107m\u00001\n\u00001,\nR2D= 0:215\u0002106\n,\u00152D= 235\u000210\u00009m and\u0015REE= 0:13\u000210\u00009m for MoS 2[17, 37].\nThe real (gr) and imaginary ( gi) parts of the spin-mixing interface conductance were\nobtained considering the complex spin-mixing interface conductance module equal to the\ne\u000bective spin-mixing conductance obtained by spin pumping measurements [17]. AfterWibson W. G. Silva and Jos\u0013 e Holanda................................................................... 7\nwe consider the dimensions of the structure, we get that the real ( gr) and imaginary ( gi)\nparts of the spin-mixing interface conductance are the same for the YIG/MoS 2structure,\nwhich is an result extremely reasonable, considering the YIG /MoS 2interface has both\nintrinsic spin-orbit coupling and proximity e\u000bect. The di\u000berent behaviors described by\n2D magnetoresistances in Fig. 2 reveal that are e\u000bects that can be measured di\u000berently\nand separately.\n/s48/s46/s48 /s50/s46/s53 /s53/s46/s48 /s55/s46/s53 /s49/s48/s46/s48/s45/s49/s52/s48/s45/s55/s48/s48/s55/s48/s49/s52/s48/s82/s40/s105/s41 /s50/s68\n/s32/s47/s32/s82\n/s50/s68\n/s108\n/s50/s68/s40/s110/s109/s41/s32/s48\n/s32/s49\n/s32/s50/s40/s105/s41\nFigure 2. (online color) 2D magnetoresistances \u0001 Ri\n2D=R2Das a function of thickness\nl2Dwith i= 0, 1, 2,gr= 2:4\u0002107m\u00001\n\u00001,gi= 2:4\u0002107m\u00001\n\u00001,R2D= 0:2\u0002106\n\n,\u00152D= 235\u000210\u00009m and\u0015REE = 0:13\u000210\u00009m for MoS 2[17, 37].\n3.2. REE length\nIn accords with Fig. 2 the REE is an e\u000bect of the order of \u00152\nREEthat becomes relevant\nonly when l2Dis su\u000eciently small. Now is important discuss the limit in which the\nspin current transverse due the spin accumulation to ^ mis completely absorbed as an\nspin transfer torque without re\rection \u0000 = gr\u001d1=(\u00152D\u0015REER2D), which occurs in 2D\ninterface. The spin current at the interface is then\nJ(FM)\nS\nJREE\nSO=\u0000\u0000!=\"\ntanh l2D\n\u00152D!\ntanh l2D\n2\u00152D!#\n^m\u0002( ^m\u0002^y); (27)\nand the maximum magnetoresistance for the FM/2DM structure is\n\u0001R(1)\n2D\nR2D= 2\u0015REE\nl2D!2 \u00152D\nl2D!\ntanh l2D\n\u00152D!\ntanh2 l2D\n2\u00152D!\n; (28)\nbut for small thickness (2D interface) l2D\u001c\u00152D, we have\n\u0015REE=\u00152D\u0011; (29)\nwhere\u0011= (\u0001R(1)\n2D=R2D)1=2. In Fig. 3, it is shown REE length \u0015REE(nm) as a function\nof the REMR \u0011for SLG and MoS 2, with spin di\u000busion length of \u0015SLG= 1:0\u000210\u00006m\n[26, 37] and \u0015MoS 2= 235\u000210\u00009m [17, 37, 38], respectively.Wibson W. G. Silva and Jos\u0013 e Holanda................................................................... 8\n/s48/s46/s48\n/s53/s46/s48/s120/s49/s48/s45/s52\n/s49/s46/s48/s120/s49/s48/s45/s51/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s32/s83/s76/s71\n/s32/s77/s111/s83\n/s50/s82/s69/s69/s32/s40/s110/s109/s41\n/s32/s40/s82/s101/s108/s46/s32/s85/s110/s105/s116/s46/s41\nFigure 3. (online color) Shown REE length \u0015REE (nm) as a function of the REMR \u0011.\nThe points (red-SLG and green-MoS 2) were obtained with the spin di\u000busion lengths\nof\u0015SLG= 1:0\u000210\u00006m [25, 34] and \u0015MoS 2= 235\u000210\u00009m [16, 34, 35], respectively\n3.3. Experimental applications\n3.3.1. YIG/SLG\nThe SLG has been considered to be very promising materials for spintronic\napplications [6, 7, 21, 22, 26]. However, due to the low atomic number of carbon,\nintrinsic graphene has a weak SOC and thus very small spin Hall e\u000bect [26]. SLG have\n2D electronic states that are expected to exhibit remarkable pseudospin. This gives\nrise to a proximity e\u000bect that results in long-range ferromagnetic ordering in graphene,\nas observed in YIG/SLG [5, 26, 38]. In fact, the SLG on the YIG \flm represent one\nexcellent example for application of the study proposal here. For SLG, it was possible\nto consider the e\u000bective thickness lSLG= 2\u000210\u000010m, \u0001R(1)\n2D=R2D= 0:5\u000210\u00008and the\nspin di\u000busion length \u0015SLG= 1:0\u000210\u00006m as in Ref. [26, 38]. Then, using the equation\n(29) is obtain for graphene REE length \u0015REE= 0:7\u000210\u000010m, which is in accord with\nthe value measured with electric spin pumping experiments [26]. In Fig. 4, it is shown\nthe REMR \u0001 R(1)\n2D=R2Das a function of graphene REE length \u0015REE(nm). The point in\nred was measured in ref. [38].\n3.3.2. YIG/MoS 2\nSeveral materials in the family of transition metal dichalcogenides (TMDs)\n[12, 13, 14, 15, 16, 17, 18, 19, 20] can also be used for spin-charge current conversion\n[17, 37]. Due to their layered structure, the TMD can be easily prepared with one\nor several atomic layers as to tailor the transport properties. One important TMD\nmaterial, molybdenum disul\fde (MoS 2), has attracted widespread attention for a variety\nof next-generation electrical and optoelectronic device applications because of its unique\nproperties [12, 13, 14, 15, 16, 17, 18, 19, 20]. For MoS 2it, was used the thickness\ntMoS 2= 2:4\u000210\u00009m, the spin di\u000busion length \u0015MoS 2= 235\u000210\u00009m [17, 37, 38] andWibson W. G. Silva and Jos\u0013 e Holanda................................................................... 9\nthe REMR, \u0001 R(1)\n2D=R2D= 30\u000210\u00008. Hence, for YIG/MoS 2we obtain with equation\n(29) one REE length of \u0015REE = 0:13\u000210\u00009m, which is also in good agreement with\nvalue measured with electric spin pumping experiments [17]. In Fig. 4, it is shown the\nREMR \u0001R(1)\n2D=R2Das a function of MoS 2REE length \u0015REE (nm). The point in green\nwas measured in ref. [37].\n/s48/s46/s48/s48 /s48/s46/s48/s55 /s48/s46/s49/s52 /s48/s46/s50/s49/s48/s52/s48/s56/s48/s82/s40/s49/s41 /s50/s68\n/s32/s47/s32/s82\n/s50/s68/s32/s40 /s120 /s32/s49/s48/s45/s56\n/s41\n/s82/s69/s69/s32/s40/s110/s109/s41/s32/s77/s111/s83\n/s50\n/s32/s83/s76/s71\nFigure 4. (online color) REMR \u0001 R(1)\n2D=R2Dversus REE length \u0015REE(10\u00009m) for\ngraphene and MoS 2. The points (red-graphene and green-MoS 2) were measured in\nreference [37].\n3.3.3. Exchange \feld acting on the spin accumulation\nOne ferromagnetic material in atomic contact with 2D material generates a\nexchange \feld. The exchange-coupling is caracterized here by HExc\n2D=EExc\n2Dgi=2e, which\nwas obtained using the equations (5) and (6). The term EExc\n2Dis the exchange energy,\nwhich for the YIG/2DM interface is EExc\n2D= 1:92\u00060:96\u000210\u000020J (orEExc\n2D= 0:12\u00060:06\neV) [17, 37, 40, 41, 42, 43]. For the YIG/SLG strtucture [22, 26, 37] with resistance\nofR2D= 9\u0002103\n, the imaginary part of the spin-mixing interface conductance\nis of the order of gi= 4:4\u0002108m\u00001\n\u00001, thus the exchange \feld \fnded was of\nHExc\n2D= 2:64\u00061:32\u0002107A/m (or\u00160HExc\n2D= 33:2\u000616:6 T). Already for the YIG/MoS 2\nstructure [17, 37], the imaginary part of the spin-mixing interface conductance is of\nthe order of gi= 2:4\u0002107m\u00001\n\u00001. In this case, the exchange \feld obtained was of\nHExc\n2D= 1:44\u00060:72\u0002106A/m (or\u00160HExc\n2D= 1:8\u00060:9 T). The intensity of exchange \feld\nacting on the spin accumulation is of the order of exchange \feld due to the proximity\ne\u000bect obtained by others methods [41, 42].\n4. Conclusion\nIn summary, we present a study that describes the Rashba-Edelstein magnetoresistance\nin 2D materials. The study was applied the measures of REMR makes at roomWibson W. G. Silva and Jos\u0013 e Holanda................................................................... 10\ntemperature in single layer graphene and in the 2D semiconductor MoS 2in contact\nwith the ferrimagnetic insulator yttrium iron garnet (YIG) measured by the modulated\nmagnetoresistance technique. In the presented discussion, the change in the electrical\nresistance is reminiscent of the magnetoresistance despite the fact that 3D SOC is not\nresponsible for the magnetoresistance in 2DM. Furthermore, the measured REE lengths\nfor these two materials are in good agreement with the study, this is, which represents\na good validation for the present analytical proposal, opening one new method to study\nthe REMR.\nAcknowledgements\nThis research was supported by the Brazilian National Council for Scienti\fc and\nTechnological Development (CNPq), Coordination for the Improvement of Higher\nEducation Personnel - Federal Rural University of Pernambuco (CAPES-UFRPE),\nFinancier of Studies and Projects (FINEP) and Foundation for Support to Science and\nTechnology of the State of Pernambuco (FACEPE).\nAuthor Contributions\nAll authors contributed to the study conception and design.\nData availability statement\nThe data will be made available on reasonable request.\nCon\ricts of interest\nAll the authors declare that there is no con\rict of interest.\nReferences\n[1] A. Soumyanarayanan, N. Reyren, A. Fert, and C. 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Center for Dynamics and Control of Materials, The University of Texas at Austin, Austin, Texas 78712, USA 3. Department of Physics, Northeastern University, Boston, MA 02115, USA 4. Department of Materials Science and Engineering, Cornell University, Ithaca, New York 14853, USA 5. Department of Mechanical Engineering, The University of Texas at Austin, Austin, Texas 78712, USA 6. Texas Materials Institute, The University of Texas at Austin, Austin, Texas 78712, USA 7. Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA * Corresponding Authors: elaineli@physics.utexas.edu, rodriguezvega@utexas.edu † Current Address: Department of Computer and Electrical Engineering, University of Maryland, College Park, MD 20740, USA Abstract: Spin-phonon interaction is an important channel for spin and energy relaxation in magnetic insulators.  Understanding this interaction is critical for developing magnetic insulator-based spintronic devices. Quantifying this interaction in yttrium iron garnet (YIG), one of the most extensively investigated magnetic insulators, remains challenging because of the large number of atoms in a unit cell. Here, we report temperature-dependent and polarization-resolved Raman measurements in a YIG bulk crystal. We first classify the phonon modes based on their symmetry. We then develop a modified mean-field theory and define a symmetry-adapted parameter to quantify spin-phonon interaction in a phonon-mode specific way for the first time in YIG.  Based on this improved mean-field theory, we discover a positive correlation between the spin-phonon interaction strength and the phonon frequency.     Introduction Magnetic insulators are of considerable interest in spintronics due to their minimal spin damping [1–3]. This low damping originates in part from the absence of low-energy electronic excitations, leaving the spins to interact primarily with other spins (magnons) and the lattice (phonons). Beyond their role in spin excitation damping, interactions between the magnons and phonons play a crucial role in developing devices based on thermally driven spin transport [4–6], spin pumping through hybrid spin-lattice excitations[7], and magnon cavity quantum electrodynamics [8, 9]. Of various magnetic insulators explored for spintronic devices, yttrium iron garnet (YIG): Y3Fe5O12 is the most widely investigated due to its remarkably low spin damping and its high transition temperature of 560 K [10, 11]. However, due to its massive unit cell (160 atoms as an inset of Fig.1a), extracting the spin-phonon interaction (SPI) of YIG from ab initio studies is remarkably difficult.  The SPI in YIG has been investigated through different types of experiments. Brillouin light scattering and spin Seebeck transport measurements of YIG have examined the interactions of magnons and phonons through quasiparticle hybridization [12–14]. Other studies have touched upon the SPI by measuring the magnon-phonon energy relaxation length and time [4, 15]. However, no study provides a direct and quantitative measurement of the strength of the SPI in YIG in a phonon-mode specific way.  2 Without knowing the SPI strength, it is difficult to develop accurate models of spin relaxation in YIG or compare YIG to other magnetic insulators for device development.  Here we report Raman spectroscopy studies of optical phonons in a YIG  bulk crystal. By analyzing their symmetry properties and temperature-dependent phonon frequency shift, we investigate if SPI changes systematically for each phonon mode. We determine that the complex unit cell precludes a direct correlation between symmetry or frequency of a phonon mode with the conventional 𝜆-model of the SPI strength [16–18]. By developing a mean-field model and defining a new parameter to describe  SPI strength, we observe a correlation between this mean-field SPI parameter and phonon frequency. These results provide crucial information and advance the understanding of how magnons and phonons interact in YIG.  Experiment  YIG (Y3Fe5O12) is an insulating ferrimagnet (FiM) with Curie temperature TC = 570 K and easy axis in the [111] direction  [19, 20]. YIG crystals exhibit symmetries described by cubic space group 𝐼𝑎3%𝑑 (No. 230) and point group Oh at the Γ point [21–23]. Inversion symmetry present in Oh implies that the phonon modes show mutually-exclusive infrared and Raman activity. The possible Raman irreducible representations in Oh are either T2g, Eg, or A1g. The crystal structure is composed of Y atoms occupying the 24c Wyckoff sites, Fe ions in the 16a and 24d positions, and O atoms in the 96h sites. The conventional unit cell has eight formula units, with 24 Y ions, 40 Fe ions, and 96 O ions for a total of 160 atoms.  Raman measurements were taken with a 532 nm laser incident on a YIG single crystal with [111] oriented along the surface normal. The scattered light was collected in a backscattering geometry and directed to a diffraction grating-based spectrometer. The observed optical phonon modes in the Raman spectra agree with previous measurements of YIG [24, 25]. Low-temperature measurements from 8.8 K to 313.65 K were performed in a closed-loop cryostat, and high-temperature measurements from 313.65 K to 631.95 K were performed with a ceramic heater. Between each temperature, the sample was allowed to equilibrate for 15 minutes or longer. A saturating magnetic field was applied in the sample plane for all measurements. Due to constraints of the experiment systems, low-temperature measurements used a 300 mT saturating field, and the high-temperature measurements used a 50 mT saturating field.  Since both fields were above the saturating field, this difference did not noticeably affect the magnetic ordering of YIG or the Raman spectra.  Raman spectra were collected with a fixed s-polarization incident on the sample. Fig.1a shows the p- and s-polarizations components of the scattered light for the sample at low temperature (8.8 K). Phonon modes of different symmetries scatter light with different polarizations. Fig. 1(b) shows the intensity of the Raman signal from the scattered light as it passed through a linear polarizer, with the polarization axis rotated in steps of 20° from 0° to 180°. Based on the results, the phonons are categorized with their respective irreducible representations: T2g, Eg, or A1g.    The temperature dependence of the phonon frequencies was determined by fitting with a Lorentzian function and extracting the central frequencies. We plot the measured Raman spectra for one T2g mode at three different representative temperatures 8.8 K, 313.65 K, and 632 K in Fig. 2 (a), (b), and (c). At low temperatures (e.g. 8.8 K), the low thermal population of the phonons reduces the Raman intensity. In contrast, the phonon modes exhibit a broader linewidth at high temperatures due to increased phonon-phonon and phonon-magnon scattering, which lowers the peak intensity. Consequently, the temperature-dependent frequency was only measurable for a  subset of the observed phonons. The temperature dependence of the peak frequencies for the two modes is shown in Fig. 2d and 2e. The temperature dependence of peak frequencies of all the measurable phonon modes can be found in Supplementary Information.   Results  3 In the absence of spin order above the transition temperature (i.e. 559 K for YIG), the temperature dependence of the optical phonon frequency 𝜔! is determined by anharmonic effects, i.e. phonon-phonon scattering. Well below the melting points, 3-phonon scattering dictates the temperature dependence of 𝜔! as follows  𝜔!(𝑇)=𝜔!(0)−𝐴/1+2Exp[𝑥]−19(1) where 𝜔!(0) is the zero temperature phonon frequency, 𝐴 is a coefficient related to the 3-phonon scattering strength and 𝑥=ℏ𝜔!(0)2𝑘\"𝑇⁄ with Planck’s constant ℏ, Boltzmann’s constant 𝑘\", and temperature 𝑇 [16–18]. We fit the peak frequency above 559 K using Eq. (1) to determine 𝜔!(𝑇) for each phonon mode. Examples of these fits are shown in Fig. 2d and 2e.  In  the magnetically ordered state, the influence of spin order on the phonon frequency can be treated as a small deviation, ∆𝜔#!, such that the optical phonon frequency is given as  𝜔!$=𝜔!(𝑇)+∆𝜔#!\t(2) where 𝜔!$ is the measured phonon frequency. Then, ∆𝜔#! can be found by taking the difference of the measured frequency and anharmonic temperature-dependent phonon frequency, i.e. 𝜔!$−\t𝜔!(𝑇), as shown in Fig. 3 for selected phonon modes.   Many previous studies of the SPI express the frequency deviation as ∆𝜔#!=𝜆〈𝑺%∙𝑺&〉, where 𝜆 is a single term capturing the SPI strength and 〈𝑺%∙𝑺&〉 represents nearest-neighbor spin correlation function [26–30]. The spin correlation function can be approximated 〈𝑺%∙𝑺&〉≈\t𝑆'(𝐵)(𝑇), where 𝐵) is the Brillouin function, which has a maximum value of 1 at 𝑇/𝑇*=0 [27]. Thus to find 𝜆 without the spin-related, temperature-dependent contribution to the frequency, ∆𝜔#! should be evaluated at  𝑇=0. Table 1 reports frequency deviation measured at 8.8 K, ∆𝜔#!+=𝜔!$−\t𝜔!(8.8\tK), the lowest temperature reached in our experiments. The high 𝑇* of YIG and slow decrease of 𝐵)(𝑇) results in 𝐵)(8.8\tK)≈1. Then, 〈𝑺%∙𝑺&〉≈\t𝑆'( and using 𝑆'=,( for the magnetic iron atoms in YIG, 𝜆 is found from ∆𝜔#!+, also reported in Table 1. Examining the results shown in Table 1, there is no clear trend for ∆𝜔#!+ and 𝜆 with either the frequency or symmetry of the mode. These results highlight the deficiency of the 𝜆 model that has been applied successfully for other materials with a simple unit cell such as FeF2 and ZnCr2O[26��30].   Discussion The simple 𝜆 model, which treats all phonon modes equally, is insufficient for describing the SPI in YIG. This is not surprising as the large unit cell leads to complicated phonon dispersions. However, a more detailed first-principles approach like density functional theory (DFT) for determining the SPI is exceedingly difficult, again due to the large unit cell of YIG, as well as the especially high precision required in the computations to accurately describe the lattice vibrations and their coupling to magnetic order. Thus, to describe spin-phonon interaction in YIG, we develop a modified mean-field model that captures the mode dependence of the SPI. We begin with the Ginzburg-Landau (GL) potential describing the magnetic order, 𝐹=\t𝐴2𝑚(+𝐵2𝑚-\t(3) where 𝑚≡𝑀/𝑀+ is the ferrimagnetic order parameter defined as the magnetization (𝑀) divided by its zero temperature value (𝑀+). The GL parameters 𝐴 and 𝐵 have units of energy and 𝐴=−𝑎(𝑇*−𝑇), where 𝑇* is the magnetic transition temperature. The temperature dependence of the order parameter agrees well with the temperature dependence of the magnetic moment of YIG reported in the literature (Supplementary Information).   This GL potential only includes magnetic order, and thus needs to be expanded to include phonon contribution to the GL potential. By including only the harmonic terms, the GL potential takes the form  4  𝐹=\t.(𝑚(+\"(𝑚-+/(𝜇𝜔𝑢(+/(𝛿#!𝑚(𝑢((4) where 𝜇 is the phonon mode reduced mass, 𝑢 is the atomic displacement, and 𝛿#! is the SPI strength [31, 32]. Note that for phonons with irreducible representation Ag and  T2g, the symmetry allows a cubic term proportional to 𝑚(𝑢, which is weak in YIG (see Supplementary Information) [33].  Equilibrium values 𝑚∗ and 𝑢∗ are found from the conditions 𝜕𝐹𝜕𝑚=0,𝜕𝐹𝜕𝑢=0\t(5)\tand the spin-dependent phonon frequency (Ω) is determined by 𝜇Ω(=\t𝜕(𝐹𝜕𝑢(V121∗323∗=\t𝜇𝜔(+\t𝛿#!𝑚∗(\t.(6) Now, using the equilibrium value of  𝑚∗=X𝑎(𝑇*−𝑇)/𝐵, Ω is approximately given by  Ω(𝑇)≈ω+\t𝛿#!2𝜇ω/1−𝑇𝑇*9\t(7) to first order in 𝛿#!. (Note: 𝛿#! is defined for angular frequencies.) Compared to the 𝜆 model, we see that the frequency deviation is determined by the frequency and reduced mass of the phonon mode, as well as the SPI strength. We use this improved mean-field theory to extract the SPI strength. Figure 3 shows ∆𝜔#! across the temperature range, with fits using Eq. (7) to extract the 𝛿#!, shown in Figure 4.  To further understand the SPI found from the modified mean-field model, we examine the atomic displacements of each phonon mode. Using group theory projection operators, we can derive a basis of eigenmodes that brings the dynamical matrix to a block-diagonal form[34]. The 739 cm-1 mode only involves the O atoms’ displacements due to its Ag symmetry (see Supplementary Information). Because it only involves O atoms, this mode has the smallest reduced mass µ compared with T2g and Eg modes. We find that this phonon mode has the largest 𝛿#!. This finding is consistent with the interpretation that the vibrations of the light O atoms are most affected by the magnetic ordering of the heavy Fe atoms. The symmetries of other phonon modes, T2g and Eg, allow motions of all three ion types (Y, Fe, O) in principle. First-principles calculations of the Raman phonon frequencies and symmetries allow us to assign 𝜇 to each Raman phonon. We find that, as expected, lower frequency phonons have larger 𝜇. (See Supplementary Information.) Using these values of 𝜇 to calculate the SPI, we find that higher frequency phonons have larger SPI as shown in Figure 4. This trend suggests that the atoms with stronger bonds (consequently higher phonon frequency) are more affected by magnetic ordering.   Conclusion In summary, we investigate SPI  associated with optical phonon modes of a YIG bulk crystal. By taking polarization-resolved Raman spectra, we analyze their symmetry. Temperature-dependent Raman spectra taken over a broad temperature range of 8.8-635 K allow us to evaluate SPI quantitatively and specific to a particular phonon mode. By developing an improved mean-field model and applying a refined analysis, we discover that the SPI increases with phonon frequency.  The Ag mode involving vibrations of only O atoms has the strongest SPI. These results provide both direct and mode-specific interaction strengths, thus, providing valuable information for advancing theories of magnetic insulators and for exploring spintronic devices such as those based on spin-caloritronic effects.   Acknowledgements This research was primarily supported (J.C., M.R.-V., B.F., J.Z., G.A.F., X.L.) by the National Science Foundation through the Center for Dynamics and Control of Materials: an NSF MRSEC under  5 Cooperative Agreement No. DMR1720595. G.K. and N.A.B. were supported by the Cornell Center for Materials Research: an NSF MRSEC under Cooperative Agreement No. DMR-1719875. G.A.F also acknowledges support from NSF Grant No. DMR-1949701. K.S.O. and J.C. performed measurements. B.F. built the experimental system. K.S.O. and J.C. analyzed experimental results. M.R.-V. and G.A.F. developed mean-field model and performed group theory analysis with input from G.K. 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(a) Raman spectra taken with s-in/s-out (colinear) and s-in/p-out (crossed) polarization configurations at 8.8 K. Solid lines connect data points for clarity. Inset shows the YIG crystal structure along the [111] direction. (b,c) Angle-dependent intensities of the representative A1g, Eg, and T2g modes. The spectra were obtained by by keeping incident polarization fixed. Panel b and c refer to colinear and crossed polarization configurations, respectively. The fit curves follow theoretical predictions from crystal lattice symmetry.  \n 10   \nFIG. 2. (a,b,c) Example spectra for different temperatures, normalized to the peak intensity at 8.8 K. Solid lines are Lorentzian fits with a linear offset to account for the background. Vertical dashed lines indicate the peak positions. (d,e) Temperature dependence of 𝜔(45 and 𝜔,6/ phonon frequencies, which have symmetries T2gand Eg, respectively. The solid curves correspond to the anharmonic phonon-phonon scattering fit, which is based on fitting to data only above the temperature Tc. The deviation from the anharmonic curve (black arrow) reflects the corresponding spin-phonon coupling strength, λ, given in Eq.(2).  \n 11   \nFIG. 3. The measured phonon frequencies is subtracted from the temperature-dependent frequency found with the anharmonic fit [see Fig. 2 (b) and (c)] to determine ∆𝜔#!. Solid lines show fits to the mean-field model which yield the spin-phonon interaction strength 𝛿#!, given in Eq.(7).    \n 12  \n  \nFIG. 4. Absolute value of the spin-phonon interaction strength evaluated with the mean-field model for the phonon modes in YIG. The measured 𝛿#! for the 𝜔748 mode is negative (purple square), while the rest of the measured 𝛿#! are positive.  \n 13 Supplementary Information: Spin-Phonon Interaction in Yttrium Iron Garnet Kevin S. Olsson1†, Jeongheon Choe1,2, Martin Rodriguez-Vega1,2,3*, Guru Khalsa4, Nicole A. Benedek4, Bin Fang1,2, Jianshi Zhou2,5,6, Gregory A. Fiete1,2,3,7, and Xiaoqin Li1,2,6*  Table of Contents SECTION 1: GROUP THEORY ASPECTS AND RAMAN ACTIVITY 13 SECTION 2: GINZBURG-LANDAU FRAMEWORK 16 SECTION 2.1: FERRIMAGNETIC ORDER GINZBURG-LANDAU POTENTIAL 16 SECTION 2.2: SPIN-PHONON INTERACTION 16 SECTION 3: FIRST-PRINCIPLES EVALUATION OF THE RAMAN PHONONS 17 SECTION 4: ANHARMONIC AND MEAN-FIELD FITS FOR ALL PHONON MODES 18  Section 1: Group theory aspects and Raman activity YIG (Y3Fe5O12) is an insulating ferrimagnet (FiM) with Curie temperature TC = 570 K, cubic space group 𝐼𝑎3%𝑑 (No. 230), and point group Oh at the Γ point [1, 2]. The crystal structure is composed of Y atoms occupying the 24c Wyckoff sites, Fe ions in the 16a and 24d positions, and O atoms in the 96h sites. The conventional unit cell has eight formula units, with 24 Y ions, 40 Fe ions, and 96 O ions. Typically, the easy axis is in the [111] direction [3–5]. We start our group theory analysis by determining the Raman activity of the crystal. First, we note that in the Oh point group, the representation of the vector is Γ9:;. = T1u. Then, we calculate the equivalence representation employing the Bilbao crystallographic server [6]. The equivalence representation is the number of atoms that remain invariant under the point group symmetry operations for each irreducible representation and is given by Γ:=.=4𝐴/>+2𝐴/3+2𝐴(>+2𝐴(3+4𝐸3+4𝐸>+5𝑇(3+5𝑇(>+3𝑇/3+5𝑇/>.(S1) Then, the representation of the lattice vibrations is Γ?@A.\t9CD.=Γ:=.⊗Γ9:;. =3𝐴/>+5𝐴(>+8𝐸>+14𝑇/>+14𝑇(>+5𝐴/3+5𝐴(3+10𝐸3+18𝑇/3+16𝑇(3,(S2) which contains the acoustic modes with symmetry T1u. Therefore, we expect 25 (first-order) Raman active modes: ΓE@F@G=3𝐴>⊕8𝐸>⊕14𝑇(>.(S3) Table S1 tabulates the Raman active modes according to the involved Wyckoff position. In Fig. 1a-c, show the atomic displacements for the three expected Ag modes obtained with the ISODISTORT [7, 8]. For the symmetric modes 𝐴/>, neither the Fe atoms nor the Y atoms participate, as Table S1 shows. Fig. 1d and 1e show two examples of atomic displacements of the modes 𝐸>, which involve Fe and Y atoms.   14  𝐴/> 𝐸> 𝑇(> Fe (16a) x x x Y (24c) x 1 2 Fe (24d) x 1 3 O (96h) 3 6 9  Table S1. Wyckoff-position-resolved Raman active modes. The sum of each column corresponds to the total number of modes expected. \nThe Raman tensors, with respect to the principal axis of the crystal, are given by \nFIG. S1. (a)-(c) The three 𝐴/> Raman active modes in YIG viewed along the [111] direction. The O (96h) atoms are shown in blue, while the Fe (16a and 24d) are shown in green. For clarity, Y atoms (24c) are not shown. For the 𝐴/> modes, the Fe and Y atoms are stationary. (d) and (e) schematically show two of the eight 𝐸> irreducible representations in YIG along the [111] direction. Y atoms are shown in yellow. The mode displayed in (d) shows motion of the Y atoms, while (e) shows motion of the Fe atoms.  15 𝑅a𝐴/>b=c𝑎000𝑎000𝑎d,𝑅e𝐸>(/)f=c𝑏000𝑏000−2𝑏d,𝑅e𝐸>(()f=h−√3𝑏000√3𝑏0000j,𝑅e𝑇(>(/)f=c00000𝑑0𝑑0d,𝑅e𝑇(>(()f=c00𝑑000𝑑00d,𝑅e𝑇(>(7)f=c0𝑑0𝑑00000d.(S4) In order to carry out mode assignment in Raman measurements, it is useful to know the Raman scattering efficiency defined as  𝑆∝hl𝑒%J%𝑅%&𝑒#&%&j((S5) where 𝑒%J/#% are the unit vectors for the incident and scattered polarization directions and 𝑅 is the Raman tensor [9]. For the degenerate modes Eg and T2g, we need to add the contributions from the two and three partner matrices, respectively.  The YIG sample used for the measurements has surface normal oriented along [111]. Therefore, we need to transform the Raman tensors such that the [111] direction corresponds to the z-direction in the new coordinate system. To determine the symmetry of the modes, we calculate the Raman intensity as a \nfunction of a rotation angle about the z-direction ([111]). In Fig. S2, we plot the Raman intensity for incident light parallel (perpendicular) to the scattered light. \nFIG. S2. Calculated Raman intensity as a function of sample rotation angle about the [111] direction for (a) parallel polarization and (b) perpendicular polarization. \n 16 Section 2: Ginzburg-Landau Framework  This section describes in further detail the Ginzburg-Landau framework used to develop the mean-field model which appears in the main text. Section 2.1: Ferrimagnetic Order Ginzburg-Landau Potential  YIG’s magnetic moment exhibits the temperature dependence shown in Fig. S3 as measured in Ref. [10, 11]. The solid line corresponds to a mean-field like fit  𝑀(𝑇)𝑀+=X1−𝑇/𝑇L.(S6) The simple mean-field temperature dependences give a good first-order approximation. Therefore, we can describe the magnetic transition with the simple Ginzburg-Landau potential  𝐹=𝐴2𝑚(+𝐵4𝑚-,(S7) where 𝑚≡𝑀/𝑀+ is the dimensionless ferrimagnetic order parameter oriented in the [111] direction, with 𝑀+=4.19361×10M-JTM//cm7. The GL parameters 𝐴=−𝑎(𝑇𝑐−𝑇)and 𝐵 have units of energy.  \nSection 2.2: Spin-Phonon Interaction In this section ISOTROPY is used to obtain invariant polynomials for constructing the Ginzburg-Landau potential to describe the coupling between phonons and the magnetic order parameters [7]. Since the magnetic order parameter 𝑚 breaks time-reversal symmetry, only even powers are allowed in the GL potential. For the phonon modes, we consider only harmonic contributions. Similar approaches have been employed before to describe spin-phonon coupling in YMnO3 [12] and MnV2O4 [13] in combination with first-principles calculations. For each of the phonon symmetries, the Ginzburg-Landau potential takes the form 𝐹.\"#=𝐹++𝐴2𝑚(+𝐵2𝑚-+12𝜇.𝜔.𝑢(+12𝑚((𝛿.N𝑢+𝛿.O𝑢()(S8) 𝐹P#=𝐹++𝐴2𝑚(+𝐵2𝑚-+12𝜇P𝜔P𝑢(+12𝛿PO𝑚(𝑢((S9) 𝐹Q$#=𝐹++𝐴2��(+𝐵2𝑚-+12𝜇Q𝜔Q𝑢(+12𝑚((𝛿QN𝑢+𝛿QO𝑢()(S10) ������������������������������������\nFIG. S3. YIG magnetic moment themperature dependence, as reported in Ref. [10], compared to the mean-field description with 𝑇L = 559 K.   17 where 𝜔% and 𝑢% are the bare frequency and reduced mass of the phonon modes. As determined in the first section, 𝐴/> modes only involve displacement of oxygen atoms, so 𝜇. = 𝜇R is the oxygen mass. We use first-principles calculations of the Raman phonons to assign 𝜇P or 𝜇Q to the phonon modes with symmetries 𝐸> and 𝑇(> in the next section. Finally, 𝛿%N and 𝛿%O  correspond to the temperature-independent spin-phonon coupling constants, reported here in units of energy per Å and Å(, respectively. In oxides, the coupling strength is usually of the order of a few cm-1 [14]. The phonon modes with symmetries Eg and T2g exhibit different coupling to the magnetic order.  The phonon modes with symmetries 𝐸> and 𝑇(> exhibit different coupling to the magnetic order. We now explore the implications of this in the temperature-dependence of the phonon frequency. The equilibrium values for the order parameters 𝑢∗ and 𝑚∗ are determined by the conditions  ∂𝐹∂𝑢=0\tand\t∂𝐹∂𝑚=0,(S11) while the phonons frequency (Ω) in the presence of coupling with spin is given by [12] µΩ(=∂(𝐹∂𝑢(V323∗121∗=µω(+δO𝑚∗((S12) in the harmonic approximation for the three symmetry modes considered. Therefore, phonon modes with smaller reduced mass are expected to be more sensitive to magnetic-ordering effects than heavier phonon modes.  The frequency shift begins by evaluating the equilibrium conditions, yielding 𝑚∗=X𝑎(𝑇*−𝑇)/𝐵. For the 𝐸> modes first, the temperature-dependent phonon frequency is  ΩP(𝑇)=|ωP(+δPOµ𝑎𝐵(𝑇*−𝑇)≈ω+δPO2µω/1−𝑇𝑇*9.(S13) The only undefined parameter is the spin-phonon coupling constant 𝛿PO , which has units of energy per Å(. The reduced mass μ is measured in kg and the angular frequency ω in cm-1. Near the critical point, this result agrees with the correction derived in Ref. [15] from an expansion of the spin Hamiltonian as a Taylor series around small phonon displacements.   For the case of 𝐴/> and 𝑇(> modes, in contrast to the 𝐸> modes, the cubic term 𝑚(𝛿Nis allowed. This term indicates the presence of a distortion (non-zero solution for the equilibrium position 𝑢+), however, from experiments, we know that distortions in YIG are very weak [16]. Therefore, we consider the limit 𝛿N≪𝛿O. Then, the same relation is obtained for the 𝐴/> and 𝑇(> modes,  Ω(𝑇)≈𝜔+𝛿O2𝜇𝜔/1−𝑇𝑇*9.(S14) Eq. S14 is the mean-field equation used to fit the anharmonic subtracted phonons frequencies (Δ𝜔) to extract the spin-phonon interaction strength 𝛿#! in the main text.   Section 3: First-principles evaluation of the Raman phonons Density functional theory (DFT) calculations were used to calculate the Raman phonon frequencies and masses. We have used the Vienna ab intio Simulation Package (VASP) with projector-augmented waves pseudopotentials [17, 18]. The generalized gradient approximation is used as implemented in the Perdew-Burke-Ernzerhof functional revised for solids with an effective Hubbard term U-J=3.7 eV implemented in the Dudarev method [19, 20]. An energy convergence threshold of 10-6 eV was used. Structural parameters were relaxed using 10-3 eV/\tÅ force convergence condition on each atom. A 6×6×6 k-point grid and 550 eV energy cut-off were found to give converged structural parameters.  18 Phonons were calculated at the 𝛤–point with the PhonoPy software [21]. Phonon masses were found by diagonalizing the generalized eigenvalue problem  𝜇𝑢=𝐾𝑢 where 𝐾 is the force-constant matrix and 𝜇 is the mass matrix.  The converged lattice constant is 𝑎=12.31\tÅ with the free paramters of the O Wyckoff site converged to 𝑥=0.05763, 𝑦=0.34949, and 𝑧=0.47321. The magnetic moments on the inequivalent a and d Fe sites were found to be 4.013 Bohr and -3.875 Bohr, respectively. The calculated Raman phonon frequencies are shown in Table S2 and change only quantitatively with reasonable variations in 𝑈−𝐽. In Table S2 the measured phonon modes are assigned to a calculated 𝜇, based on the measured mode’s symmetry and frequency. Note that the measured mode at 508 cm-1 with symmetry Eg, 𝜔,+8,  does not have a well matched calculated mode. Previous measurements of YIG have located two modes close to this frequency, one with a A1g and another with Eg symmetry [22]. Based on this, 𝜔,+8 is matched to an A1g mode close to it’s frequency instead of an Eg mode. All the calculated modes within in 50 cm-1 of 𝜔,+8 are very similar in mass, thus this assignment does not drastically change the measured SPI.   Symmetry Phonon Frequency (𝑐𝑚M/) 𝜇 Symmetry Phonon Frequency (𝑐𝑚M/) 𝜇 DFT Calculated Measured DFT Calculated Measured A1g 333  16.00 T2g 128  62.29  493 508* 16.00  168  31.19  713 739 16.00  175 174 37.50 Eg 130  48.13  189  55.56  267 276 19.87  229  21.42  287  24.02  270 276 24.54  336 346 16.54  315  18.37  400  18.51  371  16.71  446  16.19  382 378 16.83  624  18.93  429 447 16.18  659  16.04  489  17.10      586  18.11      591 591 18.71      719  17.87 Table S2. Phonon symmetry, frequency, and mass for Raman phonons from DFT, with measured frequencies corresponding to the masses used in the calculating 𝛿. *The measured mode 𝜔,+8 is matched to an A1g instead of an Eg, as no calculated Eg modes are close to this frequency. Section 4: Anharmonic and Mean-Field Fits for All Phonon Modes  This section provides the anharmonic fits (Fig. S4) and mean-field fits (Fig. S5) for all of the measured phonon modes reported in the main text Table 1.    19   \nFIG. S4. Temperature dependent phonon frequencies extracted from Lorentzian fits for  the 10 measured phonon modes. The anharmonic fit is performed using main text Eq. 1 on the frequencies above Tc. The difference between the fit and lowest temperature data point is used to determine ∆𝜔#!+ reported in the main text Table 1.   20     \nFIG. S5. Temperature dependent phonon frequency deviations found from subtracting data from the anharmonic fits shown in Fig. S3. The spin-phonon interaction strength 𝛿#! is found from fitting the mean-field model Eq. S14 to the data points. These are the 𝛿#! shown in Fig. 4 of the main text.  21 Supplementary References: [1] S. Geller and M. A. 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Mater., vol. 108, pp. 1–5, Nov. 2015. [22] W. H. Hsu, K. Shen, Y. Fujii, A. Koreeda and T. Satoh, “Observation of terahertz magnon of Kaplan-Kittel exchange resonance in yttrium-iron garnet by Raman spectroscopy,” Phys. Rev. B, vol. 102, no. 17, p. 174432, Nov. 2020.  "    },    {        "title": "1902.04608v1.Characterization_of_spin_wave_propagation_in__111__YIG_thin_films_with_large_anisotropy.pdf",        "content": "1 \n Characterization of spin wave propagation in (111) YIG thin films with \nlarge anisotropy  \nA. Krysztofik,1,b) H. Głowiński,1,a) P. Kuświk,1,2 S. Ziętek,3 L. E. Coy,4 J. N. Rychły ,5 \nS. Jurga,4 T. W. Stobiecki,3 J. Dubowik1 \n1Institute of Molecular Physics, Poli sh Academy of Sciences, M. Smoluchowskiego 17, PL -60-179 Poznań, Poland  \n2Centre of Advanced Technology, Adam  Mickiewicz University, Umultowska 89c, PL -61-614 Poznań, Poland  \n3Department of Electronics, AGH University of Science and Technology, Al. Mickiewic za 40, PL -30-059 Kraków, Poland  \n4NanoBioMedical Centre, Adam Mickiewicz University, Umultowska 85, PL -61-614 Poznań, Poland  \n5Faculty of Physics, Adam  Mickiewicz University, Umultowska 85, PL -61-614 Poznań, Poland  \n6Faculty of Physics and Applied Computer Sc ience, AGH University of Science and Technology, Al. Mickiewicza 30, PL -30-059 Kraków, \nPoland  \n \na)E-mail: hubert.glowinski@ifmpan.poznan.pl  \nb)E-mail: adam.krysztofik@ ifmpan.poznan.pl  \n \n \n \nAbstract  \nWe report on  long-range spin wave  (SW)  propagation in nanomete r-thick Yttrium \nIron Garnet (YIG) film with  an ultralow Gilbert damping. The knowledge of a wavenumber \nvalue  |𝑘⃗ | is essential  for design ing SW devices. Although determining  the wavenumber |𝑘⃗ | \nin experiments like Brillo uin light scattering spect roscopy is straightforward , quantifying \nthe wavenumber in all -electrical experiments has not been widely commented  so far. \nWe analyze magnetost atic spin wave (SW) propagation in  YIG films  in order to determine \nSW wavenumber |𝑘⃗ | excited by the coplanar waveguide . We show  that it is crucial  to \nconsider influence of magnetic anisotropy fields present in YI G thin films  for precise \ndetermination of SW wavenumber . With the proposed  methods  we find that experimentally \nderived v alues of |𝑘⃗ | are in perfect agreement with  that obtained from  electromagnetic \nsimulation only if anisotropy fields are included.  \n \n \n \n \n \n \n 2 \n  \nSpin wave (SW) propagation in magnetic thin film structures  has become intensively \ninvestigated topic in recent year s due to promising applications in modern electronics  [ 1, 2, \n3, 4 ]. The wavenumber (or equivalently – the wavelength 𝜆=2𝜋/|𝑘⃗ |) is an important \nparameter to  account for propagation characteristics.  For example, it is essential to choose \nSW wavenumber and correlate  it to certain device dimension in order to ensure observation \nof expected phenomena in SW devices e.g. i n magnonic crystals  [ 5, 6 ] or devices based on  \nwave interference such as SW transistor [  2 ], SW logic gates  [ 2 ], Mach -Zender  type \ninterferometer s [ 7 ]. The knowledge  of SW wavenumber is also very important  in the \nassessment of the  effective magnitude of Dzaloshinskii -Moriya interaction using collective \nspin-wave dynamics  [ 8 ]. \nIn propagating SW spectroscopy experiments two s horted coplanar waveguides  \n(CPW s) are commonly used as a transmitter and a receiver  [ 9 ]. Each CPW , integrated \nwithin the film , consists of a signal line and two ground lines conn ected at one end. When \na rf-current flows through the transmitter it induces  an oscillating magnetic field around the \nlines that exerts a torque and causes spin precession in the magnetic material beneath. The \ninverse  effect is then used for SW detection by the receiver . Since  the generated magnetic \nfield is not homogenous with reference to the film plane and solely depends on CPW \ngeometry , it determines the distribution of SW wavenumber that can be excited.  \nIt is assumed that the transmitter excites a broa d spectrum of SW wavevectors \nof wavenumber 𝑘 exten ding to 𝑘𝑚𝑎𝑥≈𝜋/𝑊 (𝑊 is a width of  CPW line) with a maximum \nof excitation amplitude approximately around 𝑘𝑚𝑎𝑥𝐴𝑚𝑝≈𝜋/2𝑊 [ 10 ]. The question now is : \nwhat is the actual wavenumber of the SW with the la rgest amplitude  detected by the receiver \nsituated at a certain distance f rom the transmitter. It appears that while in Brillo uin light \nscattering spect roscopy 𝑘 is easily accessible,  in all electrical spin wave spectroscopic \nexperiment s the determination of SW wavenumber  is rather challenging  [ 11 ]. \nWe aim to ans wer this question by analyzing our experimental results of  SW \npropagation  in yttrium iron garnet (Y 3Fe5O12, YIG)  thin film s. YIG films are known \nas possessing the lowe st Gilbert damping parameter enabl ing the SW transmission over  the \ndistances of several hundred micrometers  [ 2, 12 ]. However, YIG films synthesized by \npulsed laser deposition (PLD) exhibit substan tially disparate  values of  anisotropy fields and \nsaturation magnetization , depending on the growth process parameters and , consequently , \nstoichiometr y of the obtained film [ 13, 14, 15 ]. It has already been theoretically predicted 3 \n that anisotropy may significantly affect SW propagat ion and the transmission characteristics  \n[ 16, 17 ]. Therefore , for such  YIG films , SW spectra analysis requires careful consideration \nof anisotropic properties of a given film.  \nHere, we compare two methods of experimental determination of the SW \nwavenumber  which include anisotropy fields. The experimental results are then compared \nwith electromagnetic simulations.  \n \n \nFig. 1 . A θ-2θ XRD  scan of epitaxial YIG film on GGG (111) substrate near the GGG (444) reflection . \n \n \nYIG film was grown on a monocrystalline, 111-oriented Gadolinium Gallium \nGarnet substrate (Gd 3Ga5O12, GGG) by means of  PLD  technique . Substrate temperature was \nset to 650℃  and under  the 1.2×10−4 𝑚𝑏𝑎𝑟  oxygen pressure ( 8×10−8 𝑚𝑏𝑎𝑟  base \npressure)  thin film was deposited at the  0.8 𝑛𝑚/𝑚𝑖𝑛 growth rate using third harmonic \nof Nd:YAG Laser ( 𝜆=355 𝑛𝑚). After the growth, the sample was additionally ann ealed \nex situ at 800℃  for 5 𝑚𝑖𝑛. X-ray diffraction  and reflection  measurements showed that \nthe YIG film was single -phase, epitaxial with the GGG substrate  with the thickness of 82 𝑛𝑚 \nand RMS  roughness of 0.8 𝑛𝑚. XRD θ-2θ scan, presented in Fig. 1, c learly shows the high \ncrystallinity of the YIG film, displaying well defined Laue oscillations , typical for highly \nepitaxial films, which clearly point to the high quality and well textured YIG  (111) film \n[ 18 ]. Subsequently , a system of two CPW s made of 100 𝑛𝑚 thick alumin um was integrated  \nonto YIG film  (Fig. 2) using a maskless photolit hography techniqu e. The width 𝑊 of signal \nand ground lines was equal to  9.8 𝜇𝑚 and the gaps between them were 4 𝜇𝑚 wide. \nThe distance between the centers  of signal l ines was 150 𝜇𝑚. \n49.5 50.0 50.5 51.0 51.5 52.0 52.5101102103104105106Intensity [a.u.]\n2 [deg]YIG (444)\nGGG (444)4 \n  \nFig. 2. SEM  image of the integrated CPW s on the YIG film.  The distance 𝑑 between the transmitter and the \nreceiver  is equal to  150 𝜇𝑚. The depicted  Cartesian and crystallographic coordinate  system is used throughout \nthis paper. The width of signal and ground lines  is marked  with 𝑊. 𝐺 denotes the gap width between the lines.  \n \n \nTo investigate SW propagation we follow ed approach presented in Ref. [ 9 ] and \n[ 12 ]. Using a Vector  Network Analyzer  transmission signal S21 was measured for Damon -\nEshbach surface modes with  wavevector 𝑘⃗  perpendicular to the magnetization  for magnetic \nfields ranging from −310 𝑂𝑒 to +310 𝑂𝑒 (Fig.  3(a)). Exemplary S 21 signal s (imaginary \npart) , whic h are shown in Figs  3(b) and (c) , reveal  a series of oscillations as a function \nof frequency with a Gaussian -like envelope  corresponding to the excited SW wave number  \ndistribution.  Figure  3(c) shows  that frequency separation ∆𝑓 between two oscillation maxi ma \ndiffers noticeably in value  depending on the magnetic field . The decrease in signal amplitude \nis also observed  since SW decay length is inversely proportional to the frequency , so that the \nlow-frequency SWs propagate further away  [ 12, 19 ]. \n \n \n \n5 \n  \nFig. 3. (a) Color -coded SW propagation data S 21 measured at different magnetic fields.  \nWith a red line 𝑓(𝐻) dependence of the uniform excitation ( 𝑘=0) is depicted. The red line corresponds to \nthe maximum in S 11 signal  in (b). The blue dashed line represents a  dispersion relation with 𝐻𝑎=𝐻𝑢=0.     \n(b) Reflection (S 11, 𝑘=0) and transmission (S 21, 𝑘≠0) signals. The plot illustrates a magnified cross -section \nof (a) at 𝐻=−67.5 𝑂𝑒. (c) SW spectra measured  at different magnetic fields.  Color -coding in (b) and (c) \ncorresponds to the one defined in (a).  \n \n \nFor the frequencies of the highest signal am plitude, the wavenumber 𝑘𝑚𝑎𝑥𝐴𝑚𝑝  can \nbe determined according to the dispersion relation derived for (111) crystalline orientation of \nthe YIG film [ 16, 17 ]: \n6 \n  𝑓=𝜇𝐵\n2𝜋ℏ𝑔√(𝐻+2𝜋𝑀𝑠𝑡𝑘)(𝐻−1\n2𝐻𝑎−𝐻𝑢+4𝜋𝑀𝑠−2𝜋𝑀𝑠𝑡𝑘)−1\n2(𝐻𝑎sin (3𝜙))2, (1) \nwhere 𝑓 is the microwave frequency, 𝜇𝐵 – the Bohr magneton constant, ℏ – the reduced \nPlanck constant, 𝑔 – the spectroscopic splitting factor, 𝐻 – the ex ternal magnetic field, 𝑀𝑠 – \nthe saturation magnetization, 𝑡 – the film thickness, 𝑘 – the wavenumber, 𝐻𝑎 – the cubic \nanisotropy field and 𝐻𝑢 – the out -of-plane uniaxial anisotropy field. 𝐻𝑎=2𝐾𝑎\n𝑀𝑠 and 𝐻𝑢=2𝐾𝑢\n𝑀𝑠, \nwhere 𝐾𝑎 and 𝐾𝑢 are anisotropy constants. It should be highlighted that when 𝐻𝑎=𝐻𝑢=0, \nEq. 1 becomes equivalent to the one originally obtained by Damon and Eshbach  [ 20 ]. The \nazimuthal angle 𝜙 define s the in -plane orientation of magnetization direction with respect to \nthe (112̅) axis of YIG film. In our study the term −1\n2(𝐻𝑎sin (3𝜙))2 in Eq. 1 vanishes since \nmagnetic field 𝐻 is parallel to  (112̅) axis and 𝜙=0°. \nAs can be seen from Eq.  1, in order to determine wavenumber  𝑘 one need s \nto evaluate many material constants, namely 𝑔, 𝑀𝑠, 𝑡, 𝐻𝑎, 𝐻𝑢 in the first instance. \nThis problem can be partially  solved with a broadband ferromagnetic resonance measurement  \nof the film. Fo r 𝑘=0 Eq.1  simplifies to the form ula, which  allows for  the determination of \nthe spectroscopic factor 𝑔 and the effective magnetization 4𝜋𝑀𝑒𝑓𝑓∗=−1\n2𝐻𝑎−𝐻𝑢+4𝜋𝑀𝑠: \n 𝑓𝑘=0=𝜇𝐵\n2𝜋ℏ𝑔√𝐻(𝐻+4𝜋𝑀𝑒𝑓𝑓∗). (2) \nTherefore , within this approach , the film thickness and the saturation magnetization should be \ndetermined using  other experimental methods.  \nTo investigate ferromagnetic resonance of the YIG film , the reflection signal S 11 was \nmeasured. In order to avoid extrinsic  contribution to the resonance linewidth caused by non -\nmonochromatic excitation of the CPW (2𝜋∆𝑓𝑒𝑥𝑡𝑟=𝑣𝑔∆𝑘) [ 21 ] and, consequently , possible \nambiguities in  the interpret ation of  resonance peak position , it is recommended to perform \nthis measurement with the use of a wide CPW . Note that the full width a t half maximum of a \nCPW excitation spectra ∆𝑘≈𝑘𝑚𝑎𝑥𝐴𝑚𝑝  [ 21 ]. In our study  we used a CPW with signal and \nground lines  of the width equal to  450 𝜇𝑚 and with the  20 𝜇𝑚 wide gaps between  them. For \nsuch a CPW,  the simulated value of  𝑘𝑚𝑎𝑥𝐴𝑚𝑝  is equal to 49 𝑐𝑚−1 and, therefore, yields \nnegligible broadening that is of the order of a few MHz.  \nThe measured S 11 signal (imaginary part) is depicted in Fig.  3(a) with the red line. It \nappears to lie just below th e S 21 signal. Fitting to the experimental data with Eq.  2 gave  \nfollowing value  of the spectroscopic facto r  𝑔=2.010±0.001 and the effective 7 \n magnetization 𝑀𝑒𝑓𝑓∗=169±7 𝑒𝑚𝑢/𝑐𝑚3. A comparison of 4𝜋𝑀𝑒𝑓𝑓∗ with 4𝜋𝑀𝑠 (𝑀𝑠=\n120±19 𝑒𝑚𝑢/𝑐𝑚3 was measured  using  Vibrating Sample Magnetometry) gives  −1\n2𝐻𝑎−\n𝐻𝑢 of 616 𝑂𝑒, showing the substantial difference  between  obtained  values  of 𝑀𝑒𝑓𝑓∗ and 𝑀𝑠. \nThe determined value of −1\n2𝐻𝑎−𝐻𝑢 remain s in th e midst of the  range reported for  PLD -\ngrown YIG thin films , from 229 𝑂𝑒 up to 999 𝑂𝑒 [ 14, 22 ]. It is worth to mention that for \nfully stoichiometric , micrometer -thick  YIG films made by means of  liquid phase epitaxy  \n(LPE)  technique −1\n2𝐻𝑎−𝐻𝑢=101 𝑂𝑒 [ 14 ]. From  the analysis of resonance linewidth vs . \nfrequency  [ 23 ] we additionally  extracted Gilbert damping parameter of the YIG film , which \nequals to  𝛼=(5.5±0.6)×10−4 and impli es low damping of magnetization precession . \nSubstitution  of the 𝑔, 𝑀𝑒𝑓𝑓∗, 𝑀𝑠 and 𝑡 values  into Eq. 1 enabled the determination of \nwavenumber 𝑘𝑚𝑎𝑥𝐴𝑚𝑝=1980±102 𝑐𝑚−1. It sho uld be noted  that if anisotropy fields were \nneglected in the Eq.1 ( 𝐻𝑎=𝐻𝑢=0), yet only saturation magnetization was taken into \naccount , a fitting to the experimental data would  not converge . The calculated dispersion \nrelation with the derived valu e of 𝑘𝑚𝑎𝑥𝐴𝑚𝑝 , assuming 𝐻𝑎=𝐻𝑢=0 is depicted with blue \ndashed line in Fig. 3 (a). Omission of anisotropy fields in magnetization dynamic \nmeasurements  may therefore lead to the significant misinterpretation of  experimental results  \nfor YIG thin films.  \nTypical values of cubic magnetocrystalline anisotropy field 𝐻𝑎 range from  −18 𝑂𝑒 \nto −64 𝑂𝑒 for PLD grown YIG films  [ 14, 15, 22 ], what indicates that resonance \nmeasurements as well as spin wave propagation  are govern ed by the out -of-plane uniaxial \nanisotropy.  For the film employed in our study , the 𝐻𝑢 value is of about  −600 𝑂𝑒 in \nagreement with previous reports  [ 14, 15, 22 ]. For any more complex architecture  of \nmagnonic  waveguides and circuits  it is likewise  imperative t o investigate the  in-plane \nanisotropy properties  [ 24 ]. As  can be seen from Eq.  1 one would  expect  a six-fold \nanisotropy in the plane of (111) -oriented single crystals , that is common  among  rare-earth \nsubstituted YIG garnet s and LPE -YIG films  [ 18, 25, 26, 27 ]. To examine this issue , we \nperformed VSM and angular resolved ferromagnetic resonance measurements. Hysteresis \nloops for all measured in -plane directions exhibit no substantial differences regarding  \ncoercive field ( ≈1.2 𝑂𝑒), saturation field  and saturation magnetization (Fig.  4(a)). The \nangular resolved resonance measurement s confirm  this result  and show  that the (111) YIG \nfilm is isotropic in the film plane  (Fig. 4(b)). The main reason for this behavior is the low  \nvalue of cubic anisotropy field which cause s the resonance frequency modulation  by a value 8 \n of the fraction of  MHz. Such  small differences do not surpass  the experimental error, nor \nwould they  significantly affect the coherent SW propagation.  It is expecte d that t he SW \npropagation characteristics, measured for any other crystallographic orientation, would \ntherefore remain unaltered.  \n \n \n \n \nFig. 4. (a) VSM hysteresis loops measured in the film plane for three different  crystallographic directions. \nThe magneti zation is normalized to the saturation magnetization 𝑀𝑠=120±19 𝑒𝑚𝑢/𝑐𝑚3. A paramagnetic \ncontribution of the GGG substrate was subtracted for each loop.  (b) Resonance frequency as a function \nof azimuthal angle 𝜙 taken at 𝐻=150 𝑂𝑒. The red li ne depicts the calculated values of resonance frequency \naccording to Eq.1 for 𝑘=0, 𝐻𝑎=−30 𝑂𝑒 and 𝐻𝑢=−600 𝑂𝑒. \n \n \n \n-50 -40 -30 -20 -10 0 10 20 30 40 50-1.0-0.50.00.51.0M / MS\nMagnetic Field [Oe] \n \n (a)\n1.41.61.8\n0306090\n120\n150\n180\n210\n240\n2703003301.4\n1.6\n1.8\n(110)(101)(211)\no\n(112)o\n(011)o\n(121)o\noo (211)o (101)oo (110)\no (121)\no (011) Resonance frequency [GHz]\nH = 150 Oeo (112)(b)9 \n Another me thod of extracting SW wavenumber  involves the analysis of the SW \ngroup velocity  𝑣𝑔. Following Ref. [ 21 ], 𝑣𝑔 can be determined from frequency difference ∆𝑓 \nbetween two oscillation maxima in S 21 signal according to the relation:  \n 𝑣𝑔=𝑑∆𝑓, (3) \nwhere 𝑑 is the distance between two CPW s. To determine ∆𝑓 we chose two neighboring \noscillation maxima of the highes t S21 signal  amplitude  as it is shown in Fig. 3 (b) and (c) . \nIn Fig. 5 the derived values of group velocity are shown as a function of magnetic \nfield. It is found that 𝑣𝑔 reaches the  value of 7.6 𝑘𝑚/𝑠 for the field of 1.3 𝑂𝑒 (preferable in  \nmagnonic  information processing devices  of high efficiency ) and 1.4 𝑘𝑚/𝑠 for the field  of \n285 𝑂𝑒. It should be highlighted that such big difference s in 𝑣𝑔 values can be further utilized \nto design  tunable , impulse -response delay  lines  as 𝑣𝑔 changes up to five times  with the \nmagnetic field.  At a distance of 150 𝜇𝑚 between CPWs  it would allow to achieve 20 to \n110 𝑛𝑠 delay times of an impulse.  \n \n \nFig. 5. Spin wave group velocity  as a function of the external magnetic f ield. The red line represents a fit \naccording to Eq.  4. \n \n \nWith the red line in Fig. 5 a fitting is depicted according to:  \n 𝑣𝑔=2𝜋𝜕𝑓\n𝜕𝑘=𝜇𝐵\nℏ𝑔2𝜋𝑀𝑠𝑡(−1\n2𝐻𝑎−𝐻𝑢+4𝜋𝑀𝑠−4𝜋𝑀𝑠𝑡𝑘)\n2√(𝐻+2𝜋𝑀𝑠𝑡𝑘)(𝐻−1\n2𝐻𝑎−𝐻𝑢+4𝜋𝑀𝑠−2𝜋𝑀𝑠𝑡𝑘). (4) \nThe main advantage of extracting SW wavenumber  from 𝑣𝑔(𝐻) dependence is  that it does \nnot require additional measurement of 𝑀𝑠 which is often notably influenced by an error in the \nestimated  film volu me. Since  the saturation magnetization 𝑀𝑠 can be treated as a fitting \n-300 -200 -100 0 100 200 30012345678vg [km/s]\nH [Oe]10 \n parameter in Eq.  4, the derivation  of SW wavenumber  involves only S 11, S21 and thickness \nmeasurement s. The determined values of 𝑘𝑚𝑎𝑥𝐴𝑚𝑝=1690±53 𝑐𝑚−1 and 𝑀𝑠=116±\n2 𝑒𝑚𝑢/𝑐𝑚3 remain  in a good agreement with that obtained above  - directly derived from \ndispersion relation  (𝑘𝑚𝑎𝑥𝐴𝑚𝑝=1980±102 𝑐𝑚−1, 𝑀𝑠=120±19 𝑒𝑚𝑢/𝑐𝑚3). \nAs can be seen from Fig ure 5, SW group velocity attains the maximum va lue as the \nmagnetic field approaches 𝐻=0. The maximum value of 𝑣𝑔 is given by:   \n 𝑣𝑔(𝐻=0)≅𝜇𝐵\nℏ𝑔√𝜋𝑀𝑠𝑡\n2𝑘(−1\n2𝐻𝑎−𝐻𝑢+4𝜋𝑀𝑠[1−𝑡𝑘]). (5) \nThe zero -field region may therefore become the subject of interest  for magnonic applications. \nMoreover, Eq. 5 shows that the maximum value of 𝑣𝑔 depends on the anisotropy fields.  PLD -\ngrown YIG films possessing  a high anisotropy would allow faster information processing in \nSW circuits than LPE films for which  the value of  −1\n2𝐻𝑎−𝐻𝑢 is smaller (as it was pointed \nout above).  \nTo confront our experimental results with the expected, theoretical value of  \n𝑘𝑚𝑎𝑥𝐴𝑚𝑝 , we performed electromagnetic simulations in Comsol Multiphysics . Here,  CPW \nwas modeled accordin g to the geometry  of the performed CPW  (Fig.  2), assuming lossless \nconductor metallization, relative permittivity of the substrate 𝜀𝑟=12 and 50 𝛺 port \nimpedance. From the simulated in-plane  distribution of the dynamic magnetic field ℎ𝑥 (inset \nof Fig. 6) an excitation spectra of CPW was obtained  using d iscrete Fourier transformation  of \nℎ𝑥(𝑥). The highest excitation strength is observed for 𝑘𝑚𝑎𝑥𝐴𝑚𝑝=1838 𝑐𝑚−1, which \ncorrespond s well to the experimentally  obtained values  within 7% accuracy . The second \nobserved maxima is at 𝑘2=6770 𝑐𝑚−1. However,  as its amplitude is 2 0 times lower with \nrespect to  the amplitude of 𝑘𝑚𝑎𝑥𝐴𝑚𝑝  it is not observed in  the measured  S21 signal . \n \n 11 \n  \nFig. 6. Excitation spectrum  of the CPW with 9.8 𝜇𝑚 wide signal line s and 4 𝜇𝑚 gaps. The inset shows  in-plane \ncomponent of the dynamic  magnetic field excited by the CPW.  \n \n \nTo extend  our study , we performed a series of further simulations for the CPW \ndimensions , which are  achievable with electron - and photolithography.  We assumed equal \nwidths of signal and ground lines (𝑊) as well as equal widths of gaps between them  (𝐺). The \nresults are presented in Fig.  7. It is found  that for the width s 𝑊 ranging from 300 𝑛𝑚 to \n40 𝜇𝑚, the wavenumber 𝑘𝑚𝑎𝑥𝐴𝑚𝑝  vary between 70000 𝑐𝑚−1 and 250 𝑐𝑚−1, respectively , \nrevealing the CPW wavenumber probing limits. We also note that the gap width significantly \naffects  𝑘𝑚𝑎𝑥𝐴𝑚𝑝 . In order to accurately extrapolate its contribution to 𝑘𝑚𝑎𝑥𝐴𝑚𝑝 , we \ndeveloped empirical formula which incorporates width 𝐺: \n 𝑘𝑚𝑎𝑥𝐴𝑚𝑝=2.27\n𝑊+0.6 𝐺. (6) \nThe fittings , according to  the Eq. 6, are depicted in Fig.  7 with solid lines. We f ound that Eq. \n6 is valid  for gap  width 0.1 𝑊<𝐺<2 𝑊. For 𝐺=0.74 𝑊 this formula is equivalent to the \none previously proposed in Ref.  [ 10 ] (𝑘𝑚𝑎𝑥𝐴𝑚𝑝≈𝜋/2𝑊). \n \n0 4000 8000 120000.00.20.40.60.81.0\n-50 -25 0 25 50-2502550hx [Oe]\nx [m]\n-50 -25 0 25 50-2502550hx [Oe]\nx [m]\nk2= 6770 cm-1Amplitude [a.u.]\nk [cm-1]\n-50 -25 0 25 50-2502550hx [Oe]\nx [m]kmaxAmp= 1838 cm-112 \n  \nFig. 7. Wavenumber of the highest amplitude as a function of CPW signal line width. The solid lines represent \na fit according to Eq.  6. \n \nTo conclude, we  report ed on long-range spin wave propagation in the 82 𝑛𝑚 thick \nYIG film  over the distance as large as 150 𝜇𝑚. In order to precisely determine excited \nwavenumber by the coplanar antenna, it is essential to take in to account anisotropy fields \npresent in YIG films. We show ed that anisotropy significantly affect s SW propagation \ncharacteristics, namely it causes an increase in SW frequency as well as in SW group \nvelocity. The main contribution comes from  the out -of-plane uniaxial anisotropy field. T he \ncubic anisotropy field is neglig ibly small  in the YIG (111) film  and it does not affect \nmagnetization dynamics  in the film plane.  We explain ed that the wavenumber determination \nfrom group velocity vs . magnetic field depend ence requires only two types of measurement , \nthat is  broadband SW spectroscopy and the measurement of film thickness.  \n \n \nAcknowledgements  \nThis work was carried out within the Project NANOSPIN  PSPB -045/2010 supported by a \ngrant from Switzerland through the S wiss Contribution to the enlarged European Union . J. \nRychły and J. Dubowik would like to acknowledge support from the European Union’s \nHorizon 2020 MSCA -RISE -2014: Marie Skłodowska -Curie Research and Innovation Staff  \nExchange (RISE) Grant Agreement No. 644 348 (MagIC).  The authors would like to thank  \nProfessor Maciej Krawczyk  for thoughtful suggestions . We also acknowledge valuable \ncomments from Dr. Piotr Graczyk and Paweł Gruszecki .  \n1 10100010000100000\n G = W / 10\n G = W / 4\n G = W / 2\n G = W\n G = 2 WkmaxAmp [1/cm]\nW [m]13 \n References  \n \n[1] Jamali M , Kwon J H, Seo S -M, Lee K -J and Yang H 2016 Spin wave nonreciprocity \nfor logic device applications. Sci. Rep . 3, 3160  \n[2] Chumak A V, Serga A A and Hillebrands B 2014 Magnon transistor for all -magnon \ndata processing. Nat. Comun . 5, 4700  \n[3] Vogt K, Fradin F Y, Pearson J E, Sebastian T, Bader S D, Hille brands B, Hoffmann A \nSchultheiss H 2014 Realization of a spin -wave multiplexer. Nat. Comun . 5, 3727  \n[4] Gertz F, Kozhevnikov A V, Filimonov Y A, Nikonov D E and Khitun A 2015 \nMagnonic Holographic Memory: From Proposal to Device. IEEE J. Explor. Solid -State \nComputat. Devices Circuits  1, 67-75 \n[5] Serga A A, Chumak A V and Hillebrands B 2010 YIG magnonics J. Phys. 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B  77, 054425  "    },    {        "title": "2002.04284v1.Sub_micrometer_near_field_focusing_of_spin_waves_in_ultrathin_YIG_films.pdf",        "content": " 1 Sub-micrometer  near -field focusing of spin waves in ultrathin YIG \nfilms  \n \nB. Divinskiy1*, N. Thiery2, L. Vila2, O. Klein2, N. Beaulieu3, J. Ben Youssef3, S. O. \nDemokritov1, and V. E. Demidov1 \n1Institute for Applied Physics and Center for NanoTechnology , University of Muenster, \n48149 Muenster, Germany  \n2Univ. Grenoble Alpes, CNRS, CEA, Grenoble INP, IRIG -SPINTEC, F -38000 Grenoble, \nFrance  \n3LabSTICC, CNRS, Université de Bretagne Occidentale, 29238 Brest, France  \n \n \nWe experimentally demonstrate tight focusing of a spin wave beam excited in extended \nnanometer -thick films of Yttrium Iron G arnet by a simple microscopic antenna  \nfunctioning as a single -slit near -field lens . We show that the focal distance and the \nminimum transverse width of the focal spot can be controlled in a broad range by \nvarying the frequency/wavelength of spin waves  and the antenna geometry . The \nexperimental data are in good agreement with the results of numerical simulations. Our \nfindings provide a simple solution for implementation of magnonic nano -devices \nrequiring local  concentration of the spin -wave energy.  \n \n \n*Corresponding author , e-mail: b_divi01@uni -muenster.de  \n   2 The advent of high -quality nanometer -thick films of magnetic insulator Yttrium \nIron Garnet (YIG)1-3 essentially expanded  horizons for the field of magnonics4-6  \nutilizing spin waves for transmission and processing of information  on the nanoscale. \nThanks to the small thickness and ultra -low magnetic damping, these films enable \nimplementation of magnonic devices with nanometer dimensions,7-8 where the spin -\nwave losses are by several orders of magnitude smaller compared to t hose in devices \nbased on metallic fer romagnetic films .9 The large propagation length of spin waves in \nYIG is particularly beneficial for implementation of spatial manipulation of spin -wave \nbeams in the real space.  \nIt is now well established that the propagation of spin waves can be controlled  by \nusing approaches simi lar to those used  in optics .10-16 However, in contrast to light  \nwaves , the wavelength of spin waves can be as small as few tens of nanometers ,17,18 \nwhich allows one to implement efficient wave manipulation on the nanoscale.  In the  \nrecent years , particular attention was given to the possibility to controllably focus \npropagating spin waves .10-14,19,20  Such focusing allows one to concentrate  the spin-wave \nenergy  in a small spatial area, which is important, for example, for implementat ion of  \nthe efficient local detection of spin -wave signals. Provided that the position of  focal \npoint  is controllable by the spin -wave frequency, the focusing can also be utilized for \nimplementation of the frequency multiplexing .21 Additionally, the strong local \nconcentration of t he spin -wave energy can be used to stimulate  nonlinear phenomena, \nfor example, the second -harmonic generation .22  \nEfficient spin -wave focusing can be achieved relatively easily in confined \ngeometries , such as stripe waveguides ,23 where  it is governed by the interference of \nmultiple co -propagating quantized spin -wave modes .24 In the case  of extended magnetic  3 films, implementation of the focusing appears to be less straightforward.  In the recent \nyears, several approaches have been suggested  utilizing  a spatial variation of the \neffective spin -wave refraction index ,14,16 refraction  of spin waves  at the modulation of \nthe film thickness11 or the temperature ,12 diffraction from a Fresnel zone plate ,13 and \nexcitation of spin -wave beam s by curved transducer s.19,20 All these approaches are \nrather complex in terms of practical implementation, particularly on the nanoscale.  A \nmuch simpler approach known in optics25-27 relies on the utilization of Fresnel \ndiffraction patterns appearing in the near -field region  of a single slit , where the Fresnel \nnumber F=a2/(d) is of the order or larger  than 1 (Ref. 28). Here a is the length of the \nslit, is the wavelength, and d is the distance from the slit to the observation point. As \nwas experimentally  shown for light waves26 and surface plasmon polaritons ,27 such a \nslit functions as an efficient  near-field lens enabling tight focusing of the incident wave.  \nIn this work , we demonstrate experimentally that the principles of near -field \ndiffractive focusi ng are also applicable for spin waves in in -plane magnetized magnetic \nfilms, which, in contrast to light, exhibit anisotropic dispersion. By using a 2 m long \nspin-wave  antenna, which is equivalent to a single one-dimensional  slit29, we achieve \nfocusing of  the excited spin waves into a n area with the transverse width below 700 nm. \nWe show that, in agreement with the theory  developed  for light waves, the focal \ndistance increases with the decr ease of the wavelength  of the spin wave , which allows \nelectronic co ntrol of the spin-wave focusing by the frequency and/or the static magnetic \nfield. We also perform micromagnetic simulation s, which show excellent agreement \nwith the experimental data and allow us to analyze the effects of the antenna geometry  \non the focus ing characteristics.   4 Figure 1 (a) shows the schematic of the experiment. The test devices  are based on  \na 56 nm thick YIG film grown by liquid phase epitaxy on a gadolinium gallium \nsubstrate. The independently determined saturation magnetization of the film is \n4M=1.78 kG and the Gilbert damping parameter is =1.4×10-4. The YIG film is \nmagnetized to saturation by the static magnetic field H applied in the film plane. The \nexcitation of spin waves is performed by using lithographically defined 2 m long, 300 \nnm wide , and 7 nm thick spin -wave antenna contacted by 2 m wide and 30 nm thick \nAu microstrip lines30. The microwave -frequency electrical current  IMW flowing through \nthe antenna creates a dynami c magnetic field  h, which couples  to the magnetization in \nthe YI G film  and excites spin waves propagating away from the antenna. Figure 1(b) \nshows the normalized spatial distribution of the  amplitude of  the dynamic field  created \nby the antenna  in the YIG film calculated by using COMSOL Multiphysics  simulation \nsoftware ( https://www.comsol.com/comsol -multiphysics ). As seen from these data, due \nto the large difference in the width of the antenna and the  microstrip  lines, the amplitude \nof the dy namic field underneath the antenna is by an order of magnitude larger \ncompared to that underneath the line s. Therefore, the efficient spin wave excitation is \nonly possible in the 2 m long antenna region. This disbalance is further enhanced for \nspin waves with wavelengths comparable or smaller than the width of the microstrip  \nlines , due to the reduced coupling efficiency of the inductive mechanism .9 \nSpatially resolved detection of excited spin waves is performed  by micro -focus \nBrillouin light scattering (BL S) spectroscopy .9 The probing light with the wavelength of \n473 nm and the power of 0.1 mW produced by a single -frequency laser is focused into \ndiffraction -limited spot on the surface of the YIG film . By analyzing the spectrum of \nlight inelastically scatter ed from magnetic excitations, we obtain signal – the BLS  5 intensity – proportional to the intensity of spin waves at the location of the probing spot. \nBy scanning the spot over the sample surface, we obtain spatial maps of the spin-wave \nintensity. Additionally, by using the interference of the scattered light with the reference \nlight modulated at the excitation frequency ,9 we record spatial maps of the spin -wave \nphase.  \nFigures 2(a) and 2(b) show the representative  intensity and phase maps recorded \nat H=500 Oe by applying excitation current with the frequency f=3.8 GHz . The power \nof the applied signal is 10 W, which is proven to provide the linear regime of \nexcitation and propagation of spin waves.  In agreement with the above discussion, spin \nwaves are only radiated from the region of the narrow  antenna. More importantly, the \nradiated beam exhibits significant narrowing and increase of the intensity at the distance \nd=3.6 m from the center of the antenna clearly indicating focusing of the excited spin \nwaves. Qualitatively similar behaviors were  also observed for different excitation \nfrequencies in the range  f=3.2-4 GHz , although the distance d was found to change \nstrongly wit h the variation of f.  \nFrom the phase -resolved measurement s (Fig. 2(b)) we obtain the wavelength \n=0.6 m of spin waves  at f=3.8 GHz. By repeating these measurements for  different \nexcitation frequencies, we obtain the spin -wave dispersion curve (Fig. 2(c)) , which \nallows us to relate the excitation frequency to the spin -wave wavelength. Note that the \nexperimental data (symbols in Fig. 2(c)) are in perfect agreement with the results of \ncalculations (curve in Fig. 2(c)) based on the analytical theory (Ref. 31).    \nOn one side, the observed focusing is counterintuitive. Indeed, the excitation of \nwaves by a finite -length straight  antenna , as used in our experiment,  is equivalent to a \ndiffraction of a wave with a n infinite  plane front from a slit29, which is known to result  6 in a formation of a divergent beam. On the other side, it is also known25-27 that, before \nthe beam starts to diverge,  a complex focusing -like diffraction pattern is formed in the \nnear field just behind a slit. In the recent years , it was shown theoretically25 and proven  \nexperimentally ,26,27 that these near -field effects can be used for efficient focusing of \nwaves of different nature.  \nDue to the insufficient spatial resolution, the fine details of the near -field spin -\nwave pattern cannot be seen in the experimental maps (Fig. 2(b)). Therefore, we \nperform micromagnetic simulations using the software package MuMax3 (Ref. 32). We \nconsider a magnetic film with dimensions of 2 0 m 10 m 0.05 m discretized into \n10 nm  nm 50 nm cells . The standard for YIG exchange constant of 3.66 pJ/m is \nused.  The spin waves are excited by applying a sinusoidal dynamic magnetic field  with \nthe amplitude of 1 Oe , which is close to the estimated experimental value of 3 Oe.  The \nspatial distribut ion of the excitation field is taken  from COMSOL simulations (Fig. \n1(b)) . The angle of the excited magnetization precession is of the order of 0.1°.  \nThe results of simulations for the excitation frequency f=3.8 GHz and H=500 Oe \nare shown in Fig. 3(a). The simulated map of the normalized spin-wave  intensity \n<Mx2>/< Mx2 max> exhibits a narrowing of the excited beam and concentration of the \nspin-wave energy in exactly the same way, as it is observed in the experiment (compare \nwith Fig. 2(a)). Simultaneously, it shows a fine structure, which is reminiscent of that \nobtained for light diffracted on a slit .26 To further confirm the analogy , we perform \nsimulations for the case of plane spin wave s diffract ing from a slit formed by two 300 \nnm wide rectangular regions with increased  magnetic  damping ( =1) with a 2 m long \ngap between them  (Fig. 3(b)) . The close similarity between the obtained patterns shows \nthat the experimental results obtained for spin -wave excitation by the antenna are  7 equally applic able for spin -wave focusing by a slit lens. We emphasize that such a lens \ncan be easily implemented in practice , for example, by using ion implantation into \nnanometer -thick YIG  film.   \nTo additionally address the effects of the anisotropy  of the spin -wave dispersion , \nwe show in Fig. 3(c) an intensity map calculated for spin waves in an out -of-plane \nmagnetized film characterized by an isotropic dispersion33. Comparison of Fig. 3(c) \nwith Fig. 3(a) sh ows that the anisotropy makes  the near -field focusing  even better \npronounced due to the existence of the preferential direction of the energy flow \ncharacterized by the angle =17° (see Fig. 3(a)) calculated according to Ref. 34 .   \nFigure 4 shows the quantitative comparison of the experimental resu lts with those \nobtained from simulations. In Fig. 4(a) , we plot one -dimensional sections of the \nexperimental (Fig. 2(a)) and the calculated (Fig. 3(a)) intensity map s along the axis of \nthe spin-wave beam  at z=0. Both  data sets show perfect agreement. In both cases, the \nintensity increases during the first 3 m of propagation and reaches a maximum at the \ndistance y3.6 m, which can be treated as a focal distance. Transverse sections of the \nintensity maps at this distance  (Fig. 4(b))  also exhibit very similar narrowing of the \nbeam to 670 20 nm. As seen from Figs. 4(c) and 4(d) , the good agreement  between the \nexperimental data and the result of simulation s is observed in a broad range of spin -\nwave wavelengths.  \nWe note that, in contrast to the far -field focusing, in our case, the focal distance \ndepends strongly on the wavelength (Fig. 4(c)). This dependence is in agreement with  \nthe theory of the near -field diffractive focusing  of light ,25 which predicts that the focal \ndistance should increase with the decrease of the wavelength . This dependence can be \nparticularly important for magnonic applications, since it allows one to focus spin  8 waves with different frequencies at different spatial locations. Alternatively, the focal \nposition  can be controlled by the variation of the static magnetic field at the fixed spin -\nwave frequency.  \nAs seen from Fig. 4(d), the transverse width w of the spin -wave beam at the focal \nposition exhibits a monotonous decrease with the decrease of the wavelength  . \nTherefore, similarly to the far -field focusing , one can obtain stronger concentration of \nthe energy for spin waves with smaller wavelengths. Note, however, that the ratio w/ \nincreases at smaller  making the focusing of short -wavelength spin waves less \nefficient.  \nTo study the effects of the antenna geometry on the focusing efficiency , we \nperform micromagnetic simulations for different lengths of the spin -wave antenna a at \nthe fixed value of the wavelength =0.6 m. The results of these simulations show (Fig. \n5) that the focal -point width w generally reduces with decreasing  a. Therefore, more \ntight focusing can be achieved by using smaller antenna e or slit lenses. Additionally, as \ncan be seen from Fig. 5, the reduction of the antenna length a results  in a decrease of the \nfocal distance, which allows one to achieve stronger focusing in devices with smaller \ndimensions.  \nIn conclusion, we have experimentally demonstrated a simple and efficient \napproach to the focusing of spin waves  on the sub -micrometer s cale. The obtained \nresults are applicable not only to the excitation -stage focusing, but can also be used for \nimplementation of near -field lenses for plane spin waves. Our findings significantly  \nsimpl ify the implementation  of nano -magnonic devices utilizing spin -wave focusing, \nwhich is critically  important for their real -world application s. \n  9 We acknowledge support from Deutsche Forschungsgemeinschaft  (Project No. \n423113162 ) and the French ANR Maestro (No.18 -CE24 -0021).   10 References  \n1. Y. Sun, Y. Y. S ong, H. 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Rev. B 83, 054408 (2011).  \n23. V. E. Demidov, S. O. Demokritov, K. Rot t, P. Krzysteczko, and G. Reiss , Appl. \nPhys. Lett. 91, 252504 (2007).   12 24. V. E. Demidov, S. O. Demokritov, K. Rott , P. Krzysteczko, and G. Reiss, Phys. \nRev. B 77, 064406 (2008).  \n25. W. B. Case, E. Sadurni, and W. P. Schleich, Opt. Express 20, 27253 (2012).  \n26. G. Vitrant, S. Zaiba, B. Y. Vineeth, T. Kouriba, O. Ziane, O. Stéphan, J. Bosson, \nand P. L. Baldeck, Opt. Express 20, 26542 (2012).  \n27. D. Weisman, S. Fu, M. Gonçalves, L. Shemer, J. Zho u, W. P. Schleich, and A. Arie , \nPhys. Rev. Lett. 118, 154301 (2017).  \n28. E. Hecht, Optics. 5th edition. (Pearson, Harlow, 2017).  \n29. The similarity between a narrow (narrower than half of the wavelength) strip \nantenna and one -dimensional slit follows from the Huygens –Fresnel principle. In \nboth cases, the appearing patterns can be considered as a result of the interference of \nsecondary wavelets radiated by point sources located on a stra ight wavefront of finite \nlength.  \n30. We note that utilization of excitation structures based on coplanar lines with \nsmoothly changing geometrical parameters can help to improve the overall \nmicrowave -to-spin wave conversion efficiency. Such structures have been \nconsidered in P. Gruszecki , M. Ka sprzak, A. E. Serebryannikov, M. Krawczyk, and \nW. Śmigaj,  Sci. Rep. 6, 22367 (2016) and H. S. Kö rner, J. Stigloher, and C. H. Back,  \nPhys. Rev. B 96, 100401(R) (2017).  \n31. B. A. Kalinikos, IEE Proc. H 127, 4 (1980).  \n32. A. Vansteenkiste , J. Leliaert, M. Dvornik, M. Helsen, F. Garcia -Sanchez, and B. \nVan Waeyenberge , AIP Advances 4, 107133 (2014).   13 33. The calculations were performed at f=3.8 GHz. The static magnetic field was \nincreased to  H=2900 Oe to obtain the same spin -wave wavelength of 0.6 m, as in \nthe case of the in -plane magnetized film.  \n34. V. E. Demidov, S. O. Demokritov, D. Birt, B. O’Gorman, M. Tsoi, and X. Li, Phys. \nRev. B  80, 014429 (2009).  \n  \n \n \n \n   14  \n \n \n \n \n \n \n \n \n \nFigure 1. (a) Schematic of the experiment. (b)  Normalized  calculated  spatial \ndistribution of the amplitude of the dynamic  magnetic  field created by the antenna  in the \nYIG film . \n  \n  \n 15  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 2.  Representative  spatial maps of the intensity  (a) and phase  (b) of radiated spin \nwaves  recorded  by BLS  at H=500 Oe and f=3.8 GHz.  (c) Measured (symbols) and \ncalculated ( solid curve) dispersion curve of spin waves.  \n \n  \n 16  \nFigure 3.  (a) Calculated intensity map of spin waves radiated by the antenna. The \nschematic of the antenna and the connecting microwave lines is superimposed on the \nmap. (b) Calculated intensity map of spin -wave diffraction from a one -dimensional slit. \nSuperimposed rectangles mark the regions with an increased damping. (c) Similar to (a)  \ncalculated for the case of isotropic spin-wave dispersion. Calculations were performed \nfor H=500 Oe and f=3.8 GHz .   \n 17  \n \n \n \n \n \n \n \n \n \n \n \nFigure 4.  (a) One-dimensional sections of the experimental  (symbols)  and the \ncalculated  (solid curve)  intensity maps along the axis of the spin -wave beam at z=0. (b) \nTransverse section s of the experimental (symbols) and the calculated (solid curve) \nintensity maps at the y-position corresponding to the maximum intensity. (c) \nDependence of the focal distance on the wavelength. (d) Dependence of the width of the \nspin-wave beam at the focal  position on the wavelength. Curves in (c) and (d) are guides \nfor the eye.   \n  \n 18  \n \n \n \n \n \n \n \nFigure 5. Dependences of the beam width at the focal position ( squares ) and of the \nfocal distance ( diamonds ) on the length of the antenna calculated for spin waves with  \nthe wavelength of 0.6 m. Dashed c urves are guides for the eye.  \n \n"    },    {        "title": "2205.13802v3.Magnonic_Casimir_Effect_in_Ferrimagnets.pdf",        "content": "Magnonic Casimir Effect in Ferrimagnets\nKouki Nakata1,\u0003and Kei Suzuki1,y\n1Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan\n(Dated: March 2, 2023)\nQuantum fluctuations are the key concepts of quantum mechanics. Quantum fluctuations of\nquantum fields induce a zero-point energy shift under spatial boundary conditions. This quantum\nphenomenon, called the Casimir effect, has been attracting much attention beyond the hierarchy\nof energy scales, ranging from elementary particle physics to condensed matter physics together\nwith photonics. However, the application of the Casimir effect to spintronics has not yet been in-\nvestigated enough, particularly to ferrimagnetic thin films, although yttrium iron garnet (YIG) is\none of the best platforms for spintronics. Here we fill this gap. Using the lattice field theory, we\ninvestigate the Casimir effect induced by quantum fields for magnons in insulating magnets and find\nthat the magnonic Casimir effect can arise not only in antiferromagnets but also in ferrimagnets\nincluding YIG thin films. Our result suggests that YIG, the key ingredient of magnon-based spin-\ntronics, can serve also as a promising platform for manipulating and utilizing Casimir effects, called\nCasimir engineering. Microfabrication technology can control the thickness of thin films and realize\nthe manipulation of the magnonic Casimir effect. Thus, we pave the way for magnonic Casimir\nengineering.\nIntroduction. —Toward efficient transmission of infor-\nmation that goes beyond what is offered by conven-\ntionalelectronics, thelasttwodecadeshaveseenasignifi-\ncant development of magnon-based spintronics [1], called\nmagnonics. The main aim of this research field is to use\nthe quantized spin waves, magnons, as a carrier of in-\nformation in units of the reduced Planck constant ~. A\npromising strategy for this holy grail is to explore insu-\nlating magnets. Thanks to the complete absence of any\nconducting metallic elements, insulating magnets are free\nfrom drawbacks of conventional electronics, such as sub-\nstantial energy loss due to Joule heating. This is the\nadvantage of insulating magnets. Thus, exploring quan-\ntum functionalities of magnons in insulating magnets is\na central task in the field of magnonics.\nQuantum fluctuations of photon fields induce a zero-\npoint energy shift, called the Casimir energy, under spa-\ntial boundary conditions. This Casimir effect is a funda-\nmentalphenomenonofquantumphysics, andtheoriginal\nplatform for the Casimir effect was the photon field [2–\n4], which is described by quantum electrodynamics. The\nconcept can be extended to various fields such as scalar,\nvector, tensor, and spinor fields. Nowadays, the Casimir\neffect has been attracting much attention beyond the hi-\nerarchyofenergyscales, rangingfromelementaryparticle\nphysics to condensed matter physics [5, 6]. As an exam-\nple, see Refs. [7–14] for Casimir effects in magnets [15].\nHowever, the application of the Casimir effect to spin-\ntronics has not yet been studied enough, particularly to\nferrimagnetic thin films (see Fig. 1), although yttrium\niron garnet (YIG) [16] has been playing a central role in\nspintronics.\nHere we fill this gap. In terms of the lattice field the-\nory, we investigate the Casimir effect induced by quan-\n\u0003nakata.koki@jaea.go.jp; equal contribution.\nyk.suzuki.2010@th.phys.titech.ac.jp; equal contribution.\nFIG. 1. Schematic of the ferrimagnetic thin film for the\nmagnonic Casimir effect. Two kinds of magnons (circles) arise\nfrom the alternating structure of up and down spins (arrowed\nlines). Wavy lines represent quantum-mechanical behaviors\nof magnons in the discrete energy.\ntum fields for magnons in insulating magnets and refer to\nit as the magnonic Casimir effect. We study the behav-\nior of the magnonic Casimir effect with a particular fo-\ncus on the thickness dependence of thin films, which can\nbeexperimentallycontrolledbymicrofabricationtechnol-\nogy [17, 18]. Then, we show that the magnonic Casimir\neffect can arise not only in antiferromagnets (AFMs) but\nalso in ferrimagnets with realistic model parameters for\nYIG thin films. Our study indicates that YIG, an ideal\nplatform for magnonics, can serve also as a key ingredi-\nent of Casimir engineering [19] which aims at exploring\nquantum-mechanical functionalities of nanoscale devices.\nAntiferromagnets. —We consider insulating AFMs de-\nscribed by the quantum Heisenberg Hamiltonian which\nhas U (1)symmetry about the quantization axis andarXiv:2205.13802v3  [quant-ph]  1 Mar 20232\nstudy the behavior of the magnonic Casimir effect with\na focus on the thickness dependence. The AFM is a\ntwo-sublattice system, and the ground state has the Néel\nmagnetic order [20]. From the spin-wave theory with the\nBogoliubov transformation, elementary excitations are\ntwo kinds of magnons designated by the index \u001b=\u0006\nhaving the spin angular momentum \u001b~. Owing to the\nU(1)symmetry, the Hamiltonian can be recast into the\ndiagonal form with the magnon energy dispersion for the\nwave number k= (kx;ky;kz)as\u000f\u001b;kwhere the total spin\nangular momentum of magnons is conserved. Two kinds\nof magnons ( \u001b=\u0006) are in degenerate states. Hence, we\nstudy the low-energy magnon dynamics of the insulat-\ning AFM by using the quantum field theory of complex\nscalar fields, i.e., the complex Klein-Gordon field the-\nory [21, 22]. Then, we can see that there exists a zero-\npoint energy [23]. This is the origin of the Casimir effect.\nNote that throughout this study, we focus on clean insu-\nlating magnets and work under the assumption that the\ntotal spin along the quantization direction is conserved\nand thus remains a good quantum number.\nThrough the lattice regularization, the Casimir energy\nECasis defined as the difference between the zero-point\nenergyEint\n0for the continuous energy \u000f\u001b;kand the one\nEsum\n0for the discrete energy \u000f\u001b;k;nwithn2Z. In two-\nsublattice systems, such as AFMs and ferrimagnets (see\nFig. 1), the wave numbers on the lattice are replaced by\n(kja)2!2[1\u0000cos(kja)]in thejdirection for j=x;y;z,\nwhereais the length of a magnetic unit cell. Here, by\ntaking the Brillouin zone (BZ) into account, we set the\nboundary condition for the zaxis in wave number space\nk= (kx;ky;kz)askz!\u0019n=Lz, i.e.,kza!\u0019n=Nz,\nwhereLz:=aNzis the thickness of magnets, Nj2N\nis the number of magnetic unit cells in the jdirection\nforj=x;y;z, andn= 1;2;:::;2NforN2N. The\nnumberofunitcellsonthe xyplaneis 4NxNy,andthatof\nmagneticunitcellsis NxNy. Thus, themagnonicCasimir\nenergy per the number of magnetic unit cells NxNyon\nthe surface for Nz=Nis described as [24–28]\nECas:=Esum\n0(N)\u0000Eint\n0(N); (1a)\nEsum\n0(N) =X\n\u001b=\u0006Z\nBZd2(k?a)\n(2\u0019)2\"\n1\n2\u00101\n22NX\nn=1j\u000f\u001b;k;nj\u0011#\n;(1b)\nEint\n0(N) =X\n\u001b=\u0006Z\nBZd2(k?a)\n(2\u0019)2\"\n1\n2NZ\nBZd(kza)\n2\u0019j\u000f\u001b;kj#\n;\n(1c)\nwherek?:=q\nk2x+k2y,d2(k?a) =d(kxa)d(kya), the\nintegral is over the first BZ, and the factor 1=2arises\nfrom the zero-point energy for the scalar field. Since\nthe constant terms which are independent of the wave\nnumber do not contribute to the Casimir energy, we drop\nthemthroughoutthisstudy. Toseethedependenceofthe\nCasimir energy ECason the thickness of magnets Lz:=\naNz, it is convenient to introduce the rescaled Casimir\n−1−0.8−0.6−0.4−0.2 0\n 0  2  4  6  8 10  12  14  16  18  20Antiferromagnetic thin film\nNzCasimir energy ECas [meV]\nNz: Thickness of magnet in units of aδ=2.0×10−3\nδ=1.0×10−3\nδ=0 −1−0.8−0.6−0.4−0.2 0\n 0  5  10  15  20CCas[3]=Nz3ECas [meV]FIG. 2. Plots of the magnonic Casimir energy ECasand its\ncoefficient (inset) C[3]\nCas=ECas\u0002N3\nzin the AFM [see Eq. (3a)]\nas a function of Nzfor the thickness of magnets Lz=aNz.\nenergyC[b]\nCasin terms of Nb\nzforb2Ras\nC[b]\nCas:=ECas\u0002Nb\nz: (2)\nThen, we refer to C[b]\nCasas the magnonic Casimir coeffi-\ncient in the sense that ECas=C[b]\nCasN\u0000b\nz.\nHere, we consider magnons in AFMs on a cubic lattice\nwith the energy dispersion \u000f\u001b;k=\u000fAFM\n\u001b;k[29]:\n\u000fAFM\n\u001b;k=~!0r\n\u000e+\u0010ka\n2\u00112\n; (3a)\n~!0:=2p\n3JS; (3b)\n\u000e:=3h\u0010K\n6J\u00112\n+ 2\u0010K\n6J\u0011i\n; (3c)\nwherek:=jkj,J >0parametrizes the exchange interac-\ntion between the nearest-neighbor spins of the spin quan-\ntum number S, andK\u00150is the easy-axis anisotropy,\nwhich provides the magnon energy gap and ensures the\nNéel magnetic order. Two kinds of magnons ( \u001b=\u0006)\nare in degenerate states. In the absence of the spin\nanisotropy, K= 0, the energy gap vanishes \u000e= 0, and\nthegaplessmagnonmodebehaveslikearelativisticparti-\nclewiththelinearenergydispersion. Fromtheresultsob-\ntainedinRefs.[30–32],weroughlyestimatethemodelpa-\nrameter values for Cr 2O3, as an example, as follows [29]:\nJ= 15meV,S= 3=2,K= 0:03meV, anda= 0:496 07\nnm. These parameter values provide ~!0\u001877:94meV\nand\u000e\u00182\u000210\u00003.\nFigure 2 shows that the magnonic Casimir effect arises\nin the thin film of the AFM. The magnonic Casimir en-\nergyECasof the magnitude O(10\u00002)meV is induced\nforNz\u00152. Even in the presence of the magnon en-\nergy gap\u000e6= 0, the absolute value amounts to O(10\u00002)\nmeV and decreases as the magnon energy gap increases.\nThus, the magnonic Casimir energy takes a maximum3\nabsolute value in the gapless mode \u000e= 0, where the\nmagnon behaves like a relativistic particle with the lin-\near energy dispersion. We remark that in the case of\nthe gapped magnon modes, the absolute value of the\nmagnonicCasimircoefficient C[3]\nCas=ECas\u0002N3\nzdecreases\nas the thickness of the thin film increases. This behav-\nior is similar to the Casimir effect known for massive\ndegrees of freedom [33, 34]. In the case of the gapless\nmode, the magnonic Casimir coefficient C[3]\nCasapproaches\nasymptotically to a constant value, and the magnonic\nCasimir energy exhibits the behavior of ECas/1=N3\nzas\nthe thickness increases. The asymptotic value of C[3]\nCas\nfor the gapless magnon mode \u000e= 0given in the nu-\nmerical result (see Fig. 2) is estimated approximately as\n(\u0000\u00192=720)\u0002(~!0=2)\u0018\u00000:5341meV from an analyt-\nical calculation. The factor of \u0000\u00192=720is well known\nas the analytic solution for the conventional Casimir\neffect of a massless complex scalar field in continuous\nspace [34]. Thus, although the magnonic Casimir effect\nis realized on the lattice, it is qualitatively and quanti-\ntatively analogous to the continuous counterpart, except\nfora-dependent lattice effects.\nFerrimagnets. —We develop the study of AFMs into\nferrimagnets where the ground state has an alternating\nstructure of up and down spins on a cubic lattice (see\nFig. 1). In contrast to AFMs, the spin quantum num-\nber on the two-sublattice is different from each other in\nferrimagnets. Hence, the degeneracy for two kinds of\nmagnons (\u001b=\u0006) is intrinsically lifted. In ferrimagnetic\nthinfilms, dipolarinteractionsduetothenonzeromagne-\ntization play a key role. Still, at low temperatures where\nthe magnon-magnon interaction of the fourth order in\nmagnon operators is negligibly small [35], the number of\nmagnons and the total spin angular momentum are con-\nserved, and the Hamiltonian for the ferrimagnetic thin\nfilm can be diagonalized with the magnon energy disper-\nsion\u000f\u001b;k=\u000fferri\n\u001b;k.\nHere, we consider magnons in the thin film of clean\ninsulating ferrimagnets on a cubic lattice subjected to\nin-plane magnetic fields at such low temperatures. Still,\ndue to the competition between dipolar and exchange\ninteractions, the minimum energy point shifts from the\nzero mode of magnons, k= 0, to a finite wave number\nmode which is characterized by the thickness of the thin\nfilmLz=aNz(see Fig. 1).\nThe magnon energy dispersion along the in-plane di-\nrection is provided in Refs. [36, 37], whereas the disper-\nsion along the zaxis in the thin film has not yet been\nestablished [38]. Hence, taking into account the compe-\ntition between dipolar and exchange interactions in the\nthin film, we phenomenologically assume the behavior\nthat the power of kz,l2R, approaches asymptotically\ntol= 2in the bulk limit, whereas it slightly differs from\nl= 2as long as we consider the thin film (see Fig. 1).\nUsing this assumption and Refs. [36, 37], the magnon\n−120−100−80−60−40−20 0 20\n 0  2  4  6  8 10  12  14  16  18  20Ferrimagnetic thin film\nNzCasimir energy ECas [µeV]\nNz: Thickness of magnet in units of al=2.1\nl=2.0l=1.99\nl=1.9\n−100−50 0 50 100\n 0  5  10  15  20CCas[l]=NzlECas [µeV]FIG. 3. Plots of the magnonic Casimir energy ECasand its\ncoefficient (inset) C[l]\nCas=ECas\u0002Nl\nzforl= 2:1,l= 2:0,l=\n1:99, and l= 1:9in the ferrimagnetic thin film [see Eq. (4a)]\nas a function of Nzfor the thickness of magnets Lz=aNz\nwith the model parameter values for YIG of Dz=D.\nenergy dispersions are\n\u000fferri\n\u001b;k=r\n\u001bH0+ \u0001\u001b+D\na2(k?a)2+Dz\na2(jkzja)l(4a)\n\u0002r\n\u001bH0+ \u0001\u001b+D\na2(k?a)2+Dz\na2(jkzja)l+\u001b~!MFk;\nFk:=Pk?(1\u0000Pk?)\u001b~!M\n\u001bH0+ \u0001\u001b+D\na2(k?a)2+Dz\na2(jkzja)l\u0010kx\nk?\u00112\n+1\u0000Pk?\u0010ky\nk?\u00112\n; (4b)\nPk?:=1\u00001\u0000e\u0000k?Lz\nk?Lz; (4c)\nwhere the external magnetic field is applied along the y\naxis (see Fig. 1) and \u001bH0represents the resulting Zee-\nman energy, \u0001\u001b\u00150is the (intrinsic) magnon energy\ngap in ferrimagnets, D(z)>0is the spin stiffness con-\nstant,k?:=q\nk2x+k2y,~!M:= 4\u0019\rMswith the sat-\nurated magnetization density Msand the gyromagnetic\nratio\r, and the termFkis responsible for the shift of\nthe minimum energy point from the zero mode to a finite\nwave number mode due to the competition between dipo-\nlar and exchange interactions in the ferrimagnetic thin\nfilm: The first term of Fk[see Eq. (4b)] reproduces the\nDamon-Eshbach magnetostatic surface mode [39], and\nthe last term reproduces the backward volume magneto-\nstatic mode [39].\nFrom the results obtained in Refs. [40–42], the model\nparameter values for YIG thin films are estimated as fol-\nlows:D=a2\u00183:376 45meV [35] with a= 1:2376nm [43],\nH0\u00188:103 73\u0016eV [35], and ~!M\u001820:3369\u0016eV [35].\nThen, we estimate the magnon energy gap \u0001\u001bas [44]\n\u0001\u001b=\u0000\u0000\u0001\u001b=+\u001839:848 81meV with \u0001\u001b=+\u00182:131 91\nmeV and \u0001\u001b=\u0000\u001841:980 72meV, which satisfy the con-4\ndition \u0001\u001b\u001d~!M. In this condition, we study the low-\nenergy magnon dynamics of the ferrimagnetic thin film\nby using the quantum field theory of real scalar fields,\ni.e., the real Klein-Gordon field theory [21, 22]. Then,\nwe can see that there exists a zero-point energy [35].\nThe magnonic Casimir energy through the lattice reg-\nularization is given as Eq. (1a). We remark that the\nzero-point energy arises from quantum fluctuations and\ndoes exist even at zero temperature. The zero-point en-\nergy defined at zero temperature does not depend on the\nBose-distribution function [Eqs. (1b) and (1c)]. Hence,\nnot only the low-energy mode ( \u001b= +) but also the high-\nenergy mode ( \u001b=\u0000) contribute [45] to the magnonic\nCasimir energy.\nUnderthephenomenologicalassumptionthatthevalue\nofl[see Eq. (4a)] approaches asymptotically to l= 2in\nthe bulk limit, in this work focusing on the thin film,\nwe study the behavior of the magnonic Casimir effect\nby changing the value slightly from l= 2. As exam-\nples, we consider the cases of l= 2:1,l= 1:99, and\nl= 1:9. Figure 3 shows that the magnonic Casimir ef-\nfect arises in the ferrimagnetic thin film. There is the\nmagnonic Casimir energy ECasof the magnitude O(1),\nO(10), andO(10)\u0016eV forl= 1:99,l= 1:9, andl= 2:1,\nrespectively, in Nz\u00152. As the thickness increases, the\nmagnonic Casimir coefficient C[l]\nCasapproaches asymptot-\nically to a constant value, and the magnonic Casimir en-\nergy exhibits the behavior of ECas/1=Nl\nz. We also\nfind from Fig. 3 that the sign of the magnonic Casimir\ncoefficient and energy for l= 2:1is positive C[2:1]\nCas =\nECas\u0002N2:1\nz>0inNz\u00152, whereas that for l= 1:9is\nnegativeC[1:9]\nCas=ECas\u0002N1:9\nz<0. This means that the\nCasimir force works in the opposite direction.\nNote that even if l= 2, the magnonic Casimir effect\narises in the ferrimagnetic thin film. We have numer-\nically confirmed that although the value is small, there\ndoes exist the magnonic Casimir energy of the magnitude\njECasj\u0014O(0:1)neV forl= 2. This strong suppression of\nthe Casimir energy is a general property of the Casimir\neffect for quadratic dispersions on the lattice [28], and\nthe survival values originate from the dipolar interaction\nin the ferrimagnetic thin film.\nWe remark that as long as we consider thin films,\nthe value of Dzcan differ from D. Even in that case,\nthe magnonic Casimir effect arises. When the value\nofDzchanges from Dto0:8Das an example, the\nmagnonic Casimir energy ECasincreases approximately\nby0:8times. For more details about its dependence on\nthe parameters Dzandl, see the Supplemental Material.\nProposal for experimental observation. —Themagnonic\nCasimir energy of the ferrimagnetic thin film depends\nstronglyonexternalmagneticfieldsthroughZeemancou-\npling as in Eq. (4a) and contributes to magnetization of\nmagnets, whereas the photon and phonon Casimir ef-\nfects [46] do not usually. On the other hand, in the\npresenceofmagnetostriction[47–50], thephononCasimir\neffect is influenced by magnetostriction, and its correc-\ntion for the phonon Casimir energy depends on magneticfields and contributes to magnetization. However, such a\ncontribution to magnetization from the phonon Casimir\neffect should be negligibly small by the factor of 10\u00006\ncompared with that from the magnonic Casimir effect\nof ferrimagnets because the magnetostriction constant\n(i.e., the correction for the lattice constant) is known to\nbe10\u00006for YIG [47, 48]. Hence, even in the presence\nof magnetostriction, the magnonic Casimir effect can be\ndistinguished from the others. Thus, we expect that our\ntheoreticalprediction, themagnonicCasimireffectinfer-\nrimagnets, can be experimentally observed through mea-\nsurement of magnetization and its film thickness depen-\ndence. For more details, see the Supplemental Material.\nForobservation, afewcommentsareinorder. First, we\nremark on edge/surface magnon modes. The magnonic\nCasimir effect in our setup (see the thin film of Fig. 1)\nis induced by magnon fields with wave numbers kzdis-\ncretized by small Nz, and its necessary condition is a\nkz-dependent dispersion relation under the discretization\nofkz. Throughout this study, we consider thin films of\nNz\u001cNx;Ny. Even if additional edge/surface magnon\nmodesexistaswellastheDamon-Eshbachmagnetostatic\nsurface mode and the backward volume magnetostatic\nmode [see Eq. (4b)], they are confined only on the x-\nyplane. Then, their wave number in the zdirection\nis always zero, i.e., kz= 0, and its energy dispersion\nrelation is independent of kz. Since a kz-independent\ndispersion relation cannot induce the Casimir effect, the\nedge/surface modes cannot contribute to the magnonic\nCasimir effect. In this sense, our magnonic Casimir effect\nis not affected by the existence of edge/surface magnon\nmodes.\nNote that details of the edge condition, such as the\npresence or absence of disorder, may change the bound-\nary condition for the wave function of magnons, but the\nexistence of the magnonic Casimir effect remains un-\nchanged. Even if there is a change in the spectrum near\nthe edge, the magnonic Casimir effect is little influenced\nas long as one does not assume an ultrathin film such\nasNz= 1;2;3. In this sense, we expect that the fol-\nlowing size of thin films is appropriate for observation of\nour prediction: Nz\u001810, i.e., the film thickness of YIG\nisLz=aNz\u001812:376nm. Note that microfabrication\ntechnology [17, 18] can control the thickness of thin films\nand realize the manipulation of the magnonic Casimir\neffect.\nNext, we remark on the magnon band structure.\nSince our magnonic Casimir effect is induced by the kz-\ndependent dispersion, its Casimir energy of ferrimagnets\nis mainly characterized by the Dz-term in Eq. (4a), i.e.,\nDz(jkzja)l. Hence, we have investigated its dependence\non bothlandDz(see Fig. 3 and the Supplemental Mate-\nrial). Even if the magnon band structure is affected due\nto some reasons, the magnonic Casimir effect of ferrimag-\nnets is little influenced by other details of the magnon\nband structure except for landDz.\nLastly, we remark on thermal effects. At nonzero tem-\nperature, thermal contributions to the Helmholtz free en-5\nergy arise. However, at low temperatures compared to\n\u000f\u001b;k[51], the thermal contribution is exponentially sup-\npressed due to the Boltzmann factor and becomes negli-\ngibly small. Hence, at such low temperatures, the contri-\nbution of the magnonic Casimir energy given as Eq. (1a)\nis dominant.\nConclusion. —We have shown that the magnonic\nCasimir effect can arise not only in antiferromagnets but\nalso in ferrimagnets with realistic model parameters for\nYIG. Since the lifetime of magnons in YIG thin films is\nthelongestamongknownmaterials, andmagnonsexhibit\nlong-distance transport over centimeter distances [52],\nYIG is the key ingredient of magnonics [1, 16], which\nhas already realized the magnon transistor [53]. Ourresult suggests that YIG can serve also as a promis-\ning platform for Casimir engineering [19]: Because the\nmagnonic Casimir effect contributes to the internal pres-\nsure of thin films, it will provide the new principles of\nnanoscale devices such as highly sensitive pressure sen-\nsors and magnon transistors. 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Volovik, arXiv:1003.4889 .7\nSupplemental Material\nIn this Supplemental Material, first, we provide some\ndetails about the dependence of the magnonic Casimir\neffect on the parameters landDzin ferrimagnetic thin\nfilms. Next, we remark on its film thickness dependence.\nThen, we provide another point of view for its robust-\nness against disorder effects. Lastly, we comment on the\ndistinction between the Casimir effect and the thermal\nCasimir effect.\nI. THE PARAMETER l- AND Dz-DEPENDENCE\nIn the main text, we have studied the magnonic\nCasimirenergy ECasandthecoefficient C[l]\nCas=ECas\u0002Nl\nz\nforl= 2:1,l= 2:0,l= 1:99, andl= 1:9in the ferrimag-\nnetic thin film by using the model parameter values for\nYIG with fixed Dz=D. Here, we provide more details\nabout its dependence on the parameters landDz.\nFirst, we consider the cases of l= 1:5andl= 1:0with\nsettingDz=D. Figure S1 shows that the magnonic\nCasimir effect still arises in the ferrimagnetic thin film.\nThere is the magnonic Casimir energy ECasof the mag-\nnitudeO(10\u00002)meV,O(10\u00001)meV, andO(10\u00001)meV\nforl= 1:9,l= 1:5, andl= 1:0, respectively, in Nz\u00152.\nAs the value of ldecreases from l= 2and approaches\ntol= 1, the magnitude of the magnonic Casimir energy\nincreases. Note that it amounts to O(10\u00001)meV even in\nNz=O(10)forl= 1:0. As the thickness increases, the\nmagnonic Casimir coefficient C[l]\nCasapproaches asymptot-\nically to a constant value and the magnonic Casimir en-\nergy exhibits the behavior of ECas/1=Nl\nz.\nThen, we consider the cases of Dz=D= 0:3,Dz=D=\n0:5, andDz=D= 0:8by fixingl= 1:99. Figure S2 shows\nthat the magnonic Casimir effect still arises in the ferri-\nmagnetic thin film. When the value of Dzchanges from\nDto0:8Das an example, the magnonic Casimir energy\nECasincreases approximately by 0:8times. Thus, the\nvalue ofECasis approximately proportional to Dz.\nII. REMARKS ON THE THICKNESS\nDEPENDENCE OF MAGNETIZATION\nIn the main text, we have remarked that our predic-\ntion, the magnonic Casimir effect in ferrimagnets, can\nbe observed through measurement of magnetization and\nits film thickness dependence. Here, we add an expla-\nnation about it. At zero temperature, the Helmholtz\nfree energy of magnon fields in thin films (i.e., the\nsum over discrete kz) isEsum\n0(Nz)NxNy, and that un-\nder the bulk approximation (i.e., the integral with re-\nspect to continuous kz) isEint\n0(Nz)NxNy[see Eqs. (1a)-\n(1c)]. The difference between them is characterized by\nthe magnonic Casimir energy ECasasEsum\n0(Nz)NxNy=\nEint\n0(Nz)NxNy+ECasNxNy, where the magnon energy\n−1−0.8−0.6−0.4−0.2 0\n 0  2  4  6  8 10  12  14  16  18  20Ferrimagnet\nNzCasimir energy ECas [meV]\nNz: Thickness of magnet in units of al=2.0\nl=1.9l=1.5\nl=1.0\n−1−0.8−0.6−0.4−0.2 0\n 0  5  10  15  20CCas[l]=NzlECas  [meV]FIG. S1. Plots of the magnonic Casimir energy ECasand\nthe coefficient (inset) C[l]\nCas=ECas\u0002Nl\nzforl= 2:0,l= 1:9,\nl= 1:5, and l= 1:0intheferrimagneticthinfilmasafunction\nofNzfor the thickness of magnets Lz=aNzunder the model\nparameter values for YIG with fixed Dz=D.\n−14−12−10−8−6−4−2 0\n 0  2  4  6  8 10  12  14  16  18  20Ferrimagnet\n(l=1.99)\nNzCasimir energy ECas [µeV]\nNz: Thickness of magnet in units of aDz=0.3D\nDz=0.5DDz=0.8D\nDz=D\n−14−12−10−8−6−4−2 0\n 0  5  10  15  20CCas[l]=NzlECas [µeV]\nFIG. S2. Plots of the magnonic Casimir energy ECasand the\ncoefficient (inset) C[l]\nCas=ECas\u0002Nl\nzforDz=D= 0:3,Dz=D=\n0:5,Dz=D= 0:8, and Dz=D= 1:0in the ferrimagnetic thin\nfilmasafunctionof Nzforthethicknessofmagnets Lz=aNz\nunder the model parameter values for YIG with l= 1:99.\ndispersion of ferrimagnets (i.e., magnets including dipo-\nlar interactions) is Eq. (4a). Note that the magnetic-\nfield derivative (i.e., H0-derivative) of the Helmholtz free\nenergy is magnetization. Then, magnetization of thin\nfilms consists of two parts: The bulk component and the\nmagnonic Casimir energy. Since Eint\n0(Nz)/Nz, whereas\nECas/1=(Nz)l, magnetization of thin films exhibits a\ndifferentNz-dependence from the bulk component, and\nits difference is characterized by the magnonic Casimir\nenergy. In other words, magnetization of thin films ex-\nhibits a different film thickness dependence from the bulk\ncomponent due to the magnonic Casimir effect. Hence,\nour prediction, the magnonic Casimir effect in ferrimag-8\nnetic thin films (i.e., magnetic thin films including dipo-\nlar interactions), can be observed through measurement\nof magnetization and its film thickness dependence.\nNote that if dipolar interactions are relevant also in\nantiferromagnets, its low-energy magnon dynamics is es-\nsentially described by Eq. (4a) given for ferrimagnets.\nThe only difference is that the spin quantum number for\neachsublatticeisidenticalinantiferromagnets, wherethe\n(intrinsic) magnon energy gap for each mode \u001b=\u0006can\nbe identical \u0001\u001b=+= \u0001\u001b=\u0000[see Eq. (4a)]. In this sense,\nitsCasimireffectexhibitsqualitativelythesamebehavior\nas Fig. 3.\nIII. REMARKS ON DISORDER EFFECTS\nIn the main text, we have remarked that details of the\nedge condition, such as the presence or absence of dis-\norder, may change the boundary condition for the wave\nfunction of magnons, but the existence of the magnonic\nCasimir effect remains unchanged. Here, we add a com-\nment on disorder effects. Since the magnonic Casimir\nenergy does not depend on the Bose-distribution func-\ntion [see Eqs. (1a)-(1c)], not only the low-energy magnon\nmode (\u001b= +) but also its high-energy mode ( \u001b=\u0000) in\nferrimagnets contributes to the magnonic Casimir effect.\nTherefore, it can be expected that as long as disorder\neffects on the bulk are weak enough that the high-energy\nmode is little influenced, the existence of the magnonic\nCasimir effect in ferrimagnets remains unchanged.\nIV. THERMAL CASIMIR EFFECT\nIn the main text, we have explained that thermal con-\ntributions to the Helmholtz free energy arise at nonzero\ntemperature. Here, we add a remark on it. Although its\nthermal contribution is called the “thermal Casimir en-\nergy”, there is a crucial distinction between the Casimir\neffect and the thermal Casimir effect: The zero-point en-\nergy, which is the key concept of quantum mechanics and\nplays a crucial role in the Casimir effect, is absent in the\nthermal Casimir effect. It should be noted that the zero-\npoint energy is one of the most striking phenomenon of\nquantum mechanics in the sense that there are no clas-\nsical analogs. The Casimir effect arises from the zero-\npoint energy due to quantum fluctuations and is not af-\nfected by temperatures, whereas the thermal Casimir ef-\nfect arises from thermal fluctuations and is exponentially\nsuppressed at low temperatures. The thermal Casimir\neffect vanishes at zero temperature, whereas the Casimir\neffect does exist even at zero temperature. Thus, there is\na significant distinction between the Casimir effect and\nthe thermal Casimir effect."    },    {        "title": "2302.10517v2.Aluminium_substituted_yttrium_iron_garnet_thin_films_with_reduced_Curie_temperature.pdf",        "content": "arXiv:2302.10517v2  [cond-mat.mtrl-sci]  6 Jul 2023Aluminium substituted yttrium iron garnet thin films with re duced Curie temperature\nD. Scheffler,1O. Steuer,2, 3S. Zhou,3L. Siegl,4S.T.B. Goennenwein,4and M. Lammel4\n1Institute of Solid State and Materials Physics, Technische Universitaet Dresden, 01062 Dresden, Germany\n2Institute of Materials Science, Technische Universitaet D resden, 01069 Dresden, Germany\n3Institute of Ion Beam Physics and Materials Research,\nHelmholtz Zentrum Dresden-Rossendorf, 01328 Dresden, Ger many\n4Department of Physics, University of Konstanz, 78457 Konst anz, Germany\n(Dated: July 7, 2023)\nMagnetic garnets such as yttrium iron garnet (Y 3Fe5O12, YIG) are widely used in spintronic and magnonic\ndevices. Their magnetic and magneto-optical properties ca n be modified over a wide range by tailoring their\nchemical composition. Here, we report the successful growt h of Al-substituted yttrium iron garnet (YAlIG) thin\nfilms via radio frequency sputtering in combination with an e x situ annealing step. Upon selecting appropriate\nprocess parameters, we obtain highly crystalline YAlIG film s with different Al3+substitution levels on both,\nsingle crystalline Y 3Al5O12(YAG) and Gd 3Ga5O12(GGG) substrates. With increasing Al3+substitution levels,\nwe observe a reduction of the saturation magnetisation as we ll as a systematic decrease of the magnetic ordering\ntemperature to values well below room temperature. YAlIG th in films thus provide an interesting material\nplatform for spintronic and magnonic experiments in differ ent magnetic phases.\nI. INTRODUCTION\nIn the field of insulator spintronics, magnetic materials ar e\nusually used in their ordered phase, i.e. well below their re -\nspective magnetic ordering temperature [ 1–9]. Therefore, also\nalmost all spin transport experiments have been performed\nin the ferro-, ferri- or antiferromagnetic phase [ 1–9]. More-\nover, a few attempts to study spin transport in the param-\nagnetic phase have also been made, either by using param-\nagnetic materials [ 10–12], or by performing the experiments\nat elevated temperatures above the ordering temperature [ 13–\n15]. A comprehensive understanding of spin transport in the\nparamagnetic phase nevertheless is lacking, not to mention its\nevolution across the magnetic phase transition. This makes\nsystematic experiments across the magnetic phase transiti on,\ni.e. from the ferromagnetic to the paramagnetic phase, high ly\ndesirable to establish a robust understanding of spin trans fer\nas well as magnon generation and propagation processes. In\naddition, magnetic fluctuations are enhanced around the pha se\ntransition, allowing to study the impact of such fluctuation s on\nspin transport.\nThe prototype material for spin transport studies [ 2–7,13,\n16] is the magnetic insulator yttrium iron garnet Y 3Fe5O12\n(YIG), as it has a very low Gilbert damping parameter [ 17–\n20] and a small coercive field [ 17,20,21]. This makes YIG an\nideal material for magnon transport experiments and spin Ha ll\neffect driven spin transport studies [ 2–7,13,16]. However,\nYIG has proven problematic for spin transport experiments\nacross the magnetic phase transition due to its relatively h igh\nCurie temperature of Tc=559 K [ 22]. Experiments at temper-\natures close to or above Tcare hampered by a finite thermally\ninduced electrical conductivity of YIG [ 23]. In addition, a\nsignificant interdiffusion of Pt in YIG/Pt heterostructure s has\nbeen reported for T>470 K, which leads to a significant de-\nterioration of the interface and the magnetic properties of YIG\n[13].\nOne way to circumvent such high temperature issues is to\nlower Tc. As the magnetic properties of YIG are defined by the\nFe3+ions, a lower Tccan be achieved by substituting the Fe3+ions with non-magnetic ions. This has been successfully ac-\ncomplished almost exclusively for bulk crystals by substit ut-\ning with, amongst others, Al3+ions [ 24–31]. By increasing\nthe Al3+substitution, both Curie temperature Tcand satura-\ntion magnetisation Msare reduced. Sufficiently high substi-\ntution levels xallow to tailor Tcof the resulting yttrium alu-\nminium iron garnet (Y 3AlxFe5−xO12, YAlIG) from the value\nof pure YIG down to values well below room temperature. It\nhas been shown that the desired magnetic properties of YIG\nsuch as low magnetic damping and low coercivity are con-\nserved upon substitution with Al3+ions [ 24,32]. However\nthe studies of YAlIG are almost exclusively limited to bulk\nand powder materials [ 24–31], while thin films are required\nfor typical spintronic devices. So far, only YAlIG thin films\nup to x=0.783 were investigated [ 33], which is not sufficient\nto reduce Tcdown to room temperature.\nHere, we report the successful fabrication of YAlIG thin\nfilms with different Al3+substitution levels 1 .5≤x≤2 by\nradio-frequency (RF) sputtering deposition and a subseque nt\nannealing step. We evaluate the influence of the Al3+substi-\ntution level on the structural and magnetic properties of ou r\nthin films. To this end, we study the impact of different sub-\nstrate materials and annealing temperatures on the structu ral\nand magnetic quality. X-ray diffraction (XRD) is utilised t o\nshow that our sputtered and post-annealed YAlIG thin films\nhave a high crystal quality. SQUID vibrating sample mag-\nnetometry demonstrates the successful reduction of Tcto val-\nues well below room temperature, as well as coercive fields\ncomparable to pure YIG films. We utilise broadband ferro-\nmagnetic resonance measurements to investigate the dynami c\nmagnetic properties and the magnetic anisotropy of our YAlI G\nthin films.\nII. EXPERIMENTAL METHODS\nThe YAlIG films were grown on commercially available\n(111)-oriented single crystalline yttrium aluminium garn et\n(Y3Al5O12,YAG, cubic lattice parameter aYAG=1.201 nm)2\nTABLE I: Summary of nominal Al3+substitution level x, annealing temperature TA, thickness t, lattice parameters a(length) and α(angle), saturation\nmagnetisation Msat 20 K, Curie temperature Tcand the upper limit for the Curie temperature Tc,maxfor the Y 3AlxFe5−xO12films.\nsubstrate x T A(°C) t(nm) aYAlIG (nm) α(deg) Ms@ 20 K (kAm−1)Tc(K) Tc,max(K)\nYAG 1.5 as-dep. 50 amorphous amorphous <5 – –\nYAG 1.5 700 45 1.2276 90 41 270 300\nYAG 1.5 800 45 1.2276 90 41 262 300\nYAG 1.75 800 37 1.2266 90 16 156 190\nYAG 2 800 45 1.2205 90 10 102 140\nGGG 1.5 as-dep. 50 amorphous amorphous – – –\nGGG 1.5 700 45 1.2355 90.46 – – –\nGGG 1.5 800 45 1.2348 90.49 – – –\nand gadolinium gallium garnet (Gd 3Ga5O12, GGG, aGGG=\n1.240 nm) substrates by RF-sputtering in a UHV system (base\npressure below 1 ×10−8mbar). Three different 2 inch diam-\neter targets with Al3+substitution levels of x=1.5,1.75 and\n2 were used. These substitution levels were selected to obta in\nthin films with a Curie temperature Tcclose to or below room\ntemperature based on the existing literature for bulk YAlIG\n[25,28,31].\nPrior to deposition, the substrates were cleaned subse-\nquently in acetone, isopropanol and distilled water in a ul-\ntrasonic bath for 10 min each. After transferring the sub-\nstrates into the sputtering chamber, they were annealed in s itu\nat 200 °C for 1 h in vacuum to eliminate residual adsorbed wa-\nter. The YAlIG films were deposited at room temperature.\nThe thermal energy for the crystallisation of the YAlIG thin\nfilms was provided by an ex situ post-annealing in a two-zone\nfurnace. The detailed deposition and annealing parameters\nare summarised in table II. Temperatures of 700 °C and above\nwere reported to be sufficient for a complete crystallisatio n of\nYIG thin films of comparable thickness [ 34–36]. The partial\noxygen pressure was applied to counteract oxygen losses of\nthe films during the annealing [ 35]. All samples examined in\nthis study are summarised in table I.\nTABLE II: Summary of the deposition and annealing parameters for the\nfabrication of the YAlIG thin films.\nsputter deposition target-substrate distance 206 mm\nsample holder rotation rate 5 rpm\npressure (Ar) 4 .3×10−3mbar\nAr flow rate 20 sccm\nsputtering power 75 W\nannealing heating rate 15 K/min\npressure (O 2) 3 mbar\nannealing temperature 700 −800 °C\nannealing time 240 min\nVarious X-ray diffraction measurements (symmetrical 2 θ-\nωscan, ω-scan, reciprocal space mapping) were performed\nusing Cu- Kα1radiation ( λ=0.15406nm) to evaluate the\nstructural properties of the film. X-ray reflectivity was use d\nto measure the film thickness.\nRutherford backscattering spectroscopy random (RBS/R)\nexperiments at a backscattering angle of 170° were performe d\nto investigate the chemical composition of selected YAlIG\nfilms, using a 1.7 keV He+ion beam with a diameter of about1 mm generated by a 2 MV van-de-Graaff accelerator. The\nbackscattering angle was 170°. The measured data was fitted\nto a calculated spectra using the SIMNRA software [ 37]. Due\nto the low mass of oxygen, only the elemental ratios between\nFe, Al and Y are used for quantitative analysis of the chemica l\ncomposition.\nThe static magnetic properties were investigated by su-\nperconducting quantum interference device vibrating samp le\nmagnetometry (SQUID VSM). For an external magnetic field\norientation within the sample plane during the measurement\n(in-plane, ip), the sample was mounted onto a quartz rod; for\nan orientation of the external magnetic field parallel to the sur-\nface normal (out-of-plane, oop) the sample was mounted be-\ntween plastic straws. The magnetic moment of the sample was\nrecorded in dependence of the external field (maximum field\nµ0Hmax=±3 T, minimal field steps of ∆µ0H=2 mT) and\nthe temperature (2 K to 360 K, temperature steps ∆T=2 K).\nNote that due to the large paramagnetic background signal of\nthe GGG substrate and the correspondingly larger backgroun d\nsubtraction error, we focused on the YAlIG thin films on YAG\nsubstrates for the magnetisation characterization. For al l mag-\nnetometry data shown in this work, the magnetic contributio ns\nof the YAG substrate were subtracted. For temperature de-\npendent measurements, this includes the diamagnetic contr i-\nbution and an additional paramagnetic contribution from co n-\ntaminations within the YAG substrate, which combined where\nfitted to msub=a/(T+b)+cwith a,bandcas fitting param-\neters. For field dependent measurements, the substrate cont ri-\nbution is defined by the linear slope at high fields.\nBroadband ferromagnetic resonance (bb-FMR) measure-\nments were performed to investigate the dynamic magnetic\nproperties and the magnetic anisotropy of two representa-\ntive samples, both with similar nominal composition and an-\nnealing temperature( x=1.5,TA=800 °C), but on different\nsubstrates. Therefore, the samples are placed on a copla-\nnar waveguide (CPW) with the thin film facing towards the\ncenter conductor of the CPW. The CPW is connected to two\nports of a vector network analyser (VNA, PNA N5225B from\nKeysight). To control the temperature of the sample and to\napply an external magnetic field, the CPW assembly was in-\nserted into a 3D vector magnet cryostat. Using the VNA, we\nrecord the complex microwave transmission parameter S21as\na function of frequency f=1−30 GHz at several fixed static\nmagnetic fields H. The magnetic field was applied either par-\nallel (ip) or perpendicular (oop) to the sample plane. The me a-3\nsured S21was corrected for the frequency dependent and mag-\nnetic field dependent background (see moving field reference\nmethod in [ 38]) and fitted by a complex Lorentzian function to\nextract the FMR resonance frequency fresand linewidth ∆f.\nIII. RESULTS\nStructural characterisation\nFIG. 1: XRD-analysis of YAG/YAlIG films: (a)Symmetric 2 θ-ωscan for\ndifferent annealing conditions and compositions. The two g rey dashed lines\ndenote the range of values reported in literature for x=1.5 [25,30,39–42].\nThe inset shows the ωscan of the YAlIG film annealed at 800°C, x=1.5.\nThe dashed lines indicate the numerical fit of the data to two i ndependent\nPseudo-V oigt functions. (b)RSM of the asymmetric (246)-peak of a sample\nannealed at 800°C, x=1.5. The white dashed line indicates the peak\nposition for a growth of fully relaxed YAlIG and the red dashe d lines\nindicates the peak position for a growth of fully strained YA lIG.\nX-ray diffraction measurements carried out on YAlIG thin\nfilms with different compositions revealed a high crystalli ne\nquality of the respective YAlIG thin films after the anneal-\ning step. The symmetrical 2 θ-ωXRD scans of YAlIG thin\nfilms with different substitution levels grown on YAG are\nshown in Figure 1a. No diffraction peak that can be ascribedto the YAlIG thin films is observed prior to annealing, sup-\nporting the notion that the thermal energy during depositio n\nis not sufficient for crystallisation. The necessary therma l\nenergy is provided by the subsequent annealing step, which\nleads to the crystallisation of the YAlIG layer. For all chem -\nical compositions the crystalline nature of the annealed th in\nfilms is validated by the diffraction peak which corresponds\nto YAlIG (444) planes. For substituted yttrium iron garnet,\nthe lattice parameter is changing according to the size of th e\nsubstitute ion [ 39]. As Al3+is smaller than Fe3+, the lattice\nparameters of the YAlIG thin films (cp. table I) are reduced\ncompared to YIG ( aYIG=1.2376 nm [ 39]), as expected. Like-\nwise the lattice parameter is decreasing towards higher sub sti-\ntution levels, which becomes apparent by the shift towards\nhigher diffraction angles. The lattice parameter falls wel l\nwithin the range given by the literature (1 .2235 nm≤aYAlIG≤\n1.2276 nm) [ 25,30,39–42], which is illustrated by the grey\ndashed lines in Fig. 1a for x=1.5. Comparing the XRD pat-\nterns does not indicate any influence of the annealing temper a-\nture on the maximum peak position of thin films with the same\nAl3+substitution level, aside from a small change in the peak\nintensity. This might indicate a more complete crystallisa tion\nat higher temperatures.\nA slight peak asymmetry towards lower diffraction an-\ngles can be observed independently from the annealing con-\ndition or chemical composition which might originate from\na partially strained film or a gradient of chemical compo-\nsition due to diffusion of Al3+from the substrate into the\nfilm. Since the lattice mismatch εbetween the YAG substrate\nand our YAlIG thin film is relatively high ( ε=aYAlIG−asub\nasub=\n2.23%,2.14%,1.63% for x=1.5,1.75 and 2)[ 43], we antic-\nipate a relaxed growth for the given film thickness. This is\nconfirmed by reciprocal space mapping (RSM, cp. Fig. 1b),\nas the asymmetric YAlIG (246) peak is located on the cal-\nculated line for a fully relaxed crystal. Although the latti ce\nmismatch is reduced at higher substitution levels, no chang e\nof the described growth mechanism is found. Further struc-\ntural investigation via RSM to explain the peak asymmetry\nobserved in the 2 θ-ωXRD scan are not possible due to the\nlow intensity of the asymmetric (246) peak. The ω-scan of\na YAlIG film annealed at TA=800 °C, given in the inset of\nFig. 1a, shows a superposition of two diffraction peaks with\ndifferent full width at half maximum (FWHM) values centred\naround the same diffraction angle. The FWHM of each peak\nwas determined by numerically fitting of the data to the sum\nof two Pseudo-V oigt functions as 0 .04◦and 0.46◦. A simi-\nlar behaviour has been reported for different heteroepitax ially\nrelaxed thin film materials [ 44–48]. The sharp peak is asso-\nciated with areas of higher structural order, i.e. an epitax ially\nstrained crystalline layer close to the substrate/film inte rface.\nUpon further crystallisation, the influence of the substrat e de-\ncreases and the film continues to grow relaxed with the in-\ncorporation of defects like dislocations or a higher mosaic ity,\nwhich is ascribed to the broad peak feature. As 91% of the to-\ntal peak area contributes to the broad peak, we conclude that\nthe majority of the film features a higher degree of defects.\nTo evaluate the effect of a smaller lattice mismatch, YAlIG\nsamples with x=1.5 were also prepared on GGG substrates.4\nFIG. 2: XRD-analysis of GGG/YAlIG films with x=1.5:(a)Symmetric\n2θ-ωscan of films annealed at different temperatures. The ⋆indicate\nYAlIG (444) peak, the arrows indicate the first Laue oscillat ion of YAlIG.\nThe inset shows the ωscan of the YAlIG film annealed at 800°C. The\ndashed line indicates the numerical fit of the data to a Pseudo -V oigt function.\n(b)RSM of the asymmetric (246)-peak of a sample annealed at 800° C,\nwhere the white dashed line indicates the peak position for a growth of fully\nrelaxed YAlIG and the red dashed lines indicate the peak posi tion for a\ngrowth of fully strained YAlIG.\nThis reduces the lattice mismatch down to −1%. As for the\nsamples grown on YAG, a post-annealing step is necessary\nfor the crystallisation of YAlIG (444) (cp. Fig. 2a). However,\nthe peak position of YAlIG (444) in the symmetrical 2 θ-ω\nXRD scan is shifted towards higher diffraction angles com-\npared to the YAlIG samples grown on YAG, visualised by the\ngrey dashed line. This reduction of the distance between the\nYAlIG (111) lattice planes indicates a fully strained growt h\nof YAlIG on GGG due to the reduced lattice mismatch while\nkeeping the film thickness at 45 nm. This is confirmed by\nRSM analysis (cp. Fig. 2b), as the asymmetric YAlIG (246)\ndiffraction peak is located on the calculated line for a full y\nstrained film. Hence, YAlIG is strained in the film plane and\ncompressed perpendicular to the film surface in comparison\nto cubic YAlIG, resulting in a rhomboidal YAlIG crystal. The\nlattice parameters based on the RSM are summarised in ta-bleI. The high crystalline quality of the YAlIG thin films on\nGGG is confirmed by the very low FWHM of 0.04◦in the\nω-scan (cp. Fig. 2a, inset), which is in the range of the resolu-\ntion of the X-ray measuring device. In addition, the first Lau e\noscillation is visible in the symmetric 2 θ-ωscan, indicating\nhigh structural ordering of the film. The increase in lattice pa-\nrameter α, the angle between the unit cell vectors, suggests an\nenhancement of the strain for higher annealing temperature s.\nWe attribute this to a change of lattice mismatch at higher te m-\nperatures due to different thermal coefficients of substrat e and\nfilm, which was shown for different rare earth garnets [ 49].\nChemical composition\nRBS measurements were performed for selected samples to\ninvestigate the chemical composition, revealing a signific ant\ndeviation of the nominal composition. Here, we will present\nand discuss the RBS results exemplary for a YAlIG film ( x=\n1.5,TA=800 °C) on YAG as given in Fig. 3, however, similar\ncorrelations were found for all investigated samples.\nFIG. 3: RBS spectra for a YAlIG film with x=1.5 on YAG annealed at\nTA=800 °C. The shaded areas indicate the thin film contribution o f the RBS\nspectra.\nGoing from high backscattered ion energy to lower ener-\ngies of the spectrum, the elemental contributions of Y , Fe, A l\nand O (not shown in Fig. 3) can be distinguished. The first\napproximately 80 keV of each elemental contribution belong\nto the YAlIG thin film, indicated by the coloured areas in Fig.\n3. As both the film and the YAG substrate contain Y ,Al and\nO, the respective contributions of the film are superimposed\nby high mass contributions originating from the substrate t o-\nwards lower energies. The simulated spectrum (cp. solid lin e\nin Fig. 3) is in good agreement to the measurement and the el-\nemental ratios are determined from the fit as (Fe : Y)=1.01,\n(Fe : Al)=1.92 and(Y: Al)=1.90. These values differ from\nthe values for the nominal stoichiometric composition of th e\ntarget for x=1.5, i.e.(Fe : Y)target=1.17,(Fe : Al)target=2.33\nand(Y : Al)target=2. The most significant change is the de-\ncreased(Fe : Al)ratio, indicating a lowered Fe content and an\nincreased Al content. This is confirmed by the decrease of the\n(Fe : Y)ratio and the (Y : Al)ratio. Therefore we conclude5\nthat our YAlIG films have a lower Fe3+content and an in-\ncreased Al3+content compared to the chemical composition\nof the target, while there’s no significant change of the Y3+\ncontent. We attribute this difference to changes of the chem -\nical composition during the sputtering process, where it wa s\nshown that the chemical composition can be altered by dif-\nferent process conditions [ 50,51]. This has the potential to\nenable fine-tuning of the chemical composition by adjusting\nthe deposition process.\nMagnetometry\nThe SQUID VSM measurements demonstrate ferromag-\nnetism with a reduced Curie temperature that scales with the\nAl3+substitution level for YAlIG thin films grown on YAG\nsubstrates. The temperature dependent magnetisation is gi ven\nin Fig. 4a. For the as-deposited sample, no net magnetisation\nwas observed in the M(T) curve. Upon annealing and crys-\ntallising into YAlIG a magnetic phase transition emerges. T he\nCurie temperature for all substitution levels is substanti ally\nlowered with respect to that of pure YIG ( Tc=599 K) [ 22], as\nexpected by theory and previous studies in YAlIG [ 25,27,30],\nsince the ferrimagnetic coupling between the Fe3+ion sub-\nlattices is weakened by the substitution of Fe3+ions with\nnon-magnetic Al3+ions. Consequentially, Tcdecreases for\nhigher substitution levels which is also represented in our data.\nExperimentally, the ferrimagnetic-paramagnetic transit ion at\nTcis broadened by the presence of the static magnetic field\nofµ0H=0.1 T, by non-uniform temperature sweep rates and\nby inhomogeneities of composition within the sample. There -\nfore, Tcwas determined at the minimum of the first derivative\nof the magnetisation with respect to the temperaturedM\ndT(cp.\nFig. 4b). In addition, we define the upper limit for the Curie\ntemperature Tc,maxat the first temperature wheredM\ndTis below\nthe noise level of the measurement. Both, TcandTc,maxfor all\nYAG/YAlIG samples are summarised in Table I.\nRegarding the nominal ratio (Fe : Al)target, the measured\nTcof our YAlIG thin films is significantly lower than that re-\nported in the literature [ 25–29,31], as displayed in Fig. 5.\nHowever, taking into account the actual (Fe : Al)ratio of the\nYAlIG thin film, as measured by RBS, and the maximum\nCurie temperature Tc,max, the measured Tcof our YAlIG thin\nfilms is consistent with the literature, as indicated by the a r-\nrows and errorbars in Fig 5. In addition to the chemical com-\nposition, other effects might cause a reduction of the Curie\ntemperature. For bulk YAlIG, the distribution of Fe3+/Al3+\namong the a-site and d-site within the garnet structure coul d\nbe altered by different annealing conditions, which leads t o\na change of Curie temperature and saturation magnetisation\n[27]. While the latter is strongly influenced by the changed\ndistribution, only small changes for the Curie temperature of\nbelow 10 K are reported. Since our rare earth garnet thin films\nfeature thicknesses t≥37nm, dimensionality effects on the\nCurie temperature should not be relevant [ 52,53].\nThe magnetisation in dependence of the externally applied\nmagnetic field at 20 K for different Al3+substitution levels\nxis given in Fig. 6a. The observed hysteresis loop furtherFIG. 4: (a) SQUID-VSM M(T)measurements with a static field of\nµ0H=0.1 T applied in the film plane. The Curie temperature Tcwas\ndetermined at the minimum of the first derivativedM\ndT(b). To get an upper\nlimit of the Curie temperature, Tc,maxwas determined at the temperature\nwere the first derivative is lower as the noise level, which is indicated as a\ndashed black line.\ncorroborates the ferrimagnetic ordering of the YAlIG film at\nlow temperatures. Above 1 T the YAlIG thin films are sat-\nurated. The saturation magnetisation Msfor each sample\nis summarised in Table I. As already implied by the M(T)\ncurves, the field dependent measurements show a decreasing\nMstowards higher substitution levels. This is explained by\nthe predominant substitution of Fe3+on the tetrahedral sites\nin YAlIG [ 25]. Please note that the amorphous, as-deposited\nsample shows a hysteresis with Ms=5 kAm−1although no\nsign of ferrimagnetism was observed in the temperature de-\npendent measurements (cp. Fig. 4a). We attribute this appar-\nent absence of ferrimagnetic order in the M(T) measurements\nto the subtraction of the paramagnetic background, which\nis particularly delicate for samples with an inherently sma ll\nferrimagnetic magnetisation. The ferrimagnetism of the as -\ndeposited sample can be explained by iron rich areas within\nthe film which can couple ferrimagnetically. However, as the\nas-deposited YAlIG thin films exhibit no signs of a crystalli ne\nordering, the ferrimagnetism is expected to be weak compare d6\nref. 25\nref. 27\nref. 28\nref. 29\nref. 31Geller [25]\nRoeschmann [27]\nGrasset [28]\nRavi [29]\nChen [31]\nFIG. 5: Curie temperature Tcof YAlIG with different Fe:Al ratios, reported\nin the literature [ 25,27–29,31,39] and the results of this study. The shaded\narea is a guide for the eye for the expected range of Tcbased on the\nliterature. For the results of this study, the errorbars ind icate the upper limit\nTc,maxand the arrow indicates the shift due to the difference betwe en the\nnominal(Fe : Al)target ratio and the (Fe : Al)ratio measured via RBS.\nto the annealed and crystallised YAlIG samples, which is cor -\nroborated by the temperature as well as the field dependence\nof the magnetisation (cp. Fig. 4and6).\nTo extract the coercivity of the YAlIG thin films, the mag-\nnetic hysteresis curves at different temperatures and samp le\norientations at low fields µ0H<100 mT are representatively\nshown in Fig. 6b for a YAlIG thin film with x=1.5. At 300 K,\nalmost no coercivity and magnetisation can be measured as\nthe sample is in vicinity of the Curie temperature ( Tc=262 K,\nTc,max=300 K). As reported for rare earth garnet thin films,\nthe coercive field of our YAlIG thin films increases with lower\ntemperatures [ 54,55]. For an external magnetic field applied\nin the sample plane, the coercive field of µ0Hc=15 mT at\n20 K and µ0Hc=1 mT at 100 K are in the order of the co-\nercive fields of yttrium iron garnet thin films grown by mag-\nnetron sputtering on YAG [ 55]. No square loop is observed\nfor either field direction in our YAlIG thin films. The X-ray\nanalysis suggests the existence of crystal defects, which m ight\nact as pinning centres for the domain wall motion. However,\nthe absence of a distinct magnetic hard axis cannot solely be\nascribed to a pinning of the domain walls at defects within th e\nfilm during the reversal of the magnetisation. Therefore, we\nsuspect that our YAlIG thin films exhibit an additional mag-\nnetic anisotropy perpendicular to the film surface compensa t-\ning the shape anisotropy.\nFerromagnetic resonance\nThe bb-FMR measurements demonstrate that the substrate-\ndependent epitaxial deformation can induce perpendicular\nmagnetic anisotropy in the YAlIG thin films. The dynamic\nmagnetic properties of our YAlIG thin films are comparable\nto those of other rare earth garnet thin films. Two representa -\ntive samples with the same nominal chemical composition of/uni000000ed/uni00000016 /uni000000ed/uni00000015 /uni000000ed/uni00000014 /uni00000013 /uni00000014 /uni00000015 /uni00000016\nμ\n0H /uni0000000b/uni00000037/uni0000000c/uni00000017/uni00000013\n/uni00000015/uni00000013\n/uni00000013\n/uni000000ed/uni00000015/uni00000013\n/uni000000ed/uni00000017/uni00000013\nM /uni00000003/uni0000000b/uni0000004e/uni00000024/uni00000050−1\n/uni0000000c/uni0000000b/uni00000044/uni0000000c\nx = 1.5, /uni00000003/uni00000044/uni00000056/uni00000010/uni00000047/uni00000048/uni00000053/uni00000011\nx = 1.5, T\nA= 700 /uni00000083/uni00000026\nx = 1.5, T\nA= 800 /uni00000083/uni00000026\nx = 1. 75, T\nA= 800 /uni00000083/uni00000026\nx = 2.0, T\nA= 800 /uni00000083/uni00000026\n/uni000000ed/uni00000013/uni00000011/uni00000014/uni00000013 /uni000000ed/uni00000013/uni00000011/uni00000013/uni00000018 /uni00000013/uni00000011/uni00000013/uni00000013 /uni00000013/uni00000011/uni00000013/uni00000018 /uni00000013/uni00000011/uni00000014/uni00000013\nμ\n0H /uni0000000b/uni00000037/uni0000000c/uni000000ed/uni00000017/uni00000013/uni000000ed/uni00000015/uni00000013/uni00000013/uni00000015/uni00000013/uni00000017/uni00000013M /uni00000003/uni0000000b/uni0000004e/uni00000024/uni00000050−1\n/uni0000000c/uni00000052/uni00000052/uni00000053 /uni0000004c/uni00000053x = 1.5, T\nA= 800 /uni00000083/uni00000026/uni0000000b/uni00000045/uni0000000c\n/uni00000015/uni00000013/uni00000003/uni0000002e\n/uni00000014/uni00000013/uni00000013/uni00000003/uni0000002e\n/uni00000016/uni00000013/uni00000013/uni00000003/uni0000002e/uni00000015/uni00000013/uni00000003/uni0000002e\n/uni00000014/uni00000013/uni00000013/uni00000003/uni0000002e\n/uni00000016/uni00000013/uni00000013/uni00000003/uni0000002e\n/uni000000ed/uni00000016 /uni000000ed/uni00000015 /uni000000ed/uni00000014 /uni00000013 /uni00000014 /uni00000015 /uni00000016/uni00000017/uni00000013\n/uni00000015/uni00000013\n/uni00000013\n/uni000000ed/uni00000015/uni00000013\n/uni000000ed/uni00000017/uni00000013\nFIG. 6: SQUID-VSM M(H)hysteresis loops of YAlIG thin films grown on\nYAG. (a)In-plane M(H)hysteresis loops up to µ0H=±3T at 20 K for\ndifferent annealing temperatures and compositions. (b)In-plane M(H)\nhysteresis loops at different temperatures for fields up to µ0H=±0.1T for a\nsample with x=1.5 annealed at TA=800 °C. The inset shows the complete\nfield range of µ0H=±3T.\nx=1.5, which were both annealed at TA=800 °C for 4h, but\non different substrates have been used. The FMR resonance\nfrequencies fresas a function of magnetic field Hat 200 K for\nthe YAlIG thin film on YAG and GGG are shown in Fig. 7a\nand Fig. 7b, respectively. The measurement temperature of\n200 K has been chosen to be below the Curie temperature of\nYAlIG on YAG measured by SQUID VSM ( Tc=262 K). The\nexistence of the FMR signal confirms the ferrimagnetic natur e\nof the YAlIG thin film.\nTo extract the g-factor gand the effective magnetisation\nMeff, we model fresusing the approach of Baselgia et al.[ 56]\nbased on the total free energy density Fgiven by\nF=−µ0MsH(sinθMsinθHcos(φM−φH)+cosθMcosθH)\n+1\n2µ0MsMeffcos2θM,\n(1)\nwhere θM,φMandθH,φHdenote the polar and azimuthal7\n/uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000014 /uni00000013/uni00000011/uni00000015 /uni00000013/uni00000011/uni00000016 /uni00000013/uni00000011/uni00000017 /uni00000013/uni00000011/uni00000018 /uni00000013/uni00000011/uni00000019 /uni00000013/uni00000011/uni0000001a\nμ\n0H /uni00000003/uni0000000b/uni00000037/uni0000000c/uni00000013/uni00000018/uni00000014/uni00000013/uni00000014/uni00000018/uni00000015/uni00000013f\nres/uni00000003/uni0000000b/uni0000002a/uni0000002b/uni0000005d/uni0000000c\ng = 2.02 ± 0.02\nM\neff= (7.0 ± 0.4) /uni00000003/uni0000004e/uni00000024/uni00000050−1/uni0000003c/uni00000024/uni0000002a/uni0000000f/uni00000003 x = 1.5, T\nA= 800 /uni00000083/uni00000026\n/uni00000052/uni00000052/uni00000053\n/uni00000052/uni00000052/uni00000053/uni00000010/uni00000050/uni00000052/uni00000047/uni00000048/uni0000004f\n/uni0000004c/uni00000053\n/uni0000004c/uni00000053/uni00000010/uni00000050/uni00000052/uni00000047/uni00000048/uni0000004f\n/uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000014 /uni00000013/uni00000011/uni00000015 /uni00000013/uni00000011/uni00000016 /uni00000013/uni00000011/uni00000017 /uni00000013/uni00000011/uni00000018 /uni00000013/uni00000011/uni00000019 /uni00000013/uni00000011/uni0000001a\nμ\n0H /uni00000003/uni0000000b/uni00000037/uni0000000c/uni00000013/uni00000018/uni00000014/uni00000013/uni00000014/uni00000018/uni00000015/uni00000013/uni00000015/uni00000018f\nres/uni00000003/uni0000000b/uni0000002a/uni0000002b/uni0000005d/uni0000000c\ng = 2.00 ± 0.02\nM\neff= (−155 ± 5) /uni00000003/uni0000004e/uni00000024/uni00000050−1/uni0000002a/uni0000002a/uni0000002a/uni0000000f/uni00000003 x = 1.5, T\nA= 800 /uni00000083/uni00000026\n/uni00000052/uni00000052/uni00000053\n/uni00000052/uni00000052/uni00000053/uni00000010/uni00000050/uni00000052/uni00000047/uni00000048/uni0000004f\n/uni0000004c/uni00000053\n/uni0000004c/uni00000053/uni00000010/uni00000050/uni00000052/uni00000047/uni00000048/uni0000004f/uni0000000b/uni00000044/uni0000000c\n/uni0000000b/uni00000045/uni0000000c\nFIG. 7: FMR resonance fresas a function of the applied magnetic field Hat\na temperature of 200 K for a YAlIG sample grown on YAG (a) and on e\nsample grown on GGG (b). The modelled curve is based on the app roach by\nBaselgia et al.[ 56] and the total free energy density described in Eq. 1. The\ncorresponding Meffandgare displayed in the plot.\nangles of the magnetisation and the magnetic field, respec-\ntively. The first term in Eq. 1describes the Zeeman energy\nand the second term the effective demagnetisation energy. T he\neffective magnetisation Meff=Ms−Huin this model includes\nthe shape anisotropy field (equal to Ms) and a perpendicular\nuniaxial anisotropy field Hu.\nFig. 7a and 7b show that the modelled fresvalues are in\ngood agreement with the experimental data. For both sub-\nstrates, we extract a g-factor of approximately 2 which is ex -\npected for YAlIG as the magnetic moment mostly originates\nof electron spins. On the YAG substrate, Meff=7.0 kA/m\nwhich is smaller than the saturation magnetisation Ms. We\ncan therefore conclude that the shape anisotropy is partial ly\ncompensated by a perpendicular anisotropy. This is compara -\nble to the result of the SQUID VSM measurement, although\nmeasured at a different temperature. On the GGG substrate,\nMeffis negative, indicating that the perpendicular magnetic\nanisotropy is the dominant magnetic anisotropy. Thus, it is\npossible to grow YAlIG on GGG as a PMA (perpendicular\nmagnetic anisotropy) material. This large perpendicular m ag-\nnetic anisotropy, in particular compared to the sample grow n\non YAG, is caused by a magneto-elastic anisotropy. As dis-\ncussed in the XRD analysis, the YAlIG thin films grown on\nGGG are epitaxially strained so that the film is under tensile\nstrain in the film plane. In this case, previous reports also s ug-\ngest a strong perpendicular magneto-elastic contribution andthe possibility of PMA in rare earth garnet thin films [ 57–59].\nWe would anticipate a similar correlation for our YAlIG thin\nfilms grown on GGG.\nThe dynamic properties of a YAlIG thin film are eval-\nuated based on the extracted resonance linewidth ∆f. At\nfres=10 GHz, the frequency linewidth is ∆f=200MHz for\na YAlIG thin film grown on GGG, which corresponds to a\nmagnetic field linewidth of ∆B=7.1mT ( ∆B=2π¯h\ngµB∆f). This\nvalue is within the range reported for strained rare earth ga r-\nnet thin films ( ∆B=0.3−7.5mT) [ 57–60] and close to the\nrange reported for sputtered YIG thin films ( ∆B=0.35−\n7mT)[ 36,57,61,62]. The Gilbert damping αis determined\nby evaluating the frequency dependence of the extracted res o-\nnance linewidths (not shown). In our YAlIG thin films, αis of\nthe order of 10−3, which is comparable to other sputtered gar-\nnet thin films [ 59,60,62], but still higher than the best values\nreported for highest quality YIG thin films [ 19].\nCONCLUSION\nWe report the successful growth of ferrimagnetic\nY3Fe5−xAlxO12films via RF sputtering for substitution\nlevels of x=1.5,1.75,2 on single crystalline YAG (111)\nand GGG (111) substrates. A post-annealing step in reduced\noxygen atmosphere is necessary for the crystallisation of\nYAlIG. On YAG substrates with a higher lattice mismatch,\nrelaxed YAlIG(111) films are obtained with a higher degree\nof defects. The lower lattice mismatch with the GGG\nsubstrates results in the growth of fully strained perpendi cular\ncompressed YAlIG(111) layers of higher crystalline qualit y\nas compared to the samples on YAG. SQUID VSM reveals\nthat the Curie temperature of the sputtered YAlIG thin films\non YAG can be controlled by the Al3+substitution level.\nWe achieve Tcvalues down to 102 K, which is well below\nroom temperature. We also show low coercivity compa-\nrable to other sputtered YIG thin films. Broadband-FMR\nmeasurements evidence a dominant perpendicular magnetic\nanisotropy in the YAlIG thin film grown on GGG caused by\nthe epitaxial strain, whereas no such effect can be observed\nin the thin films grown on YAG. We also find that the FMR\nresonance linewidth and Gilbert damping are similar to othe r\nrare earth garnet thin films. However the chemical composi-\ntion of the YAlIG films, measured via RBS, is not equal to\nthe nominal substitution level of the sputtering targets. 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Kostylev, Effect of annealing on the structural and FMR\nproperties of epitaxial YIG thin films grown by RF magnetron\nsputtering, IEEE Transactions on Magnetics 54, 1 (2018) ."    },    {        "title": "2110.13679v1.Giant_quadratic_magneto_optical_response_of_thin_YIG_films_for_sensitive_magnetometric_experiments.pdf",        "content": "1Giant quadratic magneto-optical response of thin YIG films for sensitive magnetometric experiments \nE. Schmoranzerová1*, T. Ostatnický1*, J. Kimák1, D. Kriegner2,3, H. Reichlová2,3, R. Schlitz3, A. Baďura1, Z.\nŠobáň2, M. Münzenberg4, G. Jakob5, E.-J. Guo5, M. Kläui5, P. Němec1\n1Faculty of Mathematics and Physics, Charles University, Prague, 12116, Czech Republic\n2Institute of Physics ASCR v.v.i , Prague, 162 53, Czech Republic\n3Technical University Dresden, 01062 Dresden, Germany\n4Institute of Physics, Ernst-Moritz-Arndt University, 17489, Greifswald, Germany\n5Institute of Physics, Johannes Gutenberg University Mainz, 55099 Mainz, Germany\nAbstract\nWe report on observation of a magneto-optical effect quadratic in magnetization (Cotton-Mouton effect)\nin 50 nm thick layer of Yttrium-Iron Garnet (YIG). By a combined theoretical and experimental approach,\nwe managed to quantify both linear and quadratic magneto-optical effects. We show that the quadratic\nmagneto-optical  signal  in  the  thin  YIG  film  can  exceed  the  linear  magneto-optical  response,  reaching\nvalues  of  450  rad  that  are  comparable  with  Heusler  alloys  or  ferromagnetic  semiconductors.\nFurthermore, we demonstrate that a proper choice of experimental conditions, particularly with respect\nto the wavelength, is crucial for optimization of the quadratic magneto-optical effect for magnetometry\nmeasurement.\nI. Introduction\nYttrium Iron Garnet (Y 3Fe5O12, YIG) is a prototype ferrimagnetic insulator which represents one of the key\nsystems for modern spintronic applications [1]. It has been thoroughly studied in the last decades owing to\nits special properties, such as low Gilbert damping [2-4] and high spin pumping efficiency [5-7]. YIG has\nplayed a crucial role in fundamental spintronics experiments, revealing spin Hall magneto-resistance [9-\n10] or spin-Seebeck effect [11-13].2Many  of  the  above-mentioned  spintronic  phenomena  rely  on  high-quality  ultra-thin  YIG  films  and  on\ndetection of small changes in magnetization therein. However, YIG is a complex magnetic system with 200\nB magnetic moments per unit cell. Magnetic properties of the few monolayer systems used in spintronics\nare then vulnerable to small structural changes and relatively difficult to characterize and control [14-17].\nMoreover, reliability of conventional magnetometry tools such as Superconducting Quantum Interference\nDevice  (SQUID)  or  Vibrating  Sample  Magnetometry  (VSM)  is  limited  by  the  large  paramagnetic\nbackground and unavoidable impurity content of the gadolinium-gallium- garnet that is commonly used as\na substrate for the thin YIG layers. Direct use of magneto-transport methods for magnetic characterization\nis naturally prevented by the small electric conductivity of the insulating YIG. They can be utilized only\nindirectly in multilayers of YIG/heavy metal, via Spin Hall Magnetoresistance in the metallic layer [17].\nIn  contrast,  optical  interactions  are  not  governed  by  DC  conductivity  of  the  material.  Magneto-optics\ntherefore provides a natural tool for detection of magnetic state of ferrimagnetic insulators, and YIG in\nparticular has an extremely strong magneto-optical response that can be easily modified by doping [18].\nMagneto-optical (MO) response of a material manifests generally as a change of polarization state of a\ntransmitted or reflected light [18 ] usually detected in a form of a rotation of polarization plane of a linearly\npolarized  light.  Similar  to  the  magneto-transport  effects,  MO  effects  with  different  symmetries  with\nrespect to magnetization ( M) can occur. With certain limitations [19], an optical analogy to the anomalous\nHall effect (AHE) linear in M represents the Faraday effect in transmission geometry or the Kerr effect in\nreflection. For the anisotropic magnetoresistance (AMR) quadratic in M, the corresponding MO effect is\nmagnetic linear dichroism (MLD) [19]. As the terminology is ambiguous in magneto-optics, MLD, Q-MOKE,\nVoigt or Cotton-Moutton effect, which are sometimes used, refer all to the same phenomenon in different\nexperimental geometries. In this paper, we use the name Cotton-Moutton effect (CME) for the rotation of\npolarization plane in transmission geometry, consistently with previous works on YIG [20-21].   \nThe quadratic MO effects scale with the square of the magnetization magnitude and their symmetry is\ngiven by an axis parallel to the magnetization vector. As such, they are generally weaker than the linear\nMO response [18]. However, there are significant advantages over the linear magneto-optics that make\nthem  favourable  in  MO  magnetometry.  The  even  symmetry  with  respect  to  the  local  magnetization\nenables  to  observe  these  effects  in  systems  with  no  net  magnetic  moment,  such  as  collinear\nantiferromagnets, as the contributions from different sublattices do not cancel out [22]. Quadratic MO\neffects are sensitive to the angle between magnetization and the polarization plane [23], similarly to the\nway the AMR is sensitive to the angle between the electric current and the magnetization [19], which3enables to trace all the in-plane components of magnetization vector simultaneously in one experiment\n[23-26] There is, however, a key advantage of the MO approach. The optical polarization can be set easily\nwithout fabrication of additional structures, unlike the current direction in the case of AMR, which is given\nby the electrical contact geometry.  Variation of the probe polarization can then provide information about\nmagneto-crystalline anisotropies [25] without modification of sample properties by litography-induced\nchanges that are inherent to the methods based on electron transport. \nIn certain classes of materials for compunds with significant spin-orbit coupling, such as Heusler alloys\n[27], ferromagnetic semiconductors [23,28] or some collinear antiferromagnets [22], the quadratic MO\nresponse  can  be  strongly  enhanced.  It  has  found  important  applications  in  static  and  dynamic  MO\nmagnetometry [28-29], helping to reveal novel physical phenomena such as optical spin transfer [30] and\nspin orbit torques [31 ]. In contrast, in ferrimagnetic insulators the quadratic MO effects have been vastly\nneglected so far. The first pioneering experiments have revealed the potential of the quadratic magneto-\noptics  in  YIG  to  visualize  stress  waves  [32]  or  current-induced  spin-orbit  torque  [21],  and  the  inverse\nquadratic  Kerr  effect  has  been  even  identified  as  a  trigger  mechanism  for  ultrafast  magnetization\ndynamics in thin YIG films [21]. However, no optimization with respect to the size of the amplitude of MO\neffects was performed in these works in terms of its amplitude, spectrum, dependence on the angle of\nincidence or initial polarization.  In 1970s and 80s, limited studies were performed in the field of magneto-\noptical spectroscopy on bulk YIG crystals, demonstrating magnetic linear birefringence [33] or dichroism\n[34-35]  of  YIG  crystals  doped  by  rare-earths  and  metals,  and  on  terbium-gallium-garnet  at  cryogenic\ntemperature [36 ].  However, to our knowledge, no experiments aiming at understanding the details of the\nquadratic MO response in thin films of pure, undoped YIG have been performed so far. \n In this paper, we report on the observation of a giant CME of 50 nm thin epitaxial film of pure YIG, which\ncan  even  exceed  the  amplitude  of  the  linear  Faraday  effect.  Using  a  combined  experimental  and\ntheoretical approach, we quantify the size of CME with respect to various external parameters, such as\nwavelength, temperature or angle of incidence.  This is a key prerequisite for the quadratic magneto-\noptics  optimization  for  magnetometric  applications.  The  potential  of  CME  for  magnetometry  is\ndemonstrated on identification of magnetic anisotropy of the thin YIG film directly from the detected MO\nsignals.\nII. Experimental details and sample characterization 4We used monocrystalline 50 nm thick film of yttrium iron garnet prepared by pulsed laser deposition (PLD)\non (111)-oriented gallium gadolinium garnet (GGG). Details of the growth procedure can be found in Ref.\n37. Since the thin YIG layers are prone to growth defects and strain inhomogeneities [14],[16], the samples\nwere carefully characterized by X-ray diffraction. From the reciprocal space maps (RSM) around the YIG\nand GGG 444 and 642 Bragg peak we find that any diffraction signal of the film is aligned with the one of\nthe substrate along the in-plane momentum transfer [see Fig. 1(a) and (b)]. The RSMs therefore show the\npseudomorphic growth of the YIG film, with its in-plane lattice parameter equal to the substrate lattice\nparameter measured to be 12.385 Å, close to previously published values for GGG substrates [38 ]. Along\nthe [111] direction we find Laue thickness oscillations indicating the high crystalline quality of the YIG film.\nIn  order  to  analyze  the  out  of  plane  lattice  parameter,  a  cross-section  along  the  [111]  direction  was\nextracted from the 444 and 642 Bragg peaks and modelled using a dynamical diffraction model [see Fig.\n1(c)]. We used the rhombohedrally distorted structure of YIG with a= 12.379 Å and rhombohedral angle \n= (90.050.02) deg as an input for the model, parameters that are similar to Ref. [14]. A weak in-plane\ntensile strain occurs due to the lattice mismatch, but the resulting distortion of only 0.05 deg is unlikely to\naffect the magnetic properties of the layer in a significant way.\nFig. 1: Structural and magnetic characterization of thin YIG film.  (a), (b) Reciprocal space maps (RSM) taken on 444\nand 642 Brag peaks of YIG at room temperature. (c) Cross sections of RSM data along [111] crystallographic directions\n(points) modelled by dynamical diffraction model (line) with lattice parameter a = 12,379 Å and distortion angle =\n(90,050.02) deg. (d) Magnetic hysteresis loops measured by SQUID magnetometry in three in-plane crystallographic\ndirections  that  are  denoted  as:  A  [2-1-1],  B[01-1]  and  C  (diagonal),  and  in  the  out-of-plane  direction  D  [111]  at\ntemperature of 50 K. T\n5The  magnetic  properties  of  the  YIG  film  are  established  using  SQUID  magnetometry.  An  example  of\nhysteresis loops recorded at 50 K with external magnetic field applied along different crystallographic\ndirections is shown on Fig. 1(d). Clearly, the sample is in-plane magnetized, with a coercive field of H c = 18\nOe. Note that there is a small difference in H c between the two crystallographic directions denoted as A\nand C, indicating the presence of an in-plane magnetic anisotropy but its quantification based on our\nSQUID measurement is prevented by a large error caused by the paramagnetic background of the GGG\nsubstrate. For further details on SQUID measurements see Supplementary material, Fig. (S1). From these\nmeasurements, the room-temperature value M s = 96  kA/m was extracted.  The saturation magnetization\nis lower than that of a bulk crystal M s,bulk = 143 kA/m [39] but in very good agreement with previously\nreported values for PLD-grown ultra-thin YIG layers [17], confirming the good quality of our YIG film.\nMagneto-optical measurements on the YIG sample were performed by a home-made vectoral magneto-\noptical magnetometer, schematically shown in Fig. 2 (a). For the majority of our experiments we used a\nCW solid-state laser (Match Box series, Integrated Optics ltd.) with a fixed wavelength of 403 nm as a light\nsource. The CW laser was replaced by the second-harmonics output of a tunable titan-sapphire pulse laser\n(model Mai Tai, Spectra Physics,) to gain wider spectral range of  = 390-440 nm for the wavelength-\ndependence  measurement.  The  light  was  incident  on  the  sample  either  under  an  angle  i   =  3°  (near\nnormal incidence), or i  = 45°, as indicated in Fig. 2(c). The linear polarization of the incident light in both\ncases  was  set  by  a  polarizer  and  a  half-waveplate,  and  the  polarization  state  of  the  light  transmitted\nthrough the sample was analyzed by a differential detection scheme (optical bridge) in combination with a\nphase-sensitive (lock-in) detection [40]. \n6 Fig. 2: (a) Schematics of the experimental setup for magneto-optical magnetometry. Linearly polarized light with a\npolarization E oriented at an angle   with respect to the TM polarization mode (E s) is incident on the sample, which is\noriented under an angle  i.   with respect to the plane in which the magnetic field was applied. After being transmitted\nthrough the sample, the light polarization plane is rotated by an angle . The sample is subject to an external\nmagnetic field Hext, applied in an arbitrary direction, with the corresponding spherical angles of the Hext vector shown\nin (b) plane view projection (azimuthal angle H) and (c) side view projection (polar angle H) of the experiment\ngeometry. \nThe  sample  itself  was  mounted  in  a  closed-cycle  cryostat  (ARS  systems)  to  enable  the  temperature\nvariation in a range of T = 20 K – 300 K. The cryostat was placed between pole stages of a custom-made\ntwo-dimensional (2D) electromagnet where the external magnetic field of up to 0Hext = 205 mT could be\napplied  in  an  arbitrary  direction  in  the  plane  perpendicular  to  the  optical  beam  axis.  The  (spherical)\ncoordinate system for Hext is given in Fig. 2(b), (c). Note that the polar angle H is defined from the sample\nnormal,  and  is  equivalent  to  the  angle  of  incidence  of  the  incoming  light  i..  Utilization  of  the  2D-\nelectromagnet  allowed  for  two  different  approaches  in  our  experiments.  Firstly,  a  standard  magneto-\noptical magnetometry was used, where the magnitude of | Hext| in a fixed direction [01-1] is varied, and\nthe  resulting  hysteresis  loops  are  recorded.   Comparing  the  measured  hysteresis  loops  for  different\norientations of the light polarization with our analytical model allowed for determination of the motion of\nmagnetization during the magnetic field sweeps, as further discussed in the “Theory” section. Analysis of\nthe full polarization dependence of the hysteresis loops also enabled us to separate the contributions of\nthe  linear  Faraday  effect  (LFE)  and  the  quadratic  CME  to  the  overall  MO  signals,  and  to  extract  the\ncorresponding amplitudes (coefficients) of the CME and LFE effects. \nHowever, this method of determining the MO coefficients was inefficient and burdened by a relatively\nlarge  error  resulting  from  the  complicated  way  of  extracting  the  MO  effects  that  required  full  light\npolarization  dependence  of  the  hysteresis  loops.  For  further  systematic  study  of  the  CME  effect  we,\ntherefore,  implemented  ROT-MOKE  experiment,  where  the  external  magnetic  field  ( Hext)  of  a  fixed\nmagnitude of 205 mT was rotated in in the plane from H = 0° to 360° (see Fig. 2),  and the resulting MO\nsignal  was  recorded  as  a  function  of  H [24-25],  with  the  polarization  of  the  light  kept  fixed  to  the\nfundamental TE (s-) mode. Here, Hext was large enough to saturate magnetization of the YIG film, which\nthen exactly follows the field direction during its rotation. We can therefore neglect the effect of magnetic\nanisotropy and determine the MO coefficient simply from one field rotation curve [25], in a way very\nsimilarly to determination of AMR or Planar Hall effect coefficients from field rotations [26]. This also7directly demonstrates analogy between the magneto-optical and magneto-transport methods.\nIII.  Theory \nThe aim of our theoretical analysis is to determine the kinetics of the magnetization vector during the\nmagnetic field sweep and, based on its known orientation at each point of the experimental curves, to\nevaluate  the  magnitude  of  the  magneto-optical  coefficient.  Motion  of  the  magnetization  vector  is\nmodeled in terms of the local profile of the magnetization free energy density. Its functional F is known\nfrom  the  symmetry  considerations  (see  Eq.  (S1)  in  Supplementary)  assuming  the  lowest  terms  in\nmagnetization  magnitude  [41],  yet  the  corresponding  constants  which  appear  in  the  expression  are\nstrongly  sample-dependent.   In  the  case  of  YIG,  the  expected  order  of  magnitude  of  the  anisotropy\nconstants is known [42] and therefore, we can roughly estimate the positions of the easy magnetization\ndirections. The dominant anisotropy in high-quality thin YIG samples on GGG has its origin in the cubic bulk\ncontribution. In thin samples, there is an additional out-of-plane anisotropy (hard direction) due to the\nstress  fields  and  demagnetization  energy  which  pushes  the  magnetization  towards  the  sample  plane\n[14,15]. We therefore expect that the projection of easy directions to the crystallographically oriented\n[111] sample plane is effectively sixfold [see Fig. 3(c)] and the deflection angle of the easy directions from\nthe sample plane is only few degrees and thus will be neglected  (see the Supplementary information for\nmore details). We define an effective in-plane anisotropic energy density, assuming that the deviation\nangle of the magnetization vector from the sample plane is small:\nF\nMS=K6 sin23(φM−γ)−μ0Hext sinθH cos(φH−φM)    (1)\nwhere MS is the saturation magnetization of the sample, μ0 is the vacuum permeability, Hext is the external\nmagnetic  field  magnitude,   H  its  deflection  angle  of  from  the  sample  plane  and  H its  azimuthal\norientation  with  respect  to  one  of  the  effective  in-plane  easy  axes  (assuming  K  >  0).  The  symbol  φM\ndenotes  azimuthal  position  of  the  in-plane  magnetization  vector  and  K6 is  the  effective  anisotropy\nconstant.  γ denotes an angle between the plane of incidence and the bisectrix of the magnetization easy\naxes, resulting from an unintentional rotation of the sample in the experiment. \nWhen  interpreting  the  experimental  data,  we  theoretically  simulated  the  full  magneto-optical\nmeasurement of the hysteresis by calculating the MO response of the layered structure (nm-thick sample8on a 500 m-thick substrate), considering the optical constants of the participating materials and the\nsymmetry-breaking by the sample’s magnetization. Our calculations inherently include all effects related\nto the light propagation in the media as well as multiple reflections and resulting interferences and as such\nthey  reveal  the  sum  of  all  MO  effects  which  take  part  in  the  particular  geometry.  We  consider  the\nrefractive  index  of  the  thick  GGG  substrate  to  be  nS  =  1.96  and  the  YIG  permittivity  tensor  for\nmagnetization oriented along the x axis reads:\n(2)\nwhere we take ϵN = 6.5+3.4i [42]. Thanks to the cubic symmetry of the YIG crystal, its permittivity tensor\nfor  an  arbitrary  magnetization  orientation  in  the  sample’s  xy plane  (see  the  geometry  in  Fig.  2)  is\ncalculated  by  a  proper  rotation  of  Eq.  (2)  around  the  z  axis  corresponding  to  [111]  crystallographic\ndirection. We considered the values Q = 10i×10-3 and QA = 1.1×10-3 for the simulations of the hysteresis\nloops for best agreement with measured amplitudes of the MO effects.\nThe experimental data can be interpreted also in terms of the symmetry of the MO response: they are\ncomposed of an even and an odd component whose magnitude remain (invert) upon inversion of the\nmagnetization. The even contribution is related to the Cotton-Mouton effect and is represented by the\nparameter QA in Eq. (2). The even symmetry comes from the fact that the effect is quadratic, i.e. the\nparameter  QA is  proportional  to  the  square  of  the  magnetization.  The  odd  contribution  can  be\nphenomenologically understood as a combination of the longitudinal and the transverse Faraday effect\nand is described by the parameter Q in Eq. (2), which is linear in magnetization. Note also that the polar\nFaraday effect does not contribute to the overall MO response of the system because the projection of\nmagnetization to the polar (out-of-plane) direction is negligible.\nConsidering the incident s-polarization ( β = 0 in our geometry), the polarization rotation ∆β due to the\nlinear MO effect is zero in the transverse magnetization geometry (i.e., for magnetization lying along the x\ndirection).  It  is  therefore  reasonable  to  consider  phenomenologically  that  for  any  arbitrary  in-plane\norientation of the magnetization, the polarization rotation is solely due to the longitudinal Faraday effect,\ni.e. it is proportional to the projection of the magnetization to the plane of incidence.  We can write for the\npolarization rotation:\n∆βLFE(φM,β=0 )=PLFEsin φM  (3)9where we defined the effective LFE coefficient PLFE. Expressions for other than s-polarizations outside \nthe limit of the small angle of incidence ϑi are not convenient for practical use and therefore are not \ndiscussed here; the numerical results, however, will be presented in the text below. In magneto-optical \nexperiments, we always measure the polarization rotation change when magnetization orientation \nchanges. Therefore, we define the LFE amplitude ALFE as:\nALFE(φ1,φ2)=PLFE(sin φ1 −sin φ2 ) \n(4)\nwhere  ∆β is  the  polarization  rotation  of  incident  s-polarization,  including  all  MO  and  non-magnetic\ncontributions,  for  a  given  orientation  of  magnetization.  The  difference  in  each  of  the square brackets\nrepresents the measured change of the MO signal and the subtraction of the parentheses extracts only\nthe  linear  (odd)  LFE  component  from  the  MO  signal.  The  angles  φ1,2 denote  some  well-defined\norientations of the magnetization.  In our case, it is worth to use positions of the easy axes between which\nthe magnetization jumps during the hysteresis curve measurements.\nIn  contrast  to  LFE  (Eq.  3),  CME  is  sensitive  to  the  angle  between  magnetization  and  light  polarization\ndirection. Simple relation can be derived, describing the relation between polarization rotation ∆βCME and\nmagnetization position φM [23]\n(5)\nwhere  we  defined  the  effective  CME  coefficient  PCME.  In  the  specific  case  of  hysteresis  loops  where\nmagnetization is switched between two magnetic easy axes, we may define the CME amplitude (see Fig. 4\nand Eq. (11) in [40]) to:\nACME()=2PCMEsin ξcos 2(γ−β)  \n, (6)\nwhere =φ1−φ2  is the angle between the easy axes and γ=(φ1−φ2)/2 is the position of their bisectrix.\nThe symmetrization of the brackets ensures that only even MO signal contributes to the amplitude. When∆βCME(φM)=PCMEsin 2(φM−β) 10the angles φ1,2 are known, it is possible to extract the MO coefficients using the above expressions.\nIV.  Experimental results and discussion\nA.Hysteresis loops\nFirstly, we focused on studying the MO during external magnetic field sweeps (MO hysteresis loops). In\nFig. 3(a) we show an example of the MO hysteresis loop measured close to the normal incidence ( i = 3°)\nand at a large angle of incidence ( I = 45°) at 20 K. The character of the hysteresis loop changes significantly\nwhen deviating from the normal incidence. Close to the normal incidence, the signal displays an M-shape\nlike loop, typical for the quadratic MO effects [40]. The steps in the M-shape loops generally correspond to\na switching of magnetization between two magnetic easy axes [40]. In contrast, for I = 45° the hysteresis\nloop  gains  more  complex  shape.  Besides  the  M-shape  like  signal  which  is  still  present  with  virtually\nunchanged  size,  there  is  another  square-like  component  that  indicates  presence  of  a  signal  odd  in\nmagnetization, which can be attributed the longitudinal Faraday effect. \nFig. 3: (a) Rotation of polarization plane  as a function of the external magnetic field Hext   measured for two angles\nof incidence i = 3° and 45°, at temperature of 20 K and photon energy of 3.1 eV. The data were vertically shifted for\nclarity. The complex M-shape-like hysteresis is a clear signature of magnetization being switched between magnetic\neasy axes. (b) Simulation of the MO signal by means of the analytical model.  Based on our model, we identified 3\nequivalent easy axes (c) and extracted their mutual angle  = 120° and position of their bisectrice  = 6°. (d) The abrupt\n11changes in magneto-optical signals in (a) and (b) correspond to jumps of the magnetization between the easy axes \n1, 3 and 4 (4, 6 and 1) for the magnetic field sweep from the positive (negative) field, as schematically indicated in \n(a).  \nIn  order  to  understand  the  nature  of  the  magnetization  motion,  we  used  the  theoretical  approach\ndescribed in the Theory section to model the observed signals: we consider six effective in-plane easy\ndirections for magnetization (see Eq. (1)) and we numerically modelled the MO response considering the\nparameters of the experiment. We used four fitting parameters: two of them are related to the amplitude\nof the MO  effects (even and odd) and two describe the magnetic anisotropy. The best agreement with the\nexperimental data appears for the values Q = 10i×10-3, QA = 1.1×10-3, γ=6° and K6=61 J/m3 . As an output\nof  our  model,  the  correct  shape  of  the  MO  loops  is  obtained,  as  shown  in  Fig  3(b).   The  magnetic\nanisotropy utilized by the model is schematically depicted in Fig. 3(c), with a definition of the magnetic\neasy axes position given in Fig. 3(d). Note that the estimated magnetic anisotropy corresponds with the\nSQUID measurement [Fig. 1(d)], the diagonal orientation (denoted as “C”) being the closest to the position\nof one of the easy axes. The motion of magnetization M in the external magnetic field gained from our\nmodel is schematically indicated by the arrows in Fig. 3(a). For the large positive external field Hext, M is\noriented along the field, close to the easy axis (EA) labelled as “1”. While decreasing Hext, M is slowly\nrotated towards the direction of EA \"1\". When Hext of the opposite polarity and magnitude exceeding the\nvalue of coercive field H c is applied, M switches directly to the EA “3”. Further increase of the negative Hext\nleads to another switching, this time only by 60° to EA “4”, until, finally, M is again oriented in the direction\nof Hext. A symmetrical process takes place in the second branch of the hysteresis loop. Note that the same\nmagnetization switching occurs independently of the angle of incidence, though for larger i the shape of\nthe loop is distorted by the presence of the linear contribution to the MO signal. \nThe macrospin simulations confirmed that the full magnetization trajectory extracted from our magneto-\noptical signals corresponds to the realistic magnetic anisotropy constants for thin YIG films (see Section 2\nin Supplementary). The model allows for certain ambiguity in its parameters since we do not have access\nto the out-of-plane components of magnetization motion during the switching, to compare them with the\nexperimental data. The model thus cannot be used reliably for obtaining all magnetic anisotropy constants\nof the material without support of a complementary experimental method. However, it provides a useful\ntool for prediction of the behavior of the magneto-optical effects, as shall be shown further on.12To quantify the contributions of the linear and quadratic MO effect, we recorded the MO hysteresis loops\nfor different angles of initial light polarization . The quadratic CME and the linear LFE contributions to the\noverall MO signals were obtained by symmetrization and anitsymetrization of the hysteresis loops, as\nindicated  in  Fig.  4(a)  and  4(b)  and  Eqs.  (4)  and  (6),  for  the  two  angles  of  incidence  i  =  3°  and  45°,\nrespectively. Note that after the separation, the signal indeed splits into the square-shape hysteresis loop\ntypical for linear MO signals, and the characteristic M-shape loop of the quadratic magnetooptics. \nFig. 4: Analysis of magneto-optical (MO) signals at an angle of incidence i = 3° (left column) and i = 45° (right\ncolumn). For extracting the MO effects odd and even in magnetization, the signals were decomposed to symmetrical\n(black line) and antisymmetrical (red line) components with respect to Hext. An example of results of this procedure is\nshown on for MO signals measured for angles of incidence i = 3° (a) and i = 45° (b) using  = 0°. The original data are\nthose from Fig. 3. The amplitude of the particular MO effect A MOLFE  and A MOCME  for each polarization angle  was\ndetermined from the size of the „jumps“ in hysteresis loops, as indicated in (b). The corresponding polarization\ndependencies of LFE and CME are shown for i = 3° (c) and i = 45° (d). Points stand for the measured data. Green line\ncorresponds to fit to Eq. (3), with amplitude PLFE  =  (310 ± 20 ) rad, assuming the switching takes place between the\neasy axes separated by 1-2 = 120° . Red line is a fit to Eq. (5), where PCME = (450 ± 30) rad for i = 3°, and PCME = (320\n± 20)rad for i = 45°. Further comparison with the analytical model for polarization dependence of MO signals at the\ntwo angles of incidence for i = 3° (e) and i = 45° (f) show an excellent agreement, confirming validity of the model.\nParameters of the model are the same as for Fig. 3\n13For  further  analysis  we  need  to  determine  amplitudes  of  the  MO  effects  attributed  to  the  particular\nmagnetization switching process. The amplitude of the even component A CME is taken from the first 120°\nmagnetization switching, as indicated in Fig. 4(b). The amplitude of the odd component , is obtained from\nthe same 120°switching as the size of the square-shape signal, i.e. φ1=−φ2=60° in Eq. (5). This method\nalso  eliminates  potential  contributions  from  the  paramagnetic  GGG  substrate,  where  no  step-like\nhysteretic behavior is expected.\n The resulting amplitudes of the separated MO signals are shown as a function of the light polarization in\nFig. 4(c) and (d) for the angles of i = 3° and 45°. Points in the graphs indicate the values extracted from the\nexperiments, lines are fits by Eqs. (3) and (5), from which the values of the effective MO coefficients PCME =\n(320 ± 20) rad and PLFE  =  (310± 25 ) rad were extracted for the 45° incidence angle, and PCME =  (450 ±\n30) rad for the near-normal incidence.  Note that even for i = 45°, which is optimal for observation of the\nlongitudinal Faraday effect, the strength of the quadratic CME exceeds that of the linear LFE, and reaches\nthe values known from Heusler alloys, which are among the highest observed so far [28].\nFig. 5: Amplitudes of CME (left column) and LFE (right column) effect as a function of initial polarization angle ,\nextracted from the hysteresis loops obtained from the analytical model. The amplitudes ACME and ALFE are obtained\nby the same method as in Fig. 4.\nPolarization dependence of (a) CME and (b) LFE effects for various angles of incidence i and fixed sample thickness of\n50 nm. The angles of incidence of i = 0,30,40,60 and 80° are shown, the arrow indicates increase of i .  Clearly, the\nLFE is very strongly angle-dependent, while CME is much less affected. \nThe same feature can be observed for changing of the sample thickness d. While the polarization dependence of CME\n(c)  does  not  depend  on  the  sample  thickness,  LFE  (d)  can  vary  significantly.  The  thicknesses  of  d  =\n5,10,20,50,100,200,500 and 1000 nm are displayed for the angle of incidence i = 45° \n14The  observed  polarization  dependence  of  MO  signal  amplitudes  can  be  understood  in  terms  of  our\nanalytical  model.  Keeping  all  the  input  parameters  of  the  model  fixed,  we  calculated  polarization\ndependencies  of  the  individual  MO  amplitudes  extracted  from  the  modelled  MO  hysteresis  loops.\nResulting dependencies are presented Fig. 4.(c) and 4(d) for i = 3° and 45°, respectively. The theoretical\ncurves follow the experimental data very well even for the LFE signal close to the normal incidence, which\nproofs the validity of our analytical approach. We can therefore extend the predictions of the model to\nconditions that are not easy to systematically change in experiments, particularly the dependence on the\nangle incidence and the sample thickness. In Fig. 5 we illustrate these dependencies separately for the\nquadratic CME (graphs in the left column) and linear LFE (right column). The material parameters for each\ncurve are set such that PCME = 450 rad at near-normal incidence and PLFE = 310 rad for the 45° angle of\nincidence.  We  also  consider  sample  rotation  angle  γ=0°  for  simplicity.  We  did  not  simulate  the  full\nmagnetization  dynamics  for  the  purpose  of  Fig.  5  but  we  rather  used  a  simplified  scheme  where  we\nconsider 120° magnetization change for CME and LFE. Remarkably, there is a significant difference in how\nthe polarization dependence of the linear and the quadratic MO effect is affected by changing both the\nsample thickness and the angle of incidence. Non-intuitively, the strengths CME is only weakly affected by\nboth these  parameters. The shape of polarization dependence is modified for large angles of incidence I\n[Fig. 5(a)] but the maximum value of A CME remains virtually unchanged. The sample thickness has almost\nno effect on the CME signals [Fig. 5(b)]. In contrast, the linear MO signals are drastically modified by both\nthese parameters. As expected, the linear MO effect decreases for smaller angles of incidence [Fig. 5(c)],\neventually disappearing at normal incidence. However, not only the magnitude but also the shape of the\npolarization dependence is affected. This complex behavior of LFE results from interferences: while CME\nin our case results only from magnetic linear dichroism (i.e., the difference of absorption coefficient of the\ntwo  orthogonal  optical  polarization  eigenmodes),  LFE  is  a  consequence  of  birefrigence  of  elliptically\npolarized  eigenmodes  (i.e.,  the  difference  of  the  corresponding  effective  refractive  indexes).  As  a\nconsequence,  extrema  of  the  MO  response  appear  at  resonances.  We  observe  in  Fig.  5(d)  only  a\nmonotonous trend of the curve shaping with the increasing sample thickness because of a relatively large\nsample absorption at the used wavelength which prevents multiple wave roundtrips inside the YIG layer\neven at the position of the first resonance. The shape of the LFE response therefore relaxes from the zero-\norder resonance (small sample thickness) up to no multiple reflections for large sample thickness and\nbecomes saturated.  This complex modification of LFE response makes it difficult to optimize the sample\nthickness for magnetometry measurements, and using the quadratic CME therefore provides a significant\nadvantage.15It is important to stress that our model is independent of the magnetic anisotropy of the particular sample.\nThe conclusions drawn from the model are, therefore, universal for a series of samples with identical bulk\nmagnetic properties. \nB.ROT-MOKE measurements\nIn  order  to  further  investigate  the  nature  of  the  Cotton-Mouton  effect,  we  performed  ROT-MOKE\nmeasurements [24-25] close to normal incidence geometry to eliminate the linear MO contribution to the\nsignal.  The  ROT-MOKE  method  provides  a  more  efficient  and  sensitive  tool  for  extracting  the  MO\ncoefficients, without necessity of modification of the initial light polarization orientation.\n Examples of the as-measured ROT-MOKE signals are shown as open symbols in Fig. 6(a) and 6(b) for low\ntemperature (T= 10K) and room temperature, respectively. As we are interested in even MO signals only,\nthe small contribution of the linear MO effect was removed by symmetrization of the curve with respect to\nthe angle of the external field H [solid symbols in Fig. 6(a) and 6(b)]. The symmetrized curves display a\nclear  harmonic  behavior,  which  indicates  that  the  field  of  205  mT  was  large  enough  to  saturate\nmagnetization, which follows exactly the direction H.  We were therefore able to fit the data to equation\n(4) (red line) with angle of magnetization equal to the angle of Hext ( H = M  ). From the fits we obtain the\nMO coefficient PCME= (450±40) rad at 10 K, which is in excellent agreement with the value extracted from\nthe hysteresis loops. However, the value of PCME coefficient decreases to PCME= (230±20) when heating the\nsample to room temperature. To understand this change, we measured the temperature dependence of P\nCME. Generally, since the CME is of the second order in magnetization, scaling of PCME with a square of\nsaturation magnetization Ms is expected [22,31]. As illustrated in Fig. 6(c), the good correlation between\nthese two quantities confirms the intrinsic magnetic origin of the CME effect.16Fig. 6:  Rotation of polarization plane  as a function of the direction H of the external magnetic field of a fixed\nmagnitude 0Hext = 205 mT, measured at 20 K (a) and at room temperature (b). Polarization was set to  =0°. Open\nsquares indicate the as measured data, full squares are symmetrized in H to remove linear magneto-optical effects.\nRed line is a fit to Eq. (5) for H = M as Hext is large enough to saturate magnetization of the sample. Magneto-optical\ncoefficient for Cotton-Mouton effect obtained from the fits are PCME = 450 rad at 20 K and PCME = 230 rad at 300 K.\nThe decrease of PCME  with increasing temperature is well correlated with the reduction of square of the saturation\nmagnetization M S2 , values of which were taken from reference Ref. 44 and converted to SI units (c).  The spectral\ndependence of PCME   measured at room temperature (d) clearly shows a peak at around 3.1 eV.\nThe physical origin of the intrinsic CME effect can be unveiled by its spectral dependence. For this purpose,\nwe  extracted  PCME coefficient  from  the  room-temperature  ROT-MOKE  data  measured  at  several\nwavelengths. The obtained spectrum of PCME is presented in Fig. 6(d). The maximum of CME occurs at\naround  = 400 nm (3.1 eV), and its amplitude drops rapidly when the laser is detuned from the central\nwavelength  by  more  than  10  nm.   The  sharp  increase  of  the  CME  response  around  3.05-3.1  eV\ncorresponds energetically to transitions O-2p to Fe-3d band states of YIG [45]. Giant Zeeman shift of this\ntransition level was recently reported in a 50 nm thin [111] YIG film [45]., which is very similar to the\nsample studied in our work. Its origin was attributed to the combination of strong exchange interaction of\nFe-3d  orbitals  and  the  effect  of  spin-orbit  coupling  on  the  Fe-3d  bands.  Similar  combined  act  of  the\nexchange  of  magnetic  ions  and  spin-orbit  coupled  valence  bands  is  known  from  diluted  magnetic\nsemiconductors  [28,46],  the  systems  that  are  typical  for  their  strong  quadratic  MO  response  with  a\nsignificant peak on the Zeeman-split energy level [46 ]. Analogically, strong quadratic response of the YIG\nthin layers can be expected [46].\nApart from the intrinsic MO effect, impurity states can significantly influence the MO response of thin\nfilms. Lattice defects are known to occur during growth of the very thin YIG layers, particularly due to the\nmigration of Fe3+ and Gd3+ ions across the interface during the post-growth annealing [14,15,18]. Similarly,\ngadolinium doping can be responsible for the decrease in saturation magnetization of the PLD-grown thin\n17YIG  layers  [17].  However,  it  affects  mostly  the  interfacial  layer  of  a  few  nanometers,  which  orders\nantiferromagnetically,  reducing  the  magneto-optical  response  of  the  layer  [47],  and  cannot  thus  be\nresponsible for the origin of the observed Cotton-Mouton effect. In fact, previous works based on the\nquadratic MO response of YIG [20-21] were always performed in a spectral region close to 400 nm, even\nthough  thin  films  of  various  thickness,  prepared  by  different  methods  and  presumably  containing\ndifferent  level  of  impurities,  were  studied.  Though  the  choice  of  the  wavelength  was  not  performed\nsystematically in these works and the amplitude of CME was not evaluated, the wavelengths always lay\nclose to the optimum value that was identified from our experiments. We are thus led to a conclusion that\nthe observed strong CME response is very likely intrinsic to any YIG thin layer, and it is not related to\nunintentional doping or any type of defects. Therefore, the MO effect seems to be universally applicable.\nV.Conclusions\nIn summary, we have shown the presence of a strong Cotton-Mouton effect in a 50-nm thick epitaxial\nlayer of YIG in the spectral region close to 400 nm. We measured both magneto-optical hysteresis loops\nand ROT-MOKE data that enabled us to extract the values of CME coefficient.  The maximum PCME= 450 \nrad obtained for our YIG layer is comparable to the giant quadratic magneto-optical response of Heusler\nalloys [27] or ferromagnetic semiconductor GaMnAs [28]. Spectral and temperature dependence both\nindicate an intrinsic origin of the effect, which demonstrates its universal applicability for the magneto-\noptical magnetometry. This functionality of the MO experiment was demonstrated in determining cubic\nmagnetocrystalline  anisotropy  of  the  thin  YIG  film,  which  is  the  dominant  magnetic  anisotropy  in  the\nsample. \nThe measured signals were further analyzed using an analytical model based on a calculation of the overall\noptical  and  magneto-optical  response  of  the  thin  YIG  layer  on  GGG  substrate.  The  model  enables  to\npredict properties of longitudinal Faraday and Cotton-Mouton effect for variable sample thicknesses and\nangles  of  incidence,  which  are  parameters  crucial  for  many  thin-film  experiments.  The  calculation\nrevealed that while LFE varies strongly both with angle of incidence and sample thickness, CME has a\ncomparable magnitude but much weaker sensitivity to both the studied parameters. Therefore, using the\nquadratic  CME  provides  an  advantage  against  the  linear  LFE,  particularly  when  normal  incidence  is\ndictated by the experiment geometry, which is the case for most of the opto-spintronics experiments. Our\ncombined theoretical and experimental approach enables to optimize the condition for the experiment in18terms of choice of proper light source or measurement geometry, which can lead to a significant increase\nof signal-to-noise ratio and sensitivity in opto-spintronics experiments.\nAcknowledgmements:\nE.S. and T.O. contributed equally to the work.\nThe  authors  would  like  to  acknowledge  fruitful  discussions  with  Dr.  Jaroslav  Hamrle  and  Dr.  Eva\nJakubisová. This work was supported in part by the INTER-COST grant no. LTC20026 and by the EU FET\nOpen RIA grant no. 766566. We also acknowledge CzechNanoLab project LM2018110 funded by MEYS CR\nfor the financial support of the measurements at LNSM Research Infrastructure as well as the German\nresearch foundation (SFB TRR173 Spin+X 268565370 - projects A01 and B02).\n[1]  M.  Wu,  A.  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Grudichak, J. Sklenar, C. C. Tsai, M. Jang, Q. Yang, H. Zhang, and J. B. Ketterson, J. Appl. Phys,\n120, 033905 (2016)\n[42] D. D. Stancil, A. Prabhakar, Spin waves – theory and applications (Springer, New York, 2009)\n[44] N. Beaulieu, N. Kervarec, N. Thiery, O. Klein, V. Naletov, H. Hurdequint, G. de Loubens, J. Ben Youssef,\nand N. Vukadinovic, IEEE Magnetics Letters 9, 3706005 (2018)\n[45]  R.  Vidyasagar,  O.  Alves  Santos,  J.  Holanda,  R.  O.  Cunha,  F.  L.  A.  Machado,  P.  R.  T.  Ribeiro,  A.  R.\nRodrigues, J. B. S. Mendes, A. Azevedo, and S. M. Rezende, Appl. Phys. Lett. 109, 122402 (2016)\n[46] E. Oh, D. U. Bartholomew, A. K. Ramdas, J. K. Furdyna, and U. Debska, Phys. Rev. B 44, 10551 (1991) \n[47] E. Lišková Jakubisová, S. Višňovský. H. Chang, M. Wu, Appl. Phys. Lett. 108, 082403 (2016)1 \n Giant quadratic magneto-optical response of ultra-thin YIG films for sensitive magnetometric \nexperiments: Supplementary material \nE. Schmoranzerová1*, T. Ostatnický1*, J. Kimák1, D. Kriegner2,3, R. Schlitz3, H. Reichlová2,3, A. Baďura1, Z. \nŠobáň2, M. Münzenberg4, G. Jakob5, E.-J. Guo5, M. Kläui5, P. Němec1 \n1Faculty of Mathematics and Physics, Charles University, Prague, 12116, Czech Republic \n2Institute of Physics ASCR v.v.i , Prague, 162 53, Czech Republic \n3Technical University Dresden, 01062 Dresden, Germany \n4Institute of Physics, Ernst-Moritz-Arndt University, 17489, Greifswald, Germany \n5Institute of Physics, Johannes Gutenberg University Mainz, 55099 Mainz, Germany  \n \n1. Magnetic characterization \n \nMagnetic properties of the samples were characterized by SQUID magnetometry.  SQUID magnetic \nhysteresis loops, detected at several ambient temperatures for a fixed direction of the external magnetic \nfield along [2-11], are shown in Fig. S1(a). As expected [s1], the values of magnetic moment in saturation \nincrease with decreasing temperatures. Simultaneously, the coercive field slightly increases but remains \nwell below 20 Oe in the whole temperature range.  \n \nFig. S1: SQUID magnetometry at variable temperatures. (a) Magnetic hysteresis loops measured by SQUID \nmagnetometry in direction [2-1-1] at several ambient temperatures. As expected, the saturation magnetization M s   \nincreases when temperature is decreased, which is accompanied by a slight increase in coercive field. The \ntemperature dependence is highlighted in (b). The values of M s were recalculated to CGS units for comparison with \nvalues in Ref.  44 \nThe values of saturation magnetization M s can be extracted from the SQUID hysteresis loops by \nrecalculating to the volume of the magnetic layer. However, this parameter is burdened with a relatively \nlarge error in our experiment. The thickness of the crystalline YIG layer of d = (46.0 2.4) nm is known from \nthe XRD experiment. The sample area is estimated to S = (23 3)x10-2 cm2 , with the error resulting from \n2 \n an irregular shape of the sample.  Furthermore, the SQUID hysteresis loops were affected by a strong \nparamagnetic background of the GGG substrate, which also increased the error in estimation of Ms. These \nissues do not allow us to determine precisely the detailed temperature dependence of Ms, required for \ncomparison with the magneto-optical experiment (see Fig. 6 in main text). Instead, we measured the \nvalues of Ms only at several temperatures to verify the general trend of the dependence.  In Fig. S1(b), the \nmeasured values of saturation magnetization Ms are compared with the published results, obtained on a \nnominally similar sample (Ref. 44). Clearly, the data follow a similar trend and the values of Ms are \ncomparable in size, which justifies utilization of the published results for comparison with our magneto-\noptical data in the main text. \n \n2. Tracing the magnetization trajectory in the measurement of the hysteresis loops \n \n \nThe knowledge of the magnetization vector path in the [111] oriented YIG thin layer during the \nmeasurement of the hysteresis is essential in interpretation of the experimental data and determining the \nmagneto-optical response coefficients. We observe two distinct points in Fig. 3(a) in the main text in each \nof the branches of the hysteresis loops where the magneto-optical signal abruptly changes when smoothly \nchanging the external magnetic field magnitude. These points represent a rapid change (jump in the \nfollowing) of the magnetization vector from the vicinity of one magnetization easy axis towards another \none. For the interpretation of the data, it is then essential to know between which directions the \nmagnetization jump takes place. \n \nThe discussion of [111] oriented layers of cubic materials is not straightforward like in [001] oriented \nsamples. In the latter case, the demagnetizing field and the uniaxial out-of-plane anisotropy usually push \nthe magnetization vector to the plane of the sample where there are at most four easy directions for the \nmagnetization due to the cubic symmetry of the material. In the case of [111] orientation, on the other \nhand, all the easy axes lie out of the sample plane. The demagnetizing field can be strong enough to tilt \nthe easy axes to the close vicinity of the sample plane, however, they always provide a nonzero out-of-\nplane component. The projection of the three easy axes directions to the sample plane reveals a sixfold \nsymmetry in the contrast to [001] oriented layers with a fourfold symmetry. This behavior is illustrated in \nFig. S2 where we plot the magnetization free energy functional F as a function of the polar ( M)  and \nazimuthal ( M) angle. We consider the form of the functional including first- and second-order cubic terms \n[s3,s4] and we define the polar angle  with respect to the crystallographic axis [111] and the azimuthal \nangle  = 0 in the direction [21ത1ത] with an appropriate index referring to the magnetization position (index \nM) or the direction of the external magnetic field (index H). The resulting functional takes the form (in the \nSI units)     \n \n���= −𝜇𝐻𝑀[sin𝜃ெsin𝜃ு+cos𝜃ெcos𝜃ுcos(𝜑ு−𝜑ெ)]+ቀଵ\nଶ𝜇𝑀ଶ−𝐾uቁsinଶ𝜃ெ \n  +𝐾c1\n12ൣ7cosସ𝜃ெ−8cosଶ𝜃ெ+4−4√2cosଷ𝜃ெsin𝜃ெcos3𝜑ெ൧ \n  +𝐾c2\n108ൣ−24cos𝜃ெ+45cosସ𝜃ெ−24cosଶ𝜃ெ+4\n−2√2cosଷ𝜃ெsin𝜃ெ(5cosଶ𝜃ெ−2)cos3𝜑ெ+cos𝜃ெcos6𝜑ெ൧ ,                    3 \n     \n(S1) \n \nwhere 0 is the vacuum permeability and we consider the following values of constants [s5] of a bulk \nmaterial: magnetization M = 196 kA/m, first-order cubic anisotropy constant Kc1 = 2480 J/m3, second-\norder cubic anisotropy constant Kc2 = 118 J/m3. The external magnetic field is set to zero for the purpose \nof Fig. S2: 0H = 0 mT. The uniaxial out-of-plane anisotropy is set to compensate the effect of the \ndemagnetization Ku = 24 kJ/m3 in Fig. S2(a) while we consider Ku = 0 J/m3 in Fig. S2(b). We clearly observe \nthat the demagnetizing field causes the drag of the energy density minima towards the sample plane, \nleading to a deviation angle (relative to the sample plane) of only a few degrees. At the same time, it \nweakens the resulting total magnetic anisotropy. \n \nOur analysis of the motion of the magnetization vector in the external field is based on an estimation of \nthe magnetic anisotropic constants of the sample. These constants are determined from positions of the \njumps in the hystereses in Fig. 1(c)-(d) in the main text, which were measured using the magnetic field \ninclination angle with respect to the sample plane H = 0° and H = 45°. The value of the low-temperature \nmagnetization of our sample is M = 174 kA/m [2186 G in cgs units, see Fig. S1(b)] and, according to the \nbulk values, we set Kc2 = Kc1/21 [s5]. The out-of-plane uniaxial anisotropy is, however, uncertain and strongly \nsample-specific (see e.g., Refs. 18 and 40 of the main text). We therefore performed a fit of the \nexperimental data considering the value Ku=0, resulting in the values Kc1=4.68 kJ/m3 (kc1 = Kc1/2M = 540 \nOe). This number is higher compared to the bulk value reported in Ref. S5. On the other hand, other \npublished low-temperature values are even higher [s6] or reveal a clear tendency [s1] that the cubic \nanisotropy constant could be significantly higher than the bulk value. \n \nAs shown in Fig. S2(b), the magnetization free energy density reveals a very narrow valley. The \nmagnetization vector is constrained in the polar direction by the demagnetizing field while modulation of \nits effective potential is weak in the azimuthal direction. We may therefore regard its motion in the \nazimuthal direction in weak magnetic fields as effectively one-dimensional, similarly to the [001] oriented \nlayers. The effective one-dimensional free energy density is then calculated from Eq. (S2): \n \n     𝐹eff(𝜑ெ)=minఏಾ𝐹(𝜃ெ,𝜑ெ)    (S2) \n \nwhere min ఏಾ denotes the minimum value with respect to angle 𝜃ெ.  \nWe plot the effective free energy density for the fitted value of the cubic anisotropy Kc1= 4.68 kJ/m3 in \nFig. S3. The cosine-like curve can be fitted by an effective one-dimensional free energy functional: \n \n     𝐹inplane=𝐾sinଶ3𝜑ெ     (S3) \n \nwith the effective K 6 anisotropy constant defined in accordance to the cubic in-plane anisotropy of [001] \noriented layers. Figure S3 shows the comparison of the effective free energy density as defined by Eq. (S2) \n(black solid line) and the fitted Eq. (S3) (blue dashed line). The two curves coincide and therefore we may  \n 4 \n Fig. S2: Magnetization free energy density in [111] oriented YIG sample considering parameters of a bulk sample [s3]: \nM=196 kA/m, K c1=- 2480 J/m3, K c2= -118 J/m3. The plots represent the bulk material with (a) the demagnetizing field \nexactly compensating the out-of-plane anisotropy and (b) no effect of the demagnetization. Note that there is a \ndifferent y-scale in parts (a) and (b). \nregard the system as effectively in-plane with the fitted value of the sixfold anisotropy K6 = 245 J/m3 (k6 = \nK6/2M = 28 Oe). \nAs noted above, the strength of the uniaxial out-of-plane anisotropy is not known and the value Ku=0 kJ/m3 \nhas been used in the fitting procedure. It is therefore necessary to verify the robustness of the modelled \ncubic anisotropic constant against nonzero Ku. Using, for example, the value Ku=10.4 kJ/m3 (ku = Ku/2M=1.2 \nkOe, approx. one half of the strength of demagnetization), we fit the cubic anisotropy constant Kc1=3.48 \nkJ/m3 (400 Oe). The results are also presented in Fig. S3 (red solid line). Clearly, the effective in-plane free \nenergy density changes as compared to the situation with Ku=0 but the difference is only of the order of \n10%. We may therefore conclude that the exact knowledge of the out-of-plane anisotropy is not necessary \nfor the analysis of the in-plane magnetization dynamics and we may consider Ku=0 in the forthcoming \ncalculations.  \n Besides the cubic anisotropy, an additional stress-induced uniaxial anisotropy oriented in the plane of the \nsample is known to occur [s7]. We took this fact into consideration, performing the fit of the experimental \ndata by taking the strength and the orientation of the anisotropy field as free parameters. Our results show \nthat the effect of any eventual in-plane stress is negligible. This agrees with the reciprocal space map \nmeasurements, presented in Fig. 1(c) of the main text.  \n \n \n \n \n \n \n \n \n \n5 \n Fig. S3: Effective free energy density F as a function of in-plane (azimuthal) magnetization angle M, calculated from \nEq. (S1) by considering the parameters of Fig. (S2), and the out-of-plane uniaxial anisotropy of K U= 0 J/m3 (black solid \nline) and K U= 10.4 J/m3 (red solid line). The blue dashed line represents a fit by a sixfold in-plane anisotropy using Eq. \n(S3), giving the effective parameter K 6= 245 J/m3 \nAs a result of the analysis of the sample magnetic anisotropies, we can plot the trajectory of the \nmagnetization vector during the measurement of the hysteresis loops. We consider for this purpose no \nuniaxial out-of-plane anisotropy ( Ku=0) and the appropriate values of the cubic anisotropic constants Kc1 = \n4.68 kJ/m3 and Kc2 = Kc1/21. The resulting trajectory is compared with the experimental hysteresis curves \nshown in Fig.  S4(a). Here the points where the magnetization rapidly changes its orientation are marked \nby letters A-J. The situation is schematically depicted in Fig. S4(b), where the in-plane projections of the \neasy directions (red lines) and projections of the magnetization vector (blue arrows) at the positions \nmarked in Fig. S4(a) are presented.  The green arrows then depict the sense of the magnetization motion \nand its speed: solid lines stand for slow rotation of the magnetization vector while the dotted lines mean \nrapid changes (jumps). The magnetization motion naturally depends on the size of the uniaxial anisotropy \nthat changes the free energy density F (see Fig. S2). In Figs. S4(c)-(d) we plot the dependency of the \nazimuthal and polar angles of the magnetization vector on the external field for one branch of the \nhysteresis. The dependencies are depicted for both cases of the zero and nonzero out-of-plane uniaxial \nanisotropy and also for the effective model of the sixfold in-plane anisotropy (dashed curve). As we may \nexpect, the curves coincide in the plot of the azimuthal angle while they reveal more pronounced \ndifferences in the polar angle. The tracing of the exact magnetization path in the out-of plane direction is \nnot, however, the subject of our discussion, since it does not reflect significantly in our measurement. \nInstead, the graphs help us to confirm the correctness of the data analysis described below.  \n \nThe plots in Fig. S4 show two jumps of the magnetization vector for each of the branches of the hysteresis \nloops between the points B, C and D, E for one branch of the hysteresis loop, and G, H and I, J for the other \nbranch. We may observe in Fig. S4(c) that the in-plane orientation of the magnetization changes by \n120°(B→C) and 45°(D→E). The change of the out-of-plane orientation is zero in the first case while it is \nnonzero in the latter, as apparent from Fig. S4(d). While we could neglect the small deflection of the \nmagnetization from the sample plane in the analysis of the cubic in-plane anisotropy, it plays a significant \nrole in magneto-optical (MO) experiments, as it generates a contribution due to the polar MO effect. The \nmagnetization jumps D→E is the case in which the deflection angle rapidly changes and therefore the \nchange of the MO response during this jump is composed of both the in-plane and out-of-plane \ncomponents which cannot be further separated. The magnetization out-of-plane component stays \nconserved, on the contrary, during the B→C jump which is then the right point for extraction of the in-\nplane MO effect amplitudes, as depicted in Fig. 4(b) and explained in the main text. Note also that the \n6 \n magnetization change between any pair of labelled points in Fig. S4 reveals a nonzero out-of-plane \ncomponent change except for the B→C jump which then turns out to be the only choice for the \nexperimental determination of the magneto-optical observables.  \n \n \n \n \nFig. S4: Tracing of the magnetization vector path when external magnetic is applied at H = 45 deg in the out-of-plane \ndirection: (a) Hysteresis loop obtained from the magneto-optical experiment with highlighted important points A-F \nand F-J for the two branches of the loop, respectively. (b) Schematic representation of easy axes (red lines) and \npositions of magnetization during the switching process (blue lines). Green solid arrows represent slow motion of \nmagnetization, while dashed arrows stand for „steps“ in magnetization orientation. (c) Azimuthal and (d) polar angle \ndependency of magnetization orientation on external field magnitude, modelled for different anisotropy constants \nKu and K 6. The labelled points match the points in (a) and (b), respectively.   \n \n \n \n[s1] N. Beaulieu, N. Kervarec, N. Thiery, I. Klein, V. Naletov, H. Hurdequint, G. de Loubens, J. B. Youssef, N. \nVukadinovic, IEEE Magn. Lett. 9, 3706005 (2018).  \n[s2] C.Dubs et al.: Phys.Rev.Mat. 4, 024416 (2020) \n[s3] S. Lee, S. Grudichak, J. Sklenar, C. C. Tsai, M. Jang, Q. Yang, H. Zhang, J. B. Ketterson, J. Appl. Phys 120, \n033905 (2016).  \n[s4] A. Aharoni: Introduction to the theory of ferromagnetism (Oxford University Press, Oxford, 1996).  \n[s5] D. D. Stancil, A. Prabhakar: Spin waves – theory and applications (Springer, New York, 2009).  \n[s6] J. F. Dillion, Phys. Rev. 105, 759 (1957).  \n[s7] B. Bhoi, B. Kim, Y. Kim, M.-K. Kim, J.-H. Lee, and S.-K. Kim, J. Appl. Phys. 123, 203902 (2018).  \n"    },    {        "title": "2104.01250v1.Surface_plasmon_enhanced_photo_magnetic_excitation_of_spin_dynamics_in_Au_YIG_Co_magneto_plasmonic_crystals.pdf",        "content": "arXiv:2104.01250v1  [cond-mat.mes-hall]  2 Apr 2021Surface plasmon-enhanced photo-magnetic\nexcitation of spin dynamics in Au/YIG:Co\nmagneto-plasmonic crystals\nArtsiom Kazlou,†Alexander L. Chekhov,‡Alexander I. Stognij,¶Ilya Razdolski,†\nand Andrzej Stupakiewicz∗,†\n†Faculty of Physics, University of Bialystok, 15-245 Bialys tok, Poland\n‡Department of Physics, Freie Universit¨ at Berlin, 14195 Be rlin, Germany\n¶Scientific-Practical Materials Research Centre of the NASB , 220072 Minsk, Belarus\nE-mail: and@uwb.edu.pl\nAbstract\nWe report strong amplification of photo-magnetic spin prece ssion in Co-doped YIG\nemploying a surface plasmon excitation in a metal-dielectr ic magneto-plasmonic crys-\ntal. Plasmonic enhancement is accompanied by thelocalizat ion of theexcitation within\nthe 300 nm-thick layer inside the transparent dielectric ga rnet. Experimental results\nare nicely reproduced by numerical simulations of the photo -magnetic excitation. Our\nfindingsdemonstratethemagneto-plasmonic concept ofsubw avelength localization and\namplification of the photo-magnetic excitation in dielectr ic YIG:Co and open upa path\nto all-optical magnetization switching below diffraction li mit with energy efficiency ap-\nproaching the fundamental limit for magnetic memories.\nIn the last decade, the rapid progress of ultrafast optomagnetis m has opened up rich\npossibilities for optical data recording in magnetic materials, aiming at the heat-assisted\n1magnetic recording with 20 ×20×10 nm bit size in metallic media.1Meanwhile, the re-\nsearch has been fueled into alternative approaches, including the h ighly promising all-optical\nmagnetization switching demonstrated in multiple metallic systems in th e last few years.2,3\nYet, the all-optical switching in metals requires heating to high tempe ratures and demagne-\ntization.4Recently, based on a direct modification of the magnetic anisotropic energy bar-\nrier,5–7a non-thermal method for ultrafast photo-magnetic recording in dielectric garnets\nhas been developed8employing magnetization precession mechanism.9,10There, the key to\nfuture applications is the optics of photo-magnetic recording with t he light localization into\na∼20 nm-size spot, thus approaching the Landauer limit ( ∼0.25 aJ).11\nThis challenge of confining the photoexcitation within subwavelength volumes can be\naddressed with magneto-plasmonics, a rapidly developing branch of modern photonics.12–14\nThe high potential of magneto-plasmonics for local manipulation of m agnetic order with\nphotons is already established.15,16On the other hand, a highly interesting class of systems\nemerged recently, where magnetic dielectrics are covered with gra tings made of plasmonic\nmetals (such as Au).17–19The grating allows for the free-space excitation of surface plasmo n\npolaritons(SPPs)atbothinterfacesofthemetal,20,21whereasthetransparentdielectric layer\nensures low losses of the SPP excitations, in contrast to magneto- plasmonic systems with\ntransition metal ferromagnets.22Tailoring the electric field distribution inside the dielectric\nthrough the metal-bound SPP excitation enables novel nonlinear-o ptical and opto-magnetic\neffects.23–30\nIn this work, we employ this magneto-plasmonic grating approach fo r amplifying the\nphoto-magnetic spin precession in dielectric Co-doped yittrium iron g arnet (YIG:Co). We\nobserveastrongincreaseofthemagnetizationprecessionamplitu deinthevicinityoftheSPP\nresonance in the near-infrared. Numerical simulations of the elect ric field distribution show\nthat due to the SPP-induced light localization at the interface, the s pecific efficiency of the\nexcitation of the magnetization precession is enhanced 6-fold within the 300 nm active layer,\nas compared to the bare garnet film. Because photo-magnetic swit ching is a threshold effect,\n2our results represent an important step towards nanoscale phot o-magnetic data writing with\nfemtosecond laser pulses. They highlight the rich potential of magn eto-plasmonic approach\nfor scaling down towards nm-sized magnetic bits and further improv ing the energy efficiency\nof the all-optical magnetic recording.\nExperimental studies of the SPP-induced photo-magnetic anisotr opy were performed on\nAu/YIG:Co magneto-plasmonic crystals consisting of a 7.5 µm-thick garnet film covered\nwith Au gratings.31YIG:Co is a weakly opaque (in the near infrared spectral range) fer ri-\nmagnet with a saturation magnetization of 4 πMs= 80 Gs and a Neel temperature of 455 K.\nThe Co dopants display strong single ion anisotropy which depends on the ion’s valence\nstate. Therefore, resonant pumping of Co electronic transitions with laser pulses enables\ndirect access to the magnetic anisotropy and thus, the magnetiza tion. This results in the\nexceptionally strong photo-magnetic effect in YIG:Co,7ultimately allowing ultrafast mag-\nnetization switching with a single femtosecond laser pulse.8The Co doping also enhances\nmagnetocrystalline anisotropy and the Gilbert damping α= 0.2.8The YIG:Co garnet film\nwith a composition of Y 2CaFe3.9Co0.1GeO12was grown on a Gd 3Ga5O12(001) substrate.\nThe surface of the garnet thin film was treated with a low energy oxy gen ion beam.32A\n50 nm-thick Au grating with a 800 nm period (gap width 100 nm) was dep osited on the\ngarnet surface by ion-beam sputtering and perforated using FIB .33\nSPP-driven photo-magnetic excitation of magnetization precessio n was studied in the\ntwo-colour pump-probe transmission geometry schematically show n in Fig. 1. There, the\ntime-resolved Faraday rotation angle θFof the probe beam was monitored as a function of\nthe delay time ∆ tbetween the pump and probe pulses. The pump (probe) laser pulses with\na duration of 50 fs from a Ti-Sapphire amplifier at a 500 Hz (1 kHz) rep etition rate impinged\nat an angle of 27◦(17◦) from the sample normal, respectively, see Fig. 1. Employing the\noptical parametric amplifier, the wavelength of the pump beam λwas tuned in the near\ninfrared range within 1200-1350 nm, while the probe wavelength was set to 800 nm. The\nmore powerful pump beam was focused into a spot of about 100 µm in diameter on the\n3YIG:Co\nAu\n∆t 17\no\nprobe\npumpH\n27oθF\nzy\nxMHzy\nxM\nH\nL,z H\n∆t<0∆t>0\n50 fskspp\nFigure1: Schematics ofthe experimental pump-probetransmissio n geometry. The 50fs-long\nnear-IRp-polarized pump pulses excite a SPP resonance at the Au/YIG:Co inte rface. The\nelectromagnetic SPP field induces the photo-magnetic anisotropy HLin YIG:Co triggering\nthe magnetization precession. The latter is monitored through tra nsient Faraday rotation\nθFof the delayed probe pulses.\nsamples, resulting in an energy density of ∼4 mJ/cm2, while the spot size of the 30 times\nweaker probe beam was twice smaller. Both pump and probe beams we re polarized in their\nplane of incidence. The delay time ∆ tbetween the pump and probe pulses was controlled\nby means of a motorized delay stage.\nAn external magnetic field H= 3.2 kOe was applied in-plane of the sample along the\n[100] direction of the garnet crystal (Fig. 1) to set the magnetiza tionM∝bardblH. This enables\nmonitoring the magnetization precession through the transient Fa raday rotation θFpropor-\ntional to the out-of-plane component Mz. In this geometry, a torque Tis exerted on Mby\nthe effective field of the photo-induced magnetic anisotropy HL=/hatwideχ...EEM, whereEis\nthe electric field of light and /hatwideχis the photo-magnetic 3rd order susceptibility tensor.8This\n4torqueT ∝[HL×M] triggers the magnetization precession around its new equilibrium\ndetermined by the magneto-crystalline anisotropy field Hc,HandHL. After the relaxation\nofHL(on the scale of 20 ps7,8), the equilibrium position for the magnetization is restored\nand its precession proceeds around the original effective field direc tion. Because the incident\nlight pulse is p-polarized with the electric field E= (Ex,0,Ez) and the garnet has the cubic\n4mmsymmetry, a non-zero torque on Mxis generated by the following component of HL:\nHL,z=χzxzxExE∗\nzMx+c.c. (1)\nNotably, because the refractive index of garnets is relatively large (n/greaterorsimilar2 in the near-\ninfrared34), the normal-to-surface projection of the electric field Ezof the propagating light\nis suppressed. On the contrary, being one of the characteristic f eatures of the SPP excitation\nat a metal-dielectric interface, prominent enhancement of Ezin the dielectric promises an\namplification of the photo-magnetic anisotropy field and thus large a ngles of magnetization\nprecession.\nWe studied the magnetization precession in the spectral vicinity of t he SPP resonance at\nthe Au/YIG:Co interface ( ∼1275 nm at 27◦of incidence28,31). At each pump wavelength λ,\nwe measured the transient rotation of the probe polarization θat the two opposite directions\nofHand analyzed their difference, thus removing concomitant signal va riations of a non-\nmagnetic origin. To verify the symmetry and the magnitude of the ph oto-magnetic effect\nand enable a reference point, similar measurements were performe d on a bare YIG:Co film\nwithout an Au grating.\nThe time-resolved Faraday rotationtraces θF= [θ(+H)−θ(−H)]/2 obtained on the bare\ngarnet andthe Au/YIG:Co magneto-plasmonic crystal are shown in Fig. 2(a,b), respectively.\nIt is seen that both samples demonstrate similar magnetization dyna mics which can be rea-\nsonably well described by a single-mode precession at about 5 GHz fr equency, in agreement\nwith the previous findings.7Other temporal details of the magnetization dynamics will be\ndiscussed in a subsequent publication, and here we only focus on the precession amplitude.\n50 100 200 300 400 5000102030\n0 100 200 300 400 50004080120\n∆t (ps)Faraday rotation (mdeg)λ=1330 nm\n1300 nm\n1270 nm\n1240 nm\n1210 nm(a)\nYIG:Co Au/YIG:Co(b)\n∆t (ps)Faraday rotation (mdeg)λ=1330 nm\n1300 nm\n1270 nm\n1240 nm\n1210 nm\n1200 1240 1280 13200510\n02550\nAmplitude (mdeg)\nPump wavelength (nm)(c)\nYIG:Co Au/YIG:Co\nFigure 2: Time-resolved Faraday rotationin bare YIG:Co film (a) and A u/YIG:Co magneto-\nplasmonic crystal (b) induced by pump pulses of varied wavelength. The datasets are shifted\nvertically without rescaling. The red lines show the single-frequency damped sine function\nfitted to the data. c) Spectral dependence of the Faraday rota tion amplitude extracted from\nthe fits shown in panels (a,b).\nIt can be further seen in Fig. 2 that the absolute amplitude in the YIG :Co sample is approx-\nimately one order of magnitude stronger than that in the magneto- plasmonic Au:YIG:Co\ncrystal. However, and this will be the central point of the following d iscussion, the spectral\nbehavior of these two samples differs noticeably (Fig. 2c). Indeed, in the bare garnet the\nprecession is slightly enhanced at around λ≈1300 nm, whereas the Au/YIG:Co exhibits a\ndifferent spectral shape, with the largest precession amplitude ob served around λ≈1270 nm\n(accentuated with darker points in Fig. 2b).\nTo quantifytheSPP-induced enhancement oftheprecession amplit ude, weperformednu-\nmerical simulations of the SPP-driven excitation employing dedicated Lumerical software.35\nFrom Eq. 1 it can be shown that the photo-induced anisotropy field HLtakes the form:\nHL∝ E ≡ExE∗\nz+EzE∗\nx= 2|Ex||Ez|cosϕ, (2)\nwhereϕis the phase shift between the two components of the electric field Ex,Ez. We\ncalculated Einside the YIG:Co layer in the vicinity of the SPP resonance. Optical co nstants\nofAuweretakenfromRef. 36. Fig.3showsthecharacteristicspa tialdistributionof E(x,z,λ)\n61200 1240 1280 1320024680.0 0.5 1.00.00.10.2\n1200 1240 1280 13200102030\nPump wavelength (nm)Specific efficiency (mdeg/ m)µ\nYIG:CoAu/YIG:Co(d)0.40.60.81.0\nλ=1330 nmλ=1210 nmλ=1270 nm\nz, depth ( m) µ(c) (e)\n0.0\n0.5\n1.0z, depth ( m)µ\nx dimension ( m) µ-1 1 03\n0\n-3Nanolett b\nz-x)\nmax\n0z, depth ( m)µ0.0\n0.5\n1.0\nPump wavelength (nm)1200 1280 1240 1320(b)(a)\nPump wavelength (nm)Effective length ( m)µYIG:Co\n300 nm7.5 mµReflectivitylaser\nspectrumR\nRC\nBA\nC\nA B Cε\n1200 1280 1240 1320 1200 1280 1240 13200.4\n0.2\n0.0\n0.0 0.5 1.0\n8\n6\n4\n2\n0 01020300.40.60.81.0\nFigure 3: a) Numerically simulated spatial distribution E(x,z) inside the YIG:Co layer at\nthe SPP resonance ( λ= 1270 nm). The yellow bars indicate the Au grating. b-c) In-\ndepth profiles E(z,λ) integrated across the x-dimension. The dashed lines indicate three λ\nvalues which are shown in (c) in detail; B(λ= 1270 nm) corresponds to the resonant SPP\nexcitation. d) Calculated effective decay depth leffof the photo-magnetic excitation away\nfrom the Au/YIG:Co interface. The top dashed line at 7 .5µm indicates the total thickness\nof the YIG:Co layer. The bottom dashed line at 300 nm shows the shor test effective depth\nobtained at the resonance. e) (top) Calculated reflectivity spect ra, before ( Rc) and after\n(R) taking into account the spectral broadening due to the finite lase r pulse duration .\nFull dots: experimental Au/YIG:Co reflectivity. The shaded area illu strates the measured\nlaser spectrum at around λ= 1270 nm. (bottom) Specific efficiency of the photo-magnetic\nexcitation for the bare garnet (open dots) and Au:YIG/Co (full do ts). The solid red line is\nthe result of the numerical simulations.\ncalculated for the λ= 1270 nm, i.e. at the SPP resonance. In the Figure, we only show the\ntop1µm-thicklayerofYIG:Coalthoughthecalculationswereperformedfo rtheentiregarnet\nfilm. These data maps were collected for a set of λand for each of them, integrated across\nthex-axes to enable the spectral comparison of the in-depth distribut ionsE(z,λ). Those\nresults are shown in Fig. 3b, where the darker shaded regions illustr ate the enhancement\nand interfacial localization of the optical excitation. To highlight the latter point, we show\na few selected in-depth profiles (indicated with dashed lines in Fig. 3b) in Fig. 3c. At each\n7λ, from these profiles we calculated the effective excitation depth leff=|E ·(dE/dz)−1|z=0\ncontaining the most significant part of the excitation energy (see in Fig. 3d). It is seen there\nthat at the SPP resonance, the excitation is concentrated within t he 300 nm layer adjacent\nto the Au-garnet interface. On the contrary, away from the res onance, the effective depth\nleffincreases rapidly towards the total thickness of the YIG:Co layer ( 7.5µm).\nFinally, in order to compare the results of our calculations with the ex perimental data,\nspectral broadening due to the finite laser pulse duration has to be taken into account.\nIndeed, with 50 fs-short laser pulses, every experimental wavele ngth shown in Fig. 2 in\nfact represents a continuum of wavelengths around the indicated value. A characteristic\n50 nm-wide spectrum of a laser pulse S(λ) centered around λ= 1270 nm is exemplified in\nthe top panel of Fig. 3e with the shaded area. To verify the broade ning, we compared the\nexperimental reflectivity spectrum of the Au/YIG:Co magneto-pla smonic crystal and the\ncalculated data R(λ) convoluted with S(λ):Rc(λ)≡R(λ)∗S(λ) =/integraltext\nR(¯λ)S(¯λ−λ)d¯λ. A\nverygoodagreementwiththeexperimental dataallowsustoapplyt hesameproceduretothe\ncalculated in-depth integrated E(λ) =/integraltext\nE(z,λ)dzdata to obtainspecific excitation efficiency\nξ(λ) = [E(λ)χ(λ)]∗S(λ)/leff(λ). Here,χ(λ)accountsforthedispersionofthephoto-magnetic\ntensor/hatwideχdiscussed earlier, which can be extracted from the spectral depe ndence of the\nprecession amplitude obtained on a bare YIG:Co.\nThe results of this procedure are summarized in the bottom panel o f Fig. 3e. There,\nwe also show the specific excitation efficiency for the bare transpar ent garnet, assuming\nthat the effective depth equals its total thickness of 7 .5µm. Strong enhancement of the\nefficiency at the SPP resonance in the Au/YIG:Co sample is in a striking c ontrast with\nthe flat spectral dependence on a bare garnet. It is seen that th e SPP excitation results\nin the 6-fold enhancement of the specific amplitude of the magnetiza tion precession at the\nresonance.\nIt is worth emphasizing the similarities and differences between the sy stems studied here\nand in our recent work.28In both experiments, strong amplification of the spin dynamics\n8is inherently related to the SPP-driven localization of light at the inter face. Together with\nthe enhancement of the SPP electric fields, this effect is generic for the entire class of metal-\ndielectric plasmonic heterostructures and can be further optimize d for better performance.\nFrom the photonic point of view, another common important impact o f the SPP excitation\nconsists in the amplification of the out-of-plane projection of the e lectric field Ezwhich is\notherwise suppressed in the high- ndielectric.\nYet, the tensorial character of Eq. 1 reveals important differenc es between the two cases.\nIn general, the following form for the effective opto-magnetic field Heffcan be derived:37\nHeff,i(0)∝αijkEj(ω)E∗\nk(−ω)+χijklEj(ω)E∗\nk(−ω)Ml(0)+c.c., (3)\nwhere higher-order (in M) terms are neglected, and αijkandχijklare the antisymmetric and\nsymmetric susceptibility tensors, respectively.38The symmetry determines their dependence\non the phase shift ϕand thus on the polarization of light. The inverse Faraday effect\ncaptured by the first term in Eq. 3 requires ϕ∝negationslash= 0 which can be realized employing either\ncircularly polarized light or SPP excitation.28,39On the contrary, the photo-magnetic effect\nis present even in the absence of SPP, and moreover, the purely SP P-driven photo-magnetic\ncontribution vanishes due to ϕSPP=π/2 (cf. Eq. 2). Thus, the photo-magnetic SPP-\nmediated mechanism largely consists in the strong localization and enh ancement of the\nelectric field at the interface, whereas the excitation magnitude ∝ Eis determined by the\noptical interference of the SPP and incident fields. Similarly, the rev ersal of the precession\nphase when comparing magneto-plasmonic crystals with the bare ga rnet, as seen in Fig. 2,\ncan also be attributed to this interference of the electromagnetic fields.\nThe excitation mechanism of the spin dynamics here originates in the C o doping of YIG,\nenabling setting magnetization into motion through effective SPP-me diated photo-magnetic\nanisotropy field. The photo-magnetic mechanism offers the largest angles of magnetiza-\ntion precession available up to date at non-destructive laser fluenc es. Transient θFvalues\nobserved in our experiments correspond to about 5◦magnetization excursion from the equi-\n9librium, 1-2 orders of magnitude larger than that found in Gd,Yb-dop ed BIG28and LuIG.6\nIt can be conjectured that further amplification is feasible throug h structural optimization\nof the plasmonic geometry aimed at enhancing the excitation Eand employing numerical\nsimulations.\nFrom the perspective of magnetic data recording, the photo-mag netic switching through\nlarge-angle precession has been demonstrated exclusively in YIG:Co . The SPP photo-\nmagneticmechanismthusholdshighpotentialfortakingtheall-optic almagnetizationswitch-\ning onto the nanoscale. In our prototype system, the observed 6 -fold amplification of the\nspecific efficiency ξallows corresponding reducing of the laser fluence below the switchin g\nthreshold. The extrapolation to nm-sized bits yields about only ∼2 aJ deposited energy per\none bit, close to the fundamental thermodynamical limit for switchin g magnetic bits, thus\ncorroborating the exceptional energy efficiency of the photo-ma gnetic nanosize switching.\nAcknowledgement\nThe authors thank A. Kirilyuk (Nijmegen) and T. V. Murzina (Moscow ) for their sup-\nport. This work has been funded by the National Science Centre Po land (Grant No. DEC-\n2017/25/B/ST3/01305) and ERC Consolidator Grant TERAMAG No. 681917.\nReferences\n(1) Challener, W. A.; Peng, C.; Itagi, A. V.; Karns, D.; Peng, W.; Peng, Y.; Yang, X.;\nZhu, X.; Gokemeijer, N. J.; Hsia, Y.-T.; Ju, G.; Rottmayer, R. E.; Seigle r, M. A.;\nGage, E. C. Heat-assisted magnetic recording by a near-field tran sducer with efficient\noptical energy transfer. Nature Photonics 2009,3, 220–224.\n(2) Stanciu, C. D.; Hansteen, F.; Kimel, A. V.; Kirilyuk, A.; Tsukamoto, A.; Itoh, A.;\n10Rasing, T. 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New series, Group III, 27/e ; Springer: Berlin, 1991.\n(35) Lumerical Solutions, Inc. Vancouver, BC V6E 3L2, Canada.\n14(36) Johnson, P. B.; Christy, R. W. Optical Constants of the Noble Metals.Phys. Rev. B\n1972,6, 4370–4379.\n(37) Kimel, A. V.; Zvezdin, A. K. Magnetization dynamics induced by fe mtosecond light\npulses.Low Temperature Physics 2015,41, 682–688.\n(38) Landau, L. D.; Lifshitz, E. M. Statistical Physics. Part 1. Theoretical Physics ; Nauka:\nMoscow, 1976; Vol. 5.\n(39) Khokhlov, N.; Belotelov, V.; Kalish, A.; Zvezdin, A. Surface Plasm on Polaritons and\nInverse Faraday Effect. Magnetism and Magnetic Materials V. 2012 ; pp 369–372.\n15"    },    {        "title": "1602.01662v1.Investigation_of_the_unidirectional_spin_heat_conveyer_effect_in_a_200nm_thin_Yttrium_Iron_Garnet_film.pdf",        "content": "arXiv:1602.01662v1  [cond-mat.mtrl-sci]  4 Feb 2016Investigation of the unidirectional spin heat conveyer effe ct in a 200nm thin Yttrium\nIron Garnet film\nO. Wid1, J. Bauer2, A. M¨ uller1, O. Breitenstein2, S. S. P. Parkin2, and G. Schmidt*1,3\n1Institut f¨ ur Physik, Martin-Luther-Universit¨ at Halle- Wittenberg,\nHalle, 06120, Germany\n2Max Planck Institute of Microstructure Physics,\nHalle, 06120, Germany\n3IZM, Martin-Luther-Universit¨ at Halle-Wittenberg,\nHalle, 06120, Germany\n∗Address for correspondence: georg.schmidt@physik.uni-h alle.de\nWe have investigated the unidirectional spin wave heat conv eyereffect in sub-micron thickyttrium\niron garnet (YIG) films using lock-in thermography (LIT). Al though the effect is small in thin layers\nthistechniqueallows ustoobserveasymmetricheattranspo rtbymagnons whichleads toasymmetric\ntemperature profiles differing by several mK on both sides of t he exciting antenna, respectively.\nComparison of Damon-Eshbach and backward volume modes show s that the unidirectional heat\nflow is indeed due to non-reciprocal spin-waves. Because of t he finite linewidth, small asymmetries\ncan still be observed when only the uniform mode of ferromagn etic resonance is excited. The\nlatter is of extreme importance for example when measuring t he inverse spin-Hall effect because the\ntemperature differences can result in thermovoltages at the contacts. Because of the non-reciprocity\nthesethermovoltages reverse their sign with areversal of t hemagnetic fieldwhich is typically deemed\nthe signature of the inverse spin-Hall voltage.\nIntroduction\nIn 2013 An et al.have shown that by the excitation of nonreciprocal spin waves, so -called Damon-Eshbach modes\n(DEM), in a 400 µm thick Yttrium Iron Garnet (YIG) crystal heat can be transport ed in the absence of a temperature\ngradient [1]. The direction of the heat flow can be reversed by rever sing the applied magnetic field. In their exper-\niments they observed two different effects, both based on the non -reciprocity of the DEM, however, with different\nconsequences. On one hand the asymmetric excitation and propag ation of the spin-waves leads to an asymmetric\ntemperature profile which is dominated by the energy loss of the spin waves. These losses are highest at the point of\nexcitation and decrease with increasing distance from this point bec ause the amplitude of the spin-waves decreases.\nNonetheless, the energy transport still occurs from the point of higher temperature towards lower temperatures, how-\never, with a certain asymmetry with respect to the source becaus e of the asymmetric propagation of the spin-waves.\nIn an infinite sample no further effects would be observed. If on the other hand the sample is small enough for the\nspin-waves to actually reach the edge of the sample an additional eff ect occurs. In this case the spin waves, due to\nthe non-reciprocity cannot be reflected and thus deposit all rema ining energy as heat at the boundary. This results\nin an increase in temperature realizing actually heat transfer by mag nons into a warmer region and thus along the\ntemperature gradient as stated by An et al.(normally heat transport is against the temperature gradient). I n 2015\nmore detailed theoretical descriptions have been published based o n a phenomenological theory [2] and on micromag-\nnetic simulations [3]. The measurements reported so far were perfo rmed with an infrared camera on YIG films with\nthicknesses ranging from a few micrometers to hundreds of microm eters.\nCurrent research in magnonics concentrates more and more on th in ferromagnetic films and often uses the measure-\nment of the inverse spin-Hall-effect which relies on the measurement of very small voltages. It is thus quite important\nto know whether a similar scenario can also occur in ultra thin films ultima tely leading to temperature gradients\nwhich can lead to thermovoltages whose sign might then also depend o n the magnetic field.\nThe DEMs used in the experiment are nonreciprocal surface spin wa ves [4]. They propagate in opposite directions\n(/vectorkand−/vectork) at the top and bottom surface of the layer, respectively [1, 2]. Th e precessional amplitude of the DEM is\nmaximum at the surface and decays exponentially inside the film [5]. In order to cause unidirectional heat flow the\npopulation of /vectorkand−/vectorkmust be different. In a thick sample this condition is realized by exciting the spin waves using\na microwave antenna [1] which is in contact with one side of the layer. I n this case the spin waves at the surface close\nto the antenna are excited more strongly then at the other side re sulting in a net spin wave current in one direction.\nFor thin films with a thickness of a few hundred nm the situation is quite different. The distribution of the\nprecessional amplitude across the film thickness is almost uniform [5] as well as the excitation by the antenna at the\ntop and bottom surface described above. Nevertheless, the pop ulation and propagation of the /vectorkand−/vectorkvectors can\nstill be different. The bottom surface of a thin YIG film which is typically grown on gadolinium gallium garnet (GGG)2\nis in contact with the paramagnetic substrate while the top surface is in contact to air. Due to this fact the damping\nof the spin waves at the top and bottom surface can be different, le ading to increased damping of the spin waves\npropagating in one direction. Also the excitation by the antenna can be nonuniform for thin films. If a waveguide is\nused for excitation in the Damon-Eshbach geometry the in-plane an d the out-of-plane component of the microwave\nmagnetic field can both excite spin waves. The interference of thes e waves is destructive for k-vectors in one direction\nperpendicular to the antenna and constructive for the opposite d irection [5, 6]. Moreover in thin films the damping\nis higher and the spin-waves are less likely to reach the boundary of t he sample so we will not expect the heat pile up\nthat An et al. were able to observe.\nThus in thin YIG films the unidirectional spin wave heat conveyer effec t is expected to be present in terms of an\nasymmetric temperature profile, however, it will be small in magnitud e. Steady-state infrared cameras used in the\nmeasurements reported so far are most likely not sensitive enough to detect the effect in thin films. It is, however,\npossible to use lock-in thermography (LIT), which is well established for failure analysis in integrated circuits [7] and\nthe characterization of solar cells [8, 9]. The LIT is a dynamic method, which detects temperature modulation in\ninfrared images similar to electrical measurements using a lock-in amp lifier. With this technique the difference in\ntemperature between an excited and a non excited state is imaged. Temperature differences as small as 100 µK can\nbe resolved which is sensitive enough for the small effects described above as we will show later. Details about the\nused LIT system can be found in the Methods Section.\nExperimental Setup\nA sketch of our experimental setup is shown in Figure 1. We use a 200 nm high quality YIG film on Gadolinium\nGallium Garnet (GGG) substrate grown by liquid phase epitaxy. The da mping in ferromagnetic resonance (FMR) α\nof the layer is smaller than 1 ×10−4. To excite spin waves we use a coplanar waveguide (CPW) as an anten na, which\nis fabricated on top of the sample (size of the sample: 5 mm ×8 mm). Details about the fabrication and dimensions\nof the CPW can be found in the Methods Section.\nIn order to investigate the spin-wave spectrum FMR measurement s are performed. The corresponding experimental\nsetup is shown in Fig. 1. Measurements are done by applying a continu ous microwave to the antenna and measuring\nthe transmitted signal using a diode and a nanovoltmeter while sweep ing the magnetic field.\nLIT-\ncamerasignal\ngenerator\nRFout pulseINNano-\nVoltmeter\nRFprobe\ncoplanar\nwaveguideYIG\nGGG\n+Hdiode\nFIG. 1: Experimental setup for the FMR and the LIT measuremen ts. The FMR measurements are performed by applying a\ncontinuous microwave with a constant frequency and sweepin g the magnetic field. For the LIT measurements the microwave\nhas to be pulsed with the lock-in frequency, which is provide d by the camera.\nFor the LIT measurement the camera is placed above the sample and the lock-in reference frequency provided by\nthe camera is used to pulse the microwave excitation.\nFerromagnetic resonance (FMR) measurements\nFMR is measured at a constant frequency of 5 GHz and an excitation power of 1 dBm while sweeping the external\nmagnetic field from 990 Oe to 1260 Oe. The external field is always align ed in the plane of the layer and is either\nkept parallel to the antenna to excite the Damon-Eshbach mode wit h/vectork⊥/vectorM(DEM-geometry) or perpendicular to\nthe antenna to excite the backward volume mode with /vectork/bardbl/vectorM(BVM-geometry). The result of the two respective FMR\nmeasurements are shown in Figure 2 together with the calculated dis persion relation for an in-plane magnetized 2003\nnm YIG film. These dispersion curves for dipolar spin waves have been calculated using the equations given in [5].\nFork= 0 the uniform mode is excited at a field of 1128 Oe. For k/negationslash= 0 the dispersion relation exhibits two branches,\none for the DEM at lower fields and one for the BVM at higher fields. Co mparing the FMR spectra to the dispersion\nrelation we can see that for both geometries the uniform mode is exc ited. In the DEM geometry we also observe a\nsignal at lower /vectorHwhile for the BVM-geometry resonances at higher /vectorHappear in good agreement with the calculation.\nWe do, however, not observe the expected continuous spin-wave spectrum but several resonance lines. These lines\nappear because the geometry of the waveguide favours certain k -vectors corresponding to a fundamental k-value and\ninteger multiples with decreasing amplitude. It should be noted that t he DEM with the highest intensity overlaps\nwith the uniform mode and is not visible as a separate peak.\nFIG. 2: Result of the FMR measurement at 5 GHZ and the calculat ed dispersion relation for a 200 nm thin YIG film using\nthe following values: saturation magnetization 4 πM0=1700 Oe, gyromagnetic ratio γ=2.8MHz\nOe.\nLock-in thermography measurements\nDamon-Eshbach geometry\nFor the lock-in thermography measurements we use the knowledge obtained from the ferromagnetic resonance\nspectra. First the DEM-geometry is investigated. Using different fi eld values we excite different parts of the spin wave\nspectrum and take images with the camera. Figure 3, (a) shows an a mplitude image taken by the LIT camera at a\nmagnetic field of approx. 1120 Oe including a sketch of the rf tips con nected to the coplanar waveguide. Bright color\nin the LIT images corresponds to higher temperature than dark co lor. These measurements are repeated at different\nmagnetic fields which are marked in Figure 3,(b). These values corres pond to the maximum DEM peak which is not\noverlappingwith the uniform mode (1), the maximum DEM peak overlap pingwith the uniform mode (2), the uniform\nmode itself (3), and a position completely off resonance where no exc itation is expected at all (4), respectively. All\nmeasurements are repeated after reversing the direction of the magnetic field.\nFor the interpretation of the images it is important to first underst and what is actually visible. As a typical lock-in\ntechnique the LIT is able to eliminate backgroundsignals. In this case the absolute temperature is not measured. The\ndifference between the state without and with excitation, however , is measured with high accuracy. The temperature\ngiven by the gray scale is thus the increase in temperature arising fr om the excitation. We do not expect to measure\nnegative values because no active cooling is expected in our case. Th e background which is eliminated by the lock-\nin technique corresponds to room temperature. The accuracy of the measurements has also been determined by\ncomparing different measurements and we can show that the error is typically 0 .15mKor less. Error bars are thus\nomitted when temperature profiles are displayed in graphs. It shou ld be noted that the local emissivity can lead to\ndeviations of ±5−±10% which can locally distort the temperature profile. However, as w e will see later, we obtain\nour results mainly from the difference between two measurements in which the systematic error of the emissivity is\neliminated.\nIn order to correctly evaluate the data it is also necessary to unde rstand and eliminate possible side effects and\nartifacts. First the influence of the antenna should be discussed. As also outlined in the Methods Section the whole\nsample surface is covered by black ink to ensure uniform emission pro perties for heat radiation eliminating the low4\n+H-Hdifference\n(a)\n(b)(c)\ncoplanar\nwaveguideRFtips\n1\n2\n3\n4Tmin=-0.75mKto\nTmax=0,96mK\n+H\nTminTmaxTmin=0mKtoTmax=11mK\nFIG. 3: (a) LIT amplitude image with a sketch of the position o f the CPW and the rf tips. (b) FMR measurement in the\nDamon-Eshbach geometry (c) LIT measurements when the magne tic fields 1, 2, 3, 4 and the corresponding negative values are\napplied (left and center). Black corresponds to no increase in temperature while white indicates an increase of 11 mK. The\ncalculated difference (right) shows the expected effect. It s hould be noted that in the difference images the gray scale has been\nchanged for better visibility. The range is now between -0.7 5mK (black) or lower and 0.96 mK (white) or higher.\nradiation efficiency of a blank metal surface. The question remains, however, whether in the areas of metallization\nthe images really show the temperature of the YIG or of the metal a nd whether any heating really stems from the\nresonance in the oxide layer rather than from losses in the coplanar waveguide. The latter can easily be excluded\nby comparing pictures off-resonance and on-resonance which indic ate that the heating by the antenna observed off-\nresonance is much smaller than the heating by the resonance in the Y IG. In addition, this heating does not change\nduring a field reversaland can thus be eliminated by calculating the diff erence between imageswith opposing magnetic\nfields. Visibility of the YIG temperature through the metal is also gua ranteed because the thickness of the metal is\nonly a few hundred nm and the materials of the coplanar waveguide (A g and Au) have very high heat conductivity.\nAt the modulation frequency of 1 Hz the top of the antenna can alwa ys be considered at the same temperature as the\nYIG surface underneath. Lateral heat conduction in the thin met al film, however is much lower and will only slightly\nsmear any temperature profile originatingfrom the spin waves. The same holds for heat diffusion in the YIG. Anyway,\nboth, heat diffusion in metal and YIG are again independent from the magnetic field and can thus be eliminated by\ntaking the difference of two images with opposite field directions. The y will, however, slightly reduce the effect that\nwe intend to observe.\nFig. 3, (c), 1 shows the image taken for one H-field direction at the s maller DEM peak (position 1, Fig. 3,(b)).\nThe image itself does not allow us to directly observe any asymmetry in the temperature profile and the image\ntaken after field reversal looks quasi-identical. We now calculate th e difference between the two pictures. The grey\nscale of the pixels now no longer corresponds to an increase in tempe rature due to excitation but to a temperature\ndifference between two excited states. Negative values can occur , however they are no indication of cooling but just5\nmean that the corresponding local temperature on the subtract ed image is higher than on the image from which is\nsubtracted. For this first case the difference image already shows a small asymmetry between the two sides of the\nantenna. Moving to the maximum intensity of the DEM (position 2, Fig. 3,(b), (c)) not only shows a much bigger\nincrease in temperature but here also the difference image clearly sh ows an asymmetric temperature profile extending\nover several mm across and beyond the antenna.\nFigure 4 shows a temperature line profile obtained by line-wise averag ing the data inside the yellow region shown in\nthe inserted image for positive (black curve) and negative (grey cu rve) H-field respectively. For the diagram the grey\nvalue is converted to the temperature in mK. Already here we can ob serve that decrease in temperature away from\nthe antenna is faster on one side than on the other leading to an asy mmetric temperature profile. This asymmetry\nis reversed when the magnetic field changes sign. For better visibility we also plot the difference between the two\ngraphs showing that the maximum temperature difference between both sides of the antenna is as big as 5.2 mK\nand decays to zero far away from the antenna as expected from t heory. Negative temperatures only appear because\nthe difference has been calculated. It should be emphasized that th e fact that we can observe the asymmetrical\ntemperature distribution also outside of the region of the antenna (red lines in Fig. 4) clearly shows that the effect\nindeed originates from propagating spin waves.\nFIG. 4: Temperature profile plotted for the yellow marked reg ion. Red lines show the position of the CPW.\nFor the uniform mode at line position 3 (Fig. 3,(b) (c), 3) the heating s hould be symmetric and independent from\nthe direction of the magnetic field. Nevertheless, we still observe a small asymmetry which results from the overlap\nwith the DEM at position 2. This effect will later be discussed because it is highly relevant for measurements of\nthe inverse spin-Hall effect (ISHE) [10]. As expected we only observ e little heating and a homogeneous temperature\ndistribution for the off-resonance measurement (Fig. 3, (b), (c) , 4).\nBackward Volume geometry\nFor the BVM geometry a similar set of measurements is done as for DE M, now sampling the field range of the\nbackward volume modes including the uniform resonance mode. In th e Backward Volume geometry no non-reciprocal\nspin-waves can be excited, so that no unidirectional heat transfe r should be observed. This is confirmed by Figure 5\nwhich displays the spin wave spectrum together with the LIT images a gain for four different measurements. For the\noff-resonance case (position 1) and for the uniform mode (line 2) sy mmetric temperature profiles with a maximum\nat the antenna appear and the difference images indicate no differen ce at all. For position 3 and 4 the difference is\nnon-zero, however, we can identify this as an artifact. Although t he difference is finite, it is not asymmetric with\nrespect to the antenna. Indeed it does not originate from non-re ciprocity of the spin waves but from an absolute\nincrease in heating for one field direction. The origin of this difference is the sharpness of the resonance lines in the\nBVM-geometry. The pictures are taken at fixed field positions and it can actually happen that upon field reversal a\nslightly different field value is applied. Even a difference of 0.5 Oe (corre sponding to a relative error of 500 ppm) can\nlead to a sizeable difference in heating, explaining the observed effect .6\n+H\n1\n2\n3\n4+H-Hdifference\n(a)(b)\nTmax\nTminTmin=-2mKto\nTmax=1.66mKTmin=0mKtoTmax=11mK\nFIG. 5: (a) FMR measurement in the Backward Volume geometry ( 90◦, red curve) compared to the FMR measurement in the\nDamon-Eshbach geometry (0◦, grey curve). The peaks of the BVM appear much sharper than fo r the DEM. (b) LIT images\nfor the magnetic field positions 1 to 4.\nDiscussion\nOur results show that even for thin YIG films the unidirectional spin w ave heat conveyer effect can be observed\nalthough we do not see the heat pile-up shownby An and coworkersd ue smaller rangeoverwhich the spin-wavestravel\nin our samples. Although the temperature differences and the later al extent of the effect are smaller than for thick\nYIG films, lock-in thermography allows us to clearly identify the effect . The magnitude of the signal that is observed\nindicates that it should also be present in even thinner layers which no wadays can also be obtained with very low\ndamping (for example [11–13] ) and may possibly be detected using lon ger averaging times. The fact that this effect\nneeds to be considered also in thin films has important consequences for other research areas. In our experiments we\nobservetemperature differences ofseveralmK overa rangeof s everalmm. The DEM geometry ofexcitation is also the\none which is used when the inverse ISHE is measured. Looking at the d etails of such measurements we find that the\nheat-conveyer effect creates a temperature gradient along the direction in which the DC inverse spin-Hall voltage is\nmeasured. This temperature gradient now leads to thermovoltage s which, in contrast to other typical thermoelectric\nartifacts, exhibit the same symmetry with respect to the magnetic field as the ISHE. They can thus not be ruled out\nby comparing measurements at opposite field directions. Furtherm ore at least in our experiment the finite line width\nprevents us from completely separating the uniform mode from the DEM which would be necessary to make sure that\nno temperature gradient is created. This shows that great care m ust be taken in order to clearly discern ISHE and\npossible thermovoltages, especially when the ISHE is very small. In pa rticular when organic materials are used the\npossiblylargeSeebeckcoefficientcanresultinvoltagesashighastho seobservedfortheISHE.TheconductingPolymer\npoly(3,4-ethylenedioxythiophene) polystyrene sulfonate (PEDOT :PSS) which is often applied in organic electronics\ncan for example exhibit a Seebeck coefficient of 160 µKor higher [14, 15] resulting in a thermovoltage of 160 nV for\na temperature difference of 1 mK which we even achieve in our experim ent. For thick YIG layers where the heat\nconveyer effect is much more pronounced the expected thermovo ltages can thus be much bigger.7\nMethods\nFabrication of the CPW\nThe CPW is fabricated on top of the YIG film using electron beam lithogr aphy, metal deposition (10nm Ti / 250nm\nAg / 50 nm Au) and lift-off. The dimensions of the waveguide are as follo ws: length = 2.9 mm, width of the signal\nline = 80 µm, width of the ground planes = 125 µm, distance between signal line and ground planes = 35 µm.\nFerromagnetic resonance (FMR) measurement\nAll FMR measurements shown in this paper have been performed usin g a microwave frequency of 5 GHz and a\nexcitationpowerof-1dBm, by sweepingthe magneticfield. Theexte rnalmagneticfield isappliedbyan electromagnet\nwhich can be rotated around the sample. The microwave is provided b y a RHODE&SCHWARZ, SMF 100A signal\ngenerator, which is connected via rf probes (Cascade Microtech) to the CPW on top of the YIG sample. While\nsweeping the magnetic field at a constant frequency we measure th e absorption using a Schottky diode and an Agilent\n34420A nanovoltmeter.\nLock-in thermography technique\nForthe LITexperimentsweusedanInfraTecPV-LITsystem, whic hworkswithanInSbdetectorhavingaresolution\nof 640 x 512 pixels [16]. To perform the LIT measurement the microwa ve power is pulsed at the lock-in frequency\nsupplied by the camera. For all LIT measurements shown in this pape r we use an acquisition time of 5 minutes and\na lock-in frequency of 1 Hz. The surface of the sample is blackened w ith ink to achieve a better and uniform infrared\nemissivity. In a LIT experiment the heat sources in a device are modu lated or pulsed at a lock-in frequency lying\nwell below the frame rate of the infrared camera (here 200 Hz). Th e aquired images are evaluated synchronously\nto the heat pulses to detect the temperature modulation. Both th e in-phase and the out-of-phase modulations are\ndetected and can be converted into an amplitude and a phase signal. This is done on-line for each pixel. The result\nis equivalent to connecting each pixel to a two-phase lock-in amplifier [7–9].\nAcknowledgements\nThis work was supported by the Deutsche Forschungsgemeinscha ft in the SFB762 . We thank Georg Woltersdorf\nfor fruitful discussion.\n[1] An, T. et al.Unidirectional spin-wave heat conveyer. Nat Mater 12, 549–553 (2013).\n[2] Adachi, H. & Maekawa, S. Theory of unidirectional spin he at conveyer. Journal of Applied Physics 117, 17C710 (2015).\n[3] Perez, N. & Lopez-Diaz, L. Magnetic field induced spin-wa ve energy focusing. Phys. Rev. B 92, 014408– (2015).\n[4] Eshbach, J. R. & Damon, R. W. Surface magnetostatic modes and surface spin waves. Phys. Rev. 118, 1208– (1960).\n[5] Serga, A. A., Chumak, A. V. & Hillebrands, B. Yig magnonic s.Journal of Physics D: Applied Physics 43, 264002– (2010).\n[6] Schneider, T., Serga, A. A., Neumann, T., Hillebrands, B . & Kostylev, M. P. Phase reciprocity of spin-wave excitatio n by\na microstrip antenna. Phys. Rev. B 77, 214411– (2008).\n[7] Breitenstein, O. et al.Microscopic lock-in thermography investigation of leakag e sites in integrated circuits. Rev. Sci.\nInstr.71, 4155–4160 (2000).\n[8] Breitenstein, O., Langenkamp, M. & Warta, W. Lock-in Thermography - Basics and Use for Evaluating Electr onic Devices\nand Materials (Springer, 2010), 2 edn.\n[9] Bauer, J., Breitenstein, O. & Wagner, J.-M. Lock-in ther mography: A versatile tool for failure analysis of solar cel ls.ASM\nInternational 3, 6–12 (2009).\n[10] Qiu, Z. et al.Spin-current injection and detection in κ-(BEDT-TTF)2Cu[N(CN)2]Br. AIP Advances 5, 057167 (2015).\n[11] d’Allivy Kelly, O. et al.Inverse spin hall effect in nanometer-thick yttrium iron gar net/pt system. Applied Physics Letters\n103, 082408 (2013).\n[12] Chang, H. et al.Nanometer-thick yttrium iron garnet films with extremely lo w damping. Magnetics Letters, IEEE 5, 1–4\n(2014).8\n[13] Hauser, C., et al.Yttrium Iron Garnet Thin Films with Very Low Damping Obtaine d by Recrystallization of Amorphous\nMaterial. Scientific Reports 6, 20827– (2016).\n[14] Massonnet, N. et al.Improvement of the seebeck coefficient of pedot:pss by chemic al reduction combined with a novel\nmethod for its transfer using free-standing thin films. J. Mater. Chem. C 2, 1278–1283 (2014).\n[15] Bubnova, O. et al.Semi-metallic polymers. Nat Mater 13, 190–194 (2014).\n[16] URL www.infratec-infrared.com . (Accessed: 15th January 2016)"    },    {        "title": "1712.04074v1.Predicting_the_Spin_Seebeck_Voltage_in_Spin_polarized_Materials__A_Quantum_Mechanical_Transport_Model_Approach.pdf",        "content": "arXiv:1712.04074v1  [physics.comp-ph]  11 Dec 2017PredictingtheSpinSeebeckVoltageinSpin-polarizedMaterials: AQua ntumMechanicalTransportModelApproach\nPredicting the Spin Seebeck Voltage in Spin-polarized\nMaterials: A Quantum Mechanical Transport Model\nApproach\nAnveeksh Koneru1,a)and Terence D. Musho1,b)\nMechanical Engineering, West Virginia University, Morgan town,\nWV\n(Dated: 8 March 2022)\nThe spin Seebeck effect has recently been demonstrated as a viable method of\ndirect energy conversion that has potential to outperform ener gy conversion from\nthe conventional Seebeck effect. In this study, a computational transport model\nis developed and validated that predicts the spin Seebeck voltage in s pin-polarized\nmaterials using material parameter obtain from first principle groun d state density\nfunctional calculations. The transport model developed is based o n a 1D effective\nmass description coupled with a microscopic inverse spin Hall relations hip. The\nmodel can predict both the spin current and voltage generated in a non-magnetic\nmaterial placed on top of a ferromagnetic material in a transverse spin Seebeck\nconfiguration. The model is validated and verified with available exper imental\ndata of La:YIG. Future applications of this model include the high-th roughput\nexploration of new spin-based thermoelectric materials.\nPACS numbers: Valid PACS appear here\nKeywords: Thermoelectric, Spin Seebeck Effect, NEGF\nI. INTRODUCTION\nTougher sanctions on fossil fuel emissions and greater energy de mand around the globe is\nforcing us to rely more on renewable energy resources. Clean ener gy technologies like ther-\nmoelectric power generation provides one solution to ease our depe ndence on non-renewable\nenergy resources. However, the efficiency of thermoelectrics ha s been limited due to the\ninherent coupling of the electronic and thermal carriers. Many diffe rent approaches like\ngrain boundary scattering34, band structure engineering33,16, substitutional effects6were\nincorporated to improve the efficiency of thermoelectrics. Though these methods improved\nthe performance to certain extent, the phonon and electron inte ractions still hamper the\ncommercial applicability of thermoelectrics. More recently, an aven ue to decouple these\ninteractions has been experimental demonstrated using tempera ture gradient induced spin\ncurrents29,3,26,11,27,2. With this new discovery comes the need for new transport models\nto understand and optimize their response. In providing a solution, this research is focused\non the both the development of a 1D spin-transport model and valid ating the model using\nthe available experimental data.\nConventional thermoelectric energy conversion utilizes the princip le of Seebeck effect19\nto convert thermal gradient into electric voltage. In this effect, t he energy conversion takes\nplace when majority charge carriers drift away from the region of h igh temperature. A new\napproach to design thermoelectric modules relies on a slightly differen t principle involving\nthe electron’s spin. The pioneering research in 2008 by Uchida et.al.26has opened a new\navenue to extract additional heat energy by utilizing the intrinsic an gular momentum of\nelectrons, colloquially known as spin. This field that explores charge, spin and energy\na)Ph.D. Mechanical Engineering, West Virginia University.\nb)Corresponding author: terence.musho@mail.wvu.edu.PredictingtheSpinSeebeckVoltageinSpin-polarizedMaterials: AQua ntumMechanicalTransportModelApproach 2\ntransport due to temperature difference is called spin caloritronics3,20and is a transpiring\nfield with immense potential in heat conversion applications.\nUsing a principle called spin Seebeck effect a spin voltage can be genera ted in a metal\ncontact attached on a ferromagnetic material (FM) due to a ther mal gradient in the FM\nmaterial lattice. There are two configurations reported in literatu re to extract the spin\nvoltage,longitudinal24andtransverse25,26configurations. TheFigure1a, showsatransverse\nconfiguration arrangement, which is the configuration of interest in this research, to extract\nspin voltage from a ferromagnetic material. Due to a thermal gradie nt, spin current is\ntransferred from the FM substrate into the attached nonmagne tic metal (NM) contact.\nDue to the high spin orbital coupling associated with heavy nonmagne tic metals such as\nPt10,21, the spin carriers are separated and accumulated on the two ends of the metal. In\nresponse, a voltage is generated along the transverse direction p roportional to the amount\nof accumulated spin carriers. This effect is opposite to the Hall effec t where an applied\nelectric field in the presence of a magnetic field separates the charg e carriers while here\non the contrary the separation of spin carriers generates the vo ltage hence named as the\ninverse spin Hall effect. Only spin polarized or ferromagnetic materia ls have the capability\nto generate spin currents and transfer into the Pt metal contac ts through the inverse spin\nHall effect. Hence, this research is primarily focused on studying sp in-polarized materials.\nIn order to theoretically predict the inverse spin Hall voltage (V ISHE) of spin polarized\nmaterials, we developed a 1-D spin transport model by combining non equilibrium Green’s\nfunction formalism (NEGF)7and spin transport theory31. In this approach, first the funda-\nmental parameters of a material such as lattice constant, band g ap, Fermi energy, effective\nmass and magnetization of the material lattice, were calculated usin g density functional\ntheory (DFT). The NEGF model used this information to calculate th e surface current for\nthe spin channels independently and spin transport theory will then be used to calculate\nthe spin current injection and inverse spin Hall voltage generated in the attached metal\ncontact.\nNEGF modeling has been implemented in various studies to describe the quantum trans-\nport in different materials in the presence of thermal bias22,5. There also availablesoftwares\nlike TranSIESTA4that incorporate the application of this formalism. A combination of\nDFT and NEGF22; or NEGF and spin transport theory5, were used to describe the elec-\ntron transport but a combination of DFT+NEGF+spin transport th eory to calculate the\nVISHEin magnetic materials is near to less in literature. Hence, this researc h article aims\nto develop a 1-D model using a combination of NEGF formalism and spin t ransport theory\nby incorporating the parameters obtained from DFT calculations.\nTransverse spin Seebeck configuration, Figure 1a, which is the maj or scope of this re-\nsearch, has been experimentally verified using La:YIG27and NiFe25. Insulating magnetic\nmaterials like La:YIG displaying spin Seebeck effect is a strong indication of magnaon\ndriven spin Seebeck that is caused due to spin redistribution in a mate rial and also proves\nthe capability of spin carriers in a material lattice to generate voltag e. In addition to this,\nsemiconducting or insulating oxide magnetic materials have immense sc ope in this emerg-\ning field due to their ability to accommodate wide variety of substitutio ns and tune various\nelectronic and magnetic properties. Hence we chose La:YIG as our m aterial of study to\nvalidate and verify the developed model. LaY 2Fe5O12(La:YIG) material lattice is obtained\nby substituting one Yttrium with Lanthanum in Y 3Fe5O12(YIG). There have been few\ntheoretical studies reporting the fundamental property study using DFT to calculate lattice\nparameter, electronic band structure and effective mass data fo r YIG12,32,1but there is\nno available electronic band structure data for La:YIG. Hence, firs t principle calculations\nbased of DFT were performed on YIG to compare the effective mass and band gap data\nwith literature available for YIG, and a similar approach has been applie d to calculate the\nelectronic band structure data for La:YIG.PredictingtheSpinSeebeckVoltageinSpin-polarizedMaterials: AQua ntumMechanicalTransportModelApproach 3\n(a) A three probe system consisting of\nleft electrode with temperature T H,\nright electrode with temperature T C\nand an ISHE electrode on the\nscattering region to measure the\namount of current from the scattering\nregion into the electrode.(b) Description of the scattering region\ndivided into N lattice points.\nFIG. 1: The number of grid points N depends on length of the lattice.\nII. 1-D GREEN’S FUNCTION APPROACH\nUsing a DFT approach, a supercell arrangement or a system of isola ted molecules can\nbe solved for their electronic structure relaxation type solid state physics problems. In\ncase of quasi-particle transport phenomena calculations across t wo boundaries, the effects\nof chemical potential due an external bias or variation of charge d ensity in the scattering\nregion while considering temperature effects cannot be solved using DFT alone to replicate\nthe boundary conditions in a quantum transport. However, the co mbination of DFT with\nNEGF formalism is a powerful tool to study quantum transport phe nomena in nanoscale\nregion. The NEGF formalism is a self-consistent method, where Schr ¨ odinger’s equation and\nPoisson’s equation are solved self-consistently for a copmosite effe ctive mass system.\nA. Non-Equilibrium Green’s Function (NEGF) formalism\nA composite system modeled in this research is shown in Figure 1a which is divided into\nfour regions, i.e., the left contact, the scattering region, the righ t contact and the inverse\nspin Hall effect electrode (ISHE) which typically is Pt metal. The ISHE e lectrode is a\nconceptual floating probe used to calculate net spin flux flow betwe en the FM and NM at\nvarious locations along the lattice. These probes are conceptually u sed to extract electrons\nfrom the device or inject into the device, at the region of study, to effectively calculate the\nscattering and transmission of electrons due to the applied temper ature bias. The model\nincorporates the scattering effects that include connection of th e channel to the contacts on\nthe two ends and interactions withing the channel. To describe the s ystem, two components\nthat represent outflow [Σout]{ψ}and inflow {s}from the contacts should be added to the\nusual time-independent Schr¨ odinger equation represented by E {ψ}= [H]{ψ}which is given\nby the Equation 1.\nE{ψ}= [H]{ψ}+[Σout]{ψ}+{s} (1)\nwhereψis the many-particle wave function between the contacts.PredictingtheSpinSeebeckVoltageinSpin-polarizedMaterials: AQua ntumMechanicalTransportModelApproach 4\nFrom the above, {ψ}can be written as the following,\n{ψ}= [G]{s}. (2)\nWhere [G] is,\n[G] = [EI−H−Σout]−1,\nreplacing Σout= Σout\nl−Σout\nr,\n[G] = [EI−H−Σout\nl−Σout\nr]−1. (3)\nHere, the Σout\nland Σout\nrterms are the outflow energies at the left and right contacts re-\nspectively. In addition to the outflow and inflow terms, the electros tatic potential energy\n(U) of the channel has to be included in Equation 3 which now becomes ,\n[G] = [EI−H−U−Σout\nl−Σout\nr]−1. (4)\nThe electron density in the scattering region which can be written as {ψ}{ψ}†, generally\nrepresented by Gn, varies with the applied thermal bias. Using Equation 2, Gncan be\nwritten asGn={ψ}{ψ}†= [G]{s}{s}†[G]†. Where {s}{s}†is written as Σin(the inflow\nfrom the source), Gncan now be written as,\nGn= [G]Σin[G]†. (5)\nAs there are contacts that interfere with the system, analytical solution cannot be ob-\ntained for the Equation 5. One way to solve this equation is by expres sing the material as\nfinite discrete volumes with points at cell centers as shown in Figure 1 b, and representing\nthose points through a matrix form. In the current researchthe proposed model will assume\nisotropic behavior at the cross-section of each lattice point along t he scattering region, and\nassume a 1-dimensional model. At each lattice point (n) along the x-d irection, the amount\nof spin current generated due to the difference in the two spin popu lations at that position\ncan be calculated. This spin current is injected into the attached IS HE electrode at that\nposition which is converted into spin voltage due to the principle of inve rse spin Hall effect.\nIn order to calculate the spin current at each lattice point, the pop ulation density can be\ncalculated by applying NEGF model to solve Equation 5.\nThe Equation 5 represents the fundamental equation behind NEGF formalism. The\ninflow energy from the contacts occur due to the non-equilibrium th ermal difference at the\nhot and cold ends on the left and right contacts respectively and ca n be calculated using\nEquation 6. Calculating each of the terms on the right hand side of th e Equation 5, yields\nthe electron density in the channel,\nΣin\nl= Γl∗fl,\nΣin\nr= Γr∗fr,\nΣin= Σin\nl+Σin\nr, (6)\nwhereflandfrare temperature-dependent Fermi-Dirac distribution of the cont inuous\nenergy states ǫat left and right contacts respectively. The Fermi function can be described\nby Equation 7. Here, T l(r)is the temperature of the contact at left(right) end and ǫis\nthe energy. It is through this Fermi-Dirac distribution that an explic it temperature bias is\napplied to the system.\nfl(r)(ǫ) =1\n1+e(ǫ−µl(r))/(kBTl(r))](7)\nWhen the temperature bias is applied to the scattering region by the end contacts, the\nconduction electrons tend to move away from the hot contact. Th e two spin channels (spin-\nup (↑)andspin-down( ↓))associatedtoelectronsmoveatdifferentratesdue totheirdiffe rentPredictingtheSpinSeebeckVoltageinSpin-polarizedMaterials: AQua ntumMechanicalTransportModelApproach 5\neffective mass values. The effective mass of the ↑and↓conduction electrons depends on the\ncurvature of the respective conduction bands and can be calculat ed by taking the harmonic\nmean of the effective masses in the three reciprocal vectors direc tions in the Brillouin zone\nas given in Equation 11.\nE↓(↑)(k) =E↓(↑)cx+/planckover2pi12∗k2\nx−Γ\n2∗m∗\n↓(↑)x, (8)\nE↓(↑)(k) =E↓(↑)cy+/planckover2pi12∗k2\ny−Γ\n2∗m∗\n↓(↑)y, (9)\nE↓(↑)(k) =E↓(↑)cz+/planckover2pi12∗k2\nz−Γ\n2∗m∗\n↓(↑)z, (10)\nm∗\n↓(↑)conduction = 3∗[1\nm∗\n↓(↑)x+1\nm∗\n↓(↑)y+1\nm∗\n↓(↑)z]−1, (11)\nwhere k is the wavevector and E ↓(↑)cx, E↓(↑)cyand E ↓(↑)czare the conduction band edges\nin the x-Γ, y-Γ and z-Γ directions respectively of ↓(↑) electrons. Using the effective mass\nvalues of ↓(↑) electrons and the respective electronic band gaps, the amount o f current\ngenerated in the scattering region due to the temperature bias be tween the contacts can be\ncalculated.\nEvery electron has multiple energy levels available to accommodate th eir movement.\nDepending on the electron’s eigen energy, charge transport domin ates in certain energy\nbands which can be calculated from transmission function given by,\nΞ =Trace[ΓlGΓrG†], (12)\nwhere Γ land Γ rare NxN left and right contact anti-Hermitian matricies of Σout\nland\nΣout\nrrespectively that govern the inscattering and outscattering fro m contacts. All these\nparameters can be calculated from,\nΣout\nl=\n−to∗ei∗k∗a0. . . .0\n0. . . . . 0\n. . .\n. . .\n0. . . . . 0\n,\nΓl=i∗[Σout\nl−Σout\nl†],\nΣout\nr=\n0 0. . . . 0\n0. . . . . 0\n. . .\n. . .\n0. . . . . −to∗ei∗k∗a\n,\nΓr=i∗[Σout\nr−Σout\nr†], (13)\nwhere k is cos−1(1-E\n2to). E is the energy level in the fine spectrum of energy bands, −to∗\nei∗k∗atois analytical wavefunction given by,\nto=/planckover2pi12\n2∗me∗a2∗q, (14)\nwhere a is the distance between cell centers. The distance betwee n the cell centers is also\ncorrelated with the energy cut-off. m eis the conduction electron effective mass calculated\nfrom Equation 11 , q is the charge of an electron, /planckover2pi1is the reduced Planck’s constant.\nThe transmission function (Ξ) of an energy band when multiplied with t he Fermi func-\ntions, gives the conductance for each energy level. By integrating the quantum conductancePredictingtheSpinSeebeckVoltageinSpin-polarizedMaterials: AQua ntumMechanicalTransportModelApproach 6\nof an electron in a fine spectrum of energy bands, the overall quan tum conductance of the\nelectron with that eigen energy can be obtained. Incorporating th e same process for all\nthe eigen energy states, the overall quantum conductance in the material lattice can be\nobtained. The quantum conductance multiplied withq2\nhyields the current generated by a\nelectron of one eigen energy. As given in Equation 15, taking the sum of total current for\nall eigen energy states, gives the net current in the scattering re gion.\nItot=ǫ=Emax/summationdisplay\nǫ=Eminq2\nhΞ(ǫ)(fl−fr)dE (15)\nThe domain is divided into N lattice points, as shown in Figure 1b, can be d escribed\nby an NxN matrix called the Hamiltonian matrix. The higher value of N give s better ap-\nproximation by increasing the cut off energy while also taking a toll on t he computational\nexpense. A 1-D Hamiltonian matrix with N lattice points, [H] N∗N, can be written as given\nin Equation 16. The NxN Hamiltonian matrix has ’N’ eigen values which are the eigen\nenergy states of the scattering region. In Equation 16, E cis the conduction band edge that\ncan be obtained from DFT calculations and t ois represented in Equation 14. It can be\nobserved that the Hamiltonian matrix of the scattering region depe nds on the parameters\nobtained from DFT calculations and the fineness of the grid.\n[H] =\nEc+2to−to0 0 . . . . . 0\n−toEc+2to−to0. . . . . 0\n0−toEc+2to−to0. . . . 0\n. . .\n. . .\n. . .\n. . .\n. . .\n. . . . 0−toEc+2to−to0\n. . . . . . 0−toEc+2to−to\n. . . . . . . 0−toEc+2to\n(16)\nFor each energy level, the potential of the scattering channel (U ) and the electron density\n(N) can be calculated iteratively until self-consistency in the syste m is reached. This is\ndescribed in the flow chart in Figure 2. The self consistent procedur e accounts for the\nelectron correlation within each spin channel. After a converged se lf-consistent estimation\nof U, the electron density for that eigen energy state, which is the trace ofGnmatrix, is\nused to calculate the current generated at that state. As, isotr opic conditions are assumed\nat each lattice point, trace of Gnmatrix in Equation 5 gives electron density per unit area.\nSumming the currents over all eigen energy states gives the total current in the scattering\nregion. Using the converged value of U at each energy level, the cur rent in the channel\nat that respective energy level can be calculated. By summing the c urrents at all energy\nlevels, the total current in the scattering region can be calculated .\nIn this research the spins will be treated independent of each othe r. This commonly\nis referred to as the Stoner model15. In doing so, the currents of the spin-up, I ↑, and\nspin-down, I ↓, can be obtained from Equation 17, for the respective Ξ ↑and Ξ ↓obtained\nfrom Green’s formalism. In a ferromagnetic material there is a net s pin-polarization present\nin the lattice. This can lead to spin-polarized charge current under t emperature bias. The\ndifference in the Fermi level of spin-up and spin-down conduction ele ctrons in a material\ncauses imbalance between the populations of spin-up and spin-down electrons that can lead\nto spin-polarized charge current.PredictingtheSpinSeebeckVoltageinSpin-polarizedMaterials: AQua ntumMechanicalTransportModelApproach 7\nFIG. 2: NEGF flow chart of the self-consistent method. The self-c onsistent criteria is\nwhen consecutive energy difference reaches below 1e-6.\nItot(↑)=ǫ=Emax/summationdisplay\nǫ=Eminq2\nhΞ(↑)(ǫ)(fl(↑)−fr(↑))dE\nItot(↓)=ǫ=Emax/summationdisplay\nǫ=Eminq2\nhΞ(↓)(ǫ)(fl(↓)−fr(↓))dE (17)\nIII. CALCULATION OF INVERSE SPIN HALL VOLTAGE\nWhen a temperature gradient ( ∇T) is applied, the ↑(↓) components move away from hot\nsource at different rates and hence have different spin Seebeck co efficients, S ↑(↓). The 1-D\nmodel in this research treats the two spin channels independently. To calculate the S ↑(↓),\na voltage in the reverse bias will be applied between the left and right e lectrodes (V↑(↓)),\nsee Figure 1b. This will inject the spin current into the ferro-magne tic (FM) material\n(scattering region) that opposes the spin current generated in t he FM due to temperature\nbias. ThisV↑(↓)will be iterated until the respective spin current ( I↑(↓)) flow is zero between\nthe left contact and right contact. The respective slopes in the IV curve gives conductivity\nσ↑andσ↓of spin-up and spin-down electrons. The V↑for whichItot(↑), in Equation 17, is\nzerogivesthe Seebeckvoltageofthe ↑channelandthe spin Seebeckcoefficient for ↑channel,\nS↑, at that temperature would be S ↑=V↑\n∆T. A similar iterative procedure can yield spin\nSeebeck coefficient for ↓channel S ↓. The electronic contribution to thermal conductivity ke\ncan be calculated from Fourier’s law, ke=I↑+I↓\nA∆x\n∆T.\nThe presence of temperature gradient along the ferromagnetic m aterial changes the den-\nsity flux of spin-up and spin-down electrons. Exploring the substitu tional effects to improve\nthe difference in this spin-up and spin-down density flux will make a mat erial suitable for\nspincaloritronicapplications. Sometimesthesubstitutionsmakethe materialspin-polarized\nwhile also making the material conductive which hampers the applicatio n of the materialPredictingtheSpinSeebeckVoltageinSpin-polarizedMaterials: AQua ntumMechanicalTransportModelApproach 8\nfor thermoelectrics. Hence theoretical characterization of a ma terial performance for spin\ncaloritronic applications can be made using the developed model in this research.\nA. Spin Current Across the Magnetic and Non-Magnetic Interfac es\nAt each location along the x axis, in Figure 1b, different amount of spin current is\ninjected into the NM contact from the FM material. In a conventiona l Seebeck effect, the\ntemperature difference between the junctions of two dissimilar met al joints generates a net\nvoltage. Likewise, in case of spin Seebeck effect (SSE), the differen ce in the temperature\nbetween the magnons (electron spin waves) in the ferromagnetic fi lm (considered to be close\nto that of the FM temperature) and the electrons in the attached NM contact (considered\nto be close to that of the NM temperature), generates an inverse spin Hall voltage in the\nNM contact.\nAs SSE was observed even in magnetic insulators, the Spin Seebeck c oefficient cannot\nbe fully expressed as a sole function of conduction electrons. Henc e, a microscopic theory\nproposed by Xiao et al31was employed in this research that utilizes the calculated spin\ndensity from the quantum mechanical model to predict the spin cur rent and voltage in the\ncontact. As in conventional Seebeck effect where the temperatu re difference between the\nleft and right contacts drives the charge current, here the temp erature difference between\nthe FM (T F) and NM (T N) drives the spin current across the interface of FM and NM.\nThe model approximates that the difference between the T Fand T N, (∆T=TF−TN),\nchanges from a positive value at the hot end to a negative value at th e cold end, flipping\nthe sign at the center which in close approximation to the experiment al findings27.\nThespinpumpingcurrentfromFMtoNMoccursduetothermalnon- equilibriumbetween\nthe two, given by Equation 18. The thermal spin pumping current J spfrom FM to NM\nis proportional to T Fand spin current fluctuations J ffrom NM to FM is proportional to\nTN, given by Johnson-Nyquist8,30. Only the real component of the J spcontributes to the\nnet Jstowards the NM because the imaginary component averages to zer o. The real part\nof the spin current from FM into the NM as extracted from literatur e23is given by,\nJsp=/planckover2pi1\n4π[grm×˙m], (18)\nwhere/planckover2pi1is the reduced Plank’s constant, g ris the real part of spin mixing conductance,\nmis the unit vector that is parallel to the magnetization in the material, ˙mis the rate of\nchange of magnetization due to thermally activated dynamics in magn etization in FM. To\ncompensate the energy transfer from FM into NM in the form of spin , a fluctuating spin\ncurrent, J f, flows from NM to FM, as extracted from literature8,30is given by,\nJf=−MsV\nγ[γm×h′], (19)\nwhere M sis the saturation magnetization in the FM, V is the FM volume, γis the electron\ngyro-magnetic ratio and h′is the resultant magnetic field.\nHence, the non-equilibrium thermal difference between FM and NM ca uses the net spin\ncurrent, J s, from FM to NM in the z-direction given by J s= Jsp-Jfl\nJs=MsV\nγ[α′/an}b∇acketle{tmx˙my−my˙mx/an}b∇acket∇i}ht−γ/an}b∇acketle{tmxh′\ny−myh′\nx/an}b∇acket∇i}ht], (20)\nwhereα′is the damping enhancement due to spin pumping given by γ/planckover2pi1gr/4πMsV, the x\nand y subscripts of mand˙mare the respective components along x and y axes respectively\n(see Figure 1a). Using mean square deviation /an}b∇acketle{tm˙m/an}b∇acket∇i}htand/an}b∇acketle{tm˙h′/an}b∇acket∇i}htcan be approximated by\nEquation 21 and Equation 22 respectively as extracted from Xiao et al31.\n/an}b∇acketle{t˙mi(t)mj(0)/an}b∇acket∇i}ht=−σ2\nsd\n4πα/integraldisplay\n[χij(ω)−χ∗\nij(ω)]eiωtdω (21)PredictingtheSpinSeebeckVoltageinSpin-polarizedMaterials: AQua ntumMechanicalTransportModelApproach 9\n/an}b∇acketle{tmi(t)h′\nj(0)/an}b∇acket∇i}ht=−σ′2\nsd\n2πγ/integraldisplay\nχij(ω)eiωtdω (22)\nχ(ω) is the transverse dynamic susceptibility matrix given by Equation 23 ,\nχ(ω) =1\n(ωo−iαω)2−ω2/bracketleftbigg\nωo−iαω−iω\niω ω o−iαω/bracketrightbigg\n, (23)\nwhereωo=γHeffis the FM resonancefrequency. Heffisthe externalmagneticfield applied\nalong the direction of thermal bias that causes saturation magnet ization in the material.\nHence, by plugging Equations 22 and 21 into Equation 20 and rearran ging for the time-\naveraged spin current transferred from FM to NM in the z direction as given in literature31\nwill be,\n/an}b∇acketle{tJs/an}b∇acket∇i}ht=γ/planckover2pi1grkB\n2πMsV(TFM−TNM). (24)\nThe factorγ/planckover2pi1grkB\n2πMsVis called interfacial spin Seebeck coefficient. The Spin Seebeck voltag e\ncan be calculated based on the spin Hall current in the NM that is gene rated due to inverse\nspin Hall effect18. The DC spin Hall current in the attached NM (Pt in this case) along\ny-direction is,\nJc(x)ˆy=θH2q/an}b∇acketle{tJs/an}b∇acket∇i}ht\n/planckover2pi1Aˆz׈x, (25)\nwhereθHis the spin Hall angle of NM contact (Pt in this case). The Inverse spin Hall\nvoltage generated at location x is given by Equation 26,\nVISHE(x) =ρlJc(x) =lJc(x)\nσ, (26)\nwhereρis the electrical resistivity of Pt contact, lis the length of the contact and J c(x) is\nthe spin current from FM to NM as calculated in Equation 25.\nIV. RESULTS\nSpin Seebeck theory and NEGF transport theory combined with DFT can be used to\ntheoretically calculate the spin voltage in the attached NM metal on t he scattering region.\nAs a model has been developed in this research, validating this model with the available\nexperimental data is necessary.\nA. Computational Details\nAb-initio calculations of the electronic band structure and structu ral properties were per-\nformed based on density functional theory using the plane wave sc heme as implemented in\nQuantum Espresso package9. In all the calculations the plane wave energy cut-off of 1220\neV was used to yield high convergence in energies. Super cells were re laxed to less than\n5KPa and relative energy convergence of 10−9eV. The computed lattice constant of YIG of\n11.62˚Amatches well with the available experimental data of 12.39 ˚A1. The exchange and\ncorrelation energy was described by the GGA as presented by PBE f unctional17(QE-PBE).\nAll the calculations were spin-polarized and the atomic cores were de scribed by ultrasoft\npseudo potentials28, as they efficiently handle localised electrons and provide accurate r e-\nsults comparable to all-electron calculations. The atomic valance of 4 s13d7, 5s24d1and\n2s22p4was used for Fe, Y and O in the respective pseudopotentials. Calcula tions were\nperformed on 80 atom super cell shown in Figure 3a using a 4x4x4 Mon khorst-Pack k-point\ngrid.PredictingtheSpinSeebeckVoltageinSpin-polarizedMaterials: AQua ntumMechanicalTransportModelApproach 10\n(a)\n (b)\nFIG. 3: (a) A 80 atom super cell of YIG. The green balls represent F e+3ions, the gray\nballs represent Y+3ions and the red balls represent O-2 ions.(b) The super cell of La:YIG .\nThe four purple balls in La:YIG super cell that replace the Y+3ions represent La+3ions.\nB. YIG Model Results\nA primitive Y 3Fe5O12unit cell has 20 atoms. A 2x2x1 super cell, Figure 3a, was con-\nstructed that comprises 80 atoms. The Fe atoms with tetragonal and octahedral coordina-\ntion carry majority of the magnetic moment of YIG while Y atoms havin g dodecahedral\ncoordination and O atoms have little contribution to the magnetic mom ent. Fe atoms have\nhighest number ofunpaired d electrons that contribute to magnet icmoment in the material.\nThe electronic structure of YIG was calculated by implementing the s cheme discussed in\nthe previous section.\nThe valence and conduction bands of spin up and spin down bands of Y IG is shown in\nFigure 4a and Figure 4b. The spin down contribution to the bands is giv en in red, the spin\nup contribution to bands is given in black and the combined valence and conduction bands\nfor the YIG material is shown in Figure 4c. The k-point path was chos en along the three\nreciprocal lattice vector directions. χ(0,0,1/2), Γ(0,0,0), L(1/2,0,0) and η(0,-1/2,1/2)repre-\nsent the symmetry k-points. χto Γ, Γ to L and Γ to ηrepresent the three reciprocal vector\ndirections. The ↑and↓channels have a band gap of 1.1689eV and 2.2287eV respectively\nsuggesting that spin up channel is more conductive than the spin do wn channel. In the\ncombined bands of the YIG, the spin up channel’s valance band and sp in down channel’s\nconduction band contribute to the majority of the charge transp ort. In the presence of\nthermal gradient, the conduction band that corresponds to the ↓channel contributes to the\nelectron movement creating a non-equilibrium in spin distribution along the lattice.\nThe band gap of the YIG lattice calculated from the Ab intio calculation s was 0.3262eV\nwhich is corroborated with values reported in the literature12, where a band gap of 0.33eV\nwas reported. As discussed in the previous section, each spin chan nel is treated indepen-\ndently and the effective mass of the spin down conduction band was c alculated as shown\nin Figures 5a, 5b and 5c. The effective mass values in the three recipr ocal vector direc-\ntionsχ−Γ (mx), Γ to L (m y) and Γ to η(mz) are 0.4821*m e, 0.5317*m eand 0.5048*m e\nrespectively. The harmonic mean of the three effective mass values given in Equation 11\ngives the effective mass of the spin down conduction band to be m ↓=0.5054*m e, which is\nin close approximation to the value of 0.52*m ereported in the literature12. Compared toPredictingtheSpinSeebeckVoltageinSpin-polarizedMaterials: AQua ntumMechanicalTransportModelApproach 11\n  L     \nK points-2-101234E (eV) 1.1689 eV\n(a)  L     \nK points-2-101234E (eV)\n2.2287 eV\n(b)  L     \nK points-2-101234E (eV) 0.32625 eV\n(c)\n  L     \nK points-2-101234E (eV) 1.1546 eV\n(d)  L     \nK points-2-101234E (eV)\n2.1143 eV\n(e)  L     \nK points-2-101234E (eV) 0.35445 eV\n(f)\nFIG. 4: (a),(b) and (c) respectively correspond to the spin up cha nnel, spin down channel\nand combined valence-conduction bands of YIG material; while (d), ( e) and (f)\nrespectively correspond to the spin up channel, spin down channel and combined\nvalence-conduction bands of La:YIG material at Fermi region. Com pared to YIG, the\nband gap of individual channels decreased for La:YIG while the net ba nd gap of La:YIG\nincreased slightly.\nthe↓channel, the ↑channel has a lower effective mass leading to its high mobility but as\nthe energy gap of the ↑conduction band from the Fermi region being high 1.1689 eV in\ncomparison to 0.3262 eV of the YIG lattice, makes ↑channel less conductive. Having the\nfundamental parameters of YIG, such as the lattice constant, b and gap and effective mass\nmatch with the literature, a similar first principles approach is implemen ted for La:YIG.\nC. La:YIG Model Results\nThe Y in the garnets are large cations having dodecahedral coordin ation. Substituting\none Y atom with another large cation La yields LaY 2Fe5O12(La:YIG). Figure 3b shows\nthe super cell of La:YIG. Here 4 Y atoms are replaced with 4 La atoms and the relaxation\nof the unit cell is made following the description given in computational details section.\nAs there are 12 Y positions in an 80-atom super cell, to replace 4 Y ato ms with 4 La\natoms there are 12C4 combinations. 12C4 yields 495, meaning 495 va rious unit cells must\nbe optimized and the optimal cell that gives the least energy state m ust be chosen for\nfurther analysis. As this is computationally expensive, one means to perform this task is\nby mixing the pseudopotentials available in quantum espresso packag e. Mixing Y and La\npseudopotentials can give freedom in choosing positions to replace Y with La. Following\nthe mixing procedure embedded with quantum espresso package, a convergence in total\nenergy and forces was achieved with the energy cutoff values the s ame as discussed in\ncomputational details section. The relaxed unit cell was used to obt ain their electronic\nband structure, density of states, band gap, electron effective mass, the Fermi energy level,\nand the magnetization of the material. The lattice parameter increa sed by 0.87% compared\nto pure YIG due to La atoms being larger compared to Y and acquiring a value of 11.72 ˚A.\nThese properties were used to calculate its spin transport charac teristics in the presence of\na temperature gradient.PredictingtheSpinSeebeckVoltageinSpin-polarizedMaterials: AQua ntumMechanicalTransportModelApproach 12\n  L     \nK points-0.4-0.35-0.3-0.25E (eV)mx= 0.48215*me\n(a)  L     \nK points-0.4-0.35-0.3-0.25E (eV)my= 0.5317*me\n(b)  L     \nK points-0.4-0.35-0.3-0.25E (eV)mz= 0.50489*me\n(c)\n  L     \nK points-0.4-0.35-0.3-0.25-0.2E (eV)mx= 0.44624*me\n(d) Spin up bands  L     \nK points-0.4-0.35-0.3-0.25-0.2E (eV)my= 0.48988*me\n(e) Spin down bands  L     \nK points-0.4-0.35-0.3-0.25-0.2E (eV)mz= 0.45773*me\n(f) Combined bands\nFIG. 6: Fitting effective mass curves to the conduction band in the t hree coordinate\ndirections. 5a: effective mass of conduction band in X to Γ direction. 5b: effective mass\nof conduction band in Γ to L direction. 5c: effective mass of conduct ion band in Γ to η\ndirection.\nAs shown in Figure 4, both the ↑and↓channels of La:YIG with band gaps of 1.1546eV\n(Figure 4d) and 2.1143eV (Figure 4e) respectively, saw a reduction in band gaps when\ncompared with respective counter parts of YIG (Figures 4a and 4b ). But the overall band\ngap of the La:YIG material with 0.3544eV (Figure 4f) has a slight incre ase in the band\ngap when compared to YIG material (Figure 4c). This makes each ind ependent channel\nof La:YIG to be more conductive while the material is more insulating th an YIG. Like\nYIG, La:YIG also has spin up channel to contribute to the valence ba nd while spin down\nchannel contributes to the conduction band. The effective mass v alues of La:YIG in the\nthree reciprocal vector directions χ−Γ (mx), Γ to L (m y) and Γ toη(mz) are 0.4462*m e,\n0.4898*m eand 0.4577*m erespectively. The harmonic mean of the three the values yields\nthe effective mass of the ↓conduction band to be 0.4639*m e. The effective mass of La:YIG\nis less than that of YIG which has 0.5048*m e, thus making the conduction electrons of\nLa:YIG to have more mobility than YIG.\nD. Validation of Model\nAfter creating the Hamiltonian of the scattering region, a fine spec trum of energy bands\nin a range 0 to 5eV was created for each eigen energy level to calcula te the total current in\nthe region due to thermal bias. As the model treats the two spin ch annels independent, the\nrelative movement of the spin up and spin down electrons is different a nd leads to a spin\nredistribution in the material due to the non-equilibrium induced by th e contacts.\nUsing the fundamental parameters from IVC, the approach desc ribed in the sections II\nand III of this paper was implemented in MATLAB to calculate the spin v oltage in the\ntransversely attached Pt NM contact. To compare with the literat ure, Uchida et al27, a\ntemperature bias of 20K is assumed through the analysis. The calcu lations were performed\nat 300K temperature (of the NM contact) and linear increase of te mperature from left\ncontact (cold) to the right contact (hot), Figure 1a. Table I show s the experimental values\nand the parameters incorporated in the model. The model used the length of the scatteringPredictingtheSpinSeebeckVoltageinSpin-polarizedMaterials: AQua ntumMechanicalTransportModelApproach 13\n-4 -3 -2 -1 0 1 2 3 4\nlattice in mm (left to right is cold to hot)-2.5-2-1.5-1-0.500.511.522.5V ISHE ( V)mdn=0.4639 mup=0.0804\nExperimental data dT=20 K\nNEGF transport model data dT=20 K\nFIG. 7: Validation of the model with the experimental data available f or La:YIG. The\nVISHEvoltage inµV is calculated at the different points on the material lattice along the\nx-direction.\nregion along x-direction to be 36nm. A higher length can be incorpora ted depending on the\ncomputational power in reach. On an 8-core processor, a 36nm len gth scale took 4 hours to\ncomplete the calculation. The number of grid points, N=300, were ch osen until the relative\nchannel potential (U) reached a value 1E-6eV.\nThe model incorporates the same ratio of L x−dir:Ly−dir:Lz−dirof Pt NM contact as used\nin experimental verification. The ratio being 1:40:15E-5. Here, the le ngth of the Pt strip\nalong y direction is related to the length of the La:YIG material along t he y-direction which\nare the same values. Similarly, the length of FM material along the x-d irection is related\nto the length of the NM contact along the y-direction. The ratio of t he parameters in the\nmodel are same as those in the experimental setup. As a 1-D model was developed in this\nresearch, the quantities like spin current injected into the NM cont act along the z-direction\nhave the units A/m2and the population density of individual spin channels will have the\nunits 1/m2. Hence, the spin populations at each grid point along x-direction, Fig ure 1b,\nwerecalculatedandmultiplied with the areaofcontactbetween NM (P t) andFM (La:YIG),\nwhich is 0.45x18nm2. The difference in the populations of the spin channels causes the sp in\ncurrent to flow along the z-direction into the transversely NM cont act. This spin current\nfrom La:YIG is convertedinto inverse spin Hall voltageV ISHEin the Pt contact. The other\nconstants extracted from literature are shown in Table II.\nIncorporating the parameters from Tables I, II in the aforement ioned model, the V ISHE\nin the transversely attached Pt electrode at various locations alon g the scattering region\nis shown in Figure 7. The red line represents the trend calculated fro m the NEGF modelPredictingtheSpinSeebeckVoltageinSpin-polarizedMaterials: AQua ntumMechanicalTransportModelApproach 14\nTABLE I: Parameters used in the experimental setup of Figure 1a a s reported in\nliterature27and the parameters used in verification.\nMaterialLength\nx-directionLength\ny-directionLength\nz-direction\nLa:YIG\nexperimental8 mm 4 mm 3.9µm\nPt\nexperimental0.1 mm 4 mm 15 nm\nNEGF model\nscattering region36 nm 18 nm 17.5E-3 nm\nNEGF model\nNM contact0.45 nm 18 nm 6.75E-5 nm\nTABLE II: Constants incorporated in the model.\nConstant Value\nγ\nreference (31) 1.4x10111/T.s\nρ\nreference (25)\nelectrical resistivity of Pt at 300K15.6E-8 Ω.m\ngr/A\nreference (13) 0.1x10161/m2\nθH\nreference (14)\nHall angle of Pt0.0037\nVa\nvolume of NM0.45x18x6.75E-5 nm3\nfor a 20K temperature bias and the black line with the data points rep resented in black\ncircles is the experimental trend for the same temperature obtain ed from literature27. The\nslope of the NEGF trend is close to the experimental data with an err or of 6.5%. The\ndeveloped model can be applied to magnetic materials which have atom s with d-electrons.\nBy assuming a ±5% error in effective mass, the model predicts the curve with ±15 % close\nto that of experimental value. Hence, this model can be applicable t o materials whose\nexperimental transverse spin properties are not available.\nV. CONCLUSIONS\nThrough this research a 1-D model combining DFT, NEGF and spin tra nsport theories\nwas developed that treats both spin channels independently and ca lculates electronic and\nspin conductivities, and spin-Seebeck coefficient of the material. Va lidation was performed\non La:YIG insulator material which proves the applicability of the mode l to various other\nsemiconducting magnetic materials and opening a new avenue to theo retically evaluate the\nparameters that effect spin Seebeck coefficient. 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Here, we propose a scheme to enhance the\nspin-magnon coupling by the magnonic Kerr nonlinearity in a YIG sphere. We find that the Kerr-\nenhanced spin-magnon coupling invalidates the widely used single-Kittel-mode approximation to\nmagnons. It is revealed that the spin decoherence induced by the multimode magnons in the strong-\ncoupling regime becomes not severe, but suppressed, manifesting as either population trapping or\npersistent Rabi-like oscillation. This anomalous effect is because the spin changes to be so hybridized\nwith the magnons that one or two bound states are formed between them. Enriching the spin-\nmagnon coupling physics, the result supplies a guideline to control the spin-magnon interface.\nIntroduction.— Magnons are the elementary excitation\nof a collective spin wave in magnetic materials. The\nquantized interactions of magnons with different quan-\ntum platforms have inspired many novel applications in\nquantum technologies [1–11]. Besides quantum trans-\nduction [12], memory [13], sensing [14, 15], and uni-\ndirectional invisibility [16], using the coupling between\nmagnons and photons or phonons, the efficient couplings\nof magnons to spins have attracted much attention due\nto their potential realization of quantum networks [17]\nand quantum sensing [18–20]. The efficient spin-magnon\ncouplings via either direct interactions [21–28] or the in-\ndirect way by exchanging photons or phonons [29–32]\nhave been proposed. How to enhance the spin-magnon\ncoupling strength is a prerequisite to explore their appli-\ncations.\nThe Kerr nonlinearity of magnons in magnetic mate-\nrials supplies a useful mechanism in quantum-state engi-\nneering [33, 34]. Based on it, magnon-polariton bistabil-\nity [35–37] and tristability [38, 39], which are useful in the\nmicrowave nonreciprocal transmission [40], high-order\nsideband [41–43] and entanglement generations [44], and\nquantum phase transition [45], have been reported. A\nscheme to enhance the spin-magnon coupling using the\nKerr nonlinearity was proposed in Ref. [46]. These works\non quantum magnonics were generally based on an ap-\nproximation in which the magnons are effectively treated\nas a single first-order Kittel mode [47–50]. It has been\nrevealed that the higher-order magnonic modes are non-\nnegligible in the presence of the Kerr nonlinearity [51–\n53]. References [23–25] studied the interactions between\nspins and multimode magnons. However, they are under\nthe Markovian approximation, which is only valid in the\nweak-coupling condition.\nHere, we investigate the non-Markovian dynamics of\na spin defect coupled to magnons in a YIG sphere. A\nscheme to enhance the spin-magnon coupling by the\nmagnonic Kerr nonlinearity is proposed. We find that\nthe increasing of the coupling invalidates the widely\nused single-mode approximation of magnons to describe\n𝑎Magnetic\nemitterYIG\n𝜃𝑟\n𝜑\n𝐇𝑒𝐌𝑠\n𝑥𝑦𝑧\n𝜔𝑑FIG. 1. Schematic illustration of the system. A magnetic\nemitter interacts with the magnons in a YIG sphere with ra-\ndiusRin a static magnetic field He, and an induced magnetic\nfieldMs. A driving field with frequency ωdis applied.\nthe matter-magnon coupling. The strong coupling also\ncauses the spin to exhibit features with a suppressed de-\ncoherence, i.e., from the conventional oscillating damp-\ning to either the population trapping or the persistent\nRabi-like oscillation. Our analysis reveals that such an\nanomalous decoherence is due to the formation of dif-\nferent numbers of spin-magnon bound states. Indicating\nthat the Kerr nonlinearity endows the spin-magnon in-\nterface with a good controllability, our result paves the\nway to design quantum magnon devices.\nSystem and spectral density.— We consider a spin de-\nfect as a magnetic emitter coupled to the magnons at-\ntached to a YIG sphere in the presence of the Kerr non-\nlinearity (see Fig. 1). Its Hamiltonian reads [22, 26]\nˆHKerr=ℏω0ˆσ†ˆσ+/summationdisplay\nk/bracketleftbig\nℏωkˆb†\nkˆbk−(ℏK/2)ˆb†2\nkˆb2\nk\n−(gkˆb†\nkˆσ−Ωdeiϕdˆb†\nke−iωdt+ H.c.)/bracketrightbig\n.(1)\nHere, ˆ σ=|g⟩⟨e|is the transition operator of the spin de-\nfect with frequency ω0from the excited state |e⟩to the\nground state |g⟩,ˆbkis the annihilation operator of the kth\nmagnon mode with frequency ωk, and gk=µ0m·˜H∗\nk(r),\nwithmbeing the spin magnetic moment and ˜Hk(r) beingarXiv:2308.05927v2  [quant-ph]  28 Nov 20232\nthe vacuum amplitude of the kth magnon mode concen-\ntrated around the YIG sphere, is their coupling strength.\nAn inhomogeneous rf magnetic excited field is needed\nto trigger the multimode magnons [54]. The magnons\nare further driven by a microwave field with frequency\nωd, amplitude Ω d, and phase ϕd. The Kerr nonlinear-\nity quantified by Kis caused by the magnetocrystalline\nanisotropy of the YIG sphere. It has been used to gen-\nerate the magnon squeezing [34], which is attractive to\nthe application of the magnons [1, 55]. Rewriting the\nmagnon operators as the sum of their steady-state mean\nvalue and fluctuations, i.e., ˆbk=⟨ˆbk⟩+δˆbk, and ne-\nglecting the high-order fluctuation terms in the strong\ndriving condition, Eq. (1) in the rotating frame with\nˆH0=ωd(ˆσ†ˆσ+/summationtext\nkˆb†\nkˆbk) becomes (see Supplemental Ma-\nterial [56])\nˆHLinear =ℏ∆0ˆσ†ˆσ+/summationdisplay\nk/braceleftbig\nℏ(ωk−Πk)ˆb†\nkˆbk\n−[gkˆb†\nkˆσ+ (ℏKk/2)ˆb2\nk+ H.c.]/bracerightbig\n,(2)\nwhere ∆ 0=ω0−ωd, Πk=ωd+ 2Kk, andKk=K⟨ˆbk⟩2.\nWe have rewritten δˆbkasˆbkfor brevity. Making the\nBogoliubov transformation ˆS= exp[/summationtext\nkrk(ˆb2\nk−ˆb†2\nk)/2],\nwith rk=1\n4ln(ωk−Πk+Kk\nωk−Πk−Kk), to Eq. (2) and neglecting the\ncounter-rotating terms [46], we obtain\nˆH=ℏ∆0ˆσ†ˆσ+/summationdisplay\nk[ℏζkˆb†\nkˆbk−(Gkˆb†\nkˆσ+ H.c.)] ,(3)\nwhere ζk= (ωk−Πk)/cosh (2 rk) and Gk=erkgk/2. We\nfind that the spin-magnon coupling is exponentially en-\nhanced and the magnon frequencies are suppressed by\nthe Kerr nonlinearity assisted by the microwave driving.\nThis is one of our main results. To simplify our discus-\nsion, we approximate Π k≃ωddue to ωd≫ K kand\nrk≡rby neglecting their k-dependence, which is valid\nby properly choosing Ω dandωdin our finite magnonic\nbandwidth case [54].\nConsider that the YIG sphere is at low temperature\nsuch that the magnons are initially in the vacuum state\n[44, 57]. After tracing over the magnonic degrees of free-\ndom from the dynamics of the spin-magnon system, we\nderive an exact master equation of the spin as (see Sup-\nplemental Material [56])\n˙ρ(t) =iΩ(t)[ρ(t),ˆσ†ˆσ]+Γ(t)[2ˆσρ(t)ˆσ†−{ˆσ†ˆσ, ρ(t)}].(4)\nThe renormalized frequency is Ω( t) =−Im[˙c(t)/c(t)] and\nthe decay rate is Γ( t) =−Re[˙c(t)/c(t)], where c(t) satis-\nfies\n˙c(t) +i∆0c(t) +/integraldisplayt\n0dτc(τ)f(t−τ) = 0 , (5)\nunder c(0) = 1. The convolution in Eq. (5) makes\nthe dynamics non-Markovian with all the memory ef-\nfects incorporated in the time-dependent coefficients inEq. (4). The magnonic correlation function is f(t−τ) =/integraltextζmax\nζmindζJ(ζ)e−iζ(t−τ)and the spectral density is [22]\nJ(ζ) =ηµ0\n4ℏπIm[m∗·k2G(r,r, ζcosh(2 r) + Π) ·m].(6)\nwhere k= [ζcosh(2 r) + Π] /candη=e2rcosh (2 r). The\nGreen’s tensor reads ¯k2G(r,a,¯ω) =/summationtext\nα,β∈{r,θ,φ}[Hβ\n0,α+\nHβ\nα]eαeβ, where H=−∇∇∇ϕandH0takes the similar\nform as Hbut in the absence of the YIG sphere. The\npotential ϕcaused by the YIG satisfying ϕ=1\n4π∇∇∇a1\n|r−a|\nis subject to the boundary condition of (1 + χ)(∂2\n∂x2+\n∂2\n∂y2)ϕ+∂2ϕ\n∂z2= 0 in r≤Rand∇2ϕ= 0 in r > R [58]. χχχ\nis the magnetic susceptibility tensor and determined by\nthe Landau-Lifshitz-Gilbert equation as [59, 60]\nχxx=χyy=γ2h0Ms\nγ2h2\n0−ω2−iΓ0ω≡χ,\nχxy=χ∗\nyx=iγωM s\nγ2h2\n0−ω2−iΓ0ω≡iκ,(7)\nwhere γis the gyromagnetic ratio, Γ 0= 2γh0α, with α\nbeing the Gilbert parameter, is the damping parameter,\nh0=h0ez=He+Hd, with Hebeing the external static\nfield and Hd=−Ms/3 being the demagnetization field,\nandMsis the saturation magnetization. Putting the spin\non the equatorial plane of the YIG sphere, i.e., θ=π/2,\nand choosing m=−µB(ex+iey) =−µBeiφ(er+ieφ),\nwe have the nonzero components of the Green’s tensor as\nIm[m∗·G·m] =µ2\nB[Im(Grr+Gφφ) + Re( Grφ−Gφr)]\n[22, 24, 48]. The analytic form of G(r,a,¯ω) is given in\n(see Supplemental Material [56]). The Green’s tensor and\nthe spectral density show resonance peaks determined by\n[59, 60]\n(n+ 1−mκ)Pm\nn(ξ0) +ξ0Pm′\nn(ξ0) = 0 , (8)\nwhere Pm\nnis the associated Legendre polynomial and\nξ0= (1 + 1 /χ)1/2. The magnon modes corresponding to\nn=−m= 1, 2, and 3 in the absence of the Kerr nonlin-\nearity are the dipole or Kittel mode ωK=γ(h0+Ms/3),\nthe quadrupolar mode ωQ=γ(h0+ 2Ms/5), and the\noctupolar mode ωO=γ(h0+ 3Ms/7), respectively.\nIn the presence of the Kerr nonlinearity, they become\nζK,Q,O = (ωK,Q,O−Π)/cosh(2 r). Ranging all mandn,\nit was found that the frequency range of J(ζ) is from\nζmin= (γh0−Π)/cosh(2 r) toζmax= [γ(h0+Ms/2)−\nΠ]/cosh(2 r) [54].\nSpin dynamics.— A widely used approximation in\nstudying matter-magnon coupling is the Markovian ap-\nproximation [23–25, 47, 50, 61–63]. It is valid when their\ncoupling is weak and the time scale of f(t−τ) is much\nsmaller than the one of the matter. After replacing c(τ)\nbyc(t) and extending the upper bound of the time in-\ntegral to infinity, the Markovian approximate solution\nof Eq. (5) is cMA(t) =e−[Λ+iΥ(∆ 0)]t, where Υ(∆ 0) =3\nP/integraltext\ndζJ(ζ)/(∆0−ζ) is the Lamb shift and Λ = πJ(∆0)\nis the spontaneous emission rate. The exponential-decay\nfeature of |cMA(t)|2characterizes a unidirectional energy\nflow from the spin to the magnons and the destructive\neffect of the magnons on the spin. This approximation\ncannot reflect the energy back flow induced by the strong\nspin-magnon coupling [22, 26].\nA pseudocavity method was proposed to study the\nstrong light-matter coupling in an absorptive medium\n[64–68]. Keeping only the Kittel mode ζK, we approxi-\nmate J(ζ) as a Lorentzian form J(ω) =J(ζK)(γp/2)2\n(ω−ζK)2+(γp/2)2.\nThe system is effectively seen as a spin coherently inter-\nacting with a pseudocavity mode ˆ awith frequency ζKand\ndamping rate γpin a coupling strength g2=πJ(ζK)γp/2.\nHere, γpis relevant to the damping parameter Γ 0. Thus,\nthe spin dynamics is phenomenologically described by\n˙ρ(t) =i[ρ(t),∆0ˆσ†ˆσ+ζKˆa†ˆa+g(ˆaˆσ†+ H.c.)]\n+γp\n2[2ˆaρ(t)ˆa†− {ˆa†ˆa, ρ(t)}].(9)\nAlthough partially reflecting the energy back flow from\nthe magnons to the spin, this method misses important\nphysics from the magnonic higher-order resonant modes.\nTo fully capture the physics of the strong spin-magnon\ncoupling enhanced by the Kerr nonlinearity and uncover\nthe condition under which the pseudocavity method is\napplicable, we investigate the exact spin dynamics by\nchoosing ∆ 0=ζK. The steady-state solution of Eq. (5)\nis computable by a Laplace transform. It converts Eq.\n(5) into ˜ c(s) = [s+i∆0+/integraltextζmax\nζmindζJ(ζ)\ns+iζ]−1.c(t) is obtained\nby making an inverse Laplace transform to ˜ c(s), which\nrequires finding its poles via (see Supplemental Material\n[56])\nE\nℏ= ∆ 0+/integraldisplayζmax\nζminJ(ζ)\nE/ℏ−ζdζ≡Y(E), (10)\nwhere E=iℏs. First, the roots Eof Eq. (10)\nare exactly the eigenenergies of the total spin-magnon\nsystem. To prove this, we expand the eigenstate as\n|ϕE⟩=x|e,{0k}⟩+/summationtext\nkyk|g,1k⟩. Substituting |ϕE⟩\ninto ˆH|ϕE⟩=E|ϕE⟩, we readily obtain Eq. (10).\nSecond, because Y(E) is a decreasing function in the\nregimes E∈(−∞,ℏζmin] and [ ℏζmax,+∞), Eq. (10)\nhas one isolated root Ebin (−∞,ℏζmin] or [ ζmax,+∞)\nprovided Y(ℏζmin)<ℏζminorY(ℏζmax)>ℏζmax.\nThe eigenstate corresponding to Ebis called the bound\nstate. On the other hand, Y(E) is non-analytical in\nthe regime E∈[ℏζmin,ℏζmax] due to the singularity\nin its integration. Therefore, Eq. (10) has an infi-\nnite number of roots in this regime, which form an en-\nergy band. Using the residue theorem, we have c(t) =/summationtextM\nj=1Zje−i\nℏEb\njt+/integraltextζmax\nζminΘ(E)e−iEtdE, where Θ( E) =\nJ(E)\n[E−∆0−Υ(E/ℏ)]2+[πJ(E)]2,Mbeing the number of the\nbound states, and Zj= [1 +/integraltextζmax\nζminJ(ζ)dζ\n(Eb\nj/ℏ−ζ)2]−1the\nFIG. 2. (a) Spectral density J(ζ) in different spin-YIG dis-\ntance a. (b) Kittel-mode frequency ζKand enhancement coef-\nficient ηin different r. (c) Evolution of the excited-state pop-\nulation |c(t)|2from the Markovian approximation (red dot-\nted line), the pseudocavity method (cyan dashed line), and\nthe non-Markovian dynamics (blue line). The inset is J(ζ)\nand the fitted Lorentzian form J(ζ). We use ωd= 1 GHz,\nγ= 28 GHz ·T−1, Γ0= 8×10−3GHz, Ms= 0.178 T, h0= 0.5\nT,R= 30 nm, r= 2 in (a) and (c), and a= 26 nm in (c).\nresidue contributed by the jth bound state. Oscillating\nwith time in continuously changing frequencies E/ℏof\nthe band energies, the integrand tends to zero in the long-\ntime limit due to the out-of-phase interference. Thus, the\nsteady-state solution of Eq. (10) is [69]\nlim\nt→∞c(t) =/braceleftigg\n0, no bound state/summationtextM\nj=1Zje−i\nℏEb\njt, M bound states.(11)\nAssuming the spin is initially in |e⟩and solving Eq. (4),\nwe obtained that the excited-state population is just\n|c(t)|2(see Supplemental Material [56]). Thus, Eq. (11)\nreveals that thanks to the Kerr-nonlinearity-enhanced\nspin-magnon coupling, the formation of the bound states\nwould prevent the spin from relaxing to its ground state.\nSince it is not obtained from both the Markovian approx-\nimation and the pseudocavity method, such an anoma-\nlous decoherence manifests the distinguished role played\nby the non-Markovian effect and the feature of the en-\nergy spectrum of the total spin-magnon system in the\ndecoherence of the spin. This is another main result of\nour work.\nNumerical results.— We plot in Fig. 2(a) the spectral\ndensity J(ζ) in different spin-YIG distance aforr= 2.\nThe driving-field frequency ωdis chosen as ωd= 1 GHz\nand the Kerr coefficient Krelates to the volume Vof the4\nthree-dimensional YIG sphere, i.e., K∝V−1[34] and is\nabout kHz [46], which makes the validity of Π ≃ωd. We\nreally see that J(ζ) exhibits obvious peaks at ζ= 537,\n549, and 554 MHz irrespective of the value of a, which\nmatch well with our analytical frequencies ζK,Q,O of the\nKittel, quadrupolar, and octupolar modes evaluated from\nEq. (8). With decreasing a,J(ζ) shows an increase due\nto the near-field enhancement [26]. It signifies a strong\nspin-magnon coupling in the small- aregime. Figure 2(b)\nshows the effect of the Kerr nonlinearity on enhancing the\nspin-magnon coupling. It reveals that, with increasing r\nfrom zero to 2, ζKdecreases from 15 GHz to 537 MHz,\nwhile the prefactor ηofJ(ζ) increases from 1 to 1200.\nAn efficient increase of about four orders of magnitude\nofη/ζKmanifests a dramatic boost of the spin-magnon\ncoupling strength. It confirms that the Kerr nonlinearity\ncan be used to enhance the spin-magnon coupling [37,\n38, 46].\nFigure 2(c) shows the comparison of |c(t)|2obtained by\nthree methods when a= 26 nm. The exponential decay\nin the Markovian result entirely fails to describe the rapid\nspin-magnon energy exchanges obtained via numerically\nsolving Eq. (5), which is fully captured by the pseudo-\ncavity method. It is the signature of the non-Markovian\nmemory effect owned by the strong-coupling dynamics\n[70]. In this case, J(ζ) is dominated by the Kittel mode\nsuch that a Lorentzian fitting centered at ζKis sufficient\nand the pseudocavity method works well. However, with\nfurther decreasing a[see Fig. 2(a)] or increasing r, the\nhigh-order magnon modes become dominated, where the\npseudocavity method no longer work.\nFigure 3(a) shows the exact |c(t)|2in different awhen\nr= 2. The strong spin-magnon coupling favored by both\nthe near-field enhancement [26] and Kerr nonlinearity\ncauses |c(t)|2to exhibit rich behaviors. It is interesting\nto find that, in contrast to the damping to zero for a= 13\nnm, which has no qualitative difference from the result\npredicted by the pseudocavity method, |c(t)|2approaches\na finite value when a= 9 nm, while it exhibits a lossless\nRabi-like oscillation when a= 4 nm. It reveals an anoma-\nlous behavior in which a small spin-magnon distance with\nthe Kerr nonlinearity induces a strong spin-magnon cou-\npling, which, on the contrary, causes a suppressed deco-\nherence. It is not expected that a stronger spin-magnon\ncoupling always causes a more severe decoherence to the\nspin [32, 71]. This behavior can be explained by the\nfeatures of the energy spectrum of the total spin-magnon\nsystem. Figure 3(b) indicates that two branches of bound\nstates separate the energy spectrum into three regimes.\nWhen a≥10.4 nm, no bound state is formed and thus\n|c(t)|2decays to zero. When 4 .8 nm < a < 10.4 nm,\none bound state is present and |c(t)|2tends to finite val-\nues. When a≤4.8 nm, two bound states are present\nand|c(t)|2behaves as a persistent Rabi-like oscillation\nin a frequency |Eb\n1−Eb\n2|/ℏ. The matching of the long-\ntime behaviors of the three regimes with the analytical\nFIG. 3. (a) Evolution of |c(t)|2in different a. The black\ndashed lines are the corresponding steady-state values from\nEq. (11). The time for the cases of a= 9 and 13 nm is\nmagnified by a factor of 0.03. Energy spectrum of the whole\nsystem in different (b) aand (d) robtained by solving Eq.\n(10).|c(∞)|2from solving Eq. (5) denoted by the dots and\nfrom Eq. (11) denoted by the solid lines in different (c) aand\n(e)r. The red region covers the values during its persistent\noscillation. (f) |c(∞)|2when ∆ 0=ζK(brown dot), ζQ(purple\nsquare), and ζO(cyan rhombus). r= 2.0 in (b) and (c), a= 4\nnm in (d) and (e), and others are the same as Fig. 2(c).\nresult in Eq. (11) verifies the distinguished role played\nby the bound states and non-Markovian effect in deter-\nmining the strong-coupled spin-magnon physics; see Fig.\n3(c). Figures 3(d) and 3(e) indicate that it is just the\nKerr-nonlinearity-induced strong spin-magnon coupling\nthat causes the formation of the bound states and the\naccompanying population trapping and persistent Rabi-\nlike oscillation. Figure 3(f) shows |c(∞)|2in the dis-\ntances asupporting the formation of one bound state.\nIt demonstrates that the trapped population |c(∞)|2can\nbe controlled by choosing ∆ 0as different magnonic res-\nonant frequencies. All the results prove that the strong\nspin-magnon coupling endows the spin with rich anoma-\nlous decoherence governed by the formation of different\nnumbers of bound states. It supplies a guideline to con-\ntrol the spin coherence via engineering the feature of the\nspin-magnon energy spectrum.5\nDiscussion and conclusions.— Quantum magnonics\nexploring the efficient couplings between magnons and\ndifferent kinds of quantum matter has made great\nprogress [1, 4, 19, 31, 32, 35, 37, 72–76]. Many of these\nworks were based on the single-magnon-mode approxi-\nmation, which may be insufficient in the strong matter-\nmagnon coupling. The coupling between single spins and\nmultimode magnons was studied in Ref. [22], but the\nKerr nonlinearity was absent. The Kerr nonlinearity has\nbeen observed in cavity magnon mechanics formed by\nthe YIG [35, 37, 75]. The bound state and its distin-\nguished role in the non-Markovian dynamics have been\nexperimentally observed in both photonic crystal [77] and\nultracold-atom [78, 79] systems. The progresses give a\nstrong support that our finding is realizable in state-of-\nthe-art experiments [76, 80]. Note that, although only\nthe YIG is studied, our results are applicable to other\nmagnetic materials, such as CoFeB [19, 81]. As a fi-\nnal remark, the expectation value of the magnonic fluc-\ntuation operator described by ˆbkin Eq. (3) should be\nzero to ensure the self-consistence of our linearization\napproximation to Eq. (1). This can be proven as fol-\nlows. The evolved state of the spin-magnon system under\nour studied initial condition |Ψtot(0)⟩=|e,{0k}⟩reads\n|Ψtot(t)⟩=c(t)|e,{0k}⟩+/summationtext\nkdk(t)|g,1k⟩, which readily\nleads to ⟨Ψtot(t)|ˆbk|Ψtot(t)⟩= 0.\nIn summary, we have investigated the near-field inter-\nactions between a spin defect and magnons in a YIG\nsphere with Kerr nonlinearity. It is found that the\nKerr nonlinearity induces a dramatic enhancement to\nthe spin-magnon coupling. Contrary to the belief that\na strong coupling always causes a severe decoherence,\nsuch a strong coupling makes the magnon-induced de-\ncoherence to the spin change from complete damping to\neither population trapping or persistent Rabi-like oscil-\nlation. 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B 104, 214416 (2021).Supplemental Material for “Kerr nonlinearity induced strong spin-magnon coupling”\nFeng-Zhou Ji1and Jun-Hong An1,∗\n1Key Laboratory of Quantum Theory and Applications of MoE, Lanzhou Center for Theoretical Physics,\nand Key Laboratory of Theoretical Physics of Gansu Province, Lanzhou University, Lanzhou 730000, China\nI. MAGNON-SPIN COUPLING SYSTEM\nWITHIN KERR NONLINEARITY\nWe consider a spin defect as a magnetic emitter cou-\npled to the magnons attached to a YIG sphere in the\npresence of the Kerr nonlinearity. The Hamiltonian reads\nˆHKerr =ℏω0ˆσ†ˆσ+/summationdisplay\nk[ℏωkˆb†\nkˆbk−(ℏK/2)ˆb†2\nkˆb2\nk\n−(gkˆb†\nkˆσ−Ωdeiϕdˆb†\nke−iωdt+ H.c.)] ,(S1)\nwhere ˆ σ=|g⟩⟨e|is the transition operator of the spin\ndefect with frequency ω0,ˆbkis the annihilation operator\nofk-th magnon with frequency ωk. The gk=µ0m·˜H∗\nk(r)\nis related to the vacuum amplitude of kth magnon mode.\nIn the rotating frame with ˆH0=ωd(/summationtext\nkˆb†\nkˆbk+ ˆσ†ˆσ), Eq.\n(S1) becomes\nˆH1=ℏ∆0ˆσ†ˆσ+/summationdisplay\nk[ℏ(ωk−ωd)ˆb†\nkˆbk−(ℏK/2)ˆb†2\nkˆb2\nk\n−(gkˆb†\nkˆσ−Ωdeiϕdˆb†\nk+ H.c.)] , (S2)\nwhere ∆ 0=ω0−ωd. The Heisenberg-Langevin equation\nsatisfied by ˆbkis\ndˆbk\ndt=−i[(ωk−ωd)ˆbk−Kˆb†\nkˆb2\nk−gk\nℏˆσ+Ωdeiϕd\nℏ]\n−Γ0ˆbk+/radicalbig\n2Γ0ˆFnoise, (S3)\nwhere Γ 0is the damping rate of magnon, ˆFnoise is the\nvacuum noise with a zero expectation value. Decompos-\ningˆbkas its steady-state expectation value and fluctua-\ntion, i.e., ˆbk=⟨ˆbk⟩+δˆbk, we have\n0 =−i[(ωk−ωd)⟨ˆbk⟩+Ωdeiϕd\nℏ−KNk⟨ˆbk⟩]−Γ0⟨ˆbk⟩,(S4)\nwhere we have used the mean-field approximation by\nreplacing ˆb†\nkˆbkas its expectation value Nk=⟨ˆb†\nkˆbk⟩ ≃\n|⟨ˆbk⟩|2under the large- Nkcondition. We obtain\n⟨ˆbk⟩=−Ωdeiϕd/ℏ\n(ωk−ωd−KNk−iΓ0)≃−Ωdeiϕd/ℏ\n(ωk−ωd−iΓ0).\n(S5)\nunder the condition ωd≫K. The large- Nkcondition\nrequired by the mean-field approximation is valid in the\n∗anjhong@lzu.edu.cnregime with a large Ω dandωd≫K. Then we can use the\nlarge⟨ˆbk⟩to linearize the Hamiltonian. The equations of\nmotion of the fluctuation operators becomes\ndδˆbk\ndt=−i[(ωk−ωd−2KNk)δˆbk−gk\nℏˆσ] +/radicalbig\n2Γ0ˆFnoise\n−Γ0δˆbk+iK⟨ˆbk⟩2δˆb†\nk, (S6)\nwhere the higher-order terms of the fluctuation opera-\ntors have been neglected [1]. By properly tuning the\nphase ϕdof the driving field, we may obtain a real ⟨ˆbk⟩.\nThen ⟨ˆbk⟩2=Nk. Equation (S6) describes a dynamics\ngoverned by a linearized Hamiltonian\nˆHLinear =ℏ∆0ˆσ†ˆσ+/summationdisplay\nk[(ℏ(ωk−Πk)δˆb†\nkδˆbk\n−(gkδˆb†\nkˆσ+ (ℏKk/2)δˆb2\nk+ H.c.)] ,(S7)\nwhere Π k=ωd+ 2K⟨ˆbk⟩2andKk=K⟨ˆbk⟩2. A\nBogoliubov transformation ˆS=/producttext\nkerk\n2(δˆb2\nk−δˆb†2\nk), with\nrk=1\n4ln(ωk−Πk+Kk\nωk−Πk−Kk), converts Eq. (S7) into\nˆH=ℏ∆0ˆσ†ˆσ+/summationdisplay\nk[(ℏζkδˆb†\nkδˆbk−(Gkδˆb†\nkˆσ+ H.c.)] ,(S8)\nunder the rotating-wave approximation, where ζk=\n(ωk−Πk)/cosh(2 rk) and Gk=gkerk/2. After replacing\nδˆbkbyˆbkfor brevity, we arrive at the linearized Hamil-\ntonian in Eq. (3) of the main text.\nWe can tune ωdsuch that rkandKkexhibit a soft de-\npendence on k. We can use a relative large ωdsuch that\nωd≫ K k=K|⟨ˆbk⟩|2. It makes us roughly approximate\nKk≃ K, which reduces rkintork≃1\n4ln(ωk−ωd−K\nωk−ωd−3K).\nFurther, because ωkis within a finite band regime, for\nexample, ωkof the YIG is limited between 14 GHz to\n16.5 GHz [2], we can neglect the k-dependence of rktoo.\nThus, we phenomenologically investigate the effect of the\nKerr nonlinearity induced by a constant rfor simplifica-\ntion.\nII. SPECTRAL DENSITY\nWe derive the spectral density using the Green’s tensor\nmethod. It can be readily derived from Eq. (3) in the\nmain text that the spectral density reads\nJ(ζ) =/summationdisplay\nk|Gk|2\nℏ2δ(ζ−ζk)\n=e2rµ2\n0\n4ℏ2m∗·/summationdisplay\nk|˜Hk(r)|2δ(ζ−ζk)·m. (S9)arXiv:2308.05927v2  [quant-ph]  28 Nov 20232\nIn order to derive ˜Hk(r), we investigate a magnetic field\ninduced by a magnetic dipole ¯mataand oscillating at\nfrequency ¯ ω. The magnetic-field strength at rcan be\nrepresented by the Green’s tensor as\nH(r) =¯k2G(r,a,¯ω)·¯m, (S10)\nwhere ¯k= ¯ω/c. The process is semiclassically described\nby\nˆHsource =/summationdisplay\nk[ℏωkˆb†\nkˆbk−(gke−i¯ωtˆb†\nk+ H.c.)] ,(S11)\nwhere gk=µ0¯m·˜H∗\nk(a). According to the Heisenberg\nequation satisfied by ˆbk, i.e.,\ndˆbk\ndt=−iωkˆbk+igk\nℏe−i¯ωt. (S12)\nand the initial condition ⟨ˆbk(0)⟩= 0, we get ⟨ˆbk(t)⟩=\nigk\nℏ/integraltextt\n0dτe−i(¯ω−ωk)τ. It long-time limit reads\n⟨ˆbk(∞)⟩=gk\nℏ1\n¯ω−ωk−i0+. (S13)\nThen the magnetic field is recast into\nH(r) =/summationdisplay\nk˜Hk(r)⟨ˆbk(∞)⟩=µ0\nℏ/summationdisplay\nk˜Hk(r)˜H∗\nk(a)·¯m\n¯ω−ωk−i0+.(S14)\nComparing Eq. (S10) with Eq. (S14), we have\n¯k2G(r,a,¯ω) =µ0\nℏ/summationdisplay\nk˜Hk(r)˜H∗\nk(a)\n¯ω−ωk−i0+. (S15)\nUsing the identity1\n¯ω−ζk−i0+=iπδ(¯ω−ωk) +\nP1\n¯ω−ωk, Eq. (S15) results in Im[ ¯k2G(r,a,¯ω)] =\nπµ0\nℏ/summationtext\nk˜Hk(r)˜H∗\nk(a)δ(¯ω−ωk). Substituting ζk= (ωk���\nΠ)/cosh (2 r), with Π = ωd, into Eq. (S9), we obtain\nJ(ζ) =ηµ2\n0\n4ℏ2m∗·/summationdisplay\nk|˜Hk(r)|2δ(cosh(2 r)ζ+ Π−ωk)·m\n=ηµ0\n4ℏπIm[m∗·k2G(r,r, ζcosh(2 r) + Π) ·m],(S16)\nwhere k= [ζcosh(2 r) + Π] /candη=e2rcosh(2 r).\nIII. MAGNETOSTATIC GREEN’S TENSOR\nWe provide the detailed derivation of the magneto-\nstatic Green’s tensor induced by a point dipole near the\nYIG sphere [2, 3]. The Green’s tensor is given by\n¯k2G(r,a,¯ω) = ¯k2G0(r,a,¯ω) +/summationdisplay\nα,β∈{r,θ,φ}Hβ\nαeαeβ\n=/summationdisplay\nα,β∈{r,θ,φ}[Hβ\n0,α+Hβ\nα]eαeβ,(S17)where G0(r,a,¯ω) is the Green’s tensor describing the\nmagnetic field of a magnetic dipole in a vacuum. The\nmagnetic fields H=−∇∇∇ϕandϕ=1\n4π∇∇∇a1\n|r−a|.H0takes\nthe similar form as Hbut in the absence of the YIG\nsphere.\nThe potential of a point charge can be expressed in\nstandard spherical coordinates as [2]\n1\n|r−a|=/summationdisplay\nlm(l−m)!\n(l+m)!Rl,m(r, θ, φ )I∗\nl,m(a, θa, φa),(S18)\nwhere In,m(R) and Rn,m(R) are the solid harmonics:\nIn,m(R) =R−n−1Pm\nn(cos Θ) eimΦ\nRn,m(R) =RnPm\nn(cos Θ) eimΦ.(S19)\nThe potential of the dipole can be obtained by calcu-\nlating the derivative of Eq. (S18) with respect to the\ncoordinates {a, θa, φa}of the source. The results are\nϕα=/summationdisplay\nlmDα\nlmrlPm\nl(cosθ)eimφ, (S20)\nwhere α∈(r, θ, φ ) and\nDr\nlm=−1\n4π(l−m)!\n(l+m)!(l+ 1)\nal+2Pm\nl(cosθa)e−imφ a,\nDθ\nlm=−1\n4π(l−m)!\n(l+m)!sinθa\nal+2Pm′\nl(cosθa)e−imφ a,\nDφ\nlm=−1\n4π(l−m)!\n(l+m)!im\nal+2sinθaPm\nl(cosθa)e−imφ a.\n(S21)\nIn the presence of the YIG sphere, the boundary con-\ndition should be carefully considered. The equation of\nmotion of the potential ϕis\n/braceleftigg\n(1 +χ)(∂2\n∂x2+∂2\n∂y2)ϕ+∂2ϕ\n∂z2= 0, r≤R\n(∂2\n∂x2+∂2\n∂y2+∂2\n∂z2)ϕ= 0, r > R,(S22)\nwhere χis magnetic susceptibility tensor. χχχcan be cal-\nculated from the Landau-Lifschitz-Gilbert equation [3, 4]\nas\nχxx=χyy=γ2h0Ms\nγ2h2\n0−ω2−iΓ0ω≡χ,\nχxy=χ∗\nyx=iγωM s\nγ2h2\n0−ω2−iΓ0ω≡iκ,(S23)\nwhere γis the gyromagnetic ratio, Γ 0= 2γh0αis the\ndamping parameter, and Msis the saturation magne-\ntization. Here, the static magnetic field h0=h0ez=\nHe+Hd, where Heis the external static field and\nHd=−Ms/3 is the demagnetization field.\nIntroducing the ellipsoidal coordinates,\nx=R√−χ/radicalbig\n1−ξ2sinηcosφ,\ny=R√−χ/radicalbig\n1−ξ2sinηsinφ,\nz=R/radicalbiggχ\n1 +χξcosη,(S24)3\nthe general solution of Eq. (S22) is [2, 3]\nϕin\nα=/summationdisplay\nnmAα\nnmPm\nn(ξ)Pm\nn(cosη)eimφ, r≤R, (S25)\nϕout\nα=/summationdisplay\nnm[Bα\nnm\nrn+1+Dα\nnmrn]Pm\nn(cosθ)eimφ, r > R. (S26)\nThey satisfy the boundary conditions ( ϕin\nα)R= (ϕout\nα)R\nand\n(∂ϕout\n∂r)R= (1 + χsin2θ)(∂ϕin\n∂r)R\n+χsinθcosθ\nR(∂ϕin\n∂θ)R+iκ\nR(∂ϕin\n∂φ)R.(S27)\nNoticing η=θandξ2\n0= 1 + 1 /χat the sphere surface,\nwe obtain Bα\nnmrepresented by Dα\nnmas\nBα\nnm=Dα\nnmR2n+1[(n+κm)Pm\nn(ξ0)−ξ0Pm′\nn(ξ0)]\n(n+ 1−κm)Pmn(ξ0) +ξ0Pm′n(ξ0).\n(S28)\nWe separate ϕout\nαinto the free-vacuum and the YIG-\ninduced contributions as\nϕout\nα=ϕ(0)\nα+ϕind\nα, (S29)\nϕ(0)\nα=/summationdisplay\nnmDα\nnmrnPm\nn(cosθ)eimφ, (S30)\nϕind\nα=/summationdisplay\nnmDα\nnmR2n+1[(n+κm)Pm\nn(ξ0)−ξ0Pm′\nn(ξ0)]\n(n+ 1−κm)Pmn(ξ0) +ξ0Pm′n(ξ0)\n×r−n−1Pm\nn(cosθ)eimφ(S31)\nThen the free-vacuum contributed magnetic field can be\ncalculated via H0=−∇∇∇ϕ(0)\nαas\nHα\n0,r=−/summationdisplay\nnmDα\nnmnrn−1Pm\nn(cosθ)eimφ, (S32)\nHα\n0,θ=/summationdisplay\nnmDα\nnmrn−1Pm′\nn(cosθ) sinθeimφ, (S33)\nHα\n0,φ=−/summationdisplay\nnmDα\nnmrn−1Pm\nn(cosθ) sinθ−1imeimφ.(S34)\nThe YIG-induced magnetic filed is calculated via H=\n−∇∇∇ϕind\nαas\nHα\nr=/summationdisplay\nnmDα\nnmR2n+1[(n+κm)Pm\nn(ξ0)−ξ0Pm′\nn(ξ0)]\n(n+ 1−κm)Pmn(ξ0) +ξ0Pm′n(ξ0)\n×(n+ 1)Pm\nn(cosθ)eimφ\nrn+2, (S35)\nHα\nθ=/summationdisplay\nnmDα\nnmR2n+1[(n+κm)Pm\nn(ξ0)−ξ0Pm′\nn(ξ0)]\n(n+ 1−κm)Pmn(ξ0) +ξ0Pm′n(ξ0)\n×sinθPm′\nn(cosθ)eimφ\nrn+2, (S36)\nHα\nφ=/summationdisplay\nnmDα\nnmR2n+1[(n+κm)Pm\nn(ξ0)−ξ0Pm′\nn(ξ0)]\n(n+ 1−κm)Pmn(ξ0) +ξ0Pm′n(ξ0)\n×−imPm\nn(cosθ)eimφ\nrn+2sinθ. (S37)Substituting Eqs. (S32), (S33), (S34) and Eqs. (S35),\n(S36), (S37) into Eq. (S17), we finally obtain the Green’s\ntensor.\nWe see from Eq. (S28) that the resonance occurs when\n(n+ 1−mκ)Pm\nn(ξ0) +ξ0Pm′\nn(ξ0) = 0 . (S38)\nThe magnon modes corresponding to n=−m= 1, 2, and\n3 in the absence of the Kerr nonlinearity and the magnon\ndamping, i.e., Γ 0= 0, are the dipole or Kittel mode\nωK=γ(h0+Ms/3), the quadrupolar mode ωQ=γ(h0+\n2Ms/5), and the octupolar mode ωO=γ(h0+ 3Ms/7),\nrespectively. Rewriting Eq. (S23) for Γ 0= 0 as\nχxx=χyy=ΩH\nΩ2\nH−Ω2≡χ,\nχxy=χ∗\nyx=iΩ\nΩ2\nH−Ω2≡iκ,(S39)\nwith Ω H=h0\nMsand Ω =ω\nγMs. Walker proved a signifi-\ncant result that there is a relation 0 ≤(Ω−ΩH)≤1/2\nwhen nandmranges all their permitted values [5].\nTherefore, the magnonic frequency regime is γh0≤\nω≤γ(h0+Ms/2), which is finite in its bandwidth.\nIn our manuscript, the Kerr nonlinearity renormalizes\nthe the magnon frequency as ( γh0−Π)/cosh(2 r)≤\nζ≤(γ(h0+Ms/2)−Π)/cosh(2 r). This finite band-\nwidth is verified by our numerical result of spectral den-\nsity obtained via calculating the the Green’s tensor, see\nFig. 2(a). Thus, the integral range of the spectral\ndensity is limited by ζmin= (γh0−Π)/cosh(2 r) and\nζmax= (γ(h0+Ms/2)−Π)/cosh(2 r). The presence of\nthe magnon damping Γ 0only results in the expansion of\nthe resonance peaks in the spectral density.\nIV. ENERGY SPECTRUM\nThe eigen state of ˆHin the single-excitation subspace\ncan be expanded as |ϕE⟩=x|e,{0k}⟩+/summationtext\nkyk|g,1k⟩. Ac-\ncording to ˆH|ϕE⟩=E|ϕE⟩, we obtain\nEx=xℏ∆0−/summationdisplay\nkykG∗\nk (S40)\nEyk=ykℏζk−xGk. (S41)\nCombining these two equations, we have\nE−ℏ∆0=/summationdisplay\nk|Gk|2\nE−ℏζk. (S42)\nSubstituting J(ζ) =/summationtext\nk|Gk|2\nℏ2δ(ζ−ζk) into Eq. (S42)\nin the continuous limit of the magnonic frequencies, we\nobtain\nE/ℏ= ∆ 0+/integraldisplayζmax\nζminJ(ζ)\nE/ℏ−ζdζ, (S43)\nwhich matches with the pole equation (10) in the main\ntext.4\nV. DYNAMICS AND STEADY STATE\nConsider that the YIG sphere is at low temperature\nsuch that the magnons are initially in the vacuum state.\nTo derive the non-Markovian master equation, we con-\nsider the following special case of the initial state of the\nspin.\n1. The initial state is |Ψ1(0)⟩=|g,{0k}⟩. It is easy to\ncheck that, governed by Eq. (3) in the main text,\n|Ψ1(t)⟩=|Ψ(0)⟩.\n2. The initial state is |Ψ2(0)⟩=|e,{0k}. Its time\nevolution goverend by Eq. (3) is expanded as\n|Ψ2(t)⟩=c(t)|e,{0k}⟩+/summationdisplay\nkdk(t)|g,1k⟩. (S44)\nFrom iℏ|˙Ψ(t)⟩=H|Ψ(t)⟩, we derive\ni˙c(t) = ∆ 0c(t)−/summationdisplay\nkG∗\nk\nℏdk(t), (S45)\ni˙dk(t) = ζkdk(t)−Gk\nℏc(t). (S46)\nSubstituting the solution dk(t) =\niGk\nℏ/integraltextt\n0e−iζk(t−τ)c(τ)dτinto Eq. (S45), we\nobtain\n˙c(t) +i∆0c(t) +/integraldisplayt\n0f(t−τ)c(τ)dτ, (S47)\nunder c(0) = 1, where we have defined\nJ(ζ) =/summationtext\nk|Gk|2\nℏ2δ(ζ−ζk) and f(t−τ) =/integraltextζmax\nζmindζJ(ζ)e−iζ(t−τ).\nWith these two special cases at hand, the evolution\nof an arbitrary initial state of the spin ρtot(0) =\nρ(0)⊗ |{0k}⟩⟨{0k}|, where ρ(0) = ρee|e⟩⟨e|+ρgg|g⟩⟨g|+\n(ρeg|e⟩⟨g|+ h.c.) reads\nρtot(t) = ρee|Ψ2(t)⟩⟨Ψ2(t)|+ρgg|g,{0k}⟩⟨g,{0k}|\n+[ρeg|Ψ2(t)⟩⟨g,{0k}|+ h.c.] . (S48)\nAfter tracing over the magnonic degrees of freedom, we\nobtain\nρ(t) = Tr m[ρtot(t)]\n=ρee|c(t)|2|e⟩⟨e|+ (1−ρee|c(t)|2)|g⟩⟨g|\n+[ρegc(t)|e⟩⟨g|+ h.c.] , (S49)\nwhere ρee+ρgg= 1 has been used. In the basis formed\nby|e⟩and|g⟩, its time derivative is\n˙ρ(t) = Γ( t)/parenleftbigg\n−2ρee|c(t)|2−ρegc(t)\n−ρgec∗(t) 2ρee|c(t)|2/parenrightbigg\n−iΩ(t)/parenleftbigg\n0 ρegc(t)\n−ρgec∗(t) 0/parenrightbigg\n, (S50)where Γ( t) = −Re[˙c(t)\nc(t)] and Ω( t) = −Im[˙c(t)\nc(t)].\nRewriting/parenleftbigg\n0 ρegc(t)\n−ρgec∗(t) 0/parenrightbigg\n= [ˆσ†ˆσ, ρ(t)] and\n/parenleftbigg\n−2ρee|c(t)|2−ρegc(t)\n−ρgec∗(t) 2ρee|c(t)|2/parenrightbigg\n= 2ˆσρ(t)ˆσ†−ˆσ†ˆσρ(t)−\nρ(t)ˆσ†ˆσ, we finally obtain the exact master equation of\nthe spin as\n˙ρ(t) =iΩ(t)[ρ(t),ˆσ†ˆσ] + Γ( t)[2ˆσρ(t)ˆσ†− {ˆσ†ˆσ, ρ(t)}].\n(S51)\nIn our investigation, the initial state of the spin is\nρ(0) = |e⟩⟨e|. Thus, from Eq. (S49), we have ρ(t) =\n|c(t)|2|e⟩⟨e|+ [1− |c(t)|2]|g⟩⟨g|. The excited state popu-\nlation of the spin is just |c(t)|2.\nThe integro-differential equation (S47) becomes\n˜c(s) = [s+i∆0+/integraldisplayζmax\nζmindζJ(ζ)\ns+iζ]−1(S52)\nunder a Laplace transform. Then c(t) is obtainable by\napplying an inverse Laplace transform\nc(t) =1\n2πi/integraldisplayσ+i∞\nσ−i∞˜c(s)estds (S53)\nto Eq. (S52). Equation (S53) is evaluated by using the\nthe residue theorem. The residue is contributed by the\npoles of ˜ c(s), which is found via\nE/ℏ= ∆ 0+/integraldisplayζmax\nζmindζJ(ζ)\nE/ℏ−ζ≡Y(E), (S54)\nwhere s=−iE/ℏ. Equation (S54) has at most one iso-\nlated pole in the regime either E∈(−∞, ζmin] orE∈\n[ζmax,+∞) provided Y(ℏζmin)<ℏζminorY(ℏζmax)>\nℏζmax, respectively. Using the residue theorem, we have\nc(t) =M/summationdisplay\nj=1Zje−i\nℏEb\njt+/integraldisplayζmax\nζminΘ(E)e−iEtdE, (S55)\nwhere Θ( E) =J(E)\n[E−∆0−Υ(E/ℏ)]2+[πJ(E)]2,Mbeing\nthe number of the bound states, and Zj= [1 +/integraltextζmax\nζminJ(ζ)dζ\n(Eb\nj/ℏ−ζ)2]−1is the residue contributed by the jth\nbound state. Oscillating with time in continuously\nchanging frequencies E/ℏof the energy band, the inte-\ngrand tends to zero in the long-time limit due to the out-\nof-phase interference. Thus, the steady-state solution of\nEq. (S55) is\nlim\nt→∞c(t) =/braceleftigg\n0, no bound state/summationtextM\nj=1Zje−i\nℏEb\njt, M bound states.(S56)5\n[1] W. Xiong, M. Tian, G.-Q. Zhang, and J. Q. You, Strong\nlong-range spin-spin coupling via a Kerr magnon interface,\nPhys. Rev. B 105, 245310 (2022).\n[2] T. Neuman, D. S. Wang, and P. Narang, Nanomagnonic\ncavities for strong spin-magnon coupling and magnon-\nmediated spin-spin interactions, Phys. Rev. Lett. 125,\n247702 (2020).[3] P. C. Fletcher and R. O. Bell, Ferrimagnetic resonance\nmodes in spheres, Journal of Applied Physics 30, 687\n(1959).\n[4] L. R. Walker, Magnetostatic modes in ferromagnetic res-\nonance, Phys. Rev. 105, 390 (1957).\n[5] L. R. Walker, Resonant modes of ferromagnetic spheroids,\nJournal of Applied Physics 29, 318 (2004)."    },    {        "title": "2405.16221v1.Nonreciprocal_Multipartite_Entanglement_in_a_two_cavity_magnomechanical_system.pdf",        "content": "arXiv:2405.16221v1  [quant-ph]  25 May 2024Nonreciprocal Multipartite Entanglement in a two-cavity m agnomechanical system\nRizwan Ahmed,1Hazrat Ali,2Aamir Shehzad,3S K Singh,4Amjad Sohail,3,∗and Marcos C´ esar de Oliveira5,†\n1Physics Division, Pakistan Institute of Nuclear Science an d Technology (PINSTECH), P. O. Nilore, Islamabad 45650, Pak istan\n2Department of Physics, Abbottabad University of Science an d Technology, P.O. Box 22500 Havellian KP, Pakistan\n3Department of Physics, Government College University, All ama Iqbal Road, Faisalabad 38000, Pakistan.\n4Process Systems Engineering Centre (PROSPECT),\nResearch Institute of Sustainable Environment (RISE), Sch ool of Chemical and Energy Engineering,\nUniversiti Teknologi Malaysia, Johor Bahru 81310, Malaysi a\n5Instituto de F´ ısica Gleb Wataghin, Universidade Estadual de Campinas, Campinas, SP, Brazil\nWe propose a scheme for the generation of nonreciprocal mult ipartite entanglement in a two-\nmode cavity magnomechanical system, consisting of two cros s microwave (MW) cavities having\nan yttrium iron garnet (YIG) sphere, which is coupled throug h magnetic dipole interaction. Our\nresults show that the magnon self-Kerr effect can significant ly enhance bipartite entanglement,\nwhich turns out to be non-reciprocal when the magetic field is tuned along the crystallographic axis\n[110]. This is due to the frequency shift on the magnons (YIG s phere), which depends upon the\ndirection of magnetic field. Interestingly, the degree of no nreciprocity of entanglement depends upon\na careful optimal choice of system parameters like normaliz d cavity detunings, bipartite nonlinear\nindex ∆EK, self-Kerr coefficient and effective magnomechanical coupli ng rateG. In addition to\nbipartite entanglement, we also explored the nonreciproci ty in tripartite entanglement. Our present\ntheoretical proposal for nonreciprocity in multipartite e ntanglement may find applications in diverse\nengineering nonreciprocal devices.\nI. INTRODUCTION\nSince the early days of quantum theory, entanglement\nat macroscopic scale remained a core issue. It is now\na fundamental resource and lies at the heart of mod-\nern quantum information processing [1–3]. It is a ba-\nsic phenomenon to understand the classical to quantum\nboundary [4] and have many technological applications\nlikequantumsensing[5,6], quantumnetworksandmulti-\ntaskingquantum informationprocessing[7, 8]. Entangle-\nment has been generated in several systems, but partic-\nularly relevant for our present discussion is the theoret-\nical and experimental demonstration that entanglement\ncan be generated in nonlinear quantum systems like cav-\nity optomechanical systems [9–13] and magnomechanical\nsystems [14–18].\nRecently, nonreciprocal entanglement has gained a\nconsiderable attention in macroscopic quantum systems\ndue to wide range of applications in cloaking (invisi-\nble sensing) and noise-free information processing [19].\nIt is because entanglement can be well protected by\nLorentz reciprocity breaking [20]. Several nonrecipro-\ncal devices have been proposed and realized in cavity\noptomechanical systems [21–24]. Furthermore, Lorentz\nsymmetry can be broken by introduction of Sagnac effect\n[25, 26], which results in the nonreciprocal entanglement\n[27]. We are particularly interested in electromagnetic\nreciprocity where the use of magneto-optical materials\nlead to the breaking of Lorentz symmetry. Although\nthe physical realization of these devices is difficult be-\n∗Electronic address: amjadsohail@gcuf.edu.pk\n†Electronic address: marcos@ifi.unicamp.brcause of highly susceptible external magnetic field inter-\nference [28], there are varioussystems like optomechanics\n[29], non-Hermitian optics [30], nonlinear optics [31] and\natomic gases based systems [32] that exhibit nonrecip-\nrocal features. It is important to mention that, most of\nthese previous studies mainly addressed primarily, the\nclassical regime which is focused on the transmission\nrate non-reciprocity. However, recently, quantum devices\nare being explored based on Fizeau light dragging effect\n[33], nonreciprocalphoton blockade [34] and backscatter-\ning immune optomechanical entanglement [35]. Further-\nmore,nonreciprocalmagnonandphononlasershavebeen\nproposed, which use the similar Fizeau light-dragging ef-\nfect [36, 37].\nCavity magnomechanical systems may constitute a\nvaluable platform for investigation of nonreciprocal mul-\ntipartite entanglement, as they have drawn considerable\nattention, due to the high spin density and lower col-\nlective losses [38]. Typically, cavity magnomechanical\nsystem (CMMS) contain magnetically ordered materials\nlike yttrium-iron-garnet (YIG), which can be strongly\ncoupled to microwave (MW) cavity fields. In addition\nto magnetic interactions with magnons, magnetic modes\nin CMMS can also have magnetostrictive interactions,\nleading to coupling of magnon modes with mechani-\ncal/phonon modes [39]. These systems allow the inves-\ntigation of several quantum phenomena at microscopic\nlevel, such as, dark modes [40], entanglement [14–18],\nperfect absorption [41], unconventional magnon excita-\ntions [42] and quantum steering [16]. Apart from a\nplethora of theoretical studies and proposals in cavity\nmagnonic systems, various attempts on experimental im-\nplementations are also realized for many practical appli-\ncations [43–46].\nYIG spheres, with a typical size of 0 .1mm, offer an2\nalternate way to investigate macroscopic quantum pro-\ncessessuchasentanglementandsqueezing. Previoussug-\ngestions for analyzing nonreciprocal entanglement pri-\nmarily relied on the Sagnac effect, which generates a neg-\native or positive shift in the cavity resonance frequency\nbased on the direction of the driving force on the cav-\nity. On the other hand, the Kerr effect in magnome-\nchanical systems can also induce the positive or negative\nfrequency shift depending on the direction of the mag-\nnetic field. Unlike the Sagnac effect, the magnon Kerr\neffect generatesasupplemental two-magnoneffect, which\nchanges the optimum values of all entanglements in our\npresent configuration [47, 48]. As a result, both bipartite\nand tripartite entanglements are increased/shifted when\ncompared to the scenario without the Kerr effect. Tun-\ning the aligned magetic field along the crystallographic\naxis [100] or [110] allows for nonreciprocally formed en-\ntanglements. In the present manuscript, we are intended\nto study the nonreciprocal multipartite entanglement in\na cavity magnomechanical system (See Fig.1). Further-\nmore, it was shown that the Kerr effect modifies the\nmagnon number through frequency shift in a way that is\nadequate to strengthen bipartite entanglements between\nvarious bipartite and tripartite entanglement\nThepaperisorganizedasfollows. SectionIIintroduces\nthe basic magnomechanical system under consideration\nand the corresponding Hamiltonian. Additionally, us-\ning this Hamiltonian, quantum Langevin equations are\nderived. In Section III, the basic entanglement quan-\ntification scheme based on the logarithmic negativity is\ndiscussed. Next, in Section IV, results and discussion\nare presented for different system parameters. Finally,\nSection V concludes the paper.\nII. THEORETICAL MODEL AND\nHAMILTONIAN OF THE SYSTEM\nWe consider a magnomechanical system with a YIG\nsphere, supporting a Kittel mode, is placed inside dual\ncross-shaped MW cavities (See Fig. 1). Magnons are\nquasiparticles that show the collective excitation of spins\nwithin a ferrimagnet [49, 50]. The magnons and the cav-\nity mode are coupled via magnetic dipole interaction.\nIn our present magnomechanical system, the magnetic\nfields of the two cavity modes c1(c2) are respectively\nalong x(y)-directions. Additionally, the external mag-\nnetic field is oriented in the z-direction. In addition, the\ncoupling between the phonon and magnon modes is me-\ndiated by the magnetostrictive interaction caused by the\nYIG sphere’s geometrical deformity [51, 52]. By employ-\ning the rotating wave approximation at the driving filed\nfrequency ω0, the Hamiltonian of the dual-cavity mag-nomechanical system can then be formulated as\nH=2/summationdisplay\nj=1∆jcjc†\nj+∆m0m†m+ωb\n2/parenleftbig\nq2+p2/parenrightbig\n+Γj(cjm†+c†\njm)+gmbm†mq+Kr[m†m]2\n+iΩ/parenleftbig\nm†−m/parenrightbig\n+iEj(c†\nj−cj), (1)\nwhere ∆ m0=ωm−ω0and ∆ j=ωj−ω0(j= 1,2).\nHere,ωm(ωj) is the resonancefrequencies of the magnon\n(jth cavity) mode while ω0is the frequency of the drive\nmagnetic field. The first term in Eq. (1) reflects the free\nHamiltonian oftwothe cavitymodes, where c†\njandcjare\nthecreationandannihilationoperatorsformode j= 1,2,\nrespectively. Similarly, the secondterm is the freeHamil-\ntonian of the magnon mode, where m†andmare the\nmagnon’s creation and annihilation operators. The third\nterm is the free Hamiltonian of the phonon mode, where\nqandpbeing the dimensionless position and momentum\nquadratures of the phonon mode with vibrational fre-\nquencyωb. It is worth noting that the magnon frequency\ncan be adjusted through the bias magnetic field, H, via\nωm=γGH, whereγGis denoted the gyromagnetic ratio.\nThe fourth term represents the interaction between the\njth cavity and the magnon modes with optomagnonical\ncoupling strength Γ j[53], which is given by\nΓj=Vc\nnr/radicaligg\n2\nρspinVY ig, (2)\nwhereV,VY ig=4πr3\n3,ρspin, andnrare, respectively, the\nVerdet constant, the volume, the spin density, and the\nFIG. 1: Schematic illustration of our proposed system where\naYIG sphere (with Kerr-nonlinearity) simultaneously coup les\nwith two driven MW cavity modes, through magnetic dipole\ninteraction. The YIG sphere is positioned along its crystal -\nlographic axis ([100] or [110]) in a static magnetic field. In\naddition, vibrational motion is considered asa phonon mode ,\nwhich couples with the magnon modes owing to a magne-\ntostrictive effect.3\nrefractive index for the YIG sphere. The fifth term repe-\nsents the interaction between the magnon and the me-\nchanical modes, with The strength of the magnetostric-\ntive effect-induced magnonmechanical coupling is indi-\ncated by the parameter gmb. The sixth term in Eq. (1)\nrepresents the magnon self-Kerr nonlinear term, with the\nself-Kerr coefficient Kr, resulting in magnon squeezing\nwhich is precisely proportional to the quadratic magnon\nfield operator. Here, Kr=µ0γ2\nM2Vm, meaningthat the value\nofKrcan be increased by taking YIG sphere of small\nsize. In addition, magnon self-Kerr coefficient Krcan\nbe positive (negative) depending upon the magnetic field\nalignment along crystallographic axis [100] ([110]). Fur-\nthermore, it can be tuned from 0 .05 nHz to 100 nHz de-\npending on the diameter of the YIG sphere, which ranges\nfrom1mm to100 µm. Thefinal twotermsarethe driving\nterms on the magnonic and cavity modes.\nIII. THE QUANTUM DYNAMICS AND\nMULTIPARTITE ENTANGLEMENT\nIn this section, we calculate the quantum dynamics\nof two-cavity magnomechanical system by employing the\nstandard Langevin approach. Taking into account the\ndissipation-fluctuation process, the quantum Langevin\nequations for the two-cavity magnomechanical system,\nare as follow\n˙q=ωbp,\n˙p=−ωbq−gmbm†m−γbp+ξ,\n˙m=−i∆0\nmm−i2/summationdisplay\nj=1Γjcj−igmbmq+Ω\n−2iKrm†mm−κmm+√2κmmin,\n˙cj=−i∆jcj−iΓjm−κjcj+Ej+√\n2κccin\nj,(3)\nwhereγb,kmandkjare the decay rates of the\nphonon, magnon, and cavities modes, respectively, while\ntheir corresponding input noise operators are ξ,min\nandcin\nj. The noise operators for the cavity and\nmagnon modes must fulfill the following correlation\nfunctions [54]:/angbracketleftbig\nΛin†(t)Λin(t′)/angbracketrightbig\n=nl(ωl)δ(t−t′), and/angbracketleftbig\nΛin(t)Λin†(t′)/angbracketrightbig\n= [nl(ωl) + 1]δ(t−t′), where Λ =\nm,c1,c2,l=m,1,2, and nl(ωl) = [exp(/planckover2pi1ωl\nkBT)−\n1]−1is the thermal magnon(photon) number related to\nmagnon(cavity)modes, where Tthe temperature and kB\ndenotes the Boltzmann constant. Furthermore, the cor-\nrelation functions corresponding to the phonon damping\nratearegivenby /angbracketleftξ(t′)ξ(t)/angbracketright+/angbracketleftξ(t)ξ(t′)/angbracketright/2 =γb[2nb(ωb)+\n1]δ(t−t′).\nBelow we adopt the standard linearization method to\nderive the linearized Langevin equation. We rewrite each\noperator of the Eq.(3) as the sum of the mean value and\nthe fluctuation part, i.e., [55, 56] as, i.e., Q=/angbracketleftQ/angbracketright+\nδQ,(Q=p,q,ck,m). Therefore, we obtain the followingmean value of the operators:\nps= 0, qs=−gmb\nωb|ms|2,\ncj,s=Ej−iΓjms\nκj+i∆j,\nms=−iΓ1E1α2−iΓ2E2α1+Ωα1α2\nα1α2αm+Γ2\n1α2+Γ2\n2α1,(4)\nwhere˜∆m= ∆m+∆K, with ∆ m= ∆0\nm+gmb/angbracketleftq/angbracketrightand\n∆K= 2Kr|ms|2andαf=kf+i∆f(f= 1,2,m). As\nKrmight be positive (negative), resulting in ∆ K >0\n(∆K <0). Therefore, the steady-state magnon number\ncan be sufficiently tuned via ∆ K, resulting in nonrecip-\nrocal mean magnon number. For |∆m|,|∆j| ≫κm,κj,\nEq. (4) yields\nms=i/bracketleftbigg∆2Γ1E1+∆1Γ2E2−Ω∆1∆2\n∆1∆2∆m−∆1Γ2\n2−∆2Γ2\n1/bracketrightbigg\n,(5)\nwhichis apureimaginarynumber. Neglectinghigh-order\nfluctuation terms, the linearized equations can be ex-\ntracted as\nδ˙q=ωbδp,\nδ˙p=−ωbδq−γbδp−gmb/parenleftbig\n/angbracketleftm/angbracketrightδm†+/angbracketleftm/angbracketright∗δm/parenrightbig\n+ξ,\nδ˙m=−(i¯∆m+κm)δm−i2/summationdisplay\nk=1Γkδck−i∆K2δm†\n−igmb/angbracketleftm/angbracketrightδq+√\n2κmmin,\nδ˙ck=−(i∆k+κk)δck−iΓkδm+√2κacin\nk,(6)\nwhere¯∆m= ∆m+∆K1includes the frequency shift by\nmagnon squeezing ∆ K1= 4Kr|ms|2= 2∆K. In ad-\ndition, ∆ K2= 2Krm2\ns=−2Kr|ms|2=−∆K. Now,\nwe expand the magnon and cavity fluctuation operators\nby defining the following quadratures δx=1√\n2(δm†+\nδm),δy=i√\n2(δm†−δm),δXj=1√\n2(δc†\nj+δcj) and\nδYk=i√\n2(δc†\nj−δcj). By incorporating these quadra-\nture into the fluctuation equations, it can be written\nin a compact form as ˙̥(t) =M̥(t) +N(t), where\n̥(t) = [δq(t),δp(t),δx(t),δy(t),δX1(t),δY1(t),δX2(t),\nδY2(t)]Tis the is the vector of quadrature fluctuation op-\nerators, and N(t) = [0,ξ(t),√2kmxin\n1(t),√2kmyin\nm(t),√2k1Xin\n1(t),√2k1Yin\n1(t),√2k2(Xin\n2(t),Yin\n2(t))]Tis the\nvector of input noises. Furthermore, Mis the drift ma-\ntrix, are expressed by:\nM=\n0ωb0 0 0 0 0 0\n−ωb−γb0G0 0 0 0\n−G0−κm∆+0 Γ 10 Γ 2\n0 0 ∆ −−κm−Γ10−Γ20\n0 0 0 Γ 1−κ1∆10 0\n0 0 −Γ10−∆1−κ10 0\n0 0 0 Γ 20 0 −κ2∆2\n0 0 −Γ20 0 0 −∆2−κ2\n,\nwhere ∆ ±=±(∆m+ ∆K1)−∆K2andG=i√\n2gmbms\nis the effective magnomechanical coupling rate.4\nA. ENTANGLEMENT MEASURES\nThe stability of the current magnomechanical system\nis the primary condition and the basic criteria for the\nstability is the Routh-Hurwitz criterion [57, 58]. Accord-\ning to this criterion, the drift matrix must have negative\nreal part of the all the eigenvalues. Thus, using the sec-\nular equation i.e., |M−λMI|= 0), we must extract the\neigenvalues from the drift matrix M, and confirm the\nsystem’s stability. Our magnomechanical system’s drift\nmatrix is a 8 ×8 matrix, hence the corresponding co-\nvariance matrix will also be a 8 ×8 matrix, with the\nentriesVij(t) =1\n2/angbracketleft̥j(t′)̥i(t)+̥i(t)̥j(t′)/angbracketright, where the\nsteady-state Vcan be obtained by solving the Lyapunov\nequation [59, 60],\nMV+VMT+D= 0, (7)\nwhereD=diag[0,γb(2nb+1),κm(2nm+1),κm(2nm+\n1),κ1(2n1+1),κ1(2n1+1),κ2(2n2+1),κ2(2n2+1)].\nTheDmatrix, often known as the diffusion matrix,\ndescribes the noise correlations. We employ Simon\ncondition for Gaussian states in order to simulate the\nbipartite entanglement [60–65].\nEN= max[0,−ln2(ν−)], (8)\nwhereν−=min eig |/circleplustext2\nj=1(−σy)/tildewiderV4|defines the minimum\nsymplectic eigenvalue of the CM which is reduced to or-\nder of 4×4. Here, /tildewiderV4=̺1|2Vin̺1|2, whereVinis a 4×4\nmatrix of any bipartition. Furthermore, by removing the\nuninteresting columns and rows in V4, we can generate\nVin. Thematrix ̺1|2=diag(1,−1,1,1)=σz/circleplustextI, whereσ’s\nare the Pauli’s spin matrices and Iis the identity matrix,\nset the partial transposition at the CM level. More intu-\nitively, we introduce the bipartite nonlinear index as\n∆EK=|EN(∆K>0)−EN(∆K<0)|,(9)\nIt is crucial to mention here that there will be non-\nreprocity in bipartite entanglement if ∆ EK>0, or\n∆EK/negationslash=EN(∆K>0), or ∆EK/negationslash=EN(∆K>0).\nTo investigate the tripartite entanglement, we employ\nminimal residual contangle, as given in [64, 66, 67]\nRmin\nτ≡min[Rp|mb\nτ,Rm|pb\nτ,Rb|pm\nτ],(10)\nwhereRu|vx\nτassure the invariance of tripartite entangle-\nment under all permutations of the modes, and is given\nby\nRu|vx\nτ≡Cu|vx−Cu|v−Cu|x≥0,(u,v,x=p,m,b).\n(11)\nCu|vis the proper entanglement monotone and is equal\nto the squareoflogarithmicnegativityofthe subsystems,\ni.e.,Cu|v=E2\nu|v, wherevincorporate one or two modes.\nHowever, to compute the one-mode-vs-two-modes loga-\nrithmic negativity Eu|vx, one must redefine the definition\nofν−as given by\nη−= mineig|⊕3\nj=1(−σy)/tildewiderVj|, (12)where/tildewiderVj=Pi|jkV6Pi|jk(i/negationslash=j/negationslash=k) withV6being the\n6×6 CM of the three modes of interest. Furthermore,\nP1|23=σz⊕I⊕I,P2|13=I⊕σz⊕IandP3|12=I⊕I⊕σz,\nare the matrices that define the partial transposition at\nthe level of the CM V6. In addition, σy= [0,−i;i,0]\nandσz= [1,0;0,−1]. Moreover, the symbol ⊕, which\ndescribes the direct sum of the matrices, can expand the\ndimension to ( m+p)×(n+q) of two matrices Awith\nm×nandBwithp×qdimension. More intuitively, we\nalso introduce the tripartite nonlinear index as\n∆Rmin\nK=|Rmin\nK(∆K>0)−Rmin\nK(∆K<0)|.(13)\nThere exist nonreprocity in tripartite entanglement if\n∆Rmin\nK>0 or ∆Rmin\nK/negationslash=|Rmin\nK(∆K>0) or ∆Rmin\nK/negationslash=\n|Rmin\nK(∆K<0).\nIV. KERR-INDUCED NONRECIPROCAL\nMULTIPARTITE ENTANGLEMENT\nAt the outset of discussion, we should mention here\nthat the nonreciprocal entanglement due to the magnon\nKerr effect is quite different from the case of the Sagnac\neffect. This is because the magnetic field-mediated Kerr\neffect generates either a red or blue shift in magnon fre-\nquency. Furthermore, the primary goal of investigat-\ning entanglement in such a dual-cavity magnomechani-\ncal system with and without Kerr non-linearity is to find\noptimal detunings among the four modes to establish\ntrue bipartite and tripartite entanglement. In order to\nstudy nonreciprocal entanglement, we choose the exper-\nimentally feasible parameters: ω1=ω2=ω= 2π×10\nGHz,ωb= 2π×10 MHz, κ1=κ2=κ= 0.1ωbMHz,\nκm= 0.2ωb, Γ1= Γ2= Γ = 0 .32ωb,γb= 10−5ωb,\ngmb= 2π×0.3 Hz, and T= 10 mK [68, 69]. We\nmainly consider four bi-partitions, namely, cavity-cavity\nEc1−c2\nN, cavity-magnon Ec1−m\nN, cavity-phonon Ec2−b\nNand\nmagnon-phonon Em−b\nN. The subfigures (b) and (c) in\nFig. (2) to Fig. (5) respectively indicate the magnetic\nfieldalongthecrystallographicaxis[100]and[110],which\ncorrespond to ∆ K>0 and ∆ K<0. However, for com-\nparison, subfigures (a) in Fig. (2) to Fig. (5) correspond\nto the case of entanglement without Kerr nonlinearity,\ni.e., ∆ K= 0. Obviously, the value of Kerr-nonlinearity\nmay vary, depending upon magnon number. It is be-\ncause the parameters for both cavities are similar and\nmagnon/phonon entanglements with either cavity 1 or\ncavity 2 are identical. It is worth mention that by in-\ncorporating Kerr nonlinearity, the value of the effective\nmagnomechanical coupling strength is different from the\ncase without Kerr nonlinearity. The value of the effec-\ntive magnomechanical coupling strength is G/Γ = 1.1,\nG/Γ≈1 andG/Γ≈1.4 for ∆ K= 0, ∆ K>0, ∆K<0,\nrespectively. In addition, the chosen parameters to en-\nsurestabilityofthesysteminaccordancewiththeRouth-\nHurwitz criterion [57].5\nFIG. 2: Contour plot of Ec−c\nNversus the normalized cavity\ndetunings ∆ 1/ωband ∆ 2/ωbwhen (a) ∆ K= 0 (b) ∆ K>0\n(c) ∆K<0. The other parameters are listed in the main text.\nA. Bipartite Entanglement\nIn order to study the effects of Kerr non-linearity on\nthe bipartite entanglement, we have obtained several re-\nsults. These results incorporate the dynamics of non-\nreciprocal entanglement in the absence and presence of\nKerr non-linearity, with direction of magnon squeezing\non the different bipartition’s entanglement, as shown in\nFigs. 2, 3, 4, and 5. At first, we discuss the contour\nplot ofthe photon-photonentanglement, Ec1−c2\nN, asfunc-\ntion of the normalized detunings ∆ 1/ωband ∆ 2/ωb, re-\nspectively, keeping the magnon detuning ∆ m/ωb= 1, as\nshown in Fig. 2 (a-c). A careful analysis of these results,\nshow that the generated entanglement is nonreciprocal\nand strongly depend upon the value of self-Kerr coeffi-\ncient, ∆ K. It is clear that in the absence of Magnon self-\nKerr effect, entanglement initially has maximum values\naround ∆ 2/ωb±1. However, in the presence of self-Kerr\neffect, the maxima of entanglement is not only enhanced\nin the magnitude but also shifted towards higher (lower)\ndetuning region when ∆ K>0 (∆K<0), showing the\nnon-reciprocity of generated entanglement. This clearly\nsupportourclaimthatthemagnonnonlinearself-Kerref-\nfect has a noticeable effect upon the entanglement. This\nprovides an additional control for the manipulation of\ncavity-cavity bipartite entanglement.\nNext, in Fig. 3, we show the results for the bipartite\nFIG. 3: Contour plot of Ec1−m\nNversus the normalized cavity\ndetunings ∆ 1/ωband ∆ 2/ωbwhen (a) ∆ K= 0 (b) ∆ K>0\n(c) ∆K<0. The other parameters are listed in the main text.\nFIG. 4: Contour plot of Ec1−b\nNversus the normalized cavity\ndetunings ∆ 1/ωband ∆ 2/ωbwhen (a) ∆ K= 0 (b) ∆ K>0\n(c) ∆K<0. The other parameters are listed in the main text.\nentanglement between cavity photons and magnons, for\nthree different choices of ∆ K. It can be seen that the en-\ntanglementintheabsenceofself-Kerrnon-linearityisdif-\nferentfrom the resultsafter the inclusion ofnon-linearity.\nIt once again show that the non-reciprocal nature of en-\ntanglement (for the same system parameters used in Fig.\n2. In the similar line of action, we also studied the effects\nof self-Kerr nonlinearity on the cavity photon-phonons\n(Fig. 4) and magnon-phonons (Fig. 5) bipartitions. It\ncan be seen that all partitions bipartite entanglement is\nenhanced and has been shifted, as compared with the re-\nsults for absence of non-linearity. It is worth mentioning\nhere that the noinreciprocal entanglement is because of\nthe application of strong drive fields and then magnon\nself-Kerr nonlinearity gives rise to a shift in magnon fre-\nquency. The directionofappliedmagnetic field isrespon-\nsible for the sign of induced frequency shift. In reality,\nthe magnomechanical coupling strength can be modu-\nlated by varying the input driving fields [see Eq. (5)] in\nour approach. Therefore, it is crucial to observe the ef-\nfectofthemagnomechanicalcouplingwithorwithoutthe\nmagnon Kerr on different bipartitions. We first choose\nthe value of cavity detunings ∆ 1and ∆ 2where we found\nthe optimal entanglement without Kerr nonlinearity and\nthen observe how Kerr nonlinearity affects the effective\ncoupling rate i.e. G. So, we plot all bipartitions against\nthe effective coupling rate i.e., Gnormalized with respect\nto Γ, in the absence/presence of magnon self-Kerr non-\nFIG. 5: Contour plot of Em−b\nNversus the normalized cavity\ndetunings ∆ 1/ωband ∆ 2/ωbwhen (a) ∆ K= 0 (b) ∆ K>0\n(c) ∆K<0. The other parameters are listed in the main text.6\n0 1 200.10.2\n0 1 200.10.2\n0 1 200.10.20.3\n0 1 200.20.4\n0 1 2\nG/00.10.2\n0 1 2\nG/00.020.040.06\n0 1 2\nG/00.10.20.3\n0 1 2\nG/00.10.20.3\n(e) (f) (g) (h)(d) (c) (b) (a)\nFIG. 6: Plot of (a)(e) Ec−c\nN, (b)(f)Ec−m\nN, (c)(g)Ec���b\nN, and (d)(h) Em−b\nNversus the ratio of G/Γ with ∆ K= 0 (the green\ncurve), ∆ K>0 (the red curve), ∆ K<0 (the blue curve), and ∆ EK(the yellow curve). In (a-d), we we chose optimum value\nof entanglements when ∆ K= 0 as axis depicted from first subfigures of Fig. (2)-Fig. (5), while (e-f), we take optimum value\nof entanglements when ∆ 1=−1 and ∆ 2= 1\nlinearity. Fig. 6(a-d) shows the plot for entanglement in\nall four bipartitions for different choices of Kerr nonlin-\nearities, i.e., ∆ K= 0, ∆ K>0, ∆K<0, respectively. An\nadditional quantity ∆ EK, defined in Eq. (9) as bipartite\nnon-linear index (yellow curve), represent the reciprocity\nin system. This result clearly shows that an increase of\neffectivecouplingrate Gresultsin enhancementofentan-\nglement. Non-reciprocity of entanglement is self-evident\nfrom this plot and also, can be observed by yellow curve,\n-2 -1.5 -1 -0.5 0\n1/b00.010.020.030.040.05Rminc-m-b\n-2 -1.5 -1 -0.5 0\n1/b00.511.522.53Rminc-m-c10-3\n 7 times  6 times\nFIG. 7: Tripartite entanglement in terms of the minimum\nresidual contangle (a) Rmin\nτ,c−m−band (a) Rmin\nτ,c−m−cversus\ncavity detuning ∆ 1/ωbwith ∆ K= 0 (the blue solid curve),\n∆K>0 (the red dotted curve), ∆ K<0 (the magenta solid\ncurve) and ∆ Rmin\nK(the green dot-dashed curve).bipartite non-linear index ∆ EK. When ∆ EK= 0 or\noverlap red/blue curve represent no reciprocity. In Fig.\n6(e-h), we plot entanglement of all bipartitions and ob-\nserve the simultaneous nonreciprocity when ∆ 1=−1\nand ∆ 2= +1. These results show the smaller value of\neffective coupling rate Gresults in optimal entanglement\nfor ∆EK<0 while the larger value of Gyields opti-\nmal entanglement for ∆ EK>0, exhibit the presence of\nnon-reciprocal entanglement of all bipartitions and the\nenhancement of entanglement for non-zero self Kerr non-\nlinearity. On the basis of results obtained we can assert\nthat, the non-reciprocal entanglement can be manipu-\nlated and enhanced for required applications, using the\noptimal system parameters.\nB. Tripartite Entanglement\nApart from the Kerr-dependent bipartite entan-\nglement, the nonreciprocal photon-magnon-phonon\nRmin\nτ,c−m−band photon-magnon-photon Rmin\nτ,c−m−ctripar-\ntite entanglements could be obtained. In addition, the\nnonreciprocity of these tripartite entanglement could be\nenhanced by Kerr effects, as expected, and are numer-\nically illustrated in Fig. 8 (a-b). It is obvious from\nFig 7a that the tripartite entanglement Rmin\nτ,c−m−bcurve\nshifts to the left (for ∆ k>0) or right (for ∆ k<\n0), however, we obtained enhanced tripartite entangle-\nmentRmin\nτ,c−m−bwhen ∆ k<0. Furthermore, there ex-7\nist photon-magnon-photon Rmin\nτ,c−m−ctripartite entangle-\nment only when ∆ k>0 or ∆ k<0. Based on Eq. (13),\nwe note that the nonreciprocal tripartite entanglement\nmanifestly appears when the tripartite nonlinear index is\nnonzero , however the nonreciprocity vanishes when the\ntripartite nonlinear index become zero. In other words,\nhigher the value of ∆ Rmin\nK, the stronger non-reciprocity\nof tripartite entanglement is.\nC. Feasibility of Current Scheme\nIn this section, we discuss the experimental feasibility\nof the effective squeezing caused by the self-Kerr non-\nlinearity. Note that the coefficient of Kerr nonlinear-\nity is inversely proportional to the volume of the YIG\nsphere, so the Kerr nonlinear effect can become signifi-\ncant when using a small YIG sphere. The magnon ex-\ncitation number for magnetic materials must be much\nlower than in order to guarantee the validity of the\nmagnon description. For a 250 µm diameter YIG sphere,\n|ms|2≃1014≪5N≃1017[14], which meets the low-\nexcitation condition, guarantees that the Kerr nonlinear-\nity is considerably small. In addition, a YIG of 250 µm\ndiametergeneratesKerrcoefficient Kr= 6.4∗10−9andto\nkeep the Kerr effect negligible, Ω ≫ Kr|ms|3must fulfill.\nTherefore, Ω = 7 .1×1014≫ Kr|ms|3= 5.7×1013. It is\nvital to notethat the the self-Kerrnonlinearitycannotbe\nignored if Kr|ms|3/greaterorapproxeqlΩ. Nonetheless, we uses a 0 .1mm-\ndiameter YIG sphere to generate a strong Kerr coeffi-\ncientKr≈ ×10−7[70] and therefore, Ω = 7 .5×1014≪Kr|ms|3= 1.6×1015. Hence, the experimental val-\nues show the validity of parameter regimes considered in\ncurrent system and the above condition makes our mag-\nnomechanical system with Kerr-nonlinearity sufficiently\npracticable. In terms of experimental realization, a pla-\nnar cross-shaped cavity or coplanar waveguide can be\nused to implement the current method.\nV. CONCLUSION\nIn this study, we have proposed, theoretically, an effi-\ncient proposal for the generation of nonreciprocal entan-\nglement in a two-mode magnomechanical system. We\nexploited the magnon self-Kerr nonlinear effect, which\nleads to enhanced nonreciprocal entanglement. We have\nshown that the degree of nonreciprocity is the manipula-\ntion of optimal choice of system parameters like normal-\nizd cavityd detunings, bi-partite nonlinear index ∆ EK,\nself-Kerr coefficient and effective magnomechanical cou-\npling rate G. 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Chumak2\n1Fachbereich Physik and Landesforschungszentrum OPTIMAS,\nTechnische Universit at Kaiserslautern, D-67663 Kaiserslautern, Germany\n2Faculty of Physics, University of Vienna, A-1090 Vienna, Austria\n3Department of Physics, Oakland University, Rochester, MI, USA\n4INNOVENT e.V. Technologieentwicklung, D-07745 Jena, Germany\n(Dated: September 2021)\nPreviously, it has been shown that rapid cooling of yttrium-iron-garnet (YIG)/platinum (Pt) nano\nstructures, preheated by an electric current sent through the Pt layer, leads to overpopulation of a\nmagnon gas and to subsequent formation of a Bose{Einstein condensate (BEC) of magnons. The\nspin Hall e\u000bect (SHE), which creates a spin-polarized current in the Pt layer, can inject or annihilate\nmagnons depending on the electric current and applied \feld orientations. Here we demonstrate that\nthe injection or annihilation of magnons via the SHE can prevent or promote the formation of a rapid\ncooling induced magnon BEC. Depending on the current polarity, a change in the BEC threshold\nof -8% and +6% was detected. These \fndings demonstrate a new method to control macroscopic\nquantum states, paving the way for their application in spintronic devices.\nThe Bose{Einstein condensate (BEC) [1], often considered because of its exotic properties as the \ffth state of\nmatter, is formed when individual atoms [2], subatomic particles [3], or quasiparticles such as Cooper pairs [4] or\nquanta of molecular electric oscillations [5] coalesce into a single quantum mechanical entity existing on a macroscopic\nscale and described by a single wave function. Emerging in various physical systems from neutron stars [6] to a\ndroplet of liquid helium [7], this state leads to fascinating and valuable macroscopic quantum phenomena, such as\nsuperconductivity and super\ruidity. Recently, novel BEC applications have been proposed, including those in the\nrapidly developing \feld of quantum computing [8{13]. In contrast to existing quantum computers, which operate at\nabout 20 \u0016K [14], BEC-based computing can be performed at much higher temperatures: for example, in yttrium\niron garnet (Y 3Fe5O12, YIG) [15], a magnon condensate [16] was observed at room temperature. When using such\na magnon condensate in both quasi-quantum and classical nanoscale devices, the possibility to control it by magnon\nspintronics [17] methods via spin-polarized electric currents [18] seems particularly attractive for reducing power\nconsumption and simplifying these devices.\nThe formation of a Bose{Einstein condensate (BEC) can be achieved by a decrease in the temperature for real-\nparticle systems [2, 19], or in a quasi-particle system by the injection of bosons resulting in an increase in the\nchemical potential. The latter has been demonstrated experimentally for exciton-polaritons [20, 21], photons [22, 23]\nor magnons [16, 24{30]. In the case of magnonic systems the injection was realised by the nuclear magnetic resonance\n[24, 31], by the parametric pumping mechanism [16, 28, 29, 32], allowing for the injection of a large number of\nmagnons at a given frequency, via the spin-Seebeck e\u000bect [18] or by the mechanism of rapid cooling, as it was shown\nrecently [33]. The method of rapid cooling makes use of the application of a DC heating pulse and the subsequent\nrapid cooling of yttrium iron garnet (YIG)/Pt nano structures. The heating generates a high population of magnons\nbeing in thermal equilibrium with the phononic system. A rapid decrease in the phononic temperature results in the\nbreak of the equilibrium. Since the lifetime of magnons is larger than the phonon cooling rate in the experiment an\noverpopulation of magnons over the whole magnon spectrum is generated. This overpopulation results in a redis-\ntribution of magnons from higher to lower energies. In this way, if the temperature of the heated YIG \flm is high\nenough and the cooling process is fast enough, the magnon chemical potential is increased to the minimal energy of\nthe magnon system and the BEC formation process is triggered.\nIn the previous experiments, a Pt or Al layer was used to heat the YIG nano structure [33]. For a Pt heater an\nadditional generation of a spin-polarized current transverse to the YIG/Pt interface due to the spin Hall e\u000bect (SHE)\nis expected [34{37]. The resulting spin current is known to act on the magnetization dynamics in the YIG via the\nspin transfer torque (STT) [17, 38{43]. The SHE-STT contribution, which can be easily checked by the variation\nof the current polarity with respect to the magnetization orientation in the YIG \flm [40], was not observed in the\noriginal experiments [33]. The reason was the large YIG \flm thickness of 70 nm (STT is an interface e\u000bect) and the\nhigh quality of the Pt layer grown by molecular beam epitaxy that results in a small spin Hall angle [44].\nHere, we investigate a similar structure but with a smaller YIG \flm thickness of 34 nm [45] and with the Pt layer\ndeposited by a sputtering technique to achieve a pronounced SHE-STT e\u000bect. Using such YIG nano structures, we\nare now able to investigate the in\ruence of the SHE-STT e\u000bect on the formation of the magnon BEC by rapid cooling.arXiv:2102.13481v2  [physics.app-ph]  22 Sep 20212\nTi / Au contact padYIGPtGGG substrate\nDC pulse2 µm\nPulse\neneratorgLaser\nBLS-Fabr -P rot y é\nnterferometeri\nPhoto etector n-dTriggering\nBLS   time-\ntracesxyFocal\nspotscan-line\nTi / Au contact pad\nFIG. 1. Colored scanning-electron microscopy-image of the structure under investigation and sketch of the experimental setup.\nThe structure consists of a 2 \u0016m-wide and 34 nm-thick YIG stripe. A 3 \u0016m-long and 7 nm-thick platinum-heater is placed on\ntop and contacted by Ti/Au-leads separated by a distance of 2 \u0016m.\nAnalogously to our original studies, the experiments are conducted at room temperature. We show that the magnons\nannihilated or injected by the STT e\u000bect during the pulse application strongly in\ruence the BEC formation threshold.\nFigure 1 shows the structure under investigation and a sketch of the experimental setup. The structure consists of\na 2\u0016m-wide YIG/Pt stripe (34 nm/7 nm) on a (111) gadolinium gallium garnet (GGG) substrate. The YIG structure\nwas fabricated using electron-beam lithography with subsequent argon-ion milling [46]. Afterwards a 3 \u0016m-long Pt\nlayer was deposited on the waveguide using a RF-sputtering technique. To establish electrical contacts to the platinum\npads, Ti/Au-leads (10 nm/150 nm) with a distance of 2 \u0016m between the inner edges were fabricated by electron beam\nevaporation. In the presented experiments DC pulses of a duration of \u001cP= 100 ns are applied to the Pt pad. Standard\nferromagnetic resonance (FMR) measurements were performed on two reference pads on the same sample, one bare\nand one covered with platinum. These yielded Gilbert-damping-constants of \u000bYIG= 1:8\u000210\u00004for the bare YIG pad\nand\u000bYIGjPt= 1:7\u000210\u00003for the YIG/Pt pad, corresponding to a spin mixing conductance of g\"#= 5:1\u00021018m\u00002\n[47]. An external \feld of \u00160Hext= 110 mT magnetizes the stripe either along its short or long axis in plane. We apply\nDC-current pulses to the Pt-layer in order to trigger the SHE-STT e\u000bect based injection or annihilation of magnons\nand to heat up the structure. After pulse termination, the structure cools down rapidly since the GGG substrate and\nthe Au leads act as e\u000ecient heat sinks. In such a way, the experiments are conducted at room temperature and no\nactive cooling is required. The magnetization dynamics is measured by means of Brillouin light scattering spectroscopy\n(BLS). A laser beam with 457 nm wavelength and 5 :0\u00060:5 mW power is focused (spot size 400 nm) onto the YIG\nwaveguide through the backside of the transparent GGG substrate, allowing to measure below the platinum-covered\nYIG region [46]. The inelastically scattered light, carrying the information about the magnon intensity and frequency,\nis analyzed by a multi-pass tandem Fabry-P\u0013 erot interferometer with a time resolution of about 2 ns [33, 48]. The laser\nfocus was scanned along the platinum layer between the two contact pads on a line in the middle of the stripe (see\nscan line indicated in Fig. 1). The resulting BLS signal was integrated along this scan line to reduce any in\ruence of\nan inhomogeneous heating of the platinum region on the experimental results.\nFigures 2(a-c) show the measured color-coded BLS intensities (log-scale) as a function of time and the BLS fre-\nquency. The amplitude of the applied DC heating pulse is jUj= 1:5 V, corresponding to a current density of\njjCj= 1:6\u00021012A m\u00002.\nFigure 2(a) shows the reference experiment with an external \feld \u00160Haligned parallel to the long axis of the\nwaveguide, parallel to the direction of the current jC(\u00160HextkjC). In this geometry, no contribution of the SHE-3\n5.5\n5.0\n4.5\n4.0\n3.5\n3.0\n0 50 100 150 200 0 50 100 150 200 0 50 100 150 200DC-pulseBLS frequency (GHz)\nTime (ns) Time (ns) Time (ns)\njc\nµ0 extHjc\nµ0 extHjc\nµ0 extH\nBLS intensity (arb.units)0.051.00\n0.10(a) (b) (c)\nxy xy xy\nDC-pulse DC-pulse\nFIG. 2. BLS intensity color-coded (log-scale) as a function of the BLS frequency and time. The duration of the 100-ns long\nheating DC pulse with an amplitude of jUj= 1:5 V is marked by the black boxes. (a) Current parallel to the external \feld,\nthus no contribution of the SHE-STT e\u000bect is expected. The achieved overpopulation after pulse termination due to the rapid\ncooling e\u000bect is su\u000eciently large to trigger the formation of a magnon BEC. (b) Current is perpendicular to the external \feld.\nThe STT annihilates magnons during the pulse. The condensation of magnons is suppressed. (c) Reversed current direction in\ncomparison to the situation in (b), the STT injects magnons. After the pulse is applied the magnon BEC density is enhanced\ncompared to the situation depicted in panel (a). For the exact geometries see insets.\nSTT e\u000bect is expected and the application of the current pulse only results in a Joule heating-induced increase of\nthe YIG/Pt temperature. While the DC pulse is applied (black box in the \fgure), the BLS intensity decreases, as\noriginally investigated in Ref. [49]. Further, the spectrum of thermal magnons shifts to lower frequencies due to\nthe decrease in the saturation magnetization [15, 33]. After the DC pulse is switched o\u000b at t= 100 ns, the heat-\ndissipation-induced rapid cooling triggers the formation of the BEC at the bottom of the spectrum [33]. The BEC\nmanifests itself as a pronounced peak in the magnon intensity. The accompanying frequency increase is due to the\ncooling.\nThe contribution of the SHE can be switched on by rotating the magnetic \feld by 90\u000e, hence pointing along the\nshort axis of the stripe [see insets in Figs. 2(b,c)], i.e. perpendicular to the direction of the DC current [40].\nIn this geometry the SHE-STT contribution can be damping- or anti-damping like [34, 50]. The change in the\ne\u000bective damping can also be described by the annihilation or injection of magnons [40], or by the change in the\nmagnon chemical potential \u0016[35, 39]. In the case without a STT-contribution [Fig. 2(a), pulse duration of \u001cP= 100 ns\nand a voltage of U= 1:5 V)], the magnon and the phonon system are in thermal equilibrium at the end of the DC\nheating pulse, and both are highly populated. The SHE-STT contribution increases or decreases the number of\nmagnons at the end of the pulse with respect to the reference case. Thus, the SHE-STT is expected to change the\nBEC formation via a change of the number and distribution of excess magnons prior to the rapid cooling process [33].\nFigures 2(b), and 2(c) depict the cases for magnon annihilation and injection via the SHE-STT e\u000bect, respectively.\nThese processes can be seen in the increase or decrease of BLS intensity during the DC pulse.\nThe magnon annihilation process [Fig. 2(b)], in contrast to experiments on thicker YIG micro structures used\npreviously [33], is now large enough to compensate for the rapid cooling-induced increase of the chemical potential \u0016,\nthus suppressing BEC formation.\nIn the case of the SHE-STT-induced magnon injection process [Fig. 2(c), jC;x<0;\u00160Hext?jC] the SHE-STT e\u000bect\nenhances the magnon redistribution, which is manifested by the even higher BLS intensity measured after the DC\npulse, compared to Fig. 2(a). For a better comparison of the three cases described above, see Supplemental Material\nfor extracted BLS spectra during and after the pulse.\nNote that a purely thermal excitation of magnons takes place over the whole spectral range [Fig. 2(a)]. The\nsubsequent decrease of the saturation magnetization leads to a decrease in the BLS intensity [49]. In contrast, for\na STT-injection of magnons the BLS intensity increases [Fig. 2(c)], in spite of the heating-induced decrease of the\nBLS sensitivity (which does not depend on the \feld orientation). The reduction of the BLS sensitivity is given by\nthe decrease in the saturation magnetization that can be treated as a thermally-induced increase of the number of\nmagnons over the whole magnon spectrum [51]. Thus, the fact that the BLS intensity in Fig. 2(c) increases rather\nthan decreases during the heating process suggests that the SHE-STT mechanism injects magnons primarily into the\nlow energy part of the spectrum accessible with micro focused BLS.\nThe results presented in Fig. 2 show that the SHE-STT e\u000bect can control the BEC formation process via the\ninjection or annihilation of magnons. Here we would like to point out that the measurements presented in Fig. 2(b,c)\nare conducted for a \fxed geometry of the \feld. Thus, the di\u000berence in the BEC formation after pulse termination is4\nFit\n1/Int. BLS intenisty (arb.units)IInt30\n015\n0.5 -0.5 -1.0 0 1.0\nVoltage (V) U2025\n10\n5 Uth=-0.95 V(b) (a)\nEnd of pulse\n1.0 1.4 1.8 2.2×3⊥µjc 0 extH ⊥µjc 0 extH jc 0 ext⊥µH\nc 0 extj⊥µH(c)\nj Hc 0 ext||µjc,x>0\njc,x<0After pulse\njc 0 ext⊥µH\n1.0 1.4 1.8 2.2 1.42 1.0 1.4 1.8 2.2 1.42After pulse\nSTT-injected\nsubtracted60\n40\n20\n0Int. BLS intenisty (arb.units)IInt80Threshold\nwithout\nSTT\nThreshold\nSTT-\ninjection\nThreshold\nSTT-\nannihilation( ) d\nAbsolute voltage | (V) |U Absolute voltage | (V) |U Absolute voltage | (V) |UThreshold of\nSTT-induced\ndamping\ncompensation\nFIG. 3. (a) Inverse BLS intensity as a function of the voltage for the application of a continuous DC current. The linear \ft yields\na threshold for the damping compensation of Uth=\u00000:95 V. (b) Integrated BLS intensity at the end of the applied DC pulse\n(integration interval \u001cP\u00008 ns<t<\u001c P) as a function of the absolute value of the voltage. The external \feld of \u00160Hext= 110 mT\nis aligned perpendicular to the direction of the current. The SHE-STT e\u000bect either injects (\\+\"-sign data points) or annihilates\n(\\\u0000\"-sign data points) magnons. (c) Integrated BLS intensity after the pulse is switched o\u000b ( \u001cP<t<\u001c P+ 95 ns) as a function\nof the absolute value of the voltage. The blue curves correspond to the situation in (b), the black curves show the reference\ncurve for \u00160HextkjC, without a SHE-STT contribution. (d) Integrated intensity as in (c), the residual signal caused by the\ndecaying STT-injected magnons is subtracted. Due to the injection or annihilation of magnons via the STT the thresholds are\nshifted with respect to the case without a SHE-STT contribution (pink line).\nsolely determined by the SHE-STT e\u000bect. To investigate the threshold of the BEC formation process in the presence\nof the SHE-STT e\u000bect, we \frst characterize the state of the magnon system right before the cooling takes place.\nFigure 3(a) shows the inverse BLS intensity as a function of the applied voltage, now for an applied continuous DC\nvoltage instead of a DC pulse. The linear \ft of the inverse BLS intensity yields the threshold voltage for the STT-\ninduced damping compensation [38], which is found to be U=\u00000:95 V. Thus, for the voltages applied in the pulsed\nexperiments, the SHE-STT e\u000bect compensates the damping and should lead to the formation of auto-oscillations after\na su\u000eciently long time [50]. In the following, we limit our discussion to the voltage regime, where no auto-oscillations\nare triggered. However, we discuss measurements for higher voltages in the Supplemental materials.\nFigure 3(b) shows the BLS intensity integrated over the whole frequency range shown in Fig. 2 and integrated over\nthe last 8 ns of the applied DC pulse as a function of the voltage. This BLS intensity at the end of the pulse is\ncorresponding to the \fnal magnon intensity in the frequency range accessible with BLS, which is the low frequency\nregion of the spectrum. The \\+\"-sign data points correspond to the direction in which the STT e\u000bect injects\nmagnons, the \\\u0000\"-sign data points correspond to the case when the STT e\u000bect annihilates magnons (a negative\nvoltage corresponds to a positive current density in x-direction and results in an injection of magnons as indicated by\n\"+\"-sign data points). We observe an increasing BLS intensity at the end of the pulse with higher voltages, indicating\nan increasing STT-injection. After reaching its maximum at a voltage of U= 1:45 V the BLS intensity decreases\nagain, which is attributed to a decreased spin mixing conductance due to the heating [52], and a decreased BLS\nsensitivity at higher temperatures [49]. The decreasing BLS intensity for opposite current polarity is a superposition\nof the annihilation of magnons due to the SHE-STT e\u000bect and the decreasing BLS sensitivity.\nIn the following, the e\u000bect of the changed magnon population at the end of the pulse on the threshold voltage of\nthe BEC formation is investigated. The applied DC pulse voltage de\fnes the temperature increase achieved by the\nJoule heating, and, therefore, it de\fnes the number of excess magnons redistributed in the process of rapid cooling.\nThus, the investigation of the BLS intensity as a function of the applied voltage yields threshold information. For\nthe time evolution of the temperature, see Supplemental materials.\nFigure 3(c) shows the BLS intensity after the pulse is switched o\u000b, integrated in the time interval\n\u001cP< t < \u001c P+ 95 ns. As a reference, the black curves show the integrated BLS intensities without a SHE-STT\ncontribution ( \u00160HextkjC). These two curves (positive and negative current polarity) show a sudden increase at a\nvoltage ofU= 1:42 V, which is the threshold of the BEC formation without a SHE-STT contribution. For the case\nof a STT-annihilation (bright blue graph) a pronounced threshold at a higher voltage of U= 1:51 V is observed.\nIn the case of STT-injection, the BLS intensity after termination of the pulse (dark blue curve) is the interplay\nof the rapid cooling induced magnon redistribution and the (decaying) STT-injected magnons during the pulse. To\ninvestigate if these two contributions are a linear superposition or if the SHE-STT e\u000bect driven injection in\ruences the\nredistribution process, we need to subtract the intensity originating from an exponential decay of the STT-injected5\nmagnons. This intensity can be derived as ISTT(t) =ISTT\nt=\u001cPexp[\u00002(t\u0000\u001cp)=\u001cm], whereISTT\nt=\u001cPis the intensity at the end\nof the pulse [integrated value shown in Fig. 3(b)], and \u001cmthe magnon lifetime. Fitting the time evolution of the BLS\nintensity for the lowest voltages applied (for STT-injection, without a contribution of the rapid cooling mechanism)\nreveals\u001cm= 34 ns. Thus, by using the starting intensity of the SHE-STT e\u000bect injected magnons we can calculate the\nexpected magnon intensity after the pulse if no rapid cooling process would take place. The experimental BLS intensity\nafter the pulse with the calculated STT-contribution subtracted is shown in Fig. 3(d). For U > 1:30 V we \fnd that the\nintensity after the end of the pulse is larger than the intensity given by the \fnite lifetime of the STT-injected magnons.\nThis lowest voltage that leads to an increase of the number of magnons above the STT-injected level is identi\fed as the\nthreshold of the BEC formation. In summary, the shift of the BEC formation threshold is from U= 1:42 V down to\n1:30 V or up to U= 1:51 V, corresponding to a relative change of -8% or +6% respectively. This shift results from a rel-\native SHE-STT e\u000bect driven change of excess magnon density in the order of 0.002%, as we discuss in the Supplement.\nAt the same time, these magnons are injected directly to the bottom of the magnon spectrum and, thus, e\u000eciently\ncontribute to the BEC formation. In fact, the number of SHE-STT-e\u000bect-injected magnons compares well with the\ndensity of injected magnons in parallel parametric pumping experiments, which is in the order of 1018{1019cm\u00003. [16].\nThe decreasing intensity at higher voltages [ U > 1:65 V in Fig. 3(c)] requires further investigations but might be\nattributed to the increased temperature of the YIG. At the time, at which the magnon BEC occurs, the YIG is not\nyet cooled down to room temperature. Thus, at higher voltages the temperature at that time is increased, resulting\nin a decrease of the BLS sensitivity. The di\u000berence in the BLS intensity after the pulse between the case of jC;x>0\nandjC;x<0 decreases with higher voltages and vanishes for the highest voltages applied. The vanishing contribution\nof the SHE-STT at higher voltages U > 1:8 V is attributed to a decreasing spin mixing conductance with higher\ntemperatures [52].\nIn conclusion, we found that the SHE-STT e\u000bect injects or annihilates magnons and, thus, shifts the threshold\nvalues for the rapid cooling-induced magnon BEC up to 8%. Utilizing the SHE-STT e\u000bect opens up opportunities\nfor the control (triggering or suppression) of the magnon BEC formation when the applied voltage is close to the\nthreshold value. The comparison between the magnon populations during the application of a DC pulse for the cases\nwith and without a SHE-STT contribution has shown that the injection of magnons primarily takes place into the\nlow energy part of the spectrum, and, thus, strongly promotes the formation of a rapid cooling induced magnon BEC.\nThese \fndings suggest a new way to control the formation of a magnon BEC by the polarity of the applied current,\npaving the way for the utilization of macroscopic quantum states in spintronic devices.\nThis research was funded by the European Research Council within the Starting Grant No. 678309 \\MangonCir-\ncuits\" and the Advanced Grant No. 694709 \\SuperMagnonics\", by the Deutsche Forschungsgemeinschaft (DFG,\nGerman Research Foundation) within the Transregional Collaborative Research Center { TRR 173 { 268565370\n\\Spin+X\" (projects B01 and B04) and through the Project 271741898, and by the Austrian Science Fund (FWF)\nwithin the project I 4696-N.\n\u0003e-Mail: mi schne@rhrk.uni-kl.de\n[1] A. 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For a\n5.5\n5.0\n4.5\n4.0\n3.5\n3.0(1) (2) (3) (1) (2) (3) (1) (2) (3)\n0 50 100 150 200 0 50 100 150 200 0 50 100 150 200DC-pulseBLS frequency (GHz)\nTime (ns) Time (ns) Time (ns)\njc\nµ0 extHjc\nµ0 extHjc\nµ0 extH\nBLS intensity (arb.units)0.051.00\n0.10(a) (b) (c)\nxy xy xy\njc||µ0 extH\njc⊥µ0 extH(d) (e)\nBLS spectra application during DC-pulse (1)\nHeated,\nSTT injectionThermal (×7)\n(3)Heated\nwithout STT (×7)\nHeated\nSTT-annihilation\n(×7)\n00.51.01.52.02.53.0\nBLS intensity (arb. units)\n00.51.01.52.02.53.0\nBLS intensity (arb. units)Thermal (×7) (3)\nRapid cooling,\nSTT-annihilation\n(×7)Rapid cooling,\nwithout STTRapid cooling,\nSTT-injectionBLS spectra\nafter DC-pulse\napplication (2)\njc||µ0 extH\njc 0 ext⊥µH\n3.5 4.0 4.5 5.0 5.5 6.0 6.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 3.0\nBLS frequency (GHz) BLS frequency (GHz)DC-pulse DC-pulse\nFIG. 4. BLS intensity color-coded (log-scale) as a function of the BLS frequency and time. The black boxes mark the duration\nof the 100-ns-long heating DC pulse with an amplitude of jUj= 1:5 V. (a) Current parallel to the external \feld, thus no\ncontribution of the SHE-STT e\u000bect is expected. The achieved overpopulation after pulse termination due to the rapid cooling\ne\u000bect is su\u000eciently large to trigger the formation of a magnon BEC. (b) The current is perpendicular to the external \feld.\nThe STT annihilates magnons during the pulse. The condensation of magnons is suppressed. (c) Reversed current direction in\ncomparison to the situation in (b), the STT injects magnons. After the pulse is applied, the magnon BEC density is enhanced\ncompared to the situation depicted in panel (a). For the exact geometries, see insets. (d) BLS spectra during the application\nof the DC pulse extracted in time-frame (1), as shown in (a-c). For comparison, thermal spectra were extracted in the time\nframe (3) for the \feld pointing along the short (\flled grey curve) and long axis (\flled yellow curve) of the waveguide. (e) BLS\nspectra after the DC pulse extracted in the time frame (2), as shown in (a-c), thermal spectra as in (d).\nbetter comparison of the three cases described in the main text, Figs. 4(d,e) depict the extracted BLS spectra at the\ntime frames marked in Figs. 4(a-c). First, we consider injecting magnons via the STT and the consequent in\ruence\non the intensity during pulse application. The BLS intensity during the pulse application for the STT injecting\nmagnons [dashed red line in Fig. 4(d)] is more than one magnitude larger compared to the case without a SHE-\nSTT contribution [black curve in Fig. 4(d)]. Please note that the increase in intensity is observed, although the\nBLS sensitivity is signi\fcantly decreased at higher temperatures [49]. Furthermore, a signi\fcant shift of the magnon\nfrequency is observed for a SHE-STT-injection compared to the non-heated thermal spectra [\flled curves in Fig. 4(d),\nfor two di\u000berent directions of \u00160Hext], which is in the range of \u0001 f= 1 GHz. The shift is caused by the decrease in\nsaturation magnetization due to the large number of magnons injected via the SHE-STT mechanism. Second, in the\ncase of a SHE-STT induced magnon annihilation [solid red line in Fig. 4(d)], the BLS intensity seems to be decreased\ncompared to the case without SHE-STT contribution [black line in Fig. 4(d)]. Figure 4(e) depicts the spectra after\npulse termination during the rapid cooling process. It can be seen that the STT-annihilation of magnons suppresses\nthe BEC formation. Here, the rapid cooling process causes only a minor increase of the BLS intensity (compare the\nsolid red line with the thermal spectrum for the same geometry, gray-\flled curve). In contrast, the SHE-STT driven\ninjection of magnons (red dashed line) causes a larger BLS intensity than for the case of a suppressed SHE-STT\ncontribution (black dashed line).9\nTemperature at the end of the DC pulse\nThe phenomenon of magnon BEC formation by rapid cooling is driven by a rapid decrease in the phonon tem-\nperature. In the following, we address the temperature dynamics of the investigated YIG/Pt structure subjected to\nshort heating current pulses. Therefore, we performed COMSOL-Multiphysics-based simulations, using the Electric\nCurrents module and the Heat Transfer in Solids module included in the simulation software.\nThe material parameters used for the simulations are the same as given in the Supplement of Ref. [33], while\nthe corresponding temperature-resistance coe\u000ecient was adapted according to the actual experimental structure to\na= 2:0883\u000210\u00003K\u00001and the electric conductivity was de\fned as \u001b= 5:1020\u0002106S m\u00001. Both values were\ndetermined by electrical measurements combined with a Peltier element, which provided temperatures from room\ntemperature up to 350 K.\nFigure 5 depicts the temperature as a function of time for di\u000berent applied voltages in the simulation, corresponding\nto the threshold voltages in the experiment as indicated in the \fgure. The initial current densities were jC=\n1:39\u00021012A m\u00002,jC= 1:50\u00021012A m\u00002,jC= 1:60\u00021012A m\u00002. As seen from the \fgure, the temperature\nincrease of \u0001 T= 180 K for the BEC formation threshold voltage (without additional injection) of U\u00191:4 V is in the\nsame order as in our original study [33].\n0 20 40 60 80 100 120 140300350400450500\nTime (ns)Temperature (K) U=1.30 V\n=1.41 VU\nU=1. V50Thigh\n496 K\n477 K\n454 K\nTlow=296 KDC pulse\nFIG. 5. Temperature increase as a function of time for a 100-ns-long applied DC pulse, for three di\u000berent BEC formation\nthreshold voltages corresponding to the thresholds with additional magnon injection (blue), no SHE-STT contribution (black),\nand additional magnon annihilation (red). The temperatures were determined using COMSOL Multiphysics. The estimated\ntemperature at the end of the applied pulse is indicated in the \fgure, and the increases are found to be of the same order as\nin our original experiment [33].10\nSpin injection via the SHE-STT e\u000bect and generation of excess magnons by rapid cooling\nWe determined the critical voltage for the structure under investigation, exactly compensating for the magnon\ndecay, asU= 0:95 V [see Fig. 3(a) in the main text]. This value corresponds to an initial charge current density of\njC= 1:01\u00021012A m\u00002. We can describe the corresponding critical spin current density [43] as:\njjs;critj=\u001c\u00001\nmdMS;e\u000b\r\u00001; (1)\nwhere\u001cmis the lifetime of lowest-energy magnons, dis the thickness of the magnetic layer, MS;e\u000b= 151:2 kA m\u00001\nand \r= 28 GHz T\u00001is the gyromagnetic ratio. Hence, the spin current density, which compensates exactly for\nthe damping of the lowest-energy magnon modes, is jjs;critj= 5:41\u000210\u00006kg s\u00002or, in units of integer spin,\njj\u0003\ns;critj= 5:13\u00021028m\u00002s\u00001\u0016h. Hence, using the simplest de\fnition of a current j\u0003\ns;crit=\u0016h=N=(\u0001tA), we \fnd\nthat we inject a total number of Ninj= 2:05\u00021010spins for an applied critical voltage of U= 0:95 V. Here, we\nconsidered a pulse duration of \u0001 t=\u001cP= 100 ns and an interface area of A= 2µm\u00022µm. In good approximation,\neach of these spin moments transferred into the YIG spin system results in the excitation of one magnon. Finally,\nassuming a linear dependence on the applied voltage, we can express the total number of injected magnons for an\napplied voltage Uin our experiment as Ninj(U) = 2:16\u00021010V\u00001U, corresponding to an injected quasiparticle\ndensity ofninj(U) = 2:16\u00021016cm\u00003V\u00001U.\nConsidering the voltages applied in the experiment, we \fnd that this value (e.g., ninj= 2:81\u00021016cm\u00003for\nU= 1:3 V) is around two orders of magnitude smaller than typical values of particle densities injected in BEC\nexperiments with parametric pumping, being on the order of 1018cm\u00003[16]. Nevertheless, it should be noted that\nthe STT e\u000bect only slightly changes the BEC threshold in our experiments rather than inducing BEC formation like\nin the case of the parametric pumping experiments. Moreover, the STT e\u000bect injects magnons primarily to the lowest\npart of the magnon spectrum. In contrast, the magnons injected by the parametric pumping process typically have\nhigher energies, and a thermalization process is required before they reach the bottom of the spectrum. Furthermore,\naccounting for the relaxation of magnons with a characteristic lifetime of \u001cm= 34 ns, we can calculate the number of\ngenerated excess magnons at the end of the pulse as\nnt=\u001cP\ninj(U) =Z\u001cP\n0ninj(U)\n\u001cPexp[\u0000(\u001cP\u0000t)=\u001cm]dt; (2)\ngiving a SHE-STT driven excess magnon density of n100 ns\ninj (U= 1:3 V) = 9:04\u00021015cm\u00003at the end of the pulse.\nWe can further compare this value of SHE-STT-driven excess magnons to the number of excess magnons generated\nby the rapid cooling mechanism. To calculate the number of excess magnons generated by the rapid cooling mechanism,\nwe can approximate the excess magnon density induced for a temperature drop from Thigh= 454 K to Tlow= 296 K\n(U= 1:3 V) via:\nnBE(\u0016= 0;T= 454 K)\nnBE(\u0016= 0;T= 296 K)\u00191:6; (3)\nwhere\nnBE(\u0016;T) =ZEmax\nEminD(E)\nexp[E\u0000\u0016\nkBT]\u00001dE; (4)\nwith the density of states D(E). Here,Emin=h= 5 GHz is the bottom of the magnon spectrum and Emax=h=\n7 THz is the edge of the \frst Brillouin zone in YIG. Thus, in the process of rapid cooling, for instance, for the\ncase ofU= 1:3 V, an excess magnon density on the order of the thermal magnon density is generated, hence, of\nnRC= (nBE(\u0016=0;T=454 K)\nnBE(\u0016=0;T=296 K)\u00001)nroom\u00196\u00021020cm\u00003, assuming that the density of magnons at room temperature is\nnroom\u00191021cm\u00003[16]. However, due to their thermal origin, these magnons are distributed over the whole spectral\nrange reaching up to several THz for YIG. Thus, the high-energy magnons decay fast and, thus, weakly contribute to\nthe BEC formation, as shown in the original paper [33].\nHence, recapitulating the case of U= 1:3 V in our experiment, this simple estimation shows that approximately\n0.002 % variation of the number of excess magnons in the bottom part of the magnon spectrum results in a BEC\nformation threshold shift of about 6 % to 8 %. The reason for this comparably large e\u000bect is that the additionally\nSHE-STT-injected magnons have low energy and, thus, contribute more signi\fcantly to the BEC formation.11\nBLS spectra for di\u000berent geometries as a function of the applied voltage\nWe discussed the shift of the threshold of a rapid-cooling-induced magnon BEC formation based on the annihi-\nlation/injection of magnons via the SHE-STT e\u000bect during the pulse. For a slightly supercritical voltage range, we\nobserve an injection of magnons to the bottom of the magnon spectrum in a rather broad frequency range, promoting\nthe subsequent redistribution of the rapid cooling magnon excess to the bottom (see Fig. 2 in the main manuscript,\nand Fig. 4). However, the change of the external \feld direction changes the magnon mode pro\fle in the dipolar\nregime. Furthermore, for large enough voltages and su\u000eciently long DC pulses, the SHE-STT-e\u000bect-based injection\nof magnons is known to trigger the formation of auto-oscillations, which can a\u000bect the spectral magnon distribution\nduring the pulse. Both e\u000bects, the excitation of the auto-oscillations and the modi\fed mode pro\fle, might have\nconsequences for the resulting redistribution of magnons, and therefore they are addressed in more detail here.\nFigure 6 shows the BLS spectra as a function of time for the three di\u000berent geometries investigated and various\napplied voltages (see insets for the geometry used in the experiment). As can be seen, we observe similar spectra\nafter pulse termination just above the threshold of magnon BEC formation [compare Figs. 6(c), (d) and (e)]. The\nmain di\u000berence observed is the slightly higher frequency of the redistributed magnons for the case of magnon injection\nduring the pulse [Fig. 6(c)] compared to the other cases of slightly supercritical voltages (without any SHE-STT e\u000bect\ncontribution or for annihilation of magnons during pulse [Figs. 6(d,e)]). The reason for this is the lower temperature\nreached at the end of the pulse in the \frst case, which results in a less pronounced shift of the magnon dispersion.\nApart from that, we do not observe a signi\fcant in\ruence of the changed \feld direction, as seen from comparing the\nspectra depicted in Figs. 6(d,e) and 6(g,h).\nFor larger voltages applied to inject magnons, we observe the SHE-STT-e\u000bect-driven excitation of the magnon\nbullet mode. The reason for the excitation of the magnon bullet, which arises from nonlinear e\u000bects, is the reduced\ne\u000bective magnon relaxation rate of this particular mode. The low e\u000bective relaxation rate results from the bullet\nmode's low frequency and self-localization, preventing propagation losses.\nAs shown in Fig. 6(f), we observe the bullet mode formation at su\u000eciently high voltages for the magnon injection\ncase (starting at U\u00191:55 V). This bullet formation process changes the spectral magnon distribution during the\npulse with respect to the cases of magnon annihilation during pulse or without a SHE-STT e\u000bect contribution.\nConsequently, the spectra of redistributed magnons change after pulse termination. In particular, we \fnd that the\nmagnons are partially redistributed to this bullet mode state at f\u00193:7 GHz. The e\u000bect of the established auto-\noscillation regime on the rapid-cooling-induced Bose{Einstein condensation of magnons is out of the scope of this\nwork, and the results will be published elsewhere. However, for larger volt-ages applied, this e\u000bect vanishes again [see\nFigs. 6(i,l)], which is attributed to the fact that the magnon injection during the current pulse becomes less e\u000ecient\nwith increasing temperature and the increase of the temperature of the magnon gas.12\nFIG. 6. BLS intensities and frequencies as a function of time for di\u000berent applied voltages and for the three geometries\ninvestigated. First column: current and external \feld are parallel, no contribution of the SHE-STT e\u000bect, second column:\ncurrent and \feld are aligned perpendicularly, magnon annihilation via the SHE-STT e\u000bect during the pulse, third column:\ncurrent and \feld are aligned perpendicularly, magnon injection via the SHE-STT e\u000bect during the pulse. Insets in (a-c) depict\nthe corresponding geometries. (a-c) U= 1:40 V, only for the case of magnon injection during the pulse, the threshold of rapid\ncooling induced BEC formation is reached (c), and an increased intensity after pulse termination is observed. (d-f) U= 1:70 V,\nincreased intensity after pulse termination for all geometries. For magnon injection (f), a lower{frequency mode is populated\nduring the pulse, attributed to the excitation of a bullet mode. Rapid-cooling-induced condensation takes place partially into\nthe bullet mode state. (g-i) U= 1:90 V, analogous to (d-f) with slightly decreased intensities after pulse termination, attributed\nto an overheating of the magnon system. The additionally excited bullet mode in (i) has a minor in\ruence on the spectrum\nafter pulse termination. (j-l) U= 2:20 V, SHE-STT e\u000bect injection decreases during the pulse (l) attributed to the higher\ntemperature reached. Consequently, similar spectra after pulse termination are observed for the di\u000berent geometries."    },    {        "title": "1903.08285v1.Magnetic_properties_and_domain_structure_of_ultrathin_yttrium_iron_garnet_Pt_bilayers.pdf",        "content": "Magnetic properties and domain structure\nof ultrathin yttrium iron garnet/Pt bilayers\nJ. Mendil,1M. Trassin,1Q. Bu,1J. Schaab,1M. Baumgartner,1C. Murer,1P. T. Dao,1J.\nVijayakumar,2D. Bracher,2C. Bouillet,3C. A. F. Vaz,2M. Fiebig,1and P. Gambardella1\n1Department of Materials, ETH Zurich, CH-8093 Zurich, Switzerland\n2Swiss Light Source, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland\n3Institut de Physique et Chimie des Mat\u0013 eriaux de Strasbourg (IPCMS),\nUMR 7504 CNRS, Universit\u0013 e de Strasbourg, 67034 Strasbourg, France\n(Dated: March 21, 2019)\nWe report on the structure, magnetization, magnetic anisotropy, and domain morphology of\nultrathin yttrium iron garnet (YIG)/Pt \flms with thickness ranging from 3 to 90 nm. We \fnd that\nthe saturation magnetization is close to the bulk value in the thickest \flms and decreases towards\nlow thickness with a strong reduction below 10 nm. We characterize the magnetic anisotropy\nby measuring the transverse spin Hall magnetoresistance as a function of applied \feld. Our results\nreveal strong easy plane anisotropy \felds of the order of 50-100 mT, which add to the demagnetizing\n\feld, as well as weaker in-plane uniaxial anisotropy ranging from 10 to 100 mT. The in-plane easy\naxis direction changes with thickness, but presents also signi\fcant \ructuations among samples with\nthe same thickness grown on the same substrate. X-ray photoelectron emission microscopy reveals\nthe formation of zigzag magnetic domains in YIG \flms thicker than 10 nm, which have dimensions\nlarger than several 100 mm and are separated by achiral N\u0013 eel-type domain walls. Smaller domains\ncharacterized by interspersed elongated features are found in YIG \flms thinner than 10 nm.\nI. INTRODUCTION\nYttrium iron garnet (YIG) thin \flms have attracted\nconsiderable interest in the \feld of spintronics due to\nthe possibility of converting magnon excitations into spin\nand charge currents \rowing in an adjacent nonmagnetic\nmetal (NM) layer. Spin currents in YIG/NM bilayers\nhave been excited thermally (spin Seebeck e\u000bect)1{4, dy-\nnamically (spin-pumping)5{8or by means of the spin Hall\ne\u000bect9{13. In the latter case, a charge current in the\nNM generates a transverse spin current that is either ab-\nsorbed or re\rected at the interface with YIG. This leads\nto a variety of interesting e\u000bects such as the spin Hall\nmagnetoresistance14{17(SMR) and current-induced spin-\norbit torques18{20, which can be used to sense and ma-\nnipulate the magnetization. For the latter purpose, it is\ndesirable to work with thin magnetic \flms in order to\nachieve the largest e\u000bect from the interfacial torques.\nFor a long time, the growth of YIG has been ac-\ncomplished by liquid phase epitaxy, which o\u000bers excel-\nlent epitaxial quality and dynamic properties such as\nlow damping and a rich spin-wave spectrum. The mag-\nnetic properties of these bulk-like samples, including the\nmagnetocrystalline anisotropy21,22and magnetic domain\nstructure23{25, have been extensively characterized in the\npast. However, with rare exceptions26, samples grown\nby liquid phase epitaxy usually have thicknesses in the\nmm to mm range. Recent developments in oxide thin\n\flm growth give access to the sub- mm range by employ-\ning techniques such as laser molecular beam epitaxy27,\nsputtering28,29, and pulsed laser deposition (PLD)30{37,\nwhich allow for growing good quality \flms with thickness\ndown to the sub-100-nm range28,29,35{37and even below\n10 nm27,33,34. Since the thickness as well as structuraland compositional e\u000bects have a large in\ruence on the\nmagnetic behavior, these developments call for a detailed\ncharacterization of the magnetic properties of ultrathin\nYIG \flms. Several characteristic quantities, which are\nof high relevance for YIG-based spintronics, have been\nfound to vary in \flms thinner than 100 nm. For instance,\na reduction of the saturation magnetization29,35{40is\ntypically observed in YIG \flms with thickness down to\n10 nm, which has been ascribed to either thermally-\ninduced stress38, lack of exchange interaction partners\nat the interface36, or stoichiometric variations29,37,39,40.\nFurthermore, an increase of the damping33{35,41as well\nas a decreased spin mixing conductance42,43have been\nfound in ultrathin YIG. Finally, the emergence of un-\nexpected magnetocrystalline anisotropy was reported\nfor \flms of di\u000berent orientations grown on gadolinium\ngallium garnet (GGG) and yttrium aluminium garnet\n(YAG). The magnetic anisotropy was investigated by\nthe magneto-optical Kerr e\u000bect in GGG/YIG(111)27,36\nand by ferromagnetic resonance in GGG/YIG(111)31,\nGGG/YIG(001)32, and YAG/YIG(001)28. Whereas all\nthese studies address important magnetic characteristics\nin the sub-100 nm range, only few studies33,34explore the\nultrathin \flm regime below 10 nm. This thickness range\nis highly relevant for e\u000ecient magnetization manipula-\ntion using current-induced interfacial e\u000bects as well as\nfor strain engineering, since strain and its gradients relax\nafter 10 to 20 nm. Finally, a comprehensive knowledge\nof the domain and domain wall structure in the thin \flm\nregime is lacking. Recent studies on magnetic domains\naddress only bulk25, several micrometers44or hundreds\nof nanometers45thick YIG \flms.\nIn this work, we present a systematic investigation\nof the structure, saturation magnetization, magnetic\nanisotropy, and magnetic domains of YIG/Pt \flms grownarXiv:1903.08285v1  [cond-mat.mtrl-sci]  19 Mar 20192\non GGG substrates by PLD as a function of YIG thick-\nness fromtYIG= 3:4 totYIG= 90 nm. By combin-\ning x-ray di\u000braction (XRD), transmission electron mi-\ncroscopy (TEM), atomic force microscopy (AFM) and x-\nray absorption spectroscopy, we show that our \flms pos-\nsess high crystalline quality and smooth surfaces with no\ndetectable interface mixing throughout the entire thick-\nness range. The saturation magnetization, investigated\nusing a superconducting quantum interference device\n(SQUID), shows values close to bulk for thick \flms and\na gradual reduction towards lower thicknesses. We probe\nthe magnetic anisotropy electrically by means of SMR\nand \fnd an easy plane and uniaxial in-plane anisotropy\nwith a non-monotonic variation of the in-plane orien-\ntation of the easy axis and the magnitude of the e\u000bec-\ntive anisotropy \feld. Finally, we investigate the domain\nstructure using x-ray photoelectron emission microscopy\n(XPEEM), evidencing signi\fcant changes in the domain\nstructure above and below tYIG= 10 nm. Our results\nprovide a basis for understanding the behavior of spin-\ntronic devices based on YIG/Pt with di\u000berent YIG thick-\nness.\nII. GROWTH AND STRUCTURE\nWe prepared three sample series consisting of\nGGG/YIG( tYIG)/Pt bilayers with Pt thickness set to\n3 nm in the \frst two series for optimal SMR measure-\nments and 1.9 nm in the third series to allow for surface-\nsensitive XPEEM measurements while avoiding charg-\ning e\u000bects. The YIG thickness, tYIG, varies from 3.4\nto 90 nm. An overview of the samples and their thick-\nness in nm and YIG unit cells can be found in Table I.\nThe samples have been grown in-situ on (111)-oriented\nGGG substrates using an ultra-high vacuum PLD sys-\ntem combined with dc magnetron sputtering (base pres-\nsure 10\u00008mbar, 10\u000010mbar, respectively). The YIG\nwas grown by PLD at a growth pressure of 10\u00001mbar\nand temperature of 720\u000eC using an excimer KrF laser\n(wavelength 248 nm) at a repetition rate of 8 Hz and 1.45\nJ/cm2\ruence. Re\rection high energy electron di\u000braction\n(RHEED) was used to monitor the growth rate in-situ.\nAfter cooling down under a 200 mbar O 2atmosphere,\nthe GGG/YIG \flms were transferred to the sputtering\nchamber without breaking the ultra-high vacuum, where\nPt was deposited at room temperature under 10\u00002mbar\nAr pressure. Finally, for electrical transport measure-\nments, the YIG( tYIG)/Pt(3) series were patterned into\nHall bar structures using optical lithography followed by\nAr-ion milling. The Hall bars are oriented parallel to\nthe [1 \u001610] crystal direction and are 50 mm wide, with a\nseparation of the Hall arms of 500 mm.\nThe crystalline quality and epitaxial strain of the\nGGG/YIG(111) \flms were investigated using XRD to\nobtain\u0012\u00002\u0012di\u000braction scans and reciprocal space maps.\nThe interface quality was probed using scanning TEM\n(STEM), energy dispersive x-ray spectroscopy (EDS),tPt[nm] tYIG[nm, (unit cells)]\nSMR 1 3 3.4, 4.6, 6.2, 7.3, 9, 29, 90\n(2.7) (3.7) (5.0) (5.9) (7.3) (23.4) (72.7)\nSMR 2 3 3.7, 5.6, 6.2, 6.8, 7.4, 12.4\n(3.0) (4.5) (5.0) (5.5) (6.0) (10.0)\nXPEEM 1.9 3.7, 8.7, 12.4, 28.5, 86.7\n(3.0) (7.0) (10.0) (23.0) (70.0)\nTABLE I. Overview of the three sample series used for\nanisotropy characterization using SMR and domain struc-\nture using XPEEM where tPtandtYIGrefer to the Pt and\nYIG thickness, respectively. The numbers between paren-\ntheses refer to the thickness in unit cells of YIG (1 unit\ncell=1.238 nm [46]).\nand electron energy loss spectroscopy (EELS) in order to\nresolve the elemental composition, whereas the thin \flm\ntopography was investigated using AFM in both contact\nand tapping mode.\nThe YIG \flms are single phase epitaxial layers with\n(111) orientation, as indicated by the di\u000braction peaks\ncorresponding to the (111) planes in the di\u000braction pat-\ntern. The x-ray di\u000bractograms for three selected YIG\nthicknesses are shown in Fig. 1 (a-c). Kiessig thickness\nfringes indicate the homogeneous growth and high inter-\nface quality. The observed periodicity matches the thick-\nnesses determined by x-ray re\rectivity. To further inves-\ntigate the crystalline quality and the interfacial strain\nstates of the YIG \flms, we recorded reciprocal space\nmaps around the (664) and (486) di\u000bractions for both\nYIG and the GGG substrate. These maps allow us to\nprobe the strain state of the \flms along the two inequiva-\nlent [11 \u00162] and [1 \u001610] in-plane directions. A coherent strain\nstate of the \flms was evidenced for all thicknesses con-\nsidered. The good lattice matching between YIG and\nGGG results in high quality epitaxial growth and no de-\ntectable strain relaxation, as shown by the alignment of\nthe di\u000braction points along both [11 \u00162] and [1 \u001610] orienta-\ntions. The reciprocal space maps measured on a 99 nm-\nthick YIG \flm are shown in Fig. 1 (d) and (e).\nFigures 1 (f,g) show high resolution STEM images of\nthe two interfaces and a TEM image of the full stack\nin the inset. The corresponding chemical pro\fles across\nthe interfaces probed by EDS are shown in Figs. 1 (h,j).\nThe EDS pro\fles indicate a moderate interdi\u000busion of\nFe, Y, and Pt within a range of 2 nm at the YIG/Pt\ninterface and of Gd and Ga into YIG within a range of\nabout 4 nm at the GGG/YIG interface. Both values are\nin agreement with recent \fndings39,47. The smaller dif-\nfusion range at the YIG/Pt interface is consistent with\nthe low power and lower deposition temperature (room\ntemperature) of Pt compared to the deposition of YIG\non GGG. The moderate di\u000busion of Ga and Gd into the\nYIG \flm is also in agreement with previous reports29,39.\nThe EELS analysis, shown in Fig. 2, con\frms the moder-\nate di\u000busion of Fe in the Pt. In order to gain information\non the evolution of the local Fe environment across the3\n49 50 51 52(c) 99 nm Log. Intensity (a. u.)\n2q (o\n)(b) \nYIG (444)\nGGG (444)(a) \nGGG (444)YIG (444)\n24.8 nm 12.4 nm \nGGG (444)YIG (444)(d) (664)qz // [111] (arb.u.) \nqx // [112] (arb.u.)\n(e)     // [1 11] (arb.u.) \nqy // [1 10] (arb.u.)(486)\n0 4 8 12050100\n(f) \n Fe\n Y\n Pt(g) \nDistance (nm)Atomic ratio (%)2 nm [-110][111]YIGPt\n25 nmPt\nYIG\nGGGqz\n0 4 8 12 0 4 8 12\nDistance (nm) Distance (nm)Pt\nYIG\nGGGYIG\n050100 Atomic ratio (%) Fe\n Y\n Pt\nPt\nYIG\nGGG[110][111]\n50 nmPt\nYIG YIG\nGGG\n4 nm 4 nm\n \nFe \nY \nGd \nGa(f) (g) \n(j) (h) \nFIG. 1. (a-c) Structural characterization of GGG/YIG(111) \flms of di\u000berent thickness by \u0012\u00002\u0012scans using XRD. (d,e)\nReciprocal space maps around the (664) and (486) di\u000braction peaks. The oscillations in the \u0012\u00002\u0012andqzscans are thickness\noscillations. The thickness derived from these measurements agrees with that measured using XRD. (f,g) High resolution STEM\nimage of the two interfaces in the stack GGG/YIG(72)/Pt(9). The inset shows a cross-section of the full stack. (h,j) Elemental\npro\fles across the YIG/Pt and YIG/GGG interfaces measured by EDS.\nYIG/Pt interface, we calculated the intensity ratio of the\nFeL3andL2white lines from the Fe EELS spectra mea-\nsured within the YIG \flm (black dot in Fig. 2) and at\ntwo di\u000berent positions near the YIG/Pt interface (red\nand blue dots). In YIG, the L3=L2ratio is 5.64, which\nis consistent with an Fe3+oxidation state, as found, e.g.,\nin Fe 2O3(Ref. 48). On the Pt side of the YIG/Pt in-\nterface, the L3=L2ratio decreases to 4.24, which is in\nbetween the values found for Fe and FeO. This change\nof electronic valence con\frms the presence of Fe in the\ninterfacial Pt layer. We note that the absence of Fe3+\nin the Pt further excludes measurement artifacts due to\nsample preparation for TEM, such as Pt redeposition\nThe AFM measurements substantiate the low rough-\nness expected from the x-ray re\rectivity and the observa-\ntion of Kiessig fringes in the di\u000braction spectra. Figure\n3 (a) shows an AFM image over an area of 3 \u00023mm2of\na 28.5 nm thick GGG/YIG \flm. The measured rough-\nness is in the range of 1 \u0017A and is independent of the YIG\nthickness, as shown in Fig. 3 (b).\nIII. SATURATION MAGNETIZATION\nFigure 4 (a) shows the saturation magnetization ( Ms)\nof the unpatterned YIG( tYIG)/Pt(3) bilayers measured\nby SQUID (sample series SMR 1 and SMR 2 in red\nand grey, respectively). For each thickness, we observe a\nhysteretic in-plane magnetization with coercivity smallerthan 0.5 mT, as illustrated for YIG(9)/Pt(3) in Fig. 4\n(b). The thickest sample, YIG(90)/Pt(3), has a sat-\nuration magnetization of \u00160Ms= 153 mT, which is\nclose to the bulk value of \u00160Ms,bulk = 180 mT at room\ntemperature.49The reduction of Mscompared to the\nbulk was also observed in other studies of YIG \flms\ngrown by PLD31,36,50,51. This reduction has been as-\ncribed to the di\u000berent percentage of Fe vacancies at the\ntetrahedral or octahedral sites31, the lack of exchange\n710 720 7300.00.51.01.52.0EELS (a.u.)\nenergy (eV)\n YIG\n interface\n Pt\n10 nm\nFIG. 2. Comparison of the Fe L2;3spectra measured by\nEELS across the YIG/Pt interface. The spectra, shown after\nbackground subtraction, have been averaged on 3 nm-long\nline scans for each region of interest, as labeled in the inset\nwith colored dots. The probe beam diameter is 0.15 nm.4\nheight (pm)6 00 nm\n236( a)2\n4681012299001(b)-\n252r\noughness (Å)t\nYIG (nm)\nFIG. 3. (a) AFM image of YIG(28.5)/Pt(3). (b) Root mean\nsquare roughness as a function of YIG thickness measured by\nAFM.\ninteraction partners for atoms at the interface36, strain\nrelaxation due to a slight lattice mismatch of the sub-\nstrate and YIG50, as well as to the presence of Fe and O\nvacancies51. Furthermore, we observe a decreasing trend\nforMsat lower YIG thicknesses, with a steeper reduction\nbelow 10 nm. Our thinnest sample ( tYIG= 3:4 nm) has\n\u00160Ms= 27 mT, which is only 15% of the bulk saturation\nmagnetization. Previous studies reported a decreasing\ntrend ofMsalready for thicknesses larger than 10 nm\n(Refs. 29, 36, 39, 40, and 50).\nThe reduction of the YIG magnetization in thin \flms\nhas been often modeled as a magnetically dead layer.\nFor example, by extrapolating the areal magnetization\nas a function of thickness to the point where no surface\nmagnetization would be present, Mitra et al.39inferred a\n6 nm thick dead layer for YIG \flms in the 10-50 nm thick-\nness range. A similar extrapolation of our Msdata for\nsamples with thickness of 9 nm and above would lead to\na 4.3 nm thick dead layer. However, our data evidence\na \fnite magnetization below 4.3 nm, contradicting the\nassumption of an abrupt magnetically dead layer at the\nGGG/YIG interface. Instead, we conclude that a grad-\nual reduction of Msoccurs at thicknesses below 10 nm,\nwhich may be due to the di\u000busion of Gd from the GGG\nsubstrate into YIG observed by EDS [see Fig. 1 (j)].\nFurther information on the magnetization of\nYIG(tYIG)/Pt(1.9) was obtained using x-ray ab-\nsorption spectroscopy and x-ray magnetic circular\ndichroism (XMCD) at the L3andL2absorption edges\nof Fe. The x-ray absorption spectra of representative\nYIG(3.7,12.4,86.7)/Pt(1.9) samples present a very\nsimilar lineshape [Fig. 4 (c)], which implies that the\nchemical environment of the Fe atoms does not change\nsubstantially with thickness. The XMCD asymmetry,\nhowever, decreases in the thinner samples [Fig. 4 (d)],\nconsistent with the behavior of Msdiscussed above.\nThese data con\frm that our samples are magnetic\nthroughout the entire thickness range and suggest that\nthe reduction of Msis not due to defective Fe sites.\n710 720 7300.00.51.01.5\n710 720 730-100104681012 29 90050100150\n-1.0 -0.5 0.0 0.5 1.0-1000100XAS (a.u.)\nenergy (eV) energy (eV)(d)\n3.7 nm86.7 nmXMCD (%)m0MS (mT)\ntYIG (nm)(a)\n3.7 nm12.4 nm86.7 nm(c)(b)\nBext (mT)m0MS (mT)FIG. 4. (a) Msas a function of YIG thickness mea-\nsured by SQUID for sample series SMR 1 (red) and SMR\n2 (grey). (b) Magnetic hysteresis of YIG(9)/Pt(3) as a func-\ntion of in-plane magnetic \feld. (c) X-ray absorption spectra\nof YIG(3.7,12.4,86.7)/Pt(1.9). Each line represents the sum\nof two spectra acquired with positive and negative circular x-\nray polarization. The spectra are shifted by a constant o\u000bset\nfor better visibility. (d) XMCD spectra for the thickest and\nthinnest samples obtained by taking the di\u000berence between\ntwo absorption spectra acquired with positive and negative\ncircular x-ray polarization. The spectra were acquired on ho-\nmogeneously magnetized domains in the XPEEM setup.\nIV. MAGNETIC ANISOTROPY\nTo probe the magnetic anisotropy, we performed mea-\nsurements of the transverse SMR14,15as a function of\nmagnitude and orientation of the external magnetic \feld\nBext. Our data evidence the presence of an easy-plane\nanisotropy \feld, BK1, which adds to the demagnetization\n\feld to favor the in-plane magnetization, as well as of an\neasy axis \feld BK2, which favors a particular in-plane\ndirection that varies from sample to sample.\nA. Transverse SMR measurements\nA sketch of the Hall bar structure employed for the\nSMR measurements is presented in Fig. 5 (a). We used\nan ac current I=I0sin(!t), modulated at a frequency\n!=2\u0019= 10 Hz, and acquired the longitudinal and trans-\nverse (Hall) resistances52,53. To extract the magnetiza-\ntion orientation, we consider here only the transverse\nresistance, Rxy, which is sensitive to all three Carte-\nsian components of the magnetization due to the SMR\ne\u000bect15. We performed two types of measurements. In\nthe \frst type, which we call IP angle scan, we vary the\nin-plane angle of the applied magnetic \feld Bext; in the\nsecond type, which we call OOP \feld scan, we ramp Bext\napplied out-of-plane. For both cases, it is convenient to\nuse spherical coordinates: we de\fne 'Bas the azimuthal5\nangle between BextandIand'as the azimuthal angle\nbetween the magnetization and I, whereas the polar an-\ngles ofBextand the magnetization with respect to the\nsurface normal are \u0012Band\u0012, respectively [Fig. 5 (a)].\nIn the IP angle scan ( \u0012=\u0019=2),Rxyis determined\nsolely by the planar Hall-like (PHE) contribution from\nthe SMR, leading to\nRxy=RPHEsin(2'); (1)\nwhereRPHE is the planar Hall-like coe\u000ecient. In this\ntype of experiment, we record Rxyas a function of 'B\nfor di\u000berent Bext. For \felds large enough to saturate the\nmagnetization along the \feld direction, we can assume\n'='Band hence Rxy=RPHEsin(2'B). Conversely,\nfor external \felds small enough such that '6='B,Rxy\nwill deviate from the sin(2 'B) curve. By adopting a\nmacrospin model assuming in-plane uniaxial magnetic\nanisotropy and comparing the resulting PHE from Eq. (1)\nwith our data, we can determine quite accurately the easy\naxis direction as well as the magnitude of the in-plane\nmagnetic anisotropy energy.\nIn the OOP \feld scans, Rxydepends on the ordinary\nHall e\u000bect (OHE) and anomalous Hall-like (AHE) contri-\nbution from the SMR, which is proportional to the out-\nof-plane component of magnetization15. We thus have\nRxy=ROHEBextcos\u0012B+RAHEcos\u0012; (2)\nwhereROHE andRAHE are the OHE and AHE coef-\n\fcients, respectively. When ramping the OOP \feld,\nthe contribution due to ROHE continuously increases,\nwhereas the contribution due to RAHE saturates as the\nmagnetization is fully aligned out-of-plane. The corre-\nsponding saturation \feld Bscan be used to determine\nthe out-of-plane magnetic anisotropy knowing the value\nofMs. All measurements were performed at room tem-\nperature using a current density in the low 105A/cm2\nrange.\nB. Easy plane anisotropy\nThe easy-plane anisotropy \feld was determined by\ncomparing the hard axis (out-of-plane) saturation \feld\nBsmeasured by the Hall resistance with the demagnetiz-\ning \feld\u00160Msestimated using SQUID. Figures 5 (b) and\n(c) show the results of OOP \feld scans for the thickest\n(90 nm) and thinnest (3.4 nm) YIG/Pt(3) samples. In\nboth cases, we identify Bs(dashed line) as the \feld above\nwhich only the ordinary Hall e\u000bect contributes to the\n(linear) increase of Rxywith increasing \feld. Note that,\nfor YIG(90)/Pt(3), we observe a bell-shaped curve that\nis due to the PHE during the re-orientation of magnetic\ndomains from the initial in-plane to the \fnal out-of-plane\norientation at Bs. Figure 5 (d) reports the estimated val-\nues ofBsfor all thicknesses. In all cases, we \fnd that Bs\nis signi\fcantly larger than \u00160Msreported in Fig. 4 (a).\nWe attribute this di\u000berence to an easy-plane anisotropy\feld,BK1=Bs\u0000\u00160Ms, which favors in-plane magneti-\nzation. The magnitude of BK1varies in the range of 50-\n-1 0 11.871.881.89\n468101229900100200300\n46810122990050100-1 0 14.3654.375\n-0.1 0.0 0.14.3684.369\ntYIG= 3.4 nm tYIG= 90 nm(b)Rxy (W)\nBext,z (T)\ntYIG (nm)Bs (mT)(d) (e)BK1 (mT)\ntYIG (nm)(c)Rxy (W)\nBext,z (T)\n(a)\nFIG. 5. (a) Schematic of the Hall bar and coordinate system.\nOOP \feld scans for the thickest (90 nm) and thinnest (3.4 nm)\nYIG thickness from series SMR 1 are shown in (b) and (c),\nrespectively. From these scans, the out-of-plane saturation\n\feldBswas determined as indicated by the dashed line and\nsummarized in (d) as a function of YIG thickness. (e) Easy-\nplane anisotropy \feld BK1=Bs\u0000\u00160Ms. Data shown for\nsample series SMR 1 (red) and SMR 2 (grey).\n100 mT (except for the thinnest sample). This additional\neasy plane anisotropy is of the same order of magnitude\nas in previous studies27,36, where it was attributed to a\nrhombohedral unit cell distortion along the [111] direc-\ntion by\u00191 %. Interestingly, the thinnest sample shows\na drastically reduced BK1compared to the thicker ones,\nsuggesting that very thin YIG \flms with tYIG\u00143:4 nm\ntend to develop out-of-plane magnetic anisotropy38. This\ntendency, however, is not su\u000ecient to induce an out-of-\nplane easy axis, as veri\fed by SQUID measurements as a\nfunction of out-of-plane \feld. The reduction of BK1may\nalso explain the absence of the bell-shaped PHE contri-\nbution toRxyobserved in Fig. 5 (c).\nC. In-plane uniaxial anisotropy\nThe in-plane uniaxial anisotropy \feld BK2is deter-\nmined by measuring IP angle scans of Rxyfor di\u000berent\nconstant values of Bext. By choosing Bextbelow the in-\nplane saturation \feld and \ftting Rxyas a function of 'B\nusing a macrospin model, we determine the orientation\nof the in-plane easy axis as well as the magnitude of BK2.6\n0.410.420.43tYIG=3.4 nm7\n.2 mT0\n.480.500.52(\ne)(d)(c)( b)tYIG=9 nm1\n50 µT513 mT(a)0\n.240.28tYIG=90 nm6\n1 mT0\n90180ϕEA (deg)0\n.410.420.4360 µT0\n.480.500.520\n.240.28Rxy (Ω)1 20 µT0\n50100BK2 (µT)0\n1 803 600.410.420.432\n0 µTϕ\nB (deg)01 803 600.480.500.526\n0 µTϕ\nB (deg)01 803 600.240.286 0 µTϕ\nB (deg)468101229900510(f)K2 (J/m3)t\nYIG (nm)\nFIG. 6. Transverse resistance Rxyas a function of azimuthal orientation of the external magnetic \feld, 'B, for di\u000berent\nthicknesses in (a-c). The data are shown by black symbols, the macrospin simulations by red solid lines. (d) Easy axis\norientation, 'EA, relative to the [1 \u001610] crystal axis. (e) E\u000bective uniaxial anisotropy \feld BK2and (f) in-plane uniaxial energy\nMsBK2as a function of YIG thickness. Black and red points show the results for two di\u000berent Hall bars patterned on the same\nchips of series SMR 1; grey triangles are results obtained on Hall bars of series SMR 2.\nThe black circles in Fig. 6 (a-c) show Rxyas a function\nof'Bfor three representative YIG thicknesses (90, 9, 3.4\nnm). For all three samples, we identify the sin(2 ') be-\nhavior expected from Eq. (1) for Bext>7:2 mT [upper\npanels in Fig. 6 (a-c)]. When reducing Bext, deviations\nfrom this lineshape occur, indicating that the magnetiza-\ntion no longer follows the external magnetic \feld. This\nbehavior is most pronounced for \felds of only tens of\nmT [lower panels in Fig. 6 (a-c)]. For such low \felds,\nwe observe two prominent jumps of Rxyseparated by\n180\u000e, which we attribute to the magnetization switching\nabruptly from one quadrant to the opposite one during\nan IP angle scan in proximity to the hard axis. These\njumps, which occur at di\u000berent 'Bfor the three samples,\nindicate the presence of in-plane uniaxial anisotropy.\nIn order to quantify the in-plane uniaxial anisotropy,\nwe perform macrospin simulations of Rxyassuming the\nfollowing energy functional\nE=\u0000Ms~ m\u0001~Bext+MsBK2sin2('\u0000'EA);(3)\nwhere~ mis the magnetization unit vector and 'EAde-\n\fnes the angle of the easy axis with respect to the\n[1\u001610] crystal axis. Expressing ~ mand~Bextin spherical\ncoordinates, i.e., ~ m= (sin\u0012cos';sin\u0012sin';cos\u0012) and\n~Bext=Bext(sin\u0012Bcos'B;sin\u0012Bsin'B;cos\u0012B), and tak-\ning\u0012=\u0012B=\u0019=2, we can calculate the equilibrium po-\nsition of the magnetization as a function of Bext,'B,\nBK2, and'EA. This calculation results in a set of values\n'('B) that can be used to simulate Rxyusing Eq. (1).\nThe planar Hall constants RPHE required for the simu-\nlations are obtained by \ftting Eq. (1) to the saturated\ndata sets. Finally, we \ft Rxy('B) keepingBK2and'EA\nas free parameters. The \fts, shown as red lines in Fig. 6(a-c), reproduce the main features (lineshape and jumps)\nofRxyquite accurately, indicating that our method is ap-\npropriate to measure weak anisotropy \felds in YIG. The\nresulting values of 'EAandBK2are shown by the sym-\nbols in Fig. 6 (d) and (e), respectively.\nThe data for the series SMR 1 (red) in Fig. 6 (d) ap-\npear to follow a pattern in the orientation of the easy\naxis as a function of thickness. Very thin samples with\ntYIG\u00147:2 nm have 'EA\u001960\u000e, whereas intermedi-\nate thicknesses in the range 9 \u0014tYIG\u001429 nm have\n'EA\u0019120\u000eand the thickest sample, tYIG= 90 nm, has\n'EA\u00190\u000e. While these orientations correspond to the\nthree-fold symmetry of the (111) plane, measurements on\ndi\u000berent sets of samples show that this correspondence is\nlikely coincidental. Measurements performed on devices\npatterned on the same chip (black squares) of series SMR\n1 as well as on one device of the series SMR 2 (grey trian-\ngles) reveal uncorrelated variations of 'EArelative to the\nSMR 1 series, particularly for tYIG<10 nm. A variation\nof'EAfor samples grown on the same chip has also been\nobserved in thicker YIG \flms27and is most likely due\nto inhomogeneities of process parameters such as, e.g.,\nthe temperature of the sample surface during deposition,\nwhich could lead to local strain di\u000berences.\nFigure 6 (e) shows BK2as a function of tYIG. Most\nvalues are close to 50 mT, with minima and maxima\nof about 20 and 100 mT, respectively. Combined with\nthe thickness-dependent saturation magnetization from\nFig. 4 (a), we obtain a magnetic anisotropy energy\nK2=MsBK2in the range of 0.1 to 10 J/m3[Fig. 6\n(f)].K2is more than two orders of magnitude smaller\nthan the \frst order cubic anisotropy constant of bulk YIG\n(-610 J/m3) (Ref. [22]), which reinforces the hypothesis7\n(a) (b) (c) (d)3.7 nm\n20 µm8.7 nm\n20 µm12.4 nm\n100 µm[101]\n[011] [110]αγinγinγinγin\nFIG. 7. (a) Schematics of the direction of the incoming x-ray beam (red arrow) relative to the crystal axes in the XPEEM\nsetup. (b-d) Domain structure of YIG/Pt(1.9) bilayers with tYIG= 12.4, 8.7, and 3.7 nm observed by XPEEM. The gray scale\ncontrast corresponds to di\u000berent in-plane orientations of the magnetization.\nof an extrinsic origin of the uniaxial anisotropy reported\nhere.\nOur observation of uniaxial anisotropy agrees with\nprevious studies of YIG \flms grown by PLD on\nGGG27,31,36,54. From a crystallographic point of view,\nhowever, uniaxial anisotropy is not expected for the\nYIG(111) plane. Rather, for an ideal (111) crystal plane,\none would expect a three-fold anisotropy due to the cu-\nbic magnetocrystalline anisotropy of bulk YIG55. This\nbecomes obvious when translating the \frst order mag-\nnetocrystalline energy term of cubic crystals ( Ecubic/\n\u000b2\n1\u000b2\n2+\u000b2\n2\u000b2\n3+\u000b2\n3\u000b2\n1, where\u000b1;2;3are the directional\ncosines with respect to the main crystallographic axes)\ninto the coordinate system of the (111) plane ( \u0012with re-\nspect to the [111] direction and, for simplicity, 'with\nrespect to the [11 \u00162] direction), giving\nEcubic/1\n4sin4\u0012+1\n3cos4\u0012+p\n2\n3sin3\u0012cos\u0012cos 3':(4)\nIn order for the threefold anisotropy to appear [last term\nin Eq. (4)], the magnetization has to have a small out-\nof-plane component ( \u00126=\u0019=2). Whereas this can be\ngenerally guaranteed in bulk crystals, in thin \flms the\ndemagnetizing \feld and BK1force\u0012=\u0019=2, resulting in\nthe absence of threefold cubic anisotropy. The origin of\nthe in-plane uniaxial anisotropy thus remains to be ex-\nplained. Our XRD measurements reveal no signi\fcant\nin-plane strain anisotropy within the accuracy of the re-\nciprocal space maps in Fig. 1 (d) and (e). The small\namplitude of BK2and the intra- and inter-series varia-\ntions of'EAsuggest that this anisotropy may originate\nfrom local inhomogeneities of the growth or patterning\nparameters, which should thus be taken into account for\nthe fabrication of YIG-based devices.\nV. MAGNETIC DOMAINS\nThe domain structure in bulk YIG is of \rux-closure\ntype with magnetic orientation dictated by the easy axes\nalong the [111]-equivalent directions56,57. In the case of\na defect-free YIG crystal, the domains can be as large asthe sample itself, apart from the \rux closure domains at\nthe edges. Dislocations and strain favor the formation of\nsmaller domains57. External strain and local stress due\nto dislocations can result in a variety of magnetic con\fg-\nurations such as cylindrical domains57and complex lo-\ncal patterns56,58. Moreover, dislocations serve as pinning\nsites for domain walls59. The domain walls observed at\nthe surface of YIG single crystals are usually of the Bloch\ntype and several \u0016m wide56,60.\nIn the following, we characterize the domain struc-\nture of YIG thin \flms using XPEEM. The samples\nare YIG(tYIG)/Pt(1.9) bilayers with thickness down to\ntYIG= 3:7 nm, as described for the XPEEM series in\nTable I. The measurements were performed at the SIM\nbeamline of the Swiss Light Source61by tuning the x-ray\nenergy to obtain the maximum XMCD contrast at the\nFeL3edge (710 eV) and using a circular \feld of view\nwith a diameter of 100 mm or 50 mm. Magnetic contrast\nimages were obtained by dividing pixel-wise consecutive\nimages recorded with circular left and right polarized x-\nrays. Prior to imaging, the samples were demagnetized\nin an ac magnetic \feld applied along the surface normal.\nA. Domain structure\nFigure 7 shows representative XPEEM images of the\nmagnetic domains of 12.4, 8.7, and 3.7 nm thick YIG.\nThicker \flms ( tYIG= 86.7, 28.5 and 12.4 nm) present\nqualitatively similar domain structures such as those\nshown in Fig. 7 (b) for the 12.4 nm \flm. The domains\nin these thicker \flms extend over hundreds of mm and\nmeet head-on, separated by domain walls with charac-\nteristic zigzag boundaries. Such walls are typical of thin\n\flms with in-plane uniaxial anisotropy, where the zigzag\namplitude and period depends on the balance between\nmagnetostatic charge density and domain wall energy, as\nwell as on the process of domain formation62{64. Note\nthat the two sides of each zigzag are asymmetric, as\nfound in ion-implanted garnets with uniaxial and trigo-\nnal in-plane anisotropy65. The length and opening angle\nof the zigzags increase and decrease, respectively, with8\n(a) 86.7 nm\n10 µm 20 µm(g) 86.7 nm\n(h) 28.5 nm(b) 28.5 nm\n50 µm10 µm[101]\n[011][110]αγin\nI (a.u.)norm\n0 10 20 -10 -20\nL (µm)2.2\n2.0\n010 20 -10-2023\n14I (a.u.)norm\nL (µm)0 5 10 -5 -10\nL (µm)I (a.u.)norm2.4\n2.0\n20 µm(c) 12.4 nm\n(d) (e) (f)γin\n \n2.2\nFIG. 8. XPEEM images of domain walls in (a) 86.7 nm, (b) 28.5 nm, and (b) 12.4 nm thick YIG. The arrows indicate the\nx-ray incidence direction. (d-f) Line pro\fles of the walls corresponding to the dashed lines in (a-c). (g,h) Details of the apices\nof the zigzag domains in 86.7 nm and 28.5 nm thick YIG. The samples are oriented with the crystal axes de\fned in Fig. 7 (a).\nincreasing \flm thickness, similar to the trend reported\nfor amorphous Ga-doped Co \flms with weak uniaxial\nanisotropy65. In a quantitative model based on the men-\ntioned energy balance66, both quantities correlate with\nMsin a positive and negative manner, respectively. This\ntrend is in agreement with the increase of Msreported\nin Fig. 4 (a).\nThe domain morphology changes abruptly in \flms\nthinner than 12.4 nm. YIG \flms with tYIG= 8:7 and 3.7\nnm present much smaller domains, which extend only for\ntens of mm and have a weaving pattern, as shown in Fig.\n7 (c) and (d), respectively. In the 8.7 nm thick \flm, the\ndomains elongate along the [11 \u00162] direction, likely due to\nthe presence of in-plane uniaxial anisotropy. In the 3.7\nnm thick \flm, the domains are more irregular and do\nnot present such a strong preferred orientation, consis-\ntent with the reduced demagnetizing \feld and smaller in-\nplane anisotropy reported for this thickness [see Fig. 5 (e)\nand 6 (f), respectively].\nB. Domain walls\nXPEEM images provide su\u000ecient contrast to analyze\nthe domain walls of \flms with tYIG\u001512:4 nm, which\nconsist of straight segments along the zigzag boundary\nshown in Fig. 7 (b). Figures 8 (a-c) show details of the\n180\u000edomain walls that delimit the edges of the zigzag\ndomains in 86.7, 28.5 and 12.4 nm thick YIG. Linecuts\nof the XMCD intensity across the walls are shown in\nFigs. 8 (d-f). As the XMCD intensity scales with the\nprojection of the magnetization onto the x-ray incidence\ndirection (red arrows), the \"overshoot\" of the XMCD in-\ntensity in the central wall region compared to the sideregions in Figs. 8 (d,e) indicates that the magnetization\nrotates in the plane of the \flms. This observation is\nconsistent with the N\u0013 eel wall con\fguration expected for\nthin \flms with in-plane magnetic anisotropy64, and con-\ntrasts with the Bloch walls reported at the surface of\nbulk YIG crystals56,60. The walls appear to have a core\nregion, corresponding to the length over which the mag-\nnetic contrast changes most abruptly, which is about 1-\n2\u0016m wide, and a tail that extends over several \u0016m, which\nis a typical feature of N\u0013 eel walls in soft \flms with in-plane\nmagnetization64.\nThe domain wall in Fig. 8 (b) shows an inversion of the\nmagnetic contrast at a vortex-like singular point in the\nmiddle of the image. Such a contrast inversion reveals\nthat the wall is composed by di\u000berent segments with op-\nposite rotation mode of the magnetization, i.e., opposite\nchirality. The singular point that separates two consecu-\ntive segments is a so-called Bloch point, which can extend\nfrom the top to the bottom of the \flm, forming a Bloch\nline. Such features have smaller dimensions compared\nto the wall width and are therefore considered to favor\nthe pinning of domain walls at defects, thus acting as a\nsource of coercivity in soft magnetic materials64.\nFigures 8 (g) and (h) further show that the domain\nwalls at the apices of the zigzag domains in the 86.7 and\n28.5 nm thick \flms have a curved shape. This feature\nsuggests that the domains are indeed pinned at defect\nsites, likely of structural origin57,59. Finally, we note that\nthe domain features reported here move under the in\ru-\nence of an external in-plane magnetic \feld of the order of\nfew mT, as well as by ramping the out-of-plane \feld that\ncompensates the magnetic \feld of the XPEEM lenses at\nthe sample spot.9\nVI. CONCLUSIONS\nOur study shows that the magnetic properties and do-\nmain con\fguration of epitaxial YIG(111) \flms grown by\nPLD on GGG and capped by Pt depend strongly on the\nthickness of the YIG layer. Despite the high structural\nand interface quality indicated by XRD and TEM, the\nsaturation magnetization decreases from Ms= 122 kA/m\n(15% below bulk value) in 90 nm-thick YIG to 22 kA/m\nin 3.4 nm-thick YIG. The gradual decrease of Mssug-\ngests a continuous degradation of Msrather than the\nformation of a dead layer. All \flms except the thinner\none (tYIG= 3:4 nm) present a rather strong easy-plane\nanisotropy in addition to the shape anisotropy, of the or-\nder of 103to 104J/m3. Additionally, all \flms except the\nthinner one present weak in-plane uniaxial anisotropy,\nof the order of 3 \u000010 J/m3. This anisotropy de\fnes\na preferential orientation of the magnetization in each\nsample, which, however, is found to vary not only as\na function of thickness but also between samples with\nthe same nominal thickness and even for samples pat-\nterned on the same substrate. The origin of this vari-\nation is tentatively attributed to local inhomogeneities\nof the growth or patterning process, which could lead to\nsmall strain di\u000berences not detectable by XRD. Besides\nthese \fndings, we underline the fact that SMR measure-\nments performed for external magnetic \felds smaller or\ncomparable to the e\u000bective anisotropy \felds allow for the\naccurate characterization of both the magnitude and di-\nrection of the magnetic anisotropy of YIG/Pt bilayers,\nwith an accuracy better than 10 mT.YIG \flms with tYIG= 90\u000010 nm present large, mm\nsize, in-plane domains delimited by zigzag boundaries\nand N\u0013 eel domain walls. The apices of the zigzags are\npinned by defects, whereas the straight sections of the\nwalls incorporate Bloch-like singularities, which separate\nregions of the walls with opposite magnetization chirality.\nThe domain morphology changes abruptly in the thinner\n\flms. Whereas the 8.7 nm thick YIG presents elongated\ndomains that are tens of mm long, the domains in the\n3.7 nm thick YIG are smaller and more irregular, con-\nsistent with the reduction of the easy-plane and uniaxial\nanisotropy reported for this sample. Our measurements\nindicate that the performance of YIG-based spintronic\ndevices may be strongly in\ruenced by the thickness as\nwell as by local variations of the YIG magnetic proper-\nties.\nACKNOWLEDGMENTS\nWe thank Rolf Allenspach for valuable discussions.\nFurthermore, we acknowledge funding by the Swiss Na-\ntional Science Foundation under Grant no. 200020-\n172775. Jaianth Vijayakumar is supported by the Swiss\nNational Science Foundation through Grant no. 200021-\n153540. David Bracher is supported by the Swiss\nNanoscience Institute (Grant no. P1502). Part of this\nwork was performed at the SIM beamline of the Swiss\nLight Source, Paul Scherrer Institut, Villigen, Switzer-\nland.\n1K.-i. Uchida, H. Adachi, T. Ota, H. Nakayama,\nS. Maekawa, and E. Saitoh, Applied Physics Letters 97,\n172505 (2010).\n2M. Schreier, N. Roschewsky, E. Dobler, S. Meyer,\nH. Huebl, R. Gross, and S. T. Goennenwein, Applied\nPhysics Letters 103, 242404 (2013).\n3W. Wang, S. Wang, L. Zou, J. Cai, Z. 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Bauer3,4, and Jiang Xiao ( 萧江)5,6†\n1Department of Physics, The University of Hong Kong, Hong Kon g, China\n2Center of Theoretical and Computational Physics, Univ. of H ong Kong, Hong Kong, China\n3Kavli Institute of NanoScience, Delft University of Techno logy, Delft, The Netherlands\n4Institute for Materials Research and WPI-AIMR, Tohoku Univ ersity, Sendai, Japan\n5Department of Physics and State Key Laboratory of Surface Ph ysics, Fudan University, Shanghai, China\n6Center for Spintronic Devices and Applications, Fudan Univ ersity, Shanghai, China\n(Dated: July 26, 2018)\nWe develop a self-consistent theory for current-induced sp in wave excitations in normal\nmetal|magnetic insulator bilayer structures. We compute the spin wave dispersion and dissipa-\ntion, including dipolar and exchange interactions in the ma gnet, the spin diffusion in the normal\nmetal, as well as the surface anisotropy, spin-transfer tor que, and spin pumping at the interface.\nWe find that: 1) the spin transfer torque and spin pumping affec t the surface modes more than\nthe bulk modes; 2) spin pumping inhibits high frequency spin -wave modes, thereby red-shifting the\nexcitation spectrum; 3) easy-axis surface anisotropy indu ces a new type of surface spin wave, which\nreduces the excitation threshold current and greatly enhan ces the excitation power. We propose\nthat the magnetic insulator surface can be engineered to cre ate spin wave circuits utilizing surface\nspin waves as information carrier.\nI. INTRODUCTION\nThe rapid development of nanoscale science and tech-\nnology has opened the way for the new interdisciplinary\nresearchfieldknownasmagnonics. Magnonicdevicesuti-\nlize propagating spin waves instead of particle currents\nto transmit and process information in periodically pat-\nterned magnetic nanostructures, such as domain walls,\nmagnetic vortices and antivortices, magnetic nanocon-\ntactsetc.Magnonic devices potentially combine the ad-\nvantages of fast speed, easy and wideband tunability,\nandcompactnesswith compatibilitywith complementary\nmetal-oxide-semiconductor process.1\nA complete magnonic circuit consists of a spin wave\ninjector, a spin wave detector, and a functional medium\nthrough which the spin waves propagate and may be\nmanipulated. Due to their exceptionally low magnetic\ndamping, electrically insulating ferro or ferrimagnets are\nbelieved to be suitable for spin wave transmission line.2,3\nSpin waves can propagate much larger distances in mag-\nnetic insulator compared to both spin wave and particle-\nbased spin currentsin ferromagneticmetals. Arecent ex-\nperiment has shown that spin Hall spin currents in a nor-\nmal metal can effectively excite a wide rangeof spin wave\nmodes by the spin transfer torque in magnetic insulator\nthat is in contact with a normal metal with strong spin-\norbit coupling.4The spin wavedetection is made possible\nthrough the spin pumping and inverse spin Hall effect.5\nThe magnetic insulator functions as the spin wave trans-\nmission medium, inside which different modes of spin\nwaves can propagate. In addition to the conventional\nbulk/volume modes, a new type of surface spin wave\nmode due to easy-axis surface anisotropy (EASA) have\nbeen recently predicted6and confirmed.7The EASA sur-\nfacewavesdifferinnaturefromthemagnetostaticsurfacewaves (MSW) mode described by the Damon-Eshbach\ntheory. Because EASA surface waves are strongly local-\nized at the surface, they are strongly susceptible to the\neffects of spin transfer torques (STT) and spin pumping\n(SP), but only weakly absorb microwaves. Da Silva et al.\nindeed observed such behavior in a recent experiment.7\nIn our early study of spin wave excitation in the\nPt|YIG system,6,8we were mainly concerned with the\nmagnetization dynamics, disregarding the details of spin\ntransport in the normal metal and SP. SP affects surface\nmodes more strongly than bulk modes. In a recent theo-\nreticalstudy, itwasshownthatSPenhancesthedamping\nofYIG surfacemodesmorethan that ofthe bulk modes.9\nDue to spin-transfer torque and spin pumping, the spin\ntransport in the metal and the magnetization dynamics\nare coupled. So far, all studies have been focusing on one\nside of the story assuming the other side to be granted.\nThe spin current in the metal has been assumed to be\nfixed in order to study the magnetization dynamics in\nmagnetic insulators.6,8,9The spin transport in the metal\nwas studied in detail for a static magnetization of the\ninsulator.15In this paper, we present a complete theory\nin which the spin transport and magnetization dynamics\nare treated on equal footing.\nThis paper is organized as follows. In Section II, we\npresent the full theory of current-induced spin wave exci-\ntation in Pt |YIG system. Section III and IV are devoted\nto the analytical and numerical results for the spin wave\ndispersionand dissipation, as wellastheir dependence on\nvariousmaterialparametersincluding surfaceanisotropy,\nspin transfer torque and spin pumping etc.We conclude\nin Section V with a summary of the major results and\nreflect on the potential technological applications.2\nYIG\nm(r,t)⊗JcPt\nJsttJsp\nzx\n0\n−ddN\nFIG. 1. (Color online)Anelectrically insulatingmagnetic film\nof thickness dwith magnetization m(r,t) (/bardblˆzat equilibrium)\nin contact with a normal metal of thickness dN, with transla-\ntional symmetry in the y-zplane. A spin current Jspolarized\nalongˆzis generated in the normal metal due to the spin Hall\neffect from the applied charge current Jcand absorbed by the\nferromagnet. Jspis the SP current due to the magnetization\ndynamics at the interface.\nII. THEORY\nInthissection, wepresentourtheoryforthespintrans-\nport and spin wave excitation in a normal metal (N) -\nferromagnetic insulator (FI) bilayer structure as shown\nin Fig. 1, in which the FI is in-plane magnetized with the\nequilibrium magnetization along the ˆz-direction.\nA. Spin transport in normal metal\nWe assume an electric field E=Eyˆyapplied in N\nalongˆy.Jc=σE=Jcˆythe charge current, with σthe\nelectric conductivity of N. Due to the spin Hall effect,\na spin current polarized along ˆzflows in −ˆxdirection:\nJsH=θHJcˆzwithθHthe spin Hall angle of N. This\nspin Hall current induces a spin accumulation µ(x) in N,\nwhich satisfies the spin-diffusion equation\n∇2µ(x) =µ(x)\nλ2, (1)\nwhereλis the spin-flip length in N. The spin current\ninside N is the sum of the spin diffusion current and the\nspin Hall current\nJs(x) =−σ\n2e∂µ(x)\n∂x−θHJcˆz. (2)\nSpin-conserving boundary conditions require that Js(x)\nis continuous at the interfaces x= 0 andx=dN. Thus,\nJs(dN) = 0,Js(0) =Js0. (3)\nJs0is the spincurrentflowingthroughthe N |FI interface,\nwhich includes the STT current Jsttgenerated by thespin accumulation in N on the magnetization in FI and\nthe SP current Jspfrom FI to N:\nJs0=Jstt+Jsp\n=e\nhgr{m(0)×[m(0)×µ(0)]−/planckover2pi1m(0)×˙m(0)},(4)\nwithgrthe real part of the mixing conductance per area\nfor the N |FI interface. In Eq. (4), mandµtake the\nvalue at the interface ( x= 0). The imaginary part of the\nmixing conductance is disregarded in the following.\nThesolutionfor µ(x)satisfyingthespindiffusionequa-\ntion Eq. (1) and boundary condition Eq. (3) is given by\nµ(x) =2eλ\nσ(JsH+Js0)coshdN−x\nλ−JsHcoshx\nλ\nsinhdN\nλ.(5)\nBy plugging the above expression into the second equa-\ntion of Eq. (3), we find the interfacial value of µ(0) and\nthusJs0:\nJs0=e\nhg′\nr/bracketleftbig\nm(0)×(m(0)×µ0\ns)−/planckover2pi1m(0)×˙m(0)/bracketrightbig\n,(6)\nwhereµ0\ns= (2eλ/σ)θHJctanh(dN/2λ)ˆzis the spin ac-\ncumulation at the interface due to the spin Hall current\nalone, and\ng′\nr=gr\n1+2λe2\nhσgrcothdN\nλ(7)\nis the renormalized mixing conductance taking into ac-\ncount the effect of diffusive spin current back-flowin N.10\nThe interfacialspin current Js0exerts the STT and SP\ntorques on m:\nτstt=g′\nreλθHJc\n2πσtanhdN\n2λm×(m׈z)δ(x)\n≡τsttm×(m׈z)δ(x), (8a)\nτsp=−/planckover2pi1\n4πg′\nrm×˙mδ(x)≡ −τsp\nω0m×˙mδ(x).(8b)\nFig. 2 shows the dependence of the pre-factors of these\ntwo torques on the film thickness dNand spin diffusion\nlengthλ. In theleft panelofFig.2, weseethatforafixed\nfilm thickness dN, the STT depends non-monotonically\nonλand has a maximum value for an intermediate value\n(indicated by the dashed line). The reason for this is\nthe following: when λ→0, the spin Hall current cannot\nbuild up any spin accumulation, thus there can be no\nSTT; when, on the other hand, λ→ ∞, Eq. (1) is solved\nbyµ(x) =ax+b, which means Js(x) = const. However,\nat the top surface Js(dN) = 0, therefore the spin current\nhas to vanish everywhere. Both JsttandJspvanishes,\nbecause the above argument is valid for both ˙m= 0 and\n˙m∝ne}ationslash= 0. For the SP, the rightpanel ofFig. 2, the behavior\nis easy to understand. For λ→0, the SP is maximal\nbecauseNbecomesanidealspinsink. As λ→ ∞, thereis\nno spin flip mechanism in N, so the pumped spin current\naccumulates in N and causes a back flow spin current,\nwhich cancels the pumped spin current.3\n0102030405001020304050\nPt filmthickness: dN/LParen1nm/RParen1spin/Minusfliplength:Λ/LParen1nm/RParen1Τstt\n02.50/MuΛΤipΛy10/Minus75.00/MuΛΤipΛy10/Minus77.50/MuΛΤipΛy10/Minus71.00/MuΛΤipΛy10/Minus61.25/MuΛΤipΛy10/Minus6\n0102030405001020304050\nPt filmthickness: dN/LParen1nm/RParen1spin/Minusfliplength:Λ/LParen1nm/RParen1Τsp\n02.50/MuΛΤipΛy10/Minus85.00/MuΛΤipΛy10/Minus87.50/MuΛΤipΛy10/Minus81.00/MuΛΤipΛy10/Minus71.25/MuΛΤipΛy10/Minus7\nFIG. 2. (Color online) The contour plot of τstt(atJc=\n1011A/m2, left) and τsp(right) in Eq. (8) vs. film thickness\ndNand spin diffusion length λfor parameters given in Table I\nandgr= 1018/m2. The dashed curve on the left panel shows\nthe maximum of τsttfor fixed film thickness dN.\nB. Spin wave excitation in magnetic insulators\nThe spatially dependent dynamics of the magnetiza-\ntion unit vector m(r,t) is described by the Landau-\nLifshitz-Gilbert-Slonczewski (LLGS) equation17–19:\n˙m=−γm×Heff+αm×˙m+γ\nMs(τstt+τsp),(9)\nwhere the effective field Heff=H0+Hs+Aex\nγ∇2m+h\nincludes the external magnetic field H0, the surface\nanisotropy field Hs=2K1\nMs(m·n)n, the exchange field\nHex=Aex\nγ∇2m, and the dipolar magnetic field hdue\ntom(r,t). Here nis the outward normal as seen from\nthe ferromagnet which can be the easy or hard axis, de-\npending on the sign of the anisotropy constant K1.Aex\nandαare the exchange and Gilbert damping constants,\nrespectively.\nWe include the SP in our model thereby extending our\nearlierstudies of spin-waveexcitation in magnetic insula-\ntors by the STT.6The spin-conservation boundary con-\nditions for matx= 0 and −d:20\natx= 0 :m×∂m\n∂n−ks(m·n)m×n (10a)\n+kjm×(m׈z)+kp\nω0ˆz×˙m= 0,\natx=−d:m×∂m\n∂n= 0, (10b)\nwith∂m/∂n≡(n·∇)mandKs=/integraltext0+\n0−K1dx. We convert\nsurface anisotropy, spin current, and SP parameters into\neffective wave numbers by defining:\nks=2γKs\nAexMs, kj=γτstt\nAexMs, kp=γτsp\nAexMs.(11)Param. YIG UnitParam. Pt Unit\nMsa1.56×105A/m σe1.16×106A/Vm\nαa6.7×10−5- λe2 nm\ngrb1016∼10191/m2θH0.08 -\nKsc10−4J/m2\nAexd8.97×10−6m2/s\nγ 1.76×10111/(Ts)\nω0=γH0d17.25 GHz\nωM=γµ0Msd34.5 GHz\nd 0.61 µm dN10 nm\nTABLE I. Parameters for YIG.aRef. 4,bRef. 4, 11, and 12,\ncKs= 0.01∼0.1 erg/cm2or 10−5∼10−4J/m2, Ref. 13 and\n14,dRef. 21,eRef. 15,fRef. 4.\nCompared to our previous work,6we now establish\nthe relation between spin wave vector kjand the ex-\nperimentally controlled parameter, i.e.the charge cur-\nrent density. For example, the bulk excitation threshold\nkc=α(ω0+ωM/2)d/Aexcorrespondsto a chargecurrent\nof 6.6×1011A/m2atgr= 5.9×1017/µm.\nThe bulk magnetization inside the film ( −d < x < 0)\nsatisfies the LLG equation:\n˙m=−γm×/bracketleftbigg\nH0+Aex\nγ∇2m+h/bracketrightbigg\n+αm×˙m,(12)\nwhere the dipolar magnetic field h(r,t) obeys Maxwell’s\nequations in the quasi-static approximation:\neverywhere: 0 = ∇×h(r), (13a)\n−d≤x≤0 : 0 =∇·[h(r)+µ0Msm(r)],(13b)\nx<−dorx>0 : 0 =∇·h(r), (13c)\nwith boundary conditions\nhy,z(0−) =hy,z(0+),bx(0−) =bx(0+),(14a)\nhy,z(−d−) =hy,z(−d+),bx(−d−) =bx(−d+).(14b)\nEqs. (10 – 14) completely describe what is called dipolar-\nexchange spin waves. The method described above ex-\ntends De Wames and Wolfram’s21and Hillebrands’22by\nincluding the current-induced STT and SP.\nBecause of the translational symmetry in the lateral\ndirection, we may assume that the scalar potential is the\nplane wave:\nψ(x,y,z,t) =/summationdisplay3\nj=1/bracketleftBig\najeiq(j)\nxx+bje−iq(j)\nx(x+d)/bracketrightBig\ne−iq·seiωt\n(15)\nwheres= (y,z) is the in-plane position and q=\n(qy,qz) =q(sinθ,cosθ) withq=|q|an in-plane wave\nvector and θthe angle between the wave vector qand\nthe magnetization equilibrium ˆz.aj,bjare six coeffi-\ncients to be determined by the six boundary conditions\nin Eqs. (10, 14), which can be transformed into a set of\nlinear equations:\nM(q,ω)/parenleftBigg\naj\nbj/parenrightBigg\n= 0, (16)4\nwhereM(q,ω)isa6×6matrixdependingonthematerial\nparameters and injected spin current: ω0,α,ks,kj. The\ndipolar-exchange spin wave dispersion is determined by\nthe condition that the determinant of the coefficient ma-\ntrix vanishes: |M(q,ω)|= 0⇒ω(q). The corresponding\nsolution of Eq. (16) for aj,bjgives the spin wave ampli-\ntude profile accordingto Eq. (15), from which we also see\nthat the spin wave is amplified when\nIm[ω(q)]<0, (17)\nwhich is used as criterium for spin wave excitation with\nwave vector q.\nIII. ANALYTICAL RESULTS\nThe inclusion of the dipolar fields complicates the\nproblem significantly. Nevertheless, it is still possible to\nobtain approximate analytical expressions of the com-\nplex dispersion relation ω(q) for the dipolar-exchange\nspin waves for the few special cases: 1) the bulk modes\nforθ=π/2; 2) the magnetostatic surface wave for\nθ=π/2; 3) the surface spin wave mode induced by easy-\naxis surface anisotropy (EASA) at zero wave-length limit\nofq= 0. While the real part has been studied quite well\nbefore, the imaginary part characterizing the dispersion\nand excitation of spin waves is usually disregarded and\nfocus of the present study. All analytical expressions in\nthis section are obtained by expanding the relevant ma-\ntrixM(q,ω) to leading order in: α,ks,kj, andkp.\nA. Bulk modes for θ=π/2\nAssuming weak surface anisotropy ( Aexk2\ns≪2ω0+ωM)\nand long wavelength limits, the complex eigen-frequency\nfor thenth bulk mode reads\nωn=/radicalBig\nωnq(ωnq+ωM) (18)\n−Aexks\ndωnq\nReωn/parenleftBigg\n1−ωnq+ωM\n2ωnq+ωM/radicalBigg\nAexk2s\nωnq+ω0+ωM/parenrightBigg−1\n+i/bracketleftbigg/parenleftbigg\nα+2Aexkp\nω0d/parenrightbigg/parenleftBig\nωnq+ωM\n2/parenrightBig\n−αAexks\nd+2Aexkj\nd/bracketrightbigg\nwithωnq=ω0+Aex[q2+ (nπ/d)2] andn= 1,2,....\nReωn, the real part ofthe eigenfrequency, decreaseswith\nincreasing surface anisotropy ks. Imωngives the in-\nformation about the dissipation (or damping), which in-\ncludesthecontributionsfromGilbertdamping( αterms),\nspin current injection ( kjterm), and SP ( kpterm). For\nexample, the 2 Aexk2\np/ω0dis the enhanced damping due\nto SP effect and the 2 Aexkj/dis the effect of STT. As ex-\npected, both terms are inversely proportional to the filmthicknessdbecause both STT and SP are interfacial ef-\nfect. The spin wave excitation condition Im ωn<0 leads\nto the threshold current for exciting the bulk modes for\nθ=π/2.\nB. Magnetostatic surface wave for θ=π/2\nMagnetostatic surface wave (MSW) is a dipolar spin\nwave mode that exists for qd/lessorsimilar1 atθ=±π/2. The\ncomplex eigen frequency for MSW at θ=π/2 is\nωMSW=/radicalbigg/parenleftBig\nω0+ωM\n2/parenrightBig2\n−ω2\nM\n4e−2qd\n+i/bracketleftbigg/parenleftbigg\nα+Aexkp\nω0d/parenrightbigg/parenleftBig\nω0+ωM\n2/parenrightBig\n+Aexkj\nd/bracketrightbigg\n.(19)\nComparing Eq. (19) for the MSW and Eq. (18) for bulk\nmodes, the effect of STT and SP on the former is half of\nthat on bulk modes. It is because the MSW magnetiza-\ntion forqd/lessorsimilar1 has almost constant amplitude over the\nthickness ( i.e.a surface wave with long decay length, see\nthe thick purple curvein Fig. 4(b) below), while the mag-\nnetization for bulk modes oscillates as a cosine function\n(see the thin curves in Fig. 4(b) below). The total mag-\nnetization of MSW ( ∝dforqd/lessorsimilar1) is therefore twice as\nlargeasthetotalmagnetizationofthebulkmodes( ∝d/2\nbecause of the average of a cosine function is 1/2), which\nreduces the effect of the STT and SP by one half. As\nbefore, the threshold current for exciting the magneto-\nstatic surface wave can be derived using the spin wave\nexcitation condition Im ωMSW<0 forθ=π/2.\nC. EASA induced surface spin wave mode at q= 0\nIn Ref. 6, the EASA wasfound to induce a new type of\nsurfacespinwavemode, whosepenetrationdepth dsisin-\nversely proportional to the strength of EASA: ds∝1/ks.\nIn order to understand this EASA surface wave better,\nwe study the limit d→ ∞,i.e.the magnetic film is semi-\ninfinite and bj= 0 in Eq. (15). Focusing for simplicity\non vanishing in-plane wave-vector q= (qy,qz) = 0, the\nscalar potential can be written as:\nψ(r) =/summationdisplay2\nj=1ajeiqjxeiωt(20)\nwhere\nqj(ω) =−i/radicaltp/radicalvertex/radicalvertex/radicalbtω0+1\n2ωM±/radicalBig\nω2+1\n4ω2\nM±iαω\nAex(21)\nare negatively imaginary with |q1|≫|q2|. Imposing the\nboundary conditions from Eq. (10) at x= 0,|M(q,ω)|=\n0 leads to (up to the first order in kj):5\n0 = 2q1q2(q1+q2)+iks/bracketleftbigg\n(q1+q2)2+ωM\nAex/bracketrightbigg\n+4kjω\nAex−2kpω\nω0(q1+q2)(q1+q2+iks), (22)\nwhose solution is the complex eigenfrequencies ωSfor the EASA surface wave. By expanding Eq. (22) up to the\nleading orders in α,kj,kp, and assuming Aexk2\ns≪2ω0+ωM, we have:\nωS=/radicalbig\nω0(ω0+ωM)\n+i/parenleftBig\nω0+ωM\n2/parenrightBig/bracketleftBigg\nα+4Aexkskjω0\n(2ω0+ωM)2/parenleftBigg\n1+ks/radicalBigg\nAex(ω0+2ωM)2\n(2ω0+ωM)3/parenrightBigg\n+2Aexkskp\n2ω0+ωM/parenleftBigg\n1+ks/radicalBigg\nAex(ω0+ωM)2\n(2ω0+ωM)3/parenrightBigg/bracketrightBigg\n.(23)\nImωS<0 leads to:\nJth=−σcothdN\n2λ\n2θHλe\nαπAexMs\ng′rγ/parenleftbigg(2ω0+ωM)2\nksAexω0−ω0+2ωM\nω0/radicalbigg\n2ω0+ωM\nAex/parenrightbigg\n+/planckover2pi1\nω0+ωM\n2−ks\n2/radicalBigg\nAexω2\nM\n2ω0+ωM\n\n.\n(24)\nThe first term of Eq. (24) gives the threshold current\nthat compensates the Gilbert damping αfor the EASA\nsurface wave of penetration depth ds∝1/ks(from the\nfirst term in the first square bracket). The second term\nof Eq. (24) compensates the SP enhanced damping.\nSinceJthin Eq. (24) is the threshold current for EASA\nsurface wave at q= 0, so it actually provides a upper\nbound for the overall threshold current for the spin wave\nexcitation. However, the excitation threshold current for\nthe EASA surface wave is well below that of other spin\nwave modes in many cases ( i.e.for not too small ks),\nJthin Eq. (24) is the overall threshold current for spin\nwave excitation in a Pt |YIG bilayer. Fig. 3 shows this\nthreshold current as a function of mixing conductance\ngr. Whengris not too large (such that g′\nr≃gr), the\nthreshold current approximately decreases linearly with\ngr:Jth∝1/gr, becausethe STTapproximatelyincreases\nlinearlywith gr(seethelinearpartofleftpanelinFig.3).\nHowever, when gris large,g′\nr≃1, thenJthis indepen-\ndent ofgr, andJthreaches its lower bound (see the flat\npart of the left panel in Fig. 3). Overall, we expect Jth\ngiven by Eq. (24) to work well as the overall threshold\ncurrentforintermediate ks. It doesnotworkforsmall ks,\nbecause the penetration depth of EASA surface wave is\ntoolong, and the othermodesactuallyhavelowerthresh-\nold current. For larger ks, Eq. (24) simply does not work\nbecause it is derived assuming small ks.\nWe may also calculate the spin wave profile for the\nEASA surface wave. Using Eq. (23)\nq1=−i/radicalbigg\n2ω0+ωM\nAex(25a)\nq2=−iω0ks\n2ω0+ωM/parenleftBigg\n1+ks/radicalBigg\nAex(ω0+ωM)2\n(2ω0+ωM)3/parenrightBigg\n.(25b)Sinceq1,2areboth negativeimaginary, the corresponding\nspin wavesin Eq. (20) are localized near the surface. The\nspinwaveprofile(the xcomponent)fortheEASAsurface\nwave for a semi-infinite film is approximately given by:\nmx(x) =(q1+iks)eiq2x−(q2+iks)eiq1x\nq1−q2.(26)\nSince|q1| ≫ |q2|, the penetration depth is mostly deter-\nmined byq2:ds∝1/iq2∝1/ksfor smallks. The spin\nwave profile in Eq. (26) is compared with the numerical\ncalculation in the left panel of Fig. 3. The agreement is\nquite good except for locations near the bottom surface\n(x/d→1) because Eq. (26) is calculated for semi-infinite\nfilms, while the numerical data are computed for a thin\n/SolidCircle\n/SolidCircle/SolidCircle/SolidCircle/SolidSquare\n/SolidSquare\n/SolidSquare/SolidSquare/SolidCircleks/EquaΛ25/Slash1Μm\n/SolidSquareks/EquaΛ0\n101510161017101810191020101110121013\ngr/LParen11/Slash1m2/RParen1/MinusJth/LParen1A/Slash1m2/RParen1/MinusJthatdN/EquaΛ10nm,Λ/EquaΛ2nm\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidSquare\n/SolidSquare\n/SolidSquare\n/SolidSquare\n/SolidSquare\n/SolidSquare\n/SolidSquare\n/SolidSquare\n/SolidSquare\n/SolidSquare\n/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/MedSolidDiamond\n/MedSolidDiamond\n/MedSolidDiamond\n/MedSolidDiamond\n/MedSolidDiamond\n/MedSolidDiamond\n/MedSolidDiamond\n/MedSolidDiamond\n/MedSolidDiamond\n/MedSolidDiamond\n/MedSolidDiamond\n/MedSolidDiamond\n/MedSolidDiamond\n/MedSolidDiamond\n/MedSolidDiamond\n/MedSolidDiamond/MedSolidDiamond\n/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/SolidCircleks/EquaΛ50/Slash1Μm\n/SolidSquareks/EquaΛ25/Slash1Μm\n/MedSolidDiamondks/EquaΛ15/Slash1Μm\n0 0.5 100.51\n/Minusx/Slash1dmx/LParen1x/RParen1EASAsurfacewave /LParen1q/EquaΛ0/RParen1\nFIG. 3. (Color online) Left: Jthin Eq. (24) vs. the mixing\nconductance gr(log-log scale) for ks= 25/µm withdN= 10\nnm and λ= 2 nm. The dots are the threshold current ob-\ntained from numerical calculations below for ks= 25/µm and\nks= 0. Right: The magnetization profiles for the EASA sur-\nface wave for various ksvalues. The solid curves are plotted\nusing Eq. (26) for a semi-infinite film. The dots are obtained\nby numerical calculations for d= 0.61µm.6mx (qd = 0.09)\n00.5 1\n−x/dmx (qd = 3.78)\n10−21000.911.1\nqdRe(ω/ωM)(a) (b)ks = 0\n(c)\nFIG. 4. (Color online) Spin wave dispersion (left) and profil es\n(right) in the absence of surface anisotropy ( ks= 0) at θ=\nπ/2 (orq⊥m). Left: spin wave dispersion (a), the solid lines\n(different colors denote different bands) are calculated fro m\nthe numerical solution of Eq. (16), and the dashed lines are\nplottedusingtheanalytical expressions givenbythereal p arts\nof Eqs. (18, 19). Right: spin wave profiles ( mxcomponent)\natqd= 0.09 (b) and qd= 3.78 (c). The colors in (b, c) match\nthat in (a). The thick purple/green mode in (b)/(c) is for the\npoint enclosed with circle in (a) on the purple/green band.\nfilm of finite thickness d= 0.61µm. The deviation at\nx/d→1 reflects the bottom surface (at x=−d) in-\nfluence on the EASA surface wave localized at the top\nsurface atx= 0. Not surprisingly, the effect of the bot-\ntom surface is more obvious for the EASA surface wave\nthat is less confined (smaller ks).\nIV. NUMERICAL RESULTS\nIn this Section, we discuss the effects of the STT and\nSP on the spin wave excitation. Because of their inter-\nfacial character, both STT and SP are more effective for\nsurface spin wave modes. In the absence of STT, the\nsurface spin wave modes have larger damping compared\nto the bulk modes. When an STT is applied, the surface\nspin wave modes are easier to excite as well.\nWe show the numerical results on the spin wave dis-\npersion as well as the spin wave profiles with different\ntypes of surface anisotropy, followed by the correspond-\ning spin wave dissipation affected by the STT and SP.\nThe spin wave excitation power spectrum discussed at\nthe end shows a dramatic effect of EASA and the associ-10−21000.911.1\nqdRe(ω/ωM)\nmx (qd = 0.09)\n00.5 1\n−x/dmx (qd = 3.78)(a) (b)ks = 25/µm\n(c)\nFIG. 5. (Color online) Same as Fig. 4 but with easy-axis\nsurface anisotropy ( ks= 25/µm). Left: spin wave dispersion\n(a), the solid lines are calculated from the numerical solut ion\nof Eq. (16), while the dashed lines and the ◮symbol are\nplottedusingtheanalytical expressions givenbythereal p arts\nof Eqs. (18, 23). A new (black) band appears due to the\neasy-axis surface anisotropy. Right: spin wave profiles ( mx\ncomponent) at qd= 0.09 (b) and qd= 3.78 (c). The colors in\n(b, c) match that in (a). The thick black/red mode in (b)/(c)\nare for the points enclosed by a circle in (a) on the black/red\nband.\nated surface wave. If not stated otherwise, the numerical\nresults in this section are calculated for an in-plane mag-\nnetizedYIGthinfilmcappedwithPtaspicturedinFig.1\nwith geometry and material parameters given in Table I.\nA. Spin wave dispersion & profiles\nThespinwavedispersion, i.e.therealpartofthemode\nfrequency Re ω(q), is plotted in Fig. 4(a) for θ=π/2 (or\nq⊥m) when there is no surface anisotropy ( ks= 0).\nThe dispersion can be separated into the dipolar spin\nwave regime for qd/lessorsimilar1, where the dispersion relation is\nflat (forθ=π/2 only, non-flat for other angles), and the\nexchange spin wave regime for qd>1, where the disper-\nsion relation is approximately parabolic and increasing\nwithAexq2. In the dipolar regime ( qd/lessorsimilar1), there are\nmultiple flat bands (associate with different transverse\nmodes in the xdirection) and a magnetostatic surface\nwave(MSW) that crosseswith the lowest four flat bands.\nThese results are identical to our previous studies.21The\nspin wave profiles for the typical dipolar/exchange spin\nwaves (qd= 0.09/3.78) are shown in Fig. 4(b)/(c). For\nthe dipolar spin waves (Fig. 4(b)), the bulk modes (cor-7\n11.2α contribution\n−20STT contribution\n01SP contribution\n01α+STT+SPIm(ω/αωM)\n10−210011.2\nqd10−2100−50\nqd10−210002\nqd10−2100−20\nqdIm(ω/αωM)ks=0\nks=25/µm\nFIG. 6. (Color online) Spin wave dissipation at θ=π/2 (orq⊥m) withgr= 5.8×1017/m2(kp= 0.01/µm). Top row:\nno surface anisotropy ( ks= 0), bottom row: with easy-axis surface anisotropy ( ks= 25/µm). The 1st column is the total\ndissipation with current injection of Jc= 2.3×1011A/m2(kj= 0.35kc). The 2nd column to 4th column are the contributions\nfrom the Gilbert damping, STT, and SP, respectively. For all panels, the solid lines (different colors denote different ba nds) are\ncalculated from the numerical solution of Eq. (16), and the d ashed lines (and the ◮) are plotted using the analytical expressions\ngiven by the imaginary parts of Eqs. (18 – 23).\nresponding to the flat bands) are simply the standing\nwaves confined by the film thickness d. The MSW mode\n(thick purple curve in Fig. 4(b)) is a surface wave, but\nwithaverylongpenetrationdepth, whichmeansthatthe\nMSW mode for small qis actually more like a uniform\nmode rather than a surface mode.\nThe more interesting physics happens when including\nthe surface anisotropy ks, which can take either sign:\nks>0meansthatthesurfacespinstendtoalignwiththe\nsurface normal and is called easy-axis surface anisotropy\n(EASA), while ks<0 means that the surface spins tend\nto lie in the plane of the surface and is called hard-\naxis surface anisotropy (HASA). One effect of the sur-\nface anisotropy is to shift the bulk band frequencies as\nindicated by Eq. (18): the positive/negative ksshift the\nfrequencies downwards/upwards. For EASA ( ks>0), as\ndiscussed in our previous study,6a new type of surface\nspin wavemode (the lowest thick blackband in Fig. 5(a))\nappears. The magnetization profile for this EASA sur-\nface wave at qd= 0.09 (the mode indicated by the circle\non the thick black band in Fig. 5(a)) is plotted as the\nthick black curve in Fig. 5(b), which shows its surface\nfeature. The penetration depth dsof the EASA sur-\nface wave is inversely proportional to the strength of the\nEASA:ds∝1/ks.6B. Spin wave dissipation\nThe STT and SP mainly affect the dissipation of spin\nwavesi.e.the imaginarypart of the mode frequency, and\nleave the spin wave dispersion and profiles discussed in\nthe previous section practically unchanged.\nThe spin wave dissipation, Im ω, is plotted in the 1st\ncolumn ofFig. 6for the two casesofsurfaceanisotropyas\nthose in Fig. 4 and Fig. 5: ks= 0 (top) and ks= 25/µm\n(bottom). In both plots, STT due to current injection\nJc= 2.3×1011A/m2andSPareincluded. Theinterfacial\nmixing conductance value is taken as gr= 5.8×1017/m2.\nIn linear response regime, different mechanisms for the\nspin wave dissipation are additive. As indicated by the\nanalytical results Eqs. (18 – 23) in Section III, there are\nthree different contributions to the dissipative imaginary\npartImω: theGilbertdamping( αterm), STT( kjterm),\nand SP (kpterm). We plot these contributions to Im ω\nseparately in the 2nd-4th column in Fig. 6. The 2nd\ncolumn, the Gilbert damping contribution, is equivalent\nto the dissipation for a YIG film without Pt capping\nlayer (thus no STT or SP). The 3rd and 4th columns are\nthe contributions from STT and SP respectively, which\nshow very similar q-dependence in shape but with op-\nposite sign. Apart from an overall prefactor determined\nby the structure and material parameters ( τsttandτsp\nin Eq. (8)), the overall shape of STT and SP is deter-\nmined by the interfacial transversemagnetization m⊥(0)8\n(through the vectorial part of Eq. (8)), which is strongly\nmode dependent (or q-dependent). This common ingre-\ndient for STT and SP leads to their similarities in the\nq-dependence. The sign is governed by the polarity of\nthe charge current Jc.\nWhen surface anisotropy is absent ( ks= 0, top panels\nin Fig. 6), the green band reachesnegative dissipation for\nlargeq. This negativity is because the STT contribution\nreaches its (negative) maximum for the green mode at\nlargeq. SuchlargeSTTcontributionisduetoitslargein-\nterfacial magnetization m⊥(0) for the green mode, which\ncan be seen from its profile in the thick green curve in\nFig. 4(c). On the opposite, the m⊥(0) for the red mode\n(Fig. 4(c)) is small, therefore the STT has little effect on\nthe red mode at large q, this is why the STT contribu-\ntion for the red mode is close to zero for qd>1. The SP\ncontribution has the same feature as the STT because\nSP also depends on m⊥(0).\nFor the case with EASA ( ks= 25/µm, bottom panels\nin Fig. 6), the features of large/small STT/SP contribu-\ntions are due to the same reason as in the no surface\nanisotropy case that they all determined by the interfa-\ncialvalue m⊥(0) foraspecific mode. The maindifference\nbetween these two surface anisotropy cases is from the\nadditional EASA surface wave (the lowest thick black\nband in Fig. 5(a)). Because of its strong localization\nnearthe interface, STT and SP stronglyaffect this mode,\nand the STT/SP contribution for this mode (the black\ncurve in the bottom right two panels of Fig. 6) becomes\nlarger. For two typical modes indicated by circles on the\nblack/redbands, the largeSTT and SP contributions are\ncausedbytheirsurfacewavefeatures, asobservedintheir\nprofiles (thick black/red curves in Fig. 5(b)/(c)).\nOverall, STT and SP have a larger effect on surface\nwaves, such as the MSW (at larger q) and EASA surface\nwaves. Therefore, in the absence of an applied current,\nthe surface waves have larger damping due to larger SP\ncontribution. When a large enough charge current is ap-\nplied, theSTTcontributionovercomesthatoftheGilbert\ndamping and SP, and excites preferably surface waves.\nC. Power spectrum and threshold current\nSince there are multiple spin wave modes excited si-\nmultaneously by the STT, we study the frequency de-\npendence of the excitation power. Because the theory is\nbased on linear response, we can only predict the onset of\nthe excitation of a certain spin wave mode. Its tendency\nof being excited can be measured by the value of Im ω:\na more negative Im ωimplies more power. Therefore, we\ndefine an approximate power spectrum for the spin wave\nexcitation:\nP(ω) =/summationdisplay\nn/integraldisplay\nImωn<0|Imωn(q)|δ[ω−Reωn(q)]dq,\n(27)00.05P(ω) (a.u.)\n0.6 0.8 11.200.2P(ω) (a.u.)\nRe(ω/ωM)0.6 0.8 11.2\nRe(ω/ωM)(c)(a)\n(d)(b)\nks=25/µm\ngr=5.92×1017/m2ks=25/µm\ngr=2×1019/m2ks=0\ngr=2×1019/m2ks=0\ngr=5.92×1017/m2\nFIG. 7. (Color online) Power spectrum (resolution δω/ωM=\n0.01) for differentcombinations ofsurface anisotropyandmix -\ning conductance at ten current levels (increasing by δkj=\n0.01kc) above threshold current.\nwhich summarizes the information about the mode-\ndependent current-induced amplification as a sum over\nbands with band index n. Fig. 7 shows the power\nspectrum computed from Eq. (27) for different surface\nanisotropies and mixing conductances.\nLet us first inspect the effect of EASA. As seen in\nFig. 3(b) (the filled/empty dots are for with/without\nEASA), EASA reduces the threshold current by about\na factor of two. In addition, EASA also greatly enhances\nthe excitation power, as seen by the comparison between\nthe top and bottom panels in Fig. 7. The reason for this\neffect is the strong confinement of the EASA mode (see\nthick black profile in Fig. 5(b)) and correspondingly low\nthreshold current (given by Eq. (24)). Almost all EASA\nmodes in qphase space are excited simultaneously (see\nthe lower panels of Fig. 6). Easy excitation and the large\nexcitation phase space, lead to the large excitation power\nin the presence of EASA. In comparison, for ks= 0 the\nexcitation threshold current is higher and the modes that\ncan be excited occupy only a small area of phase space\n(only a small window of the green band can be excited\nas seen in Fig. 6).\nItisalsointerestingtocomparethepowerspectrumfor\ndifferent mixing conductances gr. Comparing Fig. 7(a -\nb) forks= 0 (or Fig. 7(c - d) for ks= 25/µm), we\nobserve that an increasing mixing conductance tends to\nshift the power spectrum to lower frequencies, or cause\na red shift. Both the STT and SP depend on (or are\nproportional to) the mixing conductance gr(see Eq. (8))\nand the interfacial value of the transverse magnetization\nm⊥(0), which dominates the q-dependence. The SP also\ndepends on the frequency ˙m(0) and is more effective for9\nthe high frequency modes, while the STT does not de-\npend explicitly on frequency. As a consequence, a large\nmixing conductance tends to suppress the excitation of\nhigh frequency modes, thereby causing a red shift of the\npower spectrum.\nV. DISCUSSIONS & CONCLUSIONS\nThe EASA induced surface wave mode for ks>0 has\nseveralpropertieswhichmakethis modesuperiorforspin\ninformation processing and transport: 1) it can be eas-\nily induced unintentionally or by engineering the surface\nanisotropy, 2) its penetration depth is controlled by the\nstrength of the surface anisotropy, 3) it can be excited by\nrelatively small currents, 4) it has a finite group veloc-\nity and can propagate long distances (in the absence of\nSP). The required surface anisotropy for this new surface\nmode is ubiquitous in magnets and sensitive to surface\ntreatments and overlayers, which can be used advanta-\ngeously, e.g.to decorate the magnetic insulator surface\nto create corridors or circuits which can accommodate\nthis surface wave mode and its propagation.\nWe find athresholdcurrentforspin waveexcitationfor\nPt|YIG structures to be in the range of 1010∼1011A/m2\nfortypicalparameters(spin Hallangle θH= 0.08, mixing\nconductance gr≃1018∼1019/m2). This value is higher\nthan the value predicted in Ref. 6, which assumesperfect\nspin current absorption at the interface and ignores the\nSP effect on the spin wave, while both tending to under-\nestimate the threshold current. The theoretical value is\nmuch higher than the experimental value for the thresh-\nold current of 109A/m2,4(even when accounting for theEASA surfacewave). Although there areuncertainties in\nthe value of surface anisotropy, spin Hall angle, spin-flip\nlength,etc.any/all of these cannot reconcile a discrep-\nancy between the experiment and the theory of almost\ntwo orders of magnitude.\nIn summary, we presented a self-consistent theory for\nthe current-induced magnetization dynamics in normal\nmetals|ferromagnetic insulators bilayer structure, includ-\ningtheeffectsofSTTandSPattheinterface.24Wefound\nthat 1) the mode dependence of the STT and SP scales\nidentically and surface wavesare more affected than bulk\nwaves, 2) the SP causes a red shift in the power spec-\ntrum, and 3) easy-axis surface anisotropy can induce\na new type of (EASA) surface wave mode, which typ-\nically has the lowest threshold current for excitation and\ncontributes most to the excitation power. We propose\nthat engineering the surface anisotropy and the EASA\nsurface waves might facilitate applications in low power\nspintronic-magnonic hybrid circuits.\nACKNOWLEDGEMENT\nWe acknowledgesupport from the University Research\nCommittee (Project No. 106053) of HKU, the Uni-\nversity Grant Council (AoE/P-04/08) of the govern-\nment of HKSAR, the National Natural Science Foun-\ndation of China (No. 11004036, No. 91121002), the\nMarie Curie ITN Spinicur, the Reimei program of the\nJapan Atomic Energy Agency, EU-ICT-7 ”MACALO”,\nthe ICC-IMR, DFG Priority Programme 1538 ”Spin-\nCaloric Transport”, and Grand-in-Aid for Scientific Re-\nsearch A (Kakenhi) 25247056..\n∗Corresponding author: yanzhou@hku.hk\n†Corresponding author: xiaojiang@fudan.edu.cn\n1V. V. Kruglyak, S. O. Demokritov, and D. Grundler, Jour-\nnal of Physics D - Applied Physics 43, 264001 (2010).\n2Y. Kajiwara, S. Takahashi, S. Maekawa, and E. Saitoh,\nIEEE Transactions on Magnetics 47, 1591 (2011).\n3A. Khitun and K. L. Wang, Journal of Applied Physics\n110, 034306 (2011).\n4Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida,\nM. Mizuguchi, H. Umezawa, H. Kawai, K. Ando,\nK. Takanashi, S. Maekawa, and E. Saitoh, Nature 464,\n262 (2010).\n5E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Applied\nPhysics Letters 88, 182509 (2006).\n6J. Xiao and G. E. W. Bauer, Physical Review Letters 108,\n217204 (2012).\n7G. L. da Silva, L. H. Vilela-Leao, S. M. Rezende, and\nA. Azevedo, Applied Physics Letters 102, 012401 (2013).\n8J. Xiao, Y. Zhou, and G. E. W. Bauer, arxiv:1305.1364\n(2013).\n9A. Kapelrud and A. Brataas, arXiv:1303.4922 (2013).10Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I.\nHalperin, Reviews of Modern Physics 77, 1375 (2005).\n11C. Burrowes, B. Heinrich, B. Kardasz, E. A. Montoya,\nE. Girt, Y. Sun, Y.-Y. Song, and M. Wu, Applied Physics\nLetters100, 092403 (2012).\n12F. D. Czeschka, L. Dreher, M. S. Brandt, M. Weiler, M. Al-\nthammer, I.-M. Imort, G. Reiss, A. Thomas, W. Schoch,\nW. Limmer, H. Huebl, R. Gross, and S. T. B. Goennen-\nwein, Physical Review Letters 107, 046601 (2011).\n13P. Yen, T. S. Stakelon, and P. E. Wigen, Physical Review\nB19, 4575 (1979).\n14O. G. Ramer and C. H. Wilts, Physica Status Solidi (b)\n73, 443 (1976).\n15Y.-T. Chen, S. Takahashi, H. Nakayama, M. Althammer,\nS. T. B. Goennenwein, E. Saitoh, and G. E. W. Bauer,\nPhys. Rev. B 87, 144411 (2013).\n16M. D. Stiles and A. Zangwill, Physical Review B 66,\n014407 (2002).\n17J. Z. Sun, Physical Review B 62, 570 (2000).\n18J. Xiao, A. Zangwill, and M. D. Stiles, Physical Review B\n72, 014446 (2005).10\n19Y. Zhou, J. Xiao, G. E. W. Bauer, and F. C. Zhang, Phys-\nical Review B 87, 020409 (2013).\n20A. G. Gurevich and G. A. Melkov, Magnetization oscilla-\ntions and waves (CRC Press, 1996).\n21R. E. De Wames, Journal of Applied Physics 41, 987\n(1970).\n22B. Hillebrands, Physical Review B 41, 530 (1990).\n23B. A. Kalinikos and A. N. Slavin, Journal of Physics C:\nSolid State Physics 19, 7013 (1986).24Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Physical\nReview Letters 88, 117601 (2002).\n25S. M. Rezende, R. L. Rodriguez-Suarez, M. M. Soares,\nL. H. Vilela-Leao, D. L. Dominguez, and A. Azevedo, Ap-\nplied Physics Letters 102, 012402 (2013).\n26X. Jia, K. Liu, K. Xia, and G. E. W. Bauer, Europhysics\nLetters96, 17005 (2011)."    },    {        "title": "2103.07307v1.Temperature_dependence_of_the_interface_spin_Seebeck_effect.pdf",        "content": "Temperature dependence of the interface spin Seebec k effect  \nFarhan Nur Kholid a,b , Dominik Hamara a, Marc Terschanski c, Fabian Mertens c, Davide \nBossini d, Mirko Cinchetti c, Lauren McKenzie-Sell a,e , James Patchett a, Dorothée Petit a, Russell \nCowburn a, Jason Robinson e, Joseph Barker f, Chiara Ciccarelli a* \n \naCavendish Laboratory, University of Cambridge, Camb ridge, CB3 0HE, United Kingdom \nbPhysics Department, Lancaster University, Lancaster , LA1 4YB, United Kingdom \ncExperimentelle Physik VI, TU Dortmund, Otto-Hahn-St rasse 4, D-44227 Dortmund, Germany \ndDepartment of Physics and Center for Applied Photon ics, University of Konstanz, Universitätstraße \n10, 78457 Konstanz, Germany \neDepartment of Materials Science and Metallurgy, 27 Charles Babbage Rd, Cambridge CB3 0FS, \nUnited Kingdom \nfSchool of Physics and Astronomy, University of Leed s, Leeds, LS2 9JT, United Kingdom \n \n*cc538@cam.ac.uk \n \nWe performed temperature-dependent optical pump – T Hz emission measurements in Y 3Fe 5O12 \n(YIG)|Pt from 5 K to room temperature in the presenc e of an externally applied magnetic field. We \nstudy the temperature dependence of the spin Seebec k effect and observe a continuous increase as \ntemperature is decreased, opposite to what is obser ved in electrical measurements where the spin \nSeebeck effect is suppressed as 0K is approached. B y quantitatively analysing the different \ncontributions we isolate the temperature dependence  of the spin-mixing conductance and observe \nfeatures that are correlated to the bands of magnon  spectrum in YIG.  \n \nThe longitudinal spin Seebeck effect (LSSE) 1 describes the transfer of a spin current from a \nmagnetic insulator driven by a temperature gradient . An adjacent heavy metal (HM) layer with large \nspin orbit coupling is typically used to convert th e spin current into an electrical signal via the in verse \nspin Hall effect (ISHE).2,3  The LSSE has been measured in a variety of differe nt materials such as \nferromagnets 1,4,5 , anti-ferromagnets 6,7  and paramagnets.8  Magnetic insulators (MI) such as Y 3Fe 5O12  \n(Yttrium Iron Garnet – YIG) are particularly intere sting for studies on the LSSE since the absence of \nelectron charge transport allows the roles of magno ns and phonons to be identified in the spin \ntransfer.1,3,9,10  Temperature, thickness and magnetic field dependen ce studies have contributed to a \nphenomenological picture of magnon-driven spin curr ent.11–15  A temperature gradient across the \nmagnetic insulator thickness leads to the diffusion  of thermal magnons that accumulate at the \ninterface with the HM.16,17  The temperature dependence of the magnon propagati on length \u0001\u0002 results \nin a characteristic peak in the SSE signal at low t emperature when the thickness of the MI is \ncomparable to \u0001\u0002.12  Low frequency magnons play a dominant role due to their large population and longer thermalisation lengths. Their contribution c an be suppressed by large magnetic fields which \nraise the energies of the magnon spectrum.14,15  \nThis picture of a bulk-like transport induced by a temperature gradient picks up the essential \nfeatures of the LSSE. However, several experimental  results raise questions on the details of how the \nspin current is transferred across at the MI|HM int erface.12  This contribution has been challenging to \nisolate in electrical measurements of the LSSE and its temperature dependence is not known.  \n Recently, ultra-fast experimental techniques using  femtosecond lasers have enabled the \nstudy of the LSSE and the underlying physical mecha nisms of spin current generation at picosecond \nand shorter timescales.18,19  In these experiments a laser pulse rapidly heats t he free electrons in the \nHM, quickly thermalising to an effective temperatur e, \u0004\u0005. The temperature of the magnons in the \ninsulator,  \u0004\u0002, is increased primarily by the spin current which p ropagates across the interface from \nthe hotter metal. This thermalisation processes is proportional to \u0004\u0005− \u0004 \u0002 and its timescale is \nultimately determined by the electron-magnon scatte ring time.18  In this ultra-short time window after \nthe laser excitation a thermal gradient is not yet established in the bulk of the MI and the spin curr ent  \ngeneration originates only at the interface between  MI and HM.19  \n In this study, we measured the LSSE in YIG|Pt on t he picosecond timescale in the low \ntemperature range from 5 K to room temperature. We observed a different temperature dependence \nof the LSSE compared to DC electrical studies carri ed out in the same temperature range 12,14,15 . Our \nsample is a 100 nm thick commercial YIG film grown by liquid phase epitaxy on a (111)-oriented \nGd 3Ga 5O12  (GGG) substrate. We cleaned the surface using pira nha etching and then sputtered a 5 nm \nthick layer of Pt on top. Fig. 1 shows the two diff erent orientations of our experiments. We pump the \nsample from either the GGG side or the Pt side with  50 fs laser pulses with a central wavelength of \n800 nm. Any spin transfer across the YIG|Pt interfa ce triggered by the pump pulse is converted into \nan electric current via the inverse spin-Hall effec t in the Pt layer. This produces a broad-band elect ric-\ndipole emission \b\t\n\u000b \f\r\u000e with a bandwidth directly related to the Fourier t ransform of the spin current \n\u000f\u0010\f\r\u000e as 20  \nE\t\n\u000b \fω\u000e=\u0014\u0015\n\u0016\u0017\u0018\u0019 \f\u001a\u000e\u001b\u0016 \u0015\f\u001a\u000e\u001b\u001c\u0014\u0015\u001d\u001e\u001f \f\u001a\u000e !\"\n\u0015#$%&' \u0005($\f\u001a\u000e\nℏ    \f1\u000e     \nwhere +, is the free space impedance in Ohms, ℏ is Planck’s constant,  - is the charge of an \nelectron, \u0001\u0010, ./0 , 1 and Θ3\n   are respectively the spin diffusion length in nm,  the electrical conductivity \nin Ohms -1 cm -1, the thickness in nm, and the spin-Hall angle of t he Pt layer. 4567 \f\r\u000e and 4,\f\r\u000e \nrepresent the refractive indices of YIG and air. We  detect the emitted radiation by electro-optic \nsampling with a 1-mm thick ZnTe crystal. The detect ed signal 8\f\r\u000e is the convolution of Eq. (1) with the detector response function, which is bandwidth limited to 0.2-2.5 THz range. We apply an external \nmagnetic field ( 9,: = ±  0.5 T) along the [100] direction (Fig. 1) during t he measurements to saturate \nthe YIG magnetisation. We extract an odd-in-magneti c field 8<= [8 \f+:\u000e− 8\f−:\u000e] 2⁄ and an even-\nin-magnetic field 8\u001b= [8 \f+:\u000e+ 8\f−:\u000e] 2⁄ contribution to the overall emission. 8\u001b is polarised in \nthe [100]-[010] plane (Fig. 2a and 2b). Its depende nce on the pump polarisation (Fig. 2a) connects its  \norigin to optical rectification. Both bulk GGG and YIG are centrosymmetric.21,22  However, their lattice \nmismatch induces elastic deformations in YIG close to the interface that gradually changes its lattice  \nparameters, breaking inversion symmetry and yieldin g a non zero value for the second order electro-\noptic constant B\fC\u000e, as also confirmed by the measurement of optical s econd harmonic generation.23  \nFrom this point forward, we focus on the 8< contribution that is due to the LSSE. Unlike  8\u001b,  8< does \nnot show any dependence on the pump polarisation an d is always polarised along the [010] axis, \nperpendicular with respect to the interface normal and the YIG magnetisation (Fig. 2b). The reversal \nof the interface normal vector with respect to the pump pulse propagation direction results in a \npolarity switching of the emitted THz radiation (Fi g. 2c). Both observations are consistent with the \nsymmetry of the ISHE for a spin current travelling across the interface with spin polarisation along t he \n[100] direction.2 As a function of the external magnetic field, the peak amplitude of 8< follows the \nhysteresis curve of the YIG magnetisation (Fig. 2d) , also in agreement with previous electrical and \noptical measurements of the LSSE.18,24   \nFig. 3a shows the temperature dependence of 8<. The continuous line represents a fitting with \nthe function \f\u0004D− \u0004\u000eE, where TD= 550 K  is the Curie temperature and  J = 2.9 ±  0.1 . This trend is \nsimilar to the temperature dependence measured abov e room temperature with both low-frequency \nelectrical 11  and ultra-fast optical methods 18 , but is remarkably different from the low temperat ure \nbehaviour of the LSSE measured in adiabatic conditi ons, where the signal diminishes towards 0 K.12,14  \nIn our experiment we detect the spin current genera ted in a time interval up to a few picoseconds \nafter the laser absorption in Pt. This interval is orders of magnitude shorter than the time needed to  \nestablish a thermal gradient in bulk YIG (1-100 nan oseconds).25,26  Thus, we are probing the electron-\nmagnon interactions localised at the interface. The  temperature difference emerges between the free \nelectron system on the HM side, \u0004\u0005, and the spin system on the MI side, \u0004\u0002. This temperature \ndifference is significantly larger than that genera ted in a DC electrical experiment. Thermalisation \nbetween these two systems occurs via direct electro n-magnon interaction which is the origin of the \nspin transfer across the interface.18,19  The interfacial spin transport parameters are summ arised by the \nspin-mixing conductance M↑↓ and the resulting spin current can be written as 17,27  \n\u000f\u0010=PℏQ RS↑↓\nCTU $V\f\u0004\u0005− \u0004 \u0002\u000e      (2) where W is the gyromagnetic ratio, XY is the Boltzmann constant, Z\u0010 is the saturation magnetisation \nof YIG and [ is the unit cell volume.  In the case of a femtose cond laser excitation, \u0004\u0005− \u0004 \u0002 is set by \nthe energy deposited in the HM layer, in other word s by the absorbed laser fluence. The Pt layer has \na strong optical absorption ( ~10 ]cm <`)28  that is enhanced by the Etalon effect 29 , while the absorption \nin GGG|YIG ( 10 cm <`) is essentially negligible. 30,31  We estimate that for an absorbed fluence of 0.15 \nmJ/cm 2 ∆\u0004\u0005,\u0002bc ~200  K at 10 K 18 , taking the electron-phonon coupling constant M\u0005<de = 10 `f  W ∙\nm<iK<` and the electron heat capacity j\u0005/\u0004\u0005= 750 J ∙ m<iK<C.32  \nTo understand the origin of the temperature depende nce of the picosecond LSSE in Fig. 3a, \nwe consider all parameters that contribute to its m agnitude, as expressed by Eq. (1) and (2). We first  \nnormalise 8< with the YIG magnetisation 1/Z \u0010 obtained from SQUID measurement (Fig. 3b), which \nshows that Z\u0010 is not accountable for the large change in the THz  emission. To experimentally verify \nhow ∆\u0004\u0005 is influenced by the ambient temperature, we perfo rm pump-probe transient reflectivity \nmeasurement on glass|Pt bilayers from 10 K to 300 K . The transient change in reflectivity ∆n/n\fo\u000e  is \nproportional to ∆\u0004\u0005\fo\u000e.33,34 As seen in Fig. 3c, The peak magnitude of ∆n/n  is weakly affected by the \nambient temperature. Similarly, the time evolution of ∆n/n  is also independent of the ambient \ntemperature with the thermalisation time between el ectrons and phonons p\u0005<de = 260 ± 10 fs, \nobtained by an exponential fit ∝ exp u−v\nwxyz{ |. This measurement indicates that we have \napproximately the same temperature difference, \u0004\u0005− \u0004 \u0002, at all ambient temperatures used in our \nexperiment. \nApart from M↑↓, which quantifies the quality of the interface in conducting spins, the other \nparameters ( \u0001\u0010, ./0  and Θ3\n ) are intrinsic to the Pt layer. To exclude the con tribution of these \ntransport parameters or any other contribution from  the set-up, we compare our LSSE results to a \nmetallic THz spintronic emitter 20,35 CoFeB (3 nm)|Pt (5 nm). In this case, the pump bea m hits the \nsample from the CoFeB side and is largely absorbed by the ferromagnet, inducing a strong \nsuperdiffusive spin current.36,37 Therefore, far from \u0004D= 1100 K38, CoFeB behaves as a temperature-\nindependent spin current source, transported to the  Pt layer by high mobility majority spin carriers. \nEq. (1) also applies to this metallic bilayer as it  relies on the spin-to-charge conversion in Pt to g enerate \nTHz emission. In agreement with a previous report 39, the amplitude of the THz pulse decreases with \ndecreasing temperature and reaches a plateau at 50 K (Fig. 3d). This behaviour, which is associated \nwith the intrinsic components of the spin Hall effe ct in Pt 39, significantly differs to what is observed in \nour YIG|Pt sample, allowing us to exclude the influ ence of the Pt layer in our measured temperature \ndependence of the LSSE. We conclude therefore that our measurement probes the temperature \ndependence of the spin mixing conductance.  The laser-excited free electrons in Pt are not spin  polarised initially. The stochastic local \nexchange field fluctuations induced by single elect ron scattering events off the interface with the MI  \nare therefore averaged to zero at timescales longer  than the interaction time (~ 4fs for YIG|Pt 18 ). \nHigher order interactions between the scattering el ectrons and the MI can lead to a net magnetic \ntorque on the MI and therefore to spin accumulation , as described in Ref. 18. An additional \ncontribution associated with the real part of the s pin-mixing conductance M} is given by inelastic spin-\nflip scattering processes that result in the excita tion of a magnon on the MI side. This contribution \ndepends on the density of states of magnons as well  as the electronic density of states at \u0004~. Using \nEqs (1) and (2) we estimate the range of the spin m ixing conductance at 10 K as M↑↓= \f1.8 −\n8.4\u000e × 10 `f  m-2, in agreement with that found in [14]. Our paramet ers are +,= 377 Ω , 4567 = 518 , \n4= 1, ./0 (10 K) = 0.03 9Ω<`cm -1 39, \u0001\u0010(10 K) = 2 − 4  nm  40,41, Θ3\n = 0.01 − 0.0223 40 , Z\u0010(10 K) = \n172 kA/m, [ = i,  =1.24 nm  42 , \u0004\u0005− \u0004 \u0002 ≈ 200 K. Note that 4567  and ./0  can be considered \nfrequency-independent within our detection bandwidt h 28,43 . We postulate that the kink in the \ntemperature dependence of the LSSE signal around 80  K (Fig. 3b) may be related to the higher energy \nmagnon bands in YIG becoming populated by the highl y energetic electrons in Pt. At an ambient \ntemperature of 100 K the first high-frequency bands  appear at ~25 meV  44-46, which coincides with the \naverage energy of the optically heated electrons in  Pt. The progressive filling of these bands at high er \nambient temperature affects the spin pumped across the interface and determines the temperature \ndependence of the LSSE.  \nIn conclusion, we characterise the low temperature behaviour of the picosecond spin Seebeck \neffect in YIG|Pt by optical pump-THz emission measu rents and show that it is substantially different \nfrom that reported in low-frequency electrical meas urements. We observe a sustained increase of the \nsignal with decreasing temperature, which is a cont inuation of the previous femtosecond SSE \nexperiment measured from room temperature to above \u0004= 550 K . This behaviour cannot be \nattributed to a variation of the temperature gradie nt at the interface or of the spin and charge \ntransport characteristics in Pt, and is instead to be associated with the spin-mixing conductance, \nproviding direct access to its temperature dependen ce.  \n   \n \nFIG . 1. Schematic illustration of the experiment performed with the femtosecond laser pulses incident \non the Pt side (left) and the GGG substrate side (r ight).  \n \n  \n \n  \n  \nFIG. 2.  (a) Time-domain THz emission resolved along the [1 00]- and [010]-axis, measured at 10 K. Odd-\nin-field signal 8< only appears along the [010]-axis component wherea s the signal along the [100]- axis \nis even-in-field 8\u001b (9,H = ±0.5 T). (b) 8\u001b (blue circle) and 8< (red diamond) dependence on the linear \npump polarisation where the angle  is relative to the [010]- axis . These measurements were carried \nout for signals along [010]-axis at room temperatur e. The orange line is a fit using ,+\n sin C\f −  ,\u000e, where , is a constant offset,  is the magnitude of the optical rectification signa l, \n, is an angle offset. This angular dependence agrees  with the 2 nd  harmonic generation measurement \nin GGG|YIG 23 . An offset of  - 0.3 V/cm is applied to  8\u001b for clarity. (c) \b\t\n\u000b  polarised along [010]- axis in \ntime-domain for Pt-side and GGG-side pumping. (d) H ysteresis curve of 8< measured at room \ntemperature.  \n   \n  \n  \nFIG. 3. (a) Temperature dependence of 8< The dashed line is a fit with a function \f\u0004D−\u0004\u000eE, =\n1.5± 1.3,  J=2.9± 0.1, and \u0004D= 550 K. (b) Temperature dependence of the normalise d 8< with \nthe YIG magnetisation Z\u0010. The grey arrow indicates the temperature at which t he slope \n18<1\u0004 ⁄  changes (c) Time-resolved transient reflectivity ( ∆n/n ) of glass|Pt measured in a \ntemperature range of 10 – 300 K at a fixed pump flu ence of 0.4 mJ/cm 2. The dashed line is an\nexponential fit ∝ exp u −v\nwy| (d) Peak THz emission from CoFeB(3 nm)|Pt(5 nm) as a function of \nambient temperature where the pump pulse hits from the CoFeB side . The error bar is comparable \nwith the symbol size. The inset shows the time-doma in data for opposite field polarities ±0.5  T. \n \n \n \n \n  \nAcknowledgement \nF.N.K. acknowledges support from the Cambridge Trus t and the Jardine Foundation. D.H \nacknowledges support from the Winton Programme for the Physics of Sustainability. J.P.P. \nacknowledges support from the EPSRC. M.C., D.B., F.M. and M.T. acknowledge support from the \nDeutsche Forschungsgemeinschaft through the Interna tional Collaborative Research Center TRR160 \n(Projects B8 and B9).  C.C. and J. 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Spin-wave spectrum of yt trium iron garnet. Physical Review  120 , 1612–1614 (1960). \n43. Hofmeister, A. M. & Campbell, K. R. Infrared sp ectroscopy of yttrium aluminum, yttrium gallium, \nyttrium iron garnet. Journal of Applied Physics  72 , 638 (1992) \n44.        Barker, J. & Bauer, G. E. W. Thermal spi n dynamics of yttrium iron garnet. Physical Review Letters  \n117 , 217201 (2016). \n45.        Princep, A. J. et al.  The full magnon spectrum of yttrium iron garnet. npj Quantum Materials  2, 63 \n(2017). \n46. Barker, J., Pashov, D., Jackson, J. Electronic structure and finite temperature magnetism of yttri um \niron garnet, Electron. Struct.  2 044002 (2020). \n \n  \n   \n "    },    {        "title": "1901.10800v1.Substrate_induced_magnetic_anisotropies_and_magneto_optical_response_in_YIG_nanosized_epitaxial_films_on_NdGG_111_.pdf",        "content": "1 \n Substrate induced magnetic anisotropies and magneto-optical response in \nYIG nanosized epitaxial films on NdGG(111)  \nB.B. Krichevtsov , V.E. Bursian , S.V. Gastev , A.M. Korovin, L.V. Lutsev, S .M. Suturin,  \nK.V. Mashkov, M.P. Volkov and N .S. Sokolov* \nIoffe Institute, 26 Politechnicheskaya str., 194021 Saint Petersburg , Russia  \n*Corresponding author, E -mail: nsokolov@fl.ioffe.ru  \nAbstract  \nNanosized Y 3Fe5O12 epitaxial films have been grown on Nd 3Ga5O12 substrates using laser molecular \nbeam epitaxy method. Magneto -optical polar Kerr effect, ferromagnetic resonance and spin wave \npropagation measurements show that the stress- related anisotropy field has an opposite sign, compared to \nthat in the YIG/GGG systems. This leads to a considerable decrease of the effe ctive magnetization that \nopens a perspective to get YIG films with perpendicular magnetization for utilizing forward volume spin \nwaves. Longitudinal magnetooptical Kerr effect magnetometry reveals a large contribution  of quadratic in \nmagnetization terms in to dielectric permittivity tensor at optical frequencies. This effect strongly \nincreases with temperature decrease and is explained by magnetization of the interface Nd3+ ions that are \nexchange coupled to the Fe3+ ions.  \nThe nanoscale heterostructures  based on yttrium iron garnet (YIG, Y 3Fe5O12) films attract \nmuch attention nowadays owi ng to the intense development of oxide spintronics and magnonics \n1, 2, 3, 4. For the realization of effective data transmission  and processi ng by spin wave packets, \none needs high quality heterostructures with low magnetic losses at GHz frequencies. A great \nnumber of recent studies has been dedicated to investigation of epitaxial growth, static and \ndynamic magnetic properties of YIG layers epitaxially grown onto gadolinium gallium garnet \n(GGG, Gd 3Ga5O12) substrates . The nanosized YIG films grown by laser molecular beam epitaxy \n(LMBE) ex hibit high crystalline quality 5, 6, 7, 8, narrow -line ferromagnetic resonance (FMR) \nand low spin wave damping 9 ,10 , 11, 12, 13. Assum ing that the damping parameter α is linear \nfunction of the resonance frequency, in Ref. 10 it was calculated that α = 6.15×10-5. On the other \nhand, the direct measurement of spin wave propagation in nanosized YIG films grown at low \ntemperatures (700 °C) shows  that the damping parameter can be even lower (<  3.6⋅10-5) 12. \nIn the absence of external magnetic field, the magnetization in YIG / GGG(111) films lies in -\nplane due to the demagnetizing field H d = 4πMs ≈ (1.2 – 1.5) kOe reinforced by the magnetic \nanisotropy Ha ≈ -1 kOe (4πMeff = 4πMs – Ha ≈ 2 kOe ) induced by t he magnetoelastic interactions \n14 ,15. In the forward volume spin wave devices the out -of-plane magnetization is necessary 16, 17, \n18 and a strong permanent magnet is required. The way to avoid the bulky magnet, is to construct 2 \n YIG films with perpendicular magnetic anisotropy. This can be done by modification of H a by \nstrain engineering (e. g. choosing a proper substrate 19, 20). It was shown in Ref. 18 that in a 115 \nnm thick YIG  film grown on neodymium gallium garnet (Nd 3Ga5O12, NdGG) substrate, the \nrhombohedral distortion is of opposite sign compared to the YIG  / GGG (111) system thus \nreducing the magnetic field required for out -of-plane magnetization orientation.  \nIn this paper, we investigate static and dynamic magnetic properties of considerably thinner \n(35 and 12nm) YIG epitaxial films grown on N dGG (111) substrat es by LMBE. The \nmagnetization reversal, ferromagnetic resonance (FMR) and spin wave propagation (SWP) in \nYIG / N dGG heterostructures are studied by magnetic and magneto -optical methods for \ndifferently oriented magnetic field.   \n The NdGG and GGG substrates  were annealed before the growth (3 hours at 1000°C, \nambient atmosphere) to make the surface atomically smooth . Excimer KrF Lambda Physics \nCOMPEX 201  laser (λ≈ 248 nm) was used to ablate the YIG target at a fluence of 3.0- 3.4 J/cm2. \nThe temperature of NdGG substrate was in the  500-850°C range (see Table 1 , samples #1- #4). \nFor the reference, YIG film has been also grown on G GG(111) substrate (sample #5). The \ngrowth mode is layer -by-layer as confirmed by specular spot intensity oscillations in reflection \nhigh energy electron diffraction (RHEED) pattern, Fig. 1a. RHEED pattern shown in Fig. 1b and  \natomic force microscopy (AFM) image, Fig. 1c , confirmed that YIG on the NdGG substrate \nforms epitaxial  films with atomically smooth surface. High qu ality YIG films were r ecently \nfabricated on GGG substrate s using s imilar  grow th conditions 8, 21.  \n  \n \nSample  Growth  \ntemperature,  \nºC Thickness,  \nnm  4πMs  \n(VSM),  \nkG  4πMeff  \n(PMOKE),  \nkG  4πMeff  \n(FMR),  \nkG  4πMeff* \n(SWP),  \nkG \n#1 500 35 1.5 1.3 [0.80 … 1.25 ] 1.24 \n#2 700 35 1.5 1.2 [0.78 … 1.23]  1.08 \n#3 850 35 1.7 0.5 [0.17  … 1.19]  0.58 \n#4 700 12 1.5 0.6 [ ≤0.08 … 0.80]  0.64 \n#5 600 35 - 2.2 1.90 - \n*In spin wave propagation experiments, 4 πMeff values were calculated using the frequency of the \nstrongest maximum in S 21 spectra.  \n \nTable.1. Growth parameters and magnetic properties of YIG films grown on NdGGG (#1 -#4) and GGG \nsubstrates (#5). Given are growth temperature, film thickness, saturation magnetization 4π Ms, measured \nby VSM  and effective magnetization 4πMeff = 4πMs – Ha, obtained from PMOKE, FMR and SWP \nexperiments.    3 \n 0 20 40 60 80 100 120 140 16035404550556065707580RHEED intensety, a.u.\nTime, sGrowth start a\n   \n    \n   \n \nFig.1 Characterization of growth mode and structural quality of YIG film #2. (a) RHEED specular beam \nintensity oscillations during the film growth, (b) RHEED pattern, (c) AFM image (3000 x 3000 x 1 nm) \nof the film surface.  \n \nIn-plane magnetization curves were measured using Quantum design PPMS -9 vibrating \nsample magnetometer (VSM). The observed hysteresis loops (not shown) were narrow and had \nsaturation  magnetization of 4 πMs ~ 1.6 kG  (Table 1) .  \nFig. 2 illustrates  static magnetic properties of the films. The out- of-plane magnetization \ncurves, measured by PMOKE ( λ = 405 nm)  in a series of YIG / NdGGG samples, saturate at \nmagnetic fields H s close or less than 4 πMs value measured by VSM. This is in contrast to the \nearlier studied YIG / GGG system in which H s > 4πMs 8, 21. From saturation field H s = 4πMeff = \n4πMs – Ha we can estimate the effective magnetization 4π Meff and anisotropy field H a. In \ncontrast to YIG/GGG system the sign of anisotropy field H a is positive, i.e. the induced magnetic \nanisotropy favors the out -of-plane  magnetization. The H a may reach a large value: e.g. H a ≈ +1 \nkOe in the sample #3.  \n -2 0 2-1,0-0,50,00,51,0normalized Popar Kerr effect\nH, kOe1,3kOe(b)\n-2 -1 0 1 2-1,0-0,50,00,51,0normalized Polar Kerr effect\nH, kOe0.51kOe (a)\n-50 -25 0 25 50-1,0-0,50,00,51,0normalized ellipticity\nH, Oe(d)\n-50 -25 0 25 50-1,0-0,50,00,51,0normalized ellipticity\nH, Oe(f)-4 -2 0 2 4-1,0-0,50,00,51,0normalized Polar Kerr effect\nH, kOe2.2 kOe\n(c)PMOKE  \nLMOKE  \n-50 -25 0 25 50-1,5-1,0-0,50,00,51,01,5normalized ellipticity\nH, Oe(e)4 \n Fig.2 Static magnetic propert ies of YIG/NdGG hetero structures. (a,b,c) - magnetization curves \nmeasured by PMOKE (polarization plane rotation) in samples #3,  #1 (YIG/NdGG) and #5 (YIG/GGG). \nMagnetization curves measured by LMOKE (ellipticity) in sample #2 (YIG/NdGG) for in- plane magne tic \nfield oriented along the in -plane easy axis (d), at angle +45o (red) and - 45o (blue) to the easy axis (e), and \nalong the in- plane hard axis (f).  \n \nThe LMOKE in -plane magnetization curves in YIG / NdGG films were obtained by \nmeasuring either ellipticity or polarization plane rotation  at λ = 405 nm . The azimuthal \ndependence of the hysteresis loop shape exhibits a 180° periodicity characteristic of the uniaxial \nin-plane magnetic anisotropy. Fig. 2d- f shows ellipticity measured in sample #2 with in- plane \nmagnetic field applied at 0o, ±45o, and 90o to the easy axis (EA) . With magnetic field along the \nEA, the hysteresis loops are narrow and rectangular similar to those observed in YIG / GGG(111)  \n8, 21. When the field direction is at an angle to the EA, the loops are strongly asymmetric, Fig. 2e . \nWith magnetic field along the hard axis (HA), peculiar jumps probably related to some \nmodification of domain structure appear in the magnetization loops, see Fig. 2f.   \nThe asymmetry of the  hysteresis loop shape depending on the in- plane magnetic field \norientation was earlier observed in Fe and Co (110) epitaxial films 22, as well as in the \nFe/GaAs (001)  23 and CoFeB /MgO (001) 24 nanostructures . The loop asymmetry is caused by the \nβijkl(ω)MkMl terms in the dielectric permittivity tensor εij(ω,M) that are quadratic in \nmagnetization at optical frequencies. In magneto -optical experiments performed in transmission  \nVoigt geometry , the real part of βijkl tensor is responsible for the Cotton- Mouton effect and the \nimaginary part – for the magnetic linear dichroism 25. In LMOKE measurements, with \nmagnetization lying in -plane, the real and imaginary parts of βijkl tensor can contribute into \nellipticity and polarization rotation, correspondingly. The polarization rotation measurements \ncarried out in sample #2 during magnetization reversal did not show any asymmetry of hysteresis \nloops. This indicates that  the observed asymmetry of the loops measured by ellipticity originates \nfrom the real part of β ijkl tensor. The linear and quadratic terms can be separated by measuring \ntwo magnetization curves with magnetic field rotated by a positive + θ and negative - θ angle with \nrespect to the EA. The half -sum and half -difference of these curves will give the linear and \nquadratic term correspondingly, Fig. 3 a,b.  \nWe have found that the quadratic contribution considerably increases at low temperatures \nand becomes higher than the linear one, Fig. 3b,c. The  temperature dependence of the linear \nterms is considerably weak er, being approximately proportional to M(T) in YIG. Temperature \ndecrease from 300K to 150K is followed by increase of quadratic contribution by a factor of ~6. This can be hardly associated with magnetization of iron sublattices within the YIG layer 5 \n becau se in this temperature range the square of iron total magnetization M2\nFe is incr eased only by \na factor of ~1.6 26. Strong increase of quadratic contribution may be associated with \nmanifestation of interface magnetization. Nd3+ ions in NdGG substrate are in paramagnetic state \nand at the interface they can be magnetized by superexchange with Fe3+ ions from tetra - or \noctahedral magnetic sublattices of YIG. During the magnetization reversal, the Nd3+ ion \nmagnetization is coupled to that of the YIG layer. Magn etization of interface Nd3+ ions should be \nproportional to product  of exchange field H e = JMFe and paramagnetic susceptibility χ  ~ 1/T. \nQuadratic effects should follow the T-2MFe2(T) dependence, which increases by a factor of ~ 6.4 \nwith temperature decreas e from 300K to 150K 26 that is close to that observed in our experiment. \nNote that similar strong increase of rare -earth magnetization has been f ound i n many rare -earth \niron garnets 27. It is also worth mentioning that in the YIG  / GGG system, the induced \nmagnetization of interface Gd3+ ions was observed at low temperature by spin polarized neutron \nreflectometry 28. Distinctive features of a nanometer thick interface layer between YIG and GGG \nwere observed b y magneto -optical spectroscopy 29. \n \nFig. 3. LMOKE magnetization curves measured in sample #2  with magnetic field applied at \ndifferent angle s to EA: a) ± 60о, T =  300 K; b) ± 45о, T = 200 K. The linear and quadratic terms are \nshown in a) by blue and magenta curves correspondingly. Quadratic contribution into ellipticity for three \ntemperatures is shown in (c).  \n \nThe pronounced manifestation of the neodymium magnetization in the reflected light \nellipticity may be related to high magnetooptical susceptibility at λ = 405 nm because this wavelength is very close to absorption lines in Nd\n3+ ions in NdGG 30.  On the contrary, much \nsmaller magnetooptic susceptibility is expected in the YIG  / GGG heterostructures, because Gd3+ \nis an S -ion and its absorption lines are much weaker than those of Nd3+. \nIt should be noted that Nd3+interface ions may be magnetized by Fe3+ ions both from tetra - or \noctahedral p ositions, which have opposite orientation of spins. Nevertheless the contribution of \nthe quadratic magnetooptical terms ~ βijkl(ω)MkMl is the same for the opposite directions of  -40 -20 0 20 40 60-400-2000200400\nEA - 45oT=200Kellipticity,  µrad\nH (Oe)EA + 45o\n(b)\n-40 -20 0 20 40-400-300-200-1000M2 terms\nH, Oe 300 K\n 200 K\n 150 K (c)\n-40 -20 0 20 40-150-100-50050100150\n H, OeEA+60o\nEA\nEA-60oellipticity, µrad\n~M2\n(a)6 \n Nd3+ magnetic moment. For this reason, the quadratic magneto- optical effects may be \npronounced even when the numbers of Nd3+ ions with opposite magnetization orientation are \ncomparable. In this case, the methods sensitive to the net magnetization (PNR or XMCD) will \ngive zero output. However the quadratic magnetooptical phenomena wi ll still be observable.   \n \nFig.4. Dynamic magnetic properties of YIG/NdGG heterostructures. (a) Experimentally measured \nFMR absorption derivative spectra in samples #1 and #4, and for comparison in YIG/GGG(111) sample \n#5. Vertical lines show the calculated resonance fields for 4 πMeff = 0 and 4π Meff = 4πMs. (b) FMR \nabsorption spectra obtained by numerical integration of experimental data. (c) FMR spectra of sample #4 \nfor magnetic field oriented in -plane, out -of-plane and at 46o to th e surface. The arrows show the \ncalculated resonance fields for 4 πMeff = 0.8 kG and  4 πMeff = 0.08 kG. (d) Amplitude -frequency \ncharacteristic (scalar gain) S 21 of the spin waves propagated in the  sample #2 in the in- plane magnetic \nfield.  \n  \nDynamic magnetic properties of YIG/NdGG heterostructures  are presented in Fig. 4. FMR \nspectra were obtained at room  temperature and F  = 9.4 GHz frequency, utilizing a conventional 7 \n ESR spectrometer with a small- amplitude modulation of  the slowly scanning magnetic field, \nwhich makes spectra to appear in the form of absorption derivative (Fig. 4a ).  The original \nspectra were then numerically integrated (Fig.  4b, c) in order to make more illustrative the nature \nof the complicated, multi -component line structure, which is discussed below. In the fi gures 4b \nand 4c also shown are  the theoretical resonance positions, calculated  with Kittel formulae for the \nuniaxially stressed film (cubic anisotropy terms or any other in -plane anisotropy are ne gligible \non the actual scale)  31: \n)H 4(H2 in\nresin\nres2\n+ =\n\n\n\neffMFπγπ \neffMFπγπ4 H2 out\nres− =\n                                                          \nwhere Hresin and H resout are resonance fields for in -plane and out -of-plane magnetic field \ncorrespondingly, 4π Meff = 4πMs – Ha is the effective magnetization . The 4πMs value is taken \nfrom VSM measurements, and γ /(2π) = 2.83 MHz/Oe   is consistent with the commonly known \ndata 31, as well as  with all spectra, presented here.   \nIt should be first of all noted that the FMR lines, obtained for the two regular directions \nof magnetic field (Fig.  4a), lie closer to each other for all YIG/NdGG samples, compared with \nthat of the YIG/GGG heterostructures . The values of 4 πMeff, evaluated from   the Hresin and  Hresout   \nresonance fields, will be evidently smaller than the average value of  4 πMs = (1.6±0.1)  kG, as \nobtained from the VSM technique. This means, that Ha is positive, in accordance with the \nPMOKE results  (Table 1) . \nOn the other hand, the FMR lines exhibit a complicated, multi -component structure \nwithin a very large total linewidth (Fig . 4 a-c). We believe, this is due to a lateral or in -depth \ninhomogeneity of the film. We also assume that this inhomogeneity can be characterized with a \ndistribution of the 4π Meff value. This assumption is supported experimentally by the fact that the \nmulti- component structure shrinks to a single narrow line (Fig. 4c , red line) at some intermediate \ndirection of magnetic field, which is consistent with the calculated angular dependences.   \nIn this case, due to the linear  relation 4πMeff = Hresout – 2πF/γ = Hresout – 3.33 kG  , the \nintegrated FMR spectra for out -of-plane magnetic field , provide the distribution density of the \n 4πMeff  parameter within the sample (Fig. 4b) . The ranges of these distributions for different \nsamples are seen from the spectra and  summarized in the Table  1. Evidently, the samples #3 and \n#4 have regions with a very small effective magnetization.   8 \n The propagation of spin waves was measured using a couple of 2 mm x 30 µm antennas \nseparated by 1.2 mm. A magnetic field of 982  Oe was applied in the film plane perpendicular to \nthe spin wave propagation direction. Figure 4 d presents amplitude -frequency characteristics \n(scalar gains) S 21 of the spi n waves propagated in sample #2. The values of 4 πMeff in different \nstructures calculated  from SWP spectra are presented in Table 1. It is noteworthy that \ncomplicated shape of FMR absorption for sample #2 in Fig. 4b correlates with that of S 21 \nspectral dependence in Fig. 4d.  \nSummarizing the above, one can conclude, that the considerable decrease of the effective \nmagnetization 4πMeff  observed in the YIG  / NdGG heterostructures grown by LMBE allows us \nto expect that by further optimization of the growth conditions and architect ure of the \nheterostructure, one can obtain YIG films with an out -of-plane easy magnetization axis. The \nmanifestation of the quadratic in magnetization contribution to the reflected light ellipticity \nduring in- plane magnetization reversal can be explained b y the induced magnetization of Nd3+ \ninterface ions caused by superexchange interaction with Fe3+ ions of the YIG layer. The \nneodymium magnetization is coupled to that of the iron in YIG, as evidenced by the anisotropy of the quadratic magnetooptical phenom ena observed upon magnetization reversal and drastic \nincrease of quadratic contribution with temperature decrease.  \n \nThe LMBE growth of YIG films and experiment on spin wave propagation w ere supported by \nRussian Science Foundation (project No 17- 12-01508). Magnetooptical, vibrating sample \nmagnetometry and ferromagnetic resonance measurements were supported by R ussian \nFoundation for Basic Research (project N o 16-02-00410). The authors thank V.N. Smelov for \nthe assistance in LMOKE measurements.  \n \nReferences  \n                                                           \n1 V.V. Kruglyak, S.O. Demokritov, D. Grundler, J. Phys. D:  Appl Phys. 43, 264001(2010).  \n2  S.A. Nikitov, D. V. Kalyabin, I. V. Lisenkov, A. N. Slavin, Yu. N. Barabanenkov, S. A. Osokin, A. V. \nSadovnikov, E. N. Beginin, M. A. Morozova, Yu. P. Sharaevsky, Yu. A. Filimonov, Yu. V. Khivintsev, \nS. L. Vysotsky, V. K. Sakharov, E. S. Pavlov, Physics -Uspekhi 58, 1002 (2015).  \n3 D. Grun dler,  J. Phys. D:  Appl Phys. 49, 391002 (2016).  \n4 A.V. Chumak, V. I. Vasyuchka, A. A.Serga, and B. Hillebrands, Nature Physics 11, 453 (2015).  \n5 B.M. Howe, S. Emori, H.- M. Jeon, T.M. Oxholm, J.G. Jones, K. Mahalingam, Y. Zhuang, N.X. Sun, \nG.J. Brown, IEEE  Magn. 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Melkov, Magnetization Oscillations and Waves (New York: CRC, 1996).  "    },    {        "title": "2304.09627v1.Nonreciprocal_ultrastrong_magnon_photon_coupling_in_the_bandgap_of_photonic_crystals.pdf",        "content": "Nonreciprocal ultrastrong magnon-photon coupling\nin the bandgap of photonic crystals\nChi Zhang,1Zhenhui Hao,1Yongzhang Shi,1Changjun Jiang,1, C. K. Ong1;2, and Guozhi Chai1\u0003\n1Key Laboratory for Magnetism and Magnetic Materials of the Ministry of Education,\nLanzhou University, Lanzhou, 730000, People's Republic of China.\n2Department of Physics, Xiamen University Malaysia,\nJalan Sunsuria, Bandar Sunsuria, 43900, Sepang, Selangor, Malaysia.\n(Dated: April 20, 2023)\nWe observe a nonreciprocal ultrastrong magnon-photon coupling in the bandgap of photonic\ncrystals by introducing a single crystal YIG cylinder into copper photonic crystals cavity as a point\ndefect. The coupling strength reaches up to 1.18 GHz, which constitutes about 10.9% of the photon\nenergy compared to the photon frequency around 10.8 GHz. It is fascinating that the coupling\nachieves unidirectional signal transmission in the whole bandgap. This study demonstrates the\npossibility of controlling nonreciprocal magnon-photon coupling by manipulating the structure of\nphotonic crystals, providing new methods to investigate the in\ruence of magnetic point defects on\nmicrowave signal transmission.\nI. INTRODUCTION\nCavity magnonics, which is based on coupling between\nmagnons and cavity photons, has become a powerful plat-\nform for studying the hybrid quantum systems [1{5]. In\nthis regime, information is carried and transmitted by\nthe polarons that are generated by the coupling between\nmagnons and photons. Since Soykal and Flatt\u0013 e proposed\nmagnon-photon coupling (MPC) in 2010 [6, 7], many\nstudies have demonstrated it in experiments and theories.\nHuebl et al. achieved a strong coupling with YIG \flms\nand coplanar waveguides at mK temperatures in 2013\n[8]. Soon after, Zhang et al. accomplished experiments\nat room temperatures with an yttrium iron garnet (YIG)\nsphere and a three-dimensional (3D) microwave cavity\n[9]. A number of researchers have also reported a lot\nof interesting and valuable studies, like dissipative cou-\nplings [10, 11], indirect couplings [12{14], nonreciprocal\ncouplings [15{18] in coupled systems. These studies show\ngreat promise in a wide range of applications in quan-\ntum information processing. Most importantly, strong\ncoupling is necessary in the coupling system for broaden-\ning the frequency range. Researchers have achieved the\nstrong MPC in di\u000berent cavity systems with the magnons\nwhich have large spin density, such as 3D rectangular\ncavity [19, 20], split-ring resonator [21{23], inverted pat-\ntern of split-ring resonator [11], cross-line cavity [15] and\nphotonic crystals (PCs) [24].\nPCs are a kind of arti\fcial inhomogeneous electromag-\nnetic structures with de\fnite refractive index or peri-\nodic dielectric constant [25{27]. Wang et al. introduced\none-way modes analogous to quantum Hall edge states\ngeneralized with gyromagnetic materials [28], and ob-\nserved the so-called chiral edge states [29] as well as in\nother research area [30{33]. In 2015, Skirlo et al. con-\nstructed a ferrimagnetic PCs with a band structure com-\n\u0003Correspondence email address: chaigzh@lzu.edu.cnprising high chern numbers and the dispersion relations\nof one-way edge modes for the \frst time in any quan-\ntum Hall e\u000bect or quantum anomalous Hall e\u000bect sys-\ntem [34]. Liu and Houck discovered an attractive ex-\nperimental phenomenon that a hybridization induced by\nthe strong MPC can create localized cavity modes that\nlive within the photonic bandgap [35]. Built on these in-\nteresting investigations, we realized an ultrastrong MPC\nwith a coupling strength of 1.05 GHz and the coupling\ne\u000eciency comes up to 11.7% in PCs with a ferrimagnetic\npoint defect in 2019 [24]. This research indicates that\nPCs is an appropriate system to investigate the cavity\nmagnonics .\nIn this work, a nonreciprocal ultrastrong coupling is\nrealized in the band gap of the PCs with a magnetic point\ndefect. Results show an ultrastrong interaction between\nthe ferromagnetic resonance (FMR) mode and the defect\nmode of the PCs. In addition to broken time-reversal\nsymmetry (TRS) of the system, microwave is allowed to\ntransmit along one direction in the bandgap by tuning\nstrength and direction of the applied magnetic \feld.\nII. EXPERIMENTS\nAs illustrated in Fig. 1, our device consists of a 2D\nPCs with a point defect by substituting a YIG cylinder\nfor a copper cylinder. A 2D chamber is constructed by\ntwo aluminum plates with 5mm of separation and sur-\nrounded by some microwave absorbing materials. All of\nthe cylinders which have a diameter and a height as 5 mm\nare placed in the chamber and the lattice constant of the\n2D simple cubic structure is de\fned as 20 mm. A vector\nnetwork analyzer (VNA, Agilent E8363B) is employed to\nfeed microwaves by connecting to two antennas. Mean-\nwhile, a static magnetic \feld along zdirection is applied\nby an electromagnet. We choose a single crystal YIG\nas the magnetic material because of its low microwave\nmagnetic-loss parameter and high-spin-density [36]. Our\nYIG has a saturation magnetization Ms= 1750 G, a gy-arXiv:2304.09627v1  [physics.app-ph]  19 Apr 20232\nromagnetic ratio \r= 2.8 MHz/Oe, and a linewidth of 10\nOe.\nFIG. 1. Sketch of the experimental setup. The two-\ndimensional copper cylinder PCs with a point defect of a YIG\ncylinder. All cylinders have a diameter of 5 mm and a height\nof 5 mm. The lattice constant is 20 mm. Two aluminum\nplates are applied to be the two-dimensional chamber with\n5 mm of separation and surrounded by some microwave ab-\nsorbing materials. Microwave signals are supplied by a VNA\nthrough two microwave cables with two antennas. The static\nmagnetic \feld is applied in the zdirection.\nThe scatter parameter Sij(i, j = 1, 2) is measured to\ncharacterize the experimental phenomena discussed sub-\nsequently.Sijrepresents the microwave transmission sig-\nnal from port jto porti. Figure 2(a) denotes the trans-\nmission coe\u000ecients S21of copper cylinder PCs and YIG\ndefect PCs as a function of frequency. Additional simu-\nlation results are introduced to understand experimental\nresults. The microwave magnetic \feld distribution at\n10.8 GHz in PCs and YIG defect PCs are illustrated as\nFigs. 2(b) and 2(c), respectively. It reveals that a strong\ngathering of the magnetic energy could be observed at\nthe position of YIG cylinder [black circle on Fig. 2(c)]. It\nimplies a localized mode within the band gap is excited.\nFocus on this eigen mode, we swept the applied static\nmagnetic \felds with a frequency range between 10.0 and\n11.5 GHz.\nIII. RESULTS AND DISCUSSION\nThe density mapping image of the magnitude of trans-\nmission coe\u000ecient is shown in Fig. 3. Figures 3(a) and\n3(b) display the density mapping images of the magni-\ntude of the microwave transmission coe\u000ecient S21and\nS12, respectively. The spectra were measured by altering\nthe static magnetic \feld Hfrom -9000 to 9000 Oe with\na step size of 18 Oe. An anti-crossing behavior arises\nowing to the coupling between magnons and photons.\nAccording to our experiments, magnons are supplied by\nFIG. 2. (a) Experiment data. Microwave transmission co-\ne\u000ecients S21of the PCs and YIG defect PCs as a function\nof frequency. (b) and (c) represents the simulation of the\nmicrowave magnetic \feld distribution of the PCs and YIG\ndefect PCs at 10.9 GHz, respectively. Black circle stands for\nthe position of the YIG cylinder.\nFMR mode and the photons are introduced by the eigen\ndefect mode of the microwave cavity. The FMR mode\n(black dashed line in Fig. 3) is calculated by the Kittel\nequation [37]:\nfK=\rq\n(H+ (Nx\u0000Nz)Ms)(H+ (Ny\u0000Nz)Ms):(1)\nThe demagnetizing factors cannot be calculated analyt-\nically as the demagnetizing \feld is not uniform in cylin-\ndrical shape magnetized bodies. We choose the experi-\nment results of the demagnetizing factors of the cylinder\nwith the same dimensional ratio in the textbook [38].\nConsequently, demagnetizing factors Nx,NyandNzare\nset a value as 0.365, 0.365 and 0.27, respectively. The\nfrequency of photon defect mode fPCis 10.8 GHz (red\ndashed line in Fig. 3). As shown in Fig. 3, the coupling\ndisplayed in the density mapping images of the trans-\nmission coe\u000ecients Sijare reversed while the direction\nof applied magnetic \feld His opposite. In addition, the\nS21andS12transmission coe\u000ecients are reversed also at\nthe same direction of H. The results indicate a nonre-\nciprocity with chirality in the system.\nUnlike the nonreciprocal microwave transmission in-\nduced by the cooperative e\u000bect of coherent and dissi-\npative coupling in an open cavity magnonic system [15],\nthe nonreciprocity arises from the gyromagnetism of YIG\ncylinder and the spatial symmetry breaking of PCs with\na defect in this work. The gyromagnetism of the YIG\ncylinder enhances in the vicinity of the resonance \feld,\nwhich reported in some previous studies [30{33]. Specif-\nically, the permeability \u0016will turn into a second-rank3\n(a)\n(b)\nFIG. 3. The density mapping image of the amplitude of the transmission coe\u000ecients through the PCs cavity as a function of\nfrequency fand the applied static magnetic \feld H. The closer the color to blue, the larger is the microwave transmission loss.\n(a)S21mapping. (b) S12mapping. Black dashed line indicates the FMR mode \ftting with the Kittel equation [Eq. 1] and red\ndots represents the photon mode. S21andS12are reversed with the same direction of Hin the region of coupled resonances.\nS21orS12is reversed with the opposite direction of Hin the region of coupled resonances.\ntensor \u0016as the the microwave magnetic \feld is perpen-\ndicular to the applied magnetic \feld H.\u0016is given by:\n\u0016=0\nBB@\u0016r\u0000i\u0016i0\ni\u0016i\u0016r0\n0 0 11\nCCA; (2)\nwhere\u0016r= 1 +f0fm=(f2\n0\u0000f2), which represents the\nin-plane permeability, \u0016i=\u0000ffm=(f2\n0\u0000f2). Here,\nfm=\rMsis the characteristic frequency [39]. The o\u000b-\ndiagonal element in the permeability tensor is induced by\nthe nonzero applied static magnetic \feld, which breaks\nthe TRS directly [40]. The degree of breakage of TRS is\ncharacterized as u=\u0016i=\u0016r[31]. For instance, ucomes\nup to a value of 98.9% as H= 2000 Oe and f= 11.0\nGHz. This implies a near complete TRS breaking.\nAdditionally, the magnetic defect is not located on the\naxis of symmetry of the PCs, spatial symmetry of the\nPCs cavity is broken, which is also helpful for breaking\nthe TRS of the system (Researchers often introduce fer-\nrites into resonant cavities with irregular shapes to breakthe TRS of the systems in many studies about quantum\nchaos.) [41{44]. Above of all, the nonreciprocity can\nbe attributed to the change in the rotational direction\nresulting from the interchange of the input and output\nchannels, which based on the properties of special cavity\nmagnonic system in this work.\nWe next calculate the coupling strength between YIG\nFMR mode and defect mode of YIG defect PCs. Firstly,\nthe microwave photon and FMR are described by the\nHamiltonian with a rotating-wave approximation (RWA)\n[9]:\nH=0\n@fK+i\u000b g\ng f P+i\f1\nA: (3)\nThe coupling curves are calculated by the two-mode\nmodel [8]:\nf\u0006=1\n2(fp+fK)\u00061\n2q\n(fp\u0000fK)2+ (2g=2\u0019)2;(4)\nwherefpis the photon frequency of the defect mode,\nf\u0006are frequencies of the coupled resonances and g=2\u00194\nis the coupling strength with a value of 1.18 GHz. As\nshown in Fig. 4, f\u0006are described by green solid lines.\nThe calculated coupling e\u000eciency g=(2\u0019fp) = 10.9% of\nthe photon energy when fp= 10.8 GHz of the photon\nmode which is quali\fed as ultrastrong coupling [45]. In\nthe ultrastrong coupling region, the stronger the coupling\nstrength, the stronger is the nonreciprocity [17].\n(a)\n(b)\nFIG. 4. The couplings in the frequency versus applied static\nmagnetic \feld with the \ftting curves. (a) S21direction; (b) S12\ndirection. The experiment data are described as blue cir-\ncles. Black dashed line indicates the FMR mode \ftting with\nthe Kittel equation [Eq. 1]; red dots indicate the photon\nfrequency of defect mode located at 10.8 GHz; green curves\nare coupling curves \ftting with the two-mode model [Eq. 4].\nBandgap is marked by dashed gray box.\nRight bottom of Fig. 4 reveals another ultrastrong\nMPC expressed in orange curve. We design another ex-\nperiment to understand the nature of the coupling. Note\nthat the magnon mode and photon mode are provided\nby the YIG cylinder simultaneously [46{48].\nAs shown in the inset of Fig. 5(a), a microwave coax-\nial connector connected to the VNA is used to feed mi-\ncrowaves for the YIG cylinder. In this system, YIG cylin-\nder is regarded as a microwave cavity itself and the \frst\neigenmode is discovered at 12.2 GHz which is evident\nfrom Fig. 5(a). Correspondingly, this eigenmode could\nbe regarded as the photon mode to interact with FMR\nmode. Subsequently, we applied static magnetic \feld H\non the YIG to discover the variation of the microwave\nre\rection coe\u000ecients S11. As anticipated, we observed\nan anticrossing phenomenon caused by FMR mode and\nphoton mode of the YIG. In Fig. 5(b) details of the\nspectrum of the coupling in the frequency versus applied\nstatic magnetic \feld, blue dots stand for the experiment\ndata, black and red dashed lines represents FMR mode\nand photon mode of YIG cylinder, respectively. The cou-\nFIG. 5. (a) Microwave re\rection coe\u000ecient S11raw data of\nthe YIG cylinder as a function of frequency at the magnetic\n\feld of 0. The inset stands for the experiment system. (b) The\ncouplings in the frequency versus applied static magnetic \feld\nwith the \ftting curves. The experiment data are described as\nblue circles. Black dashed line indicates the FMR mode \ftting\nwith the Kittel equation [Eq. 1]; red dashed line indicates the\nresonance frequency of photon mode located at 12.2 GHz;\norange curves are coupling curves \ftting with the two-mode\nmodel [Eq. 5].\npling curves could be \ft by two-mode model also:\nf0\n\u0006=1\n2(fY+fK)\u00061\n2q\n(fY\u0000fK)2+ (2g0=2\u0019)2;(5)\nwherefYis the photon frequency of the defect mode, f0\n\u0006\nare frequencies of the coupled resonances and g0=2\u0019is\nthe coupling strength with a value of 4 GHz. As shown\nin Fig. 5,f0\n\u0000is described by orange curve. It is observed\nfrom the results that polarization mode f0\n\u0000is adaptive\nfor orange solid line as shown in Fig. 4.\nIV. SUMMARY\nIn summary, we introduces a magnetic point defect\nby replacing a copper cylinder with a single crystal YIG\ncylinder, which is not located on the axis of symmetry\nin the 2D simple cubic metal PCs, breaking the spatial\nsymmetry of the system. A defect mode at a frequency\nof 10.8 GHz in the bandgap could be found. This defect\nmode can be regarded as a photon mode that couples\nwith the FMR mode of YIG, with a coupling strength5\nof 1.18 GHz. As the coupling e\u000eciency exceeding 10%,\nreaching up to 10.9%, the coupling can be considered as\nan ultrastrong MPC. Furthermore, the coupling displays\nstrong nonreciprocity, enabling unidirectional microwave\ntransmission within the bandgap of PCs. This study con-\ntributes to a better understanding of the magnetic defect\nin PCs, which is of great signi\fcance for exploring the\nfundamental principles and applications of PCs and for\ndeveloping new microwave devices. In particular, it sheds\nsome light on providing new ideas and directions for de-vices used in the \feld of microwave \fltering.\nV. ACKNOWLEDGEMENTS\nThis work is supported by the National Natural Sci-\nence Foundation of China (NSFC) (No. 12174165) and\nNatural Science Foundation of GanSu Province for Dis-\ntinguished Young Scholars (No. 20JR10RA649).\n[1] X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, Sci. Adv.\n2, e1501286 (2016).\n[2] M. Harder and C.-M. Hu, Solid State Phys. 69, 47 (2018).\n[3] B. Bhoi and S.-K. Kim, Solid State Phys. 70, 1 (2019).\n[4] C.-M. Hu, Solid State Phys. 71, 117 (2020).\n[5] B. Zare Rameshti, S. Viola Kusminskiy, J. A. Haigh,\nK. Usami, D. Lachance-Quirion, Y. Nakamura, C.-M.\nHu, H. X. Tang, G. E. Bauer, and Y. M. Blanter, Phys.\nRep.979, 1 (2022).\n[6]O. O. Soykal and M. E. Flatt\u0013 e, Phys. Rev. Lett. 104,\n077202 (2010).\n[7]O. O. Soykal and M. E. Flatt\u0013 e, Phys. Rev. B 82, 104413\n(2010).\n[8] H. Huebl, C. W. Zollitsch, J. Lotze, F. Hocke, M. Greifen-\nstein, A. Marx, R. Gross, and S. T. B. Goennenwein,\nPhys. Rev. Lett. 111, 127003 (2013).\n[9] X. Zhang, C.-L. Zou, L. 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Mater.521, 167536 (2021)."    },    {        "title": "1901.02129v1.Fabrication_of_yttrium_iron_garnet_Pt_multilayers_for_the_longitudinal_spin_Seebeck_effect.pdf",        "content": "1 \n Fabrication of yttrium–iron–garnet/Pt multilayers for the longitudinal spin Seebeck \neffect \n \nTatsuhiro Nozue1,a), Takashi Kikkawa2,3,b), Tomoki Watamura2, Tomohiko Niizeki3, \nRafael Ramos3, Eiji Saitoh2,3,4,5,6, and Hirohiko Murakami1 \n \n1Future Technology Research Laboratory, ULVAC, Inc., Tsukuba 300-2635, Japan \n2Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan \n3Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan \n4Department of Applied Physics, The University of Tokyo, Tokyo 113-8656, Japan \n5Center for Spintronics Research Network, Tohoku University, Sendai 980-8577, Japan \n6Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, \nJapan \n  2 \n (ABSTRACT)  \nFor longitudinal spin Seebeck effect (LSSE) devices, a multilayer structure comprising \nferromagnetic and nonmagnetic layers is expected to improve their thermoelectric power. \nIn this study, we developed the fabrication method for alternately stacked yttrium–iron–\ngarnet (YIG)/Pt multilayer films on a gadolinium gallium garnet (GGG) (110) substrate, \nGGG/[YIG(49 nm)/Pt(4 nm)]n (n = 1–5) based on room-temperature sputtering and ex-\nsitu post-annealing method and we evaluated their structural and LSSE properties. The \nfabricated [YIG/Pt]n samples show flat YIG/Pt interfaces and almost identical saturation \nmagnetization Ms, although they contain polycrystalline YIG layers on Pt layers as well \nas single-crystalline YIG layers on GGG. In the samples, we observed clear LSSE signals \nand found that the LSSE thermoelectric power factor (PF) increases monotonically with \nincreasing n; the PF of the [YIG/Pt]5 sample is enhanced by a factor of ~28 compared to \nthat of [YIG/Pt]1. This work may provide a guideline for developing future multilayer-\nbased LSSE devices. \n \na) tatsuhiro_nozue@ulvac.com \nb) t.kikkawa@imr.tohoku.ac.jp \n  3 \n Recent advances in the miniaturization of computing and mobile devices have increased \nthe need for small power generators. In this context, the longitudinal spin Seebeck effect \n(LSSE) has gained attention, since it enables thermoelectric power generation with a \nsmall and simple structure consisting of a bilayer of ferromagnetic material (FM) and \nnonmagnetic metal (NM).1–4 In an LSSE, when a temperature gradient is applied in the \nbilayer, a magnon flow in the FM is converted into a spin current in the NM by the \ninterfacial exchange interaction. Subsequently the spin current in the NM generates a \nthermoelectric voltage via the inverse spin Hall effect, if the spin–orbit coupling in the \nNM is strong enough (e.g., Pt).5–9 \nTheoretically, high thermoelectric efficiency is expected from LSSE devices, since their \noutput power is not limited by the Wiedemann–Franz law for conventional thermoelectric \nelements based on the (charge) Seebeck effect.2,4,10 However, in practice, the LSSE \nthermoelectric power is very small.4 This has driven a search for materials to improve the \nLSSE power.2,4 Besides, the structural design of thermoelectric elements is also a research \ntarget to increase LSSE power.2,4 Recently, LSSE power was found to be enhanced in \nalternately stacked FM/NM multilayers.4,11–15 Ramos et al. reported [Fe3O4/Pt]n \nmultilayer LSSE devices formed by repeated growth of n ferromagnetic oxide Fe3O4 and \nNM Pt bilayers, using pulsed laser deposition (PLD) and sputtering, respectively.12,13 \nThey found that the voltage as well as the power generated by the LSSE increase with \nincreasing n. They attributed the LSSE voltage enhancement to the increase in spin-\ncurrent amplitude in the multilayers, which also enhances the LSSE power in combination \nwith the increase in electrical conduction with n, because of the parallel connection of the \nconductive Pt layers.4,12 A similar experimental result was reported also in [CoFeB/Pt]n 4 \n multilayers.11 Hence, the design of the FM/NM multilayers has received attention in terms \nof fundamental physics as well as applications.4 \nIn this letter, we report the development of a fabrication method for high-quality \n[YIG/Pt]n multilayers, formed by n times repeated growth of YIG/Pt bilayers based on \nsputtering and post-annealing, and show their structural and magnetic properties as well \nas the LSSE results. Yttrium–iron–garnet (YIG: Y3Fe5O12) is a ferrimagnetic insulator \nexhibiting an exceptionally long magnon-diffusion length and is the fundamental material \nfor LSSE and magnonics research.4,16 In addition, because of its highly insulating \nproperty, the use of YIG in LSSE devices allows us to avoid a possible overlap with the \nanomalous Nernst effect of the FM layer, which may have been an issue in previous \nferromagnetic metals Fe3O4- and CoFeB-based LSSE devices.11-13 [YIG/Pt]n multilayers \nare fascinating for future LSSE-based thermoelectric application and other spintronic \ndevices based on, for instance, magnon-mediated current drag and magnon valve effects. \n17–21 However, there is no report focusing on its fabrication, the reason of which may be \ndue to the difficulty in stacking YIG/Pt bilayers with maintaining clean and flat interfaces. \nIn general, for crystallizing YIG layers, high-temperature annealing (at ~ 800℃) is \nnecessary.17 However, there are two fundamental problems in this high temperature \nprocess: (1) Pt films on YIG layers are easily deformed and may become aggregations \nand (2) the constituents such as oxygen tend to be easily desorbed from YIG layers for \nlong-time high-temperature annealing. To obtain high quality [YIG/Pt]n multilayers by \novercoming these issues, we developed the method based on in-situ n-times YIG/Pt \nrepetition sputtering and one-time rapid-thermal-annealing (RTA) process with a face-to-\nface proximity configuration. Here, each Pt (YIG) layer becomes robust against the 5 \n deformation/aggregation (desorption), since each Pt (YIG) is sandwiched by YIG (Pt) \nlayers and also the total annealing time is reduced by the RTA process, giving the solution \nto the above problems. Our versatile [YIG/Pt]n fabrication method reported here will \nexpand the scope of application of [YIG/Pt]n spintronic devices and LSSE-based \nthermoelectric power generators.  \nThe multilayer samples were deposited on 10×10×0.5 mm3 Gd3Ga5O12 (GGG) (110) \nsubstrates by a dc- and rf-magnetron sputtering system, QAM-4 STS (ULVAC Kyushu), \nat room temperature (RT). The QAM-4 is equipped with four on-axis sputtering cathodes \nwith 50-mm-diameter targets in a high-vacuum chamber. Before sputtering, GGG \nsubstrates were annealed in a furnace at 900 ºC in air for 30 min, in a face-to-face \nproximity configuration. Then, the YIG and Pt films were sputtered alternately and \nsequentially without breaking the vacuum to form GGG/[YIG/Pt]n with clean interfaces. \nHere, YIG films were deposited at an rf-power of 150 W using a sputtering gas of Ar + 2 \nvol.% O2 with a pressure of 0.10 Pa, at a deposition rate of 0.26 nm min−1. Pt films were \ndeposited at a dc-power of 20 W and Ar pressure of 0.10 Pa, at a rate of 1.14 nm min−1. \nSince the as-deposited YIG films were amorphous and nonmagnetic, post-annealing was \nnecessary to crystallize the YIG layers. A thin YIG layer (~2 nm) was deposited on the \ntop Pt layer as a protective layer to prevent deformation of the top Pt film during the post-\nannealing. The ex-situ RTA in air was performed, where the multilayered samples were \nheated up to 825 ºC by taking 60 s and were kept for only 200 s, during which the YIG \nfilms became crystallized. In the RTA process, we adopted the face-to-face proximity \nconfiguration, in which the surfaces of the top YIG layers (~2 nm) of two identical \nGGG/[YIG/Pt]n/cap-YIG samples are faced with each other to avoid possible desorption \nof the constituents, such as Y and Fe, from their surfaces. 6 \n For LSSE measurements, to apply a temperature difference, ΔT, the sample was \nsandwiched between two AlN plates. The lower AlN plate, which was in contact with the \nbottom of the GGG substrate, was heated using a Peltier module, while the temperature \nof the upper AlN plate, in contact with the top surface of [YIG/Pt]n (with a contact area \nof 5×2 mm2), was kept at RT by connecting a heat bath. To measure ΔT, thermocouples \nwere attached to the upper and bottom AlN plates.22 We note that, although the estimated \nΔT value may contain error due mainly to interfacial thermal resistance between the \nsample and AlN plates,23-25 its variation between each measurement was confirmed to be \nnegligibly small (only 1.2% variation in the SSE voltage VSSE for 6-times different \nmeasurements using the same sample). This situation allows us to compare the LSSE \ncoefficient S = (VSSE/Lsample)/(ΔT/Dsample)4 between the [YIG/Pt]n samples with different \nn number [Lsample = 5 mm (Pt length), Dsample = 0.5 mm (GGG thickness)]. With the \napplication of a constant ΔT of 3 or 6 K, the voltage, VSSE, generated in the film was \nrecorded as a function of the magnetic field H applied along the [1–10] direction of the \nGGG substrate.  \nFirst, we briefly evaluated the surface morphology of GGG, GGG/YIG, and \nGGG/YIG/Pt/thin-YIG. Figure 1(a) shows the 1×1 μm2 atomic force microscope (AFM) \nimage of the annealed GGG (110) substrate. The surface exhibits a step-and-terrace \nstructure with a root-mean-squared (RMS) roughness of 0.13 nm. We found that the \nsurface of YIG deposited on GGG (110) after the RTA at 825 ºC exhibits a similar step-\nand-terrace structure, with an RMS of 0.10 nm [see Fig. 1(b)]. The result shows that YIG 7 \n fabrication by RT sputtering and RTA can produce an extremely flat YIG film, \ncomparable to the GGG substrates. Figure 1(c) shows an AFM image of the \nGGG/YIG/Pt/thin-YIG. Although no step-and-terrace structure was confirmed, its RMS \nroughness was 0.44 nm, which is smooth enough to grow further [YIG/Pt] multilayer \nfilms. \nBy X-ray reflection (XRR) measurements (not shown), we determined the average \nthickness of the YIG, Pt, and YIG protection layers to be 49.2±0.5 nm, 4.00±0.08 nm, \nand ~2.2 nm, respectively, so that the present sample structure can be written as GGG-\nsubst./[YIG(49 nm)/Pt(4 nm)]n/YIG(2 nm), n = 1–5. By XRR, the averaged values of \ninterfacial roughness were evaluated to be 0.29, 0.30, and 0.33 nm for n =1, 2, and 3, \nrespectively. \nFigures 1(d)–(g) display the transmission electron microscope (TEM) images of a cross \nsection of n = 3 viewed along the [1–10] direction. The overall image shown in Fig. 1(d) \nreveals that each layer has a smooth surface/interface and is grown without macroscopic \ndefects. The high-resolution (HR) image around the first-YIG/GGG interface [Fig. 1(e)] \nshows that the YIG layer is single crystalline and grown epitaxially on GGG. In contrast, \nthe HR images around the first-YIG/first-Pt/second-YIG [Fig. 1(f)] and third-YIG/third-\nPt [Fig. 1(g)] interfaces show that the YIG interlayers on Pt layers are grown as \npolycrystals [see the oblique lines in the HR images of the second- and third-YIG films, \nwhich are different from the first YIG (110) film]. The result could be attributed to the \nabsence of seed GGG crystals for the YIG interlayers, unlike the first single-crystalline \nYIG layer; the as-deposited amorphous YIG layers on Pt are transformed, during the RTA, \ninto polycrystalline YIG having many single-crystal grains with various crystalline 8 \n orientations. From the HR TEM images shown in Figs. 1(f) and 1(g), we confirmed that \nstriped patterns exist in the Pt layers parallel to the interfaces. The distance between each \nstripe is ~0.23 nm, comparable to the (111) plane of Pt. This result suggests that, although \nthe Pt layers are polycrystalline, the <111> axis is oriented almost perpendicular to the \nplane.  \nTo obtain further information on the interfaces and element profile, we performed HR \nHAADF–STEM (or Z-contrast) observation. Figure 1(h) shows the Z-contrast image \naround the YIG/GGG interface of n = 3. We found that the atomic arrangements of the \nfirst YIG agree with the <110> projection of the YIG lattice and exhibit the characteristic \npattern of alternating Y and Fe atoms.26 In Fig. 1(i), we show the Z-contrast image around \nthe first-YIG/first-Pt/second-YIG interfaces. Small but finite regions of intermediate \nbrightness were observed between the Pt and YIG layers (within ~1 nm, slightly greater \nthan the interfacial roughness). From STEM–EDX line-scan measurements, we found \nthat such regions contain mixed Pt, Y, Fe, and O elements, which could be attributed to \npossible inter-diffusion due to the high-temperature annealing process.20 From STEM–\nEDX measurements, we also evaluated the chemical composition of the YIG layers and \nfound an off-stoichiometric behavior of Y : Fe : O = 3 : 4.3 : 12, similar to the sputtered \nYIG films in Ref. 27. \nFigure 2 shows the X-ray diffraction (XRD) 2θ–θ plots of the n = 1, 3, and 5 samples. \nFor all three samples, a diffraction peak was observed at 40.85º, slightly lower than that \nof GGG (440) diffraction, and its intensity (270 cps) was almost identical for all the \nsamples. We assign the peak to the (440) diffraction from the first-YIG layer epitaxially \ngrown on GGG, as confirmed in the TEM images. For n = 3 and 5, we also observed other 9 \n peaks at 28.73º (for n = 5) and 32.22º (for n = 3 and 5), which can be assigned to the YIG \n(400) and (420) diffractions, respectively, and may originate from the polycrystalline YIG \ninterlayers. For all the samples a broad peak with Laue oscillations was observed at \naround 39.7º and can be attributed to the Pt (111) diffraction, as indicated in the TEM \nimages [Figs. 1(f) and 1(g)].  \nFrom the peak position of YIG (440) diffraction, the out-of-plane lattice constant of the \n(first) single-crystalline YIG layers was determined to be 1.2486 nm, larger than that of \nboth GGG (1.2383 nm) and bulk YIG (1.2376 nm). These results are consistent with \nprevious reports on the epitaxial YIG thin films on GGG by PLD and sputtering,26,28–33 \nand can be attributed to the off-stoichiometry of YIG.29,32 \nSubsequently, we investigated the magnetization, M, and ferromagnetic resonance \n(FMR) spectrum to reveal the magnetic properties of the YIG layers at RT. Figure 3(a) \nshows the M–H curve of samples n = 1 to 5. The intensity of M was found to be \nproportional to the number of layers, which means that the saturation magnetization per \nvolume, MS, is almost equal for all the layers (MS = 108.00 ± 0.34emu cm−3) and that the \nM magnitude of each YIG layer takes almost the same value. The MS value is \napproximately 20 % smaller than that of bulk YIG (140 emu cm−3), which may be caused \nby the deficiency of Fe in our YIG films. In fact, an MS value of 103 emu cm−3 is also \nreported, via FMR, for the single-crystalline YIG films sputtered on GGG with the \nchemical composition of Y : Fe = 3 : 4.4 (Fe deficiency).27 Figure 3(b) shows the in-plane \nFMR derivative absorption spectra of n = 1, 2, and 3, measured with microwaves of 1 \nmW at 9.45 GHz. For n = 1, sharp FMR absorption was observed at 2290 Oe with a peak-\nto-peak linewidth, ΔH, of 4.25 Oe. This linewidth is comparable to the reported values 10 \n for the epitaxial YIG films prepared by PLD and sputtering.26–28,30,32–34 For n = 2 and 3, \nat the same H, we also observed sharp absorptions with values of ΔH of 4.62 and 3.97 \nOe, respectively, which can be attributed to the FMR absorption of their first single-\ncrystalline YIG layers, and indicates that the first YIG layers show similar magnetic \ndamping irrespective of the repetition-growth number n. In addition, for n = 2 and 3, we \nobserved weak and broad absorptions at 2660 Oe with a ΔH of more than 50 Oe. The \nappearance of such peaks can be interpreted as the absorption of the polycrystalline YIG \ninterlayers on Pt layers. The broadening of the absorption peak may originate from an \nenhanced magnetic damping due to grain-boundary scattering for the polycrystalline YIG \nlayers, which is absent from the first single-crystalline YIG layers. \nFinally, we investigated the LSSE for the [YIG/Pt]n multilayer. Before the LSSE \nmeasurement, we measured the resistance of the samples by the 4-probe method and \nfound that the sheet resistance, RS, of samples n = 1 to 5 is almost inversely proportional \nto n (RS = 50.8, 26.0, 17.0, 13.7, and 10.5 Ω for n = 1, 2, 3, 4, and 5, respectively). This \nmeans that the resistivity of each Pt layer is almost identical. Figure 4(a) shows the LSSE \nresults for n = 1 to 5. By applying ΔT, clear LSSE signals were observed for all the \nsamples and the LSSE curves were found to display hysteresis loops similar to the M−H \ncurve [see also Fig. 3(a)]. Although the M intensity is proportional to n, the saturated \nvalues of the LSSE coefficient, SS, exhibit a different behavior against n. The SS value of \nn = 1 [SS(1)] is 0.18 μVK−1. SS(2) = 0.38 μVK−1 for n = 2, being 210% as large as that \nof SS(1) [see also Fig. 4(b)]. For more stacked multilayers, SS(3) = 0.37 and SS(4) = 0.38 11 \n μVK−1, giving almost the same value as that of SS(2). The SS value for n = 5 is SS(5) = \n0.43 μVK−1, slightly larger than those for n = 3 and 4, and SS(5)/SS(1) = 238% [see also \nFig. 4(b)].  \nThe observed enhancement and almost saturated behavior of SS when n ≥ 2 is different \nfrom the previous report in [Fe3O4/Pt]n multilayers.12 In Ref. 12, SS(n)/SS(1) increases to \n255% at n = 2, then it monotonically increases with increasing n, and it finally takes the \nmaximum value of 370% at n = 6, greater than our present value. In Ref. 12, to interpret \nthe LSSE enhancement, the authors theoretically propose that the interfacial spin-current \ncontinuity between Fe3O4 and Pt leads to an increased amplitude of spin current in \n[Fe3O4/Pt]n multilayers, which is more than twice higher than that in a Fe3O4/Pt bilayer. \nOn the other hand, our results can be explained simply by the double transfer of the spin \ncurrents into the Pt layer from the upper and lower YIG layers. The multilayer [YIG/Pt]n \nfor n ≥ 2 has (n−1) sets of the sandwich structure of YIG/Pt/YIG. In such Pt layers, the \nspin currents are transferred from YIGs through the upper and lower interfaces. We here \ninfer that the longitudinal magnon spin-current has the same magnitude in all the YIG \nlayers, being motivated by the fact that (i) each YIG layer has an almost identical \nsaturation magnetization and (ii) the robustness of spin current against YIG crystallinity \nis reported for YIG/Pt systems through LSSE measurements.35 As a result, the spin-\ncurrent amplitude of Pt in YIG/Pt/YIG takes the value twice as large as that in YIG/Pt. In \nthis scenario, the open circuit voltage for [YIG/Pt]n may change by a factor of (2−1/n) \ncompared to n = 1, resulting in an increase in the LSSE coefficient SS with n and giving \nthe maximum SS value twice larger that of n = 1 for n ≫ 1. In Fig. 4(b), we compare the 12 \n SS values calculated by the above model (dashed line) with those obtained experimentally. \nWe found reasonable agreement between them, although the experimental results show a \nslightly rapid increase with n. Therefore, the mechanism proposed here may play an \nimportant role in the present LSSE enhancement and saturation. We attribute the \ndiscrepancy between the previous [Fe3O4/Pt]n and present [YIG/Pt]n results to the \ndifference in the samples’ interfacial condition; the interfaces of the [Fe3O4/Pt]n \nmultilayers are highly epitaxial, while those of the [YIG/Pt]n are not epitaxial. This may \ncause an increased loss of spin current at the YIG/Pt interfaces than at the Fe3O4/Pt \ninterfaces and may make the previously-assumed spin-current continuity condition in Ref. \n12 unreasonable for the present [YIG/Pt]n systems.  \nAlthough the LSSE coefficient is almost saturated when n ≥ 2, the resistance of the \nsamples decreases monotonically with increasing n, meaning that the power factor (PF) \nstill increases with n. In Fig. 4(c) we plot the PF, defined as SS2/RS.4 The PF value of n = \n5 is 0.02 pWK−2, which is 27.5 times greater than that of n = 1. The results show that the \n[YIG/Pt]n multilayers are beneficial in terms of PF. \nIn conclusion, we fabricated multilayers consisting of GGG/[YIG(49 nm)/Pt(4 \nnm)]n/YIG(2 nm) (n = 1 to 5) and investigated their structural and magnetic properties \nand their LSSE against the number of layers n. We show that well-crystallized [YIG/Pt]n \nwith flat interfaces can be grown by RT sputtering and subsequent ex-situ face-to-face \nRTA processes. Using these samples we observed clear LSSE signals and found that PF \nincreases monotonically with increasing n, although the LSSE coefficient S is almost \nsaturated when n ≥ 2. The knowledge about the [YIG/Pt]n fabrication reported here may \nbe important for industrial applications of LSSE multilayer elements. \n  13 \n (Acknowledgments) \nThis work was supported by ERATO “Spin Quantum Rectification Project” \n(JPMJER1402) from JST, Grant-in-Aid for Scientific Research on Innovative Area \n“Nano Spin Conversion Science” (JP26103005) from JSPS KAKENHI, and ULVAC, Inc. \n \n(References) \n1 K. Uchida, H. Adachi, T. Ota, H. Nakayama, S. Maekawa, and E. Saitoh, Appl. Phys. \nLett. 97, 172505 (2010). \n2 K. Uchida, M. Ishida, T. Kikkawa, A. Kirihara, T. Murakami, and E. Saitoh, J. Phys. \nCondens. Matter 26, 343202 (2014). \n3 A. 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Appl. Phys. 115, 17A501 (2014). \n32 B. M. Howe, S. Emori, H. Jeon, T. M. Oxholm, J. G. Jones, K. Mahalingam, Y. Zhuang, \nN. X. Sun, and G. J. Brown, IEEE Magnetics Lett. 6, 3500504 (2015). 16 \n 33 C. Hauser, T. Richter, N. Homonnay, C. Eisenschmidt, M. Qaid, H. Deniz, D. Hesse, \nM. Sawicki, S. G. Ebbinghaus, and G. Schmidt, Sci. Rep. 6, 20827 (2016). \n34 C. Tang, M. Aldosary, Z. Jiang, H. Chang, B. Madon, K. Chan, M. Wu, J. E. Garay, \nand J. Shi, Appl. Phys. Lett. 108, 102403 (2016). \n35 F. J. Chang, J. G. Lin, and S. Y. Huang, Phys. Rev. Materials 1, 031401 (2017). \n \n  17 \n (Figures) \n \nFIG. 1. (a)–(c) 1 μm × 1 μm AFM images of surfaces of the (a) GGG (110) substrate after \nannealing at 900 ºC, (b) YIG film deposited on GGG substrate after RTA at 825 ºC, and \n(c) YIG protection film deposited on GGG-subst./YIG/Pt(5 nm) after RTA at 825 ºC. (d)–\n(i) TEM and HAADF–STEM images of GGG/[YIG(49 nm)/Pt(4 nm)]3/YIG(2 nm), (n = \n3). (d) An overall TEM image. (e), (f), (g) HR TEM images of the interfaces indicated by \n18 \n the arrows labeled as (e), (f), and (g) in (d). (h) – (i) HR HAADF–STEM images of the \ninterfaces indicated by the arrows labeled as (h) and (i) in (d). \n \n \n 19 \n  \nFIG. 2. XRD 2θ–θ scan of n = 1, 3, and 5. The dashed lines indicate the YIG diffraction \npeaks. The spectra of n = 3 and 5 are shifted vertically for clarity. \n \n \n  \n20 \n  \nFIG. 3. (a) M–H curves of n = 1 to 5. (b) Derivative FMR absorption spectra of n = 1 to \n3. The absorption signals of n = 2 and 3 at a higher H of ~2660 Oe are magnified by \nmultiplying 60 for clarity. \n  \n21 \n  \nFIG. 4. (a) LSSE coefficient S versus H of n = 1 to 5. (b) Saturation LSSE coefficient Ss \nand (c) PF versus n. The dashed line in (b) denotes the calculated value of (2−1/n). \n"    },    {        "title": "2404.18712v1.Anomalous_Spin_and_Orbital_Hall_Phenomena_in_Antiferromagnetic_Systems.pdf",        "content": " \n \n \n \nAnomalous Spin and Orbital Hall Phenomena in Antiferromagnetic Systems   \nJ. E. Abrão ,1 E. Santos,1 J. L. Costa ,1 J. G. S. Santos,1 J. B. S. Mendes,2 and A. Azevedo1 \n \n1Department  of Physics,  Federal University  of Pernambuco , Recife, 50670 -901, Brazil   \n2Department  of Physics,  Federal  University of Viçosa,  Minas Gerais , 36570 -900, Brazil  \n \n \nWe investigate anomalous spin and orbital Hall phenomena in an tiferromagnetic (AF) materials via orbital \npumping experiments. Conducting spin and orbital pumping experiments on YIG/Pt/Ir 20Mn 80 heterostructures, \nwe unexpectedly observe strong spin and orbital anomalous signals in an out -of-plane configuration. We report \na sevenfold increase in the signal of the anomalous inverse orbital Hall effect (AIOHE) compared to \nconventional effects. Our study suggests expanding the Orbital Hall angle ( 𝜃𝑂𝐻) to a rank 3 tensor, akin to the \nSpin Hall angle ( 𝜃𝑆𝐻), to explain AIOHE. This work pioneers converting spin -orbital currents into charge \ncurrent, advancing the spin -orbitronics domain in AF materials.  \n \n \n \nOrbital Hall effect  (OHE)  provide s an intriguing \nalternative for advancing spintronics, with potential benefits \nfor non -volatile magnetic memories, sensors, microwave \noscillators, and nanodevices .1–3 Recent studies4–10 have \nhighlighted the significant potential of orbital currents in \nincreasing spin pumping signals driven by ferromagnetic \nresonance (SP -FMR) and by thermal gradient (spin Seebeck \neffect (SSE)) ,11–13 or in manipulating magnetiza tion through \norbital torque .14–17 However, understanding OHE  remains \nchallenging, with research primarily focused on light metals \nsuch as Ti, Ru, Cu ,9,11–13 2D materials ,18–20 and \nsemiconductors .21 Notably absent are discoveries concerning \norbital -to-charge conversion via inverse OHE  or inverse \norbital Rashba effects  in antiferromagnetic (AF) materials, \ndespite their unique properties and increasing interest for \nspintronic applications. AF materials,  characterized by null \nnet magnetization and insensitivity to external magnetic \nperturbations, exhibit intrinsic high -frequency magnetization \ndynamics , significant spin -orbit coupling (SOC) and \nmagneto -electric effects. They are recognized as a fertile \nground for advanced spintronics research, offering diverse \nelectrical properties and rich opportunities for both \nexperimental investigation and theoretical exploration .22–28 \nIn this letter, we investigate the intriguing \nphenomena of excitation and detection  of ordinary and \nanomalous spin and orbital Hall effects in an AF material. \nHeterostructures comprising YIG/Ir 20Mn 80(4), YIG/Pt(4), \nYIG/Pt(2)/Ir 20Mn 80(t) and YIG/Pt(2)/Ti(10), were utilized, \nwith YIG (400)  representing Yttrium Iron Garnet (Y 3Fe5O12) \nand the  AF material consists of Ir 20Mn 8 (layer thicknesses in \nnm are indicted in parenthesis). Metal lic films were \ndeposited using DC sputtering, and YIG was grown by \nLiquid phase epitaxy (LPE). Measurements were conducted \nat room temperature using the SP -FMR tec hnique.29–31 \nDuring deposition, the Ir 20Mn 80 films were submitted to a \nuniform magnetic field (~800 Oe) created by permanent \nmagnets. This procedure aligned the polycrystalline grains \ninducing an antiferromagnetic texture.32 To verify the AF \nphase of t he Ir 20Mn 80 film, we performed FMR \nmeasurements as a function of the in -plane field in \nPy(12)/Ir 20Mn 80(15), with Py denoting Permalloy (Ni 81Fe19). \nThe angular dependence of FMR field exhibited a bel l-\nshaped characteristic curve,33 typical of exchange -biased \nbilayers, thus confirming the AF nature of Ir20Mn 80. \nAdditional details on the experimental setup can be found in the supplementa ry material and References.11–13 \nIn the conventional spin -to-charge conversion \nprocess using the SP-FMR techni que, schematically shown \nin Figure  1 (a), an in -plane external field ( 𝜃=90°), pins the \nmagnetization direction. A perpendicular RF field induces \nuniform magnetization precession under FMR condition, \nthus inducing the injection of spin accumul ation across the \ninterface between the ferromagnet ( FM) and the adjacent \nlayer. This accumulation diff uses upward as a spin current 𝐽⃗𝑆 \ninto the adjacent layer. Through the inverse spin Hall effect \n(ISHE),34–38 it generates a perpendicular charge current ( 𝐽⃗𝐶), \ngoverned by,  \n                           𝐽⃗𝐶=(2𝑒ℏ⁄)𝜃𝑆𝐻(𝜎̂𝑆×𝐽⃗𝑆),  (1) \nwhere 𝜃𝑆𝐻 is the spin Hall angle, a constant that measure the \nefficiency of the spin -to-charge conversion, 𝐽⃗𝑆 is the spin \ncurrent direction and 𝜎̂𝑆 is the spin polarization direction. The \ncharge current created by the ISHE process is given by \n𝐼𝑆𝑃−𝐹𝑀𝑅 =𝑉𝑆𝑃−𝐹𝑀𝑅/𝑅, where 𝑉𝑆𝑃−𝐹𝑀𝑅 and 𝑅 are the voltage \nand electrical resistance directly measured between the \nelectrodes, respectively.  \n \n \nFigure  1. (a) illustrates the experimental setup employed, the \nconventional spin pumping measurements are performed with the \nexternal field applied in the sample plane, 𝜃=90° with 𝜙=0° or \n𝜙=180° . In (b) and (c), SP -FMR signals are depicted for \nYIG/Pt(4) and YIG/Ir 20Mn 80(4), respectively. The RF power used \nwas 13.8 mW.  \nFigure  1 (b) shows typical SP -FMR signals obtained \nfor YIG/Pt(4) in the in -plane configuration. At 𝜃=90°,𝜙=\n0° a positive voltage peak (blue symbols) is detected at the \nYIG FMR condition. When inverting 𝐻⃗⃗⃗ or rotating the \nsample to  𝜙=180° , 𝜎̂𝑆 changes its sign, while the 𝐽⃗𝑆 \ndirection remains fixed, resulting in a change in the polarity \nof the measured signal (red symbols), while the magnitude \nremains constant. The inset shows the derivative of the \nabsorption signal, where the FMR linewidth is 1.8 Oe. In \nFIG.1(c), the SP -FMR signal obtained for YIG/Ir 20Mn 80(4). \nIt is worth to point that, the ISHE in Ir 20Mn 80 has the same \npolarity as the ISHE in Pt, this result in dicates that the SOC \nin Ir 20Mn 80 is positive, i.e., 𝐿⃗⃗∙𝑆⃗>0, moreover since the \nmagnitude of the measured signal is smaller than Pt, we can \naffirm that 𝑆𝑂𝐶𝑃𝑡>𝑆𝑂𝐶𝐼𝑟20𝑀𝑛80. \nIt is important to mention that the spin current is in \nfact a rank 2 tens or. However, for practical purposes, it is \nconvenient to decompose this tensor into two distinct \nphysical quantities: its direction and its polarity. Although \nmotivated mainly by convenience, this separation proves to \nbe fundamental in the interpretation o f experimental data. In \na typical SP -FMR configuration, the direction of the spin \ncurrent ( 𝐽⃗𝑆) is always oriented out of the FM material. This \nresults in the accumulation of sp ins that diffuses through the \nadjacent  layer. On the other hand, the spin polarization vector \n𝜎̂𝑆 is always aligned with the external magnetic field  𝐻⃗⃗⃗. \nNotably, ISHE does not depend on the magnetic order of the \nmaterial.39,40 In fact, the conversion of spin to charge through \nspin Hall effe cts are due to scattering events within the bulk \nof the material, via spin -orbit interactions, be it intrinsic or \nextrinsic.35,36  \nIn recent years, groundbreaking theoretical study41 \nhas predicted the emergence of anomalous direct and inverse \nspin Hall  effect (ASHE and AISHE, respectively). This \nsignificant advance was achieved by extending the \nconventional spin Hall angle ( 𝜃𝑆𝐻) to a rank 3 tensor, taking \nin account an order parameter in the material of interest. In \nferromagnetic materials, this orde r parameter can be the \nmagnetization 𝑀⃗⃗⃗, while in antiferromagnetic materials it \ncorresponds to the Néel vector 𝑛⃗⃗. The proposed rank 3 spin \nHall angle 𝜃𝑖𝑗𝑘𝑆𝐻 can be defined as:  \n        𝜃𝑖𝑗𝑘𝑆𝐻= 𝜃0𝜖𝑖𝑗𝑘+ 𝜃1𝑛𝑖𝜖𝑖𝑙𝑛𝜖𝑗𝑛𝑘+𝜃2𝑛𝑖𝜖𝑖𝑛𝑘𝜖𝑗𝑙𝑛,  (2) \nwhere 𝜃0 represents the conventional spin Hall angle used in \nSHE/ISHE, while 𝜃1 and 𝜃2 are the anomalous spin Hall \nangles. The indexes 𝑖,𝑗,𝑘=1,2,3 correspond to the 𝑥̂, 𝑦̂ and \n𝑧̂ directions, respectively, with 𝜀𝑖𝑗𝑘 represe nting the Levi -\nCivita symbol. Consequently, by expanding the spin Hall \nangle into a rank 3 tensor, the 𝐽⃗𝑆 and 𝐽⃗𝐶 generated via SHE \nand ISHE gain an additional term which depends on the order \nparameter:  \n \n𝐽𝑘𝐶=∑(2𝑒\nℏ) 𝑖𝑗 𝜃𝑖𝑗𝑘𝑆𝐻 𝐽𝑖𝑗𝑆 and 𝐽𝑘𝐶=∑(ℏ\n2𝑒) 𝑘 𝜃𝑖𝑗𝑘𝑆𝐻 𝐽𝑖𝑗𝑆, (3) \n \nwhere 𝐽𝑘𝐶 is the charge current applied/detected along a \nspecific k  ̂ direction and 𝐽𝑖𝑗𝑆 is the spin current, a rank two \ntensor where the first index denotes the spin flow direction , \nand the second index denotes the  𝜎̂𝑆 direction.  \nIt is noteworthy that the spin Hall angle , now \nrepresented as a rank 3 tensor,  enables spin-to-charge \nconversion even when  the spin polarization aligns parallel to \nthe spin flow direction. This scenario is particu larly \nintriguing because any observed signal can be explained by \nISHE alo ne. For example , if we align the vectors  𝑗⃗𝑆 and 𝜎̂𝑆 \nalong the  𝑧̂ axis, the converted spin current will generate an \nelectrical signal, which follows:  \n                 𝐽𝑘𝐶= (2𝑒ℏ⁄)( 𝜃1+ 𝜃2 )𝛿𝑘𝑖≠3 𝑛𝑖𝐽33𝑆. (4)  \nFigure 2. (a) Out -plane SP -FMR signal for YIG/Ir 20Mn 80 (4) at 𝜃=\n0° and 𝜃=180° , where 𝜃 is the polar angle defined in Fig. 1(a). \nThe inset shows the corresponding FMR signal. (b) SP -FMR signal \nfor different RF power levels. The inset shows the peak curren t \nplotted as a function of the RF  power used to excite the FMR \ncondition, note that it presents a linear behavior. It is worth \nmentioning the high quality of our YIG films, which leads to the \ndetection of a surface ma gnetostatic mode for field values below the \nFMR field. As the excitation of the surface mode occurs very close \nto the excitation of the uniform mode, it leads to broadening of the \nFMR linewidth, as seen in all SP -FMR signals.  \n \nWhich implies that if the order parameter has components in \nthe x-y plane, a detectable signa l can be observed. This result \nis significant as it introduces the possibility of generating  \ncharge current along arbitrary directions, a phenomenon  not \npreviously antic ipated in conventional ISHE studies.   \nTo explore the AISHE in antiferromagnets, \nYIG/Ir 20Mn 80(4) samples were fabricated. While the \ntraditional ISHE is investigated by applying an in -plane \nmagnetic field  𝐻⃗⃗⃗, AISHE is investigated by applying the 𝐻⃗⃗⃗ \nin the o ut-of-plane configuration, 𝜃=0° or 𝜃=180° . In \nthis setup, the 𝐽⃗𝑆 direction will be parallel to 𝜎̂𝑆, meaning that \nwe are effectively exploring the 𝐽33𝑆 component of the spin \ncurrent tensor. See Figure 1 (a) for illustration of the spin \npumping process under out -of-plane configuration . \nFigure  2 (a) shows the SP -FMR signal in the out -\nplane configuration for YIG/Ir 20Mn 80(4). A well -defined \ncurrent  peak is d etected at around 5.05 kOe, corresponding \nto the excitation of the ferromagnetic resonance, as shown in \nthe inset. This peak corresponds to the electric current \ngenerated by the spin -pumping mechanism in the out -of-\nplane configuration. Since the directions  of 𝜎̂𝑆, and 𝐽⃗𝑆 are \nparallel, the measured signal cannot be attributed to the \nconventio nal ISHE, described by the equation (1). Moreover, \ndue to the insulating nature of YIG , no anomalous Nernst \neffect or other galvanomagnetic are present . On the other \nhand, the measured signal fits perfectly with the AISHE as \nthe N éel vector is along the x-y plane. Upon rotating the \nsample to 𝜃=180° , the orientation of 𝜎̂𝑆 changes , resulting \nin an inversion of the measured signal. This result  differs \nfrom  previous findings  where a FM was used instead of a n \nAF42. In the referenced study ,42 it was observed that  changing \nthe direction of 𝐻⃗⃗⃗  had no effect  on the detected signal. This \nwas attributed to the order parameter (magnetization) that \nchanges exclu sively in the sample plane. In the YIG/Ir 20Mn 80 \nsystem, the Néel vector serves as the order parameter, which \nexhibits much stronger rigidity compared to the \nmagnetization of a ferromagnet, thus remaining unaffected \nby 𝐻⃗⃗⃗ on the order of a few kOe. We also observed that the \nmeasured signal responds linearly  to the microwave power \nused to excite the FMR condition as  show n in Figure  2(b). \nThis result indicates that the detected signal depends linearly \non the spin current, fu rther supporting the AISHE \ninterpretation.  \nAnother attractive approach to explore spin -to-\ncharge conversion in AF involves examining orbital effects. \nIn recent years, orbital angular momentum has attracted \nsignificant attention due to its ability to impact  transport \nproperties, given that non -equilibrium orbital momentum \ndoes not suffer quenching.4 However, experimental studies \nin antiferromagnets remain scarce ,43–45 with no reports to \ndate on orbital -to-charge conversion via inverse orbital Hall \nor inv erse orbital Rashba effects in this class of materials.  \n \n \nFigure 3. (a) SP -FMR signals for: YIG/Pt(2)/Ir 20Mn 80(4) (black \nsymbols); YIG/Pt(2) (red symbols); YIG/ Ir 20Mn 80(4) (blue \nsymbols) in the in -plane configuration ( 𝜃=90°). Note that the \nweak SP -FMR signal generated by the surface mode is hardly \ndetected in YIG/ Ir 20Mn 80 and YIG/Pt, but it exhibits a strong gain \nin YIG/Pt(2)/Ir 20Mn 80(4). (b) SP -FMR signal for YIG/Pt(2)/ \nIr20Mn 80(4), for 𝜃=90°,𝜙=0° and 𝜃=270° ,𝜙=0°. (c) Pea k \nSP-FMR signals, for 𝑡=0,… 20 nm. The solid line is to guide the \neyes. The inset represents the IOHE for Ir 20Mn 80 films, and the solid \nred line was obtained as discussed in the text.  \n \nBased on previous investigation11–13 we fabricated \nsamples of YIG/ Pt(2)/Ir 20Mn 80(t) with varying thicknesses of \nthe Ir20Mn 80 layer ranging from 𝑡=0 nm to 𝑡=20 nm. The \nYIG/Pt(2) bilayer exhibits two notable characteristics: first, \ndue to the low SOC of YIG, it exclusively injects spin current \ninto Pt. Second, due to the  large SOC of Pt, a fraction of the \ninjected spin current undergoes conversion to a charge \ncurrent via ISHE, while most of the spin current transforms \ninto an entangled spin -orbital current. This entangled spin -\norbital current serves as a valuable tool for  probing orbital \neffects in different materials.  \nFigure  3 (a) shows the SP -FMR signal for YIG/Pt(2), \nYIG/Ir 20Mn 80(4) and YIG/Pt(2)/  Ir20Mn 80(4), measured in the \nin-plane configuration. Direct comparison of the measured \nsignals for the first two samples con firms the larger SOC in \nPt compared to Ir20Mn 80. However, adding a 4nm layer of \nIr20Mn 80 on top of the Pt layer almost doubles the signal \ncompared to ISHE in YIG/Pt. This observed increase cannot \nbe attributed solely to ISHE in Ir20Mn 80, so orbital Hall effect \nmust be considered. The result in Figure  3 (a) can be \nexplained by analyzing the spin Hall conductivity 𝜎𝑆𝐻 and \nthe orbital Hall conductivity 𝜎𝑂𝐻. First principles \ncalculations46 revel that Ir has 𝜎𝑂𝐻𝐼𝑟~4334  (ℏ/𝑒)(Ω∙cm)−1    \nand 𝜎𝑆𝐻𝐼𝑟~321  (ℏ/𝑒)(Ω∙cm) −1, while Mn has \n𝜎𝑂𝐻𝑀𝑛~6087  (ℏ/𝑒)(Ω∙cm)−1 and 𝜎𝑆𝐻𝑀𝑛~−37 (ℏ/𝑒)(Ω∙\ncm)−1 . Therefore, Ir20Mn 80 is anticipated to exhibit a strong \n𝜎𝑂𝐻, consequently contributing to a strong SP -FMR signal \ndue to IOHE in Ir20Mn 80 thin films.   \nIn Figure 3 (b), we present the angular dependence of \nIOHE in YIG/Pt(2)/Ir 20Mn 80(4). Upon rotating the sample, \nwe observed a change in the signal following an equation \nanalogous to the ISHE, which is mathematically described \nby: \n                         𝐽⃗𝐶=(2𝑒ℏ⁄)𝜃𝑂𝐻(𝜎̂𝐿×𝐽⃗𝐿), (5) \nwhere 𝜃𝑂𝐻 is the orbital analogous of the 𝜃𝑆𝐻, it measures the \norbital -to-charge conversion efficiency, and  𝜎̂𝐿 is the orbital \npolarization. In the approach we are using, the orientation of \n𝜎̂𝐿 is determined by the spin polarization 𝜎̂𝑆 injected into the \nPt film via the SP -FMR technique. Since the SOC in Pt is \npositive, 𝐿⃗⃗∙𝑆⃗>0 and 𝜎̂𝑆||𝜎̂𝐿. The dependence with the film thickness in Figure 3 (c) also indicates a typical diffusion -\nlike behavior;  the signal saturates for thicker films due to the \ninformation loss resulting from dissipation mechanisms \nwithin the film. The effective charge current by SP -FMR \ncomprises contributions from both ISHE in Pt(2) and IOHE \nin Ir 20Mn 80(t), given by  \n  𝐽⃗𝐶𝑒𝑓𝑓=(2𝑒/ℏ)[𝜃𝑆𝐻𝑃𝑡 (𝜎̂𝑆×𝐽⃗𝑆𝑃𝑡 )+𝜃𝑂𝐻𝐼𝑟𝑀𝑛(𝜎̂𝐿×𝐽⃗𝐿𝐼𝑟𝑀𝑛)].  (6) \nThe charge current due to IOHE in Ir 20Mn 80, represented in \nthe inset of Figure 3 (c), is given by 𝐽⃗𝐶𝐼𝑟𝑀𝑛=𝐽⃗𝐶𝑒𝑓𝑓−𝐽⃗𝐶𝑃𝑡(2) , \nwhere the theoretical fit 𝐽𝐶𝐼𝑟𝑀𝑛=𝐴𝑡𝑎𝑛 ℎ(𝑡/2𝜆𝐿), gives the \norbital diffusion length 𝜆𝐿𝐼𝑟𝑀𝑛=(3.5±0.5) nm, a value \ngreater than the spin diffusion length in Pt, 𝜆𝑆𝑃𝑡~1.6 nm .11 \nFinally, it is worth noting that each spin -to-charge \nconversion mechanism has a corresponding orb ital \ncounterpart, albeit originating from different physical \nmechanisms, but producing similar results. This raised the \nquestion of whether an Anomalous Inverse Orbital Hall \neffect (AIOHE) also exist. To explore the AIOHE, we \nconducted experiments using YI G/Pt(2)/Ir 20Mn 80(t) samples \narranged in the out -of-plane configuration and perform ed \nspin pumping measurements. Figure 4 (a) shows the spin \npumping signal for YIG/Ir 20Mn 80(4) and \nYIG/Pt(2)/ Ir20Mn 80(4) samples. While the peak signal of the \nYIG/ Ir20Mn 80 sample was around 37.5 nA, the \nYIG/Pt/ Ir20Mn 80 sample exhibited a significantly higher peak \nvalue of 271.6 nA, representing an increase in signal of more \nthan sevenfold. This surprising increase in the signal \nintensity suggests the existence of an extra s pin-orbital to \ncharge conversion mechanism beyond the traditi onal ISHE \nor IOHE, given the experimental setup employed. Moreover, \nthe signal cannot be attributed to the AISHE within the Pt \nlayer since no order parameter exists. By rotating the sample \n180º, the polarity of the signal changes indicating that the \nmeasured signal depends on the 𝜎̂𝑆 direction. Moreover, it \nhas a similar behavior to what was previously observed for \nthe AISHE in YIG/ Ir20Mn 80(4). This suggests that the signal \nis dependent on the or der parameter of the AF layer, which \nis kept fixed within the applied magnetic field range. This \nhypothesis is supported by analyzing the SP -FMR signal of \nYIG/Pt(2)/Ti(10) SP -FMR in the out -of-plane configuration, \nwhere no signal is observed, as shown in t he inset of Figure  \n4 (b). Previous experimental results have shown that \nTitanium is an excellent material to convert orbital current \ninto charge current via IOHE.12 However, Ti  does not exhibit \nan order parameter, leading to the absence of additional \ncharge current via AIOHE.  \nTo further elucidate how the behavior of the \nmeasured signals, we conducted experiments varying the \nmicrowave powe r. The results, presented in Figure  4 (c) , \nreveal a notable trend: the SP -FMR signal increases as we \nincrease the microwave power. This result indicates a direct \ncorrelation between the magnitude of the spin -orbital current \ninjected into the Ir20Mn 80 material and the observed effect. \nBy plotting the peak signal as function of the microwave \npower we observed a linear dependenc e, as shown in the inset \nof Figure  4 (c). Furthermore, we conducted a study \ninvestigating the influence of Ir20Mn 80 film thickness. As \nillustrated in Figure  4 (d), there is a clear saturation of the  \nsignal intensity for thicker films, suggestive of a \ncharacteristic diffusion -like behavior. This sa turation \nphenomenon arises from dissipati on mechanisms within the \nfilm, this behavior closely reflects observations from the \nAISHE ex periment.  \nIn summary, our findings present compelling \nevidence of spin and orbital anomalous Hall signals \ndiscovered through SP -FMR experiments in an \nantiferromagnetic material. This signal attributed to the \nAnomalous Inverse Orbital Hall effect, emerged from spin \nand orbital pumping experiments conducted at room \ntemperature. Unlike conventional ISHE and IOHE, this \nsignal demonstrates unique characteristics dependent on \nvarious parameters, including the Néel v ector of the AF \nmaterial, spin and orbital pumping configurations, external \nmagnetic field, and AF layer thickness. Comparing the \nsignals obtained from YIG/Pt(2)/Ir 20Mn 80(4) and \nYIG/Ir 20Mn 80(4) heterostructures revealed a remarkable \nseven -fold increase in the AIOHE signal. Just as 𝜃𝑆𝐻 can be \nexpanded to a rank 3 tensor if the converting layer has an \norder parameter, the 𝜃𝑂𝐻 must also be a rank 3 tensor. By \ntaking in account possible anomalous signals due to the order \nparameter, the direct and inverse orbital Hall effect will have \nadditional terms to generated/convert orbital currents, \nanalogous to their spin counterpart. Thus, the emergence of \nthe extra signal can be simply explained by the existence of \nan AIOHE. To date, no other work has explored thi s pathway \nto convert spin -orbital currents into charge current, \nexpanding the understanding of spin -orbitronics phenomena.  \n \nThe autors would like to thank D. S. Maior for helping with \nthe MOKE measurements  and with the images . This research \nis supported by  Conselho Nacional de Desenvolvimento \nCientífico e Tecnológico (CNPq), Coordenação de \nAperfeiçoamento de Pessoal de Nível Superior (CAPES) \n(Grant No. 0041/2022), Financiadora de Estudos e Projetos \n(FINEP), Fundação de Amparo à Ciência e Tecnologia do \nEstad o de Pernambuco (FACEPE), Universidade Federal de \nPernambuco, Multiuser Laboratory Facilities of DF -UFPE, \nFundação de Amparo à Pesquisa do Estado de Minas Gerais \n(FAPEMIG) - Rede de Pesquisa em Materiais 2D and Rede \nde Nanomagnetismo, and INCT of Spintroni cs and \nAdvanced Magnetic Nanostructures (INCT -SpinNanoMag), \nCNPq 406836/2022 -1. \n \n \n \n1 J. Ryu, S. Lee, K.J. Lee, and B.G. Park, “Current -Induced \nSpin–Orbit Torques for Spintronic Applications,” Advanced \nMaterials 32(35), (2020).  \n2 A. Hirohata , K. Yamada, Y. Nakatani, L. Prejbeanu, B. \nDiény, P. Pirro, and B. Hillebrands, “Review on spintronics: \nPrinciples and device applications,” J. Magn. Magn. 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The magnetic part is a mesh of magnonic waveguides with magnets placed on the waveguide \njunctions . There are amplitude and phase conditions for auto -oscillations to occur in the active ring circuit. \nThe frequency(s) of th e auto -oscillation and spin wave propagation route (s) in the magnetic part depends \non the mutual arrangement of magnets in the m esh. The propagation route is detected with a set of power \nsensors. The correlation between circuit parameters and spin wave route i s the base of memory \noperation.  The combination of input/output switches connecting electric and magnetic parts, and electric \nphase shifters constitute the memory address. The output of power sensors is the memory state.  We \npresent experimental data on th e proof -of-the-concept experiment s on the prototype with just \nthree magnets placed on top of a  single -crystal yttrium iron garnet Y 3Fe2(FeO 4)3 (YIG)  film.  The \nresults demonstrate a robust operation with On/Off ratio  for route detection  exceeding 35 dB a t \nroom temperature.  The number of propagation routes scales factorial with the size of the magnetic part. \nCoding information in propagation routes makes it possible to drastically increase the data storage density \ncompared to conventional memory devices . MCM wi th just 25 magnets can store as much as \n25!=1025 bits.  Physical limits and constraints are also discussed.  \n \n \n \n \n \n 2 \n  \nI. Introduction  \nThere is an urgent need in the increas e of data -storage density  of information storage devices  in the \ncurrent era of big data . The global data will grow to 175 zettabytes (ZB) by 2025  according to the \nInternational Data Corporation  [1].  Conventional storage systems may become unsustainable due to their \nlimited data capacity , infrastruct ure cost , and power consumption  [2].  In the traditional process of \nimproving the data -storage density (e.g., the number of bits stored per area), better performance is \nachieved by the miniaturization of the data -storage  elements.  It stimulates a quest for nanometer -size \nmemory elements  such as DNA -based [3] or sequence -defined macromolecules  [4]. At the same time,  \nmemory  architecture  and the principles of data storage remain mainly unchanged.  \nAs an example, we would like to refer to R andom Access Memory  (RAM).  In Fig.1( A), there is shown a \nhigh -level picture  of RAM  organization.  The core of the structure is an array of mem ory cells where each \ncell stores one bit of information. There is a variety of memory cells exploiting different physical \nphenomena /devices/circuits  for data storage such as static RAM [5], dynamic RAM  [6], and magnetic RAM \n[7].  In all cases, RAM contains multiplexing and demultiplexing circuitry  for cell addressing. Typically , the \naddressing of a required cell is accomplished by transistors electrica lly connecting the cell on the selected \nrow/column with the output circuit.  The correlation between the cell address and cell state is the essence \nof data storage in RAM. It is illustrated in Fig.1( B). It is shown in a table, where the first column depicts \nthe memory cell binary address.  The second column in the table depicts the cell state (i.e., 0 or 1). The \nmaximum  number of bits stored in conventional memory is limited by  the number of memory cells.  This is \nthe common property of all existing  RAM s. Regard less of the physical mechanism  of data storage (e.g., \nmechanical,  electrical, magnetic), conventional memory devices make use of the individual states of the \nmemory cells.  \nHere, we consider the possibility of building a fundamentally different data -storage devi ce, where \ninformation is stored in the signal propagation route .  It allows us to drastically increase the number of \nmemory states compared to conventional memories.  The rest of the work is organized as follows. In the \nnext Section II, we describe the principle of operation of Magnonic Combinatorial Memory (MCM). Next, \nin Section III, we present the results of numerical modeling illustrating MCM operation. E xperimental  data  \nobtained for the prototype are presented in section IV.  The Discussion and Conclusions are given in \nsections  V and VI, respectively . \n \nII. Material Structure and Principle of Operation   \nIn order  to explain th e principle of operation of combinatorial devi ces, we start with an example of a two -\npath active ring circuit schematically  shown in Fig. 2(A). It consists of two parts referred to as the active \npart the and passive part. The active part  includes a nonlinear  broadband  amplifier 𝐺 , variable phase \nshifter Ψ(𝑉), and a controllable attenuator 𝐴(𝑉). It is assumed that these components are frequency -\nindependent (i.e., provide the same amplification, the same phase shift, and the same attenuation for all \nfrequencies). The magnitude of the phase shift and atten uation level can be adjusted  by the applied \nvoltage 𝑉.  The broadband amplifier 𝐺 is the only source of energy in the system. The passive part consists \nof two paths, where each path has a bandpass frequency filter  𝐿(𝑓), phase shifter Δ(𝑓), and a power  3 \n sensor 𝑃. The filters and the phase shifters in the passive part are frequency -dependent  (i.e., provide \nattenuation and phase shift depending on the frequency of the signal) .  There are two conditions for auto -\noscillations to occur in an active ring circ uit [8]: \n 𝐺×𝐴×𝐿(𝑓)≥1,                                                                                                                                           (1.1)  \n Ψ+∆(𝑓)=2𝜋𝑘,  𝑘=1,2,3,…                                                                                                                     (1.2)  \nwhere   𝐿(𝑓) is the signal attenuation in the  passive path(s) , 𝑘 is an integer.  The first equation (1.1) states \nthe amplitude condition for auto -oscillations: the gain provided by the broadband amplifier should be \nsufficient to compensate for losses in the passive part and losses introduced by the attenuator. The second \nequation states the phase condition for auto -oscillations: the total phase shift for a signal circulating \nthrough the ring should be a multiple of 2𝜋. In this case, signals come in phase every propaga tion round.  \nA more detailed explanation of the selective signal amplification in a multi -path active ring circuit can be \nfound in Ref. [9]. \nThere are four possible scenarios for the circuit shown in Fig.2( A). (i) There may be no auto -oscillations in \nthe circuit as conditions (1.1) and/or (1.2) are not satisfied. (ii) There may be auto -oscillations in the circuit \nwhere the most of circulating power is coming through the top path. It happens when the two pat hs have \ndifferent tran smission/phase shift characteristics so the wave(s) on the resonant frequency can propagate \nonly through the top waveguide but significantly attenuated in the bottom waveguid e. (iii) The auto -\noscillation may take place and most of the power is coming  throu gh the bottom waveguide. (iv) There \nmay be auto -oscillations in the circuit with equal power flow through  both paths  in case of identical phase \nshifts and attenuation in the top and the bottom waveguides . As an example, let us consider a two -path \nactiv e ring circuit as in Fig. 2(A), where  the bandpass filters on the top and the bottom paths are set to the \ndifferent frequencies 𝑓1 and 𝑓2, respectively. The phase shifters provide the  different phase shifts ∆1=\n1.5𝜋 and ∆2=0.5𝜋. It is assumed that the amplitude condition (1.1) is satisfied for all frequencies.  There  \nare two positions of the external phase shifter  Ψ  at which the phase condition (1.2) is met. In the case  \nΨ+Δ1=2𝜋, there are auto oscillations on frequency 𝑓1  whe re the signal goes through the top \nwaveguide. In the case  Ψ+Δ2=2𝜋, there are auto oscillations on frequency 𝑓2  where the signal goes \nthrough the bottom waveguide. There are no auto oscillations for other positions of the external phase \nshifter Ψ. In Fig.2( B), the auto oscillation power is shown as a function of the external phase shifter Ψ. \nThe insets illustrate the signal propagation route. The green circle corresponds to the power sensor \ndetecting power. The grey circle depicts no power flow.  The plot in Fig.2( B) corresponds  to the  circuit with  \nideal bandpass filters transmitting signals only on the central frequencies with zero bandwidth.  \nIn our preceding experimental works, we demonstrated several electro magnetic active ring circuits \ncomprisi ng active electric and passive  multi -path  magnonic (spin wave) parts [9, 10] . There are several \nadvantages of using spin waves in the active ring circuits. (i) Spin wave is a collective oscillation of a large \nnumber of spins in magnetic lattice , where a quantum of spin wave is called a  magnon . The collective \nnature of spin wave reveals itself in a relatively long coherence length which may exceed millimeters at \nroom temper ature in ferrites [11]. Spin waves propagate much slower compared to electro -magnetic \nwaves of the same frequency , which  provid es a significant phase shift even in the micrometer -length \nwaveguides. (ii) The propagation of spin waves is affected by t he local magnetic field produced by micro \nmagnets placed on top of ferrite film. This property was used for magnetic bit read -out in a number of \nworks [12-14]. A micro magnet placed on top of ferrite film acts as a frequency filter and a phase shifter 4 \n at the same time .  The bandpass frequency  range and the phase shift can be engineered by the material, \nsize, shape,  and alignment of the magnet on top of ferrite waveguide.  \nThe most appealing property of the active -ring circuit is the ability to find or self -adjust to the resonant \nfrequency (s). The system starts with a superposition of signals of all possible frequencies. However, only \nsignal on the resonant frequency(s) 𝑓𝑟 satisfying conditions (1.1) and (1.2) will be amplified. The \npropagation paths of the signals may differ depending on the arra ngement of the frequency filters. This \nproperty  of multi -path active ring circuits  can be utilized for solving nondeterministic polynomial  (NP) \nproblem s such as Prime Factorization [9], and traveling salesman problem  [15]. In this work, we con sider \nthe multi -path active ring circuits for memory application.  \nThe schematics of the proposed MCM are shown in Fig.3 (a). It is an active ring circuit where the magnetic \npart consists of a mesh of magnonic waveguides. For simplicity, it is shown a 2D 5 ×5 me sh. The waveguides \nare shown  in the light blue  color.  There are magnets depicted by the blue -red rhombs  placed on top of \nwaveguide junctions. It is assumed that each magnet can be in one of the eight thermally stable states  \ncorresponding to the eight diff erent  direction s of magnetization  with respect to the waveguide . The \nmagnets can be of three different sizes.   Thus, t here are total 24  possible states for the junction with \nmagnet  plus one more state corresponding to the junction without a magnet. These magnets are aimed \nto act as the frequency filters and phase shifters for the propagating spin waves. Both the direction and \nthe strength of the local magnetic field  provided by the magnets  are important for spin wave transport  \n[16].  There are power sensors placed on top of the waveguides  between the junctions . These sensors are  \naimed to detect the power of the spin wave  signal and show the spin wave  propagation route(s). For \nexample, it may be Inverse Spin Hall Effect (ISHE) detectors.  Being o f a relatively simple material structure \n(e.g., Pt wire on top of YIG waveguide), I SHE pr ovide s output voltage proportional to the amplitude of spin \nwave [17]. There are (2𝑛−1)×𝑛 detectors in the mesh with 𝑛 columns and 2𝑛−1 rows (e.g., 45 \ndetectors shown in Fig.3(a)). Each sensor provides a voltage output V𝑖𝑗, where subscripts  𝑖 and 𝑗 \ncorrespond to the row and column numbers, respectively.  \nThe magnonic mesh is connected to the electric part via the input and output ports located on the left \nand right sides of the mesh. The conversion from electromagnetic wave to spin wave and vice versa may \nbe accomplished with the help of micro antennas [18].  There is a switch  (e.g., a transistor similar to one  \nused in conventional memory for r ow/column addressing) at each input and output port  to enable/disable \nthe antenna for signal generation/receiving. These switches are aimed to control  the combinati on of \ninput/output  antennas  (e.g., input ports #2 and #3, output ports #1, #2, and #5). There are  voltage -tunable \nfrequency filters  𝑓𝑖, voltage -tunable phase shifters Ψ𝑖, and voltage -tunable  attenuators 𝐴𝑖 at each output \nport,  where the subscript 𝑖 depicts  the output  number.  The phase shifters have discrete states \ncorresponding to the specific phase shifts (e.g., Ψ(0)=0𝜋,  Ψ(1)=0.25𝜋, Ψ(2)=0.5𝜋, etc.).  The \nfrequency filters and attenuators are analog devices to be used for system calibration/adjustment as will \nbe described later in the text.  There is one  broadband amplifier 𝐺 in the electric part .  \nThe principle of operation of M CM is based on the corr elation between the combination  of the \ninput/output switches , phase shifters  Ψ𝑖 , and the spin wave propagation route in the mesh  (e.g., the \noutput voltage s of the power detectors  V𝑖𝑗).  The memory address includes a binary number \ncorresponding to  the states of input switches , a binary number corresponding to the  states of  output \nswitches, and a binary number corresponding to the states of the phase shifters Ψ𝑖.  The binary number \nfor switches is an 𝑛-bit number, where 1 corresponds to the state On and 0 corresponds to the state Off.  5 \n For example, the mesh shown in Fig.3(a) has input ports #2, #3, and #4 in the position On. It corresponds \nto the binary address  01110. The o utput switches # 1, #3, #4,  and #5 a re in the position  On. It corresponds \nto the binary number  10111. There are more than two states for each phase shifter.  In this case, the \nlength of the binary number corresponding to the phase shifter states is related to 𝑧𝑛 possible \ncombinations , wher e 𝑧 is the number of states  per phase shifter. The memory state is the signal \npropagation route (i.e., the output voltages V𝑖𝑗). One may introduce a reference voltage V𝑟𝑒𝑓 to digitize \nthe output. For instance, the output state is 1 if V𝑖𝑗≥V𝑟𝑒𝑓, and 0 otherwise. In the example shown in \nFig3.(a), the memory state is a sequenc e of 45 bits (one bit for each sensor).   \nThe correlation between the memory address es and the memory state s is illustrated in Fig.3(b).   It is \nshown an example of the truth  table, where the first column is the memory address, and the second \ncolumn is the memory state. The number of possible combinations of input switches is 2𝑛−1. It excludes \none combination where all input ports are disconnected from the mesh.  There is the same number of \npossible combinations o f the output ports. In Fig.3(b), the first memory address corresponds to the case \nwhen the input port #1 and the output port #1 are connected to the mesh . The phase shifters are set to \nstate 0 . Every combination of switch states and phase shifters states constitutes an address.  The total \nnumber of addresses is given as follows:  \n# 𝑎𝑑𝑟𝑒𝑠𝑠𝑒𝑠 =(2𝑛−1)2×𝑧𝑛.                                                                                                                  (2) \nThere is one memory state that is related to the given address. The memory state is a sequence of 45 \nzeros and ones. In the left column of the truth table in Fig.3(b), these zeros and ones are arranged  in nine \nrows with five columns with five bits per row. In this case, the position of bit reflects the position of the \npowe r sensor in the mesh. For example, the  middle row in the truth table in Fig.3(b) corresponds to the \ncase shown in Fig.3(a). The top five bits are zeros corresponding to the five power sensors located in the \ntop row in the mesh.  In general, there a re (2𝑛−1)×𝑛 power sensors  in the mesh and same number of \nbits per add ress.  The total capacity of MCM is the product of the number of memory addresses and the \nnumber of bits stored per address that can be calculated  as follows:  \n# 𝑏𝑖𝑡𝑠  𝑠𝑡𝑜𝑟𝑒𝑑 =(2𝑛−1)2×𝑧𝑛×𝑛×(2𝑛−1)                                                                                      (3) \nAccording to Eq.( 3), the data storage capacity of MCM scales  according  to the power law  with the size of \nthe mesh.  It should be noted that  the number of possible combinations of row/column switches for \nconventional RA M (e.g., as shown in Fig.1) also scales according to the power law. However, only a limited \nnumber of addresses store information  (e.g., addresses including one row  and one column) . The states of \nother addresses  (e.g., addresses with two rows  and three columns)  can be computed. The information in \nMCM is stored in the mutual arrangement of magnets in the mesh.  There are  𝑛! ways to have an ordered \narrangement of 𝑛 distinct objects [19]. Considering  a set of  𝑛2 distinct magnets in the mesh with 𝑛2 \njunctions, the number of ordered arrangements (permutations) is given by  \n# 𝑜𝑟𝑑𝑒𝑟𝑒𝑑  𝑚𝑎𝑔𝑛𝑒𝑡  𝑎𝑟𝑟𝑎𝑛𝑔𝑒𝑚𝑒𝑛𝑡𝑠  =(𝑛2)!                                                                                         (4) \nIt is important that the number  of possible magnet arrangements  increases faster than the number of bits \nstored. It may be possible to f ind an arrangement  of magnets so the spin wave propagation routes match  \nthe given truth table (e.g., shown in Fig.3(b)).  For example, there are 25!=1.55×1025 possible  \narrangements for 25 magnets in  5×5 mesh as shown in Fig.3(a). That is the maximum number of bits \nthat can be stored. The increase of the number of states per phase shifter above a certain value will 6 \n increase the number of addresses but with identical states. Needless to say, that exploiting all the possible \narrangements (routes) in 5×5 mesh will cover all need s in the current information storage ( 175 \nzettabytes  =1.75×1023 bits. In this part, we restricted our consider ation by  the read -out procedure \nto emphasize  the fundamental enhancement in  data storage  density  of MCM compared to traditional \nmemories.   \n \nIII. Results of numerical modeling   \nThe s pin wave propagation route depends on the mutual arrangement of magnets in the magnonic mesh . \nThis is the keystone of MCM operation. In order to illustrate it, we present the results of numerical \nmodeling . In Fig.4, there is shown an equivalent circuit fo r MCM. The mesh of waveguides with magnets \nis replaced with a 2D mesh of impedances. The real and the imaginary part of each cell corresponds to the \nspin wave attenuation and phase shift accumulated during  propagati on through the junction. Rigorously \nspeak ing, there should be additional cells  in the mesh  (i.e., impedances) to account for spin wave damping  \nand accumulated phase shift during the propagation between the junctions. To simplify our consideration, \nwe assume that the damping and the phase shift du ring the propagation between the junction s are much \nsmaller compared to the ones during the propagation through  the junction. The real and the imaginary \nparts of the junction impedances  are frequency dependent.  They may be obtained from micromagnetic \nsimul ations [20] or from  experiment [21].  \n There are three steps in the modeling  procedure . First, one needs to find t he total  attenuation and the \nphase shift produced by the passive part for all possible frequencies .  Second, the obta ined results  are \nchecked to find the frequency(s)  at which the self -oscillation conditions (1.1) and (1.2)  are met . Final ly, \none needs to find the map of spin wave power flow through the m esh at the resonant frequency(s). The \nmost time -consuming  is the first step  as it takes a number of subsequent calculations to find the mesh \nresponse s in a wide frequency range. In order to s peed up calculations and illuminate the essence of the \nproposed memory, we make several assumptions. (i) We assume that each propagation route in the mesh \nis associated with a certain propagation frequency. (ii) The attenuation is linearly proportional to the \npropagation distance.  (iii) The junctions provide a frequency -independent phase shift. The objective is to \nshow the change in the signal propagation route depending on the arrangement of a given set of elements \nin the mesh.  \nIn Fig.5 (a), it is shown a mesh where numbers in the boxes show the phase shift accumulated by the \npropagating wave . It is assumed that the output attenuators a set to exclude the routes for more than six \njunctions (i.e., the amplitude condition is satisfied for all routes  coming  throug h five or six junctions). The \noutput phase shifters are set to Ψ=0.5𝜋. The phase condition is satisfied for the routes providing 1.5𝜋 \nphase shift. The particular frequency(s) coming through these routes are of no importance.  All the input \nand output  switches are in the On state. The set of power detectors in Fig.5(a) show the signal propagation \nroute for the given configuration of magnets. There are two routes. One  route is  from the input port #2 \nto the output port #1. The other route is from input port #5 to the output port #4. Let us change the places \nof two cells in the mesh. For example, we flip the  two adjacent cells in the second row at columns two \nand three.  It resul ts in the change of the signal propagation route. In Fig.5(b), the green circles of power \ndetector s show the routes. It is different from the one in Fig.5(a). The difference in the memory state is 7 \n eight bits. It is possible to engineer magnonic meshes where  the change  in the  position of just one magnet \nwill lead  to the tens -bit difference in the output.  \nIt should be noted that changing the mutual position of cells may or may not change the propagation \nroutes in the mesh . Fig.6(a) shows the propagation route (i.e., green circles) for the arrangement of cells \nas in Fig.5(a) but for the output phase shifter set for Ψ=0.6𝜋. The phase condition (1.2)  is satisfied for \nthe routes that provid e 1.4𝜋 phase shift. As i n the previous example, the output attenuators a set to \nexclude the routes through  more than six junctions . There is just one route from input port # 2 to  output \nport #3 that provides the required phase shift. Flipping the cells (i.e., second row, second and  third \ncolumns) does not change the propagation route. It is illustrated in Fig.6(b).  The examples shown in Figs. \n5 and 6 are aimed to illustrate the plethora of combinations in the mutual cell positions leading or not \nleading to the different outputs. The  freedom of  engineering the output  by changing the position  of cells  \nin the mesh  is important for MCM programming.   \n \nIV. Experimental data   \nIn this part, we present experimental data obtained for the prototype with just three magnets. The \nschematics of the prototype  are shown in Fig. 7(A).  It is a  multi -path  active ring circuit comprising  electric \nand magnetic parts. The electric part consists  of an amplifier ( three amplifiers Mini -Circuits, model ZX60 -\n83LN -S+ connected in series ), and a  phase shifter (ARRA, model 9418A) . The magnonic part is a ferrite film \nwith three fixed places  (i.e., depicted by the circles numbered 1,2, and 3)  for three di fferent magnets  to \nbe placed on top of the film . There are three input and three output antennas to connect electric and \nmagnetic parts. Each input/output port can be independently connected/disconnected.  There are three  \nvoltage -tunable bandpass filters at the output ports. The filters are commercially available YIG -based \nfrequency filters produced by Micro Lambda Wireless, Inc, model MLFD -40540.  The experimental data \non the filter transmission and phase delay can be found in the supplementary materials.  The filter s at \noutput port s #1, #2, and #3  are set to the central frequenc ies 𝑓1= 2.539 GHz ,  𝑓2= 2.475 GHz  and \n𝑓3= 2.590 GHz , respectively. Magnonic  and electric part s are connected via the set of splitters and \ncombiners (i.e., SPLT 1 -3, Sigatek SP11R2F 1527 ). The power at each output  port  and the total power \ncirculating in the ring circuit is measured by the spectrum analyzer  (SA) GW Instek GSP -827 \nconnected to the circuit through a  directional coupler (DC, KRYTAR, model 1850) . There are four \nplaces of connection shown in Fig.7(A). 𝑃0 is the total power in the circuit measured just after the \namplifier,  𝑃1, 𝑃2, and 𝑃3 are the powers measured at the output ports after signal propagation \nthrough the passive magnonic part.  The signal is significantly damped  after the passive part  so \nthe sum of 𝑃1, 𝑃2, and 𝑃3 is not equal to  𝑃0. SA is also used for detect ing the frequencies of the auto -\noscillations in the ring circuit .  A more detailed schematics of the experimental setup with a detailed map \nof signal attenuation through the parts can be found in the Supplementary materials.  \nThe cross -section of the passive m agnonic part is shown in Fig. 7(B). It consists  from the bottom to the top \nfrom  a permanent magnet made of NdFeB, a Printed Circuit Board (PCB) substrate with six short -circuited \nantennas, a ferrite film made of GGG substrate and YIG layer,  and a plastic pl ate with three pits for \nmagnets to be inserted.  The permanent magnet is aimed to create a constant bias magnetic field. \nHereafter, we refer to this relatively big permanent magnet (model BX8X84 by K&J Magnets, Inc., \ndimensions 1.5”x1.5”x0.25”) as a magnet in the text. Th e magnetic field produced by this magnet defines 8 \n the frequency window as well as the type of spin waves that can propagate in the ferrite film. The bias \nfield is about 375 Oe and directed in -plane on the film surface. The photo of the PCB with six antennas is \nshown in Fig. 7(c). The antennas are marked as 1,2,3,..6.   The characteristic size of the antenna is 2 mm \nlength and 0.15 mm width.  The ferrite film is made of YIG grown by liquid epitaxy on GGG substrate. YIG \nwas chosen due to the low spin wave damping.  The film is not patterned.  The thickness of the film is 42 \nμm. The saturation magnetization is close to 1750 G, the dissipation parameter  (i.e.,  the half -width of the \nferromagnetic resonance ) ΔH = 0.6 Oe. The plastic plate  is mechanically attached on top of the ferrite film. \nThere are three pits drilled in the layer for placing the micro -magnets. There are three NdFeB micro -\nmagnets of volumes 0.02 mm3, 0.03 mm3, and 0.06 mm3, respectively. The magnets are placed inside \nplastic tubes of different color. The smallest -volume magnet is placed into the tube of black color without \na sticker. The tubes with the white and the red stickers correspond to the middle -volume and l arge -\nvolume magnets respectively. Hereafter, we refer to the micromagnets as Black (B), White (W), and Red \n(R). The photo of the devices with tubes can be found in the Supplementary materials.  \nThe first set of experiments was accomplished for the case with  three input and three output antennas \nconnected . Antennas marked as # 3, #4, and #6 are used for spin wave excitation in the ferrite film. \nAntennas marked  as #1, #2, and #5  are used for detecting the inductive voltage produced by the spin \nwaves at the output. The summary of experimental data are shown in Table 1. The first column shows the \ncombination of input  and output switches . For instance, (111) means that all three input antennas are \nconnected to the electric part. The second column shows the positio n of the external phase shifter. The \nexternal phase  is set to  Ψ=0𝜋.  The third  column shows the magnet arrangement. For example, BWR \nmeans that B magnet (smallest) is inserted into the pit #1, W magnet (medium) is inserted in the pit #2, \nand R magnet (largest) is inserted into the pit #3. Combination (000) stands for the case without magnets \nplaced in the pits. The four th column shows the frequencies of the auto -oscillations (i.e., measured by \nSA). It may be one  or several  frequencies at the same time. The f ifth column shows the power of the auto -\noscillation  𝑃0 at different frequencies.  For example, the numbers in the second row (+2dBm and +4 dBm) \ncorrespond to the frequencies  2.590  GHz, and 2538  GHz, respectively. The sixth  column shows the power \nmeasured at the three output ports , where three numbers in each row correspond to  𝑃1, 𝑃2, and 𝑃3, \nrespectively.  The output power ranges from -30 dBm to -79 dBm.  The last column in the table shows \nthe logic output . It is logic 1 if the output power exceeds -45 dBm and logic 0, otherwise.  Power  below the \nreference value is shown in blue color, while the power  larger than reference power is shown in red color.  \nFor example, -27 dBm , -75 dBm , -73 dBm in the first row correspond to the memory state 100. The data \npresented in Table 1 provides a detailed picture of the active ring dynamics including the freque ncies of \nauto -oscillation, the distribution of power between the frequencies, and the analog output at each port. \nThere is no need in using  SA in a practical device. All the collected data is aimed to explain the physical \norigin of data storage in MCM.   The memory device will only  provide  binary  output for the given binary \naddress.  \nIn Tables 2,  there are shown experimental data for the case with three input and three output antennas \nconnected. The external phase  is set to  Ψ=0.63 𝜋. One can see the difference in the output power \ndistribution compared to the previous case. There three frequencie s of auto -oscillation for BWR magnet \nconfiguration.  In Table 3, there are shown experimental data for the case with three input and three \noutput antennas connected. The external phase  is set to  Ψ=1.25 𝜋. In Table 4, there are shown \nexperimental data for the case with three input and three output antennas connected. The external phase  \nis set to  Ψ=1.75 𝜋. The collected data reveal the complex dynamic of auto -oscillations in the elect ro-9 \n magnetic active ring circuit.  There may be one, two, or three different frequencies of auto -oscillations. It \nprovides all possible frequency combination at the output result ing in the different output state. There is \none just output state (000) missing in the collected data. This state appears when the amplitude condition \n(1.1) is not satisfied. We intentionally kept low attenuation and relatively high amplification to \ndemons trate the dependance output dependence on the external phase.   \nThe absence of magnet in the pit can be considered as an additional state. In Table 5, there are shown \nexperimental data for the case  when pit #1 is empty.  The other two pits were used for different \narrangement of magnets. The external phase  is set to  Ψ=0 𝜋. There is only one frequency of auto -\noscillation observed for all arrangements of magnets. It may give an insight on the importa nce of \ninformation stored in one magnet  on top of ferrite film (e.g., frequency -selective attenuation and phase \nshift) that define spin wave routing.  \nFinally, the system was studied for a different combination of input/output switches. In Table 6, there are \nshown experimental data for the case with two input (#3 and #4) and three output antennas connected. \nThe external phase  is set to  Ψ=0 𝜋. The frequ encies of auto -oscillation, frequency combination, and \nthe output states are different compared to the case with three input ports. The addition of one more \ninput lead to the disappearance of some frequencies. For example, one can compare the second and th e \nfourth rows in Table 1 and Table 5. The disappearance of auto -oscillation on some frequencies can be well \nexplained by the interference (e.g., destructive interference) of spin waves coming from different inputs. \nIn any rate, the addition of extra input/ output channel cannot be described as a simple superposition of \nroutes.     \nThe data shown in Tables 1 -5 are obtained in the active ring configuration.  It takes less than a millisecond \nfor the system to reach the self -sustained oscillations. In general, th e time required to reach the steady -\nstate regime depends on many factors including spin wave dispersion, parameters of electric broadband \namplifier, etc.  Raw data on spin wave transport in the circuit can be found in the Supplementary materials.  \n \nV. Discuss ion \nThere are two important observations we want to make based on the obtained experimental \ndata. (i) Spin wave propagation route (s) does depend on the configuration of magnets  on top of \nferrite waveguide , the combination of input and output ports, and the output phase shifter. For \ninstance, t he arrangement of three different magnets or arrangement of two different magnets \nwith one empty pit results in the different spin wave propagation routes. The reason for spin \nwave re-routing is the difference in the magnetic field  profile  on the top of ferrite fi lm that \nappears for different arrangement of magnets. Th e re-routing can be modeled consider ing \nmagnet/ferrite film as a bandpass filter and a phase shifter as illustrated in Section 3.  However, \nthe high -fidelity numerical modeling would require an enormous deal of work to link the \nmagnetic film profile to spin wave propagation routes in a wide frequency range. External phase \nis an additional parameter which affects spin wave propagation  in the active ring circuit. It makes \na fundamental difference with conventional RAM where low/high electric current is directly \nrelated on the high/low resistance states of the memory cells. In turn, the phase -dependent \ntransport allows us exploit the dif ferent combination of input/output ports. Also, t he addition of 10 \n extra ports is not equivalent to adding additional routes. It may happen that some frequencies \n(routes) disappear for a larger number of inputs due to the spin wave interference.   \n  \n(ii) Spin wave propagation routes can be recognized by the set of power sensors  with high \naccuracy . In the present ed experiments , spin wave power was measured only at the output ports \n(i.e., no sensors within the matrix). The On/Off ratio  (i.e., the difference between th e outputs \nwhere most of power flows and the outputs with minimum power)  exceeds 35 dB at room \ntemperature . It makes it possible to tolerate the inevitable structure imperfections, the \ndifference in the efficiency of input/output antennas, etc. This big rat io is achieved by the \nintroduction of  frequency filters  aimed to separate frequency response between the different \noutputs. It may be possible to achieve even bigger On/Off ration by using filters with a smaller \nbandwidth. It will take an additional compar ator-based circuit to digitize MCM output.  \n \nThese observations confirm the main idea of this work on the feasibility of data storage using \nspin wave propagation routes. It inherent the advantages of traditional magnetic -based memory  \nincluding non -volatility and a long retention time. At the same time, MCM  provide s a \nfundamental advantage in data storage density compared to the existing memory devices. To \ncomprehend this advantage, we extracted the data from Tables 1 -4 obtained for the same set of \nexternal phase shifter but with the different configuration of magnets. In Fig.7, there are shown \nthe arrangement of the magnets on the left and corresponding truth tables for four phases on \nthe right. There are three bits corresponding to the memory state in each row. Conventional \nmemory with three magnets stores just three bits. It is the same amount of data  that can be read -\nout at one given phase  in MCM  (e.g., phase = 0 𝜋).  All other rows (i.e., 3 out of 4 in each table ) \ncontain an extra  or exceed  information com pared to conventional RAM s. According to Eq.(3), the \nadvantage over the conventional data storage devices scales according to the power law  with the \nsize of magnonic mesh 𝑛. \n \nThere are several critical comments to mention . (i) The structure of the prototype is different \nfrom the general vie w MCM  shown in Fig.3(a).  There are only three output ports in the prototype. \nThe addition of power sensors within the magnetic part (e.g., between the magnets) would give \na better picture of spin wave power dist ribution. It is also not clear if the ISHE sensors would need \nfrequency filters for better spin wave route detection. (ii) The read -in procedure is not described. \nThe need to use magnets with more than two thermally stable states of magnetization (e.g., \nmagnets of different size s) significantly complicated the fabrication and initialization procedure. \nTheoretically, it can be achieved  during the fabrication or using specially engineered micro \nmagnets with after -fabrication initialization.  We want also to re fer to the recently reported \nnanomagnet reversal by propagating spin waves [22]. The switching was observed in \nferromagnet/ferrimagnet hybrid structures consisting of Ni 81Fe19 nanostripes prepared on top of \nYIG film.  It may be a convenient way for MCM programming.  \n \nThe key question regarding MCM potential advantages in data storage is associated with the \npossibility of programming. Is it possible to find a magnet arrangement for any given truth table?  \nIt is an NP -hard problem to check all possible magnet configurations with the hope to find the 11 \n desired one. On the other hand, it may be possible to utilize only a fraction of input -output \ncorrelations for data storage. This and  many other questions and co ncerns deserve special \nconsideration  combining approaches  developed in combinatorics, physics, and magnonics . This \nwork is aimed to introduce the concept of MCM and outline its potential  advantages.  \n \n \n \nVI. Conclusions  \nWe described a novel type of magnetic memory which is aimed to exploit the mutual \narrangement of magnets for data storage. The principle of operation is based on the correlation \nbetween the arrangement of magnets on top of ferrite film and the spin wave propagation \nroutes. The number of routes sc ales factorial with the number of magnets that makes it possible \nto encode more information compared to conventional magnetic memory devices exploiting the \nindividual states of magnets. We present ed experimental data on the proof -of-the-concept \nexperiments o n the prototype with just three magnets placed on top of a of single -crystal yttrium \niron garnet Y 3Fe2(FeO 4)3 (YIG)  film. The results demonstrate a robust operation with an On/Off \nratio for route detection exceeding 35 dB at room temperature.  This work is a first step toward \nthe novel type of combinatorial memory  devices which have not been explored. The material \nstructure and principle of operation of MCM are much more complicated compared to \nconventional RAMs. At the same time, MCM may pave the road to unprecedented data storage \ncapacity where a device with just 25 magnets can store as much as 25!=1025 bits.  \n \n \nCompeting financial interests  \nThe authors declare no competing financial interests.  \nData availability  \nAll data generated or analyzed during this study are included in this published article.  \nAcknowledgment  \nThis work of M. Balinskiy and A. Khitun  was supported in part by the INTEL CORPORATION, under \nAward #008635, Project director Dr. D. E. Nikonov, and by the National Science Foundation (NSF) \nunder Award # 2006290, Program Officer Dr. S. Basu. The authors would like to thank Dr. D. E. \nNikonov for  the valuable discussions.  \n \n \nFigure Captions  12 \n Figure 1 . (A) Schematics  of RAM organization.  There is a 2D mesh of memory cells arranged in cows and \ncolumns.  The addressing of a required cell is accomplished by the row/column decoders. (B) Example of \nRAM tru th table.  The first column in the table depicts the memory cell binary address.  The second column \nin the table depicts the cell state. The maximum number of bits stored is limited by the number of memory \ncells.  \nFigure 2. (A) Schematics of a two -path active ring circuit. The active part includes a nonlinear broadband \namplifier 𝐺 , a phase shifter Ψ, and a n attenuator 𝐴.  The passive part consists of two paths .  Each path \nhas a bandpass frequency filter, a phase shifter , and a power sensor 𝑃. The bandpass filters on the top \nand the bottom paths are set to the different frequencies 𝑓1 and 𝑓2, respectively. The phase shifters \nprovide t ifferent phase shifts ∆1=1.5𝜋 and ∆2=0.5𝜋. (B) The auto oscillation power as a fu nction of \nthe external phase shifter Ψ.  The auto oscillations occur on the frequencies 𝑓1 and 𝑓2when the phase \ncondition (1.2) is met.  The insets illustrate the signal propagation route. The green circle depicts the \npower flow.  \n \nFigure 3. (A) Schematics of MCM. The core of the structure is a mesh of magnonic waveguides  shown in \nthe light  blue color . There are magnets depicted by the blue -red rhombs placed on top of waveguide \njunctions.  There are three different sizes for the magnets, where each magnet has eight thermally stable \nstates of magnetization.  There are power sensors placed on top of the waveguides between the junctions.  \nThe mesh is included into an active ring circuit via switches on the left and right sides. There are voltage -\ntunable f requency filters  𝑓𝑖, voltage -tunable phase shifters Ψ𝑖, and voltage -tunable attenuators 𝐴𝑖 at each \noutput port  (right side).  There is one broadband amplifier 𝐺 in the electric part.   (B) Memory  truth table . \nThe left column shows the memory address . It includes the binary number corresponding to s the tates \nthe input switches, the binary number corresponding to the output switches, and the binary number \ncorresponding to the states of the phase shifters Ψ𝑖.  The right column shows the  memory state , which  is \nset of  digitized sensor output s.  \n \nFigure 4. Equivalent circuit for MCM. The mesh of waveguides with magnets is replaced with a mesh of \ncells with impedances  𝑍(𝑓)𝑖𝑗, where subscripts 𝑖 and 𝑗 correspond to the column and row  numbers, \nrespectively. The real and the imaginary part of each cell  are frequency dependent and  correspond to the \nspin wave attenuation and phase shift accumulated while propagating through the junction. The output \nbandpass filters, phase shifters, and at tenuators are replaced by the frequency depend impedances 𝑍(𝑓)𝑗, \nwhere subscript 𝑗 corresponds to the row number.  \nFigure 5. Schematics of the simplified circuit model.  The numbers in the boxes show the phase shift in the \ncell. The amplitude condition is satisfied for all routes coming through five or six junctions . The output \nphase shifters are set to Ψ=0.5𝜋.  All input and the output switches are in the On state. (A) The s et of \npower sensor s (green circles)  show s the signal propagation route s for the given configuration of phase \nshifters .  (B) The set of power sensors (green circles) shows the signal propagation routes where two cells \nare flipped in their positions ( two adjace nt cells in the second row at columns two and three ).   \n \nFigure 6. Schematics of the simplified circuit model. The numbers in the boxes show the phase shift in the \ncell. The amplitude condition is satisfied for all routes coming through five or six junctio ns. The output \nphase shifters are set to Ψ=0.6𝜋.  All input and the output switches are in the On state.  (A) The set of \npower sensors (green circles) shows the signal propagation routes for the given configuration of phase 13 \n shifters.  (B) The set of power sensors (green circles) shows the signal propagation routes where two cells \nare flipped in their positions (t wo adjacent cells in the second row at columns two and three).    \n \nTable 1. Experimental data  obtained for different configurations  of magnets on top of ferrite film. All three \ninput and output antennas are operating. The external phase is set to  Ψ=0𝜋.   \n \nTable  2.  Experimental data  obtained for different configuration of magnets on top of ferrite film. All three \ninput and output antennas are operating. The external phase is set to  Ψ=0.63 𝜋.    \n \nTable  3.  Experimental data  obtained for different configur ation of magnets on top of ferrite film. All three \ninput and output antennas are operating. The external phase is set to  Ψ=1.25 𝜋.    \n \nTable  4.  Experimental data obtained for different configuration of magnets on top of ferrite film. All three \ninput and output antennas are operating. The external phase is set to Ψ=1.75 𝜋.    \n \nTable  5.  Experimental data obtained for different configuration of mag nets on top of ferrite film where \npit #1 is empty. The other two pits are used for the different arrangement s of two magnets.  All three input \nand output antennas are operating. The external phase is set to Ψ=0 𝜋.    \n \nTable 6. Experimental data obtained fo r different configuration of magnets on top of ferrite film.  Three \nare two input and three output antennas con nected. The external phase is set to  Ψ=0𝜋.   \n \nTable 7. Summary of ex perimental data from Tabl es 1-4. The mutual arrangement  of magnets is shown \non the left. The tables on the right show  the output of MCM.   \n \n \n  14 \n  \n  \nFigure 1  15 \n   \nFigure 2 16 \n  \n  \nFigure 3 17 \n  \n  \nFigure 4 \n18 \n  \n  \nFigure 5 19 \n  \n  \nFigure 6 20 \n  \n  \nFigure 7 \n21 \n  Switches  \nInput  Output   Phase \n[π] Magnets  \nArrangement   Auto-oscillation \nfrequency [GHz]   𝑃0(𝑓) \n[dBm]  𝑃1, 𝑃2, 𝑃3 \n[dBm]  Memory  \n State \n 111     111   0  000  2.590  +7.5 -27, -75, -73  100 \n 111     111   0  BWR  2.590, 2 .538  +2, +4  -30, -78, -27  101 \n 111     111   0  BRW  2.475  +6  -78, -30, -71  010 \n 111     111   0  WBR  2.539  -1  -72, -71, -39  001 \n 111     111   0  WRB  2.539  +2 -75, -69, -37   001 \n 111     111   0  RBW  2.590, 2.539   +3, +1  -39, -75, -37  101 \n 111     111   0 RWB   2.539   -3   -81, -78, -36    001  \n \n  \nTable 1  22 \n  \n   Switches  \nInput  Output   Phase \n[π] Magnets  \nArrangement   Auto-oscillation \nfrequency [GHz]   𝑃0(𝑓) \n[dBm]  𝑃1, 𝑃2, 𝑃3 \n[dBm]  Memory  \n State \n 111     111   0.63  000  2.590  +6 -31, -70, -68  100 \n 111     111   0.63  BWR  2.590,2.475,2538  +2,+4, -1  -33, -28, -39  111 \n 111     111   0.63  BRW  2.590, 2.475   +6, 0  -27, -31, -71  110 \n 111     111   0.63  WBR  2.539, 2.539  -3, -1  -36, -73, -36  101 \n 111     111   0.63  WRB  2.539  +4 -77, -73, -39   001 \n 111     111   0.63  RBW  2.590, 2.475   +5, +1  -37, -39, -78  110 \n 111     111   0.63   RWB   2.475, 2.539      0, +2    -79, -36, -33    011  \nTable  2 23 \n  \n   Switches  \nInput Output   Phase \n[π] Magnets  \nArrangement   Auto-oscillation \nfrequency [GHz]   𝑃0(𝑓) \n[dBm]  𝑃1, 𝑃2, 𝑃3 \n[dBm]  Memory  \n State \n 111     111  1.25  000  2.590  +6 -31, -70, -68  100 \n 111     111   1.25  BWR   2.475 +3  -73, -31, -71  010 \n 111     111   1.25  BRW  2.590, 2.475   +5, +1  -27, -31, -71  110 \n 111     111   1.25  WBR  2.475, 2.539  0, +3  -79, -35, -31  011 \n 111     111   1.25  WRB  2.590, 2.475   +1, +2 -35, -34, -75   110 \n 111     111   1.25  RBW  2.590, 2.539   +3, +3  -30, -78, -30  101 \n 111     111   1.25   RWB   2.590,2.475, 2.539   +2,+3, -1   -33, -30, -36    111  \nTable 3 24 \n  \n   Switches  \nInput Output   Phase \n[π] Magnets  \nArrangement   Auto-oscillation \nfrequency [GHz]   𝑃0(𝑓) \n[dBm]  𝑃1, 𝑃2, 𝑃3 \n[dBm]  Memory  \n State \n 111     111   1.75  000  2.590  +5 -30, -70, -69  100 \n 111     111   1.75  BWR   2.539 +3  -75, -73, -35  001 \n 111     111   1.75  BRW  2.475,2.539   0, +3  -79, -35, -31  011 \n 111     111   1.75  WBR   2.539 +4  -77, -73, -36  011 \n 111     111   1.75  WRB  2.475, 2.539   +3, +3 -78, -33, -39   011 \n 111     111   1.75  RBW  2.590, 2.475,2.539   +2,+3,-1  -33, -30, -36  111 \n 111     111   1.75   RWB   2.475, 2.539   +2, +4  -75, -36, -33    011  \nTable 4  25 \n  \n \n   Switches  \nInput Output   Phase \n[π] Magnets  \nArrangement   Auto-oscillation \nfrequency [GHz]   𝑃0(𝑓) \n[dBm]  𝑃1, 𝑃2, 𝑃3 \n[dBm]  Memory  \n State \n 111     111   0  000  2.579  +6 -33, -81, -79  100 \n 111     111   0  0WR  2.466   +3  -80, -33, -73  010 \n111     111  0 0RW 2.466   +5 -79, -33, -78 010 \n 111     111   0  0BW   2.579 +3  -31, -81, -75  100 \n 111     111   0  0WB  2.466  +2 -81, -33, -75   010 \n 111     111   0 0BR   2.466  0  -78, -33, -79  010 \n 111     111   0  0RB   2.523  +3  -69, -72, -27    001  \nTable 5  26 \n     Switches  \nInput Output   Phase \n[π] Magnets  \nArrangement   Auto-oscillation \nfrequency [GHz]   𝑃0(𝑓) \n[dBm]  𝑃1, 𝑃2, 𝑃3 \n[dBm]  Memory  \n State \n 110     111   0  000  2.631  +3 -30, -81, -78  100 \n 110     111   0  BWR   2.631,2.492,2.522  +2,+2,+2   -32, -33, -34  111 \n 110     111   0  BRW  2.631  +6  -32, -84, -79  100 \n 110     111   0  RBW  2.631, 2.522  +4,+4  -31, -79, -34  101 \n 110     111   0  RWB  2.631  +2 -33, -81, -75   100 \n 110     111   0  RBW  2.522  +2  -81, -84, -30  001 \n 110     111   0   WRB   2.631, 2.492  +3, +3  -35, -33, -79    110  \nTable 6  27 \n  \n \n  \nTable 7  BWR \nRWB \nWBR \n28 \n References  \n[1] D. 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Parker, \"Time -resolved measurement of propagating \nspin waves in ferromagnetic thin films,\" Physical Review Letters, vol. 89, no. 23, pp. 237202 -1-4, \n2002.  29 \n [19] \"Combinatorics,\" in Mathematical Tools for Data Mining: Set Theory, Partial Orders, \nCombinatorics , (Advanced Information and Knowledge Processing, 2008, pp. 529 -549.  \n[20] A. Khitun, \"Parallel database search and prime factorization with magnonic holographic  memory \ndevices,\" Journal of Applied Physics, vol. 118, no. 24, Dec 2015, Art no. 243905, doi: \n10.1063/1.4938739.  \n[21] M. Balinskiy, D. Gutierrez, H. Chiang, Y. Filimonov, A. Kozhevnikov, and A. Khitun, \"Spin wave \ninterference in YIG cross junction,\" Aip A dvances, vol. 7, no. 5, May 2017, Art no. 056633, doi: \n10.1063/1.4974526.  \n[22] K. Baumgaertl and D. Grundler, \"Reversal of nanomagnets by propagating magnons in \nferrimagnetic yttrium iron garnet enabling nonvolatile magnon memory,\" \nhttps://arxiv.org/abs/22 08.10923 , 2022.  \n \n  30 \n  \n \n \n \nSupplementary Materials for  \nMagnonic Combinatorial Memory  \n  31 \n  \n  \nFigure [s1]: YIG-based frequency filter produced by Micro Lambda Wireless, Inc, model \nMLFD -40540 .  (A) Experimental data: S21 parameter (amplitude) of the commercial filter.  \n(B) Experimental data: S21 parameter (phase shift) of the commercial filter. (C) Results of \nnumerical fitting: transmission of the spin wave element.  (D) Results of numerical fitt ing: \nphase shift produced by the spin wave element.  The data are shown in the frequency range \nfrom 2.2 GHz to 3.8 GHz.  32 \n   \nFigure [s2]: Photo of the passive part with six antennas connected to the electric part via \ncoaxial cables. There are seen three tubes labeled as Red, Black, and White.  There are three \nNdFeB micro -magnets of volumes 0.02 mm3, 0.03 mm3, and 0.06 mm3, placed in the Black \n(B), White (W), and Red (R)  tubes, respectively.  \n 33 \n  \n  \nFigure [s3]: Schematics of the experimental setup. The passive part with three tubes with magnet is \nshown in the center. There are six antennas (e.g., three input and three output ports) connecting the \npassive and the active parts.  Magnonic and electric parts are connected via the set o f splitters and \ncombiners (i.e., SPLT 1 -3, Sigatek SP11R2F 1527 ).  Each of the output ports is equipped with a frequency \nfilter (F1, F2, F3) and an amplifier (Mini -Circuits, model ZX60 -83LN -S+) that is used for initial calibration.  \nThere is a prominent sig nal attenuation introduced by the different parts in the circuit (e.g., the device, \nsplitters, combiners, frequency filters, etc.). To comprehend the level of signal \nattenuation/amplification, there are shown numbers in pink colors. These numbers indicate the power \nlevel in the corresponding points of the schematic. There is also a programmable network analyzer \n(PNA) Keysight Technologies, model N5221A -217 for measuring S21 parameter of the circuit. The PNA \nis included in the circuit by breaking the active ring configuration.  \n \n \n \n 34 \n  \n   \n Figure [s 4]: Experimental data obtained with PNA showing spin wave transmission through \nthe film.  The bias magnetic field is about 375 Oe and directed in -plane on the film surface.  \nPlots (A) and (B) show S21 and phase shift as a function of frequency for the film wit hout \nfrequency filters . Plots (C) and (D) show S21 and phase shift as a function of frequency for \nthe film frequency filters .    \n35 \n  \n \n \n \n  \n Figure [s5]: Experimental data obtained with PNA showing spin wave transmission through the \nfilm.  The bias magnetic field is about 375 Oe and directed in -plane on the film surface.  Plots \n(A) and (B) show S21 and phase shift as a function of frequency for output #1 without and with \nthe frequency filter.  Plots (C) and (D) show S21 and phase shift as a function of frequency for  \noutput # 2 without and with the frequency filter.   Plots ( E) and ( F) show S21 and phase shift as \na function of frequency for output #2 without and with the frequency filter.    36 \n  \n \n \n \nFigure [s 6]:  Experimental data: Auto -Oscillations in the Active Ring Circuit . Magn etic mesh is connected \nto the electric part. Auto -oscillations are observed for certain levels of amplification and the external \nphase shifter. The gain in the circuit is set to +13.5 dB.  \n(a) Frequencies of the auto -oscillations depending on the external phase shifter. One div is equivalent \nto π/30 radians  (b) Total power in the act ive ring circuit depending on the external phase.  (c) Power of \nthe auto -oscillations in different propagation routes.  "    },    {        "title": "1712.08204v1.Exchange_torque_induced_excitation_of_perpendicular_standing_spin_waves_in_nanometer_thick_YIG_films.pdf",        "content": "Exchange-torque-induced excitation of\nperpendicular standing spin waves in\nnanometer-thick YIG films\nHuajun Qin1,*, Sampo J. H ¨am¨al¨ainen1, and Sebastiaan van Dijken1,*\n1NanoSpin, Department of Applied Physics, Aalto University School of Science, P .O. Box15100, FI-00076 Aalto,\nFinland\n*huajun.qin@aalto.fi, sebastiaan.van.dijken@aalto.fi\nABSTRACT\nSpin waves in ferrimagnetic yttrium iron garnet (YIG) films with ultralow magnetic damping are relevant for magnon-based\nspintronics and low-power wave-like computing. The excitation frequency of spin waves in YIG is rather low in weak external\nmagnetic fields because of its small saturation magnetization, which limits the potential of YIG films for high-frequency\napplications. Here, we demonstrate how exchange-coupling to a CoFeB film enables efficient excitation of high-frequency\nperpendicular standing spin waves (PSSWs) in nanometer-thick (80 nm and 295 nm) YIG films using uniform microwave\nmagnetic fields. In the 295-nm-thick YIG film, we measure intense PSSW modes up to 10th order. Strong hybridization between\nthe PSSW modes and the ferromagnetic resonance mode of CoFeB leads to characteristic anti-crossing behavior in broadband\nspin-wave spectra. A dynamic exchange torque at the YIG/CoFeB interface explains the excitation of PSSWs. The localized\ntorque originates from exchange coupling between two dissimilar magnetization precessions in the YIG and CoFeB layers.\nAs a consequence, spin waves are emitted from the YIG/CoFeB interface and PSSWs form when their wave vector matches\nthe perpendicular confinement condition. PSSWs are not excited when the exchange coupling between YIG and CoFeB is\nsuppressed by a Ta spacer layer. Micromagnetic simulations confirm the exchange-torque mechanism.\nIntroduction\nMagnonics aims at the use of spin waves for the processing, storage, and transmission of information1–6. With the smallest\ndamping parameter of all magnetic materials, ferrimagnetic YIG has attracted considerable interest. Several building blocks\nfor spin-wave-based technologies have been realized using YIG magnonics, including magnonic crystals6–8, logic gates9,\ntransistors10, and multiplexers11. In these experiments, coplanar waveguides (CPWs) or microstrip antennas are typically used\nto excite propagating magnetostatic spin waves. The frequency of these spin waves depends on their wave vector ( k), the\nsaturation magnetization of YIG ( Ms), and the external magnetic field ( Hext). Because the saturation magnetization of YIG is\nsmall and the wave vector is limited by the width of the antenna signal line, the spin-wave frequency is only 1\u00002GHz in weak\nmagnetic fields12, 13. Higher frequencies can be attained by the excitation of magnetostatic spin wave modes with larger wave\nvector using a grating coupler14, at the expense of emission efficiency.\nAnother spin-wave mode that can be excited in a magnetic film is the PSSW. The wave vector of this confined mode is\napproximated by k=pp=d, where dis the film thickness and pis the order number. Thus in nanometer-thick magnetic films,\nthe wave vectors of PSSW modes are large and their frequency is high. The formation of a PSSW requires a nonuniform\nexcitation across the magnetic film thickness. Laser pulses15–17, microwave magnetic fields from a miniaturized antenna18, and\neddy-current shielding in conducting films19have been used to excite PSSWs. In these experiments, the excitation field is\nnonuniform and both odd and even PSSW modes are measured. Uniform microwave magnetic fields can also excite PSSWs\nif the magnetization of the film is pinned at one or both of its interfaces20–26. Symmetrical pinning only induces odd PSSW\nmodes, whereas both odd and even modes should be detected if the magnetization is pinned at one of the interfaces. Most work\non PSSWs has focussed on metallic ferromagnetic materials such as Co/Py19, Py23–26, and CoFeB27, 28. In YIG, Klingler et al.\nused PSSWs to extract the exchange constant18and Navabi et al. demonstrated the excitation of a 1st order PSSW mode in a\n100-nm-thick YIG film on top of an undulating substrate28.\nHere, we report on efficient excitation of PSSWs in nanometer-thick YIG films. The excitation mechanism is based on\nexchange coupling between the YIG film and a CoFeB layer. We show that forced magnetization precessions in YIG and CoFeB,\ndriven by an approximately uniform CPW field, induce a dynamic exchange torque at the interface when the precessions are\ndissimilar. Consequently, the emission of spin waves into YIG is most efficient if the dynamic exchange torque is maximized\nnear the ferromagnetic resonance (FMR) frequency of either YIG or CoFeB. Because the PSSW dispersion relations cross thearXiv:1712.08204v1  [cond-mat.mtrl-sci]  21 Dec 2017FMR curve of the CoFeB layer, PSSWs with high order numbers are efficiently excited in YIG at high frequencies.\nResults\nSpin-wave spectra of YIG, CoFeB, and YIG/CoFeB films\nSingle-crystal ferrimagnetic YIG films with a thickness of 80 nm and 295 nm were grown on (111)-oriented Gd 3Ga5O12(GGG)\nsubstrates using pulsed laser deposition (PLD). To measure broadband spin-wave spectra, we placed the films face-down onto a\nCPW with a 50 mm-wide signal line. A microwave current provided by a vector network analyzer (VNA) was used to generate\na microwave magnetic field around the CPW. The main excitation strength of the CPW was at wave vector k\u00190. We recorded\nabsorption spectra in transmission by measuring the real part of the S12scattering parameter. The experiments were performed\nwith an external magnetic bias along the CPW. A schematic of the measurement geometry is shown in Fig. 1a.\nWe first discuss the spin-wave spectrum of a single 295-nm-thick YIG film. Figure 1b shows the absorption at a magnetic\nbias field of 20 mT (dashed blue line). Data as a function of magnetic field are shown in Fig. 1c. Obviously, only one\nspin-wave mode is excited in the film. The mode corresponds to uniform magnetization precession in YIG, i.e., the FMR mode.\nHigher-order PSSW modes are not detected in this sample, as expected from the near-uniform microwave magnetic field and\nthe absence of magnetization pinning at the interfaces. After characterization, we deposited a 50-nm-thick CoFeB layer onto\nthe same YIG film using magnetron sputtering. Figure 1b (solid orange line) and Fig. 1e show the spin-wave spectrum of this\nsample. Now, a large number of spin-wave modes are measured. The lowest-frequency mode, labeled as p= 0, corresponds to\nFMR in YIG (compare the blue and orange lines in Fig. 1b or the spectra in Figs. 1c and 1e). The higher-order modes ( p= 1, 2,\n...) are PSSWs in YIG (see next Section for details). The excitation of both odd and even modes implies that exchange coupling\nat the YIG/CoFeB interface produces an asymmetric pinning configuration. Finally, we note that a FMR mode is also excited in\nCoFeB ( p= 0 (CoFeB)). To support this conclusion, we show the spin-wave spectrum of a single 50-nm-thick CoFeB film on\nGGG in Fig. 1b (dashed green line) and in Fig. 1d.\nThe data in Fig. 1 clearly indicate that the excitation of PSSWs in YIG is particularly efficient near the FMR of the CoFeB\nfilm. This point is further exemplified by the spin-wave spectrum of Fig. 1f, demonstrating the formation of PSSW modes\nwith order numbers up to p= 10 at high frequencies. High-order PSSWs are only visible if their frequency approaches that of\nthep= 0 mode in CoFeB. Exchange coupling at the YIG/CoFeB interface results in mode hybridization and characteristic\nanti-crossing behavior. Consequently, the frequency of the CoFeB FMR mode in the YIG/CoFeB bilayer is slightly shifted with\nrespect to the same resonance in the single CoFeB film (orange and green curves in Fig. 1b).\nIn the following, we first analyze the PSSW modes in YIG/CoFeB using a phenomenological model. Next, we assess their\nintensity and linewidth. Results for YIG/CoFeB bilayers with a 80-nm-thick YIG film are subsequently discussed. Finally, we\nelucidate the origin of efficient PSSW excitation in exchange-coupled YIG/CoFeB bilayers using control experiments with a\nnonmagnetic spacer layer and micromagnetic simulations.\nAnalysis of PSSW mode dispersions\nTo investigate the dependence of the PSSW resonance frequency on external bias field, we first extract experimental data for p=\n0 ... 6 from the spin-wave spectra in Fig. 1. The results are plotted as symbols in Fig. 2a. We derive the saturation magnetization\nof our YIG film by fitting the frequency dependence of the p= 0 mode to the Kittel formula29:f=gm0=2pp\nHext(Hext+Ms).\nUsing g=2p= 28 GHz/T, we obtain a good fit for Ms=192kA/m. This magnetization value compares well to previous results\non YIG films30, 31. Next, we fit the PSSW modes ( p= 1 ... 6) to the following dispersion relation23, 27, 28:\nfPSSW =gm0\n2ps\u0014\nHext+2Aex\nm0Ms\u0010pp\nd\u00112\u0015\u0014\nHext+2Aex\nm0Ms\u0010pp\nd\u00112\n+Ms\u0015\n; (1)\nwhere Aexis the exchange constant. By inserting g=2p= 28 GHz/T, Ms= 192 kA/m, d= 295 nm, and p=1\u00006, we obtain\ngood fits to all PSSW modes using Aex=3:1 pJ/m (lines in Fig. 2a). This value also agrees with literature18.\nThe agreement between our experimental data and the model confirms that the higher-order resonances in the spin-wave\nspectra of the YIG/CoFeB bilayer correspond to PSSW modes in YIG. The observation that integer numbers of pand the actual\nthickness of the YIG film in Eq. 1 provide excellent fits to all dispersion curves over a large bias-field range signifies strong\nmagnetization pinning at the YIG/CoFeB interface. Weak pinning at the boundary of the YIG film would require the use of\np\u0000Dpin the fitting formula23, where correction factors Dp= 0 and Dp= 0 would correspond to full and zero magnetization\npinning, respectively. In our YIG/CoFeB bilayer, short-range exchange coupling between the two magnetic films provides a\nstrongly pinned interface.\n2/9Intensity and linewidth of PSSW resonances\nDamping of PSSW modes in magnetic films can have different origins. Besides intrinsic damping, eddy-current damping (in\nmetallic films), and radiative damping caused by inductive coupling between the sample and the microwave antenna can also\ncontribute26. To assess the damping of PSSWs in our YIG/CoFeB bilayer, we plot the full width at half maximum (FWHM)\nlinewidth of the p= 1 ... 4 modes relative to that of the p= 0 mode (Fig. 2c). The frequency evolution of this data was obtained\nfrom the spin-wave spectra in Fig. 1e for magnetic fields ranging from 0 to 30 mT. The linewidths of all PSSW modes are large\nat low frequencies. The broad resonances in YIG are caused by hybridization with the higher-loss FMR mode in CoFeB. As the\nfrequency increases, the frequency gap between the PSSWs in YIG and the FMR mode in CoFeB becomes larger. Once the\ntwo modes decouple, the linewidths of the PSSW modes decrease. In the decoupled state, the PSSW linewidths are similar to\nthat of the p= 0 mode in YIG, independent of frequency. Since eddy-current damping can be omitted in insulating YIG and\nradiative damping would cause the linewidth to increase with frequency26, the data in Fig. 2c suggest that damping of PSSWs\nis dominated by intrinsic material parameters.\nFigure 2d shows the relative intensity of the same PSSW modes. The dashed lines indicate the frequency where\nFWHM p/FWHM p=0= 1 in Fig. 2c, which we use as an indicator for dehybridization between the PSSWs in YIG and\nthe FMR mode in CoFeB. For the pure PSSW modes beyond this critical frequency, we still measure high intensities. The\nintensities of the p= 1 and p= 2 modes are up to 50% of the p= 0 resonance and this value drops to about 30% for p= 3 and p\n= 4. The large intensities of the PSSW modes demonstrate a highly efficient excitation mechanism.\nTuning of PSSW modes in YIG/CoFeB bilayers\nThe frequency of a PSSW depends on the wave vector of the confined mode and the external magnetic bias field. Since\nk=pp=d, the frequency of a PSSW could be enhanced by a reduction of the film thickness d. For an efficient excitation\nmethod, this would enable high-frequency spin waves in YIG at small magnetic fields. To test this prospect, we prepared a\n80 nm YIG/50 nm CoFeB bilayer. The spin-wave spectrum of this sample is shown in Fig. 3. In addition to the FMR modes\nin YIG and CoFeB, the first two PSSW modes are also measured. Anti-crossing behavior between the p= 1 mode and the\nCoFeB resonance and an increase of the PSSW intensity near the anti-crossing frequency are again apparent. Compared to the\n295-nm-thick YIG film, the PSSWs in thinner YIG are shifted up in frequency. At a moderate magnetic bias field of 20 mT,\nthe increase of frequency amounts to about 3 GHz for p= 1 and 8 GHz for p= 2. The data of Fig. 3 thus confirm that PSSW\nmodes are efficiently excited at high frequencies if the thickness of YIG is reduced.\nPSSW excitation mechanism\nWe explain the excitation of PSSWs in YIG/CoFeB bilayers by a dynamic exchange torque at the interface. The uniform\nmicrowave excitation field from the CPW induces forced magnetization precessions in both magnetic layers. If the amplitudes\nof these precessions are different, a dynamic exchange torque is generated, causing the emission of spin waves from the\ninterface. The efficiency of this excitation mechanism depends on the strength of the dynamic exchange torque, which is\nmaximized at the FMR frequency of YIG and CoFeB. While spin waves are emitted from the interface over a broad frequency\nrange, PSSWs only form when the wave vector of the excited spin waves matches the perpendicular confinement condition\n(k=pp=d) of the YIG film. Our experiments support this scenario. PSSWs are only measured after the YIG film is covered by\na CoFeB layer and the PSSW resonances are most intense if the induced precession of magnetization is large in one of the two\nlayers, i.e., near the FMR of YIG or CoFeB (see Figs. 1b,e,f and Fig. 3a).\nTo confirm the crucial role of exchange coupling at the YIG/CoFeB interface, we prepared a 295 nm YIG/10 nm Ta/50\nnm CoFeB trilayer on GGG. The spin-wave spectrum of this sample is shown in Fig. 4. As expected, no PSSW modes are\nmeasured in this case. The two resonances in the spectrum are identical to those in Figs. 1c and 1d and, thus, correspond to\nthe FMR mode in the YIG and CoFeB film, respectively. The Kittel formula fits the experimental data for Ms= 192 kA/m\n(YIG) and Ms= 1280 kA/m (CoFeB). The results of Fig. 4 demonstrate that the elimination of exchange coupling between\nmagnetization precessions in YIG and CoFeB by the Ta spacer layer destroys the driving force behind PSSW excitation. This\nalso implies that dipolar coupling between YIG and CoFeB is insignificant.\nWe performed micromagnetic simulations in MuMax332to further study the microscopic origin of PSSWs in YIG/CoFeB\nbilayers. In the simulations, we considered a 295-nm-thick YIG film and a 50-nm-thick CoFeB layer. The structure was\ndiscretized using finite-difference cells of size x= 54 nm, y= 54 nm and z= 2.7 nm, as schematically shown in the inset of\nFig. 5a. Two-dimensional periodic boundary conditions were applied in the film plane to mimic an infinite bilayer. We used\nthe following input parameters: Ms= 192 kA/m (YIG), Ms= 1280 kA/m (CoFeB), Aex= 3.1 pJ/m (YIG), and Aex= 16 pJ/m\n(CoFeB). The damping constant was set to 0.005 for both magnetic films. For YIG, this relatively large value was selected to\nlimit the computation time. Spin waves in the bilayer were excited by an uniform 3 mT sinc-function-type magnetic field pulse\nor a sinusoidal ac magnetic field (see Methods). The excitation field was along xand a magnetic bias field was aligned along y.\nFor comparison, we also performed micromagnetic simulations for a structure where the YIG and CoFeB films are separated by\na 10-nm-thick nonmagnetic spacer, to mimic the response of a YIG/Ta/CoFeB trilayer.\n3/9The top panel of Fig. 5a shows simulated spin-wave spectra for the YIG/CoFeB bilayer (solid orange line) and the\nYIG/Ta/CoFeB trilayer (dashed green line). The simulations were performed with a magnetic bias field of 30 mT. For\ncomparison, we plotted the measured spectra of these samples in the bottom panel of Fig. 5a. The simulations reproduce\nthe excitation of PSSW modes in the YIG film of the YIG/CoFeB bilayer ( p= 1 ... 7) and the absence of these modes in\nYIG/Ta/CoFeB. The simulated PSSW frequencies are in good agreement with the experiments, except for frequencies near\nthe CoFeB FMR mode. This discrepancy is attributed to stronger hybridization between the PSSWs and the CoFeB FMR\nmode in the simulations, caused by stronger exchange coupling at a perfectly flat interface. Mode hybridization also shifts the\nFMR mode in CoFeB. Results for YIG/Ta/CoFeB confirm this view. For this decoupled structure, the frequency of spin-wave\nresonances are the same in the simulations and experiments (dashed green curves in Fig. 5a). We note that different parameters\nare plotted in the simulated and measured spectra. In the simulations, the intensity of the resonances is proportional to the\namplitude of magnetization precession. The intensity of the modes in the experiments, on the other hand, are determined by\ninduction-related absorption of a microwave current in the CPW. For constant magnetization precession, the absorption signal\nwould increase with frequency. As a result, the relative intensity of the CoFeB resonance at higher frequency is larger in the\nlower panel of Fig. 5a. Simulated spin-wave spectra as a function of magnetic bias field for both structures are shown in Figs.\n5b and 5c.\nWe now focus on the spatial distribution of spin-wave modes in the YIG and CoFeB films. Figure 6a and 6b show simulation\nresults for YIG/CoFeB and YIG/Ta/CoFeB, respectively. Magnetization precession in YIG at the FMR frequency is reduced\nnear the CoFeB interface. This effect, which is absent in the YIG/Ta/CoFeB structure, signifies strong exchange coupling to the\nCoFeB layer. The PSSWs in YIG/CoFeB are nearly symmetric and strongly confined to the YIG film for the non-hybridized\nmodes. The number of nodes corresponds to order parameter p. Hybridization between the PSSW modes in YIG and the FMR\nmode in CoFeB at higher frequencies also induces a significant magnetization precession in the CoFeB layer.\nThe simulated time evolution of PSSW mode formation in the YIG/CoFeB bilayer is depicted in Figs. 6c and 6d. Here, we\nfocus on p= 4 at an excitation frequency of 4.9 GHz. The simulations illustrate how the magnetization responds to the onset\nof a spatially uniform sinusoidal ac magnetic field at t= 0 s. Just after the excitation field is switched on, spin waves with a\nwavelength of l\u0019150 nm ( l=2d=p) are emitted from the YIG/CoFeB interface. This excitation is triggered by a dynamic\nexchange torque originating from dissimilar magnetization precessions in the YIG and CoFeB layers. The emitted spin waves\npropagate along the thickness direction of the YIG film and reflect at the GGG/YIG interface. At the selected frequency of 4.9\nGHz, the forward and backward propagating spin waves interfere constructively. As a result, a p= 4 PSSW is formed. The\nlarge-amplitude PSSW is fully established after t\u00196 ns.\nThe simulated time evolution of magnetization dynamics in the YIG/Ta/CoFeB trilayers is shown in Figs. 6e and 6f. As\ndiscussed previously, this structure does not support the excitation of PSSWs. Instead, the ac magnetic field induces uniform\nsmall-amplitude precessions of magnetization in the YIG and CoFeB layers. Because of different precession amplitudes, a\ntime-dependent divergence of magnetization emerges at the location of the Ta insertion layer. This divergence is the source\nof the dynamic exchange torque in structures where the magnetization of YIG and CoFeB are directly coupled by interface\nexchange interactions.\nFinally, we discuss the off-resonance time evolution of magnetization dynamics in the YIG/CoFeB bilayer (Figs. 6g and\n6h). We consider an excitation frequency of 4.5 GHz, thus, in between the frequencies of the p= 3 and p= 4 PSSW modes (see\nFig. 6a). Under these circumstances, spin waves are again emitted from the YIG/CoFeB interface by the dynamic exchange\ntorque. However, since the condition for constructive spin-wave interference along the film thickness of YIG is not fulfilled,\ntheir amplitude is not amplified and a PSSW does not form.\nIn summary, we have demonstrated an efficient method for the excitation of PSSWs in nanometer-thick YIG films. The\nmethod relies on direct exchange coupling between the YIG film and a CoFeB top layer. The application of an uniform\nmicrowave magnetic field produces a strong dynamic exchange torque at the YIG/CoFeB interface. This results in short-\nwavelength spin-wave emission. A PSSW is excited if one of the perpendicular confinement conditions is met. Our findings\nopen up a new route towards the excitation high-frequency spin waves in YIG. The results can also be generalized to other\nexchange-coupled systems. The excitation of intense PSSWs with large order numbers requires crossings between their\ndispersion relations and the FMR mode in a second, exchange-coupled magnetic layer. This situation is attained if the saturation\nmagnetization is smallest in the PSSW carrying film.\nMethods\nSample fabrication\nWe grew YIG films with a thickness of 80 nm and 295 nm on single-crystal GGG(111) substrates using PLD. The GGG\nsubstrates were ultrasonically cleaned in acetone and isopropanol before loading into the PLD vacuum chamber. We degassed\nthe substrates at 550\u000eC for 15 minutes. After this, oxygen was inserted into the chamber. After setting the oxygen pressure\nto 0.13 mbar, we increased the temperature to 800\u000eC at a 5\u000eC per minute rate. The YIG films were deposited under these\n4/9conditions from a stoichiometric target. We used an excimer laser with a pulse repetition rate of 2 Hz and a laser fluence of\n1.8 J/cm2. After film growth, we annealed the YIG films at 730\u000eC in an oxygen environment of 13 mbar. The annealing time\nwas 10 minutes. This was followed by a cool down to room temperature at a rate of \u00003\u000eC per minute. The deposition process\nresulted in single-crystal YIG films, as confirmed by X-ray diffraction. The composition of CoFeB was 40%Co, 40%Fe, and\n20%B. The CoFeB and Ta layers were grown by magnetron sputtering at room temperature.\nSpin-wave spectroscopy\nWe recorded spin-wave absorption spectra in transmission by measuring the real part of the S12scattering parameter. To\nenhance contrast, a reference spectrum taken at larger magnetic field or frequency was subtracted from the measurement data.\nThe setup consisted of a two-port VNA and a quadruple electromagnet probing station. The CPW with a 50 mm-wide signal\nline and two 800 mm-wide ground lines was patterned on a GaAs substrate. The gap between the signal and ground lines was\n30mm. The CPW was designed to provided a k\u00190excitation field in the plane of the YIG film. During broadband spin-wave\nspectroscopy measurements, the sample was placed face-down onto the CPW.\nMicromagnetic simulations\nWe performed micromagnetic simulations using open-source GPU-accelerated MuMax3 software. A 6900\u00026900\u0002345nm3\nCoFeB/YIG bilayer structure was discretized into 54\u000254\u00022:7nm3cells and two-dimensional periodic boundary conditions\nwere applied in the film plane. We abruptly changed the magnetic parameters at the YIG/CoFeB interface and used the\nharmonic mean value of the exchange constants in YIG and CoFeB to simulate the interface exchange coupling. The system\nwas initialized by an external magnetic field along the yaxis followed by relaxation to the ground state. After this, a spatially\nuniform 3 mT sinc-function-type magnetic field pulse with a cut-off frequency of 20 GHz was applied along the xaxis. The\nmagnetic field pulse excited all spin-wave modes up-to the cut-off frequency with uniform excitation power (Fig. 5). To study\nthe spatial dependence of magnetization dynamics (Fig. 6), the system was driven be a sinusoidal ac magnetic field with an\namplitude of 3 mT. In these simulations, the time evolution of the perpendicular magnetization component ( mz) was recorded for\n50 ns in 3 ps time steps along the thickness direction of the system at the center of the simulation mesh. The spatially-resolved\nintensity was obtained by applying a Fourier imaging technique where the time evolution of mzwas Fourier-transformed on a\ncell-by-cell basis.\nReferences\n1.Kruglyak, V . V ., Demokritov, S. O. & Grundler, D. Magnonics. J. Phys. D: Appl. Phys. 43, 264001 (2010).\n2.Serga, A. A., Chumak, A. V . & Hillebrands, B. YIG magnonics. J. Phys. D: Appl. Phys. 43, 264002 (2010).\n3.Khitun, A., Bao, M. & Wang, K. L. Magnonic logic circuits. J. Phys. D: Appl. Phys. 43, 264005 (2010).\n4.Krawczyk, M. & Grundler, D. Review and prospects of magnonic crystals and devices with reprogrammable band structure.\nJ. Phys.: Cond. Matter. 26, 123202 (2014).\n5.Chumak, A. V ., Vasyuchka, V . I., Serga, A. A. & Hillebrands, B. Magnon spintronics. Nat. Phys. 11, 453–461 (2015).\n6.Nikitov, S. A. et al. Magnonics: a new research area in spintronics and spin wave electronics. Physics-Uspekhi 58, 1002\n(2015).\n7.Chumak, A. V ., Serga, A. A., Hillebrands, B. & Kostylev, M. P. Scattering of backward spin waves in a one-dimensional\nmagnonic crystal. Appl. Phys. Lett. 93, 022508 (2008). DOI 10.1063/1.2963027.\n8.Chumak, A. V . et al. All-linear time reversal by a dynamic artificial crystal. Nat. Commun. 1, 141 (2010).\n9.Schneider, T. et al. Realization of spin-wave logic gates. Appl. Phys. Lett. 92, 022505 (2008).\n10.Chumak, A. V ., Serga, A. A. & Hillebrands, B. Magnon transistor for all-magnon data processing. Nat. Commun. 5, 4700\n(2014).\n11.Davies, C. S. et al. Field-controlled phase-rectified magnonic multiplexer. IEEE Trans. Magn. 51, 1–4 (2015).\n12.Yu, H. et al. Magnetic thin-film insulator with ultra-low spin wave damping for coherent nanomagnonics. Sci. Rep. 4,\n6848 (2014).\n13.Collet, M. et al. Spin-wave propagation in ultra-thin yig based waveguides. Appl. Phys. Lett. 110, 092408 (2017).\n14.Yu, H. et al. Approaching soft x-ray wavelengths in nanomagnet-based microwave technology. Nat. Commun. 7, 11255\n(2016).\n15.Demokritov, S. O., Hillebrands, B. & Slavin, A. N. Brillouin light scattering studies of confined spin waves: linear and\nnonlinear confinement. Phys. Reports 348, 441–489 (2001).\n5/916.Busse, F., Mansurova, M., Lenk, B., von der Ehe, M. & M ¨unzenberg, M. A scenario for magnonic spin-wave traps. Sci.\nRep. 5, 12824 (2015).\n17.Razdolski, I. et al. Nanoscale interface confinement of ultrafast spin transfer torque driving non-uniform spin dynamics.\nNat. Commun. 8, 15007 (2017).\n18.Klingler, S. et al. Measurements of the exchange stiffness of yig films using broadband ferromagnetic resonance techniques.\nJ. Phys. D: Appl. Phys. 48, 015001 (2015).\n19.Kennewell, K. J. et al. Magnetization pinning at a py/co interface measured using broadband inductive magnetometry. J.\nAppl. Phys. 108, 073917 (2010). DOI 10.1063/1.3488618.\n20.Kittel, C. Excitation of spin waves in a ferromagnet by a uniform rf field. Phys. Rev. 110, 1295–1297 (1958).\n21.Soohoo, R. F. General exchange boundary condition and surface anisotropy energy of a ferromagnet. Phys. Rev. 131,\n594–601 (1963).\n22.Wigen, P. E., Kooi, C. F., Shanabarger, M. R. & Rossing, T. D. Dynamic pinning in thin-film spin-wave resonance. Phys.\nRev. Lett. 9, 206–208 (1962).\n23.Gui, Y . S., Mecking, N. & Hu, C. M. Quantized spin excitations in a ferromagnetic microstrip from microwave photovoltage\nmeasurements. Phys. Rev. Lett. 98, 217603 (2007).\n24.Khivintsev, Y . V . et al. Spin wave resonance excitation in ferromagnetic films using planar waveguide structures. J. Appl.\nPhys. 108, 023907 (2010).\n25.Magaraggia, R. et al. Exchange anisotropy pinning of a standing spin-wave mode. Phys. Rev. B 83, 054405 (2011).\n26.Schoen, M. A. W., Shaw, J. M., Nembach, H. T., Weiler, M. & Silva, T. J. Radiative damping in waveguide-based\nferromagnetic resonance measured via analysis of perpendicular standing spin waves in sputtered permalloy films. Phys.\nRev. B 92, 184417 (2015).\n27.Conca, A. et al. Annealing influence on the gilbert damping parameter and the exchange constant of cofeb thin films. Appl.\nPhys. Lett. 104, 182407 (2014).\n28.Navabi, A. et al. Efficient excitation of high-frequency exchange-dominated spin waves in periodic ferromagnetic structures.\nPhys. Rev. Appl. 7, 034027 (2017).\n29.Kittel, C. On the theory of ferromagnetic resonance absorption. Phys. Rev. 73, 155–161 (1948).\n30.Howe, B. M. et al. Pseudomorphic yttrium iron garnet thin films with low damping and inhomogeneous linewidth\nbroadening. IEEE Magn. Lett. 6, 1–4 (2015).\n31.Sokolov, N. S. et al. Thin yttrium iron garnet films grown by pulsed laser deposition: Crystal structure, static, and dynamic\nmagnetic properties. J. Appl. Phys. 119, 023903 (2016).\n32.Vansteenkiste, A. et al. The design and verification of mumax3. AIP Adv. 4, 107133 (2014).\nAcknowledgements\nThis work was supported by the European Research Council (Grant No. ERC-2012-StG 307502-E-CONTROL). S.J.H.\nacknowledges financial support from the V ¨ais¨al¨a Foundation. Lithography was performed at the Micronova Nanofabrication\nCentre, supported by Aalto University.\nAuthor contributions statement\nH.J.Q., S.J.H., and S.v.D. designed and initiated research. H.J.Q. fabricated the samples and conducted the measurements.\nH.J.Q. and S.J.H. performed the micromagnetic simulations. S.v.D. supervised the project. H.J.Q. and S.v.D. wrote the\nmanuscript, with input from S.J.H.\nAdditional information\nCompeting financial interests: The authors declare that they have no competing interests.\n6/9Figure 1. (a) Schematic of the measurement geometry (not to scale). Spin-wave spectra are obtained by placing the sample\nface-down on a CPW. A microwave current is injected into the CPW signal line using a VNA. This produces a nearly uniform\nspin-wave excitation field. Spin-wave absorption is measured in transmission using scattering parameter S12. The external\nmagnetic bias field is oriented parallel to the CPW. (b) Spin-wave spectra for a single YIG film (dashed blue line), a single\nCoFeB film (dashed green line), and a YIG/CoFeB bilayer (solid orange line), measured with an external magnetic bias field of\n20 mT. The YIG and CoFeB films are 295 nm and 50 nm thick. The FMR modes in YIG and CoFeB ( p= 0) and higher order\nPSSW modes in YIG ( p= 1 ... 6) are labeled. (c)-(e) Spin-wave spectra of the same samples as a function of magnetic bias\nfield: (c) YIG, (d) CoFeB, (e) YIG/CoFeB. (f) YIG/CoFeB spin-wave spectrum at higher frequency and larger magnetic bias\nfield, demonstrating the excitation of PSSWs with large order numbers.\n-150 -100 -50 0 50 100 15024681012\n0, 123456Frequency (GHz)\nMagnetic field (mT )p\n20 40 60 80 100 12081012141618202 4 6 8 10fPSSW (GHz)\nkPSSW (rad/ □m)Order number pa\ncb\nd\n1 2 3 4 50.00.51.0\np = 3 p = 4 p = 2I_p / I_p = 0\nFrequenc y (GHz)p = 1\n1 2 3 4 5012345\np = 4 p = 3 p = 2\nFrequenc y (GHz)FWHM_p / FWHM_p = 0 p = 1\nFigure 2. (a) Frequency of the FMR and PSSW modes in a 295-nm-thick YIG film as a function of magnetic bias field. The\nsymbols are extracted from the experimental spin-wave spectrum in Fig. 1e. The lines are fits to the data using the Kittel\nformula ( p= 0) and the PSSW dispersion relation (Eq. 1, p= 1 ... 6). (b) PSSW mode frequency as a function of wave vector\nfor a magnetic bias field of 185 mT. The solid and empty symbols denote experimental data and calculated values (Eq. 1),\nrespectively. (c),(d) FWHM linewidth and intensity of the PSSW resonances as a function of frequency. The properties of the p\n= 1 ... 4 modes are normalized to those of the FMR mode ( p= 0).\n7/9Figure 3. (a) Spin-wave spectrum of a 80 nm YIG/50 nm CoFeB bilayer. (b) Extracted frequency of the p= 0 ... 2 modes in\nYIG, demonstrating an up-shift in PSSW frequency compared to data for the 295-nm-thick YIG film (Fig. 2a).\nFigure 4. (a),(b) Spin-wave spectrum for a 295-nm-thick YIG film and a 50-nm-thick CoFeB layer, separated by 10 nm of Ta.\nOnly the FMR modes in YIG and CoFeB are measured in this case. The lines in (b) are fits to the two resonances using the\nKittel formula.\nFigure 5. (a) Simulated (top) and measured (bottom) spin-wave spectra for a 295 nm YIG/50 nm CoFeB bilayer (solid orange\nline) and a 295 nm YIG/10 nm Ta/50 nm CoFeB trilayer (dashed green line). The magnetic bias field is 30 mT. The inset\nillustrates the simulation geometry. (b),(c) Simulated spectra for the same structures as a function of magnetic bias field.\n8/9Figure 6. (a),(b) Simulated spatial distribution of the FMR and PSSW modes in (a) a 295 nm YIG/50 nm CoFeB bilayer and\n(b) a 295 nm YIG/10 nm Ta/50 nm CoFeB trilayer. (c)-(f) Simulated time evolution of magnetization dynamics in (c),(d) a 295\nnm YIG/50 nm CoFeB bilayer and (e),(f) a 295 nm YIG/10 nm Ta/50 nm CoFeB trilayer. The excitation frequency in these\nsimulations is 4.9 GHz, which corresponds to the frequency of the p= 4 PSSW mode. (g),(h) Simulated time evolution of\nmagnetization dynamics in a 295 nm YIG/50 nm CoFeB bilayer at an off-resonance frequency of 4.5 GHz.\n9/9"    },    {        "title": "2210.08283v2.Non_local_magnon_transconductance_in_extended_magnetic_insulating_films___Part_II__two_fluid_behavior.pdf",        "content": "Nonlocal magnon transconductance in extended magnetic insulating films.\nII: two-fluid behavior.\nR. Kohno,1K. An,1E. Clot,1V. V. Naletov,1N. Thiery,1L. Vila,1R. Schlitz,2N. Beaulieu,3J. Ben Youssef,3A. Anane,4\nV. Cros,4H. Merbouche,4T. Hauet,5V. E. Demidov,6S. O. Demokritov,6G. de Loubens,7and O. Klein1,∗\n1Université Grenoble Alpes, CEA, CNRS, Grenoble INP, Spintec, 38054 Grenoble, France\n2Department of Materials, ETH Zürich, 8093 Zürich, Switzerland\n3LabSTICC, CNRS, Université de Bretagne Occidentale, 29238 Brest, France\n4Unité Mixte de Physique CNRS, Thales, Univ. Paris-Sud, Université Paris Saclay, 91767 Palaiseau, France\n5Université de Lorraine, CNRS Institut Jean Lamour, 54000 Nancy, France\n6Department of Physics, University of Muenster, 48149 Muenster, Germany\n7SPEC, CEA-Saclay, CNRS, Université Paris-Saclay, 91191 Gif-sur-Yvette, France\n(Dated: June 13, 2023)\nThisreviewpresentsacomprehensivestudyofthespatialdispersionofpropagatingmagnonselectricallyemit-\nted in extended yttrium-iron garnet (YIG) films by the spin transfer effects across a YIG |Pt interface. Our goal\nis to provide a generic framework to describe the magnon transconductance inside magnetic films. We experi-\nmentallyelucidatetherelevantspectralcontributionsbystudyingthelateraldecayofthemagnonsignal. While\nmostoftheinjectedmagnonsdonotreachthecollector,thepropagatingmagnonscanbesplitintotwo-fluids: i)\nalargefractionofhigh-energymagnonscarryingenergyofabout 𝑘𝐵𝑇0,where𝑇0isthelatticetemperature,with\nacharacteristicdecaylengthinthesub-micrometerrange,and ii)asmallfractionoflow-energymagnons,which\nare particles carrying energy of about ℏ𝜔𝐾, where𝜔𝐾∕(2𝜋)is the Kittel frequency, with a characteristic decay\nlengthinthemicrometerrange. Takingadvantageoftheirdifferentphysicalproperties,thelow-energymagnons\ncan become the dominant fluid i)at large spin transfer rates for the bias causing the emission of magnons, ii)at\nlarge distance from the emitter, iii)at small film thickness, or iv)for reduced band mismatch between the YIG\nbelowtheemitterandthebulkduetovariationofthemagnonconcentration. Thisbroaderpicturecomplements\npart I [1], which focuses solely on the nonlinear transport properties of low-energy magnons.\nI. INTRODUCTION\nNonlocal devices, such as the geometry shown in Fig. 1,\nconsisting of two lateral circuits deposited on an extended\nmagnetic insulating film have recently attracted much atten-\ntion as novel electronic devices exploiting the spin degree of\nfreedom[2–6]. As emphasized in part I, one of their origi-\nnal features is to behave as a spin diode at large currents[1].\nThese devices rely on the spin transfer effect (STE) to elec-\ntrically modulate the magnon population in a magnetic thin\nfilm. The process alters the amplitude of thermally activated\nspin fluctuations by transferring quanta of 𝛾ℏbetween an ad-\njacentmetallicelectrodeandthemagneticthinfilmviaastim-\nulated emission process. In unconfined geometries, a wide\nenergy range of eigenmodes is available to carry the exter-\nnalflowofangularmomentum,spanningafrequencywindow\nfromGHztoTHz,asschematicallyshowninFig.1(c),which\nshows the lower branch of the spin wave dispersion over the\nBrillouin zone [7–9]. At high-energy the curve flattens out\nat about 30 meV, which corresponds to the thermal energy,\n𝐸𝑇≈𝑘𝐵𝑇0, at ambient temperature, while at low-energy it\nshows a gap, 𝐸𝑔≈ℏ𝜔𝐾≈ 30𝜇eV, around the Kittel fre-\nquency𝜔𝐾∕(2𝜋)[10]. Betweenthesetwoextremes,thespec-\ntralidentificationoftherelevanteigenmodesinvolvedinnon-\nlocal spin transport has remained mostly elusive.\nIn this review, we propose a simple analytical framework\ntoaccountforthemagnontransconductanceinextendedmag-\nnetic insulating films. We find that the observed behavior can\n∗Corresponding author: oklein@cea.frbe well approximated by a two-fluid model, which simplifies\nthespectralviewasemanatingfromtwoindependenttypesof\nmagnons placed at either end of the magnon manifold. On\nthe one hand, we have magnons at thermal energies, to be re-\nferred to as high-energy magnons[4], whose distribution fol-\nlowsthetemperatureofthelattice. Ontheotherhand,wehave\nmagnonsatthebottomofthebandneartheKittelfrequency,to\nbe referred to as low-energy magnons, whose electrical mod-\nulationathighpoweristhefocusofpartI[1]. Theresponseof\nthese two magnon populations to external stimuli is very dif-\nferent. The high-energy thermal magnons, being particles of\nhigh wavevector, are mostly insensitive to any changes in the\nexternal conditions of the sample such as shape, anisotropy\nandmagneticfield,beinginsteaddefinedbythespin-waveex-\nchange stiffness and the large k-value of the magnon[11, 12].\nIn contrast, low-energy magnons, sensitive to magnetostatic\ninteraction, depend sensitively on the extrinsic conditions of\nthesample. Itturnsoutthatnonlocaldevicesprovideaunique\nmeanstostudyeachofthesetwo-fluidsindependentlybycom-\nparingthedifferencesintransportbehaviorasafunctionofthe\nseparation,𝑑, between the two circuits, thus benefiting from\nthe spatial filtering associated with the fact that each of these\ntwo components decays very differently as a function of dis-\ntance, as schematically shown in Fig. 1(b).\nThe paper is organized as follows. After this introduction,\ninthesecondsectionwereviewthemainfeaturesthatsupport\nthe two-fluid separation. In the third section, we describe the\nanalytical framework of a two-fluid model and, in particular,\ntheexpectedsignatureinthetransportmeasurement. Thispart\nbuilds on the knowledge gained in part I[1] about the nonlin-\near behavior of the low-energy magnon. To facilitate quickarXiv:2210.08283v2  [cond-mat.mes-hall]  11 Jun 20232\nFIG. 1. Lateral geometry used for measuring the magnon transcon-\nductanceinextendedmagneticinsulatingfilms. (a)Scanningelectron\nmicroscopeimageofa4-terminalcircuit(scalebaris5 𝜇m),whose4\npolesareconnectedtotwoparallelwires,Pt1andPt2(showninpink),\ndeposited on top of a continuous YIG thin film. A continuous elec-\ntric current, 𝐼1, injected in Pt1(emitter) produces an electric mod-\nulation of the magnon population by the spin transfer effect (STE).\nThismodulationisconsequentlydetectedlaterallybythespinpump-\ning voltage−𝑅2𝐼2through a second electrode Pt2(collector) placed\natadistance 𝑑fromtheemitter. Wedefinethemagnontransmission\nratioT𝑠=𝐼2∕𝐼1andthetransconductance T𝑠∕𝑅1. Panel(b)isasec-\ntional view showing the spatial decay of propagating magnons. (c)\nSchematic representation of the spin-wave dispersion over the Bril-\nlouin zone. We consider the spin transport properties as originat-\ning from two independent fluids located at either end of the disper-\nsioncurve. Eachofthetwo-fluidshasadifferentcharacteristicdecay\nlength,𝜆𝑇and𝜆𝐾respectively, as shown in (b).\nreading of either manuscript, we point out that a summary of\nthe highlights is provided after each introduction and, in both\npapers, the figures are organized into a self-explanatory sto-\nryboard, summarized by a short sentence at the beginning of\neach caption. In the fourth section we will show the experi-\nmentalevidencethatsupportssuchapictureandfinallyinthe\nfifth section we will conclude our work by emphasizing the\nimportant results and opening to future perspectives.\nII. KEY FINDINGS\nThe purpose of this review is to present the experimental\nevidence supporting the separation of the magnon transcon-\nductanceintotwocomponents. Thisisachievedbymeasuring\nthe transmission coefficient T𝑠≡𝐼2∕𝐼1of magnons emitted\nand collected via the spin Hall effect between two parallel Pt\nwires, Pt1and Pt2, respectively. It is shown that a two-fluid\nmodel, where T𝑠=T𝑇+T𝐾is the independent sum of a\nhigh-energy and a low-energy magnon contribution, providesasimplifiedcommonframeworkthatcapturesalltheobserved\nbehaviorinnonlocaldeviceswithdifferentinter-electrodesep-\naration,differentcurrentbias,differentappliedmagneticfield,\ndifferentfilmthicknessormagneticcomposition,anddifferent\nsubstrate temperature.\nMakingaquantitativeanalysisofthetransmissionratio,we\nfindthatmostoftheinjectedspinsremainlocalizedunderthe\nemitterorpropagateinthewrongdirection(theestimatedfrac-\ntion is about 2/3), making these materials intrinsically poor\nmagnonconductors. Theremainingpropagatingmagnonsfall\ninto two distinct categories: First, a large fraction carried by\nhigh-energy magnons, which follow a diffusive transport be-\nhavior with a characteristic decay length, 𝜆𝑇, in the submi-\ncron range[13, 14]; and second, a small fraction carried by\nlow-energy magnons,which are responsible forthe asymmet-\nric transport behavior [1], and which follow a ballistic trans-\nport with a characteristic decay length, 𝜆𝐾, in the micrometer\nrange. The different decay behaviors are directly observable\nexperimentally in the change of the nonlinear spin transport\nbehavior with separation, 𝑑.\nWe also carefully study the collapse of the magnon trans-\nmission ratio with increasing temperature of the emitter, 𝑇1,\nas it approaches the Curie temperature, 𝑇𝑐. Here, the num-\nberof spin-polarizedsitesunder theelectrodebecomes ofthe\nsame order as the spin flux coming from the external Pt elec-\ntrode. Thetransitiontothisregimeofmagnetizationreduction\nleadstoasharpdecreaseinthemagnontransmissionratio. We\nreport signs of interaction between the low-energy and high-\nenergy parts of the liquid in this highly diffusive regime[15–\n17]. In addition, the collapse seemsto actually occur well be-\nforereaching 𝑇𝑐,suggestingthatthetotalnumberofmagnons\nissignificantlyunderestimatedcomparedtothatinferredfrom\nthe single temperature value of the lattice below the emit-\nter. Alternatively, this discrepancy could indicate a rotation\nof the equilibrium magnetization under the emitter[18, 19].\nSince the discrepancy actually becomes more pronounced as\nthe magnetic film gets thinner, this suggests that the culprit is\nthe amount of low-energy magnons.\nIII. ANALYTICAL FRAMEWORK\nA. Low-energy magnons\nWerecallthefindinginpartI[1],thatthetransconductance\nby low-energy magnons in open geometries can be described\nby the analytical expression:\nT𝐾∝𝑀1\n𝑀2⋅𝑘𝐵𝑇1\nℏ𝜔𝐾⋅𝑒𝜔𝐾\nth⋅1\n1−(𝐼1∕th)2,(1)\nwhere𝑒istheelectroncharge,while 𝑀1and𝑀2arethemag-\nnetizationvaluesundertheemitterandcollector,respectively.\nThe threshold current, th, is the solution of a transcenden-\ntal equation obtained by combining Eqs. (4), (6) and (7) in\nRef. [1]. In our model, its nonlinear behavior is determined\nsolely by two parameters th,0and𝑛sat, which are related to\nthe nominal value of the transmission coefficient at low cur-\nrentandthesaturationthresholdexpressedinnormalizedunits3\nFIG. 2. Current bias characteristic of the magnon transconductance\ndepending on the spectral nature of propagating magnons. Panels\n(a) and (b) compare the predicted electrical variation of T𝑠for low-\nenergymagnons(seeEq.(1),leftpanel)andforhigh-energymagnons\n(see Eq. (3), right panel), respectively, when 𝐻𝑥<0. Panel (f)\nshows the associated variation of 𝑇1=𝑇0+𝜅𝑅𝐼2\n1, the lattice tem-\nperature below the emitter. The current span exceeds 𝐼c, the current\nbias, whichraises 𝑇1to𝑇𝑐, theCurietemperature. Panels (c)and (d)\nshow the behavior when T𝑠is renormalized by 𝑇1. Panel (e) shows\nthetwo-fluidfittingfunction: theindependentsumofthelow-energy\nand high-energy magnon contributions with their respective weights\nΣ𝑇andΣ𝐾. The inset (g) shows the temperature dependence of the\nmagnetization 𝑀𝑇as measured by vibrating sample magnetometry\n(cf. Fig. S1), and the solid line is a fit with the analytical expression\n𝑀𝑇≈𝑀0√\n1−(𝑇∕𝑇𝑐)3∕2, with𝜇0𝑀0=0.21T and𝑇𝑐=550K.\nof nonlinear effects, respectively. All information about these\nfeedback effects can be found in Ref. [1].\nAsemphasizedindetailinpartI,oneofthepitfallsofnon-\nlocal devices is that the emitter electrode cannot be made im-\nmune to Joule heating due to poor thermalization in the 2D\ngeometry. This leads to a significant increase of the tempera-\nture under the emitter with current 𝐼1, which we model by\n𝑇1||𝐼2\n1=𝑇0+𝜅𝑅𝐼2\n1. (2)\nIn our notation, 𝑇0is the substrate temperature at no current\nand𝜅is the temperature coefficient of resistance for Pt. It\nis the coefficient that determines the temperature rise per de-\nposited joule power (see Fig. S1 in Appendix). We addition-\nallydefine𝐼cthecurrentrequiredtoreachtheCurietempera-\nture,𝑇𝑐=𝑇0+𝜅𝑅𝐼2\nc[see Fig. 2(f)]. This variation has pro-\nfound consequences both on the level of thermal fluctuations\nofthelow-energymagnonsandonthenumberofhigh-energy\nmagnons. In particular, the variation of 𝑇1with𝐼1expressed\nby Eq. (2) enters into the variation of T𝐾with𝐼1expressed\nby Eq. (1). The resulting variation of the magnon population\nas a function of 𝐼1is shown in Fig. 2(a). To account for thevariationof𝑇1producedbyJouleheating,whichexpressesthe\ninfluence of a varying background of thermal fluctuations on\nthe STE, we plot T𝐾∕𝑇1in Fig. 2(c). This renormalization is\nequivalent to looking at the nonlinear behavior from the per-\nspective of a thermalized background. The resulting shape of\nthe curve as a function of 𝐼1is greatly simplified. In the re-\nverse bias, marked by the symbol ◂representing the magnon\nabsorption regime, the normalized transconductance is con-\nstant up to𝐼c. In contrast, in the forward bias, denoted by the\nsymbol▸, which represents the magnon emission regime, a\npeak appears. This asymmetric peak is called the spin diode\neffect in part I[1]. The advantage of the 𝑇1normalization of\nthemagnontransmissionratioisthatitmakesthepeakachar-\nacteristic feature of the spin diode effect.\nB. High-energy magnons\nWenowassumethatthenumberofhigh-energymagnonsis\napproximatelyequaltothetotalnumberofmagnons,whichis\nthedifference 𝑀1−𝑀0,where𝑀0isthespontaneousmagne-\ntization at𝑇=0K and𝑀1is the spontaneous magnetization\nat𝑇=𝑇1, the temperature of the emitter [20]. We thus an-\nalytically express the contribution of high-energy magnons to\nthe magnon transconductance by the equation:\nT𝑇∝𝑀1\n𝑀2⋅𝑀0−𝑀1\n𝑀0, (3)\nwheretheprefactor 𝑀1representstheamountofmagneticpo-\nlarizationavailableundertheemitter. Wenotethattheanalyt-\nical form expressed by Eq. (3) has been previously proposed\nto describe spin transmission in paramagnetic materials[21].\nAs shown in the inset Fig. 2(g), we find that the tempera-\nture dependence of 𝑀1is well described by the analytical\n𝑀1≈𝑀0√\n1−(𝑇1∕𝑇𝑐)3∕2. The resulting number of ther-\nmally excited magnons contributing to the nonlocal transport\nis shown in Fig. 2(b). Repeating the same analysis developed\nin Fig. 2(c), a more revealing behavior is obtained by renor-\nmalizing T𝑇with𝑇1and the result is shown in Fig. 2(d). In\nthiscase,thecurrentdependenceof T𝑇∕𝑇1on𝐼1isaconstant\nfunction up to 𝐼c.\nC. Two-Fluid Model\nAn advantage specific to nonlocal transport measurements\nisthatthepropagationdistance, 𝑑,providesapowerfulmeans\nto spectrally distinguish different types of magnons, each of\nwhich has its characteristic decay length 𝜆𝑘along the𝑥-axis\n[13,22]. Inthefollowingwewillexaminetheexpectationfor\nthe different extrema of the dispersion curve.\nFor the high-energy magnons, the spin wave spectrum can\nsimply be approximated as 𝜔𝑘=𝜔𝑀𝜆2\nex𝑘2, where𝜔𝑀=\n𝛾𝜇0𝑀𝑠= 2𝜋×4.48GHz and𝜆ex≈ 15nm is the exchange\nlength[23]. High-energy magnons at room temperature ( 𝑇0=\n300K) have the frequency 𝜔𝑇=𝑘𝐵𝑇0∕ℏ= 2𝜋×6.25THz,4\nFIG.3. Dispersioncharacteristicoflow-energymagnons. (a)Disper-\nsion curves at the bottom of the magnon manifold of a 19 nm thick\nYIG film for two values of 𝜃𝑘= 0◦(𝑘∥𝑀) and90◦(𝑘⟂𝑀),\ntheanglebetweenthewavevectorandtheappliedmagneticfield. We\nmark with dots the Kittel mode ( 𝐸𝐾, black dot), the lowest energy\nmode(𝐸𝑔,bluedot),andthemodedegeneratetotheKittelmodewith\nthe highest wavevector ( 𝐸𝐾, orange dot). The curve is computed for\nYIG𝐴thin films. (b) Characteristic decay length calculated from the\ndispersioncurve,assumingthatthemagnonsfollowthephenomeno-\nlogical LLG equation with 𝛼LLG=4⋅10−4.\nwhich corresponds to a wavevector 𝑘𝑇=2.5nm−1. It is seri-\nously questionable whether the estimate for 𝜆𝑇from the phe-\nnomenologicalLandau-Lifshitz-Gilbert(LLG)modelisappli-\ncable to such short-wavelength magnons. Practically i)the\nGilbert damping is expected to be increased in the THz range\n[23].ii)the group velocity is reduced towards the edge of the\nBrillouin zone [7, 24], and iii)the LLG model does not con-\nsider the reduction of the characteristic propagation distance\ndue to diffusion processes. Furthermore, YIG is a ferrimag-\nnet, higher (antiferromagnetic) spin wave branches contribute\nsignificantlytothemagnontransport[7–9]. Webelievethatthe\nmost reliable estimates have been obtained experimentally by\nstudying the spatial decay of the spin Seebeck signal[13, 25]\nand have found 𝜆𝑇≈0.3𝜇m.\nIn contrast to its high-energy counterpart, the LLG frame-\nwork should provide a good basis for calculating the propa-\ngation distance of long-wavelength dipolar spin waves. This\ninteraction gives an anisotropic character to the group veloc-\nity of these spin waves. In Fig 3(a) we plot the dispersion\ncurve of a magnon propagating either along the 𝑥-axis (or-\nangeline)oralongthe 𝑦-axis(blueline). Inthefollowing,we\nwill focus our attention on the branch 𝜃𝑘= 0◦(orange line),\nwhich corresponds to the magnon propagating in the normal\ndirection of the Pt wires. As emphasized in part I[1], there\nare 3 remarkable positions on the curve, each marked by a\ncolored dot on Fig. 3. The energy minimum, 𝐸𝑔(blue dot),\ndoes not contribute to the transport because its group veloc-\nity is zero. The longest wavelength spin waves correspond\nto the Kittel mode, 𝐸𝐾(black dot). The damping rate, tak-\ning into account the ellipticity of the spin waves, is given by\nΓ𝐾=𝛼LLG(𝜔𝐻+𝜔𝑀∕2), where𝜔𝐻=𝛾𝐻0[26]. The ve-\nlocity is equal to 𝑣𝐾=𝜕𝑘𝜔=𝜔𝐻𝜔𝑀𝑡YIG∕(4𝜔𝐾), where𝜔𝐾\nis the Kittel frequency and 𝑡YIGis the YIG thickness. The re-\nsulting decay length of the spin transport carried by 𝑘→0\nmagnons is𝜆𝐾=𝑣𝐾∕(2Γ𝐾)≈2.5𝜇m for𝑡YIG=19nm. Aspointed out in part I[1], the mode that seems to be most rel-\nevant for long-range magnon transport in nonlocal devices is\nprobably𝐸𝐾, the degenerate mode with the Kittel frequency\nand the shortest wavelength. This mode is marked by an or-\nange dot in Fig. 3. For our 𝑡YIG= 19nm film, it turns out\nthat its group velocity is of the same order as that of the Kit-\ntel mode, giving a similar decay distance. We will show later\nthat this estimate is quite close to the experimental value. We\nnote, however, that the value of the decay distance at 𝐸𝐾in-\ncreaseswithincreasingfilmthicknesstobecomeindependent\nof𝑡YIGfor thicknesses above 200 nm. The saturation value is\n𝜆𝐾≈20𝜇m, assuming 𝛼LLG=4⋅10−4.\nSince𝜆𝐾≈ 10×𝜆𝑇, changing𝑑allows tuning from spin\ntransport governed by high-energy magnons to spin transport\ngoverned by low-energy magnons. One should also add that\nthecurrentintensity, 𝐼1,alsoprovidesameanstotunetheratio\nbetween the two-fluid as discussed in Ref. [1].\nLearningfromtheaboveconsiderations,wecannowputall\nthe contributions together to propose an analytical fit of the\ndata with the two-fluid function:\nT𝑠=Σ𝑇,0exp−𝑑∕𝜆𝑇T𝑇\nT𝑇,𝐼1→0+Σ𝐾,0exp−𝑑∕𝜆𝐾T𝐾\nT𝐾,𝐼1→0,\n(4)\ncombining two independent magnon contributions: one at\nthermal energy and the second at magnetostatic energy. We\nassumeherethatbothmagnonfluidsfollowanexponentialde-\ncay. To ease the notation, we shall refer below at underlined\nquantity,e.g.T𝑇≡T𝑇∕T𝑇,𝐼1→0, as the normalized quantity\nby the low current value. We define Σ𝐾||𝑑=Σ𝐾,0exp−𝑑∕𝜆𝐾\nandΣ𝑇||𝑑= Σ𝑇,0exp−𝑑∕𝜆𝑇, where the index 0represents\nthe extrapolated value at the emitter position ( 𝑑= 0): see\nFig. 1(b). Thus the parameter Σ𝐾∕(Σ𝐾+Σ𝑇)||𝑑represents\nthe variation with distance of the proportion of low-energy\npropagating magnons over the total number of propagating\nmagnons. An exemplary fit for 𝑑= 0and identical high-\nenergy and low-energy contributions is shown in Fig. 2(e).\nItshouldbeemphasizedthatthemodelproposedbyEq.(4),\nwhich assigns a fixed decay rate to each magnon category, is\ncertainlytoosimplistic. Forexample,oneshouldkeepinmind\nthatif𝑀1→0duetoJouleheating,thiscouldhaveprofound\nconsequenceson 𝜆𝑇bychangingthestiffnessoftheexchange\nconstant. This has already been discussed in the context of\nspin propagation in paramagnetic materials[21]. We will re-\nturntothisissue belowinthecontextof ourdiscussionofthe\ndiscrepancy in the values of 𝑇𝑐extracted from the transport\ndata.\nIV. EXPERIMENTS\nIn this section we present the experimental evidence sup-\nporting the two-fluid picture shown above. We focus on the\nevolutionofspintransportwithcurrent,distance,appliedmag-\nnetic field, substrate temperature and effective magnetization,\n𝑀eff. This will allow us to test the validity of our model.5\nFIG. 4. Dependence of the collected voltage on an external mag-\nnetic field. Comparison of the nonlocal voltage V2= (𝑉2,⟂−𝑉2,∥)\nbetween (a) a short-range device ( 𝑑=1.0𝜇m) and (b) a long-range\ndevice (𝑑= 2.3𝜇m). The panels show the zoom at the maximum\nand minimum of the normalized values. We interpret the detection\nof a finite susceptibility, 𝜕𝐻𝑥V2<0, as an indication of a magnon\ntransmission ratio by low-energy magnons. In contrast, a constant\nbehavior,𝜕𝐻𝑥V2≈ 0, is indicative of a magnon transmission ratio\nby high-energy magnons. Finite susceptibility is uniquely observed\nin the long-range regime when 𝐼1⋅𝐻𝑥<0,i)when the number of\nlow-energymagnonsisincreasedbyinjectingacurrentintheforward\ndirection,and ii)whenthecontributionoftherapidlydecayinghigh-\nenergy magnons becomes a minority. The data are collected on the\nYIG𝐶thinfilmdrivenbyalargecurrentamplitudeof ±𝐼1=2.0mA.\nThenormalizationvalueof V2arerespectively 10.04𝜇Vand8.66𝜇V\nin panel (a) and (b).\nA. Magnetic susceptibility of the magnon transmission ratio\nWe begin this section by first presenting some key experi-\nmentalevidencesupportingthetwo-fluidpicture. Aschematic\nof the 4-terminal device is shown in Fig. 1(a). It circulates\npure spin currents between two parallel electrodes subject to\nthespinHalleffect[27]: inourcasetwoPtstrips 𝐿Pt=30𝜇m\nlong,𝑤Pt= 0.3𝜇m wide and 𝑡Pt= 7nm thick. The experi-\nment is performed here at room temperature, 𝑇0=300K, on\na 56 nm thick (YIG𝐶) garnet thin film whose physical prop-\nerties are summarized in Table 1 of Ref. [1]. While injecting\nanelectriccurrent 𝐼1intoPt1,wemeasureavoltage 𝑉2across\nPt2,whoseresistanceis 𝑅2. Tosubtractallnon-magneticcon-\ntributions, we define the spin signal V2= (𝑉2,⟂−𝑉2,∥)as\nthevoltagedifferencebetweenthenormalandparallelconfig-\nuration of the magnetic field with respect to the direction of\nthe electric current. In practice, the measurement is obtained\nsimply by recording the change in voltage as an in-plane ex-\nternal magnetic field, 𝐻0, is rotated along the 𝑥and𝑦direc-\ntions,respectively[theCartesianframeisdefinedinFig.1(a)].\nFig.4showsthevariationof V2asafunctionof 𝐻𝑥foralarge\namplitude of |𝐼1|=2.0 mA, which corresponds to a current\ndensity of1⋅1012A/m2. To reduce the influence of Joule\nheating and also thermal activation of the electrical carriers\nin YIG[28, 29], we use a pulse method with a 10% duty cy-\ncle throughout this study to measure the nonlocal voltage[4].In the measurements, the current is injected into the device\nonly during 10 ms pulses with a 10% duty cycle. In Fig. 4 we\ncomparethemagneticfieldsensitivityofthe(normalized)spin\ntransport at two values of the center-to-center distance 𝑑be-\ntweenemitterandcollectorforpositiveandnegativepolarities\nof the current. In total, this leads to 4 possible configurations\nfor the pair ( 𝐼1,𝐻𝑥), each labeled by the symbols ◔,\n◔,◔,◔\nto match the notation of Fig. 5. There, vertical displacement\nofthemarkerdissociatesscansofopposite 𝐻𝑥-polarity,while\nhorizontaldisplacementofthemarkerdissociatesscansofop-\nposite𝐼1-polarity. Looking at Fig. 4, we recover the expected\ninversion symmetry while enhancement of the spin current is\nclearlyvisiblewhen 𝐼1⋅𝐻𝑥<0. Thesignalseemstodepend\nonthemagneticfieldonlyforlargerdistancesand 𝐼1⋅𝐻𝑥<0\n(forward bias). Considering that the two Pt wires are both\n𝑤= 0.3𝜇m wide, this corresponds to an edge to edge sep-\naration𝑠=𝑑−𝑤. In one case the distance is 𝑠≈ (2𝜆𝑇), in\nthe other case 𝑠≈ 4⋅(2𝜆𝑇), where2𝜆𝑇≈ 0.6𝜇m is the esti-\nmated amplitude decay length of the magnons at thermal en-\nergy. Itwillbeshownbelowthatundertheemitterthenumber\nofhigh-energymagnonsfarexceedsthenumberoflow-energy\nmagnons. Assuming an exponential decay of the high-energy\nmagnons, one expects in (a) an attenuation of their contribu-\ntionby50%,whilein(b)itisreducedbyalmost99%. Wethus\narriveatasituationwhereat 𝑑=0.5𝜇mthemagnontransport\nis dominated by the behavior of high-energy magnons, while\nat𝑑= 2.3𝜇m the magnon transport is dominated by the be-\nhavioroflow-energymagnons(seebelow). InFig.4weassign\nthefinitesusceptibility 𝜕𝐻𝑥V2<0asanindicationofmagnon\ntransmission through low-energy magnons. Since the energy\nofthesemagnonsaswellasthethresholdofdampingcompen-\nsation depend sensitively on the magnetic field[30, 31], the\nlow-energy magnons are significantly affected by the ampli-\ntude of the magnetic field, 𝐻𝑥[4, 32, 33]. Such a field de-\npendenceisexplainedinEq.(5)ofRef.[1]. Whatisobserved\nhere is that near the peak bias, 𝐼pk≈ 2.2mA (see definition\nin part I), the device becomes particularly sensitive to a shift\nofth. In our case, the external magnetic field shifts thby\nshifting the Kittel frequency, 𝜔𝐾=𝛾𝜇0√\n𝐻0(𝐻0+𝑀𝑠). In\ncontrast, the constant behavior, 𝜕𝐻𝑥V2≈ 0, is indicative of a\nmagnon transmission ratio by high-energy magnons: because\nof their short wavelength, their energy is of the order of the\nexchange energy, and thus independent of the magnetic field\nstrength[34]. Since these 2 plots are measured with exactly\nthe same current bias, and the only parameter changed is 𝑑, it\nshowsthatfilteringbetweenhighandlow-energymagnonscan\nbe achieved by simply changing the separation between emit-\nterandcollector. Italsodirectlysuggestsadoubleexponential\ndecay, as will be discussed later in Fig. 8.\nB. Spectral signature in nonlocal measurement.\nFig. 5 compares the variation of V2as a function of emitter\ncurrent𝐼1for two different emitter-collector separations. The\nmaximum current injected into the device is about 2.5 mA,\ncorresponding to a current density of 1.2⋅1012A/m2. The\npolarity bias for the pair ( 𝐼1,𝐻𝑥) is represented by the sym-6\nFIG.5. Measurementofthecollectedelectricalcurrent, 𝐼2,asafunc-\ntionoftheemittercurrent, 𝐼1. Wecomparethetransportcharacteris-\nticsbetweentwononlocaldevices: onewithashortemitter-collector\ndistance in the submicron range ( 𝑑= 0.5𝜇m, left column) and the\nother with a long distance of a few microns ( 𝑑= 2.3𝜇m, right col-\numn). Thefirstrow(a)and(b)shows V2at𝑇0=300Kasafunction\nof𝐼1, the injected current, for both positive and negative polarity of\n𝐻𝑥, the applied magnetic field. In our symbol notation, the marker\nposition indicates the quadrant in the plot pattern. The raw signal\nV2= −𝑅2𝐼2+V2is decomposed into an electric signal, 𝐼2, and a\nthermal background signal, V2, as shown in the third row (e,f) and\nthe second row (c,d), respectively. The background, V2, represents\nthe background magnon currents along the thermal gradients. The\nmeasurementsareperformedonYIGCthinfilms. Thedataaretaken\nat𝐻0=0.2T.\nbols◔,\n◔,◔,◔, in replication of the 4-curve pattern. We\nrecover in Fig. 5(a,b) the expected inversion symmetry with\nV◔\n2≈ −V\n◔\n2andV\n◔\n2≈ −V◔\n2, while the enhancement of\nthe spin current is visible when 𝐼1⋅𝐻𝑥<0, representing\nthe forward regime. As explained in part I [1], the raw sig-\nnalV2=V2−𝑅2𝐼2can be decomposed into i)V2|||𝐼2\n1a ther-\nmalsignalproducedbytheSpinSeebeckEffect(SSE),which\nis always odd/even with 𝐻𝑥or𝐼1and shown in panels (c,d),\nandii)−𝑅2𝐼2||𝐼1, an electrical signal produced by the spin\ntransfer effect (STE), which is in the linear regime even/odd\nwiththepolarityof 𝐻𝑥or𝐼1,respectively,andshowninpan-\nels (e,f) [35]. This decomposition is obtained by assuming\nthat in reverse bias V◔\n2= −V\n◔\n2+𝑅2T𝑠||𝐼1→0T𝑇⋅𝐼1and\nV\n◔\n2= −V◔\n2+𝑅2T𝑠||𝐼1→0T𝑇⋅𝐼1, which evaluates thenumber of absorbed magnons as a linear deviation from the\nnumber of thermally excited low-energy magnons, assuming\nC2continuityofthemagnontransmissionratioacrosstheori-\ngin. We recall that in our notation T𝑇≡T𝑇∕T𝑇||𝐼1→0. We\nthen construct V\n◔\n2=V◔\n2andV◔\n2=V\n◔\n2by enforcing that\nthesignalgeneratedbyJouleheatingisexactlyevenin 𝐼1. We\nobserve that in the short range ( 𝑑= 0.5𝜇m), we get V◓\n2≈\n(V\n◔\n2+V◔\n2)∕2and𝐼◓\n2=sign(𝐼1)(V\n◔\n2−V◔\n2)∕(2𝑅2), which\nistheexpectedsignatureforasymmetricmagnonsignal. This\nequality is not satisfied in the long range ( 𝑑= 2.3𝜇m) for\nV2due to the asymmetry of the signal between forward and\nreverse bias as explained in part I. The consistency of this\ndata manipulation is confirmed below in Fig. 6(a) and (b)\nby showing a small asymmetric enhancement of 𝐼2at high𝐼1\nby low-energy magnons at short distances and a pronounced\nenhancement at long distances as discussed in Ref. [1]. The\nfactthatamorepronouncedenhancementisobservedatlarge\ndistances is further evidence for the spatial filtering of high-\nenergy magnons.\nIt is worth noting that one can reach a situation where\n−𝑅2𝐼2= 0without necessarily having V2vanish as well, as\nshown in Fig. 5(e) and (f) at 𝐼1= 2.5mA. This is explained\nbytheformationoflateraltemperaturegradients[36]. Inother\nwords, the observation of 𝑀𝑇=0is a local problem, mostly\naffecting the region below the emitter. It does not imply that\n𝑀=0throughout the thin film.\nAs a next step, we will show how to distinguish the con-\ntributions of high-energy and low-energy magnons using the\nanalytical model in Fig. 6. Starting from Fig. 5(e,f), we will\nremovetheinfluenceofthespuriouscontributionontheelec-\ntrical spin transport signal. First, we normalize the signal by\ntheemittercurrenttoobtainthemagnontransmissionratioco-\nefficient T𝑠=𝐼2∕𝐼1as shown in Fig. 6(a,b). For small sepa-\nration, we observe that T𝑠shows a quadratic behavior that is\nsymmetricincurrentandconsequentlyweassociateitwiththe\ndevice temperature. In contrast, the device with large separa-\ntion shows an asymmetric enhancement due to the spin diode\neffect [1]. The influence of the increase of the emitter tem-\nperature𝑇1due to the Joule heating of 𝐼1can be removed\nby normalizing with 𝑇1∕𝑇0. This normalization removes the\nsymmetric enhancement of the magnon transmission ratio as\nreported in previous studies[3, 29, 37, 38], where the justi-\nfication will be discussed later in Fig. 7 [39]. The obtained\ntraces are shown in Fig. 6(c,d) and can be compared with the\ntheoretical expectation given by Eq. (4), which is graphically\nsummarized in Fig. 2(e). The solid lines are fit curves with\nourmodelrepresentingthesumofthecontributionfromlow-\nenergymagnonsandthebackgroundcontributionsfromhigh-\nenergymagnons,withtheparametersofthefitgiveninTableI.\nThe dashed line and the gray shaded area represent the latter\nΣ𝑇Δ𝑛𝑇. Fromthefitswecanobtaintheratio Σ𝑇∕(Σ𝑇+Σ𝐾)for\nthetwomagnonfluids,wherethecontributionofhigh-energy\nmagnons decreases from 95% at 0.5 𝜇m to 50% at 2.3 𝜇m, in\naccordance with the spatial filtering proposed above.\nTo illustrate Eq. (3) experimentally, we repeated the mea-\nsurement for different values of the substrate temperature 𝑇07\nFIG. 6. Dependence of the magnon transmission ratio on the sepa-\nration between the electrodes. Starting from the extraction of 𝐼2in\nFig. 5, the first row compares the variation of the ratio T𝑠=𝐼2∕𝐼1\nbetweenshort-range(leftcolumn)andlong-range(rightcolumn)de-\nvices. In the short range, the behavior shows a symmetrical signal\nof the magnon transmission ratio with respect to the current polar-\nity𝐼1, while in the long range, the behavior is asymmetrical. We\ninterpret the difference to be due to two different types of magnons:\ndominantly high-energy magnons in the short range and dominantly\nlow-energymagnonsinthelongrange. Toeliminatenonlineardistor-\ntionscausedbyJouleheating, T𝑠isrenormalizedby 𝑇1||𝐼2\n1,theemit-\nter temperature variation produced by Joule heating (see text). The\nsolidlinesarefittedwithEq.(4),wheretheshadedregionshowsthe\nbackground contribution from high-energy magnons Σ𝑇T𝑇, where\nΣ𝑇∕(Σ𝑇+Σ𝐾)||𝑑represent their relative weight at this distance. In\n(c) this ratio is about 0.95, while in (d) it drops to about 0.5.\nat small separation. Fig. 7(a) shows the experimental result\nfor five different values of 𝑇0when𝐼1varies on the same\n[−2.5,2.5]mA span. Note that the data are plotted as a func-\ntion of𝑇1=𝑇0+𝜅𝐴𝑅Pt𝐼2\n1, the emitter temperature. The ra-\ntionale for this transformation of the abscissa is apparent in\nFig. 7(b) and (c), which show that the nonlinear current de-\npendence of both the SSE and STE signals originates from\nthe enhancement of 𝑇1. In particular, Fig. 7(c) shows the rise\nof the SSE signal V2as a function of 𝐼1for different values\nof𝑇0. We find that all curves almost overlap on the same\nparabola, suggesting an identical thermal gradient of the Pt1\nelectrode through 𝐼1independently of 𝑇0, with a small devia-\ntionforsmaller 𝑇0duetothedecreaseof 𝑅Pt. Inaddition,Fig.\n7(d) shows(T𝑠)−1≡(T𝑠∕T𝑠|𝐼1→0)−1, the inverse transmis-\nsion ratio of the spin current generated by the STE normal-\nized by its low current value[40]. The data from the different\ncurvesoverlapand,similartotheSSE,showaparabolicevolu-\ntion(seedottedline). Thissuggeststhattheprimarysourceof\nthe symmetric nonlinearity between 𝐼2and𝐼1is simply Joule\nheating. It therefore justifies the transformation of the current\nabscissa𝐼1into a temperature scale 𝑇1in Fig. 7(a). Focusing\nnow on the remarkable features of Fig. 7(a), one could no-tice that the low current data taken at 𝑇0= 300K fall on a\nstraight line intercepting the origin, as predicted by Eq. (1),\nwhich is𝐼2∕𝐼1∝𝑇1. Another notable feature, as previously\nreported[3, 38], is that the transmission ratio reaches a maxi-\nmum at high temperature.\nTo support this picture with experimental data, we have\nplotted in the inset of Fig. 7 the behavior of 𝑀𝑇(𝑀0−𝑀𝑇)\nsuggestedbyEq.(3). Thisshouldrepresentthemagnontrans-\nmission ratio by the high-energy fraction, i.e. the number of\navailable high-energy magnons multiplied by the amount of\nspin polarization available in the film. We find that the ob-\nserved variation of T𝑠with𝑇1follows the expected behavior\nderivedfromthesingletemperaturevariationofthetotalmag-\nnetization shown in the inset Fig. 7(b). This provides exper-\nimental evidence that the short range behavior is dominated\nby high-energy magnons and that the density change follows\nthe analytical expression in Eq. (3). Furthermore, it is con-\nfirmed that the drop in the magnon transmission ratio above\n440Kisassociatedwithadropinthesaturationmagnetization\nas one approaches 𝑇𝑐, precisely where high-energy magnons\nreachtheirmaximumoccupancy. Thedropsuggeststhathigh-\nenergy magnons actually prevent STE spin transport. This is\nthenonlineardeviationexpectedforadiffusivegas: thehigher\nthenumberofparticles,themorethetransportisinhibited(see\nalsoRef.[1]). Whatitshowshereisthatthemagnontranscon-\nductance is dominated by high-energy magnons around the\nemitter. This confirms the initial finding of Cornelissen et\nal.[2] whodrew thisconclusion based onthe similarityof the\ncharacteristic decay of SSE and STE as a function of 𝑑.\nC. Double decay of the magnon transmission ratio\n1. Thin films with anisotropic demagnetizing effect\nHaving established that the spin current is carried by the\ntwo-fluidsandthatthefitallowstoextracttherespectivecon-\ntributions of high and low-energy magnons, we took a series\nofexperimentaldataof T𝑠⋅𝑇0∕𝑇1withdifferentseparations 𝑑\nrangingfrom 0.5𝜇mto6.3𝜇m. TheresultsareshowninFig.8.\nWe see directly in Fig. 8 that the decay length of the magnon\ntransmission ratio at small 𝐼1is much shorter than the decay\nlengthofthemagnontransmissionratioatlarge 𝐼1(spindiode\nregime). Thisshowsexperimentallythateachofthetwo-fluids\nhas a different decay length with 𝜆𝑇≪ 𝜆𝐾. These are ad-\njusted by varying Σ𝐾∕(Σ𝐾+Σ𝑇)||𝑑while keeping the other\nparametersinEq.(1-3). Thefitsareshownasthesolidlinein\nFig. 8(a,c). The fit parameters are set according to the values\ngiven in Table I.\nBymeansoftheanalysis,weobtainedtheamplitudeandthe\nfractionofhigh-energyvs. low-energymagnonsasafunction\nof𝑑,whicharesummarizedinpanel(b)andextractthetwode-\ncaylengths𝜆𝐾=1.5𝜇mand𝜆𝑇=0.4𝜇m,respectively. This\nconfirms the short-range nature of the high-energy magnons\nand the much longer range of the low-energy magnons. We\nnote that since the shortest decay length is of the same order\nof magnitude as the spatial resolution of standard nanolithog-\nraphy techniques, the regime of magnon conservation could8\nFIG. 7. Dependence of the magnon transmission ratio on the sub-\nstrate temperature, 𝑇0. Short-range measurement ( 𝑑= 0.5𝜇m) of\nnonlocal spin transport in YIGA. (a) Variation of T𝑠as the emitter\ncurrent𝐼1isvaried inthe range [−2.5,2.5]mAat differentvalues of\nthe substrate temperature 𝑇0. The data are plotted as a function of\n𝑇1=𝑇0+𝜅𝐴𝑅Pt𝐼2\n1, the emitter temperature. The resulting temper-\nature dependence of T𝑠observed in (a) corresponds to the variation\nof𝑀𝑇(𝑀0−𝑀𝑇)shown in inset (b), where 𝑀𝑇is the temperature\ndependence of the saturation magnetization. The dots are the exper-\nimental points, while the blue solid line is the expected behavior as-\nsuming𝑀𝑇≈𝑀0√\n1−(𝑇∕𝑇𝑐)3∕2. Thistransformationissupported\nby the observation in (c) and (d) that both the SSE signal V2and the\nnormalized inverse transmission ratio T𝑠vs.𝐼1scale on the same\nparabolicbehavior(dashedline),suggestingthattherelevantbiaspa-\nrameter is𝑇1.\nprobably never be achieved in lateral devices. Note that there\nis the discrepancy that the vanishing of 𝐼2occurs slightly be-\nfore𝑇𝑐. Wewillshowthatthisoccurssystematicallyonallour\nsamples (see subsection 3). The same analysis applied to the\nYIGfilmwithlargerthickness(panels(c,d))revealsanidenti-\ncalbehaviorofthehigh-energymagnons,whereasthedecayof\nthelow-energymagnonsisslightlyslowerwith 𝜆𝐾=1.9𝜇m.\nWe do not see an obvious increase in the transmission ra-\ntio in thinner films (YIG𝐴), although Eq. (5) of Ref. [1] pre-\ndicts inverse proportionality as previously observed experi-\nmentally[41],whichcanbeattributedtothedifferenceinma-\nterial quality. Nevertheless, an interesting feature observed\nwhencomparingFig.8(a)and(c)isthattheratiooflow-energy\nmagnons to high-energy magnons increases with decreasing\nfilmthickness. Thiscanbeattributedtoanincreaseinthecut-\noffwavevector,wherethemagnonsbehavetwo-dimensionally,\nandthusthespectralrange,wherethedensityofstateremains\nconstant, which favors the exposure of the increasing occu-\npancyoflow-energymagnons. Thelongerdecaylengthinthe\nthickerfilmisalsoconsistentwiththelongerpropagationdis-\ntanceexpectedforballisticlow-energymagnons,whoseprop-\nagation range is determined by the film thickness. However,\ntheenhancementisnotproportionaltothethickness,suggest-\ning that some other undefined process is also involved in this\ndecay.\nWe emphasize that the shape of the decay observed in\nFig. 8(b) and (d) corresponds to a double exponential decay\nwith two different decay lengths in unprocessed data. This\nreinterprets the double decay behavior reported in previous\nFIG.8. Doubleexponentialspatialdecayofthemagnontransmission\nratio. (a,c) Current dependence of the magnon transmission ratio for\n(a) the 19 nm thick YIG𝐴and (c) the 56 nm thick YIG𝐶thin films.\nThe solid lines are a fit by Eq. (4), where the only variable parame-\nter is the value of Σ𝐾∕(Σ𝐾+Σ𝑇)||𝑑. For the YIG𝐶sample, we have\nadded in panel (c) the variation of the spin magnetoresistance (right\naxis), which corresponds to the conductivity at 𝑑=0. Spatial decay\nofthemagnontransmissionratiofor(b)YIG𝐴and(d)YIG𝐶,respec-\ntively. In both cases, the decay of high-energy magnons follows an\nexponentialdecaywithcharacteristiclength 𝜆𝑇≈0.5±0.1𝜇m. The\ndecayoflow-energymagnons,ontheotherhand,followsanexponen-\ntialdecaywithcharacteristiclength 𝜆𝐾=1.5𝜇mforthethinnerfilm\n(b)andanexponentialdecaywithcharacteristiclength 𝜆𝐾=1.9𝜇m\nfor the thicker film (d).\nnonlocaltransportmeasurements[2,22,41,42]. Theinterpre-\ntationpresentedinthisworkisdifferentfromtheoneproposed\nbyCornelissen etal.,whereitwasrelatedtotheboundarycon-\ndition of the diffusion problem[2]. We note that while chang-\ning the current bias 𝐼1can affect the ratio between the two-\nfluids, it does not change the decay length, as shown by the\npurple lines in panel (b,d). This is consistent with the notion\nthat the bias affects the mode occupation of the transported\nmagnons but not their character. The obtained decay lengths\nareinroughagreementwiththeexpecteddecaylengthofthese\ntwo populations as discussed in Sect.III. Note also that the\nhigh- and low-energy magnon length scales appear to be sim-\nilar to the energy and spin relaxation length scales observed\nin the Spin Seebeck effect as proposed by A. Prakash et al.9\nTABLE I. Fitting parameters by Eq. (4).\n𝑡YIG(nm) 𝑛sat𝑇⋆\n𝑐(K) th,0(mA) 𝜆𝐾(𝜇m)𝜆𝑇(𝜇m)Σ𝐿→𝑅\n𝐾,0Σ𝐿→𝑅\n𝑇,0\nYIG𝐴 19 4 495 8 1.5 0.4 5 % 37%\n(Bi-)YIG𝐵 25 11 480 3 3.8 0.5 4 % 39%\nYIG𝐶 56 4 515 8 1.9 0.6 3 % 15%\nYIG𝐷 65 4 545 8\n[43],thecorrelationbetweenthelengthscalesisacomplexis-\nsuethatwarrantsamorerigoroustheoreticalinvestigation(see\nalso conclusion below).\nWe note that our value of 𝜆𝐾appears to be dependent on\nthicknessandanisotropy(seeTableI).Thiscontradictsthebe-\nhavior observed for thicker films ( 𝑡YIG>200nm), where the\nvalue was reported to be independent of film thickness[41].\nThe latter observation may be consistent with the assignment\nof the dominant low-energy propagating magnons to the 𝐸𝐾mode(orangedotinFig.3). Webelievethatthegroupvelocity\nthereis weaklydependent on 𝑡YIG, atleastfor thickfilms(see\ndiscussion above). We should emphasize here that our report\ndoes not cover the same dynamic range as those reported in\nthickerfilms,duetothelowersignal-to-noiseratio. Itispossi-\nble that a third exponential decay could appear at much lower\nsignallevels. Apossibleexplanationforthelongrangebehav-\niorcouldbethattheangularmomentumiscarriedbycircularly\npolarized phonons, which have been found to have very long\ncharacteristic decay lengths in the GHz range[44, 45].\nFinally, it is useful to quantify the spin current emitted by\nthe STE, as shown in panel (b,d). Renormalizing the trans-\nmissionratiocoefficient T𝑠bytheproductofthespintransfer\nefficiency at both the emitter and collector interfaces, 𝜖1⋅𝜖2\n(see Table 1 of Ref. [1]), we observe that only 10% of the\ngenerated magnons reach a collector placed at 𝑑= 0.2𝜇m\naway. This percentage increases to 15% by extrapolating the\ndecay to𝑑=0, which is the proportion of itinerantmagnons\namong the total generated, and there are about an order of\nmagnitude(×14)morehigh-energymagnonsthanlow-energy\nmagnons below the emitter. Taking into account the fact that\nmagnons can escape from both sides of the emitter, while we\nmonitoronlyoneside,wecanestimatethat70%ofthegener-\nated magnons remain localized. This localization is the con-\nsequence of three combined effects, which mainly affect the\nlow-energy magnons: i)STE primarily favors an increase in\ndensityatthebottomofthemagnonmanifold,whichhaszero\ngroup velocity ii)STE, as an interfacial process, efficiently\ncouples to surface magnetostatic modes [46], The nonlinear\nfrequency shift associated with the demagnetizing field [47]\nproduces a band mismatch at high power between the region\nbelowtheemitterandtheoutside,whichpreventsthepropaga-\ntionofmagnons(seepartI[1]). Thespatiallocalizationcould\nbe induced either by the thermal profile of the Joule heating\n[13] or by the self-digging ball modes [18, 48, 49]. This ra-\ntional concerns mainly the magnons whose wavelengths are\nshorter than the width of the Pt electrode.\nAnother confirmation is the variation of the ratio between\nlow-energy magnons and high-energy magnons with the uni-\naxial anisotropy. When the latter compensates the out-of-plane depolarization field, we observe a suppression of the\nlow-energy magnon confinement, and the transmitted signal\natlargedistances(10 𝜇m)fullyreplicatesthevariationoflow-\nenergy magnons under the emitter.\n2. Thin films with isotropically compensated demagnetizing effect\nIn this section, we will clarify the influence of self-\nlocalization on the saturation threshold 𝑛satthat we intro-\nduce in our analytical model. For this purpose, we have re-\npeated the experiment on a Bi-YIG𝐵sample. This mate-\nrial has a uniaxial anisotropy corresponding to the saturation\nmagnetization (see Table 1 in Ref. [1]). As a consequence,\nthe Kittel frequency follows the paramagnetic proportional-\nity relation𝜔𝐾=𝛾𝐻0(similar to the response of a sphere),\nwhere the value of 𝜔𝐾is independent of 𝑀𝑇and the cone\nangle of precession, and therefore exhibits a vanishing non-\nlinear frequency shift[32, 47, 50] (see further discussion in\nRef.[1]). Werefertothisasanisotropicallycompensatedma-\nterial. We emphasize, however, that although the nonlinear\nfrequency shift is zero, the system is still subject to satura-\ntion effects[10]. Compensation of the out-of-plane demagne-\ntization factor eliminates only the ellipticity of the trajectory\ncaused by the finite thickness, but not the self-depolarization\neffect of the magnons on themselves. The latter depends on\nthe angle between the propagation direction and the equilib-\nrium magnetization direction and is the origin of the magnon\nmanifold broadening.\nAs shown in Fig. 9(a), the nonlinear behavior of T𝑠ob-\nserved in the Bi-YIG𝐵sample is qualitatively similar to that\nof YIG𝐶. Quantitatively, however, the magnitude of the spin\ndiode effect is more pronounced in the former case. This is\nespecially noticeable at long distances. Comparing Fig. 9(b)\n(𝑑=0.70𝜇m)withFig.9(c)( 𝑑=10.3𝜇m),fortheformerthe\nconductivitycanonlybeincreasedbyafactorof3withrespect\nto its initial value, while for the latter it can be increased by a\nfactorof15. Thisisagainduetothefilteringoutoftheback-\ngroundofhigh-energymagnons: inthecaseoflargedistances,\nthe contribution of low-energy magnons is more pronounced.\nRecalling that in YIG𝐶the conductivity was enhanced by a\nfactor of 7 by low-energy magnons (see 𝑑 >4.3𝜇m data in\nFig. 8(c) or Fig. 7 of Ref. [1]), here a larger fitting parameter\nof𝑛sat=11isusedinBi-YIG𝐵while𝑛sat=4isusedinYIG𝐶,\nindicating a larger threshold for saturation. This is consistent\nwiththesuppressionofthenonlinearfrequencyshiftaffecting\nthelongwavelengthspinwaveintheYIG𝐶sample. Thisresult\nsuggeststhatremovingtheselfnonlinearityonthelongwave-\nlength magnons improves the ability to generate more propa-10\nFIG. 9. Two-fluid behavior in thin films with isotropically compen-\nsated demagnetization effect ( 𝑀eff= 0). (a) Variation of the spin\ndiode signal T𝑠measured in BiYIG𝐵for different emitter-collector\nseparations𝑑. The main panel (a) shows the normalized magnon\ntransmission ratio as a function of 𝑇1, while the right panels show\nthe corresponding current dependence for (b) 𝑑= 0.7𝜇mand (c)\n𝑑=10.3𝜇m. ThesolidlinesarefitsbyEq.(4),withtheonlyvariable\nparameters,Σ𝐾andΣ𝑇, representing the fraction of low and high-\nenergymagnons. (d)Spatialdecayofthetwo-fluidmodelseparating\nthe contributions of high-energy and low-energy magnons. The ob-\nserved decay can be explained by a short decay 𝜆𝑇≈0.5𝜇m of the\nhigh-energymagnoncontribution( 𝑘𝐵𝑇,blackline)andalongdecay\n𝜆𝐾≈4.0𝜇mofthelow-energymagnoncontribution( ℏ𝜔𝐾,magenta\nandbluelines). Thedataat 𝐼1=1.3mAshowthedecaybehaviorin\nthecondensedregime. (e)Magneticfielddependenceofthenormal-\nized magnon transmission ratio at different currents.\ngating magnons. It can also be understood as the removal of\nthe self-digging process under the emitter in pure YIG sam-\nples. The fit parameters are listed in Table I. Note that the\ndiscrepancybetween 𝑇𝑐and𝑇⋆\n𝑐,whichmarksthedropof T𝑠,\nisevenmorepronouncedinthissystem. Thedropoccurs70K\nbelow𝑇𝑐. We will return to this point in the last subsection.\nInFig.9(d)weplotthespatialdecayof T𝑠renormalizedby\n𝜖2, obtained from fits with Eq. (4) in percent for high-energy\nmagnonsinblack,low-energymagnonsat 𝐼1=0.4mA(𝜇𝑚≪\n𝐸𝑔) in blue, and 𝐼1=1.3mA (𝜇𝑚≈𝐸𝑔) in purple. The two\ndecay lengths are 𝜆𝑇≈ 0.4𝜇m for high-energy magnons, in\nagreement with the results in YIG, and a much larger value\nof𝜆𝐾= 4𝜇m for low-energy magnons. The latter value is\nsimilar to the decay length of low-energy magnons observed\nFIG.10. Dependenceof 𝑇⋆\n𝑐onthethicknessofYIGfilms. Compar-\nison of nonlocal devices with approximately the same ratio of high-\nenergy magnons to low-energy magnons at 𝐼1→0. We observe an\nincreasein𝑇𝑐−𝑇⋆\n𝑐withdecreasingfilmthickness,suggestinganin-\ncreasing influence of low-energy magnons at high power 𝐼1→𝐼𝑐\nwith decreasing film thickness.\nby BLS in these films[50]. Moreover, it is in good agreement\nwith the estimate made in Sect. III.\nForthesakeofcompleteness,weplotthemagneticfieldde-\npendencefordifferent 𝐼1inFig.9(e). Thedecreaseofthesig-\nnal at zero field is due to the residual out-of-plane anisotropy,\nwhich forces the magnetization to be along the film normal,\nresulting in no STE applied by Pt. The magnon transmission\nratiobecomesmaximumnear0.05T,whichisthesaturationof\nthe effective magnetization for BiYIG. The field dependence\nat a larger field than 0.05 T becomes significant for the cur-\nrent values near the appearance of the peak in (a) at 𝐼1=1.3\nmA, where the conductivity of low-energy magnons reaches\nthe highest. As noted in a previous study[4], the fact that we\nsee a dependence with magnetic fields is direct evidence that\nwearedealingherewithlow-energymagnons. Heretheextra\nsensitivityof T𝑠tochangesin thnear𝐼pk,asdiscussedabove\nin the context of describing the behavior of Fig. 4, is clearly\nillustrated here with the BiYIG sample.11\n3. Discrepancy between 𝑇𝑐and𝑇⋆\n𝑐\nFinally, we discuss the disappearance of the magnon trans-\nmission ratio already at 𝑇⋆\n𝑐far below the experimentally de-\ntermined𝑇𝑐(see Fig. S1). We note in Fig. 8 and Fig. 9 that\nall curves collapse at the same value independent of 𝑑. This\nclearly points to a problem that only concerns the region be-\nlow the emitter, since there is a lateral temperature gradient.\nTo this end, we summarize the normalized magnon transmis-\nsionratioforYIGsamplesasafunctionofemittertemperature\n𝑇1inFig.10withdifferentthicknesses. Toavoidanyinfluence\nof thermal gradients, we have chosen devices whose spacing\n𝑑leads to a similar ratio between Δ𝑛𝑇andΔ𝑛𝐾. This re-\nquires𝑑toincreasewithincreasingfilmthickness,suggesting\na decreasing contribution of low-energy magnons. We spec-\nulate that the collapse can be caused either by the onset of\nstrong electron-magnon scattering as the YIG film becomes\nconducting[28, 29], or by a reversal of the equilibrium mag-\nnetization below the emitter, which becomes aligned with the\ninjectedspindirection[18,19]. Inthelattercase,themagneti-\nzation below the emitter and collector are opposite, suppress-\ning any spin transport. This process is consistent with the as-\nsumption that a large fraction of the injected spins remain lo-\ncalized. Thisprocessisalsoconsistentwiththedecreaseof V2\nobservedatlarge 𝐼1,wherenowtheelectriccurrentdecreases\ntheeffectivetemperatureofthespinsystem(decreasefluctua-\ntions) despite the fact that 𝐼1⋅𝐻𝑥<0.\nWeexaminetheothercluesthatsupportthispicture. Ifone\ncomparesthediscrepancybetween 𝑇⋆\n𝑐and𝑇𝑐betweenthedif-\nferent samples, one can clearly see on the data in Fig. 10 that\nthe discrepancy increases with decreasing film thickness, as\nexpected for an increased surface effect of STE and reduced\nvolume of polarized spins. Another indication is the fact that\nthe largest discrepancy is observed on films with large uniax-\nialanisotropy,asshowninFig.9(a). Thisisinagreementwith\ntheobservationmadeonnano-devicesontheswitchingofthe\nmagnetizationdirectionbythespinHalleffect[51]. Neverthe-\nless,thediscrepancydoesnotseemtoscalesimplywith 𝑡YIGin\nour observation, suggesting that there may be additional phe-\nnomena at play that are responsible for the vanishing magnon\ntransmissionratioathightemperaturewhilethesystemisstill\ninitsferromagneticphase(seealsothediscussionofFig.5of\nRef. [1]).\nWe have tentatively calculated 𝐼𝑓, the critical current re-\nquired to flip the magnetization. We call 𝑛sat=𝑉𝑀1∕(𝛾ℏ)\nthetotalnumberofspinsthatremainpolarizedundertheemit-\nter. We compare this to the number of injected spins within\nthe spin-lattice relaxation time, which is 𝐼𝑓𝜖∕(2𝑒𝛼LLG𝜔𝐾).\nEqualizing the two quantities, we find that 𝐼𝑓= 2.5mA for\nYIG𝐴samples. According to the upper scale of Fig. 8, 𝑇𝑐is\nreached when 𝐼= 2.7mA. Using Fig. S1, we can calculate\nthetemperaturedifferenceproducedbyJouleheatingbetween\nthesetwovalues,andtheresultisabout65K.Thisisveryclose\nto the shift of 50 K observed experimentally on this sample.\nWhilethereareindicationsthatashiftoccurs,andthenum-\nber roughly matches the expected numbers, the above para-\ngraphisstillratherspeculativeatthisstage,andadirectproof\nisstillmissing. Forthesakeofcompleteness,itisworthmen-tioningthattheremaybealternativeexplanations. Onepossi-\nbilityisadecreaseof 𝑇𝑐intheregionbelowthePt. Theorigin\nofsuchaneffectcouldbeinterdiffusionofPtatomsinsidethe\nYIG at the interface. More thorough systematic studies will\nbe required to clarify this point.\nV. CONCLUSION\nThrough these two consecutive reviews, we present a com-\nprehensive picture of magnon transport in extended magnetic\ninsulating films, covering a wide range of current and mag-\nnetic field bias, substrate temperature, as well as nonlocal ge-\nometrieswithvaryingpropagationdistance. Thepictureofthe\ntwo-fluid model expressed in this part II, complemented by a\npicture of the nonlinear behavior of the low-energy magnon\nexpressed in part I, is formulated analytically and it is sup-\nported by a series of different experiments that include non-\nlocal transport on different thicknesses YIG thin films with\ndifferent garnet composition, different interfacial efficiency,\nas well as different thermalization. While providing a com-\nprehensive study of these materials, our model accounts for\nalmost all the experimental observations within this common\nframework.\nWhat the analytical model allows to do is:\ni) to describe the expected signal in the linear regime\n[Eq. (6) in part I]\nii) to fit the nonlocal transport data well on the whole cur-\nrentrangeandfordifferentseparationbetweentheelec-\ntrodes using very few parameters ( th,0,𝑛sat,𝑇⋆\n𝑐,𝜆𝑇,\n𝜆𝐾,Σ𝑇andΣ𝐾)\niii) to incorporate all relevant physical effects: effect of\nJoule heating on 𝑀1, divergent form of magnon-\nmagnon relaxation.\nWhat it doesn’t do, but could be important:\ni) to take into account the propagation properties (propa-\ngationangle,groupvelocity,modeselectionbytheelec-\ntrodegeometry,spatialvariationofthesepropertiesdue\ntothetemperaturegradient)ofthemagnonsexcitedun-\ndertheemittertoknowhowtheycontributetothesignal\nunder the collector.\nii) to take into account nonlinear magnon localization ef-\nfects under the emitter (for YIG in particular).\niii) to take into account the effects of high power (change\nin temperature or change in low energy magnon occu-\npancy) on damping, exchange constant (and thus group\nvelocity), pumping, and detection efficiency.\nThefactthatthesepointsarenotdirectlyconsideredandthat\nthe fits are excellent means that these effects are effectively\nused in the other components of the model. In particular, Eq.\n(6) of the relaxation in part I is very general and can absorb\nmany different physical effects, hence the effectiveness of the\nmodel.12\nIn this paper, we assume that low-energy magnons propa-\ngating in the ballistic regime lead to a magnon transconduc-\ntancethatfollowsanexponentialspatialdecayinthinfilmge-\nometries. This argument follows from the experimental find-\ningthatinallBLSexperimentsmonitoringthelow-energypart\nofthemagnonmanifold,theamplitudeofthesignalfollowsan\nexponentialdecay. Nevertheless,thetransportbehaviorinthe\ncleanlimit,wherethemagnonmeanfreepathislargerthanthe\nsampleboundary,isinitselfaveryinterestinglineofresearch.\nAnother open question concerns the premature collapse of\nthesignalat𝑇⋆\n𝑐. Wehavetentativelyexplainedthisasapoten-\ntialswitchingofthemagnetizationdirectionbelowtheemitter.\nHowever, direct evidence for such a process remains elusive.\nWe think that spin transport in materials with low magnetiza-\ntionorclosetotheparamagneticphasearebothveryinterest-\ning topics.\nFinally, we summarize the main result of our two-fluid\nmodel, which separates the low-energy magnons from the\nhigh-energy ones. This allows us to propose an alternative\nexplanation for the measured variation of the magnon trans-\nmission ratio with distance, due to a double exponential de-\ncay. Each of the fluids has its own transport characteristics,\nwhich are expressed by two different propagation lengths. A\ndecay length in the submicron range is assigned to the high-\nenergy magnon and a decay length above the micron range\nis assigned to the low-energy magnon. This explanation im-\nplies that even in the short-range regime, the magnon number\nisnotaconservedquantity,andthusanyanalogytoelectronic\ntransport should take this rapid decay into account. Despite\nthe fact that the model includes several parameters, there are\nstill open questions. The similarity of the decay of SSE and\nSTE currents with 𝑑must be reconciled with our results. A\npossible reason is that low-energy magnons participate in the\nSSE transport in the long range[52]. Although the amount of\nquanta carried is clearly 𝐸𝑇∕𝐸𝐾∼ 103against the latter, we\nshouldkeepinmindthatwearedealingwithatinysignal. The\nroleofacousticphonons[44,45]inthisprocessisstillunclear.\nRecentexperimentshaveshownthattheyarestronglycoupled\nto low-energy magnons and also benefit from a very low de-\ncay length. Of particular interest is the contribution of circu-\nlarly polarized acoustic phonons, which have been shown to\nbe strongly coupled to long-wavelength spin waves while al-\nlowing angular momentum transfer over large distances.\nACKNOWLEDGMENTS\nThis work was partially supported by the French Grants\nANR-18-CE24-0021 Maestro and ANR-21-CE24-0031\nHarmony; the EU-project H2020-2020-FETOPEN k-\nNET-899646; the EU-project HORIZON-EIC-2021-\nPATHFINDEROPEN PALANTIRI-101046630. K.A.\nacknowledges support from the National Research Founda-\ntion of Korea (NRF) grant (No. 2021R1C1C201226911)\nfundedbytheKoreangovernment(MSIT).Thisworkwasalso\nsupported in part by the Deutsche ForschungsGemeinschaft\n(Project number 416727653).\nFIG. S1. Characterization of garnet thin films. The left column\n(a,b,c) shows the variation of the Pt resistance as a function of the\ninjected current for YIG𝐴, (Bi-)YIG𝐵and YIG𝐶without and with\nAl coating, respectively (see Table 1 of Ref. [1]). The right ordinate\nallows to convert the current bias into a temperature increase in the\nrange [300,600] K due to Joule heating. The upper abscissa gives\nthe corresponding current density in Pt. The right column (d,e,f)\nshowsthecorrespondingvariationofthesaturationmagnetizationin\nthe [300,600] K range.\nVI. ANNEX\nA. Sample characterisation\nThe4magneticgarnetfilms(seeTableI)usedinthisstudy\nhavebeengrownby2differentmethods: liquidphaseepitaxy\ninthecaseofYIG𝐴,𝐶,𝐷andpulsedlaserdepositioninthecase\nof (Bi-)YIG𝐵. Their macroscopic magnetic properties have\nbeencharacterizedusingacommercialvibratingsamplemag-\nnetometer,wherethesampletemperaturecanbecontrolledby\na flow of argon gas from room temperature to 1200K. Curves\nof magnetization versus temperature in the range of 300K to\n600KareshowninFig.S1(d-f). Theyhighlightthevalueofthe\nCurietemperature( 𝑇𝑐)foreachsamplesummarizedinTable1\nin Ref. [1]. Similarly, the Pt metal for the middle electrode\nwas deposited by 2 different techniques: e-beam evaporation13\nin the case of YIG𝐴and YIG𝐶and sputtering in the case of\n(Bi-)YIG𝐵.\nIn this work we convert the Joule heating associated with\nthe circulation of an electric current 𝐼1in the emitter into a\ntemperature increase, which we plot on the abscissa of Fig. 7,\nFig.10andFig.6,Fig.7ofRef.[1]. Thisisdonebycalibrating\n𝑅Pt||𝐼1: the variation of the resistance Pt1with the injectedelectric current 𝐼1. We introduce the calibration factor\n𝜅𝐴,𝐵or𝐶=𝜅Pt𝑅Pt∕𝑅0−1\n𝑅Pt𝐼2\n1, (5)\nfor the conversion coefficient, with 𝑅0≡𝑅Pt||𝐼1=0=\n𝜌Pt𝐿Pt∕(𝑤1𝑡Pt)is the nominal value of the Pt wire resistance\nandthecoefficient 𝜅Pt=𝑅Pt∕𝜕𝑇𝑅Ptisobtainedbymonitoring\nthe variation of the Pt resistance at low current vs. substrate\ntemperature. The obtained values of 𝜅Ptand𝜌Ptare given in\nTable 1 in Ref. [1]. 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Saitoh, Physical Review B 92, 064413 (2015)."    },    {        "title": "1512.08890v2.Non_equilibrium_thermodynamics_of_the_spin_Seebeck_and_spin_Peltier_effects.pdf",        "content": "Non-equilibrium thermodynamics of the spin Seebeck and spin Peltier e\u000bects\nVittorio Basso, Elena Ferraro, Alessandro Magni, Alessandro Sola, Michaela Kuepferling and Massimo Pasquale\nIstituto Nazionale di Ricerca Metrologica, Strada delle Cacce 91, 10135 Torino, Italy\n(Dated: October 4, 2018)\nWe study the problem of magnetization and heat currents and their associated thermodynamic\nforces in a magnetic system by focusing on the magnetization transport in ferromagnetic insulators\nlike YIG. The resulting theory is applied to the longitudinal spin Seebeck and spin Peltier e\u000bects.\nBy focusing on the speci\fc geometry with one YIG layer and one Pt layer, we obtain the optimal\nconditions for generating large magnetization currents into Pt or large temperature e\u000bects in YIG.\nThe theoretical predictions are compared with experiments from the literature permitting to derive\nthe values of the thermomagnetic coe\u000ecients of YIG: the magnetization di\u000busion length lM\u00180:4\u0016m\nand the absolute thermomagnetic power coe\u000ecient \u000fM\u001810\u00002TK\u00001.\nPACS numbers: 75.76.+j, 85.75.-d, 05.70.Ln\nI. INTRODUCTION\nThe recent discovery of the longitudinal spin Seebeck\ne\u000bect in ferromagnetic insulators has raised a renewed in-\nterest in the non equilibrium thermodynamics of spin or\nmagnetization currents1. Experiments have shown that\na temperature gradient applied across an electrically in-\nsulating magnetic material is able to inject a spin current\ninto an adjacent metal, where the spin polarization is re-\nvealed by means of the inverse spin Hall e\u000bect (ISHE)2,3.\nTypical experiments have been performed by using ferri-\nmagnets, like the yttrium iron garnet (Y 3Fe5O12, YIG)\nas insulating magnetic material and Pt or other noble\nmetals, as conductors2,3. In analogy to thermoelectrics,\nthe reciprocal of the spin Seebeck e\u000bect has been called\nspin Peltier e\u000bect4. This reciprocal e\u000bect has been re-\ncently observed by using the spin Hall e\u000bect of Pt as\nspin current injector and observing the thermal e\u000bects\non YIG4. All these experiments show that the magneti-\nzation current can propagate along di\u000berent media using\ndi\u000berent type of carriers. While spin currents in metals\nare associated to the unbalance in the spin polarization\nof conduction electrons, in magnetic insulators the mag-\nnetization transport is due to spin waves or magnons5.\nSpin Seebeck and spin Peltier experiments reveal that the\nmagnetization current carried by magnons in the mag-\nnetic insulator can be transformed into a spin current\ncarried by electrons and viceversa. The mechanism of\nthis conversion is seen as the interfacial s-d coupling be-\ntween the localized magnetic moment of the ferromagnet\n(which is often due to d shell electrons) and the con-\nduction electrons of the metal (which are often s shell\nelectrons)6{8.\nThe thermodynamics of thermo-magneto-electric ef-\nfects, i.e. spin caloritronics, has been already developed\nfor metals by adding the spin degree of freedom to the\nthermo-electricity theory9,10. However, spin caloritronics\ncannot be directly applied to electrical insulating mag-\nnetic materials like YIG. Therefore it is necessary to\ndevelop a more general theory which could be applied\nto both conductors and insulators. The formulations\nof the problem present in the literature often focus onthe microscopic origin without paying much attention\nto the formal thermodynamic theory that is expected\nas a result. Refs.5,11{14describe the non equilibrium\nmagnon distribution through an e\u000bective magnon tem-\nperature di\u000berent from the lattice temperature. How-\never from an experimental point of view in Ref.15it was\nobserved a close correspondence between the spatial de-\npendencies of the exchange magnon and phonon temper-\natures. The Boltzmann approach for magnon transport\nwas used in Ref.8,16{19, combined by a YIG/Pt interface\ncoupling7,8. Within these approaches the spin accumu-\nlation and the magnon accumulation take the role of an\ne\u000bective force able to drive the magnetization current.\nThe use of di\u000berent quantities between the two sides of\na junction requires therefore the introduction of a spin\nconvertance to account for the magnon current induced\nby spin accumulation and the spin current created by\nmagnon accumulation20.\nThe aim of the present paper is to de\fne the macro-\nscopic non-equilibrium thermodynamics picture for the\nproblems related to magnetization currents that could\nbe used independently of the speci\fc magnetic moment\ncarrier. To this aim we start from the results of the ther-\nmodynamic theory of Johnson and Silsbee9. The main\ndi\u000berence with respect to the classical theories of the\nthermoelectric e\u000bects is that the magnetization current\ndensityjMis not continuous. The magnetic moment can\nboth \row through a magnetization current but also can\nbe locally absorbed and generated by sinks and sources.\nHere, by limiting the analysis to the scalar case, we state\nthe simplest possible continuity equation for the magne-\ntization. As a result we \fnd that the potential for the\nmagnetization current is the di\u000berence H\u0003=H\u0000Heqbe-\ntween the magnetic \feld Hand the equilibrium \feld Heq.\nThe gradient of the potential rH\u0003is the thermodynamic\nforce to be associated to the magnetization current.\nWith this de\fnition it is then possible to state the con-\nstitutive equations for the joint magnetization and heat\ntransport and to identify the absolute thermomagnetic\npower coe\u000ecient \u000fMrelating the gradient of the poten-\ntial of the magnetization current \u00160rH\u0003with the tem-\nperature gradient rT, in analogy with thermoelectricity.arXiv:1512.08890v2  [cond-mat.mtrl-sci]  5 May 20162\nThe same coe\u000ecient also determines the spin Peltier heat\ncurrent\u000fMTjMwhen the system is subjected to a mag-\nnetization current.\nIn the present work we apply the previous arguments to\ndescribe the generation of a magnetization current by the\nspin Seebeck e\u000bect and the heat transport caused by the\nspin Peltier e\u000bect. To this end we have to complement\nthe constitutive equations for the thermo-magnetic active\nmaterial (YIG) with the equations for the spin Hall active\nlayer (Pt). Once the equations for the two materials are\nwritten by using the same thermodynamic formalism, one\ncan apply the theory to solve speci\fc problems of magne-\ntization current traversing di\u000berent layers. The di\u000busion\nlength for the magnetization current lM= (\u00160\u001bM\u001cM)1=2\nis related to intrinsic properties of each material: the\nmagnetization conductivity \u001bMand the time constant\n\u001cM, describing how fast the system is able to absorb the\nmagnetic moment in excess. We are also able to show\nthat the passage of the magnetization current from one\nlayer to the other is governed by the ratio between lM=\u001cM\nof the two layers.\nBy focusing on the speci\fc geometry with one YIG\nlayer and one Pt layer, we obtain the optimal conditions\nfor generating large magnetization currents into Pt in the\ncase of the spin Seebeck e\u000bect and for generating large\nheat current in YIG in the case of spin Peltier e\u000bects.\nIn both cases we \fnd that e\u000ecient injection is obtained\nwhen the thickness of the injecting layer is larger than the\ncritical thickness lMas recently experiments con\frm21.\nWe \fnally determine the values of the thermomagnetic\ncoe\u000ecients of YIG by comparing the theory to recent\nexperiments4,22.\nThe paper is organized as follows. In Section II we\n\frst discuss the thermodynamic properties of an out-of-\nequilibrium but spatially uniform magnetic system23and\non that basis we introduce, for non spatially uniform sys-\ntem, the currents and the thermodynamic forces in anal-\nogy with the non equilibrium thermodynamics of ther-\nmoelectric e\u000bects24. In Section III we set the constitu-\ntive equations for the magnetization and heat transport\nin both an insulating ferrimagnet and a metal with the\nspin Hall e\u000bect. Section IV is devoted to the solutions\nof the magnetization current problem. In Section V we\nfocus on the speci\fc longitudinal spin Seebeck geometry\nand on the spin Peltier e\u000bect. Finally some conclusive\nremarks are drawn in Section VI.\nII. THERMODYNAMICS OF\nMAGNETIZATION CURRENTS\nA. Thermodynamics of uniform magnetic systems\nWe consider a magnetic system that can be described\nby a scalar magnetization M. Suitable systems can be\nferromagnetic or ferrimagnetic materials where an easy\naxis is present, due for example to an anisotropic crys-\ntal structure, along which all the vector quantities arelying. We take spatially uniform quantities and all ex-\ntensive quantities as volume densities. The derivative of\nthe internal energy density u(s;M) with respect to the\nmagnetization at constant entropy density s, gives the\nequilibrium state equation\nHeq=1\n\u00160@u\n@M\f\f\f\f\ns(1)\nwhere\u00160is the magnetic permeability of vacuum. In\nequilibrium the magnetic \feld His equal to the state\nequationH=Heq(M;s). WhenHis di\u000berent from its\nequilibrium value Heqthe system state will try to reach\nthe equilibrium by the action of dissipative processes. In\na generic out-of-equilibrium situation the variation of in-\nternal energy must take into account that dissipative pro-\ncesses correspond to an entropy production. The energy\nbalance then reads\ndu=Tds+\u00160HdM\u0000T\u001bsdt (2)\nwhereT=@u=@s is the temperature, \u00160HdM is the in-\n\fnitesimal work done on the system, \u001bsis the entropy\nproduction rate, which has to be a de\fnite positive term\nandtis time. When approaching equilibrium, the mag-\nnetizationMwill change until the equilibrium condition\nH=Heq(M) will be reached. The typical situation is\nsketched in Fig.1 showing two processes connecting the\nequilibrium states (1) and (2). The equilibrium path\n(solid line) corresponds to the slow variation of \feld H\nfromH1toH2through the equilibrium state equation\nH=Heq(M). The out-of-equilibrium path (dashed line)\npasses through the out-of-equilibrium state (10) and cor-\nresponds to the sudden variation of the \feld from H1to\nH2and to the subsequent time relaxation. As the initial\nand \fnal states are always equilibrium states, the \fnal\ninternal energy variation must be the same for any pro-\ncess. This is obtained by assuming that the part of the\nwork going into the internal energy is always the equi-\nlibrium one. Then, by inserting du=\u00160HeqdM(from\nEq.(1)) into Eq.(2) at constant entropy ( ds= 0) we \fnd\nthe expression for the entropy production rate\n\u001bs=\u00160H\u0000Heq\nTdM\ndt: (3)\nAs expected, the entropy production rate is the product\nof a generalized force, or a\u000enity, represented by the term\n\u00160(H\u0000Heq)=T, times a generalized \rux, or velocity, rep-\nresented by dM=dt24. If the distance from equilibrium is\nnot too large one can consider the linear system approx-\nimation and assume the velocity to be proportional to\nthe a\u000enity. It is appropriate to describe this fact by in-\ntroducing a typical time constant \u001cMfor the process by\nde\fning\ndM\ndt=H\u0000Heq\n\u001cM; (4)3\nwhere the temperature Tand\u00160appearing in the gener-\nalized force, have been incorporated into the de\fnition of\nthe time constant. Eq.(4) provides a kinetic equation for\nthe magnetization describing the time relaxation from a\ngeneric out-of-equilibrium state by showing that the ve-\nlocitydM=dt depends on the distance from equilibrium\nH\u0000Heq(see Fig.1).\nH\nMHeq(M)H-HeqH2M1H1(1)(1')Δu(2)M2\nFIG. 1. Equilibrium path (solid line) and out-of-equilibrium\npath (dashed line) connecting the equilibrium states (1) and\n(2) in the HversusMdiagram of a magnetic material.\nHeq(M) is the equilibrium state equation at constant en-\ntropy. (10) is an out-of-equilibrium state obtained by the\nsudden change of the \feld from H1toH2. In the relaxation\npath from (10) to (2) the work is \u00160HdM , the internal en-\nergy change is du=\u00160HeqdMand the entropy production is\nT\u001bsdt=\u00160(H\u0000Heq)dM. The relaxation equation is Eq.(4)\nThe interesting physics behind Eq.(4) is that it also\nexpresses the non conservation of the magnetic moment\nwith the presence of sources and sinks, although the total\nangular momentum for an isolated system is conserved.\nAs a matter of fact in the solid state there is a huge\nreservoir of angular momentum available (electrons, nu-\nclei, etc) and only a very small part of it is associated\nto the magnetic moment. As a result, the magnetization\ncan be easily varied by exchanging angular momentum\nwith the reservoir constituted by non magnetic degrees\nof freedoms. With this in mind, the physical meaning of\nEq.(4) is to express how fast the angular momentum from\nthe magnetization subsystem can be exchanged with the\nreservoir.\nFinally, as it happens in many problems involving a\nnon conserved magnetization, also the internal energy is\na non conserved quantity. To avoid the problem, we pass\nto the enthalpy potential ue=u\u0000\u00160HM which contains\nthe magnetic \feld Hand the entropy sas independent\nvariables. Dealing with out-of-equilibrium processes, the\npotentialueis also a non equilibrium one which depends\non the magnetization Mas an internal variable. From\nEq.(2), the enthalpy variation is\ndue=Tds\u0000\u00160MdH\u0000\u00160(H\u0000Heq)dM (5)\nwhere we have used the de\fnition of the entropy produc-\ntion of Eq.(3). The expression for the variation of the en-\nthalpy potential (5), together with the kinetic equation(4), constitutes the out-of-equilibrium thermodynamics\nof the system23and can be employed to build up the\nthermodynamics of \ruxes and forces.\nB. Thermodynamics of \ruxes and forces\nWe now pass from the out-of-equilibrium thermody-\nnamics of a spatially uniform magnetic system to the\nproblem of having a non uniform situation involving cur-\nrents of the extensive variables, entropy and magnetiza-\ntion, and the associated thermodynamic forces24. Both\nthe extensive and intensive variable are now allowed to\nvary as a function of space coordinates r. In the case\nof extensive variables the volume densities are intended\nas moving averages over a small volume \u0001 Varound the\npoint r. As the magnetization is a non conserved quan-\ntity, we need to explicitly express the fact that any mag-\nnetization change dMis in part drawn from the reser-\nvoir of angular momentum, which is external to the ther-\nmodynamic system, and in part exchanged between the\nsurrounding regions of the thermodynamic system itself,\ngiving rise to a current of magnetic moment jM. The\nsources and sinks of the magnetic moment are exactly\nthose described in the previous Section by Eq.(4), then\nwe can immediately write a continuity equation for the\nmagnetization by extending Eq.(4), obtaining\n@M\n@t+r\u0001jM=H\u0000Heq(M)\n\u001cM: (6)\nNext, as it is usually done in the non equilibrium theory\nof \ruxes and forces24, we use Eq.(5) to pass to the entropy\nrepresentation by writing the entropy variation\nds=1\nTdue+\u00160M\nTdH+\u00160(H\u0000Heq)\nTdM: (7)\nAs we aim to de\fne the entropy current as a function of\nthe other currents, we have to look at the previous equa-\ntion in search for the variations of the extensive variables.\nEq.(7) contains the variation of the enthalpy dueand the\nmagnetization dMwhich both have associated currents,\nwhile the variation of the magnetic \feld dHdoes not\ncorresponds to any current and has not to be taken into\naccount in the de\fnition of the entropy current. Then\nwe de\fne\njs=1\nTjue+\u00160(H\u0000Heq)\nTjM (8)\nwhere jsis the entropy current and jueis the enthalpy\ncurrent which obeys the following continuity equation\n@ue\n@t+r\u0001jue=\u0000\u00160M@H\n@t; (9)4\nfrom which one notices that the enthalpy is conserved if\nthe \feldHis constant in time. The continuity equation\nfor the magnetization is Eq.(6) and \fnally the entropy\nobeys the continuity equation\n@s\n@t+r\u0001js=\u001bs: (10)\nAs it is done in the classical treatment24,25, one expresses\nthe entropy production rate \u001bsin terms of a sum of prod-\nucts of each current times its thermodynamic force. By\nusing Eqs.(7)-(9) into Eq.(10) and introducing the heat\ncurrent as jq=Tjs, after a few passages one obtains\n\u001bs=r\u00121\nT\u0013\n\u0001jq+FM\u0001jM+1\n\u001cM\u00160(H\u0000Heq)2\nT(11)\nwhere we have de\fned the thermodynamic force associ-\nated to the magnetization current\nFM=1\nT\u00160r(H\u0000Heq): (12)\nIn Eq.(11) we see the products of the heat current jq\ntimes its forcer(1=T), of the magnetization current\njMtimes its forceFMand the last term which is ex-\nactly the entropy production associated with the out-of-\nequilibrium homogeneous processes and not to the \ruxes.\nThe last term can be also recognized as entropy produc-\ntion of Eq.(3) where the a\u000enity is \u00160(H\u0000Heq)=Tand the\nmagnetization change dM=dt is (H\u0000Heq)=\u001cMas given\nby Eq.(4).\nAs a main result we have found that the gradient of\nthe distance from equilibrium Eq.(12) is the generalized\nforce associated with the magnetization current jM. For\nsimplicity we de\fne H\u0003=H\u0000Heqto specify the dis-\ntance from equilibrium and we observe that the driving\nforce of the magnetization current appears as soon as\nthe system is brought out-of-equilibrium. In that case\nthe system may \fnd more e\u000bective to draw magnetiza-\ntion from the surroundings rather than from the local\nspin reservoir. The strength of this e\u000bect is given by a\nfurther parameter, the magnetization conductivity \u001bM,\nwhich establishes the relationship between the magneti-\nzation current jMand the gradient of H\u0003\njM=\u001bM\u00160rH\u0003: (13)\nH\u0003can be di\u000berent from zero in stationary situation ev-\nery time the material experiences the accumulation of\nmagnetization (i.e. spin accumulation in the case of\nmetallic conductors). We have to notice that even if H\u0003\nhas the units of a magnetic \feld, it is not a magnetic\n\feld in the sense of the Maxwell equations of electro-\nmagnetism. Its status is analogous to the exchange \feld\nor the anisotropy \feld of ferromagnets whose origins is inthe quantum mechanics of the solid. H\u0003represents the\nthermodynamic reaction of the system for \fnding itself in\nan out-of-equilibrium situation. In the following we refer\ntoH\u0003as the potential for the magnetization current.\nIII. CONSTITUTIVE EQUATIONS\nHaving de\fned the potential H\u0003associated with the\nmagnetization current, we are ready to write the consti-\ntutive equations for the two materials of interest for the\nspin Seebeck and spin Peltier e\u000bects: a magnetic insu-\nlating material with a spin Seebeck e\u000bect and a metallic\nconductor with the spin Hall e\u000bect.\nA. Thermomagnetic e\u000bects in magnetic insulators\nIn analogy with the thermoelectric e\u000bects24, we can\nwrite the constitutive equation for the joint transport of\nmagnetization and heat by using the potential associated\nwith the magnetization current derived in the previous\nSection. The general case which includes the presence\nof electric current is reported in Appendix A. Here we\nlimit to insulators and we take currents and forces in one\ndimension (rx=@=@x ). The equations for the thermo-\nmagnetic e\u000bect reads\njM=\u001bM\u00160rxH\u0003\u0000\u001bM\u000fMrxT (14)\njq=\u000fM\u001bMT\u00160rxH\u0003\u0000(\u0014+\u000f2\nM\u001bMT)rxT (15)\nwhere\u001bMis the spin conductivity, \u000fMis absolute ther-\nmomagnetic power coe\u000ecient, jqis the heat current den-\nsity and\u0014is the thermal conductivity under zero mag-\nnetization current. Since the magnetization is not con-\nserved, the magnetization current is not continuous and\nwe have always to add the continuity equation (6). In\nnon-equilibrium stationary states we always ask the con-\ndition@M=@t = 0 to be true, so Eq.(6) becomes\nrxjM=H\u0003\n\u001cM: (16)\n1. Uniform temperature gradient\nIf we disregard for the moment the heat currents,\nthe solution of magnetization current problems will cor-\nrespond to \fnd solutions to the system composed by\nEqs.(14) and (16). Under a uniform temperature gra-\ndient, whererTis a constant, the second term at the\nright hand side of Eq.(14) is just a magnetization current\ndensity source jMS=\u0000\u001bM\u000fMrxT. Then the solution\nof\njM=jMS+\u001bM\u00160rxH\u0003(17)5\ntogether with Eq.(16), considering constant coe\u000ecients,\nleads to a di\u000berential equation for the potential\nl2\nMr2\nxH\u0003=H\u0003(18)\nwhere\nlM= (\u00160\u001bM\u001cM)1=2(19)\nis a material dependent di\u000busion length. The di\u000berential\nequation (18) has general solutions in the form\nH\u0003(x) =H\u0003\n\u0000exp(\u0000x=lM) +H\u0003\n+exp(x=lM) (20)\nwhereH\u0003\n\u0000andH\u0003\n+are coe\u000ecients to be determined on\nthe base of the boundary conditions. By looking at\nEqs.(14) and (16) we have that if the conduction process\nis present in di\u000berent materials, the solution is made by\ntaking Eq.(20) for each material and \fnally joining the\nsolutiosn by requesting the continuity in both jMand\nH\u0003.\n2. Adiabatic conditions\nWhen the temperature is not externally controlled, we\nhave to formulate the thermal problem by writing the\nheat di\u000busion equation. To this aim we need to write\nthe continuity equations for the entropy. In stationary\nconditions Eq.(10) becomes rxjs=\u001bswhere the term\nat the left hand side is written by using js=jq=Tand\nEq.(15) rewritten as\njq=\u000fMTjM\u0000\u0014rxT (21)\nwhile the term at the right hand side is given by Eq.(11).\nAfter a few passages, we obtain\nrxjq=\u00160rxH\u0003jM+\u00160(H\u0003)2\n\u001cM(22)\nwhere the terms at the right hand side are due to the en-\nergy dissipation of the magnetization current and to the\nlocal damping, respectively. Both terms are quadratic in\nthe force and the potential, therefore if we assume small\ncurrents and forces we are allowed to neglect them in a\n\frst approximation. In this case we obtain the condi-\ntionrxjq= 0 which, in one dimension, corresponds to\na constant heat \rux traversing the material. Moreover\nwe choose here to study the adiabatic condition corre-\nsponding to jq= 0 in which the two terms at the right\nhand side of Eq.(21), the spin Peltier term \u000fMTjMand\nthe heat conduction caused by the temperature pro\fle\nT(x), counterbalance each other, giving no net heat \row\nthrough the layer. The pro\fle T(x) will be stable if tem-\nperature of the thermal baths at the boundaries of thematerial are let free to adapt at the temperatures of the\ntwo ends. By using the adiabatic condition jq= 0 in\nEq.(15) we immediately obtain\nrxT=1\n^\u000fM\u00160rxH\u0003(23)\nwhere ^\u000fMis the thermomagnetic power coe\u000ecient in adi-\nabatic conditions\n1\n^\u000fM=1\n\u000fM\u0014M\n\u0014+\u0014M(24)\nand\u0014M=\u000f2\nM\u001bMT. From Eq.(23) we see that the tem-\nperature pro\fle depends on the pro\fle of the potential\nH\u0003. This last one is determined by inserting Eq.(23)\ninto Eq.(14). We have \fnally\njM= ^\u001bM\u00160rxH\u0003(25)\nthat has to be solved with the continuity equation (16)\ngiving again the di\u000busion equation (18) of the previous\nsection. However now the di\u000busion length is the adiabatic\nvalue ^lM= (\u00160^\u001bM\u001cM)1=2where\n^\u001bM=\u001bM\u0014\n\u0014+\u0014M(26)\nis the conductivity for the magnetization current in adi-\nabatic conditions.\nB. Spin Hall e\u000bect in non-magnetic metals\nThe spin Hall e\u000bect is due to the spin orbit interac-\ntion for conduction electrons. This e\u000bect is particularly\nrelevant for noble metals with high atomic number. Be-\ncause of the spin orbit interaction, a spin polarized elec-\ntric current is de\rected by an angle which is called the\nspin Hall angle \u0012SH. To include spin Hall e\u000bects into the\ntheory of Section III A one should \frst extend the equa-\ntions for the thermo-magnetic e\u000bects to the presence of\nan additional electric current. This is straightforward\nand the formal result is reported in Appendix A. How-\never to state the equation for the spin Hall e\u000bect, the\nequations must be further extended for two dimensional\n\row. The complete constitutive equations are character-\nized by six force variables, namely: the derivative along\nxandyof the three driving forces for magnetic, elec-\ntric and heat currents. Here we simplify the problem by\njust disregarding the thermal e\u000bects. For our \fnal aims\nthis is a reasonable approximation, since the contribu-\ntion arising from the thermomagnetic coe\u000ecients of Pt\nis smaller than the other contributions involved in the\nfull matrix of the thermo-magneto-electric e\u000bects26. The\ngeneral constitutive equations for the joint electric and\nmagnetic transport are reported in Appendix B. Here we6\nanalyze in more detail the case of a non magnetic conduc-\ntor with negligible Hall e\u000bect. We select the conditions\nin which the electric current jeis always along y, and\nthe magnetization current jMalongx. We have then the\nequations for the spin Hall and the inverse spin Hall ef-\nfects from Eqs.(B5) and (B6). By converting to magnetic\nunits one obtains\njey=\u0000\u001b0ryVe+\u001b0\u0012SH\u0010\u0016B\ne\u0011\n\u00160rxH\u0003(27)\njMx=\u001b0\u0012SH\u0010\u0016B\ne\u0011\nryVe+\u001bM\u00160rxH\u0003(28)\nwhere\u001bM=\u001b0(\u0016B=e)2is the conductivity for the mag-\nnetization current, \u001b0is the electric conductivity, Veis\nthe electric potential, eis the elementary charge and \u0016B\nis the Bohr magneton. The equations contain the spin\nHall e\u000bects in the non diagonal terms which couples dif-\nferent directions and di\u000berent currents. It is worthwhile\nto notice that the e\u000bects are fully described by the spin\nHall angle \u0012SHwhich for metals is a de\fnite negative\nquantity.\n1. Spin Hall e\u000bect\nIn the spin Hall e\u000bect a magnetization current is gen-\nerated in the parallel direction xbecause of an electric\ncurrent in the perpendicular one y. By eliminatingryVe\nby Eq.(27) and Eq.(28) we \fnd that the magnetization\ncurrent is related to the electric current density by\njMx=\u0000\u0012SH\u0010\u0016B\ne\u0011\njey+\u001b0\nM\u00160rxH\u0003(29)\nwhere\u001b0\nM=\u001bM(1 +\u00122\nSH). If the electric current density\nis uniform, the spin Hall e\u000bect corresponds to a mag-\nnetization current source jMS=\u0000(\u0016B=e)\u0012SHjey. The\npro\fle of the magnetization current jMxwhich is actu-\nally traversing the layer also depends on the boundary\nconditions posed by the adjacent layers. Then, to \fnd\nthe pro\flejMx(x), Eq.(29) must be solved together with\nthe continuity equation (16) giving a di\u000berential equation\nfor the driving potential H\u0003(x) which has the same from\nof Eq.(18) but with lM= (\u00160\u001b0\nM\u001cM)1=2.\n2. Inverse spin Hall e\u000bect\nIn the con\fguration corresponding to the inverse spin\nHall e\u000bect one has a magnetization current in the parallel\ndirection which generates an electric e\u000bect perpendicular\nto it. The electric equation in the ydirection is\njey=\u0000\u001b0\n0ryVe+\u0012SH\u0012e\n\u0016B\u0013\njMx (30)where\u001b0\n0=\u001b0(1 +\u00122\nSH). The magnetization current\ntraversing the layer is not constant and it will be given\nby the solution of Eq.(29) if the electric current jeyis\nconstrained or by the solution of Eq.(28) if the electric\npotentialryVeis constrained. In both cases the constitu-\ntive equation must be solved together with the continuity\nequation (16), giving again the di\u000berential equation (18).\nIV. SOLUTIONS OF THE MAGNETIZATION\nCURRENT PROBLEM\nA. Single active material\nFor an active material both the spin Seebeck e\u000bects\nand the spin Hall e\u000bect results in a magnetization cur-\nrent source and the pro\fle of the magnetization current\nwill be due to the boundary conditions. In presence of\nboundaries blocking the \row of the magnetization cur-\nrent, the magnetic moments accumulate giving rise to\nthe potential H\u0003. The magnetization current close to a\nboundary is therefore absorbed by the materials itself as\nthe potential H\u0003is also the driving force for the non con-\nservation of the magnetic moment (Eq.(6)). As it was\nshown in the previous Section, both spin Seebeck and\nspin Hall e\u000bects are characterized by constitutive equa-\ntions that have the same functional form. Then we can\nwork out the solution for the pro\fle of the magnetization\ncurrent independently of the speci\fc e\u000bect and consider-\ning boundary conditions only. The speci\fc solution will\ncorrespond to use as the current source jMSthe expres-\nsion derived from the spin Seebeck Eq.(14) or to the spin\nHall Eq.(29). We initially consider a single material with\ngeneric boundary conditions. The solution of the magne-\ntization current problem with several layers will then be\nobtained by applying appropriate boundary conditions\nand joining the solutions for di\u000berent layers. We take a\nmaterial from x=d1tox=d2with a uniform source\nof magnetization current jMS. Starting from the formal\nsolution Eq.(20), we derive the magnetization current by\nEq.(17) and we \fx arbitrary values of the current at both\nboundaries, i.e. jM(d1) andjM(d2). The expression for\nthe current is\njM(x) =jMS\u0000(jM(d1)\u0000jMS)sinh((x\u0000d2)=lM)\nsinh(t=lM)+\n+ (jM(d2)\u0000jMS)sinh((x\u0000d1)=lM)\nsinh(t=lM)\n(31)\nand for the potential is\nH\u0003(x) =\u0000(jM(d1)\u0000jMS)1\n(lM=\u001cM)cosh((x\u0000d2)=lM)\nsinh(t=lM)+\n+ (jM(d2)\u0000jMS)1\n(lM=\u001cM)cosh((x\u0000d1)=lM)\nsinh(t=lM);\n(32)7\nwheret=d2\u0000d1. Figs.2 and 3 shows the pro\fles of the\nmagnetization current and the e\u000bective \feld along the\nmaterial for di\u000berent thicknesses t=lM. The spin accu-\nmulation close to the boundaries generates, as a reaction,\nan e\u000bective \feld which counteracts the e\u000bect considered\n(e.g. the spin Seebeck e\u000bect) in order to let the current\nto go to zero at the interface.\nFIG. 2. Magnetization current pro\fles for a single active ma-\nterial. Curves are Eq.(31) with d1=\u0000t=2 andd2=t=2,\nboundary conditions \fxed to zero ( jM(\u0000t=2) =jM(t=2) = 0)\nand show di\u000berent thicknesses t=lM. The curves are normal-\nized tojMS.\nFIG. 3. Magnetization potential pro\fle H\u0003for a single active\nmaterial. Curves are Eq.(32) with d1=\u0000t=2 andd2=t=2,\nboundary conditions \fxed to zero ( jM(\u0000t=2) =jM(t=2) = 0)\nand show di\u000berent thicknesses t=lM(same as Fig.2). The\ncurves are normalized to H\u0003\n0=jMS=(lM=\u001cM).\nB. Injection of a magnetization current\nWe consider the injection of a magnetization current\nfrom an active material which is acting as current gen-\nerator, or current injector, into a passive material which\nis acting as a conductor. It is known that the quality\nof the interface plays an important role in the injection\nof the spin currents27. In Ref.27the condition of thePt/YIG interface was intentionally modi\fed by creating\na thin amorphous YIG layer varying from 1 to 14 nm and\nit was shown that the spin Seebeck e\u000bect is depressed as\nthe thickness of the amorphous layer increases. The max-\nimum value is obtained with a fully crystalline interface\nand the typical decay length of the e\u000bect with thickness is\n2:3 nm. In the present theory this kind of interlayer inter-\nface can be taken into account by introducing a third ef-\nfective layer, with degraded properties, between the two.\nIn the present paper we consider ideal interfaces between\ninjector and conductor which is appropriate for spin See-\nbeck experiments characterized by crystalline interfaces.\nTo analyze the injection of a magnetization current, we\nsimplify the notation by dropping the Msubscript and\nemploying subscripts describing the role of the material:\n(1) for the injector and (2) for the conductor. The mag-\nnetization current source is that of the active material (1)\nand is denoted jMS. The connection between the two me-\ndia is set at x= 0. The boundary conditions for the mag-\nnetization current is j1(0) =j2(0) =j0and the bound-\nary condition for the potential is H\u0003\n1(0) =H\u0003\n2(0) =H\u0003\n0.\nAppendix C reports the formal solutions in the case in\nwhich each layer has \fnite width. These solutions will be\nemployed in the comparison with real experiments per-\nformed in bilayers. Here we discuss how the e\u000eciency of\nthe injections is determined by intrinsic parameters. To\nthis aim we take the solutions of Appendix C in the limit\nof semi in\fnite width and we obtain\nj1(x) =jMS\u0000(jMS\u0000j0) exp(x=l1) (33)\nand\nj2(x) =j0exp(\u0000x=l2) (34)\nfor the currents and\nH\u0003\n1(x) =j0\u0000jMS\n(l1=\u001c1)exp(x=l1) (35)\nand\nH\u0003\n2(x) =\u0000j0\n(l2=\u001c2)exp(\u0000x=l2): (36)\nBy setting the boundary condition at the interface be-\ntween the two media H\u0003\n1(0) =H\u0003\n2(0) we \fnd the value of\nthe current at the interface\nj0=jMS\n1 +r12(37)\nwherer12= (l1=\u001c1)=(l2=\u001c2). Ifr12\u001c1 the current is\ne\u000eciently injected, while if r12\u001d1 the magnetization\ncurrent is not transmitted into the conductor. In terms\nof intrinsic parameters we have8\nr12=r\u001b1\n\u001b2\u001c2\n\u001c1: (38)\nSo a junction with an e\u000ecient injection from (1) to (2)\nshould have a conductor (2) with a magnetization con-\nductivity much larger than the injector \u001b2\u001d\u001b1and a\ntime constant much smaller \u001c2\u001c\u001c1.\nV. SPIN SEEBECK AND SPIN PELTIER\nEFFECTS\nIn this Section we apply the theory previously devel-\noped to the spin Seebeck and spin Peltier e\u000bects.\nA. Spin Seebeck e\u000bect\nThe spin Seebeck e\u000bect consists in a magnetization\ncurrent generated by a temperature gradient across a\nferromagnetic material. We study the longitudinal spin\nSeebeck e\u000bect (LSSE) where the magnetization current\nand the temperature gradient are along the same direc-\ntion. We consider experiments in which the active layer\nis YIG, the injector, labeled as (1) and the sensor layer\nis Pt, the conductor, labeled as (2). The geometry of the\nexperiment is schematically shown in Fig.4. The YIG\ninjector has thickness t1=tY IGwhile the Pt conductor\nhas thickness t2=tPt. The interface is set at x= 0.\nxyzme-jeyHYIGPt\njMxt1=tYIGt2=tPtcoldhotM\nFIG. 4. Geometry of the longitudinal spin Seebeck e\u000bect.\nThe temperature gradient is applied along x, the mag-\nnetic \feld is along z, the electric e\u000bects (ISHE voltage)\nare measured along y. We consider a constant tempera-\nture gradientrxT, therefore the magnetization current\nsource of YIG is jMS=\u0000\u001bY IG\u000fY IGrxTgiven by the\nequations of Section III A. The solutions of the magneti-\nzation current problem are Eqs.(C1) and (C2) reported\nin Appendix C and the magnetization current at the in-\nterface is given by Eq.(C5) in which l1=lY IG,\u001c1=\u001cY IG\nandl2=lPt,\u001c2=\u001cPt. As the thickness of the Pt layer\nis generally of the same order of the spin di\u000busion length\n(tPt\u0018lPt\u001810 nm), we can approximate Eq.(C2) forthe case of t2\u0018l2and \fnd that the pro\fle of the mag-\nnetization current is, at a good approximation, a linear\ndecay from j0at the interface x= 0 to zero at the bor-\nderx=t2. The average magnetization current in the Pt\nlayer is therefore hjMxix=j0=2 wherej0is the magne-\ntization current injected at the interface. If the experi-\nments are performed by measuring the ISHE voltage, by\ntaking Eq.(30) with jey= 0, we obtain the relation be-\ntween the magnetizations current along xand the electric\npotential along y. We assume the relation to be valid for\nthe average values along xover the thickness t2. The\naverage potential is then\nhryVeix=\u0012SH\n\u001be\u0012e\n\u0016B\u0013\nhjMxix: (39)\nwhere\u001be, corresponding to \u001b0\n0in Eq.(30), is the electric\nconductivity of Pt. The current injected at the interface\nj0can therefore be estimated by the gradient of the ISHE\nvoltageryVISHE =hryVeix,\nj0= 2\u001be\n\u0012SH\u0010\u0016B\ne\u0011\nryVISHE: (40)\nIn experiments, the spin Seebeck coe\u000ecient is determined\nasSLSSE =ryVISHE=rxT. The magnetization current\nat the interface can be calculated by Eq.(40) where the\nspin Hall angle is evaluated as \u0012SH=\u00000:1 from Ref.28.\nIn turn, the relation between the spin Seebeck current\njMSandj0at the interface, given by Eq.(C5), will de-\npend on the intrinsic parameters of both layers and their\nthickness. Once the current jMSis calculated, one can\nestimate the spin Seebeck coe\u000ecient as\n\u000fY IG=1\n\u001bY IG\u0012jMS\n\u0000rT\u0013\n: (41)\nIn Pt the magnetization di\u000busion length is known to be\nlPt= 7:3 nm28. The spin conductivity can be estimated\nby assuming that in a normal metal the scattering acts\nindependently of the spin29. Then, by converting the\nelectrical conductivity of Pt \u001be= 6:4\u0001106\n\u00001m\u00001, into\nthe conductivity for the magnetization current, we obtain\n\u00160\u001bPt= 2:6\u000110\u00008m2s\u00001. The time constant is \fnally\ncalculated and results \u001cPt=l2\nPt=(\u00160\u001bPt)'2\u000110\u00009s.\nIn YIG the estimations of the magnetization di\u000bu-\nsion length present in literature, range from micron to\nmillimeter30{32for the transverse experiment (in which\ncurrent and magnetization are parallel) to much lower\nvalue (i.e.<1\u0016m)33for the longitudinal e\u000bect (in which\ncurrent and magnetization are perpendicular). From\nRef.3the LSSE coe\u000ecient measured on 1 mm of YIG,\nSLSSE'4\u000110\u00007VK\u00001, results to be larger than the one\nmeasured on a 4 \u0016m sample22SLSSE'2:8\u000110\u00007VK\u00001,\nbut of the same order of magnitude. Therefore we can\nguess thatlY IGis of the same order of magnitude of the\nthinner sample (4 \u0016m) in order to allow for an e\u000ecient9\ninjection in both cases. In a more recent study, the de-\npendence of the spin Seebeck e\u000bect on the thickness of\nYIG was investigated21. It has been reported that the\ntypical di\u000busion length is below lY IG= 1:5\u0016m. We set\nin the following lY IG= 1\u0016m. For the evaluation of the\nabsolute thermomagnetic power coe\u000ecient \u000fY IG we use\nthe result of Ref.22where the thermal conditions were\nproperly taken into account. These experiments were\nperformed by using a YIG layer of 4 \u0016m and a Pt layer\nof 10 nm.\nBy using the LSSE coe\u000ecient estimated at the sat-\nuration magnetization of YIG we obtain j0=(\u0000rxT)'\n2\u000110\u00003As\u00001K\u00001m22. The only missing intrinsic param-\neter is the magnetization conductivity of the YIG, \u001bY IG.\nTo have an order of magnitude we suppose a reasonable\ninjection from YIG into Pt (i.e 50%, with j0= 0:5jMS).\nThen we set r12= 1, i.e. l1=\u001c1=l2=\u001c2. By using\nthe resulting value for the magnetization conductivity\nof YIG\u00160\u001bY IG\u00184\u000110\u00007m2s\u00001, we \fnally obtain\nan order of magnitude for the absolute thermomagnetic\npower coe\u000ecient as \u000fY IG\u001810\u00002TK\u00001. In analogy with\nthe thermoelectric e\u000bects where the absolute thermoelec-\ntric power coe\u000ecient is compared to the classical value\n\u000fe=\u0000kB=e'\u000086\u000110\u00006VK\u00001, the value found here can\nbe compared with the ratio kB=\u0016B'1:49 TK\u0000117. Fur-\nthermore, as the experiments show that ryVISHE and\nthereforejMS, changes sign when the magnetization of\nthe YIG layer is inverted, this means that \u000fY IGchanges\nsign when inverting the magnetization M. The value re-\nported before corresponds to the absolute value when the\nmagnetization of YIG is at saturation.\nB. Spin Peltier e\u000bect\nIn the spin Peltier experiments a magnetization cur-\nrent is generated by the spin Hall e\u000bect in a Pt layer,\nlabeled as (1) and is injected into a YIG layer, labeled\nas (2). The injection of the magnetization current into\nthe YIG, generates thermal e\u000bects. The geometry of the\nexperiment is schematically shown in Fig.5.\nxyzme-jeyHYIGPt\njMxt2=tYIGt1=tPtMcoldhot\nFIG. 5. Geometry of the spin Peltier e\u000bect.\nThe interface is set at x= 0, the electric current isalongy, the magnetization current is along xand the\nmagnetic \feld is along z. The magnetization current\nsource is now jMS=\u0000\u0012SH(\u0016B=e)jeygiven by the spin\nHall e\u000bect in Pt discussed in Section III B. When the\nmagnetization current di\u000buses inside YIG, it also gen-\nerates a heat current because of the spin Peltier e\u000bect\ndescribed in Section III A 2. The solution of the magne-\ntization conduction problem is mathematically identical\nto the spin Seebeck one, but with the role of YIG and\nPt inverted. For this reason we have employed label (1)\nfor the injector, which is now Pt, and label (2) for the\nconductor which is now YIG. The solutions of the mag-\nnetization current problem are again Eqs.(C1) and (C2)\nreported in Appendix C and the magnetization current\nat the interface is given by Eq.(C5). With respect to the\nprevious spin Seebeck case, the di\u000busion length of YIG\nis the adiabatic value ^lY IG = (\u00160^\u001bY IG\u001cY IG)1=2. In the\nspin Peltier experiment the temperature pro\fle in YIG\nis given by the integration of Eq.(23)\nT(x)\u0000T(0) =1\n^\u000fY IG\u00160(H\u0003\n2(x)\u0000H\u0003\n2(0)) (42)\nwhereH\u0003\n2(x) is given by Eq.(C4). The result is shown in\nFig.6.\n!!\"#!\"$!\"%!\"&'!!\"'!\"#!\"(!\"$!\")\n*+,#\u0001-+\u0001-./)$(#0#+,#1'\nx/tYIGlYIG/tYIG=0.10.5125\nFIG. 6. Temperature pro\fle of YIG for the spin Peltier e\u000bect.\nCurves are \u0001 T=T(x)\u0000T(0) from Eq.(42) normalized to\n\u0001TSH=\u00160H\u0003\nSH=^\"Y IG andH\u0003\nSH=jMS=(lY IG=\u001cY IG). The\nparameters are r12= 1; lPt=tPt= 0:1.\nBy looking at the magnetization current pro\fle (Fig.7),\nwe see, as in the spin Seebeck experiment, that in order to\nhave a good e\u000eciency, the thickness of each layer should\nbe larger than its di\u000busion length ( t1>l1andt2>l2) to\npermit to the magnetization current to develop. More-\nover the e\u000eciency of the injection is regulated by the\nratio of intrinsic parameters r12= (l1=\u001c1)=(l2=\u001c2), where\n(1) is the injector Pt and (2) is the conductor YIG. Again\nthe magnetization current at the interface is large if the\nratior12is small. However it should be noticed that\ngiven the two materials in the junction (i.e. Pt,YIG)\nwe have that rPt!Y IG = 1=rY IG!Pt. So, the value\nrPt!Y IG =rY IG!Pt'1 is the value which permits10\nrelatively e\u000ecient injection both from Pt into YIG and\nfrom YIG into Pt.\nFinally from the temperature pro\fle Fig.6 obtained in\nadiabatic conditions we can reach information about the\ncoe\u000ecient of the absolute thermomagnetic power in adi-\nabatic conditions ^ \"Y IG. The pro\fle T(x) is normalized\nto the temperature \u0001 TSHwhich gives the typical scale\nof the e\u000bect\n\u0001TSH=1\n^\"Y IG\u00160H\u0003\nSH (43)\nwhereH\u0003\nSH=jMS=(lY IG=\u001cY IG). From the litera-\nture the thermal conductivity of YIG is \u0014= 6 W\nK\u00001m\u00001. From Section V A, \"Y IG'10\u00002TK\u00001and\nthe parameter \u0014Y IG'10\u00002W K\u00001m\u00001 34. Moreover\nthe potential H\u0003\nSHis related to the spin Hall current\njMS=\u0000(\u0016B=e)\u0012SHjeyinjected from Pt. Using the val-\nues from34lY IG=\u001cY IG = 3 ms\u00001and\u0012SH=\u00000:1 we\nare able to give an order of magnitude estimate of the\ntemperature change, obtaining \u0001 TSH=jey= 4\u000110\u000013K\nA\u00001m2.\nExperimental values are taken from Ref.4, where in\ncorrespondence to an electric current density of 3 \u00011010\nA m\u00002in Pt, the temperature di\u000berence measured by a\nthermocouple in YIG was 2 :5\u000110\u00004K, considering that\nthe Joule heating of the electric current in Pt was already\nsubtracted. The parameter \u0001 TSHresults 1:2\u000110\u00002K\nwhich is of the correct order of magnitude. Consequently\nby usingt1=tPt= 5 nm and t2=tY IG = 0:2\u0016m\nin Eqs.(C4) and (42), we \fnd an adiabatic tempera-\nture change of T(tY IG)\u0000T(0)'2:5\u000110\u00004K with\nlY IG = 0:4\u0016m. This value re\fnes the upper limit of\n1\u0016m which was found in Section V A, however the phe-\nnomenology of the spin Peltier e\u000bect in YIG seems coher-\nent with the absolute thermomagnetic power coe\u000ecient\nderived previously.\nVI. CONCLUSIONS\nIn this paper the problem of magnetization and heat\ncurrents is investigated through a non equilibrium ther-\nmodynamics approach. Based on the constitutive equa-\ntions of a ferromagnetic insulator and a spin Hall active\nmaterial we are able to solve the problem of the pro-\n\fles of the magnetization current and of the potential\nin the geometry of the longitudinal spin Seebeck and of\nthe spin Peltier e\u000bects. By focusing on the speci\fc ge-\nometry with one YIG layer and one Pt layer, we obtain\nthe optimal conditions for generating large magnetization\ncurrents into Pt in the case of the spin Seebeck e\u000bect and\nfor generating large heat current in YIG in the case of\nspin Peltier e\u000bects. In both cases we \fnd that e\u000ecient\ninjection is obtained when the thickness of the injecting\nlayer is larger than the di\u000busion length lM. The the-\nory predictions are compared with experiments and this\npermits to determine the values of the thermomagneticcoe\u000ecients of YIG: the magnetization di\u000busion length\nlM\u00180:4\u0016m and the absolute thermomagnetic power\ncoe\u000ecient\u000fM\u001810\u00002TK\u00001.\nACKNOWLEDGMENTS\nThis work has been carried out within the Joint Re-\nsearch Project EXL04 (SpinCal), funded by the Eu-\nropean Metrology Research Programme. The EMRP\nis jointly funded by the EMRP participating countries\nwithin EURAMET and the European Union.\nAppendix A: Constitutive equations of the\nthermo-magneto-electric e\u000bects\nThe equations for the thermo-magneto-electric e\u000bects\nrelates the current densities of the electric charge je, the\nmagnetic moment jM, and the heat jq, with the gradients\nof the electric potential Ve, the magnetization potential\nH\u0003, and the temperature T. In one dimension ( rx=\n@=@x ) the equations are\nje=\u0000\u001berxVe+\u0011\u00160rxH\u0003\u0000\u001be\u000ferxT (A1)\njM=\u0000\u0011rxVe+\u001bM\u00160rxH\u0003\u0000\u001bM\u000fMrxT (A2)\njq=\u0000\u000fe\u001beTrxVe+\u000fM\u001bMT\u00160rxH\u0003\u0000\u0014isorxT:\n(A3)\nwhere\u001beis the electrical conductivity, \u000feis the abso-\nlute thermoelectric power coe\u000ecient, \u0011represents the\nmagneto-electric conductivity, \u001bMis the magnetic con-\nductivity,\u000fMis the absolute thermomagnetic power coef-\n\fcient and\u0014isois the thermal conductivity with rxVe=\n0 andrxH\u0003= 0. By de\fning the heat current as\njq=Tjswe obtain from Eqs.(10) and (11)\nT@s\n@t+rxjq=\u00160rxH\u0003jM+\u00160(H\u0003)2\n\u001cM\u0000rxVeje:(A4)\nBy solving the previous equation together with the con-\nstitutive equation (A3), one can obtain the generalized\nheat di\u000busion equations.\nAppendix B: Magnetization current carried by\nelectrons\nWe consider the speci\fc case of metals in which the\nmagnetic and electric current are due to the same type\nof carriers (electrons or holes) with di\u000berent spin. The\ntheory can be equivalently formulated in terms of mag-\nnetic moment (up or down). One subdivides the particle\ncurrentjn=jn++jn\u0000into the sum of moment up jn+\nand moment down jn\u0000. The electric current is je=qjn\nwhereqis the charge of the carrier, while the magneti-\nzation current is \u0016B(jn+\u0000jn\u0000), where\u0016Bis the Bohr11\nmagneton. As it is somehow customary to de\fne a mag-\nnetization current jmmeasured in the same units of the\nelectric currents, we have then jm= (e=q)(je+\u0000je\u0000)\nwhereeis the elementary charge. Electrons, moving in\nthe opposite direction of the charge current, with a mag-\nnetic moment up will give a negative jm, while holes with\nmoment up, will give a positive jm. One is allowed to as-\nsume di\u000berent conductivities among the two sub-bands\nas a function of the gradients of the potentials rVe\u0006rel-\native to each sub-band. The equations are\nje+=\u0000\u001b+rVe+\u0000\u001bmixrVe\u0000 (B1)\nje\u0000=\u0000\u001bmixrVe+\u0000\u001b\u0000rVe\u0000 (B2)\nwhere one has to introduce both the individual channel\nconductivities \u001b+and\u001b\u0000and the spin mixing conduc-\ntivity\u001bmix. One obtains\n\u0012\nje\njm\u0013\n=\u0000\u001b0\u0012\n1 +\u000b \f\n\f 1\u0000\u000b\u0013\u0012\nrVe\nrVm\u0013\n(B3)\nwithVe=Ve++Ve\u0000andVm= (e=q)(Ve+\u0000Ve\u0000)\nwhere\u001b0= (\u001b++\u001b\u0000)=2 is the electric conductivity,\n\u000b=\u001bmix=\u001b0is the spin mixing coe\u000ecient ( \u000b\u00141) and\n\f= (\u001b+\u0000\u001b\u0000)=(2\u001b0) represents the spin unbalance of\nthe conductivities. Vmis a potential for the current jm\nwith the same units of Ve. The electric conductivity is\n\u001be=\u001b0(1+\u000b) and the conductivity for the magnetization\ncurrentjmis\u001bm=\u001b0(1\u0000\u000b). It is often the case that\nthe spin mixing conductivity is very small (i.e. \u000b= 0\ninto Eq.(B3)) because the spin \rip events are much more\nrare than the normal scattering conserving the spin, so\n\u001bm=\u001be. This leads to the Mott's two current model.\nIn that case the spin unbalance coe\u000ecient \fis a number\nbetween 1 and -1.\nThe previous equations form also the basis to describe\nthe Hall and the spin Hall e\u000bects. We need to extend\nthe equations for the magneto-electric e\u000bects to two di-\nmensions. We consider the case in which the spin mixing\nconductivity is zero and \u001be=\u001bm=\u001b0. The equations\nread\n0\nB@jex\njey\njmx\njmy1\nCA=\u0000\u001b00\nB@1\u0000\u0012H\f\u0000\u0012SH\n\u0012H 1\u0012SH\f\n\f\u0000\u0012SH 1\u0000\u0012H\n\u0012SH\f \u0012 H 11\nCA0\nB@rxVe\nryVe\nrxVm\nryVm1\nCA\n(B4)\nwhere\u0012His the Hall angle and \u0012SHis the spin Hall an-\ngle. It is important to notice that the Hall angle depends\non the magnetic \feld while the spin Hall angle is a con-\nstant that is determined by the spin orbit interaction for\nconduction electrons.\nWe analyze in more detail a non magnetic conductor\nwith\f= 0 for which the Hall angle is negligible \u0012H= 0.\nFurthermore we select conditions in which the electric\ncurrent is always along yand the magnetic current alongx. We have \fnally the equations for the spin Hall and\nthe inverse spin Hall e\u000bects\njey=\u001b0=\u0000ryVe\u0000\u0012SHrxVm (B5)\njmx=\u001b0=\u0012SHryVe\u0000rxVm: (B6)\nTo convert to magnetic units of Section III B one simply\nuses\nrVm=\u0000\u0010\u0016B\ne\u0011\n\u00160rH\u0003(B7)\nand\njm=\u0012e\n\u0016B\u0013\njM: (B8)\nAppendix C: One junction\nLet us consider a bilayer of two materials: the injector\n(1) fromx=\u0000t1tox= 0 which contains a magneti-\nzation current source jMSand the conductor (2) from\nx= 0 tox=t2. The connection between the two me-\ndia is put at x= 0 and the boundary conditions on the\nmagnetization current are: j1(\u0000t1) = 0,j2(t2) = 0 and\nj1(0) =j2(0) =j0. The solutions for the magnetization\ncurrents, where only the injector (1) is an active material,\nare\nj1(x) =jMS+jMSsinh(x=l1)\nsinh(t1=l1)+\n+ (j0\u0000jMS)[sinh(x=l1) coth(t1=l1) + cosh(x=l1)] (C1)\nand\nj2(x) =\u0000j0[sinh(x=l2) coth(t2=l2)\u0000cosh(x=l2)] (C2)\nand for the potentials\nH\u0003\n1(x) =jMS\n(l1=\u001c1)cosh(x=l1)\nsinh(t1=l1)+\n+j0\u0000jMS\n(l1=\u001c1)[cosh(x=l1) coth(t1=l1) + sinh(x=l1)] (C3)\nand\nH\u0003\n2(x) =\u0000j0\n(l2=\u001c2)[cosh(x=l2) coth(t2=l2)\u0000sinh(x=l2)]:\n(C4)\nBy setting the boundary condition at the interface be-\ntween the two media H\u0003\n1(0) =H\u0003\n2(0) we \fnd the value of\nthe current at the interface12\nj0=jMScosh(t1=l1)\u00001\ncosh(t1=l1) +r12sinh(t1=l1) coth(t2=l2)(C5)\nwherer12= (l1=\u001c1)=(l2=\u001c2). 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Pasquale, Physics Procedia 75, 939 (2015)."    },    {        "title": "1811.12514v1.Structural_and_Magnetic_Study_of_Metallo_Organic_YIG_Powder_Using_2_ethylhexanoate_Carboxylate_Based_Precursors.pdf",        "content": "1 \n   Structural and Magnetic Study  of Metallo -Organic YIG Powder Using 2 -ethylhexanoate \nCarboxylate  Based P recursors  \nS. Hosseinzadeha, P.Elahib,  M. Behboudnia*, a, M.H. Sheikhic, S.M. Mohseni*, d \naDepartment of Physics, Urmia University of Technology, Urmia, Iran  \nbDepar tment of Material Science, The U niversity of Utah, Utah, U.S.  \ncDepartment of Communications and Electronics, School of Electrical and Computer \nEngineering, Shiraz, Iran  \ndFaculty of Physics, S hahid Beheshti University , Evin,  19839,  Tehran,  Iran \n \nAbstract  \nThe crystallization and magnetic behavior of yttrium iron garnet  (YIG)  prepared by metallo -\norganic  decomposition  (MOD)  method  are discussed . The chemistry and physics related to \nsynthesis  of iron and yttrium carboxylates based on 2 -ethylhexanoic acid  (2EHA)  are studied , \nsince no literature was found which elucidate s synthesi s of metallo -organic precursor of YIG  in \nspite of the literatures of doped YIG samples such as Bi -YIG. Typically, the  metal carboxylates \nused in preparation of ceramic oxide materials are 2 -ethylhexanoate  (2EH)  solvents. Herein , the \nsynthesis , thermal behavior and solubility of  yttrium and iron  2EH used in synthesis of  YIG \npowder by MOD  are reported . The crystallization and magnetic parameters , including saturation \nmagnetization and coercivity of these samples , smoothly  change as a function  of the annealing \ntemperature. It is observed that high sintering temperature  of 1300 to 1400 °C promotes the \ndiffraction peaks of YIG , therefore, we can conclude  that the formation of YIG in MOD method  \nincreases the crystallization  temperature . The maximum value of  saturation magnetization  and \nminimum value of coercivity and remanence are observed fo r the sample sintered at 1200°C  \nwhich  are 13.7 emu/g, 10.38 Oe and 1.5 emu/g , respectively . This study  cites the drawbacks in \nchemical synthesis of  metallo -organic based YIG production.  2 \n  \nKeywords  \nYIG; MOD ; metallo -organic precursor ; crystallization ; magnetic  particles  \n*Corresponding author. E -mail Address: m-mohseni@sbu.ac.ir , majidmohseni@gmail.com  (Seyed Majid Mohseni) . \nmbehboudnia@gmai l.com  (Mahdi Behboudnia)  \n \n1. Introduction  \nMagnetic t hin film of Yttrium Iron Garnet (Y 3Fe5O12 – YIG) has found great attention in \nemerging spintronic devices. Such a thin film with low Gilbert damping constant is suitable for \nmagnonics and beside its insulator behavior ( insulating nature) , it gains a great deal of attention \nin generation of spin current. With the advent of spintronics, the demand for synthesis and \nprocessing of YIG films has surged  forward greatly.  \nIn general, t he characteri stics and  surface morphology of the thin films strongly depends  on \ndeposition techniques.  Pulsed laser deposition (PLD) has emerged as a preferred technique to \ndeposit complex  oxide thin films, heterostructures, and superlattices  with high quality  in \ncomparison with the  other deposition techniques such as e -beam evaporation and sputtering  [4, \n5].  Contrarily , chemica l solution  deposition (C SD) techni ques are cost effective  synthesis \nprocess es in which  the precursor solution is deposited by  variety of relatively simple techniques  \nsuch as  spin or dip coating followed by  post treatments of  drying and anneal ing in case needed . \nMore r ecently , metallo -organic decomposition (MOD) recognized as a chemical technique  has \nbeen  growing  due to  its extreme ly good  molecular level composition controllability  in the \nfabrication process  of garnet thin films . Kirihara et.al  [6] reported generation of spin -current -\ndriven thermoelectric conversion by using B i-YIG thin layer prepared  by MOD. There are  also 3 \n further  literature s [7-9] which report deposition of  doped YIG and YIP  (yttrium iron perovskite, \nYFeO 3) by MOD  technique  using  metal carboxylate precursors in organic solvents. In majority \nof the publications , the effect of the annealing temperature and duration  of doped YIG powders \nhas been investigated on the physical and magnetic properties . It is shown  that at higher \ntemperature s, the observed  physical and magnetic properties  of garnet are similar to that of solid \nstate reaction techniques  [10]. \nThe MOD procedure introduced by Azevedo et. al  is used  for thin film deposition  [11-14]. Their  \nprocedure consist s of preparation of metal -organic carboxylates, which exhibit  high \ncompatibility to organic solvents , result in  better final thin film properties . They succeeded in \npreparing polycrystalline Gd 2BiFe 5O12 and (DyBi) 3(Fe, Ga) 5O12 thin films on glass substrates. In \ntheir work , large carboxylate compounds of 2 -ethylhexanoate (2EH) precursors were \nsynthesized, and then  a stoichiometric ratio of precursors was dissolved in xylene followed by \nspin coating of the solution  on preferred substrate.  \nTypically, the metal carboxylates used in the preparation of ceramic oxide materials are  2EH \nsalts which  are dispersed in aromatic  solvents [10, 12, 13, 15, 16 ]. Metal 2EH have found wide \napplication such as metal –organic precursors , catalysts for ring  opening poly merizations and \ndrier agent in paint  industries  [17]. Realization of the MOD advantages require s in-depth \nunderstanding of the precursor solution chemistry such as precursor species, solute \nconcentration, and solvent system and its relation to the final material properties. A detailed \nstudy of the fundamental  properties  of 2EH yttrium 2-ethylhexanoate  (Y-2EH)  and iron-\n2ethylhexanoate  (Fe-2EH) precursors used in  synthesis of YIG thin films  has not been reported \nyet. The primary objective of this work is to investigate  the fundamental  properties  of yttrium 2-\nethylhexanoate  (Y-2EH)  and iron-2-ethylhexanoate  (Fe-2EH)  precursor s needed to synthesis 4 \n YIG thin films, such as  chemi cal properties  (including solvent/solute concentration ratios, \nsolution structure and internal bonding) a nd on process optimization methodolo gies in order to \nobtain optimum YIG properties .  \n2. Materials and methods  \n2.1.Reagents  \nIron(III) nitrate nonahydrate  (Fe(NO 3)3, Yttrium nitrate hexahydrate (Y(NO 3)3), Y 2O3, 2EHA , \nXylene, Toluene, n -Hexane, n -octane, benzene e, THF, Isopropanol, propioni c acid, and glacial \nacid acetic are of analytical reagent grade.  \n2.2.Synthesis of Yttrium and Iron carboxylates  \nA metal carboxylate compound is defined as the  central metal atoms linked to ligands through a \nhetero –atom s [14]. \n2.2.1 Fe-2EH : \nThe iron carboxylates were prepared by double decomposition from ammonium soaps obeying  \nthe following reactions [18] \nNH 4OH + C7H15COOH 1           C7H15COONH 4 + H 2O \nFe (NO 3)3 + 3 C7H15COONH 4          Fe (C7H15COO) 3 + 3 (NH 4) NO 3 \nThe first reaction involves the preparation of the ammoni um soap of 2-ethylhexanoic acid  \n(2EHA) . Then the soap from second reaction  was mixed with the aqueous solution of Fe(NO 3)3. \nAfter stirring for 10 minutes , the solution was separate d into Fe(RCOO) 3 and (NH 4) NO 3. In a \nfunnel separator, a solvent  (xylene ) was added to  this solution and the carboxylate which \ndissolves in xylene was separated  and filtered by 0.22 μ -filter  and was kept away  until xylene \nevaporated under fume hood  to reach  a reddish brown powder of Fe-2EH. By decomposing a 5 \n certain  amount of the Fe-2EH into the metal oxide and weighing the oxide, the equivalent \namount of iron oxide in Fe -2EH was determined.  \n2.2.2 Y-2EH : The Y-2EH is prepared according to the following reaction  [19] \nY2O3 + 6 C7H15COOH 1           2 (C7H15COO) 3Y + 3 H 2O \nYttrium oxide ( Y2O3) powder was added gradually  to 2EHA  while  gently  stirring and it was kept \nat 100°C till all liquid has been  evaporated.  The product was stirred with an excess of toluene for \n24h at room temperatu re. Thereupon , it was  filtered and was kept under fume hood  until the \ntoluene was evaporated and a white solid is achieved . This white so lid is Y-2EH. By \ndecomposing a known amount  of the Y -2EH into the metal oxide and weighing the oxide, the \nequivalent amount of Y2O3 in Y-2EH was determined.  \n2.3.Synthesis of YIG powder  \nMetallo -organic YIG was prepared using Y-2EH and Fe- 2EH in stoichiometric ratio of 3:5. \nFirstly, The Y - 2EH and Fe -2EH were dissolved into toluene and xylene , respectively. Secondly, \nthe solutions were mixed together with respect  to the weight percentage of each carboxylate  to \nachieve the desired stoichiometry. The process was followed by adding glacial acetic acid un til a \nhomogeneous solution was reached with no precipitation . The powders  of MOD  solution  were  \nprepared  by drying  for 72 hours at 150 °C and  then they were grinded  in a mortar . Figure 1 \nillustrate s the schematic diagram of metallo -organic decomposition of YIG.  \n \n 6 \n  \nFigure 1: Schematic diagram for the metallo -organic decomposition process of YIG . \n \n2.4.Characterization  of samples  \nTo investigate the pyrolysis and crystallization process of the YIG prepared by the MOD \nmethod, a thermo -gravimetric -differential thermal  analysis (TG A-DTG -DTA ) (model mettle \nToledo C1600 analyzer) was carried out  from controlled room  temperature to 1400 °C in an air \natmosphere with the  heating rate of 10°C/min. The crystalline structure of samples was \ncharacterized using X -ray diffractometer (STOE -STADI) with Cu Kα (λ = 0.154 nm) radiation. \nRoom temperature magnetization measurements were performed  using  vibrating sample \nmagnetomete r (Meghnatis Daghigh  Kavir  Co.) \n3. Results  and discussions  \nAs previously reported  by Beckel [20] and Neagu [21], the solvent influences the boiling point of \nthe solution  and determines  the speed of evaporation during heating of the droplets  which  affects  \n7 \n the film roughness. The solvent also impacts  the maximum metal carboxylate solubility and \nspreading behavior of the droplets. The deposition temperature is primarily  influenced by the \nsolvent  and additives . These organic solvents and additives  help in gelation and polymerizations, \nand modif ication o f the solution properties  [21-24], such  as viscosity, solubility of metal \ncarboxylate and spreading of the droplets.  The propionic acid and mixture of ethanol and 2EHA \nwere used as additives as mentioned in literatures beyond the number. Besides, glacial acetic \nacid (GAA) were used as additive.  This study reveals that GAA improves YIG formation and \ndecreases YIG crystallization temperature . \nFigure  2 demonstrates  the TGA -DTG and DTA curves of the MOD precursor . It show s thermal \ndecomposition proceeds via six-step process es in an air atmosphere . The broad  endothermic peak \nfrom room temperature to about 200 ° C with a total weight loss of about 3% corresponds  to the \nevaporation of residual solvents including  xylene  with boiling point  (bp) of 138 °C  and glacial \nacetic acid  with boiling point of  118 °C [25]. Three exothermic peaks from 200  to about 480 °C  \nwith total weight loss of 51.5 % are ascribe d to the volatilization of 2 EHA  (bp~228 °C ) and the \npyrolysis of possible metal -organic compounds such as Fe-2EH are expected [25, 26] . The three \nexothermic peaks from 480  to 820 °C represent removal of the three 2 -EH groups from the Y-\n2EH molecule to  form Y2O3 with total weight loss of 20.7% [27]. The following  two exothermic \npeaks at temperature range of 820 to 920 °C with a weight loss of 2.6% correspond to  \ncrystallization of YIG and YIP.  The weight loss curve then approaches plateau  from temperature \nrange 920  to 1300 °C. The two exothermic  peaks observed within  this temperature range  are \nattributed to the conversion of YIP to YIG and crystallization of YIG.  This crystallization \ntemperature is higher than other chemical solution decomp osition methods  [28, 29] . On the other \nhand, the main advantage of the carboxylate -based -routines is the comparably low  crystallization  8 \n temperature. This is  due to  the educt molecules that are mixed at the molecular level. Thus, the \ndiffusion paths of metal -and-oxygen -ions are sho rt compared to classical powder -based \nsyntheses of ceramic bulk materials  [18, 30] .  However, The formation of YIG in the MOD \nmethod increases the crystallization temperature which is  mentioned  previously by Lee et  al. \n[13]. The above results  suggest s that the 2 EHA  may not serve as an excellent ligan d for yttrium \nprecursor, since  the decomposition  of the organics for Y -2EH occurs at temperatures higher than \n500°C as reported in literatures  [27]. \n  \nFigure 2: TGA -DTG (a), DTA (b)  characteristics for YIG powder  \n \nFigure 3 shows the XRD patterns of YIG particles annealed  at 1000 -1400 °C for 2hrs. The XRD \npattern of the YIG particles calcined at 1000 °C is associated with the formation of YIG with \nalready formed yttrium oxide (Y 2O3) around 2θ=29° and  hematite (α -Fe2O3) around 2θ=33° as  \nthe major phases in addition to the traces of  YIP around 2θ=48, 33° . As the solid state reaction \nmethod suggested, the crystallization process could be  described by the following equation s, \nY2O3+Fe 2O3          2YFeO 3                      at 800 -1200 °C  \n3YFeO 3+Fe 2O3          Y3Fe5O12              at 1000 -1300 °C  \n9 \n which indicates that  the Y 2O3 phase would  first transform into YIP phase  at low temperatures  \nand then convert s to YIG phase by combining with α-Fe2O3 at higher temperatures . The color of \nthe particles calcined at 1000 °C temperature  was reddish brown which is due to the presen ce of \nα-Fe2O3 and YIP as the major  phases. When the YIG particles were  calcined at 1100 °C  and \n1200 °C, the color converted to brownish green , indicating  the conversion of YIP to YIG phase.  \nFor the sample calcined at these two temperatures , the diffraction peak around 2θ=33° became \nstronger , the Y 2O3 and YIP peaks became weaker, but still there is excess amount of α -Fe2O3 \nphase. As the annealing  temperature is increased  to 1300 &  1400 °C there is no change in  \ndiffraction peaks of YIG and α -Fe2O3 and the peak related to the α -Fe2O3, located near 2θ= 33°, \nstill remain s the same  and the  intensity of  garnet peaks are not  increased  as sintering temperature  \nis risen. The remained  α-Fe2O3 phase can be explained due to the insufficient amount of Y-2EH \nduring the reaction  because  of precipitation of Y -precursor. The metal carboxylat e must be \nsoluble and stable at room  temperature in the solution, if not  precipitation will occur and lead to \ninhomogeneity  repartition of cations in the obtained gel.  \nThe successful application of the MOD process significantly depends on  the metallo -organic \ncompounds used as precursors for a variety of  elements. The ideal compounds should satisfy \nsome requirements  such as  high solubility in a common solvent. The solutions of individual \nmetallo -organic compounds should mix  in the appropriate ratio to give the desired stoichiometry \nfor final formation.  The main conclusion is that there is no theoretical database  for selecting the \noptimum solvent suitable for MOD process . In order to explore  the interactions between  solute  \nand solvent system , the polarity  of the solvent and solute should be considered due to evaluation \nof the effect of dipole -dipole interactions .  10 \n Generally, to have a good solubility , the polarity of solute and solvent should be close to each \nother. The longer chain acid like 2EHA can be solved in low polar solvents (eg. Xylene, alcohol \netc.). In case of unknown solubility parameter of a compound, a successful approach is to first \ntry a non -polar solvent which has low solubility parameter. If the approach is not successful, then \na moderately polar solvent with intermediate solubility paramet er should be tried.  \nIn order to find  an adequate solubility in the desired  solvent s which were compatible with each \nother, we tested some solvent s recommended by literatures for yttrium and iron 2EH. Therefore, \nwe tested xylene, toluene, benzene, n -hexane, THF for both Fe-2EH and Y-2EH and found that \nthe homogeneity of toluene and xylene are the best for yttrium and iron , respectively.  However , \nthe solubility and homogeneity of Y-2EH tends to be much less than that of the Fe-2EH. The Y-\n2EH showed precipitation and was not as homogenous as Fe-2EH. As reported  by Ishibashi et al. \n[12], the Y -2EH cann ot dissolve in the solvent introduced by Azevedo  et al.[14]. As a result, we \nsuggest that synthesis of Y-2EH is not an easy and homogenous  synthesis  approach . 11 \n \n20 30 40 50 60 70 80\n\n\n\n\nO\nGGG OGGGG\nGH G1400 C \n1300 C \n1200 C \n1100 C  Intensity (a.u. )\n2 Theta (degree)1000 C YG\nH\nOG\nGG\n  \nFigure 3. XRD pattern of the YIG powder annealed  at 1000 -1400 °C. Assignment of diffraction peaks are \nindicated as following: G: YIG, O: Y IP, H: α -Fe2O3, Y: Y2O3  \nFigure 4 (a) show s the sintering temperature dependence of  magnetization . Parameters such as \nMS, H c and M r are shown in Figure 4 (b,c). The MS of the powders sintered at 1000 -1400 °C \nwere 9  to 13 emu/g, and a maximum value of 13.7 emu/g was observed for the powder sintered \nat 1200°C . From the XRD results, we observed  that the intensity of garnet phases around 2θ=32, \n45° are strongest  at 1000 -1400° C, which  can be deduced that  the magnetic behavior of sintered \npowders was strongly  determined  by the garnet phases due to the weak ferromagnetic properties \nof the α-Fe2O3 and Y IP phases. The magnetic result s is similar to the magnetic result s report ed \nby Lee et al. [10]. In the range of 1000 -1400°C, the H c and Mr decrease whereas the MS show s a \nrise. The decrease in Mr and H c versus  the increase in MS are explained due to  an increase in the \nparticle size  [31]. \n 12 \n \n-10 -8 -6 -4 -2 0 2 4 6 8 10-15-10-5051015\n-0.1 0.0 0.1-0.10.00.1M (emu/g)\nH (kOe) 1000C\n 1100 C\n 1200 C\n 1300 C\n 1400 C\nM (emu/g)\nH (kOe) \n1000 1100 1200 1300 140091011121314 \nsintering temperature ( C)Ms (emu/g)\n1020304050607080\n Hc (Oe)\n1000 1100 1200 1300 14001.41.61.82.02.22.42.62.83.03.2 \nsintering temperature ( C)Mr (emu/g)\n91011121314\n Ms (emu/g)A\n \nFigure 4. (a) Room temperature magnetization hysteresis loops of powders sintered at 1000 -1400 °C, (b) Variation \nof MS, Hc, and (c) Mr of YIG powders as a f unction of sintering temperature  \n4-Conclusion : \nMetallo -organic precursors of yttrium and iron metal -carboxylates were  synthesized and the \nchemistry and physics related  to various fabrication  steps were investigated. The metallo -organic \n(a) \n(b) \n (c) 13 \n compound  in work  can be dissolved in proper  solvent s such as  toluene and xylene with the  GAA \nused as an additi ve, to achieve the desired stoichiometry for preparing  the YIG powder. \nCrystallization and magnetic behavior of the YIG was studied. It is observed that the Y-2EH \nshow s precipitation and is not as stable as Fe-2EH and also Y-2EH is not homogenously \nsynthesized . Our results can be valuable to revive useful materials for chemical solution \nprocessing of YIG family thin films . \nAcknowledgments  \nS.M. Mohseni acknowledges support from Iran Scie nce Elites Federation (ISEF),  Iran \nNanotechnology Initiative Council (INIC) and Iran’s National Elites Foundation (INEF)  \nReferences  \n1. Kajiwara, Y., et al., Transmission of electrical signals by spin -wave interconversion in a magnetic insulator.  Nature, \n2010. 464(728 6): p. 262.  \n2. 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Duta, Photocatalytic zinc oxide thin films obtained by surfactant assisted spray pyrolysis \ndeposition.  Applied Surface Science, 2014. 306: p. 80 -88. \n24. Denayer, J., et al., Surfactant -assisted ultrasonic spray pyrolysis of nickel oxide and lithium -doped nickel oxide thin \nfilms, toward electrochromic applications.  Applied Surface Science, 2014. 321: p. 61 -69. \n25. Han, J. -P., J. Gu, and T. Ma, SrBi2Ta2O9 (SBT) thin films prepared by electrostatic spray.  Integrated Ferroelectrics, \n1997. 14(1-4): p. 229 -235.  \n26. Xue, S., W. Ousi -Benomar, and R. Lessard, Α-Fe2O3 thin films prepared by metalorganic deposition (MOD) from Fe (III) \n2-ethylhexanoate.  Thin Solid Films, 1994. 250(1-2): p. 194 -201.  \n27. Apblet t, A.W., et al., Metal organic precursors for Yttria.  Phosphorus, Sulfur, and Silicon and the Related Elements, \n1994. 93(1-4): p. 481 -482.  \n28. Nguyet, D.T.T., et al., Temperature -dependent magnetic properties of yttrium iron garnet nanoparticles prepared b y \ncitrate sol –gel. Journal of Alloys and Compounds, 2012. 541: p. 18 -22. \n29. Vajargah, S.H., H.M. Hosseini, and Z. Nemati, Preparation and characterization of yttrium iron garnet (YIG) \nnanocrystalline powders by auto -combustion of nitrate -citrate gel.  Jour nal of Alloys and Compounds, 2007. 430(1): p. \n339-343.  \n30. Schneller, T. and D. Griesche, Carboxylate Based Precursor Systems , in Chemical Solution Deposition of Functional \nOxide Thin Films . 2013, Springer. p. 29 -49. \n31. Moreno, E., et al., Preparation of narrow size distribution superparamagnetic γ -Fe2O3 nanoparticles in a sol− gel \ntransparent SiO2 matrix.  Langmuir, 2002. 18(12): p. 4972 -4978.  \n \n \n \n "    },    {        "title": "2404.16182v1.Optomagnetic_forces_on_YIG_YFeO3_microspheres_levitated_in_chiral_hollow_core_photonic_crystal_fibre.pdf",        "content": "Optomagnetic forces on YIG/YFeO3 microspheres levitated in chiral hollow-core photonic crystal fibre SOUMYA CHAKRABORTY,1,2 GORDON K. L. WONG,2 FERDI ODA,3,4 VANESSA WACHTER,5,2 SILVIA VIOLA KUSMINSKIY,5,2 TADAHIRO YOKOSAWA6, SABINE HÜBNER,6 BENJAMIN APELEO ZUBIRI,6 ERDMANN SPIECKER,6 MONICA DISTASO,3,4 PHILIP ST. J. RUSSELL,2 AND NICOLAS Y. JOLY1,2  1Department of Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg, 91058 Erlangen, Germany 2Max Planck Institute for the Science of Light, Staudtstraße 2, 91058 Erlangen, Germany 3Institute of Particle Technology, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstraße 4, 91058 Erlangen, Germany 4Interdisciplinary Center for Functional Particle Systems, Friedrich-Alexander-Universität Erlangen-Nürnberg, Haberstraße 9a, 91058 Erlangen, Germany 5Institute for Theoretical Solid-State Physics, RWTH Aachen University, 52074 Aachen, Germany 6Institute of Micro- and Nanostructure Research and Center for Nanoanalysis and Electron Microscopy, Friedrich-Alexander-Universität Erlangen-Nürnberg, Interdisciplinary Center for Nanostructured Films, Cauerstraße 3, 91058 Erlangen, Germany *Corresponding author: soumya.chakraborty@mpl.mpg.de / nicolas.joly@fau.de Received XX Month XXXX; revised XX Month, XXXX; accepted XX Month XXXX; posted XX Month XXXX (Doc. ID XXXXX); published XX Month XXXX  We\texplore\ta\tmagnetooptomechanical\tsystem\tconsisting\tof\ta\tsingle\tmagnetic\tmicroparticle\toptically\tlevitated\twithin\tthe\tcore\tof\ta\thelically\ttwisted\tsingle-ring\thollow-core\tphotonic\tcrystal\tfibre.\tWe\tuse\tnewly-developed\tmagnetic\tparticles\tthat\thave\ta\tcore\tof\tantiferromagnetic\tyttrium-ortho-ferrite\t(YFeO3)\tand\ta\tshell\tof\tferrimagnetic\tYIG\t(Y3Fe5O12)\tapproximately\t50\tnm\tthick.\tUsing\ta\t632.8\tnm\tprobe\tbeam,\twe\tobserve\toptical-torque-induced\trotation\tof\tthe\tparticle\tand\trotation\tof\tthe\tmagnetization\tvector\tin\tpresence\tof\tan\texternal\tstatic\tmagnetic\tfield.\tThis\tone-of-a-kind\tplatform\topens\ta\tpath\tto\tnovel\tinvestigations\tof\toptomagnetic\tphysics\twith\tlevitated\tmagnetic\tparticles.\t\tThe\toptical\ttweezer\ttechnique\thas\trevolutionized\tour\tability\tto\ttrap\tand\tmanipulate\tmesoscopic\tparticles.\t[1–3]\tMagnetic\tfields\tprovide\tan\tadditional\ttool\tfor\tmanipulating\ttweezered\tmagnetic\tparticles.\t[4]\tStrong\t trapping\t of\tmicro/nanoparticles\tin\tfree-space\trequires\ta\ttightly-focused\tlaser\tbeam,\twhose\tdepth\tof\tfocus\tis\tlimited\tby\tthe\tRayleigh\tlength;\t to\t maintain\t a\t linear\t trap\t over\t long\t distances,\t the\ttrapping\tbeam\tmust\tbe\tboth\ttightly\tfocused\tand\tdiffraction-free,\t which\t are\t conflicting\t requirements.\t[5]\tHollow-core\tphotonic\tcrystal\tfibre\t(HC-PCF)\tuniquely\tprovides\ta\tmeans\tof\tachieving\ta\t linear\t trap\t[5–8],\tlight\tbeing\tconfined\t to\tthe\thollow\tcore\t either\t by\ta\tphotonic\t bandgap\t[9]\tor\t by\t anti-resonant\t reflection.\t[10]\tThe\thollow\t core\talso\tprovides\t a\tprotected\tenvironment\tthat\tcan\t be\tadjusted\tthrough\tthe\taddition\tof\tgases\tor\tliquids,\tor\tby\tevacuation.\t\t[11,12]\tThis\tis\tan\tasset\tin\tstudies\tof\tthe\teffects\tof\tparticle\tbirefringence\ton\tthe\toptical\tforces\t[13]\tand\tin\tmore\tapplied\texperiments\tsuch\tas\tliving\t cell\t delivery\t[14]\tor\t thermal\t sensing\t using\t doped\tparticles.\t[15]\t\tIn\t this\t paper\twe\treport\toptical\t trapping\t of\tmagnetic\tmicroparticles\tand\tinvestigate\ttheir\tresponse\tto\tan\texternal\tmagnetic\tfield.\tWe\tfocus\ton\tparticles\tformed\tfrom\tyttrium-iron-garnet\t(YIG),\twhich\tis\ta\tdielectric\twith\ta\tstrong\tmagnetic\tresponse.\t[16]\tAlthough\tit\tis\ttransparent\tin\tthe\tinfrared,\tits\trelatively\t high\trefractive\t index\t(2.2\tat\t 1064\t nm)\t makes\tit\tchallenging\t to\t trap\t optically.\t[17,18]\tThe\ttypical\t core\tdiameter\tof\t HC-PCF\tis\t a\t few\t tens\t of\t µm,\t so\t that\tµm-sized\tparticles\tcan\tbe\tconveniently\taccommodated.\t[5,7]\tAlthough\tit\tis\tpossible\tto\ttrap\tsmaller\t(~100\tnm-scale)\tparticles,\tthe\toverlap\tbetween\tlight\tand\t particle\tis\t very\t small,\t making\texperiments\tdifficult.\tVarious\ttechniques\thave\tbeen\tused\tto\tsynthesize\tnm-scale\tYIG\t particles,\t such\t as\t co-precipitation,\t[19]\tsol-gel,\t[20]\tmicro-emulsion,\t[21]\tmicrowave\tirradiation,\t[22]\tand\ttraditional\t solid–state\treaction\t methods.\t[23] Although current\tsynthesis\ttechniques\tpermit\ta\tdegree\tof\tcontrol\tof\tparticle\tsize,\tthey\tdo\tnot\tyet\tallow\tcontrol\tof\tparticle\tshape\tand\tsurface\troughness,\twhich\t is\t irregular\t and\t unpredictable.\tThis\t prevents\t the\tformation\t of\t high-Q\t internal\t optical\t resonances\twhich\t are\tneeded\tto\tenhance\tthe\tweak\tphoton-magnon\tcoupling.\t[4]\t\tThe\tparticles\tused\tin\tthe\texperiments\thave\ta\thybrid\tcore-shell\tstructure,\tthe\tshell\tbeing\ta\tlayer\tof\tcubic\tferrimagnetic\tYIG\t (Y3Fe5O12)\t approximately\t50\tnm\t thick,\t and\t the\tcore\ta\tsphere\t of\tbiaxial\tantiferromagnetic\t yttrium-ortho-ferrite\t(YFeO3).\tThey\tare\tpropelled\tinto\ta\tchiral\t[24]\tsingle-ring\tHC-PCF\t[25,26]\tusing\ta\tdual-beam\ttrapping\tscheme\t[7].\tSince\tthe\thybrid\tparticles\t are\ton\t average\toptically\tbiaxial,\tthey\texperience\ta\ttorque\twhen\tsubject\tto\ta\tlinearly\tpolarized\tlight\tbeam,\tcausing\tthem\tto\talign\talong\tthe\telectric\tfield\tof\tthe\tlight.\t\tSubsequent\t application\t of\t a\t static\t external\t magnetic\t field\tresults\tin\tanisotropic\tchanges\tin\tmagnetic\tpermeability\tthat\tin\tturn\tcause\tanisotropic\tchanges\tin\tcomplex\trefractive\tindex\t(the\tVoigt\teffect\t[27–29]) that are probed using HeNe laser light at 632.8 nm. \n\tParticle\tsynthesis\twas\tcarried\tout\tin\ttwo\tstages\t(Fig.\t1a\tand\tmethods\t in\tSM).\tFirst,\t the\t yttrium\t and\t iron\t molecular\tprecursors\t were\t solubilized\t in\t N,\tN-dimethylformamide\t(DMF)\t and\t after\t the\t addition\t of\t the\t nonionic\t surfactant\tsorbitan\tmonooleate-\tSpan\t80,\tthe\tsolution\twas\taged\tunder\tsolvothermal\tconditions\tat\t200°C\tfor\t6\thours.\tThe\tsurfactant\tSpan\t80\tis\twell\tknown\tto\tform\tmicelles\tin\tpolar\tsolvents\tand\thas\tbeen\tused\tto\tsynthesize\tporous\talumina\tmicroparticles\twith\tinterconnected\tpores,\t[30]\tTiO2\tmicrospheres,\t[31]\tgold\tnanoparticles,\t[32]\tBaTiO3,\t[33]\tand\tto\tcontrol\tthe\tgrowth\tof\tCaCO3.\t[34]\tThe\t solid\tobtained\twas\tthen\tisolated,\t washed,\tdried,\tresulting\tin\tamorphous\tspherical\tparticles.\tIn\ta\tsecond\tstep\tthe\t particles\t were\tcalcined\tin\t air\t at\t temperatures\tbetween\t700°C\tand\t1000°C\tfor\t8\thours\t(Fig.\t1a).\tThe\tfinal\tparticles\t were\t characterized\t by\tpowder\tx-ray\t diffraction\t(XRD),\tscanning\tand\ttransmission\telectron\tmicroscopy\t(SEM\tand\t TEM),\thigh\tresolution-TEM,\tselected\tarea\telectron\tdiffraction\t(SAED)\tand\tenergy-dispersive\tX-ray\tspectroscopy\t(EDS).\tIn\tFig.\t1b\tpowder\tdiffraction\tof\tthe\tparticles\tcalcined\tat\t 1000°C\treveals\t the\t co-presence\t of\t three\t phases:\torthorhombic\tYFeO3\t(ICSD\t43260),\tcubic\tYIG\t(ICSD:14342),\tand\t minor\t quantities\t of\t cubic\t phase\t Y2O3\t(ICSD:33648).\t It\twas\tobserved\tthat\tby\tincreasing\tthe\tcalcination\ttemperature,\tthe\tproportion\tof\t the\t YIG\t phase\t increases\t (see\t Methods\tin\tSM).\tSEM\t characterization\t of\t particles\t isolated\t at\t 1000°C\tshowed\tthat\tthe\tmajority\tof\tthem\thad\ta\tspherical\tshape\twith\tmedian\tdiameter\tx50,0\tof\t1.33±0.5\tµm\t(Fig.\t1b,\tc).\tIndividual\tparticles\tappear\tto\thave\ta\trough\tand\tporous\tsurface\t(Fig.\t1b,\tc).\tWe\tanalysed\tparticles\tcalcined\tat\t700°C\tby\tspin-coating\t\nthem\ton\tto\ta\tstandard\tsilicon\twafer\tand\tsputtering\tthem\twith\t200\tnm\tthick\tgold\tlayer.\tWe\tthen\tcut\tthem\topen\tusing\ta\tGa\tfocused\tion\tbeam\t(FIB)\tin\ta\tZeiss\tNVision\t40.\tThe\tFIB\tprobe\twas\tset\tto\t30\tkV-80\tpA.\tFIB-cut\tSEM\timages\tof\tthe\tcut-open\tparticles\treveal\ta\tcomplex\t morphology\t(Fig.\t1d).\t Their\tstructure\tis\tegg-like,\t consisting\tof\tan\t external\tshell\tand\ta\tporous\tinterior\tcore.\tThe\tcomposition\tof\tthe\tdifferent\tparts\tof\tthe\tparticles\twere\tanalysed\tusing\tEDX\tin\tconjunction\twith\tSAED\tand\tHR-TEM,\trevealing\tthat\tthe\tthin\tshell\tis\tmade\tof\tYIG\t and\tthe\tcore\t contains\t YFeO3\t(Fig.\t1g-n).\t There\t is\t no\tspecific\t crystal\t orientation\trelationship\t between\t the\t shell\tand\tcore.\tIn\tthe\toptical\texperiments\twe\ttrapped\tparticles\tcalcined\tat\t700°C\tin\ta\tconventional\tdual-beam\ttrap\t(Fig.\t2).\tLight\tfrom\ta\tcontinuous-wave\t ytterbium\tfibre\tlaser\t (Keopsys\t CYFL-KILO)\tdelivering\t3W\tat\t1064\tnm\twas\tused\tfor\tthe\ttrapping\tbeams.\tThe\tlight\twas\tsplit\tat\ta\tpolarizing\tbeam\tsplitter\t(PBS)\tand\tcoupled\tinto\tthe\tLP!\"-like\tmode\tat\tboth\tends\tof\ta\t7-cm-long\tchiral\t“single-ring”\tHC-PCF\twith\tcore\tdiameter\t44\tμm.\tThis\t fibre\thas\tweak\t circular\tbirefringence\t𝐵#~10$%\tand\tis\toptically\tactive,\ti.e.,\tthe\telectric\tfield\tof\ta\tlinearly\tpolarized\tsignal\trotates\tslowly\twith\tdistance\twhile\tremaining\tlinearly\tpolarized,\t travelling\t around\t the\t equator\t of\t the\t Poincaré\tsphere.\t[24]\tOver\tthe\t7-cm\tlength\tof\tthe\tfibre\tthis\trotation\tis\t\nFig.\t1:\tFabrication\tand\tcharacterization\tof\tmagnetic\tYIG\tmicron-sized\tparticles.\t(a)\tSchematic\tof\tthe\tsynthesis\tof\tthe\tparticles.\t(b,c)\tSEM\tof\tthe\tfabricated\tparticles\tcalcined\tat\t1000°C\tand\t700°C\trespectively,\t(d)\tFIB-cut\tof\ttwo\tparticles\tcalcined\tat\t700°C\trevealing\ta\tconsistent\tcore-shell\t structure.\t (e)\tPowder\tx-ray\t diffraction\t (XRD)\t for\t particles\t fabricated\tat\t1000°C\t revealing\t co-presence\t of\t three\t different\tcrystalline\tmaterials.\t(f)\tHR-TEM\timage\tof\ta\tsingle\tparticle\tcalcined\tat\t700°C.\t(g)\tHigh-resolution\tzoom\tof\tthe\tshell\tregion\tin\tthe\tHR-TEM\timage\tand\t(h)\tthe\tcorresponding\tfast\tFourier\ttransform\t(FFT)\trevealing\tthe\tcubic\tphase\tof\tYIG.\t(i)\tSAED\tpattern\tof\tthe\tshell\tdemonstrating\tcubic\tphase\tof\tYIG,\t(j)\tSAED\tof\tthe\tcore\tshowing\torthorhombic\tphase\tof\tYFeO3. however\tvery\tsmall\tso\tcan\tbe\tneglected.\tPrecise\tpreservation\tof\tlinear\tpolarization\tstate\tis\tessential\tsince\twe\twish\tto\tprobe\tsmall\tmagnetically-induced\tanisotropic\tchanges\tin\tcomplex\tdielectric\t constant.\tThe\tfibre\twas\tmounted\tin\ta\t V-groove\tinside\t a\t custom-built\tlow-pressure\tchamber\t with\t a\ttransparent\t acrylic\t lid\t so\t as\t to\tallow\taccess\t to\t light\tside-scattered\tby\tthe\tparticle\tas\tit\tis\tpropelled\talong\tthe\tfibre.\t\t\n Fig.\t2:\tSchematic\t of\t the\t experimental\t setup.\t HWP:\t half-wave\tplate,\t PBS:\t polarizing\t beam\tsplitter,\t DM:\t dichroic\t mirror,\t NF:\tnotch\t filter\t at\t 1064\tnm,\t BP:\t 1\tnm\t bandpass\t filter\tcentred\tat\t632.8\tnm.\tThe\tdistance\tbetween\tthe\tend\tof\tthe\tmagnet\tand\tthe\tfibre\tis\t𝑑.\tInset:\t Scanning\t electron\t micrograph\t showing\t the\tcross-section\tof\tthe\tsingle-ring\tHC-PCF.\tThe\tcore\tdiameter\tof\tthe\tfiber\t is\t 44\t μm\t and\t the\t outer\t diameter\t ~270\t μm.\t The\t motion\tsensor\tis\ta\tquadrant\tphotodiode. Particles\twere\t launched\tusing\t the\taerosol\tmethod.\t[7]\tThe\tparticles\tare\tfirst\tdispersed\tin\ta\t50-50\tmixture\tof\ta\tspan-80\tand\twater.\tA\tmedical\tnebulizer\twas\tused\tto\tproduce\tsmall\taerosol\tdroplets\twhich\twere\tthen\tdelivered\tthrough\tan\tinlet\tplaced\tabove\tthe\tfibre\tinput\tface\tuntil\tone\tof\tthe\tparticles\twas\ttrapped\tin\tfront\tof\tthe\tfibre.\tOnce\ttrapped\tat\tthe\tentrance\tof\tthe\t HC-PCF,\ta\t particle\t could\t be\t held\tthere\tlong-term\t and\tpropelled\tinto\tthe\tcore\tby\tmomentarily\tlowering\tthe\tpower\tof\t the\tcounter-propagating\ttrapping\tbeam.\tThe\ttrapping\tsuccess\t rate\twith\t these\tparticles\twas\tclose\t to\t100%.\tThe\tmotion\tof\t the\t bound\t particle\t along\t the\tfibre\twas\timaged\tusing\ta\thigh-speed\tcamera\t(Mikrotron\tEoSens\tmini2)\tplaced\tabove\t the\t chamber\tand\ta\t quadrant\t detector\t (Thorlabs\tPDQ30C)\tconnected\tto\tan\toscilloscope\t(PicoScope\t3406B).\tA\tmagnet\twas\tmounted\ton\ta\ttranslation\tstage\tso\tas\tto\tallow\tthe\tmagnetic\tfield\tstrength\tto\tbe\tvaried.\t\t Fig.\t3:\t(a)\t Snapshot\t of\t an\t optically\t trapped\t magnetic\t particle\tinside\tthe\tcore\tof\ta\tHC-PCF\tcaptured\twith\ta\thigh-speed\tcamera.\t(b)\t Spectrum\t of\t the\t damped\t mechanical\t motion\t of\t the\t bound\tmagnetic\tparticle\tat\ta\tpressure\tof\t2\tmbar.\tThe\tlaser\tpower\tis\t3\tW.\t(c)\t Measured\t spectral\t linewidth\t (FWHM)\t as\t a\t function\t of\t the\tenvironment\t pressure\tat\ta\t fixed\t laser\t power\t of\t 3\tW.\t (d)\tMeasured\t central\tfrequency\t as\t a\t function\t of\t laser\t power\t at\t6\tmbar\t pressure.\t The\t red\t line\tis\tthe\t theoretical\t prediction\taccording\tto\tEq.\t(1).\tFirst,\twe\ttested\tthe\t limits\t of\t the\t levitated\t system\t by\tevacuating\tthe\tchamber\twith\ta\tparticle\talready\ttrapped\tin\tthe\tfibre\tcore,\tas\tshown\tin\tFig.\t3a.\tAt\tpressures\tbelow\t~1\tmbar\tthe\tparticle\tescaped\tfrom\tthe\toptical\ttrap\tand\twas\tlost,\twhich\twe\tattribute\tto\tthe\tonset\tof\tthe\tballistic\tregime\tcaused\tby\tthe\tincreased\tmolecular\tmean-free\tpath.\t[35]\tIn\tthe\ttransverse\tdirection\tthe\ttrapped\t particle\tbehaves\t like\t a\tdamped\tmechanical\toscillator,\tdriven\tby\tBrownian\tmotion.\t[12]\tFrom\ttime-domain\t data\t recorded\t with\t the\t position-sensitive\tquadrant\tdetector\twe\textracted\tthe\tLorentzian\tspectrum\tof\tthe\tparticle\tmotion\t(Fig.\t3b).\tAs\tthe\tgas\tpressure\tdecreases,\tthe\t viscosity\t falls,\t and\t the\tlinewidth\tnarrows.\t The\trelationship\t between\t the\t spectral\t linewidth\tΓ\tand\t the\t air\tpressure\t𝑝\tfollows\tthe\tKnudsen\trelation:\tΓ=𝛾𝑚=12𝜋𝑅𝑚𝜂!21+𝐾&(𝑝)6𝛽\"+𝛽'𝑒$(!/*\"(,)9:;\t\t\t\t\t\t(1) where\t𝛾\t\tis\tthe\tdamping\tcoefficient\tcaused\tby\tviscosity,\t2𝑅\tis\ta\tcharacteristic\tlength\t(the\tparticle\tdiameter),\tand\tm\tis\tthe\tparticle\tmass\t(densities\tof\tYIG\tand\tYFeO3\tare\t5.11\tand\t5.47\tg/cm3\trespectively)\t and\t𝐾&(𝑝)=\t66×10$.(𝑝𝑅)⁄\tis\t the\tKnudsen\t number,\t𝜂!=18.1\tµPa×s\tis\t the\t viscosity\t of\t air\t at\tatmospheric\tpressure\tand\t𝛽\"=1.231,\t𝛽'=0.469\tand\t\t𝛽/=1.178\tare\tdimensionless\tconstants.\t[7]\tFigure\t 3(c)\tplots\tthe\tmeasured\tspectral\t linewidth\tas\t a\tfunction\tof\tpressure,\tand\tthe\tred\tline\tis\ta\tfit\tto\tEq.\t(1).\tThe\ttrap\tstiffness,\t which\t governs\t the\t resonant\t frequency,\tis\tcontrolled\tby\tthe\ttrapping\tlaser\tpower\t𝑃.\tAt\ta\tfixed\tpressure\t(6\tmbar\tin\tthe\texperiment)\tthe\tresonant\tfrequency\tincreases\tlinearly\twith\tthe\tlaser\tpower,\tas\texpected\t(Fig.\t3d).\t\nCrystalline\tYFeO3\tis\torthorhombic\tand\tbiaxial,\tdisplaying\toptical\t birefringence,\twhich\t means\t that\t the\t linearly\tpolarized\t trapping\t beam\t can\t be\t used\t as\t an\t optical\tspanner,\t[13]\tpermitting\t measurements\t to\t be\t made\t as\t a\tfunction\tof\tparticle\torientation.\tThe\tshell\tof\tthe\tparticles\tis\tformed\t from\tYIG,\t which\tis\t cubic\t and\t isotropic,\t becoming\toptically\tbiaxial\twhen\ta\tmagnetic\tfield\tis\tapplied\tparallel\tto\tthe\t(110)\tcrystallographic\tplane\t(for\t details\t refer\t to\t SM).\tBoth\tcrystals\tare\tstrongly\tabsorbing\tat\t632.8\tnm,\toffering\ta\tsimple\t means\t of\tprobing\tmagnetically\tinduced\tchanges\t in\tcomplex\t refractive\t index\tby\tmonitoring\t the\t power\t and\tpolarization\tstate\tof\tthe\ttransmitted\tprobe\tlight\t[27,29,36].\t\tThe\ton-axis\tmagnetic\tflux\tof\t the\tNdFeB\t permanent\tmagnet\t(N35,\t cross-section\t4×4\tmm)\t used\t in\t the\texperiments\tis\tplotted\tin\tFig.\t6\tas\ta\tfunction\tof\tthe\tdistance\tfrom\tthe\tmagnet’s\tend-face.\tThe\tmagnet\twas\tplaced\twith\tits\tN-S\taxis\toriented\tperpendicular\tto\tthe\tfibre\taxis\tand\tcentred\ton\tthe\ttrapped\tparticle\t(Fig.\t1),\tand\ta\tmotorized\ttranslation\tstage\twas\tused\tto\tvary\tthe\tdistance\t𝑑\tbetween\tthe\tmagnet\tand\t the\tparticle.\tProbe\t light\twas\t provided\t by\ta\tlinearly\tpolarized\tHeNe\tlaser\t(2\t mW,\t 632.8\t nm).\t The\ttransmitted\ttrapping\tbeam\tlight\twas\tfiltered\tout\tusing\ta\tcombination\tof\tdichroic\t mirror,\t1\tnm\t bandpass\t filter\tcentred\tat\t632.8\tnm,\tand\t2\tnm\tnotch\t filter\tcentred\tat\t1064\tnm\t (Fig.\t1).\tIn\tthe\texperiments,\tboth\tthe\tpower\tand\tthe\tpolarization\tstate\tof\tthe\tprobe\tbeam\twas\tmonitored.\t\n Fig.\t4:\tTransmitted\tprobe\tpower\tas\tthe\tparticle\tis\trotated\tinside\tthe\tHC-PCF\tand\tsubject\tto\ta\tconstant\tmagnetic\tfield\tof\t29\tmT.\tThe\tred\tdotted\tcurve\tis\ta\theuristic\tfit\tto\tEq.\t(3).\t\tThe\t opto-magnetic\t response\twas\tfirst\tinvestigated\tby\tapplying\ta\tconstant\tmagnetic\tfield\t(B\t=\t29\tmT)\tand\trotating\tthe\tparticle\tby\trotating\tthe\ttrapping\tbeam\tpolarization\t[13].\tThe\ttransmitted\tprobe\tpower\twas\tdirectly\tmonitored\tusing\tboth\ta\tpolarimeter\tand\ta\tlock-in\tamplifier.\tFigure\t4\tplots\tthe\ttransmitted\tprobe\tpower\tas\ta\tfunction\tof\tthe\torientation\tof\tthe\ttrapping\telectric\tfield\t𝜃,\twhere\t𝜃=0°\twhen\tthe\ttrapping\tand\tprobe\tbeams\tare\tco-polarized.\t\tFor\teach\tvalue\tof\t𝜃\twe\tmade\trepeated\tmeasurements\tof\tthe\ttransmitted\tpower\tand\tevaluated\tthe\tmean\t(blue\tdot)\tand\tstandard\tdeviation\t(error\tbar).\tOver\t180°\tthe\tdata\tshows\ttwo\tdistinct\tpeaks,\twhich\twe\tattribute\tto\tthe\tcomplex-valued\tbiaxial\trefractive\tindex\tof\tthe\tparticle.\t\tIn\tthe\tcase\tof\tpure\tYIG\tcrystal,\tassuming\tits\tmagnetization\tvector\tpoints\tin\tthe\t(110)\tplane,\tthe\timaginary\tpart\tof\tthe\tdielectric\tsusceptibility\tcan\tbe\twritten\tin\tthe\tform\t[27,28]:\tΔ𝜒0(𝜃)=𝑀1'𝑛2Q𝑔33+Δ𝑔16(3+2cos2𝜃+3cos4𝜃)U\t\t\t(2)\twhere\t𝜃\tis\t angle\t between\t the\t magnetization\t vector\t𝑀VV⃗\tand\t[001]\tcrystal\taxis\t(see\tSM\tfor\tdetail),\t𝑛2\tis\tthe\twavelength-dependent\treal\tpart\tof\tthe\trefractive\tindex\tof\tYIG,\t𝑀4\tis\tits\tsaturation\t magnetization,\tΔ𝑔=𝑔\"\"−𝑔\"'−2𝑔33,\tand\t𝑔\"\",𝑔\"'\tand\t𝑔33\tare\tcomplex\tnumbers\trepresenting\tthe\tnon-vanishing\ttensor\telements\tof\tthe\tdielectric\ttensor\tinduced\tby\ta\tmagnetic\tfield\tand\tcausing\tbiaxial\tlinear\tbirefringence\tand\t(in\tthe\tvisible)\tdichroism.\tSince\tin\tour\tcase\tthe\tparticle\tis\ta\tcomplex\thybrid\tof\tYIG\tand\t YFeO3,\t the\t system\t cannot\t be\t so\t easily\t modelled.\tMoreover,\twhen\ta\tnew\tparticle\tis\tlaunched\tinto\tthe\ttrap,\tthe\tinitial\t orientations\t of\t its\t optical\t axes\t as\t well\t as\t the\tmagnetisation\taxis\tare\tunknown.\tHowever,\ta\theuristic\tfit\tto\tthe\tpower\tmeasurement\tcan\tbe\tmade\tusing\ta\tsimilar\tfunction\twith\tan\tadded\tphase\tshift\tof\t𝜓:\t𝑃(𝜃)∝a\t[1+bcos2(𝜃−𝜓)+ccos4(𝜃−𝜓)]\t\t\t\t\t\t\t\t\t(3)\twhere\tthe\tphase\t𝜓=−50°\tis\tadded\tto\tthe\tangle\t𝜃,\twhich\tin\tour\texperiment\tis\tthe\tangle\tbetween\tprobe\tand\tpump\tbeam\tpolarization.\tWe\tnote\tthat\t𝑎\trepresents\tthe\taverage\tpower\tof\tour\t dataset\t which\t is\t 70.8\tµW.\tThe\tother\tcoefficients\tare\trespectively\t𝑏=2.12×10$/\tand\t𝑐=4.1×10$/.\tThe\tEq.\t(3)\tqualitatively\tfits\tto\tthe\tdata,\tas\tseen\tin\tthe\tred\tdashed\tcurve\tin\tFig.\t4.\tThe\tindividual\tmagnetooptomechanic\tresponse\tof\teach\tparticle\tis\tslightly\tdifferent\tdue\tto\tits\tinitial\torientation,\tthough\tthey\tall\tfollow\tthe\tsame\tgeneral\ttrend,\texhibiting\ttwo\tmaxima\t (Fig.\t4).\tThe\t first\t peak\t occurs\t at\t𝜃≃50°\t(Fig.\t 4),\twhich\tis\tin\treasonable\tagreement\twith\tthe\tvalues\tfor\tpure\tYIG\t(50°)\tand\tYFeO3\t(45°)\t(see\tSM\tfor\tmore\tdetails).\t\t[37]\t\n \n\tFig.\t5:\tTransmitted\tpower\t(blue\taxis)\tas\tthe\tmagnet\tis\tmoved\tinwards\ttowards\tthe\tparticle.\tThe\tangle\tbetween\tthe\torientation\tof\t the\t linearly\t polarized\t probe\t beam\tand\t the\t applied\t external\tfield\t B\t is\t 19°\t in\t (a)\t and\t 45°\t in\t (b).\tThe\t red\t dashed\t line\t is\t a\theuristic\tfitting\tusing\tEq.\t(3)\tNext,\t we\t kept\t the\t particle\t stationary\t and\t moved\t the\tmagnet\t inwards,\t keeping\t the\t angle\t between\t the\t magnetic\tfield\tand\tthe\tprobe\tbeam\tpolarization\t(𝐸,2567)\t\tfixed\tto\teither\t19°\tor\t45°(Fig.\t5\tinset).\tIn\tboth\tcases,\tthe\tmagnetic\tfield\tis\torthogonal\tto\tthe\tpolarization\tof\tthe\ttrapping\tbeam\t(Fig.\t2).\tAt\t𝑑=23\tmm,\t the\t magnetic\t field\t𝐵V⃗\tat\t the\t particle\t is\t very\tweak\t and\t the\t magnetization\t vector\t𝑀VV⃗\tis\t unaffected.\t As\t the\tmagnet\tapproaches\tthe\tparticle,\t𝑀VV⃗\tgradually\trotates\tuntil\tit\taligns\t almost\t parallel\t to\t𝐵V⃗\tat\t𝑑=3\tmm.\t At\t present,\t we\tcannot\t distinguish\t between\t the\t physical\t rotation\t of\t the\tparticle\tfrom\tthe\trotation\tof\tonly\tthe\tmagnetization\tvector,\thowever\tthey\twill\tgive\trise\tto\tsame\texperimental\tresult.\tFigure\t5(a)\t shows\t the\t variation\t of\t the\t transmitted\t probe\tbeam\t power\t with\t respect\t to\t𝑑\twhen\t the\t angle\t between\tmagnetic\tfield\tand\tthe\tprobe\tbeam\tpolarisation\tis\t19°\tand\tFig.\t5(b)\tshows\tthe\tvariation\twhen\tthis\tangle\tis\t45°\tas\tshown\tin\tthe\tinsets\tof\tthe\tfigures.\tThe\tangle\t𝜃\tby\twhich\tmagnetization\tvector\t rotates\t depends\t inversely\t on\t𝑑\tallowing\t us\t to\theuristically\tfit\tthe\texperimental\tdata\twith\t𝑃(𝜃).\tFor\tfitting\tof\tthe\tdata\tin\tFig.\t5(a),\tthe\tadded\tphase\tshift\t𝜓\tto\t𝑃(𝜃)\tin\tEq.\t(3)\tis\t𝜓=90°\tand\t𝑎=96\tµW,\t𝑏=5.73×10$/\tand\t𝑐=0.098.\tFor\tFig.\t5(b),\tthe\tadded\tphase\tshift\tis\t𝜓=\t−22.5°\tand\tthe\t parameters\t𝑎=98\tµW,\t𝑏=1.53×10$/\tand\t𝑐=3.98×10$/\trespectively.\tThese\tplots\tshow\ta\tbehaviour\tsimilar\tto\tthat\tin\tFig.\t4,\tfrom\twhich\twe\tdeduce\tthat\tthe\timaginary\tpart\tof\tthe\tdielectric\tsusceptibility\tis\tbeing\tprobed\tas\ta\tfunction\tof\tthe\t rotation\t angle\t𝜃\tand\t distance\td.\tWe\t also\t observed\t that\tonce\tthe\texternal\tmagnetic\tfield\twas\tstrong\tenough\tto\talign\tthe\t magnetization\t vector\t parallel\t to\t itself,\t the\ttransmitted\tprobe\tpower\tand\tpolarisation\tstate\tno\tlonger\tresponded\tto\tchanges\t in\t magnetic\t field\t strength\t(see\t SM).\tA\t similar\tresponse\tcould\tbe\trecovered\tby\tmoving\tthe\tparticle\talong\tthe\tfibre\tout\tof\tthe\tmagnetic\tfield\tand\tthen\treturning\tit.\t\tIn\tsummary,\tspheroidal\tµm-sized\tmagnetic\tparticles\twith\ta\t~50\tnm\tshell\tof\tYIG\tand\ta\tcore\tof\tFeO3\twere\tsynthesized.\tThe\trelative\t proportion\tof\tthe\t two\tmaterials\tcould\tbe\tadjusted\tby\t running\tthe\t calcination\t process\tat\tdifferent\ttemperatures.\tThe\t particles\t could\t be\t reproducibly\t trapped\tlong-term\tin\tthe\tevacuated\tHC-PCF\tcore.\tMeasurements\twith\ta\t 632.8\t nm\t probe\t beam\t and\ta\t single\tµm-diameter\tparticle\treveal\tdetectible\tchanges\tin\tthe\ttransmitted\tpower\tand\tthe\tpolarization\tstate.\tThe\tsystem\tis\tsuitable\tfor\ta\twide\tvariety\tof\tdifferent\tapplications,\tsuch\t as\tremote\tmagnetic\t field\tsensing\t[36],\tinteractions\tbetween\t waveguide\t modes,\tand\tthe\tstudy\tof\trotational\tdegrees\tof\tfreedom\tand\tspin\twaves\tin\toptomechanically\t cooled\tresonators\t\t[38].\tThe\t reported\tresults\tsuggest\tnew\tpossibilities\tfor\texperiments\tin\tparticle-based\t magneto-optomechanical\tphysics.,\t including\tcooling\tdown\tto\tthe\t single\t quantum\t regime\t[38],\t possibly\t at\t room\ttemperature.\tMethods The\tparticles\twere\tfabricated\tin\ttwo\tsteps.\tAt\tfirst,\tan\tyttrium\tmolecular\tprecursor\t(Y(NO3)3·6H2O)\t(1.15\tg,\t3\tmmol)\tand\tan\tiron\tmolecular\tprecursor\t(Fe(acac)3)\t(1.8\tg,\t5.10\tmmol)\twere\tsolubilized\t at\t room\t temperature\t in\t 50\t mL\t N,N-dimethylformamide\t (DMF)\t and\t the\t surfactant\t sorbitan\tmonooleate\t-\tSpan\t 80\twas\tadded\t during\t stirring.\t The\t as-obtained\tsolution\twas\ttransferred\tin\ta\tTeflon\tliner\tand\taged\tin\t a\t stainless-steel\t autoclave\t at\t 200°C\t for\t 6\thours.\t Upon\tcooling\t to\t room\t temperature,\t toluene\t was\t added\t to\t the\treaction\t mixture\t to\t induce\t precipitation.\t The\t solid\t was\tisolated\tby\tcentrifugation\tand\twashed\tby\tthree\tredispersion\tand\tcentrifugation\tcycles.\tFinally,\tit\twas\tdried\tin\tair\tat\t60°C\tfor\t20\thours.\tThe\tproduct\tat\tthis\tstage\tcomprised\tamorphous\tspherical\t particles.\t The\t second\t step\t of\t the\t fabrication\tprocedure\t comprised\t a\t calcination\t process\t in\t air\t at\ttemperature\t between\t 700\t and\t 1000\t °C\t for\t 8\t hours.\t This\tsecond\t part\t leads\t to\t an\t amorphous\tto\tcrystalline\t phase\ttransition\tduring\twhich\tthe\tparticles\tbecome\tmagnetic.\tThe\tmagnetic\tfield\twas\tproduced\tby\teight\t4×4×3\tmm/\tNdFeB\tN35\tpermanent\tmagnets\tplaced\tin\ta\trow.\tThe\ton-axis\tmagnetic\tflux\tdensity\twas\tmeasured\twith\ta\tGaussmeter\tas\ta\tfunction\tof\td,\tthe\tdistance\tfrom\tthe\tend-face\tof\tthe\tmagnet\t(Fig.\t6).\t\n Fig.\t6:\tMeasured\t magnetic\t flux\t 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Turchinovich, \"Single-pulse terahertz coherent control of spin resonance in the canted antiferromagnet YFeO3 mediated by dielectric anisotropy,\" Phys. Rev. B 87, 094422 (2013). 38.  D. Grass, J. Fesel, S. G. Hofer, N. Kiesel, and M. Aspelmeyer, \"Optical trapping and control of nanoparticles inside evacuated hollow core photonic crystal fibers,\" Appl. Phys. Lett. 108, 221103 (2016).   "    },    {        "title": "1804.07023v1.Effect_of_magnons_on_interfacial_phonon_drag_in_YIG_metal_systems.pdf",        "content": "1 \n Effect of magnons  on interfacial phonon drag  in YIG/metal systems  \nArati Prakash1, Jack Brangham1, Sarah J. Watzman2, Fengyuan Yang1, Joseph P. Heremans1,2,3 \n1 Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA  \n2 Department of Mechanical Engineering, The Ohio State University, Columbus, Ohio 43210, \nUSA  \n3 Department of Materials Science and Engineering, The Ohio State University, Columbus, Ohio \n43210, USA  \n \nAbstract  \nWe examine substrate -to-film interfacial phonon dra g on typical spin Seebeck heterostructures, in \nparticular studying the effect of ferromagnetic magnons on the ph onon -electron drag dynamics at \nthe interface . We investigate  with high precision  the effect of magnons  in the Pt|YIG  \nheterostructure by designing a magnon drag thermocouple; a hybrid sample with both a Pt|YIG \nfilm and Pt|GGG interf ace accessible isothermally via a 6 nm Pt film patterned in  a rectangular U \nshape with one arm on the 250 nm YIG film and the other on GGG.  We measure t he voltage \nbetween the isothermal ends  of the U, while applying a temperature gradient parallel to the arms \nand perpendicular to the bottom connection.  With a uniform applied temperature gradient, the Pt \nacts as a differential thermocouple. We conduct temperature -dependent longitudinal thermopower \nmeasurements on this sample. Results show that the YIG interface actually decreases the \nthermopower of the film, implying that magnons impede phonon drag. We repeat the experiment \nusing metals with lo w spin Hall angles, Ag and Al, in place of Pt. We find that the phonon drag \npeak in thermopower is killed in samples where the metallic interface is with YIG. We also \ninvestigate magneto -thermopower and YIG film thickness dependence. These measurements \nconfirm our findings that magnons impede the phonon -electron drag interaction at the metallic \ninterface in these heterostructures.   2 \n  \nIntroduction  \nIn this study, we focus on the longitudinal thermopower , αxxx, of a typical spin Seebeck \nheterostructure, i.e. P t|YIG . In this configuration, a temperature bias is applied in the direction \nparallel to the direction that voltage is measured on the metal, i.e. along the direction of the \ninterface.  In comparison to an SSE measurement, in these studies we will essentia lly turn the \nsample on its side relative to the applied heat flux (see Figure 1).  \nSpecifically, we are interested in studying the influence of magnons  on the thermopower \nof the Pt|YIG  heterostructure. We s peculate that magnons at the Pt|YIG  interface coul d exert a \ndrag-like force on the electrons in the Pt, either directly or via an interaction mediated by phonons \nin YIG, and we endeavor to measure and discern this effect.  In fact, when we aim to elucidate the \nparameter space of the SSE, which arises from  magnon -phonon interactions, the question of drag \nbetween magnons, phonons and electrons in the Pt|YIG heterostructure is highly pertinent and is \nmotivated by several contemporary studies, both experimental and theoretical.  \nNonlocal drag (i.e. interfacial,  between substrate and thin film) has been studied both in \ntheory and experiment.1,2 Ref. [ 1] examines nonlocal phonon -electron drag between an insulating \nsapphire substrate and a Bi 2Te3 thin film , show ing that the  temperature dependent thermopower of \nthe Bi 2Te3 follows that of the t hermal conductivity of the sapphire substrate, which implies  that \nelectronic transport in the film is enhanced by phononic transport in the sapphire substrate \n(phonon -electron drag). This demonstrates that substrate -to-thin-film phonon -electron drag can \noccur even when the two layers have dissimilar crystal structure. From this study, we would expect \nthat there could be strong phonon -electron drag affecting the thermopower on either Pt|GGG or 3 \n Pt|YIG films. Furthermore, Ref. [ 2] predicts nonlocal magnon -magnon drag in a FM bilayer, \narising from dipolar interactions across the interface.  \nLocal drag effects (i.e. bulk, not interfacial) have also been studied: local magnon -electron \ndrag (MED), or the advective transport of electrons dragged by magnons, has been examined in \nbulk metallic ferromagnets, where a thermal gradient drives magnons and electrons along with \nphonons.3 While all metals contain drag  and diffusive contributions to their thermopower,  it was \nshown that the MED contribution actually dominated the thermopower in ferromagnetic metals \n(Fe, Co). The findings of thermopowers enhanced by magnon drag in Ref. [ 3] and of phonon drag \ndominating the thermopower even in dissimilar substrate -to-thin film heterostructures in Ref. [ 1], \ncombined with the theoretical predictions in Ref. [ 2] that magnons can also participate in such \nnonlocal, interfacial drag effects, together inspire a look at magnon effects on the interfacial \nphonon drag on the Pt|YIG heterostructure.  \nInstead of an out of plane  temperature gradient, which drive s a nonloca l spin flux across \nthe interface, we apply an in plane temperature gradient longitudinally as the driving force and \nexamine magnon  transport along  this direction. We probe these dynamics by measurements of the \nlongitudinal thermopower on the Pt film of Pt|YIG and similar  heterostructures . As per Ref. [ 1], \none would expect this thermopower as measured on the thin film to follow the thermal conductivi ty \nof the substrate (YIG or GGG) due to the interfacial phonon -electron drag. However, the question \nof interest here is what is the effect  of magnons on this phonon electron drag?  \nTo study this, we compare the thermopower of Pt films grown on ferrimagneti c YIG to that \ngrown on paramagnetic GGG , where the only difference in substrate dynamics can be attributed \npurely to magnons in the YIG.  In order t o isolate the hypothetical drag contribution from the \nmagnons in YIG into the adjacent Pt film, we design a thermocouple device using a hybrid sample 4 \n with half Pt|GGG and half Pt|YIG(250nm)|GGG  (see Figure 2) . With a uniform applied \ntemperature gradient, t he Pt acts as a differential thermocouple. The effective voltage at the \nisothermal ends of the Pt provides a direct measure of the difference in thermopower of the two \nsystems, which we attribute to magnon dynamics in YIG and their interactions at the Pt|Y IG \ninterface. The effective voltage at the isothermal ends of the Pt provides a direct measure of the \ndifference in thermopower of the two systems, which we attribute to magnon dynamics in YIG \nand their interactions with phonons and electrons at the Pt|YIG  interface.   \nSince  the Pt|YIG (or Pt|YIG|GGG) heterostructure is the typical system used to examine \nspin thermal effects , we primarily focus on that system here. However, the large spin orbit coupling \nin Pt (which is precisely what makes it advantageous as  the ISHE layer for SSE devices) could \nraise the question of contributions to the thermopower from SHE signals, especially in the presence \nof a magnetic field. To isolate any contribution from spin Hall effects, we repeat the experiment \non similar heterost ructures where the Pt metal is replaced by metals with rather low spin Hall \nangles. To examine  magnonic effects on phonon electron drag (phonons in the YIG, electrons in \nthe m etallic thin film), we choose metals with relatively simple, clean Fermi surfaces  as far as \npossible. We choose p-type Ag and n -type Al . When examining these metals in their thin film \nform to see how they interact with a substrate, we consider previous knowledge regarding the \nthermopower of these materials. Temperature dependent thermo power data for these bulk metals \nwere measured decades back .4,5 Although n -type, with a negative diffusion thermopower of -5 \nμV/K at 300 K, Pt exhibits a sign change in its thermopower around 200 K, with a positive  phonon \ndrag thermopower peak at 6 μV/K.4 In contrast to this, Ag has a consistently positive (p -type) \nthermopower with a phonon drag peak near 1 μV/K and Al has a consistently negative ( n-type) 5 \n thermopower with a phonon drag peak near -2 μV/K.5 These values inform our interpretation of \ndata in the context of relative strength of t he thermopower  measure d here.  \nExperiment  \nIn order to circumvent the influence of sample to sample variability on our measurements, \nwe devise a hybrid h eterostructure on which we can make  a differential measurement α Pt|YIG vs. \nαPt|GGG on one sample in situ . We name this hybrid heterostructure the magnon drag thermocouple \n(MDT). The MDT consists of 3 layers: a GGG substrate, a YIG film (250 nm) grown on half of \nthe substrate with a gradually stepped edge at the longitudinal center fold of the sample, and a P t \nfilm (6 nm) deposited across the entire structure then patterned into a squared -U shape with four \ncorners (A, B, C, and D). In addition to the MDT for Pt, we construct an identical MDT for Ag \n(10 nm) and Al (6 nm). A list of MDT samples created can be fo und in Table 1.  \nAll samples were measured for steady state, zero field thermopower, αxxx, in the static -\nheat-sink configuration using a Quantum Design PPMS as in Ref. [ 3], with thermometry and gold \nplated copper leads attached to the back face of each substrate. Voltage probes to measure \nthermopower were attached via small (~25 μm) Ag epoxy contacts placed di rectly on the thin film. \nWith a temperature gradient applied longitudinally,  the two ar ms (AB and DC) comprise  the \ntherm ocouple on which a differential voltage  (Vad) can be measured . The hybrid heterostructure \nacts effectively as a thermocouple for the Pt interface: because both sides of the bottom bar (B and \nC) are isothermal with an applied longitudinal temperature g radient, any voltage measured across  \nAD would be due to a difference in the voltage drop across arm AB vs arm DC, i.e., a difference \nin the i nterfacial thermopower depending on whether or not YIG is present.  6 \n One can reasonably ask the question of the influence of the bottom bar of the U on the \nsignal. Depending on the direction of the applied magnetic field, this would correspond to a Nernst -\nlike configuration or transverse spin Seebeck effect (see Figure 3 ). Upon the addition of an applied \nmagnetic field, such a point becomes  relevant. This question is simply addressed by a measurement \nacross the bar, revealing little -to-none signal on  the V bc channel, a result which could also have \nbeen predicted noting that the Nernst effect is ten times smaller (often 1 in 2000) than the Seebeck \neffect in metals.6  \nSample  Film Deposition  \nPt(6 nm)|half YIG(250nm)|GGG  U \nAl(6 nm)|half YIG(250 nm)|GGG  U \nAg(6 nm)|half YIG(250 nm)|GGG  U \nPt(6 nm)|GGG (control)  U (no half -YIG)  \n \nTable 1.  Directory of magnon drag thermocouple (MDT) sample s. \n \n Results  \nIn order to characterize our P t films, we measure temperature dependence of the resistivity using \nthe standard AC Transport option in the PPMS. We measure in zero field and at 7 Tesla applied \nmagnetic field; results show no anomalies and the resistance behaves as expected  (see Figure 4) . \nWe also measure the thermal conductivity of every substrate used in t his study, in situ with the \nSeebeck measurements. An example of thermal conductivi ty of GGG is shown in Figure 5 . These \nmeasurements help to keep track of sample quality and check for induced defects as the study goes 7 \n on. As is characteristic for phonon th ermal conductivity, there is a low temperature drop off, where \nthe density of carriers decreases as T3. The high temperature drop is attributed to intrinsic phonon -\nphonon Umklapp scattering, and t he low temperature drop is attributed to phonon -boundary \nscattering.7 The intermediate temperature range is where phonon transport is maximum and where \nthe phonon drag peak in the thermopowers is expected to be maximal.   \nTo test of the validity of the MDT, we measure a control sample of a Pt U deposited on a \nGGG su bstrate, with no half -YIG film. Although in principle there should clearly be no signal on \nVad of such a sample, this measurement demonstrates the validity of the assumption of the U as a \nreliable thermocouple in practice. Measurements confirm there is no spurious signal from the \nbottom bar, and that V ad is isovoltaic in the absence of YIG. The low temperature measurements \nfrom this control reveal the baseline noise level of the experiment, on the order of 1 µV/K below \n6 K.  \nSeebeck measurements from the MDT in differential mode (V ad) are shown in Figure 6 . \nThe data show a temperature dependence roughly following that of the thermal conductivity of the \nsubstrate GGG and a positive peak near 8 μV/K. It is worth noting carefully that measurements of \nthe MDT in differential mode actually give the thermopower of Pt on YIG subtracted from the \nthermopower of Pt on GGG, considering the cold side to be the positive voltage terminal, as is \nconventional in Seebeck measurements. This means that the positive signal  on the Pt MDT  in \nFigure 6 implies that the magnons lower  the thermopower of the Pt on YIG relative to the Pt on \nGGG. This result is surprising, as it implies that magnons at the interface may actually be \nsuppressing the drag effects across the Pt interface.  \nNext, we examine the differential thermopower of the Ag and Al MDTs. Two observations \nare immediately evident : 1) the thermopower magnitudes of each of the films exceed t hose of their 8 \n bulk counterparts and 2) t heir temperature dependence roughly follows the rmal conductivity of \nsubstrate, implying phonon electron drag, substrate to film. A positive low temperature peak is \naround 4 μV/K on the Ag sample, and a negative thermopower with a peak is around -14 μV/K in \nthe Al. Given that Ag is p type and Al is n ty pe, we find that the polarity of the effect matches that \nof the sign of the carrier in the metal. This verifies that the measured thermopower voltage is not \nrelated to spin Hall physics, but rather electron drag by phonons. Thus, t hese results are consiste nt \nwith the results on the Pt sample; the YIG yields a lower signal than the GGG interface, implying \nthat the magnons interfere with phonon drag.  \nAs a follow up to  this observation, we conduct magnetic field -dependent measurements of \nthe the rmopower  on th e Pt and Ag MDT s, as shown in Figure 7 . Applied magnetic fields are \nexpected to suppress or “freeze out” magnon dynamics,8 with a more pronounced effect at low \ntemperatures.9 At lowest temperatures, where one would expect the effect of the magnetic field to \nbe strongest (6 to 9 K) the interfacial thermopower itself is difficult to resolve, so that a field \ndependent study is difficult to obtain. At moderately low temperatures w here signal is strongest, \nnear the phonon drag peak in thermopower, this magnetic field effect is measurable. As the  \nmagnetic field is increased from 0 to 9 Tesla  at 10 K, we observe a decrease in thermopower on \nboth Pt and Ag films . A decrease in thermopo wer implies a decrease in signa l as magnon dynamics \nare suppressed  at large magnetic fields so that both YIG and GGG exert th e same amount of drag \non the Pt. T his effect is more pronounced at low temperatures, but where signal is still well \nresolvable from  the noise floor, which becomes difficult below around 7 K.  By contrast, we also \nmeasure the magneto -Seebeck coefficient of a  Pt|YIG (250)  sample. Here, we observe an increase  \nin thermopower as magnetic field is increased from 0 to 9 Tesla  below 30 K,  as shown in Figure \n8. An increase in thermopower implies a recovery  of signal  as magnon dynamics are suppressed  9 \n out at large magnetic fields; this result is consistent with the results in Refs. [ 8,9] and differential \nmeasurements from the Pt MDT , supporting the conclusion that magnons interfere with phonon \ndrag in these heterostructures .  \nHaving isolated that there is a magnonic impedance to the phonon drag effect, we explore \nthe length scale of this effect. In particular, as we decrease YIG thickness, there should be less \nmagnons available, so t hat the Pt|YIG thermopower should increase and ultimately for very thin \nYIG, match that of Pt|GGG. We now measure a series of Pt|YIG samples  with YIG o f varied \nthickness (bulk, 1 μm, 250 nm, 100 nm, 40 nm). The bulk sample behaves much like the 250 nm \nYIG sample. At 100 nm, the signal increases, and at 40 nm, the phonon drag peak in the Pt|YIG \nthermopower is nearly equivalent to that of the Pt|GGG. In the 1 μm samples, the phonon drag \npeak is killed altogether, but the diffusion thermopower (high temperatur es) equilibrates for all \nsamples above around 100 K.  \nThe implications of these data  are summarized as follows. Substrate -to-thin film phonon \nelectron drag is observed on Pt|YIG and Pt|GGG with equal magnitudes at high temperatures. The \nphonon drag peak in  thermopower is significantly attenuated in the metal when YIG is present, so \nthat this attenuation is attributed to magnons in the YIG. The thicker the YIG film, the larger the \nmagnon scattering volume, which effectively acts as a barrier for the phonons which otherwise \nwould drag electrons in the neighboring conducting film. At the smallest YIG thicknesses, we \nrecover results very similar in magnitude to the signals on GGG. This length scale dependence on \nYIG thickness complements the identification of th e magnon energy relaxation length from Ref. \n[10]. In fact the magnon -to-phonon energy relaxation, or a difference between magnon and phonon \ntemperatures, could reasonably affect scattering rates between magnons and phonons and \nconsequently interrupt the pho non-electron drag effects that occur in the absence of magnons.  10 \n With an applied magnetic field at temperatures below 30 K , we observe a recovery of the \nphonon -electron drag as magnon dynamics are partially suppressed. The effect manifests as an \nincrease or  recovery in signal on the isolated YIG system, and a decrease in differential signal  on \nthe MDT. These observations are consistent with the magnetic field -dependent “freeze out” of \nmagnon dynamics established in Refs. [ 8,9]: as magnons are suppressed with high magnetic fields,  \nthe Pt|YIG interface behaves more closely like the Pt| GGG.  \nThis series of measurements support our conclusion that magnons in fact inte rfere with the \nphonon -electron drag interaction at the metallic interface in these heterostructures.  Further work \nshould focus on developing a quantitative theoretical model for such an effect, accounting for \nscattering rates of magnons with phonons and el ectrons in the YIG and at the Pt interface.  \nAcknowledgements  \nFunding for this work was provided by the OSU Center for Emergent Materials, an NSF MRSEC, \nGrant DMR -1420451  and the U.S. Department of Energy, Office of Science, Basic Energy \nSciences, Grant  No. DE-SC0001304.  \nFigures  \n \n11 \n Figure 1.  Schematic of longitudinal thermopower measurement, where voltage is measured in the \ndirection of the temperature bias, along the direction of the interface on Pt|YIG heterostructure.  \n  \nFigure 2 . Schematic of magnon drag thermocouple with measured voltage  in differential mode \n(Vad) (left ), and photograph of actual Pt|YIG(/GGG) U shaped sample (right ). \n \nFigure 3. The direction of the applied magnetic field yields either a Nernst configuration (blue) or \na transverse spin Seebeck configuration (red) on the bottom bar of the MDT.  \n12 \n  \nFigure  4.  Temperature dependent resistivity of Pt on GGG in 7 T field (orange) and in zero -field \n(purple) shows no anomalous features and serves as an experimental check of the Pt film. \n \nFigure 5.  Temperature dependent thermal conductivity of bulk single crystal GGG  substrate . \n13 \n   \nFigure 6.  Temperature dependent thermopower of Pt, Ag, and Al magnon drag thermocouples.  \n  \nFigure 7.  Magnetic field dependence of thermopower at various temperatures indicated on the \ngraphs in the Pt and Ag magnon drag thermocouples in differential mode (left). The relative signal \ncan be seen as a relative decrease from the zero field value (right).  \n14 \n     \nFigure 8.  Magnetic field dependence of thermopower, at various base temperatures indicated on \nthe graphs for the Pt|YIG interface (left). The strength of this effect can be seen as a relative \nincrease from zero -field thermopowe r (right).  \n \nFigure 9. Temperature dependence of interfacial Pt|YIG thermopower for various YIG thicknesses \nas shown on the graph.  \n15 \n  \n1 G.Wang , L. Endicott, H. Chi, P. Lost’ak, and C. Uher, Phys. Rev. Lett. 111, 046803 (2013).   \n2 T. Liu, G. Vignale, M. E. Flatte, Phys. Rev. Lett. 116, 237202 (2016).   \n3 S.J. Watzman, R.A. Duine, Y. Tserkovnyak, H. Jin, A. Prakash, Y. Zheng, and J. P. Heremans, \nPhys. Rev. B. 94, 144407 (2016).  \n4 R. P. Huebner, Phys. Rev. 140, 5A (1965).  \n5 R. J. Gripshover, J. B. VanZytveld, and J. Bass, Phys. Rev. 163, 3 (1967).  \n6 S. R. Boona, R. C. Myers and J. P. Heremans, Energy Environ. Sci. 7, 885-910 (2014).  \n7 J. M. Ziman, Principles of the Theory of Solids, Cambridge University Press (1972).  \n8 T. Kikkawa, K. Uchida, S. Daimon, Z. Qiu, Y. Shiomi, and E. Saitoh, Phys. Rev. B. 92, (6), \n064413 (2015).  \n9 H. Jin, S. R. Boona, Z. Yang, R. C. Myers, and J. P. Heremans, Phys. Rev. B . 92, 054436 (2015).  \n10 A. Prakash, B. Flebus, J. Brangham, F. Yang, Y. Tserkovnyak, and J.P. Heremans, Phys. Rev. \nB. 97, (2), 020408(R ) 2018.                                                             "    },    {        "title": "2104.09592v1.Magnetic_coupling_in_Y__3_Fe__5_O___12___Gd__3_Fe__5_O___12___heterostructures.pdf",        "content": "Magnetic coupling in Y 3Fe5O12/Gd 3Fe5O12heterostructures\nS. Becker,1,\u0003Z. Ren,1, 2, 3, †F. Fuhrmann,1A. Ross,4, 1S. Lord,1, 2, 5\nS. Ding,1, 2, 6R. Wu,1J. Yang,6J. Miao,3M. Kläui,1, 2, 7and G. Jakob1, 2, ‡\n1Institute of Physics, Johannes Gutenberg-University Mainz, Staudingerweg 7, 55128 Mainz, Germany\n2Graduate School of Excellence “Materials Science in Mainz” (MAINZ), Staudingerweg 9, 55128 Mainz\n3School of Materials Science and Engineering, University of Science and Technology Beijing, Beijing 100083, China\n4Unité Mixte de Physique CNRS, Thales, Université Paris-Saclay, 91767 Palaiseau, France\n5Department of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom\n6State Key Laboratory for Mesoscopic Physics, School of Physics, Peking University, Beijing 100871, China\n7Center for Quantum Spintronics, Norwegian University of Science and Technology, 7491 Trondheim, Norway\n(Dated: April 21, 2021)\nFerrimagnetic Y 3Fe5O12(YIG) is the prototypical material for studying magnonic properties due to its ex-\nceptionally low damping. By substituting the yttrium with other rare earth elements that have a net magnetic\nmoment, we can introduce an additional spin degree of freedom. Here, we study the magnetic coupling in epitax-\nial Y 3Fe5O12/Gd 3Fe5O12(YIG/GIG) heterostructures grown by pulsed laser deposition. From bulk sensitive\nmagnetometry and surface sensitive spin Seebeck effect (SSE) and spin Hall magnetoresistance (SMR) mea-\nsurements, we determine the alignment of the heterostructure magnetization through temperature and external\nmagnetic field. The ferromagnetic coupling between the Fe sublattices of YIG and GIG dominates the overall\nbehavior of the heterostructures. Due to the temperature dependent gadolinium moment, a magnetic compen-\nsation point of the total bilayer system can be identified. This compensation point shifts to lower temperatures\nwith increasing thickness of YIG due the parallel alignment of the iron moments. We show that we can control\nthe magnetic properties of the heterostructures by tuning the thickness of the individual layers, opening up a\nlarge playground for magnonic devices based on coupled magnetic insulators. These devices could potentially\ncontrol the magnon transport analogously to electron transport in giant magnetoresistive devices.\nI. INTRODUCTION\nA major challenge in information technology is solving the\nissue of Joule heating due to charge currents. One approach\nis to move away from electron-based to magnon-mediated in-\nformation transport and processing [1]. This requires the de-\nvelopments of new logic devices, such as magnon valves that\nallow for the manipulation of spin currents [2]. Material can-\ndidates require an insulating character, magnetic ordering and\nlow magnetic damping. One promising candidate is ferrimag-\nnetic Y 3Fe5O12(YIG = yttrium iron garnet), which is a well-\nknown material in magnetism as it shows ultra-low magnetic\ndamping and low magnetic anisotropy. The net ferrimagnetic\nmoment originates from an antiparallel alignment of Fe3+mo-\nments on different crystallographic sites. Each minimal unit\ncell consists of 12 trivalent Fe3+ions that are tetrahedrally\ncoordinated with oxygen atoms ( dsites) and 8 trivalent Fe3+\nions that are octahedrally coordinated ( asites). The dominant\ncoupling is antiferromagnetic between iron atoms on minor-\nityaand majority dsites. By substituting Y3+with Gd3+, an\nadditional moment appears aligned antiparallel to the d-site\nFe atoms [3, 4]. Due to the strong temperature dependence\nof the Gd net magnetic moment, Gd 3Fe5O12(GIG) shows a\nmagnetization compensation at temperature T\u0019295 K [5].\nFor low power information processing, magnons (the\nquanta of magnetic excitation) are exciting candidates. The\nmagnon spectra of YIG has been the subject of both experi-\n\u0003svenbecker@uni-mainz.de\n†zengyaoren@163.com; S.B. and Z.R. contributed equally to this work\n‡jakob@uni-mainz.demental and theoretical investigation [6, 7]. In heterostructures,\nmagnon-magnon coupling and magnetic coupling play a deci-\nsive role for the propagation of magnons [2, 8–13]. Here, we\ninvestigate the coupling between two different iron based gar-\nnets YIG and GIG as candidates for an all insulator magnon\nspin valve. Both YIG and GIG are important ferrimagnetic\ninsulators that can be grown epitaxially on isostructural but\nparamagnetic Gd 3Ga5O12(GGG) substrates. Recently, mag-\nnetic coupling of YIG to an ultrathin GIG layer was reported\n[13, 14], where GIG is formed by an interdiffusion process\nwhile preparing YIG on GGG. In our work, we realize con-\ntrolled growth of epitaxial YIG/GIG heterostructures on GGG\nsubstrates, where the individual layers have sufficient mag-\nnetic moment to be detected by magnetometry. The bilay-\ners show high crystalline quality and a magnetic compensa-\ntion of the entire bilayer, which shifts to lower temperature\nwith increasing thickness of YIG, indicating interlayer mag-\nnetic coupling of YIG and GIG. The alignment of the mag-\nnetization of bilayers with temperature and field can be char-\nacterized by SQUID magnetometry, which measures the sum\nsignal of all the layers. To identify the magnetization direc-\ntion, we utilize the surface sensitive spin Hall magnetoresis-\ntance (SMR), which has proven to be a suitable tool to inves-\ntigate the magnetic properties of magnetic insulators [15–18].\nWe further conduct spin Seebeck (SSE) measurements, which\nhave previously been used to investigate pure GIG samples\n[18–23]. SSE measurements show dominating sensitivity to\nthe top layer of the heterostructure. Our results demonstrate\nthat the respective Fe sublattices of YIG and GIG are ferro-\nmagnetically coupled across the interface. This allows for an\nunprecedented tunability of the magnetic properties of the bi-\nlayer system by choosing the relative thickness of the YIG andarXiv:2104.09592v1  [cond-mat.mtrl-sci]  19 Apr 20212\nGIG layers.\nII. EXPERIMENTAL DETAILS\nY3Fe5O12(YIG) and Gd 3Fe5O12(GIG) are deposited on\n(001)-oriented Gd 3Ga5O12(GGG) substrates by pulsed laser\ndeposition (PLD) in an ultrahigh vacuum chamber with a base\npressure lower than 2 \u000210\u00008mbar. For ablation, a KrF ex-\ncimer laser (248 nm wavelength) with a nominal energy of\n130 mJ per pulse is used at a pulse frequency of 10 Hz. The\nfilms are grown under a stable atmosphere of 0.026 mbar of\nO2at 475\u000eC substrate temperature. After deposition, the films\nare subsequently cooled down to room temperature at a rate of\n\u000025 K/min. The crystalline structure of the films was deter-\nmined by x-ray diffraction (XRD). The magnetic moment was\nmeasured by using a superconducting quantum interference\ndevice magnetometer (SQUID, Quantum Design MPMS II).\nFor spin Seebeck effect (SSE) measurements, the samples are\ncovered with a continuous layer of 4 nm Pt deposited by mag-\nneton sputtering. The measurements are performed in the lon-\ngitudinal geometry, where the heat gradient is perpendicular\nto the sample surface [19]. For spin Hall magnetoresistance\n(SMR) measurements, a 4 nm thick Pt bar of around 0.3 mm\nwidth is defined on the surface along the crystallographic axes.\nThe Pt is deposited ex-situ using magneton sputtering in an Ar\natmosphere through a shadow mask.\nIII. RESULTS\nFig. 1(a) and 1(b) present the XRD patterns of GGG/YIG,\nGGG/GIG, and GGG/YIG/GIG bilayer films measured with\nthe scattering vector normal to the (001) oriented cubic sub-\nstrate. As the films grow coherently on the substrate surface,\ntheaandbaxes are equally strained and the (004) peaks and\n(008) peaks, indicating the length of the c-axis, are evident\nin the corresponding high resolution XRD patterns, respec-\ntively. Near the respective (004) (Fig. 1(a)) and (008) (Fig.\n1(b)) diffraction peaks, the XRD patterns show Laue oscilla-\ntions, indicating a smooth surface and interface. While the\nYIG reflex is partly shadowed by the substrate peak, we can\nclearly identify the reflex of the GIG top layer. From the GIG\npeak position, we determine the out-of-plane lattice param-\neter to around c=1:26 nm, independent of the thickness of\nthe underlying YIG layer, indicating strained growth for every\nsample. The absence of a structural alteration of the top GIG\nlayer indicates that the YIG interlayer does not influence the\nGIG growth. It is therefore expected, that the magnetic prop-\nerties of the GIG layer are similarly unaffected. The rocking\ncurve of each reflex is around Dw=0:04\u000efurther showing\nwell aligned unit cells for every bilayer. Together, these XRD\nmeasurements indicate the high-quality growth of YIG/GIG\nheterostructures.\nTo characterize the magnetic properties of the heterostruc-\ntures, the magnetization vs field ( m\u0000H) dependence was\nmeasured with a magnetic field applied within the sample\nplane. The m\u0000Hof YIG/GIG for magnetic fields up to\n（a） （b）\n2Θ (deg) 2Θ (deg)\nlog(intensity) (arb. units) log(intensity) (arb. units)FIG. 1. Out-of-plane 2 Q=wmeasurements of YIG, GIG and\nYIG/GIG bilayer films near the (004) peaks (a) and (008) peaks (b)\nshown in a logarithmic intensity scale. The thicknesses of the indi-\nvidual layers are detailed in nanometers.\n50 mT are obtained by measuring the GGG/YIG/GIG sample\nand subtracting a linear fit to compensate for the paramag-\nnetic contribution of the substrate. Fig. 2 (a) shows the m\u0000H\ncurves for single layer GIG and YIG measured at a temper-\nature of T=100 K. Note that the magnetic moment of the\nYIG layer of around 0 :6\u000210\u00007Am2corresponds to a mag-\nnetization of 133 kA/m, which is similar to other YIG thin\nfilms and bulk samples [24], confirming the high quality of\nthe thin films. Since heterostructures of varying thickness are\ninvestigated, in the following we will focus on the total mag-\nnetic moment of the layers rather than on the magnetization.\nFig. 2(b-d) show the m\u0000Hcurves measured at the same tem-\nperature for YIG/GIG heterostructures of varying YIG thick-\nness. All samples generally show sharp switching features,\nhowever, the YIG(36 nm)/GIG(30 nm) (Fig. 2 (d)) shows seg-\nmented switching features in this magnetic field range. The\ncurve displays an ‘inner hysteresis’ and additionally a larger\nhard axis loop. Secondly, the total magnetization of the bi-\nlayers at m0H=20 mT decreases with increasing YIG layer\nthickness. This behavior indicates that the net moments of\nYIG and GIG are antiparallel in the low-field region.\nIn order to further understand this behavior, the tempera-\nture dependence of the m\u0000Hcurves was measured. The\nnet magnetic moments mof the YIG/GIG samples at 50 mT\nwere obtained. As shown in the top of Fig. 3 (a), the mag-\nnetic moment of the pure YIG layer is only weakly dependent\non temperature, while the pure GIG sample is ferrimagnetic\nwith a compensation temperature ( Tcomp ;G) of 280 K, which\nis close to the bulk value (295 K [5]). The GIG magnetiza-\ntion is strongly temperature dependent due to the Gd moment\nincreasing towards lower temperatures. Above Tcomp ;G, the\ndirection of the magnetization of the GIG sample is given by\nthe direction of the d-Fe moments, while below Tcomp ;G, the\nmagnetization direction is given by the direction of the c-Gd\nmoments. Note that the antiferromagnetic coupling between\nthea-Fe and d-Fe sublattices within one layer cannot be bro-\nken by magnetic fields accessible in our labs. We therefore3\n@100 K（a） （b）\n（c） （d）\n‘inner hysteresis’\nFIG. 2. SQUID measurements ( m\u0000H-loops) for different YIG/GIG\nbilayers measured at a temperature of 100 K with a maximum applied\nfield of 50 mT.\nsimplify the description of the magnetic properties by intro-\nducing one magnetic Fe lattice for YIG and GIG as the sum\nof minority a-Fe and majority d-Fe, respectively.\nHaving established the compensation temperature of pure\nGIG, we observe that when grown in a bilayer with YIG, the\ncompensation temperature shifts to lower temperatures with\nincreasing YIG thickness. We indicate in Fig. 3 three regions\nof the bilayer, corresponding to: above the compensation of\nthe pure GIG film (grey, zone I), temperatures below the com-\npensation of the bilayer (blue, zone III), and an intermedi-\nate region (orange, zone II). The shifting of the compensa-\ntion temperature of the bilayer Tcomp ;Bindicates that there is\na coupling between the YIG and GIG layer that is compen-\nsating for the change in the Gd orientation that occurs. The\nmagnitude of the magnetic moments at a magnetic field of\n20 mT at different temperatures is summarized in Fig. 3 (b).\nAt 20 K, the total magnetic moment decreases with increas-\ning YIG thickness, similar to the 100 K measurements shown\nin Fig. 2). At 300 K, the total magnetization increases with\nincreasing thickness of YIG. Taking into account the rever-\nsal of the magnetization direction of the magnetic sublattices\nin GIG, this indicates an interlayer ferromagnetic coupling in\nthe whole temperature range of the Fe sublattices of YIG and\nGIG, i.e. considering the Fe moments and Fe-O bonds only,\nthe bilayer has a coherent magnetic structure at low fields at\nall temperatures.\nIn order to support our claim of a coherent magnetization\nstructure of the Fe sublattices, we perform a simple simula-\ntion taking into account the temperature dependence of the\nGd and Fe magnetic moments [4]. We model the net Fe mag-\nnetic moment as m(Fe) =jm(d-Fe)\u0000m(a-Fe)j, so the magnetic\nmoment of GIG is equal to jm(Fe)\u0000m(Gd)j. Above Tcomp ;G,\nthe modulus m(Gd) = m(Fe)\u0000m(GIG), below Tcomp ;G,m(Gd)\n=m(GIG) +m(Fe). Since the total magnetic moment of the\nFe ions sublattices is the same in YIG and GIG, the m(Fe)\ncan be seen as m(YIG). The magnetic moment of Gd in GIG\ncan be extracted from the magnetization of individual YIG\n1E-61E-51E-4\n1E-61E-51E-4\n1E-61E-51E-4\n20 60 100 140 180 220 260 3001E-61E-51E-4\nYIG(18 nm)\nGIG(30 nm)\nYIG/GIG (9/30)\n-3\n2\nYIG/GIG (18/30)\nYIG/GIG (36/30)\nT (K)\n（a） （b）\nIII II I\nTcomp,B Tcomp,G20 K\n300 K\nTcomp,B\nTemperature (K)FIG. 3. (a) M\u0000Tcurves of the YIG, GIG and YIG/GIG films in low\nmagnetic field measured by m\u0000H-loops at different temperatures.\nThe shaded regions indicate T>Tcomp ;G(zone I), Tcomp ;G>T>\nTcomp ;B(zone II), T<Tcomp ;B(zone III) for the respective samples.\n(b) The dependence of the magnetic moment m(300 K), m(20 K) and\nthe bilayer compensation temperature on the YIG thickness.\n(18 nm) and GIG (30 nm) layers shown in Fig. 3(a) top panel.\nThe magnetization of YIG is only weakly temperature de-\npendent, so we can model its magnetic moment as constant\nm(Fe,18 nm) = 5 :4\u000210\u00008Am2. We note that the YIG mag-\nnetization should follow T3=2from the Curie temperature TC\nto 0 K, however, we stay far below TC. The m(Gd) can be ex-\ntracted from the temperature dependent magnetization curves\nand the data points are fitted by a third order polynomial used\nfor the simulations. More precise modeling of m(Gd) can be\ndone in principle using a self consistent molecular field acting\non the Gd spin moments as input for the Brillouin function\n[25]. This would require knowledge of the detailed tempera-\nture dependence of the Fe sublattice magnetization. However,\nthe exact behavior of the sublattice magnetization is out of\nscope of this work and the parameters from the bulk cannot\nbe transferred to the thin films that usually have somewhat re-\nduced Curie temperatures. Therefore, we describe m(Gd) only\nphenomenologically in the temperature range of our measure-\nments to facilitate the analysis. The temperature dependencies\nof the magnetic moments of Gd3+, Fe3+and simulated GIG\nare plotted in Fig. 4(a).\nWe model three different cases for the interlayer cou-\npling of YIG/GIG bilayers: the respective magnetic\nmoments m(Fe,GIG) of GIG and m(Fe,YIG) of YIG are\nferromagnetically coupled, antiferromagnetically coupled\nand without coupling. For the ferromagnetic Fe-Fe cou-\npling case, the overall magnetic moment of the system\nis given as m=jm(Fe,GIG) +m(Fe,YIG)\u0000m(Gd,GIG)j;\nFor the antiferromagnetic Fe-Fe coupling, m=\njm(Fe,GIG)\u0000m(Fe,YIG)\u0000m(Gd,GIG)j; Without coupling\nm=jm(Fe,GIG)\u0000m(Gd,GIG)j+m(Fe,YIG). The respective\nresults are plotted in Fig. 4. As shown in Fig. 4(b), for\nferromagnetic coupling, Tcomp ;Bis decreasing with increasing\nthickness of YIG. In addition, the total magnetic moment\nincreases with the thickness of YIG at 300 K and the total\nmagnetic moment decreases with the thickness of YIG at4\nGd3+Fe3+\nGd3+Fe3+Gd3+Fe3+（a） （b）\n（c） （d）Exp. Sim.\nFIG. 4. (a) Modeled m\u0000Tcurve of the m(Gd) and m(Fe). Modeled\nm\u0000Tcurves of GIG and YIG/GIG bilayers, where m(Fe,GIG) of\nGIG and m(Fe,YIG) of YIG are (b) ferromagnetically coupled, (c)\nantiferromagnetically coupled and (d) without coupling. The insets\nshow the magnetic sublattice alignment at low magnetic fields. The\nblue arrow indicate the presence of coupling.\n20 K. These features are consistent with the experimental\nresults and support our claim that the layers are indeed\nferromagnetically coupled with respect to iron atoms. For an\nassumed antiferromagnetic Fe-Fe coupling at the YIG-GIG\ninterface (Fig. 4 (c)), the bilayers do not possess compensa-\ntion temperatures below 300 K for the simulated thicknesses.\nWithout coupling between the YIG and GIG magnetization\n(Fig. 4 (d)), the bilayers never have a compensation point, but\nonly a total magnetization minimum at Tcomp ;G.\nThe above discussion was on magnetization measurements\nat low magnetic field and we used simulations based on the\nindividual materials to compare to the measured SQUID data.\nWe have seen that these investigations show a ferromagnetic\nFe-Fe coupling across the YIG-GIG interface. The double\nswitching in the YIG(36nm)/GIG(30nm) sample (Fig. 2(d))\nindicates that a sufficiently large magnetic field can break\nthe interlayer coupling. To investigate this further, we per-\nform m\u0000Hmeasurements exploiting larger magnetic fields\nat various temperatures. These m\u0000Hof YIG/GIG can be\nobtained via measuring the entire GGG/YIG/GIG sample and\nthen subtracting the paramagnetic contribution of GGG de-\ntermined in a separate measurement. As shown in Figs. 5\n(a)-(c), we observe double switching over a large tempera-\nture range. Further increase of the external field leads to a\nsaturation of the overall magnetic moment. To describe the\ncurves, we introduce Amp 2 as the maximum amplitude of\nthe hysteresis. Here, the Fe-Fe interlayer coupling is broken\nand the net moments of YIG and GIG layers align parallel so\nthatm(YIG) +m(GIG) =Amp 2. Approaching the compensa-\ntion temperature of the GIG layer Tcomp ;G, the signal from the\nGIG layer becomes too weak to perform this type of analysis\nwith respect to the signal resolution of the SQUID and in the\npresence of the strong substrate background. Therefore, we\n20 60 100 140 180 220 260 300012345\nT(K)Amp2Amp1\nµ0H0III II I\nYIGGIG\n-1,0 -0,5 0,0 0,5 1,0-4-2024\n160 K\n-7\n2\nµ0H (T)-1,0 -0,5 0,0 0,5 1,0-8-4048\n-7\n2\nµ0H (T)20 K\nYIG/GIG (36/ 30)\n-1,0 -0,5 0,0 0,5 1,0-4-2024\n120 K\n-7\n2\nµ0H (T)(a)\n(b)\n(c)(d)\n-7\n2\nFIG. 5. m\u0000Hcurves of a YIG(36)/GIG(30) sample measured at\na temperature (a) 20 K (b) 120 K and (b) 160 K in high magnetic\nfield. (d) m\u0000Tcurve of YIG(36)/GIG(30) with parallel state and\nantiparallel state and the alignment of the YIG and GIG layer in high\nand low magnetic fields in different temperature regions\nevaluate Amp 2 only up to T = 180 K. The ‘inner hysteresis’,\nas already seen in Fig. 2 (d) and in Fig. 5 (a), which indicates\na switching at low magnetic fields has the amplitude Amp 1.\nIn this low-field region, the Fe-Fe coupling is not broken and\nthe magnetic moments of the whole bilayer stack reverses by\nreversing the small magnetic field keeping the relative orien-\ntation of the Fe and Gd moments intact. The amplitude of\nthis hysteresis is given by the difference of the net magnetic\nmoments of the individual layers jm(YIG)\u0000m(GIG)j=Amp 1.\nThe preservation of the relative orientation of the individual\nmoments implies that the orientation of the net magnetic mo-\nments of the YIG and GIG layers changes going from zone\nII to zone III in Fig. 3. In zone II, where the Gd moment is\nsmaller than the net Fe moment of the coupled layers, the YIG\nlayer aligns parallel to the field while the GIG layer aligns an-\ntiparallel to the field. If the magnitude of the Gd moment\novercomes that of the net bilayer Fe moment in zone III, GIG\naligns parallel to the field and YIG antiparallel. This is illus-\ntrated in Fig. 5(d). The relative alignment of Fe (orange) and\nGd (red) moments is depicted. In the high temperature region\nabove the compensation temperature of the GIG layer (grey\nsketched zone I in Fig 5(d)) the magnetization of GIG is dom-\ninated by the net iron moment. Here the magnetization of the\nYIG layer and GIG layer are always parallel to the external\nfield at low and high field.\nWe have seen in SQUID measurements that the magnetiza-\ntion of the YIG and GIG layers aligns differently depending\non the temperature and relative thickness. Tuning the thick-\nness of the respective layers allows for choosing the magnetic\nproperties of the heterostructures. The larger the YIG:GIG ra-\ntio, the lower the bilayer compensation temperature Tcomp ;B.\nWe can thereby functionalize the coupled layers and build de-\nvices that have defined relative orientation of the magnetic5\n-50 0 50 100\n(deg)-1-0.500.51RL/RL(0)10-3\n0.1 T\n0.5 T\n2.0 T\n-50 0 50 100\n(deg)-4-3-2-101RL/RL(0)10-4\n0.1 T\n0.5 T\n2.0 T\n-2 0 2-2.5-2-1.5-1-0.50RL/RL(0)10-4\n30 K\n50 K\n-2 0 2\nµ0H (T) µ0H (T)-4-20246RL/RL(0)10-4\n100 K\n200 K(a) (b)\n(c) (d)\nHI\nx\ny30 K 200 Kzone III zone II\nI\nFIG. 6. Uniaxial SMR measurements at a GGG/YIG(36)/GIG(30)/Pt\nsample at various temperatures (a-b) with the field applied in the\nsample plane perpendicular to the current as depicted in the inset\nof (a). ADMR for two different temperatures (c-d) where the field is\nrotated within the sample plane as illustrated in the inset of (c).\nmoments. However, SQUID measurements do not distinguish\nwhich layer switches (top or bottom). In order to disentan-\ngle this, we conduct surface-sensitive methods to probe the\ntop surface layer individually to then, in combination with\nSQUID, identify which layer switches at which field.\nA suitable tool to investigate the magnetic properties of the\ntop layer is spin Hall magnetoresistance (SMR) [15, 26]. The\nspin accumulation in a heavy metal in close contact with a\nferrimagnetic insulator interacts with the magnetic order pa-\nrameter only at the very interface. The resistance of a Pt bar\ndefined on the surface of a GGG/YIG(36)/GIG(30) sample is\nmodulated by DRLµ(1\u0000m2\ny)[26], where myis the magne-\ntization component of the GIG layer in plane perpendicular\nto the Pt bar. We normalize the change of resistance to the\nzero-field value. We perform uniaxial measurements with the\nmagnetic field applied in-plane perpendicular to the Pt bar.\nThe uniaxial field measurements at 30 K and 50 K describe\na sharp drop of resistance at low magnetic fields followed\nby negligible further resistance changes, which is shown in\nFig. 6 (a). The sharp decrease of resistance is likely due to\nthe annihilation of differently aligned domains, going from\na multidomain state to a monodomain state when the GIG\nnet magnetic moment aligns with the field. In the absence\nof an external field, the sample symmetry allows for domains\nwith magnetization aligned along the Pt bar. At temperatures\nabove Tcomp ;B, double switching is observed in the SMR data\nwith increasing magnetic field (see Fig. 6 (b)). This indicates\nthe rotation of the GIG magnetic moment after the interfacial\ncoupling between YIG and GIG is broken. The continuous\nincrease of resistance in the switching process indicates thatthe top layer switching is in fact not an abrupt process, but a\ngradual rotation since only the alignment of the GIG magne-\ntization away from the y-direction can increase the resistance\ndue to SMR. We confirm this by performing rotation mea-\nsurements, where the field rotates within the sample plane\n(a-plane) from around \u000055\u000eto 125\u000e, relative to the current\ndirection, and back. The experiment is performed at 30 K,\nwhich is in zone III (Fig. 6 (c)) and at 200 K, which is in\nzone II (Fig. 6 (d)). The longitudinal resistance is given as\nrelative to the resistance value at 0° (along the Pt bar). At\nlow temperatures, where no top-layer switching is observed\nin the uniaxial SMR measurements, we also do not observe\nany changes in the phase of the angular-dependent magnetore-\nsistance (ADMR) for magnetic fields of different magnitude.\nHowever, increasing the temperature to 200 K, the ADMR be-\ncomes strongly field-dependent. At low magnetic field and at\nhigh magnetic field, the phase of the ADMR is close to 0\u000e,\nindicating the alignment of the GIG magnetization axis with\nthe external magnetic field. At 0.5 T, however, in the region\nof the GIG switching, a phase shift of almost 90\u000eis observed,\nindicating the perpendicular alignment of GIG magnetization\nto the magnetic field in the switching region.\nA complementing tool to investigate the magnetic proper-\nties of the bilayer system is the longitudinal spin Seebeck ef-\nfect (SSE). The SSE can be used to investigate the magnetic\nproperties close to Tcomp ;Gat which SQUID measurements\ncannot resolve the magnetization of the GIG layer. For the\nSSE measurement, the sample is exposed to an out-of-plane\ntemperature gradient by sandwiching it between a tempera-\nture sensor and a resistive heater (see inset of Fig. 7 (a)). The\ntemperature gradient is estimated by monitoring the resistance\nof sensor and heater. The external magnetic field is applied in\nthe sample plane, perpendicular to the temperature gradient.\nMeasuring the thermal excitation of spin waves in such a bi-\nlayer can potentially lead to superposing spin currents origi-\nnating from YIG and GIG, measured as a voltage VISHE via\nthe inverse spin Hall effect (ISHE) in a heavy metal top layer\n[22, 27]. The SSE of YIG(36)/GIG(30)/Pt(4) was measured at\ndifferent temperatures. As shown in Fig. 7, a hysteretic volt-\nage signal VISHE is obtained by sweeping the magnetic field.\nAt the lowest temperature of 35 K, we observe a negative am-\nplitude of the SSE as shown in Fig. 7(a). Going to 60 K, the\nsign of the measured voltage changes its sign to positive (Fig.\n7(b)). For 110 K, the shape of the SSE signal fundamentally\nchanges as seen in Fig. 7(c). At low magnetic fields, a nega-\ntive switching is observed before the signal changes sign again\nat around 40 mT. At 306 K, only one switching event is visible\nat low magnetic fields, having a negative sign (Fig. 7(d)).\nThese features emphasize the SSE signal to have it’s ori-\ngin in the GIG layer. We investigate these features in the fol-\nlowing, by taking into account the magnetic properties of the\nbilayer system determined by SQUID and the complex behav-\nior of SSE signal measured at pure GIG samples [22, 23]. For\ntemperatures below the compensation temperature of the bi-\nlayer Tcomp ;B, we can compare the SSE measurement with the\nm\u0000Hloop. In the SQUID measurement at 60 K (zone III), a\nswitching of YIG is expected at approximately 30 mT. How-\never, we do not observe this feature at 30 mT in SSE mea-6\nFIG. 7. VISHE\u0000Hloops of a GGG/YIG(36)/GIG(30)/Pt sample at\nvarious temperatures (a) 35 K, (b) 60 K, (c) 110 K and (d) 306 K.\nFor (d) a larger field range is displayed. VISHE is normalized by the\nestimated temperature gradient and a constant offset of the signal is\nsubtracted.\nsurements. Thus, a possible contribution to the spin current\noriginating from YIG seems to be damped in the GIG layer in\nthis sample. At temperatures above Tcomp ;Band below Tcomp ;G\n(zone II), a segmented switching is observed. This is expected\nfor the GIG sensitive SSE measurement, as the GIG layer\nswitches at higher fields in this temperature range (thus the\nsecond step in the field sweep). The sign change of the SSE\nsignal VISHE at low temperatures also suggests the GIG layer\nto be the main spin current source. For even lower temper-\natures ( T=35 K), the SSE signal undergoes a sign change,\nwhich can be explained by the change of spectral weight and\ntemperature dependence of occupied magnon modes in GIG\n[22, 23]. For pure GIG, another sign change is observed at\ncompensation temperature Tcomp ;G, because of the reversal of\nthe sublattice magnetization for a constant external applied\nmagnetic field [22]. As the SSE signal follows the GIG mag-\nnetization, we can investigate the orientation of the GIG at\nhigher temperatures. When changing the temperature from\nbelow to above Tcomp ;G, there is no sign change observed in\nthe YIG/GIG bilayer. For a coupling of YIG and GIG via the\nnet moment of each film, a reversal of the sign of VISHE would\nbe expected at Tcomp ;G, which is not supported by our data.\nThis is in line with our claim, that the Fe sublattice moments\nof YIG and GIG dominate the coupling, thus when changing\nthe temperature across Tcomp ;Gthe GIG orientation stays the\nsame and no reversal of the sign of VISHE is observed.\nIV . DISCUSSION\nTogether, our measurements indicate that we have robust\nFe-Fe exchange coupling between defined YIG and GIG lay-\ners. While SQUID gives a basis of the interpretation of the\nmagnetic behavior of the heterostructures, both SSE and SMR\nselectively show the behavior of the top GIG layer only. Our\nhigh quality thin films access a phase space of samples witharbitrarily aligned magnetization of the bilayers, depending\nof the relative thicknesses. Up to now, the exchange cou-\npling between YIG and GIG has only been described as an\ninterface effect in GGG/YIG samples [12–14]. The artificial\nstacks grown here give much better control over the bilayer\nproperties and allow for fine tuning of these by choosing the\nrelative thickness of YIG and GIG. We identify three temper-\nature regions, where the samples have fundamentally differ-\nent responses to an external magnetic field. Above the GIG\ncompensation point Tcomp ;G, the coupled net iron moments are\nalways parallel and show in the direction of the external mag-\nnetic field. Reducing the temperature below Tcomp ;G, we still\nobserve ferromagnetic Fe-Fe coupling across the YIG/GIG in-\nterface at low magnetic fields. Increasing the field, we break\nthis coupling across the interface and the Gd moment of the\nGIG layer orients with the external field. Uniaxial SMR mea-\nsurements show the switching of the top GIG layer and rota-\ntion measurements indicate a continuous rotation of the mag-\nnetic moments in the reorientation regime. Decreasing the\ntemperature further, we observe a second compensation point,\nwhere the net GIG layer moment equals the YIG layer mo-\nment. We label this bilayer compensation point Tcomp ;B. We\nshow that Tcomp ;Bcan be tuned by choosing the relative thick-\nness of the YIG and GIG layers. This allows for devices with\ncontrollable magnetization direction. Below Tcomp ;B, we ob-\nserve again a double switching in SQUID, indicating the re-\norientation of one of the layers. Both SMR and SSE measure-\nments do not show this switching, indicating that the bottom\nYIG moment rotates, leaving the top GIG moment aligned\nwith the field. SSE measurements thereby prove to be sen-\nsitive to the top GIG layer only, since the switching of the\nbottom YIG spin current source is not observed in our mea-\nsurements. Moreover, SSE measurements show a change of\nsign at certain temperature, which is not associated with the\ndirection of the net magnetic moment, but with the magnon\npopulation as observed in pure GIG layers [22, 23]. Recently,\nit was reported that magnon hybridization may occur in YIG-\nGIG heterostructures leading to a reduction of the temperature\nof the sign change. We note that here, the change of sign\noccurs at around 35 K for a YIG(36 nm)/GIG(30 nm) sam-\nple. SSE measurements performed at pure GIG layers have\nshown the sign change to occur at around 44 K to 72 K [22],\ndepending on several parameters like magnetic compensation\npoint, heavy metal material and GIG-heavy metal interface\nquality. The here observed sign change temperature of 35 K\nis remarkably low compared to these reports, which might in-\ndicate magnon hybridization. However, a direct comparison\nwith literature values is difficult due to the strong dependence\non the surface quality of the samples. A simultaneous depo-\nsition of the HM layer excludes a different GIG/HM interface\nfor different samples and will be target of futures studies.\nFor the interlayer coupling strength between YIG and GIG,\nwe assume that the coupling energy is equal to the gain of\nZeeman energy to align both YIG and GIG magnetization in\nzones II and III. We calculate the energy at the low temper-\nature state of sample YIG(36 nm)/GIG(30 nm). From Fig.\n5 (a) we extract the saturation field at 20 K, which is the\nfield at which Amp 2 is reached as m0H=0:1 T. Rotating7\nm(YIG) from antiparallel to parallel leads to a gain in Zee-\nman energy by 2\u0001m(YIG )\u0001H.m(YIG) is given by m(YIG)=\n1\n2(Amp 2\u0000Amp 1)\u00191:5\u000210\u00007Am2. Taking into account\nthe interface area of 0.25 cm2, we estimate an effective inter-\nface coupling energy of 0.0012 J/m2. With 8 Fe atoms (d or a\nsite) on the surface of a (001) oriented unit cell this relates to\n1.4 meV per Fe atom at the interface. Comparing this to the\ndominant exchange parameter J1in YIG, which is found to\nbe 6.8 meV [7] and J1in GIG of 4.0 meV [14], we observe a\nqualitative agreement, but the interlayer coupling strength ap-\npears to be smaller than the direct exchange coupling in pure\nYIG and pure GIG but of the same order of magnitude as the\ntheoretical limits. For interfacial coupling of YIG-GIG at the\nYIG/GGG interface, a coupling strength of 0.14 meV was dis-\ncussed by Gomez et al. based on the c-Gd antiferromagnetic\ninteraction Jcdto d-Fe [14]. Our value is significantly larger\nthan this. This can be understood by the fact that they in-\nvestigate an ultra-thin layer formed by interdiffusion with the\nsubstrate. In our case, however, the Jcdinteraction energy is\nnot a limiting factor as it needs to be integrated over the vol-\nume/thickness of the Gd layer and the resulting energy is then\nmuch larger than the interfacial exchange coupling between\nYIG and GIG layers that is limited by Fe-Fe interactions at\nthe interface. In reality, the competition between different\nexchange interactions can easily lead to more complicated\nspin structures than in our fully collinear toy model discussed\nabove and we see hints for this in the continuous rotation of\nthe magnetization observed in SMR measurements. Also in\nour samples we have the YIG-GGG interface at the substrate\nand therefore the coupling effects induced by intermixing with\nthe substrate should be present. However, as this interface is\nultrathin and contributes very little to the total sample magne-\ntization it can be neglected to first order in our analysis and it\nwas only observed at very low temperatures by Gomez at al.\n[14]. Traces of this intermixing GIG at the substrate interface\nmight be seen in the low amplitude SQUID curves shown in\nFig. 2, where the pure YIG as well as the bilayers with a thin\nYIG layer appear slightly exchange-biased. In spite of the\nlimitations of the simplified toy model for the analysis we can\nsafely state that the largest interfacial coupling dominating the\nproperties of our bilayers stems from the upper YIG/GIG in-\nterface.V . CONCLUSION\nYIG/GIG bilayers were fabricated by PLD and the mag-\nnetic coupling of the samples was investigated by SQUID,\nSSE and SMR. It is found that the YIG/GIG bilayers show\nferrimagnetic features. The compensation temperature of the\nbilayer system shifts to lower temperatures with increasing\nthickness of the YIG layer, which originates from the effec-\ntive ferromagnetic coupling of the iron magnetic sublattices\nat the YIG-GIG interface, i.e. in the epitaxial bilayer, the iron\natoms spin subsystem on respective d-Fe and a-Fe sites is co-\nherent over the boundary of the two materials in zero mag-\nnetic field. Below the compensation temperature of the GIG\nlayer, an external magnetic field can break the coupling, lead-\ning to a parallel magnetization of YIG and GIG layers at high\nfield. At low magnetic fields, the orientation of the YIG and\nGIG magnetic moment depends on the temperature (Gd mo-\nment). SMR and SSE measurements reveal the behavior of\nthe top GIG layer, from which the temperature and field de-\npendence of alignment of YIG and GIG can be obtained. Our\nresults demonstrate that the magnetic coupling in insulating\nYIG/GIG heterostructures can be manipulated analogously to\nthat of metallic spin valves and open the pathway to manipu-\nlate the magnon transport.\nACKNOWLEDGMENTS\nThe authors gratefully acknowledge funding by Deutsche\nForschungsgemeinschaft (DFG, German Research Founda-\ntion) Project No. 358671374. This work was supported by\nthe Max Planck Graduate Center with the Johannes Guten-\nberg–Universität Mainz (MPGC) as well the Graduate School\nof Excellence Materials Science in Mainz (GSC266). This\nwork was funded by the Deutsche Forschungsgemeinschaft\n(DFG, German Research Foundation) Spin+X (A01+B02)\nTRR 173 - 268565370. This work is partially supported\nby the National Key R&D Program of China (Grant No.\n2018YFB0704100), the National Science Foundation of\nChina (Grants No. 11974042, No. 51731003, No. 51927802,\nNo. 51971023, No. 11574027, and No. 61674013). 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Goennenwein,6, 7and Mathias Weiler1, 2\n1Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany\n2Physik-Department, Technische Universit¨ at M¨ unchen, 85748 Garching, Germany\n3Center for Nanoscale Science and Technology, National Institute of\nStandards and Technology, Gaithersburg, Maryland 20899-6202, USA\n4Institute for Research in Electronics and Applied Physics,\nUniversity of Maryland, College Park, MD 20742\n5Nanosystems Initiative Munich, 80799 Munich, Germany\n6Institut f¨ ur Festk¨ orperphysik, Technische Universit¨ at Dresden, 01062 Dresden, Germany\n7Center for Transport and Devices of Emergent Materials,\nTechnische Universit¨ at Dresden, 01062 Dresden, Germany\nWe investigate yttrium iron garnet (YIG)/cobalt (Co) heterostructures using broadband ferro-\nmagnetic resonance (FMR). We observe an efficient excitation of perpendicular standing spin waves\n(PSSWs) in the YIG layer when the resonance frequencies of the YIG PSSWs and the Co FMR line\ncoincide. Avoided crossings of YIG PSSWs and the Co FMR line are found and modeled using mutual\nspin pumping and exchange torques. The excitation of PSSWs is suppressed by a thin aluminum\noxide (AlOx) interlayer but persists with a copper (Cu) interlayer, in agreement with the proposed\nmodel.\nIn magnonics, information is encoded into the electron\nspin-angular momentum instead of the electron charge\nused in conventional CMOS technology [ 1–10]. Magnon-\nics based on exchange spin waves is particularly appeal-\ning, due to isotropic spin-wave propagation with small\nwavelengths and large group velocities [ 5]. With its long\nmagnon propagation length, yttrium iron garnet (YIG)\nis especially interesting for this application. However, an\nexcitation of exchange spin waves by microwave magnetic\nfields requires nanolithographically defined microwave an-\ntennas [ 11] that have poor efficiency due to high ohmic\nlosses and impedance mismatch.\nHere, we show that exchange spin waves can be ex-\ncited by interfacial spin torques (ST) in YIG/Co het-\nerostructures. These STs couple the YIG and Co mag-\nnetization dynamics by microwave frequency spin cur-\nrents. Phenomenological modeling of the coupling reveals\na combined action of exchange, damping-like and field-like\ntorques that are localized at the YIG/Co interface. This is\nin contrast to the previously observed purely damping-like\nST in all-metallic multilayers [12].\nWe study the magnetization dynamics of YIG/Co thin\nfilm heterostructures by broadband ferromagnetic res-\nonance (FMR) spectroscopy. From our FMR data we\nfind an efficient excitation of perpendicular standing spin\nwaves (PSSWs) in the YIG when the YIG PSSW reso-\nnance frequency is close to the Co FMR line. We ob-\nserve about 40 different PSSWs with wavelengths down\ntoλPSSW≈50 nm.\nClear evidence for the coupling is provided by avoided\ncrossings and corresponding characteristic changes of the\n∗stefan.klingler@wmi.badw.delinewidths of the YIG PSSW and the Co FMR line. This\ncoupling and the excitation of PSSWs is also observed\nwhen a copper (Cu) layer separates the YIG and the\nCo films. However, the insertion of an insulating AlOx\ninterlayer completely suppresses the excitation of YIG\nPSSWs. This allows us to exclude dipolar coupling as the\norigin of the PSSW excitation and is in agreement with\nthe mediation of the coupling by spin currents. Our data\nare well described by a modified Landau-Lifshitz-Gilbert\nequation for the Co layer, which includes direct exchange\ntorques and spin torques from mutual spin pumping at\nthe YIG/Co interface. Simulations of our coupled systems\nreveal the strong influence of spin currents on the coupling\nof the different layers.\nWe investigate a set of four YIG/Co samples, which\nare YIG/Co(50), YIG/Co(35), YIG/Cu(5)/Co(50) and\nYIG/AlOx(1.5)/Co(50), where the numbers in brackets\ndenote the layer thicknesses in nanometers. The YIG\nthicknessd2is = 1µm for all samples. The FMR mea-\nsurements are performed at room temperature using a\ncoplanar waveguide (CPW) with a center conductor width\nofw= 300µm. The CPW is connected to the two ports\nof a vector network analyzer (VNA) and we measure the\ncomplexS21parameter as a function of frequency fand\nexternal magnetic field H(for details of the sample prepa-\nration and the FMR setup see Supplemental Material S1\nand S2 [13]).\nFig. 1 (a) shows the background-corrected field-\nderivative [ 14] of the VNA transmission spectra\n|∂DS21/∂H|for the YIG/Co(50) sample as a function\nofHandfas explained in S3 [ 13] and we clearly observe\ntwo major modes. The low frequency mode corresponds\nto the YIG FMR line, whereas the high-frequency mode\ncorresponds to the Co FMR line. Within the broad Co\nFMR line, we find several narrow resonances, of whicharXiv:1712.02561v1  [cond-mat.mes-hall]  7 Dec 20172\nYIG/Cu(5 nm)/Co(50 nm) YIG/Co(50 nm) YIG/AlOx(1.5 nm)/Co(50 nm)\n5 10 15 20 25 5 10 15 20 25 5 10 15 20 250.1\n00.20.30.4 1\n0.5\n0µ0H (T)\nf(GHz) f (GHz) f (GHz)(c) (b) (a)\n|∂DS21/∂H| (arb.u.)\nYIG Co\nCo PSSW\nexchange \nmode\nFIG. 1. (Color online) Field-derivative of the Vector Network Analyzer (VNA) transmission spectra for three different\nsamples as a function of magnetic field and frequency. All samples show two modes corresponding to the YIG (low-frequency\nmode) and Co (high-frequency mode) FMR lines. The color scale is individually normalized to arbitrary values. (a) The\nYIG/Co(50) sample additionally reveals YIG PSSWs and pronounced avoided crossings of the modes for small frequencies.\n(b) The YIG/Cu(5)/Co(50) sample also shows the YIG PSSWs, but the frequency splittings of the modes are much smaller than\nin (a). (c) The YIG/AlOx(1.5)/Co(50) sample does not show any PSSWs in the Co FMR line as expected if the YIG and the\nCo films are magnetically uncoupled.\nthe dispersion is parallel to the YIG FMR. These lines\nare attributed to the excitation and detection of YIG\nPSSWs with wavelengths down to 50 nm (for details see\nFig. S5 [ 13]). We find avoided crossings between these\nYIG PSSWs and the Co FMR line (inset), where the\nfrequency splitting geff/2π≤200 MHz (see S4 [ 13] for\ndetails). This is a clear indication that the YIG and Co\nmodes are coupled to each other. Furthermore, an addi-\ntional low-frequency mode with lower intensity is observed\nin Fig. 1 (a). This line is attributed to an exchange-spring\nmode of the coupled YIG/Co system. A qualitatively sim-\nilar transmission spectrum is observed for the YIG/Co(35)\nsample (for details see Fig. S6 [ 13]). Furthermore, we\nobserve the first Co PSSW at around f= 22 GHz and\nµ0H= 0.1 T for samples with a 50 nm thick Co layer.\nFig. 1 (b) shows|∂DS21/∂H|for the YIG/Cu(5)/Co(50)\nsample as a function of Handf. Again, we observe\nthe YIG FMR, YIG PSSWs and the Co FMR lines.\nHowever, the frequency splitting between the modes (in-\nset) is much smaller in comparison to the YIG/Co(50)\nsample,geff/2π≤40 MHz. This strongly indicates\nthat the coupling efficiency is reduced in comparison to\nFig. 1 (a). We attribute this mainly to the suppression\nof the static exchange coupling by insertion of the Cu\nlayer. This is also in agreement with the vanishing of the\nexchange mode. Fig. 1 (c) displays |∂DS21/∂H|for the\nYIG/AlOx(1.5)/Co(50) sample as a function of Hand\nf. No YIG PSSWs are observed within the Co FMR line\n(inset Fig. 1 (c)). This provides strong evidence that the\ninsertion of the thin AlOx layer suppresses the coupling\nbetween the YIG and Co magnetization dynamics. An\nanalysis of the Co FMR linewidth (for details see S7 [ 13])\nalso demonstrates that the AlOx layer eliminates anycoupling between the YIG and Co layers. From Fig. 1,\nwe conclude that any magneto-dynamic coupling is sup-\npressed by insertion of a thin insulator between the two\nmagnetic layers. This provides strong evidence against\na magnetostatic coupling by stray fields, and is in agree-\nment with a dynamic coupling mediated by spin currents,\nwhich can pass through the Cu layer, but are blocked by\nthe AlOx barrier.\nFig. 2 shows the magnetic hysteresis loops of the\nYIG/Co samples recorded by Superconducting Quantum\nInterference Device (SQUID) magnetometry. The hys-\nteresis loop of the YIG/Co(50) sample (solid blue line in\nFig. 2) exhibits a sharp switching at the YIG coercive\nfield of about 0.1 mT. However, no sharp switching of the\nCo layer is visible but a smooth increase of the measured\nmagnetic moment until the bilayer magnetization is satu-\nrated. This can be explained by a direct, static exchange\ncoupling between YIG and Co magnetizations (inset), as\nknown from exchange springs [ 15,16]. The form of the\nhysteresis loop suggests an antiferromagnetic coupling,\nas comparably large magnetic fields are required to force\na parallel alignment of the layers. However, without a\ndetailed examination of the remnant state, we cannot rule\nout any ferromagnetic coupling. By inserting a Cu or\nAlOx layer between the YIG and the Co (dash-dotted and\ndashed lines in Fig. 2) we find a sharp switching at the Co\ncoercive field µ0Hc≈1 mT. This switching is in agree-\nment with the behavior expected for statically uncoupled\nmagnetic layers [ 17]. However, we still observe a dynamic\ncoupling in Fig. 1 (b) in the YIG/Cu(5)/Co(50) sample.\nSince we expect no static exchange coupling between Co\nand YIG in this sample, this observation requires a differ-\nent mechanism as the origin of the excitation of the YIG3\nPSSWs.\nWe model the data of Fig. 1 with a modified Landau-\nLifshitz-Gilbert approach, which includes finite mode cou-\npling between the YIG and the Co magnetizations at\nthe YIG/Co interface at z=d2. We model the Co mag-\nnetization M1as a macrospin, which is fixed primarily\nalong they-direction with small transverse parts and the\nYIG magnetization M2(z) as a vector that depends on\nthe distance zfrom the YIG/Co interface (for detailed\ncalculations see S8, S9 [ 13]). In the limit that the trans-\nverse parts are small, the equation of motion for the Co\nmacrospin then reads:\n˙M1=−γ1ˆy×/bracketleftbigg\n−µ0HM1−α1\nγ1˙M1−µ0Ms,1M1,zˆz\n−J\nd1Ms,1(M1−M2(d2))−µ0h/bracketrightbigg\n−γ1\nd1Ms,1/bracketleftbig\n(τF−τDˆy×)(˙M1−˙M2(d2))/bracketrightbig\n.\n(1)\nHere,α1is the Gilbert damping parameter for Co, γ1and\nMs,1its gyromagnetic ratio and saturation magnetization,\nrespectively, ˆzis the unit vector in z-direction, and d1is\nthe thickness of the Co layer. The magnetic driving field\nfrom the CPW is denoted by h. In our model, his as-\nsumed to be spatially uniform, to reflect the experimental\nsituation where the CPW center conductor width is much\nlarger than either the YIG or Co thickness. The exchange\ncoupling constant between the YIG and the Co is given\nbyJ. The torques due to spin currents pumped from\nnormalized magnetization1\n-1-0.50.5\n0\n-4 4 2 2- 0YIG/C o(50)\nYIG/C u(5)/C o(50)\nYIG/A lOx/Co(50)\nYIGCo\nYIGCuCo\nµ0H (mT)1 3 -1 -3\nFIG. 2. Magnetization of YIG/Co(50) (solid),\nYIG/Cu(5)/Co(50) (dash-dotted) and YIG/AlOx(1.5)/Co(50)\n(dashed) normalized to the magnetization at µ0H= 4 mT.\nThe magnetic hysteresis loops of YIG/Co show an enhance-\nment of the Co coercive field as well as a rather smooth\nswitching. The samples with a Cu or AlOx interlayer reveal a\nsharp switching of the magnetization at the Co coercive field\nµ0Hc≈1 mT. The inset shows a possible static magnetization\ndistribution in a exchange coupled (left) and an uncoupled\n(right) heterostructure.one layer and absorbed in the other have field-like τFand\ndamping-like τDcomponents. The YIG magnetization\ndirection at the YIG/Co interface is given by M2(d2).\nThe YIG magnetization obeys two boundary conditions.\nFirst, the total torque at the YIG/Co interface at z=d2\nhas to vanish:\n0 =2Aˆy×∂zM2(z)|z=d2−Jˆy×(M1−M2(d2))\n+ (~/e)(τF−τDˆy×)/parenleftbig˙M1−˙M2(d2)/parenrightbig\n.(2)\nHere,Ais the exchange constant of YIG. Second,\nwe assume an uncoupled boundary condition at the\nYIG/substrate interface\n0 = 2Aˆy×∂zM2(z)|z=0, (3)\nwhere the torque vanishes as well. The Co susceptibility\nχ1is then derived using the ansatz for the transverse YIG\nmagnetization m2(z,t) = (m2,x(z,t),m2,z(z,t)):\nm2(z,t) = Re/bracketleftbig\nc+m2+cos(kz) exp(−iωt)\nc−m2−cos(κz) exp(−iωt)/bracketrightbig\n.(4)\nHere,m2±are the complex eigenvectors of the uncoupled\ntransverse YIG magnetization, c±are complex coefficients,\nω= 2πfis the angular frequency, kandκare complex\nwavevectors of the undisturbed YIG films. The transverse\nCo magnetization follows a simple elliptical precession:\nm1= Re [ m1,0exp(−iωt)] (5)\nwhere m1= (m1,x,m1,z), and m1,0≈(m1,0,x,m1,0,z) is a\ncomplex precession amplitude. After finding the complex\ncoefficients c±, the Co susceptibility χ1can be obtained\nfrom Eq. (1).\nFig. 3 (a-c) show the simulated microwave signal\n|∂DS21/∂H| ∝ |∂χ1/∂H|(for details see S3, S9 [ 13]).\nFor all simulations we take the same material parame-\nters, namely µ0Ms,1= 1.91 T,A= 3.76 pJ/m,α1=\n7.7×10−3,α2= 7.2×10−4,γ1= 28.7 GHz/T and\nγ2= 27.07 GHz/T, as extracted in S4, S5, S7 [ 13]. The\nthicknesses are d1= 50 nm and d2= 1µm. For the YIG\nsaturation magnetization we take the literature value\nµ0Ms,2= 0.18 T [ 18]. In Fig. 3 (a) we show the simula-\ntions for the YIG/Co(50) sample using τF= 30 A s/m2,\nτD= 15 A s/m2andJ=−400µJ/m2. The interfacial\nexchange constant J < 0 models an antiferromagnetic\ncoupling as suggested by the SQUID measurements. The\nsign of the damping-like torque is required to be positive,\nas it depends on the real part of the spin mixing con-\nductance of the interface. The simulation reproduces all\nsalient features observed in the experiment, in particu-\nlar the appearance of the YIG PSSWs and their avoided\ncrossing with the Co FMR line. Note that the simulations\ndo not reproduce the YIG FMR, as we only simulate\nthe Co susceptibility. However, we can obtain a similar\ncolor plot for a ferromagnetic coupling and a negative\nfield-like torque (see for example S6, S10 [ 13]). The com-\nbination of exchange torques with the field-like torques4\n \nτF = 30 As/m2, τD = 15 As/m2\nJ = 0τF =0, τD = 0, J = 0\n5 10 15 20 25 5 10 15 20 251\n0.5\n0\nf (GHz) f (GHz)(c) (b) (a)τF =30 As/m2, τD = 15 As/m2\nJ = -400 µJ/m2\n5 10 15 20 25µ0H (T)\nf(GHz)0.1\n00.20.30.4\n|∂DS21/∂H| (arb.u.)\nFIG. 3. (Color online) Calculated |∂DS21/∂H|of the simulated transmission spectra. Simulation of the (a) YIG/Co(50) sample,\n(b) YIG/Cu(5)/Co(50) sample, (c) YIG/AlOx(1.5)/Co(50) sample.\nat the FM1/vextendsingle/vextendsingleFM2interface complicates the analysis of the\ntotal coupling because both torques affect the coupling\nin very similar ways. Hence, the signs of the field-like\ntorque and the exchange torque cannot be determined\nunambiguously for the YIG/Co(50) sample.\nIn Fig. 3 (b) we show the simulations for the\nYIG/Cu(5)/Co(50) sample. Here, τFandτdare un-\nchanged compared to the values used for the simulation of\nthe YIG/Co(50) sample, but we set J= 0, as no static cou-\npling was observed for YIG/Cu(5)/Co(50) in the SQUID\nmeasurements. The simulation is in excellent agreement\nwith the corresponding measurement shown in Fig. 1 (b).\nThe elimination of the static exchange coupling results in\na strong reduction of the coupling between the YIG and\nCo magnetization dynamics. However, the Cu layer is\ntransparent to spin currents mediating the field-like and\ndamping-like torques, as the spin-diffusion length of Cu is\nmuch larger than its thickness [ 19]. We note that a finite\nfield-like torque is necessary to observe the excitation of\nthe PSSWs for vanishing exchange coupling J. Further-\nmore, the field-like torque is required to be positive to\nmodel the intensity asymmetry in the mode branches of\nthe YIG/Cu(5)/Co(50) sample (cf. Fig. S10 [13]).\nIn Fig. 3 (c) we use τF=τD=J= 0, which reproduces\nthe experimental observation for the YIG/AlOx/Co(50)\nsample. Importantly, no YIG PSSWs are observed in\neither the experiment or the simulation for this case.\nIn summary, the simulations are in excellent qualitative\nagreement with the experimental observation of spin dy-\nnamics in the coupled YIG/Co heterostructures.\nWe attribute small quantitative discrepancies between\nthe simulation and the experiment to the fact that we do\nnot take any inhomogeneous linewidth and two-magnon\nscattering into account, which is, however, present in our\nsystem (see S7 [ 13] for details). This results in an under-\nestimated linewidth of the Co FMR line, in particular\nfor small frequencies. As |∂DS21/∂H|is inversely propor-tional to the linewidths, this causes small quantitative\ndeviations of the simulations and the experimental data.\nFurthermore, the exchange modes in Fig. 1 (a) are not\nfound in the simulations. We attribute this to the fact\nthat the simulations only represent the Co susceptibil-\nity. However, as shown in Fig. S10 [ 13], similar exchange\nmodes can also be found in the Co susceptibility from our\nsimulations.\nIn conclusion, we investigated the dynamic magnetiza-\ntion coupling in YIG/Co heterostructures using broad-\nband ferromagnetic resonance spectroscopy. We find ex-\nchange dominated PSSWs in the YIG, excited by spin\ncurrents from the Co layer, and static interfacial exchange\ncoupling of YIG and Co magnetizations. An efficient\nexcitation of YIG PSSWs, even with a homogeneous ex-\nternal magnetic driving field, is found in YIG/Co(35),\nYIG/Co(50) and YIG/Cu(5)/Co(50) samples, but is sup-\npressed completely in YIG/AlOx(1.5)/Co(50). We model\nour observations with a modified Landau-Lifshitz-Gilbert\nequation, which takes field-like and damping-like torques\nas well as direct exchange coupling into account.\nOur findings pave the way for magnonic devices which\noperate in the exchange spin-wave regime. Such devices\nallow for utilization of the isotropic spin-wave dispersion\nrelations in 2D magnonic structures. An excitation of\nshort-wavelength spin waves by an interfacial spin torque\ndoes not require any microstructuring of excitation an-\ntennas but is in operation in simple magnetic bilayers.\nRemarkably, this spin torque scheme allows for the cou-\npling of spin dynamics in a ferrimagnetic insulator to that\nin a ferromagnetic metal. The coupling is qualitatively dif-\nferent to that found for all-metallic heterostructures [ 12].\nFurthermore, the excitation of magnetization dynamics\nby interfacial torques should allow for efficient manipula-\ntion of microscopic magnetic textures, such as magnetic\nskyrmions.\nFinancial support from the DFG via SPP 1538“Spin5\nCaloric Transport” (project GO 944/4 and GR 1132/18) is\ngratefully acknowledged. S.K. would like to thank Meike\nM¨ uller for fruitful discussions. V.A. acknowledges support\nunder the Cooperative Research Agreement between theUniversity of Maryland and the National Institute of\nStandards and Technology, Center for Nanoscale Science\nand Technology, Award 70NANB14H209, through the\nUniversity of Maryland.\n[1]A. Khitun, M. Bao, and K. Wang, IEEE Transactions\non Magnetics 44, 2141 (2008).\n[2]A. A. Serga, A. V. Chumak, and B. Hillebrands, Journal\nof Physics D: Applied Physics 43, 264002 (2010).\n[3]A. V. Chumak, A. A. Serga, and B. 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Cespedes1* \n1School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom.  \n2Scientific Computing Department, Science and Technology Facilities Council, Didcot OX11 0QX, United \nKingdom \n3Beijing Computational Science Research Center, 100193 Beijing, China \n4Stephenson Institute for Renewable Energy, Department of Chemistry, University of Liverpool, L69 3BX \nLiverpool, United Kingdom \n5School of Chemistry, University of Southampton, Highfield, SO17 1BJ Southampton, United Kingdom \n6School of Physics, Trinity College Dublin, The University of Dublin, Dublin 2, Ireland \n7Centre for Research on Adaptive Nanostructures and Nanodevices (CRANN) and the SFI Advanced \nMaterials and Bio-Engineering Research Centre (AMBER), Dublin 2, Ireland \n8SuperSTEM, SciTech Daresbury Science and Innovation Campus, Keckwick Lane, Daresbury WA4 4AD, \nUnited Kingdom \n9Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge CB3 0HE, United Kingdom \n \n5d metals are used in electronic architectures because of their high spin-orbit coupling (SOC) leading \nto efficient spin ↔ electric conversion and strong magnetic interactions. When C 60 is grown on a \nmetal, the electronic structure is altered due to hybridisation and charge transfer. The spin Hall \nmagnetoresistance for Pt/C 60 and Ta/C 60 at room temperature are up to a factor 6 higher than for \nthe pristine metals, with the spin Hall angle increased by 20-60%. At low fields of 1-30 mT, there is \nan anisotropic magnetoresistance, increased up to 700% at room temperature by C 60. This is \ncorrelated with non-collinear Density Functional Theory simulations showing changes in the \nacquired magnetic moment of transport electrons via SOC. Given the dielectric properties of \nmolecules, this opens the possibility of gating the effective SOC of metals, with applications for spin \ntransfer torque memories and pure spin current dynamic circuits. \n                                                           \n* o.cespedes@leeds.ac.uk 2 \n The spin-orbit interaction is perhaps the most crucial mechanism in the design of magnetic \nstructures and metal device physics. It determines the magnetocrystalline anisotropy, is key to the \npropagation and electrical conversion of spin currents, determines the magnitude of interfacial \nmechanisms such as the Dzyaloshinskii–Moriya interaction and guides new paths of research, such \nas the generation of Majorana fermions and energy band engineering of topological insulators[1-5]. \nThe SOC also controls the efficiency of spin - charge conversion in the spin Hall, spin torque and spin \nSeebeck effects. All of these are key to reducing the power consumption and energy dissipation of \ncomputing and electronic devices, an issue that is quickly coming to the forefront of technology. \nHowever, currently we can only tune the SOC by static means, such as doping, preventing the design \nof architectures where spin, charge and magnetic interactions can be reversibly modified to enhance \ndevice performance or to acquire new functionalities. \nThe Spin Hall magnetoresistance (SHMR) can quantify the SOC in thin (~nm) heavy metal layers \ndeposited on a magnetic insulator such as the yttrium iron garnet Y 3Fe5O12 (YIG) [6-8]. When an \nelectric current Jc flows in the metal, the spin Hall effect (SHE) induces a perpendicular spin current \nJs, with the spin polarization s parallel to the film surface. If the YIG magnetization M is parallel to s, \nJs cannot flow into the magnet and a spin accumulation forms. The resistance is the same as a bare \nPt wire. When M is not parallel to s, the transverse component exerts a torque on the YIG magnetic \nmoments, injecting spin current into the magnet. This opens a channel of dissipation for the spin \ncurrent, reducing the inverse SHE contribution to Jc so that the resistance of Pt appears to have \nincreased [5,9]. The largest dissipation takes place when M is perpendicular to s and the maximum \nSHMR should occur. The SHMR is measured by rotating the angle  in Fig. 1a, with the applied H field \n(and therefore M) always orthogonal to the electrical current, but varying from in-plane to out-of-3 \n plane, and therefore from parallel (R min) to perpendicular to the spin polarization (R max) [5,10,11]. \nThe ratio of the spin to charge current is known as the spin Hall angle: 𝜃ௌு=|𝑱𝒔|/|𝑱𝑪| [12-14]. 𝛩ௌு \nhas technological relevance, as it is correlated with the torque exerted on ferromagnets in spin \ntransfer torque memories [15]. A larger SHMR and therefore increased 𝜃ௌு can result in lower power \nor smaller switching currents for such devices. By tuning the SOC in conventional magnetic \ninsulator/metal structures with a molecular layer, we can also differentiate spin transport effects \nbased on their physical origins [16,17]. \nAt metallo-molecular interfaces, the electronic and magnetic properties of both materials change \ndue to charge transfer and hybridisation [18-21]. This can lead to the emergence of spin ordering \nand spin filtering [22-25], or change the magnetic anisotropy [21,26,27]. Even though composed of \nlight carbon, fullerenes with large curvature can produce a large spin-electrical conversion [2,28-30]. \nHere, we study the effect of metal/C 60 interfaces on the SHMR and anisotropic magnetoresistance \n(AMR) of YIG/Pt and YIG/Ta. We aim to: investigate the mechanisms behind spin orbit scattering at \nhybrid metal/C 60 interfaces, maximise technologically-relevant parameters, and open new paths of \nresearch towards tunable SOC.  \nUsing shadow mask deposition, we grew two metal wires simultaneously on the same YIG \nsubstrate and, without breaking vacuum, covered one wire with 50 nm of C 60 –modifying the density \nof states (DOS) and transport properties of the metal. According to our density functional theory \n(DFT) calculations for Pt/C 60, 0.18-0.24electrons per C 60 molecule are transferred [31], and the first \nmolecular layer is metallised. This reduces the electron surface scattering, improving the residual \nresistance ratio (RRR) –Figs. 1b and SI [31]. Our Ta wires have a resistivity (~1-2 ·m) and a negative 4 \n temperature coefficient (~-500 ·10-6 K-1), consistent with a sputtered -Ta phase [32]. Opposite to Pt, \nC60 increases the resistivity of Ta –see Fig. 1b.  \nThe change in resistivity as the magnetic field is rotated is fitted to a cosଶ(𝛽) function, and the \namplitude is taken to be the SHMR [31]. The SHMR saturates once the applied field saturates the \nmagnetisation out-of-plane, at 0.1-0.15 T for a YIG film 170 nm thick at 290 K, and no higher than \n0.5 T for any measured condition. However, above this field range, other contributions such as \nKoehler MR, localisation and the Hanle effect can result in significant linear and parabolic \ncontributions to the MR that would artificially enhance the SHMR ratio and SH (Fig 1c)[31,33]. For a \nYIG/Pt(2nm) sample, the C 60 layer increases the MR due to spin accumulation by about a factor 3, \nbut reduces the polynomial contributions because of the increased effective (conducting) thickness \nof the Pt/C 60 bilayer. In YIG/Ta(4nm), where C 60 increases the resistance rather than reducing it, both \nthe SHMR up to 0.15 T and the polynomial MR at higher fields are enhanced (Fig. 1d). \n \n 5 \n  \nFIG. 1 (a) Schematic of the experiment. There are three possible orientations of the magnetic field \n(H) w.r.t. the electrical current and the YIG film. To measure only the SHMR without AMR effects, we \nrotate H from perpendicular to transverse (change in ). (b) Typical resistivity of thin Pt (3 nm) and \nTa (4 nm) wires on YIG. With C 60 on top, the Pt resistivity is about 40% lower, the RRR factor increases \nand the upturn at low T is absent. With Ta, we observe the opposite effect, an increase in the \nresistivity with the molecular interface. Inset: Resistance with different applied fields as a function \nof the angle . The data is fitted to a cos2( function. We take the amplitude at the lowest field of \n0.5 T, when the YIG substrate is saturated but the polynomial contributions are small, as the SHMR \nvalue. (c) MR in a Pt wire with H perpendicular . The spin Hall contribution to the MR at 300 K reaches a \nmaximum at ~0.1-0.15 T, where the YIG film is saturated out of plane. (d) MR in a Ta wire with \nHperpendicular  at 75 K. The maximum in the spin Hall contribution at this temperature is reached at ~0.2 \nT.  \n6 \n The SHMR values at 0.5 T for Pt and Pt/C 60 are plotted in Fig. 2a, and the ratios with and without a \nmolecular overlayer in Fig. 2b.  The temperature dependence of the SHMR reproduces observations \nin RF-sputtered YIG/sputtered Pt wires [33]. For Pt grown by evaporation on thicker, liquid epitaxy \nor pulsed laser deposition YIG, the SHMR has a gentler drop at high temperatures. This is attributed \nto a smaller temperature dependence of the spin diffusion length [34,35], which could be due to a \ndifferent resistivity of Pt and different magnetic behaviour of YIG films depending on the growth \nmethod. It is possible that the larger SHMR observed in metallo-molecular wires could be due to a \nchange in the spin mixing conductance ( G↑↓) induced by C 60 [6,36]. However, G↑↓ is related to the \nspin transparency of the YIG/Pt interface, where the effect of the molecular interface is small [31]. \nAlso, we do not observe an increase in the ferromagnetic resonant damping , proportional to G↑↓, \nof YIG/Pt with C 60 interfaces (Fig. 2c) [31,37]. Furthermore, the SHMR versus temperature results \ncannot be fitted by changing G↑↓ without also changing 𝛩ௌு [31]. Fig. 2d shows the 𝛩ௌு values \ntaking G↑↓=4×1014 -1m-2 [8]; see the SI for other fitting values [34,38]. For Pt wires of ≤5 nm, there \nis an increase in 𝛩ௌு with C 60. This effect disappears for thick wires (> 10nm), where the molecular \ninterface does not significantly change the spin Hall angle. A similar molecular enhancement of the \nSHMR and 𝛩ௌு is observed for Ta wires [31]. Molecules may affect the Rashba effect and spin texture \nof the metal, leading to changes in the effective SOC of the hybrid wire [39-41]. In our simulations, \nwe consider the perpendicular dipole formed due to charge transfer at the Pt/C 60 interface and its \nassociated potential step breaking symmetry [42]. However, this dipole is maximum at 2.5 nm, \nwhere the experiments show a local minimum. Our calculations point rather towards a mechanism \nmediated by the magnetic moment acquired by the transport electrons, resulting in spin-dependent \ncharge flow [31].  7 \n  \nFIG. 2 (a) SHMR for Pt and equivalent Pt/C 60 wires of different thicknesses on GGG/YIG(170 nm) \nfilms. (b) SHMR Ratios between Pt/C 60 and Pt. The maximum effect of the molecular layer (factor 4 \nto 7 change) take place for thin films (1.5 nm) at low temperatures or thick films (5 nm) at room \ntemperature. (c) The magnetic resonance damping  is not increased by the C 60 interface; here a \ncomparison of YIG/Pt/Al/C 60 and YIG/Pt/C 60 shows similar or even higher damping values for the \ndecoupled Pt/Al/C 60 sample. (d) For wires ≤5 nm, SH obtained from the SHMR data fits is significantly \nhigher with the molecular overlayer. Inset: Top view of the optimized C 60/Pt(111)-(2√3x2√3)R30° \ninterface DFT model. The C 60 molecules are adsorbed on top of one Pt-vacancy. The black polygon \nmarks the in-plane periodicity of the system. Pt: silver, C: cyan.  (e) DFT simulations of the electrical \ncurrent-induced, in-plane magnetic moments (| mxy|) and experimental SHMR, normalized to the \nlargest calculated (| mxy|) or measured value (SHMR) as a function of the Pt thickness.  \n \n \n8 \n Non-collinear band structure calculations enable analysis of the atom Projected (energy-\ndependent) Magnetization Density (PMD) for different Pt and Pt/C 60 film thicknesses. In all cases, we \nfind the PMD for the in-plane (x,y) magnetic moment components ( mx,y) to be larger than for the \nout of plane one ( mz). It is also possible to observe an enhancement of the PMD oscillation \nmagnitudes due to the adsorption of C 60. The effect becomes smaller as the Pt thickness increases \nfrom 1.1 nm to 2.5 nm and 3.9 nm, correlated with the SHMR values in Pt/C 60 (Fig. 2e). The \ndifferences in PMD between the C 60/Pt and Pt systems document the role of the Pt/C 60 interfacial \nre-hybridization, and ensuing changes in the electronic structure, for enhancing SOC-related \nanisotropies and spin transport in Pt-based systems. \nThe fabrication of YIG films can lead to elemental diffusion and defects that change the magnetic \nproperties of the ferrimagnet and the interpretation of transport measurements [43]. Figs. 3a-b \nshow atomic-resolution aberration corrected cross-sectional scanning transmission electron \nmicroscopy (STEM) images and electron energy loss spectroscopy (EELS) chemical maps. It is possible \nto observe, in addition to a certain level of surface roughness of the YIG film, an area close to the \nYIG surface and below the sputtered Pt wire into which some Pt metal may have diffused and formed \na low density of nm-sized clusters (see also Fig. S4 in [31]). This diffusion can affect the magnetization \nand anisotropy direction at the surface of the YIG layer, originating the minor loops we observe in \nthe perpendicular field direction in some YIG films [31,43].  \nFor Pt grown on YIG, an additional change in resistance is observed at low magnetic fields <5-20 \nmT when the direction of an applied magnetic field is changed with respect to the electrical current. \nThe origin of this AMR is controversial. It has been attributed to a proximity-induced magnetization \nof Pt, which is close to the Stoner criterion, but it is also claimed that there is no evidence for this 9 \n induced magnetization [16,17]. The same effect is also seen in YIG/Ta. This low field AMR (LF-AMR) \nis characterized by the presence of peaks, positive or negative depending on the field direction, \nresembling the AMR observed in magnetic films with domain wall scattering [44,45]. Due to the SOC, \nin most magnetic materials domain walls reduce the resistance for in-plane fields, and increase it for \nout of plane fields. This domain wall AMR peaks at the coercive field Hc of the magnet, for the \ngreatest magnetic disorder and domain wall density. In YIG/Pt, the position of out-of-plane LF-AMR \npeaks coincides with the coercivity of the perpendicular minor YIG loops (Fig. 3c and [31]), which \ncould point to a YIG surface layer with an out-of-plane easy axis. We find that the LF-AMR has the \nsame shape and peak position with or without a molecular overlayer. However, the magnitude of \nthe LF-AMR is larger when C 60 is present. This molecular effect is stronger for the perpendicular \nconfiguration (Fig. 3d), which may be due a larger perpendicular magnetic anisotropy induced by C 60, \nas reported for Co [21]. A larger LF-AMR is also observed in YIG/Ta when C 60 is deposited on top [40]. \nFor YIG films grown on YAG substrates, the in-plane coercivity is increased by 1-2 orders of \nmagnitude, and the LF-AMR peaks appear at higher fields, supporting the correlation between the \nAMR in Pt and the surface YIG magnetisation (Figs. S5-S7 in [31]).  \n \n \n \n 10 \n  \nFIG. 3 (a) Cross-sectional high angle annular dark field (HAADF) image of the YIG/Pt interface \nobtained using a scanning transmission electron microscope (see methods for details). (b) Elemental \nchemical analysis of the interface using EELS: the relative intensity maps of the Y, Fe and Pt ionization \nedges are presented with a simultaneously acquired HAADF image of the region, indicated by a white \nbox in (a). Bright clusters immediately below the YIG surface, indicated by white arrows in the Pt \nmap and the overview HAADF image, contain a higher Pt concentration and may be due to Pt \ndiffusion into the YIG. (c) Low field MR and minor hysteresis loop with the field in the perpendicular \norientation at 200 K. The full loop uncorrected and other examples can be found in [31]. (d) Room \ntemperature LF-AMR comparison between YIG/Pt and YIG/Pt/C 60.  The curves are qualitatively the \nsame, but the magnitude of the effect is enhanced by the molecules. \n \n11 \n  The LF-AMR peak position (coercivity of the YIG surface) and peak width (saturation field of the \nYIG surface), increase as the temperature is lowered (Figs. 4a-b). Typically, the AMR of YIG/Pt \nmeasured at high fields is reported to vanish above 100-150 K. If measuring at 3 T, where quantum \nlocalisation and other effects are strong, we observe this same decay with temperature. However, \nthe LF-AMR can be observed up to room temperature. C 60 not only increases the LF-AMR value, but \nit also makes it less temperature dependent, so that the LF-AMR ratio can be up to 700% higher for \nPt/C60 at 290 K. This supports our suggestion from DFT simulations of a mechanism based on C 60-\ninduced re-hybridization enhancing the magnetic moment acquired by transport electrons via SOC \n(Fig. 4c). \nThe LF-AMR depends on the Pt thickness, 𝑡, as (𝑡−𝑥)ିଵ (Fig. 4d). We identify the value of 𝑥, \napproximately 1 nm, as the magnetised Pt region contributing to the AMR. This relationship is not \naffected by the C 60 layer, although the magnitude is uniformly higher with molecules.  \n \n \n \n 12 \n  \nFIG. 4 (a) Perpendicular LF-AMR for GGG/YIG(170)/Pt(2)/C 60(50). (b)  As the sample is cooled, the \nperpendicular LF-AMR peak position and width are increased in steps, rather than monotonic \nfashion.  (c) Temperature dependence of the maximum LF-AMR, calculated as the change in \nresistance from the peak in the perpendicular orientation to the minima in the longitudinal. There is \na faster temperature drop in the MR values for Pt when compared with Pt/C 60. This may be due to \nthe acquired magnetic moment in Pt/C 60 leading to a more stable induced magnetisation up to \nhigher temperatures. (d) The LF-AMR for Pt and Pt/C 60 can be fitted to a (𝑡−𝑥)ିଵ function, where \n𝑡  is the Pt wire thickness and x is a constant of 1 nm that we identify with the magnetically active Pt \nregion.    \n \n13 \n Our results show that molecular overlayers can enhance the spin orbit coupling of heavy metals, \nas observed in SHMR and AMR measurements. Additionally, the molecular layers aid in \ndistinguishing the origin of spin scattering mechanisms, such as the coupling with YIG surface \nmagnetisation and a LF-AMR measurable at high temperatures. The enhancement of the effective \nSOC with molecular interfaces has a wide range of applications, e.g. to reduce the current densities \nin spin transfer torque memories. Given the dependence on surface hybridisation and charge \ntransfer, the effect could be controlled via an applied electrical potential. This is an important \ndevelopment, as nearly all other methods to alter the spin-orbit coupling of a material are static. The \ninverse SHE can be modified by gating with ionic liquids, but changes to the SOC are undetermined \nand the electrical conversion may only be quenched [46]. Materials can be doped during fabrication \nto increase the spin-orbit effect, but that becomes fixed in a circuit, i.e. static.  Using UHV grown \nnanoscale molecular films that can be gated offers a dynamic response – the transport properties of \nan active circuit, e.g. to control the direction and magnitude of pure spin currents. \nACKNOWLEDGMENTS \nThis work was supported by Science Foundation Ireland [19/EPSRC/3605] and the Engineering and Physical \nSciences Research Council (EPSRC) UK through grants EP/S030263, EP/K036408, EP/M000923, EP/I004483 \nand EP/S031081. 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Krawczyk3∗\n1Faculty of Physics, University of Bia/suppress lystok, Bia/suppress lystok, Poland,\n2Institute of Metal Physics, Ural Division of Russian Academ y of Science, Yekaterinburg, Russia,\n3Faculty of Physics, Adam Mickiewicz University in Poznan, U multowska 85, Pozna´ n, Poland.\n4Scientific-Practical Materials Research Center at Nationa l Academy of Sciences of Belarus, Minsk, Belarus\n(Dated: June 10, 2021)\nAbstract\nThe magnonic band gaps of the two types of planar one-dimensi onal magnonic crystals comprised of the periodic array of\nthe metallic stripes on yttrium iron garnet (YIG) film and YIG film with an array of grooves was analyzed experimentally\nand theoretically. In such periodic magnetic structures th e propagating magnetostatic surface spin waves were excite d and\ndetected by microstripe transducers with vector network an alyzer and by Brillouin light scattering spectroscopy. Pro perties\nof the magnonic band gaps were explained with the help of the fi nite element calculations. The important influence of the\nnonreciprocal properties of the spin wave dispersion induc ed by metallic stripes on the magnonic band gap width and its\ndependence on the external magnetic field has been shown. The usefulness of both types of the magnonic crystals for potent ial\napplications and possibility for miniaturization are disc ussed.\nPACS numbers: 75.75.+a,76.50.+g,75.30.Ds,75.50.Bb\nI. INTRODUCTION\nThe spatial periodicity determines the rule of conser-\nvation of the the quasi-momentum for excitations in arti-\nficial crystals, similar to the conservation of momentum\nin homogeneous material. In the frequency domain this\nperiodicity causes the formation of the pass bands and\nband gaps, i.e., frequency regions in which there are no\navailable excitation states and the wave propagation is\nprohibited. Magnetic structures with artificial transla-\ntional symmetry are investigated to design new materials\nwith properties that otherwise do not exist in nature, so\ncalled metamaterials. In particular, artificial ferromag-\nnetic materials with periodicity comparable to the wave-\nlength of spin waves (SWs), known as magnonic crystals\n(MCs),1–3have recently focused attention of the physics\ncommunity. The typical example of the exploitation of\nMCs is control of the propagation and scattering of SWs.\nThefirstexperimentalstudy ofthemagnetostaticSWsin\nferromagnetic thin film with periodic surface was made\nby Sykes et al. already in 1976.4Nowadays, the number\nof studies about MCs has surged and continues to grow\nat a fast pace due to interesting physics and potential\nnew applications.5–8\nIn this study we present the complementary experi-\nmental and theoretical investigation of SWs in two types\nof MCs having the same period in dependence on the\nexternal magnetic field amplitude. The first type is a\nsystem of periodically arranged grooves etched in the\nyttrium iron garnet (YIG) film, the second is an uni-\nform YIG crystal with placed atop metallic stripes. Both\ntypes of structures have already been studied.9–14The\nfirst one was proposed as a SW waveguide and exper-\nimentally tested as delay line or filter for microwave\napplications4,15,16and more recently as a basic element\nof the purely magnonic transistor17or microwave phase(a) 80 μm 150 μm\nGGGYIG10 μm\n1.5μm10μmGGGYIG1μmAuSample A\n(b)\nSample B\nyx\nzUnit cell\nUnit cellH0\nFIG. 1. Geometry of the two MCs investigated in the paper.\n(a) Sample A, 1D MC created by the array of grooves etched\nin the YIG film on the GGG substrate. The thickness of the\nfilm is 10 µm, grooves width and depth is 80 µm and 1.5\nµm, respectively. (b) Sample B, 1D MC formed on the basis\nof homogeneous YIG film of 10 µm thickness by deposition\nof the array of Au stripes. The stripe width is 80 µm and\nthickness 1 µm. The same lattice constant 150 µm is kept in\nboth samples.\nshifter.18Thestructuresofthesecondtypewerealsocon-\nsidered as delay lines and filters16,19but recently have\nalso been proposed as room temperature magnetic field\nsensors.20–22\nWe perform comparative study of these two types of\nMCs magnetically saturated by the external magnetic\nfield along grooves and metallic stripes. For measure-\nments we use the passive delay line with a network ana-\nlyzer and Brillouin light scattering (BLS) measurements.\nThe SW dynamic in these MCs is modeled with finite\nelement method (FEM) in the frequency domain. Ac-\nquired information from calculations is used to explain\nexperimental data and to discus properties of magnonic\nband gap formation in these two kinds of MCs and their\nusefulness for recently proposed applications.\n1The paper is composed as follows, in Sec. II we briefly\ndescribe the experimental methods used in the study:\nmeasurements of the transmission of the SWs with mi-\ncrowave transducers and BLS measurements. In Sec. III\nwe introduce FEM used for calculation of the magnonic\nband structure. In the next section, Sec. IV we discuss\nthe results obtained in two types of MCs. The paper is\nended with Sec. V where the summary of the paper is\npresented.\nII. EXPERIMENTS\nThe fabrication process of the artificial periodic struc-\ntures with characteristic dimensions in deep nanoscale is\nvery hard to control and so far mainly theoretical stud-\nies are available in this scale.23,24Especially it concerns\nthe quality of edges and the interfaces between adjacent\nmaterials which makes up MC and can significantly in-\nfluence magnonic band structure.25–27However, in larger\nscalewithaperiod startingfromhundrednm the fabrica-\ntion technology is already well established for thin ferro-\nmagnetic metallic films.28,29The YIG films of the thick-\nness of tens nm have been fabricated only recently and\nthe quality of these films increases systematically. The\ndamping comparable to the value in thick YIG (of or-\nder smaller than in the best metallic ferromagnetic film)\nhas already been achieved.30,31However, the YIG thin\nfilms with patterning in nanoscale is not yet investigated.\nHere, we study dielectric YIG films structurized in larger\nscale where edge properties have minor influence on the\nSW dynamics. 10 µm thick YIG films were epitaxially\ngrown on gallium gadolinium garnet (GGG) substrates\nin a (111) crystallographic plane and serve to fabricate\none-dimensional (1D) MCs. The MCs used in our exper-\niment had been produced in the form of the waveguide\nof 3.5 mm width and 50 mm length with (i) an array of\nparallel grooves chemically etched [sample A, Fig. 1(a)]\nand (ii) an array of Au microstripes placed on the top\nof the film [sample B, Fig. 1(b)]. The grooves and Au\nmicrostripes were perpendicularly oriented with respect\nto the SW propagation direction and include nine lines\nof 80µm width which are spaced 70 µm from each other,\nso that the period ais 150µm.\nThe external magnetic field H0is applied along the\ngrooves and microstripes in order to form conditions\nfor propagation of the magnetostatic surface spin wave\n(MSSW), alsocalledasDamon-Eshbachwave. Thiswave\nhas asymmetric distribution of the SW amplitude across\nthe film, which depends on the direction of the magnetic\nfield with respect to the direction of the wave vector, and\nthis asymmetry increases with increasing wavenumber.\nBy putting metal on the ferromagnetic film the nonre-\nciprocal dispersion relation of MSSW is induced.32\nThe MSSW were excited and detected in garnet film\nwaveguide using two 30 µm wide microstripe transduc-\ners connected with the microwave vector network ana-\nlyzer (VNA), one placed in the front and another one be-hind the periodic structure. An external magnetic field\n(µ0H0= 0.1 T) was strong enough to saturate the sam-\nples. Microwave power of 1 mW used to the input trans-\nducer was sufficiently small in order to avoid any non-\nlinear effects. A VNA was used to measure amplitude-\nfrequency characteristics collected for the second trans-\nducer. The transmission spectra of SWs measured in the\nreference sample, i.e., a thin YIG film, is shown in Fig. 2.\nThe transmission of the microwave signal above 10 dB is\nin the band from 4.46 to 5.14 GHz.\n4 4 . 4 8 . 5 2 .-70-50-30-10\nFrequency (GHz)Transmission (dB)\nFIG. 2. Transmission spectra of MSSW in the reference sam-\nple, i.e., uniform YIG film of 10 µm thickness in the external\nmagnetic field 0.1 T.\nIn BLS measurements the SWs were excited with the\nsingle 30 µm wide microwave transducer located in front\nof the array of Au microstripes (sample B). The SWs\nscattered on the line of waveguide were detected by\nspace-resolved BLS spectroscopy in the forward scatter-\ning configuration.33The probe laser beam was scanned\nacross the sample (in the areas between Au stripes) and\nthe BLS intensity, which is proportional to the square\nof the dynamic magnetization amplitude, was recorded\nat various points. This technique allows for a two-\ndimensional (2D) mapping of the spatial distribution of\nthe SW amplitude with step sizes of 0.02 mm.\nIII. THEORETICAL MODELING\nIn our calculations we assume the stripes and grooves\nhave infinite length (i.e., they are infinite along zaxis).\nThis is reasonable assumption taking into account that\nlength is around 23 times longer than the period of the\nstructure. The structures under investigation remain in\na magnetically saturated state along the z-axis due to\nthe static external magnetic field pointing in the same\ndirection.\nTo obtain insight into the formation of the magnonic\nband structure and opening magnonic band gaps the nu-\nmerical calculations of the dispersion relation were per-\nformed. For SWs from the GHz frequency range, due to\n10µm thickness of YIG film and a small value of the\nexchange constant in YIG, the exchange interactions can\n2be safely neglected. In order to calculate the SW disper-\nsion relation in magnetostatic approximation we solved\nthe wave equation for the electric field vector E:34\n∇×/parenleftbigg1\nˆµr(r)∇×E/parenrightbigg\n−ω2√ǫ0µ0/parenleftbigg\nǫ0−iσ\nωǫ0/parenrightbigg\nE= 0,(1)\nwhereω= 2πf,fisaSWfrequency, µ0andǫ0denotethe\nvacuum permeability and permittivity, respectively, and\nσisconductivity, different fromzeroonlyin sampleB.To\ndescribe the dynamics of the magnetization components\nin the plane perpendicular to the external magnetic field,\nit is sufficient to solve Eq. (1) for the zcomponent of the\nelectric field vector Ewhich depends solely on xandy\ncoordinates: Ez(x,y).35\nThe permeability tensor ˆ µ(r) in Eq. (1) can be ob-\ntained from Landau-Lifshitz (LL) equation.32The as-\nsumption that the magnetization is in the equilibrium\nconfiguration allows us to use the linear approximation\nin SW calculations, which implies small deviations of the\nmagnetization vector M(r,t) from its equilibrium orien-\ntation. Thus, for the MCs saturated along z-axis the\nmagnetization vector can split into the static and dy-\nnamic parts: M(r,t) =Mzˆz+m(r,t), and we can ne-\nglect all nonlinear terms with respect to dynamical com-\nponents of the magnetization vector m(r,t) in the equa-\ntion of motion defined below. Since |m(r,t)| ≪Mz, we\ncan assume also Mz≈MS, whereMSis the saturation\nmagnetization. We consider only monochromatic SWs\npropagating along the direction of periodicity, thus we\ncan write m(r,t) =m(x,y)exp(iωt). Under these as-\nsumtions the dynamics of the magnetization vector m(r)\nwith negligible damping is described by stationary LL\nequation:\niωm(r) =γµ0(MSˆz+m(r))×Heff(r),(2)\nwhereγis the gyromagnetic ratio (we assume γ= 176\nrad GHz/T) and Heffdenotes the effective magnetic field\nacting on the magnetization. The effective magnetic field\nis in general a sum of several components, here we will\nconsidertwoterms, the static externalmagneticfield and\nthe dynamic magnetostatic field:\nHeff(r,t) =H0ˆz+hms(r,t). (3)\nThe permeability tensor ˆ µ(r) in Eq. (1) obtained from\nthe linearized damping-free LL Eq. (2) for ferromagnetic\nmaterial takes following form:\nˆµr=\nµxxiµxy0\n−iµyxµyy0\n0 0 1\n, (4)\nwhere\nµxx=γµ0H0(γµ0H0+γµ0MS)−ω2\n(γµ0H0)2−ω2,(5)\nµxy=γµ0MSω\n(γµ0H0)2−ω2, (6)\nµyx=µxy, µyy=µxx, (7)in non-magnetic areaspermeability is an identity matrix.\nEquation (1) with the permeability tensor defined in\nEq. (4) in the periodic structure has solutionswhich shall\nfulfill Bloch theorem:\nEz(x,y) =E′\nz(x,y)eiky·y, (8)\nwhereE′\nz(x,y) is a periodic function of y:E′\nz(x,y) =\nE′\nz(x,y+a).kyis a wave vector component along yand\nais a lattice constant. Due to considering SW propaga-\ntion along ydirection only, we assume ky≡k. Eq. (1)\ntogether with Eq. (8) can be written in the weak form\nand the eigenvalue problem can be generated, with the\neigenvalues being frequencies of SWs or in the inverse\neigenproblem with the wavenumbers as eigenvalues. The\nformer eigenproblem is used to obtain magnonic band\nstructure, the later to calculate the complex wavenumber\nof SW inside the magnonic band gaps. This eigenequa-\ntion is supplemented with the Dirichlet boundary condi-\ntions at the borders of the computational area placed far\nfrom the ferromagnetic film along xaxis (bold dashed\nlines in Fig. 1).\nIn FEM the equations are solved on a discrete mesh in\nthe two-dimensional real space [in the plane ( x,y)] lim-\nited due to Bloch equation to the single unit cell (marked\nby the gray box in Fig. 1). In this paper we use one\nof the realizations of FEM developed in the commercial\nsoftware COMSOL Multiphysics ver. 4.2. This method\nhas already been used in calculations of magnonic band\nstructure in thin 1D MCs, and their results have been\nvalidated by comparing with micromagnetic simulations\nand experimental data.36–38The detailed description of\nFEM in its application to calculation of the SW spectra\nin MCs can be found in Refs. [38 and 39].\nIncalculationswehavetakennominalvaluesoftheMC\ndimensions and the saturation magnetization of YIG as\nMS= 0.14×106A/m. The conductivity of the metal is\nassumed as σ= 6×107S/m, which is a tabular value for\nAu.\nIV. RESULTS AND DISCUSSION\nIn Fig. 3(a) and (b) we present the results of the SW\ntransmission measurements with the use of microstripe\nlines in the external magnetic field 0.1 T for sample A\nand sample B, respectively. We can see a clear evidence\nof three (centered at 4.81, 4.97 and 5.05 GHz) and two\nmagnonic band gaps (at 4.88 and 5.05 GHz) in sample\nA and B, respectively. The transmission band in both\nsamples is approximately the same as in the reference\nsample (Fig. 2), however at high frequencies in sample\nB a large decrease of the transmission magnitude is ob-\nserved. Thus, in MC with metallic stripes the second\nmagnonic band gap [marked with blue square in Fig. 3\n(b)] is already at the part of the low transmission. The\nestimation of its position and width will be loaded with\nadditional errors and some ambiguity, thus in further in-\nvestigations we will not consider this band gap.\n34 75 . 4 85 .w(a) (b) Sample A Sample B\n-70 -70-50 -50-30 -30-10 -10\nTransmission (dB) Transmission (dB) 4 4 . 4 8 . 5 2 .\nFrequency (GHz)4 4 . 4 8 . 5 2 .\nFrequency (GHz)\nFIG. 3. Transmission spectra of SWs in (a) sample A and\n(b) in sample B measured with microstripe lines in external\nmagnetic field 0.1 T. The magnonic band gaps are marked by\nsolid symbols: in sample A there are three gaps, in sample\nB there are two gaps, however the second gap is at the part\nof low transmission and will not be considered in the paper.\nIn the inset of the figure (a) the enlargement of the spectra\naround of the first band gap is shown, the width of this gap\nisw.\nThe calculated magnonic band structures are pre-\nsented in Fig. 4 with blue dashed and red solid lines\nfor sample A and B, respectively. For MC with grooves\nthe dispersion relation is symmetric and magnonic band\ngaps are opened at the Brillouin zone (BZ) border (first\nand third gap) and in the BZ center (the second gap).\nThe frequencies of gaps obtained in calculations agree\nwell with the gaps found in transmission measurements\n[Fig. 3 (a)]. For sample B, the magnonic band struc-\nture is nonreciprocal, i.e., f(k)/negationslash=f(−k).35,40Moreover,\nthe first band has large slope (larger than for sample\nA), especially in + kdirection has significantly increased\ngroup velocity. These effects are results of conducting\nproperties of the Au stripes, which cause fast evanescent\nof the dynamic magnetic field generated by oscillating\nmagnetization in the areas occupied by metallic stripes.\nDue to this nonreciprocity in the dispersion relation the\nmagnonic band gap opens inside the BZ and it is an in-\ndirect band gap. Also for sample B we have found good\nagreement between calculations and measured data.\nIn the measured data shown in Fig. 3 there is visi-\nble difference between the width and depth of the first\nband gap in sample A and B. In order to estimate the\ndepth ofthe gap fromcalculations we need to solvean in-\nverse eigenproblem, i.e., to fix the frequency as a param-\neter and search for a complex wavenumber as an eigen-\nvalue. In Fig. 5 the calculated imaginary part of the\nwavenumber (Im[ k]) as a function of frequency around\nthe first band gap is presented. In figures (a), (b) the\nexternal magnetic field was set on 0.1 T, in figures (c),\n(d) it was enlarged to the value 0.15 T. For sample A\n[Fig. 5(a) and (c)] the Im[ k] has zero value outside of the\ngap since the Gilbert damping is neglected in the calcula-\ntions. However, forsampleB [Fig. 5(b) and(d)] the func-\ntion Im[k](f) is nonzero outside of the gap, it is because\nthe metal stripes induce attenuation of SWs. Outside of\nthe gap regions in sample B the Im[ k] increases with the\nfrequency and this behavior is observed in the transmis--1 1 04.64.85.0\nWavevector ( / ) k a/c112Frequency (GHz)\nFIG. 4. Magnonic band structure in the first Brillouin zone\ncalculated for sample A (blue dashed line) and sample B (red\nsolid line) with magnetic field µ0H0= 0.1 T. Magnonic band\nstructure is symmetric and asymmetric with respect of the\nBrillouin zone center (marked by vertical black dashed line )\nin sample A and B, respectively.\nsion spectra as decrease of the signal at large frequencies\n(still in the transmission band of the reference sample)\nin sample B [Fig. 3(b)].\nIt is observed that the maximal value of Im[ k] in the\nfirst band gap is significantly larger for sample B than A\n(0.113 and 0.062, respectively for 0.1 T magnetic field).\nBecause an inverse of Im[ k] describes the decaying length\nof SWs, it correlates with the magnonic band gap depth\nin the transmission measurements. Indeed this finds con-\nfirmation in the experimental data, where the minimal\ntransmission magnitude in the band gap is -18 dB at\n4.81 GHz and -39 dB at 4.89 GHz in sample A and B,\nrespectively. This significant suppression of the trans-\nmission of SW signal in the first magnonic band gap in\nsample B is confirmed also in BLS measurements pre-\nsented in Fig. 6, where two excitation frequencies were\nset to (a) 4.64 GHz and (b) 4.89 GHz. These frequencies\nwere chosen to visualize the SW propagation at frequen-\ncies from the band and from the band gap, respectively.\nIn both cases the decrease of the SW amplitude with in-\ncreasing the distance form the transducer is found, how-\never in the band gap this decrease is more pronounced.\nNevertheless, some signal is still observed at the end of\nMC for frequencies from the band gap. We suppose that\nthis is due to limited number of Au stripes used in the\nexperiment and direct excitation of SWs from the trans-\nducer.\nThere is also another difference between function\nIm[k](f) for both samples. This is an asymmetry be-\ntween the bottom and top part of the gap in sample B,\nwhile in sample A the function Im[ k](f) is almost sym-\nmetric with respect to the magnonic band gap center.\nTo have some measure of this asymmetry we have cal-\nculated the derivatives ∂Im[k]/∂fat the points where\nIm[k] is half of its maximum value, i.e., at points a-d\nmarked in Fig. 5(a) and (b). For the sample A these val-\nues are: 2 .23×10−4s/m and −2.28×10−4s/m (points\n4Sample A\n. .b a\n4.77 4.78 4.790\n00.04\n0.040.08\n0.080.12\n0.12Sample BSample B\n. .d c\n4.87 4.89 4.91 4.93\n6.34 6.35 6.36Sample A(a) (b)\n(c) (d)00.040.080.12\nIm[ ] ( / )k a/c112\n00.040.080.12\nIm[ ] ( / )k a/c112Im[ ] ( / )k a/c112\nIm[ ] ( / )k a/c112f(GHz)\nf(GHz)f(GHz)\nf(GHz)6.46 6.42 6.44\nFIG. 5. The imaginary part of the wavenamber around the\nfirst band gap in sample A (a), (c) and in sample B (b), (d)\nfor the two values of the magnetic field µ0H0= 0.1 T (a), (b)\nand 0.15 T (c), (d). The calculation were done for the inverse\neigenproblem with FEM. The letters a-d indicate points wher e\nthe value of Im[ k] takes a half of its maximum. These points\nmay indicate the borders of the band gap extracted from the\ntransmission measurements.\na and b, respectively) and for sample B: 1 .50×10−4s/m\nand−1.97×10−4s/m (respectively points c and d).41\nWe attribute this difference in Im[ k] between MCs to the\ndifferent group velocities of SWs around the gaps, i.e.,\nthe symmetric and asymmetric dispersion curves of the\nfirst (and second) band near the edge of the band gap for\nthe sample A and B, respectively (Fig. 4). We point out\nthat this asymmetry in Im[ k] might appear as asymmet-\nric slope in the transmission spectrum (Fig. 3) and it can\nbe of some importance for applications in magnetic field\nsensors and magnonic transistors.17,21\n0.30.91.52.1\n0 00.6 0.6 1.2 1.2y(mm)\nz(mm) z(mm)(a)   = 4.64 GHzf (b)   = 4.89 GHzf\nminmax\nH0k\nMagnonic\ncrystal\nFIG. 6. Maps of the SW intensity acquired with BLS from\nsample B at two frequencies (a) 4.64 GHz and (b) 4.89 GHz\nrelated to the transmission band and the band gap. The mi-\ncrostripe transducer aligned along zaxis used to excite SWs\nis located below presented area.\nFinally, we study magnonic band gap widths in depen-\ndence on amplitude of the external magnetic field. The\nresults are presented in Fig. 7(a) and (b) for sample A\nand B, respectively. In this figure there are points (full\ndots and squares) extracted from the transmission mea-(a) (b) Sample A Sample B\n0.08 0.08 0.12 0.12 0.16 0.1651525\nWidth of gap (MHz)w\nWidth of gap (MHz)w\nMagnetic f T ield ( )/c1090H0 Magnetic f T ield ( )/c1090H020304050\nFIG. 7. Width of the magnonic band gap as a function of\nthe external magnetic field (a) in sample A and (b) in sam-\nple B. The experimental data are marked by full dots and\nsquares, while the results of calculations are shown with so lid\nand dashed lines for the first and second band gap, respec-\ntively. The horizontal lines at some selected values of H0show\nerrors of the measured magnonic band gap width.\nsurements and lines (red-solid and blue-dashed) obtained\nfrom FEM calculations (first and second band gap, re-\nspectively). Overall, we have found good agreement be-\ntween theory and measurements, the calculation results\narealwaysin the rangeofthe experimentalerrorsmarked\nin figuresby solid verticallines. The decreaseofthe band\ngapwidthwithincreaseofthemagneticfieldweattribute\nto the decrease of the band width (i.e., decrease of the\ngroup velocity) of the MSSW.42The steeper decrease of\nthe band gap width is observed for the MC with metal-\nlic stripes. These dependencies find also reflection in the\nvalues of Im[ k] shown in Fig. 5, where the Im[ k] drops\ndown by 23% in sample B, while for sample A the Im[ k]\nremains almost the same with the increase of the mag-\nnetic field by 0.05 T.\nThis different dependencies for sample A and B are\nrelated to the larger sensitivity of the group velocity of\nMSSW on changes of the magnetic field in metallized\nfilm then in unmetallized, but also to the nonreciprocal\nmagnonic band structure and the presence of the indi-\nrect band gap in the case of sample B. The sensitiv-\nity of the group velocity around the gap might be es-\ntimated analytically. In Fig. 8 the analytical dispersion\nrelation of MSSW in 10 µm thick YIG film with metal\noverlayeris presented in the empty lattice model (ELM).\nThe periodicity was taken the same as a periodicity of\nthe samples. The crossing point between dispersions of\nthe MSSW propagating in opposing directions, + k(with\nmaximum of the amplitude close to the metal) and −k\n(with amplitude on the opposite surface, in the figure\nthis dispersion is shifted by the reciprocal lattice vector\n2π/a) indicates the Bragg condition, i.e., the condition\nfor opening magnonic band gap. It means that the Bragg\ncondition takes wavenumbers from the first and second\nBZ for waves propagating into positive (+ k) and nega-\ntive (−k) direction of the wavevector, respectively.39,43\nThe group velocities were calculated at crossing points\nat field values 0.1 T and 0.15 T for structures with and\nwithout metal overlayer. Based on these values, we have\nfound that the ratio of group velocity changes with the\nfield is almost twice higher in structure with metal layer\n5than without this.\nAlthough, the change of the dispersion slope around\nBragg condition is larger for + kwave, the magnitude of\nthe wavevector |+k|is smaller than | −k|and small\nchange of the group velocity of −kwave might also have\nimpact on changes in the band gap position. In our case,\nthe increase of the magnetic field from 0.1 T to 0.15 T\nresults also in the shift of the Bragg condition towards\nBZ center (see Fig. 8). However, in general, the posi-\ntion of the Bragg condition might shift towards center or\nedge of the BZ with the increase of the field. To which\ndirection will shift the Bragg condition is determined by\nboth groupvelocitychangeand wavenumberdifferenceof\nMSSW propagating in opposite directions. We note also,\nthat for both samples, the width of the second band gap\nis less sensitive to the magnetic field amplitude than the\nwidth of the first gap.\n1.0 0.05.06.07.0\nWavevector ( / ) k a/c112Frequency (GHz)\n0.2 0.6 0.4 0.85.56.5\n/c1090 0H=0.10 T/c1090 0H=0.15 T\n+k\n-k\nFIG. 8. Analytical estimation of the Bragg condition for\nmagnonic band gap opening in 10 µm thick metalized YIG\nfilm with periodicity a= 150µm at two values of the ex-\nternal magnetic field: 0.1 T and 0.15 T. Dispersion relation\nof the propagating MSSW waves with maximum of the am-\nplitude close to the metal ( k+) and on opposite side ( k−) of\nthe film are marked with solid and dashed lines, respectively .\nThe dispersion of k−wave is shifted by the reciprocal lattice\nvector 2π/afrom its original position. The Bragg condition\nis fulfilled at the cross-section of the k+andk−lines.\nIt is also sample B which has a wider band gap then\nsample A in the considered magnetic field values [see\nFig. 7(a) and (b)]. It was already shown that cover-\ning bi-component MC or ferromagnetic film with lat-\ntice of grooves by a homogeneous metallic overlayer shall\nincrease the band gap width of the MSSW due to in-\ncreased group velocity of MSSW propagating along met-\nalized surface.43,44However, the influence of metal with\nfinite conductivity depends on the wavenumber(and film\nthickness), and disappears for large k.35Thus, band gap\nwidthwilldependonthewavenumberatwhichtheBragg\ncondition is fulfilled, i.e., will depend on the lattice con-\nstant. For sufficiently large k(small period) the influence\nofmetal disappearsand band gapswill not form.43In the\nhomogeneous YIG film of 10 µm thickness the influence\nof the homogeneous Au overlayer on the dispersion rela-\ntion of MSSW disappears for k≈1.57×106rad m−1,\ni.e., for the MC with a period a= 2µm at the BZ border\nthe effect of metallization will be absent. The influence\nof metal will disappear also when the separation between\nmetallic stripes and YIG will be introduced, howeverthiscan be avoided by proper fabrication technique.\nIn sample A an influence of the corrugation shall pre-\nserve also for small a, thus it is expected that for small\nlattice constant the band gap in sample A will be wider\nthan for sample B. However, we note also that the band\ngap width depends also on the grooves depth in sample\nA, thus there is an additional parameter to be taken into\naccount. In the caseofsmallsurfaceperturbations(small\nratioofthe grovedepth to the film thickness) the coupled\nmode theory shows that the width ofthe gap and the gap\ndepth (maximal Im[ k] in the gap) are proportionalto the\nperturbation.15Nevertheless, the structure with larger\ngroves has not been found very promissing for magnonic\nband gaps applications so far, because suppressed trans-\nmission in the bands due to excitations of the standing\nspin waves.11,12,45This can change when the very thin\nYIG samples will be used for MCs, then the frequency\nof standing exchange SW modes will moved to high fre-\nquencies. However, the fabrication regularmodulation of\nthe film thickness in deep nanoscale remain challenging\ntask.\nV. CONCLUSIONS\nIn summary, the SW spectrum of the two planar 1D\nMCs comprised of a periodic array of etched grooves in\nYIG film and an array of metallic stripes on homoge-\nneous YIG film has been fabricated and studied experi-\nmentally and theoretically. The properties of propagat-\ning SWs and magnonic band gaps were in focus of our\ninvestigations. The two different kinds of MCs elaborate\nthe fundamental differences in the magnonic band spec-\ntrum, and also their band gap properties are different.\nTo study of band gap widths and depths we used numer-\nical method, which is based on FEM in the frequency.\nWe obtained these values according with the measured\ndata. We haveshownthat MC formed bymetallic stripes\nposses wider magnonic band gap with larger depth than\nthe second MC. Moreover, the fabrication of arrays of\nmetallic stripes is much more feasible than etching of\ngrooves in the dielectric slab. However, the influence of\nthe metal overlayer on the band gaps of magnetostatic\nwavesis limited to relatively small wavenumbersand this\nlimits the miniaturizing prospective for these MCs. In\ncontrast the MC based on the lattice of grooves does not\nhave such limit, nevertheless its band width and depth is\nlimiting by the excitation of the standing exchange SWs.\nIn both types of MCs, the magnonic band gap width de-\ncreases with increasing external magnetic field, we have\nidentified mechanisms responsible for these changes.\nThe results obtained here should have impact on the\napplications of MCs, because we have shown the influ-\nence of different types of periodicity on the magnonic\nband gaps. These properties shell be especially impor-\ntant for magnonic devices, like magnetic field sensors21\nor full magnonic transistors17which functionality were\nalready experimentally demonstrated. 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Smith, and J. M. Owens, IEEE Trans.\nMagn.16, 1159 (1980).\n7"    },    {        "title": "1712.03322v1.Observation_of_spin_orbit_magnetoresistance_in_metallic_thin_films_on_magnetic_insulators.pdf",        "content": "Science Advances                                                Manuscript Template                                                                           Page 1 of 12 \n   \nObservation of spin-orbit magnetoresistance in metallic thin films on magnetic insulators \n \nLifan Zhou,1,† Hongkang Song,2,3,† Kai Liu,3 Zhongzhi Luan,1 Peng Wang,1 Lei Sun,1 \nShengwei Jiang,1 Hongjun Xiang,3,4 Yanbin Chen,1,4 Jun Du,1,4 Haifeng Ding,1,4 Ke Xia,2 \nJiang Xiao,3,4,5,* and Di Wu1,4,* \n \n1National Laboratory of Solid State Microstructures and Department of Physics, Nanjing \nUniversity, Nanjing 210093, P. R. China. \n2Department of Physics, Beijing Normal University, Beijing 100875, P. R. China. \n3Department of Physics and State Key Laboratory of Surface Physics, Fudan University, \nShanghai 200433, P. R. China. \n4Collaborative Innovation Center of Advanced Microstructures, Nanjing 21 0093, P. R. \nChina. \n5Institute for Nanoelectronics Devices and Quantum Computing, Fudan University, \nShanghai 200433, P. R. China.  \n† These authors contributed equally to this work. \n*Corresponding author: xiaojiang@fudan.edu.cn, dwu@nju.edu.cn. \n \nAbstract \nA magnetoresistance effect induced by the Rashba spin -orbit interaction was predicted, but \nnot yet observed, in bilayers consisting of normal metal and  ferromagnetic insulator. Here, \nwe present an experimental observation of  this new type of spin-orbit magnetoresistance \n(SOMR) effect in a bilayer structure Cu[Pt]/Y 3Fe5O12 (YIG), where the Cu/YIG interface \nis decorated with nanosize Pt islands. This new MR is apparently not caused by the bulk \nspin-orbit interaction because of the negligible spin-orbit interaction in Cu and the \ndiscontinuity of the Pt islands. This SOMR disappears when the Pt islands  are absent or \nlocated away from the Cu/YIG interface, therefore we can  unambiguously ascribe it to the \nRashba spin-orbit interaction at the interface enhanced by the Pt decoration. The numerical \nBoltzmann simulations are consistent with the experimental SOMR results in the  angular \ndependence of magnetic field  and the Cu thickness dependence. Our finding demonstrates \nthe realization of the spin manipulation by interface engineering. \n \nIntroduction \nRelativistic spin-orbit interaction (SOI) plays a critical role in a variety of interesting phenomena, including the spin Hall effect (SHE) ( 1-3), topological insulators (4), the \nformation of skyrmions (5, 6). In SHE, a pure spin current transverse to an electric current \ncan be generated in conductors with str ong SOI, such as Pt, Ta etc (7, 8). The inverse SHE \n(ISHE) is generally used to detect the spin current electrically by converting a pure spi n \ncurrent into a charge current (9, 10). It was recently discovered that the interplay of the SHE \nand ISHE in a nonmagnetic heavy metal (NM) with strong SOI in contact with a ferromagnetic insulator (FI) leads to an unconventional magnetoresistance (MR) - the spin \nHall magnetoresistance (SMR), in which the resistance of the NM layer depends on the \ndirection of the FI magnetization M (11-13). SMR has been observed in several NM/FI \nsystems and even in metallic bilayers  (14-17). However, it has been argued that SMR may \noriginate from the magnetic moment in the NM layer induced by the magnetic proximity \neffect (MPE) (18). These two mechanisms were proposed to be distinguished by the an gular \ndependent MR measurements (11, 13). Very recently, another type of MR, the Hanle MR \n(HMR), is demonstrated in a single metallic film with strong SOI owing to the combined actions of SHE and Hanle effect (19). HMR depends on the direction and the strength of the Science Advances                                                Manuscript Template                                                                           Page 2 of 12 \n  external magnetic field H, rather than that of M in SMR. Within the framework of SMR, \nbecause of the negligible SOI in Cu (20), one would not expect any MR effect in a Cu/FI \nbilayer. \nRecently, Grigoryan et al. predicted a new type of MR effect in the NM/FI systems when \na Rashba type SOI is present at the interface between NM and FI ( 21). This new spin-orbit \nMR (SOMR) works even with light metals such as Cu or Al with negligible bulk SOI, \nprovided that the Rashba SOI is present at the NM/FI interface. Because of the identical \nangular dependence on M direction for SOMR and SMR, however, it is difficult to \ndistinguish SOMR from SMR in systems like Pt/Y 3Fe5O12 (YIG), where both SOMR and \nSMR are present in principle. In this work, we report the first observation of SOMR in a \nCu/YIG bilayer, where the Rashba SOI at Cu/YIG interface is enhanc ed by an ultrathin Pt \nlayer (< 1 nm). We also confirmed that SOMR almost disappears when Pt is placed inside or on the other side of the Cu layer, indicating that SMR from the ultrathin Pt layer cannot \nbe the origin of the observed MR and the Pt-decoration of the Cu/YIG interface is crucial \nfor SOMR. The observed SOMR has the same angular dependence as the SMR in Pt/YIG, in agreement with the SOMR prediction (21). The monotonous Cu-thickness dependence \nof SOMR is clearly different from the non-monotonous dependence of SMR (13, 18). Both \nthe angular- and Cu-thickness- dependence of the observed MR are in good agreement with \nour Boltzmann simulations based on the SOMR mechanism.  In addition, the MR shows two \nmaxima as the Pt layer thickness increases, in sharp contrast with that of SMR (13, 22).\n \n Result s and discussions \nSample morphology and structure  The YIG films used in this study are 10 nm thick, unless otherwise stated, grown by pulsed \nlaser deposition (PLD) on Gd\n3Ga5O12 (GGG) (111) substrates. The surface morphology of \nthe YIG films was characterized by atomic force microscopy (AFM), as shown in Fig. 1A. \nThe film is fairly smooth with the root-mean-square (rms) roughness of 0.127 nm and the \npeak-to-valley fluctuation of 0.776 nm. The 0.4-nm-thick Pt layer, thinner than the peak-to-\nvalley value of the YIG film, deposited on YIG by magnetron sputtering forms the nanosize \nislands with the rms roughness of ~ 0.733 nm, shown in Fig. 1B. This discontinuous Pt layer \nis non-conductive with the resistance over the upper limit of a multimeter. The surface \nroughness is reduced after the deposition of Cu onto Pt, as shown in Fig. S1. Figure 1C \npresents the cross-section high-resolution transmission electron microscope (HRTEM) \nimage of the Au(3)/Cu(4)[Pt(0.4)]/YIG films, where the numbers are the thicknesses in the \nunit of nanometer. The YIG film is clearly single-crystalline and smooth. The lattice \nconstant of the YIG film is determined to be 1.2234 nm, to be compared to 1.2366 nm for \nthe bulk YIG. A clear interface is observed between the metallic films and the YIG film. \nThe metallic films are polycrystalline.  \n \nField-dependent magnetization and transport measurements  \nIn this work, all the measurements were performed at room temperatu re. The YIG film is \nalmost isotropic in the film plane with the coercivity of about 0.4 Oe, shown in Fig. 2A. Due \nto the large paramagnetic background of the GGG substrate, it is difficult to measure the \nmagnetization of a thin YIG/GGG film in the out -of-plane geometry. We measured a 400 -\nnm-thick YIG/GGG(111) film instead. As shown in Fig. 2B, the magnetization is saturated at ~ 1800 Oe. The saturation magnetization M\ns of our YIG film is determined to be 164.5 \nemu/cc measured by ferromagnetic resonance (FMR ) (see the Supplementary Materials ). In \ncomparison, M s of bulk YIG is 140 emu/cc.  \nFigures 2C and 2D present the resistivity r as a function of H for Cu(2)[Pt(0.4)]/YIG(10) \nsample. In experiments, H was applied along i) the direction of the current I (x-axis), ii) in Science Advances                                                Manuscript Template                                                                           Page 3 of 12 \n  the sample plane and perpendicular to the current direction ( y-axis), and iii) perpendicular \nto the sample plane (z-axis), respectively. The MR effects are clearly present in all \nmeasurements. For H along x- and y-directions, r shows two peaks around the coercive \nfields of YIG. For H along z-direction, r shows a minimum at H = 0 and remains almost a \nconstant value above the saturation field. These features indicate that the MR effects are \nintimately correlated with M, meaning that the observed MR effects are not HMR.  \n \nAngular dependent MR measurements  \nTo further study the anisotropy of the MR effects in Cu[Pt]/YIG, we performed the angular \ndependent MR measurements. Figure 3A shows Dr/r of Cu(3)[Pt(0.4)]/YIG(10) sample \nwith rotation of H in the xy- (a-scan), yz- (b-scan) and xz- (g-scan) planes, where a, b and g \nare the angles between H and x-, z- and z-directions, respectively, as defined in the inset of \nFig. 3A. The applied magnetic field strength ( H = 1.5 T) is large enough to align M with H. \nThe MR effect is clearly anisotropic. The MR ratio, defined as Dr/r = [r(angle) - r(angle = \n90o)]/r(angle = 90o)], in a- and b-scans is about 0.012%, comparable to the SMR ratio in \nPt/YIG (see Fig. S3) (11, 13, 23). \nNext, we investigated the origin of the observed MR effect. Considering that Pt on YIG \nmay suffer from the MPE induced ferromagnetic moment and the corresponding anisotropic MR (AMR) (24), we replaced Pt by a 0.4-nm-thick Au layer, which is well-known to have \na negligible MPE (25). The MR effect of 0.002% still appears as shown in Fig. 3B, \ncomparable to the SMR ratio in Au/YIG (see Fig. S 4), ruling out MPE as the origin of the \nobserved MR. Furthermore, the MR ratios of the Cu(3)[Pt(0.4)]/YIG(10) sample in \na- and \nb-scans are comparable and almost one order of magnitude larger than that in g-scan. This \nis different from AMR of a ferromagnetic metal, where the MR ratio in a- and g-scans is \nmuch larger than that in b-scan  (11, 14, 24). Therefore, the MPE-induced AMR can be ruled \nout.  \nIn fact, the behaviors of the MR angular dependence follow the SMR scenario well (11, \n13-15, 17). However, with several control experiments, we can unambiguously exclude \nSMR as the explanation for our observations. \nFirst, the observed MR amplitude cannot be explained by SMR. In our samples, the 0.4 -\nnm-thick ultrathin Pt layer is non -conductive and the conductivity of bulk Pt is about one \norder of magnitude smaller than that of bulk Cu, meaning that the current mainly  passes \nthrough the Cu layer. We prepared a 3-nm-thick single layer Cu on YIG without interface \ndecoration and performed the MR angular dependent measurement in a-scan. MR is not \nobserved, as shown in Fig. 3B, evidencing that the Pt-decorated interface is indispensable. \nA conductive 0.4-nm-thick Pt layer is not available experimentally. Considering that a small \nfraction of current may flow in the Pt islands, there is a possibility of the occurrence of SMR \nfrom the Pt islands. According to the reported SMR results in Pt/YIG bilayers, the SMR \nratio in Pt/YIG decreases rapidly with decreasing Pt thickness when the Pt th ickness is less \nthan about 3 nm (13, 18). The SMR ratio of Pt(0.4)/YIG is extrapolated to be well below \n0.01% from the previously reported Dr/r versus Pt thickness data (13, 18). Considering the \npronounced shunting current of the highly conductive Cu layer, the SMR ratio should be significantly reduced in Cu[Pt]/YIG, i.e., much less than 0.01%. In comparison, the MR \nratio is as large as ~ 0.012% in Cu(3)[Pt(0.4)]/YIG (see Fig. 3A). Therefore, the SMR \nmechanism cannot explain our observations.  \nSecond, the potential enhancement of SMR caused by intermixing or alloying between  a \nstrong SOI material and a weak SOI material  can be excluded (17, 26, 27). For this purpose, \nwe prepared two types of control samples with the 0.4 -nm-thick Pt layer either on top of or \ninserted inside the Cu layer: [Pt(0.4)]Cu(3)/YIG and Cu(1)[Pt(0.4)]Cu(3)/YIG. Since both \nsamples are fabricated under the same condition as the Cu[Pt]/YIG samples, the intermixing Science Advances                                                Manuscript Template                                                                           Page 4 of 12 \n  of Pt and Cu should be similar. The MR vanishes in the [Pt(0.4)]Cu(3)/YIG and \nCu(1)[Pt(0.4)]Cu(3)/YIG samples, shown in Fig. 3B. These results rule out the Pt-Cu \nalloying induced SMR. Thus, we conclude that the observed MR effect is not SMR. \n    \nCu-thickness dependent transport measurements  To identify the physical origin of the observed unusual MR, we carried out the Cu -thickness \ndependent measurements. Figure 4A presents the angular dependent MR measurements of \nCu(t\nCu)[Pt(0.4)]/YIG in a-scans for various Cu thickness (t Cu). Obviously, the MR ratio \nsteadily decreases with increasing tCu, highlighting the importance of the Pt-decorated \nCu/YIG interface. This monotonous NM-thickness dependence of this MR is in sharp \ndifference with the non-monotonous behavior of SMR, which peaks at ~ 3 nm for Pt/YIG \n(13, 18). The Cu-thickness dependence of r and the MR ratio extracted from Fig. 4A are \nshown in Fig. 4B. For very thin Cu film (t Cu £ 5 nm), r dramatically increases with \ndecreasing t Cu, indicating that r is dominated by the interface/surface scatterings.  \nBesides SMR, there is another type MR predicted recently possessing the same angular \ndependence as we found (see Fig. 3A) (21).  It originates from the Rashba SOI at the \ninterface of a NM/FI bilayer. By comparing the samples of Cu[Pt]/YIG, [Pt]Cu/YIG and \nCu[Pt]Cu/YIG, one can see that only the Cu[Pt]/YIG samples exhibit a significant MR (see \nFig. 3B). It strongly suggests that the MR observed in our experiments is the SOMR predicted in Ref. 21, and the Pt-decoration enhances the Rashba SOI at the Cu/YIG \ninterface. \n  \nFirst principles calculations and Boltzmann simulations  In order to prove that the Pt-decoration can indeed induce Rashba SOI at the Cu/YIG \ninterface, we carried out first principles band structure calculations based on i) a Cu ultra -\nthin film of 14 monolayers, ii) the same Cu film as i) but covered by Au on surfaces on both \nsides, iii) the same Cu film as i) but covered by Pt on both surfaces, iv) the Pt layer inside \nthe Cu film. By comparing these four different scenarios, we can see that there is no clear \nRashba effect in the bare Cu film and the one covered by Au. A strong Rashba effect appears \nonly for Pt on the Cu film surface (the details of calculations are given in the Supplementary Materials). \nFor a quantitative analysis, we employ a Boltzmann formalism to calculate the charge \nand spin transport in a NM/FI bilayer structure. We solve the following spin-dependent \nBoltzmann equation in the NM layer: \n() ( )( )00 ,0 ,FS0, , ,()() () ( )\nxyzfR ef d P fa\naa\naa aa a d¢¢\n¢==-¶¢ ¢¢ ×- × +\n¶åòr,kvk E vk r k  ,k k k,k r,k\nr，     (1) \nwhere fa=0,x,y,z(r,k)  is the four-component distribution function denoting the charge/spin \noccupation at position r and wavevector k. The interface at FI z = z+ contains a Rashba type \nSOI described by the Hamiltonian: ( )( )RHz z hd=× ´ -+ σzp! !!, where h is the strength of the \nRashba SOI, z! is the normal direction of the interface, p! is the momentum operator. HR \ngives rise to an anomalous velocity localized at the interface. The Boltzmann equation is \nsolved by discretizing the spherical Fermi surface of Cu and the real space in z direction of \nCu film. With the full distribution function, we calculate all charge/spin transport properties, \nincluding the longitudinal and transverse conductivities. This method extends the earlier \nBoltzmann method developed for current-perpendicular-to-plane structure like spin valves \nto current-in-plane structure like NM/FI bilayers (28-31) by taking into account the surface \nroughness and Rashba SOI at the interface. The detai ls of the simulations are given in the \nSupplementary Materials . Science Advances                                                Manuscript Template                                                                           Page 5 of 12 \n  In the numerical Boltzmann calculation, ther e are only two fitting parameters, the surface \nroughness and the Rashba coupling constant. All other parameters are either given by the \nexperiment (such as the film thickness) or can be determined otherwise (such the bulk \nrelaxation time). By employing the quantum description of rough surface (32-34), we are \nable to fit the thickness dependence of r in the ultra-thin Cu film to a reasonably good \nprecision as shown in Fig. 4B. It is quite surprising considering that there is only one fitting parameter – the surface roughness. Once surface roughness is determined, we calculate the \nmagnetization angular dependence of \nr, in a good agreement with the experimental results \n(see Fig. 3A), from which we obtain the SOMR ratio. The calculated SOMR ratio is shown \nin Fig. 4B, which shows monotonic decreasing behavior as a function of Cu film thickness, \nconsistent with our experiment results but very different from the non -monotonic behavior \nobserved in SMR (13, 18). \n \nPt-thickness dependent MR measurements  \nFinally, to further differentiate SOMR from SMR, we carried out the Pt thickness tPt \ndependent measurements. To reduce the sample fluctuation, we fabricated the YIG films successively under the same condition. Figure 4A shows the angular dependent MR \nmeasurements of Cu(3)/Pt(t\nPt)/YIG in a-scans with H = 2000 Oe. The MR ratio extracted \nfrom Fig. 4A exhibits non-monotonous behavior with increasing tPt as shown in Fig. 4C. \nTwo separate regimes can be identified: 1) the SOMR regime for tPt < 1 nm and 2) the \nconventional SMR regime for tPt > 2.2 nm (see Fig. 4C). For tPt < ~0.6 nm, r and Dr/r \nincrease with increasing tPt because the Pt islands not only introduce the interface scattering \nbut also enhance the Rashba SOI. For ~0.6 nm < tPt < ~1 nm, the Pt islands start to form a \ncomplete layer, leading to the reduction of the interface roughness and the rapid decrease of \nr as seen in Fig. 4C. The MR ratio continues to increase in this region because of the \nenhanced Rashba SOI with increasing Pt coverage on YIG. For ~1 nm < tPt < ~2 nm, r is \nsmaller than the resistivity of Cu/YIG, suggesting that the interface scattering has minor \ncontribution to r. Since SOMR is caused by the interface scattering, the MR ratio rapidly \ndrops in this region. A sizable SMR ratio only appears when tPt > 2 nm in Pt/YIG (13, 18, \n22). Therefore, around tPt ~ 2 nm, both SOMR and SMR are small, resulting in a minimum \nin MR. In the SMR regime, the SMR ratio exhibits a maximum, as expected for SMR (13, \n18, 22). This result demonstrates the differences between the SOMR and the SMR.  \n    A theoretical calculation shows that a rough interface can enhance SHE ( 34). To \nunderstand the role of the roughness to the SOMR, we fabricated a control sample of \nCu(3)[Ag(0.7)]/YIG. The rms roughness of Ag(0.7)/YIG is 0.797 nm, as shown in Fig. \nS9(A), similar as that of Pt(0.4 nm)/YIG. Owing to the weak SOI in Ag, the Rashba SOI in \nCu[Ag]/YIG is expected to be weak. We do not observe any MR effect down to 5´10-6 in \nCu(3)[Ag(0.7)]/YIG, shown in Fig. S9(B). This result means that the rough surface alone \ncannot cause the SOMR. \n \nConclusions \nIn conclusion, we report the first observation of the SOMR effect predicted recently ( 21) at \nroom temperature in Cu/YIG films with the Pt decoration at interface. We show that this MR effect is caused by the enhanced Rashba SOI at the Pt-decorated interface. The angular \ndependence of SOMR is similar to that of SMR, but all other features are different, such as \nthe increasing MR with decreasing Cu thickness. The amplitude of the SOMR ratio is comparable to that of the SMR ratio in Pt/YIG, highlighting the importance of the NM/FI \ninterfaces. Our finding demonstrates the possibility of realizing spin manipulation by \ninterface decoration. \n Science Advances                                                Manuscript Template                                                                           Page 6 of 12 \n  Materials and Methods \nThe single crystalline YIG films were epitaxially grown on GGG (111) substrates by PLD \ntechnique using a KrF excimer laser with wavelength of 248 nm. The PLD  system was \noperated at a laser repetition rate of 4 Hz and an energy density of 10 J/cm2. The distance \nbetween the substrate and the target is 50 mm. Before films deposition, the chamber was \nevacuated to a base pressure of 1 × 10−7 torr. The YIG films were deposited at ~ 730 oC in \nan oxygen pressure of 0.05 Torr. The growth of the YIG films was monitored by in situ \nreflection high-energy electron diffraction (RHEED). The structure was further examined \nby X-ray diffraction and HRTEM. The magnetic properties of all YIG films were \ncharacterized using a vibration sample magnetometer (VSM). Then we used magnetron \nsputtering to deposit polycrystalline metallic films onto the YIG films via dc sputtering at \nroom temperature with a shadow mask to define 0.3 -mm-wide and 3-mm-long Hall bars. \nThe deposition rate was calibrated by X -ray reflectivity. After the metallic film deposition, \nthe samples were immediately mounted and transferred into a vacuum chamber for the \ntransport measurements to minimize the metal oxidation. The resistance was  measured by \na Keithley 2002 multimeter in a four-probe mode. For the angular dependent MR \nmeasurements with the magnetic field less than 5000 Oe, the resistance was monitored  as \nthe magnet was rotated. The angular dependent MR measurements with the magnetic field larger than 5000 Oe were performed in a physical property measurement system (PPMS) \nequipped with a rotatory sample holder.  \n H2: Supplementary Materials \nsection S1. AFM images of Cu(t\nCu)[Pt (0.4)]/YIG(10)/GGG(111) \nsection S2. Magnetic properties of the YIG films \nsection S3. SMR in Pt/YIG  \nsection S4. SMR and AFM image of Au/YIG \nsection S5. First principles calculations  \nsection S6. Boltzmann simulations  \nfig. S1. AFM images of Cu(t Cu)[Pt(0.4)]/YIG(10)/GGG (111). \nfig. S2. FMR of the YIG films. fig. S3. SMR in Pt/YIG. \nfig. S4. SMR and AFM image of Au/YIG. \nfig. S5. The band structures of Cu, Au/Cu/Au and Pt/Cu/Pt. \nfig. S6. The spin textures of outer band and inner band. \nfig. S7. The Rashba splitting in Cu/Pt/Cu. \nfig. S8. 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Lett. \n103, 191602 (2013). \n51.  Yukta P Timalsina, Andrew Horning, Robert F Spivey, Kim M Lewis, Tung-Sheng Kuan, \nGwo-Ching Wang and Toh-Ming Lu, Effects of nanoscale surface roughness on the resistivity of ultrathin epitaxial copper films. Nanotechnology 26, 075704 (2015). \n52.  S. Takahashi and S. Maekawa, Spin current, spin accumulation and spin Hall effect. \nScience and Technology of Adv. Mat. 9, 014105 (2008). \n Acknowledgments: Funding: L.Z. and D.W. are supported by National Key R&D Program of \nChina (2017YFA0303202), NSF of China (11674159, 51471086 and 11727808), National \nBasic Research Program of China (2013CB922103). H.S. and J.X. acknowledge the support \nby NSF of China (11474065 and 11722430) and National Key R&D Program of China \n(2016YFA0300702). Author contributions: J.X. and D.W. designed and supervised the \nproject. L.F.Z. and Z.Z.L. prepared the samples. L.F.Z. performed the transport measurements with support from Z.Z.L., P.W., S.W.J. H.K.S. performed the Boltzmann \nsimulations under supervision of J.X. K.L. performed the first principles calculations under \nsupervision of H.J.X. L.S. and Y.B.C. were responsible for the HRTEM characterization. J.X., D.W. and L.F.Z. wrote the manuscript and J.D., H.J.X., H.F.D. and K.X. commented \non the manuscript. All authors discussed the results and reviewed the manuscript. \nCompeting interests: The authors declare no competing financial interests. Data and \nmaterials availability: All data needed to evaluate the conclusions in the paper are present \nin the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors. \n  \n \n \n \n \n \n \n \n \n Science Advances                                                Manuscript Template                                                                           Page 10 of 12 \n \nFigures and Tables \n \n \nA                                         B                                         C           \n \n \n \n \n \n \n \n \n \nFig. 1. Sample characterization. (A) AFM image of YIG(10)/GGG, the rms roughness is \n0.127 nm. (B) AFM image of Pt(0.4)/YIG(10)/GGG, the rms roughness is 0.733 nm. \n(C) HRTEM image of Au(3)/Cu(4)[Pt(0.4)]/YIG heterostructure where Au is used \nto prevent the oxidation. \n \n           A                                                                B \n \n \n \n \n \n \n C                                                                D \n \n \n \n \n \n \n \n \nFig. 2. Field-dependent magnetization and transport measurements. Magnetic \nhysteresis loops of (A) YIG(10)/GGG with field in-plane and (B) YIG(400)/GGG \nwith field out-of-plane. r measured on the Cu(2)[Pt(0.4)]/YIG(10)/GGG sample for \nH applied (C) along x-axis, y-axis and (D) z-axis, respectively. \n \n \n Science Advances                                                Manuscript Template                                                                           Page 11 of 12 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 3. Angular dependent MR measurements. (A) Angular dependent MR measurements \nin the xy, yz, and xz planes for Cu(3)[Pt(0.4)]/YIG. The solid lines are the Boltzmann \nsimulation results. (B) Angular dependent MR measurements in the xy plane for \nseveral control samples. \n Science Advances                                                Manuscript Template                                                                           Page 12 of 12 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 4. Cu- and Pt-thickness dependent transport measurements. Angular dependent \nMR measurements in the xy plane for (A) Cu(t Cu)[Pt(0.4)]/YIG samples and  \nCu(3)/Pt(t Pt)/YIG samples. (B) Cu thickness dependence of the MR ratio and r, \nrespectively, for Cu(t Cu)[Pt(0.4)]/YIG. The solid lines are the Boltzmann simulation \nresults. (C) The Pt layer thickness dependence of the MR ratio and r, respectively, \nfor Cu(3)/Pt(t Pt)/YIG. The solid lines are guide to the eyes. \n\t\n\t\n1 \n\t\n\tSupplementary Materials for  \nObservation of spin -orbit magnetoresistance in  metallic thin  films  on \nmagnetic insulators  \nLifan  Zhou, Hongkang  Song, Kai Liu, Zhongzhi  Luan, Peng  Wang, Lei Sun, Shengwei  Jiang,  \nHongjun Xiang, Yanbin  Chen, Jun Du, Haifeng  Ding, Ke Xia, Jiang  Xiao , and Di Wu \nThis PDF file includes:  \n• section S1. AFM images of Cu(t Cu)[Pt (0.4)]/YIG(10)/GGG(111) \n• section S2. Magnetic properties of the YIG films  \n• section S3. Spin Hall magnetoresistance in Pt/YIG  \n• section S4. SMR and AFM image of Au/YIG \n• section S5. First principles calculations  \n• section S6. Boltzmann simulations  \n• fig. S1. AFM images of Cu(t Cu)[Pt(0.4)]/YIG(10)/GGG (111). \n• fig. S2. FMR of the YIG films. \n• fig. S3. Spin Hall magnetoresistance in Pt/YIG. \n• fig. S4. SMR and AFM image of Au/YIG. \n• fig. S5. The band structures of Cu, Au/Cu/Au and Pt/Cu/Pt. \n• fig. S6. The spin textures of outer band and inner band.  \n• fig. S7. The Rashba splitting in Cu/Pt/Cu. \n• fig. S8. Specular and diffusive interface scattering in the NM/FI bilayer.  \n• fig. S9. AFM image of Ag(0.7)/YIG and MR of Cu(3)[Ag(0.7)]/YIG. \n• References (35–52) \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \t\n\t\n2 \n\t\n\tsection S1.  AFM images of Cu( tCu)[Pt (0.4)]/YIG(10)/GGG(111)  \nThe discontinuous 0.4-nm-thick Pt layer is insulating. The surface morphology gets \nsmoother after the deposition of Cu onto Pt. The rms roughness of \nCu(t Cu)[Pt(0.4)]/YIG(10)/GGG(111) increases with increasing tCu, shown in Fig. S1. \nFor the thin Cu film, resistivity r increases with decreasing Cu thickness tCu, \nindicating that r is dominated by the interface/surface scatterings.  \nsection S2.  Magnetic  properties of the YIG films  \nThe saturation magnetization Ms is measured by ferromagnetic resonance (FMR) with an \nin-plane magnetic field and in an X-band microwave cavity operated at a frequency of f = \n9.7798 GHz. Fig. S2A shows the FMR absorption derivative spectrum of the YIG (10 nm) \nfilm measured at room temperature . The resonance field Hr is at 2608.8 Oe. Ms is \ndetermined to be 164.5 emu/cc by using the Kittel formula : (4rr S fH HMgp=+  where \ng is the gyromagnetic ratio. In comparison, M s of bulk YIG is 140 emu/cc.  \nsection S3.  Spin Hall magnetoresistance in Pt/YIG \nPt/YIG is a typical system with the SMR. We fabricated a sample of Pt(3.4)/YIG to \ncompare with the magnetoresistance of Cu[Pt]/YIG. Fig. S3 presents the angular dependent \nmagnetoresistance of our Pt(3.4)/YIG sample measured in b-scan  at room temperature. The \napplied magnetic field H = 1.5 T is much larger than the demagnetization field to align the \nmagnetization along H. We determined the SMR ratio of about 4.5´10-4, comparable to \nprevious reports (11, 14).  \nsection S4.  SMR and AFM image of Au /YIG  \nIt would be better to compare the MR ratio of the Cu[Au]/YIG sample with Au/YIG sample. \nFig. S4A shows the angular dependent magnetoresistance of the Au(6)/YIG sample \t\n\t\n3 \n\t\n\tmeasured in a-scan  at room temperature. A SMR ratio of ~1.2´10-5 is observed, consistent \nwith the previous reports (17). The SMR ratio in Au/YIG should be larger for the optimized \nAu thickness. The surface of the Au film is smoother than that of the Pt film, shown in Fig. \nS4B. Therefore, the observed magnetoresistance ratio of about  2´10-5 in \nCu(3)[Au(0.4)]/YIG is reasonable.  \nsection S5.  First principles calculations  \nThe calculations are performed within density -functional theory (DFT) using the projector \naugmented wave (PAW) method (35) encoded in the Vienna ab initio simulation pa ckage \n(VASP) (36, 37). The exchange-correlation potential is treated in the generalize d-gradient \napproximation (GGA) (38). The plane-wave cutoff energy is set to be 400 eV. For \ngeometry optimization, all the internal coordinates are relaxed until the Hellmann -\nFeynman forces are less than 1meV/Å and SOI is not included. For the band structure \ncalculation, the SOI is included.  \nWe build three models. The first model is a pure Cu ultra-thin film of 14 monolayers. \nThe second (third) one is the same film covered by Au (Pt) at surfaces on both sides to \nkeep the inversion symmetry of the whole system. The thickness of the bare Cu film, the \nvacuum layer and the surface lattice constant are 27 Å, 20 Å and 2.56 Å, respectively.  \nThe band structures of the three models are s hown in Fig. S5. There is no obvious \nRashba splitting in the bare Cu film and in the film covered by Au, as shown in Fig. S5A \nand S5B, respectively. While in the Cu film covered by Pt, there is a Rashba splitting, and \nthe splitted bands are highlighted by green bold lines, as shown in Fig. S5C. Near the \nGamma point, the bands highlighted by green bold lines are very similar to the parabolic energy dispersion of a two-dimensional-gas in a structure inversion asymmetric \nenvironment, characteristics of the k-linear Rashba effect. To further confirm the nature of \nthe Rashba splitting in Fig. S5C, we calculate the spin textures of outer band and inner \nband around -0.35 eV and -0.10 eV iso-energy surface, respectively, shown in Fig. S6. The \t\n\t\n4 \n\t\n\tinverse rotation of spin orientations of outer band and inner ba nd is characteristic of a pure \nRashba splitting. This indicates that Pt can indeed induce a strong Rashba effect at Cu \nsurfaces. \nIn Fig. S7, we show that when Pt is placed inside Cu film away from the surface, \nthe Rashba splitting decreases significantly, and vanishes when Pt is in the middle of \nthe film. This is consistent with our experimental data that when Pt is placed inside Cu, the SOMR disappear. \nsection S6.  Boltzmann simulations  \nBased on the Boltzmann method developed for CPP (current-perpendicular-to-plane) \nstructure like spin valves  (29-31, 39, 40), we made modifications for the CIP (current-in-\nplane) structure like the bilayer systems used in SMR/HMR/SOMR. \n6.1 Basic formalism of Boltzmann calculation  \nWe use four-component distribution function  ()0, , ,xyz fa= r,k  to denote the charge/spin \noccupation at position  r and wavevector k: f0 is the electric charge distribution and  \n( ) ,,xyzfff=f  is the pure spin (no net charge) distribution. Thus , the majority/minority \nspin distribution 0 ff±=± f, where ± denotes the majority/minority spin along  ˆ±f \ndirection. These four-component distribution function satisfies the generalized spin -\ndependent Boltzmann equations (31, 39-41), \n() ( )( )00 ,0 ,FS0, , ,()() () ( )\nxyzfR ef d P fa\naa\naa aa a d¢¢\n¢==-¶¢ ¢¢ ×- × +\n¶åòr,kvk E vk r k  ,k k k,k r,k\nr，\t(S1) \nwhere E is the applied external electric field and 0m=vk /!  is the velocity in free electron \nmodel, and the right-hand side is the scattering-out and scattering-in collision terms and\nFSd ¢ òkdenotes the integral over Fermi surface. ( ),FS() , Rd Paa a a¢ ¢¢¢ =åòkk k k  is the total \nrelaxation rate for ( ) far,k . And ( ),Paa ¢ ¢k,k  describes the  ¢®kk  scattering \t\n\t\n5 \n\t\n\t\n probability from charge/spin -a¢ to the charge/spin-α, e.g ., ( )0,0P ¢k,k  is the scattering \nprobability for electric charge, ( ),xxP ¢k,k  is the spin-conserved scattering probability for \nspin-x, ( ),yxP ¢k,k  is the scattering probability with spin flip spin -x → spin-y,  ( )0,xP ¢k,k  \nis the scattering probability with spin Hall effect converting pure spin -x to charge, and \n( ),0xP ¢k,k  is the scattering probability with ISHE converting charge to pure spin-x. In the \ncase of normal metal like Cu without bulk SHE, ( ) ()1\n,F S ,/ PAaa aadt-\n¢¢ ¢¢= k,k k in the \nrelaxation time approximation, where AFS is the area of Fermi surface and  () t ¢k is the \nspin-conserved relaxation time for electrons at ¢k.() ()( ) (),0 sf 1/ 1 / Raa td t +- k= k k，where \n()sftk is the spin-flip relaxation time. All scatterings are assumed to be elastic, i.e.,  \n¢=kk . \nWe study an NM/FI bilayer structure, as shown in Fig. S8, whose interfaces/surfaces \nare in x-y plane and locate at /2 zd±=± . The boundary condition for the upper interface \nat zz+=  is given by a surface scattering matrix S+ that connects the impinging \ndistribution function ( 0zk>) and the reflected distribution function  ('0zk>): \n( ) ()( ),FS,, 0 , , 0zz fz k d S f z kaa a a+\n¢¢ ++¢¢ ¢ ¢ >= < òkk k ,k k .         (S2) \nWe regard the interface/surface scattering as specular when  \n( ) ( )( )( )()(),, xx yy zz z z Sk k k k kk k kaa aadd d d+\n¢¢ ¢¢ ¢ ¢ ¢=- - +Q Q - k,k ,   (S3) \nand as diffusive when \n( ) ( )()()1\n,, FS zz SAk kaa aadd+-\n¢¢ ¢¢ ¢=- Q Q - k,k k k ,          (S4) \nwhere the factor ,aad¢ means that the surface scattering is spin-conserving. Similar \nboundary condition can be written down at zz-=. Due to conservation of charge, we have \nthe following identity \t\n\t\n6 \n\t\n\t( ) ( )0,0 0,0FS FS1 dS d S±±¢¢ ¢ òòkk , k =  k k , k =  .              (S5) \nSince spin is generally not conserved, there is no constraint on the spin related boundary  \nscattering matrix. \nOnce the distribution function has been found by solving Eq. (S1), all transport \nproperties can be calculated  accordingly: \ncharge/spin accumulation:\n()()3\n3FS()\n2defaaµ\np=-òkr r,k,                   (S6a) \ncharge current density ( 0a=):()\n()()() ()3\n00 0 3FS,, 2 xyzdje v f v fbb b\naa\na p =éù=- + êú\nëûå òkrk r,k r ,k,(S6b) \nspin current density ( ,,xyza= ):()\n()()() ()3\n00 3FS2dje v f v fbb b\naa apéù =- +ëû òkr k r,k r,k ,(S6c) \nwhere aµ is the charge accumulation when 0a= and spin-a accumulation when \n,,xyza= , 0jb is the charge current flowing in b-direction, jb\na and is the spin-α \ncurrent flowing in b direction when ,,xyza= . The two contributions in 0j and ja \nare due to the fact that different spins may have different velocities, e.g., the \nmajority/minority spin- a has velocity 0 a¢±vv , where a¢v is the anomalous velocity \ndue to the spin-orbit coupling [see Eq. (S10) below]. In the bilayer structure in Fig. S7 with \ntranslational invariance in x-y plane, () ()jj zbb\naa=r , and the film conductivities can be \ncalculated from the current density as  \n()1 z\nzdzj zEbb\naas+\n-=ò.                       (S7) \nFor electric field applied in x direction, the longitudinal conductivity is  0xs, and the \ntransverse Hall conductivity is  0ys. \nTo carry out the Boltzmann calculation numerically, we discretize the Fermi surface in \nk-space and the real space (in the out-of-plane z-direction only) simultaneously: \t\n\t\n7 \n\t\n\t{}{}1 1,z kn n\niji jz= =k . The Boltzmann equation then becomes a set of linear equations, which is \nsolved by matrix inversion. \n6.2 Surface roughness \nFor metallic thin films, the rough surface becomes an important or even dominate factor \non the transport. There are various models in dealing with a rough surface, including the \nFuchs-Sondheimer model (42, 43), the Mayadas-Shatzkes model (44), and the Namba \nmodel (45). All these models are phenomenological, and work only in certain \ncircumstances (46-51). \nTo deal with ultrathin films, we adopt the quantum description of a rough surface as \ndeveloped in Ref. 32-34, in which the relaxation rate becomes channel dependent:  \n()2\n00 ,111 1 1\nnn mnn\ntt t t t tº= + = +¢¢ k with 2\n2214\n3F\ncES\nnd\nta=¢ !,        (S8) \nwhere 0t \tis the bulk impurity relaxation time and nt¢ is the channel-dependent surface \nrelaxation time, and 23\n'13' / 1cn\nc\nnSn n\n==»å . Here a is the lattice constant and δ parameterizes \nthe magnitude of the surface roughness. In Eq. (S8), the relaxation due to the surface \nroughness is built into the Boltzmann equation via total relaxation rate as the bulk impurity  \nscattering, rather than a simple surface scattering. The reason for this is the following: for \nultrathin film, one cannot view the electron as a classical point particle bouncing back and \nforth between the two surfaces, and only feels the surface as the electron hits the surface. \nInstead, the electronic wave function spreads out in the thickness direction and is in contact \nwith the surface all the time, thus the rough surface becomes a ‘bulk’ effect and is felt \nconstantly by the electron and causes scattering from ¢k to k. \n6.3 Interfacial Rashba spin-orbit interaction \nWhen considering the interfacial Rashba spin-orbit interaction at the NM/FI interface, we \nassume a Rashba Hamiltonian of the following form (21) \t\n\t\n8 \n\t\n\t( )( )RHz z hd=× ´ -+ σzp! !!,\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t(S9) \nwhere ˆp is the momentum operator, ( ) ˆ ˆ ˆ ˆ ,,xyzsssσ=  are the Pauli matrices, η is the \nstrength of the Rashba SOI, and ˆz is the normal direction of the interface. Eq. (S9) gives \nrise to a spin-dependent anomalous velocity at the interface: (21, 52) \n[ ] ( )( ) ( )( )ivH z z z z mb\naa a a bb bsh d sh d++¢=- = - ´ » - ´Rr, σzm z!! ! ! !\n\",  (S10)  \nwhere operator ˆσ is approximated by the magnetization direction m at the top surface in \ncontact with FI. vb\na¢ is the anomalous velocity in b direction for spin- a. Therefore, the \nvelocity used in the Boltzmann equation Eq. (S1) is modified with replacement \n0, 0 0, 0 , 'aa g a g ddd®+ vv v  at the top surface at zz+=. The anomalous velocity also \ncontributes to the evaluation of charge/spin currents in Eq. (S6). Such an anomalous \nvelocity would not change the drift term in the Boltzmann equation (first term in Eq. (S1)) \nin the bilayer system studied in this paper. The reason is that the anomalous velocity in Eq. \n(S10) is in-plane and perpendicular to ˆz, while the spatial dependence of fa is in the \nˆz direction, therefore the dot product in the drift term with vanishes identically. \n6.4 Boltzmann simulation results \nWith the numerical Boltzmann method described above, we are ready to calculate the  \nlongitudinal conductivity and the transverse Hall conductivity as function of magneti zation \norientation, NM film thickness, and temperature.  \nWe adopt the conventional magnetic field scanning scheme as show in Fi g. 3A of main \ntext. We calculate both the longitudinal and transverse resistivity as function of the \nmagnetization angel in the a-, b-, g-scans as in Fig. 3A of main text. Then MR is calculated \nas \n()() o90MR  with = , , .rq rqq abgr-=               (S11)  \t\n\t\n9 \n\t\n\twhere r is the average value of ρ(θ), and o90r is the resistivity value at 90o. Fig. 3A  of \nmain text shows the angular dependent of ρ and MR for a Cu film of thickness t = 3 nm, \nwhere δ is chosen to match the average resistivity. It is seen that all three angular \ndependences are in agreement with the experimental data. The small oscillation in the \nexperimental data for the γ-scan might be caused by the weak anisotropic MR, which is  not \nincluded in our simulation. It should note that these angular dependence  of r and MR ratio \nin SOMR are identical to that in the SMR. Therefore, it is impossible to tell SOMR from \nSMR from this angular dependence. The more interesting part is the NM film thickness \ndependence. As we know for sure that r must decrease monotonically as increasing t \nbecause of the reducing surface scattering. This is exactly what has been observed \nexperimentally and calculated  using the Boltzmann method, as  shown in Fig. 4C of main \ntext. In the Boltzmann simulation, we have chosen τ0 as the bulk Cu relaxation time at room \ntemperature. And δ is a varying fitting parameter to account for the surface scattering. \nUsing δ as the only fitting parameter (δ = 6), we find that the longitudinal resistivity can \nbe fitted with a s atisfactory level (Fig. 4C), considering the large error bar and extremely \nthin film. We also note that the Rashba coupling strength has  little effect on the longitudinal \nresistivity as expected. \nIt is well-known that the magnitude of SMR depends on the NM film thickness in a non-\nmonotonic fashion, i.e., there is a peak when the film thickness is comparable to the spin \ndiffusion length of NM. However, the SOMR observed in this work shows a monotonic \ndecreasing behavior as increasing NM thickness. We should note that Cu has very long \nspin diffusion length, much longer than the film thickness. Such monotonic MR ratio can \nalso be fitted using the Boltzmann simulation with only one fit ting parameter, i.e., the \nRashba coupling constant η (η = 0.174). Similar to the SMR effect, the SOMR effect also \ndepends quadratically on the spin-orbit coupling strength, therefore 2MR hµ .  \n \n \t\n\t\n10 \n\t\n\t\n \n\t\n\t\n\t\n\t  \n           \n \n \n \n \n \n \n \n \n \n\t\n \n \nfig. S1. AFM images of Cu(t\nCu)[Pt(0.4)]/YIG(10)/GGG (111). \n \n \n\t\n\t\n11 \n\t\n\t \nfig. S2. FMR of the YIG films. FMR absorption derivative spectrum of YIG(10 \nnm)/GGG(111) film with field in sample plane measured at room temperature.  \n \n \nfig. S3. Spin Hall magnetoresistance in Pt/YIG. The angular dependent \nmagnetoresistance of our Pt(3.4)/YIG sample measured in b-scan  at room temperature with \nthe magnetic field of 1.5 T. \n\t\n\t\n12 \n\t\n\t \nfig. S4. SMR and AFM image of Au/YIG. (A) The angular dependent MR of the \nAu(6)/YIG sample measured in a-scan with H = 2000 Oe at room temperature. (B) AFM \nimage of Au(0.4)/YIG. The rms roughness is 0.135 nm. \n \n \n \nfig. S5. The band structures of Cu, Au/Cu/Au and Pt/Cu/Pt. The band structures of (A) \nthe Cu ultra-thin film of 14 monolayers, (B) the same film covered by Au, (C) the same \nfilm covered by Pt. The bands marked by green bold lines indicate a R ashba splitting. \n \n \n\t\n\t\n13 \n\t\n\t \nfig. S6. The spin textures of outer band and inner band. Spin texture of (A) the outer \nband around -0.35 eV iso-energy surface and (B) the inner band around -0.10 eV iso-energy \nsurface. The outer band and inner band are hig hlighted by green bold lines in Fig. S5C. \n \n \n \nfig. S7. The Rashba splitting in Cu/Pt/Cu. The band structures of (A) the Cu ultra-thin \nfilm of 14 monolayers with Pt located 4 monolayers away from the surface, and it s hows \na weak Rashba splitting; (B) the same film with Pt located in the middle, and there is no \nRashba splitting. \n \n \n\t\n\t\n14 \n\t\n\t\n    \nfig. S8. Specular and diffusive interface scattering in the NM/FI bilayer. \n \n(A)                           (B)   \n \n   \nfig. S9. AFM image of Ag(0.7)/YIG and MR of Cu(3)[Ag(0.7)]/YIG. (A) AFM \nimage of Ag(0.7)/YIG. The rms roughness is 0.797 nm. (B) The angular dependent \nMR of the Cu(3)[Ag(0.7)]/YIG sample measured in \na-scan with H = 2000 Oe at room \ntemperature. \n"    },    {        "title": "1709.01890v1.Complex_THz_and_DC_inverse_spin_Hall_effect_in_YIG_Cu___1_x__Ir___x___bilayers_across_a_wide_concentration_range.pdf",        "content": "Complex THz and DC inverse spin Hall effect in YIG/Cu 1−xIrx\nbilayers across a wide concentration range\nJoel Cramer,1, 2Tom Seifert,3Alexander Kronenberg,1Felix Fuhrmann,1\nGerhard Jakob,1Martin Jourdan,1Tobias Kampfrath,3, 4and Mathias Kl¨ aui1, 2,∗\n1Institute of Physics, Johannes Gutenberg-University Mainz, 55099 Mainz, Germany\n2Graduate School of Excellence Materials Science in Mainz, 55128 Mainz, Germany\n3Department of Physical Chemistry,\nFritz Haber Institute of the Max Planck Society, 14195 Berlin, Germany\n4Department of Physics, Freie Universit¨ at Berlin, 14195 Berlin, Germany\n(Dated: September 7, 2017)\nAbstract\nWe measure the inverse spin Hall effect of Cu 1−xIrxthin films on yttrium iron garnet over a\nwide range of Ir concentrations (0 .056x60.7). Spin currents are triggered through the spin\nSeebeck effect, either by a DC temperature gradient or by ultrafast optical heating of the metal\nlayer. The spin Hall current is detected by, respectively, electrical contacts or measurement of the\nemitted THz radiation. With both approaches, we reveal the same Ir concentration dependence\nthat follows a novel complex, non-monotonous behavior as compared to previous studies. For small\nIr concentrations a signal minimum is observed, while a pronounced maximum appears near the\nequiatomic composition. We identify this behavior as originating from the interplay of different\nspin Hall mechanisms as well as a concentration-dependent variation of the integrated spin current\ndensity in Cu 1−xIrx. The coinciding results obtained for DC and ultrafast stimuli show that the\nstudied material allows for efficient spin-to-charge conversion even on ultrafast timescales, thus\nenabling a transfer of established spintronic measurement schemes into the terahertz regime.\n1arXiv:1709.01890v1  [cond-mat.mtrl-sci]  6 Sep 2017INTRODUCTION\nSpin currents are a promising ingredient for the implementation of next-generation,\nenergy-efficient spintronic applications. Instead of exploiting the electronic charge, transfer\nas well as processing of information is mediated by spin angular momentum. Crucial steps\ntowards the realization of spintronic devices are the efficient generation, manipulation and\ndetection of spin currents at highest speeds possible. Here, the spin Hall effect (SHE) and\nits inverse (ISHE) are in the focus of current research [1] as they allow for an interconversion\nof spin and charge currents in heavy metals with strong spin-orbit interaction (SOI). The\nefficiency of this conversion is quantified by the spin Hall angle θSH.\nIn general, the SHE has intrinsic as well as extrinsic spin-dependent contributions. The\nintrinsic SHE results from a momentum-space Berry phase effect and can, amongst others,\nbe observed in 4 dand 5dtransition metals [1–3]. The extrinsic SHE, on the other hand,\nis a consequence of skew and side-jump scattering off impurities or defects [4]. It occurs\nin (dilute) alloys of normal metals with strong SOI impurity scatterers [5–8], but can also\nbe prominent in pure metals in the superclean regime [9]. As a consequence, the type of\nemployed metals and the alloy composition are handles to adjust and maximize the SHE.\nRemarkably, it was recently shown that the SHE in alloys of two heavy metals (e.g. AuPt)\ncan even exceed the SHE observed for the single alloy partners [10]. Pioneering work within\nthis research field covered the extrinsic SHE by skew scattering in copper-iridium alloys [5].\nHowever, previously the iridium concentration was limited to 12 % effective doping of Cu\nwith dilute Ir. The evolution of the SHE in the alloy regime for large concentration thus\nremains an open question and the achievable maximum by an optimized alloying strategy\nis unknown.\nThe potential of a metal for spintronic applications (i.e. θSH) can be quantified by inject-\ning a spin current and measuring the resulting charge response. This can be accomplished\nby, for instance, coherent spin pumping through ferromagnetic resonance [11–13] or the spin\nSeebeck effect (SSE) [14, 15]. The SSE describes the generation of a magnon spin cur-\nrent along a temperature gradient within a magnetic material. Typically, such experiments\ninvolve a heterostructure composed of a magnetic insulator, such as yttrium iron garnet\n(YIG), and the ISHE metal under study [see Fig. 1(a)]. A DC temperature gradient in the\nYIG bulk is induced by heating the sample from one side. On the femtosecond timescale,\n2however, a temperature difference and thus a spin current across the YIG-metal interface\ncan be induced by heating the metal layer with an optical laser pulse [Fig. 1(b)] [16–19].\nThis interfacial SSE has been shown to dominate the spin current in the metal on timescales\nbelow∼300 ns [16].\nFor ultrafast laser excitation, the resulting sub-picosecond ISHE current leads to the\nemission of electromagnetic pulses at frequencies extending into the terahertz (THz) range,\nwhich can be detected by optical means [20]. Therefore, femtosecond laser excitation offers\nthe remarkable benefit of contact-free measurements of the ISHE current without any need\nof micro-structuring the sample. The all-optical generation as well as detection of ultra-\nfast electron spin currents [20, 21] is a key requirement for transferring spintronic concepts\ninto the THz range [22]. So far, however, characterization of the ISHE was conducted by\nexperiments including DC spin current signals as, for instance, the bulk SSE [Fig. 1(a)].\nFor the use in ultrafast applications, it thus remains to be shown whether alloying yields\nthe same notable changes of the spin-to-charge conversion efficiency in THz interfacial SSE\nexperiments [Fig. 1(b)] and whether alloys can provide an efficient spin-to-charge conversion\neven at the ultrafast timescale.\nIn this work, we study the compositional dependence of the ISHE in YIG/Cu 1−xIrx\nbilayers over a wide concentration range (0 .056x60.7), exceeding the dilute doping\nphase investigated in previous studies [5]. The ISHE response of Cu 1−xIrxis measured as a\nfunction of x, for which both DC bulk and THz interfacial SSE are employed. Eventually,\nwe compare the spin-to-charge conversion efficiency in the two highly distinct regimes of DC\nand terahertz dynamics across a wide alloying range.\nEXPERIMENT\nThe YIG samples used for this study are of 870 nm thickness, grown epitaxially on\n(111)-oriented Gd 3Ga5O12(GGG) substrates by liquid-phase-epitaxy. After cleaving the\nGGG/YIG into samples of dimension 2 .5 mm×10 mm×0.5 mm, Cu 1−xIrxthin films (thick-\nnessdCuIr= 4 nm) of varying composition ( x= 0.05,0.1,0.2,0.3,0.5 and 0.7) are deposited\nby multi-source magnetron sputtering. To prevent oxidation of the metal film, a 3 nm Al\ncapping layer is deposited, which, when exposed to air, forms an AlO xprotection layer. For\nthe contact-free ultrafast SSE measurements, patterning of the Cu 1−xIrxfilms into defined\n3(a)\n(b)Figure 1. (a) Scheme of the setup used for DC SSE measurements. The out-of-plane temperature\ngradient is generated by two copper blocks set to individual temperatures T1andT2. An external\nmagnetic field is applied in the sample plane. The resulting thermovoltage Vtis recorded by\na nanovoltmeter. (b) Scheme of the contact-free ultrafast SSE/ISHE THz emission approach.\nThe in-plane magnetized sample is illuminated by a femtosecond laser pulse, inducing a step-like\ntemperature gradient across the YIG/Cu 1−xIrxinterface. The SSE-induced THz spin current in\nthe CuIr layer is subsequently converted into a sub-picosecond in-plane charge current by the ISHE,\nthereby leading to the emission of a THz electromagnetic pulse into the optical far-field.\nnanostructures is not necessary. In the case of DC SSE measurements, the unpatterned film\nis contacted for the detection of the thermal voltage.\nThe DC SSE measurements are performed at room temperature in the conventional\nlongitudinal configuration [15]. While an external magnetic field is applied in the sample\nplane, two copper blocks, which can be set to individual temperatures, generate a static\nout-of-plane temperature gradient, see Fig. 1(a). This thermal perturbation results in a\nmagnonic spin current in the YIG layer [23], thereby transferring angular momentum into\nthe Cu 1−xIrx. A spin accumulation builds up, diffuses as a pure spin current and is eventually\nconverted into a transverse charge current by means of the ISHE, yielding a measurable\nvoltage signal. The spin current and consequently the thermal voltage change sign when the\n4YIG magnetization is reversed. The SSE voltage VSSEis defined as the difference between\nthe voltage signals obtained for positive and negative magnetic field divided by 2. Since\nVSSEis the result of the continuous conversion of a steady spin current, it can, applying the\nnotation of conventional electronics, be considered as a DC signal.\nFor the THz SSE measurements, the same in-plane magnetized YIG/Cu 1−xIrxsamples\nare illuminated at room temperature by femtosecond laser pulses (energy of 2 .5 nJ, duration\nof 10 fs, center wavelength of 800 nm corresponding to a photon energy of 1 .55 eV, repetition\nrate of 80 MHz) of a Ti:sapphire laser oscillator. Owing to its large bandgap of 2 .6 eV\n[24], YIG is transparent for these laser pulses. They are, however, partially (about 50 %)\nabsorbed by the electrons of the Cu 1−xIrxlayer. The spatially step-like temperature gradient\nacross the YIG/metal interface leads to an ultrafast spin current in the metal layer polarized\nparallel to the sample magnetization [19]. Subsequently, this spin current is converted into\na transverse sub-picosecond charge current through the ISHE, resulting in the emission of a\nTHz electromagnetic pulse into the optical far-field. The THz electric field is sampled using\na standard electrooptical detection scheme employing a 1 mm thick ZnTe detection crystal\n[25]. The magnetic response of the system is quantified by the root mean square (RMS) of\nhalf the THz signal difference Sfor positive and negative magnetic fields.\nRESULTS\nFigure 2(a)-(f) shows DC SSE hysteresis loops measured for YIG/Cu 1−xIrx/AlOx multi-\nlayers with varying Ir concentration x. The temperature difference between sample top and\nbottom is fixed to ∆ T= 10 K with a base temperature of T= 288.15 K. In the Cu-rich\nphase, we observe an increase of the thermal voltage signal with increasing x, exhibiting\na maximum at x= 0.3. Interestingly, upon further increasing the Ir content VSSEreduces\nagain. This behavior is easily visible in Fig. 2(g), in which the SSE coefficient VSSE/∆Tis\nplotted as a function of x. The measured concentration dependence shows that VSSE/∆T\nexhibits a clear maximum in the range from x= 0.3 to 0.5. Thus, as a first key result the\nmaximum spin Hall effect is obtained for the previously neglected alloying regime beyond\nthe dilute doping. For comparison, the resistivity σ−1of the metal film is also shown in\nFig. 2(g). We see that the resistivity of the Cu 1−xIrxlayer follows a similar trend as the DC\nSSE signal.\n50.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7\nx0.20.40.60.81.0\n246810\n−10−50510\n−10−50510VSSE(µV)\n−10 0 10\nµ0H(mT)−10−50510\n−10 0 10\nµ0H(mT)(a)\n(b)\n(c)(d)\n(e)\n(f)x = 0.05\nx = 0.1\nx = 0.2x = 0.3\nx = 0.5\nx = 0.75.0 µV\n5.7 µV\n8.1 µV18.1 µV\n15.7 µV\n9.1 µVVSSE /ΔT µVK−1) )(g)\nVSSE/∆T\n10−7Ωm) ) σ−1σ−1Figure 2. (a)-(f) Measured DC SSE voltage in YIG/Cu 1−xIrx/AlOx stacks for different Ir con-\ncentrations xin ascending order. The temperature difference between sample top and bottom is\nfixed to ∆T= 10 K. (g) SSE coefficient VSSE/∆T(red squares) and resistivity σ−1(blue circles)\nas a function of Ir concentration x.\nTypical THz emission signals from the YIG/Cu 1−xIrx/AlOx samples are depicted in\nFigs. 3(a)-(f). The THz transients were low-pass filtered in the frequency domain with\na Gaussian centered at zero frequency and a full width at half maximum of 20 THz. The\nRMS of the THz signal odd in sample magnetization is plotted in Fig. 3(g) as a function of x.\nAfter an initial signal drop in the Cu-rich phase, the THz signal increases with increasing Ir\nconcentration, indicating a signal maximum in the range between x= 0.3 and 0.5. Further\nincrease of the Ir content leads to a second reduction of the THz signal strength.\nDISCUSSION\nIn the following, a direct comparison of the signals obtained from the DC and the ultrafast\nTHz measurements is established. To begin with, the emitted THz electric field right behind\nthe sample is described by a generalized Ohm’s law, which in the thin-film limit (film is much\n6(a)\n−4−2024\n−4−2024THz signal (arb. u. )\n−1.0 −0.5 0.0 0.5\nt (ps)−4−2024\n−1.0 −0.5 0.0 0.5(b)\n(c)(d)\n(e)\n(f)x = 0.05\nx = 0.1\nx = 0.2x = 0.3\nx = 0.5\nx = 0.7\n0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.60.70.80.91.01.1\nx(g)THz signal strength (norm. ) t (ps)Figure 3. (a)-(f) Signal waveforms (odd in the sample magnetization) of the THz pulses emitted\nfrom YIG/Cu 1−xIrx/AlOx stacks for different Ir concentrations xin ascending order. (g) THz\nsignal strength (RMS) as a function of Ir concentration x.\nthinner than the wavelength and attenuation length of the THz wave in the sample) is in\nthe frequency domain given by [21]\n˜E(ω)∝θSHZ(ω)/integraldisplayd\n0dzjs(z,ω), (1)\nwhereωis the angular frequency. The spin-current density js(z,ω) is integrated over the\nfull thickness dof the metal film. The total impedance Z(ω) can be understood as the\nimpedance of an equivalent parallel circuit comprising the metal film (Cu 1−xIrx) and the\nsurrounding substrate (GGG/YIG) and air half-spaces,\n1\nZ(ω)=n1(ω) +n2(ω)\nZ0+G(ω). (2)\nHere,n1andn2≈1 are the refractive indices of substrate and air, respectively, Z0=\n377 Ω is the vacuum impedance, and G(ω) is the THz sheet conductance of the Cu 1−xIrx\nfilms. Considering the Drude model and a velocity relaxation rate of 28 THz for pure Cu\nat room temperature as lower boundary [26], the values of G(ω) vary only slightly over the\n7detected frequency range from 1 to 5 THz (as given by the ZnTe detector crystal). Therefore,\nthe frequency dependence of the conductance can be neglected, i.e. G(ω)≈G(ω= 0).\nImportantly, the metal-film conductance ( G≈8×10−3Ω−1) is much smaller than the shunt\nconductance ([ n1(ω) +n2(ω)]/Z0≈4×10−2Ω−1) for the investigated metal film thickness\n(d= 4 nm) and can be thus neglected. Therefore, the Ir-concentration influences the THz\nemission strength only directly through the ISHE-induced in-plane charge current flowing\ninside the NM layer.\nThe measured DC SSE voltage, on the other hand, is given by an analogous expression\nrelated to the underlying in-plane charge current by the standard Ohm’s law,\nVSSE\n∆T∝θSHR/integraldisplayd\n0dzjs(z). (3)\nHere,Ris the Ohmic resistance of the metal layer between the electrodes, which is inversely\nproportional to the metal resistivity σ, andjs(z) is the DC spin current density. Therefore,\nin contrast to the THz data, the impact of alloying on VSSEthroughσ−1is significant. For a\ndirect comparison with the THz measurements, we thus contrast the RMS of the THz signal\nwaveform with the DC SSE current density jSSE=VSSE·σ/∆T.\nIn Fig. 4, the respective amplitudes are plotted as a function of the Ir concentration.\nRemarkably, DC and THz SSE/ISHE measurements exhibit the very same concentration\ndependence. This agreement suggests that the ISHE retains its functionality from DC up to\nTHz frequencies, which vindicates the findings and interpretations of previous experiments\n[21]. Small discrepancies may originate from a varying optical absorptance of the near-\ninfrared pump light, which is, however, expected to depend monotonically on xand to only\nvary by a few percent [21]. Furthermore, as discussed below, these findings imply that for\nDC and THz spin currents comparable concentration dependences of spin-relaxation lengths\nmay be expected.\nTo discuss the concentration dependence of the DC and THz SSE signals (Fig. 4), we\nconsider Eqs. (1) and (3). According to these relationships, the THz signal and the SSE\nvoltage normalized by the metal resistivity result from a competition of (i) the spin Hall\nangleθSHand (ii) the integrated spin-current density/integraltextd\n0dzjs(z,ω).\nAt first, we consider the local spin signal minimum at small, increasing Ir concentration\nx(dilute regime) that appears for both jSSEand the THz signal. In fact, with regard to\n(i)θSHone would expect the opposite behavior as for the dilute regime the skew scattering\n8THz signal strength (norm. )\n0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7\nx0.080.100.120.14\n0.60.70.80.91.01.1107µV(KΩm)−1) ) jSSE/ΔTjSSE/∆T\nTHzFigure 4. Ir concentration dependence of the thermal DC spin current (red squares) and the RMS\nof the THz signal (green diamonds).\nmechanism has been predicted [4] and experimentally shown [5] to yield the dominant ISHE\ncontribution. With increasing SOI scattering center density ( ρimp∝σ−1), a linear increase\nof the spin signal should appear. In this work, this trend is observed for VSSE[Fig. 2(g)].\nThe significantly deviating signal shapes of jSSEand the THz signal, however, suggest that\nthe converted in-plane charge current is notably governed by additional effects. An explana-\ntion can be given by (ii), considering a spatial variation of the spin current density that, as\nwe discuss below, can be influenced by both electron momentum- and spin-relaxation. The\ninitial electron momenta and spin information of a directional spin current become random-\nized over length scales characterized by the mean free path /lscriptand the spin diffusion length\nλsd, yielding a reduction of the spin current density. For spin-relaxation, the integrated spin\ncurrent density is given by [27]:\n/integraldisplaydCuIr\n0dzjs(z)∝λsdtanh/parenleftBiggdCuIr\n2λsd/parenrightBigg\nj0\ns (4)\nwithdCuIrbeing the thickness of the Cu 1-xIrxlayer. According to Niimi et al. [5] the spin-\ndiffusion length λsddecreases exponentially from λsd≈30 nm forx= 0.01 toλsd≈5 nm for\nx= 0.12. This exponential decay implies that the integrated spin current density is nearly\nconstant for both small and large x, but undergoes a significant decline in the concentration\nregion where λsd≈dCuIr. This effect possibly explains the observed reduction of the signal\namplitude from x= 0.05 tox= 0.2. Furthermore, we interpret the fact that for DC\nand THz SSE signals similar trends are observed as an indication of similar concentration\ndependences of λsdin the distinct DC and THz regimes. This appears reasonable when\nconsidering that spin-dependent scattering rates are of the same order of magnitude as\nthe momentum scattering [28] (e.g. Γmom.\nCu = 1/36 fs≈28 THz [26]) and thus above the\n9experimentally covered bandwidth.\nIn addition to spin-relaxation, the integrated spin current density is influenced by mo-\nmentum scattering. As shown in Fig. 2, alloying introduces impurities and lattice defects\nin the dilute phase, such that enhanced momentum scattering rates occur. Assuming that\nthe latter increase more rapidly than θSH, the appearance of the previously unexpected local\nminimum near x≈0.2 can be thus explained.\nWe now focus on the subsequent increase of the spin signal at higher x(concentrated\nphase). It can be explained by a further increase of extrinsic ISHE as well as intrinsic\nISHE contributions, as pure Ir itself exhibits a sizeable intrinsic spin Hall effect [2, 3]. A\nquantitative explanation of the intrinsic ISHE, however, requires knowledge of the electron\nband structure (obtainable by algorithms based on the tight-binding model [2] or the density\nfunctional theory [29]), which is beyond the scope of this work. The decrease of jSSEand\nthe THz Signal at x= 0.7 may then be ascribed to an increase of atomic order and thus a\ndecrease of the extrinsic ISHE.\nIn conclusion, we compare the spin-to-charge conversion of steady state and THz spin\ncurrents in copper-iridium alloys as a function of the iridium concentration. We find a clear\nmaximum of the spin Hall effect for alloys of around 40 % Ir concentration, far beyond\nthe previously probed dilute doping regime. While the detected DC spin Seebeck voltage\nexhibits a concentration dependence different from the raw THz signal, very good qualitative\nagreement between the DC spin Seebeck current and the THz emission signal is observed,\nwhich is well understood within our model for THz emission. Ultimately, our results show\nthat tuning the spin Hall effect by alloying delivers an unexpected, complex concentration\ndependence that is equal for spin-to-charge conversion at DC and THz frequencies and allows\nus to conclude that the large spin Hall effect in CuIr can be used for spintronic applications\non ultrafast timescales.\nACKNOWLEDGMENTS\nThis work was supported by Deutsche Forschungsgemeinschaft (DFG) (SPP 1538 “Spin\nCaloric Transport”, SFB/TRR 173 ”SPIN+X”), the Graduate School of Excellence Materi-\nals Science in Mainz (DFG/GSC 266), and the EU projects IFOX, NMP3-LA-2012246102,\nINSPIN FP7-ICT-2013-X 612759, TERAMAG H2020 681917.\n10∗Klaeui@uni-mainz.de\n[1] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, Rev. Mod. Phys.\n87, 1213 (2015).\n[2] T. Tanaka, H. Kontani, M. Naito, T. Naito, D. S. Hirashima, K. Yamada, and J. Inoue, Phys.\nRev. B 77, 165117 (2008).\n[3] M. Morota, Y. Niimi, K. Ohnishi, D. H. Wei, T. Tanaka, H. Kontani, T. 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Lett. 100, 096401 (2008).\n12"    },    {        "title": "1608.08043v1.Sub_micrometer_yttrium_iron_garnet_LPE_films_with_low_ferromagnetic_resonance_losses.pdf",        "content": "Sub-micrometer yttrium iron garnet LPE \flms with low ferromagnetic\nresonance losses\nCarsten Dubs,1Oleksii Surzhenko,1Ralf Linke,1Andreas Danilewsky,2Uwe Br uckner,3and Jan Dellith3\n1)INNOVENT e.V., Technologieentwicklung, Pr ussingstr. 27B, 07745 Jena, Germany\n2)Kristallographie, Albert-Ludwigs-Universit at Freiburg, Hermann-Herder-Str. 5, 79104 Freiburg,\nGermany\n3)Leibniz-Institut f ur Photonische Technologien (IPHT), Albert-Einstein-Str. 9, 07745 Jena,\nGermany\n(Dated: 30 August 2016)\nUsing liquid phase epitaxy (LPE) technique (111) yttrium iron garnet (YIG) \flms with thicknesses of \u0019100 nm\nand surface roughnesses as low as 0.3 nm have been grown as a basic material for spin-wave propagation\nexperiments in microstructured waveguides. The continuously strained \flms exhibit nearly perfect crys-\ntallinity without signi\fcant mosaicity and with e\u000bective lattice mis\fts of \u0001 a?=as\u001910\u00004and below. The\n\flm/substrate interface is extremely sharp without broad interdi\u000busion layer formation. All LPE \flms ex-\nhibit a nearly bulk-like saturation magnetization of (1800 \u000620) Gs and an `easy cone' anisotropy type with\nextremely small in-plane coercive \felds <0.2 Oe. There is a rather weak in-plane magnetic anisotropy with a\npronounced six-fold symmetry observed for saturation \feld <1.5 Oe. No signi\fcant out-of-plane anisotropy is\nobserved, but a weak dependence of the e\u000bective magnetization on the lattice mis\ft is detected. The narrowest\nferromagnetic resonance linewidth is determined to be 1.4 Oe @ 6.5 GHz which is the lowest values reported\nso far for YIG \flms of 100 nm thicknesses and below. The Gilbert damping coe\u000ecient for investigated LPE\n\flms is estimated to be close to 1 \u000210\u00004.\nPACS numbers: 81.15.Lm, 75.50.Gg, 76.50.+g\nI. INTRODUCTION\nMagnonics is an increasingly growing new branch\nof spin-wave physics, speci\fcally addressing the use of\nmagnons for information transport and processing1{4.\nSingle crystalline yttrium iron garnet (YIG), which is a\nferrimagnetic insulator with the smallest known magnetic\nrelaxation parameter5, appears to be a superior candi-\ndate for this purpose6{8. As bulk or as thick \flm mate-\nrial, which is commonly grown by liquid phase epitaxy\n(LPE)9, it has a very low damping coe\u000ecient and allows\nmagnons to propagate over distances exceeding several\ncentimeters6. However YIG functional layers for practi-\ncal magnonics should be nanometer-thin with extremely\nsmooth surfaces in order to achieve optimum e\u000eciency\nin data processing and dramatic reduction in energy con-\nsumption of sophisticated spin-wave devices. Therefore,\nhigh-quality thin and ultra-thin YIG \flms were grown us-\ning di\u000berent growth techniques such as LPE, pulsed laser\ndeposition (PLD) and rf-magnetron sputtering to inves-\ntigate diverse spin-wave e\u000bects and to design YIG waveg-\nuides as well as nanostructures for spin wave excitation,\nmanipulation and detection in prospective magnonic cir-\ncuits.\nFrom previous reports about sub-micrometer YIG\n\flms with thicknesses between 100 and 20 nm10{15avail-\nable microwave and magnetic key parameters were taken\nand summarized in Table I. Thus, ferromagnetic reso-\nnance (FMR) data were included which have been ex-\ntracted from measurements of the absorption curves or\nabsorption derivative curves versus sweeping magnetic\nin-plane \feld Hat a \fxed frequency for vs. sweep-ing rf-exciting \feld hrfwith an applied in-plane static\nmagnetic bias \feld. The reported FMR linewidths \u0001 H\nand converted peak to peak linewidths \u0001 Hp\u0000pof the\n\feld derivative values (\u0001 H=p\n3\u0001Hp\u0000p), which will\nbe given during the further paper as full-width at half-\nmaximum \u0001 HFWHM , varied between 3 Oe and 13 Oe.\nThe Gilbert damping coe\u000ecient \u000bwere found in the\nrange from 2\u000210\u00004to 8\u000210\u00004. Only the lowest given \u000b\nvalue of 0:9\u000210\u00004was obtained for a very short \ft range\nof about 4 GHz without any given data in the low fre-\nquency range below 10 GHz13and is therefore not really\ncomparable with the other reported values. From this\ncompilation it is obvious that neither \u0001 HFWHM nor\u000bis\nsigni\fcantly in\ruenced by the YIG \flm thickness down\nto 20 nm. The di\u000berences are probably resulted from\nadditional ferromagnetic losses due to contributions of\nhomogeneous and/or inhomogeneous broadening by mi-\ncrostructural imperfections or magnetic inhomogeneities.\nIn this report we present microstructural, magnetic\nand FMR properties of LPE-grown 100 nm thin YIG and\nLanthanum substituted (La:YIG) \flms with low ferro-\nmagnetic resonance losses. Film thicknesses were deter-\nmined by X-ray re\rectometry (XRR) and surface rough-\nness by atomic force microscopy (AFM) measurements.\nCrystalline perfection and compositional homogeneity\nwere investigated by high-resolution X-ray di\u000braction\n(HR-XRD) and X-ray photoelectron spectroscopy (XPS)\nas well as by secondary ion mass spectroscopy (SIMS).\nStatic and dynamic (microwave) magnetic characteriza-\ntions were carried out by vibrating sample magnetometry\n(VSM) and by Vector Network Analysis (VNA), respec-\ntively.arXiv:1608.08043v1  [cond-mat.mtrl-sci]  29 Aug 20162\nTABLE I: Key parameters reported for thin/ultrathin YIG \flms on (111) GGG substrates\nGrowth method Thick- RMS- 4 \u0019MsaHca\u0001Haf0 \u0001H0a\u000b\n(Reference) ness roughness FWHM FWHM \u000210\u00004\n(nm) (nm) (kGs) (Oe) (Oe) (GHz) (Oe)\nLPE10100 - 1.81 - 3.0 7 1.6 2.8\nLPE (this study) 83{113 0.3{0.8 1.78{1.82 \u00140.2 1.4{1.6 6.5 0.5{0.7 1.2{1.7\nPLDb 1179 0.2 1.72 <2 3.0 10 1.4 2.2\nPLD1223 - 1.60 <1 3.5c9.6 3.5{7c2{4\nSputtering1322 0.13 1.78 0.4 12c16.5 6.4c0.9\nSputtering1420 0.2 - 0.4 13c9.7 7c8\nPLD1520 0.2{0.3 2.10 0.2 3.3c6 2.4c2.3\naMeasurements at RT with the in-plane external magnetic \feld H\nbYIG \flms grown on the (100) GGG substrates\ncPeak-to-peak value \u0001 Hp\u0000pof the derivative of FMR absorption transformed into \u0001 HFWHM = \u0001Hp\u0000p\u0002p\n3\nII. RESULTS\nA. Microstructural properties\nSelected microstructural and magnetic properties of\nliquid phase epitaxial grown YIG (sample A-C) and\nLa:YIG (sample D) \flms are given in Table II. The con-\nsistent magnetic as well as microwave properties obtained\nfor \flms deposited during di\u000berent growth runs demon-\nstrate a high reproducibility of the LPE growth tech-\nnique. Fig. 1a shows XRR plots of \flms with thicknesses\nof about 100 nm which are smaller than the previously re-\nported thinnest LPE YIG \flms16{18. The smallest root-\nmean-square (RMS) surface roughness of about 0.25 nm\nobtained for the sample B in Fig. 1b is nearly compa-\nrable with epi-polished GGG substrate quality of \u00190.15\nnm and with the best PLD and sputtered YIG \flms (see\ne.g. Table I). Besides, \flms with slightly rougher surfaces\n(see Table II)) were obtained as a result of additional\ndendritic aftergrowth and/or due to plateau formation,\nso called \\mesas\", if any solution droplet adheres to the\nsample surface.\nHR-XRD studies of our thin epitaxial LPE \flms have\nbeen found to be di\u000ecult because of the nearly super-\nimposed di\u000braction pattern of YIG \flm and GGG sub-\nstrate. Although the angle distances between \flm and\nsubstrate Bragg re\rections were above the resolution\nlimit of our HR-XRD equipment, the di\u000braction inten-\nsity of the \flm re\rection was very low and results only\nin a broadening of the GGG Bragg re\rection. Fig. 2a\nshows a!-scan (rocking curve) with a Gaussian-like \ft-\nted GGG substrate 444 re\rection and a second \ftted\npeak at the right shoulder which corresponds to the YIG\n444 \flm re\rection. This indicates a tensile stressed YIG\n\flm because of the smaller \flm lattice parameter com-\npared to the commercially available Czochralski-grown\nGGG substrate ( as=1.2382 nm). For La:YIG \flms we ob-\nserved a perfect pseudo-Voigt \ftted substrate peak with-\nout any additional shoulder (not shown) which indicates\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s49/s48/s49/s49/s48/s51/s49/s48/s53/s49/s48/s55\n/s100/s32/s61/s32/s57/s55/s32/s110/s109/s100/s32/s61/s32/s56/s51/s32/s110/s109/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s99/s112/s115/s41\n/s80/s111/s115/s105/s116/s105/s111/s110/s32 /s32 /s40/s67/s117/s45/s75/s76\n/s50/s44/s51/s41/s83/s97/s109/s112/s108/s101/s32/s67\n/s83/s97/s109/s112/s108/s101/s32/s68(a)\n(b)\nFIG. 1: (a) XRR plots of sub-micrometer-thick YIG\nLPE \flms. (b) 5\u00025\u0016m2AFM surface topography of\nsample B with RMS roughness of 0.25 nm.\na perfect lattice match between substrate and LPE \flm.\nThis is in remarkable contrast to YIG \flms deposited\nby various gas phase techniques such as PLD and rf-3\nTABLE II: YIG/La:YIG \flm properties grown on (111) GGG substrates by LPE technology\nThick- RMS- Relative lattice VSMaFMRa\nSample ness roughness mis\ft \u0001 a?=as 4\u0019MsHc 4\u0019Me\u000b \u0001HFWHMb\u0001H0\u000b\n(nm) (nm) \u000210\u00004(kGs) (Oe) (kGs) (Oe) (Oe) \u000210\u00004\nA 113 0.8 4.7 1.82 0.10 1.637 1.4 0.5 1.4\nB 106 0.3 1.8 1.78 0.20 1.658 1.5 0.7 1.2\nC 83 0.6 0.3 1.82 0.16 1.672 1.4 0.5 1.6\nDc97 0.8 0.0 1.78 0.18 1.712 1.6 0.7 1.7\nAccuracy \u00061 \u00060:1 \u00060:3 \u00060:04 \u00060:03 \u00060:010 \u00060:1 \u00060:1 \u00060:1\naVSM and FMR measurements at room temperature with applied in-plane magnetic \feld\nbFMR linewidth value at frequency f=6.5 GHz\ncLa:YIG LPE \flm\nsputtering11{15,19,20on GGG substrates. For those \flms\nthe YIG re\rection has always been detected at consider-\nably lower Bragg angles compared to the GGG substrate\nindicating a signi\fcant distortion of the cubic YIG garnet\ncell with signi\fcantly enlarged lattice parameters (com-\npressive stress)19,21.\nThe relative e\u000bective mis\ft \u0001 a?=as= (as\u0000a?\nYIG)=as\nobtained from strained \flm lattice parameter in growth\ndirectiona?\nYIGand the substrate lattice parameter ascan\nbe used as a measure for epitaxial induced in-plane ten-\nsion or strain. Due to YIG Poisson's ratio of \u0017P= 0:29\npseudomorphously grown, fully strained YIG \flms with\nan ideal YIG bulklattice parameter aYIG= 1:2375 nm22\nshould have a relative e\u000bective mis\ft of \u0001 a?=as=\n+11\u000210\u00004(tensile stress). In the case of our sub-\nmicrometer YIG \flms \u0001 a?=ashas been determined to\nbe in the range between zero and +5 \u000210\u00004(see Ta-\nble II) compared to PLD-grown YIG \flms with up to\n\u0001a?=as=\u0000100\u000210\u00004(see e.g. Ref.12). Hence, our\nLPE \flms are under tension but not to the extent which\nwe expected for nominally pure YIG material without\nadditional lattice expansion by lattice defects or impu-\nrities. To \fnd the reason for this, high-resolution re-\nciprocal space map (HR-RSM) and XPS investigations\nwere performed. Fig. 2b shows a HR-RSM plot around\nthe symmetrical 444 Bragg re\rection with symmetrical\ndi\u000bracted intensity for the GGG substrate and asym-\nmetric di\u000bracted intensity toward higher scattering an-\ngles along Qz(2\u0012-!-Scan) which we attribute to the\nYIG 444 \flm re\rection. Broadening of the \flm re\rec-\ntion along Qzis due to the \fnite coherence lenght of\nthe sub-micrometer thin \flm in growth direction and\nother broadening mechanisms as for example heteroge-\nneous strain. The extension of the \flm re\rection up\nto the substrate peak position suggests that the \flm is\ncontinuously strained due to an existing compositional\nand/or strain gradient. No peak broadening along the\nQxdirection (!-scan) indicates single crystalline perfec-\ntion parallel to the \flm plane without signi\fcant mosaic-\nity due to tilts of epitaxial regions with respect to one\nanother.\nTo evaluate the compositional homogeneity along the\n25.34 25.36 25.38 25.401000200030004000\n \nYIG 444\nFWHM=0.011□Intensity (cps)\nAngle ω (deg) Sample B\n Gaussian fit GGG\n Gaussian fit YIG\n Cumulative fit\nFWHM=0.0058□GGG 444(a)\n/s45/s48/s46/s48/s48/s50 /s45/s48/s46/s48/s48/s49 /s48/s46/s48/s48/s48 /s48/s46/s48/s48/s49 /s48/s46/s48/s48/s50/s53/s46/s53/s57/s48/s53/s46/s53/s57/s50/s53/s46/s53/s57/s52/s53/s46/s53/s57/s54/s53/s46/s53/s57/s56/s53/s46/s54/s48/s48/s53/s46/s54/s48/s50\n/s81\n/s88/s32/s40/s49/s47/s110/s109/s41/s81\n/s90/s32/s40/s49/s47/s110/s109/s41\n/s49/s50/s52/s56/s49/s54/s51/s50/s54/s52/s49/s50/s56/s50/s53/s54/s53/s49/s50/s49/s48/s50/s52/s50/s48/s52/s56\n/s52/s52/s52\n/s83/s97/s109/s112/s108/s101/s32/s65\n(b)\nFIG. 2: (a) HR-XRD !-scan around substrate/\flm 444\nBragg re\rection of sample B. By \ftting procedures the\nYIG \flm peak has been extracted. (b) HR-RSM scans\naround substrate/\flm 444 reciprocal point reveal\nasymmetric di\u000bracted intensities towards higher Q z\nvalues for sample A.\ngrowth direction of the \flms and to detect expected\nimpurities (e.g. Pb from solvent) depth pro\fle analy-4\n0 500 1000 1500 2000 2500 30000246810\nsubstrate\n Intensity (a.u.)\nEtch time (s) Y 3p3\n Fe 2p\n O 1s\n Gd 3d5\n Ga 2p3\n Pb 4f< 5 nm\nYIG/GGG\ninterfaceYIG film\n(a)\n0 250 500 750 1000 1250 15001234\n  \n GasubstrateLa:YIG film La:YIG/GGG\n     interfaceIntensity 139La, 69Ga (a.u.)\nEtch time (s) La< 11 nm\nx 100\n(b)\nFIG. 3: (a) XPS depth pro\fle of sample B reveals a\nvery narrow interface between \flm and substrate. The\nPb 4f signal could not be detected within the detection\nlimit of about 0.1 at-%. (b) SIMS depth pro\fle analysis\ndetects the139La signal of the \flm as well as the69Ga\nsignal of the substrate (sample D) and their changes at\nthe \flm/substrate interface.\nses were carried out by XPS. Fig. 3a shows a homo-\ngeneous distribution of the YIG matrix elements along\nthe \flm growth direction and a sharp transition at the\n\flm/substrate interface. The obtained width of the tran-\nsition layer for sample B is below 5 nm. But the obtained\ndepth pro\fle consists of a convolution of the true concen-\ntration pro\fle with the depth resolution of the XPS sys-\ntem under the concrete measuring conditions and should\nbe narrower. Therefore, these pro\fles demonstrate that\nno broad interdi\u000busion layer is formed by element in-\ntermixing at the interface at an early state of epitaxial\ngrowth or by di\u000busion of substrate ions into the epitaxial\nlayer and vice versa during the subsequent growth pro-\ncess.\nWhereas XPS surface analysis of the very \frst atomic\nlayers (not shown) gives a Pb content of about 0.2 at-%,\nno Pb signal could be observed during the depth pro\fleanalyses within the detection limit of 0.1 at-%23. There-\nfore, it is assumed that the Pb signal corresponds to\na surface contamination of condensed PbO vapor from\nhigh temperature solution and this contamination is com-\npletely removed by the \frst argon-ion etching step. For\nYIG \flms grown in La 2O3containing solution no La sig-\nnal could be detected by XPS that give indicates that\nthe La content must be below 0.5 at-%24. In order to\nimprove the detection capability additional qualitative\nSIMS measurements were carried out. Due to the result-\ning sputtering e\u000bect and by time-dependent detection of\nthe sputtered sample ions one obtains depth pro\fles of\nthe \flm elements as shown for139La in Fig. 3b. Here, the\ncounts of two separate measurements taken under identi-\ncal measuring conditions at neighboring sample positions\nwere added up in order to enhance the statistical signi\f-\ncance. It is clearly visible that the lanthanum signal de-\ncreases at the \flm/substrate interface whereas substrate\nsignals like69:7Ga simultaneously increase.\nB. Static magnetic measurements\nThe vibrating sample magnetometry was used to mea-\nsure the net magnetic moment mof the YIG/GGG sam-\nples at room temperature. As a thickness of GGG sub-\nstrates\u00195000 times exceeded these of the studied YIG\n\flms, a proper calculation of the YIG parameters re-\nquired us (i) to extract the GGG contribution that lin-\nearly increased with the external \feld Hand (ii) to prefer\nthe in-plane sample orientation that ensured considerably\nlower \felds Hsfor the YIG \flms to attain the saturation.\nFig. 4a presents a typical dependence of the total mag-\nnetic moment mvs the in-plane magnetic \feld Hand\nillustrates the method allowing us to separate the m'\ncomponents produced by the YIG \flm and the GGG\nsubstrate. Being subsequently normalized to the \flm\nvolume, the YIG component loops yield the following\nmaterial parameters { a saturation magnetization Ms, a\ncoercivityHcand a saturation \feld Hs, i.e. the \feld (av-\neraged over ascending H\"and descending H#branches\nof hysteresis loops) where the YIG \flm magnetization\napproaches 0.9\u0002Ms. In order to estimate the in-plane\nanisotropy, we have repeated this procedure for the sam-\nples rotated around the h111iaxis perpendicular to \flm\nsurfaces. Fig. 4b demonstrates such results as polar\nsemi-log plots vs the azimuthal angle '. A saturation\nmagnetization Msin Fig. 4b seems independent of '.\nThe obtained 4 \u0019Msvalues cluster around 1800 Gs usu-\nally reported25for bulk YIG single crystals. Within an\nexperimental error (mostly de\fned by the YIG volume\nuncertainity of\u00062%), the same is valid for the 4 \u0019Msval-\nues in other LPE \flms listed in Table II. The obtained\ncoercivity ( Hc\u00140:2 Oe) in studied LPE \flms is among\nthe best values reported for gas phase epitaxial \flms (see\nTable I). No distinct in\ruence of the crystallographic ori-\nentation on the Hcvalues is also registered. In contrast,\nthe azimuthal dependence of the saturation \feld Hsob-5\n-150 -100 -50 0 50 100 150-800-600-400-2000200400600800\n-2 0 2-2000200m (µemu)\nH (Oe) YIG+GGG\n YIG\n GGG  m (µemu)\n H (Oe)\n(a)\n/s48/s46/s49/s49/s54/s48 /s49/s50/s48\n/s49/s56/s48\n/s50/s52/s48 /s51/s48/s48/s48/s46/s49\n/s49/s32/s72\n/s67/s32/s44/s32/s72\n/s83/s32/s40/s79/s101/s41/s32/s32/s32/s32/s32/s32/s32/s32/s52 /s77\n/s83/s32/s40/s107/s71/s115/s41\n/s32/s32/s52 /s77\n/s115\n/s32/s72\n/s115\n/s32/s72\n/s99\n(b)\nFIG. 4: (a) The net VSM magnetic moment mof the\nsample D as well as its components induced by the YIG\n\flm and the GGG substrate vs the in-plane magnetic\n\feldHparallel to theh110idirection. (b) Azimuthal\nangle dependencies for the VSM loop parameters of\nsample D, i.e. a saturation magnetization Ms, a\nsaturation \feld Hsand a coercivity Hc. TheHssix-fold\nsymmetry with the mimima along the h112i`easy axes'\nand the maxima along the h110i`hard axes' indicates\nthe cubic magnetocrystalline anisotropy.\nviously reveals the six-fold symmetry which matches the\ncrystallographic symmetry of YIGs. The Hsmaxima co-\nincide with the in-plane h110iprojections of the hard\nmagnetization axes, whereas the Hsminima correspond\nto theh112icrystallographic directions. The h112i`easy\naxes' orientation suggests an `easy cone' anisotropy after\nUbizskii26. He has also demonstrated27that relatively\nsmall in-plane magnetic \felds lead to single-domain YIG\n\flms, although a deviation of magnetization vector from\nthe \flm plane still remains due to \fnite values of the\ncubic anisotropy constants.\nIn conclusion, as the demagnetizing factor at the out-\nof-plane YIG \flm orientation is 1, the out-of-plane satu-\nration \feld has to be close to the in-plane 4 \u0019Msvalues.\nThis fact is qualitatively con\frmed by our out-of-plane\nmeasurements. Unfortunately, the GGG component ofthe total VSM signal at \felds H?\u00191:8 kOe is much\nlarger than magnetic moments of YIG \flms with a thick-\nness of\u0019100 nm (see, for instance, Fig. 4a) and, hence,\na reasonable accuracy of \u00060.5 % at the GGG signal elim-\nination inevitably results in too large errors for the YIG\nparameters. One may conclude that the out-of-plane con-\n\fguration may provide reliable results when the ratio of\nYIG to GGG thickness exceeds, at least, 10\u00003.\nC. FMR absorption\nFMR absorption spectra for each of studied YIG \flms\nwere recorded at several values ( H\u00145 kOe) of the in-\nplane magnetic \feld. The inset in Fig. 5 shows such a\nspectrum at H= 1:6 kOe that looks like the Lorentz\nfunction with a linewidth \u0001 fFWHM\u00194 MHz centered\nnear the FMR frequency f\u00196:5 GHz. Since the FMR\nlinewidth is mostly expressed in units of magnetic \feld,\nwe, at \frst, used the centers fof measured spectra and\nthe corresponding in-plane \felds Hto estimate the gy-\nromagnetic ratio \rand the e\u000bective magnetization Me\u000b\nin the Kittel formula28\nf=\rp\nH(H+ 4\u0019Me\u000b): (1)\nThen, the best \ftting pair of \randMe\u000ballowed us (i) to\nconvert every frequency spectrum into the magnetic \feld\nscale, (ii) to \ft rescaled spectra with the Lorentz function\nand (iii) to evaluate, thereby, the corresponding linewidth\n\u0001HFWHM . The selected results of the described proce-\ndure { namely, 4 \u0019Me\u000band \u0001HFWHM at the reference\nfrequencyf= 6:5 GHz { are listed in Table II, while\nthe whole summary of the obtained \u0001 HFWHM values is\n0 5 10 15012\n012\n6.48 6.49 6.500.0000.0050.0100.015\n  \n  A: 1.4 x10-4; 0.5 Oe\n  B: 1.2 x10-4; 0.7 Oe\n  C: 1.6 x10-4; 0.5 Oe\n  D: 1.7 x10-4; 0.7 Oe\n ∆HFWHM (Oe)\nf (GHz)Sample: α ; ∆H0 \n f (GHz)1-S21\nFIG. 5: Frequency dependence of FMR absorption\nlinewidth \u0001 HFWHM for YIG LPE \flms A{D at various\nvalues of the in-plane magnetic \feld ( H\u00145 kOe).\nStraight lines are linear \fts that the Gilbert damping\nfactors\u000bare obtained from. Inset shows an example of\nFMR absorption spectrum measured for the sample A\natH= 1:6 kOe.6\n0 5 10 15 200.00.51.01.52.02.5\n α = 1.2×10-4\nα = 0.7×10-4\nα = 0.4×10-4 d = 106 nm\n d = 410 nm\n d = 3.0 µm\n d = 300 µm∆HFWHM (Oe)\nf (GHz)α = 0.5×10-4\nFIG. 6: Frequency dependencies of the FMR linewidth\n\u0001HFWHM for YIG LPE \flms of various thickness dand\nthe YIG sphere with diameter d= 300\u0016m. The Gilbert\ndamping factors \u000bare calculated from slopes of the best\nlinear \fts according to Eq. (2).\npresented in Fig. 5 vs the FMR frequency. The plots in\nFig. 5 are known10{14to provide data about the Gilbert\ndamping coe\u000ecient \u000band the inhomogeneous contribu-\ntion \u0001H0to the FMR linewidth that are mutually related\nby\n\u0001HFWHM = \u0001H0+2\u000bf\n\r(2)\nAs the FMR performance of thin YIG \flms strongly\ndepends on the working frequency of future magnonic\napplications, we have included various quality parame-\nters in Table II, viz. i) the Gilbert damping coe\u000ecient\n\u000bwhich is mostly responsible for the FMR losses at\nhigh magnetic \felds ( H\u001d4\u0019Me\u000b), ii) the inhomoge-\nneous contribution \u0001 H0that dominates at small \felds\n(H\u001c4\u0019Me\u000b) as well as iii) the FMR linewidth at the\nreference frequency f=6.5 GHz which approximately cor-\nresponds to the case H\u00194\u0019Me\u000b. The latter is estimated\ndown to \u0001HFWHM =1.4 Oe that is to our knowledge the\nnarrowest value reported so far for YIG \flms with a thick-\nness of about 100 nm and smaller. The Gilbert damping\ncoe\u000ecients are estimated to be close to \u000b\u00191\u000210\u00004\nwhich is comparable to the best values reported so far\n(compare with Table I). The zero frequency term \u0001 H0is\nfound almost the same for all YIG \flms including the La\nsubstituted one. The obtained value \u0001 H0\u00190:5\u00000:7 Oe\nappears as well appreciably lower than that for gas phase\nepitaxial \flms (see Table I).\nIn summary, optimized LPE growth and post-\nprocessing conditions improve FMR linewidths and\nGilbert damping coe\u000ecients (compare this study and\nRef.29 with Ref.10). However, the improved values are\nstill far from these in bulk YIGs and relatively thick YIG\n\flms (see Fig. 6) due to the decreasing volume to inter-\nface ratio in sub-micrometer \flms. For example, imper-\nfections at the \flm interface of thin \flms should have astronger in\ruence on the magnetic losses in contrast to\nthe dominating volume properties of perfect thick \flms.\nIt requires us to undertake further attempts to mini-\nmize the FMR performance deterioration with a decrease\nof the YIG \flm thickness. These attempts will be fo-\ncused on avoiding the most probable sources of FMR\nlosses such as contributions due to homogeneous broad-\nening (interface roughness, homogeneously distributed\ndefects and impurities) and inhomogeneous broadening\n(geometric and magnetic mosaicity, single surface de-\nfects) and, thus, on approaching the \\target\" parameters\nof \u0001HFWHM = 0:3 Oe at 6.5 GHz and \u000b= 0:4\u000210\u00004\nreported by R oschmann and Tolksdorf30for bulk discs\nmade of single YIG crystals.\nIII. OUTLOOK AND CONCLUSIONS\nBesides the e\u000borts to avoid growth defects as well as\ninterface roughness and to reduce impurity incorpora-\ntion during the LPE deposition process further high-\nresolution investigations are necessary to gain more in-\nsight into the YIG microstructure and to identify the\nproperties which play an essential role for its FMR per-\nformance. Therefore, in future studies we will carry out\nHR-RSM scans with asymmetrical re\rections to deter-\nmine in-plane and axial strain, respectively, the Time-\nof-Flight (ToF) SIMS analysis technique using element\nstandards to precisely quantify the La substitution con-\ncentration as well as to detect impurity elements from the\nhigh-temperature solutions in our sub-micrometer LPE\n\flms. Furthermore, angular dependent measurements of\nthe resonance \feld and of the FMR linewidth will be in-\ntended to determine the in\ruence of uniaxial magnetic\nanisotropies on the ferromagnetic resonance losses.\nIn conclusion, liquid phase epitaxy has the potential to\nprovide sub-micrometer YIG \flms with outstanding crys-\ntalline and magnetic properties to meet the requirements\nfor future magnon spintronics with ultra-low e\u000bective\nlosses if a drastic miniaturization down to the nanometer\nscale is possible. First sub-100 nm lateral sized structures\nhave presently been prepared31which could be the next\nstep to LPE-based microscaled spintronic circuits. The\ndevelopment of YIG LPE \flms with thicknesses below\n100 nm is now in progress and remains a big challenge\nfor the classical thick-\flm LPE technique.\nIV. METHODS\nA. Sample fabrication\nYIG \flms were grown from PbO-B 2O3based high-\ntemperature solutions resistively-heated in a platinum\ncrucible at about 900\u000eC using standard dipping LPE\ntechnique. During di\u000berent growth runs nominally pure\nYIG \flms were grown on one-inch (111) gadolinium gal-\nlium garnet (GGG) substrates to check the reproducibil-7\nity of the sub-micrometer liquid phase epitaxial growth.\nFor La substituted \flms La 2O3was added to the al-\nready used high-temperature solution. To remove solu-\ntion remnants from the sample surfaces the holder had\nto be stored in a hot acidic solution after room tem-\nperature cooling. Afterwards the reverse side layer was\nremoved by mechanical polishing from the double-side\ngrown samples. Chips of di\u000berent sizes were prepared by\na diamond wire saw and sample surfaces were cleaned\nusing ethanol, distilled water and acetone. The LPE \flm\nthickness was determined by X-ray re\rectometry using a\nPANanalytical/X-Pert Pro system.\nB. Microstructural properties\nThe root-mean-square surface roughness was deter-\nmined by AFM measurements for each sample at three\ndi\u000berent regions over 25 \u0016m2ranges using a Park Scien-\nti\fc Instruments, M5. HR-XRD studies were performed\nby a \fve-crystal di\u000braction spectrometer of Seifert (3003\nPTS HR) equipped with a four-fold Ge 440 asymmetric\nmonochromator using CuK \u000bradiation. The resolution\nlimit was 1\u000210\u00004deg. GGG substrate lattice param-\neters were obtained by the Bond method. Depth pro-\n\fle analyses were carried out by an Axis UltraDLDXPS\nsystem (Kratos Analytical Ltd.) using a mono-atomic\nargon-ion etching technique. Qualitative SIMS (Hiden\nAnalytical) measurements were carried out. Here, a \flm\narea of 500\u0002500\u0016m2is irradiated by 5 keV oxygen ions.\nC. Magnetic properties\nThe vibrating sample magnetometer (MicroSense\nLLC, EZ-9) was used to register the in-plane hystere-\nsis loops of the YIG/GGG samples at room tempera-\nture. The external magnetic \feld Hwas controlled with\nan error of\u00140.01 Oe. To estimate the magnetization of\nthe YIG \flms we removed a contribution of the GGG\nsubstrates from the total VSM signal. To monitor the\nin-plane anisotropy as a function of the crystallographic\norientation, the hysteresis loops at the azimuthal angles\n0\u000e\u0014'\u0014360\u000ewere measured with an angular step of\n3\u000e. The FMR absorption spectra were registered with a\nvector network analyzer (Rohde & Schwarz GmbH, ZVA\n67) attached to a broadband stripline. The sample was\ndisposed face-down over a stripline and the transmission\nsignals (S21&S12) were recorded. During the measure-\nments, a frequency of microwave signals with the input\npower of\u000010 dBm (0.1 mW) was swept across the res-\nonance frequency, while the in-plane magnetic \feld H\nwas constant and measured with an accuracy of 1 Oe.\nEach recorded spectrum was \ftted by the Lorentz func-\ntion and allowed us to de\fne the resonance frequency\nand the FMR linewidth \u0001 HFWHM corresponding to the\napplied \feld H.1V. V. Kruglyak and R. J. Hicken, \\Magnonics: Experiment to\nprove the concept,\" J. Magn. Magn. Mater. 306, 191{194 (2006),\ncond-mat/0511290.\n2S. Neusser, B. Botters, and D. Grundler, \\Localization, con\fne-\nment, and \feld-controlled propagation of spin waves in Ni 80Fe20\nantidot lattices,\" Phys. Rev. B 78, 054406 (2008).\n3V. V. Kruglyak, S. O. Demokritov, and D. Grundler, \\Magnon-\nics,\" J. Phys. D: Appl. Phys. 43, 264001 (2010).\n4R. L. Stamps, S. Breitkreutz, J. \u0017Akerman, A. V. Chumak,\nY. Otani, G. E. W. Bauer, J.-U. Thiele, M. Bowen, S. A.\nMajetich, M. Kl aui, I. Lucian Prejbeanu, B. Dieny, N. M.\nDempsey, and B. Hillebrands, \\The 2014 Magnetism Roadmap,\"\nJ. Phys. D: Appl. Phys. 47, 333001 (2014), arXiv:1410.6404\n[cond-mat.mtrl-sci].\n5R. C. LeCraw, E. G. Spencer, and C. S. Porter, \\Ferromagnetic\nResonance Line Width in Yttrium Iron Garnet Single Crystals,\"\nPhys. Rev. 110, 1311{1313 (1958).\n6A. A. Serga, A. V. Chumak, and B. Hillebrands, \\YIG magnon-\nics,\" J. Phys. D: Appl. Phys. 43, 264002 (2010).\n7A. V. Chumak, A. A. Serga, and B. Hillebrands, \\Magnon tran-\nsistor for all-magnon data processing,\" Nat. Commun. 5, 4700\n(2014).\n8A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands,\n\\Magnon spintronics,\" Nat. Phys. 11, 453{461 (2015).\n9E. A. Giess, J. D. Kuptsis, and E. A. D. White, \\Liquid phase\nepitaxial growth of magnetic garnet \flms by isothermal dipping\nin a horizontal plane with axial rotation,\" J. Cryst. Growth 16,\n36{42 (1972).\n10P. Pirro, T. Br acher, A. V. Chumak, B. L agel, C. Dubs,\nO. Surzhenko, P. G ornert, B. Leven, and B. Hillebrands, \\Spin-\nwave excitation and propagation in microstructured waveguides\nof yttrium iron garnet/Pt bilayers,\" Appl. Phys. Lett. 104,\n012402 (2014), arXiv:1311.6305 [cond-mat.mes-hall].\n11M. C. Onbasli, A. Kehlberger, D. H. Kim, G. Jakob, M. Kl aui,\nA. V. Chumak, B. Hillebrands, and C. A. Ross, \\Pulsed laser\ndeposition of epitaxial yttrium iron garnet \flms with low Gilbert\ndamping and bulk-like magnetization,\" APL Mater. 2, 106102\n(2014).\n12B. M. Howe, S. Emori, H.-M. Jeon, T. Oxhol, J. G. Jones, K. Ma-\nhalingam, Y. Zhuang, N. X. Sun, and G. J. Brown, \\Pseudomor-\nphic yttrium iron garnet thin \flms with low damping and inho-\nmogeneous linewidth broadening,\" IEEE Magn. Lett. 8, 3500504\n(2015).\n13H. Chang, P. Li, W. Zhang, T. Liu, A. Ho\u000bmann, L. Deng,\nand M. Wu, \\Nanometer-thick yttrium iron garnet \flms with\nextremely low damping,\" IEEE Magn. Lett. 5, 6700104 (2014).\n14H. Wang, Understanding of Pure Spin Transport in a Broad\nRange of Y3Fe5O12-based Heterostructures , Ph.D. thesis, The\nOhio State University (2015).\n15O. d'Allivy Kelly, A. Anane, R. Bernard, J. Ben Youssef,\nC. Hahn, A. H. Molpeceres, C. Carr\u0013 et\u0013 ero, E. Jacquet, C. Der-\nanlot, P. Bortolotti, R. Lebourgeois, J.-C. Mage, G. de Loubens,\nO. Klein, V. Cros, and A. Fert, \\Inverse spin Hall e\u000bect in\nnanometer-thick yttrium iron garnet/Pt system,\" Appl. Phys.\nLett. 103, 082408 (2013), arXiv:1308.0192 [cond-mat.mtrl-sci].\n16V. Castel, N. Vlietstra, B. J. van Wees, and J. B. Youssef, \\Fre-\nquency and power dependence of spin-current emission by spin\npumping in a thin-\flm YIG/Pt system,\" Phys. Rev. B 86, 134419\n(2012), arXiv:1206.6671 [cond-mat.mtrl-sci].\n17C. Hahn, G. de Loubens, O. Klein, M. Viret, V. V. Naletov, and\nJ. Ben Youssef, \\Comparative measurements of inverse spin Hall\ne\u000bects and magnetoresistance in YIG/Pt and YIG/Ta,\" Phys.\nRev. B 87, 174417 (2013), arXiv:1302.4416 [cond-mat.mes-hall].\n18L. J. Cornelissen, J. Liu, R. A. Duine, J. B. Youssef, and B. J.\nvan Wees, \\Long-distance transport of magnon spin information\nin a magnetic insulator at room temperature,\" Nat. Phys. 11,\n1022{1026 (2015), arXiv:1505.06325 [cond-mat.mes-hall].\n19S. A. Manuilov, R. Fors, S. I. Khartsev, and A. M. Grishin, \\Sub-\nmicron Y 3Fe5O12Film Magnetostatic Wave Band Pass Filters,\"\nJ. Appl. Phys. 105, 033917{033917 (2009).8\n20Y. Sun, Y.-Y. Song, H. Chang, M. Kabatek, M. Jantz, W. Schnei-\nder, M. Wu, H. Schultheiss, and A. Ho\u000bmann, \\Growth and\nferromagnetic resonance properties of nanometer-thick yttrium\niron garnet \flms,\" Appl. Phys. Lett. 101, 152405 (2012).\n21S. A. Manuilov, S. I. Khartsev, and A. M. Grishin, \\Pulsed laser\ndeposited Y 3Fe5O12\flms: Nature of magnetic anisotropy I,\" J.\nAppl. Phys. 106, 123917{123917 (2009).\n22R. Hergt, H. Pfei\u000ber, P. G ornert, M. Wendt, B. Keszei, and\nJ. Vandlik, \\Kinetic Segregation of Lead Impurities in Garnet\nLPE Films,\" Phys. Stat. Sol. (a) 104, 769{776 (1987).\n230.1 at-% Pb corresponds to xPb\u00190:02 formular units in stoi-\nchiometric Y 3\u0000xPbxFe5O12.\n240.5 at-% La corresponds to yLa\u00190:1 formular units in stoichio-\nmetric Y 3\u0000yLayFe5O12.\n25G. Winkler, \\Magnetic garnets,\" in Vieweg tracts in pure and\napplied physics; Volume 5 (Friedrich Vieweg & Sohn Verlag,\nBraunschweig, Wiesbaden, 1981) Chap. 2, pp. 75{79.\n26S. B. Ubizskii, \\Orientational states of magnetization in epitaxial\n(111)-oriented iron garnet \flms,\" J. Magn. Magn. Mater. 195,\n575{582 (1999).\n27S. B. Ubizskii, \\Magnetization reversal modelling for (111)-\noriented epitaxial \flms of iron garnets with mixed anisotropy,\"\nJ. Magn. Magn. Mater. 219, 127{141 (2000).\n28C. Kittel, \\On the Theory of Ferromagnetic Resonance Absorp-\ntion,\" Phys. Rev. 73, 155{161 (1948).\n29V. Lauer, D. A. Bozhko, T. Br acher, P. Pirro, V. I. Vasyuchka,\nA. A. Serga, M. B. Jung\reisch, M. Agrawal, Y. V. Kobljanskyj,\nG. A. Melkov, C. Dubs, B. Hillebrands, and A. V. Chumak,\n\\Spin-transfer torque based damping control of parametrically\nexcited spin waves in a magnetic insulator,\" Appl. Phys. Lett.\n108, 012402 (2016), arXiv:1508.07517 [cond-mat.mes-hall].\n30P. R oschmann and W. Tolksdorf, \\Epitaxial growth and anneal-\ning control of FMR properties of thick homogeneous Ga substi-\ntuted yttrium iron garnet \flms,\" Mat. Res. Bull. 18, 449{459\n(1983).31T. L ober, A. V. Chumak, and B. Hillebrands, Unpublished re-\nsults.\nV. ACKNOWLEDGEMENTS\nWe acknowledge the partial \fnancial support by\nDeutsche Forschungsgemeinschaft (DU 1427/2-1). We\nthank M. Frigge for EPMA analysis, Ch. Schmidt for\nXRR measurements and R. Meyer and B. Wenzel for\ntechnical support.\nVI. AUTHOR CONTRIBUTIONS STATEMENT\nC.D. conceived the experiments, prepared all samples\nand analyzed the data. O.S. performed VSM and FMR\nmeasurements and analyzed the data. R.L. performed\nthe XPS experiments. J.D. and U.B. performed the SIMS\nexperiments. A.D. conducted the XRD experiments and\nanalyzed the data. C.D. and O.S. wrote the manuscript.\nAll authors contributed to scienti\fc discussions and the\nmanuscript review.\nVII. ADDITIONAL INFORMATION\nA. Competing \fnancial interests\nThe authors declare no competing \fnancial interests."    },    {        "title": "1607.03409v1.Effect_of_Quantum_Tunneling_on_Spin_Hall_Magnetoresistance.pdf",        "content": "arXiv:1607.03409v1  [cond-mat.mes-hall]  12 Jul 2016Effect of Quantum Tunneling on Spin Hall Magnetoresistance\nSeulgi Ok,1Wei Chen,1Manfred Sigrist,1and Dirk Manske2\n1Institut f¨ ur Theoretische Physik, ETH-Z¨ urich, CH-8093 Z ¨ urich, Switzerland\n2Max-Planck-Institut f ¨ur Festk¨orperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, G ermany\n(Dated: September 5, 2018)\nWe present a formalism that simultaneously incorporates th e effect of quantum tunnelingand spin\ndiffusion on spin Hall magnetoresistance observed in normal metal/ferromagnetic insulator bilayers\n(such as Pt/Y 3Fe5O12) and normal metal/ferromagnetic metal bilayers (such as Pt /Co), in which\nthe angle of magnetization influences the magnetoresistanc e of the normal metal. In the normal\nmetal side the spin diffusion is known to affect the landscape o f the spin accumulation caused by\nspin Hall effect and subsequently the magnetoresistance, wh ile on the ferromagnet side the quantum\ntunneling effect is detrimental to the interface spin curren t which also affects the spin accumulation.\nThe influence of generic material properties such as spin diff usion length, layer thickness, interface\ncoupling, and insulating gap can be quantified in a unified man ner, and experiments that reveal the\nquantum feature of the magnetoresistance are suggested.\nPACS numbers: 75.76.+j, 75.47.-m, 85.75.-d, 73.40.Gk\nI. INTRODUCTION\nThe electrical control of magnetization dynamics has\nbeen a central issue in the field of spintronics1,2, owing\nto its possible applications in magnetic memory devices\nwith low power consumption. A particularly promis-\ning mechanism for the electrical control is to utilize the\nspin Hall effect3–6(SHE) in a normal metal (NM), such\nas Pt or Ta, to convert an electric current into a spin\ncurrent, and subsequently to magnetization dynamics\nin an adjacent magnet via mechanisms such as spin-\ntransfer torque7,8(STT). In reverse, the inverse spin\nHall effect9,10(ISHE) can convert the spin current gen-\nerated by certain means, for instance spin pumping11,12,\ninto an electric signal. A particularly intriguing phe-\nnomenon that involves both SHE and ISHE is the spin\nHall magnetoresistance13–22(SMR), in which a charge\ncurrent in an NM causes a spin accumulation at the edge\nof the sample due to SHE, yielding a finite spin current\nat the interface to a ferromagnet. Through ISHE, the\nspin current gives an electromotive force along the orig-\ninal charge current, effectively changing the magnetore-\nsistance of the NM.\nThe two major ingredients that determine SMR are\nthe spin diffusion25in the NM and the spin current at\nthe NM/ferromagnet interface. The spin diffusion part\nhas been addressed in detail by Chen et al.for the\nNM/ferromagnetic insulator (NM/FMI) bilayer, such as\nPt/Y3Fe5O12(Pt/YIG), and FMI/NM/FMI trilayer22.\nThis approach solves the spin diffusion equation in the\npresence of SHE and ISHE in a self-consistent manner,\nwhere the spin current at the NM/FMI interface serves\nas a boundary condition. However, the interface spin\ncurrent remains an external parameter for which exper-\nimental or numerical input is needed23,24. On the other\nhand, a quantum tunneling formalism has emerged re-\ncently as an inexpensive tool to calculate the interface\nspin current from various material properties such as\nthe insulating gap of the FMI and the interface s−dcoupling26. The quantum tunneling theory also success-\nfully explains27the reduced spin pumping spin current\nwhen an additional oxide layer is inserted between NM\nandFMI28. It is thenoffundamental importanceto com-\nbine the spin diffusion approach with the quantum tun-\nneling formalism for the interface spin current to give a\ncomplete theoretical description of the SMR, in particu-\nlar to quantify how various material properties influence\nthe SMR.\nIn this article we provide a minimal formalism that\nbridges the quantum tunneling formalism to the spin dif-\nfusion approach. We focus on the SMR in NM/FMI\nbilayer realized in Pt/YIG, and the NM/ferromagnetic\nmetal (NM/FMM) bilayer realized in Pt/Co and\nTa/Co14. The spin diffusion in the NM is assumed to\nbe described by the same formalism of Chen et al.22,\nwhereas the interface spin current is calculated from the\nquantum tunneling formalism26,27. In the NM/FMM bi-\nlayer, we consider an FMM that has long spin diffusion\nlength and a small thickness, such that the spin diffu-\nsion effect is negligible and the spin transport is predom-\ninately of quantum origin26. This is presumably ade-\nquate for the case of ultrathin Co films29, but not for\nmaterials with very short spin diffusion length such as\npermalloy30,31. Within thisformalism,theeffectofmate-\nrial properties including spin diffusion length of the NM,\ninterfaces−dcoupling, insulating gap of the FMI, and\nthe thickness of each layer can all be treated on equal\nfooting. In particular, we reveal the signature of quan-\ntum interference in SMR in NM/FMM bilayer, and dis-\ncuss the situation in which it can be observed.\nThe structure of the article is arranged in the follow-\ning manner. In Sec. II, we detail the quantum tunneling\nformalism for the interface spin current in the NM/FMI\nbilayer, and how it is adopted into the spin diffusion ap-\nproachthat describestheNM. Section III generalizesthis\nrecipe to the NM/FMM bilayer, and discuss the observ-\nability ofthe predicted signatureof quantum interference\nin SMR. Section IV gives the concluding remark.2\nII. NM/FMI BILAYER\nA. Interface spin current\nWe start with the quantum tunneling formalism that\ncalculates the interface spin current in the NM/FMI bi-\nlayer, which later serves as the boundary condition for\nthe spin diffusion equation that determines SMR. The\nquantum tunneling formalism describes the NM/FMI bi-\nlayer shown in Fig. 1 (a) by the Hamiltonian\nHN=p2\n2m−µσ\nx(−lN≤x<0), (1)\nHFI=p2\n2m+V0+ΓS·σ(0≤x≤lFI),(2)\nwhereµσ\nx=±µx·ˆz/2 is the spin voltage of σ={↑,↓}\nproduced by an in-plane charge current Jc\nyˆy,ǫFis the\nFermi energy, V0−ǫFis the insulating gap, and S=\nS(sinθcosϕ,sinθsinϕ,cosθ) is the magnetization. We\nchoose Γ<0 such that the magnetization has the ten-\ndency to align with the conduction electron spin σ. The\nwave function near the interface is\nψN= (Aeik0↑x+Be−ik0↑x)/parenleftbigg\n1\n0/parenrightbigg\n+Ce−ik0↓/parenleftbigg\n0\n1/parenrightbigg\n,(3)\nψFI= (Deq+x+Ee−q+x)/parenleftbigg\ne−iϕ/2cosθ\n2\neiϕ/2sinθ\n2/parenrightbigg\n+(Feq−x+Ge−q−x)/parenleftbigg\n−e−iϕ/2sinθ\n2\neiϕ/2cosθ\n2/parenrightbigg\n,(4)\nwherek0σ=/radicalbig\n2m(ǫF+µσ\n0)//planckover2pi1andq±=/radicalbig\n2m(V0±ΓS−ǫF)//planckover2pi1. The amplitudes B∼E\nare solved in terms of the incident amplitude Aby\nmatching wave functions and their first derivative at the\ninterface. The x<−lNandx>lFIregions are assumed\nto be vacuum or insulating oxides that correspond to\ninfinite potentials such that the wave functions vanish\nthere for simplicity. We identify the incident flux with\n|A|2=NF|µ0|/a3whereNFis the density of states per\na3witha= 2π/kF=h/√2mǫFthe Fermi wave length.\nThe spin current inside the FMI at position xis calcu-\nlated from the evanescent wave function\njx=/planckover2pi1\n4im/bracketleftbig\nψ∗\nFIσ(∂xψFI)−(∂xψ∗\nFI)σψFI/bracketrightbig\n.(5)\nAngularmomentum conservation8,26dictatesthat thein-\nterface spin current to be equal to the STT exerts on the\nmagnetization\nj0−jlFI=j0=τ\na2\n=ΓSNF\n/planckover2pi1/bracketleftBig\nGrˆS×/parenleftBig\nˆS×µ0/parenrightBig\n+GiˆS×µ0/bracketrightBig\n,(6)\nwhich defines the field-like Giand dampling-like Grspin\nmixing conductance that in turn can be calculated fromthe interface spin current26\nΓSNF\n/planckover2pi1Gr=2jx\n0cosϕ\n|µ0|sin2θ+2jy\n0sinϕ\n|µ0|sin2θ=−jz\n0\n|µ0|sin2θ,\nΓSNF\n/planckover2pi1Gi=jx\n0sinϕ\n|µ0|sinθ−jy\n0cosϕ\n|µ0|sinθ. (7)\nA straight forward calculation yields\nGr,i=−4\na3|γθ|2/parenleftbiggq+cothq+lFI−q−cothq−lFI\nq2\n+−q2\n−/parenrightbigg\n×(Im,Re)/parenleftBig\nn∗\n↓+n↓−/parenrightBig\n, (8)\nwhereσx,yisx,ycomponent of Pauli matrix, and\nnσ±=k0σ\n(k0σ+iq±cothq±lFI),\nγθ=n↓+\nn↑+cos2θ\n2+n↓−\nn↑−sin2θ\n2. (9)\nEquation (8) describes the spin mixing conductance in\nSTT, as well as that in spin pumping since the Onsager\nrelation32is satisfied in this approach26. BothGrand\nGihave very weak dependence (at most few percent)\non the angle of magnetization θthroughγθ, which may\nbe considered as higher order contributions26. In the\nnumerical calculation below we set θ= 0.3πwithout loss\nof generality.\nNumerical results of the spin mixing conductance Gr,i\nare shown in Fig. 1, plotted as a function the FMI thick-\nnesslFIand at different strength of the interface s−d\ncoupling ΓS/ǫF. BothGrandGiincrease with lFIini-\ntially and then saturate to a constant as expected, since\nthey originatefrom the quantum tunneling ofconduction\nelectrons that only penetrate into the FMI over a very\nshort distance. At a FMI thickness small compared to\nFermi wave length lFI≪a, we found that Gr∝l6\nFIand\nGi∝l3\nFI, therefore the damping-like to field-like ratio\nis|Gr/Gi| ≪1. In most of the parameter space, the\ntorque is dominated by field-like component |Gr/Gi|<1\nthroughout the whole range of lFI. Only when the mag-\nnitude ofs−dcoupling is large compared to the insu-\nlating gap ( V0−ǫF)/ǫFis the torque dominated by the\ndamping-like component |Gr/Gi|>1, consistent with\nthat found previously26and also in accordance with the\nresult from first principle calculation23. The magnitude\nofGr,igenerally increases with the s−dcoupling, yet\nmore dramatically for Gr. Note that GrandGido not\ndepend on the NM thickness in this quantum tunneling\napproach.\nB. SMR\nWe adopt the spin diffusion approach of Chen et al.22\nto address the effect of the interface spin current in\nEq. (6) on SMR, which is briefly summarized below. The3\nFIG. 1: (color online) (a) Schematics of the bilayer con-\nsists of an NM with thickness lNand an FMI with thickness\nlFI. (b) The spin mixing conductance Gr,iversus the FMI\nthickness lFI, at different values of interface s−dcoupling\nstrength −ΓS/ǫF. The insulating gap strength is fixed at\n(V0−ǫF)/ǫF= 1.5. The absolute units for Gr,iise2//planckover2pi1a2\nwhich is about 1014∼1015Ω−1m−2depending on the Fermi\nwave length a.\nspin diffusion approachis based on the following assump-\ntions for the spin transport in the NM: (1) The spin cur-\nrent in NM consists of two parts, one from the spatial\ngradient of spin voltage and the other the bare spin cur-\nrent caused directly by SHE,\njx=−σc\n4e2∂xµx+θSHσcEy\n2eˆz, (10)\nwhereθSHis spin Hall angle, σcis the conductivity of\nNM,Eyis applied external electric in ydirection, and\n−eis electron charge. (2) The spin voltage obeys the\nspin diffusion equation ∇2µx=µx/λ2, whereλis the\nspin diffusion length. (3) Spin current vanishes at the\nedge of NM ( x=−lN), which serves as one boundary\ncondition. (4) The spin current at the NM/FMI interface\nisdescribed byEq.(6), whichservesasanotherboundary\ncondition. The self-consistent solution satisfying (1) ∼(4)\nis22\njx·ˆx\njSH=βxsinθ/bracketleftBig\ncosθcosϕRe/parenleftbig/tildewideG/parenrightbig\n+sinϕIm/parenleftbig/tildewideG/parenrightbig/bracketrightBig\n,\njx·ˆy\njSH=βxsinθ/bracketleftBig\ncosθsinϕRe/parenleftbig/tildewideG/parenrightbig\n−cosϕIm/parenleftbig/tildewideG/parenrightbig/bracketrightBig\n,\njx·ˆz\njSH= 1−cosh(2x+lN\n2λ)\ncosh(lN\n2λ)−βxsin2θRe/parenleftbig/tildewideG/parenrightbig\n,(11)\nwhere\nβx=sinh(x+lN\nλ)\nsinh(lN\nλ)tanh(lN\n2λ),\n/tildewideG=αGc\n1−αGccoth(lN\nλ),\nα=4ΓSNFe2λ\n/planckover2pi1σc, (12)\nandjSH=θSHσcEy/2eis the bare spin current. Here\nα <0 is a negative parameter (because we assume theinterfaces−dcoupling Γ<0) that bridges our tunneling\nformalism to the spin diffusion equation, and Gc=Gr+\niGiis the complex spin mixing conductance.\nThrough ISHE, the spin currents in Eq. (11) is con-\nverted back to a chargecurrent in the longitudinal (along\nˆy) and transverse (along ˆz) direction\n∆jc\nlong(x) =−2eθSH/parenleftBig\njx−θSHσcEy\n2eˆz/parenrightBig\n·ˆz,(13)\n∆jc\ntrans(x) = 2eθSH/parenleftBig\njx−θSHσcEy\n2eˆz/parenrightBig\n·ˆy.(14)\nTheconductivityaveragedovertheNMlayerthenfollows\nσlong=σ+1\nlNEy/integraldisplay0\n−lNdx∆jc\nlong(x),(15)\nσtrans=1\nlNEy/integraldisplay0\n−lNdx∆jc\ntrans(x). (16)\nUsingθ2\nSH∼0.01≪1, the longitudinal and transverse\ncomponent of SMR read\nρlong=σ−1\nlong≈ρ+∆ρ0+sin2θ∆ρ1,\nρtrans=−σtrans/σ2\nlong\n≈cosθsinθsinϕ∆ρ1−sinθcosϕ∆ρ2,(17)\nwhere\n∆ρ0/ρ=−θ2\nSH2λ\nlNtanh/parenleftbigglN\n2λ/parenrightbigg\n,\n∆ρ1/ρ=−θ2\nSHλ\nlNtanh2/parenleftbigglN\n2λ/parenrightbigg\nRe/parenleftbig/tildewideG/parenrightbig\n,\n∆ρ2/ρ=θ2\nSHλ\nlNtanh2/parenleftbigglN\n2λ/parenrightbigg\nIm/parenleftbig/tildewideG/parenrightbig\n.(18)\nClearly the FMI thickness lFIaffects ∆ρ1and ∆ρ2only\nthrough /tildewideG=/tildewideG(lFI).\nTo perform numerical calculation of Eq. (18), we make\nthe following assumption on the parameter αin Eq. (12)\nthat connects the quantum tunneling formalism with the\nspin diffusion equation. Firstly, αcontains the density\nof state per a3at the Fermi surface, which is assumed to\nbe the inverse of Fermi energy NF= 1/ǫF. The com-\nbined parameter Γ SNF= ΓS/ǫFtherefore represents\nthe strength of s−dcoupling. Other parameters that\ninfluenceαare the spin diffusion length assumed to be\nλ≈10nm, the conductivity of the NM film taken to be\nσc≈5×106Ω−1m−1, and Fermi wave length assumed to\nbe roughly equal to the lattice constant a≈0.4nm, all of\nwhich are the typical values for commonly used materials\nsuch as Pt. These lead to the dimensionless parameter\nαGc≈10×(ΓS/ǫF)×/parenleftbig\nGc/(e2//planckover2pi1a2)/parenrightbig\nin Eq. (12) be-\ning expressed in terms of the relative strength of s−d\ncoupling and the spin mixing conductance divided by its\nunit. Inwhatfollows, weexaminethe effectofFMI thick-\nness, NM thickness, insulating gap, and interface s−d\ncoupling on SMR. On the contrary, the spin Hall angle,4\nFIG. 2: (color online) The longitudinal ∆ ρ1/ρand transverse −∆ρ2/ρcomponent of SMR in the NM/FMI bilayer, plotted\nagainst the FMI thickness in units of Fermi wave length lFI/aand NM thickness in units of the spin diffusion length lN/λ,\nat various strength of s−dcoupling −ΓS/ǫFand the insulating gap ( V0−ǫF)/ǫF. Note that the color scale of each plot is\ndifferent.\nspindiffusionlength, andconductivityaretreatedascon-\nstants, although in reality they may also depend on the\nlayer thickness or on each other in such thin films33.\nThenumericalresultofSMRisshowninFig.2, plotted\nas a function of the FMI thickness lFIand NM thickness\nlNat several values of insulating gap ( V0−ǫF)/ǫFand\ns−dcoupling ΓS/ǫF. As a function of the FMI thick-\nnesslFI, both the longitudinal ∆ ρ1/ρand the transverse\n∆ρ2/ρcomponent initially increase and then saturate at\naroundlFI/a∼2, which is expected since conduction\nelectrons only tunnel into the FMI over a short depth,\nso the interface spin current saturates once the FMI is\nthicker than this tunneling depth. The insulating gap\n(V0−ǫF)/ǫFobviouslyaffects the tunneling depth, and is\nparticularly influential on the magnitude of longitudinal\n∆ρ1/ρ, as can be seen by comparing plots with different\n(V0−ǫF)/ǫFin Fig. 2. The magnitude of ∆ ρ1/ρalso\ngenerally increases with the s−dcoupling ΓS/ǫF, while\nthe transverse component ∆ ρ2/ρat large ΓS/ǫFdisplays\na nonmonotonic dependence on the FMI thickness. On\nthe other hand, as a function of NM thickness lN, both\nSMR components increase and peak at around lN/λ∼1\nand then decrease monotonically for large lN. This can\nbe understood because both ∆ ρ1and ∆ρ2are interfaceeffects that become less significant compared to bulk re-\nsistivityρwhen NM thickness increases, and the spin\nvoltage is known to be maximal when the NM thickness\nis comparable to the spin diffusion length25lN/λ∼1.\nIII. NM/FMM BILAYER\nA. Interface spin current and SMR\nWe proceed to address the SMR in the NM/FMM bi-\nlayer, with the assumption that the FMM film is much\nthinner than its spin diffusion length lFM≪λsuch that\nquantum tunneling is the dominant mechanism for spin\ntransport in the FMM, while the spin diffusion inside the\nFMM can be ignored. The calculation of the spin cur-\nrent at the NM/FMM interface starts with the model\nschematically shown in Fig. 3 (a). The NM and FMM\noccupy−lN≤x<0 and 0≤x≤lFM, respectively. The\nNM region is described by Eqs. (1) and (3), while the\nFMM layer is described by HFM=p2/2m+ΓS·σand5\nFIG. 3: (color online) (a) Schematics of an NM/FMM bi-\nlayer with finite thickness. (b) The ratio of spin mixing con-\nductance (c) Grand (d)−Giin this system, plotted against\nthe thickness lFMof the FMM and s−dcoupling −ΓS/ǫF,\nin units of e2/planckover2pi1/a2whereais the Fermi wave length.\nthe wave function\nψFM= (Deik+x+F−ik+x)/parenleftbigg\ne−iϕ/2cosθ\n2\neiϕ/2sinθ\n2/parenrightbigg\n+(Eeik−x+Ge−ik−x)/parenleftbigg\n−e−iϕ/2sinθ\n2\neiϕ/2cosθ\n2/parenrightbigg\n,(19)\nwherek±=/radicalbig\n2m(ǫF∓ΓS)//planckover2pi1. The wave functions out-\nside of the bilayer in x > l FMandx <−lNare as-\nsumed to vanish for simplicity. The coefficients A∼I\nare again determined by matching wave functions and\ntheir first derivative at the interface. The interface spin\ncurrent and the spin mixing conductance are calculated\nfrom Eqs. (5) to (7), with replacing ψFItoψFMandlFI\ntolFM, resulting in\nGr,i=1\na3|γ′\nθ|2(Im,Re)/bracketleftBigg\nZ∗\n↓−+Z↓++\n×/parenleftBig\nu+−−u++−u−−+u−+/parenrightBig/bracketrightBigg\n,(20)\nwhere\nuαβ=iei(αk++βk−)lFM\nαk++βk−, Wσαβ=k0σ+βkα\n2k0σ,\nZσαβ=Wσαβe−ikαlFM−WσαβeikαlFM,\nγ′\nθ=Z↑++Z↓−+cos2θ\n2+Z↓++Z↑−+sin2θ\n2,(21)withβ=−β. Apart from a change in magnitude, the\npattern of the spin mixing conductance as a function of\ns−dcoupling and the FMM thickness shown in Fig. 3\nis almost indistinguishable from that reported in Fig. 2\nof Ref. 26, which shows clear signals of quantum inter-\nference with respect to both s−dcoupling and FMM\nthickness. This similarity is expected, since the only dif-\nference between the formalism here and in Ref. 26 is the\ninsulating gap V0−ǫFof the substrate or vacuum in the\nx > lFMregion in Fig. 3 (a), which is assumed to be\ninfinite here for simplicity but finite in Ref. 26. The in-\nsulating gap is spin degenerate and essentially does not\ninfluence the spin transport.\nFIG. 4: (color online)The longitudinal ∆ ρ1/ρandtransverse\n∆ρ2/ρcomponent of SMR in the NM/FMM bilayer, plotted\nagainst the FMM thickness in units of Fermi wave length\nlFM/aand NM thickness in units of the spin diffusion length\nlN/λ, at various strength of s−dcoupling −ΓS/ǫF.6\nTo get SMR, we use Eq. (17) ∼(18) while taking the\nGc=Gr+iGiobtainedfromEq.(20). Theresultsforthe\nlongitudinal ∆ ρ1/ρand transverse ∆ ρ2/ρcomponent of\nSMR as functions of FMM thickness lFMand NM thick-\nnesslNare shown in Fig. 4, for several values of s−d\ncoupling ΓS/ǫF. As a function of NM thickness, both\ncomponents reach a maximal at around the spin diffu-\nsion length lN/λ∼1 and then decrease monotonically,\nsimilartothatreportedinFig.2forNM/FMIbilayerand\nis due to the spin diffusion effect explained in Sec. IIB.\nOn the other hand, asa function of FMM thickness, both\ncomponents show clear modulations with an average pe-\nriodicity that decreases with increasing s−dcoupling, a\ntrend similar to that of GrandGishown in Fig. 3 and is\nattributed to the quantum interference of spin transport.\nIntuitively, a larger s−dcoupling renders a faster preces-\nsion of conduction electron spin when it travelsinside the\nFMM, hence more modulations appear for a given FMM\nthickness. The transverse component of SMR is found\nto be generally one order of magnitude smaller than the\nlongitudinal component.\nB. To observe the predicted oscillation in SMR\nThe experimental detection of the oscillation of SMR\nwith respect to FMM thickness lFMshown in Fig. 4\nwould be a direct proof of our approach. In a typical\nNM/FMI set up, however, there are other sources that\ncontribute to the total resistance measured in experi-\nments, therefore it is important to investigate whether\nthere is a situation in which the predicted oscillation of\nSMR can manifest. To explore this possibility, we use\na three-resistor model to characterize the total longitu-\ndinal resistance14, which contains the resistor that rep-\nresents the NM layer ( N), the FMM layer ( F), and the\ninterface layer ( I) connected in parallel, each denoted by\nRi=R0\ni+δRiwithi={N,I,F}. HereR0\niis the contri-\nbution to the longitudinal resistance in layer ithat does\nnot depend on the angle of the magnetization, and δRi\nis the part that depends on the angle which is generally\nmuch smaller δRi≪R0\ni. Expanding the total longitudi-\nnal resistance to leading order in δRiyields\nRtot≈R0\ntot+/parenleftbiggR0\nIR0\nF\nB/parenrightbigg2\nδRN\n+/parenleftbiggR0\nNR0\nF\nB/parenrightbigg2\nδRI+/parenleftbiggR0\nNR0\nI\nB/parenrightbigg2\nδRF,\nR0\ntot=R0\nNR0\nIR0\nF\nB,\nB=R0\nNR0\nI+R0\nIR0\nF+R0\nNR0\nF. (22)\nEachresistanceisassumed to satisfythe usualrelation to\nthe sample size/braceleftbig\nR0\ni,δRi/bracerightbig\n=/braceleftbig\nρ0\ni,δρi/bracerightbig\n×L/lih, whereL\nandhare the length and the width of the sample, respec-\ntively,ρ0\niandδρiare the corresponding resistivity, and li\nis the thickness of layer i. The thickness of the interfacelIis assumed to be intrinsically constant, in contrast to\nlNandlFthat can be varied experimentally14. The per-\ncentage change of the total resistance due to the angle of\nthe magnetization is\nRtot−R0\ntot\nR0\ntot≈/parenleftbiggρ0\nIρ0\nF\nlIlFC/parenrightbiggδρN\nρ0\nN+/parenleftbiggρ0\nNρ0\nF\nlNlFC/parenrightbiggδρI\nρ0\nI\n+/parenleftbiggρ0\nNρ0\nI\nlNlIC/parenrightbiggδρF\nρ0\nF,\nC=ρ0\nNρ0\nI\nlNlI+ρ0\nNρ0\nF\nlNlF+ρ0\nFρ0\nI\nlFlI.(23)\nNote that the ρ0\niρ0\nj/liljCfactors are monotonic functions\nof the layer thickness {lN,lI,lF}, and are independent\nfrom the angle of the magnetization.\nThe contribution to the angular dependent part of RF\ncomes from the anisotropic magnetoresistance (AMR)\nwhich takes the form34,35δρF∝(jc·ˆm)2∝(my)2\nsince the in-plane charge current jcruns along ˆyas\nshown in Fig. 3 (a), and we denote ˆm=S/S=\n(sinθcosϕ,sinθsinϕ,cosθ) as the unit vector along the\ndirection of the magnetization. In addition, Zhang et\nal.35showed that the interface resistance has a quadratic\ndependence on both myandmz, a result of surface spin-\norbit scattering. On the other hand, the SMR in the\nNM has the angular dependence22described by Eq. (17).\nThese considerations lead to the parametrization of re-\nsistivity by\nρ0\nF+δρF=ρ0\nF+∆ρb\nF(my)2,\nρ0\nI+δρI=ρ0\nI+∆ρs\nI,y(my)2+∆ρs\nI,z(mz)2,\nρ0\nN+δρN= (ρ+∆ρ0)+∆ρ1/bracketleftbig\n(mx)2+(my)2/bracketrightbig\n.\n(24)\nCombinig this with Eq. (23) motivates us to propose the\nfollowing experiment that should isolate the effect of lon-\ngitudinal SMR represented by δρN. From Eq. (24), we\nseethatδρFandδρIvanishifthe magnetizationdoesnot\nhave an in-plane component, i.e., my=mz= 0, while\nδρNremains finite as long as the out-of-plane component\nis nonzeromx∝ne}ationslash= 0. Thus we propose to fix the magne-\ntization of the FMM film to be out-of-plane mx∝ne}ationslash= 0,\nin which case the percentage change of total longitudinal\nresistance as a function of FMM thickness takes the form\nRtot−R0\ntot\nR0\ntot≈l1\nlF+l1+l2×∆ρ1\nρ+∆ρ0(mx)2,\n(formy=mz= 0) (25)\nwhereρ, ∆ρ0, and ∆ρ1are those in Eqs. (17) and (18),\nl1=lNρ0\nF/ρ0\nNandl2=lIρ0\nF/ρ0\nIare two length scales\nthat can be treated as fitting parameters in experiments.\nEquation (25) indicates that, for the case of only out-\nof-plane magnetization, the percentage change of magne-\ntoresistance decays with the FMM thickness lFdue to\nthel1/(lF+l1+l2) factor, but also oscillates with lFdue\nto the ∆ρ1/(ρ+ ∆ρ0)≈∆ρ1/ρfactor as quantified in7\nEq. (18) and shown in Fig. 4. Thus varying FMM thick-\nness while keeping its magnetization out-of-plane may\nbe a proper set up to observe the predicted oscillation of\nlongitudinal SMR, provided the FMM thickness remains\nthinner than its spin relaxation length lF≪λ. Finally,\nwe remark that the convention of labeling coordinate in\nSMR or STT experiments is that the charge current is\ndefined to be along ˆ xand the direction normal to the\nfilm is along ˆ z. Therefore the coordinate in our tun-\nneling formalism ( x,y,z) corresponds to ( z,x,y) in the\nexperimental convention.\nIV. CONCLUSION\nIn summary, the quantum tunneling formalism for the\ninterface spin current is incorporated into the spin diffu-\nsionapproachtostudytheeffectofvariousmaterialprop-\nerties on SMR, in particular the effect of layer thickness,\ninsulating gap, and interface s−dcoupling. The advan-\ntage of combining the quantum and diffusive approachis that the effects of all these material properties can be\ntreated on equal footing. For the NM/FMI case, we re-\nveal an SMR that saturates at large FMI thickness since\nthe conduction electrons only tunnels into the FMI over\na short distance, whereas the longitudinal and transverse\nSMR display different dependence on the insulating gap\nand interface s−dcoupling. For the NM/FMM case, we\npredict that SMR may display a pattern of oscillation as\nincreasing FMM thickness due to quantum interference,\nand propose an experiment to observe it by using fixed\nout-of-plane magnetization to isolate SMR from other\ncontributions. We anticipate that our minimal model\nthat combines the quantum and diffusive approach may\nbe usedto guidethe searchforsuitable materialsthat op-\ntimize the SMR, and help to predict novel spin transport\neffects in ultrathin heterostructures in which quantum\neffects shall not be overlooked.\nThe authors acknowledge the fruitful discussions with\nP. Gambardella, F. Casanova, and J. Mendil. W. C. and\nM. 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Jacobsen\nCenter for Quantum Spintronics, Department of Physics,\nNTNU Norwegian University of Science and Technology, NO-7491 Trondheim, Norway\nA recent proof of concept showed that cavity photons can mediate superconducting (SC) signatures to a\nferromagnetic insulator (FI) over a macroscopic distance [Phys. Rev. B, 102, 180506(R) (2020)]. In contrast\nwith conventional proximity systems, this facilitates long-distance FI–SC coupling, local subjection to different\ndrives and temperatures, and studies of their mutual interactions without proximal disruption of their orders.\nHere we derive a microscopic theory for these interactions, with an emphasis on the leading effect on the FI,\nnamely, an induced anisotropy field. In an arbitrary practical example, we find an anisotropy field of 14–16µT,\nwhich is expected to yield an experimentally appreciable tilt of the FI spins for low-coercivity FIs such as Bi-\nYIG. We discuss the implications and potential applications of such a system in the context of superconducting\nspintronics.\nI. INTRODUCTION\nEnabling low-dissipation charge and spin transport, super-\nconducting spintronics presents a pathway to reducing en-\nergy costs of data processing, and provides fertile ground for\nexploring new fundamental physics [1–3]. Conventionally,\nsuperconducting and spintronic systems are coupled by the\nproximity effect, with properties of adjacent materials trans-\nported across an interface. The superconducting coherence\nlength thus limits the extent to which superconducting prop-\nerties can be harnessed in proximity systems, to a range of\nnm–\u0016m near interfaces [4–8].\nBy contrast, cavity-coupled systems offer mediation across\nmacroscopic distances [9–13]. They also offer interaction\nstrengths that relate inversely to the cavity volume [14, 15],\nwhich is routinely utilized experimentally to achieve strong\ncoupling in e.g. GHz–THz cavity set-ups [16–20]. Further-\nmore, research on the coupling of magnets and cavity photons\nshows that the effective interaction strengths scale with the\nnumber of spins involved [9, 20–28], which has been utilized\nexperimentally to achieve effective coupling strengths far ex-\nceeding losses [11, 13, 20, 26–36].\nTheoretically, a number of methods have been employed\nto extract mediated effects in cavity-coupled systems. This\nincludes, but is not limited to, classical modelling for cou-\npling two ferromagnets [37], and a ferromagnet to a super-\nconductor [10]; application of Jaynes–Cummings-like models\nfor coupling a ferromagnet and a qubit [12, 13, 32, 38], and\ntwo ferromagnets [39]; perturbative diagonalization by the\nSchrieffer–Wolff transformation for coupling a ferro- and an-\ntiferromagnet [9, 21, 28], and a normal metal to itself [14, 20];\nand perturbative evolution of the density matrix, as well as\nperturbative diagonalization by the non-equilibrium Keldysh\npath integral formalism, for coupling a mesoscopic circuit to\na cavity [40].\nIn this paper, we will employ the Matsubara path inte-\ngral formalism [41–45] to derive a microscopic theory for\nthe cavity-mediated coupling of a ferromagnetic insulator (FI)\nwith a singlet s-wave superconductor (SC). In particular, we\n\u0003Corresponding author: henning.g.hugdal@ntnu.no\nFIG. 1. Illustration of the set-up. A thin ferromagnetic insulator\nand thin superconductor are placed spaced apart inside a rectangular,\nelectromagnetic cavity. The FI is subjected to an aligning external\nmagnetic field Bext. The cavity is short along the zdirection, and\nlong along the perpendicular xydirections, causing cavity modes to\nseparate into a band-like structure. The FI and the SC are respectively\nplaced in regions of maximum magnetic ( z=Lz) and electric ( z=\nLz=2) cavity field of the `z= 1modes, as defined in Sec. II B 1 and\nillustrated above by the colored field cross-section on the right wall.\nconsider the Zeeman coupling to the FI, and the paramag-\nnetic coupling to the SC. We show that with this approach,\nwe may exactly integrate out the net mediated effect by the\ncavity photons. This is in contrast to the Schrieffer–Wolff ap-\nproach, which would limit the integrating-out of the cavity to\noff-resonant regimes [21]. For instance, a pairing term analo-\ngous to the one found via the Schrieffer–Wolff transformation\nin Ref. [14] also appears in our calculations, without the lim-\nitation to an off-resonant regime. Furthermore, unlike many\npreceding works which single out the coupling to the uniform\nmode of the magnet [9, 10, 13, 22, 29, 30], we retain the influ-\nence of a range of modes in our model. Their non-negligible\ninfluence when the magnet exceeds a certain size relative to\nthe cavity, has been emphasized by both experimentalists [30]\nand theorists [22].\nThe Matsubara path integral approach was very recently ap-\nplied to construct a general effective theory of cavity-coupled\nmaterial systems of identical particles [45], highlighting some\nof the same advantages of this approach as above. By contrast,\nwe consider the cavity-mediated coupling of lattices of two\ndistinct classes of quasiparticles, specifically magnons and SC\nquasiparticles.\nBy a careful choice of cavity dimensions and the place-\nment of subsystems, we couple the insulator to the momen-arXiv:2209.09308v2  [cond-mat.mes-hall]  31 Jan 20232\ntum degrees of freedom of the superconductor. In this case,\nthe cavity acts as an effective spin–orbit coupling. Here, we\nemphasize the leading effect of the superconductor on the in-\nsulator, namely, the induction of an anisotropy field. In an\narbitrary, practical example, we achieve a field of 14–16µT,\nwhich is expected to yield an experimentally appreciable tilt\nof the FI spins for an insulator of sufficiently low coercivity\nsuch as Bi-YIG. Since the cavity facilitates coupling across\nunconventionally long distances, it enables the FI and SC to\nbe held at different temperatures, be subjected separately to\nexternal drives, and have them interact without the same mu-\ntual disruption of their orders associated with the proximity\neffect [2, 10], such as the breaking of Cooper pairs by mag-\nnetic fields from the FI. In practical applications, our system\nmay be used to bridge superconducting and other spintronic\ncircuitry.\nThe article is organised as follows. In Section II A we\npresent the set-up: A cavity with an FI and SC film placed\nat magnetic and electric antinodes as shown in Fig. 1, with\nno overlap in the xyplane. In Section II B we cover theoret-\nical preliminaries: The quantized gauge field, the magnon-\nbasis Hamiltonian for the insulator, and the Bogoliubov\nquasiparticle-basis Hamiltonian for the superconductor. The\nsystem Hamiltonian is subsequently constructed. In Sec. II C–\nII E, we construct an effective magnon theory using the path\nintegral formalism. Here we exactly integrate out the cav-\nity, and perturbatively the superconductor. In Section III, we\nextract from the effective theory the leading effect of the su-\nperconductor on the insulator, namely, the induced anisotropy\nfield. In a practical example, we calculate this field numer-\nically, and find here an induced field on the order of µ Tin\nmagnitude. Finally, in Section IV, we give concluding re-\nmarks, discussing the results and their significance, and an\noutlook. In the appendices, we affirm the mathematical con-\nsistency of the effective theory with an alternative derivation,\nexplore a variation of the set-up with the SC placed at the\nopposite magnetic antinode, and elaborate on the interpreta-\ntion of certain quantities in the effective action as an effective\nanisotropy field.\nII. THEORY\nA. Set-up\nOur set-up is illustrated in Fig. 1. We place two thin layers,\none of a ferromagnetic insulator (FI) and one of a supercon-\nductor (SC), spaced apart inside a rectangular electromagnetic\ncavity. The dimensions of the cavity are Lx;Ly\u001dLz, with\nLzon the\u0016m–mm scale, and Lx;Lyon the cm scale. The\naspect ratios render photons more easily excited in the xydi-\nrections. The FI is placed at the upper magnetic antinode of\nthe`z= 1 modes (cf. Sec. II B 1), and the SC at the corre-\nsponding electric antinode, as illustrated in Fig. 1. Because\nthe layers are thin in comparison to Lz, the local spatial vari-\nation of the modes in the zdirection is negligible, i.e., the\nmodes are treated as uniform in the zdirection.\nThe FI is locally subjected to an aligning and perpendicularuniform, external magnetostatic field, which vanishes across\nthe SC. This was achieved experimentally with external coils\nand magnetic shielding in Tabuchi et al. [13]. Furthermore,\nthe SC is subjected to an in-plane supercurrent. This may be\nrealized by passing a direct current (DC) through small elec-\ntric wires, entering the cavity via small holes in the walls and\nconnecting along the sides of the SC, similarly to Ref. [46].\nProvided the wires and holes are sufficiently small, their influ-\nence on the cavity modes are negligible. Provided the sample\nwidth does not exceed the Pearl length \u00152=dSC[46–48], the\nleading effect of the DC is to induce an equilibrium supercur-\nrent with a Cooper pair center-of-mass momentum 2P, with\nthe magnitude of Pdetermined by the current. Here \u0015is the\neffective magnetic penetration depth, and dSCis the sample\ndepth. For Nb thin films, we expect the Pearl length crite-\nrion to be met at widths of up to 0:1 mm for adSCdown to\n1 nm [49].\nB. Hamiltonian\nIn the following, we deduce a Hamiltonian\nH\u0011H FI+Hcav\n0+HSC: (1)\nfor the system illustrated in Fig. 1. We begin by quantizing\nthe cavity gauge field, and introducing the cavity Hamiltonian\nHcav\n0. Following this, we deduce a Hamiltonian HFIfor the\nFI in the magnon basis, including the Zeeman coupling to the\ncavity. Finally, we deduce a Hamiltonian HSCfor the SC in\nthe quasiparticle basis, including the paramagnetic coupling\nto the cavity.\n1. Cavity gauge field\nWe begin by presenting the expression for the quantized\ncavity gauge field Acav[15]. Starting from the Fourier de-\ncomposition of the classical vector potential, we impose the\ntransverse gauge and quantize the field. We employ reflect-\ning boundary conditions at the cavity walls in the zdirection,\nand periodic boundary conditions at the comparatively distant\nwalls in thexydirections. The gauge field is thus\nAcav\u0011X\nQ&s\n~\n2\u000f!Q(aQ&\u0016 uQ&+ay\nQ&\u0016 u\u0003\nQ&): (2)\nAbove,\nQ\u0011(Qx;Qy;Qz)\u0011(2\u0019`x=Lx;2\u0019`y=Ly;\u0019`z=Lz)(3)\nare the momenta of each photonic mode, with `x;`y=\n0;\u00061;\u00062;::: and`z= 0;1;2;:::. The discretization of Qz\ndiffers from that of QxandQydue to the different bound-\nary conditions in the transverse and longitudinal directions.\nFurthermore, &= 1;2labels polarization directions, \u000fis the\npermittivity of the material filling the cavity, and\n!Q=cjQj (4)3\nis the cavity dispersion relation, with cthe speed of light. ay\nQ&\nandaQ&are photon creation and annihilation operators, satis-\nfying\n[aQ&;ay\nQ0&0] =\u000eQQ0\u000e&&0; (5)\nwhere the factors on the right-hand side are Kronecker delta\nfunctions.\nLastly, the mode functions\n\u0016 uQ&\u0011X\nD^eDOQ\n&DuQD (6)\nencapsulate the spatial modulation of the modes. Here, ^eDis\nthe unit vector in the D=x;y;z direction.OQ\n&Dare elements\nof a matrix that rotates the original xyz basis of unit vectors\nto a new basis labeled 123, with the 3direction aligned with\nQ(see Fig. 2):\n0\n@^eQ\n1\n^eQ\n2\n^eQ\n31\nA=OQ0\n@^ex\n^ey\n^ez1\nA; (7)\nOQ\u00110\n@cos\u0012cos'cos\u0012sin'\u0000sin\u0012\n\u0000sin' cos' 0\nsin\u0012cos'sin\u0012sin'cos\u00121\nA: (8)\nHere\u0012=\u0012Qand'='Qare the polar and azimuthal angles\nillustrated in Fig. 2. OQoriginates from the implementation\nof the transverse gauge, which amounts to neglecting the lon-\ngitudinal 3component of the gauge field. Finally, uQDare\nthe mode functions in the xyzbasis, given by\nuQx=uQy=r\n2\nVeiQxx+iQyyisinQzz; (9)\nuQz=r\n2\nVeiQxx+iQyycosQzz; (10)\nwhereVis the volume of the cavity [50].\nOur set-up facilitates coupling to the `z= 1 band of cav-\nity modes, as the FI and SC are placed in field maxima as\nillustrated in Fig. 1. We will only consider variations of the\nin-plane part qof the general momenta Q, defined via\nQ\u0011q+\u0019^ez=Lz: (11)\nFor this reason we will use the subscript qfor functions of Q\nwhere thezcomponent is locked to the `z= 1mode, e.g.\n!q\u0011!Q\f\f\nQ=q+\u0019^ez=Lz=cs\u0012\u0019\nLz\u00132\n+q2: (12)\nThe cavity itself contributes to the system Hamiltonian with\nthe term\nHcav\n0\u0011X\nq&~!qay\nq&aq&; (13)\nwhere we have disregarded the zero-point energy, since it does\nnot influence our results.\nFIG. 2. Illustration of the 123coordinate system. Qis the photon\nmomentum vector, and qis its component in the xyplane.\u0012(single\nline) is the polar, and '(double line) the azimuthal angle associated\nwithQin relation to the xyz basis. The 123 axes results from a\nrotation of the xyzaxes by an angle \u0012about theyaxis, followed by a\nrotation by an angle 'about the original zaxis. In the illustration, the\n1axis points somewhat outwards and downwards, the 2axis points\nsomewhat inwards and is confined to the original xyplane, and the\n3axis aligns with Q.\n2. Ferromagnetic insulator\nThe Hamiltonian of the FI in the cavity is\nHFI\u0011H ex+Hext+HFI\u0000cav; (14)\nwith\nHex\u0011\u0000JX\nhi;jiSi\u0001Sj; (15a)\nHext\u0011\u0000g\u0016B\n~BextX\niSiz; (15b)\nHFI\u0000cav\u0011\u0000g\u0016B\n~X\niSi\u0001Bcav(ri): (15c)\nThe first term is the exchange interaction: J > 0is the ex-\nchange interaction strength for a ferromagnetic insulator, Si\nis the spin at lattice site i, and only nearest neighbor interac-\ntions are taken into account, as indicated by the angle brack-\nets. The next two terms are Zeeman couplings: gis the gyro-\nmagnetic ratio, \u0016Bis the Bohr magneton, Bextis a strong\n(i.e.jBextj \u001d j Bcavj) and uniform external magnetostatic\nfield aligning the spins in the zdirection, and Bcav(ri)is the\nmagnetic component of the cavity field at lattice site i. The\ncorresponding position vector is ri.\nIt is convenient to transition from the spin basis\nfSix;Siy;Sizgto a bosonic magnon basis f\u0011i;\u0011y\nig. This\nis achieved with the Holstein–Primakoff transformation [51],\nwhich is covered in detail in Refs. [21, 52].\nEach FI lattice site carries spin S. The aligning field Bext\nregulates the excitation energy of magnons (cf. Eq. (21)),\nhence a sufficiently strong field implies few magnons per lat-\ntice site, i.e.\nh\u0011y\ni\u0011ii\u001c2S: (16)4\nWe can therefore Taylor-expand the Holstein–Primakoff\ntransformation, leading to the relations\nSiz=~(S\u0000\u0011y\ni\u0011i); (17)\nSid\u0019~p\n2S\n2(\u0017d\u0011i+\u0017\u0003\nd\u0011y\ni); (18)\nwhered=x;yandf\u0017x;\u0017yg=f1;\u0000ig.\nNow, upon Fourier-decomposing the magnon operators\n\u0011ri\u00111pNFIX\nk\u0011keik\u0001ri; (19)\nwe obtain the conventional expression for Hex+Hextin the\nmagnon basis [52]:\nHex+Hext\u0019HFI\n0\u0011X\nk~\u0015k\u0011y\nk\u0011k; (20)\nwhere we have introduced the magnon dispersion relation\n\u0015k\u00112~JN\u000eS \n1\u00001\nN\u000eX\n\u000eeik\u0001\u000e!\n+g\u0016B\n~Bext:(21)\nAbove,NFIis the total number of FI lattice points, N\u000e=\n6is the number of nearest-neighbor lattice sites on a\ncubic lattice (neglecting edges and corners), and \u000e=\n\u0006aFI^ex;\u0006aFI^ey;\u0006aFI^ezare nearest-neighbor lattice vectors.\nThe magnon momenta are\nk\u0011(2\u0019mFI\nx=lFI\nx;2\u0019mFI\ny=lFI\ny;0)\u0011(kx;ky;0); (22)\nwheremFI\nd=\u0000j\nNFI\nd\u00001\n2k\n;:::;NFI\nd\u00001\u0000j\nNFI\nd\u00001\n2k\ncovers\nthe first Brillouin zone (1BZ), with NFI\ndthe number of FI lat-\ntice points in direction d, andb\u0001cthe floor function. Here we\nneglect thekzcomponent; only the kz= 0 modes enter our\ncalculations due to the thinness of the FI film (cf. Eq. (26)).\nNote that the set of magnon momenta generally does not over-\nlap with that of photon momenta in Eq. (3). Observe further-\nmore that the magnon energies (21) can easily be regulated\nexperimentally by adjusting Bext.\nProceeding to the interaction term, we deduce the magnetic\ncavity field Bcav(ri)across the FI, which is the curl of the\ngauge field at z\u0019Lz:\nBcav(ri)\f\f\nFI=r\u0002Acav(ri)\f\f\nFI\n=\u0000X\nqdi\u00172\ndq\u0016d^edsin\u0012qs\n~\n\u000f!qVeiq\u0001ricos\u0019zi\nLz(aq1+ay\n\u0000q1):\n(23)\nAbove, \u0016d“inverts”dsuch that \u0016x=yand\u0016y=x, andzi\nis thezposition of lattice site i. Note that the photon mo-\nmentum component q\u0016denters the sum with an inverted lower\nindex. Observe that only the 1direction enters the expression,\nbecause Acavatz\u0019Lzpoints purely along the zdirection.The2direction is by definition locked to the xyplane, and\ndoes therefore not contribute at z\u0019Lz.\nInserting Eqs. (17)–(19) and (23) into Eq. (15c), we find\nHFI\u0000cav\u0019X\nkdX\nq&gkq\nd(\u0017d\u0011\u0000k+\u0017\u0003\nd\u0011y\nk)(aq1+ay\n\u0000q1);\n(24)\nand hence a complete FI Hamiltonian HFI\u0019HFI\n0+HFI\u0000cav.\nAbove, we defined the coupling strength\ngkq\nd\u0011\u0000g\u0016Bq\u0016di\u00172\ndsin\u0012qs\nS~NFI\n2\u000f!qVDFI\nkqeiq\u0001rFI\n0: (25)\nDFI\nkqquantifies the degree of overlap between magnonic and\nphotonic modes. An analogous quantity appears in the cavity–\nSC coupling in Sec. II B 3, so we define it via the general ex-\npression\nDM\nlMq\u0011ei(lM\u0000q)\u0001rM\n0\nNMX\ni2Me\u0000i(lM\u0000q)\u0001ri\n\u0002\u001a\u0000cos\u0019zi\nLz; M = FM\nsin\u0019zi\nLz; M = SC\u001b\n\u0019\u000elM;z0Y\ndsinc\u0014\n\u0019NM\nd\u0012mM\nd\nNM\nd\u0000`daM\nLd\u0013\u0015\n:(26)\nHereM=fFI;SCgis a material index, lMrepresents ei-\nther a magnon or an SC quasiparticle momentum, rM\n0is the\ncenter position of lattice Mrelative to the origin, and the\nphoton momentum numbers `d=`x;`ywere defined under\nEq. (3). The latter, along with other SC quantities, are defined\nin Sec. II B 3. The sum over iis taken over either FI or SC\nlattice points, as indicated by M, and the last equality holds\nforNM\nd\u001d1.\nDFI\nkqreduces to a Kronecker delta \u000ekqonly whenLd=\nld=aFINFI\nd, i.e. when the FI and the cavity share in-\nplane dimensions [53]. At the other end of the scale, when\nthe FI becomes infinitely small, DFI\nkqreduces to\u000ek0, im-\nplying all cavity modes couple exclusively to the uniform\nmagnon mode, which is often assumed in cavity implemen-\ntations [9, 10, 13, 29]. We assume this uniform coupling only\nin thezdirection, hence the factor \u000elM;z0in Eq. (26) (thus\nkz= 0); the condition is that \u0019dM=2Lz\u001c1, withdMthe\nthickness of film M[54].\n3. Superconductor\nThe SC Hamiltonian is\nHSC=Hsing+HBCS+Hpara; (27)\nwith\nHsing\u0011X\np\u0018pcy\np\u001bcp\u001b0; (28a)5\nHBCS\u0011\u0000X\np\u0010\n\u0001pcy\np+P;\"cy\n\u0000p+P;#+ \u0001\u0003\npc\u0000p+P;#cp+P;\"\u0011\n;\n(28b)\nHpara\u0011X\ndX\njjd(rj)Ad\u0012rj+Id+rj\n2\u0013\n; (28c)\nHsingis the single-particle energy, where \u0018pis the lattice-\ndependent electron dispersion, and cp\u001bandcy\np\u001bare fermionic\noperators for an electron of lattice momentum pand spin\u001b.\nThe momenta are discretized as\np\u0011(2\u0019mSC\nx=lSC\nx;2\u0019mSC\ny=lSC\ny;2\u0019mSC\nz=lSC\nz)\u0011(px;py;pz);\n(29)\nwheremSC\ndandmSC\nzare defined analogously to mFI\nd(see be-\nlow Eq. (22)), covering the 1BZ of the SC with NSC\nd(NSC\nz)\nthe number of SC lattice points in direction d(z).\nHBCS is the BCS pairing term, with \u0001pthe pairing po-\ntential. The leading order effect of applying an in-plane DC\nacross the SC is to shift the center of the SC pairing poten-\ntial from p=0top=P, where 2Pis the generally finite\ncenter-of-mass momentum of the Cooper pairs [46, 55, 56].\nThe maximum value of Pis limited by the critical current of\nthe superconductor.\nHparais the paramagnetic coupling. jd(rj)is thedcompo-\nnent of the discretized electric current operator at lattice site j\nwith the position vector rj, and is defined as [14]\njd(rj)\u0011iaSCet\n~X\n\u001b(cy\nj+Id;\u001bcj\u001b\u0000cy\nj\u001bcj+Id;\u001b): (30)\nThezcomponentjzdoes not contribute to our Hamiltonian\nbecause the cavity gauge field is in-plane at z\u0019Lz=2. Above,\naSCis the lattice constant, eis the electric charge, tis the\nlattice hopping parameter, and cj\u001bandcy\nj\u001bare real-space\nfermionic operators for electrons with spin \u001bat lattice site j.\nThey relate to cp\u001bandcy\np\u001bvia\ncj\u001b=1pNSCX\npcp\u001beip\u0001rj; (31)\nwithNSCthe total number of SC lattice points. Further-\nmore,Idrepresents a unit step in the ddirection with re-\nspect to lattice labels. For instance, if j= (1;1), then\nj+Ix= (1 + 1;1) = (2;1).\nInserting Eqs. (2), (30) and (31) into Eq. (28c) yields\nHpara=X\npp0\u001bX\nq&gqpp0\n&(aq&+ay\n\u0000q&)cy\np\u001bcp0\u001b:(32)\nHere, we have introduced the coupling strength\ngqpp0\n&\u0011\u0000aSCet\n~s\n~\n\u000f!qVDSC\np\u0000p0;qeiq\u0001rSC\n0\u0001X\nd\u0010\ne\u0000i(p\u0000q=2)\u0001\u000ed\u0000ei(p0+q=2)\u0001\u000ed\u0011\nOq\n&d;(33)\nwhere \u000ed\u0011aSC^edare in-plane primitive lattice vectors.\nDSC\np\u0000p0;qis defined in Eq. (26), quantifying the degree of over-\nlap between two electron modes and a photon mode. It re-\nduces to\u000ep\u0000p0;qonly when the cavity and the SC share in-\nplane dimensions, as is the case in Ref. [14].\nAs we move onto the imaginary-time (Matsubara) path in-\ntegral formalism in the next sections, it becomes convenient\nto eliminate creation–creation and annihilation–annihilation\nfermionic operator products. To this end, we absorb the BCS\nterm (28b) into the diagonal term (28a) by a straight-forward\ndiagonalization:\nHsing+HBCS\n=X\np\u0012cp+P;\"\ncy\n\u0000p+P;#\u0013y\u0012\u0018p+P\u0000\u0001p\n\u0000\u0001\u0003\np\u0000\u0018\u0000p+P\u0013\u0012cp+P;\"\ncy\n\u0000p+P;#\u0013\n=X\np\u0012\n\rp0\n\rp1\u0013y\u0012\nEp00\n0Ep1\u0013\u0012\n\rp0\n\rp1\u0013\n: (34)\nHere we introduced the Bogoliubov (SC) quasiparticle basis\nf\rpm;\ry\npmg, withm= 0;1and dispersion relations\nEpm=1\n2\u0014\n\u0018p+P\u0000\u0018\u0000p+P\n+ (\u00001)mq\n(\u0018p+P+\u0018\u0000p+P)2+ 4j\u0001pj2\u0015\n:(35)\nThe elements upandvpof the basis transformation matrix are\ndefined through [48]\ncp+P;\"\u0011u\u0003\np\rp0+vp\rp1; cy\n\u0000p+P;#\u0011\u0000v\u0003\np\rp0+up\rp1:\n(36)\nInserting the above into Eq. (34), one finds the relations\n\u0001\u0003\npvp\nup=1\n2[(Ep0\u0000Ep1)\u0000(\u0018p+P+\u0018\u0000p+P)]; (37a)\njvpj2= 1\u0000jupj2=1\n2\u0012\n1\u0000\u0018p+P+\u0018\u0000p+P\nEp0\u0000Ep1\u0013\n;(37b)\nwhich determine upandvp. RecastingHparain terms of this\nbasis yields\nHpara=X\npp0X\nq&X\nmm0gqpp0\n&mm0(aq&+ay\n\u0000q&)\ry\npm\rp0m0;(38)\nwhere the coupling strength is now\ngqpp0\n&mm0\u0011 \ngq;p+P;p0+P\n&upu\u0003\np0+gq;p\u0000P;p0\u0000P\n&vpv\u0003\np0gq;p+P;p0+P\n&upvp0\u0000gq;p\u0000P;p0\u0000P\n&vpup0\n\u0000gq;p\u0000P;p0\u0000P\n&u\u0003\npv\u0003\np0+gq;p+P;p0+P\n&v\u0003\npu\u0003\np0gq;p\u0000P;p0\u0000P\n&u\u0003\npup0+gq;p+P;p0+P\n&v\u0003\npvp0!\nmm0: (39)6\nThis concludes the derivation of the terms entering the sys-\ntem Hamiltonian in terms of the various (quasi)particle bases.\nWe now turn our focus to the construction of an effective FI\ntheory.\nC. Imaginary time path integral formalism\nWe now seek to extract the influence of the SC on the FI, in\nparticular the anisotropy field induced across the FI. Diagonal-\nizing the Hamiltonian directly, as was done in Eq. (34), would\nin this case be very challenging, as it couples many more\nmodes, and furthermore contains trilinear operator products.\nSince the external drives ( Bextand the DC) only give rise to\nequilibrium phenomena in our system, the Matsubara path in-\ntegral formalism of evaluating thermal correlation functions is\nvalid [41]. This translates the evaluation into a path integral\nproblem, which is very convenient for our purposes. The path\nintegral approach facilitates aggregation of the influences of\nspecific subsystems into effective actions, without explicit di-\nagonalization. On this note, for comparison, Cottet et al. [40]\nanalyze a scenario in which the non-equilibrium Keldysh path\nintegral formalism is used to analyze the net influence of a\nQED circuit on a cavity.\nThe starting point is the imaginary time action\nS\u0011SFI\n0+Scav\n0+SSC\n0+SFI\u0000cav\nint +Scav\u0000SC\nint\n=Z\nd\u001c\u0014X\nk\u0011y\nk~@\u001c\u0011k+X\nq&ay\nq&~@\u001caq&\n+X\npm\ry\npm~@\u001c\rpm+H\u0015\n: (40)\n\u001cis a temperature parameter treated as imaginary time,\nwhich relates to the thermal equilibrium density matrix\nexp(\u0000\fH=~), with\f\u0011~=kBTthe inverse temperature Tin\nunits of time, andHthe system Hamiltonian. The dependence\nof the field operators on temperature ( \u001c) is implied. In formu-\nlating the path integral, the magnon, photon and Bogoliubov\nquasiparticle operators have been replaced by eigenvalues of\nthe respective coherent states [41]; i.e. the bosonic operators\nhave been replaced by complex numbers, and the fermionic\noperators by Graßmann numbers. The magnons, photons and\nBogoliubov quasiparticles are furthermore taken to be func-\ntions of\u001c[41]. The integral over \u001cis taken over the interval\n(0;\f]. Note that we assume the gap to be fixed to the bulk\nmean field value, and therefore do not include a gap action or\nintegration in the partition function.\nWe now replace the integral over \u001cby an infinite sum over\ndiscrete frequencies by a Fourier transform of the magnon,\nphoton and Bogoliubov quasiparticle operators with respect\nto\u001c. The conjugate Fourier parameters are Matsubara fre-\nquencies:\n\nn=2n\u0019\n\f(41)for bosons, and\n!n=(2n+ 1)\u0019\n\f(42)\nfor fermions, with n2Z. The transforms read\n\u0011k=1p\fX\n\nm\u0011\u0000\nm;ke\u0000i\nm\u001c; (43a)\naq&=1p\fX\n\nna\u0000\nn;q&e\u0000i\nn\u001c; (43b)\n\rpm=1p\fX\n!n\r\u0000!n;pme\u0000i!n\u001c: (43c)\nTo avoid clutter, we introduce the 4-vectors\nk\u0011(\u0000\nm;k); (44a)\nq\u0011(\u0000\nn;q); (44b)\np\u0011(\u0000!n;p); (44c)\nand the generally complex energies\n~\u0015k\u0011\u0000i~\nm+~\u0015k; (45a)\n~!q\u0011\u0000i~\nn+~!q; (45b)\nEpm\u0011\u0000i~!n+Epm: (45c)\nThe actions in (40) then become\nSFI\n0=X\nk~\u0015k\u0011y\nk\u0011k; (46a)\nScav\n0=X\nq&~!qay\nq&aq&; (46b)\nSSC\n0=X\npmEpm\ry\npm\rpm; (46c)\nSFI\u0000cav\nint =X\nkdX\nq&gkq\nd&(\u0017d\u0011\u0000k+\u0017\u0003\nd\u0011y\nk)(aq&+ay\n\u0000q&);\n(46d)\nScav\u0000SC\nint =1p\fX\nq&X\npmX\np0m0gqpp0\n&mm0(aq&+ay\n\u0000q&)\ry\npm\rp0m0;\n(46e)\nwhere we introduced the coupling functions\ngkq\nd&\u0011gkq\nd\u000e&1\u000e\nm;\nn; (47)\ngqpp0\n&mm0\u0011gqpp0\n&mm0\u000e!n0;!n\u0000\nn: (48)\nWe additionally introduced a redundant Kronecker delta func-\ntion\u000e&1to the coupling (47), which will facilitate the gather-\ning of interaction terms in Eq. (51). We will use the notation\ng\u0011andg\rfor the magnitudes of the FI–cavity and cavity–SC\ncoupling, respectively.\nWe are now equipped to construct effective actions by in-\ntegrating out the photonic and fermionic degrees of freedom,7\nto which end we will consider the imaginary-time partition\nfunction [41, 43]\nZ\u0011hvac;t=1jvac;t=\u00001i\n=Z\nD[\u0011;\u0011y]Z\nD[a;ay]Z\nD[\r;\ry]e\u0000S=~; (49)\nwhere e.g.\nZ\nD[\r;\ry]\u0011Y\npmZ\nD[\rpm;\ry\npm] (50)\nis to be understood as the path integrals over every Bogoliubov\nquasiparticle mode.\nD. Integrating out the cavity photons\nThe order in which we integrate out the cavity and the SC is\ninconsequential. We will begin with the cavity, which can be\nintegrated out exactly. We show that interchanging the order\nof integrations leads to identical results in Appendix A.\nWe gather the interactions between the cavity and FI and\nSC,\nScav\nint=X\nq;&[Jq&aq&+J\u0000q&ay\n\u0000q&]; (51)\nwhere we have defined\nJq&=X\nksgkq\nd&(\u0017d\u0011\u0000k+\u0017\u0003\nd\u0011y\nk)\n+1p\fX\npp0X\nmm0gqpp0\n&mm0\ry\npm\rp0m0: (52)\nThese interaction terms are illustrated by the diagrams in the\ntop panel of Fig. 3. Integrating out the cavity modes [41], we\ntherefore get the effective action\nSe\u000b=\u0000X\nq&Jq&J\u0000q&\n~!q: (53)\nInserting the expression for Jq&we get three different terms,\nSe\u000b=SFI\n1+SSC\n1+Sint, shown diagrammatically in the\nbottom panel of Fig. 3. The first term,\nSFI\n1=\u0000X\nqkk0X\n&dd0gkq\nd&gk0\u0000q\nd0&\n~!q\n\u0002(\u0017d\u0011\u0000k+\u0017\u0003\nd\u0011y\nk)(\u0017d0\u0011\u0000k0+\u0017\u0003\nd0\u0011y\nk0); (54)\nis a renormalization of the magnon theory due to interactions\nwith the cavity, resulting in a non-diagonal theory. The second\nterm,\nSSC\n1=\u00001\n\fX\nqpp0\noo0X\n&mm0\nnn0gqpp0\n&mm0g\u0000qoo0\n&nn0\n~!q\ry\npm\rp0m0\ry\non\ro0n0;\n(55)\nScav\nint:agη\nη + agγ\nγγ\nSFM\n1 :Gcav\nSSC\n1 :Gcav\nSint :GcavFIG. 3. Feynman diagrams [57] of the bare cavity coupling to the\nFI and SC, and the resulting terms in the FI and SC effective actions\nafter integrating out the cavity photons, where Gcavis the photon\npropagator.\nis an interaction term coupling four quasiparticles, similar to\nthe term found in Ref. [14] for a normal metal coupled to a\ncavity, leading to superconducting correlations. Note that un-\nlike the pairing term found in Ref. [14] via the Schrieffer–\nWolff transformation, the term above is not limited to an off-\nresonant regime. In principle it could also lead to renormaliza-\ntion of the quasiparticle spectrum and lifetime. Since we are\nhere concerned with the effects of the cavity and SC on the\nFI, we will neglect this term as it only leads to higher order\ncorrections.\nFinally, we have the cavity-mediated magnon-quasiparticle\ncoupling,\nSint=\u00001p\fX\nkpp0X\ndmm0Vkpp0\ndmm0(\u0017d\u0011\u0000k+\u0017\u0003\nd\u0011y\nk)\ry\npm\rp0m0;\n(56)\nwhere we have defined the effective FI-SC interaction\nVkpp0\ndmm0=X\nq&gkq\nd&g\u0000qpp0\n&mm0\u00141\n~!q+1\n~!\u0000q\u0015\n: (57)\nThis term is generally nonzero, and we therefore see that the\ncavity photons lead to a coupling between the FI and SC, po-\ntentially over macroscopic distances. This means that the FI\nand SC will have a mutual influence on each other, possibly\nleading to experimentally observable changes in the two mate-\nrials. We therefore integrate out the Bogoliubov quasiparticles\nand calculate the effective FI theory below. We reiterate that\nthe interaction is exact at this point, not a result of a perturba-\ntive expansion.8\nE. Integrating out the SC quasiparticles — effective FI theory\nThe full effective SC action comprises the sum SSC\n0+SSC\n1+\nSint. The second term is second order in g\r, but does not\ncontain FI operators, and will therefore only have an indirect\neffect on the effective FI action. In a perturbation expansion of\nthe effective FI action, the term SSC\n1will therefore contribute\nhigher order correction terms compared to Sint. We therefore\nneglect this term in the following, leading to the SC action\nSSC\u0019\u0000X\npp0X\nmm0\ry\npm(G\u00001)pp0\nmm0\rp0m0; (58)\nwhere we have defined G\u00001=G\u00001\n0+ \u0006, with\n(G\u00001\n0)pp0\nmm0=\u0000Epm\u000epp0\u000emm0; (59)\n\u0006pp0\nmm0=1p\fX\nkdVkpp0\ndmm0(\u0017d\u0011\u0000k+\u0017\u0003\nd\u0011y\nk): (60)\nIntegrating out the SC quasiparticles results in the effective\nFI action [41]\nSFI=SFI\n0+SFI\n1\u0000~Tr ln(\u0000\fG\u00001=~): (61)\nThe Green’s function matrix G\u00001contains magnon fields, and\nwill be treated perturbatively in order to draw out the lowest\norder terms in the effective FI theory. We expand the loga-\nrithm to second order in the FI–SC interaction,\nln\u0012\n\u0000\fG\u00001\n~\u0013\n\u0019ln\u0012\n\u0000\fG\u00001\n0\n~\u0013\n+G0\u0006\u00001\n2G0\u0006G0\u0006;\n(62)\nwhereG0is the inverse of G\u00001\n0. This expansion is valid when\njG0\u0006j\u001c 1, meaningjg\u0011g\r=~!qEpmj\u001c 1, where we use\nshorthand notation for the couplings g\u0011andg\rbetween cav-\nity photons and \u0011and\rfields respectively. The first term in\nEq. (62) does not contain magnonic fields, and therefore does\nnot contribute to the FI effective action [58]. The third term\ncontains bilinear terms in magnonic fields, and gives a correc-\ntion to the magnon dispersion of order j[g\u0011g\r=~!q]2=Epmj,\na factor ofj(g\r)2=~!qEpmjsmaller than the corrections con-\ntained inSFI\n1, and will therefore also be neglected. Keeping\nonly the second term, and using the fact that G0is diagonal in\nboth quasiparticle type mand momentum p, we therefore get\nthe effective FI action to leading order,\nSFI=X\nk~\u0015k\u0011y\nk\u0011k\u0000g\u0016BX\nkdhk\nd\u0001r\nS\n2(\u0017d\u0011\u0000k+\u0017\u0003\nd\u0011y\nk)\n+X\nkk0dd0Qkk0\ndd0(\u0017d\u0011\u0000k+\u0017\u0003\nd\u0011y\nk)(\u0017d0\u0011\u0000k0+\u0017\u0003\nd0\u0011y\nk0);\n(63)\nwhere we have defined the anisotropy field due to the coupling\nto the superconductor,\nhk\nd=\u0000~\ng\u0016Br2\nS\fX\npmVkpp\ndmm\nEpm; (64)and a function\nQkk0\ndd0\u0011 \u0000X\nq&gkq\nd&gk0\u0000q\nd0&\n~!q: (65)\ndescribing the cavity-mediated self-interaction in the ferro-\nmagnetic insulator.\nIII. RESULTS\nThe main result of our work is the effective magnon ac-\ntion (63). The interaction with the cavity and the SC gives rise\nto linear and bilinear correction terms to the diagonal magnon\ntheory, corresponding to an induced anisotropy field and cor-\nrections to the magnon spectra.\nTo extract a specific quantity, we consider the leading order\neffect of coupling the FI to the SC via the cavity, namely the\nlinear magnon term. Physically this can be understood as a\ncontribution from an additional magnetic field trying to reori-\nent the FI in a direction other than along the zaxis. We can\nsee this explicitly if we Fourier transform the linear magnon\nterm back to real space and imaginary time,\nSFI\nlin=\u0000g\u0016B\n~Z\nd\u001cX\nriX\ndhd(ri;\u001c)Sid(\u001c); (66)\nwhere we have used the definition of the in-plane spin compo-\nnents in Eq. (18), and defined the real space anisotropy field\ncomponents due to the interaction with the superconductor\nhd(ri;\u001c) =1pNFI\fX\nkhk\ndeik\u0001ri: (67)\nAbove, we introduced the 4-vector\nri\u0011(\u001c;ri): (68)\nIn order for the anisotropy field components to be real, we re-\nquirehk\nd= (h\u0000k\nd)\u0003. Inserting the expressions for Epmand\nVkpp\ndmmfrom Eqs. (45c) and (57) into Eq. (64), and performing\nthe sum over the Matsubara frequencies [41], we get the fol-\nlowing expression for the Fourier transposed anisotropy field\ncomponents,\nhk\nd=\u0000p\nNFI\f\u000e\nm0X\nq;d04\u0019aSCet\n~\u000f!2qVLzq\u0016dqd0\njQj2\u00172\ndeiq\u0001(rFI\n0\u0000rSC\n0)\n\u0002DFI\nk;qDSC\n0;\u0000qe\u0000iqd0aSC=2\u0005Pd0; (69)\nwhere the dependence on the supercurrent comes in through\nthe factor\n\u0005Pd=X\np\b\nsin[(pd+Pd)aSC]jupj2\n+ sin[(pd\u0000Pd)aSC]jvpj2\t\ntanh\fEp0\n2~: (70)\nNotice that the field is finite only for zero Matsubara fre-\nquency, meaning that it is time-independent (magnetostatic).9\nIt is possible to show that hk\nd= (h\u0000k\nd)\u0003by letting q!\u0000qin\nthe sum in Eq. (69), and using DFI\nk;q= (DFI\n\u0000k;\u0000q)\u0003,DSC\n0;\u0000q=\n(DSC\n0;q)\u0003from the definition in Eq. (26). Observe that in the\ncase of no DC (i.e. P=0), the summand in Eq. (70) is odd\ninp, and the sum therefore zero, i.e., \u0005Pd= 0 ifPd= 0.\nHence there is no anisotropy field induced across the FI in\nthe absence of a supercurrent. This stresses the necessity of\nbreaking the inversion symmetry of the SC in order to induce\nan influence on the FI.\nA. Special case: small FM\nFIG. 4. Illustration of the set-up used in the example given in\nSec. III A. A small, square FI and SC are placed spaced apart in the\nyandzdirections inside a comparatively large cavity. Only a small\nportion of the cavity length in yis utilized as the contributions by the\nvarious mediating cavity modes add constructively only over short\ndistances. The FI and SC are nevertheless separated by hundreds\nof µm, 2–5 orders larger than typical effectual lengths in proximity\nsystems.\nThe anisotropy field (67) generally gives rise to compli-\ncated, local reorientation of the FI spins. However, there are\nspecial cases in which it takes on a simple form. In partic-\nular, assume the FI to be very small relative to the cavity,\ni.e.`xlFI\nx;`ylFI\ny\u001cLx;Ly. In this case, the FI sum (26) be-\ncomes highly localized around k=0for the relevant ranges\nof`xand`y, which are limited by the other factors DSC\n0qand\n(!qjQj)\u00002found in Eq. (69). We may therefore set k=0.\nFor a specified set of material parameters and dimensions, the\nvalidity is confirmed numerically. In this case, Eq. (67) thus\nreduces to\nhd=h0\ndpNFI\f; (71)\nrepresenting a uniform anisotropy field across the FI. In this\nlimit we can simplify the expression for the anisotropy field\ncomponents,\nhd=\u0000X\nq;d02\u0019aSCet\n~\u000f!2qVLz\u00172\ndDFI\n0;qDSC\n0;\u0000q\u0005Pd0q\u0016dqd0\njQj2\n\u0002\u0002\ncosqxLsep\nxcosqyLsep\ny\u0000sinqxLsep\nxsinqyLsep\ny\u0003\n;\n(72)\nwhere we have assumed e\u0000iqd0aSC=2\u00191, which is a good ap-\nproximation as long as the cavity dimensions far exceed thelattice constant and only low jqjcontribute to the sum, and\nused the fact that DM\n0;q[Eq. (26)] is an even function in q. We\nhave also defined the separation length Lsep\nd= (rFI\n0\u0000rSC\n0)\u0001^ed.\nAssuming a finite separation between the FI and SC only in\none direction, the last term in the above equation vanishes,\nmaking every remaining factor even in qd, except the product\nq\u0016dqd0for\u0016d6=d0. The sum over qtherefore picks out terms\nsuch that \u0016d=d0. In order to get a finite hdwe must, there-\nfore, have \u0005P\u0016d6= 0, i.e., the supercurrent momentum must be\nfinite in the direction \u0016d. Hence, in the case that the separation\nbetween the FI and SC is finite in only one direction, applying\na supercurrent in the xdirection can only induce an anisotropy\nfield in theydirection, and vice versa.\nWe consider the specific case of a small, square FI and SC\ndisplaced along yandz(Fig. 4). In Fig. 5 we show numer-\nically how the effective anisotropy field varies with the su-\npercurrent momentum in this special case, using Nb and YIG\nas material choices for the FI ( lFI\nx=lFI\ny= 10 µm) and SC\n(lSC\nx=lSC\ny= 50 µm,dSC= 10 nm ) films, respectively; see\nTable I. We use Python with the NumPy andMatplotlib li-\nbraries for the numerics. We furthermore use the interpolation\nformula [59]\n\u0001 = 1:76kBTc0tanh(1:74p\nTc0=T\u00001) (73)\nfor the superconducting gap, and a simple cubic tight-binding\nelectron dispersion. With the FI and SC center points sepa-\nrated by 140µmin theydirection (meaning they are separated\nedge-to-edge by 115µmin-plane), we find an anisotropy field\nwith a magnitude of .14µT(Fig. 5a). If the constraint on\nseparating the FI and SC in-plane is eased, the maximum mag-\nnitude increases to 16µTin our specific example (Fig. 5b).\nWe discuss the latter case in the concluding remarks.\nTwo factors determine the inhomogeneous distribution of\nthe responses seen in Fig. 5. First, the anisotropy field is\nnearly linear in the components Pdof the supercurrent mo-\nmentum, which is seen by expanding the anisotropy field\n(see Eq. (70)) around PdaSC= 0(note thatPcaSC\u00190:001).\nThis generally makes the response stronger for larger jPj,\nwhich is as expected, since it relies on breaking the p-\ninversion symmetry. This dependency is evident in Fig. 5.\nSecond, the factor eiq\u0001(rFI\n0\u0000rSC\n0)renders the anisotropy field\nvery sensitive to the separation of the FI and SC center points\nin the in-plane directions. This factor expresses that cavity\nmodes associated with a range of different in-plane momenta\nq(i.e., spatial oscillations) with a coherent amplitude at no\nin-plane separation ( rFI\n0\u0000rSC\n0= 0), become increasingly\ndecoherent with increasing separation. Eventually, this deco-\nherence causes states in the SC to contribute oppositely, hence\ndestructively, to the effective anisotropy field. The destructive\naddition at finite separation is limited by the range of low- q\ncavity modes that contribute to the mediated interaction un-\ntil the coupling is suppressed by the factor DFI\n0qDSC\u0003\n0q=!2\nqQ2,\nwhich in turn is determined by the dimensions of the three\nsubsystems. For sufficiently small separations (determined by\nthe contributing range of q), this oscillation is mild, and can\nbe used to change the polarity of the anisotropy field without\nextinguishing the response. This is why the polarity of the\nresponse component hxchanges between Figs. 5a and 5b.10\nIt is furthermore clear by inspection of Eq. (70) that the\nmain contributions to the anistotropy field come from states\nnear the Fermi surface. Series-expanding the expression\ninP, most terms are seen to cancel due the odd symme-\ntry in pthat was remarked below Eq. (70). The strongest\nasymmetry caused by Pis seen to originate from the factor\nsin [(pd0+Pd0)aSC]jupj2+ sin [(pd0\u0000Pd0)aSC]jvpj2in the\nsummand, due to the step-like nature of jupj2andjvpj2near\nthe Fermi surface. This is as expected, since we consider in-\nteractions involving the scattering of SC quasiparticles, hence\nthe low-energy events are concentrated near the Fermi surface.\n(a)\n(b)\nFIG. 5. The magnitude and direction (arrows) of the effective\nanisotropy field [Eq. (72)] at T= 1 K as a function of the super-\ncurrent momentum P, for the simple case of a small FI ( lFI\nx=lFI\ny=\n10µm) relative to the cavity ( Lx=Ly= 10 cm ,Lz= 0:1 mm ).\nThe SC dimensions are lSC\nx=lSC\ny= 50 µm, with a depth of\ndSC= 10 nm . The FI and SC center points are separated by\n(a)Lsep\ny= 140 µmand (b) nothing (placed directly over each\nother). Observe the change in both the strength and direction of the\nanisotropy field. The plots were produced using Python with the\nNumPy andMatplotlib libraries.TABLE I. Table of numerical parameter values.\nYIG (FI) Nb (SC)\naFI 1:240 nm [60] aSC 0:330 nm [61]\nTc0 6 K[49]\nt 0:35 eVa\nPc 3:1\u0002107m\u00001b\nEF 5:32 eVc[61]\naBased on the tight-binding expression t=~2=2ma2\nSC[14], with\nmthe effective electron mass.\nbBased onPc=jcm=~ens[46], with an estimated critical\ncurrentjc= 4 MA=cm2[62], and a superfluid density ns=\nm=\u0016 0e2\u00152[48] with a penetration depth \u0015= 200 nm [49].\ncFermi energy for Nb. Does not appear explicitly in Eq. (72), but is\nused in the electron dispersion.\nIV . CONCLUDING REMARKS\nIn this paper, we have calculated the cavity-mediated cou-\npling between an FI and an SC by exactly integrating out the\ncavity photons. The main result is the effective FI action (63),\nin which linear and bilinear magnon terms appear in addition\nto the diagonal terms. These respectively correspond to an\ninduced anisotropy field, and corrections to the magnon spec-\ntra. In contrast to conventional proximity systems, the cavity-\nmediation allows for relatively long-distance interactions be-\ntween the FI and the SC, without destructive effects on order\nparameters associated with proximity systems, such as pair-\nbreaking magnetic fields. The separation furthermore facili-\ntates subjection of the FI and the SC to separate drives and\ntemperatures. In contrast to common perturbative approaches\nto cavity-mediated interactions involving the Schrieffer–Wolff\ntransformation [9, 14, 21] or Jaynes–Cummings-like mod-\nels [12, 13, 39], the path-integral approach allows for an ex-\nact integrating-out of the cavity, without limitations to off-\nresonant regimes. This carries the additional advantage of\nallowing for magnon–photon hybridization; that is, we are\nnot theoretically limited to regimes of weak FI–cavity Zeeman\ncoupling. We furthermore take into account that the finite and\ndifferent FI, cavity and SC dimensions enable interactions be-\ntween large ranges of particle modes, which is neglected in\nvarious preceding works [9, 10, 13, 14, 22, 29, 30], although\nits importance has been emphasized by both experimentalists\n[30] and theorists [22].\nIn an arbitrary practical example, we estimate numerically\nthe effective anisotropy field induced by leading-order inter-\nactions across a small YIG film (FI) ( lFI\nx=lFI\ny= 10 µm)\ndue to mediated interactions with an Nb film (SC) ( lSC\nx=\nlSC\ny= 50 µm,dSC= 10 nm ). We find it is .14µT, medi-\nated across 130µmedge-to-edge accounting for both in-plane\nand out-of-plane separation, inside a 10 cm\u000210 cm\u00020:1 mm\ncavity (Fig. 5a). With out-of-plane coercivities in nm-thin Bi-\ndoped YIG films reportedly as low as 300µT[63], this result\nis expected to yield an experimentally appreciable tilt in the FI\nspins. The separation is 2–5 orders of magnitude greater than\nthe typical length scales of influence in proximity systems,\nand facilitates local subjection to different drives and temper-\natures. The main contributions from the SC originate from a11\nnarrow vicinity of the Fermi surface determined by the Cooper\npair center-of-mass momentum 2P. The response is very sen-\nsitive to the in-plane separation of the FI and SC center points\ndue to the spatial decoherence of the mediating cavity modes\nover distances, which in turn depends on the dimensions of\nthe FI, cavity and SC. For this reason, the in-plane separation\nof FI and SC was much smaller than the cavity width.\nIn Appendix B we have included the calculation of the\nanisotropy field when placing the SC at the magnetic antin-\node atz= 0. Since the vector potential is purely out of plane\nin this case, the paramagnetic coupling is zero, and we there-\nfore couple the cavity to the SC via the Zeeman coupling. As\nshown in the appendix, this results in a much weaker cou-\npling and therefore much smaller anisotropy field. This can\nbe understood by comparing the effective fields the SC cou-\nples to in the two cases. The strength of the Zeeman cou-\npling is proportional to q\u0002A, which for the lowest cavity\nmodes gives a field strength proportional to jAj=L. How-\never, for the paramagnetic coupling, the effective field is pro-\nportional to p\u0001A. In both cases, the main contribution to\nthe anisotropy field originates from a narrow vicinity of the\nFermi level, the extent of which is determined by the magni-\ntude of the symmetry-breaking supercurrent (electric antin-\node) or applied field (magnetic antinode). Thus, we have\na paramagnetic coupling proportional to pFjAj, wherepF\nis the Fermi momentum. A Fermi energy of 5:32 eV gives\npF\u00181010m\u00001\u001d1=Lfor cavities with lengths in the mm\ntocmrange. Together with the fact that the contributing com-\nponents of Aare larger for low jqjat the electric antinode\ncompared with the magnetic antinode, the difference in length\nscales leads to a much larger paramagnetic coupling between\ncavity and SC compared to the Zeeman coupling, resulting in\na much larger effective FI–SC coupling and anisotropy field.\nOne important constraint in our model that can potentially\nbe eased, is that the FI and the SC cannot overlap in-plane. In\nthis case, we found a stronger response (cf. Fig. 5b). This was\nassumed in order to enable the FI to be subjected to the align-\ning magnetostatic field Bextwithout affecting the SC, analo-\ngously to the experimental set-up in Refs. [12, 13]. Combined\nwith the eventually destructive contributions of various cav-\nity modes over finite in-plane distances that limited us to us-\ning only a fraction of the cavity width in our example, this\nleads to significant constraints on the dimensions and rela-\ntive placements of the FI and SC. However, Ref. [64] reports\nout-of-plane critical fields of nm-thin Nb films of roughly 1–\n4 T, while Ref. [63] reports out-of-plane coercivities in nm-\nthin Bi-doped YIG films of roughly 3\u000210\u00004T. An aligning\nfield can therefore be many orders of magnitude smaller than\nthe SC critical field with appropriate material choices. One\nwould then expect the effect of Bexton the SC to be negli-\ngible. However, we have not considered here the subsequent\neffect of the SC on the spatial distribution of Bext, which was\ntaken to be uniform across the FI.\nMoreover, the Pearl length criterion, which greatly lim-\nits SC dimensions, can potentially be disregarded if the odd\npsymmetry of the anisotropy field (64) is broken by other\nmeans than a supercurrent. A candidate for this is taking into\naccount spin–orbit coupling on the SC and subjecting it to aweak (non-pair breaking) magnetostatic field.\nFurthermore, in our set-up, we have considered coupling to\nthe quasiparticle excitations of the SC. This has partly been\nmotivated by the prospect of using the FI to probe detailed\nspin and momentum information about the SC gap, which\nwould require an extension of our present model. Another in-\nteresting avenue to explore is coupling directly to the gap by\nconsidering fluctuations from its mean-field value. This has\nbeen explored for an FI–SC bilayer, where the Higgs mode of\nthe SC couples linearly to a spin exchange field [65]. This has\na significant impact on the SC spin susceptibility in a bilayer\nset-up.\nDespite coupling to the quasiparticles, we find that the\nanisotropy field magnitude nearly constant at low tempera-\ntures, and rapidly decreases to zero near the critical tempera-\nture. This can be understood from the fact that the symmetry-\nbreaking supercurrent momentum enters the system Hamilto-\nnian via the gap (cf. Eq. (28b)). Hence, when the gap van-\nishes, so does the quantity that breaks the symmetry. On the\nother hand, for temperatures substantially below Tc0, the gap\nvaries little with temperature; the anisotropy field becomes\nclose to constant, with a magnitude depending on the momen-\ntum associated with the inversion symmetry-breaking current\nP.\nIn the normal state, the DC through the SC induces a sur-\nrounding magnetostatic field, by the Biot–Savart law. This\ndiffers from the response in the superconducting state by in-\nstead being appreciable above Tc0, and by its spatial distribu-\ntion; for instance, the magnetostatic field cannot reverse the\nfield direction as observed between Fig. 5a and 5b.\nLastly, it is seen from Eq. (64) that the SC quasiparticle\nmodes uniformly affect the anisotropy field in our current set-\nup, as the sum over fermion momenta pcan be factored out\nfrom the sum over photon momenta q. This limits the reso-\nlution of SC features in the anisotropy field, and by extension\nthe FI. However, to higher order in the calculations, the quan-\ntityGqq0\n&&0defined in Eq. (A6) enters, with sums over fermion\nmomenta pandp0that are inseparable from the cavity mo-\nmenta qandq0. This quantity is a candidate for extracting\nmore features of the SC via the FI.\nACKNOWLEDGMENTS\nWe acknowledge funding via the “Outstanding Academic\nFellows” programme at NTNU, the Research Council of Nor-\nway Grant number 302315, as well as through its Centres of\nExcellence funding scheme, project number 262633, “QuS-\npin”.\nAppendix A: Integrating out the SC first\nThe order in which we integrate out the cavity and the SC\nis inconsequential. We show this here by integrating out the\nSC first, starting from the partition function (49).12\nWe introduce the interaction matrix Gwith elements\nGpp0\nmm0\u00111p\fX\nq&gqpp0\n&mm0(aq&+ay\n\u0000q&); (A1)\nand furthermore the diagonal matrix Ewith elements\nEpp0\nmm0\u0011Epm\u000epp0\u000emm0: (A2)\nHence the action involving the SC can be written as\nSSC\n0+Scav\u0000SC\nint =X\npmX\np0m0(E+G)pp0\nmm0\ry\npm\rp0m0:(A3)\nThe part of the partition function (49) which depends on the\nSC is a Gaussian integral, and can now be written as [41]\nZSC\u0011Z\nD[\r;\ry] exp2\n4\u00001\n~X\npmX\np0m0(E+G)pp0\nmm0\ry\npm\rp0m03\n5\n\u0019exp\u0002\nTr\u0002\nE\u00001G\u0000E\u00001GE\u00001G=2\u0003\u0003\n:\n(A4)\nIn the last line, we neglected a factor exp Tr ln (\fE=~)that\nis constant with respect to the integration variables, and ex-\npanded another logarithm to second order in jE\u00001Gj. Hence,\nintegrating out the SC to second order in the cav–SC coupling\nyields an effective action\nScav\n1\u0011\u0000~Tr\u0002\nE\u00001G\u0000E\u00001GE\u00001G=2\u0003\n=\u0000~p\fX\nq&X\npmgqpp\n&mm\nEpm(aq&+ay\n\u0000q&)\n+X\nq&X\nq0&0Gqq0\n&&0(aq&+ay\n\u0000q&)(aq0&0+ay\n\u0000q0&0);(A5)\nwhere we introduced the coefficient\nGqq0\n&&0\u0011~\n2\fX\npmX\np0m0gqpp0\n&mm0gq0p0p\n&0m0m\nEpmEp0m0: (A6)We now proceed to isolate the photonic terms and integrate\nout the cavity, i.e., we will perform the integral\nZcav\u0011Z\nD[a;ay]e\u0000Scav=~; (A7)\nwhere the effective cavity action is\nScav\u0011Scav\n0+Scav\n1+SFI\u0000cav\nint: (A8)\nTo this end, we introduce the current operator\nJq&\u0011\u0000X\nkdGkq\nd&(\u0017d\u0011\u0000k+\u0017\u0003\nd\u0011y\nk) +sq&; (A9)\nand perform a shift of integration variables\naq&!aq&+J\u0000q&=~!q; (A10a)\nay\nq&!ay\nq&+Jq&=~!q: (A10b)\nThe quantities Gkq\nd&(to be distinguished from Gqq0\n&&0) andsq&\nare coefficients of linear photon terms to be determined.\nWe now require that the shifts (A10a)–(A10b) absorb the\nexplicit linear photon terms in the action (A8), leaving only\nbilinear and constant terms in the shifted variables. This leads\nto self-consistency equations for Gkq\nd&andsq&. However, to\nsecond order injE\u00001Gj, it can be shown that only the lowest-\norder expressions for Gkq\nd&andsq&affect the anisotropy field\nto be extracted at the end, cf. Sec. III. These are\nGkq\nd&=gkq\nd&; (A11)\nsq&=~p\fX\npmgqpp\n&mm\nEpm: (A12)\nHence, the action (A8) can be written as\nScav=Scav\nbil+Scav\ncon (A13)\nwhere\nScav\nbil\u0011X\nq&~!qay\nq&aq&+X\nq&X\nq0&0Gqq0\n&&0(aq&+ay\n\u0000q&)(aq0&0+ay\n\u0000q0&0);(A14)\nScav\ncon\u0011X\nq&Jq&J\u0000q&\n~!q+X\nq&X\nq0&0Gqq0\n&&0J\u0000q&J\u0000q0&0\u00141\n~!q+1\n~!\u0000q\u0015\u00141\n~!q0+1\n~!\u0000q0\u0015\n: (A15)\nScav\nbilcontains all bilinear terms with respect to the shifted vari-\nables, andScav\nconall constant terms.\nReturning to the integral (A7), by Eq. (A13), we now have\nZcav=Z\nD[a;ay]e\u0000Scav=~=e\u0000Scav\ncon=~Z\nD[a;ay]e\u0000Scav\nbil=~:\n(A16)The integrand is now independent of magnons, and therefore\ninconsequential to the physics of the ferromagnetic insulator.\nWe can therefore neglect the integral, leaving only the expo-\nnential prefactor. We are thus left with an effective FI partition13\nfunction\nZFI\u0011Z\nD[\u0011;\u0011y]e\u0000SFI=~; (A17)\nwhere the effective FI action is\nSFI\u0011SFI\n0+Scav\ncon: (A18)Neglecting magnon-independent terms, SFIreads, after some\nrewriting,\nSFI=X\nk~\u0015k\u0011y\nk\u0011k+X\nkdX\nk0d0Qkk0\ndd0(\u0017d\u0011\u0000k+\u0017\u0003\nd\u0011y\nk)(\u0017d0\u0011\u0000k0+\u0017\u0003\nd0\u0011y\nk0)\u0000g\u0016BX\nkdhk\nd\u0001r\nS\n2(\u0017d\u0011\u0000k+\u0017\u0003\nd\u0011y\nk): (A19)\nAbove, we introduced\nQkk0\ndd0\u0011\u0000X\nq&2\n4gkq\nd&gk0\u0000q\nd0&\n~!q+X\nq0&0Gqq0\n&&0\u00141\n~!q+1\n~!\u0000q\u0015\u00141\n~!q0+1\n~!\u0000q0\u0015\ngkq\nd&gk0q0\nd0&03\n5; (A20)\nhk\nd=\u0000~\ng\u0016Br2\nS\fX\npmVkpp\ndmm\nEpm; (A21)\nwhich to leading order in the paramagnetic coupling are indeed the same as Eqs. (64) and (65).\nAppendix B: SC at magnetic antinode\nFIG. 6. Illustration of the set-up with the SC placed at the mag-\nnetic antinode. The SC is subjected to an aligning external in-plane\nmagnetic field BSC\next. This set-up is otherwise identical to the one\nillustrated in Fig. 1.\nTo compare our results for the FI-SC coupling with the\nSC placed at the electric antinode, we examine what happens\nwhen we place the superconductor at a magnetic maximum at\nz\u00190, cf. Fig. 6. In this case the vector potential Apoints\npurely in the zdirection, and therefore does not couple to the\nSC via the paramagnetic coupling term used above. We there-\nfore couple the SC to the cavity via the Zeeman coupling, and\ncalculate the resulting anisotropy field across the FI. For the\nsetup considered in the main text, it was necessary to break the\ninversion symmetry to get a finite anisotropy field, achieved,\nfor instance, by applying a DC current. For the present setup,\nit is necessary to break the in-plane spin rotation symmetry,\nwhich can be achieved by applying an in-plane magnetic field\nto the SC. This becomes evident when considering the cou-\npling between the cavity and SC. Placing the SC at z\u00190, thecavity magnetic field is purely in-plane, pointing in the oppo-\nsite direction to the field at z=Lz[Eq. (23)], resulting in a\ncoupling term,\nSZeeman =X\nqpp0X\n\u001b\u001b0gqpp0\n\u001b\u001b0(aq1+ay\n\u0000q1)cy\np\u001bcp0\u001b0; (B1)\nwith interaction matrix\ngqpp0\n\u001b\u001b0=\u000e\nn;!n\u0000!0n\n\u0002s\n~\u00162\nB\n\u000f!qVDSC\np\u0000p0;qeiq\u0001rSC\n0isin\u0012q(\u001b\u0002q)\u001b\u001b0\u0001^ez:\n(B2)\nThis interaction alone would lead to a SC-cavity coupling that\nis off-diagonal in quasiparticle basis. The anisotropy field,\ncorresponding to the diagram for Sintin Fig. 3 with connected\nquasiparticle lines will therefore be exactly zero unless one\nbreaks the spin-rotation symmetry by an in-plane magneto-\nstatic field BSC\next. The latter can for example be experimen-\ntally realized using external coils, as suggested for Bext. In\nthat case the quasiparticle bands are spin-split, resulting in the\nSC term\nSSC\n0=X\npn(\u0000i~!n+Epn)\ry\npn\rpn; (B3)\nwith the four quasiparticle bands\nEpn= (\u00001)bn=2cEp+ (\u00001)nH; (B4)\nwithEp=q\n\u00182p+j\u0001pj2,n2[0;1;2;3]andH=j\u0016BBSC\nextj.\nThe bands are independent of in-plane direction of the field14\nBSC\next, with the directional dependence entering through the\ncoupling between the quasiparticles and the cavity photons,\nSSC\u0000cav\nint =1\n2p\fX\nqppX\nnn0gqpp0\nnn0(aq1+ay\n\u0000q1)\ry\npn\rp0n0;(B5)where we have defined the interaction matrix in the Bogoli-\nubov quasiparticle basis\ngqpp0\nnn0=\u00001\n2gqpp0\n\"#ei\u001e \n[uy\npup0+vpvy\np0][\u001bz+i\u001by] [uy\npvp0\u0000vpuy\np0][\u001b0\u0000\u001bx]\n[vy\npup0\u0000upvy\np0][\u001b0+\u001bx] [vy\npvp0+upuy\np0][\u001bz\u0000i\u001by]!\nnn0\n\u00001\n2gqpp0\n#\"e\u0000i\u001e \n[uy\npup0+vpvy\np0][\u001bz\u0000i\u001by] [uy\npvp0\u0000vpuy\np0][\u001b0+\u001bx]\n[vy\npup0\u0000upvy\np0][\u001b0\u0000\u001bx] [vy\npvp0+upuy\np0][\u001bz+i\u001by]!\nnn0; (B6)\nwhere\u001b0is the 2\u00022identity matrix, and \u001eis the angle of the\nin-plane field relative to the xaxis. We have also defined the\nfunctions\nup=ei\u0012ps\n1\n2\u0012\n1 +\u0018p\nEp\u0013\n; (B7a)\nvp=ei\u0012ps\n1\n2\u0012\n1\u0000\u0018p\nEp\u0013\n; (B7b)\nwhich satisfyjupj2+jv2\npj= 1. Here 2\u0012pis the phase of the\norder parameter.\nFollowing the same procedure of integrating out the cav-\nity photons and quasiparticles in the SC, we get an expres-\nsion identical to Eq. (63), with the only change coming in the\nanisotropy field, which is now defined as\nhk\nd\u0011\u0000~p2S\fg\u0016BX\npnVkpp\ndnn\nEpn; (B8)with\nVkpp0\ndnn0=X\nqgkq\nd1g\u0000qpp0\nnn0\u00141\n~!q+1\n~!\u0000q\u0015\n: (B9)\nThe additional factor of 1=2in the definition of hk\ndis due to the\nfield integral resulting in the Pfaffian of the antisymmetrized\nGreen’s function in this case, which is the square root of the\ndeterminant [66]. The reason for this is the necessity of an ex-\npanded Nambu spinor, which contains both creation and anni-\nhilation operators of both types of quasiparticles when includ-\ning an in-plane field [67].\nInserting Eqs. (B6) and (B9) into Eq. (B8) and performing\nthe sum over fermionic Matsubara frequencies [41], we get\nhk\nd=p\f\u000e\nm0p\n2Sg\u0016BX\nqpgkq\nd\n~!q[g\u0000qpp\n\"#ei\u001e+g\u0000qpp\n#\"e\u0000i\u001e]\n\u0002\u0014\ntanh\f(Ep+H)\n2~\u0000tanh\f(Ep\u0000H)\n2~\u0015\n;(B10)\nwhere we have used the fact that !qis even in q. Here it is\nclear that the anisotropy field is exactly zero when the in-plane\nfield is zero, since the last two terms exactly cancel in that\ncase. Moreover, since the anisotropy field is independent of\nthe frequency \nm, we define the time-independent anisotropy\nfieldhk\nd=P\n\nmhk\nde\u0000i\nm\u001c=p\f. Inserting the expressions\nforgkq\ndandg\u0000qpp\n\u001b\u001b0from Eqs. (25) and (B2) we get\nhk\nd=\u0000\u0016BpNFI\n\u000fVX\nqeiq\u0001(rFI\n0\u0000rSC\n0)DFI\nkqDSC\u0003\n0qsin2\u0012q\n!2qq\u0016d\u00172\nd[qycos\u001e\u0000qxsin\u001e]X\np\u0014\ntanh\f(Ep+H)\n2~\u0000tanh\f(Ep\u0000H)\n2~\u0015\n:\n(B11)\nWe focus on the anisotropy field averaged across the FI, hhdi=P\nihd(ri;\u001c)=NFI=P\niP\nkhk\ndeik\u0001ri=N3=2\nFI=h0\nd=pNFI\n(cf. Eq. (67)), rewrite the first sum such that it becomes dimensionless, and transform the second sum into an integral using a15\nfree electron gas dispersion \u0018k=~2p2=2m\u0000\u0016. Assuming cavity dimensions Lx=Ly=Land ans-wave gap, we get\nhhdi=\u0000\u0016BVSC(m\u00010)3=2\np\n2\u00192~3\u000fc2VX\nqeiq\u0001(rFI\n0\u0000rSC\n0)DFI\n0qDSC\u0003\n0q`\u0016d\u00172\nd[`ycos\u001e\u0000`xsin\u001e][`2\nx+`2\ny]\n\u0014\n`2x+`2y+\u0010\nL\n2Lz\u00112\u00152\n\u0002\u0018max=\u00010Z\n\u0000\u0016=\u00010dxr\nx+\u0016\n\u00010\u0014\ntanh1:764Tc\u0010p\nx2+j\u0001=\u00010j2+H=\u00010\u0011\n2T\u0000tanh1:764Tc\u0010p\nx2+j\u0001=\u00010j2\u0000H=\u00010\u0011\n2T\u0015\n:\n(B12)\nHereVSCand\u00010are the volume and zero temperature gap of\nthe superconductor, respectively, and mthe electron mass. `x\nand`yare integer indexes corresponding to cavity momen-\ntumq. From the above expression we expect terms even\nin`dto dominate, resulting in the anisotropy field and ex-\npectation values of the in-plane spin components to have a\n\u001edependence given by hk\nx\u0018 hSixi / \u0000 cos\u001eandhk\ny\u0018\nhSiyi / \u0000 sin\u001e. This is in good agreement with numer-\nical solutions of Eq. (B12) in an arbitrary practical exam-\nple, as shown in Fig. 7. The results were obtained using the\nPython libraries NumPy andMatplotlib , and sub-package\nscipy.integrate . Notice, however, that the magnitude\nof the anisotropy field is very small, on the order of 10\u00009T.\nThis is several orders of magnitude smaller than the previously\nconsidered setup, and we do not expect this to be a measurable\neffect. Here we have neglected the effect of an in-plane finite\nseparation between the SC and FI by placing them directly\nabove each other. A finite separation would further reduce the\nanisotropy field.\nAt zero temperature the two hyperbolic tangent functions\nin Eq. (B12) are always equal to one, as long as H < \u00010.\nSince the field must be below the critical field Hc0= \u0001 0=p\n2\nin the superconducting state, the two terms in the integral al-\nways cancel exactly at zero temperature. On the other hand,\nin the case of temperatures just above the critical tempera-\nture,T&Tc, and\u0016;\u0018max> H , we get the analytical re-\nsult4Hp\u0016=\u00013=2\n0for the integral, assuming that the main\ncontribution to the integral comes from energies close to the\nFermi level. Hence we expect the anisotropy field to in-\ncrease from zero to the normal state value as temperature in-\ncreases towards Tc, and thathhdiincreases linearly with ap-\nplied field in the normal state. This is found to be in good\nagreement with numerical results, see the inset in Fig. 7 for\njHj> Hc. In the numerical calculations we have assumed\n\u0016;\u0018max\u001d\u00010, and that the gap’s dependence on temper-\nature and applied field is described by Eq. (73) multiplied\nwithp\n1\u0000(H=Hc)2[59, 68], and the critical field depends\non temperature as Hc=Hc0[1\u0000(T=Tc0)2][48], where Tc0\nis the critical temperature for zero field. Below the critical\ntemperature and field, the field-dependence of the anisotropy\nfield is more complicated due to the additional effect of re-\nducing the superconducting gap, see inset in Fig. 7. The dif-\nference in temperature and applied field-dependence of the\nanisotropy field between the normal and superconducting state\n-10 -5 0 5 10\nHx[T]-10-50510Hy[T]\n0.000.250.500.751.001.251.50\n|⟨h⟩|[T]×10−9\n1010−9HcFIG. 7. Absolute value (contour plot) and direction (arrows) of the\naveraged anisotropy field as a function of applied field strength and\ndirection. The anisotropy field points opposite the applied field over\nthe SC, following a cos\u001eandsin\u001edependence for the xandycom-\nponent respectively. The inset shows the absolute value of the in-\nplane projection as a function of the field strength. The temperature\nis set toT= 0:5Tc0. The cavity dimensions are Lx=Ly=L=\n10 cm andLz= 1 mm , and the FI and SC have sides of length\n0:001Lin thexandydirections, and are placed at the center of the\ncavity. The thickness of the SC is dSC= 10 nm .\ncould therefore in principle be a way of detecting the onset\nof superconductivity without directly probing the supercon-\nductor, though the anisotropy field calculated in this arbitrary\nexample is too small to be detectable.\nAppendix C: Linear terms as an anisotropy field\nIn this appendix, we take a closer look at the interpretation\nof the linear magnon terms as interactions with an effective\nanisotropy field. Consider an FI in an inhomogeneous applied\nfield,\nH=\u0000JX\nhi;jiSi\u0001Sj\u0000X\niHi\u0001Si: (C1)\nAbove, Hi= (Hx\ni;Hy\ni;Hz)is the inhomogeneous external\nfield, withHzassumed homogeneous and much larger than16\nHx\ni;Hy\ni. We therefore assume ordering in the zdirection\nwhen performing the Holstein–Primakoff transformation, re-\nsulting in the Fourier-transformed Hamiltonian\nH=E0+X\nkh\n~\u0015k\u0011y\nk\u0011k\u0000hk\u0011y\nk\u0000h\u0003\nk\u0011ki\n: (C2)\nHere~\u0015kis the dispersion defined in Eq. (21), the classical\nground state energy is\nE0=\u0000~SNFI[J~SN\u000e+Hz]; (C3)\nand the momentum-dependent in-plane magnetic energy\nhk=r\nS\n2NFI~X\ni(Hx\ni+iHy\ni)e\u0000ik\u0001ri: (C4)\nSince the applied field has in-plane components, the zdi-\nrection is not the exact ordering direction in the ground state,\nleading to a non-diagonal Hamiltonian with linear terms. To\nget rid of these terms, we translate the fields according to\n\u0011k!\u0011k+tk;\n\u0011y\nk!\u0011y\nk+t\u0003\nk;(C5)\nand require that linear terms cancel. Translating the fields\nleads to the Hamiltonian\nH!E0+X\nkn\n~\u0015k\u0011y\nk\u0011k+ [~\u0015ktk\u0000hk]\u0011y\nk\n+ [~\u0015kt\u0003\nk\u0000h\u0003\nk]\u0011k+~\u0015kt\u0003\nktk\u0000hkt\u0003\nk\u0000h\u0003\nktko\n;\n(C6)and we therefore require\ntk=hk\n~\u0015k: (C7)\nThe resulting diagonal Hamiltonian is\nH=E0+X\nk[~\u0015k\u0011y\nk\u0011k\u0000~\u0015kt\u0003\nktk]: (C8)\nThe last term in the above equation results in a renormal-\nization of the classical ground state,\nE0!E0\u0000X\nk~\u0015kt\u0003\nktk\n=E0\u0000X\ni;j;kS~2(Hx\ni+iHy\ni)(Hx\nj\u0000iHy\nj)2eik\u0001(rj\u0000ri)\n2NFI~\u0015k:\n(C9)\nIn the case of constant in-plane components, this simplifies to\nE0=\u0000~SNFI\u001a\nJ~SN\u000e+\u0014\nHz+(Hx)2+ (Hy)2\n2Hz\u0015\u001b\n\u0019 \u0000~SNFI[J~SN\u000e+jHj]; (C10)\nwhere the approximation in the last line is valid in the limit\njHxj;jHyj \u001c jHzj. This is as expected, since the classi-\ncal ground state is generally oriented along H, notHz. 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Douglass, Physical Review Letters 6, 346 (1961)."    },    {        "title": "1905.12523v3.Coherent_long_range_transfer_of_angular_momentum_between_magnon_Kittel_modes_by_phonons.pdf",        "content": "Coherent long-range transfer of angular momentum between\nmagnon Kittel modes by phonons\nK. An,1A.N. Litvinenko,1R. Kohno,1A.A. Fuad,1V. V. Naletov,1, 2L. Vila,1\nU. Ebels,1G. de Loubens,3H. Hurdequint,3N. Beaulieu,4J. Ben Youssef,4N.\nVukadinovic,5G.E.W. Bauer,6A. N. Slavin,7V. S. Tiberkevich,7and O. Klein1,\u0003\n1Univ. Grenoble Alpes, CEA, CNRS,\nGrenoble INP, Spintec, 38054 Grenoble, France\n2Institute of Physics, Kazan Federal University, Kazan 420008, Russian Federation\n3SPEC, CEA-Saclay, CNRS, Universit\u0013 e Paris-Saclay, 91191 Gif-sur-Yvette, France\n4LabSTICC, CNRS, Universit\u0013 e de Bretagne Occidentale, 29238 Brest, France\n5Dassault Aviation, Saint-Cloud 92552, France\n6Institute for Materials Research and WPI-AIMR and CSRN,\nTohoku University, Sendai 980-8577, Japan\n7Department of Physics, Oakland University, Michigan 48309, USA\n(Dated: March 13, 2020)\nAbstract\nWe report ferromagnetic resonance in the normal con\fguration of an electrically insulating mag-\nnetic bi-layer consisting of two yttrium iron garnet (YIG) \flms epitaxially grown on both sides of\na 0.5 mm thick non-magnetic gadolinium gallium garnet (GGG) slab. An interference pattern is\nobserved and it is explained as the strong coupling of the magnetization dynamics of the two YIG\nlayers either in-phase or out-of-phase by the standing transverse sound waves, which are excited\nthrough the magneto-elastic interaction. This coherent mediation of angular momentum by circu-\nlarly polarized phonons through a non-magnetic material over macroscopic distances can be useful\nfor future information technologies.\n1arXiv:1905.12523v3  [cond-mat.mes-hall]  12 Mar 2020The renewed interest in using acoustic oscillators as coherent signal transducers [1{3]\nstems from the extreme \fnesse of acoustic signal transmission lines. The low sound atten-\nuation factor \u0011abene\fts the interconversion process into other wave forms (with damping\n\u0011s) as measured by the cooperativity, C= \n2=(2\u0011a\u0011s) [4, 5], leading to strong coupling as\nde\fned by C>1 even when the coupling strength \n is small. Here we present experimen-\ntal evidence for coherent long-distance transport of angular momentum via the coupling to\ncircularly polarized sound waves that exceeds previous benchmarks set by magnon di\u000busion\n[6{8] by orders of magnitude.\nThe material of choice for magnonics is yttrium iron garnet (YIG) with the lowest mag-\nnetic damping reported so far [9, 10]. The ultrasonic attenuation coe\u000ecient in garnets is\nalso exceptional, i.e.up to an order of magnitude lower than that in single crystalline quartz\n[11, 12]. Spin-waves (magnons) hybridize with lattice vibrations (phonons) by the magnetic\nanisotropy and strain dependence of the magneto-crystalline energy [13{18]. Although often\nweak in absolute terms, the magneto-elasticity leads to new hybrid quasiparticles (\\magnon\npolarons\") when spin-wave (SW) and acoustic-wave (AW) dispersions (anti)cross [19{21].\nThis coupling has been exploited in the past to produce microwave acoustic transducers\n[22, 23], parametric acoustic oscillators [24] or nonreciprocal acoustic wave rotation [25, 26].\nRecent studies have identi\fed their bene\fcial e\u000bects on spin transport in thin YIG \flms by\npump-and-probe Kerr microscopy [27, 28] and in the spin Seebeck e\u000bect [29]. The adiabatic\nconversion between magnons and phonons in magnetic \feld gradients proves their strong\ncoupling in YIG [30].\nBut phonons excited by magnetization dynamics can also transfer their angular momen-\ntum into an adjacent non-magnetic dielectrics [32, 33]. When the latter acts as a phonon\nsink, the \\phonon pumping\" increases the magnetic damping [34]. The substrate of choice for\nYIG is single crystal gadolinium gallium garnet (GGG) which in itself has very long phonon\nmean-free path [35, 36] and small impedance mismatch with YIG [37], raising the hope of a\nphonon-mediated dynamic exchange of coherence through a non-magnetic insulating layer\n[34].\nHere we report ferromagnetic resonance experiments (FMR) of a \\dielectric spin-valve\"\nstack consisting of half a millimeter thick single-crystal GGG slab coated on both sides\nby thin YIG \flms. We demonstrate that GGG can be an excellent conductor of phononic\nangular momentum currents allowing the coherent coupling between the two magnets over\n2FIG. 1. (Color online) a) Schematic and picture of the ferromagnetic resonance (FMR) setup. A\nbutter\ry shaped stripline resonator [31] with 0.3 mm wide constriction is in contact with the bottom\nlayer of the YIG1( d= 200 nm)jGGG(s= 0:5 mm)jYIG2(d= 200 nm) \\dielectric spin-valve\" stack.\nThe microwave antenna can be tuned in or out of its fundamental resonance (5 :11 GHz) as shown\nin the re\rectivity spectrum b). c) Schematic of the coupling between the top (red) and bottom\n(blue) YIG layers by the exchange of coherent phonons: the magnetic precession m+generates\na circular shear deformation u+of the lattice that can be tuned into a coherent motion of all\n\felds. Constructive/destructive interference between the dynamics of the two YIG layers occurs\nfor even/odd mode numbers ncausing d) a contrast \u0001 in the absorbed microwave power ( Pabs)\nbetween tones separated by half a phonon wavelength. e) Density plot of the spectral modulation of\nPabsproduced by Eq.(1) when magnetic bi-layers are strongly coupled to coherent phonon modes.\nThe orange/green dots indicate the spectral position of the even/odd acoustic resonances.\nmillimeter distance. Figure 1a illustrates the experimental setup in which an inductive\nantenna monitors the coherent part of the magnetization dynamics. The spectroscopic\nsignature of the dynamic coupling between the two YIG layers is a resonant contrast pattern\nas a function of microwave frequency (see intensity modulation along FMR2 in Figure 1e).\nBefore turning to the experimental details, we sketch a simple phenomenological model\nthat captures the dynamics of the \felds as described by the continuum model for magneto-\nelasticity with proper boundary conditions [34]. The perpendicular dynamics of a trilayer\n3with in-plane translational symmetry can be mapped on three coupled harmonic oscillators,\nviz. the Kittel modes of the two magnetic layers mi=1;2and then-th mechanical mode, un,\nin the dielectric, which obey the coupled set of equations\n(!s\u0000!1+j\u0011s)m+\n1= \n 1u+\nn=2 +\u00141h+(1a)\n(!s\u0000!2+j\u0011s)m+\n2= \n 2u+\nn=2 +\u00142h+(1b)\n(!s\u0000!n+j\u0011a)u+\nn= \n 1m+\n1=2 + \n 2m+\n2=2 (1c)\nHere!n=(2\u0019) =v=\u0015n, wherevis the AW velocity and \u0015n=2 = (2d+s)=nis a half wavelength\nthat \fts into the total sample thickness 2 d+s, withnbeing an integer (mode number). The\ndynamic quantities m+\ni= (mx+jmy)iare circularly polarized magnetic complex amplitudes\n(jbeing the imaginary unit) precessing anti-clockwise around the equilibrium magnetization\nat Kittel resonance frequencies !16=!2. In our notation \u0011s=aare the magnetic/acoustic\nrelaxation rates [38] and the constants \n iand\u0014iare the magneto-elastic interaction and\ninductive coupling to the antenna, respectively. Coherence e\u000bects between m1andm2can\nbe monitored by the power Pabs=\u0014iIm(h?mi) as a function of the microwave frequency\n!sof the driving \feld with circular amplitude h+[39]. Note that Eq.(1) holds when the\ncharacteristic AW decay length exceeds the \flm thickness (see below).\nThe acoustic modes with odd and even symmetry couple with opposite signs, i.e. \n 2=\n(\u00001)n\n1(see Figure 1c), which a\u000bects the dynamics as sketched in Figure 1d. When nis odd\n(even), the top layer returns (absorbs) the power from the electromagnetic \feld, because the\nphonon amplitude is out-of(in) phase with the direct excitation, corresponding to destructive\n(constructive) interference. In other words, the phonons pumped by the dynamics of the\nlayer 1 are re\rected vs. absorbed by layer 2. According to Eq.(1), a contrast \u0001 should emerge\nbetween tones separated by half a wavelength. This is illustrated in Figure 1e by plotting\nthe calculated modulation of the magnetic absorption when two Kittel modes with slightly\ndi\u000berent resonance frequencies and di\u000berent inductive coupling to the antenna interact via\nstrong coupling to coherent phonons (see below values in Table.(I)). In the \fgure the e\u000bect is\nmore visible around the resonance of the layer with weaker coupling \u00142<\u0014 1to the antenna\n(FMR2, red dashed line) since, according to the model, the amplitude of the contrast is\nproportional to the amplitude ratio of the microwave magnetic \felds felt by the two YIG\nlayers: \u0001/\u00141=\u00142. We employ here a stripline with width (0 :3 mm) that couples strongly\nto the lower layer YIG1, while still allowing to monitor the FMR absorption of YIG2.[40]\n4FIG. 2. (Color online)a) Microwave absorption spectra of a YIG(200 nm) jGGG(0.5 mm) crystal,\nrevealing a periodic modulation of the intensity interpreted as the avoided-crossing between the\nFMR mode (see blue arrow) at !1=\r\u00160(H0\u0000M1), and thenthstanding (shear) AW nresonances\nacross the total thickness (horizontal dash lines in orange and green) at !n=n\u0019v= (d+s) . The\nright panels (b,c,d) show the intensity modulation for 3 di\u000berent cuts (blue, magenta and red)\nalong the gyromagnetic ratio ( i.e.parallel to the resonance condition). The solid lines in the 4\npanels are \fts by the oscillator model (cf. Eq.(1) with \ft values in Table.(I)).\nFigure 1a is a picture of the bow-tie \u0015=2-resonator (with re\rectivity spectrum shown in\nFigure 1b) with which we perform spectroscopy around 5 GHz. The later ful\flls the \\half-\nwave condition\" of the phonon relative to the YIG thickness that maximizes the phonon\npumping [34]. The sample was grown by liquid phase epitaxy, i.e. by immersing a GGG\nmonocrystal substrate with thickness s= 0:5 mm and orientation (111) into molten YIG.\nThe concomitant growth leads to nominally identical YIG layers, with thickness d= 200 nm\non both sides of the GGG. The Gilbert damping parameter \u000b\u00199\u000210\u00005, measured as the\nslope of the frequency dependence of the line width, is evidence for the high crystal quality.\nAll experiments have been carried out at room temperature and on the same sample. Because\nof that, the results shall be presented in inverse chronological order.\nHaving removed YIG2 by mechanical polishing, we \frst concentrate on the dynamic\nbehavior of a single magnetic layer. Figure 2a shows the FMR absorption of YIG1 jGGG\nbilayer [41{44] around 5 :56 GHz i.e.for a detuned antenna having weak inductive coupling.\n5These spectra are acquired in the perpendicular con\fguration, where the magnetic precession\nis circular, by magnetizing the sample with a su\u000eciently strong external magnetic \feld, H0,\napplied along the normal of the \flms. Figure 2a provides a detailed view of the \fne structure\nwithin the FMR absorption that is obtained when one sweeps the \feld/frequency in tiny\nsteps of 0.01 mT/0.1 MHz, respectively.\nThe FMR mode (see arrow) follows the Kittel equation !1\u0019\r\u00160(H0\u0000M1) [45], with\n\r=(2\u0019) = 28:5 GHz/T, the gyromagnetic ratio and \u00160M1= 0:1720 T, the saturation magne-\ntization, but its intensity vs. frequency is periodically modulated [42, 46] which we explain\nby the hybridization with the comb of standing shear AWs described by Eq.(1) truncated\nto one magnetic layer.\nWe ascribe the periodicity of 3.50 MHz in the signal of Figures 2 to the equidistant\nsplitting of standing phonon modes governed by the transverse sound velocity of GGG along\n(111) ofv= 3:53\u0002103m/s [42, 46, 47] via v=(2d+ 2s)\u00193:53 MHz [48]. This value thus\nseparates two phononic tones, which di\u000ber by half a wavelength. At 5.5 GHz, the intercept\nbetween the transverse AW and SW dispersion relations occurs at 2 \u0019=\u0015 n=!s=v\u0019105cm\u00001,\nwhich corresponds to a phonon wavelength of about \u0015n\u0019700 nm with index number\nn\u00181400. The modulation is strong evidence for the high acoustic quality that allows elastic\nwaves to propagate coherently with a decay length exceeding twice the \flm thickness, i.e.\n1 mm. For later reference we point out that the absorption is the same for odd and even\nphonon modes, whose eigen-values are indicated here by green and orange dots.\nIn Figures 2bcd we focus on the line shapes at detunings parallel to the FMR resonance\nas a function of \feld and frequency indicated by the blue, magenta, and red cuts in Fig-\nure 2a. The amplitude of the main resonance (blue line) in Figure 2b dips and the lines\nbroaden at the phonon frequencies [42, 46]. The minima transform via dispersive-looking\nsignal (magenta in 2ac) into peaks (red 2ad) once su\u000eciently far from the Kittel resonance\nas expected from the complex impedance of two detuned resonant circuits, illustrating a\nconstant phase between miandunalong these cuts. The miare circularly polarized \felds\nrotating in the gyromagnetic direction, that interact only with acoustic waves unwith the\nsame polarity, as implemented in Eq. (1) [30].\nThe observed line shapes can be used to extract the lifetime parameters in Eq. (1).\nWe \frst concentrate on the observed 0 :7 MHz full line width of the acoustic resonances in\nFigure 2d. Far from the Kittel condition, the absorbed power is governed by the sound\n6attenuation. According to Eq. (1), the absorbed power at large detuning reduces to Pabs/\n((!s\u0000!n)2+\u00112\na)\u00001. The AW decay rate \u0011a=(2\u0019) = 0:35 MHz is obtained as the half line width\nof the acoustic resonance, leading to a characteristic decay length \u000e=v=\u0011a\u00192 mm for AW\nexcited around 5.5 GHz. The acoustic amplitude therefore decays by \u001820% over the half\nmillimeter \flm thickness. The sound amplitude in both magnetic layers are therefore roughly\nthe same, as assumed in Eq.(1). This \fgure is consistent with the measured ultrasonic\nattenuation in GGG: 0.70 dB/ \u0016s at 1GHz [36, 49], i.e., a lifetime of about 0.5 \u0016s at 5GHz.\nThe SW lifetime 1 =\u0011sfollows from the broadening of the absorbed power at the Kittel\ncondition which contains a constant inhomogeneous contribution and a frequency-dependent\nviscous damping term. When plotted as function of frequency, the former is the extrapola-\ntion of the line widths to zero frequency, in our case \u00185.7 MHz (or 0.2 mT). On the other\nhand, the Gilbert phenomenology (see above) of the homogeneous broadening \u0011s=\u000b!scor-\nresponds to a \u0011s=(2\u0019) = 0:50 MHz at 5.5 GHz. The dominantly inhomogeneous broadening\nis here caused by thickness variations, a spatially dependent magnetic anisotropy, but also\nby the inhomogeneous microwave \feld.\nConspicuous features in Figure 2a are the clearly resolved avoided-crossing of SW and\nAW dispersion relations, which prove the strong coupling between two oscillators. Fitting\nby hand the dispersions of two coupled oscillators through the data points (white lines),\nwe extract a gap of \n =(2\u0019) = 1 MHz and a large cooperativity C\u00193. From the overlap\nintegral between a standing shear AW con\fned in a layer of thickness \u0018sand the Kittel\nmode con\fned in a layer of thickness d, one can derive the analytical expression for the\nmagneto-elastic coupling strength [42, 50]:\n\n =Bp\n2r\r\n!sM1\u001asd\u0012\n1\u0000cos!sd\nv\u0013\n(2)\nwhere [35]B= (B2+ 2B1)=3 = 7\u0002105J/m3, withB1andB2being the magneto-elastic\ncoupling constants for a cubic crystal, and \u001a= 5:1 g/cm3is the mass density of YIG.\nFrom Eq.(2) we infer that coherent SW excited around !s=(2\u0019)\u00195:5 GHz have a dynamic\ncoupling to shear AW of the order of \n =(2\u0019) = 1:5 MHz, close to the value extracted from\nthe experiments.\nThe material parameters extracted for our YIG jGGG are summarized in Table (I). Nu-\nmerical solutions of Eq. (1) using these values are shown as solid lines in Figure 2bcd. The\nagreement with the data is excellent, con\frming the validity of the model and parameters.\n7TABLE I. Material parameters used in the oscillator model (all values are expressed in units of\n2\u0019\u0002106rad/s).\n!1\u0000!2 !n+1\u0000!n \n \u0011s \u0011a\n40 3.50 1.0 0.50 0.35\nFIG. 3. (Color online) FMR spectroscopy of the YIG1 jGGGjYIG2 trilayer. Panel a) is a transpar-\nent overlay of magnetic \feld sweeps for frequencies in the interval 5 :101\u00060:008 GHz by 0.1 MHz\nsteps. Dark lines reveal two acoustic resonances marked by orange and green dots. Panel b) and\nc) show the frequency modulation of the FMR amplitude for respectively the bottom YIG1 layer\nand the top YIG2 layer, in which a contrast \u0001 appears between neighboring acoustic resonances.\nThe solid lines show the modulation predicted by Eq.(1).\nThe other needed parameter for solving Eq.(1) in the general case is the attenuation ratio\n\u00142=\u00141\u00197 deducted from a factor of 50 decreased power when \ripping the single YIG layer\nsample upside down on the antenna. The layer is then separated 0.5 mm from the antenna,\nand the observed reduction agrees with numerical simulations using electromagnetic \feld\nsolvers.\nWe turn now our attention to the magnetic sandwich in which YIG1 touches the antenna\nand the nominally identical YIG2 is 0.5 mm away, where a slight di\u000berence in uniaxial\nanisotropy causes separate resonance frequencies. Since we want to detect also the resonance\n8of the top layer, we have to compensate for the decrease in inductive coupling by tuning\nthe source frequency to the antenna resonance at 5.11 GHz (see Figure 1b). This enhances\nthe signal by the quality factor Q\u001830 of the cavity at the cost of an increased radiative\ndamping of the bottom layer signal [51].\nFigure 3a is a transparent overlay of \feld sweeps for frequency steps of 0.1 MHz in the\ninterval 5:101\u00060:008 GHz. We attribute the two peaks separated by 1.4 mT (or 40 MHz)\nto the bottom and top YIG Kittel resonances, the later shifted due to a slight di\u000berence\nin e\u000bective magnetization \u00160M2=\u00160M1+ 0:0014 T. Note that the detuning between the\ntwo Kittel modes is large compared to the strength of the magneto-elastic coupling \n. In\nFigure 3b and Figure 3c we compare the measured modulation of the resonance amplitude\nfor respectively the bottom YIG1 layer and top YIG2 layers. This corresponds to performing\n2 cuts at the resonance condition FMR1 and FMR2 in the same fashion as Figure 2b. The\ntop YIG2 signal is modulated with a period of 7.00 MHz (Figure 3c) with a contrast \u0001\nbetween even and odd modes. This agrees with the prediction of Eq.(1) (see solid lines) due\nto constructive/destructive couplings mediated by even/odd phonon modes, the modulation\nperiod of the absorbed power doubles along the resonance of the top layer (FMR2), when\ncompared to the case of a single YIG layer (Figure 2). Figure 3b illustrates also that the\nstrong coupling \u00141to the antenna hinders clear observation of this modulation in the bottom\nYIG1 layer resonance. Nevertheless, the anticipated sign change of \u0001 (by the inverted phase\nofunrelative tom2in Eq.(1)) between FMR1 and FMR2 remains observable.\nWe now address the acoustic resonances revealed by the dark lines in Figure 3a for\nodd/even indices labeled by green/orange circles in the wings. The phonon line with even\nindex (orange marker) progressively disappears when approaching the YIG2 Kittel resonance\nfrom the low \feld (left side) of the resonance, while the opposite behavior is observed for\nthe odd index feature (green marker), which disappears when approaching the YIG2 Kittel\nresonance from the high \feld (right side). This behavior agrees with the model in Figure 1e.\nThe contrast in the acoustic resonance intensity mirrors the contrast of the amplitude of the\nFMR resonance.\nFigure 4a shows the observed FMR absorption spectrum around 5.11 GHz measured at\n\fxed \feldH0= 0:3453 T. We enhance the \fne structure in Figure 4b by subtracting the\nFMR envelope and progressively amplifying the weak signals in the wings. The orange/green\ncolor code emphasizes the constructive/destructive interference of the even/odd acoustic\n9FIG. 4. (Color online) a) Frequency sweep at \fxed \feld performed on the magnetic bi-layer. The\n\fne regular modulation within the FMR envelop is ascribed to the excitation of acoustic shear\nwaves resonances. The acoustic pattern is enhanced in panel b) by subtracting the FMR envelop\nemphasizing the constructive/destructive interferences of the even/odd acoustic resonances in the\nvicinity of the YIG2 FMR mode. Panel c) shows on a logarithmic scale the predicted modulation\nusing the experimental parameters of Table.(I). Panel d) shows on a linear scale the corresponding\npower absorbed by the top magnetic layer only.\nresonances in the top-layer signal. This feature can be explained by Eq. (1), as shown by\nthe calculated curves in Figure 4cd. The acoustic modes change character from even to odd\n(or vice versa) across the FMR frequency, which is caused by the associated phase shift by\n180\u000eof the acoustic drive, again explaining the experiments. The absorption by the YIG2\ntop layer in Figure 4d may even become negative so the phonon current from YIG1 drives\nthe magnetization in YIG2. This establishes both angular momentum and power transfer\nof microwave radiation via phonons.\nIn summary, we report interferences between the Kittel resonances of two ferromagnets\nover macroscopic distance through the exchange of circularly polarized coherent shear waves\npropagating in a nonmagnetic dielectric. We show that magnets are a source and detector for\nphononic angular momentum currents and that these currents provide a coupling, analogous\nto the dynamic coupling in metallic spin valves [52], but with an insulating spacer, over much\nlarger distances, and in the ballistic/coherent rather than di\u000buse/dissipative regime. This\n10should lead to the creation of a dynamical gap between collective states when the two Kittel\nresonances are tuned within the strength of the magneto-elastic coupling. Our \fndings might\nhave implications on the non-local spin transport experiments [53], in which phonons provide\na parallel channel for the transport of angular momentum. While the present experiments\nare carried out at room temperature and interpreted classically, the high acoustic quality of\nphonon transport and the strong coupling to the magnetic order in insulators may be useful\nfor quantum communication.\nThis work was supported in part by the Grants No.18-CE24-0021 from the ANR of France,\nNo. EFMA-1641989 and No. ECCS-1708982 from the NSF of the USA, by the Oakland\nUniversity Foundation, the NWO and Grants-in-Aid of the Japan Society of the Promotion\nof Science (Grant 19H006450). V.V.N. acknowledges support from UGA through the invited\nProf. program and from the Russian Competitive Growth of KFU. We would like to thank\nSimon Streib for illuminating discussions.\n\u0003Corresponding author: oklein@cea.fr\n[1] A. Bienfait, K. J. Satzinger, Y. P. Zhong, H.-S. Chang, M.-H. 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Saitoh, Physical Review B 96(2017), 10.1103/physrevb.96.104441.\n14"    },    {        "title": "1912.10432v1.First_principles_study_of_magnon_phonon_interactions_in_gadolinium_iron_garnet.pdf",        "content": "First-principles study of magnon-phonon interactions in gadolinium iron garnet\nLian-Wei Wang,1, 2, 3Li-Shan Xie,1Peng-Xiang Xu,3and Ke Xia1, 2, 3, \u0003\n1The Center for Advanced Quantum Studies and Department of Physics, Beijing Normal University, Beijing, 100875, China\n2Shenzhen Institute for Quantum Science and Engineering, and Department of Physics,\nSouthern University of Science and Technology, Shenzhen 518055, China\n3Center for Quantum Computing, Peng Cheng Laboratory, Shenzhen 518005, China\n(Dated: December 24, 2019)\nWe obtained the spin-wave spectrum based on a first-principles method of exchange constants, calculated the\nphonon spectrum by the first-principles phonon calculation method, and extracted the broadening of the magnon\nspectrum, \u0001!, induced by magnon-phonon interactions in gadolinium iron garnet (GdIG). Using the obtained\nexchange constants, we reproduce the experimental Curie temperature and the compensation temperature from\nspin models using Metropolis Monte Carlo (MC) simulations. In the lower-frequency regime, the fitted positions\nof the magnon-phonon dispersion crossing points are consistent with the inelastic neutron scattering experiment.\nWe found that the \u0001!and magnon wave vector khave a similar relationship in YIG. The broadening of the\nacoustic spin-wave branch is proportional to k2, while that of the YIG-like acoustic branch and the optical\nbranch are a constant. At a specific k, the magnon-phonon thermalization time of \u001cmpare approximately\n10\u00009s,10\u000013s, and 10\u000014s for acoustic branch, YIG-like acoustic branch, and optical branch, respectively.\nThis research provides specific and effective information for developing a clear understanding of the spin-wave\nmediated spin Seebeck effect and complements the lack of lattice dynamics calculations of GdIG.\nI. INTRODUCTION\nThe collinear multi-sublattice compensated ferrimagnetic\ninsulator gadolinium iron garnet ( Gd3Fe5O12, GdIG) has the\nsame crystalline structure as YIG1–4, only if yttrium is re-\nplaced by the magnetic rare-earth element, gadolinium.5–9In\ncomparison with YIG10,11, GdIG also has a low Gilbert damp-\ning constant of nearly 10\u00003,12but has three sublattices, where\nthe 12 Gd sublattice moments (dodecahedrals) are ferromag-\nnetically coupled to the 8 Fe moments (octahedrals) and an-\ntiferromagnetically coupled to the 12 Fe moments (tetrahe-\ndrals),4–9so that GdIG has more complex spin-wave modes\nthan YIG, which have been obtained by first-principles study\nof exchange interactions, indicating that the accurate calcula-\ntion method can improve and compensate for the abnormality\nin the spin-wave spectrum caused by exchange constants.4,13\nGdIG has high compensation temperatures T comp =286-\n295 K,14–17which is close to room temperature. Recently,\nthe heterostructures consisting of YIG18–21and heavy metals\n(FMI/NM) have been frequently used to study the spin See-\nbeck effect (SSE)22–24and spin Hall magnetoresistance effect\n(SMR).25–27. Similar to YIG, GdIG has been frequently used\nto study the SSE in FMI/NM heterostructures.22–24SSE ex-\nperiments have shown two sign changes of the current signal\nupon decreasing temperature.28,29One can be explained by the\ninversion of the sublattice magnetizations at T comp , where the\nnet magnetization vanishes and the other can be attributed to\nthe contributions of Ferrimagnetic resonance mode ( \u000b-mode)\nand a gapped optical magnon mode ( \f-mode).28–30The SMR\nexperiments shows that GdIG has a canted configuration31and\na sign change of SMR signal32at around T comp . Unlike in\nSSE,28,29the sign change of SMR is decided by the orientation\nof the sublattice magnetic moments associated with exchange\ninteraction.32Thus, these experiments28,32indicate that mul-\ntiple magnetic sublattices in a magnetically ordered system\nhave different individual contribution and highlight the im-\nportance of the multiple spin-wave modes determined by ex-change interactions. However, the microscopic mechanisms\nresponsible for these spin current associated effect are still\nunder investigation. A major question is whether the high-\nfrequency magnons play an important role in the SSE, and the\nfitting exchange parameters used in the literature through lim-\nited experimental data7,33,34are always physically credible.\nIn addition to the pivotal magnon-driven18,30,35,36effect,\nphonon-drag37,38effect plays non-negligible roles in the SSE\nthrough magnon-phonon interactions,39–41which play an im-\nportant role in YIG based spin transport phenomena.22,30,39–42\nThus, the understanding of the scattering process of magnon-\nphonon interactions is important and meaningful. In fact,\nthe magnon-phonon thermalization (or spin-lattice relaxation)\ntime,\u001cmp,39,43,44is an important parameter used to describe\nthe magnon-phonon interactions and calculate magnon dif-\nfusion length30,39. We have extracted the \u001cmp(\u001810\u00009s)\nfrom the broadening of magnon spectrum quantitatively13, in\ngood agreement with reported data39,44,45, however, the value\nis three orders of magnitude lower than the reported \u001cmp\u0018\n10\u00006s30,43,46,47. For the spin-wave spectrum and phonon spec-\ntrum to aid our understanding of the magnon-phonon scatter-\ning mechanism, the temperature-dependent magnon spectrum\nand lattice dynamic properties of GdIG have still not been\ncompletely determined. Here, we investigate these charac-\nteristics of GdIG based on the operable and effective method\nused in YIG.4,13\nTo computationally reveal the microscopic origin of SSE\nin these hybrid nanostructures, the magnon spectrum, phonon\nspectrum and magnon-phonon coupling dominant effect in\nGdIG will also be investigated step by step. First, we use\ndensity functional theory (DFT) technology to study the elec-\ntronic structure and exchange constants, and using Metropolis\nMonte Carlo (MC) simulations, we obtain the Curie temper-\nature (TC) and compensation temperature ( Tcomp ). Second,\nwe obtain the spin-wave spectrum using numerical methods\ncombined with exchange constants. Then, the phonon spec-\ntrum is studied using first-principles calculations, allowing usarXiv:1912.10432v1  [cond-mat.mtrl-sci]  22 Dec 20192\nto extract intersecting points of magnon branch and acoustic\nphonon branch. In the end, we study the temperature depen-\ndence of spin moment, exchange constants, and magnon spec-\ntrum, and calculated broadening of the spin-wave spectrum of\nGdIG is used to extract the magnon-phonon thermalization\ntime.\nII. COMPUTATIONAL DETAILS AND RESULTS\nIn this study, we investigate GdIG, which belongs to the\ncubic centrosymmetric space group, No. 230 Ia3d.6,7The\ncubic cell contains eight formula units, as shown in Fig. 1,\nwhere rare-earth gadolinium ions occupy the 24c Wyckoff\nsites (green dodecahedrals), the FeOand FeToccupy the 16a\nsites (blue octahedrals) and 24d sites (yellow tetrahedrals), re-\nspectively, and the O ions occupy the 96h sites (red balls).\nThe atomic sites from the experimental structural parameters\n(TABLE I)6–8are used in the study.\n/s40/s98/s41/s40/s97/s41\n/s70/s101/s79\n/s71/s100\n/s70/s101/s84/s79\n/s74\n/s100/s99/s74\n/s97/s99\n/s74\n/s97/s97/s74\n/s100/s100\n/s74\n/s97/s100\nFigure 1. (a) 1/8 of the GdIG unit cell. The dodecahedrally co-\nordinated Gd ions (green) occupy the 24c Wyckoff sites, the octa-\nhedrally coordinated FeOions (blue) occupy the 16a sites, and the\ntetrahedrally coordinated FeTions (yellow) occupy the 24d sites. (b)\nThe dashed lines denote the nearest-neighbor (NN) exchange interac-\ntions. Subscripts aa,dd,ad,acanddcstand for FeO-FeO,FeT-FeT,\nFeO-FeT,FeO-GdandFeT-Gd interactions, respectively.\nTo calculate the electronic structure and total energy of\nGdIG, we use DFT, as implemented in the Vienna ab initio\nsimulation package (V ASP).48,49The electronic structure is\ndescribed by the generalized gradient approximation (GGA)\nof the exchange correlation functional. Projector augmented\nwave pseudopotentials50are used. By using a 500 eV plane-Table I. Atomic positions in the GdIG unit cell. The lattice constant\nisa= 12:465Å.\nWyckoff Position x y z\nFeO16a 0.0000 0.0000 0.0000\nFeT24d 0.3750 0.0000 0.2500\nGd 24c 0.1250 0.0000 0.2500\nO 96h 0.9731 0.0550 0.1478\n-1.00-0.75-0.50-0.250.000.250.500.751.00-\n1.00-0.75-0.50-0.251.61.822.22.42.62.833.23.4-\n1.00-0.75-0.50-0.251.61.822.22.42.62.833.23.4 \n0.0(a)( b)( c) \n0.0  E-Ef (eV) \nMajority spin \nMinority spin \nΓP H (U-J)Fe = 5.7 eVGGA+U(\nU-J)Gd = 6.3 eV  E-Ef (eV) \nΓP H  \n E-Ef (eV) \nΓP H GGA+U(\nU-J)Fe = 5.7 eVG GA\nFigure 2. The energy band structure of the GdIG ground state under\ndifferent calculation conditions. (a) GGA calculation results. (b)\nGGA + U , thedorbital of the Featom plusU, where theU\u0000J\nvalue is 5:7eV . (c) GGA + U , thedorbital of the Featom plusU,\nwhere theU\u0000Jvalue is 5:7eV; theforbital of the Gd atom plus\nU, where theU\u0000Jvalue is 6:3eV . The green lines represent 0 eV .\nwave cutoff and a 6\u00026\u00026Monkhorst-Pack k-point mesh we\nobtain results that are well converged.\nA. Electronic structure\nThe calculated energy band structures of the ferrimagnetic\nground-state structure, are shown in Fig. 2. The apparent band\ngap indicates the properties of the insulator. The total moment\n(including Fe, Gd and O ions) per formula unit is consistently\n16\u0016B, which is consistent with experimental data9,51. The Fe\nand Gd sublattice contribute the majority of the spin moments\nwithin the unit cell. In the DFT-GGA calculation, the spin\nmoments of the Fe ions are \u00003.69\u0016Bfor FeOand 3.63\u0016B\nfor FeT, which are lightly larger than the computational data9,\nbut spin moments for the Gd and O ions are the similar values\n\u00006.85\u0016Band 0.08\u0016Brespectively, and the electronic band\ngap is 0.55 eV , as shown in Fig. 2(a). And just like we did\nin YIG,4because DFT is not good at predicting the energy\ngap of insulators, DFT-GGA+ Ucalculations with U\u0000Jford\norbital of Fe in the range of 2:2–5:7eV andU\u0000Jforforbital\nof Gd in the range of 0:3–6:3eV are conducted to determined\nthe Hubbard Uand Hund’s Jparameters. The variation in3\n/s50 /s51 /s52 /s53 /s54/s51/s46/s55/s52/s46/s48/s54/s46/s56/s55/s46/s48/s55/s46/s50\n/s48 /s50 /s52 /s54/s52/s46/s48/s52/s46/s49/s52/s46/s50/s54/s46/s56/s55/s46/s48/s55/s46/s50\n/s83/s40/s70/s101/s79\n/s41\n/s32/s32/s32/s83/s40/s70/s101/s84\n/s41\n/s32 /s83/s40/s71/s100/s41\n/s32/s32/s83/s112/s105/s110/s32/s109/s111/s109/s101/s110/s116/s40 /s41\n/s40/s85/s45/s74 /s41\n/s70/s101/s32/s40/s101/s86/s41/s40/s85/s45/s74 /s41\n/s71/s100/s61/s32/s51/s46/s51/s101/s86/s40/s97/s41/s40/s98/s41\n/s32/s32/s83/s112/s105/s110/s32/s109/s111/s109/s101/s110/s116/s32/s40 /s41\n/s40/s85/s45/s74 /s41\n/s71/s100/s32/s40/s101/s86/s41/s83/s40/s70/s101/s79\n/s41\n/s32/s32/s32/s83/s40/s70/s101/s84\n/s41\n/s32 /s83/s40/s71/s100/s41/s40/s85/s45/s74 /s41\n/s70/s101/s32/s61/s52/s46/s55/s101/s86\nFigure 3. Variation in the spin moments of Fe and Gd ions accord-\ning to different GGA+U calculations of the forbital of Gdandd\norbital of FeplusU. (a) Variation in the fixed (U\u0000J)Gd= 3:3\neV calculations and (b) Variation in the fixed (U\u0000J)Gd= 4:7eV\ncalculations.\nthe spin magnetic moments of different atoms under different\nconditions is shown in Fig. 3.\nAs shown in Figs. 2(b) and (c) and in Fig. 3, the electronic\nenergy gap and the spin moments slightly increase with U\u0000J.\nFor the GGA+U calculations(Fig. 2(b)), when the U\u0000Jvalue\nfor the Fe atom is constant, the band structure of the GdIG\nnear the Fermi energy is similar to that of the YIG. When the\nGd atom have (U\u0000J)Gd= 6:3eV , the energy band of Gd\nmoves up, as shown in Fig. 2(c). For the largest values of\nU\u0000J, the spin moments of FeO, FeT, and Gd are\u00004.26\u0016B,\n4.18\u0016Band\u00007.05\u0016B, respectively, and the electric band gap\nis approximately 2.08 eV . Even for the largest values of U\u0000J,\nthe moments are much smaller than expected for the pure\nFe3+, electronic spin S= 3=2state [\u0016s=gp\nS(S+ 1) =\n5:916\u0016B]and for the pure Gd3+, electronic spin S= 7=2\nstate [\u0016s=gp\nS(S+ 1) = 7:937\u0016B]. Compared with the\nelectronic structure calculation for YIG, the results of the spin\nmoments of Fe and the energy gap have been found to be sim-\nilar.4\nB. Exchange constants\nTo obtain the five independent nearest-neighbor(NN) ex-\nchange constants, Jaa,Jdd,Jad,JacandJdccovering the\ninter- and intra-sublattice interactions, as shown in Fig. 1(b).\nIn TABLE II, we map ten different collinear spin configura-\ntions(SCs) a-j on the Heisenberg model without external mag-\nnetic field energy or anisotropic energy. The calculation de-\ntails can be found in Ref. 4\nIn the NN model, with Eac=JacSaScandEdc=\nJdcSdSc,Eaa,Edd, andEadare just as the work in Ref. 4,\nwhereSa,Sd, andScare the +=\u0000directions of the FeO, FeT\nand Gd ions, the total energies, Etotof the Heisenberg model\nare determined as listed in TABLE II. Here Ecalare the calcu-\nlated total energies for fixed (U\u0000J)Fe= 3:4eV and different\n(U\u0000J)Gdvalues relative to the ground state of SC (a). When\nall or part of the magnetic moment directions of Gd atoms are\nflipped at (U\u0000J)Gd= 0 eV , SC (e), (g), and (j) have lower\ntotal energies than SC (a), which is in contrast to the experi-\n/s50 /s52 /s54/s48/s46/s48/s48/s46/s50/s48/s46/s52/s50/s51/s52\n/s50 /s52 /s54/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50\n/s48 /s50 /s52 /s54/s48/s46/s48/s48/s46/s49/s48/s46/s50/s51/s46/s49/s51/s46/s50/s51/s46/s51\n/s48 /s50 /s52 /s54/s45/s48/s46/s51/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54/s48/s46/s55/s48/s46/s56/s69/s110/s101/s114/s103/s121/s32/s40/s109/s101/s86/s41\n/s40/s85 /s32/s45/s32 /s74 /s41\n/s70/s101/s32/s40/s101/s86/s41/s32/s74\n/s97/s97\n/s32/s74\n/s100/s100\n/s32/s74\n/s97/s100\n/s69/s110/s101/s114/s103/s121/s32/s40/s109/s101/s86/s41\n/s40/s85 /s32/s45/s32 /s74 /s41\n/s70/s101/s32/s40/s101/s86/s41/s32/s74\n/s97/s99\n/s32/s74\n/s100/s99/s32/s32/s69/s110/s101/s114/s103/s121/s32/s40/s109/s101/s86/s41\n/s40/s85 /s32/s45/s32 /s74 /s41\n/s71/s100/s32/s40/s101/s86/s41/s32/s74\n/s97/s97\n/s32/s74\n/s100/s100\n/s32/s74\n/s97/s100\n/s69/s110/s101/s114/s103/s121/s32/s40/s109/s101/s86/s41\n/s40/s85 /s32/s45/s32 /s74 /s41\n/s71/s100/s32/s40/s101/s86/s41/s32/s74\n/s97/s99\n/s32/s74\n/s100/s99/s40/s97/s41/s40/s98/s41\n/s40/s99/s41\n/s40/s100/s41Figure 4. (a) and (b) Five different exchange interactions change with\ndifferent values of (U\u0000J)Fefor fixed (U\u0000J)Gd= 3:3eV . (c) and\n(d) Five different exchange interactions change with different values\nof(U\u0000J)Gdfor fixed (U\u0000J)Fe= 3:4eV. The negative value of\nthe exchange constants indicates that the magnetic moment tends to\nbe aligned in the same direction.\nmental result, and the energy differences between these three\nSCs and SC (a) decrease as (U\u0000J)Gdincreases. Furthermore,\nin SC (g), the static magnetic moment of the Gd sublattice is\n0\u0016B, and the total magnetic moment of GdIG unit molecular\nformula is 5 \u0016B, indicating that it is necessary to add U for\nGd ions. Through the differences of \u0001Ebetween the Ecal\nandEtot, we also find that \u0001Eincrease when (U\u0000J)Gdis\ntoo small or too large. For (U\u0000J)Gd= 3:3eV , the maximum\nj\u0001Ejis 0.63 %, which is acceptable.\nThe exchange constants shown in Fig. 4 are obtained by the\nleast-squares of six linear equations using the SCs a-g listed\nin TABLE II. SCs h-j are selected to check whether the results\nare reasonable. The exchange constants Jaa,JddandJadare\npositive (antiferromagnetic), whereas the exchange constants\nJacandJdcdepend on the value of U\u0000J. In Figs. 4(a)\nand (b),Jaa,JddandJaddecrease as (U\u0000J)Feincreases\nwhen (U\u0000J)Gdis kept constant 3.3 eV , which is similar to\nthe situation for YIG.4The values of Jadare approximately\n4 % and 2 % lower compared with the ones for YIG when\n(U\u0000J)Feis 4.7 eV and 5.7 eV , respectively. JacandJdc\nwith different signs decrease slightly as (U\u0000J)Feincreases\nand forjJdcj>jJacj. In Fig. 4(c) and (d), when (U\u0000J)Feis\nkept constant at 3.4 eV , JaaandJddmaintain almost the same\nvalues, whereas Jadincrease slightly as (U\u0000J)Gdincreases.\nJacandJdcdecrease to zero and then change their signs as\n(U\u0000J)Gdincreases. Among all the results, Jadis one order\nof magnitude larger than the other interactions, whereas Jaais\napproximately half of Jddand the absolute value of Jacis al-\nways smaller than that of Jdc. Thus, the strong inter-sublattice\nexchange interaction, Jad, dominates the other smaller ener-\ngies and helps maintain the ferrimagnetic ground state of the4\nTable II. Comparison of the calculation of the total energies for different SCs in the NN models. The b \u0000j are obtained by changing the\nmagnetization directions of part of the magnetic ions based on the ferrimagnetic ground state SC a. Etotis the total energy fitting formula.\nEcalis the total energy (in units of meV) calculated via ab initio with different (U\u0000J)Gdat fixed (U\u0000J)Fe= 3:4(in units of eV). \u0001Eis\nthe difference between EtolandEcal.Ecalof SC a is denoted as zero.\nSC EtotEcal \u0001E\n0.0 1.3 3.3 5.3 0.0 1.3 3.3 5.3\naE0+ 32Eaa+ 24Edd+ 48Ead+ 48Eac+ 24Edc 0.00 0.00 0.00 0.00 \u00000.01 0.00\u00000.01\u00000.02\nbE0+ 32Eaa+ 24Edd\u000048Ead\u000048Eac+ 24Edc3957.97 5036.60 5145.78 5236.59 0.00 0.00 \u00000.01\u00000.03\nc E0+ 32Eaa\u000024Edd\u000048Eac 1729.27 2661.94 2582.27 2506.26 \u00000.01 0.00\u00000.01\u00000.03\nd E0\u000032Eaa+ 24Edd+ 24Edc 1364.90 2399.29 2454.38 2500.18 \u00000.01 0.00\u00000.01\u00000.03\neE0+ 32Eaa+ 24Edd+ 48Ead\u000048Eac\u000024Edc\u0000187.34 599.72 331.46 88.54 \u00000.01 0.00\u00000.01\u00000.02\nf E0+ 32Eaa\u000024Edd 1770.30 2662.68 2532.24 2412.53 0.00 0.00 \u00000.01\u00000.02\ng E0+ 32Eaa+ 24Edd+ 48Ead\u0000589.27 299.52 165.63 43.94 0.96 0.34 0.09 0.31\nh E0\u000032Eaa+ 24Edd 1807.52 2700.43 2570.38 2450.69 \u00000.63\u00000.54\u00000.31 0.00\ni E0+ 32Eaa\u000024Edd+ 48Eac 1813.35 2664.33 2482.58 2319.38 \u00002.02\u00000.90\u00000.38\u00000.60\njE0+ 32Eaa+ 24Edd+ 48Ead\u000032Eac\u000016Edc\u0000327.56 496.21 274.47 71.62 6.56 3.56 1.74 2.14\nbulk.7,52,53Moreover, with a change in the (U\u0000J)Gdvalue,\nJacandJdcmay change signs, which implies that it is possi-\nble to change the direction of the Gd atomic magnetic moment\nin the ground state.\nA comparison of our exchange constants with those found\nin prior studies is provided in TABLE III. We find that dif-\nferent methods provide different exchange constants. Us-\ning limited experimental data, neither the magnetization fit-\nting7nor the molecular field approximation52,53can effec-\ntively determine whether the interaction between the inter-\nand the intra-sublattice is ferromagnetic or antiferromagnetic\ncoupling. Although our calculated value is smaller than the\nvalue provided in the TABLE III and the obtained exchange\nconstants between Fe atoms in GdIG are smaller than those\nin YIG4, the relative size relationship is Jad> Jdd> Jaa,\nJad> Jdc> Jac. Here, we can well determine the type\nof exchange constants between sublattices, and use the ex-\nchange constants to obtain a reasonable experimental Curie\ntemperature and compensation temperature. Therefore, the\nfirst-principles method of exchange constants4,13undoubtedly\nprovides an effective way to calculate the interaction parame-\nters in GdIG.\nC. Magnetization, Curie temperature, and compensation\ntemperature\nTo obtain the temperature dependence of the magnetiza-\ntion, Curie temperature ( TC), and compensation temperature\n(Tcomp ), we use the spin models by Metropolis MC simula-\ntions on a 32\u000232\u000232 supercell with a unit cell containing\n32 spins under periodic boundary conditions. The computa-\ntional details can be found in Ref. 4. The results are shown in\nFig. 5.\nWith the parameters of (U\u0000J)Gd= 3:3eV and (U\u0000\nJ)Fe= 4:7eV , the temperature dependence of magnetiza-\ntion,Ma,Md, andMcof FeO, FeT, and Gd, respectively,\nand the total magnetization ( M=Ma+Mb+Mc) of a for-\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48 /s53/s48/s48 /s54/s48/s48/s45/s50/s53/s45/s50/s48/s45/s49/s53/s45/s49/s48/s45/s53/s48/s53/s49/s48/s49/s53\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s32/s32/s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s32/s40 /s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s32/s84/s111/s116/s97/s108\n/s32/s70/s101/s79\n/s32/s70/s101/s84\n/s32/s71/s100/s84\n/s99/s111/s109/s112/s32/s61/s32/s51/s49/s48/s32/s75/s40/s97/s41\n/s40/s98/s41\n/s32/s32/s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s40/s85/s45/s74 /s41\n/s71/s100/s32/s32\n/s32/s32/s48/s32/s101/s86\n/s32/s32/s50/s46/s51/s32/s101/s86\n/s32/s32/s51/s46/s51/s32/s101/s86\n/s32/s32/s52/s46/s51/s32/s101/s86\n/s32/s32/s82/s101/s102Figure 5. (a) Temperature dependence of magnetization of FeO, FeT,\nand Gd and the total magnetization of a formula unit with exchange\nconstants fitted to the ab initio energies for (U\u0000J)Gd= 3:3eV\nand(U\u0000J)Fe= 4:7eV . The black arrow represents the position\nof the compensation temperature, Tcomp = 310 K (b) The absolute\nvalue of the total magnetization is jMj=jM(T= 0 K)jfor different\n(U\u0000J)Gdat different temperatures. The reference curves (green\nlines) are calculated using the exchange constants in Ref. 7.5\nTable III. Exchange constants taken from the literature and our study. In calculation, (U\u0000J)Gd= 3:3eV . The unit for the interaction\ncoefficient is meV . (U\u0000J)Fe= 4:7eV is used for comparing with the result in YIG.4\nJaaJddJadJacJdc Method Reference\n0.78 0.78 3.94 0.22 0.87 Magnetization fit Ref.7\n\u00001.05\u00001.47 3.14\u00000.11 0.58 Molecular field approximation Ref.52\n0.56 1.04 2.59 0.05 0.16 Molecular field approximation Ref.53\n0.081 0.137 2.487 \u00000.032 0.157 Ab initio GGA+U ( (U\u0000J)Fe= 4.7eV) This paper\n0.103 0.185 3.018 \u00000.035 0.170 Ab initio GGA+U ( (U\u0000J)Fe= 3.7eV) This paper\nmula unit are determined, as shown in Fig. 5(a). The crossing\npoint of the total magnetization curve (black) and the hori-\nzontal dash line shows that Tcomp (310 K) and TC(550 K),\nwhich are in good agreement with the experimental values of\n290 K16,17and 560 K5,54, respectively. Through the Fig. 5(a),\nwe can find that the change of the total spin moments of\nFeO-sublattice and FeT-sublattice is considerably flat. How-\never, the total spin moment of Gd-sublattice rapidly declines\nwith increasing temperature until approximately 200 K; As\nthe temperature continued to increase, owing to competition\nbetween the Gd and Fe magnetic moments, the total spin mo-\nments undergoes a transition dominated by Gd to Fe. The\ndirection of the total magnetization changes from Gd (FeO)\nto FeTand the value first decreases and then increases; then,\nTcomp emerges16,17, wherein one sign change of SSE signal\nappears.28With a further increase in temperature, the decreas-\ning trend of Gd-sublattice spin moments slows down. Ad-\nditionally, the decreasing trend of the spin moments of FeO-\nsublattice and FeT-sublattice becomes steeper, and the total\nmagnetic moment slowly increases and then decreases to 0 \u0016B\nat transition temperature TC. The temperature dependence of\nthe magnetization of GdIG is similar to that reported in the\nliterature.28\nAs shown in Fig. 5(b), we determine the absolute values\nof the total magnetization, jMj, normalized by its value at\nzero temperature,jM(T= 0 K)j, for different (U\u0000J)Gd\nat the fixed (U\u0000J)Fe= 3:4eV . There is no Tcomp with\n(U\u0000J)Gd= 0 eV , whereas the calculated Tcomp decreases\nas(U\u0000J)Gdincreases.Tcomp of the reference curve(green\nline) calculated using the exchange constants in Ref. 7 is\nalso approximately 310 K. Compared with Fig. 4, when the\n(U\u0000J)Gdvalue is small, JacandJdcin Fig. 4 are positive\nandJac<Jdc. Thus, the magnetization direction of Gd with\nFeOand FeTis anti-parallel, the latter is dominant, the ground\nstate corresponds to SC (g) in TABLE II, and half of the Gd\nhas an inverted magnetic moment. With increasing (U\u0000J)Gd,\nJacandJdcreverse so that Gd tended to be parallel to FeOand\nanti-parallel to FeT; thus, the ground state corresponds to SC\n(a) in TABLE II. Under this condition, the Tcomp of the system\ndecreases gradually with increasing (U\u0000J)Gdvalues. There-\nfore, with appropriate parameters of (U\u0000J)Gd= 3:3eV and\n(U\u0000J)Fe= 4:7eV , we can reproduce the experimental com-\npensation temperature and transition temperature.D. Magnon spectrum\nUsing the exchange interaction obtained under conditions\nof(U\u0000J)Gd= 3:3eV and (U\u0000J)Fe= 4:7eV , as shown in\nTABLE III, we obtain the spin-wave spectrum at zero temper-\nature, as shown in Fig. 6. The details of the calculation can be\nfound in Ref. 4.\nIn Fig. 6(a), the special ferrimagnetic resonance \u000b-mode,\nlower-frequency optical modes with a slight gap, YIG-like\nacoustic\f-mode and lower-frequency optical \r-mode are\nmarked by red, orange, yellow, and green curves. Other high\noptical modes are marked by blue curves. To contrast with\nthe lowest optical mode of YIG in Ref. 4, a black dash line\nindicates the frequency. Fig. 6(b) clearly shows the lower-\nfrequency branches below 0.5 THz. The flat part (orange)\nhas two modes at approximately 425 GHz and two modes\nat approximately 440GHz, which are dominated by the Gd\nprecessing.7The second derivative of the \u000b-mode at the \u0000-\npoint is 19\u000210\u000041J\u0001m2, which is approximately one quarter\nof the spin-wave stiffness of YIG, D= 77\u000210\u000041J\u0001m2.4\nThe\f-mode around 1.4 THz has a similar parabolic branch\nwith the acoustic branch of YIG in which the second deriva-\ntive of the mode at the \u0000point is 62\u000210\u000041J\u0001m2. The gap\nbetween\f- and\u000b-mode at the \u0000point depends on JdcandJac\ninteraction,28and the gap between the second parabolic low-\nest\r-mode and the \f-mode is approximately 5THz, which is\nconsistent with the conditions of the NN model in YIG.4As\nthe temperature increases, the \f-mode will red-shift to gain\na sufficient thermal magnon population below KbT, then the\nSSE signal changes sign.28,35However, the \r-mode will also\nred-shift below 6.25 THz (T =300 K), which may also have\nsome indispensable effects in the SSE.\nThe precession patterns of these special low-frequency\nmodes at \u0000are shown in Fig. 7. In Fig. 7(a), for the \u000b-mode,\nthe magnetization of FeOis parallel to Gd, anti-parallel to\nFeT, and near the \u0000point, where !andksatisfy the square\nrelationship, which is similar to the acoustic mode of YIG13.\nFor the\f-mode, the pattern is different from the acoustic\nbranch modes of YIG, the magnetization of 8 FeOions, 12\nGd ions and 12 FeTions have different directions with re-\nspect to the\u000b-mode, and the magnetization of Gd has a small\nangle (\u00190:12) with the z-axis. For the \r-mode, the mag-\nnetization of FeOhas different directions with respect to Gd\nand FeT, and the magnetization of FeOand Gd have small\nangles (\u00190:11) and (\u0019\u00000:16) with respect to the z-axis, re-6\n024681012140\n.00.10.20.30.40.5ΗΝ   ω (ΤΗz) \nΓ(a)(\nb) \nα βγ  \n ω (ΤΗz)N\nΓ  H 1100 011\n100 01(b)(c)(d)(e)\nFigure 6. Spin-wave spectrum at zero temperature in the first Bril-\nlouin zone at (U\u0000J)Gd= 3:3eV and (U\u0000J)Fe= 4:7eV .\n(a) The entire spin wave spectrum. The black dash line represents\nthe position of the lowest optical branch frequency at 4.8 THz for\nYIG calculated in the NN model from Ref. 4. The notations \u000b(red),\n\f(yellow), and \r(green) mark the three main spin-wave modes,\nindicating positive, negative and positive polarization, respectively.\nThe orange line marks the two nearly clearance modes at approxi-\nmately 0.4 THz. These low-frequency optical modes are the Gd mo-\nments precession dominant. (b) The partial enlarged details of the\nlow-frequency modes that are red- and orange- marked in (a) around\n0.4 THz whereas (b), (c), (d), and (e) mark the modes at approxi-\nmately 0.4 THz. The directions in the k-space use the standard labels\nfor a bcc reciprocal lattice.\nspectively. The \u000b-mode has different polarizations with the\n\f-mode, but the same as \r-mode. The polarizations of \u000b- and\n\f-mode switch at Tcomp , which is related to the other sign\nchange at a lower temperature in the SSE.28However, the two\nmodes induce the same sign in the detected SMR signal in a\nmagnetic canted phase of GdIG.26,32. Therefore, spin-wave\nmodes need to be verified in greater detail by experiments in\nthe future.\nFor the 425 GHz case, two patterns with two and three\ndegenerated modes have FeOthat spins lie along the z-axis,\nwhile FeTspins precess at small angles, as shown in Fig. 7(b)\nand (c). For the 440GHz case, for the two patterns with three\nand three degenerated modes, FeTspins align along the z-\naxis, and FeOspins precess at small angles or take the op-\nposite direction as the FeTspins, as shown in Fig. 7(d) and\n(e). In both cases, Gd spins precess at a larger angle than\nthe Fe spins in the exchange field of the Fe spins.7,28The gap\nFigure 7. Precession patterns of the low-frequency modes color\nmarked except for blue in Fig. 6 at the \u0000-point. (a) The patterns\nmark the\u000b-,\f-, and\r-mode. The red arrows represent the differ-\nent chiral patterns. (b) and (c) show two patterns at approximately\n425 GHz with two and three degenerated modes, respectively. (d)\nand (e) show two patterns at approximately 440 GHz with three and\nthree degenerated modes, respectively. ( \u0006)a, b, c and d denote the\ndifferent precession angles. The lower optical modes indicate that\nthe Gd moments precess around the exchange field induced by Fe\nmoments.\n/s48/s49/s50/s51\n/s32/s32\n/s32/s32\n/s32/s74\n/s97/s99/s48/s59/s32/s74\n/s100/s99\n/s32/s74\n/s97/s99/s48/s59/s32/s74\n/s100/s99\n/s32/s74\n/s97/s99/s48/s59/s32/s74\n/s100/s99\n/s32/s74\n/s97/s99/s48/s59/s32/s74\n/s100/s99/s40 /s122 /s41\n/s49/s49/s48 /s32/s78\nFigure 8. The spin-wave spectrum is affected by the change in ex-\nchange constants at (U\u0000J)Gd= 3:3eV and (U\u0000J)Gd= 4:7eV .\nJaa,Jdd, andJadare unchanged. For black ball curves JacandJdc\nare both 0 meV; for green star curves, Jacis 0 meV and Jdcis the\noriginal value; for blue triangle, Jdcis 0 meV and Jacis the original\nvalue; for red triangle curves, JacandJdcare both original values.\nbetween these modes at the \u0000point and the \f-mode is ap-\nproximately 1THz, which is dominated by the interactions of\nFe and Gd. To show that the gap is primarily derived from\nthe exchange interaction between the Gd atoms and Fe atoms,\nwe fixJaa,JddandJad, then change JacandJdcto show\nthe change in spectrum along the highly symmetric direction7\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48/s48/s53/s49/s48/s49/s53/s50/s48\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48/s48/s53/s49/s48/s49/s53/s50/s48\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53/s32/s32/s80/s68/s79/s83\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s84/s72/s122/s41/s32/s79\n/s89\n/s70/s101/s79/s70/s101/s84\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s84/s72/s122/s41\n/s32/s32/s84/s68/s79/s83/s32/s89/s73/s71/s32/s32/s32/s80/s68/s79/s83\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s84/s72/s122/s41/s71/s100/s79\n/s70/s101/s84\n/s70/s101/s79/s40/s100/s41/s40/s99/s41/s40/s98/s41\n/s32/s32/s84/s68/s79/s83\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s84/s72/s122/s41/s32/s71/s100/s73/s71/s40/s97/s41\nFigure 9. First-principles phonon calculation of YIG and GdIG at\nzero temperature. (a) Projected density of states (PDOS) for YIG.\n(b) Total density of states (TDOS) for YIG. (c) PDOS for GdIG. (d)\nTDOS for GdIG. FeOrepresents the octahedral atom sites, and FeT\nrepresents the tetrahedral atom positions, the same representation as\nshown in Fig. 1.\n(110), as shown in Fig. 8. We find that with the reduction of\nJac, the intersection point between the branch with the low-\nest frequency and the boundary of Brillouin region decreases\nand the band gap becomes narrower. As Jdcdecreases, the in-\ntersection point shows a more obvious reduction, the spectral\nlines near 0:4THz degenerate, and the frequency decreases.\nWhenJacandJdcsimultaneously decrease and the reduction\neffect is superimposed, the spectrum near 0:4THz completely\ndisappears.\nE. Phonon spectrum\nDifferent ab initio techniques and methods can be em-\nployed to calculate the phonon spectrum.55–59The density\nfunctional perturbation theory method is typically used to ob-\ntain the real-space force constants of GdIG whose DFT+U\nground state is used for self-consistent linear-response cal-\nculations in V ASP as above. Phonon band structures, partial\ndensity of states (PDOS), total density of states (TDOS) and\nthe phonon velocity of GdIG are investigated using the force\nconstants via the Phonopy code.56,60We also use the same cal-\nculation method to obtain the phonon spectrum of YIG, where\nthe calculation parameters come from Ref. 4.\nIn the calculation, we first obtain PDOS and TDOS of GdIG\nand YIG, as shown in Fig. 9. Figs. 9(a) and (c) show the PDOS\nfor different atoms. Figs. 9(b) and (d) show the TDOS. The\nclear results for YIG are similar to the results from Ref. 58.\nThe phonon gap in GdIG is approximately 2THz, which is\nconsistent with the phonon spectrum for YIG. There is a dif-\nference between YIG and GdIG in the low frequency region\n(0-5 THz).\n051015200\n24680\n51015200\n2468(d)( c)(b)  Frequency (THz)Γ\n H N P ΓYIG \n Frequency (ΤΗz)(\n110)( 001)TALA(a)N\nΓ  H   Frequency (THz) \nΓH N P ΓGdIG \n \n N\nΓ  H ( 110)( 001)TALA \nFigure 10. (a)Phonon spectrum of YIG along the \u0000-H-N-P- \u0000high-\nsymmetry lines. (b) Comparison between the phonon and spin-wave\nspectra along the \u0000-N and \u0000-H high-symmetry directions in YIG. (c)\nPhonon spectrum of GdIG along the same high-symmetry lines with\n(a). (d) Comparison between phonon and spin-wave spectra along\nthe same directions with (b) in GdIG. The longitudinal acoustic (LA)\nand transverse acoustic (TA) phonons are marked on (b) and (d). The\npartial spin-wave spectra in (b)(red dots) are obtained from Fig. 5 in\nRef. 4, and the representations in (d) are the same as Fig. 6(a), the\npartial phonon spectra are extracted from (a) and (c), respectively.\nThe phonon spectrum along the path of \u0000-H-N-P- \u0000in the\nBrillouin zone of the bcc lattice for YIG and GdIG are shown\nin Figs. 10(a) and (c), which cover 240phonon branches. The\nphonon spectrum of YIG is consistent with the results calcu-\nlated by the finite-displacement method in Ref. 58. We are in-\nterested in the low-frequency phonon branches, labeled as lon-\ngitudinal acoustic (LA) and transverse acoustic (TA) phonons\nin Figs. 10(b) and (d). The frequency of the special branches\nshows a linear kdependence in the lower frequency region and\nthe TA modes are double degenerate. The slope of the TA(LA)\nphonon dispersion is presented in TABLE IV. For YIG, the\nvelocity of TA (LA), v= 3:8kms\u00001(6.74 kms\u00001), is consis-\ntent with experiment results.42For GdIG, the TA (LA) veloc-\nityv= 3:3kms\u00001(6.08 kms\u00001), is almost consistent with\nthe transverse (longitudinal) sound velocity found via experi-\nments.61,62\nIn Fig. 10(b) for YIG, the spin-wave acoustic branch (red)\ntaken from the Ref. 4 has an intersecting point at a very lower-\nenergy approximately 1:38(5:13) meV with the TA (LA)\nphonon branch. These fitting magnon-phonon intersecting\npoints are almost consistent with experiment results.42\nIn Fig. 10(d) for GdIG, the intersection points are more\ncomplicated than for YIG. The LA phonons and TA phonons\nhave only one cross point with the \u000b-modes (red) at the\n\u0000point, and not intersection points with the \f-modes (yel-\nlow). However, they have many more intersection points with\nflat lower-frequency optical branches (orange) because many\nmultiple degenerated branches stay here as shown in Fig. 6(b),8\nTable IV . Comparison between the calculated and reported values of\nthe phonon velocities for YIG and GdIG\nSystemLA velocity TA velocitySource(105cm/s) (105cm/s)\nYIG 7.200 3.900 Ref.42\nYIG 7.209 3.843 Ref.63\nGdIG 6.500 3.390 Ref.62\nYIG 6.740 3.800 This paper\nGdIG 6.080 3.300 This paper\nsuch as eight crossing points along the \u0000-Hpath and eleven\ncrossing points along the \u0000-Npath. We speculate that in\nthe lower-frequency region, low-frequency lattice vibrations\n(phonons) can couple with magnons and there may be com-\nplicated and interesting magnon-phonon40,42,64and magnon-\nmagnon65coupling effects. The results are useful for un-\nderstanding the scattering process of magnon-phonon inter-\nactions in the SSE.\nF. Magnon-phonon coupling\nTo investigate the variation of frequency and linewidth in\nthe spin-wave spectrum at room temperature ( T= 300 K),\nthe temperature-induced atomic vibration is considered. The\nstatistical mean square of the displacements, ui, of thei-th\natom with its mass, Mi, are determined by the Debye model,13\nhjuij2i=9~2\nMikB\u0012D\u00121\n4+T2\n\u00122\nDZ\u0012D\nT\n0x\nex\u00001dx\u0013\n;(2.1)\nwheremGd= 157:25amu,mFe= 55:85amu,mO=\n15:99amu and the Debye temperature is \u0012D= 655:00K.9\nHere, the change in atomic displacement does not cause sig-\nnificant lattice deformation. The atomic vibration displace-\nments modeled in Eq. 2.1 are added to the experimental struc-\nture shown in TABLE I. Forty atomic configurations, which\nare denoted as cf01 \u0018cf40, respectively, are used to ob-\ntain the spin-wave spectrum. We chose the parameters of\n(U\u0000J)Gd= 3:3eV and (U\u0000J)Fe= 4:7eV for the total\nenergy calculations because they provide reasonable TCand\nTcomp . The magnon-phonon relaxation time can be extracted\nfrom the broadening spin-wave spectrum. For calculation de-\ntails, we refer to Ref. 13.\nThe spin moments of Fe and Gd ions for the ferrimagnetic\nground-state structure with these 40 configurations are shown\nin Fig. 11 (a). The average moments of the \u0000S(FeO),S(FeT),\nandS(Gd) ions are marked as black squares, red dots, and\nblue triangles, respectively. In comparison with the zero tem-\nperature values (marked as dash lines), the average moments\nof the Fe ions are lower for all configurations, whereas the\nones for Gd ions showed no signigicant difference. The error\nbars denote the minimum and maximum range for each con-\nfiguration. The spin moments of the Fe ions have a variation\n/s51/s46/s57/s48/s51/s46/s57/s53/s52/s46/s48/s48/s52/s46/s48/s53/s52/s46/s49/s48/s52/s46/s49/s53/s52/s46/s50/s48/s54/s46/s57/s53/s55/s46/s48/s48\n/s51/s46/s57/s48\n/s51/s46/s57/s53\n/s52/s46/s48/s48\n/s52/s46/s48/s53\n/s52/s46/s49/s48\n/s52/s46/s49/s53\n/s52/s46/s50/s48\n/s54/s46/s57/s53\n/s55/s46/s48/s48/s32/s45 /s83 /s40/s70/s101/s79\n/s41\n/s32/s32 /s83 /s40/s70/s101/s84\n/s41\n/s32/s45 /s83 /s40/s71/s100/s41/s99/s102/s51/s50 /s99/s102/s51/s48/s99/s102/s50/s50/s99/s102/s50/s48/s99/s102/s49/s50 /s99/s102/s49/s48\n/s99/s102/s52/s48\n/s99/s102/s51/s49/s99/s102/s50/s49/s99/s102/s49/s49\n/s99/s102/s48/s50/s83/s112/s105/s110/s32/s109/s111/s109/s101/s110/s116/s32/s40\n/s66/s41\n/s32\n/s99/s102/s48/s49\n/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53\n/s45/s48/s46/s53\n/s48/s46/s48\n/s48/s46/s53\n/s50/s46/s48\n/s50/s46/s53\n/s51/s46/s48\n/s51/s46/s53/s32/s74\n/s97/s97\n/s32/s74\n/s100/s100\n/s32/s74\n/s97/s100\n/s32/s74\n/s97/s99\n/s32/s74\n/s100/s99/s32/s74\n/s105/s106/s32/s40/s109/s101/s86/s41/s99/s102/s50/s48\n/s99/s102/s50/s49\n/s99/s102/s50/s50/s99/s102/s49/s50/s99/s102/s49/s49\n/s99/s102/s49/s48\n/s99/s102/s48/s50\n/s99/s102/s48/s49\n/s99/s102/s52/s48\n/s99/s102/s51/s50\n/s99/s102/s51/s49/s99/s102/s51/s48/s40/s98/s41/s40/s97/s41Figure 11. Spin moments and exchange constants of 40 atomic con-\nfigurations (cf01-cf40). (a) Spin moments of the ions (in units of\n\u0016B) for these ground-state structures. The \u0000S(FeO) (black square),\nS(FeT) (red dot), and\u0000S(Gd) (blue triangle) curves represent differ-\nent spin moments. The error bars represent the magnitude of moment\nchange. (b) ab initio calculation of exchange constants (in units of\nmeV) in the NN models as like in TABLE.II. The calculation param-\neters for (U\u0000J)Gd= 3:3eV and (U\u0000J)Fe= 4:7eV . The results\nat zero temperature are indicated by dashed lines.\nrange of approximately 0.1 \u0016B, which is much wider than the\nGd ions at room temperature. The calculated exchange con-\nstants for each configuration are shown in Fig. 11(b). The re-\nsults show that the antiferromagnetic exchange constants, Jad,\nstill dominate. Exchange constants Jaa,Jdd,Jac, andJdcmay\nchange their signs, where Jdchas the largest variation range\nfrom\u00000.6 meV to 0.6 meV . The ground state of GdIG is still a\nferrimagnetic configuration, in which the moments of the FeO\natoms are arranged anti-parallel to the FeTatoms and parallel\nto the Gd atoms. We can see that magnon-phonon coupling\ncan induce small fluctuation of magnetic moment and vari-\nation of exchange constants, so that the broadening of spin-\nspectrum can be shown.\nAs shown in Fig. 12, at room temperature, spin-wave modes9\n024681012140\n2468101214 ω (THz) \nN         110          Γ               001              Hα β  γ \nΓ\n            111            P(b)( a)\nFigure 12. (a) Spin-wave spectrum in the first Brillouin zone derived\nfrom ab initio calculations of exchange constants with (U\u0000J)Gd=\n3:3eV and (U\u0000J)Fe= 4:7eV for 40 atomic configurations. The\nentire spin-wave spectrum at zero temperature is denoted using gray\nlines, and the picked modes for \u000b,\f,\rare marked in red, yellow, and\ngreen. The blue curves with error bars denote the range changes of\nthe spectrum induced by atomic vibration. (b) Spin-wave spectrum\nin highly symmetric direction 111. The color means the same as in\n(a). The directions in the k-space have the standard labels for a bcc\nreciprocal lattice.\nare plotted as the blue curves with error bars governed by the\nNN exchange constants in Fig.11(b). At zero temperature, the\nlowest frequency \u000b-mode and two slightly higher frequency\nparabolic\f- and\r-mode are shown by the red, yellow, and\ngreen curves, respectively, which is the same as Fig.6. Other\nmodes are marked by gray curves. We can see that the blue\ncurves can superimpose with other modes and show a signifi-\ncant spread in energy at room temperature. For the \u000b-modes,\nthe frequency of different phonon configurations is nearly the\nsame as red curves in different directions, and the spectral\nline had a slightly larger distribution range at the Brillouin\nzone boundary. For the \f-modes, the spectral lines distribute\naround the yellow curves, and the distribution range increase\nas thekvalue increases in all directions. Compared with the\nacoustic branch of YIG,13the spectrum shows a smaller dis-\ntribution range. For the \r-modes, the spectral lines also dis-\ntribute around green curves and disperse larger than the \f-\nmode. However, the distribution in all directions decreases\nthen increases with increasing k, which is not the case for the\nYIG.13. So the spin-wave spectrum using the phonon con-\nfigurations at room temperature shows a noticeable broaden-\ning. As shown in Fig. 13, the broadening of the spectrum,\n\u0001!, for the\u000b-,\f-, and\r-modes are extracted from 40room-\ntemperature configurations by using the method in Ref. 13.\nIn Fig. 13(a), \u0001!has a strong dependence on the value of\nk. When the k-value is small, \u0001!in the three high symme-\ntry directions are very close to each other, and have differ-\nent trends with increasing k. For the\u000b-mode, \u0001!increase\nslowly with increasing k, but in the (111) direction, there is\na slight decrease when the kvalue approaches the Brillouin\nzone boundary. Compared with the three directions, we find\n/s48 /s49 /s50/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53/s52/s46/s48\n/s48 /s52 /s56 /s49/s50/s48/s49/s50\n/s48/s46/s48/s49 /s48/s46/s49 /s49/s49/s69/s45/s52/s49/s69/s45/s51/s48/s46/s48/s49/s48/s46/s49/s49/s49/s48/s32/s48/s48/s49\n/s32/s49/s49/s48\n/s32/s49/s49/s49/s32/s40/s84/s72/s122/s41\n/s107 /s32/s40 /s47/s97/s41/s45/s109/s111/s100/s101/s45/s109/s111/s100/s101/s45/s109/s111/s100/s101/s32/s40/s84/s72/s122/s41\n/s32/s40/s84/s72/s122/s41\n/s32/s48/s48/s49\n/s32/s49/s49/s48\n/s32/s49/s49/s49/s32/s40/s84/s72/s122/s41\n/s107 /s32/s40 /s47/s97/s41/s40/s97/s41\n/s40/s98/s41\n/s45/s109/s111/s100/s101\n/s45/s109/s111/s100/s101\n/s45/s109/s111/s100/s101Figure 13. (a) Calculated broadening of the spin-wave spectrum of\nGdIG, \u0001!, at room temperature as a function of k. Inset: \u0001!replot-\nted as a function of the spin-wave frequency, !. (b)\u0001!(k)replotted\non a log-log scale. The black lines indicate a constant \u0001!and a\nquadratic dependence on kfor the optical branch ( \f-,\r- modes) and\nthe acoustic branch ( \u000b-mode), as shown in Figs. 6 and 12, respec-\ntively.\nthe relationship of \u0001!(001)>\u0001!(110)>\u0001!(111) in the\nregion ofk > \u0019=a . For the\f-mode, \u0001!increases with in-\ncreasingk. Upon comparing the three directions, we find the\nrelationship of \u0001!(001)>\u0001!(111)>\u0001!(110) in the re-\ngion ofk > \u0019=a . For the\r-mode, \u0001!decrease as kin-\ncrease; however, in the (001) direction, there is a small in-\ncrease when kapproaches the Brillouin zone boundary. Upon\ncomparing the three directions, we find the relationship of\n\u0001!(001)>\u0001!(110)>\u0001!(111) in the region of k >\u0019=a .\nThe trend for the three modes can also be obtained from the in-\nset in Fig. 13(a), where the curves in each direction are almost\nexactly the same, indicating that the anisotropy plays a negli-\ngible role in the broadening \u0001!. Combined with Fig.7(a), we\nfind that the \u000b- and\r- modes have the same positive polar-\nization direction, and the trend of broadening is consistent as\nthe wave vector changes in different directions. However, the\n\f- mode has a negative polarization direction, and the trend is10\ndifferent.\nUsing the broadening \u0001!of the spin-wave spectra at room\ntemperature and the uncertainty relationship of \u0001!\u0001\u001cmp=~,\nwe calculate the magnon-phonon thermalization time, \u001cmp\nor spin-lattice relaxation time to explore the magnon-phonon\ninteractions. In Fig. 13(b), \u0001!is replotted on a logarith-\nmic scale for observing the asymptotic behavior in the long-\nwavelength region, where we find a quadratic dependence\nonkof\u0001!for the\u000b- modes and constants \u0001!\f= 2:07\nmeV and \u0001!\r= 9:14meV for the \f- and\r- modes, re-\nspectively, corresponding to \u001c\f\nmp= 3:18\u000210\u000013s and\n\u001c\r\nmp= 7:19\u000210\u000014s as illustrated by the black solid lines.\nAs shown in Fig. 11, the lattice vibrations can induce fluctu-\nations of magnetic moment and the exchange constants. Addi-\ntionally, the phonon-induced fluctuation of the exchange con-\nstants has an obvious effect on the magnon spectrum and can\ninduce broadening of the spin-wave spectrum at room temper-\nature (as shown in Fig. 12). At the long-wavelength limit ( k\n!0), the acoustic phonon represents the centroid motion of\natoms in the same unit cell, so that the change in atomic dis-\nplacement caused by the temperature has little effect on the\nlattice; so for the \u000b-mode, lattice vibration induced spin-wave\nbroadening is approximately zero. In addition, the decay rate\nof the spin-wave is found to be proportional to the square of\nkat the long-wavelength limit, as shown in the hydrodynamic\ntheory for spin-wave.66,67Thus\u001cmpis proportional to k\u00002for\nthe acoustic \u000b-mode. For \f- and\r- modes, as the optical\nphonon represents the reverse motion of the positive and neg-\native ions in the unit cell, the temperature causes the fluctua-\ntion of the average displacement of atoms, which can induce\na constant spin-wave broadening, so \u001cmpis constant for the\noptical modes.\nTo compare with YIG, we also chose a specific wave vector,\nk= 5:67\u0002105cm\u00001, from Ref. 13 and 45, and values for the\n\u0001!of three modes are 6:49\u000210\u00005THz, 4:86\u000210\u00001THz,\nand1:70THz. We obtain \u001cmp= 2:45\u000210\u00009s,3:27\u000210\u000013s,\nand9:36\u000210\u000014s for the\u000b-,\f-, and\r-modes, respectively,\nwhich are approximately 4:3times, 0:6\u000210\u00003times, and 0:1\ntimes the values for the acoustic branch and lowest-frequency\noptical branch of YIG. As shown in Fig. 10(d), for the YIG-\nlike\f-mode, the sufficient density of state of the phonons\ncan induce larger magnon-phonon scattering rate in the long-\nwavelength region so that the magnon-phonon thermalization\ntime\u001cmpis rather small, which is similar to the case of YIG.13.\nFor the optical \r-mode, it also has a relatively high frequency,\nwhere the phonons have a large density of state so that the\nmagnon-phonon scattering rate is quite large, which can re-\nsult in a smaller magnon-phonon thermalization time.\nIII. CONCLUSION\nIn conclusion, we investigate the NN exchange interaction\ncoefficient using a more reliable and accurate method, which\nhas been applied to YIG. We obtained the Curie temperature\nand magnetic compensation temperature that matched the ex-\nperiment well. We found that the spin-wave spectrum ob-tained by numerical methods using the exchange constants\ncan explain the experimental phenomena in SSE well. We\nreveal the spin-wave precession mode in the low frequency\nregion, which indicates that the acoustic branch \u000b-modes and\nYIG-like optical branch \f-modes have different chiral charac-\nteristics, but the same as the lower optical \r-modes. A first-\nprinciples phonon calculation method was used to obtain the\nphonon spectrum of GdIG and YIG at zero temperature. We\nreproduce the fitting intersecting point of the spin-wave and\nphonon branches(LA, TA) that are in good agreement with\nexperiment results in the very low-energy region. We discuss\nthe interaction between magnons and phonons in GdIG by in-\ntroducing temperature-dependent lattice shifts. Three special\nspin-wave modes ( \u000b,\f, and\r) are found to exhibit differ-\nent broadening of the spin-wave spectrum, \u0001!of GdIG. In\na small wave vector region, the \u0001!of the\u000b- modes have a\nsquare relationship with wave vector k ( \u0001!\u0018k2). For the\n\f- modes, the \u0001!are nearly a constant, which is similar to\nthe lower optical branch of YIG.13A higher optical branch\n\r-mode also exists below KbT\u00186:25THz at room tempera-\nture, which may play an indispensable role in magnon-phonon\ncoupling, and the \u0001!has also a constant relationship with k.\nAt a particular wave vector, the magnon-phonon thermaliza-\ntion time,\u001cmp, for these branches at room temperature is also\ndifferent from that of YIG. \u001cmp\u001810\u00009s for the\u000b-mode is\nbigger than the acoustic branch of YIG, the \u001cmpof the\f- and\n\r-mode (\u001810\u000013and\u001810\u000014s) are smaller than the acous-\ntic branch and lower optical branch of YIG, respectively. The\nmagnon-phonon coupling effect may play more central role in\nhigher spin-wave modes compared with lower modes.\nAdditionally, we also do ab initio phonon calculations us-\ning the finite-displacement method in the packages V ASP,48,49\nABACUS,68and QUANTUM ESPRESSO(QE) package69\ncombined with Phononpy56,60to obtain the phonon spectrum\nof YIG and GdIG, and the results are consistent with those\npresented in this paper (not shown here). A well-known prob-\nlem with most of the theories of magnon-phonon coupling is\nthat they do not take into account the magnon-magnon cou-\npling or magnon-phonon coupling directly. 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Tiersten’s equations for  small fields superposed on finite initial fields \nin a saturated ferromagnetoelastic material are employed, with their quasistatic magnetic field \nextended to dynamic electric and magnetic fields governed by Maxwell’s equations for \nelectromagnetic waves. Dispersion relations of the plane waves are obtained. The cutoff \nfrequenc ies and long -wave approximation  of the dispersion cur ves are determined. Results show \nthat acousti c, electromagnetic and magnetic spin waves are coupled in such a material. For YIG \nwhich is a cubic crystal without piezoelectric coupling, the acoustic and electromagnetic waves \nare not directly coupled but they can still interact indirectly through spin waves.   \n \n1. Introduction  \nIn saturated ferromagnetic solids, the  magnetization vector has a fixed magnitude (saturation \nmagnetization) and can change its direction only in a precessional motion. Below the Curie \ntemperature, n eighboring magnetization v ectors align themselves in a certain direction called an \neasy axis of the material to form a distribution of spontaneous magnetization. A disturbance of \nthe magnetization field propagates as spin waves with various applications [1]. Spin waves can \ninteract wit h acoustic waves through  magnetoelastic couplings such as piezoma gnetic and \nmagnetostrictive effects, which is called magnon -phonon interaction for which many references \ncan be found in [2], a recent review article. Obviously, as a motion of magnetic moments, spin \nwaves interact with electromagnetic waves directly as go verned by Maxwell’s equations which is \nreferred to as photon -magnon coupling (see [3] and the references therein). Thus, in deformable \nferromagnetic solids, acoustic waves and  electromagnetic waves can interact indirectly through \nspin waves. If the materia l is piezoelectric, acoustic waves and electromagnetic  waves are also \ncoupled piezoelectrically . It has been reported recently [4] that the couplings among surface \nacoustic w aves (SAW), spin waves and electromagnetic waves can be used for making \nelectromag netic antenna at SAW frequencies  (low-frequency antenna) . This has motivated our  2 study below on the propagation of coupled acoustic, electromagnetic and spin waves (p honon -\nphoton -magnon interaction ). \n \n2. Governing Equations  \nConsider the widely -used Yttrium  Iron G arnet (Y3Fe5O12) or YIG. In Gaussian units, the \ngoverning equations are  [5,6] \n0\n,,M\nij i i j i jM h u\n,                                                           (1) \n0M d\n,                                                                 (2) \n0M b\n,                                                               (3)  \n10M\nM\nCt  be\n,                                                         (4)  \n1M\nM\nCt dh\n,                                                            (5) \n00\n,1()ML\nijk j k lk l k ijk j k iM h a h m H m   \n,                                         (6) \nwhere τ is the stress tensor. ρ is the mass density. u is the displacement vector. eM, dM, bM and hM \nare the Maxwellian electric field, electric displacement, magnetic induction and magnetic field. C \nis the speed of light in a vacuum. M0 and H0 are the initial m agnetization and initial magnetic \nfield which are static . The initial electric and polarization fields are assumed to be zero. hL is an \neffective local magnetic field which describes the interaction between the magnetic spin and the \nlattice  [5]. a describe s the exchange interaction between neighboring magnetic spins  [5]. m is the \nincremental magnetization  vector . γ is the gyromagnetic ratio which is a negative number. (1) is \nthe linear momentum equation . (2)-(5) are Maxwell’ s equations . (6) is the angular momentu m \nequation of the magnetic spin . (2) and (3) are essentially implied by (4) and (5). We also have the \nfollowing relationship s:  \n0\n,4,\n4 4 ,MM\ni i i\nMM\ni i i i j jd e p\nb h m M u\n\n  \n                                                   (7) \nwhere p is the electric polarization vector. Magne toelectric coupling , if present,  is not considered .  \nYIG is a cubic crystal  of class (m3m) . Let the  spontaneous magnetization M0 (and H0) be \nalong the x3 axis. In this case m3=0 because of the saturation condition \nMM =(M0)2 which \nimplies that M0·m=0 where M=M0+m and m is small . The constitutive relations are  [6] \n1 11 11 1,1 12 2,2 12 3,3\n2 22 12 1,1 11 2,2 12 3,3\n3 33 12 1,1 12 2,2 11 3,3,\n,\n,c u c u c u\nc u c u c u\nc u c u c u\n\n   \n   \n   \n                                               (8) \n0\n4 23 44 2,3 3,2 44 2\n0\n5 31 44 1,3 3,1 44 1\n6 12 44 1,2 2,1( ) 2 ,\n( ) 2 ,\n( ),c u u b M m\nc u u b M m\nc u u\n\n   \n   \n  \n                                         (9) \nore M M M\ni i i ip e d e\n,                                               (10) \n0 2 0\n1 1 44 1,3 3,1\n0 2 0\n2 2 44 2,3 3,2\n3( ) 2 ( ),\n( ) 2 ( ),\n0,L\nL\nLh M m b M u u\nh M m b M u u\nh\n  \n  \n\n                                     (11) \n  3 \n0 2 0\n1 1 44 1,3 3,1\n0 2 0\n2 2 44 2,3 3,2\n3( ) 2 ( ),\n( ) 2 ( ),\n0,L\nL\nLh M m b M u u\nh M m b M u u\nh\n  \n  \n                                     (12) \n11 ,2ib b iam\n,                                                           (13) \nwhere  \n3 11 2\n11\n11 2 11 2\n12 44\n22\n11 12 44\n5 2 11 2\n4 12 4 11 11\n7 2 05.172 g/cm , 26.9 10 dyn/cm ,\n10.77 10 dyn/cm , 7.64 10 dyn/cm ,\n1.66 10 , 1.66 10 ,\n3 3.36 10 Oe , 1.87 10 cm ,\n1.76 10 Oe-cm /dyn-sec, 1750 / 4 G.c\ncc\nb b b\nM\n   \n    \n   \n    \n     \n  \n                      (14) \nYIG is nonpiezo electric and nonpiezomagnetic in its natural state without any fields. Due to the \nspontaneous magnetization and magnetostriction, it becomes effectively piezomagnetic.  \n        \n3. Antiplane Motion  \nConsider cubic crystals of class (m3m)  such as YIG in Gaussi an units . With  the initial \nmagnetization M0 and magnetic field H0 along the x3 axis, for antiplane problems [7] with \nu1=u2=0 and ∂/∂x3 = 0, the relevant fields are  \n1 2 3 3 1 2\n1 2 3 3 1 2\n1 2 3 3 1 2\n1 1 1 2 2 2 1 2 3\n1 1 1 2 2 2 1 2 30, ( , , ),\n0, 0, ( , , ),\n0, 0, ( , , ),\n( , , ), ( , , ), 0,\n( , , ), ( , , ), 0.M M M M\nM M M M\nM M M M M\nM M M M Mu u u u x x t\ne e e e x x t\nd d d e x x t\nb b x x t b b x x t b\nh h x x t h h x x t h  \n\n\n\n\n                                   (15) \nIn this case (2) is trivially satisfied. (4) and (5) reduce to  \n12\n3,2 3,1110, 0MM\nMM bbeeC t C t    \n,                                            (16) \n3\n2,1 1,21M\nMM dhhCt\n.                                                              (17) \nWe also have:  \n0\n44 3,11 3,22 44 1,1 2,2 3\n0 0 0 3 0 2 0\n2 11 2,11 2,22 2 44 3,2 2 1\n0 0 0 3 0 2 0\n1 11 1,11 1,22 1 44 3,1 1 2\n1,1 2,2 1,1 2,2( ) 2 ( ) ,\n12 ( ) ( ) 2 ( ) ,\n12 ( ) ( ) 2 ( ) ,\n4 ( ) 0,M\nM\nMMc u u b M m m u\nM h M m m M m b M u H m m\nM h M m m M m b M u H m m\nh h m m\n\n\n   \n      \n     \n   \n     (18)       \nwhere (18) 4 is essentially implied by (16). Then (16), (17) and (18) 1-3 can be written as six \nequations for u3, \n3Me , \n1Mh , \n2Mh , \n1m and \n2m .  \n \n \n \n \n  4 4. Propagation of Plane Waves  \nLet \n3 1 1 3 2 1\n1 3 1 2 4 1\n1 5 1 2 6 1exp[ ( )], exp[ ( )],\nexp[ ( )], exp[ ( )],\nexp[ ( )], exp[ ( )].M\nMMu A i x t e A i x t\nh A i x t h A i x t\nm A i x t m A i x t   \n   \n      \n   \n   \n                       (19) \nThe substitution of (19) into (16), (17) and (18) 1-3 results in a system of six linear and \nhomogeneous equations for A1 through A6. For nontrivial solutions, the determinant of the \ncoefficient matrix has to vanish. This leads to the following equation that determines the \ndispersion relatio n of the wave: \n \n 2 2 2 6 2 2 2 4\n44 44\n2 2 2 44\n44 2 0 2\n24\n2 0 2\n2 2 6 2 2 2 4\n44 44\n2 2 2 44\n44 2 0 2(2 4 )\n( 4 ) (2 4 )()\n( 4 )()\n2 ( 4 )\n( 4 ) ( 4 ) 2 ( 4 )()C c c P e\nce P c P P PM\nPPM\nc c P e\nce P c P PM        \n    \n   \n        \n        \n        \n   \n    \n       2\n2 2 4\n2 0 2( 4 ) 0,()PM\n    \n  \n    \n                (20) \nwhere  \n0 0 0 0 2\n44 112 , 2 , / , ( )e b M P H M K K M        \n.                        (21) \nIn the special case of b44=0, the magnetoelastic coupling disappears and (20) reduces to th e \nproduct of two factors. One is for uncoupled acoustic waves:  \n22\n44 0 c \n.                                                   (22) \nThe other is for coupled electromagnetic and spin waves : \n22\n2 2 2 2 2 2 2\n2 0 2 2 0 2( )( 4 ) ( 4 ) 0( ) ( )C P P PMM                        \n.  (23 ) \nWhen C→∞, (23 ) reduces to the following dispersion relation  for uncoupled spin waves : \n2\n2 4 2\n2 0 2(2 4 ) ( 4 )()P P PM         \n.                               (24) \nWhen α=0 and γ→∞, (23 ) reduces to the following dispersion relation for uncoupled \nelectromagnetic waves  \n22\n2( 4 )CP\nP\n  \n.                                                         (25)\n \nWhen ε=1 and M0→0, we have P→∞ and (25 ) reduces to  ω/ξ=C for electromagnetic waves in a \nvacuum.  \nWhen H0=1500 Oe and ε=14, the disper sion relatio ns of the uncoupled waves in (22), (24) \nand (25) are shown in Fig. 1 in logarithmic scales for both the coordinate and the abscissa. The \nacoustic and electromagnetic waves are represented by straight lines  and are nondispersive , with  5 the electromagnetic waves at higher frequencies. The spin wave is represented by a curve. Each \nstraight li ne intersects with the curve at two points.  \n Fig. 1. Uncoupled  acoustic, electromagnetic and spin  waves. \n \nWhen H0=1500 Oe  and ε=14, the dispersion  relations of the coupled waves determined by (20) \nare shown in Fig. 2 . Near B and C there are strong couplings between acoustic and spin waves. \nNear A and D there are strong couplings between electromagnetic and spin waves. Since the \nmaterial is nonpiezoelectric, there is no direct coupling between acoustic and electromagnetic \nwaves.   \n \nFig. 2 . Dispersio n curves of coupled waves when H0=1500 Oe  and ε=14.  \n \n \n \n  6 H0 is an independent parameter. If it is varied a little, its effect s on the d ispersion curves are  \nshown in Fig. 3. The spin waves are sensitive to H0 but the two other waves are not, which is \nreasonable.  \n \nFig. 3.  Effects of H0 (in Oe) on dispersion relations of coupled waves.  ε=14. \n \nThe numerical value of ε for YIG in the literat ure ran ges from 14 to 18. The effect of slightly \ndifferent values of ε on the dispersion curves of the coupled waves is shown in Fig. 4 where ε is \ndenoted by εr which mainly affects the electromagnetic waves as expected.  \n \nFig. 4 . Effects of εr=ε on disper sion curves of coupled waves . H0=1500  Oe. \n \n \n  7 5. Conclusion s \nIn saturated ferromagnetoelastic solids such as YIG, acoustic, electromagnetic and magnetic \nspin waves are coupled. Thus it is possible to manipulate one wave by another  or design \ntransducers usin g the couplings of these waves . This offers more possibilities for new devices. At \npresent, the literature on the three -wave coupling of photons, phonons and magnons are limited, \nwith an absence of the mechanics community. Since the equations of elasticity  represent a maj or \npart of the coupled theory for  these three waves, mechanics researchers can play an important role \nin this interdisciplinary area.  \n \nAcknowledgment  \nThis work was supported by the National Natural Science Foundation of China (Nos. \n12072167  and 11972199), the Zhejiang Provincial Natural Science Foundation of China ( No. \nLR12A02001 ), and the K. C. Wong Magana Fund through Ningbo University . \n \nReferences  \n[1] D.D. Stancil , A. Prabhakar , Spin Waves : Theory and Applications , Springer, New York, \n2009  \n[2] D.A. Bozhko, V.I. Vasyuchka, A.V. Chumak, A.A. Serga, Magnon -phonon interactions in \nmagnon spintronics (Revi ew article), Low Temp. Phys., 46, 383 -399, 2020.  \n[3] B. Bhoi, S. -K. Kim, Roadmap for photon -magnon coupling and its applications, Solid State \nPhysics , Volume 71, Chapter 2, 39 -71, Elsevier, 2020.  \n[4] R. Fabiha, J. Lundquist, S. Majumder, E. Topsakal, A. Ba rman , S. Bandyopadhyay , Spin \nwave electromagnetic nano -antenna enabled by tripartite phonon -magnon -photon coupling, \nAdv. Sci., 9, 2104644 , 2022 .  \n[5] H.F. Tiersten, Coupled magnetomechanical equation for magnetically saturated insulators, J. \nMath. Phys., 5 , 1298 -1318, 1964.  \n[6] H.F. Tiersten, Thickness vibrations of saturated m agneto elastic p lates, J. Appl. Phys.,  36, \n2250 -2259, 1965.  \n[7] Q.G. Xia, J.K. Du, J.S. Yang, Antiplane problems of saturated ferromagnetoelastic solids, \nActa Mech., under review.  "    },    {        "title": "1505.01882v1.Formation_of_Bright_Solitons_from_Wave_Packets_with_Repulsive_Nonlinearity.pdf",        "content": " 1Formation of Bright Solitons from Wave Packets with \nRepulsive Nonlinearity  \n \nZihui Wang ,1 Mikhail Cherkasskii,2 Boris A. Kalinikos ,2 Lincoln D. Carr ,3 and Mingzhong Wu1* \n1Department of Physics, Colorado  State  University, Fort Collins, Colorado  80523 , USA \n2St.Petersburg Electrotechnical  University, 197376, St.Petersburg, Russia  \n3Department of Physics, Colorado School of Mines, Golden, Colorado 80401, USA  \n \nFormation of bright envelope solitons from wave packets with a repulsive nonlinearity was \nobserved for the first time.  The e xperiments used surface spin -wave  packets  in magnetic yttrium \niron garnet (YIG) thin film strips.  When the wave packet s are  narrow and ha ve low power, they \nundergo self -broadening during the propagation.  When the wave packet s are relatively wide or \ntheir power is relatively high, they can experience  self-narrowing or even evolve into bright \nsoliton s.  The experimental results were reproduced by numerical si mulations based on a modified \nnonlinear Schrödinger equation model . \n \n   \n 2Solitons are a universal  phenomenon in nature, appearing in systems as diverse as water, optical fibers, \nelectromagnetic transmission lines, deoxyribonucleic acid, and ultra -cold quantum gases .1,2,3,4,5  The formation of \nsolitons from large -amplitude waves  can be described by paradigmatic  nonlinear equations, one of which is the \nnonlinear Schrödinger equation (NLSE).  In the terms of the NLSE model, two classes of envelope solitons, bright \nand dark, can be excited  in nonlinear media.  A bright envelope soliton  is a localized excit ation on the envelope of a \nlarge -amplitude carrier wave.  It typically takes a hyperbolic secant shape and has a constant phase across its width.6  \nA dark envelope soliton  is a dip or null in a large -amplitude wave background.  When the dip goes to zero, one has a \nblack soliton.  When the amplitude at the dip is nonzero, one has a gray soliton.   A dark soliton has  a jump in phase \nat its center.  For a black soliton, such a phase jump equals to  π. For a gray soliton, the phase jump is between 0 and \nπ.  The envelope of a dark soliton can be described by a unique function.3  For a black soliton, this function is typically \na hyperbolic tangent function.  \nAccording to the NLSE model, the formation of a bright soliton from a large -amplitude wave packet  is possible in \nsystems with an attractive (or self -focusing) nonlinearity  and is prohibited in systems with a repulsive (or defocusing) \nnonlinearity .  The un derlying physics is as follows .  The attractive nonlinearity produces a pulse self -narrowing effect ; \nat a certain power level the self -narrowing can balance the dispersion -induced pulse self-broadening and give rise to \nthe formation of a bright envelope so liton.   In contrast, in systems with a repulsive nonlinearity  the nonlinearity \ninduces self -broadening of the wave packet, just as the dispersion does , and thereby disables the formation of a bright \nsoliton .  Previous experiments show good agreement s with these theoretical predictions:  the formation of bright \nsolitons from wave packets has been demonstrated in different systems  with an attractive nonlinearity ,3,7 while the \nself-broadening has been observed for wave packets in systems with a repulsive nonlinearity.8 \nThis letter reports  on the first observation of the formation of bright solitons from  wave packets with a repulsive  \nnonlinearity.  The experiments made use of spin waves traveling along long and narrow magnetic yttrium iron garnet \n(Y3Fe5O12, YIG)9 thin film strip s.  The YIG strips  were magnetized by static magnetic field s applied in the ir plane s \nand perpendicular to the ir length  direction s.  This film/field configuration supports the propagation of surface spin \nwaves with a repulsive nonlinearity.10,11  To excite a spin wave  packet in the YIG strip, a microstrip line transducer \nwas placed on  one end of the YIG strip  and was fed with a microwave pulse .  As the spin wave  packet propagates \nalong  the YIG strip, it was measured by either a secondary microstrip line or a magneto -dynamic inductive probe \nlocated above  the YIG strip. When the input microwave pulse is relatively narrow and has relatively low  power,  one  \n 3observes the broadening of the spin wave  packet  during its propagation .  At certain large input pulse width s and high \npower level s, however, the spin wave  packet undergoes self -narrowing and evolves into a bright  envelope soliton.  \nThe formation of this soliton is contradictory to the prediction of the standard NLSE model , but was reproduced by \nnumerical simulations with a modified NLSE model that took into account damping and saturable  nonlinearity . \nFigure 1 shows representative data on the formation of bright solitons from surface spin -wave  packets.  Graph (a) \nshows the experimental configuration.  The YIG film strip was cut from a 5.6-mm-thick (111) YIG wafer grown on a \ngadolinium gallium garnet substrate.  The strip was 30 mm long and 2 mm wide.  The magnetic field was set to 910 \nOe.  The input and output transducers were 50 -mm-wide striplines and were 6.3 mm apart.  The input microwave \npulses had a carrier frequency of 4.51 GHz.   Note that, in Fig. 1 and other figures as well as the discussions below, \nPin denotes the nominal microwave pulse power applied t o the input transducer, tin denotes the half-power width of \nthe input microwave pulse, Pout is the power of the output signal, and tout represents the half -power width of the output \npulse.  In Fig. 1, g raphs (b), (c), (d), (f), and (g) give the power profiles  of the output signals measured with different \n \nFIG. 1.  Propagation of spin -wave packets in a 2.0-mm-wide YIG strip.  (a) Experimental setup.  (b), (c), (d), (f), and (g) Envelopes \nof output signals obtained at different input pulse power levels ( Pin) and widths ( tin).  (e) Phase ( q) profile for the signal shown in \n(d).  (h) Width of output pulse ( tout) as a function of Pin.   (i) Width of output pulse as a function of tin. \n \n 4Pin andtin values , as indicated .  The circles  in (d) shows a fit to the hyperbolic secant squared function.1,3  Graph (e) \nshows the corresponding phase ( q) profile of the  signal shown in (d) .  Here , the profile shows the phase relative to a \nreference continuous wave whose frequency equals to the carrier frequency of the input microwave.6  Graph (h) shows \nthe change of tout with Pin for a fixed tin, as indicated, while graph (i) shows the change of tout with tin for a fixed  Pin, \nas indicated.  \nThe data in Fig. 1 show three important results.  (1) The data in Figs. 1 (b) -(e) and (h) show the change of the \noutput signal with the input power  Pin.  One can see that the output pulse  is broader than the input pulse when Pin=13 \nmW, as shown in (b), and is significantly narrower when Pin>30 mW , as shown in (c), (d), and (h).  This indicates that \nthe spin-wave  packet undergoes self -broadening at low power and self -narrowing at relatively high power.   (2) The \ndata in Figs. 1 ( d), (f), (g),  and ( i) show the change of the output signal with the input pulse width  tin.  It is evident \nthat the width of the output pulse increases with tin when tin<50 ns and then saturates to about 19.5 ns when tin>50 ns.  \nThese results indicate that the spin-wave  packe t experiences strong self -narrowing when it is relatively broad.  (3) The \npulses shown in (d) and (g) are indeed bright solitons.  As shown representatively in (d) and (e), they have a hyperbolic \nsecant shape and a constant phase profile at their centers, which are the two key signatures of bright solitons.1,6 \nThe data from Fig. 1 clearly demonstrate the formation of bright solitons from surface spin -wave  packets when \nthe energy of the initial signals (the product of Pin and tin) is beyond a certain level.  This result is contradictory to the \npredictions of the NLS E model.  One possible argument is that the width of the YIG strip might play a role in the \nobserved formation of bright solitons.  To rule out this possibility, similar measurements were carried out with a n YIG \nstrip that is an order of magnitude narrower.  The main data are as fo llows.  \nFigure 2 gives the data measured with a 0.2 -mm-wide YIG strip .  This figure is shown in the same format as in \nFig. 1.  In contrast to the data in Fig. 1, the data here were measured by a 50 -W inductive prob e,12 rather than a \nsecondary microstrip transducer.  The distance between the input transducer and the inductive probe was about 2.6 \nmm.  The magnetic field was set to 1120 Oe.  The input microwave pulse had a carrier frequency of 5.07 GHz.   \nThe data in Fig. 2 show results very similar to those shown in Fig. 1.  Specifically, the low -power, narrow spin-\nwave  packets undergo self -broadening, as shown in (b), (c), (f), and (h); as the power and width are increased to certain \nlevels, the spin-wave  packets  experience self -narrowing, as shown in (h) and (i), and can also evolve into solitons, as \nshown in (d), (e), and (g).  Therefore, the data in Fig. 2 clearly confirm the results from Fig. 1.  This indicate s that the \nformation of solitons reported here is n ot due to any effects associated with the YIG strip width.   Note that the solitons  \n 5shown in Fig. 2 are narrower than those shown in Fig. 1.  This difference results mainly from the fact that the spin-\nwave amplitudes and dispersion properties  were different in the two experiments .  The spin -wave dispersion differed \nin the two experiments because the magnetic field s were  different and the wave number s of the excited spin -wave \nmode s were also not the same.       \nTurn now to the spatial formation of solitons from surface spin -wave packets.  F igure 3 shows representative data.  \nGraph (a) gives the profile  of an inpu t signal.  The power and carrier frequency of the input signal were  700 mW and \n5.07 GHz, respectively.  Graphs (b) -(f) give the corresponding output signals measured with the same experimental \nconfiguration as depicted in Fig. 2(a).  The signals were measured by placing the inductive probe at different distances \n(x) from the input transducer, as indicated.   The red curves in ( b)-(f) are the corresponding phase profiles.  \nThe data in Fig. 3 show the spatial evolutio n of a spin-wave packet.  At x=1.1 mm, the packet has a width similar \nto that of the input pulse.  As the packet propagates to x=2.1 mm, it develops into a soliton, which is not only much  \nFIG. 2.  Propagation of spin -wave packets in a 0.2 -mm-wide YIG strip.  (a) Experimental setup.  (b), (c), (d), (f), and (g) Envelopes \nof output signals obtained at different input pulse power levels ( Pin) and widths ( tin).  (e) Phase ( q) profile for the signal shown in \n(d).  (h) Width of output pulse ( tout) as a function of Pin.   (i) Width of output pulse as a function of tin. \n \n 6narro wer than both the initial pulse and the packet at x=1.1 mm but also has a constant  phase at its center portion, as \nshown in (c).  At x=2.6 mm, the packet has a lower amplitude due to the magnetic damping but still maintains its \nsolitonic  nature, as shown in (d).  As the packet continues to propagate further, it loses its solitonic properties and \nundergoes self -broadening, as shown in (e) and (f) , due to significant reduction in amplitude .  Note that the phase \nprofiles for all the signals in (b), (e), and (f) are not constant .  These results support the above -drawn conclusion, \nnamely, that it is possible to produce a bright soliton from a surface spin -wave packet.   \nThe data in Fig. 3  also indicate the other two important results.  (1) The development of a soliton takes a certain \ndistance, about 2 mm for the above -cited conditions, due to the fact that the nonlinearity effect needs a certain \npropagation distance to develop.  (2) The soliton exists on ly in a relatively short range, about 1 -2 mm for the above -\ncited conditions , due to the damping  of carrier spin waves.  To increase the \"life\" distance  or lifetime of a spin-wave \nsoliton, one can take advantage of parametric pumpin g13 or active feedback9 techniques.  \nAs mentioned above, the  soliton formation  presented here is contradictory to the standard NLSE model.  However, \nit can be reproduced by numerical simulations based on the equation   \nFIG. 3.  Spatial formation of a spin -wave soliton in a 0.2 -mm-wide YIG strip.  (a) Profile of an input signal.  (b) -(f) Profiles of  \noutput signals measured by an inductive probe placed at different distances ( x) from the input transducer.  The red curves in ( b) \nand ( f) are the corresponding phase profiles.  800 850 900 950 10000.00.20.40.60.8Power (W)(a) Input pulse\n800 850 900 950 10000.000.020.040.06 (b) x=1.1 mmPower (mW)\n800 850 900 950 10000.000.010.020.030.04\n(f) x=4.6 mm (e) x=3.6 mm(d) x=2.6 mm (c) x=2.1 mmPower (mW)\n800 850 900 950 10000.000.010.02Power (mW)\n800 850 900 950 10000.000.010.02Power (mW)\nTime (ns)800 850 900 950 10000.000.010.02Power (mW)\nTime (ns)\nPhase\n180 ºPhase\n180 º\nPhase\n180 º180 º\nPhase\n180 º\nPhase \n 7( )22 4\n2102gu u ui v u D N u S u ut x xh¶ ¶ ¶é ù+ + - + + =ê ú¶ ¶ë û ¶        (1) \nwhere u is the amplitude of a spin -wave packet, x and t are spatial and temporal coordinates, respectively, vg is the \ngroup velocity, h is the damping coefficient, D is the dispersion coefficient, and N and S are the cubic and quintic \nnonlinearity coefficients, respectively.   The quantic nonlinearity term is included because the cubic nonlinearity i s \ninsufficient to capture the experimental observations presented abov e.  This additional term is an expansion to the \nlowest order of saturable nonlinearity.  The simulations used the split -step method to solve the derivative terms with \nrespect to x and used the  Runge -Kutta method  to solve the equation with the rest of the terms.14,15  A high-order \nGaussian profile  was taken in simulations for the input pulse because it is much closer to the experimental situation \nthan a squared pulse.   The use of a square pulse as in the input pulse gave rise to numerical noise due to the \ndiscontinuity at the pulse's edges.  The use of a fundamental Gaussian function  did not onsiderably change the \nsimulation results.   It sho uld be noted that both the standard and modified NLSE  models are for nonlinear waves in \none-dimensional (1D) systems, and previous work had demonstrated the feasibility of using the 1D NLSE models  to \ndescribe nonlinear spin waves in quasi -1D YIG film strips.16,17   \nFigure 4 shows representative results obtained for different initial pulse amplitudes ( u0), as indicated.  In each \npanel, the left and right diagrams show the power and phase profiles, res pectively.  The simulations were carried out \nfor a  20-mm-long 1D film strip and a  total propagation time of 250 ns.  The film strip was split into 9182 steps, and \nthe temporal evolution step was set to 0.05 ns.  The input pulse was a high -order Gaussian profile with an order number \nof 20 and a half-power width of 15 ns.   The other parameters used are as follows: vg=3.8×106 cm/s, h=3.1×106 rad/s, \nD=-4.7×103 rad×cm2/s, N=-10.1×109 rad/s,  and S=1.8×1012 rad/s.  Among these parameters,  vg, D, h, and N were \ncalculated according to the properties of the YIG film, 9 and the S was optimized for the reproduction of the \nexperimental responses.      \nThe profiles in Fig. 4 indicate that, at low initial power , the pulse is broader than  the initial pulse and has a phase \nprofile which is not constant  at the pulse center, as shown in (a) and (b); at relatively high power, however, the pulse \nis not only  significantly narrower than the initial pulse but also has a constant phase across its center portion, as shown \nin (c).  These results agree with the experimental results presented  above.   \nThe reproduction of the experimental responses with the modified NLSE model indicates  the underlying physical  \n 8processes for the formation of bright solitons from surface spin -wave packets.  In comparison with the standard NLSE, \nthe additional terms  in the modified equation  are uh\n and 4S u u .  The term uh accounts for the damping of spin \nwaves in YIG films, while the term 4S u u  is needed for the reproduction of the experimental responses.  Since the \nsign of S was opposite to that of N, the term 4S u u  played a role opposite to 2N u u and caused nonlinearity \nsaturation.  In particular, for the configuration cited for  Fig. 4(c) the term 4S u u  overwhelm ed the term 2N u u , \nresulting in a repulsive -to-attractive nonlinearity transition and the formation of a bright soliton.  Thus, one can see \nthat the saturable nonlinearity played a critical role in the formation of the bright soliton s from surface spin -wave \npackets .  It should be noted that t he saturable nonlinearity has been known as a critical factor for the formation of \nsolitons in optical fibers.18  \nIn summary, this letter reports the first observation of the formation of bright solitons from surface spin -wave \npackets  propagating in YIG thin films .  The formation of such soliton s was observed in YIG film strips with \nsignificantly different widths.  The spatial evolution of the solitons was measured by placing an inductive probe at \ndifferent po sitions along the YIG strip.  The experimental observation was reproduced by numerical simulations based  \nFIG. 4.  Power (left) and phase (right) profiles of spin -wave packets propagati ng in a YIG strip.  The profiles were obtained from \nsimulations with different initial pulse amplitudes, as indicated, for a propagation distance of 4.9 mm. 150160170 1801902002100.00.20.40.60.81.0Power (a.u.)\nTime (ns)160 180 200-180-90090180Phase (degree)\nTime (ns)(a) Initial pulse amplitude u0=0.0005\n150160170 1801902002100.00.20.40.60.81.0Power (a.u.)\nTime (ns)150 160170180 190200210-180-90090180Phase (degree)\nTime (ns)\n150160170 1801902002100.00.20.40.60.81.0Power (a.u.)\nTime (ns)150 160170180 190200210-180-90090180(c) Initial pulse amplitude u0=0.071(b) Initial pulse amplitude u0=0.005Phase (degree)\nTime (ns) \n 9on a modified NLSE model.  The agreement between the experimental and numerical results indicates that the \nsaturable nonlinearity played important role s in the soliton formation.  \nThis work was supported in part by U. S. National Science Foundation (DMR -0906489 and ECCS -1231598)  and \nthe Russian Foundation for Basic Research . \n \n*Corresponding author.   \n  E-mail: mwu@lamar.colostate.edu  \n   \n 10 \n1 M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform  (SIAM, Philadelphia, 1985).  \n2 A. Hasegawa and Y. Kodama, Solitons in Optical Communications  (Oxford, New York, 1995).  \n3 M. Remoissenet, Waves Called Solitons: Concepts and Experiments  (Springer -Verlag, Berlin, 1999).  \n4 Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystal s (Academic, New York, 2003).  \n5 P. G. Kevrekidis, D. J. Frantzeskakis, and R. Carretero -González, Emergent Nonlinear Phenomena in Bose -Einstei n \nCondensates  (Springer -Verlag, Berlin, 2008).  \n6 J. M. Nash, P. Kabos, R. Staudinger, and C. E. Patton, J. Appl. Phys. 83, 2689 (1998).  \n7 M. Chen, M. A. Tsankov, J. M. Nash, and C. E. Patton, Phys. Rev. B 49, 12773 (1994) . \n8 M. Chen, M. A. Tsankov, J. M. N ash, and C. E. Patton, Phys. Rev. Lett. 70, 1707 (1993) . \n9 Mingzhong Wu, “Nonlinear Spin Waves in Magnetic Film Feedback Rings, ” in Solid State Physics  Vol. 62, Edited \nby Robert Camley and Robert Stamps (Academic Press, Burlington, 2011), pp. 163 -224. \n10P. Kabos and V.  S. Stalmachov, Magnetostatic Waves and Their Applications  (Chapman and Hall, London, 1994) . \n11D. D. Stancil and A. Prabhakar, Spin Waves – Theory and Applications  (Springer, New York, 2009).  \n12 M. Wu, M. A. Kraemer, M. M. Scott, C. E. Patton, an d B. A. Kalinikos,  Phys. Rev. B 70, 054402 (2004) . \n13 A. V. Bagada, G. A. Melkov, A. A. Serga, and A. N. Slavin, Phys. Rev. Lett. 79, 2137 (1997) . \n14 J. A. C. Weideman and B. M. Herbst, SIAM Journal on  Numerical Analysis 23, 485 (1986) . \n15 D. Pathria  and J. L. Morris, J. Comp. Phys. 87, 108 (1990).  \n16 H. Y. Zhang, P. Kabos, H. Xia, R. A. Staudinger, P. A. Kolodin, and C. E. Patton, J. Appl. Phys. 84, 3776 (1998).  \n17 Z. Wang, A . Hagerstrom, J . Q. Anderson, W . Tong, M . Wu, L . D. Carr, R . Eykholt, and B . Kalinikos, Phys. Rev. \nLett. 107, 114102 (2011).  \n18 S. Gatz and J. Herrmann, J. Opt. Soc. Am. B  8, 2296 (1991) .                                                         "    },    {        "title": "1901.01207v1.How_to_accurately_determine_a_saturation_magnetization_of_the_sample_in_a_ferromagnetic_resonance_experiment_.pdf",        "content": "How to accurately determine a saturation magnetization of the sample\nin a ferromagnetic resonance experiment?\nP. Tomczak\u0003\nQuantum Physics Division\nFaculty of Physics, Adam Mickiewicz University ul. Umultowska 85,\n61-614 Pozna\u0013 n, Poland\nH. Puszkarski\nSurface Physics Division\nFaculty of Physics, Adam Mickiewicz University ul. Umultowska 85,\n61-614 Pozna\u0013 n, Poland\n(Dated: March 11, 2022)\nThe phenomenon of ferromagnetic resonance (FMR) is still being exploited for determining the\nmagnetocrystalline anisotropy constants of magnetic materials. We show that one can also deter-\nmine accurately the saturation magnetization of the sample using results of FMR experiments after\ntaking into account the relationship between resonance frequency and curvature of the spatial dis-\ntribution of free energy at resonance. Speci\fcally, three examples are given of calculating saturation\nmagnetization from FMR data: we use historical Bickford's measurements from 1950 for bulk mag-\nnetite, Liu's measurements from 2007 for a 500 mn thin \flm of a weak ferromagnet (Ga,Mn)As, and\nWang's measurements from 2014 for an ultrathin \flm of YIG. In all three cases, the magnetization\nvalues we have determined are consistent with the results of measurements.\nIntroduction | For a very long time, the ferromag-\nnetic resonance has been used successfully to \fnd mag-\nnetocrystalline anisotropy constants and spectroscopic\nsplitting factors of ferromagnets, see e.g., Ref. [1]. Re-\ncently, we have shown [2] that using this classic experi-\nmental technique completed with a cross-validation of the\nnumerical solutions of Smit-Beljers equation [3{6], not\nonly makes it possible to determine very precisely the\nspectroscopic splitting factor gand magnetocrystalline\nanisotropy constants, e.g., up to fourth order for the di-\nluted magnetic semiconductor (Ga,Mn)As, but also to\nstate which anisotropy constants are necessary to prop-\nerly describe the spatial distribution of the energy which\nis related to magnetocrystalline anisotropy. In what fol-\nlows we present that one can interpret the results of FMR\nexperiment in such a way that it is possible to determine\nthe saturation magnetization of the sample under inves-\ntigation, i.e, that it is possible to \fnd the spatial depen-\ndence of magnetocrystalline energy stored in a ferromag-\nnet using data collected in a single FMR experiment.\nWe will show how to determine saturation magneti-\nzation for three cases: for bulk magnetite, Fe 3O4, we\nwill use the historical Bickford's measurements [7] from\n1950, for a 500 mn \flm of a weak diluted ferromagnet,\n(Ga,Mn)As, we will use Liu's measurements [8] from 2007\nand for an ultrathin \flm of YIG we will use Wang's mea-\nsurements [9] from 2014.\nWhat does one measure while performing FMR\nexperiment? | The most simple answer is that the\nspatial distribution of resonance \feld Hris measured,\ni.e., its dependence on angles #Hand'H, see Fig. 1.\nHowever, let us consider two possible ways of performing\nthe FMR experiment. First, while \fxing a frequency !of\nan alternating microwave \feld one measures the depen-\nFIG. 1: The geometry of the FMR experiment. The\norientation of the applied magnetic \feld His usually\ndescribed in the coordinate system attached to the\ncrystallographical axes of the sample. The \feld\ndirection is represented by angles #Hand'Hand\nequilibrium direction of the sample magnetization Mis\nrepresented by angles #and'.\ndence of the static magnetic \feld Hron angles#Hand\n'Hat resonance, and the second | a static magnetic\n\feld is \fxed and one changes the frequency of microwave\n\feld to \fnd a frequency at which the precession of the\nmagnetic moment of the sample occurs. The resonance\ncondition, for both measurement methods, is given by\nthe following Smit-Beljers equation [3{6]\n\u0012~!\ng\u0016B\u00132\n=1\nsin2#(f##f\u001e\u001e\u0000f2\n#\u001e); (1)\nwherefis free energy of the sample divided by its satu-\nration magnetization M, i.e., free energy is expressed in\nterms of real (Zeeman term) and \fctitious (demagnetiz-\ning and anisotropy terms) \felds. f#\u001e=@f\n@#@f\n@',gis the\nspectroscopic splitting factor, \u0016B{ the Bohr magneton\nand~{ the Planck constant.\nLet us recall, that the Gaussian curvature of the sur-\nfaceS(#;') at one of its points, being the product of twoarXiv:1901.01207v1  [cond-mat.str-el]  4 Jan 20192\nprincipal curvatures \u00141and\u00142, is the determinant of the\nHessian matrix calculated at this point. One obtains in\nspherical coordinates\n\u00141\u00142= det\"\n@2S\n@#21\nsin#@2S\n@S#@'\u0000cot#\nsin#@S\n@'\n1\nsin#@2S\n@'@#\u0000cot#\nsin#@S\n@'1\nsin2#@2S\n@'2+ cot#@S\n@##\n:\n(2)\nLet us now assume that we are interested in the curva-\nture of the the free energy landscape of the sample at\nresonance and put S=fin Eq. (2). Remembering that\nMprecesses around its eqilibrium position, that is, the\nequations\n@f\n@#= 0;@f\n@'= 0 (3)\nare satis\fed, we arrive at Eq. (1) if we interpret\u0000~!\ng\u0016B\u00012\nas a Gaussian curvature of the spatial distribution of free\nenergy.\nReturning to the two ways of measuring the resonance\n\feld we see, that by \fxing a static \feld and subsequent\nmeasuring the frequency !of a microwave \feld for di\u000ber-\nent angles at resonance, we see directly a spatial distri-\nbution of the curvature of the free energy landscape. If,\nhowever, we set the microwave \feld, and then by chang-\ning the static \feld we hit the resonance ( Hr) for di\u000ber-\nent angles#Hand'Hthen we will maintain a constant\nGaussian curvature of the energy landscape during the\nmeasurement.\nThe answer to the question posed at the beginning\nof this section is, that while performing a typical FMR\nexperiment by \fxed !, we measure such a \feld Hrat\nwhich the Gaussian curvature \u00141\u00142of the free energy\nlandscape is constant, i.e., does not depend on #Hand\n'H. As we shall see, this observation will not only allow\n\fnding contributions from various types of anisotropy to\nthe free energy but also will enable the determining of\nsaturation magnetization of the sample.\nWhat determines the spatial distribution of the\nGaussian curvature of the free energy at res-\nonance? | Assume that the free energy density\nF(#H;'H) of a ferromagnet placed in a magnetic \feld,\nexpressed in terms of magnetic \felds, is given by\nF(#H;'H)\nM=f(#H;'H) =\u0000HX\n\u000bn\u000bnH\n\u000b\n+MX\n\u000b;\fN\u000b\fn\u000bn\f+1\n2HcX\n\u000b6=\fn2\n\u000bn2\n\f:(4)\n\u000b;\f =x;y;z . The \frst term on the right-hand side of\nEq. (4) is the Zeeman energy, the second term is the\ndemagnetizing energy, with N\u000b\fbeing the demagnetiza-\ntion tensor. The last term describes the energy related\nto cubic magnetocrystalline anisotropy. The components\nof vectorsnH\n\u000b=H=Handn\u000b=M=Mare de\fned as\nusuall, see e.g., Eqs. (3a) and (3b) in Ref. [2] and Fig. 1.Note that with ful\flled relations (3), the free energy of a\nsample with a speci\fc geometry and saturation magneti-\nzation in a magnetic \feld does not depend on angles #;'\nexplicitly.\n(a)\n (b)\n (c)\n (d)\nFIG. 2: Terms in Eq. (4) representing \fctitious\nmagnetic \felds and their Gaussian curvatures\n(a)MP\n\u000b;\fN\u000b\fn\u000bn\f(ellipsoid) (b) its curvature\n(c)1\n2HcP\n\u000b6=\fn2\n\u000bn2\n\f(d) its curvature.\nEach component of Eq. (4) describes a spatially vary-\ning magnetic \feld with a certain curvature. For exam-\nple the spatial dependencies of the \frst two terms and\ntheir curvatures are shown in Fig. 2. Surface (a) - el-\nlipsoid - represents the \fctitious demagnetizing \feld for\nthe sample oriented along axes [100], [010] and [001]. Its\ncurvature is shown in Fig. 2b. Note that it is non-zero\nin the (001) plane, although this is not visible due to\nthe used scale. The surface (c) represents the \frst order\ncubic magnetocrystalline \fctitious \feld and in Fig. 2d\nis shown its curvature. The curvature of the Zeeman\nterm in Eq. (4) depends on the static magnetic \feld and\ncan be changed during the measurement while the cur-\nvatures associated with the \fctitious demagnetizing and\ncubic anisotropy \felds remain constant during the mea-\nsurement.\nHow to \fnd saturation magnetization and\nanisotropy \felds numerically? | During the FMR\nexperiment the total Gaussian curvature of the free en-\nergy remains constant: contributions of curvatures from\nvarious \fctitious magnetic \felds (anisotropy, demagne-\ntizing), related to the sample itself, are compensated by\nthe curvature of the Zeeman term. To \fnd numerically\nvalues of those \felds, g-factor and saturation magnetiza-\ntion we apply a procedure to some extent reverse to the\nmeasurement: for a given set of measured Hr(#H;'H)\ndata, collected in a single experiment, the constant g,M\nand anisotropy and demagnetizing \felds should be cho-\nsen so that Eq. (1) should be met for all measured static\nresonance \felds Hr(#H;'H). It means, that minimizing\nan appropriate objective function, which measures the\ndeviation from the constant curvature with respect to un-\nknown anisotropy \felds, gandMlets us \fnd them. This\nprocedure is described in detail in Ref. [2]. The key point\nfor \fnding saturation magnetization is that its change\nleads to the change of the curvature of the demagnetizing\n\feld. But to ensure that the determined magnetization\nvalue is unambiguous, the constraint Nx+Ny+Nz= 1\nshould be met during the minimization procedure.\nTo carry out the minimization procedure, it is neces-3\nsary to know the functional form of free energy. Usu-\nally one assumes its speci\fc form taking into account the\nsymmetry of the system | free energy should be invari-\nant under relevant symmetry transformations. Then it is\npossible to expand it into basis functions (orthogonal or\nnon-orthogonal) with the same symmetry. Typically this\nexpansion is limited to some low-order terms of system-\natically decreasing basis functions. It should be remem-\nbered, however, that in the case of examination of cur-\nvature, the higher-order basis functions may give greater\ncontributions to total curvature of free energy than the\nlower-order ones.\nNumerical example I: Saturation magnetization\nof bulk magnetite | Bickford [7] measured the reso-\nnance \felds for disk-shaped samples of magnetite cut in\n(100) and (110) planes. The results of his measurements\nare shown as blue squares in Figs 3a and 3b, respectively.\nEach sample was oriented di\u000berently with respect to the\ncrystallographic planes and thus it has di\u000berent demag-\nnetization tensor. We will assume, however, that both\ntypes of samples have the same g-factor and the same\nsaturation magnetization. The demagnetizing tensor is\nassumed to be diagonal for samples cut in (100) plane\nN(100) =2\n4Nx0 0\n0Ny0\n0 0Nz3\n5: (5)\nThe curvature of free energy distribution, given by\nEq. (1), depends now on six parameters which we denote\ncollectively by the vector hB= (g;M;Nx;Ny;Nz;Hc):\nUsing the procedure described in Ref. [2], we get the\nfollowing coordinates of the vector hB\nhB= (2:220(88);5:906(150);0:2216(267);0:2212(268);\n0:5572(535);\u00000:2392(95)):\n(6)\nBoth,MandHcare given in kOe. The errors (in brack-\nets) refer to the last signi\fcant digits and were calculated\nassuming that the measurement errors are subject to nor-\nmal distribution. Having determined the vector hB, we\nsolve Eq. (4) numerically with respect to Hrand get the\nresonance \feld dependence on the #Hangle (black line in\nFig. 3a). To consider measurements of samples cut in the\nplane (110) let us rotate the coordinate system in such a\nway, that the zaxis is perpendicular to the (110) plane.\nThe demagnetizing tensor is given by\nN(110)=2\n641\n4(Nxy+ 2Nz)1\n4(Nxy\u00002Nz)p\n2\n4(Ny\u0000Nx)\n1\n4(Nxy\u00002Nz)1\n4(Nxy+ 2Nz)p\n2\n4(Ny\u0000Nx)p\n2\n4(Ny\u0000Nx)p\n2\n4(Ny\u0000Nx)1\n2Nxy3\n75;\n(7)\nNxystands forNx+Ny. Solving again Eq. (4) for such a\ntensorN(110), we reach the dependence of Hr(#H) shown\nin Fig. 3b. In Table I the relevant anisotropy constants\nfound by Bickford are presented compared with those\nobtained in the way as described in the text.\nFIG. 3: Resonance \feld versus out-of-plane angle #H\nfor bulk magnetite for samples cut in (100) { (a) and\n(110) planes { (b). Squares - Bickford's measurements\n[7], black line - solution of Smit-Beljers equation for the\nvector hBgiven by Eq. (6). Figs (a) and (b) correspond\nto Figs 3 and 4 in Ref. [7], respectively.\nTABLE I: Comparison of g-factor, saturation\nmagnetization Mand cubic anisotropy constant Kc\nvalues found by Bickford [7]with those obtained by the\nmethod described in the text. The errors (in brackets)\nrefer to the last signi\fcant digits.\ng M [kA/m] Kc[kJ/m3]\n2.220(88) 470(12) -14.12(92)\nRef. [7] 2.07 472a-11\naMeasurement result, [7].\nTo summarize this example, let us stress two things.\nFirst, we observe a fairly large statistical errors that are\nthe result of the scattering of original measurements re-\nported in Ref. [7]. Second, in fact, the Bickford's samples\nhad the shapes of a circular \rattened disks, since in (100)\ngeometryNx\u0019Ny6=Nz.\nNumerical example II: Saturation magnetization\nof the (Ga,Mn)As thin \flm | In Ref. [2] we showed,\nanalyzing Liu's measurements [8] of resonance \felds for\nthe weak ferromagnet (Ga,Mn)As, that in order to prop-\nerly describe the free energy spatial distribution at reso-\nnance, one should expand it up to the fourth order with\nrespect to cubic anisotropy \felds. Here we add to this ex-\npansion demagnetizing \felds, and, since the sample was\noriented along crystallographical axes of (Ga,Mn)As we\nuse a diagonal form od the demagnetizing tensor. The\nvector hLdepends now on eleven parameters\nhL\u0011(g;M;Nx;Ny;Nz;Hc1;:::;Hc4;H[001];H[110]):(8)\nAnisotropy \felds are denoted like in Ref. [2]. The mini-\nmization procedure leads to (\felds are given in Oe)\nhL= (1:984(3);383:5(2:2);0:08104(90);0:06796(95);\n0:8510(20);76:86(0:42);\u0000539:5(10:0);42:86(0:92);\n1412(70);4213(24);68:08(0:64)):\n(9)\nThe numerical solutions of Eq. (1) for such an hLvector\nis shown in Fig. 4. They are equivalent to those obtained4\nin Ref. [2] but now the use of the demagnetizing \felds\nmade it possible to determine saturation magnetization\nM= 30:52(0:18) [emu/cm3].\nFIG. 4: Resonance \feld versus out-of-plane angle #H\n(a) and versus in-plane angle 'H(b) for (Ga,Mn)As\n\flm. Squares - Liu measurements [8], black line -\nsolution of Smit-Beljers equation for the vector hL\n(Eq. 9). Figs (a) ad (b) correspond to Figs 5 and 6 in\nRef. [8], respectively.\nNumerical example III: Saturation magnetization\nof the YIG ultrathin thin \flm | The measurements\nof the resonance \feld of the 9.8 nm thin \flm are taken\nfrom Ref. [9] and shown in Figs 5a and 5b, respectively, as\nblue squares. To determine the anisotropy \felds let us as-\nsume that the free energy is given by Eq. (1) from Ref. [9]\nand that the demagnetizing \felds, as in the (Ga,Mn)As\ncase, are calculated using a diagonal demagnetizing ten-\nsor. Then the vector hHdepends on nine parameters\nhH= (g;M;Nx;Ny;Nz;H2?;H4?;H2k;H4k):(10)\nFactorg= 2 for YIG [10], while saturation magnetization\nM= 1851 [Oe] was determined for this sample using\na vibrating sample magnetometer [9].\nThe components of the vector hHcorresponding to the\nconstant curvature of free energy, i.e., for the sample at\nresonance, are collected in Table II for four cases. In the\n\frst case, we take gandMvalues as measured experi-\nmentally and \fnd remaining ones. In the second case we\n\fx only experimental value of M, in the third case - only\nvalue ofg, and \fnally, we treat both gandMas unknown\n(last row of in Table II). In the last column of Table II the\nerror function E1\nRMS(hH) is given, as de\fned in Ref. [2].\nIts value informs us about the quality of the prediction\nof new experimental results of resonance \felds calculated\nfrom a given free energy formula with speci\fc values (row\nof Table II) of anisotropy \felds. We see that analysis\nof FMR measurements of the YIG ultrathin \flm givesworse results when we treat measured gandMvalues\nas known. This points out that the form of free energy\nused for this analysis is not chosen optimally, because it\nwas not possible to reproduce the experimental values g\nandMwith satisfactory accuracy. Thus the presented\nmethod may also serve as a test for the correctness of\nthe assumed free energy form. Note also, that although\nthe investigated \flm was ultrathin, we obtained non-zero\nvalues of the demagnetization \felds in the (001) plane.\nThe dependencies of resonance \felds on angles #Hand\n'Hare shown, for hHfrom the \frst row of Table II, in\nFig. 5.\nFIG. 5: Resonance \feld for 9.8 nm ultrathin YIG \flm\nversus out-of-plane angle #H(a) and in-plane angle \u001eH\n(b). Squares - Wang's measurements [9], black line -\nsolution of Smit-Beljers equation for the vector hH\ngiven by Eq. (10). Figs (a) and (b) correspond to\nFigs 2(b) and 2(c) in Ref. [9], respectively .\nConclusion | We presented in this article one more ap-\nproach to the interpretation of the classical FMR experi-\nmental data. It is based on the analysis of the curvature\nof the spatial dependence of free energy of the sample at\nresonance and makes it possible, after assuming the func-\ntional form of free energy, to determine (with an accu-\nracy of 0.5-4% in the presented numerical examples) the\nsample saturation magnetization and, consequently, the\nspatial dependence of magnetocrystalline energy stored\nin the sample in a single FMR experiment. The method\npresented here can also be a test for the correctness of the\nassumed form of free energy of the sample at resonance.\nThis may be important for people working in the \feld of\nmagnetic resonance.\nAcknowledgment | This study is a part of the project\n\fnanced by Narodowe Centrum Nauki (National Science\nCentre of Poland) No. DEC-2013/08/M/ST3/00967. Nu-\nmerical calculations were performed at Pozna\u0013 n Super-\ncomputing and Networking Center under Grant No. 284.\n\u0003e-mail: ptomczak@amu.edu.pl\n1M. Farle, Reports on Progress in Physics 61, 755 (1998).\n2P. Tomczak and H. Puszkarski, Phys. Rev. B 98, 144415(2018).\n3J. Smit and H. G. Beljers, Philips Res. Rep. 10, 113 (1955).\n4J. O. Artman, Phys. Rev. 105, 74 (1957).5\nTABLE II: Comparison of g-factor, saturation magnetization Mand and anisotropy \feld values found in Ref. [9]\nwith those obtained by the method described in the text. The errors (in brackets) refer to the last signi\fcant digits.\ng M [Oe] Nx Ny Nz H2?[Oe] H4?[Oe] H2k[Oe] H4k[Oe] E1\nRMS(hH)\n2.000a1851b0.08429(10) 0.08721(11) 0.8285(10) -497.7(2.0) 372.3(1.1) 62.23(0.27) 49.02(0.22) 0.815\n2.014(1) 1851b0.08795(17) 0.08985(10) 0.8222(15) -466.5(1.5) 276.9(0.9) 61.83(0.23) 47.27(0.18) 0.691\n2.000a1800(11) 0.08102(15) 0.08408(11) 0.8349(16) -538.9(1.7) 372.6(1.0) 62.16(0.11) 49.05(0.16) 0.815\n2.013(1) 1772(12) 0.08263(8) 0.08467(13) 0.8327(13) -526.2(1.6) 277.5(0.9) 61.82(0.09) 47.42(0.18) 0.691\nRef. [9] 1851(37) -1253(25) 60.4(1.2) 42.0(1.2)\naMeasurement result, see Apendix A of Ref. [10].\nbMeasurement result, Ref. [9].\n5L. Baselgia, M. Warden, F. Waldner, S. L. Hutton, J. E.\nDrumheller, Y. Q. He, P. E. Wigen, and M. Mary\u0014 sko,\nPhys. Rev. B 38, 2237 (1988).\n6A. Morrish, The Physical Principles of Magnetism (Wiley-\nIEEE Press, 2001).\n7L. R. Bickford, Phys. Rev. 78, 449 (1950).8X. Liu, Y. Y. Zhou, and J. K. Furdyna, Phys. Rev. B 75,\n195220 (2007).\n9H. Wang, C. Du, P. C. Hammel, and F. Yang, Phys. Rev.\nB89, 134404 (2014).\n10D. Stancil, A. Prabhakar, Spin Waves. Theory and Appli-\ncations (Springer, 2009)."    },    {        "title": "2304.13553v2.Critical_Cavity_Magnon_Polariton_Mediated_Strong_Long_Distance_Spin_Spin_Coupling.pdf",        "content": "Critical Cavity-Magnon Polariton Mediated Strong Long-Distance Spin-Spin Coupling\nMiao Tian,1Mingfeng Wang,1Guo-Qiang Zhang,2,\u0003Hai-Chao Li,3,yand Wei Xiong1,z\n1Department of Physics, Wenzhou University, Zhejiang 325035, China\n2School of Physics, Hangzhou Normal University, Hangzhou, Zhejiang 311121, China\n3College of Physics and Electronic Science, Hubei Normal University, Huangshi 435002, China\n(Dated: April 28, 2023)\nStrong long-distance spin-spin coupling is desperately demanded for solid-state quantum infor-\nmation processing, but it is still challenged. Here, we propose a hybrid quantum system, consisting\nof a coplanar waveguide (CPW) resonator weakly coupled to a single nitrogen-vacancy spin in di-\namond and a yttrium-iron-garnet (YIG) nanosphere holding Kerr magnons, to realize strong long-\ndistance spin-spin coupling. With a strong driving \feld on magnons, the Kerr e\u000bect can squeeze\nmagnons, and thus exponentially enhance the coupling between the CPW resonator and the squeezed\nmagnons, which produces two cavity-magnon polaritons, i.e., the high-frequency polariton (HP) and\nlow-frequency polariton (LP). When the enhanced cavity-magnon coupling approaches the critical\nvalue, the spin is fully decoupled from the HP, while the coupling between the spin and the LP is sig-\nni\fcantly improved. In the dispersive regime, a strong spin-spin coupling is achieved with accessible\nparameters, and the coupling distance can be up to \u0018cm. Our proposal provides a promising way\nto manipulate remote solid spins and perform quantum information processing in weakly coupled\nhybrid systems.\nI. INTRODUCTION\nSolid spins such as nitrogen-vacancy centers in dia-\nmond [1], having good tunability [2] and long coher-\nence time [3{5], are regarded as promising platforms for\nquantum information science [14, 15]. However, direct\nspin-spin coupling is weak due to their small magnetic\ndipole moments [16{21]. Moreover, the coupling dis-\ntance is directly determined by their separation. To over-\ncome these, the natural ideal is to look for quantum in-\nterfaces [6{13] as bridges to couple long-distance spins,\nforming diverse hybrid quantum systems [14, 15].\nRecently, the emerged low-loss magnons (i.e., the\nquanta of collective spin excitations) in ferromagnetic\nmaterials [22{25] have shown great potential in me-\ndiating distant spin-spin coupling [26{31]. For ex-\nample, magnons in the Kittle mode of a nanometer-\nsized yttrium-iron-garnet (YIG) sphere have been used\nto strongly couple spins with tens of nanometers dis-\ntance [26{28], via enhancing the local magnetic \feld.\nTo further improve the coupling distance between two\nspins from nanometer to micronmeter, magnons with\nKerr e\u000bect as quantum interface are proposed [32]. Also,\nthe YIG nanosphere can be used to realize strong spin-\nphoton coupling in a microwave cavity [33]. Besides\nthese, magnons in a bulk material [29, 30] and thin ferro-\nmagnet \flm [31] have been suggested to coherently cou-\nple remote spins. However, achieved strong coupling is\nseverely limited by the distance between two spins.\nMotivated by this, we propose a hybrid spin-cavity-\nmagnon system to realize a strong spin-spin coupling\n\u0003zhangguoqiang@hznu.edu.cn\nyhcl2007@foxmail.com\nzxiongweiphys@wzu.edu.cnwith coupling distance \u0018centimeter . In the proposed sys-\ntem, the spin in diamond is located at tens of nanometers\nfrom the central line of the CPW resonator, and weakly\ncoupled to the CPW resonator. The nanometer-sized\nYIG sphere supporting Kerr magnons (i.e., magnons\nwith Kerr e\u000bect) is employed but weakly coupled to the\nCPW resonator. Experimentally, strong and tunable\nmagnon Kerr e\u000bect, originating from the magnetocrys-\ntalline anisotropy, has been demonstrated [34], giving rise\nto bi- and multi-stabilities [35{39], nonreciprocity [40],\nsensitive detection [41], quantum entanglement [42] and\nquantum phase transition [43, 44]. Under a strong driv-\ning \feld, this Kerr e\u000bect can squeeze magnons, and thus\nthe coupling between magnons and the CPW resonator\nis exponentially enhanced to the strong coupling regime.\nThe strong magnon-cavity coupling generates two polari-\ntons, i.e., the high-frequency polariton (HP) and the low-\nfrequency polariton (LP). When the enhanced magnon-\ncavity coupling strength approaches to the critical value,\nthe LP becomes critical. Then the coupling between\nthe spin and the HP is fully suppressed in the polari-\nton representation, while the coupling between the spin\nand the LP is greatly enhanced. By further consider the\ncase of two spins dispersively coupled to the LP, an in-\ndirect and strong spin-spin coupling can be induced by\nadiabatically eliminating the degrees of freedom of LP.\nMoreover, the coupling strength is not limited by the\nseparation between two spins, it is actually determined\nby the length of the CPW resonator. Experimentally, the\ncentimeter-sized CPW resonator has been fabricated [45].\nTherefore, the achieved strong spin-spin coupling can be\nup to\u0018cm. Our proposal privides an alternative path\nto remotely manipulate solid spin qubits and perform-\ning quantum information processing in weakly coupled\nspin-cavity-magnon systems.arXiv:2304.13553v2  [quant-ph]  27 Apr 20232\n/uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000018 /uni00000014/uni00000011/uni00000013 /uni00000014/uni00000011/uni00000018\n/uni00000030/uni00000044/uni0000004a/uni00000051/uni00000048/uni00000057/uni0000004c/uni00000046/uni00000003/uni00000029/uni0000004c/uni00000048/uni0000004f/uni00000047/uni00000013/uni00000014/uni00000015/uni00000016/uni00000017/uni00000028/uni00000051/uni00000048/uni00000055/uni0000004a/uni0000005cms=1\nms=1\nms=0/uni0000000b/uni00000045/uni0000000c\n/uni00000013/uni00000011/uni00000013/uni00000013 /uni00000013/uni00000011/uni00000014/uni00000018 /uni00000013/uni00000011/uni00000016/uni00000013\n/uni00000035/uni00000003/uni0000000b/uni00000050/uni0000000c\n/uni00000013/uni00000014/uni00000015/uni00000016/uni0000004am/2/uni00000003/uni0000000b/uni00000030/uni0000002b/uni0000005d/uni0000000c\n/uni0000000b/uni00000046/uni0000000c\nFIG. 1: (a) Schematic diagram of a hybrid quantum system.\nThe single nitrogen vacancy (NV) center spin, located ddis-\ntance away from the central line, is weakly coupled to the\nCPW resonator. The YIG sphere is driven by a microwave\n\feld through a microwave antenna. (b) The level structure\nof the triplet ground state of the NV center, ms= 0 and\nms=\u00001 are selected to form a spin qubit. (c) The cavity-\nmagnon coupling verus the radius Rof the YIG nanosphere.\nII. MODEL AND HAMILTONIAN\nWe consider a hybrid quantum system consisting of a\ncoplanar waveguide (CPW) resonator weakly coupled to\nboth a single NV spin in diamond and a nanometer-sized\nYIG sphere, as shown in Fig. 1(a). The spin is fabri-\ncated far away from the YIG sphere to avoid their direct\ncoupling. In addition, the magnon Kerr e\u000bect, stemming\nfrom the magnetocrystallographic anisotropy [35, 36], is\ntaken into account. Thus, the total Hamiltonian of the\nhybrid system can be written as (setting ~= 1),\nHtot=HNV+HCM+HCS+HK+HD; (1)\nwhereHNV=1\n2!NV\u001bz, with the transition frequency\n!NV=D\u0000ge\u0016BBexbetween the lowest two levels of the\ntriplet ground state of the NV [see Fig. 1(b)], is the free\nHamiltonian of the NV spin. Here, D= 2\u0019\u00022:87 GHz\nis the zero-\feld splitting, ge= 2 is the Land\u0013 efactor,\u0016B\nis the Bohr magneton, and Bexis the external magnetic\n\feld to lift the near-degenerate stats jms=\u00061i. The\nsecond term\nHCM=!caya+gm\u0000\naym+amy\u0001\n(2)\nrepresents the Hamiltonian of the coupled magnon-cavtiy\nsusbsytem, where !cis the frequency of the CPW res-\nonator and gmis the coupling strength [33], nearly propo-\ntional to the radius Rof the YIG sphere [see Fig. 1(c)].Obviously, strong coupling can be obtained by using\nmicronmeter-sized sphere, which is widely employed in\nexperiments [46{49]. For the nanometer-sized sphere\nsuch asR\u001850 nm, we have gm\u00182\u0019\u00020:2 MHz, which\nis much smaller than the typical decay rates of the cav-\nity (\u0014c=2\u0019\u00181 MHz) [50] and Kittle mode ( \u0014m=2\u0019\u00181\nMHz) [51], i.e., gm< \u0014 c;\u0014m. This indicates that the\ncoupling between the Kittle mode of the nanosphere and\nthe cavity is in the weak coupling regime, consistent with\nour assumption.\nThe Hamiltonian HCSin Eq. (1) describes the inter-\naction between the spin qubit and the cavity. With\nthe rotating-wave approximation, HCScan be governed\nby [19, 20]\nHCS=\u0015\u0000\n\u001b+a+ay\u001b\u0000\u0001\n; (3)\nwhere\u0015= 2ge\u0016BB0;rms(d) [19] is the coupling strength,\nwithB0;rms(d) =\u00160Irms=2\u0019d,Irms=p\n~!c=2La, andd\nbeing the distance between the spin and the center con-\nductor of the CPW resonator. To estimate \u0015,!c\u00182\u0019\u00022\nGHz andLa\u00182 nH [51] are chosen. For d\u00185\u0016m,\n\u0015\u00182\u0019\u000270 Hz, and d\u001850 nm,\u0015\u00182\u0019\u00027 kHz [19],\nleading to\u0015<\u0014 c. This shows that the coupling between\nthe spin qubit and the cavity is also in the weak coupling\nregime. Due to this fact, we here assume that the spin\nqubit is placed close to the central line of the CPW res-\nonator to obtain a moderate coupling strength, althoght\nit is still weakly coupled to the cavity. Experimentally,\nsuch weak spin-cavity weak couplings can be measured.\nThe Hamiltonian HKin Eq. (1) denotes the magnon\nKerr e\u000bect, characterizing the coupling among magnons\nin the YIG sphere and provides the anharmonicity of the\nmagnons, which is given by [36]\nHK=!mmym+Kmymymm; (4)\nwhere!m=\rB0\u00002\u00160Kan\r2s=M2Vm+\u00160Kan\r2=M2Vm\nis the frequency of the Kittle mode, with the gyromag-\nnetic ration \r=2\u0019=ge\u0016B=~(\u0016Bis the Bohr magneton),\nthe vacuum permeability \u00160, the \frst-order anisotropy\nconstant of the YIG sphere Kan, the amplitude of a bias\nmagnetic \feld B0, the saturation magnetization M, and\nthe volume of the YIG sphere Vm.K=\u00160Kan\r2=M2Vm\nis the coe\u000ecient. Apparently, the Kerr coe\u000ecient is in-\nversely proportional to the volume of the YIG sphere,\ni.e.,K/1=Vm, the Kerr e\u000bect can become signi\fcantly\nimportant for a YIG nanosphere. For example, when\nR\u001850 nm,K=2\u0019\u0018128 Hz, but K=2\u0019\u00180:05 nHz\nforR\u00180:5 mm (the usual size of the YIG sphere used\nin various previous experiments). Obviously, Kis much\nsmaller in the latter case. Because our proposal mainly\nrelies on the Kerr e\u000bect, we here use the nanometer-sized\nYIG sphere to obtain strong Kerr e\u000bect. The last term\nHD= \nd\u0000\nmye\u0000i!dt+mei!dt\u0001\n: (5)\nin Eq. (1) describes the interaction between the Kittle\nmode and the driving \feld, where \n dis the Rabi fre-\nquency and !dis the frequency of the driving \feld.3\n/uni00000013/uni00000011/uni00000013/uni00000013 /uni00000013/uni00000011/uni00000013/uni00000018 /uni00000013/uni00000011/uni00000014/uni00000013 /uni00000013/uni00000011/uni00000014/uni00000018\n/uni00000035/uni00000003/uni0000000b/uni00000050/uni0000000c\n/uni00000013/uni00000016/uni00000013/uni00000019/uni00000013/uni0000001c/uni00000013/uni0000002a/uni00000012/uni00000015/uni00000003/uni0000000b/uni00000030/uni0000002b/uni0000005d/uni0000000c\n/uni0000000b/uni00000044/uni0000000crm=0\nrm=3\nrm=5\n/uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000015 /uni00000013/uni00000011/uni00000017 /uni00000013/uni00000011/uni00000019 /uni00000013/uni00000011/uni0000001b /uni00000014/uni00000011/uni00000013\nG/s\n/uni00000014\n/uni00000013/uni00000014/uni00000015/uni000000162\n±/2\ns\nGc\n/uni0000000b/uni00000045/uni0000000c2\n+\n2\nFIG. 2: (a) The coupling strength between the squeezed\nmagnons and the CPW resonator versus the radius of the YIG\nnanosphere with di\u000berent squeezing parameters rm= 0;3;5.\n(b) The square of polariton frequencies versus the coupling\nbetween the squeezed magnons and the CPW resonator.\nIn the rotating frame with respect to the driving fre-\nquency (!d), the total Hamiltonian in Eq. (1) becomes\nHsys=1\n2\u0001NV\u001bz+ \u0001 caya+H0\nK+ \nd\u0000\nmy+m\u0001\n+\u0015\u0000\n\u001b+a+ay\u001b\u0000\u0001\n+gm\u0000\naym+amy\u0001\n;(6)\nwhere \u0001 NV=!NV\u0000!dis the frequency detuning of the\nspin qubit from the driving \feld, \u0001 c=!c\u0000!dis the fre-\nquency detuning of the cavity \feld from the driving \feld,\nandH0\nK=\u000emmym+Kmymymm[32], with\u000em=!m\u0000!d\nbeing the frequency detuning of the Kittle mode from the\ndriving \feld. Due to the large \n d, the Hamiltonian Hsys\nin Eq. (6) can be linearized by writting each system op-\nerator as the expectation value plus its \ructuation [52].\nBy neglecting the higher-order \ructuation terms, Eq. (6)\nis linearize as\nHlin=1\n2\u0001NV\u001bz+ \u0001 caya+HK\n+\u0015\u0000\n\u001b+a+ay\u001b\u0000\u0001\n+gm\u0000\naym+amy\u0001\n;(7)\nwith\nHK= \u0001 mmym+Ks\u0000\nm2+my;2\u0001\n; (8)\nwhere the e\u000bective magnon frquency detuning \u0001 m=\n\u000em+ 4Kjhmij2is induced by the Kerr e\u000bect, which has\nbeen demonstrated experimentally [35, 36]. The ampli-\n\fed coe\u000ecient Ks=Khmi2is the e\u000bective strength of\nthe two-magnon process, which can give rise to squeeze\nmagnons in the Kittle mode. Aligning the biased mag-\nnetic \feld along the crystalline axis [100] or [110] of the\nYIG sphere [34, 36], Kcan be positive or negative, and\nwe can have Ks>0 orKs<0. There we choose Ks<0\nwhenK < 0. The linearized Kerr Hamiltonian HKin\nEq. (8) describes the two-magnon process, which can give\nrise to the magnon squeezing.\nBelow we operate the proposed hybrid system in the\nmagnon-squeezing frame by diagnolizing the Hamilto-\nnianHKwith the Bogoliubov transformation m=\nmscosh (rm) +my\nssinh (rm), whererm=1\n4ln\u0001m\u00002Ks\n\u0001m+2Ksis the squeezing parameter. After diagnolization, HK\nbecomes\nHKS= \u0001 smy\nsms (9)with \u0001 s=p\n\u00012m\u00004K2sbeing the frequency of the\nsqueeze magnon, and Eq. (7) is transformed to\nHS=1\n2\u0001NV\u001bz+HCMS+\u0015\u0000\n\u001b+a+ay\u001b\u0000\u0001\n; (10)\nwhere\nHCMS= \u0001 caya+ \u0001 smy\nsms+G\u0000\nay+a\u0001\u0000\nmy\ns+ms\u0001\n(11)\nis the e\u000bective Hamiltonian of the CPW resonator cou-\npled to the squeezed magnons, G=1\n2gmermis the\nexponentially enhanced coupling strength between the\nsqueezed magnons and the CPW resonator. Because\nboth the parameters \u0001 mandKscan be tuned, so rmcan\nbe very large when \u0001 m\u0018\u00002Ks, leading to the strong\nGeven for nanometer-sized YIG sphere [see curves in\nFig. 2(a)]. Speci\fcally, when rm= 0, i.e., magnons in\nthe Kittle mode is not squeezed, the coupling strength\nbetween the CPW resonator and the Kittle mode is un-\nampli\fed, giving rise to weak G[see the black curve\nin Fig. 2(a)]. When magnons in the Kittle mode are\nsqueezed but with moderate squeezing parameters such\nasrm= 3 andrm= 5, we \fnd the coupling strength\nGcan be signi\fcantly improved for the YIG nanosphere.\nFor example, R\u001850 nm and rm\u00183, we haveG=2\u0019= 2\nMHz, which is comparable with the decay rates of the\nCPW resonator ( \u0014c) and the Kittle mode ( \u0014m). But\nwhenrm\u00185,G=2\u0019= 17 MHz, which is much larger\nthan both\u0014cand\u0014m. These indicates that indicates that\nstrong coupling between the squeezed magnons and the\nCPW resonator can be realized by tuning the squeezing\nparameterrm. In addition, Gcan be further enhanced\nby using the larger radius of the YIG sphere when rm\nis \fxed. Once the strong coupling between the squeezed\nmagnons and the CPW resonator is achieved, the coun-\nterrotating terms /aymy\nsandamsin Eq. (11) are related\nto two-mode squeezing, while rotating terms /aymsand\namy\nsallow quantum state transfer between the squeezed\nmagnons and the CPW resonator. By combining these,\npolaritons with criticality can be formed, as shown below.\nIII. STRONG COUPLING BETWEEN THE\nSINGLE NV SPIN AND THE LOW-FREQUENCY\nPOLARITON\nBy further diagnolizing the Hamiltonian HCMS in\nEq. (11), two polaritons with eigenfrequencies\n!2\n\u0006=1\n2\u0014\n\u00012\nc+ \u00012\ns\u0006q\n(\u00012c\u0000\u00012s)2+ 16G2\u0001c\u0001s\u0015\n(12)\ncan be obtained. This is owing to the fact of the\nachieved strong coupling between the squeezed magnons\nand the CPW resonator. For convenience, we call two\npolaritons with frequencies !+and!\u0000as the high- and\nlow-frequency polaritons (HP and LP). The diagnolized\nHCMS reads\nHdiag=!+ay\n+a++!\u0000ay\n\u0000a\u0000; (13)4\n/uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000014 /uni00000013/uni00000011/uni00000015\n/uni00000037/uni0000004c/uni00000050/uni00000048/uni00000003/uni0000000b/uni00000056/uni0000000c\n/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013/uni00000032/uni00000046/uni00000046/uni00000058/uni00000053/uni00000044/uni00000057/uni0000004c/uni00000052/uni00000051/uni0000000b/uni00000044/uni0000000c /uni00000036/uni00000053/uni0000004c/uni00000051\n/uni00000033/uni00000052/uni0000004f/uni00000044/uni00000055/uni0000004c/uni00000057/uni00000052/uni00000051\n0.0 0.2 0.4 0.6 0.8 1.0\nTime (s)\n0.00.51.0Occupation(b) Spin\nPolariton\nFIG. 3: The occupation of the LP and spin qubit versus the\nevolution time at G!Gcand \u0001 c\u001d\u0001s(a) without and (b)\nwith dissipations. The spin decay rate is \r?\u00181 kHz and the\nLP dacay rate is \u0014\u0000\u00181 MHz. In both (a) and (b), the spin\nqubit is initially prepared in the excited state and the LP is\nin the ground state, and the coupling strength is gr=2\u0019\u00183:5\nMHz.\nwhere\na=cos\u0012\n2p\n\u0001c!\u0000h\na\u0000(\u0001c+!\u0000) +ay\n\u0000(\u0001c\u0000!\u0000)i\n+sin\u0012\n2p\n\u0001c!+h\na+(\u0001c+!+) +ay\n+(\u0001c\u0000!+)i\n:(14)\nSubstituting Eqs. (13) and (14) into Eq. (10), the Hamil-\ntonian of the coupled spin-CMP can be given by\nHCMP =1\n2\u0001NV\u001bz+!+ay\n+a++!\u0000ay\n\u0000a\u0000 (15)\n+gr\u0010\n\u001b+a\u0000+\u001b\u0000ay\n\u0000\u0011\n+gcr\u0010\n\u001b+ay\n\u0000+\u001b\u0000a\u0000\u0011\n+g0\nr\u0010\n\u001b+a++\u001b\u0000ay\n+\u0011\n+g0\ncr\u0010\n\u001b+ay\n++\u001b\u0000a+\u0011\n;\nwheregc(cr)=\u0015cos\u0012(\u0001c\u0006!\u0000)=2p\n\u0001c!\u0000denote the ef-\nfective coupling strength between the NV spin and the\nLP,g0\nc(cr)=\u0015sin\u0012(\u0001c\u0006!+)=2p\n\u0001c!+represent the ef-\nfective coupling strength between the NV spin and the\nHP. Obviously, both gc(cr)andg0\nc(cr)can be tuned by the\ndriving \feld on the Kittle mode of the YIG sphere. The\nparameter\u0012is de\fned by tan(2 \u0012) = 4Gp\u0001c\u0001s=(\u00012\nc\u0000\n\u00012\ns). To show the behavior of two polaritons with thecoupling strength G, we plot the square of polariton fre-\nquencies versus the coupling strength Gin Fig. 2(b).\nClearly, one can see that !2\n+increases with G, but!2\n\u0000de-\ncreases. When !2\n\u0000= 0,Greduces to the critical coupling\nstrengthGc, i.e.,\nG=Gc\u00111\n2p\n\u0001c\u0001s; (16)\nwhich means !\u0000is real forG < G c, while!\u0000is imagi-\nnary (the low-polariton is unstable) for G > G c. When\nwe operate the coupled cavity-magnon subsystem around\nthe critical point (i.e., G!Gc) and \u0001 c\u001d\u0001sis satis\fed,\nwe havegr\u0019gcr!1\n2\u0015p\n\u0001c=!\u0000,g0\nr\u0019g0\ncr!0. Due to\nthe large \u0001 cand the extremely small !\u0000,gr\u0019gcr\u001d\u0015.\nThese indicate that coupling between the NV spin and\nthe HP is completely decoupled, while the coupling be-\ntween the NV spin and the LP is signi\fcantly enhanced.\nBy choosing \u0001 c= 106!\u0000,gr=gcr\u0018103\u0015are esti-\nmated. Obviously, three orders of magnitude of the spin-\nlow-polariton coupling is improved. Using d= 50 nm,\n\u0015= 2\u0019\u00027 kHz is obtained, resulting in gr=2\u0019= 3:5\nMHz, which is larger than the decay rates of the CPW\nresonator and the Kittle mode, i.e., gr(cr)>\u0014c;\u0014m. This\nsuggests that the coupling between the spin and the LP\ncan be in the strong coupling regime. In principle, gr(cr)\ncan be further enhanced by using the larger \u0001 cor much\nsmaller!\u0000. In the strong coupling regime, the rotating-\nwave approximation is still valid, and the counterrotating\nterm related to gcrin Eq. (15) can be safely ignored, so\nEq. (15) reduces to\nHJC=1\n2\u0001NV\u001bz+!\u0000ay\n\u0000a\u0000+gr\u0010\n\u001b+a\u0000+\u001b\u0000ay\n\u0000\u0011\n;(17)\nwhich is the so-called Janes-Cumming model with the\nstrong coupling, allowing quantum state exchange be-\ntween the spin and the LP, as demonstrated in Fig. 3(a),\nwhere the spin is initially prepared in the excited state\nand the the LP is in the ground state.\nWhen dissipations are included, the dynamics of the\nsystem can be described by the master equation,\nd\u001a\ndt=\u0000i[HJC;\u001a] +\u0014\u0000D[a\u0000]\u001a+\r?D[\u001b\u0000]\u001a; (18)\nwhereD[o]\u001a=o\u001aoy\u00001\n2\u0000\noyo\u001a+\u001aoyo\u0001\n, and\r?is the\ntransversal relaxation rate of the NV spin [53], \u0014\u0000is\nthe decay rate of the LP. In Fig. 3(b), we use the qutip\npackage in python [54, 55] to numerically simulate the\ndynamics of the spin and LP governed by Eq. (18). The\nresults show that state exchange between the spin and\nthe LP can be realized in the presence of dissipations\nsuch as\u0014\u0000\u00181 MHz and \r?\u00181 kHz [58], althougth\nthe occupation probability decreases with long evolution\ntime.5\n0 10 20 30 40 50\nTime (s)\n0.00.51.0Occupation\n(a)spin 1\nspin 2\nPolariton\n0 20 40 60 80 100\nTime (s)\n0.00.51.0Occupation\n(b)spin 1\nspin 2\nPolariton\nFIG. 4: The occupation of two spins and the LP versus the\nevolution time in the dispersive regime (a) without and (b)\nwith dissipations. The parameters are the same as in Fig. 3.\nIV. THE EFFECTIVE STRONG COUPLING\nBETWEEN TWO SINGLE NV SPINS\nHere, we further consider the case that two identical\nNV spins are symmetrically placed away from the YIG\nsphere in the CPW resonator. Thus, two spins interact\nwith the CPW resonator with the same coupling strength\n\u0015. By operating the cavity-magnon subsystem around\nthe critical point, the couplings between two spins and\nthe HP can be fully suppressed, while the couplings be-\ntween two spins and the LP is greatly enhanced, similar\nto the single spin case. Therefore, the Hamiltonian of the\nhybrid system with two identical spins can be e\u000bectively\ndescribed by Tavis-Cumming model,\nHTC=!\u0000ay\n\u0000a\u0000+1\n2\u0001NV\u0010\n\u001b(1)\nz+\u001b(2)\nz\u0011\n+grh\u0010\n\u001b(1)\n++\u001b(2)\n+\u0011\na\u0000+ h:c:i\n: (19)\nIn the dispersive regime, i.e., j\u0001NV\u0000!\u0000j\u001dgr, the LP\ncan be as an interface to induce an indirect coupling be-\ntween two spins by using the Fr ohlich-Nakajima transfor-\nmation [56, 57]. By adiabatically eliminating the degrees\nof freedom of the LP, we can obtain the e\u000bective spin-spin\nHamiltonian as\nHe\u000b=1\n2!e\u000b\u0010\n\u001b(1)\nz+\u001b(2)\nz\u0011\n+ge\u000b\u0010\n\u001b(1)\n+\u001b(2)\n\u0000+\u001b(1)\n\u0000\u001b(2)\n+\u0011\n;\n(20)where!e\u000b= \u0001 NV+ 2ge\u000bn\u0000+ge\u000bis the e\u000bective tran-\nsition frequency of the NV spin, depending on the mean\noccupation number n\u0000=hay\n\u0000a\u0000iof the LP, ge\u000b=\n\u0000g2\nr=\u0001NVis the e\u000bective spin-spin coupling strength in-\nduced by the LP. To estimate ge\u000b, we assume the dis-\ntance between the spin and the central line of the CPW\nresonatord= 50 nm, so gr=2\u0019= 3:5 MHz, thus we have\nge\u000b=2\u0019= 12:7 kHz when \u0001 NV=2\u0019= 960 MHz. Obvi-\nously,ge\u000b\u001d\r?\u00181 kHz, i.e., the strong spin-spin cou-\npling is achieved. This can be directly demonstrated by\nsimulating the dynamics of the e\u000bective system, governed\nby Eq. (20) or Eq. (19) in the dispersive regime, with the\nmaster equation. The simulating results are presented in\nFig. 4. One can see that quantum states of two spins can\nbe exchanged each other with [see Fig. 4(a)] and with-\nout [see Fig. 4(b)] dissipations, while the LP is always in\nthe initial state. Note that the achieved strong spin-spin\ncoupling is not limited by the separation between two\nspins, it is only determined by the length of the CPW res-\nonator. Experimentally, the centimeter-sized cavity has\nbeen fabricated, so the distance of the strong spin-spin\ncoupling can be improved to centimeter level. Compared\nto previous proposals of directly coupled spins to a YIG\nnanosphere [26{28], the distance here is nearly enhanced\nbysixorders of magnitude.\nV. CONCLUSIONS\nIn summary, we have proposed a hybrid system con-\nsisting of a CPW resonator weakly coupled to NV spins\nand a YIG nanosphere supporting magnons with Kerr\ne\u000bect. With the strong driving \feld, the Kerr e\u000bect\ncan squeeze magnons, giving rise to exponentially en-\nhanced strong cavity-magnon coupling, and thus CMPs\ncan be formed. By approaching the cavity-magnon cou-\npling strength to the critical value, the spin-LP coupling\nis greatly enhanced to the strong coupling regime with\nthe accessible parameters, while the coupling between the\nspins and the HP is fully suppressed. Using the LP as\nquantum interface in the dispersive regime, strong long-\ndistance spin-spin coupling can be achieved, which allows\nquantum state exchange between two spins. With cur-\nrent fabricated technology of the CPW resonator, the dis-\ntance of the strong spin-spin coupling can be up to \u0018cm\nlevel, which is improved about six orders of magnitude\nthan the previous proposal of directly coupled spins to a\nYIG nanosphere. 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Appl. 11, 044026\n(2019)."    },    {        "title": "1811.05801v1.Tunable_space_time_crystal_in_room_temperature_magnetodielectrics.pdf",        "content": "Tunable space-time crystal in room-temperature magnetodielectrics\nAlexander J. E. Kreil,\u0003Halyna Yu. Musiienko-Shmarova, Dmytro A. Bozhko,\nSebastian Eggert, Alexander A. Serga, and Burkard Hillebrands\nFachbereich Physik and Landesforschungszentrum OPTIMAS,\nTechnische Universit at Kaiserslautern, 67663 Kaiserslautern, Germany\nAnna Pomyalov and Victor S. L'vov\nDepartment of Chemical and Biological Physics, Weizmann Institute of Science, Rehovot 76100, Israel\nWe report the experimental realization of a space-time crystal with tunable periodicity in time and\nspace in the magnon Bose-Einstein Condensate (BEC), formed in a room-temperature Yttrium Iron\nGarnet (YIG) \flm by radio-frequency space-homogeneous magnetic \feld. The magnon BEC is pre-\npared to have a well de\fned frequency and non-zero wavevector. We demonstrate how the crystalline\n\\density\" as well as the time and space textures of the resulting crystal may be tuned by varying the\nexperimental parameters: external static magnetic \feld, temperature, thickness of the YIG \flm and\npower of the radio-frequency \feld. The proposed space-time crystals provide a new dimension for\nexploring dynamical phases of matter and can serve as a model nonlinear Floquet system, that brings\nin touch the rich \felds of classical nonlinear waves, magnonics and periodically driven systems.\nSpontaneous symmetry breaking is a fundamental\nconcept of physics. A well known example is the\nbreaking of spatial translational symmetry, which leads\nto a phase transition from \ruids to solid crystals. By\nanalogy, one can think about a \\time crystal\" as the\nresult of breaking translational symmetry in time. More\ngenerally, one expects the appearance of a \\space-time\ncrystal\" as a consequence of breaking translational\nsymmetry both in time and in space. If a time crystal\nexists it should demonstrate time-periodic motion of its\nground state [1]. In addition to the time periodicity, the\nspace-time crystals should be periodic in space, similar\nto an ordinary crystal.\nIt was recently argued that time- and space-time\ncrystals cannot be realized in thermodynamic equilib-\nrium [2, 3]. This led to a search of space-time symme-\ntry breaking in a wider context, for example in a sys-\ntem with \rux-equilibrium, rather than in the thermo-\ndynamic equilibrium. Needless to say that oscillatory\nnon-equilibrium states are well known already. One can\nremember, for example, gravity waves exited on a sea\nsurface under windy conditions, quasi-periodic current\ninstabilities in some semiconductors carrying strong DC\nelectric current (a Gunn e\u000bect [4] in A 3B5semiconduc-\ntors, such as n-type GaAs, used in police radar speed\nguns), instabilities of an electron beam in plasma. More\nrecent similar example is microwave generation by nano-\nsized magnetic oscillators driven by a spin-polarized DC\nelectric current [5]. On the other hand new physical phe-\nnomena can be observed in time-crystals which do not\nabsorb or dissipate energy from external pumping, but\ninstead build up a coherent time-periodic quantum state\ndue to the complexity in the interactions. Possibly a\n\frst nontrivial example, found in Ref. [6], was a periodi-\ncally driven Floquet quantum disordered system, demon-\nstrating subharmonic behavior. Non trivially, this system\ndoes not actually absorb and dissipate any of the exter-nally pumped energy because the disorder in the system\nmakes the energy states isolated from one another, see\nalso Refs. [7, 8]. The possibly most recent observation of\na time crystal with very long energy relaxation time com-\npared to energy-conserving interaction processes is the\nBose-Einstein condensate (BEC) of magnons in a \rexible\ntrap in super\ruid3He-Bunder periodic driving by an ex-\nternal magnetic \feld [9]. Other important requirements\nfor the existence of time crystals include the robustness|\nthe independence of their features to the perturbations of\nthe physical system (e.g. level of disorder) and appear-\nance of the soft modes [10{12]. The problems related to\nspace-time crystals [13] are more involved and up to now\nwere explored mostly theoretically [14{18].\nIn this paper we report the experimental realization\nof a space-time crystal with tunable periodicity in time\nand space in the magnon BEC [19{24], formed in a room-\ntemperature Yttrium Iron Garnet (YIG, Y 3Fe5O12) \flm.\nThe condensate spontaneously arises as a result of scat-\ntering of parametrically injected magnons to the bot-\ntom of their spectrum. The scattered magnons initially\nform spectrally localized groups, which can be best de-\nscribed as a time-polycrystal with partial coherence. Af-\nter switching o\u000b the pumping, we observe an interaction-\ndriven condensation into two coherent spatially extended\nspin waves|magnon BECs|which are best character-\nized by a space-time crystal. We show that this coherent\nstate has the hallmark of non-universal relaxation times,\nwhich are much longer than the intrinsic time scales and\nthe crystallization time. We consider the magnon BEC\nas a model object in studies of Floquet nonlinear wave\nsystems, subject to intensive periodical in time impact.\nThe frequency spectrum of magnons in this system,\nshown in Fig. 1, has two symmetric minima with non-\nzero frequency and wave-vectors !min=!(\u0006qmin). The\npossible BEC has accordingly two components with the\nwave vectors\u0006qminand the frequency !min. The simplestarXiv:1811.05801v1  [cond-mat.quant-gas]  14 Nov 20182\nFIG. 1. Magnon spectrum of the \frst 48 thickness modes in\n5.6-\u0016m-thick YIG \flm magnetized in plane by a bias magnetic\n\feldH= 1400 Oe, shown for the wavevector qkH(lower\npart of the spectrum, blue curves) and for q?H(upper\npart, magenta curves). The red arrow illustrates the magnon\ninjection process by means of parallel parametric pumping.\nTwo orange dots indicate positions of the frequency minimum\n!min(\u0006qmin) occupied by the BECs of magnons.\nform of their common wave function is a standing wave:\nC(r;t) =C0cos(qmin\u0001r) exp(\u0000i!mint): (1)\nIn this experiment we create a BEC by microwave radia-\ntion of frequency !p'2\u0019\u000113:6 GHz that can be consid-\nered space homogeneous with wave number qp\u00190. The\ndecay instability of this \feld with the conservation law\n!p=)!(q) +!(\u0000q) = 2!(q) (2)\nexcites \\parametric\" magnons with frequency\n!(q) =!p=2 and wave vectors \u0006q. These para-\nmetric magnons further interact mainly via 2 ,2\nscattering with the conservation laws:\n!(q1) +!(q2) =!(q3) +!(q4);q1+q2=q3+q4;(3)\nthat preserves the total number af magnons and their\nenergy. It is known from the theory of weak wave\nturbulence [25] (see also Ref. [26]) that the scattering\nprocess (3) results mostly in a \rux of energy towards\nlargeq, which leads to a nonessential accumulation of\nenergy at large q, and to a \rux of magnons toward small\nq. This in turn results in an accumulation of magnons\nnear the bottom of the frequency spectrum !min. The\nsame 2,2 processes (3) lead to e\u000bective thermalization\nof the bottom magnons during some time \u001cth.50\u000070 ns\nand the subsequent creation of the BEC state [24, 27].\nThe described processes that lead to the creation of\nBEC, are an experimental manifestation of the space-\nSwitchPower\namplifierYIG filmProbing laser \nbeam and back-\nscattered light\nMicrowave \nresonatorHθǁ\nSuper-\ncurrents\nz\nyx\nMicrowave\nsource\nAttenuator    Pulsed \n  microwave \npumpingPulse\ngeneratorLaser AOMBeam \nsplitter\nFabry-Pérot\ninterferometerPhoto-\ndetector\nTime-resolved\nanalysis\nLens\nqBECFIG. 2. Experimental set-up. The lower part of the \fgure\nshows the microwave circuit, consisting of a microwave source,\na switch and an ampli\fer. This circuit drives a microstrip res-\nonator, which is placed below the in-plane magnetized YIG\n\flm. Light from a solid-state laser ( \u0015= 532 nm) is chopped\nby an acousto-optic modulator (AOM) and guided to the YIG\n\flm. There it is scattered inelastically from magnons, and the\nfrequency-shifted component of the scattered light is selected\nby the tandem Fabry-P\u0013 erot interferometer, detected, and an-\nalyzed in time.\ntime crystal (STC): a system, driven away from ther-\nmodynamic equilibrium by a space-homogeneous, time-\nperiodic (with frequency !p) pumping \feld, sponta-\nneously chooses a space-time periodic state (1) with the\nfrequency!minand non-zero wavevectors \u0006qmin. Impor-\ntantly, the parameters !minand\u0006qminare fully deter-\nmined by intrinsic interactions in the system and are in-\ndependent of the pumping frequency in a wide range of its\nvalues. By varying the strength and direction of the ex-\nternal time-independent homogeneous magnetic \feld H,\nthe temperature Tand the thickness of the YIG \flm, we\ncan change the magnon spectrum !(q) and consequently\n!minand\u0006qminindependently of !p. Note that the life-\ntime\u001cBECof the condensate is much longer than \u001cth, en-\nabling the observation of the magnon BEC state and the\nstudy of related e\u000bects, such as magnon supercurrent [22]\nand Nambu-Goldstone modes|the Bogolyubov second\nsound [28]. All these meet the presently accepted crite-\nria of a space-time crystal, i.e the spontaneous symmetry\nbreaking in time and in space, manifested by long-range\norder and soft modes [17] (in our case the Bogolyubov\nsecond sound [28]).\nThe BEC is created from the gaseous incoherent\nmagnons, that accumulate in a relatively narrow fre-\nquency band \u0001 fSTPC near the bottom of the spectrum.\nTo keep in line with the crystal analogy, we will re-\nfer to this state as a space-time-polycrystal (STPC).\nIn our measurements, the autocorrelation time of these\nmagnons (1 =\u0001fSTPC\u00152 ns, see Fig. 4) signi\fcantly ex-\nceeds the wave period 2 \u0019=! min\u00190:15\u00000:3 ns, similarly3\nto the autocorrelation length in polycrystals that spans\nmany unit cell sizes.\nIn our experiments, the magnon BEC in the room-\ntemperature YIG \flms was detected by means of pulsed\nBrillouin light scattering (BLS) spectroscopy. Here the\nfocused laser beam acts both as a probe of the magnon\ndensity and as a heating source, which induces a thermal\ngradient across the probing light spot. The temperature\nin the spot, and thus the value of thermal gradient, was\ncontrolled by the duration of a probing laser pulse. The\nthermal gradient locally changes the saturation magne-\ntization and induces a frequency shift between di\u000berent\nparts of the magnon condensate [29]. Consequently, a\nphase gradient in the BEC wavefunction is gradually cre-\nated and a magnon supercurrent [22, 23], \rowing out of\nthe hot region of the focal spot is excited. Such a process\nreduces the number of magnons in the heated area and\nresults in the disappearance of the condensate and in the\nsubsequent disappearance of the supercurrent. The con-\nventional relaxation dynamics of the magnons is then re-\ncovered. More details about our experimental techniques\none \fnds in a sketch of the experimental setup shown in\nFig. 2, in Ref. [22] and in the supplementary material.\nWe demonstrate here how to change all three param-\neters of the STC, Eq. (1): the BEC magnon density\njC0j2=NBEC, the frequency !minand the wave number\n\u0006qmin. The STC lifetime can be controlled as well. The\nmost interesting information may be obtained by varying\nNBEC. We succeeded to change NBECby more than an\norder of magnitude by tuning the power of the pumping\n\feld. The measured BLS intensity is shown in Fig. 3a as\na function of time for selected pumping powers Ppump\nand two probing laser pulse durations \u001cL. The pumping\npulse acts during the time interval from \u00002000 ns to\n0 ns. Clearly, a decrease in the pumping power from\nPpump = 31 dBm to 19 dBm and a consequent reduction\nin the number of parametric magnons, which are injected\nat!(q) =!p=2, leads to a weakening of 2 ,2 magnon\nscattering and, thus, to an increasing delay in the\nappearance of these magnons near the bottom of the\nenergy spectrum, as observed by BLS. The density of\nthe bottom magnons, proportional to the intensity of the\nmeasured BLS signal, decreases as well (see the yellow\nshaded area in Fig. 3a, labeled \\Polycrystalline phase\").\nAfter the pumping pulse is switched o\u000b (for t>0 ns),\nthe magnons condense in the energetic minimum of the\nspectrum, creating the STC. In case of strong heating\n(\u001cL= 80\u0016s), this process results in the appearance of\na magnon supercurrent, which only involves the con-\ndensed and therefore coherent magnons. This out\row\nof magnons (blue shaded area in Fig. 3a labeled space-\ntime crystal ) results in a higher decay rate of the magnon\ndensity in the laser focal point. This e\u000bective decay rate,\nwhich is in\ruenced by the inherent damping of both co-\nherent and incoherent magnons to the phonon bath and\nby the supercurrent-related leakage of the magnon BEC,\nFIG. 3. Transition from the polycrystalline magnon phase\nto the space-time crystal phase and back. (a) Temporal\ndynamics of the measured magnon density for a few pumping\npower values Ppump at di\u000berent temperatures of the probing\nspot. The bias magnetic \feld H= 1400 Oe. The top BLS\nwaveform measured for Ppump = 31 dBm corresponds to\nthe case of the weakly heated YIG sample (duration of the\nprobing laser pulse \u001cL= 6\u0016s) and, therefore, is not a\u000bected\nby a supercurrent magnon out\row. In all other cases, the\nnon-uniform heating of the YIG sample ( \u001cL= 80\u0016s) creates a\nmagnon supercurrent \rowing out from the heated area result-\ning in a higher decay rate of the magnons in the BEC phase.\nThis e\u000bective decay rate falls with the pumping power. Below\na critical magnon density Ncr, characterizing the transition\nfrom the space-time crystal phase back to the polycrystalline\nphase, the decay rate is approximately the same for all cases.\n(b) The decay times \u001cdecof the space-time crystal phase\n(open and solid circles) and polycrystalline phase (squares)\nas functions of the pumping power Ppump. The space-crystal\nphase does not exist at pumping powers below 21 dBm.\nis strongly dependent on the pumping power. This de-\npendence stems from the fact, that a lower pumping\npower leads to a reduced magnon density and therefore\nto a smaller fraction of BEC magnons. Below a certain\nthreshold density Ncr, when the majority of condensed\nmagnons are \rown out of the measured region of the\nBEC, the observed decay rate approaches the same value\nfor all di\u000berent pumping powers. This decay rate corre-\nsponds to the inherent decay rate of a narrow package of\nthe remaining polycrystalline magnon phase.\nIt is worth noting that the same decay rate is observed4\nFIG. 4. BLS intensity (color code) measured for q=qminas\na function of the magnon frequency fand of the bias magnetic\n\feldH. Film thickness 5 :6\u0016m, pumping power 40 W, pump-\ning frequency 14 GHz, pumping pulse duration 1 \u0016s, pump-\ning period 200 \u0016s. The dashed line represents the analytical\ndependence of the frequency of the spin wave spectrum bot-\ntom onH:fqmin(H) =\rH, where the gyromagnetic ratio\n\r= 2:8 MHz/Oe.\nduring the entire decay time when heating of the YIG\nsample can be neglected, and therefore there is no su-\npercurrent that takes away the coherent magnons. For\nexample, the black top waveform in Fig. 3a was measured\nat shorter heating times \u001cL= 6\u0016s. The pumping power\nis the same as for the red waveform (hot spot, Ppump =\n31 dBm), therefore it corresponds to a well-formed BEC.\nHowever, it is not possible to distinguish between the\nBEC and the incoherent magnons via the decay rate mea-\nsurements in this case. The latter fact contradicts a pre-\nvious interpretation of similar dynamics of the magnon\nBEC and the incoherent magnon phase in Ref. [30] as be-\ning a result of the sensitivity of the BLS technique to the\ndegree of coherence of the scattering magnons. Further-\nmore, two di\u000berent lifetimes of BEC observed at the same\npumping power prove our ability to control the lifetime\nof the magnon BEC by a thermal gradient.\nThereby, the density (Fig. 3a) and the lifetime\n(Fig. 3b) of the magnon space-time crystal are tunable\nby the parametric pumping power and by the proper\nadjustment of a spatial temperature pro\fle.\nThe time periodicity 1 =!minof the STC can be easily\nchanged by variation of the bias magnetic \feld. Fig-\nure 4 shows the BLS intensity from the bottom of the\nmagnon spectrum ( q=qmin) as a function of the fre-\nquency and the magnetic \feld. The color coded inten-\nsity of the BLS re\rects the e\u000eciency of the parametric\nmagnon transfer to the bottom of the frequency spec-\ntrum during the pumping pulse [23]. The dependence\nfqmin(H) is well described by the analytical dependence\nFIG. 5. Theoretical dependencies of the energy minimum\nwavelength \u0015min(black line) and frequency !min(red line) on\nthe YIG \flm thickness for T= 300 K and H= 1400 Oe.\n!min= 2\u0019fqmin(H) = 2\u0019\rH , where\ris the gyromag-\nnetic ratio.\nThe spatial periodicity of the STC can be changed in a\nwide range from'0:5\u0016m to'4\u0016m by a proper choice\nof the YIG \flm thickness, see Fig. 5. Note that, except\nfor very thin \flms, the !minis insensitive to the \flm\nthickness.\nThe tunable magnon space-time crystal, realized by\na periodically driven room-temperature YIG \flm, rep-\nresents an example of a nonlinear Floquet system and\ntherefore serves as a bridge between magnonics and clas-\nsical nonlinear wave physics from one side and the Flo-\nquet time-crystal description of the periodically driven\nsystems from another. Joining these two perspectives\nmay give birth to a new \feld of physical research: \\Flo-\nquet (or periodically driven) nonlinear wave physics\".\nThe advantage of a macroscopic system that may be\nstudied at room temperature as compared to small sam-\nples at milli-Kelvin temperatures, is obvious. Moreover,\nstrong nonlinearity, non-reciprocity, topology, local ma-\nnipulation via external electric and magnetic \felds and\nsample patterning, available in a magnonic system, com-\nbined with tunability and space-, time-, wave-vector- and\nfrequency-resolved measurements using BLS, makes the\nsuggested system a good experimental basis for the newly\nproposed \feld. On the other hand, concepts discussed\nin the framework of the Floquet systems such as quasi-\nenergy, umklapp scattering, forbidden bands in quasi-\nmomentum space, once applied to magnon space-time\ncrystals, may give new insight into the rich physics of\nthis system, creating new physical ideas and paving a\nway to new engineering applications.\nFinancial support by the European Research Coun-\ncil within the Advanced Grant 694709 \\SuperMagnon-\nics\" and by Deutsche Forschungsgemeinschaft (DFG)\nwithin the Transregional Collaborative Research Center\nSFB/TR49 \\Condensed Matter Systems with Variable\nMany-Body Interaction\" as well as by the DFG Project\nINST 248/178-1 is gratefully acknowledged.5\n\u0003kreil@rhrk.uni-kl.de\n[1] F. Wilczek, Quantum time crystals , Phys. Rev. Lett. 109,\n160401 (2012).\n[2] P. Bruno, Comment on \\Quantum time crystals\" , Phys.\nRev. Lett. 110, 118901 (2013).\n[3] H. Watanabe and M. Oshikawa, Absence of quantum time\ncrystals , Phys. Rev. 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Hillebrands, and A.N.\nSlavin, Bose-Einstein condensation of quasi-equilibriummagnons at room temperature under pumping , Nature\n443, 430 (2006).\n[20] P. Nowik-Boltyk, O. Dzyapko, V.E. Demidov, N.G.\nBerlo\u000b, and S.O. Demokritov, Spatially non-uniform\nground state and quantized vortices in a two-component\nBose-Einstein condensate of magnons , Sci. Rep. 2, 482\n(2012).\n[21] T. Giamarchi, C. R uegg, and O. Tchernyshyov, Bose-\nEinstein condensation in magnetic insulators , Nat. Phys.\n4, 198{204 (2008).\n[22] D.A. Bozhko, A.A. Serga, P. Clausen, V.I. Vasyuchka, F.\nHeussner, G.A. Melkov, A. Pomyalov, V.S. L'vov, and B.\nHillebrands, Supercurrent in a room-temperature Bose-\nEinstein magnon condensate , Nat. Phys. 12, 1057{1062\n(2016).\n[23] A.J.E. Kreil, D.A. Bozhko, H.Yu. Musiienko-Shmarova,\nV.S. L'vov, A. Pomyalov, B. Hillebrands, and A.A. Serga,\nFrom kinetic instability to Bose-Einstein condensation\nand magnon supercurrents , Phys. Rev. Lett. 121077203\n(2018).\n[24] A.A. Serga, V.S. Tiberkevich, C.W. Sandweg, V.I.\nVasyuchka, D.A. Bozhko, A.V. Chumak, T. Neumann,\nB. Obry, G.A. Melkov, A.N. Slavin, and B. Hillebrands,\nBose-Einstein condensation in an ultra-hot gas of pumped\nmagnons , Nat. Commun. 5, 3452 (2014).\n[25] V.E. Zakharov, V.S. L'vov, and G.E. Falkovich, Kol-\nmogorov Spectra of Turbulence I (Wave Turbulence)\n(Springer, Berlin, 1992).\n[26] Yu.M. Bunkov, E.M. Alakshin, R.R. Gazizulin, A.V.\nKlochkov, V.V. Kuzmin, V.S. L'vov, and M.S. Tagirov,\nHigh-Tcspin super\ruidity in antiferromagnets , Phys.\nRev. Lett. 108, 177002 (2012).\n[27] P. Clausen, D.A. Bozhko, V.I. Vasyuchka, B. Hillebrands,\nG.A. Melkov, and A.A. Serga, Stimulated thermalization\nof a parametrically driven magnon gas as a prerequisite\nfor Bose-Einstein magnon condensation , Phys. Rev. B\n91, 220402 (2015).\n[28] D.A. Bozhko, A.J.E. Kreil, H.Yu. Musiienko-Shmarova,\nA.A. Serga, A. Pomyalov, V.S. L'vov, and B. Hille-\nbrands, Long-distance supercurrent transport in a room-\ntemperature Bose-Einstein magnon condensate , arXiv:\n1808.07407 (2018).\n[29] M. Vogel, A.V. Chumak, E.H. Waller,T. Langner, V.I.\nVasyuchka, B. Hillebrands, and G. von Freymann, Op-\ntically recon\fgurable magnetic materials , Nat. Phys. 11,\n487{491 (2015).\n[30] V.E. Demidov, O. Dzyapko, S.O. Demokritov, G.A.\nMelkov, and A.N. Slavin, Observation of spontaneous co-\nherence in Bose-Einstein condensate of magnons , Phys.\nRev. Lett. 100, 47205 (2008).6\nSupplemental material: Experimental technique\nFigure 2 provides a sketch of the experimental setup. The\nsample is placed between the poles of an electromagnet,\nwhich creates a homogeneous magnetizing \feld Hly-\ning in the plane of YIG \flm. In order to reach a high\nenough density of the magnon gas to form a BEC phase,\na rather strong microwave pulse with the peak power\nPmax= 41 dBm is applied to the half-wave microstrip\nresonator at a carrier frequency fpump = 13:6 GHz. The\nresonator creates an Oersted \feld q(t)kHin the YIG\n\flm to excite magnons by means of parallel paramet-\nric pumping [31, 32]. Additionally, a variable attenua-\ntor is implemented in the microwave circuit to allow for\npower reduction of the ampli\fed microwave pulse to a\nlower value Ppump\u0014Pmax. The microwave pulse dura-\ntion is kept constant at \u001cp= 2\u0016s with a repetition rate\noffrep= 1 kHz to ensure that microwave heating e\u000bects\nare negligible.\nBoth the detection of the excited magnons and the\nheating of the YIG \flm were made by using the probing\nlaser beam with a power of 30 mW. The beam is chopped\nby an acusto-optic modulator (AOM) to control the en-\nergy input into the YIG sample. A duration of the laser\npulse of\u001cL= 80\u0016s is used for local heating, while the\npulse duration \u001cL= 6\u0016s is used to avoid the heatingof the probing point. The probing laser pulse is syn-\nchronized with the microwave pumping and has the same\nrepetition rate frep. The laser pulses of both durations\nare switched on before the application of the microwave\npulse and are switched o\u000b 3 \u0016s after its end. From the\nsample backscattered laser light is collected and sent to\na Tandem-Fabry-P\u0013 erot interferometer for frequency and\ntime-of-\right analysis with a frequency and time resolu-\ntion of approximately 100 MHz and 1 ns.\nIn order to selectively detect only the magnons con-\ndensed in the lowest energy state of the magnonic sys-\ntem with a wave number qk\u00194:5 rad\u0016m\u00001(qkkH)\nand a frequency fmin= 4 GHz (see Fig. 2, the angle\nof incidence \u0012kof the probing laser beam has to be\nchosen accordingly. The condition to detect a magnon\nwith a speci\fc wavevector qSWlying in a \flm plane is\nqSW= 2qlightsin(#). Therefore the angle of the incident\nlight has been chosen to be \u0012k= 12\u000e.\n\u0003kreil@rhrk.uni-kl.de\n[31] A.G. Gurevich and G.A. Melkov, Magnetization Oscilla-\ntions and Waves (CRC Press, Boca Raton, 1996).\n[32] V.S. L'vov, Wave Turbulence Under Parametric Excita-\ntion (Applications to Magnets) (Springer, Berlin, 1994)."    },    {        "title": "2303.14843v1.Compact_localised_states_in_magnonic_Lieb_lattices.pdf",        "content": "Compact localised states in magnonic Lieb lattices\nGrzegorz Centa la and Jaros law W. K los\u0003\nInstitute of Spintronics and Quantum Information,\nFaculty of Physics, Adam Mickiewicz University, Pozna\u0013 n,\nUniwersytetu Pozna\u0013 nskiego 2, Pozna\u0013 n 61-614, Poland\n(Dated: March 26, 2023)\nLieb lattice is one of the simplest bipartite lattices where compact localized states (CLS) are\nobserved. This type of localisation is induced by the peculiar topology of the unit cell, where the\nmodes are localized only on one sublattice due to the destructive interference of partial waves.\nThe CLS exist in the absence of defects and are associated with the \rat bands in the dispersion\nrelation. The Lieb lattices were successfully implemented as optical lattices or photonic crystals.\nThis work demonstrates the possibility of magnonic Lieb lattice realization where the \rat bands\nand CLS can be observed in the planar structure of sub-micron in-plane sizes. Using forward\nvolume con\fguration, we investigated numerically (using the \fnite element method) the Ga-dopped\nYIG layer with cylindrical inclusions (without Ga content) arranged in a Lieb lattice of the period\n250 nm. We tailored the structure to observe, for the few lowest magnonic bands, the oscillatory\nand evanescent spin waves in inclusions and matrix, respectively. Such a design reproduces the Lieb\nlattice of nodes (inclusions) coupled to each other by the matrix with the CLS in \rat bands.\nKeywords: \rat bands, compact localized states, Lieb lattice, spin waves, \fnite element method\nI. INTRODUCTION\nThere are many mechanisms leading to wave localiza-\ntion in systems with long-range order, i.e. in crystals\nor quasicrystals. The most typical of these require (i)\nthe local introduction of defects, including the defects in\nthe form of surfaces or interfaces [1] (ii) the presence of\nglobal disorder [2], (iii) the presence of external \felds [3]\nor (iv) the existence of many-body phenomena [4]. How-\never, since at least the late 1980s, it has been known\nthat localization can occur in unperturbed periodic sys-\ntems in the absence of \felds and many-body e\u000bects, and\nis manifested by the presence of \rat, i.e., dispersion-free\nbands in the dispersion relation. The pioneering works\nare often considered to be the publications of B. Suther-\nland [5] and E. H. Lieb [6], who found the \rat bands of\nzero energy [7] for bipartite lattices with use of the tight-\nbinding model Hamiltonians, where the hoppings occur\nonly between sites of di\u000berent sublattices. The simplest\nrealization of this type of system is regarded as the Lieb\nlattice [6, 8], where the nodes of one square sublattice, of\ncoordination number z= 4, connect to each other only\nvia nodes with a coordination number z= 2 from other\ntwo square sublattices (Fig. 1). In the case of extended\nLieb lattices [9, 10], the nodes of z= 2 form chains:\ndimmers, trimmers, etc.(Fig. 2). An intuitive explana-\ntion for the presence of the \rat bands is the internal\nisolation of excitations located in one of the sublattices.\nThe cancelling of excitations at one sublattice is the re-\nsult of forming destructive interference and local symme-\ntry within the complex unit cell [11]. When only one of\nthe sublattices is excited, the other sublattice does not\nmediate the coupling between neighbouring elementary\n\u0003klos@amu.edu.plcells, and the phase di\u000berence between the cells is irrel-\nevant to the energy (or the frequency) of the eigenmode\non the whole lattice - i.e. the Bloch function. Modes\nof this type are therefore degenerated for di\u000berent wave\nvector values in in\fnite lattices. We are dealing here\nwith the localization on speci\fc arrangements of struc-\nture elements, which are isolated from each other. Such\nkinds of modes are called compact localized states (CLS)\n[12{16] and show a certain resistance to the introduc-\ntion of defects [17, 18]. The \rat band systems with CLS\nare the platform for the studies of Anderson localization\n[19], and unusual properties of electric conductivity [20].\nA similar localization is observed in the quasicrystals,\nwhere the arrangements of the elements composing the\nstructure are replicated aperiodically and self-similarly\nthroughout the system [21, 22] and the excitation can be\nlocalized on such patterns. The CLS in \fnite Lieb lattices\nhave a form of loops (plaquettes) occupying the majority\nnodes (z= 2). These states are linearly dependent and\ndo not form a complete basis for the \rat band. Therefore\noccupancy gaps need to be \flled (for in\fnite lattice) by\nstates occupying only one sublattice of majority nodes,\nlocalizes at lines, called noncontractible loop states (NLS)\n[14, 15, 23].\nThe topic of Lieb lattices and other periodic struc-\ntures with compact localization and \rat bands was re-\nnewed [8] about 10 years ago when physical realizations\nof synthetic Lieb lattices began to be considered for elec-\ntronic systems [24, 25], optical lattices [26, 27], supercon-\nducting systems [28, 29], in phononics [30] and photonics\n[14, 31]. In a real system, where the interaction cannot\nbe strictly limited to the nearest elements of the struc-\nture, the bands are not perfectly \rat. Therefore, some\nauthors use the extended de\fnition of the \rat band to\nconsider the bands that are \rat only along particular di-\nrections or in the proximity of high-symmetry Brillouin\nzone points [32]. In tight-binding models, this e\u000bect canarXiv:2303.14843v1  [cond-mat.mes-hall]  26 Mar 20232\nbe included by considering the hopping to at least next-\nnearest-neighbours [33, 34]. Similarly, the crossing of the\n\rat band by Dirac cones can be transformed into anti-\ncrossing and lead to opening gaps, separating the \rat\nband from dispersive bands. This e\u000bect can in induced\nby the introduction of spin-orbit term to tight-binding\nHamiltonian (manifested by the introduction of Peierls\nphase factor to the hopping) or by dimerization of the\nlattice (by alternative changes of hoppings or site ener-\ngies) [33{37]. The later scenario can be easily observed in\nreal systems where the position of rods/wells (mimicking\nthe sites of Lieb lattice) and contrast between them can\nbe easily altered [38]. Opening the narrow gap between\n\rat band and dispersive bands for Lieb lattice is also fun-\ndamentally interesting because it leads to the appearance\nof so-called Landau-Zener Bloch oscillations [39].\nThe isolated and perfectly \rat bands for Lieb lattices\nare topologically trivial { their Chern number is equal to\nzero [40]. For weakly dispersive (i.e. almost \rat) bands\nthe Chern numbers can be non-zero [41]. However, when\nthe \rat band is intersected by dispersive bands then it\ncan exhibit the discontinuity of Hilbert{Schmidt distance\nbetween eigenmodes corresponding to the wave vectors\njust before and just after the crossing. Such an e\u000bect is\ncalled singular band touching [16]. This limiting value\nof Hilbert{Schmidt distance is bulk invariant, di\u000berent\nfrom the Chern number.\nOne of the motivations for the photonic implementa-\ntion of systems with \rat, or actually nearly \rat bands\n[42], was the desire to reduce the group velocity of light\nin order to compress light in space, which leads to the\nconcentration of the optical signal and an increase in\nthe light-matter interaction, or the enhancement of non-\nlinear e\u000bects. Another, more obvious application is the\npossibility of realizing delay lines that can bu\u000ber the sig-\nnal to adjust the timing of optical signals [43]. A promis-\ning alternative to photonic circuits are magnonic sys-\ntems, which allow signals of much shorter wavelengths\nto be processed in devices several orders of magnitude\nsmaller [44, 45]. For this reason, it seems natural to seek\na magnonic realization of Lieb lattices.\nIn this paper, we propose the realization of such lat-\ntices based on a magnonic structure in the form of\na perpendicularly magnetized magnetic layer with spa-\ntially modulated material parameters or spatially vary-\ning static internal \feld. Lieb lattices have been studied\nalso in the context of magnetic properties, mainly due to\nthe possibility of enhancing ferromagnetism in systems\nof correlated electrons [46], where the occurrence of \rat\nbands with zero kinetic energy was used to expose the in-\nteractions. There are also known single works where the\nspin waves have been studied in the Heisenberg model in\nan atomic Lieb lattice, such as the work on the magnon\nHall e\u000bect [47]. But the comprehensive studies of spin\nwaves in nanostructures that realize magnonic Lieb lat-\ntices and focus on wave e\u000bects in a continuous model have\nnot been carried out so far. In this work, we demonstrate\nthe possibility of realization of magnonic lattices in pla-nar structure based on low spin wave damping material:\nyttrium iron garnet (YIG) where the iron is partially sub-\nstituted by gallium (Ga). We present the dispersion rela-\ntion with a weakly depressive (\rat) band exhibiting the\ncompact localized spin waves. The \rat is almost inter-\nsected at the Mpoint of the 1stBrillouin zone by highly\ndispersive bands, similar to Dirac cones. We discuss the\nspin wave spectra and compact localized modes both for\nsimple and extended Lieb lattices.\nThe introduction is followed by the section describing\nthe model and numerical method we used, which pre-\ncedes the main section where the results are presented\nand discussed. The paper is summarized by conclusion\nand supplemented with additional materials where we\nshowed: (A) the results for extended Lieb-7 lattice, (B)\nan alternative magnonic Lieb lattice design via shaping\nthe demagnetizing \feld, and (C) an attempt of formation\nmagnonic Lieb lattice by dipolarly coupled magnetic na-\nnoelements, (D) discussion of small di\u000berences in the de-\nmagnetizing \feld of majority and minority nodes respon-\nsible for opening a small gap in the Lieb lattice spectrum.\nII. STRUCTURE\nMagnonic crystals (MCs) are regarded as promis-\ning structures for magnonic-based device applications\n[45, 48]. In our studies, we consider planar MCs to de-\nsign the magnonic Lieb lattice, owing to the relative ease\nof fabrication of such structures and their experimental\ncharacterization [49{51]. We proposed realistic systems\nthat mimic the main features of the tight-binding model\nof Lieb lattice [16, 33].\nInvestigated MCs consist of yttrium iron garnet doped\nwith gallium (Ga:YIG) matrix and yttrium iron gar-\nnet (YIG) cylindrical inclusions arranged in Lieb lattice\nFig. 1. Doping YIG with Gallium is a procedure where\nmagnetic Fe3+ions are replaced by non-magnetic Ga3+\nions. This method not only decreases saturation mag-\nnetizationMSbut, simultaneously, arises uni-axial out-\nof-plane anisotropy, that ensures the out-of-plane orien-\ntation of static magnetization in Ga:YIG layer at a rel-\natively low external \feld applied perpendicularly to the\nlayer. Discussed geometry, i.e. forward volume magne-\ntostatic spin wave con\fguration, does not introduce an\nadditional anisotropy in the propagation of spin waves,\nrelated to the orientation of static magnetization.\nThe design of the Lieb lattice requires the partial local-\nization of spin wave in inclusions, which can be treated\nas an approximation of the nodes from the tight-binding\nmodel. Furthermore, the neighbouring inclusions in the\nlattice have to be coupled strongly enough to sustain\nthe collective spin wave dynamics, and weakly enough\nto minimize the coupling between further neighbours.\nTherefore, the geometrical and material parameters were\nselected to ensure the occurrence of oscillatory excita-\ntions in the (YIG) inclusions and exponentially evanes-\ncent spin waves in the (Ga:YIG) matrix. The size of3\nBAa)\nb)\nFIG. 1. Basic magnonic Lieb lattice. The planar magnonic\nstructure consists of YIG cylindrical nanoelements embedded\nwithin Ga:YIG. Dimensions of the ferromagnetic unit cell are\nequal to 250x250x59 nm and the unit cell contains three in-\nclusions of 50 nm diameter. (a) The structure of basic Lieb\nlattice, and (b) top view of the Lieb lattice unit cell where the\nnode (inclusion) from minority sublattice Aand two nodes\n(inclusions) from two majority sublattices Bare marked.\ninclusions was chosen small enough to separate three\nlowest magnonic bands with almost uniform magnetiza-\ntion precession inside the inclusion from the bands of\nhigher frequency, where the spin waves are quantised in-\nside the inclusions. Also, the thickness of the matrix and\ninclusion was chosen in a way that there are no nodal\nlines inside the inclusion. The condition which guaran-\ntee the focussing magnetization dynamics inside the in-\nclusions is ful\flled in the frequency range below the ferro-\nmagnetic resonance (FMR) frequency of the out-of-plane\nmagnetized layer made of Ga:YIG (matrix material):\nfFMR;Ga:YIG =4.95 GHz and above the FMR frequency\nof out-of-plane magnetized layer made of YIG (inclusions\nmaterial):fFMR;YIG= 2:42 GHz. These limiting values\nwere obtained using the Kittel formula for out-of-plane\nmagnetised \flm: fFMR =\r\n2\u0019j\u00160H0+\u00160Hani\u0000\u00160MSj,\nwhere we used the following values of material param-\neters [52] for YIG: gyromagnetic ratio \r= 177GHz\nT,\nmagnetization saturation \u00160MS= 182:4 mT, exchange\nsti\u000bness constant A= 3:68pJ\nm, (\frst order) uni-axial\nanisotropy \feld \u00160Hani=\u00003:5 mT, and for Ga:YiG:\n\r= 179GHz\nT,\u00160MS= 20:2 mT,A= 1:37pJ\nm,\u00160Hani=\n94:1 mT. Since the greatest impact of the \frst order\nuniaxial anisotropy \feld ( \u00160Hani), we decided to ne-\nglect higher order terms of uni-axial anisotropy and cubic\nanisotropy of (Ga:)YIG. Due to the presence of out-of-\nplane anisotropy and relatively low saturation magneti-\nzation, we could consider a small external magnetic \feld\n\u00160H0= 100 mT to reach saturation state.\nIt is worth noticing that without the evanescent spin\nwaves in the ferromagnetic matrix, the appropriate cou-\nBAa)\nb)FIG. 2. Extended magnonic Lieb lattice { Lieb - 5. Dimen-\nsions of the unit cell are 375x375x59 nm and contain 5 inclu-\nsions of size 50 nm in diameter. Also, we maintain the same\nseparation (distance between centres of neighbouring sites is\n125 nm) as for considered basic Lieb lattice { Fig. 1. (a) The\nstructure of Lieb-5 lattice, and (b) top view on Lieb-5 lattice\nunit cell where the node (inclusion) from minority sublattice\nAand four nodes (inclusions) from two majority sublattices\nBare marked.\npling between inclusions would not be possible. There-\nfore the realization of the Lieb lattice in form of the array\nof ferromagnetic nanoelemets embedded in air/vacuum\nseems to be very challenging { see the exemplary results\nin Supplementary Information C.\nWe also tested the possibility of other realizations of\nmagnonic Lieb lattices. One solution seemed to be the\ndesign of a structure in which the concentration of the\nspin wave amplitude in the Lieb lattice nodes would be\nachieved through an appropriately shaped pro\fle of the\nstatic demagnetizing \feld { Supplementary Information\nB. However, the obtained results were not as promising\nas for YIG/Ga:YIG system.\nIn the main part of the manuscript, we present the re-\nsults for the basic Lieb lattice (showed in Fig. 1) and\nextended Lieb-5 lattice (showed in Fig. 2), based on\nYIG/Ga:YIG structures. The further extension of the\nLieb lattice may be realized by increasing the number of\nBnodes between neighbouring Anodes. Supplementary\nInformation A presents the results for Lieb-7, where for\neach site (inclusion) from minority sublattice A, we have\nsix nodes (inclusions), grouped in three-element chains,\nfrom majority sublattices B.\nIII. METHODS\nThe spin waves spectra and the spatial pro\fles of\ntheir eigenmodes were obtained numerically in a semi-\nclassical model, where the dynamics of magnetization4\nvector M(r;t) is described by the Landau-Lifshitz equa-\ntion [53]:\ndM\ndt=\u0000\r\u00160[M\u0002He\u000b+\u000b\nMSM\u0002(M\u0002He\u000b)]:(1)\nThe symbol He\u000b(r;t) denotes e\u000bective magnetic \feld.\nIn numerical calculations, we neglected the damping\nterm since \u000bis small both for YIG and for YIG with\nFe substituted partially by Ga (for \u000bGa:YIG = 6:1\u000210\u00004\nand\u000bYIG= 1:3\u000210\u00004[52]). The e\u000bective magnetic \feld\nHe\u000bcontains the following components: the external \feld\nH0, exchange \feld Hex, bulk uniaxial anisotropy \feld\nHaniand dipolar \feld Hd:\nHe\u000b(r;t) =H0^ z+2A\n\u00160M2\nSr2M(r;t)+Hani(r)^ z\u0000r'(r;t);\n(2)\nwhere the z\u0000direction is normal to the plane of the\nmagnonic crystal. We assume that the sample is satu-\nrated inz\u0000direction and magnetization vector precesses\naround this direction. The material parameters ( MS,A,\n\u000band\r) are constant within matrix and inclusions.\nUsing the magnetostatic approximation the dipolar\nterm of the e\u000bective magnetic \feld may be expressed as\na gradient of magnetic scalar potential:\nHd(r;t) =\u0000r'(r;t) (3)\nBy using the Gauss equation magnetic scalar potential\nmay be associated with magnetisation as follows:\nr2'(r;t) =r\u0001M(r;t) (4)\nSpin-wave dynamics is calculated numerically using\nthe \fnite-element method (FEM). We used the COMSOL\nMultiphysics [54] to implement the Landau-Lifshitz equa-\ntion (Eq. 1) and performed FEM computation for the de-\n\fned geometry of magnonic Lieb lattices. The COMSOL\nMultiphysics is the software used for solving a number of\nphysical problems, since many implemented modules it\nbecomes more and more convenient. Nevertheless, all the\nequations were implemented in the Mathematics module\nwhich contains di\u000berent forms of partial di\u000berential equa-\ntions. Eq. 1 was solved by using eigenfrequency study, on\nthe other hand, to solve Eq. 4 we used stationary study.\nTo obtain free decay of scalar magnetic potential in the\nmodel we applied 5 \u0016m of a vacuum above and under-\nneath the structure. At the bottom and top surface of the\nmodel with vacuum, we applied the Dirichlet boundary\ncondition. We use the Bloch theorem for each variable\n(magnetostatic potential and components of magnetiza-\ntion vector) at the lateral surfaces of a unit cell. We\nselected the symmetric unit cell with minority node Ain\nthe centre to generate a symmetric mesh which does not\nperturb the four-fold symmetry of the system { this ap-\nproach is of particular importance for the reproduction of\nthe eigenmodes pro\fles in high-symmetry points. In our\nnumerical studies, we used 2D wave vector k=kx^x+ky^y\nas a parameter for eigenvalue problem which was selectedalong the high symmetry path \u0000 \u0000X\u0000M\u0000\u0000 to plot the\ndispersion relation. We considered the lowest 3, 5 and\n7 bands for basic Lieb lattice, Lieb-5 lattice and Lieb-7\nlattice, respectively.\nIV. RESULTS\nThe tight-biding model of the basic Lieb lattice, with\nhopping restricted to next-neighbours gives three bands\nin the dispersion relation. The top and bottom bands are\nsymmetric with respect to the second, perfectly \rat band,\nand intersect with this dispersionless band at Mpoint of\n1stBrillouin zone, with constant slope forming two Dirac\ncones[8, 27]. In a realistic magnonic system, the spin\nwave spectrum showing the particle-hole symmetry with\na zero energy \rat band is di\u000ecult to reproduce because\n(i) the dipolarly dominated spin waves, propagating in\nmagnetic \flm, experience a signi\fcant reduction of the\ngroup velocity with an increase of the wave vector (this\ntendency is reversed for much larger wave vectors were\nthe exchange interaction starts to dominate) [53], (ii) the\ndipolar interaction is long-range. The \frst e\u000bect makes\nthe lowest band wider than the third band, and the latter\none { induces the \fnite width of the second band [33].\nWe are going to show, that this weakly dispersive band\nsupports the existence of CLS. Therefore, we will still\nrefer to it as \rat band , which is a common practice for\ndi\u000berent kinds of realization of Lieb lattices in photonics\nor optical lattices.\nThe results obtained for the basic magnonic Lieb lat-\ntice, (Fig. 1), are shown in Fig. 3. As we predicted, three\nlowest bands form a band structure which is similar to the\ndispersion relation known from the tight-binding model\n[10]. However, in a considered realistic system there is an\nin\fnite number of higher bands, not shown in Fig. 3(a).\nFor higher bands, spin waves can propagate in an oscil-\nlatory manner in the matrix hence the system does not\nmimic the Lieb lattice where the excitations should be\nassociated with the nodes (inclusions) of the lattice.\nDue to the fourfold symmetry of the system, the dis-\npersion relation could be inspected along the high sym-\nmetry path \u0000\u0000X\u0000M\u0000\u0000. Frequencies of the \frst three\nbands are in the range fFMR;YIG\u0000fFMR;Ga:YIG . Their\ntotal width is about \u00190:78 GHz. The \frst and third\nband form Dirac cones at Mpoint, separated by a tiny\ngap\u001915 MHz. The possible mechanism responsible for\nopening the gap is a small di\u000berence in the demagnetizing\n\feld in the areas of inclusions A(from the minority lat-\ntice) and inclusions B(from two majority sublattices) {\nsee Supplementary Information D. Inclusions A(B) have\nfour (two) neighbours of type B(A). Although inclusions\nAandBhave the same size and are made of the same\nmaterial, the static \feld of demagnetization inside them\ndi\u000bers slightly due to the di\u000berent neighbourhoods. This\ne\u000bect is equivalent to the dimerization of the Lieb lattice\nby varying the energy of the nodes in the tight-binding\nmodel, which leads to the opening of a gap between Dirac5\na)\nb)BA\nFIG. 3. Dispersion relation for the basic magnonic Lieb lat-\ntice, containing three inclusions in the unit cell: one inclusion\nAfrom minority sublattice and two inclusions Bfrom ma-\njority sublattices (see Fig. 1). (a) The dispersion relation\nis plotted along the high symmetry path \u0000-X-M-\u0000 (see the\ninset). The lowest band (blue) and the highest band (red)\ncreate Dirac cones almost touching (b) in the M point. The\nmiddle band (green) is relatively \rat in the vicinity of the M\npoint.\ncones and parabolic \rattening of them in very close prox-\nimity to the M-point. It is worth noting that in the in-\nvestigated system, the gap opens between the \frst and\nsecond bands, while the second and third bands remain\ndegenerated at point M, with numerical accuracy.\nThe middle band can be described as weakly disper-\nsive. The band is more \rat on the X\u0000Mpath and,\nin particular, in the vicinity of Mpoint { see Fig. 3(b).\nThe small width of the second band can be attributed\nto long-range dipolar interactions which govern the mag-\nnetization dynamics in a considered range of sizes and\nwave vectors. It is known that even the extension of the\nrange of interactions to next-nearest-neighbours in the\ntight-binding model induces the \fnite width of the \rat\nband for the Lieb lattice.\nTo prove that the second band supports the CLS re-\ngardless of its \fnite width, we plotted the pro\fles of spin\nwave eigenmodes at Mpoint and in its close vicinity. The\nresults are presented in Fig. 4. The pro\fles were shown\nfor in\fnite lattice and are presented in the form of square\narrays containing 3x3 unit cells, where the dashed lines\nBA\nM1\nM2\nM3\nM1←\nM2←\nM3←CLS NLS\n+-\n-\n+ - -NLS++\n-\n-FIG. 4. The spin wave pro\fles obtained for the basic\nmagnonic Lieb lattice, composed of three inclusions in the\nunit cell (see Fig. 1). The modes are presented for each band\nexactly at M(left column) and in its proximity ( M ) on the\npath M\u0000\u0000 (right column). In the presented pro\fles, the sat-\nuration and the colour denote the amplitude and phase of the\ndynamic, in-plane component of magnetization. The compact\nlocalized states (CLS) are presented at the point M for the\nsecond band { right column. The CLS do not occupy minority\nsublattice A. The inclusions B, in which the magnetization\ndynamics is focused, are quite well isolated from each other.\nOne can easily notice that the lattice is decorated by loops\n(marked by grey patches) where the phase of the precessing\nmagnetization \rips between inclusions (+ and \u0000signs). Ex-\nactly at point M{ left column, we observe the degeneracy\nof the second and third bands. The spin waves occupy Bin-\nclusions only in one majority sublattice, i.e. along vertical or\nhorizontal lines, \flliping the phase from inclusion to inclusion\nwhich gives the pattern characteristic to noncontractible loop\nstates (NLS) - marked by grey stripes.\nmark their edges. It is visible that the spin waves are\nconcentrated in the cylindrical inclusions, where the am-\nplitude and phase of precession is quite homogeneous. In\ncalculations, we used the Bloch boundary conditions ap-\nplied for a single unit cell, which means that at Mpoint\nthe Bloch function is \ripped after translation by lattice\nperiod, in both principal directions of the lattice and we6\nwill not see the single closed loops of CLS or lines of NLS.\nExactly at Mpoint, all three bands have zero group ve-\nlocity. Therefore, the corresponding modes (left column)\nare not propagating. The lowest band ( M1) occupy only\ninclusionsAfrom the minority sublattice where the static\ndemagnetizing \feld is slightly lower than inside inclu-\nsionsB(see Supplementary Information D), which jus-\nti\fes its lower frequency and lifting the degeneracy with\ntwo higher modes M2andM3of the same frequency.\nEach of the modes M2andM3occupy only one of two\nsublattices B, therefore they can be interpreted as NLS.\nTo observe the pattern typical for CLS, we need to move\nslightly away from Mpoint. The \frst and third modes\nhave then the linear dispersion with high group velocity\nand the second band remains \rat. We selected the point\nM shifted from Mpoint toward \u0000 point by 5% of M\u0000\u0000\ndistance (right column). We can see that the \frst and\nthird modes M \n1,M \n3occupy now all inclusions and\nthe modeM \n2from the \rat band has a pro\fle typical for\nCLS, predicted by tight-binding models [8{10, 55, 56]:\njmk>= [\u0000eiky\n2|{z}\nB;0|{z}\nA;eikx\n2|{z}\nB] (5)\nwheremkis the complex amplitude of the Bloch function\nin the base of unit cell (i.e. on two inclusion Bfrom ma-\njority sublattices and one inclusion Afrom minority sub-\nlattice), k= [kx;ky] is dimensionless wave vector. From\n(Eq. 5), we can see that (i) CLS do not occupy the mi-\nnority nodes Aand (ii) close to Mpoint the phases at\ntwo nodesB, from di\u000berent majority sublattices, are op-\nposite. These two features are reproduced for M \n2mode\nin investigated magnonic Lieb lattice. In the pro\fle of\nthis mode, we marked (by a grey patch) the elementary\nloop of CLS which is easily identi\fed in \fnite systems.\nHere, in an in\fnite lattice with Bloch boundary condi-\ntions, the loops are in\fnitely replicated with \u0019phase\nshift after each translation x\u0000andy\u0000direction. The lo-\ncalization at the inclusions Band the absence of the spin\nwave dynamics in inclusions Ais observed regardless of\nthe wave vector. Therefore, the coupling can take place\nonly between the next neighbours (inclusions B), i.e. on\nlarger distances and mostly due to dipolar interactions,\nthat makes the second band not perfectly \rat.\nLet's discuss now the presence of \rat bands and CLS\nin an extended magnonic Lieb lattice (Lieb-5), contain-\ning \fve inclusions in the unit cell: one inclusion Aform\nminority sublattice and four inclusions Bfrom majority\nsublattices, as it is presented in Fig. 2. In the considered\nstructure, we add two additional inclusions Binto the\nunit cell in such a way that neighbouring inclusions A\nare linked by the doublets of inclusions B. The sizes of\ninclusions, distances between them, the thickness of the\nlayer and the material composition of the structure re-\nmained the same as for the basic Lieb lattice, discussed\nearlier (Fig. 1).\nThe dispersion relation obtained for the magnonic\nLieb-5 lattice can be found in Fig. 5(a). The proper-\nties of the extended Lieb lattices are well described in\na)\nb)\nBAFIG. 5. Dispersion relation for the extended magnonic Lieb\nlattice Lieb-5, containing \fve inclusions in the unit cell: one\ninclusion Afrom minority sublattice and four inclusions B\nfrom majority sublattices (see Fig. 2). (a) The dispersion\nrelation is plotted along the high symmetry path \u0000-X-M-\u0000\n(see the inset). The \frst, third and \ffth bands (dark blue,\nred and cyan) are strongly dispersive bands, while the second\nand fourth bands (green and magenta) are less dispersive and\nrelated to the presence of CLS. The system does not support\nthe appearance of Dirac cones, even in case when the inter-\naction is \fctitiously limited only to inclusions, according to\ntight-binding model. (b) The zoomed regions in the vicinity\nof \u0000 (in dark green frame) and Mpoints show the essential\ngaps with relatively low, parabolic-like curvatures for top and\nbottom bands.\nthe literature [10, 57{59]. The tight-binding model de-\nscription of Lieb-5 lattices, with information about their\ndispersion relation and the pro\fles of the eigenmodes,\nare presented in numerous papers[10, 16, 55]. Therefore,\nit is possible to compare the obtained results with the\ntheoretical predictions of the tight-binding model.\nThe tight-binding model of Lieb-5 lattice predicts two\n\rat bands with CLS: the second (green) and fourth (ma-\ngenta) band in the spectrum. The \rat bands in the tight-\nbinding model are not intersected by Dirac cones but\nthey are degenerated at \u0000 and Mpoint with the third\nband (red). These features are reproduced in investigated\nmagnonic Lieb-5 lattice (Fig. 2). The dispersion relation\nfor this system is presented in the Fig. 5(a). Also, we have\nmarked, with two rectangles (dark green and violet), the\nvicinities of \u0000 and Mpoints, where the \rat bands (the7\nΓ3\nΓ4\nΓ5BA\nM1\nM2\nM3\nM1←\nM2←\nM3←\nΓ3←\nΓ4←\nΓ5←-\n+-+-\n++- -+\n-+-++-\nFIG. 6. The pro\fles obtained for the extended Lieb lattice consisted of 5 inclusions in the unit cell. The modes are presented\nfor bands No. 3-5 in \u0000 point and its proximity \u0000 (the \frst and second column). In the third and fourth columns, we presented\nthe pro\fles for bands No. 1-3 at Mpoint and its vicinity M . Each pro\fle of eigenmode is presented on a grid composed of\n3x3 unit cells - dashed lines mark the edges of unit cells. The scheme of the unit cell is presented in top-left corner. Exactly at\n\u0000 (and M) point the bands No. 3 and 4 (No. 2 and 3) are degenerated and pro\fles: \u0000 3and \u0000 4(M2andM3) have non-standard\n(for CLS) complementary form { i.e. their combinations \u0000 3\u0006i\u00004(M2\u0006iM3) gives NLS. To obtain proper pro\fles of CLS,\nwhere the phase of procession \rips around CLS loop, we need to explore the vicinity of \u0000 ( M) point { see the grey patches for\nthe mode \u0000 \n4(M \n2) with + and \u0000signs.\nfourth and second bands) become degenerated with the\nthird, dispersive band { Fig. 5(b). It is easy to notice the\nessential frequency gaps ( \u001933 MHz and\u001984 MHz at\n\u0000 andMpoints, respectively), which qualitatively corre-\nsponds to the prediction of the tight-binding model. It\nis worth noting that although the low dispersion bands\n(the second and fourth band) are in general not perfectly\n\rat. Nevertheless, around the point \u0000 and Mpoints the\nbands are \rattened and the \u0000 \u0000XandX\u0000Msections\nare very \rat for the fourth and second band, respectively.\nThe spin wave pro\fles of CLS at the high symmetry\npoints: \u0000 and Mare presented in Fig. 6. Exactly at \u0000\nandM(the \frst and third column), we can see the pairs\nof degenerated mods \u0000 3, \u00004andM2,M3which exhibit\nfeatures of CLS predicated by the tight-binding model(see the loops of sites on grey patches): (i) modes oc-\ncupy only the inclusions Bfrom majority sublattices, (ii)\ndoublets of inclusions Bin the loops of CLS have oppo-\nsite (the same) phases at \u0000 ( M) point. The signi\fcant\ndi\u000berence is that; once we switch one to another B-B\ndoublet, circulating the CLS loop the phase of preces-\nsion charges by\u0006\u0019=2 not by 0 or \u0019. However, when we\nmake combinations of degenerated modes: \u0000 3\u0006i\u00004or\nM2\u0006iM3, then we obtain the NLS occupying the hor-\nizontal or vertical lines, where the precession at exited\nBinclusion will be in- or out-of-phase. The CLS modes\nare clearly visible when we move slightly away from the\nhigh symmetry point where the degeneracy occurs. In\nthe proximity of \u0000 and Mpoint, one can see the CLS\nmodes \u0000 \n4andM \n2for which the phase of precession8\ntakes the relative values close to 0 or \u0019. The small dis-\ncrepancies, are visible as a slight change in the colours\nrepresenting the phase, resulting from the fact that we\nare not exactly in high symmetry points but shifted by\n5% on the path \u0000 \u0000M.\nThe extension of the presented analysis to magnonic\nLieb - 7 lattice, where the inclusions Aare liked by the\nchains composed of three inclusions B, is presented in\nSupplementary Information A.\nV. CONCLUSIONS\nWe proposed a possible realisation of the magnonic\nLieb lattices where the compact localized spin wave\nmodes can be observed in \rat bands. The presented\nsystem qualitatively reproduces the spectral properties\nand the localization features of the modes, predicted by\nthe tight-binding model and observed for photonic andelectronic counterparts. 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Double extended Lieb lattice\nWe can generate further extensions of the magonic Lieb\nlattice by adding more inclusions B, i.e. by introducing\nadditional majority sublattices. We considered here a\ndoubly extended Lieb lattice (Lieb-7) to check to what\nextent the magnonic system corresponds to the tight-\nbinding model. The mentioned lattice consists of seven\nnodes; six belong to majority sublattices Band one be-\nlongs to minority sublattice A(Fig. 7). The magnetic\nparameters were kept as for basic and Lieb-5 lattices,\nconsidered in the manuscript. The geometrical param-\neters have changed only as a result of the introduction\nof additional inclusions B. Therefore, the unit cell has\nincreased to the size 500x500 nm.\na)\nb)\nBA\nFIG. 7. Doubly extended magnonic Lieb lattice: Lieb-\n7. Dimensions of the ferromagnetic unit cell are equal to\n500x500x59 nm. The unit cell contains seven inclusions of 50\nnm diameter. (a) The structure of extended Lieb lattice, and\n(b) top view on Lieb-7 lattice unit cell where the node (inclu-\nsion) from minority sublattice Aand two nodes (inclusions)\nfrom two majority sublattices Bare marked.\nIn the case of a doubly extended Lieb lattice (Lieb-\n7), we expect (according to the works [10, 59]) to obtain\nseven bands in the dispersion relation. The tight-binding\nmodel predicts that the bands will be symmetric with\nrespect to the fourth band, exhibiting particle-hole sym-\nmetry. However, due to the dipolar interaction, we did\nnot expect such symmetry. Another feature which one\nmay deduce from the tight-binding model is that bands\nNo. 2, 4 and 6 should be \rat while bands No. 1, 3, 5 and\n7 are considered dispersive. Moreover, bands No. 3 and\n5 suppose to form a Dirac cone intersecting \rat band No.\n4 at the \u0000 point.\nWe calculated the dispersion relation for magnonic\nLieb-7 lattice (Fig. 8(a)), which share many properties\nwith those characteristic for the tight-binding model [59]:\n(i) third and \ffth bands form the Dirac cones which al-most intersect the \ratter forth band at \u0000 point; (ii) the\nthird (and \ffth) band has a parabolic shape at Mpoint\nwhere it is degenerated with the second (and six) band\nwhich is weakly dispersive. The mentioned regions of\ndispersion are presented as 3D plots in Fig. 8(b). Also,\nwe are going to discuss shortly the pro\fles of spin wave\neigenmodes (including CLS) in these two regions of the\ndispersion relation, which are presented in Fig. 9.\nDirac cones appear at the \u0000 point for bands No. 3\nand 5. At this point, as for the basic magnonic Lieb lat-\ntice (Fig. 3), there is a very narrow gap of the width\n\u00192 MHz. The pro\fles \u0000 4and \u0000 5(left column in\nFig. 9) represent the degenerated states originating from\n\rat and dispersive bands. Both of them do not occupy\nthe inclusions Aand are more focused on two inclusions\na)\nb)BA\nFIG. 8. Dispersion relation for the double extended magnonic\nLieb lattice (Lieb-7) containing seven inclusions in the unit\ncell: one inclusion Afrom minority sublattice and six in-\nclusions Bfrom majority sublattices (see Fig. 2). (a) The\ndispersion relation is plotted along the high symmetry path\n\u0000-X-M-\u0000 (see the inset). The \frst, third, \ffth and seven\nbands (dark blue, red, cyan and orange) are dispersive, while\nthe second, fourth and sixth bands (green, magenta and dark\ngreen bands) are the \ratter bands, supporting the magnonic\nCLS. Dirac cones occur at the \u0000 point and almost interact\nwith the \ratter fourth band, while at Mpoint, we observe\nthe degeneracy of the dispersive parabolic third (\ffth) band\nwith a \ratter second (six) band. (b) The zoomed vicinity of \u0000\npoint (dark green frame) and Mpoint (violet frame) regions\nare presented in 3D.11\nM5\nM7Γ3\nΓ4\nΓ5M6BA\n3coor \nYIG inclusions in Ga:YIG Matrix\nDone \nreformulation Hani\ngamma Symmetrized\nM. not\nFIG. 9. The pro\fles of eigenmodes were obtained for\nmagnonic Lieb-7. The modes are presented for bands No.\n3-5 at \u0000 point and 5-7 at Mpoint. The modes denoted as\n\u00003and \u0000 4are degenerated whereas the \u0000 5is separated from\nthem by extremely small gap \u00192 MHz. At Mpoint, we\nshowed the pro\fles for bands No. 5, 6 and 7. The modes M5\nandM6are degenerated and separated from M7by essential\ngap { predicted by the tight-binding model.\nBarranged in horizontal (\u0000 4) and vertical lines (\u0000 5) {\nsee grey stripes. Therefore, their pro\fles are similar\nto NLS, where the \frst and third inclusion Bin each\nthree-element chain, linking inclusions A, precesses out-\nof-phase and the second (central) inclusion Bremains\nunoccupied.\nAt theMpoint, the M5andM6bands are degen-\nerated. For these bands, the spin waves are localized\nin all inclusions Band do not occupy inclusions A(see\nright column of Fig. 6) The \frst and third inclusion B\nin each three-element chain, linking inclusions A, pre-\ncess in-phase, whereas the second (central) inclusion B\nprecesses out-of-phase with respect to the \frst and third\none. This pattern of occupation of inclusions and the\nphase relations between them is similar to one observed\nfor CLS (see grey patches marking the loops of inclu-\nsions in the left column of Fig. 6), but has one signi\f-\ncant di\u000berence. The phase di\u000berence between successivethree-element chains of inclusions B, in the loop, is equal\nto\u0006\u0019=2. However, the linear combination of the modes\nM5\u0006iM6produces, similarly to the case of the Lieb-5\nlattice, the NLS. To observe the proper pro\fles of CLS\nor NLS, we need to shift slightly from the high symmetry\npoints \u0000 and Mto cancel the degeneracy.\nB. Realization of Lieb lattice\nby shaping demagnetizing \feld\nWe have considered also an alternative realisation\nmethod for a magnonic Lieb lattice in a ferromagnetic\nlayer. This approach is based on shaping the internal de-\nmagnetizing \feld. The structure under consideration is\npresented in Fig. 10. It consists of a thin (28.5 nm) and\nin\fnite CoFeB layer on which a Py antidot lattice (ADL),\nof 28.5 nm thickness, is deposited. The cylindrical holes\nin ADL are arranged in shape of the basic Lieb lattice.\nThe size of the unit cell and diameter of holes remains\nthe same as for the basic Lieb lattice proposed in the\nmain part of the manuscript (see Fig. 1). Due to the ab-\nsence of perpendicular magnetic anisotropy (PMA), we\ndecided to apply a much larger external magnetic \feld\n(H0= 1500 mT) to saturate the ferromagnetic material\nin an out-of-plane direction.\na)\nb)\nBA\nFIG. 10. Basic magnonic Lieb lattice where spin wave exci-\ntations in the CoFeB layer are shaped by demagnetizing \feld\nfrom Py antidot lattice. Dimensions of the ferromagnetic unit\ncell are equal to 250x250x59 nm and contain 3 inclusions of 50\nnm diameter. (a) structure of basic Lieb lattice, (b) top view\non basic Lieb lattice unit cell and di\u000berentiation to nodes of\nsublattice AandB.\nWe assumed the same gyromagentic ratio for both ma-\nterials\r= 187GHz\nT, the following values of material\nparameters for CoFeB [60]: saturation magnetization -\nMS= 1150kA\nm, exchange sti\u000bness constant - A= 15pJ\nm.\nFor Py, we used material parameters [61]: saturation\nmagnetization - MS= 796kA\nm, exchange sti\u000eness con-\nstant -A= 13pJ\nm.12\na)\nb)BA\nFIG. 11. The dispersion relation obtained for basic Lieb\nlattice formed by demagnetizing \feld of antidot lattice (see\nFig. 10). (a) The dispersion relation, (b) the 3D plot of dis-\npersion relation in the region marked with the green frame\nin (a). Results were obtained for H0= 1500 mT applied\nout-of-plane.\nThe deposition of the ADL made of Py (material of\nlowerMS) above the CoFeB layer (material of higher\nMS) is critical for spin wave localization in CoFeB below\nthe exposed parts (holes) of the ADL. The demagnetiza-\ntion \feld produced on CoFeB/Air interface creates wells\npartially con\fning the spin waves. However, this pattern\nof internal demagnetizing \feld becomes smoother with\nincreasing distance from the ADL.\nThe obtained dispersion relation is shown in Fig. 11.\nIt is worth noting that the lowest band is very disper-\nsive, while the highest band is \rattened more than in the\ncase of the structure presented in the main part of the\nmanuscript (see Fig. 3). The middle band, which sup-\npose to support CLS, varies in extent similar to the third\nband. For this structure, Dirac cones in the Mpoint\ncannot be clearly unidenti\fed.\nC. Lieb lattice formed by YIG inclusions\nin non-magnetic matrix\nThe periodic arrangement of ferromagnetic cylinders\nsurrounded by nonmagnetic material (e.g. air) seems to\nbe the simplest realization of the Lieb lattice. To refer\na)\nb)FIG. 12. Dispersion relations for basic Lieb lattice. (a) The\nresults obtained for YIG inclusions in Ga:YIG matrix (dashed\nlines) and YIG inclusions without matrix (solid lines). (b)\nThe zoomed dispersion relation obtained for YIG inclusions\nwithout matrix, marked in (a) by the frame.\nthis structure to the bi-component system investigated in\nthe main part of the manuscript, we assumed the same\nmaterial and geometrical parameters for inclusions as for\nthe structure presented in Fig. 1.\nThe advantage of this system is that the con\fnement\nof spin waves within the areas of inclusions is ensured for\narbitrarily high frequency. We are not limited here by the\nFMR frequency of the matrix, as it was for bi-component\nLieb lattices (Figs. 1, 2). However, the coupling of mag-\nnetization dynamics between the inclusions is here pro-\nvided solely by the dynamical demagnetizing \feld, i.e.\nthe evanescent spin waves do not participate in the cou-\npling. Therefore, the interaction between inclusions is\nmuch smaller in general, which leads to a signi\fcant nar-\nrowing of all magnonic bands (Fig. 11). The widths of the\nsecond and third band can be even smaller than the gap\nseparating from the \frst bands { Fig. 11(b). Such strong\nmodi\fcation of the spectrum makes the applicability of\nthe considered system for the studies of magnonic CLS\nquestionable.13\nD. Demagnetizing \feld\nin YIG|Ga:YIG Lieb lattice\nThe di\u000eculty in designing the magnonic system is not\nonly due to the adjustment of geometrical parameters of\nthe system but also due to the shaping of the internal\nmagnetic \feld He\u000b.\nThe components of the e\u000bective magnetic \feld can be\ndivided into long-range and short-range. The realiza-\ntion of our model is inseparably linked to the long-range\ndipole interactions through which the coupling between\ninclusions is possible. This kind of interaction is sensitive\nto the geometry of the ferromagnetic elements forming\nthe magnonic system.\nIn Lieb lattice, the nodes of minority sublattice Ahave\nfour neighbours and the nodes of majority sublattice B\nhave two. As a result, identical inclusions (in terms\nof their shapes and material parameters) become dis-\ntinguishable, because of slightly di\u000berent values of the\ninternal demagnetising \feld. This has consequences for\nthe formation of a frequency gap between Dirac cones at\npointMin the dispersion relation obtained for the basic\nLieb lattice. In the literature, this phenomenon has been\ndescribed for the tight-binding model and is called node\ndimerisation of the lattice [37].\nIn Fig. 13 we have shown the pro\fle of the z-component\nof the demagnetising \feld. For each inclusion through\nwhich the cut line passes, we have marked the minimum\nvalue of the demagnetising \feld. The slightly lower value\nof internal \fled for inclusions Ais responsible for a tiny\nlowering of the frequency for the mode M1(concentrated\nin inclusions A) respect the degenerated modes M2and\nM3(con\fned in inclusions B).\na)\nb)BA\nFIG. 13. Pro\fle of static demagnetizing \feld plotted at cut\nthrough (a) Lieb lattice unit cell. (b) The z-component of the\ndemagnetizing \feld along the cut line is shown in (a). In the\nplot, we have marked peaks for the areas of inclusions Aand\nB. Please note the slightly di\u000berent values of demagnetizing\nin the centre of AandBinclusion due to di\u000berent the number\nneighboring of nodes: four for inclusion A, two for inclusion\nB."    },    {        "title": "1510.09007v1.Pure_spin_Hall_magnetoresistance_in_Rh_Y3Fe5O12_hybrid.pdf",        "content": "arXiv:1510.09007v1  [cond-mat.mes-hall]  30 Oct 2015Pure spin-Hall magnetoresistance in Rh/Y 3Fe5O12\nhybrid\nT. Shang1,Q.F.Zhan1,*,L.Ma2, H.L.Yang1, Z.H.Zuo1, Y. L.Xie1,H. H.Li1, L.P. Liu1,B.\nM. Wang1,Y. H.Wu3,S. Zhang4,†,and Run-WeiLi1,‡\n1Key Laboratory of Magnetic Materials andDevices& Zhejiang ProvinceKey Laboratory ofMagnetic Materials\nand Application Technology,Ningbo Institute ofMaterial T echnology and Engineering, Chinese Academy of\nSciences,Ningbo, Zhejiang 315201, P.R.China\n2Department of Physics,Tongji University,Shanghai, 20009 2, P.R.China\n3Department of Electrical andComputer Engineering, Nation al Universityof Singapore, 4Engineering Drive3\n117583, Singapore\n4Department of Physics,Universityof Arizona,Tucson,Ariz ona85721, USA\n*zhanqf@nimte.ac.cn\n†zhangshu@email.arizona.edu\n‡runweili@nimte.ac.cn\nABSTRACT\nWe report an investigation of anisotropic magnetoresistan ce (AMR) and anomalous Hall resistance (AHR) of Rh and Pt\nthin films sputtered on epitaxial Y 3Fe5O12(YIG) ferromagneticinsulator films. For the Pt/YIG hybrid, large spin-Hall magne-\ntoresistance (SMR) along with a sizable conventional aniso tropic magnetoresistance (CAMR) and a nontrivial temperat ure\ndependenceof AHR were observed in the temperaturerange of 5 -300 K. In contrast, a reduced SMR with negligibleCAMR\nand AHR was found in Rh/YIG hybrid. Since CAMR and AHR are char acteristics for all ferromagnetic metals, our results\nsuggest that the Pt is likely magnetized by YIG due to the magn etic proximity effect (MPE) while Rh remains free of MPE.\nThustheRh/YIGhybridcouldbeanidealmodelsystem toexplo rephysicsanddevicesassociatedwithpurespincurrent.\nIntroduction\nThestudiesofmagneticinsulator-basedspintronicshaver ecentlygeneratedgreatinterestduetotheirsegregatedch aracteristic\nof spin current from charge current.1The interplay between spin and charge transports in nonmagn eticmetal/ferromagnetic\ninsulator(NM/FMI)hybridsgivesrise tovariousinteresti ngphenomena,suchasspin injection,2,3spinpumping,4–6andspin\nSeebeck.7,8The previous investigations on NM/FMI hybrids, e.g., Pt/Y 3Fe5O12(Pt/YIG), also demonstrated a new-type of\nmagnetoresistance9–13in whichtheresistivityoffilms, ρ,hasanunconventionalangulardependence,namely,\nρ=ρ0−Δρ/bracketleftbigˆm·(ˆz׈j)/bracketrightbig2(1)\nwhereˆmandˆjareunitvectorsinthedirectionsofthemagnetizationandt heelectriccurrent,respectively,and ˆzrepresentsthe\nnormal vector perpendicular to the plane of the film; ρ0is the zero-field resistivity. The above angular-dependent resistivity\nhas been named as the spin-Hall magnetoresistance (SMR) in o rder to differentiate from the conventional anisotropic ma g-\nnetoresistance (CAMR) in which ρ=ρ0+Δρ(ˆm·ˆj)2. A theoretical model outlined below has been proposed to exp lain the\nSMR.Anelectriccurrent( je)inducesaspincurrentduetothespin-Halleffectandintur n,theinducedspincurrent,viainverse\nspin-Hall effect, generates an electric current whose dire ction is opposite to the original current.14–21Thus , the combined\nspin-Hallandinversespin-Halleffectsleadtoanaddition alresistanceinbulkmaterialswithspin-orbitcoupling(S OC).How-\never, in an ultra-thin film, the spin current could be either r eflected back at the interface or absorbed at the interface th rough\nspin transfer torque. In the former case, the total spin curr ent in the metal layer is reduced and thus the additional resi stance\nis minimized. The spin current reflection is strongest when t he magnetization direction ˆmof the ferromagnetic insulator is\nparallelto thespinpolarization ˆz׈jofthespincurrent,leadingtothe resistiveminimumasdesc ribedin Eq.(1).14,15\nHowever, the magnetic proximity effect (MPE), in which a non -magnetic metal develops a sizable magnetic moment in\nthe close vicinityof a ferromagneticlayer,maycomplicate the interpretationofthe SMR. Pt is neartheStonerferromag netic\ninstability and could become magnetic when in contact with f erromagnetic materials, as experimentally shown by x-ray\nmagnetic circular dichroism (XMCD), anomalous Hall resist ance (AHR), spin pumping, and first principle calculations o f\nthe Pt/YIG hybrid.22–26In order to separate the MPE form the pure SMR, many attempts h ave been made. By inserting a\n1/s53/s46/s48 /s109/s48/s46/s48\n/s45/s48/s46/s53/s110/s109/s48/s46/s53/s110/s109\n/s40/s99/s41 /s89/s73/s71/s48 /s57/s48 /s49/s56/s48 /s50/s55/s48 /s51/s54/s48\n/s32/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s46/s117/s46/s41\n/s40/s100/s101/s103/s114/s101/s101/s41/s32/s71/s71/s71/s32 /s32/s89/s73/s71/s40/s98/s41\n/s45/s52 /s45/s50 /s48 /s50 /s52/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s32/s32/s77/s47/s77\n/s83\n/s70/s105/s101/s108/s100/s32/s40/s75/s79/s101/s41/s32/s105/s110/s45/s112/s108/s97/s110/s101\n/s32/s111/s117/s116/s45/s111/s102/s45/s112/s108/s97/s110/s101/s40/s100/s41/s52/s57/s46/s53 /s53/s48/s46/s48 /s53/s48/s46/s53 /s53/s49/s46/s48 /s53/s49/s46/s53 /s53/s50/s46/s48 /s53/s50/s46/s53/s89/s73/s71/s32/s40/s52/s52/s52/s41\n/s32/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s46/s117/s46/s41\n/s50 /s40/s100/s101/s103/s114/s101/s101/s41/s71/s71/s71/s32/s40/s52/s52/s52/s41\n/s40/s97/s41\nFigure1. (Coloronline)(a)A representative2 θ-ωXRD patternsforYIG/GGG filmnearthe(444)peaksofGGGsubst rate\nandYIGfilm. (b)The ϕ-scanofYIG/GGGfilm. (c)AFM surfacetopographyofa represe ntativeYIGfilm. (d)Thefield\ndependenceofnormalizedmagnetizationforYIG/GGGfilm mea suredatroomtemperature. Forthein-plane(out-of-plane)\nmagnetization,the magneticfieldisappliedparallel(perp endicular)tothefilm surface.\nlayer of Au or Cu between NM and FMI, the MPE can be effectively screened, but the SMR amplitude is largelysuppressed\nas well.9,10Furthermore, the insertion of an extra layer would introduc e an additional interface whose quality is not easily\naccessible. An alternative approach to pursue the pure SMR i s to find proper NM metals in direct contact with YIG, but\nwithout the MPE. The Au has a SOC strength comparableto Pt or P d and it is freeof the MPE, but it hasan extremelyweak\ninverse spin-Hall voltage and SMR.26,27According to the theoretical calculation,28the 4dmetal Rh also possesses a large\nSOC strengthandspin-Hall conductivity,and a small magnet icsusceptibility,implyinganinsignificantMPE in the Rh me tal\nwhenincontactwithferromagneticmaterials. ThusRh might be anexcellentmaterialforthe pureSMRstudy.\nInthisarticle,theanisotropicmagnetoresistance(AMR)a ndAHRofRh/YIGandPt/YIGhybridswereinvestigatedinthe\ntemperaturerangeof5-300K.Indeed,weshowthatthediffer encesinmagneto-transportpropertiesbetweenthesetwohy brids\nare attributed to the strong (Pt) and weak (Rh) MPE, and thus, Rh/YIG providesan ideal model system for purespin-current\ninvestigations.\nResults\nFigure1(a)plotsarepresentativeroom-temperature2 θ-ωXRDscanofepitaxialYIG/GGGthinfilmnearthe(444)reflecti ons\nofgadoliniumgalliumgarnet(GGG)substrateandYIGfilm. Cl earLaueoscillationsindicatetheflatnessanduniformityo fthe\nepitaxial YIG film. The epitaxial nature of YIG film was charac terizedby ϕ-scan measurementswith a fixed 2 θvalue at the\n(642)reflections,asshowninFig. 1(b). Inthisstudy,theth icknessesofYIGandRhorPtfilms,determinedbyfittingthex- ray\nreflectivity(XRR) spectra,areapproximately50nmand3nm, respectively. TheAFM surfacetopographyofarepresentati ve\nYIG film in Fig. 1(c) reveals a surface roughnessof 0.15 nm, in dicating atomically flat of the epitaxial YIG film. As shown\nin Fig. 1(d),thein-planeandout-of-planecoercivitiesof theYIGfilm are <1Oe and60Oe, respectively. Theparamagnetic\nbackgroundoftheGGGsubstratehasbeensubtractedandthem agnetizationisnormalizedtothesaturationmagnetizatio nMs.\nThe out-of-planemagnetization saturates at a field above 2. 2 kOe, which is consistent with previousresults.12,22The above\npropertiesindicatetheexcellentqualityofourepitaxial YIGfilm.\nFigures 2(a)-(c) plot the room-temperature AMR for the Rh/Y IG (open symbols) and Pt/YIG (closed symbols) hybrids.\nAs shown in the rightpanels, the Rh/YIG and Pt/YIG hybridsar e patternedinto Hall-bargeometryandthe electriccurrent is\napplied along the x-axis. The AMR is measuredin a magnetic field of 20 kOe, which i s sufficiently strong to rotate the YIG\nmagnetization in any direction. Here the total AMR is defined asΔρ/ρ0= [ρ(M/bardblI) -ρ(M⊥I)]/ρ0. We note that when the\nmagnetic field scanswithin the xyplane [Fig. 2(a)],boththe CAMR and SMR contributeto the tot al AMR, and it is difficult\nto separate them from each other; for the xzplane [Fig. 2(b)], the magnetization of YIG is always perpen dicularto the spin\n2/7Figure2. Anisotropicmagnetoresistanceforthe Rh/YIG(opensymbol s)andPt/YIG (closedsymbols)hybridswith the\nmagneticfieldscanningwithinthe xy(a),xz(b),andyz(c)planes. TheAMRismeasuredatroomtemperatureina fieldo f\nµ0H =20kOe. Thesolidlinesthroughthedataarefitsto cos2θwitha 90degreephaseshift. Therightpanelsshowthe\nschematicplotsoflongitudinalresistance andtransverse Hall resistancemeasurementsandnotationsofdifferentfie ldscans\nin thepatternedHallbarhybrids. The θxy,θxz,andθyzdenotetheanglesoftheappliedmagneticfieldrelativeto th ey-,z-,\nandz-axes,respectively.\npolarizationof the spin currentand the SMR is absent, and th e resistance changesare attributed to the MPE-inducedCAMR .\nFortheyzplane[Fig. 2(c)],theelectriccurrentisalwaysperpendic ularto themagnetization,theCAMRiszero,andonlythe\nSMR survives. According to Eq. (1), the amplitudes of CAMR or SMR (Δρ/ρ0) oscillate as a function of cos2θ, as shown\nby the solid black lines in Fig. 2. Both the Rh/YIG and Pt/YIG h ybrids display clear SMR at room temperature, with the\namplitudes reaching 7.6 ×10−5and 6.1×10−4, respectively [see Fig. 2(c)]. On the other hand, the CAMR al so exists in\nthe Pt/YIG, and its amplitude of 2.2 ×10−4is comparableto the SMR. However, as shown in Fig. 2(b), for R h/YIG hybrid,\ntheθxzscan shows negligible AMR and the resistivity is almost inde pendent of θxz, indicating the extremely weak MPE at\ntheRh/YIG interfaceincontrasttothesignificanteffectat thePt/YIGinterface. TheMPEat Pt/YIGinterfacewasprevio usly\nevidencedfromthemeasurementsofXMCD,AHR, andspinpumpi ng.22,24,25\nUpondecreasingtemperature,theSMRpersistsdownto5Kinb oththeRh/YIGandPt/YIGhybrids[seeFig. 3]. However,\nthereisnosizableCAMRintheRh/YIGhybriddowntothelowes ttemperature[seeFig. 3(b)],indicatingtheextremelywea k\nMPE at the interface even at low temperature. While for the Pt /YIG hybrid, as shown in Fig. 3(e), the amplitude of CAMR\nis almost independentof temperaturefor T>100K, andthen decreasesby furtherloweringtemperature,w ith theamplitude\nreaching 1.0 ×10−4at 5 K. The above features are quite different from the Pd/YIG hybrid, where the amplitude of CAMR\nincreases as the temperature decreases, showing a comparab le value to the SMR at 3 K.12The reason for these different\nbehaviors is unclear, and further investigations are neede d. Since the CAMR is negligible in Rh/YIG, the SMR dominates\nthe AMR when the magnetic field is varied within the xyplane. Figures 3(g) and (h) plot the temperature dependence of\nSMR amplitude for the Rh/YIG and Pt/YIG, respectively. The S MR amplitudes exhibit strong temperature dependence,\nreachingamaximumvalueof1.1 ×10−4(Rh/YIG)and6.9 ×10−4(Pt/YIG)around150K.Suchnonmonotonictemperature\ndependenceofSMRamplitudewaspreviouslyreportedin Pt/Y IG hybrid,whichcan bedescribedbya single spin-relaxatio n\nmechanism.29It is noted that the hybridswith different Rh thicknesses ex hibit similar temperaturedependentcharacteristics\nwith different numerical values compared to the Rh(3 nm)/YI G hybrid shown here. For example, the Rh(5 nm)/YIG hybrid\nreachesitsmaximumSMRamplitudeof0.8 ×10−4around100K.\nIn order to characterize the MPE at the NM/FMI interface, we a lso carried out the measurements of transverse Hall\nresistance R xywith a perpendicular magnetic field up to 70 kOe, as shown in Fi gs. 4(a) and (b). In both Rh and Pt thin\nfilms, the ordinary-Hall resistance (OHR), which is proport ional to the external field, is subtracted from the measured R xy,\n3/7/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s51/s48/s54/s48/s57/s48\n/s47\n/s48/s40/s49/s48/s45/s53\n/s41\n/s84/s32/s40/s75/s41/s32/s72/s47/s47/s121/s122/s40/s104/s41/s48/s52/s56/s49/s50\n/s45/s50/s48/s50\n/s48 /s57/s48 /s49/s56/s48 /s50/s55/s48 /s51/s54/s48/s45/s56/s45/s52/s48/s48/s50/s48/s52/s48/s54/s48\n/s45/s50/s48/s45/s49/s48/s48\n/s48 /s57/s48 /s49/s56/s48 /s50/s55/s48 /s51/s54/s48/s45/s54/s48/s45/s52/s48/s45/s50/s48/s48/s40/s97/s41\n/s32/s32\n/s51/s48/s48/s75\n/s50/s53/s48/s32\n/s50/s48/s48/s32/s32/s32\n/s49/s53/s48/s32/s32\n/s49/s48/s48/s32/s32\n/s53/s48\n/s49/s48/s32/s32/s32/s32\n/s53/s82/s104/s47/s89/s73/s71\n/s32/s47\n/s48/s32/s40/s49/s48/s45/s53\n/s41\n/s40/s98/s41\n/s40/s99/s41\n/s32\n/s32/s40/s100/s101/s103/s114/s101/s101/s41/s80/s116/s47/s89/s73/s71\n/s40/s100/s41\n/s40/s101/s41\n/s47\n/s48/s40/s49/s48/s45/s53\n/s41\n/s40/s102/s41\n/s32/s40/s100/s101/s103/s114/s101/s101/s41\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s52/s56/s49/s50\n/s72/s47/s47/s120/s121\n/s72/s47/s47/s121/s122/s47\n/s48/s40/s49/s48/s45/s53\n/s41\n/s84/s40/s75/s41/s40/s103/s41\nFigure3. Anisotropicmagnetoresistanceforthe Rh/YIGhybridat var ioustemperaturesdownto 5K forthe θxy(a),θxz(b),\nandθyz(c)scans. TheresultsofPt/YIG areshownin(d)-(f). TheAMR ismeasuredina fieldof µ0H= 20kOe. (g)and(h)\nplotthe temperaturedependenceofSMRamplitudefortheRh/ YIGandPt/YIG hybrids,respectively. Thecubicandtriangl e\nsymbolsstandforthe θxyandθyzscans,respectively.\n4/7/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s45/s50/s48/s48/s50/s48/s52/s48\n/s40/s101/s41/s82\n/s65/s72/s82/s32/s40/s109 /s41\n/s84/s32/s40/s75/s41\n/s82\n/s65/s72/s82/s32/s40/s109 /s41\n/s84/s32/s40/s75/s41/s40/s102/s41\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s49/s48/s50/s48/s51/s48/s45/s56/s45/s52/s48/s52/s56\n/s45/s56/s48 /s45/s52/s48 /s48 /s52/s48 /s56/s48/s45/s49/s46/s50/s45/s48/s46/s54/s48/s46/s48/s48/s46/s54/s49/s46/s50/s45/s54/s48/s45/s51/s48/s48/s51/s48/s54/s48\n/s45/s56/s48 /s45/s52/s48 /s48 /s52/s48 /s56/s48/s45/s52/s48/s45/s50/s48/s48/s50/s48/s52/s48/s49/s48/s75\n/s32/s82\n/s120/s121/s32/s40/s109 /s41\n/s51/s48/s48/s75\n/s40/s97/s41\n/s40/s99/s41\n/s32/s32/s82\n/s65/s72/s82/s32/s40/s109 /s41\n/s70/s105/s101/s108/s100/s32/s40/s75/s79/s101/s41/s40/s98/s41\n/s82\n/s120/s121/s32/s40/s109 /s41/s53/s75\n/s51/s48/s48/s75\n/s40/s100/s41\n/s82\n/s65/s72/s82/s32/s40/s109 /s41\n/s70/s105/s101/s108/s100/s32/s40/s75/s79/s101/s41\nFigure4. TransverseHall resistanceR xyfortheRh/YIG (a)andPt/YIG (b)hybridsasafunctionofmagn eticfieldupto70\nkOe atdifferenttemperatures. TheanomalousHall resistan ceRAHRfortheRh/YIG(c)andPt/YIG(d)at different\ntemperatures. TheR AHRcanbederivedbysubtractingthelinearbackgroundofOHR. T emperaturedependenceofR AHRfor\nthe Rh/YIG(e)andPt/YIG (f). All R AHRareaveragedby[R AHR(70kOe)-R AHR(-70kOe)]/2. Theerrorbarsarethe results\nofsubtractingOHRindifferentfieldranges.\ni.e.,RAHR=Rxy-ROHR×µ0H,RAHRistheanomalousHallresistance. TheresultingR AHRasafunctionofmagneticfieldfor\nthe Rh/YIG and Pt/YIG hybrids are shown in Figs. 4(c) and (d), respectively. The AHR is proportional to the out-of-plane\nmagnetization,andthusprovidesanotionforMPEattheNM/F MIinterface. Atroomtemperature,fortheRh/YIGhybrid,th e\nRAHR= 0.57mΩ,which is 22 timessmaller than the Pt/YIG hybrid,implyingt he extremelyweak MPE at Rh/YIG interface,\nbeing consistent with the AMR results in Fig. 2(b). We note th at the R AHRof Rh/YIG hybridswith different Rh thicknesses\nwas also measured. For example, the R AHRreaches 1.65 m Ωand 0.26 m Ωin Rh(2 nm)/YIG and Rh(5 nm)/YIG at room\ntemperature,respectively. The temperaturedependenceof RAHRfor the Rh/YIG and Pt/YIG hybridsare summarizedin Figs.\n4 (e) and (f), respectively. As can be seen, the R AHRexhibits significantly different behaviors: the R AHRroughly decreases\non lowering temperature in Rh/YIG. However, in Pt/YIG, the m agnitude of R AHRdecrease with temperature for T>50 K\nand then it suddenly increases upon further decreasing temp erature. Moreover, the R AHRof Pt/YIG changes sign below 50\nK, while it is stays positive for Rh/YIG. Similar non-trivia l AHR were also observed in Pt/LCO hybrids,30but there is no\nexisting quantitativetheory to comparethese results, fur thertheoretical and experimentalinvestigationsare need edto clarify\nthe dominatingmechanisms.\nSummary\nIn summary, we carried out measurements of angular dependen ce of magnetoresistance and transverse Hall resistance in\nRh/YIG and Pt/YIG hybrids. Both hybrids exhibit SMR down to v ery low temperature. The observed AHR and CAMR\nindicateasignificantMPEatthePt/YIGinterface,whileiti snegligibleattheRh/YIGinterface. Ourfindingssuggestth atthe\nabsenceoftheMPE makestheRh/YIGbilayersystem anidealpl aygroundforpurespin-currentrelatedphenomena.\nMethods\nThe Rh/YIG and Pt/YIG hybrids were prepared in a combined ult ra-high vacuum (10−9Torr) pulse laser deposition (PLD)\nand magnetron sputter system. The high quality epitaxial YI G thin films were grown on (111)-orientated single crystalli ne\nGGG substrate via PLD technique at 750◦C. The thin Rh and Pt films were deposited by magnetron sputter ing at room\ntemperature. All the thin filmswere patternedintoHall-bar geometry. The thicknessand crystalstructurewere charact erized\n5/7by using Bruker D8 Discover high-resolution x-ray diffract ometer (HRXRD). The thickness was estimated by using the\nsoftware package LEPTOS (Bruker AXS). The surface topograp hy and magnetic properties of the films were measured in\nBruker Icon atomic force microscope (AFM) and Lakeshore vib rating sample magnetometer (VSM) at room temperature.\nThe measurements of transverse Hall resistance and longitu dinal resistance were carried out in a Quantum Design physic al\npropertiesmeasurementsystem(PPMS-9T)witha rotationop tionina temperaturerangeof5-300K.\nAcknowledgments\nWe acknowledgethefruitfuldiscussionswithS.M.Zhou. Thi sworkisfinanciallysupportedbytheNationalNaturalScien ce\nFoundation of China (Grants No. 11274321, No. 11404349, No. 11174302, No. 51502314, No. 51522105) and the Key\nResearch Program of the Chinese Academy of Sciences (Grant N o. KJZD-EW-M05). S. Zhang was partially supported by\nthe U.S. NationalScienceFoundation(GrantNo. ECCS-14045 42).\nAuthor contributions\nQ. F. Z., S. Z.,and R. W. L. plannedthe experiments. T.S., L. M ., andY. L. X. synthesizedthe hybrids. Structurecharacter i-\nzation,magneticandtransportmeasurementswereperforme dbyT.S.,H.L.Y.,Z.H.Z.,H.H.L.,andL.P.L.Thedatawere\nanalysed by T. S., H. L. Y., Y. H. W., B. M. W., Q. F. Z., S. Z., and R. W. L. T. S., Q. F. Z., and S. Z. wrote the paper. All\nauthorsparticipatedin discussionsandapprovedthe submi ttedmanuscript.\nAdditionalinformation\nCompetingfinancialinterests: Theauthorsdeclarenocompetingfinancialinterests.\nReferences\n1.Wu, M. Z. and Hoffmann, A. 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Rev.B\n92,165114(2015).\n7/7"    },    {        "title": "2211.03510v1.Design_of_X_Band_Bicontrollable_Metasurface_Absorber_Comprising_Graphene_Pixels_on_Copper_Backed_YIG_Substrate.pdf",        "content": "Design of X-Band Bicontrollable Metasurface Absorber Comprising Graphene\nPixels on Copper-Backed YIG Substrate\nGovindam Sharma1, Akhlesh Lakhtakia2, and Pradip Kumar Jain1\n1Department of Electronics and Communication Engineering, National Institute Technology\nPatna, Patna 800005, India\n2Departtment of Engineering Science and Mechanics, The Pennsylvania State University,\nUniversity Park, PA 16802, USA\nAbstract\nThe planewave response of a bicontrollable metasurface absorber with graphene-patched\npixels was simulated in the X band using commercial software. Each square meta-atom is a\n4\u00024 array of 16 pixels, some patched with graphene and the others unpatched. The pixels are\narranged on a PVC skin which is placed on a copper-backed YIG substrate. Graphene provides\nelectrostatic controllability and YIG provides magnetostatic controllability. Our design delivers\nabsorptance\u00150:9 over a 100-MHz spectral regime in the X band, with 360 MHz kA\u00001m mag-\nnetostatic controllabity rate and 1 MHz V\u00001\u0016m electrostatic controllability rate. Notably,\nelectrostatic control viagraphene in the GHz range is novel.\nKeywords: Bicontrollability, Magnetostatic controllability, Electrostatic controllability, Pixe-\nlation, Graphene, Yttrium iron garnet, Meta-atom, Metasurface, GHz.\n1 Introduction\nMetasurfaces are thin compared to the operational wavelength, accounting for their popularity\nin the R&D arena. The use of materials that respond electromagnetically to a stimulus allows\ncontrollable metasurfaces to be designed for beam-steering re\rectors/\flters [1], mirrors/lenses with\nvariable focus [2], and absorbers/\flters [3, 4] in a wide spectrum beginning with the microwave\nfrequencies and ending with the visible frequencies.\nTypically, controllable metasurfaces are designed to operate at high frequencies [5]. Direct\nscaling [6, 7] of controllable metasurface absorbers from THz frequencies to GHz frequencies is\nnot always feasible, since constitutive parameters are frequency dependent. Generally, at GHz\nfrequencies, metal is used to design the top layer of the metasurface, but using materials such as\nferrites [8], graphene [9], and conductive rubber [10] allow control of metasurface absorbers.\nHuang et al. experimentally demonstrated a magnetostatically controllable (or tunable) X-band\nabsorber containing a ferrite slab, with a 300-MHz controllability range for absorptance A>0:9 [8].\nFallahi et al. design an electrostatically controllable metasurface absorber containing patterned\ngraphene|but only the maximum absorptance Amax, not the maximum-absorptance frequency\nfmaxA, can be controlled with that design [11]. Yi et al. used shape memory polymers to thermally\ncontrolfmaxA2[11:3;13:5] GHz [12]. None of these metasurface absorbers covers the complete X\nband with absorptance in excess of 0 :9, which is an important requirement for wide use.\n1arXiv:2211.03510v1  [physics.app-ph]  1 Nov 2022Bicontrollable X-band metasurface absorbers are desirable for weather radar, police speed radar,\nand direct broadcast television. With that in mind, Sharma et al. designed a pixellated metasur-\nface absorber with coarse magnetostatic and \fne thermal controllability of fmaxA over the entire X\nband\n[13]. The meta-atoms in this design comprise yttrium iron garnet (YIG)-patched pixels and con-\nductive rubber (CR)-patched pixels on a metal-backed silicon substrate.\nContinuing in the same vein, we are now reporting a pixelated metasurface absorber with\nfmaxA controllable both magnetostatically and electrostatically in the entire X band, while keeping\nA\u00150:9 over a 100-MHz spectral regime. Each meta-atom is a Nr\u0002Nrarray of pixels, some patched\nwith graphene and the others unpatched. In contrast to numerous designs [14,15], the patches are\nnot metallic. The pixels are arranged on a PVC skin which is placed on top of a copper-backed\nYIG substrate. Graphene provides electrostatic controllability and YIG provides magnetostatic\ncontrollability. Pixel size as well as the con\fguration of patched pixels were decided by examining\nthe absorptance spectrums of many designs.\nThe plan of this paper is as follows. Section 2 provides information on the metasurface geometry,\nthe relative permeability dyadic of YIG, the surface conductivity of graphene, and theoretical\nsimulations. Numerical results are presented and discussed in Sec. 3. Some remarks in Sec. 4\nconclude the paper.\nAn exp(j!t) dependence on time tis implicit, with j=p\u00001,!= 2\u0019fas the angular frequency,\nandfas the linear frequency. The free-space wavenumber is denoted by k0=!p\"0\u00160= 2\u0019=\u0015 0,\nwhere\u00150is the free-space wavelength, \"0is the free-space permittivity, and \u00160is the free-space\npermeability. Vectors are denoted by boldface letters; the Cartesian unit vectors are denoted by ^x,\n^y, and ^z; and dyadics are double underlined.\n2 Materials and Methods\n2.1 Device Structure\nThe metasurface extends to in\fnity in all directions in the xyplane, but it is of \fnite thickness\nalong thezaxis, as depicted in Fig. 1. The metasurface is a biperiodic array of square meta-atoms\nwhose sides are aligned along the xandyaxes.\nEach meta-atom is of side a. The front surface of the meta-atom is an array of Nr\u0002Nrsquare\npixels of side b, each pixel separated from every neighboring pixel by a distance d\u001ca, so that\nNr=a=(b+d). Some of the pixels are patched with graphene, but others are not. Underneath the\npixels is a polyvinyl-chloride (PVC) skin of thickness LPVC, a YIG substrate of thickness Lsub, and\na copper sheet of thickness Lmserving as a back re\rector.\nWe \fxedNr= 4,LPVC= 0:08 mm,Lsub= 0:2 mm, and Lm= 0:07 mm. In addition, we \fxed\na= 6 mm,b= 1:45 mm, and d= 0:05 mm, after multiple iterations of parameter sweeps.\n2.2 YIG\nThe relative permeability dyadic \u0016\nYIGof YIG depends on the magnitude and direction of the\nexternal magnetostatic \feld H0. With this \feld aligned along the xaxis (i.e, H0=H0^x), we\nhave [16]\n\u0016\nYIG=^x^x+\u0016yy(^y^y+^z^z) +j\u0016yz(^y^z\u0000^z^y); (1a)\n2Figure 1: Schematics of four meta-atoms: (a) copper-backed YIG substrate; (b) graphene on top\nof a PVC skin overlaying a copper-backed YIG substrate; (c) the same as (b) but with graphene\npartitioned as a 4 \u00024 array of graphene patches; and (d) the same as (c) but with only ten pixels\npatched with graphene. The Cartesian coordinate system is also shown.\nwhere\n\u0016yy= 1 +4\u0019\u00162\n0\rMs\u0010\nH0+j4H\n2\u0011\n(\u00160\r)2\u0010\nH0+j4H\n2\u00112\n\u0000!2(1b)\nand\n\u0016yz=4\u0019!\u0016 0\rMs\n(\u00160\r)2\u0010\nH0+j4H\n2\u00112\n\u0000!2: (1c)\nIn these equations, \r= 1:76\u00021011C kg\u00001is the gyromagnetic ratio, 4H= 1:98 kA m\u00001is\nthe resonance linewidth, and Ms= 0:18 Wb m\u00002is the saturation magnetization. The relative\npermittivity scalar of YIG is \"YIG= 15. Note that H0^xcan be applied by placing the metasurface\nbetween two magnets, so long as the lateral extent of the metasurface is in excess of 10 \u00150.\n2.3 Graphene\nGraphene is not a\u000bected signi\fcantly by H0^x, because that magnetostatic \feld is wholly aligned\nin the plane containing the carbon atoms [17]. It is, however, a\u000bected by the external electrostatic\n\feldE0=E0^zaligned normal to that plane, which can be applied using transparent electrodes\nsigni\fcantly above and below the metasurface.\n3The surface conductivity of graphene \u001bgrcomprises an intraband term and an interband term,\nthe latter being negligibly small compared to the former in the X band [17,18]. Accordingly,\n\u001bgr=\u0000jq2\nekBT\n\u0019~2(!\u00002j\u001c\u00001gr)\u0002\n\u001a\u0016c\nkBT+ 2 ln\u0014\n1 + exp\u0012\n\u0000\u0016c\nkBT\u0013\u0015\u001b\n; (2)\nwhereqe= 1:602 177\u000210\u000019C is the elementary charge, kB= 1:380 649\u000210\u000023J K\u00001is the\nBoltzmann constant, and ~= 1:054 572\u000210\u000034J s is the reduced Planck constant. All calculations\nwere made for temperature T= 300 K. We \fxed the momentum relaxation time \u001cgr= 0:4 ps after\nexamining values of the maximum absorptance Amaxand the controllability rate @fmaxA=@E 0for\n\u001cgr2[0:01;1] ps. This relaxation time can be controlled by impurity level [3].\nThe value of the chemical potential \u0016cdepends on E0as well as on the d.c. relative permittivity\n\"PVC= 2:7 of PVC [19]. Thus [4,17],\n\u0019\"0~2\u001d2\nF\nqek2\nBT2\"PVCE0= Li 2\u0014\n\u0000exp\u0012\n\u0000\u0016c\nkBT\u0013\u0015\n\u0000Li2\u0014\n\u0000exp\u0012\u0016c\nkBT\u0013\u0015\n; (3)\nwhere\u001dF= 106m s\u00001[20] is the Fermi speed for graphene and Li \u0017(\u0010) is the polylogarithm function\nof order\u0017and argument \u0010[21]. The Newton{Raphson technique [22] was used to determine \u0016cas\na function of E0.\n2.4 Theoretical Simulations\nThe pixels of the metasurface were taken to be illuminated by a normally incident, linearly polarized\nplane wave whose electric \feld phasor can be written as\nEinc=\u000b^xexp(\u0000jk0z); (4)\nwith\u000bas its amplitude.\nAs the metasurface is periodic along the xandyaxes, the re\rected \feld must be written as\na doubly in\fnite series of Floquet harmonics [23]. Since a < \u0015 0=4 in the entire X band, only\nspecular components of the re\rected \feld are non-evanescent. Therefore, the re\rected electric \feld\nasz!\u00001 may be written as\nEref'\u000b(\u001axx^x+\u001ayx^y) exp(jk0z); (5)\nwhere\u001axx2Cis the co-polarized re\rection coe\u000ecient and \u001ayx2Cis the cross-polarized re\rection\ncoe\u000ecient. The transmitted \feld in the region beyond the metallic back re\rector was negligibly\nsmall in magnitude, because Lmis much larger than the penetration depth in copper. Hence, the\nabsorptance was calculated as\nA= 1\u0000(j\u001axxj2+j\u001ayxj2): (6)\nNormal incidence on several con\fgurations of the pixelated-metasurface absorber was simulated\nusing the commercial tool CST Microwave Studio ™2020. Periodic boundary conditions were applied\n4along thexandyaxes. The option open was chosen for the zaxis and the planewave condition\napplied. The meta-atom was partitioned into as many as 10,026 tetrahedrons for each simulation\nin order to achieve convergent results. The absorptance Awas calculated for f2[8;12] GHz,\nH02[180;270] kA m\u00001, andE02[0;100] V\u0016m\u00001.\n3 Numerical Results\nWe begin by discussing the response of the copper-backed YIG substrate shown in Fig. 1(a). Fig-\nure 2(a) shows the computed spectrums of AforH02f180;210;240;270gkA m\u00001, this metasurface\nbeing una\u000bected by E0. The maximum-absorptance frequency fmaxA blueshifts as the magneto-\nstatic \feld H0increases, but the maximum absorptance Amax\u00140:8. Hence, the copper-backed\nYIG substrate does not satisfy the requirement of Amax2[0:9;1] in any spectral regime within the\nX band.\nFigure 2: Absorptance spectrums of (a) the YIG/copper structure of Fig. 1(a) for H02\nf180;210;240;270gkA m\u00001and (b) the graphene/PVC/YIG/copper structure of Fig. 1(b) for\nE02f0;50;100gV\u0016m\u00001andH0= 240 kA m\u00001.\nCovering the YIG substrate on the top, \frst by a PVC skin and then by graphene, as in\nFig. 1(b), certainly a\u000bects the absorptance. Graphene makes this structure susceptible to E0, in\naddition to the YIG-mediated susceptibility to H0. The spectrums of Aare shown in Fig. 2(b) for\nE02f0;50;100gV\u0016m\u00001andH0= 240 kA m\u00001. Now,Amaxbecomes a decreasing function\nofE0, although the controllability of fmaxA byH0(results not shown) is maintained. Therefore,\nthe copper-backed YIG substrate with or without the graphene/PVC bilayer is inadequate as the\ndesired bicontrollable metasurface absorber.\nFor the next set of simulations, we partitioned the graphene in Fig. 1(b) into 16 patches\nper meta-atom, as shown in Fig. 1(c). The absorptance spectrums in Fig. 3(a) for E02\nf0;50;100gV\u0016m\u00001andH0= 240 kA m\u00001clearly indicate that pixelation can increase Amaxand\nmake it less susceptible to variations in E0, when compared with the spectrums in Fig. 2(b). The\n5Figure 3: Absorptance spectrums of the pixelated metasurface of Fig. 1(c), with all 16 pixels per\nmeta-atom patched with graphene. (a) E02f0;50;100gV\u0016m\u00001andH0= 240 kA m\u00001. (b)\nH02f180;210;240;270gkA m\u00001andE0= 50 V\u0016m\u00001.\nabsorptance spectrums in Fig. 3(b) for H02f180;210;240;270gkA m\u00001andE0= 50 V\u0016m\u00001con-\n\frm the magnetostatic controllability of fmaxA.\nFinally, we present the absorptance spectrums calculated for the metasurface of Fig. 1(d),\nwhich has ten graphene-patched and six unpatched pixels. The speci\fc con\fguration of unpatched\npixels was selected after studying the absorption spectrums for many other con\fgurations. The\nspectrums in Fig. 4(a) for E02f0;50;100gV\u0016m\u00001andH0= 240 kA m\u00001and Fig. 4(b) for\nH02f180;210;240;270gkA m\u00001andE0= 50 V\u0016m\u00001indicate that a bicontrollable spectral\nregime with A\u00150:9 andAmax\u00190:99 can be achieved with 360 MHz kA\u00001m magnetostatic\ncontrol and 1 MHz V\u00001\u0016m electrostatic control of fmaxA. Coarse control is possible through H0\nand \fne control through E0. The bandwidth 4fA\u00150:9of this absorber is about 100 MHz, which is\nsuitable for many X-band applications.\nTable 1 compares the proposed metasurface absorber with previously reported absorbers. Yuan\net al. [24] designed a voltage-controlled metasurface absorber containing varactor diodes, for X-band\noperation with fmaxA controlled in a 440-MHz range. Huang et al. [8] incorporated a meta-atom\nwith a metal resonator printed on FR4 and a\u000exed to a metal-backed ferrite substrate. Their meta-\nsurface absorber has a wider bandwidth than the proposed absorber redhas, but the controllability\nrange is smaller than of the proposed absorber. Sharma et al. [13] reported a meta-atom with a\nsquare array of pixels patched with conductive rubber and YIG on a metal-backed silicon substrate.\nThis bicontrollable metasurface has a wider bandwidth with stable maximum absorptance in the\nentire X band, and \fne control is thermal rather than electrostatic as for the proposed absorber.\nI\n6Figure 4: Absorptance spectrums of the pixelated metasurface of Fig. 1(d), with only 10 pixels per\nmeta-atom patched with graphene. (a) E02f0;50;100gV\u0016m\u00001andH0= 240 kA m\u00001. (b)\nH02f180;210;240;270gkA m\u00001andE0= 50 V\u0016m\u00001.\n4 Concluding Remarks\nWe conceived, designed, and investigated a electrostatically and magnetostatically controllable\nmetasurface absorber for operation in the entire X band. The meta-atom comprises ten graphene-\npatched pixels and six unpatched pixels in a 4 \u00024 array on a PVC skin that is a\u000exed to a metal-\nbacked YIG substrate. Graphene provides electrostatic controllability and YIG provides magneto-\nstatic controllability. Electrostatic control of the maximum-absorptance frequency using graphene-\npatched pixels in the GHz range is novel. The con\fguration of graphene-patched and unpatched\npixels was optimized to achieve stable maximum absorptance of 0 :99, with pixelation performing\nbetter than continuous graphene. According to our simulations, the chosen design delivers absorp-\ntance\u00150:9 over a 100-MHz band, with 360 MHz kA\u00001m magnetostatic controllabity rate and\n1 MHz V\u00001\u0016m electrostatic controllability rate. The proposed X-band absorber can be used to\nimprove the performance of radar systems.\nReferences\n[1] Wu PC, Pala RA, Shirmanesh GK, Cheng W-H, Sokhoyan R, Grajower M, Alam MZ, Lee D,\nAtwater HA. Dynamic beam steering with all-dielectric electro-optic III{V multiple-quantum-\nwell metasurfaces. Nat. Commun. 2019;10(1):3654.\n[2] Ding P, Li Y, Shao L, Tian X, Wang J, Fan C. Graphene aperture-based metalens for dynamic\nfocusing of terahertz waves. Opt. Exp. 2018;26(21):28038{28050.\n[3] Kumar P, Lakhtakia A, Jain PK. Tricontrollable pixelated metasurface for stopband for tera-\nhertz radiation. J. Electromag. Waves Appl. 2020;34(15):2065{2078.\n7Table 1: Structure, type of control, controllability range of maximum-absorptance frequency\n(fmaxA), bandwidth (4fA\u00150:9), and controllability rate of reported metasurface absorbers and the\nproposed metasurface absorber.\nRef. Structure Control fmaxA4fA\u00150:9 Controllability\nmethod(s) (GHz) (MHz) rate\n8 Metal resonator/FR4/ magnetostatic 9.3{9.7 150 3 MHz kA\u00001m\nferrite/metal sheet\n24 Metal pads separated by\nvaractor diodes/FR4 electrical 8.25{9.25 400 100 MHz V\u00001\nsheet/metal sheet\n13 YIG- and CR-patched pixels/ magnetostatic 8{13 200 360 MHz kA\u00001m\nsilicon/metal sheet and thermal and 1 MHz K\u00001\nThis Graphene pixels/PVC magnetostatic 8{12 100 360 MHz kA\u00001m\nwork skin/YIG/metal sheet and electrostatic and 1 MHz V\u00001\u0016m\n[4] Kumar P, Lakhtakia A, Jain PK. Tricontrollable pixelated metasurface for absorbing terahertz\nradiation. Appl. Opt. 2019;58(35):9614{9623.\n[5] He Q, Sun S, Zhou L. Tunable/recon\fgurable metasurfaces: physics and applications. Re-\nsearch. 2019;2019:1849272.\n[6] Sinclair G. Theory of models of electromagnetic systems. Proc. IRE. 1948;36(11):1364{1370.\n[7] Lakhtakia A. 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IEEE Trans.\nAntennas Propagat. 2022;70(8):6804{6812.\n[13] Sharma G, Kumar P, Lakhtakia A, Jain PK. Pixelated bicontrollable metasurface absorber\ntunable in complete X band. J. Electromag. Waves Appl. 2022;36(17):2505-2518.\n8[14] Mahabadi RK, Goudarzi T, Fleury R, Sohrabpour S, Naghdabadi R. E\u000bects of resonator\ngeometry and substrate sti\u000bness on the tunability of a deformable microwave metasurface.\nAEU Int. J. Electron. Commun. 2022;146:154123.\n[15] Yousaf A, Murtaza M, Wakeel A, Anjum S. A highly e\u000ecient low-pro\fle tetra-band meta-\nsurface absorber for X, Ku, and K band applications. AE U Int. J. Electron. Commun.\n2022;154:154329.\n[16] Pozar DM. Microwave engineering. USA: Wiley;2011.\n[17] Hanson GW. Dyadic Green's functions for an anisotropic, non-local model of biased graphene.\nIEEE Trans. Antennas Propagat. 2008;56(3):747{757.\n[18] Geng M-Y, Liu Z-G, Wu W-J, Chen H, Wu B, Lu W-B. A dynamically tunable microwave\nabsorber based on graphene. IEEE Trans. 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Phys. 2015;117(17):173103.\n9"    },    {        "title": "1703.08752v1.Unexpected_structural_and_magnetic_depth_dependence_of_YIG_thin_films.pdf",        "content": "1 \n UNEXPECTED\tSTRUCTURAL\tAND\tMAGNETIC\t\nDEPTH\tDEPENDENCE\tOF\t YIG\tTHIN\tFILMS \t\nJ.F.K. Cooper, C.J. Kinane, S. Langridge \nISIS Neutron and Muon Source, Rutherford Appleton L aboratory, Harwell Campus, Didcot, OX11 0QX \nM. Ali, B.J. Hickey \nCondensed Matter group, School of Physics and Astro nomy, E.C. Stoner Laboratory, University of \nLeeds, LS2 9JT \nT. Niizeki \nWPI Advanced Institute for Materials Research, Toho ku University, Sendai 980-8577, Japan \nK. Uchida \nNational Institute for Materials Science, Tsukuba 3 05-0047, Japan \nInstitute for Materials Research, Tohoku University , Sendai 980-8577, Japan \nCenter for Spintronics Research Network, Tohoku Uni versity, Sendai 980-8577, Japan  \nPRESTO, Japan Science and Technology Agency, Saitam a 332-0012, Japan \nE. Saitoh \nWPI Advanced Institute for Materials Research, Toho ku University, Sendai 980-8577, Japan \nInstitute for Materials Research, Tohoku University , Sendai 980-8577, Japan \nCenter for Spintronics Research Network, Tohoku Uni versity, Sendai 980-8577, Japan  \nWPI Advanced Institute for Materials Research, Toho ku University, Sendai 980-8577, Japan \nAdvanced Science Research Center, Japan Atomic Ener gy Agency, Tokai 319-1195, Japan \nH. Ambaye \nNeutron Sciences Directorate, Oak Ridge National La boratory, Oak Ridge, Tennessee 37831, USA \nA. Glavic  \nLaboratory for Neutron Scattering and Imaging, Paul  Scherrer Institut, Villigen PSI, Switzerland \nNeutron Sciences Directorate, Oak Ridge National La boratory, Oak Ridge, Tennessee 37831, USA \n \nPACS: 75.70.-I, 75.47.Lx, 68.55.aj 2 \n  \nAbstract\t\nWe report measurements on yttrium iron garnet (YIG)  thin films grown on both gadolinium gallium \ngarnet (GGG) and yttrium aluminium garnet (YAG) sub strates, with and without thin Pt top layers. \nWe provide three principal results: the observation  of an interfacial region at the Pt/YIG interface, \nwe place a limit on the induced magnetism of the Pt  layer and confirm the existence of an interfacial \nlayer at the GGG/YIG interface. Polarised neutron r eflectometry (PNR) was used to give depth \ndependence of both the structure and magnetism of t hese structures. We find that a thin film of YIG \non GGG is best described by three distinct layers: an interfacial layer near the GGG, around 5 nm \nthick and non-magnetic, a magnetic ‘bulk’ phase, an d a non-magnetic and compositionally distinct \nthin layer near the surface. We theorise that the b ottom layer, which is independent of the film \nthickness, is caused by Gd diffusion. The top layer  is likely to be extremely important in inverse spi n \nHall effect measurements, and is most likely Y 2O3 or very similar. Magnetic sensitivity in the PNR t o \nany induced moment in the Pt is increased by the ex istence of the Y 2O3 layer; any moment is found \nto be less than 0.02 uB/atom. \nIntroduction\t\nYttrium iron garnet (YIG) has long been known to be  a ferrimagnetic insulator and is used widely as a \ntuneable microwave filter or, when doped with other  rare earth elements, for a variety of optical \nand magneto-optical applications. However, since th e discovery of the spin Seebeck effect in \ninsulators 1,2 , YIG, particularly when grown on gadolinium galliu m garnet (GGG), has become the \nmodel system for investigating the physics of the s pin Seebeck effect. The spin Seebeck effect \ncombines two future technologies: the pure spin cur rents of spintronics promise to eliminate Joule \nheating in computing and many other industries 3–5, whereas energy recovery materials seek to \nharvest waste heat and movement to reduce energy lo sses, either actively 6–8, or passively from the \nconventional Seebeck effect 9 or otherwise 10 . The combination of these into a single material \nprovides a great opportunity for energy efficiency and a new generation of future devices. \nIt is therefore extremely important that both the i nterfacial physics of GGG/YIG and the physical \nsystem, with all possible imperfections, are well u nderstood. Much work has been done on the \ntheoretical understanding of the spin Seebeck effec t 11–14 , with a general consensus that the effect is \nmagnon driven, with non-equilibrium phonons also pl aying a role. This paper seeks to explore the \nmaterial science aspect of the GGG/YIG system, and t o understand the effects of interfacial structure \non high quality epitaxial films. \nThin films of YIG, with different annealing times, were grown on GGG by sputtering and these films \nwere characterised using polarised neutron reflecti vity (PNR) to extract a magnetic depth profile, as \nwell as x-ray reflectivity (XRR) and magnetometry. Films were measured both with and without thin \nPt layers on top of the YIG; these layers are conve ntionally used for inverse spin Hall effect (ISHE) \nmeasurements, to quantify the strength of the spin Seebeck effect. Additional films were grown on \nyttrium aluminium garnet (YAG) in order to investig ate the effects of the substrate on the films. \n 3 \n Methods\t\nSamples were grown in both Leeds and Tohoku Univers ities with common growth methodologies \nwith the exception of the annealing time. The Leeds  samples were sputtered in a RF magnetron \nsputter chamber with a base pressure of 2x10 -8 Torr, with oxygen and argon flow rates of 1.2 and \n22.4 sccm respectively. They were deposited onto ei ther GGG or YAG substrates, 1” in diameter. The \nsamples were then removed from the vacuum and annea led in air at 850 oC for 2 hours. They were \nthen sputtered with a thin layer of Pt ~ 27 Å thick , a typical thickness for ISHE measurements. The \nfilms grown in Tohoku were prepared according to th e methods detailed in the work by Lustikova et \nal 15 , where the same annealing temperature was used, bu t for 24 hours instead of 2.  \nA total of eight samples were measured: six from Le eds, four grown on GGG substrates and two on \nYAG, and two from Tohoku, both on GGG. The Leeds fil ms on GGG were either 300 Å or 800 Å thick \nwith and without a thin Pt layer on top. Both of th e YAG films were 800 Å, and one had a Pt top \nlayer. The Tohoku films were both around 1500 Å thi ck, one with a Pt layer (135 Å thick) on top and \none without. \nPNR measurements record the neutron reflectivity as  a function of the neutron’s wave-vector \ntransfer and spin eigenstate. Modelling of the resu ltant data allows the scattering length density \n(SLD) to be extracted and provides a quantitative d escription  of the depth dependent structural and \nmagnetic profile 16 . \nPNR measurements were taken on the Polref and Offsp ec 17  beamlines at the ISIS neutron and muon \nsource, as well as the Magnetism Reflectometer beam line at the Spallation Neutron Source, in Oak \nRidge. X-ray measurements and magnetometry was carr ied out in the R53 Characterisation lab at \nISIS. \nFitting to both the neutron reflectivity and the x- ray reflectivity data was performed in the GenX \nfitting package 18 . The Pt cap layers thickness were determined by XR R since the scattering contrast \nbetween Pt and YIG or GGG is very good  for x-rays and reduced for neutrons, whereas the contrast \nbetween YIG and GGG is poor for x-rays and good for  neutrons. Magnetometry was performed using \na Durham Magneto Optics NanoMOKE3 and showed that a ll of the films presented here had a \ncoercivity of <5 Oe and generally ~1-2 Oe, indicati ng high quality YIG. \nResults and Discussion \nFigure 1 presents the polarised neutron reflectivit y and resultant SLD for the YIG layer on GGG and \nYAG substrates. From the nominal structure of the s puttered samples the neutron reflectivity can be \ncalculated. This simple model does not accurately d escribe the observed data. To describe the GGG \nsystem an additional interfacial YIG-like layer was  required, see layer (a) in Figure 1. This layer wa s \neither non-magnetic, or had a very small moment (~< 0.1 µ B /unit cell), and was ~50 Å thick, \nirrespective of the total YIG film thickness.  This  layer was not present in the films grown on a YAG \nsubstrate. The large roughness of the interface bet ween the GGG and the YIG in the models suggests \nthat a diffusion process created this layer. This l ayer was not formed at the interface with the YAG \nsubstrates indicating that this process must be eit her Ga or Gd diffusion. A recent temperature \ndependent study of this interface also using neutro ns showed that it is Gd 19 . 4 \n Beyond the initial 50 Å non-magnetic layer, the str uctural and magnetic properties of the sputtered \nYIG films of differing thickness were very similar and did not have any thickness dependences. This \nwas true of the films grown on both GGG and YAG, mea ning that the effects of the substrate, for \nthese at least, are minimal beyond its ability to d iffuse during annealing. \nFrom the magnetic SLD the moment of the bulk YIG (a way from both interfaces) was found to be \nconsistently 3.8(1) µ B / unit cell, this value did not depend on the film  thickness. This value compares \nwell with literature values 20  at room temperature, though very slightly higher.  \nThe measurements on the YIG films with extended ann ealing times were less conclusive. Models \nboth with and without the non-magnetic layer at the  GGG interface gave similar fits to the PNR data. \nThese films were significantly thicker than the fil ms with a 2 hr anneal, ~1500 Å, and as such the \nsensitivity to the GGG/YIG interface is reduced. As  a results it is not possible to conclusively ident ify \nthe presence of an interfacial layer. Since the ann ealing procedures for both sets of films are very \nsimilar we can assume that the Gd diffusion will al so be similar, and a common feature of the \nGGG/YIG interface. \nIn addition to the substrate interface layer, an ad ditional layer was discovered for all of the sample s, \nlabelled as layer (b) in Figure 1. This layer is ar ound 15(5) Å in thickness, with little variation, a cross \nall samples measured. This layer was distinct from both the Pt and the YIG, as it had a markedly \nlower scattering length density than either of them . Figure 2 shows datasets for both thin Leeds (300 \nÅ) and thick Tohoku (1500 Å) YIG on GGG, with the b est fit to the data. Since the SLD of YIG and Pt is  \nsimilar, the low frequency oscillations in both dat asets results from the low SLD layers contrast \nbetween the Pt and the YIG. Analysing x-ray reflect ivity curves of the same samples, with and \nwithout the Pt cap also require this layer. Examini ng the two scattering length densities (neutrons \nand x-rays) of the layer involved we can elucidate its composition. Pt alloys would generate a strong \nx-ray contrast, and the layer would not appear in u ncapped samples and can therefore be ruled out. \nBoth iron and all forms of its oxide have too large  an SLD for neutrons so it can be ruled out. This \nleaves yttrium based compounds: pure Y, Y 2O3, (yttria) and YN (which is possible, though unlike ly, \ndue to the annealing of the sample in air). The x-r ay SLD of Y 2O3 is a close match with the SLD of the \nlayer required for a good fit, as shown in Figure 3 . In addition to the matching SLD, we remark that \nY3Fe 5O12  has an oxidation state of +3 for Y, which is the s ame as Y 2O3 and both have similar oxygen \nco-ordination. The Y-O bond length in Y 2O3 is between 2.225 and 2.323 Å 21 , which represents a slight \ncontraction with respect to the bond length in YIG,  which is between 2.37 and 2.43 Å 22 . \nThe magnetic signal from the YIG decays across this  layer (see Figure 3) and as it is likely that the \nlayer is predominantly yttria (a non-magnetic insul ator), the electrical resistivity at some point in the \nfilm becomes that of the Pt. This means that any IS HE effects are likely sampling a lot more of the \nyttria, than the YIG. Several studies have investig ated the influence of the interface quality 23–25  and \nhave determined that, as might be expected, a high quality interface yields better ISHE results. Qiu \net al .24  found that, the ISHE voltage varied from around 3. 5 µV/K for a minimal interface yttria \nregion, to nearly 0 for regions over 7 nm thick. Th ey also found that, at least for samples grown by \nliquid phase epitaxy, optimisation of after growth annealing could minimise this layer’s formation. \nThis interface has also been investigated by Song et al. 26  using electron microscopy, though they \nattribute the layer to being oxygen deficient iron (whose magnetic moment is then reduced).  5 \n Knowledge of the existence and the information abou t the nature of this layer extracted here gives \nan opportunity to eliminate it in all cases; either  by appropriate annealing, or selective etching. \nAs a result of the low SLD of this layer and its co ntrast to the Pt layer, we have gained unusual \nsensitivity to any induced magnetism in the Pt laye r. Proximity effects in the Pt have been widely \nstudied, with many works extracting the origins of the observed voltages from the iSHE 27–29 . \nTheoretical studies such as Liang et al. 30  show how a non-magnetic, or reduced magnetic layer  would \nbe important in proximity effects in this system. F igure 4 shows a portion of the spin asymmetry \n(difference in reflectivity of the two spin states normalised by their sum) from \nGGG/YIG(8000)/Pt(27), at high momentum transfer. Th e spin asymmetry is sensitive to the \nmagnetism, and is, to a first approximation, indepe ndent of the exact structure. The reflectivity can \nbe approximated as the Fourier transform of the lay er structure, so at high Q we are sensitive to \nthinner layers, e.g. the top Pt layer.  \nThe best model line is shown in grey and is clearly  a good fit, even by this, more highly processed, \nmeasure. In addition to the best fit in Figure 4 (s hown in grey) are two models (blue and red), with \nthe same structural parameters and bulk YIG magneti sm, but with an induced moment added to the \nPt of ±0.05 µ B/Pt atom to show how this would affect the fit. Fro m the deviation of these models \nfrom the data, we can clearly see that the average magnitude of any induced moment in the Pt is \ncertainly less than 0.05 µ B/Pt atom, and likely less than 0.02 µ B/Pt atom, within a 1σ error bound. \nThis result is in line with previous experiments 31  which specifically tried to measure an induced \nmoment in Pt, albeit with the YIG grown by a differ ent method. It is worth noting here that it may \nstill be possible that a more magnetic sub-region o f the Pt may exist, since PNR cannot probe \ninfinitely thin layers. However, the magnitude of t he total magnetism would still have to remain \nbelow our upper bound, e.g. if half was non-magneti c and half was polarised, then it would have to \nbe below 0.04 µ B/Pt atom, etc. \nConclusion \n \nWe have used polarised neutron reflectivity to dete rmine the magnetic depth dependence of \nyttrium iron garnet thin films grown on gadolinium gallium garnet and yttrium aluminium garnet \nsubstrates, with and without a Pt layer on the surf ace. It was found that if the YIG is grown on a GGG  \nsubstrate there can be a ~50 Å non-magnetic layer a t the substrate interface, this does not depend \non the YIG film thickness. This is likely to be cau sed by Gd diffusion during annealing, since this la yer \ndoes not appear when the YIG is grown on YAG, this is in line with recent investigations 19 . The effect \nof growing YIG on YAG, other than the absence of th is interface layer was minimal, with roughnesses \nand magnetic moments extremely similar to those gro wn on GGG. \nWe also see an additional layer at the YIG/Pt inter face, roughly 15 Å for all samples measured; \nfurther investigation and cross referencing with x- ray measurements identifies this layer as Y 2O3. \nWhile the existence of this (non-magnetic and usual ly insulating) layer may have large repercussions \nfor the interpretation of ISHE measurements on this  model system, knowledge of its existence and \ncomposition means it may be possible to eliminate i t.  6 \n Our measurements also give us unusual sensitivity t o any induced magnetism in the Pt layer, and \nallow us to give an upper bound on the magnitude of  the moment of ±0.02 µ B/Pt atom. \n \nAcknowledgements  \nThe neutron work in this paper was performed at bot h the Spallation Neutron Source in the Oak \nRidge National Laboratory (IPTS-13192), USA, and at  the ISIS Pulsed Neutron and Muon Source, \nwhich were supported by a beamtime allocation from the Science and Technology Facilities Council \n(RB1410610 and RB1510146). 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Interfaces 8, 8175 \n(2016). \n31  S. Geprägs, S. Meyer, S. Altmannshofer, M. Opel, F . Wilhelm, A. Rogalev, R. Gross, and S.T.B. \nGoennenwein, Appl. Phys. Lett. 101 , 262407 (2012). \n \n 8 \n  \nFigure 1 Structural (blue) and magnetic (green)  scattering length densities of fitted models for a Y IG/Pt \nbilayer grown on a GGG substrate (top) and YAG subs trate (bottom). The top shows a thin ~300 Å  YIG \nlayer, and the bottom shows a thick ~800 Å  YIG lay er, however, the data are representative of all the  \nsamples grown on each substrate irrespective of YIG  thickness. The GGG substrate has two extra \nlayers: (a), at the substrate interface, and (b), a t the YIG/Pt interface. (a) is non-magnetic at room  \ntemperature and around 50 Å thick irrespective of t he thickness of the YIG layer. This layer is not \npresent at the YAG/YIG interface. The nature of lay er (b) discussed later and is absolutely required f or \na good fit in all samples with a Pt layer. 9 \n  \nFigure 2 Polarised Neutron Reflectivity data (points) and mo delled fit (line) of thin (~300 Å ) GGG/YIG/Pt from  Leeds, (a), \nand thicker (~1500 A) Tohoku, (b), samples, both of  which have Pt top layers. The low frequency oscill ations (which is the \nmajority of the curve in (a) since it has a thinner  Pt layer) are visible due to the low scattering le ngth density layer between \nthe YIG and the Pt and is required in models for al l samples with Pt on to get a reasonable fit.  10 \n  \nFigure 4 A selected portion of the spin asymmetry of a thick  (~800 Å ) YAG/YIG/Pt structure, with a zoom inset.  The higher \nQ range is where we are sensitive to any induced ma gnetism in the Pt layer. The grey curve shows the a symmetry \nproduced by the optimal fit with no induced magneti sm in the Pt. The red and blue curves show the same  model with \n+0.05 µ B/Pt atom and -0.05 µ B/Pt atom of induced magnetism. A similar result is found for all samples measured, \nirrespective of YIG thickness or substrate.  Figure 3 X-ray scattering length density of the interface be tween YIG and air, with x-ray reflectivity data and  fit inset. The \nsubtle step in the SLD is required in order to corr ectly model the slow oscillation in the reflectivit y data. By using both the \nx-ray and neutron scattering length densities we ca n deduce that the top layer in this, and all other samples measured in \nthis study, is extremely likely to be yttria (Y 2O3), whose bulk SLD is indicated by the dotted red li ne. "    },    {        "title": "1812.09766v1.Temperature_dependence_of_the_effective_spin_mixing_conductance_probed_with_lateral_non_local_spin_valves.pdf",        "content": "Temperature dependence of the e\u000bective spin-mixing conductance probed\nwith lateral non-local spin valves\nK. S. Das,1,a)F. K. Dejene,2B. J. van Wees,1and I. J. Vera-Marun3,b)\n1)Physics of Nanodevices, Zernike Institute for Advanced Materials, University of Groningen, 9747 AG Groningen,\nThe Netherlands\n2)Department of Physics, Loughborough University, Loughborough LE11 3TU,\nUnited Kingdom\n3)School of Physics and Astronomy, University of Manchester, Manchester M13 9PL,\nUnited Kingdom\nWe report the temperature dependence of the e\u000bective spin-mixing conductance between a normal metal\n(aluminium, Al) and a magnetic insulator (Y 3Fe5O12, YIG). Non-local spin valve devices, using Al as the\nspin transport channel, were fabricated on top of YIG and SiO 2substrates. By comparing the spin relaxation\nlengths in the Al channel on the two di\u000berent substrates, we calculate the e\u000bective spin-mixing conductance\n(Gs) to be 3:3\u00021012\n\u00001m\u00002at 293 K for the Al/YIG interface. A decrease of up to 84% in Gsis observed\nwhen the temperature ( T) is decreased from 293 K to 4.2 K, with Gsscaling with ( T=T c)3=2. The real part\nof the spin-mixing conductance ( Gr\u00195:7\u00021013\n\u00001m\u00002), calculated from the experimentally obtained\nGs, is found to be approximately independent of the temperature. We evidence a hitherto unrecognized\nunderestimation of Grextracted from the modulation of the spin signal by rotating the magnetization direction\nof YIG with respect to the spin accumulation direction in the Al channel, which is found to be 50 times smaller\nthan the calculated value.\nThe transfer of spin information between a normal\nmetal (NM) and a magnetic insulator (MI) is the crux\nof electrical injection and detection of spins in the\nrapidly emerging \felds of magnon spintronics1and an-\ntiferromagnetic spintronics2,3. The spin current \row-\ning through the NM/MI interface is governed by the\nspin-mixing conductance4{7,G\"#, which plays a cru-\ncial role in spin transfer torque8{10, spin pumping11,12,\nspin Hall magnetoresistance (SMR)13,14and spin See-\nbeck experiments15. In these experiments, the spin-\nmixing conductance ( G\"#=Gr+iGi), composed of a\nreal (Gr) and an imaginary part ( Gi), determines the\ntransfer of spin angular momentum between the spin ac-\ncumulation ( ~ \u0016s) in the NM and the magnetization ( ~M) of\nthe MI in the non-collinear case. However, recent experi-\nments on the spin Peltier e\u000bect16, spin sinking17and non-\nlocal magnon transport in magnetic insulators18,19neces-\nsitate the transfer of spin angular momentum through\nthe NM/MI interface also in the collinear case (~ \u0016sk~M).\nThis is taken into account by the e\u000bective spin-mixing\nconductance ( Gs) concept, according to which the trans-\nfer of spin angular momentum across the NM/MI inter-\nface can occur, irrespective of the mutual orientation be-\ntween~ \u0016sand~M, via local thermal \ructuations of the\nequilibrium magnetization (thermal magnons20) in the\nMI. The spin current density ( ~js) through the NM/MI\ninterface can, therefore, be expressed as17,21,22:\n~js=Gr^m\u0002(~ \u0016s\u0002^m) +Gi(~ \u0016s\u0002^m) +Gs~ \u0016s;(1)\nwhere, ^mis a unit vector pointing along the direction\nof~M. WhileGrandGihave been extensively studied\na)e-mail: K.S.Das@rug.nl\nb)e-mail: ivan.veramarun@manchester.ac.ukin spin torque and SMR experiments23{25, direct experi-\nmental studies on the temperature dependence of Gsare\nlacking.\nIn this letter, we report the \frst systematic study of\nGsversus temperature ( T) for a NM/MI interface. For\nthis, we utilize the lateral non-local spin valve (NLSV)\ngeometry, which provides an alternative way to study\nthe spin-mixing conductance using pure spin currents in\na NM with low spin-orbit coupling (SOC)17,26,27. A low\nSOC of the NM in the NLSV technique also ensures that\nthe spin-mixing conductance is not overestimated due\nto spurious proximity e\u000bects in NMs with high SOC or\nclose to the Stoner criterion, such as Pt28{30. We exclu-\nsively address the temperature dependence of Gsfor the\naluminium (Al)/Y 3Fe5O12(YIG) interface, which is ob-\ntained by comparing the spin relaxation length ( \u0015N) in\nsimilar Al channels on a magnetic YIG substrate and a\nnon-magnetic SiO 2substrate, as a function of tempera-\nture.Gsdecreases by about 84% when the temperature is\ndecreased from 293 K to 4.2 K and scales with ( T=T c)3=2,\nwhereTc= 560 K is the Curie temperature of YIG, con-\nsistent with theoretical predictions19,31{33. The real part\nof the spin-mixing conductance ( Gr) is then calculated\nfrom the experimentally obtained values of Gsand com-\npared with the modulation of the spin signal in rotation\nexperiments, where the magnetization direction of YIG\n(~M) is rotated with respect to ~ \u0016s.\nThe NLSVs with Al spin transport channel were fab-\nricated on top of YIG and SiO 2thin \flms in multiple\nsteps using electron beam lithography (EBL), electron\nbeam evaporation of the metallic layers and resist lift-o\u000b\ntechnique, following the procedure described in Ref. 34.\nThe 210 nm thick YIG \flm on Gd 3Ga5O12substrate and\nthe 300 nm thick SiO 2\flm on Si substrate were obtained\ncommercially from Matesy GmbH and Silicon Quest In-\nternational, respectively. Permalloy (Ni 80Fe20, Py) hasarXiv:1812.09766v1  [cond-mat.mes-hall]  23 Dec 20182\n(b) (a) (c)\nAlI\nL\nPy1\nYIG\nM\nxz\ny\nPy2V+_\nPS\nMagnons\n500 nmIPy\nV+\n_\nAlPy\nxy B\nθ\nL\nFIG. 1. (a)Schematic illustration of the experimental geometry. The spin accumulation ( ~ \u0016s), injected into the Al channel\nby the Py injector, has an additional relaxation pathway into the (insulating) magnetic YIG substrate due to local thermal\n\ructuations of the equilibrium YIG magnetization ( ~M) or thermal magnons. (b)SEM image of a representative NLSV device\nalong with the illustration of the electrical connections for the NLSV measurements. An alternating current ( I) was sourced\nfrom the left Py strip (injector) to the left end of the Al channel and the non-local voltage ( VNL) was measured across the\nright Py strip (detector) with reference to the right end of the Al channel. An external magnetic \feld ( B) was swept along the\ny-axis in the non-local spin valve (NLSV) measurements. In the rotation measurements, Bwas applied at di\u000berent angles ( \u0012)\nwith respect to the y-axis in the xy-plane. (c)NLSV measurements at T= 293 K for an Al channel length ( L) of 300 nm on\nthe YIG substrate (red) and on the SiO 2substrate (black).\nbeen used as the ferromagnetic electrodes for injecting\nand detecting a non-equilibrium spin accumulation in the\nAl channel. A 3 nm thick Ti underlayer was deposited\nprior to the evaporation of the 20 nm thick Py electrodes.\nThe Ti underlayer prevents direct injection and detection\nof spins in the YIG substrate via the anomalous spin Hall\ne\u000bect in Py35,36.In-situ Ar+ion milling for 20 seconds\nat an Ar gas pressure of 4 \u000210\u00005Torr was performed,\nprior to the evaporation of the 55 nm thick Al chan-\nnel, ensuring a transparent and clean Py/Al interface. A\nschematic of the device geometry is depicted in Fig. 1(a)\nand a scanning electron microscope (SEM) image of a\nrepresentative device is shown in Fig. 1(b). A low fre-\nquency (13 Hz) alternating current source ( I) with an\nr.m.s. amplitude of 400 \u0016A was connected between the\nleft Py strip (injector) and the left end of the Al channel.\nThe non-local voltage ( VNL) due to the non-equilibrium\nspin accumulation in the Al channel was measured be-\ntween the right Py strip (detector) and the right end of\nthe Al channel using a standard lock-in technique. The\nmeasurements were carried out under a low vacuum at-\nmosphere in a variable temperature insert, placed within\na superconducting magnet.\nIn the NLSV measurements, an external magnetic \feld\n(B) was swept along the y-axis and the corresponding\nnon-local resistance ( RNL=VNL=I) was measured. In\nFig. 1(c), NLSV measurements for an Al channel length\n(L) of 300 nm at T= 293 K are shown for two devices,\none on YIG (red) and another on SiO 2(black). The\nspin signal, Rs=RP\nNL\u0000RAP\nNL, is de\fned as the di\u000berence\nin the two distinct states corresponding to the parallel\n(RP\nNL) and the anti-parallel ( RAP\nNL) alignment of the Py\nelectrodes' magnetizations. The Rswas measured as a\nfunction of the separation ( L) between the injector andthe detector electrodes for several devices fabricated on\nYIG and SiO 2substrates, as shown in Fig. 2(a). To de-\ntermine the spin relaxation length ( \u0015N) in the Al chan-\nnels on YIG ( \u0015N, YIG ) and SiO 2(\u0015N, SiO 2) substrates, the\nexperimental data in Fig. 2(a) were \ftted with the spin\ndi\u000busion model37for transparent contacts:\nRs=4\u000b2\nF\n(1\u0000\u000b2\nF)2RN\u0012RF\nRN\u00132e\u0000L=\u0015N\n1\u0000e\u00002L=\u0015N; (2)\nwhere,\u000bFis the bulk spin polarization of Py, RN=\n\u001aN\u0015N=wNtNandRF=\u001aF\u0015F=wNwFare the spin resis-\ntances of Al and Py, respectively. \u0015N(F),\u001aN(F),wN(F)\nandtNare the spin relaxation length, electrical re-\nsistivity, width and thickness of Al (Py), respectively.\nAt room temperature, \u0015N, YIG = (276\u000630) nm and\n\u0015N, SiO 2= (468\u000620) nm were extracted, with \u000bF\u0015F=\n(0:84\u00060:05) nm.\nThe NLSV measurements were carried out at di\u000ber-\nent temperatures, enabling the extraction of \u0015N, YIG and\n\u0015N, SiO 2, as shown in Fig. 2(b). From this tempera-\nture dependence, it is obvious that \u0015N, YIG is lower than\n\u0015N, SiO 2throughout the temperature range of 4.2 K to\n293 K. The corresponding electrical conductivities of the\nAl channel ( \u001bN) on the two di\u000berent substrates were also\nmeasured by the four-probe technique as a function of T,\nas shown in Fig. 2(c). The similar values of \u001bNfor the Al\nchannels on both YIG and the SiO 2substrates suggests\nthat there is no signi\fcant di\u000berence in the structure and\nquality of the Al \flms between the two substrates. There-\nfore, considering the dominant Elliott-Yafet spin relax-\nation mechanism in Al38, di\u000berences in the spin relax-\nation rate within the Al channels cannot account for the\ndi\u000berence in the e\u000bective spin relaxation lengths between\nthe two substrates.3\n(a) (b) (c)\nFIG. 2. (a)The spin signal ( Rs) plotted as a function of the Al channel length ( L) for NLSV devices on YIG (red circles) and\nSiO2(black square) substrates at 293 K. The solid lines represent the \fts to the spin di\u000busion model (Eq. 2). (b)The e\u000bective\nspin relaxation length in the Al channel ( \u0015N) extracted at di\u000berent temperatures ( T).\u0015Nis smaller on the YIG substrate as\ncompared to the SiO 2substrate. (c)The electrical conductivity ( \u001bN) of the Al channels on the YIG and the SiO 2substrates\nas a function of temperature. The close match between the two conductivities suggests similar quality of the Al \flm grown on\nboth substrates.\nThe smaller values of \u0015N, YIG as compared to \u0015N, SiO 2\nsuggest that there is an additional spin relaxation mech-\nanism for the spin accumulation in the Al channel on the\nmagnetic YIG substrate. This is expected via additional\nspin-\rip scattering at the Al/YIG interface, mediated by\nthermal magnons in YIG and governed by the e\u000bective\nspin-mixing conductance ( Gs). As described in Ref. 17,\n\u0015N, YIG and\u0015N, SiO 2are related to Gsas\n1\n\u00152\nN, YIG=1\n\u00152\nN, SiO 2+1\n\u00152r; (3)\nwhere,\u0015r= 2Gs=(tAl\u001bN). Using the extracted values\nof\u0015Nfrom Fig. 2(b) and the measured values of \u001bNfor\nthe devices on YIG from Fig. 2(c), we calculate Gs=\n3:3\u00021012\n\u00001m\u00002at 293 K. At 4.2 K, Gsdecreases by\nabout 84% to 5 :4\u00021011\n\u00001m\u00002.\nThe temperature dependence of Gsis shown in\nFig. 3(a). Since the concept of the e\u000bective spin-mixing\nconductance is based on the thermal \ructuation of the\nmagnetization (thermal magnons), Gsis expected to\nscale as (T=T c)3=2, whereTcis the Curie temperature\nof the magnetic insulator6,19,31,32. UsingTc= 560 K\nfor YIG, we \ft the experimental data to C(T=T c)3=2,\nwhich is depicted as the solid line in Fig. 3(a). The\ntemperature independent prefactor, C, was found to be\n8:6\u00021012\n\u00001m\u00002. The agreement with the experimental\ndata con\frms the expected scaling of Gswith tempera-\nture. Note that the deviation from the ( T=T c)3=2scaling\nat lower temperatures could be in part due to slightly\ndi\u000berent quality of the Al \flm on the YIG substrate.\nNevertheless, the small di\u000berence of \u001910% in the elec-\ntrical conductivities of the Al channel on the two di\u000berent\nsubstrates at T < 100 K in Fig. 2(c) cannot account for\nthe di\u000berences in \u0015N. On the other hand, we note that\nquantum magnetization \ructuations39,40in YIG can also\nplay a role at low T, leading to an enhanced Gs.\nNext, we investigate the temperature dependence of\nthe real part of the spin-mixing conductance ( Gr). Forthis, we \frst calculate Grfrom the experimentally ob-\ntainedGs, using the following expression19:\nGs=3\u0010(3=2)\n2\u0019s\u00033Gr; (4)\nwhere\u0010(3=2) = 2:6124 is the Riemann zeta function cal-\nculated at 3 =2,s=S=a3is the spin density with total\nspinS= 10 in a unit cell of volume a3= 1:896 nm3, and\n\u0003 =p\n4\u0019Ds=kBTis the thermal de Broglie wavelength\nfor magnons, with Ds= 8:458\u000210\u000040Jm2being the spin\nwave sti\u000bness constant for YIG19,41. The temperature\ndependence of the calculated Gris shown in Fig. 3(b).\nKeeping in mind that Eq. 4 is not valid in the limits of\nT!TcandT!0, we ignore the data points below\n100 K. Above this temperature, Gris almost constant at\n\u00195:7\u00021013\n\u00001m\u00002, represented by the dashed line in\nFig. 3(b). This is consistent with Ref. 25, where Grwas\nfound to be T-independent. Moreover, the magnitude\nofGris comparable with previously reported values for\n(a) (b)\nFIG. 3. (a)Temperature dependence of the e\u000bective\nspin-mixing conductance (black symbols). Gsscales with\nthe temperature as ( T=T c)3=2(solid line). (b) The real\npart of the spin-mixing conductance ( Gr) is calculated from\nEq. 4 by using the experimentally obtained values of Gs.Gr\n(\u00195:7\u00021013\n\u00001m\u00002) is essentially found to be constant\n(dashed line) for T >100K.4\n(a) (b) (c)\nFIG. 4. (a)NLSV measurement for a device on the YIG substrate with L= 300 nm at 150 K. (b)Rotation measurement\nfor the same device with B= 20 mT applied at di\u000berent angles ( \u0012) with respect to the y-axis. The black and the red symbols\ncorrespond to the average of ten rotation measurements carried out with the magnetization of the Py electrodes in the parallel\n(P) and the anti-parallel (AP) con\fgurations, respectively. (c)The spin signal ( Rs=RP\nNL\u0000RAP\nNL) exhibits a periodic modulation\nof magnitude \u0001 Rswhen the angle \u0012between the magnetization direction in YIG ( ~M) and the spin accumulation direction in\nAl (~ \u0016s) is changed. The black symbols represent the experimental data at 150 K, while the red line is the numerical modelling\nresult corresponding to Gr= 1\u00021012\n\u00001m\u00002.\nAl/YIG17and Pt/YIG19,42interfaces.\nAn alternative approach for extracting Grfrom the\nNLSVs fabricated on the YIG substrate, is by the rota-\ntion of the sample with respect to a low magnetic \feld\nin thexy-plane. We have also followed this method, de-\nscribed in Refs. 17 and 26. In the rotation experiments,\nthe angle\u0012between the magnetization direction in YIG\n(~M) and the spin accumulation direction in Al ( ~ \u0016s) is\nchanged, which results in the modulation of the spin sig-\nnal in the Al channel due to the transfer of spin angu-\nlar momentum across the Al/YIG interface, as described\nin Eq. 1, dominated by the Grterm. First, the NLSV\nmeasurement for a device with L= 300 nm was carried\nout at 150 K, as shown in Fig. 4(a). In the next step,\nB= 20 mT was applied in the xy-plane and the sample\nwas rotated, with the magnetization orientations of the\nPy electrodes set in the parallel (P) or the anti-parallel\n(AP) con\fguration. For improving the signal-to-noise\nratio, ten measurements were performed for each of the\ncon\fgurations (P and AP). The average of these measure-\nments is shown in Fig. 4(b). The spin signal is extracted\nfrom Fig. 4(b) and plotted as a function of \u0012in Fig. 4(c).\nRsexhibits a periodic modulation with the maxima at\n\u0012= 0\u000eand minima at \u0012=\u000690\u000e, consistent with the\nbehaviour predicted in Eq. 1. The modulation in the Rs,\nde\fned as(R0\u000e\ns\u0000R\u000690\u000e\ns)\nR0\u000e\ns=\u0001Rs\nR0\u000e\ns, was found to be 2 :8%. A\nsimilar modulation of 2 :9% was reported in Ref 26 for an\nNLSV with a Cu channel on YIG with L= 570 nm at\nthe same temperature.\nGris estimated from the rotation measurements us-\ning 3D \fnite element modelling, as described in Ref. 17.\nFrom the modelled curve for the spin signal modula-\ntion, shown as the red line in Fig. 4(c), we extract\nGr= 1\u00021012\n\u00001m\u00002. This value is comparable to that\nreported in Ref. 26, within a factor of 2, for an evaporated\nCu channel on YIG. However, this value is more than 50\ntimes smaller than our estimated value from Eq. 4, andalso that reported in Ref. 17 for a sputtered Al chan-\nnel on YIG. One reason behind the small magnitude of\nGrextracted from the rotation measurements can be at-\ntributed to the thin \flm deposition technique used. In\nRef. 14, it was shown that the SMR signal for a sputtered\nPt \flm on YIG is about an order of magnitude larger\nthan that for an evaporated Pt \flm. Moreover, during\nthe fabrication of our NLSVs, an Ar+ion milling step\nis carried out prior to the evaporation of the NM chan-\nnel for ensuring a clean interface between the NM and\nthe ferromagnetic injector and detector electrodes17,26.\nConsequently, this also leads to the milling of the YIG\nsurface on which the NM is deposited, resulting in the\nformation of an\u00192 nm thick amorphous YIG layer at\nthe interface43. Since an external magnetic \feld of 20 mT\nis not su\u000ecient to completely align the magnetization di-\nrection within this amorphous layer parallel to the \feld\ndirection44, the resulting modulation in the spin signal\nwill be smaller. This might lead to the underestima-\ntion ofGr. Note that since the e\u000bect of Gsdoes not\ndepend on the magnetization orientation of YIG (Eq. 1),\nthe milling does not a\u000bect the estimation of Gs. Our\nobservations are consistent with a similarly small value\nofGr\u00194\u00021012\n\u00001m\u00002reported in Ref. 26 for the\nCu/YIG interface, where the Cu channel was evaporated\nfollowing a similar Ar+ion milling step. Using the re-\nported values of \u0015N= 522 nm (680 nm) on YIG (SiO 2)\nsubstrate for the 100 nm thick Cu channel at 150 K in\nRef. 26, we extract Gs= 2\u00021012\n\u00001m\u00002, which is 5\ntimes larger than their reported Grextracted from rota-\ntion measurements.\nIn summary, we have studied the temperature depen-\ndence ofGsandGrusing the non-local spin valve tech-\nnique for the Al/YIG interface. From NLSV measure-\nments, we extracted Gsto be 3:3\u00021012\n\u00001m\u00002at 293 K,\nwhich decreases by about 84% at 4.2 K, approximately\nobeying the ( T=T c)3=2law. While Grremains almost\nconstant with the temperature, the value extracted from5\nthe modulation of the spin signal (1 \u00021012\n\u00001m\u00002)\nwas around 50 times smaller than the calculated value\n(5:7\u00021013\n\u00001m\u00002). 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Goennenwein1, 3\n1)Walther-Mei\u0019ner-Institut, Bayerische Akademie der Wissenschaften,\n85748 Garching, Germany\n2)Physik-Department, Technische Universit at M unchen, 85748 Garching,\nGermany\n3)Nanosystems Initiative Munich, 80799 M unchen, Germany\n(Dated: 8 April 2015)\nWe measure the ordinary and the anomalous Hall e\u000bect in a set of yttrium iron\ngarnetjplatinum (YIGjPt) bilayers via magnetization orientation dependent magne-\ntoresistance experiments. Our data show that the presence of the ferrimagnetic in-\nsulator YIG leads to an anomalous Hall e\u000bect like voltage in Pt, which is sensitive\nto both Pt thickness and temperature. Interpretation of the experimental \fndings in\nterms of the spin Hall anomalous Hall e\u000bect indicates that the imaginary part of the\nspin mixing conductance Giplays a crucial role in YIG jPt bilayers. In particular, our\ndata suggest a sign change in Gibetween 10 K and 300 K. Additionally, we report a\nhigher order Hall e\u000bect contribution, which appears in thin Pt \flms on YIG at low\ntemperatures.\na)Electronic mail: sibylle.meyer@wmi.badw.de\n1arXiv:1501.02574v3  [cond-mat.mtrl-sci]  7 Apr 2015The generation, manipulation and detection of pure spin currents are fascinating chal-\nlenges in spintronics . In normal metals with large spin orbit interaction, the spin Hall e\u000bect\n(SHE)1{4and its inverse (ISHE)5enable the generation viz. detection of spin currents in\nthe charge transport channel. In this context, the spin Hall angle \u0012SHand the spin di\u000busion\nlength\u0015are key material parameters1,2. Additionally, the spin mixing conductance Gwas\nproposed as a measure for the number of spin transport channels per unit area across a nor-\nmal metal (NM)jferromagnet (FM) interface, in analogy to the Landauer-B uttiker picture\nin ballistic charge transport6,7. Here,G=Gr+{Giis introduced as a complex quantity8{12.\nThe real part Gris linked to an in-plane magnetic \feld torque13,14and accessible e.g. from\nspin pumping experiments5{7,15,16. The imaginary part Giis related to the spin precession\nand interpreted as a phase shift between the spin current in the NM and the one in the\nFM.Githus can be either positive or negative7. As suggested recently, the spin Hall mag-\nnetoresistance (SMR)17{19based on the simultaneous action of SHE and ISHE allows for\nquantifying Gifrom measurements of anomalous Hall-type e\u000bects (AHE) in ferromagnetic\ninsulatorjNM hybrids, referred to as spin Hall anomalous Hall e\u000bect (SH-AHE)19.\nHere, we present an experimental study of ordinary and anomalous Hall-type signals ob-\nserved in yttrium iron garnet (Y 3Fe5O12, YIG)jplatinum (Pt) bilayers. We discuss the \flm\nthickness and temperature dependence of the AHE signals in terms of the SH-AHE. While\nthe AHE voltage observed in metallic ferromagnets usually obeys VH/Mn\n?withn= 1 and\nM?the component of the magnetization along the \flm normal, we observe a more complex\nAHE-type response with higher order terms VH/Mn\n?at low temperatures in YIG jPt sam-\nples with a Pt \flm thickness tPt\u00145 nm. The higher order contributions are directly evident\nin our experiments, since we measure the magneto-transport response as a function of exter-\nnal magnetic \feld orientation, while conventional Hall experiments are typically performed\nas a function of \feld strength in a perpendicular \feld arrangement. For comparison, we also\nstudy thin Pt \flms deposited directly onto diamagnetic substrates. In these samples, we\nneither \fnd a temperature dependence of the ordinary Hall-e\u000bect (OHE), nor an AHE-type\nsignal, not to speak of higher order AHE contributions.\nWe investigate two types of thin \flm structures, YIG jPt bilayers and single Pt thin \flms on\nyttrium aluminum garnet (Y 3Al5O12, YAG) substrates. The YIG jPt bilayers are obtained by\ngrowing epitaxial YIG thin \flms with a thickness of t\u001960 nm on single crystalline YAG or\ngadolinium gallium garnet (Gd 3Ga5O12,GGG) substrates using pulsed laser deposition20,21.\n2In an in situ process, we then deposit a thin polycrystalline Pt \flm onto the YIG via electron\nbeam evaporation. We hereby systematically vary the Pt thickness from sample to sample in\nthe range 1 nm\u0014tPt\u001420 nm. In this way, we obtain a series of YIG jPt bilayers with \fxed\nYIG thickness, but di\u000berent Pt thicknesses. For reference, we furthermore fabricate a series\nof YAGjPt bilayers, depositing Pt thin \flms with thicknesses 2 nm \u0014tPt\u001416 nm directly\nonto YAG substrates. We employ X-ray re\rectometry and X-ray di\u000braction to determine tPt\nand to con\frm the polycrystallinity of the Pt thin \flms22. For electrical transport measure-\nments, the samples are patterned into Hall bar mesa structures (width w= 80\u0016m, contact\nseparation l= 600\u0016m)23[c.f. Fig. 1(a)]. We current bias the Hall bars with Iqof up to\n500\u0016A and measure the transverse (Hall like) voltage Vtranseither as a function of the mag-\nnetic \feld orientation (angle dependent magnetoresistance, ADMR21,24) or of the magnetic\n\feld amplitude \u00160H(\feld dependent magnetoresistance, FDMR), for sample temperatures\nTbetween 10 K and 300 K. For all FDMR data reported below, the external magnetic \feld\nwas applied perpendicular to the sample plane ( \u00160Hkn, c.f. Fig. 1(a)). For the ADMR\nmeasurements, we rotate an external magnetic \feld of constant magnitude 1 T \u0014\u00160H\u00147 T\nin the plane perpendicular to the current direction j23. Here,\fHis de\fned as the angle be-\ntween the transverse direction tand the magnetic \feld H. In all ADMR experiments, we\nchoose\u00160Hlarger than the anisotropy and the demagnetization \felds of the YIG \flm. As\na result, the YIG magnetization Mis always saturated and oriented along Hin good ap-\nproximation. The transverse resistivity \u001atrans(\fH;H) =Vtrans(\fH;H)tPt=Iqof the Pt layer\nis calculated from the voltage Vtrans(\fH) along t.\nFigure 1(b-d) show FDMR measurements carried out at 300 K in YIG jPt bilayers with\ntPt= 2:0;6:5 and 19:5 nm. Extracting the ordinary Hall coe\u000ecient \u000bOHE=@\u001atrans(H)=@(H)\nfrom the slope, we obtain \u000bOHE(19:5 nm) =\u000025:5p\nm=T for the thickest Pt layer [see\nFig. 1(b)], close to the literature value for bulk Pt25. Additionally, we observe a small\nsuperimposed S-like feature around \u00160H= 0 T, indicating the presence of an AHE like\ncontribution. To quantify this contribution, we extract the full amplitude of the S-shape\ncorresponding to an AHE like contribution \u000bAHEfrom linear \fts to \u00160H= 0 T,as indicated\nin Fig. 1(d). In the sample with tPt= 6:5 nm [Fig. 1(c)], \u000bOHEdecreases to\u000023:1p\nm=T and\nwe \fnd an increased \u000bAHE(tPt= 6:5 nm) = (\u00006\u00061)p\nm. For tPt= 2:0 nm [see Fig. 1(d)],\nwe observe \u000bOHE= 7 p\nm=T, i.e. an inversion of the sign of the OHE. Additionally, we \fnd\n\u000bAHEequal to (\u000012\u00061) p\nm. The presence of an AHE like behavior in YIG jPt samples\n3(a)\nIqn\nt \njH\nVtrans\nβH\n-2 0 2YIG|Pt(6.5nm)\nµ0H (T)-2 0 2-80-4004080\nYIG|Pt(19.5nm)ρtrans(pΩm)\nµ0H (T)-2 0 2YIG|Pt(2.0nm)\nµ0H (T)(b) (c) (d)\n2αAHE\n-80-404080\n0FIG. 1. (a)Sample and measurement geometry. (b)-(d):Transverse resistivity \u001atrans taken from\nFDMR measurements for YIG jPt bilayers with (b)tPt= 19:5 nm, (c)6:5 nm and (d)2:0 nm,\nrespectively. All data are taken at 300 K. The dashed red lines in panel (d)indicate the extraction\nof\u000bAHEfrom linear \fts to \u001atrans(H) extrapolated to \u00160H= 0 T.\ncoincides with recent reports18,26{30. However, our study of \u000bAHEas a function of platinum\nthickness and temperature in addition reveals a pronounced thickness dependence of \u000bAHE\nfortPt\u001410 nm that will be addressed below [c.f. Fig. 3(b))]. For reference, we also per-\nformed FDMR measurements on Pt thin \flms deposited directly onto diamagnetic YAG\nsubstrates. In these samples, we \fnd a similar thickness dependence of the ordinary Hall-\ne\u000bect (OHE), but no AHE-type signal22. Thus, the sign inversion of the OHE is intimately\nconnected to the Pt thin \flm regime18.\nComplementary to the FDMR experiments, we further investigate \u001atransas a function of the\nmagnetic \feld orientation (ADMR). In Fig. 2(a) we show ADMR data for a YIG jPt(2:0 nm)\nhybrid recorded at 10 K. In ADMR experiments, the OHE is expected to depend only on\nthe component H?=Hsin(\fH), i.e.,\u001a(\fH)/sin(\fH). However, our experimental data\nreveals additional higher than linear order contributions of the form Vtrans/Mn\n?, with\n\u001atrans/Asin(\fH) +Bsin3(\fH) +\u0001\u0001\u0001. A fast Fourier transformation22of the ADMR data\nsuggests the presence of sinn(\fH) contributions up to at least n= 522. However, a quantita-\ntive determination of corresponding higher order coe\u000ecients is di\u000ecult, since the amplitudes\nof the contributions for n\u00155 are below our experimental resolution of 1 p\nm. A behavior\nsimilar to that shown in Fig. 2(a) is found in all YIG jPt samples with tPt\u00145 nm, but not\nin plain Pt \flms on YAG22. To allow for simple analysis, we use\n\u001atrans=Asin(\fH) +Bsin3(\fH) (1)\n4in the following. Fits of the ADMR curves measured at di\u000berent \feld magnitudes according\nto Eq. (1) are shown as solid lines in Fig. 2(a). The magnetic \feld dependence of the \ft\nparameters AandBis shown in Figs. 2(c),(d) for two samples with tPt= 3:1 nm22and\ntPt= 2:0 nm. We disentangle magnetic \feld dependent (OHE like) and \"\feld independent\"\n(AHE like) contributions to Aby \ftting the data to A(\u00160H) =AOHE\u00160H+AAHE. As evi-\ndent from Fig. 2, the \u000bOHEand\u000bAHEvalues derived from FDMR and ADMR measurements\nare quantitatively consistent.\nTheAOHEas a function of tPtis shown in Fig. 3(a). Obviously, AOHEdeviates from the bulk\nOHE literature value25in YIGjPt bilayers with tPt\u001410 nm and also exhibits a tempera-\nture and thickness-dependent sign change for small tPt. A thickness-dependent behavior of\nthe OHE without sign change has also been reported in Ref. 18. However, these authors\nfound an increase of the OHE coe\u000ecient in the thin \flm regime, which could be due to\nthe formation of a thin, non-conductive \\dead\" Pt layer at the interface as, e.g., reported\nfor NijPt31. In contrast, we attribute the thickness dependence of the OHE in our samples\n23456750100150\n234567-40-200\nYIG|3.1nm PtYIG|2.0nm Pt\nB(pΩm)(a) (b)A(pΩm)\nµ0H (T)(c) (d)ρtrans(pΩm)\nβH -6-4-20246 0°90°180°270°360°-150-100-50050100150\n \nYIG|Pt (2.0nm) 1T\n 2T\n 4T\n 7T\nµ0H (T)µ0H (T)\nFIG. 2. (a)ADMR and (b)FDMR data of a YIG jPt sample with tPt= 2:0 nm, taken at 10 K for\ndi\u000berent\u00160H(open symbols). The dashed horizontal lines are intended as guides to the eye, to\nshow that the \u001atrans values inferred from FDMR and ADMR are consistent for identical magnetic\n\feld con\fgurations.The \fts of Eq. (1) to the data are shown as solid lines. (c)and(d)show the \ft\nparameters AandBobtained from Eq. (1) for YIG jPt(3:1 nm) (black) and YIG jPt(2:0 nm) (red)\natT= 10 K. Linear \fts to the magnetic \feld dependence of AandBare shown as solid lines.\n5solely to a modi\fcation of the Pt properties in the thin \flm regime. Further experiments\nwill be required in the future to clarify the origin of the temperature dependence of the OHE\nin YIGjPt hybrids.\nThe anomalous Hall coe\u000ecient AAHE, present only in YIG jPt hybrids, i.e., when a magnetic\ninsulator is adjacent to the NM, is depicted in Fig. 3(b). We observe a strong dependence\nofAAHEontPtsimilar to the thickness dependent magnetoresistance obtained from lon-\ngitudinal transport measurements reported earlier21, but with a sign change in AAHEbe-\ntween 100 K and 10 K. This observation agrees with recent reports of AAHE= 54 p\nm for\nYIGjPt(1:8 nm)30andAAHE= 6 p\nm for YIG jPt(3 nm)29, both taken at 10 K. Our study\nsuggests a maximum in AAHEaroundtPt= 3 nm, compatible with a complete disappearance\nofAAHEfortPt!0. This observation however is at odds with the attribution of the AHE\nin YIGjPt to a proximity MR as postulated in Ref. 29. In this case one would expect a\nmonotonous increase of the AHE signal with decreasing Pt layer thickness, and eventually a\nsaturation when the entire nonmagnetic layer is spin polarized. The absence of a proximity\nMR in our Hall data is consistent with XMCD data on similar YIG jPt samples20as well as\nother ferromagnetic insulator jNM hybrids32. However, we want to point out that a magnetic\nproximity e\u000bect has been reported in some YIG jPt samples33,34.\nWe now model our experimental \fndings in terms of the SH-AHE theory19\n\u001atrans=\u00002\u00152\u00122\nSH\ntPtGitanh2\u0000tPt\n2\u0015\u0001\n(\u001b+ 2\u0015Grcoth\u0000tPt\n\u0015\u0001\n)2mn; (2)\nwhere\u001b=\u001a\u00001is the electric conductivity of the Pt layer and mnthe unit vector of the\nprojection of the magnetization orientation monto the direction n(c.f. Fig. 1). To \ft the\nnonlinear behavior of AAHE(tPt), we combine this expression with the thickness dependence\nof the sheet resistivity for thin Pt \flms35as discussed in Ref. 23. We use the parameters\n\u0015= 1:5 nm,Gr= 4\u00021014\n\u00001m\u00002,\u0012SH(300 K) = 0 :11 and\u0012SH(10 K) = 0 :07 obtained\nfrom longitudinal SMR measurements on similar YIG jPt bilayers23. As obvious from the\nsolid lines in Fig. 3(b), Eq. (2) reproduces our thickness dependent AHE data upon using\nGi= 1\u00021013\n\u00001m\u00002for 300 K and Gi=\u00003\u00021013\n\u00001m\u00002for 10 K. For 300 K, the value\nforGinicely coincides with earlier reports21as well as theoretical calculations36. In the\nSH-AHE model, the only parameter allowing to account for the sign change in \u001atransas a\nfunction of temperature is Gi. In this picture, our AHE data thus indicate a sign change in\nGibetween 300 K and 10 K.\n6FIG. 3. (a)-(c) Field dependent (OHE-like) and \feld independent (AHE-like) Hall coe\u000ecients\nAandBproportional to sin ( \fH) and sin3(\fH), respectively, plotted versus the Pt thickness for\nT= 300 K (blue), T= 100 K (red) and T= 10 K (black). The data is obtained from ADMR\nmeasurements for YAG jPt (open symbols) and YAG jYIGjPt (full symbols). AOHEdepicted in (a)\ndescribes the conventional Hall e\u000bect, the olive dashed line corresponds to the literature value for\nbulk Pt25.(b)Thickness dependence of AAHE. The solid lines show \fts to the SH-AHE theory\nusingGi= 1\u00021013\n\u00001m\u00002forT= 300 K (blue) and Gi=\u00003\u00021013\n\u00001m\u00002forT= 10 K (black).\nPanel (c)shows the thickness dependence of the \feld independent coe\u000ecient BAHEof the sin3(\fH)\nterm.\nWe \fnally address the thickness and temperature dependence of the sin3(\fH) contribution\nparametrized by B=BAHE+BOHE\u00160H, that cannot straightforwardly be explained in a\nconventional Hall scenario. As evident from the linear \fts in Fig. 2(c), Bis nearly \feld\nindependent. A slight \feld dependence BOHE\u00141 p\nm=T might arise due to \ftting errors\ncaused by neglected higher order terms ( n\u00155). Therefore, we focus our discussion on\nthe \feld independent part BAHEin the following. BAHEexhibits a strong temperature and\nthickness dependence as shown in Fig. 3(c), suggesting a close link to AAHEand therefore\nthe SH-AHE. However, we do not observe a temperature-dependent sign change in BAHE.\nExpanding the SMR theory19to include higher order contributions of the magnetization\ndirectionsmi(i=j;t;n ) in analogy to the procedure established for the AMR of metal-\n7lic ferromagnets24,37, sin3(\fH) terms appear in \u001atrans, but with an amplitude proportional\nto\u00124\nSH. Assuming \u0012SH(Pt)\u00190:1, this would lead to BAHE=AAHE\u00190:01, which disagrees\nwith our experimental \fnding BAHE=AAHE\u00150:2. Additionally, we study the in\ruence\nof the longitudinal resistivity on \u001aAHE. For metallic ferromagnets, one usually considers\n\u001aAHE/M(H)\u001a\u000b\nlongwith 1\u0014\u000b\u0014238,39. Applying this approach to Vtransof the YIGjPt\nsamples discussed here is not possible: Since the longitudinal resistance is modulated by\nthe SMR with \u001a1=\u001a0\u001410\u0000321,\u001aAHE/\u001a\u000bwould imply BAHE=AAHE\u001410\u00003. This is in\ncontrast to our experimental \fndings. Thus, a dependence of the form \u001aAHE/\u001a\u000b\nlongcannot\naccount for our experimental observations. Finally, a static magnetic proximity e\u000bect26,33,34\nalso cannot explain BAHE, since the thickness dependence of BAHEshown in Fig. 3 (c) clearly\nindicates a decrease for tPt\u00142:5 nm. Consequently, within our present knowledge, neither a\nspin current related phenomenon (SMR, SH-AHE), nor a proximity based e\u000bect can explain\nthe origin or the magnitude of this anisotropic higher order anomalous Hall e\u000bect. We also\nwould like to point out that the higher order sin3(\fH) term can be resolved only in ADMR\nmeasurements. In conventional FDMR experiments, such higher order contributions cannot\nbe discerned.\nIn summary, we have investigated the anomalous Hall e\u000bect in YIG jPt heterostructures\nfor di\u000berent Pt thicknesses, comparing magnetization orientation dependent (ADMR) and\nmagnetic \feld magnitude dependent (FDMR) measurements at temperatures between 10 K\nand 300 K. In Pt thin \flms on diamagnetic (YAG) substrates, we observe a Pt thickness\ndependent ordinary Hall e\u000bect (OHE) only. However, in YIG jPt bilayers, an AHE like signal\nis present in addition. The AHE e\u000bect changes sign as a function of temperature and can\nbe modeled using a spin Hall magnetoresistance-type formalism for the transverse transport\ncoe\u000ecient. However, we need to assume a sign change in the imaginary part of the spin\nmixing interface conductance to describe the sign change in the anomalous Hall signal ob-\nserved experimentally. Finally, we identify contributions proportional to sin3(\fH) and higher\norders in the ADMR data for YIG jPt. The physical mechanism responsible for this behavior\ncould not be clari\fed within this work and will be subject of further investigations. The\nobservation of higher order contributions to the AHE in angle dependent magnetotransport\nmeasurements con\frms the usefulness of magnetization orientation dependent experiments.\nClearly, magnetotransport measurements as a function of the magnetic \feld magnitude only,\ni.e. for a single magnetic \feld orientation (perpendicular \feld), as usually performed to study\n8Hall e\u000bects, are not su\u000ecient to access all transverse transport features.\nWe thank T. Brenninger for technical support and M. Schreier for fruitful discussions. Fi-\nnancial support by the Deutsche Forschungsgemeinschaft via SPP 1538 (project no. GO\n944/4) is gratefully acknowledged.\nReferences\n1M. Dyakonov and V. Perel, \\Current-induced spin orientation of electrons in semiconduc-\ntors,\" Phys. Lett. A 35, 459{460 (1971).\n2J. E. Hirsch, \\Spin hall e\u000bect,\" Phys. Rev. Lett. 83, 1834{1837 (1999).\n3Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, \\Observation of the spin\nhall e\u000bect in semiconductors,\" Science 306, 1910 {1913 (2004).\n4S. O. Valenzuela and M. Tinkham, \\Direct electronic measurement of the spin hall e\u000bect,\"\nNature 442, 5 (2006).\n5E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, \\Conversion of spin current into charge\ncurrent at room temperature: Inverse spin-Hall e\u000bect,\" Appl. Phys. Lett. 88, 182509\n(2006).\n6Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, \\Spin pumping and magnetization\ndynamics in metallic multilayers,\" Physical Review B 66, 10 (2002).\n7K. Xia, P. Kelly, G. E. W. Bauer, A. Brataas, and I. Turek, \\Spin torques in\nferromagnetic/normal-metal structures,\" Phys. Rev. B 65, 220401 (2002).\n8D. Huertas Hernando, Y. Nazarov, A. Brataas, and G. E. W. Bauer, \\Conductance modu-\nlation by spin precession in noncollinear ferromagnet normal-metal ferromagnet systems,\"\nPhys.l Rev. B 62, 5700 (2000).\n9A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, \\Finite-element theory of transport in\nferromagnet\u0015normal metal systems,\" Phys. Rev. Lett. 84, 2481{2484 (2000).\n10M. D. Stiles and A. Zangwill, \\Anatomy of spin-transfer torque,\" Phys. Rev. B 66, 014407\n(2002).\n11Z. Wang, Y. Sun, M. Wu, V. Tiberkevich, and A. Slavin, \\Control of spin waves in a thin\n\flm ferromagnetic insulator through interfacial spin scattering,\" Phys. Rev. Lett. 107,\n146602 (2011).\n912E. Padron-Hernandez, A. Azevedo, and S. M. Rezende, \\Ampli\fcation of spin waves in\nyttrium iron garnet \flms through the spin hall e\u000bect,\" Appl. Phys. Lett. 99, 192511 (2011).\n13D. C. Ralph and M. D. Stiles, \\Spin transfer torques,\" J. MMM 320, 1190{1216 (2008).\n14Z. Wang, Y. Sun, Y.-Y. Song, M. Wu, H. Schulthei\u0019, J. E. Pearson, and A. Ho\u000bmann,\n\\Electric control of magnetization relaxation in thin \flm magnetic insulators,\" Appl. Phys.\nLett. 99, 162511 (2011).\n15Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa,\nH. Kawai, K. Ando, K. Takanashi, S. Maekawa, and E. Saitoh, \\Transmission of electrical\nsignals by spin-wave interconversion in a magnetic insulator.\" Nature 464, 262{266 (2010).\n16F. D. Czeschka, L. Dreher, M. S. Brandt, M. Weiler, M. Althammer, I. Imort, G. Reiss,\nA. Thomas, W. Schoch, W. Limmer, H. Huebl, R. Gross, and S. T. B. Goennenwein,\n\\Scaling behavior of the spin pumping e\u000bect in Ferromagnet-Platinum bilayers,\" Phys.\nRev. Lett. 107, 046601 (2011).\n17H. Nakayama, M. Althammer, Y.-T. Chen, K. Uchida, Y. Kajiwara, D. Kikuchi, T. Ohtani,\nS. Gepr ags, M. Opel, S. Takahashi, R. Gross, G. E. W. Bauer, S. T. B. Goennenwein, and\nE. Saitoh, \\Spin hall magnetoresistance induced by a nonequilibrium proximity e\u000bect,\"\nPhys. Rev. Lett. 110, 206601 (2013).\n18N. Vlietstra, J. Shan, V. Castel, J. Ben Youssef, G. E. W. Bauer, and B. J. v. Wees,\n\\Exchange magnetic \feld torques in yig/pt bilayers observed by the spin-hall magnetore-\nsistance,\" Appl. Phys. Lett. 103, 032401 (2013).\n19Y.-T. Chen, S. Takahashi, H. Nakayama, M. Althammer, S. T. B. Goennenwein, E. Saitoh,\nand G. E. W. Bauer, \\Theory of spin hall magnetoresistance,\" Phys. Rev. B 87, 144411\n(2013).\n20S. Gepr ags, S. Meyer, S. Altmannshofer, M. Opel, F. Wilhelm, A. Rogalev, R. Gross, and\nS. T. B. Goennenwein, \\Investigation of induced pt magnetic polarization in pt/y3fe5o12\nbilayers,\" Appl. Phys. Lett. 101, 262407 (2012).\n21M. Althammer, S. Meyer, H. Nakayama, M. Schreier, S. Altmannshofer, M. Weiler,\nH. Huebl, S. Gepr ags, M. Opel, R. Gross, D. Meier, C. Klewe, T. Kuschel, J.-M. Schmal-\nhorst, G. Reiss, L. Shen, A. Gupta, Y.-T. Chen, G. E. W. Bauer, E. Saitoh, and S. T. B.\nGoennenwein, \\Quantitative study of the spin hall magnetoresistance in ferromagnetic\ninsulator/normal metal hybrids,\" Phys. Rev. B 87, 224401 (2013).\n22See supplemental material at [URL will be inserted by publisher] for details.\n1023S. Meyer, M. Althammer, S. Gepr ags, M. Opel, R. Gross, and S. T. B. Goennenwein,\n\\Temperature dependent spin transport properties of platinum inferred from spin hall\nmagnetoresistance measurements,\" App. Phys. Lett. 104, 242411 (2014).\n24W. Limmer, M. Glunk, J. Daeubler, T. Hummel, W. Schoch, R. Sauer, C. Bihler, H. Huebl,\nM. S. Brandt, and S. T. B. Goennenwein, \\Angle-dependent magnetotransport in cubic\nand tetragonal ferromagnets: Application to (001)- and (113)A-oriented (Ga,Mn)As,\"\nPhys. Rev. B 74, 205205 (2006).\n25P. Gehlho\u000b, E. Justi, and M. Kohler, \\Der hall-e\u000bekt von iridium,\" Z. Naturforschg. 5a,\n16{18 (1950).\n26S. Y. Huang, X. Fan, D. Qu, Y. P. Chen, W. G. Wang, J. Wu, T. Y. Chen, J. Q. Xiao,\nand C. L. Chien, \\Transport magnetic proximity e\u000bects in platinum,\" Phys. Rev. Lett.\n109, 107204 (2012).\n27S. Shimizu, K. S. Takahashi, T. Hatano, M. Kawasaki, Y. Tokura, and Y. Iwasa, \\Electri-\ncally tunable anomalous hall e\u000bect in pt thin \flms,\" Phys. Rev. Lett. 111, 216803 (2013).\n28D. Qu, S. Y. Huang, B. F. Miao, S. X. Huang, and C. L. Chien, \\Self-consistent deter-\nmination of spin hall angles in selected 5 dmetals by thermal spin injection,\" Phys. Rev.\nB89, 140407 (2014).\n29B. F. Miao, S. Y. Huang, D. Qu, and C. L. Chien, \\Physical origins of the new magne-\ntoresistance in pt/yig,\" Phys. Rev. Lett. 112, 236601 (2014).\n30Y. Shiomi, T. Ohtani, S. Iguchi, T. Sasaki, Z. Qiu, H. Nakayama, K. Uchida, and\nE. Saitoh, \\Interface-dependent magnetotransport properties for thin pt \flms on ferri-\nmagnetic y3fe5o12,\" Appl. Phys. Lett. 104, 242406 (2014).\n31S.-C. Shin, G. Srinivas, Y.-S. Kim, and M.-G. Kim, \\Observation of perpendicular mag-\nnetic anisotropy in ni/pt multilayers at room temperature,\" Appl. Phys. Lett. 73, 393{395\n(1998).\n32D. K. Satapathy, M. A. Uribe-Laverde, I. Marozau, V. K. Malik, S. Das, T. Wagner,\nC. Marcelot, J. Stahn, S. Br uck, A. R uhm, S. Macke, T. Tietze, E. Goering, A. Fra~ n\u0013 o, J. H.\nKim, M. Wu, E. Benckiser, B. Keimer, A. Devishvili, B. P. Toperverg, M. Merz, P. Nagel,\nS. Schuppler, and C. Bernhard, \\Magnetic proximity e\u000bect in yba2cu3o7=la2=3ca1=3mno 3\nandyba2cu3o7=lamn 3+\u000esuperlattices,\" Phys. Rev. Lett. 108, 197201 (2012).\n33Y. M. Lu, Y. Choi, C. M. Ortega, M. X. Cheng, J. W. Cai, S. Y. Huang, L. Sun, and\nC. L. Chien, \\Pt magnetic polarization on y3fe5o12 and magnetotransport characteristics,\"\n11Phys. Rev. Lett. 110, 147207 (2013).\n34Y. M. Lu, J. W. Cai, S. Y. Huang, D. Qu, B. F. Miao, and C. L. Chien, \\Hybrid\nmagnetoresistance in the proximity of a ferromagnet,\" Phys. Rev. B 87, 220409 (2013).\n35G. Fischer, H. Ho\u000bmann, and J. Vancea, \\Mean free path and density of conductance\nelectrons in platinum determined by the size e\u000bect in extremely thin \flms,\" Phys. Rev. B\n22, 6065{6073 (1980).\n36X. Jia, K. Liu, K. Xia, and G. E. W. Bauer, \\Spin transfer torque on magnetic insulators,\"\nEurophys. Lett. 96, 17005 (2011).\n37W. Limmer, J. Daeubler, L. Dreher, M. Glunk, W. Schoch, S. Schwaiger, and R. Sauer,\n\\Advanced resistivity model for arbitrary magnetization orientation applied to a series of\ncompressive- to tensile-strained (Ga,Mn)As layers,\" Phys. Rev. B 77, 205210 (2008).\n38R. Karplus and J. M. Luttinger, \\Hall e\u000bect in ferromagnetics,\" Phys. Rev. 95, 1154{1160\n(1954).\n39L. Berger, \\Side-jump mechanism for the hall e\u000bect of ferromagnets,\" Phys. Rev. B 2,\n4559{4566 (1970).\n40O. Panchenko, P. Lutsishin, and Y. G. Ptushinskii, \\Galvanomagnetic e\u000bects in thin \flms\nof some transition metals,\" JETP 29, 134 (1969).\n12Supplemental Materials: Anomalous Hall e\u000bect in YIG jPt\nbilayers\nI Reference measurements in YAG jPt bilayers\nHere, we discuss the reference samples consisting of plain Pt thin \flms on single-crystalline\ndiamagnetic Yttrium Aluminum Garnet (YAG). Figures S 4(b),(d),(f) show the character-\nistic linear behavior for \u001atrans(H)/H, i.e., an ordinary Hall e\u000bect, without any AHE\ncontribution. Extracting the ordinary Hall coe\u000ecient \u000bOHE =@\u001atrans(H)=@(H) from the\nslope, we obtain \u000bOHE(tPt= 2:0 nm) =\u00003:1 p\nm=T,\u000bOHE(tPt= 3:5 nm) =\u000015:9 p\nm=T,\nand\u000bOHE(tPt= 15:6 nm) =\u000023:1 p\nm=T with a systematic error of \u0001 \u000bOHE= 0:1 p\nm=T.\nWhile\u000bOHEof the thickest Pt \flm with tPt= 15:6 nm is consistent with the literature value\n\u000bOHE=\u000024:4 p\nm=T25, we \fnd signi\fcantly smaller OHE coe\u000ecients for the 3 :5 nm and\nthe 2:0 nm thick Pt \flm. This behavior in the thin \flm regime ( tPt\u001410 nm) agrees with\nearlier reports40. Measurements of \u000bOHE(T) show aTindependent \u000bOHEand the absence\nof any AHE like contribution. Complementary to the FDMR experiments, we investigate\n\u001atransas a function of the magnetic \feld orientation (ADMR). As evident from Fig. S 4(c)\nand (e), we obtain that the OHE depends only on the component H?=Hsin(\fH), i.e.,\n\u001atrans(\fH)/sin(\fH).\nII Magnetization Orientation and Field Magnitude Dependent Measurements for\nYIGjPt(3:1 nm )\nIn Fig. S 5(a) we show a set of ADMR data for a YIG jPt (3:1 nm) sample taken at 10 K as\na reference to Fig. 2 in the main text. This additional data substantiates the reproducibility\nof the observation of higher order contributions to \u001atrans up to at least n= 3 in a set\nof ADMR measurements [see Fig. S 5]. Please note that the data shown in Fig. S 5(a) is\ntaken at 10 K, while the FDMR measurements performed on similar samples shown in Fig. 1\nwere taken at 300 K and thus have a di\u000berent OHE and AHE behavior. In particular, for\nT= 10 K, we observe an almost vanishing OHE signal in this sample, \u000bOHE= 6 p\nm=T and\ntherefor the sin3(\fH) contribution becomes prominent even for the 7 T data, which otherwise\nwould be overwhelmed by the sin( \fH) characteristic of the OHE. The \ftting parameters A\n13(a)\nIqn\nt \njH\nVtrans\nβH(c) (d)\n(e) (f)-80-4004080ρtrans(pΩm)\n  \n1T3T\nYAG|Pt(3.5nm)\n-2 0 2 0° 180°360°-80-4004080ρtrans(pΩm)\nµ0H (T)  \nβH YAG|Pt(15.6nm)1T3T\n-80-4004080ρtrans(pΩm)\n \nYAG|Pt(2.0nm)(b)\n-2 0 2\nµ0H (T)FIG. 4. (a)Sample and measurement geometry. (b)Transverse resistivity \u001atrans taken from a\nFDMR measurement for YAG jPt (2:0 nm). (c)\u001atrans as a function of \fHfor YAGjPt (3:5 nm).\n(d)Corresponding FDMR data for \fH= 90\u000e.(e, f) : ADMR and FDMR measurements for\ntPt= 15:6 nm on YAG. The colored, horizontal, dashed lines in panels (c,d) and (e,f) are intended\nas guides to the eye, to show that the \u001atransvalues inferred from FDMR and ADMR are consistent\nfor identical magnetic \feld con\fgurations. All data taken at 300 K.\n0° 180° 360°-20-1001020\n  ρtrans (pΩm)\n 2T\n 4T\n 7TYIG|3.1nm Pt\n02468100.1110\nFrequenc y (1/360° )Amplitude (pΩm)\n7T\n10K\nβH (a) (b)\nFIG. 5. (a)Transverse resistivity \u001atrans as a function of \fHfor a YIGjPt bilayer with tPt= 3:1 nm,\ntaken at 10 K. (b)Fast Fourier transform (FFT) of the ADMR data taken at T= 10 K with \u00160H=\n7 T for the YIGjPt(3:1 nm) sample shown in (a). The dashed line indicates the experimental noise\nlevel of 1 p\nm.\nandBobtained from \fts of Eq.(1) to the ADMR data shown in S 5(a) for YIG jPt (3:1 nm)\nare represented by the black data points in Fig. 2(c) and (d) in the main article. For a full\n14picture of the temperature dependence of OHE and AHE contributions to the parameters\nAandB, we refer to Fig. 3 in the main text.\nIII Fast Fourier Transform\nAs shown in Fig. S 5(a) and in Fig. 2(a) in the main text, our magnetization orientation\ndependent measurements on YIG jPt bilayers reveal additional higher order contributions\nto\u001atrans, such that we can formulate \u001atrans/Asin(\fH) +Bsin3(\fH) +\u0001\u0001\u0001. To specify the\nparticular contributions, we perform fast Fourier transformations (FFT) of the ADMR data\nas exemplarily shown in Fig. S 5(b) for YIG jPt(3:1 nm) taken at 10 K [see Fig. S 5(a)]. For\nthe FFT, we use a rectangular window with amplitude correction. The amplitude spectrum\nof the FFT for this set of data reveals the presence of sin( n\fH) contributions up to at least\nn= 5. Possibly occurring higher order contributions could not be quanti\fed, since the\namplitude for the n= 5 contribution is already comparable to our experimental resolution\nof 1 p\nm.\nPlease note that the FFT results depicted in S 5(b) are not sign-sensitive and can not\nstraightforwardly be compared to results for AandBobtained from \fts using Eq. (2). The\nFFT algorithm speci\fes frequency components proportional to sin( n\fH), while our approx-\nimation in Eq. (2) is a power series proportional to sinn(\fH). However, both expressions\nrepresent the same phenomenology and can be transformed into the respective other by\nfundamental algebra.\n15IV Table of Samples\nA detailed information on the \flm thicknesses for both types of thin \flm structures\nused in our study is listed in Tab. S I. The parameter hrepresents the surface roughness of\nPt obtained from high-resolution X-ray re\rectometry. For YIG jPt bilayers, we determine\nan averaged surface roughness of h= (0:7\u00060:2) nm, while for plain Pt on diamagnetic\nsubstrate, we obtain a slightly lower value of h= (0:5\u00060:1) nm. However, within the\nestimated errors, the interface roughnesses of both types of samples are comparable and\nthus we expect no in\ruence of the surface roughnesses on our OHE and AHE data.\nsubstratetYIG(nm)tPt(nm)h(nm)\nYAG 34 0.8 0.7\nYAG 56 3.1 1.0\nYAG 38 1.2 0.9\nYAG 63 6.5 0.9\nYAG 57 2.0 0.8\nGGG 61 11.1 0.6\nYAG 49 2.0 0.6\nYAG 61 19.5 1.0\nYAG 58 2.5 1.1\nYAG 0 2.0 0.4\nYAG 0 15.6 0.6\nYAG 0 3.5 0.5\nTABLE I. Substrate material, YIG thickness tYIG, platinum thickness tPtand platinum roughness\nhfor all samples investigated in this work.\nReferences\n1M. Dyakonov and V. Perel, \\Current-induced spin orientation of electrons in semiconduc-\ntors,\" Phys. Lett. A 35, 459{460 (1971).\n2J. E. Hirsch, \\Spin hall e\u000bect,\" Phys. Rev. Lett. 83, 1834{1837 (1999).\n163Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, \\Observation of the spin\nhall e\u000bect in semiconductors,\" Science 306, 1910 {1913 (2004).\n4S. O. Valenzuela and M. Tinkham, \\Direct electronic measurement of the spin hall e\u000bect,\"\nNature 442, 5 (2006).\n5E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, \\Conversion of spin current into charge\ncurrent at room temperature: Inverse spin-Hall e\u000bect,\" Appl. Phys. Lett. 88, 182509\n(2006).\n6Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, \\Spin pumping and magnetization\ndynamics in metallic multilayers,\" Physical Review B 66, 10 (2002).\n7K. Xia, P. Kelly, G. E. W. Bauer, A. Brataas, and I. Turek, \\Spin torques in\nferromagnetic/normal-metal structures,\" Phys. Rev. B 65, 220401 (2002).\n8D. Huertas Hernando, Y. Nazarov, A. Brataas, and G. E. W. Bauer, \\Conductance modu-\nlation by spin precession in noncollinear ferromagnet normal-metal ferromagnet systems,\"\nPhys.l Rev. B 62, 5700 (2000).\n9A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, \\Finite-element theory of transport in\nferromagnet\u0015normal metal systems,\" Phys. Rev. Lett. 84, 2481{2484 (2000).\n10M. D. Stiles and A. Zangwill, \\Anatomy of spin-transfer torque,\" Phys. Rev. B 66, 014407\n(2002).\n11Z. Wang, Y. Sun, M. Wu, V. Tiberkevich, and A. Slavin, \\Control of spin waves in a thin\n\flm ferromagnetic insulator through interfacial spin scattering,\" Phys. Rev. Lett. 107,\n146602 (2011).\n12E. Padron-Hernandez, A. Azevedo, and S. M. Rezende, \\Ampli\fcation of spin waves in\nyttrium iron garnet \flms through the spin hall e\u000bect,\" Appl. Phys. Lett. 99, 192511 (2011).\n13D. C. Ralph and M. D. Stiles, \\Spin transfer torques,\" J. MMM 320, 1190{1216 (2008).\n14Z. Wang, Y. Sun, Y.-Y. Song, M. Wu, H. Schulthei\u0019, J. E. Pearson, and A. Ho\u000bmann,\n\\Electric control of magnetization relaxation in thin \flm magnetic insulators,\" Appl. Phys.\nLett. 99, 162511 (2011).\n15Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa,\nH. Kawai, K. Ando, K. Takanashi, S. Maekawa, and E. Saitoh, \\Transmission of electrical\nsignals by spin-wave interconversion in a magnetic insulator.\" Nature 464, 262{266 (2010).\n16F. D. Czeschka, L. Dreher, M. S. Brandt, M. Weiler, M. Althammer, I. Imort, G. Reiss,\nA. Thomas, W. Schoch, W. Limmer, H. Huebl, R. Gross, and S. T. B. Goennenwein,\n17\\Scaling behavior of the spin pumping e\u000bect in Ferromagnet-Platinum bilayers,\" Phys.\nRev. Lett. 107, 046601 (2011).\n17H. Nakayama, M. Althammer, Y.-T. Chen, K. Uchida, Y. Kajiwara, D. Kikuchi, T. Ohtani,\nS. Gepr ags, M. Opel, S. Takahashi, R. Gross, G. E. W. Bauer, S. T. B. Goennenwein, and\nE. Saitoh, \\Spin hall magnetoresistance induced by a nonequilibrium proximity e\u000bect,\"\nPhys. Rev. Lett. 110, 206601 (2013).\n18N. Vlietstra, J. Shan, V. Castel, J. Ben Youssef, G. E. W. Bauer, and B. J. v. Wees,\n\\Exchange magnetic \feld torques in yig/pt bilayers observed by the spin-hall magnetore-\nsistance,\" Appl. Phys. Lett. 103, 032401 (2013).\n19Y.-T. Chen, S. Takahashi, H. Nakayama, M. Althammer, S. T. B. Goennenwein, E. Saitoh,\nand G. E. W. Bauer, \\Theory of spin hall magnetoresistance,\" Phys. Rev. B 87, 144411\n(2013).\n20S. Gepr ags, S. Meyer, S. Altmannshofer, M. Opel, F. Wilhelm, A. Rogalev, R. Gross, and\nS. T. B. Goennenwein, \\Investigation of induced pt magnetic polarization in pt/y3fe5o12\nbilayers,\" Appl. Phys. Lett. 101, 262407 (2012).\n21M. Althammer, S. Meyer, H. Nakayama, M. Schreier, S. Altmannshofer, M. Weiler,\nH. Huebl, S. Gepr ags, M. Opel, R. Gross, D. Meier, C. Klewe, T. Kuschel, J.-M. Schmal-\nhorst, G. Reiss, L. Shen, A. Gupta, Y.-T. Chen, G. E. W. Bauer, E. Saitoh, and S. T. B.\nGoennenwein, \\Quantitative study of the spin hall magnetoresistance in ferromagnetic\ninsulator/normal metal hybrids,\" Phys. Rev. B 87, 224401 (2013).\n22See supplemental material at [URL will be inserted by publisher] for details.\n23S. Meyer, M. Althammer, S. Gepr ags, M. Opel, R. Gross, and S. T. B. Goennenwein,\n\\Temperature dependent spin transport properties of platinum inferred from spin hall\nmagnetoresistance measurements,\" App. Phys. Lett. 104, 242411 (2014).\n24W. Limmer, M. Glunk, J. Daeubler, T. Hummel, W. Schoch, R. Sauer, C. Bihler, H. Huebl,\nM. S. Brandt, and S. T. B. Goennenwein, \\Angle-dependent magnetotransport in cubic\nand tetragonal ferromagnets: Application to (001)- and (113)A-oriented (Ga,Mn)As,\"\nPhys. Rev. B 74, 205205 (2006).\n25P. Gehlho\u000b, E. Justi, and M. Kohler, \\Der hall-e\u000bekt von iridium,\" Z. Naturforschg. 5a,\n16{18 (1950).\n26S. Y. Huang, X. Fan, D. Qu, Y. P. Chen, W. G. Wang, J. Wu, T. Y. Chen, J. Q. Xiao,\nand C. L. Chien, \\Transport magnetic proximity e\u000bects in platinum,\" Phys. Rev. Lett.\n18109, 107204 (2012).\n27S. Shimizu, K. S. Takahashi, T. Hatano, M. Kawasaki, Y. Tokura, and Y. Iwasa, \\Electri-\ncally tunable anomalous hall e\u000bect in pt thin \flms,\" Phys. Rev. Lett. 111, 216803 (2013).\n28D. Qu, S. Y. Huang, B. F. Miao, S. X. Huang, and C. L. Chien, \\Self-consistent deter-\nmination of spin hall angles in selected 5 dmetals by thermal spin injection,\" Phys. Rev.\nB89, 140407 (2014).\n29B. F. Miao, S. Y. Huang, D. Qu, and C. L. Chien, \\Physical origins of the new magne-\ntoresistance in pt/yig,\" Phys. Rev. Lett. 112, 236601 (2014).\n30Y. Shiomi, T. Ohtani, S. Iguchi, T. Sasaki, Z. Qiu, H. Nakayama, K. Uchida, and\nE. Saitoh, \\Interface-dependent magnetotransport properties for thin pt \flms on ferri-\nmagnetic y3fe5o12,\" Appl. Phys. Lett. 104, 242406 (2014).\n31S.-C. Shin, G. Srinivas, Y.-S. Kim, and M.-G. Kim, \\Observation of perpendicular mag-\nnetic anisotropy in ni/pt multilayers at room temperature,\" Appl. Phys. Lett. 73, 393{395\n(1998).\n32D. K. Satapathy, M. A. Uribe-Laverde, I. Marozau, V. K. 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Jia, K. Liu, K. Xia, and G. E. W. Bauer, \\Spin transfer torque on magnetic insulators,\"\nEurophys. Lett. 96, 17005 (2011).\n37W. Limmer, J. Daeubler, L. Dreher, M. Glunk, W. Schoch, S. Schwaiger, and R. Sauer,\n\\Advanced resistivity model for arbitrary magnetization orientation applied to a series of\ncompressive- to tensile-strained (Ga,Mn)As layers,\" Phys. Rev. B 77, 205210 (2008).\n1938R. Karplus and J. M. Luttinger, \\Hall e\u000bect in ferromagnetics,\" Phys. Rev. 95, 1154{1160\n(1954).\n39L. Berger, \\Side-jump mechanism for the hall e\u000bect of ferromagnets,\" Phys. Rev. B 2,\n4559{4566 (1970).\n40O. Panchenko, P. Lutsishin, and Y. G. Ptushinskii, \\Galvanomagnetic e\u000bects in thin \flms\nof some transition metals,\" JETP 29, 134 (1969).\n20"    },    {        "title": "1706.04373v1.Nonlocal_magnon_polaron_transport_in_yttrium_iron_garnet.pdf",        "content": "Nonlocal magnon-polaron transport in yttrium iron garnet\nL.J. Cornelissen,1,\u0003K. Oyanagi,2,\u0003T. Kikkawa,2, 3Z. Qiu,3T.\nKuschel,1G.E.W. Bauer,1, 2, 3, 4B.J. van Wees,1and E. Saitoh2, 3, 4, 5\n1Physics of Nanodevices, Zernike Institute for Advanced Materials,\nUniversity of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlandsy\n2Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan\n3WPI Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan\n4Center for Spintronics Research Network, Tohoku University, Sendai 980-8577, Japan\n5Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan\nThe spin Seebeck e\u000bect (SSE) is observed in magnetic insulator jheavy metal bilayers as an inverse\nspin Hall e\u000bect voltage under a temperature gradient. The SSE can be detected nonlocally as\nwell, viz. in terms of the voltage in a second metallic contact (detector) on the magnetic \flm,\nspatially separated from the \frst contact that is used to apply the temperature bias (injector).\nMagnon-polarons are hybridized lattice and spin waves in magnetic materials, generated by the\nmagnetoelastic interaction. Kikkawa et al. [Phys. Rev. Lett. 117, 207203 (2016)] interpreted a\nresonant enhancement of the local SSE in yttrium iron garnet (YIG) as a function of the magnetic\n\feld in terms of magnon-polaron formation. Here we report the observation of magnon-polarons in\nnonlocal magnon spin injection/detection devices for various injector-detector spacings and sample\ntemperatures. Unexpectedly, we \fnd that the magnon-polaron resonances can suppress rather\nthan enhance the nonlocal SSE. Using \fnite element modelling we explain our observations as a\ncompetition between the SSE and spin di\u000busion in YIG. These results give unprecedented insights\ninto the magnon-phonon interaction in a key magnetic material.\nWhen sound travels through a magnet the local dis-\ntortions of the lattice exert torques on the magnetic\norder due to the magnetoelastic coupling1. By reci-\nprocity, spin waves in a magnet a\u000bect the lattice dy-\nnamics. The coupling between spin and lattice waves\n(magnons and phonons) has been intensively researched\nin the last half century2,3. Yttrium iron garnet (YIG) has\nbeen a singularly useful material here, because it can be\ngrown with exceptional magnetic and acoustic quality2.\nMagnons and phonons hybridize at the (anti)crossing of\ntheir dispersion relations, a regime that has attracted\nrecent attention4{10. When the quasiparticle lifetime-\nbroadening is smaller than the interaction strength, the\nstrong coupling regime is reached; the resulting fully\nmixed quasiparticles have been referred to as magnon-\npolarons6,7.\nIn spite of the long history and ubiquity of the magnon-\nphonon interaction, it still leads to surprises. Evidence of\na sizeable magnetoelastic coupling in YIG was recently\nfound in experiments on spin caloritronic e\u000bects, i.e. the\nspin Peltier11and spin Seebeck e\u000bect12,13(SPE and SSE\nrespectively). Recently, Kikkawa et al. showed that the\nhybridization of magnons and phonons can lead to a res-\nonant enhancement of the local SSE in YIG9. Bozhko\net al. found that this hybridization can play a role\nin the thermalization of parametrically excited magnons\nusing Brillouin light scattering. They observed an ac-\ncumulation of magnon-polarons in the spectral region\nnear the anticrossing between the magnon and trans-\nverse acoustic phonon modes14. However, these previous\nexperiments did not address the transport properties of\nmagnon-polarons.\nNonlocal spin injection and detection experiments are\nof great importance in probing the transport of spin inmetals15, semiconductors16and graphene17. Varying the\ndistance between the spin injection and detection con-\ntacts allows for the accurate determination of the trans-\nport properties of the spin information carriers in the\nchannel, such as the spin relaxation length18. Recently,\nit was shown that this kind of experiments are not limited\nto (semi)conducting materials, but can also be performed\non magnetic insulators19, where the spin information is\ncarried by magnons. Such nonlocal magnon spin trans-\nport experiments have provided additional insights in the\nproperties of magnons in YIG, for instance by studying\nthe transport as a function of temperature20{23or exter-\nnal magnetic \feld24. Finally, the nonlocal magnon spin\ninjection/detection scheme can play a role in the develop-\nment of e\u000ecient magnon spintronic devices, for example\nmagnon based logic gates25,26. In this study, we make\nuse of nonlocal magnon spin injection and detection de-\nvices to investigate the transport of magnon-polarons in\nYIG.\nMagnons can be excited magnetically using the oscil-\nlating magnetic \feld generated by a microwave frequency\nac current25, or electrically using a dc current in an ad-\njacent material with a large spin Hall angle, such as\nplatinum19. Finally, they can be generated thermally\nby the SSE27{30, in which a thermal gradient in the mag-\nnetic insulator drives a magnon spin current parallel to\nthe induced heat current.\nThe generation of magnons via the SSE can be de-\ntected in several con\fgurations: First, the heater-induced\ncon\fguration (hiSSE)31, which consists of a bilayer\nYIGjheavy metal sample that is subject to external\nPeltier elements to apply a temperature gradient nor-\nmal to the plane of the sample. The SSE then gener-\nates a voltage across the heavy metal \flm (explained inarXiv:1706.04373v1  [cond-mat.mes-hall]  14 Jun 20172\nmore detail below), which can be recorded. Second, the\ncurrent-induced con\fguration (ciSSE)28,32in which the\nheavy metal detector used to detect the SSE voltage is si-\nmultaneously used as a heater. A current is sent through\nthe heavy metal \flm, creating a temperature gradient in\nthe YIG due to Joule heating. Due to this temperature\ngradient, the SSE generates a voltage across the heavy\nmetal \flm, which can again be recorded. Third, the non-\nlocal SSE (nlSSE)19,33, in which a current is sent through\na narrow heavy metal strip to generate a thermal gradi-\nent via Joule heating as well. However, the SSE signal\nresulting from this thermal gradient is detected in a sec-\nond heavy metal strip, located some distance away from\nthe injector.\nIn the nlSSE, the magnons responsible for generating a\nsignal in the detector strip are generated in the injector\nvicinity and then di\u000buse through the magnetic insula-\ntor to the detector. The temperature gradient under-\nneath a detector located several microns to tens of mi-\ncrons from the injector does not contribute signi\fcantly\nto the measured voltage23,34. In contrast, the hiSSE and\nciSSE always have a signi\fcant temperature gradient di-\nrectly underneath the detector. The hiSSE and ciSSE are\ntherefore local SSE con\fgurations, contrary to the nlSSE\nwhich is nonlocal.\nIn all three con\fgurations, the resulting voltage across\nthe heavy metal \flm is due to magnons which are ab-\nsorbed at the YIG jdetector interface, causing spin-\rip\nscattering of conduction electrons and generating a spin\ncurrent and spin accumulation in the detector. Due to\nthe inverse spin Hall e\u000bect35, this spin accumulation is\nconverted into a charge voltage that is measured.\nAt speci\fc values for the external magnetic \feld, the\nphonon dispersion is tangent to that of the magnons and\nthe magnon and phonon modes are strongly coupled over\na relatively large region in momentum space. At these\nresonant magnetic \feld values, the e\u000bect of the magne-\ntoelastic coupling is at its strongest and magnon-polarons\nare formed e\u000eciently. If the acoustic quality of the YIG\n\flm is better than the magnetic one, magnon-polaron\nformation leads to an enhancement in the hiSSE signal\nat the resonant magnetic \feld9. This enhancement is at-\ntributed to an increase in the e\u000bective bulk spin Seebeck\ncoe\u000ecient\u0010, which governs the generation of magnon\nspin current by a temperature gradient in the magnet.\nThis was demonstrated experimentally by measuring the\nspin Seebeck voltage in the hiSSE con\fguration9, estab-\nlishing the role of magnon-polarons in the thermal gen-\neration of magnon spin current.\nHere we make use of the nlSSE con\fguration to di-\nrectly probe not only the generation, but also the trans-\nport of magnon-polarons. We show that in the YIG sam-\nples under investigation not only \u0010, but also the magnon\nspin conductivity \u001bmis resonantly enhanced by the hy-\nbridization of magnons and phonons, which leads to sig-\nnatures in the nonlocal magnon spin transport signals\nclearly distinct from the hiSSE observations. Notably,\nresonant features in nonlocal transport experiments havevery recently been theoretically predicted by Flebus et\nal.10, who calculated the in\ruence of magnon-polarons\non the YIG transport parameters such as the magnon\nspin and heat conductivity and the magnon spin di\u000bu-\nsion length.\nI. RESULTS\nA. Sample characteristics\nOur nonlocal devices consist of multiple narrow, thin\nplatinum strips (typical dimensions are 100 \u0016m\u0002100\nnm\u000210 nm [l\u0002w\u0002t]) deposited on top of a YIG thin\n\flm and separated from each other by a centre-to-centre\ndistanced. We have performed measurements of non-\nlocal devices on YIG \flms from Groningen and Sendai,\nboth of which are grown by liquid phase epitaxy on a\nGGG substrate. The YIG \flm thickness is 210 nm (2.5\n\u0016m) for YIG from Groningen (Sendai). In Sendai, \fve\nbatches of devices where investigated (sample S1 to S4)\non pieces cut from the same YIG wafer. In Groningen,\ntwo batches of devices where investigated (G1 and G2).\nThe platinum strips are contacted using Ti/Au contacts\n(see Methods for fabrication details) Figure 1a shows an\noptical microscope image of a typical device, with the\nelectrical connections indicated schematically. The cen-\ntral strip functions as a magnon injector while the two\nouter strips are magnon detectors, measuring the nonlo-\ncal signal at di\u000berent distances from the injector.\nB. Experimental results\nA nonlocal signal is generated in the devices by passing\na low frequency ac current I(!) (typically w=(2\u0019)<20\nHz andIrms= 100\u0016A) through the injector. This leads\nto both thermal and electrical generation of magnons,\nas outlined above. The voltage that is due to the ther-\nmally generated magnons is proportional to the excita-\ntion current squared, and hence can be directly detected\nin the second harmonic detector response V(2!) (i.e. the\nvoltage measured at twice the excitation frequency). Si-\nmultaneously, the voltage due to electrically generated\nmagnons can be measured in the \frst harmonic response\nV(1!)19. The sample is placed in an external magnetic\n\feldH, under an angle \u000b= 90\u000eto the injector/detector\nstrips.\nFigure 1b shows the results of two typical nonlo-\ncal measurements at di\u000berent distances, in which \u00160H\nis varied from\u00003:0 to 3:0 T. Several distinct features\ncan be seen in these results. As the magnetic \feld is\nswept through zero, the YIG magnetization and hence\nthe magnon spin polarization change direction, since a\nmagnon always carries a spin opposite to the majority\nspin in the magnet. This causes a reversal of the polar-\nization of the spin current absorbed by the detector and3\nconsequently the voltage VnlSSE changes sign. Addition-\nally,VnlSSE for short distance d(Figure 1b bottom pan-\nels) shows an opposite sign compared to VnlSSE for long\ndistance (Figure 1b top panels). This sign-reversal for\nshort distances is a characteristic feature of the nlSSE19\nthat has so far been observed to depend on both the\nthickness of the YIG \flm tYIG(roughly speaking, at room\ntemperature when d<t YIGthe sign will be opposite to\nthat ford > t YIG33) as well as the sample temperature,\nwhere a lower temperature reduces the distance at which\nthe sign-change occurs21,22.\nThe sign for short distances corresponds to the sign\none obtains when measuring the SSE in its local con\fg-\nurations (hiSSE, indicated schematically in Figure 1c or\nciSSE). The results for a hiSSE measurement on sample\nS3 as a function of Hare shown in Figure 1d, and VhiSSE\nclearly shows the same sign as VnlSSE for short distance.\nWe will discuss the origin of this sign-change in more de-\ntail later in this manuscript. The data shown in Figure 1\nare from samples with tYIG= 2:5\u0016m, hence the di\u000berent\nsigns ford= 2\u0016m andd= 6\u0016m.\nResonant features can be observed in the data for\nj\u00160Hj=\u00160HTA\u00192:3 T, where the subscript TA sig-\nni\fes that these features stems from the hybridization of\nmagnons with phonons in the transverse acoustic mode,\nrather than the longitudinal acoustic mode (LA) which is\nexpected at larger magnetic \felds. The rightmost panels\nof Figure 1 show a close-up of the data around H=HTA.\nFor smalldthe magnon-phonon hybridization causes a\nresonant enhancement (the absolute value is increased)\nofVnlSSE , while for large da resonant suppression (the\nabsolute value is reduced) occurs.\nFigure 2 shows the results of a magnetic \feld sweep\nfrom sample G1 for both electrically generated magnons\n(\frst harmonic) and thermally generated magnons (sec-\nond harmonic). A feature at jHj=HTAcan be resolved\nboth in the \frst and second harmonic voltage. This sug-\ngests that magnon-phonon hybridization does not only\na\u000bect the YIG spin Seebeck coe\u000ecient, as the \frst har-\nmonic signal is generated independent of \u0010. It indicates\nthat not only the generation, but also the transport of\nmagnons is a\u000bected by the hybridization. In the second\nharmonic, the signal is clearly suppressed at the resonant\nmagnetic \feld. Unfortunately, because the feature in the\n\frst harmonic is barely larger than the noise \roor in the\nmeasurements (see Fig. 2a and inset), we cannot conclude\nwhether the signal due to electrical magnon generation\nis enhanced or suppressed at the resonance. Due to the\nfact that the e\u000bect in the \frst harmonic is so small, in the\nremainder of this paper we present a systematic study of\nthe e\u000bect in the second harmonic, the nlSSE.\nThe resonant magnetic \felds are di\u000berent for the TA\nand LA modes ( HTAandHLA, respectively). Due to the\nhigher sound velocity in the LA phonon mode, HTA<\nHLA, and the resonance due to magnons hybridizing with\nphonons in this mode can also be observed in our nonlo-\ncal experiments. In the Supplementary Material (section\nA) we show the results of a magnetic \feld scan over anextended \feld range, and it can be seen that the res-\nonance atHLAalso causes a suppression of the nlSSE\nsignal, similar to the HTAresonance. This is compara-\nble to the case for the hiSSE con\fguration, in which the\nHLAandHTAresonances both show similar behaviour in\nthe sense that they both enhance the hiSSE signal. For\nthe nlSSE case at distances larger than the sign-change\ndistance, both resonances suppress the signal.\nWe now focus on the resonance at HTAin the nlSSE\ndata and carried out nonlocal measurements as a function\nof magnetic \feld for various temperatures and distances.\nFigure 3a (b) shows the distance (temperature) depen-\ndent results, obtained from sample S1 (S2). The regions\nwhere the sign of the nlSSE equals that of the hiSSE are\nshaded blue. From Figure 3a the sign-change in VnlSSE\ncan be clearly seen to occur between d= 2 andd= 5\n\u0016m, as atd= 2\u0016m the nlSSE sign is equal to that of\nthe hiSSE for any value of the magnetic \feld, whereas for\nd= 5\u0016m it is opposite. Additionally, when comparing\ntheVnlSSE\u0000Hcurves for 300 K and 100 K in Figure 3b,\nthe e\u000bect of the sample temperature on the sign-change is\napparent: At 100 K, the nlSSE sign is opposite to that of\nthe hiSSE over the whole curve. Furthermore, Figure 3b\ndemonstrates the in\ruence of the magnetic \feld on the\nsign change, for instance in the curve for T= 160 K. At\nlow magnetic \felds, the nlSSE sign still agrees with the\nhiSSE sign (inside the blue shaded region), but around\nj\u00160Hj= 1:5 T the signal changes sign.\nIn addition, Figure 3a shows that the role of the\nmagnon-polaron resonance changes as the nlSSE signal\nundergoes a sign change. For d\u00142\u0016m, magnon-phonon\nhybridization enhances VnlSSE atH=HTA, whereas for\nd\u00155\u0016mVnlSSE is suppressed at the resonance mag-\nnetic \feld. Similarly, from Figure 3b we observe that at\ntemperatures T > 160 K, magnon-phonon hybridization\nenhances the nlSSE signal at H=HTA, while atT\u0014160\nK the nlSSE is suppressed at HTA. Since the thermally\ngenerated magnon spin current is related to the thermal\ngradient by jm/\u0000\u0010rT, a resonant enhancement in \u0010\nshould lead to an enhancement of the nlSSE signal at all\ndistances and temperatures, which is inconsistent with\nour observations. This is a further indication that not\nonly the generation, but also the transport of magnons\nis in\ruenced by magnon-polarons.\nThe temperature dependence of the low-\feld ampli-\ntude of the nlSSE V0\nnlSSE and the magnitude of the reso-\nnanceVTA(de\fned in Figure 1b) are shown in Figure 4a\nand 4b respectively. The curve for V0\nnlSSE atd= 6\u0016m\nagrees well with an earlier reported temperature depen-\ndence of the nlSSE at distances which are larger than the\n\flm thickness23, while that at d= 2\u0016m qualitatively\nagrees with earlier reports for distances shorter than the\nYIG \flm thickness21,22. Moreover, from the distance de-\npendence of V0\nnlSSE we have extracted the magnon spin\ndi\u000busion length \u0015mas a function of temperature, which is\nshown in the Supplementary Material (section B). \u0015m(T)\nobtained from the Sendai YIG approximately agrees with\nthat for Groningen YIG23for temperatures T > 30 K,4\nbut di\u000bers in the low temperature regime. For further\ndiscussion we refer to the Supplementary Material of this\nmanuscript. The temperature dependence of VTAis dif-\nferent from that of V0\nnlSSE , since \frst of all no change in\nsign occurs here even for d= 2\u0016m and furthermore a\nclear minimum appears in the curve around T= 50 K.\nThis indicates that the resonance has a di\u000berent origin\nthan the nlSSE signal itself, i.e. magnon-polarons are\na\u000bected di\u000berently by temperature than pure magnons.\nThe resonant magnetic \feld HTAdecreases with in-\ncreasing temperature, reducing from \u00160HTA\u00192:5 T at\n3 K to\u00160HTA\u00192:2 T at room temperature as shown\nin Figure 4c. In earlier work by some of us regarding\nthe magnetic \feld dependence of the nonlocal magnon\ntransport signal at room temperature, structure in the\ndata at\u00160H= 2:2 T was indeed observed24, but not\nunderstood at that time. It is now clear that this struc-\nture can be attributed to magnon-phonon hybridization.\nHTAdepends on the following three parameters9: The\nYIG saturation magnetization Ms, the spin wave sti\u000b-\nness constant Dexand the TA-phonon sound velocity\ncTA.Dexis approximately constant for T < 300 K36\nand bothMsandcTAdecrease with temperature. The\nreduction of HTAas temperature increases from 3 K to\n300 K can be explained by accounting for a 7 % decrease\nofcTAin the same temperature interval, taking the tem-\nperature dependence of Msinto consideration37. The\nresults regarding the behaviour of the magnon-polaron\nresonance qualitatively agree for the Sendai and Gronin-\ngen YIG (see Supplementary Material (section C) for the\ntemperature dependent results for sample G2).\nMoreover, we performed measurements of the nlSSE\nsignal as a function of the injector current, and found\nthat the nlSSE scales linearly with the square of the\ncurrent at high temperatures, as expected. However, at\nlow temperatures ( T < 10 K) and su\u000eciently high cur-\nrents (typically, I > 50\u0016A), this linear scaling breaks\ndown (see Supplementary Material (section D)). This\ncould be a consequence of the strong temperature depen-\ndence of the YIG and GGG heat conductivity at these\ntemperatures38,39. The injector heating causes a small in-\ncrease in the average sample temperature which increases\nthe heat conductivities of the YIG and GGG, thereby\ndriving the system out of the linear regime. However, it\nmight also be related to the bottleneck e\u000bect which is ob-\nserved in parametrically excited YIG14. A more detailed\ninvestigation is needed in order to establish the origin of\nthe nonlinearity.\nFinally, we have investigated the ciSSE con\fguration,\nmeaning that current heating of the Pt injector is used to\ndrive the SSE and the (local) voltage across the injector is\nmeasured. The sign of the ciSSE voltage corresponds to\nthat obtained in the hiSSE con\fguration. However, no\nresonant features were observed in the ciSSE measure-\nments, contrary to the hiSSE and nlSSE con\fgurations.\nWe believe that this is due to the low signal-to-noise ratio\nin the ciSSE con\fguration, which could cause the feature\nto be smaller than the noise level in our ciSSE measure-ments. We refer to the Supplementary Material (section\nE) for further discussion.\nC. Modelling\nThe physical picture underlying the thermal genera-\ntion of magnons has been a subject of debate in the\nmagnon spintronics \feld recently. Previous theories ex-\nplain the SSE as being due to thermal spin pumping,\ncaused by a temperature di\u000berence between magnons in\nthe YIG and electrons in the platinum13,40,41. However,\nthe recent observations of nonlocal magnon spin trans-\nport and the nlSSE give evidence that not only the inter-\nface but also the bulk magnet actively contributes and\neven dominates the spin current generation. At elevated\ntemperatures the energy relaxation should be much more\ne\u000ecient than the spin relaxation, which implies that the\nmagnon chemical potential (and its gradient) is more im-\nportant as a non-equilibrium parameter than the temper-\nature di\u000berence between magnons and phonons. A model\nfor thermal generation of magnon spin currents based\non the bulk SSE42which takes into account a non-zero\nmagnon chemical potential has been proposed in order\nto explain the observations34.\nThis model has been reasonably successful in explain-\ning the nonlocal signals (due to both thermal and electri-\ncal generation) in the long distance limit23,33, yet is not\nfully consistent with experiments in the short distance\nlimit for thermally generated magnons33. The model is\nexplained in detail in Refs. 33 and 34, and is described\nconcisely in the Methods section of this manuscript. The\nphysical picture captured by the model is explained in\nFigure 5a and b, where for this study we focus on the\nthermally generated magnons driving the nlSSE. In Fig-\nure 5a a schematic side-view of the YIG jGGG sample\nwith a platinum injector strip on top is shown. A cur-\nrent is passed through the injector, causing it to heat up\nto temperature TH. The bottom of the GGG substrate\nis thermally anchored at T0. As a consequence of Joule\nheating, a thermal gradient arises in the YIG, driving a\nmagnon current Jm\nQ=\u0000\u0010=TrTparallel to the heat cur-\nrent, i.e. radially away from the injector. This reduces\nthe number of magnons in the region directly below the\ninjector (magnon depletion).\nIn Figure 5b the same schematic cross-section is shown,\nbut now the colour coding refers to the magnon chemical\npotential\u0016m. Directly below the injector contact \u0016m\nis negative due to the magnon depletion in this region\n(\u0016\u0000). At the YIGjGGG interface, magnons accumulate\nsince they are driven towards this interface by the SSE\nbut are re\rected by the GGG, causing a positive magnon\nchemical potential \u0016+to build up. Note that the \u0016\u0000\nand\u0016+regions are not equal in size since part of the\nmagnon depletion is replenished by the injector contact,\nwhich acts as a spin sink. Due to the gradient in magnon\nchemical potential, a di\u000buse magnon spin current Jm\ndnow\narises in the YIG given by Jm\nd=\u0000\u001bmr\u0016m.5\nThe combination of these two processes leads to a typ-\nical magnon chemical potential pro\fle as shown in Fig-\nure 5c, which is obtained from the \fnite element model\n(FEM) at room temperature. The sign change from \u0016\u0000\nto\u0016+occurs at a distance of roughly dsc= 2:6\u0016m from\nthe injector, comparable to the YIG \flm thickness.\nHere we used the e\u000bective spin conductance of the\nPtjYIG interface gsas a free parameter in order to get\napproximate agreement between the modelled and exper-\nimentally observed sign-change distance dsc(see Methods\nfor the further details of the model). The value for gsis\napproximately a factor 30 lower than what we calculated\nfrom theory34and used in our previous work23. When\nusinggs= 9:6\u00021012S/m2as in previous work, dsc\u0019300\nnm which is much shorter than what we observe in the\nexperiments. This discrepancy between the models for\nelectrically and thermally generated magnon transport\nmight indicate that some of the material parameters such\nas spin or heat conductivity and spin di\u000busion length (for\nboth YIG and platinum) we use are not fully accurate.\nHowever, it is also conceivable that the models are not\ncomplete and need to be re\fned further33, for instance\nby including temperature di\u000berence at material interfaces\nwhich are currently neglected.\nThe value of dscdepends mainly on four parameters:\nThe thickness of the YIG \flm tYIG, the transparency\nof the platinumjYIG injector interface, parameterized in\nthe e\u000bective spin conductance gs, the magnon spin con-\nductivity of the YIG \u001bmand \fnally the magnon spin\ndi\u000busion length \u0015m. At high temperatures (i.e. close\nto room temperature), the thermal conductivities \u0014GGG\nand\u0014YIGare similar in magnitude43and a\u000bectdsconly\nweakly, allowing us to focus here on the spin transport.\nIncreasingtYIGor\u001bmincreasesdscsince this reduces\nthe spin resistance of the YIG \flm, allowing the depleted\nregion to spread further throughout the YIG. However,\nincreasinggsor\u0015mcauses the opposite e\u000bect and re-\nducesdscsince this increases the amount of \u0016\u0000which is\nabsorbed by the injector contact compared to that which\nrelaxes in the YIG. The precise dependency of dscon\nthese parameters is nontrivial but can be explored using\nour \fnite element model. Ganzhorn et al. and Zhou et\nal.in Refs. 21 and 22 observed that dscbecomes smaller\nwith lower temperatures. This indicates that the ratio of\nthe e\u000bective spin resistance of YIG to that of the Pt con-\ntact increases, causing spins to relax preferentially into\nthe contact and thereby reducing the extend of \u0016\u0000.\nFlebus et al. developed a Boltzmann transport theory\nfor magnon-polaron spin and heat transport in magnetic\ninsulators10. Here we implement the salient features of\nmagnon-polarons into our \fnite element model. We ob-\nserve that when the combination of gs,\u0015m,\u001bm,tYIG\nanddis such that the detector is probing the depletion\nregion, i.e. \u0016\u0000, the magnon-polaron resonance causes\nenhancement of the nlSSE signal. Conversely, when the\ndetector is probing \u0016+the resonance causes a suppres-\nsion of the signal. This cannot be explained by assuming\nthat the only e\u000bect of the magnon-polaron resonance isthe enhancement of \u0010, as this would simply increase the\nthermally driven magnon spin current Jm\nQand hence en-\nhance both \u0016\u0000and\u0016+. To understand this behaviour,\nwe have to account for the enhancement of \u001bmby the\nmagnon-polaron resonance as well.\nA resonant increase in \u001bmleads to an increased di\u000bu-\nsive back\row current Jm\nd, which can lead to a reduction\nof the magnon spin current reaching the detector at large\ndistances. We model the e\u000bect of the magnon-phonon hy-\nbridization by assuming a \feld-dependent magnon spin\nconductivity \u001bm(H) and bulk spin Seebeck coe\u000ecient\n\u0010(H), which are both enhanced at the resonant \feld HTA.\nNote that the \feld-dependence only includes the con-\ntribution from the magnon-polarons10, and does not in-\nclude the e\u000bect of magnons being frozen out by the mag-\nnetic \feld24,44{46since this is not the focus of this study.\nThe parameter values used in the model are given in the\nMethods section of this paper. The model is used to cal-\nculate the spin current \rowing into the detector contact\nas a function of magnetic \feld, from which we calculate\nthe voltage drop over the detector due to the inverse spin\nHall e\u000bect. We then vary the ratios of enhancement for\n\u001bmand\u0010, i.e.f\u001b=\u001bm(HTA)=\u001b0\nmandf\u0010=\u0010(HTA)=\u00100,\nwhere\u001b0\nmand\u00100are the zero \feld magnon spin conduc-\ntivity and spin Seebeck coe\u000ecient and \u001bm(HTA); \u0010(HTA)\nare these parameters at the resonant \feld. The ratio of\nenhancement \u000e=f\u0010=f\u001bis crucial in obtaining agreement\nbetween the experimental and modelled data. To change\ndelta, we \fx f\u0010= 1:09 and vary f\u001b. The value for f\u0010\nis comparable to the enhancement in \u0010calculated from\ntheory for low temperatures10.\nD. Comparison between model and experiment\nFigure 6 shows a comparison between the distance de-\npendence of V0\nnlSSE andVTAobtained from experiments\n(Fig. 6a) and the \fnite element model (Fig. 6b and c) at\nroom temperature. In Figure 6a, V0\nnlSSE shows a change\nin sign around d= 4\u0016m, whileVTAhas a positive sign\nover the whole distance range. Fig. 6b shows the model\nresults for V0\nnlSSE (red), and the voltage measured at\nH=HTAfor\u000e= 2 (green) and \u000e= 0:5 (purple). While\nthe voltage obtained from the model is approximately\none order of magnitude lower than in experiments, the\nqualitative behaviour of the experimental data is repro-\nduced. In particular, the modelled dscapproximately\nagrees with the experimentally observed distance.\nFor\u000e= 2, the modelled voltage at HTAis always en-\nhanced with respect to V0\nnlSSE (ford < dsc,V(HTA)<\nV0\nnlSSE and ford > dsc,V(HTA)> V0\nnlSSE ). This is\nnot consistent with the experiments as it leads to a sign\nchange in VTA, which is de\fned as VTA=V0\nnlSSE\u0000\nV(HTA), as can be seen from Fig. 6c.\nHowever, for \u000e= 0:5,V(HTA) is enhanced with respect\ntoV0\nnlSSE ford < dscbut suppressed for d > dsc. This\nresults in a positive sign for VTAover the full distance\nrange, comparable to the experimental observations. The6\nfull magnetic \feld dependence obtained from the model\ncan be found in the Supplementary Material (section F).\nAs can be seen from the inset in Fig. 6c, \u000e= 0:5 results in\na decay ofVTAwith distance which is comparable to the\nexperimentally observed VTA(d) (inset Fig. 6a). We \ftted\nthe data for VTAobtained from both the experiments and\nthe simulations to VTA(d) =Aexp\u0000d=`TA, whereAis\nthe amplitude and `TAthe length scale over which VTA\ndecays. From the \fts, we obtain `exp\nTA= 6:3\u00061:2\u0016m\nand`sim\nTA= 10:6\u00060:1\u0016m at room temperature, where\nwe have \ftted to the model results for \u000e= 0:5. From\nthe simulations, we \fnd that `TAis in\ruenced by the\nvalue used for \u000e, where a smaller \u000eleads to a longer `TA.\nThis could indicate that \u000ehas to be increased slightly to\nobtain better agreement between `exp\nTAand`sim\nTA.\nTherefore, in order to explain the observations, 0 :5<\n\u000e < 1, i.e. the relative enhancement due to magnon-\nphonon hybridization in \u001bmhas to be larger than that of\n\u0010.`exp\nTAis enhanced at low temperatures (see Supplemen-\ntary Material (section B) for the distance dependence of\nVTAat low temperatures). This could indicate that \u000e\ndecreases with decreasing temperatures. For further dis-\ncussion we refer to the Supplementary Material (section\nB).\nThe model results depend sensitively on gs. A largergs\nreduces the dscobserved in the model, so that our model\nno longer qualitatively \fts the distance dependence of\nVnlSSE obtained in experiments. As a consequence, the\n\u000eneeded to model the resonant suppression of the signal\natHTAfor long distances decreases further, which would\nimply that the enhancement in \u001bmis much stronger than\nthat in\u0010. Such a strong enhancement in \u001bmshould result\nin a clear magnon-polaron resonance in the electrically\ngenerated magnon spin signal, whereas we observed only\na small e\u000bect here (see Fig. 2a). This is an indication\nthat our choice of reducing gscompared to our previous\nwork is justi\fed.\nII. DISCUSSION\nWe report resonant features in the nlSSE as a function\nof magnetic \feld, which we ascribe to the hybridization\nof magnons and acoustic phonons. They occur at mag-\nnetic \felds that obey the \\touch\" condition at which the\nmagnon frequency and group velocity agree with that\nof the TA and LA phonons. The signals are enhanced\n(peaks) for short injector-detector distances and high\ntemperatures, but suppressed (dips) for long distances\nand/or low temperatures. The temperature dependence\nof the TA resonance di\u000bers from that of the low-\feld\nnlSSE voltage, indicating that di\u000berent physical mechan-\nsims are involved (this in contrast to the local SSE con-\n\fguration). The sign of the nlSSE signal corresponds\nto that of the signal in the hiSSE con\fguration for dis-\ntances below the sign-change distance. In this regime the\nmagnon-polaron feature causes signal enhancement, sim-\nilar to the hiSSE con\fguration. For distances longer thanthe sign-change distance, the nlSSE signal is suppressed\nat the resonance magnetic \feld.\nThese results are consistent with a model in which\ntransport is di\u000buse and carried by strongly coupled\nmagnons and phonons10(magnon-polarons). Theory\npredicts an enhancement of all transport coe\u000ecients\nwhen the acoustic quality of the crystal is better than the\nmagnetic one. Simulations show that the dip observed in\nthe nlSSE is not caused by deteriorated acoustics, but\nby a competition between the thermally generated, SSE\ndriven magnon current and the di\u000buse back\row magnon\ncurrent which are both enhanced at the resonance. More\nexperiments including thermal transport as well as an ex-\ntension of the Boltzmann treatment presented in Ref. 10\nto 2D geometries are necessary to fully come to grips with\nheat and spin transport in YIG.\nAdditionally, we observed features in the electrically\ngenerated magnon spin signal at the resonance magnetic\n\feld. This is further evidence that not only the gener-\nation of magnons via the SSE, but additional transport\nparameters such as the magnon spin conductivity are af-\nfected by magnon-polarons.\nThe nonlocal measurement scheme provides an excel-\nlent platform to study magnon transport phenomena and\nopens up new avenues for studying the magnetoelastic\ncoupling in magnetic insulators. Finally, these results are\nan important step towards a complete physical picture of\nmagnon transport in magnetic insulators in its many as-\npects, which is crucial for developing e\u000ecient magnonic\ndevices.\nIII. METHODS\nSample fabrication. The YIG \flms used in this\nstudy were all grown on gadolinium gallium garnet\n(GGG) substrates by liquid phase epitaxy (LPE) in the\n[111] direction. The samples from the Sendai group have\na thickness of 2.5 \u0016m, the samples used in Groningen are\n210 nm thick. The Sendai samples were grown in-house,\nwhereas the Groningen samples were obtained commer-\ncially from Matesy GmbH. In Sendai, \fve batches of de-\nvices where fabricated from the same YIG wafer (S1 to\nS4). The fabrication method and platinum strip geome-\ntry are the same for all batches, but they were not fabri-\ncated at the same time, which might lead to variations in\nfor instance the interface quality from batch to batch. In\nGroningen, two batches of devices were investigated (G1\nand G2). The nonlocal devices fabricated in Groningen\nare de\fned in three lithography steps: the \frst step was\nused to de\fne Ti/Au markers on top of the YIG \flm via\ne-beam evaporation, used to align the subsequent steps.\nIn the second step, Pt injector and detector strips were\ndeposited using magnetron sputtering in an Ar+plasma.\nIn the \fnal step, Ti/Au contacts were deposited by e-\nbeam evaporation. Prior to the contact deposition, a\nbrief Ar+ion beam etching step was performed to remove\nany polymer residues from the Pt strip contact areas to7\nensure optimal electrical contact to the devices. The non-\nlocal devices fabricated in Sendai were de\fned in a single\nlithography step. Two parallel Pt strips and contact pads\nwere patterned using e-beam lithography followed by a\nlift-o\u000b process, in which 10-nm-thick Pt was deposited\nusing magnetron sputtering in an Ar+plasma.\nMeasurements. Electrical measurements were car-\nried out in Groningen and in Sendai, using a current-\nbiased lock-in detection scheme. A low frequency ac\ncurrent of angular frequency !(typical frequencies are\n!=(2\u0019)<20 Hz, and the typical amplitude is I= 100\n\u0016Arms) is sent through the injector strip, and the voltage\non the detector strip is measured at both the frequencies\n!(the \frst harmonic response) and 2 !(the second har-\nmonic response). This allows us to separate processes\nthat are linear in the current, which govern the \frst\nharmonic response, from processes that are quadratic in\nthe current which are measured in the second harmonic\nresponse19,28,47.\nThe measurements in Sendai were carried out in a\nQuantum Design Physical Properties Measurement Sys-\ntem (PPMS), using a superconducting solenoid to apply\nthe external magnetic \feld (\feld range up to \u00160H=\n\u000610:5 T). The measurements in Groningen were carried\nout in a cryostat equipped with a Cryogenics Limited\nvariable temperature insert (VTI) and superconducting\nsolenoid (magnetic \feld range up to \u00160H=\u00067:5 T).\nElectronic measurements in Groningen are carried out us-\ning a home built current source and voltage pre-ampli\fer\n(gain 104) module galvanically isolated from the rest of\nthe measurement electronics, resulting in a noise level\nof approximately 3 nV r:m:s:at the output of the lockin\nampli\fer for a time constant of \u001c= 3 s and a \flter\nslope of 24 dB/octave. The electronic measurements in\nSendai were carried out by means of an ac and dc cur-\nrent source (Keithley model 6221) and a lockin ampli\fer\nusing a time constant of \u001c= 1 s and a \flter slope of\n24 dB/octave. The data shown in Figure 1b and Fig-\nure 3 is the asymmetric part of the measured voltage\nwith respect to the magnetic \feld. The antisymmetriza-\ntion procedure includes both the forward and backward\nmagnetic \feld sweep, and the voltage shown in the \fg-\nures is given by VH+= (Vbackward (H)\u0000Vbackward (\u0000H))=2\nandVH\u0000= (Vforward (H)\u0000Vforward (\u0000H))=2, whereVH+\nis the voltage at postive magnetic \feld values and VH\u0000\nthat at negative magnetic \feld values.\nSimulations. The two-dimensional \fnite element\nmodel is implemented in COMSOL MultiPhysics (v4.4).\nThe linear response relation of heat and spin transport\nin the bulk of a magnetic insulator reads\n\u00122e\n~jm\njQ\u0013\n=\u0000\u0012\n\u001bm\u0010=T\n~\u0010=2e \u0014\u0013\u0012\nr\u0016m\nrT\u0013\n; (1)\nwhere jmis the magnon spin current, jQthe total\n(magnon and phonon) heat current, \u0016mthe magnon\nchemical potential, Tthe temperature (assumed to be\nthe same for magnons and phonons by e\u000ecient thermal-\nization),\u001bmthe magnon spin conductivity, \u0014the total(magnon and phonon) heat conductivity and \u0010the spin\nSeebeck coe\u000ecient. We disregard temperature di\u000ber-\nences arising from the Kapitza resistances at the Pt jYIG\nor YIGjGGG interfaces. \u0000eis the electron charge and ~\nthe reduced Planck constant. The di\u000busion equations for\nspin and heat read\nr2\u0016m=\u0016m\n\u00152m; (2)\nr2T=j2\nc\n\u0014\u001b; (3)\nwherejcis the charge current density in the injector con-\ntact,\u001band\u0014the electrical and thermal conductivity and\n\u0015mthe magnon spin di\u000busion length. Eq. (3) represents\nthe Joule heating in the injector that drives the SSE.\nIn the simulations, tYIG= 2:5\u0016m andwYIG= 500\u0016m\nare the thickness and width of the YIG \flm, on top of\na GGG substrate that is 500 \u0016m thick.wYIGis much\nlarger than \u0015mand \fnite size e\u000bects are absent. The\ninjector has a thickness of tPt= 10 nm and a width\nofwPt= 300 nm. The spin and heat currents normal\nto the YIGjvacuum, Ptjvacuum and GGG jvacuum inter-\nfaces vanish. At the bottom of the GGG substrate the\nboundary condition T=T0is used, i.e. the bottom of\nthe sample is taken to be thermally anchored to the sam-\nple probe. Furthermore, a spin current is not allowed to\n\row into the GGG. The spin current across the Pt jYIG\ninterface is given by jint\nm=gs(\u0016s\u0000\u0016m), wheregsis the\ne\u000bective spin conductance of the interface, \u0016sis the spin\naccumulation on the metal side of the interface and \u0016m\nis the magnon chemical potential on the YIG side of the\ninterface. The nonlocal voltage is then found by calculat-\ning the average spin current density hjsi\rowing in the\ndetector, which is then converted to non-local voltage\nusingVnlSSE =\u0012SHLhjsi=\u001b, where\u0012SHis the spin Hall\nangle in platinum and Lis the length of the detector\nstrip. The spin current in the platinum contact relaxes\nover the characteristic spin relaxation length \u0015s.\nThe parameters we use for platinum in the model are\n\u0012SH= 0:11,\u001b= 1:9\u0002106S/m,\u0015s= 1:5 nm and\u0014=\n26 W/(m K). For YIG, we use \u001bm= 3:7\u0002105S/m,\n\u0015m= 9:4\u0016m which was obtained in our previous work23.\nFurthermore, we use \u0014= 7 W/(m K), based on YIG\nthermal conductivity data from Ref. 39. For the bulk\nspin Seebeck coe\u000ecient at zero \feld we use \u00100= 500\nA/m, based on our previous work in which we gave an\nestimate for \u0010at room temperature33. For GGG, the spin\nconductivity and spin Seebeck coe\u000ecient are set to zero.\nFor the GGG thermal conductivity we use \u0014= 9 W/(m\nK), based on data from Refs. 38 and 43. Finally, for the\ne\u000bective spin conductance of the interface we use gs=\n3:4\u00021011S/m2. We note that this is roughly a factor 30\nsmaller than in our earlier work23. This variation of the\ninterface transparency in di\u000berent experiments indicates\nthe presence of physical processes that are not taken into\naccount in the modeling.8\nIV. ACKNOWLEDGEMENTS\nWe thank H. M. de Roosz, J.G. Holstein, H. Adema\nand T.J. Schouten for technical assistance and R.A.\nDuine, B. Flebus and K. Shen for discussions. This\nwork is part of the research program of the Nether-\nlands Organization for Scienti\fc Research (NWO) and\nsupported by NanoLab NL, EU FP7 ICT Grant No.\n612759 InSpin, the Zernike Institute for Advanced Mate-\nrials, Grant-in-Aid for Scienti\fc Research on Innovative\nArea \"Nano Spin Conversion Science\" (Nos. JP26103005\nand JP26103006), Grant-in-Aid for Scienti\fc Research\n(A) (No. JP25247056) and (S) (No. JP25220910) from\nJSPS KAKENHI, Japan, and ERATO \"Spin Quantum\nRecti\fcation Project\" (No. JPMJER1402) from JST,\nJapan. Further support by the DFG priority program\nSpin Caloric Transport (SPP 1538, KU3271/1-1) is grate-fully acknowledged. K.O. acknowledges support from\nGP-Spin at Tohoku University. T.Ki. is supported by\nJSPS through a research fellowship for young scientists\n(No. JP15J08026).\nV. 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C. Myers, and J. P.\nHeremans, Physical Review B 92, 054436 (2015).\n46T. Kikkawa, K. I. Uchida, S. Daimon, and E. Saitoh,\nJournal of the Physical Society of Japan 85, 065003 (2016).\n47F. L. Bakker, A. Slachter, J.-P. Adam, and B. J. van Wees,\nPhysical Review Letters 105, 136601 (2010).10\nFIG. 1. Experimental geometries and main results. Figure ais an image of a typical device, with schematic current\nand voltage connections. The three parallel lines are the Pt injector/detector strips, connected by Ti/Au contacts. \u000bis the\nangle between the Pt strips and an applied magnetic \feld H(inb-d\u000b= 90\u000e).bThe nonlocal spin Seebeck (nlSSE) voltage\nfor an injector-detector distance d= 6\u0016m (top) and d= 2\u0016m (bottom) as a function of \u00160H. Atj\u00160Hj=j\u00160HTAj\u00192:3\nT, a resonant structure is observed that we interpret in terms of magnon-polaron formation (indicated by blue triangles as a\nguide to the eye). The right column is a close-up of the anomalies for H > 0. The results can be summarized by the voltages\nV0\nnlSSE andVTAas indicated in the lower panels. cSchematic geometry of the local heater-induced hiSSE measurements. Here\nthe temperature gradient rTis applied by external Peltier elements on the top and bottom of the sample. dThe hiSSE\nvoltage measured as a function of magnetic \feld. The close-up around the resonance \feld (right column) focusses on the\nmagnon-polaron anomaly. All results were obtained at T= 200 K. The results for d= 6,d= 2 andd= 0\u0016m were obtained\nfrom sample S1, S2, S3, respectively (see Methods for sample details).11\nFIG. 2. Nonlocal voltage due to electrically and thermally generated magnons as a function of magnetic \feld.\nFigure ashows the nonlocal voltage generated by magnons that are excited electrically (\frst harmonic response to an oscillating\ncurrent in the injector contact). An anomaly is observed at H=jHTAj(the \feld that satis\fes the touching condition for magnons\nand transverse acoustic phonons). The inset shows a second set of data from the same sample, taken with a higher magnetic\n\feld resolution ( \u00160\u0001H= 15 mT), sweeping the magnetic \feld both in the forward (black) and backward (red) directions.\nFigure bshows the nlSSE voltage (second harmonic response) for the same device. VnlSSE is suppressed at H=jHTAj. The\ninset shows the corresponding second harmonic data of the high resolution \feld sweep. The results were obtained on sample\nG1 (thickness 210 nm) with d= 3:5\u0016m andI= 150\u0016Ar:m:s:, at room temperature. A constant background voltage Vbg= 575\nnV was subtracted from the data in Fig. a.12\nTemperature dependence for d = 2 μm Distance dependence for T = 300 K a b\nnonlocal sign  = \n local sign\nFIG. 3.VnlSSE vs magnetic \feld as a function of distance and temperature. Figure ais a plot of VnlSSE vsHfor\nvarious injector-detector separations at T= 300 K, while Figure bshowsVnlSSE vsHfor di\u000berent temperatures and d= 2\n\u0016m. The data in Figs. aandbare from sample S1 and S2, respectively. The magnon-polaron resonance is indicated by the\nblue arrows. The blue shading in the graphs indicates the region in which the sign of the nlSSE signal agrees with that of the\nhiSSE. The right column in both aandbshows close-ups of the data around the positive resonance \feld (blue triangles). The\ndata in the close-ups has been antisymmetrized with respect to H, i.e.V= (V(+H)\u0000V(\u0000H))=2. Fig. ashows that when the\ncontacts are close ( d\u00142\u0016m), the magnon-polaron resonance enhances VnlSSE , while for long distances VnlSSE is suppressed at\nthe resonance magnetic \feld. For very large distances ( d\u001520\u0016m), the resonance cannot be observed anymore. Similarly in\nFig.b, for temperatures T\u0015180 K, the magnon-polaron resonance enhances the nlSSE signal, while for lower temperatures\nthe nlSSE signal is suppressed. The excitation current I= 100\u0016Ar:m:s:for all measurements.13\nFIG. 4. Temperature dependence of V0\nnlSSE,VTAandHTA. adisplays the temperature dependence of the low-\feld V0\nnlSSE ,\nford= 2\u0016m andd= 6\u0016m. For 2\u0016m, the signal changes sign around T= 143 K. The blue shading in the graph indicates the\nregime in which the sign agrees with that of the hiSSE. The temperature dependence of the magnon-polaron resonance VTAis\nshown in Figure b. Here, no sign change but a minimum around T= 50 K is observed, which is absent in Figure a. Figure\ncshows the temperature dependence of the resonance \feld HTA. Error bars in bandcre\rect the peak-to-peak noise in the\ndata used to extract VTAand the step size in the magnetic \feld scans ( \u00160\u0001H= 20 mT), respectively.\nFIG. 5. Physical concepts underlying the nlSSE signal and simulated magnon chemical potential pro\fle. Figure\nasketches the e\u000bects of Joule heating in the injector, heating it up to temperature TH, which leads to a thermal gradient in the\nYIG. The bulk SSE generates a magnon current Jm\nQantiparallel to the local temperature gradient, spreading into the \flm away\nfrom the contact. When the spin conductance of the contact is su\u000eciently small, this leads to a depletion of magnons below\nthe injector, indicated in Figure bas\u0016\u0000. When the magnons are re\rected at the GGG interface, Jm\nQaccumulates magnons at\nthe YIGjGGG interface, shown in Figure bas\u0016+. The chemical potential gradient induces a backward and sideward di\u000buse\nmagnon current Jm\nd. Both processes in Figure aandbare included in the \fnite element model (FEM). Its results are plotted\nin Figure cin terms of a typical magnon chemical potential pro\fle. \u0016mchanges sign at some distance from the injector, also at\nthe YIG surface, where it can be detected by a second contact. The magnon-polaron resonance enhances both the spin Seebeck\ncoe\u000ecient\u0010and the magnon spin conductivity \u001bm. The increased back\row of magnons to the injector causes a suppression of\nthe nonlocal signal at long distances (see Figure 6).14\nFIG. 6. Comparison of the experimental and simulated V0\nnlSSE andVTA.Figure ashows the distance dependence of\nV0\nnlSSE andVTA(inset) measured at room temperature. The dashed line in the inset is an exponential \ft to the data. V0\nnlSSE\nchanges sign around d= 4\u0016m, whileVTAremains positive. Figure bis a plot of the calculated distance dependence of V0\nnlSSE\nat zero magnetic \feld (red) and at the resonant \feld for \u000e= 2 (green) and \u000e= 0:5 (purple). Here \u000eis a parameter that\nmeasures the relative enhancement of the spin Seebeck coe\u000ecient compared to the magnon spin conductivity, as explained in\nthe main text. The inset shows the signal decay at long distances on a logarithmic scale. Figure cshows the modelled distance\ndependence of VTAfor various values of \u000eon a linear scale (inset for logarithmic scale). \u000e= 0:5 results in a positive sign for\nVTAover the full distance range with a slope that roughly agrees with experiments (cf. insets of Figure aandc). Reducing \u000e\nfurther leads to a more gradual slope for VTA. In the simulations, the SSE enhancement is f\u0010= 1:09, whilef\u001bis varied with \u000e."    },    {        "title": "1910.04046v1.Magnetic_field_dependence_of_the_nonlocal_spin_Seebeck_effect_in_Pt_YIG_Pt_systems_at_low_temperatures.pdf",        "content": "arXiv:1910.04046v1  [cond-mat.mtrl-sci]  9 Oct 2019Magnetic field dependence of the nonlocal spin Seebeck effect in\nPt/YIG/Pt systems at low temperatures\nKoichi Oyanagi,1,a)Takashi Kikkawa,1,2and Eiji Saitoh1,2, 3, 4,5\n1)Institute for Materials Research, Tohoku University, Send ai 980-8577, Japan\n2)WPI Advanced Institute for Materials Research, Tohoku Univ ersity, Sendai 980-8577,\nJapan\n3)Center for Spintronics Research Network, Tohoku Universit y, Sendai 980-8577, Japan\n4)Department of Applied Physics, University of Tokyo, Hongo, Tokyo 113- 8656, Japan\n5)Advanced Science Research Center, Japan Atomic Energy Agen cy, Tokai 319-1195, Japan\n(Dated: 10 October 2019)\nWe report the nonlocal spin Seebeck effect (nlSSE) in a later al configuration of Pt/Y 3Fe5O12(YIG)/Pt systems as a\nfunction of the magnetic field B(up to 10 T) at various temperatures T(3 K<T<300 K). The nlSSE voltage\ndecreases with increasing Bin a linear regime with respect to the input power (the applie d charge-current squared I2).\nThe reduction of the nlSSE becomes substantial when the Zeem an energy exceeds thermal energy at low temperatures,\nwhich can be interpreted as freeze-out of magnons relevant f or the nlSSE. Furthermore, we found the non-linear power\ndependence of the nlSSE with increasing Iat low temperatures ( T<20 K), at which the B-induced signal reduction\nbecomes less visible. Our experimental results suggest tha t in the non-linear regime high-energy magnons are over\npopulated than those expected from the thermal energy. We al so estimate the magnon spin diffusion length as functions\nofBandT.\nSpin caloritronics1is an emerging field to study the inter-\nconversion between spin and heat currents. The spin Seebeck\neffect (SSE) is one of the fundamental phenomena in this field ,\nreferring to the spin-current generation from a heat curren t.\nThe SSE is well studied in a longitudinal configuration2,3,\nwhich consists of a heavy metal(HM)/ferromagnet(FM) bi-\nlayer system, typically Pt/Y 3Fe5O12(YIG) junction. When a\nthermal gradient is applied perpendicular to the interface , a\nmagnon spin current is generated in FM and converted into\na conduction-electron spin current in HM via the interfacia l\nexchange interaction4, which is subsequently detected as a\ntransverse electric voltage via the inverse spin Hall effec t\n(ISHE)5,6. Recent studies of the longitudinal SSE (LSSE)7–9\nsuggest that magnon transport in FM plays a key role in SSEs.\nNonlocal experiment is a powerful tool to investigate the\ntransport of spin currents in various magnetic insulators10–20.\nEspecially, when a spin current is excited via a thermal grad i-\nent, it is called the nonlocal SSE (nlSSE)10,11. A typical non-\nlocal device consists of two HM wires on top of a magnetic\ninsulator, which are electrically separated with the dista nce\nd. In nlSSE measurements, one of the HM wires is used as a\nheater; the Joule heating of an applied charge current ( I) drives\nmagnon spin currents in the magnetic insulator. Some of the\nmagnons reach the other HM wire and inject a spin current,\nwhich is converted into a voltage via the ISHE. By changing\nthe injector-detector separation distance d, we can address the\ntransport property of magnon spin currents.\nIn this paper, we report the high magnetic field ( B) depen-\ndence of the nlSSE in lateral Pt/YIG/Pt systems at various\ntemperatures from T=300 K to 3 K and up to |B|= 10 T. We\nobserved that the nlSSE signal V2ωdecreases with increasing\nB, but the feature turns out to depend on the amplitude of the\napplied I. In a linear regime ( V2ω∝I2), substantial B-induced\na)Electronic mail: k.oyanagi@imr.tohoku.ac.jpsuppression of the nlSSE was observed below 10 K, which is\nconsistent with the previous LSSE results in Pt/YIG8,21. In a\nnon-linear regime ( V2ω/negationslash∝I2), however, the nlSSE signal re-\nmains almost unchanged under high Bat low Ts. By mea-\nsuring the ddependence, we estimate the magnon diffusion\nlength λas functions of TandB.\nWe prepared series of nonlocal Pt/YIG/Pt devices, schemat-\nically shown in Fig. 1(a). A 2.5- µm-thick YIG film\nwas grown by liquid phase epitaxy on a Gd 3Ga5O12(111)\nsubstrate22. On top of the YIG film, we fabricated two Pt\nwires using e-beam lithography and the lift-off process19. The\ndimension of the Pt wires is 200 µm length, 100 nm width,\nand 10 nm thickness. The Pt wires were deposited by mag-\nnetron sputtering in Ar+atmosphere. We investigated four\nbatches of samples (S1-S4) cut from the same YIG wafer.\nThe ddependence was studied in S1 ( d=5,6,8,13,15,\nand 20 µm) and S2 ( d=9µm), while the Idependence at\nlowTs in S3 ( d=2µm) and S4 ( d=8µm). We measured a\nnonlocal voltage using a lock-in detection technique; we ap -\nplied an a.c. charge current, I, of 13.423 Hz in frequency to\nthe injector Pt wire and measured a second harmonic nonlocal\nvoltage V2ωacross the detector Pt wire11.\nFirst, we confirmed that the obtained nonlocal voltage sat-\nisfies the features of the nlSSE at room temperature. Figure\n1(b) shows typical V2ωas a function of in-plane B(θ=90◦)\nat 300 K in the d=9µm sample. A clear V2ωappears,\nwhose sign changes with respect to the Bdirection. V2ωdis-\nappears when Bis applied perpendicular to the plane ( θ=0).\nThis symmetry is consistent with that of the SSE3. We de-\nfine the low-field amplitude of the voltage signal as VnlSSE=\n[V2ω(0.18 T)−V2ω(−0.18 T)]/2, at which the magnetization\nof YIG is fully saturated along B. As shown in Fig. 1(c),\nVnlSSE is proportional to I2, indicating that VnlSSE appears due\nto the Joule heating. With increasing B,V2ωgradually de-\ncreases, and at around ±2.2 T, sharp dip structures show up,\nwhich are induced by magnon polarons due to magnon −TA-2\nBV\nJs\nYIG Joule heat I\nθ(a)\nVnlSSE ɹ\t7\n \n0.010.101.00\n0 10 \nd (µm)5 20 15 T = 300 K\nI = 100 µAExponential fitExperiment\nB = 0.18 T(c)\nV2ωɹ\t7\n 0\nB (T)(b)\n-0.50.5\n2 -2 0T = 300 K\nd = 9.0 µmθ = 0 θ = 90 °\n-1 1 3 -3 VnlSSE Pt \n0 100 200\nI (µA)0.20.4VnlSSE  (µV) (d)\nT = 300 K\nd = 9.0 µm\nB = 0.18 TFittingExperiment\nFIG. 1. (a) A schematic illustration of the nlSSE measuremen t in a\nlateral Pt/YIG/Pt system. B,θ,I, and Jsdenote the external magnetic\nfield, angle between Band sample surface normal, charge current\nthrough the Pt injector, and spin current at the Pt/YIG detec tor inter-\nface, respectively. An a.c. charge current is applied to the Pt injector,\nand the second harmonic voltage V2ωis measured across the Pt de-\ntector. (b) The Bdependence of the nonlocal voltage V2ωin the d=\n9µm sample. V2ωatθ=90◦(θ=0) is measured with I=200µA\n(100 µA).VnlSSE=[V2ω(0.18 T)−V2ω(−0.18 T)]/2 represents the\namplitude of the nlSSE. (c) VnlSSE(I)in the d=9µm sample. The\nsolid red line shows a I2fitting to data. (d) Semi logarithmic plot of\nVnlSSE(d). The red line is fit with VnlSSE=Cexp(−d/λ). The error\nbars represent the 68% confidence level ( ±s.d.).\nphonon hybridization16,23,24.\nBy changing the injector-detector separation distance d, we\nestimate the length scale of the magnon spin current25. As\nshown in Fig. 1(d), VnlSSE decreases with increasing d. A\none-dimensional spin diffusion model11,26describes the de-\ncay, which reads\nVnlSSE=Cexp/parenleftbigg\n−d\nλ/parenrightbigg\n, (1)\nwhere λis the magnon spin diffusion length and Cis the d-\nindependent constant. We fit Eq. (1) to the ddependence of\nVnlSSE and obtain λ=6.76±0.16µm at 300 K. Similar values\nare reported in previous studies in both thin (200 nm)11, and\nthick (50 µm)27YIG films.\nNext, we measured the Tdependence of V2ωwith I=\n100µA. As shown in Fig. 2(a), at 300 K negative voltages\nare observed for the d=0.5 and 1.5µm samples, while the\npositive ones show up for the d=8 and 15 µm samples. With\ndecreasing T, the d=8 and 15 µm samples exhibit a mono-\ntonic increase of VnlSSE . On the other hand, with decreasing T\nthe negative voltages observed for the d=0.5 and 1.5µm\nsamples at 300 K change their sign at several tens Kelvin.\nThe sign change of VnlSSE with changing dandThas been\nobserved in previous nlSSE experiments and explained as a\nresult of a spatial profile of the magnon chemical potential\nµmthat governs the sign and amplitude of VnlSSE ; a negatived = 0.5 µm\n1.5 µm\n8.0 µm\n15.0 µm\n100 10 T (K)VnlSSE ɹ\t7\n \n-10010 \n-5 5I = 100 µA\nB = 0.18 T(a)\n2 1B (T)0 310 \n6λ (µm) 8\n4\n2T = 20 K\n50 K100 K\n300 KI = 100 µA(b)\n10 5\nB (T)-5 -10 0V2ωɹ\t7\n \n-1 0\n-2 12 T = 300 K\nd = 1.5 µm\nI = 100 µA(c)\n10040 0\nT (K)20 \n60 \n80 -20\n100 10 (d)\nd = 0.5 µm\n1.5 µm\n8.0 µm\n15.0 µmδɹ\t%\nFIG. 2. (a) Semi logarithmic plot of VnlSSE(T)for various dwith\nI=100µA. (b) λ(B)at various Ts. We obtained λbyCexp(−d/λ)\nfitting to the ddependence of VnlSSE . (c) V2ω(B)in the d=1.5µm\nsample with I=100 µA at 300 K. (d) δ(T)for different d.δis\ndefined by Eq. (2).\nµmcreated beneath the Pt injector exponentially decays apart\nfrom the injector and above a certain distance a positive one\nmanifests due to the presence of YIG/GGG interface. The\noverall µmprofile varies with T14–16,25–27. Furthermore, we\nfound a second sign change for the d=0.5µm sample at 3 K,\nwhich is unclear at this moment.\nWe now focus on the magnetic field Bdependent features of\nV2ω. Figure 2(b) shows the Bdependence of λat various Ts\nobtained by fitting Eq. (1) to VnlSSE(d). At 300 K, λdecreases\nwith increasing Bby 30 % up to 3 T [from λ=6.8µm at\nB=0.18 T to 4 µm at 3 T, see blue filled circles in Fig. 2(b)].\nA similar field-induced decrease of λhas been observed in\nthe time-resolved LSSE28, nlSSE29, and electrically excited\nmagnon transport experiment29at room temperature. On the\nother hand, at lower Ts,λwas found to be less sensitive to B\n[see Fig. 2(b)].\nTo further investigate the effect of high Bon the nlSSE, we\napplied larger magnetic fields up to 10 T. Figure 2(c) shows a\ntypical V2ω−Bresult for |B|<10 T in the d=1.5µm sample\nwith I=100 µA at 300 K. High B-induced suppression of\nV2ωis clearly observed. In Fig. 2(d) we plot the degree of\nB-induced V2ωsuppression up to 8 T, defined as\nδ=100×/parenleftbigg\n1−V8T\n2ω\nV0.18T\n2ω/parenrightbigg\n(2)\nas a function of Tfor the d=0.5, 1.5, 8.0, and 15 µm sam-\nples. At 300 K, all the samples show the substantial high B-\ninduced V2ωreduction; 65 % <δ<75 % for the d=1.5, 8.0,\nand 15 µm samples and δ=39 % for the d=0.5µm sam-\nple. For the d=8.0 and 15 µm samples, with decreasing T,\nδgradually decreases in the range of 20 K <T<300 K and\nslightly increases below 20 K. For the d=0.5 and 1.5 µm\nsamples, more complicated Tdependences were observed,\nwhich may be related to the non-monotonic Tresponses of3\n(a)\n0 50 100\nI (µA)2\n1S (kΩ/A) 3\nd = 2 µm\n8 µmT = 3 K\nB = 0.18 T\n9 µmLinear regime\n10 5\nB (T)02\n1S (kΩ/A) 3\nI = 100 µA\n10 µA\n5 µA\n3 µA50 µAd = 2.0 µm\nT = 3 K(c) (b)\n2\n1S (kΩ/A) 3\n20 10 2\nT (K)0d = 8.0 µm\nB = 0.5 TLinear  \nNon-linear \n(I = 100 µA) \n100 40 20 \n60 \n80 0\nδ (%) \nT (K) d = 8.0 µm \n20 10 2(d) \nLinear  \nNon-linear\nLSSE  (I = 100 µA) \nFIG. 3. (a) SatB=0.18 T and T=3 K for different d.Sis given as S=V2ω/I2.The gray shading represents the linear regime, where Sshows\nthe linear dependence of I2. (b) Semi logarithmic plot of Sin the linear (red circles) and non-linear (blue circles) re gimes in the d=8.0µm\nsample at B=0.5 T. (c) Swith various Iin the d=2.0µm sample at T=3 K. (d) Semi logarithmic plot of δ(T)of the linear (red circles)\nand non-linear (blue circles) regimes in the d=8µm sample. The triangles are δof the LSSE from Ref. 8. The error bars represent the 68%\nconfidence level ( ±s.d.).\nV2ωas shown in Fig. 2(a). The T−δbehavior above 20 K for\nthed=8.0 and 15 µm samples qualitatively agrees with the\nprevious LSSE result in Pt/YIG-bulk systems8,30. However,\nbelow 20 K, the present nlSSE and previous LSSE results are\ntotally different; δof the LSSE becomes more outstanding\nwith decreasing Tand reaches δ∼100% at∼3 K21, much\ngreater than the present nlSSE results.\nSignificantly, we found that the disagreement at low tem-\nperatures is relevant to the applied current intensity I. So far,\nthe nlSSE experiments were carried out with I=100µA. Be-\nlow 20 K, however, V2ωturned out to deviate from the I2scal-\ning in this Irange. To see this, we introduce the normalization\nfactor,\nS=V2ω\nI2. (3)\nIfV2ωis proportional to I2,Skeeps a constant with I, which\nwas indeed confirmed above 20 K for I<100µA. Figure 3(a)\nshows the Idependence of Sat 3 K at the low Bof 0.18 T for\nthed=2,8, and 9 µm samples. Stakes almost the same\nvalue for I/lessorsimilar5µA [see the gray colored area in Fig. 3(a)],\nbut for I/greaterorsimilar5µA,Sdecreases with increasing I. We refer the\nformer region to the linear regime ( V2ω∝I2), while the lat-\nter to the non-linear regime ( V2ω/negationslash∝I2). In Fig. 3(b), we plot\ntheTdependence of Sin the linear and non-linear regimes at\nB=0.5 T for the d=8µm sample. The difference in Sbe-\ntween the linear and non-linear regimes becomes significant\nwith decreasing T, and at 3 K Sin the linear regime is about 4\ntimes greater than that in the non-linear regime. Important ly,\ntheBdependence of V2ωandδalso vary between the linear\nand non-linear regimes. In Fig. 3(c), we show representativ e\nresults on V2ωversus Bwith several Ivalues at 3 K for the\nd=2.0µm sample. In the linear regime (for I=3µA), clear\nB-induced V2ωsuppression was observed ( δ=78 %). By in-\ncreasing Iand entering into the non-linear regime, however,\ntheB-induced V2ωreduction becomes less visible and, when\nI=100µA,V2ωis almost flat against B(δ=−0.1 %). In Fig.\n3(d), we summarize the δvalues as a function of Tobtained\nin the linear (red filled circles) and non-linear (blue filled cir-\ncles) regimes for the d=8.0µm sample and compare them\nto the previous LSSE result (gray filled triangles)21. Interest-\ningly, the Tdependence of δfor the nlSSE agrees well withthat for the LSSE.\nThe matching of the T−δresults in the low- Trange be-\ntween the nlSSE in the linear regime and the LSSE indicates\nthat the same mechanism governs the B-induced suppression.\nIn Ref. 8 and 21, the Tdependence of δfor the LSSE at\nlowTs was well reproduced based on a conventional LSSE\ntheory in which the effect of the Zeeman-gap opening in a\nmagnon dispersion ( ∝gµBB, where gis the g-factor and µBis\nthe Bohr magneton) was taken into account; the competition\nbetween thermal occupation of the magnon mode relevant for\nthe LSSE (whose energy is of the order of kBT) and the Zee-\nman gap ( gµBB) dominates the B-induced LSSE reduction.\nWhen kBT≪gµBB(≈10 K at 8 T), magnons cannot be ther-\nmally excited, leading to the suppression of the LSSE (see\nFig. 3(d)). Our results indicate that the same scenario is va lid\nalso for the nlSSE in the linear regime.\nFinally, we discuss the non-linear feature of the nlSSE.\nBoth the Sandδvalues of the nlSSE in the non-linear regime\ngradually increase with decreasing T[see Figs. 3(b) and 3(d)].\nHowever, their increasing rates are much smaller than those\nfor the linear regime; both Sandδat 3 K in the non-linear\nregime are ∼4 times smaller than those at the same Tfor\nthe linear regime and also comparable to those at 12 K for\nthe linear regime. These results suggest that the energy sca le\nof magnons driving the nlSSE in the non-linear regime at 3\nK may be much higher than the thermal energy kBTat 3 K\nand the Zeeman energy gµBBat 8 T. We note that, in the\nnon-linear regime, the system temperature at least remains un-\nchanged during the measurements, indicting that temperatu re\nrise due to the Joule heating is negligible. Furthermore, we\nfound that, in the non-linear regime of I=100 µA, the in-\ntensity of magnon-polaron dips at 3 K at B=2.5 T (9.2 T) is\nsmaller (larger) than that in the linear regime of I=3µA at\nthe same T[see the dip structures marked by blue (red) trian-\ngles in Fig. 3(c)]. Here, the dip at the low B(high B) originates\nfrom the spin currents carried by hybridized magnon −TA-\nphonon (magnon −LA-phonon) modes with the fixed energy\nofEMTA≈6 K ( EMLA≈26 K). The dip intensity should\nthereby be maximized when the magnon mode at the energy\nofEMTA(EMLA) is most significantly occupied under the con-\ndition of kBT≈6 K (26 K), and apart from this temperature\nthe intensity of magnon-polaron dip decreases. Therefore,4\nthe small (large) magnon-polaron dip at B=2.5 T (9.2 T)\nat 3 K in the non-linear regime also indicates the over occu-\npation of high-energy magnons than that expected from the\nthermal energy kBTat 3 K, as with the Sandδresults dis-\ncussed above. Future work should address the origin of such\nhigh-nonequilibrium state realized in this regime.\nIn summary, we systematically investigated the nonlocal\nspin Seebeck effect (nlSSE) in the lateral Pt/YIG/Pt system s\nas functions of separation distance ( d), magnetic field ( B),\ntemperature ( T), and excitation current ( I). We found that\nbelow 20 K, the nlSSE voltage V2ωdeviates from the conven-\ntional I2scaling for I/greaterorsimilar5µA. In this non-linear regime, the\namplitudes of V2ωandB-induced signal reduction δbecome\nsmaller than those in the linear regime, where V2ω∝I2and\nI<5µA. In the linear regime, the Tdependence of δof the\nnlSSE agrees well with that of the longitudinal SSE (LSSE),\nwhich can be attributed to the suppression of magnon excita-\ntion by the Zeeman effect. Our results provide an important\nclue in unraveling the B-induced suppression of the nlSSE and\nuseful information on the non-linear effect in nonlocal spi n\ntransport at low temperatures.\nWe thank G. E. W. Bauer, B. J. van Wees, L. J. Cornelis-\nsen, J. Shan, T. Kuschel, F. Casanova, J. M. Gomez-Perez,\nS. Takahashi, Z. Qiu, Y . Chen, and R. Yahiro for fruitful dis-\ncussion, and K. Nagase for technical help. This work is a\npart of the research program of ERATO Spin Quantum Rec-\ntification Project (No. JPMJER1402) from JST, the Grant-\nin-Aid for Scientific Research on Innovative Area Nano Spin\nConversion Science (No. JP26103005), the Grant-in-Aid for\nScientific Research (S) (No. JP19H05600), and Grant-in-\nAid for Research Activity Start-up (No. JP19K21031) from\nJSPS KAKENHI, JSPS Core-to-Core program, the Interna-\ntional Research Center for New-Concept Spintronics Device s,\nWorld Premier International Research Center Initiative (W PI)\nfrom MEXT, Japan. 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You2,\u0003\n1Quantum Physics and Quantum Information Division,\nBeijing Computational Science Research Center, Beijing 100193, China\n2Interdisciplinary Center of Quantum Information and Zhejiang Province Key Laboratory of Quantum Technology and Device,\nDepartment of Physics and State Key Laboratory of Modern Optical Instrumentation, Zhejiang University, Hangzhou 310027, China\n(Dated: March 12, 2019)\nWe develop a theory for the magnon Kerr e \u000bect in a cavity magnonics system, consisting of magnons in a\nsmall yttrium iron garnet (YIG) sphere strongly coupled to cavity photons, and use it to study the bistability in\nthis hybrid system. To have a complete picture of the bistability phenomenon, we analyze two di \u000berent cases in\ndriving the cavity magnonics system, i.e., directly pumping the YIG sphere and the cavity, respectively. In both\ncases, the magnon frequency shifts due to the Kerr e \u000bect exhibit a similar bistable behavior but the corresponding\ncritical powers are di \u000berent. Moreover, we show how the bistability of the system can be demonstrated using\nthe transmission spectrum of the cavity. Our results are valid in a wide parameter regime and generalize the\ntheory of bistability in a cavity magnonics system.\nI. INTRODUCTION\nOwing to the fundamental importance and promising ap-\nplications in quantum information processing, hybrid quan-\ntum systems consisting of di \u000berent subsystems have recently\ndrawn considerable attention [1, 2]. Among them, the spin\nensemble in a single-crystal yttrium iron garnet (YIG) sam-\nple coupled to a cavity mode was theoretically proposed [3–\n5] and experimentally demonstrated [6–11] in the past few\nyears. In contrast to spin ensembles in dilute paramagnetic\nsystems, e.g., nitrogen-vacancy centers in diamond [12], the\nferromagnetic YIG material possesses a higher spin density\n(\u00182:1\u00021022cm\u00003) and essentially is completely polarized\nbelow the Curie temperature ( \u0018559 K) [13]. It is found that\na strong coupling between the microwave cavity mode and\nthe spin ensemble in a small YIG sample with a low damp-\ning rate can be achieved [6–10], which is a challenging task\nfor spin ensembles in paramagnetic materials. In this cav-\nity magnonics system, many exotic phenomena, such as cav-\nity magnon-polaritons [14–16], magnon Kerr e \u000bect [17–19],\nbidirectional microwave-optical conversion [20], ultrastrong\ncoupling [21, 22], magnon dark modes [23], cavity spintron-\nics [24, 25], optical manipulation of the system [26], synchro-\nnized spin-photon coupling [27], strong interlayer magnon-\nmagnon coupling [28], cooperative polariton dynamics [29]\nand non-Hermitian physics [30–32] have been investigated.\nMoreover, the coupling of magnons to other quantum sys-\ntems, e.g., the superconducting qubit [33, 34], phonons [35]\nand optical whispering gallery modes [36–42] was also imple-\nmented.\nThe cavity magnon polaritons are new quasiparticles re-\nsulting from the strong coupling of magnons to cavity pho-\ntons [14–16]. In Ref. [17], the bistability of the cavity magnon\npolaritons was experimentally demonstrated by directly driv-\ning a small YIG sphere placed in a microwave cavity, and\nthe conversion from magnetic to optical bistability was also\nobserved. However, a special case was focused on there by\n\u0003jqyou@zju.edu.cnconsidering the situation that only the lower-branch polari-\ntons were much generated [17]. In fact, to have a complete\npicture of the bistability phenomenon, one needs to study the\nmore general case with both lower- and upper-branch polari-\ntons considerably generated and also consider the coupling\nbetween them. Moreover, one can use a drive tone supplied by\na microwave source to pump the cavity [43] instead of the YIG\nsphere and tune the drive-field frequency from on-resonance\nto far-o \u000b-resonance with the magnons. These important issues\nwere not studied in Ref. [17].\nIn this work, we develop a theory to study the bistability\nof the cavity magnonics system in a wide parameter regime,\nwhich applies to the di \u000berent cases mentioned above. In\nsect. II, we present a theoretical model to describe the cav-\nity magnonics system. This hybrid system consists of a mi-\ncrowave cavity strongly coupled to the magnons in a small\nYIG sphere which is magnetized by a static magnetic field. In\ncomparison with the model of two strongly-coupled harmonic\noscillators [7–9], there is an additional Kerr term of magnons\nin the Hamiltonian of the system, resulting from the magne-\ntocrystalline anisotropy in the YIG [44, 45]. In sect. III, we\ndevelop the theory for the bistability of the cavity magnonics\nsystem. We analyze two di \u000berent cases of driving the hybrid\nsystem corresponding to two experimental situations [18, 43],\ni.e., directly pumping the YIG sphere and the cavity, respec-\ntively. In both cases, the magnon frequency shifts due to the\nKerr e \u000bect exhibit a similar bistable behavior but the corre-\nsponding critical powers are di \u000berent. Here the positive (neg-\native) Kerr coe \u000ecient corresponds to the blue-shift (red-shift)\nof the magnon frequency. When the cavity and Kittel modes\nare on-resonance (o \u000b-resonance), the critical power for driv-\ning the cavity is approximately equal to (much larger than) the\ncritical power for driving the YIG sphere. Finally, in sect. IV,\nwe derive the transmission coe \u000ecient of the cavity with the\nsmall YIG sphere embedded and show how the bistability of\nthe system can be demonstrated via the transmission spectrum\nof the cavity.\nOur results bring the studies of cavity magnonics from the\nlinear to nonlinear regime. Compared with other hybrid sys-\ntems, the cavity magnonics system owns good tunabilities\nwith, e.g., the magnon frequency, the cavity-magnon interac-arXiv:1903.03754v1  [quant-ph]  9 Mar 20192\ntion [31], the drive power, and the drive-field frequency. The\neasily controllable bistability of the cavity magnonics sys-\ntem may have promising applications in memories [46, 47],\nswitches [48, 49], and the study of the dissipative phase tran-\nsition [50, 51]. In the future, more nonlinear phenomena such\nas auto-oscillations and chaos [52] may be explored by us-\ning an even stronger drive field and a smaller YIG sphere to\nenhance the nonlinearity of the cavity magnonics system.\nII. THE HAMILTONIAN OF THE SYSTEM\nAs schematically shown in Fig. 1, we study a system con-\nsisting of a small YIG sphere (with the order of submilimeter\nor milimeter in size) coupled to a three-dimensional (3D) rect-\nangular microwave cavity via the magnetic field of the cavity\nmode. Here we focus on the case in which the YIG sphere is\nuniformly magnetized to saturation by a bias magnetic field\nB0=B0ezin the z-direction, where ei,i=x;y;z, are the\nunit vectors in the rectangular coordinate system. This cor-\nresponds to the Kittel mode of spins in the YIG sphere, i.e.,\nthe uniform procession mode with homogeneous magnetiza-\ntion [18]. In this mode, the Heisenberg-type exchange cou-\npling and the dipole-dipole interaction between spins can be\nneglected since their contributions to the Hamiltonian of the\nsystem become constant in the considered long-wavelength\nlimit [53]. For instance, the Heisenberg interaction between\nany two neighboring spins becomes Ji jsi\u0001sj=Ji js2(i.e., a con-\nstant) for the Kittel mode, because all spins uniformly precess\nin phase together. Here Ji jis the exchange coupling strength\nandsi(sj) is the spin operator of the ith (jth) spin in the YIG\nsphere with the spin quantum number s=~=2. As given in\nAppendixes A and B, this hybrid system can be described us-\ning a nonlinear Dicke model (setting ~=1)\nHs=!caya\u0000\rB0Sz+DxS2\nx+DyS2\ny+DzS2\nz\n+gs(S++S\u0000)(ay+a);(1)\nwhere aandayare the annihilation and creation operators of\nthe cavity mode at the frequency !c,\r=ge\u0016B=~is the gy-\nromagnetic ration with the g-factor geand the Bohr magne-\nton\u0016B,S=P\njsj\u0011(Sx;Sy;Sz) and S\u0006\u0011Sx\u0006iSyare the\nmacrospin operators with the summationP\njover all spins in\nthe YIG sphere, and gsdenotes the coupling strength between\neach single spin and the cavity mode. The nonlinear terms\nDxS2\nx+DyS2\ny+DzS2\nzin Eq. (1) originate from the magne-\ntocrystalline anisotropy in the YIG [44, 45] and their coe \u000e-\ncients Direly on the crystallographic axis of the YIG, along\nwhich the external magnetic field B0is applied. When the\ncrystallographic axis aligned along B0is [110], the nonlinear\ncoe\u000ecients Diread (see Appendix A)\nDx=3\u00160Kan\r2\n2M2Vm;Dy=9\u00160Kan\r2\n8M2Vm;Dz=\u00160Kan\r2\n2M2Vm; (2)\nwhere\u00160is the vacuum permeability, Kan(>0) is the first-\norder anisotropy constant of the YIG, Mis the saturation mag-\nnetization, and Vmis the volume of the YIG sample. The YIG\nYIG Sphere\nYIG Sphere3D cavity\nB0hc\nx\nyzhcmax\nminΩdωd\nωpεpFIG. 1. Upper panel: schematic diagram of a YIG sphere coupled\nto a 3D microwave cavity. Lower panel: the simulated magnetic-\nfield distribution of the fundamental mode of the cavity, where the\nmagnetic-field amplitude and direction are indicated by the colors\nand blue arrows, respectively. The YIG sphere, which is magnetized\nto saturation by a bias magnetic field B0aligned along the z-direction,\nis mounted near the cavity wall, where the magnetic field hcof the\ncavity mode is the strongest and polarized along x-direction to ex-\ncite the magnon mode in YIG. Either the cavity mode or the magnon\nmode is driven by a microwave field with frequency !dand Rabi\nfrequency \nd. A weak probe field with frequency !pand its cou-\npling strength \"pto the cavity mode is also applied for measuring the\ntransmission spectrum of the cavity.\nsphere is here required to be in the macroscopic regime to\ncontain a su \u000ecient number of spins. Usually, the diameter of\nthe YIG sphere used in the experiment varies from 0.1 mm to\n1 mm.\nDirectly pumping the YIG sphere with a microwave field\nof the frequency !d, the interaction Hamiltonian is (see Ap-\npendix B)\nHd= \n s(S++S\u0000)(ei!dt+e\u0000i!dt); (3)\nwhere \nsis the drive-field Rabi frequency. In the experiment,\na drive coil near the YIG sample goes out of the cavity through\none port of the cavity connected to a microwave source [18].\nAlso, a probe field at frequency !pacts on the input port of\nthe cavity, which can be described by the Hamiltonian\nHp=\"p(ay+a)(ei!pt+e\u0000i!pt); (4)\nwhere\"pis the coupling strength between the cavity and the\nprobe field. In the experiment, compared with the drive field,\nthe probe tone is usually extremely weak, and the probe-field\nfrequency!pis tuned to be o \u000bresonance with the drive-field\nfrequency!d, so as to avoid interference between them [17].\nNow, we can write the total Hamiltonian H=Hs+Hd+Hp3\nof the hybrid system in Fig. 1 as\nH=!caya\u0000\rB0Sz+DxS2\nx+DyS2\ny+DzS2\nz\n+gs(S++S\u0000)(ay+a)+ \n s(S++S\u0000)(ei!dt+e\u0000i!dt)\n+\"p(ay+a)(ei!pt+e\u0000i!pt):\n(5)\nUsing the Holstein-Primako \u000btransformation [54],\nS+=p\n2S\u0000bybb;\nS\u0000=byp\n2S\u0000byb;\nSz=S\u0000byb;(6)\nwe can convert the macrospin operators to the magnon opera-\ntors, where by(b) is the magnon creation (annihilation) opera-\ntor,S=\u001asVmsis the spin quantum number of the macrospin,\nand\u001as=2:1\u00021022cm\u00003is the net spin density of the YIG\nsphere. Under the condition of low-lying excitations with\nhbybi=2S\u001c1,p\n2S\u0000bybcan be expanded, up to the first\norder of byb=2S, asp\n2S\u0000byb\u0019p\n2S(1\u0000byb=4S), so\nS+\u0019p\n2S\u0012\n1\u0000byb\n4S\u0013\nb;\nS\u0000\u0019p\n2S by\u0012\n1\u0000byb\n4S\u0013\n:(7)\nSubstituting the expression Sz=S\u0000bybin Eq. (6) and Eq. (7)\ninto Eq. (5), as well as neglecting the constant terms and\nthe fast oscillating terms via the rotating-wave approximation\n(RWA) [55], we can reduce the total Hamiltonian Hto\nH=!caya+!mbyb+Kbybbyb+gm \n1\u0000byb\n4S!\n(ayb+aby)\n+ \n d \n1\u0000byb\n4S!\n(bye\u0000i!dt+bei!dt)\n+\"p(aye\u0000i!pt+aei!pt);\n(8)\nwhere\n!m=\rB0+13\u00160\u001assKan\r2\n8M2(9)\nis the angular frequency of the magnon mode,\nK=\u000013\u00160Kan\r2\n16M2Vm(10)\nis the Kerr nonlinear coe \u000ecient, gm\u0011p\n2S gsis the collec-\ntively enhanced magnon-photon coupling strength and \nd\u0011p\n2S\nsis the Rabi frequency.\nHowever, when the crystallographic axis aligned along B0\nis [100], the nonlinear coe \u000ecients Diin Eq. (2) become (see\nAppendix A)\nDx=Dy=0;Dz=\u00160Kan\r2\nM2Vm: (11)\n0306090120150gm/2π (MHz)(a)\n0.0 0.2 0.4 0.6 0.8 1.010-1100101102|K/2π| (nHz)\nd (mm) [100]\n [110](b)FIG. 2. (a) The coupling strength gmwith gs=2\u0019=39 mHz and\n(b) the Kerr coe \u000ecient K(log scale) as a function of the diameter\ndof the YIG sphere. The black solid (red dashed) curve in (b) cor-\nresponds to the case with the crystalline axis [100] ([110]) aligned\nalong B0. Other parameters are \u00160Kan=2480 J=m3,M=196 kA /m,\nand\r=2\u0019=28 GHz /T.\nIn the RWA, the Hamiltonian Hin Eq. (5) is also converted to\nthe same form as in Eq. (8) using Eq. (7) and the expression\nSz=S\u0000bybin Eq. (6), but the magnon frequency is\n!m=\rB0\u00002\u00160\u001assKan\r2\nM2; (12)\nand the Kerr coe \u000ecient is\nK=\u00160Kan\r2\nM2Vm: (13)\nIt is worth noting that the magnon frequency !mis irrelevant\nto the volume Vmof the YIG sphere, but the Kerr coe \u000ecient\nis inversely proportional to Vm, i.e., K/V\u00001\nm. Thus, the\nKerr e \u000bect of magnons can become important for a small YIG\nsphere. Moreover, the Kerr coe \u000ecient becomes positive (neg-\native) when the crystallographic axis [100] ([110]) of the YIG\nis aligned along the static field B0.\nIn the experiment, instead of using a drive tone supplied\nby a microwave source to directly pump the YIG sphere, one\ncan also apply a drive field with frequency !ddirectly on the\ncavity [43]. In this case, the total Hamiltonian of the hybrid\nsystem under the RWA is written as\nH=!caya+!mbyb+Kbybbyb+gm \n1\u0000byb\n4S!\n(ayb+aby)\n+ \n d(aye\u0000i!dt+aei!dt)+\"p(aye\u0000i!pt+aei!pt):\n(14)4\nNote that in both cases, we use the same symbols \ndand!d\nfor simplicity.\nHere we estimate the collective coupling strength gmand\nthe Kerr coe \u000ecient K. As shown in Fig. 2, we plot gm\nand Kversus the diameter dof the YIG sphere, where\nwe choose the experimentally obtained single-spin coupling\nstrength gs=2\u0019=39 mHz [7]. From Fig. 2, it can be seen that\nwhen the diameter dis reduced from 1 mm to 0.1 mm (the\nusual size of the YIG sphere used in experiments), the cou-\npling strength gmdecreases one order of magnitude but the\nKerr coe \u000ecient Kincreases from 0.05 nHz to 100 nHz, i.e., a\nthree orders of magnitude increase. Thus, it is vital to choose\na YIG sphere of suitably small size, so as to have strong non-\nlinear e \u000bect of magnons but still maintain the hybrid system\nin the strong coupling regime.\nIII. THE NONLINEAR EFFECT ON THE HYBRID\nSYSTEM\nA. Pump the YIG sphere\nWhen directly pumping the YIG sphere with a drive field,\nconsiderable magnons are usually generated in the YIG\nsphere. The magnon number operator bybcan be expressed\nas a sum of the mean value hbybiand the fluctuation \u000ebyb,\ni.e.,byb=hbybi+\u000ebyb, so\nbybbyb=(hbybi+\u000ebyb)(hbybi+\u000ebyb)\n=(hbybi)2+2hbybi\u000ebyb+(\u000ebyb)2:(15)\nWhen a considerable number of magnons are generated in the\nYIG sphere by the drive field, i.e., hbybi\u001dh\u000ebybi, we can\nneglect the high-order fluctuation term and have\nbybbyb\u0019(hbybi)2+2hbybi\u000ebyb\n=\u0000(hbybi)2+2hbybibyb:(16)\nUnder this mean-field approximation (MFA), the Hamiltonian\nin Eq. (8) can then be written as\nH=!caya+(!m+2Khbybi)byb\n+ \n1\u0000hbybi\n4S!\ngm(ayb+aby)\n+ \n1\u0000hbybi\n4S!\n\nd(bye\u0000i!dt+bei!dt)\n+\"p(aye\u0000i!pt+aei!pt):(17)\nNote that the generated magnons may yield an appreciable\nshift\u0001m=2Khbybito the magnon frequency [17, 18]. How-\never, if the drive field is not too strong, the condition hbybi\u001c\n2Scan easily be satisfied owing to the very large number of\nspins in the YIG sphere. Therefore, we can take the approxi-\nmation 1\u0000hbybi=(4S)\u00191 in Eq. (17), and then the Hamilto-\nnian becomes\nH=!caya+(!m+ \u0001 m)byb+gm(ayb+aby)\n+ \n d(bye\u0000i!dt+bei!dt)+\"p(aye\u0000i!pt+aei!pt):(18)With the Heisenberg-Langevin approach [55], we can de-\nscribe the dynamics of the coupled hybrid system by the fol-\nlowing quantum Langevin equations:\nda\ndt=\u0000i(!c\u0000i\u0014c)a\u0000igmb\u0000i\"pe\u0000i!pt+p\n2\u0014cain;\ndb\ndt=\u0000i(!m+ \u0001 m\u0000i\rm)b\u0000igma\u0000i\nde\u0000i!dt+p\n2\rmbin;\n(19)\nwhere\u0014c=\u0014i+\u0014o+\u0014intis the decay rate of the cavity mode,\nwith\u0014i(\u0014o) being the decay rate of the cavity mode due to the\ninput (output) port and \u0014intbeing the intrinsic decay rate of\nthe cavity mode, \rmis the damping rate of the Kittel mode,\nandainandbinare the input noise operators related to the\ncavity and Kittel modes, whose mean values are zero, i.e.,\nhaini=hbini=0. These input noise operators result from\nthe respective environments of the cavity and Kittel modes,\nwhich include both quantum noise and thermal noise. If we\nwrite a=hai+\u000eaandb=hbi+\u000eb, wherehai(hbi) is the\nexpectation value of the operator a(b) and\u000ea(\u000eb) is the corre-\nsponding fluctuation, it follows from Eq. (19) that the steady-\nstate valueshaiandhbisatisfy\ndhai\ndt=\u0000i(!c\u0000i\u0014c)hai\u0000igmhbi\u0000i\"pe\u0000i!pt;\ndhbi\ndt=\u0000i(!m+ \u0001 m\u0000i\rm)hbi\u0000igmhai\u0000i\nde\u0000i!dt:(20)\nExperimentally, the drive field is much stronger than the probe\nfield, i.e.,\"p\u001c\nd, so the probe field can be treated as a\nperturbation. We assume that the expectation values haiand\nhbican be written as\nhai=A0e\u0000i!dt+A1e\u0000i!pt;\nhbi=B0e\u0000i!dt+B1e\u0000i!pt;(21)\nwhere the amplitudes A0andB0are the expectation values of\noperators aandbin the absence of the probe field, and the\namplitudes A1andB1result from the perturbation (i.e., probe\nfield). A1andB1are significantly smaller than A0andB0.\nIn this case, the magnon frequency shift \u0001mcan be written\nas\u0001m=2KjB0j2. At the steady states for both A0andB0\n(A1andB1),dA0=dt=0 and dB0=dt=0 (dA1=dt=0 and\ndB1=dt=0). Then, we have\n(\u000ec\u0000i\u0014c)A0+gmB0=0;\n(\u000em+ \u0001 m\u0000i\rm)B0+gmA0+ \n d=0;(22)\nand\n\u0002(!c\u0000!p)\u0000i\u0014c\u0003A1+gmB1+\"p=0;\u0002(!m+ \u0001 m\u0000!p)\u0000i\rm\u0003B1+gmA1=0;(23)\nwhere\u000ec(m)\u0011!c(m)\u0000!dis the frequency detuning of the cavity\nmode (Kittel mode) relative to the drive field. The first equa-\ntion in Eq. (22) can be expressed as A0=\u0000gmB0=(\u000ec\u0000i\u0014c).\nBy inserting this expression of A0into the second equation in\nEq. (22), we obtain\n(\u000e0\nm+ \u0001 m\u0000i\r0\nm)B0+ \n d=0; (24)5\n0 100 200 30005101520\n(a) ∆=0, K>0\n  \n0 100 200 30005101520\n(b) ∆=3gm, K>0\n  \n0 100 200 300-20-15-10-50\n(c) ∆=0, K<0\n  \n0 100 200 300-20-15-10-50\n(d) ∆=3gm, K<0\n  ∆/2π (MHz) /2π (MHz)\nP (mW) P (mW)m ∆m\nd d\nFIG. 3. The magnon frequency shift \u0001mversus the drive power Pd\nfor di \u000berent \u0001andK, where \u0001 =!c\u0000!mis the frequency de-\ntuning of the cavity from the magnon. (a) Frequency shift \u0001mver-\nsusPdwhen \u0001=0 and K>0. Here\u000em=2\u0019=36:2 MHz for the\n(black) solid curve, \u000em=2\u0019=35 MHz for the (red) dashed curve, and\n\u000em=2\u0019=34 MHz for the (blue) dotted curve. (b) Frequency shift\n\u0001mversus Pdwhen \u0001 = 3gmandK>0. Here\u000em=2\u0019=9 MHz for\nthe (black) solid curve, \u000em=2\u0019=4 MHz for the (red) dashed curve,\nand\u000em=2\u0019=1 MHz for the (blue) dotted curve. (c) Frequency\nshift \u0001mversus Pdwhen \u0001=0 and K<0. Here\u000em=2\u0019=43 MHz\nfor the (black) solid curve, \u000em=2\u0019=45 MHz for the (red) dashed\ncurve, and\u000em=2\u0019=47 MHz for the (blue) dotted curve. (d) Fre-\nquency shift \u0001mversus Pdwhen \u0001 = 3gmand K<0. Here\n\u000em=2\u0019=15 MHz for the (black) solid curve, \u000em=2\u0019=18 MHz for\nthe (red) dashed curve, and \u000em=2\u0019=21 MHz for the (blue) dotted\ncurve. The constant is c=(2\u0019)3=2 MHz3=mW in both (a) and (b),\nandc=(2\u0019)3=\u00002 MHz3=mW in both (c) and (d). Other parameters\naregm=2\u0019=40 MHz, and \u0014c=2\u0019=\rm=2\u0019=2 MHz.\nwhere the e \u000bective frequency detuning \u000e0\nmand the e \u000bective\ndamping rate \r0\nmof the Kittel mode are given, respectively, by\n\u000e0\nm=\u000em\u0000\u0011\u000ec; \r0\nm=\rm+\u0011\u0014c; (25)\nwith\n\u0011=g2\nm=(\u000e2\nc+\u00142\nc): (26)\nUsing Eq. (24) and its complex conjugate expression, we ob-\ntain\n\u0014\u0000\u000e0\nm+ \u0001 m\u00012+\r0\nm2\u0015\n\u0001m\u0000cPd=0; (27)\nwhere 2 Kj\ndj2=cPd, with Pdbeing the drive power and c\na coe \u000ecient characterizing the coupling strength between the\ndrive field and the Kittel mode.\nNote that Eq. (27) is a cubic equation for the magnon fre-\nquency shift \u0001m. Under specific parameter conditions, \u0001mhas\ntwo switching points for the bistability, at which there must be\ndPd=d\u0001m=0, i.e.,\n3\u00012\nm+4\u000e0\nm\u0001m+\u000e0\nm2+\r0\nm2=0: (28)\n9.96 9.98 10.00 10.02 10.0405101520\n(a) K>0 ∆m/2π (MHz)\nωm/2π (GHz)\n Pd=80 mW\nPd=140 mW\nPd=200 mW\n9.98 10.00 10.02 10.04 10.06-20-15-10-50\nωm/2π (GHz)∆m/2π (MHz)(b) K<0 Pd=80 mW\nPd=140 mW\nPd=200 mWFIG. 4. The magnon frequency shift \u0001mversus!mfor di \u000berent val-\nues of the drive power Pdin the cases of (a) K>0 and (b) K<0.\nHere Pd=80 mW for the (black) solid curve, Pd=140 mW for\nthe (red) dashed curve, and Pd=200 mW for the (blue) dotted\ncurve. The constant is c=(2\u0019)3=2 MHz3=mW in (a) and c=(2\u0019)3=\n\u00002 MHz3=mW in (b);!c=2\u0019=10 GHz and \u000ec=2\u0019=35 MHz in both\n(a) and (b). Other parameters are the same as in Fig. 3(a).\nAccording to the root discriminant of the quadratic equation\nwith one unknown, if Eq. (28) has two real roots (correspond-\ning to the two switching points), \u000e0\nmand\r0\nmmust satisfy the\nrelation 4\u000e0\nm2\u000012\r0\nm2>0, i.e.,\n\u000e0\nm<\u0000p\n3\r0\nm;K>0;\n\u000e0\nm>p\n3\r0\nm;K<0:(29)\nWhen 4\u000e0\nm2\u000012\r0\nm2=0, Eq. (28) has only one real so-\nlution and the two switching points coalesce to one point,\nwhich means that the bistability disappears. In the case of\n4\u000e0\nm2\u000012\r0\nm2=0, the corresponding power Pd, called the\ncritical power, is given by\nPm= +(\u0000)8p\n3\r0\nm3\n9c; (30)\nwith cbeing positive (negative) for K>0 (K<0). For\n4\u000e0\nm2\u000012\r0\nm2<0, Eq. (28) has no real solution and the\nmagnon frequency shift \u0001mincreases monotonically with the\ndrive power Pd.\nIn Fig. 3(a), the magnon frequency shift \u0001mversus the driv-\ning power Pdis plotted for several di \u000berent values of detuning\n\u000emwhen \u0001=0 and K>0, where \u0001\u0011!c\u0000!mis the frequency6\ndetuning of the cavity from the magnon. In a certain parame-\nter regime, \u0001mexhibits a bistable behavior. It is clearly shown\nthat the value of the detuning \u000embetween the Kittel mode and\nthe drive field is crucial for the bistability of \u0001m. Moreover,\nthe frequency shift \u0001mversus the driving power Pdin the case\nof\u0001 =3gmandK>0 is shown in Fig. 3(b) for di \u000berent values\nof\u000em. We also see hysteresis loops. In both the on-resonance\nand o \u000b-resonance cases, we further study the relationship be-\ntween the magnon frequency shift \u0001mand the drive power Pd,\nas shown in Figs. 3(c) and 3(d), when K<0. We also ob-\nserve the similar bistability, but the magnon frequency shift is\nnegative because the Kerr coe \u000ecient is negative in this case.\nFrom the cubic equation in Eq. (27), we can further study\nthe magnon frequency shift \u0001mversus the e \u000bective frequency\ndetuning\u000e0\nm. In the experiment, \u000e0\nmcan be tuned by either\nsweeping the magnon frequency !m(i.e., the bias magnetic\nfield B0) or sweeping the drive-field frequency !d. Because\n\u0001mhas similar behaviors when sweeping !mor!d, here we\nonly focus on the magnon frequency shift \u0001mversus!m. Fig-\nure 4(a) displays the magnon frequency shift \u0001mversus!mfor\ndi\u000berent values of the drive power Pdwith a fixed !dwhen\nK>0. With a small drive power, \u0001mdepends nonlinearly\non!mbut has no bistable behavior [see the black solid curve\nin Fig. 4(a)]. When increasing the drive power Pd,\u0001mversus\n!mshows the bistability and the hysteresis-loop area increases\nwith Pd[see the red dashed curve and the blue dotted curve in\nFig. 4(a)]. In the case of K<0, we plot \u0001mversus!min\nFig. 4(b). With appropriate parameters, there is also the bista-\nbility but \u0001mis negative.\nB. Pump the cavity\nWhen a microwave field is applied to directly pump the cav-\nity rather than the YIG sphere, linearizing the nonlinear terms\nvia the MFA, the total Hamiltonian in Eq. (14) becomes\nH=!caya+(!m+ \u0001 m)byb+gm(ayb+aby)\n+ \n d(aye\u0000i!dt+aei!dt)+\"p(aye\u0000i!pt+aei!pt);(31)\nwhere we have also used the approximation 1 \u0000hbybi=(4S)\u0019\n1. When directly driving the cavity, the dynamics of the cou-\npled hybrid system follows the quantum Langevin equations:\nda\ndt=\u0000i(!c\u0000i\u0014c)a\u0000igmb\u0000i\nde\u0000i!dt\u0000i\"pe\u0000i!pt+p\n2\u0014cain;\ndb\ndt=\u0000i(!m+ \u0001 m\u0000i\rm)b\u0000igma+p\n2\rmbin: (32)\nIn this case, the evolution equation of the expectation value\nhai(hbi) is given by\ndhai\ndt=\u0000i(!c\u0000i\u0014c)hai\u0000igmhbi\u0000i\nde\u0000i!dt\u0000i\"pe\u0000i!pt;\ndhbi\ndt=\u0000i(!m+ \u0001 m\u0000i\rm)hbi\u0000igmhai: (33)\n-400 -200 0 200 400\n∆/2π (MHz) Pm\n Pc\n100101102103Critical power (mW)FIG. 5. The critical powers PmandPc(log scale) versus the detun-\ning\u0001when the crystalline axis [100] is aligned along the external\nmagnetic field B0. Other parameters are the same as in Fig. 3(a).\nSubstituting Eq. (21) into Eq. (33), A1andB1also satisfy\nEq. (23), but the steady-state equations of A0andB0become\n(\u000ec\u0000i\u0014c)A0+gmB0+ \n d=0;\n(\u000em+ \u0001 m\u0000i\rm)B0+gmA0=0:(34)\nEliminating A0in Eq. (34), we have\n(\u000e0\nm+ \u0001 m\u0000i\r0\nm)B0\u0000\ne\u000b=0; (35)\nwhere \ne\u000b=gm\nd=(\u000ec\u0000i\u0014c) is the e \u000bective driving strength\non the YIG sphere, which depends not only on the Rabi fre-\nquency \ndbut also on the coupling strength gmand the fre-\nquency detuning \u000ecbetween the cavity mode and the drive\nfield. From Eq. (35), it is straightforward to obtain a cubic\nequation for \u0001m,\n\u0014\u0000\u000e0\nm+ \u0001 m\u00012+\r0\nm2\u0015\n\u0001m\u0000\u0011cPd=0; (36)\nwith\u0011given in Eq. (26). Comparing Eq. (36) with Eq. (27),\n\u0011Pdis the e \u000bective drive power on the YIG sphere. By substi-\ntuting the drive power Pdin Eq. (27) with the e \u000bective drive\npower\u0011Pd, the bistable condition in Eq. (29) is still valid, but\nthe critical power PcforK>0 (K<0) now becomes\nPc\u0011Pm\n\u0011= +(\u0000)8p\n3\r0\nm3\n9\u0011c: (37)\nAlso, cis positive (negative) when K>0 (K<0).\nBecause the values of Pm(Pc) are approximately equal for\na specific value of \u0001in both cases of aligning the crystalline\naxes [100] and [110] of the YIG along the external magnetic\nfield B0, we only study the critical powers PmandPcver-\nsus the detuning \u0001when the axis [100] is aligned along B0\n(K>0). As shown in Fig. 5, PmandPcare approximately\nequal in the near-resonance region jgm=\u0001j>1, but Pcis much\nlarger than Pmin the dispersive regime jgm=\u0001j\u001c1. The un-\nderlying physics is that in the case of j\u0001j\u001dgm, the magnon\nand cavity are nearly decoupled, so directly driving the cavity\nhas weak influence on the magnon subsystem and then it be-\ncomes hard to observe the nonlinear e \u000bect in the hybrid sys-\ntem. In the experiment, it is di \u000ecult to apply an extremely7\n(a)\n9.909.9510.0010.0510.10ωp/2π (GHz)\n9.909.9510.0010.0510.10ωp/2π (GHz)(b)\n9.90 9.95 10.00 10.05\nωm/2π (GHz)9.90 9.95 10.00 10.05\nωm/2π (GHz)9.90 9.95 10.00 10.05 10.10\nωm/2π (GHz)(d)(c) (e)\n(f)|S21(ωp)|2\n(dB)-80-400\nFIG. 6. Transmission spectrum of the cavity-magnon system versus the probe-field frequency !pand the magnon frequency !mwhen the\ndrive-field frequency is fixed at \u000ec=35 MHz. (a) The transmission spectrum when Pd=0. (b) The transmission spectrum when Pd=80 mW\nandK>0. (c) and (d) the transmission spectrum in the case of Pd=200 mW and K>0 when sweeping the external field B0up and down.\n(e) and (f) the transmission spectrum in the case of Pd=200 mW and K<0 when sweeping the external field B0up and down. The sweep\ndirections and the switching points of the bistability are indicated, respectively, by the black arrows and the vertical black dashed lines. Here\nwe choose\u0014i=2\u0019=\u0014o=2\u0019=0:7 MHz, and other parameters are the same as in Fig. 4.\nstrong microwave field to pump a cavity. Therefore, in the\ndispersive regime, it is better to directly pump the magnon to\nobserve the nonlinear e \u000bect of the hybrid system. In the case\nof aligning the crystalline axis [110] along B0(K<0), the\nabove conclusions are still valid.\nIV . TRANSMISSION SPECTRUM\nIn the experiment, one can probe the bistability via the\ntransmission spectrum of the cavity. In this section, we show\nthe e\u000bect of the magnon frequency shift \u0001m(due to the Kerr\nnonlinearity) on the transmission spectrum of the cavity. From\nEq. (23), the amplitude A1of the cavity field due to the probe\nfield reads\nA1=\u0000i\"p\ni(!c\u0000!p)+\u0014c+ \u0006(!p); (38)\nwhere\n\u0006(!p)=g2\nm\ni(!m+ \u0001 m\u0000!p)+\rm: (39)\nAccording to the input-output theory [55], because there is no\ninput field on the output port, the output of the cavity field\nfrom the output port is\nha(out)\npi=p\n2\u0014ohai=p\n2\u0014oA0e\u0000i!dt+p\n2\u0014oA1e\u0000i!pt;(40)where the first (second) term of the output field ha(out)\npiis due\nto the drive (probe) field. The probe field to be input into\nthe cavity via the input port can be written as [55] ha(in)\npi=\n\u0000i\"pe\u0000i!pt=p2\u0014i. Then, we obtain the transmission coe \u000ecient\nS21(!p) of the cavity at frequency !p,\nS21(!p)\u0011p2\u0014oA1\u0000\u0000i\"p=p2\u0014i\u0001=2p\u0014i\u0014o\ni(!c\u0000!p)+\u0014c+ \u0006(!p);(41)\nwhere the self-energy \u0006(!p), as given in Eq. (39), includes\nthe contribution from the magnon frequency shift \u0001m. Note\nthat the transmission coe \u000ecient given in Eq. (41) is valid in\nboth cases of the drive field applied on the YIG sphere and the\ncavity.\nLet us consider the case of directly driving the YIG sphere\nfor an example. In Fig. 6, using Eqs. (41) and (27), we plot the\ntransmission spectrum for the cavity magnonics system ver-\nsus the probe-field frequency !pand the magnon frequency\n!m(which is related to the bias magnetic field B0) for di \u000ber-\nent values of the drive power Pdwhen fixing the drive-field\nfrequency!d. The corresponding \u0001mversus!mcan be found\nin Fig. 4. When the drive field is o \u000b, i.e., Pd=0, a pro-\nnounced avoided crossing of energy levels resulting from the\nstrong coupling between magnons and cavity photons can be\nobserved [see Fig. 6(a)]. Sweeping the magnon frequency !m\nup and down at Pd=80 mW, we obtain a similar transmis-\nsion spectrum [Fig. 6(b)] but it looks di \u000berent from Fig. 6(a) at\naround!m=2\u0019=10 GHz, due to the magnon Kerr e \u000bect. We\nfurther study the transmission spectrum in the case of K>08\n(K<0) in Figs. 6(c) and 6(d) [Figs. 6(e) and 6(f)] when\nPd=200 mW. The arrows indicate the sweep directions of\nthe bias magnetic field B0(i.e.,!m) and the vertical dashed\nlines indicate the switching points of the bistability. Clearly,\nthe transmission spectrum depends on the sweep directions,\ndisplaying the bistability of the system. Therefore, one can\nextract the unique information of the magnon frequency shift\n\u0001mby measuring the cavity transmission spectrum in the ex-\nperiment.\nV . DISCUSSIONS AND CONCLUSIONS\nIn our work, the temperature e \u000bect is not explicitly shown.\nWhen the frequencies of the cavity mode and the magnon\nmode are chosen to be a few gigahertz (the usual values of\n!cand!min the experiment), the numbers of cavity photons\nand magnons excited by the thermal field are about 1 \u0002103\neven at the Curie temperature ( \u0018559 K) of the YIG mate-\nrial [13]. However, when pumping either the YIG sphere or\ncavity, the pumping field generates magnons and cavity pho-\ntons up to 1\u00021016[17] for observing the bistability in cavity\nmagnonics. Therefore, the approximation of neglecting the\ntemperature e \u000bect is reasonable, and our theoretical predic-\ntions are valid below the Curie temperature.\nThe bistability of a cavity magnonics system was experi-\nmentally investigated by directly driving a small YIG sphere\ncoupled to a cavity mode [17] in a special case with only the\nlower-branch polaritons much generated. However, the theory\nused in Ref. [17] fails to accurately describe the bistability\nin the cavity magnonics system when di \u000berent experimental\nconditions are used (e.g., both lower- and upper-branch po-\nlaritons are considerably generated, the cavity [43] rather than\nthe YIG sphere is directly pumped, and the drive-field fre-\nquency is swept from on-resonance to far-o \u000b-resonance with\nthe magnons). It is the limitation of the theory using the po-\nlariton basis in Ref. [17], because the coupling between the\nlower- and upper-branch polaritons is neglected when deriv-\ning the equation for bistability. In these more general cases,\nwe can use the theory developed here.\nIn conclusion, we have studied the Kerr-e \u000bect-induced\nbistability in a cavity magnonics system consisting of a small\nYIG sphere strongly coupled to a microwave cavity and de-\nveloped a theory for it which works in a wide regime of the\nsystem parameters. We analyze two di \u000berent cases of driv-\ning this hybrid system which correspond to the two typical\nexperimental situations [18, 43], i.e., directly pumping the\nYIG sphere and the cavity, respectively. In both cases, the\nmagnon frequency shifts due to the Kerr e \u000bect exhibit a simi-\nlar bistable behavior, but the corresponding critical powers are\ndi\u000berent. Specifically, it is shown that directly driving the cav-\nity needs a larger critical power than directly driving the YIG\nsphere when the magnons are o \u000b-resonance with the cavity\nphotons. Furthermore, we show how the bistability of the cav-\nity magnonics system can be probed using the transmission\nspectrum of the cavity. Our results provide a complete picture\nfor the bistability phenomenon in the cavity magnonics sys-\ntem and also generalize the theory of bistability in Ref. [17].ACKNOWLEDGMENTS\nThis work is supported by the National Key Re-\nsearch and Development Program of China (Grant\nNo. 2016YFA0301200) and the National Natural Science\nFoundation of China (Grant Nos. 11774022 and U1530401).\nAppendix A: The uniformly magnetized YIG sphere\nAs shown in Fig. 1, the YIG sphere used is magnetized to\nsaturation by an externally applied magnetic field B0=B0ez\nalong the z-direction, where ei,i=x;y;z, are the unit vectors\nalong three orthogonal directions. For the magnetized YIG\nsphere, the internal magnetic field Hinin the YIG sphere is\nHin=Hex+Hde+Han; (A1)\nwhere the exchange field Hexis caused by the exchange in-\nteraction, the demagnetization field Hderesults from the mag-\nnetic dipole-dipole interaction, and the anisotropic field Han\nis induced by the magnetocrystalline anisotropy of the YIG.\nWhen Zeeman energy is included, the Hamiltonian of the YIG\nsphere reads [56] (setting ~=1)\nHm=\u0000Z\nVmM\u0001B0d\u001c\u0000\u00160\n2Z\nVmM\u0001Hind\u001c; (A2)\nwhere\u00160is the vacuum permeability, Vmis the volume of the\nYIG sample and M=(Mx;My;Mz) is the magnetization of\nthe YIG sphere.\nFor the uniformly magnetized YIG sphere with a uniform\nmagnetization M, the exchanged field, i.e., the molecular field\nin Weiss theory, is [44, 45] Hex=\u0000\u0003M, with the molecu-\nlar field constant \u0003. The induced demagnetizing field is [57]\nHde=\u0000M=3 for a YIG sphere, but the anisotropic field Han\ndepends on which crystallographic axis of the YIG is aligned\nalong the externally applied static field B0. When the crystal-\nlographic axis [110] is aligned along B0, the anisotropic field\ncan be written as [58]\nHan=\u00003KanMx\nM2ex\u00009KanMy\n4M2ey\u0000KanMz\nM2ez; (A3)\nwhere we only consider the dominant first-order anisotropy\nconstant Kan(>0) and Mis the saturation magnetization.\nThen, the Hamiltonian of the YIG sphere in Eq. (A2) takes\nthe form\nHm=\u0000B0MzVm+\u00160KanVm\n8M2(12M2\nx+9M2\ny+4M2\nz);(A4)\nwhere a constant term (1 +3\u0003)\u00160M2Vm=6, which includes the\ndemagnetization energy and the exchange energy, has been\nignored.\nFor the jth spin in the YIG sphere, the magnetic moment\nismj\u0011\rsj, where\r=ge\u0016B=~is the gyromagnetic ration,\ngeis the g-factor,\u0016Bis the Bohr magneton, and sjis the spin\noperator with the spin quantum number s=1=2. The YIG\nsphere acting as a macrospin has the magnetization [3, 4]\nM=P\njmj\nVm\u0011\rS\nVm; (A5)9\nwhere we have introduced the macrospin operator S=P\njsj\u0011\n(Sx;Sy;Sz), with the summationP\njover all spins in the\nsphere. Substituting Eq. (A5) into Eq. (A4), we have\nHm=\u0000\rB0Sz+DxS2\nx+DyS2\ny+DzS2\nz; (A6)\nwhere the nonlinear coe \u000ecients are\nDx=3\u00160Kan\r2\n2M2Vm;Dy=9\u00160Kan\r2\n8M2Vm;Dz=\u00160Kan\r2\n2M2Vm:(A7)\nHowever, when the crystalline axis [100] is aligned along\nthe bias magnetic field B0, the exchange field and the demag-\nnetization field remain unchanged, but the anisotropic field\nbecomes [58]\nHan=\u00002KanMz\nM2ez: (A8)\nUsing the expressions in Eqs. (A2) and (A5), we can write\nthe Hamiltonian Hmin the same form as in Eq. (A6) but the\nnonlinear coe \u000ecients become\nDx=Dy=0;Dz=\u00160Kan\r2\nM2Vm; (A9)\nwhere we have omitted the constant demagnetization and ex-\nchange energies.\nAppendix B: The YIG sphere coupled to a 3D cavity\nSo far, the Hamiltonian Hmof the YIG sphere has been\nobtained. Then, we derive the Hamiltonian of the cavity\nmagnonics system.\nThe 3D microwave cavity is usually machined from high-\nconductivity copper to have a high Qfactor. When focusing\nonly on one cavity mode (e.g., the fundamental mode), this\n3D resonator can be described by the Hamiltonian\nHc=!c\u0012\naya+1\n2\u0013\n; (B1)where a(ay) denotes the annihilation (creation) operator of\nthe cavity mode with frequency !c.\nTo achieve a strong coupling between magnons and cav-\nity photons, we can place the small YIG sphere near a wall\nof the cavity (see Fig. 1), where the magnetic field hcof\nthe microwave cavity mode becomes the strongest and is po-\nlarized along the x-direction [8]. Also, the static magnetic\nfield B0is aligned perpendicular to hc. The field hcinduces\nthe spin-flipping and excites the magnon mode. In com-\nparison with the microwave cavity, the small dimensions of\nthe YIG sphere permit us to regard the cavity field as be-\ning nearly uniform around the YIG sample. Thus, we can\nwrite hc=\u0000h0(ay+a)ex, with h0=p\n~!c=(\u00160Vc) being the\nmagnetic-field amplitude and Vcthe volume of the cavity. 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A 64, 968 (1951)."    },    {        "title": "2206.04237v1.Magnetically_tunable_zero_index_metamaterials.pdf",        "content": "1 \n Magnetically tunable zero -index metamaterials  \n \nYucong Yang1,2, Yueyang Liu3, Jun Qin1,2, Songgang Cai1,2, Jiejun Su1,2, Peiheng Zhou1,2, Longjiang Deng1,2*,  \nYang Li3* and Lei Bi1,2* \n \n \n1National Engineering Research Centre of Electromagnetic Radiation Control Materials, University \nof Electronic Science and Technology of China, Chengdu 610054, China  \n2State Key Laboratory of Electronic Thin -Films and Integrated Devices, University of Electronic \nScience and Technology of China, Chengdu 610054, China  \n3State Key Laboratory of Precision Measurement Technology and Instrument, Department of \nPrecision Instrument, Tsinghua University, Beijing 100084, China  \n \nCorresponding author  \ndenglj@uestc.edu.cn, yli9003@mail.tsinghua.edu.cn , bilei@uestc.edu.cn  \n \n \n  \n \n    \n 2 \n Abstract  \nZero -index metamaterials (ZIMs) feature a uniform electromagnetic mode over a large area in arbitrary \nshapes, enabling many applications including high- transmission supercouplers with arbitrary shapes, \ndirection- independent phase matching for nonlinear opt ics, and collective emission of many quantum \nemitters. However, most ZIMs reported till date are passive, with no method for the dynamic modulation \nof their electromagnetic properties. Here, we design and fabricate a magnetically tunable ZIM consisting \nof yttrium iron garnet (YIG) pillars sandwiched between two copper clad laminates  in the microwave \nregime. By harnessing the Cotton- Mouton effect of YIG, the metamaterial was successfully toggled \nbetween gapless and bandgap states, leading to a \"phase transition\" between a zero -index phase and a single \nnegative phase of the metamaterial. Using an S -shaped ZIM supercoupler, we experimentally demonstrated \na tunable supercoupling state with a low intrinsic loss of 0.95 dB and a high extinction ratio of up to 30.63 \ndB at 9 GHz. Our work enables dynamic modulation of the electromagnetic chara cteristics of ZIMs, \nenabling various applications in tunable linear, nonlinear, quantum and nonreciprocal electromagnetic \ndevices.  \n \nIntroduction  \nZero -index materials are materials or composite structures that exhibit an effective refractive index of \nzero at a given frequency1-4, resulting in an infinite spatial wavelength. This effect can be leveraged to \novercome the limitations imposed by the finite spatial wavelength of electromagnetic waves, thereby \nenabling various novel physical phenomena and applications in  linear5-7, nonli near8-10 and quantum \nelectromagnetic systems11,12. Recently,  zero-index materials , such as indium tin oxide10,13,14, waveguides at \ncut-off frequencies15-17, fishnet metamaterials18, and doped ε-near-zero (ENZ) media19, have garnered \nsignificant research  interest2. However, the aforementioned materials and artificial structures exhibit  large \nohmic losses because of their metallic components20. In contrast, zero- index metamaterials ( ZIMs) based \non all -dielectric photonic crystals exhibit zero ohmic loss, enabling the realization of ZIMs over a large \narea of arbitrary shapes. A photonic crystal -based ZIM was first realized in the microwave regime based \non Al 2O3 pillars embedded in a parallel metal  waveguide21. Subsequently, photonic -crystal -based ZIM \nranging from acoustic22,23 to photonic regimes24,25 have been reported, demonstrating fascinating physical \nphenomena and applications , such as supercoup ling1,26, leaky- wave antennas27, cloaking21,28, \nsuperradiance12,25, and direction- independent phase matching for nonlinear optics9. \nDespite this progress, most of the ZIMs reported thus far are passive, with constant post -fabrication \nelectromagnetic properties, limiting their applications in passive devices. Active ZIMs, whose magnetic \n(µeff) and electric ( εeff) properties can be tuned using external stimuli , may allow the dynamic tuni ng of 3 \n delicate photonic band structure s, thereby inducing \"phase transitions \" in metamaterials. In turn, this  unique \nmechanism is expected to enable energy- efficient modulation of electromagnetic wave propagation with \nlow insertion loss, high extinction ratio, and compact device footprint —which are all essential factors in \nmicrowave and optical communication applications4,19,29,30. \nIn this study, we design  and experimentally investigate a magnetically tunable ZIM consisting of an \narray of gyromagnetic pillars embedded within a parallel -plate copper waveguide. Under an applied \nmagnetic field of 430  Oe, the photonic band structure of the proposed ZIM changes from a zero -index state \nto a photonic bandgap state, corresponding to a transition from a “zero -index phase” to a “single negative \nphase”. Based on this property, we propose a magnetic field-induced on- off switch of the supercoupling \nstate in a n S-shaped ZIM waveguide, resulting in a low intrinsic loss of 0.95 dB and a high extinction ratio \nof 30.63 dB at 9 GHz. We also demonstrate a magnetic  field-controlled switch to effect transitions between \na zero -index state and a nontrivial topological boundary state in the magnetic ZIM. These results \ndemonstrate the potential  of applying active ZIMs in  electromagnetic wave modulators and nonreciprocal \ndevices.  \n \nResults  \nFirst, we  design ed a Dirac- like cone -based zero -index metamaterial (DCZIM) consisting of a square \narray of dielectric pillars embedded in a parallel -plate copper waveguide21. At the Γ point, the accidental \ndegeneracy of two linear dispersion bands and a quadratic dispersion band form ed a Dirac -like cone \ndispersion, corresponding to an impedance -matched zero effective index21. In contrast to conventional \npassive ZIMs, we achieved  active modulation  by fabricating  a square lattice of pillars constructed using a \nmagnetic dielectric material,  yttrium iron garnet (YIG). By applying a magnetic field to the YIG pillars \nalong the direction perpendicular to the wave vector of the transverse magnetic (TM) mode, an effective \npermeability modulation that is quadratically proportional to the YIG magneti zation was observed owing \nto the Cotton- Mouton effect ( see Supplementary Information S1 for further details ). In turn, this effect \nenable d the modulation of  the photonic band structure and the effective index of metamaterials.  \nWe implemented the proposed design using the structure depicte d in Figure 1 . Figure 1a illustrates the \nexperimentally fabricated metamaterial consisting of a square lattice of  gyromagnetic  YIG pillars with a \n3.53- mm radius and a 17.9- mm lattice constant. The dielectric constant31 and permeability of the YIG \nmaterial were characterized  (see the Supplementary Information S2 for further deta ils). The YIG pillars \nwere placed in a waveguide consisting of  two parallel copper clad laminate s separated by 4 mm. A magnetic \nfield was applied to each YIG pillar by placing a neodymium iron boron (NdFeB) permanent magnet under \neach pillar and behind the copper back plate. A uniform magnetic field along the z-direction was observed, 4 \n whose intensity reached 430 Oe in the middle of the two copper clad laminates . This was  sufficient to \nsaturate the YIG pillars (see Supplementary Information S3  for further details ). \n \nFigure 1 . Schematic  diagram of the active DCZIM structure.  \n(a) The structure  of an active DCZIM based  on a gyromagnetic photonic crystal . (b) Schematic diagram of \na unit cell. The YIG pillars were placed in a parallel -plate copper -clad waveguide with height , h1 = 4 mm. \nThe thickness of the waveguide plates was  h2 = 2 mm. Permanent  magnets with 5 -mm diameter and height \nh3 = 5 mm we re placed in an a crylic matrix  underneath the waveguide and we re aligned with the YIG pillars.  \nTo characterize the properties of DCZIM, we first calculated and then experimentally measured the \nphotonic band structure  in the plane of the YIG array . The gray dotted line in Figure 2a represents the \ncalculated band structure for the TM modes (the electric field is polarized in the z direction) of this  photonic \ncrystal , as observed based on  a simulat ion using COMSOL  MULTIPHYSICS. We observed  a clear Dirac-\nlike cone dispersion at 9 GHz . When a magnetic field of +430 Oe was applied along the z direction, the off -\ndiagonal component induced the Cotton- Mouton effect in YIG, changing the frequencies of the three \nphotonic bands forming the Dirac -like cone. Owing to time reversal symmetry (TRS) breaking, t he photonic \ncrystal transitioned from  symmetry to  symmetry (see Supplementary Information  S4 for further \ndetails ). As a result, the degeneracy was broken, resulting in two bandgaps, as indicated by the gray dotted \nlines in the right panel of Figur e 2a.  \nThe modes supported by this structure  were analyzed by considering those supported by a square array \nof two-dimensional YIG pillars. As depicted in Figure 2b –d, this structure supported three  modes for TM \npolarization near the 9 GHz frequency—a  monopole mode, a transverse magnetic dipole mode, and a \nlongitudinal magnetic dipole mode. When a magnetic field was applied, as illustrated in the right part of \nthe panel, these three modes were displaced to different frequencies  (8.95 GHz, 9.52 GHz, 11.04 GHz , \nrespectively ). Because of the broken TRS , all three modes required to be  rotated through 180  ° to coincide \n5 \n with each other.  \nWe further  calculated the Chern number of each photonic band from low frequency to high frequency \nnear the Dirac point to be 0, 1, and -1, respectively. Based on the Chern number of each band, we calculated \nthe Chern numbers of the bandgaps , 1 and 2 (right panel of Figure 2a) , to be  and \n, respectively.  Based on the Chern numbers of the bandgaps , 1 and 2, we can \ndetermine  their topological nature —bandgaps 1 and 2 were topologically trivial and nontrivial, respectively.  \n \nFigure 2 . Theoretical and experimental demonstration of an active ZIM. ( a). Measured and calculated \n(grey dots) photonic bands  of the active ZIM using Fourier transform  field scan (FTFS) of the TM modes \ncorresponding to applied magnetic fields of 0 (left p anel) and 430 Oe (right panel). ( b)–(c). Simulated  three-\ndimensional dispersion surfaces near the Dirac -point frequency, depicting the relationship between the \nfrequency and the wave vectors ( kx and k y) (d). COMSOL -computed R e(Ez) on the cross- section of a ZIM \nunit cell at the frequencies  indicated by dashed arrows , depicting an electric monopole mode , a transverse \nmagnetic dipole mode , and a longitudinal magnetic dipole . The black circles indicate the boundaries of the \nYIG pillar s. \n6 \n  \nWe experimentally characterized this metamaterial using the setup proposed by Zhou et al.31 The sizes \nof all the samples were designed to be 10 × 10 periods, as illustrated in Figure 1a. The parallel -plate \nwaveguide consisting of two copper clad laminate s separated by  4 mm supported only the fundamental \nTEM mode in a parallel -plate waveguide below  37.5 GHz. To facilitate the  excitation and measurement of  \nelectromagnetic fields inside the waveguide, a square array of holes,  with a lattice constant of 5 mm , was \ndrilled through the top surface of the copper -clad laminate, enabling the insertion of a probe for  field \nmeasurement. The thickness of the copper -clad laminate s was taken to be 2 mm to tune the  uniform \ndistribution of the magnetic flux.  \nFirst, we measured the photonic band structure of the metamaterial under an applied magnetic field  of \n0, as illustrated using  the intensity plot in the left panel of Figure 2a. T he band structure of the TM bulk \nstates was obtained by applying two -dimensional discre te Fourier transform  (2D-DFT)  to the  measured \ncomplex field distribution over the metamaterial (see Supplementary Information  S5 for further details ). \nThe measured photonic band structure exhibited good agreement with the simulation results ( represented \nby gray dots). Both the measured and computed band structures exhibited a bandgap between 5 GHz and 7 \nGHz as well as bulk modes between 7 GHz and 12 GHz, indicating a Dirac -like cone dispersion near 9 \nGHz. The nondegeneracy of the photonic bands was also recorded after applying a 430 Oe magnetic field \nalong z-direction , as depicted in the right panel in Figure 2a. The bandgaps were experimentally measured \nto be at 8.95–9.60 GHz and 10.24–11.04 GHz, corroborating  the simulation results.  \n \nFigure 3 . Magnetic  field -induced phase transition of active ZIM . Real and imaginary parts of the effective \npermittivity ( εeff) and permeability tensor elements ( µ and κ) (a) under an applied magnetic field  of 0, and (b) \nunder an applied magnetic field of 430 Oe in the bandgap frequency range of 8.95–9.60 GHz  and (c) the bandgap \nfrequency range of 10.24 –11.04 GHz  \n7 \n The magnetic  field-induced band structure and transmittance modulation can be regarded to be  a phase \ntransition process from a material perspective. This phenomenon can be observed by retrieving the effective \npermittivity and permeability tensor elements of the metamaterial in the presence and absence of  an applied \nmagnetic field, as illustrated in Figure 3. We used the boundary effective  medium approach (BEMA)  to \ncalculate the effective constitutive parameters , as proposed in a previous study on ZIM32. In Figure 3a –c, \nεeff denotes the effective permittivity,  µ  denotes t he diagonal of the effective permeability tensor, and κ  \ndenotes  the off -diagonal component of the effective permeability tensor , .  \nFigure 3a  depicts the results corresponding to an applied magnetic field of 0. The real parts of εeff and \nµ cross zero simultaneously and linearly at 9 GHz, exhibiting ε-and-µ-near-zero ( EMNZ ) behavior  \ncorresponding to the zero- index phase . The imaginary parts of εeff and µ  were both close to 0. This low loss \nwas attributed to  the small loss tangent (0.0002) of the YIG material in this frequency range . When a \nmagnetic field of 430 Oe was applied,  phase transitions  occurred from the zero -index phase to the µ-\nnegative (MNG) or the ε−negative (ENG) phase, as depicted  in Figure s 3b and 3c , respectively . In the \nbandgap , 8.95 GHz –9.6 GHz (Figure 3b), εeff is positive , whereas µeff ( )33 is negative, which \ncorresponds to the MNG phase. T he impedance  at 9 GHz was tuned from 1.84 to 0.62i  after applying the \nmagnetic field , leading to a total reflection of the incident EM wave owing to impedance mismatch. In the \nbandgap , 10.24 –11.04 GHz  (Figure  3c), εeff is negative , whereas µeff is positive, which corresponds to the \nENG phase. Corresponding to both frequency ranges , the real parts of εeff and µeff decreased as the frequency \nincreased, exhibiting anomalous dispersion . The incident EM wave was still reflected due to impedance \nmismatch. Additionally, as depicted in the inset of  Figure 3c in the ENG frequency regime, a nontrivial \ntopological boundary state was observed  at the metamaterial edge because of the difference between the \nChern numbers of the upper and lower structure s.  \nThe complex phase transition phenomena discussed above were attributed to the location of  the ZIM \nat the origin of the metamaterial phase diagram, which allowed it to reach all quadrants of the phase diagram \nvia appropriate  tuning of the constitutive parameters33,34. Further discussion on the attainment of  other \nphases based on the phase diagram  is depicted  in Figure s S5–7 of Supplementary Information S6–7. Such \nunique properties of an active ZIM are indicative of its potential  with respect to the  modulation of the \npropagation of EM waves.  8 \n  \nFigure 4 Structure and characterization of a microwave switch based on  active DCZIM. ( a) \nPhotograph of the microwave switch sample. (b) The measured transmissions in the absence and presence \nof an applied magnetic field of 430 Oe. (c) The real part of the Ez distribution observed at 9 GHz inside the \nmetamaterial (each pillar  is indicated in gray) in the absence of a magnetic field. (d) The real part of  the Ez \ndistribution observed at 9 GHz inside the metamaterial (each pillar is indicated in gray) in the presence of \na 430 Oe magnetic field.  \nTo showcase a microwave switch based on the phase transition effect of active ZIM, we fabricated a ZIM \nwaveguide switch by leveraging the contingency of the supercoupling state on the applied magnetic field, as \nillustrated in Figure 4a. First, a ZIM waveguide co mprising  top and bottom metal plates and 100 YIG pillars was \nfabricated, forming two sharp 90-degree bends , to verify the supercoupling effect  experimentally (see \nSupplementary Information  S8 for further details ). The waveguide was coupled to the coaxial line via SMA \nconnectors. Linear taper ed sections were used to sustain the TE 10 mode and induce its gradual evolution into the \nTM mode of the metamaterial during propagation from the source to the waveguide. P erfect magnetic conductor \n(PMC) boundary condition s were realized using  aluminum alloy walls at a distance of λ /4 from the metamaterial \nas the lateral boundaries26 (see Supplementary Information S8 for further details). Using  the device depicted in \n9 \n Figure 4a, we successfully switch ed between the bulk state ( supercoupling state) and the photonic bandgap state \n(off state) by applying a ppropriate  magnetic field s to the YIG pillars.  \nFigure 4b depicts the measured transmission spectra corresponding to both states. A large transmission \ncontrast was observed over 8.9–9.4 GHz, w hich was consistent with the bandgap frequencies calculated via \nnumerical simulation in Figure 2a. The difference in device transmission under zero and 430 Oe magnetic field s \nwas larger than 30 dB at approximately 9 GHz  and t he device insertion loss was 3.75 dB. Considering that the \ncoupling loss induced by the SMA connectors was 2.8 dB, the intrinsic loss of the ZIM waveguide was as low \nas 0.95 dB, primarily  induced by the absorption of YIG materials.  \nWe verified the supercoupling behavior in the absence of an applied magnetic field by measuring the Ez \ncomponent of the electric field at each point of  the metamaterial, as illustrated in Figure 4c. A t 9 GHz, the electric \nfield tunnel ed through the metamaterial with almost no phase ch ange in the form of a bulk mode, verifying \nsupercoupling behavior. In contrast, in the absence of YIG pillars in the waveguide, the wave was  reflected back \nto the incident port, as described in Supplementary Information S9. This confirmed that the supercoupling \nbehavior was induced by the DCZIM. As depicted in Figure 4d, in the presence of an applied magnetic field , the \nelectromagnetic wave decayed exponentially in the metamaterial owing to the photonic bandgap 1 depicted in \nFigure 2a, leading to a high ext inction ratio. We also constructed a switch between the supercoupling state and \nthe topological one -way transmission state by probing at the upper bandgap frequency of 10.6 GH z (see \nSupplementary Information S10 for further details) . These unique properties are indicative of the potential of \nactive ZIMs in novel active electromagnetic devices.  \n \nDiscussion  \nIn this study, we propose d and experimentally operated  a magnetically tunable ZIM. The metamaterial \nwas operated by leveraging the Cotton -Mouton effect of the constitutive YIG pillars under applied magnetic \nfields, which alter the symmetry and bandgap opening of the Dirac -like cone -based ZIM. From a material \nperspective, the proposed metamaterial exhibited a phase transition from the zero -index phase to a single \nnegative phase, leading to an effective index change from 0 to 0.09i at 9 GHz. Based on this property, we \nconstructed and verified the function of  a microwave switch by manipulating the supercoupling effect , \nthereby reducing the  intrinsic loss to 0.95 dB and achieving a high extinction ratio of  30.63 dB at 9 GHz .  \nWe believe that this study  introduces a new approach to active ZIMs, particularly with respect to the \ndevelopment of efficient  active elect romagnetic and nonreciprocal devices. By appropriately engineering \nthe slots in parallel- plate copper waveguide s27, continuous beam steering over the broadside can be \nachieved by varying the applied magnetic field. Moreover, our design can be extended to the optical regime \nby embedding YIG pillars in a polymer matrix with gold films cladded24, enabling the dynamic modulation \nof optical DCZIM. Further,  such a magnetically tunable DCZIM  can be used to modulate the four -wave 10 \n mixing process in DCZIM by manipulating the zero -index phase -matching condition9. Additionally , we \nwere able to modulate DCZIM- based large -area single -mode photonic crystal surface- emitting lasers with \nhigh output power35. Finally, we also successfully modulate d the extended superradiance by tuning the \neffective index of the DCZIM, in which many quantum emitters were embedded12. \n \nMethods  \nNumerical simulation . The permittivity of the YIG used during numerical simulation was taken from that \npublished by Zhou et al.36 The band structure was calculated using the COMSOL software. The results were \nobtained by calculating the modes with periodic boundary conditions along the x and y directions. The TM \npolarization mode was selected by considering the electric field to be  an out -of-plane vector. Under \nmagnetic fields, t he DCZIM was simulated to obtain frequency -dependent electromagnetic field profiles. \nThe transitions between the different metamaterial phases in the presence and absence of magnetic field s \nwere simulated by considering the off-diagonal elements o f the YIG permeability tensor. Home -made \nMATLAB codes based on BEMA and 2D -DFT were used to calculate the effective permittivity, \npermeability tensor, and photonic band structure.  \nDevice Fabrication . The active DCZIM consisted of a square array of  YIG pill ars with a radius of 3.53 \nmm, a height of 4 mm , and a lattice constant of 17.9 mm ( as indicated in Figure 1a). Th e waveguide was \nformed using two 500 mm × 500 mm copper -clad laminates. During the measurement of the nontrivial \nboundary state, a copper bar w ith a length of 40 cm and a width of 2 cm was placed at the edge of the \nmetamaterial to form its boundary.  \nThe YIG pillars were fabricated from  YIG bulk crystals using an ultrahigh- accuracy computer \nnumerical control ( CNC) machine (Mazak V ARIAXIS i -700) with a dimensional accuracy exceeding 0.05 \nmm. As depicted in Figure 1a, an external magnetic field was introduced to maintain the saturation \nmagnetization of the garnet. To affix the YIG pillars to the copper clad lamin ate substrate tightly, double -\nsided tape with a radius of 3.53 mm was applied to one side of each pillar. The position of each YIG pillar  \nwas precisely defined by placing the other side in an acrylic mold with 10 × 10 holes with a radius of 3.54 \nmm and a l attice constant of 17.9 mm. The copper clad laminate substrate was pressed onto the side of the \nYIG pillars with tape, which affixed the YIG pillars tightly and precisely onto the copper clad laminate \nsubstrate . The acrylic mold was removed carefully  to prevent extrusion and damage to the YIG pillars.  \nEach YIG pillar  was properly magnetized  by aligning the NdFeB magnet used to apply the magnetic \nfield precisely underneath each YIG pillar . To this end, a 2- mm- thick acrylic sheet was first covered with a \ndouble -sided tape to form the substrate. Then, w e fixed a 5- mm-thick acrylic sheet including 10  × 10 11 \n perforations with a radius of 5 mm and a lattice constant of 17.9 mm onto a 2 -mm-thick acrylic substr ate. \nFinally, with respect to  the designated direction of the magnetic field, we placed the NdFeB magnets into \nthe holes of the 5 -mm- thick acrylic sheet, achieving a good alignment with the array of the YIG pillars \n(Figure 1b).  \nCharacterization Setup . We measure d the photonic band structure, transmission spectra, and near -field \ndistribution of the active metamaterial sample  using two dipole antennas as the transmitter and receiver. \nBoth antennas were inserted into the waveguide via holes drilled using an ul trahigh -precision CNC machine \nand they were connected to a vector network analyzer ( Rohde & Schwarz ZNB 20) to measure the S \nparameters. Prior to measurement, 3.5 -mm 85052D through- open- short -load calibrations were performed. \nAs a result , the measured S parameters included only the insertion loss of the tapered waveguides and the \nmetamaterial.  \nReferences  \n1.  Li, Y., Chan, C. T. & Mazur, E. Dirac -like cone -based electromagnetic zero -index metamaterials. Light \nSci. Appl. 10 , 203 (2021).  \n2. Kinsey, N. et al.  Near -zero-index materials for photonics. Nat. Rev. Mater.  4, 742- 760 (2019).  \n3. Liberal, I. & Engheta, N. Near -zero refractive index photonics. Nat. 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Z., Leon, I. D. & Boyd, R. W. Large optical nonlinearity of indium tin oxide in its epsilon-\nnear-zero region. Science 352 , 795- 797 (2016).  \n11. Xu, J. et al.  Unidirectional single -photon generation via matched zero- index metamaterials. Phys. Rev. \nB 94, 220103 (2016).  \n12. Mello, O. et al.  Extended many- body superradiance in diamond epsilon near -zero metamaterials. Appl. \nPhys. Lett.  120 (2022).  \n13. Yang, Y. et al.  High -harmonic generation from an epsilon- near-zero material. Nat. Phys. 15, 1022-\n1026 (2019).  \n14. Jia, W.  et al.  Broadband terahertz wave generation from an epsilon- near-zero material. Light Sci. Appl.  \n10, 11 (2021).  \n15. Vesseur, E. J.  et al.  Experimental verification of n = 0 structures for visible light. Phys. Rev. Lett.  110, \n013902 (2013).  \n16. Zhou, Z. & Li, Y. Effective Epsilon- Near -Zero (ENZ) Antenna Based on Transverse Cutoff Mode. \nIEEE Trans. Antennas Propag. 67, 2289- 2297 (2019).  12 \n 17. Qin, X.  et al.  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Moitra, P. et al.  Realization of an all -dielectric zero -index optical metamaterial. Nat. P hotonics  7, 791-\n795 (2013).  \n26. Camayd -Muñoz, P. Integrated zero -index metamaterials. PhD thesis thesis, Harvard University, \n(2016).  \n27. Memarian, M. & Eleftheriades, G. V. Dirac leaky -wave antennas for continuous beam scanning from \nphotonic crystals. Nat. Commun. 6, 5855 (2015).  \n28. Chu, H. et al.  A hybrid invisibility cloak based on integration of transparent metasurfaces and zero-\nindex materials. Light Sci. Appl. 7, 50 (2018).  \n29. Engheta, N. Pursuing Near -Zero Respons. Science  340, 286- 287 (2013).  \n30. Maas, R.  et al.  Experimental realization of an epsilon- near-zero metamaterial at visible wavelengths. \nNat. Photonics  7, 907- 912 (2013).  \n31. Zhou, P. et al.  Observation of Photonic Antichiral Edge States. Phys. Rev. Lett.  125, 263603 (2020).  \n32. Wang, N. et al.  Effective medium theory for a photonic pseudospin- 1/2 system. Phys. Rev. B  102, \n094312 (2020).  \n33. Davoyan, A. R. & Engheta, N. Theory of wave propagation in magnetized near -zero-epsilon \nmetamaterials: evidence for one -way photonic states and magnetically switched transparency and \nopacity. Phys . Rev. Lett. 111 , 257401 (2013).  \n34. Smith, D. R., Pendry, J. B. & Wiltshire, M. C. K. Metamaterials and Negative Refractive Index. \nScience 305 , 788- 792 (2004).  \n35. Chua, S.- L. et al.  Larger -area single -mode photonic crystal surface -emitting lasers enabled  by an \naccidental Dirac point. Opt. Lett.  39, 2072- 2075 (2014). \n36. Zhou, P. et al.  Photonic amorphous topological insulator. Light Sci. Appl. 9, 133 (2020).  \nAcknowledgements  \n  The authors appreciate the discussions with Hengbin Cheng from the Institute of Physics, Chinese \nAcademy of Science , and their assistance . The authors are also grateful for the support received from the \nMinistry of Science and Technology of the People’s Republic of China (MOST) (Grant No. \n2018YFE0109200, 2021YFA1401000, a nd 2021YFB2801600), National Natural Science Foundation of \nChina (NSFC) (Grant Nos. 51972044, 52021001, and 62075114), Sichuan Provincial Science and \nTechnology Department (Grant No. 2019YFH0154), the Fundamental Research Funds for the Central \nUniversities (Grant No. ZYGX2020J005), the Beijing Natural Science Foundation (Grant No. 4212050), 13 \n and the Zhuhai Industry- University Research Collaboration Project (ZH22017001210108PWC). This work \nwas supported by the Center of High- Performance Computing, Tsinghua Un iversity.  "    },    {        "title": "2101.04405v1.Temperature_dependence_of_the_mean_magnon_collision_time_in_a_spin_Seebeck_device.pdf",        "content": "Temperature dependence of the mean magnon collision\ntime in a spin Seebeck device\nVittorio Basso, Alessandro Sola, Patrizio Ansalone and Michaela\nKuepferling\nIstituto Nazionale di Ricerca Metrologica, Strada delle Cacce 91, 10135, Torino, Italy\nAbstract\nBased on the relaxation time approximation, the mean collision time for\nmagnon scattering \u001cc(T) is computed from the experimental spin Seebeck\ncoe\u000ecient of a bulk YIG / Pt bilayer. The scattering results to be composed\nby two processes: the low temperature one, with a T\u00001=2dependence, is at-\ntributed to the scattering by defects and provides a mean free path around 10\n\u0016m; the high temperature one, depending on T\u00004, is associated to the scat-\ntering by other magnons. The results are employed to predict the thickness\ndependence of the spin Seebeck coe\u000ecient for thin \flms.\nKeywords: spin Seebeck e\u000bect, yttrium iron garnet, magnon scattering\n1. Introduction\nA spin Seebeck device is a bilayer composed by a ferromagnetic insulator\n(i.e. the ferrimagnet YIG) and a metal with a strong spin Hall e\u000bect (i.e. Pt).\nThe spin Seebeck e\u000bect is obtained by applying a temperature gradient to\nthe YIG that generates a current of magnetic moment. Part of the magnetic\nmoment current from YIG is injected at the interface into the side metallic\nlayer where it is carried by electrons. The use of Pt as a metallic layer per-\nmits to detect the magnetic moment current because of the inverse spin Hall\ne\u000bect that laterally de\rects polarized electrons and creates an electric volt-\nage in the transverse direction [1]. In the last decade the spin Seebeck e\u000bect\nhas attracted attention as an alternative thermoelectric generator and as a\nsource of a magnetic moment current for spintronic devices without involv-\ning a charge current [2, 3]. However, to optimize the e\u000bect, the underlying\nphysics needs to be appropriately understood. At any \fnite temperature the\nPreprint submitted to Elsevier January 13, 2021arXiv:2101.04405v1  [cond-mat.mes-hall]  12 Jan 2021saturation magnetization of the ferromagnet is decreased with respect to its\nspontaneous value because of the thermal excitation of spin waves (magnons).\nIt is believed that the temperature gradient across the ferromagnet forces the\ndi\u000busion of the thermal magnons in the direction of the gradient, therefore\ngenerating a current of magnetic moment [4, 5, 6]. A crucial point is therefore\nto understand the transport properties of magnons and their scattering [7].\nExperiments of the temperature dependence of the spin Seebeck coe\u000ecient\n[8, 9] revealed non obvious features: in bulk YIG single crystals a peak is\nobserved at temperatures around 70 K. The peak is shifted at higher tem-\nperatures and partially suppressed by both decreasing the thickness [8] or\ndecreasing the grain size in polycrystalline materials [10, 11]. The interpre-\ntation of this e\u000bect has stimulated a debate on the possible sources of the\nscattering of the magnons: impurities, phonons, other magnons and so on\n[10, 7, 9]. One of the problems is to disentangle the temperature dependence\nof the magnon scattering processes from the temperature dependence other\nparameters such as the conductance and the di\u000busion length of Pt.\nUnlike the classical Seebeck e\u000bect of metals, in which the ratio voltage\nover temperature gives the absolute thermo-electric power coe\u000ecient, in the\nspin Seebeck e\u000bect the coe\u000ecient given by the ratio between the voltage\ngradient over the temperature gradient (in the geometry of Fig.1) depends\nnot only on the absolute thermomagnetic power coe\u000ecient of the YIG, \u000fYIG,\nand on the spin Hall angle of the platinum, \u0012SH, but also on the magnetic\nmoment conductances, the di\u000busion lengths and the thicknesses of the two\nlayers [12]. Even if certain parameters can be obtained by independent ex-\nperiments, the expression of the spin Seebeck coe\u000ecient still contains at least\ntwo free parameters: \u000fYIGand the magnetic moment conductivity \u001bM;YIGof\nthe ferromagnet. To simplify the problem in this work we set \u000fYIG=\u00000:956\nT/K as predicted by the Boltzmann approach to the transport of magnons\nin the relaxation time approximation[13, 14]. Then the magnetic moment\nconductivity can be easily deduced from the experimental data and the re-\nlaxation time \u001cc(T) computed as a function of temperature. \u001cc(T) is an\nestimate of the typical scattering time and gives insights on the evolution of\nthe scattering processes as a function of the temperature. The main result\nof this paper is to show that the scattering is composed by two processes.\nAt low temperature the active process changes with temperature as T\u00001=2, a\ndependence that can be attributed to the scattering by defects. At high tem-\nperature it depends strongly on TasT\u00004a variation that could be associated\nto the scattering by other magnons.\n2xyzme-HYIGPt\ndYIGdPtcoldhotM-∇xT∇yVFigure 1: Spin Seebeck bilayer composed by a ferromagnetic insulator (i.e. the ferrimagnet\nYIG) and a metal with a strong spin Hall e\u000bect (i.e. Pt). The temperature gradient along\nxon YIG injects a current of magnetic moment into Pt where it is revealed as a voltage\ngradient along ybecause of the inverse spin Hall e\u000bect.\n2. Spin Seebeck e\u000bect\nThe spin Seebeck coe\u000ecient is given by the ratio between the voltage\ngradient in platinum and the temperature gradient in YIG (see Fig.1)\nSSSE=ryVe\nrxT(1)\nTo derive a theoretical expression for the spin Seebeck coe\u000ecient one has\nto use a thermodynamic theory describing the transport of the magnetic\nmoment in the ferromagnet caused by the temperature gradient and the\ntransverse electric e\u000bect of the inverse spin Hall e\u000bect in the metal [15]. For\nboth materials one has also to take into account that the magnetic moment is\na non conserved quantity and therefore the transport of the magnetic moment\nis characterized by a di\u000busion length lMand by a typical time constant \u001cM.\nThe two values and their ratio vM=lM=\u001cM, a parameter that assumes\nthe meaning of a magnetic moment conductance, are characteristic values\nof each material (M = Pt, YIG) and can be temperature dependent. From\nthe thermodynamic theory by exploiting the reciprocity of both the intrinsic\nthermal and magnetic transport in the ferromagnet and the spin Hall e\u000bect\nin the metal [16], one derives the following expression (see appendix A for a\nderivation)\n3SSSE=\u0000\u0012SH\u0010\u0016B\ne\u0011lYIGvYIG\ndPtv\u000fYIG (2)\nwhere\u0012SHis the spin Hall angle of the metal (the angle for the magnetic\nmoment is opposite with respect to the one for the spin i.e. negative for Pt\nand positive for W and Ta), \u000fYIGis the thermomagnetic power coe\u000ecient\nof the ferromagnet, lYIGandvYIGare the di\u000busion length and the magnetic\nmoment conductance of the ferromagnet. The product lYIGvYIG=\u00160\u001bM;YIG\nis proportional to the magnetic moment conductivity of the ferromagnet\n\u001bM;YIG.vis an e\u000bective conductance of the bilayer and summarizes the e\u000bects\nof the ratio between the thicknesses of the layers and their own intrinsic\ndi\u000busion lengths and contains the sum of the e\u000bective conductances. Its\nexpression is\n1\nv=f(dPt=lPt)f(dYIG=lYIG)\nvPttanh(dPt=lPt) +vYIGtanh(dYIG=lYIG)(3)\nwheref(x) = tanh(x) tanh(x=2). in the case of a thick YIG with dYIG\u001dlYIG\nit is simpli\fed as\n1\nv=f(dPt=lPt)\nvPttanh(dPt=lPt) +vYIG(4)\nand if the conductance of the metal is much larger than the one of the\nferromagnet, vPttanh(dPt=lPt)\u001dvYIG, it is directly given by the properties\nof the metal v=vPtcoth(dPt=(2lPt)).\nThe aim of this paper is to compute lYIGandvYIGof YIG as a function of\ntemperature by using Eq.(2) and the experimental data for the spin Seebeck\ncoe\u000ecient. In Eq.(2) many parameters are known from other experiments.\nThe missing one is the intrinsic parameter \u000fYIG. In this work we employ\nthe result of the transport theory of magnons which predicts a temperature\nindependent constant \u000fYIG=\u00000:956 T/K [14].\nWe initially simplify the problem by considering a bulk YIG single crystal\nat constant temperature. For a bulk YIG we have dYIG\u001dlYIGand, expecting\nvPt\u001dvYIG, we estimate v'vPtcoth(dPt=(2lPt)). The magnetic moment\nconductivity of Pt is \u001bM;Pt= (\u0016B=e)2\u001be;Pt. Then, with \u001be;Pt= 6:4\u0002106\n\n\u00001m\u00001we obtain\u00160\u001bM;Pt= 2:6\u000210\u00008m2s\u00001. As\u00160\u001bM;Pt=lPtvPt, with\nlPt= 7:3 nm [17], we get vPt= 3:5 m/s,\u001cPt= 2:1 ns and \fnally v'11:2 m/s.\nThe spin Hall angle is taken as \u0012SH=\u00000:1 [17]. From recent experiments\n4resistivity (Ωm)temperature (K)Pt\ntemperature (K)lM(nm)conductance vPtlenght lPtvM(m/s)Figure 2: Left: standard resistivity of Pt (from Platinum Metals Rev., 28 (4) 164 (1984))\nand \ftting function %(T) =x%L+(1\u0000x)%Hwith%L= 6\u000110\u000014T3,%H= 4:1\u000210\u000010(T\u000032),\nx= [1 + tanh[( T\u000040)=10]]=2. Right: estimated lPt= (\u00160\u001bM;Pt\u001cPt)1=2andvPt=lPt=\u001cPt\nwith\u001cPt= 2:1 ns,\u001bM;Pt= (\u0016B=e)2\u001be;Pt,\u001be;Pt= (%(T) +%0)\u00001with residual resistivity\n%0= 0:43\u000210\u00007\n m.\nat room temperature with a bulk YIG single crystal ( dYIG= 0:5 mm) we\nfoundSSSE=\u00004:8\u000210\u00007V K\u00001[16]. Then we compute the only missing\nparameter, the product lYIGvYIG, resulting 5 :3\u000210\u00009m2/s. By estimating the\ntime constant of the YIG as \u001cYIG= (\u00160\rLM0\u000b)\u00001where\u000bis the damping,\nwe get\u001cYIG= 1\u000210\u00006s with\u000b= 2:5\u000210\u00005(M0= 1:95\u0002105A/m is the\nspontaneous magnetization of YIG). Therefore, at room temperature one\n\fnds a di\u000busion length lYIG'70 nm and a conductance vYIG'0:07 m/s.\nIt must be remarked that the di\u000busion length computed in this way is very\nsimilar to the one deduced from the YIG thickness dependence in Ref.[18].\nThe same calculation can be performed by considering the tempera-\nture dependence of the spin Seebeck coe\u000ecient for the YIG single crystal\n(dYIG= 1 mm) of Ref.[8] (we take from now on the SSSEas an absolute value\nwithout the minus sign given by our choice of the reference system). For\nplatinum, the spin Hall angle is not expected to change signi\fcantly with\nT[19], therefore we assume \u0012SH=\u00000:1, while the temperature dependence\nof the di\u000busion length and of the conductance of Pt are estimated on the\nbasis of the temperature dependence of the resistivity of Pt (see Fig.2 left)\nas shown in Fig.2 right. For YIG, \u000fYIGis expected from the di\u000busion theory\nto be temperature independent [14], while the temperature dependence of\nthe time constant \u001cYIGis attributed to the temperature dependence of \u000b. In\nRef.[20] it was found that \u000bis approximately linear with T. With these as-\nsumptions one derives lYIG(T) andvYIG(T) shown in Fig.3 right. Again here\n5it is remarkable that, by decreasing the temperature, the di\u000busion length\nreaches a plateau at around 1 \u0016m that matches the low temperature estimate\nof Ref.[18].\nNow, having the temperature dependence of lYIG(T) andvYIG(T) it is\npossible to employ Eq.(2) with Eq.(3) to compute the spin Seebeck coe\u000ecient\nas a function of both YIG thickness and temperature. The result is shown\nin Fig.4 left. The set of curves fully captures the phenomenology of the\nexperimental curves of Refs.[8] and [9] but the peak starts to decrease at\na YIG thickness ( \u00182\u0016m) much smaller than the one seen in experiments\n(\u001820\u0016m). To understand more we have therefore to go into the details of\nthe scattering processes.\nlYIG(m)temperature (K)τc (s)1/T41/T1/2temperature (K)∇yV/∇xT (V/K)\ntemperature (K)\nvYIG(m/s)temperature (K)bulk YIG / Pt 1/T3/2\n1/T1/2T\nFigure 3: Top left: spin Seebeck coe\u000ecient for a bulk (1 mm) YIG from Ref.[8]. Top right:\ndi\u000busion length lYIG(T). Bottom right: magnetic moment conductance vYIG(T). Bottom\nleft: relaxation time \u001cc. Points are computed from the data of Ref.[8]. Dashed lines are\n\ftted behavior in terms of the powers of temperature marked on the graphs. Full line\n\u001c\u00001\nc=\u001c\u00001\nc;L+\u001c\u00001\nc;Hwith\u001cc;L= 1:8\u000210\u00009(T=Tm)\u00001=2s,\u001cc;H= 3:4\u000210\u000012(T=Tm)\u00004s.\n63. Magnon collision time\nThe previous results have been obtained by \fxing the thermomagnetic\npower coe\u000ecient of the ferromagnet \u000fYIGto the constant -0.956 T/K which\nis the result of the Boltzmann transport theory in the relaxation time ap-\nproximation [13, 14]. With the relaxation time approximation one assumes\nthat the di\u000busing magnons relax to the equilibrium distribution with a typi-\ncal time constant \u001cc. From the theory the magnetic moment conductivity is\ngiven by\n\u001bM;YIG=(2\u0016B)2nm\u001cc\nm\u0003\nm(5)\nwhere 2\u0016Bis the magnetic moment carried by the magnon, m\u0003\nmis its e\u000bective\nmassm\u0003\nm=}2=(2D), whereDis the spin wave exchange sti\u000bness, \u001ccis the\nrelaxation time and nmis the number of magnons. The number of magnons\nis expected to change signi\fcantly with temperature and nmis given by\nnm=n\n8\u00193=2\u0012T\nTm\u00133=2\n(6)\nwherenis the volume density of localized magnetic moments in the system\nn= 1=a3,Tm=D=(a2kB) andkBis the Boltzmann constant. ais typical\ndistance between localized magnetic moments and is related to the sponta-\nneous magnetization by M0=\u0016B=a3. For YIG one has D= 8:6\u000210\u000040J m2,\nm\u0003\nm= 2:5\u000210\u000028kg (about 280 times the electron mass) and Tm= 480 K\n[21]. As the temperature dependence of the magnetic moment conductivity\n\u00160\u001bM;YIG=lYIGvYIGhas been estimated from the spin Seebeck coe\u000ecient,\nwe can deduce the temperature dependence of the relaxation time \u001cc(T) de-\nscribing the collision processes. The result is shown in Fig.3 bottom left.\nIt appears that the scattering of the magnons is the superposition of two\nmechanisms. If we apply the Matthiessen's rule\n1\n\u001cc=1\n\u001cc;L+1\n\u001cc;H(7)\nwe can separate two di\u000berent time constants. At low temperature the active\nprocess\u001cc;Lappears to change with temperature as T\u00001=2while at high tem-\nperature\u001cc;Hdepends strongly on TasT\u00004. The low temperature process\ncan be attributed to the scattering by defects. If we take the velocity of the\nmagnons to vary with temperature as vm=p\nkBT=m\u0003and compute a mean\n7free path due to intrinsic defects as \u0015=vm\u0001\u001cc;Lwe get a temperature inde-\npendent value of \u0015'10\u0016m. We recall that the mean free path was obtained\nfrom the bulk data. In the case of thin \flms with dYIG<\u0015therefore we have\nto introduce an additional extrinsic time constant \u001cc;L;ewhich is thickness\ndependent as \u001cc;L;e=dYIG=vmin order to add the scattering at the inter-\nface. The curves calculated with this additional scattering mechanism can\nbe seen in Fig.4 right. It must be observed that the plot at the right shows\nthe decrease of the signal at thicknesses much larger with respect to the case\nof the plot of the left and in better agreement with the experiments. Con-\nsidering that all the parameters have been derived only from the data of the\nbulk, the agreement is reasonably good. The high temperature time constant\n\u001cc;Hdecreasing as T\u00004is exactly what is expected for the magnon-magnon\nscattering process [22].\ntemperature (K)  scattering at interfaces30 µm10 µm3 µm1 µm0.3 µmtemperature (K)∇yV/∇xT (V/K)bulk YIG / Pt 2 µm1 µm0.6 µm0.3 µm0.1 µmno scattering at interfaces\nFigure 4: Spin Seebeck coe\u000ecient as a function of thickness and temperature. Lines\nare theoretical predictions. Left: computed by using the scattering corresponding to the\nbulk for all thicknesses. Right: including a an additional scattering contribution due to\ninterfaces a bulk. In both cases the points corresponds to the data of the spin Seebeck\ncoe\u000ecient YIG from Ref.[8] (black squares: 1mm, red triangles 10 \u0016m, blue stars 1 \u0016m,\npurple diamonds 0.3 \u0016m).\n4. Conclusions\nIn the paper we have performed a study of the temperature dependence\nof the spin Seebeck coe\u000ecient of the YIG/Pt bilayer. We have interpreted\nthe literature data [8, 9] by using the thermodynamic theory of Refs.[15, 14]\n8in which the magnon transport occurs by di\u000busion in presence of a gradi-\nent of the thermodynamic temperature and in the gradient of the potential\nfor the magnon di\u000busion. The resulting temperature dependence of the re-\nlaxation time describes two sources of scattering: the scattering by defects\nor interfaces, changing with temperature as T\u00001=2, and mainly active at low\ntemperature and a scattering by other magnons, depending on T\u00004and active\nat high temperature.\nThe classical literature investigating the contributions of magnons to the\nthermal conductivity of ferromagnetic insulators, has always considered that\nthe magnon-magnon process was very di\u000ecult to identify. This is because\nit is relevant only at high temperatures, a range in which the contribution\nof magnons to the transport of heat is irrelevant with respect to the phonon\none. The spin Seebeck e\u000bect represents an case for the detailed study of\nmagnon-magnon scattering because it reveals the transport of magnetic mo-\nment. In reference to the recent literature claiming a magnon-phonon drag\ne\u000bect [23], the results obtained in this study are indicating that the magnon-\nphonon process as a main source of scattering can be excluded. However it\ncannot be a priori excluded that, especially in materials with a strong mag-\nnetoelastic interaction, the magnon-phonon processes, by means of a phonon\ndrag e\u000bect, could give an enhancement of the e\u000bective \u000fYIG[23]. From our\nstudy it appears that such a mechanism is not necessary to explain the peak\nof the spin Seebeck signal of YIG. Future works could clarify the limits and\npossibilities of these two approaches.\nA possible interesting extension of the present approach to the magnon\ntransport is at high temperatures close to the Curie point. This extension\nis not straightforward because the spin wave spectrum (especially the long\nwavelengths) develops over a background saturation magnetization Mswhich\nis less than the spontaneous M0and such a decrease is due to the spin wave\nthemselves (especially the short wavelengths). Possible advancements of a\nBoltzmann transport approach are possible in terms of a self consistent ap-\nproach as suggested by Robert Brout with his random phase approximation\ntechnique [24].\nAppendix A. Spin Seebeck coe\u000ecient\nThe spin Seebeck coe\u000ecient of Eq.(2) is calculated as follows [15]. The\ntransverse voltage gradient in Pt is given by the formula\n9ryVe=1\n\u001be;Pt\u0012SH\u0012e\n\u0016B\u0013\n<jM>Pt (A.1)\nwhere<jM>Ptis the average magnetic moment current in Pt. By solving\nthe di\u000busion equation for the magnetic moment in Pt, <jM>Ptis estimated\nas a function of the magnetic moment current injected at the interface by\nthe ferromagnet, jM;0, by assuming that the current at the other end of Pt\nis zero. The result is\n<jM>Pt=lPt\ndPttanh(dPt=(2lPt))jM;0 (A.2)\nSimilarly by solving the di\u000busion equation also in the YIG and imposing\nthe boundary conditions between the metal and the ferromagnet, jM;0is\ncomputed as a function of the magnetic moment current source jMSdue to\nthe intrinsic spin Seebeck e\u000bect in YIG. jM;0is given by the current divider\nequation\njM;0=vPttanh(dPt=lPt)\nvPttanh(dPt=lPt) +vMtanh(dM=lM)jMS (A.3)\nwhere the expressions of the type vMtanh(dM=lM) are the e\u000bective conduc-\ntances (depending on the ratio dM=lM) andvMis the intrinsic magnetic mo-\nment conductance which is a property of each material. For a ferromagnet\nof \fnite thickness one has that the magnetic moment source is\njMS=\u0000tanh(dYIG=(2lYIG)) tanh(dYIG=lYIG)\u000fYIG\u001bM;YIGrxT (A.4)\nBy joining the previous equations one obtains expression (2) with vgiven by\nEq.(3).\nReferences\n[1] K. Uchida, H. Adachi, T. Ota, H. Nakayama, S. Maekawa, and E. Saitoh.\nObservation of longitudinal spin-seebeck e\u000bect in magnetic insulators.\nAppl. Phys. Lett. , 97:172505, 2010.\n[2] K.I. Uchida, H. Adachi, T. Kikkawa, A. Kirihara, M. Ishida, S. Yorozu,\nS. Maekawa, and E. Saitoh. Thermoelectric generation based on spin\nseebeck e\u000bects. Proceedings of the IEEE , 104:1946, 2016.\n10[3] Xiao-Qin Yu, Zhen-Gang Zhu, Gang Su, and A.-P. Jauho. Spin-\ncaloritronic batteries. Phys. Rev. Applied , 8:054038, Nov 2017.\n[4] Jiang Xiao, Gerrit E. W. Bauer, Ken-chi Uchida, Eiji Saitoh, and\nSadamichi Maekawa. Theory of magnon-driven spin seebeck e\u000bect. Phys.\nRev. B , 81:214418, Jun 2010.\n[5] Steven S.-L. Zhang and Shufeng Zhang. Magnon mediated electric\ncurrent drag across a ferromagnetic insulator layer. Phys. Rev. Lett. ,\n109:096603, Aug 2012.\n[6] S. M. Rezende, R. L. Rodriguez-Suarez, R. O. Cunha, A. R. Rodrigues,\nF. L. A. Machado, G. A. Fonseca Guerra, J. C. Lopez Ortiz, and\nA. Azevedo. Magnon spin-current theory for the longitudinal spin-\nseebeck e\u000bect. Phys. Rev. B , 89:014416, 2014.\n[7] Stephen R Boona and Joseph P Heremans. Magnon thermal mean free\npath in yttrium iron garnet. Physical Review B , 90(6):064421, 2014.\n[8] Takashi Kikkawa, Ken-ichi Uchida, Shunsuke Daimon, Zhiyong Qiu,\nYuki Shiomi, and Eiji Saitoh. Critical suppression of spin seebeck e\u000bect\nby magnetic \felds. Phys. Rev. B , 92:064413, Aug 2015.\n[9] Er-Jia Guo, Joel Cramer, Andreas Kehlberger, Ciaran A. Ferguson,\nDonald A. MacLaren, Gerhard Jakob, and Mathias Kl aui. In\ruence\nof thickness and interface on the low-temperature enhancement of the\nspin seebeck e\u000bect in yig \flms. Phys. Rev. X , 6:031012, Jul 2016.\n[10] K. Uchida, T. Ota, H. Adachi, J. Xiao, T. Nonaka, Y. Kajiwara,\nG. E. W. Bauer, S. Maekawa, and E. Saitoh. Thermal spin pump-\ning and magnon-phonon-mediated spin-seebeck e\u000bect. J. Appl. Phys. ,\n111:103903, 2012.\n[11] Asuka Miura, Takashi Kikkawa, Ryo Iguchi, Ken ichi Uchida, Eiji\nSaitoh, and Junichiro Shiomi. Probing length-scale separation of ther-\nmal and spin currents by nanostructuring yig. ArXiv , page 1704.07568,\n2017.\n[12] Vittorio Basso, Michaela Kuepferling, Alessandro Sola, Patrizio Ansa-\nlone, and Massimo Pasquale. The spin seebeck and spin peltier reciprocal\nrelation. IEEE Magnetics Letters , 9:1{4, 2018.\n11[13] Kouki Nakata, Pascal Simon, and Daniel Loss. Wiedemann-franz law\nfor magnon transport. Phys. Rev. B , 92:134425, Oct 2015.\n[14] Vittorio Basso, Elena Ferraro, and Marco Piazzi. Thermodynamic trans-\nport theory of spin waves in ferromagnetic insulators. Phys. Rev. B ,\n94:144422, Oct 2016.\n[15] V. Basso, E. Ferraro, A. Sola, A. Magni, M. Kuepferling, and\nM. Pasquale. Nonequilibrium thermodynamics of the spin seebeck and\nspin peltier e\u000bects. Phys. Rev. B , 93:184421, 2016.\n[16] Alessandro Sola, Vittorio Basso, M. Kuepferling, Carsten Dubs, and\nMassimo Pasquale. Experimental proof of the reciprocal relation be-\ntween spin peltier and spin seebeck e\u000bects in a bulkyig/pt bilayer. Sci-\nenti\fc reports , 9:2047, 2019.\n[17] H. L. Wang, C. H. Du, Y. Pu, R. Adur, P. C. Hammel, and F. Y.Yang.\nScaling of spin Hall angle in 3d, 4d, and 5d metals from Y 3Fe5O12/metal\nspin pumping. Phys. Rev. Lett. , 112:197201, 2014.\n[18] A. Kehlberger, U. Ritzmann, D. Hinzke, E.J. Guo, J. Cramer, G. Jakob,\nM. C. Onbasli, D. H. Kim, C. A. Ross, M. B. Jung\reish, B. Hillebrands,\nU. Nowak, and M. Kl aui. Length scale of the spin seebeck e\u000bect. Phys.\nRev. Lett. , 115:096602, 2015.\n[19] Miren Isasa, Estitxu Villamor, Luis E. Hueso, Martin Gradhand, and\nFelix Casanova. Temperature dependence of spin di\u000busion length and\nspin Hall angle in Au and Pt. Phys. Rev. B , 91:024402, 2015.\n[20] Hannes Maier-Flaig, Stefan Klingler, Carsten Dubs, Oleksii Surzhenko,\nRudolf Gross, Mathias Weiler, Hans Huebl, and Sebastian TB Goennen-\nwein. Temperature-dependent magnetic damping of yttrium iron garnet\nspheres. Physical Review B , 95(21):214423, 2017.\n[21] D. D. Stancil and A. Prabhakar. Spin Waves. Theory and Applications .\nSpringer, New York, 2009.\n[22] P Erd\u0015 os. Low-temperature thermal conductivity of ferromagnetic insu-\nlators containing impurities. Physical Review , 139(4A):A1249, 1965.\n12[23] H. Adachi, K. Uchida, E. Saitoh, J. Ohe, S. Takahashi, and S. Maekawa.\nGigantic enhancement of spin Seebeck e\u000bect by phonon drag. Applied\nPhysics Letters , 97:252506, 2010.\n[24] Rober Brout. Statistical mechanics of ferromagnetism. In Rado and\nSuhl, editors, Magnetism . Academic Press, 1965.\n13"    },    {        "title": "1504.01512v1.Generation_of_coherent_spin_wave_modes_in_Yttrium_Iron_Garnet_microdiscs_by_spin_orbit_torque.pdf",        "content": "Generation of coherent spin-wave modes in Yttrium Iron Garnet microdiscs\nby spin-orbit torque\nM. Collet,1X. de Milly,2O. d'Allivy Kelly,1V. V. Naletov,3, 4R. Bernard,1P. Bortolotti,1V. E. Demidov,5S.\nO. Demokritov,5, 6J. L. Prieto,7M. Mu~ noz,8V. Cros,1A. Anane,1G. de Loubens,2and O. Klein3\n1Unit\u0013 e Mixte de Physique CNRS/Thales and Universit\u0013 e Paris Sud 11, 1 av. Fresnel, 91767 Palaiseau, France\n2Service de Physique de l' \u0013Etat Condens\u0013 e (CNRS UMR 3680), CEA Saclay, 91191 Gif-sur-Yvette, France\n3INAC-SPINTEC, CEA/CNRS and Univ. Grenoble Alpes, 38000 Grenoble, France\n4Institute of Physics, Kazan Federal University, Kazan 420008, Russian Federation\n5Department of Physics, University of Muenster, 48149 Muenster, Germany\n6Institute of Metal Physics, Ural Division of RAS, Yekaterinburg 620041, Russian Federation\n7Instituto de Sistemas Optoelectr\u0013 onicos y Microtecnolog\u0013 \u0010a (UPM), Madrid 28040, Spain\n8Instituto de Microelectr\u0013 onica de Madrid (CNM, CSIC), Madrid 28760, Spain\n(Dated: August 13, 2018)\nSpin-orbit e\u000bects1{4have the potential of radically changing the \feld of spintronics\nby allowing transfer of spin angular momentum to a whole new class of materials.\nIn a seminal letter to Nature5, Kajiwara et al. showed that by depositing Platinum\n(Pt, a normal metal) on top of a 1.3 \u0016m thick Yttrium Iron Garnet (YIG, a magnetic\ninsulator), one could e\u000bectively transfer spin angular momentum through the interface\nbetween these two di\u000berent materials. The outstanding feature was the detection of\nauto-oscillation of the YIG when enough dc current was passed in the Pt. This \fnding\nhas created a great excitement in the community for two reasons: \frst, one could\ncontrol electronically the damping of insulators, which can o\u000ber improved properties\ncompared to metals, and here YIG has the lowest damping known in nature; second,\nthe damping compensation could be achieved on very large objects, a particularly\nrelevant point for the \feld of magnonics6,7whose aim is to use spin-waves as carriers\nof information. However, the degree of coherence of the observed auto-oscillations\nhas not been addressed in ref.5. In this work, we emphasize the key role of quasi-\ndegenerate spin-wave modes, which increase the threshold current. This requires to\nreduce both the thickness and lateral size in order to reach full damping compensation8,\nand we show clear evidence of coherent spin-orbit torque induced auto-oscillation in\nmicron-sized YIG discs of thickness 20 nm.\nWhen spin transfer e\u000bects were \frst introduced by\nSlonczweski and Berger in 19969,10, the authors immedi-\nately recognized that the striking signature of the trans-\nfer process would be the emission of microwave radia-\ntion when the system is pumped out of equilibrium by a\ndc current. Since the spin transfer torque on the mag-\nnetisation is collinear to the damping torque, there is an\ninstability threshold when the natural damping is fully\ncompensated by the external \row of angular momentum,\nleading to spin-wave ampli\fcation through stimulated\nemission. Using analogy to light, the e\u000bect was called\nSWASER10, where SW stands for spin-wave. Until 2010,\nall SWASER devices required a charge current perpen-\ndicular to the plane to transfer angular momentum be-\ntween di\u000berent magnetic layers9,10. This implied that the\ne\u000bect was restricted to conducting materials. The situa-\ntion has radically changed since spin-orbit e\u000bects such as\nthe spin Hall e\u000bect (SHE)11,12are used to produce spin\ncurrents in normal metals. Here a right hand side rule\nlinks the de\rected direction of the electron and the orien-\ntation of its spin. This allows the creation of a pure spin\ncurrent transversely to the charge current, with an e\u000e-\nciency given by the spin Hall angle \u0002 SH. Using a metal\nwith large \u0002 SH, such as Pt, a charge current \rowing in\nplane generates a pure spin current \rowing perpendic-\nular to the plane, which can eventually be transferredthrough an interface with ferromagnetic metals, result-\ning in the coherent emission of spin-waves13, but also\nwith non-metals such as YIG5.\nThe microscopic mechanisms of transfer of angular mo-\nmentum between a normal metal and a ferromagnetic\nlayer are quite di\u000berent depending on the latter being\nmetallic or not. In the \frst case, electrons in each layer\nhave the possibility to penetrate the other one, whereas\nin the second case the transfer takes place exactly and\nsolely at the interface. It is thus much more sensitive to\nthe imperfection of the interface. Still, a direct experi-\nmental evidence that spin current can indeed cross such\nan hybrid interface is through the so-called spin pump-\ning e\u000bect14: adding a normal metal on top of YIG in-\ncreases its ferromagnetic resonance (FMR) linewidth15,\nwhich is due to the new relaxation channel at the inter-\nface through which angular momentum can escape and\nget absorbed in the metal. This e\u000bect being interfacial,\nthe broadening scales as 1 =tYIG, wheretYIGis the thick-\nness of YIG. Even for YIG, whose natural linewidth is\nonly a few Oersted at 10 GHz, it is hardly observable\niftYIGexceeds a couple hundreds of nanometers. For\nthese thick \flms though, the spin pumping can still be\ndetected through inverse spin Hall e\u000bect (ISHE). In a nor-\nmal metal with strong spin-orbit interaction, the pumped\nspin current is converted into a transverse charge current.arXiv:1504.01512v1  [cond-mat.mtrl-sci]  7 Apr 20152\nThis generates a voltage proportional to the length of the\nsample across the metal, which can easily reach several\ntens of microvolts in millimeter-sized samples. Since the\n\frst experiment by Kajiwara et al.5, many studies re-\nported the ISHE detection of FMR using di\u000berent metals\non YIG layers16{18, hereby providing clear evidence of at\nleast partial transparency of the hybrid YIG jmetal inter-\nfaces to spin currents. Due to Onsager relations, these\nresults made the community con\fdent that a spin cur-\nrent could thus be injected from metals to YIG and lead\nto the SWASER e\u000bect.\nFrom the beginning it was anticipated that the key to\nobserve auto-oscillations in non-metals was to reduce the\nthreshold current. The \frst venue is of course to choose\na material whose natural damping is very low. In this re-\nspect YIG is the optimal choice. The second thing is to\nreduce the thickness since the spin-orbit torque (SOT)\nis an interfacial e\u000bect. This triggered an e\u000bort in the\nfabrication of ultra-thin \flms of YIG of very high dy-\nnamical quality19,20. For 20 nm thick YIG \flms with\ndamping constant as low as \u000b= 2:3\u000110\u00004, a striking re-\nsult was that there were no evidence of auto-oscillations\nin millimeter-sized samples at the highest dc current pos-\nsible in the top Pt layer20,i.e., before it evaporates. It\nis worth mentioning at this point that reducing further\nthe thickness or the damping parameter of such ultra-\nthin YIG \flms21does not help anymore in decreasing the\nthreshold current, as the relevant value of the damping\nis that of the YIG jPt hybrid, which ends up to be com-\npletely dominated by the spin-pumping contribution.\nBut most notably, none of these high-quality ultra-\nthin YIG \flms display a purely homogeneous FMR line.\nThe reason for that is well known. In such extended\n\flms, there are many degenerate modes with the main,\nuniform FMR mode, which through the process of two-\nmagnon scattering broaden the linewidth22,23. A striking\nevidence of these degenerate modes can be obtained by\nparametrically pumping the SW modes. It reveals an\nuncountable number of modes which are at the same\nenergy as the FMR mode24. Any threshold instabil-\nity will be a\u000bected by the presence of those modes, as\nlearnt from LASERs where mode competition is known\nto have a strong in\ruence on the emission threshold25.\nThus, the next natural step was to reduce as well the\nlateral size in order to lift the degeneracy between modes\nthrough con\fnement. The \frst microstructures of YIG\nappeared revealing that the patterning indeed narrowed\nthe linewidth through a decrease of the inhomogeneous\npart26. The e\u000bect is clear in the perpendicular geometry,\nwhere magnon-magnon processes are suppressed owing\nto the fact that the FMR mode lies at the bottom of the\nSW dispersion relation. However, this is not the case in\nthe parallel geometry where the FMR mode is not the\nlowest energy SW mode. Even then, we showed that the\nlinewidth in a micron-sized YIG jPt disc could be still\ntuned thanks to SOT8. In the following, we describe the\ndirect electrical detection of auto-oscillations in similar\nsamples and show that the threshold current is increased\nFIG. 1. Inductive detection of auto-oscillations in a\nYIGjPt microdisc. a , Sketch of the sample and measure-\nment con\fguration. The bias \feld His oriented transversely\nto the dc current Idc\rowing in Pt. The inductive voltage Vy\nproduced in the antenna by the precession of the YIG mag-\nnetisation is ampli\fed and monitored by a spectrum anal-\nyser. b{e, Power spectral density maps measured at \fxed\njHj= 0:47 kOe and variable Idc. The four quadrants corre-\nspond to di\u000berent possible polarities of HandIdc.\nby the presence of quasi-degenerate SW modes.\nWe study magnetic microdiscs with diameter 2 \u0016m\nand 4\u0016m which are fabricated based on a hybrid\nYIG(20 nm)jPt(8 nm) bilayer. The 20 nm thick YIG\nlayer is grown by pulsed laser deposition20and the 8 nm\nthick Pt layer is sputtered on top of it27. Their physical\nparameters are summarized in Table I. We stress that\nthe extended YIG \flm is characterized by a low Gilbert\ndamping parameter \u000b0= (4:8\u00060:5)\u000110\u00004and a remark-\nably small inhomogeneous contribution to the linewidth,\n\u0001H0= 1:1\u00060:3 Oe. Each microdisc is connected to\nelectrodes enabling the injection of a dc current Idcin\nthe Pt layer, and a microwave antenna is de\fned around\nit to obtain an inductive coupling with the YIG magneti-\nsation, as shown schematically in Fig. 1a.\nFirst, we monitor with a spectrum analyser the voltage\nproduced in the antenna by potential auto-oscillations of\nthe 4\u0016m YIG disc as a function of the dc current Idcin-\njected in Pt. The in-plane magnetic \feld H= 0:47 kOe3\nTABLE I. Physical parameters of the Pt and bare YIG layers, and of the hybrid YIG jPt bilayer.\nPt tPt(nm) \u001a(\u0016\n.cm) \u0015SD(nm) \u0002 SH\nfrom ref.278 17 :3\u00060:6 3 :4\u00060:4 0 :056\u00060:010\nYIG tYIG(nm) \u000b0 4\u0019Ms(G) \r(107rad.s\u00001.G\u00001)\nthis study 20 (4 :8\u00060:5)\u000110\u000042150\u000650 1 :770\u00060:005\nYIGjPt tYIGjtPt(nm) \u000b g \"#(1018m\u00002) T\nthis study 20j8 (2 :05\u00060:1)\u000110\u000033:6\u00060:5 0 :2\u00060:05\nFIG. 2. ISHE-detected FMR spectroscopy in YIG jPt microdiscs. a , Sketch of the sample and measurement con\fgu-\nration. The bias \feld His oriented perpendicularly to the Pt electrode and to the excitation \feld hrfproduced by the antenna\nat \fxed microwave frequency. The dc voltage Vxacross Pt is monitored as a function of the magnetic \feld. b, ISHE-detected\nFMR spectra of the 4 \u0016m and 2\u0016m YIG(20 nm)jPt(8 nm) discs at 1 GHz and 4 GHz, respectively. c, Dispersion relation of\nthe main FMR mode of the microdiscs. The continuous line is a \ft to the Kittel law. d, Frequency dependence of the FMR\nlinewidth in the two microdiscs. The vertical bars show the mean squared error of the lorentzian \fts. The continuous lines are\nlinear \fts to the data. The dashed line shows the homogeneous contribution of the bare YIG.\nis applied in a transverse direction with respect to Idc,\nas shown in Fig. 1a. This is the most favorable con-\n\fguration to compensate the damping and obtain auto-\noscillations in YIG, as spins accumulated at the YIG jPt\ninterface due to SHE in Pt will be collinear to its magneti-\nsation. Color plots of the inductive signal measured as a\nfunction of the relative polarities of HandIdcare pre-\nsented in Fig. 1b{e. At H < 0, we observe in the power\nspectral density (PSD) a peak which starts at around\n2.95 GHz and 13 mA and then shifts towards lower fre-\nquency asIdcis increased (Fig. 1b), a clear signature that\nspin transfer occurs through the YIG jPt interface. The\nlinewidth of the emission peak, lying in the 10{20 MHz\nrange for 13 < I dc<17 mA, also proves the coherent\nnature of the detected signal. An identical behaviour is\nobserved at H > 0 andIdc<0 (Fig. 1d). In contrast,\nthe PSD remains \rat in the two other cases (Fig. 1c{d).\nTherefore, an auto-oscillation signal is detected only if\nH\u0001Idc<0, in agreement with the expected symmetry of\nSHE.\nIn order to characterize the \row of angular momen-\ntum across the YIG jPt interface, we now perform ISHE-\ndetected FMR spectroscopy on our microdiscs. The con-\n\fguration of this experiment is similar to the previous\ncase, but now the antenna generates a uniform microwave\n\feldhrfto excite the FMR of YIG while the dc voltage\nacross Pt is monitored at zero current (see Fig. 2a). In\nother words, we perform the reciprocal experiment of theone presented in Fig. 1. As described in the introduction,\na voltageVISHE develops across Pt when the FMR con-\nditions are met in YIG. This voltage changes sign as the\n\feld is reversed, which is expected from the symmetry of\nISHE, and shown in Fig. 2b, where the FMR spectra of\nthe 4\u0016m and 2\u0016m microdiscs are respectively detected\nat 1 GHz and 4 GHz. We also note that for a given\n\feld polarity, the product between VISHE andIdcmust\nbe negative to compensate the damping8, which enables\nto observe auto-oscillations in Fig. 1.\nFrom these ISHE measurements, the dispersion rela-\ntion and frequency dependence of the full linewidth at\nhalf maximum of the main FMR mode can be deter-\nmined, as shown in Figs. 2c and 2d, respectively. The\ndispersion relation follows the expected Kittel law. The\ndamping parameters of the 4 \u0016m and 2\u0016m microdiscs,\nextracted from linear \fts to the data, \u0001 H= 2\u000b!=\r +\n\u0001H0(continuous lines in Fig. 2d, !is the pulsation fre-\nquency and \rthe gyromagnetic ratio), are found to be\nsimilar with an average value of \u000b= (2:05\u00060:1)\u000110\u00003.\nThe small inhomogeneous contribution to the linewidth\nobserved in both microdiscs, \u0001 H0= 1:3\u00060:4 Oe and\n\u0001H0= 0:7\u00060:4 Oe, respectively, decreases with the\ndiameter and is attributed to the presence of several un-\nresolved modes within the resonance line8. In order to\nemphasize the increase of damping due to Pt, we have\nreported in Fig. 2d the broadening produced by the ho-\nmogeneous contribution of the bare YIG using a dashed4\nline. The observed increase of damping is due to spin\npumping14,15,\n\u000b\u0000\u000b0=g\"#\r~\n4\u0019MstYIG; (1)\nwhere ~is the reduced Planck constant, Msthe satu-\nration magnetisation and g\"#the spin-mixing conduc-\ntance of the YIGjPt interface. This allows us to extract\ng\"#= (3:6\u00060:5)\u00011018m\u00002, which lies in the same window\nas previously reported values26,28. From the spin-mixing\nconductance g\"#, the spin di\u000busion length \u0015SDand the\nresistivity\u001aof the Pt layer, we can also calculate the\ntransparency of the YIG jPt interface to spin current29,\nT= 0:2\u00060:05. The physical parameters extracted for\nthe YIGjPt hybrid bilayer are summarized in the last raw\nof Table I.\nTo gain further insight about the origin of the\nauto-oscillation signal, we now monitor how the auto-\noscillations of the 4 \u0016m disc evolve as the angle \u001ebe-\ntween the in-plane bias \feld \fxed to H= 0:47 kOe and\nthe dc current Idcis varied from 30\u000eto 150\u000e. The re-\nsults are summarized in Fig. 3. Pannels b{d show the\nauto-oscillation voltages detected in the antenna ( Vy) and\nacross the Pt electrode ( Vx). At\u001e= 90\u000e, the auto-\noscillation signal is only visible in the Vychannel. At\n\u001e= 60\u000e, bothVxandVychannels exhibit the auto-\noscillation peak. At \u001e= 40\u000e, it almost vanishes in Vy,\nwhile it slightly increases in Vx. The normalized signals\nas a function of \u001eare plotted in Fig. 3e. The Pt elec-\ntrode and antenna loop being oriented perpendicularly to\neach other (see Fig. 3a), the ac \rux due to the precession\nof magnetisation picked up by each of them respectively\nvaries as cos \u001eand sin\u001e(dashed lines in Fig. 3e).\nMore importantly, this study of angle dependence also\nallows us to extract the threshold current for auto-\noscillations as a function of \u001e. As\u001edeviates from the\noptimal orientation 90\u000e, the absolute value of the thresh-\nold current rapidly increases, see Fig. 3f. In fact, the\nSOT acting on the oscillating part mof the magneti-\nsation scales as m\u0002s\u0002m/sin\u001e, where sis the spin\npolarisation of the dc spin current produced by SHE in Pt\nat the YIGjPt interface. Therefore, the expected thresh-\nold current scales as 1 =sin\u001e, which is plotted as a dashed\nline in Fig. 3f, in very good agreement with the data.\nIn summary, the results reported in Figs. 1 and 3 un-\nambiguously demonstrate that the auto-oscillations ob-\nserved in our hybrid YIG jPt discs result from the ac-\ntion of SOT produced by Idc. We have also shown that\nthey correspond to the reverse e\u000bect of the spin pump-\ning mechanism illustrated in Fig. 2d and its detection\nthrough ISHE in Fig. 2b.\nIn the last part of this letter, we analyse quantitatively\nthe main features of auto-oscillations, which allows us\nto determine their nature and to understand the role of\nquasi-degenerate SW modes in the SOT driven dynamics.\nFor this, we compare the auto-oscillations observed in\nthe 4\u0016m and 2\u0016m microdiscs. Figs. 4a and 4b re-\nspectively present the inductive signal Vydetected in the\nFIG. 3. Auto-oscillations as a function of the angle\nbetween the dc current and the bias \feld. a , Sketch\nof the sample and measurement con\fguration. The bias \feld\nHis oriented at an angle \u001ewith the dc current Idcin the\nPt. The precession of the YIG magnetisation induces volt-\nagesVxin the antenna and Vyacross Pt, which are ampli\fed\nand monitored by spectrum analysers. b{d,VxandVyat\nH= 0:47 kOe for three di\u000berent angles \u001e.e, Dependence of\nthe normalized signals in both circuits and f, of the thresh-\nold current for auto-oscillations on \u001e. Dashed lines show the\nexpected angular dependences.\nantenna coupled to these two discs as a function of Idc.\nThe con\fguration is the same as depicted in Fig. 1a,\nwith a slightly larger bias \feld set to H= 0:65 kOe.\nOne can clearly see a peak appearing in the PSD close\nto 3.6 GHz in both cases, at a threshold current of about\n\u000013:5 mA in the 4 \u0016m disc and\u00007:4 mA in the 2 \u0016m\ndisc. These two values correspond to a similar threshold\ncurrent density in both samples of (4 :4\u00060:2)\u00011011A.m\u00002,\nin agreement with our previous study8. As the dc cur-\nrent is varied towards more negative values, the peaks\nshift towards lower frequency (Fig. 4c), at a rate which\nis twice faster in the smaller disc. This frequency shift is\nmainly due to linear and quadratic contributions in Idcof\nOersted \feld and Joule heating, respectively8(from the\nPt resistance, the maximal temperature increase in both5\nFIG. 4. Quantitative analysis of auto-oscillations in YIG jPt microdiscs .a, Inductive voltage Vyproduced by auto-\noscillations in the 4 \u0016m and b, 2\u0016m YIGjPt discs as a function of the dc current Idcin the Pt. The experimental con\fguration\nis the same as in Fig. 1a, with the bias \feld \fxed to H= 0:65 kOe. c, Auto-oscillation frequency, d, linewidth and e,\nintegrated power vs.Idc.f, Dependence of the onset frequency and g, of the threshold current on the applied \feld in both\ndiscs. Expectations taking into account only the homogeneous linewidth or the total linewidth are respectively shown by dashed\nand continuous lines.\nsamples is estimated to be +40\u000eC). At the same time,\nthe signal \frst rapidly increases in amplitude, reaches\na maximum, and then, more surprisingly, drops until it\ncannot be detected anymore, as seen in Fig. 4e, which\nplots the integrated power vs.Idc. The maximum of\npower measured in the 4 \u0016m disc (2.9 fW) is four times\nlarger than the one measured in the 2 \u0016m disc (0.7 fW),\nwhich is due to the inductive origin of Vy. The lat-\nter can be estimated from geometrical considerations,\nVy=\u0011(!\u00160DtYIGMssin\u0012)=2. Here\u00160is the magnetic\nconstant,Dthe diameter of the disc and \u0012the angle of\nuniform precession (the prefactor \u0011'0:1 accounts for\nmicrowave losses and impedance mismatch in the mea-\nsured frequency range with our microwave circuit). For\nthe same\u0012, the inductive voltage produced by the 4 \u0016m\ndisc is thus twice larger than by one produced by the\n2\u0016m disc, hence the ratio four in power. Moreover, the\nmaximal angle of precession reached by auto-oscillations\nis found to be about 1\u000ein both microdiscs8. Finally,\nthe disappearance of the signal as Idcgets more nega-\ntive is accompanied by a continuous broadening of the\nlinewidth, which increases from a few MHz to several\ntens of MHz (Fig. 4d). This rather large auto-oscillation\nlinewidth is also consistent with a small precession angle,\ni.e., a small stored energy in the YIG oscillator30.\nBy repeating the same analysis as a function of H,\nwe can determine the bias \feld dependence of the auto-\noscillations in both microdiscs. The onset frequency\nand threshold current at which auto-oscillations start are\nplotted in Figs. 4f and 4g, respectively. The onset fre-\nquency in the 4 \u0016m and 2\u0016m microdiscs is identical and\nclosely follows the dispersion relation of the main FMR\nmode plotted as a continuous line. The small redshift to-\nwards lower frequency, which increases with the applied\n\feld, is ascribed to the Joule heating and Oersted \feld in-\nduced byIdc(the Kittel law in Figs. 2c and 4f is obtained\natIdc= 0 mA). We also note that the main FMR mode\nis the one which couples the most e\u000eciently to our induc-\ntive electrical detection, because it is the most uniform.Hence, we conclude that the detected auto-oscillations\nare due to the destabilisation of this mode by SOT.\nIn order to reach auto-oscillations, the additional\ndamping term due to SOT has to compensate the natural\nrelaxation rate \u0000 rin YIG. Given the transparency Tof\nthe YIGjPt interface and the spin Hall angle \u0002 SHin Pt,\nthis condition writes:\nT\u0002SH~\n2e\r\ntYIGMsIth\ntPtD=\u0000\u0000r; (2)\nwheretPtDis the section of the Pt layer. The homoge-\nneous contribution to \u0000 ris given by the Gilbert damping\nrate, which for the in-plane geometry is:\n\u0000G=\u000b\r(H+ 2\u0019Ms): (3)\nWe remind that this expression is obtained by convert-\ning the \feld linewidth to frequency linewidth through\n\u0001!= \u0001H(@!=@H ). If only the homogeneous contribu-\ntion to the linewidth is taken into account, the threshold\ncurrentIthis thus expected to depend linearly on H, as\nshown by the dashed lines plotted in Fig. 4g using Eqs.\n2 and 3, and the parameters listed in Table I (the only\nadjustment made is for the 4 \u0016m disc, where the calcu-\nlatedIthhas been reduced by 20% in order to reproduce\nasymptotically the experimental slope of Ithvs.H). It\nqualitatively explains the dependence of Ithat large bias\n\feld in both microdiscs, but underestimates its value and\nfails to reproduce the optimum observed at low bias \feld.\nTo understand this behaviour, the \fnite inhomoge-\nneous contribution to the linewidth \u0001 H0measured in\nFig. 2d should be considered as well. In fact, this contri-\nbution dominates the full linewidth at low bias \feld. In\nthat case, the expression of the relaxation rate writes:\n\u0000r= \u0000G+\r\u0001H0\n2H+ 2\u0019Msp\nH(H+ 4\u0019Ms): (4)\nThe form of the last term in Eq. 4 is due to the Kittel\ndispersion relation and is responsible for the existence6\nof the optimum in IthatH6= 0. Using the value of\n\u0001H0= 0:7 Oe extracted in Fig. 2d for the 2 \u0016m disc in\nEq. 4 in combination with Eq. 2, the continuous blue\nline of Fig. 4g is calculated, in very good agreement with\nthe experimental data. To get such an agreement for the\n4\u0016m disc, \u0001H0has to be increased by 25% compared to\nthe value determined in Fig. 2d. In this case, both the\nposition of the optimum (observed at H'0:3\u00000:5 kOe)\nand the exact value of Ithare also well reproduced for\nthe 4\u0016m disc, as shown by the continuous red line in\nFig. 4g.\nHence, it turns out that quasi-degenerate SW modes,\nwhich are responsible for the inhomogeneous contribu-\ntion to the linewidth, strongly a\u000bect the exact value and\ndetailed dependence vs.HofIth. In fact, it is the to-\ntal linewidth that truly quanti\fes the losses of a mag-\nnetic device regardless of the nature and number of mi-\ncroscopic mechanisms involved. Even in structures with\nmicron-sized lateral dimensions, there still exist a few\nquasi-degenerate SW modes as evidenced by the \fnite\n\u0001H0observed in Fig. 2d. Due to magnon-magnon scat-\ntering, they are linearly coupled to the main FMR mode,\nwhich as a result has its e\u000bective damping increased,\nalong with the threshold current. The presence of these\nSW modes is also known to play a crucial role in SOT\ndriven dynamics. The strongly non-equilibrium distri-\nbution of SWs promoted by SOT in combination with\nnonlinear interactions between modes can lead to mode\ncompetition, which might even prevent auto-oscillations\nto start31. We believe that the observed behaviours of\nthe integrated power (Fig. 4d) and linewidth (Fig. 4e)\nvs.Idcare reminiscent of the presence of these quasi-\ndegenerate SW modes. A meaningful interpretation of\nthese experimental results is that as the FMR mode\nstarts to auto-oscillate and to grow in amplitude as the\ndc current is increased above the threshold, its coupling\nto other SW modes { whose amplitudes also grow due to\nSOT { becomes larger, which makes the \row of energy\nout of the FMR mode more e\u000ecient. This reduces the\ninductive signal, as non-uniform SW modes are poorly\ncoupled to our inductive detection scheme. At the same\ntime, it enhances the auto-oscillation linewidth, which\nre\rects this additional nonlinear relaxation channel.\nThe smaller inhomogeneous linewidth in the 2 \u0016m disc\n(Fig. 2d) results in a \feld dependence of the threshold\ncurrent closer to the one expected for the purely homo-\ngeneous case (Fig. 4g). This indicates that reducing\nfurther the lateral size of the microstructure will allow\nto completly lift the quasi-degeneracy between spin-wave\nmodes26, as predicted by micromagnetic simulations,\nwhich show that this is obtained for lateral sizes smaller\nthan 1\u0016m. This could extend the stability of the auto-\noscillation for the FMR mode, and experimental tech-\nniques capable of detecting SWs in nanostructures8,31\nshould be used to probe this transition. Very importantly\nfor the \feld of magnonics, it was recently shown that this\nconstraint on con\fnement could be relaxed in one dimen-sion such as to produce a propagation stripe32. Other\nstrategies might consist in using speci\fc non-uniform SW\nmodes or to engineer the SW spectrum using topological\nsingularities such as vortices, or bubbles, which could be\nmost relevant to design active magnonics computational\ncircuits.\nMETHODS\nSamples { Details of the PLD growth of the YIG\nlayer can be found in ref.20. Its dynamical proper-\nties have been determined by broadband FMR measure-\nments. The transport parameters of the 8 nm thick Pt\nlayer deposited on top by magnetron sputtering have\nbeen determined in a previous study27. The YIGjPt mi-\ncrodiscs are de\fned by e-beam lithography, as well as the\nAu(80 nm)jTi(20 nm) electrodes { separated by 1 \u0016m\nfrom each other { which contact them. This electrical\ncircuit is insulated by a 300 nm thick SiO 2layer, and a\nbroadband microwave antenna made of 250 nm thick Au\nwith a 5\u0016m wide constriction is de\fned on top of each\ndisc by optical lithography.\nMeasurements { The samples are mounted between\nthe poles of an electromagnet which can be rotated to\nvary the angle \u001eshown in Fig. 3a. Two 50 \n matched\npicoprobes are used to connect to the microwave antenna\nand to the electrodes which contact the Pt layer. The lat-\nter are connected to a dc current source through a bias-\ntee. To perform ISHE-detected FMR measurements, a\nmicrowave synthesizer is connected to the microwave an-\ntenna, and the output power is turned on and o\u000b at a\nmodulation frequency of 9 kHz. The voltage across Pt\nis measured by a lock-in after a low-noise preampli\fer\n(gain 100). For the detection of auto-oscillations, high-\nfrequency low-noise ampli\fers are used (gain 33 dB to\n39 dB, depending on the frequency range). Two spec-\ntrum analysers simultaneously monitor in the frequency\ndomain the voltages VxandVyacross the Pt layer and\nin the microwave antenna, respectively (Fig. 3a). The\nresolution bandwidth employed in the measurements is\nset to 1 MHz.\nACKNOWLEDGMENTS\nWe acknowledge E. Jacquet, R. Lebourgeois and A. H.\nMolpeceres for their contribution to sample growth, and\nM. Viret and A. Fert for fruitful discussion. This research\nwas partially supported by the ANR Grant Trinidad\n(ASTRID 2012 program). V. V. 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Saitoh1, 2, 4, 5, 6\n1Advanced Institute for Materials Research,\nTohoku University, Sendai 980-8577, Japan\n2Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan\n3Fachbereich Physik and Landesforschungszentrum OPTIMAS,\nTechnische Universit at Kaiserslautern, 67663 Kaiserslautern, Germany\n4Department of Applied Physics, The University of Tokyo, Tokyo 113-8656, Japan\n5Center for Spintronics Research Network,\nTohoku University, Sendai 980-8577, Japan\n6Advanced Science Research Center,\nJapan Atomic Energy Agency, Tokai 319-1195, Japan\n(Dated: November 10, 2021)\nAbstract\nResonant enhancement of spin Seebeck e\u000bect (SSE) due to phonons was recently discovered\nin Y 3Fe5O12(YIG). This e\u000bect is explained by hybridization between the magnon and phonon\ndispersions. However, this e\u000bect was observed at low temperatures and high magnetic \felds,\nlimiting the scope for applications. Here we report observation of phonon-resonant enhancement of\nSSE at room temperature and low magnetic \feld. We observed in Lu 2BiFe 4GaO 12an enhancement\n700 % greater than that in a YIG \flm and at very low magnetic \felds around 10\u00001T, almost\none order of magnitude lower than that of YIG. The result can be explained by the change in the\nmagnon dispersion induced by magnetic compensation due to the presence of non-magnetic ion\nsubstitutions. Our study provides a way to tune the magnon response in a crystal by chemical\ndoping with potential applications for spintronic devices.\n1arXiv:1903.09007v1  [cond-mat.mtrl-sci]  21 Mar 2019Heat is an ubiquitous and highly underexploited energy source, with about two-thirds\nof the used energy being lost as wasted heat1, thus representing an opportunity for ther-\nmoelectric conversion devices2. Recently, a new thermoelectric conversion mechanism has\nemerged: the spin Seebeck e\u000bect (SSE)3,4, driven by thermally induced magnetization dy-\nnamics in magnetic materials, which generates a spin current. The spin current is then\ninjected into an adjacent metal layer, where it is converted into an electric current by the\ninverse spin Hall e\u000bect5,6.\nLately, it has been shown that the magnetoelastic coupling can improve the conversion ef-\n\fciency of SSE, as demonstrated by the observation of a resonant enhancement of the SSE\nvoltage in YIG \flms7,8. The observed SSE enhancement has been explained by the magnon-\nphonon hybridization at the crossing points of the magnon and phonon dispersions due to\nmomentum ( k) and energy (\u0016 h!) matching, at certain magnetic \feld values the dispersions\ntangentially touch each other and the magnon-phonon hybridization e\u000bects are maximised\n[see Figure 1(a)], resulting in peak structures due to the reinforcement of the magnon lifetime\na\u000bected by the phonons. However, this e\u000bect has only been observed at low temperatures\nand high magnetic \felds7{14.\nEngineering the magnon dispersion o\u000bers the possibility to tune the magnetic \feld at which\nthe magnon-phonon hybridization e\u000bects are maximised, one possible approach is modi\fca-\ntion of the magnetic compensation of a ferrimagnet15,16: intuitively, it can be expected that\nas the magnetic compensation of the system is introduced, the magnon dispersion gradually\nevolves from a parabolic k-dependence in the non-compensated state (similar to a ferro-\nmagnet) towards a linear dispersion when the system becomes fully compensated (i.e. zero\nnet magnetization, antiferromagnetic case), schematically shown in \fgure 1(b). Ideally, in\nthe antiferromagnetic state in which spin-wave velocity is same as the sound velocity the\nmagnon-phonon coupling e\u000bects can be maximised even further as a consequence of poten-\ntially larger overlap between the magnon and phonon dispersions, however this is yet to be\nexperimentally observed.\nHere, in order to investigate the in\ruence of increased magnetic compensation on the\nmagnon-phonon coupling e\u000bects, we use a Lu 2BiFe 4GaO 12(BiGa:LuIG) \flm as a model\nsystem and study the magnetic \feld and temperature dependences of the SSE. This system\nis a garnet ferrite, similar to YIG: it has two magnetic sublattices, where all the ions carrying\nspin angular momentum are Fe3+. The ferrimagnetic order originates from the di\u000berent site\n2occupation between the two di\u000berent magnetic sublattices, with the 3:2 ratio of tetrahedral\n(d) to octahedral (a) sites, resulting in non-zero magnetization. Substitution of Fe with Ga\nreinforces the magnetic compensation of the system due to the preferential occupation of\nthe tetrahedral sites by the Ga ions17, resulting in the reduction of the magnetic moment\nand ordering temperature (see methods). The use of BiGa:LuIG \flm allows systematic\nevaluation of spin wave dispersion using magneto-optical spectroscopy18,19.\nI. RESULTS\nRoom temperature resonant enhancement of the SSE. We performed SSE mea-\nsurements in the longitudinal SSE con\fguration, as schematically depicted in Fig. 2a. The\nmagnetic-\feld dependence of the SSE voltage measured at 300 K is shown in Fig. 2b. We\nobserved clear peaks in the voltage at magnetic \feld values of \u00160HTA\u00180.42 T and \u00160HLA\u0018\n1.86 T, as shown in the data blown ups in Fig.2c and Fig.2d. These correspond to the hy-\nbridization of the magnons with the transversal acoustic (TA) and longitudinal acoustic\n(LA) phonons, respectively. The observation of the clear enhancement of the SSE voltage at\nroom temperature is in a stark contrast to previous observations of the magnon-polaron SSE\nin YIG and other ferrite systems7,9{11,13. In fact, the observed enhancement in BiGa:LuIG\nis about 700% greater than that observed in a YIG \flm at room temperature (see Fig. 2e).\nMoreover, the features of the resonant SSE enhancement are already present at very low\n\feld values, as visible in the inset of Fig. 2b, where we can see a larger background SSE\nsignal forH <H TA.\nLet us consider possible reasons for the observation of clear magnon-polaron SSE peaks\nat room temperature. The energy transfer between phonon (heat) and spin system via\nmagnon-phonon coupling is resonantly enhanced at the crossing of the magnon and phonon\ndispersion relations. In the theory of the magnon-polaron SSE8, the peaks in the voltage\nare understood in terms of increased magnon-phonon hybridization at the magnetic \feld\nvalues where the magnon and phonon dispersions just tangentially touch each other. At\nthese positions the overlap over momentum space between the magnon and the TA or LA\nphonon dispersions is maximised. This results in a resonant enhancement of the SSE when\nthe acoustic quality of the crystal is larger than its magnetic quality, with the enhancement\nproportional to the ratio \u0011=j\u001cph=\u001cmagj>1, where\u001cmagand\u001cphare the values of magnon\n3and phonon lifetime7,8, respectively. In the case of a crystal having a better magnetic than\nacoustic quality ( \u0011 <1), dip structures instead of peaks in the SSE voltage are expected8.\nTherefore, according to the picture above, there is two possible scenarios to explain the\nlarger magnon-polaron SSE peaks observed at room temperature: (1) an increased overlap\noverk-space at the touching \felds ( HTA, LA ) or (2) a larger ratio between the magnetic\nand acoustic quality of the crystal ( \u0011\u001d1). The \frst scenario can be discarded in our\nsystem, since the e\u000bect of larger magnetic compensation increases the curvature of the\nmagnon dispersion (obtained in the next section), which results in a rather reduced overlap\nbetween the magnon-phonon dispersions at the touching points (see Supplementary Note\n3). Then we must look at the scenario (2): it has been shown that the Bi substitution\nresults in a decrease in the magnon lifetime in YIG20. As a consequence, the larger ratio\nbetween the impurity scattering potentials, or \u0011, can be expected, and therefore a greater\nenhancement of the lifetime of magnons by hybridization with the phonons at the touching\npoints, resulting in a greater SSE enhancement. Detailed knowledge of the magnon and\nphonon lifetimes, \u001cmag, ph , is required in order to quantitatively discuss the magnitude of the\nresonant enhancement of the SSE.\nWe will now centre the rest of our discussion mainly on the magnetic \feld dependence of\nthe magnon-polaron peaks and its relation to the dispersion characteristics. Let us now\ntry to understand the magnitude of the magnetic \felds required for the observation of the\nSSE peaks by considering the magnon and phonon dispersions. First, we consider linear\nTA and LA phonon dispersions !p=cTA, LAk, where the phonon velocities of the sample\nhave been determined using optical spectroscopy18, obtaining: cTA= 2.9\u0002103ms\u00001and\ncLA=6.2\u0002103ms\u00001for TA and LA modes, respectively. If we now assume the conventional\nmagnon dispersion for a simple ferromagnet: !m=\r\u00160H+Dexk2, as previously used for\nYIG7, we can never explain the observed results without assuming an exceedingly large spin\nwave sti\u000bness parameter, with a value about four times higher than previously reported for\nYIG (D ex= 7.7\u000210\u00006m2s\u00001)7,21,22and LuIG23. Even if we take into account the e\u000bect of\nBi-doping, this can only explain a 1.4 times increase of the spin wave sti\u000bness magnitude,\nas shown in previous reports for similar Bi-doping in YIG20. Moreover, if we consider the\nexpression of the spin wave sti\u000bness for a ferromagnet Dex/JSa2(hereJcorresponds\nto the exchange constant, Sthe spin of the magnetic atoms and athe distance between\nneighboring spins), this estimation would imply an increased magnitude of Jdespite the\n4presence of non-magnetic ion substitutions, contrary to expectations. This shows that the\nmagnon dispersion of simple ferromagnets cannot capture the microscopic features of our\nsystem, therefore in the following we will focus our attention on the ferrimagnetic ordering:\ne\u000bects of the Ga-doping on the magnetic compensation, and its impact on the magnon\ndispersion characteristics of our system.\nMagnon dispersion: theoretical model and experimental determination. We\nwill now discuss the magnon dispersion characteristics of a ferrimagnetic material as a func-\ntion of the degree of magnetic compensation. As previously explained, the ferrimagnetic\norder in iron garnet systems typically arises from the di\u000berent Fe3+occupation between the\ntwo magnetic sublattices, with the 3 to 2 ratio of tetrahedral (d) to octahedral (a) sites,\nas shown in Fig. 1b. Therefore, the degree of magnetic compensation can be modi\fed\nby substitution of the magnetic Fe3+ions in the tetrahedral sites by non-magnetic ones\n(i.e. Ga). To describe this e\u000bect, we consider the conventional Heisenberg Hamiltonian\nfor a ferrimagnetic system15,24, and express it as a function of the occupation numbers in\nthe d and a sites. The expression of the Hamiltonian with the exchange and the Zeeman\ninteraction terms is given below, where we have neglected dipolar and magnetic anisotropy\ninteractions for simplicity:\nH=Jad\n\u0016h2NaX\ni;\u000eSa;iSd;i+\u000e\u0000\r\u00160HNaX\niSz\na;i\n+Jad\n\u0016h2NdX\nj;\u000eSa;j+\u000eSd;j\u0000\r\u00160HNdX\njSz\nd;j: (1)\nHere, the upper (lower) part of the Hamiltonian accounts for the octrahedral (tetrahedral)\nsites, with Na(Nd) magnetic ions per unit volume. Jadis the nearest-neighbour inter-\nsublattice exchange (positive in the above expression), \u000erepresents a vector connecting the\nnearest-neighbour a-d sites, Sx(x = a, d) are the spin operators (for a, d sites), \u00160His the\nexternal magnetic \feld, and \r=g\u0016B=\u0016his the gyromagnetic ratio, with gthe spectroscopic\nsplitting factor, \u0016Bthe Bohr magneton, \u0016 hthe reduced Planck constant. Then, using the\nstandard Holstein-Primako\u000b approximation24,25, we can obtain the magnon dispersion rela-\ntion as a function of the ratio of Fe3+ions occupying a and d sites (see Supplementary Note\n51 for details of the derivation):\n!m=\r\u00160H+JadS(zda\u0016+zad\u0015)\n2\u0016h8\n<\n:\u0000\u0012\u0016\u0000\u0015\n\u0015\u0016\u0013\n\u0006\"\u0012\u0016\u0000\u0015\n\u0015\u0016\u00132\n+4k2a2\n3\u0015\u0016#1=29\n=\n;;(2)\nwhereS= 5=2 for Fe3+,ais the nearest-neighbor a-d distance, zad(zda) and\u0015=Na=N\n(\u0016=Nd=N) correspond to the number of nearest-neighbours and occupation ratio of mag-\nnetic ions in octahedral (tetrahedral) sites, respectively. Where Nis the total number of\nmagnetic Fe3+ions per unit volume. Using the above expression we calculate the magnon\ndispersion in the two sublattice ferrimagnet: Lu 2Bi[AxFe2\u0000x](DyFe3\u0000y)O12, where A (D) de-\nnotes the non-magnetic ions in octahedral (tetrahedral) sites with concentrations x(y), the\noccupation ratio of the a (d) sites can be expressed as a function of x(y) as:\u0015= 0:4\u00002\u0000x\n2\u0001\n(\u0016= 0:6\u00003\u0000y\n3\u0001\n)26. Equation 2 can explain the magnetic \feld values of previously observed\nmagnon-polaron SSE in YIG7withx=y= 0 (no substitutions). Comparing our model\nwith magnon-polaron SSE measurements in YIG at room temperature10, we estimated the\nvalue of the exchange constant at 300 K, obtaining Jad= (4:3\u00060:2)\u000210\u000022J (see Sup-\nplementary Note 2), which shows reasonable agreement with recently reported values by\nneutron scattering measurements ( Jad= 4:65\u000210\u000022J)27.\nWe now evaluate the e\u000bect of the non-magnetic ion substitutions on the magnon spectrum,\nwe can see that as the magnetic compensation of the system increases (larger y) the dis-\npersion becomes gradually steeper (see Fig. 3a), with higher magnon frequencies for the\nsame wave-vector values. When the system is fully compensated ( x= 0,y= 1) a linear\ndispersion is obtained, as expected for an antiferromagnetic system.\nTo further test the validity of our model, we also performed wave-vector-resolved Brillouin\nlight scattering (BLS) spectroscopy measurements19,28,29to obtain the spin wave dispersion\nrelation of our system and compare it with our model. As shown in Fig. 3b, the peak\nfrequency in BLS spectra for di\u000berent wavenumbers measured at \u00160H= 0:18 T can be\nexplained with the calculated magnon dispersion using x= 0:101 andy= 0:909. This com-\nposition is in close agreement with the one estimated by x-ray spectroscopy (see Methods).\nThis result further proves the good agreement between the experiment and the theoretical\nmodel. Note that the small deviation of BLS plots at small kregion is due to dipole-dipole\ninteraction, which is not considered in our model.\nNow we are in a position to look into the magnetic \feld dependence of the SSE voltage and\n6the conditions for the observation of magnon-polaron in our system. The low magnetic \feld\nvalue for the observation of the magnon polaron SSE is attributed to an increased magnetic\ncompensation which makes the magnon dispersion steeper compared to the non-doped case.\nThis results in the touching condition for the magnon and phonon dispersions at lower\nmagnetic \felds than that of YIG. Figures 3c and 3d show the comparison of the magnon\nand phonon dispersions with the above estimated composition at the magnetic \felds HTA\nandHLAfor which the SSE peaks are observed. In Fig. 3c, we can distinguish three\ndi\u000berent regions of the SSE response with respect to the magnitude of the magnetic \feld:\nforH < H TAthe SSE has both magnonic and phononic contributions (possibly from the\ncrossing points of the dispersions), at H=HTAthe SSE is resonantly enhanced showing\npeak structures, due to the increased e\u000bect of magnon-phonon hybridization at the touching\ncondition between the dispersions and at H >H TA, a shift of the SSE voltage background\nsignal can be observed, which is likely due to the fact that the magnon and TA-phonon\ndispersions do not cross at any point of the !\u0000kspace and the contribution from the TA\nphonons is suppressed. The presence of this shift in the SSE voltage indicates that the\ncontribution from magnon-phonon coupling e\u000bects can be already sensed at magnetic \felds\nvalues much lower than that of the touching condition (see inset of Fig. 2b).\nTemperature dependence. Let us now investigate the temperature dependence of the\nmagnon-polaron SSE. Magnetization measurements as a function of temperature show that\nthe increased magnetic compensation, due to the presence of Ga substitution, results in the\nreduction of the saturation magnetization and ferrimagnetic ordering temperature ( Tc) in\nLuIG with Tc= 401.6\u00060.2 K (see Fig.4b). Figure 4a shows the result of the magnetic-\n\feld dependent SSE voltages at di\u000berent temperatures, we can see that the magnitude of\nthe SSE decreases close to the transition temperature as previously reported on YIG30, and\neventually, at T\u0019400 K, the magnon-driven SSE is suppressed, showing only a weak voltage\nwith a paramagnetic-like magnetic \feld dependence. The magnon-polaron SSE is gradually\nweakened as the temperature increases toward the transition temperature. The peaks are\nstrongly suppressed at temperatures well below the transition temperature (the maximum\ntemperature for observation of the peaks is T = 380 K for TA and T = 315 K for LA\nphonon). Here, we will now focus on the temperature dependence of the magnetic \feld at\nwhich the magnon-polaron peaks are observed, which is shown in Fig. 4c: we can see that the\n7magnitude of the magnetic \felds required for the magnon-polaron to appear increase with the\ntemperature. This trend can be understood by the softening of the magnon dispersion upon\nincreasing temperature; in the previously obtained magnon dispersion (Eq. 2), most of the\nparameters are temperature independent except for the intersite exchange energy Jad. The\nobserved temperature dependence of the magnon polaron magnetic \feld can be explained\nin terms of the magnitude of Jadwhich decreases around the transition temperature, in\nagreement with the previous observations in YIG and other ferrimagnets31{33.\nThe temperature dependence of Jadestimated from the touching condition between magnon\nand phonon dispersions is shown in Fig. 4d, the obtained dependence can be understood in\nterms of a temperature dependent exchange energy, with an expression similar to that used\nin Ref. 38: Jad=J0(1\u0000\u0011T5=2), withJ0= (6:41\u00060:05)\u000210\u000022J, which is obtained from\nconsidering the e\u000bect of magnon-magnon interactions in a ferromagnet34. These results\npossibly suggest that the magnon-magnon interactions might play a role in the magnon-\npolaron SSE in the temperature region studied here.\nII. DISCUSSION.\nIn this study, we reported the resonant enhancement of SSE in a partially compensated\nferrimagnet. Sharp peaks were observed in SSE voltage at room temperature and low mag-\nnetic \felds. The resonant enhancement of SSE is 700 % greater than that observed in YIG\n\flms, atributable to reduced magnon lifetime of BiGa:LuIG in comparison to YIG, which\nresults in larger reinforcement of the magnon lifetimes a\u000bected by the phonon system via\nthe hybridization.\nThe observed resonant enhancement of SSE at low magnetic \felds is attributable to steeper\nmagnon dispersion caused by the increased magnetic compensation of the system. Our\nresults show the possibility to tune the spin wave dispersion by chemical doping, which\nallows exploring magnon-phonon coupling e\u000bects at di\u000berent regions of the spin-wave spec-\ntrum. The value of the intersite exchange parameter ( Jad) was also estimated with values in\nreasonable agreement with previous studies. This fact shows the potential of the SSE as a\ntable-top tool to investigate the spin wave dispersion characteristics in comparison to other\nmore expensive and less accessible techniques, such as inelastic neutron scattering9,35.\n8III. METHODS\nA. Sample\nWe used epitaxial Lu 2BiFe 4GaO 12(3\u0016m) and YIG(2.5 \u0016m) \flms grown by liquid phase\nepitaxy on Gd 3Ga5O12[001] and [111]-oriented substrates, respectively. The composition\nof the Lu 2BiFe 4GaO 12\flm was determined using wavelength dispersive x-ray spectroscopy\n(WDX). Magnetization measurements show a saturation magnetization \u00160MS= 0.024 T at\n300 K and a transition temperature of TC\u0019402 K. A 5 nm Pt layer was deposited at room\ntemperature by (DC) magnetron sputtering in a sputtering system QAM4 from ULVAC,\nwith a base pressure of 10\u00005Pa. The sample dimensions for the SSE measurements were\nLy= 6 mm,Lx= 2 mm and Lz= 0:5 mm.\nB. Measurements\nThe SSE measurements were performed in a physical property measurement (PPMS)\nDynacool system of Quantum Design, Inc., equipped with a superconducting magnet with\n\felds of up to 9 Tesla. The system allows for temperature dependent measurements from 2\nto 400 K. For the SSE measurements the sample is placed between two plates made of AlN\n(good thermal conductor and electrical insulator): a resistive heater is attached to the upper\nplate and the lower plate is in direct contact with the thermal link of the cryostat, providing\nthe heat sink. The temperature gradient is generated by applying an electric current to the\nheater, while the temperature di\u000berence between the upper and lower plate is monitored by\ntwo E-type thermocouples connected di\u000berentially. The samples are contacted by Au wire\nof 25\u0016m diammeter. To minimize thermal losses, the wires are thermally anchored to the\nsample holder. The thermoelectric voltage is monitored with a Keithley 2182A nanovolt-\nmeter.\nThe magnetization measurements were performed using the VSM option of a PPMS system\nby Quantum Design, Inc, the temperature dependent magnetization measurements were\nobtained by performing isothermal M-H loops at each temperature and extracting the sat-\nuration magnetization value for each of the temperatures.\nThe Brillouin light scattering (BLS) measurements were performed using an angle-resolved\nBrillouin light scattering setup. Brillouin light scattering is an inelastic scattering of light\n9due to magnons. As a result, some portion of the scattered light shifts in the frequency\nequivalent to that of magnon. The scattered light is introduced to multi-pass tandem Fabry-\nPerot interferometer to determine the frequency shift. Wavenumber resolution was realized\nby collecting the back-scattered light from the sample by changing the incident angle ( \u0012in).\nAs a result of conservation of wavenumber of magnon ( km) and light ( kl), the wavenum-\nber of magnon is determined as km= 2klsin(\u0012in). All the spectrum was obtained at room\ntemperature.\nIV. ACKNOWLEDGMENTS\nWe thank J. Barker for fruitful discussions. This work was supported by ERATO \\Spin\nQuantum Recti\fcation Project\" (Grant No. JPMJER1402) and Grant-in-Aid for Scienti\fc\nResearch on Innovative Area, \\Nano Spin Conversion Science\" (Grant No. JP26103005),\nGrant-in-Aid for Research Activity Start-up (No. JP18H05841) from JSPS KAKENHI,\nJSPS Core-to-Core program \\the International Research Center for New-Concept Spintron-\nics Devices\" Japan, the NEC Corporation and the Noguchi Institute.\nV. AUTHOR CONTRIBUTIONS\nR.R. performed the measurement, analysed the data and developed the theoretical model\nwith input from T.H., Y.H., T.K. and E.S. P.F., A.J.E.K., T.H., V.I.V. and A.A.S. performed\nthe BLS measurements. T.K. performed the SSE measurements in YIG. R.R., Y.H. and\nE.S. planned and supervised the study. R.R. wrote the manuscript with review and input\nfrom T.H., Y.H., T.K. and E.S. All authors discussed the results and commented on the\nmanuscript.\n\u0003ramosr@imr.tohoku.ac.jp\n1K. Biswas, J. He, I. D. Blum, C-I. Wu, T. P. Hogan, D. N Seidman, V. P. Dravid, and M. G.\nKanatzidis, \\High-performance bulk thermoelectrics with all-scale hierarchical architectures,\"\nNature 489, 414 (2012).\n102J. He and T. M. Tritt, \\Advances in thermoelectric materials research: Looking back and\nmoving forward,\" Science 357, eaak9997 (2017).\n3K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. 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Prabhakaran, and A. T. Boothroyd,\n\\The full magnon spectrum of yttrium iron garnet,\" npj Quantum Materials 2, 63 (2017).\n36S. Geller, J. A. Cape, G. P. Espinosa, and D. H. Leslie, \\Gallium-substituted yttrium iron\ngarnet,\" Phys. Rev. 148, 522 (1966).\n37P. Hansen, P. Rschmann, and W. Tolksdorf, \\Saturation magnetization of gallium-substituted\nyttrium iron garnet,\" J. Appl. Phys. 45, 2728 (1974).\n13kmagnon\nkphonon x\nyz(a) (b) \nH = HTA \nMagnetic \ncompensation\nCompensated Non-compensated\nnon-magnetic \nsubstitution [ ]a: spin-down ion \n)d: spin-up ion (\n)d: non-magnetic ion (ω ωm\nωp\nk\nkω\nωFIG. 1. Lattice and spin waves in a magnetic material. (a) Schematic representation\nof a propagating phonon (lattice) and magnon (spin) excitation in a magnetic medium. Magnon-\npolarons are formed due to magnon-phonon hybridization when the magnon and phonon dispersions\ntangentially touch each other (i.e. they have coincident wavevector kmagnon =kphonon , and group\nvelocity@!\n@kjmagnon =@!\n@kjphonon ), as schematically depicted by the dispersion relation shown in the\ninset of (a) representing the magnon dispersion (blue curve) and phonon dispersions (dashed line)\nunder an applied magnetic \feld with magnitude HTA, in this condition a resonant enhancement\nof the SSE can be observed. (b) Schematic representation of the e\u000bect of magnetic compensa-\ntion on the magnon dispersion of a ferrimagnetic system: a parabolic dispersion is expected when\nthe system is non-magnetically compensated with non-zero magnetization (black curve), and as\nthe magnetic compensation increases the magnon dispersion gradually evolves towards a linear\ndependence when the system reaches magnetic compensation (antiferromagnet). This can be ex-\nperimentally achieved by introduction of non-magnetic ion substitutions into the lattice as shown\nin the bottom part of (b). Red and blue circles represent magnetic ions in two magnetic sublattices\nwith oppositely oriented spins and the orange circle represents the non-magnetic ion substitution.\n14T\n(a)\n(Lu 2Bi)Fe 4GaO 12 (b) (c)\n(d)(d)\n(e)x\nzy\nPt\nS0δSFIG. 2. Spin Seebeck e\u000bect measurement and magnon-polaron peaks in Ga-doped\ngarnet system (a) Schematic of the SSE and ISHE mechanism and measurement geometry. (b)\nMagnetic \feld dependence of the SSE thermopower measured at T= 300 K in a (Lu 2Bi)Fe 4GaO 12\n(BiGa:LuIG) \flm (normalized by the sample geometry: Ey=rT=S= (V=\u0001T)(Lz=Ly)). Inset\nshows detail of the SSE in the 0 < \u0016 0H < 0:7 T range, showing that the signal increase due\nto magnon-phonon hybridization is already present at very small \felds. c, d Detail of the SSE\nsignal in the vicinity of magnetic \felds \u00160HTA= 0.42 T and \u00160HLA= 1.86 T, depicting the\nresonant enhancement of the SSE voltage, resulting from the magnon and phonon hybridization at\nthe magnetic \felds when the magnon dispersion just tangentially touches the transversal acoustic\n(TA) (c)and the longitudinal acoustic (LA) (d)phonon dispersions, respectively. (e)Comparison\nof the resonant enhancement of the SSE centred around the peak position ( HTA) for YIG and\nBiGa:LuIG at 300 K (the enhancement at the peak position is estimated as: \u000eS=S 0, whereS0is\nthe extrapolated background SSE coe\u000ecient at the peak position and \u000eS=S(HTA)\u0000S0).\n15(c) (d)(b) (a)\nHTA HLA \n0.2 T0.6 T\n0.42 TFIG. 3. E\u000bect of magnetic compensation on the spin wave dispersion (a) Results of\nthe magnon dispersion calculated using our model for di\u000berent concentrations of non-magnetic ion\nsubstitutions ( x,y), illustrating the e\u000bect of magnetic compensation in a ferrimagnetic system.\nA 90 % preferential tetrahedral site occupation by Ga ions is assumed according to previous\nreports.36,37(b)Comparison between the magnon dispersion measured experimentally by Brillouin\nlight scattering and the theoretical dispersion for a ferrimagnet with non-magnetic ion substitution\nofx= 0:101 (octahedral site occupation) and y= 0:909 (tetrahedral site occupation) (theoretical\ndispersion is vertically shifted to account for demagnetizing and anisotropy \feld contributions).\n(c, d) Magnon and phonon dispersions at magnetic \felds of \u00160H= 0:42 T and\u00160H= 1.86 T\ndepicting the condition for the observation of magnon-polaron SSE enhancement by hybridization\nof magnon with TA (c)and LA (d)phonons in our system (insets show detail of the measured\nSSE voltage enhancement at the peak positions). In (c)the magnon dispersion for three di\u000berent\n\feld values, corresponding to the arrows in the inset, is shown.\n16(a) (b) \n(c)\n(d)0.1FIG. 4. Temperature dependent magnetization and heat-driven spin transport prop-\nerties of BiGa:LuIG \flm (a) Magnetic \feld dependence of the measured spin Seebeck voltage\nat di\u000berent temperatures (b)Saturation magnetization ( \u00160MS) as a function of temperature. Red\nline shows \ftting to \u00160MS/(Tc\u0000T)\f, withTc= 401:6\u00060:2 K and\f= 0:305\u00060:007.(c)Temper-\nature dependence of the magnetic \feld magnitude for the observation of the magnon-polaron SSE,\nHTA(black squares) and HLA(red circles) (lines interconnecting experimental points are for visual\nguide). (d)Temperature dependence of the intersite exchange integral ( Jad) estimated from the\ncondition for tangential touching of the magnon and phonon dispersions at di\u000berent temperatures.\nRed line shows \ftting to Jad=J0(1\u0000\u0011T5=2), withJ0= (6:41\u00060:05)\u000210\u000022J.\n17"    },    {        "title": "2405.02212v1.Piezoelectric_microresonators_for_sensitive_spin_detection.pdf",        "content": "Piezoelectric microresonators for sensitive spin detection\nCecile Skoryna Kline, Jorge Monroy-Ruz, and Krishna C. Balram∗\nQuantum Engineering Technology Labs and Department of Electrical and Electronic Engineering,\nUniversity of Bristol, Woodland Road, Bristol BS8 1UB, United Kingdom\n(Dated: May 6, 2024)\nPiezoelectric microresonators are indispensable in wireless communications, and underpin radio\nfrequency filtering in mobile phones. These devices are usually analyzed in the quasi-(electro)static\nregime with the magnetic field effectively ignored. On the other hand, at GHz frequencies and\nespecially in piezoelectric devices exploiting strong dimensional confinement of acoustic fields, the\nsurface magnetic fields ( B1) can be significant. This B1field, which oscillates at GHz frequen-\ncies, but is confined to µm-scale wavelengths provides a natural route to efficiently interface with\nnanoscale spin systems. We show through scaling arguments that B1∝f2for tightly focused acoustic\nfields at a given operation frequency f. We demonstrate the existence of these surface magnetic\nfields in a proof-of-principle experiment by showing excess power absorption at the focus of a sur-\nface acoustic wave (SAW), when a polished Yttrium-Iron-Garnet (YIG) sphere is positioned in the\nevanescent field, and the magnon resonance is tuned across the SAW transmission. Finally, we out-\nline the prospects for sensitive spin detection using small mode volume piezoelectric microresonators,\nincluding the feasibility of electrical detection of single spins at cryogenic temperatures.\nI. INTRODUCTION\nPiezoelectric microresonators [1, 2] have revolutionized\nwireless communication by enabling small form-factor,\nhigh performance radio frequency (RF) filters that can\nbe compactly packaged into mobile phones. In addition,\nthese devices have had a broad impact on areas ranging\nfrom sensing [3] to quantum communication [4]. A piezo-\nelectric material enables conversion of RF electromag-\nnetic fields into acoustic fields, which have wavelengths\n≈µm at GHz frequencies, 105smaller than the cm-scale\nwavelengths of the RF fields. This deeply-subwavelength\nconfinement is the key driver for the majority of applica-\ntions involving piezoelectric devices.\nThe constitutive relations for a piezoelectric device [5]\nrelate the stress ( ⃗T) induced by an applied electric field\n(⃗E). While this is strictly true for applied DC fields,\nthe equations are extended to RF fields under the quasi-\n(electro)static approximation [6], wherein the Poisson\nequation for electrostatics is substituted for Maxwell’s\nequations for the electromagnetic field, and is solved\nalong with the elastic wave equation to propagate acous-\ntic fields in piezoelectric devices. The quasistatic ap-\nproximation is usually justified because the deeply sub-\nwavelength confinement provided by the acoustic field\nensures that the far-field electromagnetic radiation com-\nponent is minimal. By definition, using this approxima-\ntion forces the magnetic field to be strictly zero.\nOn the other hand, it is known that piezoelectric de-\nvices radiate electromagnetically [7] and that this radia-\ntion presents a limit on the achievable mechanical qual-\nity factor ( Qm) in piezoelectric resonators [8]. While the\nfar-field radiation efficiency is low for the reasons out-\nlined above, it can be significantly enhanced in a reso-\nnant geometry and provides significant size advantages\n∗krishna.coimbatorebalram@bristol.ac.ukin the design of very low frequency antennas [9]. In this\nwork, we focus on the near-field (surface) component of\nthis oscillating magnetic field ( B1), and ask if these B1\nfields can be exploited for improving the spin detection\nsensitivity of nanoscale electron spin resonance (ESR) ex-\nperiments [10, 11]. Our aim is to apply ideas from cavity\nquantum electrodynamics (cQED) [12] to nanoscale spin\nsystems [13, 14], with the key sensitivity enhancement\nbeing provided by the vastly reduced mode volume ( Vm)\nof piezoelectric microresonators in comparison to their\nelectromagnetic counterparts.\nII. SURFACE CURRENT DENSITY SCALING\nIN PIEZOELECTRIC DEVICES\nIn a piezoelectric material, the propagating acoustic\ndisplacement field is accompanied by a surface polariza-\ntion ( ρ). The evanescent electric fields ( ⃗E) generated at\nthe material-air boundary curl as depicted in Fig.1(a).\nThe plot shows a finite element method (FEM) simula-\ntion of a surface acoustic wave (SAW) mode propagating\non a Scandium Aluminum Nitride (ScAlN) on Si sub-\nstrate. The accompanying electric fields are shown by an\narrow plot. This curling of surface fields is one instance\nof the universal phenomenon of spin-momentum locking\n[15] that applies to all evanescent fields (cf. Appendix A\nfor a discussion of the analogy between the surface fields\nin surface plasmons and surface acoustic waves). The ac-\ncompanying surface magnetic field ( B1) orientation can\nbe directly obtained from the ( ⃗E) fields, and in this case\nwould form loops that come periodically into and out\nof the plane (shown schematically in Appendix A). One\ncan verify the ⃗B1orientation by noting the direction of\nthe surface polarization currents ( J=∂ρ\n∂t), plotted in\nFig.1(b) and invoking Ampere’s law. For focusing acous-\ntic field geometries as would be necessary for increasing\nthe local field strength, the field orientation is shown inarXiv:2405.02212v1  [physics.app-ph]  3 May 20242\n(a) (b)\n(c)(d)Air\nSiScAlN  (1 µm)\n1 µm750 nm\n1 µm\n0 +\nFIG. 1. (a) FEM simulation showing the surface displace-\nment of a surface acoustic wave propagating on a ScAlN on\nSi substrate. The accompanying ≈2.9 GHz electric field is\nshown using an overlaid arrow plot. As can be seen there\nexists an evanescent field in the air whose orientation (helic-\nity) is determined by spin momentum locking [15], (b) FEM\nsimulation, same as (a) but showing the overlaid oscillating\npolarization current density using an arrow plot. This os-\ncillating current density which exists at both the interfaces\n(air-ScAlN and ScAlN-Si) is responsible for the surface ⃗B1\nfields we investigate in this work. (c) Focusing the SAW using\ncurved electrodes in AlN (d) At the focus, the displacement,\nand corresponding evanescent field and current densities, are\nenhanced by the focusing ratio.\nFig.1(c), with a zoomed-in plot of the focus region shown\nin Fig.1(d). We would like to note here that while the\n⃗Efield lines can be computed using an FEM solver like\nCOMSOL, due to the quasistatic approximation being\nimposed on the solver, the magnetic field is strictly zero.\nTherefore, current FEM solvers cannot be used to visu-\nalize this ( B1) field directly from taking the curl of the ⃗E\nfield. The results presented in Figs.1(a-d) make certain\nsimplifications. We model (Sc)AlN films using aluminum\nnitride’s material parameters, and in Fig.1(c,d), we simu-\nlate focusing on a thin AlN film to reduce the simulation’s\nmemory constraints.\nTo estimate the scaling of this surface B1field, we\ntherefore start with the oscillating surface current density\n(J, [A m−2]) instead [3] following prior work on SAW\nbased sensors [16]:\nJ2= 2K2ωk2(ϵc+ϵs)P (1)\nwhere K2is the material’s piezoelectric coefficient and\nrepresents the fraction of the acoustic wave energy that\nis stored in the electric field. ωis the wave frequency,\nk= 2π/λ the wave vector, ϵc,s[F m−1] represent the\ndielectric constants of the cladding (air) and substrate\nrespectively, and Pis the power density of the acoustic\nwave expressed in power per unit beam width [W m−1].\nWe can see that for a tightly focused acoustic beam withfocus width ≈λ,P∝1/λ, and hence B2\n1∼J2∝f4, or the\nsurface field B1∝f2. With K2= 0.05, λa= 1µm,f=\n3 GHz, ϵc+ϵs≈10, beam width at focus ≈λaand\nPa=1 mW, we expect a surface current density at the\nfocus of ≈81.15 MA m−2. The scaling of Jwith ϵcan\nbe visualized directly in Fig.1(b), where the current den-\nsity at the (Sc)AlN-Si interface is stronger than at the\n(Sc)AlN-air interface.\nAssuming the current flows uniformly in a (semi-\ncircular) loop of size ≈λa, one can roughly estimate\ntheB1field as B1≈µ0Jλa/4 = 25 .5µT, with µ0the\nvacuum permeability. By trapping the acoustic field in\nwavelength scale microcavities [17], the surface current\ndensity and therefore the accompanying B1field can be\nenhanced by the cavity mechanical quality factor Qm,\nwith Qm≈104feasible in crystalline media at ambi-\nent conditions [18]. Alternately, one can also estimate\nthe spin-resonator single photon coupling strength gin a\ncavity-QED framework [10, 12] with:\ng=µB\nℏr\n2µ0ℏω0\nVm(2)\nwhere µBis the Bohr magneton, ℏthe reduced Planck\nconstant, µ0vacuum permeability and Vmis the cavity\nmode volume. Assuming an acoustic cavity mode with\ndimensions of 5 λ3\naat 3 GHz with λa=1µm, this gives us\na coupling strength g≈2π∗14 kHz. We would like to em-\nphasize that we are interested in the near-field B1here.\nDue to the alternating signs of the surface current den-\nsity, the far-field radiation is relatively weak.\nSuch piezoelectric approaches to enhancing spin de-\ntection sensitivity are in many ways complementary to\nsuperconducting cavity approaches which have recently\nachieved single spin sensitivity [11, 19] at mK temper-\natures. Both approaches rely on a Purcell-like [12] en-\nhancement of the spin detection sensitivity which scales\n∝p\nQc/Vc, where Qcis the quality factor and Vcis\nthe mode volume of the cavity. The superconducting\napproaches support high Qfactors ( >105) and small\nmagnetic mode volume ( ≈10−12λ3\nRF) [20] by exploiting\nlumped element inductor-capacitor ( LC) cavity designs,\nwhere the B1field is strongly localized in close proximity\nto the inductor. Piezoelectric resonators, on the other\nhand provide moderate Qm(≈104), but vastly reduced\ncavity mode volumes ( ≈10−15λ3\nRF) with the additional\nadvantage of enabling experiments in ambient conditions,\nwhich makes it feasible to use this technique with chem-\nical and biological samples which might deteriorate ap-\npreciably in cryogenic environments.\nWe would like to distinguish our work from previous\nresults that have studied the interaction between acoustic\nfields and spin systems (both nanomagnets and magnetic\nthin films) that primarily exploit the magnetostrictive ef-\nfect [21–23] or the strain field to perturb spin systems\n[24–26]. In this work, we instead focus on using piezo-\nelectric devices for generating the oscillating magnetic\nfield ( B1) directly and the methods developed here can\nbe applied in principle to any spin resonance experiment.3\n(a)\n<110>\n(b) (c) 200 µmSide ViewTop View\nYIG\nIDT\nFIG. 2. (a) Schematic of the experimental setup used to\nprobe the B1. Curved IDTs are used to launch and detect\nSAWs on a ScAlN on Si substrate. A polished YIG sphere is\nbrought in proximity (non-contact) of the evanescent field at\nthe focus of the SAW and the IDT transmission is monitored\nas the magnon mode is tuned across the SAW transmission\nresonance. The magnitude and phase of the transmission is\nmonitored using a vector network analyzer and the magnon\nmode is tuned by a Bzfield applied using an electromagnet\nunderneath the sample. (b) and (c) show images from the side\nand top view cameras of the experiment during operation.\nGiven the field generated is on the surface and evanes-\ncent, physical contact between the sample and the spin\nsystem can be completely avoided, as discussed in the\nexperiments below.\nIII. USING SPIN RESONANCE ABSORPTION\nTO PROBE THE EVANESCENT B1FIELD\nWe perform spin resonance measurements by position-\ning a polished Yttrium-Iron-Garnet (YIG) sphere (diam-\neter≈150µm) in the evanescent field ( < λa≈1µm) of\nthe SAW devices. YIG spheres support high-Q ( ≈104\nat 10 GHz) collective spin wave excitations (magnons),\nwhich have been used for a wide range of applications,\nranging from tunable filters and low noise oscillators for\nwireless communication [27] to hybrid quantum trans-\nduction, wherein quantum states from superconducting\nqubits are mapped back and forth from the magnon\nmodes [28]. By mounting the YIG sphere on a mov-\nable three-axis translation stage and positioning it in the\nevanescent field of the acoustic wave, we can spatially\nprobe the surface B1field by monitoring the SAW trans-\nmission and looking for changes in the amplitude and\nphase response of the received SAW signal as the magnon\nmode frequency is tuned across the SAW resonance.\nThe schematic of our experimental setup is illustrated\nin Fig.2(a). Fig.2(b,c) show the side and top view of the\nYIG sphere positioned at the centre of the SAW transmit-receive circuits as captured by the two zoom lenses indi-\ncated in Fig.2(a). SAWs are launched and detected us-\ning a vector network analyzer (VNA) by patterning a set\nof interdigitated transducers (IDT) [2] on a piezoelectric\n(c-axis oriented) Sc 0.06Al0.94N film deposited on a [100]\noriented silicon substrate. We pattern IDTs with both\nstraight fingers to launch quasi plane waves of sound,\nand curved fingers [29] to focus the acoustic field down\nto a beam width of 1-5 λa. The curved IDT devices\nare designed as confocal transmit-receive pairs [17, 30].\nTo achieve the highest spin detection sensitivity, focusing\ndevices are necessary as they significantly enhance the lo-\ncal power density ( Pin equation 1) by the focusing ratio\n(≈20−50x) and the strongest B1fields are therefore al-\nways generated at the focus. The size of the YIG sphere\nin our experiments presents two constraints. It requires\nus to separate the focusing IDTs by ≈100−125λa, which\nmagnifies the effect of wave diffraction and the anisotropy\nof the underlying silicon substrate with the net result be-\ning a relatively low acoustic throughput.The second issue,\nwhich can be seen from Fig.2(c) is the difficulty of deter-\nmining the precise acoustic focus location while aligning\nthe YIG sphere for maximum SAW extinction.\nThe magnon frequency can be tuned by applying a\nstatic DC magnetic field ( Bz) using an electromagnet,\nas shown in Fig.2(a). Due to the frequency of opera-\ntion and the orientation (shown in Fig.2(b)), the YIG\nsphere is not saturated, and the magnon mode relaxes to\nlower frequencies with time. Therefore, there is a finite\ntime offset between the setting of the voltage on the elec-\ntromagnet and the data acquisition on the VNA, which\nwould result in a lower effective Bzdue to relaxation. In\nprinciple, one can park the magnon mode on the higher\nfrequency side of the SAW transmission and let the mode\nrelax and down-shift in frequency across the SAW reso-\nnance, while recording VNA traces as a function of time.\nIn practice, we found this method did not give us the\nresolution needed to capture the exact crossing between\nthe magnon mode and the SAW. Therefore, we rely on\nincrementing the voltage in small steps and acquiring the\ndata quickly to minimize the effects of mode relaxation.\nAs the Bzfield is tuned, ferromagnetic resonance\n(FMR) in the YIG sphere induces an excess absorption\nwhich shows up as a reduced transmission (attenuation)\nat the SAW frequency. This excess absorption is maxi-\nmized when the frequencies of the SAW and the magnon\nmode are identical. This is shown in Fig.3. Fig.3(a,c)\ncorrespond to datasets taken from two distinct curved\nIDT devices with the same period but different curva-\nture. The colorbar on the 2D plot indicates the mag-\nnitude of the (normalized) transmitted signal ( S21) as a\nfunction of both Bzand frequency. As the plot shows,\nwe observe a variety of magnon modes [31] in our exper-\niment, with varying coupling strengths and quality fac-\ntors. Without mode imaging [31], it is hard to ascertain\nthe mode symmetry from a pure microwave transmission\nexperiment. Here, we focus on the mode that gives us\nthe strongest signal and generically label it as magnon4\n(a)\n(b)MagnonSAW\nMagnon\nSAW\n (c)\n(d)\n0 +\nFIG. 3. The SAW-magnon interaction is probed by monitoring the VNA transmission ( S21) as a function of Bz. The 2D\ncolorplots in (a,c) plot |S21|as a function of frequency and Bzfor two different curved IDT devices with the same period, but\ndifferent curvature. The SAW mode frequency is constant with Bz, whereas the magnon mode frequency shifts linearly with\nBz. When the magnon mode frequency crosses the SAW transmission resonance, one expects an excess attenuation in the SAW\ntransmission due to magnon mode excitation and dissipation. (b,d) show representative linecuts from (a,c) respectively for\nfm< fs(red), fm≈fs(black), and fm> fs(blue). The excess attenuation when fm≈fsis especially clear in (b), and the same\ntrend can also be observed, albeit with lower magnitude in (d). The plotted data are time gated with a 40 ns. Uncorrected\ndatasets are available in Appendix B, and the bare IDT reflection and transmission spectra are plotted in Appendix D.\nin Fig.3(a,c). As the 2D color plots show, the SAW fre-\nquency stays relatively independent of Bz, whereas the\nmagnon mode linearly tunes to higher frequency with in-\ncreasing Bz. When the magnon mode frequency ( fm)\ncrosses the SAW mode ( fs), one can clearly see an ex-\ncess absorption which is stronger in Fig.3(a) compared\nto the dataset in Fig.3(c). One can see this excess ab-\nsorption effect more clearly by taking 1D cuts through\nthe 2D data corresponding to the cases when fm< fs,\nfm≈fsandfm> fs. These 1D cuts are shown respec-\ntively in Fig.3(b) and (d), with the cases colored red,\nblack and blue respectively. The background SAW trans-\nmission when the magnon mode is way off resonance\n(fm≪fs, labelled fs) is indicated in magenta. Especially,\nin Fig.3(b) the excess attenuation as the magnon mode\npasses through the SAW frequency is clear. While we\ndo observe a similar effect in Fig.3(d), the magnitude is\nmuch weaker which we believe is due to a combination\nof the mode drifting with time and the difficulty of posi-\ntioning the YIG sphere at the focus, given the geometry\nshown in Fig.2(c).\nOur analysis here is complicated by the fact that there\nis significant electromagnetic crosstalk between the two\nports and one needs to distinguish between the localmagnon-SAW interaction occurring on the sample and\npossible interference effects occurring at the VNA. In\nparticular, the electromagnetic radiation from the probes\nand the IDT, which act as inefficient antennas [32], can\nexcite the magnon mode and the scattered signal picked\nup by the receiving port will interfere with the acous-\ntic transmission. We refer to this second pathway as a\nnonlocal interaction due to the phase sensitive detection\nemployed by the VNA. In our experiments, the crosstalk\nsignal is larger than the acoustic transmission because\nof the diffraction effects mentioned above. One can see\nthis in action, by noting that in the case of fm> fsand\nfm< fs, the magnon signal appears as a transmission\npeak rather than a dip, which is clear signature of a non-\nlocal interference pathway [33]. One way to reduce this\nbackground is to exploit the significantly lower speed of\nsound compared to light and time-gate the VNA trans-\nmission [34]. The datasets shown in Fig.3 have been time\ngated with a notch of 40 ns. While this helps to improve\nthe signal to noise ratio (cf. the raw datasets in Appendix\nB), the high quality factor of the YIG sphere makes the\ncross talk persist for significantly longer than the electro-\nmagnetic transit time. This residual cross-talk makes it\nchallenging for us to extract the SAW-magnon interac-5\ntion strength from the experiments, although the excess\nlocal interaction with the SAW through the surface B1\nfield is clear from Fig.3(a,b). As a control, we repeat the\nexperiment with a straight IDT device and do not ob-\nserve an excess attenuation at the magnon-SAW crossing\n(cf. Appendix C).\nGiven that the SAW-magnon mode interaction is ob-\nserved with the YIG sphere not touching the sample, this\nexperiment provides strong preliminary evidence that\nGHz frequency, localized B1fields can be generated on\nthe surface of piezoelectric devices and can be used to\ninterface with spin systems. While this is encouraging,\nwe would like to emphasize that the experimental modal-\nity has a few limitations obvious with hindsight and the\nresults should be interpreted within these constraints. In\nparticular, the size of commercially available YIG spheres\nlimits the sensitivity of the experiment by physically re-\nquiring the IDTs be separated by >100λa, and making\nit challenging to determine the focus in real-time. The\nsize of the YIG sphere is also responsible for the mag-\nnitude of the crosstalk, which makes it challenging to\ninfer the coupling strength and the coupling dynamics\nfrom our experiments. In particular, one of the key ef-\nfects we were hoping to confirm was the frequency de-\npendence of the interaction, which should scale ∝f2.\nAlthough we made devices with varying SAW frequency,\nwe were unable to quantify this effect. Moving forward,\nby impedance matching the transducers (to reduce elec-\ntromagnetic radiation and reflections) and working with\nhigh Q YIG samples with dimensions <5µm, one can\npotentially scan the surface and verify the spatial extent\nof the B1field both in-plane ( x, y) and in z. Mapping the\nspatial confinement of the B1field is critical to the spin\nsensitivity enhancement experiments and this is some-\nthing we are unable to do with our current setup. Finally,\nmoving to higher acoustic frequencies (and higher Bz)\nwould enable us to saturate the YIG sphere and avoid\nthe relaxation drift with time, which is another source\nof error in our experiments. In passing, we would like\nto note that the B1field description provides a compli-\nmentary route towards understanding the spin rotation\neffects in nanoparticles interacting with SAWs [35], and\ninterpret the switching of magnetization observed in pre-\nvious experiments [36].\nIV. PROSPECTS FOR SINGLE SPIN\nELECTRICAL READOUT\nAs noted in a recent review [10], the problem of improv-\ning spin detection sensitivity in electron spin resonance\nexperiments boils down to focusing the magnetic field\nto deeply sub-wavelength geometries while maintaining\nhigh-Q. In effect, piezoelectric microresonators are ideal\nin that they naturally provide both strong confinement\nand high Q, and therefore provide a natural complement\nto traditional electromagnetic approaches [37]. The key\nissue is whether the magnetic field strengths can be sub-stantial in these devices. As we have shown above us-\ning both scaling arguments and proof-of-principle exper-\niments, the surface current density has a ∝f2and can\nbe further enhanced by working with stronger piezoelec-\ntric materials / orientations ( K2\neff>0.2) and designing\nsmall mode volume acoustic cavities [17] to exploit the\npower scaling. With advances in materials and device ge-\nometries pushing acoustic device operation to ever-higher\nfrequencies ( >50 GHz) [38], the prospect of acoustics en-\nabled X-Band ESR is within reach. There is still the open\nquestion of how one can efficiently load the near field\nof these devices efficiently to separate the pure field in-\nduced effects from strain effects. One possible route could\nbe to employ suspended membranes (similar to [39], but\nmade with an insulator like alumina) in close proximity\n(<50 nm) to the piezoelectric substrate.\nWe can estimate the minimum spin detection sensitiv-\nity, following [40]:\nNmin=κ\n2gprnw\nκ2(3)\nwhere Nminis the single-shot spin detection sensitiv-\nity (per echo), κis the total cavity decay rate given by\nκ=ω0/QLwith ω0= 2πfbeing the operating frequency\nandQLthe loaded Q factor of the cavity. We assume the\ncavity is operated at critical coupling with external cou-\npling rate κ2≈κ/2,nis the average number of noise pho-\ntons, given by n=kBT/ℏω0,pis the spin polarization\np=1−e−ℏω0\nkBT\n1+e−ℏω0\nkBT, and the spin resonance width w= 2/T2\nwith T2the average spin dephasing time. gis the spin\ncavity coupling rate defined above in eqn.2. For oper-\nating frequencies of 10 GHz, temperature 4 K, mode vol-\nume 5 µm3andT2≈50 ms, achieving Nmin≈1 requires\nQL≈105, which is challenging for mechanical systems,\nbut feasible given recent results [18] and the favourable\nscaling of acoustic dissipation with temperature. In any\ncase, this shows that provided the B1field in piezoelectric\nmicroresonators has a spatial extent comparable to that\nof the acoustic field which requires experimental confir-\nmation, then detecting individual magnetic nanoparticles\nat room temperature ( Nspins≥105) is well-within reach\nusing current devices.\nThe exquisite spin detection potential of these res-\nonators can be understood from a different perspective by\ntreating them as the high frequency analogs of mechani-\ncal cantilever based spin sensors [41], where the piezoelec-\ntric effect enables inductive detection. It was noted [42]\nthat the signal to noise ratio for both inductive and me-\nchanical detection of spin resonance scales ∝p\nω0Q/k m,\nwith ω0is the operating frequency, Qthe cavity qual-\nity factor and kman effective magnetic spring constant\nwhich scales with the cavity mode volume Vmfor in-\nductive detection. The sensitivity enhancement there-\nfore derives from achieving high quality factors in deeply\nsub-wavelength mode volumes, which gives an effective\nPurcell enhancement ∝p\nQ/V mto the sensitivity [12].6\nWe would like to conclude by noting that while in this\nwork, we have primarily focused on using piezoelectric\ndevices for improving spin detection sensitivity in ESR,\nthe strong field confinement is also of interest in scenarios\ninvolving large-scale closely packed efficient spin address-\ning [43] without deleterious crosstalk effects, as would be\nnecessary in future spin-based quantum computing.\nV. ACKNOWLEDGEMENTS\nWe would like to thank John Rarity, Joe Smith, Vivek\nTiwari, Hao-Cheng Weng, Alex Clark, Rowan Hoggarth\nand Edmund Harbord for valuable discussions and sug-\ngestions. We acknowledge funding support from the\nUK’s Engineering and Physical Sciences Research Coun-\ncil (EP/N015126/1, EP/V048856/1) and the European\nResearch Council (ERC-StG SBS 3-5, 758843).\nAppendix A: Spin Momentum Locking of\nEvanescent fields in surface acoustic waves and\nsurface plasmons\n++\n++++\n++++\n++--\n----\n--\nMetalAir𝐸 (𝐸𝑥,𝐸𝑦)\n𝐻𝑧𝑘Surface plasmon on a metal\n(THz)\n--\n--++\n++--\n----\n--++\n++𝐸 (𝐸𝑥,𝐸𝑦)𝐻𝑧\n(b)Surface acoustic wave on a piezoelectric\n𝑘\nSurface polarization(GHz)\nxy(a)\nFIG. 4. (a) Schematic illustration of surface plasmon propa-\ngating on a metal-air interface. The evanescent electric field\nand the magnetic field orientation are shown, alongwith the\nsurface charges that terminat the electric field on the metal\nsurface. (b) Illustration of a surface acoustic wave propagat-\ning on a piezoelectric material. The evanescent electric field\norientation is similar to that in (a), but terminated by bound\npolarization charges induced in the piezoelectric. By analogy,\nthe orientation of the B1field can be determined, as shown.\nOne can see the universality of spin-momentum lock-\ning [15] as applied to all evanescent fields by lookingat two very different surface waves: a surface plas-\nmon propagating at a metal air interface (at >100 THz)\nand a surface acoustic wave ( <50 GHz) propagating at\na piezoelectric-air interface. The respective cases are\nshown in Fig.4(a,b). As can be seen in both cases, the\nevanescent electric fields curl with an orientation deter-\nmined by spin-momentum locking. The main difference\nbetween the two scenarios is the termination of the fields\non free charges in the metal and on bound polarization\ncharges in the piezoelectric case. Given that the surface\nplasmon dispersion relation is traditionally derived by\nsolving for the magnetic field, the B1orientation can be\nderived by analogy as shown in Fig.4(b).\nAppendix B: Uncorrected datasets without\ntime-gating\n(a)\n(b)\n0 +\nFIG. 5. (a) Uncorrected data sets for the data correspond-\ning to Fig.3(a,b). Without the time-gating, the background\nelectromagnetic crosstalk makes it impossible to observe the\nexcess attenuation during the SAW-magnon mode crossing,\nalthough even with the raw dataset, a net attenuation (cf.\nblack curve) is clearly visible.\nFigure 5 plots the raw data sets corresponding to the\ntime-gated datasets plotted in Fig.3(a,b). As discussed\nin the main text, the presence of the background electro-\nmagnetic crosstalk makes it challenging to infer the SAW-\nmagnon interaction, which can be best seen by comparing\nthe black curves ( fm≈fs) in Fig.3(b) and Fig.5(b). On\nthe other hand, even with the raw datasets, we can clearly\nobserve an overall attenuation as the magnon mode fre-\nquency comes close to the SAW frequency. This is in\ncontrast to what we observe with the straight IDT de-7\nvices, as discussed in the next section.\nWe would like to note that the choice of the notch\ngate time (40 ns) was not optimized for this analysis and\nneither was the gate shape. Given the separation between\nIDTs was 200 µm, and a speed of sound of ≈3750 m /sec,\nthe acoustic wave takes ≈53 ns to reach the receiving\nIDT. The choice of gate time was made as a rough trade-\noff between minimizing the background crosstalk signal\nand ensuring minimal attenuation of the acoustic signal\nin the transmitted spectrum. We would like to note again\nthat time gating does not fully eliminate the background\ncrosstalk because of the Q factor of the YIG sphere.\nAppendix C: Straight IDT control results\n(a)\n(b)\n0 +\nFIG. 6. (a) 2D colorplot of the time-gated |S21|for a straight\nIDT transmit-receive device with the YIG sphere positioned\nin between. The YIG was mounted vertically in this exper-\niment to align the 110 axis with z, which shifts the modes\nto higher frequencies in the 3 .4 GHz range (b) 1D datasets\nfrom (a) corresponding to the three different cases ( fm< fs,\nfm≈fs, and fm> fs), along with the background SAW trans-\nmission (magenta). The excess attenuation at the magnon-\nSAW crossing is not observable here.\nThe power dependence of J2in equation 1 can be\ntested by measuring the relative performance of IDT with\ncurved and straight fingers in inducing magnon absorp-\ntion. Given the local field intensity in curved devices at\nthe focus is increased by the focusing ratio, the straight\nIDT devices here serve as a control experiment to sepa-\nrate the local SAW-magnon interaction from the nonlocal\nEM crosstalk-SAW interaction occuring in the VNA, dis-\ncussed in the main text.Experiments identical to that reported in Fig.3 were\ndone with a straight IDT device. The YIG sphere in these\nexperiments was mounted vertically with a view towards\nsaturating the sphere and avoiding the time drift in the\nmagnon modes. This moves the magnon modes to higher\nfrequencies in the 3.4 GHz range, and the IDT period\nwas reduced correspondingly to shift the SAW response\nto higher frequencies. Fig.6(a) plots the 2D colorplot of\nthe transmitted |S21|as a function of frequency and Bz.\nTo achieve the higher Bz, we use a permanent magnet\nin combination with our electromagnet. Fig.6(b) shows\nlinecuts from 6(a) that correspond to the three different\ncases fm< fs,fm≈fsandfm> fs. Here, we don’t ob-\nserve an excess attenuation as the magnon mode crosses\nthrough the SAW resonance, which can be interpreted as\na signature of the dependence of the local current density\non the local power density.\nWhile mounting the YIG sphere vertically makes the\nmagnon mode frequency stable due to field saturation, it\nmakes it very challenging to determine the positioning of\nthe sphere with respect to the beam focus using imaging\ncameras. In particular, the top view camera, shown in\nFig.2(a) can not be used anymore and one has to rely\nmore on the side view camera with associated parallax\nerrors. 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Smoluchowskiego 17, 60-179 Pozna´ n, Poland; and\n6Department of Applied Physics, Eindhoven University of Technology,\nP.O. Box 513, 5600 MB Eindhoven, The Netherlands\n(Dated: January 23, 2024)\nSpin dynamics is usually described as massless or, more precisely, as free of inertia. Recent\nexperiments, however, found direct evidence for inertial spin dynamics. In turn, it is necessary to\nrethink the basics of spin dynamics. Focusing on a macrospin in an environment (bath), we show\nthat the spin-to-bath coupling gives rise to spin inertia. This bath-induced spin inertia appears\nuniversally from all the high-frequency bath modes. We expect our results to provide new insights\ninto recent experiments on spin inertia. Moreover, they indicate that any channel for spin dissipation\nshould also be accompanied by a term accounting for bath-induced spin inertia. As an illustrative\nexample, we consider phonon-bath-induced spin inertia in a YIG/GGG stack.\nIntroduction. —Spin dynamics is usually considered to\nbe free of inertia and, in turn, it is described by first-order\ndifferential equations; for example, the Bloch equation\n[1] for small quantum-mechanical spins or the Landau-\nLifshitz-Gilbert (LLG) equation [2–5] for larger quasi-\nclassical spins. Based on the broad success of those de-\nscriptions, one might be tempted to conclude that spin\ndynamics is free of any inertia, as spin inertia would be\ndescribed by second-order time derivatives. Recent ex-\nperiments, however, find direct evidence for spin inertia\nclose to THz-frequencies [6, 7]; see also [8]. First and\nforemost, the discovery of spin inertia is of immense inter-\nest for our fundamental understanding of spin dynamics.\nFor example, spin inertia leads to a redshift in the ferro-\nmagnetic resonance peak [9, 10] and, more interestingly,\ngives rise to nutation spin waves [11–13]. In addition,\nspin inertia might also prove relevant in applications as\nthe related nutation dynamics takes place on short time\nscales [6–8, 11, 14]; for example, spin inertia might al-\nlow for faster and more efficient spin switching similar\nto antiferromagnets [15]. For a recent review on inertial\neffects, see reference [16].\nSpin inertia and spin nutation have already been de-\nrived for several systems with various approaches: from\nan adiabatic expansion of a dissipation kernel [17]; for a\nspin in a superconducting Josephson junction [18]; from\nmesoscopic nonequilibrium thermodynamics [19]; with\na classical Lagrangian approach [20] analogous to the\nderivation of Gilbert damping [5]; for the spins coupled\nto an electron bath [21–24]; from higher-order relativistic\nterms [25, 26]; for environments with a Lorentzian bathspectral density [27]; and by modeling a magnetization in\nterms of a current-carrying loop [28]. Despite the many\ncrucial insights provided by these approaches, a complete\nunderstanding of spin inertia is arguably still missing [7].\nHere, in the quest to contribute to a more general and\nunified understanding of spin inertia, we show that spin\ninertia arises universally from the interaction with an en-\nvironment.\nIn this letter, we consider a macrospin that is ex-\nposed to an effective magnetic field and coupled to its\nFIG. 1. The bath spectral density J(ϵ) contains the infor-\nmation about how the environment (bath) affects the spin\ndynamics. We separate J(ϵ) into a low-frequency approxima-\ntionJlf(ϵ) (green solid line) and the remaining high-frequency\npart J(ϵ)−Jlf(ϵ) (red shaded area). This separation sets an\nenergy scale, ϵlf, up to which Jlf(ϵ) is a good approximation\ntoJ(ϵ). Assuming the typical frequencies of the spin dynam-\nics to be small, ω≪ϵlf, the low-frequency bath modes give\nrise to damping, whereas all the high-frequency bath modes\ngive rise to spin inertia. While our results are general, for the\nfigure we assumed Jlf(ϵ) to be linear, which leads to Gilbert\ndamping.arXiv:2310.05621v2  [cond-mat.mes-hall]  22 Jan 20242\nenvironment. The interaction with the effective mag-\nnetic field is described by the Zeeman energy. Using the\nCaldeira-Leggett approach [29, 30], analogous to [31], we\nmodel the environment as a bath of harmonic oscillators\nand, for simplicity, assume a linear coupling between the\nmacrospin and the bath modes. In the quasi-classical\nlimit, after integrating out the bath modes, we find an\ninertial LLG equation. More explicitly, we show that\nthe environmental degrees of freedom (bath modes) af-\nfect spin dynamics in two ways: the low-frequency bath\nmodes give rise to Gilbert damping (or fractional Gilbert\ndamping if the bath is non-ohmic), while all the high-\nfrequency bath modes give rise to spin inertia (indepen-\ndent of the bath details); see Fig. 1.\nFinally, to show that the Caldeira-Leggett approach\nwith linear coupling is not just a toy model but of real\nuse, we discuss its application to a macrospin that is\ncoupled to a phonon bath and we derive the phonon-bath-\ninduced spin inertia for a magnetic plane (YIG) that is\nsandwiched between two nonmagnetic insulators (GGG).\nModel Hamiltonian. —We consider a macrospin that is\ncoupled to its environment and exposed to an (effective)\nmagnetic field. The corresponding Hamiltonian can be\nseparated into three parts, ˆH=ˆHs+ˆHc+ˆHb, where the\nsystem Hamiltonian ˆHsdescribes the macrospin in the\nmagnetic field, the bath Hamiltonian ˆHbdescribes the en-\nvironmental degrees of freedom, and the coupling Hamil-\ntonian ˆHcdescribes the coupling between the macrospin\n(system) and its environment (bath). Explicitly, the\nmacrospin in the magnetic field is described by the Zee-\nman energy with the Hamiltonian ˆHs=−B·ˆS, where\nˆS= (ˆSx,ˆSy,ˆSz) is the macrospin operator with standard\nspin operators ˆSx,ˆSy,ˆSzandBdenotes the (effective)\nmagnetic field that may dependent on time and can point\nin any direction. Using the Caldeira-Leggett approach\n[29, 30], we model the environment as a bath of harmonic\noscillators; in turn, the bath Hamiltonian is given by\nˆHb=P\nn(ˆ p2\nn/2mn+mnω2\nnˆ q2\nn/2), where mn,ωn,ˆ qn,ˆ pn\nare mass, eigenfrequency, position operator, and momen-\ntum operator of the n-th bath oscillator. For simplicity,\nwe assume a linear system-to-bath coupling, which is de-\nscribed by the coupling Hamiltonian ˆHc=−P\nnγnˆ qn·ˆS,\nwhere γnis the coupling coefficient describing the cou-\npling strength between the macrospin and the n-th bath\noscillator respectively. The rationale behind the linear\ncoupling is that, close to the ground state of the full sys-\ntem (macrospin + environment), a Taylor expansion is\nlikely to lead to a linear coupling as the leading order\nterm. Thinking in more physical terms, the linear cou-\npling might, for example, be an sd-like coupling between\nthe macrospin and electron-hole pairs in metallic mag-\nnets. However, as we will show below, even in the case of\nmagnetoelastic coupling, which is quadratic in the spin\nvariables, the linear-coupling approach is still useful to\nunderstand the inertial macrospin dynamics close to the\nmagnetic ground state.Effective action for spin dynamics. —Starting from the\nmodel, we use the Keldysh formalism in its path-integral\nversion [32–34] with spin coherent states |g⟩, as in refer-\nence [33], to derive an action for the combined dynamics\nof the macrospin and the bath oscillators. Because we\nmodel the environment as a bath of harmonic oscilla-\ntors and assume linear coupling, the action is (at most)\nquadratic in ˆ pnandˆ qn. In turn, we can integrate out\nthe bath modes by performing Gaussian path integrals;\nfirst in pn, then in qn. As result, we obtain the Keldysh\npartition function Z=R\nDg eiSwith the effective action\nfor the macrospin dynamics\nS=I\nKdt[−i⟨˙g|g⟩+B·S] +1\n2I\nKdtI\nKdt′S(t)α(t−t′)S(t′),\n(1)\nwhere |˙g⟩=∂t|g⟩andS=⟨g|ˆS|g⟩. Information about\nthe coupling to the bath is contained in the kernel func-\ntionα(t−t′) that, when represented in Keldysh space,\nis a matrix containing as elements the retarded and\nadvanced parts αR/A(ω) and a Keldysh part αK(ω).\nThe noiseless effects (damping and inertia) are de-\nscribed by the retarded and advanced parts αR/A(ω) =\n−P\nnγ2\nn/{mn[(ω±iη)2−ω2\nn]}, where ηis an infinites-\nimal level broadening or decay rate. The information\nabout fluctuations (noise) is contained in the Keldysh\npartαK(ω) = coth( ω/2kBT) [αR(ω)−αA(ω)] with Boltz-\nmann constant kBand bath temperature T, which is\nsimply the fluctuation-dissipation theorem, because we\nassume the bath to be in (local) equilibrium [34].\nTo be more explicit, as in reference [33], we use\nthe Euler-angle representation of spin coherent states,\n|g⟩= exp( −iϕˆSz) exp(−iθˆSy) exp(−iψˆSz)|⇑⟩, where\n|⇑⟩is the eigenstate to ˆSzwith the maximal eigen-\nvalue S; that is, ˆSz|⇑⟩=S|⇑⟩. This representa-\ntion is convenient, as the macrospin takes the intu-\nitive form of a vector in spherical coordinates S=\nS(sinθcosϕ,sinθsinϕ,cosθ), where Sis its length and\nθandϕdescribe its orientation. The Berry-phase term\nbecomes −i⟨˙g|g⟩=S(˙ψ+˙ϕcosθ), where the angle ψis\na gauge freedom [33]. Properly fixing this gauge on the\nKeldysh contour is nontrivial and can be crucial [35, 36].\nHere, however, a simple choice of ˙ψ=−˙ϕon the Keldysh\ncontour will be just fine; for a detailed discussion, see the\nSupplemental Material.\nGeneralized Landau-Lifshitz-Gilbert equation. —From\nthe effective action, we derive an effective quasi-classical\nequation of motion for the macrospin along the following\nlines: first, to retain information about fluctuations, we\nuse the Schmid trick [33, 34, 37–39] and “decouple” the\npart of the action that is quadratic in quantum compo-\nnents by a Hubbard-Stratonovich transformation; then,\nwe vary the action with respect to the quantum com-\nponents θqandϕq; finally, the resulting quasi-classical\nequations of motion for θcandϕccan be recast into a\nsingle vectorial equation of motion [40]. As result, we3\nobtain a generalized LLG equation\n˙S=S×(B+ξ) +S×Z∞\n−∞dt′˜α(t−t′)S(t′),(2)\nwhere ξis a fluctuating field with ⟨ξm(t)⟩= 0 and\n⟨ξm(t)ξm′(t′)⟩=−(i/2)δmm′αK(t−t′); the indices\nm, m′∈ {x, y, z}denote Cartesian components corre-\nsponding to S= (Sx, Sy, Sz). Furthermore, we defined\n˜α(ω) =αR(ω)−αR(0), where the ω= 0 part is sub-\ntracted for regularization [41]. More explicitly, the kernel\nfunction is given by\n˜α(ω) =−2\nπZ∞\n−∞dϵω2J(ϵ)\n[(ω+iη)2−ϵ2]ϵ, (3)\nwhere J(ϵ) = πP\nn(γ2\nn/2mnωn)δ(ϵ−ωn) is the bath\nspectral density that contains two pieces of information:\nin the delta function δ(ϵ−ωn), it contains the informa-\ntion at which energies the bath modes can be found; in\nthe pre-factor ( γ2\nn/2mnωn) it contains the information of\nhow strongly the macrospin couples to the bath modes.\nSeparating low- and high-frequency bath modes. — In an\nenvironment with many bath modes that are close in en-\nergy, the bath spectral density J(ϵ) will be a continuous\nfunction; see Fig. 1. For a general bath spectral den-\nsity, it is hard to proceed analytically. However, inspired\nby reference [39] but without introducing a cutoff, we\nmake analytical progress by separating the contributions\nof low-frequency and high-frequency bath modes. Explic-\nitly, we rewrite J(ϵ) =Jlf(ϵ)+[J(ϵ)−Jlf(ϵ)], where Jlf(ϵ)\nis a low-frequency approximation of J(ϵ) and we refer to\nJ(ϵ)−Jlf(ϵ) simply as high-frequency contribution, even\nthough non-low-frequency contribution might be a more\naccurate name. Accordingly, we split the kernel function\n˜α(ω) = ˜αlf(ω) + ˜αhf(ω) into a low-frequency contribu-\ntion ˜αlf(ω) and the high-frequency contribution ˜ αhf(ω).\nExplicitly, the low-frequency contribution is given by\n˜αlf(ω) =−2\nπZ∞\n−∞dϵω2Jlf(ϵ)\n[(ω+iη)2−ϵ2]ϵ, (4)\nwhile the high-frequency contribution is given by\n˜αhf(ω) =−2\nπZ∞\n−∞dϵω2[J(ϵ)−Jlf(ϵ)]\n[(ω+iη)2−ϵ2]ϵ. (5)\nNow, the key point to realize is that we are able to\ndetermine the high-frequency contribution under a quite\ngeneral assumption: we assume that typical frequency of\nthe spin dynamics ωis much smaller than ϵlf, which is the\nenergy up to which Jlf(ϵ)≈J(ϵ); see Fig. 1. Under this\nassumption, we can disregard the ωdependence in the\ndenominator of Eq. (5). The reason is that ϵ < ϵ lfthe\nnumerator vanishes, J(ϵ)−Jlf(ϵ)≈0, while for ϵ > ϵ lf\nthe denominator can be approximated, ( ω+iη)2−ϵ2≈\n−ϵ2. In turn, we can approximate the high-frequency\ncontribution to the kernel function,\n˜αhf(ω)≈I ω2, (6)and we arrive at our central result: the bath-induced spin\ninertia\nI=2\nπZ∞\n−∞dϵJ(ϵ)−Jlf(ϵ)\nϵ3. (7)\nSo, if the bath spectral density J(ϵ) is known, it can be\nused to determine its low-frequency approximation Jlf(ϵ)\nand, in turn, to find the spin inertia Iby straightforward\n(potentially numerical) integration. Note that the sign\nof the spin inertia is not fixed; depending on the form of\nJ(ϵ), spin inertia can be positive or negative.\nInertial Landau-Lifshitz-Gilbert equation. — To recover\nthe inertial LLG equation used in experimental analysis\n[6, 7], we assume the bath spectral density to be approxi-\nmately linear at low frequencies. This assumption allows\nus to use an Ohmic bath, which is a bath with linear bath\nspectral density, for the low-frequency approximation\nJlf(ϵ); for non-Ohmic low-frequency approximations, see\nreference [31]. Explicitly, we assume Jlf(ϵ) =α0ϵΘ(ϵ),\nwhere Θ( ϵ) is the Heaviside Θ-function and α0is some\nconstant that will turn out to be the Gilbert-damping\ncoefficient. We then find ˜ αlf(ω) = iα0ωfor the low-\nfrequency kernel function and I= (2/π)R∞\n0dϵ[J(ϵ)−\nα0ϵ]/ϵ3for the bath-induced spin inertia, where we used\nthat bath spectral densities vanish for negative energies.\nIt is now straightforward to derive the inertial LLG\nequation. First, transforming the low- and high-\nfrequency parts of the kernel function back into time\nspace, we find ˜ αlf(t−t′) =−α0δ′(t−t′) and ˜ αhf(t−t′) =\n−I δ′′(t−t′), where δ′(t) and δ′′(t) are respectively the\nfirst and second derivative of the Dirac δ-function. Then,\ninserting them back into the generalized LLG equation\n(2), we obtain the inertial LLG equation\n˙S=S×h\nB+ξ−α0˙S−I¨Si\n, (8)\nwhere α0is the Gilbert-damping coefficient, Iis the spin\ninertia, and ξis a fluctuating field with ⟨ξm(t)⟩= 0 and\n⟨ξm(t)ξm′(t′)⟩ ≈2kBTα0δmm′δ(t−t′) for which we as-\nsumed the temperature to be large [42]. Note that only\nGilbert damping—but not spin inertia—contributes to\nfluctuations. On the formal level, spin inertia does not\ncontribute to fluctuations, as even-frequency contribu-\ntions of αR(ω) and αA(ω) cancel out in αK(ω). In more\nphysical terms, spin inertia is not dissipative and, based\non the general idea of the fluctuation-dissipation theorem\n[39, 43], should therefore also not contribute to fluctua-\ntions.\nThe Gilbert-damping coefficient α0and the spin inertia\nIdepend on the bath spectral density J(ϵ); see Eqs. (4)\nand (5) respectively. Next, to illustrate how our general\nresults can be applied, we consider a macrospin that is\ncoupled to a phonon bath.\nA macrospin coupled to a phonon bath. — The coupling\nbetween spins and phonons can be described by magne-\ntoelastic coupling, as in reference [44], where they derived4\na generalized LLG equation analogous to Eq. (2) but for\na spin lattice. So, to identify the bath-induced spin in-\nertia for phonon baths, we do not need to rederive the\ngeneralized LLG equation. Instead, we start from their\ngeneralized LLG equation [44], recast it into the form of\nour Eq. (2), identify the bath spectral density J(ϵ), and\nfinally determine the spin inertia from Eq. (7).\nNote that magnetoelastic coupling is derived in the\ncontinuum limit of long-wavelength phonons. So, strictly\nspeaking, one would have to rederive the spin-phonon\ncoupling to include short-wavelength (high-frequency)\nphonon modes, which are responsible for spin inertia.\nNevertheless, we believe that the following illustrative\nexample based on magnetoelastic coupling provides valu-\nable insights into the physics of spin inertia and paves the\nway for more detailed derivations from microscopic mod-\nels.\nMagnetoelastic coupling is quadratic in the spin vari-\nables [44]. Close to the ground state, however, we are\nallowed to linearize the spin dynamics and with it the\nmagnetoelastic coupling. Combining this linearization\nwith a macrospin approximation, we relate the phonon\nbath [44] to the Caldeira-Leggett approach above; for de-\ntails, see Supplemental Material. Explicitly, we find that\nthe kernel function (3) becomes a tensor ˜ αmm′(ω), as also\nthe bath spectral density takes a tensorial form [45],\nJmm′(ϵ) =πX\niX\nkλeikRiBmzBm′z\n2MNS 1S ωkλ\n×(kmz·ekλ) (km′z·e−kλ)δ(ϵ−ωkλ),(9)\nwhere S1is the effective spin length on an individual\nlattice site with spin, Bm˜mandBm′˜mare the magne-\ntoelastic coupling coefficients (here ˜ m=z, as we chose\nthez-direction for the ground state), the scalars Nand\nMare respectively the number and effective (oscillat-\ning) mass of lattice sites, the function ωkλis the phonon\ndispersion relation with the phonon momentum k, the\nvector ekλis the polarization vector and λis the index\ndenoting longitudinal and transversal polarization, the\nvector kmm′is defined by kmm′=kmem′+km′emwith\nCartesian unit vectors emandem′, and the sum iruns\nover the lattice-positions Riof the individual spins; the\nnotation is analogous to reference [44].\nKnowing the phonon-bath spectral density (9), we then\nfind a low-frequency approximation Jlf,mm′(ϵ) and, in\nturn, determine the phonon-bath-induced spin inertia\nfrom the difference Jmm′(ϵ)−Jlf,mm′(ϵ) as in Eq. (7).\nTo be specific, let us consider a layer of yttrium iron\ngarnet (YIG) that is sandwiched between two bulk lay-\ners of gadolinium gallium garnet (GGG); afterwards, we\nconsider implications for a layer of YIG on top of a bulk\nof GGG. We model the system as a planar spin lat-\ntice (in the z= 0 plane) that is embedded in a three-\ndimensional lattice with lattice constant a. In this geom-\netry, only phonons travelling in z-direction contribute tothe macrospin damping and inertia. Focusing on acous-\ntic phonons, we use the dispersion relation for a simple\ncubic lattice ωkzλ= (2/a)vλ|sin(kza/2)|, where vλis the\nsound velocity for transversal ( λ∈ {x, y}) or longitudinal\n(λ=z) polarization. At low energies ϵthe bath spec-\ntral density (9) is governed by long wavelength (small k)\nacoustic phonons, which have an approximately linear\ndispersion relation, ωkzλ=vλ|kz|. In turn, Jmm′(ϵ) be-\ncomes linear in ϵat low energies and we can use the low-\nfrequency approximation Jlf,mm′(ϵ) =αmm′ϵΘ(ϵ) with\na tensorial Gilbert-damping coefficient αmm′; for YIG on\nGGG, a tensorial Gilbert damping has been found before\n[46]. The phonon-bath-induced spin inertia is now found\nfrom the high-frequency modes as in Eq. (7); it also\ntakes a tensorial form Imm′= (2/π)R∞\n−∞dϵ[Jmm′(ϵ)−\nJlf,mm′(ϵ)]/ϵ3. Explicitly, we find the Gilbert-damping\ncoefficient αmm′=δmm′(1 +δmz)2B2\nmza/2MS 1Sv3\nmand\nthe phonon-bath-induced spin inertia\nImm′=0.9\nπa\nvmαmm′, (10)\nwhere, after making the ϵ-integral dimensionless, we eval-\nuated and approximated it numerically to 0 .9; a detailed\ncalculation is provided in the Supplemental Material. For\nYIG on top of GGG (not sandwiched) only about half\nthe phonon modes are present, such that we expect the\nGilbert-damping coefficient, and with it the spin inertia,\nto be only half as large.\nBath-induced spin inertia and Gilbert damping are\nclosely related, as becomes clear from Eq. (10); namely,\nthe spin inertia is proportional to the Gilbert-damping\ncoefficient. Thus, since the Gilbert-damping tensor αmm′\nis diagonal, also the spin-inertia tensor Imm′is diago-\nnal. The proportionality constant τm=Imm/αmm=\n0.9a/πv mis typically used in experimental works to char-\nacterize spin inertia or nutation dynamics [6–8]. For the\nGGG phonon bath, with lattice spacing a= 1.2383 nm\n[47] and phonon velocities vz= 6411 m /s (longitudinal)\nandvx=vy= 3568 m /s (transversal) [46, 48], we find\nτz≈55 fs and τx=τy≈99.4 fs. These results, com-\npared to previous measurements on metallic magnets, are\nroughly in the same order of magnitude [7, 8] or about\ntwo to three orders of magnitude lower [6].\nNote that the frequency of the nutation resonance\npeak may deviate from previous experimental results\non metallic magnets, as it also depends on the Gilbert-\ndamping coefficient [49]. Using the effective (oscillating)\nmass of GGG M=ρ a3with the GGG density ρ=\n7.07·103kg/m3[46, 48], the magnetoelastic coupling coef-\nficients Bzz= 6.6·10−22J and Bxz=Byz= 13.2·10−22J\nwith the YIG-lattice-site spin length S1= 14.2ℏ[50], we\nfindαzz= 2·10−7/Sandαxx=αyy= 1.2·10−6/Sfor\nthe elements of the tensorial Gilbert-damping coefficient.\nFor those numbers, using the “weak coupling” approxi-\nmation of reference [49], the nutation resonance peaks\nwould be in the order of X-ray frequencies. While proba-5\nbly impractical in for experiments, it does not affect the\nillustration purposes of our simple example.\nDiscussion and Conclusion. —Using the Caldeira-\nLeggett approach, we have shown that the\nhigh-frequency modes of an environment (bath)\nshould—universally—lead to bath-induced spin inertia.\nThe low-frequency bath modes, if Ohmic, lead to the\nusual Gilbert-damping term. This has two important\nconsequences. First, our results suggest that the ap-\npearance of bath-induced spin inertia is more robust (or\nuniversal) than the form of Gilbert damping; while for\nnon-Ohmic baths Gilbert damping turns into fractional\nGilbert damping [31], the bath-induced spin inertia\nretains its form. Second, our results suggest that,\nwhenever a dissipation channel (environment/bath) is\nadded to some spin system, the spin dynamics will also\nacquire an additional contribution to its spin inertia.\nIn short, spin relaxation is always accompanied by spin\ninertia.\nTo show how our general derivation based on the\nCaldeira-Leggett approach applies to more realistic mod-\nels, we considered the dynamics of a macrospin in a\nphonon bath; specifically, we considered a heterostruc-\nture based on YIG and GGG. We expect that our ap-\nproach can be applied, along similar lines, to many other\nspin environments as well; for example, to electron baths\nin metallic magnets and to metallic leads in contact with\ninsulating magnets. Furthermore, we believe that our ap-\nproach can be generalized from the macrospin case con-\nsidered here to the more general case of spin lattices and,\nin turn, also to continuum theories of spin textures.\nBecause the high-frequency bath modes affect the ac-\ntion, Eq. (1), already before the quasi-classical approx-\nimation, we also expect small-spin systems, as investi-\ngated in [51, 52], to be affected by bath-induced spin\ninertia from high-frequency bath modes.\nAcknowledgements. —We thank M. Cherkasskii,\nE. Di Salvo, A. Kamra, A. Semisalova, A. Shnirman,\nD. Thonig, R. Willa, and X. R. Wang for fruitful dis-\ncussions. This work is part of the research programme\nFluid Spintronics with project number 182.069, financed\nby the Dutch Research Council (NWO). R. C. V. is\nsupported by (NWO, Grant No. 680.92.18.05). R. A. 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Greilich, Spin inertia of resident and photoexcited car-\nriers in singly charged quantum dots, Physical Review B\n98, 121304 (2018).\n[52] D. Smirnov, E. Zhukov, E. Kirstein, D. Yakovlev,\nD. Reuter, A. Wieck, M. Bayer, A. Greilich, and\nM. Glazov, Theory of spin inertia in singly charged quan-\ntum dots, Physical Review B 98, 125306 (2018).1\nSUPPLEMENTAL MATERIAL: BATH-INDUCED SPIN INERTIA\nA SIMPLE CHOICE OF GAUGE IS SUFFICIENT\nBefore choosing a gauge, the action, Eq. (1) in the main text, still contains the gauge freedom ψin the Berry-phase\nterm, which becomes −i⟨˙g|g⟩=S(˙ψ+˙ϕcosθ) in the Euler-angle representation. In principle, we can fix ψas we like.\nHowever, we should respect “boundary conditions” on the Keldysh contour; otherwise, we could choose ˙ψ=−˙ϕcosθ,\nwhich would eliminate the Berry-phase term/contintegraltext\nKdt S(˙ψ+˙ϕcosθ) from the action. It can be crucial to be careful\nwith the boundary conditions of the chosen gauge [1, 2]. Here, however, a simple choice of ˙ψ=−˙ϕon the Keldysh\ncontour will be just fine; as shown next, it leads to the same quasi-classical dynamics as the more elaborate choice of\nreferences [1, 2].\nTo compare the gauge choice of references [1, 2] with the simple choice ˙ψ=−˙ϕ, we rewrite ψ=χ−ϕ. Then, the\ngauge choices differ only in χ; references [1, 2] choose ˙ χc=˙ϕc(1−cosθc) and χq=ϕq(1−cosθc), while the simple\nchoice corresponds to χ= 0. For the Berry-phase term, there is no relevant difference at finite times:\n/contintegraldisplay\nKdt(˙ψ+˙ϕcosθ) =−/contintegraldisplay\nKdt˙ϕ(1−cosθ) +/contintegraldisplay\nKdt˙χ; (1)\nbut with χ±=χc±χq/2 and χq=ϕq(1−cosθc), we find\n/contintegraldisplay\nKdt˙χ=/integraldisplay\ndt[˙ϕq(1−cosθc) +ϕq˙θcsinθc] = 0 , (2)\nwhere, in the last step, we used integration by parts and disregarded potentially arising boundary terms that are\ndynamically irrelevant in the quasiclassical approximation.\nDERIVATION OF BATH SPECTRAL DENSITY FOR A PHONON BATH\nSpins on a lattice will be coupled to the phonons that arise from oscillations of the atoms/ions at the lattice sites. In\nspintronics, this coupling is often described by the magnetoelastic coupling. For example, see [3], where they derived\nthe coupled equations of motions for spins Sion a lattice (with lattice sites denoted by i) that are coupled to a phonon\nbath,\n˙Si=Si×H+Si×hi+Si×/summationdisplay\njKijSj−Si×/integraldisplayt\n0dt′/summationdisplay\njGij(t, t′)˙Sj(t′), (3)\nwhere His the external magnetic field, Kijaccounts for dipolar and nearest-neighbor exchange interactions, and the\ncoupling to the phonon system gives rise to the induced magnetic field hiand the generalized Gilbert damping with\nthe kernel function Gij(t, t′). For the components of that kernel function, they find\nGmm′\nij(t, t′) =1\nNS4\n1/summationdisplay\n˜m˜m′Bm˜mBm′˜m′S˜m\ni(t)S˜m′\nj(t′)/summationdisplay\nkλeik·(Ri−Rj)(km˜m·ekλ)(km′˜m′·e−kλ)cos[ωkλ(t−t′)]\nMω2\nkλ,(4)\nwhere S1is the spin length of the effective spin on a single lattice site with spin, Nis the number of lattice sites,\nkdenotes the phonon momentum, λits polarization (longitudinal/transversal), and the corresponding dispersion\nrelation and polarization vector are given by ωkλandekλrespectively. Further, Nis the number of lattice sites, Mis\ntheir effective ionic mass, and Bm˜mare the magnetoelastic coupling coefficients. We also adopt their short notation\nforkm˜m=kme˜m+k˜memwith Cartesian unit vectors emform∈x, y, z .\nNow, to find the bath spectral density for the phonon bath, our intermediate goal is to relate the spin-lattice\nequation of motion (3) to the macrospin equation of motion of the main text (2); more specifically, we need to relate\nthe generalized Gilbert-damping terms of both equations. This immediately leads us to a key problem: the kernel\nfunction for the phonon bath, Gmm′\nij(t, t′), depends on the lattice spins via S˜m\ni(t)S˜m′\nj(t′), whereas the kernel function\nfor the harmonic-oscillator bath of the main text, ˜ α(t−t′), does not depend on the macrospin. This key difference\narises from the different spin-to-bath couplings. While the magnetoelastic coupling is nonlinear in the spin [3], in the\nmain text we used the Caldeira-Leggett approach with a simple linear spin-to-bath coupling. However, if the spins2\nremain close to the z-direction, such that we can linearize the equation of motion in deviations from the z-direction,\nthen we may simply approximate Si≈S1ezinGmm′\nij(t, t′). In turn, the kernel function simplifies to\nGmm′\nij(t−t′) =1\nNS2\n1BmzBm′z/summationdisplay\nkλeik·(Ri−Rj)(kmz·ekλ)(km′z·e−kλ)cos[ωkλ(t−t′)]\nMω2\nkλ. (5)\nWith this simplification, we can now Fourier-transform equation (3) with/summationtext\nie−iqRi...and use the macrospin approx-\nimation with Sq=0=SandSq̸=0= 0. For the generalized Gilbert damping, we then find\n−Si×/integraldisplayt\n0dt′/summationdisplay\njGij(t, t′)˙Sj(t′) −→ −1\nNsS×/integraldisplayt\n−∞dt′G0(t−t′)˙S(t′), (6)\nwhere Nsis the number of lattice sites with spin and, disregarding transient effects, we set the lower integral boundary\nfrom 0 to −∞. The kernel function G0(t−t′) is the q= 0 case of the Fourier-transform\nGq(t−t′) =/summationdisplay\nieiq(Ri−Rj)Gij(t−t′). (7)\nNow, the generalized Gilbert damping, Eq. (6), can be integrated by parts to find\n−1\nNsS×/integraldisplayt\n−∞dt′G0(t−t′)˙S(t′) =−1\nNsS×/bracketleftbigg/integraldisplayt\n−∞dt′˙G0(t−t′)S(t′) + [G0(t−t′)S(t′)]t\n−∞/bracketrightbigg\n, (8)\nwhere the boundary term [ G0(t−t′)S(t′)]t\n−∞only contains a term that renormalizes the magnetic field and some\ninformation about the initial state, which would only affect transient dynamics that we are not considering anyways.\nSo, we can finally relate the generalized Gilbert damping for the phonon bath to the generalized Gilbert damping\nfor the Caldeira-Leggett approach in the main text. We find that the relation between the two kernel functions is\ngiven by\n˜α(t−t′) =−1\nNs˙G0(t−t′) Θ(t−t′). (9)\nSubtracting the zero-frequency part ˜ α(ω= 0) for regularization, as in the main text, and using the relation\n/integraldisplay∞\n−∞dω\n2πe−iωt 1\n(ω+iη)2−ϵ2=−sin(ϵ t)\nϵΘ(t), (10)\nwe find the (tensorial) bath spectral density for the phonon bath,\nJmm′(ϵ) =π/summationdisplay\ni/summationdisplay\nkλeikRiBmzBm′z\n2MNS 1S ωkλ(kmz·ekλ) (km′z·e−kλ)δ(ϵ−ωkλ), (11)\nwhere we used that the macrospin length Sis given by the sum over all the individual spin lengths S1; so,S=NsS1.\nThe resulting bath spectral density, Eq. (11), is used in the main text.\nEXPLICIT CALCULATION FOR A PLANAR SIMPLE-CUBIC SPIN LATTICE IN A\nTHREE-DIMENSIONAL SIMPLE-CUBIC LATTICE\nWe consider a three-dimensional simple cubic lattice with site length a, where the z= 0-plane (YIG) contains spins\nof length S1on every site, while there is no spin on all the other sites (GGG).\nStarting from the bath spectral density, Eq. (11), we can now use that the sum over lattice sites with spin yields/summationtext\niexp[ikRi] =/summationtext\nnxnyexp[i(kxa nx+kya ny)] =Nxδkx,0Nyδky,0, where nxandnynumerate, respectively, the lattice\nsites in x-direction and y-direction; and NxandNyare the number of lattice sites in x-direction and y-direction\nrespectively. Consequently, the bath spectral density becomes\nJmm′(ϵ) =π/summationdisplay\nkλNxδkx,0Nyδky,0BmzBm′z\n2MNS 1S ωkλ(kmz·ekλ) (km′z·e−kλ)δ(ϵ−ωkλ), (12)3\nand, after carrying out the sums over kxandky, it simplifies to\nJmm′(ϵ) =δmm′π(1 +δmz)2B2\nmz\n2MS 1S/summationdisplay\nkzλk2\nz\nNzωkzλδmλδ(ϵ−ωkzλ), (13)\nwhere we used that the number of lattice sites is given by N=NxNyNzwith the number of lattice sites in z-direction\nNz. Carrying out the summation over the phonon polarization yields,\nJmm′(ϵ) =δmm′π(1 +δmz)2B2\nmz\n2MS 1S/summationdisplay\nkzk2\nz\nNzωkzmδ(ϵ−ωkzm). (14)\nInstead of directly carrying out the sum over kz, which runs over all kz-states in the Brillouin zone (BZ), we switch\nto the continuum case and turn it into an integral; namely, we replace/summationtext\nkz∈BZ...= (Lz/2π)/integraltextπ/a\n−π/adkz..., where Lz\nis the length of the system in z-direction. Further, using that Lz=aNz, we find\nJmm′(ϵ) =δmm′(1 +δmz)2B2\nmza\n4MS 1S/integraldisplayπ/a\n−π/adkzk2\nz\nωkzmδ(ϵ−ωkzm). (15)\nTo carry out the remaining integral, we need to know the dispersion relation. For a simple cubic lattice, focusing\non acoustic phonons, the dispersion relation is given by ωkzm= (2/a)vm|sin(kza/2)|, where vmis the sound velocity\nfor transversal ( m∈ {x, y}) or longitudinal ( m=z) polarization; see [4, 5] for example. In terms of microscopic\nparameters, the sound velocities are given by vm=/radicalbig\nKm/Mwith the force constants Kx=Ky=K2andKz=\nK1+ 2K2, where K1andK2are respectively the force constants for nearest neighbor and next-nearest neighbor\nrestoring forces [4, 5]. Now, the momentum integral becomes\n/integraldisplayπ/a\n−π/adkzk2\nz\nωkzmδ(ϵ−ωkzm)=/integraldisplayπ/a\n−π/adkzk2\nz\n2vm\na/vextendsingle/vextendsinglesin/parenleftbigkza\n2/parenrightbig/vextendsingle/vextendsingleδ/parenleftbig\nϵ−2vm\na/vextendsingle/vextendsinglesin/parenleftbigkza\n2/parenrightbig/vextendsingle/vextendsingle/parenrightbig\n=\n\n24\na2arcsin2a ϵ\n2vm\nvmϵ/radicalig\n1−(a ϵ\n2vm)2forϵ∈/bracketleftbig\n0,2vm\na/bracketrightbig\n0 otherwise.\n(16)\nIn turn, we find the bath spectral density in explicit form\nJmm′(ϵ) =δmm′(1 +δmz)2B2\nmz\nMS 1S4\na2arcsin2a ��\n2vm\n2vm\naϵ/radicalig\n1−/parenleftbiga ϵ\n2vm/parenrightbig2Θ(ϵ) Θ/parenleftbig2vm\na−ϵ/parenrightbig\n. (17)\nAt low energies, ϵ≪2vm\na, we can approximate arcsin/parenleftbiga ϵ\n2vm/parenrightbig\n≈a ϵ\n2vmand/radicalig\n1−/parenleftbiga ϵ\n2vm/parenrightbig2≈1, such that we find the\nlow-energy or low-frequency approximation for the bath spectral density\nJlf,mm′(ϵ) =δmm′(1 +δmz)2B2\nmza\n2MS 1S v3mϵΘ(ϵ), (18)\nwhich is linear in ϵ. So, following the derivation of the main text, we can simply read off the (tensorial) Gilbert-damping\ncoefficient as\nαmm′=δmm′(1 +δmz)2B2\nmza\n2MS 1S v3m. (19)\nKnowing the bath spectral density Jmm′(ϵ) and its low-frequency approximation Jlf,mm′(ϵ), we can now determine\nthe bath-induced spin inertia from Eq. (7) from the main text,\nImm′=2\nπ/integraldisplay∞\n−∞dϵJmm′(ϵ)−Jlf,mm′(ϵ)\nϵ3. (20)\nInserting Eqs. (17) and (18), and using Eq. (19), we find\nImm′=2\nπαmm′/integraldisplay∞\n0dϵ\n4v2\nm\na2arcsin2a ϵ\n2vm\nϵ4/radicalig\n1−/parenleftbiga ϵ\n2vm/parenrightbig2Θ/parenleftbig2vm\na−ϵ/parenrightbig\n−1\nϵ2\n. (21)4\nTheϵ-integral can be made dimensionless by substituting ϵ= 2vmϵ′/a, which yields\nImm′=1\nπa\nvmαmm′/integraldisplay∞\n0dϵ′/bracketleftbiggarcsin2ϵ′\nϵ′4√\n1−ϵ′2Θ(2vm\na−2vm\naϵ′)−1\nϵ′2/bracketrightbigg\n. (22)\nTo deal with the Θ-function, it is now convenient to split the integral from 0 to ∞into two parts: one from 0 to 1\nand another one from 1 to ∞. Then, we find\n/integraldisplay∞\n0dϵ′/bracketleftbiggarcsin2ϵ′\nϵ′4√\n1−ϵ′2Θ(2vm\na−2vm\naϵ′)−1\nϵ′2/bracketrightbigg\n=/integraldisplay1\n0dϵ′/bracketleftbiggarcsin2ϵ′\nϵ′4√\n1−ϵ′2−1\nϵ′2/bracketrightbigg\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\n≈1.9+/integraldisplay∞\n1dϵ′/bracketleftbigg\n−1\nϵ′2/bracketrightbigg\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\n=−1≈0.9, (23)\nwhere we evaluated the integral from 0 to 1 numerically. So, overall, for a layer of YIG sandwiched between two bulk\nlayers of GGG, we find the phonon-bath-induced spin inertia\nImm′=0.9\nπa\nvmαmm′, (24)\nas given in the main text.\n[1] A. Shnirman, Y. Gefen, A. Saha, I. S. Burmistrov, M. N. Kiselev, and A. Altland, Geometric quantum noise of spin, Physical\nReview Letters 114, 176806 (2015).\n[2] T. Ludwig, I. S. Burmistrov, Y. Gefen, and A. Shnirman, Thermally driven spin transfer torque system far from equilibrium:\nEnhancement of thermoelectric current via pumping current, Physical Review B 99, 045429 (2019).\n[3] A. R¨ uckriegel and P. Kopietz, Rayleigh-jeans condensation of pumped magnons in thin-film ferromagnets, Physical review\nletters 115, 157203 (2015).\n[4] O. Madelung, Introduction to solid-state theory , Vol. 2 (Springer Science & Business Media, 1996).\n[5] J. S´ olyom, Fundamentals of the Physics of Solids (Springer, 2007)."    },    {        "title": "1702.06994v2.Is_spin_superfluidity_possible_in_YIG_films_.pdf",        "content": "Is spin super\ruidity possible in YIG \flms?\nE. B. Sonin\nRacah Institute of Physics, Hebrew University of Jerusalem, Givat Ram, Jerusalem 91904, Israel\n(Dated: August 17, 2021)\nRecently it was suggested that stationary spin supercurrents (spin super\ruidity) are possible in\nthe magnon condensate observed in yttrium-iron-garnet (YIG) magnetic \flms under strong external\npumping. Here we analyze this suggestion. From topology of the equilibrium order parameter in\nYIG one must not expect energetic barriers making spin supercurrents metastable. However some\nsmall barriers of dynamical origin are possible nevertheless. The critical phase gradient (analog of\nthe Landau critical velocity in super\ruids) is proportional to intensity of the coherent spin wave\n(number of condensed magnons). The conclusion is that although spin super\ruidity in YIG \flms is\npossible in principle, the published claim of its observation is not justi\fed.\nThe analysis revealed that the widely accepted spin-wave spectrum in YIG \flms with magne-\ntostatic and exchange interaction required revision. This led to revision of non-linear corrections,\nwhich determine stability of the magnon condensate with and without spin supercurrents.\nI. INTRODUCTION\nSpin super\ruidity has already been discussed from 70s\nof the last century1,2. Its investigation continues nowa-\ndays (see recent reviews in Refs. 3 and 4). The inter-\nest to this phenomenon was revived after emergence of\nspintronics. Manifestation of spin super\ruidity is a sta-\nble spin supercurrent. Experimental observation of it in\nmagnetically ordered solids would be an essential break-\nthrough in the condensed matter physics. A spin super-\ncurrent is proportional to the gradient of the phase '\n(spin rotation angle in a plane) and is not accompanied\nby dissipation, in contrast to a dissipative spin di\u000busion\ncurrent proportional to the gradient of spin density.\nIn general the spin current proportional to the gradi-\nent of the phase 'is ubiquitous and exists in any spin\nwave or domain wall, although in these cases variation\nof the phase 'is small (very small in weak spin waves\nand not more then on the order \u0019in domain walls and\nin disordered materials). Analogy with mass and charge\npersistent currents (supercurrents) arises when at long\n(macroscopical) spatial intervals along streamlines the\nphase variation is many times larger than 2 \u0019. The super-\ncurrent state is a helical spin structure, but in contrast\nto equilibrium helical structures is metastable.\nAn elementary process of relaxation of the supercur-\nrent is phase slip. In this process a vortex with 2 \u0019phase\nvariation around it crosses streamlines of the supercur-\nrent decreasing the total phase variation across stream-\nlines by 2\u0019. Phase slips are suppressed by energetic bar-\nriers for vortex creation, which disappear when phase\ngradients reach critical values determined by the Landau\ncriterion.\nRecently Sun et al.5suggested (see also Ref. 6) that\nspin super\ruidity is possible in a coherent magnon con-\ndensate created in yttrium-iron-garnet (YIG) magnetic\n\flms by strong parametric pumping7, and Bozhko et al.8\ndeclared experimental detection of spin supercurrent in a\ndecay of this condensate. Although the experimental evi-\ndence of spin super\ruidity was challenged9,10(see discus-\nsion in the end of the paper) the very idea of spin super-\ruidity in YIG \flms deserves a further analysis. In YIG\nthe equilibrium order parameter in the spin space was\nnot con\fned to some easy plane analogous to the order\nparameter complex plane in super\ruids. The easy-plane\norder parameter topology providing a barrier stabilizing\na supercurrent was considered as a necessary condition\nfor spin super\ruidity.3,4However, one cannot rule out\nthat metastability of supercurrent states is provided by\nbarriers not connected with topology of the equilibrium\norder parameter. The goal of the present paper was to in-\nvestigate this possibility and to determine critical values\nof possible supercurrents at which metastability is lost.\nThe critical supercurrents are determined from the\nprinciple similar to that of the Landau criterion for super-\n\ruids: any weak perturbation of the current state always\nincreases the energy, and therefore the current state is\nmetastable. This requires an analysis of nonlinear correc-\ntions to spin waves in the Landau{Lifshitz{Gilbert (LLG)\ntheory. But calculation of nonlinear corrections is based\non knowledge of the wave pattern and the wave spectrum\nin the linear theory. It was revealed that the commonly\naccepted and used up to now the linear theory of spin\nwaves in YIG \flms5,11{13must be revised. This was done\nproperly taking into account boundary conditions at \flm\nsurfaces. For \flms now used in experiments on coherent\nmagnon condensation the boundary problem in the pres-\nence of exchange and magnetostatic interaction has an\naccurate analytical solution, which gives the wave pat-\ntern and the wave spectrum di\u000berent from known before.\nThis is important for the stability analysis of the magnon\ncondensate with and without spin supercurrents.\nSection II discusses connection of metastability of cur-\nrent states and topology of the order parameter space,\nwhich is a continuum of all degenerate ground states\nemerging from continuous symmetry (gauge symmetry in\nsuper\ruids, rotational symmetry of the spin space in fer-\nromagnets). It is also demonstrated how non-equilibrium\nstate of spin precession supported by magnon pumping\ncreates an e\u000bective \\easy plane\" for the order parameter,\nwhich allows metastable spin supercurrents. Section III\nreviews the LLG theory and the dispersion relation of lin-arXiv:1702.06994v2  [cond-mat.other]  10 Apr 20172\near plane spin waves in YIG bulk. Section IV considers\nlinear spin waves in YIG \flm and determines their pat-\ntern and dispersion relation solving the boundary prob-\nlem in the presence of the exchange and the magneto-\nstatic interaction. The result di\u000bers from known before,\nand the origin of this di\u000berence is discussed. Section V\nanalyzes nonlinear corrections, which determine stabil-\nity of the coherent magnon condensate and the distri-\nbution of magnons between two energy minima in the k\nspace. The quasi-equilibrium approach \fxing the total\nnumber of magnons does not predict stable condensate,\nin con\rict with observation of the coherent condensate\nin experiments. It was suggested to modify the quasi-\nequilibrium approach \fxing magnon numbers condensed\nin any of two minima, but not only their total number.\nFinally Sec. VI derives critical gradients in supercurrents\nfrom the Landau criterion generalized on spin super\ru-\nidity. The last section VII discusses and compares the\nresults of the present work with results of previous inves-\ntigations.\nII. TOPOLOGY AND SUPERFLUID SPIN\nCURRENTS\nA knowledge on why super\ruid currents can be\nmetastable is provided by the analysis of topology of the\norder parameter space. Spin super\ruidity was suggested\nby the analogy with the more commonly known mass su-\nper\ruidity, and we start from discussion of the latter. At\nthe equilibrium the order parameter of a super\ruid is a\ncomplex wave function  = 0ei', where the modulus  0\nof the wave function is a positive constant determined by\nminimization of the energy and the phase 'is a degener-\nacy parameter since the energy does not depend on 'be-\ncause of gauge invariance. Any from degenerate ground\nstates in a closed annular channel (torus) maps on some\npoint at a circumference j j= 0in the complex plane  ,\nwhile a current state with the phase change 2 \u0019naround\nthe torus maps onto a circumference (Fig. 1a) winding\naround the circumference ntimes. It is evident that it is\nimpossible to change nkeeping the path on the circum-\nferencej j= 0all the time. In the language of topology\nstates with di\u000berent nbelong to di\u000berent classes, and n\nis atopological charge . Only a phase slip can change it\nwhen the path in the complex plane leaves the circumfer-\nence. This should cost energy, which is spent on creation\nof a vortex crossing the cross-section of the torus channel\nand changing nton\u00001. The state with a vortex in the\nchannel maps on the full circle j j\u0014 0.\nIf we consider transport of spin parallel to the axis zthe\nanalog of the phase of the super\ruid order parameter is\nthe rotation angle of the spin component in the plane xy,\nwhich we note also as '. Here we neglect the processes,\nwhich break rotational invariance in spin space (analog of\ngauge invariance in. super\ruids) and violate the conser-\nvation law for the total spin. These processes can be of\nprinciple importance and were thoroughly investigated.3\nReψImψψa)\nb)c)d)\nMzzMzzxyHxyHxyHFIG. 1. Mapping of current states on the order parameter\nspace.\na) Mass currents in super\ruids. The current state in torus\nmaps on a circumference of radius j jon the complex plane\n .\nb) Spin currents in an isotropic ferromagnet. The current\nstate in torus maps on an equatorial circumference on the\nsphere of radius M(top). Continuous shift of mapping on\nthe sphere (middle) reduces it to a point at the northern pole\n(bottom), which corresponds to the ground state without cur-\nrents.\nc) Spin currents in an easy-plane ferromagnet. Easy-plane\nanisotropy contracts the order parameter space to an equato-\nrial circumference in the xyplane topologically equivalent to\nthe order parameter space in super\ruids.\nd) Spin currents in an isotropic ferromagnet in a magnetic\n\feld parallel to the axis zwith nonequilibrium magnetization\nMzsupported by magnon pumping. Spin is con\fned in the\nplane parallel to the xyplane but close to the northern pole.\nThis plane is an \\easy plane\" of dynamical origin.\nBut in the present discussion their e\u000bect can be ignored\nfor the sake of simplicity.\nIn isotropic ferromagnets the order parameter space is\na sphere of radius equal to the absolute value of the mag-\nnetization vector M(Fig. 1b). All points on this sphere\ncorrespond to the same energy of the ground state. Sup-\npose we created the spin current state with monotonously\nvarying phase 'in a torus. This state maps on the\nequatorial circumference on the order parameter sphere.\nTopology allows to continuously shift the circumference\nand to reduce it to the point of the northern pole. Dur-\ning this process shown in Fig. 1b the path remains on\nthe sphere all the time and therefore no energetic barrier\nresists to the transformation. Thus metastability of the\ncurrent state is not expected.\nIn a ferromagnet with easy-plane anisotropy the order\nparameter space contracts from the sphere to an equa-\ntorial circumference in the xyplane. This makes the\norder parameter space topologically equivalent to that in\nsuper\ruids (Fig. 1c). Now transformation of the equa-3\ntorial circumference to the point shown in Fig. 1b costs\nanisotropy energy. This allows to expect metastable spin\ncurrents (supercurrents). They relax to the ground state\nvia phase slips events, in which magnetic vortices cross\nspin current streamlines. States with vortices maps on a\nhemisphere of radius Meither above or below the equa-\ntor.\nUp to now we considered states close to the equilibrium\n(ground) state. In a ferromagnet in a magnetic \feld the\nequilibrium magnetization is parallel to the \feld. How-\never, by pumping magnons into the sample it is possible\nto tilt the magnetization with respect to the magnetic\n\feld. This creates a nonstationary state, in which the\nmagnetization precesses around the magnetic \feld. Al-\nthough the state is far from the true equilibrium, but it,\nnevertheless, is a state of minimal energy at \fxed mag-\nnetizationMz. Because of inevitable spin relaxation the\nstate of uniform precession requires permanent pumping\nof spin and energy. However, if these processes violat-\ning the spin conservation law are weak, one can ignore\nthem and treat the state as a quasi-equilibrium state.\nThe state of uniform precession maps on a circumference\nparallel to the xyplane, but in contrast to the easy-plane\nferromagnet (Fig. 1c) the plane con\fning the precessing\nmagnetization is much above the equator and not far the\nnorthern pole (Fig. 1d). One can consider also a current\nstate, in which the phase (the rotation angle in the xy\nplane) varies not only in time but also in space with a\nconstant gradient. The current state will be metastable\ndue to the same reason as in an easy-plane ferromagnet:\nin order to relax via phase slips the magnetization should\ngo away from the circumference on which the state of\nuniform precession maps, and this increases the energy.\nThen the plane, in which the magnetization precesses,\ncan be considered as an e\u000bective \\easy plane\" originat-\ning not from the equilibrium order parameter topology\nbut created dynamically. Further the concept of dynami-\ncal easy plane will be applied to YIG magnetic \flms with\nsome modi\fcations. They take into account that the spin\nconservation law is not exact due to magnetostatic energy\nand the precession is not uniform since spin waves in YIG\n\flms have the energy minima at non-zero wave vectors.\nIn contrast to the equilibrium state, stability of the dy-\nnamically supported non-equilibrium state even without\ncurrent is not for granted and must be checked.\nIn our discussion of topology we assumed that phase\ngradients were small and ignored the gradient-dependent\n(kinetic) energy. At growing gradient and gradient-\ndependent energy, we reach the critical gradient at which\nbarriers making the supercurrent stable vanish. For su-\nper\ruids the critical gradient (critical velocity) is deter-\nmined from the famous Landau criterion. The analo-\ngous criterion was also known for spin super\ruidity in\neasy-plane anti- and ferromagnets.3In the present paper\nwe derive this criterion for possible spin supercurrents\nin YIG magnetic \flms with the easy plane of dynamical\norigin.III. LANDAU{LIFSHITZ{GILBERT THEORY\nAND LINEAR SPIN WAVES IN YIG BULK\nThe coherent state of magnons is nothing else but a\nclassical spin wave, and one can use the classical equa-\ntions of the LLG theory. In the LLG theory the abso-\nlute value of the magnetization vector Mdoes not vary\nin space and time, and the classical LLG equations are\nreduced to two equations for only two independent mag-\nnetization components:\n_Mx=\u0000\rMz\u000eH\n\u000eMy;_My=\rMz\u000eH\n\u000eMx; (1)\nwhere\ris the gyromagnetic ratio and \u000eH=\u000eM xand\n\u000eH=\u000eM yare functional derivatives of the hamiltonian H.\nThe third LLG equation for _Mzis not independent and\ncan be derived from two equations (1). Instead of two\nreal functions MxandMyone can introduce one com-\nplex function  =Mx+iMy. The equation for  directly\nfollows from and fully equivalent to the LLG equations\n(1). By analogy with the theory of super\ruids they call\nit the Gross{Pitaevskii equation.\nThere is another form of the equations in the LLG the-\nory especially convenient for the analysis of spin trans-\nport. Magnetization dynamics is described in the terms\nof two independent variables, Mzand the angle 'of the\nmagnetization rotation around the zaxis:\n_Mz=\u0000\u000eH\n\u000e'=\u0000@H\n@'+ri@H\n@ri'=\u0000r\u0001j+Tz;(2)\n_'=\u000eH\n\u000eMz(3)\nThese are the Hamilton equations for the pair of conju-\ngated canonical variables \\moment{angle\" analogous to\nthe conjugated pair \\momentum{coordinate\". The \frst\nequation is the balance equation for magnetization along\nthe axiszproportional to the zcomponent of spin den-\nsity, and we introduced the magnetization current jand\nthe torqueTz:\nj=@H\n@r'; Tz=\u0000@H\n@': (4)\nThere was decades-long discussion of ambiguity in def-\ninition of the spin current. Ambiguity emerges because\nthe continuity equation for Mzcontains the torque Tz,\nwhich violates the spin conservation law. Indeed, one can\nadd any vector bto the magnetization current jand com-\npensate it by adding the divergence r\u0001bto the torque\nTz. This does not a\u000bect the \fnal balance. There were\nnumerous attempts to \fnd a proper de\fnition of the spin\ncurrent. It was argued3that no de\fnition is more proper\nthan others. But some de\fnition can be more convenient\nthan others, and the convenience criterion may vary from\ncase to case. The choice of de\fnition should not a\u000bect4\n\fnal physical results like the choice of gauge in electro-\ndynamics.\nYIG is a ferrimagnet with complicated magnetic struc-\nture consisting of numerous sublattices.14However at\nslow degrees of freedom relevant for our analysis one can\ntreat it simply as an isotropic ferromagnet15with the\nspontaneous magnetization Mdescribed by the hamil-\ntonian\nH=Z\u0014\n\u0000H\u0001M+DriM\u0001riM\n2\u0015\ndr\n+Zr\u0001M(r)r\u0001M(r1)\n2jr\u0000r1jdrdr1: (5)\nHere the \frst term is the Zeeman energy in the mag-\nnetic \feld H, the second term /Dis the inhomogeneous\nexchange energy, and the last one is the magnetostatic\n(dipolar) energy. Let us consider a spin wave in a YIG\nbulk propagating in the plane xzin a magnetic \feld H\nparallel to the axis z. In a weak spin wave\nMz\u0019M\u0000M2\n?\n2M;r\u0001M\u0019r xMx; (6)\nwhereM?=q\nM2x+M2y, and the linearized equations\nof motion (1) are\n_Mx=\u0000\rHM y+\rDM (r2\nxMy+r2\nzMy);\n_My=\rHM x\u0000\rDM (r2\nxMx+r2\nzMx)\n\u0000\rMrx\u0012ZrxMx(r1)\njr\u0000r1jdr1\u0013\n: (7)\nThe equations look as integro-di\u000berential equations be-\ncause of the magnetostatic term in the equation for My.\nBut applying the Laplace operator r2to this equation\nmakes this term purely di\u000berential. After exclusion of\nany of two component MxorMyone receives a di\u000beren-\ntial equation of the 6th order.\nFor the plane wave with the frequency !and the wave\nvector k(kx;0;kz) Eqs. (7) become\n\u0000i!M x=\u0000\rMy(H+DMk2);\n\u0000i!M y=\rMx\u0012\nH+DMk2+4\u0019Mk2\nx\nk2\u0013\n:(8)\nA solution of linear equations is an elliptically polarized\nrunning spin wave,\nMx=m0cos(k\u0001r+!t);\nMy=s\n1 +4\u0019Mk2x\n(H+DMk2)k2m0sin(k\u0001r+!t);(9)\nwith the wave vector k(kx;0;kz) and the frequency\n!(k) =\rs\n(H+DMk2)\u0012\nH+DMk2+4\u0019Mk2x\nk2\u0013\n:\n(10)The energy density in the spin wave mode is\nE=m2\n0\n2M\u0012\nH+DMk2+4\u0019Mk2\nx\nk2\u0013\n\u0019!(M\u0000hMzi)\n\r;\n(11)\nwherehMziis the averaged magnetization. In quantum-\nmechanical description the magnon density nm=E=~!\ndi\u000bers from the di\u000berence of M\u0000hMzionly by a constant\nfactor.\nIV. SPIN WAVES IN FILMS, BOUNDARY\nCONDITIONS\nA spin wave propagating in the \flm of thickness dpar-\nallel to the plane yz(Fig. 2) must satisfy the boundary\nconditions at two \flm surfaces x=\u0006d=2. Neglecting\nthe exchange interaction /Dthe spin wave reduces to\na magnetostatic wave investigated in the past by Damon\nand Eshbach16. The boundary conditions are imposed\non the magnetostatic magnetic \feld induced by magnetic\ncharges 4\u0019r\u0001M(r) and determined from the equation\nr\u0001(h+ 4\u0019M) = 0: (12)\nThe magnetostatic \feld is curl-free and is given by\nh=r ;  (r) =Zr\u0001M(r1)\njr\u0000r1jdr1: (13)\nAt any \flm surface the tangential component of the mag-\nnetic \feld hand the normal component of the magnetic\ninduction h+ 4\u0019Mmust be continuous. For the mag-\nnetostatic mode Mx/m0coskxxeikzz\u0000i!tthe magneto-\nstatic potential inside the \flm is\n =\u00004\u0019Mk x\nk2sinkxxeikzz\u0000i!t: (14)\nOutside the \flm at x>d= 2 there is no magnetic charges\nand the magnetostatic potential must satisfy the Laplace\nequation \u0001  = 0. Continuity of the tangential compo-\nnent of the magnetic \feld hz=rz at the \flm boundary\nrequires continuity of  , and atx>d= 2\n =\u00004\u0019Mk x\nk2sinkxd\n2ekz(d=2\u0000x)+ikzz\u0000i!t:(15)\nHdx\nyz\nFIG. 2. The YIG \flm of thickness din a magnetic \feld H\nparallel to the axis z.5\nContinuity of the normal component of the magnetic in-\nductionhx+ 4\u0019Mx=rx + 4\u0019Mxalso takes place if\ntankxd\n2=kz\nkx: (16)\nThis equation determines discrete values of kxfor mag-\nnetostatic modes of Damon and Eshbach16.\nIn our case the exchange interaction cannot be ignored,\nand this imposes additional boundary conditions. One\ncannot satisfy all boundary conditions by a single plane\nwave and must consider a superposition of plane waves\nwith the same frequency !and the wave number kzbut\nwith di\u000berent values of kx.\nThe di\u000berential equations are of the 6th order in space.\nCorrespondingly the dispersion relation (10) at \fxed !\nandkzis a characteristic equation of the 6th order with\nrespect tokxbut is tri-quadratic (cubic with respect to\nk2\nx). The roots of the characteristic equation determinekxin the superposition. This approach was used in the\npast17. The \frst root of the cubic equation for k2\nxyields\na small real kx, which determines the bulk mode with the\nfrequency!. Other two roots can be found analytically if\nthe relevant wave number k=p\nk2z+k2xis much smaller\nthan 1=ld, whereld=p\nD=\u0019 is a small scale determined\nby the exchange energy. The values k2\n\u0006of two additional\nroots of the cubic equation for k2\nxare negative and k\u0006\nare imaginary and very large (on the order of 1 =ld):\nk2\n\u0006\u00191\nD \n\u00002\u0019\u0000H\nM\u0006s\n4\u00192+!2\n\r2M2!\n\u00191\n\u0019l2\nd \n\u00002\u0019\u0000H\nM\u0006r\n4\u00192+H2\nM2!\n:(17)\nThese values correspond to evanescent modes con\fned to\nsurface layers of rather small width ld.\nClose to the surface x=d=2 the boundary conditions\nare satis\fed by a superposition of three modes:\nMx/h\ncoskxx+a+e\u0000p+(d=2\u0000x)+a\u0000e\u0000p\u0000(d=2\u0000x)i\neikzz\u0000i!t;\nMy/\"r\n1 +4\u0019Mk2x\nH+DMk2coskxx+a+s\n1 +4\u0019M\nH\u0000DMp2\n+e\u0000p+(d\n2\u0000x)+a\u0000s\n1 +4\u0019M\nH\u0000DMp2\n\u0000e\u0000p\u0000(d\n2\u0000x)#\neikzz\u0000i!t;\n(18)\nwherep\u0006=ik\u0006are real and positive and a\u0006are ampli-\ntudes of two evanescent modes.\nThe exchange boundary condition for unpinned spins11\narerxMx=rxMy= 0. They are satis\fed if\nkxsinkxd\n2\u0000a+p+\u0000a\u0000p\u0000= 0;\nkxr\n1 +4\u0019Mk2x\nH+DMk2sinkxd\n2\n\u0000a+p+s\n1 +4\u0019M\nH\u0000DMp2\n+\n\u0000a\u0000p\u0000s\n1 +4\u0019M\nH\u0000DMp2\n\u0000= 0: (19)\nRepeating derivation of the magnetostatic boundary con-\ndition done above for magnetostatic modes one obtains\nk2\nx\nk2coskxd\n2+a++a\u0000\n=kxkz\nk2sinkxd\n2+a+kz\np++a\u0000kz\np\u0000: (20)\nEquation (19) shows that the amplitudes of evanescent\nmodes are of the order a\u0006\u0018kxsinkxd\n2=p\u0006. Then their\ncontribution to the magnetostatic boundary condition\n(20) by a small factor kz=p\u0006\u0018kzldless than the otherterms and can be ignored. Eventually we return back\nto the equation (16) for kxobtained for magnetostatic\nwaves of Damon and Eshbach16without e\u000bects of ex-\nchange interaction. Thus even though evanescent modes\nare indispensable for satisfying all boundary conditions\nthey do not a\u000bect the shape of the wave in the most of\nthe bulk.\nAt largekzdEq. (16) yields kx=\u0019=d, and in the\nbulk the magnetization components Mx\u0018coskxxand\nMy\u0018coskxxvanish at the \flm surfaces. This au-\ntomatically satis\fes the exchange boundary conditions\nMx=My= 0 for pinned spins without adding evanes-\ncent modes. Ignoring narrow surface layers where evanes-\ncent modes can be important, the plane wave propagat-\ning in the \flm plane is\nMx=p\n2m0cos\u0019x\ndcos(kzz+!t);\nMy=p\n2\u0012\n1 +2\u00193M\nHk2zd2\u0013\nm0cos\u0019x\ndsin(kzz+!t)(21)\nindependently from the exchange boundary conditions.\nThe wave frequency is\n!(kz)\u0019\r\u0012\nH+DMk2\nz+2\u00193M\nk2zd2\u0013\n: (22)\nThis dispersion relation di\u000bers from the spin-wave spec-\ntrum derived for YIG \flms by Kalinikos and Slavin11and6\nwidely used in the past, in particular, in articles address-\ning Bose{Einstein condensation and spin super\ruidity in\nYIG \flms5,12,13. Kalinikos and Slavin11received a dis-\npersion relation, in which the term 2 \u0019M(1\u0000e\u0000kzd)=kzd\nreplaces the magnetostatic contribution in our dispersion\nrelation (22) (the third term /1=k2\nzd2). Instead of solv-\ning di\u000berential equations Kalinikos and Slavin11approx-\nimately solved the integro-di\u000berential equations. They\napproximated the magnetization distribution in space\nby a superposition of functions, which do not satisfy\ndi\u000berential equations in the bulk. This is easily seen\nin the recent simpli\fed derivation of their spectrum by\nRezende13. Rezende approximated a spin wave in the\n\flm bulk by a superposition of plane-wave modes with\ndi\u000berent values of kxas in our solution (our axis xcor-\nrespond to the axis yof Rezende and vice versa). But\nRezende'skxwere not roots of the characteristic equa-\ntion of the relevant system of di\u000berential equations. As\na result, frequencies of modes in his superposition di\u000ber\none from another and from the frequency given by the\ndispersion relation. In particular, two of his modes have\nvalueskx=\u0006ikz, for which k2=k2\nx+k2\nzvanishes and\nthe spectrum (10) of a single plane spin wave gives an\nin\fnite frequency! Thus Rezende's superposition does\nnot describe a proper monochromatic eigenmode at all.\nCorrespondingly the spectrum of Kalinikos and Slavin\nfollowing from this superposition is invalid.\nThe spectrum of Kalinikos and Slavin and the spec-\ntrum Eq. (22) are compared in Fig. 3. Quantitate dif-\nference between two spectra is not so dramatic. More\nimportant is that our analysis predicts an essentially dif-\nferent distribution of magnetization across the \flm. The\ncomponent Mxnormal to the \flm approaches to zero\nclose to the \flm surface (but still outside narrow bound-\nary layers, where evanescent modes are important). On\nthe other hand, according to Rezende13, in the approxi-\n1520253035400.150.200.250.300.35!\u0000!L⇡\u0000Mkzd!\u0000!L⇡\u0000Mkzd12\nFIG. 3. Comparison of the linear spin-wave spectrum in a\nYIG \flm calculated by Kalinikos and Slavin11(curve 1) and\nin the present paper (curve 2). Here !L=\rHis the Larmor\nfrequency.mation of Kalinikos and Slavin11variation of Mxacross\nthe \flm is negligible. This is important for evaluation\nof nonlinear corrections, which determine stability of su-\npercurrent states investigated further in the paper.\nInaccuracy of the theory of Kalinikos and Slavin11has\nalready been noticed by Kreisel et al.18. They calculated\nnumerically the linear spin-wave spectrum in the micro-\nscopic theory and revealed that the numerically calcu-\nlated spectrum lies lower than the spectrum of Kalinikos\nand Slavin as curve 2 in Fig. 3 calculated in the LLG\ntheory. Agreement between the microscopic and macro-\nscopic LLG theory is not surprizing since all scales rele-\nvant for our analysis are larger than atomic.\nV. COHERENT MAGNON CONDENSATE AND\nITS STABILITY\nBy strong parametric pumping Demokritov et al.7\nwere able to create a coherent state of magnons con-\ndensed at states with lowest energies with non-zero wave\nvectors, which was called a magnon Bose{Einstein con-\ndensate. A condition for emerging of the magnon conden-\nsate is that magnon-magnon interactions violating the\nspin conservation law are much weaker than interactions\nthermalizing the magnon gas. Despite the magnon gas\nrequired at least weak pumping for compensation of lost\nspin (magnons) it was treated as a quasi-equilibrium gas\nwith \fxed total number of magnons (see below).\nThe energy and the frequency !(kz) given by Eq. (22)\nhave two degenerate minima15at \fnitekz=\u0006k0where\nmagnons can condense (Fig. 4). Here\nk0=\u00122\u00193\nDd2\u00131=4\n=\u00122\u00192\nl2\ndd2\u00131=4\n: (23)\nIn the linear theory the distribution of magnons between\ntwo condensates is arbitrary and does not a\u000bect the total\nenergy (at \fxed magnetization hMzi, i.e., at \fxed con-\ndensate magnon density). But non-linear corrections lift\nthis degeneracy.19Let us consider the e\u000bect of a non-\nlinear term/M4\n?in the expansion for hMzi:\nhMzi=M\u0000hM2\n?i\n2M\u0000hM4\n?i\n8M3: (24)\nThe energy density of the condensate spin wave as a func-\ntion ofM\u0000hMziis\nE=H(M\u0000hMzi)\n+\u0012\nDMk2\nz+2\u00193M\nk2zd2\u0013\u0012\nM\u0000hMzi\u0000hM4\n?i\n8M3\u0013\n:(25)\nFor the running wave given by Eq. (21) (all magnons con-\ndensate in one minimum) hM4\n?i= 6(M\u0000hMzi)2. The\nsign of the nonlinear correction is negative. This cor-\nresponds to attraction between magnons, and the con-\ndensate is unstable. For the running wave (21) all other\nnonlinear corrections are smaller and cannot a\u000bect this\nconclusion.7\n!\nkz\nk0\nk0+K\nk0K!\nkz\nk0\nk0+K\nk0K\n!\nkz\nk0\nk0+K\nk0K!\nkz\nk0\nk0+K\nk0K!\nkz\nk0\nk0+K\nk0K!\nkz\nk0\nk0+K\nk0\u0000K!\nkz\nk0\nk0+K\nk0K!\nkz\nk0\nk0+K\nk0\u0000K\nFIG. 4. The spin-wave spectrum in a YIG \flm. In the ground\nstate the magnon condensate occupies two minima in the k\nspace with kz=\u0006k0(large circles). In the current state two\nparts of the condensate are shifted to k=\u0006k0+K(small\ncircles).\nHowever, if magnons condense in two minima there is\nanother nonlinear term arising from the magnetostatic\nenergy:\nEms=ZrzMz(r)rzMz(r1))\n2jr\u0000r1jdrdr1: (26)\nFor the running wave this term is negligible compared to\nthe term considered above, because zvariation of Mzis\nweak. But the nonlinear magnetostatic term is maximal\nfor the standing wave (two energy minima are equally\npopulated by magnons):\nMx= 2m0cos\u0019x\ndcoskzzcos!t;\nMy= 2\u0012\n1 +2\u00193M\nHk2zd2\u0013\nm0cos\u0019x\ndcoskzzsin!t:(27)\nIn the standing wave\nMz=M\u0000m2\n0\nM(1 + cos 2kzz)(1 + cos 2kxx); (28)\nand\nEms=3\u0019m4\n0\n8M2=3\u0019(M\u0000hMzi)2\n2: (29)\nNow magnon interaction is repulsive. But this does not\nmean that the standing-wave condensate is absolutely\nstable, because the interaction energy at \fxed hMzide-\ncreases when the distribution of magnons between twocondensates becomes more and more asymmetric. Even-\ntually the condensate spin wave transforms to the run-\nning wave in which magnon-magnon interaction is at-\ntractive and the interaction energy is negative. Thus\nthe magnon condensate cannot be stable! Then in-\nevitably a question arises why a relatively stable long-\nliving magnon condensate was observed. Instability of\nthe magnon condensate in YIG \flms was already re-\nvealed earlier by Tupitsyn et al.12. In order to explain\nthe paradox that the magnon condensate was observed\ndespite its expected instability, they referred to size ef-\nfects. Another scenario is also possible. Apparently the\nquasi-equilibrium approach determining distribution of\nmagnons between two energy minima from the condition\nof the minimal energy at \fxed hMzi, i.e., at \fxed to-\ntal magnon number, is not satisfactory, and instead the\nmagnon distribution between two minima must be re-\nceived from the dynamical balance taking into account\nspin pumping and spin relaxation. There is no evident\nreason why pumped magnons prefer to condensate in one\nminimum rather than in another, and R uckriegel and\nKopietz20numerically investigated the dynamical pro-\ncess of the magnon condensate formation in the LLG\ntheory assuming that the two minima are \flled symmet-\nrically. Malomed et al.21solved numerically the Gross{\nPitaevskii equation with added spin pumping and relax-\nation and found that sometimes asymmetric magnon dis-\ntributions emerge, but only at asymmetric boundary con-\nditions. Experimentally Nowik-Boltyk et al.22revealed\nspatial periodic oscillations of magnon density, which are\npossible only if magnons condense in the both energy\nminima.\nApparently possible asymmetry of magnon distribu-\ntion in the process of formation of the magnon conden-\nsate still deserves further investigations similar to those\nin Refs. 20 and 21, but it is beyond the scope of this\nwork. Studying stability of current states (the next sec-\ntion) we shall use a modi\fed quasi-equilibrium approach\nassuming that dynamical processes (spin pumping and\nrelaxation) \fx not only the total number of magnons but\nalso distribution of them between two energy minima.\nWe shall focus on a pure standing wave with symmetric\nmagnon distribution in the kspace for which critical gra-\ndients are higher than for asymmetric distribution. Thus\nwe look for the upper bound for critical gradients.\nVI. SPIN-SUPERCURRENT STATE AND ITS\nSTABILITY (LANDAU CRITERION)\nThe phase variation in space in the magnon condensate\ndepends on distribution of magnons between two energy\nminima. In the running wave (21)\n'= arctanMy\nMx=!t+kzz+\u00193M\nk2zd2sin 2(!t+kzz);(30)8\nwhile in the standing wave\n'=!t+\u00193M\nk2zd2sin 2!t: (31)Thus apart from nonessential small periodical oscilla-\ntions the phase gradient is rz'=kzin the running wave\nbut vanishes in the standing wave.\nIn the standing wave the magnetization (spin) current\nappears if the wave numbers kzof two condensates di\u000ber\nfrom\u0006k0(Fig. 4), and neglecting weak ellipticity\nMx=m0coskxx[cos(k0z+Kz+!t) + cos(k0z\u0000Kz\u0000!t)] = 2m0coskxxcosk0zcos(Kz+!t);\nMy=m0coskxx[sin(k0z+Kz+!t)\u0000sin(k0z\u0000Kz\u0000!t)] = 2m0coskxxcosk0zsin(Kz+!t): (32)\nThusrz'=K=kz\u0000k0\u001ck0. Keeping the mag-\nnetizationhMzi\fxed as before and taking into account\nthe nonlinear magnetostatic term (29) the energy in the\nspin-current state apart from some constant terms is\n\u0001E=d2!(k0)\ndk2zM\u0000hMzi\n\r(rz')2\n2+3\u0019(M\u0000hMzi)2\n2;\n(33)\nwhere\nd2!(k0)\ndk2z=\rM\u0012\n2D+12\u00193\nk4\n0d2\u0013\n=16\u00193\rM\nk4\n0d2: (34)\nStability of the spin-current state can be checked fol-\nlowing the principal idea of the Landau criterion of\nsuper\ruidity3: If weak perturbations of the current state\n(creation of a quasiparticle in the Landau case) always in-\ncrease energy, the current state is metastable. If there are\nperturbations decreasing the energy super\ruid transport\nwith suppressed dissipation is impossible. Let us con-\nsider slowly varying in space weak perturbations mz=\nMz\u0000hMziandrz'0=rz'\u0000K. Quadratic in mzand\nrz'0terms in expansion of the energy (33) are\n\u0001E0=d2!(k0)\ndk2z\u0014M\u0000hMzi\n\r(rz'0)2\n2\n\u0000Kmz\n\rrz'0\u0015\n+3\u0019m2\nz\n2: (35)\nFor stability of the supercurrent the quadratic form in\nperturbations mzandrz'0must be always positive.\nThis takes place as far as rz'=Kis less than the\ncritical value\n(rz')cr=vuut3\u0019\r(M\u0000hMzi)\nd2!(k0)\ndk2z=r\n3(M\u0000hMzi)\nMk2\n0d\n4\u0019:\n(36)\nThis corresponds to the critical group magnon velocity\nvcr=d2!(k0)\ndk2z(rz')cr=4\u00192\rM\nk2\n0dr\n3(M\u0000hMzi)\nM:\n(37)\nNote that applying our course of derivation to super-\n\ruid hydrodynamics one obtains exactly the Landau crit-\nical velocity equal to the sound velocity (see Sec. 2.1 in\nRef. 3).We conclude this section by estimation of the magneti-\nzation supercurrent jusing the canonical expression (4).\nClose to the energy minimum the magnetization current\nalong thezaxis is\njz=@E\n@kz=M\u0000hMzi\n\rd!\ndkz\n\u0019M\u0000hMzi\n\rd2!(k0)\ndk2z(kz\u0000k0): (38)\nAt our de\fnition of the current it is proportional to the\ngroup velocity d!=dk zof magnons3and therefore van-\nishes in the ground state of the condensate both for the\nrunning and the standing wave.\nVII. DISCUSSION AND CONCLUSIONS\nThe derived critical gradient is essentially lower than\nobtained by Sun et al.5who determined the critical su-\npercurrent equating the kinetic energy to the high Zee-\nman energy. Our analysis demonstrates that the Zeeman\nenergy does not a\u000bect the stability condition at all. The\nmagnetostatic term (29) stabilizing supercurrents plays\nthe same role as easy-plane anisotropy in easy-plane mag-\nnets, but the former is of dynamical origin and much\nsmaller than the latter being proportional to the wave\nintensity (density of condensed magnons).\nA byproduct of our analysis was revision of the widely\naccepted spin-wave spectrum in YIG \flms, which took\ninto account proper magnetostatic and exchange bound-\nary conditions on \flm surfaces. This in\ruenced estima-\ntions of non-linear corrections to spin waves crucial for\nmetastability of the magnon condensate with and with-\nout spin supercurrents.\nLet us make some numerical estimations. Accord-\ning to Dzyapko et al.23the magnon density can reach\n1018cm\u00003. Assuming that 10 % of magnons are in\nthe coherent state, this corresponds to rather small ratio\n(M\u0000hMzi)=M\u00180:32\u000210\u00004. Then Eq. (37) yields for\nk0= 5:5 104cm\u00001andd= 5 10\u00004cm the critical veloc-\nityvcrabout 3.6 m/sec (instead of 420 m/sec found by\nSunet al.5).\nIn the light of the presented analysis let us discuss the\nreport by Bozhko et al.8on detection of spin supercur-9\nrents in observation of a decaying magnon condensate\nprepared in a YIG magnetic \flm by magnon pumping.\nThe major problem with this claim is small total phase\nvariation along streamlines of the supposed current re-\nalized in the experiment. Bozhko et al. applied a tem-\nperature gradient to the magnon BEC cloud, which led\nto a di\u000berence \u000e!of the frequency of magnetization pre-\ncession (phase rotation velocity) across the condensate\ncloud. This produced a total phase variation \u000e'=\u000e!t\nacross the BEC cloud growing linearly with time tand\ngenerating spin currents. For the maximal \u000e!= 2\u0019\u0002550\nrad/sec and the maximal life time t= 0:5\u0016sec of the con-\ndensate in the experiment of Bozhko et al.8(see their\nFig. 5) one can conclude that the total phase variation\n\u000e'never exceeded about 1/3 of the full 2 \u0019rotation. As\ndiscussed in introduction, only currents with large num-\nber of full 2 \u0019rotations along streamlines deserve the title\nof \\supercurrent\" manifesting spin super\ruidity.\nOne might consider it as a purely semantic issue. But\ncalling any current / r'supercurrent demonstrating\nspin super\ruidity would reduce spin super\ruidity to a\ntrivial ubiquitous phenomenon. Currents produced by\nsuch small phase variations cannot relax via phase slips\nand are trivially stable. They emerge in any spin wave\nor domain wall. Any inhomogeneity produces them, and\nthey must present in the experiment of Bozhko et al.8but in contrast to authors' claim they have nothing to do\nwith the macroscopic phenomenon of super\ruidity.\nIn summary, spin super\ruidity in YIG \flms is possible\nin principle, although the recent report on its experimen-\ntal observation8is not founded. Metastability of spin\nsupercurrents in this material is provided by energetic\nbarriers not of topological but of dynamic origin, which\ndepend on intensity of a nonlinear spin wave describing\nthe coherent magnon condensate.\nIt is worth noting that at growing magnetic \feld in\nYIG \flms the orientational phase transition takes place\nfrom the state with the total and sublattice magnetiza-\ntions along the magnetic \feld to the state, in which mag-\nnetizations deviate from the magnetic \feld direction and\nhave large components in the plane normal to the mag-\nnetic \feld.15This is a state with easy-plane anisotropy,\nfor which spin super\ruidity have been predicted. But\nthis requires magnetic \felds \u0018105G, which are orders\nof magnitude larger than \felds nowadays used in exper-\niments on magnon condensation.\nACKNOWLEDGMENTS\nThe author thanks S.O. Demokritov, V.S. L'vov, V.L.\nPokrovsky, and A.A. Serga for useful discussions.\n1E. B. Sonin, Zh. Eksp. Teor. 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However , it is \ndifficult to expect that a new switch will outperform CMOS in \nall figures of merit, and more importantly, will be able to \nprovide multiple generations of improvement as was the case \nfor CMOS [4]. An alternative route to the computational \npower enhancement is via the development of novel \ncomputing devices aimed not to replace  but to complement \nCMOS  by special task data processing [5].  Spin wave \n(magnonic) logic devices are one of the alternative approaches \naimed to take the  advantages of the wave interference at \nnanometer scale and utilize phase in addition to amplitude for \nbuilding logic units for parallel data processing.  \n A spin wave is a collective oscillation of spins in a \nmagnetic lattice, analogous to phonons, the col lective \noscillation of the nuclear lattice. The typical propagation speed \nof spin waves does not exceed 107cm/s, while the attenuation \ntime at room temperature  is about a nanosecond in the \nconducting ferromagnetic materials (e.g. NiFe, CoFe) and may \n \n be hu ndreds of nanoseconds in non -conducting materials (e.g. \nYIG).  Such a short attenuation time explains the lack of \ninterest in spin waves as a potential information carrier in the \npast. The situation has changed drastically as the technology \nof integrated l ogic circuits has scaled down to the deep sub -\nmicrometer scale, where the short propagation distance of spin \nwaves (e.g. tens of microns at room temperature) is more than \nsufficient for building logic circuits.  At the same time, spin \nwaves have several in herent appealing properties making them \npromising for building wave -based logic devices. For instance, \nspin wave propagation can be directed by using magnetic \nwaveguides similar to optical waveguides. The amplitude and \nthe phase of propagating spin waves c an be modulated by an \nexternal magnetic field.   Spin waves can be generated and \ndetected by electronic components (e.g. multiferroics [6]), \nwhich make them suitable for integration with conventional \nlogic circuits. Finally, the coherence length of spin waves at \nroom temperature may exceed tens of micr ons, which allows \nfor the utilization of spin wave interference for logic \nfunctionality.  It makes spin waves much more prone to \nscattering than a single electron and resolves one of the most \ndifficult problems of spintronics associated with the necessity \nto preserve spin orientation while transmitting information \nbetween the spin -based units.      \nDuring the past decade, there have been a growing number \nof theoretical and experimental works exploring spin wave \npropagation in a variety of magnetic structure s [7, 8], the \npossibility of spin wave propagation modulation by an \nexternal magnetic field  [9, 10], and spin wave interference  \nand diffraction [11-15]. The collected experimental data \nrevealed interesting and unique properties of spin wave F. Gertz1, A. Kozhevnikov2, Y. Filimonov2, D.E. Nikonov3 and A. Khitun1 \n1)Electrical Engineering Department, University of California -  Riverside, Riverside, CA, USA, 92521  \n2)Kotel’niko v Institute of Radioengineering  and Electronics of Russian Academy of Sciences, Saratov Branch, Saratov, Russia, \n410019  \n3)Technology & Manufacturing Group Intel Corp. , 2501 NW 229th Avenue, Hillsboro, OR, USA, 97124  Magnonic Holographic Memory: from Proposal to Device  \nT   In this work, we present recent developments in m agnonic holographic memory devices exploiting spin waves for information \ntransfer. The devices comprise a magnetic matrix and spin wave generating/detecting elements placed on the edges of the waveg uides. \nThe matrix consists of a grid of magnetic waveguide s connected via cross junctions. Magnetic memory elements are incorporated \nwithin the junction while the read -in and read -out is accomplished by the spin waves propagating through the waveguides.  We \npresent experimental data on spin wave propagation throu gh NiFe and YIG magnetic crosses. The obtained experimental data show \nprominent spin wave signal modulation (up to 20 dB for NiFe and 35 dB for YIG) by the external magnetic field, where both the  \nstrength  and the direction of the magnetic field define the  transport between the cross arms.  We also present experimental data on \nthe 2 -bit magnonic holographic memory built on the double cross YIG structure with micro -magnets placed on the top of each cross. \nIt appears possible to recognize the state of each ma gnet via the interference pattern produced by the spin waves with all experiments \ndone at room temperature. Magnonic holographic devices aim to combine the advantages of magnetic data storage with wave -based \ninformation transfer. We present estimates on th e spin wave holographic devices performance, including power consumption and \nfunctional throughput. According to the estimates, magnonic holographic devices may provide data processing rates higher than  \n1×1018 bits/cm2/s while consuming 0.15mW. Technologic al challenges and fundamental physical limits of this approach are also \ndiscussed.  \n \nIndex Terms — spin waves, holography, logic device.   \n  \n2 \n transport (e.g. non -reciprocal spin wave propagation [16]) for \nbuilding magnonic logic circuits.  The first working spin -wave \nbased logic device has been experimentally demonstrated by \nKostylev et al in 2005 [17]. The authors used the Mach –\nZehnder -type current -controlled spin wave interferometer to \ndemonstrate output voltage modulation as a result of spin \nwave interference. Later on, exclusive -not-OR (NOR) and not -\nAND (NAND) gates were experimentally demonstrated \nutilizing a similar structure [18].  The idea of using Mach –\nZehnde r-type spin wave interferometers has been further \nevolved  by proposing a spin wave interferometer with a \nvertical current -carrying wire [19]. With zero applied current, \nthe spin waves in two branches interfere constructively and \npropagate through the structure. The waves interfere \ndestructively and do not propagate through the structure if a \ncertain electric current is  applied. At some point, these first \nmagnonic logic devices resemble the classical field effect \ntransistor, where the magnetic field produced by the electric \ncurrent modulates the propagation of the spin wave —an \nanalogue to the electric current.  Then, it was proposed to \ncombine spin wave with nano -magnetic logic aimed to \ncombine the advantages of non -volatile data storage in \nmagnetic memory and the enhanced functionality provided by \nthe spin wave buses [20]. The use of spin wave interference \nmakes it possible to realize Majority gates (which can be used \nas AND or OR gates) and NOT gates with a fewer number of \nelements than is required for transistor -based circuitry, \npromising the further reduction of the size of the l ogic gates.  \nThere were several experimental works demonstrating three -\ninput spin wave Majority gates  [21,22]. Ho wever, the \nintegration of the spin wave buses with nano -magnets in a \ndigital circuit, where the magnetization state of the nano -\nmagnet is controlled by a spin wave has not yet been realized.   \nAn alternative approach to spin wave -based logic devices is \nto build non -Boolean logic gates for special task data \nprocessing. The essence of this approach is to maximize the \nadvantage of spin wave interference. Wave -based analog logic \ncircuits are potentially promising for solving problems \nrequiring parallel operatio n on a number of bits at time (i.e. \nimage processing, image recognition).  The concept of \nmagnonic holographic memory (MHM) for data storage and \nspecial task data processing has been recently proposed [23].  \nHolographic devices for dat a processing have been extensively \ndeveloped in optics during the past five decades. The \ndevelopment of spin wave -based devices allows us to \nimplement some of the concepts developed for optical \ncomputing to magnetic nanostructures utilizing spin waves \ninstead of optical beams. There are certain technological \nadvantages that make the spin wave approach even more \npromising than optical computing.  First, short operating \nwavelength (i.e. 100nm and below) of spin wave devices \npromises a significant increase of the data storage density (~λ2 \nfor 2D and ~λ3 for 3D memory matrixes). Second, even more \nimportantly, is that spin wave bases devices can have voltage \nas an input and voltage as an output, which makes them \ncompatible with the conventional CMOS circuitry. Th ough \nspin waves are much slower than photons, magnonic holographic devices may possess a higher memory capacity \ndue to the shorter operational wavelength and can be more \nsuitable for integration with the conventional electronic \ncircuits. In this work, we p resent recent experimental results \non magnonic holographic memory and discuss the advantages \nand potential shortcomings of this approach. The rest of the \npaper is organized as follows. In Section II, we describe the \nstructure and the principle of operation  of magnonic \nholographic memory. Next, we present experimental data on \nthe first 2 -bit magnonic holographic memory in Section III. \nThe advantages and the challenges of the magnonic \nholographic devices are discussed in the Sections IV. In \nSection V, we pres ent the estimates on the practically \nachievable performance characteristics.  \nII. MATERIAL STRUCTURE AN D THE PRINCIPLE OF O PERATION  \nThe schematics of a MHM device are shown in Figure 1(A). \nThe core of the structure is a magnetic matrix consisting of the \ngrid of  magnetic waveguides with nano -magnets placed on top \nof the waveguide junctions. Without loss of generality, we \nhave depicted a 2D mesh of orthogonal magnetic waveguides, \nthough the matrix may be realized as a 3D structure \ncomprising the layers of magnetic  waveguides of a different \ntopology (e.g. honeycomb magnetic lattice). The waveguides \nserve as a media for spin wave propagation – spin wave buses.  \nThe buses can be made of a magnetic material such as yttrium \niron garnet Y 3Fe2(FeO 4)3 (YIG) or permalloy  ( Ni81Fe19) \nensuring maximum possible group velocity and minimum \nattenuation for the propagating spin waves at room \ntemperature.  The nano -magnets placed on top of the \nwaveguide junctions act as memory elements holding \ninformation encoded in the magnetizatio n state. The nano -\nmagnet can be designed to have two or several thermally \nstable states of magnetization, where the number of states \ndefines the number of logic bits stored in each junction.  The \nspins of the nano -magnet are coupled to the spins of the \njunction magnetic wires via the exchange and/or dipole -dipole \ncoupling affecting the phase of the propagation of spin waves. \nThe phase change received by the spin wave depends on the \nstrength and direction of the magnetic field produced by the \nnano -magnet. At  the same time, the spins of the nano -magnet \nare affected by the local magnetization change caused by the \npropagating spin waves.  \nThe input/output ports are located at the edges of the \nwaveguides.  These elements are aimed to convert the input \nelectric si gnals into spin waves, and vice versa, convert the \noutput spin waves into electrical signals. There are several \npossible options for building such elements by using micro -\nantennas [24, 25], spin torque oscillators [26], and multiferroic \nelements [6].  For example, the micro -antenna is a conducting \ncontour placed in the vicinity of the spin wave bus. An electric \ncurrent passed through the contour generates a magnetic field \naround the current -carrying wires, which excites spin waves in \nthe magnetic material, and vice versa, a propagating spin wave \nchanges the magnetic flux from the magnetic waveguide and \ngenerates an inductive vo ltage in the antenna contour.  The \nadvantages and shortcomings of different input/output  \n3 \n elements will be discussed later in the text.  \nSpin waves generated by the edge elements are used for \ninformation read -in and read -out. The difference among these \ntwo modes of operation is in the amplitude of the generated \nspin waves.   In the read -in mode, the elements generate spin \nwaves of a relatively large amplitude, so two or several spin \nwaves coming in -phase to a certain junction produce magnetic \nfield sufficien t for magnetization change within the nano -\nmagnet. In the read -out mode, the amplitude of the generated \nspin waves is much lower than the threshold value required to \novercome the energy barrier between the states of nano -\nmagnets.  So, the magnetization of the junction remains \nconstant in the read -out mode. The details of the read -in and \nread-out processes are presented in Ref. [23].  \nThe formation of the hologram occurs in the following way.  \nThe incident spin wave beam is produced by the number of \nspin wave generating elements (e.g. by the elements on the left \nside of the matrix as illustrated in Figure 1(B)).  All the \nelements are biased by the same RF generator exciting spin \nwaves of the same frequency, f, and amplitude, A0, while th e \nphase of the generated waves are controlled by DC voltages \napplied individually to each element.  Thus, the elements \nconstitute a phased array allowing us to artificially change the \nangle of illumination by providing a phase shift between the \ninput waves . Propagating through the junction, spin waves \naccumulate an additional phase shift, Δ, which depends on \nthe strength and the direction of the local magnetic field \nprovided by the nano -magnet, Hm:   \n \n,                                    (1) \n \n \nwhere the particular form of the wavenumber k(H) \ndependence varies for magnetic materials,  film dimensions, \nthe mutual direction of wave propagation and the external \nmagnetic field [27].  For example, spin waves propagating \nperpendicular to the external magnetic field (magnetostatic \nsurface spin wave – MSSW) and spin waves propagating \nparallel to the direction of the external field (backward volume \nmagnetostatic spin wave – BVMSW) may obtain significantly \ndifferent phase shifts for the same field strength. The phase \nshift  produced by the external magnetic field variation H \nin the ferromagnetic film can be expressed as follows [17]: \n2 22 2\n2)(\nHMH\ndl\nHS \n  (BVMSW), \n) 4 () 2 (\n2 22\nSS\nM HHM H\ndl\nH  \n\n\n        (MSSW),        (2)                                \nwhere  is the phase shift produced by  the change of the \nexternal magnetic field H, l is the propagation len gth, d is the \nthickness of the ferromagnetic film,  is gyromagnetic ratio, \nω=2πf , 4πM s is the saturation magnetization of the \nferromagnetic film. The output signal is a result of \nsuperposition of all the excited spin waves traveling through \nthe different paths of the matrix. The amplitude of the output spin wave is detected by the voltage generated in the output \nelement (e.g. the inductive voltage produced by the spin waves \nin the antenna contour).  The amplitude of the output voltage \nis corresponding to t he maximum when all the waves are \ncoming in -phase (constructive interference), and the minimum \nwhen the waves cancel each other (destructive interference).  \nThe output voltage at each port depends on the magnetic states \nof the nano -magnets within the matri x and the initial phases of \nthe input spin waves. In order to recognize the internal state of \nthe magnonic memory, the initial phases are varied (e.g. from \n0 to ).  The ensemble of the output values obtained at the \ndifferent phase combinations constitute a hologram which \nuniquely corresponds to the internal structure of the matrix.     \nIn general, each of the nanomagnets can have more than 2 \nthermally stable states, which makes it possible to build a \nmulti -state holographic memory device (i.e. zN possible \nmemory states, where z is the number of stable magnetic states \nof a single junction and N is the number of junctions in the \nmagnetic matrix). The practically achievable memory capacity \ndepends on many factors including the operational \nwavelength, coherence  length, the strength of nano -magnets \ncoupling with the spin wave buses, and noise immunity. In the \nnext Section, we present experimental data on the operation of \nthe prototype 2 -bit magnonic holographic memory.  \nIII. EXPERIMENTAL DATA \nThe set of experiments st arted with the spin wave transport \nstudy in a single cross structure, which is the elementary \nbuilding block for 2D MHM as depicted in Fig.1.  Two types \nof single cross devices made of Y 3Fe2(FeO 4)3 (YIG) and  \nPermalloy (Ni 81Fe19) were fabricated. Both of t hese materials \nare promising for application in magnonic waveguides due to \ntheir high coherence length of spin waves. At the same time, \nYIG and Permalloy differ significantly in electrical properties \n(e.g. YIG is an insulator, permalloy is a conductor) and  in \nfabrication method.  YIG  cross structures were made from \nsingle crystal YIG films epitaxially grown on top of \nGadolinium Gallium Garnett (Gd 3Ga5O12) substrates using the \nliquid -phase transition process.  After the films were grown, \nmicro -patterning was  done by laser ablation using a pulsed \ninfrared laser ( λ≈1.03 μm), with a pulse duration of ~256 ns. \nThe YIG cross junction has the following dimensions:   the \nlength of the whole structure is 3mm; the width of the arm in \n360µm; thickness is 3.6um. Permalloy crosses were fabricated \non top of oxidized silicon wa fers.  The wafer was spin coated \nwith a 5214E Photoresist at 4000 rpm and exposed using a \nKarl Suss Mask Aligner.  After development, a permalloy \nmetal film was deposited via Electron -Beam Evaporation with \na thickness of 100nm and with an intermediate seed  layer of \n10 nm of Titanium to increase the adhesion properties of the \nPermalloy film.  Lift -off using acetone completed the process. \nPermalloy cross junction has the following dimensions:  the \nlength of the whole structure is 18um; the width of the arm in  \n6µm; thickness is 100 nm.  \nSpin waves in YIG and Permalloy structures were excited \nand detected  via micro -antennas that were placed at the edges \nr\nmdrHk\n0)(\n \n4 \n of the cross arms.  Antennas were fabricated from gold wire \nand mechanically placed directly at the top of the  YIG cross.  \nIn the case of  permalloy, the conducting cross was insulated \nwith a 100nm layer of SiO 2 deposited via Plasma -Enhanced \nChemical Vapor Deposition (PECVD) and gold antennas were \nfabricated using the same photolithographic and lift -off \nprocedure as with the permalloy cross structures. A Hewlett -\nPackard 8720A Vector Network Analyzer (VNA) was used to \nexcite/detect spin waves within the structures using RF \nfrequencies.  Spin waves were excited by the magnetic field \ngenerated by the AC electric curre nt flowing through the \nantenna(s).  The detection of the transmitted spin waves is via \nthe inductive voltage measurements as described in Ref. [28].  \nPropagat ing spin waves change the magnetic flux from the \nsurface, which produces an inductive voltage in the antenna \ncontour. The VNA allowed the S -Parameters of the system to \nbe measured; showing both the amplitude of the signals as \nwell as the phase of both the transmitted and reflected signals. \nSamples were tested inside a GMW 3472 -70 Electromagnet \nsystem which allowed the biasing magnetic field to be varied \nfrom -1000 Oe to +1000 Oe.  The schematics of the \nexperimental setup for spin wave transport study in the  single \ncross structures are shown in Figure 2.  \nFirst, we studied spin wave propagation between the four \narms of the permalloy cross -structure as shown in Fig.3(A -B)  \nunder different bias magnetic field.  The input/output ports are \nnumbered from 1 to 4 sta rting at the 9 O’ clock position and \nthen enumerated sequentially in along a clockwise direction.  \nIn order to define the angle between the external magnetic \nfield and the direction of signal propagation, we define the X \naxis along the line from port 1 to port 3, and the Y axis along \nthe line from port 4 to port 2 propagating as depicted in Fig.2.  \nSpin waves were excited on port 2 (the top of the magnetic \ncross) and read out from port 4 (the bottom of the cross) (see \nfigure 1).  The graph in Fig.3 shows th e change of the \namplitude of the transmitted signal as a function of the \nstrength of the external magnetic field directed perpendicular \nto the propagating spin waves as depicted in the inset to Fig.3. \nHereafter, we show the relative change of the amplitude  in \ndecibels normalized to some value (e.g.  to the maximum \nvalue). The normalization is needed as the input power varies \nsignificantly for permalloy and YIG structures as well as for \nthe type of experiment. The reference transmission level is \ntaken at 300  Oe, where the S 12 parameter is at its absolute \nmaximum.  At small magnetic fields below 100 Oe a very \nsmall amplitude signal was observed.  At approximately 150 \nOe there is a noticeable increase in the amplitude followed by \na plateau in the response as th e field is increased to 500 Oe.  \nAlso of interest is the response of the signal as a function of \nthe applied magnetic field direction. In Fig.3(D), we present  \nan example of the experimental data  showing the influence of \nthe direction of the bias field on  spin waves transport from \nport 2 to port 4. The results demonstrate prominent change in \nthe amplitude of the transmitted signal [18dB] when the field \nis applied between 20° and 30°.  The main observations of \nthese experiments are the following. (i) Spin w ave propagation \nthrough the cross junction can be efficiently controlled by the external magnetic field. (ii) Both the amplitude and the \ndirection of the magnetic field can be utilized for spin wave \ncontrol.     \nWe conducted similar experiments on the YIG single cross \ndevice as shown in Figure 4.  It was observed that prominent \nsignal modulation could be determined by the direction and \nthe strength of the external magnetic field. In Fig.4, there is \nshown an example of experimental data on the spin wave \ntransport between ports 2 and 1. The maximum transmission \nbetween the orthogonal arms occurs when the field is applied \nat 68°, while the minimum is seen when the field is applied at \n0°. The On/Off ratio for the YIG cross reaches 35dB. Of \nnoticeable interest is  also the effect of non -reciprocal spin \nwave propagation. The two curves in Figure 4(D) show signal \npropagation from port 2 to port 4, and in the opposite direction \nfrom port 4 to port 2. The measurements are done at the same \nbias magnetic field of 998 Oe.  There is a difference of about \n5dB for the signals propagating in the opposite direction. The \neffect is observed in a relatively narrow frequency range (e.g. \nfrom 5.2GHz to 5.4GHz). The effect of non -reciprocal spin \nwave propagation may be of some practic al interest for \nbuilding magnonic diodes, though a more detailed study is \nrequired.  \nConcluding on the spin wave transport in the permalloy and \nYIG single cross structures, prominent signal modulation has \nbeen observed in both cases. For the chosen paramete rs, the \noperation frequency is slightly higher for YIG structure \n(~5GHz) than for permalloy  (~3GHz). The speed of signal \npropagation is slightly faster in permalloy (3.5×106 cm/s) than \nin YIG (3.0×106 cm/s).  The difference in the spin wave \ntransport can be attributed to the differences between the \nintrinsic material properties of YIG and Permalloy as well as \nthe difference in the cross dimensions. It is important to note, \nthat in both cases the level of the power consumption was at \nthe microwatt scale (e. g.  0.1µW -1.0µW for permalloy and \n0.5µW -5.0µW for YIG) with no feasible effect of micro \nheating on the spin wave transport. The summary of the \nexperimental findings for permalloy and YIG single cross \njunctions cab be found in Table I.  \nNext, we carried out  experiments on spin wave transport \nand interference in the double -cross structure made of YIG as \nshown in Figure 5. The choice of material is mainly due to the \nlarger size of the structure and spin wave detecting antennas, \nwhere the larger the area of the  detecting contour results in \nhigher the observed output inductive voltage.  The multi -port \ndouble -cross YIG structure is suitable for the study of spin \nwave interference.  In this study, several coherent spin wave \nsignals were excited by ports 2,3,4 and 5  connected to one port \nof the VNA. The output is detected at port 6.  The phase \nshifters were employed to vary the phase difference between \nthe ports as shown in Figure 5(B).   Figure 6 show the \nexperimental data on the output voltage collected in the \nfrequency range from 5.3GHz to 5.5GHz.  The curves of the \ndifferent color correspond to the different phase shifts between \nthe spin wave generated ports.  Phase 1 represents a change in \nthe phase of ports 4 and 6 and Phase 2 represents a change in \nthe phase of  ports 3 and 5.  Figures 6(B -D) show the slices of  \n5 \n data taken at a frequency of 5.385GHz, 5.410GHz and \n5.45GHz, respectively. The black markers depict the \nexperimentally obtained data, and the red markers depict the \ntheoretical output for the ideal case of  the interfering waves of \nthe same frequency and amplitude.  The theoretical data is \nnormalized to have the same maximum value as the \nexperimental data at phase difference zero (constructive \ninterference). Taking l=1.1mm, d=360μm, H=1000 Oe, \n4πM s=1750G, and H=20Oe, we estimated possible phase \nshift by Eqs. (1 -2) to be about π/2, which is in good agreement \nwith the experimental data.  This fact implies the dominant \nrole of wave interference in the output signal formation.  \nDiscrepancies in the amplitude can be attributed to parasitic \nnoise which raises the base amplitude of the signal to greater \nthan nonzero value even when the phase should be perfectly \ndestructive.  Also, it should be noted that Eqs. (1 -2) are \nderived for sp in wave propagating in a homogeneous magnetic \nfield, while the magnetic field produced by the micromagnets \nin the experiment may be inhomogeneous across the thickness \nand in lateral dimensions.  \nThe data presented in Figure 7 are collected in the \nexperimen ts where Phase 2 (ports 3 and 5) was changed, while \nPhase 1 (ports 2 and 4) was kept constant. The ability to \nindependently change the initial phases of the spin waves is \nequivalent to changing the angle of illumination for building a \nholograms as illustra ted in Fig.1(B).  In Figure 7, we present \nexperimental data showing the holographic image of the \ndouble -cross structure without memory elements.  The surface \nis a computer reconstructed 3 -D plot showing the output \nvoltage as a function of Phase 1 and Phase 2. The excitation \nfrequency is 5.40 GHz, the bias magnetic field is 1000 Oe \ndirected from port 1 toward port 6.  In this case, antennas on \nports 2 and 4 generated spin waves with the initial Phase 1, \nand antennas on ports 3 and 5 generated spin waves with \ninitial Phase 2.  No signal is applied to port 1. The output is \ndetected at port 6. The change of the output inductive voltage \nis a result of spin wave interference. It has maximum values in \nthe case of the constructive interference (i.e. Phase 1=Phase 2, \n(0,0) or (π,π)), and shows minimum output signal when the \nwaves are coming out -of-phase ((0,π) or (π,0)).   \nFinally, we conducted experiments to demonstrate the \noperation of a prototype 2 -bit magnonic holographic memory \ndevice. Two micro -magnets made of co balt magnetic film \nwere placed on  top of the junctions of the double -cross YIG \nstructure.  As mentioned in Section II, these magnets serve as \na memory element, where the magnetic state represents logic \nzeroes and ones. The schematic of the double -cross st ructure \nwith micro -magnets attached are shown in Fi gure 8(A). The \nlength of each magnet is 1.1mm, the width is 360μm and each \nhas a coercivity of 200 -500 Oersted (Oe).  For the test \nexperiments, we used four mutual orientations of micro -\nmagnets, where the magnets are oriented parallel to the axis \nconnectin g ports 1 -6, or the axis connecting 2 -4; and two cases \nwhen the micro -magnets are oriented in the orthogonal \ndirections. Holographic images were collected for each case. \nFig.8 shows the collection of data corresponding to output \nvoltage obtained for differ ent magnetic configurations. The phases of the input elements are the same as in the previous \nexperiment. Markers of different shape and color in the legend \nof figure 8 represent the direction of the “north” end of the \nmicro magnet. The output from the sam e structure varies \nsignificantly for different phase combinations. In some cases, \nthe magnetic states of the magnets can be recognized by just \none measurement (e.g. (0,0) phase combination). It is also \npossible that different magnetic states provide almost  the same \noutput (e.g. parallel and orthogonal magnet configurations \nmeasured at (π,0) phase combination). The main observation \nwe want to emphasize is the feasibility of parallel read -out and \nreconstruction of the magnetic state via spin wave \ninterference .  As one can see from the data in Figure 8(B), it is \npossible to distinctly identify the magnetic states by changing \nthe phases of the interfering waves, which is similar to \nchanging the angle of observation in a conventional optical \nhologram. We would li ke to emphasize that all experiments \nreported in this Section are done at room temperature.  \nIV. DISCUSSION  \nThe obtained experimental data show the practical \nfeasibility of utilizing spin waves for building magnonic \nholographic logic devices and helps to illust rate the \nadvantages and shortcomings of the spin wave approach. Of \nthese results there are several important observations we wish \nto highlight.  \nFirst, spin wave interference patterns produced by multiple \ninterfering waves are recognized for a relatively lo ng distance \n(more than 3 millimeters between the excitation and detection \nports) at room temperature. Despite the initial skepticism [29], \ncoding information into the phase of the spin waves appears to \nbe a robust instrument for information transfer showing a \nnegligible effect to thermal noise  and immunity  to the \nstructures imperfections. This immunity to the thermal \nfluctuations  can be explain ed by taking into account that the \nflicker noise level in ferrite structures usually does not exceed \n-130 dBm [30]. At the same time, spin waves are not sensitive \nto the structure’s imperfections which have dimensions much \nshorter than the wavelength. These facts explain the good \nagreement between the experimental and theo retical data (e.g. \nas shown in Figure 6).   \nSecond, spin wave transport in the magnetic cross junctions \nis efficiently modulated by an external magnetic field. Spin \nwave propagation through the cross junction depends on the \namplitude  as well as the directi on of the external field. This \nprovides a variety of possibilities for building magnetic field -\neffect logic devices for general and special task data \nprocessing. Boolean logic gates such as AND, OR, NOT can \nbe realized in a single cross structure, where an  applying \nexternal field exceeding some threshold stops/allows spin \nwave propagation between the selected arms. The ability to \nmodulate spin wave propagation by the direction of the \nmagnetic field is useful for application in non -Boolean logic \ndevices.  It  is important to note that in all cases the magnitude \nof the modulating magnetic field is of the order of hundreds of \nOersteds, which can be produced by micro - and nano -\nmagnets.   \n6 \n Finally, it appears possible to recognize the magnetic state \nof the magnet pla ced on the top of the cross junction via spin \nwaves, which introduces an alternative mechanism for \nmagnetic memory read out.  This property itself may be \nutilized for improving the performance of conventional \nmagnetic memory devices. However, the fundament al \nadvantage of the magnonic holographic memory is the ability \nto read -out a number of magnetic bits in parallel though the \nobtained experimental data demonstrates the parallel read -out \nof just two magnetic bits. In the rest of this Section, we \ndiscuss the  fundamental limits and the technological \nchallenges of building multi -bit magnonic holographic devices \nand present the estimates on the device performance.  \nWe start the discussion with the choice of magnetic material \nfor building spin waveguides. Spin wa ve transport in \nnanometer scale magnetic waveguides has been intensively \nstudied during the past decade [28, 31-33].  There are two \nmaterials that have become predominant, permalloy (Ni 81Fe19) \nand YIG, for spin wave devices prototyping.  The coherence \nlength of spin waves in permalloy is about tens of microns at \nroom temperature [28, 31], while the coherence length in a \nnon-conducting YIG exceeds millimeters [34].  The attenuation \ntime for spin waves at room temperature is about a \nnanosecond in permalloy and a hundreds of nanoseconds in \nYIG[34]. However, the fabrication of YIG waveguides require \na special  gadolinium gallium garnet (GGG) substrate. In \ncontrast, a permalloy film can be deposited onto a silicon \nplatform by using the sputtering technique. Though YIG has \nbetter properties in terms of the coherence length and a lower \nattenuation, permalloy is mo re convenient for making \nmagnonic devices on a silicon substrate.      \nThere are two major physical mechanisms affecting the \namplitude/phase of the spin wave propagating under the \njunction magnet: (i) interaction with magnetic field produced \nby the magnet,  and (ii) damping due to the presence of the \nconducting material. The effect of conducting films on spin \nwave propagation has been studied for MSSWs in the ferrite -\nmetal structures [35, 36].  It was found that the strength of the \nspin wave dispersion modification is defined by the critical \nparameter G given as following:  \n                                                                                                                                                                    \n(3) \n \nwhere t is the thickness of the conducting film, q is the \nwave number, and lsk is the skin depth . The presence of a \nmetallic film results in a prominent spin wave dispersion \nmodi fication for G>3, if the is width of the gap between the \nferrite and metallic film is less than the wavelength. Spin wave \nis completely damped in the range 1/3<G<3 due to the \nexcessive absorption by the conducting electrons. The effect \nvanishes for G<1/3. In our exper iments, we used junction \nmagnets made of cobalt with the thickness of 50nm (bulk \nelectric conductivity σ 1.6107 S/m), which correspond to lsk \n1.5 μm. The range of wave numbers q is restricted by the size \nof the sample L q >π/L>10 cm-1 , and by the width o f the \nmicro -antennas W 30μm, q <π/W<1000 cm-1.   In all \nexperiments, the range of the wave numbers is confined within the following range: 10 cm-1<q<1000 cm-1, which, in turn, \ncorresponds to the range for parameter G: 0.002< G<2.  It \nshould be noted that Eq. 3 has been derived for an infinite \nferrite waveguide, which ignores the effect of space \nconfinement. Taking into account the real dimensions of the \nYIG substrate 3.6μm, we obtain the minimum boundary for  \nq≈80 cm-1. Thus, we have G<1/3 (negligible effect of  spin \nwave dispersion modification due to the losses) for all \nexperiments. In general, there may be other physical \nphenomena contributing to the spin wave dispersion \nmodification in a magnetic cross -structure. The development \nof MHM devices will require a great deal of efforts in the \ntheoretical study and numerical modeling of spin wave \ntransport in magnetic nanostructures.  \nIn order to make a multi -bit magnonic holographic devices, \nthe operating wavelength should be scaled  down below \n100nm [23]. The main challenge with shortening the operating \nwavelength is associated with the building of nanometer -scale \nspin wave generating/detecting elements.  There are several \npossible ways of building input/output elements by using \nmicro -antennas [28], spin torque oscillators [26], and multi -\nferroic elements [6]. So far, micro -antennas are the most \nconvenient and widely used tool for spin wave excitation and \ndetection in ferromagnetic films [31]. Reducing the size of the \nantenna will lead to the reduction of the detected inductive \nvoltage. This fact limits the practical application of  any types \nof conducting contours for spin wave detection. The utilization \nof spin torque oscillators makes it possible to scale down the \nsize of the elementary input/output port to several nanometers \n[37].   The main challenge for the spin torque o scillators \napproach is to reduce the current required for spin wave \ngeneration. More energetically efficient are the two -phase \ncomposite multiferroics comprising piezoelectric and \nmagnetostrictive materials [38]. An electric field applied \nacross the piezoelectric produces stress, which, in turn, affects \nthe magnetization of the magnetoelastic mater ial. The \nadvantage of the multiferroic approach is that the magnetic \nfield required for spin wave excitation is produced via \nmagneto -electric coupling by applying an alternating electric \nfield rather than an electric current. For example, in Ni/PMN -\nPT synt hetic multiferroic reported in Ref. [39], an electric \nfield of 0.6MV/m has to be applied acro ss the PMN -PT in \norder to produce 90 degree magnetization rotation in Nickel. \nSuch a relatively low electric field required for magnetization \nrotation translates in ultra -low power consumption for spin \nwave excitation [20]. At the same time, the dynamics of the \nsynthetic multiferroics, especially at the nanometer scale, \nremains mostly unexplored.  \nTo benchmark the performance of the magnonic \nholographic devices, we apply the charge -resistance approach \nas developed in R ef. [40] The details of the estimates and the \nkey assumpt ions are given in Appendix A.  According to the \nestimates, MHM device consisting of 32 inputs, with a 60nm \nseparation distance between the inputs would consume as low \nas 150µW of power or 72fJ per computation.  At the same \ntime, the functional throughput o f the MHM scales \nproportional to the number of cells per area/volume and \n2\nsklqtG \n7 \n exceeds 1.5×1018 bits/cm2/s for a 60nm feature size. It is \ninteresting to note, that holographic logic units can be used for \nsolving certain nondeterministic polynomial time (NP) clas s of \nproblems (i.e. finding the period of the given function). The \nefficiency of holographic computing with classical waves is \nsomewhere intermediate between digital logic and quantum \ncomputing, allowing us to solve a certain class of problems \nfundamentall y more efficient than general -type processors but \nwithout the need for quantum entanglement [41].  Image \nrecognition and processing are among the most promising \napplications of magnonic holographic device exploiting its \nability to process a large number of bits/pixels in parallel \nwithin a single core.  \nTher e are many questions on spin wave transport (e.g. in \nmagnetic crosses), which remain mostly unexplored. For \ninstance, it is not clear the mechanism responsible for spin \nwave splitting between the orthogonal arms. To the best of our \nknowledge, there is no t heoretical work explaining the \nobserved spin wave propagation in magnetic crosses. It would \nbe of great interest to study the dynamics of spin wave \nredirection depending on the geometry of the cross. It may be \nexpected that the amplitude/phase of the redir ected (bended) \nspin wave depends on the wavelength/size ratio, the material \nproperties of the cross, and magnetic field produced by the \nnano -magnet. Also, in this work, we attribute the change in the \ninterference pattern to the different phase shifts accum ulated \nby spin waves propagating under the nano -magnets of \ndifferent orientation (i.e. Eqs. 1 -2).  A real picture may be \nmuch more complicated due to the difference in amplitudes, \nwhich may arise to the different factors.  There is no doubt \nthat the develo pment of magnonic holographic memory \ndevices will take a great deal of efforts including experimental \nas well as theoretical studies.   \nV. CONCLUSIONS  \nThe collected experimental data show rich physical \nphenomena associated with spin wave propagation in single - \nand double -cross structures.  Prominent signal modulation by \nthe direction, rather than the amplitude of magnetic field and \nthe low effect of thermal noise on spin wave propagation at \nroom temperature are among the many interesting findings \npresented her e.  The effect of spin wave redirection between \nthe cross arms by the external magnetic field may be further \nexploited for building a variety of logic devices.  Besides, spin \nwaves appear to be a robust instrument allowing us to sense \nthe magnetic state of  micro -magnets by the change in the \ninterference pattern.  Quite surprisingly, it is possible to \nrecognize the unique holographic output for the different \norientations of micro -magnets in a relatively long device at \nroom temperature. Overall, the obtained data demonstrates the \npractical feasibility of building magnonic holographic devices.  \nThese holographic devices are aimed not to replace but to \ncomplement CMOS in special type data processing such as \nspeech recognition and image processing.  According to \nestimates, scaled magnonic holographic devices may provide \nmore than 1×1018 bits/cm2/s data processing rate while \nconsuming less than 0.2mW of energy.  The main \ntechnological challenges are associated with the scaling down the operating wavelength and buil ding nanometer scale spin \nwave generating/detecting elements with spin torque \noscillators and multiferroics being among the most promising \nsolutions. At the same time, it is expected that reducing the \noperating wavelength will make spin waves more sensitiv e to \nstructure imperfections. The development of scalable \nmagnonic holographic devices and their incorporation with \nconventional electronic devices may pave the road to the next \ngenerations of logic devices with functional capabilities far \nbeyond current C MOS.  \nAPPENDIX  \nThis section contains the details on the power consumption \nestimates. We assume that the input to each magnetoelectric \n(ME) cell is an AC voltage with amplitude \ninV  created in a \nRLC oscillator. Then the power dissipation on  resonance is  \nRVPin\nin22\n\n. (1) \nThe amplitude of the electric field in the piezoelectric of \nthickness \npzt  is \npz in in tV E /\n. (2) \nThe amplitude of strain created in the piezoelectr ic of the ME \ncell is  \nin xx Ed31\n, (3) \nwhere the piezoelectric coefficient is \n31d . Hereafter, it is \nassumed that spin wave is propagating along the X axis as \nshown in the inset to Fig2. The stress transferre d to the \nferromagnet is  \nxxY\n, (4) \nwhere the  Young’s modulus of the piezoelectric is \nY . The \nchange in the magnetic anisotropy due to magnetostriction is  \n23msU\n, (5) \nwhere the magnetostriction coefficient of the ferromagnet is \n\n. Then the maximum amplitude of magnetization change is  \nsms\nxMUM\n02\n\n, (6) \nwhere the permeability of vacuum is , and the saturation \nmagne tization is \nsM . This can be expressed via the \nmagnetoelectric coefficient  \ns inx\nMYd\nEM31 0 3  \n. (7) \nThen the generated dimensionless amplitude of the spin wave \ncan be approximated as follows  \npzsin\nsx\nintMV\nMMA\n0\n, (8) \nThe spin waves interact and attenuate as they propagate. If the \ndistance between inputs is \nL , the number of inputs is \nN , and \nthe attenuation length is \natl , then the propagation distance is   \n8 \n \nNL Ltot (9) \nand let the minimum amplitude needed to be detected is a \nquarter of the average output amplitude  \n\n\n\n\nattot in\nlL AA exp4min\n. (10) \nIf the group velocity of the spin waves is \nsw totcL/ , then \nthe time needed for one holographic imaging is  \nswcNL/\n. (11) \nAssuming that the detection occurs by the inverse of the ME \neffect, and that its coefficients are the same as for the direct \nME effect, we obtain that the connection between the \nmagnetization amplitude and the amplitude of the gene rated \nelectric field  \nMEpz\nmin0\n. (12) \nThus the minimum output voltage is  \n0min\nmin min\npzpz s\npztAMtE V \n. (13) \nCombining expressions (10) and (13), we have the following \nfor voltages  \n002\nmin exp4 \npz atin\nlNL VV\n\n\n\n. (14) \nThen the total driving power for N inputs is  \n4422\nmin22exp8\n2 \nc lNL\nRNV\nRNVPpz\natin\ntot \n\n\n\n, (15) \nwhere \nc is the speed of light.  For minimum detectability, the \noutput voltage should exceed the Johnson noise voltage by \n5X. The spectral density of noise is  \nTRk VB n42\n. (16) \nThe required power within the bandwidth \nB  (approximately \nequal to the ac voltage frequency) is  \n4422exp 800\nc lNLT NBk Ppz\natB tot \n\n\n\n. (17) \nAnd the total energy for one imaging is  \ntot totP E\n. (18) \nUsing the magnetostriction parameters for the most \nadva ntageous case of Terfenol -D and PMN -PT, the \nmagnetoelectric coefficient  \nc mns /17 / 57\n. 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Wu, A. Bur, P. Zhao, K. P. Mohanchandra, K. Wong, K . L. \nWang, C. S. Lynch, and G. P. Carman, \"Giant electric -field-\ninduced reversible and permanent magnetization reorientation on \nmagnetoelectric Ni/(011) [Pb(Mg1/3Nb2/3)O3](1−x) –[PbTiO3]x \nheterostructure,\" Applied Physics Letters, vol. 98, pp. 012504 -7, \n2011. \n[40] A. Sarkar, D. E. Nikonov, I. A. Young, B. Behin -Aein, and S. \nDatta, \"Charge -resistance approach to benchmarking performance \nof beyond -CMOS information processing devices,\" Ieee \nTransactions on Nanotechnology, vol. 13, pp. 143 -50, Jan. 2014.  \n[41] L. G. Valiant, \"Holographic algorithms,\" Proceedings. 45th \nAnnual IEEE Symposium on Foundations of Computer Science, pp. \n306-15, 2004 2004.  \n \n Figure Captions  \n \nFigure 1. (A) Schematics of Magnonic Holographic Memory \nconsisting of a 4×4 magnetic matrix and an  array of spin wave \ngenerating/detecting elements.  For simplicity, the matrix is \ndepicted as a two -dimensional grid of magnetic wires with just \n4 elements on each side. These wires serve as a media for spin \nwave propagation. The nano -magnet on the top of the junction \nis a memory element, where information is encoded into the \nmagnetization state. The spins of the nano -magnet are coupled \nto the spins of the magnetic wires via the dipole -dipole or \nexchange interaction. (B) Illustration of the principle of \noperation. Spin waves are excited by the elements on one or \nseveral sides of the matrix (e.g. left side), propagate through \nthe matrix and detected on the other side (e.g. right side) of the \nstructure. All input waves are of the same amplitude and \nfrequency. The initial phases of the input waves are controlled \nby the generating elements. The output waves are the results \nof the spin wave interference within the matrix.  The \namplitude of the output wave depends on the initial phases and \nthe magnetic states of th e junctions.  \nFigure 2. Schematics of the experimental setup for single cross \nstructures testing.  The input and the output micro -antennas \nare connected to the Hewlett -Packard 8720A Vector Network \nAnalyzer (VNA). The VNA generates input RF signal and \nmeasur es the S parameters showing the amplitude and the \nphases of the transmitted and reflected signals. The device \nunder study is placed inside a GMW 3472 -70 Electromagnet \nsystem which allows the biasing magnetic field to be varied \nfrom -1000 Oe to +1000 Oe.  T he in -plane axes X and Y are \ndefined along the lines from port 1 to port 3, and from port 4 \nto port 2, respectively.  \nFigure 3. (A) Microscope image of Permalloy single cross \nstructure with antennas placed on the edges of the structure.  \nThe length of the w hole structure is 18um; the width of the \narm in 6µm; thickness is 40 nm. (B) Photo of the packaged \ndevice with microwave input/output ports used for connection \nto VNA.  (C) Experimental data showing the relative change \nof the output signal (inductive volta ge) as a function of the \nstrength of the external magnetic field. The output is \nnormalized to the maximum output detected at 300 Oe.  The \nsignal is transmitted from port 2 to port 4, the bias magnetic \nfield is along the X axis as depicted in the inset. The  input \nfrequency is 3.16GHz. (D)  Experimental data showing the \nrelative change of the output signal amplitude as a function of  \n10 \n the direction of the external magnetic field.  The output is \nnormalized to the maximum amplitude at zero degrees \n(parallel to X axis). The signal is transmitted from port 2 to \nport 4. The measurements are taken at the different angles α of \nthe bias magnetic field of 148Oe, where α is defined as the \nangle to the X axis as depicted in the inset.   \nFigure 4.  Microscope image of YIG s ingle cross structure. \nThe length of the whole structure is 3mm; the width of the arm \nin 360µm; thickness is 3.6um.  (B) Experimental data showing \nthe relative change of the output signal amplitude as a function \nof the direction of the external magnetic fie ld.  The signal is \ntransmitted from port 2 to port 1. (C) Experimental data on the \nsignal transmission from port 4 to port 2 (black curve) and \nfrom port 2 to port 4 (red curve).  The data show about 5dB \ndifference for the signal propagating in the opposite directions \nin the frequency range from 5.2GHz to 5.5GHz.   \n \n \nFigure 5. (A) Microscope image of YIG double cross \nstructure. The length of the whole structure is 3mm; the width \nof the arm in 360µm; thickness is 3.6um. (B) Schematics of \nthe double -cross device with six micro -antennas fabricated on \nthe edges  under study.  (C) Schematics of the experimental \nsetup. The input and the output micro -antennas are connected \nto the Hewlett -Packard 8720A Vector Network Analyzer \n(VNA).  There is asset of splitters (depicted as S), attenuators \n(depicted as A), and phase shifters  (depicted as Ph) used for \nthe connections with VNA. The device under study is placed \ninside a GMW 3472 -70 Electromagnet  system.  \nFigure 6. Experimental data on the output voltage at port 1 as \na result of spin wave interference. The data are collected in the \nfrequency range from 5.3GHz to 5.5GHz.  The bias magnetic \nfield is 1000 Oe directed from port 1 toward port 6.  The \ncurves of the different color correspond to the different phase \nshifts between the spin wave generated ports.  Phase 1 \nrepresents a change in the phase of ports 4 and 6 and Phase 2 \nrepresents a change in the phase of ports 3 and 5. (B -D) Slices \nof data taken  at the frequencies of 5.385GHz, 5.410GHz and \n5.45GHz, respectively. The black markers depict the \nexperimentally obtained data, and the red markers depict the \ntheoretical data for the ideal case of the interfering waves of \nthe same frequency and amplitude.   The theoretical data is \nnormalized to have the same maximum value as the \nexperimental data at phase difference zero (constructive \ninterference).  \nFigure 7.  Holographic image of the double -cross structure \nwithout memory elements. The cyan surface is a com puter \nreconstructed 3 -D plot based on the experimental data: output \nvoltage as a function of the phases of the interfering spin \nwaves. The output is detected at port 6. The excitation \nfrequency is 5.40 GHz, the bias magnetic field is 1000 Oe \ndirected from port 1 toward port 6.  No signal is applied to \nport 1. The legend and schematics on the right side explain the \nphases of the spin waves generated at the six ports. Phase 1 \nand Phase 2 correspond to the phases generated at the ports \n2,4 and 3,5, respectivel y.    \nFigure 8.  Collection of experimental data showing the output \nof the  double -cross structure with micro -magnets placed on \nthe top of the junctions. The phase coordinates show the \ncombination of the initial phases of the spin waves, where Phase 1 and P hase 2 are defined the same way as in Fig.7 (i.e. \n0,)means that the spin waves generated in the ports 2,4 and \n3,5 have a - difference in the initial phase.  The markers of \ndifferent shape and color correspond to the different magnetic \nconfigurations as il lustrated on the right side.  All results are \nobtained at room temperature.  \n   \n11 \n   \nFigure 1   \n12 \n \nFigure 2  \n13 \n \nFigure 3  \n14 \n   \nFigure 4  \n15 \n   \n Permalloy  YIG \nCross \ndimensions  L=18µm, \nw=6µm, \nd=100nm  L=3mm, \nw=300µm, \nd=3.8µm  \nOperational \nFrequency  3GHz -4GHz  5GHz -6GHz  \nSW group \nvelocity  3.5×106 \ncm/s  3.0×106 \ncm/s  \nMaximum  \nOn/Off ratio  20dB  35dB  \nPower \nconsumption  0.1µW -1µW  0.5µW -5µW  \nCompatibility \nwith Silicon  Yes No Table I. Summary on the experimental d ata collected \nfor permalloy and YIG single cross structures   \n16 \n   \nFigure 5  \n17 \n   \nFigure 6  \n18 \n   \nFigure 7  \n19 \n  \n \nFigure 8 "    },    {        "title": "1612.07239v1.Pure_spin_current_transport_in_gallium_doped_zinc_oxide.pdf",        "content": "Pure spin current transport in gallium doped zinc oxide\nMatthias Althammer,1, 2, a)Joynarayan Mukherjee,3Stephan Gepr ¨ags,1Sebastian T. B. Goennenwein,1\nMatthias Opel,1M.S. Ramachandra Rao,3and Rudolf Gross1, 2, 4\n1)Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching,\nGermany\n2)Physik-Department, Technische Universit ¨at M¨unchen, 85748 Garching, Germany\n3)Department of Physics, Nano Functional Materials Technology Centre and Materials Science Research Centre,\nIndian Institute of Technology Madras, Chennai, Tamil Nadu 600036, India\n4)Nanosystems Initiative Munich (NIM), 80799 M ¨unchen, Germany\n(Dated: 12 August 2018)\nWe study the flow of a pure spin current through zinc oxide by measuring the spin Hall magnetoresistance (SMR) in\nthin film trilayer samples consisting of bismuth-substituted yttrium iron garnet (Bi:YIG), gallium-doped zinc oxide\n(Ga:ZnO), and platinum. We investigate the dependence of the SMR magnitude on the thickness of the Ga:ZnO\ninterlayer and compare to a Bi:YIG/Pt bilayer. We find that the SMR magnitude is reduced by almost one order\nof magnitude upon inserting a Ga:ZnO interlayer, and continuously decreases with increasing interlayer thickness.\nNevertheless, the SMR stays finite even for a 12 nm thick Ga:ZnO interlayer. These results show that a pure spin current\nindeed can propagate through a several nm-thick degenerately doped zinc oxide layer. We also observe differences in\nboth the temperature and the field dependence of the SMR when comparing tri- and bilayers. Finally, we compare our\ndata to predictions of a model based on spin diffusion. This shows that interface resistances play a crucial role for the\nSMR magnitude in these trilayer structures.\nThe generation and detection of pure spin currents, i.e. of\nnet flows of angular (spin) momentum without accompany-\ning charge currents, has been intensively studied in theory\nand experiments. Prominent spin current based phenomena\nare spin pumping1,2, the spin Seebeck effect3–5and the spin\nHall magnetoresistance6–10(SMR). The SMR manifests itself\nin ferromagnetic insulator (FMI) / normal metal (NM) bilayer\nsamples, as a dependence of the electric resistance of the NM\non the orientation of the magnetization in the FMI. The mag-\nnitude of the SMR depends on the size of the spin Hall an-\ngle in the NM. In a FMI/NM bilayer system, the longitudinal\n(rlong) and transverse resistivity ( rtrans) of the NM can be writ-\nten as6,8,9,11\nrlong =r0+r1(mt)2; (1)\nrtrans=r2mn\u0000r1(mjmt)2; (2)\nwhere mj,mt, and mnare the projections of the net magnetiza-\ntion unit vector m=M=Mdetermined by the magnetization\norientation in the FMI on the directions jandtparallel and\nperpendicular to the current direction, respectively, and the\nfilm normal n(cf. Fig. 1). The resistivity parameters ride-\npend on the material parameters of the hybrid structure. A\ndetailed analysis allows to extract the spin Hall angle qSH;NM\nand the spin diffusion length lsf;NMof the NM, if the spin mix-\ning conductance g\"#at the FMI/NM interface is known.9,12\nUp to now, most SMR studies were based on FMI/NM bi-\nlayer structures. Only in a few experiments an additional con-\nductive interlayer was inserted at the FMI/NM interface to rule\nout the contribution of a proximity-polarized NM layer.9,12–14\nHowever, trilayer structures also allow to study the transport\nof pure spin currents in interlayer materials with a negligible\na)Electronic mail: matthias.althammer@wmi.badw.despin Hall angle. In particular, the effect of interlayer resistiv-\nity and spin diffusion length onto the SMR has not yet been\ninvestigated. Moreover, a quantitative comparison between a\nspin diffusion theory model and experiment is still missing\nfor such trilayer systems. In this letter, we present SMR ex-\nperiments conducted on bismuth-substituted yttrium iron gar-\nnet (Bi 0:3Y2:7Fe5O12, Bi:YIG) / platinum (Pt) bilayers and\nBi:YIG / gallium doped zinc oxide (Ga 0:01Zn0:99O, Ga:ZnO)\n/ Pt trilayers. Our results demonstrate pure spin current trans-\nport across a degenerately doped, several nm-thick ZnO inter-\nlayer, resulting in a sizeable SMR effect. In addition, we ex-\nploit the tunability of the resistivity and spin diffusion length\nin Ga:ZnO with temperature to quantitatively compare our ex-\nperimental data with the predictions of a model based on spin\ndiffusion. This comparison suggests that interface resistances\nand their possible spin selectivity may play a crucial role for\nthe SMR in these trilayer structures.\nThe samples have been grown in-situ , without breaking the\nvacuum, on (111)-oriented yttrium aluminium garnet (YAG)\nsubstrates in an ultra high vacuum deposition system. Bi:YIG\nand Ga:ZnO layers were deposited via laser-molecular beam\nepitaxy (laser-MBE) from stoichiometric targets in an oxy-\ngen atmosphere. For Bi:YIG we applied an energy density at\nthe target of ED L=1:5 J=cm2, an oxygen partial pressure of\npO2=25mbar, and a substrate temperature of Tsub=450\u000eC\n(see Ref. 9 for more details). For Ga:ZnO we used: ED L=\n1 J=cm2,pO2=1mbar,Tsub=400\u000eC (See Ref. 15 for more\ndetails). Pt was deposited via electron-beam evaporation at\na base pressure of 1 \u000210\u00008mbar at room temperature. The\nin-situ deposition process ensures very clean interfaces.\nFrom the structural and magnetic characterization of a\nBi:YIG(54)/Pt(4) bilayer and a Bi:YIG(54)/Ga:ZnO(8)/Pt(4)\ntrilayer, where the numbers in parentheses give the film thick-\nnesses in nm, we conclude that the structural and the magnetic\nproperties of the bi- and trilayer sample are nearly identical\n(see supplemental materials). This suggests that the influencearXiv:1612.07239v1  [cond-mat.mes-hall]  21 Dec 20162\nof the Ga:ZnO deposition on the magnetic properties of the\nBi:YIG layer is negligible.\nFor electrical transport measurements, the samples have\nbeen patterned into Hall bar-shaped mesastructures (80 mm\nwide, 800 mm long) via photolithography and Ar-ion milling,\nand then mounted in a superconducting magnet ( m0H\u00147 T)\ncryostat (5 K\u0014T\u0014300 K). Resistivity data have been taken\nusing a DC current reversal technique. From the measured\nlongitudinal and transverse voltage we then calculated rlong\nandrtrans.9,16We performed angle-dependent magnetoresis-\ntance (ADMR) measurements, where the direction of the ap-\nplied magnetic field of constant magnitude is rotated within\nthree orthogonal planes as illustrated in the left column of\nFig. 1: in the film plane (ip), in the plane perpendicular to\nthej-direction (oopj), and in the plane perpendicular to the\nt-direction (oopt). Although the smallest field magnitude is\nbelow the full saturation field of Bi:YIG of 2 T, it is sufficient\nto investigate the main angular dependence.\nWe first discuss the results obtained from ADMR exper-\niments at T=10 K and m0H=1 T for our bilayer and\ntrilayer sample, as shown in Fig. 1. For the bilayer we\nmeasure the typical fingerprint expected for SMR9: For the\nip rotation plane we observe a sin2(a)-dependence of rlong\nand a sin (a)cos(a)-dependence of rtrans(see Fig. 1(a)), for\nthe oopj rotation plane we observe a sin2(b)-dependence of\nrlong(see Fig. 1(c)), while we do not observe any sizeable\nangular-dependence of rlongin the oopt rotation plane (see\nFig. 1(e)). Furthermore, rtransonly shows a very small cosine\ndependence in the oopj and oopt rotation planes, due to the\nnearly vanishing ordinary Hall coefficient (OHC) of Pt thin\nfilms at low temperatures.17We note that the remaining sin2-\ndependencies, for rlongin oopt and for rtransin oopj and oopt\nconfiguration, can be explained by a non-vanishing SMR con-\ntribution in these rotation planes due to a small tilting ( \u00143\u000e)\nof the actual rotation plane with respect to the surface normal\nbecause of experimental limitations in the sample mounting.\nFor the trilayer sample the angular dependence looks qualita-\ntively the same (Fig. 1(b,d,f)). As the observed ADMR data\nreflects the symmetry expected for SMR, we can safely as-\nsume the SMR as the only cause for the magnetoresistance in\nthe bilayer as well as for the trilayer.\nQuantitatively, however, there are differences. In the tri-\nlayer, rlongis about 2 times larger than in the bilayer, which\ncan be explained by the one order of magnitude larger resis-\ntivity of the ZnO layer as compared to the bare Pt layer and\nthe effective average of resistivity from Pt and Ga:ZnO for the\ntrilayer (see Fig. 3(b) for the extracted resisitivity of the ZnO\nlayer). For further quantitative comparison, we simulated the\nSMR response for rlongandrtrans(red lines in the graphs) by\nusing Eqs. (1) and (2), while assuming that the magnetization\norientation is always parallel to the external magnetic field\nand including the ordinary Hall effect as a contribution pa-\nrameterized by a field dependence of r2. From the simulation\nwe extract the SMR amplitude SMR =jr1=r0j. For the bi-\nlayer we obtain SMR =4:0\u000210\u00004, which agrees nicely with\nour previous results in YIG/Pt hybrids.9,18For the trilayer we\nfind SMR =2:2\u000210\u00005, which is about an order of magni-\ntude smaller. There are two obvious reasons for the decreasein SMR amplitude upon the insertion of a Ga:ZnO interlayer.\nFirst, in the trilayer the Ga:ZnO layer acts as a resistive shunt\nfor the Pt layer thereby reducing the SMR amplitude. Sec-\nond, part of the spin current generated in the Pt layer is lost\nwhen diffusing across the Ga:ZnO layer. In other words, the\nGa:ZnO only acts as a parallel and spin Hall-inactive resistor\nfor the SMR. Moreover, when comparing the transverse resis-\ntivity for both samples, the amplitude of the ADMR signal is\ndifferent in all rotation planes for the two different samples\n(please note that the same scale has been used for rtransin all\ngraphs of Fig. 1). For the ip rotation plane the bilayer has a\nmuch larger amplitude than for the trilayer. This is expected\ndue to the larger SMR in the bilayer, which is the only contri-\nbution to the ADMR for this rotation plane. For the oopj and\noopt rotation planes, however, the rtransamplitude is larger\nfor the trilayer. This can be traced back to the contribution of\nthe ordinary Hall effect to the ADMR data: for the bilayer\nthe OHC is rather small17(\u00190:1 mW=T), while for the tri-\nlayer it is an effective average of the Pt layer and the Ga:ZnO\ninterlayer, resulting in a larger OHC. From the field depen-\ndence of r2for the trilayer, we extract an OHC of 30 m W=T\nfor the Ga:ZnO, corresponding to a carrier concentration of\n2\u00021021cm\u00003. This value nicely agrees with control measure-\nments on blanket Ga:ZnO layers (see supplemental material).\nTo get a deeper insight into the SMR of the trilayer we\nmeasured the temperature and field dependence and com-\npare it to the results for the bilayer sample. To this end, we\nconducted ADMR measurements in the ip rotation plane for\n5\u0014T\u0014300 K and m0H=1;3;5;and 7 T and extracted the\nSMR by fitting the data using Eq.(1). The result is shown in\nFig. 2. We first discuss the temperature dependence for the bi-\nlayer sample in Fig. 2(a). The SMR shows a maximum around\nT=225 K and then decreases with decreasing temperature.\nThis observation nicely agrees with results on YIG/Pt hy-\nbrids18,19and can be attributed to the temperature dependence\nofqSH;Pt.18However, the temperature dependence is clearly\ndifferent for the trilayer sample (Fig. 2(b)). Here, the SMR\nis only weakly changing with temperature, displays an upturn\ntowards low temperatures and reaches its maximum value for\n5 K. We may explain this upturn by two contributions. On the\none hand, the spin diffusion length for Ga:ZnO increases with\ndecreasing temperature as it is dominated by the D’yakonov-\nPerel’ mechanism for spin dephasing.15,20This leads to an in-\ncrease of the SMR and a saturation at low temperatures. On\nthe other hand, due to the laser-MBE deposition of Ga:ZnO\nan intermixing of Bi:YIG and Ga:ZnO at the interface can\noccur, which may lead to the formation of isolated paramag-\nnetic moments at the interface between Ga:ZnO and Bi:YIG.\nThese isolated paramagnetic moments align parallel to the ex-\nternal magnetic field at low temperatures, which would also\nresult in an increase of SMR with decreasing temperatures.\nThe temperature dependent data of the bilayer also shows a\nweak plateau at low temperatures, which could be related to\nisolated paramagnetic moments. However, more detailed in-\nvestigations of the interface properties will be necessary in the\nfuture in order to really unravel the underlying physics.\nThe evolution of the magnetic field dependence of the\nSMR with temperature for the bilayer and the trilayer sam-3\n-90° 0° 90° 180°270°700.26700.27700.28\nαρlong(nΩm)\n-5051015\nρtrans(nΩcm)\n-90° 0° 90° 180°270°699.79699.80699.81\nβρlong(nΩm)\n-5051015\nρtrans(nΩcm)\n-90° 0° 90° 180°270°679.59679.60679.61\nγρlong(nΩm)\n-5051015\nρtrans(nΩcm)-90° 0° 90° 180°270°322.4322.5322.6322.7ρlong(nΩm)\nα-5051015\nρtrans(nΩcm)\n-90° 0° 90° 180°270°322.4322.5322.6322.7ρlong(nΩm)\nβ-5051015\nρtrans(nΩcm)\n-90° 0° 90° 180°270°322.4322.5322.6322.7ρlong(nΩm)\nγ-5051015\nρtrans(nΩcm)h||t\nγh\njtn\nhjtn\nα\nβh\njtnh||-t h||j h||t h||-t h||j\nh||t h||-t h||n h||t h||-t h||n\nh||-j h||j h||n h||-j h||j h||nip\noopj\nooptρlong\nρtrans\nρlong\nρtrans\nρlong\nρtransρlong\nρtrans\nρlong\nρtrans\nρlong\nρtrans1 T, 10 K\n1 T, 10 K\n1 T, 10 K1 T, 10 K1 T, 10 K1 T, 10 K(b) (a)\n(d) (c)\n(f)(e)\nFIG. 1. ADMR data of a Bi:YIG(54)/Pt(4) bilayer (panels (a), (c), and (e)) and a Bi:YIG(54)/Ga:ZnO(8)/Pt(4) trilayer (panels (b), (d), and\n(f)) sample grown on YAG (111) substrates. The data has been recorded at 10 K. The three orthogonal rotation planes for the magnetic field\nare sketched in the left column. In the plot, black and blue symbols represent the experimental data of the longitudinal rlongand transverse\nresistivity rtrans, respectively. The red lines are simulations using Eqs. (1),(2).\n0 2 4 6\n456789µ0H(T)SMR (x10-4)0 100200300\n46810T(K)SMR (x10-4)\n0 1002003000123SMR (x10-4)\nT(K)0 2 4 6012SMR (x10-4)\nµ0H(T)7 T\n5 T300 K\n3 T\n1 T7 T\n5 K\n300 K5 K(a) (c)\n(d) (b)5 T\n3 T\n1 T\nFIG. 2. Temperature dependence of the SMR signal extracted from\nADMR measurements for (a) a Bi:YIG(54)/Pt(4) bilayer and (b) a\nBi:YIG(54)/Ga:ZnO(8)/Pt(4) trilayer at magnetic fields of 1 ;3;5;7 T.\nMagnetic field dependence of SMR from ADMR measurements for\nthe bilayer (c) and the trilayer (d) at 5 K (black squares) and 300 K\n(red circles).\nple is also different. For the bilayer the relative increase\nof the SMR from 1 T to 7 T is 37% at 5 K and 27% at\n300 K(Fig. 2(c)). For the trilayer we obtain 600% at 5 K\nand 1500% at 300 K(Fig. 2(d)). The much stronger field de-\npendence observed for the trilayer sample again might be ex-\nplained by the presence of isolated paramagnetic moments at\nthe Bi:YIG/Ga:ZnO interface. From our bilayer data it is not\ndirectly possible to pin point the origin for the observed fielddependence. Hanle magnetoresistance as proposed by V ´elez\net al.21can be ruled out, as this should yield a quadratic field\ndependence of the SMR, while we observe a linear/square root\ndependence in our data. Clearly, more detailed studies are\nnecessary to explain the observed field dependence.\nTo analyze the pure spin current transport in Ga:ZnO we\nfirst extract the resistivity of Pt ( rPt(T)) from the rlong(T)\nof the bilayer and then determine the resistivity of Ga:ZnO\n(rZnO(T)) from the measured rlong(T)of the trilayer by as-\nsuming a parallel conductance model and the same rPt(T)as\nfor the bilayer (see Fig. 3(a) and (b)). For both materials we\nobserve a decrease in resistivity with decreasing temperature,\nwhile rZnOis about an order of magnitude larger than rPt.\nFrom this we conclude that the Ga:ZnO layer just behaves\nlike a dirty metal due to degenerate doping.\nIn a next step we model the dependence of the SMR on\nthe Ga:ZnO thickness ( dZnO) using the SMR theory approach8\nand adding the spin diffusion through the Ga:ZnO to the\ntheory model, while neglecting any interface resistance ef-\nfects, i.e. assuming continuity of the spin-dependent electro-\nchemical potential at the Pt/Ga:ZnO interface (see supplemen-\ntal materials). We note that ADMR measurements on bare\nBi:YIG/Ga:ZnO reference samples yield no SMR response\nwithin our experimental resolution (for a 8 nm thick Ga:ZnO\nlayer on Bi:YIG, we find SMR \u00148\u000210\u00006for 5 K\u0014T\u0014\n300 K and m0H\u00147 T), such that we assume qSH;ZnO=0.\nFrom this model, two parameters define the dZnOdependence\nof the SMR: rZnOand the spin diffusion length lsf;ZnOin the\nGa:ZnO layer. The resistivity rZnOhas a twofold implica-\ntion. On the one hand, it determines the amount of charge\ncurrent flowing through Pt and, hence, the amount of spin4\ncurrent generated via the spin Hall effect in Pt. On the other\nhand, it parameterizes the amount of spin current diffusing\nthrough Ga:ZnO due to the gradient in the spin-dependent\nelectrochemical potential. To test whether or not this sim-\nple model explains our experimental findings, we simulated\nSMR at 5 K and 300 K and compared it to our experimen-\ntal data as shown in Fig. 3(c) and (d). We also included\nADMR data obtained for Bi:YIG(54)/Ga:ZnO(12)/Pt(9) and\nBi:YIG(54)/Ga:ZnO(4)/Pt(7) trilayers, which showed a sim-\nilar surface roughness determined from x-ray reflectometry\nand thus similar interface properties. For better comparison,\nwe renormalized the resistivity data to a Pt thickness of 4 nm\nusing our previous results18. For the simulation we used rPt\nandrZnOfrom Fig. 3(a) and (b), while we chose temperature-\nindependent values of g\"#=1\u00021019m\u00002andlsf;Pt=1:5 nm\nfrom Refs. 9, 12, and 18. For qSH;Ptwe used 0 :11 at 300 K and\n0:07 at 5 K from Ref. 18 as well as 4 nm at 300 K and 12 nm at\n5 K for lsf;ZnOfrom Ref. 15. Our very simple model can only\nreproduce the general trend of the measurements (see Fig. 3(c)\nand (d)). The differences between simulation and data evident\nfrom Fig. 3 most likely originate from the crucial role of a spin\ndependent interface resistance15at the Pt/Ga:ZnO interface\nand a possible change of g\"#when going from a Bi:YIG/Pt\nto a Bi:YIG/Ga:ZnO interface. However, to extract these pa-\nrameters from our measurements, a more systematic study is\nnecessary, which goes beyond the scope of this paper.\nIn summary, we experimentally investigated the flow of a\npure spin current through a Ga:ZnO layer by utilizing the\nSMR in Bi:YIG/Ga:ZnO/Pt trilayer thin film samples. Our\nresults show the possibility to transfer a pure spin current\nthrough a degenerately doped ZnO interlayer. Our results also\nhighlight the importance of interface quality for the SMR. Fi-\nnally, using a spin diffusion model for the SMR response, we\nachieve reasonable agreement between simulation and exper-\n0 5 1010-210-1100101102SMR (x10-4)\ndZnO(nm)0 5 1010-1100101102SMR (x10-4)\ndZnO(nm)0100200300300350400ρPt(nΩm)\nT(K)0 100200300200030004000ρZnO(nΩm)\nT(K)(a)\n(c)(b)\n(d) T=300 K T=5 K\n7 T\n5 T\n3 T\n1 T7 T5 T3 T\n1 T\nFIG. 3. Extracted temperature dependence of the resistivity of Pt\n(a) from the bilayer and the resistivity of Ga:ZnO (b) extracted from\nthe trilayer ADMR data by using the resistivity of Pt extracted from\nthe bilayer. For Pt and ZnO the resistivity increases with increasing\ntemperature. SMR amplitude as a function of Ga:ZnO thickness at\n300 K (c) and 5 K(d). The red line is a spin diffusion based simulation\nof the SMR amplitude.iment. Our results demonstrate how SMR experiments in tri-\nlayers can be used to study pure spin currents in materials with\nvanishing spin Hall angle qSH;NM.\nSUPPLEMENTARY MATERIAL\nSee supplementary online material for the structural and\nmagnetic characterization of the bi- and trilayer samples, the\ncarrier concentration of the Ga:ZnO layer, and a more elabo-\nrate discussion of the SMR trilayer simulation.\nACKNOWLEDGMENTS\nM.S.R. and J.M. would like to thank for funding from De-\npartment of Science and Technology, New Delhi, that facil-\nitated the establishment of Nano Functional Materials Tech-\nnology Centre (Grant: SR NM/NAT/02-2005). J.M. would\nlike to thank UGC for SRF fellowship. We acknowledge fi-\nnancial support by the German Academic Exchange Service\n(DAAD) via project no. 57085749.\n1K. Ando, Y . Kajiwara, K. Sasage, K. Uchida, and E. Saitoh, IEEE Trans.\nMagn. 46, 3694 (2010).\n2F. D. Czeschka, L. Dreher, M. S. Brandt, M. Weiler, M. Althammer, I.-M.\nImort, G. Reiss, A. Thomas, W. Schoch, W. Limmer, H. Huebl, R. Gross,\nand S. T. B. Goennenwein, Phys. Rev. Lett. 107(2011).\n3K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando,\nS. Maekawa, and E. Saitoh, Nature 455, 778 (2008).\n4K. ichi Uchida, H. Adachi, T. Ota, H. Nakayama, S. Maekawa, and\nE. Saitoh, Appl. Phys. Lett. 97, 172505 (2010).\n5M. Weiler, M. Althammer, F. D. Czeschka, H. Huebl, M. S. Wagner,\nM. Opel, I.-M. Imort, G. Reiss, A. Thomas, R. Gross, and S. T. B. Goen-\nnenwein, Phys. Rev. Lett. 108(2012).\n6H. Nakayama, M. Althammer, Y .-T. Chen, K. Uchida, Y . Kajiwara,\nD. Kikuchi, T. Ohtani, S. Gepr ¨ags, M. Opel, S. Takahashi, R. Gross,\nG. E. W. Bauer, S. T. B. Goennenwein, and E. Saitoh, Phys. Rev. Lett.\n110, 206601 (2013).\n7C. Hahn, G. de Loubens, O. Klein, M. Viret, V . V . Naletov, and J. B.\nYoussef, Phys. Rev. B 87(2013).\n8Y .-T. Chen, S. Takahashi, H. Nakayama, M. Althammer, S. T. B. Goennen-\nwein, E. Saitoh, and G. E. W. Bauer, Phys. Rev. B 87, 144411 (2013).\n9M. Althammer, S. Meyer, H. Nakayama, M. Schreier, S. Altmannshofer,\nM. Weiler, H. Huebl, S. Gepr ¨ags, M. Opel, R. Gross, D. Meier, C. Klewe,\nT. Kuschel, J.-M. Schmalhorst, G. Reiss, L. Shen, A. Gupta, Y .-T. Chen,\nG. E. W. Bauer, E. Saitoh, and S. T. B. Goennenwein, Phys. Rev. B 87,\n224401 (2013).\n10N. Vlietstra, J. Shan, V . Castel, B. J. van Wees, and J. B. Youssef, Phys.\nRev. B 87(2013).\n11K. Ganzhorn, J. Barker, R. Schlitz, B. A. Piot, K. Ollefs, F. Guillou, F. Wil-\nhelm, A. Rogalev, M. Opel, M. Althammer, S. Gepr ¨ags, H. Huebl, R. Gross,\nG. E. W. Bauer, and S. T. B. Goennenwein, Phys. Rev. B 94(2016).\n12M. Weiler, M. Althammer, M. Schreier, J. Lotze, M. Pernpeintner,\nS. Meyer, H. Huebl, R. Gross, A. Kamra, J. Xiao, Y .-T. Chen, H. Jiao,\nG. E. W. Bauer, and S. T. B. Goennenwein, Phys. Rev. Lett. 111, 176601\n(2013).\n13S. Gepr ¨ags, S. Meyer, S. Altmannshofer, M. Opel, F. Wilhelm, A. Rogalev,\nR. Gross, and S. T. B. Goennenwein, Appl. Phys. Lett. 101, 262407 (2012).\n14B. F. Miao, S. Y . Huang, D. Qu, and C. L. Chien, Phys. Rev. Lett. 112\n(2014).\n15M. Althammer, E.-M. Karrer-Muller, S. T. B. Goennenwein, M. Opel, and\nR. Gross, Appl. Phys. Lett. 101, 082404 (2012).\n16M. Schreier, N. Roschewsky, E. Dobler, S. Meyer, H. Huebl, R. Gross, and\nS. T. B. Goennenwein, Appl. Phys. Lett. 103, 242404 (2013).5\n30°35°40°45°50°101102103104105\n-0.4°0.0°0.4°020406080I(cps)\n2θI(cps)\n∆ω\n30°35°40°45°50°101102103104105\n-0.4°0.0°0.4°01020304050I(cps)\n2θI(cps)\n∆ω\n-6 -3 0 3 6-100-50050100M(kA/m)\nµ0H(T)-6-3 0 3 6-100-50050100M(kA/m)\nµ0H(T)0.025°\nBi:YIG (444)**\n*\n*YAG (444)\nBi:YIG (444)\nYAG (444)0.331° 0.494°0.024°(b)\n(c)\nMs=106 kA/m Ms=108 kA/m(d)(a)\nT=300 K T=300 K Bi:YIGGa:ZnOPt\n54 nm8 nm4 nm\nBi:YIGPt 54 nm4 nm\nFIG. S1. Structural and magnetic properties of the laser-MBE grown\nBi:YIG(54)/Pt(4) bilayer and Bi:YIG(54)/Ga:ZnO(8)/Pt(4) trilayer\nsamples grown on YAG (111) substrates. 2 q\u0000wscan of bilayer\n(a) and trilayer (b) samples, the insets of (a) and (b) are the rocking\ncurve of the Bi:YIG (444) reflection, numbers represent the FWHM\nof the corresponding Gaussian fit. Peaks marked with asterisks are\nbackground reflections from the sample holder. In-plane magnetiza-\ntion versus applied magnetic field curve of bilayer (c) and trilayer (d)\nsamples measured at 300 K.\n17S. Meyer, R. Schlitz, S. Gepr ¨ags, M. Opel, H. Huebl, R. Gross, and S. T. B.\nGoennenwein, Appl. Phys. Lett. 106, 132402 (2015).\n18S. Meyer, M. Althammer, S. Gepr ¨ags, M. Opel, R. Gross, and S. T. B.\nGoennenwein, Appl. Phys. Lett. 104, 242411 (2014).\n19A. Aqeel, N. Vlietstra, J. A. Heuver, G. E. W. Bauer, B. Noheda, B. J. van\nWees, and T. T. M. Palstra, Phys. Rev. B 92(2015).\n20S. Ghosh, V . Sih, W. H. Lau, D. D. Awschalom, S.-Y . Bae, S. Wang,\nS. Vaidya, and G. Chapline, Appl. Phys. Lett. 86, 232507 (2005).\n21S. V´elez, V . N. Golovach, A. Bedoya-Pinto, M. Isasa, E. Sagasta, M. Aba-\ndia, C. Rogero, L. E. Hueso, F. S. Bergeret, and F. Casanova, Phys. Rev.\nLett.116(2016).\nI. STRUCTURAL AND MAGNETIC CHARACTERIZATION\nWe first compare the results obtained for a\nBi:YIG(54)/Pt(4) bilayer and a Bi:YIG(54)/Ga:ZnO(8)/Pt(4)\ntrilayer, where the numbers in parentheses give the film\nthicknesses in nm. The structural quality of the thin films was\nanalyzed by x-ray diffraction (XRD). The magnetic proper-\nties were obtained by superconducting quantum interference\ndevice magnetometry.\nThe XRD results presented in Fig. S1 for the bilayer (panel\na) as well as for the trilayer (panel b) indicate excellent struc-\ntural quality. In both samples we only observe reflections that\ncan be assigned to the substrate or the layers itselfthemselves.\nIn the trilayer sample no reflections of the Ga:ZnO layer could\nbe observed, which we attribute to the small thickness of the\nlayer. The XRD rocking curves of the Bi:YIG (444) reflec-\ntion (insets in Fig. S1(a),(b)) consist in both samples of one\nrather broad peak superimposed on a second narrow peak.\nWe attribute this to partial strain relaxation in the film due to\nthe large lattice mismatch of 3 %, between YAG and Bi:YIG.\nNevertheless, the full width at half maximum (FWHM) ex-\ntracted from Gaussian fits to the data yield nearly identical\n0501001502002503001.51.61.7n(x1021cm-3)\nT(K)FIG. S2. Carrier concentration of a 8 nm thick Ga:ZnO grown on\nsapphire. The carrier concentration is independent of temperature\nindicating the degenerate doping of the Ga:ZnO layer.\nvalues of 0 :025\u000efor the bilayer and 0 :024\u000efor the trilayer\nsample. However, we obtain different values for the FWHM\nof the broader peak, 0 :331\u000efor the bilayer and 0 :494\u000efor the\ntrilayer sample. The broad peak in the Bi:YIG (444) rock-\ning curve can be attributed to the strain relaxation layer in the\nBi:YIG forming at the YAG substrate interface. Taking this\ninto account, we attribute the change in width to a possible\nchange in strain relaxation by defect formation in the Bi:YIG\nlayer, due to the different thermal treatment by the additional\nGa:ZnO deposition in the trilayer. However, further investiga-\ntions are necessary to confirm this assumption.\nThe magnetization curves recorded at 300 K are shown in\nFig. S1(c) for the bilayer and (d) for the trilayer sample, where\na diamagnetic background from the substrate has been sub-\ntracted from the raw data. We extract a saturation magneti-\nzation Ms=106 kA =m for the bilayer and Ms=108 kA =m\nfor the trilayer sample. Both numbers are smaller than the\nbulk value?Ms=144 kA =m, which we attribute to defects\nin our layers. These defects can either be structural defects\nin the strain relaxation layer present at the YAG substrate in-\nterface?, effectively reducing the total saturation magnetiza-\ntion over the whole film thickness, or iron and oxide vacancies\npresent in the whole Bi:YIG film?. The coercive field for both\nsamples is 0 :6 mT. Taken together, the structural and the mag-\nnetic properties of the bi- and trilayer sample are nearly identi-\ncal. This suggests that the influence of the Ga:ZnO deposition\non the magnetic properties of the Bi:YIG layer is negligible.\nII. ORDINARY HALL EFFECT IN GA:ZNO\nWe used ordinary Hall effect (OHE) measurements on a\n8 nm thick Ga:ZnO layer grown via pulsed laser deposition\non a (0001)-oriented sapphire substrate under identical depo-\nsition conditions as for the trilayer structures investigated for\nthe SMR experiments. From the OHE measurements, carried\nout at various temperatures, we then extracted the n-type car-\nrier concentration nas a function of temperature, the results\nare shown in Fig. S2.6\nThe carrier concentration is independent of temperature\nwith n\u00191:6\u00021021cm\u00003. This indicates that the Ga:ZnO is\ndegenerately doped.\nIII. SIMULATION OF THE SMR IN TRILAYER-SYSTEMS\nFor modelling the SMR in trilayer structures we follow the\napproach outlined in Ref. 8. For the calculations, we use the\ncoordinate system depicted in Fig. S3. Our system consists\nof two conductive layers (NM1, NM2), where the pure spin\ncurrent in both layers is carried by the spin angular momen-\ntum of the charge carriers. In our calculations we assume that\nonly the charge current jq;NM1(z)flowing in NM1 gets con-\nverted into a spin current js;NM1(z), while there is no spin cur-\nrent generation via the spin Hall effect in NM2(spin Hall in-\nactive layer). At the NM1/NM2 interface we assume that the\nspin-dependent electrochemical potentials s;NM1(z),ms;NM2(z)\nand the spin current flowing across the interface are contin-\nuous, and thus neglecting any interface resistance contribu-\ntions. This leads to the following boundary conditions:\nms;NM1(dNM2) =ms;NM2(dNM2);\njs;NM1(dNM2) =js;NM2(dNM2);\njs;NM1(dNM2+dNM1) =0;\njs;NM2(0) =1=e[Grm\u0002(m\u0002ms;NM2(0))+Gi(m\u0002ms;NM2(0))];\nwhere GrandGiare the real and imaginary part of the spin\nmixing conductance per unit area. We then solve the spin\ndiffusion equation with the above boundary conditions as de-\ntailed in Ref. 8. The analytical expressions obtained from this\nprocedure are lengthy and therefore not written down in the\ntext here. With these solutions, we then calculate the spin and\ncharge currents as outlined in Ref. 8. Averaging over the film\nthicknesses and expanding up to the second order in spin Hall\nangle we then obtain an expression for rlongof the whole tri-\nlayer stack. Using this expression we can then determine the\nSMR amplitude by determining rlongformkjandmkt. We\nnote that similar calculations can also be carried out for a spin\nHall active NM2 layer using the very same approach.\nUsing these result we can then calculate the NM2 thickness\ndependence of the SMR amplitude for different resisitivity ra-\ntiosrNM2=rNM1, while using the following values for the re-\nmaining parameters: g\"#=1\u00021019m\u00002,lsf;NM1=1:5 nm,\nqSH;NM1=0:11,lsf;NM2=12 nm; The result of these calcu-\nlations is shown in Fig. S3(b). For rNM2=rNM1=1, the SMR\ngives the largest values for a finite thickness of NM2. For\nrNM2=rNM1=0:1, we find a lower SMR amplitude as com-\npared to rNM2=rNM1=1 as now a large part of the electri-\ncal current runs through the spin Hall inactive layer and thus\nis not contributing to the SMR. For rNM2=rNM1=10 and\nrNM2=rNM1=100 the SMR gets enhanced for ultrathin NM2\nlayers, which can effectively understood as an enhancement\nof the interface resistance at the FMI/NM2 interface which\nboosts the SMR effect similar to the experiments on match-\ning spin mixing conductance in spin pumping experiments as\nelaborated in Ref. ?.\nIn our experiments we used Ga:ZnO as the spin Hall inac-\ntive NM2 to vary the spin diffusion length and resistivity of\n0 5 10 1510-610-510-410-310-2SMR\ndNM2(nm)NM2NM1 \nFMI dNM2dNM1z\ndNM1+dNM2\ndNM2\n0(a)\n(b)\nρNM2/ρNM1=\n1\n10\n1000.1FIG. S3. Simulation of the SMR in a trilayer structure with two\nconductive layers and one FMI layer. (a) Schematic drawing of\nthe coordinate system used for the calculations. (b) Dependence\nof the SMR on the thickness of the spin Hall inactive interlayer for\nrNM2=rNM1=0:1 (black line), 1 (red line), 10 (blue line), and 100\n(green line). Similar to the conductivity mismatch problem for spin\ninjection into semiconductors, the SMR effect is reduced.\nthe NM2 layer as a function of temperature. Similar effects\nwill occur if the carrier concentration is changed in the ZnO\nlayer, as this will again affect the spin diffusion length and the\nresistivity of the NM2 layer.\n1K. Ando, Y . Kajiwara, K. Sasage, K. Uchida, and E. Saitoh, IEEE Trans.\nMagn. 46, 3694 (2010).\n2F. D. Czeschka, L. Dreher, M. S. Brandt, M. Weiler, M. Althammer, I.-M.\nImort, G. Reiss, A. Thomas, W. Schoch, W. Limmer, H. Huebl, R. Gross,\nand S. T. B. Goennenwein, Phys. Rev. Lett. 107(2011).\n3K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando,\nS. Maekawa, and E. Saitoh, Nature 455, 778 (2008).\n4K. ichi Uchida, H. Adachi, T. Ota, H. Nakayama, S. Maekawa, and\nE. Saitoh, Appl. Phys. Lett. 97, 172505 (2010).\n5M. Weiler, M. Althammer, F. D. Czeschka, H. Huebl, M. S. Wagner,\nM. Opel, I.-M. Imort, G. Reiss, A. Thomas, R. Gross, and S. T. B. Goen-\nnenwein, Phys. Rev. Lett. 108(2012).\n6H. Nakayama, M. Althammer, Y .-T. Chen, K. Uchida, Y . Kajiwara,\nD. Kikuchi, T. Ohtani, S. Gepr ¨ags, M. Opel, S. Takahashi, R. Gross,\nG. E. W. Bauer, S. T. B. Goennenwein, and E. Saitoh, Phys. Rev. Lett.\n110, 206601 (2013).\n7C. Hahn, G. de Loubens, O. Klein, M. Viret, V . V . Naletov, and J. B.\nYoussef, Phys. Rev. B 87(2013).\n8Y .-T. Chen, S. Takahashi, H. Nakayama, M. Althammer, S. T. B. Goennen-\nwein, E. Saitoh, and G. E. W. Bauer, Phys. Rev. B 87, 144411 (2013).\n9M. Althammer, S. Meyer, H. Nakayama, M. Schreier, S. Altmannshofer,\nM. Weiler, H. Huebl, S. Gepr ¨ags, M. Opel, R. Gross, D. Meier, C. Klewe,\nT. Kuschel, J.-M. Schmalhorst, G. Reiss, L. Shen, A. Gupta, Y .-T. Chen,\nG. E. W. Bauer, E. Saitoh, and S. T. B. Goennenwein, Phys. Rev. B 87,\n224401 (2013).\n10N. Vlietstra, J. Shan, V . Castel, B. J. van Wees, and J. B. Youssef, Phys.\nRev. B 87(2013).\n11K. Ganzhorn, J. Barker, R. Schlitz, B. A. Piot, K. Ollefs, F. Guillou, F. Wil-\nhelm, A. Rogalev, M. Opel, M. Althammer, S. Gepr ¨ags, H. Huebl, R. Gross,\nG. E. W. Bauer, and S. T. B. Goennenwein, Phys. Rev. B 94(2016).\n12M. Weiler, M. Althammer, M. Schreier, J. Lotze, M. Pernpeintner,\nS. Meyer, H. Huebl, R. Gross, A. Kamra, J. Xiao, Y .-T. Chen, H. Jiao,\nG. E. W. Bauer, and S. T. B. Goennenwein, Phys. Rev. Lett. 111, 176601\n(2013).\n13S. Gepr ¨ags, S. Meyer, S. Altmannshofer, M. Opel, F. Wilhelm, A. Rogalev,\nR. Gross, and S. T. B. Goennenwein, Appl. Phys. Lett. 101, 262407 (2012).\n14B. F. Miao, S. Y . Huang, D. Qu, and C. L. Chien, Phys. Rev. Lett. 112\n(2014).\n15M. Althammer, E.-M. Karrer-Muller, S. T. B. Goennenwein, M. Opel, and\nR. Gross, Appl. Phys. Lett. 101, 082404 (2012).\n16M. Schreier, N. Roschewsky, E. Dobler, S. Meyer, H. Huebl, R. Gross, and\nS. T. B. Goennenwein, Appl. Phys. Lett. 103, 242404 (2013).7\n17S. Meyer, R. Schlitz, S. Gepr ¨ags, M. Opel, H. Huebl, R. Gross, and S. T. B.\nGoennenwein, Appl. Phys. Lett. 106, 132402 (2015).\n18S. Meyer, M. Althammer, S. Gepr ¨ags, M. Opel, R. Gross, and S. T. B.\nGoennenwein, Appl. Phys. Lett. 104, 242411 (2014).\n19A. Aqeel, N. Vlietstra, J. A. Heuver, G. E. W. Bauer, B. Noheda, B. J. vanWees, and T. T. M. Palstra, Phys. Rev. B 92(2015).\n20S. Ghosh, V . Sih, W. H. Lau, D. D. Awschalom, S.-Y . Bae, S. Wang,\nS. Vaidya, and G. Chapline, Appl. Phys. Lett. 86, 232507 (2005).\n21S. V´elez, V . N. Golovach, A. Bedoya-Pinto, M. Isasa, E. Sagasta, M. Aba-\ndia, C. Rogero, L. E. Hueso, F. S. Bergeret, and F. Casanova, Phys. Rev.\nLett.116(2016)."    },    {        "title": "2305.11629v1.Quantum_transduction_of_superconducting_qubit_in_electro_optomechanical_and_electro_optomagnonical_system.pdf",        "content": "Quantum transduction of superconducting qubit in electro-optomechanical and\nelectro-optomagnonical system.\nRoson Nongthombam,\u0003Pooja Kumari Gupta,yand Amarendra K. Sarmaz\nDepartment of Physics, Indian Institute of Technology Guwahati, Guwahati-781039, India\nWe study the quantum transduction of a superconducting qubit to an optical photon in electro-\noptomechanical and electro-optomagnonical systems. The electro-optomechanical system comprises\na \rux-tunable transmon qubit coupled to a suspended mechanical beam, which then couples to\nan optical cavity. Similarly, in an electro-optomagnonical system, a \rux-tunable transmon qubit\nis coupled to an optical whispering gallery mode via a magnon excitation in a YIG ferromagnetic\nsphere. In both systems, the transduction process is done in sequence. In the \frst sequence, the qubit\nstates are encoded in coherent excitations of phonon/magnon modes through the phonon/magnon-\nqubit interaction, which is non-demolition in the qubit part. We then measure the phonon/magnon\nexcitations, which reveal the qubit states, by counting the average number of photons in the optical\ncavities. The measurement of the phonon/magnon excitations can be performed at a regular intervals\nof time.\nI. INTRODUCTION\n`Quantum network' is a rapidly developing area owing\nto its potential applications in scaling up quantum\ncomputers by connecting multiple quantum processors.\nRecently, much research has been initiated on developing\na modular quantum computer based on linking multiple\nsuperconducting chips where each chip has a few high-\nquality qubits. Instead of cramping more qubits onto\na single chip, which will result in high error rates\nand complex hardware, creating a network of modules\ncontaining few high-quality qubits on a single chip is\nbetter. This modular quantum computing approach has\nlower error rates and lesser hardware constraints.\nFor connecting the modules, optical \fbers, which have\nlow propagation loss in a noisy thermal environment,\nare employed. So, the qubit operations must \frst be\ntransferred to the \rying optical photons in the optical\n\fber. The transduction of the qubit to the optical\nphoton cannot be achieved directly due to the vast\nseparation of the frequencies between the two (qubit\nin GHz and optical photon in THz). One way to\nachieve transduction is by introducing a bosonic system\nas a mediator that couples both the qubit and the\noptical photon, forming a hybrid qubit-boson-optical\nsystem. In this work, we discuss transduction in two\nsuch hybrid systems, namely, electro-optomechanical\nand electro-optomagnonic systems. The electro-\noptomechanical system consists of a superconducting\nmicrowave circuit coupled to a mechanical resonator\nwhich in turn is connected to an optical cavity. In\nrecent years, this hybrid system has been extensively\nstudied experimentally[1{5] and theoretically[6{12] for\nmicrowave-to-optical photon transduction. There are\nseveral ways of coupling a transmon qubit, formed by\n\u0003n.roson@iitg.ac.in\nypooja.kumari@iitg.ac.in\nzaksarma@iitg.ac.ina superconducting microwave circuit, to a mechanical\nresonator [13{17]. Here, we consider a \rux tunable\ntransmon qubit that is coupled to a suspended\nmechanical beam [18]. The mechanical beam is then\nintegrated as an end mirror of an optomechanical cavity\nforming the required hybrid system[10].\nThe hybrid electro-optomagnonic system consists\nof a superconducting microwave circuit coupled to\na ferromagnetic magnon excitation [19{22], which is\ncoupled to an optical photon [23{27]. This hybrid system\nis less explored. It is mainly due to the weak coupling\nbetween the magnon and the optical photon. However,\nthere has been some progress recently [28]. For example,\nenhancement of magnon-photon coupling under the triple\nresonance condition of input photon, magnon and output\nphoton is demonstrated in [25, 29]. By implementing\nthe triple resonant condition, a microwave-to-optical\nconversion based on multiple magnon mode interaction\nwith the optical photon mode is demonstrated in [30].\nAnother theoretical study to improve the magnon-optical\ncoupling based on optical optical whispering gallery\nmode (WGM) coupled to localized vortex magnon mode\nin a magnetic microdisk is done in [31].\nIn this work, we construct the hybrid electro-\noptomagnonical system by merging the scheme proposed\nin [26], where a \rux-tunable transmon qubit is coupled\nto a magnon mode formed in a \u0016m size YIG sphere, and\nthe optomagnonical setup experimentally demonstrated\nin [29], where an optical WGM interacts with magnon\nmode in a YIG sphere of radius having few hundred \u0016m.\nOne main di\u000eculty in realizing this hybrid system is the\nsize gap in the YIG spheres. However, the possibility\nof reducing the sphere used in the optomagnonic case is\npointed out in [29]. Assuming this to be possible, we\nconsider a YIG sphere of few \u0016m size radius that couples\nboth the superconducting qubit and the optical WGM\npresent in the sphere.\nThe technique used here for measuring the qubit states\nfrom the optical photon is similar to the one employed\nin [3]. The idea is to \frst associate or encodes the\nqubit states in the magnon/phonon coherent excitationsarXiv:2305.11629v1  [quant-ph]  19 May 20232\nand then measure these excitations by counting the\naverage number of the photon in the optical cavity.\nAlthough measuring the qubit states by detecting the\noptical photon count is demonstrated in [3], our scheme\nexhibits two distinct features: (1) The interaction of\nthe qubit and the magnon/phonon commutes with the\nintrinsic Hamiltonian of the qubit. In other words, the\ninitial state of the qubit remains the same during the\ninteraction. (2) Due to the coherent and oscillatory\nevolution of the magnon/phonon and photon states\nduring the interaction, we can perform measurements\nof the qubit states from the photon count at regular\nintervals of time.\nThe paper is organized as follows. We describe the two\nhybrid systems under study in Sec.II. The \frst sequence\nof the transduction process, i.e., encoding the qubit state\nin the magnon/phonon coherent excitation is studied in\nSec.III A. The measurement of the qubit state from the\noptical photon count is done in Sec.III B. Finally, we\nconclude by summarizing our work in Sec. IV.\nII. THE HYBRID SYSTEM\nWe \frst consider the hybrid electro-optomechanical\nsystem. This hybrid system comprises a \rux-tunable\ntransmon (formed by a SQUID loop ( EJ, \bJ)), coupled\nto a mechanical resonator (realized by suspending one\narm of another SQUID loop ( EM, \bM))[18] which\ncan oscillate out of plane. The suspended mechanical\nmembrane is then integrated as an end mirror of an\noptical cavity forming an optomechanical cavity [10], as\nshown in Fig. 1(a). The Hamiltonian of the system is\ndescribed by\n^Heom=^H0+^Htm+^Hom+^Hd (1)\nwhere,\n^H0=~\u0001c^ay^a+~!m^by^b+~!t^cy^c\u0000Ec\n2^cy^cy^c^c;(2a)\n^Htm=~gtm^cy^c(^b+^by); (2b)\n^Hom=~gom^ay^a(^b+^by); (2c)\n^Hd=~E0\u0000\n^a+ ^ay\u0001\n: (2d)\nHere, ^a(^ay),^b(^by), and ^c(^cy) are the annihilation\n(creation) operators of the optical photon, the\nmechanical phonon and the transmon, respectively. ^H0\nis the Hamiltonian of the individual components of the\nhybrid system in the absence of any interactions. The\ntransmon-mechanical resonator interaction is described\nby^Htm, and the optomechanical interaction by ^Hom.\nThe strength of the coupling constant gomis generally\nquite small (\u00191Hz). So, we drive the optomechanical\ncavity to increase the coupling strength. This drive is\nincluded in the Hamiltonian of the hybrid system as ^Hd.\nThe Hamiltonian ^Homis written in the optomechanical\ndrive frame (\u0001 c=!c\u0000!d, where!cis the cavity\nC\nE\nM\nɸ\nM\nɸ\nJ\nE\nJ\nB\nz\nX\nω\nd\nω\nc \nY\nX\nZ\n(a)\nE\nJ\nE\nJ\n \nC \nɸ\nJ\nB\nz\nX\nZ\nY\nTM\nTE\n (b)FIG. 1. (Color online) (a) Schematic diagram of an electro-\noptomechanical system. On application of an in-plane\nmagnetic \feld Bz, the transmon qubit formed by a SQUID\nloop (EJ;\bJ) is coupled to a mechanical beam suspended at\none arm of the loop ( EM;\bM) [18]. The mechanical resonator\nis integrated as a movable plate of an optomechanical cavity\nwhose resonance frequency is !c. The optomechanical\ncavity is shined on by a red-detuned laser light. (b) A\nYIG ferromagnetic sphere that supports both the magnon\nexcitation and optical WGM is placed near a \rux-tunable\ntransmon qubit formed by the loop ( EJ;\bJ) [26]. An optical\nchannel is mounted on top of the YIG sphere. A TM-\npolarized light given at the input of the channel passes the\nYIG sphere in a clockwise direction, and a TE-polarized light\nemerges at the channel output. An in-plane magnetic \feld\nBz, responsible for the transmon-magnon coupling, is passed\nthrough the ferromagnetic sphere.\nfrequency and !dis the drive frequency.). The qubit-\nmechanical and optomechanical interactions arise from\nthe displacement of the mechanical resonator. On\napplication of an in-plane magnetic \feld Bz, as shown in\nFig. 1(a), the displacement of the mechanical resonator\npicks up a \rux in the Josephson energy. This motional\ndependent Josephson energy leads to the transmon-\nmechanical resonator interaction ^Htm[18]. Note that\nin addition to the third order non-linear interaction\nterm, ^Htm, higher order non-linear interaction terms\nare present as demonstrated in [18]. But, here, we\nhave excluded this higher order corrections due to their\nnegligible contribution to the system dynamics. The\nmechanical motion of the resonator also simultaneously\nalters the resonator frequency of the optomechanical\ncavity, which results in the optomechanical interaction\n^Hom. We have considered a small SHM (simple harmonic3\nmotion) displacement of the resonator.\nWe next consider the hybrid electro-optomagnonic\nsystem. In this system, a YIG sphere having a diameter\nin the\u0016m range is placed near a \rux-tunable transmon\nformed by a symmetric SQUID loop ( EJ,\bJ) [26]. The\nYIG sphere is then mounted on an optical \fber placed\njust above the plane of the SQUID loop [29], as shown\nin Fig. 1(b). Just like in the previous system, an in-\nplane magnetic \feld Bzis applied. This \feld magnetizes\nthe magnetic sphere along the z-direction. Because of\nthis magnetization, a uniform magnetostatic mode or\nkittle mode is excited on the YIG sphere, whose magnetic\nmoment produces a stray \feld and traverse along the\ntransmon SQUID loop. Subsequently, the stray \feld\npicks up an additional \rux in the loop thereby changing\nthe Josephson energy and frequency of the transmon and\neventually leading to a transmon-magnon interaction.\nOn the other hand, a TM (transverse magnetic \feld)\npolarized light is pumped at the input of the optical\nwaveguide. When in resonance, this pumped light or\nsignal is con\fned in the YIG sphere forming WGM\n(whispering gallery mode) in the clockwise direction. The\nTM input signal then interacts with the magnons in the\nmagnetic sphere. This interaction has two signi\fcant\nfeatures. One is that it changes the TM-polarized input\nsignal to a circulating TE WGM and then comes out as\na TE-polarized signal at the output. The other feature is\nthat it changes the input and output signal frequencies.\nThe amount of change in the input and output signal\nfrequencies is equivalent to that of the magnon frequency.\nThe above two features are the outcomes of satisfying\nthe triple-resonance condition. The triple resonance\ncondition is experimentally demonstrated in a 100-\n300\u0016m size ferromagnetic sphere [29]. In our case,\nthe size of the sphere is considered to be around 3\n\u0016m. Although the size we have considered here is\nnot yet experimentally realized for the triple resonance\ninteraction, the possibility of sizing down the sphere is\npointed out in [29]. The e\u000bective Hamiltonian of the\nhybrid electro-optomagnonic system is described by\n^H0\neom=^H0\n0+^H0\nt+^H0\ntm+^H0\nom+^H0\nd (3)\nwhere,\n^H0\n0=~!v^ay\nv^av+~!h^ay\nh^ah+~!0\nm^my^m; (4a)\n^H0\nt=~!0\nt^cy^c\u0000Ec\n2^cy^cy^c^c; (4b)\n^H0\nom=~g0\nom(^ay\nh^av^m+ ^ah^my^ay\nv); (4c)\n^H0\ntm=~g0\ntm^cy^c( ^m+ ^my); (4d)\n^H0\nd=~Ev\u0000\n^ave\u0000i!Lt+ ^ay\nvei!Lt\u0001\n: (4e)\nHere, ^av(^ay\nv), ^ah(^ay\nh), ^m( ^my), and ^c(^cy) are the\nannihilation (creation) operators of the input TM optical\nphoton, the output TE optical photon, the magnon\nand the transmon, respectively. ^H0\n0and ^H0\ntare\nthe Hamiltonian of the individual components of the\nhybrid system in the absence of any interactions. Thetransmon-magnon interaction is described by ^H0\ntmand\nthe optomagnonic interaction by ^H0\nom. Here, the\ninteraction term ^H0\ntmis for the symmetric SQUID loop.\n^H0\ndis the optical drive of the TM mode.\nIII. TRANSDUCTION\nHere, we discuss the quantum transduction of qubit\nstates to optical photons via mechanical phonons or YIG\nsphere magnons. The transduction process is realized\nin sequence. First, we encode the qubit states to the\nphonon/magnon excitations, and next, we measure these\nexcitations by counting the average number of photons\nin the optical cavity. This section is divided into two\nparts. The \frst part discusses the process of encoding\nthe qubit states in the mechanical phonon states, and in\nthe second part, we discuss how the phonon states, and\nhence the qubit states, are determined from the optical\nphoton number.\nA. Qubit-phonon/magnon transduction\nWe \frst consider the qubit-mechanical interaction in\nthe hybrid electro-optomechanical system and show how\nqubit states can be encoded to the phonon excitations.\nThe qubit-mechanical coupling rate gtmis much larger\nthan the single-photon optomechanical coupling rate\ngom. So, if there is no optomechanical cavity drive,\nwe can neglect the optomechanical interaction in the\nHamiltonian given by Eq. 2. Now we are left with just\nthe electromechanical part of the hybrid system.\n^Hem=^H0+~gtm^cy^c(^b+^by): (5)\nHere, the coupling constant gtmis dependent on the\nexternal \rux bias \b mas [26],\ngtm=g0sin(\u001eb); (6)\nwhere,\u001eb=\u0019\bb\n\b0and \b 0=h\n2eis the \rux quantum. g0is\ncoupling constant. Next, we enhance the coupling rate\nby modulating it parametrically by applying a weak ac\nbias\u001eb=\u001eaccos(!act) (\u001eac<<1) as done in [26].\ngtm=g0\u001eaccos(!act): (7)\nBy substituting this modulated time dependent coupling\nconstant in the Hamiltonian 5, and then transforming\nthe resultant Hamiltonian in the reference frame of the\nac drive (U=ei!actbyb), we get\n^H0\nem=^H0+~g0\u001eac^cy^c(^b+^by)\u0000!ac^by^b: (8)\nHere, we have ignored the fast rotating terms since\n2!ac>>g 0\u001eac. To do the qubit transduction, we convert\nthe transmon to a transmon qubit by considering only4\nthe \frst two energy levels. We then let the system evolve\nunder resonant modulation ( !m=!ac). If the qubit\nis initially in the ground state jgiand the mechanical\nresonator is in the vacuum state j0bithen after some\ntime t, the qubit will remain in the ground state and\nthe mechanical resonator will change to a coherent state\nj\fb=ig0\u001eacti. Similarly, if the qubit is initially in\nthe excited state jeiand the resonator in the vacuum\nstate, then the qubit will remain in the excited state, and\nthe mechanical resonator will evolve to another coherent\nstatej\fb=\u0000ig0\u001eactiafter some time t.\njg;0bi0\u0000!jg;\fb=g0\u001eactit\nje;0bi0\u0000!je;\fb=\u0000ig0\u001eactit\nAn overall phase term induced from the intrinsic qubit\nHamiltonian is not included as it does not contribute\nto the transduction process. We see that as the system\nevolves, the mechanical resonator changes from a vacuum\nstate to a coherent state, whereas the qubit state remains\nas it is. It is because the interaction between the qubit\nand the mechanical resonator commute with the intrinsic\nHamiltonian of the qubit. In other words, the interaction\nis `non-demolition' in the qubit part.\nWe next consider transduction in the electro-\noptomagnonic case and analyze how qubit states can be\nencoded to magnon excitations. Just like in the previous\ncase, we can neglect the optomagnonic part and consider\nonly the electro-magnonic part since the single magnon-\nphoton coupling g0\nomis much less than the transmon-\nmagnon coupling g0\ntmfor no optical drive. The electro-\nmagnonic part is described by\n^H0\nem=^H0\n0+~g0\ntm^cy^c( ^m+ ^my); (9)\nwhere,\ng0\ntm=g0\n0sin(\u001em)p\njcos(\u001em)j; (10)\nwhere,\u001em=\u0019\bm\n\b0and \b 0=h\n2eis the \rux quantum.\ng0is coupling constant. Similar to the previous system,\nhere also we enhance the coupling rate by modulating\nit parametrically by applying a weak ac bias \u001em=\n\u001e0\naccos(!0\nact) (\u001e0\nac<<1) as done in [26].\ng0\ntm=g0\n0\u001e0\naccos(!0\nact): (11)\nSubstituting Eq. 11 in Eq. 9 and then transforming in\ndrive frame ( U=ei!0\nactmym) gives\n^H0\nem=^H0+~g0\n0\u001e0\nac^cy^c( ^m+ ^my)\u0000!0\nac^my^m: (12)\nThe fast rotating terms are neglected for 2 !0\nac>>g0\n0\u001e0\nac.\nWe now take the \frst two levels of the transmon and\nallow the system to evolve. For resonance modulation\n(!0\nm=!0\nac), we obtain results similar to that of the\nelectro-mechanical case, i.e.,jg;0mi0\u0000!jg;\fm=ig0\n0\u001e0\nactit\nje;0mi0\u0000!je;\fm=\u0000ig0\n0\u001e0\nactit\nHere,jg;0mi0andje;0mi0are the initial states, and\njg;\fm=ig0\n0\u001e0\nactandje;\fm=\u0000ig0\n0\u001e0\nactitare the \fnal\nstates of the qubit-magnonic system after time t. An\noverall phase is not included.\nSo far, we have not included the noise factor while\nevolving the system. To include the noisy environment,\nwe allow the system to evolve under the Lindblad master\nequation. For the electro-mechanical system\n_^\u001aem=\u0000i[^Hem;^\u001aem] + \u0000L[ ^\u001bz] + \u0000L[^\u001b\u0000]\n+\rb(nth+ 1)L[^b] +\rbnthL[^by]; (13)\nand for the electro-magnonic system\n_^\u001a0\nem=\u0000i[^H0\nem;^\u001a0\nem] + \u0000L[ ^\u001bz] + \u0000L[^\u001b\u0000]\n+\rm(n0\nth+ 1)L[ ^m] +\rmn0\nthL[ ^my];(14)\nwhereL[^o] = (2^o^\u001a^oy\u0000^oy^o^\u001a\u0000^\u001a^oy^o)=2. Here, \u0000 is the\ndecay rate of the transmon qubit, \rb(\rm) is the decay\nrate of phonon (magnon), nth(n0\nth) is the thermal phonon\n(magnon) number, and ^ \u001aem(^\u001a0\nem) is the density operator\nof the qubit-mechanical (qubit-magnonic) system. To\nobserve the coherent excitations of phonon and magnon\nin the dissipating environment, we plot the Wigner\nfunctions in Fig 2. Here, we observe that the Wigner\nfunctions of the phonon and magnon at some time \u001c=\n(3=2\u0019)\u0016sand for coupling constants g0\u001eac=g0\n0\u001e0\nac= 2\u0019\nMHz show coherent state pro\fle. The amplitude of the\ncoherent states when the qubit is in the ground state\nisj\fb=ig0\u001eac\u001ci=j3iifor the phonon and j\fm=\ng0\n0\u001e0\nac\u001ci=j3iifor the magnon, as shown in the \fgure.\nOn the other hand, when the qubit is in the excited state,\nthe coherent amplitudes are j\fb=\u0000ig0\u001eac\u001ci=j3ii\nandj\fm=\u0000ig0\n0\u001e0\nac\u001ci=j3ii. These changes in the\namplitude of the coherent states corresponding to the\nqubit ground and excited states are similar to the ones\nthat are observed in the non-dissipative case.\nSo, in both the dissipative and non-dissipative qubit-\nmechanical and qubit-magnonic systems, we observe that\nthe ground state of the qubit is encoded or associated\nwith a coherent excitation of both the phonon and\nmagnon and the excited state of the qubit is encoded\nin another coherent excitation of the same magnon and\nphonon having amplitudes exactly opposite to that of the\nexcitation associated with qubit ground state.\nB. Qubit-optical photon transduction\nWe have seen that the state of the qubit can be encoded\nin the coherent excitations of phonon and magnon. Here,\nwe will complete the qubit transduction sequence by\ntransferring the phonon and magnon states to the optical\nphoton. In the phonon case, this can be achieved through\nthe optomechanical interaction, and in the magnon case,5\nFIG. 2. (Color online) Wigner function representation of\nthe coherent states of phonon and magnon. In (a) and (b),\nthe phonon and the magnon excites to the coherent states\nj\fb= 3iitandj\fm= 3iitwhen the qubit is in the ground\nstatejgi. The coherent states of the phonon j\fb=\u00003iit\nand the magnonj\fm=\u00003iitwhen the qubit is in the exited\nstatejeiis shown in (c) and (d), respectively. The coherent\nstates are taken at time t=\u001c= (3=2\u0019)\u0016sfor coupling\nconstantsg0\u001eac=g0\n0\u001e0\nac= 2\u0019MHz. The other parameters\naregamma b=2\u0019= 1 Hz,nth= 400,gamma m=2\u0019= 0:1 GHz,\nn0\nth= 0:5, \u0000=2\u0019= 0:1 GHz\nit can be achieved through the optomagnonic interaction\nsatisfying the triple-resonant condition.\nFirst, we consider the optomechanical transfer.\nWe have previously seen from the electro-mechanical\ninteraction that the mechanical resonator can be\ncoherently excited with di\u000berent amplitudes depending\non the initial states of the qubit. So, we \frst excite the\nmechanical resonator to coherent states j\fb=\u0006ig0\u001eacti\nand then switch o\u000b the interaction g0\u001eacby turning o\u000b\nthe \rux bias \u001eac. The interacting system remaining is\nthen the optomechanical system.\n^H=~\u0001c^ay^a+~!m^by^b+~gom^ay^a(^by+^b) +~E0(^ay+ ^a):\n(15)\nSincegom\u00191Hz is very weak, we drive the cavity with\nan intense laser. Because of this strong drive, we can\nseparate the amplitudes of the mechanical resonator and\noptical cavity into a semi-classical coherent part ( \f;\u000b)\nand a small quantum \ructuation ( \u000e^a;\u000e^b) around it, i.e.,\n^a!\u000e^a+\u000band^b!\u000e^b+\f. We substitute this separation\nin Eq.15. By retaining only the interacting term, which\nis multiplied by the factor \u000b(j\u000bj\u0019103), the Hamiltonian\nreads\n^Hom=~\u00010^ay^a+~!m^by^b+~Gom(^ay+ ^a)(^by+^b);(16)\nwhere \u0001 = \u0001 c\u0000(\f+\f\u0003)gomandGom=gomj\u000bjfor\na constant phase preference of alpha. For simplicity we\nhave rewritten \u000e^ato ^aand\u000e^bto^b. Note that while\nwriting Eq.16, we have ignored all the constant terms\nand all the linear terms containing ^ a, ^ay,^band^byare\nequated to zero [32].The coherent state of the mechanical resonator\nprepared from the electro-mechanical interaction is in\nthe mechanical frame !m=!ac. So, we transform the\nHamiltonian 16 in the mechanical frame. We further\ntransform the system in the cavity detuning frame \u0001.\nTherefore, for a red-detuned laser drive \u0001 = !m, Eq.16\nbecomes\n^Hom=~Gom(^ay^b+^by^a): (17)\nHere, the fast rotating terms are ignored provided\nGom<< 2!m. For studying the state transfer from\nmechanical phonon to optical photon, we write down the\ndynamics of average number of photon and phonon in\nthe presence of dissipation.\ndh^ay^ai\ndt=\u0000i(h^ay^bi\u0000h^by^ai)Gom\u0000\u0014h^ay^ai(18a)\ndh^by^bi\ndt=\u0000i(h^by^ai\u0000h^ay^bi)Gom\u0000\rbh^by^bi\n+\rbnth (18b)\ndh^by^ai\ndt=\u0000(\u0014+\rb)\n2h^by^ai\u0000iGom(h^ay^ai\n\u0000h^by^bi) (18c)\nBy choosing the initial state of the mechanical resonator\nas the coherent state j\fb(0)iprepared from the electro-\nmechanical interaction, the average number of photon in\nthe absence of dissipation is given by\nh^ay^a(t)i= (g0\u001eac\u001c)2(1\u0000sin(2Gomt)) (19)\nwhen the qubit is in the ground state ( j\fb(0)i=j+\nig0\u001eac\u001ci), and\nh^ay^a(t)i= (g0\u001eac\u001c)2(1 +sin(2Gomt)) (20)\nwhen the qubit is in the excited state ( j\fb(0)i=j\u0000\nig0\u001eac\u001ci). Here, we have taken the initial state of the\ncavity photon to be j\u000b(0)i=jg0\u001eac\u001ci. The reason\nfor choosing this particular initial state is discussed in\nthe Appendix A. The evolution of the average photon\nnumber is shown in Fig. 3(a). From the \fgure, we\nobserve that if we measure the average photon number in\nthe cavity at the interval of t=\u0019=2Gom(starting from\nt=\u0019=4Gom) , then we either detect or do not detect\nthe presence of photons depending on the state of the\nqubit. If we detect photons in the cavity at the interval of\nt= (2n+ 1)\u0019=4Gom, wheren= 0;2;4;:::, then we know\nthat the qubit is in the ground state, and if at the same\ninterval, we do not detect any photons then the qubit\nis in the excited state. Similarly, if we detect photons\nin the cavity at the interval of t= (2n+ 1)\u0019=4Gom,\nwheren= 1;3;5;:::, then we know that the qubit is in\nthe excited state, and if at the same interval, we do not\ndetect any photons then the qubit is in the ground state.\nWe have chosen the above particular intervals because\nthe average photon numbers at these intervals are at\nthe maximum separation, and the qubit states can be\ndetermined more e\u000eciently than the other intervals.6\nFIG. 3. (Color online) Evolution of the average number of\nphotonh^ay^aiin the optical cavity. (a) In the absence of\ndissipation, the oscillatory evolution of h^ay^aikeeps on going.\nWhen the qubit is in the ground state, the oscillation is\nrepresented by the red colour, and when the qubit is in the\nground state, the oscillation is represented by the black colour.\nThe evolution of average photon number in the presence of\ndissipation is shown in (b) for \u0014= 2Gom, (c) and (d) for\n\u0014=Gom. In (a), (b), and (c), \u0014=2\u0019= 0:01 GHz is used, and\nin (d),\u0014=2\u0019= 1 MHz is used. The other common parameters\nare\r= 1 Hz and nth= 400\nIn the presence of dissipation, the oscillatory nature of\nh^ay^a(t)idecays with time, and in order to know the state\nof the qubit by counting the photon number we require\nthat the optomechanical coupling rate Gomshould be\ncomparable to the decay rate \u0014of the cavity. In Fig.\n3(b), we show the decay of cavity photon number for \u0014=\n2Gom, a moderate coupling strength. At this coupling\nstrength, we are able to make an e\u000ecient measurement\nof qubit states at just two intervals, t= 0:02\u0016sand\nt= 0:075\u0016s, before the number of average photon decay\nto zero. At a coupling strength lower than this, we will\nnot be able to identify the qubit states from the optical\nphoton. We also plot the case when the coupling strength\nis equal to the decay rate in Fig. 3(c). Here, more\noscillations can be seen, and hence more time intervals to\nmeasure the qubit states. Furthermore, we can increase\nthe time period for a same number of oscillations by\ndecreasing the decay rate \u0014as shown in Fig. 3(d).\nOne could go on and \fnd out the \fdelity of state\ntransfer of coherent state from the mechanical phonon\nto the optical photon. However, in our case, it is not\nnecessary since our purpose of determining the qubit\nstate is achieved by simply counting the cavity photon\nnumber. Since we are dealing with coherent states,\nwe can quantify how well the measured photon number\nindicates that the qubit is in a particular state. In Fig.\n4, we plot the probability distribution of the coherent\nstate for the \u0014= 2Gomcoupling case (Fig. 3(b)) at\nthe measurement time \u001c= 0:075\u0016s. We see that\neven when the qubit is in the excited state, there is\nstill some probability of not \fnding any photons in the\nFIG. 4. (Color online) Probability distribution of coherent\nstates of the optical photon number in the presence of\ndissipation. (a) and (b) shows the distribution when the qubit\nis in the ground and excited state, respectively. The coherent\nstates are measured at time \u001c= 0:075\u0016s. The average photon\nnumber in (a) is 3.4, and 0 in (b)\ncavity. The di\u000berence in the probability of not \fnding\nphotons in the cavity when the qubit is in the excited\nstate (Pe= 0:035) and when it is in the ground state\n(Pg= 0:999) gives the e\u000eciency of determining the qubit\nstate,P=Pe\u0000Pg= 0:964. This e\u000eciency decreases for\nless average photon number and vice versa. So, we need\nto repeat the counting measurement several times before\nconcluding the nature of the qubit state.\nWe now move on to the optomagnonic state transfer.\nJust like in the optomechanical case, we \frst excite\nthe magnon to coherent state j\fm=\u0006ig0\nom\u001e0\nomtifor\nsome time t, and then switch o\u000b the interaction g0\nom\u001e0\nom\nby turning o\u000b the \rux bias \u001e0\nom. The remaining\noptomagnonic system in the drive frame then reads\n^H0\nom=~\u000ev^ay\nv^av+~\u000eh^ay\nh^ah+~!0\nm^my^m+~Ev\u0000\n^av+ ^ay\nv\u0001\n~g0\nom(^ay\nh^av^m+ ^ah^my^ay\nv); (21)\nwhere\u000eh=!h\u0000!Land\u000ev=!v\u0000!L. We write down\nthe dynamics of the system.\nd^m\ndt=\u0000(\rm\n2+i!0\nm) ^m+ig0\nom^ah^ay\nv; (22a)\nd^av\ndt=\u0000(\u0014v\n2+i\u000ev)^av+ig0\nom^ah^my+iEv;(22b)\nd^ah\ndt=\u0000(\u0014h\n2+i\u000eh)^ah+ig0\nom^av^m: (22c)\nHere,\rm,\u0014vand\u0014hare the decay rates of magnon,\ninput TM \feld, and output TE \feld, respectively. The\nintrinsic magnon-photon coupling rate g0\nomis of the\norder of 10Hz, which is relatively very weak. We can\nenhance this coupling strength up to the order of MHz\nby performing an optical drive ( Ev) to the ferromagnetic\nsphere. After the drive, we can separate the input\n\feld into semi-classical mean amplitude ( \u000bv) and small\nquantum \ructuation around it ( \u000e^av), i.e., ^av!\u000bv+\u000e^av.\nSubstituting this separation in Eq. 22 and writing the7\nquantum and classical parts separately, we have\nd^m\ndt=\u0000(\rm\n2+i!0\nm) ^m+ig0\nom(^ah^ay\nv+ ^ah\u000b\u0003\nv);(23a)\nd^av\ndt=\u0000(\u0014v\n2+i\u000ev)^av+ig0\nom(^ah^my+\u000bh^my);(23b)\nd^ah\ndt=\u0000(\u0014h\n2+i\u000eh)^ah+ig0\nom^av^m; (23c)\nand\nd\u000bv\ndt=\u0000(\u0014v\n2+i\u000ev)\u000bv+iEv: (24)\nThe linear coupling terms in Eq. 23 are multiplied by a\nfactor of\u000bvor\u000b\u0003\nv(j\u000bvj\u0019103) compared to the non-linear\ncoupling terms. Therefore, we can neglect the non-linear\ncoupling terms and retain only the linear coupling terms.\nThe corresponding linear Hamiltonian reads\n^H0\nom=~\u000ev^ay\nv^av+~\u000eh^ay\nh^ah+~!0\nm^my^m+\n~G0\nom(^ay\nh^m+ ^my^ah); (25)\nwhereG0\nom=g0\nomj\u000bvj.j\u000bvjis given by the steady value\nof Eq. 24.\nSince the initial coherent state of the magnon prepared\nfrom the electro-magnonic interaction is in the magnon\nframe, we transform the Hamiltonian 25 in the magnon\nframe. Thus, for a resonant optical drive \u000ev= 0, the\nresultant Hamiltonian of the system in the magnon as\nwell as the output TE \feld frame of reference yields\n^H0\nom=~G0\nom(^ay\nh^m+ ^my^ah); (26)\nHere, since the interaction satis\fes the triple resonance\ncondition, we have taken \u000eh=!0\nmor!h=!v+!0\nm\nand ignored the fast-rotating terms provided 2 !0\nm>>\nG0\nom. We see that Hamiltonian 26 and 17 are identical.\nTherefore, the analysis that we have done for determining\nthe qubit states in the optomechanical system is also\napplicable here. The dissipative and non-dissipative\ndynamics studied in the optomechanical system and\nall the plots in Fig. 3 and 4 will be similar. The\noptomagnonic parameters that produce similar plots in\nFig. 3 areG0\nom= 0:5\u0014h,\rm= 0:1 Mhz,\u0014h= 0:01 GHz\nandn0\nth= 0:5.\nIV. CONCLUSION\nIn conclusion, we have studied quantum transduction\nof superconducting \rux-tunable transmon qubit in\ntwo hybrid systems: electro-optomechanical and\nelectro-optomagnonical system. The realization and\nadvancement of quantum transduction in such hybrid\nsystems are very crucial for the development of quantum\nnetwork, quantum internet, etc. The transduction is\ndone in two stages. First, we encode the qubit states\nin the coherent excitations of mechanical phonon or\nferromagnetic sphere magnon without disturbing the\nqubit state (non-demolition interaction) and in the next\nstage, we identify these excitations by counting the\naverage number of photon in the optomechanical oroptomagnonic WGM cavity. Because of the coherent\ninteraction between the phonon/magnon and the optical\nphoton, the average photon number oscillates with time.\nThe oscillation when the qubit is in the ground state\nand when in the excited state is exactly opposite. As a\nresult, we can make multiple measurements of the photon\nnumber at a regular interval of time and hence know\nthe state of the qubit at each interval. In the presence\nof dissipation, the optomechanical and optomagnonical\ncoupling strength should be atleast moderately strong in\norder to perform any measurements before the photon\nnumber altogether decays to zero. The required coupling\nstrength in the optomechanical system is extensively\nstudied. But, in the optomagnonic system, the required\ncoupling regime to perform the transduction is not yet\nexplored. However, the possibility of optomagnonic\ncoupling strength going upto 10 MHz is disscued in [29].\nOne of the ways to reach such coupling magnitude is to\nreduce the size of the YIG sphere to few \u0016m, which is\ncomparable to the size considered in the hybrid system\nproposed in this work.\nACKNOWLEDGEMENT\nRN gratefully acknowledges support of a research\nfellowship from CSIR, Govt. of India.\nAppendix A: Non-dissipative dynamics.\nThe analytical solution of Eq. 18 in the absence of\ndissipation is given by\nh^ay^ai=1\n2fj\u000b0j2+j\f0j2+ (j\u000b0j2\u0000j\f0j2)cos(2Gomt)\n\u0000i(\u000b\u0003\n0\f0\u0000\u000b0\f\u0003\n0)sin(2Gomt)g (A1)\nHere,j\u000b0j2=h^ay^ai0andj\f0j2=h^by^bi0are the initial\nvalues of photon and phonon/magnon. The above\nequation can be further simpli\fed by simply choosing\nj\u000b0j2=j\f0j2.\nh^ay^ai=j\u000b0j2\u0000Im(\u000b\u0003\n0\f0)sin(2Gomt) (A2)\nThe initial coherent amplitudes of phonon/magnon is\n\fxed at\f0=\u0006ig0\u001eac\u001c. To keep the oscillatory part\nin Eq. A2, which is necessary for the transduction, we\nrequire that the initial coherent amplitude of the cavity\nphoton\u000b0should have a non-zero real part. Therefore,\nwe choose\u000b0=g0\u001eac\u001c. The oscillation of Eq. A2 then\nbecomes\nh^ay^ai= (g0\u001eac\u001c)2f1\u0000sin(2Gomt)g; (A3)\nwhen the qubit is in the excited state ( \f0=\u0000ig0\u001eac\u001c),\nand\nh^ay^ai= (g0\u001eac\u001c)2f1 +sin(2Gomt)g; (A4)\nwhen the qubit is in the ground state ( \f0=ig0\u001eac\u001c).8\n[1] W. Jiang, C. J. Sarabalis, Y. D. Dahmani, R. N. Patel,\nF. M. Mayor, T. P. McKenna, R. Van Laer, and A. H.\nSafavi-Naeini, Nature communications 11, 1166 (2020).\n[2] M. Forsch, R. Stockill, A. Wallucks, I. 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However, an integrated magnonic circuit with several  separate magnonic elements has \nyet not been reported  due to the lack of a magnonic amplifier to compensate  for transport and \nprocessing  losses . The magnon transistor reported in [Nat. Commun. 5, 4700, (2014)] could only \nachieve a gain of 1.8, which is insufficient in many practical cases. Here, we use the stimulated \nthree -magnon splitting phenomenon to numerically propose a concept of magnon transistor in which \nthe energy of the gate magnons at 14.6 GHz is directly pumped into the energy of the source \nmagnons at 4.2 GHz, thus achieving the gain of 9. The st ructure is based on the 100 nm wide YIG \nnano -waveguides, a directional coupler is used to mix the source and gate magnons, and a dual-\nband magnon ic crystal is used to filter out the gate and idler magnons at 10.4 GHz frequency. The \nmagnon transistor preser ves the phase of the signal and the design allows integration into a magnon \ncircuit.  \n \n \nSpin wave s (magnons) , having  low intrinsic losses , high-frequency  range  (gigahertz  to \nterahertz) , and short wavelengths (down to several nanometers) , are promising candidates for data \ntransport and processing devices  [1-5]. In co ntrast  to sound and light waves, spin waves exhibit \nstronger  and more diverse  intrinsic nonlinear phenomena  [6-9], making  it easier to construct all -\nmagnonic integrated circuits, in which the magnons are controlled by the magnons themselves \nwithout any intermediate conversion  to electr ic currents . In the last decade, s everal individual \nmagnonic devices have been proposed such as  spin-wave logic gate [ 10,11], magnon  transistor  \n[12,13], majority gate  [14,15], magnon valve [1 6,17], and direction al coupler  [18,19]. However,  the \nrealization of an integrated magnonic network [20,21] with several separate magnonic devices is \nstill a challenge . The main reason is that spin wave amplitude decreases after passing through the \nupper -level device due to magnetic damping and could not reach the nonlinear threshold of the next -\nlevel device. The key element to solve this problem  is the magnonic amplifier, which compensates \nthe loss and brings the spin -wave amplitude back to the initial state.   \nIn recent years, researchers have explor ed effective ways to amplify propagating spin waves. \nOne of the  method s is based on the  reduction of  magnetic damping, which enhances  spin-wave  \nsignals by spin transfer torque and spin  orbit torque  generated by a DC current  [22,23]. Recent ly, \nMerbouche et al. [24] reported a true amplification of spin waves based on spin currents. In this \n \n*Author to whom correspondence should be addressed: williamqiwang@hust.edu.cn   \n research, the direction of the external field and the composition of the materials were  precisely tuned \nto avoid the occurrence of nonlinear magnon scattering and auto -oscillations.  Another common  \napproach  is paral lel parametric pumping,  in which one microwave photon  splits into two magnon s \nat half the frequency  under the conservation of energy and momentum  to amplify the propagating \nspin waves [25-27]. V ery  recently, Breitbach et al.  [28] proposed a spin -wave amplifier based on \nrapid cooling, which is a purely thermal effect. The magnon system is brought into a state of local \ndisequilibrium with an  excess of magnons. A propagating spin -wave packet reaching this region \nstimulates the subsequent redistribution process, and is in turn amplified.  \nIn modern CMOS electronics, transistors are used to amplify electrical signals. The magnon \ntransistor reported in [1 2] was designed to suppress one magnon flux by another in order to perform \nall-magnon logic operations. Since the suppression efficiency was very high, it was shown that using \na special interferometric scheme, the transistor could also be used for amplifica tion, but with a rather \nsmall gain of 1.8. Another magnon transistor concept suitable for direct amplification of the magnon \nfluxes is need ed. \nIn this letter, we propose a magnon transistor  concept based on stimulated three -magnon \nsplitting  and validate it using micromagnetic simulations . First, we study three -magnon scattering \nin a nanoscale straight magnonic waveguide with different external field  directions . A pronounced \nthree -magnon scattering is observed, in which one gate (pump) magnon  (14.6 GHz)  splits into two \nmagnons  (10.4  GHz and 4.2 GHz)  according to the laws of energy and momentum conservation . It \nis found that the additional injection of the source magnons at the frequency of 4.2  GHz leads to a \ndrastic enhancement of the splitting of the gate magnons, a phenomenon we refer to as stimulated \nmagnon splitting. Based on this mechanism, we propose a magnon transistor design that employs a \ndirectional coupler to mix gate and source magnons. In addition, a dual -band magnon crystal is used \nto filter out the gate (14.6 GHz) and idler (10.4 GHz) magnons from the transistor drain. The \ntransistor allows the drain magnon density to be 9 times higher than the source magnon density.  \nAs shown in Fig. 1 (a), we consider a yttrium iron garnet (YIG) waveguide  of length l = 20 m, \nwidth w = 100 nm , and thickness t = 50 nm . To investigate the effects of three -magnon splitting, \nmicromagnetic simulations are preformed using Mumax3 [29] with the following parameters of YIG  \n[30]: saturation magnetization  Ms = 1.4  105 A/m, exchange constant A = 3.5  10−12 J/m, and \ndamping coefficient  = 2  10−4. An external bias magnetic field Bext = 100 mT  is applied in the xy \nplane forming an angle H with the negative x-axis, as illustrated  in Fig. 1(a) . The maximum  splitting \nefficiency is obtained at approxi mately H = 60°, which correspond s to M = 45° (the angle between \nthe magnetization direction and the negative x-axis). It is a known feature of the three -magnon \ninteraction in an effectively one -dimensional system [31], which is additionally explained in the \nsupplementary materials.  The magnetization direction is not aligned with the direction of the \nexternal field due to the non negligible  demagnetic field along the width direction in  the nanoscale \nwaveguide . In the following studies, the external field is fixed at the optimal value H = 60°. The \nprorogating spin wave  in the waveguides is excited by an alternating field as a sinusoidal  function \nof time: hrf = bsin(2ft)ez with the oscillation amplitude  b and the excitation  frequency f. The applied \nfield is in the center of waveguide over an area of 30 nm in length  [blue area shown in Fig. 1(a)]. \nTo avoid spin wave  reflection, the damping coefficient has an exponential  increase to 0.5 at both \nends of the waveguide  [32]. In addition , all the simulations are performed with room temperature \nT = 300 K. We recorded the time-dependent magnetization data from t = 0 ns to 50 ns  across  the \nentire  wave guide  and then obtained the spin wave dispersion curves by using two-dimensional  fast  \n Fourier transform  [33,34].  \n \nFig. 1. (a)  A sketch of  spin-wave  excitation  system:  a 30-nm wide excitation region  \nis used to generate  oscillation magnetic  field hrf, and the in-plane magnetic field is  \napplied at H = 60° . Dispersion curve  of the propagating spin wave s and spin-wave \nintensity  as a function of frequency for  the different  amplitude s of excitation field  are \nshown in (b) b = 20 mT and  (c) b = 90 mT. \n \nIn the current study , the excitation  frequency fG is considered  to be  14.6 GHz.  The left part of \nFig. 1(b) and 1(c) shows the dispersive curves corresponding to different excitation field amplitudes . \nAt b = 20 mT, the dispersive curve exhibits only one weak bright spot [Fig. 1(b) left], and it can also \nbe seen that there is only one weak peak in the  spin-wave intensity  spectrum , corresponding to  the \ndirectly pump ed magnon  of frequency 14.6 GHz [Fig. 1(b) right]. When the amplitude b is increas ed \nto 90 mT, two other stronger bright spots at lower frequencies appear in the dispers ion curve \n[Fig.  1(c) left] . These  peaks , observed at 14.6, 10.4, and 4.2 GHz , correspon d to three distinct \nmagnons: the gate magnon fG (kG = 93.5 rad/m) and the scattering idler magnons  fI \n(kI = 73.1 rad/m) and source  magnons  fS (kS = 20.4 rad/m), as shown in  the right of  Fig. 1(c). This \nobserved process adheres to the energy -momentum conservation  laws, satisfying  fG = fI+ fS, kG = kI \n+ kS. This phenomenon is nothing else th an three -magnon splitting [ 31,35,36]. \nNow we consider stimulated three -magnon scattering, by exciting both the gate and one of \nthe split ting (source) magnons. Namely, we considered the excited field as hrf = bSsin(2fSt)ez + \nbGsin(2fGt)ez with one more excitation frequency fS = 4.2 GHz, and  collected the spin-wave  \nintensity  of drain magnons having frequency equal to the frequency of the source magnons  fD = \n \n fS = 4.2 GHz for the varying amplitude of excitation field bG in the range from 1 mT to 130 mT. As \nshown in Fig.  2, the spontaneous three -magnon splitting process appears when bG is greater than \n30 mT, see black line  (rigorously speaking, it is not absolute three -wave instability, but a convective \none [3 7, 38]). The red and blue lines  correspond to the stimulated process, and were obtained by \napplying two excitation frequencies of fG = 14.6 GHz and fS= 4.2 GHz . The spin-wave  intensity  \nspectra  of the red and blue lines are obtained by subtracting the energy of the directly excited \nmagnons at a frequency of 4.2 GHz, and thus show the same ground intensity  as the black line.  First, \nwe see, that stimulated process takes place at a lower gate power than the  spontaneous one and, in \nfact, does not demonstrate any threshold. This feature is expected for three -wave processes and, for \nmagnon system, in particular, was demonstrated in [3 6] by inspecting idler magnon density . \nFor the same excitation amplitude bG, the red and blue lines show stronger spin-wave  \nintensity  in contrast to the black line. This indicates that the existing magnon of frequency \nfS = 4.2 GHz can stimulate  not only the appearance of idler magnons (as shown in [3 6]), but also \nincreases the population of the source magnons itself , and the scattering efficiency depends on the \nsource magnon density.  In addition, the stronger density  of scattering magnon is achieved when the \namplitude bS of fS = 4.2 GHz is increased. This enhancement of the three -magnon splitting by \nintroducing one of the splitting magnons can be used for the amplification of propagation spin waves.  \n \nFiG. 2.  Intensity  of the scatter ing spin waves  at frequency of  fD = 4.2 GHz  as a function \nof the gate excitation field bG, which is proportional to  gate magnon density . The black  \nline corresponds to the case when only the gate magnons are directly excited at \nfG = 14.6 GHz. The red and blue  lines depict  the case when gate and source  spin wave s \nare simultaneously excited by  alternating field at fG = 14.6 GHz and fS = 4.2 GHz for \ntwo different densities of source magnons , driven by source excitation field  bS = 4 mT \n(red line) and bS = 10 mT (blue line).  \n \nEstimation of the amplification efficiency can be done in a way, similar to that used in \nnonlinear optics [3 7]. The maximum  amplification rate can be calculated in the conservative \napproximation, i.e. neglecting the damping. As derived in the supplementary materials, the \nmaximum  amplitude  of the drain magnons AD is given by:  \n \nG\nS2 2 2\nD,ma G x\nSvA A Av−= ,  (1) \nwhere AS and AG are initial amplitudes of source and gate magnons at fS and fG, respectively, while \nvS and vG are their group velocities ; amplitudes are defined as standard canonical amplitudes [ 31]. \nThis linear dependence of the gain on the gate power \n()22\nD,max SAA− ~ bG2 is nicely reproduced in the \nsimulations; only at high gate powers, above bG > 50 mT, the gain saturates due to the influence of \n \n other nonlinear effects, in particular, self - and cross - nonlinear frequency shift . We see, that the \namplification rate AD/AS is larger for smaller source amplitude AS, since the total power, which  can \nbe transferred to the source  wave, is limited by the gate magnon  power. Also, amplification \nefficiency increases with the ratio vG/vS. In the case of spin waves in waveg uides, higher -frequency \n(gate ) magnon almost always has larger  group veloc ity than lower -frequency (source ) magnon, \nimproving the amplification mechanism  (In our case, t he ratio vG/vS, estimated to be approximately \n6.8, is derived from the dispersive curve depicted in Fig. 1(c) ). Of course, damping decreases the \namplification efficiency and introduces a dependence on the transferred power on the initial power \nof source  magnons. Detailed consideration of this process lie s beyond the scope of this work . \nA magnon transistor is design ed as shown in Fig. 3(a) consisting of one magnonic directional  \ncoupler  [39] and a dual-band magnon ic crystal  with different periods  [40]. The directional coupler  \n[41] is employed for combining  spin wave  signals with different frequencies from two separated \nwaveguides into one waveguide  via frequency -dependent dipolar coupling strength [19,39]. The \ndirectional coupler is carefully design ed with the following  geometric dimensions:  the length of \ncoupled waveguides ( L1) is 390 nm , the angle between waveguides ( 𝛷) is 10° , the gap between \ncoupled waveguides ( δ) is 10 nm, and the edge -to-edge distance ( d1) is 200 nm , to make sure that \ngate magnon s can pass through it without significant  energy loss, and the source magnon s will be \ncompletely guided from the bottom waveguide to the top one as shown by the black and red arrows \nin Fig. 3(a) . Once the source magnon is coupled into the top waveguide and mixed with the gate \nmagnon, the  three -magnon splitting  is enhanced and the gate magnon efficiently splits into an idler \nmagnon (10.4  GHz)  and a drain magnon with the frequency of 4.2 GHz resulting in an amplification \nof the source magnon.   \nFigure  3(b) shows the working principle of the magnon  transistor . When only the gate \nmagnon ( fG = 14.6 GHz, bG = 40 mT) is applied, the two -dimensional colormap shows the weak  \nmagnon density  of 4.2 GHz due to the low three magnon splitting efficiency  [top of Fig. 3(b) ]. The \nmiddle  panel of Fig.  3(b) shows the case of only the source magnon is injected from the bottom \nwaveguide. It clearly shows that all the source magnons are guided from the bottom waveguide to \ntop waveguide (transistor’s drain) by the directional coupler  as expected . Once both gate and source \nmagnons are simultaneously excited , the drain magnon  density  at 4.2 GHz is dramatically increased \ndue to the stimulation of the three magnon splitting  [botto m of Fig. 3(b) ]. In order to get only the \nflux of drain magnon s at the output, a specially designed dual -band  magnonic crystal , in the form \nof two in -line connected magnonic crystals of different periodicities, is used to filter out the gate \nand idler magnon s produced by three magnons  splitting  [42]. The design principle of magnonic \ncrystals is discussed in detail in supplementary materials. Figure  3(c) shows the magnetization \noscillations spectr a extracted from the end of the top waveguide  as marked by the red dashed \nrectangular regions  in Fig. 3(a) . It shows two properties: (1) The source magnon has a gain factor \nof 9. (2) The gate magnon and idler magnon have been efficiently suppressed by the dual-band \nmagnon ic crystal . Importantly, that such a large gain is observed for moderate source magnon power, \nwhich is required for logic applications  – amplification of nonlinear spin waves is much more \nnontrivial task than of small -amplitude ones [ 43]. Furthermore, the amplification is not s ensitive  to \nthe relative phase between the gate and source magnons (see the supplementary materials). \nTherefore, the output signal can be directly used to connect the next logic gate without any phase \nmodulations and the proposed magnonic transistor is suitable for further integration.   \n  \nFig. 3. (a) Schematic of magnon  transistor  designed with  a directional coupler with  \ndual-band magnon ic crystal . (b) The intensity distributions  of spin waves  at 4.2 GHz \nand (c) the magnetization oscillation  spectrum extracted at the end of the top \nwaveguide marked by red dashed rectangular regions  for different  cases.  \n \nIn summary, we numerically  demonstrate magnon transistor based on the phenomenon of \nstimulated three -magnon splitting within a nano -scaled waveguide with an in -plane inclined static  \nexternal  field. The three -magnon  splitting efficiency  of the gate magnons is substantially  enhance d \nby directly introducing one of the signal magnons . A magnon directional coupler is used to mix the \ngate and source magnons, and a dual -band magnon crystal is used to filter the idler and gate magnons \nfrom the drain of the transistor. The transistor has a gain of 9, measured as the ratio of the drain \nmagnon density normalized to the source magnon density. The phase of the drain magnons depends \nonly on the phase of the source magnons,  and the design of the transistor with separate source, gate \nand drain magnon conduits makes it suitable for further integration into a complex magnonic \nnetwork.  This device represents a step forward in the amplification of propagating spin waves in \nnanoscal e waveguides, offering promising avenues for the advancement of spintronic applications.  \n \nThis work was  support ed from the National Key Research and Development Program of China \n(Grant No. 2023YFA1406600) and National Natural Science Foundation of China, the startup grant \nof Huazhong University of Science and Technology (Grant  No. 3034012104).  R.V . Acknowledge \nsupport of the MES of Ukraine.  \n \n \n \n \n Reference  \n[1] A. V . Chumak et al., IEEE Trans. Magn. 58, 1 (2022) . \n[2] P. Pirro, V . I. V asyuchka, A. A. Serga, and B. Hillebrands, Nat. Rev. Mater.  6, 1114 (2021).  \n[3] B. Dieny, et al. , Nat. Electron.  3, 446 (2020).  \n[4] A. Mahmoud, F. Ciubotaru, F. V anderveken, A. V . Chumak,  S. Hamdioui, C. Adelmann, and S. \nCotofana, J. Appl. Phys.  128, 161101  (2020).  \n[5] H. Qin, R. B. Hollander, L. Flajsman, F. Hermann, R. Dreyer, G. 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Beck, and B. Hillebrands, Appl. Phys. Lett.  95, 262508 (2009).  \n[41] A. V . Sadovnikov, S. A. Odintsov, E. N. Beginin, S. E.  Sheshukova, Y . P . Sharaevskii, and S.A. \nNikitov, Phys. Rev. B  96, 144428 (2017).  \n[42] K.-S. Lee, D. -S. Han, and S. -K. Kim, Phys. Rev. Lett. 102, 127202 (2009).  \n[43] R. Verba, M. Carpentieri, G. Finocchio, V . Tiberkevich, A. Slavin, Appl. Phys. Lett.  112, \n042402 (2018).  \n  \n Supplementary Materials for  \n“Nanoscaled magnon transistor based on stimulated three -magnon \nsplitting ” \nXu Ge1, Roman V erba2, Philipp Pirro3, Andrii V . Chumak4, Qi Wang1 \n1 School of Physics, Huazhong University of Science and Technology, Wuhan, China  \n2 Institute of Magnetism, Kyiv, Ukraine  \n3 Fachbereich Physik and Landesforschungszentrum OPTIMAS, Rheinland -Pfälzische Technische \nUniversität Kaiserlautern -Landau, Kaiserslautern, Germany  \n4 Faculty of Physics, University of Vienna, Vienna, Austri a \n \nS1. Identification the inclination angle H based  on the scattering intensity  \n \nFigure S1. The scattering spin -wave intensity as function of the angle M the angel \nbetween the magnetization direction and negative x-axis (bottom transverse axis ). \n \nThe three -magnon Hamiltonian, governing considered process, is derived as   \n \n( ( )(3) *\n12,3 1 2 3 1 2 3\n123 c.c.) V c c c= +  + − k k k H    (S1) \nwhere ki = kiex is spin-wave  wave vector  (1, 2, and 3  denote  the three magnons) , directed along the \nwaveguide  (x direction). Spin-wave amplitudes ci are defined as usual canonic variables [1], i.e. \nhave an order of dimensionless dynamic magnetization. In our case, magnons 1, 2, and 3 are source, \nidle and gate magnons, respectively.  \n The three -magnon coefficient  V12,3 could  be found using scalar [1] or vectorial [2] spin-wave  \nHamiltonian formalism. General expression is quite cumbersome and is  not presented here. \nHowever, using a simplified expression for the dynamic demagnetization tensor equal to one of a  \nfilm, and neglecting minor ellipticity -related term, three -magnon coefficient is derived as  \n \n()() ( ) 12,3 1 2cos sin\n22M M MV f k t f k t  + ,  (S2) \nwhere  𝜔𝑀=𝛾𝜇0𝑀𝑆, with 𝛾 representing  the gyromagnetic ratio, 𝜇0 as the vacuum permeability , \nand 𝑀𝑆  as the saturation magnetization . Additionally , \n( ) 1 (1 exp[ | |])/ | |f x x x= − − −   is the so -\n \n called thin -film function.  Consequently, the calculated value of  M = 45° corresponds to the \nmaximum strength of three -magnon splitting determined by the above theoretical equation of the \nscattering coefficient  [Eq. ( S2)]. Furthermore , as seen in Fig. S1, we extracted the magnetization \ndirection angle M and the intensity of lower frequency scattering magnons by varying the direction \nof the in -plane external field from H = 0° to H = 90°. The strong est peak is observed at around \nM = 45°, which  is consistent with  the above theoretical analysis , in turn, the corresponding angle  \nH is found to be around 60° (Fig. S1 ), and hence , a tilt angle  H = 60° of the external field is \nutilized to investigate the magnon  scattering in the main text.  \n \nS2 Spin -wave amplification by three -magnon scattering  \nUsing well -developed approach from nonlinear optics [3], one can derive equations for spatial -\ntemporal evolution of  spin waves  envelope amplitudes ai = ai (x, t): \n \n( )\n( )\n( )*\n1 1 1 2 3\n*\n2 2 2 1 3\n3 3 3 1 22\n2\n2,\n,\n.tx\ntx\ntxv a iVa a\nv a iVa a\nv a iVa a +  + =−\n +  + =−\n +  + =−   (S3) \nHere vi is spin-wave  group velocity,  i the damping rate, \n12,3 VV , and  equations are written for \nthe resonant case, i.e.  when f3 = f1+ f2.  \nEq. (S3) does not allow for exact analytical solution. Let’s first look on the amplification \nconditions.  For this, assume the pumping wave amplitude a3 to be constant, signal wave initial value  \na1 (x = 0) = A1 and absent  idler wave a2 (x = 0) = 0. Also, stationary regime (\n0t→ ) is considered.  \nThen, solution of the first two equations of  Eq. (S3) is a simple sum of exponents  \n1 2\n1 1 2~xxa c e c e+ , \nwhere   \n \n( )( )( )2 2\n1,2 1 2 2 2 1 2 2 2 1 2 3 1 2\n12144 v v v v v v VAvv= −  +    +  + − .  (S4) \nIt is clear, that if  \n3 1 22VA   , one of the exponent becomes positive,  κ1 > 0, meaning \namplification of signal  wave. This condition is independent of initial amplitude of the signal spin-\nwave  and is common for any parametric pumping  process . \nTo estimate amplification rate, we look on conservative approximation of Eq. ( S3), i.e. neglect \nthe damping. Then,  introducing new variables  \n2/i i j lb Va v v= , j, l\ni, this system is reduced to \nstandard form:   \n \n*\n1 2 3\n*\n2 1 3\n3 1 2,\n,\n.x\nx\nxb ib b\nb ib b\nb ib b =−\n =−\n =−    (S5) \nExact solution of this system is expressed via elliptic functions and could be found in [3]. Here we \nnote only that this system possesses three integrals of motion (strictly speaking, only two of which \nare independent), called the Manley -Rowe relations:  \n22\n12( ) ( )b x b x−  = const, \n22\n13( ) ( )b x b x+  = \nconst, and \n22\n23( ) ( )b x b x+  = const. It is clear, that the  powers of signal and idler wave change  \n synchronously – either both increase taking energy from the pumping, either  both decrease when \nthe energy flows back to the pumping wave. Also, the power of signal wave cannot be smaller  than \nthe the initial power, as the initial power of idler wave is zero. Thus, signal wave is always amplified \nin the  conservative approximation. The maximal power is  \n222\n1 1 3max(0) (0) b b b=+ . Returning to \nthe initial variables, we  get the gain  \n \n2 22 3\n1 1 3max\n1va A Av−= ,   (S6) \nwhich is presented in the main text [Eq. (1) ] accounting for the notation of gate, source and drain \nspin-wave amplitudes.  \nS3. Frequency filtering with nanostrip magnonic crystals  \n \nFigure S2 . (a) Schematic of magnonic  filter: spin waves are generated in the blue \narea of magnonic waveguide, and filtered  out in the dual-band magnonic crystals \n(labeled as MC 1 and MC 2) region.  (b) Intensities of propagating spin  wave s at \ndifferent positions along the waveguide for the indicated frequenc ies of 14.6 , 10.4, \nand 4.2 GHz . \n \nThe propagation characteristics of spin waves can be controlled through spatially periodic \nmodulation in magnonic crystals, effectively serving as spin -wave filters due to band gaps  [4,5]. As \nseen in Fig. S2 (a) and Fig. 3 (a) of the main text , we demonstrate  the dual-band magnonic filter \ncomposed of two parts . One is the 100 -nm wide spin-wave propagation waveguide  in the left and \nthe other parts are two  magnonic crystals (MC 1 and MC 2) with periodicity P1 = 40 nm and P2 = 50 \nnm, respectively.  Fig. S2 (b) presents  the calculated  intensities of spin waves  propagating along the \nwaveguide . After traveling  through MC 1 and MC 2, the spin -wave intensities with frequencies of  \n14.6 and 10.4 GHz  (black and blue lines) are reduced by more than 70% , while spin wave  with \nfrequency of 4.2 GHz (red line) can propagate  through the two magnonic crystals without additional \nloss. \n \nS4. Dependence of the output spin -wave intensity on the relative phase between the pumping \nand source magnons  \nFor construction of an integrated magnonic circuit, it is hard to accurately control the relative \nphase between the pumping and source magnons. T o elucidate the relationship between the output \nspin-wave intensity and the relative phase between the pumping and source magnons, we introduced \n \n a variable, denoted as , representing the relative phase between two magnons . Subsequently, we \nrecorded the output spin -wave intensity at 4.2 GHz for various  values spanning from 0 �� to 360°. \nThe results , depicted in Fig. S3 , indicate that the output spin -wave intensity exhibits relatively low \nsensitivity to alterations in the relative phase between the gate and source magnons.  The results \nsuggest that the magnonic amplifier based on three -magnon scattering has potential to help to \nconstruct an integrated magnonic circuits.  \n \nFigure S3 The output spin -wave intensity  with frequency of  4.2 GHz  as function of  \nthe varied relative phase between the gate and source magnons  \n \nReference:  \n[1] P. Krivosik and C. E. Patton, Phys. Rev. B  82, 184428  (2010).  \n[2] V . Tyberkevych, A. Slavin, P . Artemchuk, and G. Rowlands,  ArXiv :2011.13562.  \n[3] N. Bloembergen, Nonlinear optics  (Addison -Wesley Pub. Co., 1965).  \n[4] K.-S. Lee, D. -S. Han, and S. -K. Kim, Phys. Rev. Lett.  102, 127202  (2009 ). \n[5] S.-K. Kim, K. -S. Lee, and D. -S. Han, Appl. Phys. Lett.  95, 082507  (2009).  \n"    },    {        "title": "1301.3266v2.Spin_Hall_Magnetoresistance_in_Platinum_on_Yttrium_Iron_Garnet__Dependence_on_platinum_thickness_and_in_plane_out_of_plane_magnetization.pdf",        "content": "arXiv:1301.3266v2  [cond-mat.mes-hall]  8 May 2013Spin-Hall Magnetoresistance in Platinum on Yttrium Iron Ga rnet: Dependence on\nplatinum thickness and in-plane/out-of-plane magnetizat ion\nN. Vlietstra, J. Shan, V. Castel, and B. J. van Wees\nUniversity of Groningen, Physics of nanodevices, Zernike In stitute for Advanced Materials,\nNijenborgh 4, 9747 AG Groningen, The Netherlands.\nJ. Ben Youssef\nUniversit´ e de Bretagne Occidentale, Laboratoire de Magn´ etisme de Bretagne CNRS, 6 Avenue Le Gorgeu, 29285 Brest, Fra nce.\n(Dated: September 24, 2018)\nThe occurrence of Spin-Hall Magnetoresistance (SMR) in pla tinum (Pt) on top of yttrium iron\ngarnet (YIG) has been investigated, for both in-plane and ou t-of-plane applied magnetic fields and\nfor different Pt thicknesses [3, 4, 8 and 35nm]. Our experimen ts show that the SMR signal directly\ndepends on the in-plane and out-of-plane magnetization dir ections of the YIG. This confirms the\ntheoretical description, where the SMR occurs due to the int erplay of spin-orbit interaction in the\nPt and spin-mixing at the YIG/Pt interface. Additionally, t he sensitivity of the SMR and spin\npumping signals on the YIG/Pt interface conditions is shown by comparing two different deposition\ntechniques (e-beam evaporation and dc sputtering).\nPACS numbers: 72.25.Ba, 72.25.Mk, 75.47.-m, 75.76.+j\nI. INTRODUCTION\nPlatinum (Pt) is a suitable material to be used as a\nspin-current to charge-currentconverterdue to its strong\nspin-orbit coupling.1A spin current injected into a Pt\nfilm will generate a transverse charge current by the In-\nverse Spin-Hall Effect (ISHE), which can then be electri-\ncally detected. The ISHE has been used to detect for ex-\nample spin pumping into Pt from various materials such\nas permalloy2(Py) and yttrium iron garnet (YIG).3–5\nFor the opposite effect, to use Pt as a spin current in-\njector, a charge current is sent through the Pt, creating\na transverse spin accumulation by the Spin-Hall Effect\n(SHE).6–8\nRecently, Weiler et al.9and Huang et al.10observed\nmagnetoresistance (MR) effects in Pt on YIG and re-\nlated those effects to magnetic proximity. These MR ef-\nfects havebeen further investigatedby Nakayamaet al.11\nand they found and explained a new magnetoresistance,\ncalledSpin-HallMagnetoresistance(SMR).11,12Achange\nin resistance due to SMR can be explained by a com-\nbination of the Spin-Hall Effect (SHE) and the Inverse\nSpin-Hall Effect (ISHE), acting simultaneously. When\na charge current /vectorJeis sent through a Pt strip, a trans-\nverse spin current /vectorJsis generated by the SHE following\n/vectorJe∝/vector σ×/vectorJs,13–16where/vector σis the polarization direction\nof the spin current. Part of this created spin current is\ndirected towards the YIG/Pt interface. At this interface\nthe electrons in the Pt will interact with the localized\nmoments in the YIG as is shown in Fig. 1. Depend-\ning on the magnetization direction of the YIG, electron\nspins will be absorbed ( /vectorM⊥/vector σ) or reflected ( /vectorM∝bardbl/vector σ). By\nchanging the direction of the magnetization of the YIG,\nthe polarization direction of the reflected spins, and thus\nthedirectionoftheadditionalcreatedchargecurrent,can\nbe controlled. A charge current with a component in thedirection perpendicular to /vectorJecan also be created, which\ngenerates a transverse voltage.\nIn this paper, the angular dependence of the SMR in\nPt on YIG is investigated for different Pt thicknesses\n(3, 4, 8 and 35nm) and different deposition techniques\n(e-beam evaporation and dc sputtering), for applied in-\nplane as well as out-of-plane magnetic field sweeps, re-\nvealing the full magnetization behaviour of the YIG.17\nAll measurements are performed at room temperature.\nThe magnitude of the SMR is shown to be dependent on\nthe magnetization direction of the YIG, as well as on the\nPt thickness, indicating its relation to the spin diffusion\nlength. Also the used deposition technique is found to be\nan important factor for the magnitude of the measured\nsignals.\nMJe\nMJabsJreflPt Pt \nYIG YIGa) b) \nMPt \nYIG JabsJreflc) \nJe\nJe\nFIG. 1. Schematic drawing explaining the SMR in a YIG/Pt\nsystem. (a) When the magnetization /vectorMof YIG is perpen-\ndicular to the spin polarization /vector σof the spin accumulation\ncreated in the Pt by the SHE, the spin accumulation will be\nabsorbed ( /vectorJabs) by the localized moments in the YIG. (b) For\n/vectorMparallel to /vector σ, the spin accumulation cannot be absorbed,\nwhich results in a reflected spin current back into the Pt,\nwhere an additional charge current /vectorJreflwill be created by\nthe ISHE. (c) For /vectorMin any other direction, the component\nof/vector σperpendicular to /vectorMwill be absorbed and the component\nparallel to /vectorMwill be reflected, resulting in a current /vectorJrefl\nwhich is not collinear with the initially applied current /vectorJe.2\nII. SAMPLE CHARACTERISTICS\nPt Hall bars with thicknesses of 3, 4, 8, and 35nm were\ndeposited on YIG by dc sputtering. Similar Pt Hall bars\nwerealsodepositedonaSi/SiO 2substrate,asareference.\nFinallyasamplewasfabricatedwherealayerofPt(5nm)\nwas deposited on YIG by e-beam evaporation. Fig. 2(a)\nshows the dimensions of the Hall bars. The thickness\nof the deposited Pt layers was measured by atomic force\nmicroscopy with an accuracy of ±0.5nm. The used YIG\n(single-crystal) has a thickness of 200nm and is grown\nby liquid phase epitaxy on a (111) Gd 3Ga5O12(GGG)\nsubstrate. By using a vibrating sample magnetometer,\nthe magnetic field dependence of the magnetization was\ndetermined, as shown in Fig. 2(b). The magnetic field\ndependence shows the same magnetization behaviour for\nall in-plane directions, indicating isotropic behaviour of\nthe magnetization in the film plane, with a low coercive\nfield of only 0.06mT. To saturate the magnetization of\nthisYIGsampleintheout-of-planedirection, anexternal\nmagnetic field higher than the saturation field ( µ0Ms=\n0.176T)5has to be applied.\nYIG(111) [200nm] Pt [3, 4, (5), 8, 35 nm]\n800µm20 µm\n100 µm 100 µma) b) \n-0.4 -0.2 0 0.2 0.4 -1.0 -0.5 00.5 1.0 \n-B c\n  M / Ms\nB [mT] +Bc\nFIG. 2. (a) Schematics of the used Pt Hall bar geometry. (b)\nIn-plane magnetic field dependence of the magnetization M\nof the pure single-crystal of YIG. Bcindicates the coercive\nfield of 0.06mT.\nIII. RESULTS AND DISCUSSION\nA. In-plane magnetic field dependence\nFirst, the longitudinal resistance of the Pt strip was\nmeasured (using a current I= 100µA) while sweeping\nan externally applied in-plane magnetic field. For subse-\nquent measurements the magnetic field was applied for\ndifferentin-planeangles α, asdefinedinFig. 3(a). Asthe\nin-plane magnetization of YIG shows isotropic behaviour\nwith a coercive field Bcof only 0.06mT, its magnetiza-\ntion will easily align with the applied in-plane magnetic\nfield. It was observed that the measured longitudinal re-\nsistanceisdependent onthe directionofthe appliedmag-\nnetic field, and thus of the magnetization direction of the\nYIG, as can be seen in Fig. 3(c) for the YIG/Pt [4nm]\nsample. For clarity, a background resistance R0of 1007-\n1008Ω was subtracted in the plots (the small change in\nR0between different measurementsoccurreddue to ther-\nmal drift). A maximum in resistance was observed whenthe magnetic field was applied parallel to the direction of\nthe charge current Je(α= 0◦). The resistance was min-\nimized for the case where BandJewere perpendicular\n(α= 90◦). These results are consistent with the SMR\nas described by Fig. 1 and as observed by Nakayama\net al.11. The measured resistivity for the longitudinal\nconfiguration can be formulated as11\nρL=ρ0−∆ρmy2(1)\nwhereρ0is a constant resistivity offset, ∆ ρis the mag-\nnitude of the resistivity change, which can be calculated\nfrom the measurements, giving ∆ ρ= 2×10−10Ωm, and\nmyis the component of the magnetization in the y-\ndirection.\nThesameexperimentswererepeatedforthetransverse\nresistance, where the resistance wasmeasured perpendic-\nular to the current path as shown in Fig. 3(b). Also in\nthis configuration it was found that the measured resis-\ntance depends on the direction of the applied in-plane\nmagnetic field, as shown in Fig. 3(d) for the YIG/Pt\n[4nm] sample. Here a maximum resistance is observed\nforα= 45◦, and a minimum for α= 135◦. The observed\nSMR resistivity for the transverse configuration can be\nformulated as11\nρT= ∆ρmxmy (2)\nwheremxis the component of the magnetization in\nthex-direction. From the shown measurements, a ratio\n∆RL/∆RT≈7 is found, which is close to the expected\nratio of 8 following from equations (1) and (2).\nFor both the longitudinal and the transverse configu-\nration, there is a peak and/or dip observed around + Bc\nfor all measurements. This can also be explained by the\nabove described SMR. While sweeping the magnetic field\n(here from negative to positive B), the magnetization of\ntheYIG willchangedirectionwhenpassing+ Bc(seeFig.\n2(b)). Due to its in-plane shape anisotropy, the magne-\ntization of the YIG will rotate fully in-plane towards B.\nThis rotation of Mresults in a change in measured resis-\ntance, passing the maximum and/or minimum possible\nresistance,whichisobservedasapeakand/ordiparound\n+Bc(when sweeping the field from positive to negative\nB, a peak/dip will occur at - Bc). Similar features were\nnot observed by Huang et al.10and Nakayama et al.11.\nThey do observe some peaks and dips, but these do not\ncover the maximum and minimum possible resistances,\nand thus do not show the full rotation of the magnetiza-\ntion in the plane. The absence of the full peaks and dips\ncan be explained by different magnetization behaviour\nof their YIG samples, showing higher coercive fields and\nswitching of the magnetization which is probably domi-\nnated by non-uniform reversal processes.\nTheresistancemeasurementsforthe in-planemagnetic\nfields were repeated for all different samples. A sum-\nmary of these measurements is shown in Fig. 3(e). Here\n∆RLis defined as the difference between the maximum\n(α= 0◦) and minimum ( α= 90◦) measured longitudinal3\na) b) \nYIG(111) [200nm] V\n-B \nα\nYIG(111) [200nm] JeV\n-B \nα\nα = 0°\nα = 45°\nα = 90°\nα = 135°\nα = 180°α = 0°\nα = 45°\nα = 90°\nα = 135°\nα = 180°d) c) \ne) 400420440460\n400420440460\n400420440460\n400420440460\n-3 -2 -1 0 1 2 3400420440460 \n RT [mΩ]   \nB [mT] ∆RT∆RL\n0 5 10 15 20 25 30 35 40 4502468 \nSiO2/Pt:\n Pt Sputtered\nYIG/Pt:\n Pt Sputtered\n Pt Evaporated∆R L / R0\nPt thickness [nm]x 10 - 4-0.4-0.3-0.2-0.10.0\n-0.4-0.3-0.2-0.10.0\n-0.4-0.3-0.2-0.10.0\n-0.4-0.3-0.2-0.10.0\n-3 -2 -1 0 1 2 3-0.4-0.3-0.2-0.10.0 \n R L - R0 [Ω]   \nB [mT]xy\nJe\nFIG. 3. Results of the in-plane magnetic field dependence of\nthe resistance of the Pt strip with a thickness of 4nm. Config-\nuration for (a) longitudinal and (b) transverse resistance mea-\nsurements. (c) and (d) show the measured resistance of the\nPt strip while applying an in-plane magnetic field for differe nt\nanglesα, for thelongitudinal and transverse configuration, re-\nspectively. R0has a magnitude of 1007-1008Ω. (e) Thickness\ndependence of the measured magnetoresistance for YIG/Pt\nand SiO 2/Pt samples. ∆ RLis defined as the maximum differ-\nence in longitudinal resistance ( RL(α= 0◦)−RL(α= 90◦))\nandR0isRL(α= 0◦). The solid red line is a theoretical\nfit.11,12\nresistance and R0isRL(α= 0◦). The shown thickness\ndependent measurements are in agreement with data as\npublished by Huang et al.10, though they do not relate\ntheir results to SMR. The red line shows a theoretical\nfit11,12of the SMR signal. The position and width of the\npeak are mostly determined by the spin relaxationlength\nλofPt, andthemagnitudeofthesignalbyacombination\nof the spin-Hall angle θSHand the spin-mixing conduc-tanceG↑↓of the YIG/Pt interface. For the shown fit,\nλ= 1.5nm,θSH= 0.08,G↑↓= 1.2×1014Ω−1m−2and\na thickness dependent electrical conductivity as used in\nRef18, were used.\nWhen YIG is replaced by SiO 2, the SMR signal to-\ntally disappears, showing the effect is indeed caused by\nthe magnetic YIG layer. More notable, the e-beam evap-\norated Pt layer on YIG did show only a very low SMR\nsignal (≈10−5). This suggests that the spin-mixing con-\nductance (which is determined by the interface)19is an\nimportant parameter for the occurrence of SMR.\nB. Out-of-plane magnetic field dependence\nTo further investigate the characteristics of the Pt\nlayer, also the transverse resistance was measured while\napplying an out-of-plane magnetic field, as shown in\nFig. 4(a). The Pt layers on the Si/SiO 2substrate\nshowed linear behaviour with transverse Hall resistances\nof 1.3, 0.9 and 0.3 ±0.05mΩ for Pt thicknesses of 4,\n8 and 35nm, respectively, at B= 300mT. These re-\nsults, due to the normal Hall effect, are in agreement\nwith the theoretical description RHall=RHB/d, where\nRH=−0.23×10−10m3/C is the Hall coefficient of Pt20\nanddis the Pt thickness.\nFor the YIG/Pt samples, results of the out-of-plane\nmeasurements are shown in Fig. 4(b). At fields lower\nthan the saturation field, a large magnetic field depen-\ndence is observed. The magnitude of this dependence de-\ncreases with Pt thickness and disappears for the thickest\nPt layer of 35nm. The occurrence of this magnetic field\ndependence can be explained by the SMR, using the re-\nsults of the in-plane measurementsas shownin Fig. 3(d),\nbecause for applied fields lower than the saturation field,\nthe magnetization of the YIG will still have an in-plane\ncomponent. To investigate its effect on the transverse\nresistance measurements, the direction of the in-plane\nmagnetization in the YIG should be known. To achieve\nthis, the external magnetic field was applied with a small\nintended deviation φfrom the out-of-plane z-direction\ntowards the - y-direction as defined in Fig. 4(a). This\nsmall deviation results in a small in-plane component of\nthe applied field, which controls the magnetization di-\nrection of the YIG. Using this configuration the sign of\nthe signal due to the SMR can be checked according to\nFig. 3(d) by varying the direction of the in-plane com-\nponent of the applied magnetic field. Fig. 4(c) shows\nresults applying an external field fixing φ=−1◦for var-\nious angles θ, whereθis an additional rotation from the\nz- towards the x-direction. According to the theory of\nthe SMR and also comparing the results shown in Fig.\n3(c), a maximum additional resistance due to SMR is\nexpected for an in-plane magnetic field with α= 45◦,\nwhich is the direction of the in-plane component when\napplying a magnetic field choosing φ=−1◦andθ= 1◦.\nSimilarly, for φ=θ=−1◦, the in-plane component of\nthe field will be α= 135◦, resulting in a minimum addi-4\na) b) \nc) YIG(111) [200nm] θ-B V\nJeϕ\nd) \n-300 -200 -100 0 100 200 300-30-20-100102030\nϕ = -1 ο \n  -10 ο\n  -1 οRT [mΩ]\nB [mT]  1 ο\n  15 ο\n  45 ο\n  90 οθ = YIG/Pt [4nm]z\n- yx\n-300 -200 -100 0 100 200 300-10010203040506070θ = 1 ο\nϕ = -1 ο \n  YIG/Pt:\n 3 nm\n 4 nm\n 8 nm\n 35 nmRT [mΩ]\nB [mT]\n-300 -200 -100 0 100 200 300-30-20-100102030\nθ =-1 ο  ϕ = -1 ο  RT [mΩ]\nB [mT]θ =1 ο  YIG/Pt [4nm]\nFIG. 4. Results of the out-of-plane magnetic field dependenc e\nof the transverse resistance. (a) Configuration for the tran s-\nverse resistance measurements. φis definedas a rotation from\nthez- towards the - y-direction, whereas θgives a rotation\nfrom the z- towards the x-direction. (b) Magnetic field de-\npendence of the transverse resistance for different thickne sses\nof Pt on top YIG, for φ=−1◦andθ= 1◦. (c) Dependence of\nthe transverse resistance on θ, fixingφ=−1◦, pointing out\nthe effect of the direction of the in-plane component of the ap -\nplied magnetic field on the observed signal. (d) Theoretical\nfitsoftheSMRsignal forout-of-planeappliedfieldslowerth an\nthe saturation field, assuming a linear background resistan ce,\nas shown by the dotted red line. For all shown measurements,\na constant background resistance of 10-900mΩ is subtracted .\ntional resistance. Results as shown in Fig. 4(c) confirm\nthat the sign and magnitude of the magnetic field depen-\ndence are consistent with the SMR observed for in-plane\nfields. The shape of the curve can be explained by the\ndependence of the resistance on the direction of M, as\nonly the component of σparallel to M(σM) will be re-\nflected. For out-of-plane applied fields, σMis given by\nσM=σcosβcosα, whereβis the angle by which Mis\ntilted out of the x/y-plane. Using the Stoner-Wohlfarth\nModel,21for an applied field in the z-direction, it was\nderived that β= arcsin( b), where b=B/BsandBsis\nthe saturation field. Assuming that the transverse re-\nsistivity change due to SMR scales linearly with the in-\nplane component of σM(σM,in−plane=σMcosβ), this\ngives (for applied fields close towards the z-direction and\nφ=θ=±1)\nρT=±1\n2∆ρ(1−b2) (3)\nTwo fits using this equation are shown in Fig. 4(d).\nFor both fitted curves, an assumed linear background re-\nsistance,asindicatedbythedottedredline, isalsoadded.\nThederivedfitsareingoodagreementwiththemeasured\ndata for applied fields below the saturation field, which\nconfirms the presence of SMR and its dependence on the\nmagnetization direction,12,22,23also for out-of-plane ap-\nplied fields.\nAlso for the out-of-plane measurements a peak and/or\na dip is observed at zero applied field. These peaks anddips have the same origin as those observed for the in-\nplane measurements, which is the rotation of the mag-\nnetization in the plane towards the new magnetic field\ndirection.\nFor applied magnetic fields above the saturation field\nno in-plane component of M is left, but still a small mag-\nneticfielddependenceisobserved. At B= 300mT,trans-\nverse resistances of 10.1, 5.1, 1.5 and 0.3 ±0.05mΩ were\nmeasured for Pt thicknesses of 3, 4, 8 and 35nm, respec-\ntively. So for thin Pt layers, at applied fields above the\nsaturation field, an increased transverse resistance is ob-\nserved compared to the SiO 2/Pt sample. Possible origins\nof this difference might be related to the imaginary part\nof the spin-mixing conductance, or to the (spin-) anoma-\nlous Hall effect.\nC. Comparison of e-beam evaporated and dc\nsputtered Pt\nAdditional to the thickness and angular dependence\nof the SMR signal, also the difference in signal for two\ndeposition techniques, e-beam evaporation and dc sput-\ntering was investigated. It was observed that the e-beam\nevaporated Pt layer did show very low SMR effects com-\npared to the sputtered layers. To compare, Fig. 5(a)\nshows the out-of-plane transverse measurement for both\nthe sputtered [4nm] and evaporated [5nm] Pt layers. The\nvalue of the signal at applied fields higher than the satu-\nration field is the same, but the additional signal which\nis described to SMR is lowered by a factor 7.\nAs the evaporated Pt layer showed lower SMR signals\ncompared to the sputtered Pt layers, the effect of using a\ndifferentdepositiontechniqueonthe spinpumping/ISHE\nsignalwas also investigated. By using a rf-magnetic field,\nthe magnetization of the YIG was brought into reso-\nnance. During resonance, a spin current will be pumped\ninto the Pt layer where it will be converted in a charge\ncurrent by the ISHE. A more detailed description of the\nused measurement technique can be found in ref.5. Fig.\n5(b) shows a measurement of the spin pumping voltage\nfor both e-beam evaporated Pt and dc sputtered Pt on\nYIG. A rf-frequency and power of 1.4GHz and 10mW,\nrespectively, were used to excite the magnetization pre-\ncession in the YIG. The same measurement was repeated\nfor different rf-frequencies between 0.6 and 4GHz, all at a\npower of 10mW (not shown). For all measurements, the\nspin pumping signalofthe evaporatedPt layerwasfound\nto be a factor 12 smaller than the signal of the sputtered\nlayer. This change in magnitude of the signal shows the\ndifference of the YIG/Pt interface between both deposi-\ntion techniques, determining a probable difference in the\nspin-mixing conductance. As e-beam evaporation is a\nmuch softer deposition technique compared to dc sput-\ntering, the spin-mixingconductance at the YIG/Pt inter-\nface might be lower in case of evaporation, resulting in\nless spin pumping.19Also the structure of the Pt layers\nmight be different, resulting in different spin-Hall angles5\nand/or different spin diffusion lengths.\nYIG/Pt:\n 4 nm (S) \n 5 nm (E) \n-0.5 0 0.5 1024681012141618\n VISHE [µV]\nB - Bres [mT] -300 -200 -100 0 100 200 300-5051015202530θ = 1 ο\nϕ = -1 ο\n YIG/Pt:\n 4 nm (S) \n 5 nm (E) RT [mΩ]\nB [mT]a) b) \nFIG. 5. Comparison of (a) transverse resistance for an out-o f-\nplane applied magnetic field, and (b) spin pumping/ISHE sig-\nnal (using an rf-frequency of 1.4GHz with a power of 10mW)\nfor Pt on top of YIG, deposited by e-beam evaporation (E)\nand dc sputtering (S).\nIV. SUMMARY\nIn summary, the SMR in Pt layerswith different thick-\nnesses [3, 4, 8 and 35nm], deposited on top of YIG, was\ninvestigated for both in-plane and out-of-plane applied\nmagnetic fields. In-plane magnetic field scans clearly\nshow the presence of SMR for the transverse as well as\nthe longitudinal configuration. Out-of-plane measure-\nments present a magnetic field dependence which canalso be assigned to the SMR. The sign and magnitude\nof the SMR signal are shown to be determined by the\nmagnetization direction of the YIG. Further, thickness\ndependence experiments show that the SMR signal de-\ncreases in magnitude when increasing the Pt thickness.\nNo SMR signals were observed for SiO 2/Pt samples. For\nPt layers deposited by e-beam evaporation, in stead of\ndc sputtering, the found SMR signals are decreased by a\nfactor 7. Also spin pumping experiments show reduced\nsignals for e-beam evaporated Pt compared to sputtered\nPt. The difference in spin pumping signals and SMR sig-\nnals show the possible importance of the YIG/Pt inter-\nface, and connectedtothis, the spin-mixingconductance,\nfor this kind of experiments.\nACKNOWLEDGEMENTS\nWe would like to acknowledge B. Wolfs, M. de Roosz\nand J. G. Holstein for technical assistance and prof. dr.\nir. G. E. W. Bauer for useful comments regardingthe ex-\nplanation of the measurements. This work is part of the\nresearchprogram(Magnetic Insulator Spintronics) of the\nFoundation for Fundamental Research on Matter (FOM)\nandis supported byNanoNextNL, a microandnanotech-\nnology consortium of the Government of the Netherlands\nand 130 partners, by NanoLab NL and the Zernike Insti-\ntute for Advanced Materials.\n1K. Ando, Y. Kajiwara, K. Sasage, K. Uchida, and\nE. Saitoh, IEEE Transactions on Magnetics 46, 3694\n(2010).\n2E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara,\nApplied Physics Letters 88, 182509 (2006).\n3K. Ando, S. Takahashi, J. Ieda, Y. Kajiwara,\nH. 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Saitoh,\nNature (London) 464, 262 (2010).\n17Nakayama et al.11also investigated the out-of-plane be-\nhavior of the SMR, but only for saturated magnetization\ndirections, which are fully aligned to the applied field.\n18V. Castel, N. Vlietstra, J. Ben Youssef, and B. J. van\nWees, Applied Physics Letters 101, 132414 (2012).\n19C. Burrowes, B. Heinrich, B. Kardasz, E. A. Mon-\ntoya, E. Girt, Y. Sun, Y.-Y. Song, and M. Wu,\nApplied Physics Letters 100, 092403 (2012).6\n20C. M. Hurd, The Hall Effect in Metals and Alloys (Plenum\nPress, New York, 1972).\n21E. Stoner and E. Wohlfarth,\nIEEE Transactions on Magnetics 27, 3475 (1991).\n22M. Althammer, S. Meyer, H. Nakayama, M. Schreier,\nS. Altmannshofer, M. Weiler, H. Huebl, S. Gepr¨ ags,\nM. Opel, R. Gross, D. Meier, C. Klewe, T. Kuschel,J.-M. Schmalhorst, G. Reiss, L. Shen, A. Gupta,\nY.-T. Chen, G. E. W. Bauer, E. Saitoh, and\nS. T. B. Goennenwein, ArXiv e-prints (2013),\narXiv:1304.6151 [cond-mat.mes-hall].\n23C. Hahn, G. De Loubens, O. Klein, M. Viret, V. V.\nNaletov, and J. Ben Youssef, ArXiv e-prints (2013),\narXiv:1302.4416 [cond-mat.mes-hall]."    },    {        "title": "1612.07610v2.Imaging_Magnetization_Structure_and_Dynamics_in_Ultrathin_YIG_Pt_Bilayers_with_High_Sensitivity_Using_the_Time_Resolved_Longitudinal_Spin_Seebeck_Effect.pdf",        "content": "1 \nImaging Magnetization Structure and Dynamics in Ultrathin YIG/Pt Bilayers with \nHigh Sensitivity Using the Time -Resolve d Longitudinal Spin Seebeck Effect  \nJason  M. Bartell1, Colin L. Jermain1, Sriharsha V. Aradhya1, Jack T. Brangham2, Fengyuan  \nYang2, Daniel C. Ralph1,3, Gregory  D. Fuchs1 \n1Cornell University, Ithaca , NY 14853 , USA  \n2Department of Physics, The Ohio State University, Columbus , OH 43016, USA  \n3Kavli Institute at Cornell for N anoscale Science, Ithaca, NY  14853, USA \nAbstract   \nWe demonstrate an instrument  for time -resolved magnetic imaging  that is highly sensitive to the \nin-plane magnetization  state and dynamics of thin-film bilayers of yttrium iron garnet \n(Y3Fe5O12,YIG)/Pt: the time -resolved longitudinal s pin Seebeck (TRLSSE) effect microscope.   \nWe detect  the local , in-plane magnetic  orientation within the YIG by focusing  a picosecond  laser \nto generate thermally -driven spin current  from the YIG into the Pt by the spin Seebeck effect, \nand then  use the inverse spin Hall effect in the Pt  to transduce this spin current to an output \nvoltage.   To establish the time resolution of TRLSSE , we show that pulsed optical  heating  of \npatterned YIG (20 nm)/ Pt(6 nm)/Ru (2 nm) wires  generates a  magnetization -dependent  voltage \npulse of less than 100 ps. We demonstrate TRLSSE microscopy  to image  both static magnetic  \nstructure and gigahertz -frequency magnetic resonance  dynamics  with sub-micron spatial \nresolution and a sensitivity to magnetic  orientation  below  0.3 deg/ √𝐻𝑧  in ultrathin YIG.   \n  2 \nMain Text \nUltrathin bilayers  of the magnetic insulator YIG interfaced  with a heavy , non-magnetic \nmetal (NM) such at Pt are being intensely studied  for the development of high -efficiency \nmagnetic memory and logic devices operated by spin -orbit torque  [1,2] , for magnon generation \nand propagation  [3–5], and as a model system for understanding spin -current generation by the \nlongit udinal spin Seebeck effect (LSSE)  and spin pumping  [6–9].  For all of these research areas , \nit would be useful to have a high -sensitivity  and local  probe of magnetization dynamics in the \nYIG layer , especially for the ultrathin films required in many devices .  This has proven \nchallenging , and although m agneto -optical techniques such as Brillouin light scattering and  the \nmagneto -optical Kerr effect (MOKE) have proven valuable  [3,10 –14], they have not enabled \ndirect time -resolved imaging  of magnetic precession  or direct imag ing of in-plane magnetization  \nof ultra -thin YIG films (20 nm and below) .  An alternative approach  that enables in -plane \nimaging of YIG/Pt bilayer devices was demonstrated by Weiler et al. [15].  In that work, the \nauthors use laser heating to image the in -plane magnetic structure of YIG , but not its dynamics .  \nHere we extend the approach into the time  domain to perform high sensitivity  imaging of the in -\nplane magnetic orientation (< 0.3°/√𝐻𝑧) with sub -micron spatial resolution and sub -100 ps \ntemporal resolution. Using TR LSSE microscopy we can observe, for example, that the resonance \nfield in ultra -thin YIG films can vary by up to 30 Oe within micron -scale regi ons of a YIG/Pt \ndevice .  Our results d emonstrate that TR LSSE microscopy is a powerful tool to characterize \nstatic and dynamic magnetic properties in ultrathin YIG.  \nThe principle behind the TRLSSE microscope , shown schematically in Fig. 1,  is the \ngeneratio n and detection of a thermally generated local spin current  [16]. For the case of YIG/Pt , \na local thermal gradient perpendicular to the film plane is generated by laser heating of Pt. The 3 \ngradient creates a thermally -induced spin current that is  proportional to th e local \nmagnetization  [17–19]. The spin current that flows into the Pt is detected with the  ISHE  [20,21]  \nin which spin -orbit coupling leads to a  spin-dependent  transverse electric field . For this work, the \nresulting voltage can be described as  [17,19]  𝑉𝐿𝑆𝑆𝐸 ∝ − 𝜉𝑆𝐻 𝑆𝐌(𝒙,𝑡)\n𝑀𝑠×𝛁𝐓(𝒙,𝑡), where, 𝜉𝑆𝐻 is \nthe spin Hall efficiency, S is the spin -Seebeck coefficient, M is the local magnetization, Ms is the \nsaturation magnetization and 𝛁𝐓 is the thermal gradient .  The LSSE has been attributed to both \nthermal gradien ts across the thickness of YIG  and to interfacial temperature differences between \nYIG and Pt  [17–19,22,23] . Our experiment cannot  definiti vely distinguish between these two \nmechanisms . Thus , here we discuss only 𝛁𝐓 as single quantity for simplicity and for consistenc y \nwith our prior work using the anomalous Nernst effect , however, this question requires further \nstudy . VLSSE is a read -out of the local magnetization my because the  electric  field is generated in \nresponse to  the spatially  local  z-component of the thermal gradient, ∇𝑇𝑧 (coordinates as defined \nin Fig. 1)  [15,24] .   \nTo extend LSSE imaging into the time -domain, we use picosecond laser  heating to \nstroboscopic ally sample  magnetization. We have previously shown , in metallic ferromagnets,  \nthat picosecond heati ng can be used for stroboscopic  magnetic microscopy using the time-\nresolved anomalous Nernst effect  (TRANE)  [25]. In TRANE  microscopy , the temporal \nresolution  is set by the  excitation and  decay of a thermal gradient  within a single material that \nboth absorbs the heat from the laser  pulse and produces a TRANE voltage from internal spin -\norbit interactions  [26,27] . In the LSSE  however, the timescale of spin current generation can \ndepend on both the timescale of  the thermal gradient  and the timescale  of energy transfer \nbetween the  phonons and magnons. Recent experiments indicate that in the  qausi -static regime  \nthe magnon -phonon relaxation rate may play a dominant role  [28–31]. Using picosecond heating 4 \nand time -resolved electrical det ection to move beyond the quasi -static regime , we show a \nTRLSSE in agreement with a  recent all -optical experiment  [22].  \nWe grew our samples using  off-axis sputtering onto (110) -oriented gadollinum gallium \ngarnet (Gd 3Ga5O12, GGG),  [32–34] followed by ex situ . deposition  of 6 nm of Pt with a 2 nm Ru \ncapping layer. Photolithography and ion milling were used to pattern wires  and contacts for \nwirebonding . We present measurements of  a 2 µm × 10 µm wire and a 4 µm × 10 µm wire with \nDC resistances of 296 Ω and 111 Ω respectively. In this room temperature study , we neglect  the \npotential anomalous Nernst effect of interfacial  Pt with induced magnetization  [35,36] , and we \nneglect a poss ible photo -spin voltaic effect  [37], neither of which can be distinguished from \nTRLSSE in presented measurements.  \nOur TRLSSE  measurement consists of pulsed laser heating and homodyne  electrical  \ndetection as shown in Fig. 2a. We use a  Ti:Sapphire laser  pulse  to locally heat the s ample w ith 3 \nps pulses of 780 nm light at a repetition rate of 25.5 MHz . The electrical signal produced at  the \nsample is the sum of the LSSE dependent voltage, 𝑉𝐿𝑆𝑆𝐸 (∇𝑇𝑧,𝐌), and a voltage, 𝑉𝐽(Δ𝑇,𝐽), which \nis generated when a  current  density J is passing through the local region of Pt with increased \nresistance due to laser heating  [38]. To reject noise and recover the signal of the resulting  \nelectrical  pulse s, we use a time-domain h omodyne  technique in which  we mix  the VLSSE + VJ \npulse  train with a synchronized reference pulse  train, Vmix, in a broadband (0.1 -12 GHz) electrical \nmixer.  The mixer output is the convolution of the two pulse  trains  given by  [38]  \n𝑉𝑠𝑖𝑔(𝒙,𝜏)=𝛫∫(𝑉𝐿𝑆𝑆𝐸 (𝛻𝑇𝑧(𝒙,𝑡),𝐌(𝒙,𝑡))+ 𝑉𝐽(𝛥𝑇(𝒙,𝑡),𝐽(𝒙,𝑡)) 𝑉𝑚𝑖𝑥(𝜏−𝑡)𝑑𝑡Γ\n0, (1) 5 \nwhere x(x,y) is the laser spot position  in the sample plane , Γ is the period of the laser pulses, 𝛫 is \nthe transfer coefficient , and 𝜏 is the relative delay. A relative delay of zero corresponds to the  \nmaximum of both  pulse train s arriving at the mixer simultaneously.  \nWe study the timescale of the  LSSE signal generated by a picosecond pulse by measurin g \nVsig as a function of mixer delay 𝜏. Fig. 2b shows the result of this measurement using a  100 ps \nmixing pulse  reference,  Vmix, at a saturating magnetic field, H, perpendicular to the wire  at H = \n+414 Oe and – 414 Oe , respectively. In Figure 2 c we plot  the difference  between these two \nvoltage traces to reject non -magnetic contributions.  We find that the full-width at half -maximum \n(FWHM) is 100±10 ps, which is followed by electrical oscillations  that we attribute to non -\nidealities in the detection circuit (see the SI for further discussion.)   Because the duration  of the \nmagnetic component of Vsig is experimentally  indistinguishable from the FWHM of Vmix, we \nconclude that  100 ps is an experimental  upper bound for the TR LSSE signal  duration . To our \nknowledge, this is the first direct electrical measurement of picosecond duration  LSSE  voltages.   \nTo calibrate  the local  change in  the Pt temperature , ΔTPt, due to picosecond  heating  and to \nquantify  the rate of thermal relaxation , we measur e VJ in the presence of a DC current , which \nuses the local Pt resistivity  as an ultra -fast thermometer. Figure 2 d shows VJ as a f unction of \nmixer  delay, VJ(τ) = Vsig(τ, J = 4.2 MA/cm2) – Vsig(τ, J = -4.2 M A/cm2), for applied currents of  \n±0.5 mA.  VJ (τ) is proportional to ΔTpt through VJ, but it is not proportional to  either the \nmagnetic state of the sample or ∇𝑇𝑧. We observe that VJ relaxes to zero faster than the laser \nrepetition period , indicating that the sample  thermally recovers between pulses. To quantitatively \nconsider the  spatiotemporal thermal evolution , we performed a time-domain finite element  \n(TDFE) calculation of focused laser heating in the wire . Additional details are available in the SI, \nand se e Ref . [25] for a lengthier discussion of the procedure.   The comparison of the 6 \nspatiotemporal profile of the calculation and the known temperature dependence of resistivity  \nenable us to calibrate the spatiotemporal temperature rise due to laser heating.  We find that the \npeak film temperature changes by  ~50 K in the platinum and ~ 10 K in the YIG for a laser \nfluence of 5.8 mJ/cm2, which is the maximum fo r the presented measurements. Note that  we \nassume all laser heating  is mediated by optical absorption in Pt because YIG  and GGG  are \ntransparent at 780 nm  [39,40] . The TDFE calculation reveals that , in agreement wi th experiment,  \n∇𝑇𝑧 across the YIG thickness decays more quickly  than the full thermal relaxation of the Pt back \nto the ambient temperature  (e.g. ΔTpt = 0). This difference in timescales between  ∇𝑇𝑧 and ΔTpt is \nimportant because  the magnetic signal in our experiment is sensitive to only ∇𝑇𝑧(𝑡), not ΔTpt (t) \nof the Pt . \nThe s ub-100 ps  spin current lifetime  in our experiment  is short enough  that the TRLSSE is \nuseful  for stroboscopic measurements of resonant YIG magnetization dynamics.  To confirm this \nidea, we use TRLSSE microscopy to measure ferromagnetic resonance  (FMR)  by driving  a \ngigahertz -frequency  a.c. current into the  Pt, which generates magnetic torques on YIG from both \nthe Oersted magnetic field and from spin currents generate d by the spin Hall effect  [41–43]. The \ncurrent is generated with an arbitrary waveform generator  (AWG ) that is phase -locked to the \nlaser repetition rate and coupled to the YIG/Pt device  through  a circulator  (see schematic in  Fig. \n3a). Synchronizing  the a.c. current and the laser  repetition  rate ensures  a constant  but \ncontrollable  phase between the precessing magnetization and the sensing heat pulse for a given \ndriving frequency and magnetic field . In our FMR measurements , we fix 𝜏 = 0 and align the wire \naxis parallel to the external magnetic field . In this configuration,  the TRLSSE signal is \nstroboscopically sensitive to  the magnetic projection  my at a particular phase of the magneti c \nprecession  about the x-axis. In addition to VLSSE, Vsig contains a contribution from VJ that is 7 \nproportional to  the local a.c. current amplitude and phase  [38].  We separate  the magnetic  VLSSE \nfrom  the non -magnetic  VJ by measuring Vsig with a lock -in amplifier referenced t o a 383 Hz , 7.6 \nOe RMS  modulation of the  external magnetic field. Fig. 3 b shows LSSE FMR spectra as a \nfunction of field  that is excited  using  a 0.5 mA  a.c. current at  4.1 and 4.9 GHz. In the limit that \nthe modulation  magnetic  field is small compared  to the FMR linewidth, we can interpret the  \nresulting  signal  Vmod as a derivative signal that contains  a linear combination of the real  and \nimaginary  parts of the dynamic susceptibility , 𝜒, 𝑉𝑚𝑜𝑑(𝐻)∝𝑑𝜒′\n𝑑𝐻𝑆𝑖𝑛(𝜃)+𝑑𝜒′′\n𝑑𝐻𝐶𝑜𝑠(𝜃). This \nrelation is used to fit the FMR spectra to extract the amplitude, phase , linewidth, and resonant \nfield. For mo re details on fitting see refs  [25,38] .  To demonstrate  that the TRLSSE microscope \nis a phase -sensitive stroboscope, we rotate d the phase of the microwave current by 180°  and re -\nmeasure FMR .  As expected, inverting the phase of the drive in verts the phase of the FMR \nlineshape  (Fig. 3 c). \nNext, we quantify the sensitivity of  TRLSSE microscopy  for our ultra-thin YIG/Pt samples . \nFigure 4 shows representative LSSE measurements of the YIG magnetization versus magnetic \nfield perpendicular to the wire  at several optical powers . In this geometry, t he positive and \nnegative  saturation value s of VLSSE quantify  the full range of magnetization,  +M to –M. The n, \nusing the standard  deviation of the  noise in the LSSE voltage, 𝜎𝐿𝑆𝑆𝐸, we can quantify the angular \nsensitivity noise floor assuming small angle  magnetic  deviations from the wire axis , such as  for \nstroboscopic FMR measurements . The sensitivity is calculated using  [25] 𝜃min=\n𝜎LSSE\nsin(𝜃o)(𝑉LSSEmax−𝑉LSSEmin)/2√𝑇𝐶 where TC is the lock-in time constant . We find a sensitivity  of 0.3 \ndeg/√Hz  for an optical  power of 0.6 mW, corresponding to a laser fluence of  5.8 mJ/cm2. It is 8 \nimportant to note that the sensitivity is sample dependent  through both  sample geometry and the \nimpedance match with the detection circuit  [25].  \nThe i nterface  quality  of the sample plays a key role in determining the sensitivity . As spin \ncurrent diffuses into the platinum , it is subject to loss at the interface. A good indication of \ninterfacial spin transparency is the spin Hall magnetoresistance ( SMR ) [44,45] , which is \nsensitive to the  spin mixing conductance at the interface. For the data presented here, the devices \nshow a SMR of 0.063%, which is the largest value by a factor of 2 from the other devices we \npatterned . This is consistent with a number of recent SMR reports  [44–48], and we expect the \nhigh SMR value indicates strong spin transparency at the YIG/Pt interface . We also studied \nYIG/Pt samples with no measureable SMR  which we expect to have  a significantly reduced \nLSSE induced ISHE voltage .  We  found that the LSSE signal in these devices is approximately \nan order of magnitude lower  for the same laser fluence . Additional details are in the SI.    \nHaving placed upper bounds on the time resolution and quantified the sensitivity,  next we \ndemonstrate the application of TRLSSE microscopy  for imaging of static magnetization . We \nacquire i mages by  scanning the laser focus and making a point -by-point measurement of the \nTRLSSE voltage  and reflect ed light .  Figures 5a and 5b show a reflected light image and \nsaturated LSSE  image, respectively , for a  4 μm wide YIG/Pt device .  In the ref lection image , we \nsee the structure of the wire and the contact pads at both ends . We acquired the  TRLSSE image  \nat H =–405 Oe  and shifted the  background level for clarity  of the color scale.  N o other image \nprocessing  was performed . We observe a uniform magnetization state of the YIG/Pt device , as \nexpected from the previously presented ma gnetic hysteresis measurements (Fig. 4 ). When we \nreduc e the field to near zero ( H = 4 Oe ) and re-imag e the wire (Fig. 5c ), magnetic texture  is \nrevealed that indicates  non-uniform canting of the  device  magnetization. To more clearly show 9 \nthe variation in contrast between images , we plot l ine cuts of Figs. 5a -c in Fig. 5d. Despite the \ninhomogeneous remanence  that is evident in Fig. 5c,  we were not able to  observe  domains with \noppositely aligned magnetization ; possibly because  once a reversal domain is nucleated, the  \ndomain wall propagate s without strong pinning .  \nWithout a 180o domain wall the spatial resolution of TRLSSE cannot be directly evaluated . \nNevertheless , we use the reflected light image and TDFE simulat ions to study the possibility that \nlateral thermal spreading degrades the res olution. To approximate the lateral point spread \nfunction of the laser, we fit a scan of the wire step edge to a Gaussian  point spread function . This \nyields a spot FWHM  of 0.606  μm. Calculations of the heating indicate that the thermal gradient \ndoes not spread laterally in the Pt, thus we expect that the resolution of the TRLSSE is the same \nas the diffraction -limited optical resolution in this experiment .  \nWe now  demonstrate that TRLSSE microscopy has the sensitivity to image dynamic \nmagnetization in the 4 μm YIG/Pt device , which provides quantitative and spatially localized \ninformation about dynamical properties of ultrathin YIG materials .  As described above , for \nFMR characterization  we orient the external magnetic field parallel to the wire axis a nd drive a \n1.1 mA , 4.9 GHz current into the wire. We image dynamical magnetization at  a series of  \nmagnetic  fields near the resonance field, from  H = 896 Oe to 1105  Oe, and pl ot a selection of the \nunprocessed images in  Figs. 5e-g. The data show that  at H far from resonance  (Fig. 5 e) where \nprecession amplitudes are tiny , the TRLSSE  signal  at the center of the wire  is well below the \ndetection noise  floor.  There is a small , current -induced , non-magnetic signal artifact at the edge s \nof the wire  which we discuss further in the supplementary information. For H near the resonant \nfield, Hres, the device has a  strong , position -dependent TRLSSE response . To quantitatively \nanalyze the data, images are correct ed for background offset and sample drift  before fitting a 10 \nresonance field curve for  each pixel .  We plot  a selection of curves from individual pixels i n Fig. \n6a. We then construct a spatial map of each  fitting parameter : Hres, relative phase, 𝜙, amplitude, \nA, and linewidth,  ΔH, and offset, all of which are shown in Fig. 6b-f. We immediately notice \nspatial variation in these images that is qualitatively similar to the non-uniform magnetic \nremanence texture shown in Fig . 5c.  Together, these measurements confirm the presence of \nvarying local magnetic anisotropy and quantif y both static and dynamic magnetic properties in \neach region.  The ability to quantitatively relate the spatial variation  of static  and dynamic \nproperties  in ultrathin YIG/Pt devices  is a unique capability of our microscope .  \nIn conclusion, w e have demonstrated  sensitive  and high-resolution  TRLSSE microscopy  of \nultrathin YIG/Pt devices  that we expect  will prove useful for developing spintronic applications .  \nUsing picosecond heating, we demonstrate that TRLSSE  microscopy  is a sub-100 picosecond  \nprobe  of ultra-thin YIG/Pt device  magnetization , both for static magnetic configurations and for \ndynamical measurements at gigahertz frequencies .  We have demonstrated  an angular sensitivity \nof 0.3 °/√𝐻𝑧, which to our knowledge is the most sensitive experimental probe of ultra-thin YIG \nmagnetic orientation reported to date .  \nAcknowledgments  \nWe thank J. Kimling and D. G. Cahi ll for helpful comments on an early version of the \nmanuscript, and for providing the interface thermal resistance of YIG/Pt. This research was \nsupported by the U.S. Air Force Office of Scientific Research , under Contract No. FA9550 -14-1-\n0243 , and by U.S. National Science  Foundation under Grant s No. DMR -1406333  and DMR -\n1507274  and through the Cornell Center for Materials Research (CCMR) (DMR -1120296). This \nwork made use of the CCMR  Shared Facilities and the Cornell NanoScale Facility, a member of 11 \nthe Nation al Nanotechnology Coordinated Infrastructure, which is supported by the NSF (Grant \nNo. ECCS -1542081) . \n  12 \nReferences  \n[1] B. Behin -Aein, D. Datta, S. Salahuddin, and S. 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The pulsed heating  drives a pulsed magnon \nflux, Js, from the YIG into the Pt where it is transduced into a pulsed voltage via the ISHE .   \n17 \n \nFIG. 2   (a) Schematic of the LSSE detection circuit  used for time -resolved voltage \nmeasurements .  (b) Time -domain  measurement of the LSSE generated voltage  in the 2 µm  wide  \nwire. The time -varying LSSE signal is measured by electrically mixing  the pulsed laser \ngenerated voltage with a 100 ps voltage pulse from the AWG. Comparing measurements of the \nYIG at +414 Oe (filled blue circles) and –414 Oe (open orange circles) shows that the signal \ndepends on the orientation of the magnetic moment.  Here d.c. level noise and has been removed. \nThe data was acquired with a lock-in time constant of 500 ms and integration time of 2 s  per \npoint . (c) The solid blue circles show  the difference between the two curves i n (b), The orange  \nline is a model , normalized by the data amplitude,  of the signal determined by numerically \nconvolving the calculated thermal gradient with the measured mixing pulse . (d) Difference signal \nof the temperature dependent voltage VJ measured using +/ – 0.5 mA  and a 600 ps mixing pulse . \nIn (b -d) we report the voltage as detected at the lock-in after passing through the r.f. mixer , not \nthe LSSE signal at the sample itself.   \n18 \n \nFIG. 3   Stroboscopic detection of ferromagnetic resonance  a) Schematic of measurement circuit \nfor detection of ma gnetization dynam ics in the 2 µm wide wire . b) TRLSSE detected FMR for \n4.1 GHz (blue, closed circles) and 4.9 GHz (orange, open circles) excitation. The solid lines are a \nfit to the data using a modified Lorentzian. c) Demonstration of stroboscopic FMR det ection in \nwhich we measure the response of the YIG driven at phases that differ by 180 degrees. The data \nwas acquired with a lock-in time constant of 1s and integration time of 5 s per point.   \n19 \n \nFIG. 4   Measurement of YIG magnetization with LSSE  measuring VLSSE versus external \nmagnetic field for different laser powers  and wire widths. For these curves, a DC background \nwas subtracted . The inset shows the wire geometry. We define the signal size to be one -half of \nthe difference in voltage when the magnetization is saturated in opposing directions.  The data \nwas acquired with a lock-in time constant of 500 ms and integration time of 2 s per point.  \n  \n20 \n \nFIG. 5   Images of the 4 µm  wide YIG/Pt wire  (a) Reflected light image of the YIG/Pt wire \nmeasured with a photodiode at the same time as the LSSE voltage. (b) Background subtracted \nLSSE voltage at sa turated magnetization  and (c) remnant magnetization at 4 Oe after saturation. \n21 \n(d) Line cuts of the 2D scans. The normalized reflection signal is shown with black squares, blue \ncircles represent the saturated magnetization, and the orange triangles represent the \nmagnetization of the remnant state. Note, that in the line cuts the low field line cut is normalized \nwith respect to the saturation magnetization. The  right side of the figure represents the raw \nimages of the 4 μm wire at different fields around the resonance : (e) 896 Oe.  (f) 1007 Oe, ( g) \n1025 Oe. Images (e -g) share the same color scale. Line cuts of the images are shown in (h) black \nsquares, blue circl es, and orange triangles correspond to the boxed regions of (e), (f), and (g) \nrespectively. For (e -g) the data was acquired with a lock-in time constant of 200 ms and an \nintegration time of 2 s.   22 \n \nFIG. 6   Spatial maps of FMR fitting parameters  for the 4 µm wide wire.  (a) Traces are the pixel \nvalues of three points on the sample as a function of magnetic field . b-f) Spatial maps of the \nFMR fitting parameters made by fitting of the FMR curves at each pixel in the sequence of \nimages measured with LSSE. Before fitting, we correct for image -to-image offset and use a 3x3 \npixel moving average to smooth the data. (b) Resonance field, the symbols mark the pixels \ncorresponding to the FMR spectra shown in (a).  (c) Resonance amplitude , (d) resonance phase , \n(e) resonance linewidth  (f) offset used in the fit.  \nImaging Magnetization Structure and Dynamics in Ultrathin YIG/Pt Bilayers with \nHigh Sensitivity Using the Time -Resolved L ongitudinal Spin Seebeck Effect  \nSupplemental information  \nJason M. Bartell1, Colin L. Jermain1, Sriharsha V. Aradhya1, Jack T. Brangham2, Fengyuan \nYang2, Daniel C. Ralph1,3, Gregory D. Fuchs1,3 \n1Cornell University, Ithaca, NY 14853, USA  \n2Department of Physics, The Ohio State University, Columbus, OH 43016, USA  \n3Kavli Institute at Cornell for Nanoscale Science, Ithaca, NY 14853, USA  \nOptica l path \nTo heat the YIG/Pt bilayers, we use a  Ti:Sapphire laser tuned to 780 nm and pulse durations of 3 \nps at 76.5 MHz.  An electro -optic modulator referenced to the laser pulses is used to reduce the \nrepetition rate  to 25.5 MHz , which allows time for thermal recovery . Next, a photoelastic \nmodulator and a polarizer are used to modulate  the optical amplitude  at 100 kHz  for lock-in \ndetection.  The resulting vertically polarized light is focused on the sample with a 0.9 NA \nobjective. A fast -steering mirror with a 4 -f lens pair is used to scan the laser focus  across the \nsample. The light reflected from the sample is detected with a photodiode bridge . \nFIG. S1. Schematic of TRLSSE microscope.  \nModel of TRLSSE temporal convolution  \nWe develo p a model of the detection circuit  to clarify  the impact of circuit bandwidth and \nelectrical artifacts on the TRLSSE trace s shown in Figs. 2b and 2c . The time domain \nmeasurements shown i n Fig. 2  show that  the duration of Vsig matches the ~100 ps duration of the \nmixing pulse. This implies that thermal gradient induced VLSSE must be sufficiently short -lived  to \nsample the mixing pulse , and thus it is suitable for stroboscopic measurement of GHz frequency \ndynamics.  In addition to t he main pulse,  we also observe oscillations  that can be attributed to  \nnon-idealities in the mixing reference pulse produced by the arbitrary waveform gene rator \n(AWG) and the RF mixer  itself . To account for these effects , we develop a  phenomenological \nmodel  of the signal, which we describe as the convolution of the TRLS SE-induced electrical \npulse from the sample and the reference pulse from the AWG as a function of relative delay, \n𝞽 [1]. We account for  bandwidth contributions and  the realistic profile of  the mixing reference \npulse .  \nThe model  consist s of a  12 GHz  low-pass filter leading to the radio frequency and local \noscillator inputs of an idealized mixer  (Fig. S 2a). The output of the circuit is described by    \n𝑉𝑠𝑖𝑔(𝑡) ~ ℱ−1[ℱ(𝑉𝑚𝑖𝑥)∗ℱ(𝑉𝐿𝑆𝑆���� )∗𝐿𝑃(𝑓)2] (S1) \nWhere LP(f)  is a first -order low -pass filter 𝐿𝑃(𝑓)=1\n1+𝑓/𝑓𝑐 for frequency f and cut -off frequency \nfc = 12 GHz. The Fourier transform ℱ(𝑉) is given by  ℱ(𝑉𝛿𝑡 )= 1\n√𝑇∑ 𝑉𝛿𝑡  e2 𝜋 𝑖 (𝛿𝑡 −1)(𝑓−1)/𝑇 𝑇\n𝛿𝑡 =1  \nwhere T = 12.9 ns is the duration of the kernel,  𝛿𝑡 = 2.5 ps is the time step, and f is frequency.  In \nthe experiment, the mixing pulse Vmix is generated by an arbitrary wave form generator (AWG) \nsynchronized  to the laser repetition rate  with a sampling rate of 9.98 GSamples /s. For mixing \nvoltage pulse Vmix we use the output of the AWG  measured using a LeCroy SDA 11000 \nOscilloscope (Fig. S2b ). To model the signal from the sample, VLSSE, we use the normalized \nthermal gradient determined from time -domain finite element  (TDFE)  calculations (further \ndiscussion below). In the main text, we use a 100 ps mixing  pulse to acquire  the data presented in \nFig. 2b,c. and a 600 ps mixing pulse to acquire  the other data.  \nFigure 1 of the main text shows Vsig calculated via Eq. S1 normalized to the measured data   \nalong with the measured convolution . The model qualitativel y captures the oscillations at delay times greater than 100 ps. This model, together with the lack of magnetic field dependence, \nsupports the idea that the oscillations  in the data  are electrical artifacts , not magnetic  oscillations . \n \nFIG. S 2. (a) Schematic of  circuit model for interpretation of time -domain circuit. The arbitrary \nwaveform generator (AWG) creates a mixing pulse that goes through a 12 GHz low -pass filter \nbefore being mixed  with the pulse from the sample  that has also been sent th rough  a 12 GHz  \nlow-pass filter. (b) Oscilloscope measurements of the mixing  pulses used in the experiments.   \nDetermination of temperature change from laser heating  \nAlthough we know the laser fluence, we do not know the film absorbance for this thin -film \nlimit in which the Pt film is much thinner than the optical skin depth. To determine the \ntemperature change in our experiment we use the following methodology: (1) we numerically \ncalculate the spatiotemporal thermal response to focused laser heating assuming  the peak \nabsorbed power is 1 W (an absorbed fluence of 0.7 mJ/cm2).  We take the model’s predictions \nfor the spatiotemporal thermal evolution to be correct but the total temperature change amplitude \nas being unc alibrated.   (2) We calibrate and measure VJ, which is equivalent to using the sample \nresistivity change as a thermometer.   (3) We calculate the VJ from our spatiotemporal thermal \nmodel  calculations  and compare it to the measured VJ.  We assume there is  linear response \nbetween the amplitude of the absorbed laser energy and the maximum temperature increase , \ntherefore the ratio of the measured to the calculated values of VJ determines the scale factor of \nthe absorbance . This also  scales the temperature inc rease from the model to a value that agrees \nwith our electrical measurement.  Additional details have been described previously in the \nsupporting information of Ref.  [1]. \nWe base our model on TDFE  calculations  of thin -film thermal diffusion  to determine the \nspatiotemporal profile of the  thermal gradient  temperature distri bution. We consi der a  \nGGG/YIG(20 nm)/Pt(6 nm) trilayer  with material parameters given by Table S1. The YIG/Pt \nlayer s are modeled as a 2 µm x 10 µm  bar to match the measured device. Heat transfer in the \nstructure is calculated using the diffusion equation  \n𝜌 𝐶𝑝𝛿𝑇(𝐱,𝑡)\n𝛿𝑡−𝜅∇2𝑇(𝐱,𝑡)=𝑄(𝐱,𝑡) (S2) \nwith the COMSOL Multiphysics® software package. In Eq. S2 𝜌 is the material density, Cp,  is \nthe specific heat, 𝜅, is the  thermal conductivity, Q is the heat source, x is the 3D spatial \ncoordinate, and t is time.  We assume the YIG/ Pt interfacial thermal conductance  is \n170 W m- 2  K- 1 [2].  \nWe also assume that laser heating only takes place in the Pt layer because of the negligible \noptical absorption in the YIG  [3] and GGG  [4]. Thus, the laser is effectively a  radially symmetric  \nheat source , with radius r, in the platinum with a spatial  temporal distribution,  for positive z, \ngiven by,  𝑄(𝐱,𝑡)=𝐸𝑥𝑝 (−𝑧\n𝜀)∗(1\n2𝜋 𝑑2)∗𝐸𝑥𝑝 (−𝑟2\n2 𝑑2)∗𝐸𝑥𝑝 (−(𝑡−𝑡0)2\n2 𝑤2), where d = 257 nm is \nthe focused laser spot size (see “determination of optical spot size ” below ), 𝜀 = 12 nm is the skin \ndepth  [5,6] , w = 1.27 ps is the laser pulse width for a 3 ps FWHM Gaussian pulse, t0 = 100 ps is \nthe time that the heat source is at  the maximum.  The heat source is applied every 39.6 ns and the \nsimulation runs from time t = 0 ns to t = 42 ns to capture two pulses.  \nFigure S3  shows the result of the  model  calculation in the space and time  domain s. The z -\ncomponent of the thermal gradien t within the YIG decays to 1/e in 92 ps and the t emperature \ndifference between the Pt and YIG decays in 91ps, time scales that are experimentally \nindistinguishable in our measurement and consistent with the time domain measurement shown \nin Fig s 2b,c of the main text. T he overall temperature  increase within the laser heated region  takes longer to relax to room temperature, 295 ps, consistent with Fig. 2d . These calculations \nsupport that the TRLSSE signal originates from ∇𝑇𝑧(𝑡) (or indistinguis hably  in this work , the \ntemperature difference between YIG and Pt) and that it  is localized in time making it  suitable for \nstroboscopic measurements .  \nThe model calculation predicts about a 400 K change in the Pt, however , as discussed above, \nwe calculated  the amplitude of the laser -induced temperature change  without experimental \nknowledge of the absorbed fluence.  Therefore, the true temperature change in the Pt may be \nscaled up or down  to account fo r correct value of the  absorbed laser power. To establish  the \nabsorbance experimentally, w e compar e the measured  values of VJ, which originates from the  \nresistance change of the metal  due to laser heating, with a model calculated value of VJ, which is \ndetermined from the resistance change expected from our model  calculation. Specifically, we \ncalculate VJ using the 3D temperature distribution  created from laser heating to determine  the \nsample resistance increase .  We use  the linear relationship between the resistance and the \ntemperature , 𝑅(𝑇)=𝑅𝑜(1+𝛼 𝑇), with  the resistance correction factor α = 1.3 × 10-3 K-1 \nmeasured for the Pt films used in our experiment . To compare the calculated value to the \nexperimentally measured VJ, we also determine  the electrical circuit transfer function in which \nwe account for the measurement  bandwidth and gain (see Ref. [1] for further discussion) . From \nthis analysis we find that our experimentally measured V J is 0.12 times the calculated V J, \nindicating the peak temperature change in the Pt is 5 0 K, corresponding  to a peak absorbed \nfluence of 0. 09 mJ/cm2, 1.6% of the incident laser energy. The uncertainty in the temperature is \nestimated to be on the order of 25% based on uncertainties in the circuit calibration . \nTABLE  S1 M aterial parameters used in the TDFE simulations of laser heating  \naReference  [7] \nbReference  [8] \ncReference  [9]  Specific Heat, C p \n(J/kg*K)  Density, 𝜌 \n(kg/m3) Thermal conductivity, \n𝜅 (W/m*K)  \nPt 133a 21500a 71.6a \nYIG 570b  5170c 6b \nGGG  400b 7080b  7.94b   \n \nFIG. S 3. Time -domain fini te element calculations of the temperature and thermal gradient using \nCOMSOL. (a) Time -domain  thermal profiles at the YIG/Pt interface calculated with COMSOL \nassuming an absorbed fluence of 0.7 mJ/cm2 and showing the z -component of thermal gradient \nin the YIG (orange curve), change in temperature of the Pt (blue curve), and temperature \ndifference between the Pt  and the YIG across the interface (black dashed line). The laser turns on \nat 100 ps in the calculation. (b) Calculated t emperature vs. z -axis positio n showing heating as a \nfunction  of film depth at the maximum temperature difference (orange curve) and 16 ps later \n(blue curve) . (c,d) The curves from (a) and (b) scaled by the correction factor.  \nEffect of interface spin transparency   \nThe spin Hall magnet oresistance (SMR) is the change in resistance due to spin-dependent \ntransport in a heavy , nonmagnetic metal that shares an interface with a ferromagnet  [10]. Thus, \nfor bilayers of the same materials but different spin mixing conductance , measuring SMR \nprovides insight into the efficiency with which spins can cross the interface. The efficiency of \ninterfacial spin transport is important for TR LSSE measurements because in order for the \nmagnetization to be transduced into a voltage, the thermally driven spins must cross the \ninterface.  \nFor the data presented in the main text we find a SMR of 0.063%. We compare the signal \nfrom this wire with a rel atively strong SMR to the TRLSSE signal from a wire without  \ndetectable SMR above  the 0.003%  noise floor of our lock -in measurement . Both wires were 2 \nμm x 10 μm with resistances of 296 Ω and 220 Ω for the sample with and without SMR \nrespectivly. The sample without SMR had a thinner  YIG film ( 8 nm ), however this is not \nexpected to effect the SMR since SMR is an interfacial effect  [11].  \nFigure S4 shows representative plots of the TRLSSE signal versus field for the different \nwires at similar laser powers. We find that the sample with SMR has a signal approximately an \norder of magnitude greater than the sample without. The d ifference is consistent with the model  \nof TRLSSE driving spin current across the YIG/Pt interface. We also note that even though the \nsignal is reduced, it is still measurable in both samples, enabling measurement of YIG \nmagnetization even in systems that c annot be measured electrically.  \nFIG. S 4. TRLSSE signal as a function of applied external field for a sample with 0.063% \nSMR (blue triangles) and a sample with no measurable SMR (orange squares). The applied laser \nfluences are 5.4 mJ/cm2 and 6.7 mJ/cm2 for the blue an orange curves respectively. For the data \npresented here, the laser repetition rate was 76.5 MHz and no amplifier was used between the \nsample and the RF mixer.  \n Determination of optical spot size  \nWe determine the diameter of the illuminated area by modeling  a Guassian laser focus  and \nfitting the traces of the image shown in Fig . 5a. Fig. S4  shows a y -axis cross section of the \nimage. The trace shows an approximately flat region on the wire surface a nd a sigmoidal  edge \ndue to the convolution of the sharp wire edge with the point -spread function of the laser focus . \nTo fit the reflection signal, I, at the edge,  we use the convolution of  a Guassian with a step \nfunction ,  \n𝐼=1\n𝑏 √2 𝜋∫ exp (−(𝑥−𝑎)2\n2 𝑏2)∞ \n−∞ Θ(𝑥−𝑎)𝑑𝑥,  (S3) \nin which 𝑏 determines  the Guassian width , a is the center of the peak , and Θ is the step function \ndefined as Θ(𝑥−𝑎)={0 ,𝑥<𝑎\n1 ,𝑥≥𝑎. The fit of the data yields b = 0.240  ± 0.007  µm and  b = 0.274  \n± 0.010  µm for the left and right edges respectively . We take the average to be the optical spot \nsize. We attribute t he difference between the two edges to a slight out-of-plane tilt of the sample \nleading to asymmetry in the reflection.    \nAs a comparison, we fi t a y-axis scan of the TRLSSE signal to Eq. S3. The result gives b = \n0.380 ± 0.006 µm and b = 0.381  ± 0.009 µm for the left and right edges respectively. This \ndifference corresponds to a difference  of ~1 pixel between the  rise-width of the  reflection  signal  \nand TRLSSE signal .  \nFIG. S 5. Fit of step edge signal for determination of optical spot size. (a) Line cut in y -axis \ndirection of the reflected light image, shown in Fig. 5a, and the TRLSSE image of the static \nsaturated moment, shown in F ig. 5b. (in set) schematic representation of the sample tilt  that can \nlead to the observed anisotropy .     \nAnalysis of dynamic TRLSSE images  \nTo image the ferromagnetic resonance of YIG in the 4 µm wide wire a series of images was \ntaken at fields ranging from 896 to 1 105 for an applied RF power of 1.1 mA . A selection of \nunprocessed images is shown in Fig. 5e -g of the main text. Although the signal is quite clear, we  \naccount for sample drift and noise, before fitting the FMR curves.  \nWe correct for sample drift using autocorrelation  to find the image overlap. The kernel for \nthe autocorrelation  is a 5 ×12.5 µm region from the center of the reflected light image at H = 896 \nOe (the first image in the series).  We determine t he drift of subsequent images by finding the  \ndistance between the centers  of the kernel  and the minimum of the autocorrelation. M ost of the \nsample drift is on the order of a pixel (0.25  µm) with a maximum sample drift of Δy = 0.75  µm \nand Δx = 0.25  µm. We correct for t he offset by shifting the images  and then cropping the \nborders. The scans cover a large enough area that the cropped region is well away from the wire. \nAfter correcting for the sample drift, we remove the background from the vibration edge artifacts  \nby subtract ing the TRLSSE signal of the wire at 896 Oe from the subsequent images. Finally, we \nreduce random pixel to pixel noise, smoothing the signal with a 3x3 pixel moving average. The \n3x3 pixel window is approximately the sampling spot size  (see determination of optical spot \nsize).   \nWe attribute t he small signal features at the edges of the wires in Fig . 5 of the main text to \nmagnetic field modulation  induced relative motion  between the microscope objective and the \nsample . As mentioned in the main text, we separate VJ  (which is in princ iple non -magnetic) \nfrom VTRLSSE  (which is magnetic) by adding a modulation magnetic field  (7.6 Oe RMS, ω H = 383 \nHz) to the d.c. magnetic field.  We then demodulate Vsig with respect to ω H using a lock -in \namplifier . Although  this procedure is effective  for isolating VTRLSSE  from VJ when we focus in the \ncenter of the  wire (away from the wire edge) , the modulation  field induces a tiny “wobble” in the \nlaser focus on the sample . When the  laser is focused  on the  sample  edge and a current is applied \nto the sample, the wobble introduces a slight modulation of VJ at ωH because  𝑑𝑉𝐽\n𝑑𝐻=(𝑑𝑉𝐽\n𝑑𝑦)(𝑑𝑦\n𝑑𝐻), \nwhere 𝑑𝑦\n𝑑𝐻 is due to field-induced mechanical motion and 𝑑𝑉𝐽\n𝑑𝑦 is large at the sample edge. We note \nthat these edge signals are independent of external field but  that they are  sensitive to the current \namplitude and phase , both of which are consistent with this interpretation of the artifact . In Fig. \nS6 we plot both the profile of the externally modulated fi eld signal in the y -direction and the \nnumerical derivative of VJ measured by the lock-in referenced to the 100 kHz laser modulation \nrate, which demonstrates their correspondence.  \nFIG. S 6. (a) Spatial variation of the  TRLSSE in a 4 × 10 μm YIG/Pt wire at 911 Oe. The signal \nmeasured by a lock-in amplifier referenced  to the frequency of an a.c. magnetic field. (b) Profile \nof the TRLSSE signal shown in (a) (blue circles) and the derivative of V J from the same area  of \nthe wire  (orange triangle s). The trace is the average of twenty -six y-axis line  scans  from along \nthe length of the wire .    \nReferences  \n[1] J. M. Bartell, D. H. Ngai, Z. Leng, and G. D. Fuchs, Towards a table -top microscope for \nnanoscale  magnetic imaging using picosecond thermal gradients, Nat. Commun. 6, 8460 \n(2015).  \n[2] J. Kimling and D. G. Cahill (private communication)  \n[3] S. H. Wemple, S. L. Blank, J. A. Seman, and W. A. Biolsi, Optical properties of epitaxial \niron garnet thin films, Phys. Rev. B 9, 2134 (1974).  \n[4] D. L. Wood and K. Nassau, Optical properties of gadolinium gallium garnet, Appl. Opt. \n29, 3704 (1990).  \n[5] J. H. Weaver, Optical properties of Rh, Pd, Ir, and Pt, Phys. Rev. B 11, 1416 (1975).  \n[6] A. D. Rakić, A. B. Djurišić, J. M. Elazar, and M. L. Majewski, Optical properties of \nmetallic films for vertical -cavity optoelectronic devices, Appl. Opt. 37, 5271 (199 8). \n[7] E. W. M. Haynes, editor , CRC Handbook of Chemistry and Physics , 97th Editi (CRC \nPress/Taylor & Francis, Boca Raton, FL., n.d.).  \n[8] A. M. Hofmeister, Thermal diffusivity of garnets at high temperature, Phys. Chem. Miner. \n33, 45 (2006).  [9] A. E. C lark and R. E. Strakna, Elastic Constants of Single -Crystal YIG, J. Appl. Phys. 32, \n1172 (1961).  \n[10] M. Weiler, M. Althammer, M. Schreier, J. Lotze, M. Pernpeintner, S. Meyer, H. Huebl, R. \nGross, A. Kamra, J. Xiao, Y. -T. Chen, H. Jiao, G. E. W. Bauer, and  S. T. B. \nGoennenwein, Experimental Test of the Spin Mixing Interface Conductivity Concept, \nPhys. Rev. Lett. 111, 176601 (2013).  \n[11] N. Vlietstra, J. Shan, V. Castel, B. J. van Wees, and J. Ben Youssef, Spin -Hall \nmagnetoresistance in platinum on yttrium i ron garnet: Dependence on platinum thickness \nand in -plane/out -of-plane magnetization, Phys. Rev. B 87, 184421 (2013).  \n "    },    {        "title": "2109.05901v1.Control_of_magnetization_dynamics_by_substrate_orientation_in_YIG_thin_films.pdf",        "content": "1 \n Control of magnetization dynamics by substrate orientation in  YIG thin films \nGanesh Gurjar1, Vinay Sharma3, S. Patnaik1*, Bijoy K. Kuanr2,* \n1School of Physical Sciences, Jawaharlal Nehru University, New Delhi, INDIA 110067  \n2Special Centre for Nanosciences, Jawaharlal Nehru University, New Delhi, INDIA 110067  \n3Morgan State University, Department of Physics, Baltimore, MD, USA 21251  \n \n \nAbstract  \nYttrium Iron Garnet (YIG) and b ismuth (Bi)  substituted YIG (Bi 0.1Y2.9Fe5O12, BYG) films are \ngrown  in-situ on single crystalline Gadolinium Gallium Garnet (GGG) substrates  [with (100) and \n(111) orientation s] using  pulsed laser deposition  (PLD ) technique .  As the orientation of the Bi-\nYIG film changes from (100) to (111) , the lattice constant  is enhanced  from 12.384  Å to 12.401 Å \ndue to  orientation dependent distribution of Bi3+ ions at dodecahedral  sites in the lattice cell. \nAtomic force microscopy (AFM) images show smooth  film surfaces  with roughness 0.308 nm in \nBi-YIG (111) . The change in substrate orientation leads to the modification of Gilbert damping \nwhich , in turn, gives rise to the enhancement of  ferromagnetic resonance (FMR) line width . The \nbest value s of Gilbert damping are found to be  (0.54±0.06 )×10-4, for YIG (100)  and \n(6.27±0.33) ×10-4, for Bi-YIG (111)  oriented  films . Angle variation ( ) measurements of the H r are \nalso performed, that shows  a four -fold symmetry for the resonance field  in the  (100) g rown film. \nIn addition, the value of  effective magnetization  (4πM eff) and extrinsic linewidth  (ΔH 0) are \nobserved to be  dependent on  substrate orientation . Hence PLD growth can assist single -crystalline \nYIG and BY G films with  a perfect interface that can be used for  spintronics  and related device \napplications.  \n \n \n \nKeyword s: Pulse Laser Deposition, Epitaxial YIG thin films, lattice strain, ferromagnetic \nresonance, Gilbert damping, inhomogeneous broa dening  \n \nCorresponding authors:  bijoykuanr@mail.jnu.ac.in , spatnaik@mail.jnu.ac.in  2 \n 1. Introduction  \nOne of the most widely studied material s for the realization of spintronic devices appears to be the \niron garnets , particularly the  yttrium iron garnet (YIG , Y3Fe5O12) [1,2] .  In thin film form  of YIG  \nseveral potential applications have been envisaged that include spin-caloritronics  [3,4] , magneto -\noptical (MO) devices, and microwave resonators, circulators, and  filters [5–8]. The attraction of \nYIG over other ferroic materials is primarily due to their strong magnet o-crystalline  anisotropy \nand low magnetization damping  [2]. Furthermore, towards  high frequency  applications, YIG’s \nmain advantage s are its electrically insulating behavior  along with low ferromagnetic resonance \nline-width (H) and low Gilbert damping  parameter [9–11]. These are important parameters for \npotential  use in high fr equency filters and actuators [12–14]. In this paper, we report optimal \ngrowth parameters for pure and Bi -doped YIG on oriented subs trates and identify the conditions \nsuitable for their  prospective  applications.  \n \nIn literature, YIG is known to be a room temperature  ferrimagnetic insulator with a  Tc near 560 K \n[15]. It has a cubic structure (space group Ia3̅d). The y ttrium  (Y) ions occupy the  dodecahedral  \n24c sites ( in the Wyckoff notation), two Fe ions  at octahedral 16a and three at tetrahedral 24d sites, \nand oxygen the 96h sites  [16,17] . The d site is resp onsible for the ferri magnetic  nature of YIG. It \nis already reported that substitution of Bi in place of Y in YIG leads to substantial improvement in \nthe magneto -optical response [7,18 –25]. It was also observed that MO performance increa ses \nlinearly with Bi/Ce doping concentration [22]. Furthermore, substitution of  Bi in YIG  (BYG)  is \ndocumented to provide growth -induced anisotropy  that is useful in applications such as magnetic \nmemory and logic devices [26–30]. The study of basic properties of  Bi -substituted YIG materials \nis of great current interest  due to their applications  in magneto -optical devices , magnon -3 \n spintronics , and related fields such as caloritronics  due to its high uniaxial anisotropy and faraday \nrotation  [21,31 –35]. The structural and magnetic pr operties can be changed via change in Bi3+ \nconcentration in YIG or via choosing a proper  substrate orientation. Therefore, t he choice of \nperfect substrate orientation is crucial for the identification of the growth of Bi  substituted YIG \nthin films.  \nIn this  work, we have studied the structural and magnetic properties  of Bi-substituted YIG \n[Bi0.1Y2.9Fe5O12 (BYG)] and YIG thin films with two different single crystalline Gadolinium \nGallium Garnet (GGG)  substrate orientation s: (100) and (111) . The YIG and BYG films of  \nthickness  ~150 nm  were grown by pulsed laser deposit ion (PLD) method  [23,36,37]  on top of  \nsingle -crystalline GGG  substrates . The structural and magnetic properties of all grown films were \ncarried out using x -ray diffraction (XRD), surface morphology by atomic force microscopy \n(AFM) , and magnetic properties via vibrating sample magnetometer (VSM) and ferromagnetic \nresonance (FMR) techniques.  The FMR is the most useful  technique to study the magnetization \ndynamics by measuring the  properties of magnetic materials through evaluation of their damping \nparameter and linewidth .  Furthermore, it provides insightful information on the static magnetic \nproperties such as the saturation magnetization and the anisotropy field. FMR is also extremely \nhelpful to study fundamentals of spin wave dynami cs and towards characte rizing the relaxation \ntime and L ande g factor of  magnetic material s [11]. \n \n2. Experiment  \n \nYIG a nd BY G target s were synthesized  via the solid -state reaction method . Briefly, y ttrium oxide \n(Y2O3) and iron oxide (Fe 2O3) powder s were ground for ~14 hours  before calcination at 1100 oC. 4 \n The calcined  powders were pressed into pellets and sintered at 1300 oC.  Using thes e YIG and \nBYG targets,  thin films of thickness ~150 nm  were grown  in-situ on (100) - and (111) -oriented \nGGG  substrate s by the PLD technique . The prepared samples have been labeled as YIG (100) , \nYIG (111) , BYG (100), and BYG (111).  GGG substrates were cleaned using acetone and \nisopropanol.  Before deposition, the deposition chamber  was thoroughly cleaned and evacuated to \na base vacuum of 2 ×10-6 mbar. We have used KrF excimer laser (248  nm), with pulse frequency \n10 Hz to ablate the target s at 300mJ  energy . During deposition , target to substrate distance, \nsubstrate temperature , and oxygen pressure w ere kept at  ~4.8 cm, 825 oC, and 0.15 mbar , \nrespectively.  Best films were grown at a rate  of 6 nm/min . The as -grown thin film s were annealed \nin-situ for 2 hours at 825  oC and cooled down to 300 oC in the presence of oxygen  (0.15 mbar)  \nthroughout the process . The structural properties of the thin film were determined  by XRD  using \nCu-Kα radiation  (1.5406  Å) and surface morphology as well as  the thickness of the film were \ncalculated with atomic force microscopy  by WITec  Gmb H, Germany . Magnetic properties were \nstudied using a 14 tesla PPMS (Cryogenic) . FMR  measurements were carried out  by the  Vector \nNetwork Analyzer ( VNA ) (Keysight , USA)  using a coplanar waveguide ( CPW ) in a flip -chip \ngeometry with dc magnetic field applied parallel  to the film plan e. \n \n3. Results and Discussion  \n3.1 Structural properties  \nThe room temperature XRD  data for the polycrystalline targets of YIG and BYG are plotted in \nfigure  1 (a) and 1  (b) respectively.  Rietveld refinement patterns after fitting XRD data are also \nincluded in the panel s. XRD peaks are indexed according to the JCPDS card no. ( # 43-0507) . Inset 5 \n of figure  1 (a) shows crystallographic sub -lattices  of YIG that elucidates  Fe13+ tetrahedral  site, \nFe23+ octahedral site , and Y3+ dodecahedral site. Inset (i) of figure  1 (b)  shows evidence for \nsuccessful incorporation of Bi into YIG ; the lattice constant  increases when Bi is substituted into \nYIG due to larger ionic radii of Bi (1.170 Å) as compared with Y (1.019 Å) [19]. From Rietveld \nrefinement  we estimate the lattice constant of YIG and BYG to be  12.377 Å [38] and 12.401 Å  \nrespectively .  \n \nFigure  2 (a) and 2  (b) show the XRD pattern of  bare (100) and (111) oriented GGG substrates . \nThis is followed by figure  2 (c)  & 2(d) for YIG and figure  2 (e)  & 2 (f) for BYG as grown thin \nfilms. XRD patterns confirm  the single -crystalline grow th of YIG and BYG thin film s over GGG \nsubstrates . The l attice  constant, lattice mismatch  (with respect to substrate) , and lattice volume \nobtain ed from XRD  data are listed in Table  1. Lattice cons tant a for the cubic structure is evaluated \nusing the [39]. \n𝒂=𝜆√ℎ2+𝑘2+𝑙2\n2sin𝜃                                                        (1) \n Where  𝜆 is the wavelength of Cu -Kα radiation , 𝜃 is the diffraction angle , and [h , k, l ] are the \nmiller indices of the corresponding XRD peak. Lattice  misfit  (𝛥𝑎\n𝑎) is evaluated using equation 2  \n[24,38] . \n𝛥𝑎\n𝑎=(𝑎𝑓𝑖𝑙𝑚− 𝑎𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 )\n𝑎𝑓𝑖𝑙𝑚 100               (2) \nWhere  𝑎𝑓𝑖𝑙𝑚 and 𝑎𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒  are the lattice constant of film and substrate respectively.  Lattice \nconstant of pure YIG bulk  is 12.377  Å, whereas we have observed  a larger value of lattice constants \nof YIG and BY G films  than th at of bulk YIG as shown in T able 1 . Sim ilarly, to these  obtained \nresults, a larger value of lattice constants than that of bulk YIG has been reported as well [40–44]. 6 \n The obtained values of the lattice constant are in agreement with the previous reports \n[18,21,25,34,45] . In the case of  BYG (111), the value of lattice constant slightly increases \ncompared to BYG (100) because the distribution of Bi3+ in the dodecahedral site depends on the \norientation of the substrate [28,46] . Inset (ii) of figure  1 (b) shows  plane (111) has more \ncontribution of Bi3+ ions [(ionic radius of bismuth (1.170 Å) is larger as compared with YIG (1.019 \nÅ) [19]]. This slight increase in the lattice constant  (in 111 direction)  implies a more lattice \nmismatch (or strain ) in BYG films . Positive  value of lattice mismatch indicates the slightly larger \nlattice constant of films (YIG and BYG) were observed as compared to substrates (GGG). We \nwould like to emphasize  that lattice plane dependence growth  is important to signify  the changes \nin the struc tural and magnetic properties.  \n3.2 Surface morphology study  \nRoom temperature AFM images  with roughness are  shown in figure 3 (a)-(d). Roughness plays \nan important role from the application prospective as it is related to  Gilbert  damping factor α. \nLower roughness (root mean square height) is observed for the (111) oriented films of YIG and \nBYG compared to those grown on (100) oriented substrates. Available literature [61] indicate that \nroughness would depend more on variation on growth parameters ra ther than on substrate \norientation.  In this sense further study is needed to clarify substrate dependence of roughness. No \nsignificant change in the roughness is observed between YIG and BYG films  [38,47] . Table 1  \ndepicts a comparison between the roughne ss measured in YIG and the BY G thin films.  7 \n 3.3 Static magnetization p roperties  \nVSM  magnetization  measurements were performed  at 300 K with magnetic field appl ied parallel \nto the film plane  (in-plane) .  Figure 3 (e) and 3  (f) show the magnetization plot s for YIG and BYG \nrespectively after careful subtraction of paramagnetic contr ibution  that is assigned to the substrate.  \nThe m easured saturation magnetization ( 4πM S) data are given in Table 1 which are in general \nagreement with the reported values [11,40,48] . Not much change in the measured  4πM S value of \nYIG and Bi -YIG films  are observed . The ferrimagnetism nature of YIG arises from super -\nexchange interaction between the non -equivalent Fe3+ ions at octahedral  and tetrahedral  sites [49]. \nBismuth located at dodecahedral site does not affect the tetrahedral and octahedral Fe3+ ions. So, \nBismuth does not show a significant change in saturation magnetization at  room temperature. I t is \nreported in literature th at Bi addition leads to increase  in Curie temperature, so in t hat sense there \nis an decreasing  trend in saturation magnetization in BYG films in contrast to YIG films [50,51] . \nError bars in saturation magnetization relate to uncertainty in sample volume.  \n3.4 Ferromagnetic r esonance properties  \nThe FMR absorption spectroscopy  is shown in figure 4. These measurements were  performed  at \nroom temperature .  The external dc magnetic field  was appli ed parallel to the plane of the film . \nLorentzian  fit of the calibrated experimental data are used to calculate t he FMR linewidth (∆H) \nand resonance  magnetic  field (H r). From the e nsemble of all the FMR data at different resonance \nfrequencies  (f = 1 GHz -12 GHz ), we have calculated the gyromagnetic ratio (γ) , effective \nmagnetization field ( 4𝜋𝑀𝑒𝑓𝑓) from the  fitting  of Kittel’s  in-plane equation  [52].  \n 8 \n In general, t he uniform precession of magnetization can be described by the Landau -Lifshitz -\nGilbert (LLG) equation of motion;  \n𝜕𝑀⃗⃗ \n𝜕𝑡=−𝛾(𝑀⃗⃗ ×𝐻⃗⃗ 𝑒𝑓𝑓)+𝐺\n𝛾𝑀𝑠2[𝑀⃗⃗ ×𝜕𝑀⃗⃗ \n𝜕𝑡]   (3) \nHere, t he first term corresponds to the precessional torque in the effective magnetic field and the \nsecond term is the Gilbert damping torque.  The gyromagnetic ratio  is given by 𝛾=𝑔𝜇𝐵/ℏ , where  \n𝑔 is the Lande’s factor, 𝜇𝐵 is Bohr magnetron and ℏ is the Planck’s constant. Similarly, 𝐺=𝛾𝛼𝑀𝑠 \nis related to the intrinsic relaxation rate in the nanocomposites and 𝛼 represents the Gilbert \ndamping constant. Ms (or 4πMs) is the saturation magnetization. It can be shown that t he solution \nfor in -plane resonance frequency can be written as;  \n𝑓𝑟=𝛾′√(𝐻𝑟)(𝐻𝑟+4𝜋𝑀𝑒𝑓𝑓)     (4), \nWhere  𝛾′=𝛾/2𝜋, 4𝜋𝑀𝑒𝑓𝑓=4𝜋𝑀𝑠−𝐻𝑎𝑛𝑖 is the effective field and 𝐻𝑎𝑛𝑖=2𝐾1\n𝑀𝑠 is the anisotropy \nfield.  Following through, we have obtained Gilbert damping parameter (α) and inhomogeneous \nbroadening (∆H 0) linewidth from the fitting of Landau –Lifshitz –Gilbert equation (LLG) [53] \n𝛥𝐻(𝑓)=𝛥𝐻0+4𝜋𝛼\n√3𝛾𝑓                                      (5) \n \n \n \n \nDerived parameters from the FMR study are listed in T able 2 . The obtai ned Gilbert damping (α) \nis in agreement with the  reported thin films used for the study  of spin-wave  propagation  \n[2,27,54,55] .  In the case of YIG no t much change  in the value of α is seen . However , a substantial \nincrease is observed in case of BYG with (111) orientation. Qualitatively this could be assigned to  9 \n the presence of  Bi3+ ions which induce s spin-orbit coupling (SOC)  [56–58] and also  due to  electron \nscattering inside the lattice as lattice  mismatch (or  strain ) increases  [59]. We have seen  more \ndistribution of Bi3+ ions along (111) planes ( see inset (ii) of figure  1 (b) ) and also slightly larger \nlattice mismatch  in BYG (111) from our XRD  results. These results also explain higher  value of \nGilbert damping  and ΔH 0 in case of BYG (111).  The change in 4𝜋𝑀𝑒𝑓𝑓 could be attributed to \nuniaxial in -plane magnetic anisotropy . This is because no change in 4πM S is observed from \nmagnetization measurements and 4𝜋𝑀𝑒𝑓𝑓=4𝜋𝑀𝑠−𝐻𝑎𝑛𝑖 [38,40,60] . The uniaxial inplane magnetic \nanisotropy  is induced due to lattice mismatch between films and GGG substrates  [38,40] . The \ncalculated gyromagnetic ratio (γ)  and ΔH 0 are also included in Table 2 . The magnitude of  ΔH 0 is \nclose to reported values for same  substrate orientation  [38]. In summary we find that YIG with \n(100) orientation yields lowest damping fact or and extrinsic contribution to  linewidth.  These are \nthe r equired optimal parameters for spintronic s application with high spin diffusion length.  \nHowever,  MOKE signal is usual ly very low in bare YIG thin films because of its lower magnetic \nanisotropy and strain [61]. But previous reports suggest that magnetic anisotropy and magnetic \ndomains formation can be achieved in YIG system by doping rare earth materials like Bi and Ce \n[18,61]. We have shown that anisotropic characteristic  with Bi doping in YIG is more pron ounced \nalong <111> direction which can lead to the enhanced MOKE signal in Bi -YIG films on <111> \nsubstrate.  \nWe have also recorded polar angle  () data of resonance field ( Hr) versus magnetic field \n(H) at frequency 12  GHz  for the BYG (100) and BYG (111)  films (figure  5 (c) & 5(d) respectively  \nwhere inset shows the azimuthal angle ( ) variation of Hr measured at frequency of 3 GHz ). The \ndata are fitted with modified Kittel equation . From figure 5 (c) & (d),  we can see that Hr increases \nup to 2.5 kOe in BYG (100) and 3.0 kOe in BYG (111)  by varying the  direction  of H from 0 to 90 10 \n degree with respect to sample surface  (inset of Fig 5 (a)) . Obtained parameters from angular \nvariation of FMR magnetic field H r (θH) are listed in the inset of figure 5 (c) & (d) . From  variation \ndata (by varying the  direction  of H from 0 to 18 0 degree with respect to sample  edge (Fig 5 (a)  \nInset ), we see clear four -fold and two -fold in-plane anisotropy in BYG (100) and BYG (111)  films  \n[61,62] . This further consolidates single -crystalline characteristics of our films. The change \nobserved in Hr with respect to  variation  is 79.52 Oe in BYG (100)  (H=0 to 45)  and 19.25 Oe in \nBYG (111) ( H=0 to 45). Thus,  during  in-plane rotation, higher change in FMR field is observed \nalong  (100) orientation . \n \n4. Conclusion  \nIn conclusion , we have  grown high quality YIG and B i-YIG thin film s on GGG substrates with \n(100)  and (111)  orientation . The films were gr own by pulsed laser deposition. The optimal \nparameters  i.e. target to substrate distance, substrate temperature, and oxygen pressure are \ndetermined to be  ~ 4.8 cm, 825 oC, and 0.15 mbar, respectively. The as  grown thin films have \nsmooth surfaces and  are found to be  phase pure  from AFM and XRD characterizations.  From FMR \nmeasurements , we have found  lower value of damping parameter in (100) YIG that indicates \nhigher spin diffusion length for potential spintronics application.  On the other -hand bismuth \nincorporation to YIG leads to dominance of anisotropic characteristics  that augers well for \napplication in magnetic bubble memory and  magneto -optic  devices . The enhanced value of α in \nBi-YIG films is ascribed to the spin orbit coupled Bi3+ ions. We also ta bulate the values of \nmagnetic parameters such as linewidth ( ∆H0), gyromagnetic ratio ( γ), and effective magnetization  \n4𝜋𝑀𝑒𝑓𝑓 with respect to substrate orientation.  Unambiguous four-fold in -plane anisotropy is \nobserved in (100) oriented films. 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YIG (111)  (1.05±0.13)  26.51±0.21  2331 .38±65.78  2.86±0.02  \n3. BYG (100)  (1.66±0.10)  26.52 ±0.17  1701.67±31.87  2.89±0.11  \n4. BYG (111)  (6.27±0.33)  29.28 ±0.62  2366 .85±62.60  2.86±0.02  \n \n  21 \n Figure Captions  \nFigure 1: XRD with Rietveld refinement pattern of (a) YIG target ( inset shows  crystallographic \nsub-lattices, Fe 13+ tetrahedral  site, Fe 23+ octahedral site and Y3+ dodecahedral site ) (b) BYG target  \n(inset (i) shows effect of Bi doping into YIG , inset  (ii) shows contribution of the Bi3+ along the \n(100) and (111) planes ). \n \nFigure 2: XRD pattern of  (a) GGG (100) , (b) GGG (111), (c)  YIG (100) , (d) YIG (111) , (e) BYG \n(100) , and (f) BYG (111) . \n \nFigure 3: AFM images of  (a) YIG  (100), (b ) YIG (111) , (c) BYG (100), (d) BYG (111) and  static \nmagnetization graph of (e ) YIG (100), YIG (111);  and (f) BY G (100), BYG (111) . \n \nFigure 4: FMR absorption spectra of (a) YIG (100), (b) YIG (111), (c) BYG (100), and (d) BYG \n(111).  \n \nFigure 5: (a) FMR magnetic field Hr is plotted as a function of frequency f.  Experiment data fitted \nwith Kittel equation for YIG and BYG oriented films. Inset shows how the applied field  angle is \nmeasured from sample surface  (b) Frequency -dependent FMR linewidth data fitted with LLG \nequation for YIG and BYG oriented films.  Inset shows the magnified version to illustrate  the effect \nof Bi doping in YIG . (c) and (d) show angular variation   of FMR magnetic field (Hr (θH)) fitted \nwith modified Kittel  equation at 12  GHz frequency for BYG (100) and  BYG (111)  films .  Insets \nshow the FMR magnetic  field (H r) as a function of azimuthal angle ( ). \n 22 \n  \nFigure 1  \n \n23 \n  \nFigure 2 \n \n \n \n \n \n24 \n  \nFigure 3 \n \n \n \n \n25 \n  \n \n \n \n \nFigure 4 \n \n \n \n \n \n \n26 \n  \n \n \n \nFigure 5 \n \n \n"    },    {        "title": "1706.02154v1.Spin_Seebeck_effect_and_magnon_magnon_drag_in_Pt_YIG_Pt_structures.pdf",        "content": "arXiv:1706.02154v1  [cond-mat.other]  6 Jun 2017Spin Seebeck effect and magnon-magnon drag in Pt/YIG/Pt stru ctures\nI.I. Lyapilin1,2, M.S. Okorokov1∗\n1Institute of Metal Physics, UD RAS, Ekaterinburg, 620137, R ussia\n2Ural Federal University after the first President of Russia B .N. Yeltsin, Yekaterinburg, 620002 Russia\n(Dated: November 5, 2021)\nThe formation of the two: injected (coherent) and thermally excited, different in energies magnon\nsubsystems and the influence of its interaction with phonons and between on drag effect under spin\nSeebeck effect conditions in the magnetic insulator part of t he metal/ferromagnetic insulator/metal\nstructure is studied. An approximation of the effective para meters, when each of the interacting\nsubsystems (”injected”, ”thermal” magnons, and phonons) i s characterized by its own effective\ntemperature and drift velocities have been considered. The analysis of the macroscopic momentum\nbalance equations of the systems of interest conducted for d ifferent ratios of the drift velocities of\nthe magnon and phonon currents show that the injected magnon s relaxation on the thermal ones\nis possible to be dominant over its relaxation on phonons. Th is interaction will be the defining\nin the forming of the temperature dependence of the spin-wav e current under spin Seebeck effect\nconditions, and inelastic part of the magnon-magnon intera ction is the dominant spin relaxation\nmechanism.\nI. INTRODUCTION\nThe influence of non-equilibrium phonons on kinetic\ncoefficients in electron-phonon or spin-phonon systems\nhas been theoretically studied chiefly by the Boltzmann\nkinetic equation method or using the formalism of the\nKubo response theory. The present work employs the\nmethodofthenon-equilibriumstatisticaloperator(NSO)\nto analyze how interactions between three flows (two\nmagnon and phonon ones) affect the drag effect and the\ntemperature dependence of the spin Seebeck effect (SSE)\nwithin the above model.\nThe concept of magnon spintronics, i.e., the genera-\ntion, detection and manipulation of pure spin currents\nin the form of spin wave quanta1, the magnons, has at-\ntracted growing interest in the recent years2,3. Magnons\nare quasiparticles representing a low-energy excited state\nofferromagnets. Aquantized magnonis a boson and car-\nries basic spin angular momentum quanta of ¯ h4. Similar\nto spintronics and electronics, magnonics refers to using\nmagnons for data storage and information processing5.\nUp to now, magnons in the field of spintronics have been\ninvestigated within the content of magnetostatic spin\nwaves which describe the nonuniform spatial and tem-\nporal distribution of the classical magnetization vector.\nRecently, a great deal of attention is devoted to the\ninvestigation of thermally excited magnons, particularly\nin studies of the spin Seebeck effect5–9in Pt/YIG/Pt\nstructure. In SSE effect, an applied electric current in\none Pt layer accompanies an electron spin current due\nto the spin Hall effect (SHE)8,9. When the spin cur-\nrent flows to the boundary between the Pt and the YIG,\nnonequilibrium spins are accumulated and, consequently,\ndue to the s-d exchange interaction between conduction\nelectrons in normal metal (NM) and magnetic moments\nin ferromagnetic insulator (FI), magnons are created at\nthe interface10,11. The induced magnons subsequently\ndiffuse in FI to the other interface where the magnon\ncurrent converts back to an electron spin current in theother NM layer, leading to a charge current due to the\ninverse spin Hall effect (ISHE)12–14. Thus, the induced\nelectric current in the second NM layer which is electri-\ncally insulated from the current-flowing NM layer by a\nFI would elucidate the magnons as spin information car-\nriers. One of the key advantages of magnon spin currents\nis their large damping length, which can be several or-\ndersofmagnitude higherthan the spindiffusion length in\nconventional spintronic devices based on spin-polarized\nelectron currents15.\nThe propagation of magnons in a magnetic insula-\ntor is described by two characteristic quantities: mean\nfree path and spin diffusion length that are governed, in\nturn, by various magnon relaxation mechanisms. A se-\nries of experiments determine the range of the diffusion\nlengths as being quite wide: from 4 µm to 120 µm16–20.\nTo explain so large values of the spin diffusion lengths,\nthe number of papers has put forward several concepts\nof appearance, along with ”thermal” magnons, of long-\nwave (”subthermal”) magnons in a magnetic insulator.\nUnder SSE conditions, the former are characterized by\nshort wavelength, the latter are weakly coupled with the\nlattice17,20. Forinterpretingtheexperimentalresults,the\nworks17,20,21haveadoptedthehypothesisoftheexistence\nofthetwomagnonsubsystemswithdifferentenergies. As\ntothetemperaturedependence oftheSeebeckcoefficient,\nit is non-monotonic and reaches its maximum within the\nrangeof50- 100K.And asthe investigationshaveshown,\nit is affected by strength of a magnetic field, dimensions\nof the samples, and quality of the interface20,22.\nTo explain the low temperature enhancement23pro-\nposed the phonon-drag SSE scenario based on a theoret-\nical model24–26. Back in 1946, in the context of ther-\nmoelectricity, Gurevich pointed out that thermopower\ncan be generated by nonequilibrium phonons driven by\na temperature gradient, which then drag electrons and\ncause their motions24. It was suggested first by Bailyn27\nthat the theory of magnon-drag should be analogous to\nthat of phonon-drag24. Following this work the magnon-2\ndrag component of the thermopower has been calculated\nby Grannemann and Berger28. In this phenomenologi-\ncal model, the temperature dependence of the phonon\nlife time is involved, which reaches a maximum at low\ntemperatures. Based on this, they propose a strong in-\nteraction between the phonons and magnons, which are\nresponsible for the heat transport in the system. The\nphonons flow along the thermal gradient and interact\nwith thermally excited magnons. These phonons drag\nthe magnons. Thus, the phonon-magnon coupling is sug-\ngested to explain the observed enhancement of SSE sig-\nnal at low temperature23,24. The first investigation of\nthe phonon-magnon interaction in magnetic insulators\nwas conducted by Adachi et al., reporting a giant en-\nhancementofSSEin LaY2Fe5O12atlowtemperatures23.\nHowever, the observed transport of magnons over a long\ndistance of up to millimeters in magnetic insulators im-\nplies a relative weak interaction with phonons and impu-\nrities, and the measurements of the temperature depen-\ndent thermal conductance of YIG single crystals show\nthat the phonon contribution to the thermal conductiv-\nity reaches its maximum at around 25 K16, which is 50\nK lower than the observed peak in the SSE. In28the\ntemperature dependence of the spin Seebeck effect was\nmeasured in ( Ga,Mn)As, and the data showed a pro-\nnounced peak at low temperatures.\nHere, we study how the formation of two interact-\ning magnon subsystems with distinguished energies af-\nfects the SSE17,19–21. We assume that the first group of\nmagnons is ”thermal” ones subjected to a non-uniform\ntemperature field applied to the magnetic insulator. The\nenergy of such magnons is of the order of the tempera-\nturekBT. Further, under the SSE, inelastic scattering\nof spin-polarized electrons of the metal by localized spins\nlocated near the interface causes the magnons to inject\ninto the magnetic insulator. The energy of the injected\n(”coherent”) magnons is of the order of the spin accu-\nmulation energy ∆ sof conduction electrons of the metal.\nIt can be generated, for example, by the spin Hall effect\nwhen passing a direct electric current through the metal\n(Pt) [1].\nUnder the SSE conditions, the injection of magnons\ninto the magnetic insulator dominates scattering pro-\ncesseswithmagnonabsorptionprovidedthattheinequal-\nity ∆s> kBTis fulfilled. Thus, it can be said that the\nmagnetic system of the insulator forms another subsys-\ntem of ”injected”(”coherent”) magnons that are actu-\nally responsible for the SSE. As a consequence, in the\npresence of a non-uniform temperature field, there are\nthreeflowsinsidethemagneticinsulator, namely, phonon\nand two magnon ones. The evolution of the magnon and\nphonon subsystems to equilibrium occurs due to the re-\nlaxation of both their energy and their moment. These\nsubsystemstend tobecomebalancedwith differentveloc-\nities. Obviously, the interaction between the flows gives\nrise to the drag effect29–31.\nIt is worth noting that the paper32has already dis-\ncussed the influence ofmutual draggingbetween phononsand spin excitations on thermal conductivity of a spin\nsystem. Once the magnon subsystems are thermalized in\nenergy, the magnon system can be characterized solely\nby a temperature Tm. As to the moment relaxation pro-\ncesses, this time is defined, as well as in the event of spin-\nflip electron scattering, by inelastic magnon scattering.\nThe time relaxation is large enough, which, apparently,\nprovides the existence of the SSE over long distances16.\nThe paper is structured in the following manner. The\nfirst part is devoted to the splitting of the ”injected”\nmagnonsflow responsiblefor the spin Seebeck effect from\nthemagnoncurrent. Inthesecondpartthebalanceequa-\ntions for the magnons and phonons in the approximation\nof effective parameters when each subsystem is charac-\nterized by its effective temperature and drift velocity are\nbuilt and analysed.\nSPIN CURRENT\nThe density of the spin current Js(r) can be repre-\nsented as the sum of two terms: collisional ˙ sz\n(sm)(r) and\ncollisionless Isz(r). The former is controlled by the in-\nelastic spin-flip electron scattering by localized moments\nat the interface, the latter is due to the flows of electrons\nwith different spin orientation33\nJs(r) =d\ndtsz(r) =1\ni¯h[sz(r),H] =−∇Isz(r)+ ˙sz\n(sm)(r),\nIsz(r) =/summationdisplay\nisz\ni{pi/m,δ(r−ri)},\n˙sz\n(sm)(r) =1\ni¯h[sz(r),Hsm]. (1)\nHereHis the Hamiltonian of the system considered,\nHsmis the density of the exchange interaction energy\nbetween conduction electrons and localized moments at\nthe interface34.\nHsm=−J0/summationdisplay\nj/integraldisplay\ndrs(r)S(Rj)δ(r−Rj),(2)\nJ0is the exchange integral, S(Rj) is the operator of a\nlocalized spin with the coordinate Rjat the interface.\ns(r) is the spin density of the electrons in the metal. Ac-\ncompaniedby the creationofmagnons, the inelastic scat-\ntering of spin-polarized conduction electrons by localized\nimpurity centers dominates other processes. This leads\nto magnon accumulation ( δN(r) =N(r)−N0(r))33,\nN(r), N0(r) are the non-equilibrium and equilibrium\nmagnon distribution functions near the interface in the\nmagnetic insulator. The macroscopic spin current can\nbe found by averaging the expression with the non-\nequilibrium statistic operator ρ(t)35:\n/an}b∇acketle{tJs(r)/an}b∇acket∇i}htt=−∇ /an}b∇acketle{tIsz(r)/an}b∇acket∇i}htt+/angbracketleftBig\n˙sz\n(ms)(r)/angbracketrightBigt\n,(3)\nwhere/an}b∇acketle{t.../an}b∇acket∇i}htt=Sp(ρ(t)...).3\nGiven that [ sz\ni,s±\nj] =±s±\niδij, we have\n/angbracketleftBig\n˙sz\n(ms)(r)/angbracketrightBigt\n= (−J0/2)/summationdisplay\nj/integraldisplay\ndr/angbracketleftbig\n(s+(r)S−(Rj)−\n−s−(r)S+(Rj))δ(r−Rj)/angbracketrightbigt(4)\nRestricting ourselves to the linear approximation in\nthe interaction Hsm, we omit the interaction Hsmin the\noperator ρ(t) in which the averaging is performed. In\nthis case < ... >t=< ... >t\ne< ... >t\nm, i.e. the averaging\nin the electron system and the localized spin system is\ncarried out separately:\n/angbracketleftbig\nsα(r)Sβ(Rj)/angbracketrightbigt→ /an}b∇acketle{tsα(r)/an}b∇acket∇i}htt\ne/angbracketleftbig\nSβ(Rj)/angbracketrightbigt\nm.\nThus, we arrive at,\n/angbracketleftBig\n˙sz\n(ms)(r)/angbracketrightBigt\n= (−J0/2)/summationdisplay\nj/integraldisplay\ndr{/angbracketleftbig\ns+(r)/angbracketrightbigt\ne/angbracketleftbig\nS−(Rj)/angbracketrightbigt\nm−\n−/angbracketleftbig\ns−(r)/angbracketrightbigt\ne/angbracketleftbig\nS+(Rj)/angbracketrightbigt\nm}δ(r−Rj).(5)\nLet us calculate S±(R). We write down the equations of\nmotion for the transverse components\n˙S±(R) = (i¯h)−1[S±(R), Hm+Hk\nm+Hmp+Hms],\nwhereHm, Hk\nm, Hmp, Hmsare the energy operators\nof the magnetic subsystem (Zeeman and kinetic), the\nmagnon-phonon( mp) and exchange( sm) interaction, re-\nspectively. Computing the commutators, we obtain\n˙S±(R) =∓iωmS±(R)−∇IS±(R)+˙S±\n(mp)(R)+˙S±\n(ms)(R),\n˙A(ik)(R) = (i¯h)−1[A(R),Hik] (6)\nand the density of the magnon flows at the interface\nIS±(R) =/summationdisplay\njS±\nj{Pj/M,δ(R−Rj)}(7)\nPj,Mare the magnon momentum and the effective\nmagnon mass. The last two summands in the right-\nhand side of (6)describe the scattering of the magnons\nby phonons and electrons at the interface. Conducting\nthe averaging, in the stationary case we come to\n∓iωm/angbracketleftbig\nS±(R)/angbracketrightbig\nm=∇/an}b∇acketle{tIS±(R)/an}b∇acket∇i}htm−\n−/angbracketleftBig\n˙S±\n(mp)(R)/angbracketrightBig\nm−/angbracketleftBig\n˙S±\n(ms)(R)/angbracketrightBig\nm.(8)\nFurther, weinsertthe expression(8) intothe equationfor\nthe spin current/angbracketleftBig\n˙sz\n(ms)(r)/angbracketrightBig\nand estimate the summands.\nThefirsttermintheright-handsideoftheexpression(5),\napproximatelyproportionalto ∼J0, governsthe magnon\nflow excited at the interface due to electron scattering by\nlocalized moments. The second term is proportional to\n∼J0Upand sets forth the magnon scattering by phonons\n(Upcharacterizesthe intensity of the magnon-phonon in-\nteraction). Finally, the last term in the right-hand sideof the expression (5) ∼J2\n0. Putting that Up≫J0, we\nleave this term aside.\nLet us unravel the evolution of the magnetic subsys-\ntem.\n˙Sz(R) = (i¯h)−1[Sz(R), Hm+Hk\nm+H(mp)+H(ms)]\nThen we have\n˙Sz(R) =−∇ISz(R)+˙Sz\n(ms)(R)+˙Sz\n(mp)(R).(9)\nThe first term in the right-hand side of (9) involves the\nspin-density flow of localized spins (magnons), and the\nterms˙Sz\n(mp)(R),˙Sz\n(ms)(R), described the scattering of\nthe localized spins by phonons at the interface.\nThus, the macroscopic spin-wave current realized in\nthe magnetic insulator can be written as\n/an}b∇acketle{tIS(R)/an}b∇acket∇i}ht=−∇/an}b∇acketle{tISz(R)/an}b∇acket∇i}ht+/angbracketleftBig\n˙Sz\n(ms)(R)/angbracketrightBig\nm+/angbracketleftBig\n˙Sz\n(mp)(R)/angbracketrightBig\nm+\n+(−J0)/summationdisplay\nj/integraldisplay\ndr{/angbracketleftbig\ns+(r)/angbracketrightbig\ne/angbracketleftbig\nS−(Rj)/angbracketrightbig\nm)−\n−/angbracketleftbig\ns−(r)/angbracketrightbig\ne/angbracketleftbig\nS+(Rj)/angbracketrightbig\nm}δ(r−Rj),(10)\nwhere\n/angbracketleftbig\nS±(R)/angbracketrightbig\nm=iω−1\nm{−∇/an}b∇acketle{tIS±(R)/an}b∇acket∇i}htm−/angbracketleftBig\n˙S±\n(mp)(R)/angbracketrightBig\nm/bracerightbig\n.(11)\nIn (11), we have omitted the summand that describes\nthe magnon scattering at the interface ( ∼J2\n0). It can be\nseen from (10), (11) that the magnetic subsystem real-\nizes two magnon flows. The first is due to a non-uniform\ntemperature perturbation of the magnetic subsystem. It\nis a flow of ”thermal” magnons. The mean energy of\nthese magnons is of the temperature. The second is\n∼J0∇IS±(R) and isbroughtabout bymagnonsinjected\ninto the magnetic subsystem as a result of inelastic scat-\ntering of conduction electrons by localized moments at\nthe interface. The energy of such magnons is of the or-\nder of the spin-accumulation energy of the conduction\nelectrons and is equal to ∆ s≫kbT.\nMACROSCOPIC MOMENTUM BALANCE\nEQUATIONS\nThe influence of non-equilibrium phonons on kinetic\ncoefficients in electron-phonon or spin-phonon systems\nhas been theoretically studied chiefly by the Boltzmann\nkinetic equation method or using the formalism of the\nKubo response theory32. The present work employs the\nmethodofthenon-equilibriumstatisticaloperator(NSO)\nfor analyzing how interactions between three flows (two\nmagnon and phonon one) affect the drag effect and the\ntemperaturedependence ofthe spinSeebeck effect within\nthe above model. In constructing macroscopic momen-\ntum balance equations for the system at hand, we should\nuse the Hamiltonian\nH=HM+HP+HV (12)4\nHereHMis the Hamiltonian of the magnetic sys-\ntem that consists of two magnetic subsystems: of\n”injected”(”coherent”) ( Hm1) and ”thermal”( Hm2)\nmagnons and their mutual interaction\nHM=/integraldisplay\ndr(/summationdisplay\niHmi(r)+Hmimi(r)), i= 1,2 (13)\nThe integration is performed over the volume occupied\nby the magnetic insulator FI. Hmi(r) is the energy den-\nsity operator of the (i) magnetic subsystem. Hmimi(r) is\nthe Hamiltonian of the magnon-magnon interaction in-\nside each the subsystems\nSuppose the magnon gas to be free: Hmimi=/summationtext\nkε(k)b+\nkbk, ε(k) =P2/(2M) is the sum of the en-\nergies of quasi-particles, ferromagnons having a quasi-\nmomentum P= ¯hkwith their effective mass Mand\nmagnetic momentum34).b+\nk, bkare the creation and an-\nnihilationoperatorsforthemagnonswiththewavevector\nk.\nHpis the lattice Hamiltonian\nHp=/integraldisplay\ndr(Hp(r)+Hpp(r)), (14)\nwhereHp(r) is the energy density operator for the\nphonon subsystem. Hpp(r) is the phonon scattering by\nnon-magnon relaxation mechanisms (scattering by the\nboundaries of the sample, impurities and defects of the\nlattice, etc.)\nHV(r) =Hmimj(r)+Hmip(r)+Hmis(r) (15)\nis the energy density operator of interaction between the\nphonons. Hmip(r) is the energydensity operatorofinter-\naction between the phonon and magnetic (i) subsystems.\nHmimj(r) describes the interaction between ”thermal”\nand ”coherent” (injected) magnons. Hmis(r) is the en-\nergy density operator of exchange interaction between\nconduction electrons and localized magnetic moments at\nthe interface.\nUnder the influence ofa non-uniform temperature field\n(a temperature gradient) applied to the system, the\nmagnonsandphononsbegintravelling; theirmacroscopic\ndrift affects the propagation of the spin-wave current.\nObviously, the drag effects that may arise in the sys-\ntem considered are governed by both magnon-phonon\ncollision frequencies and phonon relaxation mechanisms\nby other mechanisms of their scattering. The prob-\nlem to be solved reduces to constructing and analyz-\ning a set of macroscopic momentum balance equations\n˙Pi(r) = (i¯h)−1[Pi(r),H] for the magnon ( i= 1,2 ) and\nphonon ( i=p) subsystems.\nIn writing the Hamiltonian, we have omitted the ex-\nchange interaction between localized spins and conduc-\ntion electrons at the interface. In doing so, we have put\nthat it is the exchange interaction that is responsible for\nthe magnon injection into the magnetic insulator and\nmakes no significant contribution to the momentum re-\nlaxation of the magnons and phonons.The equations of motion for the magnon and phonon\nmomenta have the form:\n∂\n∂tPmi(r) =−∇IPmi(r)+˙P(mi,v)(r),(i/ne}ationslash=j= 1,2)\n∂\n∂tPp(r) =−∇IPp(r)+˙P(p,pp)(r)+˙P(p,v)(r),(16)\nwhere\n˙A(i,v)= (i¯h)−1[Ai,Hv].\nThe first terms in the right-hand sides of (16) are the\nflows of appropriate momenta IP(r) =/summationtext\ni{Pi/M,δ(r−\nri)}. The rest of the terms in the right-hand side of these\nequations describe the relaxation processes: magnon-\nphonon and magnon-magnon scattering.\nTo derive the macroscopic equations (16)\n/angbracketleftBig\n˙Pi(r)/angbracketrightBigt\n=Sp{˙Pi(r)ρ(t)},(i=m1,m2,p),\nthe expression for the NSO needs to be sought. Accord-\ning to35,36, forρ(t) we have:\nρ(t) =ǫ0/integraldisplay\n−∞dt′eiǫt′eit′Lρq(t+t′), ǫ→+0,\nρq(t) =e−S(t), eitLA=e−itH/¯hAeitH/¯h,(17)\nHereρq(t) is the quasi-equilibrium statistical operator.\nThe non-equilibrium state of the system considered cor-\nrespondsintermsofaveragedensityvaluestotheentropy\noperator\nS(t) =S0+δS(t) =\n= Φ(t)+/integraldisplay\ndr{βmi(r,t)[Hmi(r)+Hmimi(r)+\n+Hmimj(r)]−βµmi(r,t)Nmi(r)+\n+βp(r,t)[Hp(r)+Hpp(r)+Hpmi(r)]−\n−βmi(r,t)Vmi(r,t)Pmi(r)+βp(r,t)Vp(r,t)Pp(r)}.(18)\nHereS0is the entropy operator for the equilibrium sys-\ntem.δS(t) describes the deviation of the system from\nits equilibrium state. Φ( t) is the Massieu-Plank func-\ntional.βmi(r,t) are local-equilibrium values of the in-\nverse temperatures of the magnon ( i= 1,2) and phonon\nsubsystems βp(r,t).µmi(r,t)is a local equilibrium value\nof the chemical potential of the magnons. N(r) =\nNm1(r) +Nm2(r) is the magnon number density oper-\nator.Vmi,Vpare the drift velocities of the magnons\n(i= 1,2) and the phonons respectively.\nMagnons, as well as phonons are Bose particles; their\ndistribution function is the Bose-Einstein function with\na zero chemical potential. However, the situation be-\ncomes quite different if magnons are non-equilibrium. In\nour case, the non-equilibrium magnon system may be\ndescribed by introducing the non-equilibrium chemical\npotential of magnons37–39.5\nForρ(t), in the linear approximation in deviation from\nequilibrium, we arrive at\nρ(t) =ρq(t)−0/integraldisplay\n−∞dt′eǫt′eit′L1/integraldisplay\n0dτ ρτ\n0˙S(t+t′)ρ−τ\n0ρ0.\n(19)\n˙S(t) =∂S(t)/∂t+(i¯h)−1[S(t),H] is the entropy produc-\ntion operator. ρ0= exp{−S0}.Thus, the problem boils\ndown to finding the entropy production operator.\nWe write down the equations of motion for the opera-\ntors involved in the entropy operator. Then, we have\n˙Hmi(r) =−∇IHmi(r)+˙H(mi,v)(r)\n˙Hp(r) =−∇IHp(r)+˙H(p,pp)(r)+˙H(p,v)(r),\n˙Nmi(r) =−∇INmi(r)+˙N(mi,v)(r). (20)\nThe first terms in the right-hand sides of these equations\ncontrol the flows of appropriate quantities: the energy\nand number of magnons, phonons, meanwhile, the rest\nof the terms describe relaxation processes. INm(r) is the\ndensity of the magnon flow.\nSubstituting the equations of motion into the entropy\nproduction operator, we come to\nδ˙S(t)=∆/integraldisplay\ndr{−δβmi(r,t)∇IHmi(r)+\n+βµmi(r,t)∇INmi(r)−δβp(r,t)∇IHp(r)+\n+βmi(r,t)Vmi(r,t)∇IPmi(r)+βp(r,t)Vp(r,t)∇IPp(r)+\n+δβmimj(r,t)˙H(mi,mimj)(r)−βµmi(r,t)˙N(mi,v)(r)−\nβmi(r,t)Vmi(r,t)˙P(mi,v)(r)−βp(r,t)Vp(r,t)˙P(p,v)(r)},(21)\nwhereδβmimj=βmi−βmj,∆A=A−< A > 0.\nWe integrate by parts the terms containing the flow\ndivergences. Then, we ignore the surface integrals and\nwrite down the entropy operator as\n˙S(t)=∆/integraldisplay\ndr{−βINmi(r)∇µmi(r,t)+\n+δβmimj(r,t)˙H(mi,mimj)(r)+I∗\nmi(r)∇βmi(r,t)+\n+I∗\np(r)∇βp(r,t)−βµmi(r,t)˙N(mi,v)(r)−\n−βVmi(r,t)˙P(mi,v)(r)−βVp(r,t)[˙P(p,pp)+˙P(p,v)(r)]}.(22)\nHere\nI∗\nmi(r)=[IHmi(r)+IPmi(r)Vmi(r)],\nI∗\np(r)=[IHp(r)+Vp(r)IPp(r)]\nand we have taken into account that\n∇(βk(r,t)Vk(r,t))∼Vk(t)∇βk(r,t).\nBefore going over to the macroscopic momentum\nbalance equations, we should find a relation between\nthe chemical potential and effective temperature of themagnon subsystem. From the quasi-equilibrium distri-\nbutionρq(t) it follows that\nδ/an}b∇acketle{tNm1(r)/an}b∇acket∇i}ht=−/integraldisplay\ndr′{δβm1(r′,t)(Nm1(r),Hm1(r′))−\nβµm1(r′,t)(Nm1(r),Nm1(r′))−βm1Vm1(r′,t)(Nm1(r),Pm1(r′))},\n(23)\nwhere\nδ/an}b∇acketle{tA/an}b∇acket∇i}ht=/an}b∇acketle{tA/an}b∇acket∇i}ht−/an}b∇acketle{tA/an}b∇acket∇i}ht0,(A,B) =1/integraldisplay\n0dλSp{Aρλ\n0∆Bρ1−λ\n0}.\nIf one admits that Nm1(r) =constin a non-equilibrium\nbut steady-state case, (23) implies that\nµm1≃(βm1/β−1)R−(βm1/β)R1\nR=(Nm1,Hm1)\n(Nm1,Nm1)R1=Vm1(Nm1,Pm1)\n(Nm1,Nm1).(24)\nNote that as βm1→β, Vm1= 0, the chemical potential\nof magnons tends to zero: µm→0.\nMACROSCOPIC EQUATIONS\nInsertingthe entropyproductionoperator(22) intothe\nexpression for the NSO (19), we average the operator\nequations (16) for momenta of the subsystems under dis-\ncussion. Then we have\n/angbracketleftBig\n˙Pmi(r)/angbracketrightBigt\n=\n=−0/integraldisplay\n−∞dt′eǫt′/integraldisplay\ndr′{β(∇IPmi(r),INmj(r′,t′))∇µmj(r′,¯t)+\n+(∇IPmi(r),I∗\nmj(r′,t′))∇βmj(r′,¯t)+\n+(˙P(mi,v)(r),˙P(mj,v)(r′,t′))βVmj(r′,¯t)}, (25)\nhere¯t≡t+t′. The first summand in the right-hand side\nof (25) describes the diffusion and drift of magnons due\nto the gradients of the chemical potential and the tem-\nperature, the last summand the magnon-magnonscatter-\ning processes both inside each of the magnon subsystems\nand the ”coherent” magnon scattering by the ”thermal”\nmagnons.\nAnalogously, we characterize the relaxation processes\nin the phonon subsystem:\n/angbracketleftBig\n˙Pp(r)/angbracketrightBigt\n=\n=0/integraldisplay\n−∞dt′eǫt′/integraldisplay\ndr′{(∇IPp(r),I∗\np(r′,t′))∇βp(r′,¯t)+\n−(˙P(p,v)(r),˙P(mi,v)(r′,t′))βVi(r′,¯t)−\n(˙P(p,v)(r),˙P(p,v)(r′,t′))βVp(r′,¯t)}.\n(26)6\nGiven that the chemical potential and the effective tem-\nperature are related as in (24), we introduce the general\ndiffusion coefficient Dmimj(r,r′,t′) :\nβ(∇IPmi(r),INmj(r′,t′))∇µmj(r′,¯t)+\n+(∇IPmi(r),I∗\nmj(r′,t′))∇βmj(r′,¯t) =\n=Dmimj(r,r′,t′)∇µmj(r′,¯t) (27)\nwhere\nβDmimj(r,r′,t′)=(∇IPmi(r),INmj(r′,t′)) +\n+(∇IPmi(r),I∗\nmj(r′,t′))/(R−R1).(28)\nThe above equations represent the temperature gradi-\nent as a driving force. Therefore, the entropy operator\ninvolves the additional summands such as βi(r,t)Viin-\nstead of βVi(r,t).\nNow, revealing explicitly the correlation functions de-\nscribing the relaxation processes and appearing in the\nmomentum balance equations (25), (26), we have\n(˙P(mi,v)(r),˙P(mj,v)(r′,t′))=\n=(˙P(mi,mp)(r),˙P(mi,mp)(r′,t′))+\n+(˙P(mi,mimj)(r),˙P(mi,mimj)(r′,t′)),(29)\nThe first summand in the right-hand side of (29) de-\nscribes the magnon-phonon scattering, the second the\nmagnon-magnon scattering processes both inside each of\nthe magnon subsystems and the injected” magnon scat-\ntering by the ”thermal” magnons.\n(˙P(p,v)(r),˙P(p,v)(r′,t′)) =\n= (˙P(p,pp)(r),˙P(p,pp)(r′, t′))+\n+(˙P(p,pm)(r),˙P(p,pm)(r′,t′)).(30)\nThe first term describes the processes of non-magnon re-\nlaxation of phonons; the second one governsthe magnon-\nphonon scattering. Introducing the notation\nL(k,v)(r,r′,t′) = (˙P(k,v)(r),˙P(k,v)(r′,t′)),(31)\nwe re-write down the momentum balance equations in a\nconvenient form for further analysis\n/angbracketleftBig\n˙Pm1(r)/angbracketrightBigt\n=\n−0/integraldisplay\n−∞dt′eǫt′/integraldisplay\ndr′β{Dm1m1(r,r′,t′)∇µm1(r′,¯t) +\n+L(m1,m1p)(r,r′,t′)δVm1,p(r′,¯t)+\n+L(m1,m1m2)(r,r′,t′)δVm1,m2(r′,¯t)},(32)\n/angbracketleftBig\n˙Pm2(r)/angbracketrightBigt\n=\n−0/integraldisplay\n−∞dt′eǫt′/integraldisplay\ndr′β{Dm2m2(r,r′t′)∇µm2(r′,¯t) +\n+L(m2,m2p)(r,r′,t′)δVm2,p(r′,¯t)+\n+L(m2,m1m2)(r,r′,t′)δVm2,m1(r′,¯t)},(33)/angbracketleftBig\n˙Pp(r)/angbracketrightBigt\n=\n−0/integraldisplay\n−∞dt′eǫt′/integraldisplay\ndr′β{−Dpp(r,r′,t′)∇βp(r′,¯t)+\n+L(p,m1p)(r,r′,t′)δVp,m1(r′,¯t)+\n+L(p,m2p)(r,r′,t′)δVp,m2(r′,¯t)+\n+L(p,pp)(r,r′,t′)Vp(r′,¯t)}.(34)\nHereδVik=Vi−Vk, Dpp(r,r′,t′) =β(∇IPp(r),I∗\np(r′,t′)).\nEquations (32) - (34) allow conducting the analysis of\nhow the interaction between the subsystems at hand af-\nfects the implementation ofthe drageffect. We introduce\nthe average values of the forces induced by the chemical\npotential and temperature gradients:\nFmi(r)=0/integraldisplay\n−∞dt′eǫt′/integraldisplay\ndr′Dmimj(r,r′,t′)∇µmj(r′,¯t)\nFp(r)=0/integraldisplay\n−∞dt′eǫt′/integraldisplay\ndr′Dpp(r,r′,t′)∇βp(r′,¯t).\nBesides, introduce the inverse times of the magnon\nand phonon momentum relaxation caused by interac-\ntion with phonons processes of non-magnon relaxation\nof phonons. Let us designate them as ω(mp), andω(pp),\nrespectively40,41\nω(γ,v)= (Pγ,Pγ)−10/integraldisplay\n−∞dt′eǫt′(˙P(γ,v),˙P(γ,v)(t′)),\nγ=m1,m2,p(35)\nWe restrict ourselves to the discussion of a stationary\ncase. For this purpose, we average the balance equa-\ntions over time t. To start the analysis, we consider the\nsimplest case when the drift velocities of the magnon sys-\ntems are equal: Vm1=Vm2≡Vmandβm1=βm2. This\nactually means that we deal with one magnon and one\nphonon systems. In addition, we shall assume that the\nphonon momentum is maintained from the outside by\nan unchanged. Then, the momentum balance equations\nappear as\nFm=Pmω(m,mp)(Vm−Vp), (36)\n0 =Ppω(p,mp)(Vp−Vm)+Ppω(p,pp)Vp.(37)\nThe balance equation for the magnon momentum ac-\nquires the form\nFm=Vmω(p,pp)ω(m,mp)\nω(m,mp)+ω(p,pp)Pm. (38)7\nwherePm≡(Pm,Pm),Pp≡(Pp,Pp). Finally, the\ndraggingleads to the change in frequency of the magnon-\nphonon collisions, and the quantity\nΩ =ω(p,pp)ω(m,mp)\nω(m,mp)+ω(p,pp)\nis the inverse relaxation time of the magnon momentum\nby non-equilibrium phonons.\nFrom the expression (38) it follows that the drag ef-\nfect has an influence on the magnon-phonon collision fre-\nquency. The phonon subsystem almost always remains\nin equilibrium, and the inverse relaxation time is defined\nby the frequency ω(m,mp)provided that the inequality\nω(p,pp)> ω(m,pm)is fulfilled. The latter means that the\nphonon momentum gained quickly relaxes in the pro-\ncesses of non-magnon relaxation. If the opposite inequal-\nityω(p,pp)< ω(m,pm)holds, the leakage of the phonon\nmomentum occurs slower than the gain momentum rate\nin the magnon-phonon collisions. In this case, the mech-\nanism of the non-magnon phonon relaxation mainly con-\ntributes to the drag effect. In addition,\n¯ω(m,mp)≃ω(p,pp)(ω(m,mp)/ω(m,pm))=\n=ω(p,pp)(Pp/Pm)=Pm0/integraldisplay\n−∞dteǫt(˙P(p,pp),˙P(p,pp)).(39)\nThus, the criterion of realizing the drag effect consists in\nthe requirement ω(m,pm)>ω(p,pp)that coincides with the\nsolution of the kinetic equation41. It is worth emphasiz-\ning that to calculate the correlation function in the for-\nmulafor ω(p,pp), it isnecessaryto knowparticularmecha-\nnisms of the non-magnon phonon momentum relaxation.\nForconsideringthedrageffectstherearetwomechanisms\nsuch as the Herring mechanism ω(p,pp)∼(kBT)3and the\nSimons mechanism ω(p,pp)∼(kBT)4leading to a rather\nstrongtemperaturedependenceoftherelaxationfrequen-\ncies.\nAnother limiting case corresponds to the situation\nwhen the drift velocities of thermal magnons and\nphonons are equal to Vm2=Vp(βm2=βp). In this\ncase, thermal magnonsandphonons formone subsystem.\nFrom balance equations (32), (33) we obtain\nFm1=Vm1ω(p,pp)[ω(p,mp)+ω(m,m1m2)]\nω(p,mp)+ω(p,pp).(40)\nFrom the expression (41) it follows that if ω(p,pp)≫\nω(p,mp)thenF1∼ω(p,mp)andF1∼ω(m,m1m2)if\nω(p,mp)≪ω(m,m1m2).If the opposite inequality, when\nω(p,pp)≪ω(p,mp)thenF1∼ω(p,pp)[1+ω(m,m1m2)/ω(p,mp)]\nandF1∼ω(p,pp)ifω(m,m1m2)≪ω(p,mp)andF1∼\nω(p,pp)ω(m,m1m2)/ω(p,mp)whenω(m,m1m2)≫ω(p,mp).\nNow we look into the drag effect in the event of two\nmagnon and one phonon systems. Then, the momentum\nbalance equations can be written as follows. The set of\nthe equations (36), (37) implies\nFm1={ω(m1,mp)+ω(m,m1m2)−ω(m1,mp)ω(m1,mp)\nΩ}Vm1−−{ω(m1,mp)ω(m2,mp)\nΩ+ω(m,m1m2)}·\n·{Fm2+(��(m,m1m2)+ω(m1,mp)ω(m2,mp)/Ω)Vm1\nω(m2,mp)+ω(m,m1m2)−ω(m2,mp)ω(m2,mp)/Ω},(41)\nwhere Ω = ω(m1,mp)+ω(m2,mp)+ω(p,pp).\nLet the energy transfer channels from the magnonsub-\nsystems to the phonon subsystem be equal ω(m1,mp)=\nω(m2,mp)=ω(m,mp), Vm1=Vm. In this case we have\nFm1={ω(m,mp)+ω(m,m1m2)−ω(m,mp)/Ω}Vm−\n−{ω(m,mp)/Ω+ω(m,m1m2)}×\n×{Fm2+(ω(m,m1m2)+ω(m,mp)/Ω)Vm\nω(m,mp)+ω(m,m1m2)−ω(m,mp)/Ω}.(42)\nHere Ω = 2+ ω(p,pp)/ω(m,mp). Ifω(p,pp)≫ω(m,mp), then\nFm1={ω(m,mp)+ω(m,m1m2)}Vm−\n−ω(m,m1m2)·{Fm2+ω(m,m1m2)\nω(m,mp)+ω(m,m1m2)}.(43)\nThe expression (43) claims that the spin-wave current\n∼F1is determined by the relations between the correla-\ntion functions ω(m,m1m2)andω(m,mp). As it follows from\nthe expression (43) that if ω(m,m1m2)≪ω(m,mp)then\nF1∼ω(m,mp). In this case magnon-phonon interaction\nis the dominant channel of a magnon relaxation. If we\nhave the opposite inequality ω(m,m1m2)≫ω(m,mp)then\nF1∼ω(m,m1m2). In this case the interaction between ”in-\njected” and ”thermal” magnons is the dominant channel\nof a magnon relaxation. Moreover, the inelastic scatter-\ning of the ”injected” magnons by ”thermal” ones can\nbe regarded as scattering by impurity centers whose con-\ncentration is temperature-varied. This interaction will\ndetermine the temperature-field behaviour of the spin-\nwave current under the conditions of the Seebeck spin\neffect.\nBecause of the existence of two relaxation channels\n(magnon-phonon and magnon-magnon), the inelastic\nscattering of the ”injected” magnons by ”thermal” ones\nmay give rise to the bottleneck effect and heating of the\n”thermal” magnons. Such a situation emerges if the\n”thermal”-magnon subsystem gains energy through the\nmagnon-magnon channel faster than loses it along the\nmagnon-phonon channel, i.e. ω(m,m1m2)≫ω(m,mp).\nCONCLUSION\nThe formation of the two: ”injected” (coherent) and\nthermally excited, different in energies magnon subsys-\ntems and the influence of its interaction with phonons\nand between on drag effect under spin Seebeck ef-\nfect conditions in the magnetic insulator part of the\nmetal/ferromagnetic insulator/metal structure is stud-\nied. The analysis of the macroscopic momentum bal-\nance equations of the systems of interest conducted for\ndifferent ratios of the drift velocities of the magnon and8\nphonon currents show that the injected magnons relax-\nation on the thermal ones is possible to be dominant\nover its relaxation on the phonons. This interaction\nwill be the defining in the forming of the temperature\ndependence of the spin-wave current under SSE condi-\ntions, and inelastic part of the magnon-magnon inter-\naction is the dominant spin relaxation mechanism. The\nexistenceofthe tworelaxationchannels(magnon-phonon\nand magnon-magnon) in the case of inelastic scattering\nof the injected magnons on the thermal ones is shown tobe leading to the warming of the letter and to Narrow-\nneck effect. 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Karenowska1\n1Clarendon Laboratory, Department of Physics, University of Oxford, Oxford OX1 3PU, United Kingdom\n2Center for Emergent Matter Science, RIKEN, Wako, Saitama 351-0198, Japan\n3Fachbereich Physik and Landesforschungszentrum OPTIMAS,\nTechnische Universitaet Kaiserslautern, 67663 Kaiserslautern, Germany\n4School of Engineering, University of Glasgow, Glasgow G12 8LT, United Kingdom\n5Department of Physics and Energy Science, University of Colorado at Colorado Springs, Colorado Springs CO 80918, USA\n(Dated: October 28, 2019)\nMagnon systems used in quantum devices require low damping if coherence is to be maintained.\nThe ferrimagnetic electrical insulator yttrium iron garnet (YIG) has low magnon damping at room\ntemperature and is a strong candidate to host microwave magnon excitations in future quantum\ndevices. Monocrystalline YIG \flms are typically grown on gadolinium gallium garnet (GGG) sub-\nstrates. In this work, comparative experiments made on YIG waveguides with and without GGG\nsubstrates indicate that the material plays a signi\fcant role in increasing the damping at low tem-\nperatures. Measurements reveal that damping due to temperature-peak processes is dominant above\n1 K. Damping behaviour that we show can be attributed to coupling to two-level \ructuators (TLFs)\nis observed below 1 K. Upon saturating the TLFs in the substrate-free YIG at 20 mK, linewidths of\n\u00181:4 MHz are achievable: lower than those measured at room temperature.\nMicrowave magnonic systems have been subject to ex-\ntensive experimental studies for decades. This work is\nmotivated not only by an interest in their rich basic\nphysics, but also by their potential application as infor-\nmation carriers in beyond-CMOS electronics [1, 2]. Re-\ncently, enthusiasm has grown for the study of magnon\ndynamics at millikelvin (mK) temperatures, the tempera-\nture regime in which solid-state microwave quantum sys-\ntems operate [3{12]. This work o\u000bers the possibility to\nexplore the dynamics of microwave magnons in the quan-\ntum regime and to study novel quantum devices with\nmagnonic components [13{15].\nArguably the most important material in the context of\nroom-temperature experimental magnon dynamics is the\nferrimagnetic insulator yttrium iron garnet (Y 3Fe5O12,\nYIG). Pure monocrystalline YIG has the lowest magnon\ndamping of any known material at room temperature\n[16] and is produced in the form of bulk crystals and\n\flms. Films suitable for use as waveguides in conjunction\nwith micron-scale antennas are grown by liquid-phase\nepitaxy to a thickness of between 1 and 10 \u0016m on gadolin-\nium gallium garnet (Gd 3Ga5O12, GGG) substrates. The\nuse of GGG is motivated by the need for tight lattice\nmatching to assure a high crystal quality. Recently, YIG\n\flms were recognised as promising media for the study of\nmagnon Bose-Einstein condensation and related macro-\nscopic quantum transport phenomena [17{20]. In the\ncontext of quantum measurements and information pro-\ncessing, YIG \flms hold noteworthy promise, however, if\nthey are to be practical, they must be shown to exhibit\nthe same (or better) dissipative properties at mK tem-\nperatures as they do at room temperature.\nMagnon linewidths in YIG at mK temperatures have\nthus far only been characterised in bulk YIG resonators\n\u0003sandoko.kosen@physics.ox.ac.uk\nB\n(Top View)(Cross Section)B\nSample Spacer Signal line0\n-1\nSignal (dB)\nFrequencyf0=4GHzAbsorption Spectrum\nd\nw∆f A\n(b)\nInputOutput(a)\n-20dB-20dB\n(20mK - 9K)100mK4K70K\n-20dB BPF50Ω\nSampleBPF\nLODAQ\nLPF\n50ΩRF\nLOQ IRF\nroom\ntemperature50ΩFIG. 1. (a) The measurement con\fguration used to charac-\nterise the sample's damping. The sample and the microstrip\n(signal line) are separated from each other by a spacer. When\nthe microwave drive is resonant with the magnons in the sam-\nple, a decrease in the transmitted signal is observed. (b) The\nlow-temperature setup and its corresponding data acquisition\nsystem at room temperature.\n(speci\fcally, spheres) [4, 6, 7, 10, 21]. Bulk YIG has\nbeen shown to retain its low magnon damping at mK\ntemperatures. However, in the case of YIG \flms grown\non GGG, the story is more complex. GGG is a geomet-\nrically frustrated magnetic system [22] and it has long\nbeen known that at temperatures below 70 K, it exhibits\nparamagnetic behaviour that has been reported to in-\ncrease damping in \flms grown on its surface [23{25].\nThe behaviour of GGG at mK temperatures is yet to\nbe thoroughly characterised [26{28], but recent results\nat mK temperatures have suggested that magnon damp-\ning in YIG \flms grown on GGG is higher than expected\nif the properties of the YIG system alone are consideredarXiv:1903.02527v3  [cond-mat.mes-hall]  25 Oct 20192\n4\n5\n6\n7\nΔf (MHz) Δf (MHz)\nfo (GHz)T = 300K\nT = 20mK, Pb = -65dBm\nT = 20mK, Pb = -100dBmSubstrate-free YIG\nYIG/GGG\nT = 300K\nT = 20mK, Pb = -65dBm\nT = 20mK, Pb = -100dBm1234\n371115\nFIG. 2. Magnon linewidths (\u0001 f) versus resonance frequency\n(f\u000e) for a YIG/GGG \flm and a substrate-free YIG \flm.\nDatasets at room temperature (300 K, \u000f) are obtained with\nan input power of -25 dBm. Datasets at 20 mK are obtained\nfor two input powers Pb=\u000065 dBm (N), and Pc=\u0000100 dBm\n(\u0004). Each error bar represents the standard deviation of\nthe linewidth values obtained from repeated measurements.\nDashed lines are linear \fts, the details of these \fts are sum-\nmarised in table I. Note the di\u000berent scaling of the vertical\naxes of the plots.\n[9, 11, 29]. In this work we report a comparative set of ex-\nperiments on YIG \flms with and without GGG and move\ntoward a more complete understanding of the damping\nmechanisms involved.\nWe present data from the measurement of two YIG\nsamples: a 11 \u0016m-thick \flm and a substrate-free 30 \u0016m-\nthick \flm. Both samples are grown using liquid phase\nepitaxy with the surface normal of the YIG \flm (and the\nsubstrate) parallel to the h111icrystallographic direction.\nThe substrate-free YIG is obtained by mechanically pol-\nishing o\u000b the GGG until a 30 \u0016m-thick pure YIG \flm\nis obtained [25]. The corresponding lateral size of each\nsample can be found in table I.\nWe measure the damping in both \flms using the\nmicrostrip-based technique illustrated in \fg. 1(a) [30].\nThe sample is positioned above a microstrip and magne-\ntised by an out-of-plane magnetic \feld ( B). Continuous-\nwave microwave signals transmitted through the mi-\ncrostrip probe the ferromagnetic resonance of the sam-\nple. In the room-temperature experiments, the trans-\nmitted signals are measured by connecting the two ends\nof the microstrip directly to a commercial network anal-\nyser. In our low-temperature experiments, the sample is\nmounted on the mixing-chamber plate of a dilution re-\nfrigerator as shown in \fg. 1(b), similar to that used in\nRef. [11]. A microwave source is used to generate the\ninput microwave signal. At the input line, three 20 dB\nattenuators are used to ensure an electrical noise tem-\nperature that is comparable to the temperature of theTABLE I. Comparing results at 300 K and at 20 mK\nYIG/GGG Substrate-free YIG\nSize 2 mm\u00023 mm\u000210\u0016m \u00181 mm\u00021 mm\u000230\u0016m\nw=d 1.7 mm / 70 \u0016m 0.9 mm / 540 \u0016m\n300 K \u000b1a= (22\u00064)\u000210\u00005\u000b2a= (8:9\u00060:5)\u000210\u00005\n\u0001f\u000e;1a= (0:7\u00060:4) MHz \u0001f\u000e;2a= (0:9\u00060:1) MHz\n20 mK \u000b1b= (74\u00065)\u000210\u00005\u000b2b= (2:3\u00060:7)\u000210\u00005\nPb=-65 dBm \u0001f\u000e;1b= (1:7\u00060:6) MHz \u0001f\u000e;2b= (1:1\u00060:1) MHz\n20 mK \u000b1c= (85\u00066)\u000210\u00005\u000b2c= (9:3\u00061:0)\u000210\u00005\nPc=-100 dBm \u0001f\u000e;1c= (2:6\u00060:6) MHz \u0001f\u000e;2c= (2:0\u00060:1) MHz\nsample. The output signals then pass through two cir-\nculators, a bandpass \flter, and an ampli\fer, before they\nare down-converted to a 500 MHz signal at room temper-\nature. A DAQ card then digitises the transmitted signal\nat a 2.5 GHz sampling frequency. Signals are usually av-\neraged about 50000 times before being digitally down-\nconverted in order to obtain a signal similar to the one\nshown in \fg. 1(a). The magnon linewidth is given by the\nfull-width at half maximum (FWHM) of the Lorentzian\n\ft to the transmitted signal. The measurement frequency\nrange is between 3.5 GHz and 7 GHz. The low-frequency\nlimit is imposed by the limited bandwidth of the circu-\nlators used for our low-temperature measurement setup\nand the top of the measurement band is determined by\nthe maximum external \feld that can be applied by our\nmagnet.\nThe measured damping comprises contributions from\nthe sample and from radiation damping caused by its\ninteraction with the microstrip. In our experiments, ra-\ndiation damping originates from eddy currents excited\nin the microstrip by the magnetic \feld of the magnons\n[31, 32] and can be decreased by increasing the separation\nbetween the sample and the microstrip ( d) at the expense\nof reducing the measured absorption signal strength ( A).\nThere is therefore a tradeo\u000b to be made between be-\ning able to measure linewidths very close to the intrinsic\nlinewidth of the sample (thick spacer, negligible radiation\ndamping) and being able to achieve su\u000ecient signal-to-\nnoise (SNR) ratio (thin spacer, non-negligible radiation\ndamping). Table I lists the microstrip-sample spacings\n(d) in our experiments. Since earlier experiments sug-\ngested that the YIG/GGG linewidth would be higher at\nlow temperature, the YIG/GGG sample is closer to the\nmicrostrip to maintain su\u000ecient SNR.\nWithin the YIG \flm itself, the primary contributions\nto magnon damping are: intrinsic processes [33, 34],\ntemperature-peak processes [35, 36], two-level \ructuator\n(TLF) processes [4] and two-magnon processes [35, 36].\nIntrinsic processes are due to interactions with optical\nphonons and magnons; they are expected to decrease\nwith reducing temperature. Temperature-peak processes\noriginating from interactions with rare-earth impurities\nare only signi\fcant at low temperature (above 1 K). TLF\nprocesses are due to damping sources that behave as two-\nlevel systems; they are dominant below 1 K. Two-magnon\nprocesses have their origins in inhomogeneities in the ma-3\n0.02 0.1 1 1003691210152025Δf (MHz) Δf (MHz)\nT (K)YIG/GGG\nSubstrate-free YIGfo = 4GHz\nfo = 7GHz\nfo = 4GHz\nfo= 7GHz\nFIG. 3. Temperature ( T) dependent magnon linewidths (\u0001 f)\nfor both YIG/GGG \flm and substrate-free YIG, measured\nwith input power Pc=\u0000100 dBm. Note the di\u000berent scaling\nof the vertical axes of the plots.\nterial; in our experiments, they are minimized by mag-\nnetising the sample out of plane [37, 38].\nFigure 2 compares the magnon linewidth (\u0001 f) of each\nsample at 300 K (room temperature) and at 20 mK as a\nfunction of the ferromagnetic resonance frequency ( f\u000e).\nResults at 300 K are obtained by sweeping the input\nmicrowave frequency under constant B-\feld. Results\nat 20 mK are obtained by sweeping the B-\feld at con-\nstant input microwave frequency. In the latter case, the\nlinewidths are measured in terms of magnetic \feld (\u0001 B)\nand converted to units of frequency (\u0001 f) via the rela-\ntion \u0001f= (\r=2\u0019)\u0001B, where\ris the gyromagnetic ra-\ntio. Note that there is no conversion factor other than\n\r=2\u0019that is used to translate the low-temperature \feld-\ndomain data into the frequency domain. A linear \ft to\n\u0001f= 2\u000bf\u000e+ \u0001f\u000egives the characteristic Gilbert damp-\ning constant \u000b(unitless) and the inhomogeneous broad-\nening contribution \u0001 f\u000e. Table I summarises the results\nof linear \fts to data in \fg. 2.\nWe \frst compare the results at 300 K and 20 mK\nobtained at relatively high input drive level ( Pb=\n\u000065 dBm). The substrate-free YIG shows a measured\nlinewidth decreasing from the room temperature value\nto approximately 1 :4 MHz at 20 mK. The reduction in\ndamping is as anticipated by existing models that de-\nscribe the intrinsic damping of YIG [33{35]. The ra-\ndiation damping contribution to the linewidth for the\nsubstrate-free YIG is small due to the large spacing from\nthe microstrip ( d= 540 \u0016m).\nThe YIG/GGG sample is substantially closer ( d=\n70\u0016m) to the microstrip and its measured \u000btherefore\nincludes a non-negligible radiation damping contribution\n\u000br. In our raw data, uncorrected for this e\u000bect, we mea-\nsure a damping constant at 20 mK ( \u000b1b) that is 3:4 timeslarger than the room temperature value ( \u000b1a). Following\nRef. [31], the radiation damping can be modelled with an\nequivalent Gilbert damping constant \u000br=CgMs, where\nCgdepends on the geometry of the system and Msis\nthe saturation magnetisation of the sample. As both\n300 K and 20 mK measurements are performed with iden-\ntical sample geometry, it is reasonable to expect that\nthe change in \u000bras the temperature is lowered is due to\nthe change in Ms. Therefore, the increase in \u000brbetween\n20 mK and 300 K is determined by the ratio of the sat-\nuration magnetisation, i.e. Ms(20 mK)=Ms(300 K)\u00191:4\n[39]. The fact that we see a signi\fcantly larger damp-\ning increase ( \u000b1b=\u000b1a\u00193:4) in the YIG/GGG and a\ndecrease (\u000b2b=\u000b2a\u00190:26) in the substrate-free YIG in-\ndicates that the GGG plays an important role in increas-\ning the magnon linewidth of the YIG/GGG sample at\n20 mK.\nThe parameters \u000band \u0001f\u000ein both samples increase\nas the input drive level ( Pc) reduces as shown in Table I.\nThis behaviour can be explained by the TLF model upon\nwhich we shall elaborate later.\nFigure 3 shows the temperature dependence of the\nmagnon linewidths for both samples measured at low in-\nput power ( Pc=\u0000100 dBm). For the YIG/GGG results\nin \fg. 3, the radiation damping contribution ( \u000br=CgMs\n[31]) across the examined temperature range can be con-\nsidered to be an approximately constant vertical shift to\neach dataset. This is due to the small change (less than\n0.07%) inMsof YIG between 20 mK and 9 K [39].\nAbove 1 K, linewidths of both samples increase as the\ntemperature is increased up to 9 K. In this temperature\nrange, damping is dominated by temperature-peak pro-\ncesses caused by the presence of rare-earth impurities in\nthe YIG [25, 35, 39{41]. When temperature-peak pro-\ncesses are dominant, the linewidth of the sample peaks\nat a characteristic temperature ( Tch) determined by the\ndamping mechanism and the type of impurity.\nTemperature-peak processes at low temperatures fall\ninto two categories [35, 36]: those associated with (1)\nrapidly relaxing impurities, and (2) slowly relaxing impu-\nrities. Rapidly relaxing impurities produce a Gilbert-like\ndamping and a characteristic temperature Tchthat is in-\ndependent of the magnon resonance frequency f\u000e. Slowly\nrelaxing impurities produce a non-Gilbert-like damping\nand a corresponding characteristic temperature that de-\ncreases as the resonance frequency f\u000eis lowered. The\nbehaviour observed in \fg. 3 at 9 K, with the linewidth for\nthef\u000e= 4 GHz being higher than that at f\u000e= 7 GHz, in-\ndicates that impurities of slowly relaxing type dominate\nin this temperature range.\nAs the temperature is decreased below 1 K, linewidths\nfor the substrate-free YIG start to increase and eventu-\nally saturate as shown in \fg. 3. This can be explained\nby the presence of two-level \ructuators (TLFs) and has\nbeen previously observed in a bulk YIG [4]. In the TLF\nmodel, the damping sources are modelled as an ensem-\nble of two-level systems with a broad frequency spectrum4\n-120 -100 -80 -601234\n-120 -100 -80 -600.02 0.1 0.3 1.0 3.012345Δf (MHz) Δf (MHz)T (K)\nP (dBm) P (dBm)Substrate-free YIG\n20mK\n300mK\n1K20mK\n300mK\n1KPsat(20mK)\nPsat(300mK)(a)\n(b)low\ndrive\nhigh\ndrivehigh drive level: Pb = -65dBm\nlow drive level: Pc = -100dBm\nfo = 4GHz fo = 7GHz\nPsat(20mK)4GHz\n4GHz7GHz\n7GHz\nSubstrate-free YIG Substrate-free YIG\nFIG. 4. Magnon linewidths (\u0001 f) in the substrate-free YIG\n\flm for various temperatures ( T) and input powers ( P). (a)\nTemperature dependent linewidths for two di\u000berent input\npowers Pb=\u000065 dBm and Pc=\u0000100 dBm. (b) Power de-\npendent linewidths at 20 mK, 300 mK, and 1 K. The dashed\nlines are the \fts to the data.\n[42, 43]. The linewidth contribution can be expressed as\n\u0001fTLF=CTLF!tanh ( ~!=2kBT)p\n1 + (P=P sat)(1)\nwhereCTLF is a factor that depends on the TLF and\nthe host material properties. The power-dependent term\ncan be rewritten as P=P sat= \n2\nr\u001c1\u001c2, where \n ris the\nTLF Rabi frequency, \u001c1and\u001c2are respectively the TLF\nlongitudinal and transverse relaxation times [44].\nAt high temperatures ( kBT\u001dhfTLF), thermal\nphonons saturate the TLFs and therefore the material\nbehaves as if the TLFs were not present. At low temper-\natures (kBT\u001chfTLF) and low drive levels ( P\u001cPsat),\nmost of the TLFs are unexcited. Under these condi-\ntions, the TLFs increase the damping of the material\nby absorbing and re-emitting magnons or microwaves at\nrates set by their lifetimes, coupling strength, and den-\nsity. When the drive level is increased past a certain\nthreshold (P\u001dPsat), the TLFs are once again saturated\nand therefore do not contribute to the damping.\nEvidence for the presence of the TLFs is shown in\n\fgs. 2 and 4. The datasets for 20 mK in \fg. 2 show that\nthe linewidths for both samples are lower when the drive\nlevel is higher ( PbvsPc). Figure 4(a) shows a similar be-\nhaviour in the substrate-free YIG. Above 1 K, linewidths\nfor both drive levels are similar: an indication that the\nrelevant TLFs are saturated by thermal phonons. Thedi\u000berences in linewidths for the two drive levels begin to\nappear as the temperature is lowered below 1 K.\nFigure 4(b) shows the linewidths of the substrate-free\nYIG as a function of drive level ( P) at three di\u000berent tem-\nperatures (1 K, 300 mK, and 20 mK). At 1 K, there is no\nobservable power dependence as the relevant TLFs have\nbeen saturated by the thermal phonons. At 20 mK and\n300 mK, the linewidth increases as the power decreases,\nsaturating at mK temperatures. This is in agreement\nwith the theory previously articulated and the \fts shown\nby dashed lines in \fg. 4(b). The data are \ftted using\neq. (1) with an additional y-intercept to account for non-\nTLF linewidth contributions.\nFor thef\u000e= 7 GHz dataset in \fg. 4(b), Psatat 300 mK\nis clearly larger than at 20 mK. This is in-line with ex-\npectations: \u001c1and\u001c2are anticipated to decrease as the\ntemperature is increased, leading to a higher Psat(recall\nthatPsat/1=\u001c1\u001c2) [44{46]. The exact temperature de-\npendence of 1 =\u001c1\u001c2is not clear; in previous experiments,\na phenomenological model was suggested with the quan-\ntity 1=\u001c1\u001c2varying from T2toT4[45]. This places the\nratioPsat(300 mK)=Psat(20 mK) in the range of 23.5 dB\nto 47 dB. The \ftted Psatvalues from our data correspond\nto a ratio of approximately 22.5 dB, suggestive of a T2\nbehaviour.\nIt should be noted that the f\u000e= 4 GHz,T= 300 mK\ndataset shows a very weak TLF e\u000bect since there are su\u000e-\ncient thermal phonons to saturate the TLFs with central\nfrequencies around 4 GHz; this is not the case for higher\nfrequency datasets taken at the same temperature. A\nhigherPsatis also observed at 300 mK for f\u000e= 5 GHz\nandf\u000e= 6 GHz (data not shown).\nFigure 4(a) shows that the input power Pb=\u000065 dBm\nused in our experiments is not enough to saturate the\nrelevant TLFs for temperatures between 100 mK and\n1 K. The datasets obtained with high drive level ( Pb)\nin \fg. 4(a) show that the linewidth di\u000berence \u000ef=\nj\u0001f(f\u000e= 7 GHz)\u0000\u0001f(f\u000e= 4 GHz)jbroadens as the\ntemperature is increased from 100 mK to 300 mK, nar-\nrowing back as the temperature reaches 1 K. If a higher\ndrive level is used, \u000efis expected to be smaller at tem-\nperatures between 100 mK and 1 K.\nIn conclusion, the substrate GGG on which typical\nYIG \flms are grown signi\fcantly increases the magnon\nlinewidth at mK temperatures. However, if the substrate\nis removed, it is possible to obtain YIG linewidths at mK\ntemperatures that are lower than the room-temperature\nvalues. Measured linewidths of both YIG/GGG and\nsubstrate-free YIG systems above 1 K are consistent with\nthe temperature-peak processes typically observed in\nYIG containing rare earth impurities. Damping due to\nthe presence of unsaturated two-level \ructuators is ob-\nserved in both YIG/GGG and substrate-free YIG \flms\nbelow 1 K. We observe the TLF saturation power to be\nhigher at higher temperatures in agreement with the ex-\nisting literature. We further verify that using high drive\nlevel reduces the linewidths of the substrate-free YIG\n\flms down to\u00181:4 MHz (f\u000e= 3:5 GHz to 7.0 GHz)5\nat 20 mK.\nLooking forward, our measurements suggest that|in\nthe context of the development of magnonic quantum\ninformation or measurement systems|it may be worth-\nwhile to investigate the possibility of growing YIG \flms\non substrates other than GGG, or techniques which cir-\ncumvent the use of a substrate entirely [47{50]. It should\nbe emphasised that the current experimental con\fgura-\ntion does not allow us to pinpoint the origin of the TLFs;\nfurther investigations into TLFs in YIG would be useful\nin obtaining high-quality YIG magnonic devices that op-\nerate in the quantum regime.\nNote added - A pre-print by P\frrmann et al. [21]\nrecently reported experiments concerning the e\u000bect of\ntwo-level \ructuators on the linewidth of bulk YIG. 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Bailey, S. Finizio, E. Josten,\nD. Meertens, C. Dubs, D. A. Bozhko, H. Stoll, G. Di-\neterle, N. Tr ager, J. Raabe, A. N. Slavin, M. Weigand,\nJ. Gr afe, and G. Sch utz, , 1 (2019), arXiv:1903.00498."    },    {        "title": "1512.04146v1.Nonlocal_Anomalous_Hall_Effect.pdf",        "content": "Nonlocal Anomalous Hall E\u000bect\nSteven S.-L. Zhang and Giovanni Vignale\nDepartment of Physics and Astronomy, University of Missouri, Columbia, MO 65211\n(Dated: May 27, 2022)\nThe anomalous Hall e\u000bect is deemed to be a unique transport property of ferromagnetic metals,\ncaused by the concerted action of spin polarization and spin-orbit coupling. Nevertheless, recent\nexperiments have shown that the e\u000bect also occurs in a nonmagnetic metal (Pt) in contact with a\nmagnetic insulator (yttrium iron garnet (YIG)), even when precautions are taken to ensure there is\nno induced magnetization in the metal. We propose a theory of this e\u000bect based on the combined\naction of spin-dependent scattering from the magnetic interface and the spin Hall e\u000bect in the bulk\nof the metal. At variance with previous theories, we predict the e\u000bect to be of \frst order in the\nspin-orbit coupling, just as the conventional anomalous Hall e\u000bect { the only di\u000berence being the\nspatial separation of the spin orbit interaction and the magnetization. For this reason we name this\ne\u000bect nonlocal anomalous Hall e\u000bect and predict that its sign will be determined by the sign of the\nspin Hall angle in the metal. The AH conductivity that we calculate from our theory is in good\nagreement with the measured values in Pt/YIG structures.\nIntroduction.\u0000The anomalous Hall (AH) e\u000bect is the\ngeneration of an electric current perpendicular to the\nelectric \feld in a ferromagnetic metal [1]. At variance\nwith the ordinary Hall e\u000bect, which arises from the ac-\ntion of a magnetic \feld on the orbital\nmotion of the electrons, the AH e\u000bect is ascribed to\nstrong spin-orbit coupling in concert with spin-polarized\nitinerant electrons. The spin orbit coupling plays a cen-\ntral role in inducing a left-right asymmetry (with respect\nto the direction of the electric \feld) in the scattering of\nelectrons of opposite spins. It is this asymmetry that\ngenerates a transverse charge current from a longitudi-\nnal spin current. The same scattering process generates a\npure transverse spin current for systems with spin unpo-\nlarized electrons, which is known as spin Hall e\u000bect [2{5].\nBased on this picture, the conventional AH e\u000bect appears\nat\frst order in spin orbit coupling, no matter which kind\nof microscopic mechanisms predominates.\nRecently, an AH signal has also been detected in a\nPlatinum (Pt) layer in direct contact with a YIG layer [6{\n8]. The former is a non-magnetic heavy metal with strong\nspin orbit coupling and the latter is a well-known ferro-\nmagnetic insulator. In view of the two aforementioned\ningredients for the AH e\u000bect in ferromagnets, it is puz-\nzling that an AH current would arise in Pt in the absence\nof spin polarized conduction electrons. In a \frst attempt\nto solve the puzzle, Huang et. al. [6] showed that the Pt\nlayer in close proximity with YIG acquires ferromagnetic\ncharacteristics, which essentially subsumes the novel AH\ne\u000bect under the conventional AH e\u000bect for ferromagnetic\nmetals. This explanation ran into di\u000eculties when it\nwas found that the AH e\u000bect persists in Pt/Cu/YIG tri-\nlayers [9] where the Cu layer is deliberately inserted to\neliminate the magnetic proximity e\u000bect.\nAn alternative explanation was then proposed [9, 10],based on the physical mechanism depicted in panel (a)\nof Fig. 1. In this mechanism the applied charge cur-\nrentjxgenerates, via the spin Hall e\u000bect, a spin current\nQy\nzpropagating in the z\u0000direction with spin along the\ny\u0000direction. When those electrons carrying Qy\nzare re-\n\rected back from the magnetic interface, spin rotation\noccurs and gives rise to a spin current of Qx\nz, which in\nturn induces a transverse charge current jyvia the in-\nverse spin Hall e\u000bect [11, 12]. Based on this picture, the\ntransverse electric current is of second order in the spin\norbit coupling or spin Hall angle, which is qualitatively\ndi\u000berent from a conventional AH current. It is worth\nmentioning that a \ft to the experimental data based on\nthis model [7, 10], requires a spin di\u000busion length on the\norder of 1 nm. Such a short spin di\u000busion length, an\norder of magnitude smaller than the room-temperature\nelectron mean path of Pt [13], casts doubt on the internal\nconsistency of the spin di\u000busion model.\nIn this paper, we propose a di\u000berent mecha-\nnism for the AH current observed in hybrid heavy-\nmetal/ferromagnetic-insulator structures. The essential\nnew ingredient is the scattering of electrons from the\n(rough) metal-insulator interface. Because the insulator\nis magnetic, the scattering rate is spin-dependent (see\nAppendix B for a proof). This means that a charge cur-\nrent \rowing parallel to the interface is partially converted\nto a spin current, while a spin current \rowing parallel to\nthe interface is partially converted to a charge current.\nThe surface-induced conversion of charge to spin current\nand viceversa conspires with the spin Hall e\u000bect in the\nbulk of the metal to produce the observed AH current.\nThis may happen in two ways: in the \frst process, (b1),\nthe charge current jxgenerates, via spin-dependent inter-\nfacial scattering a spin current Qz\nx, which subsequently\ngives rise to the transverse spin polarized current jyvia\nthe inverse spin Hall e\u000bect; in the second process, (b2),arXiv:1512.04146v1  [cond-mat.mes-hall]  14 Dec 20152\nM-𝑸𝑧𝑦\n jy \n-𝑸𝑧𝑥\n-jx \n𝑥𝑦𝑧\n(a) Spin Hall AH e\u000bect ( /\u00122\nsh)\nM\njy \n-𝑸𝑥𝑧\n \n-jx \n(b1)\nM\n-𝑸𝑦𝑧\n \n-jx jy \n(b2) (b) Nonlocal AH e\u000bect ( /\u0012sh)\nFIG. 1: Schematics of two di\u000berent mechanisms of the AH e\u000bect in heavy-metal (HM)/ferromagnetic-insulator (FMI) bilayers:\n(a) the spin Hall AH mechanism and (b) nonlocal AH mechanism with two coexisting physical processes depicted separately in\npanels b1 and b2. The curved arrows represent the trajectories of electrons upon spin orbital scattering and the dotted arrows\nstand for spin dependent scattering at the magnetic interface.\nthe applied charge current jx\frst generates, via spin Hall\ne\u000bect, a transverse spin current Qz\ny, which is then turned\ninto a spin polarized current jydue to spin dependent in-\nterfacial scattering. Both physical processes involve spin\norbit scattering only once (through the spin Hall e\u000bect)\nand hence the resulting AH current is of \frst order in\nthe spin orbit coupling or spin Hall angle. As a matter\nof fact, this AH e\u000bect has the same physical nature as\nits conventional counterpart in bulk ferromagnets, and\ndi\u000bers from the latter only in the spatial separation of\nthe spin orbit interaction and the magnetization: it is\nfor this reason that we name it nonlocal AH e\u000bect .\nCompared to the double spin Hall e\u000bect mechanism\nproposed in Refs. [9, 10], our proposal replaces one of\nthe spin Hall steps, the \frst or the second, by a spin-\ndependent interfacial scattering. This leads to a good\nquantitative description of the transverse current with-\nout the need of introducing an exceedingly small spin\ndi\u000busion length, as we will show in details in the remain-\nder of the paper. In fact, our mechanism survives in the\nlimit of in\fnite spin di\u000busion length, while the double\nspin Hall e\u000bect mechanism vanishes in that limit [10].\nIn addition, the new mechanism has distinctive features\nthat can be tested experimentally, the most striking one\nbeing the sign of the e\u000bect, which we predict to track the\nsign of the bulk spin Hall angle.\nLinear response theory \u0000Let us consider a\nmetal/insulator bilayer as shown in Fig. 1 with an\nexternal electric \feld applied in the x\u0000direction (i.e.,\nEext=Eext^x) and with the magnetization of the insula-\ntor layer pointing in the zdirection, i.e., m=^z. We also\nassume that both surfaces of the metal are rough, but\non the average translational invariance is recovered so\nthat the transport properties are independent of xandy\ncoordinates. The linear response of current densities tospin dependent electric \felds can be written as follows\nj(z) =C0E(z) +CsEk(z)\nQk(z) =C0Ek(z) +CsE(z)\nQ?(z) =C0\nrE?(z) +C00\nr^z\u0002E?(z) (1)\nwhere j(z) = (jx;jy) is the in-plane current density (note\nthatjz= 0 everywhere in the metal layer due to the\nopen boundary conditions), Qk= (Qz\nx;Qz\ny) is the in-\nplane spin-current density (with spin in the zdirection),\nandQ?= (Qx\nz;Qy\nz) is the perpendicular-to-plane spin\ncurrent density with Qx\nzandQy\nzcarrying the xandy\ncomponents of the spin. The corresponding \felds are\nE= (Ex;Ey),Ek= (Ez\nx;Ez\ny) and E?= (Ex\nz;Ey\nz). Notice\nthatCkis de\fned as the integral operator with kernel\nck(z;z0), i.e.,Ckf(z)\u0011R\ndz0ck(z;z0)f(z0). WhileC0is\nan ordinary in-plane conductivity, Csdescribes the gen-\neration of an in-plane spin current from an electric \feld\nin the presence of surface scattering. As we show below,\nCsis the essential ingredient of our theory, producing\nan AH current of \frst order in the spin Hall angle. On\nthe other hand, C0\nrandC00\nr{ respectively the real and\nthe imaginary part of the spin-mixing conductance [14]\n{ contribute only to second order. In particular, C00\nris\nthe essential ingredient of the spin Hall mechanism of the\nAH e\u000bect [10].\nIn the presence of the spin-orbit scattering, the driving\nelectric \felds E;Ek;E?are self-consistently determined\nby the internal current densities as follows\nE=Eext+\u001ash^z\u0002(Qk\u0000Q?)\nEk=\u001ash^z\u0002j\nE?=\u0000\u001ash^z\u0002j (2)\nwhere\u001ash\u0011\u001a0\u0012shwith\u001a0being the Drude resistivity\nand\u0012shthe spin Hall angle of the metal layer. Solving3\nthe system of linear equations (1) and (2), we obtain\na general expression for the AH current density up to\nO\u0000\n\u00122\nsh\u0001\njy(z) =\u0002\n\u001ashfC0;Csg\u0000\u001a2\nshC0C00\nrC0\u0003\nEext (3)\nwheref;grepresents the anticommutator of the two in-\ntegral operators. Note that with \fnite Csthe AH e\u000bect\nappears already at the \frst order of \u0012sh. The two or-\nderings ofC0andCsin the anticommutator of Eq. (3)\ncorrespond to the processes b1 and b2 of Fig. 1. The sec-ond term on the right hand side of Eq. (3) corresponds\nto the spin Hall AH e\u000bect which is of second order in \u0012sh\nand is proportional to the imaginary part of the spin-\nmixing conductivity kernel. In what follows, we employ\nthe Boltzmann transport theory to explicitly construct\nthe integral kernels C0andCsin the presence of a rough\nmagnetic interface.\nBoltzmann theory \u0000To quantitatively describe the non-\nlocal AH e\u000bect in a heavy metal thin layer with an\nexternal electric \feld applied in the x\u0000direction (see\nFig. 1(b)), we make use of the spinor Boltzmann equation\nin the relaxation time approximation [3, 15{17]\nvz@^f(k;z)\n@z\u0000eEextvx \n@^f0\n@\"k!\n+\u001b\u0001[ek\u0002^\u0013(k;z)]\n\u001cso=\u0000^f(k;z)\u0000^\u0016f(k;z)\n\u001c+2^\u0016f(k;z)\u0000^ITr\u001b^\u0016f(k;z)\n\u001csf(4)\nwhere ^f0and ^f(k;z) are 2\u00022 matrices repre-\nsent respectively the equilibrium and nonequilibrium\nspinor distribution functions, v=d\"k=~dkis conduc-\ntion electron velocity,^\u0016f(k;z)\u0011(1=4\u0019)R\nd\nk^f(k;z)\nis the angular average of the distribution and\n^\u0013(k;z)\u0011(1=4\u0019)R\nd\nkek^f(k;z) is its dipolar moment,\nwithekthe unit vector of k. Non-spin-\rip and spin-slip\nprocesses are included, with \u001cand\u001csfbeing the momen-\ntum and spin relaxation times respectively. The addi-\ntional source term \u001c\u00001\nso\u001b\u0001[ek\u0002^\u0013(k;z)], where\u001c\u00001\nsois the\nspin-orbit scattering rate, is responsible for the spin Hall\ne\u000bect [18{20]. It is this term that generates the current-\ndependent \felds in Eq. (2).\nThe crucial step in our theory is the description of spin-\ndependent interfacial scattering via boundary conditions\nfor the distribution function. For the interface (at z= 0)\nbetween the heavy metal and the ferromagnetic insulator,\nwe impose the following generalized Fuchs-Sondheimer\nboundary condition [21],\n^f+(k;0) =1\n2^s^Ry^f\u0000(k;0)^R+1\n2\u0010\n^I\u0000^s\u0011D\n^f\u0000(k;0)E\n+h:c:\n(5)\nwhereh:c:represents hermitian conjugate which ensures\n^f+to be an hermitian, ^Iis the 2\u00022 identity matrix,D\n^fE\n= (2\u0019)\u00001R\nd\u001ek^fwith\u001ekthek-space azimuthal an-\ngle, and both ^ sand ^Rare 2\u00022 matrices in spin space\nwhich are responsible for spin dependent specular re\rec-\ntion and spin rotation of incident electrons.\nThe matrix ^R, satisfying ^Ry^R=^I, is the re\rection\namplitude matrix which captures the spin rotation of\nelectrons that are specularly re\rected from the magnetic\ninterface (Note that we assume such a coherent spin ro-\ntation does not occur for the di\u000busively scattered elec-trons). The explicit form of ^Rcan be determined by\nelectron wave function matching subject to the following\nspin-dependent potential barrier\n^V(z) =\u0010\nVb^I\u0000Jex^\u001bz\u0011\n\u0002 (\u0000z) (6)\nwhereVbis the averaged potential barrier of the insulator,\nJexmeasures the spin splitting of the energy barrier, ^ \u001bz\nis thez\u0000component of the Pauli spin matrices, and \u0002 ( z)\nis the unit step function. Explicitly, ^Rtakes the following\nform (see Appendix B for the derivation)\n^R=\u0012R\"+R#\n2\u0013\n^I+\u0012R\"\u0000R#\n2\u0013\n^\u001bz (7)\nwhereR\u001b=\u0000(\u0014\u001b+ikz)=(\u0014\u001b\u0000ikz) withkzthe\nz\u0000component of the electron wave vector, \u0014\u001b\u0011p\n2m\u0003e(Vb\u0000\u001bJex)\u0000k2z(we have let ~= 1 for notation\nconvenience) and m\u0003\nebeing the electron e\u000bective mass.\nThe matrix ^ s, on the other hand, is introduced to de-\nscribe the averaged e\u000bects of spin dependent scattering\nat the magnetic interface due to roughness, impurities,\netc. In general, we write [22, 23]\n^s=s0\u0010\n^I+ps^\u001bz\u0011\n(8)\nwheres0\u0011\u0000\ns\"+s#\u0001\n=2 is the average of the specular re-\n\rection coe\u000ecients s\"ands#for spin-up and spin-down\nelectrons with \\up\" and \\down\" de\fned with respect to\nm(=^z), andps\u0011\u0000\ns\"\u0000s#\u0001\n=\u0000\ns\"+s#\u0001\nis their asymme-\ntry. A simple model calculation for the rough interface\nyields (see Appendix B for the detailed calculation), to\nthe lowest order in Jex=Vb, the specular re\rection asym-\nmetryps' \u00002Jex\nVb(1\u0000s0) fors0.1. Note that ps4\nisnegative , meaning that more spin-down electrons are\nspecularly scattered than spin-up electrons, for the for-\nmer encounter a higher energy barrier. Also, we notice\nthat a rough magnetic interface is essential for the spin\nasymmetry of the specular re\rection coe\u000ecients: for an\nideally \rat interface, both s\"ands#are exactly equal to\none, and no charge/spin conversion can occur.\nFor the outer surface at z=d, we assume, for simplic-\nity, that the scattering is di\u000busive, i.e.,\n^f\u0000(k;d) =D\n^f+(k;d)E\n(9)\nNote that the boundary conditions given by Eqs. (5)\nand (9) demand that both charge and spin currents \row-\ning along the z-direction vanish at the outer (non mag-\nnetic) surface, whereas only the charge current and the\nz-component of the spin current \rowing along the z-\ndirection vanish at the magnetic surface.\nBy solving the Boltzmann equation (4) with the\nboundary conditions given by Eqs. (5) and (9), we have\ncalculated the current densities in the heavy-metal layer.\nUp to \frst order in \u0012sh(\u0011\u001c=\u001cso), the Hall current density\ncan be expressed as follows\njah\ny(z) =\u001ashEextZd\n0dz0\nle[cs(z;z0) \u0016c0(z0) +c0(z;z0) \u0016cs(z0)]\n(10)\nwhereleis the electron mean free path, the nonlocal in-\ntegral kernels cs(z;z0) andc0(z;z0) are given by\nc0(z;z0) =3\n4Z1\n0d\u0018\u0000\n\u0018\u00001\u0000\u0018\u0001\u0012\ns0e\u0000z+z0\nle\u0018+e\u0000jz\u0000z0j\nle\u0018\u0013\n(11)\nand\ncs(z;z0) =3\n4psZ1\n0d\u0018\u0000\n\u0018\u00001\u0000\u0018\u0001\ns0e\u0000z+z0\nle\u0018 (12)\nwith their spatial averages de\fned as \u0016 c0(z)\u0011Rd\n0dz0\nlec0(z;z0) and \u0016cs(z)\u0011Rd\n0dz0\nlecs(z;z0). The non-\nlocality of the AH e\u000bect, i.e., the spatial separation of\nthe spin-orbit scattering and the magnetization, is clearly\nre\rected in the structure of these integral kernels which\ndepend on the relative distance between the current and\n\feld points as well as the distance of their center of mass\ncoordinate from the interface. Equations (10)-(12) are\nthe main results of this paper.\nOne of the most remarkable features of the nonlocal\nAH e\u000bect is that it appears at the \frst order of the spin\nHall angle, which is distinctly di\u000berent from the spin Hall\nAH e\u000bect which occurs at the second order. Since psis\nnegative, the directions of the nonlocal AH and the spin\nHall AH currents would be the same for positive \u0012shbut\ntheopposite for negative \u0012sh, as can be seen from Eq. (3).\nFurthermore, the nonlocal AH is independent of spin dif-\nfusion and thus is present in both ballistic and di\u000busive\n10-210-11001011020.00.51.01.5 \n  \ns0 = 0.6  \ns0 = 0.7 \ns0 = 0.8 \ns0 = 0.9Ιahy\n / Ι0 ( 10-5 )d\n / leFIG. 2: The ratio of total AH current Iah\nytoI0(=c0Eextwd)\nas a function of the thickness of the heavy metal layer for\nseveral specular re\rection parameters. Other Parameters:\n\u0012sh= 0:05,Jex= 0:01eVandVb= 12eV.\nregimes, whereas the spin Hall AH e\u000bect vanishes as the\nthickness of the metal layer becomes much smaller than\nthe spin di\u000busion length [10].\nThe total AH current can be calculated from Eq. (10)\nby integrating the AH current density over the thickness\nof the layer, i.e., Iah\ny(d)\u0011wRd\n0dzjah\ny(z) withwbe-\ning the width of the metal bar. By doing so, we \fnd\nIah\ny(d) = 2\u001ashEextwRd\n0dz0\nle\u0016cs(z0) \u0016c0(z0) where the fac-\ntor of 2 shows that the two physical processes that we\ndescribed in Fig. 1b contribute equally to the total AH\ncurrent. In Fig. 2, we show the thickness dependence of\nthe total AH current for several values of the specular\nre\rection coe\u000ecient. We \fnd that Iah\nybegins to saturate\nwhen the thickness reaches the electron mean free path.\nAlso, we note that the saturation current is smaller for a\nsmoother surface (larger s0), as expected from the above\ndiscussions.\nExperimentally, a most relevant quantity is the ratio of\nthe spatially averaged AH resistivity to the longitudinal\nresistivity, i.e., \u0012ah\u0011\u0016\u001aah\nxy(d)=\u0016\u001axx(d). The AH resistiv-\nity can be obtained by inverting the conductivity tensor.\nSincepss0\u0012sh.10\u00001, to a good approximation, we can\ntake \u0016\u001aah\nxy'\u0016cah\nxy=\u0016c2\nxxwhere \u0016cah\nxy\u0011d\u00001Rd\n0dzjah\ny(z)=Eext\nwithjah\ny(z) given by Eq. (10). In Fig. 3, we show the\nthickness dependence of \u0012ahfor several values of the spec-\nular re\rection coe\u000ecient s0. Ford\u001cle,\u0012ahtends\nto zero, because \u0016 \u001axx(d) increases with decreasing layer\nthickness. In the opposite limit of d\u001dle,\u0012ahalso dimin-\nishes since the nonlocal AH e\u000bect is essentially an inter-\nface e\u000bect, which saturates for thicknesses larger than the\nelectron mean path. By choosing the following parame-\nters for a Pt (7 nm)/YIG bilayer at room temperature:5\n10-310-210-11001011020.00.51.01.5 \n AH angle θah ( 10-5 )d\n / le s0 = 0.6  \ns0 = 0.7 \ns0 = 0.8 \ns0 = 0.9\nFIG. 3: The AH angle \u0012ah(\u0011\u0016\u001aah\nxy(d)=\u0016\u001axx(d)) as a function\nof thickness of the heavy metal layer for several values of the\nspecular re\rection coe\u000ecient. Other Parameters: \u0012sh= 0:05,\nJex= 0:01eVandVb= 12eV.\n\u0012sh= 0:05 [24],s0= 0:6,Jex= 0:01eV[25],Vb= 12\neVandle= 20nm[13], we estimate the AH angle arising\nfrom our mechanism to be about 1 :3\u000210\u00005, which is in\ngood agreement with experimental observations [6, 7].\nAs a \fnal point, we suggest a crucial veri\fcation of our\nmechanism by contrasting the directions of the Hall cur-\nrent (or the signs of Hall voltages) of two trilayer struc-\ntures Pt/Cu/YIG and \f-Ta/Cu/YIG. Since the spin Hall\nangles of Pt and \f-Ta are of opposite signs [26{28] we\npredict that the Hall current directions in these two tri-\nlayers will be opposite. A Cu layer, thinner than the\nelectron mean free path, may be inserted between the\nheavy-metal and the magnetic insulator in order to elim-\ninate the magnetic proximity e\u000bect, while the nonlocal\nAH e\u000bect will still be operative.\nAcknowledgement. \u0000It is a pleasure to thank O.\nHeinonen, S. Zhang, A. Ho\u000bmann, W. Jiang and W.\nZhang for various stimulating discussions. One of the\nauthor, S. S.-L. Zhang is deeply indebted to O. Heinonen\nfor his hospitality at Argonne National Lab, where part\nof the work was done. This work was supported by NSF\nGrants DMR-1406568.\nAppendix A: Spinor re\rection amplitude at a\nmetal/magnetic-insulator interface\nConsider the following free electron Hamiltonian for a\nmetal/magnetic-insulator interface\n^H=^p2\n2m\u0003e+\u0010\nVb^I\u0000Jex^\u001bz\u0011\n\u0002 (\u0000z) (A1)whereVbis the spin-averaged barrier for electrons to go\nfrom the metal to the insulator, Jexis the exchange cou-\npling which is responsible for the spin-splitting of energy\nlevels in the insulator, and \u0002 ( z) is the unit step func-\ntion. Here we have chosen the spin quantization axis to\nbe parallel to the magnetization m(=^z). For an inci-\ndent electron (from the metal side, z >0) with its spin\npointing in the direction ( \u0012;\u001e) with respect to m, we can\nwrite the scattering wave function as follows\n^ (r) = cos\u0012\u0012\n2\u0013\ne\u0000i\u001e=2'\"(r)j\"i+sin\u0012\u0012\n2\u0013\nei\u001e=2'#(r)j#i\n(A2)\nwhere the spatial parts of the spinor wave function are\n'\u001b(r) =\u001a\u0000\ne\u0000ikzz+R\u001beikzz\u0001\neiq\u0001\u001a,z>0\nT\u001be\u0014\u001bzeiq\u0001\u001a, z<0(A3)\nwhere\u001b=\"(#),R\u001bandT\u001bare the corresponding\nre\rection and transmission amplitudes, k= (q;kz) and\nr= (\u001a;z) are the wave vector and spatial coordinates\nrespectively, and \u0014\u001b=p\nk2\nb\u0000\u001bk2\nJ\u0000k2zwithkb\u0011p\n2m\u0003eVb=~2andkJ\u0011p\n2m\u0003eJex=~2. By matching the\nwave functions and their derivatives at z= 0, we \fnd\nR\u001b=\u0000\u0014\u001b+ikz\n\u0014\u001b\u0000ikz(A4)\nand\nT\u001b= 1 +R\u001b=\u00002ikz\n\u0014\u001b\u0000ikz. (A5)\nAppendix B: Spin dependent specular re\rection\ncoe\u000ecient\nIn this section, we prove that the specular re\rection co-\ne\u000ecientsis spin-dependent for a rough metal/magnetic-\ninsulator interface. In the absence of interface roughness,\nthe bilayer can be modeled as a simple spin-dependent\nstep potential as given in Eq. (A1), the corresponding\nfree electron Green's function (setting ~= 1) reads\ng\u001b\nq(z;z0;E) =m\u0003\ne\nikzh\neikzjz\u0000z0j+R\u001be\u0000ikz(z+z0)i\n(B1)\nwherez <0 andz0<0,kz=p\n2m\u0003eE\u0000q2withEthe\ntotal kinetic energy and qthe in-plane momentum, and\nthe re\rection amplitude for electron with spin \u001bis given\nby Eq. (A4).\nNow we model a rough interface by a set of randomly-\ndistributed impurities localized at the interface ( z= 0)\nwith\u000e-correlated potential Vimp(r) satisfying the follow-\ning properties [29{31]\nhVimp(r)i= 0 (B2)\nand\nhVimp(r)Vimp(r0)i=\r\u000e(\u001a\u0000\u001a0)\u000e(z)\u000e(z0) (B3)6\nwherehidenotes the impurity ensemble average and \rde-\nscribes the amplitude of the \ructuation. Up to \frst order\nin\r, the impurity-averaged Green's function reads [30]\n\nG\u001b\nq(z;z0;E)\u000b\n=g\u001b\nq(z;z0;E)\n+\rg\u001b\nq(z;0;E)g\u001b\nq(0;z0;E)N\u001b(0;E) (B4)\nwhereN\u001b(0;E)\u0011Rdq0\n(2\u0019)2g\u001b\nq0(0;0;E). By placing\nEq. (B1) into Eq. (B4), we \fnd\n\nG\u001b\nq(z;z0;E)\u000b\n=m\nikzn\neikzjz\u0000z0j\n+e\u0000ikz(z+z0)R\u001b\u0014\n1\u00002i\rN\u001b(0;E)kz\nU\u001b\nb\u0015\u001b\n(B5)\nwhereU\u001b\nb\u0011Vb\u0000\u001bJexis the spin-dependent barrier.\nComparing Eq. (B5) with Eq. (B1), we identify the ef-\nfective re\rection amplitude in the presence of the surface\nroughness as\n\u0016R\u001b=R\u001b\u0014\n1\u00002i\rN\u001b(0;E)kz\nU\u001b\nb\u0015\n(B6)\nUp toO(\r), the re\rection coe\u000ecient is\nr\u001b=jR\u001bj2\u0014\n1\u00002\rA\u001b(0;E)kz\nU\u001b\nb\u0015\n(B7)\nwith the surface spectral function de\fned as A\u001b(0;E) =\n\u00002=mN\u001b(0;E). We thus identify the specular re\rection\ncoe\u000ecient as\ns\u001b= 1\u00002\rA\u001b(0;E)kz\nU\u001b\nb(B8)\nBy placing Eq. (B1) into Eq. (B8) and carrying out the\nintegration over in-plane momentum q, we obtain an ex-\nplicit expression for s\u001b\ns\u001b= 1\u0000\r2kz(2m\u0003\neE)3=2\n3\u0019(U\u001b\nb)2'1\u0000\r2kzk3\nF\n3\u0019V2\nb\u0012\n1 +\u001b2Jex\nVb\u0013\n(B9)\nwhere we have replaced the total kinetic energy Eby the\nFermi energy and kept term up to O(Jex=Vb). Therefore,\nwe have shown that the specular re\rection coe\u000ecient is\nindeed spin-dependent. We also note that s\u001bis in gen-\neral dependent on the direction of the incident momen-\ntum. For brevity, we shall work with an angle-averaged\nspecular re\rection coe\u000ecient, i.e.,\n\u0016s\u001b=Z\nd\nks\u001b(q)=4\u0019= 1\u0000\rk4\nF\n3\u00192V2\nb\u0012\n1 +\u001b2Jex\nVb\u0013\n(B10)It follows that the spin averaged specular re\rection co-\ne\u000ecient as well as the spin symmetry of the specular\nre\rection can be expressed as (up to O(Jex=Vb;\r))\ns0\u0011\u0016s\"+ \u0016s#\n2= 1\u0000\r\u0001k4\nF\n3\u00192V2\nb(B11)\nand\nps\u0011\u0016s\"\u0000\u0016s#\n\u0016s\"+ \u0016s#'\u0000\r\u0001Jex\nVb\u00012k4\nF\n3\u00192V2\nb(B12)\nInterestingly, we note the pshas a negative sign; in other\nwords, the specular re\rection coe\u000ecient for spin-up elec-\ntrons is smaller than that of the spin-down electrons as\nthe latter encounter a higher barrier.\nEliminating the parameter \rfrom Eqs. 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Banerjee1* \n1Department of Physics, Indian Institute of Technology, Kanpur, Uttar Pradesh, India, \n2Department of Nuclear and Atomic Physics, Tata Institute of Fundamental Research, \nHomi Bhabha Road, Mumbai, India.  \n3Department of Theo retical Physics, Tata Institute of Fundamental Research, Homi \nBhabha Road, Mumbai, India.  \n \nCorresponding authors’ email: #grk@tifr.res.in, *satyajit@iitk.ac.in.  \n \nAbstract: Study of the formation and evolution of large scale, ordered structures is an \nenduring theme in science. The generation, evolution and control of large sized magnetic \ndomains are intriguing and challenging task s, given the complex nature of competing \nintera ctions present in any magnetic system. Here, we demonstrate large scale non -\ncoplanar ordering of spins, driven by picosecond, megagauss magnetic pulses derived \nfrom a high intensity, femtosecond laser. Our studies on a specially designed Yttrium Iron \nGarne t (YIG)/dielectric/m etal film sandwich target , show  the creation of complex , large, \nconcentric, elliptical shaped magnetic domains  which resembl e the layered shell  structure \nof an onion . The largest  shell has a major axis of over hundreds of micrometers, in stark \ncontrast to conventional sub micrometer scale polygonal, striped or bubble shaped \nmagnetic domains found in magnetic materials, or the large dumbbell shaped domains \nproduced in magnetic films irradiated with accelerat or based relativistic electron beams. \nThrough micromagnetic simulations , we show that the giant magnetic field pulses create \nultrafast terahertz (THz) spin wave s. A snapshot  of these fast propagating spin waves is 2 \n stored as the layered onion shell shaped domains in the YIG film.  Typically , information \ntransport via spin waves in magnonic devices occurs in the gigahertz ( GHz) regime, where \nthe devices are susceptible to thermal disturbance s at room temperature. Our intense laser \nlight pulse - YIG sandwich target combination, paves the way for room temperature table -\ntop THz spin  wave device s, which operate just above or in the range of the thermal noise \nfloor. This dissipation -less device off ers ultrafast control of spin information over distances \nof few hundreds  of microns.   \n \nConventional condensed matter systems display a diverse variety of static magnetization \nconfigurations like Bloch or Neel domain walls, magnetic vortices  (1,2), stripe domains  (3) and \nskrymions  (4). These domains  arise out of a competition between different magnetic exchange \nand magneto static energies  (5,6).  Over the past few decades,  a great deal of  effort has gone into \nusing light for p robing different aspects of condensed matter physics  (7,8,9,10,11,12). Effects of \nlight interacting with magnetism ha ve been primarily studied with low intensity (I ~ 105-106 \nWcm-2) femtosecond  (fs) lasers  (7,8,9,10) for exploring demagnetization processes occurring on \ntime scales of a few picosec onds  (ps) (10,13,14,15). Recent ly, optical coupling of angular \nmomentum of light with  spins in magnetizable media  has been shown to create micron -sized \ndomains  on fs timescales  (8,16,17,18). The use of high intensity femtosecond laser pulses (I~ 1014 \n- 1018 Wcm-2) for such studies has however not been attempted so far.  This may be attributed to \nthe apprehension  that the enormous energy scale associated with such excitation  would \noverwhelm  the spin - spin interaction energy scale and the thermal damage induced by such \nintense laser pulse would obliterate the possibility of seeing any ordered spin configuration . \nIndeed, d irect irradiation with such intense pulses typically ablates the material creating a high \ntemperature plasma.  However, interaction of an intense fs laser pulse with t he plasma is \ninteresting as  it is known to produce  giant megagauss (MG) magnetic field pulses of p icosecond  3 \n duration  (19,20,21,22,23). It is therefore worthwhile studying  the response of magnetic materials \nto such intense magnetic pulses .  In this paper, with an innovative  target design and careful \ncontrol of experimental conditions, we demonstrate the creation of novel, unusual spin structures \ncreated by this magnetic pulse.  \nHere we study the response  of Yttrium Iron Garnet (YIG) film subjected to megagauss \nmagnetic field pulse s produced by the interaction of  a few hundred petawatt/cm2 intensity , 30 fs  \nlaser pulse  with a solid target . YIG is a well -known ferrimagnetic insulator film with very low \ndamping , which in recent times has  become an attractive material for studying magnon dynamics  \n(24,25,26) (magnons are quasi particles associated with spin waves). Low damping  of \nmagnetization dynamics coupled with  large magnon diffusion lengths  reaching several microns  \n(27), make YIG an important material for applications in magnonics  (28,29,30), spin caloritronics  \n(31,32,33) and magnon -based microwave application s (34,35). A careful target design is \nhowever, crucial  for eliminating the ablative degradation of YIG due to laser induced ionization \nand subsequent heating.  We therefore implement a novel sandwich target geometry of metal film \n(Al)-dielectric -YIG (Fig. 1 A), where the laser irradiates the top Al layer , leavi ng the lower YI G \nlayer unaffected by the laser induced damage . Magneto -optical microscopy (MOM) of the YIG  \nsamples exposed to the laser generated giant magnetic field shows  the creation  of novel, large, \nconcentric, elliptically  shaped magnetic domains exte nding up to a few hundreds of microns  from \nthe projected irradiation location . The shape resembles layered shells of an onion.  Furthermore, \nwe see that the local magnetic field direction flips up and down periodically  across the se elliptical  \ndomain structures  and its magnitude also has a periodic variation with distance from the center of \nthe irradiation . Micro magnetic simulati on of a YIG film subjected to megagauss field pulse  \nshow s the excitation of ripples of spin waves travelling  across the low d amping YIG film,  a few \npicoseconds  after the pulse . The spin waves cause moments to gradually rotate out of the film \nplane periodically resulting in the observed behavior of the measured local field. These fast spin \nwaves diffuse up  to a few hundred s of microns in the YIG film from the projected laser 4 \n irradiation site , giving rise to a  non-collinear spin configuration , which  in turn we propose,  gives \nrise to an additional Dzyloshinskii –Moriya type interaction contribution to the magnetic energy of \nYIG. This interaction together with  pinning  effects , results in the spin waves getting stored as the \nlayered onion shell like magnetic domain structure  in YIG .  \nEach target (Fig. 1 A) consists of a 16 m thick Al film suspended over a GGG (Gallium \nGadolinium Garnet ) substrate  in a sandwic h configuration. The lower side of the GGG substrate \nhas a B ismuth doped YIG film grown  on it (36,37). We use single pulses of 25  femtosecond (fs) \nlaser (p-polarized , center wavelength 800 nm) having 20 m beam diameter to irradiate identical \npoints at different locations on the Al layer at an angle of incidence of 45 . The laser intensities \nused are be tween 3 × 1017 to 1 × 1018 Wcm-2 (details of setup in Supplementary section ). The YIG \nfilm is isolated from the optical field of the intense laser as well the heating effects it generate s. \nThe sandwich configuration  provides a two level protection to the YIG film. Firstly, it  eliminate s \nlaser induced  ionization of the YIG film and the resulting thermal heat load  that could lead to \ndirect damage of the film , since the intense laser pulse ablates the sacrificial Al layer which takes  \naway the se deleterious effects . Secondly, t he magnetic YIG layer has additional  shield ing from \nthe heat ing effects  provided  by the intervening 200 m thick dielectric air gap  and the GGG layer \npresent between Al and YIG film layer. The YIG film was devoid of any micron sized magnetic \ndomains prior to irradiation.  As late as four days after irradiation, the irradiated region is imaged \nusing a high sensitivity  magneto -optic microscope  (MOM setup details in Supplementary \ninformation  and Ref. 23). The magneto -optical intensity is  \n2\nzB, where Bz is the component of \nlocal field perpendicular to the surface. The Bz(x,y) distribution is determined from the MOM \nintensity distribution by suitable calibration  (23) (the x and y axes are in the film plane while z \naxis is perpendicular to the film) .  5 \n  \nFig. 1, MOM Images of  laser Irradiated YIG  films.  (A), Schematic shows the incidence of the \nfemtosecond laser pulse on a 16 m thick Al film suspended 100 m above the GGG substrate,  \non the lower side of which is the YIG film. A dielectric layer of thickness ~ 200 m composed of \nair and the GGG substrate, exists between the Al and the YIG film. The  mark is the estimated \nprojected location of the laser spot on the YIG film . (B,D,F ) MOM images of  concentric  elliptical \nmagnetic domains generated in YIG film are shown after irradiation at laser intensities (I) of  3  \n1017 Wcm-2, 6  1017 Wcm-2 and 1018 Wcm-2, respectively.  (C) Color coded three -dimensional map \nof the Bz(x,y) distribution in the elliptical domains of 1( B). The colors represent the magnitude of \nthe Bz values as shown in the color -bar scale.  (E) A one-dimensional map of Bz profile ( viz., Bz(r)) \nmeasured along the red line in Fig. 1( D), show s periodic oscillations in Bz as one traverses the  \nconcentric rings of the elliptical domain pattern . The maximum amplitude of the oscillating Bz(r) is \n 40 G. (F) Shows two domain patterns generated in YIG wit h laser pulse irradiation of intensity \nof 1018 Wcm-2. The smaller upper domain  structure is more circular than the one below . Also seen \nis a defect in the YIG film which was present before irradiation. ( G) MOM image of the region \nsame as that in ( F), imaged after 10 days of laser irradiation, with the black defect as the \nidentifier.  Here we see layered onion shell like magnetic domain patterns have disappeared.  \n \nThe MOM images in Figs. 1  B, D and F show the creation of magnetic domain patterns \nrecorde d in the YIG film layer , after irradiat ion by single pulses of laser intensities of 3  1017 \nWcm-2, 6  1017 Wcm-2 and 1018 Wcm-2, respectively.  All images show the formation of \nconcentric black and white elliptical shaped onion shell -like magnetic domains  in the YIG film. \nThe overall sizes of the elliptic domain patterns at different  laser intensities are comparable and \n6 \n do not scale with  laser intensity. The number of concentric rings however increases with \nincreasing laser intensity. The full extent of the concentric domain patterns is at least an order of \nmagnitude larger than the laser beam diameter of 20 m. The peaks and troughs in the color -\ncoded three -dimensional view of Bz (x,y) in Fig.  1C, correspond to periodic modulation of l ocal \nfield Bz across the bright and dark regions of the pattern in Fig. 1 B. A comparison of the \nconcentric domains in Fig. 1 B with Fig. 1 D shows the number of concentric rings increasing with \nintensity ( I) of the laser. Furthermore, Fig. 1 F shows that at a high intensity of 1018 Wcm-2 there is \na large elliptical domain pattern with concentric rings, and an adjacent satellite concentric domain \nwith a relatively lower eccentricity.  Fig. 1 G shows a MOM image of the same region of the YIG \nfilm a s in Fig. 1 F recorded 10 days after irradiation. Note that image of the defect in the YIG film \nlayer of Fig. 1F is retained in Fig. 1G, however the magneto -optical contrast of the domain \npatterns ha s diminished such that the patter ns are no longer discerni ble. This disappearance of the \npattern is  natural ly expected,  as the remnant magnetization of the film at room temperature \ndecays out to zero with time.  This feature suggests the domain patterns  are not permanent and \nirreversible, i.e. , they are not related to laser induced heating damage. The magnetized regions are \ngenerated by the action of the megagauss magnetic pulse created by the laser. The following \nsections explore the origin of this quasi -stable domain feature observed in YIG fi lm. \nIt is well established in intense laser -solid interaction studies  that a high intensity, p -\npolarized femtosecond laser pulse incident  on a target  at a non -normal angle sets up electron \nwaves in the generated plasma, which grow to large amplitude before  breaking. This breaking \nunleashes a giant current pulse ( ~mega -ampere) that travels normally into the planar target and \nthe entire process is known as resonance absorption (RA ) (22,38). The current pulse is due to RA \ngenerated single or multiple collimated relativistic electron jets  (39). These jets generate giant , \nazimuthal magnetic fields  (B), having peak pulse height of few hundreds of Megagauss with 7 \n typical pulse  widths of a  few ps  (19-21) (Schematic in Fig. 3 C and d etails on RA in \nsupplementary )  \n \nFig. 2.  Simulation of YIG films using MuMax . (A) Schematic of the magnetic precession of the in -\nplane moments of magnetic film due to the megagauss magnetic field (see text for details). (B,C) \nSimulation of YIG films showing r ipples in the magnetic moment configurations are generated in \nthe YIG film with magnetic pulse s, B0 = 7 megagauss and 35 megagauss, recorded at time ( t) = \n0.5 secs (see text for details) . The ripples spread out as a function of increasing time. ( D) Plot of \n1tanz\nxM\nM\n across the red solid line in (D) shows periodic modulations of the magnetic \nmoment configuration in the film at t = 0.5 ps.  \n \n \nTo explore the effect of these RA generated giant magnetic field pulses on YIG films , we \nmodel t he temporal evolution of in -plane local magnetization\nM\n in the YIG film under the \ninfluence of a magnetic field pulse (refer Fig. 2 A) through the Landau -Lifshitz -Gilbert (LLG) \nequation using the MuMax  software  (40,41) (see Methods for simulation details)  \n8 \n \n eff eff\nsdMB M M ( M B )dt M    \n…………………….. eqn. 1 \nThe first term on the right in eqn.1 is the torque on \nM\n due to the effective magnetic field (\neffB = \nApplied in plane azimuthal field (\nB) + Demagnetizing Field (\ndB ) + Anisotropy field (\naniB )), \nwhere  is the gyromagnetic ratio.  The in -plane azimuthal field pulse (see Fig.  2A) is \nˆB B  \nwhere  \nˆis the  azimuthal  unit vector in film plane . The second term is a damping term with \ndamping constant . The \nB pulse  first causes  \nMto flip out of the film plane by an angle 0 due \nto a torque , \n BM . For YIG a large B pulse of 7.5 MG (discussed subsequently) gives  a \n0( ) tB\n= 10.5 in a pulse duration t ~ 0.5 ps (YIG has  = 28 GHzT-1 (42)). The flipped \nM\ngenerates a demagnetization field (\ndB\n0 ~ M sin\nzˆ , \nzˆis  to film ) (see Fig. 2A) around \nwhich  \nM precesses . At this stage,  spin waves are excited in the magnetic material. \nSimultaneously, damping ( second term in eqn.1) comes into play, leading to the decay  of the \nwaves as the precessing \nM gradually loses energy and falls towards the film plane (see blue \ndashed trajectory in Fig. 2A), until its energy is just lower than the anisotropy energy barrier (  \nKu sin(),  - angle between \nM and \naniB ). Note  that the anisotropy energy is minimum  for \nparallel  ( = 0) or antiparallel ( = ) orientation . Here \nM performs a  damped precess ion around \naniB\nbefore falling back into the film plane , with \nM oriented either along or opposite to\naniB . \nFinally , minimizing the free energy leads to domains with in-plane  \nM, where the in-plane \nM\norientation periodically flips by  across the domain s. \nThere appears to be only one other group that has studied  the influence of ultra-strong , \nultrashort magnetic pulses  on magnetic films  (43,44) and it is therefore interesting to make a \ncomparison  with their results . Their studies subject ed films  of high Ku (Ku  ~ 0.1 MJ m-3) viz., 9 \n high \naniB to (a) magnetic field pulses  of the order of a few tens of Tesla and also (b) electric  (E)  \nfields ~109 V/m. The se fields were generated by irradiating the film  directly with relativistic \nelectron (e-) bunches with 28 and 40 GeV energies at SLAC  (43,44). The presence of a strong \nanisotropy field in the magnetic film material resulted in anisotropic dumbbell like domain \npattern (43,44). The in -plane \nM across adjacent domains  in the dumbbell pattern were anti-\nparallely aligned . A comparison with our work and shows the distinct spatial symmetry of our \ndomain shapes , viz., layered onion shell structure,  which is in contrast to the anisotropic dumbbell \nshaped domain s found in the SLAC studies . Another distinguishin g feature of our domains is t hat \n- as one moves along a line cutting across the domains the \nM  periodically twist s out of the plane \nleading in turn to a periodic modulation of local Bz (see Fig. 1E), while in the SL AC study , \nM \nalways remains in -plane . These contrasting results suggest significant differences in the physical \nprocesses  leading to the distinct domain shapes  seen in our experiments . We would like to \nsuggest that our use of YIG has properties which are completely different from those used in the \nSLAC study.  YIG has a time independent damping constant (), which is nearly two orders of \nmagnitude smaller compared to that of the material used in the SLAC study  (see supplementary \ninformation) . Furthermore , the domain structures seen in the SLAC study are only explained by \nconsidering that for the material s used in their study, the  increase exponentially with time until, \nthe spin wave carries away all the energy from the precessing\nM . For YIG, not only is  low but \nalso we do not need to consider any time dependence of  to explain the domain feature we \nobserve. Due to low  value, for our YIG simulations the damping (second) term in eqn.1 has a \nrelatively small effect (compared to that in the SLAC studies of refs. 43,44) resulting in stable \nspin waves excited by the field pulse  which propagate a cross  the YIG film. Also note that  for \nYIG, Ku = 6.1 x 10-4 MJ m-3 is nearly two orders smaller compared to the strong magnetic \nanisotropy material  used at SLAC  (43,44), hence the domains formed in YIG are more 10 \n symmetrical compared to the asymmetrical dumbbell pattern found in the SLAC  study . It is also \nimportant to mention the differences in the method s used to generate the field pulses  in both \ncases . Our intense laser generated  field pulses are essential ly magnetic while both \nB and\nE pulses \nare generated by the relativistic e - bunches in the SLAC experiments  (43). The electron beam in \nthe SLAC study traverse s the film and  cause s film damage, while such a deleterious effect is \ncompletely avoided in our study . Lastly,  our B pulse is larger by at least an order of magnitude  \n(19,20) (~ 100 T) compared to that in the SLAC study ( ~ 10 T)  (43).  \nFor our simulation , the azimuthal magnetic field distribution experienced by the YIG film \nis approximated using an expression similar to that for the field distribution  at positions located \naway from the relativistic electron bunch  (45), viz.,  \n0( , , ) ( )\n()pBB x y t t tr \n  for r> , where , r \nis the distance from the projected center of laser irradiation on the  YIG film and  is the diameter  \nof region around irradiation center within which there are the RA generated current jet(s). We use \n as the diameter of the irradiating laser beam.  In the region r <  , we use a uniform B = B0. The \ntemporal behavior of the pulse is \n( ) 1ptt  for 0  t  tp = 1 ps and \n( ) 0ptt for t > tp. In \nthe above expression ,\n0\n0 3/22\n(2 )pneBt\n , is the field (45) present outside the boundary  of radius \n/2, where n is the number of electrons in the current jet and e is the electron charge . Figures 2 B-\nC show results of the LLG simulations in YIG with a time independent damping constant , at \ntime ( t) = 0.5 ps after the application of magnetic field pulse with peak field B0 of 7 MG and 35 \nMG respectively. Note that B0 = 7.5 MG (n =  1.2 x 1012) and 35 MG (n = 5.6 x 1012) correspond \nto an order of magnitude larger number of electrons in our intense laser pulse generated electron \njet compared to those in the relativistic electron bunch at SLAC .  The Zeeman energy associated \nwith our giant B pulse in YIG  is ~ 14296 MJ  m-3 (see supplementary section) , is sufficiently \nlarge to completely overwhelm the low magnetic anisotropy energy  of YIG ( Ku = 6.10 x 102 J m-11 \n 3).  Figures 2B and C show that the B pulse excites circular  spin wave ripples in the YIG film \naround  \nor\n (center of the pulse). With increasing time,  the ripples spread out across the film (see \nmovie ( 46)), until the y reach the film edge  which occurs within 1 ns. At long time s (~ 100 ms) , a \ncomplex rectangular multi -domain configuration  is stabilized in the Y IG film , which are unlike \nthe domains we have recorded in Fig. 1 (see movie at link Ref. 46, and supplementary \ninformation section ). We do not observe any dependence of  our simulation results on the lateral \nfilm dimensions as long as they are larger than the ripple wavelength. The rippling feature seen in \nthe simulations closely resemble s our observed elliptical layer ed onion shell domain structure of \nFig. 1. Further similarit y between the simulated features and  our experiment is seen i n Fig. 2 D, \nwhere we calculate the orientation (  ) of local \nM\n in the film w.r.t the film plane viz.,\n1tanz\nxM\nM\n,  measured along the radial direction drawn as a red line in Fig. 2 C. The \ncontrast modulations seen in Figs. 2 B and C clearly correspond to the periodic flipping of z \ncomponent of \nM\n , which  is evident from the (r) behavior in Fig. 2 D. The periodic flipping of \nM\n, viz., the behavior of (r) in Fig. 2C is similar to the behavior of Bz(r) in Fig. 1 E.  \nFigures 2 B-C show that at 0.5 ps, the diameter of the outer edges of the rippling structure \nare comparable  for B0 = 7.5 MG and 35 MG . The number of rings inside the concentric circular \nstructures increases with B0. These observation s from the simulations match with that  of Fig. 1 - \nwhile the overall size of the domain is independent of the laser intensity the number of concentric \nrings (N) increase with intensity  (cf. Figs. 1 B and 1 F). The simulations show that increase in N is \nrelated to increase in the pulse peak field B0 (cf. Figs. 2B and 2 C). By comparing the number of \nrings ( N), determined as a function of laser intensity ( I) (experiments , like Figs. 1 B, D and F) and \nas a function of B0 (determined from simulations  like those in Fig. 2), an empirical relation \nbetween B0 and I is obtained (details in s upplementary).  Our experiments reveal that for I  1017 \nWcm-2, no elliptical domain s form . Using the empirical relation,  this intensity  correspond s to a 12 \n peak field of B0 ~ 0.1 MG . Our simulations also confirm that there are no discerni ble magnetized \nripples  excited in YIG  below 0.1 MG field.  \nSimulating eqn. 1 using a general form of B with multi -peak s (19-22), viz.,\n0,\n, ( , , ) ( )\n()i\npi\ni iBB x y t t tr \n\n with i = 1 to 2, we observe in Fig.  3B two well separated rippling \nstructures  (after 0.5 ps) , which are similar to the features in Fig. 1 F. By reducing the spacing \nbetween the field pulses down to 18 m we observe (Fig.  3C) a single rippling structure produced \nby the overlap of two spin wave ripples. The resulting ripple is  not circular but elliptical  in shape . \nIt is these elliptical shapes  that we observe in our experiment s (Fig. 1 ). The  multi -peaked field \nstructure may result from multiple closely spaced e-jets generated during RA process as shown in \nthe schematic of Fig. 3 A.  \n \nFig. 3 . Correlation of simulations with experiments.  (A) Schematic of the Resonance \nAbsorption processes in which fs laser hits the Al film to form a plasma plume and \ninteractions with the laser plasma creates multiple electron (e) jets. These e -jets moving \nat relativistic speeds carrying mega -amps of current generate magnetic field around it \n(B) which is shown only around one of the e -jet paths for the sake of cla rity. These \nelectron jets give rise to megagauss azimuthal magnetic fields inside the YIG. (B,C) \nSimulations done on the YIG film using two sharply peaked magnetic field pulses having \nB0 of 15 MG and 7 MG, separated by 30 m and 18 m. The ripples created in the YIG \nfilm are well separated when the separation between the field pulses is significant while \nthe two ripples merge into an elliptical shaped ripple when the pulses are closer to each \nother. The ripple patterns shown are obtained by stopping the sim ulation at t = 0.5 ps.   \n \n13 \n From our simulations,  we determine the phase velocity ( vp) of the spin waves generated \nby the giant field pulse, as the ratio of the distance covered by the crest of a ripple to the time \ntaken. The vp turns out to be in the range of ~ 107 m s-1 which is four orders of magnitude greater \nthan the typical limit of domain wall velocity (~ 1000 ms-1) reported for YIG  films  (47). Note that \nvp is not related to physical motion of domain walls. Using  vp and the measured wavelength of the \nripple () excited at different energies,  we obtain \n2()mpkv for our spin wave to be in the \nrange of ~ 30 to 100  THz (depending on the laser intensity) . Some earlier m easurements \nsuggested that spin waves with frequencies\nm ~ few THz  (48,49) can be excited in YIG films . \nWe show that one to two order of magnitude higher THz frequency spin waves can be excited \nusing our intense laser and the novel metal/dielectric/YIG film sandwich target combination.  \nThe cl ose match between our simulations  and experiments  in YIG  demonstrate s a unique \neffect, viz., excitation of fast propagating spin waves excited in the YIG film . Furthermore, t he \nYIG effectively capture s a snapshot of the propagating spin wave . It is pertinent  to ask by what \nmechanism  is the spin wave transform ing into a static  magnetic domain pattern ? To get a glimpse \ninto the answer , we det ermine the amount of non -collinear \nM\n configuration generated at the \ndomain site i in YIG by the giant field pulse viz., the average value of \n||ijMM\n within a region \nof 10 m  10 m inside the simulated ripple of magnetic disturbance  in Fig. 2. The average \nvalue of \n||ijMM\n per unit area associated with each ripple  turns out to be ~ 0.9 m-2 compared \nto zero m-2 prior to the pulse.  We propose  that the spin waves excited in YIG generate a non \ncollinear twisted moment configuration which leads to an enhanced contribution of magnetic \nenergy associated with the spin configuration in YIG coming from the Dzyloshinskii -Moriya \n(DM)  (5,6) interaction energy , where  DM \n| |j iM M\n . The addition al DM interaction in YIG \nexcited by the spin wave we believe stabilizes the non -collinear magnetization configuration  \nresulting in the spin wave ripple pattern being frozen into YIG as the elliptical, concentric onion 14 \n shell -like domains as seen in Fig.  1. It may  be recall ed that DM interactions in conventional \ncondensed matter systems are crucial for stabilizing the small chiral magnetic structures like \nskrymions  (4). The ever present pinning in the YIG film would also contribute to stabilizing the \nM configuration excited by the spin wave. Pinning of the outermost spin waves ripples by \nmicroscopic defects in the film is responsible for the irregular shape of the out ermost ring seen in \nthe images of Fig. 1.  \nFrom the layered onion shell domains recorded in YIG film we show that f emto second  \nlaser pulse excites THz spin waves in YIG with diffusion  lengths upto hundred s of microns . This \nresult has potential applications  for spin wave based devices . In recent times, the generation, \ncontrol and manipulation of spin waves ha ve emerged as an important research area  (50,51) as \nthey can be utilized in dissipation -less transfer of information across a device . In conventional \nelectron transport based devices , increasing electrical resistance with miniaturization significantly \nincreases the dissipation losses , thereby limit ing the processing speed of these device s. However \nin magnonic devices , spin waves with frequencies, \nm < 10 GHz , are launched via spin Hall \neffect or spin torque effect  using an electrical current  (27,34). The electrical contacts in these \nmagnonic devices for current injection  result in unavoidable Joule heating dissipation  at the \ncontacts . These devices also need to be operated at low temperatures as the spin waves are \nsusceptible to thermal noise , since for \nm < 10 GHz  the \nmB kT\n (=0.025 eV at 300 K). \nHowever , the spin waves excited in our metal/dielectric/ YIG sandwich using indirect irradiation \nof YIG with the intense laser pulses , completely eliminat e the need for using electric currents for \nexciting the waves and hence limit  Joule heating dissipation . Furthermore, the fast propagating \nspin waves with long diffusion lengths reaching up  to hundreds of microns  have frequenc ies that \ncan be varie d between 10 to 100 THz by varying the laser intensity . Hence magnonic devices \nemploying our method are suitable f or room temperature operation, as the energy of spin waves \nexcited in YIG by intense laser pulse is far above the thermal noise floor ( as for \nm~10 to 100 15 \n THz,\nmBkT\n ). We also re-emphasize that our design for launch ing spin wave s has multiple \nadvantages compared to doing the same with an accelerator based relativistic electron beam  \n(43,44). We have a much more compact table top design, less complexity in operation and the \nstrength of the field pulses and pulse durations can be conve niently controlled by varying the \nlaser intensity , the target design  and the interaction geometries.      \n \nConclusions   \nWe have demonstrated novel , macroscopi c, concentric elliptical onion shell -like \nmagnetic domains created in Yttrium Iron Garnet (YIG)  film via giant magnetic field pulses of \npicosecond duration , generated from femtosecond, intense laser pulses . Spin waves excited in the \nYIG film by the field pulse generate non -collinear spin configurations  in the YIG film which \ngives rise to DM interacti ons which initially w as very weak in the YIG film. These interactions \ntogether with pinning in these films cause the spin waves  to stabilize into surprisingly long lived  \nelliptical domain  patterns.  Our innovative metal/dielectric/YIG structure irradiated with intense \nlaser pulse offers a new way to excite ultrafast spin waves with frequency in the few tens to \nhundred s THz range with large diffusion lengths. These high frequency spin waves are \nundistu rbed by thermal fluctuations . Hence our tabletop route offers a new way for developing a \ndissipation  less high frequency magnonic devices which are capable of operat ing at room \ntemperature  (52).  \n \nAcknowledgements:  SSB would like to acknowledge the funding support from IIT Kanpur and   \nthe Department of Science and Technology, Government of India, New Delhi. GRK \nacknowledges partial support from the Science and Engineering Board (SERB), Government of \nIndia, New Delh i through a J C Bose National Fellowship (JCB -2010/037).  GRK acknowledges \nlate Predhiman K. Kaw for stimulating  discussions.  16 \n  \nAuthor Contribution statement:  \nGRK and SB conceived and guided the study. KN prepared the targets. MS, KJ, ADL, KN and \nDS were involved in the laser irradiation experiments. KN conducted simulations  and MOM \nexperiments . SSB, GRK, ADL, KN , interpreted the data and wrote  the paper, with the approval of \nother authors.  \n \nMethods:  \nSimulations using MuMax: We simulate  YIG films with dime nsions 150 × 150 µm2 and \nthickness 3 µm. The material parameters used in the calculations are as follows: Ms = 1.4 × 105 \nA/m, exchange constant A = 3.6 × 10−12 J/m, α = 0.0005 and anisotropy constant K = 610 J/m3. \nThe film is discretized cells with grid lengths of 1 µm × 1 µm × 1 µm. 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Phys. Rev. Lett. 95, 025005 (2005).  \n \n39. I. Dey et al., Intense femtosecond laser driven collimated fast electron transport in a dielectric medium –\nrole of intensity contrast. Optics Express, 24, 28419 (2016).  \n \n40. A. Vansteenkiste  and J. Leliaert, The design and verification of MuMax3. AIP Adv. 4 107133 ( 2014) . 19 \n                                                                                                                                                  \n \n41. K. Nath et al., Flipping anisotropy and changing magnetization reversal modes in nano -confined Cobalt \nstructures. Journal of Magnetism and Magnetic Materials 476, 412–416 (2019).  \n \n42. Y. Sun et al., Growth and ferromagnetic resonance properties of nanometer -thick yttrium iron garnet \nfilms .  Appl. Phys. Lett. 101, 152405 (2012).  \n  \n43. S. J. Gamble  et al., Electric Field Induced Magnetic Anisotropy in a Ferromagnet. Phys. Rev. Lett. 102, \n217201 (2009).  \n \n44. C. H. Back et al., Minimum Field Strength in Precessional Magnetization Reversal. Science 285, 864 \n(1999).     \n \n45. J. Stöhr, H.C. Siegmann, (Chapter 4) Magnetism From Fundamentals to Nanoscale Dynamics, Springer \nSeries in Solid State Sciences, ISBN -10 3-540-30282 -4 Springer Berlin Heidelberg (2006 ).  \n \n46. Movie: https://1drv.ms/u/s!Ak12kEMznzYya4Rz7fu3zw5NV40?e=79BtTi  \n47. M. Yan et al., Fast domain wall dynamics in magnetic nanotubes: Suppression of Walker breakdown \nand Cherenkov -like spin wave emission. Appl. Phys. Lett.  99, 122505 (2011).  \n \n48. K. Shen,  Finite temperature magnon spectra in yttrium iron garnet from a mean field approach i n a \ntight-binding model. New J. Phys. 20, 043025 (2018).   \n \n49. J. S. Plant,  Spin Wave dispersion curves for Yittrium Iron garnet. J. Phys. C: Solid State Phys.  10, 4805 \n(1977).  \n \n50. Y. Wang  et al., Magnetization switching by magnon -mediated spin torque through an  antiferromagnetic \ninsulator . Science 366, 1125 (2019) . \n \n51. J. Han et al., Mutual control of coherent spin waves and magnetic domains in a magnonic device. \nScience 366, 1121 -1125 (2019).  \n   \n52. A.S. Sandhu et al. , Real time study of fast electron transport inside dense, hot plasmas, Phys. Rev. E , \n73, 036409 (2006)  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 20 \n                                                                                                                                                  \nFigure 1   \n \n \n \nFig. 1, MOM Images of  laser Irradiated YIG  films.  (A), Schematic shows the incidence of the \nfemtosecond laser pulse on a 16 m thick Al film suspended 100 m above the GGG substrate,  \non the lower side of which is the YIG film. A dielectric layer of thickness ~ 200 m composed of \nair and the GGG substrate, exists between the Al and the YIG film. The  mark is the estimated \nprojected location of the laser spot on the YIG film . (B,D,F ) MOM images of  concentric  elliptical \nmagnetic domains generated in YIG film are shown after irradiation at laser intensities (I) of  3  \n1017 Wcm-2, 6  1017 Wcm-2 and 1018 Wcm-2, respectively.  (C) Color coded three -dimensional map \nof the Bz(x,y) distribution in the elliptical domains of 1( B). The colors represent the magnitude of \nthe Bz values as shown in the color -bar scale.  (E) A one-dimensional map of Bz profile ( viz., Bz(r)) \nmeasured along the red line in Fig. 1( D), show s periodic oscillations in Bz as one traverses the  \nconcentric rings of the elliptical domain pattern . The maximum amplitude of the oscillating Bz(r) is \n 40 G. (F) Shows two domain patterns generated in YIG with laser pulse irradiation of intensity \nof 1018 Wcm-2. The smaller upper domain  structure is more circular than the one below . Also seen \nis a defect in the YIG film which was present before irradiation. ( G) MOM image of the region \nsame as that in ( F), imaged after 10 days of laser irradiation, with the black defect as the \nidentifier.  Here we see layered onion shell like magnetic domain patterns have disappeared.  \n \n \n \n \n \n21 \n                                                                                                                                                  \n \n \n \nFigure  2 \n \n \n \n \nFig. 2.  Simulation of YIG films using MuMax . (A) Schematic of the magnetic precession of the in -\nplane moments of magnetic film due to the megagauss magnetic field (see text for details). (B,C) \nSimulation of YIG films showing r ipples in the magnetic moment configurations are generated in \nthe YIG film with magnetic pulse s, B0 = 7 megagauss and 35 megagauss, recorded at time ( t) = \n0.5 secs (see text for details) . The ripples spread out as a function of increasing time. ( D) Plot of \n1tanz\nxM\nM\n across the red solid line in (D) shows periodic modulations of the magnetic \nmoment configuration in the film at t = 0.5 ps.  \n \n \n \n \n22 \n                                                                                                                                                  \nFigure  3 \n \n \n \n \n \nFig. 3 . Correlation of simulations with experiments.  (A) Schematic of the Resonance \nAbsorption processes in which fs laser hits the Al film to form a plasma plume and \ninteractions with the laser plasma creates multiple electron (e) jets. These e -jets moving \nat relativistic speeds carrying mega -amps of current generate magnetic field around it \n(B) which is shown only around one of the e -jet paths  for the sake of clarity. These \nelectron jets give rise to megagauss azimuthal magnetic fields inside the YIG. (B,C) \nSimulations done on the YIG film using two sharply peaked magnetic field pulses having \nB0 of 15 MG and 7 MG, separated by 30 m and 18 m. The ripples created in the YIG \nfilm are well separated when the separation between the field pulses is significant while \nthe two ripples merge into an elliptical shaped ripple when the pulses are closer to each \nother. The ripple patterns shown are obtained  by stopping the simulation at t = 0.5 ps.   \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n23 \n                                                                                                                                                  \n \n \nSupplementary Materials for  \nMacroscopic , layered onion shell like magnetic domain  structure  \ngenerated in YIG film using ultrashort , megagauss magnetic pulses  \n \nKamalika Nath1, P. C. Mahato1, Moniruzzaman Shaikh2, Kamalesh Jana2, Amit D . \nLad2, Deep Sarkar2, Rajdeep Sensarma3, G. Ravindra Kumar2#, S. S. Banerjee1* \n \n  1Department of Physics, Indian Institute of Technology, Kanpur, Uttar Pradesh, India  \n2Department of Nuclear and Atomic Physics, Tata Institute of Fundamental Research, \nHomi Bhabha Road, Mumbai, India  \n3Department of Theoretical Physics, Tata Institute of Fundamental Research, Homi \nBhabha Road, Mumbai, India  \nCorresponding authors ’ Email: #grk@tifr.res.in, *satyajit@iitk.ac.in  \n \nMateri als and Methods  \nSimulations using MuMax  \nWe simulate  YIG films with dimensions 150 × 150 µm2 and thickness 3 µm. The material \nparameters used in the calculations are as follows: Ms = 1.4 × 105 A/m, exchange constant A = 3.6 \n× 10−12 J/m, α = 0.0005 and anisotropy constant K = 610 J/m3. The film is discretized cells with \ngrid lengths of 1 µm × 1 µm × 1 µm. At the vertex of each cell a magnetic moment is placed. We \nuse large grid lengths in our simulation as we are interested to capturing long wavelength \nmagnetic modes which maybe excited on the YIG films by the application of giant magnetic field \npulses. W e are not interested in the excitation of small (submicron) wavelength modes. The initial \nmagnetic configuration of YIG in the simulations is in -plane. We have used magnetic field pulse \nof pulse height varying between 0.07 MG (half -maxima) and 350 MG , pul se width σ = 10 m \nand pulse time, t = 1 ps (see text for details).  24 \n                                                                                                                                                  \n \nExperimental set -up \nThe laser irradiation experiments are performed in experimental chamber with base vacuum of \n10-5 mbar. The experimental schematic is shown in Fig. S1 (a). To irradiate  a fresh portion of \nsample every laser shot, the sample is mounted on precise X -Y-Z-θ stage assembly. P-polarized \n25 fs, 800 nm laser pulses were focused to 20 µm diameter spot by off -axis parabolic mirror \n(OAP) on to a sample to create a plasma. The inten sity on the sample is varied from 1017 to 1018 \nWcm-2, by changing the laser energy appropriately. The angle of incidence on sample is \nmaintained to 45o to maximize resonance absorption (RA) (discussed later).  \n \nFigure S1: (a)  Schematic of intense, femtosecond laser irradiation set -up. M1, M2: Mirrors, OAP: off -axis \nparabolic mirror. (b) Schematic of Magneto -optical Imaging set -up. \n \n The laser irradiated samples are imaged by high sensitivity magneto -optical imaging technique, \nwhich is based on the principle of Faraday effect. Figure S1(b) shows a schematic of the \nmagneto -optical setup where the components refer to: un -polarized Light source (L), Polarizer \n(P), Beam Splitter (BS), Analyzer (A). In Faraday rotation, the plane of polarization of a linearly \n25 \n                                                                                                                                                  \npolarized light undergoes rotation by an angle  in presence of magnetic field in the direction of \nlight propagation. The Faraday rotated angle (  ) is given by,  = VBd, where V is the Verdet \nconstant (depends on material and th e wavelength of light), B is the magnetic field and d is the \npath length in the sample. Therefore, we get information about the magnitude and direction of the \nlocal magnetic field from the sample by knowing the degree of rotation . The Faraday rotated \nintensity distribution is captured in a CCD camera and henceforth calibrated to the corresponding \nB values.  In a MOI, the intensity of the Faraday rotated light I(x,y) is proportional to the local \nfield \n2\nzB  . We calibrate the I(x,y) vers us well calibrated magnetic field by applying known fields \nto the YIG film. Using this information, we convert the I(x,y) in the MO images of Fig. 1 (main \ntext) into Bz(x,y). The bright and dark contrasts in the MO images represent the local Bz(x,y) \nfields and hence Mz pointing either out or into the YIG film respectively. The shade in the image \ncorresponds to the magnitude of Bz.  \n \nResonance absorption  \nWe give a short summary of the generation and effects of magnetic fields as a result of reson ance \nabsorption (RA) in laser -matter interaction. During the RA process, the incoming p-polarized ( E-\nfield in the plane of incidence) high intensity laser pulse impinges on the sample at oblique \nincidence and generates dense plasma. The preformed plasma cr eated by the laser pre -pulse \nexpands away from the target surface as shown in the schematic (Fig. S2). The laser light \npropagating into such plasma encounters a density gradient in the plasma, with the density being \nhighest close to the site of irradiation . The electrons of mass me and charge e in the plasma with a \ndensity ne oscillates with a frequency\np , known as the plasma frequency. If the incident light \nfield has a frequency \np  then the  light is reflected  from the plasma. Inside the plasma dome \nthe density increases as one approach the surface of the material. Hence, the femtosecond laser 26 \n                                                                                                                                                  \nbeam is being reflected from the ‘critical surface’ (where the local plasma frequency matches that \nof the light wave). However, an evanescent wave continues deeper into the plasma.  \n \nFigure S2: Schematic of Resonance Absorption mechanism during fs laser -matter interaction.  \n \nup to a region inside the plasma with a critical density. If the frequency of the evanescent light \nfield (ω) matches the plasma frequency (\np ) in the dense region of the plasma then large \nresonant amplitude oscillations are excited in the plasma. Due to the collapse of the resonant \nwaves in the critical layer of the plasma, energ y is transferred from the plasma wave energy to the \nelectrons in the plasma, thereby generating a collimated jet of hot electrons channeling through \nthe target material at relativistic speeds. Consequently giant, megagauss range azimuthal \nmagnetic field ( viz., fields in the plane of the target material and perpendicular to the electron jet) \nare generated from the strong mega ampere currents associated with the hot electrons jet. While \nthese magnetic fields ( B) are very large, their typical lifetime is a fe w picoseconds. Thus \nresonance absorption leads to generation of megagauss magnetic field pulses for picosecond time \ninterval.  \n \n \nplasma\nHot e-Jetω\nωp\nMagnetic\nlayerInsulator\nBPoint of collapse of the \nresonant waves set up \ninside plasma\nAzimuthal \nmagnetic fields27 \n                                                                                                                                                  \nRelation between P and B0 : \nThe log -log plots in Figs. S3(I), (II), show a linear relation between N and P and N with B0. The \ntrends of the experimental data and simulation results of N vs P and B0 behave linearly in the \ndouble log plot suggesting N = C1P and N = C2B0. From the same N using the values of P and \nB0 deduced from panels I and II we plot on a double log scale the linear relationship between B0 \nand P which is displayed on a log – log plot in Fig. S3(III). From the fit in the panel we obtain an \nempirical relation             and this empirical relation is approximately valid until the ripples \nreach the edges of  the film. Using this empirical relationship, we estimate that our laser intensity \nP ~ 7 × 1017 W cm-2 corresponds to generating a gigantic field pulse B0 ~ 100 MG experienced by \nthe YIG. We would like to mention that we did not observe formation of any of  the elliptical \ndomain structures below a lower threshold of P = 1017 W cm-2, which from backward \nextrapolation in Fig. S3(III) corresponds to B0 ~ 0.1 MG. This estimate concurs well with our \nsimulations which show no excitation of long wavelength ripples for B0 < 0.07 MG.  \n \nFigure  S3: Panel I shows a plot of N vs P from experiments and panel II shows a plot of N vs B0 obtained \nfrom simulations (both on log -log scale). Panel III shows the correlation between P and B0. \n \n28 \n                                                                                                                                                  \n \n \nA comparison of material parameters between YIG and the material studied in Ref. \n[i] (Ref. [43] in main MS), viz., Co70Fe30 film.  \n \nMaterial Parameters  YIG  Co70Fe30 \nMagneto -crystalline Anisotropy ( Ku) 6.10 × 102 J.m-3 7.6 ×  104  J.m-3 \nDamping Constant ( α) 0.0005  0.015  \n \nA comparison from the above table shows that the Ku and  of the Co 70Fe30 films used in the \nSLAC study  [i] is two orders of magnitude larger than those in YIG.   \nYIG has a cubic lattice with lattice parameter of a = 12.37 A, magnetic moment of  = 40 B per \nunit cell [ ii]. The Zeeman energy density of YIG associated with giant magnetic field pulses, B ~ \n7.5 MG can be written as, \n314296B\na MJ.m-3. This Zeeman energy ( Ez), generated either by the \nrelativistic electron bunch at SLAC [43] or via intense FS laser pulses, overwhelm the magnetic \nanisotropy energy barrier of materials (for YIG Ez ~ 14296 MJ.m-3 >> its magnetic anisotropy \nenergy, Ku = 6.10 x 102 J.m-3).  \n \nShape of domains formed in YIG after a long time (~ 100 ms after field pulse)  \n \nIn YIG, the equilibrium (100 msecs after B pulse) domain configuration is rectangular shaped, \nwhich minimizes the free energy (Fig. S4). These are the natural equilibrium shapes of the \ndomains in YIG which minimize the free energy of the domains. Note the layered onion shell like \nmagnetic domains in YIG film (Fig. 1 of our main MS) are completely different from these \nequilibrium rectangular shaped domains. The layered onion shell lik e magnetic domain structure \nis a snapshot of a rippling spin wave excited in YIG film by the intense fs laser pulse.  \n 29 \n                                                                                                                                                  \n \nFigure S4 : Showing the simulated magnetization configuration of YIG, recorded 100 msecs after the \nmagnetic field pulse.  \n \n \n[i] S. J. Gamble  et al., Electric Field Induced Magnetic Anisotropy in a Ferromagnet. Phys. Rev. Lett. 102, \n217201 (2009).  \n[ii] Sahalos , John N, Kyriacou , George A. , Tunable Materials with Applications in Antennas and \nMicrowaves , (Morgan & Claypool Publishers ), 2019.  \n \n \n \n \n \n \n \n \n \n"    },    {        "title": "1803.05545v1.Synthetic_antiferromagnetic_coupling_between_ultra_thin_insulating_garnets.pdf",        "content": " 1 Synthetic a ntiferromagnetic  coupling between  ultra -thin insulating garnets  \n \n \nJuan M. Gomez -Perez1, Saül Vélez1, †, Lauren  McKenzie -Sell2, Mario Amado2, Javier \nHerrero -Martín3, Josu López -López1, S. Blanco -Canosa1,4, Luis E. Hueso1,4, Andrey \nChuvilin1,4, Jason  W. A. Robinson2, Fèlix Casanova1,4, * \n \n1CIC nanoGUNE, 20018 Donostia -San Sebastian, Basque Country, Spain  \n2Department of Materials Science and Metallurgy, University of Cambridge, 27 Charles \nBabbage Road, Cambridge  CB3 0FS, United Kingdom  \n3ALBA Synchrotron Light Source, Carrer de la Llum 2 –26, 08290 Cerdanyola del Vallès, \nCatalonia, Spain  \n4 IKERBASQUE, Basq ue Foundation for Science, 48013  Bilbao, Basque Country, Spain  \n \n†Present address: Department of Materials, ETH Zürich, 8093 Zürich, Switzerland.  \n*Email: f.casanova@nanogune.eu  \n \nAbstract  \n \nThe use of magnetic insulators is attracting a lot of interest due to a rich variety of  spin-\ndependent phenomena with potential applications  to spintronic devices . Here w e report ultra -\nthin yttrium iron garnet (YIG)  / gadolinium  iron garnet (GdIG) insulating bilayer s on \ngadolinium iron garnet (GGG) . From spin Hall magnetoresistance  (SMR) and X-ray \nmagnetic circular dichroism  measurements , we show that the YIG and  GdIG  magnetically \ncouple antiparallel even in moderate  in-plane magnetic fields . The results demonstrate an all-\ninsulating  equivalent of a  synthetic antiferromagnet in a garnet -based thin film \nheterostructure and could open new venues for insulators in  magnetic devices . As an \nexample, we demonstrate a memory element  with orthogonal magnetization switching that \ncan be read by SMR.  \n \nI. Introduction  \n \nSpintronics  is an emerging field that involves the  manipulat ion of  not only electron charge \nbut also electron spin, and is seen as a promising alternative to conventional charge -based \nelectronics.  The application of magnetic insulators for spintronics is gaining interest because \nsuch materials offer advantages  over metals such as long spin transmission length s1 and the \nabsence of energy dissipation due to Ohmic losses2. Heavy metal (HM)/ferromagnetic \ninsulator (FMI) heterostructures  are an interesting  platform where a plethora of novel \nspintronics phenomena ha s been discover ed, including  spin pumping1,3,4, spin Hall \nmagnet oresistance (SMR)4,5, spin Seebeck effect4,6 and many others1,7–15. SMR is based on \nthe interaction between the spin-Hall-induced spin accumulation at the HM layer and the \nmagnetization of the FMI at the HM/FMI interface16. SMR is thus a good candidate to \nexplore the magnetic properties of surfaces17–19  because it is only sensitive to the first atomic \nplanes of the FMI20. The most extensively used FMI in insulating spintronics is y ttrium iron \ngarnet  or YIG (Y3Fe5O12), due to its low Gilbert damping, soft ferrimagnetism and negligible \nmagnetic anisotropy1,4–7,10–15,17,18,21 –30. Alternative magnetic insulator s include  \nantiferromagnet s31–33, non-collinear magnets34–36, hexagonal ferrites8, ferrima gnetic \nspinel s19,37–39 and other ferrimagnetic  garnets9,40–45.  \n \nDownscaling is a n important factor for spintronic devices  and so maintaining magnetic  2 properties of the FMI at reduced dimensions is considered key for deterministic  \nmagnetization reversal due to spin-orbit torque 8,9 or for guiding magnon s2. Since a top -down \napproach to nano fabrication requires the use of thin film  materials, there is  much  effort \nfocused on obtain ing high quality YIG thin films. Standard  growth  techniques such as liquid \nphase epitaxy ( LPE) are being pushed towards the 100 nm thickne ss46, but sub-100-nm-thick \nfilms still require  alternative  techniques such as pulsed laser deposition  (PLD)47–49 or \nmagnetron sputtering50–53. However, material quality in these cases is not as consistently  high \nas seen in LPE-based YIG . For example, r ecent works  report  unusual magnetic anisotropy \nrelated to Fe3+ vacancie s in PLD -grown  YIG47,48, and either exceptionally high \nmagnetization53 or a magnetization suppression52 in sputtered films that could be related to \nthe interface between YIG and the used substrate Gd3Ga5O12 (GGG ). The variety in the \nresults and interpretations that can be found in the literature calls for an in -depth \ncharacterization of those thin YIG films . \n \nIn this paper , we report  ultra-thin (13 nm thick) epitaxial YIG on GGG. Structural and \ncompositional analysis by transmis sion electron microscopy (TEM)/ scanning TEM (STEM) \nreveal a well-defined GdIG interlayer at the YIG/GGG interface.  The magnetic properties of \nthe top YIG layer , characterized by SMR ( using Pt as the spin -Hall material ) and X-ray \nmagnetic circular dichroism  (XMCD)  measurements , are dramatically  modified with the YIG \nmagnetization pinned antiparallel to  the GdIG  one. The results demonstrate  the presence of a  \nnegative exchange interaction between YIG and GdIG  that constitutes a  novel  insulating  \nsynthetic antiferromagnet ic state, with a potential use in insulating spintronic devices54. For \ninstance , we show  that the complex  interplay between  the negative exchange interaction and \nthe demagnetizing fields of the layers induce  a memory effect that could  be exploited as a \ndevice.  \n \nII. Experimental details  \n \nEpitaxial YIG (13 nm thick) is grown on (111) oriented GGG  by pulse laser deposition \n(PLD)  in an ultra-high vacuum chamber with a base pressure of better than 5×10-7 mbar . \nPrior to film growth,  the GGG  is rinsed with deionised water, acetone and is opropyl alcohol \nand annealed ex situ in a constant flow of O2 at 1000C for 8 hours.  The YIG is deposited \nusing KrF excimer laser (248 nm wa velength ) with a nominal energy of 4 50 mJ and fluence \nof 2.2 W/cm2. The films are grown under a stable atmosphere of 0.12 mbar of O 2 at 750C \nand fixed  frequency of 4 Hz for 20 minutes. An in-situ postannealing  at 850C is performed \nfor 2 hours in 0.5 mbar partial pressure of static O 2 and subsequently cooled down  to room \ntemperature  at a rate of  -5 C/min . A 5-nm-thick Pt layer was magnetron -sputtered ex situ  (80 \nW; 3 mtorr of Ar ) and a Hall bar (w idth 450  nm, length  80 m) was patterned by negative e-\nbeam lithography and Ar -ion milling. Unpatterned s amples for TEM/STEM and XMCD were  \ncapped with a 2-nm-thick la yer of sputtered Pt. \n \nTEM/STEM was performed on a Titan 60 -300 electron microscope (FEI Co., The \nNetherlands) equipped with EDAX detector (Ametek Inc., USA), high angular annular dark \nfield ( HAADF )-STEM detector and imaging Cs corrector. High resolution TEM (HR -TEM) \nimages were obtained at 300  kV at negative Cs imaging conditions55 so that atoms look \nbright. Composition profiles were acquired in STEM mode with drift correction utilizing \nenergy dispersive X -ray spectroscopy ( EDX ) signal. Geometrical  Phase Analysis (GPA) was \nperformed on HR -TEM images using all strong reflections for noise suppression56. \nMagnetotransport measurements were performed in a liquid -He cryostat (with a temperature \nT between 2 and 300 K, externally applied magnetic field H up to 9 T and 360º sample  3 rotation) using a  current source  (I=100 μA) and a nanovoltmeter operating in the dc -reversal \nmethod57–59. XMCD measurements were performed across t he Fe -L2,3 absorption edges at the \nBL29 -BOREAS beamline60 of the ALBA Synchrotron Light Source (Barcelona, Spain), \nusing surface -sensitive total electron yield (TEY) detection.   \n \nIII. Results  and Discussion  \n \nIII.a. Structural  characterization  \n \nFigure 1 shows the structural and compositional analysis of a Pt/YIG film by TEM/STEM. \nFigure 1(a) shows a HR-TEM  cross -sectional micrograph where t he top layer  corresponds to \nPt (polycrystalline) , with epitaxial  YIG on single crystal GGG  beneath . The YIG/GGG \ninterface reveals an  extended region with visually different contrast. Comparison of high -\nresolution contrast in the  YIG, interfacial and GGG  regions  [averaged unit cells are shown in \nthe insets in Fig. 1(a)]  show that within the same crystallographic structure there is a \nvariation in distribution of heavy atoms from region to region . To confirm the nature of this \nmiddle region , we performed EDX analysis of a spatial distribution of the elements  along the \nout-of-plane direction [ see Fig. 1(b), the scan line is indicated  in Fig.  1(a)] . From this \nanalysis , we confirm that the film consists of  2-nm-thick Pt on t he top surface, followed by a \n12-nm-thick YIG layer that is  Ga-doped. The i nterface between Pt  and YIG is assumed to be \natomically sharp, thus the inclination of Y curve and declination of Pt curve give the \nestimation of spatial resolution of composition measurement, which is of the order of 1  nm. \nAt the depth of 1 2 nm, Y concentration decreases to  zero, though the slope of declination is \nlower than at the upper interface, indicating a smooth change of concentration in this case. \nGd concentration in the same region increases complementar ily to Y. At the same time the \ndecrease of Fe concentration is delayed by ~3 nm relative to Y, and Ga concentration changes \ncomplementar ily to Fe. Thus , it may be  concluded that , starting from a depth of 12 nm, Gd \ngradually (within a range of ~2  nm) substitutes Y in the lattice ; similarly Ga substitutes Fe , \nbut with ~ 3 nm delay  in depth . This d elay results in the formation of a 2.8-nm-thick  \ninterlayer with a nominal composition corresponding to gadolinium iron garnet (GdIG).  The \ndetailed analytical deconvolution of the concentration profiles  gives a thickness of the “pure” \nGdIG as 2.2 nm61.  \n \nFurther insight into the nature of the layers can be obtained from the analysis of the \ninterplanar distances in the direction normal to the surface. This is done by generalized GPA \non the base of HR -TEM images56. Variations of the interplanar  distance are calculated in \nterms of strain with respect to the GGG lattice . The obtained strain profile is presented as a \nblack line in Fig. 1(b) , and shows that the region corresponding to GdIG  composition is \nexpanded by 1.1% with respect to GGG. This is lower than the  2.3% theoretically expected  in \nepitaxial GdIG  on GGG62, which could be explain ed by the presence of an inter-diffusion \nlayer betwee n GGG and GdIG that reduces the  strain  as compared to a sharp interface . The \nYIG layer shows an unexpected  0.2% expansion of the lattice (on average) with respect to \nGGG , in spite of the very similar lattice constant62,63. The out -of-plane expansion of the YIG \nlattice, which may be attributed to the presence of vacancies47,48, is consistent with the X -ray \ndiffraction measu rements (0.35 -0.6%  expansion of the lattice ) in the same deposition batch. \nThis detailed analysis confirms that we have a magnetic garnet bilayer. The presence of a Gd -\ndoped YIG interlayer in YIG/GGG films after similar postannealing treatments has been \nrecently reported51,52, but in ou r case we can confirm a well -defined, 2 .2-nm-thick GdIG \nlayer, and the fact  that the YIG layer is Ga -doped.  \n  4  \nFIG. 1. (a) HR-TEM micrograph of Pt (2 nm)/ YIG (13 nm) (thickness es are nominal) on  GGG (111) . \nInset: average d unit cells obtained in the different regions shown corresponding to YIG, GdIG and \nGGG from top to bottom. (b) S patial distribution of the elements extracted  along the white arrow in \n(a) by spatially resolved EDX . The strain , extracted  from the HR-TEM image  as a v ariation  of the \ninterplanar distance with respect to the GGG lattice, is also plotted as a black line. Strain of +0.01 \nmeans a lattice expansion by 1% in out -of-plane direction.  \n \nIII.b. Spin Hall magnetoresistance  measurements  \n \nThe magnetic properties of th e ultra-thin magnetic garnet bilayer  cannot be extracted  using \nstandard magnetometry,  because the 500-m-thick GGG substrate shows  a dominating  \nparamagnetic background that masks the magnetic signal from the  bilayer . We performed \nlongitudinal SMR  measurements , which only probe  the top surface magnetization17–20, and \nthus the magnetization  of the GdIG  interlayer at the bottom interface is not expected to \ninfluence  the SMR signal52. SMR depends on the relative angle of the surface magnetization \nin the FMI and the spin  accumulation in  the HM. When the spin accumulation  and the \nmagnetization are parallel (perpendicular) the longitudinal resistance state is low (high ). A \ntwo-point measurement  on the films confirmed that  the YIG is insulat ing at room \ntemperature61,64. The patterned Hall bar on the YIG (see section II)  corresponds to  a Pt/Y IG \nstructure  widely measured before5,17,21,22,24,30. Figure 2 shows the longitudinal resistance RL \nfrom  a 4-point configuration at 2 K vs H applied  along the three main axes of the sample [see \nFig. 2(a) ]. These field-dependent magnetoresistance (FDMR)  curves are expected to show the \nfeatures of SMR: i) a low resistance when the magnetic field satur ates the magnetization in \nthe y-direction (i.e., parallel to the spin -Hall-induced spin accumulation in Pt) with a peak at \nlow H corresponding to the magnetization reversal of the YIG film; ii) a high resistance value \nwhen H saturates the magnetization in the x- or z- direction (i.e., perpendicular to the spin \naccumulation in Pt) with a dip at low fields due to the magnetization rev ersal . However, t he \nFDMR curves are very different from the ones observed so far in  YIG5,17,21,22. A high H ~ 8 T \nis needed to saturate the magnetization of the film  [see FDMR curve  along the y–direction  in \nFig. 2(a) ], while Y IG is expected to saturate  within  a few mT in plane22. This result already \nsuggests that the top  surface magnetization of the 12 -nm-thick YIG is strongly influenced by \nthe 2.2-nm-thick GdIG at the bottom . Moreover, at relativ ely low H (below ~1.5  T) and at \nlow temperature (below ~100 K , see Ref. [61] for high temperature behavior ) the FDMR \ncurves along the three main  axes show unexpected crossings  [see Fig. 2(b) ], indicat ing \ncomplex magnetic behavior  with the magnetization  being non -collinear  with the applied H.  \n \n 5  \nFIG. 2. (a) Longitudinal FDMR  measurements at 2 K along the three main axes (sketch indicates the \ndefinition of the axes, color  code of the magnetic field  direction , and the measurement configuration) . \n(b) Zoom of the FDMR curves at low magnetic field s. Three  different zones associate d to the \nmagnetization behavior  are indicated (see text for details) .  \n \nTo understand better the magnetic properties  and to confirm the non -collinear magnetization \nbehavior  of th e bilayer , we performed angular -dependent magnetoresistance  (ADMR) \nmeasurements  in –, – and –planes (see sketches  in Fig. 3 ) at 2 K . The AD MR curves have \nthree distinct behaviors  depending on the applied H [zones 1 -3 indicated in Fig. 2(b )]. At \nhigh H [above ~1.5 T, zone 3,  Fig. 3(c) ], we have a sin2 dependence  with the angle for – \nand –planes , and no modulation  for the –plane , which  is the expected dependence for \nSMR16–19,38 when the magnetization  is saturated and  collinear with H. The same angular \nbehavior  is expected  for Hanle magnetoresistance22 (HMR) , which has a common origin with \nSMR and is only relevant at ver y large fields . At low  H [below ~0.25 T, zone 1, Fig. 3(a)]  we \nstill have the sin2 dependence in –plane, but the amplitude is smaller because the bilayer is \nnot saturated  (as evidenced in Fig. 2 (a)). However, we have an unusual ADMR  for – and – \nplanes . In –plane , the ADMR curve does not follow a sin2 dependence, indicat ing that the \nmagnetization and H are not collinear. When H is perfectly out -of-plane (=0º and 180º ) the \nmagnetization also points ou t-of-plane. As soon as H rotates away from the out -of-plane into \nthe y–direction, the magnetization switches abruptly to the in -plane y-direction  (=90º and \n270º). This effect can not be simply explained by  the demagnetization field due to  the strong \nshape anisotropy expected in the ultra -thin film65,66. As we will s ee below, the presence of the \nGdIG layer also plays a role  in this behavior . Accordingly, the same abrupt switch ing from \nthe out -of-plane (=0º and 180º) into the in -plane x–direction ( =90º  and 270º ) when \n 6 rotating H along the  –plane  should not give any ADMR modulation; however, the dip in \nADMR at =0º and 180º shows  that a small net contribution of the magnetization along y \nexists , probably because the YIG film  breaks  into domains . \n \n \nFIG. 3. Longitudinal ADMR measurements at 2 K along  the three relevant H-rotation planes (  ) \nfor different applied magnetic fields : (a) 0.1 T ( zone 1 ), (b) 0.5 T ( zone 2 ), and (c) 9 T ( zone 3 ). A \ndifferent background RL0 is subtracted for the ADMR curves at  each field. Sketches indicate the \ndefinition of the angles, the axes, and the measurement configuration.  Dotted line at each sketch \ncorresponds to 0º.  \n \nAt intermediate magnetic fields  (0.25 T ≤  H ≤1.5 T , zone 2 , Fig. 3(b)), we can see an extra \nmodulation in the ADMR curves for – and –plane s. In the case of –plane (plane) when \nH rotates from out -of-plane  to in-plane parallel (perpendicular) to the y-direction , we observe  \na high (low) resistance  state, suggest ing that the magnetization and H are not collinear in the \nplane of the sample . This is confirmed by the ADMR curve for –plane  and H = 0.5 T , where \nthe sin2 dependence is maintained, but with a phase shift 0 which can be either ~112º or \n~68º and should correspond to the angle between H and the surface magnetization . To study \nwith more detail the behavior  of 0, we performed ADMR measurements for different \napplied magnetic fields  (from 20 m T to 2 T) . Figure  4(a) show s how the phase of the ADMR \ncurves shift with increas ing H. Figure  4(b) plots 0 as a function of H, showing a monotonic \nchange between 0º and 180º. Although we cannot , in principle , determine if the phase shift at \nlow fields corresponds to  0º or 180º, we assume that 0 goes from 180º at low fields to 0º at \nhigh fields because it is physically more plausible. The three different zones  already \ndescribed  can be distinguished  in this plot : (i) zone 1, where  the surface magnetization is \nantiparallel to the applied H; (ii) zon e 2, where the surface magnetization has certain angle  \nwith H; (iii) zone 3, where the surface magnetization is almost align ed with H.  \n \n 7  \nFIG. 4. (a) Longitudinal ADMR measurement s for different applied magnetic fields in  the –plane at \n5 K. (b) Phase shift ( 0) as a function of magnetic field taken from data in (a). 0 corresponds to the \neffective angle between the magnetization vector and the applied magnetic field . (c) Hysteresis loop \nmeasured by XMCD  with the magnetic field applied in plane at 2 K . \n \nIII.c. X-ray magnetic circular dichroism measurements  \n \nTo confirm this unconventional behavior  that suggests that the surface magnetization of YIG \nopposes  a low external field and only aligns parallel under a high enough field (> 1.5 T) , we \nperform ed XMCD , a technique that extract s information of the magnetization associated to \neach atomic species. The sample is oriented with its surface forming a grazing angle with \nrespect to the propagation direction of incident x -rays (in -plane configuration ), H is applied \nparallel to the x -ray beam  and TEY  detectio n is used, which is sensitive to the surface . We \nobtained the typical XMCD spectrum for standard YIG at Fe L 2,3 absorption edges61. From \nthese data, and applying the sum rules for XMCD spectra at different H values, we can \nestimate the magnetization per Fe ion and plot the  hysteresis loop (see Fig. 4(c) ). The loop \nclearly confirms our scenario: a negative net magnetization is measured at low applied  H, i.e., \nthe magnetization vector of the YIG surface is aligned antiparallel  to H. The net \nmagnetization is reduced  with incre asing H because the magnetization vector starts rotating \nmonotonously towards the applied H and, at certain value of H, becomes perpendicular to H, \nleading to no net magnetization. At higher H, the net magnetization  becomes positive while \nthe magnetization vector  approaches a  collinear  configuration with H, finally saturating at \nvery high fields.  Note that the saturation  magnetization (3-3.5 μ B/Fe) is lower than expected \nin YIG (5 μ B/Fe), which can be explained by th e presence of Ga substituting Fe along our \nYIG film67.  \n \nThe behavior  of the surface magnetization of YIG observed both via SMR and XMCD can be \nexplained if we consider that  YIG is in fact coupled  antiparallel to the GdIG  interlayer . A \nhysteresis loop similar to the one in Fig. 4(c) has been recently observed in Ni/Gd layers  and \nattributed  to the negative exchange coupling between the transition metal and rare -earth \nferromagnets68. YIG has two magnetic sublatti ces [3 tetrahedrally coordinated (“FeD”) and 2  \noctahedrally coordinated (“FeA”) Fe3+ ions per formula unit ] which are antiferromagnetically  \ncoupled, leading to its ferri magnetism , with the magnetization dominated by the FeD \nsublattice . GdIG has the same iron garnet crystal structure, with  a third magnetic sublattice ( 3 \ndodecahedrally coordinated Gd3+ ions per formula unit ), which is ferro magnetically coupled \nto the FeA  sublattice.  The strong variation  of the magnetization of the Gd sublattice  with \ntemperature makes GdIG  a compensated ferrimagnet , with the magnetization dominated by \nthe Gd and FeA sublattices below room temperature43,45,69. We hypothesize that t he perfect \nepitaxy of the crystal structure at the YIG/GdIG interface (Fig. 1 (a)) would favor  the \ncontinuity of the FeA and FeD s ublattices across the interface . Such continuity leads  to an \n 8 antiferromagnetic coupling betwe en the net magnetization of the GdIG (dominated by Gd ) \nand the net magnetization of the YIG (dominated by FeD ). This very same coupling has been \ndeduced from recent magnetooptical spectroscopy51 and polarized neutron reflectivity52 \nexperiments  in YIG/GGG interfaces . In our case, t he Gd magnetization at low T is so high \nthat a 2.2-nm-thick GdIG layer can pin the whole  12-nm-thick YIG layer  antiparallel to H. \nWhen increasing H in plane above ~0.25 T, the surface magnetization of YIG coherently \nrotates  (see Fig. 4(b)) , becoming parallel above ~1.5 T.  Note that the behavior  of this bilayer \nis equivalent to that of a synthetic antiferromagnet70, although we are not aware of previous \nreports of such man-made system with insulating materials.  \n \nIII.d. Memory effect  \n \nThe presence of the negative exchange  coupling between YIG and GdIG would  also explain \nthe sharp switch ing from the out -of-plane to in -plane magnetization deduced from th e shape  \nof the ADMR in –plane (Fig. 3a and 3b). The strong and opposing demagnetization field s \nexpected from the YIG and  GdIG layer s combined with the antiferromagnetic  coupling \nbetween them  would favor the switching  of the entire bilayer  magnetization to the plane,  \nmuch sharper than the case of a single YIG layer of similar thickness65. This effect  is \nconfirmed with detailed FDMR measurements sweeping at low magnetic fields along  z-\ndirection (Fig. 5): the higher resistance state corresponds to the YIG magnetization point ing \nout-of-plane and the lower resistance state around  zero field corresponds to in -plane \nmagnetization. Interestingly, the sw itching has a clear hysteretic behavior , which is probably \ndue to the complex interp lay between antiparallel coupling  and the opposing demagnetizing \nfields  of each layer . Switching between two metastable states with orthogonal  magnetic \nconfigurations can b e used  in a memory device  which is written with a very low magnetic \nfield and read by longitudinal SMR.  This w ould be an advantage with respect to p revious \nproposal s of a memory device based on magnetic insulators  with perpendicular magnetic \nanisotropy , because they use the transverse SMR to read  the magnetization state, which has a \nresistance change almost three  order s of magnitude  smaller 8,9. \n \n \nFIG. 5. Longitudinal FD MR measurement (trace and retrace)  at 2 K with the magnetic field applied  \nalong the z-direction (out -of-plane) .  \n \n 9  \n \n \nIV. Conclusions  \n \nWe structurally and magnetically characterized ultra-thin epitaxial YIG films on GGG , which \nreveal an atomically well-defined interlayer  of GdIG  at the YIG/GGG interface.  From SMR \nand XMCD we demonstrate that t he YIG magnetization opposes moderate external magnetic \nfields. This unconventional behavior occurs because YIG /GdIG couple  magnetically \nantiparallel , forming the equivalent to a synthetic antiferromagnet, with the exceptional fact \nof being insulating. Furthermore, we observe a memory effect between orthogonal \nmagnetization orientations, which can  be read with an adjacent Pt film via longitudinal SMR \nmeasurements. This bilayer system could be further engineered to optimize the functionalities \nexploited in  insulating spintronic devices, such as  writing operation s with spin-orbit torque  \nand reading operation s with SMR in insulating magnetic memories8,9, or in envisioned \ndevices where the application of antiferromagnets71 and their synthetic versions54 is \nadvantageous . \n \n \nAcknowledgments  \n \nThe work  was supported by the Spanish MINECO under the Maria de Maeztu Units of \nExcellence Programme (MDM -2016 -0618) and under Project No. MAT2015 -65159 -R and \nby the Regional Council of Gipuzkoa  (Project No. 100/16). J.M.G. -P. thanks the Spanish \nMINECO for a Ph.D. fellowship (Grant No. BES -2016 -077301) . J.L.-L. thanks the Basque \nGovernment for a Ph.D. fellowship (Grant No.  PRE-2016 -1-0128 ). J.W.A.R., M.A., and \nL.M-S. a cknowledge funding from the Royal Society and the EPSRC through the \n“International network to explore novel superconductivity at advanced oxi de \nsuperconductor/magnet inter faces and in nanodevices ” (EP/P026 311/1) and the Programme \nGrant “ Superspin”  (EP/N017242/1). L.M -S. also acknowle dges funding the Winton Trust. \nJ.W.A.R  and M.A. also acknowledges support from the MSCA -IFEF -ST Marie Curie Grant \n656485 -Spin3.  XMCD measurements were performed at BL29 -BOREAS beamline with the \ncollaboration of ALBA staff.  \n \nReferences  \n \n \n1. Kajiwara, Y. et al.  Transmission of electrical signals by spin -wave interconversion in a \nmagnetic insulator. Nature  464, 262–266 (2010).  \n2. Chumak, A. V., Vasyuchka, V. I., Serga, A. A. &  Hillebrands, B. Magnon spintronics. \nNat. Phys.  11, 453–461 (2015).  \n3. Saitoh, E., Ueda, M., Miyajima, H. & Tatara, G. 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Effect of the interface resistance in non -\nlocal Hanle measurements. J. Appl. Phys.  117, 223911 (2015).  \n59. Pietrobon, L. et al.  Weak Delocalization in Graphene on a Ferromagnetic Ins ulating \nFilm. Small  11, 6295 –6301 (2015).  \n60. Barla, A. et al.  Design and performance of BOREAS, the beamline for resonant X -ray \nabsorption and scattering experiments at the ALBA synchrotron light source. J. \nSynchrotron Radiat.  23, 1507 –1517 (2016).  \n61. See Supplemental Material at [URL will be inserted by publisher] for a detailed \nanalysis of the concentration profile, electrical characterization, FDMR curves at \ndifferent temperatures and XMCD data of the samples.  \n62. The comparison of the lattice consta nt of GGG (12.375 Å, very close to YIG, 12.376 \nÅ) with the one in GdIG (12.471 Å) translates into a change of volume that, due to \nepitaxial conditions, gives a 2.3% change in strain in the out -of-plane direction. Values \nextracted from [ref. 6 3]. \n63. 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Ferrimagnetic order in the mixed garnet (Y1 -xGd x)3Fe5O12. \nPhilos. Mag.  85, 1819 –1833 (2005).  \n70. Parkin, S. et al.  Magnetically engineered spintronic sensors and memory. IEEE  91, \n661–679 (2003).  \n71. Jungwirth, T., Marti, X., Wadley, P. & Wunderlich, J. Antiferromagnetic spintronics. \nNat. Nanotechnol.  11, 231–241 (2016).  \n \n \n \n \n \n  13 SUPPLEMENTAL MATERIAL  \n \nS1. Analytical deconvolution of the concentration profiles obtained by TEM  \n \nConcentration profiles were fitte d by sigmo id functions [Fig.  S1(a)]. This fit gives a precise \nmeasure of the thickness of YIG layer (the distance between the surface and position of 50% \nY concentration) as 12.0  nm, and the thickness of Fe -Gd (measured as the distance between \n50% Gd and 50% Fe concentrations) mixing zon e as 2.8  nm. Assuming that Pt and top Y \ninterfaces should be abrupt step functions from the parameters of sigmo id function , we can \nextract a transfer function of EDX mapping and make an analytical deconvolution of \nconcentration p rofiles at the interfaces. Figure S1(b) shows deconvoluted profiles at the \ninterface region normalized to 0 -1 interval in order to compare their relative sharpness.  \n \nFrom these deconvolutions we can estimate the “true” widths of concentrations \ndecay/increase. In terms of 25 -75% inte rval this will be: for Y – 0.7 nm, for Gd – 0.9 nm, for \nGa – 0.8 nm, for Fe – 0.5 nm. Deconvoluted profiles also give the estimation of the thickness \nof “pure” GdIG layer (in terms of >75% of both Gd and Fe), which is 2.2  nm [shown i n Fig. \nS1(b) ]. \n \n \nFIG. S1. (a) Concentration profiles ( solid lines ) fitted by sigmo id functions (dashed  lines) . (b) \nDeconvoluted fitted profiles in the GdIG layer region. Inte nsities are normalized to 0 -1 interval in \norder to compare the sharpness of the transitions for differen t elements.  \n \n \nS2. Transport properties  of the ultra -thin iron garnet bilayer  \n \nIn order to check the electrical behavior of the fabricated film, we applied  a voltage between \n60-m-long Pt strips (separated by 24 m) and detected the charge current flowing through  \nthe YIG/G dIG bilayer , see Fig.  S2. The high resistance measured (~1012 ), similar to thick \ncrystalline YIG,  confirms that our YIG /GdIG ultra-thin film behaves as an insulator . This \ncontrol experiment rules out leakage currents through the YIG/GdIG bilayer as the origin of \nthe magnetoresistance effects observed in the longitudinal resistance of the Pt  strips. \n \n 14  \nFIG. S2. Current -voltage  characteristics  of our YIG/GdIG ultrathin film  measured between two Pt \nstrips  at 300 K . \n \n \nS3. Field -dependent magnetoresistance measurements at high er temperature s \n \nWe show the FDMR curves obtained at higher temperatures  corresponding to the same \nsample used in the main text .  The unexpected crossings shown by the FDMR curves along \nthe three main axes remain up to 100 K. Above this  temperature , no signature of SMR is \nobserved, suggesting that the surface of our YIG ultra -thin film  is non-magnetic . The features \nand symmetry of t he observed magnetoresistance correspond to  Hanle magnetoresistance , an \neffect related to SMR occurring solely at the Pt thin film1. \n \n \nFIG. S3. Longitudinal FDMR measurements performed at (a) 30 K, (b) 50 K, (c) 70 K, (d) 100 K, (e) \n200 K and ( f) 300 K, along the three different main axes (sketch in ( f) indicates the definition of the \naxes, colour  code of the magnetic field direction, and the measurement configuration).  Insets in (a), \n(b), (c) and (d) are a zoom between -2 T and 2 T.  \n \n \nS4. Magnetic characterization , XMCD measurements  \n \nFigure S4(a) shows the spectra for circular polarization and the corresponding x-ray \nabsorption spectr um (XAS) of the sample  YIG (13 nm)/Pt (2 nm) from where we extract ed \n 15 the XMCD curve shown in Fig. S4(b). This XMDC spectrum is consistent with the Fe L 2,3 \nedge in thick YIG.  We obtained  the hysteresis loops by sweeping t he magnetic field between  \n6 T and –6 T in in-plane and out -of-plane  configuration  and measuring  the difference of the  \nXMCD absorption peak  (710.2 eV, Fe L 3 peak ) at 2 K . The in -plane hysteresis loop is shown \nin the main text, whereas the out -of-plane hysteresis loop is shown in Fig. S4(c), confirming \nthe hard magnetization behavior of our sample due to the strong shape anisotropy . \n \n \nFIG. S4.  (a) Absorption spectra for positive (black line) and negative (red line) circular ly polarize d \nlight and X -ray absorption spectrum (blue line) at 2 K and 6 T. (b) XMCD spectrum extracted from \nthe XAS measurements.  (c) Hysteresis loop measured by XMCD with the magnetic field applied o ut \nof plane at 2 K.  \n \n \nReferences  \n \n1. Vélez, S. et al.  Hanle Magnetoresistance in Thin Metal Films with Strong Spin -Orbit \nCoupling. Phys. Rev. Lett.  116, 16603 (2016).  \n \n"    },    {        "title": "2402.14553v2.Unraveling_the_origin_of_antiferromagnetic_coupling_at_YIG_permalloy_interface.pdf",        "content": "Unraveling the origin of antiferromagnetic coupling at YIG/permalloy interface\nJiangchao Qian,1Yi Li,2Zhihao Jiang,1Robert Busch,1Hsu-Chih Ni,1\nTzu-Hsiang Lo,1Axel Hoffmann,1Andr´ e Schleife,1and Jian-Min Zuo1,∗\n1Materials Research Laboratory and Department of Materials Science and Engineering,\nUniversity of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA\n2Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA\n(Dated: March 26, 2024)\nWe investigate the structural and electronic origin of antiferromagnetic (AFM) coupling in the\nYttrium iron garnet (YIG) and permalloy (Py) bilayer system at the atomic level. Ferromagnetic\nResonance (FMR) reveal unique hybrid modes in samples prepared with surface ion milling, in-\ndicative of antiferromagnetic exchange coupling at the YIG/Py interface. Using atomic resolution\nscanning transmission electron microscopy (STEM), we found that AFM coupling appears at the\nYIG/Py interface of the tetrahedral Fe terminated YIG surface formed with ion milling. The EELS\nmeasurements suggest that the interfacial AFM coupling is predominantly driven by an oxygen-\nmediated super-exchange coupling mechanism, which is confirmed by the density functional theory\n(DFT) calculations to be energetically favorable. Thus, the combined experimental and theoreti-\ncal results reveal the critical role of interfacial atomic structure in determining the type magnetic\ncoupling in a YIG/ferromagnet heterostructure, and prove that the interfacial structure can be\nexperimentally tuned by surface ion-milling.\nYttrium iron garnet (Y 3Fe5O12) is well-known for its\nlow magnetic damping [1–7], making it the material of\nchoice for efficient spin interactions in magnonics [8–11],\nspin transport [12], cavity spintronics [13–16], and quan-\ntum information science [17, 18]. Considerable atten-\ntion has been directed toward building YIG-based thin\nfilm heterostructures for spin-based information process-\ning by taking advantage of interfacial spin interactions.\nOne example is the YIG/Pt bilayers, with experimental\nobservations of spin pumping [19], nonlocal spin injection\n[12], and spin Hall magnetoresistance [20], where the in-\nterlayer exchange coupling has significantly enhanced the\nspin transmission across the YIG-Pt interface.\nRecently, another YIG heterostructure with a ferro-\nmagnetic (FM) layer, i.e. YIG/FM bilayer, has aroused\nincreasing interests owing to its potential in hybrid\nmagnonics from the interlayer magnon-magnon coupling\n[21–24]. The ferromagnetic resonance mode in the FM\nlayer can form strong coupling with the perpendicular\nstanding spin wave modes in YIG due to the interfa-\ncial exchange interaction, leading to new physical phe-\nnomena such as coherent spin pumping [24], magneti-\ncally induced transparency [25], and efficient excitations\nof short-wavelength spin waves [26]. However, the physi-\ncal mechanisms underlying the interfacial exchange cou-\npling, crucial for coupling spin excitations between the\ntwo magnetic systems, remain not fully understood. Par-\nticularly, recent works have revealed pronounced antifer-\nromagnetic coupling across YIG/CoFeB, YIG/Co, and\nYIG/Py interfaces, where the origin of antiferromag-\nnetic coupling between YIG and FM layers has been a\ntopic of considerable debate. Previous studies, reported\neither ferromagnetic coupling or antiferromagnetic cou-\npling in different prepared YIG/FM bilayers [21, 27, 28].\nSpecifically, Fan et al. [27], Quaterman et al. [28], andKlingler et al. [21] have posited theories of direct ex-\nchange coupling, while the possibility of super-exchange\ncoupling also remains. These findings hint at the signif-\nicant role of interfacial structure in determining bilayer\ncoupling mechanisms, yet tuning magnetic interactions\nthrough interface engineering remains insufficiently ex-\nplored. Here we elucidate the interface structure between\nYIG and Py and their magnetic coupling. By integrat-\ning FMR with STEM/EELS and DFT calculations, our\nstudy reveals the significant role of surface treatments us-\ning ion-milling in enhancing antiferromagnetic coupling\nthrough promoting the oxygen mediated super-exchange\ncoupling mechanisms at the YIG/Py interface.\nIn line with the preparation of YIG/Py bilayer in our\nprior work [24], we first deposited YIG (100 nm) onto\ntwo (111)-oriented Gd 3Ga5O12substrates by magnetron\nsputtering. Then the amorphous YIG films were an-\nnealed in air at 850 °C for 3 hours, and slowly cooled\ndown to room temperature by 0.5 °C/min, yielding epi-\ntaxial YIG films with light yellow color. To study the\nformation of antiferromagnetic interfacial exchange cou-\npling, we slightly ion milled one YIG film in the sput-\ntering chamber under Ar environment by applying an\nRF bias voltage through the substrate holder, where the\nholder acts as an effective sputtering gun, and trigger\nAr+ion bombarding of the substrate surface.The milling\nrate was 3 nm/mins and the milling process lasted for 1\nmin 30 s. A Py (10 nm) thin film was subsequently sput-\ntered on the milled YIG film without breaking the vac-\nuum, ensuring efficient interfacial exchange coupling. A\ncontrol YIG/Py bilayer sample was also prepared with-\nout ion milling the YIG surface. Figure 1a shows the\ncross-sectional film structure. The quality of the bilayers\nwas checked using X-Ray Diffraction. Clear (444) peaks\nof YIG and GGG were measured along with Laue os-arXiv:2402.14553v2  [cond-mat.mtrl-sci]  22 Mar 20242\nFIG. 1. The YIG/Py bilayer structure and FMR character-\nization. (a) STEM image together with the designed bilayer\nstructure. (b) Illustration of the FMR setup: The copla-\nnar waveguide is located underneath the YIG/Py bilayer thin\nfilms. (c)-(d): FMR induced magnetization excitations in\nYIG/Py bilayer samples on top of the coplanar waveguide.\nThe Line shapes of (c) without ion-milling and (d) with ion-\nmilling samples for the detected resonance mode (n=2) of\nYIG and the uniform mode (n=0) of Py.\ncillations, indicating that both samples possess high film\nquality and maintain epitaxial relationships (Suppl. FIG.\n1). The two samples we prepared will be named subse-\nquently as IM for with ion milling and WoIM for without\nion milling.\nWe first conducted a FMR measurement on the two\nsamples we prepared with and without ion milling. Fig-\nure 1b details this experimental arrangement using the\nsame setup with the coplanar waveguide beneath the\nYIG/Py bilayer as in our previous work [24]. The FMR\nresults are depicted in FIG. 1c and 1d, showing FMR-\ninduced magnetization excitations in the YIG/Py bi-\nlayer samples. Notably, two hybrid modes are present in\nthe ion-milled samples, but absent in those without ion-\nmilling. As illustrated in FIG. 1d, these observed hybrid\nmodes in the YIG/Py bilayer system’s spin pumping ex-\nperiment signify antiferromagnetic exchange coupling at\nthe interface. The broader linewidth mode, exhibiting a\nhigher resonance field than the narrower linewidth mode,\nacts as a key indicator of this antiferromagnetic coupling.\nThis finding is discussed in detail in our previous work\n[24].\nTo investigate the structural origin of the magnetic dif-\nferences between samples with and without ion-milling,we performed cross-sectional STEM of the YIG/Py bi-\nlayer films using a high-angle annular dark-field detec-\ntor (HAADF) for Z-contrast (details in Suppl. Notes).\nFigure 2a shows the STEM-HAADF images of the two\nsamples, providing an atomic-scale inspection of their in-\nterfacial structural differences. The YIG surface termi-\nnation in the IM sample, as revealed in FIG. 2a, com-\nprises a plane of visible Fe and Y atoms, and invisible\nO atoms, leading to an interface layer that sharply con-\nnects to the polycrystalline Py. In contrast, the interface\nin the WoIM sample exhibits roughness and features an\namorphous layer approximately 0.7 nm in width (Fig.\n2b). The direct adjacency of the termination plane to\nthe Py layer in the IM sample eliminates the observed\ngap in the WoIM sample. This distinction in interfacial\nstructure between the two samples is further substanti-\nated by the normalized intensity line profiles depicted in\nFIG. 2c. The YIG surface termination in the IM sample\nis further examined in FIG. 2d, with line profiles marked\nin FIG. 2a. These profiles demonstrate that the termina-\ntion plane ends with a combination of Y and Fe contain-\ning atomic columns, with Py atoms directly adjoining the\ncrystalline garnet structure in the IM sample.\n(a)\n(d)(b)\n(c)\nInterfacial \nlayer \nwithout \nsurface \nion-milling\n#1\n#2With Surface Ion -Milling Without Surface Ion -Milling \nInterfacial layer \nwith surface \nion-milling\nFIG. 2. Interfacial structure as seen by atomic-resolution\nSTEM-HAADF along the YIG [110] zone axis. STEM-\nHAADF images for the with ion-milling (a) and without ion-\nmilling (b) samples. The scale bars are both 0.5 nm. (c) The\nmarked horizontal line profiles of HAADF images in (a) and\n(b) show the different YIG/Py interface width. (d) The line\nprofiles of HAADF intensity across the dashed lines in (a),\nshowing the termination of Y and Fe, with Py atoms adja-\ncent to YIG.\nFrom our STEM-HAADF observations, the garnet\nstructure appears consistent up to the interface, sug-3\ngesting a minimal structural modification. Electron en-\nergy loss spectroscopy (EELS) analysis was performed in\nthe STEM mode to further examine the chemical sharp-\nness of the YIG/Py interface in the IM sample. Figures\n3a,b display the oxygen K-edge fine structure, alongside\na HAADF survey image (FIG. 3c) with a sampling reso-\nlution of 0.5 nm. This combination provides insights into\nboth the oxygen content and the nature of its chemical\nbonding. The oxygen K-edge fine features are observed\nup to the interface. A drop in the K-edge intensity is ob-\nserved near within the distance of 1 nm to the interface,\nwhich can be attributed to the electron probe spreading\neffect [29]. In addition to the pre-peak for oxygen, the\nfine features in the range of 535 eV to 540 eV also exhibit\nchanges near the interfaces, indicative of slightly altered\nyttrium-oxygen (Y-O) bonding. FIG. 3d,e showcase the\nEELS fine structures for the Fe L 3-edge, coupled with a\nHAADF survey image (FIG. 3f). The marked peaks of\n708.7 eV and 710 .5 eV in FIG. 3d are features for Fe0\nand Fe3+respectively. The lower layer consists of Fe0\nfrom Py, while the upper layer contains Fe3+from YIG.\nThe evolution of the oxygen K-edge and Fe L 3edge are\nfurther highlighted in FIG. 3b,e, which plot the hyper-\nspectral EELS data versus the STEM probe position. At\nthe interface, the Fe L 3EELS signal is approximately\na composite of Fe0and Fe3+, while O2−is detected up\nto the interface. The presence of O2−and Fe3+at the\ninterface and slightly beyond, and its sharper transition\nthan the L 3edge of Fe, suggests an oxygen-terminated\nYIG surface. The fine feature of L 3edge shows the domi-\nnant peak is still 710 .5 eV with a wider shoulder left.The\ncoexistence of Fe0and Fe3+in the interfacial transition\nregion suggests a small amount of interfacial defects, pos-\nsibly Fe interstitial atoms inside a rather open YIG struc-\nture. The EELS map shows a oxidized YIG surface with\nremaining garnet structures directly adjacent to the Py\nlayer.\nTo uncover the origin of experimentally observed an-\ntiferromagnetic coupling at the ion-milled interface of\nYIG/Py theoretically, we employed density functional\ntheory (DFT) calculations to simulate the YIG/Py in-\nterfacial configurations. The interface structural models\nwe proposed based on the experimental observations are\ndepicted in FIG. 4a,b. The YIG (111) surface is termi-\nnated with yttrium, coordinated by 8 oxygen ions, and\ntetrahedral coordinated Fe, respectively. The superposi-\ntion of the YIG structural motifs on the magnified atomic\nresolution image from FIG. 2a shows good agreement be-\ntween the model and the observed interfacial atomic ar-\nrangements. The Py atoms, which are not individually\nresolved in FIG. 4a, appear slightly disordered due to in-\nteractions with the YIG surface. For simplicity, we have\nassumed two perfect layers of Py atoms parallel to the\nYIG surface in our model.\nTwo potential magnetic arrangements can exist at the\nYIG/Py interface: antiferromagnetic (AFM), where bulk\n(a) (b) (c)\nO K\nFe L3(d)\n (e) (f) Fe0Fe3+FIG. 3. With surface ion-milling sample: the EELS near-\nedge fine structures for (a) O K-edge; and (d) Fe L3-edge,\nacross the YIG/Py interface in the corresponding regions\nshown in the HAADF images at right, (c) and (f), respec-\ntively. The spectra are integrated horizontally. The 0 nm po-\nsition is set at the YIG/Py interfaces. Hyperspectral images\nfor O K-edge (b) and (e) Fe L3-edge shows the fine structure\nevolution. The scale bars are both 1 nm in (c) and (f). The\nresults here directly evident of the presence of O2−and Fe3+\nat the interface.\nYIG and Py have opposite magnetic moments, and ferro-\nmagnetic with same magnetic moments. Using the struc-\nture model in FIG.4b, we performed spin-polarized DFT\ncalculation with different Fe and Ni arrangements in Py\n(details in Suppl. Notes). The results indicate that the\nAFM arrangement is energetically more favorable than\nFM arrangement with an average energy difference of 2.2\nmeV per atom. However, when we placed the Py layers\non the YIG slab with another different surface termina-\ntion, the simulation results show no preference between\nFM and AFM interficial coupling (detailed structures of\nsimulation cells of different interface terminations can be\nseen in FIG. S2 and FIG. S3 in SI). Within the YIG, oc-\ntahedral Fe and tetrahedral Fe are antiferromagnetically\ncoupled, while the tetrahedral Fe provides the dominant\nspin. In the AFM arrangement, the spins of tetrahe-\ndral Fe are also antiferromagnetically coupled with Py\natoms. The DFT simulation results thus supports our\nexperimental observation of AFM in the ion-milled sam-\nple where the tetrahedral Fe ions interact with Py atoms\nwith the mediation of oxygen ions as highlighted by cir-\ncles in Fig. 4b. The surface on the YIG without the\nion-milling in comparison are rough with a mixture of\nsurface terminations in addition to the amorphous-like\nlayer that breaks the AFM coupling between bulk YIG\nand Py.\nGiven the coexistence of Fe0, Fe3+, and oxygen at\nthe YIG/Py interface, we propose that oxygen-mediated\nsuper-exchange coupling could be the predominant mech-4\nanism for the antiferromagnetic interaction between Py\nand YIG, conceptualized as < Fe0| ↑>−−O−−<\nFe3+| ↓>. The substitution of Ni0for Fe0likely does\nnot alter the direction of the magnetic moments. This\ninsight is crucial in our discussion, as it suggests that the\nantiferromagnetic coupling between Py and YIG is pre-\ndominantly driven by oxygen-mediated super-exchange\ncoupling, a key finding of this study.\nPermalloy\nYIG\nFe Y\n Py(b) (a)\nO\nFIG. 4. YIG surface termination and structure model. (a)\nZoomed-in HAADF image in FIG. 2a showing the over-\nlap with YIG motif structures at surface terminations. (b)\nYIG/Py slab model for Density Functional Theory (DFT)\nsimulation. Highlighted white hexagons represent YIG mo-\ntifs. Both (a) and (b) are illustrated at [110] projection. The\ninset illustrates the coupling between Py atom and tetrahe-\ndral coordinated Fe\nIn conclusion, we have identified interfacial atomic\nstructure as a key factor in determining the type of mag-\nnetic coupling between YIG and Py. AFM coupling,\nas implied from our FMR measurements, is associated\nwith a sharp YIG/Py interface with tetrahedral Fe as\nthe first magnetic layer on YIG side. DFT calculations\nquantitatively confirm that AFM coupling is an energeti-\ncally favorable magnetic state for this particular interface\nstructure. The origin of AFM is explained by oxygen-\nmediated super-exchange interaction between tetrahedral\nFe in YIG and magnetic atoms (Fe and Ni) in Py. 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Zuo, Micron 42, 539 (2011)."    },    {        "title": "1903.12130v2.Acoustic_excitation_and_electrical_detection_of_spin_waves_and_spin_currents_in_hypersonic_bulk_waves_resonator_with_YIG_Pt_system.pdf",        "content": " \nAcoustic excitation and electrical detection of spin waves and spin currents in \nhypersonic  bulk waves resonator with YIG/Pt system  \nN. I. Polzikovaa,*, S. G. Alekseeva, V. A. Luzanovb, A. O. Raevskiyb \naKotel'nikov Institute of Radio Engineering and Electronics of Russian Academy of Sciences,  \nMokhovaya str. 11, build. 7, Moscow, 125009, Russia  \nbFryazino branch Kotel'nikov Institute of Radio Engineering and Electronics of Russian Academy of Sciences  \nVvedenskiy sq.,1, Fryazino, Moscow Region, 141190, Russia  \nAbstract  \nWe report  on the self-consisted semi -analytical theory of magnetoelastic excitation  and electrical \ndetection of  spin waves and spin currents in hypersonic  bulk acoustic waves resonator with ZnO -\nGGG -YIG/Pt layered structure. Electrical detection of acoustically  driven spin waves occurs due \nto spin pumping from YIG to Pt and inverse spin Hall  (ISHE ) effect in Pt as well as due to \nelectrical response of ZnO piezotransducer . The frequency -field dependences of the resona tor \nfrequencies and  ISHE voltage UISHE are correlated with experimental ones observed previously.  \nTheir fitting allows to determine some magnetic and magnetoelastic  parameters of YIG. The \nanalysi s of the YIG film thickness influence on UISHE gives the  possibility to find the optimal \nthickness for maximal  UISHE value.   \n \nKey words : magnetoelastic interaction, acoustic spin pumping, bulk acoustic waves, resonator, YIG/Pt, ZnO  \n1.Introduction  \nIn recent years, acoustically driven  spin waves (ADSW) are of great interest in \nconnection with the key objectives  of next -generation spin -based technologies [1-14].  The \npiezoelectric  generation of ADSW  in composite magnetoelastic  structures  [8 - 14] is promising \nfor use in low power consumption devices free from energy dissipation due to o hmic losses .  In \nparticular, acoustic spin pumping - the generation of spin -polarized electron currents from \nADSW  [7-9] - is promising for microwave spintronics and attracts much attention of the \nresearchers.   \nThe acoustic waves (AW) and spin waves (SW)  coupling , caused by linear \nmagnetostriction, is most significant under the condition of the phase syn chronism, i.e. at \nmagnetoelastic resonance (MER) . As far as  MER  frequency  generally lies in the gigahertz range  \n[15] the generation of  hypersonic AW  is required .  At present, it is considered that bulk AW  are promising for applications at frequencies above 2.5 -3 GHz. One of the ways  to excite  bulk AW \nwith  the frequencies up to 20 GHz is  the use of a high overtone  (n ~ 102 ÷103) bulk acoustic \nwave resonator (HBAR)   [16]. Previously , in [13,14] we demonstrated the piezoelectric  \nexcitation  of ADSW at 2 GHz by means of  HBAR   containing ferrimagnetic  yttrium iron garnet  \n(YIG) and piezoelectric  zinc oxide ( ZnO) films . Quite  recently, the resonant  acoustic spin \npumping in HBAR  containing YIG/ Pt system  was proposed and  implemented in our works [17, \n18]. \nThis paper  presents  a theoretical  consideration for acoustic spin pumping  in HBAR and \ndetection of the ADSW through the inverse spin Hall effect (ISHE) in Pt.   Accounting for the \nback action of ADSW in YIG on the elastic system in  all layers of the structure (in non -magnetic \nlayers through boundary conditions) makes it possible  to determine and compare  the frequency \nand magnetic field dependences of HBAR resonance  frequ encies ,  fn , and dc ISHE voltage, \nUISHE  . Comparison of the se theoretical and experimental [18] dependences shows a qualitative \nagreement and allows us to determine a number of magnetic parameters of the  YIG films. The \ncalculation a lso shows that there is the optimal  YIG film thickness for acoustic spin pumping  \nefficienc y, which may be an order of magnitude high er than observed previously  by means of \nHBAR .  \n2. HBAR s tructure  \nIn Fig.1 the HBAR  structure is shown. It contains a gadolinium -gallium garnet (GGG) \nsubstrate 4 and   two YIG films 3, 5 on both sides of the substrate [19]. A  bulk AW transducer \nconsisting of a piezoelectric  ZnO film 1, sandwiched between thin -film Al electrodes 2, is \ndeposited  on one side of the YIG -GGG -YIG structure. To excite the  bulk AW propagating \nalong the x-axis, the rf voltage Ũ(f) with frequency  f  is applied across the transducer. A thin Pt \nstrip 6 is attached to the YIG film 5 underneath the acoustic resonator aperture.  Below we will \nuse for the layer  with the index  i =1…6 the notations l(i) for the thickness and xi for the \ncoordinate of the lower surface.  The external magnetic field H lies in the plane of the structure \nalong the z-axis and magnetizes YIG films up to uniform  saturation  magnetization  M0|| H.  \nIt is assumed that ZnO film with an inclination of piezoelectric  c-axis excites shear bulk \nAW polarized along the z-axis [20].  In YIG layers , this wave drives magnetization dynamics \ndue to the magnetoelastic interaction. The  AW and SW interaction    results in the shift  Δfn(H) = \nfn(H) - fn(0) of HBAR resonance frequencies   in the magnetic field  [13, 14]. The resonance  \nfrequencies itself correspond to the extrem a in the frequency response of the transducer's \nelectrical impedance.  Thus, the  SW excitation and detection  are performed  electrically by the \nsame  piezo transducer .  These AD SW establish a spin current ( js)x   from YIG into the Pt strip  [21].  The ISHE  converts the spin current in the Pt film to a conductivity current  (short circuit)  or \nan electrostatic dc field EISHE along the y-axis (idle circuit) [22].  \n \n \n \nFig. 1. HBAR structure: 1 — ZnO film, 2 —Al electrodes, 3, 5 —YIG films, 4 — GGG substrate, 6 — \nthin film Pt strip with direction a perpendicular to the figure plane . The typical layer thicknesses: l(1) = 3 \nμm, l(2) = 200 nm, l(3,5) = s = 30 μm, l(4) = 500 μm, l(6) = 12 nm. The overlapping area of the top and bottom \nelectrodes  2 has a diameter \naa = 170 μm. \n 3. Theory    \n Further, we assume that all the thicknesses of the layers l(i) are much smaller than \ntransverse dimensions in the plane ( y, z). In this case , in linear approximation , all variables \ndepend on coordinate   x  and time t  as exp[j(k(i)x – ωt)], where \n1j ,  ω=2πf , k(i)  is  a wave  \nnumber .  \nFor each nonmagnetic  layers ( i=1,4) Newton equation of motion  for the elastic displacement u= \nuz along with Hooke’s law  lead to the  relation ships  k(i) = ω/V(i).  Here  \n)( )( )(/i i iC V   is AW \nvelocity , ρ(i) is the mass  density,  C(i)  is the effect ive elastic modulus account ing for \npiezoelectric stiffening  in layer 1  [23]. Using two roots ± k(i), one can obtain the general \nsolutions for u(i). The normal stress  can be represented as   T (i)=T(i)zx= C(i)(u(i)/x)+ \neIδi1/(jωl(1)C0), where second term exists only in the piezoelectric layer 1. Here e and C0 are the \npiezoelectric  modulus  and capacity  of the layer , I is the displacement current flowing in the layer  \n[24]. \n For YIG layers  (i=3, 5) from the Newton equation and Landau -Lifshitz equation for the \nprecession of magnetization vector  \n),,(0Mmmy xM   together with   Maxwell equations ,  we \nobtain the secular equation in the form  \n0 ) )( (22 2\n02 22 2   VkωωωωVkωM H\n,                                           (1)  \nwhere  we omit the upper indices ( i). Here , \n] )( )[( )(2 2 2 2\n02\n0 M H H k k k    , \n)( )(2\neff2k H kH\n, \neff 4MM , Heff (k2)= H + Dk2 and Meff ≈M0 are the  uniform effective \nmagnetic field   and magnetizatio n,  D is the  exchange stiffness, \n is the  gyromagnetic ratio, \n) 4/(2\neff2CM b\n is the dimensionless coupling parameter , and b is  the magnetoelastic  constant  \n[15, 25, 26 ].  Herei nafter, the system of Gaussian -CGS  units  is used , but for convenience, the  \nlayer thicknesses are given in the SI system : micro - and nanometers . \nAs it is known the crossover of two independent solutions (1) in case of  ξ = 0 determine s \nMER frequency and wave number: ωMER(H)=2πfMER(H) and kMER(H). In case of  ξ≠0 the \nformation of coupled waves and the repulsion of the  solutions in  the vicinity  of ωMER take place . \nAs one can see, for  a given real, positive ω, there are three real roots of the secular equation , k2p \n(ω) (p= 1,2,3 ). Using six roots ± k1,2,3, one can obtain the general solutions for u, mx,y, and normal \nstress component  T =Tzx=C(u/x)+bmx/M0.  \nThe solutions obtain ed for all layers should  satisfy the elastic and electrodynamic \nboundary conditions  at the interfaces.  At the magnetic layers’  interfaces , the additional \nmagnetization boundary conditions for ac magnetization should be taken into account.  Here  the \ncase of free spins  is considered :  mx,y/x =0.  \nMagnetic and acoustic losses are taken into account phenomenologically with the help of \nthe following substitutions: \n)( )( )( i i ii C C   and \nHiH H  , where, \n)(i and \nH  are  \nviscosity factor  and ferromagnetic resonance (FMR)  line width [23, 27] .  \nFurther  we use the  impedance method for calculating the multilayer resonator structure  \ncharacteristics  [23]. The input electric impedance of piezo transducer  may be represented as  [28] \nZE = \nIU/~  (1+ ZAW)/(jωC 0),                                                                    (2) \nwhere,  ZAW is the function of  transducer parameters  (material and geometrical) and of the \nacoustic impedance Z of transducer l oad (layers 2-6). It follows from the impedance continuity \ncondition that the load impedance of layer i-1 is equal to the input impedance of layer i: \n)/) (/() ()( )( )(\nin t lxu lxT zi\nii\nii\n. Since the influence of 150–200 nm thickness  electrodes on the \nproperties of HBAR is negligible, we can assume that \n)3(\ninzZ .  In the absence of magnetoelastic  \ninteraction ( ξ = 0), the load impedance  is calculated by the sequential application of the  \nimpedance transformation formula  [23] \n) sin cos /() sin cos ()( )1(\nin)( )( )( )( )( )1(\nin)( )(\nini i i i i i i i i ijz z jz zz z       \n,                                    ( 3) \nwhere  φ(i) = k(i)l(i) and \n)()( )( i i iV z are the phase  shift and the material acoustic impedance . \nFor magnetic layers  with magnetoelastic  interaction , the expression ( 3) is not applicable, \nbecause all three roots k21,2,3 of the secular equation  should be taken into account  in the general solution. By matching boundary conditions  at YIG surfaces we ob tain for input acoustic  \nimpedance s zin(3,5) the formula analogous to ( 3) with the corresponding substitutions:  \n2 12 1 )5,3(\n2 121 )5,3( 21 )5,3(cos,2sin,~zzzz\nzzzz zzjz   \n.                            (4) \nHere \n)2/ tg(3\n11 sk zp p\npp\n , \n)2/ ctg(3\n12 sk zp p\npp\n ,   \n\n3\n1~\npp , s = l(3,5), and \np ,\np are the \ncoefficients determined in [ 28].  Thus, the relations ( 2) - (4) allow us to describe the HBAR \nspectrum, and  determine (f,H) dependences  of fn and resonance Qn factor s.   \nLet us now consider the features of acoustic spin pumping in our structure. The time -\naveraged spin current polarized along z, \nn mm j2\nr5)/ ( xx s t g , flows  from YIG layer 5 \ninto Pt layer 6 [21].  Here \nrg is the real part of  spin mixing conductance, \n] /)()( Im[2\n0 5 5*Mxmxmy x\n is the magnetization precession cone angle at YIG/Pt interface x5, \nand n is the normal to the interface. The ISHE in Pt leads to an electrostatic field\n) ( ) (2\nSH ISHE zn zj E   s\n, where  \nSH  is the spin  Hall angle of Pt [22].   For a rectangular \nPt strip, the dc voltage between its ends in the direction a is   \n \n)) (( ) (2\nISHE ISHE azn a E   U .                                                              ( 5) \nIn (5 ), the constants \nrg  and \nSH  are omitted, since their values are considered to be independent \nof the field, frequency, and thickness of YIG and Pt . We also omitted the factor resulting from \nthe current density averaging across the Pt thickness.  \nAfter substitution the general solutions  in magnetic layer 5 to magnetic and elastic \nboundary conditions we obtain  \n\n\n\n\n\n\n\n\n\n qp qpqp qp\nqppq\nxxyxNzzxujmm\n\n3\n1, 2 14\n) (~)(\n5\n,                        (6) \nwhere \n)]2/ (tg)2/ (tg/[)]2/ (tg)2/ ([tg2 2sk sk sk sk Nq p q p qp pq   . An explicit view of the \namplitude coefficients \np  and\np  is given in [2 8] . \n For allowance of the ADSWs back action  on all elastic subsystems , the displacement \nu(x4) should be expressed via an electrical parameter of the transducer, for example, the voltage \nU~\napplied to  the electrodes. The transformation  ) sin cos /( ) ( )()( )1(\nin)( )( )(\n1i i i i i\ni i jz z zxu xu  \n                                       (7) \nis used to express  u(x4) in terms of u(x3).  For the transformation of u(x3) to u(x2) via ( 7) the \nsubstitution rules ( 4) are needed. Finally, from the equations for the  piezoelectric  layer 1 [24], \nwe obtain   \n              \n)] 1)( cos sin ( /[)2/ (sin~2)(AW)1( )1( )1( )1( )1( 2\n2 Z Z izl Uje xu       .                  (8) \nSubstituting ( 4), (6) - (8) in (5) we can get  an analytical expression for UISHE, which is then \nanalyzed numerically.  \n4. Results  and discussions  \nFigures 2 (a),(b) demonstrate the  frequency dependences  of Re[ k1,2,3(f)]  and Im[ k1,2,3(f)] \nfor infinite magnetic media  at H= 740 Oe. We attribute k1, k2 to the continuous magneto elastic  \nbranches, which for f < fMER are quasi AW and SW , but for f > fMER are quasi SW and AW.  The \nthird  root, k3(f), is always imaginary and plays a certain role for the satis faction of boundary \nconditions a s well as the root k2(f) in case  f < fFMR= ω0(0)/2π  [4, 29, 30].  \n \n \nFig. 2. Frequency dependences  of: real (a) and imaginary (b) parts of wave numbers k1 (blue lines 1), k2 \n(red lines 2 ), and k3 (green  lines 3 ); normalized voltages for structures II (c) and I (d) at fixed magnetic \nfield H = 740 Oe. The elastic and magnetic parameters: (111) oriented YIG – V(3,5) = 3.9 ×105 cm/s, ρ(3,5) = \n5.17 g/cm3 [15], b = 4 × 106 erg/cm3, D =4.46 ×10–9 Oe cm2, 4πMeff = 955 G [30],  \nH =0.7Oe; GGG – \nV(4) = 3.57×105 cm/s, ρ(4)  = 7.08 g/cm3; ZnO – V(1)  = 2.88×105 cm/s, ρ(1) = 5.68 g/cm3. The layer \nthicknesses  are given  in caption  to Fig.1  \n \nFigures 2 (c), (d) show the frequency dependence of normalized UISHE(f) at H= 740 Oe \nfor two structures, I and II: with two YIG films (II) (Fig.2(c)) and with only one film 5 (I) \n(Fig.2(d)) . The parameters of the YIG / Pt interface and the Pt itself are not considered in this \ncase and the voltage was normalized to the maximum for the structure I.  As one can see from \nFig. 2(c), (d) the dependences for both struc tures, I and II are similar in the form and differs only \nby the scale.  Note that the absolute maximum of UISHE(f) corresponds not to fMER , but to a lower \nfrequency near fFMR. The local maxima frequencies, as it was shown in [28], coincide with \nHBAR resona nce frequencies.   \nNext, we consider the frequency dependences in a varying magnetic field and compare \nthem with the experimental ones observed previously in [18].  Figure  3(a) shows for the structure \nII the calculated dependence UISHE(f,H), which is in a good agreement with the experimental \nresult s, shown in Fig. 3 (b). The voltage  magnitudes  follow the change in the positi on of the \nresonance frequencies  fn(H) - the red lines in Fig. 3 (a) and the red points in Fig. 3 (b).  Both \ncalculated and experimental UISHE magnitudes have different behavior above and below the line \nfMER(H) (line 1),  which is a consequence of excitation of SW wave s with basically different  \nwavenumbers . Higher than fMER(H) line , the sho rt SW s with wavenumbers more than 5×104 cm-1 \nare excited, whereas below  the line the wavenumbers of excited modes are essentially smaller \n(see Fig.2 (a)).  Fitting of the theoretical and experimental dependences allows  us to evaluate   \nmagnetic parameters b, 4πMeff, and D listed in the Fig.2 (a) caption [31 ].  \n  \n \nFig. 3. 3D colour plot  UISHE(f,H) for the structure II: (a) – calculated, (b) – experimental adopted from \n[18] (http://creativecommons.org./licenses/by/4.0/) . The red lines (a) and points (b) correspond to \npositions of HBAR resonant frequency fn positions. The calculated frequencies fMER(H) and fFMR(H) are \nshown by line 1 and 2 in (a). For calculation the same parameters as listed in Fig.2 were used. They \ncorrespond to the experimental ones.  \nUp to this point, the consideration concerned  rather thick YIG films of about 30 μm.  But \ntoday , much attention is paid to the technology and the study of submicron and nanometer -sized \nYIG films. In particular, the effective magnetoelastic interaction in such films was observed \n[32]. Let us consider  the effect of YIG thickness  l(5) = s on the UISHE  for the structure I. Figure 4 \nshows the  calculated  UISHE(f0 max, s) dependence for a maximum  located at frequency f0 max ≈ \nclosest to fFMR.  With the decrease of s from a few tens microns to a micron, the effect increases \nby an order of magnitude , oscillating with the period ~  0.65 µ m, which  corresponds  to the AW \nhalf-length. In this case, the local  minima  correspond to s = (p+1)V/(2f) and maxima – to s = \n(p+0.5)V/(2f), where p is an integer .  At s ~ 3÷2 µm other excitation  zones  appear at higher \nfrequencies ft max, corresponding to the SW resonance conditions for free spins : k2s ≈ πt, where t \n= 1, 2,  3, ... A detailed description of the behavior of these high order SW  resonance is beyond \nthe scope of this paper.   We just note that at certain s the value  UISHE  (ft max,s) induced by ADSW \nresonances with t ≤ 3 become s larger than the UISHE(f0 max, s)  (as it is shown in the inse rtion). \nWith the decrease s up to 120 nm the frequencies ft max beco me so high  that are shifted out of the \nMER influence region. Finally,  at the s ~ 100 nm only one signal  UISHE  (f0 max, s) remains  in the \nspectrum. With a further thickness decrease, the signal reaches a maximum and then begins to \ndrop.  \nSince in these calculations we did not take into account the additional magnetic damping \ndue to the spin pumping , the dependence on thickness is entirely determined by the method of \nthe SW excitation. It should be noted that at thicknesses s< 200 nm the additional  magnetic \ndamping becomes noticeable .  Assuming that the  FMR linewidth  broadening  ΔH sp~ \nrg /s [21, \n33] one can obtain a steeper curve for the small thicknesses.    \n \n \nFig. 4.  The voltage UISHE  (f0 max) dependence on YIG thickness s at the fixed magnetic field H = \n740 Oe. The insertion:  the spectrum of the signal UISHE  (f, s=0.2 μm) with two zones of maximal \nexcitation near  f0 max   and  f1 max . \n5. Conclusion  \nA semi -analytic  theory  for ADSW in hypersonic composite HBAR  with ZnO -GGG -\nYIG/Pt layered structure is developed . The theoretical and experimental dependenc es of the \nelectric voltage UISHE(f, H) in Pt  are in  good agreement : the significant asymmetry of the \nUISHE(fn(H)) value in reference to the magnetoelastic resonance  line fMER(H) position , \nexperimentally observed  previously , manifest s itself also  in the theoretical calculations. This \nasymmetry is due to the SW spectrum  governed by nonuniform exchange:  near the fFMR the \nefficiency of quasiuniform SW excitation is higher than the efficiency for the frequencies \nexceeding fMER. The theory  involved takes  into account the self -consistent mutual influence of \nthe AW and SW and gives the possibility to evaluate  some magnetic parameters of the YIG films \nincluding exchange stiffness.  The analysis of YIG film thickness inf luence on UISHE at the main \nfrequency f0 max show s that this value reaches a  maximum for thicknesses about 100  nm. Also , \nUISHE maxima due to the high SW resonances at higher frequencies can be detected. So we \nbelieve  that acoustic spin pumping  created by means of HBAR is a sensitive spectroscopic \ntechnique  for the investigation of  magnetic films properties.  \n  \nAcknowledgments  \nThis work was partially supported by grants 16 -07-01210 and 17 -07-01498 from the \nRussian Foundation  for Basic Research.  \nREFERENCES  \n[1] A. S. Salasyuk, A. V. Rudkovskaya, A. P. Danilov, B. A. Glavin, S. M. Kukhtaruk,  M. \nWang,  A. W. Rushforth, P. A. Nekludova,  S. V. Sokolov, A. A. Elistratov,  D. R. Yakovlev,  \nM. Bayer, A. V. 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Luzanov,  S. G. Alekseev, N. I. Polzikova,   Deposition process optimization of zinc  \noxide films with inclined texture axis,   J. Commun. Technol. Electron., 63 (2018) 1076 –1079.      \nDOI: 10.1134/S1064226918090127    \n[21]Y. Tserkovnyak, A. Brataas, G.E.W. Bauer, Enhanced Gilbert damping in thin ferromagnetic \nfilms, Phys. Rev. Lett.,  88   (2002) 117601.  https://doi.org/10.1103/PhysRevLett.88.117601  \n[22] E. Saitoh, M. Ueda, H. Miyajima,  G. Tatara,  Conversion of spin current into charge current \nat room temperature: inverse spi n-Hall effect, Appl. Phys. Lett., 88  (2006) 182509. \n https://doi.org/10.1063/1.2199473  [23] B. A. Auld, “Acoustic Fields and Waves in Solids,” Vol. I, Wiley, New York, 1973 , 423  p. \n[24] D. Royer, E. Dieulesaint, Elastic Waves in Solids II. Generation. Acousto -optic Interaction, \nApplications, Springer -Verlag, Berlin Heidelberg, 2000.  \n[25] In the geometry involved, the demagnetization field \n)0,0, 4(d xmh  contributes only to \nthe ac component of the effective field. The effect of the YIG crystalline anisotropy is not \nconsidered in detail in this work, but it can be taken into account both in Meff and Heff. For (111) \nYIG films orientation, as in Ref. 18, ani sotropy contributes mainly to the Meff, rather than to the \nH eff [26]. In this case t he coupling  constant b is determined by a linear combination of two cubic \nconstants b2 and b1 with coefficients depending on the field direction in the (111) plane [15].  \n[26] S. Klingler, A. V. Chumak, T. Mewes, B. Khodadadi, C. Mewes, C. Dubs, O. Surzhenko, B. \nHillebrands, and A. Conca, Measurements of the exchange stiffness of YIG films using \nbroadband ferromagnetic resonance techniques  Journal of Physics D: Applied Phys ics 48, (2015)  \n015001 https://doi.org/10.1088/0022 -3727/48/1/015001  \n[27]  A. G. Gurevich and G. A. Melkov, Magnetization Oscillations and Waves  (CRC -Press, \nBoca Raton, 1996), p. 464 . \n[28] N.I.Polzikova, S.U.Alekseev, V.A.Luzanov, A.O.Raevskiy, Electroacoustic excitation of \nspin waves and their detection via inverse spin Hall effect, Phys. Solid State, 60 ( 2018) 2211. \nhttps://doi.org/10.1134/S1063783418110252   \n[29] N.M.Salanskii, M.S.Yerukhimov, The Physical Properties and Applications of Magnetic \nFilms, Nauka, Novosibirsk, 1975 (in Russian).  \n[30] H.F.Tiersten, Thickness vibrations of saturated magnetoelastic plates, J.  Appl. Phys., 36 \n(1965) 2250 -2259. https://doi.org/10.1063/1.1714459   \n[31] Note that the reduced magnetization 4π Meff = 955 G is characteristic for La, Ga -substituted \nYIG epitaxial films used in the experiment.  \n[32] Yu. V. Khivintsev, V. K. Sakharov, S. L. Vysotskii, Yu. A. Filimonov, A. I. Stognii, \nS. A. Nikitov, Magnetoelastic waves in submicron yttrium –iron garnet films manufactured by \nmeans of ion -beam sputtering onto gadolinium –gallium garnet sub strates, Technical Phys ., 63 \n(2018) 1029 -1035. https://doi.org/ 10.1134/S1063784218070162  \n[33] M. B. Jungfleisch,  A. V. Chumak,  A. Kehlberger,  V. Lauer,  D. H. Kim,  M. C. Onbasli,  C. \nA. Ross,  M. Kl äui, and B. Hillebrands  Thickness and power dependence of the spin -pumping \neffect in Y3Fe5O12/Pt heterostructures measured by the inverse spin Hall effect , Phys. Rev. B  \n91, (2015)  134407 https://do i.org/10.1103/PhysRevB.91.134407   "    },    {        "title": "2003.13760v4.Tunable_multiwindow_magnomechanically_induced_transparency__Fano_resonances__and_slow_to_fast_light_conversion.pdf",        "content": "arXiv:2003.13760v4  [quant-ph]  23 Sep 2020Tunable multiwindow magnomechanically induced transpare ncy, Fano resonances, and\nslow to fast light conversion\nKamran Ullah,∗M. Tahir Naseem,†and¨Ozg¨ ur E. M¨ ustecaplıo˘ glu‡\nDepartment of Physics, Ko¸ c University, Sarıyer, ˙Istanbul, 34450, Turkey\nWe investigate the absorption and transmission properties of a weak probe field under the in-\nfluence of a strong control field in a cavity magnomechanical s ystem. The system consists of two\nferromagnetic-material yttrium iron garnet (YIG) spheres coupled to a single cavity mode. In addi-\ntion to two magnon-induced transparencies (MITs) that aris e due to magnon-photon interactions,\nwe observe a magnomechanically induced transparency (MMIT ) due to the presence of nonlinear\nmagnon-phonon interaction. We discuss the emergence of Fan o resonances and explain the splitting\nof a single Fano profile to double and triple Fano profiles due t o additional couplings in the pro-\nposed system. Moreover, by considering a two-YIG system, th e group delay of the probe field can\nbe enhanced by one order of magnitude as compared with a singl e-YIG magnomechanical system.\nFurthermore, we show that the group delay depends on the tuna bility of the coupling strength of\nthe first YIG with respect to the coupling frequency of the sec ond YIG, and vice versa. This helps\nto achieve larger group delays for weak magnon-photon coupl ing strengths.\nKeywords: Magnon induced transparency; magnomechanical i nduced transparency; Fano resonances; sublu-\nminal and superluminal effects.\nI. INTRODUCTION\nStoring information in different frequency modes of\nlight has attracted much attention due to its critical role\ninhigh-speed, long-distancequantumcommunicationap-\nplications [1–3]. The spectral distinction of optical sig-\nnals eliminates their unintentional coupling to the sta-\ntionary information or memory nodes in a communica-\ntion network. For that aim, multiple transparency win-\ndow Electromagnetically Induced Transparency (EIT)\nschemes have been considered for multiband quantum\nmemory implementations mainly in the medium ofthree-\nlevel cold atoms. Experimental demonstrations of three\nEIT windows have been reported [4], and extended to\nseven windows using external fields [5]. Observation of\nnineEITwindowshasbeenexperimentallydemonstrated\nquite recently, using an external magnetic field in a va-\npor cell of Rubidium atoms [6]. A practical question is\nif such results can be achieved at higher temperatures,\nfor example, for a room temperature multiband quan-\ntum memory.\nIn recent years, remarkable developments have been\nachieved to strongly couple spin ensembles to cavity pho-\ntons, leading to the emerging field of cavity spintron-\nics. Quanta of spin waves, magnons, are highly ro-\nbust against temperature [7–11], and hence significant\nmagnon-photon hybridization and magnetically induced\ntransparency(MIT) havebeen successfullydemonstrated\neven at room temperature [11]. Tunable slow light and\nits conversion to fast light based upon room tempera-\nture MIT has been theoretically shown recently [12]. Be-\n∗Electronic address: kamran@phys.qau.edu.pk\n†Electronic address: mnaseem16@ku.edu.tr\n‡Electronic address: omustecap@ku.edu.trsides, at strong magnon-photon interaction, a wide tun-\nability of slow light via applied magnetic field has been\nshown in [13]. These results demonstrate the promising\nvalue of these systems for practical quantum memories\n[12]. Here we explore how to split such a MIT window\ninto multiple bands for a room temperature multimode\nquantum memory. Our idea is to exploit the coupling of\nmagnons to thermal vibrations, which is known to yield\nmagnomechanically induced transparency (MMIT) [14],\nin combination with multiple spin ensembles to achieve\nmultiple bands in MIT. We also discuss the emergence of\nFano resonance in the output spectrum and explore the\nsuitable system parameters for its observation. Fano res-\nonance was first reported in the atomic systems [15], and\nit emerges due to the quantum interference of different\ntransition amplitudes which give minima in the absorp-\ntion profile. In later years, it has been discussed in differ-\nent physical systems, such as photonic crystal [16], cou-\npled microresonators [17], optomechanical system [18].\nRecently, Fano-like asymmetric shapes have been exper-\nimentally reported in a hybrid cavity magnomechanical\nsystem [14].\nOur model consists of two ferromagnetic insulators,\nspecifically yttrium iron garnets (YIGs), hosting long-\nlived magnons at room temperature, placed inside a\nthree-dimensional (3D) microwave cavity; we remark\nthat another equivalent embodiment of our model could\nbe to place the YIGs on top of a superconducting co-\nplanar waveguide, which can have further practical sig-\nnificance being an on-chip device [19]. Specific benefits\nof YIG as the host of spin ensemble over other systems,\nsuch as paramagneticspin ensembles in nitrogen-vacancy\ncenters is due to its high spin density of 2 .1×1022µB\ncm−3(µB is the Bohr magneton) and high room tem-\nperature spin polarization below the Curie temperature\n(559 K). In addition to multimode quantum memories,\nour results can be directly advantageous for readily in-2\ntegrated microwave circuit applications at room temper-\nature such as multimode quantum transducers coupling\ndifferentsystemsatdifferent frequencies[20], tunable fre-\nquency quantum sensors [21] or fast light enhanced gyro-\nscopes [22]. In addition to the magnetic dipole interac-\ntion between the cavity field and the spin ensemble, we\ntake into account coupling between the magnons and the\nquanta of YIG lattice vibrations, phonons, arising due\nto the magnetostrictive force [14]. We only consider the\nKittel mode [23] of the ferromagnetic resonance modes\nof the magnons. Such three-body quantum systems can\nbe of fundamental significance to examine macroscopic\nquantum phenomena towards thermodynamic limit and\nquantum to classical transitions [24].\nIn our model, tunable slow and fast light emerges\nas a natural consequence of tunable splitting of MIT\nwindow. Slow-light propagation at room temperature\nhas been investigated recently in a cavity-magnon sys-\ntem and the group delays are found to be in the ∼µs\nrange[12]. InasingleYIGmagnomechanicalsystemwith\nstrong magnon-photon coupling strength, slow-light has\nachieved with a maximum group delay of <0.8 ms [13].\nIn this paper, we discuss the slow and fast light in a two\nYIGs magnomechanical system. Further, we exlain the\ngroup delay depends on the tunability of the magnon-\nphoton coupling of the first YIG (YIG1) with respect\nto magnon-photon coupling of the second YIG (YIG2).\nThisnotonlyhelpstoachievelargergroupdelaysatweak\nmagnon-photon coupling, but also increase the group de-\nlay of the transmitted probe field by one order of magni-\ntude, which is not possible with a single YIG system [13].\nThe rest of the paper is organized as follows: We de-\nscribe the model system in Sec. II and present dynam-\nical equations with steady-state solutions. The results\nand discussions for MMIT are presented in the Sec. III.\nWe discuss the emergence and tunability of the multiple\nFano resonances in Sec. IV. Next, in Sec. V, we present\nthe transmission of the probe field and discuss the group\ndelays for slow and fast light propagation. Finally, in\nSec. VI, we present the conclusion of our work.\nII. SYSTEM HAMILTONIAN AND THEORY\nWe consider a hybrid cavity magnomechanical sys-\ntem that consists of two YIG spheres placed inside a\nmicrowave cavity, as shown in Fig. 1. A uniform bias\nmagnetic field (z-direction) is applied on each sphere,\nwhich excites the magnon modes and these modes are\ncoupled with the cavity field via magnetic dipole inter-\naction. The excitation of the magnon modes inside the\nspheres leads to the variation magnetization that results\nin the deformation of their lattice structures. The mag-\nnetostrictive force causes vibrations of the YIGs which\nestablishes magnon-phonon interaction in these spheres.\nThe single-magnon magnomechanical coupling strength\nis very weak [14], and it depends on the spheres diam-\neters and external bias field directions. Either by con-\nFIG. 1: (color online) A schematic illustration of a hybrid\ncavity magnomechanical system. It consists of two ferromag -\nnetic yttrium iron garnet (YIG) spheres placed inside a mi-\ncrowave cavity. A Bias magnetic field is applied in the zdi-\nrection on each sphere, which excites the magnon modes, and\nthese modes are strongly coupled with the cavity field. The\nbias magnetic fields activate the magnetostrictive (magnon -\nphonon) interaction in both YIGs. The single-magnon mag-\nnomechanical coupling strength is very weak [14], and it de-\npends on the spheres diameters and external bias field direc-\ntions. Either byconsidering a larger YIG1sphere or adjusti ng\nthe direction of the bias field on it, the magnomechanical cou -\npling of this sphere can be ignored. Here, we assume the di-\nrection of the bias field on YIG1 such that the single-magnon\nmagnomechanical interaction becomes very weak and can be\nignored [14]. However, the magnomechanical interaction of\nYIG2 is enhanced by directly driving its magnon mode via a\nmicrowave drive (y direction). This microwave drive plays t he\nrole of a control field in our model. Cavity, phonon, magnon\nmodes are labeled as a,b, andmi(i= 1,2), respectively.\nsidering a larger YIG1 sphere or adjusting the direction\nof the bias magnetic field on it, the magnomechanical\ncoupling of this sphere can be ignored [24]. Here, we\nassume the direction of the bias field on YIG1 such that\nthesingle-magnonmagnomechanicalinteractionbecomes\nvery weak and can be ignored [14]. However, the mag-\nnomechanicalinteractionofYIG2 is enhancedbydirectly\ndriving its magnon mode via a external microwave drive.\nThis microwave drive plays the role of a control field in\nour model. In addition, the cavity is driven by a weak\nprobe field.\nIn this work, we consider high quality YIG spheres, each\nhas a 250 µm diameter, and composed of ferric ions Fe+3\nof density ρ= 4.22×1027m−3. This causes a total spin\nS= 5/2ρVm= 7.07×1014, whereVmis the volume ofthe\nYIG and Sis the collective spin operator which satisfy\nthe algebra; [ Sα,Sβ] =iεαβγSγ. The Hamiltonian of the3\nsystem reads [24]\nH//planckover2pi1=ωaˆa†ˆa+ωbˆb†ˆb+2/summationdisplay\nj=1[ωjˆm†\njˆmj+gj(ˆm†\njˆa+mjˆa†)]\n+gmbˆm†\n2ˆm2(ˆb+ˆb†)+i(Ωdˆm†\n2e−iωdt−Ω⋆\ndˆm2eiωdt)\n+i(ˆa†εpe−iωpt−ˆaε⋆\npeiωpt)\n(1)\nwherea†(a) andb†(b) are the creation (annihilation) op-\nerators of the cavity and phonon modes, respectively.\nThe resonance frequencies of the cavity, phonon and\nmagnon modes are denoted by ωa,ωbandωj, respec-\ntively. Moreover, mjis the bosonic operator of the Kit-\ntle mode of frequency ωjand its coupling strength with\nthe cavity mode is given by gj. The frequency ωjof\nthe magnon mode mjcan be determined by using gy-\nromagnetic ratio γjand external bias magnetic field Hj\ni.e.,ωj=γjHjwithγj/2π= 28 GHz. The Rabi fre-\nquency Ω d=√\n5/4γ√\nNB0[23], represents the coupling\nstrength of the drive field of amplitude B0and frequency\nωd. Furthermore, in Eq. (1), ωpis the probe field fre-\nquency having amplitude εpwhich can be expressed as;\nεp=/radicalbig\n2Ppκa//planckover2pi1ωp.\nNote that in Eq. (1), we have ignored the non-linear\ntermKˆm†\njˆm†\njˆmjˆmjthat mayarisedue to stronglydriven\nmagnon mode [25, 26]. To ignore this nonlinear term,\nwe must have K|/angbracketleftm2/angbracketright|3≪Ω, and for the system pa-\nrameters we consider in this work, this condition always\nsatisfies. The Hamiltonian in Eq. (1) is written after\napplying the rotating-wave approximation in which fast\noscillating terms gj(ˆaˆmj+ ˆa†ˆm†) are dropped. This is\nvalid for ωa,ωj≫gj,κa,κmjwhich is the case to be\nconsidered in the present work. Where κaandκmjare\nthe decay rates of the cavity and magnon modes, respec-\ntively. In the frame rotating at the drive frequency ωd,\nthe Hamiltonian of the system is given by\nH//planckover2pi1=∆aˆa†ˆa+ωbˆb†ˆb+2/summationdisplay\nj=1[∆mjˆm†\njˆmj+gj(ˆm†\njˆa+\nmjˆa†)]+gmbˆm†\n2ˆm2(ˆb+ˆb†)+i(Ωdˆm†\n2−Ω⋆\ndˆm2)+\ni(ˆa†εpe−iδt−ˆaε⋆\npeiδt),\n(2)\nhere, ∆ a=ωa−ωd, ∆mj=ωj−ωd, andδ=ωp−ωd.\nThe quantum Heisenberg-Langevin equations based on\nthe Hamiltonian in Eq. (2) can be written as\n˙ˆa=−i∆aˆa−i2/summationdisplay\nj=1gjˆmj−κaˆa+εpe−iδt+√2κaˆain(t),\n˙ˆb=−iωbˆb−igmbˆm†\n2ˆm2−κbˆb+√2κbˆbin(t),\n˙ˆm1=−i∆m1ˆm1−ig1ˆa−κm1m1+√2κm1ˆmin\n1(t),\n˙ˆm2=−i∆m2ˆm2−ig2ˆa−κm2m2−igmbˆm2(ˆb+ˆb†)\n+Ωd+√\n2κm2ˆmin\n2(t).\n(3)Whereκbis the dissipation rate ofthe phonon mode, and\nˆbin(t), ˆmin\nj(t)and ˆain(t)arethevacuuminputnoiseoper-\nators which have zero mean values i.e., /angbracketleftˆqin/angbracketright= 0 [27, 28],\nand (q=a,m,b). The magnon mode m2is strongly\ndriven by a microwave drive that causes a large steady-\nstate amplitude |/angbracketleftm2s/angbracketright| ≫1ofmagnonmode, and due to\nbeam splitter interaction, this leads to the large steady-\nstate amplitude of the cavity mode |/angbracketleftas/angbracketright| ≫1. Conse-\nquently, we can linearize the quantum Langevin equa-\ntions around the steady-state values and take only the\nfirst-order terms in the fluctuating operator: /angbracketleftˆO/angbracketright=\nOs+ˆO−e−iδt+ˆO+eiδt[29], here ˆO=a,b,m j. First,\nwe consider the zero-order solution, namely, steady-state\nsolutions which are given by\nas=−i/summationdisplay\n1,2gjmjs\nκa+i∆a,\nbs=−igmb|m2s|2\nκb+iωb,\nm1s=−ig1as\nκm1+i∆m1,m2s=Ωd−ig2as\nκm2+i˜∆m2,\n˜∆m2= ∆m2+gmb(bs+b⋆\ns).(4)\nWe assume that the coupling of the external microwave\ndrive on magnon mode m2is much stronger than the\namplitude ǫpof the probe field. Under this assumption,\nthe linearizedquantum Langevinequations can be solved\nby considering the first-order perturbed solutions and ig-\nnoring all higher order terms of ǫp. The solution for the\ncavity mode is given by\na−=εp\nA′+C′\n1+g2\n2\nβ′+α⋆α′\nβ⋆β′+A⋆−C⋆\n1+g2\n2\nβ⋆\n−1\n,\n(5)\nwhere\nA=κa+i(∆a+δ),B=G2\nmbωb\nω2\nb−δ2+iδκb,\nC1=g2\n1\nκm1+i(∆m1+δ),C2=g2\n2\nκm2+i(˜∆m2+δ),\nA′=κa+i(∆a−δ),B′=G2\nmbωb\nω2\nb−δ2−iδκb,\nC′\n1=g2\n1\nκm1+i(∆m1−δ),C′\n2=g2\n2\nκm2+i(˜∆m2−δ),\nα=g2\n2B\nC2+iB,α′=g2\n2B′\nC′\n2+iB′,\nβ=C2−iC′⋆\n2B\nC′⋆\n2+iB,β′=C′\n2−iC⋆\n2B′\nC⋆\n2+iB′.\nHereGmb=i√\n2gmbm2sis the effective magnon-phonon\ncoupling. We use the input-output relation for the cavity\nfieldεout=εin−2κa/angbracketlefta/angbracketright[30], and the amplitude of the\noutput field can be written as4\nFIG. 2: (Color online). AbsorptionRe[ εout] profilesare shown\nagainst the normalized probe field detuning δ/ωb. (a)g1=\ngmb= 0,g2/2π= 1.2 MHz and (b) g1= 0,g2/2π= 1.2 MHz,\nGmb/2π= 2.0MHz(c) g1/2π=g2/2π= 1.2MHz,Gmb/2π=\n2 MHz and (d) g1/2π=g2/2π= 1.2 MHz, Gmb/2π= 3.5\nMHz. The other parameters are given in Sec. III.\nεout=2κaa−\nεp. (6)\nThe real and imaginaryparts of εoutaccount for in-phase\n(absorption) and out of phase (dispersion) output field\nquadratures at probe frequency.\nIII. MMIT WINDOWS PROFILE\nFor the numerical calculation, we use parameters from\na recent experiment on a hybrid magnomechanical sys-\ntem [14], unless stated differently. Frequency of the cav-\nity fieldωa/2π= 10GHz, ωb/2π= 10MHz, κb/2π= 100\nHz,ω1,2/2π= 10 GHz, κa/2π= 2.1 MHz, κm1/2π=\nκm2/2π= 0.1 MHz, g1/2π=g2/2π= 1.5 MHz,\nGmb/2π= 3.5 MHz, ∆ a=ωb, ∆mj=ωb,ωd/2π= 10\nGHz.\nWe first illustrate the physics behind the multiband\ntransparency by systematically investigating the role of\ndifferent couplings in the model. Fig. 2 displays the\nresponse of the probe field in the absorption spectrum\nof the output field for different coupling strengths. In\nFig. 2(a), we assume the magnon-phonon coupling ( gmb)\nand magnon mode m1coupling ( g1) with the cavity are\nabsent. Therefore, only magnon mode m2is coupled\nwith the cavity. Under these considerations, we observe\na magnon induced transparency (MIT) in which a typi-\ncal Lorentzian peak of the output spectrum of the simple\ncavity splits into two peaks with a single dip, as shown\nin Fig. 2(a). The width of this transparency window can\nbe controlled via microwave driving field power and the\nmagnon-photon coupling g2. On increasing the coupling\nstrength g2the width of the window increases, and vice\nversa.FIG. 3: (Color online) Dispersion Im[ εout] profiles are shown\nagainstthenormalizedprobedetuning δ/ωb. (a)g1=gmb= 0\nandg2/2π= 1.2 MHz and (b) g1= 0,g2/2π= 1.2 MHz,\nGmb/2π= 2.0 MHz (c), (d) g1/2π=g2/2π= 1.2 MHz, and\n(c)Gmb/2π= 2 MHz and (d) Gmb/2π= 3.5 MHz. The other\nparameters are given in Sec. III.\nWe observe two transparency windows in the absorp-\ntion as we switch on the magnon-phonon coupling ( gmb)\nand keeping g1= 0. Due to the non-zero magnetostric-\ntive interaction, single MIT window in Fig. 2(a) splits\ninto double window shown in Fig. 2(c). The right trans-\nparency window in Fig. 2(c) is associated with magnon-\nphonon interaction, and this is so called magnomechani-\ncallyinducedtransparency(MMIT)[14]window. Wecan\nobserve double MIT by removing magnon-phonon cou-\nplinggmb, and considering non-zero couplings between\nthe magnon modes and the cavity field.\nFinally, if we consider all three couplings simultane-\nously non-zero, then the transparency window splits into\nthree windows consist of four peaks and three dips, this\nis shown in Fig. 2(c). In this case, one window is as-\nsociated with the magnomechanical interaction, and the\nrest of the two are induced by magnon-photon couplings.\nThe width and peaks separation of these windows in-\ncreases and broadens, respectively, at higher values of\nmagnon-phonon coupling Gmb, which can be seen in\nFig. 2(d). Moreover, we have a symmetric multi-window\ntransparency profile where the splitting of the peaks oc-\ncurs at side-mode frequencies ωp=ωb±ωd.\nIn Figs. 3(a-d), we plot the dispersion spectrum of the\noutput field versus normalized frequency of the probe\nfield. The single MIT dispersion spectrum in the absence\nof YIG1 and magnon-phonon coupling gmbis shown in\nFig. 3(a). The dispersion spectra for the case of g1= 0,\ng2/negationslash= 0 and gmb/negationslash= 0 is plotted in the Fig. 3(b). In the\npresence of all three couplings, the dispersion spectrum\nof the output field is given in the Figs. 3(c-d). It is clear\nfromFigs.3(c-d), bytheincreaseintheeffectivemagnon-\nphononcoupling Gmb, the transparencywindowsbecome\nwider. We like to point out that the magnomechanically\ninduced amplification (MMIA) of the output field, in5\nour system, can be obtained in the blue detuned regime;\n∆m2=−ωb.\nFIG. 4: (Color online) Fano line shapes in the asymmetric\nabsorption Re[ εout] profiles are shown against the normalized\nprobe frequency δ/ωb. (a) ∆ m2= 0.7ωb,g2= 1.5 MHz,\ng1=gmb= 0, and (b) ∆ m2= 0.7ωb,g1= 0,g2= 1.5 MHz,\nGmb= 3.5 MHz. (c) ∆ m1,2= 0.7ωb,g1=g2/2π= 1.5\nMHz and Gmb/2π= 3.5MHz, and (d) ∆ m1,2=ωb,g1=\ng2/2π= 1.5 MHz and Gmb/2π= 3.5 MHz. In all panels,\ng1=g2/2π= 1.5 MHz,Gmb/2π= 3.5 MHz, and rest of the\nparameters are give in Sec. III.\nIV. FANO RESONANCES IN THE OUTPUT\nFIELD\nIn the following, we discuss the emergence and phys-\nical mechanism of the Fano line shapes in the output\nspectrum. The shape of the Fano resonance is distinctly\ndifferentthanthesymmetricresonancecurvesintheEIT,\nMIT, optomechanically induced transparency (OMIT)\nand MMIT windows [14, 31]. Fano resonance has ob-\nservedinthesystemsinwhichEIThasreportedbyasuit-\nable selection of the system parameters [14, 31–36]. The\nphysical origin of Fano resonance in the systems having\noptomechanical-likeinteractionshasexplained dueto the\npresence of non-resonant interactions. For example, in a\nstandard optomechanical system, if the anti-Stokes pro-\ncess is not resonant with the cavity frequency, asymmet-\nric Fano shapes appear in the spectrum [31–33]. In our\nsystem, thiscorrespondsto∆ m1/negationslash=ωb, becauseinsteadof\na cavity mode, magnon mode m1is coupled with phonon\nmodeviaoptomechanical-likeinteraction. Theasymmet-\nric Fano shapes can be seen in Figs. 4(a-c) for differ-\nent non-resonant cases, where the absorption spectrum\nof the output field as a function of normalized detuning\nδ/ωbis shown. In Fig. 4(a), we consider g1=gmb= 0,\nand coupling of the magnon mode m2with the cavity is\nnon-zero. Due to the presence of non-resonant process\n(∆m2= 0.7ωm), the absorption spectrum of the sym-\nmetric MIT (Fig. 2(a)) profile changes into asymmetricFIG. 5: (Color online). The transmission |tp|2spectrum as a\nfunction of normalized probe field frequency δ/ωbis shown for\ndifferent values of g1. (a)g1/2π= 0.5 MHz (b) g1/2π= 0.8\nMHz (c) g1/2π= 1.2 MHz (d) g1/2π= 1.5 MHz. In all\npanels,g2/2π= 1.5 MHz,Gmb/2π= 3.5 MHz and the other\nparameters are given in Sec. III.\nwindow profile, as shown in Fig. 4(a). Such asymmetric\nMIT band can be related to Fano-like resonance, emerg-\ning frequently in optomechanical systems [31–35]. If we\nremove YIG1 and consider only YIG2 is coupled with\nthe cavity mode, and ∆ m2= 0.7ωm. We observe double\nFano resonance in the output spectrum, which is shown\nin Fig. 4(b). Similarly, in the presence of all three cou-\nplings and ∆ m1,m2= 0.7ωm, the double Fano resonance\ngoes over to a triple Fano profile, as shown in Fig. 4(c).\nThis is because the cavity field can be build up by three\ncoherent routes provided by the three coupled systems\n(the magnons, cavity, and phonon modes), and that can\ninterferewith eachother. TheFanoresonancesdisappear\nwhen we consider a resonant case ∆ m1= ∆m2=ωb, as\nshown in Fig. 4(d).\nV. NUMERICAL RESULTS FOR SLOW AND\nFAST LIGHTS\nHere we investigate the transmission and group delay\nof the output signal, and show the effect of the magnon-\nphoton and magnon-phonon couplings on the transmis-\nsion spectrum. From Eq. (6), the rescaled transmission\nfield corresponding to the probe field can be expressed as\ntp=εp−2κaa−\nεp. (7)\nIn Figs. 5(a-d), we plot the transmission spectrum of the\nprobe field against the scaled detuning δ/ωb, for different\nvalues of g1. It is clear from Fig. 5(a), the transmis-\nsion peak associated with the magnon-photon coupling\nof YIG1 is smaller than the other two peaks. This is be-\ncause in Fig. 5(a) g1coupling is weaker than the other6\ntwointeractions g2andGmbpresentinthesystem. Byin-\ncreasing the coupling strength g1, the peak of the middle\ntransparency profile grows up in height and reaches close\nto unity, as shown in Figs. 5(b-c). In addition, Fig. 5(d)\nshows that the width of the transparency window can be\nincreased at higher higher values of the magnon-photon\ncoupling g1.\nFIG. 6: (Color online). The transmission |tp|2spectrum as a\nfunction of normalized probe field frequency δ/ωbis shown.\n(a)Gmb/2π= 0.5 MHz (b) Gmb/2π= 1.0 MHz. In (c)\ng2/2π= 0.4 MHz, and (d) g2/2π= 0.8 MHz. The other\nparameters are same as in Fig. 5.\nIn Figs. 6(a-b), the transmissionspectrum ofthe probe\nfield as a function of dimensionless detuning is shown for\ndifferent values of Gmb. In Figs. 6(a-b), we consider both\ng1andg2to be the same in the strong coupling regime.\nHowever, the effective coupling ˜ g2=g2αsdepends on\nthe steady-state amplitude of the cavity field αswhich\ndepends on the m2s. Consequently, ˜ g2andGmbare re-\nlated and it can be seen from Eq. (4). For a smaller value\nofGmbin Fig. 6(a), we have two small peaks associated\nwithg2andGmb, in addition, the third-highest peak is\nassociatedwith g1. Forafixedvalueof gmb, ifweincrease\nGmb, it increases ˜ g1, and the peaks associated with these\ntwocouplingsbecome morevisible, asshownin Fig. 6(b).\nSimilarly, in Fig. 6(c-d), we observe a similar increase in\nthe height of two peaks associated with g2andGmb, for\nthe variation in g2.\nThe phase φtof the transmitted probe field tpis given\nby the relation φt= Arg[tp]. The plot of φtas a func-\ntion of normalized detuning δ/ωbis shown in Fig. 7.\nIn the inset of Fig. 7(a), we consider both g1andgmb\nare switched off, and only g2is non-zero. This gives a\nconventional phase of the transmitted field with a sin-\ngle MIT curve, which appears similar to the standard\nsingle OMIT curve [31]. In Fig. 7(b), we switch-off the\nYIG1 coupling with the field ( g1= 0), and the other two\ncouplings are present ( gmb/negationslash= 0,g2/negationslash= 0), due to which\nthe single transparency window splits into a double win-\ndow. Ifwekeepallthree couplingsnon-zero,wegettripletransparency window which is shown in Fig. 7(c).\nFIG. 7: (Color online) The phase φtof the transmitted probe\nfield versus normalized detuning δ/ωbfor different coupling\nstrengths. (a) g1=gmb= 0, (b) g1= 0,g2/2π= 1.5\nMHz,Gmb/2π= 4 MHz (c) g1/2π=g2/2π= 1.5 MHz, and\nGmb/2π= 4 MHz. Rest of parameters are given in Sec. III.\nFIG. 8: (Color online) Group delay τgof the output probe\nfield against the amplitude of the magnetic field B0for (a)\ng1= 0, and (b) g1/2π= 1.5 MHz. The other parameter are\ng2/2π= 1.5,Gmb/2π= 3.5 MHz,κb/2π= 100 Hz, κm1/2π=\nκm2/2π= 0.1 MHz,κa/2π= 2.1 MHz and Ω d= 1.2 THz.\nThe transmitted probe field phase is associated with\nthe group delay τgof the output field and it is defined as\nτg=∂φ(ωp)\n∂ωp, (8)\nwhich means a more rapid phase dispersion leads to a\nlarger group delays and vice versa. In addition, a nega-\ntive slope of the phase represents a negative group delay\nor fast light ( τg<0) whereas, a positive slope of the\ntransmitted field indicates positive group delay or slow\nlight (τg>0). From Fig. 7, we observethat in the regime\nof the narrowtransparency window, there is a rapid vari-\nation in the probe phase, and this rapid phase dispersion\ncan lead to a significant group delay.\nFig. 8 shows the group delay τgcan be tuned by the\nvariation of the bias magnetic field B0applied on YIG2.\nIn the absence of YIG1 (Fig. 8(a)), we have a lower slope\nof Eq. (8), as a result, a maximum group delay of τg= 1\nms is achieved. This group delay can be enhanced by one\norder of magnitude once second YIG is introduced see\nFig. 8(b). The slope of Fig. 8(b) become steeper and the\ntime delay for slow light is increased up to 13.8 ms. This\nshows the two YIGs system is a good choice to observe a\nlonger group delay in a magnomechanical system while a\nsingleYIG systemcannotdoso. Moreover,thenumerical\nvalue of the group delay τgcan be tuned from positive7\n(slow light) to negative (fast light) by tuning the magnon\nfielddetuning∆ m1=ωbto∆m1=−ωb. Here, it isworth\nmentioning that Fig. 8(b) can be switched into fast light\nwith a maximum group delay in the order of τg≈ −1.4\nms in the presence of both YIGs and we not show in\nthe figure. This negative group delay for the fast light\npropagation is one order of magnitude greater than a\nsingle YIG magnomechanical system [13].\nFIG. 9: (Color online) The group delay τgof the transmitted\nprobe field as a function of the driving power Pdfor several\nvalues of the magnon-photon couplings. (a) ∆ m1=ωb, (b)\n∆m1=−ωb, and the other parameter are same as given in\nFig. 8.\nFinally, we investigate the control of group delay\nwith the external microwave driving power and magnon-\nphoton couplings. For this purpose, in Figs. 9(a-b), we\nplotτgagainst the driving power for different strengths\nof the magnon-photon coupling of YIG1 with respect to\nthe coupling frequency of YIG2. Fig. 9(a) shows that the\nmagnitude of the group delay increases with the increase\nofg1corresponding to g2, which indicates an enhanced\ngroup delay of the transmitted probe field in a two-YIG\nsystem. We tuned the coupling strength of YIG1 ( g1)\nfor different values via keeping the coupling strength of\nYIG2 (g2) constant. This shows increasing the magnon-\nphoton coupling strength increases the group delay of\nthe transmitted probe field and vice versa. This helps us\nto obtain larger group delays at relatively weak magnon-\nphotoncouplingstrengthswhichisnototherwisepossible\nwith a single YIG magnomechanical system [13]. Similar\nresults can also be obtained by increasing the magnon-\nphoton coupling g2and fixing g1. For the blue detuned\nregime∆ m1=−ωb, groupdelaybecomes negative. How-\never, the effect of magnon-photon couplings remains the\nsame, as shown in Fig. 9(b). From Fig. 8 and Fig. 9,\nwe see that two YIGs magnomechanical system providesnot only extra tunability, but also drastically enhances\nthe group delays compared to single YIG system studied\nin Ref. [13]. Our system can be used as a tunable switch,\nwhich can be controlled via different system parameters,\nand our results are comparable with the existing propos-\nals based on various hybrid quantum systems [37–40].\nVI. CONCLUSION\nWe have investigated the transmission and absorption\nspectrum of a weak probe field under a strong control\nfield in a hybrid magnomechanical system in the mi-\ncrowave regime. Due to the presence of a nonlinear\nphonon-magnon interaction, we observed magnomechan-\nically induced transparency (MMIT), and the photon-\nmagnon interactions lead to magnon induced trans-\nparency (MIT). We found single MMIT, a result of the\nsingle-phonon process, and found two MIT windows in\nthe output probe spectra due to the presence of two\nmagnon-photon interactions. This is demonstrated by\nplotting the absorption, dispersion, and transmission of\ntheoutput field. WediscussedtheemergenceofFanores-\nonances in the output field spectrum of the probe field.\nThese asymmetric line shapes appeared due to the pres-\nenceofanti-Stokesprocessesinthesystem. We examined\nconditions of slow and fast light propagation in our sys-\ntem, which can be controlled by different system param-\neters. It has shown that in a two YIGs magnomechan-\nical system, the tunability of the first coupling strength\n(YIG1) corresponding to the coupling of the second YIG\n(YIG2) has an immense effect on the slow and fast light\nand vice versa. This not only helped to investigate larger\ngroup delays at a weak magnon-photon coupling but also\nenhanced the group delay of the transmitted probe field,\nwhich is not possible in a single YIG system. 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Eltsov1,c)\n1)Low Temperature Laboratory, Department of Applied Physics, Aalto University,\nPOB 15100, FI-00076 AALTO, Finland\n2)Department of Physics, Lancaster University, Lancaster LA1 4YB,\nUK\n(Dated: 19 December 2023)\nUnder suitable experimental conditions collective spin-wave excitations, magnons,\nform a Bose-Einstein condensate (BEC) where the spins precess with a globally co-\nherent phase. Bose-Einstein condensation of magnons has been reported in a few\nsystems, including superfluid phases of3He, solid state systems such as Yttrium-\niron-garnet (YIG) films, and cold atomic gases. Among these systems, the super-\nfluid phases of3He provide a nearly ideal test bench for coherent magnon physics\nowing to experimentally proven spin superfluidity, the long lifetime of the magnon\ncondensate, and the versatility of the accessible phenomena. We first briefly recap\nthe properties of the different magnon BEC systems, with focus on superfluid3He.\nThe main body of this review summarizes recent advances in application of magnon\nBEC as a laboratory to study basic physical phenomena connecting to diverse areas\nfrom particle physics and cosmology to new phases of condensed matter. This line\nof research complements the ongoing efforts to utilize magnon BECs as probes and\ncomponents for potentially room-temperature quantum devices. In conclusion, we\nprovide a roadmap for future directions in the field of applications of magnon BEC\nto fundamental research.\na)jere.makinen@aalto.fi\nb)s.autti@lancaster.ac.uk\nc)vladimir.eltsov@aalto.fi\n1arXiv:2312.10119v1  [cond-mat.quant-gas]  15 Dec 2023I. INTRODUCTION\nSpin waves are a general feature of magnetic materials. Their quanta are called magnons,\nspin-1 quasiparticles that obey bosonic statistics. At sufficiently large number density and\nlow temperature, magnons form a Bose-Einstein condensate (BEC), akin to neutral atoms in\nsuperfluid4He or ultracold gases. The magnon BEC is manifested as spontaneous coherence\nof spin precession across a macroscopic ensemble in both frequency and phase even in the\npresence of incohering forces resulting from, e.g., magnetic field gradients.\nBose-Einstein condensation of magnons (in the form of a homogeneously precessing do-\nmain , HPD) was first observed in 19841in nuclear magnetic resonance (NMR) experiments\nin the superfluid B phase of3He. The experiments discovered a spontaneous formation\nof coherent precession of magnetization within a superfluid with antiferromagnetic ground\nstate. In the early experiments, spontaneous coherence of magnons was manifested in pulsed\nNMR measurements2in a very peculiar manner. At first the amplitude of the free induction\ndecay signal dropped, as expected for a dephasing set of local oscillators in an inhomoge-\nneous magnetic field. However, shortly after the signal amplitude was restored and, even\nmore surprisingly, the observed ringing time was a few orders of magnitude longer than the\ninitial dephasing time! These observations were explained using the terminology of spin\nsuperfluidity that acted as the mechanism establishing spontaneous coherence.\nLater experiments demonstrated spin supercurrent between two homogeneously precess-\ning domains connected by a channel3, and showcased the spin current analog of the DC\nJosephson effect4. Further signatures of spin superfluidity and macroscopic coherence in-\nclude the observation of topological objects, spin vortices5, and the collective Goldstone\nmodes, i.e. oscillations of the phase of precession6,7. These modes can be treated as phonons\nin a time crystal, discussed in Sec. III B.\nTreating the coherent spin precession as a Bose-Einstein condensate of magnons allows\nfor a simplified description of the system. The condensate wave function amplitude describes\nthe density of magnons and the phase the precession of magnetization. The magnon BEC\ndiffers from the atomic BECs in one important respect: magnons are quasparticles and,\nthus, their number is not conserved. In a thermodynamic equilibrium, magnon chemical\npotential is always zero, µeq≡0, and thus no equilibrium BEC of magnons can exist.\nIf the lifetime of magnons τNis much larger than the thermalization time τEwithin the\n2Pump\n...Thermalization\nBECDecayMagnonFIG. 1. Creation of the magnon BEC. In a typical scheme quanta of spin-wave excitations\n(magnons) are pumped into a higher energy level by e.g. a radio-frequency pulse. The pumped\nmagnons then thermalize with time constant τE. The magnon decay from the ground state is\ncharacterized by the decay time τN. Under sufficiently strong pumping, or if the magnon decay\ntime is much longer than the thermalization time, τN≫τE, a macroscopic number of magnons\noccupy the ground state of the system, forming a BEC.\nmagnon subsystem, i.e., τN≫τE, or if magnons are continuously pumped into the system,\nthe magnon condensate obtains a nonzero µand forms a lowest-energy-level condensate\nanalogous to a BEC as illustrated in Fig. 1. These conditions are well met e.g. in3He-B,\nwhere the thermalization time is a fraction of a second while the lifetime of the free coherent\nprecession and the corresponding magnon BEC can reach tens of minutes at the lowest\ntemperatures8.\nMagnon condensation in the lowest energy level(s) occurs when the system-specific critical\nmagnon density ncis exceeded. Roughly speaking, nccorresponds to the point when the\ninter-magnon separation becomes comparable to the thermal de Broglie wavelength λdB,\ni.e. when n−1/3\nc∼λdB. At this point the chemical potential of the magnon system µ\nasymptotically approaches the ground state energy ϵ0and the ground state population\nn0=1\ne(ϵ0−µ)/kBT−1(1)\ndiverges. Here kBis the Boltzmann constant and Tis temperature. As a result, the ground\nstate becomes populated by a macroscopic number of constituent particles that sponta-\n3neously form a macroscopically coherent state that is stable against decohering perturba-\ntions.\nBose-Einstein condensates of non-conserved quasiparticles are ubiquitous in nature; sim-\nilar phenomenology is used to describe systems consisting of phonons9, rotons10, photons11,\nexcitons12, exciton-polaritons13, etc. To date BECs consisting of magnons, in particular,\nhave been reported in various superfluid phases of3He14–16, in cold atomic gases17,18, and in\na few solid-state systems19,20. Moreover, the antiferromagnetic hematite ( α-Fe2O3) has been\nput forwards as a promising candidate system for condensation of magnons21,22. We note\nthat while the concept of magnon BEC is also useful for describing the onset of magnetic-\nfield-induced magnetic order in spin-dimer compounds23, such as TlCuCl 3, the excitations\nin such systems are in thermal equilibrium and therefore the chemical potential is always\nzero, i.e. µ= 0. While the phenomenology is similar24, such systems are outside the scope\nof this Perspective.\nThe nature and experimental realization of the magnon BEC varies widely between differ-\nent physical systems. In1H, a magnon BEC was created at high magnetic field of multiple\ntesla by initially preparing a dense cloud of cold gas in a higher-energy low-field-seeking\nspin state, while magnons are created by pumping a number of atoms to a lower-energy,\nhigh-field-seeking state17. The magnons become collective excitations via an effect known\nas ”identical spin rotation”, where a single spin-flip is carried through multiple atom-atom\ncollisions25–27. In the87RbF= 1 spinor condensate the magnon (quasi-) BEC results from\nspin-exchange collisions between different internal spin states in the same hyperfine mani-\nfold at low magnetic fields18. In superfluid phases of3He, magnons are collective excitations\nstemming from the Nambu-Goldstone modes of the underlying order parameter28. Finally,\nout of the solid state systems where magnon BEC has been realized, perhaps the most\npromising systems are thin Yttrium-iron-garnet (YIG) films, where the magnon condensate\nis created either by continuous radio frequency pumping19or by laser-induced spin currents29\nat room temperature in highly non-equilibrium state and, moreover, in the momentum space\ninstead of in real space as for the other examples.\nRegardless of the system, a magnon BEC can be viewed as the ground state of the\nrelevant subsystem, which is an essential viewpoint in the context of time crystals. For\nexample, if one ignores the weak non-conservation of magnons in3He-B caused by the spin-\norbit interaction, the ground state of the subsystem would evolve in time, realizing the time\n4crystal as originally suggested by Wilczek30. However, due to conservation of particles the\ntime evolution of such a system is unobservable31–34. A similar scenario is realized in atomic\nBECs as well as in superconductors (in superconductors the coherent precession is that of\nAnderson pseudospins), where the evolution of the superfluid phase is unobservable. In\nprinciple, for long enough measurement times, the phase should become observable due to\nproton decay. In some cases, such as in1H and in3He, the absolute value of the phase can be\ndirectly monitored in real time as the phase of the precessing magnetization can be measured\nby oriented pickup coils which provide the necessary loss channel. Such direct measurement\nof the absolute value of the phase of the macroscopic wave function is rather uncommon and\ncan be exploited for a variety of purposes, as discussed later in this Perspective.\nThe spontaneously formed coherent precession of magnetization has many faces: spin\nsuperfluidity, off-diagonal long-range order (ODLRO), the Bose-Einstein condensation of\nnonequilibrium (pumped) quasiparticles, and time crystal. In Section II, we briefly describe\nthe basic properties of the magnon BEC. Section III focuses on magnon-BEC time crystals.\nMagnon BECs can be utilized to simulate different processes and objects in particle physics,\nsuch as spherical charge solitons (Section IV), the light Higgs particle (Section V) and\nanalogue event horizons (Section VI). Section VII concentrates on probing topological defects\nusing magnon BEC and, finally, Section VIII contains an outlook on future prospects of\nmagnon BECs in various systems.\nII. COHERENT PRECESSION AND SPIN SUPERFLUIDITY\nA. Off-diagonal long-range order\nThe phenomenon of Bose-Einstein condensation was originally suggested by Einstein for\nstable particles with integer spin. Under suitable conditions, the BEC gives rise to macro-\nscopic phase coherence and superfluidity, first observed in liquid4He. This a consequence\nof the spontaneous breaking of the global U(1) gauge symmetry related to the conservation\nof the particle number N, e.g. of4He atoms.\nAs distinct from many other systems with spontaneously broken symmetry, such as crys-\ntals, liquid crystals, ferro- and antiferromagnets, the order parameter in superfluids and\nsuperconductors is manifested in the form known as the off-diagonal longe-range order\n50 0.5 1 1.5 2-0.2-0.100.10.2Mx(arb. un.)\ntime (s)\nba\ncpumping\npulseFIG. 2. Observing magnon Bose-Einstein condensation: (a) Magnons are pumped to the\nsystem with a radio-frequency pulse at zero time, seen as the sharp peak in the data. As illus-\ntrated in the central panel on a colored background, the pumping is followed by dephasing of the\nprecession. If the magnon density is high enough, a BEC emerges after τE, manifest in coherent\nprecession of magnetization Mx+iMy∝D\nˆS+E\n=√\n2S⟨ˆa0⟩=S⊥eiωt. This is picked up by the\nNMR coils and measured as an oscillating voltage. Magnetic relaxation in superfluid3He is very\nslow, and the number of magnons Ndecreases with time constant τN(here τN∼10 s), seen as a\nslow decrease in the signal amplitude, shown in panel ( b). Panel ( c) shows a further zoom-in into\nthe band indicated by the green line. Here, the sinusoidal pick-up signal generated by the preces-\nsion of magnetization is clearly seen. Data shown in this figure was measured at 0 bar pressure and\n131µK temperature35.\n6(ODLRO). In bosonic superfluids (such as liquid4He) the manifestation of the ODLRO\nis that the average values of the creation and annihilation operators for the particle number\nare nonzero in the superfluid state, i.e.,\nΨ =D\nˆΨE\n,Ψ∗=D\nˆΨ†E\n. (2)\nIn conventional (i.e. not superfluid or superconducting) states the creation or annihilation\noperators have only the off-diagonal matrix elements, such asD\nN|ˆΨ†|N+ 1E\n, describing\nthe transitions between states with different number of particles. In the thermodynamic\nlimit N→ ∞ , the states with different numbers of particles in the Bose condensate are\nnot distinguished, and the creation or annihilation operators acquire the nonzero average\nvalues. In superconductors and fermionic superfluids such as superfluid3He, the ODLRO is\nrepresented by the average value of the product of two creationD\na†\nka†\n−kE\nor two annihilation\n⟨aka−k⟩operators, which reflects the Cooper pairing in fermionic systems. In quantum the-\nory, states with nonzero values of the creation or annihilation operators are called squeezed\ncoherent states.\nB. ODLRO and coherent precession\nThe magnetic ODLRO can be represented in terms of magnon condensation, applying\nthe Holstein-Primakoff transformation. The spin operators are expressed in terms of the\nmagnon creation and annihilation operators\nˆa0s\n1−¯ha†\n0a0\n2S=ˆS+\n√\n2S¯h,s\n1−¯ha†\n0a0\n2Sˆa†\n0=ˆS−\n√\n2S¯h, (3)\nˆN= ˆa†\n0ˆa0=S − ˆSz\n¯h. (4)\nEq. (4) relates the number of magnons Nto the deviation of spin Szfrom its equilibrium\nvalueS(equilibrium)\nz =S=χHV/γ , where χandγare spin susceptibility and gyro-magnetic\nratio, respectively. Pumping Nmagnons into the system (e.g. by a RF pulse) reduces the\ntotal spin projection by ¯ hN, i.e.Sz=S −¯hN. The ODLRO in magnon BEC is given by:\n⟨ˆa0⟩=N1/2eiωt+iα=r\n2S\n¯hsinβ\n2eiωt+iα, (5)\nwhere βis the tipping angle of precession. The role of the chemical potential µis played by\nthe global frequency of the coherent precession ω, i.e. µ≡¯hωand the phase of precession\n7αplays the role of the phase of the condensate, i.e. Φ ≡α. A typical experimental signal\nshowing an exciting pulse, the formation of the BEC and the slow decay is shown and\nanalysed in Fig. 2. The experimental setup used in this particular experiment is shown in\nFig. 3. Note that the analogy with atomic BECs is valid only for the dynamic states of the\nmagnetic subsystem and not e.g. for static magnets with ω= 0.\nC. Gross-Pitaevskii and Ginzburg-Landau description\nAs for atomic Bose condensates, the magnon BEC is described by the Gross-Pitaevskii\nequation. The local order parameter is obtained by extension of Eq. (5) to the inhomoge-\nneous case, ˆ a0→ˆΨ(r, t), and is determined as the vacuum expectation value of the magnon\nfield operator:\nΨ(r, t) =D\nˆΨ(r, t)E\n, n=|Ψ|2,N=Z\nd3r|Ψ|2. (6)\nwhere nis the magnon density.\nIf the dissipation and pumping of magnons are ignored, the corresponding Gross-\nPitaevskii equation has the conventional form:\n−i¯h∂Ψ\n∂t=δF\nδΨ∗, (7)\nwhere F{Ψ}is the free energy functional forming the effective Hamiltonian of the spin\nsubsystem. In the coherent precession, the global frequency is constant in space and time\nΨ(r, t) = Ψ( r)eiωt, (8)\nand the Gross-Pitaevskii equation transforms into the Ginzburg-Landau equation with ¯ hω=\nµ:\nδF\nδΨ∗−µΨ = 0 . (9)\nThe free energy functional reads\nF −µN=Z\nd3r\u00121\n2gik∇iΨ∗∇kΨ + ¯h(ωL(r)−ω)|Ψ|2+Fso(|Ψ|2)\u0013\n, (10)\nwhere ωLis the local Larmor frequency ωL(r) =γH(r) and gikdescribes rigidity of the\nmagnon system. The spin-orbit interaction energy Fsois a sum of contributions proportional\n8NMR excitation\nand pick upMagnon BEC\nIminH\nM3HeB\nˆntexture\nspin orbit\nenergyrz\nZeeman energyβ\nN1N2\nω2\nω1N1>N2\nω1< ω 2FIG. 3. Magnon BEC in magneto-textural trap in superfluid3He. The magnetization M\nof the condensate is deflected by an angle βfrom the direction of magnetic field Hand precesses\ncoherently around the field direction with the frequency ω. Magnons are confined to the nearly\nharmonic 3-dimensional trap formed by the spatial variation of the field H(r) via Zeeman energy\nand by the spatial variation (texture) of the orbital anisotropy vector ˆn(r) of3He-B via spin-orbit\ninteraction energy. The orbital texture is flexible and yields with increasing number of magnons\nNin the trap, resulting in lower radial trapping frequency. As a result, the chemical potential,\nobserved as the precession frequency, decreases. This leads to inter-magnon interaction (Sec. II C)\nand eventually to Q-ball formation (Sec. IV A).\nto|Ψ|2and|Ψ|4, see e.g. Ref. 36. Thus, the free energy functional can be compared with\nthe conventional Ginzburg-Landau free energy of an atomic BEC:\nF −µN=Z\nd3r\u00121\n2gik∇iΨ∗∇kΨ + ( U(r)−µ)|Ψ|2+b|Ψ|4\u0013\n(11)\nwith the external potential U(r) formed by the magnetic field profile and a part of the spin-\norbit interaction energy. The fourth order term, which describes the interaction between\nmagnons, originates from the rest of spin-orbit interaction Fso(|Ψ|2). Fig. 3 illustrates the\nappearance of the inter-magnon interaction via flexible orbital texture in the case of trapped\nmagnon BEC.\n9The gradient energy in Eq. (11) is responsible for establishing coherence across the sample.\nIn the London limit, it can be expressed via gradients of the precession phase α\nFgrad=1\n2gik∇iΨ∗∇kΨ =1\n2Kik∇iα∇kα=1\n2n\u0000\nm−1\u0001\nik∇iα∇kα . (12)\nA necessary condition for spin superfluidity and phase coherence is that the gradient energy is\npositively determined. This condition is not universally valid in all systems with magnons,\nbut is applicable, e.g., in3He-B. The spin superfluid currents are then generated by the\ngradient of the phase. The rigidity tensor Kikcan be further expressed via magnon mass\nmik(n), which in general depends on the magnon density nand is anisotropic due to applied\nmagnetic field37.\nIII. MAGNON BEC AS A TIME CRYSTAL\nOriginally, time crystals were suggested as class a quantum systems for which time trans-\nlation symmetry is spontaneously broken in the ground state30. It was quickly pointed\nout that the original concept cannot be realized and observed in experiments, essentially\nbecause that would constitute a perpetual motion machine31–34. That is, if the system is\nstrictly isolated, i.e. when the number of particles Nis conserved, there is no reference\nframe for detecting the time dependence38. This no-go theorem led researchers to search for\nspontaneous breaking of the time-translation symmetry on more general grounds, turning to\nout-of-equilibrium phases of matter (see e.g. reviews 39–41). With this adjustment, feasible\ncandidates of time-crystal systems include those with off-diagonal long range order, such as\nsuperfluids42, Bose gases43, and magnon condensates34.\nAny system with ODLRO can be characterized by two relaxation times34: the lifetime of\nthe corresponding (quasi)particles τNand the thermalization time τEduring which the BEC\nis formed. If τN≫τE, the system has sufficient time to relax to a minimal energy state with\n(quasi-)fixed N(i.e. to form the condensate). During the intermediate interval τN≫t≫τE\nthe system has finite µcorresponding to spontaneously-formed uniform precession that can\nbe directly observed as shown in Fig. 2. In3He-B τNcan reach tens of minutes at the lowest\ntemperatures8– this is the closest an experiment has got to a time crystal in equilibrium.\nFinally, we point out that in the grand unification theory extensions of Standard Model\nthe conservation of the number of atoms is absent due to proton decay44. Therefore, in\n10principle, the oscillations of an atomic superfluid in its ground state can be measured, albeit\nthe time scale for the decay is at least in the ∼1036years range44.\nA. Discrete and continuous magnon time crystals\nTime crystals are commonly divided into two broad categories based on their symmetry\nclassification. If, relative to the Hamiltonian before the phase transition to the time crystal\nphase, the system spontaneously breaks the discrete time translation symmetry, it is called a\ndiscrete time crystal . Such a system is realized e.g. in parametric pumping scenarios, when\nthe periodicity of the formed time crystal differs from that of the drive. On the other hand,\nif the spontaneously broken symmetry is the continuous time translation symmetry (the\npreceding Hamiltonian is not periodic in time), the system is a continuous time crystal . Note\nthat both discrete and continuous time crystals may still possess a discrete time translation\nsymmetry.\nBoth types of time crystals have been observed in magnon BEC experiments35,45–48. Dis-\ncrete time crystals are realized under an applied RF drive, when the frequency of the co-\nherent spin precession deviates from that of the drive. If the induced precession frequency\nis incommensurate with the drive, the system obtains the characteristics of a discrete time\nquasicrystal. On the other hand, if the magnon number decay is sufficiently slow, i.e.,\nτ−1\nN≪ω, where ωis the frequency of motion of the time crystal, the coherent precession\ncan be observed for a long time after the drive has been turned off. This is a continuous time\ncrystal, which can generally be formed in magnon BEC materials as τ−1\nE< ωby definition.\nIn3He-B continuous time crystals reach life times longer than 107periods35.\nB. Phonon in a time crystal\nSpontaneous breaking of continuous time translation symmetry in a regular crystal results\nin the appearance of the well-known Nambu-Goldstone mode – a phonon. Similarly, the\nspontaneous breaking of time translation symmetry in a continuous time crystal should lead\nto a Nambu-Goldstone mode, manifesting itself as an oscillation of the phase of the periodic\nmotion of the time crystal, Fig. 4a. This mode can be called a phonon in the time crystal.\nIn time crystals formed by magnon BECs, the phononic mode is equivalent to the Nambu-\n110 0.1 0.2 0.3 0.4 0.5 0.6\nHrf (µT)024681012141618M2\nNG (103 Hz2)\nexperiment\ntheory no fit0.43 Tc\n0.64 Tc60 80 100 120\nModulation freq. (Hz)-50510Modulation\nresp. (arb.un.)\nMNGabsorption\ndispersion\nωNG|k=π\nLa\nb\nccrystal\ntime\ncrystalx\nt\nωωNGMFIG. 4. Phonon in a magnon-BEC time crystal. (a) In a crystal in a ground state, atoms\noccupy periodic locations in space (empty circles), while phonon excitation results in a periodic\nshift from these positions (filled circles). A time crystal is manifested by a periodic process (thin\nline), and a phonon excitation leads to periodic variation of the phase of that process (thick line).\n(b) In a magnon-BEC time crystal, the periodic process is the precession of magnetization Mat\nfrequency ω. The phonon excitation modulates the precession phase at the frequency ωNG(right).\nThis mode can be excited by applying modulation to the rf drive of the condensate and observed by\ndetecting response in the induction signal from the pick-up coil at the modulation frequency (left).\nTwo standing-wave modes in the sample are visible (marked with the vertical dashed lines). ( c)\nAs the measurements on the panel (b) are done with finite rf excitation of the amplitude Hrf, the\nphonon becomes a pseudo-Nambu-Goldstone mode with the mass MNG, Eq. (13). Extrapolation to\nthe freely evolving time crystal at zero Hrfshows that the phonon becomes massless, as expected\nfor spontaneous symmetry breaking. The measurements are done at two temperatures at a pressure\nof 7.1 bar in the polar phase of3He15.\n12Goldstone mode related to spin-superfluid phase transition49. It is easier to excite in exper-\niments when the spin precession is driven by a small applied rf field Hrfby modulating the\nphase of the drive, Fig. 4b. In this case the time translation symmetry is already broken\nexplicitly by the drive, and the phonon becomes a pseudo-Nambu-Goldstone mode with the\nmass (gap) MNG. Its dispersion relation connecting the wave vector kand the frequency\nωNGbecomes\nω2\nNG=M2\nNG+c2\nNGk2. (13)\nFor the sample size of L, standing-wave resonances can be seen for k=nπ/L , where nis\nan integer, Fig. 4b, and the mass MNGand the propagation speed cNGcan be determined.\nAccording to the theory MNG∝Hrf– experiments in the polar phase of3He demonstrate\nexcellent agreement without fitting parameters15, Fig. 4c. This mode was also observed in\ntime crystals formed by magnon BEC in the B phase of3He6,7. Extrapolation of the mass to\nthe case of a freely evolving time crystal at Hrf= 0 leads to a true massless Nambu-Goldstone\nmode – a phonon in a time crystal.\nC. Interacting time crystals\nInteracting time crystals have been realized in3He-B by creating two continuous time\ncrystals with close natural frequencies close to each other45,46. In a magneto-textural trap\nsuch as used in Refs. 35,45,46 the radial trapping potential is provided by the spin-orbit\ninteraction via the spatial order parameter distribution. The magnetic feedback of the\nmagnon number to the order parameter texture means the time crystal frequency (period)\nis regulated internally, ωB(NB).50The frequency increases as the magnon number slowly\ndecreases (the decay mechanism is not important but details can be found in Refs. 51,52).\nThe second time crystal is created against the edge of the superfluid where such feedback is\nsuppressed. The result is a macroscopic two-level system described by the Hamiltonian\nH= ¯h\nωB[NB(t)]−Ω\n−Ω ωS\n (14)\nwhere the coupling Ω is determined by the spatial overlap of the time crystals’ wave functions.\nIn this configuration, the two time crystals may interact by exchanging the constituent\nquasiparticles. The exchange of magnons results in opposite-phase oscillations in the re-\n13spective magnon populations of the two time crystals (Fig. 5) which is equivalent to the AC\nJosephson effect in spin superfluids53.\nFurther two-level quantum mechanics can be accessed by making the precession frequen-\ncies of two time crystals cross during the experiment using the dependence ωB(NB). The\nresult is magnons moving from the ground state to the excited state of the two-level Hamil-\ntonian in a Landau-Zener transition, see Fig. 6. Remarkably, these phenomena are directly\nobservable in a single experimental run, including the chemical potentials and absolute\nphases of the two time crystals, implying that such basic quantum mechanical processes are\nalso technologically accessible for magnonics and related quantum devices.\nIV. MAGNON BEC AND COSMOLOGY\nIn the context of particle physics and cosmology, the magnon BEC provides a laboratory\ntest bench for otherwise inaccessible or convoluted theoretical concepts. This phenomenology\nmay eventually play a major role in the technological toolbox for magnonics, albeit potential\napplications cannot yet be predicted. Here we will discuss the analogs between trapped\nmagnons and two cosmological concepts: the Q-ball and the MIT bag model.\nA. Magnonic Q-ball\nSelf-bound macroscopic objects encountered in everyday life are made of fermionic matter,\nwhile bosons mediate interactions between and within them. Compact (self-bound) objects\nmade purely from interacting bosons may, however, be stabilized in relativistic quantum field\ntheory by conservation of an additive quantum number Q54–56. Spherically symmetric non-\ntopological Q-charge solitons are called Q-balls. They generally arise in charge-conserving\nrelativistic scalar field theories.\nObserving Q-balls in the Universe would have striking consequences beyond support-\ning supersymmetric extensions of the Standard Model57,58– they are a candidate for dark\nmatter58–61, may play a role in the baryogenesis62and in formation of boson stars63, and\nsupermassive compact objects in galaxy centers may consist of Q-balls64. Nevertheless, un-\nambiguous experimental evidence of Q-balls has so far not been found in cosmology or in\nhigh-energy physics. Analogues of Q-balls have been speculated to appear in atomic BECs\n14a\nMs\nMb\n1 2 3 4 5\ntime (s)242628303234frequency (Hz)time (s)b\n1 2 3 4 5160180200220240260280chemical potential / h (Hz)\ncsurface (MS)bulk (MB)side band\nside bandFIG. 5. Josephson (Rabi) population oscillations between two magnon-BEC time crys-\ntals. (a) We can create two local minima for magnon-BEC time crystals, one in the bulk and\none against a free surface of the superfluid. ( b) Both are populated with a pulse at zero time,\nafter which the bulk frequency is slowly changing due to changes in the trap shape as the magnon\nnumber slowly decreases. The two levels are coupled, resulting in Josephson population oscillations\nbetween them, observed as the side bands above and below the main traces. The side band fre-\nquency separation (green arrow), shown by the green line in panel c, corresponds to the separation\nof the main traces (red arrow), shown by the red line in panel c. This ties the population oscillation\nto the chemical potential difference of the time crystals and thus to Josephson oscillations. The\noscillations the bulk population and the surface population are shown to take place with opposite\nphases in Ref. 45. The Josephson oscillations become Rabi oscillations if the two-level frequencies\nare brought close to one another as explained in Ref. 46.\n15Tunneling through \navoided crossingFIG. 6. Landau-Zener tunneling between two magnon-BEC time crystals. One of the\ntwo levels is populated at time zero with an exciting pulse (framed out to emphasise the rest\nof the signal). The chemical potential of this state increases gradually as the magnon number\ndecays, crossing the second level after 3 s. As the avoided crossing is traversed at finite rate\n(not adiabatically), a part of the magnon population tunnels to the excited level at the avoided\ncrossing46.\nin elongated harmonic traps65and possibly play a role in the3He A-B transition puzzle66.\nAdditionally, the properties of bright solitons in 1D atomic BECs67and Pekar polarons in\nionic crystals68bear similarities with Q-balls.\nThe trapped magnon BEC in3He-B provides a one-to-one implementation of the Q-ball\nHamiltonian. The charge Qis the number of magnons and the BEC precession frequency\ncorresponds to the frequency of oscillations of the relativistic field within the Q-ball. Above\na critical magnon number, the radial trapping potential for the magnons changes from\nharmonic to a ”Mexican-hat” potential. The modification is eventually limited by the un-\nderlying profile of the magnetic field (see Fig. 7). Here, the systems’ Hamiltonian mimics\nthat of the Q-ball. All essential features of Q-balls, including the self-condensation of bosons\ninto a spontaneously formed trap, long lifetime, and propagation in space across macroscopic\ndistances (here several mm) have been demonstrated experimentally as shown in Fig. 7.69\n16a\nb\ncFIG. 7. Magnon BEC as a self-propagating Q-ball soliton. (a) The magnon BEC is created\nwith a pulse (not shown) with a very large initial magnon number. The time-dependent frequency\nspectrum of the recorded signal is shown here in such a way that dark corresponds to no magnons\nand yellow to large magnon population. In the beginning (red dot and panel b) the BEC (orange\nline and orange blob) is located in a side minimum, where the trapping potential (blue line, blue\nsurface) is modified down to the unyielding trap component controlled by the magnetic field (black\nline, dark surface). As the magnon number decreases due to slow dissipation, the trapping potential\nevolves and the BEC gradually moves across several millimeters to the symmetric central position\n(magenta dot and panel ( c)). Here the BEC is illustrated by the green line and blob.69\nB. Magnons-in-a-box – The MIT Bag model\nThe confinement mechanism of quarks in colorless combinations in quantum chromody-\nnamics (QCD) is an open problem. One of the most successful phenomenological models,\n17u\nd da b\nTrue vacuumFIG. 8. Magnon BEC as a bosonic analogue to the MIT bag. a In the limit of large\nmagnon density the magnon BEC carves a potential well (white), described by the charge field\nΦ(r), in the neutral field ζwhich plays the role of the true vacuum. bIn the context of QCD, the\nquarks carve a potential well (false vacuum, white) in the true QCD vacuum, illustrated here for\nthe charge-neutral neutron consisting of two down-quarks (labeled d) and one up-quark (labeled\nu).\ncoined MIT bag model70as per the affiliation of its inventors, assumes a step change from\nzero potential within the confining region to a positive value elsewhere, a cavity surrounded\nby the QCD vacuum. The cavity is filled with false vacuum, in which the confinement is\nabsent and quarks are free, thus creating the asymptotic freedom of QCD. Outside the cavity\nthere is the true QCD vacuum, which is in the confinement phase, and thus a single quark\ncannot leave the cavity. Within the cavity quarks occupy single-particle orbitals, and there\nthe zero point energy compensates the pressure from true vacuum.\nA similar situation is realized for a magnon BEC, if the magnetic maximum applied in the\nQ-ball experiment discussed in the previous Section is removed. Under these conditions the\nmagnon BEC forms a self-trapping box analogously to the MIT bag model50,71, c.f. Fig. 8.\nThe flexible Cooper pair orbital momentum distribution ˆlplays either the role of the pion\nfield or the role of the non-perturbative gluonic field, depending on the microscopic structure\nof the confinement phase.\nMuch like quarks, magnons dig a hole in the confining ”vacuum”, pushing the orbital field\naway due to the repulsive interaction. The main difference from the MIT bag model is that\nmagnons are bosons and may therefore macroscopically occupy the same energy level in the\ntrap, forming a BEC, while in MIT bag model the number of fermions on the same energy\nlevel is limited by the Pauli exclusion principle. The bosonic bag becomes equivalent to the\n18fermionic bag in the limit of large number of quark flavors due to bosonization of fermions.\nThis phenomenon has been observed in cold gas experiments for SU( N) fermions72.\nV. LIGHT HIGGS BOSONS: PARTICLE PHYSICS IN MAGNON BEC\nBoth in the Standard Model (SM) of particle physics and in condensed matter physics the\nspontaneous symmetry breaking during a phase transition gives rise to a variety of collective\nmodes. This is how the Higgs boson arises from the Higgs field in the Standard Model, for\nexample. The gapless phase modes related to the breaking of continuous symmetries are\ncalled the Nambu-Goldstone (NG) modes, while the remaining gapped amplitude modes are\ncalled the Higgs modes. In superfluid3He, we can make the magnon BEC interact with\nother collective modes, implementing scenarios that in the Standard Model context may\nrequire years of measurements using a major collider facility.\nThe superfluid transition in3He takes place via formation of Cooper pairs in the L= 1,\nS= 1 channel, for which the corresponding order parameter is a complex 3 ×3 matrix\ncombining spin and orbital degrees of freedom. Thus,3He possesses 18 bosonic degrees of\nfreedom, both massive (amplitude or Higgs modes) and massless (phase or Nambu-Goldstone\nmodes), see Fig. 9. The 14 Higgs modes have masses (energy gaps) of the order of the\nsuperfluid gap ∆ /h∼100 MHz, where his the Planck constant, while the four Nambu-\nGoldstone modes (a sound wave mode and three spin wave modes) are massless at this\nenergy scale. Higgs modes have been investigated for a long time theoretically73–75and\nexperimentally76–78in3He-B as well as s-wave superconductors79,80and ultracold Fermi\ngases81.\nAt low energies, the superfluid B phase of3He breaks the relative orientational freedom\nof the spin and orbital spaces, and the resulting order parameter (at zero magnetic field)\nbecomes\nAαj= ∆eiφRαj(ˆn, θ), (15)\nwhere the rotation matrix Rαjdescribes the relative orientation of the spin and orbital\nspaces; the spin space is obtained from the orbital space by rotating it by angle θwith\nrespect to vector ˆn. If the spin-orbit interaction is neglected or, equivalently, one considers\nenergy scales of the order of the superfluid gap ∆, the order parameter obtains an additional\n(”hidden”) symmetry with respect to spin rotation. That is, the energy is degenerate with\n19(a)\n(b) (c)\n(d)FIG. 9. Collective modes and decay channels. (a) The collective mode spectrum in3He-B\ncontains six separate branches of collective modes. The 14 gapped Higgs modes (orange) are: four\ndegenerate pair-breaking modes with gap 2∆ /h∼100 MHz, five imaginary squashing modes with\ngapp\n12/5∆, and five real squashing modes with gapp\n8/5∆. The gapless modes are a sound\nwave mode (oscillations of Φ, yellow), a longitudinal spin wave mode (oscillations of θ, purple), and\ntwo transverse spin wave modes (oscillations of ˆn, green). (b) The longitudinal spin wave mode\nacquires a gap of Ω L/h∼100 kHz due to spin-orbit interaction and becomes a light Higgs mode.\nThe transverse spin wave modes are split by the Zeeman effect in the presence of a magnetic field\ninto optical and acoustic magnons. The arrows indicate possible decay channels. (c) The spatial\nextent of the optical magnon BEC in a typical experiment is of the order of a millimeter, and can\nbe used as a probe for quantized vortices. (d) In a container of fixed size R, the spin wave modes\nform standing wave resonances.\nrespect to ˆnandθ.\nThe spin-orbit interaction lifts the degeneracy with respect to θ, and the minimum energy\ncorresponds to a rotation between the spin and orbital spaces by the Leggett angle θL=\narccos( −1/4)≈104◦. Due to this broken symmetry, one of the Nambu-Goldstone modes (the\nlongitudinal spin wave mode) obtains a gap with magnitude equal to the Leggett frequency\nΩL/h∼100 kHz. In the B-phase the longitudinal spin wave mode therefore becomes a\nlight Higgs mode. Additionally, the presence of the magnetic field breaks the degeneracy\nof the transverse spin wave modes, one of which becomes gapped by the Larmor frequency\nωL=|γ|H, where γis the gyromagnetic ratio in3He. The gapped transverse spin wave\nmode is called optical and the gapless mode acoustic. Throughout the manuscript, the term\n”magnon BEC” in the context of3He-B refers to a BEC of optical magnons.\n20Magnons in the BEC can be converted into other collective modes in the system. For\nexample, the decay of the optical magnons of the BEC into light Higgs quasiparticles has\nbeen observed via a parametric decay channel in the absence of vortices82, and via a direct\nchannel in their presence83(see Sec. VII) as illustrated in Fig. 9. The parametric decay\nchannel is directly analoguous to the production of Higgs modes in the Starndard Model.\nThe separation of the Higgs modes in3He-B into the heavy and light Higgs modes poses\na question whether such a scenario would be realized in the context of the Standard Model\nas well. In particular, we note that the observed 125 GeV Higgs mode84–86is relatively light\ncompared to the electroweak energy scale and, additionally, later measurements at higher\nenergies87show another statistically significant resonance-like feature at the electroweak\nenergies of ≈1 TeV related by the authors to possible Higgs pair-production. Entertaining\nthe possibility of a3He-B-like scenario, the observed feature could stem from formation\nof a ”heavy” Higgs particle; in this case the 125 GeV Higgs boson would correspond to a\npseudo-Goldstone (or a ”light” Higgs) boson, whose small mass results from breaking of\nsome hidden symmetry (see e.g. Ref. 88 and references therein).\nVI. CURVED SPACE-TIME: EVENT HORIZONS\nThe properties of the magnon BECs have also been utilized to study event horizons. In\nthe conducted experiment89, two magnon BECs were confined by container walls and the\nmagnetic field in two separate volumes connected by a narrow channel, Fig. 10. The channel\ncontains a restriction, controlling the relative velocities of the spin supercurrents traveling\nin the bulk fluid and the spin-precession waves traveling along the surfaces of the magnon\nBEC.\nThe magnitude and direction of the spin superflow is controlled by the phase difference of\nthe two magnon BECs, both of which are driven continuously by separate phase-locked volt-\nage generators. The phase difference controls the spin supercurrents, while spin-precession\nwaves are created by applied pulses. For a sufficiently large phase difference, spin-precession\nwaves propagating opposite to the spin superflow are unable to propagate between the two\nvolumes and instead are blocked by the spin superflow. This situation is analogous to a\nwhite-hole event horizon.\n21HPD HPDSpin\nsuperflow\nSpin-\nprecession\nwavesSource DetectionFIG. 10. Magnon BECs and event horizon. In the experiment two volumes filled with\nsuperfluid3He-B, in which the magnetization precesses uniformly (HPD) are connected by a narrow\nchannel. An imposed phase difference between the precessing HPDs creates a spin supercurrent\nproportional to the phase difference. For sufficiently large magnitude of the spin superflow, counter-\npropagating spin-precession waves (surface-wave-like excitations of the HPD) can not propagate\nbetween the two volumes, analogously to the white-hole horizon.\nVII. MAGNON BEC AS A PROBE FOR QUANTIZED VORTICES\nMagnon BEC has proved itself as a useful probe of topological defects, especially quan-\ntized vortices. Vortices affect both the precession frequency of the condensate through\nmodification of the trapping potential for magnons90and the relaxation rate of the con-\ndensate through providing additional relaxation channels83,91–93. Trapped magnon BECs\nprovide a way to probe vortex dynamics locally and down to the lowest temperatures94–96,\nwhere there are still many open questions related to vortex dynamics, see e.g. Refs. 97,98.\nThe effect of vortex configuration on the textural energy, which determines the radial\nmagnon BEC trapping frequency, may be written as90\nFv=2\n5amH2λ\nΩZ\nd3r\u0010\nωv·ˆl\u00112\nωv, (16)\nwhere amis the magnetic anisotropy parameter, Ω is the angular velocity, λis a dimensionless\nparameter characterizing vortex contribution to textural energy, and ω=1\n2⟨∇ × vs⟩is the\nspatially averaged vorticity.\nThe vortex contribution to the textural energy may be introduced via a dimensionless\nparameter λ, which is contains the contributions from the orienting effect related to the\nsuperflow and the vortex core contribution. While the equilibrium vortex configuration is\n22well understood99, the mechanism of dissipation in the zero-temperature limit, in particular,\nremains an open problem.\nIf the equilibrium vortex configuration is perturbed, i.e. ω:=Ω+ω′, where Ωis the\nequilibrium vorticity and ω′is a random contribution with ⟨ω′⟩= 0, parameter λis replaced\nby an effective value\nλeff=λ1 + (ωv∥/Ω)2−(ωv⊥/Ω)2\np\n1 + (ωv∥/Ω)2+ 2(ωv⊥/Ω)2. (17)\nHere ωv∥andωv⊥are the random contributions along the equilibrium orientation and per-\npendicular to it, respectively. This effect has been observed in experiments95by introducing\nvortex waves via modulation of the angular velocity around the equilibrium value and mon-\nitoring the precession frequency (i.e. the ground state energy) of the magnon BEC, see\nFig. 11. Vortex core contribution can be extracted separately from the measured magnon\nenergy levels by comparing measurements with and without vortices90.\nBased on numerical 1D calculations using the uniform vortex tilt model from Ref. 100,\nwhere all vortices are tilted relative to the equilibrium position by the same angle, the\nmagnon BEC ground state frequency, see Fig 12 a, is found to scale as\n∆f≈ −f0sin2θ , (18)\nwhere the sensitivity f0∼100 Hz is found to scale linearly with the vortex core size, see\nFig. 12 b. Using Eq. (18) one can then extract the average tilt angle of vortices within the\nvolume occupied by the magnon BEC from the measured frequency shift. This method has\nbeen utilized for probing transient vortex dynamics95.\nWhen quantized vortices penetrate the magnon BEC, like in Fig. 9c, they also contribute\nto enhanced relaxation of the condensate. Distortion of the superfluid order parameter\naround vortex cores opens direct non-momentum-conserving conversion channels of optical\nmagnons from the condesate to other spin-wave modes, predominantly light Higgs101, see\nFig. 9b. The decay rate of macroscopic (mm-sized) condensate depends on the internal\nstructure of microscopic (100 nm-sized) vortex core. This effect has been used in experiments\nto distinguish between axially symmetric and asymmetric double-core vortex structures91\nand to measure the vortex core size83.\n23FIG. 11. Probing vortex dynamics with magnon BEC. a Vortex waves can be excited by\napplying perturbation in the form of angular modulation (top) on a steady vortex lattice. The\nvortex configuration is monitored locally with two separate magnon BECs (’Upper BEC’ and\n’Lower BEC’), which allows extracting time scales relevant for turbulence buildup. The angular\ndrive results in decreased trapping frequency of the magnon BEC due to the ωv⊥term in Eq.(17). b\nAs expected, the extracted value for λeffdecreases monotonously with increasing drive amplitude.\nBoth data are measured under the same experimental conditions with the same relative drive\nfrequency ω/Ω0, where Ω 0is the mean angular velocity during the drive. Inset shows the definition\nof the tilt angle θwith respect to the axis of rotation. cSchematic illustration of the vortex\n(red) array evolution in response to the angular drive and the eventual relaxation after the drive\nis stopped.\nVIII. OUTLOOK\nBose-Einstein condensates of magnon quasiparticles have been realized experimentally in\ndifferent systems including solid-state magnetic materials, dilute quantum gases, and super-\n2400.20.40.60.81\nVortex tilt angle, sin2100200300400500Radial trap frequency fr, Hz\n100 150 200 250\nVortex core size, nm100200300400500Tilt sensitivity f0, Hza b\nfr = (466 -303, sin2), Hz\nP = 4.1 barFIG. 12. Magnon BEC as a probe for vortex configuration. a The radial trapping potential\nfor a magnon BEC, originating from the textural configuration in a cylindrical trap, scales with the\nvortex tilat angle roughly as fr=ωr/2π∝sin2θ, where θis the tilt angle of vortices relative to the\naxis of rotation. The dashed line is a linear fit to the numerically calculated frequency shift using\nexperimentally determined value for ( λ/Ω)|θ=0at 4.1 bar pressure (vortex core size ∼170 nm).\nThe numerical model assumes uniform vortex tilt. bThe tilt sensitivity f0, calculated using the\nmeasured ( λ/Ω)|θ=0at all pressures, is found to scale with roughly linearly with the vortex core\nsize (1 + Fs\n1/3)ξ0, where Fs\n1is the first symmetric Fermi-liquid parameter and ξ0is the T= 0\ncoherence length.\nfluid3He. Spin superfluidity of those condensates and phenomena such as the Josephson\neffect may be viewed as analogs of superconductivity in the magnetic domain. Supercon-\nducting quantum electronics, based on Josephson junctions, is one of the most important\nplatforms for quantum technologies, however requiring dilution refrigerator temperatures\nof about 10 mK. Coherent magnetic phenomena are generally more robust to temperature\nthan superconductivity with existing room-temperature demonstrations of magnon BEC,\nspin supercurrents and the magnon Josephson effect19,102,103. Thus one of the strong axes of\nresearch on magnon condensates is the development of practically useful (super-)magnonic\ndevices operating at ambient conditions22,104–106. In this Perspective, we have shown that\nthere is another important dimension of magnon BEC applications as a laboratory to study\nfundamental questions in various areas of physics from Q-balls and Higgs particles to time\ncrystals and quantum turbulence. These fundamental phenomena can and should be utilized\nin future magnon-based devices.\n25Advances outlined in this Perspective have triggered suggestions for further applications\nof magnon BEC for fundamental research. In particular, magnon BEC has been proposed to\nbe used as a dark matter axion detector107,108via the coupling between axions and coherently\nprecessing spins. The coupling gives rise to a term in the Hamiltonian of the BEC that looks\nlike an effective magnetic field, oscillating with the frequency depending on the axion mass.\nWhen the frequency of precession matches the frequency of the axion field, potentially\nobservable glitches in the magnon BEC decay are expected. The change of the frequency of\nthe magnon BEC during decay due to inter-magnon interactions (see an example in Fig. 7a)\nallows to probe a continuous range of the axion masses.\nMagnon BECs could also be used as a source and a detector of spin currents in particular\nto probe composite topological matter at the interface of superfluid3He and graphene109.\nAtoms of3He do not penetrate through a graphene sheet, but coupling of the spin degrees\nof freedom of two superfluids on the opposite sides of the sheet immersed in the liquid is\nnevertheless possible through the excitations of the graphene itself (electronic or ripplons)\nor via magnetic coupling of the quasiparticles living at the interface between graphene and\nhelium superfluid, including Majorana surface states. As in the original observation of\nthe spin Josephson effect in3He4, two magnon condensates separated by a channel can be\nmaintained at the controlled phase difference of the magnetization precession, which drives a\nspin current through the channel similar to Fig. 10. In this case, instead of the constriction,\none would place a graphene membrane across the channel and find whether the Josephson\ncoupling is nevertheless observable.\nEven a single magnon BEC placed in contact with the surface of topological superfluid\ncan provide valuable information on the topological surface states. Trapping a condensate\nwith the magnetic field profile, like in Fig. 3, allows to move the BEC around the fluid\nsimply by adjusting the trapping field. Preliminary measurements show that magnon loss\nfrom BEC is significantly enhanced when the condensate is brought from bulk to the surface\nof3He-B sample110. Future experiments should clarify whether this relaxation increase is\ncaused by Majorana surface states111,112. The ability to manipulate Majorana states would\ncome with applications in quantum information processing.\nDevelopment of optical lattices opened many new areas of physics for probing in cold\natom experiments113. So far experiments on magnon BECs were limited to one or two\ncondensates. An exciting development will be to form magnon condensates on a lattice to\n26probe solid-state physics in magnetic domain, perhaps utilizing spinor cold gases114on a\n2D lattice of elongated trapping tubes18created by superimposing two orthogonal standing\nwaves of attractive lase beams, or in solid-state systems where one may similarly utilize\noptical beamshaping techniques to directly print a 2D lattice and impose spin currents29.\nFor magnon BEC in superfluid3He, multiple regularly arranged traps can potentially be\nformed using orbital part of the trapping potential, Fig. 3. Array of the quantized vortices,\ncreated by rotation, modulate the orbital degrees of freedom of superfluid3He and at certain\nconditions may form individual magnon traps around the vortex cores115. The spacing of the\nlattice sites can then be regulated by rotation velocity. This will change coupling between\ncondensates at individual sites which will allow to observe, for example, a superfluid-Mott\ninsulator transition116but for spin superfluid.\nExcitations of the magnon BEC, in particular its Nambu-Goldstone mode (phonon of a\ntime crystal) provide ample possibilities to model propagation of particles in curved space in\nacoustic-metric type experiments117. In such models, effective metric is created by the fluid\nflow which is externally controlled, while the dispersion relation of the propagating modes is\nusually nature-given, like gravity waves on water118–120. For magnon BEC, remarkably, the\nspectrum of Goldstone bosons can typically be adjusted in a wide range by external magnetic\nfield, while the spin flow is formed by the phase of the coherent precession controlled with\nrf pumping. Thus non-trivial metrics can be realized even without using geometrical flow\nconditioners (like a channel in Fig. 10) and cases impossible for phonons or ripplons in\nclassical fluids can be achieved. For example, in a magnon BEC in the polar phase of\n3He, the propagation speed of the Goldstone boson is controlled by the angle of the static\nmagnetic field with the orbital anisotropy axis of the superfluid. The mode can be brought\nto a complete halt at a critical angle, and beyond this angle the metric changes signature\nfrom Minkowski to Euclidean121. Using magnon BEC one can potentially study instability\nof quantum vacuum in such signature transition.\nMagnon BECs make one of the most versatile implementations of time crystals that also\ncomes the closest to the ideal time crystal of all systems in the laboratory. Expanding on the\nexperiments summarized in this Perspective, one may pose fundamental questions such as is\nit possible to melt a time crystal into a time fluid, is it possible to seed time crystallization122,\nor how time crystals interact with different types of matter. The time crystal description\nof magnon BECs also emphasizes potential for quantum magnonics applications: the mag-\n27nitude and phase of the wave function of a single magnon-BEC time crystal, or that of a\nmulti-level composite system of time crystals, is directly accessible in experiments, revealing\nbasic quantum mechanical processes such as Landau-Zener transitions and Rabi oscillations\nin a non-destructive measurement in real time. These can therefore be harnessed unimpeded\nfor also technological applications.\nAdditionally, physics similar to (and beyond!) that outlined in this perspective can per-\nhaps be studied in systems for which the experimental realization is yet to come. One\npromising system is the superfluid fermionic spin-triplet quantum gas, which could be real-\nized by synthetic gauge fields e.g. through Rashba-coupling scenarios123, by tuning into a\np-wave Feshbach resonance124, or perhaps by induced interactions125. In the context of the\nweak-coupling theory, the B-like phase is always expected to be the lowest energy superfluid\nphase28. Therefore, it is expected that the spin-orbit interaction opens up a gap for one of\nthe Goldstone modes, giving rise to the light Higgs mode. We note that such a scenario\nallows for unique research directions as the spin-orbit coupling strength is likely controllable\ne.g. via the Rashba coupling strength, via the amplitude of magnetic field, or via density of\nthe inducing component, depending on the experimental setup.\nAnother interesting research project would be to study the properties of magnon BECs\nin a (putative) spin-triplet superconductor, such as UTe 2, see e.g. Ref. 126 and references\ntherein. The order parameter of UTe 2may take multiple forms, including the one whose\nd-vector representation is ˆd(k)∝(0, ky, kz). Such an order parameter corresponds to the\nB3girreducible representation of the D2hpoint symmetry group in UTe 2and to the so-called\nplanar phase28in the context of3He. In3He the planar phase is predicted to never be the\nlowest energy phase, as its energy always lies between the B phase and the polar phase28.\nDue to the presence of the discrete point symmetry group, similar argumentation may not\napply in UTe 2, making this a unique possibility to study yet another novel topological phase,\nincluding its collective modes such as spin waves and by extension the magnon BEC. The\nstrength of the spin-orbit coupling in UTe 2remains an open question126, but it is expected\nto be non-zero and quite possibly significant (bare uranium has a large spin-orbit interaction\nstrength). As long as the spin-orbit coupling is non-zero, a gap opens in the longitudinal\nmagnon spectrum which then becomes a light Higgs mode. In principle, the respective order\nparameter is also unique in that it supports isolated monopoles, i.e. monopoles that do not\nact as termination points for linear objects such as Dirac monopoles127.\n28To conclude, magnon BECs are interesting systems in their own right, as they form\nanalogies to various fields of physics, from time crystals and particle physics to QCD, and\nprovide non-invasive methods for probing the dynamics and structure of quantized vortices.\nMoreover, magnon BECs hold enormous future potential for accessing novel physics, as\nreplacements for electronic components, or perhaps for detecting dark matter.\nDATA AVAILABILITY\nData sharing is not applicable to this article as no new data were created or analyzed in\nthis study.\nAUTHOR DECLARATIONS\nThe authors have no conflicts to disclose.\nACKNOWLEDGMENTS\nWe thank Grigory Volovik for stimulating discussions. 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The method is applied to a ferrimagnetic insula-\ntor/heavy metal bilayer, Y 3Fe5O12(YIG)/Pt. Typically, SSE measurements use an AC current to\nproduce an alternating temperature gradient and measure the voltage generated by the inverse spin-\nHall e\u000bect in the heavy metal at twice the AC frequency. Here we show that when Joule heating\nis associated with AC and DC bias currents, the SSE response occurs at the frequency of the AC\ncurrent drive and can be larger than the second harmonic SSE response. We compare the \frst and\nsecond harmonic responses and show that they are consistent with the SSE. The \feld dependence\nof the voltage response is used to characterize the damping-like and \feld-like torques. This method\ncan be used to explore nonlinear thermoelectric e\u000bects and spin dynamics induced by temperature\ngradients.\nA central theme in spintronics is the interconversion\nof charge and spin currents [1]. Recently, a focus has\nbeen on magnetic insulators where spin transport occurs\nthrough spin-wave propagation and spin currents can be\ngenerated by either spin injection [2] or by thermal gra-\ndients [3, 4]. These phenomena can be studied in simple\nbilayer \flms consisting of a ferrimagnetic (FIM) insula-\ntor, such as Y 3Fe5O12(YIG), and a heavy metal (HM)\nwith large spin-orbit coupling such as Pt. Spin to charge\ncurrent conversion in such bilayers occurs by the inverse\nspin-Hall [5{7] and Rashba-Edelstein e\u000bects [8, 9].\nSpin to charge conversion enables determination of the\nspin Seebeck e\u000bect (SSE)[10{13]. A thermal gradient\nacross the FIM \flm produces a spin current into a neigh-\nboring heavy metal \flm, resulting in a transverse charge\ncurrent or a voltage across the heavy metal \flm in an\nopen circuit situation. This leads to a convenient route\nto characterize the spin transport as well as a means to\nstudy the inverse e\u000bects, such as the spin torque on the\nFIM magnetization in response to spin currents associ-\nated with charge current \row in the HM. In fact, the SSE\nalso enables detection of the FIM magnetization direction\nby relatively simple electrical measurements.\nIn this article, we present \frst harmonic measurements\nof the SSE by a technique that combines AC and DC\ncurrents in a YIG/Pt bilayer. The temperature gradient\nis created by Joule heating in a Pt strip and both the\nlinear and nonlinear responses in the longitudinal and\ntransverse voltages are determined as a function of the\nangle between the AC current and an in-plane external\nmagnetic \feld. An analysis of the responses shows that\nthe SSE accounts for the main component of the second\nharmonic voltage response, corroborating results in the\nliterature [14]. However, when a DC current is present,\n\u0003These two authors contributed equally\nyElectronic address: andy.kent@nyu.eduseveral new features are observed. First, we detect \feld-\ninduced switching of the YIG magnetization in both the\n\frst and second harmonic longitudinal voltage measure-\nments. Second, both the \frst and second harmonic trans-\nverse voltages show a step when the magnetization re-\nverses. Interestingly, the step height in the \frst harmonic\nresponse has a linear dependence on DC current density\nand cosine dependence on in-plane \feld angle. Of par-\nticularly interest is that the presence of a DC current\nsuperposed on the AC current enables measurements of\nthe SSE in the \frst harmonic response with an increase in\nsignal amplitude relative to the second harmonic signal.\nThe samples we studied consist of a 20 nm thick epi-\ntaxial YIG \flm grown on a gadolinium gallium garnet\n(Gd 3Ga5O12) substrate by RF sputtering [15] and a 5\nnm thick Pt \flm grown by DC sputtering in separate de-\nposition systems. The YIG \flm is transferred in air and\nAr+plasma cleaning is performed prior to the deposi-\ntion of the Pt \flm. A Hall bar with a width of 4 \u0016m and\na length between the voltage contacts of 90 \u0016m is fab-\nricated using e-beam lithography and ion milling. The\ncurrent \rows in the x-direction and the voltage is mea-\nsured both along the current direction (V xx) and trans-\nverse to the current direction (V xy) with separate elec-\ntrical contacts (Fig. 1(a)). Lock-in ampli\fers are used to\nmeasure the \frst harmonic and second harmonic voltages\nwith phases \u001e1= 0oand\u001e2=\u000090\u000eand a time constant\nof 300 ms. The AC current frequency is 953 Hz and its\nrms amplitude is indicated in the \fgures. All the angu-\nlar dependent data are averaged 50 times to improve the\nsignal-to-noise ratio. The measurements are conducted\nat room temperature.\nFigure 1(b) and (c) show the second harmonic longitu-\ndinalV2!\nxxand transverse V2!\nxyvoltage, respectively, as a\nfunction of the in-plane angle of a 400 mT magnetic \feld,\na \feld su\u000ecient to saturate the magnetization of the YIG\nlayer. To con\frm that the second harmonic signal is as-\nsociated with the SSE, measurements were repeated as a\nfunction of the applied magnetic \feld magnitude [16, 17].arXiv:1808.07813v1  [cond-mat.mes-hall]  23 Aug 20182\nFIG. 1: (a) Measurement setup. ~jcis the charge current\ndensity along the x direction. VxxandVxyare voltages mea-\nsured in the longitudinal and transverse directions, respec-\ntively, while 'is the angle between the applied \feld and the\ncurrent. (b) Angular dependence of second harmonic longitu-\ndinal voltage V2!\nxxat a \fxed current density of jac= 1:5\u00021010\nA/m2with an applied \feld of \u00160H = 400 mT. The curve is a\n\ft toV2!\nxx(0) sin('). (c) Angular dependence of second har-\nmonic transverse voltage V2!\nxyat the same current density,\njac= 1:5\u00021010A/m2. The curve is a \ft to V2!\nxy(0) cos(').\n(See the Supplementary section [19].) It is important to\nnote that there are contributions to the second harmonic\nsignal from the damping-like (DL) torque, \feld-like (FL)\ntorque and Oersted (Oe) \felds. By characterizing the\n\feld dependence of the second harmonic response these\ne\u000bects can be separated, particularly at small applied\n\felds at which these torques and Oersted \felds intro-\nduce additional structure in the angular dependence of\nthe second harmonic signal. This is discussed in the sup-\nplementary section [19], where the relative contributions\nof SSE, DL, FL and Oe \feld torques are determined [16{\n18]. We \fnd that for an applied \feld of 400 mT, the\nsecond harmonic signal is dominated by the SSE.\nFigure 2(a) and (b) show the \feld dependence of sec-\nond harmonic V2!\nxy(Fig. 2(a)) and \frst harmonic V!\nxy\n(Fig. 2(b)) response with the \feld applied along the cur-\nrent direction ( '= 0\u000e) at a \fxed AC current density of\n1:5\u00021010A/m2as the DC component of the current\ndensity is varied, jdc= 0;\u00062:5;\u00065:0\u00021010A/m2. The\nSSE response is expected to change sign when the mag-\nnetization direction reverses, which is evident in Fig. 2(a)\nin the step change in V2!\nxynear zero \feld at the coerciv-\nity of the YIG ( \u00160Hc'10 mT). The step in voltage in\nthe second harmonic signal is nearly independent of the\nDC current. Interestingly, the \frst harmonic response\ndepends systematically on the DC current. At zero DC\ncurrent there is virtually no response, only small signal\nvariations near zero \feld. However, when the DC current\ndensity is non-zero, a clear voltage step is evident near\nzero \feld, with a change in voltage that depends on the\nFIG. 2: Field dependent measurement of second and \frst\nharmonic transverse voltage with \fxed AC current jac= 1:5\u0002\n1010A/m2and varying DC current. (a) Field dependence of\nV2!\nxyat'= 0\u000eandjdc= 0;\u00062:5;\u00065:0\u00021010A/m2. (b) Field\ndependence of V!\nxywith'= 0\u000eandjdc= 0;\u00062:5;\u00065:0\u00021010\nA/m2.\nDC current.\nThe magnitude of the \frst harmonic signal is about\none order of magnitude larger than the second harmonic\nsignal. In addition, the step in the \frst harmonic sig-\nnal changes sign when the DC current is reversed. Fig-\nure 3(a) and (b) show how the steps in voltage depends\non DC current. The step in the second harmonic signal\n\u0001V2!\nxy(Fig. 3(a)) is slightly modi\fed due to DC current,\nwhereas there is a clear linear relation between the step\nin the \frst harmonic signal \u0001 V!\nxy(Fig. 3(b)) and the DC\ncurrent.\nIn order to understand this behavior one needs to con-\nsider Joule heating by the AC and DC current through\nthe Pt. This leads to a power dissipation given by:\nP= [p\n2jaccos(!t) +jdc]2RA2\n= [j2\naccos(2!t) + 2p\n2jacjdccos(!t) +j2\nac+j2\ndc]RA2;\n(1)\nwherejacis the rms AC current density, Ris the resis-\ntance of the Pt and Aits cross sectional area, the \flm\nthickness times the width of the current line. The tem-\nperature gradient rTzis proportional to the power dis-\nsipation. It follows that the SSE voltage generated has\nthe following form:\nVISHE/j2\naccos(2!t)+2p\n2jacjdccos(!t)+j2\nac+j2\ndc:(2)\nThere is thus an SSE response at two times the oscillation\nfrequency of the current, the second harmonic, 2 !, as\nexpected, as well as a signal at frequency, !, the \frst\nharmonic. Thus the combination of AC and DC currents\nprovides a technique to measure the SSE voltage as a \frst\nharmonic response. The relative magnitude of the \frst\nand the second harmonic signals is given by V!\nxy=V2!\nxy=\n2p\n2jdc=jac. The \frst harmonic response is thus about\n2.8 times larger than the second harmonic response when\nthe AC and DC currents are the same. The linear relation\nbetween \u0001V!\nxyandjdcin Fig. 3(b) con\frms this model.\nFurther, we experimentally verify the symmetry and\nmagnitude of the SSE \frst harmonic response in com-\nparison to the conventional second harmonic signal. We\nhave performed \feld dependent measurements of the3\nFIG. 3: Dependence of the second and \frst harmonic trans-\nverse voltage amplitudes on the DC current density with jac\n\fxed at 1:5\u00021010A/m2. (a) Second harmonic transverse volt-\nage versus DC current. (b) First harmonic transverse voltage\nversus DC current.\n\frst harmonic transverse voltage by sweeping the ex-\nternal magnetic \feld between -400 mT and +400 mT\nat di\u000berent in-plane angles 'from 0\u000eto 360\u000eat \fxed\njac= 1:5\u00021010A/m2andjdc=\u00065:0\u00021010A/m2. Using\nthese results, we have determined \u0001 V!\nxyusing the proce-\ndure mentioned in the preceding section and plot its vari-\nation with'(Fig. 4). The SSE voltage is proportional to\nthe projection of the magnetization on the axis perpen-\ndicular to the voltage probes. The temperature gradient\nis along the z-axis, whereas the spin polarization is along\nthe YIG magnetization direction. Therefore the angular\ndependence of the \frst harmonic and second harmonic\ntransverse response are V!\nxy/2p\n2jacjdccos(');V2!\nxy/\nj2\naccos('), as seen experimentally. Measurements of\n\u0001V!\nxycan be \ftted well with \u0001 V!\nxy(0) cos', denoted by\nthe solid line in Fig. 4. The second harmonic transverse\nvoltage was measured with varying 'at a \fxed \feld of\n+400 mT and a \fxed jac= 1:5\u00021010A/m2(Fig. 1(c)).\nEquation 2 predicts that for jac= 1:5\u00021010A/m2and\njdc=\u00065:0\u00021010A/m2, the relative magnitudes of the\n\frst and second harmonic signals should be 9 :4. We have\nextracted the maximum \u0001 V!\nxy(0) = 0:187\u00060:053\u0016V\nand \u0001V2!\nxy(0) = 0:0227\u00060:0007\u0016V by \ftting the data\nin Fig. 4 and Fig. 1(d) respectively. The experimentally\nobtained ratio of the \frst and second harmonic signals\nis 8:2\u00062:7. The experimentally obtained ratio is thus\nconsistent with our simple AC and DC current heating\nmodel. The data in Fig. 4 clearly indicates that the \frst\nharmonic response has a much higher signal-to-noise ra-\ntio than that of the second harmonic voltage (Fig. 1(c)).\nIn summary, we have determined the SSE-produced\nlinear and nonlinear voltage responses in a YIG/Pt bi-\nlayer system. The second harmonic longitudinal voltage\nhas a sine relation with respect to the in-plane \feld angle\n'when the YIG is saturated. Angular dependence mea-surement of the longitudinal and transverse voltages as a\nfunction of the applied \feld magnitude enabled estima-\ntion of the contributions from SSE, DL, FL and Oe \feld\ntorques. It was found that the SSE dominates over the\nother contributions when the applied \feld is su\u000ecient to\nsaturate the YIG layer. In addition, by applying an AC\ncurrent with DC bias, we determined that SSE can be\nmeasured by a \frst harmonic lock-in technique, and can\n0 90 180 270 360\nφ(o)−0.2−0.10.00.10.2ΔVω\nxy\nFIG. 4: Angular dependence of \u0001 V!\nxymeasured with an AC\ncurrent of 1 :5\u00021010A/m2and DC current 5 :0\u00021010A/m2.\nThe curve is a \ft to the data of the form \u0001 V!\nxy(0)cos(').\nbe more sensitive and have higher signal to noise than the\nconventional second harmonic metthod. This technique\ncan be used to characterize the SSE in ferromagnetic (or\nferrimagnetic) and non-magnetic bilayer systems as well\nas to study nonlinear thermoelectric e\u000bects and spin dy-\nnamics induced by temperature gradients.\nAcknowledgements\nThe instrumentation used in this research was support\nin part by the Gordon and Betty Moore Foundations\nEPiQS Initiative through Grant GBMF4838 and in part\nby the National Science Foundation under award NSF-\nDMR-1531664. This work was supported partially by the\nMRSEC Program of the National Science Foundation un-\nder Award Number DMR-1420073. ADK received sup-\nport from the National Science Foundation under Grant\nNo. DMR-1610416. At CSU, \flm growth was sup-\nported by the U.S. National Science Foundation (EFMA-\n1641989), and \flm characterization was supported by the\nU.S. Department of Energy, O\u000ece of Science, Basic En-\nergy Sciences (DE-SC-0018994).\n[1] A. Brataas, A. D. Kent, and H. 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SUPPLEMENTAL MATERIALS: FIRST\nHARMONIC MEASUREMENTS OF THE SPIN\nSEEBECK EFFECT\nSeparation of the Spin Seebeck E\u000bect, anti-damping\nspin-orbit torque, \feld-like torque and Oersted \feld\ncontributions\nIn the main text we state that the origin for both the\nlongitudinal and transverse second-harmonic signals for\na 400 mT applied \feld are due to the spin Seebeck ef-\nfect arising from current-induced Joule heating in the Pt\nstrip. Here, we estimate the relative contributions to\nV2!\nxyandV2!\nxxfrom SSE, anti-damping spin-orbit torque\n(AD), \feld-like torque (FL) and Oersted \feld contribu-\ntions (Oe).\nAngular dependence measurements with applied mag-\nnetic \feld ranging from 2 mT to 400 mT are used to\nseparate there contributions using the following rela-\ntions [S1,S2]:\nV2!\nxx(') = \u0001V2!;'\nxx sin('+ \u0001')\n+ \u0001V2!;3'\nxx sin('+ \u0001') cos2('+ \u0001')\n+A0(S1)\n\u0001V2!;'\nxx is the SSE and AD contributions and \u0001 V2!;3'\nxxis the FL and Oe contributions. A0is the o\u000bset of the\nsignal.\nV2!\nxy(') = \u0001V2!;'\nxy cos('+ \u0001')\n+ \u0001V2!;3'\nxy cos('+ \u0001') cos[2('+ \u0001')]\n+B0(S2)\n\u0001V2!;'\nxy is the SSE and AD contributions and \u0001 V2!;3'\nxy\nis the FL and Oe contributions. B0is the o\u000bset of the\nsignal.\nWe plot \u0001V2!;'\nxx=xyversus the inverse applied \feld, \u00160H:\n\u0001V2!;'\nxx=xy=kxx;'\n\u00160H+C'\nxx=xy(S3)\nto determine the slope kxx;'and intercept Cxx=xy .\nWe then plot \u0001 V2!;3'\nxx=xyversus the inverse applied \feld,\n\u00160H:\n\u0001V2!;3'\nxx=xy=kxx;3'\n\u00160H+D3'\nxx=xy(S4)\nto determine the slope kxx;3'and intercept Dxx=xy .\nFig. S1 (a) and Fig. S1 (b) show the angular depen-\ndence ofV2!\nxxandV2!\nxyversus applied magnetic \feld rang-\ning from 2 to 400 mT. When \u00160H= 400 mT, V2!\nxxand\nV2!\nxy\ft well to sin( ') and cos('), with negligible 3 '-\ncontributions. For \u00160Hsmaller than 25 mT, clear 3 '-\nsymmetry can be observed due to the non-negligible FL +\nOe e\u000bects, indicating that the applied \feld torque is com-\nparable to the AD, FL and Oe torques. By \ftting V2!\nxx\nFig. S 1: Second harmonic measurements of V2!\nxxandV2!\nxy\nwith di\u000berent applied magnetic \felds. The AC current den-\nsityjac= 1:5\u00021010A=m2and applied magnetic \feld \u00160H\nranges from 2 to 400 mT. Solid lines denotes the \fts. (a)\nAngular dependence of V2!\nxx, data \fts to equation (S1); (b)\nAngular dependence of V2!\nxy, data \fts to equation (S2); (c)\nField dependence of \u0001 V2!;'\nxx and \u0001V2!;3'\nxx with the inter-\nceptsC'\nxx= 0:59\u0016VandD'\nxx= 3:8\u000210\u00003\u0016V; (d) Field\ndependence of \u0001 V2!;'\nxy and \u0001V2!;3'\nxy with the intercepts D'\nxy\n= 22:2\u000210\u00003\u0016VandD'\nxx= 2:1\u000210\u00003\u0016V.5\nwith Eqn. S1 and V2!\nxywith Eqn. S2, we can extracted\nthe relative SSE, AD, FL and Oe contributions. \u0001 V2!;'\nxx\nand \u0001V2!;'\nxy are denoted as the '-contributions while\n\u0001V2!;3'\nxx and \u0001V2!;3'\nxy indicate the 3 '-contributions.\nFig. S1(c) and Fig. S1(d) show the \feld dependence of\n'and 3'-contributions. The 'contributions are almostindependent of the applied magnetic \felds, showing neg-\nligible AD. However, the 3 'contributions are inversely\nproportional to the applied magnetic \felds with negligi-\nble intercepts. When the applied \feld is in the range of\n\u00183 mT, the 3 'contributions surpass the 'contributions,\nshowing increasing FL and Oe e\u000bects at low \felds.\n[S1] N. Vlietstra, J. Shan, B. J. van Wees, M. Isasa, F.\nCasanova, and J. Ben Youssef, Phys. Rev. B 90, 17443\n[S2] M. Hayashi, J. Kim, M. Yamanouchi, and H. Ohno, Phys.\nRev. B 89, 144425 (2014)."    },    {        "title": "1609.07891v2.Magnon_Kerr_effect_in_a_strongly_coupled_cavity_magnon_system.pdf",        "content": "Magnon Kerr e \u000bect in a strongly coupled cavity-magnon system\nYi-Pu Wang,1,\u0003Guo-Qiang Zhang,1,\u0003Dengke Zhang,1Xiao-Qing Luo,1Wei\nXiong,1Shuai-Peng Wang,1Tie-Fu Li,2, 1,yC.-M. Hu,3and J. Q. You1,z\n1Quantum Physics and Quantum Information Division,\nBeijing Computational Science Research Center, Beijing, 100193 China\n2Institute of Microelectronics, Tsinghua National Laboratory of Information\nScience and Technology, Tsinghua University, Beijing 100084, China\n3Department of Physics and Astronomy, University of Manitoba, Winnipeg R3T 2N2, Canada\n(Dated: December 8, 2016)\nWe experimentally demonstrate magnon Kerr e \u000bect in a cavity-magnon system, where magnons in a small\nyttrium iron garnet (YIG) sphere are strongly but dispersively coupled to the photons in a three-dimensional\ncavity. When the YIG sphere is pumped to generate considerable magnons, the Kerr e \u000bect yields a perceptible\nshift of the cavity’s central frequency and more appreciable shifts of the magnon modes. We derive an analytical\nrelation between the magnon frequency shift and the drive power for the uniformly magnetized YIG sphere and\nfind that it agrees very well with the experimental results of the Kittel mode. Our study paves the way to explore\nnonlinear e \u000bects in the cavity-magnon system.\nPACS numbers: 71.36. +c, 42.50.Pq, 76.50. +g, 75.30.Ds\nI. INTRODUCTION\nHybridizing two or more quantum systems can harness the\ndistinct advantages of di \u000berent systems to implement quan-\ntum information processors (see, e.g., Ref.[1, 2]). Recently, a\ncavity-magnon system has attracted considerable attention [3–\n9], because of the enhanced coupling between magnons in a\nyttrium iron garnet (YIG) single crystal and microwave pho-\ntons in a high-finesse cavity. This hybrid system involves\nmagnon polaritons [10, 11]. Thus, a series of phenomena re-\nalized in other polariton systems [12, 13], including the Bose-\nEinstein condensation of exciton polaritons [14, 15] and the\noptical bistability in semiconductor microcavities [16], can be\nexplored using the magnon polaritons. Based on the strongly\ncoupled cavity-magnon system, coherent interaction between\na magnon and a superconducting qubit was realized [17], and\nmagnon dark modes in a magnon gradient memory [18] were\nutilized to store quantum information. When combined with\nspin pumping techniques, this cavity-magnon system provides\na new platform to explore the physics of spintronics and to\ndesign useful functional devices [7, 9]. Potentially acting as\na quantum information transducer, microwave-to-optical fre-\nquency conversion between microwave photons generated by\na superconducting circuit and optical photons of a whisper-\ning gallery mode supported by a YIG microsphere was also\nexplored [19–22]. Furthermore, coherent phonon-magnon in-\nteractions relying on the e \u000bect of magnetostrictive deforma-\ntion in a YIG sphere was demonstrated [23]. Now, a versatile\nquantum information processing platform based on the coher-\nent couplings among magnons, microwave photons, optical\nphotons, phonons, and superconducting qubits is being estab-\nlished.\n\u0003These authors contributed equally.\nylitf@tsinghua.edu.cn\nzjqyou@csrc.ac.cnIn this paper, we report an experimental demonstration of\nthe magnon Kerr e \u000bect in a strongly coupled cavity-magnon\nsystem. The magnons in a small YIG sphere are strongly\nbut dispersively coupled to the microwave photons in a three-\ndimensional (3D) cavity. When considerable magnons are\ngenerated by pumping the YIG sphere, the Kerr e \u000bect gives\nrise to a shift of the cavity’s central frequency and yields more\nappreciable shifts of the magnon modes, including the Kit-\ntel mode [24], which holds homogeneous magnetization, and\nthe magnetostatic (MS) modes [25–27], which have inhomo-\ngeneous magnetization. We derive an analytical relation be-\ntween the magnon frequency shift and the pumping power for\na uniformly magnetized YIG sphere and find that it agrees\nvery well with the experimental results of the Kittel mode. In\ncontrast, the experimental results of MS modes deviate from\nthis relation, which confirms the deviation of the MS modes\nfrom homogeneous magnetization. To enhance the magnon\nKerr e \u000bect, the pumping field is designed to directly drive the\nYIG sphere and its coupling to the magnons is strengthened\nusing a loop antenna. Moreover, this pumping field is tuned\nvery o \u000b-resonance with the cavity mode to avoid producing\nany appreciable e \u000bects on the cavity. Our paper is a convinc-\ning study of a cavity-magnon system with magnon Kerr e \u000bect\nand paves the way to experimentally explore nonlinear e \u000bects\nin this tunable cavity-magnon system.\nII. EXPERIMENTAL SETUP\nThe experimental setup is diagrammatically shown in\nFig. 1. The 3D cavity is made of oxygen-free copper with\ninner dimensions of 44 :0\u000220:0\u00026:0 mm3and contains three\nports labeled as 1, 2 and 3 (here ports 1 and 2 are used for\ntransmission spectroscopy and port 3 is for loading the drive\nfield). The frequency of the cavity mode TE 102that we use is\n!c=2\u0019=10:1 GHz. A samll YIG sphere of diameter 1 mm\nis glued on an inner wall of the cavity at the magnetic-field\nantinode of the TE 102mode (c.f., the magnetic-field intensityarXiv:1609.07891v2  [quant-ph]  7 Dec 20162\nB\nxz\ny\nRoom Temp.\nAmplifier\nHEMT  \nAmplifier\nAttenuator\nIsolator\nMagnetic FieldYIG Sphere\n Magnet300K\n50K\n4K\n500mK\n100mK\n22mK\nB0\nYIG  Sphere\nCavity\u0012\n\u0013\u0014-23dB 60dB\n10dB\n10dB\n6dB\n3dB\n10dBmax\nminSMA  Connector\nBc\ndMW VNA\nFIG. 1. (color online). Schematic of the experiment setup. The 3D\ncavity is placed in the uniform magnetic field created by a supercon-\nducting magnet. Ports 1 and 2 are used for transmission spectroscopy\nand port 3 is for driving the YIG sphere via a superconducting mi-\ncrowave line with a loop antenna at its end. The total attenuation of\nthe input port is 99dB. The right part shows the magnetic-field distri-\nbution of the cavity mode TE 102. The bias magnetic field, the drive\nmagnetic field and the magnetic field of the TE 102mode are mutually\nperpendicular at the site of the YIG sphere, where the magnetic field\nof the TE 102mode is maximal.\ndistribution of this mode marked by colored shades in Fig. 1).\nWe apply a static magnetic field generated by a superconduct-\ning magnet to magnetize the YIG sphere. This bias magnetic\nfield is tunable in the range of 0 to 1 T, so the given frequency\nof the Kittel mode (i.e., the ferromagnetic resonance mode)\nranges from several hundreds of megahertz to 28 GHz. The\ncavity is placed in a BlueFors LD-400 dilution refrigerator\nunder a cryogenic temperature of 22 mK. The spectroscopic\nmeasurement is carried out with a vector network analyzer by\nprobing the transmission of the cavity. A drive tone supplied\nby a microwave source can directly drive the YIG sphere via a\nsuperconducting microwave line going through port 3. More-\nover, a loop antenna is attached to the end of the superconduct-\ning microwave line near the YIG sphere (see Appendix A), so\nas to strengthen the coupling between the drive field and the\nYIG sphere. Here the driving magnetic field Bd, the bias mag-\nnetic field B0(which is aligned along the hard magnetization\naxis [100] of the YIG sphere), and the magnetic field Bcof\nthe TE 102mode are orthogonal to each other at the site of the\nYIG sphere. Also, a series of attenuators and isolators is used\nto prevent thermal noise from reaching the sample and the\noutput signal is amplified by two low-noise amplifiers at the\nstages of 4 K and room temperature, respectively.\nIII. STRONG COUPLING REGIME\nWe first measure the transmission spectrum of the cavity\ncontaining the YIG sphere, without applying a drive field on\nthe YIG sphere. The transmission spectrum as a function\nof the probe microwave frequency and B0are recorded by\nvector network analyzer [see Fig. 2(a)]. At the point where\nFIG. 2. (color online). (a) Transmission spectrum for the normal-\nmode splitting measured as a function of the bias magnetic field\nand the probe microwave frequency. The large anticrossing indicates\nstrong coupling between the Kittel mode and the cavity mode TE 102.\nThe small splittings are due to the MS modes coupled with the cavity\nmode. (b) Transmission spectrum at three values of the bias magnetic\nfield. The curves are o \u000bset vertically for clarity.\nthe Kittel mode is resonant with the cavity mode TE 102, a\ndistinct anti-crossing of the two modes occurs, indicating\nstrong coupling between them. Some other small splittings\nare due to the couplings between the cavity mode and the\nMS modes in the YIG sphere. The coupling strength be-\ntween the Kittel mode and the cavity mode TE 102is found\nto be gm=2\u0019=42 MHz from the magnon polariton splitting\nat the resonance point [see the red cure in Fig. 2(b)]. By\nfitting the measured transmission spectrum [8], the cavity-\nmode linewidth \u0014=2\u0019\u0011(\u00141+\u00142+\u0014int)=2\u0019and the Kittel-\nmode linewidth \rm=2\u0019are determined to be 2 :87 MHz and\n24:3 MHz, respectively. Here \u00141(\u00142) is the loss rate due to the\nport 1 (2) and \u0014intis due to the intrinsic loss of the cavity. The\nobvious increase in the Kittel mode damping rate compared\nwith the previous work [4, 5] is due to the antenna close to the\nYIG sphere, which acts as an additional decay channel. Note\nthat all the linewidths throughout the paper are defined as the\nfull width at half maximum (FWHM). Because gm> \u0014;\r m,\nthe hybrid system falls in the strong-coupling regime with a\ncooperativity C\u00114g2\nm=\u0014\r m=101:\nIV . RESULTS AND ANALYSIS OF THE DISPERSIVE\nMEASUREMENT\nA. Dispersive measurement\nWe tune the static bias magnetic field B0to 346:8 mT, yield-\ning about 9.55 GHz for the frequency of the Kittel mode. As\nshown in Fig. 3(a), we first measure the transmission spec-\ntrum of the cavity (i.e., the black curve) by tuning the fre-\nquency of the probe field, but without the drive field on the\nYIG sphere. The measured central frequency of the cavity\nmode is 10.1035 GHz, which has a frequency shift of about 3\nMHz compared with the intrinsic frequency of 10.1003 GHz\nof the TE 102mode of an empty cavity. This cavity mode has a\ndetuning \u0001=2\u0019\u0019550 MHz from the Kittel mode. Because\n\u0001>10gm, the coupled hybrid system is in the dispersive3\n(a)\nKittel mode-26\n-27\n-28\n9.3 9.4 9.6 9.9 9.8 9.7 9.5Drive power : 1 1 dBm|S      21  | \nDrive Frequency (GHz)(b)-27\n-29\n-31\n-33\n-35|S  21  | \n10.1035 GHz 10.1042 GHz\n10.101 10.106 10.105 10.104 10.103 10.1020.7MHz\nMW Drive OFF\nMW Power : 1 1 dBm, Freq : 9.5915GHz\nProbe Frequency (GHz)\n(c)\nIncreasing drive power|S  21  | -26\n-27\n-28\nDrive Frequency (GHz)9.3 9.4 9.6 9.9 9.8 9.7 9.5MS mode 1\nMS mode 22 2 2(dB) (dB) (dB)\nFIG. 3. (color online). (a) Central frequency shift of the cavity mode\nTE102when the drive field is on (red curve) and o \u000b(black curve),\nrespectively. (b) Transmission spectrum of the cavity measured as a\nfunction of the drive-field frequency. The blue arrow indicates the\nresponse of the Kittel mode, whereas the orange and purple arrows\nindicate the MS modes 1 and 2, respectively. (c) Transmission spec-\ntrum of the cavity measured as a function of the drive frequency by\nsuccessively increasing the driving power. The probe field is fixed at\n10:1035 GHz in both (b) and (c).\nregime. We then measure the transmission spectrum of the\ncavity by both tuning the frequency of the probe field and ap-\nplying a drive field on the YIG sphere in resonance with the\nKittel mode. The measured red curve corresponds to the drive\npower of 11 dBm. This transmission spectrum has a central\nfrequency of 10 :1042 GHz, with a frequency shift of about\n0:7 MHz from the measured central frequency without apply-\ning a drive field.\nFigures 3(b) and 3(c) show the measured transmission spec-\ntra by tuning the frequency of the drive field, where the fre-\nquency of the probe field is fixed at the central frequency of\n10:1035 GHz of the cavity containing the YIG sphere. Theprobe field power is -129 dBm. The corresponding average\ncavity probe photon number can be estimated by [28]\n¯n=\u00141Pp\n~!p[\u00012p+(\u0014=2)2]; (1)\nwhere Ppis the probe field power and \u0001p=!p\u0000!c. In our ex-\nperiment, it is measured that \u00141=2\u0019=0:70 MHz, i.e., \u00141\u0018\u0014=4.\nAlso, the probe field frequency !pis tuned in resonance with\nthe cavity mode TE 102. Then, the average cavity probe photon\nnumber is reduced to ¯ n=Pp=(~!p\u0014)\u00191. Here the probe tone\nis chosen extremely weak, so as to avoid producing any ap-\npreciable e \u000bects on the system. In Fig. 3(b), the power of the\ndrive field is 11 dBm. It can be seen that when the frequency\nof the drive microwave field is resonant with the Kittel mode,\nthe transmission coe \u000ecient has a large decrease at 9 :59 GHz\n(see the main dip indicated by a blue arrow), caused by the\nshift of the central frequency of the cavity mode. The dips\nindicated by orange and purple arrows correspond to two dif-\nferent MS modes. In addition, we vary the power of the drive\nfield from -5 to 10 dBm in Fig. 3(c) and observe two interest-\ning features, i.e., when increasing the drive power, the main\ndip becomes deeper successively and it simultaneously shifts\nrightwards. This reveals that the Kittel mode has a blue shift\nwith the increase in the drive power. The responses of MS\nmodes are similar.\nB. Origin of the Kerr term\nFor a YIG sphere uniformly magnetized by an external\nmagnetic field along the zdirection, when the magnetization\nis saturated, the induced internal magnetic field includes the\ndemagnetizing field [29] Hde=\u0000M=3 and the anisotropic\nfield [30, 31] Han=\u0000(2Kan=M2)Mz, where M\u0011(Mx;My;Mz)\nis the magnetization, Mis the saturation magnetization and\nKanis the first-order magnetocrystalline anisotropy constant\nof the YIG sphere. When both the Zeeman energy and the\nmagnetocrystalline anisotropic energy are included (see Ap-\npendix B), the Hamiltonian of the YIG sphere in the magnetic\nfield B0is given by (setting ~=1)\nHm=\u0000\rB0Sz\u0000\u00160\r2Kan\nM2VmS2\nz; (2)\nwhere\r=2\u0019=28 GHz /T is the gyromagnetic ratio, \u00160is\nthe vacuum permeability and Sz=MzVm=\ris a macrospin\noperator of the YIG sphere, with Vmbeing the volume of\nthe YIG sample. The macrospin operator Szis related to\nthe magnon operators via the Holstein-Primako \u000btransforma-\ntion [32]: Sz=S\u0000byb, where by(b) is the magnon creation\n(annihilation) operator.\nWhen including the drive field, the cavity mode, and the in-\nteraction between the cavity photon and the magnon, the total\nHamiltonian of the coupled hybrid system is (see Appendix B)\nH=!caya+!mbyb+Kbybbyb\n+gm(ayb+aby)+ \n d(bye\u0000i!dt+bei!dt); (3)4\nwhere ˆ ay(ˆa) is the creation (annihilation) operator of the cav-\nity photons at frequency !c,Kbybbybrepresents the Kerr ef-\nfect of magnons owing to the magnetocrystalline anisotropy in\nthe YIG sphere, with K=\u00160Kan\r2=(M2Vm),\nd(i.e., the Rabi\nfrequency) denotes the strength of the drive field, and !dis the\ndrive field frequency. Thus, our experimental setup provides\na strongly coupled cavity-magnon system with the magnon\nKerr e \u000bect, which is an extension of the cavity-magnon sys-\ntem without the nonlinear e \u000bect [33]. Note that Kis inversely\nproportional to Vm, so the Kerr e \u000bect can become important\nwhen using a small YIG sphere.\nC. Cavity and magnon frequency shifts\nBelow we study the case of considerable magnons gener-\nated by the drive field. Because the coupled hybrid system\nis in the dispersive regime, its e \u000bective Hamiltonian can be\nwritten as (see Appendix C)\nHe\u000b=\"\n!c+g2\nm\n\u0001+2g2\nm\n\u00012Khbybi#\naya\n+\"\n!m\u0000g2\nm\n\u0001+ \n1\u00002g2\nm\n\u00012!\nKhbybi#\nbyb\n+\n0\nd(bye\u0000i!dt+bei!dt);] (4)\nwith the e \u000bective Rabi frequency \n0\ndgiven by\n\n0\nd=\u0014\n1\u00001\n2(!c\u0000!d)\u0012g2\nm\n\u0001+2g2\nm\n\u00012Khbybi\u0013\u0015\n\nd; (5)\nwhere \u0001 =!c\u0000!m. Due to the coupling between the cav-\nity and the YIG sphere, the cavity frequency shifts from the\nintrinsic cavity mode frequency !cto!c+g2\nm=\u0001, with g2\nm=\u0001\nbeing the dispersive shift. The measured central frequency\nof 10.1035 GHz corresponds to !c+g2\nm=\u0001. When pumping\nthe YIG sphere with a drive field, the magnon number hbybi\nincreases. Then the cavity frequency has an additional blue\nshift of \u0001c=(2g2\nm=\u00012)Khbybidue to the Kerr e \u000bect. Also,\nthe Kerr e \u000bect yields a blue shift to the magnon frequency,\n\u0001m=(1\u00002g2\nm=\u00012)Khbybi\u0019Khbybi. Both cavity frequency\nshift and magnon frequency shift due to the Kerr e \u000bect have a\nsimilar trend depending on hbybi, which is related to the drive\npower.\nV . RELATION BETWEEN THE MAGNON FREQUENCY\nSHIFT AND THE DRIVE POWER\nIn Fig. 4, we extract the Kerr-e \u000bect-induced frequency\nshifts of the magnon as well as the central frequency shift of\nthe cavity mode at each given drive power P. From Fig. 4(a),\nit is clear that both the Kittel-mode frequency shift and the\ncavity central frequency shift indeed have similar behaviors\ndepending on the drive power, as predicted above. We also\nsee that all the frequency shifts exhibit nonlinear dependence\non the drive power. As given in Appendix D, we derive an an-\nalytical relation between the magnon frequency shift \u0001mand\n0 2 4 6 8 10010203040MagnonFrequencyShift(MHz)\n0.00.40.81.21.6CavityFrequencyShift(MHz)Kittel mode (Left)\nCavity mode (Right)\n0 2 4 6 8 10010203040\nMagnonFrequencyShift(MHz)\nDrivePower(mW)MS  mode 1\nMS  mode 2 \n(a)\n(b)FIG. 4. (color online). (a) Frequency shift of the Kittel mode (blue\nsquare) and the central frequency shift of the cavity mode TE 102(red\ncircle) measured at various values of the drive power. The blue fitting\ncurve for the Kittel mode is obtained using Eq. (6). (b) Frequency\nshifts of MS mode 1 (orange up-triangle) and MS mode 2 (purple\ndown-triangle) measured at various values of the drive power. The\ncorresponding orange and purple fitting curves also are obtained us-\ning Eq. (6). The frequency shifts of the Kittel mode, MS mode 1 and\nMS mode 2 are here referenced from 9.5526, 9.4758, 9.6174GHz,\nrespectively.\nthe drive power Pusing a Langevin equation approach,\n\u0014\n\u00012\nm+\u0012\rm\n2\u00132\u0015\n\u0001m\u0000cP=0; (6)\nwhere cis a characteristic parameter reflecting the coupling\nstrength of the drive field with the magnon mode. For the\nKittel mode, we have already measured its linewidth \rm=2\u0019=\n24:3 MHz. We use Eq. (6) to fit the experimental results of the\nKittle mode. As shown in Fig. 4(a), the obtained theoretical\n(blue) curve fits very well with the experimental data, where\nc=(2\u0019)3\u00024:7\u00021024kg\u00001m\u00002.\nFor the MS modes, we have two unknown parameters, the\nMS mode linewidth \rmand the parameter c. We manage to\nfit the experimental data in Fig. 4(b) with \rm=15 MHz and\nc=(2\u0019)3\u00021:35\u00021024kg\u00001m\u00002for MS mode 1 (orange curve),\nand with\rm=30 MHz and c=(2\u0019)3\u00026\u00021024kg\u00001m\u00002for\nMS mode 2 (purple curve). Note that the theoretical curves\ndo not fit the experimental data of the MS modes so well as\nthose of the Kittel mode, especially in the region around the\nthreshold power [see the region of 1-3 mW in Fig. 4(b)]. In\nfact, as a collective mode of spins with a zero wavevector,5\nthe Kittel mode is the uniform precession mode with homo-\ngeneous magnetization, whereas the MS modes are nonuni-\nform precession modes holding inhomogeneous magnetiza-\ntion and have a spatial variation comparable to the sample\ndimensions [26, 27, 31]. The appreciable deviations of the\nexperimental data from the theoretical fitting curves are due\nto the inhomogeneous magnetization of the MS modes.\nNote that when the drive power is small, \rm\u001d\u0001m, so\nEq. (6) reduces to\n\u0012\rm\n2\u00132\n\u0001m\u0000cP=0; (7)\ni.e., the magnon frequency shift depends linearly on the drive\npower in the small drive power limit. When the drive power\nbecomes su \u000eciently large, \u0001m\u001d\rm, and then Eq. (6) reduces\nto\n\u00013\nm\u0000cP=0: (8)\nIt yields\n\u0001m=(cP)1=3; (9)\ni.e., in the large drive power limit, the magnon frequency shift\ndepends linearly on the cubic root of the drive power. These\nlimit results are consistent with the previous work in Ref. [34],\nwhere there is a threshold power separating the small and large\ndriving power regions.\nA dispersive measurement on the cavity transmission was\nimplemented at room temperature in Ref. [35], but the cav-\nity’s central frequency shift due to the magnon Kerr e \u000bect\nwas not observed. In Ref. [35], the drive field was applied\non the cavity rather than the YIG. This is di \u000berent from our\nsetup in which the YIG sphere is directly pumped by the drive\nfield and the nonlinear e \u000bect of large-amplitude spin waves\ncan be induced [36]. Moreover, choosing a suitable angle be-\ntween the external magnetic field and the crystalline axis is\nalso important to observe the magnon Kerr e \u000bect because the\nvalue of the Kerr coe \u000ecient Kstrongly depends on this an-\ngle [37]. In our case, the bias magnetic field B0is aligned\nalong the hard magnetization axis [100] of the YIG sphere,\nwhich gives rise to the largest K. Furthermore, our exper-\niment is implemented at a cryogenic temperature where the\nmagnetocrystalline anisotropy constant Kan(so the Kerr coef-\nficient K) is several times larger than that at room tempera-\nture [3, 31]. These may be the reasons why appreciable Kerr\ne\u000bect was not observed in Ref. [35].\nVI. CONCLUSION\nWe have realized a strongly coupled cavity-magnon system\nwith magnon Kerr e \u000bect. By directly pumping the YIG sphere\nwith a drive field, we have demonstrated the Kerr-e \u000bect-\ninduced central frequency shift of the cavity mode as well as\nthe frequency shifts of the Kittel mode and MS modes. An an-\nalytical relation between the magnon frequency shift and the\npumping power for a uniformly magnetized YIG sphere is de-\nrived, which agrees very well with the experimental results ofthe Kittel mode. In contrast, the experimental results of MS\nmodes deviate from this relation owing to the spatial varia-\ntions of the MS modes over the sample. We can use this rela-\ntion to characterize the degrees of deviation of the MS modes\nfrom the homogeneous magnetization. Our setup can provide\na flexible and tunable platform to further explore nonlinear\ne\u000bects of magnons in the cavity-magnon system. Moreover,\nthis coupled hybrid system involves magnon polaritons. It can\nbe used to explore a series of phenomena realized in other po-\nlariton systems [12, 13].\nACKNOWLEDGMENTS\nThis work was supported by the National Key Re-\nsearch and Development Program of China (Grant\nNo. 2016YFA0301200), the MOST 973 Program of\nChina (Grant No. 2014CB848700), and the NSAF (Grant\nNo. U1330201 and No. U1530401). C.M.H. was supported\nby the NSFC (Grant No. 11429401).\nAppendix A: CA VITY USED IN THE EXPERIMENT\nFigure. 5 shows the three-dimensional (3D) rectangular\ncavity used in our experiment. It has inner dimensions of\n44:0\u000220:0\u00026:0 mm3and contains three ports. The Port\n1 (2) is used for the probe field into (out of) the cavity and\nthe Port 3 is used for inputting the drive field [Fig. 5(a)]. The\ndrive antenna is placed just beside the YIG sphere [Figs. 5(b)\nand 5(c)] and it is connected to a superconducting microwave\nline that goes into the cavity via port 3. This makes it e \u000ecient\nto pump the YIG sphere with a drive field.\nFIG. 5. (a) The 3D cavity, which is made of oxygen-free copper\nand plated with gold. (b) The drive antenna, which is connected to\na superconducting microwave line that goes into the cavity via port\n3. (c) The small YIG sphere, which has a diameter of 1 mm and is\nplaced near the antenna and glued on the inner wall of the cavity.6\nAppendix B: HAMILTONIAN OF THE COUPLED HYBRID\nSYSTEM\nThe hybrid system shown in Fig. 5 consists of a small YIG\nsphere coupled to a 3D rectangular cavity and driven by a\nmicrowave field. Its Hamiltonian can be written as (setting\n~=1)\nH=Hc+Hm+Hint+Hd: (B1)\nHere Hc=!cayais the Hamiltonian of the cavity mode TE 102\nused in our experiment, with !canday(a) being the frequency\nand creation (annihilation) operator of the cavity mode, re-\nspectively. When Zeeman energy, demagnetization energy\nand magnetocrystalline anisotropy energy are included, the\nHamiltonian of the YIG sphere, which has a volume Vm, can\nbe written as [29]\nHm=\u0000Z\nVmM\u0001B0d\u001c\n\u0000\u00160\n2Z\nVmM\u0001(Hde+Han)d\u001c(B2)\nwhere\u00160is the magnetic permeability of free space, B0=\nB0ezis the static magnetic field applied in the zdirection\nwhich is aligned along the crystalline axis [100] of the YIG\nsphere in our experiment, Mis the magnetization of the\nYIG sphere, Hdeis the demagnetizing field induced by the\nstatic magnetic field, and Hanis the anisotropic field caused\nby the magnetocrystalline anisotropy in YIG. For a uni-\nformly magnetized YIG sphere, the induced demagnetizing\nfield is [29] Hde=\u0000M=3, and the anisotropic field is [30]\nHan=\u0000(2Kan=M2)Mz, where only the dominant first-order\nanisotropy constant Kanis taken into account and Mis the\nsaturation magnetization. Then, the Hamiltonian in Eq. (B2)\nbecomes\nHm=\u0000B0MzVm+\u00160\n6M2Vm+\u00160Kan\nM2M2\nzVm: (B3)\nThe YIG sphere can act as a macrospin S=MVm=\r\u0011\n(Sx;Sy;Sz), where\r=g\u0016B=~is the gyromagnetic ration [33],\nwith gbeing the g-factor and \u0016Bthe Bohr magneton. With the\nmacrospin operator introduced, the Hamiltonian Hmreads\nHm=\u0000\rB0Sz+\u00160Kan\r2\nM2VmS2\nz: (B4)\nwhere we have neglected the constant term \u00160M2Vm=6. The\ninteraction Hamiltonian between the macrospin and the cavity\nmode is\nHint=gs(S++S\u0000)(ay+a)\u00112gsSx(ay+a); (B5)\nwhere gsdenotes the coupling strength between the macrospin\nand the cavity mode, and S\u0006\u0011Sx\u0006iSyare the raising\nand lowering operators of the macrospin, respectively. In our\nexperiment, the YIG sphere (i.e., the macrospin) is directly\npumped by a drive field with frequency !d. The interaction\nbetween the macrospin and the drive field is\nHd= \n s(S++S\u0000)(ei!dt+e\u0000i!dt)\u00114\nsSxcos(!dt);(B6)where \nscharacterizes the coupling strength of the drive field\nwith the macrospin.\nThe macrospin operators are related to the magnon opera-\ntors via the Holstein-Primako \u000btransformation [32]:\nS+=\u0012p\n2S\u0000byb\u0013\nb;\nS\u0000=by\u0012p\n2S\u0000byb\u0013\n; (B7)\nSz=S\u0000byb;\nwhere Sis the total spin number of the macrospin opera-\ntor and by(b) is the creation (annihilation) operator of the\nmagnon with frequency !m. For the low-lying excitations\nwithhbybi=2S\u001c1, one has S+\u0019bp\n2S, and S\u0000\u0019byp\n2S.\nThen, the Hamiltonian in Eq. (B1) becomes\nH=!caya+!mbyb+Kbybbyb\n+gm(a+ay)(b+by)\n+ \n d(b+by)(ei!dt+e\u0000i!dt);(B8)\nwhere!m=\rB0\u00002\u00160Kan\r2S=(M2Vm) is the frequency of\nthe magnon mode, K=\u00160Kan\r2=(M2Vm) is a coe \u000ecient\ncharacterizing the strength of the nonlinear magnon e \u000bect,\ngm=p\n2S gsdenotes the magnon-photon coupling strength,\nand\nd=p\n2S\nsdenotes the coupling strength of the drive\nfield with the magnon mode. In the rotating-wave approxima-\ntion, the Hamiltonian is reduced to\nH=!caya+!mbyb+Kbybbyb+gm(ayb+aby)\n+ \n d(bye\u0000i!dt+bei!dt):(B9)\nNote that because the YIG sphere contains a very large num-\nber of spins, the condition hbybi=2S\u001c1 for the low-lying\nexcitations can be easily satisfied [8], even when considerable\nmagnons are generated by the drive field.\nAppendix C: EFFECTIVE HAMILTONIAN IN THE\nDISPERSIVE REGIME\nFor convenience of calculations, we first transform the\nHamiltonian Hin Eq. (B9) to a rotating reference frame with\nrespect to the frequency of the drive field by the unitary trans-\nformation\nR1=exp(\u0000i!dayat\u0000i!dbybt); (C1)\ni.e.,\nH0=Ry\n1HR 1\u0000iRy\n1@R1\n@t\n=!caya+!mbyb+Kbybbyb+gm(ayb+aby)\n+ \n d(by+b)\u0000(!daya+!dbyb)\n=\u000ecaya+\u000embyb+Kbybbyb+gm(ayb+aby)\n+ \n d(by+b);(C2)7\nwith\u000ec(m)\u0011!c(m)\u0000!d. Here the coupled hybrid system is in\nthe strong coupling regime, i.e., gm\u001d\u0014;\r m, where\u0014(\rm) is\nthe decay rate of the cavity (magnon) mode. The Hamiltonian\n(C2) can be divided into two parts, H0=H0+HI, with the\nfree part\nH0=\u000ecaya+\u000embyb+Kbybbyb+ \n d(by+b); (C3)\nand the interaction part\nHI=gm(ayb+aby): (C4)Below we use a Fr ¨ohlich-Nakajima transformation to re-\nduce the Hamiltonian H0. It needs to find a unitary transfor-\nmation U=exp(V), where Vis an anti-Hermitian operator\nVy=\u0000Vand satisfies [ H0;V]+HI=0. Up to the second\norder, the reduced Hamiltonian is given by\nH0\ne\u000b=UyH0U\u0019H0+1\n2[HI;V]: (C5)\nWe choose V=\u00151(ayb\u0000aby)+\u00152(ay\u0000a). Then,\n[H0;V]+HI=h\n\u000ecaya+\u000embyb+Kbybbyb+ \n d(by+b);\u00151(ayb\u0000aby)+\u00152(ay\u0000a)i\n+gm(ayb+aby)\n=\u00151(\u000ec\u0000\u000em)(ayb+aby)\u0000\u00151Kh\n(2byb+1)ayb+aby(2byb+1)i\n\u0000\u00151\nd(ay+a)\n+\u00152\u000ec(ay+a)+gm(ayb+aby):(C6)\nIn our experiment, we use a drive field to directly pump the\nYIG sphere, so as to generate considerable magnons. In this\ncase, the mean-field approximation can be applied to the term\nbybin Eq. (C6). Because hbybi\u001d 1, Eq. (C6) can approxi-\nmately be written as\n[H0;V]+HI\u0019h\n\u00151(\u000ec\u0000\u000em)\u00002\u00151Khbybi+gmi\n(ayb+aby)\n+(\u0000\u00151\nd+\u00152\u000ec)(ay+a):\n(C7)\nUsing the relation [ H0;V]+HI=0, we get\n\u00151(\u000ec\u0000\u000em)\u00002\u00151Khbybi+gm=0;\n\u0000\u00151\nd+\u00152\u000ec=0;(C8)\nwhich give\n\u00151=\u0000gm\n\u000ec\u0000\u000em\u00002Khbybi\n=\u0000gm\n\u0001\u00002Khbybi;\n\u00152=\nd\n\u000ec\u00151=\u0000\nd\n\u000ecgm\n\u0001\u00002Khbybi;(C9)\nwhere \u0001 =!c\u0000!m. Therefore, Vhas the form\nV=\u0000gm\n\u0001\u00002Khbybi(ayb\u0000aby)\n\u0000\nd\n\u000ecgm\n\u0001\u00002Khbybi(ay\u0000a):(C10)\nAlso, we apply the mean-field approximation to the Kerr term\nin Eq. (C5). Then, the Hamiltonian (C5) becomes\nH0\ne\u000b\u0019H0+1\n2[HI;V]\n\u0019\u0014\n\u000ec+g2\nm\n\u0001+2g2\nm\n\u00012Khbybi\u0015\naya+\u0014\n\u000em\u0000g2\nm\n\u0001+ \n1\u00002g2\nm\n\u00012!\nKhbybi\u0015\nbyb\n+\n0\nd(by+b); (C11)\nwith\n\n0\nd=\u0014\n1\u00001\n2\u000ec\u0012g2\nm\n\u0001+2g2\nm\n\u00012Khbybi\u0013\u0015\n\nd: (C12)\nFinally, we further rotate the reduced Hamiltonian H0\ne\u000bus-\ning the unitary transformation\nR2\u0011Ry\n1=exp(i!dayat+i!dbybt); (C13)\nwhich is the inverse transformation of R1in Eq. (C1). The\nderived Hamiltonian is given by\nHe\u000b=Ry\n2H0\ne\u000bR2\u0000iRy\n2@R2\n@t\n=\u0014\n!c+g2\nm\n\u0001+2g2\nm\n\u00012Khbybi\u0015\naya\n+\u0014\n!m\u0000g2\nm\n\u0001+ \n1\u00002g2\nm\n\u00012!\nKhbybi\u0015\nbyb\n+\n0\nd(bye\u0000i!dt+bei!dt): (C14)\nThis is the e \u000bective Hamiltonian of the coupled hybrid sys-\ntem obtained in the dispersive regime [i.e., Eq. (4)]. In our\nexperiment, the drive field is tuned to be in resonance with the\nmagnon mode,\n!d=!m\u0000g2\nm\n\u0001+ \n1\u00002g2\nm\n\u00012!\nKhbybi: (C15)8\nAppendix D: RELATION BETWEEN THE MAGNON\nFREQUENCY SHIFT AND THE DRIVE POWER\nWith Hamiltonian (C2), we can obtain the quantum\nLangevin equations for the coupled hybrid system,\nda\ndt=\u0000i\u000eca\u0000igmb\u0000\u0014\n2a;\ndb\ndt=\u0000i\u000emb\u0000i(2Kbyb+K)b\n\u0000igma\u0000i\nd\u0000\rm\n2b:(D1)\nHere we write the operator a(b) as a sum of the steady-state\nvalue and the fluctuation, i.e., a=A+\u000eaandb=B+\u000eb. It\nfollows from Eq. (D1) that AandBsatisfy\ndA\ndt=\u0000i\u000ecA\u0000igmB\u0000\u0014\n2A;\ndB\ndt=\u0000i\u000emB\u0000i(2KjBj2+K)B\n\u0000igmA\u0000i\nd\u0000\rm\n2B:(D2)\nFrom Eq. (C15), we have \u000em\u0011!m\u0000!d\u0019g2\nm=\u0001\u0000KjBj2,\nbecause \u0001\u001dgmin the dispersive regime. Also, \u000ec\u0011!c\u0000\n!d= \u0001 +\u000em\u0019\u0001. At the steady states for both AandB,dA=dt=0 and dB=dt=0. Then, it follows from Eq. (D2) that\n\u0000i\u0001A\u0000igmB\u0000\u0014\n2A=0;\n\u0000i\u0012\nKjBj2+g2\nm\n\u0001\u0013\nB\u0000igmA\u0000i\nd\u0000\rm\n2B=0:(D3)\nEliminating Ain Eq. (D3), we get\n\u0014\nKjBj2+g2\nm\n\u0001\u0000g2\nm\n\u0001\u0000i(\u0014=2)\u0000i\rm\n2\u0015\nB+ \n d=0: (D4)\nBecause \u0001\u001d\u0014and\rm\u001d\u0014, Eq. (D4) is reduced to\n\u0012\nKjBj2\u0000i\rm\n2\u0013\nB+ \n d=0: (D5)\nUsing Eq. (D5) and its complex conjugate expression, we ob-\ntain\n\u0014\u0010\nKjBj2\u00112+\u0012\rm\n2\u00132\u0015\njBj2\u0000\n2\nd=0: (D6)\nIn our experiment, the measured frequency shift of THE\nmagnons is\n\u0001m=\u0012\n1\u00002g2\nm\n\u00012\u0013\nKhbybi\u0019Khbybi: (D7)\nNote thathbybi=jBj2for small fluctuation \u000eb, corresponding\nto the case with considerable magnons generated in the YIG\nsphere. 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Melkov, Magnetization Oscillations\nand Waves (CRC, Boca Raton,FL, 1996)."    },    {        "title": "2107.06508v3.Sublattice_spin_reversal_and_field_induced__Fe__3____spin_canting_across_the_magnetic_compensation_temperature_in__Y__1_5_Gd__1_5_Fe__5_O__12___rare_earth_iron_garnet.pdf",        "content": "arXiv:2107.06508v3  [cond-mat.mtrl-sci]  13 Sep 2021Sublattice spin reversal and field induced Fe3+spin-canting across the magnetic\ncompensation temperature in Y 1.5Gd1.5Fe5O12rare-earth iron garnet\nManik Kuila,1Jose Mardegan,2Akhil Tayal,2Sonia Francoual,2and V.Raghavendra Reddy1,∗\n1UGC-DAE Consortium for Scientific Research, University Cam pus, Khandwa Road, Indore 452001, India.\n2Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, 2 2607 Hamburg, Germany.\n(Dated: September 14, 2021)\nIn the present work Fe3+sublattice spin reversal and Fe3+spin-canting across the magnetic\ncompensation temperature (T Comp) are demonstrated in polycrystalline Y 1.5Gd1.5Fe5O12(YGdIG)\nby means of in-field57FeM¨ossbauer spectroscopy measurements. Corroborating in-fiel d57Fe\nM¨ossbauer measurements, both Fe3+& Gd3+sublattice spin reversal has also been manifested\nwith x-ray magnetic circular dichroism (XMCD) measurement in hard x-ray region. Moreover from\nin-field57FeM¨ossbauer measurements, estimation and analysis of effective internal hyperfine field\n(Heff), relative intensity of absorption lines in a sextet elucid ated unambiguously the signatures of\nFe3+spin reversal, their continuous transition and field induce d spin-canting of Fe3+sublattices\nacross T Comp. Further, Fe K- (Gd L 3-) edge XMCD signal is observed to consist of additional spec -\ntral features, those are identified from Gd3+(Fe3+) magnetic ordering, enabling us the extraction\nof both the sublattices (Fe3+& Gd3+) information from a single edge analysis. The evolution of\nthe magnetic moments as a function of temperature for both ma gnetic sublattices extracted either\nat the Fe K- or Gd L3-edge agree quite well with values that are extracted from bulk magnetization\ndata of YGdIG and YIG (Y 3Fe5O12). These measurements pave new avenues to investigate how\nthe magnetic behavior of such complex system acts across the compensation point.\nI. INTRODUCTION\nRare-earth iron garnets (RIG), R 3Fe5O12, where R\nis rare-earth (Y, & La-Lu) have become an important\nclass of ferrimagnetic oxide materials finding a signif-\nicant role in many microwave, bubble memories and\nmagneto-optical device applications [1–4]. Remarkable\nintriguing magnetic properties and their chemical sta-\nbility with a large variety of elemental substitutions are\none of the prime reasons for exploring these materials\nby various groups since their discovery [5–11]. Among\ntheseRIGsystems, yttrium irongarnet(Y 3Fe5O12, YIG)\nreceived a renewed attention in recent years owing to\nits low damping, low optical absorption, magneto-optical\nswitching, thermoelectric generation in spin Seebeck in-\nsulators and other spintronics, magneonic based applica-\ntion [12–15]. Doping of magnetic light rare-earths (e.g.,\nCe, Nd andGd) orevenheaviermetals, suchasBiatoms,\non the yttrium sites yield a significantly enhanced visi-\nble to near-infraredFaraday /Kerrrotation and magneto-\noptical (MO) figure of merit, without losing their mag-\nnetic insulator character [16–20].\nRIG consist of three different magnetic ions viz., two\nin-equivalent Fe3+ions located at tetrahedral ( d-) and oc-\ntahedral ( a-) oxygen polyhedron and the third one is the\nR3+ion situated at dodecahedral ( c) oxygen polyhedron\n[5]. The d- anda- site Fe3+are always coupled anti-\nparallel and the resultant moment of Fe3+is also coupled\nanti-parallel with the R3+. These two magnetic sublat-\ntices exhibit quite a different temperature dependance\nand as a result there exists a point in temperature at\n∗varimalla@yahoo.com; vrreddy@csr.res.inwhich the resultant Fe3+magnetic moments are equal\nand opposite to the R3+magnetic moments resulting in\nzero total magnetization [6], known as magnetic com-\npensation temperature (T Comp). Therefore, a thorough\nunderstanding of the macroscopic magnetic properties in\nRIG compounds is achieved from the knowledge of the\ndifferent sublattices magnetization [21–27].\nUsually at low temperatures i.e., T <TCompthe R3+\nmoments dominate the macroscopic magnetic prop-\nerties and align along the externally applied mag-\nnetic field (H ext), whereas for intermediate temperature,\nTComp<T<TC(Curie temperature) it is the resultant\nFe3+moments which will dominate and align along the\nHext. The net magnetization (M) always align along the\nHext. Therefore, the magnetic sub-lattices rearrange ac-\ncordingly whether the temperature of the system is be-\nlow or above the T Compas shown schematically in Fig-\nure. 1. Now, if the R-atom is non-magnetic (like as in\nYIG systems) or for temperatures above the T Comp, the\ndominant Fe3+atd- sites decide the resultant magneti-\nzation direction, whereas for temperatures below T Comp,\nthed- site Fe3+moments will be in opposite direction\nto Hext. However, the competition between H extthat\nalways tend to align all the moments parallel to it and\nthe strong anti-ferromagnetic super-exchange interaction\nbetween the sublattices result field induced spin-canted\nphase (between magnetic R3+and resultant Fe3+) close\nto TComp, which is shown to be a second-order phase\ntransition in literature [21–23].\nIn this context, the in-field57FeM¨ossbauer spec-\ntroscopy is an ideal method to track the evolution of\nFe3+sublattice magnetization, demonstrate their inver-\nsion acrossT Compand also probe the spin-canting unam-\nbiguously by analyzing the variation of effective internal\nfield (H eff), which is the vector sum of internal hyper-2\nFIG. 1. Schematic diagram showing the relation between in-\nternal (H int), effective (H eff) hyperfine and externally applied\n(Hext)fieldsbelowandabovemagneticcompensation (T Comp)\ntemperature. M’s represent the magnetic moments and the\ncorresponding arrows are direction of the moments of the re-\nspective sublattice.\nfine field (H int), Hext& the relative line intensities in a\ngiven six-line pattern [28, 29]. However, there seems to\nbe no systematic in-field M¨ ossbauer study across T Comp\nin RIG systems in literature. Most of the M¨ ossbauer lit-\nerature focused on the estimation of cation distribution\nin RIG systems, except the work of Stadnik et al., and\nSeidel et al [30, 31]. Seidel et al., reported that there\nis no spin-canting in Gd 3Fe5O12across T Compwith zero-\nfield M¨ossbauer measurements[30] and Stadnik et al., re-\nported canting of the Fe3+spins in Sc doped Eu 3Fe5O12\nwith in-field M¨ ossbauer measurements [31].\nThe in-field57FeM¨ossbauer measurements reported\nin this paper, are further complimented by x-ray mag-\nnetic circular dichroism (XMCD) measurements. In RIG\nsystems, most of the XMCD investigation is focused on\nthe transition metal (TM) L 2,3(2p→3d) and R- M 4,5\n(3d→4f) absorption edges, since in both cases the mag-\nnetism is directly probed via electric-dipole transitions\n[24, 25]. Recently studies have shown that due to signifi-\ncant improvement in the 3rd generation synchrotron, the\nXMCD method is also possible to be carried out with\nmuch higher signal to noise ratio even at hard x-rays\nregime i.e., across the Fe-K and R L-edges [32]. Fur-\nther it is shown in the previous literature that Fe K-edge\n(R L-edge) XMCD signal is strongly influenced by the\nR-atom (Fe-atom) magnetic ordering in rare-earth ironcompounds such as RFe 2, NdFeB and RIGs. Therefore,\nby performing XMCD measurements in hard x-ray re-\ngion across a single edge, one would be able to get the\nelement specific magnetic information from both Fe and\nR sublattices in these type of compounds [21, 22, 33–35].\nThe present work reports the temperature dependent\nin-field57FeM¨ossbauer spectroscopy and XMCD mea-\nsurements on polycrystalline Y 1.5Gd1.5Fe5O12(YGdIG)\nand Y 3Fe5O12(YIG) samples with the aim of tracing\nsublattice spin across T Comp, looking for the possible\nspin-canting across T Compand decomposition of XMCD\ndata into Gd-like and Fe-like spectra from a single edge\n(i.e., Gd L 3-edge or Fe K-edge) measurement.\nII. EXPERIMENTAL\nPolycrystalline Y 1.5Gd1.5Fe5O12and Y 3Fe5O12sam-\nples are prepared with conventional solid-state-reaction\nmethod starting with high purity ( ≥99.9%) oxide pre-\ncursors. The structural characterization of the prepared\nsamples is carried out with Brucker D8-Discover x-ray\ndiffraction system equipped with LynxEye detector and\nCu-Kαsource.57FeM¨ossbauer measurements are car-\nriedout in transmissionmode usinga standardPC-based\nM¨ossbauer spectrometer equipped with a WissEl veloc-\nity drive in constant acceleration mode. The velocity\ncalibration of the spectrometer is done with natural iron\nabsorber at room temperature. For the low temperature\nhighmagneticfield M¨ ossbauermeasurements,the sample\nis placed inside a Janis superconducting magnet and an\nHextis appliedparalleltothe γ-rays. Bulkmagnetization\nmeasurementsare performed with vibrating sample mag-\nnetometer (VSM). X-ray absorption spectroscopy (XAS)\nmeasurements are carried out at beamline P09 at PE-\nTRA III (DESY) at low temperature at the Gd L 3and\nFe K-absorptionedges in transmissiongeometryin which\nboth incident and transmitted x-ray beams are recorded\nusing silicon photo-diodes. The pellets, prepared from\nsampleswithsuitableamountofboronnitrateadd-mixer,\nare cooled down by an ARS cryostat with temperature\nrange between 5 to 300 K. XAS /XMCD measurements\nwere performed fast-switching the beam helicity between\n/s40/s98/s41/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s32/s89/s111/s98/s115\n/s32/s89/s99/s97/s108 /s99\n/s32/s89/s111/s98/s115/s45/s89/s99/s97/s108 /s99\n/s32/s66/s114/s97/s103/s103/s95/s112/s111/s115/s105/s116/s105/s111/s110/s89/s73/s71/s40/s97/s41\n/s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48/s89/s71/s100/s73/s71\n/s50 /s113 /s32/s40/s100/s101/s103/s41/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48/s77/s32/s40/s101/s109/s117/s47/s103/s109/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s32/s89 /s71/s100/s73/s71\n/s32/s89 /s73/s71\n/s84\n/s67/s111/s109/s112\nFIG. 2. (a) X-ray diffraction data of Y 1.5Gd1.5Fe5O12and\nY3Fe5O12(b) M-T data measured in field-cooled protocol\nwith 500 Oe field. The magnetic compensation (T Comp) is\nclearly seen for YGdIG as indicated by black dash line.3\nleft and right circular polarization to improve the signal-\nto-noise ratio. The dichroic signal is revealed when the\ndifference in the absorption spectra for left and right cir-\ncularly polarized radiation is performed [36]. In order to\nalign the domains and to correct for nonmagnetic arti-\nfacts in the XMCD data, an external magnetic field of\napproximately 1 T was applied parallel and antiparallel\nto the incident beam wave vector→\nk.\nIII. RESULTS\nA. Structural and magnetic characterization\nFigure. 2(a) shows the XRD patterns of the YIG and\nYGdIG samples and the data confirms the phase purity\nof the prepared samples. Further, the XRD data is an-\nalyzed with FullProf Rietveld refinement [37] program\nconsidering the Ia-3dspace-group for the estimation of\nlattice parameters and the obtained lattice parameters\nare 12.374(1) ˙Aand 12.424(1) ˙Afor YIG and YGdIG,\nrespectively, which also match with the literature [38].\nFigure. 2(b) shows the temperature dependent (M-T)\nmagnetizationdataoftheYIG andYGdIG samples. One\ncan clearly see the magnetic compensation temperature\n(TComp)forYGdIGatabout126Kandnosuchsignature\nis seen for YIG as expected. Since, the bulk magnetiza-\ntion data of YIG is considered to be coming only from\nFe3+moments this gives a rough idea about the con-\ntribution of Fe3+sublattice to the total magnetization\nin YGdIG. So, one can get an estimated contribution of\nGd3+sublattice tothe overallmomentin YGdIG bysuit-\nably subtracting the bulk magnetization data of YGdIG\nfrom YIG data. This method is employed to cross-check\nthe reliability of the XMCD data analysis results as dis-\ncussed in the later sections.\n/s45/s49/s48 /s45/s53 /s48 /s53 /s49/s48/s48/s46/s57/s56/s49/s46/s48/s48/s48/s46/s57/s56/s49/s46/s48/s48/s48/s46/s57/s56/s49/s46/s48/s48/s48/s46/s57/s56/s49/s46/s48/s48\n/s86/s101/s108/s111/s99/s105/s116/s121 /s32/s40/s109/s109/s47/s115/s41/s82/s101/s108/s97/s116/s105/s118/s101/s32/s84/s114/s97/s110/s115/s109/s105/s115/s115/s105/s111/s110/s32/s32/s70/s101/s51/s43\n/s40/s100 /s45/s115/s105/s116/s101/s41\n/s32/s70/s101/s51/s43\n/s40/s97 /s45/s115/s105/s116/s101/s41\n/s49/s53/s48/s32/s75/s49/s50/s55/s32/s75/s49/s49/s54/s32/s75/s53/s32/s75\nFIG. 3.57FeM¨ossbauer data of Y 1.5Gd1.5Fe5O12at the se-\nlected temperatures. Black symbols represent the experime n-\ntal data and the black solid line is the best fit to the data.B.57FeM¨ossbauer spectroscopy results\nFigure. 3 shows the temperature dependent57Fe\nM¨ossbauer spectra of YGdIG measured across T Comp.\nOne can clearly see two components corresponding to\nFe3+sublattices located at d- anda- sites and the ob-\ntained hyperfine parameters match with literature val-\nues of garnet [30, 40, 41]. Values of H int, area fraction\nand A 23are shown in Table-1. Further, the area frac-\ntion of Fe3+atd- anda- sites is found to remain same\nat all the temperatures and the values of A 23(area ra-\ntio of second and third lines in a given sextet) is fixed\nas 2.0 in accordance with the random distribution of θ\ncorresponding to the powder samples measured in zero-\nfield conditions [28]. However, the in-field M¨ ossbauer\nmeasurements across T Compwith 5 T external magnetic\nfield applied parallel to the γ-rays, as shown in Figure. 4,\nFIG. 4. In-field57FeM¨ossbauer data of Y 1.5Gd1.5Fe5O12at\nthe selected temperatures under 5 T magnetic field applied\nparallel to the γ-rays. Black symbols represent the exper-\nimental data and the black solid line is the best fit to the\ndata. Blue arrow represents Fe phase1aligned anti-parallel to\nHext, orange arrow represents Fe phase2aligned parallel to H ext\nand the red arrow represents the resultant of these two. H ext\nis the applied external field which is parallel to the inciden t\nγ-rays.4\nTABLE I. Hyperfine parameters obtained from the analysis\nof zero field57FeM¨ossbauer spectroscopy data as shown in\nFigure. 3. δis centre shift, ∆E is quadrupole shift, H intis the\ninternal hyperfine field, A 23is the area ratio of second and\nthird lines of the respective sextet.\nT (K)δ(mm/s) ∆E(mm/s) H int(T) A 23% Area Fe3+\n±0.01 ±0.02 ±0.1 ±2 Site\n5 0.29 0.01 49.5 2.00 58 d\n0.57 0.11 56.9 2.00 42 a\n116 0.27 0.01 48.5 2.00 58 d\n0.56 0.10 56.4 2.00 42 a\n125 0.26 -0.01 48.3 2.00 57 d\n0.56 0.13 56.4 2.00 43 a\n127 0.26 0.02 48.4 2.00 57 d\n0.55 0.08 56.3 2.00 43 a\n136 0.26 0.01 48.4 2.00 58 d\n0.54 0.09 56.3 2.00 42 a\n150 0.25 0.01 48.2 2.00 59 d\n0.55 0.08 56.3 2.00 41 a\nTABLE II. Representative hyperfine parameters obtained\nfrom the analysis of in-field57FeM¨ossbauer spectroscopy\ndata as shown in Figure. 4. H effis the effective field which\nis the vector sum of H ext, Hintand demagnetizing fields [39].\nA23is the area ratio of second and third lines of the respec-\ntive sextet. Fe phase1denotes the phase aligned anti-parallel to\nHextand Fe phase2denotes the phase aligned parallel to H ext\nas shown in Figure. 4.\nT (K) H eff(T) A 23% Area Fe3+Phase\n±0.1±0.1 ±2 Site\n5 53.2 0.00 58 dFePhase1\n54.9 0.00 42 aFePhase1\n130 51.6 1.22 25 dFePhase1\n53.8 1.22 17 aFePhase1\n45.3 1.22 34 dFePhase2\n61.1 1.22 24 aFePhase2\n150 43.6 0.10 59 dFePhase2\n61.1 0.10 41 aFePhase2\nreveal a very interesting information regarding the rever-\nsal of Fe3+moments, spin-canting etc., as elaborated in\nsection-IV.\nC. Hard x-ray magnetic circular dichroism\n(XMCD) results\nFigures. 5 and 6 show the XMCD data measured at\nthe Fe K- and Gd L 3- edges of YGdIG sample at differ-\nent temperatures. For the comparison purpose, the Fe\nK-edge XMCD data of YIG sample is also shown in Fig-\nure. 5. As a representative, XAS data collected at the FeK-edge and at the Gd L 3- edges for the YGdIG sample\nare also shown in Figure. 5 (a) and 6 (a), respectively.\nA clear reversal of XMCD signals at Fe K- and Gd L 3-\nedges are observed in YGdIG sample when measured at\n5 and 290 K i.e., far below and above the T Comp. Addi-\ntional features that are only observed in the Fe K-edge\nspectra of YGdIG in between 7141 and 7154 eV photon\nenergy, could be due to induced signal of Gd magnetic\nmoments. It may be noted that such a clear signal of\nrare-earth contribution in Fe K-edge XMCD data was\nnot detected previously in RIG systems, even though the\nopposite case i.e., Fe contribution in rare-earth L-edge\nXMCD is observed in many RIG systems [22, 42–44].\nHowever, unlike RIG systems, the rare-earth contribu-\ntion wasobservedat Fe K-edgeXMCD data in rare-earth\ntransition metal intermetallics (RTI), which could be due\nto the fact that the R(5d) electronic orbitals hybridize\nwith the outermost states of absorbing Fe(3d) in RTI\nsystems whereas it is mediated via oxygen in case of RIG\nsystems [21, 33, 34]. The direct subtraction method from\na reference spectrum (YIG, where Y nonmagnetic at c-\nsite) is employed to separate out the Gd contribution at\nevery temperature, as shown in Figure. 5(i)-(p), and is\nfurther discussed in the following sections.\nGd L3- edge XMCD spectrum can broadly be charac-\nterized by the structures labeled as P1, P2, and P3 (Fig-\nure. 6). The peak P1, in the lower energy region that is\neventuallyburied in the profile ofpeak P2 at low temper-\natures, can be ascribed to the Fe contribution that be-\ncomes visible at temperatures higher than T Compwhere\nGd magnetic moments are weak. Whereas, P2 and P3\nmainly originate from the Gd electronic states [42–44].\nThe Gd L 3- edge XMCD spectra are analyzed with sin-\ngular value decomposition (SVD) method to extract the\nFe3+contribution as discussed in the following section.\nIV. DISCUSSIONS\nThe in-field57FeM¨ossbauer data (Figure. 4) is ana-\nlyzed considering two Fe sites at temperatures well below\nand above T Comp(5, 100 and 150 K) which correspond\ntod- anda- sites of Fe3+. Representative H effvalues for\nsomeofthetemperaturesareshowninTable-II..H extwill\nbe added (subtracted) to the H intofa- site above (be-\nlow) the T Compand it will be vice versa for the d- site as\nshown schematically in Figure. 1. Hence, the H extfor the\ntwo spectral components corresponding to d- anda- sites\nwill be very close to each other below T Comp, whereas\nsignificant difference will be observed above T Comp. As\na result of this, one would observe well resolved two sex-\ntets corresponding to d- anda- sites above T Compand\na single broad sextet due to the overlapping components\nbelow T Compas shown in Figure. 4, specifically at tem-\nperatures of 150 K and 5 K. Therefore, the obtained H eff\nvalues which is the vector sum of H intand H extunam-\nbiguously demonstrates the reversal of Fe3+sublattice\nacross T Compin YGdIG, which can be generalized to all5\n/s32/s89/s71/s100/s73/s71/s32/s88/s65/s83\nFIG. 5. (a)-(h) Temperature variation of Fe K-edge X-ray mag netic circular dichroism data of Y 1.5Gd1.5Fe5O12(YGdIG) and\nY3Fe5O12(YIG) at the indicated temperatures. (i)-(p) shows the enla rged view of region of interest depicting the development\nof Gd contribution at low temperatures. The Gd contribution is extracted by subtracting the YGdIG data from YIG data and\nits temperature variation is shown in Figure. 9\n/s32/s88/s65 /s83/s32/s100/s97/s116/s97\n/s88/s65/s83/s32/s40/s65/s114/s98/s46/s32/s85/s110/s105/s116/s115/s41\nFIG. 6. Temperature variation of Gd L 3-edge X-ray magnetic\ncircular dichroism data of Y 1.5Gd1.5Fe5O12. Symbols repre-\nsent the experimental data, and red lines are reconstructed\nspectrausingthefirsttwodominating contributionswhich a re\nshown by shaded areas. Deep green represents Gd- contribu-\ntion and cyan associated with the Fe- contribution. Labelin g\nof the peaks and the data fitting carried out using singular\nvalue decomposition (SVD) method are discussed in the text.\nX-ray absorption spectra of the sample measured at 5 K is\nalso shown as a representative data.RIG systems exhibiting magnetic compensation.\nHowever, an inspection of the in-field M¨ ossbauer data\ncloseto T Comprevealthe presenceofmorethan twocom-\nponents. This is very clearly seen at 127, 130, 136 K and\na careful inspection also reveals the presence of these\ncomponents at 116 and 120 K as shown in Figure. 4.\nThe spectra could be resolved into four sites correspond-\ning to four Fe3+sublattices of two ferrimagnetic phases\nof YGdIG i.e., the phases with resultant Fe3+moment\naligned along (Fe phase1) and opposite (Fe phase2) to H ext\nacrossT Comp. Asmentionedintheprecedingsection, the\nsignificant contrast between these two phases in terms of\nHeffenable the quantitative study of their evolution with\ntemperature. In view of this, the data at these temper-\natures is fitted with four sextets and the obtained phase\nfractions of these two phases (Fe phase1, Fephase2) is plot-\nted in Figure. 7. As the temperature is changing, one can\nsee that there is a gradual change in the area fraction of\nthese phases indicating a continuous spin reversal across\nTComp.\nIn addition to this information, the inspection of in-\nfield M¨ossbauer data close to T Compreveal the presence\nof considerable intensity for the second and fifth lines\n(corresponding to ∆m=0 transitions) of a given sextet\nunlike the data of 5, 100 and 150 K i.e., well below and\nabove T Comp. Quantitatively this is estimated by A 23\nparameter for given sextet and the value of A 23is found\nto be zero at temperatures 5, 100 and 150 K indicat-\ning a collinear magnetic structure of Fe3+[28]. How-\never, the in-field M¨ ossbauer data close to T Compis fit-\nted keeping A 23parameter as variable and is constrained\nto be same for all the sites. The obtained variation of6\nA23as a function of temperature is also shown in Fig-\nure. 7 and it is interesting to note that the magnetic\nstructure deviates from collinear configuration as one ap-\nproaches T Comp. This is considered to be a signature of\nspin-cantingasshownrecentlyfromthemagneticcircular\ndichroism [22] and spin Hall magnetoresistance [45] ex-\nperiments. Considering two-sublattice model (Fe3+and\nR3+sublattices), simulated magnetic field versus tem-\nperature (H-T) phase diagram of compensated RIG sys-\ntemsclearlyshowtheregionofcollinear,cantedferrimag-\nnetic and aligned phases [21–23, 45]. The present in-field\nM¨ossbauer study indicates the presence of such canted\nphases across T Compunambiguously. Across this region\nof temperatures, the two Fe phase1and Fe phase2phases\nare mixed up with forming similar but opposite angle at\na given temperature with respect to H ext. As a conse-\nquence, resultant Fe moment exhibits a continuous ro-\ntation across this canted region in accordance with the\ntwo sublattice model as shown schematically in Fig. 4.\nIt may be emphasized here that even the XMCD data\nmight not be able to distinguish the mixing of these two\ncomponents as it measures resultant projection of the Fe\nmagnetic moments along the beam direction.\nApart from showing a clear reversal of the magnetic\nsignals at the Fe K- and the Gd L 3- edges in YGdIG\nacorss the T Comp, the XMCD spectra reported in Fig-\nure. 5and 6 are employed to extract quantitative infor-\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48/s48/s53/s48/s49/s48/s48\n/s40/s99/s41/s65\n/s50/s51\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s32/s32/s70/s101\n/s80 /s104 /s97 /s115/s101 /s49 /s37/s32/s65/s114/s101/s97/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48/s48/s49/s48/s50/s48/s51/s48\n/s40/s98/s41/s77/s32/s40/s101/s109/s117/s47/s103/s109/s41/s32/s70/s67\n/s84\n/s67/s111 /s109/s112/s40/s97/s41\n/s48/s53/s48/s49/s48/s48\n/s32/s70/s101\n/s80 /s104 /s97 /s115/s101 /s50 \n/s37/s32/s65/s114/s101/s97\nFIG. 7. (a) M-T data, reproduced from Figure. 2(b). Tem-\nperature variation of (b) Phase fraction (c) area ratio of se c-\nond and third line intensities (A 23) as obtained parameters\nfrom the analysis of in-field57FeM¨ossbauer data as shown in\nFigure. 4./s55/s50/s51/s48 /s55/s50/s52/s48 /s55/s50/s53/s48 /s55/s50/s54/s48 /s55/s50/s55/s48/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s88/s77/s67/s68/s32/s40/s65/s114/s98/s46/s32/s85/s110/s105/s116/s115/s41\n/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s32/s71/s100/s32/s108/s105/s107/s101\n/s32/s70/s101/s32/s108/s105/s107/s101\nFIG. 8. Eigen or basis spectra of Gd-like and Fe-like com-\nponents calculated by singular value decomposition (SVD)\nmethod for the Gd L 3-edge XMCD data set.\nmation about both the sublattices magnetization from\na single absorption edge measurement. Mainly two ap-\nproachesareusedinliteraturetoextractthespectralcon-\ntributions of different sublattices from such complicated\nXMCD data [33, 44]. Subtraction method in which the\nmagnetic dichroic signal from the investigated system is\nsubtracted from a reference sample has been extensively\nexploited and has provided very reliable interpretation\n[33].In our present work, XMCD spectra obtained at the\nFe K-edge for the YIG sample are considered as reference\nto estimate the magnetic contribution from the Gd ions\nat the Fe K-edge. Using this simple methodology, the Gd\nmagnetic contribution can be obsevred via XMCD at the\nFe K-edge as shown in Figure. 5(i)-(p).\nIn order to check the reliability of thus obtained Gd3+\ncontribution, the total area of these features observed at\nthe Fe K post-edge are compared with the temperature\nvariation of Gd3+moment, as calculated from tempera-\nture dependent bulk magnetization data of YGdIG and\nYIG.The results using this approach are summarized in\nFig. 9(a). The bulk magnetization data of YIG is consid-\nered to have its origin basically from the Fe3+ions since\nwecanassumethatbothY3+andO2-ionshavenegligible\nmagnetic moments and their hybridization with the Fe3+\nions does not play an important role to the total mag-\nnetism of the system. Therefore, subtracting bulk mag-\nnetization data of YGdIG from the YIG data (Fig. 2),\nresults magnetic signal which is primarily from the Gd3+\nions. The sign of the obtained sublattice magnetization\nasafunctionoftemperatureischangedaccordingtotheir\nmagnetization direction w.r.t H extacross T Comp(Fig. 9\ndashed line). The extracted Gd3+XMCD amplitude\n(blue dot Fig. 9(a)) from Fe K edge XMCD data (us-\ning direct subtraction method) almost follow the trend\nof Gd3+moment variation as a function of temperature\nbelowthe T Comp. However,aboveT CompGd3+magnetic\ncontribution to Fe K-edge XMCD data is weaker and7\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s45/s52/s48/s45/s50/s48/s48/s50/s48/s52/s48/s54/s48\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s45/s52/s48/s45/s50/s48/s48/s50/s48/s52/s48/s54/s48\n/s40/s98/s41/s70/s114/s111/s109/s32/s70/s101/s32/s75/s32/s101/s100/s103/s101/s32/s88/s77/s67/s68/s32/s115/s112/s101/s99/s116/s114/s97/s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s32/s40/s101/s109/s117/s47/s103/s109/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101 /s40/s75/s41/s32/s89 /s71/s100/s73/s71\n/s32/s70/s101/s32/s109/s111/s109/s101/s110/s116/s32/s40/s89 /s73/s71/s41\n/s32/s71/s100/s32/s109/s111/s109/s101/s110/s116/s32 /s40/s89 /s71/s100/s73/s71/s45/s89 /s73/s71/s41\n/s40/s97/s41\n/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s32/s70/s101\n/s88 /s77/s67 /s68 \n/s70/s101\n/s88 /s77 /s67 /s68 /s32/s40/s65/s114/s98/s46/s32/s85/s110/s105/s116/s115/s41\n/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s32/s71/s100\n/s88 /s77/s67 /s68 \n/s71/s100\n/s88 /s77 /s67 /s68 /s32/s40/s65/s114/s98/s46/s32/s85/s110/s105/s116/s115/s41/s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s32/s40/s101/s109/s117/s47/s103/s109/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101 /s40/s75/s41/s32/s89 /s71/s100/s73/s71\n/s32/s70/s101/s32/s109/s111/s109/s101/s110/s116/s32/s40/s89 /s73/s71/s41\n/s32/s71/s100/s32/s109/s111/s109/s101/s110/s116/s32 /s40/s89 /s71/s100/s73/s71/s45/s89 /s73/s71/s41/s70/s114/s111/s109/s32/s71/s100/s32/s76\n/s51/s32/s101/s100/s103/s101/s32/s88/s77/s67/s68/s32/s115/s112/s101/s99/s116/s114/s97\n/s45/s52/s45/s50/s48/s50/s52/s54/s32/s70/s101\n/s88 /s77/s67 /s68 \n/s70/s101\n/s88/s77/s67 /s68 /s32/s40/s65/s114/s98/s46/s32/s85/s110/s105/s116/s115/s41\n/s45/s52/s48/s45/s50/s48/s48/s50/s48/s52/s48/s54/s48\n/s32/s71/s100\n/s88 /s77/s67 /s68 \n/s71/s100\n/s88/s77/s67 /s68 /s32/s40/s65/s114/s98/s46/s32/s85/s110/s105/s116/s115/s41\nFIG. 9. The two sublattice (Fe3+and Gd3+) magnetization\nvalues (proportional to XMCD amplitude) obtained from the\n(a) Fe K- and (b) Gd L 3- edge XMCD data analysis. The\nvalues are shown by symbols. Temperature variation of bulk\nmagnetization data of YGdIG sample (Fig. 2) and the calcu-\nlated contribution of Fe3+and Gd3+sublattices to the bulk\nmagnetization are shown in both frames. M-T data of YIG is\nconsidered as due to Fe3+, except that the sign of the data is\nchanged accordingly across T Comp.\nhence giving discrepancy to the comparison. We have\nalso compared Fe XMCD value (red symbols in Fig. 9(a),\npeak height of the XMCD dispersion spectra located at\nthe pre-edge region ( ≈7115 eV) with the temperature\nvariationoftheFe-onlymagnetizationdatafromtheYIG\nsample and it was observed similar trend (red dashed\nline) as shown in Fig. 9(a).\nAs mentioned above the direct subtraction method or\nlinear combination fit according to individual magnetic\nmoment value as function of temperature are mostly em-\nployed in literature to decompose the XMCD spectra of\nRIG and RTI systesm [33, 44]. However, recently it is\nshown that the application of singular value decomposi-\ntion (SVD) rationalizes previous approaches in a more\ngeneral framework and simplifies the exploration of mag-\nnetic phase diagramofsuch compounds[22, 35]. One can\nanalyze the shape and amplitude of the hidden compo-\nnentsin XMCDdatafromcorrelateddataset. Cornellius\net al., successfully analyzed and separated Fe contribu-\ntion from the Er L 2,3edges using SVD method [22]. The\nsame procedure is adopted in the present work also to\nde-convolute Fe3+contribution from temperature depen-dence Gd L 3edge XMCD data set (Figure. 6).\nAccording to SVD theorem any data matrix A(m×n)\ncan be decomposed into three matrices as A=UΣVT\nwhereU(m×m) andV(n×n) are orthogonal matrices\nand Σ(m×n) is diagonal matrix of singular values. We\nhave used eight XMCD spectra of Gd L 3- edge over the\ntemperature range (5-300 K) to form data matrix Aand\nMATLAB software is used to perform the SVD on the\ndata matrix Ato find the U, V,Σ. Only the first two\neigenvectors or basis which are dominated over the noise\nlevelareused to reconstructthe originaldataas shownin\nFigure. 8. The reconstructed plot along with separated\nFe and Gd like components to the original Gd L 3- edge\nXMCD data are shown in Figure. 6. It is to be men-\ntioned that spectral features of Fe contribution from Gd\nL3- edge XMCD obtained by SVD in the present work\nmatch with previous literature[22, 44] and the XMCD\nsignal amplitude (red symbols), as shown in Fig. 9(b), is\nalso following the similar temperature variation as of Fe\nsublatticemagnetizationdata(reddashedline)fromYIG\nin terms of its magnitude and direction. The SVD sepa-\nrated Gd XMCD amplitudes (blue symbols) at indicated\ntemperature and corresponding temperature variation of\nGd only sublattice magnetization (blue dashed line) are\nalso shown in the Fig. 9(b).\nV. CONCLUSIONS\nIn conclusion, in the present work in-field57Fe\nM¨ossbauer spectroscopy is employed to demonstrate\nthe Fe3+spin reversal and signatures of spin-canting\nacross the magnetic compensation temperature (T Comp)\nin Y1.5Gd1.5Fe5O12. The M¨ ossbauer data also clearly\ndemonstrate the continuous rotation of Fe3+moment\nacross the T comp, which is nothing but a second order\nfield induced phase transition. Inversion of sublattice\nspin across the T Compis also confirmed by Fe K- and\nGd L3- edge XMCD data. The quantitative estimation\nof the two sublattice contribution viz., Fe3+and Gd3+\nto the net magnetization is separated out from XMCD\nspectra collected from either Fe K- or Gd L 3- edge using\ndirect subtraction method from reference spectrum and\nSVD method, respectively i.e., in RIG systems one can\nprobe both Fe3+and Gd3+magnetism from only single\nedge XMCD measurements in the hard x-ray region.\nVI. ACKNOWLEDGMENTS\nWe acknowledge DESY (Hamburg, Germany), a mem-\nber of the Helmholtz Association HGF, for the provision\nof experimental facilities. Parts of this research were car-\nried out at beamline P09. 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The rich variety of \fndings encompass dynamic regimes\nhosting localized, non-propagating solitons, a turbulent chaotic regime, which condenses into a\nquasi-static phase featuring a non-collinear spin texture. Eventually, at largest spin current, a\nhomogeneously switched state is established.\n1arXiv:2009.10628v1  [cond-mat.mes-hall]  22 Sep 2020INTRODUCTION\nRecent advancements[1{5] in the art of thin-\flm growth allows nowadays to prepare YIG\n\flms with nanometer thickness. These \flms in particular feature magnetic losses comparable\nor lower than metallic ferromagnets like the widely used Permalloy or amorphous CoFeB\nalloys. Such YIG nano-\flms are of great interest to implement functionalities based on\nwave interference in magnon spintronic applications [6{9]. Being electrically insulating, YIG\nallows to completely disentangle spin and charge current related physics, which makes this\nmaterial in particular attractive for studies on spin-related transport phenomena [10{16]. In\naddition to its appearance in these topical research \felds, YIG is since its discovery a great\nmedium to study highly nonlinear spin dynamics. Turbulence [17], parametric instabilities\n[18{21], and even Bose-Einstein-condensation (BEC) [21{23] have been studied in YIG since\nquite a few decades. But so far many of these intriguing phenomena could only be realized\non rather macroscopic scales, rendering them less attractive for practical application. To\naddress these e\u000bects, YIG samples are usually exposed to strong, monochromatic microwave\nradiation, whose magnetic part can directly drive magnetization dynamics.\nOn the other hand, if single-frequency excitation is not a prerequisite, spin currents can\nbe considered as a convenient method, realizing a broad-band excitation. Spin currents can\nbe generated by a charge current when being lead through a spin-Hall material [24{27],\nwhich are for example the very common heavy metals platinum, or tungsten. In a simple\npicture, one can relate the appearance of spin currents in patterned \flms consisting of these\nmaterials to spin-orbital coupling (SOC): If a lateral charge current carried by a priori\nnot-spin-polarized electrons, experiences scattering with SOC, this scattering gives rise to\na vertical spin imbalance, building up between top and bottom surfaces of the conducting\n\flm. When deposited on top of a YIG nano-\flm, the spin accumulation at the interface can\ninteract with the magnetic moments in the YIG. This in particular can result in an e\u000bective\nreduction of magnetic losses of magnons. A critical current can in this context be de\fned\nas the magnitude at which the mode with lowest losses reaches the point of full damping\ncompensation. The spin current-induced instability of a particular mode is the essential\nmechanism behind spin-Hall oscillators, which have been realized with metallic Permalloy\n[28, 29], as well as with insulating YIG [30, 31] as active magnetic media.\nThe \fndings presented in the following in particular shed light on the question what\n2happens if one exceeds the instability threshold, in a situation when the injection of spin\ncurrents is only con\fned in one lateral dimension, or even not at all. Thereby the inves-\ntigations presented here complement recent \fndings [12, 14, 15] about magnon transport\nphenomena in YIG nano-\flms, and theoretical investigations, predicting BEC in such an\nexperimental situation [32, 33]. The here pursued micromagnetic approach provides a view\ninside the \flm, circumventing spatial and temporal resolution limitations encountered in\ncommon experimental approaches like Brillouin Light scattering [34], used to image mag-\nnetization dynamics. When con\fning the spin current, I have found \frst the nucleation of\nso-called spin wave bullets, whose density quickly increases, leading into a chaotic regime.\nAt even larger spin current a novel quasi-static phase condenses out of these turbulent \ruc-\ntuations. This phase is characterized by a stripe-like, non-collinear magnetization texture.\nAt higher current, this texture gradually disappears, and a fully switched, homogeneous\nmagnetic state is established.\nThis paper is organized as follows. First, I provide details about the numerical method.\nIn particular, I will explain how I take temperature-related e\u000bects into account. Then,\nI present results obtained for the case of con\fned spin-current injection. Subsequently, I\npresent the \fndings for the case of unrestricted injection. In the \fnal discussion, I \frst\nexplain the magnitude of the numerically found threshold current density, and explain why\nsubsequently turbulence arises. Finally, a tentative interpretation for the emerging quasi-\nstatic texture is presented.\nEXPERIMENTAL DETAILS\nTo simulate the spin-current injection into a YIG nano-\flm with a thickness of tYIG=\n20 nm, the micromagnetic simulation code MuMax3 [35] was used. In this \fnite-di\u000berences\nnumerical code, the magnetic \flm is divided into cubic cells of size 5 nm by 5 nm by\n20 nm. Each cell hosts a magnetic moment with a \fxed vectorial length, interacting via\nmicromagnetic exchange and dipolar \felds with its surroundings. A total lateral area of\n2560 nm by 2560 nm was considered. For the YIG \flm at 285 K a saturation magnetiza-\ntion ofM0= 0:11 MA/m, an exchange constant of A= 3:7 pJ/m2, a gyromagnetic ratio of\n\r= 1:7588\u000110111/Ts, and a Gilbert damping constant of \u000b= 0:001 were assumed. The\nspin torque generated via the spin-Hall-e\u000bect in a tPt= 3:5 nm thick Pt layer was taken into\n3Figure 1. In\ruence of Joule heating on static and dynamic magnetization. (a) Temperature\ndependence of magnetization according to [38], solid line is a power law \ft. Dashed vertical line\nmarks the Curie temperature TC. (b) Current density dependence of the temperature underneath\nthe Pt stripe according to [15], solid line is a quadratic \ft. Dashed horizontal line marks the Curie\ntemperature TC. (c) Derived current density dependence of magnetization M0. (d) Derived current\ndensity dependence of exchange constant A.\naccount by adding the Slonczewski torque term [36, 37] to the equation of motion of the\nmagnetization. As a conversion factor between charge and spin current, a spin-Hall angle\nof\u0012SHE= 0:11, and an interface transparency of about \u001ci= 0:47 were employed. These\nmaterial parameters resemble typical experimental values, as used in reference [15]. The\nMuMax3 script \fle in the supplement to this article provides all information to reproduce\nthe simulations.\nNote that, I did not consider the Oersted \feld created by the charge current. In the\nSupplementary Information I show that, due to its small magnitude, the Oersted \feld does\nnot in\ruence the dynamics. In contrast, the in\ruence of sample temperature on the mag-\nnetization and exchange, enhanced by Joule heating is taken into account. For simplicity, I\nhave assumed homogeneous heating. Laterally inhomogeneous temperature pro\fles do not\nimpact the dynamics, as discussed in the Supplementary Information. In the simulation,\n4a static reduction of the magnetization as well as temperature driven \ructuations, imple-\nmented by means of a \ructuating thermal \feld [35], are taken into account by the method\ndescribed in the following. I assume the temperature dependence of the magnetization\nshown in Figure 1(a), which was published in [38]. Note that, these data are well described\nby a phenomenological power law with exponent 0 :511(5) (red curve). Figure 1(b) shows\nexperimental temperature calibration data from [15], which extrapolates quadratically (red\ncurve) toTC= 560 K at about j= 8\u00011011A/m2. Combining both data sets and \ftting\ncurves, I have constructed the current dependence of the magnetization shown in Figure\n1(c). This curve is taken for rescaling of the e\u000bective magnetization at a given current and\ntemperature in the simulation: this means that in practice the length of the magnetization\nvector in each simulated cell is adjusted accordingly. Note that, in the simulation also long\nrange / low frequency \ructuation are included, stochastically excited by the thermal \feld.\nSuch \ructuations further reduce the e\u000bective magnetization. Across the whole temperature\nrange (285 K to 560 K) valid here, I have found the necessity to increase the magnetization\nby about 1 percent in order to take the additional reduction of the e\u000bective magnetization\nby such \ructuations into account. For the exchange constant I have assumed the classical\nmicromagnetic expectation A(T)/M0(T)2[39, 40]. The resulting current dependence is\nshown in 1(d).\nFigure 2 shows the experimental sample designs considered in this work. In Fig. 2(a)\nthe case of a spatially con\fned spin current injection is depicted, realized by patterning the\ncharge current carrying Pt layer to a stripe with a width of w= 500 nm. In the simulation\nthe Pt stripe is considered only implicitly, by enabling the Slonczewski torque only in the\ninjection region beneath the conductor. Absorbing boundary conditions (ABC) were applied\nto the edges parallel to the wire, and periodic boundary condition (PBC) were applied to\nthe perpendicular edges. Thereby an in\fnitely long wire was simulated. The external \feld\nhas a magnitude of \u00160H= 50 mT, and is oriented in the \flm plane, perpendicular to the\nwire. The detection stripe included in Figure 2(a) is depicted in order to graphically de\fne\nthe region underneath in the YIG \flm. This region is used for probing dynamics outside\nthe actively excited region.\nIn Fig. 2(g) the case of homogeneous spin current injection is depicted. Here, the PBCs\nare applied to all edges.\n5Figure 2. Sketch of the experimental situations and snapshots of simulated magnetization dynamics\nin terms of the normalized magnetic vector \feld m(x;y). (a) Sketch for con\fned spin current\ngeneration and injection. (g) Sketch for homogeneous spin current generation and injection. Edges\nmarked by PBC and ABC refer to periodic and absorbing boundary conditions. Note that both\nsketches include color-coded maps of m, referring to a current density below the onset of bullet\nformation, at \u0000 = \u00000:27 as indicated. (b) to (f) show snapshots of mfor increasing \u0000 as indicated\nfor the case of con\fned spin current injection. Dashed lines mark the boundaries of the Pt stripe.\n(h) to (l) depict analogously snapshots of mfor uncon\fned spin current injection.\nRESULTS\nFigure 2 depicts snapshots of the dynamics obtained for the two cases of con\fned, and\nrestricted spin current injection, at current densities above and below a certain critical\nthresholdjth. Note that I quantify this threshold later from the data, and use it to de\fne\nthe overcriticality \u0000 by\n\u0000 =j\njth\u00001: (1)\n6Figure 3. Spectral characterization of dynamics in the injection and detection area. (a) Part\nof typical transient dynamics in terms of the magnetic component mi\ny(md\ny), spatially averaged\nover the injection (detection) area, obtained at \u0000 = 7. (b) Corresponding Fourier power spectra\ncalculated from a in total 50 ns long transient, featuring a dominant peak at frequency fb, marked\nby the vertical blue dashed line (close to the frequency of Ferromagnetic Resonance f0, marked\nby the vertical green dashed line). (c) and (d) Dependency of power spectra in the injection and\ndetection area on the current density. The green dashed line marks the calculated f0(j). The blue\ndashed line serves as a guide to the eye for fb(j). Blue and orange arrows indicate the spectra\nshown in (b).\nCon\fned spin current injection\nLet us begin the inspection of the results by analyzing the case of spin current injection\ncon\fned to a stripe. Figure 2(a) to (f) shows snapshots of the magnetization after dynamic\nequilibrium has been established. For a current density below a certain threshold j < j th\n(\u0000<0, see Fig. 2(a)), no dynamic response can be seen. When increasing the current to\nj > j th(\u0000>0), this situation changes. Now, the simulation features localized hot spots,\nwhere the \flm is strongly excited (see Fig. 2(b)).\n7The normalized magnetization component mi\ny=hMyiinjection\nM0averaged across the injection\narea provides quantitative access to these dynamics. A representative time series obtained\nat \u0000 = 7 is shown in Figure 3(b). The Fourier transform power spectrum shown in Fig.\n3(c) is dominated by a strong peak at the frequency fb= 2:4 GHz. Note that this value\nis smaller than the bottom of the linear spin wave spectrum at about f0= 2:7 GHz (green\ndashed line in Fig. 3(c)). Both spectral and spatial features are typical for so-called spin-\nwave bullet modes [41]. Such a bullet is a nonlinear, non-propagating solitonic solution of\nthe gyromagnetic equation of motion. On the other hand, the dynamics in the detection\narea captured by md\ny=hMyidetection\nM0(see Fig. 3(b) and (c)), shows oscillations at a frequency\nclose to the frequency of Ferromagnetic Resonance (FMR)\n!0= 2\u0019f0=p\n!H(!H+!M(j)); (2)\nwhere!H=\r\u00160H, and!M(j) =\r\u00160M0(j).[42] When increasing the current density,\nthe number of simultaneously existing bullets in the injection area increases, as Fig. 2(c)\nillustrates. Simultaneously, their frequency fbdecreases, as shown in Fig. 3(c). This down-\nshift in frequency is well-known for bullets in in-plane magnetized magnetic \flms. In the\ndetection area, the frequency f0of the dominating FMR mode follows the thermally driven\ndecrease of the magnetization due to Joule heating (see for a plot of Eq. (2) the green dashed\nline in Fig. 3(d)). When reversing the current polarity, the dynamics in the injection as well\nas in the detection area are progressively suppressed and dominated by the FMR mode, as\ndemonstrated by the good agreement of the spectral maxima with the calculated dependence\nof the FMR frequency f0onjshown in Fig. 3(c) and (d).\nThe emergence of the bullets in the injection area can be characterized by an order\nparameter\n\t =1\u0000mi\nx\n2; (3)\nwheremi\nx=hMxii\nM0. The order parameter \t in essence captures how strong the magnetiza-\ntion deviates from the equilibrium orientation in the absence of a spin current, when MkH.\nFigure 4 shows a plot of the dependence of \t on j. One can see a quick initial growth,\nfollowed by an intermediate slowing down of the growth, which then speeds up again to\nreach values \t >0:5. Let us take a closer look at the initial growth. For a continuous phase\n8Figure 4. Dependence of the order parameter \t (blue circles) and of the magnon emission \u0006\n(orange rectangles) on the current density j. The red dashed line is a \ft of Eq. (3). The inset\nmagni\fes the behavior close the threshold current density, marked by the horizontal green dashed\nline. The horizontal green and blue dashed lines mark the onset of spin wave bullet formation, and\nthe emergence of the quasi-static texture, respectively. Orange dashed line is a guide to the eye.\ntransition one can expect according to Landau [43] a generic dependence\n\t =\u0012j\njth\u00001\u0013\"\n= \u0000\": (4)\nIndeed, \ftting Eq.(4) to the data yields a critical exponent of \"= 0:72(3), and a threshold\ncurrent density of jth= 0:17(1)\u00011011A/m2(see also inset in Fig. 4). Fig. 4 clearly shows that,\nat around \u0000 = 32, further evolution of the order parameter deviates from Eq. (4). Indeed,\nthe order paramter soon exceeds \t = 0 :5, which implies that on average, the magnetization is\nthen aligned antiparallel to the external \feld. Before this switching is completely achieved,\na quasi-static magnetic texture emerges (see Fig.2(e)). The spin-torque induced magnon\nemission from the injection area can be captured by\n\u0006(j) =hMx(js= 0)i2\nd\u0000hMx(j)i2\nd; (5)\nwhere the spatial average across detection area hMx(js= 0)idrefers to a simulation\n9conducted at \fnite temperature T(j), as caused by Joule heating, but without taking into\naccount the spin current js\rowing from the Pt stripe into the YIG \flm. In contrast\nhMx(j)idrefers to a simulation including the action of the spin current. By construction, \u0006\nis proportional to the number of magnons emitted from the injection area, which are caused\nonly by the spin injection, without compromising the thermal background. The current\ndependence \u0006( j) is included in Fig. 4. It displays a quick initial growth, followed by a\nsaturation around \u0000 = 30. Thereafter, \u0006 quickly decreases down to zero emission.\nUnrestricted spin current injection\nIn this section the situation sketched in Figure 2(g) is analyzed, where no spatial restric-\ntions are imposed on the spin current injection (see Fig. 2(g) to (l) for typical snapshots).\nAlso here, spin wave bullets appear, albeit chaos sets in earlier. The motivation for this\nexperiment is to analyze and better understand the transition from bullets to the emer-\ngence of the quasi-static stripe-like texture. The evolution of this transition is elucidated in\nFigure 5 in terms of 2d spatial, and spatio-temporal Fast Fourier transform (FFT) power\nmapsPFFT(kx;ky) andPFFT(kx;f)ky=0ofmz. In the left panel of Figure 5(a) one can see\nPFFT(kx;ky) of an already chaotic state, obtained at \u0000 = 1. The magnetization displays\nno clear structure, as the quite isotropic Fourier spectrum demonstrates. The agreement\nbetween the computed dispersion of plane spin waves [44] with the maxima of the spatio-\ntemporal Fourier spectrum PFFT(kx;f)ky=0depicted in the right panel shows that, the \ruc-\ntuations here still mainly correspond to linear spin waves. At \u0000 = 6 :4 (Fig. 5(b)), short\nwave length \ructuations strongly increase. Secondly, one can see a signature of the bullets\nappearing in the spatio-temporal Fourier spectrum. Namely, the largest spectral weight ap-\npears around kx= 0 at frequencies below the computed spin wave dispersion (dashed line).\nThis deviation is even more pronounced at \u0000 = 21 (see Fig.5(c)). At \u0000 = 43, the short wave\nlength \ructations are suppressed, and the spectrum displays a peculiar anisotropy, corre-\nsponding to the stripe-like magnetic texture shown in Figure 2(k). The static behaviour of\nthis state is re\rected by the spatio-temporal Fourier spectrum, which shows two maximima\nat frequency f= 0. Now, these maxima can not be related to linear spin waves at all\n(dashed white curve).\n10Figure 5. Spatio-temporal spectral characterization of spin dynamics in case of uncon\fned spin\ninjection. The left panels show 2d spatial FFT power maps PFFTfmz(x; y)g(kx;ky) of snapshots\nof the magnetization component mz. The right panels shows spatio-temporal FFT power maps\nPFFT(kx;f) alongkxforky= 0. The di\u000berent sub\fgures refer to speci\fc overcriticalites \u0000 as\nindicated.\n11DISCUSSION\nBullet dynamics\nIn the simulations considering a spatially restricted spin current injection, one sees the\nappearance of localized modes above a current density of jth= 0:17\u00011011A/m2. This number\ncan be compared with a simple expectation. In case of YIG nano-\flms, the mode with lowest\nlosses is the FMR mode. Without spin currents, its relaxation rate reads [45]\n!R=\u000b(!H+ 0:5!M): (6)\nThe spin torque pumps energy into the magnetic oscillations with a rate [36, 37]\n\f=j\u0001\r~\n2eM0tYIG\u0002SHE\u001ci (7)\nExact compensation, that is !R=\f, leads to a theoretical critical current density of 0 :16\u0001\n1011A/m2in the Pt stripe. Only when exceeding this value, the magnetization can become\nunstable. Indeed, the observed threshold almost exactly coincides with this theoretical\nexpectation. All properties derived from inspecting the current dependency of the dynamics\ncomply with the interpretation that, the unstable mode is a spin-wave bullet [41].\nTurbulence\nAs more and more bullets appear with increasing current, the dynamics quickly becomes\nchaotic. Note that, this chaos is deterministically driven by the spin current injection. As\nsignature of deterministic chaos, I \fnd that, in all spectra discussed in this article, the phases\nare random, and react sensitively on small perturbation of the initial state. This sensitivity\nis maintained, when excluding the thermal \ructuation \feld.\nIn the Supplementary Information accompanying this article, an analysis of spectral\nproperties of this chaotic state is shown. Chaos appears, because with increasing current,\nfor a larger and larger part of the spin-wave spectrum losses are compensated. Therefore,\ndissipation can only occur when three-magnon or higher-order scattering pushes energy into\nhigher-frequency modes, whose losses are not yet overcompensated by the injected spin\ncurrent. These nonlinear processes inevitably set in when the unstable modes have achieved\n12large enough amplitudes. Such an energy cascade is indeed prototypical for turbulence\n[46, 47]: energy is injected into the low wave number, low frequency part of the spectrum,\nand energy is dissipated as it reaches the large wave number, high frequency part.\nFurthermore, there is an interesting connection to classical pipe \row experiments. There,\nso-called pu\u000bs appear as precursors to turbulence.[48, 49] At a \frst glance, pu\u000bs and bullets\nseem to have a lot in common, as both appear prior to the onset of turbulence, and both\ndynamics are nonlinear and localized. Similar to the pu\u000bs in pipes, the bullets have a \fnite\nlifetime. How far does the analogy hold? I would like to emphasize that, in contrast to\npu\u000bs, the bullets do not move. They remain stationary inside the injection region. Note\nthat, this re\rects a quite di\u000berent experimental situation: in pipe \row experiments, one\ninduces turbulence locally, by placing objects in the \row, or by a nozzle. Here, my focus is\non a spatially extended injection region for the spin current injection, giving rise to chaotic\ndynamics in this region. The data presented in Figure 3 shows that, outside this region,\nthe magnetic \flms behaves mainly like a normal, thermally excited system. Secondly, with\nincreasing spin current injection, turbulence evolves, and the lifetime of the bullets decreases.\nAt even larger current density, the turbulence disappears again, in favor of a quasi-static\ntexture. In contrast, pu\u000bs moving down-stream have an increasing lifetime as a function\nof the Reynolds number. As I explain in the Supplementary Information, the latter can\nbe regarded as e\u000bectively controlled by the spin current injection. To further investigate\nsimilarities and di\u000berences, one could envisage a di\u000berent sample design, in which the Pt\ninjection stripe consists of two adjacent sections with large and small width, with a metallic\nferromagnetic \flm below. Then, one can locally induce bullets (=pu\u000bs) below the small\nwidth part (=reservoir under pressure). In addition, one may be able to push the bullets\ninto the large width part (=pipe) by means of the spin torque due to the current \row inside\nthe ferromagnet, similar to a moving domain wall.\nNon-collinear spin texture\nAt larger overcriticality, the progressive softening of the bullet mode culminates in a quasi-\nstatic pattern. Note that, besides softening, also the local switching of the magnetization\ndrives the condensation into the stripe pattern: wherever Mk\u0000H, the injected spin exerts\na damping-like torque. Only at small overcriticality, MkHstill holds on average, and the\n13torque is anti-damping like.\nRegarding the quasi-static texture, one may recall that Bender et al. [32] proposed in\n2014 that Bose-Einstein condensation of magnons should set in under spin current injection.\nIn their theory, a phase diagram is derived under the assumption of small angle dynamics.\nI here emphasize that, in case of a strongly excited YIG nano-\flm, the nonlinear spin-\nwave bullets have to be considered as dynamic modes undergoing condensation. Their local\noscillation angle is large. Therefore the theory of reference [32] cannot be applied directly. To\nfurther understand the classical condensation phenomenon observed in this micromagnetic\nsimulation work, I suggest to \frst-of-all consider the dispersion of bullets, which I here\napproximate by\n!b(k) =p\n(!H\u0000a!Mk2) (!H\u0000a!Mk2+!M); (8)\nwherea=2A\n\u00160M2\n0. Note that, in this expression the wave number k/1\ndbcharacterizes\nthe diameter of the non-propagating bullet [41]. Comparing the maxima of the Fourier\npower in Figure 5(d) with the overlaid dispersion curve Eq.(8) (dashed green line), I \fnd\nan intersection approximately at the point of vanishing frequency. To rule out that this\nis a mere coincidence, I have repeated the simulations for di\u000berent external \felds between\n\u00160H= 25 mT and 400 mT. At all \felds I have found at a current density of j= 7:5\u0001\n1011A/m2(corresponding to \u0000 = 43) the stripe texture, and determined the corresponding\ncharacteristic wave number k0. The \feld dependence of k0is plotted in Figure 6, on top of a\ncoloured map encoding the \feld and wave number dependence of !b(H;k). For all selected\n\feldsH, the characteristic wave numbers k0lie approximately on the isocontour \u0010!=0of\nvanishing frequency. This re\rects the \fnding that the emerging texture is a quasi-static\nfeature.\nWhy should this particular mode be chosen? Recall that the conventional dissipation\nargument for spin-wave instabilities implies that, the mode with smallest losses is selected\n[17]. For a magnon BEC, this is also the mode with the lowest frequency. Here, this\nargument fails, because the spin torque anyway compensates the direct dissipative losses.\nBut by pushing the bullets as far away from the linear spin-wave spectrum as possible, the\nsystem minimizes nonlinear losses, which occur due to multiple-magnon scattering. Such\nprocesses redistribute energy from the bullets into high frequency magnons, whose losses\nare not compensated by the spin current. This indirect route remains as active dissipation\n14Figure 6. Field dependence of quasi-static texture. The colored map in the background shows the\n\feld and wave number dependence of Equation (8). The characteristic wave numbers k0(open\ncircles) lie on the isocontour \u0010!=0(red line).\nchannel as long as the bullet frequency does not vanish.\nSUMMARY\nTo summarize, the overall picture for spin current induced magnetization dynamics in\nYIG nano-\flms obtained from micromagnetic simulation is like this: when exceeding the\nthreshold current density jth= 0:17\u00011011A/m2, \frst-of-all single spin-wave bullets ap-\npear, whose number quickly increases with increasing current. The bullets then give rise to\ndeterministic chaos. This turbulent state eventually freezes out, in favor of a quasi-static,\nnon-collinear magnetic texture, which \fnally gradually turns over into a completely switched\nstate. Note that, combining materials with large spin-Hall angles like \f-tungsten [50], with\noptimally grown YIG nano-\flms, displaying Gilbert damping constants as small as only\n7\u000210\u00005[2, 4], opens up a realistic, and fruitful perspective for studying samples with large\nactive areas ( w\u001dk\u00001\nb). Then, one might be able to observe turbulent dynamics, as well as\nthe novel, quasi-static texture. While so far, no experimental reports about the emergence\nof such a texture exist, the possibility to establish a connection to experimental work (see\nSupplementary Information of this article) further supports this chance. In addition to such\nexperimental opportunities, the \fndings presented in this paper also open in interesting\nperspective for the application of spin hydrodynamic theory [51{54]. 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Lett.\n123, 117203 (2019).\n19"    },    {        "title": "1708.00593v1.Strongly_exchange_coupled_and_surface_state_modulated_magnetization_dynamics_in_Bi2Se3_YIG_heterostructures.pdf",        "content": "Strongly e xchange -coupled and surface -state -modulated m agnetization \ndynamics in Bi 2Se3/YIG heterostruct ures \n \nY . T.  Fanchiang1, K. H.  M. Chen2, C. C.  Tseng2, C. C.  Chen2, C. K.  Cheng1, C. N.  Wu2, S. F.  \nLee3*, M. Hong1*, and J. Kwo2* \n \n1Department of Physics, National Taiwan University, Taipei 1 0617, Taiwan  \n2Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan  \n3Institute of Physics, Academia Sinica, Taipei 11529, Ta iwan  \n \nAbstract  \nWe report strong interfacial exchange coupling in Bi 2Se3/yttrium iron garnet (YIG) \nbilayers manifested as large in-plane  interfacial magnetic anisotropy (IMA)  and enhancement \nof damping probed by ferromagnetic resonance (FMR) . The IMA and spin mixing \nconductance reached a maximum when Bi 2Se3 was around 6 quintuple -layer (QL) thick. The \nunconventional Bi 2Se3 thickness dependence of the IMA and spin mixing conductance  are \ncorrelated w ith the evolution of surface band structure of  Bi2Se3, indicating that topological \nsurface states  play an important  role in the magnetization dynamics  of YIG. Temp erature -\ndependen t FMR of Bi2Se3/YIG  revealed signatures of magnetic proximity effect of Tc as high \nas 180 K , and an effective field parallel to the YIG magnetization direction at low temperature . \nOur study shed s light o n the effects of topological insulators on magnetization dynamics, \nessential for dev elopment of TI -based spintronic  devices.   \n \n \n  Topological insulato rs (TIs) are emergent quantum materials hosting topologically \nprotected surface states, with dissipationless trans port prohibiting backscattering [1,2] . Strong \nspin-orbit coupling (SOC) along with time reversal symmetry (TRS) ensures that the electrons \nin surface states have thei r direction of motion and spin \"locked\" to each other [1,3,4] . When \ninterfaced with a magnetic layer, the interfacial excha nge coup ling can induce magnetic order \nin TIs and break TRS [5-8]. The resulting gap opening of the Dirac state  is necessa ry to realize \nnovel phenomena such as topological magneto -electric effect  [9] and quantum anomalous Hall \neffect [10,11] . Another appro ach of studying a TI/ferromagnet (FM) system focus es on spin -\ntransfer c haracteristic at the interface and intend s to exploit  the helical spin texture of \ntopological surface states (TSSs). Attempts have b een made to estimate the spin -charge \nconversion (SCC) effici ency, either by using microwave -excited d ynamical method  [12-16] \n(e.g. spin p umping and  spin-torque FMR) or  thermally induced spin injection [17]. Very large \nvalues of SCC ratio have been reported [13,15,16] . Recently, TIs are shown to be excellent \nsources of spin -orbit torques for efficient magnetization switching  [18]. \n Since the magnetic proximity effect (MPE) an d spin -transfer process rel y on interfacial \nexchange coupling of TI/FM , understanding the magnetism at the interface has  attracted strong \ninterest s in recent years.  Several technique s have  been adopted to investigate the interfacial \nmagnetic properties, including spin -polarized neutron reflectivity  [7,19] , second harmonic \ngeneration [20], electrical transport  [6,21] , and magneto -optical Kerr effect  [6]. All these \nstudies clearly indicate the existence of MPE resulting from exchange coupling and strong \nspin-orbit interaction in TIs. For device application,  however, it is equally important to \nunderstand how the  interfacial exchange coupling affect s the magnetization dyn amic of FM. \nFor example, TIs  can introduce additional magnetic damping and greatly alter the dynamical \nproperties of FM layer, as  commonly ob served in FM/heavy metals systems  [22-24]. The \nenhanced damping is visualized as larger linewidth  of FMR spectra  [22-24]. In TI/FM, t he \npresence of TSS and, possibly, MPE complicate the system under study, and is still an open question to answer.  \n In this work w e systematically investigate d FMR characteristic of the ferrimagnetic \ninsulator YIG under the influence  of the prototypical three -dimensional TI Bi 2Se3 [25]. We \nchoose YI G as the  FM layer because of its  technological  importa nce, with high 𝑇𝑐 ~550 K \nand extremely low damping  coefficient α [26]. When YIG is interfaced with TIs, its good \nthermal stability  minimize s interdiffusion of materials . Through Bi 2Se3 thickness dependence \nstudy, w e observed strong modulation of FMR properties , attributed  to the TSS of Bi 2Se3. \nTemperature -dependent study unraveled an effective field parallel to the magnetization \ndirection existing in Bi 2Se3/YIG. Such effective field built up as temperature decreased, which \nwas utilized to demonstrate the zero -applied -field FMR of YIG . Furthermore, we identified the \nsignature  of MPE of 𝑇𝑐 as high as 180 K manifested as enhanced spin pumpin g in a fluctuating \nspin system.  \nBi2Se3 thin films were grown by molecular beam epitaxy (MBE)  [27] on magnetron -\nsputtered YIG films.  The high quality of Bi 2Se3/YIG samples in this work is verified by high \nresolution X -ray diffraction  (XRD)  and transmission electron microscope  (TEM) [28]. To \ninvestigate the magnetic properties of Bi 2Se3/YIG, room -temperature angle - and frequency -\ndependent FMR measurements were performed independently using a cavity and co -planar \nwaveguide (CPW), respective ly (Figure 1( a) and (b)). For the temperat ure-dependent FMR , \nthe CPW was mounted in a cryogenic probe station  (Lake  Shore, CPX -HF), which enable \nsamples to be cooled as low as 5 K.  The external field is modulated for lock -in detection in all \nof the measurements.  \n The FMR spectra in Figure 1( c) are compared for single layer YIG(12) and \nBi2Se3(25)/YIG(12) bilayer (digits denote  thickness in nanometer) , showing  a large shift of \nresonance field ( 𝐻𝑟𝑒𝑠) ~317 Oe after the Bi2Se3 growth  plus a markedly broadened  peak -to-\npeak width  ∆𝐻 for Bi 2Se3/YIG.  Figure  1(d) shows 𝐻𝑟𝑒𝑠 vs applied field angle with respect \nto surface normal 𝜃𝐻 for YIG(12) and Bi 2Se3(25)/YIG(12). The larger variation  of 𝐻𝑟𝑒𝑠 with 𝜃𝐻 indicate s stronger magnetic anisotropy in the bilayer sample. When the applied field was \ndirected in the film plane, clear negative 𝐻𝑟𝑒𝑠 shifts induced by Bi 2Se3 were observed at all \nmicrowave frequency f in Figure 1 e. The data can be fitted in the scheme of magnetic thin films \nhaving  uniaxial perpendicular  magnetic anisotropy  [28]. Since the XRD results show th at the \nYIG films did not gain additional strain  after growing Bi2Se3, the enhanced anisotropy cannot \nbe attributed to changes of magnetocrstalline anisotropy. Defining eff  ective demagnetization \nfield 4𝜋𝑀𝑒𝑓𝑓=4𝜋𝑀𝑠−𝐻𝑎𝑛−𝐻𝑖𝑛𝑡, where 4𝜋𝑀𝑠, 𝐻𝑎𝑛 and 𝐻𝑖𝑛𝑡 are the demagnetization \nfield of bare YIG , the magnetocrystalline  anisotropy field of YIG,  and the anisotropy field \ninduced by Bi 2Se3, we obtain  𝐻𝑖𝑛𝑡 = -926 and -1005 Oe fr om Figure 1 (d) and 1 (e), \nrespectively.  The minus sign indicates the additional anisotropy points in the film plane.  \nThe above observations suggest ed the presence of IMA in Bi 2Se3/YIG. To ve rify this \nassumption, we systematically varied the thickness of YIG 𝑑𝑌𝐼𝐺 while  fixing the thickness of \nBi2Se3. Figure 2( a) presents the 𝑑𝑌𝐼𝐺 dependence of 4𝜋𝑀𝑒𝑓𝑓 for single  and bilayer samples. \nThe 4𝜋𝑀𝑒𝑓𝑓 of single layer YIG was independent of 𝑑𝑌𝐼𝐺 varying from 12 to 30 nm. In \nsharp contrast, 4𝜋𝑀𝑒𝑓𝑓 of Bi 2Se3/YIG became significant ly larger , especially at thinner YIG,  \nwhich is a feature of an interfacial  effect. The f- and 𝜃𝐻-dependent FMR were performed \nindependently to doubly confirm the trends. The IMA can be further characterized by defining \nthe effective anisotropy constant   𝐾𝑒𝑓𝑓= (1/2)4𝜋𝑀𝑒𝑓𝑓𝑀𝑠 =(1/2)(4𝜋𝑀𝑠−𝐻𝑎𝑛)+𝐾𝑖/\n𝑑𝑌𝐼𝐺, with the interfacial anisotropy constant 𝐾𝑖=𝑀𝑠𝐻𝑎𝑛𝑑𝑌𝐼𝐺/2.  The  𝐾𝑒𝑓𝑓𝑑𝑌𝐼𝐺 vs 𝑑𝑌𝐼𝐺 \ndata in Figure 2(b) is well fitted by a linear function, indicating  that the 𝑑𝑌𝐼𝐺 dependence \npresented in Figure  2(a) is suitably described by  the current form  of  𝐾𝑒𝑓𝑓. The intercept  \nobtained by extrapolating the linear function corresponds to 𝐾𝑖=−0.075  erg/cm2. \n To further investigate the physical origin of the IMA , we next varied the thick ness of \nBi2Se3 (𝑑𝐵𝑆) to see how 𝐾𝑖 evolved  with 𝑑𝐵𝑆. Figure 2( c) shows the 𝑑𝐵𝑆 dependence of 𝐾𝑖. \nStarting from 𝑑𝐵𝑆=40 nm sample, the magnitude of 𝐾𝑖 went up as 𝑑𝐵𝑆 decreased. An \nextremum  of 𝐾𝑖 −0.12±0.02 erg/cm-2 was reached at 𝑑𝐵𝑆=7 nm. An abrupt upturn  of Ki occurred in the region 3 nm <𝑑𝐵𝑆<7 nm. The Ki magnitude dropped drastically and \nexhibited a sign change in the interval. The Ki value of 0.014 erg/cm2 at 𝑑𝐵𝑆=3 nm \ncorresponds to  weak interfacial perpendicu lar anisotropy. Based on previous investigation on \nsurface band structure of ultrathin Bi2Se3 [29], 𝑑𝐵𝑆=6 nm was identified  as the 2D quantum \ntunneling limit of Bi 2Se3. When 𝑑𝐵𝑆<6 nm, the hybridization of top and bottom TSS \ndeveloped a ga p of the surface states. Spin -resolved photoemission study later showed that the \nTSS in this regime exhibited decre ased in-plane  spin polarization . The modulated spin texture  \nmay lead to the weaker IMA than that in  3D regime  [30,31] . We thus  divide  Figure 2( c) into \ntwo regions  and correlate  the systematic magnetic properties with the surface state band \nstructure. The sharp change of 𝐾𝑖 around 𝑑𝐵𝑆=6 nm strongly suggests  that the IMA in \nBi2Se3/YIG  is of topological origins . \n The ∆𝐻 broadening in FMR spectra after growing Bi 2Se3 on YIG indicates additional \ndamping arising from spin-transfer  from YIG to Bi 2Se3, which is a s ignature of spin pumping \neffect  [24]. The spin pump ing efficiency of an interface can be evaluated by spin mixing \nconduct ance 𝑔↑↓, using the following relation  [24], \n𝑔↑↓=4𝜋𝑀𝑠𝑑𝑌𝐼𝐺\n𝑔𝜇𝐵(𝛼𝐵𝑆/𝑌𝐼𝐺−𝛼𝑌𝐼𝐺)                        (1) \n, where  𝑔, 𝜇𝐵, 𝛼𝐵𝑆/𝑌𝐼𝐺 and 𝛼𝑌𝐼𝐺 are the Landé g factor, Bohr magneton,  the damping \ncoefficient of Bi 2Se3/YIG  and YIG, respectively.  Figure 2( d) displays the dBS dependence of \neffective spin mixing conductance of Bi 2Se3/YIG . Similar to Ki in Figure 2( c), 𝑔↑↓ had its \nmaximum  at 𝑑𝐵𝑆=7 nm with a very large  value  ~2.2×1015 cm-2, about four times larger \nthan that of our Pt/YIG sample  indicated  by the red dashed line . The inset shows ∆𝐻 vs 𝑓 \ndata for Bi 2Se3 (7)/YIG(13) and YIG(13) fitted by linear functions. One can clearly see a \nsignificant change of slope, from which we determined 𝛼𝐵𝑆/𝑌𝐼𝐺−𝛼𝑌𝐼𝐺 to be 0.014. The large \n𝑔↑↓ of Bi 2Se3/YIG  implies an efficient spin pumping to an excel lent spin sink of Bi 2Se3. We \nagain divide  Figure 2( d) into two regions according to the surface state band structure. Upon crossing the 2D limit from 𝑑𝐵𝑆=7 nm, 𝑔↑↓ dropped remarkably to 1.7×1014 at 𝑑𝐵𝑆=\n3 nm. The trend in Figure 2( d) is distinct from that of no rmal metal (NM)/FM structures . In \nNM/FM,  the 𝑔↑↓ increase s with increasing  NM thickness as a result  of vanishing spin \nbackflow in thicker NM  [32]. It is worth noting that the conducting bulk of Bi 2Se3 can dissipate \nthe spin-pumping -induced spin accumulation at the interface [12,33] . In this regard, the 𝑑𝐵𝑆=\n7 nm sample has the largest weight of surface state contribution to  𝑔↑↓. Such  unconventional \n𝑑𝐵𝑆 dependence of 𝑔↑↓ suggests that TSS  play a  dominant  role in  the damping enhancement . \n    Since the effects of TSS are expected to enhance  at low temperature , we next performed \ntemperature -dependent FMR on  Bi2Se3/YIG . Two bilayer samples Bi2Se3(25)/YIG(15) and  \nBi2Se3(16)/YIG(17) , and a single layer YIG(23) were measured for comparison. Figure 3( a) \nand (b) show  the 𝐻𝑟𝑒𝑠 vs 𝑓 data at various T for YIG(23) and Bi 2Se3(25)/YIG( 15). The 𝐻𝑟𝑒𝑠 \nof both samples show negative shifts at all f with decreasing T. The data of YIG(23) can be  \nreproduced by  the Kittel equation with increasing 𝑀𝑠 of YIG at low T. In sharp contrast, \nBi2Se3(25)/YIG (15) exhibited negative  intercepts  at 𝐻𝑟𝑒𝑠, and the intercepts  gained its \nmagnitude when the sample was cooled dow n. This behavior  of non -zero intercept is common \nfor all of our Bi 2Se3/YIG  samples . To account for the behavior, a phenomenological effective \nfield 𝐻𝑒𝑓𝑓 is add ed to the Kittel equation, i.e.   \n𝑓=𝛾\n2𝜋√(𝐻𝑟𝑒𝑠+𝐻𝑒𝑓𝑓)(𝐻𝑟𝑒𝑠+𝐻𝑒𝑓𝑓+4𝜋𝑀𝑒𝑓𝑓).                (2) \nThe solid lines in F igure 3( b) generated by the modified Kittel equation fitted the experimental \ndata very well . \nFigure s 3(c) and (d) presents the T dependence of 𝐻𝑟𝑒𝑠 and ∆𝐻 for the YIG(23) and \ntwo Bi2Se3/YIG  samples. As we lowered T, all of the samples had decreasing 𝐻𝑟𝑒𝑠, which was \nviewed as effects of the concurrent increasing 𝑀𝑒𝑓𝑓 and 𝐻𝑒𝑓𝑓 as seen in Figure 3( a) and (b). \nOn the other hand, ∆𝐻 built up with decreasing T. We first examine ∆𝐻 of the YIG(23) \nsingle layer . The ΔH remained relatively unchanged with T decreasing  from RT , and dramatically incr eased below 100 K. The pronounced T dependence of ∆𝐻 or α has been \nexplored in vario us rare -earth iron garnet and was  explained by the low T slow -relaxation \nprocess via rare -earth elements or Fe2+ impuritie s [34]. For sputtered YIG films, specifically, \nthe increase of  ∆𝐻 was less prominent in thicke r YIG, indicating  that the dominant impurities \nlocate d near the YIG surface [35]. Distinct from that of YIG(23) , the ∆𝐻 progressive ly \nincrease d for the bilayer samples.  We were  not able to detect FMR signals with ∆𝐻 beyond \n100 Oe due to our instrumental limits. However, one can clearly see that, for \nBi2Se3(25)/YIG(15) and Bi 2Se3(16)/YIG(17), the  ∆𝐻 broaden ed due to increased spin \npumping at first . For Bi 2Se3(25)/YIG(15) , the ∆𝐻 curve gradually leveled off, and intersect ed \nwith that of YIG(23)  at T ~40 K . The seeming ly \"anti -damping\" by Bi 2Se3 at low T may be \nrelated to the modification of the  YIG surface  chemistry  during the Bi 2Se3 deposition. \nAdditional analyses  are needed to verify t he scenario , which  is, however,  beyond the scope of \nthis work.  \nIn both 𝐻𝑟𝑒𝑠 and ∆𝐻 curves , bump-like feature s located at T = 140 and 180 K (indicated \nby the arrows) were revealed for Bi 2Se3(25)/YIG(15) and Bi 2Se3(16)/YIG(17), respectively.  \nThe bumps are reminiscent of spin pumping in the case of fluctuating magnets , where  an \nenhancement of spin pumping is expected as the spin sink is closed to its magnetic phase \ntransition  point  [36-38]. In our system, a possibl y newly formed magnetic phase would be the \ninterfacial magnetization  driven by the proximity effect , namely, Tc = 140 and 180 K for our \nBi2Se3(25)/YIG(15) and Bi 2Se3(16)/YI G(17) , respectively . In fact, the Tc values of our samples \nare in good agreement with the reported ones  of MPE  in TI/YIG systems  [6,21] . We did not \ndetect anomalous Hall effect in our sample s, which might be o bscured by the bulk conduction \nof Bi 2Se3 in the magneto -transport measurements . \nUsing Eq. (2 ), we further determine  the T dependence of 4𝜋𝑀𝑒𝑓𝑓 and 𝐻𝑒𝑓𝑓 of YIG(23) \nand the two  bilayer s samples, as shown in Figures 3(e)  and (f) [28]. The 4𝜋𝑀𝑒𝑓𝑓 of YIG(23)  \nwent larger  monotonically as previously discussed, while the 4𝜋𝑀𝑒𝑓𝑓 of bilayer samples increased before reaching a maximum of 4000 Oe  at T around 150 K, and then  decreased \nslightly at low T. With 4𝜋𝑀𝑒𝑓𝑓𝐵𝑆/𝑌𝐼𝐺−4𝜋𝑀𝑒𝑓𝑓𝑌𝐼𝐺≈−𝐻𝑖𝑛𝑡, such T dependence corresponds to \ndecreasing in-plane IMA below 150 K. Note that 150 K is close d to our assumed 𝑇𝑐 from  \nMPE at 140 and 180 K. Calculations of total electronic energy at an TI/FM interface show that \nperpendicular anisotrop y is in favor  [39], which, in our case, may effectively weaken  the in -\nplane IM A. The decreasing in -plane IMA can also be viewed as a result of modified spin texture \nby MPE. The analyses thus provide  another clue suppor ting the existence of MPE in our \nsamples. The 𝐻𝑒𝑓𝑓 of bilayer samples, again, show  different T evolution  than that of the single \nlayer  in Figure 3( f). 𝐻𝑒𝑓𝑓 built up with decreasing T in bilayers while the 𝐻𝑒𝑓𝑓 of the YIG \nsingle layer was T-independent and closed to zero. The positive 𝐻𝑒𝑓𝑓 corresponds to an \neffective field parallel to  the magnetization vector M and resembles the exchange bias field. \nHowever, we did not observe  shifts of magnetization hysteresis loop  characteristic of an \nexchange bias effect  [28,40] . The 𝐻𝑒𝑓𝑓 present in FMR m easurement suggests  it may come  \nfrom spin imbalance at the interface  [41]. \nFinally, we demonstrated that the large IMA and 𝐻𝑒𝑓𝑓 in Bi 2Se3/YIG are strong enough \nto ind uce FMR without an 𝐻𝑒𝑥𝑡, which we term zero-field FMR . Figure 4( a) displays T \nevolution of FMR first derivative spectra of Bi 2Se3(25)/YIG(15) at 𝑓=3.5 GHz. The spectral \nshape started to deform when the 𝐻𝑟𝑒𝑠 was approaching zer o. The sudden twists at 𝐻𝑒𝑥𝑡 ~ \n+30 ( -30) for positive (negative ) field sweep arose from magnetization switching of YIG, and \ntherefore led to hysteric spectra. The two spectra merged at 25 K and then separated again when \nT was further decreased. Figure 4( b) shows the microwave absorption intensity I spectra with \npositive field sweep s. We traced the position of I spectrum 𝐻𝑝𝑒𝑎𝑘 using the red dashed line, \nand found it coincided with zero 𝐻𝑒𝑥𝑡 at the zero -field FMR temperature  𝑇0 ~25 K. Below \n25 K, 𝐻𝑝𝑒𝑎𝑘 moved across the origin and one needed to reverse 𝐻𝑒𝑥𝑡 to counte r the internal \neffective field comprised of the demagnetization field 4𝜋𝑀𝑠, 𝐻𝑖𝑛𝑡 and 𝐻𝑒𝑓𝑓 (Figure 4( e)). Note that the presence of 𝐻𝑖𝑛𝑡 alone would be inadequate to  realize zero -field FMR. Only \nwhen 𝐻𝑒𝑓𝑓 is finite would the system exhibit non -zero intercepts as we have seen in Figure \n3(b). We further calculate  𝑇0 as a function of microwave excitation frequency f (Figure 4 (f)) \nusing Eq. (2) and the extracted 𝐻𝑒𝑓𝑓 of Figure 3(f).  We obtain  that, with f inite 𝐻𝑒𝑓𝑓 \npersisting  up to room temperature, zero -field FMR can be realized at high T provided f is \nsufficiently low. However, we emphasize that it’s advantageous for YIG to be microwave -\nexcited above 3 GHz. When 𝑓<3 GHz, parasitic effects such as th ree-magnon splitting  \n[42,43]  take place and significantly decrease the microwave absorpti on in YIG. Here, we \ndemonstrate  that the strong exchange coupling between Bi 2Se3 and YIG gave rise to zero -field \nFMR in the feasible high frequency operation  regime of YIG. Further improvement of  interface \nquality of Bi 2Se3/YIG is expected to raise 𝐻𝑒𝑓𝑓 and 𝑇0 for room temperature, field free \nspintron ic application.  \nIn summary , we investigate the magnetization dynamics of YIG in the presence of \ninterfacial exchange coupling and TSS of  Bi2Se3. The significantly modulated magnetization \ndynamics at room temperature is shown to be TSS -originated through the Bi 2Se3 thickness \ndependence study. It can be expected that the strong coupling between Bi 2Se3 and YIG will \nmodify interface band structures an d even the spin texture of TSS. The underlying mechanism \nof the c oexisting large in -plane IMA and pronouncedly enhanced damping calls for further \ntheoretical  modeling and understanding . 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(c) FMR spectra of Bi 2Se3(25)/YIG(12) and YIG(12) measured by the cavity. \n(d), (e) 𝜃𝐻 and f dependence of 𝐻𝑟𝑒𝑠 of Bi 2Se3(25)/YIG(12) and YIG(12), respectively.  \n  \nFigure 2.  \n \nFIG. 2. (a) The 𝑑𝑌𝐼𝐺 dependence of 4𝜋𝑀𝑒𝑓𝑓 of Bi 2Se3/YIG (solid triangles) and YIG \n(hollow squares) obtained from 𝜃𝐻 (red) and f (blue) dependent FMR. (b) The 𝐾𝑒𝑓𝑓𝑑𝑌𝐼𝐺 vs \n𝑑𝑌𝐼𝐺 plot for determining 𝐾𝑖 using a linear fit. The intercept of the y -axis corresponds to the \n𝐾𝑖 value. (c) 𝑑𝐵𝑆 dependence of 𝐾𝑖. The figure is divided into two regions. For  𝑑𝐵𝑆>6 nm, \nthe Dirac cone of TSS is intact, with the Fermi level located in the bulk conduction band. For \n𝑑𝐵𝑆<6 nm, a gap and quantum well states form. (d) The 𝑑𝐵𝑆 dependence of spin mixing \nconductance  𝑔↑↓. The inset shows ∆𝐻 as a function of Bi 2Se3(7)/YIG(13) a nd YIG(13) for \ncalculating 𝑔↑↓. The red dashed line indicates the typical value of 𝑔↑↓ of Pt/YIG.  \n \n \n \n \n  \nFigure 3.  \n \nFIG. 3. (a), (b) 𝑓 vs 𝐻𝑟𝑒𝑠 data for various T for YIG(23) and Bi 2Se3(25)/YIG(15), \nrespectively. Solids lines are fitted curves using Eq. (2). (c), (d), (e), and (f) T dependence of \n𝐻𝑟𝑒𝑠, ∆𝐻, 4𝜋𝑀𝑒𝑓𝑓 and 𝐻𝑒𝑓𝑓 of the one YIG (23) single layer and two Bi 2Se3/YIG  bilayer \nsamples, Bi 2Se3(25)/YIG(15) and Bi 2Se3(16)/YIG(17), respectively. The arrows in (c) and (d) \ndenote the position of the \"bumps\". Solids line are guide of eyes obtained by properly \nsmoothing the experiment al data.  \n \n \n \n  \nFigure 4.  \n \n \nFIG. 4. (a), (b) FMR first derivative and microwave absorption s pectra for variou s T, \nrespectively. The arrows indicate the 𝐻𝑒𝑥𝑡 sweep direction. The dashed line in (b) traces the \nT evolution of the absorption peak 𝐻𝑝𝑒𝑎𝑘. (c), (d), an d (e) Schematics of the Bi 2Se3/YIG \nsample when T = 100, 25, and 5 K, respectively. (f) Zero -field FMR temperature 𝑇0 as a \nfunction of f. \n \nSupplemental materials  \nGrowth and F MR characteristics of YIG films  \n \n \n \n The YIG thin films were deposited on (111) -oriented GGG substrates  by off -axis \nsputtering at room temperature. The GGG(111) substrates were first ultrasonically \ncleaned in order of acetone, ethanol an d DI -water before mounted in a  sputtering \nchamber  with the base pressure of 2×10−7 mTorr. For  YIG deposition, a 2 -inch YIG \ntarget was sputtered with the following conditions: an applied rf power of 75 W , an Ar \npressure of 50 mtorr  and a growth rate of 0.6 nm/min. The samples were then annealed \nat 800oC with O2 pressure of 11.5 mtorr for 3 hours.  Fig. S1(a) displays  the atomic force \nmicroscopy (AFM) image of the YIG surface, showing a flat surface with roughness of \n0.19 nm. Fig . S1(b) shows the high-angle annular dark -field (HAADF) image of \nYIG/GGG. The YIG thin film was epitaxially grown on the GGG substrate with \nexcellent crystallinity. No crystal defects were observed at the YIG bulk and YIG/GGG \ninterface.  \n \n \nFIG. S1. (a) and (b) AFM surface image and HAADF -STEM image of YIG/GGG.   \n \n \n \n \n Fig. S2(a) shows the representative FMR data of our YIG film  measured by \ncoplanar wav eguide. The FMR spectra exhibit Lorentzian lineshape at all measured \nfrequencies ranging from 3 to 9.76 GHz. To determine the 4𝜋𝑀𝑒𝑓𝑓 and 𝛼, the \nresonance fields 𝐻𝑟𝑒𝑠 and peak -to-peak widths ∆𝐻 of these spectra were plotted as a \nfunction of 𝑓 as shown in Fig. S2(b) and (c), respectively.  We obtain the 4𝜋𝑀𝑒𝑓𝑓 of \nour 23 nm YIG to be 1630 Oe. We note that the 4𝜋𝑀𝑒𝑓𝑓 value is lower th an the \nreported values of the YIG prepared by  either sputtering or pulsed laser deposition  [S1-\n3]. The difference may come from growth conditions dependent on different systems. \nFor determinati on of 𝛼, the data in Fig. S2(c) is fitted to the following equation,  \n∆𝐻=∆𝐻0+4𝜋𝑓𝛼\n√3𝛾.                    (S1) \nHere, ∆𝐻0 and 𝛾 are the inhomogeneous broadening and gyromagnetic ratio. The \nlinear fit in Fig. S2(c) corresponds to 𝛼=1.1×10−3. FIG. S2. (a) FMR first -derivative spectra of 23 nm YIG at various frequencies. (b) 𝑓 vs 𝐻𝑟𝑒𝑠 data fitted \nto the Kittel equation (red line) . (c) ∆𝐻 as a function of 𝑓 data. From the linear fit (red line) the 𝛼 value \nof the sample is obtained.  \nStructural characterizations of Bi 2Se3/YIG heterostructures  \n \nThe YIG/GGG samples were annealed at 450oC in the MBE growth chamber for \n30 min prior to Bi 2Se3 growth at 280oC [27]. The base pressure of the system was kept \nabout 2×10−10 Torr. Elemental Bi (7N) and Se (7N) were evaporated from regular \neffusion cells. As shown in Fig . S3a, streaky reflection high -energy electron diffraction \n(RHEED) patterns of Bi 2Se3 were observed. Fig . S3b displays the s urface morphology  \nof 7 QL  Bi2Se3 taken by atomic force microscopy (AFM). The image shows layer -by-\nlayer growth of Bi 2Se3 with the step heights ~ 1nm, which corresponds to the thickness \nof 1 QL. The l ayer structure of Bi 2Se3 was also revealed by the HAADF image  shown \nin Fig . S3c. Despite the high quality  growth of  Bi2Se3, an amorphous interfacial layer \nof ~ 1 nm formed.  The excellent crystallinity of our samples was verified by clear \nFIG. S3 . (a) RHEED patterns of MBE grown 7  QL Bi 2Se3 on YIG/GGG(111) substrates. (b) AFM image \nof a 7 QL Bi 2Se3. (c) HAADF -STEM image of Bi 2Se3/YIG/GGG heterostructures. (d) SR-XRD of our \nBi2Se3(25)/YIG(12) sample. Clear Pendellösung fringes  of YIG and Bi 2Se3 indicates excellent crystallinity. \nThe inset shows the radial scans of YIG before and after Bi2Se3 growth.  Pendellösung fringes of the synchrotron radiation x -ray diffraction (SR -XRD) data \nshown in Fig . S3(d). In particular, the radial scans data of YIG/GGG(22 -4) before and \nafter growing Bi 2Se3 are perfectly matched, indicating the absence of Bi 2Se3-induced \nstrains in YIG t hat might contribute additional magnetic anisotropy  [S4]. \n \nAnalyses of magnetic anisotropy  \nWe express the free energy density E of the system as  \n𝐸=−𝑴∙𝑯+1\n2𝑀𝑠(4𝜋𝑀𝑠−𝐻𝑎𝑛−𝐻𝑖𝑛𝑡)cos2𝜃𝑀     (S2) \n, where  𝑴, 𝑯, 𝑀𝑠, 𝐻𝑎𝑛and 𝜃𝑀 are magnetization vector, applied field vector, \nsaturation magnetization, the anisotropy field induced by Bi 2Se3 and magnetization \nangle with respect to the surface normal, respectively. We further define the effective \ndemagnetization field 4𝜋𝑀𝑒𝑓𝑓=4𝜋𝑀𝑠−𝐻𝑎𝑛−𝐻𝑖𝑛𝑡, where 𝐻𝑎𝑛 is the \nmagnetocrystalline anisotropy field of sputtered YIG and 𝐻𝑖𝑛𝑡 is the interfacial \nanisotropy field. We have  𝐻𝑖𝑛𝑡=0 for the YIG single layer  by definition . The first \nterm of Eq. ( S2) is the Zeeman energy and the second term accounts for uniaxial out -\nof-plane anisotropy. Here, we neglect higher order terms that are  relatively small for a \nstrain -free cubic system. The 𝐻𝑟𝑒𝑠 can be calculated by minimizing 𝐸 and, at the \nequilibrium angle of 𝑴, solving the Smit -Beljers equation  [S5],  \n(𝜔\n𝛾)2\n=1\n𝑀2sin2𝜃𝑀[𝜕2𝐸\n𝜕𝜃𝑀2𝜕2𝐸\n𝜕𝜑𝑀2−(𝜕2𝐸\n𝜕𝜃𝑀2𝜕𝜑𝑀2)2\n].                                                (S3) \nAs 𝜃𝐻=𝜋/2, the FMR conditions reduce to the Kittel equation 𝑓=\n𝛾\n2𝜋√𝐻𝑟𝑒𝑠(𝐻𝑟𝑒𝑠+4𝜋𝑀𝑒𝑓𝑓). \nWe can safely assume that the 𝐻𝑎𝑛 did not change before and after the growth of \nBi2Se3 based on Figure S3(d). With this in mind, w e further notice that 𝐾𝑖 can be \nalternatively expressed as (1/2)(4𝜋𝑀𝑒𝑓𝑓𝑌𝐼𝐺−4𝜋𝑀𝑒𝑓𝑓𝐵𝑆/𝑌𝐼𝐺)𝑀𝑠𝑑𝑌𝐼𝐺 , where 4𝜋𝑀𝑒𝑓𝑓𝑌𝐼𝐺 (4𝜋𝑀𝑒𝑓𝑓𝐵𝑆/𝑌𝐼𝐺) represents the 4π𝑀𝑒𝑓𝑓 of YIG (Bi 2Se3/YIG) for a specific \n𝑑𝑌𝐼𝐺. The calculated 𝐾𝑖 using this expression for 𝑑𝐵𝑆 = 25 nm gives an average of -\n0.068  erg/cm-2, in good agreement with a 𝐾𝑖 of -0.075 erg/cm2 obtained from the linear \nfit in Figure 2(b).  \n \nTemperature -dependent FMR spectra of YIG and Bi 2Se3/YIG  \n \n \nExtraction of 𝟒𝝅𝑴𝒆𝒇𝒇 and 𝑯𝒆𝒇𝒇 from temperature dependence of 𝑯𝒓𝒆𝒔 \n \n Since the effective damping constant of YIG and Bi 2Se3/YIG increased \npronouncedly at low 𝑇, the weakened FMR signal was inevitably accompanied by \nlarger uncertainties of measured 𝐻𝑟𝑒𝑠. The uncertainties are  even more serious when \nFIG. S4 . Temperature -dependent FMR first-derivative spectra of (a) YIG(23) and (b) \nBi2Se3(25)/YIG(15). The Δ𝐻 increased with decreasing 𝑇, accompanied by decreas ed \n𝑑𝐼/𝑑𝐻𝑒𝑥𝑡 peak magnitude s due to enhanced damping.  The 𝐻𝑒𝑥𝑡 scale are fixed  to clearly \nshow  the pronounce d change s of 𝐻𝑟𝑒𝑠 and Δ𝐻 induced by Bi2Se3. \nFIG. S5 . (a) 𝐻𝑟𝑒𝑠 vs 𝑇 data of Bi 2Se3(25)/YIG (15) measured at 4 and 4.5 GHz. (b) and (c) \nComparison of extracted 4𝜋𝑀𝑒𝑓𝑓 and 𝐻𝑒𝑓𝑓 by the fitting and solving method.  𝑇<150  K and 𝑓>5 GHz for our instruments . Fitting the data  including points in \nthe 𝑓>5 GHz region to the Kittel equation gives large error s of 4𝜋𝑀𝑒𝑓𝑓 and 𝐻𝑒𝑓𝑓, \nwhich  obscure the temperature dependency  of these two quantities. To reduce the \nuncertainties in the process of extracting 4𝜋𝑀𝑒𝑓𝑓 and 𝐻𝑒𝑓𝑓, we focused on the FMR  \ndata for 𝑓=4 and 4.5 GHz , from which we obtained 𝐻𝑟𝑒𝑠 with satisfactory \naccuracy (Fig . S5(a)). The Kittel equation can be arranged in the following form,  \n𝐻𝑒𝑓𝑓2+(2𝐻𝑟𝑒𝑠+4𝜋𝑀 𝑒𝑓𝑓)𝐻𝑒𝑓𝑓+(4𝜋𝑀 𝑒𝑓𝑓𝐻𝑟𝑒𝑠+𝐻𝑟𝑒𝑠2−4𝜋2𝑓2\n𝛾2)=0.         (S3) \nWith 𝛾=1.77×1011 t−1s−1, the two sets of data in Fig . S5a provided us with \nsufficient information to explicitly solve the second -order equation. Fig . S5b and S5 c \ncompare the results of \"fitting\" and \"solving\" the Kittel equation. For 𝑇>150  K, \nwhere FMR can be accurately measured up to 7 GHz, the 4𝜋𝑀𝑒𝑓𝑓 and 𝐻𝑒𝑓𝑓 obtained \nfrom solving and fitting met hod agree well, de monstrating the reliability of the solving \nmethod.  \n \n \n \n \n \n \n \n \n \n \n \n Temperature dependence of magnetization hysteresis loop  \n \n  \nFIG. S6 . Temperature dependence of magnetization hysteresis loop  of Bi2Se3(16)/YIG (17) \nmeasured by a SQUID magnetometer. The paramagnetic background of the GGG substrate has \nbeen subtracted. No shifts of hysteresis loops were observed.  References:  \n[S1] O. d'Allivy Kelly, A. Anane, R. Bernard, J. Ben Youssef, C. Hahn, A. H. \nMolpeceres, C. Carrétéro, E. Jacquet, C. Deranlot, P. Bortolotti, R. Lebourgeois, \nJ. C. Mage, G. de Loubens, O. Klein, V . Cros, and A. Fert, Appl. Phys. Lett. 103, \n082408 (2013).  \n[S2] J. C. Gallagher, A. S. Yang, J. T. Brangham, B. D. Esser, S. P. White, M. R. Page, \nK.-Y . Meng, S. Yu, R. Adur, W. Ruane, S. R. Dunsiger, D. W. McComb, F. Yang, \nand P. C. Hammel, Appl. Phys. Lett. 109, 072401 (2016).  \n[S3] H. Chang, P. Li, W. Zhang, T. Liu , A. Hoffmann, L. Deng, and M. Wu, IEEE \nMagn. Lett. 5 (2014).  \n[S4] H. L. Wang, C. H. Du, P. C. Hammel, and F. Yang, Phys. Rev. B 89 (2014).  \n[S5] J. Smit and Beljers, Philips Res. Repts. 10, 113 (1955).  \n "    },    {        "title": "1911.11311v1.Ultrastrong_coupling_between_a_microwave_resonator_and_antiferromagnetic_resonances_of_rare_earth_ion_spins.pdf",        "content": "Ultrastrong coupling between a microwave resonator and antiferromagnetic\nresonances of rare earth ion spins\nJonathan Everts,1Gavin G. G. King,1Nicholas Lambert,1Sacha Kocsis,2Sven Rogge,2and Jevon J. Longell1\n1The Dodd-Walls Centre for Photonic and Quantum Technologies,\nDepartment of Physics, University of Otago, Dunedin, New Zealand\n2Centre for Quantum Computation and Communication Technology, The\nUniversity of New South Wales, Sydney, New South Wales 2052, Australia\nQuantum magnonics is a new and active re-\nsearch \feld, leveraging the strong collective cou-\npling between microwaves and magnetically or-\ndered spin systems [1]. To date work in quan-\ntum magnonics has focused on transition met-\nals and almost entirely on ferromagnetic reso-\nnances in yttrium iron garnet (YIG) [2, 3]. An-\ntiferromagnetic systems have gained interest as\nthey produce no stray \feld, and are therefore\nrobust to magnetic perturbations and have nar-\nrow, shape independent resonant linewidths [4].\nHere we show the \frst experimental evidence of\nultrastrong-coupling between a microwave cav-\nity and collective antiferromagnetic resonances\n(magnons) in a rare earth crystal. The combina-\ntion of the unique optical and spin[5, 6] properties\nof the rare earths and collective antiferromagnetic\norder paves the way for novel quantum magnonic\napplications.\nSuperconducting qubits have provided a new set of ca-\npabilities in quantum computing, control and measure-\nment. In turn, this has generated much interest in \\hy-\nbrid quantum systems\" [7]. In such approaches the super-\nconducting qubits capabilities are enhanced by coupling\nthem to another type of system. Amongst the challenges\nthat the hybrid approach tries to address is the long term\nstorage of quantum information by using spins, as well\nas long distance, room temperature, quantum communi-\ncation by microwave to optical conversion [8].\nRare earth ions o\u000ber exciting possibilities for quantum\ninformation because of the long coherence times available\nfor both their optical [9] and optically addressed spin\ntransitions [10]. Quantum memories based on ensembles\nof rare earth ions have demonstrated large bandwidths\n[11], high e\u000eciencies and very long storage times [10].\nRare earth ions in solids are also being investigated\nfor microwave to optical conversion [12{16]. When using\nrare earth doped samples for these applications, there\nis a tradeo\u000b when it comes to the concentration of the\ndopant. Higher dopant concentrations cause a desirable\nincrease in the collective coupling strength to electro-\nmagnetic waves but an undesirable increase in both ho-\nmogeneous and inhomogeneous linewidths. For optical\ntransitions in low concentration rare earth dopants, this\nincrease is often due to Stark shifts or an increase in crys-\ntal strain from dopants degrading the crystal quality [17].\nFor electron spin transitions it is often due to magnetic\ndipole-dipole interactions [18].A way out of this concentration-linewidth trade o\u000b is to\ninstead use fully concentrated rare earth crystals where\nthe rare earth is part of the host crystal. In the case\nof optical transitions it has been shown that very narrow\ninhomogeneous linewidths (25 MHz)[6] can be achieved in\nfully concentrated samples, linewidths comparable with\nthe narrowest seen in doped samples [5, 19].\nHere we investigate the coupling between antiferro-\nmagnetic magnon modes in the fully concentrated rare\nearth crystal gadolinium vanadate (GdVO 4) and a mi-\ncrowave cavity. We show that not only can ultrastrong-\ncoupling be achieved but that the linewidth of the\nmagnon resonances are narrow, an important property\nfor their use in hybrid quantum systems.\nThe magnetic and thermal properties of fully concen-\ntrated rare earth phosphates and vanadates have been\ninvestigated before [20, 21], and collective resonances\nwere observed in gadolinium aluminate (GdAlO 4) and\nGdVO 4. These were observed however, at much higher\ntemperatures and showed much larger linewidths than\nwe observe here. In general antiferromagnetic resonance\nin rare earth systems is a largely unexplored area.\nGdVO 4has a tetragonal crystal lattice with space\ngroupD19\n4h(I4=amd ). There are four Gd3+ions per\nunit cell with site symmetry D2d. At room tempera-\nture GdVO 4is paramagnetic but upon cooling at zero-\nmagnetic \feld to T\u00142.495 K, it orders antiferromagnet-\nically as a simple two sublattice system parallel to the\ncrystallographic c-axis [22]. The low ordering tempera-\nture is a characteristic of rare earth spins as the exchange\ninteraction between the highly shielded 4 felectrons is\nweak, and often the magnetic dipole-dipole interaction\ndominates [23]. The Gd3+ions are in an8S7=2ground\nstate, a predominantly L= 0 state; crystal \feld e\u000bects\nare therefore small and the g-factor is approximately\nisotropic with a value of 2. GdVO 4is iso-structural with\nthe ubiquitous laser host material yttrium vanadate and\nis a good laser host in its own right. The sample we used\nwas commerically grown and available o\u000b the shelf.\nThe antiferromagnetic resonance [24] seen in our sys-\ntem can be understood by modelling each sublattice by a\nsingle spin-1 =2 and using the following interaction Hamil-\ntonian\nH=ASz\n1Sz\n2\u0000\u0016BgB\u0001(S1+S2): (1)\nThe \frst term describes the (predominantly) magnetic\ndipole-dipole interaction that causes antiferromagneticarXiv:1911.11311v1  [quant-ph]  26 Nov 20192\nordering and the second term describes the interaction\nof the spins with an applied magnetic \feld.\nWe takeSz\n1to be our upward pointing spin ( S1\nz\u00191\n2,\nS2\nz\u0019\u00001\n2), and apply a static magnetic \feld of strength\nB0along thez-axis. Assuming only small excitations\nof the spins we make the Holstein-Primako\u000b transforma-\ntions [25]\nSz\n1=1\n2\u0000^ay^a (2)\nSz\n2=\u00001\n2+^by^b (3)\nS+\n1\u0019^a (4)\nS\u0000\n2\u0019^b; (5)\nwhere ^aand^bare bosonic annihilation operators for a\nspin excitation (spin \rip) on sublattice 1 and 2 respec-\ntively. Substituting into Eq.1 and keeping terms up to\nsecond-order in creation/annihilation operators leads to\nthe Hamiltonian\nH= \u0016h!0(^ay^a+^by^b) +g\u0016BB0(^ay^a\u0000^by^b); (6)\nwhere!0=A=2\u0016h. Thus in our simple model there are\ntwo antiferromagnetic resonance modes each involving\nLarmor precession of just one sublattice. At zero mag-\nnetic \feld, the two modes are degenerate at a frequency\nthat is directly related to the strength of the ordering\ninteraction. The frequency of mode ^ a, which sees the\napplied magnetic \feld in addition to the internal \feld,\nincreases linearly with applied magnetic \feld, where as\nthe frequency of mode ^b, which sees the applied \feld op-\nposing the internal \feld, decreases linearly with applied\nmagnetic \feld.\nAntiferromagnetic resonance in GdVO 4has been seen\nin electron spin resonance experiments, with a predicted\nlow temperature zero-\feld frequency of \u001834 GHz [26].\nUsing a g-factor of 2 the antiferromagnetic resonance fre-\nquencies predicted by Eq. 6 are plotted in Fig. 1(a). At\n\u00181:2 T the lower antiferromagnetic resonance branch\nintercepts the zero frequency axis. At this point a transi-\ntion occurs between the antiferromagnetic phase and the\nspin-\rop phase. In the spin \rop phase, the energy penalty\nassociated with having one of the sublattices anti-aligned\nwith the external \feld is relieved by the spins switching\nfrom the crystal c-axis to the crystal a-axis, and tilting\na small angle \u0012towards the external \feld. As the ex-\nternal \feld is increased, \u0012increases, eventually returning\nthe crystal to the paramagnetic phase. The magnetic\nphase in GdVO 4is shown as a function of temperature\nand magnetic \feld in Fig. 1(b).\nThe experimental setup is shown in Fig. 5 the GdVO 4\nsample was placed inside a loop-gap resonator and cooled\nwithin a dilution refrigerator. Figure 2(a) shows the\ntransmission through the cavity as the cavity is cooled\nfrom room temperature to 4 K, with zero applied mag-\nnetic \feld. With decreasing temperature the resonance\n~FIG. 1. (a) The antiferromagnetic resonance frequencies\nof GdVO 4as a function of magnetic \feld. Beyond the spin-\n\rop transition Eq.6 no longer holds, spin resonance theory in\nthe spin-\rop state is required instead [27]. (b) The magnetic\nphase diagram of GdVO 4as a function of applied magnetic\n\feld strength and temperature [28].\n50100 150 200\nT (K)11.411.611.8f (GHz)\n-70-60-50S21(dB)\n0 1 2 3\nB (T)11.111.211.311.411.5f (GHz)\n-60-50-40-30S21(dB)\nAbs. Abs. Abs. Abs.T=300 K\nB=0 T\nT=4 K\nB=0 T\nT=4 K\nB=3 T\nT<2.5 K\nB=0 T\nf (GHz) 11.25  34\nc\nFIG. 2. (a) The cavity transmission as cooled from room\ntemperature to 4 K. (b) Cavity transmission as a function\nof applied magnetic \feld with a \fxed temperature of T=\n4 K. (c) A sketch showing the relative locations of the cavity\n(blue) and spin (orange) resonances at di\u000berent \felds and\ntemperatures.\nfrequency of the cavity is seen to increase and broaden\nuntil the resonance completely disappears at T\u001435 K.\nThis occurs due to the interaction between the cavity and\nthe spins. At 300 K, there is a very broad paramagnetic\nresonance centered on 0 Hz. Because it is so broad there\nis very little absorbtion at the cavity frequency. As the\ntemperature decreases, spin lattice relaxation decreases\nand the spins resonance narrows, leading to more loss at3\nFIG. 3. T= 25 mK, transmission through the cavity as a\nfunction of magnetic \feld and frequency. The red line indi-\ncates the onset of the spin-\rop phase.\nthe cavity frequency. Once the temperature reaches 4 K\nthe attenuation has increased to the point that the cavity\nresonance disappears.\nKeeping the temperature \fxed at 4 K an external \feld\nis applied (along the crystals c-axis) to shift the param-\nagnetic resonance of the ions away from the cavity. In\nFig. 2(b) we see that as the magnetic \feld increases and\nthe resonance of the ions is moved, the source of loss\nto the cavity is reduced allowing the cavity resonance to\nreappear.\nThe cavity resonance also reappears when the sample is\ncooled past the transition temperature T\u00142.495 K. Be-\nlow the transition temperature the spins become locked\ntogether in a long-range order giving a narrow resonant\npeak at the zero \feld antiferromagnetic resonance fre-\nquency,\u001834 GHz. The cavity is well detuned from this\npeak and hence resonates as if it were empty.\nTo investigate the coupling between the antiferromag-\nnetic resonant mode and the microwave cavity, the cavity\ntransmission is measured as the lower antiferromagnetic\nresonant branch (shown in Fig. 1) is pulled through the\ncavity resonance via sweeping the applied magnetic \feld.\nWith the temperature \fxed at 25 mK Fig. 3(a) shows\nthe cavity transmission as a function of applied magnetic\n\feld. As the lower antiferromagnetic resonant branch\npasses through the cavity resonance a clear avoided cross-\ning is seen centered at \u00190:9 T. The shape is not sym-\nmetric, there is a signi\fcant bump in the dressed state\nfrequency at\u00191:1 T due to the onset of the spin-\rop\ntransition. A detailed investigation of this phenomenon\nwill be left to future work.\nWe \ft to our data (for B < 1:1 T) the eigenvalues of a\nsimple coupling Hamiltonian\nH=!c^cy^c+!m^my^m+G\u0000\n^cy^m+ ^my^c\u0001\n; (7)\n0 100 200 300 400 500 600 700\nTmxc (mK)050100150200250γ (MHz)\n00.20.40.60.811.21.41.6\nG (GHz)FIG. 4. Blue - polariton linewidth measured at a \fxed fre-\nquency of 15.6 GHz as a function of the mixing chamber tem-\nperature, Orange - coupling strength Gobtained from \ftting\nthe coupling Hamiltonian Eq.(7). The grey points indicate\nthe region where we haven't ruled out the possibility that the\nmixing chamber and the sample are di\u000berent temperatures.\nwhere!c(!m) is the resonant frequency of the cavity\nmode ^c(magnon mode ^ m) andG=p\nNgis the cou-\npling strength, proportional to the number of spins N\nand single ion coupling strength g. From the \ft we ob-\ntain a coupling strength of G= 1.72 GHz, which divided\nby the central frequency of the cavity ( f0= 11:245 GHz)\ngives a coupling \fgure of G=f 0= 0.15 putting the system\nin the ultrastrong coupling regime.\nMeasuring the linewidth of the polariton modes at\npoints far away from the central cavity frequency gives\nus an estimate for the linewidth of the antiferromag-\nnetic resonant mode. In order to avoid the background\nfrequency dependence of the experimental components\n(cables/attenuators/ampli\fer) we measure the magnetic\nlinewidth of the polariton. The linewidth measured in\nTesla is then converted into the frequency domain via\nthe relation \r(!) =\r(B)g\u0016B, where\r(!) and\r(B) are\nthe polariton linewidths in the frequency and magnetic\ndomains respectively. Fixing the frequency at 15.6 GHz\nthe linewidth of the upper branch was measured as a\nfunction of the mixing chamber temperature ( Tmxc), as\nshown in Fig. 4. As the temperature of the mixing\nchamber is lowered the magnon linewidth reduces un-\ntil a temperature of Tmxc\u0014200 mK where the linewidth\nremains roughly constant at \u001835 MHz. The reduction\nin linewidth with temperature is expected because of a\nreduction in multimagnon processes [29]. The constant\nlinewidth we see for Tmxc\u0014200 mK could be due to some\nnon-magnetic broadening mechanism, however it could\nalso be explained by imperfect thermalisation between\nthe mixing chamber and our sample leaving the sam-4\nple temperature constant at \u0018200 mK despite the lower\nmixing chamber temperature. The smooth line shown\nbehind the linewidth data points is a \ft to the equation\n\r=A+BT4, where the coe\u000ecients were calculated as\nA= 34:8 MHz and B= 8:6\u000210\u000010MHz/K4.\nThe coupling strength Gmeasured via \ftting Eq. (7) is\nalso shown as a function of Tmxcin Fig. 4. The coupling\nstrength follows a similar trend to the linewidth data, re-\nmaining constant at \u00181:72 GHz up until Tmxc\u0019200 mK,\nafter which it starts decreasing proportional to T4. Fit-\nting the expression G=A\u0000BT4gives coe\u000ecients\nA= 1:7 GHz and B= 9:39\u000210\u000012GHz/K4.\nOur results put us in the ultrastrong coupling regime,\nG=f 0= 0:15>0:1, the region where counter rotating\nterms in the coupling Hamiltonian start to become sig-\nni\fcant. With a lower frequency resonator ( \u00141:7 GHz)\nthe deep strong coupling regime could be reached. The\nchallenge will be keeping narrow magnon linewidths in\nspite of this low frequency which will require low spin\ntemperatures to maintain high spin order.\nOur results have signi\fcant implications for microwave\nto optical conversion, however GdVO 4is unlikely to be\nthe best material for this. The Gd3+ions have half \flled\n4forbitals and as a result the lowest energy 4 f\u00004ftran-\nsition from the ground-state is in the ultraviolet spectrum\naround 315 nm. Unfortunately the VO3\u0000\n4ions absorb\nstrongly at this wavelength. Di\u000berent rare earth crys-\ntals with better optical properties for upconversion will\nbe the subject of future work.\nAnother exciting prospect provided by rare earth ions\nis the large spectroscopic gfactors available, particularly\nin erbium and dysprosium where g\u001915 is common.\nWith this the collective coupling, which is linear in g,\ncould be further improved.\nOur measurements show that narrow collective mag-\nnetic resonances occur in rare earth crystals. In com-\nparison to rare earth doped samples the concentration to\nlinewidth ratio seen here is orders of magnitude higher.\nQuantum magnonics using rare earth crystals has an ex-\nciting future. The remarkable properties of the rare earth\n4f\u00004ftransitions, the easy accessibility of antiferomag-\nnetic resonances as well as the large gfactors, open up\nmany possibilities for improvements to hybrid quantum\nsystems.I. METHODS\nThe experimental setup used to investigate the cou-\npling between antiferromagnetic modes and a microwave\ncavity is shown in Fig. 5. A loop-gap microwave res-\nonator [30], with a central frequency of 11.245 GHz and\na Q factor of 1300, is mounted on the coldest stage of a\ndilution fridge (BlueFors LD-250) and inside the bore of\na 3 T superconducting magnet. The magnetic \feld was\napplied along the crystal c-axis, which was identi\fed by\nobserving the crystal through crossed polarisers.\nMicrowaves are coupled into the resonator using two\nwire antennas attached to SMA connectors. A vector\nnetwork analyser is used as the microwave signal source\nand detector. To reduce the electronic noise the input\nsignal is attenuated at various temperature stages of the\ndilution fridge, while on the output line a cryoampli\fer is\nused to improve the signal-to-noise ratio. An attenuator\nis also added between the ampli\fer and the cavity; this\nis to suppress heating due to backaction from the input\nof the ampli\fer.\n36+\n22\nGdVO4\nFIG. 5. Schematic of the experimental setup.\n[1] D. Lachance-Quirion, Y. Tabuchi, A. Gloppe, K. Usami,\nand Y. Nakamura, Applied Physics Express 12, 070101\n(2019).\n[2] Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Us-\nami, and Y. Nakamura, Physical Review Letters 113,\n083603 (2014).[3] M. Goryachev, W. G. Farr, D. L. Creedon, Y. Fan,\nM. Kostylev, and M. E. Tobar, Phys. Rev. Applied 2,\n054002 (2014).\n[4] H. Y. Yuan and X. R. Wang, Applied Physics Letters\n110, 082403 (2017), https://doi.org/10.1063/1.4977083.\n[5] C. W. Thiel, T. B ottger, and R. L. Cone, Journal of5\nLuminescence Selected papers from DPC'10, 131, 353\n(2011).\n[6] R. Ahlefeldt, M. Hush, and M. Sellars, Physical Review\nLetters 117, 250504 (2016).\n[7] G. Kurizki, P. Bertet, Y. Kubo, K. M\u001clmer, D. Pet-\nrosyan, P. Rabl, and J. Schmiedmayer, Proceedings of\nthe National Academy of Sciences 112, 3866 (2015).\n[8] N. J. Lambert, A. Rueda, F. Sedlmeir, and H. G. L.\nSchwefel, arXiv:1906.10255 [physics, physics:quant-ph]\n(2019), arXiv: 1906.10255.\n[9] T. B ottger, Y. Sun, C. W. Thiel, and R. L. Cone, Phys-\nical Review B 74, 075107 (2006).\n[10] M. Zhong, M. P. Hedges, R. L. Ahlefeldt, J. G.\nBartholomew, S. E. Beavan, S. M. Wittig, J. J. Longdell,\nand M. J. Sellars, Nature 517, 177 (2015).\n[11] C. Clausen, I. Usmani, F. Bussi\u0012 eres, N. Sangouard,\nM. Afzelius, H. d. Riedmatten, and N. Gisin, Nature\n469, 508 (2011).\n[12] C. O'Brien, N. Lauk, S. Blum, G. Morigi, and M. Fleis-\nchhauer, Physical Review Letters 113, 063603 (2014).\n[13] L. A. Williamson, Y.-H. Chen, and J. J. Longdell, Phys-\nical Review Letters 113, 203601 (2014).\n[14] X. Fernandez-Gonzalvo, S. P. Horvath, Y.-H. Chen, and\nJ. J. Longdell, Physical Review A 100, 033807 (2019).\n[15] S. Welinski, P. J. Woodburn, N. Lauk, R. L. Cone, C. Si-\nmon, P. Goldner, and C. W. Thiel, Physical Review\nLetters 122, 247401 (2019).\n[16] J. R. Everts, M. C. Berrington, R. L. Ahlefeldt, and J. J.\nLongdell, Physical Review A 99, 063830 (2019).\n[17] M. J. Sellars, E. Fraval, and J. J. Longdell, Journal of Lu-\nminescence Proceedings of the 8th International Meeting\non Hole Burning, Single Molecule, and Related Spectro-\nscopies: Science and Applications, 107, 150 (2004).\n[18] R. S. d. Biasi and A. A. R. Fernandes, Journal of Physics\nC: Solid State Physics 16, 5481 (1983).\n[19] R. M. Macfarlane, R. S. Meltzer, and B. Z. Malkin,\nPhysical Review B 58, 5692 (1998).[20] B. Bleaney, Applied Magnetic Resonance 19, 209 (2000).\n[21] G. J. Bowden, Australian Journal of Physics 51, 201\n(1998).\n[22] J. Cashion, A. Cooke, L. Hoel, D. Martin, and M. Wells,\nin\u0013Elements des terres rares , Vol. 2, Centre National de\nla Recherche Scienti\fque (Paris) ( \u0013Editions du Centre Na-\ntional de la Recherche Scienti\fque, Paris, 1970) pp. 417{\n426.\n[23] E. Lagendijk, H. W. J. Bl ote, and W. J. Huiskamp,\nPhysica 61, 220 (1972).\n[24] C. Kittel, Physical Review 82, 565 (1951).\n[25] T. Holstein and H. Primako\u000b, Phys. Rev. 58, 1098\n(1940).\n[26] M. M. Abraham, J. M. Baker, B. Bleaney, J. Z. Pfe\u000ber,\nand M. R. Wells, Journal of Physics: Condensed Matter\n4, 5443 (1992).\n[27] W. Yung-Li and H. B. Callen, Journal of Physics and\nChemistry of Solids 25, 1459 (1964).\n[28] B. Mangum and D. Thornton, AIP Conference Proceed-\nings5, 311 (1972).\n[29] S. M. Rezende and R. M. White, Physical Review B 14,\n2939 (1976).\n[30] J. R. Ball, Y. Yamashiro, H. Sumiya, S. Onoda,\nT. Ohshima, J. Isoya, D. Konstantinov, and Y. Kubo,\nApplied Physics Letters 112, 204102 (2018).\nII. ACKNOWLEDGEMENTS\nThe authors would like to thank Rose Ahlefeldt, Matt\nBerrington and Matt Sellars for valuable discussions.\nThis work was supported by Army Research O\u000ece\n(ARO/LPS) (CQTS) grant number W911NF1810011,\nthe Marsden Fund (Contract No. UOO1520) of the Royal\nSociety of New Zealand, the ARC Centre of Excellence\nfor Quantum Computation and Communication Tech-\nnology (Grant CE170100012) and the Discovery Project\n(Grant DP150103699)."    },    {        "title": "1702.05226v3.Spin_conductance_of_YIG_thin_films_driven_from_thermal_to_subthermal_magnons_regime_by_large_spin_orbit_torque.pdf",        "content": "Spin conductance in extended thin \flms of YIG driven from thermal to subthermal\nmagnons regime by large spin-orbit torque\nN. Thiery,1A. Draveny,1V. V. Naletov,1, 2L. Vila,1J.P. Attan\u0013 e,1C. Beign\u0013 e,1G. de\nLoubens,3M. Viret,3N. Beaulieu,3, 4J. Ben Youssef,4V. E. Demidov,5S. O. Demokritov,5, 6\nA. N. Slavin,7V. S. Tiberkevich,7A. Anane,8P. Bortolotti,8V. Cros,8and O. Klein1,\u0003\n1SPINTEC, CEA-Grenoble, CNRS and Universit\u0013 e Grenoble Alpes, 38054 Grenoble, France\n2Institute of Physics, Kazan Federal University, Kazan 420008, Russian Federation\n3SPEC, CEA-Saclay, CNRS, Universit\u0013 e Paris-Saclay, 91191 Gif-sur-Yvette, France\n4LabSTICC, CNRS, Universit\u0013 e de Bretagne Occidentale, 29238 Brest, France\n5Department of Physics, University of Muenster, 48149 Muenster, Germany\n6Institute of Metal Physics, Ural Division of RAS, Yekaterinburg 620041, Russian Federation\n7Department of Physics, Oakland University, Michigan 48309, USA\n8Unit\u0013 e Mixte de Physique CNRS, Thales, Universit\u0013 e Paris-Saclay, 91767 Palaiseau, France\n(Dated: October 26, 2021)\nWe report a study on spin conductance through ultra-thin extended \flms of epitaxial Yttrium\nIron Garnet (YIG), where spin transport is provided by propagating spin waves, that are generated\nand detected by direct and inverse spin Hall e\u000bects in two Pt wires deposited on top. While at low\ncurrent the spin conductance is dominated by transport of exchange magnons, at high current, the\nspin conductance is dominated by magnetostatic magnons, which are low-damping non-equilibrium\nmagnons thermalized near the spectral bottom by magnon-magnon interaction, with consequent a\nsensitivity to the applied magnetic \feld. This picture is supported by microfocus Brillouin Light\nScattering spectroscopy.\nFIG. 1. (Color online) a) Top view of the lateral device. Two\nvertical Pt wires (grey) are placed at a distance d= 1:2\u0016m\napart on top of a 18 nm thick YIG \flm (scale bar is 5 \u0016m).\nThe non-local conductance I-V(injector-detector) is mea-\nsured using negative and positive current pulses while rotat-\ning the magnetic \feld H0in-plane by an angle \u0012. Panel b)\nshows the temperature elevation produced in the Pt injector\nby Joule heating while increasing the pulse amplitude I.\nThe recent demonstrations that spin orbit torques\n(SOT) allow one to generate and detect pure spin cur-\nrents [1{8] has triggered a renewed e\u000bort to study\nmagnons transport in extended magnetic \flms. This\ntopic is currently recognized as one of the important\nemerging research direction in modern magnetism [9].\nThis is because, in contrast to spin transfer process in\ncon\fned geometries ( e.g.nano-pillars or nano-contacts)\nwhere usually the uniform magnon mode dominate the\ndynamics, very little is known about spin transfer in\nextended geometries, which have continuous spin-wavespectra containing many modes which can take part in\nthe magnon-magnon interactions. A large e\u000bort has con-\ncentrated so far on yttrium iron garnet (YIG), a magnetic\ninsulator, which is famous for having the lowest known\nmagnetic damping parameter [10]. From a purely funda-\nmental point of view, the studies of magnon transport in\nYIG by means of the direct and inverse spin Hall e\u000bects\n(ISHE) [2, 11{19] are very interesting, as they provide\nnew means to alter e\u000eciently the energy distribution of\nmagnons and, potentially, even to trigger condensation\n[20].\nContrary to the case of magnons excited coherently,\ne.g.by means of a ferromagnetic resonance or parametric\npumping, when the frequencies of the excited magnons\nare fully determined by the frequencies of the external\nsignals, the excitation of magnons by means of spin trans-\nfer process lacks frequency selectivity [21], and, therefore,\ncan lead to their excitation in a broad frequency range.\nThis poses a challenge for the identi\fcation of the nature\nof magnons modes excited by SOT. It has been already\nshown in [22], that it is convenient and useful to introduce\nthe concepts of subthermal (having energy close to the\nbottom of the spin wave spectrum) and thermal (having\nenergy close to kBT) magnons. On one hand, it has been\nwell established [23, 24] that subthermal magnons can be\nvery e\u000eciently thermalized near the spectral bottom (re-\ngion of so-called magnetostatic waves) by the intensive\nmagnon-magnon interaction, whose decay rate between\nquasi-degenerate modes increases with power, to reach a\nquasi-equlibrium state by a non-zero chemical potential\n[23{25] and an e\u000bective temperature [26]. On the other\nhand, it has been shown, both experimentally and the-arXiv:1702.05226v3  [cond-mat.mtrl-sci]  17 Sep 20172\noretically [26], that the groups of subthermal and ther-\nmal magnon are e\u000bectively decoupled from each other, as\nunder the intensive parametric pumping one can reach\na state where the e\u000bective temperature of subthermal\nmagnons exceeds the real remperature characterizing the\nthermal magnons by a factor of 100.\nUnder spin transfer process, whose e\u000eciency is known\nto increase with decreasing magnon frequency, in con-\n\fned geometries with localized spin-current injection ( i.e.\nwhen there are no quasi-degenerate modes), it has been\nshown, that one can reach current induced coherent GHz-\nfrequency magnon dynamics in YIG [16, 17, 27]). In\nextended thin-\flms, the recently discovered non-local\nmagnon transport [28{32] suggests that the magnon\ntransport properties of YIG \flms subjected to small SOT\nare dominated by thermal magnons, whose number over-\nwhelmingly exceeds the number of other modes at any\nnon-zero temperature. The interesting challenge is to\nelucidate what will happen to this spectrum (in partic-\nular the interplay between the thermal and subthermal\npart [33]) when one applies large SOT to a magnon con-\ntinuum.\nWe propose herein to measure the room temperature\nspin conductance of YIG \flms when the driving current is\nvaried in a wide range magnitudes [34, 35] creating, \frst,\na quasi-equilibrium transport regime, and, then, driving\nthe system to a strongly out-of-equilibrium state. To\nreach this goal the spin current injected in the YIG by\nSOT shall be increased by more than one order of magni-\ntude compared to previous works, while simultaneously\nreducing the \flm thickness by also an order of magnitude,\nusing ultra-thin \flms of YIG grown by liquid phase epi-\ntaxy (LPE) [14, 36]. A series of lateral devices have been\npatterned on a 18 nm thick YIG \flms. Ferromagnetic res-\nonance (FMR) characterization of the bare \flm are sum-\nmarized in Table 1. On these \flms, we have deposited Pt\nwires, 10 nm thick, 300 nm wide, and 20 \u0016m long. The\nmeasured resistance of the Pt wire at room temperature\nisR0= 1:3 k\n. The lateral device geometry is shown\nin FIG.1a. One monitors the voltage Valong one wire\nas a current I\rows through a second wire separated by\na gap of 1.2 \u0016m. Here the Pt wires are connected by\n50 nm thick Al electrodes colored in yellow. Since large\namount of electrical current needs to \row in the Pt, a\npulse method is used to reduce signi\fcantly Joule heat-\ning. In the following the current is injected during 10 ms\npulses series enclosed in a 10% duty cycle. Temperature\nsensing is provided by the change of relative resistance of\nthe Pt wire during the pulse. In FIG.1b, we have plotted\n\u0014Pt(RI\u0000R0)=R0as a function of the current I, where\nthe coe\u000ecient \u0014Pt= 254 K is speci\fc to Pt. We observe\nthat the pulse method allows to keep the absolute tem-\nperature of our YIG below 340 K [37] at the maximum\ncurrent amplitude of 2.5 mA. Avoiding excessive heating\nof the YIG is crucial because, in a joint review paper\n[38], it is shown that epitaxial YIG \flms grown by LPETABLE I. Summary of the physical properties of the materials\nused in this study.\nYIGtYIG(nm) 4\u0019M s(G)\u000bYIG \u0001H0(Oe)\n18 1 :6\u00021034:4\u000210\u000043.7\nPttPt(nm)\u001a(\u0016\n.cm) \u000bYIGjPtg\"#(m\u00002)\n10 19.5 2 :4\u000210\u000033\u00021018\nFIG. 2. (Color online) Angular dependence of the non-local\nvoltagesV\u0006\n\u0007measured while inverting the polarity of the ap-\nplied \feldH0=\u00062kOe (red/blue) respectively for a) nega-\ntive and b) positive current pulses I=\u00071:5mA. The mea-\nsured signal can be decomposed c) and d) in three compo-\nnents: \u0006 (green): the signal sum, \u0001 (orange): the signal dif-\nference and Vk: the o\u000bset; respectively even/odd, odd/even\nin \feld/current, and an independent contribution (dashed).\nPanel e) shows the current dependence of the amplitude \u0006\nand \u0001.\nbehave as a large gap semiconductor, with an electrical\nresistivity that decreases exponentially with increasing\ntemperature following an activated behavior. As shown\nin Ref.[38], at 340 K, however, the electrical resistivity of\nYIG remains larger than 106\n\u0001cm and thus the YIG can\nstill be considered a good insulator ( R> 30 G\n) over the\ncurrent range explored herein.\nThe lateral device is biased by an in-plane magnetic\n\feld,H0set at a variable polar angle \u0012, with respect\nto the perpendicular of the Pt wires. FIG.2a-d displays\nthe results when I= 1:5 mA andH0= 2 kOe [39]. For\neach value of \u0012, 4 measurements V\u0006\n\u0007are performed corre-\nsponding to the 4 combinations of the polarities of \u0006H0\nand\u0007I(the polarity convention is de\fned in FIG.1a).3\nFIG.2a and 2b show the raw data obtained respectively\nfor negative and positive current pulses. Clearly the non-\nlocal voltage oscillates around an o\u000bset, Vk, de\fned as the\nvoltage measured at \u0012=\u000690\u000e. This o\u000bset is indepen-\ndent of the current polarity and its amplitude scales with\nthe temperature elevation of Pt produced by Joule heat-\ning (see FIG.2c in [38]). We ascribe it to thermoelectric\ne\u000bects produced by a small temperature di\u000berence at the\ntwo PtjAl contacts of the detector circuit [40]. By con-\ntrast, the anisotropic part of the voltage is ascribed to\nmagnons transport.\nTo gain more insight, the data are sorted according to\ntheir symmetry with respect to the ( H0;I)-polarity. This\nis done in FIG.2c and 2d by constructing the signal sum\n\u0006\u0007= (V+\n\u0007+V\u0000\n\u0007)=2\u0000Vk(green tone) even in \feld and\nthe signal di\u000berence \u0001 \u0007= (V+\n\u0007\u0000V\u0000\n\u0007)=2 (orange tone)\nodd in \feld. This separation is exposed in their angular\ndependences, which follow two di\u000berent behaviors, one\nin cos2\u0012, the other one in cos \u0012, respectively. The solid\nlines in FIG.2c and in 2d are a \ft by these two functions.\nComparing the behavior, we observe that the signal \u0006\nis odd in current, while the signal \u0001 is even in current.\nAs noticed in ref [28], these symmetries of \u0006 and \u0001 are\nthe hallmark of respectively SOT [17] and spin Seebeck\ne\u000bects [41, 42]. Hereafter, we shall use the \ft of the\nwhole angular dependence as a mean to extract precisely\nthe amplitude of \u0006 and \u0001 at \u0012= 0.\nFIG.2e shows their evolution as a function of current.\nOne observes that the correspondence between the sym-\nmetries of \u0006 and of \u0001 with respect to the polarities of\nH0andIis respected (within our measurement accuracy)\non the whole current range. While \u0001 approximately fol-\nlows the parabolic increase of the Pt temperature (cf.\nFIG.1a), as expected for thermal e\u000bects, the interesting\nnovel feature is the fact that \u0006 deviates from a purely\nlinear transport behavior at large I. It is important also\nto notice that, when the high/low binding posts of the\ncurrent source and voltmeter are biased in the same ori-\nentation (cf. FIG.1a), the sign of (\u0006 \u0001I)<0. This is a\nsignature that the observed non-local voltage is produced\nby ISHE and not by leakage electrical currents inside the\nYIG [38], although these e\u000bects are only expected to oc-\ncur at much higher temperatures ( >370 K[43]). While\nin both scenarii the induced electrical current \rows in the\nsame direction in the two parallel Pt wires, for ISHE, the\nYIG acts as a source and the potential increases along\nthe current direction [44], in contrast, for Ohmic loss,\nthe YIG acts as a load and the potential drops along the\ncurrent direction ( i.e.(\u0006\u0001I)>0 cf. FIG2b in [38]). We\nshould though add that selecting the component of the\nnon-local voltage that is \u0012-dependent is another e\u000bective\nmean to eliminate Ohmic contribution, since the later are\nindependent of the in-plane orientation of H0. We have\nrepeated this measurement on other devices either on the\nsame \flm or on di\u000berent LPE YIG \flms of similar thick-\nness. On all the devices, we observe an up-turn of \u0006 at\nFIG. 3. (Color online) Current dependence of the absolute\nsum signal \u0006 averaged over the two current polarities. The\ndashed line is a linear \ft of the low current regime and the ar-\nrow marks the onset at which the spin transport deviates from\nlinear behavior. Spin transport excited by SOT can be sepa-\nrated in two independent channels: a linear contribution \u0006(t)\ntaken on by magnons transporting thermal heat and \u0006(s), an\nadditional magnons' conduction channel that emerges above\nIc. Variation of \u0006 as a function of magnetic \feld for two\ndi\u000berent current intensities b) above and c) bellow Ic.\nthe same current density. While the sign of (\u0006 \u0001I) is al-\nways negative, this is not the case for the sign for (\u0001 \u0001I),\nwhich depends on the \flm quality (the same (\u0001 \u0001I) sign\nis observed for all devices on the same \flm but could\nchange depending on the \flm quality). Further progress\non the later issue requires a better understanding of the\ndi\u000berent phenomena contributing to thermal e\u000bects (\u0001-\nsignal) and the means to separate them. In the following,\nwe shall concentrate exclusively on the non-linear behav-\nior of \u0006 which measures the number of magnons created\nby SOT relatively to the number of magnons annhilated\nby SOT while being immune to e\u000bects caused by Joule\nheating.\nUsing open dots, we have plot in FIG.3a both the vari-\nation ofj\u0006\u0000jandj\u0006+jas a function of the current inten-\nsity. Both data set show clearly the emergence of a new\nspin transport channel at large current densities, as ev-\nidence by the deviation from a purely linear conduction\nregime. Since both quantities j\u0006\u0007jfollow the same be-\nhavior on the whole current range, for the sake of simplic-\nity we shall call simply \u0006 (dark green) their averaged. At\nlow current, the SOT signal follows \frst a linear behavior\n\u0006(t)which has been shown to be dominated by thermal\nmagnons transport [28] [45]. We shall de\fne \u0006(s)the ad-\nditional conduction contribution, i.e.the deviation from\nthe extrapolated linear behavior.\nQuite remarkably the enhancement of the conductance\ndue to \u0006(s)occurs very gradually. We emphasize that\nsuch a low rise is very di\u000berent from the sudden surge of\ncoherent magnons observed at the critical thresold in con-\n\fned geometries [46, 47]. Fitting a straight line through\nthe low current regime, \u0006 I2[0;0:9]mA, and the high cur-\nrent deviation, \u0006 I2[1:6;2:3]mA, the intersection provides4\nan estimation of the onset current of this new conduction\nchannel,Ic\u00191:5 mA, which corresponds to a current\ndensityJc\u00195\u00021011A/m2. This value is very close to the\nthreshold current for damping compensation of coherent\nmodes observed at the same applied \feld ( H0= 2 kOe)\nin micron-sized disks [17] and stripes [48] with similar\ncharacteristics.\nMore insight about the nature of the magnons excited\naboveIccan be obtained by studying the \feld depen-\ndence of \u0006 [49]. The results are shown in FIG.3b and\n3c for two values of the current I= 0:4 and 2.5 mA, re-\nspectively below and above Ic. While in the \feld range\nexplored, the signal is almost independent of H0when\nI <I c, it becomes strongly \feld dependent when I >I c.\nThis di\u000berent behaviors are consistent with assigning the\nspin transport to thermal magnons below Icand mainly\nto subthermal magnons above Ic. In the former case, the\nmagnons' energy is of the order of the exchange energy,\nwhich is much larger than the Zeeman energy, while in\nthe latter case, because of their long wavelength, their\nenergy is of the order of the magnetostatic energy. In\nconsequence, \u0006 is expected to increase with decreasing\n\feld at \fxed I, because of the associated decrease of\nIc. The behavior scales well with the reduced quan-\ntityI=Ic. This is shown by the solid line in FIG.3b,\nwhich displays the expected \feld dependence of 1 =Ic(H0)\n[17]) where Ic/(!H+!M=2) (\u000b+\r\u0001H0=(2!K)), where\n!H=\rH0and!M= 4\u0019\rM s,\rbeing the gyromagnetic\nratio, and!K=p\n!H(!H+!M) is the Kittel's law. We\nhave used here the amount of inhomogeneous broaden-\ning, \u0001H0= 1:5G (probably position dependent), as an\nadjustable parameter, while the value of the other pa-\nrameters are those extracted from Table.1.\nThe above interpretation has been checked by preform-\ning microfocus Brillouin light scattering ( \u0016-BLS) in the\nsub-thermal energy range. For this measurement, we\nhave used a second series [50] of non-local devices, where\nthe Pt thickness has been reduced to 7 nm (thus compar-\nison of the results between the 2 series should be done by\njuxtaposing data obtained with indentical current densi-\nties, cf. upper scale). FIG.4a and 4b show on a logarith-\nmic scale the spectral distribution of the BLS intensity, J,\na) underneath the injector and b) underneath the detec-\ntor, which are here separated by d= 0:7\u0016m. The distri-\nbution is measured at I=\u00062 mA ( i.e.9:5\u00021011A/m2)\nwhile the \feld is set to H0= +2 kOe and \u0012= 0\u000e. In\nboth cases, an enhancement of the subthermal magnons\npopulation is observed when ( I\u0001H0)<0 (blue), which\ncorresponds to the con\fguration where the SOT com-\npensates the damping (cf convention in FIG1a). The\nmeasurement for the opposite case ( I\u0001H0)>0 (red) pro-\nvides a reference about the out-of-equilibrium state pro-\nduced by Joule heating. The maximum intensity of the\nred curve indicates the resonance frequency of the Kittel\nmode,!K=2\u0019at the corresponding temperature. This is\nbecause the \u0016-BLS response function is centered around\nFIG. 4. (Color online) Micro-BLS studies of the subthermal\nmagnons spectrum at H0= +2 kOe a) underneath the in-\njector and b) underneath the detector 0.7 \u0016m away. The left\ncolumn show the spectral distribution of the BLS intensity J\nmeasured at I=\u00062mA. The arrows indicate the Kittel fre-\nquency. The di\u000berence J\u0006(grey area) indicate the spectral\ndistribution of the magnons excited by SOT relative to the\nones annihilated. The panel c) plots the current evolution of\nthe integrated intensity J\u0006underneath the injector.\nthe long wavelength magnons. Indeed, the detected sig-\nnal decreases once the magnon wavelength is smaller than\nthe spot size (approximately 0.4 \u0016m: di\u000braction limited).\nIn order to isolate the contribution produced by SOT,\nwe substract the spectral contribution measured at + I\nto the one measured at \u0000I(grey shaded area). This al-\nlows to cancel out the spectral deformation produced by\nJoule heating but, as for the \u0006-signal, this only measures\nthe enhancement of the magnons created by SOT rela-\ntive to the magnons annihilated by SOT. One can clearly\nsee on the shaded data that SOT enhances the magnons\npopulation in a spectral window between the Kittel fre-\nquency and the bottom of the magnon manifold. Next,\nwe have plotted in FIG4c how the spectral integration\nof this di\u000berential signal J\u0006=R\nJ\u0006d!varies as a func-\ntion of the current amplitude underneath the injector.\nOne observes a regime of linear rise at small current, fol-\nlowed by a growth above Jc\u00195\u00021011A/m2in a similar\nfashion as the one reported in FIG3a. The \u0016-BLS exper-\niment thus provides a direct evidence that an additional\nspin conduction channel has indeed emerged in the GHz\nfrequency range (subthermal) at large current when SOT\nis in the range to compensate the damping. It also shows\nthat the magnons newly created are spread at the bottom\nof the magnon manifold.\nIn summary, we have shown that while at low values\nof the spin current the main contribution to the spin\nconductance comes from thermal magnons, the subther-\nmal magnons mainly determine the magnon transport at\nhigh values of the spin current, comparable to the criti-\ncal magnitude at which damping compensation of coher-\nent magnons takes place. We believe that our current\n\fndings are not only important from the from the funda-\nmental point of view, but might be also useful for future5\napplications. While transport of thermal magnons are\ndi\u000ecult to control due to their relatively high energies,\nthe subthermal magnons could be e\u000eciently controlled\nby variation of relatively weak magnetic \felds.\nThis research was supported in part by the CEA pro-\ngram NanoScience (project MAFEYT), by the priority\nprogram SPP1538 Spin Caloric Transport (SpinCaT) of\nthe DFG and by the program Megagrant 14.Z50.31.0025\nof the Russian ministry of Education and Science.The\nwork at Oakland University was supported by the Grants\nNos. EFMA-1641989 and ECCS-1708982 from the NSF\nof the USA, by the CND, NRI and by DARPA. VVN ac-\nknowledges fellowship from the emergence strategic pro-\ngram of UGA, and Russian competitive growth program.\nWe thank G. Zhand, T. van Pham, A. Brenac for their\nhelp in the fabrication of the lateral devices.\n\u0003Corresponding author: oklein@cea.fr\n[1] S. O. Valenzuela and M. Tinkham, Nature 442, 176\n(2006).\n[2] Y. Kajiwara, K. Harii, S. Takahashi, J. 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B 93,\n020403 (2016).\n[50] This second series has been used to investigate the elec-\ntrical properties of YIG thin \flms at high temperature\n[38]."    },    {        "title": "1308.0192v1.Inverse_Spin_Hall_Effect_in_nanometer_thick_YIG_Pt_system.pdf",        "content": "1 Inverse Spin Hall Effect in nanometer -thick  YIG/Pt system  \nO. d’Allivy Kelly1, A. Anane1*, R. Bernard1, J. Ben Youssef2, C. Hahn3, A-H. Molpeceres1, C. \nCarrétéro1, E. Jacquet1, C. Deranlot1, P. Bortolotti1, R. Lebourgois4, J-C. Mage1, G. de Loubens3, O. \nKlein3, V. Cros1 and A. Fert1 \n1Unité Mixte de Physique CNRS/Thales and Université Paris -Sud, 1 avenue Augustin Fresnel, \nPalaiseau, France  \n2Université de Br etagne Occidentale, LMB -CNRS, Brest, France  \n3Service de Physique de l’Etat Condensé, C EA/CNRS, Gif-sur-Yvette, France  \n4Thales Research and Technology, 1 avenue Augustin Fresnel, Palaiseau, France  \n \nKey words:  YIG, FMR, Inverse Spin Hall Effect, spin waves  \n \nAbstract:  \nHigh  quality nanometer -thick (20 nm , 7 nm and 4 nm) epitaxial YIG films have been grown on GGG \nsubstrates using pulsed laser deposition. The Gilbert damping coefficient for the 20 nm thick films is \n2.3 x 10-4 which is the lowest value reported for sub -micrometric thick films. We demonstrate Inverse \nspin Hall effect (ISHE) detection of propagating spin waves using Pt. The amplitu de and the lineshape \nof the ISHE voltage correlate  well to the increase of the Gilbert damping when decreasing thickness of \nYIG. Spin Hall effect based loss -compensation experiments have been conducted but no change in the \nmagnetization dynamics could be d etected.  \n \n \n \n* Contact author :  \nAbdel madjid Anane  \nabdelmadjid.anane@thalesgroup.com  \n  2  \nAmong all magnetic materials, Yttrium Iron Garnet Y 3Fe5O12 (YIG) has been the one that had the most \nprominent role in understanding high frequency magnetization dynamics. Because of its unique \nproperties, bulk YIG crystal was the prototypal material for ferromagnetic resonance (FMR) studies in \nthe mid -twentieth ce ntury. The attractive properties of YIG include: high Curie temperature , ultra low \ndamping (the lowest among all materials at room temperature), electrical insulation, high chemical \nstability and easy synthesis in single crystalline form. Micrometer thick films of YIG were first grown \nusing liquid phase epitaxy (LPE)1, and paved the way for the emergence of a large variety of \nmicrowave devices for high -end analogue electronic applic ations throughout the 1970’s  2. \nMore recently, the interest in emerging large -scale integrated circuit technologies for beyond CMOS \napplications has fostered  new paradigms for data processing . Many of them are based on state \nvariables other than the electron charge and may eventually allow for unforeseen functionalities. \nCoding the information in a spin wave (SW) is among the most promising routes under investigation  \nand has been referred to as magnonics  3. Exciting and detecting spin waves has been ma inly achieved  \nthrough inductive coupling with radiofrequenc y (rf) anten nas but this technology remains \nincompatible with large scale integration4. Disruptive  solution s merging magnonics and spintronics \nhave been recently proposed  where spin transfer torque  (STT)  and magnetoresistive effects  would be \nused to couple to the SW s. For instance, using a STT -nano -oscillator  in a nanocontact geometry,  \ncoherent SWs emission in Ni81Fe19 (Py) thin metallic layer has been recently demonstrated and probed \nby micro -focused  Brilloui n light scattering (BLS)  5.  \nYIG is often considered as the best medium for SW propagation because of its very small  Gilbert \ndamping coefficient (2x10-5 for bulk YIG) . Being an electrical insulator, electron mediated angular \nmomentum transfer can only  occur at the  interface between YIG and a metallic layer . In that context , \nmetals with  large Spin Orbit Coupling (SOC)  like Pt  where  a pure spin current can be generated \nthrough Spin Hall Effect (SHE) 6 have been used to excite7 or amplify8,9  propagating SWs  through \nloss compensation  in YIG . Moreover, d etection of SW  can be achie ved using the  Inverse Spin Hall \nEffect (ISHE) . In ISHE , the flow of a pure spin current from the YIG into the large SOC metal \ngenerates a dc voltage . The ISHE voltage is proportional to the Spin Hall angle () and the effective \nspin mixing conductance  ( ) that is in play in the physics of spin pumping . As for the SOC \nmaterials, u p to recently, mainly Pt has been used , however it can be observed that  other 5 d heavy \nmetals such as Ta 10, W 11 or CuBi 12 are also very promising . \nAs the  amount of angular momentum transferred from (to) the YIG magnetic film per unit volume \nscales with 1/  (where t is the YIG thickness) , it is necessary to reduce the YIG thickness as much as \npossible while keeping its magnetic properties . Indeed , the threshold current density in the SOC metal \nfor the macrospin  mode excitation is expressed as13 :  (Eq. 1) where  is the spin \nHall angle, the electron charge,  the gyromagnetic ratio ,  the FMR frequency and  the YIG’s 3 saturation magnetization . Furthermore, a better understanding of the physics involved in spin \nmomentum transfer at the YIG/metal interface would be achieved by reducing the YIG film thickness \nbelow the exchange length (~ 10 nm)  14 . Up to now, sub micrometer -thick YIG films have been \nmainly grown by  LPE but the ultimate thickn ess are around  200 nm15. To further reduce the \nthickness, other growth  methods are to be considered . Pulsed laser deposition (PLD) is the most \nversatile technique for oxide films epitax y. Several  groups have worked on PLD grown YIG16,17,18,19 \nbut it is only recently that the films quality is approaching that of LPE20,21. \nIn this letter , we present PLD growth of ultrathin YIG films with various thickness (20 nm, 7 nm and 4 \nnm) on Gadolinium Gallium  Garnet  (GGG)  (111) substrates . Structural and magnetic  characterizations  \nand FMR  measurements  demonstrate the high quality of our nanometer -thick YIG films  comparable to \nstate of the art LPE films . The growth has been performed using a frequency tripled (\n = 355 nm) \nNd:YAG laser  and a stochiometric polycrystalline YIG pellet . The pulse rate was 2.5  Hz and the \nsubstrate -target distance was 4 4 mm. Prior to the YIG deposition, the GGG substrate  is annealed  at \n700°C under an oxygen pressure of 0.4 mbar. Growth temperature is then set to be 650 °C, and oxygen \npressure to 0.25 mbar. After the film deposition , samples are cooled down to room temperat ure under \n300 mbar of O 2. The YIG thickness is measured for each sample  using X -ray reflectometry which \nyield a precision better than 0.3 nm. The surface morphology and roughness have been studied by \natomic force microscopy ( AFM ). RMS roughness has been measured over 1 µm2  ranges between 0.2 \nnm and 0.3 nm  for all films  (Fig 1a) . As often with PLD growth, d roplets  are present  on the film \nsurface, here the ir lateral sizes  are below 100 nm and their density is very low (~ 0.1 µm-2). X-Ray \nDiffraction  (XRD)  spectr a using Cu K\n 1 radiation  show that the growth is along the (111) direction . \nOnly peaks characteristic of YIG and the GGG substrate are observed ( see Fig 1b ). The YIG lattice \nparameter is very close to that of the substrate and can only be resolved ev entually at large diffraction \nangle s. For the 20 nm YIG  film (Fig 1b ,1a), refinement using EVA software on the 888 reflection \nyields a cubic lattice parameter of 1.2459 nm, to be compared to 1.2376  nm for the bulk YIG22. For \nthinner films (between 4 and 15 nm) , it was not possible to distinguish the YIG peaks from those of \nthe substrate  (Fig 1d, 1e). This  sharp variation of the XRD spectra  with respect to the film thickness  \ntends to point  towards a critical thickness  for strain relaxation. It is however worth noting that the \ncubic lattice parameter of the 20 nm thick film is larger than the bulk lattice parameter  but also of that \nto the GGG substrate (1.2383 nm) . A slight o ff-stoichiometry (either oxygen vaca ncies or cation \ninterstitials)  is probably at the origin of this observation . Pole figure  measurements  have been \nperformed to gain insight s into the in -plane crystal structure, but it was not possible to resolve , at this \nstage,  the film s peaks from the substrate peaks  from w hich we infer that  the growth is epitaxial and the \nfilm single crystalline . \nFrom SQUID magnetometry with in -plane  magnetic field , we measure  a magnetization of 4\nMs = \n2100 G \n 50 G at room temperature  for both the 20 and 7 nm films . This value is independently  4 confirmed by out -of plan e FMR resonance  while the tabulated bulk value  for YIG  is 4\nMs = 1760 G. \nA similar increase of the PLD grown YIG magnetization have been reported and attributed to an  off-\nstoichiome try19. The coercitive  fields are extremely small , about 0.2 Oe  (which is the experimental \nresolution ) and the saturation field is 5 Oe. There is no evidence for in -plane magnetic anisotropy . The \noverall magnetic signature is that of an ultra -soft material . Note that for the thinnest films (4 nm) we \nmeasure a decrease of the saturation  magnetization  to roughly 1700 G . Finally, we emphasize that the \nstructural and magnetic properties of the samples are well reproducible with respect to the elaboration \nconditions.  \nFMR fields and linewidths were measured at frequencies in the range 1 -40 GHz using high sensitive \nwideband r esonance spectrometer with a nonresonant microstrip transmission line.  The FMR is \nmeasured via the derivative of microwave power absorption using a small rf exciting field. Resonance \nspectra were recorded with the applied static magnetic field  oriented in plane . During the magnetic \nfield sweeps , the amplitude of the modulation field was appreciably smaller than the FMR linewidth. \nThe amplitude of the excit ation field h rf is about 1 mOe , which corresponds to the linear response \nregime. A phase -sensitive detector with lock -in detection was used. The field derivative of the \nabsorbed power  is proportional to the field derivative of the imaginary part of the rf susceptibility: \n (where  and \n″ is the imaginary part of the susceptibility of the uniform \nmode) . Typical resonance curve s are  plotted in Fig. 2a 2b . In Fig. 2c, we  show the frequency \ndependence of the peak -to-peak linewidth  for three different  YIG thicknesses , i.e., 20, 7 and 4 nm . As \nfor the 20 nm  YIG film, we find a linear dependence of the FMR linewidth with rf frequency  while for \nthe thinnest films , we do find an almost linear increase in the low frequency range (< 12 GHz) and \nthen a saturation  of the linewidth with frequency . Such qualitative difference depending on the \nthickness is reminiscent of the qualitative difference discussed earlier in the X -ray diffraction data  (Fig \n1). The linear dependence  of the resonance linewidth  is expected with in the frame of the Landau -\nLifshitz Gilbert equation and allow for a straightforward  calculation  of the intrinsic Gilbert damping \ncoefficient ( for the 20 nm thick film). The zero frequency intercept of the fitting line , \nusually referred to a s the extrinsic lin ewidth  23,24, is found to be  \nH0  =1.4 Oe. We emphasize that our \nvalue for the  intrinsic damping on the 20 nm thick film is among the best ever reported  independ ently \nof the growth technique  and is only outperformed by the 1.3 µm film used by Y. Kajiwara  et al. 7. As \nfor the extrinsic damping, our value s are still a bit larger than those obtained for 200 nm thick films \ngrown by  LPE (\n H0= 0.4 Oe) 10. The saturation of the linewidth with increasing  excitation  frequency  \nobserved for  thinnest films ( t = 7 and 4 nm) is usually  ascribed to two -magnon s scattering  due to the \ninterfaces 25. An estimation of the intrinsic damping in such thin films is  thus not correct. \nNevertheless , and only for the sake of comparison, considering frequencies under 6 GHz , we can \nrough ly estimate the low frequency Gilbert damping  to be  1.610-3 for the 7 nm and 3.810-3 the 4nm \nYIG films . However, it is  worth mentioning that for those  two thinnest films;  samples  sliced  from the 5 same substrate  can give different linewidths ( up to a factor of 3) with for some of them up to 2 \nabsorption s lines. This observation point s to a slight lateral non-homogeneity  in the chemical \ncomposition24. The data presented in figure 2 are tho se of the best samples showing a single absorption \nline. For the 20  nm thick films , all samples have only one resonance line and the dispersion of \nlinewidths is within 5%.   \nIn order to characterize the conversion of propagating SWs in YIG into a charge current in a normal \nadjacent metal with large SOC , we perform ISHE detection of SW  with the strip geometry used by \nChumak et al.26. In our sample design (see Fig 3a) , SWs  excitation is achieved using a patterned 100 \nµm wide  Au stripline  antenna  whereas the ISHE voltage is measured on a 13 nm thick Pt strip ( 0.2 \nmm x  5 mm ) located at  100 µm away from the Au stripe  and parallel to it . The metallic Pt strip is \ndeposited using dc magnetron sputtering  and lift -off. Prior to  the Pt deposition , an in -situ O2/Ar-\nplasma is used to  remove  the photo -resist residue s and increase the ISHE voltage.  This cleaning step \nhas been shown recently to improve the ISHE signal by one order of magnitude27,28. Measurement s of \nthe ISHE signal is performed either using a lock -in (with a 5 kHz TTL modulation  of the rf power ) or \na nano -voltmeter.  In order to increase the output ISHE signa l, we chose a specific configuration with a \nmagnetic field at 45° from  to the SW propagation direction  (cf Fig. 3a). This configuration  has the \nadvantage of provid ing a good coupling of the YIG film  to the rf field under the antenna while still \nhaving a sig nificant spin polarization  () that is orthogonal  to the measured ISHE electrical field . We \nshould point out  that our choice of magnetic field direction implies that the propagating SWs are \nneither Damon -Eshbach modes  where k \n M nor backward volume modes where  k // M.  \nIn Fig 3b, we display the  ISHE voltage (without any geometrical correction) as a function of the in \nplane magnetic field  measured  on a 20 nm thick YIG under a 10 mW rf excitation at 1 GHz. The sign \nof voltage peak (occurring at magnetic fields that resonantly excite the magnetization) reverses when \nthe applied field is reversed as expected from the ISHE symmetry 29 : where  \nis the pure spin current that flows through the YIG/Pt interface , is the spin polarization vector \nparallel to the dc magnetization direction  and  is the unit vector parallel to the Pt strip.  A close -up \non the ISHE signal lineshape for the different thickness es is plotted in Fig . 3c 3d 3e. In accordance \nwith FMR study, the linewidth increases with decreasing YIG thickness. The spectral  lineshap es are \nalmost symmetrical confirming previous reports on YIG films  grown using LPE  30. The ISHE \nmaximum voltage decreases when the film thickness  is decrease d. Going from 20 nm to 4 nm this \ndecrease is as large as two order s of magn itudes  and correlates to the increase of the Gilbert damping . \nA decrease of the ISHE signal when increasing damping is expected.  Indeed, a s we are in the weak \nexcitation regime, the ISHE voltage is expected to scale with 1/\n2, see for instance supplementary  \nmate rials of Ref [7]. If we consider the Gilbert damping obt ained from  FMR on the bar e YIG samples,  \na more dramatic decrease of the I SHE signal  than the one observed is expected.  In fact here, one \nshould consider the effective damping of the Pt/YIG stack as spin -pumping will increase the YIG 6 effective damping under the Pt strip. We should ag ain point out  that our measurement geometry relies  \non propagating SWs  and therefor e they are subject to an exponential decay with distance . The length \nscale of this exponential decay is different for the three  thicknesses  owing to the difference in the \ndamping parameter ; hence a quantitative interpretation of the voltage amplitude is to be avoided here.  \nWe thus succeeded  to grow high quality ultrathin YIG film and demonstrated that a n efficient  spin \nangular momentum transfer from the YIG film toward the Pt layer . The nex t objective is to induce an \nmodification on the effective YIG damping coefficient via interfacial spin injection using SHE , as it \nhas been  reported in Py/Pt31,32 system . Using YIG/Pt,  Kajiwara et al.  have shown that even without rf \nexcitation, spin injection induced by  a dc current in Pt can generate propagating SWs ; the threshold  \ncurrent density has been estimated to be 4.4 x  108 A.m-2. Such unexpectedly low value has been \nattributed to the presence of an easy –axis surface anisotropy13. Hence, w e have performed experiments  \nwhere  a large  dc bias current is applied on the Pt electrode  while pumping spin waves  in the 20 nm \nthick YIG film  with TTL modulated rf excitation field . The VISHE linewidth  is measured at the TTL \nmodulation frequency using a lock -in. We  expected to increase or decrease  this linewidth  depending \non the dc current  polarity  but w e have not been able to see any sizable effect even for current densities \nas large as 6 x 109 Am-2, within a 0.2 Oe resolution, the measured linewidth remains absolutely \nunchanged . Theory  predicts that th e threshold dc current for the onset of magnetic excitations scales \nwith the Gilbert  damping and the thickness  of the ferromagnetic insulator  (Eq. 1), the smaller the \n product the lower  the threshold current for SW excitation  is. In Kajiwara et al. ’s experiment  \n nm; in our case  nm (considering the spin pumping contribution  to the \ndamping ). We therefore applie d a dc current that is roughly 2  orders of magnitude  larger than the \nexpected threshold current . One possible explanation  is that in our films , surface anisotropy is absent \nand therefore the pumped angular momentum is spread over many excitations modes33. Further \ninvestigations are under progress to clarify this point.  \nIn summary,  we have fabricated PLD grown YIG on GGG (111) substrates with t hickness as low as 4 \nnm. We present here a comparative study on three  different thicknesses : 4, 7 and 20 nm . The Gilbert \ndamping coefficient for the 20 nm thick films is 2.3 x 10-4 which is the lowest value reported for sub -\nmicrometric thick films. We demonstrate ISHE detection of SWs for ultra thin YIG film. The \namplitudes of the ISHE voltage correlate s well to the increase of the Gilbert damping when decreasing \nthickness. Owing to extremely low product  of the 20 nm film that is almost 10 time s smaller \nthan the one reported by Kajiwara et al.  we expected to observe  compensation of the damping by spin \ncurrent injection through the SHE  but our preliminary results on the VISHE  linewidth did not reveal any \neffect on the magnetization dynamics.  \n \nAcknowledgement:  \nThis work has been supported by ANR -12-ASTR -0023 Trinidad   7  \nReferences:  \n \n1 J. E. Mee, J. L. Archer, R.H. Meade, and T.N. Hamilton,  Applied Physics Letters  10 \n(1967).  \n2 J. P. Castera,  Journal of Applied Physics 55 (6), 2506 (1984).  \n3 A. Khitun, M. Q. Bao, and K. L. Wang,  Journal of Physics D -Applied Physics 43 \n(26), 10 (2010).  \n4 V. Vlaminck and M. Bailleul,  Physical Review B 81 (1) (2010).  \n5 Vladislav E. Demidov, Sergei Urazhdin, and Sergej O. Demokritov,  Nature  Materials \n9 (12), 984 (2010).  \n6 J. E. Hirsch,  Physical Review Letters 83 (9), 1834 (1999).  \n7 Y. Kajiwara, K. Harii, S. 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Tatara,  Applied Physics Letters 88 (18) \n(2006).  \n30 T. Tashiro, R. Takahashi, Y. Kajiwara, K. Ando, H. Nakaya ma, T. Yoshino, D. \nKikuchi, and E. Saitoh, presented at the Conference on Spintronics V, San Diego, CA, \n2012 (unpublished).  \n31 K. Ando, S. Takahashi, K. Harii, K. Sasage, J. Ieda, S. Maekawa, and E. Saitoh,  \nPhysical Review Letters 101 (3), 4 (2008).  \n32 V. E. Demidov, S. Urazhdin, H. Ulrichs, V. Tiberkevich, A. Slavin, D. Baither, G. \nSchmitz, and S. O. Demokritov,  Nature Materials 11 (12), 1028 (2012).  \n33 M. Madami, S. Bonetti, G. Consolo, S. Tacchi, G. Carlotti, G. Gubbiotti, F. B. \nMancoff, M. A. Yar, a nd J. Akerman,  Nature Nanotechnology 6 (10), 635 (2011).  9 Figure Captions :  \nFigure 1 : \n(color online). (a) 1 µm x 1µm AFM surface topography of a 20nm YIG film on GGG (111) \n(RMS roughness=0.23  nm). (b) XRD  : \n-2\n scan of the same film using Cu -K\n1 radiation. \nThis spectrum shows that the growth is along the (111) direction with no evidence of parasitic \nphases. The YIG peak are best resolved at high diffraction angle, a zoom around the 888 GGG \npeak is shown in panel (c). \n(c) (d) (e) XRD diffraction s pectrum centered on GGG (888) reflection for different YIG \nthicknesses: 20, 7 and 4 nm. For the 2 thinnest films, the YIG peak is masked by the substrate \npeak . The cubic lattice parameter of the 20 nm thick YIG film (obtained from 888 peak) is \nslightly lar ger than that of the substrate ( aYIG=1.2459 nm , aGGG=1.2383  nm) \n \nFigure 2 : \n(color online) ( a) (b). FMR absorption derivative spectra of 20 and 4 nm thick YIG films at an \nexcitation frequency of 6 GHz.  \n(c) rf excitation frequency dependence of FMR absorption linewidth measured on different \nYIG film thicknesses with an in -plane oriented static field. The black continuous line is a \nlinear fit on the 20  nm thick film from w hich a Gilbert damping coefficient of 2. 3 x 10-4 can \nbe inferred  ( . The damping of the 7 nm and 4 nm films is \nsignificantly larger but must off all the frequency dependence is not linear (see text for \ndiscussion).   \n \nFigure 3 : \n(a) Schematic illustration of the experimental setup: spin waves are excited through the YIG \nwaveguide with a microstrip antenna. The detection is performed by measuring ISHE voltage \ndc signal on a ~200µm wide Pt stripe. (b) External dc magnetic field dependen ce of ISHE \nvoltage measured on Pt electrode showing the polarity inversion when the magnetic field is \nreversed. The peaks occur at the FMR conditions as verified through S 11 spectroscopy on the \nAu antenna (not shown here).  \n(c). (d). (e). ISHE voltage for d ifferent YIG film thicknesses around the resonance magnetic \nfield; microwave frequency is f=3GHz. The peaks can be fitted to a lorentzian shape and the \nextracted linewidth are respectively:  19.2 , 13.2 and 4.6 Oe. The dramatic increase of the \nISHE signal w ith thickness is discussed in the text.   10  \n \nFig 1  \n \n \n  \n11  \nFig 2  \n  \n12  \nFig 3  \n"    },    {        "title": "1702.00846v2.Magnon_Condensation_and_Spin_Superfluidity.pdf",        "content": "arXiv:1702.00846v2  [cond-mat.quant-gas]  11 Dec 2017Magnon Condensation and Spin Superfluidity\nYury M. Bunkovaand Vladimir L. Safonovb,c\n(a)Kazan Federal University, Kremlevskaya 18, 420008 Kazan, R ussia\n(b)Mag and Bio Dynamics, Inc., Granbury, TX 76049, USA\n(c)Physical Science Department, Tarrant County\nCollege - South Campus, Fort Worth, TX 76119, USA\nAbstract\nWe consider the Bose-Einstein condensation (BEC) of quasi- equilibrium magnons which leads\nto spin superfluidity, the coherent quantum transfer of magn etization in magnetic material. The\ncritical conditions for excited magnon density in ferro- an d antiferromagnets, bulk and thin films,\nare estimated and discussed. It was demonstrated that only t he highly populated region of the\nspectrum is responsible for the emergence of any BEC. This fin ding substantially simplifies the\nBEC theoretical analysis and is surely to be used for simulat ions. It is shown that the conditions\nof magnon BEC in the perpendicular magnetized YIG thin film is fulfillied at small angle, when\nsignals are treated as excited spin waves. We also predictth at the magnon BEC should occur in the\nantiferromagnetic hematite at room temperature at much low er excited magnon density compared\nto that of ferromagnetic YIG. Bogoliubov’s theory of Bose-E instein condensate is generalized to\nthe case of multi-particle interactions. The six-magnon re pulsive interaction may beresponsible for\nthe BEC stability in ferro- and antiferromagnets where the f our-magnon interaction is attractive.\nPACS numbers: 75.45.+j\nKeywords: Bose-Einstein condensation, magnons, YIG, hematite\n1INTRODUCTION\nSpin deviations from the magnetic order in a magnetic material (ferr omagnet, antifer-\nromagnet or ferrites) are manifested by spin waves and their quan ta, magnons. Magnons\nare quasiparticles which represent a very useful quantum theore tical tool to describe various\ndynamic and thermodynamic processes in magnets in terms of magno n gas. Since magnons\nhave magnetic moments, the external alternating magnetic field ca n excite extra magnons\nand increase the disorder in the magnetic system. However, in cert ain conditions, the in-\ncrease of magnon density leads to a new state, so-called, magnon c ondensate, in which a\nmacroscopic number of magnons forms a coherent quantum state (see, e.g., [1], [2]). This\nmacroscopic state can significantly change the properties of magn on gas, its dynamics and\ntransport. An example is the phenomenon of quasi-equilibrium Bose- Einstein condensation\n(BEC) of excited magnons on the bottom of their spectrum as a sing le-particle long-range\ncoherent state of quantum liquid. This state generate an uniform lo ng-lived precession of\nspins formed by quantum specificity of the magnon gas when the mag non density exceeds\ncertain critical value. The spatial gradients of this state exhibit a s pin superfluidity, the\nnon-potential transport of deflected magnetization. The spin su perfluidity is an extremely\ninteresting phenomenon for both fundamental and applied studies . It should be emphasized\nthat the main paradigm of magnetic dynamics, the Landau-Lifshitz- Gilbert equation, does\nnot contain complete information about the Bose-Einstein condens ate of magnons. BEC is\nthe principal result of quantum statistics and for magnons it can ex ist at room and even\nhigher temperatures.\nFor the first time the existence of quasi-equilibrium Bose condensat e was demonstrated\nin the experiment with nuclear magnons in the superfluid antiferroma gnetic liquid crystal\n3He-B in 1984 [3]. The theoretical explanation of this phenomenon [4] was developed on the\nbasis of global Ginzburg-Landau energy potential. A similar approac h was later developed\nto explain the atomic BEC [5]. In the experiments with an antiferromag netic3He-B, the\nfollowing phenomena were observed: a) transport of magnetizatio n by spin supercurrent\nbetween two cells with magnon BEC; b) phase-slip processes at the c ritical current; c)\nspin current Josephson effect; d) spin current vortex formation ; d) Goldstone modes of\nmagnon BEC oscillations. Comprehensive reviews of these studies ca n be found in Refs.[6–\n8]. Currently magnon BEC found in different magnetic systems: i) in an tiferromagnetic\n2superfluid3He-A [9, 10]; ii) in in-plane magnetized yttrium iron garnet Y 3Fe5O12(YIG)\nfilm (with two minima in the magnon spectrum) [11, 12] and in normally ma gnetized YIG\nfilm [13]; iii) in antiferromagnets MnCO 3and CsMnF 3with Suhl-Nakamura indirect nuclear\nspin-spin interaction [14–16]. An explanation of analogy between the atomic and magnon\nBEC is given in Ref. [17].\nA microscopic theory of quasi-equilibrium magnon BEC was developed in Refs.[18]-[21]\n(”KS theory”). It was predicted that the external strong pump ing of magnons leads to a\nrapid growth of magnon density and saturation. This state can be c onsidered in terms of\nweakly non-ideal gas of ”dressed” magnons in a thermodynamic qua si-equilibrium with an\neffective chemical potential µand effective temperature T. The dressed magnon energy is\ndefined by εk=ε(0)\nk+δεk, whereε(0)\nk=/planckover2pi1ωkis the energy spectrum of bare magnons and δεk\nis the energy shift due to magnon gas nonlinearities. Magnon-magno n scattering processes\nretainthetotalnumberofdressedmagnonsinthesystemandhold theirdistributionfunction\nof the form\nnk=/parenleftbigg\nexpεk−µ\nkBT−1/parenrightbigg−1\n. (1)\nThe instability at µ= minεkin the quasi-equilibrium magnetic system is an analog of BEC\nphenomenon for the bottom dressed magnons. The distribution (1 ) seems to underlie the\nphenomenon of spin superfluidity, since it nullifies the integral of fou r-magnon collisions\nI(4){nk} ∝/integraldisplay\nd3k1d3k2d3k3|Φ4(k,k1;k2,k3)|2(2)\n×[(nk+1)(nk1+1)nk3nk4−nknk1(nk2+1)(nk3+1)]\n×δ(εk+εk1−εk2−εk3)∆(k+k1−k2−k3)\nand thus this energy loss channel vanishes.\nKS theory qualitatively explained the parallel pumping experiments ([2 2],[23] (YIG at\nroom temperature) and [24] (nuclear magnons in CsMnF 3), where the accumulation of\nmagnons at the bottom of the spin wave spectrum was observed. O ne and a half decade\nlater, purposeful experiment [11] directly demonstrated BEC of quasi equilibrium magnons\nin the thin film of YIG. Subsequent experimental studies have shown qualitative correspon-\ndence with the predicted distribution of excited magnons [25] and ag reement with the BEC\nunder noisy pumping [26].\n3In this paper we analyze critical conditions of quasi-equilibrium magno n BEC in ferro-\nandantiferromagnets, bulkandthinfilms, andevaluatethepossibilit iesoftheirexperimental\nachievements.\nBEC OF BOSE PARTICLES\nLet us first briefly discuss BEC of real bose particles. Their distribu tion is defined by\nEq.(1), where εk= (/planckover2pi1k)2/2mis the kinetic energy of particle with the wave vector kand\nmassm. The total number of bosons in the system is\nN(µ,T) =N=Vs/integraldisplay\nnkd3k\n(2π)3, (3)\nwhereVsis the volume of the system. For the critical condition µ= minεk, from (3) follows\nwell-known formula for the BEC critical temperature versus the de nsity of bosons:\nTBEC=κ0/planckover2pi12\nkBm/parenleftbiggN\nVs/parenrightbigg2/3\n, κ0=2π\n/bracketleftbig\nζ/parenleftbig3\n2/parenrightbig/bracketrightbig2/3≃3.31. (4)\nIt is interesting to note that the BEC is formed mainly by bosons with h igh populations\nwhen Eq.(1) can be written as\nnk≃kBT\nεk−µ. (5)\nLet us prove it by direct calculation. Substituting the high temperat ure population (5) into\nEq.(3) and cutting the upper integral limit by the thermal energy εT≃kBT, one obtains:\nTBEC≃˜κ0/planckover2pi12\nkBm/parenleftbiggN\nVs/parenrightbigg2/3\n,˜κ0=π4/3\n21/3≃3.65. (6)\nWe see that the only difference between Eqs.(4) and (6) is a slightly diff erent (∼10%)\nnumerical factor.\nThe fact that the high population Eq.(5) is dominant does not mean th at the BEC\nphenomenon is a classical one. The criterion of classical Maxwell-Bolt zmann statistics\nexp(µ/kBT)≪1 in this case can be written as (see, e.g., [27]):\nexp(µ/kBT) =/bracketleftbiggVs\nN/integraldisplay\nexp/parenleftbigg\n−εk\nkBT/parenrightbiggd3k\n(2π)3/bracketrightbigg−1\n≪1, (7)\nor\n4N\nVsλ3\nT=N\nVs/parenleftbigg2π/planckover2pi12\nmkBT/parenrightbigg3/2\n≪1, (8)\nwhereλTis the thermal de Broglie wavelength. Substituting BEC temperatur e Eq.(6) into\n(8), we obtain the opposite relation: 2 .26>1, which obviously corresponds to a degenerate\nbose gas.\nBEC OF MAGNONS\nNow let us consider a Bose-Einstein condensation of so-called, “dre ssed” magnons as an\ninstability in the externally pumped quasi-equilibrium magnon gas. The t otal number of\nmagnons N(µ,T)is equal to thenumber of thermal magnons N(0,T) ata giventemperature\nTand the number of magnons Npcreated by external pumping. So far as the energy shift\nof dressed magnons is usually much less than the energy of bare mag nonsδεk≪minε(0)\nk,\nfor simplicity we can approximate εk≃ε(0)\nk.\nBEC in a ferromagnet\nConsider a ferromagnet with the quadratic spectrum (we neglect d etails of the dipole-\ndipole interactions):\nεk=ε0+εex(ak)2. (9)\nHereεexis the exchange interaction constant and ais the elementary cell linear size. The\nquasi-equilibrium BEC will be mainly determined by pumping if the number o f pumped\nmagnons is much greater than the thermal magnon number Np≫N(0,T). In this case we\nobtain an analog of Eq.(4):\nTBEC=κ02εex\nkB/parenleftbigg\na3Np\nVs/parenrightbigg2/3\n, (10)\nor,\nTBEC≃˜κ02εex\nkB/parenleftbigg\na3Np\nVs/parenrightbigg2/3\n(11)\nin the high-population approximation.\n5The above formula, however, does not work for a BEC estimate if Np/lessorsimilarN(0,T). Using\na high-population approximation, we write\nNp=N(µ,T)−N(0,T)\n≃Vs/integraldisplayεT\nε0/parenleftbiggkBT\nεk−µ−kBT\nεk/parenrightbiggk2dk\n2π2, (12)\nand obtain at µ=ε0\nNp\nVs≃kBTBEC\n4πa3ε1/2\n0\nε3/2\nex,\nTBEC≃4πεex\nkB/parenleftbiggεex\nε0/parenrightbigg1/2/parenleftbigg\na3Np\nVs/parenrightbigg\n. (13)\nThese formulas coincide with the accuracy of notations with the exa ct calculation given in\nRef.[2]. This is one more direct proof that BEC is formed by the high-po pulated part of\nspectrum.\nAn estimate for YIG, where εexa2//planckover2pi1= 0.092 cm2s−1forε0//planckover2pi1= 2π×2.5 GHz gives\nTBEC≃2.14×10−17(Np/Vs) cm3K. Thus, we obtain a room-temperature BEC TBEC≃300\nK at the pumped magnon density Np/Vs= 1.41×1019cm−3that in order of magnitude\ncorresponds to the experiment [11].\nAs in the above case of particles, the opposition of high density of ma gnons to their high\npopulation makes the classical criterion exp[( µ−ε0)/kBT]≪1 inapplicable to the case\nof condensation when µ=ε0. As in the previous section, we can write the criterion for\nMaxwell-Boltzmann statistics\nexp[(µ−ε0)/kBT] =/bracketleftBigg\nVs\nN(µ,T)/integraldisplay\nexp/parenleftBigg\n−εex(ak)2\nkBT/parenrightBigg\nd3k\n(2π)3/bracketrightBigg−1\n≪1,(14)\nor\nN(µ,T)\nVsλ3\nT=N(µ,T)\nVs/parenleftbigg4πεexa2\nkBT/parenrightbigg3/2\n≪1. (15)\nIn the case of high temperatures we have N(µ,T)≃N(0,T). Substituting the thermal\ndensity\nN(0,T)\nVs=/parenleftbiggkBT\n2κ0εexa2/parenrightbigg3/2\n(16)\n6into (15), we obtain the opposite relation: 2 .61>1. In other words, magnon gas which\nundergoes Bose-Einstein condensation is always a degenerate bos e gas. Thus the assertion\nof recent publication [28] “that the experimentally observed conde nsation of magnons in\nyttrium-iron garnet at room temperature is a purely classical phen omenon” is untenable.\nBEC in an antiferromagnet\nConsider now the magnon energy of the form\nεk=/radicalBig\nε2\n0+ε2\nex(ak)2. (17)\nThis is typical for magnons in the ”easy-plane” (or, canted) antife rromagnets. Taking into\naccount that\nk=/radicalbig\nε2\nk−ε2\n0\naεexandkdk=εkdεk\nε2exa2,\nin the high-population approximation, one can write\nN(µ=ε0,T)\nVs≃kBT\n2π21\na3ε3ex/integraldisplayεT\nε0/radicalbiggε+ε0\nε−ε0εdε. (18)\nIfNp≫N(0,T), forkBT≫ε0we obtain\nTBEC≃(2π)2/3εex\nkB/parenleftbigg\na3Np\nVs/parenrightbigg1/3\n. (19)\nIfNp/lessorsimilarN(0,T), one can rewrite Eq.(12) in the form:\nNp\nVs≃kBT\n2π2ε0\na3ε3ex/integraldisplayεT\nε0/radicalbiggε+ε0\nε−ε0dε. (20)\nAfter integration, at kBT≫ε0we obtain\nNp\nVs≃(kBTBEC)2\n2π2ε0\na3ε3\nex,\nor,\nTBEC≃√\n2πεex\nkB/parenleftbiggεex\nεo/parenrightbigg1/2/parenleftbigg\na3Np\nVs/parenrightbigg1/2\n. (21)\nNote that the BEC temperature for antiferromagnet has lower po wer dependence on small\nparameter a3Np/Vs≪1 and therefore one can expect much lower densities of pumped\nmagnons to achieve condensation. An estimate for α−Fe2O3(hematite), where εexa//planckover2pi1≈\n24×105cm/s forε0//planckover2pi1= 2π×2.5 GHz gives TBEC≈10−6(Np/Vs)1/2cm3/2K. Thus we\nobtain a room-temperature BEC, T= 300 K at Np/Vs= 0.89×1017cm−3. This estimate is\nmore lower than corresponding estimate for a ferromagnetic YIG. This means that hematite\nis a very attractive object to observe BEC of magnons experiment ally.\n7BEC IN A FERROMAGNETIC FILM\nLet us now consider an ultra-thin ferromagnetic film. There are two principal cases: 1)\nexternal magnetic field His parallel to the the film surface and 2) His perpendicular to this\nsurface. In the first case, the BEC condition µ= min/planckover2pi1ωkgives us two minima at ±kmin∝ne}ationslash= 0.\nThis case was demonstrated experimentally for the YIG film in Ref.[11], where the critical\ndensity of pumped magnons was estimated numerically. Later, in Ref .[1] it was considered\nanalytically. Here we focus on the second case with just one energy minimum at k= 0.\nThe magnon spectrum of the perpendicular magnetized ultra-thin f erromagnetic film can\nbe written as [29]:\nωk=/braceleftbig\n[ωH+ωex(ak)2][ωH+ωex(ak)2+ωMf(kτ)]/bracerightbig1/2, (22)\nwhereγ= 2π2.8 MHz/Oe is the gyromagnetic ratio, ωH=γHi,Hi=He−4πMs+H⊥is\nan effective internal magnetic field, H⊥is the perpendicular anisotropy field, Ms= 139 Oe\nis the saturation magnetization, ωM= 4πγMs= 2π×4.9 GHz,ωexa2= 2π×1.09×10−2\nHz cm2is the exchange constant, f(kτ) = 1−[1−exp(−kτ)]/kτ,τis the thickness of the\nfilm. All numerical parameters are given for YIG. Spin waves are ass umed to be propagated\nonly in the film plane and there is a uniform magnetization along the dept h.\nThe critical density of pumped magnons at temperature Tcan be estimated by the\nfollowing equation:\nNp\nAs=N(µ,T)\nAs−N(0,T)\nAs, (23)\nwhere the sample volume isreplaced by thefilm area As. The Eq.(23) inthe high-population\napproximation has the form:\nNp\nAs≈kBT\n2π/integraldisplaykT\n0/parenleftbigg1\n/planckover2pi1ωk���µ−1\n/planckover2pi1ωk/parenrightbigg\nkdk, (24)\nwherekTcorresponds to the frequency ωk≃kBT//planckover2pi1. This integral at µ→/planckover2pi1ω0has a\nlogarithmic divergence for an infinitely large film. However, for a finite film we have a\nmagnetostatic mode on the bottom, which can be separated from t he spin-wave spectrum\nby a small gap ∆ ω. Thus, we can estimate Eq.(24) as\nNp\nAs≈kBT\n4π/planckover2pi1ωexa2ln/parenleftBigω0\n∆ω/parenrightBig\n. (25)\nNote that this formula corresponds to the model of ultra-thin film, in which the magnon\ndynamics is considered in two dimensions. Magnetic excitations acros s the film plane are\n8discrete and they interact weakly with magnons in the plane of the film . For this reason\nthey practically do not affect the quasi-equilibrium in the two-dimensio nal system. Formula\n(25) is convenient for simple estimates, which can be refined by nume rical calculations with\naccounting for all magnetic degrees of freedom.\nCritical angle\nLet us now find a critical angle of the magnetic moment deviation from the equilibrium,\nwhich is assumed to correspond to the critical number of excited ma gnons. This angle is\ndefined by the ratio of perpendicular spin component to its longitudin al component tan θ=\nS⊥/Sz. The perpendicular component is equal to\nS⊥=/radicalBig\nS2x+S2y=/radicalbigg\nS+S−+S−S+\n2(26)\n≈√\n2Sa∗a=/radicalbig\n2SNp.\nSubstituting Sz≃S, for small angles one obtains\nθ≈/radicalbigg\n2Np\nS=/radicalbigg\n2/planckover2pi1γ\nMsNp\nAsτ. (27)\nFor the film thickness τ= 1µm,ω0= 2π×2.5 GHz, ∆ ω= 2π×1 Hz at room temperature\nT= 300 K we have θfilm≈0.044 (2.5◦). An estimate for a bulk material at the same\nconditions gives θbulk≈0.061 (3.5◦). Taking into account finite thickness of the film, one can\nexpect the experimental value of the angle will be within θfilm< θ < θ bulk. For comparison,\nthe BEC deflection angle in the antiferromagnetic liquid crystal3He-B was about 10−3[1].\nBEC STABILITY\nThe critical density is required but not a sufficient condition for a unif orm magnon BEC.\nFor the BEC stability it is important to consider the interaction betwe en magnons. The\nHamiltonian of magnon system described by creation ( b†\nk) and annihilation ( bk) bose oper-\nators can be written in the form:\nH=/summationdisplay\nk(εk−µ)b†\nkbk+H4+H6+H8+... (28)\n9where the magnon-magnon interaction terms are\nH4=1\n2/summationdisplay\n1,2;3,4Φ4(k1,k2;k3,k4)b†\n1b†\n2b3b4∆(k1+k2−k3−k4),\nH6=1\n3/summationdisplay\n1,2,3;4,5,6Φ6(k1,k2,k3;k4,k5,k6)b†\n1b†\n2b†\n3b4b5b6\n×∆(k1+k2+k3−k4−k5−k6)\nand so fourth. According to Bogoliubov’s theory (see, e.g.,[2],[30]), we have to single out\nclassical condensate amplitudes with k=0:b†\n0=b0=√N0,N0=N−N′. HereN=\n/summationtext\nkb†\nkbkis the total magnon number and N0is the magnon number in the condensate.\nAssuming that N′/N≪1, we can reduce the Hamiltonian (28) to H=H0+H2, where\nH0= (ε0−µ)N0+1\n2T4(0)N2\n0+1\n3T60N3\n0+... (29)\nis the condensate energy and\nH2=/summationdisplay\nk/negationslash=0/bracketleftbigg\nAkb†\nkbk+Bk\n2/parenleftBig\nbkb−k+b†\nkb†\n−k/parenrightBig/bracketrightbigg\n. (30)\nHere\nAk=εk−µ+2T4(k)N0+3T6(k)N2\n0+... (31)\nT4(k) = Φ 4(k,0;k,0),\nT6(k) = Φ 6(k,0,0;k,0,0)...\nand\nBk=S4(k)N0+S6(k)N2\n0+... (32)\nS4(k) = Φ 4(k,−k;0,0),\nS6(k) = Φ 6(k,−k,0;0,0,0)...\nDiagonalizing the quadratic form (30) by the linear canonical transf ormation bk=ukck+\nvkc†\n−k, we find that\nH2=U2+/summationdisplay\nk/negationslash=0/tildewideεkd†\nkdk, (33)\nwhere\nU2=1\n2/summationdisplay\nk/negationslash=0(/tildewideεk−Ak) (34)\n10and\n/tildewideεk= sign(Bk)/parenleftbig\nA2\nk−B2\nk/parenrightbig1/2(35)\nis the spectrum of quasiparticles. From this formula obviously follows the following criterion\nof the condensate stability:\nBk>0, A2\nk−B2\nk>0. (36)\nFor the theory with four-magnon interactions we obtain\nS4(k)>0,[2T4(k)]2−[S4(k)]2>0. (37)\nIf this condition does not work (for an attractive interaction, S4(k)<0), the six-magnon\ninteractions which arerepulsive inferro-andantiferromagnets (d ueto specificity ofHolstein-\nPrimakoff representation of spin operators by the bose operator s) can make the system\nstable:\nS4(k)+S6(k)N >0, (38)\n[2T4(k)+3T6(k)N]2−[S4(k)+S6(k)N]2>0.\nIn this case one can expect a sharp appearence of the uniform con densate at N >\n−S4(k)/S6(k). A detailed analysis will be published elsewhere.\nDISCUSSION\nThe Bose-Einstein condensate of magnons with k= 0 is a uniform precession of the\nmagnetic moment in the effective magnetic field. How does this preces sion differ from the\nusual precession of the magnetic moment? It is known that the unif orm precession of a\nmagnetic moment deviated from equilibrium precesses in an effective fi eld and is gradually\ndamped due to losses in the magnetic material and radiation damping. In this case we have\nan excited coherent state of magnons with k= 0, and the magnons of the entire spectrum\nare in thermodynamic equilibrium with the chemical potential µ= 0. In the case of BEC,\nwe also have an excited coherent state of magnons with k= 0, but this state arose as a\nresult of a change in the density of magnons and their chemical pote ntial becomes µ=ε0.\nIn this case, the losses of the precessing condensate in the magne tic material disappears, the\nspin superfluidity arises. There remains only a weak radiation damping [19, 21], which leads\n11to a long-lived coherent precession. This long-lived state was first o bserved experimentally\nin antiferromagnetic3He-B [3].\nInthispaperwehavefocusedoncriticalconditionsofBECinferro- andantiferromagnets,\nbulk and thin films. Recent similar analysis [31] demonstrated a good a greement with\nexperiments for nuclear magnon BEC. Let us list main results.\n1. Wehaveshownthatingeneraltheonlyhighlypopulatedregionoft hespectrum(which\nhas been previously considered for particular cases in Refs.[1],[28]) is r esponsible for the for-\nmation of any BEC. The high population approximation, nk≃kBT/(εk−µ), however, does\nnot mean that we deal with the classical physics, this is an expressio n for the Bose-Einstein\ndistribution if nk/greaterorsimilar1. The opposition of high density of particles or quasiparticles to the ir\nhigh population makes the classical criterion of Maxwell-Boltzmann st atistics inapplicable\nwhich means that the Bose-Einstein condensation condition µ= minεkalways occurs in the\ndegenerate bose gas. The high population approximation substant ially simplifies analysis\nof the critical conditions and can simplify magnetic dynamics simulation s of systems with\nBEC by the use of classical variables instead of operators. So far a s the BEC occurs in the\nk-space, the most convenient form of simulation is the use of kinetic e quations for nkwith\nthe integrals of magnon-magnon collisions (see, e.g., (2)), sources of pumping and relaxation\n[19],[12].\n2. We have found that the condition of magnon BEC in one of the most interesting\nmaterials, perpendicular magnetized YIG thin film, is fulfilled at a small a ngle, when signals\nare usually treated as excited spin waves [32].\n3. We have estimated that in hematite, high-temperature antiferr omagnet, the BEC\nshouldoccur at much lower level ofthemagnonexcitation compared tothat offerromagnetic\nYIG. We believe that this theoretical prediction can open a new direc tion of purposeful\nstudies for fundamental and applied research.\n4. We have generalized the Bogoliubov’s BEC theory to the case of mu lti-particle inter-\nactions. According to this theory the uniform Bose-Einstein conde nsate is unstable if the\nfour-particle interaction is attractive. This situation sometimes ta kes place in ferro- and\nantiferromagnets. The account of six-magnon interactions (whic h are repulsive in magnets)\ncan resolve the problem of BEC stability. In principle, there are also f actors of time, the\nsample size, and relaxation for the BEC instability has to be developed in the system.\nOne more fundamental problem that remains in this field is to connect microscopic and\n12phenomenological description of bose system with Bose-Einstein co ndensate. This problem\nexists for a long time and it’s solution, say, on the base of magnon sys tems, will advance the\nunderstanding and progress in all areas where the BEC is manifeste d. The solution of this\nproblem will help understand the spin superfluidity as a property of t he magnetic system\nthat accompanies BEC of quasi-equilibrium magnons.\nIn conclusion, we emphasize that the Bose-Einstein condensation o f quasi-equilibrium\nmagnons is a fundamental law of physics. BEC appears due to quant um statistics of quasi-\nparticles in magneto-ordered systems and can exist at room, or ev en higher temperatures.\nOne of the most intriguing properties of the BEC is a superfluid spin cu rrent, a coherent\nquantum flow of energy and information. This understanding of mag non BEC in different\nmagnetic materials can be very useful for spin transport and magn onic quantum devices. An\ninterest in the spin currents, the magnetization projection trans fer in magnetic materials, is\ngrowing every year.\nTheauthorswishtothankGrigoryVolovikandanonimousreviewer fo rhelpfulcomments.\nFor Yu. M. B. this work was financially supported by the Russian Scien ce Foundation (grant\nRSF 16-12-10359).\n[1] Yu. M. Bunkov, G. E. Volovik, Spin superfluidity and magnon BEC , Chapter IV of the book\n”Novel Superfluids”, eds. K. H. Bennemann and J. B. Ketterson (Oxford, University press,\n2013)\n[2] V. L. Safonov, Nonequilibrium Magnons (Wiley-VCH, Weinheim, 2012).\n[3] A. S. 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V. Sh olom, J. Magn. Magn. Mater.\n132 (1994) 180.\n[24] S. A. Govorkov and V. A. Tulin, Sov. Phys. JETP 68 (1989) 8 07.\n[25] V. E. Demidov, O. Dzyapko, S. O. Demokritov, G. A. Melkov , and A. N. Slavin, Phys. Rev.\nLett. 99 (2007) 037205.\n[26] A. V. Chumak, G. A. Melkov, V. E. Demidov, O. Dzyapko, V. L . Safonov, and S. O. Demokri-\ntov, Phys. Rev. Lett. 102 (2009) 187205.\n[27] K. Huang, Introduction to statistical physics (Taylor and Francis, New York, 2001).\n[28] A. R¨ uckriegel and P. Kopietz, Phys. Rev. Lett. 115 (201 5) 157203.\n[29] G. N. Kakazei, P. E. Wigen, K. Yu. Guslienko, V. Novosad, A. N. Slavin, V. O. Golub, N. A.\nLesnik, and Y. Otani, Appl. Phys. Lett. 83 (2004) 443.\n14[30] N. N. Bogolubov and N. N. Bogolubov Jr., Introduction to quantum statistical mechanics\n(World Scientific Publishing Company, Singapore, 2009)).\n[31] R. R. Gazizulin, Yu. M. Bunkov, V. L. Safonov, JETP Lette rs 102 (2015) 766.\n[32] Yu. K. Fetisov, C. E. Patton, and V. T. Synogach, IEEE Tra ns. Magn. 35 (1999) 4511.\n15"    },    {        "title": "1903.12498v1.Quantum_Simulation_of_the_Fermion_Boson_Composite_Quasi_Particles_with_a_Driven_Qubit_Magnon_Hybrid_Quantum_System.pdf",        "content": "Quantum Simulation of the Fermion-Boson Composite Quasi-Particles with a Driven\nQubit-Magnon Hybrid Quantum System\nYi-Pu Wang,1, 2Guo-Qiang Zhang,1, 2Da Xu,1Tie-Fu Li,3, 4,\u0003Shi-Yao Zhu,1J. S. Tsai,5, 6and J. Q. You1,y\n1Zhejiang Province Key Laboratory of Quantum Technology and Device,\nDepartment of Physics and State Key Laboratory of Modern Optical\nInstrumentation, Zhejiang University, Hangzhou 310027, China\n2Quantum Physics and Quantum Information Division,\nBeijing Computational Science Research Center, Beijing 100193, China\n3Institute of Microelectronics, Tsinghua University, Beijing 100084, China\n4Beijing Academy of Quantum Information Sciences, 100193 Beijing, China\n5Department of Physic, Tokyo University of Science,\nKagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan\n6Center for Emergent Matter Science, RIKEN, Wako-shi, Saitama 351-0198, Japan\n(Dated: April 1, 2019)\nWe experimentally demonstrate strong coupling between the ferromagnetic magnons in a small\nyttrium-iron-garnet (YIG) sphere and the drive-\feld-induced dressed states of a superconducting\nqubit, which gives rise to the double dressing of the superconducting qubit. The YIG sphere and the\nsuperconducting qubit are embedded in a microwave cavity and the e\u000bective coupling between them\nis mediated by the virtual cavity photons. The theoretical results \ft the experimental observations\nwell in a wide region of the drive-\feld power resonantly applied to the superconducting qubit and\nreveal that the driven qubit-magnon hybrid quantum system can be harnessed to emulate a particle-\nhole-symmetric pair coupled to a bosonic mode. This hybrid quantum system o\u000bers a novel platform\nfor quantum simulation of the composite quasi-particles consisting of fermions and bosons.\nBy exploiting the advantages of di\u000berent components,\nhybrid quantum systems can provide outstanding archi-\ntectures for quantum information processing (see, e.g.,\nRefs. [1{4]). Recently, a qubit-magnon hybrid quan-\ntum system was implemented by strongly coupling both\na superconducting qubit and the Kittel mode of magnons\nin a small yttrium-iron-garnet (YIG) sphere to a three-\ndimensional (3D) microwave cavity [5, 6], where an ef-\nfective appreciable qubit-magnon coupling was achieved\nby exchanging virtual cavity photons between the qubit\nand magnons. For the Kittel mode in the YIG sphere,\nthe magnons are collective excitations of spins with zero\nwave number (i.e., in the long-wavelength limit) where\nall exchange-coupled spins in the sample precess uni-\nformly [7]. When hybridized with microwave cavity\nphotons [8, 9], optical photons [10], and phonons [11],\nthese magnons can provide a platform to implement var-\nious novel phenomena and applications [12{26], including\nmagnon gradient memory [27, 28], cavity spintronics [16,\n29], quantum transducer [30{33], parity-time symme-\ntry [34], bistability of cavity magnon-polaritons [35, 36],\nand the generations of magnon-photon-phonon entan-\nglement [37] and squeezed magnon-phonon states [38].\nMoreover, ultrstrong-coupling regime between magnons\nand cavity photons have also be shown to be imple-\nmentable [15, 39, 40], in addition to the strong cou-\nplings [12{17].\nThe superconducting qubit as a quantum informa-\ntion processing unit can be used as the core of a solid-\nstate hybrid quantum system [1, 2]. Coherent dress-\ning of the qubit can provide access to a new quantum\nsystem with improved properties (e.g., longer coherencetimes [41, 42]) and o\u000ber various applications in quantum\ninformation (see, e.g., Refs. [43{45]). Quantum mechan-\nically, magnons behave similar to other bosons, so their\nstrong coupling to the superconducting qubit can yield a\ncoherent dressing of the qubit, as analogous to the dress-\ning of a two-level system via photons [46].\nIn this Letter, we study a driven qubit-magnon hy-\nbrid quantum system consisting of a 3D transmon qubit\nand a small YIG sphere embedded in a rectangular\n3D microwave cavity. When driven in resonance by a\nmonochromatic microwave \feld, the transmon qubit is\ndressed by the microwave \feld. Owing to the qubit-\nmagnon coupling mediated by the virtual cavity photons,\nthe 3D transmon qubit is further dressed by the magnons.\nFrom the dispersive readout results, we clearly show these\nnewly-formed doubly dressed states of the superconduct-\ning qubit by demonstrating their frequency splittings\nversus the drive power. The theoretical results \ft the\nexperimental observations well in a wide region of the\ndrive power and reveal that this driven qubit-magnon hy-\nbrid system can be harnessed to emulate a particle-hole-\nsymmetric pair coupled to a bosonic mode. These doubly\ndressed states behave like composite quasi-particles con-\nsisting of fermions and bosons and can be more fermion-\nlike or not, depending on the portion of magnons in the\nhybridized normal modes. Our work opens a new avenue\nto quantum simulation of the fermion-boson composite\nquasi-particles with a hybrid quantum system.\nFigure 1(a) schematically show the experimental setup\nof the hybrid quantum system. The rectangular 3D mi-\ncrowave cavity has inner dimensions of 58 \u000232\u00026 mm3.\nIt consists of two parts which are made of aluminum andarXiv:1903.12498v1  [quant-ph]  29 Mar 20192\nffMagnon modeTransmon\n    qubit\nYIG sphere\n6.95\n6.907.007.05\n240 245 250 255 260\nFIG. 1. (color online). (a) Schematic of the qubit-magnon\nhybrid quantum system in a rectangular 3D microwave cavity.\nA small YIG sphere is placed in the part of the cavity made\nof oxygen-free copper at the magnetic-\feld antinode of the\ncavity mode TE 102. To mount the transmon qubit, the part\nof the cavity made of aluminum is further separated into two\nparts that clamp the qubit at the electric-\feld antinode of\nthe cavity mode TE 102. (b) Transmission spectrum of the\ncavity when the Kittel mode of magnons in the YIG sphere\nis magnetically tuned to be near resonance with the cavity\nmode TE 102.\noxygen-free copper, respectively. The cavity is placed in a\nBlueFors LD-400 dilution refrigerator at a base cryogenic\ntemperature of\u001820 mK. At such a low temperature,\nthe aluminum is superconducting and becomes diamag-\nnetic. The 3D transmon qubit [47, 48] mounted in the\nsuperconducting aluminum part of the cavity can be pro-\ntected from the magnetic \feld generated by the outside\nsuperconducting magnet. In this superconducting qubit,\ntwo aluminum pads are attached to the small Joseph-\nson junction, which, together with the cavity, provide a\nlarge shunt capacitor to suppress the charge noise in the\nqubit, as in the capacitively-shunted \rux qubit [49, 50],\n2D transmon [51], and Xmon [52]. A small YIG sphere of\ndiameter 1 mm is placed in the copper part of the cavity\nwhich is not superconducting at the base temperature of\nthe dilution refrigerator. The static magnetic \feld pro-\nduced by the outside superconducting magnet can go into\nthe copper part of the cavity to adjust the frequency of\nthe Kittel mode in the YIG sphere. The static magnetic\n\feld is aligned along the hard magnetization axis [100]\nof the YIG sphere and both the 3D transmon qubit and\nthe Kittel mode are strongly coupled to the cavity mode\nTE102. The frequency of this cavity mode TE 102, whichis measured to be !c=2\u0019\u00196:99 GHz, is designed to have\na large detuning from the transition frequency of the 3D\ntransmon qubit. When the Kittel mode is tuned to be\nnearly resonant with the qubit by the static magnetic\n\feld (i.e., the cavity mode TE 102is also largely detuned\nfrom the Kittel mode), an e\u000bective qubit-magnon cou-\npling can be achieved via the exchange of virtual cavity\nphotons between the qubit and magnons [5, 6].\nBefore exhibiting the coupling between the 3D trans-\nmon qubit and the Kittel mode of magnons, we \frst mea-\nsure the coupling between the Kittel mode and the cavity\nmode TE 102by tuning the frequency of the Kittel mode\nin resonance with this cavity mode. Owing to the strong\ncoupling between them, two branch of magnon polaritons\noccur around the anticrossing point [Fig. 1(b)], with a\nlevel splitting\u001886 MHz at the anticrossing point. The\ntransition frequency between ground state jgiand \frst\nexcited statejeiof the 3D transmon qubit is measured to\nbe!q=2\u0019\u00196:49 GHz. Because the qubit is now largely\ndetuned from both the Kittel mode and the cavity mode\nTE102, its e\u000bect on the magnon polaritons can be ne-\nglected at around the anticrossing point. Therefore, from\nthe measured level splitting at the anticrossing point, the\ncoupling strength between the Kittel mode and the cavity\nmode TE 102can be obtained as gm\u001943:0 MHz.\nTo achieve an e\u000bective coupling between the qubit and\nthe Kittel mode, we then tune the frequency !mof the\nKittel mode to be near the transition frequency !qof\nthe qubit. Now, the cavity mode TE 102is largely de-\ntuned from both the qubit and the Kittel mode, and an\ne\u000bective qubit-magnon coupling can be achieved via the\nvirtual photons of the cavity mode TE 102[5, 6]. At the\ncryogenic temperature T\u001820 mK,kBT\u001c!q;!m, so\nboth the qubit and the Kittel mode stay nearly in their\nground statesjgiandj0i, respectively, and a transition\nfromjeitojgiinduces a transition from the vacuum to\nsingle-magnon states (i.e., from j0itoj1i). As in Ref. [5],\nthe spectroscopic measurement is carried out with a vec-\ntor network analyzer (VNA) by probing the transmis-\nsion of the cavity mode TE 103. During measurement,\nthe probe \feld is applied at the resonant frequency of\nthe cavity mode TE 103. Largely detuned from the cavity\nmode, the qubit is usually in the ground state. When\nthe qubit is excited, the cavity mode has an observable\nfrequency shift in this dispersive regime and the probed\n\feld transmission thereby changes. Also, a series of at-\ntenuators and isolators are used to prevent thermal noise\nfrom reaching the sample and the signal going out from\nthe output port of the cavity is ampli\fed by two low-noise\nampli\fers at the stages of 4K and room temperature, re-\nspectively. Moreover, two microwave \felds from di\u000berent\nsources are applied to the input port of the cavity. One\nis used as an excitation \feld and tuned to excite the hy-\nbridized normal modes of the system, and the other has\na \fxed frequency near or in resonance with the 3D trans-\nmon qubit and is harnessed to drive the qubit via the3\nExcitation field□ |,0□ |,1 □ |,0\n0(a)\n(b)\n2302312322332342352366.446.466.486.506.526.546.56\nMagneticfield(mT)Excitationfrequency(GHz)\n-89.00-84.00-79.00-74.00-69.00-64.00S212 (dB) Magnonmode\nQubit2gqm/2π>\n>>\nFIG. 2. (color online) (a) Energy levels of the qubit-magnon\nsystem with only the vacuum and single-magnon states in-\nvolved for the Kittel mode. The coupling between jg;1iand\nje;0iinduces the vacuum Rabi splitting \n 0. (b) The vacuum\nRabi splitting of the qubit-magnon system measured via the\ntransmission spectrum of the cavity by tuning both the exci-\ntation \feld and the static magnetic \feld. The probe \feld is\napplied in resonance with the cavity mode TE 103.\nlarge capacitor shunted to the Josephson junction.\nIn Fig. 2(a), we show the energy levels of the qubit-\nmagon hybrid quantum system at the cryogenic temper-\nature when the drive \feld on the qubit is turned o\u000b.\nThe interaction between jg;1iandje;0igives rise to the\nvacuum Rabi splitting \n 0. Owing to the exchange of\nvirtual cavity photons between the qubit and the Kit-\ntel mode, the e\u000bective qubit-magnon coupling is [53]\ngqm=1\n2gqgm(1=\u0001q+ 1=\u0001m), where \u0001 q\u0011!c\u0000!q\n(\u0001m\u0011!c\u0000!m) is the frequency detuning of the cavity\nmode TE 102from the qubit (Kittel mode). The e\u000bec-\ntive Hamiltonian of the coupled qubit-magnon system is\ngiven by (we set ~= 1)\nHqm=!q\n2\u001bz+!mbyb+gqm(\u001b+b+\u001b\u0000by); (1)\nwhere\u001bzand\u001b\u0006are Pauli operators of the qubit and by\n(b) is the creation (annihilation) operator of the magnons.\nWhen the Kittel mode is tuned to be in resonance with\nthe 3D transmon qubit, gqmis reduced to [5] gqm=\ngqgm=\u0001, with \u0001 q= \u0001 m= \u0001. The coupling between\nthe qubit and the Kittel mode is measured in Fig. 2(b)\nby tuning the static magnetic \feld and scanning the fre-\nquency of the excitation \feld. From the vacuum Rabi\nsplitting measured at the anticrossing point (\n 0= 2gqm),\nwe obtaingqm=2\u0019\u001920:1 MHz. Comparing the linewidth\nof the mode at around 230 and 237 mT, we see that the\nlinewidth becomes broader at a stronger magnetic \feld,indicating that some static magnetic \feld penetrates into\nthe cavity to a\u000bect the quantum coherence of the qubit.\nActually, in our experiment, the static magnetic \feld\nwas already designed to be parallel to the chip surface\nof the qubit, so as to reduce its in\ruence on the qubit\nas much as possible. With the measured \u0001 \u00190:5 GHz,\ngm\u001942:0 MHz and gqm=2\u0019\u001920:1 MHz, we can de-\nduce using gqm=gqgm=\u0001 thatgq\u0019239 MHz, which\nreveals that the interaction between the qubit and the\ncavity mode TE 102is in the strong-coupling regime.\nWhen turning on the monochromatic drive \feld of fre-\nquency!dappied to the 3D transmon qubit, the Hamil-\ntonian of the driven qubit-magnon hybrid system is writ-\nten asHd=Hqm+1\n4\nd(\u001b+e\u0000i!dt+\u001b\u0000ei!dt), where \n d\nrepresents the coupling strength between the drive \feld\nand the qubit. In the rotating frame with respect to the\ndrive frequency !d, the Hamiltonian of this driven qubit-\nmagnon system becomes H=1\n2\u000eq\u001bz+1\n2\nd\u001bx+\u000embyb+\ngqm(\u001b+b+\u001b\u0000by), where\u000eq\u0011!q\u0000!d(\u000em\u0011!m\u0000!d)\nis the frequency detuning between the qubit (magnon)\nand the drive \feld. The qubit is now dressed by a clas-\nsical drive \feld and the corresponding Rabi frequency is\ne\nd=q\n\n2\nd+\u000e2q. Due to the coupling between the qubit\nand magnons, the qubit is then further dressed by the\nmagnons. Therefore, the driven qubit-magnon hybrid\nsystem can be described as a doubly dressed qubit . Below\nwe focus on the resonant case with \u000eq=\u000em= 0. The\nHamiltonian Hbecomes\nH=\nd\n2\u001bx+gqm(\u001b+b+\u001b\u0000by): (2)\nEven for this simple model, one cannot exactly solve it,\nbecause the model of a double dressed qubit is not ex-\nactly solvable [54]. With the new basis states j\u0006i=\n(jgi\u0006jei)=p\n2 and the mean-\feld approximation, the\nHamiltonian (2) can be reduced to [53] H=Hp+Hh,\nwith\nHp=\nd\n2aya+ ~gqm(ayb+aby);\nHh=\u0000\nd\n2hyh\u0000~gqm(hyb+hby); (3)\nwherea=hy\u0011j\u0000ih +j, and ~gqm=1\n2gqm(A+ 1), with\nA=hai. Becausefa;ayg= 1 andfa;ag=fay;ayg= 0,\nthese newly-de\fned operators are fermionic. Here Hhhas\nthe same form as Hp, but its two parameters \n dand ~gqm\nchange the signs. This means that the Hamiltonian H=\nHp+Hhhas the particle-hole symmetry . Thus, we can\nuse the driven qubit-magnon hybrid system to emulate\na particle-hole-symmetric pair of fermions coupled to a\nbosonic mode.\nThe two eigenvalues of the Hamiltonian Hp(de-\nnoted as!4and!2) are\u0015\u0006=1\n2[(\nd=2)\u0006[(\nd=2)2+\n(2~gqmp\nN+ 1)2]1=2, whereNis the number of magnons\nexcited. The eigenstates correspond to the particle-\nmagnon composite quasi-particle states. The two eigen-\nvalues ofHh(denoted as !1and!3) are\u0015\u0006=\u0000\u0015\u0006, and4\n6.45 6.50 6.553dBm(b)\n5dBm(c)1dBm(a)\n7dBm\nExcitationfrequency(GHz)(d)11dBm\n(e)\n(h)13dBm\n(g)(f)\n15dBm\n6.40 6.45 6.50 6.55 6.6017dBm\nExcitationfrequency(GHz)-86-84-82-80\n-86-84-82-80\n-86-84-82-80\n-86-84-82-80-94-92-90\n-94-92-90\n-94-92-90\n-94-92-9012 3 4\n1 2 3 4\n1 2 3 4\n1 2341 2 3 4\n1 4\n1 4\n1 4S212 (dB)23\nFIG. 3. (color online) Dispersive readout of the hybridized\nnormal modes of the driven qubit-magnon system. An ex-\ncitation \feld is tuned to excite the hybridized normal modes\nand a probe \feld is applied in resonance with the cavity mode\nTE103. The power of the microwave \feld to drive the super-\nconducting qubit is tuned to be (a) 1 dBm, (b) 3 dBm, (c) 5\ndBm, (d) 7 dBm, (e) 11 dBm, (f) 13 dBm, (g) 15 dBm, and\n(h) 17 dBm, respectively.\nthe eigenstates correspond to the hole-magnon composite\nquasi-particle states. By tuning \n d, these hybridized nor-\nmal modes can be more fermion-like or not, depending on\nthe portion of magnons in these modes. The frequency\nsplitting between modes 4 and 1 and the frequency split-\nting between modes 3 and 2 are\n!4\u0000!1=\nd\n2+h\n(\nd=2)2+ (2~gqmp\nN+ 1)2i1=2\n;\n!3\u0000!2=\u0000\nd\n2+h\n(\nd=2)2+ (2~gqmp\nN+ 1)2i1=2\n:(4)\nIn Fig. 3, we display the transmission spectrum of the\ndriven hybrid system measured in the resonant case of\n\u000eq=\u000em= 0 by successive increasing the power Pdof\nthe drive \feld. The spectrum exhibits a clear mirror\nsymmetry owing to the particle-hole symmetry in the\nHamiltonian. When increasing Pd, two absorption dips\nare split into four, with the outer two dips (labelled as 1\nand 4) departing away. Accordingly, the inner two dips\n(labelled as 2 and 3) gradually approach and then merge\ntogether. The Rabi frequency \n dis proportional to the\ndrive-\feld amplitude \", so \n d/pPd. From Eq. (4)\nit follows that !4\u0000!1increases and !3\u0000!2decreases\nwhen increasing Pd. In particular, for a su\u000eciently strong\ndrive \feld with \n d\u001d4~gqmp\nN+ 1,!3\u0000!2!0. These\nbehaviors agree with the observations in Fig. 3.\nFor zero \n d(i.e., when turning o\u000b the drive \feld),\nEq. (4) gives !4\u0000!1=!3\u0000!2= 2~gqmp\nN+ 1. In\nfact, when \n d= 0, the model in Eq. (2) becomes exactly\nsolvable, with the Rabi splitting \n N= 2gqmp\nN+ 1. In\nthis limiting case, we obtain ~ gqm\u00111\n2gqm(A+ 1) =gqm,\ni.e.,A= 1. In Fig. 4, we display the frequency split-\ntings!4\u0000!1and!3\u0000!2versus the drive power \n d,\n0.51 .01 .54060801001201401600\n.250 .500 .751020304050(\nb)(ω4−ω1)/2π (GHz)D\nrive Power (µW)(a)(\nω3−ω2)/2π (GHz)D\nrive Power (µW)FIG. 4. (color online) (a) Fitting the experimental data of the\nfrequency splitting between hybridized normal modes 1 and 4\nversus the drive power Pd. (b) Fitting the experimental data\nof the frequency splitting between hybridized normal modes\n2 and 3 versus the drive power Pd. To \ft the data, we use\nEq. (4), where ~ gqm=gqm,N= 0, and \n d=kpPd, with\nk= 103 MHz/mW1=2.\nwith the data extracted from Fig. 3. Fitting these data,\nwe use Eq. (4), where \n d=kpPdand ~gqmis replaced\nbygqm. WithN= 0, we obtain the only \ftting pa-\nrameterk= 103 MHz/mW1=2. Here the attenuation of\nthe drive \feld from the microwave source to the input\nport of the cavity is 45 dB. In the region of a weaker\ndrive \feld, the numerical results with N= 0 agree well\nwith the experimental results. This reveals that for a\nweaker drive \feld, the Kittel mode still stays nearly in\nthe ground state and the drive \feld does not appreciably\na\u000bect it. However, when the drive \feld becomes strong,\nthe numerical results with N= 0 deviate from the exper-\nimental observations, i.e., smaller (larger) than the data\nfor!4\u0000!1(!3\u0000!2). This is what we expect because\nmore magnons may be excited when the drive \feld is\nstrengthened to raise the temperature in the cavity.\nIn conclusion, we have convincingly demonstrated\nstrong coupling between the Kittel-mode magnons in\na small YIG sphere and the drive-\feld-induced dressed\nstates of a 3D transmon qubit. The coupling between the\nmagnons and the dressed states of the superconducting\nqubit is mediated by the virtual cavity photons. The the-\noretical results \ft the experimental observations well in a\nwide region of the drive power of the microwave \feld res-\nonantly applied to the transmon qubit. 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Chernyshev1\n1Department of Physics, University of California, Irvine, California 92697, USA\n(Dated: September 13, 2021)\nWe discuss threshold \feld-dependence of the decay rate of the uniform magnon mode in yttrium\niron garnet (YIG) thin \flms. We demonstrate that decays must cease to exist in YIG \flms of\nthickness less than 1 \u0016m, the lengthscale de\fned by the exchange length. We show that due to the\nsymmetry of the three-magnon coupling the decay rate is linear in \u0001 H= (Hc\u0000H) in the vicinity\nof the threshold \feld Hcinstead of the step-like \u0000 k=0/\u0002 (\u0001H) expected from the two-dimensional\ncharacter of magnon excitations in such \flms. For thicker \flms, the decay rate should exhibit\nmultiple steps due to thresholds for decays into a sequence of the two-dimensional magnon bands.\nFor yet thicker \flms, such thresholds merge and crossover to the three-dimensional single-mode\nbehavior: \u0000 k=0/j\u0001Hj3=2.\nPACS numbers: 75.10.Jm, 75.40.Gb, 75.30.Ds\nIntroduction. |Extensive experimental and theoretical\nresearch on the ferromagnetic insulator, yttrium iron gar-\nnet [Y 3Fe2(FeO 4)3or YIG] was started more than half\na century ago has bene\fted from this material's excep-\ntional purity, high Curie temperature, and relative sim-\nplicity of the low-energy magnon spectrum.1Recent dis-\ncovery of the Bose-Einstein condensation of the highly\noccupied magnon states created by microwave pumping\nin YIG thin \flms has attracted substantial interest.2\nRecently, threshold e\u000bects due to the so-called three-\nmagnon splitting have been reported in a quasi-two-\ndimensional (2D) thin \flms of YIG under microwave\npumping and as a function of external \feld.3In this, as\nwell as in the other recent works,4control over the spin\ncurrent enhancement in layered metal-ferromagnet struc-\ntures by the three-magnon processes is sought. This calls\nfor a deeper theoretical insight into the decay dynamics\nof such structures. Fundamentally, given its outstand-\ning properties,1YIG may o\u000ber a fertile playground in\nthe studies of threshold phenomena, because the decay\nconditions in it can be varied continuously by both \flm\nthickness and external magnetic \feld.\nDecays. |In this work, we discuss the decay rate of the\nuniform mode ( k=0 magnon) in an insulating ferromag-\nnetic thin \flm as a function of external magnetic \feld and\n\flm thickness. In particular, using magnon dispersion in\nthe lowest-mode approximation, we outline the ranges\nof the \feld and \flm thickness that favor decays in YIG.\nSpeci\fcally, we show that kinematic conditions for three-\nmagnon decays cannot be met for the YIG \flms of thick-\nnessdmin\u00191\u0016m or less and for the \felds Hmax\nc\u0019600 Oe\nor higher. The upper limit on the external \feld is de\fned\nsolely by the magnetization of a ferromagnet, in agree-\nment with Ref. 3. Less obvious is the existence of the\nlimit on the \flm thickness, which is \fxed by another fun-\ndamental characteristics of a ferromagnet: its exchange\nlength. The physical reason for that limit is the decreas-\ning role of the long-range dipolar interactions with the\ndecrease of the \flm thickness. The presence of such a\nlimit must be important for the control of relaxation and\ntransfer of the spin current in layered structures, whichrely on the three-magnon processes in YIG.4\nWe \fnd that the threshold \feld-dependence of the de-\ncay rate for the k= 0 magnon near the threshold \feld\nHcis \u0000k=0/j\u0001Hj\u0001\u0002(\u0001H), where \u0001 H= (Hc\u0000H),\nbecause the three-magnon interaction vanishes along the\ndirection of the \flm magnetization, which is also precisely\nthek-direction where magnon band minima are located.\nThis leads to a reduction of the phase space for decays\nand results in a vanishing decay rate near the threshold,\ncontrary the na \u0010ve expectation of the step-like increase of\n\u0000k=0/\u0002(\u0001H), when the symmetry of the three-magnon\ninteraction is ignored.5We have supported our considera-\ntion by explicit calculations of both T= 0 andT= 300K\nrelaxation rate dependencies on the \feld for several rep-\nresentative YIG \flm thicknesses.\nThe \fnite-size quantization in the \flm thickness direc-\ntion leads to the formation of multiple magnon bands.6\nWe argue that the \feld-dependence for the decay rate\nshould exhibit multiple steps linear in jHci\u0000Hj, corre-\nsponding to the Hcithresholds for decays into a sequence\nof magnon bands. For thick \flms, these multiple thresh-\nolds should merge in a continuum, and the decay rate will\ncrossover to a three-dimensional (3D) single-band behav-\nior: \u0000 k=0/j\u0001Hj3=2. These results should be helpful\nin \fnding an optimal set of parameters for spin-current\nenhancement.3\nDispersion, density of states. |Despite its fairly com-\nplicated crystal structure, at low energies YIG can be\ndescribed with great success as an e\u000bective large-spin\nHeisenberg ferromagnet on a cubic lattice with nearest-\nneighbor exchange, long-range dipolar interactions, and\nnegligible spin anisotropy.1,7Thus, at long wavelength,\nmagnon energy in YIG is determined by a competition\nbetween three couplings: exchange, dipolar, and Zeeman.\nFor a ferromagnetic crystal of the \flm geometry and\nexternal \feld directed in-plane where it co-aligns with\nthe magnetization direction, as is done most commonly\nin experiments, the lowest-mode approximation for thearXiv:1208.0831v1  [cond-mat.str-el]  3 Aug 20122\nkydkzdEk(GHz)xyzdH||M\n012345E\n0DoS (arb. units)θ = 0°(b) d = 5 µm\n050100150|k|d12345Ek1000 Oe700 Oe300 Oe1 Oe(a)kmkmkmkm(a)\nFIG. 1. (Color online) (a) 2D plot of magnon dispersion in\nYIG in the lowest-mode approximation, Eq. (1), for H= 1000\nOe,d= 5\u0016m, and inkz>0 sector. Inset: axes conventions\nrelative to the \flm and the \feld/magnetization direction. (b)\n2D magnon density of states (DoS) for several representative\n\feld values, Eis in GHz.\nmagnon energy yields:5,6,8,9\nEk=r\u0010\nh+\u001ak2+e\u0001ksin2\u0012k\u0011\u0010\nh+\u001ak2+ \u0001 k\u0011\n;(1)\nwhere \u0001 k=fk\u0001 ande\u0001k= (1\u0000fk)\u0001 andh=\u0016H,\u001a=\nJSa2, and \u0001 = 4 \u0019\u0016M are the energy scale parametriza-\ntions of the external \feld, exchange, and dipolar interac-\ntions, respectively. The form factor fk= (1\u0000e\u0000jkjd)=jkjd\nis from the dipolar sums in the direction of the \flm thick-\nnessd\u001da, and\u0012kis the angle between the ferromagnet's\nmagnetization (directed in-plane, zaxis) and magnon's\nin-plane 2D wave vector k, see Fig. 1(a). We use \u0016=g\u0016B\nwhereg= 2 is an e\u000bective gfactor and \u0016Bis the Bohr\nmagneton. In this work we adhere to the notations and\nunits of Ref. 6, which has provided a detailed micro-\nscopic spin-wave theory of YIG in 1 =Sapproximation,\nand we use experimental parameters for YIG, magnetiza-\ntion 4\u0019M= 1750 G, exchange sti\u000bness \u001a=\u0016= 5:17\u000110\u000013\nOe m2, and lattice constant a= 12:376\u0017A. In Fig. 1(a),\nthe magnon dispersion from Eq. (1) for a representative\n\feldH= 1000 Oe and \flm thickness d= 5\u0016m is shown.\nIt should be noted that the dipolar interactions are re-\nsponsible for the nontrivial structure of magnon band\nwith the minima at \fnite wave vectors, which allow\nthe decay conditions to occur. Although approximate,\nEq. (1) provides a close quantitative description of the\nlowest 2D magnon energy band,6quantized due to \f-\nnite \flm thickness din thexdirection [Fig. 1(a), inset].\nAt small k, dipolar interactions dominate over the ex-\nchange and result in a steep decrease from the energy of\nthe uniform mode, E0=p\nh(h+ \u0001), for k's along the\nmagnetization direction. At larger k, exchange energy\ndominates, giving Ek\u0019h+\u001ak2; while at intermediate\nk, competition between exchange and dipolar terms re-\nsults in peculiarly shaped minima at \u0006km= (0;\u0006km),\nsee Fig. 1(a).\n0.1 1 10 100\nd (µm)0100200300400500600700\nH (Oe) 200 400 600\nH (Oe)0.511.52\n2Emin/E0d = 1 µm\nd = 2 µm\nd = 5 µm\nd = 100 µm\nHcHcHcdminHcmaxFIG. 2. (Color online) d\u0000Hdecay diagram for YIG, shaded\narea is where decays are allowed. Solid (dashed) boundary is\nthe numerical (analytical) solution for the threshold bound-\nary, see text. Hmaxanddminare shown. Inset: 2 Emin=E0vs\nHfor severald's, upper threshold \felds are indicated.\nThe 2D density of magnon states from Eq. (1) is shown\nin Fig. 1(b) for several representative \feld values. Pre-\ndictably, 2D DoS exhibits a step-like increase at the band\nminimum, which is followed by an unusual, almost linear\nincrease, reminiscent of the similar behavior for the rela-\ntivistic dispersion. The latter is due to a nonparabolicity\nof the long-wavelength magnon dispersion, see Fig. 1(a).\n\\Decay diagram\". |For the decays to take place the\nkinematic conditions must be ful\flled. For the two-\nmagnon decay (three-magnon splitting) of the uniform\nmode, the condition to be satis\fed is simply E0= 2Eq.\nWith the microscopic parameters, such as exchange sti\u000b-\nness and magnetization, \fxed, some other parameters\ncan be varied to allow or to forbid decays altogether.\nAs is clear from Eq. (1), the external \feld increases the\nenergies of both the uniform mode and the minimum,\nmaking decays kinematically impossible at some higher\n\feld value.3Another parameter is the \flm thickness d,\nwhich enters Eq. (1) through the form factor fk. While\nthe manner in which din\ruences decays is not a priori\nclear, both trends, vs \feld and vs thickness, can be ex-\namined numerically. Such an examination is exempli\fed\nin Fig. 2, which shows 2 Emin=E0vs \feld for several \flm\nthicknesses. Clearly, when the plotted quantity is <1,\ndecays are allowed, and the crossing of 1 corresponds to\na threshold \feld for decays.\nOur Fig. 2 gives the complete d\u0000H\\decay diagram\" for\nYIG with the shaded area showing the parameter space\nwhere decays are allowed. Two key results are clear in\nFig. 2: (i) the upper threshold \feld does not exceed some\nHmax\nceven for large values of d, and (ii) there exists a\nlower limit on the \flm thickness dmin, below which the\ndecays of the uniform mode may not occur at all. The\nsolid boundary is the numerical solution of Eq. (1) for the\nenergy minimum and the decay conditions. The dashed\nline is an approximate analytical solution, which turns\nout to be very precise. The latter also gives us a deeper3\ninsight into the nature of Hmax\ncanddmindiscussed next.\nLarge-jkmjdapproximation |At large enough dthe\nwavevector of the magnon energy minimum satis\fes\njkmjd\u001d1. Then the formfactor fkm\u00191=jkmjdis small,\nre\recting the reduced role of dipolar interactions in the\nenergy of the km-magnon. One can then show10that\nboth the exchange and the dipolar energies for the km-\nmagnon scale as/d\u00002=3and thus are\u001chfor any reason-\nable \feld. This implies that the energy of the magnon\nband minimum is Emin\u0019h, expected for the uniform\nmode in the absence of dipolar interactions. Then, the\ndecay threshold equation, 2 Emin=E0, trivially gives\nhc= \u0001=3, relating saturated value of the decay thresh-\nold \feld to the material's magnetization: Hmax\nc=4\u0019M= 3\n(= 583 Oe for YIG in Fig. 2(b)), see also Ref. 3.\nThe physical question is: what parameter of the ferro-\nmagnet de\fnes dmin? One can extend the large- jkmjd\napproach and \fnd10that the energies of the dipolar\nand exchange interactions must be related by jkmjd=\u0000\n\u0001d2=4\u001a\u00011=3. With this, the decay threshold condition\n2Emin=E0leads to an algebraic equation in Hcvsd,\nwhich can be resolved in a compact form.10It is plot-\nted as a dashed line in Fig. 2(b), which coincides almost\nexactly with the decay boundary obtained from Eq. (1)\nnumerically. From the same equation we \fnd the mini-\nmal thickness to be dmin\u0019Cp\n\u001a=\u0001, explicitly related to\nthe exchange length of the ferromagnet, `ex=p\n\u001a=\u0001, al-\nbeit with a large numerical coe\u000ecient C\u001962:04.10Using\nparameters for YIG, the exchange length is `ex\u001913:9a\nand the minimum thickness dmin=1:067\u0016m, remarkably\nclose to the numerical result dmin=1:017\u0016m.\nThe physical reason for the very existence of dminis the\ndecreasing role of long-range dipolar interactions in the\nmagnon's energy with the decrease of \flm thickness, as\nthe relation of dminto exchange length implies. The fact\nthat the decay boundary in dexceeds the exchange length\nby a large numerical factor is due to the rather stringent\nrequirements of the decay conditions. We emphasize that\nthe provided d\u0000Hdiagram should apply equally to the\nother thin-\flm ferromagnets.\nDecay rate. |Transitions that involve changing the\nnumber of magnons, such as decays, recombination, or\ncoalescence, originate from the dipolar interactions that\ncouple longitudinal and transverse spin components and\ntherefore do not conserve magnetic1,7as well as mechani-\ncal angular momentum.3Microscopically, dipolar interac-\ntions result in anharmonic couplings of magnons, which,\nfor the decay processes of the k= 0 uniform mode into\ntwo magnons at qand\u0000q, can be written as\nH(3)=1\n2X\nqV(3)\n0;q;\u0000q\u0010\nay\nqay\n\u0000qa0+ H:c:\u0011\n: (2)\nThe three-magnon coupling in Eq. (2) has an angular de-\npendence:V(3)\n0;q;\u0000q=V0sin 2\u0012q, withV0= \u0001=p\n2S.1,11,12\nThis angular dependence is essential since it re\rects the\nsymmetry of dipolar coupling of the transverse, Sx(Sy),\nand longitudinal, Sz, spin components: Vxz/xz=r3\n0 100 200 300 400 500\nH (Oe)0.00.51.01.52.02.53.0Γk=0 (KHz)d = 10 µm\nd = 5 µm\nd = 2 µm\nHc Hc Hc1 _\n5 × Γ0FIG. 3. (Color online) The T= 0 decay rate \u0000 k=0vsHfor\nd= 2, 5, and 10 \u0016m. Dotted line is1\n5\u0000k=0ford= 10\u0016m\nwith the angular-dependence of the three-magnon coupling\nomitted,V(3)\n0;q;\u0000q)V0.\n(Vyz/yz=r3). In particular, it is natural for this cou-\npling to vanish for the spin-wave propagating with the\nmomentum qalong the direction of magnetization M[z\naxis, Fig. 1(a)].11We would like to point out that it is\nalso precisely the direction along which the minima of\nthe magnon band are located.13Therefore, the ampli-\ntude of the decay of k= 0 magnon into two magnons at\nthe band minima qmand\u0000qmis zero. This will lead to\na rather spectacular violation of the na \u0010ve expectation:\nwhile kinematic conditions for the decay of k=0 magnon\ninto\u0006qmare just met at Hc, the corresponding decay\namplitude is vanishing. Thus, the decay rate must in-\ncrease gradually from the threshold, not in a jump-like\nfashion as in the DoS, Fig. 1(b).\nAtT= 0 only spontaneous magnon decays are\nallowed.14The three-magnon recombination processes\nhave to obey the same kinematic constraints as the de-\ncay, having therefore the same threshold conditions. The\nthree-magnon coalescence processes involving the k= 0\nmode correspond to the \\vertical\" transitions, which are\nforbidden either kinematically as in the single mode case\nor by the quantum number of the interband transition\nin the multi-band situation. The four-magnon scattering\namplitude from the exchange interaction vanishes iden-\ntically for the uniform mode, while the remaining four-\nmagnon interactions from the dipole-dipole interaction\ntogether with impurity scattering should be providing\na background with weak HandTdependence, distinct\nfrom the threshold behavior discussed here.\nWith this in mind, using the kinetic approach,1,7which\ntakes into account the balance between decay and recom-\nbination processes in the relaxation time approximation,\nthe decay rate15of the uniform mode is\n\u0000k=0=\u0019X\nq\f\fV(3)\n0;q;\u0000q\f\f2\u0002\n2nq+ 1\u0003\n\u000e(E0\u00002Eq) (3)\nwherenq= [e~Eq=kbT\u00001]\u00001is the Bose occupation fac-\ntor. It is clear from Eq. (3) that while the magnitude of4\n0 100 200 300 400 500\nH (Oe)010203040506070Γk=0 (MHz)d = 10 µm\nd = 5 µm\nd = 2 µm\nHcHc\nHc1 _\n4 × Γ0\nFIG. 4. (Color online) Same as in Fig. 3, but for T= 300 K.\nthe decay rate at \fnite Tcan be substantially modi\fed\nfrom theT= 0 result by the Bose-occupation factors, the\nqualitative threshold behavior must remain the same.\nOne can investigate the threshold behavior of Eq. (3)\nanalytically and obtain \u0000 k=0/jE0\u00002EminjforH!Hc.10\nGiven the proportionality between ( E0\u00002Emin) and\n\u0001H=(Hc\u0000H) demonstrated in Fig. 2, this yields linear\ndependence of the decay rate on the \feld relative to the\nthreshold: \u0000 k=0/j\u0001Hj\u0001\u0002(\u0001H), see Figs. 3 and 4. Once\nagain, this is the consequence of an e\u000bective suppression\nof the phase space for decays due to the discussed angular\ndependence of the three-magnon coupling.\nIn Figs. 3 and 4 we show \u0000 k=0for three di\u000berent \flm\nthicknesses and for T= 0 andT= 300 K, respectively.\nWhile the overall scale in the two \fgures is in a com-\npletely di\u000berent frequency range,10the shapes of \u0000 vs H\nare qualitatively very similar, especially concerning their\nthreshold behavior vs \feld, in agreement with the above\nanalysis. The relative di\u000berence between the curves for\ndi\u000berentd's re\rects the smaller phase space for decays in\nthinner \flms. In the same Figs. 3 and 4 we demonstrate a\ndramatic contrast with the results of Eq. (3) if the angu-\nlar dependence of the three-magnon coupling in Eq. (2) is\nneglected,V(3)\n0;q;\u0000q)V0. The latter results exhibit jumps\natHc's and a linear increase after that, similar to the\n2D magnon DoS in Fig. 1(b). One should also note that\nthe overall decay rate is also markedly overestimated if\nthe angular dependence of the three-magnon interactionis ignored.\nIn thicker \flms, the decay rate will be further modi\fed\nby the multiple magnon bands that occur due to \fnite-\nsize quantization in the \flm thickness direction.6It will\nexhibit a sequence of steps linear in jHci\u0000Hj, whereHci\nis the threshold \feld for decay into an ith band, increas-\ning the decay rate every time the corresponding kine-\nmatic conditions are met. Strictly speaking, the angle\n\u0012qinV(3)\n0;q;\u0000qis between the 3D qvector and magnetiza-\ntion vector M. In the quasi-2D geometry with the levels\nquantized in the xdirections,qxhas discrete values and\nthe minimal value of \u0012min\nqim\u0019qmin\nx;i=jqjis zero only for\nthe lowest magnon band. Thus, the thresholds in Hci's\nwill have some small step-like behavior. However, this\ne\u000bect must be negligibly smaller compared to the steps\nin Figs. 3 and 4 when the angular dependence in V(3)is\nignored altogether. Extending our analysis to the limit\nof thicker \flms where multiple bands merge into a single\n3D band, using Eq. (3) we obtain10that the decay rate\nnear the threshold \feld will crossover to \u0000 k=0/j\u0001Hj3=2.\nConclusions. | In this work, we have discussed the\n\feld-dependence of the decay rate of the uniform magnon\nmode in YIG thin \flms and the e\u000bects of \flm thickness\nin it. As a result of our analysis, we have established that\nthe two key characteristics of a ferromagnet, its magneti-\nzation and exchange length, de\fne the extent of the d\u0000H\nparameter range that favors decays in thin \flm geome-\ntry. Our calculations of the decay rate should provide an\nimportant guidance for the experimentalists in designing\nthe optimal conditions for the control of spin current and\nits relaxation in thin \flms. A particularly intriguing sug-\ngestion to pursue is to study the properties of a thin \flm\nwith varying thickness, which may permit decays and, as\na consequence, the spin current enhancement in one part\nof the \flm and forbid it in the other.\nAcknowledgments. |I am deeply indebted to Doug\nMills for introducing me to this problem, for many illumi-\nnating discussions and constant encouragement. I thank\nIlya Krivorotov for numerous enlightening conversations,\nhis genuine interest and support, as well as generosity\nwith his time spent on educating me in this \feld. I also\nacknowledge useful conversation with Andreas Kreisel.\nThis work was supported by the U.S. DoE under grant\nDE-FG02-04ER46174.\n1V. Cherepanov, I. Kolokolov, and V. L'vov, Phys. Rep.\n229, 81 (1993).\n2S. O. Demokritov, V. E. Demidov, O. Dzyapko, G. A.\nMelkov, A. A. Serga, B. Hillebrands, and A. N. Slavin,\nNature 443, 430 (2006); V. E. Demidov, O. Dzyapko, S.\nO. Demokritov, G. A. Melkov, and A. N. 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C 19, 7013\n(1986).\n10See supplemental material for details of the calculations of\nthed\u0000Hdecay boundary and of the decay rate threshold\n\feld-dependence.\n11M. Sparks and C. Kittel, Phys. Rev. Lett. 4, 232, 320\n(1960).\n12M. Sparks, R. Loudon, and C. Kittel, Phys. Rev. 122,\n791 (1961); E. Schl omann, Phys. Rev. 121, 1312 (1961);\nB. Lemaire, H. Le Gall, and J. L. Dormann, Solid State\nCommun. 5, 499 (1967); A. V. Syromyatnikov, Phys. Rev.\nB82, 024432 (2010).\n13J. Hick, F. Sauli, A. Kreisel, and P. Kopietz, Eur. Phys. J.\nB78, 429 (2010).\n14M. E. Zhitomirsky and A. L. Chernyshev, arXiv:1205.5278.\n15Our \u0000 k=0corresponds to half-width at half-maximum, i.e.,\n\u0000k=0=1\n2\u001c\u00001\nk=0.\nI. SUPPLEMENTAL MATERIAL\nA. large-jkmjdapproximation\nAssuming that for large enough dthe magnon en-\nergy minimum satis\fes jkmjd\u001d1, makes the formfactor\nfk= (1\u0000e\u0000jkmjd)=jkmjd\u00191=jkmjdsmall and allows\nto simplyEkin Eq. (1) to an algebraic expression in\nreduced energy scales of the dipolar and exchange inter-\nactions, \u0016\u0001 = \u0001=jkmjdand \u0016\u001a=\u001ajkmj2:\nEkm=r\u0010\nh+ \u0016\u001a\u0011\u0010\nh+ \u0016\u001a+\u0016\u0001\u0011\n; (1)\nwhere we also used the fact that the minimum is located\nalong thez-axis, so\u0012km= 0. The minimum condition\n@Ek=@k= 0 yields:\nh+ \u0016\u001a+\u0016\u0001\n4=h\u0001\u0016\u0001\n4\u0016\u001a; (2)\nwhich, in the limit of both reduced exchange and dipolar\nenergy scales being small compared to the \feld, \u0016 \u001a;\u0016\u0001\u001ch,\nyields a straightforward relation between the two:\n\u0016\u0001 = 4\u0016\u001a: (3)\nThis relation gives the \feld-independent answer for\njkmjd:\njkmjd=\u0012\u0001d2\n4\u001a\u00131=3\n; (4)\nwhich justi\fes the above approximation \u0016 \u001a;\u0016\u0001\u001chfor\nlarge enough dbecause:\n\u0016\u0001 = 4\u0016\u001a=\u00124\u001a\u00012\nd2\u00131=3\n: (5)Having the dipolar and exchange scales related by\nEq. (3), we can substitute it into Eq. (1) for Eminand\nuse it to solve the decay threshold condition 2 Emin=E0\nwithE0=p\nh(h+ \u0001). This leads to a quadratic equa-\ntion in the threshold \feld hc(=\u0016Hc) vs \u0016\u0001 (which, in\nturn, is a function of dby Eq. (5)), that can be resolved\nin a compact form:\nhc1;2=\u0012\u0001\n6\u0000\u0016\u0001\u0013\n\u0006s\u0012\u0001\n6\u0000\u0016\u0001\u00132\n\u00005\n12\u0016\u00012;(6)\nwith two solutions being the upper and the lower thresh-\nold \felds. They give the (approximate) threshold bound-\nary in Fig. 2(b), shown by the dashed line. From the\nsame solution one can \fnd the \\maximal\" \u0016\u0001mthat cor-\nresponds to the lower boundary on d=dminfor which\nthe solution of the threshold condition exists. This is also\nthe point at which hc1=hc2in Eq. (6) and the content\nof the square-root is zero, solving for which gives\n\u0016\u0001m=\u0016\u0001(dmin) = \n6\u0000p\n15\n21!\n\u0001: (7)\nUsing Eq. (5) we \fnally obtain:\ndmin= 2\u001221\n6\u0000p\n15\u00133=2r\u001a\n\u0001\u001962:04r\u001a\n\u0001:(8)\nAlthough the following result can be trivially obtained\nfrom the solution for hc1in Eq. (6) taking the d!1\nlimit ( \u0016\u0001!0), one can derive the large- dlimit forHc1!\nHmax\ncvia an observation that at d!1 energy minimum\nin Eq. (1) is Emin\u0019h, as \u0016\u0001;\u0016\u001a!0. This reduces the\ndecay condition to a simple one:\n2hc=p\nhc(hc+ \u0001); (9)\nwith two solutions: hc2= 0 andhc1=\u0016Hmax\nc= \u0001=3,\nthus relating the saturated magnetization of the material\nto the upper limit of the threshold \feld for decays.\nB. threshold \feld-dependence of \u0000k=0\nSince we are interested in the threshold behavior, only\nspontaneous ( T= 0) decay rate is considered. Near the\nthreshold, decays are happening into the magnons in the\nvicinity of the magnon band minima \u0006qm= (0;\u0006qm).\nExpanding in ( q\u0000qm) near the minima gives:\n\u0000k=0=\u0019X\nq\f\fV(3)\n0;q;\u0000q\f\f2\u000e(E0\u00002Eq) (10)\n=V2\n0\n2\u0019Z\u0019\n\u0000\u0019dqyZ\u0019\n0dqzsin22\u0012q\u000e\u0012\n\u0001E\u0000\u0001q2\nm\u0013\n;\nwhere \u0001 q= (q\u0000qm), \u0001E= (E0\u00002Emin) andmis the\ne\u000bective magnon mass. Rewriting sin 2 \u0012qas 2qyqz=q2\nand shifting integration in qzbyqmyields\n\u0000k=0=2V2\n0\n\u0019jqmj2Z\nq2\nydqyZ\nd\u0016qz\u000e\u0012\n\u0001E\u0000\u0016q2\nm\u0013\n;(11)6\n200 300 400\nH (Oe)0123456Γk=0 (arb. units)∆H3/2\nHc0Hc1Hc2\nFIG. 5. (Color online) A sketch of \u0000 k=0vsHfor the case of\nmultiple magnon bands. Dotted lines show linear in jHci\u0000Hj\ncontributions of each band, solid line with the shading is the\ntotal e\u000bect, dashed line is the 3D limit /\u0001H3=2.\nwhere \u0016qz=qz\u0000qm,\u0016q2= \u0016q2\nz+q2\ny, and in approximating\nq2\u0019q2\nmthe smallness of \u0016 qz's andqy's in comparisonwithjqmjwas used. It \fnally leads to:\n\u0000k=0=V2\n0m2\njqmj2\f\fE0\u00002Emin\f\f: (12)\nSincejE0\u00002Eminj/jHc\u0000Hj, this yields \u0000 k=0/\u0001H.\nAt \fniteT\u001dEmin, the decay rate near the threshold\nis\n\u0000k=0=2T2\nE2\nminV2\n0m2\njqmj2\f\fE0\u00002Emin\f\f: (13)\nFor the 3D case, when the 2D bands merge into a con-\ntinuum, the above consideration results in:\n\u00003D\nk=0/V2\n0\njqmj2Z\nq3\n?dq?d\u0016qz\u000e\u0012\n\u0001E\u0000\u0016q2\nm\u0013\n;(14)\nwhere \u0016q2= \u0016q2\nz+q2\n?now and possible mass anisotropies,\nwhile neglected, are not going to change the result qual-\nitatively:\n\u00003D\nk=0/V2\n0m2\njqmj2\f\fE0\u00002Emin\f\f3=2; (15)\nand thus, \u00003D\nk=0/\u0001H3=2."    },    {        "title": "2210.08304v2.Non_local_magnon_transconductance_in_extended_magnetic_insulating_films____Part_I__spin_diode_effect.pdf",        "content": "Nonlocal magnon transconductance in extended magnetic insulating films.\nI: spin diode effect.\nR. Kohno,1K. An,1E. Clot,1V. V. Naletov,1N. Thiery,1L. Vila,1R. Schlitz,2N. Beaulieu,3J. Ben Youssef,3A. Anane,4\nV. Cros,4H. Merbouche,4T. Hauet,5V. E. Demidov,6S. O. Demokritov,6G. de Loubens,7and O. Klein1,∗\n1Université Grenoble Alpes, CEA, CNRS, Grenoble INP, Spintec, 38054 Grenoble, France\n2Department of Materials, ETH Zürich, 8093 Zürich, Switzerland\n3LabSTICC, CNRS, Université de Bretagne Occidentale, 29238 Brest, France\n4Unité Mixte de Physique CNRS, Thales, Univ. Paris-Sud, Université Paris Saclay, 91767 Palaiseau, France\n5Université de Lorraine, CNRS Institut Jean Lamour, 54000 Nancy, France\n6Department of Physics, University of Muenster, 48149 Muenster, Germany\n7SPEC, CEA-Saclay, CNRS, Université Paris-Saclay, 91191 Gif-sur-Yvette, France\n(Dated: June 13, 2023)\nThis review presents a comprehensive study of the nonlinear transport properties of magnons electrically\nemitted or absorbed in extended yttrium-iron garnet (YIG) films by the spin transfer effect across a YIG |Pt\ninterface. Ourgoalistoexperimentallyelucidatethepertinentpicturebehindtheasymmetricelectricalvariation\nof the magnon transconductance, analogous to an electric diode. The feature is rooted in the variation of the\ndensity of low-lying spin-wave modes (so-called low-energy magnons) via an electrical shift of the magnon\nchemical potential. As the intensity of the spin transfer increases in the forward direction (magnon emission\nregime), the transport properties of low-energy magnons pass through 3 distinct regimes: i)at low currents,\nwherethespincurrentisalinearfunctionoftheelectriccurrent,thespintransportisballisticanddeterminedby\nthefilmthickness; ii)foramplitudesoftheorderofthedampingcompensationthreshold,itswitchestoahighly\ncorrelated regime, limited by the magnon-magnon scattering process and characterized by a saturation of the\nmagnontransconductance. Herethemainbiascontrollingthemagnondensityisthermalfluctuationsbelowthe\nemitter.iii)AsthetemperatureundertheemitterapproachestheCurietemperature,scatteringwithhigh-energy\nmagnons starts to dominate, leading to diffusive transport. We find that such a sequence of transport regimes\nis analogous to the electron hydrodynamic transport in ultrapure media predicted by Radii Gurzhi. This study,\nrestricted to the low-energy part of the magnon manifold, complements part II of this review, which focuses on\nthe full spectrum of propagating magnons.\nI. INTRODUCTION.\nDiodesarekeycomponentsintheartofelectronics1. Their\ndistinctive function is to create an asymmetric electrical con-\nductance that facilitates transport in the forward direction\nwhile blocking it in the reverse direction. This asymmetry is\nexploited in rectification or clipping devices. Controlling the\nforward threshold voltage is the basis of the bipolar junction\ntransistor. Afewofthesediodesalsooffertheunusualfeature\nof negative differential resistance, which is exploited in oscil-\nlators and active filters2,3. Until recently, it was believed that\nsolid state diodes could only be realized with semiconductor\nmaterials. Their band structure allows strong modulation of\ncarrier density by an electrical shift of the chemical potential\nbetween the valence and conduction bands.\nRecent advances in the field of spintronics have shown\nthat it is possible to design new types of diode devices that\nrely on the transport of the electron’s spin instead of its\ncharge4–7. Inthisnewparadigm,electricalinsulatorsaregood\nspin conductors by allowing spin to propagate between lo-\ncalized magnetic moments via spin-waves (or their quanta\nmagnons) to carry spin information within the crystal lattice\nwithoutanyJouledissipation8–11. TheabsenceofJouledissi-\npationusuallygivesthedielectricmaterialsverylowmagnetic\ndamping, which is associated with a high propensity of the\nmagnons to behave nonlinearly. In ultra-low damping materi-\nals, a variation of 0.1% from the thermal occupancy is usu-\nally sufficient to drive the magnetization dynamics into thenonlinear regime12. This effect could be exploited for spin\ndiodes10,13,spinamplifiers14–16orspinrectifiersofmicrowave\nsignals17–21.\nAnimportantmilestoneinthisnascentoffshootofspintron-\nics using electrical insulators, called insulatronics22, was the\ndiscoveryoftheinterconversionprocessbetweenthespinand\ncharge degrees of freedom (e.g., by the spin Hall effect23–29),\nwhich allows the electrical control of 𝜇𝑀, the magnon chem-\nical potential, via the spin transfer effects (STE) from an ad-\njacent metal electrode14,30–32. The expected benefit is the re-\nalization of a new form of solid state spin diode (see Fig. 1)\nobtainedbyelectricallyshifting 𝜇𝑀relativeto𝐸𝑔,theenergy\ngap in the magnon band diagram (see Fig. 2). The lever-arm\nis set here by the magnetic damping parameter, 𝛼LLG30,33–36:\nthe smaller is the damping, the larger is the electrical shift of\n𝜇𝑀atagivencurrent. Therefore,theferrimagnetyttrium-iron\ngarnet(YIG),withthelowestmagneticdampingknowninna-\nture, is expected to lead to the highest benchmark in asym-\nmetric transport performance37. While the spin diode effect\ninlaterallyconfinedgeometriesisfairlywellunderstood38–40,\nits generalization to extended thin films has been largely\nelusive41. The corresponding difficulty is the collective dy-\nnamics that emerges inside the continuum of magnons when\nthe level splitting between eigenmodes becomes smaller than\ntheir linewidth34,42. Given that magnons are bosons, the pre-\ndicted appearance of new condensed phases at high powers\nis ofparticular interest. These includea superfluid phasepro-\nducedbyaBose-Einsteincondensate(BEC)43–49,whichcouldarXiv:2210.08304v2  [cond-mat.mes-hall]  11 Jun 20232\nlead to novel coherent transport phenomena15,50–52.\nThe manuscript is organized as follows. In the following\nsection, we emphasize the highlights of our transport study\nobserved at high currents on lateral devices deposited on ex-\ntended thin films. In the third section, we present the relevant\nanalytical framework to account for the electrical variation of\nthe magnon transconductance. This framework is based on a\ntwo-fluidmodelintroducedinpartII53,whichsplitstheprop-\nagatingmagnonsintoeitherlow-orhigh-energymagnons. To\nfacilitatequickreadingofeithermanuscript,wepointoutthat\na summary of the highlights is provided after each introduc-\ntion and, in both papers, the figures are organized into a self-\nexplanatorystoryboard,summarizedbyashortsentenceatthe\nbeginning of each caption. In this first part, we focus mainly\nonthenonlinearpropertiesthatleadtothespindiodeeffect. It\nwillbeshownthatthesearemainlycontrolledbythevariation\nofthespin-spinrelaxationratesbetweenlow-energymagnons.\nIn the fourth section, we present the experimental evidence\nsupporting our interpretation. Finally, we conclude the paper\nby expressing what we believe to be the interesting future di-\nrections of this field.\nII. KEY FINDINGS.\nThemainexperimentalresultpresentedinthisreviewisthat\nthenonlineargrowthofthemagnonconductivityobservedfor\nthe forward bias (magnon emission regime) is rapidly capped\nby a saturation threshold in the case of extended geometries.\nSuch behavior can be understood by focusing on the non-\nlinear properties of low-energy magnons, where the limited\ngrowth is reminiscent of the saturation effect observed in the\nferromagnetic resonance of magnetic films at high power55.\nThere, the amplitude of the uniform precession mode, the so-\ncalled Kittel mode, is constrained not to exceed a maximum\ncone angle by a power dependent magnon-magnon scatter-\ning rate, which increases as the mode amplitude approaches\nsaturation. This process involves the increase in parametric\ncoupling via the magnetostatic interaction between pairs of\ncounter-propagatingmagnonmodes,alloscillatingattheKit-\ntel frequency. This process is also known as the Suhl second-\norder instability or the four-magnon process. We believe that\nthe same process is at work here between our low-energy\nmagnons. We therefore propose a model in which Γ𝑚, the\nmagnon-magnonrelaxationrate,alsoincreasesstronglyas 𝜇𝑀\napproaches𝐸𝑔30,41. This conclusion is drawn from the ex-\nperimental observation that in extended thin films the regime\n𝜇𝑀≥𝐸𝑔is never reached despite the use of very large cur-\nrent densities. The fact that it is never reached cannot be at-\ntributedtoathresholdcurrent, 𝐼th,whichwouldnominallybe\noutsidethecurrentrangeexplored. Infact,thecurrentrangeis\nlargeenoughtoreachtheparamagneticstatebelowtheemitter\ndue to Jouleheating. Assuming that the dampingrate is inde-\npendentofthecurrentvalue,thevanishingmagnetizationisin\nprinciplesufficienttobring 𝐼thintotheexploredcurrentrange,\nsincethethresholdcurrentdropstozeroattheCurietempera-\nture. Despitethisreductionofthemagnetizationvalue,wedo\nnot detect nonlocally any evidence of full damping compen-\nFIG. 1. Spin diode effect in a lateral device. (a) Perspective and (b)\nsectionalviewofthemagnoncircuit: anelectriccurrent, 𝐼1,injected\ninto Pt1emits or absorbs magnons via the spin transfer effect (STE).\nThe change in density is consequently sensed nonlocally in Pt2(col-\nlector)bythespinpumpingcurrentdefinedasb,−𝐼2=(𝒱2−𝒱2)∕𝑅2,\nwhere𝒱2is a background signal generated by magnon migration\nalongthermalgradients. (c)Deviationfromthenominalthermaloc-\ncupation of low-energy magnons when 𝐻𝑥<0. We have shaded in\nblue the deviation in the magnon emission regime corresponding to\nthe forward bias ( ▸) and in red the magnon absorption regime cor-\nresponding to the reverse bias ( ◂). The inset (d) shows the behav-\nior expressed as a transmission ratio 𝒯𝑠≡Δ𝐼2∕𝐼1, whereΔ𝐼2=\n(𝐼▸\n2−𝐼◂\n2)∕2. Asshownin(c),anasymmetricnonlineargrowthof 𝐼2\noccursintheforwardbias,correspondingtotheso-calledspindiode\neffect. However,thisgrowthislimitedby 𝑛sat,aneffectivesaturation\nthreshold due to nonlinear coupling between low-energy modes, as\nexpressed by Eq. (7). The solid and dashed lines show the behavior\nfor two values of the saturation level, 𝑛sat. The colored current scale\nin (c) distinguishes 3 transport regimes, ➀(green window): a linear\nregime,𝐼2∝𝐼1atlowcurrents,whichalsoincludesthereversebias;\n➁(blue window): a nonlinear asymmetric increase above a forward\nbias of the order of 𝐼𝑐∕2b; and➂(red window): above 𝐼pk, a col-\nlapse at𝐼cof the spin conduction as the temperature of the emitter,\n𝑇1→𝑇𝑐. When||𝐼1||>||𝐼𝑐||all magnon transport disappears (black\nwindow).\naThe negative sign in front of 𝐼2is a reminder that the origin of the spin\nHall effect is an electromotive force whose polarity is opposite to the\nohmic losses54\nbAccording to Eq. (4) and Eq. (5), 𝒯−1\n���follows a parabola that intersects\nthe abscissa at 𝐼thand the current that reaches a 25% drop10in signal is\nthe landmark of 𝐼th∕2.\nsation. We therefore conclude that the increase of 𝜇𝑀by the\nSTE is eventually compromised by the concomitant increase\nofthedamping,whichcontinuouslypushes 𝐼thtoanunattain-\nable higher level. The analytical model proposed to describe\nthe asymptotic approach of 𝜇𝑀to𝐸𝑔is governed by a Γ𝑚,\nwhose power dependence is enhanced by a Lorentz factor to\nlimit the growth of mode amplitude below a saturation value.3\nSuchananalyticalformwasfirstproposedtodescribethesat-\nuration effects of dipolarly coupled spin waves56,57. We show\nthat this simple model captures well the high-power transport\nregime of each of our nonlocal devices, where we have var-\nied the aspect ratio of the electrodes, the magnetic properties,\nandthefilmthickness 𝑡YIG(seeTable.I).Theotheradvantage\nofthis simplifiedmodel isthat itreproduces theexperimental\nbehaviorwithaverylimitedsetofeffectivefittingparameters.\nComplementing the transport results with Brillouin light\nscattering (BLS) experiments, we show experimentally that\nthe asymmetric transport behavior, i.e., the spin diode effect,\nis indeed predominantly caused by low-energy magnons with\nenergies around 𝐸𝐾=ℏ𝜔𝐾≈30𝜇eV, where𝜔𝐾∕(2𝜋)is the\nKittel frequency58. This picture is further confirmed experi-\nmentallybytheenhancementofthespindiodeeffectinmate-\nrials with isotropically compensated demagnetization factors,\nwhere the uniaxial perpendicular anisotropy 𝐾𝑢compensates\nthe out-of-plane demagnetization field. Here the vanishing\nsmalleffectivemagnetization 𝑀eff=𝑀1−2𝐾𝑢∕(𝜇0𝑀1)≈0,\nwhere𝜇0is the vacuum permeability, reduces the limiting\nnonlinear magnon-magnon interaction and thus increases the\nsaturation threshold. Nevertheless, even in the case of these\nthin films, we observe that the growth of the spin diode effect\nis capped by a saturation threshold, indicating that the rapid\ngrowth ofΓ𝑚as𝜇𝑀approaches𝐸𝑔is a generic process. In\nfact,reducing 𝑀effonlyallowstoreducethestaticcomponent\nofthedipolarinteraction,butlow-energymodescanstillinter-\nactthroughthedynamiccomponentofthedipolarinteraction.\nThepicturethatemergesfromourstudyofnonlocalmagnon\ntransport in extended thin films at high power59is thus very\ndifferent from the frictionless many-body condensate around\nthelowestenergymode,indicatinga reduction ofitsrelaxation\nrate. On the contrary, our results indicate an increase in the\nmagnon-magnon relaxation rate, which prohibits the appear-\nance of BEC phenomena. In other words, the reported signal\nsuggests coupled dynamics between numerous modes rather\nthan the BEC picture of a single dominant mode43–46.\nFinally, we report a collapse of the magnon transconduc-\ntance as the emitter temperature, 𝑇1, approaches the Curie\ntemperature, 𝑇𝑐. In this limit, the population of high-energy\nmagnons becomes significantly larger than the number of po-\nlarized spins, spin conduction by low-lying spin excitations\nis perturbed by collisions with their high-energy counterpart,\nleading to a sharp decrease in the transmission ratio.\nWestudyhowthemagnondensityenhancementvarieswith\nthe physical properties of the nonlocal device itself. By sys-\ntematically varying the transparency of the emitter-collector\ninterface and the thermal gradient near the emitter, we show\nthat the density of low-energy magnons is dominated by ther-\nmal fluctuations rather than by the effective damping value,\nwhich always remains finitely positive ( >0). The process for\nthe forward polarity (magnon emission regime) leads to the\nfinding that the density of low-energy magnons seems to be\ndeterminedbythetemperatureoftheemitter, 𝑇1,withaparti-\ncledensitythatincreaseswithincreasingtemperature. Thisis\nin contrast to the usual (ground state) condensation phenom-\nenathatnormallyoccurasthetemperaturedecreases. Thebe-\nhavior of our system is similar to that of a free electron gasinsideultrapure2Dmaterialssuchasmonolayergrapheneen-\ncapsulated by two layers of hexagonal boron nitride60,61. In\nthe latter, electron transport becomes hydrodynamic at high\ntemperature and can thus be described by the Navier-Stokes\nequation62. This leads to unusual transport behavior, as first\npredicted by Radii Gurzhi in the 1960s63. Our results thus\nsuggest that very similar processes are at work in ultra-low\ndamping magnon conductors64–66, the consequence of which\nis the emergence of a magneto-hydrodynamic regime at high\npower.\nIII. ANALYTICAL FRAMEWORK.\nSince the discovery of spin injection into an adjacent mag-\nnetic layer67, there have been extensive efforts to describe the\nhigh-power regime of magnets excited by STE from an exter-\nnal source. For highly confined geometries, such as nanopil-\nlars, the pertinent picture that emerged is that of a dynam-\nical state dominated by a single eigenmode whose damp-\ning is reduced or enhanced by the external flow of angu-\nlar momentum34–36,38. Its peculiarity is the emergence of\nan auto-oscillation regime, which characterizes the damping\ncompensation threshold, 𝐼th. The selection of the dominant\neigenmode is mostly based on the relaxation rate criterion,\nwhich favors the lowest possible value36,38. It usually coin-\ncideswiththeKittelmode,whosecharacteristicistohavethe\nlongestwavelengthorsmallestwavevector(acriterionmostly\nvalid for strong confinement). In pillars based on ultrathin\nfilms, the eigenfrequency is roughly determined by the so-\ncalled Kittel frequency68, which has a simple analytical form\n𝜔𝐾≡𝛾𝜇0√\n𝐻0(𝐻0+𝑀𝑠)for an in-plane magnetized sam-\nple, where𝛾is the gyromagnetic ratio. In nanopillars, the\ndegeneracy with other magnons modes is removed by con-\nfinement, preventing them from playing a significant role in\nthe dynamics38,69. An additional advantage of working with\nnanopillarsisthatthefreemagneticlayerisefficientlythermal-\nized with the substrate and any Joule effect produced by high\ncurrent densities is reduced. In the case of current perpen-\ndicular to the plane (CPP), the value of the threshold current\nfor damping compensation is set by 𝐼th=2Γ𝐾𝑒N∕𝜖1, where\nΓ𝐾=𝛼LLG𝜔𝐾is the relaxation rate of the Kittel mode, 𝛼LLG\nis the damping parameter, N=𝑉1𝑀1∕(𝛾ℏ)is the effective\nnumber of spins that can flip within 𝑉1=𝑡YIG⋅𝑤1⋅𝐿Pt, the\nmagnetic volume fed by the spin emitter, and 𝜖1is the overall\nefficiency of both the spin-to-charge interconversion and the\ninterfacialspintransferprocessthatdependsonspindiffusion\nlength of Pt and spin mixing conductance at the Pt |YIG in-\nterface [see below Eq. (10)]33,38. While the introduction of a\nwell-defined 𝐼th,asdescribedabove,successfullyallowstode-\nscribe the high-power regime of STE in confined geometries,\nwherethemagnonmodesarediscrete,itwillbeshownbelow\nthat the application to extended thin films, where the magnon\ndispersioniscontinuous,requirestheadditionalconsideration\nofcouplingbetweendegeneratemodes. Thisisdonebyintro-\nducingapowerdependentdampingrateleadinginthiscaseto\na threshold current that increases as 𝐸𝑔−𝜇𝑀(𝐼1)vanishes.4\nFIG. 2. Magnon transport in extended magnetic thin films. (a)\nAt the bottom of the magnon manifold, magnons behave as a two-\ndimensional gas with a step-like density of states, 𝐷(𝐸)(blue). The\nnon-equilibrium magnon distribution, 𝑛(𝐸)(orange), can be modu-\nlatedbyshifting 𝜇𝑀,themagnonchemicalpotential. (b)Inourcase,\nthe shift of𝜇𝑀is produced by an electric current, 𝐼1, injected into\nPt1(emitter). (c) Magnon dispersion for different in-plane propaga-\ntiondirections, 𝜃𝑘. TheplotisshownfortheYIG𝐶filmat300K.The\ngrayshadingemphasizesthedegeneracyweightofthemagnons,i.e.,\ntheir tendency to excite parametrically degenerate modes. The black\ndot denotes𝐸𝐾, the energy of the Kittel mode ( 𝑘→0), the blue dot\ndenotes𝐸𝑔,theminimumofthemagnonband,andtheorangedotde-\nnotes𝐸𝐾,themodedegeneratewith 𝐸𝐾withthehighestwavevector.\nThesaturationinstabilitydescribedinFig.1(b)producesastrongen-\nhancement of Γ𝑚, the nonlinear coupling between degenerate modes\nintheenergyrange 𝐸𝐾±(𝐸𝐾−𝐸𝑔). Thisforces 𝜇𝑀toasymptotically\napproach𝐸𝑔at large currents, as shown in (b).\nA. Magnon transport in extended thin films.\nExtending the zero-dimensional (0D) model38,42,70to\neither the one-dimensional (1D)71–73or two-dimensional\n(2D)10,35,74casesystemshasprovenchallenging. Experimen-\ntally,thisextensionhasbeenmadepossiblebythediscoveryof\nregenerativespintransfermechanisms,suchastheSpinHallor\nRashba effects, which provide STE while allowing the charge\ncurrent to flow in-plane (CIP). The nonlocal device used to\nmeasuremagnontransconductanceisshowninFig.1(a). Ithas\ntwo contact electrodes deposited on YIG and subjected to the\nspin Hall effect (in our case, two Pt strips 𝐿Pt=30𝜇m long,\n𝑤Pt=0.3𝜇mwideand𝑡Pt=7nmthick). Thetwostripesare\nseparated by a center-to-center distance, 𝑑. To reach the high\npower regime, current densities up to 1012A/m2are injected\nin Pt1.\nTo characterize the transport properties, we propose to fo-\ncusonthedimensionlesstransmissioncoefficient 𝒯𝑠≡𝐼2∕𝐼1,\nwhich corresponds to the ratio of the emitted and collected\nspin Hall currents circulating in the two Pt strips, Pt1and\nPt2. Thequantity 𝒯𝑠∕𝑅1canbelooselyrelatedtothemagnon\ntransconductance75.\nNotethatsinceSTEisatransverseeffect,thedifferentcross\nsections of the spin and charge currents must be taken into\naccount,resultingintheSTEefficiency 𝐼CIP=𝐼CPP⋅𝐿Pt∕𝑡Pt.\nAssuming that all injected spins remain localized below theemitter, one can obtain an estimate for the amplitude of the\ncriticalcurrentthatcompensatesforthe Γ𝐾attenuationforthis\nconfined mode:\n𝐼th\n𝑒=2Γ𝐾\n𝜖1𝑡YIG⋅𝑤1⋅𝑡Pt⋅𝑀1\n𝛾ℏ, (1)\nwhere𝑒istheelectroncharge. Anumericalcalculationyields\n𝐼th≈ 1.25mA for YIG𝐶films (see Table. I). As will be\nshown below, the corresponding expression for the threshold\ncurrent in extended thin films ,Ith, differs significantly from\nthe simplistic estimate above. First, the magnon continuum\nwith the absence of discrete energy levels in unconfined ge-\nometries allows for nonlinear interaction between degener-\nate eignemodes56,76, whose signature is a power dependent\nmagnon-magnonscatteringtime77–80. Second,theJouleheat-\ning and the less efficient thermalization of the emitter lead\nto a significant increase of the emitter temperature 𝑇1||𝐼2\n1=\n𝑇0+𝜅𝑅𝐼2\n1forlargecurrents. Inournotation 𝜅representsthe\nthermalization efficiency of our Pt stripe. It is the coefficient\nthatdeterminesthetemperatureriseperdepositedjoulepower\n(seeFig.S1in53). Heretheriseisproducedrelativeto 𝑇0,the\ntemperature of the substrate. This introduces additional com-\nplexityduetoasignificantvariationof 𝑀1,themagnetization\nunder the emitter. These difficulties are large spectral shifts\nof the magnon manifold on the scale of 𝐸𝐾−𝐸𝑔: the rele-\nvantenergyrangeforlargechangesinthelow-energymagnon\ndensityinthinfilms. Inthefollowingwewillpresentamodel\nthatincludesalltheseeffectsandallowstomodelthemagnon\ntransmission ratio [see Fig. 1(c)]. In this model, Ithdepends\non a low current nominal estimate of Ith,0whose value is to\nbe extracted from the fit to the data. Although the estimate\nprovided by Eq. (1) gives the correct order of magnitude, the\nfitvalueissystematicallylarger,indicatingthatthevolumeaf-\nfectedbySTEismuchlargerthanjusttheYIGvolumecovered\nby the Pt1electrode.\nB. Spin transfer effect in extended thin films.\n1. Magnon chemical potential.\nOn the analytical side, the single mode picture38is no\nlonger relevant and should be replaced by a statistical distri-\nbution filled by an integration over all possible wavevectors\n∫𝑑𝐤∕(2𝜋)314. Theappropriateframeworktodescribetheout-\nof-equilibriumregimeofthemagnongasistheBose-Einstein\nstatistics [see Fig. 2(a)]:\n𝑛(𝜔𝑘)=1\nexp[(ℏ𝜔𝑘−𝜇𝑀)∕(𝑘𝐵𝑇1)]−1,(2)\nwhere𝜇𝑀is the electrically controlled chemical potential of\nthe magnons whose value follows the expression30:\n𝜇𝑀=𝐸𝑔𝐼1∕Ith, (3)\nwhere the analytical expression of Ithin extended thin films\nwillbedefinedlaterinEq.(8). Adrasticchangeinthedensity5\noflow-energymagnonsisexpectedwhen 𝜇𝑀approaches𝐸𝑔,\nindicated by a blue dot in Fig. 2(c)46. This energy level cor-\nrespondstothespin-waveeigenmodewiththelowestpossible\nenergy in the dispersion relation of in-plane magnetized thin\nfilms. Itoccursforspin-wavespropagatingalongthemagneti-\nzationdirectionwithwavevectors 𝑘𝑔≈2𝜋⋅105cm−1,orwave-\nlength𝜆𝑔≈600nm. Suchawavelengthisstillverylargecom-\npared to the film thickness, and for all practical purposes we\nwillassumethatthemagnonsbehaveasa2Dfluidinthelow-\nestspectralpartofthemagnonmanifoldasshowninFig.2(c),\nand thus the density of states is a step function as shown in\nFig. 2(a)81. In addition,in thin films,the gap between 𝐸𝑔and\ntheenergyoftheKittelmode, 𝐸𝐾(blackdot),issmall,sothat\nfor all practical purposes 𝐸𝑔≈𝐸𝐾can be approximated by\nthe analytical expression of ℏ𝜔𝐾. It will be shown below that\nin extended thin films both Ithand𝐸𝑔depend on𝐼1, the cur-\nrentbias,fortwodifferentreasons: i)asmentionedabove,the\npoorthermalizationoftheelectrodesleadstoadecreaseof 𝑀1\nwithincreasingthermalfluctuationproducedbyJouleheating\n(∝𝐼2\n1), andii)the finite degeneracy of the magnon bands al-\nlowsnonlinearcouplingbetweeneigenmodesviadipolarcou-\npling [see Fig. 2(b)]. As a consequence, Γ𝑀, the magnon-\nmagnonrelaxationratewhichdefines Ith,increasesforhigher\ndensities of low-energy magnons.\nA drastic simplification can be made by noting that for bo-\nsonstatisticstherearetwoingredientsthatgiverisetoachange\nin the magnon occupation described by Eq. (2): one is the\nchemical potential, 𝜇𝑀; the second is the temperature of the\nemitter,𝑇1. The former dominantly affects the low-lying spin\nexcitations,whilethelatteraffectsthewholespectrum,mostly\nweightedbythehigh-energypart. Itisthereforenaturaltosim-\nplify the problem as a competing two-fluid problem: one of\nmagnetostaticnatureatenergyaround 𝐸𝐾andasecondofex-\nchange nature at energy around 𝐸𝑇∼𝑘𝐵𝑇1. The segregation\nwithinthetwo-fluidsispreciselythefocusoftherelatedwork\ninpartII53. Thus,todescribethemagnontransconductancein\nournonlocaldevices,weproposetosplitthetransmissionratio\n𝒯𝑠=𝒯𝐾+𝒯𝑇into two separate components, each account-\ningforthecontributionofthelow-energyandthehigh-energy\nmagnon53.2. Transconductance by low-energy magnons.\nWe now concentrate on deriving an analytical expression\nforthelow-energymagnontransmissionratio 𝒯𝐾inthelinear\nregime(𝐼1≪𝐼th). StartingfromEq.(2),wederivethe linear\nvariationofthelow-energymagnonsaroundtheKittelenergy,\n𝐸𝐾, which is measured by Δ𝑛𝐾=||𝑛𝐾(+𝐼1)−𝑛𝐾(−𝐼1)||∕2,\nwherethenumberofmagnonsemittedbytheSTE( 𝐼1⋅𝐻𝑥<0)\ninFig. 1(b)ismeasuredrelativetothenumberofmagnonsab-\nsorbedwhilereversingthecurrent(ormagnetization)direction\n(𝐼1⋅𝐻𝑥>0)82. InFig.1(c)thesetwobiasesaresymbolizedby\n▸, for the forward bias (magnon emission), and by ◂, for the\nreverse bias (magnon absorption). The subtraction allows to\ndistinguishelectricallyproducedmagnons( 𝑛(𝜔𝑘)isoddin𝐼1)\nfrom magnons produced by pure Joule heating ( 𝑛(𝜔𝑘)is even\nin𝐼1)8,83–85. Followingtheexpressionof 𝑛(𝜔𝑘)inEq.(2),one\nobtainsananalyticalexpressionforthevariationofthenumber\nof low-energy magnons82:\nΔ𝑛𝐾||𝐼1≈𝑘𝐵𝑇1\nℏ𝜔𝐾𝐼1\nIth1\n1−(𝐼1∕Ith)2, (4)\nwhereIthwas introduced in Eq. (3). The application of cur-\nrent causes a relative decrease of the magnetization accord-\ning to the expression Δ𝑀1= Δ𝑛𝐾𝛾ℏ∕𝑉1, where𝑉1is the\neffective propagation volume of these magnons. These fluc-\ntuations are then detected nonlocally by the change in the\nspin pumping signal generated in the collector: Δ𝐼2∕𝑒=\n𝜖2𝜔𝐾(𝜎Pt𝑤2∕𝐺0)Δ𝑀2∕𝑀2, where𝜖2is the total interfacial\nefficiency of spin-to-charge interconversion at the collector,\n𝑤2isthewidthofthecollector, 𝐺0=2𝑒2∕ℎisthequantumof\ntheconductance,and 𝑀1and𝑀2arethemagnetizationvalues\nunder the emitter and collector, respectively. Since the latter\ntwo values differ when the emitter and collector are at differ-\nent temperatures, we have Δ𝑀2=𝜁 𝑀1Δ𝑀1∕(2𝑀2), where\n𝜁=e−𝑑∕𝜆𝐾≪1is the attenuation ratio of the spin signal\nasmagnonspropagatefromtheemittertothecollector,where\n𝜆𝐾is the characteristic decay length and the factor 1/2 takes\ninto account that an equal flux of magnons will propagate un-\ndetectedintheoppositedirection. Thus,thetransmissionratio\nof low-energy magnons\n𝒯𝐾||𝐼1≡Δ𝐼2\n𝐼1∝𝜖2⋅𝑒𝜔𝐾\n𝐼1⋅𝑀1\n𝑀2⋅Δ𝑛𝐾||𝐼1(5)\nfinds an analytical expression, which in the linear regime\n(𝐼1≪Ith) simplifies as:\n𝒯𝐾||𝐼1→0∝e−𝑑∕𝜆𝐾⋅𝜖1𝜖2⋅𝑘𝐵𝑇1\n𝛼LLGℏ𝜔𝐾⋅𝜎Pt𝑤2\n𝐺0⋅𝑀1\n𝑀2⋅𝛾ℏ\n𝑀1𝑡YIG𝑤1𝑡Pt, (6)\nwhere the value of Ith,0has been replaced by the analytical\nestimation of 𝐼thas expressed by Eq. (1). The above ex-\npression predicts an increase in the magnon transmission ra-tio with decreasing film thickness ∝𝑡−1\nYIG, consistent with re-\ncentresults86. Thisfollowsdirectlyfromthefactthatthespin\npumpingsignal, 𝐼2,isproportionaltochangesinmagnonden-6\nsity. Relating this to an external flow of spins from the in-\nterface decaying at a fixed rate, the result simply translates to\ntheconcentrationbeinginverselyproportionaltothemagnetic\nfilm thickness, all else being equal87. This raises the ques-\ntion of how far the transmission can go when 𝑡−1\nYIG→0. It\nwill be shown below that the answer is disappointing because\nthe nonlocal geometry inherently prevents efficient transport\noflow-energymagnons. Theothergeometricalinfluencesare\nconfirmed in the appendix [see Fig. S1]. Note that at large\ncurrents,Jouleheatingincreasesthetemperatureofthelattice\nbelow the emitter, as described in Fig. 3(a). The correspond-\ninglateraltemperaturegradientsmayalsointroduceadditional\ndependencies88,89, which are not considered in Eq. (6). Also,\nwedonotconsiderheretheeffectsofthelocalgradientofthe\nexternalmagneticfieldneartheemittercausedbytheOersted\nfield generated by the current 𝐼190.\nC. High power regime.\nHaving established the linear response, this section re-\nviewsthephenomenathataffectthepropagationoflow-energy\nmagnonsinthestronglyout-of-equilibriumregime. Wedistin-\nguishbetweenthermaleffectsandnonlineareffectswithinthe\nlow-energy magnon gas.\n1. Joule heating.\nAs mentioned above, the 2D geometry prevents efficient\nthermalization of the magnetic material. Therefore, at high\ncurrent densities, one should expect a significant variation of\n𝑇1. We define 𝑇1=𝑇0+𝜅𝑅𝐼2\n1, the temperature rise of\nthe emitter caused by joule heating when the substrate is at\n𝑇0, and again𝜅is the coefficient that determines the temper-\nature rise per deposited joule power. Fig. 3(a) shows the typ-\nical parabolic increase of 𝑇1produced by passing high cur-\nrent densities through a very thin Pt strip of 300 nm width.\nThe collateral damage is a decrease in the saturation mag-\nnetization, which decreases with increasing temperature [see\nFig. 3(b)], approximately along the analytical form 𝑀𝑇≈\n𝑀0√\n1−(𝑇∕𝑇𝑐)3∕2, where𝑀0is the saturation magnetiza-\ntion at𝑇= 0K53,91. These temperature changes induce a\nlarge spectral shift of the magnon manifold. The relevant en-\nergyscalehereissetbythedifferencebetween 𝐸𝐾−𝐸𝑔,which\nislessthanhalfaGHzinthesethinfilmsasshowninFig.2(c).\nFurthermore, the rise can easily exceed 𝑇𝑐, the Curie temper-\nature,evenwhentryingtousethepulsemethodasameansto\nreduce the duty cycle54. We define the current 𝐼cneeded to\nreach𝑇𝑐=𝑇0+𝜅𝑅𝐼2\nc. In this case, we expect 𝑀1=0at𝐼c,\nas shown by the arrow in Fig3(b).\nIn turn, the decrease of 𝑀1should cause a collapse below\n𝐼cof any threshold currents at the emitter site. At this stage\nwe still assume that the magnons remain non-interacting. We\ndefine𝐼pk=Ith,0⋅𝑀𝑇pk∕𝑀𝑇082, the conjectured threshold\ncurrentbelowtheemitterheatedbytheJouleeffect,while Ith,0\nis the nominal threshold current estimated from the magnetic\npropertiesat 𝐼1=0. Inournotation, 𝑇pkisthetemperatureof\nFIG. 3. Electrical variation of magnetic properties at high power.\n(a) Temperature rise at the emitter due to Joule heating, 𝑇1||𝐼2\n1. As\nshown in (b), this results in a reduction of the magnetization below\nthe emitter, 𝑀1||𝑇1. We define 𝐼c, the critical current to reach 𝑇𝑐,\nthe Curie temperature. (c) Variation of the presumed threshold cur-\nrent for damping compensation. The reduction of 𝑀1causes a col-\nlapse below 𝐼cof𝐼pk=Ith,0𝑀𝑇pk∕𝑀𝑇0, the expected onset of auto-\noscillation for non-interacting magnons. The orange arrow in (c) in-\ndicates𝐼pkwhen the normal threshold Ith,0=5 mA. Due to para-\nmetric instability, the occupancy of any eigenmode is capped at 𝑛sat\nbyasharpincreasein Γ𝑚,thenonlinearcouplingbetweendegenerate\nmodes, as shown in (d). This translates in (c) as a sharp increase of\nIth=Ith,0Γ𝑚∕(𝛼LLG𝜔𝐾).\ntheemitterat 𝐼pk,while𝑇0isitstemperatureat 𝐼1=0. Thepo-\nsitionof𝐼pkcanbeobtainedgraphicallybylookingattheinter-\nsectionofthedashedcurve,whichrepresents Ith,0⋅𝑀𝑇1∕𝑀𝑇0and a straight line of slope 1 crossing the origin, shown by\nthedottedpointsinpanelFig.3(c). Theverticalorangearrow\nindicates the expected position of the auto-oscillation onset,\nassuming that the nominal value is Ith,0= 5mA92. It is im-\nportant to note that the position of 𝐼pkis very weakly depen-\ndent on the nominal value if Ith,0≫ 𝐼𝑐because of the rapid\ndecrease of𝑀1near𝐼c[see Fig. 3(b)]. We will return to this\nobservationwhendiscussingbelowtherelevantbiastorenor-\nmalize the data.\nThe important conclusion at this stage is that, taking into\naccount the Joule heating, the damping compensation should\nalwaysbeachievedwithintherange [−𝐼𝑐,𝐼𝑐],regardlessofthe\nvalueof Ith,0. Thus,aslongasthesinglemodepictureremains\nvalid, one should always observe a divergent increase of the\nnumber of magnons within the currently explored range. We\nwillshowbelowthatthisisnotthecase,andthattheculpritis\nthe increased magnon-magnon scattering, which prevents the\ndivergent growth of the magnon density.\n2. Self-localization effect.\nWe begin this discussion by emphasizing a trivial observa-\ntionaboutthemagnondispersioncurve,whichconcernsnon-\nlocal devices, i.e., magnon transport outside the volume be-\nlow the Pt |YIG interface. Lateral geometries are notappro-7\npriatetostudycondensation,whichselectivelyfavorsthelow-\nestlyingenergymode,suchasBEC.Forin-planemagnetized\nthin films, the minimum in the dispersion relation, 𝐸𝑔, corre-\nsponds to a mode with vanishing group velocity, i.e., a non-\npropagatingmode93: cf. bluedotinFig.2(c). Thismeansthat\nnonlocal devices are inherently insensitive to changes in the\nmagnon population that occur in a localized mode. This sit-\nuation is exacerbated when changes in bias or design end up\nincreasingthemagnonconcentrationbecauseofthenonlinear\nredshiftofthemagnonspectrum. Thispushestheentirespec-\ntral range of low-lying spin fluctuations at the emitter below\nthe magnon bandgap for outside the emitter. This prevents\nthese magnons from reaching the collector and further pro-\nmotes self-localization90,94,95. The latter is further enhanced\nby temperature variations caused by Joule heating when large\ncurrents are circulated in the emitter.\nTheseeffectscanbemitigatedbyloweringthenonlinearfre-\nquencyshift74,96–98. Afirstpossibilityisbytiltingthesample\nout-of-plane. There is a peculiar angle where the depolarisa-\ntioneffectvanishes99. Asecondpossibilityistouseamaterial\nwhose uniaxial anisotropy, 𝜇0𝐾𝑢, compensates for the out-of-\nplane demagnetization factor, 𝜇0𝑀𝑠, leading to a vanishing\neffective magnetization. Note that full redshift cancellation\nrequirestuningtheuniaxialanisotropytoaprecisionoftheor-\nder of(𝐸𝐾−𝐸𝑔)∕(𝛾ℏ)(see below). In this case, the Kittel\nfrequencysimplyreducesto 𝜔𝐾=𝛾𝐻0andisindependentof\nthe magnetization amplitude or direction. It has already been\nnotedthatthiseliminatestheself-localizationeffectofthede-\npolarizationfactorandthuspromotesspinpropagationoutside\ntheemitterregion74. ItwillbeshowninpartIIthatevenifthe\nredshift is extinguished and the product 𝜖1⋅𝜖2≪1is omit-\nted,thetransmissionratioremainswellbelow50%. Thisupper\nlimitisactuallyexpected,consideringthatlessthanhalfofthe\nmagnonspropagateinadirectioncapturedbythecollector. Fi-\nnally,weaddthatthisdifficultyofefficienttransmissionaffects\nnotonlylow-energymagnons,butalsohigh-energymagnons,\nwhich suffer from very short decay lengths (see part II).\n3. Lorentz factor enhanced magnon-magnon decay rate.\nWenowfocusontheinter-magnonnonlinearitythatoccurs\nat high power. We are mainly concerned with saturation ef-\nfects. Thisinstabilityarisesfromthenon-isochronouspreces-\nsion of the magnetization (inherent to elliptical orbits), which\nradiates at harmonics of the eigenfrequency, allowing para-\nmetric excitation of other modes100. This problem was first\ndescribed by Harry Suhl57,101. It is interpreted that the num-\nber of magnons that can fill a particular mode is limited by\nits nonlinear coupling with other magnon modes, by intro-\nducingamagnon-magnonrelaxationtimethatdependsonthe\nmode occupation. The usual requirement is to find a degener-\nate eigenmode within the linewidth. Since this effect depends\non the level of degeneracy, it becomes dominant in extended\nthin films due to the increase in mode density. Moreover, the\npeculiarshapeofthebandstructureofthemagnonsatlowen-\nergy levels introduces a discrimination between the different\nfrequencies in terms of the number of degenerate modes. Itturns out that the energy level with the largest number of de-\ngenerate modes occurs precisely at the Kittel frequency. This\nisemphasizedinFig.2(c)byshadingthedegeneracyweightin\ngray. It should also be noted that among all the modes being\ndegenerate at 𝐸𝐾, the mode with the highest group velocity,\nwhichalsopropagatesalongthenormaldirectiontothewires,\nisthepoint𝐸𝐾markedbyanorangedotinFig.2(c). Interest-\ningly, the wavelength at the orange dot here is of the order of\n1∕𝑤1, the lateral size of the Pt1electrode. This suggests that\nlow-energymagnontransconductanceispreferentiallycarried\nbymagnonsatthisparticularpositioninthedispersioncurve.\nTo describe the nonlinear interaction between magnons we\nintroduce a saturation occupancy Nsat, which marks the max-\nimum number of magnons that one can put in one mode be-\nfore decay to degenerate energy levels starts to kick in. Near\nthis threshold, we assume that the damping rate follows the\nequation102:\nΓ𝑚||𝐼1=Γ𝐾√\n1−(\nΔ𝑛𝐾||𝐼1∕Nsat)2, (7)\nwithΓ𝑚−Γ𝐾representing the nonlinear enhancement of the\nrelaxationratecausedbymagnon-magnonscattering. Thede-\npendence ofΓ𝑚on𝐼1is plotted in Fig. 3(d).\n4. Current threshold in extended thin films.\nToaccountfortheincreaseofcorrelationbetweenmagnons\ndiscussed above, we replace 𝐼thin Eq. (1) by\nIth||𝐼1=Ith,0𝑀𝑇1\n𝑀𝑇0Γ𝑚||𝐼1\nΓ𝐾. (8)\nIntroducing this new expression of Ithinto Eq. (4) gives a\ntranscendental equation for Ith, whose dependence on 𝐼1is\nshown in Fig. 3(c). As Δ𝑛𝐾(𝐼1)approaches Nsatby increas-\ning𝐼1,Ithrises sharply together with the damping Γ𝑚(𝐼1)as\nshown in Fig. 3(d) according to Eq. (7-8). The consequence\nof this increase is that Ithremains unreachable due to a redis-\ntribution of the injected spin among an increasing number of\ndegenerateeigenmodes. Theconsequenceforthedependence\nof𝜇𝑀,thechemicalpotentialofthemagnons,on 𝐼1isshown\nin Fig. 2(b). Near the origin, the linear dependence of 𝜇𝑀on\n𝐼1is set by the intrinsic damping parameter. As Joule heat-\ning begins to decrease the magnetization 𝑀1below the emit-\nter, it shifts the curve upward. In the same way, the decrease\nin magnetization pushes 𝐸𝑔to a lower energy in accordance\nwith the red-shift nonlinear frequency coefficient. The sharp\nriseinΓ𝑚(𝐼1)near𝐼cstopstheriseand 𝜇𝑀,whicheventually\napproaches𝐸𝑔asymptotically at high currents.\nFig.1(c)showstheexpectedbehaviorproducedat 𝐼2fortwo\nvaluesof𝑛sat=5(solidline)or10(dashedline). Hereweuse\nas parameter 𝑛sat=Nsat∕NNLthe value expressed relative to\nNNL=NΓ𝐾∕𝜔𝑀, which marks the onset when the change\nin magnetization becomes of the order of the linewidth, with\n𝜔𝑀=𝛾𝜇0𝑀1≈2𝜋×4.48GHz58. In these data we assume8\nthat𝐼𝑐= 2.5mA and𝐼pk= 2.2mA, which is equivalent to\nassuming that Ith,0=5mA [see Fig. 3(c)].\nDepending on the current values, we observe 3 regimes of\ntransport: ➀:𝐼1< 𝐼𝑐∕2, where we have a linear behavior\n𝐼2∝𝐼1;➁:𝐼1∈ [𝐼𝑐∕2,𝐼pk], where we have an asymmetric\npolynomial increase 𝐼2∝ (1−𝐼2\n1∕𝐼2\n𝑐)−1, the regime for the\nspindiodeeffectinanextendedfilm; ➂:𝐼1∈[𝐼pk,𝐼𝑐],where\nwe have a drop of 𝐼2∝ (1−𝐼3\n1∕𝐼3\n𝑐)1∕2. In the following we\nwilluseEq.(4)combinedwithEq.(8)tofitthedata. Notethat\ntheresultisobviouslyinvertedbyreversingthefielddirection\n(not shown).\nWe conclude this section by emphasizing that the suscepti-\nbility of low-energy magnons to capture external angular mo-\nmentum flux in a thermally changing environment predicts a\ndistinctive nonlinear shape for the current dependence of the\nmagnontransmissionratio,whichwillbeconfirmedbyexper-\nimental data in the following sections.\nIV. EXPERIMENTS.\nIn this section, we present the experimental evidence sup-\nportingthephysicalpicturepresentedabove. Wefocusonthe\nnonlinear and asymmetric transport properties, our so-called\nspindiodeeffect,andtheextractionoftherelevantparameters\nthat govern it.\nA. Nonlocal magnon transport.\n1. Measurement of the magnon transconductance.\nThe experiment is performed here at room temperature,\n𝑇0=300K, on a 56 nm thick (YIG𝐶) garnet thin film whose\nphysical properties are summarized in Table. I. As explained\nin part II, we deliberately choose a device with a large sep-\naration between the two Pt electrodes ( 𝑑= 2.3𝜇m) to al-\nlow the low-energy magnons to dominate the transport prop-\nerties. Byinjectinganelectriccurrent 𝐼1intoPt1,wemeasure\na voltage𝑉2across Pt2, whose resistance is 𝑅2. To subtract\nall non-magnetic contributions, we define the magnon signal\n𝒱2=(𝑉2,⟂−𝑉2,∥)asthevoltagedifferencebetweenthenormal\nandparallelconfigurationofthemagnetizationwithrespectto\nthe direction of current flow in Pt. Fig. 4(a) shows the mea-\nsuredvariationof 𝒱2foralargespanof 𝐼1. Themaximumcur-\nrentinjectedintothedeviceisabout2.5mA,correspondingto\na currentdensity of 1.2⋅1012A/m2. TheJoule heatingat this\nintensity is large enough to reach 𝑇𝑐, the Curie temperature.\nThe resulting voltage 𝒱2is shown in Fig. 4 for both positive\n(𝐻𝑥pointing to+𝑥) and negative ( 𝐻𝑥pointing to−𝑥) polar-\nity of the applied field, whose amplitude is 𝜇0𝐻0=0.2T. In\nFig.4(a),theexpectedinversionsymmetrywithrespecttothe\nfield polarity has been folded between 𝐻𝑥>0(left ordinate\nlabel in pink) and 𝐻𝑥<0(right ordinate label in blue) to di-\nrectlyemphasizetheSTEinduceddeviationbetweenmagnon\nemission and absorption. The measured signal decomposes\ninto two contributions 𝒱2= −𝑅2𝐼2||𝐼1+𝒱2|||𝐼2\n1: one is𝐼2,\nFIG. 4. Experimental observation of the spin diode signal. Panel (a)\nshows the𝐼1dependence of the nonlocal voltage 𝒱2=(𝑉2,⟂−𝑉2,∥):\nvoltage difference between the normal and parallel configuration of\nthe magnetization with respect to the direction of current flow. The\ndataareshownforbothpositive(leftaxisinpink)andnegative(right\naxis in cyan) polarity of the applied magnetic field, 𝐻𝑥. The sign\nof𝒱2is reversed when the field is inverted. The nonlocal voltage\n𝒱2= −𝑅2𝐼2+𝒱2is decomposed into an electric signal, 𝐼2, and\na thermal signal, 𝒱2(black). Panel (b) shows the variation of 𝐼2at\n𝐻𝑥<0in forward and reverse bias. The blue and red shaded areas\nin panels (a) and (b) highlight respectively the regimes of magnon\nemissionandabsorptionrespectivelyfortheforwardandreversebias.\nPanel (c) plots the magnon transmission ratio 𝒯𝑠= Δ𝐼2∕𝐼1, where\nΔ𝐼2≡(𝐼▸\n2−𝐼◂\n2)∕2. The solid line is a fit with Eq. (4), (7) and (8)\nusing𝑛sat=4andIth,0=6mA. The black shaded region shows the\nassumedbackgroundcontributiontospinconductionbyhigh-energy\nmagnons,Σ𝑇(cf.53). The data are collected on the YIG𝐶thin film\nat ambient temperature, 𝑇0=300K, using a device operating in the\nlong-range regime ( 𝑑=2.3𝜇m).\ntheelectricalsignalproducedbytheSTE,andtheotheris 𝒱2,\nabackgroundvoltageassociatedwithmagnontransportalong\nthermal gradients. The latter voltage corresponds to the Spin\nSeebeck Effect (SSE). One expects 𝒱2≈ 0in well thermal-\nized devices. We emphasize the minus sign in front of 𝐼2,\nwhich accounts for the fact that the spin-charge conversion is\nanelectromotiveforce,sothecurrentflowsintheoppositedi-\nrection to the voltage drop. It is thus a reminder that the re-\nsultingpolarityisoppositetotheohmiclosses54. Inthelinear\nregime(𝐼1→0),theelectricalsignaliseven/oddwiththepo-\nlarityof𝐻𝑥or𝐼1,whilethethermalsignalisalwaysodd/even\nwith𝐻𝑥or𝐼110.\nFocusingonthenonlinearbehaviorobservedforthe 𝐻𝑥<0\nconfiguration(bluedata),103,welabeltheforwardbiasas 𝒱▸\n2as the nonlocal voltage for 𝐼1>0and the reverse bias as\n𝒱◂\n2as the nonlocal voltage for 𝐼1<0. The cancellation of\nJoule effects can be obtained simply by calculating the dif-9\nTABLE I. Physical properties of the magnetic garnet films (values at 𝑇0=300K).\n𝑡YIG(nm)𝜇0𝑀𝑠(T)𝐻𝐾𝑢(T)𝑇𝑐(K)𝛼YIG(×10−4)𝜌Pt(𝜇Ω.cm)𝑡Pt(nm)𝜅Pt(K)𝐺↑↓(×1018m−2)𝜖(×10−3)\nYIG𝐴 19 0.167 +0.005 545 3.2 27.3 7 480 0.6 2.1\n(Bi-)YIG𝐵25 0.147 +0.174 560 4.2 42.0 6 890 2 .4 7.8\nYIG𝐶 56 0.178 -0.001 544 2.0 19.5 7 476 ◦:0.64/•:1.9◦:1.3/•:3.6\nference𝒱◂\n2−𝒱▸\n2, which eliminates any contribution that is\neven in current. The latter quantity represents the number\nof magnons produced by the STE relative to the number of\nmagnons absorbed by the STE at a given current bias |𝐼1|.\nThe inset (c) of Fig. 4 shows the observed averaged behav-\nior,Δ𝐼2≡(𝐼▸\n2−𝐼◂\n2)∕2 = (𝒱◂\n2−𝒱▸\n2)∕(2𝑅2), expressed as\na renormalized quantity 𝒯𝑠=Δ𝐼2∕𝐼1, a strictly positive pa-\nrameter, as suggested by Eq. (6). However, the subtraction\noperationdoesnotallowtoseparatethebehaviorbetweenthe\nforward and the backward direction. In fact, the asymmetry\nof the electrical signal cannot be obtained from the transport\ndata alone. It requires additional input information. This will\nbe provided below by the measurement of the integral inten-\nsityoftheBLSsignal,whichdirectlymonitorsthelow-energy\npartofthemagnonspectrum. BasedontheexperimentalBLS\nobservation in Fig. 5(e), we expect the magnon transmission\nratiotoshowcontinuityovertheorigin( 𝐼1=0)andtobehave\nasastepfunctionofamplitudeinthereversebiasupto 𝐼c. This\nleadstothefollowingexpression 𝒯◂\n𝑠≈𝒯𝑠||𝐼1→0𝑇1∕𝑇0forthe\nmagnontransmittanceinthereversedbiasbelow 𝐼𝑐. Theratio\n𝑇1∕𝑇0takesintoaccountthatthenumberofthermallyexcited\nlow-energymagnonsinthereversepolarizationvarieswith 𝐼1\nduetoJouleheating. Thisintroducesasecondorderdistortion\nwhich will be discussed in more detail in the next section and\ninpartII.Soweconstruct 𝒱2=−𝒱◂\n2+𝑅2𝒯◂\n𝑠⋅𝐼1inthere-\nversebiasfromtheoppositemagneticconfiguration. Wethen\nforce𝒱2to be even in current to get the forward bias behav-\nior. TheresultisshownasblackdotsinFig.4(a). Wecanthen\nderive𝐼▸\n2=(𝒱2−𝒱▸\n2)∕𝑅2and𝐼◂\n2=(𝒱2−𝒱◂\n2)∕𝑅2,which\nis shown in Fig. 4(b).\nIn Fig. 4(b) we observe 3 different transport regimes as\npredicted above. We have ➀with𝐼1∈ [−2,1]mA: the\nspin conductance is approximately constant; ➁with𝐼1∈\n[1,2.2]mA:thespinconductanceincreasesgraduallyandsat-\nurates quickly; and ➂with𝐼1∈ [2.2,2.5]mA: the spin con-\nductance decreases abruptly to vanish at 𝐼c. Only the regime\n➀is anti-symmetric in current. We emphasize at this stage\nthat the sequence of behavior is reminiscent of the 3 regimes\npredicted in Fig. 1(c). We can also repeat the same analysis\nto construct the data when 𝐻𝑥>0(see part II). As expected,\nthepolarityoftheasymmetryreverseswhenthemagnetization\ndirection is changed. In all cases, the forward regime occurs\nonly when𝐼1⋅𝐻𝑥<0, which corresponds to the polarity of\nthe damping compensation.\nInthenextsection,wewillconfirmtheabovebehaviorusing\nBLS spectroscopy. Throughout the rest of the paper, we will\nalwaysdiscussthedataasshowninFig.4(c)intheformof Δ𝐼2\nas a function of |𝐼1|and properly renormalized by 𝐼1and𝑇1\nto allow comparison between different devices (see part II53).2. Spin diode effect.\nWe now discuss in more detail the amplitude of the asym-\nmetric rise of the 𝐼2signal in Fig. 4(c). A striking feature of\nFig. 4(c) is the limited growth of the spin diode signal, which\nis capped by a meager factor of 3 rise compared to the value\nat small currents. This variation is significantly smaller than\nthe changes in cone angles observed at the damping compen-\nsation threshold in nanopillars, which reach several orders of\nmagnitude. Suchinefficiencyisfurtherconfirmedbyprevious\nreports aiming at modulating the transport by damping com-\npensationwithanadditionalheavymetalelectrodeplacedbe-\ntween the emitter and the collector, which increases the con-\nductionbyafactorof6atmost14–16,98,104. AsshowninFig.2,\nthis cannot be explained by a threshold current larger than\nthe current explored window [−𝐼𝑐,+𝐼𝑐], but rather indicates\na strong coupling between the magnons that prevents a large\ngrowthofthemagnondensity. Theasymmetriccontributionis\nwellaccountedforbyEq.(4). Thebestfitisobtainedbyusing\n𝑇⋆\n𝑐=515K,Ith,0=6mAand𝑛sat=4andisindicatedinthe\nplot by the solid line. The low-energy magnon contribution\nis added to a background indicated by the dashed line. This\nbackground accounts for the competing contribution of high-\nenergy thermal magnons to the electrical transport. Its origin\nand analytical expression can be found in Ref.53. In our fit, it\nrepresentsa50%additionalcontribution Σ𝑇∕(Σ𝐾+Σ𝑇)=0.5\ninEq.(4)ofref.53. Wealsonotethatthevalueof 𝑇⋆\n𝑐usedfor\nthe fit is significantly different from the Curie temperature of\nthisfilm(seeFig.S1ofref.53). Thispointwillbeinvestigated\nin more detail in part II53.\nB. Brillouin Light Scattering.\nItisusefulatthisstagetoconfirmspectroscopicallythatthe\nspindiodeeffectreportedaboveisindeedduetoanasymmet-\nricmodulationofthelow-energymagnons. BLSisatechnique\nof choice for this purpose, since it is specifically designed to\nmonitor spectral shifts in the magnon population at GHz en-\nergies. Furthermore, its high sensitivity allows the detection\nof fluctuations down to the thermal level. While in the past\nwe have performed comparative studies of the transport and\nBLS behavior on exactly the same device10,54, in this partic-\nular case we will introduce in the discussion a thinner YIG𝐴\ngarnet film, whose physical properties are also given in Table\nI.Thechangeofsamplesisnotintentionalandispurelyrelated\nto the chronological context of the experiments. For all prac-\ntical purposes, the only relevant difference is a change in the\nvalueof𝐼cfrom𝐼𝑐=2.5mAto2.1mAforYIG𝐶andYIG𝐴,\nrespectively,duetodifferencesinPtresistivity. Wehaveveri-10\nFIG.5. Experimentalevidencethatthespindiodeeffectisdominated\nbylow-energymagnons. Panels(a)and(c)showthemicrofocusBril-\nlouin light scattering ( 𝜇-BLS) spectra as a function of 𝐼1performed\nunderthetwoPtelectrodes(seeblackdotontheinset)atPos1(emit-\nter) and Pos2 (collector) for 𝐻𝑥<0. Each BLS spectrum is renor-\nmalizedbytheamplitudeoftheKittelpeakat 𝐼1=0. Panels(b)and\n(d) show two sections at fixed current 𝐼1= ±2mA near𝐼pk. The\ngreenmarkerindicatesthespectralpositionof 𝐸𝐾at𝐼1=2mA.The\nspectralpositionoftheself-localizedspinfluctuations, 𝐸𝐾1,(seecir-\ncles)at𝐼1=+2mA(seeorangemarker)occurwellbelow 𝐸𝑔2≈𝐸𝐾2\nat Pos2 (see green marker). The shift 𝐸𝐾1−𝐸𝑔2is about 0.7 GHz.\nTheparamagneticlimit, 𝛾𝐻0∕(2𝜋),isindicatedbytheyellowvertical\ndotted line at 5.8 GHz. Panel (e) shows the integrated BLS intensity\nat Pos1 as a function of 𝐼1. The solid blue line is a guide for the\neye. Theblackdashedlineshowstheexpectedvariationofthermally\nactivated low-energy magnons produced by Joule heating. In echo\nwith Fig. 4, the blue and red shaded areas highlight the variation of\nmagnonemissionandabsorptionwithrespecttothermalfluctuation.\nTheinsetpanel(f)showstheevolutionof 𝑀1inducedbyJouleheat-\ning. ThebluepointsareinferredfromtheevolutionoftheKittelfre-\nquency (dashed parabola in panel (a)). The dashed lines are inferred\nfromtheSQUIDmeasurementsshowninFig.S1ofRef.53. Thedata\narecollectedontheYIG𝐴thinfilmat𝑇0=300Kusingadevicewith\na distance𝑑=1.0𝜇m between the two Pt electrodes.\nfiedthatallotherpropertiesdiscussedherearegenerictoboth\nsamples.\nSinceweareinterestedinlocalchangesinthemagnonpop-\nulation,weusemicro-focusBrillouinlightscattering( 𝜇-BLS)\nspectroscopy105. Theprobelightwithawavelengthof532nm\nandapowerof0.1mWisgeneratedbyasingle-frequencylaser\nwith a spectral linewidth of <10MHz. It is focused through\nthe sample substrate into a submicron diffraction-limited spot\nusing a 100×corrected microscope objective with a numeri-cal aperture of 0.85. The scattered light was collected by the\nsamelensandanalyzedbyasix-passFabry-Perotinterferome-\nter. Thelateralpositionoftheprobespotwascontrolledusing\na custom-designed high-resolution optical microscope. The\nmeasured signal (the BLS intensity at a given magnon fre-\nquency)isproportionaltothespectraldensityofthemagnons\nat that frequency and at the position of the spot. By moving\nthe focal spot across the film surface, we can obtain informa-\ntion about the spatial variations of the magnon spectral dis-\ntribution. Note that only low-energy magnons contribute to\ntheBLSintensity,sincetheBLStechniqueissensitiveonlyto\nthe wavevectors smaller than 2.4⋅105cm−1. However, since\nthe frequency of magnetostatic magnons also depends on the\ntotal number of magnons in the sample, the spectral position\noftheKittelpeakallowstoderiveinformationaboutthelocal\ntemperature from the BLS data106,107.\nFig.5comparesthecurrentmodulationofthespectraloccu-\npationbelowtheemitterandcollectorelectrodeslabeledPos1\n(𝑥= 0, emitter) and Pos2 ( 𝑥= +𝑑, collector). The position\nof the spot is indicated by a black circle in the inset images in\nFig. 5(a) and (c). To allow a quantitative comparison of the\ndifferent locations, all curves have been renormalized by the\nvalue of the thermal fluctuations at 𝐼1=0, which is assumed\ntobeconstantthroughoutthethinfilms. Inallthesemeasure-\nments, the field is fixed at 𝜇0𝐻𝑥= −0.2T: a value identical\nto that used in the transport measurement shown in Fig. 4.\nWe start the analysis by concentrating first on panel (a) of\nFig. 5, which shows in a density plot the actual variation of\nthemagnonspectraatPos1. ThespectralvariationoftheBLS\nsignal at𝐼1= 0leads experimentally to a peak instead of\nthepredictedstepfunctionshowninFig.2(a)for2Dsystems.\nThe attenuation of the spectral sensitivity at high frequencies\nis an experimental artifact related to the extreme focus of the\noptical beam, which renders the scattered light insensitive to\nspin-waves whose wavevectors are larger than the inverse of\nthe beam waist. The decrease of the signal above the Kit-\ntel frequency is thus directly related to the transfer function\nof the detection scheme, which attenuates short wavelength\nmagnons35,108,109.\nThe black dashed lines in the plots Fig. 5(a) show the shift\noftheKittelfrequencyasafunctionof 𝐼1. Atlowcurrent,the\nshift follows a parabolic behavior as expected for Joule heat-\ning. Note that the curvature of the parabola increases as one\nmoves away from the heat source, as shown in Fig. 5(c) taken\nat Pos2, which we associate with the lateral thermal gradient\n𝜕𝑥𝑇1<086,91. ItisalsoworthnotingthatforthesignalatPos1\nin Fig. 5(a), the local curvature increases dramatically as one\napproaches𝐼c,whichweattributetothedecreasein 𝑀1as𝑇1\napproaches𝑇𝑐. Interestingly, the effect is more pronounced at\n𝐼1>0than at𝐼1<0, suggesting a self-digging effect due to\nasymmetric excitation of low-energy magnons. Also visible\ninFig.5(a)istheextinctionoftheBLSintensityintheregion\n➂when𝐼1≥2.1mA. Finally, the BLS signal decays at large\ncurrents atPos1 (below theemitter), while remainingfinite at\nPos2 (below the collector).\nThe density plot in Fig. 5(c) further confirms the enhance-\nment / attenuation of the spin fluctuations depending on the\npolarity of the current. Starting from thermal fluctuations at11\n𝐼1= 0, the signal decreases for 𝐼1⋅𝐻𝑥>0and increases\nfor𝐼1⋅𝐻𝑥<0. A more detailed analysis at low current am-\nplitude(notshown)confirmsalinearvariationin 𝐼1inthere-\ngion➀. BLS spectra at large currents are shown in Fig. 5(b).\nThey compare the magnon distribution observed in Fig. 5(a)\nat𝐼1=2mA for both negative (green horizontal dash-dotted\ncut) and positive (orange horizontal dash-dotted cut) polar-\nity of the current. At 𝐼1= −2mA, the maximum ampli-\ntude in Kittel mode (green vertical marker at 6.7 GHz) has\na normalized amplitude below 1, i.e., a lower amplitude than\nat𝐼1= 0. At𝐼1= +2mA the density plot shows a sig-\nnificant enhancement of the signal, highlighted by the dashed\ncircles in Fig. 5(a). The corresponding amplitude of the sig-\nnal [orange dots in panel (b)] shows an enhancement of more\nthananorderofmagnitude. Thisenhancementcorrespondsto\nthe increase in spin conduction in the region ➁. In addition,\nthe BLS data taken at Pos1 show the self-localization of the\nasymmetric excitation due to the red shift that arises for large\ncone angles30,97. It is important to note that the induced shift\nisalmostaslargeasitcanbe,sincetheKittelfrequencyalmost\nreachestheparamagneticlimit 𝜔𝐻=𝛾𝐻0,visibleonthepan-\nels(a)and(c)asalightverticaldotedlineatabout5.8GHz110.\nNotethattheparamagneticlimitisreachedonlyfor 𝐼1>0and\nnot for𝐼1<0. This suggests that the film is still in its ferro-\nmagnetic phase when the signal disappears at 𝐼𝑐≈ 2.1mA.\nThis feature will be discussed in connection with the discrep-\nancy between 𝑇⋆\n𝑐and𝑇𝑐in part II. The excitation pocket cre-\nated under the emitter within the circular area also appears as\napeakatPos2inFig.5(d),centeredaroundtheorangevertical\nmarker at 6.7 GHz, i.e., well below the position of the Kittel\nmodeatthatposition(greenverticalmarkerat7.4GHz). This\nsuggests that the collector could still probe magnetic fluctua-\ntionsthatarespectrallybelow(about0.7GHz) 𝐸𝑔,theenergy\nbandgapatthisposition. Althoughtheamplitudeofthepeakat\nthe orange marker is significantly reduced as one moves from\nPos1toPos2,confirmingnumerousindicationsthatthemajor-\nityoflow-energymagnonsremainlocalizedbelowtheemitter\nelectrode(seebelow),thesespectralfluctuationsarenotcom-\npletelysuppressed. Weinterpretthisapparentcontradictionas\na signature that the wavelength of the mode excited under the\nemitter(𝜆around0.6𝜇m)remainslargecomparedtothewidth\noftheemitterwell,andtheratiocorrespondstotheevanescent\ndecaybetween Pos1andPos2. Theissueof spatialdecaywill\nbe investigated in more detail in part II53.\nTo gain further insight, we plot the spectral integration of\nthe BLS signal as a function of 𝐼1in Fig. 5(e). The first key\nfeature directly observed in the plot is the continuity of the\nsignal across the origin for both polarity of the current 𝐼1.\nThis observation is used above to extract the asymmetry of\nthe magnon transmission ratio. It will be shown below that\nthe BLS measurement confirms some other key features ob-\nserved in the magnon transport properties shown in Fig. 4.\nThe second key feature is the limited growth of the intensity\nat𝐼pk. However, the size of the peak is larger at Pos1 than at\nPos2, suggesting the importance of localized magnons under\nthe emitter. The third key feature is the idea that there are 3\ntransportregimes. Inparticular,weobservethecollapseofthe\nBLS signal at |𝐼1|>|𝐼𝑐|under the emitter.To emphasize the similarity with transport, in Fig. 5(e) we\ntentatively plot the evolution with 𝐼1of the number of ther-\nmally activated low-energy magnons produced by Joule heat-\ning with a dashed line. The behavior follows the curve Δ𝑛𝑇1\nintroduced in part II53. The underlying parabolic increase of\nthis background signal is directly visible in the evolution of\nthemaximumBLSintensityalongthedashedlineinFig.5(e).\nThe blue and red shaded areas show the deviation of the BLS\nintensity from the nominal thermal occupancy. They indicate\nthe amount of magnons emitted and absorbed by the STE in\nforward and reverse bias, respectively, and the deviation is\nanalogous to the blue and red shaded areas in Fig. 4(b). For\n𝐼1< 𝐼𝑐∕2the curves deviate equally from the dashed curve,\nindicating that equal amounts of magnons are transferred be-\ntweenthemetalelectrodeandtheYIGfilminthelinearregime\nbetween forward and reverse bias. The curvature at the origin\nshould scale as the variation of 𝑀1with𝐼1. To this end, we\nplottheevolutionof 𝑀1undertheemitterintheinsetFig.5(f).\nThebluepointsarederivedfromthevariationwithcurrentof\nthespectralpositionoftheKittelfrequency,whichissensitive\nto temperature. The observed behavior is consistent with the\nexpected evolution of 𝑀1with𝐼1, which is derived from the\ndependence of 𝑇1on𝐼1as shown in Fig. S1 of Ref.53. The\nresultisshownasadashedlineandtheagreementensuresthe\nvalidity of the evolution with 𝐼1of the thermal background\nshown in Fig. 5(e). In this data treatment, we do not attempt\nto correct for the use of different pulse duty cycles, which re-\nsults in a small discrepancy in the value of 𝐼c.\nTo conclude this section, we interpret the fact that all the\ndistinct transport signatures observed in Fig. 4(b) are present\nin the BLS intensity shown in Fig. 5(e) as a strong indication\nthattheyarepredominantlydrivenbythechangeindensityof\nlow-energymagnonsbelowtheemitter. However,weobserve\na difference in the scaling of the effect, which we attribute to\nthe self-localization of these low-energy magnons. This will\nbe the subject of the band mismatch below. A more rigorous\nquantificationofthelocalizationfractionispartoftheanalysis\nin part II.\nC. Influence of the lattice temperature.\nInthissectionweinvestigatetherelevantparametersthatin-\nfluencethespindiodeeffect. Tothisend,weproposeinFig.6\ntocompare theasymmetricregimewhile independentlyvary-\ningeither𝜖,theinterfacialefficiencyofspin-to-chargeconver-\nsion, or𝜅, the efficiency of device thermalization. It builds\non our two recent studies91and111, where more details can\nbe found. The device was patterned on YIG𝐶, which is the\nsame film as the one studied in Fig. 4, but twice as thick as\nthe YIG𝐴film studied in Fig. 5 (see Table. I). In this batch,\nlocal annealing111can be used to apparently increase the spin\nmixing conductance 𝐺↑ ↓of a single electrode. The term spin\nmixingconductanceshouldbeinterpretedhereasaneffective\nfittingquantity. AsexplainedbyKohno etal.111,thephysicsat\nplayinthisannealingmechanismdoesnotinvolveachangein\nintrinsicproperties,butratherachangeinextrinsicproperties\nvia an expansion of the contact area. We denote 𝜖◦and𝜖•as12\nFIG. 6. Experimental evidence that the spin diode effect is mainly\ncontrolled by thermal fluctuations below the emitter. Spin diode ef-\nfect as a function of emitter temperature raised by Joule heating,\n𝑇1=𝑇0+𝜅𝐶𝑅Pt𝐼2\n1, when either (a) 𝜖, the spin-transfer efficiency,\nor (d)𝜅𝐶, the heat dissipation coefficient, are independently varied.\nTheupperscaleshowsthecorrespondingcurrentbias, 𝐼1. Thelegend\nof panel (a) shows with the symbols ◦/•which electrode of the pair\n(emitter,collector)hasbeensubjectedtolocalannealing111resulting\nin𝜖•/𝜖◦≈3asshowninTable.I.Panel(d)showsthebehaviorwhen\ntheannealeddeviceiscoveredbyanAlheatsink,whichreducesthe\nefficiencyoftheJouleheating 𝜅′\n𝐶<𝜅𝐶. Inpanels(a)and(d),allsolid\nlineshavethesameshapeasinFig.4(b),albeitwithdifferentscaling\nfactors. Therightpanelshowsthecorrespondingcurrentdependence\nof the SSE voltage 𝒱2in panels (b) and (e) and the normalized in-\nverse transmission coefficient of the spin current 𝒯−1\n𝑠=𝐼1∕𝐼2in\npanels (c) and (f). The dashed lines are parabolic fits.\ntheSTEefficiencybeforeandafterannealing. Wehaveshown\nthatapplyinglocalannealingfor60minsat560Kcanreduce\n𝜖•∕𝜖◦≈3without changing the Pt resistance, i.e., 𝜅𝐶111.\nWe first investigate changes in the spin transport when al-\nternatively the collector and emitter are annealed using the 4\npossiblecombinatorialconfigurationspossibleassummarized\ninthelegendofFig.6(a). WefindinFig.6(c)thatthenormal-\nized current ratio\n𝒯𝑠−1≡(\n𝒯𝑠/\n𝒯𝑠|𝐼1→0)−1\n(9)\nof the 4 curves fall on the same parabola. Note that we have\nintroduced the notation of underlined symbols, which will be\nreferred to in the following as the quantity normalized to thelow current value. The parabolic shape is consistent with the\nbehaviorreportedinFig.7(c)ofpartII53,whichsuggestsade-\ncreaseas1−(𝐼1∕𝐼𝑐)2inthe➁regime. Thisalreadysupports\nthatthevalueof 𝜖playsverylittleroleintherelativeamountof\nelectrically excited low-energy magnons. If the sample were\nwell thermalized 𝑇1≈𝑇0, the zero point of the parabolic fit\nshiftssignificantlytohighercurrentvalues,asshownbelowin\nFig. 6(f), suggesting that in this case the extrapolated decay\n𝒯𝑠−1pointstotheamplitudeof Ithassuggestedbytheinver-\nsionofEq.(4). Theinterceptvaluedecreasesastheinfluence\nofJouleheating,or 𝜅𝐶,increasesandisboundedby 𝐼casdis-\ncussedinFig.3. Werecallthatlocalannealinghasnotchanged\nthePtresistanceandthusinFig.6(a-c)theefficiencyofJoule\nheating𝜅𝐶is identical for all 4 cases. These conclusions are\nalsoconsistentwiththebehavioroftheSSEsignal 𝒱2shown\nin Fig.6(b). The 4data sets areexactly divided intotwo pairs\nofcurves,whicharescaledbytheratio (𝜖•∕𝜖◦). Thedifference\nis solely determined by the value of the efficiency coefficient\n𝜖2on the collector side. As expected for SSE, changes in the\nvalue of𝜖1on the emitter side are irrelevant.\nWe now plot in Fig. 6(a) the variation of the renormal-\nized STE transmission coefficient 𝒯𝑠⋅𝑇0∕𝑇1as a function of\n𝑇1=𝜅𝐶𝑅Pt𝐼2\n1+𝑇0, with𝑇0=300K. The normalization by\n𝑇1allows to view the magnon density changes from the point\nofviewofawellthermalizeddevice,assuggestedbyEq. (6).\nTheupperabscissascalecanbeusedasanabacustoconvert 𝑇1\nbackto𝐼1. Onalldeviceswefindthat 𝒯𝑠⋅𝑇0∕𝑇1increasesnon-\nlinearly above 360 K. Above 460 K the spin transmission de-\ncreaseswithincreasingtemperature,asexplainedinFig.4(b).\nAs expected, we find that the amplitude of 𝒯𝑠⋅𝑇0∕𝑇1scales\nas the product of 𝜖1⋅𝜖2. This observation confirms that the\nPt is weakly coupled to the YIG, i.e., only a small part of the\nspincurrentcirculatingintheYIGisdetectedbythecollector.\nIf this were not the case, the signal would not depend on the\ncollector efficiency 𝜖2. This is further confirmed by the width\ndependence of 𝒯𝑠(see Fig. S1).\nThe most interesting feature is the approximate superposi-\ntionofthe2curves( ◦,•)and( •,◦),whichconsistsininverting\nemitterandcollectorinadevicewhereoneelectrodeis3times\nmore efficient at emitting magnons. While the superposition\nis almost perfect at low currents, at high currents the magnon\nconductance with emitter and collector inverted (orange dots)\nleads to a significantly larger magnon conductance than the\nnormal configuration (green dots). However, the difference is\nsmallcomparedtothefactorof3producedbyannealing. Thus\nfor all practical purposes, we interpret the data as somewhat\nconfirming that the spin current circulating between the con-\ntact electrodes is proportional to 𝜖1⋅𝜖2, as Eq. (6) explains it.\nThesuperpositionofthetwocurvesinFig.6(a)showsthatthe\ndensityofmagnonsintheYIG(hereitisvariedbyafactorof\n3)seemstohavenoeffectontheshapeof 𝒯𝑠⋅𝑇0∕𝑇1,andthe\npertinent parameter that determines its behavior remains the\nemitter temperature 𝑇1. Moreover, if one compares in Fig. 8\nof53𝒯𝑠versus𝑇1(as opposed to as versus 𝐼1) taken on two\ndifferentYIGthicknesses,itseemsthatthethicknessplaysno\nroleindeterminingthenonlinearbehaviorbetween 𝐼2and𝐼2\n1.\nTo further support this notion, the coefficient 𝜅𝐶can be\nmodified while keeping 𝜖constant. As explained in91, one13\ncan cover the annealed device with a 105 nm thick Al layer\nthat acts as a heat sink (the heat sink is separated from the Pt\nelectrode by a 20 nm thick protective layer of Si3N4). The in-\nfluenceoftheheatsinkcanbeseenbycomparingtheabacusin\nFig.6(a)andFig.6(d). TheAllayerhasreducedtheefficiency\noftheJouleheating 𝜅𝐶by27%. Thechangesaremostvisible\nin Fig. 6(e), which shows a completely different nature of the\nSSE signal compared to Fig. 6(b) (different curvature, differ-\nentpolarity91). However,once 𝐼1isconvertedto 𝑇1,Fig.6(d)\nscales with Fig. 6(a), even though a larger amount of current\n𝐼1is circulating in the former.\nThis leads to the conclusion that the nonlinearity of the\nmagnontransmissionratio 𝒯𝑠⋅𝑇0∕𝑇1producedbySTEseems\nto be governed mainly by the lattice temperature. This obser-\nvationaloneseemstocontradictourpreviousconclusionfrom\nthe BLS experiment, which attributes the spin diode effect to\nlow-energy magnons. The apparent discrepancy is explained\ninFig.3(c). Knowingthatthedensityoflow-energymagnons\nis also sensitive to temperature, the disappearance of 𝑀1at\n𝐼cdue to Joule heating shifts the position of 𝐼pkbelow𝐼c,\nwhich then becomes weakly dependent on the room temper-\nature value of the threshold current, Ith,0. The process still\nrequires𝜇𝑀toapproach𝐸𝑔byinjectingspins,butonceinthe\nvicinityandthenonlinearprocessisfullyinplace,thetemper-\nature change is dominant. Thus, this artifact is a consequence\noftheinabilitytoefficientlythermalizethePtelectroderather\nthananintrinsicphenomenon. Nevertheless,itemphasizesthe\nimportanceofthermalfluctuationsinnonlinearphenomena. It\nunderlinesthattheparametricthresholdisalsodeterminedby\nthe initial magnon thermal fluctuation in the sample, bearing\nin mind that all nonlinearities are suppressed at absolute zero\ntemperature.\nAll experimental data in Fig. 6(a) and Fig. 6(d) are fitted\nwith the same curve as shown in Fig. 4(b). The only param-\neter that varies is the vertical scaling factor. The fact that all\nthe curves have exactly the same shape also supports the sug-\ngestionthat𝑇1determinesthenatureof 𝒯𝑠⋅𝑇0∕𝑇1,wherethe\nshaded region shows the background contribution from high-\nenergythermalmagnons 𝒯𝑇,whereΣ𝑇∕(Σ𝑇+Σ𝐾)≈0.5rep-\nresents the relative weight at this distance.\nThe Al capping also changes the spatial profile of 𝑇(𝐫),\nwhichdefinestheconfinementpotentialof 𝑀𝑇neartheemit-\nter, as shown in Ref.91. Thus, the absence of a large discrep-\nancybetweenFig.6(a)andFig.6(d)indicatesthatthedepthof\nbandshiftproducedbyJouleheatingisthedominantparame-\nter, while the spatial extent of the confinement region defined\nhere by𝜕𝑥𝑇is not significant.\nD. Magnon band mismatch.\nInthissectionwewillfurtherelucidatewhichnonlinearef-\nfects lead to the suppression of spin propagation in the high\npower regime.\nFIG. 7. Experimental evidence that the spin diode effect involves a\nnarrow spectral window of magnons around the Kittel energy. The\nupper panels illustrate the band mismatch in the GHz spectral range\nbetween two regions at different temperatures. In panel (a), calcu-\nlatedforYIG𝐶thinfilms,apropagationgapofalmost0.7GHzrises\nbetween the low-lying magnon modes, when the temperature differ-\nenceattains150K.There,thefrequenciesofthelow-energymagnons\nbelow the emitter are below the bulk bandgap and thus prevented\nfrompropagatingandinsteadencouragedtoself-localize,limitingthe\ngrowthofthenonlocalsignal. Inpanel(b),calculatedfor(Bi)-YIG𝐵,\nthis barrier vanishes when 𝐾𝑢−𝜇0𝑀𝑠→0. There, the low-energy\nmagnonsareallowedtocontributesignificantlytothelong-rangespin\ntransport. Comparison of the propensity of low-energy magnons to\ntravel long distances ( 𝑑= 4.3𝜇m) between (c) YIG𝐶and (d) (Bi-\n)YIG𝐵. We observe a significant increase in the spin diode signal\nbetween(c)YIG𝐶and(d)(Bi-)YIG𝐵. Weexplainthisbythesuppres-\nsion of self-localization effects produced by the nonlinear frequency\nshift.\n1. Nonlinear frequency shift.\nWe begin this discussion by examining the issue of non-\nlinear frequency shift. We recall that this effect comes from\nthe depolarization factor along the film thickness 𝑁𝑧𝑧= −1,\nwhich introduces a correction to the Kittel frequency that de-\npendsonthemagnetization, 𝑀𝑠. Thiscoefficientcorresponds\nto a red shift, i.e., a decrease of 𝑀𝑠causes a red shift of the\nKittel mode and thus of 𝐸𝑔. The maximum shift that can\nbe induced is(𝜔𝐾−𝛾𝐻0)∕(2𝜋) ≈ 2GHz, which is the en-\nergydifferencebetweentheKittelmodeandtheparamagnetic\nlimit at𝜇0𝐻0= 0.2T. Fig. 5 shows experimental evidence\nfor shifts as large as 0.7 GHz, produced either by Joule heat-\ning of the emitter [see the dashed line in Fig. 5(a) and (c)]\nor by self-localization of the mode (see the dashed circle in\nFig. 5). An increase in the precession angle 𝜃also produces a\nredshiftoftheresonanceduetoadecreaseintheinternalfield,14\nwhich depends on the time-averaged magnetization following\n𝑀𝑠cos𝜃77,80. The latter effect is responsible for the foldover\nof the main resonance. As shown in the inset of Fig. 7(c),\nsuch a shift is strong enough to push the entire spectral win-\ndowoflow-energymagnonsfluctuatingbelowtheemitterun-\nderneaththeenergygap 𝐸𝑔oftheYIGfilmoutside. Thissug-\ngeststhatspinfluctuationsproducedbySTEcanstillbesensed\nbelow the collector despite having energies below the propa-\ngationbandwindowofmagnonsatthisposition,andtheprop-\nagation length 𝜆𝐾is the decay length of the evanescent spin-\nwaveoutsidetheenergywell. Thisfavorstheself-localization\neffect90,94,95as reported in the analysis of Fig. 8 of Ref.53.\nWe want to compare this behavior with the transport prop-\nerties in (Bi-)YIG𝐵thin films (see material parameters in Ta-\nble. I). The peculiarity of sample B is that the YIG sample is\ndoped with Bi. For the right concentration of Bi it is possible\nto match the uniaxial anisotropy with the saturation magneti-\nzation (see Table. I), which leads to 𝑀eff≈ 0. This corre-\nspondstoathinfilmthathasanisotropicallycompensatedde-\nmagnetization effect. In this case, the Kittel mode becomes\nisochronous and independent of the precession cone angle.\nThe Kittel frequency simply follows the paramagnetic pro-\nportionality relation 𝜔𝐾=𝛾𝐻0(similar to the response of\na sphere). In particular, the value of 𝜔𝐾is independent of\n𝑀𝑇and therefore the nonlinear frequency shift is null74,97.\nThis implies that the conduction band of the magnons be-\ntween emitter and collector remains aligned as schematically\nshown in Fig. 7(d), which then prevents the self-localization\neffect of the nonlinear frequency shift. The improvement of\nthemagnontransmissionratioisclearlyvisiblewhencompar-\ningFig.7(c)andFig.7(d). Furtherevidenceforthereduction\nofself-localizationinthelattercaseistheobservationthatthe\nvariationofthetransmissionratioinFig.7(d)nowmimicsthe\nobserved variation of low-energy magnons under the emitter\nby BLS, as shown in Fig. 5(e). We will return to this impor-\ntant point in PartII53while discussing the spatial decay of the\nmagnon transmission ratio.\nOn the quantitative side, we find that the ratio of initial to\nmaximum values is 15.1 for (Bi-)YIG𝐵compared to 7.2 for\nYIG𝐶). A first important observation is that while the sup-\npression of the elliptical precession and the temperature de-\npendence of 𝜔𝐾by the uniaxial anisotropy compensating the\ndipolarfieldfavorsthepopulationoflow-energymagnons,the\nsignal still saturates in the case of (Bi-)YIG𝐵. The solid line\nin Fig. 7(c) and (d) is a fit with Eq. (4) using 𝑛satas a free pa-\nrameter. While for the YIG𝐶sample we find that the best fit\nis obtained by using 𝑛sat=4as explained above, the value of\nthe saturation threshold increases to 𝑛sat= 11in the case of\nBi-YIG𝐵. This is also a direct experimental evidence that the\ncontribution of low-energy magnons to the spin diode effect\nconcerns a rather narrow spectral window around the Kittel\nenergy,withanupperlimitwidthofabout1GHz. Thisbroad-\neningshouldbecorrelatedwiththeenhancedmagnon-magnon\nscattering time indicated by Γ𝑚in Fig. 2(c). Another relevant\nenergy scale is the difference 𝐸𝐾−𝐸𝑔, which varies as 𝑡Y1G.\nThisimplies thatinultrathin filmsofgarnet thisphenomenon\nof localization of low-energy magnons is enhanced.2. Saturation of the magnon density.\nWe emphasize, however, that although the nonlinear fre-\nquency shift is zero in the case of (Bi-)YIG𝐵, the system is\nstillsubjecttothesaturationeffect. Forexampletheobserved\nfactor15.1isstillsignificantlysmallerthanthechangeincone\nangle observed in nanopillars at the damping compensation\nthreshold. This implies that the aforementioned saturation ef-\nfects and the rapid growth of the magnon-magnon relaxation\nrate is a general phenomenon that is not suppressed by reduc-\ning the nonlinear frequency shift. The vanishing nonlinear\nfrequency coefficient only eliminates the ellipticity of the tra-\njectoryforthelongwavelengthmagnonswhosewavevectoris\nsmallerthantheinverseofthefilmthickness. Itdoesnotelimi-\nnatetheself-depolarizationeffectofthespin-wave. Thisvalue\ndepends mainly on 𝜃𝑘, the angle between the propagation di-\nrectionandtheequilibriummagnetizationdirection,thelatter\nbeingtheoriginofthemagnonmanifoldbroadening. Wenote\nthat for the wavevectors around 𝐸𝑔this is the dominant ori-\ngin of the ellipticity, since the broadening almost reaches the\nmaximum value of 𝛾𝜇0𝑀𝑠, as shown in Fig. 1(d).\nThisshowsthatthetuningof 𝑀eff,whichallowstoremove\nthe non-isochronicity of the long wavelength magnons, is re-\nsponsible for an increase of the saturation threshold, which is\nfound to be almost three times higher.\nV. CONCLUSION.\nIn this work we draw a comprehensive picture of the role\nof low-energy magnons in the electrical transconductivity of\nextended magnetic thin films. While spin conduction at low\nintensitiesappearstobehavelargelyasexpected,thebehavior\nat high intensities is markedly different from that observed in\nhighly confined geometries such as nanopillars.\nThe main difference is related to the tendency of the in-\njected spin to spread between different degenerate eigen-\nmodes. Thus, there are phenomena related to the two-\ndimensionality that intrinsically prevent a single mode from\ndominating the others (as is possible in 0D and 1D45). In a\nconfinedgeometry,theenergygapcreatedbyconfinementbe-\ntween different eigenmodes protects the main fluctuator from\nrelaxationintoothermodes. Inanextendedthinfilm,thisbar-\nrier is removed and degeneracies arise, leading to an efficient\nredistribution of energy between degenerate modes. Even if\none mode is pumped more efficiently than the others (e.g. the\nmode with wavelength 1∕𝑤1), nonlinear saturation phenom-\nena quickly come into play, so that the critical current can\nnever be reached. This leads to the magnon-magnon relax-\nation rate becoming power dependent, and in particular to a\nsharpincreaseaboveacertainmodeoccupationthreshold. We\nuse this to paint a picture of a condensate that appears to be-\nhave like a liquid. This picture is supported by a number of\ndifferent experiments, including nonlocal transport on differ-\nent thicknesses of YIG thin films as well as different garnet\ncompositions, Brillouin light spectroscopy, independent vari-\nation of the spin mixing conductance or the thermal gradient\nnear the emitter.15\nWe have shown that doping with Bi improves the nonlo-\ncal signal. The first reason is that we avoid the nonlinear red\nshift under the emitter, which produces localization, which is\nobviously detrimental to the nonlocal geometry. The second\nreasonisthatthelateraltemperaturegradienthasnoeffecton\nthe magnon spectrum, since 𝑀eff≈ 0regardless of the tem-\nperature. So the magnons excited under the emittor have no\nproblem propagating to the collector. And the third reason is\nthatsincetheprecessionisquasi-circular,theparametricexci-\ntationofothermodesofmagnonsisstronglylimited,allowing\n𝑛satto become larger.\nWehavealsoshownthattheinabilitytothermalizetheemit-\nter electrode plays a crucial role in the decrease of 𝑀1under\ntheemitter. Thisisaverystrongeffectin2Dgeometrywhich,\ncombined with the fact that a single mode cannot be excited\nand the critical current goes to infinity, means that we reach\nthe Curie temperature before it self-oscillates. Although this\nreduction in 𝑀1may seem favorable for reaching the critical\ncurrent (which tends to 0 as 𝑀1→0), the nonlinear effects\nmentionedabove(couplingbetweenmodes,locationunderthe\nemitter, etc.) make it inaccessible.\nWehavenotfoundanydirectsignatureofBECinourtrans-\nportstudiesonnonlocaldevices. Allofourexperimentaldata\npoint to strong interaction between degenerate modes rather\nthan fluctuation of a single mode. This problem plagues the\nnonlocalgeometry,whereoneonlyobservesthemagnonprop-\nagatingoutsidethePtelectrode,which,representsonlyasmall\nfraction of the total injected spin. In this respect, our work\nabove does not provide answers about what happens directly\nunder the emitter.\nAlthoughthereisnoBECcondensationoutsidetheareabe-\nlow the emitter, the analogy with the Gurzhi effect in an ul-\ntrapureelectronconductorseemsappropriate. Furtherstudies\nwouldberequiredtoprovidedirectevidenceforsuchmagneto-\nhydrodynamicfluidbehaviorathighpower. Theuniquesigna-\nturewouldbefeaturesthatcanonlybeattributedtotheNavier-\nStokes transport equation62.\nACKNOWLEDGMENTS\nThis work was partially supported by the French Grants\nANR-18-CE24-0021 Maestro and ANR-21-CE24-0031\nHarmony; the EU-project H2020-2020-FETOPEN k-\nNET-899646; the EU-project HORIZON-EIC-2021-\nPATHFINDEROPEN PALANTIRI-101046630. K.A.\nacknowledges support from the National Research Founda-\ntion of Korea (NRF) grant (No. 2021R1C1C201226911)\nfundedbytheKoreangovernment(MSIT).Thisworkwasalso\nsupported in part by the Deutsche Forschungs Gemeinschaft\n(Project number 416727653).\nFIG. S1. Impact of collector width, 𝑤2. The change of 𝑤2occurs\nwhile keeping both the emitter width, 𝑤1= 300nm, and the edge\nto edge distance between the two Pt strips, 𝑠=𝑑−(𝑤1+𝑤2)∕2 =\n0.7𝜇m, where𝑑is the center to center distance. (a) shows 𝒯𝑠as a\nfunctionof𝐼1for3differentvaluesof 𝑤2and(b)shows 𝒯𝑠at𝐼1=1\nmA as a function of 𝑤2. The dashed line is a linear fit through the\ndatapoint. Theinsetisaschematicsideviewofthedevices,defining\nthe various dimensions used throughout the paper.\nFIG. S2. Impact of emitter width, 𝑤1. The change of 𝑤1occurs\nwhile keeping both the collector width, 𝑤2=300nm, and the edge\nto edge distance between the two Pt strips, 𝑠=𝑑−(𝑤1+𝑤2)∕2 =\n0.7𝜇m, constant. (a) shows 𝒯𝑠as a function of the current density\n𝐽1=𝐼1∕(𝑤1𝑡Pt)for 2 different values of 𝑤1and (b) shows the 𝒱2\nvoltage (∝SSE) as a function of 𝑇1.\nVI. ANNEX\nA. Sample characterization\nAllofthemagneticgarnetfilmsusedinthisstudyhavetheir\nmacroscopic magnetic properties fully characterized. Curves\nof magnetization versus temperature are shown in the ap-\npendixof53. ThesameistrueforthePtmetalelectrode,whose\nresistivity and its dependence on temperature are given in the\nsame appendix. All values are summarized in Table. I.\nCharge to spin (or vice versa) interconversion is provided16\nby the Spin Hall effect. Its efficiency process is described by\nthe relation\n𝜖≡𝐺↑ ↓𝜃SHEtanh[𝑡Pt∕(2𝜆Pt)]\n𝐺↑ ↓coth(𝑡Pt∕𝜆Pt)+𝜎Pt∕(𝐺0𝜆Pt),(10)\nwhere𝜃SHEisthespinHallangle, 𝐺↑ ↓isthespinmixingcon-\nductance,𝑡Ptis the thickness of the Pt layer, 𝐺0= 2𝑒2∕ℎis\nthequantumofconductance,and 𝜆Ptistheinternalspindiffu-\nsion length. To calculate the value of 𝜖, we assume that 𝜆Pt=\n3.8nm42and𝜆Pt𝜃SHE=0.18nm111–113. FromTable.Iwesee\nthatthevalueof 𝐺↑ ↓≪𝜎Pt∕(𝐺0𝜆Pt)andthusEq.(10)reduces\nto𝜖≈𝜃SHE𝑇withproportionality 𝑇=𝐺↑ ↓⋅𝐺0𝜆Pt∕𝜎Pt≈0.1\nwhich is maximized when 𝑡Pt≈7nm. Numerical evaluations\nof𝜖are found in Table. I. The maximum achieved value is\n𝜖≈0.08observed in (Bi-)YIG𝐵.\nNotethatforPt, 𝜃SHE>0. Thismeansthattheinterconver-\nsion is governed by the right-hand rule. Using the convention\nof Fig. 1, a positive current (i.e., circulating along −̂ 𝑦) injects\nspinspolarizedalong +̂ 𝑥,intoanadjacentlayer. Thus,theam-\nplificationofspinfluctuationsrequiresthat 𝑀𝑠isalignedwith\n−̂ 𝑥or that𝐻𝑥<0, as indicated in the figure.\nB. Linear regime\nFinally, it seems relevant to emphasize that the data pre-\nsentedaboveprovidesomecross-checkingofthedimensional\ndependenceofthemagnontransmissionratiowithexternalpa-\nrameters,assuggestedbyEq.(6). ThedashedlineinFig.7of\nRef.53showstheproportionaldependence of 𝒯𝑠on𝑇1. Com-\nparison of the nonlocal signals using local annealing of the\nPt electrode as shown in Fig. 6(a) shows the proportional de-\npendence of 𝒯𝑠on𝜖1⋅𝜖2. The Groeningen group has exten-\nsivelystudiedthethicknessdependenceoftheconductivityof\nelectrically excited magnons (low-energy magnons) and theyobserved the monotonous decrease with increasing thickness,\nconfirming the 1∕𝑡YIGbehavior86.\nEq. (6) also predicts the linear and inverse linear relation-\nship of the magnon transmission ratio with the width of the\ncollector𝑤2and the emitter 𝑤1. Fig. S1 shows the influence\nof𝑤2on𝒯𝑠with different 𝑤2as a function of 𝐼1for (a) and\nof𝑤2for (b). Note that the width of the emitter and the edge\nto edge distance between the two Pt strips remain constant as\n𝑤1=300nmand𝑠=𝑑−(𝑤1+𝑤2)∕2=0.7𝜇m. Thesede-\nvicesarefabricatedatthesametimeastheYIG𝐶devicesand\nweverifythatthespinmixingconductance 𝐺↑ ↓foreach𝑤2is\nrelativelysimilar. 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M.19\nBlanter, and T. van der Sar, Advanced Quantum Technologies 4, 2100094 (2021)."    },    {        "title": "2302.00585v1.Temperature_independent_ferromagnetic_resonance_shift_in_Bi_doped_YIG_garnets_through_magnetic_anisotropy_tuning.pdf",        "content": "Temperature-independent ferromagnetic resonance shift in Bi doped YIG garnets through magnetic anisotropy tuning.  Diane Gouéré1, Hugo Merbouche1, Aya El Kanj1, Felix Kohl1, Cécile Carrétéro1, Isabella Boventer1, Romain Lebrun1, Paolo Bortolotti1, Vincent Cros1, Jamal Ben Youssef2, Abdelmadjid Anane1 1 Unité Mixte de Physique CNRS, Thales, Université Paris-Saclay, 91767, Palaiseau, France 2 Lab-STICC, UMR 6285 CNRS, Université de Bretagne Occidentale, 29238, Brest, France Abstract : Thin garnet films are becoming central for magnon-spintronics and spin-orbitronics devices as they show versatile magnetic properties together with low magnetic losses. These fields would benefit from materials in which heat does not affect the magnetization dynamics, an effect known as the non-linear thermal frequency shift. In this study, low damping Bi substituted Iron garnet (Bi:YIG) ultra-thin films have been grown using Pulsed Laser Deposition. Through a fine tuning of the growth parameters, the precise control of the perpendicular magnetic anisotropy allows to achieve a full compensation of the dipolar magnetic anisotropy. Strikingly, once the growth conditions are optimized, varying the growth temperature from 405 °C to 475 °C as the only tuning parameter induces the easy-axis to go from out-of-plane to in-plane. For films that are close to the dipolar compensation, Ferromagnetic Resonance measurements yield an effective magnetization µ0Meff (T) that has almost no temperature dependence over a large temperature range (260 K to 400 K) resulting in an anisotropy temperature exponent of 2. These findings put Bi:YIG system among the very few materials in which the temperature dependence of the magnetic anisotropy varies at the same rate than the saturation magnetization. This interesting behavior is ascribed phenomenologically to the sizable orbital moment of Bi3+.          diane.gouere@cnrs-thales.fr madjid.anane@universite-paris-saclay.fr   I. INTRODUCTION  The recent discoveries in the field of insulating spintronics [1] have triggered an interest for the growth of ultra-thin insulating film based on garnets [2]. The ultra-fast magnetic domain walls (DW) driven by spin-orbit-torques in Bi doped YIG films [3] (BixY3-xFe5O12) as well as the coherent emission of spin-waves (SW) [4] has made such materials a favorable ground for new physical phenomenon to be explored. Even if garnets have been investigated since the 60’s, the emergence of new growth approaches such as Pulsed Laser Deposition (PLD) and off-axis sputtering [5] have proven that garnet ultra-thin films can possess dynamical properties comparable to that of micron-thick films traditionally grown using Liquid Phase Epitaxy (LPE)  [6]. Interface transparency to pure spin currents that take place in adjacent heavy metal layers (most often Pt) has allowed for efficient modulation of the SW life-time through the Damping-like spin-orbit-torque [7]. For instance, a fivefold increase of the SW attenuation length in 20 nm thick YIG films has been observed and even auto-oscillations of the magnetization have been reached [8]. Nevertheless, the required electrical current flow for the generation of the torques in Pt also induces a Joule heating in the Pt/YIG bilayers. As a result, detrimental effects on the spin-wave spectrum appear, since heating-induced reduction of the saturation magnetization in turn generates a self-localization of the induced magnetization dynamics that inhibits SW emission and propagation [9]. This non-linear frequency shift is reminiscent, even if the physical origin is different, of the solitons formation in all-metallic spin-Hall nano-oscillators [10]. A solution to this issue has been proposed using the out-of-plane magnetic anisotropy of BixY3-xFe5O12 system [11]. In parallel to these lines of investigations, the study of spin-orbit-torque induced magnetic motion of DW in garnets thin films (such as Tm3Fe5O12 or BixY3-xFe5O12) has pointed out the existence of a weak, but necessary, interfacial Dzyaloshinskii-Moriya Interaction(DMI) [12]. The BixY3-xFe5O12 system hence appears as being the most versatile material platform as it combines tunable magnetic anisotropy and ultra-low magnetic losses [13]. A precise control of this system will eventually lead to practical magnon-spintronics devices that could perform, for instance, in the static regime for memory application or in the dynamical regime for spin-waves based information processing.  Up to now there are no reports establishing a clear correlation between growth parameters and magnetic anisotropy of doped garnets. Indeed, the magnetic anisotropy parameter for garnets, which is of importance for magnon-spintronics applications, is more usually tuned by extrinsic approaches such as the nature of the substrate, the nature of the dopant atom, or the content of the dopant atom [5,6,13]. In this study, we investigate the temperature dependence of the magnetic anisotropy of low-loss Bi substituted thin YIG films grown by PLD. First, the growth of thin films with optimal structural and magnetic properties has been optimized. Then we show that by tailoring the growth temperature and only the growth temperature, the film magnetic anisotropy can be tuned from out-of-plane to in-plane, yielding a Ferromagnetic Resonance (FMR) frequency’s temperature shift that is also tunable. We demonstrate that the FMR response can be finely adjusted and that we can even reach the conditions for which no temperature dependence on a large temperature range (260 K to 400 K) is observed. This provides a route for material properties engineering to solve a long standing issue of SW devices that up to now needed temperature regulation to perform correctly. By using suitable Bi:YIG, such devices can be made much more compact and less bulky by relieving the temperature stability constraints. Furthermore, temperature independent magnetization dynamics will be instrumental for the development of future spin-orbitronics and magnonics radiofrequency processing devices. II. EXPERIMENTS Thin films have been grown with a PLD process using a tripled frequency YAG laser (λ = 355 nm) pulsed at 2.5 Hz. We use (111)-oriented substituted-Gd3Ga5O12 (sGGG) substrate and a single Bi1Y2Fe5O12 (Bi:YIG) target. The lattice parameters are a = 12.497 Å for the sGGG substrate and a = 12.45 Å for the Bi:YIG target. The laser fluence (1.3 J.cm-2), target-substrate distance (44 mm), oxygen pressure, growth rate, and all the other growth parameters have first been optimized. The oxygen pressure was set at 0.26 mbar during the growth process and Bi:YIG films were grown at temperatures ranging from 405 °C to 475 °C at a growth-rate of 0.41 Å.s-1. Note that this temperature range is lower than what most studies report for the growth of garnets. Going from YIG, for which the growth temperature is reported to be superior to 650 °C, to Bi:YIG has indeed an effect on the value of the optimal growth temperature. To preserve 2D like RHEED patterns, the growth temperature needs to be lowered.  The films thicknesses have been determined by X-Ray-Reflectivity (XRR) and the structural characterizations have been performed by X-Ray-Diffraction (XRD) using a 5-axes Empyrean Panalytical set-up with a Kα1 monochromator (λ = 1.54056 Å). The static magnetic properties have been characterized by Magneto-Optical Kerr Effect (MOKE) measurements with a magnetic field applied perpendicular or parallel to the film plane. In addition, we employ the « Polar-mode » to image the domain configuration for our samples with Perpendicular Magnetic Anisotropy (PMA) and the « Pure Transversal-mode » for films with in-plane easy magnetization axis. The characterization of the magnetization dynamics has been done using broadband FMR with frequencies ranging from 4 up to 40 GHz. The FMR measurements have been performed between 260 K and 400 K in order to probe the temperature dependence of the magnetic properties.  All the elaborated films have a thickness of 22 ± 2 nm with a typical root-mean square (RMS) roughness of 0.4 nm over 5x5 µm2 (see  [14]). The present study focuses on a series of six films in which a clear correlation is established between the growth temperature and the temperature dependence of the magnetic properties, such as magnetic anisotropy, effective magnetization, and non-linear frequency shift. We use the Kerr-effect to record the hysteresis cycle of the films. In Figure 1, we show that films A to D grown at the lowest temperatures (400 °C < TGrowth < 440 °C) exhibit an out-of-plane easy magnetization axis with characteristic loop-shapes and low saturation fields (< 10 mT). Insets of the Polar-Kerr images confirm the uniaxial magnetic anisotropy; they show meander-like domains after demagnetization with characteristic domain size that decreases when increasing the growth temperature. For films E and F grown at higher temperature (TG = 460 °C and TG = 475 °C) the hysteresis cycles have been obtained using an in-plane magnetic field in the Pure-Transversal-Kerr mode, as their saturation in the Polar field configuration is reached at much higher magnetic fields than films A-D. These films E-F have an in-plane easy magnetic axis and, as shown by the insets in Figure 1, highly contrasted in-plane domains with zigzag boundaries. These results indicate that modifying the growth temperature alone allows changing the PMA. Easy magnetization axis is found to switch from out-of-plane to in-plane at TG ≈ 450 °C.  To characterize more precisely the magnetic anisotropy, we use broadband FMR in the field-in-plane configuration. Measurements have been performed from 4 to 20 GHz at room temperature. The Kittel formula is used to extract the value of the effective magnetization defined as 𝑀!\"\"≝𝑀#−𝐻$ where 𝑀# is the saturation magnetization and 𝐻$ is the uniaxial out-of-plane anisotropy field. The frequency dependence ( 𝑓%!#) of the FMR resonance field (𝐻%!#) is fitted for each film to : 𝑓%!#=µ&𝛾*𝐻%!#(𝐻%!#+𝑀!\"\")        (Equation 1) where 𝛾 is the electron gyromagnetic ratio, fixed to its free-electron value : 28 GHz/Tesla.  In Figure 2a, we show that the room temperature effective magnetization (µ&𝑀!\"\") for Bi-doped YIG thin films can vary over hundreds of mT (from -155 mT to 220 mT) when the target composition, the substrate, the growth temperature, the laser fluence and the oxygen pressure are varied  [13] (films labeled 1, 2 and 3). However, for the samples of interest here, labeled from A to F, we kept all parameters constant except for the growth temperature indicated on the x-axis (Table 1). It is observed for these films that the effective magnetization value changes within a much smaller range (from - 46 mT to +33 mT) and increases almost linearly with TG (the red line is a guide for the eye). Films grown below TG = 450 °C have a negative effective magnetization. For films E and F, the effective magnetization is positive. The compensation (𝑀!\"\"=0)\tfor which we observe the out-of-plane to in-plane spin reorientation transition is reached for TG » 428 °C (µ&𝑀!\"\"=−\t1.5\t mT). These results obtained by FMR are hence consistent with those obtained by Kerr-imaging shown in Figure 1: only films that show meander domains and out-of-plane easy magnetization axis have negative effective magnetizations. The inset in Figure 2a displays the peak-to-peak resonance linewidth values of films A to F. Note that the Gilbert damping parameters are comparable for all films and is about 𝛼=7.10'( at room temperature. Optimization in the measurement geometry could eventually lead to a lower value such that for that of Film F : α=4\t10'( (see  [14]).  The absorption FMR spectra of films E (blue, Figure 2b) and B (red, Figure 2c) are reported for f = 12 GHz: both the peak-to-peak linewidth (µ&𝛥𝐻))) and the resonance field (µ&𝐻%!#) are indicated. We note that the resonance fields are shifted from µ&𝐻%!#=413.1 mT to µ&𝐻%!#=439.2 mT when the growth temperature decreases by 40 °C. Moreover, the resonance fields are lower (respectively higher) than the expected resonance field µ&𝐻%!#(𝑀!\"\"=0)=428.6 mT calculated using Equation 1 for compensated thin films. The growth temperature thus appears as a control knob for fine tuning of the perpendicular magnetic anisotropy in a continuous way.  In order to obtain a more thorough characterization of the perpendicular magnetic anisotropy properties, FMR measurements from 5 to 40 GHz have been performed in an extended range of temperature (from 260 K to 400 K). These measurements were done for thin films A-F, but also for films reported on Figure 2a (films 1, 2, and 3). These films are presented to illustrate the extent on which the effective magnetization can vary in the Bi:YIG system, they have been grown using a different set of growth parameters (see Table 2). In Figure 3a, we show effective magnetization as a function of the measurement temperature (TMEAS) for thin films 1 and 3 with strong out-of-plane anisotropy. At TMEAS = 260 K, µ&𝑀!\"\"\tvalues are −156\tmT and −102\tmT respectively. As reported in Figure 3b, the effective magnetization gradually increases until TMEAS = 400 K (to  µ&𝑀!\"\"=−127\tmT and µ&𝑀!\"\"=−77\tmT respectively). The opposite behavior is observed for film 2 (with in-plane anisotropy) for which µ&𝑀!\"\"  decreases from +232\t mT at TMEAS = 260 K to +166\tmT at TMEAS = 400 K. This range of variation is to be compared to what is measured on films C and F that are closer to compensation, for which  µ&𝑀!\"\" varies only by +6 mT and -6 mT between 260 K and 400 K. We note that the absolute thermal shift value is comparable for films C and F (+0.039 mT.K-1 and -0.046 mT.K-1, respectively). It is observed that films with the largest <µ&𝑀!\"\"< values have large temperature dependence of their effective magnetization. But the temperature independence of <µ&𝑀!\"\"<, observed for films C and F, can only be explained by a specific power law between 𝑀𝑠 and K𝑢 (see Discussion and  [14]). Absorption FMR spectra acquired at 260 K, 300 K and 340 K are shown in Figure 3c for the film closest to compensation (film C) for which the resonance field shift (µ&δH*+,) in this range of temperature is only µ&δH*+,≈+4\tmT. It is also observed that the absorption linewidth is sharper at TMEAS = 400 K than at lower temperatures as usually seen in thin garnet films [15]. X-Ray Diffraction (XRD) has been used to investigate the structural properties of the Bi:YIG thin films. Inset in Figure 4 displays the entire diffraction spectra (2θ angle varies from 10 to 130 °) of film E in which several orders of diffractions are observed. The ones associated to sGGG substrate are identified for (444) and (888) together with forbidden reflections, (222) and (666). We observe a systematic shoulder on the side of (444) and (888) peaks that is present even on the bare substrate originating, most probably, from a secondary sGGG phase. As shown in Figure 4, the substrate’s (888) diffraction peak is at 2θ888 = 117.3 ° and its shoulder 2θ888 = 117.7 °; Bi:YIG films diffract at 2θ444 = 50.8 ° and 2θ888 = 118.5 °. Diffraction spectra of thin films grown between TG = 405 °C and TG = 475 °C are plotted in the main panel of Figure 4, 2θ888 = 118.5 ° corresponds to a pseudo-cubic out-of-plane lattice constant of 1.242 nm. The relaxed Bi1Y2IG unit cell parameter is expected to be 1.245 nm (against 1.249 nm for sGGG). As a consequence the 2θ-ω scan points toward tensile strained Bi:YIG films at all growth temperatures. It is shown that the diffraction peaks of all films perfectly overlap leading to a growth-temperature independent strain for Bi:YIG thin films. The identical shape of the films peaks also indicated that the Bi content is constant among all films. The presence of Laue fringes at the (444) reflection is reminiscent of a pseudomorphic growth. III. DISCUSSION One of the important results is that the magnetic anisotropy in Bi:YIG system has two peculiarities, that up to our knowledge have not been previously reported. First it can be finely tuned by the variation of the growth temperature. Second, its temperature dependence has a non-usual behavior in the vicinity of the dipolar compensation (µ&𝑀!\"\"≈0). In Bi doped YIG the out-of-plane magnetic anisotropy has two origins: a magneto-elastic and a growth induced term. The former results from epitaxial strain, for which the pseudomorphic in-plane elastic deformation yields: either a negative or a positive magneto-strictive effect. For garnets deposited along the (111) axis, the magneto-strictive parameter (l111) is negative. A compressive strain favors in-plane magnetic anisotropy, while a tensile strain favors an out-of-plane easy magnetization axis. In this study with a doping level of x = 1 (BixY3-xFe5O12) and the sGGG substrate, we are in the former case i.e. a tensile strain as the relaxed parameter (𝑎&) of Bi:YIG is smaller than that of the substrate. Hence using the magneto-elastic theory, the out-of-plane misfit 𝛥𝑎=𝑎#$-−𝑎& (where 𝑎#$-is the lattice parameter of the substrate) is positive and yields a positive magneto-elastic anisotropy constant (𝐾./) [16,17]: 𝐾./=−B\t01×2345×\tD6!7\"E×l333F\t (Equation 2) and\t𝑎&=J\t(7#$%&''7()*)345×\t(1−µ)+𝑎#$-K (Equation 3) Where 𝑎:\";<==1.242\tnm is the experimentally obtained out-of-plane lattice parameter, 𝐸=2.055×1033\t J.m-3 and µ=0.29 are the Young modulus and the Poisson coefficient respectively, and l333=\t−5.0575×10'> is the magneto-strictive parameter [18,19]. Considering the XRD measurements performed on the Bi:YIG thin films, we calculated 𝑎&=1.245 nm, 𝛥7=4.7×10'0\tnm and \t𝐾./=4.56×100\tJ.m-3 which favors an out-of-plane anisotropy. This value of 𝐾./ is constant for all films of the study (A, B, C, D, E, and F) as no change in the (888) diffraction peak is observed (see Figure 4). For Bi:YIG thin films, the film’s saturation magnetization (𝑀?) has been measured and its value is 𝑀𝑆=116\t kA.m-1 (µ0𝑀𝑆=145.8\tmT) as previously  reported by Lin et al.  [20]. Note that this value is lower than that of un-doped YIG thin films for which  𝑀𝑆=140\t kA.m-1. We can then deduce the magneto-elastic anisotropy field: µ0𝐻𝑀𝑂=µ0\t2\t𝐾𝑀𝑂𝑀𝑆=98 mT. This value is roughly 2/3 of µ0𝑀𝑆, alone it is yet far from sufficient to overcome the shape anisotropy. One needs to consider an extra-term introduced five decades ago by H. Callen [21], known as the growth induced anisotropy. It arises from the preferential occupation by Bi atoms of nonequivalent dodecahedral sites of the unit cell resulting in a symmetry lowering along the growth direction. This term contributes to the magnetic free energy of the system as 𝐾G%HIJK\tsin1𝜃, where q is the polar angle relative to surface normal and\t𝐾G%HIJK\t  the associated anisotropy constant. 𝐾G%HIJK\tis positive under either a compressive or a tensile strain. In the present case, the fact that a vanishing effective magnetization is found, implies that  𝐾G%HIJK is of the same order than  𝐾./. Hence, the sum of both contributions compensates the dipolar term, whereas the cubic magnetic anisotropy is more than an order of magnitude smaller [16] and can be neglected at first order. Our first main observation is the correlation between TGrowth and the magnetic anisotropy. The relevant question is thus to identify which one of the anisotropy terms is varying with the growth temperature (TGrowth). From X-ray diffraction, we obtain that the (888) diffraction peak remains identical for all films. Assuming a pseudomorphic growth for films A to F, we conclude that the relaxed cell parameter (a0) is identical for all films. This has two consequences: (i) the Bi content and (ii) the strain state are identical for films A to F. We can therefore discard any significant variation of the magneto-elastic anisotropy among these films.  As a consequence the only remaining anisotropy term affected by TGrowth is the growth induced anisotropy. In fact, we deduce a very strong variation of KGrowth than span from 4327 J.m-3 to 667 J.m-3 over a growth temperature range of only 75 °C. Such a large effect of TGrowth on Kgrowth is largely unexpected. We speculate that the effect of growth temperature is due to a change in the Bi atoms distribution over the nonequivalent six dodecahedral sites [22]. Yet difficult to evidence experimentally, this explanation is supported by the fact that increasing TGrowth induces a monotonic decrease of growth anisotropy as expected from entropy arguments. The second main observation is that the effective magnetization can be tuned to be almost temperature independent over a wide temperature range for films close to dipolar compensation, yielding an FMR frequency that is stable within 100 MHz (\t𝛥𝑓≈L×6..$$1 ) over a 140 K temperature range. As a comparison, pure YIG films grown by PLD with similar thickness, exhibits about an order of magnitude larger frequency shift over the same temperature range. Such a temperature dependence of the effective magnetization is the second peculiarity of the films under investigation. To have this effect, having a small value of |µ&𝑀!\"\"|, yet necessary, is not the only condition. In the system under investigation, 𝐻𝑢 and 𝑀𝑠 should have the same temperature dependence to keep |µ&𝑀!\"\"|constant. Callen and Callen have proposed the following power law to relate the temperature dependence of the anisotropy to the temperature dependence of the magnetization  [23] : !N(#)!N(%)=\"&O(#)&O(%)#'(Equation 4) with 𝐾$ is the uniaxial magnetic anisotropy, 𝑀# the saturation magnetization and 𝑛 an exponent that is related to the crystal symmetry. It has been predicted by Van Vleck and Zener to be 𝑛=3 in case of uniaxial anisotropy [24,25]. In our case, for a film with vanishing effective magnetization, we observe that 𝑀!\"\"(𝑇)≈0, i.e. 𝑀#(𝑇)≈𝐻$(𝑇)\t. The uniaxial anisotropy field is therefore equal to the saturation magnetization at first order, for all explored temperatures (from 260 K to 400 K). Using Equation 4, the uniaxial magnetic anisotropy 𝐾$(𝑇)=P\"1𝑀#(𝑇)×𝐻$(𝑇) is thus simply proportional to the square of the saturation magnetization. This corresponds to an exponent 𝑛=2 over a temperature range spanning at least from 260 K to 400 K, while the films Curie temperature is expected to be 559 K. A similar exponent 𝑛=2 has been found while plotting 𝐾𝑢(T) as a function of 𝑀𝑠(T) and fitting the slope with Equation 4, (Fig. 3d) (see  [14]). The authors want to stress out here that such exponent is not expected by the theory that has been developed for many system including garnets. As a matter of fact, many systems in the literature deviate from their theoretical n exponent (given for cubic or uniaxial anisotropy). Indeed interestingly, there are  few systems that deviate from the standard value of 𝑛=3  [26–28]. Chatterjee et al. found an exponent 𝑛=2.6 for the NiFe2O4 system, Wang et al. found an exponent of 𝑛=1 for BaFe12O19, after fitting experimental data. Thus, although the deviation of the existent model is then not specific to Bi:YIG system, the explanation may still differ from one system to the other. However, a value of 𝑛=2 has only been reported for ordered Pt-Fe alloys of L10-type  where this specific temperature dependence has been ascribed to a two-ion anisotropy related to the induced  moments on the Pt  [29]. For our films, even if Bi+3 ions in Bi:YIG  are not expected to carry  a spin magnetic moment,  M1-edge X-ray Circular magnetic dichroism (XMCD) experiments  have unambiguously revealed that Bi atoms bear a sizable orbital moment [30]. This explains the fact that we observe a significantly smaller magnetization saturation than that of YIG confirming previous observations (as Vertruyen et al. [31]). Additionally, the presence of the orbital moment may also explained a two-ions like magnetic anisotropy behavior in Bi:YIG that would then be related to the complex [5d,6p] Bi orbitals hybridization with Fe. We hence conclude that our observations are resulting from two effects that are growth induced anisotropy sensitivity to growth temperature and the magnetic state of Bismuth. A first-principal investigation could give enlightening insights on the inter-links between these two effects, which is beyond this work. IV. CONCLUSION In conclusion, we show (i) how to finely tune the magnetic anisotropy in garnets and (ii) how to obtain a temperature-free variation of their uniform magnetization dynamics characteristics when achieving a vanishing effective magnetization while preserving a low damping (𝛼≈7\t.10'(). We explain these observations, based on the growth-induced anisotropy for the former and on the importance to consider Bismuth orbital-momentum for the latter. Growth induced anisotropy is found to be very sensitive to growth temperature whereas no significant effect is observed on the magneto-elastic anisotropy. The anisotropy temperature exponent is found to change with the anisotropy; its value is 𝑛 ~ 2 when the out-of-plane anisotropy compensates the shape anisotropy. The practical implications of these findings for the field of magnonics and spin-orbitronics are numerous. At first, they solve a long standing issue of YIG based radiofrequency devices [32],  that most often need thermally regulated packaging. Given the much smaller thermal sensitivity of optimized Bi:YIG, such technical constraints can now be lifted and more compact and less power hungry devices could be developed. Furthermore, we explain why using Bi:YIG has been very important for the physics of spin-orbit-torque in magnonics, where Joule heating is unavoidable. Not only compensating exactly the dipolar contribution is necessary to avoid non-linear effects such as non-linear magnon-magnon interactions  [4], but it is equally important  to suppress any temperature dependence of FMR frequency to avoid spin wave localization. As the only temperature dependent term in Equation 1 is the effective magnetization, for low damping materials, such condition can be reached in Bi:YIG and as far as the authors are aware of, only in the Bi:YIG system. Finally, we provide perspectives on how magnons dispersion relation can be engineered to have the desired temperature dependence. Such latitude can open a new field for magnonics lenses where the magnetic medium has reconfigurable effective indexes for spin waves optical elements [33].  ACKNOWLEDGMENTS We acknowledge support from the French National Research Agency (ANR), ANR MARIN ANR-20-CE24-0012. This project has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No 861300. and under FET-Open Grant No. 899646 (k-NET), and as part of the “Investissements d’Avenir” program (Labex NanoSaclay, reference: ANR-10-LABX-0035 “SPiCY”.   References  [1] Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi, S. Maekawa, and E. Saitoh, Transmission of Electrical Signals by Spin-Wave Interconversion in a Magnetic Insulator, Nature 464, 262 (2010). [2] G. Schmidt, C. Hauser, P. Trempler, M. Paleschke, and E. T. Papaioannou, Ultra Thin Films of Yttrium Iron Garnet with Very Low Damping: A Review, Phys. Status Solidi Basic Res. 257, (2020). [3] L. Caretta, S. H. Oh, T. Fakhrul, D. K. Lee, B. H. Lee, S. K. Kim, C. A. Ross, K. J. Lee, and G. S. D. Beach, Relativistic Kinematics of a Magnetic Soliton, Science (80-. ). 370, 1438 (2020). [4] V. E. 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Table 1: Calculated magneto-elastic anisotropy constant (KMO) for films grown between 405 °C and 475 °C, all other growth condition being identical. The deduced relaxed cell parameter(a0) is given.  Film Laser Fluence (J.cm-2) Growth Temperature (°C) Growth Rate (Å.s-1) Thickness (nm) a0 (nm) KMO (J.m-3) A   405         B   420         C 1.3 428 0.41 22 1.245 4560 D   442         E   460         F   475                                Table 2: Growth parameters of thin films 1, 2 and 3 compared with the one of films A and F of the study. The main difference is the laser fluence varying from 1 to 1.7 J.cm-2.   Film Chemical composition Growth temperature (°C) O2 Pressure (mbar) Thickness (nm) Laser Fluence (J.cm-2) 1 Bi1-YIG//sGGG 470  21 1.6 2 Bi0.7-YIG//GGG 510   23 1.7 3 Bi1-YIG//sGGG 510 0.25 25 1 A Bi1-YIG//sGGG 405   22 1.3 F Bi1-YIG//sGGG 475   22 1.3       Figures:  \n  Figure 1: Magneto-optical Kerr Effect (MOKE) measurements in Polar-mode (A, B, C, and D) and in the Pure-Transversal mode (E and F)  for the 22 nm thick Bi:YIG//sGGG films. Films grown between 405 °C and 442 °C have meander-like domain structure at remanance and an out-of-plane easy magnetization axis. Films E and F grown at 460 °C and 475 °C are in-plane magnetized as evidenced by in-plane hystersesis cycles and the in-plane domains having characterestic zigzag boundaries. The saturation field is also indicated.   \n40 µm\n40 µm\n100 µm\n10 µm\n20 µm20 µm\n          Figure 2: Room-temperature Ferromagnetic Resonance (FMR) measurements in the in-plane field configuration. (a)The effective magnetization obtained through fitting to the Kittel formula  (Equation 1) is ploted as a function of the growth temperature. The red line is a guide-for-the-eye for the present series of six films (A, B, C, D, E, and F). Note the quasi-linear temperature dependence of the effective magnetization, that is negative for films with out-of-plane easy magnetization axis (A, B, C, and D) and positive for films with in-plane easy magnetization axis (E,F). The linewidth at 12 GHz corresponding to the broadband FMR measurements is reported as an inset. Films labeled 1, 2, and 3 corespond to Bi:YIG films with larger effective magnetizations obtained either by changing the Bi content (1 and 2) or by changing the laser power (3). (b) and (c) FMR absorption spectra  for films E and B showing the lower resonance field for in-plane magnetized films.    \n    \n   Figure 3: Temperature dependence of the Ferromagnetic Resonance (FMR) measurements. (a)  The effective magnetization is ploted as a function of the measurement temperature for films (C,F) of the study and three other films (1-3). For films with large effective magnetizations (1, 2, and 3) we observe a significant variation of its value when sweeping the measurement temperature. (b) Whereas for films with much smaller effective magnetization (C,F) the temperature dependence is much smaller. (c) FMR absorption spectra for film C at three measurements temperatures at 5 GHz. The shift in the resonance field does not exceed 4 mT. (d) Inferred dependence of the out-of-plane anisotropy constant as a function of the saturation magnetization, including a fit which corresponds to an anisotropy exponent 𝑛=2.   \n \n \n(d) \n(c) \n             Figure 4: q-2q X-Ray diffraction (XRD) spectra for Bi:YIG//sGGG. The main panel (with several films grown between 405 °C and 475 °C) in which (888) reflections show that the out-of-plane interlayer distance is identical for all films (curves have been offseted for clarity). The inset displays a typical XRD pattern in 10°-130° range. No parasitic phase is observed.    \n1  Supplementary material  Temperature-independent ferromagnetic resonance shift in Bi doped YIG garnets through magnetic anisotropy tuning.  Diane Gouéré1, Hugo Merbouche1, Aya El Kanj1, Felix Kohl1, Cécile Carrétéro1, Isabella Boventer1, Romain Lebrun1, Paolo Bortolotti1, Vincent Cros1, Jamal Ben Youssef2, Abdelmadjid Anane1 1 Unité Mixte de Physique CNRS, Thales, Université Paris-Saclay, 91767, Palaiseau, France 2 Lab-STICC, UMR 6285 CNRS, Université de Bretagne Occidentale, 29238, Brest, France  1. Atomic force microscopy characterization (AFM):  The typical root-mean-square (RMS) roughness of the films is 0.4 nm over a surface of 25 µm². The atomic force microscopy images of sample A, B and F are shown in Figure A. The largest RMS roughness, for sample B, is no more than 0.623 nm. This is more than 30 times lower than the film thickness, we can therefore neglect any shape induced variation of the effective magnetization.   \n Figure A: Atomic Force Microscopy (AFM) imaging of (a) film A (b) film B and (c) film F and their 3D view over a surface of 25 µm² recorded in tapping mode. The associated RMS are indicated on the figures.   (a) (b) RMS : 384 pm RMS : 623 pm \n(c) RMS : 508 pm 2  2. Ferromagnetic resonance characterization (FMR):           Figure B: Plot of the derivative of absorption spectra at fres=16 GHz for film A. The red line is the fit used to extract both:  𝜇!𝐻\"#$ (mT) and 𝜇!𝛥𝐻%% (mT) parameters for each frequency. The film is probed by FMR in the in-plane configuration at room temperature. The resonance field (𝜇!𝐻\"#$) and the peak-to-peak linewidth (𝜇!𝛥𝐻%%) are extracted for each resonance frequency (fres) from the fits. The Gilbert damping parameter α is fitted using: µ!𝛥𝐻\"\"=µ!𝛥𝐻!+4𝜋𝛼𝑓#$%𝛾µ!√3 The first term corresponds to an inhomogeneous broadening that may be important while measuring PMA thin films (like film A) in the in-plane configuration. The damping parameter α is extracted from the slope when plotting 𝜇!𝛥𝐻%%(𝑓\"#$). For films A and F, we show the variation over 8 or 12 GHz:        Figure C: Plot of 𝜇!𝛥𝐻𝑝𝑝 versus 𝑓\"#$ and extracted Gilbert damping parameter for (a) film A where 𝛼=0.0007 and (b) film F where 𝛼=0.00045. Both films are probed by FMR in the in-plane configuration at room temperature. In the main text, we indicated a Gilbert damping of α=7∗10&' that is the extracted value for film A with PMA. This value is an upper bound. Optimization in the measurement geometry could eventually lead to a lower value such that for that of film F : α=4∗10&'.    \n(b) \n(a) \n• Data - Fit \nIntensity (arb. units) 3  3. Magnetic Anisotropy exponent analysis Ferromagnetic Resonance (FMR) measurement performed between 260 K and 400 K yields the temperature dependence of the effective magnetization 𝑀$(((𝑇) (see Figure 3 in the main), the effective magnetization value of films C and F is stable (temperature coefficient ~ 4.10-2 mT.K-1). We also measure independently the temperature dependence of the magnetization 𝑀%(𝑇) using SQUID magnetometry over the same temperature range. The combination of these two measurements allows to obtain the uniaxial anisotropy constant 𝐾)(𝑇) using: 𝑀$(((𝑇)=𝑀%(𝑇)−𝐻)(𝑇) and 𝐻)\t(𝑇)=*+!(-)/\"0#(-) We then plot 𝐾)(𝑇) versus 𝑀%(𝑇) and fit to Equation 1. The result is shown in Figure D (Figure 3d in the main). The best fit for the exponent n is 2.02 very close to the claimed value of 2. This procedure however can be subject to some criticisms. There are additive errors and the dynamic range may be viewed as insufficient even if it allows to easily rule out the n=3 as a possible value (see green curve in Figure D). .1(/).1(0)=\"12(/)12(0)#2 (Equation 1)           Figure D: Plot of 𝐾$(𝑇) extracted from the FMR temperature dependent measurement as a function of 𝑀%(𝑇) extracted from SQUID magnetometry measurements. Fits give a n exponent value of n=2.02 close to n=2.  The second approach relies on the fact that we do observe a vanishing temperature dependence of the effective magnetization :   Temperature scaling argument : We observe that for films that are close to anisotropy compensation:  𝑀$(((𝑇)=𝑀%(𝑇)−𝐻)(𝑇)≈0\t  over all the temperature range  As a result:  𝐻)(𝑇)≈𝑀%(𝑇)\t(Equation\t2) 𝐻)(𝑇)𝐻)(0)≈𝑀%(𝑇)𝑀%(0) \n4  𝐻)(0) and 𝑀%(0) refer to the values of the anisotropy field and the saturation magnetization at the lower bound of the temperature range of interest. By replacing the anisotropy field 𝐻)(𝑇) with the anisotropy constant 𝐾)(𝑇) defined as:  𝐻)(𝑇)=2𝐾)(𝑇)𝜇!𝑀%(𝑇) We obtain: 𝐾)(𝑇)∙𝑀%(0)𝑀%(𝑇)∙𝐾)(0)≈𝑀%(𝑇)𝑀%(0) And finally: 𝐾)(𝑇)𝐾)(0)≈E𝑀%(𝑇)𝑀%(0)F* Identifying this result to Equation 1 yields an exponent of n=2. Mathematical derivation of the temperature stability condition: It is worth noting that having a small 𝑀$(( alone, as it can be obtained by anisotropy engineering is not enough to guaranty a vanishing temperature dependence of 𝑀$((. This can be mathematically demonstrated within the Callen and Callen model. A vanishing temperature dependence of the effective magnetization can only be obtained for 𝑛=2 : According to the definition:  𝑀$(((𝑇)=𝑀%(𝑇)−𝐻)(𝑇)\t (Equation 3 ) The uniaxial anisotropy field is linked to 𝐾)(𝑇) and 𝑀%(𝑇) by: 𝐻)(𝑇)=*+!(-)/\"0#(-) (Equation 4) From Callen and Callen power law we have: 𝐾)(𝑇)=𝐾)(0)H0#(-)0#(!)I3(Equation 5) By replacing successively 𝐻)(𝑇) (using Equation 4) and 𝐾)(𝑇)\t(using Equation 5) in the in Equation 3, we obtain: 𝑀$(((𝑇)=𝑀%(𝑇)−2𝐾)(0)µ!𝑀%(𝑇)E𝑀%(𝑇)𝑀%(0)F3 𝑀$(((𝑇)=𝑀%(𝑇)−2µ!𝐾)(0)∙\t𝑀%(𝑇)3&4∙𝑀%(0)&3 And finally: 𝑀$(((𝑇)=𝑀%(𝑇)\tH1−\t*/\"𝐾)(0)\t∙𝑀%(𝑇)3&*\t∙𝑀%(0)&3I\t=\t𝑀%(𝑇)\t∙𝐴3,-  The temperature dependence is obtained by calculating : 𝜕𝑀$(((𝑇)𝜕𝑇=\t𝑀%(𝑇)∙𝜕𝐴3,-𝜕𝑇+𝐴3,-∙\t𝜕𝑀%(𝑇)𝜕𝑇 All terms involved in defining 𝐴3,- being temperature independent but the saturation magnetization, a vanishing temperature dependence of the effective magnetization can only be obtained by having 𝑛=2 : 5  𝜕𝐴3,-𝜕𝑇=\t\t𝜕𝜕𝑇L1−2µ!\t𝐾)(0)∙\t𝑀%(𝑇)!\t∙𝑀%(0)&*M=0 Furthermore, fixing 𝑛=2 allows to write : 𝐴*,-=1−6!(!)0#(!)≈0 when we achieve vanishing effective magnetization. As a conclusion, obtaining a temperature independent effective magnetization requires two strong conditions : a Callen and Callen exponent that is equal to 2 and a magnetic anisotropy that compensates almost exactly the dipolar field. "    },    {        "title": "2007.08908v3.Coherent_coupling_between_multiple_ferrimagnetic_spheres_and_a_microwave_cavity_in_the_quantum_limit.pdf",        "content": "Coherent coupling between multiple ferrimagnetic spheres\nand a microwave cavity at millikelvin temperatures\nN. Crescini,1, 2,\u0003C. Braggio,2, 3G. Carugno,2, 3A. Ortolan,1and G. Ruoso1\n1INFN-Laboratori Nazionali di Legnaro, Viale dell'Universit\u0012 a 2, 35020 Legnaro (PD), Italy\n2Dipartimento di Fisica e Astronomia, Via Marzolo 8, 35131 Padova, Italy\n3INFN-Sezione di Padova, Via Marzolo 8, 35131 Padova, Italy\n(Dated: March 7, 2022)\nThe spin resonance of electrons can be coupled to a microwave cavity mode to obtain a photon-\nmagnon hybrid system. These quantum systems are widely studied for both fundamental physics\nand technological quantum applications. In this article, the behavior of a large number of ferrimag-\nnetic spheres coupled to a single cavity is put under test. We use second-quantization modeling\nof harmonic oscillators to theoretically describe our experimental setup and understand the in\ru-\nence of several parameters. The magnon-polariton dispersion relation is used to characterize the\nsystem, with a particular focus on the vacuum Rabi mode splitting due to multiple spheres. We\ncombine the results obtained with simple hybrid systems to analyze the behavior of a more complex\none, and show that it can be devised in such a way to minimize the degrees of freedom needed to\ncompletely describe it. By studying single-sphere coupling two possible size-e\u000bects related to the\nsample diameter have been identi\fed, while multiple-spheres con\fgurations reveal how to upscale\nthe system. This characterization is useful for the implementation of an axion-to-electromagnetic\n\feld transducer in a ferromagnetic haloscope for dark matter searches. Our dedicated setup, con-\nsisting in ten 2 mm-diameter YIG spheres coupled to a copper microwave cavity, is used for this aim\nand studied at mK temperatures. Moreover, we show that novel applications of optimally-controlled\nhybrid systems can be foreseen for setups embedding a large number of samples.\nI. INTRODUCTION\nIn the fruitful and renowned \feld of light-matter\ninteraction1, the study of hybrid quantum systems based\non magnonics gave outstanding results2,3. Magnons,\nquanta of spin excitations, can coherently couple to pho-\ntons through a magnetic dipole interaction4,5. The elec-\ntron spin resonance of a magnetic material can thus be\ncoupled to the rf magnetic \feld of a microwave cavity\nmode, to form a photon-magnon hybrid system (HS)6{9\nat GHz frequencies. This kind of HSs are employed\nin \felds including, but not limited to, the develop-\nment of quantum computers10, quantum networks11,12,\nand quantum sensing13. In these areas, the coherent\ncoupling of spin ensembles, microwave cavities6{9,14{16\nand superconducting quantum circuits17{19is used to\ncreate quantum memories20,21, to convert optical pho-\ntons to microwaves22and vice versa23{25, and to de-\ntect single quanta of magnetization26. In spintronics,\nmagnons27are suitable carriers of spin information, and\nhave several advantages over electrical currents used in\nmodern electronics. Furthermore, the investigation of\nnon-Hermitian quantum mechanics28is currently pur-\nsued with HSs. Exceptional points are the signature\nof non-Hermitian physics29,30, and their observation was\nrecently proposed31, and experimentally veri\fed32in\nphoton-magnon HSs. These results were followed by the\ndemonstration of exceptional surfaces33, and the measure\nof a third order exceptional point has been proposed in\nsystems of multiple ferrimagnetic spheres34. Besides the\nstudy of non-Hermitian quantum mechanics, di\u000berent ap-\nplications can be envisioned for these points, for example\nas sensitive probes of magnetic \felds35. These and manyother studies36{40maintain that photon-magnon HSs are\nan outstanding testbed for the study of many physical\nphenomena.\nThe study of HSs comprising a large amount of\nmagnetic samples has been explored to create gradient\nmemories9, or to detect faint pseudo-magnetic \felds41.\nIn precision physics, rf spin-magnetometers based on HSs\nare used as axion haloscopes under the name of ferro-\nmagnetic haloscopes41,42(FH). In these devices, a large\nnumber of spins increases the sensitivity to magnetic os-\ncillations induced by dark matter axions. As the axion\ncouples to magnons43in a FH the HS acts as an axionic-\nto-electromagnetic \feld transducer.\nThis work reports on the coherent coupling of ten\n2.1 mm-diameter ferrimagnetic spheres to a cylindrical\ncopper cavity, with the aim of implementing the result-\ning HS in a FH. Our results, however, go beyond this\nsole purpose, and investigate the dynamics of such large\nand complex systems, which are promising for multiple\nusages9,39,44{47. In our setup, the strong coupling regime\nis largely reached already with a single sphere, thus we\ncan study the scaling without qualitatively changing the\nbehavior of the HS. We singularly tested spheres with\ndiameters ranging from 0.5 mm to 2.5 mm and observed\ntwo di\u000berent size-e\u000bects, a\u000becting the g-factor and the\nsphere's zero-\feld splitting. By choosing a diameter of\n2.1 mm, we focus on the coherent coupling of multiple\nspheres. The multi-samples design is preferred to a sin-\ngle rod of magnetic material for practical reasons, like\nthe material availability and geometric demagnetization.\nThe in\ruence of the dimension and relative distance of\nthe samples are consistently accounted for in a model\nwhich precisely reproduces the experimental data. OurarXiv:2007.08908v3  [quant-ph]  3 Mar 20222\nstudy thus demonstrates that it is feasible to scale up a\nHS while preserving its optimal control.\nIn Sec. II we detail the model used to describe the ex-\nperiment. Sec. III illustrates the setup, and is further di-\nvided into two parts: the \frst (III A) explains the room\ntemperature tests used to understand the behavior of sin-\ngle spheres, while the second (III B) reports on the cou-\npling of many multiple spheres to the cavity, and on the\nroom temperature and ultra cryogenic tests of the result-\ning HS. The last part of the paper, Sec. IV, is dedicated\nto the possible improvements of the setup, in terms of\nincreasing the quantity of material coupled to the cavity,\nand to a summary of the obtained results.\nII. SECOND-QUANTIZATION MODEL\nThis section deals with the model used to study the\nHS dynamics, which essentially consists in a system of\ncoupled harmonic oscillators in the second-quantization\nformalism48,49that e\u000eciently describes the number of\n(quasi-)particle in a state by using creation and annihi-\nlation operators to add or remove a quanta. Similar ap-\nproaches have been used to study the interaction of mul-\ntiple qubits in a cavity50, and test the Tavis-Cummings\nmodel51for a low number of two-level systems. Since\nwe are interested in describing resonant quanta, raising\nand lowering operators share the same algebra of a col-\nlection of interacting quantum harmonic oscillators. Letus consider Nphoton modes X=fx1;x2;:::;x Ng, and\nMmagnon modes Y=fy1;y2;:::;y Mg, and label with\n!andgtheir frequencies and couplings, respectively. Ev-\nery modeXcan couple to any of the modes Y, so the\nHamiltonian of the system in the rotating wave approxi-\nmation reads\nH=~X\nx2X!xxyx+~X\ny2Y!yyyy\n+X\nx2XX\ny2Ygxy(xyy+yyx)\n+X\ni6=jgyiyj(yy\niyj+yy\njyi);(1)\nwhere~is the reduced Planck constant, and xy,yy(x,\ny) are the creation (annihilation) operators of the corre-\nsponding mode quanta. Our interest lies in the evolution\nof the mean values of xandyoperators that can be cal-\nculated by Heisenberg-Langevin equations. The e\u000bects of\ndissipations in a resonant system due to its coupling with\na thermal reservoir can be easily taken into account by\nadding an imaginary part to the mode frequencies which\ncorrespond to their linewidths \rs. Thus the equations of\nmotion for the HS evolution can be recast as a \frst or-\nder system of di\u000berential equations (see Appendix A for\nthe complete derivation), and its solution can be readily\nfound in Fourier space. The associated HS matrix reads\nH=0\nBBBBBBBBBBBBBBBB@!x1\u0000i\rx1=2 0 ::: 0\n0!x2\u0000i\rx2=2::: 0\n............\n0 0 ::: ! xN\u0000i\rxN=2gx1y1gx1y2::: g x1yM\ngx2y1gx2y2::: g x2yM............\ngxNy1gxNy2::: g xNyM\ngx1y1gx2y1::: g xNy1\ngx1y2gx2y2::: g xNy2\n............\ngx1yMgx2yM::: g xNyM!y1\u0000i\ry1=2gy2y1::: g y1yM\ngy1y2!y2\u0000i\ry2=2::: g y2yM............\ngy1yMgy2yM::: ! yM\u0000i\ryM=21\nCCCCCCCCCCCCCCCCA:(2)\nThe top-left block stands for the cavity modes, which are\nconsidered uncoupled to each other, as typically their\nresonant frequency di\u000berence is much larger than their\nlinewidths. The lower-right block shows the material\nmagnetic modes, which may auto-interact mainly thanks\nto the dipole coupling between di\u000berent spheres (see Sec-\ntion III). The o\u000b-diagonal blocks are the couplings be-\ntween the cavity and magnetic modes. The matrix in\nEq. (2) completely describes the system dynamics, and\ncan be used to calculate the dispersion relation of themagnon-polariton.\nIn our study we variate the static magnetic \feld to\nunderstand how it a\u000bects the transmission function of\nthe HS. The spin precession frequency of a free elec-\ntron in a generic magnetic \feld Bis!0(B) =\rB, with\n\r= (2\u0019)28 GHz=T. This basic element can be used in\nthe theoretical model to account for a variation of the\nmagnetostatic mode frequencies !Y. In particular, we\nwrite the frequencies of the Ymodes as\n!Y(B) =\r(B+bY); (3)3\nwhere the zero \feld shift \rbYis a constant depending on\nthe magnetostatic mode under consideration. In Eq. (3)\nwe choose a linear dependence between !and the ap-\nplied static \feld, but it is straightforward to include\nnon-linear magnetic \feld dependencies if any are present.\nThe poles of the cavity magnon-polariton dispersion rela-\ntion can be found by solving the characteristic equation\ndet(K(!;!Y)) = 0, where\nK(!;!Y) =!IN+M\u0000H(!Y); (4)\nandIN+Mis the identity matrix of dimension N+M.\nHere!can be seen as the frequency of a monochromatic\nprobe signal. Experimental spectra are collected through\ntwo antennas coupled to the resonant cavity. In princi-\nple, we should consider the transmission of all the cavity\nmodes (see Appendix A), but, as the model does not ac-\ncount for the antenna coupling strength, we can restrict\nto one mode and neglect other transmission channels. To\ncalculate the transmission spectra of the HS, we can con-\nsider the monochromatic system excitation coupled to\nthe \frst cavity mode x1\nsx1(!;!Y) =\rx1!2\n2j(K\u00001(!;!Y)\u0001^e1)1j2; (5)\nwhere ^e1is the unit vector of x1. Note that the param-\neters ofHcan be determined experimentally by \ftting\nthe transmission spectra to Eq. (5).\nIn the following, to indicate a particular HS con\fgu-\nration we will write the Hamiltonian of the system, and\ntake for granted that to get the dispersion relation one\nfollows Eq. (4) and (5).\nTo reduce the complexity of the HS, and understand\nhow the di\u000berent couplings a\u000bect its dynamics, we re-\nduce the number of considered modes to four: two cavity\nmodescandd, and two magnetic modes mandn. The\nmodec\u0011x1corresponds to the one coupled to the an-\ntennas, and m\u0011y1is the Kittel mode with frequency\n!m. The resulting Hamiltonian is\nH4=0\nBB@!c\u0000i\n2\rc 0gcmgcn\n0!d\u0000i\n2\rdgdmgdn\ngcmgdm!m\u0000i\n2\rmgmn\ngcngdngmn!n\u0000i\n2\rn1\nCCA:\n(6)\nThe resonant frequency of the cavity modes are set\nto!c= (2\u0019)10:65 GHz and !d= (2\u0019)10:9 GHz, and\nthe linewidths to \rc=\rd= (2\u0019)1:0 MHz, while\nthe linewidths of the magnon modes \rm=\rn=\n(2\u0019)2:0 MHz. The e\u000bects of the di\u000berent terms are stud-\nied by \frst setting to zero all the couplings but gcm, set to\n(2\u0019)90 MHz to be close to the value measured with a sin-\ngle 2.1 mm diameter sphere. Then, the other couplings\nare selectively turned on one by one, and the resulting\ndispersion relations are compared in Fig. 1.\nPlot 1(a) is a usual anticrossing curve, where the cou-\npling 2gcmis the frequency splitting of the two HS modes\nwhen!=!m. This HS is the starting point for all the\nfollowing considerations.\n(a) (b)\n(c) (d)c-m c-m-d\nn-c-m c-m-n\nFIG. 1: Plots of the transmission spectra sx1(!;! 0) illustrat-\ning the dispersion relations of the magnon-polariton systems.\nThe color scale is in logarithmic arbitrary units, where blue\ncorresponds to low and light green to high transmission. The\ndash between the labels of two modes indicates that the two\nare coupled. Upper plots: normal anticrossing curve (a), and\ncoupling of the Kittel mode with the two cavity modes cand\nd(b). Lower plots: (c) c-mode coupled to two magnetostatic\nmodesmandn, withbn6= 0; (d)c-mode coupled to m, cou-\npled ton, withbn= 0 andgmn= 25 MHz. One can obtain\nthe same plot with gmn= 0 andbn\u00180. See text for further\ndetails.\nThe interaction of the Kittel magnetic mode mwith\ntwo cavity modes c;dis considered in Fig. 1(b), where we\nset equal couplings gcm=gdm, producing a combination\nof two anticrossing curves. The second cavity mode d,\nat the frequency !d= (2\u0019)10:9 GHz, is not coupled to\nthe antennas, as is shown by Eq. (5). For this reason no\ntransmission happens at !dif not mediated by the m\nmode.\nFig. 1(c) shows a single cavity mode ccoupled to two\nindependent magnetostatic modes, mandn. The fre-\nquency of the mode nis obtained by shifting !mof\n\rbn= (2\u0019)0:4 GHz [see Eq. (3)], corresponding to a \feld\nbias of about 15 mT. This o\u000bset \feld may describe higher\norder modes52,53or two spheres coupled to the same\nmode but through slightly di\u000berent \felds. In fact, as we\nshall see in the following Section, a size e\u000bect related to\nthe diameter of the spheres54{58is the presence of a bias\n\feld. The coupling of this mode to the cavity mode is\narbitrarily set to gcn= (2\u0019)25 MHz to roughly resemble\nthe coupling to higher order modes53.\nThe dispersion relation (d) of Fig. 1 is obtained by\nadding a magnon-magnon coupling gmn= (2\u0019)25 MHz\nbetween the magnetostatic mode nand the Kittel mode\nm, while keeping nuncoupled from the cavity mode c. A\nthird hybrid mode appears, its resonance frequency lies4\nsampleE-\feld\nhigh\nlowc d\nFIG. 2: Section of the cavity body geometrical shape, and\npro\fle of the canddmodes. The sample lies on a maximum\nof the rf magnetic \feld for both modes.\nbetween the two ones of the c-msystem, and dispersively\nshifts them. It is interesting to note that a similar result\ncan be obtained by using a small detuning bnin the pre-\nviousn-c-mcon\fguration but with no gmncoupling and\nsuch that\rm\u001c\rbn<2p\ng2cm+g2cn. This means that an\nidentical dispersion relation results from hybridizing, un-\nder the same static \feld, a cavity and two spheres with\na di\u000berent o\u000bset \feld bY. The central mode is usually\nreferred to as dark mode, which was studied e. g. for\nqubits50and for magnons9,59. The fact that the e\u000bect of\na magnon-magnon coupling results similar to the one of\nan-c-msystem with a small bias bnis further discussed\nand experimentally explored in Sec. III B.\nWhenMmagnon modes are degenerate (i. e. have the\nsame resonant frequency) they couple with a cavity mode\nas a single oscillator, and the vacuum Rabi splitting, the\nfrequency di\u000berence of the two hybrid modes, scales as\nthe root of the sum of the squared couplings. In this case\nthe two-modes system c-mholds. Ideally, the addition\nof more spheres gives the same result, and the splitting\nscales as the square root of the total number of spins.\nHowever, a large increment of the number of spheres in\na single cavity forces us to face the dynamics outlined\nin Fig. 1. These e\u000bects can be combined to obtain more\nconvoluted dispersion relations, which can be explained\nthanks to the understanding of their basic elements pro-\nvided by this Section.\nIII. EXPERIMENTAL RESULTS\nThe photonic resonators of our setup are copper cavi-\nties with a cylindrical body and cone-shaped end caps60.\nThe TM110 mode of a perfectly cylindrical cavity is de-\ngenerate for rotations about its axis, complicating the\ncoupling to the sample. To lift the degeneracy, the sym-\nmetry of the cavity body is broken by two \rat walls\nwhich reduce its diameter, as shown in Fig. 2. As a conse-\nquence, the two polarizations have frequencies which dif-\nfers about 200 MHz, and we identify the two correspond-\ning modes with the canddappearing in Eq. (6). Themagnetic material is Yttrium Iron Garnet (Y 3O5Fe12), a\nferrimagnetic insulator61{64widely known due to its ap-\nplications in microwave electronics65, mainly related to\nits narrow linewidth66{68. The values of resonance fre-\nquencies, linewidths and couplings used in the previous\nSection were chosen to resemble the measured ones of\ncopper cavities and of YIG.\nWe tested several spheres of di\u000berent diameter to infer\nthe maximum size that can be used and characterized\nwithout taking into account perturbative e\u000bects. Except\nfor 0.5 mm and 1 mm-diameter spheres, the others were\nmanufactured on site from a large YIG single crystal.\nAfter cutting the crystal into small cubes, spheres are\nobtained by grinding them with the procedure described\nin Appendix B. In the measurement setup, a fused silica\npipe and PTFE cups hold the spheres in a position close\nto the cavity center and let them rotate freely, guarantee-\ning alignment of their easy axis to the external magnetic\n\feld. Such condition is necessary in order to coherently\nmaximize the coupling strength as prescribed in [41]. In\nthe following tests this condition is always ful\flled, and\nin Sections III A and III B it is analyzed and understood.\nThe magnetic \feld is provided by a superconducting\nmagnet, and care is taken to place the mounted spheres\nat its center: within a cylindrical region of 0.5 cm diame-\nter and 7 cm height the homogeneity of the \feld is better\nthan 7 ppm, guaranteeing no inhomogeneous broadening\nof the electron spin resonance. The linearity and ab-\nsence of o\u000bsets for the relation between the power supply\ncurrent and the magnetic \feld has been veri\fed using a\nparamagnetic BDPA sample, whose g-factor is remark-\nably close to the one of the free electron. Further details\non the experimental apparatus can be found in Appendix\nC.\nThe aim of this analysis is to build an optimally-\ncontrolled HS which embeds the largest possible quantity\nof magnetic material. The measurements in Sec. III A\nand in the \frst part of Sec. III B are obtained at room\ntemperature, while the test of the haloscope transducer\nwas performed at T= 90 mK. At this temperature, quan-\ntum e\u000bects prevail over thermal ones, as kBT <~!,\nwherekBis the Boltzman constant.\nA. Single sphere coupling\nIn the \frst study, single spheres of diameters rang-\ning from 0.5 mm to 2.5 mm have been mounted in two\ndi\u000berent cavities. The TM110 mode frequency is 14.09\nGHz for the \frst cavity and 10.65 GHz for the second,\nwith wavelengths of the rf radiation of 21 mm and 28 mm,\nrespectively. Such values allow us to test the system\nin a regime where the diameter-to-wavelength ratio is\nlarger than 1/10, and dimensional e\u000bects should come\ninto play54.\nFig. 3 shows some results in the form of anticrossing\ncurves. We have veri\fed that the hybridization scales\nas the square root of sample volume, con\frming that all5\n(a) 0.5 mm (b) 1.5 mm\n(c) 2.5 mm (d) 2.5 mm\nFIG. 3: Anticrossing curve of spheres with di\u000berent diameter.\n(a) and (b) show the dispersion relation of single spheres of\ndi\u000berent diameters coupled to the same cavity mode, the cou-\npling strength is obtained by the \ft (dashed lines), and scales\nas expected. (c) and (d) are the anticrossings of the same\nsphere coupled to the TM110 mode of two di\u000berent cavities,\nand the dashed line is the lower frequency hybrid mode. In\nplot (d) one can see a discrepancy between the theory and\nthe result, possibly due to the fact that the sphere diameter\nis larger than one tenth of the microwave \feld wavelength.\nthe spins are behaving coherently, in agreement with the\nTavis-Cummings model. The same behavior is expected\nwhen the number of spheres is increased.\nOne can note that the dispersion relation of Fig. 3(c)\nclosely resembles the one of Fig. 1(b), as the coupling\nstrength between the 2.5 mm sphere and the 10.65 GHz\nmode of the cavity is large enough to involve both the\ncanddpolarizations of this TM110 mode. As expected\nfrom the symmetry of the system, the coupling strengths\ngcmandgdmare about the same. In Fig. 1(b) the cou-\npling to the dmode is much weaker because of the char-\nacteristic of the antennas, discussed in Section II.\nA \frst dimensional e\u000bect is observed when the largest\ndiameter YIG sphere is coupled to the higher frequency\ncavity, i.e. the one with smaller wavelength. It consists\nin an apparent variation of the g-factor, as was \frst ev-\nidenced in [55] for various ferrites. The 2.5 mm sphere's\nelectrons, placed in the 14 GHz cavity, exhibit a gyro-\nmagnetic ratio of 26 GHz/T and not of 28 GHz/T, as re-\nported in Fig. 3(d), which corresponds to an apparent\ng-factor of 1.86. In the \fgure, the dashed line shows\nthe expected behavior of the hybrid mode's resonant fre-\nquency for g= 2, which signi\fcantly di\u000bers from the\nmeasured spectra. The e\u000bect is not present if the same\nsphere is coupled to the lower frequency cavity, as shown\nin Fig. 3(c), were the dashed line is superimposed to the\nmeasured hybrid frequency. The variation of the g-factor\nbmFIG. 4: Relation between the sphere's diameter and the full\nhybridization \feld Bfh, measured with the 14 GHz cavity. The\npositive o\u000bset follows from the zero-\feld splitting of the YIG.\nwas observed only when the sample size exceeds one tenth\nof the radiation \feld wavelength, in agreement with the-\noretical suggestions54.\nA second size e\u000bect has also been evidenced in our\nmeasurements, and it relates the dimension of the ferrite\nto its resonant frequency. Such e\u000bect was \frst observed\nin the \ffties55and described ten years later55,57,58, and\nstates that larger spheres need higher magnetic \felds to\nreach the same resonance frequency. Here, Bfhis the ap-\nplied magnetic \feld value that realizes full hybridization,\nmeasured through the current \rowing in the magnet. In\nperfectly spherical samples, the related theory55,57,58in-\ntroduces a diameter dependent o\u000bset frequency\nbm(\u001e) =!c=\r\u0000Bfh=\u00002\u00192M0\n45\u00152m(5 +\u000fr)\u001e2+b0(7)\nwhereM0'178 mT and \u000frare the YIG saturation\nmagnetization and relative dielectric constant, \u0015mis the\nwavelength associated to the Larmor frequency !m,\u001eis\nthe sphere's diameter, and b0is the zero-diameter o\u000bset.\nFig. 4 shows the measured values for our set of spheres,\ntogether with a linear \ft of the data. Using Eq. (7), the\nmagnetic \feld o\u000bset for null sphere diameter is found to\nbeb0= 14\u00069 mT, compatible with the zero-\feld splitting\ncalculations performed for YIG69. The resulting slope is\n6:2 mT=mm2, from which we extract a value of \u000fr'30,\nwhich di\u000bers by a factor about 2 from the one found in\nthe literature \u000fr'1570,71. It is true that other measure-\nments of the YIG permittivity yielded values higher72or\nlower73than 15, however this discrepancy could be ex-\nplained by some other diameter-dependent size-e\u000bect, for\nwhich indications may already be present in the X-band\nmeasurements of a previous work55. Furthermore, as we\ndo not have a precise control over the sample shape, it is\nnot possible to exclude that our spheres are a\u000bected by\na size-dependent ellipticity that reduces the goodness of\nEq. (7).\nWhilst the second size-e\u000bect is not expected to in\ru-\nence the magnon-to-photon transduction, it is not clear6\nwhether the variation of the gyromagnetic ratio does. In\nthe forthcoming measurements, to avoid this issue, we\nconservatively choose to engineer our HS-based trans-\nducer employing spheres almost with the same diameter,\nnamely\u001e'2:1 mm.\nB. Multiple spheres acting as one\nTo move forward in the realization of the transducer,\none needs to grasp the coupling of two spheres before\nscaling up the system to an arbitrarily large number of\nsamples. We focus on understanding the system disper-\nsion relation in the light of the interactions shown in\nFig. 1, to maintain the optimal control of the HS, and\ndemonstrate its applicability as an axion to electromag-\nnetic transducer.\nFirst of all, as it is clear from Fig. 4, two spheres with\ndi\u000berent diameters will exhibit di\u000berent Larmor frequen-\ncies even when subjected to the same magnetic \feld. An\nHS with two such spheres would have a dispersion re-\nlation as the one described in Fig. 1(d), i.e. equivalent\nto the situation of two samples with di\u000berent zero \feld\nsplitting and with the presence of a dark mode. It follows\nthat, to preserve the control of the HS and the coherent\ncoupling of a large number of spheres, the samples' di-\nameter should be as akin as possible.\nSecondly, the chance of a magnon bouncing through\nmagnetic modes before being converted to photons has\nto be minimum, otherwise it will result in a non-negligible\nsignal loss. Magnons in di\u000berent samples are assumed to\ninteract with each other through a dipole-like potential,\nhence we studied the sphere-sphere interaction by varying\nthe distance between two YIGs as shown in Fig. 5, which\ncan be compared with the theoretical results of Fig. 1(d).\nThe spheres are placed along the cavity axis in both rf\nand dc uniform magnetic \felds. Being the dc \feld aligned\nwith the cavity axis, when this is turned on all the spheres\nwill have their easy axis along the same line, i.e. the\ncavity axis again. Along this direction, the transmission\nspectra are independent of the sphere position provided\nthat this is kept in a central range of about 5 cm length.\nIn our measurements, we will not care about the ab-\nsolute positions of the spheres, albeit in their relative\nseparation, which, as we will see, is strongly in\ruenc-\ning the results. Following the plots of Fig. 5, in (a) and\n(b) we obtain the usual hybridization with one sphere\nby turning on the static \feld. Then, in (c), a second\nsphere is introduced in the system with less than 0.5 mm\nof spacing from the \frst one: the vacuum Rabi splitting\nincreases and extra peaks appear between the two hy-\nbrid modes. By increasing the separation between the\nspheres to 4 mm \u00182\u001e, we obtain the plot (d) showing\na reduced amplitude of the peaks which are not hybrid\nmodes and a slightly smaller Rabi splitting, now closer to\n\u0018p\n2 times the single sphere case. Relying on the model,\ndescribed essentially by Figure 1(d), the plots 5(c) and\nB=0\nB=0.5 T\nB=0.5 T\nB=0.5 T(a)\n(d)(c)(b)YIG65 dBFIG. 5: Coherent coupling of two spheres, and e\u000bect of their\nrelative distance. On the left side of the plots four schematic\ndrawings show the YIGs (black sphere) position inside the\npipe (blue). The pipe is placed on the cavity axis, parallel to\nthe static magnetic \feld. Plots (a) and (b) show the single\nsphere con\fguration without and with magnetic \feld, that\nproduces the hybrid modes indicated by the small vertical\narrows. In (c) one sphere is placed close to the \frst one, with\nthe e\u000bect of increasing the coupling, i.e. the hybrid mode\nseparation, but also of generating a number of dark modes\nin the intermediate frequency interval. It is evident from (d)\nthat the strength of such modes is reduced by tens of dB\nthanks to the distance between the two YIGs, in agreement\nwith what is expected from the model.\n(d) are evidencing the e\u000bects of a magnon-magnon cou-\npling for close spheres: an increase of the splitting due\nto dispersive shifts of the hybrid resonances and a high\ntransmission through the dark modes. These observa-\ntions bolster the assumption that the sphere-sphere cou-\npling is dipole-like, as it depends on their distance and,\nspeci\fcally, it is much weakened when the samples are\nseparated by more than their diameter.\nThe scaling of the system to a larger number of spheres,\nnecessary to realize a more sensitive ferromagnetic halo-\nscope, is straightforward once the behavior of the two-\nYIGs HS is understood. The conditions considered to\nengineer such an enhanced HS are summarized as fol-\nlows: (i) the single sphere diameter is smaller than 1/10\nof\u0015m; (ii) the spheres are free to rotate and align with the\nexternal \feld; (iii) each bare sphere is far in the strong\ncoupling regime with the cavity mode, and is positioned\non the cavity axis; (iv) all the spheres must have the same\ndiameter; (v) the relative distance between the spheres\nshould be about 2 \u001e. Following these constraints we \flled\nthe volume having uniform magnetic \felds, rf of the cav-\nity and dc of the magnet, with ten 2.1 mm diameter YIG\nspheres. Each one placed along a cylindrical fused silica\npipe and separated by thin PTFE caps (see Fig. 7 for a\nsketch and a picture of the system).\nWe now note that the experimental errors of the mea-\nsurements are mostly systematic, and possibly due to\nvariations of the samples positioning in the di\u000berent tests.7\nFIG. 6: Ten spheres HS: simulated (left, where blue-to-light green corresponds to low-to-high transmission) and measured\n(right, bright colors correspond to higher transmission) anticrossing dispersion relation, obtained with the model based on H4\nand a room temperature transmission measurement, respectively. The parameters used to obtained the left anticrossing curve\n!\u0000are detailed in the text.\nAs a consequence, the uncertainties in determining the\neigenfrequencies of the HS are not related to the fre-\nquency resolution, and we estimate them by confronting\nrepeated measurements of the same HS con\fguration,\nwhich are found to be consistent within an 8% error.\nThe dispersion relations of Fig. 6, comparing the the-\nory to the experiment, demonstrate that the resulting HS\ncan be completely described by means of our model based\non the Hamiltonian H4in Eq. (6). The theoretical spec-\ntrum is calculated considering only an external coupling\nto the cavity mode cat 10.65 GHz (see Section II). The\nexperimental spectrum in Fig. 6 (right) was collected\nwith the use of a movable antenna allowing for a good\ncoupling with mode c, and, for symmetry reasons, a much\nweaker one with mode d. Our model yields a sound agree-\nment with the measurements for what concerns mode fre-\nquencies and their relative transmission amplitude. For\nexample, the measured transmission of the dmode is ex-\ntremely low, and the one of the dark mode almost van-\nishes at higher frequencies. In the experimental plot, a\n\ft of the lower frequency hybrid mode !\u0000is also shown\nas a dashed line. The analysis of the anticrossing curve\nshows a Rabi splitting of 528 MHz, correctly scaling as\nthe square root of the number of spheres from the single\nsphere value of about 180 MHz. The resonant frequen-\ncies of the cavity modes are the ones reported in Sec-\ntion II, and remaining parameters are gdm= 264 MHz\nandgmn= 28 MHz. The local discrepancies between the\ntwo plots (i. e. the anticrossing of !\u0000around 0.4 T) can\nbe imputed to spurious magnon modes. This mode pre-\nserves a good antenna output, and is less a\u000bected by the\npresence of other resonances. In fact, the closer cavity\nmodes are the TM11 z, forz= 1;2;:::, which lie at fre-\nquencies higher than !c, making!\u0000well isolated. This\nmode originates from the coherent coupling of all the ten\nYIG spheres, and has an adequate coupling to the dipole\nantenna used to extract the signal, and hence its signal\ntransduction is e\u000ecient.This system, with minor modi\fcations for thermaliza-\ntion (see Appendix C), was then operated at ultra cryo-\ngenic temperatures. The anticrossing pattern obtained\nat 90 mK is reported in Fig. 7(b) and closely resembles\nthe one of Fig. 6, with the increase of the coupling gcm\nas expected from the higher value of the YIG saturation\nmagnetization at such temperatures. The single sphere\nsplitting increases by about 17%74and becomes 210 MHz.\nA \ft with the model based on H4on the experimental\ndispersion relation gives 2 gcm= (2\u0019)638 MHz, to be com-\npared with the value 210p\n10 MHz = 664 MHz. Within\nthe experimental uncertainties, the ten YIG spheres are\ncoherently coupled to the cavity mode, as they e\u000bectively\nreact as a single oscillator. Furthermore, the HS behaves\nas expected also at temperatures where quantum e\u000bects\ndominate over thermal \ructuations7.\nSome other traits of the HS dynamics can be seized by\nlooking at the dispersion relations in Fig. 6 and Fig. 7(b):\nthe coupling of the magnetic m-mode with the d-mode\nof the cavity, and the existence of a dark mode related\nto a magnetic mode n. Evidence for these facts comes\nby looking at the two models described in Fig. 1(b) and\nFig. 1(d), respectively. The explanation for the dark\nmode is not straightforward. In addition to the Kit-\ntel mode of uniform precession, a ferromagnetic sphere\nhas several higher order magnetostatic modes. Following\nthe mode classi\fcation of Walker62, we identify the uni-\nform mode m= (1;1;0), which is degenerate with the\n(4;3;0) one. The latter could be the n-mode producing\nthe dark mode, since sphericity is not perfect in our home\nmade samples, and mandncould actually be slightly de-\ntuned from each other62. However, we are actually not\nfavoring such explanations since single spheres dispersion\ncurves only show standard behavior. The presence of a\nmodencould also be due to the use of spheres with dif-\nferent diameters, as it was discussed in Section III (see\nFig. 4). Also for this case, we believe this is not the cor-\nrect explanation. In fact, by performing measurements8\nPTFE supports\nMicrowave cavity\nAntennas\nYIG spheres\nNbTi magnet\n(a)\n(b) (c)\nFIG. 7: (a) Picture (left) and scheme (right) of the HS tested\nat a temperature of 90 mK and used in a FH75. See text and\nAppendix C for a detailed description of the apparatus. (b)\nAnticrossing curve obtained at 90 mK with the !\u0000frequency\nevidenced with a dashed line, and (c) tuning of the hybrid\nmode frequency !\u0000with a \feld variation of 3 mT.\nwith two spheres we have veri\fed that, for our set, co-\nherent interaction with all the available spins is not real-\nized only when the di\u000berences in sphere diameters satisfy\n\rbm(\u001e)\u001d\rm(see discussion of Fig. 1(d) and in Section\nII). Since all the YIGs used for the ultra cryogenic mea-\nsurement have much close diameters, we believe that the\ndark mode in our anticrossing curves of Fig. 6 and 7(b)\nresults from some residual magnon-magnon coupling. A\nway to certify this could be testing the presence of the\ndark mode with almost identical spheres. In Fig. 7(b) are\npresent two low frequency modes which are absent in the\nroom temperature anticrossing of Fig. 6. This unidenti-\n\fed cavity resonances are visible also at 300 K, but their\nfrequency shifts with the temperature. Even if they ap-\npear in the spectra, they are not coupled to magnon\nmodes and so does not interfere with the FH operation.\nAs the apparatus behavior is consistently described bythe model based on H4, we can now use it to \fnd the\nmaximal transduction bandwidth of the realized set-up\nfor an input on the magnetic mode mand readout on\nthe cavity mode cwith matched antenna. This is the\ngeneral situation of a ferrimagnetic haloscope described\nin the introduction, where a pseudo-magnetic \feld due\nto a dark matter axion excites a magnetic resonance in\nthe material, producing an rf power on the coupled cav-\nity mode. By tuning the Larmor frequency !mthrough\nthe magnetic \feld it is possible to search for dark matter\n\felds oscillating at various frequencies !\u0000, considering\na transduction e\u000eciency given by the relative change of\nmagnon-to-photon conversion probability. We can de-\n\fne the detection bandwidth as the range of frequencies\nfor which transduction e\u000eciency is within 3 dB from the\nmaximum; we mention that the veri\fcation of this ap-\nproach validity will be presented in a forthcoming work76.\nFor the apparatus described by Fig. 7(b) this bandwidth\nis about 400 MHz. In Fig. 7(c) a close up of the tuning\nobtained with a variation of 3 mT of the magnetic \feld is\nshown. The possibility to achieve large and simple tun-\nability for the input frequency is a major advantage of a\nferromagnetic haloscope75, and can be used to simplify\nmany HS-based applications2,40.\nIV. FURTHER DEVELOPMENTS AND\nCONCLUSIONS\nDetection sensitivity of FH is directly related to the\namount of sensitive material. To improve sensitivity,\nwhile keeping the same central working frequency, longer\ncylindrical cavities holding more spheres can be used, and\no\u000b axis loading of the spheres can be implemented. In or-\nder to coherently use all the available spins, each sphere\nmust be strongly coupled to the cavity mode76. This\nresults in the production of ultra strong vacuum Rabi\nsplitting for the complete system, with the drawback of\npossible interference with other cavity modes or between\nhigher order magnetic modes. To avoid such problems,\nwe leverage the fact that as long as the single sphere\ncoupling is larger than the losses the transduction ef-\n\fciency is preserved. Hence, magnetic spheres can be\nplaced in a region of lower rf magnetic \feld of the cavity\nmode, thereby reducing the single sphere coupling to a\nfew times the magnon or cavity linewidth, with the re-\nsult of a smaller total splitting. Moreover, some prelim-\ninary measurements that we have performed show that\nmagnon-magnon interaction is kept small when two or\nmore spheres are placed in a plane perpendicular to the\ncavity axis, thus allowing a small gap separation between\nthe spheres in such direction. It is then possible to fore-\nsee a single microwave cavity with a volume of active\nmaterial one order of magnitude larger than the one em-\nployed in this article, just by \flling the cavity with planes\nof spheres, each plane separated by the required distance\nto avoid mutual interaction. If carefully devised, these\nfuture HSs may still be described by a few oscillators9\nmodels, preserving their optimal control, and remarkably\nincreasing their possible applications.\nIn conclusion, we outlined a model composed of an ar-\nbitrary number of photon and magnon oscillators. The\nanalysis was limited to the experimental case of two cav-\nity modes and two magnetic modes, and the various cou-\nplings among them were separately studied. Afterwards,\nwe make use of a \rexible HS to experimentally reproduce\nthe theoretical results and understand the constraints to\ntake into account when devising a HS with a considerable\nmaterial quantity. The outcome is a recipe that we use to\nbuild the largest optimally-controlled HS to date, which\nwe test at room and at millikelvin temperature. This de-\nvice is used as axion-to-electromagnetic \feld transducer\nin a FH75, paving the way to the use of these devices\nto search for Dark Matter with a cosmologically relevant\nsensitivity. Our results benchmark a starting point for\nfuture large HS designs, and demonstrates that the de-grees of freedom of such a complex system can be reduced\nto an acceptable number.\nAcknowledgments\nThe authors would like to deeply acknowledge Enrico\nBerto for the help in the fabrication of the spheres, and\nRiccardo Barbieri for the initial formulation of the model.\nWe are thankful to Andrea Benato, Fulvio Calaon and\nMario Tessaro for the work on the cryogenics and elec-\ntronics of the setup, and to David Alesini and Mario Zago\nfor the cavity design and realization. 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B , 95:214423, Jun 2017.\nAppendix A: Langevin equations and the e\u000bective\nHamiltonian\nThe dynamics of the system can be described by the\nquantum Langevin equations\n_x=\u0000i(!x\u0000i\rx)x\u0000iX\ny2Ygxy\n2y\n_y=\u0000i(!y\u0000i\ry)y\u0000iX\nx2Xgxy\n2x\u0000iX\ny\u00036=ygyy\u0003\n2y\u0003;(A1)\nwhich are valid for every cavity mode x2Xand magnon\nmodey2Y, and where iis the imaginary unit and\ny\u00032Y. For the sake of this work it su\u000eces considering\nonly the intrinsic decay rates of the cavity modes, as they\ndominate over the ones introduced by external couplings\nif we assume that the setup's antennas are weakly cou-\npled. Eq.s (A1) may be recast in their matrix form as\n_W=\u0000iHW; (A2)\nwhereW=fX;Yg=fx1;x2;:::;x N;y1;y2;:::;y Mg,\nandHis the matrix reported in Eq. (2) and used through-\nout this work, in particular to determine sx1(!;!m) in\nEq. (5). Here we note that it is experimentally challeng-\ning to have an antenna coupled to a single mode, mak-\ning our calculated transmission spectra an incomplete de-\nscription of the apparatus. Nevertheless, as is detailed in\nSection III and Fig. 2, this approximation holds thanks to\nthe geometric con\fguration of the two considered cavity\nmodes. We stress that one can get a more realistic de-\nscription of the HS spectra by summing over the possible\ntransmissions due to external couplings to other modes\nas\nstot(!;!m) =X\nx2X\u0011xsx(!;!m); (A3)\nwhere\u0011xis a constant accounting for the coupling\nstrength between the antenna and the di\u000berent cavity\nmodes. This is an e\u000bective approximation which resists\nif all the external couplings are weak, and do not in\ruence\nthe linewidths of the modes. An even better description\nconsists in the addition of a mode-dependent dissipation\nin Langevin equations to account for the coupling of the\nantennas, see for example Ref. [34], which naturally yields\nthe increase of the modes' decay rate.12\nFIG. 8: The instrument used to produce the spheres (left).\nImprovement of linewidth with subsequent steps for a batch\nof six spheres (right).\nAppendix B: Spheres production\nSuper polished YIG spheres show the sharpest mag-\nnetic resonance linewidth66{68. Material purity and sur-\nface roughness are the two key elements to be cured\nin order to obtain the best linewidth values at all\ntemperatures77. Our spheres have been produced on site\nto have better control of all the relevant parameters. We\nbuy large single crystal high purity YIG cylinders, nor-\nmally a few cm length and 5 to 7 mm diameter, and\ncut them into 3 mm size cubes. Each cube then follows\na grinding procedure to obtain spheres of about 2.1 mm\ndiameter. The grinder is based on a high frequency rotat-\ning plate with replaceable silicon carbide (SiC) sandpa-\nper foils where the YIG piece get abraded while tumbling\ninside a plastic holder (see the left plot in Fig. 8). Start-\ning from a P800 SiC paper, \fner and \fner grit size are\nused in sequence up to the \fnal P4000. The duration\nof each step has been optimized after several trials. The\nlast step consists of four passages with P4000 virgin SiC\npaper, each lasting half an hour. At the end of this pro-\ncess, we get spheres with the nominal diameter within a\nfew percent and linewidths of about 2 MHz. A \fnal 24\nhours polishing with Alumina-based suspension on a low\nfrequency rotating system results in a linewidth slightly\nabove 1.3 MHz at room temperature (see the right plot\nof Fig. 8).\nAppendix C: Hot-bore superconducting magnet and\nmillikelvin temperature hybrid system\nThe setup used for testing the single and multiple-\nspheres con\fgurations includes a superconducting mag-\nnet, described in Section III, with a hot bore in which\nthe HS is inserted. This custom setup was built to test\nseveral HSs per-day, which is incompatible with multiple\nrefrigeration cycles. The cavity is mounted on a plastic\nsupport that aligns its axis with the one of the magnet,\nand a PTFE support holds the pipe which positions the\nYIG spheres on this same axis.\nAt milli-Kelvin temperatures, the cavity is anchored\nto the mixing chamber of a dilution refrigerator with two\nlarge copper bars. Its temperature is monitored witha thermometer attached to the cavity body. The setup\nmust ensure a proper thermalization of the YIG spheres,\nwhich for this reason are contained in a pipe \flled with\ngaseous helium. The preparation of the fused silica pipe\nis done at room temperature as follows. A vacuum sys-\ntem is designed in such a way to remove the air from the\npipe, which is then immersed in a 1 bar helium controlled\natmosphere and closed. In this way the pipe is \flled with\nhelium, and can be sealed by using a copper plug glued\nwith Stycast. First the sealing is tested without the sam-\nples by measuring the frequency shift of the TM110 mode\nof the cavity-pipe system with and without helium. The\nfrequency is measured with the helium-\flled pipe, which\nis then immersed in liquid nitrogen and again placed in\nthe cavity. Re-measuring the same frequency excludes\nthe presence of leaks. The plug used to close the pipe\nis eventually anchored to the cavity body with a copper\nrod, granting a good thermalization of the whole system."    },    {        "title": "2209.14643v2.Strong_to_ultra_strong_coherent_coupling_measurements_in_a_YIG_cavity_system_at_room_temperature.pdf",        "content": "Strong to ultra-strong coherent coupling measurements in a YIG/cavity system at\nroom temperature\nGuillaume Bourcin,1, 2,\u0003Jeremy Bourhill,3Vincent Vlaminck,1, 2and Vincent Castel1, 2,y\n1IMT Atlantique, Technopole Brest-Iroise, CS 83818, 29238 Brest Cedex 3, France\n2Lab-STICC (UMR 6285), CNRS, Technopole Brest-Iroise, CS 83818, 29238 Brest Cedex 3, France\n3Quantum Technologies and Dark Matter Labs, Department of Physics,\nUniversity of Western Australia, 35 Stirling Hwy, 6009 Crawley, Western Australia.\n(Dated: April 27, 2023)\nWe present an experimental study of the strong to ultra-strong coupling regimes at room tem-\nperature in frequency-recon\fgurable 3D re-entrant cavities coupled with a YIG slab. The observed\ncoupling rate, de\fned as the ratio of the coupling strength to the cavity frequency of interest, ranges\nfrom 12% to 59%. We show that certain considerations must be taken into account when analyzing\nthe polaritonic branches of a cavity spintronic device where the RF \feld is highly focused in the\nmagnetic material. Our observations are in excellent agreement with electromagnetic \fnite element\nsimulations in the frequency domain.\nKeywords: cavity spintronics, cavity-spintronics, Yttrium Iron Garnet, ultra-strong coupling\nI. INTRODUCTION\nCavity spintronics is an emerging research \feld that\ninvestigates light-matter interactions within magnetism,\nspeci\fcally the interactions between cavity photons and\nthe quanta of spin waves based on the magnetic dipole\ninteraction { magnons. At the core of cavity spintron-\nics are cavity-magnon polaritons (CMPs) which are the\nassociated bosonic quasiparticles, i.e., hybridized cavity-\nmagnon-photon states in the strong coupling regime.\ncavity spintronics has drawn a growing interest since the\n\frst theoretical prediction in 2010 [1], and then shortly\nafter experimental demonstration of CMPs at both mil-\nlikelvin (mK) temperatures [2, 3] and room temperature\n(RT) [4]. Cavity spintronics display a broad range of ap-\nplicability for quantum information systems and RF de-\nvices such as adjustable sensitive \flter [5{7], isolators or\ncirculators [8], gradient memories [9] and for engineering\nchiral states of electromagnetic radiation [10, 11].\nIn a cavity{magnon system, when the magnon fre-\nquency is tuned by an externally applied static magnetic\n\feld towards the cavity resonance frequency, the sys-\ntem undergoes hybridization (e.g. forms a CMP) with\na characteristic anti-crossing signature in the dispersion\nspectrum. The interaction is quanti\fed by the coupling\nstrengthg=2\u0019and by its ratio g=!with the cavity fre-\nquency!=2\u0019. When the coupling g=2\u0019is larger than\nthe systems losses, there exist three di\u000berent coupling\nregimes. These have commonly been referred to as: (i)\nStrong coupling when g=! < 0:1, (ii) Ultra-Strong Cou-\npling (USC) for 0 :1< g=! < 1, and (iii) Deep-Strong\nCoupling (DSC) for g=!> 1, a regime that still remains\nlargely unexplored. The value of g=!= 0:1 is considered\nas a threshold between the SC and USC regimes, but\n\u0003guillaume.bourcin@imt-atlantique.fr\nyvincent.castel@imt-atlantique.frthis is only a historical convention, supposedly indicat-\ning the cuto\u000b beyond which the coupling rate grepresents\na \\sizeable fraction\" of the system energy and therefore\ncannot be deemed to be a slowly rotating term in the\nrotating wave approximation.\nThe USC regime was predicted theoretically in inter-\nsubband cavity polaritons in 2005 [12] and \frst observed\nin 2009 [13] in n-doped GaAs quantum wells embedded\nin a microcavity, with g=! = 0:11. Since this experi-\nmental observation, several research groups have exper-\nimentally achieved the USC regime [14, 15] in di\u000berent\nsystems such as superconducting circuits [16], polaritons\n[17], and optomechanics [18]. So far, the USC regime in\ncavity spintronics has been experimentally achieved at\nlow temperature [19{25] and investigated theoretically\n[26, 27].\nVery recently, Golovchanskiy et al. [25] proposed an\napproach to achieve on-chip USC hybrid magnonic sys-\ntems reaching g=! = 0:6 and based on superconduct-\ning/insulating/ferromagnetic multilayered microstruc-\ntures operating below 10 K. They highlighted in particu-\nlar the drastic failure of currently adopted models in the\nUSC regime.\nHere, we present measurements and simulations of a\nrecon\fgurable hybrid system that allows the study of the\ntransition from the SC to USC regimes at room temper-\nature in the 0 :1\u000015 GHz frequency range. We utilize a\nmagnetic \feld-focusing double-post re-entrant cavity \frst\ndescribed by Goryachev et al. [19]. A set of three dif-\nferent resonators (by their dimensions and posts shape)\nallow us to follow the evolution of the coupling strength\nthrough USC regime (starting from the SC/USC limit).\nWith these results, we con\frm that it is necessary to add\nan extra term in the expression of the Ferromagnetic Res-\nonance (FMR) frequency equation to accurately describe\nthe observed hybridization (measurements and simula-\ntions) with the commonly used Dicke model [28]. We\nshow that this additional term does not depend on the\ncoupling rate but on the level of con\fnement of the RFarXiv:2209.14643v2  [quant-ph]  26 Apr 20232\nmagnetic \feld in the magnetic material. Moreover, this\nadded term can be negligeable in the SC regime, while it\nis essential in the USC regime.\nII. HYBRID SYSTEM DESCRIPTION\nThe hybrid system presented here is made of a com-\nmercial single crystal of YIG (Yttrium Iron Garnet,\nY3Fe5O12) and a modi\fed re-entrant cavity. The YIG\nis a slab of 3.82\u00026.09\u00020.61 mm3.\nThe multiple post re-entrant cavity [19] is a unique\ntype of microwave cavity. There are two \frst-order reso-\nnant modes, termed the Dark Mode (DM) and the Bright\nMode (BM). Both contain the electric \feld of the mode\nbetween the top of the post and the lid of the cavity.\nFor the DM (as shown in Fig. 1 (a)), the RF electric\n\felds ( e-\felds) focused above the two posts are in-phase,\nresulting in the circulating RF magnetic \felds ( h-\felds)\ndestructively interfering in the region between the posts\n(hence \\dark\"), whilst the opposite is true for the BM\n(as shown in Fig. 1 (b)). The advantages of such a\ncavity are three-fold: \frst, the highly localized electric\n\feld results in extremely large frequency sensitivity to\nany perturbations inside this region (displacement of the\ncontainment area or modi\fcation of the dielectric mate-\nrial). Secondly, the physical separation of the electric and\nmagnetic \felds permits separate interaction with both\nmagnetic and electrically sensitive devices at di\u000berent\nlocations, potentially simultaneously. Finally, the mag-\nnetic \feld focusing between the posts results in extremely\nstrong interactions with any magnetically susceptible ma-\nterial placed there.\n/vectorh\n/vectore\n/uni0000000b/uni00000044/uni0000000c\n/vectorh\n/vectore\n/uni0000000b/uni00000045/uni0000000c\n/uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000015 /uni00000013/uni00000011/uni00000017 /uni00000013/uni00000011/uni00000019 /uni00000013/uni00000011/uni0000001b /uni00000014/uni00000011/uni00000013|h|2/uni00000003/uni0000003e/uni00000044/uni00000011/uni00000058/uni00000011/uni00000040/uni0000005b/uni0000005c\n/uni0000005d/vectorH0\n/uni00000013/uni00000015/uni00000017/uni00000019/uni0000001bω/2π/uni00000003/uni0000003e/uni0000002a/uni0000002b/uni0000005d/uni00000040\n/uni0000000b/uni00000046/uni0000000c\n/uni00000025/uni00000030\n/uni00000027/uni00000030\n/uni00000013 /uni00000015/uni00000013 /uni00000017/uni00000013 /uni00000019/uni00000013 /uni0000001b/uni00000013 /uni00000014/uni00000013/uni00000013\nd/uni00000003/uni0000003e/uni00000097/uni00000050/uni00000040/uni00000013/uni00000011/uni00000017/uni00000013/uni00000011/uni00000018/uni00000013/uni00000011/uni00000019/uni00000013/uni00000011/uni0000001a/uni00000013/uni00000011/uni0000001bg/ω\n/uni0000000b/uni00000047/uni0000000c\n/uni00000013/uni00000011/uni0000001a/uni0000001a/uni00000013/uni00000011/uni0000001a/uni0000001b/uni00000013/uni00000011/uni0000001a/uni0000001c/uni00000013/uni00000011/uni0000001b/uni00000013\nη\nFIG. 1. Re-entrant cavity with electromagnetic simulation\noverlay where jhj2is displayed for the \frst two photonic\nmodes: (a) the DM; and (b) the BM.\nThe interaction between a single cavity mode and the\nFMR can be described by two coupled harmonic oscilla-\ntors, for which the Hamiltonian is read as:\n^H=^Hc+^Hm+^Hint; (1)\nwhere ^Hc=~!^cy^crepresents the photonic mode, ^Hm=\n~!m^by^bthe magnon mode, and ^Hintis the Zeeman inter-action [22], which describes the coupling between the two\noscillators for this system. ~is the reduced Planck con-\nstant,!c(m)=2\u0019is the cavity (magnon) frequency, and ^ cy\nand ^c(^byand^b) are the creation and annihilation cavity\n(magnon) operators, respectively.\nFollowing [22], and as demonstrated in appendix A, the\nphysics system is described by the Dicke model reading\nas:\n^H=~=!^cy^c+!b^by^b+g(^cy+ ^c)(^by+^b): (2)\nAn easy way to solve eigenvalues of the Dicke Hamilto-\nnian is to use the Rotating Wave Approximation (RWA)\nwhere the counter-rotating terms, ^ cy^byand ^c^b, are ne-\nglected. In the case of a system being in the USC regime,\nit is well known [14, 15] that this approximation no longer\ndescribes this system. Using the Hop\feld-Bogoliubov\ntransformation allows one to solve for the system eigen-\nfrequencies whilst considering co-rotating, ^ cy^band ^c^by,\nand counter-rotating terms:\n!\u0006=1p\n2r\n!2+!2m\u0006q\n(!2\u0000!2m)2+ 16g2!!m:(3)\nMoreover, the coupling strength is de\fned as [29]:\ng\n2\u0019=\r\n4\u0019\u0011r\n\u00160S~!\nVm=\u0011p!\r\n4\u0019r\u0016\ngl\u0016B\u00160~ns; (4)\nwhere\r= 2\u001928 GHz.T\u00001is the gyromagnetic ratio for\nYIG,gl= 2 is the Land\u0013 e g-factor for an electron spin, \u00160\nis the vacuum permeability, \u0016Bis the Bohr magneton,\n\u0016= 5\u0016Bis the magnetic moment of the sample, ns=\n4:22\u00021027m\u00003is the spin density for YIG [29], and \u0011\nis the \flling factor, where\n\u0011=vuuut\u0010R\nVmh\u0001^xdV\u00112\n+\u0010R\nVmh\u0001^ydV\u00112\nVmR\nVcjhj2dV: (5)\nThe \flling factor describes the proportion of the h-\feld\n(x- and y-axis components), perpendicular to the static\nmagnetic \feld ( H-\feld), named H0in Fig. 1 compared\nto the h-\feld for all directions inside the entire cavity\nvolumeVc.\nIII. OPTIMIZATION\nAn appropriate optimization of the cavity allows\none to maximize the coupling and to obtain a quasi-\nhomogeneous h-\feld inside the YIG slab. With the use of\nFinite Element Modeling (FEM) and following the pro-\ncedure described by Bourhill et al. [29], we were able\nto precisely predict and therefore optimize prior to con-\nstruction, the cavity frequency, frequency tuning range,3\nand the coupling strength considering equation (4).\nThe optimization of the cavity design was based on\nthe maximization of the \flling factor \u0011and the h-\feld\nhomogeneity at the \frst BM inside the YIG slab. For a\ncorrect distribution of the RF \feld inside the cavity (seen\nas a Perfect Electric Conductor, PEC), it is necessary\nto consider the electrical property of the YIG, namely\na relative dielectric permittivity of 15. Dynamic mag-\nnetic properties are not useful at this stage and instead\nof considering the magnetic permeability with the Polder\ntensor, we consider it as that of vacuum.\nThere exist only three free parameters for the optimiza-\ntion of the hybrid system, two for the size of the posts,\nthe widthWand the length L, and one for the cavity, the\nradiusR. The other parameters such as the height of the\ncavity and the distance between the posts were \fxed by\nthe constraints imposed by the YIG dimension and the\ncavity manufacturing accuracy. The optimization step is\ndescribed in appendix B, and the optimized values are\nW= 0:6 mm,L= 6 mm, and R= 12 mm.\n/uni00000013 /uni00000015/uni00000013 /uni00000017/uni00000013 /uni00000019/uni00000013 /uni0000001b/uni00000013 /uni00000014/uni00000013/uni00000013\nd/uni00000003/uni0000003e/uni00000097/uni00000050/uni00000040/uni00000013/uni00000011/uni00000017/uni00000013/uni00000011/uni00000018/uni00000013/uni00000011/uni00000019/uni00000013/uni00000011/uni0000001a/uni00000013/uni00000011/uni0000001bg/ω\n/uni0000000b/uni00000044/uni0000000c\n/uni00000013/uni00000011/uni00000013 /uni00000017/uni00000011/uni00000013 /uni00000019/uni00000011/uni00000013 /uni0000001b/uni00000011/uni00000013 /uni00000014/uni00000013/uni00000011/uni00000013 /uni00000014/uni00000011/uni0000001a/uni00000014/uni00000011/uni00000014\nω/2π/uni00000003/uni0000003e/uni0000002a/uni0000002b/uni0000005d/uni00000040/uni00000013/uni00000011/uni00000013/uni00000013/uni00000013/uni00000011/uni00000015/uni00000018/uni00000013/uni00000011/uni00000018/uni00000013/uni00000013/uni00000011/uni0000001a/uni00000018/uni00000014/uni00000011/uni00000013/uni00000013/uni00000014/uni00000011/uni00000015/uni00000018/uni00000014/uni00000011/uni00000018/uni00000013g/ω\n/uni00000036/uni00000026/uni00000038/uni00000036/uni00000026/uni00000027/uni00000036/uni00000026/uni0000000b/uni00000045/uni0000000cη/uni00000003/uni00000020/uni00000003/uni00000014\nη/uni00000003/uni00000020/uni00000003/uni00000013/uni00000011/uni0000001a/uni0000001c\n/uni00000028/uni00000030/uni00000003/uni00000036/uni0000004c/uni00000050/uni00000058/uni0000004f/uni00000044/uni00000057/uni0000004c/uni00000052/uni00000051/uni00000013/uni00000011/uni0000001a/uni0000001a/uni00000013/uni00000011/uni0000001a/uni0000001b/uni00000013/uni00000011/uni0000001a/uni0000001c/uni00000013/uni00000011/uni0000001b/uni00000013/uni00000013/uni00000011/uni0000001b/uni00000014/uni00000013/uni00000011/uni0000001b/uni00000015\nη\n/uni00000013 /uni00000015/uni00000018 /uni00000018/uni00000013 /uni0000001a/uni00000018 /uni00000014/uni00000013/uni00000013\nd/uni00000003/uni0000003e/uni00000097/uni00000050/uni00000040/uni00000013/uni00000015/uni00000017/uni00000019/uni0000001bω/2π/uni00000003/uni0000003e/uni0000002a/uni0000002b/uni0000003d/uni00000040\n/uni00000025/uni00000030\n/uni00000027/uni00000030\nFIG. 2. (a) Evolution of the ratio g=!(with!=!BM) in\nblue and\u0011in red versus dfor Eigen-Mode simulations (EM).\nInset: Evolution of simulated DM and BM frequencies versus\nd. (b) Evolution of g=!versus the BM frequency for \u0011= 1\nand\u0011= 0:79.\nThe simulated evolution of the two eigenmodes (DM\nand BM) are shown in the inset of Fig. 2 (a) with respect\nto the distance dbetween posts and the lid of the cavity,\nwith a range from 1 to 100 \u0016m. Decreasing dwill decrease\nthe frequency of the eigenmodes and the frequency dif-\nference between the BM and the DM. Fig. 2 (a) shows\nelectromagnetic simulation results for \u0011(right y-axis) andg=!(left y-axis) versus dfor a cavity with the optimized\ndimensions, where !=!BMthe frequency mode of in-\nterest in our study. \u0011is maximized for d= 9\u0016m. The\nvariation of \u0011over this range of dvalues is only is 2.7%,\ntherefore we may consider it more or less invariant. The\ntuneability of the distance dplays a role on the g=!ratio\nas shown in Fig. 2 (a). Indeed, !=2\u0019is decreasing with\nd, and\u0011is remaining almost constant. Considering Eq.\n(4),g=2\u0019is a function of \u0011and the square root of !.\nTherefore, the ratio g=!will increase with the inverse of\nthe square root of !from 36.8 to 80.5% as ddecreases\nfrom 100 to 1 \u0016m.\nFig. Fig. 2 (b) illustrates the SC to DSC transition for\nYIG with the frequency dependence of g=!. The blue\ndots correspond to the values extracted from EM simu-\nlation already discussed in Fig. 2 (a) and the solid line\ndependencies are based on equation (4) for two constant\nvalues of\u0011, 0.79 (blue) and 1 (green). The magnetic\nproperties of YIG require working in a speci\fc frequency\nrange in order to explore the DSC. For the maximum\nreachable value of \u0011(green line), which corresponds to\nthe entire h-\feld perpendicular to H0and fully con\fned\ntoVm, DSC is possible when the magnons are coupled\nto a microwave mode below 1.72 GHz [29]. In our case\n(with\u0011close to 0.79), DSC is achievable but at a smaller\nresonant frequency (1.07 GHz). Note that the optimized\ncavity con\fguration of this work does not allow to reach\nthe DSC due to the presence of the dark mode which con-\ntaminates the low frequency response and the di\u000eculty\nto control distance dlower than 3 \u0016m.\nIV. RESULTS AND DISCUSSION\nA. Simulation details\nTo compare the experimental results, simulations in\nthe frequency domain (FD), solving for the S21scatter-\ning parameter were conducted for di\u000berent values of d\nfrom 2 to 100 \u0016m. For these simulations, we consid-\nered the excitation probes and hence the coupling losses.\nLosses due to \fnite conductivity of the cavity walls are\nalso taken into consideration.\nThe static and dynamic magnetic properties of YIG\nare used to solve the frequency response of the entire\nsystem as a function of the applied magnetic \feld. The\nspin dynamics of ferrimagnetic systems can be described\nby the Landau-Lifshitz-Gilbert (LLG) equation and the\nfrequency dependence of the coupled dynamics can be\naccurately estimated by using a linear solution of the\nLLG equation in solving Maxwell's equations. Some con-\nsideration regarding the shape of the YIG sample must\nbe taken into account. The FMR dispersion for a rela-\ntively thick slab geometry requires careful consideration.\nBased on the works of Kittel [30], R. I. Joseph and E.\nSchl omann [31], the demagnetizing \feld expression has\nbeen adapted to our non-ellipsoidal sample of YIG (as\ndescribed in Appendix B). From these results, it is de-4\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\n/uni0000000b/uni00000044/uni0000000c\ng//uni00000003/uni00000020/uni00000003/uni00000013/uni00000011/uni00000016/uni00000019\n/uni00000030/uni00000048/uni00000044/uni00000056/uni00000058/uni00000055/uni00000048/uni00000050/uni00000048/uni00000051/uni00000057/uni00000056\n/uni00000015/uni00000017/uni00000019/uni0000001b/uni00000014/uni00000013/uni0000000b/uni00000046/uni0000000c\ng//uni00000003/uni00000020/uni00000003/uni00000013/uni00000011/uni00000017/uni00000019\nd/uni00000003/uni00000020/uni00000003/uni00000014/uni00000013/uni00000003/uni00000097/uni00000050\n/uni00000015/uni00000017/uni00000019/uni0000001b/uni0000000b/uni00000048/uni0000000c\ng//uni00000003/uni00000020/uni00000003/uni00000013/uni00000011/uni00000018/uni0000001c\nd/uni00000003/uni00000020/uni00000003/uni00000016/uni00000003/uni00000097/uni00000050\n/uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000014 /uni00000013/uni00000011/uni00000015 /uni00000013/uni00000011/uni00000016 /uni00000013/uni00000011/uni00000017 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\n/uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000014 /uni00000013/uni00000011/uni00000015 /uni00000013/uni00000011/uni00000016H0/uni00000003/uni0000003e/uni00000037/uni00000040/uni00000015/uni00000017/uni00000019/uni0000001b/uni00000014/uni00000013/uni0000000b/uni00000047/uni0000000c\n/uni00000013/uni00000011/uni00000013/uni00000013 /uni00000013/uni00000011/uni00000013/uni00000018 /uni00000013/uni00000011/uni00000014/uni00000013 /uni00000013/uni00000011/uni00000014/uni00000018H0/uni00000003/uni0000003e/uni00000037/uni00000040/uni00000015/uni00000017/uni00000019/uni0000001b/uni0000000b/uni00000049/uni0000000c\n/uni00000014/uni00000017/uni00000013\n/uni00000014/uni00000015/uni00000013\n/uni00000014/uni00000013/uni00000013\n/uni0000001b/uni00000013\n/uni00000019/uni00000013\n/uni00000017/uni00000013\n/uni00000015/uni00000013\n/uni00000036/uni00000003/uni0000003e/uni00000047/uni00000025/uni00000040\nfDM\nfBM\nfFMR\nf±\nFIG. 3. Transmission spectra versus frequency and H0for (a), (c), (e) measurements at RT, and (b), (d), (f) simulations.\nComparison spectra between measurement and simulations are shown for di\u000berent distances das labelled. A \ft with the Dicke\nmodel with a shifted magnon frequency is shown superimposed on (c) and (d) where the FMR frequency ( fFMR =!FMR=2\u0019)\nis shown in black, the DM frequency ( fDM=!DM=2\u0019) in red, the BM frequency ( fBM=!BM=2\u0019) in orange and the two\npolariton frequencies ( f\u0006=!\u0006=2\u0019) in white.\ntermined that the demagnetizing \feld is signi\fcantly dif-\nferent from the thin-\flm form, and therefore for accurate\nsimulations proper consideration of this di\u000berence must\nbe taken into consideration. Hence, the e\u000bective static\nmagnetic \feld in the YIG is di\u000berent from the applied\none and read as:\nHi=H0\u0000Nzz(x;y;z )M0 (6)\nwhereHiis the internal static magnetic \feld along the\nz-axis, and Nzzis the spatially dependent demagnetizing\ncomponent along the z-axis, and is describe in Eq. C3 in\nappendix C.\nB. Experimental set-up\nTo reach the speci\fcations described above, an alu-\nminum cavity with an accuracy of 20 \u0016m has been ma-\nchined.\nFor the applied static magnetic \feld, we used an elec-\ntromagnet where the produced \feld is aligned along the\nz-axis (see Fig. 1), in the direction of the height of the\nposts. H0aligns all the spin moments along the z-\naxis and to saturate the macroscopic YIG magnetization.\nWith the shape of the cavity, the h-\feld for the BM, con-\nsidered as the perturbative \feld, is only along the x-axis\ninside the YIG slab between the two posts, as shown in\nFig. 1 (b) due to the constructive interference of the twoh-\felds around each post. A gaussmeter allows one to\nmeasure in situH0magnitudes. S-parameters are mea-\nsured with a two-port Vector Network Analyzer (VNA),\nwith the magnitude and phase of the scattering parame-\nters recorded between 0.1 to 15 GHz with an input power\nof\u000010 dBm. All measurements are conducted at RT\nThe magnitude of the S21transmission spectra as a\nfunction of H0are displayed in Fig. 3 for measurement\nand simulation with di\u000bering sized gaps between the top\nof the posts and the roof of the cavity. Experimentally,\nthis is varied by using di\u000berent cavity lids which had\nrecesses of di\u000bering heights machined into them.\nC. Results\nMeasurement and simulation results of magnetic spec-\ntroscopy of the cavity magnon system are shown in Fig. 3\nas the \frst and second row, respectively, for di\u000berent val-\nues ofd. Each column represents a comparison between\na measurement and a simulation with a distance close to\nthe measured value. The latter can be determined by the\nunperturbed value of fDM, which acts as a calibration for\nd.\nThe external magnetic \feld was always applied sym-\nmetrically for negative and positive values. This allows\nto improve the \ft accuracy on measurements, because we\nhave twice as many data points. All measurements with5\ncomplete frame are shown in appendix F.\nWe can easily distinguish the two hybrid eigenfrequen-\nciesf+=!+=2\u0019(for the higher branch) and f\u0000=!\u0000=2\u0019\n(for the lower branch) from either side of the BM fre-\nquency. It should be noted that at low H0values the BM\nis not visible, whilst we can clearly see the DM which is\nthe lowest frequency mode and has a negligible coupling\nwith the magnon mode, hence is constant versus H0.\nSome minor discrepancies between simulation and ex-\nperiment should be pointed out: ( i) an in\rection point\non the curvature of the upper CMP in frequency at low\nH0(observed only in the USC regime) for measurements,\nappearing neither in simulation nor analytic \fts; ( ii) anti-\nresonances only appearing either in measurement, the\nhorizontal one around 4.3 GHz in Fig. 3 (a) and (e),\nor in simulation with a S-like shape, around 10, 4, and\n2 GHz in respectively Fig. 3 (b), (d) and (f). Let us\nnotice that this anti-resonance does not appear in mea-\nsurements when a cavity mode is overlapping with this\ntransmission dip, as shown for d= 10 µm in Fig. 3 (c) and\nFig. 9 (e), and for d= 116 µm in Fig. 9 (a); ( iii) another\nmagnon mode exists near the upper CMP in simulations.\nIt is clear that it is another magnon mode because its\nH-\feld's frequency dependence does not change as dis\nvaried.\nDi\u000berences given in the two last point could be explained\nby the fact that the YIG sample is a perfect rectangu-\nlar prism in simulation whereas the real sample is not.\nThe imperfections of the YIG geometry could result in a\nweak transmission, which could be not detected in mea-\nsurement.\nDespite these minor deviations, the agreement between\nsimulations and measurements on the magnon-photon\ncoupling and the resulting CMPs is excellent. In particu-\nlar, we validated the spatial distribution of the demagne-\ntizing \feld, hence the expression of the FMR for a slab,\nand the ability of the Maxwell's equations to describe the\nsystem. This permits one to conduct a simulation with a\nmagnetic \feld larger than experimentally possible in or-\nder to extract the BM frequency. Indeed, it is impossible\nto measure the unperturbed BM frequency fBMin the\nUSC regime even when applying a high magnetic \feld\nnear to 2 T.\nD. Model Description\nIn the USC regime, the Tavis-Cummings model be-\ncomes no longer applicable [13, 32], as g=!> 0:1 leads to\na failure of the rotating wave approximation as the inter-\naction term of the Hamiltonian can no longer be assumed\nto be \\slowly rotating\" compared to the system terms.\nThe standard model for cavity magnonics is the Dicke\nmodel (see Eq. (2)). However, we have noticed that in\nthe coupling regime of our system, even the Dicke model\ncannot describe observed polariton frequency dispersion\nfor measurements and simulations, as shown in Fig. 7 in\nappendix D. Another standard model describing light-matter interactions is the Hop\feld model [33], similar\nto the Dicke model with an additional diamagnetic term.\nThis well known model neither \ft the measured data with\nthe use of the Hop\feld model, as shown in appendix E.\nTo remedy these issues, it has been proposed to modify\nthe Dicke model with the addition of a H-\feld in the\nFMR dispersion equation [24]. We also modi\fed the term\nof the FMR frequency dependence in equation (3) to:\n!m!!m+ \u0001m (7)\nwhere \u0001m= 2\u0019f\u0001is a frequency shift, which will be\nfurther discussed in section IV E. This modi\fed Dicke\nmodel was found to \ft best the experimental and simu-\nlation spectra, as seen in the white dash lines of Fig. 3 for\nd= 75\u0016m in (a) and (b), d= 10\u0016m in (c) and (d) and\nd= 3\u0016m in (e) and (f). Measurement \ft, shown in Fig. 3\n(a), (c), and (e), is achieved with the BM frequency fBM\n(in orange), the coupling strength g=2\u0019, and the added\nfrequencyf\u0001as \ftting parameters. For simulation \ft,\nshown in Fig. 3 (b), (d), and (f), the BM frequency is\nconsidered as \fxed parameter. Indeed, simulations were\nperformed at an arti\fcial high H-\feld (H0= 10 T), in\norder to tune the magnon mode many orders of coupling\nstrength away, and clearly distinguish the two photonic\nmodes.\nAn o\u000bset far detuned from the BM frequency at a zero\nH-\feld arises for high g=! when the FMR is shifted.\nWhen the FMR is not shifted, in the standard Dicke\nmodel, no BM frequency detuning at zero H-\feld exists\nfor anyg=!value. See appendix G for more details about\nthe frequency detuning at zero H-\feld.\nAll values of \ft parameters, for measurements and sim-\nulations, are available in appendix F, and are pooled in\nFig. 4. For the measurements (shown in blue), the dis-\ntancedhas been estimated from the measured DM fre-\nquency. The \ftted BM frequencies of the measurements\nare in good agreement with simulations (shown in black\nin the inset of Fig. 4). Regarding the coupling strength\ng=!, we achieve a ratio g=!ranging from 0.35 to 0.59,\ncorresponding to d= 116\u0016m tod= 4\u0016m, respectively.\nAs mentioned in section III, the values of g=!are di\u000ber-\nent from the optimization step ones (dotted red curve),\nmainly due to the di\u000berent estimated frequencies, shown\nin the inset. Once again, the correlation between \ftted\nsimulations and measurements for the ratio g=!are also\ngood. This clearly demonstrates the validity of the sim-\nulations.\nE. Discussion\nWe discuss here the physical meaning of the frequency\nshift in the modi\fed Dicke model. For a deeper un-\nderstanding of the behavior of this added term, we in-\nvestigated the transition between the SC and the USC\nregimes. In order to study a wide range or g=!values,\nwe have used two other cavities with the same YIG sam-6\n/uni00000013 /uni00000015/uni00000013 /uni00000017/uni00000013 /uni00000019/uni00000013 /uni0000001b/uni00000013 /uni00000014/uni00000013/uni00000013 /uni00000014/uni00000015/uni00000013\nd/uni00000003/uni0000003e/uni00000097/uni00000050/uni00000040/uni00000013/uni00000011/uni00000017/uni00000013/uni00000011/uni00000019/uni00000013/uni00000011/uni0000001b/uni00000014/uni00000011/uni00000013g/ω\n/uni00000028/uni00000030/uni00000003/uni00000036/uni0000004c/uni00000050/uni00000058/uni0000004f/uni00000044/uni00000057/uni0000004c/uni00000052/uni00000051\n/uni00000029/uni00000027/uni00000003/uni00000036/uni0000004c/uni00000050/uni00000058/uni0000004f/uni00000044/uni00000057/uni0000004c/uni00000052/uni00000051/uni00000056\n/uni00000030/uni00000048/uni00000044/uni00000056/uni00000058/uni00000055/uni00000048/uni00000050/uni00000048/uni00000051/uni00000057/uni00000056/uni00000025/uni00000030 /uni00000027/uni00000030\n/uni00000013 /uni00000015/uni00000013 /uni00000017/uni00000013 /uni00000019/uni00000013 /uni0000001b/uni00000013 /uni00000014/uni00000013/uni00000013 /uni00000014/uni00000015/uni00000013\nd/uni00000003/uni0000003e/uni00000050/uni00000040\n/uni00000013/uni00000015/uni00000017/uni00000019/uni0000001bω/2π/uni00000003/uni0000003e/uni0000002a/uni0000002b/uni0000005d/uni00000040\nFIG. 4.g=! versusdfor \ftted FD simulations (in black)\nand for \ftted measurements (in blue). The simulation trend\nis plotted in the black, dashed line. Inset: DM and BM fre-\nquencies versus the distance d, in the black dashed line are\nshown DM and BM reading values from simulations at ex-\ntremely high applied H-\feld. Eigen-Mode (EM) simulations\nare shown as the red dashed line.\nple. The \frst described machined cavity will be named\n\\CAV 01\" in the following. This cavity operates in a g=!\nrange from 0.35 to 0.59, as mentioned in the table II in\nappendix F.\nThe second cavity, \\CAV 02\", has been 3D printed and\nhas the same shape as CAV 01, but with smaller height\nposts. This cavity is performing in a certain range of\ng=!, from 0.28 to 0.32 (see table III in appendix F).\nThe third cavity, \\CAV 03'\", is also a double re-entrant\n3D printed cavity with cylinder posts, adjustable in\nheight. This cavity was used in a previous work [29]\nto experimentally verify a reworked theory that predicts\ncoupling values from simulations alone. The cavity has\nradiusRcav= 20 mm and height Hcav= 4.6 mm, whilst\nthe posts have radius Rpost= 2.05 mm and are spaced to\n2.7 mm. The operating ratio g=!is lower than the two\nothers cavities and enables to have experimental results\nat the SC/USC threshold, with g=!comprised between\n0.12 and 0.25 (see table IV in appendix F).\nThe operating range in BM frequencies, coupling\nstrengths, and added frequencies for the three cavities\nare summarized in Table I.\nTABLE I. Operating range of the cavities\nCavityfBM[GHz]g=2\u0019[GHz] \u0001 m=2\u0019[GHz]\nCAV 01 2.80 - 7.65 1.64 - 2.68 2.27 - 2.59\nCAV 02 7.63 - 9.79 2.42 - 2.72 1.63 - 1.74\nCAV 03 2.35 - 5.53 0.58 - 0.69 0.29 - 0.50\nThanks to the validation of the FD simulations, we\nwere able to simulate the USC CAV 01design for di\u000ber-\nent dimensions of the YIG slab, while keeping the aspect\nratio of the slab constant. Since the demagnetizing com-\nponents described in Eq. (C3) are only dependent on this\naspect ratio, the FMR remains unchanged. However, stilldecreasing the YIG slab dimensions decreases the \flling\nfactor\u0011, therefore the coupling strength and g=! from\n63 to 5 % with d= 50\u0016m.\n/uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000015 /uni00000013/uni00000011/uni00000017 /uni00000013/uni00000011/uni00000019 /uni00000013/uni00000011/uni0000001b\ng/ω/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013/uni00000014/uni00000011/uni00000018∆m/ω\n/uni00000036/uni00000026 /uni00000038/uni00000036/uni00000026/uni0000000b/uni00000044/uni0000000c\nCAVsimu\n01\nCAVmeas\n01\nCAVmeas\n02\nCAVmeas\n03\n/uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000015 /uni00000013/uni00000011/uni00000017 /uni00000013/uni00000011/uni00000019 /uni00000013/uni00000011/uni0000001b /uni00000014/uni00000011/uni00000013 /uni00000014/uni00000011/uni00000015 /uni00000014/uni00000011/uni00000017\ng2/2πω/uni00000003/uni0000003e/uni0000002a/uni0000002b/uni0000005d/uni00000040/uni00000013/uni00000014/uni00000015/uni00000016/uni00000017∆m/2π/uni00000003/uni0000003e/uni0000002a/uni0000002b/uni0000005d/uni00000040\n∝η2/uni0000000b/uni00000045/uni0000000c\nFIG. 5. (a) \u0001 m=!versusg=!and (b) \u0001 m=2\u0019versusg2=2\u0019.\nShown are the FD simulations on CAV 01in red and measure-\nments in blue, in green for CAV 02, and in purple for CAV 03.\nIn (a) and (b), \ftted values for a reduced CAV 01with an as-\npect ratio equal to 0.025 as the black square. The two data\npoints circled in (a) corresponds to the same value of \u0001 m=2\u0019\nin (b).\nWe plotted \u0001 m=!versusg=!in Fig. 5 (a) which clearly\ndisplay a quadratic dependence. For g=!\u00140:1, \u0001m=!\nis more or less negligible. This description agrees with\nthe commonly situated transition point (shown as the\nred dotted line) between the SC and USC regimes where\nall models converge. Our simulations show the need for\nthe \u0001m=2\u0019parameter to properly \ft the data. Fig. 5\n(b) shows \u0001 m=2\u0019versusg2=2\u0019!which is proportional\nto the square of the \flling factor, \u00112. According to this\nobservation and the de\fnition of \u0011, we noticed that the\nmore this energy is con\fned in the YIG, the larger the\nshift in the magnon frequency will be. In the literature,\nthe parameter \u0011is not so often considered or estimated.\nIn ref [29], we had the opportunity to test the model of\nequations (4) and (5) on multiple published experimental\nresults, and \u0011rarely exceeds 0.05 in any of them. As a\nreminder, and in view of the description in Fig. 2, our\nsystem (CAV 01) proposes a \u0011of about 0.79.\nIn Fig. 5 is represented by square marker a simulation\nwhere dimensions of the cavity and the YIG are reduced\nby a ratio equal to 0.025 for d= 50\u0016m. By decreasing\nthe dimensions of the entire cavity CAV 01by this ratio,\nthe BM frequency is increased to 275 GHz. Then, this\ncavity operates in the SC regime. However, the propor-7\ntion of the h-\feld in the YIG remains the same, hence\nalso\u0011.\nIn (a), are circled the reduced system performing in\nthe SC regimes, and the unmodi\fed cavity in the USC\nregime presenting the same \u0011value. In (b), it is shown\nthat the frequency shift \u0001 m=2\u0019is the same for both cav-\nities, and for the same value of g2=2\u0019!. Considering eq.\n4, \u0001m=2\u0019depends of the magnetic properties of the YIG\nand\u00112. It is then important to note that this e\u000bect is not\nbounded to the coupling strength and hence to the cou-\npling regime, but instead to the \flling factor, something\nthat has never been discussed so far.\nAs physical mechanism of \u0001 m=2\u0019, the nonlinear op-\ntical processes having similar behavior, would be a\ngood candidate. Among them, two di\u000berent e\u000bects at-\ntracted our attention: the multi-photon Rabi oscillations\n[14, 34, 35] for its e\u000bective coupling being proportional\ntog2=!. When the coupling between an arti\fcial sin-\ngle atom and a cavity is in the USC regime, the sys-\ntem can exchange several photons (and undergo multi-\nphoton Rabi oscillations) instead to a single one (com-\nmonly known as Rabi oscillation); and the self-Kerr, and\nthe cross-Kerr e\u000bects [36{40] presenting a frequency shift\nof the magnon, due to magnetocrystalline anisotropy and\nmagnon-magnon interactions, respectively.\nV. CONCLUSION\nIn conclusion, we proposed a double re-entrant cav-\nity design to achieve USC magnon/photon coupling at\nmicrowave frequencies, which was supported by both ex-\nperimental data and electromagnetic simulations. To the\nbest of our knowledge, this is the only demonstration of\nUSC magnon/photon coupling at room temperature so\nfar. Noteworthily, reaching the USC without cryogenic\ntemperatures is promising for the development of RF ap-\nplications based on cavity spintronics.\nWe explained the importance of optimizing the \flling-\nfactor\u0011for reaching the USC, aside from just the fre-quency of the resonator and the spin density. Impor-\ntantly, the cavity we proposed is parametrized by the\ndistancedbetween the posts and the lid. We showed\nthat tuning this parameter allowed to continuously go\nfrom the regular SC to the USC regime. The ability to\nstudy the transition from the SC to USC regime is a sig-\nni\fcant step towards understanding the physics of USC\nmagnon/photon coupling.\nIndeed, we showed that the standard models describ-\ning the coupling of a single resonator mode to many\ndipoles (e.g. the Dicke and Hop\feld models) failed to\nproperly decsribe our experimental data. Nevertheless,\nthanks to the validation of our electromagnetic simula-\ntions, we showed that a frequency shift in the magnon\nfrequency adequately modelled our data, which we note\nis fully captured by the classical Maxwell's equations.\nFurthermore, we showed that this frequency shift only de-\npended on the \flling-factor \u0011, highlighting its importance\nfor hybrid magnon/photon systems. While the physical\norigin of the magnon's frequency shift is still unknown,\nwe hope that its relation with \u0011will motivate further re-\nsearch into deriving a proper theoretical model for USC\nmagnon/photon coupling.\nACKNOWLEDGMENTS\nThis work is part of the research program supported\nby the European Union through the European Regional\nDevelopment Fund (ERDF), by Ministry of Higher Ed-\nucation and Research, Brittany and Rennes M\u0013 etropole,\nthrough the CPER Project SOPHIE/STIC & Ondes, by\nthe CPER SpaceTechDroneTech, by Brest M\u0013 etropole,\nand the ANR project MagFunc. 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Bistability of squeezing\nand entanglement in cavity magnonics. Physical Review\nResearch , 3(2), May 2021.\n[37] M. X. Bi, C. J. Dai, Jun-Ling Che, Ming-Liang Hu, and\nX. H. Yan. Bistability of cavity magnon polaritons be-\nyond the holstein{primako\u000b transformation. Journal of\nApplied Physics , 130(24):243902, December 2021.\n[38] Zhi-Bo Yang, Wei-Jiang Wu, Jie Li, Yi-Pu Wang, and\nJ. Q. You. Steady-entangled-state generation via the\ncross-kerr e\u000bect in a ferrimagnetic crystal. Physical Re-\nview A , 106(1), July 2022.9\n[39] Wei-Jiang Wu, Da Xu, Jie Qian, Jie Li, Yi-Pu Wang,\nand J. Q. You. Observation of magnon cross-kerr e\u000bect\nin cavity magnonics, 2021.\n[40] GuoQiang Zhang, YiPu Wang, and JianQiang You.\nTheory of the magnon kerr e\u000bect in cavity magnonics.\nScience China Physics, Mechanics &amp; Astronomy ,\n62(8), March 2019.[41] F. Rana. Quantum optics lectures ECE5310, Cornell Uni-\nversity.\n[42]Spin Waves . Springer US, 2009.\n[43] Klaus Hepp and Elliott H Lieb. On the superradi-\nant phase transition for molecules in a quantized radi-\nation \feld: the dicke maser model. Annals of Physics ,\n76(2):360{404, April 1973.10\nAppendix A: Physics Description\nThe system under consideration is best described by the Hamiltonian of two coupled harmonic oscillators. The\noscillators represent the cavity photonic mode ^Hc=~!^cy^c, and the uniformly precessing Kittel magnon mode ^Hm=\n~!m^by^b, where ~is the reduced Planck constant, !=2\u0019and!m=2\u0019are respectively the cavity and magnon frequencies,\nand ^cy(^by) and ^c(^b) are the creation and annihilation cavity (magnon) operators. The coupling is then read as a an\ninteraction ^Hint, and the entire system Hamiltonian can then be written as:\n^H=^Hc+^Hm+^Hint (A1)\nThe quantization of the Maxwell's equation leads to the expression of the vector potential:\n^A(r;t) =X\nn^qn(t)p\u000f0\u000fr;nUn(r) (A2)\nwhere\u000f0the vacuum permittivity, \u000fr;nthe relative permittivity experienced by the cavity mode n, ^qn(t) is the temporal\nterm, and Un(r) is the space dependent operator. This expression is generalized for all modes in a cavity. In the\nfollowing, we will concern ourselves with only a single mode.\nThis potential vector in a cavity is comparable to a simple harmonic oscillator, where radiation modes are de\fned\naccording to annihilation and creation operators:\n^c=1p\n2~!\u0010\n!^q\u0000i^_q\u0011\n^cy=1p\n2~!\u0010\n!^q+i^_q\u0011 (A3)\nThen, the RF magnetic \feld ( h-\feld) bounded to the cavity mode is[41]:\n^h=1\n\u00160r\u0002 ^A=1\n\u00160s\n~\n2!\u000f0\u000fr;c\u0000\n^cy+ ^c\u0001\nr\u0002U (A4)\nwhere\u00160is the vacuum permeability, \u000f0the vacuum permittivity, \u000fr;cthe relative permittivity experienced by the\ncavity mode, and Uis the space dependent operator of the potential vector. The component of this \feld perpendicular\nto the sample's magnetization direction will couple to the Kittel magnon mode.\nFor such a uniform precession of the magnetic sample, we introduce the macrospin operator considering the entire\nsample, as:\n^S=Vm\n\r^M (A5)\nwhere ^Mis the magnetization operator, Vmthe volume of the magnetic sample, and \rthe gyromagnetic ratio.\nWe consider a saturated magnetization by the use of an applied static magnetic \feld ( H-\feld) in the z-axis di-\nrection. It is then useful to introduce spin raising ^S+and lowering ^S\u0000operators. Following the Holstein-Primako\u000b\ntransformation[42] and considering low excitation numbers versus the total spin number of the macrospin operator,\nwe obtain:\n^S+=^Sx+i^Sy=q\n2S\u0000^by^b^b\u0019p\n2S^b\n^S\u0000=^Sx\u0000i^Sy=^byq\n2S\u0000^by^b\u0019p\n2S^by\nSz=S\u0000^by^b(A6)\nwhereS=\u0016\ngl\u0016BNsis the total spin number of the macrospin, \u0016Bis the Bohr magneton, \u0016is the magnetic moment of\nthe sample, glis the Land\u0013 e g-factor, and Ns=nsVmis the number of spins in the sample, with nsthe spin density.\nThe interaction term corresponds in this case to the Zeeman energy:\n^Hint=\u0000\u00160Z\nVm^M\u0001^hdV (A7)11\nSubstituting ^hand ^Min equation (A7) by their expressions in equations (A4) and (A5), and replacing cartesian\nmacrospin values by raising and lowering ones, with neglecting z-axis terms, we arrive at:\n^Hint=~=gx(^c+ ^cy)(^b+^by) +igy(^c+ ^cy)(^b\u0000^by) (A8)\nwhere the coupling strengths are de\fned as:\ngx=\u0000\r\n2Vms\n~S\n!\u000fr;c\u000f0Z\nVm(r\u0002U)\u0001^xdV\ngy=\r\n2Vms\n~S\n!\u000fr;c\u000f0Z\nVm(r\u0002U)\u0001^ydV(A9)\nIn order to consider the integration of the r\u0002Uterm in the magnetic sample volume in x- and y-axis, it is needed\nto rewrite it considering the entire h-\feld \\seen\" by the sample. Therefore, it is convenient to normalize the h-\feld\nagainst that of the entire cavity. The classical expression for the h-\feld energy for a single cavity mode is:\nE=\u00160\n2Z\nh\u0001hdV=\u00160\n2\u000f0\u000fr;cq\u0001qZ\n(r\u0002U)\u0001(r\u0002U) dV (A10)\nwhereR\nVc(r\u0002U)\u0001(r\u0002U) =\u000fr;c!2\nc2[41].\nRegarding the ratio of the h-\feld energy in the magnetic sample versus the one in the whole cavity, we get:\nEVm\nEVc=R\nVmh\u0001hdVR\nVcjhj2dV=R\nVm(r\u0002U)\u0001(r\u0002U) dV\n\u000fr;c!2=c2(A11)\nUsing the center and right terms of the above equation, deriving numerators and applying the square root, we\n\fnally read the in\fnitesimal normalized energy amplitude of the h-\feld:\nr\u0002U=p\u000fr;c!\nchqR\nVcjhj2dV(A12)\nEquation (A1), with the use of equation (A8), can be rewritten over a matrix form as:\n^H=1\n2\u0002\n^cy^by^c^b\u0003\nH\u0002\n^cy^by^c^b\u0003y+const\nH=2\n64! gx+igy 0gx\u0000igy\ngx\u0000igy!mgx\u0000igy 0\n0gx+igy! gx\u0000igy\ngx+igy 0gx+igy!m3\n75(A13)\nUsing Hop\feld-Bogolubov transformation [24], the solution of the problem is to \fnd polariton operators ^ p\u0006, ex-\npressed as a linear combination of ^ c, ^cy,^b, and ^by. Being bosonic operators, they should obey the Hop\feld formulation\n[33]:\nh\n^p\u0006;^Hi\n=!\u0006^p\u0006 (A14)\nwhere!\u0006=2\u0019are frequency eigenvalues associated with the eigen-operators ^ p\u0006.\nAs previously, the Hamiltonian in the polaritonic basis can be rewritten as:\n^H=1\n2\u0002\n^py\n\u0000^py\n+^p\u0000^p+\u0003\nM\u0002\n^py\n\u0000^py\n+^p\u0000^p+\u0003y(A15)\nIn order to respect equation A14, the Hop\feld matrix Mhas to be read as:\nM=Hdiag (1;1;\u00001;\u00001) (A16)12\nSolving eigenvalues of the matrix Mleads to:\n!\u0006=1p\n2r\n!2+!2m\u0006q\n(!2\u0000!2m)2+ 16g2!!m (A17)\nwhere the coupling strength g=q\ng2x+g2yis de\fned as:\ng\n2\u0019=\r\n4\u0019\u0011r\n\u00160S~!\nVm=\u0011p!\r\n4\u0019r\u0016\ngl\u0016B\u00160~ns (A18)\nwith the \flling factor:\n\u0011=vuuut\u0010R\nVmh\u0001^xdV\u00112\n+\u0010R\nVmh\u0001^ydV\u00112\nVmR\nVcjhj2dV(A19)\nIt is important to note that eigenfrequencies are solution of the Dicke model[14, 15]. Finally the Hamiltonian can\nbe rewritten over the easier form:\n^H=~=!^cy^c+!b^by^b+g(^cy+ ^c)(^by+^b) (A20)\nAppendix B: Cavity Optimization\nFig. 6 is a representation of the optimization of the \flling factor \u0011for two of the variable parameters; the width\n(W) and the length ( L) of the posts, with dchosen equal to 50 µm. The cavity radius ( R) has been chosen at its\noptimized value. The containment of the h\u0000field inside the YIG is at its maximum when the post dimensions are\nof the slab dimensions. Hence, the width of the posts has been optimized over a range from 0.1 mm to 2 mm, and\ntheir lengths from 4 mm to 8 mm. The radius of the cavity does not have a big impact on \u0011. The cavity radius has\nbeen optimize over a range from 10 mm to 14 mm.\nEach contour represents the value of \u0011with respect to WandL. The hashed contour delimits the surface where\n\u0011\u001578:5%. For better feasibility, we choose the largest values of WandL. This leads to an optimal value of \u0011for\nW= 0:6 mm,L= 6 mm, and R= 12 mm.\n/uni00000017 /uni00000018 /uni00000019 /uni0000001a /uni0000001b\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\n/uni00000013/uni00000011/uni00000019/uni00000013/uni0000003a/uni0000004c/uni00000047/uni00000057/uni0000004b/uni00000003/uni0000003e/uni00000050/uni00000050/uni000000400.70\n0.72\n0.740.76\n0.77\n0.78\n0./uni00000019/uni000000190.680.700.720.740.760.78\nη\nFIG. 6. \flling factor \u0011function of the width ( W) and the length ( L) of the two posts13\nAppendix C: FMR model\nUsing the Landau-Lifshitz equation of the magnetization with the proper approximations leads us to the FMR\npulsation for all types of ferromagnet shapes [30]:\n!0=\rs\u0012!e\n\r\u00132\n\u0000[(Nxy+Nyx)Ms]2(C1)\nwhereMsis the saturation magnetization, Ni;jis a component of the demagnetizing tensor at the ithcolumn and the\njthrow, and!eis the FMR pulsation for an ellipsoidal body read as:\n!e=\rq\n[jHzj+ (Nxx\u0000Nzz)Ms] [jHzj+ (Nyy\u0000Nzz)Ms] (C2)\nUsing the perturbation theory with a small perturbation on the H-\feld\u000f=Mz\nHzat the \frst order, it is shown that\nthe demagnetizing components for a rectangular prism are for the diagonal components[31]:\nN(1)\nkk=1\n4\u00198\n><\n>:cot\u00001[f(xi;xj;xk)] +cot\u00001[f(\u0000xi;xj;xk)] +\ncot\u00001[f(xi;\u0000xj;xk)] +cot\u00001[f(xi;xj;\u0000xk)] +\ncot\u00001[f(\u0000xi;\u0000xj;xk)] +cot\u00001[f(\u0000xi;xj;\u0000xk)] +\ncot\u00001[f(xi;\u0000xj;\u0000xk)] +cot\u00001[f(\u0000xi;\u0000xj;\u0000xk)]9\n>=\n>;(C3)\nwith :\nf(xi;xj;xk) =q\n(ai\u0000xi)2+ (aj\u0000xj)2+ (ak\u0000xk)2(ak\u0000xk)\n(ai\u0000xi) (aj\u0000xj)(C4)\nand for the o\u000b-diagonal terms:\nN(1)\nik=\u00001\n4\u0019log\u001aG(rjai;aj;ak)G(rj\u0000ai;\u0000aj;ak)G(rj\u0000ai;aj;\u0000ak)G(rjai;\u0000aj;\u0000ak)\nG(rj\u0000ai;aj;ak)G(rjai;\u0000aj;ak)G(rjai;aj;\u0000ak)G(rj\u0000ai;\u0000aj;\u0000ak)\u001b\n(C5)\nwith:\nG(rjai;aj;ak) = (aj\u0000xj) +q\n(ai\u0000xi)2+ (aj\u0000xj)2+ (ak\u0000xk)2(C6)\nLet us notice that the demagnetizing components are spatially dependent and were averaged to x, y, and z equal\nto zero for analytical equations. For the YIG dimensions mentioned in the manuscript, the o\u000b-diagonal components\nof the demagnetizing tensor are equal to zero, then the slab as he same FMR frequency as read in Eq. C2.\nAppendix D: Dicke model\n1. Normal phase\nThe Dicke model is the simplest model to describe the magnon-photon interaction. It consider each Hamiltonian\nof the cavity photonic mode and magnon as well as the interaction Hamiltonian[24]:\n^H=!^cy^c+!m^by^b+g(^cy+ ^c)(^by+^b) (D1)\nFrom this equation, we can easily solve for the eigenmodes, that is to say the polaritronic modes described in Eq.\n(A17). This equation is only valid when the ratio g=!is less than 0.5. For a description of a system with a ratio\nhigher than 0.5, it is necessary to use the Dicke superradiant phase [43].14\n2. Superradiant phase\nThe superradiant phase is a quantum transition in the Dicke model and represents the displacement of bosonic\nmodes [24]: ^ cy!^ay+p\u000band^by!^dy\u0000p\f, where\u000band\frepresent averaged values of the displaced ground states\nfor the photon and the magnon, respectively. Using Holstein-Primako\u000b transformation in the Dicke Hamiltonian, the\neigen-frequencies become:\n!\u0006=1p\n2r\n!2+ ~g4!2m\u0006q\n(!2\u0000~g4!2m)2+ 4!2!2m (D2)\nwhere ~g= 2g\n!.\nIn this case, Fig. 7 (b) shows \ftted measurement with the superradiant Dicke model. For this \ft, we do not need\nto add a frequency term on the FMR and we found that != 4:75 GHz and g= 2:58 GHz.\nWith comparing BM frequencies, DM frequencies, and gas done in section IV C, the BM frequency and gshould\nbe higher than those obtained from simulation. Because of this mismatch, it seems that the superradiant phase is not\nreached.\n/uni00000014/uni00000011/uni00000013\n /uni00000013/uni00000011/uni00000018\n /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000018 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\n/uni00000013/uni00000011/uni0000001a/uni00000018\n /uni00000013/uni00000011/uni00000018/uni00000013\n /uni00000013/uni00000011/uni00000015/uni00000018\n /uni00000013/uni00000011/uni00000013/uni00000013 /uni00000013/uni00000011/uni00000015/uni00000018 /uni00000013/uni00000011/uni00000018/uni00000013 /uni00000013/uni00000011/uni0000001a/uni00000018H0/uni00000003/uni0000003e/uni00000037/uni00000040/uni0000000b/uni00000045/uni0000000cfDM\nfBM\nfFMR\nf±\n/uni00000014/uni00000017/uni00000013\n/uni00000014/uni00000015/uni00000013\n/uni00000014/uni00000013/uni00000013\n/uni0000001b/uni00000013\n/uni00000019/uni00000013\n/uni00000017/uni00000013\n/uni00000015/uni00000013\n/uni00000036/uni00000003/uni0000003e/uni00000047/uni00000025/uni00000040\nFIG. 7. Transmission spectra versus the RF frequency and the H-\feld. Fitted polariton branches are shown in white. The\nBM frequency (in orange) and the coupling strength are variables. The FMR is shown in black and the DM in red. Fits were\nbe done with the normal phase of the Dicke model. In (a) the measurement for d= 50\u0016m and a \ft with the Dicke model. In\n(b) the measurement for d= 4\u0016m and a \ft with the Dicke superradiant phase model.\nFig. 7 shows transmission spectra with respect to the frequency and the H-\feld. A \ft has be done with the\nstandard Dicke model. The two eigen-modes of the \ft shown in dotted white line which are not consistent with the\nmeasurement prove the inability to \ft with the standard Dicke model.\nAppendix E: Hop\feld model\n1. Standard\nThe Hop\feld model is equivalent to the Dicke one with a supplementary term: the diamagnetic one.\nConsidering a carried particle in a magnetic \feld, we rede\fne the impulse of the system[24]:\n^p!^p\u0000q^A (E1)\nwith ^Athe vector potential and associated to the photonic mode: ^A/(^cy+ ^c)\nWe \fnally have the Hop\feld Hamiltonian of the system:\n^H=!^cy^c+!m^by^b+g(^cy+ ^c)(^by+^b) +D(^cy+ ^c)2(E2)15\nwhereDis the diamagnetic term where the Thomas-Reiche-Kuhn sum rules gives D=g2\n!.\nWith using Hop\feld-Bogolubav transformation and rede\fning gasgp!m\n!we have:\n!\u0006=1p\n2r\n!2+!2m+ 4g2\u0006q\n(!2+!2m+ 4g2)2\u00004!2!2m (E3)\n2. Modi\fed\nFollowing ref. [17], where a prefactor dis added before the diamagnetic term in Eq. E4, we tried to \ft with the\nmodi\fed Hop\feld model by varying this prefactor.\n!\u0006=1p\n2r\n!2c+ 4dD!c+!2c\u0006q\n(!2c+ 4dD!c\u0000!2m)2+ 16g2!c!m (E4)\nwhereD=g2=!m\n/uni00000014/uni00000011/uni00000013\n /uni00000013/uni00000011/uni00000018\n /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000018 /uni00000014/uni00000011/uni00000013H0/uni00000003/uni0000003e/uni00000037/uni00000040/uni00000015/uni00000011/uni00000018/uni00000018/uni00000011/uni00000013/uni0000001a/uni00000011/uni00000018/uni00000014/uni00000013/uni00000011/uni00000013/uni00000014/uni00000015/uni00000011/uni00000018/uni00000014/uni00000018/uni00000011/uni00000013/uni00000029/uni00000055/uni00000048/uni00000054/uni00000058/uni00000048/uni00000051/uni00000046/uni0000005c/uni00000003/uni0000003e/uni0000002a/uni0000002b/uni0000005d/uni00000040/uni0000000b/uni00000044/uni0000000c\n/uni00000014/uni00000011/uni00000013\n /uni00000013/uni00000011/uni00000018\n /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000018 /uni00000014/uni00000011/uni00000013H0/uni00000003/uni0000003e/uni00000037/uni00000040/uni0000000b/uni00000045/uni0000000c fFMR\nfDM\nfBM\nf±\n/uni00000014/uni00000011/uni00000013\n /uni00000013/uni00000011/uni00000018\n /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000018 /uni00000014/uni00000011/uni00000013H0/uni00000003/uni0000003e/uni00000037/uni00000040/uni0000000b/uni00000046/uni0000000c\n/uni00000014/uni00000017/uni00000013\n/uni00000014/uni00000015/uni00000013\n/uni00000014/uni00000013/uni00000013\n/uni0000001b/uni00000013\n/uni00000019/uni00000013\n/uni00000017/uni00000013\n/uni00000015/uni00000013\n/uni00000036/uni00000003/uni0000003e/uni00000047/uni00000025/uni00000040\nFIG. 8. Transmission spectra versus the RF frequency and the H-\feld. Fitted polariton branches are shown in white. The\nBM frequency (in orange) and the coupling strength are variables. The FMR is shown in black and the DM in red. Fits were\nbe done with the modi\fed Hop\feld model where the prefactor is: (a) d<1; (b)d= 1; and (c) d>1.\nFig. 8 shows the \ft with the modi\fed Hop\feld model when the prefactor is less, equal, or more than 1 in respectively\n(a), (b), and (c). Let us notice that the standard Hop\feld model is for d= 1. Finally the only e\u000bect of this prefactor\nis equivalent to increase (for d<1) or decrease (for d>1) the BM frequency whereas it is needed to have a model\nwhich a\u000bects the FMR.16\nAppendix F: Measurements\n1. CAV 01\n/uni00000014\n /uni00000013 /uni00000014/uni00000015/uni00000011/uni00000018/uni00000018/uni00000011/uni00000013/uni0000001a/uni00000011/uni00000018/uni00000014/uni00000013/uni00000011/uni00000013/uni00000014/uni00000015/uni00000011/uni00000018/uni00000014/uni00000018/uni00000011/uni00000013/uni00000029/uni00000055/uni00000048/uni00000054/uni00000058/uni00000048/uni00000051/uni00000046/uni0000005c/uni00000003/uni0000003e/uni0000002a/uni0000002b/uni0000005d/uni00000040/uni0000000b/uni00000044/uni0000000c\n/uni00000014/uni00000011/uni00000013\n /uni00000013/uni00000011/uni00000018\n /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000018 /uni00000014/uni00000011/uni00000013/uni0000000b/uni00000047/uni0000000c\n/uni00000014/uni00000011/uni00000013\n /uni00000013/uni00000011/uni00000018\n /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000018 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\n/uni00000013/uni00000011/uni00000018\n /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000018/uni0000000b/uni00000048/uni0000000c\n/uni00000014/uni00000011/uni00000013\n /uni00000013/uni00000011/uni00000018\n /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000018 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\n/uni00000013/uni00000011/uni00000018\n /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000018H0/uni00000003/uni0000003e/uni00000037/uni00000040/uni0000000b/uni00000049/uni0000000c\n/uni00000014/uni00000017/uni00000013\n/uni00000014/uni00000015/uni00000013\n/uni00000014/uni00000013/uni00000013\n/uni0000001b/uni00000013\n/uni00000019/uni00000013\n/uni00000017/uni00000013\n/uni00000015/uni00000013\n/uni00000036/uni00000003/uni0000003e/uni00000047/uni00000025/uni00000040\nfDM\nfBM\nfFMR\nf±\nFIG. 9. Transmission spectra versus the RF frequency and the H-\feld. Fitted polariton branches are shown in white. The\nBM frequency (in orange) and the coupling strength are variables. The FMR is shown in black and the DM in red. Fitted\nparameters are shown in Table II.17\nTABLE II. Cavity parameters from Fig. 9\nNumbering d[\u0016m]fDM[GHz]fBM[GHz]g=2\u0019[GHz]g=! g2=2\u0019![GHz] \u0001 m=2\u0019[GHz]fgap[GHz]\n(a) 116 3.75 7.65 2.68 0.35 0.94 2.35 0.58\n(b) 75 3.19 7.31 2.62 0.36 0.94 2.29 0.54\n(c) 65 3.05 7.16 2.56 0.36 0.92 2.31 0.54\n(d) 36 2.40 6.44 2.41 0.37 0.90 2.27 0.64\n(e) 10 1.38 4.46 2.03 0.46 0.92 2.39 0.87\n(f) 3 0.81 2.80 1.64 0.59 0.96 2.59 1.22\n2. CAV 02\n/uni00000015/uni00000011/uni00000018/uni00000018/uni00000011/uni00000013/uni0000001a/uni00000011/uni00000018/uni00000014/uni00000013/uni00000011/uni00000013/uni00000014/uni00000015/uni00000011/uni00000018/uni00000014/uni00000018/uni00000011/uni00000013/uni00000029/uni00000055/uni00000048/uni00000054/uni00000058/uni00000048/uni00000051/uni00000046/uni0000005c/uni00000003/uni0000003e/uni0000002a/uni0000002b/uni0000005d/uni00000040/uni0000000b/uni00000044/uni0000000c\nfDM\nfBM\nfFMR\nf±\n/uni0000000b/uni00000046/uni0000000c\n/uni00000013/uni00000011/uni00000017\n /uni00000013/uni00000011/uni00000015\n /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000015 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\n/uni00000013/uni00000011/uni00000017\n /uni00000013/uni00000011/uni00000015\n /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000015 /uni00000013/uni00000011/uni00000017H0/uni00000003/uni0000003e/uni00000037/uni00000040/uni0000000b/uni00000047/uni0000000c\n/uni00000014/uni00000017/uni00000013\n/uni00000014/uni00000015/uni00000013\n/uni00000014/uni00000013/uni00000013\n/uni0000001b/uni00000013\n/uni00000019/uni00000013\n/uni00000017/uni00000013\n/uni00000015/uni00000013\n/uni00000036/uni00000003/uni0000003e/uni00000047/uni00000025/uni00000040\nFIG. 10. Transmission spectra versus the RF frequency and the H-\feld. Fitted polariton branches are shown in white. The\nBM frequency (in orange) and the coupling strength are variables. The FMR is shown in black and the DM in red. Fitted\nparameters are shown in Table III.\nTABLE III. Cavity parameters from Fig. 10\nNumbering fDM[GHz]fBM[GHz]g=2\u0019[GHz]g=! g2=2\u0019![GHz] \u0001 m=2\u0019[GHz]fgap[GHz]\n(a) 4.06 9.79 2.72 0.28 0.76 1.63 0.24\n(b) 3.26 8.76 2.59 0.30 0.77 1.71 0.30\n(c) 3.01 8.32 2.52 0.30 0.76 1.74 0.31\n(d) 2.64 7.63 2.42 0.32 0.77 1.69 0.3418\n3. CAV 03\n/uni00000013/uni00000011/uni00000015\n /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000015/uni00000015/uni00000017/uni00000019/uni0000001b/uni00000014/uni00000013/uni00000029/uni00000055/uni00000048/uni00000054/uni00000058/uni00000048/uni00000051/uni00000046/uni0000005c/uni00000003/uni0000003e/uni0000002a/uni0000002b/uni0000005d/uni00000040/uni0000000b/uni00000044/uni0000000c\nfDM\nfBM\nfFMR\nf±\n/uni00000013/uni00000011/uni00000015\n /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000015/uni00000015/uni00000017/uni00000019/uni0000001b/uni00000014/uni00000013/uni0000000b/uni00000046/uni0000000c\n/uni00000013/uni00000011/uni00000015\n /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000015H0/uni00000003/uni0000003e/uni00000037/uni00000040/uni00000015/uni00000017/uni00000019/uni0000001b/uni00000014/uni00000013/uni00000029/uni00000055/uni00000048/uni00000054/uni00000058/uni00000048/uni00000051/uni00000046/uni0000005c/uni00000003/uni0000003e/uni0000002a/uni0000002b/uni0000005d/uni00000040/uni0000000b/uni00000045/uni0000000c\n/uni00000013/uni00000011/uni00000014/uni00000013\n /uni00000013/uni00000011/uni00000013/uni00000018\n /uni00000013/uni00000011/uni00000013/uni00000013 /uni00000013/uni00000011/uni00000013/uni00000018 /uni00000013/uni00000011/uni00000014/uni00000013H0/uni00000003/uni0000003e/uni00000037/uni00000040/uni00000014/uni00000015/uni00000016/uni00000017/uni00000018/uni0000000b/uni00000047/uni0000000c\n/uni00000014/uni00000017/uni00000013\n/uni00000014/uni00000015/uni00000013\n/uni00000014/uni00000013/uni00000013\n/uni0000001b/uni00000013\n/uni00000019/uni00000013\n/uni00000017/uni00000013\n/uni00000015/uni00000013\n/uni00000036/uni00000003/uni0000003e/uni00000047/uni00000025/uni00000040\nFIG. 11. Transmission spectra versus the RF frequency and the H-\feld. Fitted polariton branches are shown in white. The\nBM frequency (in orange) and the coupling strength are variables. The FMR is shown in black and the DM in red. Fitted\nparameters are shown in Table IV.\nTABLE IV. Cavity parameters from Fig. 11\nNumbering fDM[GHz]fBM[GHz]g=2\u0019[GHz]g=! g2=2\u0019![GHz] \u0001 m=2\u0019[GHz]fgap[GHz]\n(a) 3.02 5.53 0.65 0.12 0.08 0.33 0.01\n(b) 2.29 4.36 0.69 0.16 0.11 0.29 0.02\n(c) 1.44 2.92 0.63 0.22 0.14 0.37 0.02\n(d) 1.30 2.35 0.58 0.25 0.14 0.50 0.0519\nAppendix G: Gap Study\n/uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000014 /uni00000013/uni00000011/uni00000015 /uni00000013/uni00000011/uni00000016 /uni00000013/uni00000011/uni00000017 /uni00000013/uni00000011/uni00000018 /uni00000013/uni00000011/uni00000019 /uni00000013/uni00000011/uni0000001a /uni00000013/uni00000011/uni0000001b\ng/ω/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000015/uni00000013/uni00000011/uni00000017/uni00000013/uni00000011/uni00000019∆g/ω\n/uni0000000b/uni00000044/uni0000000cCAVsimu\n01\nCAVmeas\n03\nCAVmeas\n02\nCAVmeas\n01\n/uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000015 /uni00000013/uni00000011/uni00000017 /uni00000013/uni00000011/uni00000019 /uni00000013/uni00000011/uni0000001b /uni00000014/uni00000011/uni00000013 /uni00000014/uni00000011/uni00000015 /uni00000014/uni00000011/uni00000017\n∆m/ω\n/uni0000000b/uni00000045/uni0000000c\nFIG. 12. \u0001 g=!versus (a)g=!; (b) \u0001 m=!. Shown are the FD simulations on CAV 01in red and measurements in blue, in green\nfor CAV 02, and in purple for CAV 03.\nWithout adding \u0001 mto the FMR in the Dicke model, and without applied static magnetic \feld, the frequency of\nthe upper polariton is equal to the cavity one. However, when the FMR is shifted, an observable forbidden gap in\nfrequency appears. Considering Fig. 12, \u0001 g=!is not observable when g=!is equal or lower to 0.2. For higher g=!\nvalues, \u0001 g=!is quadratic, as shown in (a). In (b) is shown the evolution of \u0001 g=!versus \u0001m=!."    },    {        "title": "2401.16857v1.Entropy_production_rate_and_correlations_of_cavity_magnomechanical_system.pdf",        "content": "Entropy production rate and correlations of cavity magnomechanical system\nCollins O. Edet,1, 2Muhammad Asjad,3,∗Denys Dutykh,3, 4Norshamsuri Ali,5and Obinna Abah6,†\n1Institute of Engineering Mathematics, Universiti Malaysia Perlis, 02600 Arau, Perlis, Malaysia\n2Department of Physics, University of Cross River State, Calabar, Nigeria\n3Department of Mathematics, Khalifa University, Abu Dhabi 127788, United Arab Emirates\n4Causal Dynamics Pty Ltd, Perth, Australia\n5Advanced Communication Engineering (ACE) Centre of Excellence,\nUniversiti Malaysia Perlis, 01000 Kangar, Perlis, Malaysia\n6School of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon Tyne NE1 7RU, United Kingdom\nWe present the irreversibility generated by a stationary cavity magnomechanical system composed of a yttrium\niron garnet (YIG) sphere with a diameter of a few hundred micrometers inside a microwave cavity. In this system,\nthe magnons, i.e., collective spin excitations in the sphere, are coupled to the cavity photon mode via magnetic\ndipole interaction and to the phonon mode via magnetostrictive force (optomechanical-like). We employ the\nquantum phase space formulation of the entropy change to evaluate the steady-state entropy production rate\nand associated quantum correlation in the system. We find that the behavior of the entropy flow between the\ncavity photon mode and the phonon mode is determined by the magnon-photon coupling and the cavity photon\ndissipation rate. Interestingly, the entropy production rate can increase/decrease depending on the strength of the\nmagnon-photon coupling and the detuning parameters. We further show that the amount of correlations between\nthe magnon and phonon modes is linked to the irreversibility generated in the system for small magnon-photon\ncoupling. Our results demonstrate the possibility of exploring irreversibility in driven magnon-based hybrid\nquantum systems and open a promising route for quantum thermal applications.\nI. INTRODUCTION\nHybrid quantum systems play a crucial role in the advance-\nment of quantum technologies [ 1–6]. In the last decade, re-\nmarkable progress has been made in this field with applications\nspanning through quantum computing [ 5,7], quantum simu-\nlation [ 8], quantum communication [ 6], quantum sensing [ 9],\nand quantum thermodynamics [ 10]. A typical example is the\ncavity optomechanical system, which combines mechanical\ndegrees of freedom with electromagnetic (EM) cavities [ 11].\nRecently, hybrid quantum systems based on magnons, quanta\nof collective spin excitations in ordered ferrimagnetic materi-\nals, e.g. yttrium iron garnet (YIG), have attracted considerable\nattention due to great frequency tunability and very good co-\nherence [ 4]. Magnons can be coherently coupled to different\ndegrees of freedom, such as phonons via the magnetostrictive\nforce [ 12], microwave photons via the magnetic dipole interac-\ntion [ 4,13,14], optical photons [ 15,16] and superconducting\nqubits [ 17–19]. The realization of this cavity magnomechan-\nical system, a system of photon-magnon-phonon interaction\nwith YIG spheres interacting with microwave cavities has open\nup possible applications in the preparation of macroscopic\nquantum states [ 20], generation of squeezed states [ 21], the\ngeneration of high-performance detectors [ 22], entanglement\ngeneration [ 23,24], ground state cooling [ 25], and quantum\ninformation processing [26–29].\nQuantum technological and nanofabrication advancements\nhave motivated the design of microscopic and coherent thermo-\ndynamic machines – quantum thermal machines [ 10], as well\nas investigating the interplay between quantum information\nand thermodynamic processes [ 30]. Optomechanical thermal\n∗asjad qau@yahoo.com\n†obinna.abah@newcastle.ac.ukmachines have been proposed in different configurations [ 31–\n37]. The irreversibility (such as, friction, disorder) that in-\nfluences the machine thermodynamic processes/performance\ncan be quantified by entropy production [ 38,39]. Thus, quan-\ntifying the degree of irreversible entropy generated in a dy-\nnamic process is useful for the distinctive description of the\nnon-equilibrium processes, and decreasing it, enhances a ther-\nmal machine efficiency [ 40]. Based on quantum phase-space\nmethod, the measure of quantifying the irreversible entropy pro-\nduction of quantum systems that interact with nonequilibrium\nreservoirs has been formulated in [ 41–43] and experimentally\nverified in two distinct setup - an optomechanical system and a\ndriven Bose-Einstein condensate coupled to a high finesse cav-\nity [44]. The effect of self-correlation on irreversible entropy\nproduction rate in a parametrically driven dissipative system\nhas been investigated [ 45]. It has been recently shown that the\npresence of nonlinearity via an optical parametric oscillator\nplaced inside the cavity optomechanical system influences the\nstationary state entropy production rate [ 46]. Moreover, it has\nbeen established that the entropy produced in a bipartite quan-\ntum system is related to the amount of correlations shared by\nits subsystems [41].\nHere, we investigate the generation of irreversibility in a\nhybrid cavity magnomechanical setup comprising a microwave\ncavity and a YIG sphere. In this system, magnons are simul-\ntaneously coupled to the phonons of the vibrational sphere\nvia magnetostrictive interaction and to the cavity photons via\nmagnetic-dipole interaction, while there is no direct interaction\nbetween the cavity mode and the mechanical mode. We find\nthat the magnon-photon coupling and cavity photon dissipation\nrate influence the entropy production rate. Furthermore, we\ndemonstrate that the amount of correlations in the cavity mag-\nnomechanical system deviates from the steady-state entropy\nproduction rate for large magnon-photon coupling.\nThe rest of this paper is organized as follows. In Section II,arXiv:2401.16857v1  [quant-ph]  30 Jan 20242\nFIG. 1. Diagrammatic representation of a cavity magnomechani-\ncal system consisting of photon, magnon, and phonon modes. The\nmagnon-photon interaction strength and the magnon-phonon interac-\ntion of coupling strength are denoted by gamandgmbrespectively.\nwe present the physical model of the cavity magnomechanical\nsystem. We derive the linearized Hamiltonian of the system\nvia quantum Langevin equations of motion and standard lin-\nearization techniques. In Section III, the stationary entropy\nproduction rate and quantum correlation quantified by mutual\ninformation are presented using the experimentally feasible\nparameters. Finally, the conclusions are summarized in Section\nIV.\nII. CA VITY MAGNOMECHANICAL MODEL\nWe consider a hybrid cavity magnomechanical system,\nwhich consists of a microwave cavity and a small sphere (a\nonemm-diameter, highly-polished YIG sphere is considered\nin Ref. [ 47,48]). The YIG sphere is positioned close to the\nmaximal microwave magnetic field of the cavity mode, and\na variable external magnetic field Hin the z-axis is added to\nestablish the magnon–photon coupling [ 47,49]. The coupling\nrate can be tailored by adjusting the position of the sphere. The\nmagnons couple to phonons via the magnetostrictive effect.\nThe vibrational modes (phonons) result from the geometric\ndeformation of the YIG sphere because the magnon excita-\ntion inside the YIG sphere induces a varying magnetization.\nThe magnomechanical coupling can be enhanced by directly\ndriving the magnon mode with a microwave source [ 12]. In\naddition, the size of the sphere considered is much smaller\nthan the wavelength of the microwave field, such that the in-\nteraction between cavity microwave photons and phonons can\nbe neglected (i.e. the radiation pressure effect is negligible).\nThus, the system has three modes: cavity photon, magnon,\nand phonon modes, which can be schematically depicted by\nthe equivalent coupled harmonic oscillator model as shown in\nFig. 1.\nThe Hamiltonian of the hybrid quantum system under\nrotating-wave approximation in a frame rotating with the fre-\nquency ωdof the driving field can be expressed as ( ℏ= 1)[23, 48]:\nˆH:= ∆ aˆa†ˆa+ Ω bˆb†ˆb+ ∆ mˆm†ˆm+gam(ˆaˆm†+ ˆa†ˆm)\n+gmbˆm†ˆm(ˆb+ˆb†) + i(Ω dˆm†−Ω∗\ndˆm), (1)\nwhere the bosonic annihilation (creation) operators ˆa(ˆa†),\nˆb(ˆb†)andˆm( ˆm†)denote, respectively, cavity photon, phonon\nand maganon modes whose resonant frequencies are taken to\nbeωa,Ωb, and ωmcorrospondingly. The detuning param-\neters ∆a: =ωa−ωd, where ωa/2π= 10 GHz [ 12], and\n∆m: =ωm−ωd. The uniform magnon mode frequency in\nthe YIG sphere is ωm=γgH, where γg/2π=28 GHz/T is the\ngyromagnetic ratio, and we set ωmat the Kittel mode frequency\n[50], which can lead to cavity polaritons by strongly coupling\nmagnon and cavity photons. The parameters gamandgmb\nare the optomagnon (photon-magnon) and magnomechanical\n(magnon-phonon) interaction coupling strengths respectively.\nThe last term, (Ωdˆm†−Ω∗\ndˆm)in the Hamiltonian describes\nthe external driving of the magnon mode. The Rabi frequency\nΩd≡√\n5\n4γg√NtB0(assuming low-lying excitations) denotes\nthe coupling strength of the drive magnetic field [ 23]. The\namplitude and frequency of the drive field are B0andωd, re-\nspectively, and the total number of spins Nt=ρ V, where Vis\nthe volume of the sphere and ρ=4.22×1027m−3is the spin\ndensity of the YIG. Each of the modes is coupled to an indepen-\ndent noise reservoir, their energy decay rates are γa,γm, and\nγb, for photon, magnon and phonon respectively. Experimen-\ntally, gmbis extremely weak [ 12], but the magnomechanical\ninteraction can be enhanced by driving the magnon mode with\na strong microwave field [ 47,49]. The magnon-photon cou-\npling rate gamcan be larger than the dissipation rates of the\ncavity and magnon modes, γaandγm, entering into the strong\ncoupling regime, gam>max{γa, γm}[13, 14].\nAs a result of strong driving, the Hamiltonian in Eq. (1)\ncan be linearized around the coherent steady-state amplitude:\nˆO→Os+ˆO(O∈ {a,b,m}), where Osand the operators\nˆO, represent the steady-state amplitudes and quantum fluctu-\nations of the corresponding mode. We have the steady-state\namplitudes\nms=Ωd(i∆a+γa)\ng2am+ (i∆ m+γm)(i∆ a+γa), (2)\nandbs=−igmb|ms|2/(i Ωb+γb). Then, the linearized\nHamiltonian can be derived as\nˆHlin= ∆ aˆa†ˆa+ Ω bˆb†ˆb+˜∆mˆm†ˆm+gam(ˆaˆm†+ ˆa†ˆm)\n+ (G∗\nmbˆm+Gmbˆm†)(ˆb+ˆb†), (3)\nwhere the enhanced magnon-phonon coupling Gmb=gmbms,\nand˜∆m= ∆ m−gmb(bs+b∗\ns)is the effective magnon de-\ntuning incorporating the magnetostriction. For the considered\nparameters, gmb(bs+b∗\ns)≪∆m, so we can have, ˜∆m≈∆m.\nSince the driving field affects ms, we can improve Gmbby\nadjusting the external driving field Ωd.\nFrom the Hamiltonian in Eq. (3), we obtain the quantum3\nLangevin equations as;\n˙ˆa=−(i∆a+γa) ˆa−igamˆm−p\n2γaˆain,\n˙ˆm=−\u0010\ni˜∆m+γm\u0011\nˆm−igamˆa−iGmb(ˆb+ˆb†)−p\n2γmˆmin,\n˙ˆb=−(iΩb+γb)ˆb−i(Gmbˆm†+G∗\nmbˆm)−p\n2γbˆbin, (4)\nwhere ˆfin∈ {ˆain,ˆmin,ˆbin}are input noise operators for\nthe cavity, magnon and mechanical modes, respectively,\nwhich are zero mean and characterized by the follow-\ning correlation functions [ 51]:⟨ˆfin(t)ˆf†\nin(t′)⟩= (Nk+\n1)δ(t−t′)and⟨ˆf†\nin(t)ˆfin(t′)⟩=Nkδ(t−t′), where Nk=\n1/\u0000\neℏωk/kBT−1\u0001\n(k∈{a,m}), are the equilibrium mean\nthermal photon, and magnon number, respectively, while\nNb= 1/\u0000\neℏΩb/kBT−1\u0001\nis the equilibrium mean thermal\nphonon number, Tis the environmental temperature and kBis\nthe Boltzmann constant . Eq. (4) represents the evolution of the\nfluctuation incorporating the interplay with the environment\nvia the noise operators [ 25]. The photon number and magnon\noccupation number are approximately zero, i.e., Na,m≈0,\ndue to the high frequencies of their modes.\nSince the nature of the noise is Gaussian, all the infor-\nmation is contained in the first and second-order moments\nof the operators. In particular, it is convenient to intro-\nduce the quadratures ˆxandˆyof the photon, magnon, and\nphonon modes by using the relation ˆO= (ˆxO+ iˆyO)/√\n2\nwithO∈ {a,m,b}, and elements of the corresponding co-\nvariance matrix are defined as Vij=1\n2⟨Ri(∞)Rj(∞) +\nRj(∞)Ri(∞)⟩, (i, j∈ {1,2,3,4,5,6}). Here, Ri(t) :=\n[ˆxa,ˆya,ˆxm,ˆym,ˆxb,ˆyb]⊤is the column vector of quadratures\nand the stationary covariance matrix is obtained by solv-\ning the algebraic equation A⊤V+A V +D= 0, where\nD=diag(γa, γa, γm, γm, γb(2Nb+ 1), γb(2Nb+ 1)) and\nthe drift matrix Ais expressed as:\nA=\n−γa∆a 0 gam 0 0\n−∆a−γa−gam 0 0 0\n0gam−γm˜∆m−Gmb 0\n−gam 0−˜∆m−γm 0 0\n0 0 0 0 0 Ω b\n0 0 0 Gmb−Ωb−γb\n.(5)\nThe system must be stable for a steady state to exist, to as-\ncertain this, the Routh-Hurwitz criterion [ 52] is employed to\ncharacterize the stability of the system. To achieve this, the\nreal part of the spectrum of the drift matrix A, Eq. (5), must be\nnegative, this means that all the eigenvalues of the drift matrix\nAhave non-positive real parts.\nIII. ENTROPY PRODUCTION RATE\nThe basic thermodynamics principle asserts the entropy of\nan open system, (classical or quantum) evolves as:\ndS\ndt= Π−Φ, (6)where Π⩾0is the irreversible entropy production rate and\nΦis the entropy flow from the system to the reservoir. In\nthermal equilibrium, the steady states are characterized by\ndS/dt=Π=Φ=0 . However, when the system is connected\nto multiple reservoirs or being externally driven, it may instead\nreach a nonequilibrium steady states where dS/dt= 0 but\nΠ=Φ⩾0. In this nonequilibrium steady state case, the system\nis characterized by the continuous production of entropy, all of\nwhich flows to the reservoir.\nWe now move to study the entropy production rate from a\nmultipartite system. In analogy with the bipartite case [ 41], we\ncombine quantum phase-space methods and the Fokker-Planck\nequation to characterise the irreversible entropy production of\nquantum systems interacting with reservoirs. In general, the\nentropy production rate of the quantum system described in\nSection II is given by (see, Appendix A):\nΠs:=3X\ni=12γi\u0012V2i−1,2i−1+V2i,2i\n2Ni+ 1−1\u0013\n, (7)\nwhere Γi∈ {Γa,Γm,Γb}withΓ∈ {γ,N}.\nHere, we focus on characterizing the entropy production\nrates in a cavity magnomechanical system. Without loss of\ngenerality, we consider the effective entropy flow between the\nmagnon mode and the mechanical resonator. Thus, the rate of\nentropy production Πsat a steady-state reads\nΠs= 2γm(V33+V44−1) + 2 γb\u0012V55+V66\n2Nb+ 1−1\u0013\n.(8)\nWe remark, when the system is in the equilibrium state, we\nhaveV11+V22= 1,V33+V44=2Nb+ 1, and hence, Πs≡0.\nTo proceed, we study the entropy production rate in a\nmagnon-phonon-photon system at a steady state in the resolved\nsideband, where the magnon dissipation rate is comparable\nto or well below the mechanical resonance frequency (i.e.,\nγm<Ωb). We assume the following parameters close to\nthose employed in the experimental realizations [ 12], as the\nphonon frequency Ωb/2π=10 MHz, the cavity dissipation rate\nγa/2π=3MHz, the magnon dissipation rate γm/2π=1MHz,\nthe phonon damping rate γb= 300 Hz, the phonon-magnon\ncoupling gmb/2π≃1Hz, and the temperature T= 10−100\nmK (i.e., Nb≃200). In the following analysis, we will uti-\nlize dimensionless quantities; that is, the quantities will be\nexpressed in units of the phonon frequency, Ωb.\nIn Fig. 2, we present the entropy production rate Πsis plot-\nted as a function of the normalized magnon detuning ∆m/Ωb\nfor various values of the photon-magnon coupling gam. In\nFig. 2(a) and (b), we consider the dissipation rate of the cavity\nγa= 10−1Ωbfor distinct occupation number Nb= 10 and\nNb= 100 , respectively. It can be seen that in the absence of\nmagnon-photon interaction gam= 0, the entropy production\nrateΠspeaks at ∆m/Ωb=±1. The two peaks in the rate of\nentropy production for positive/negative detuning imply the\ncooling/heating processes behave differently in two distinct\nregimes of the system. For non-zero gam, the smaller peak\nsmears out with reduced maximum Πs. We observe that for\ngam=Ωb(gam=2 Ω b), the maximum Πsis in the red (blue)4\n-6 -4 -2 0 2 4 60.010.050.100.501510\n-6 -4 -2 0 2 4 60.010.10110100\n-6 -4 -2 0 2 4 60.010.050.100.501510\n-6 -4 -2 0 2 4 60.010.10110100\nFIG. 2. Plot of entropy Πs(blue dashed) as a function of normalized\nmagnon detuning ∆m/Ωbfor different values of magnon-photon\ncoupling gam= 0(black curve), gam= Ω b(blue curve) and gam=\n2Ωb(red curve), where (a)Nb= 10 (b)Nb= 100 ,(c),Nb= 10 ,\nand(d)Nb= 100 . In(a)&(b):γa= 0.1Ωb, andγa= Ω bin(c)&\n(d). The other parameters are Gmb= 10−1Ωb,γb= 10−2Ωband\nγm=Ωb/2.\nsideband region. The maximum entropy flow between the\nphonon mode and the effective magnon mode occurs when\ngam=2Ω b, at∆m=ωb. Fig 2 (c) and (d) show more clearly\nthe effect and interplay between the cavity photon dissipation\nrate and the magnon-photon coupling. For the case of γb=Ωb,\nFig. 2(c) and (d), the value of Πsfurther decreases with a\nbroaden peak as magnon-phonon coupling gamincreases. In\nthe blue-sideband, for gam̸= 0, the value of Πsis always\nreduced compare to the gam=0scenario. It can be seen that\nΠscould be enhanced close to the negative detuning region\n∆m<0. The effects of thermal fluctuations of the environ-\nment on the entropy production rate is shown in Fig. 2. The\nFig. 2 shows that increasing the phonon thermal excitation\n(occupation number) Nb, increases Πsin magnitude (see the\nmagnitude of Πsin Fig. 2(a) and (c) compare to (b) and (d)\nrespectively). The non-uniform behaviour when varying gam\nfor different γais a direct implication of the complex nature of\nthe interaction between the dissipation processes in the cavity\nmagnomechanical system.\nIn Fig. 3, the effects of the cavity photon dissipation rate\nof the environment on the entropy production rate is explored.\nWe present in Fig.3 (a), the plot of the entropy production rate\nas a function of magnon-photon coupling with various values\nof the cavity decay rate, while Fig. 3 (b) shows the entropy\nproduction rate as function of cavity decay rate with different\nvalues of the magnon-photon coupling. For large thermal exci-\ntations, Nb=100 , and ∆m=Ωb, the entropy production rates\ndecrease in oscillatory form as the magnon-photon coupling\nincreases. For γa= 10−1Ωb, the maximum entropy produc-\ntion rate corresponds to the gam= 2Ω b. It can be seen that\nthe entropy production rate can be increase/decrease by tuning\nthe cavity decay value γaand the magnon-photon coupling,\nseegam≃0.2−3 Ωb. The entropy production rate Πslinearly\ndecreases with respect to the magnon-photon coupling when\n0 1 2 3 4 50.010.10110100\n0 1 2 3 4 50.10.5151050100FIG. 3. (a)The entropy production rate Πsas a function of magnon-\nphoton coupling gam/Ωbfor different values of cavity photon decay\nrateγa= 10−1Ωb(black curve), γa= Ω b(blue curve) and γa=\n2 Ωb(red curve). (b)Entropy production rate Πsas a function of\ncavity photon decay rate γa/Ωbfor different values of magnon-photon\ncoupling gam= 10−1Ωb(black curve), gam= Ω b(blue curve) and\ngam= 2 Ω b(red curve). Other parameters are Nb= 100 ,∆m= Ω b,\nGmb=10−1Ωb,γb=10−2Ωbandγm=Ωb/2.\ngam>3/Ωbbut the value slightly increase when we increase\nγa. This non-monotonic behaviour (increase/decrease) of the\nentropy production with respect to the magnon-photon cou-\npling can be attributed to the imbalance in populations between\nthe interacting modes induced by the environment. Further-\nmore, Fig. 3 (b) shows that the entropy production rate can\nincrease/decrease when γa< gambut increases slightly after\nγa=gam. In the region, γa≫gam, the entropy production rate\nsaturates to a quasi-constant value with decreasing value as the\nmagnon-photon gamincreases. This is due to the fact that the\nindividual modes are far from resonance and they are effec-\ntively decoupled, in such a way that the individually thermalize\ntheir own bath.\nNext, we analyze the behaviour of the entropy production\nrate with respect to the amount of correlations shared between\nthe magnon and phonon modes. For coupled quantum system,\nit has been demonstrated that the irreversibility generated by\nthe dissipative system at steady state and the total amount of\ncorrelations shared between the subsystems are closely related\nin small coupling limit as [ 41];I ≈Πs/(2γtot), where Iis\nthe quantum mutual information between the modes at the\nstationary state and γtotis the sum of the dissipation rates to\nthe local baths. Since the quantum noises are Gaussian, the\nmutual information between the magnon and phonon modes\ncan be computed as [53]:\nI(Va:b) =1\n2ln\u0012det(Va)det(Vb)\ndet(Vab)\u0013\n, (9)\nwhere Va(Vb)is the covariance matrix of magnon (mechani-\ncal) mode.\nIn Fig. 4, we show a comparative plot of the entropy pro-\nduction rate ΠSto the correlations established by the cavity\nmagnomechanical system, as quantified by the mutual informa-\ntionI. Considering small thermal phonon excitation, Nb=10 ,\nFig. 4 (a) shows a good similarity in both ΠsandIcurves\nfor|∆m/Ωb|⩽2. As the magnon-photon coupling increases,\nboth the entropy production rate and mutual information are de-\ncreasing as well as an increase in both quantities deviation, see\nFig. 4 (b) and (c). The increase in deviation between ΠsandI\nfor increasing detuning ∆m, clearly demonstrates the interplay5\n-4 -2 0 2 40.010.050.100.501510\n-4 -2 0 2 40.0010.0050.0100.0500.1000.5001\n-4 -2 0 2 40.0010.0050.0100.0500.1000.5001\nFIG. 4. Entropy rate Πs(blue curve) and mutual information I(red curve) as a function of normalized magnon detuning ∆m/Ωbfor\ndifferent values of magnon-photon coupling (a) gam=0, (b)gam=Ωband (c) gam=2 Ω b. Here, other parameters are γa= Ω b,Nb= 10 ,\nGmb=10−1Ωb,γb= 10−2Ωbandγm= Ω b/2.\namong magnon-photon coupling and dissipation rates. This\nshows that degree of irreversibility induced by the stationary\nprocess is directly related to the amount of correlations shared\nin the system.\nIV . CONCLUSIONS\nWe have investigated the entropy production rate in a hy-\nbrid magnomechanical system where a microwave cavity mode\nis coupled to a magnon mode in a YIG sphere, and the latter\nis simultaneously coupled to a phonon mode via magnetostric-\ntive force. Specifically, within the quantum phase space for-\nmulation of the entropy change, we evaluate the steady-state\nentropy production rate and associated quantum correlation\nin the hybrid system. We have shown that the entropy flow\nbetween the effective magnon mode and the phonon mode\nis influenced by the magnon-photon coupling and the cavity\nphoton dissipation rate. Our numerical analysis shows non-\nuniform behavior of irreversibility resulting from the complex\nand competing nature of the interactions in the cavity mag-\nnomechanical system. We find that the entropy production rate\ncan be increased/decreased when the cavity decay rate is less\nthan the magnon-photon coupling. Furthermore, we studied\nthe range of validity of the link between irreversibility and\nthe amount of correlations in a mesoscopic quantum system.\nThese results provide insight into the impact of magnonics\non the thermodynamics processes of hybrid quantum systems.\nFinally, we anticipate that our study will open new perspectives\nfor quantum thermodynamics applications, such as realizing\nthermal machines in the deep quantum regime and quantum\nthermal transport.\nACKNOWLEDGEMENT\nCOE and NA have been supported by LRGS Grant\nLRGS/1/2020/UM/01/5/2 (9012-00009) provided by the Min-\nistry of Higher Education of Malaysia (MOHE). MA and DD\nhave been supported by the Khalifa University of Science and\nTechnology under Award No. FSU- 2023-014.Appendix A: Entropy production\nIn this appendix, we provide the explicit expression for\nthe entropy production rate presented in Eq. 7. Considering\nthe Gaussian nature of the quantum system, their states are\ncharacterized by the Wigner function of the form\nW(R) =1\nπn√\ndetσexp\u0012\n−1\n2(R−¯R)⊤σ−1(R−¯R)\u0013\n,\n(A1)\nwhere nis the number of the bosonic mode and σis the\ncovariance matrix (CM), whose entries are given by σij=\n1\n2⟨RiRj+RjRi⟩ − ⟨Ri⟩⟨Rj⟩. The entropy of the Gaussian\nsystem is associated to the corresponding Shannon entropy\n(Wigner entropy) of the Wigner distribution.\nFollowing the quantum Langenvin equations for the system\ndynamics described in Section II, we can recast the Fokker-\nPlanck equation for the Wigner function W(u, t)as a local\nconservation equation\n∂tW=−∂uJ(u, t), (A2)\nwhere u= (xa, ya, xm, ym, xb, xb)⊤is a point in phase space,\n∂uis the phase-space gradient, and Jis the total probability\ncurrent vector that reads\nJ(u, t)=AuW(u, t)−1\n2D∂uW(u, t). (A3)\nFor the three mode bosonic system model that we consider,\nthe drift matrix Aand the diffusion matrix Dare defined in\nSection II (Eq. 5). Then, introducing the time-reversal operator\nE=diag(1,−1,1,−1,1,−1), the dynamical variables can\nbe split according to their symmetry. The drift matrix A\ncan be rewritten by splitting it into the irreversible part Airr\nwhich is even under time reversal and the reversible part Arev\nthat is odd [ 41]. They can be evaluated as Airr:=1\n2(A+\nEAE⊤)andArev:=1\n2(A−EAE⊤). The irreversible part\nis associated with the damping rates, while the reversible part\ncomes from the Hamiltonian part of the dynamics. The drift\nmatrix separation results in splitting of the probability currents\nasJ(u, t)≡ Jrev(u, t) +Jirr(u, t), where\nJrev(u, t) :=ArevuW(u, t), (A4)6\nand\nJirr(u, t) :=AirruW(u, t)−1\n2D∂uW(u, t). (A5)\nTaking the derivative of the Wigner entropy with respect to\ntime and integrating by parts, we get\nSW=Z\nd2α(JirrW)⊤\u0012∂uW\nW\u0013\n. (A6)\nEquation (A6) can be rewritten in the usual form in terms of\nthe entropy production rate [41]:\nΠ =1\n2Z\ndu1\nW(u, t)Jirr(u, t)⊤D−1Jirr(u, t), (A7)\nand entropy flux rate\nΦ =−1\n2Z\nduJirr(u, t)D−1Airru. (A8)\nMoreover, for a Gaussian state, ∂uWσ(u, t) =\n−Wσ(u, t)σ−1(t)u, then the irreversible component of\nprobability current Jirr(u, t) =Wσ(u, t)(Airru+D)σ−1(t)u.\nTherefore, the explicit integration of Eq. (A7) gives theentropy production rate as [41]\nΠ =1\n2tr\u0002\nσ−1D\u0003\n+ 2tr[Airr] + 2tr\u0002\n(Airr)⊤D−1Airrσ\u0003\n.\n(A9)\nFor the case of three mode bosonic system, in stationary state,\nwe obtain\nΠs= 2γa\u0012V11+V22\n2Na+ 1−1\u0013\n+ 2γm\u0012V33+V44\n2Nm+ 1−1\u0013\n+ 2γb\u0012V55+V66\n2Nb+ 1−1\u0013\n(A10)\n=3X\ni=12γi\u0012V2i−1,2i−1+V2i,2i\n2Ni+ 1−1\u0013\n(A11)\nwhere Γi∈ {Γa,Γm,Γb}withΓ∈ {γ,N}.\nWithin this framework, the rate of entropy production Πs\nat steady-state for two coupled oscillators mode is expressed\nas follows [41, 44],\nΠs= Tr\u0000\n2AirrD−1AirrV+Airr\u0001\n= 2γm(V33+V44−1) + 2 γb\u0012V55+V66\n2Nb+ 1−1\u0013\n,\n(A12)\nwhereAirr=diag{0,0,−γm,−γm,−γb,−γb}. This is the\nexpression given in the main text.\n[1]Z.-L. Xiang, S. Ashhab, J. Q. You, and F. Nori, Hybrid quan-\ntum circuits: Superconducting circuits interacting with other\nquantum systems, Rev. Mod. Phys. 85, 623 (2013).\n[2]B. Bauer, D. Wecker, A. J. Millis, M. B. Hastings, and M. 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Lett. 109, 190502 (2012)."    },    {        "title": "1404.2360v2.Current_induced_spin_torque_resonance_of_magnetic_insulators.pdf",        "content": "arXiv:1404.2360v2  [cond-mat.mes-hall]  20 Aug 2014Current-induced spin torque resonance of magnetic insulat ors\nTakahiro Chiba1, Gerrit E. W. Bauer1,2,3, and Saburo Takahashi1\n1Institute for Materials Research, Tohoku University, Send ai, Miyagi 980-8577, Japan\n2WPI-AIMR, Tohoku University, Sendai, Miyagi 980-8577, Jap an and\n3Kavli Institute of NanoScience, Delft University of Techno logy, Lorentzweg 1, 2628 CJ Delft, The Netherlands\n(Dated: August 10, 2018)\nWe formulate atheory ofthe ACspinHall magnetoresistance ( SMR)in abilayer system consisting\nof a magnetic insulator such as yttrium iron garnet (YIG) and a heavy metal such as platinum (Pt).\nWe derive expressions for the DC voltage generation based on the drift-diffusion spin model and\nquantum mechanical boundary condition at the interface tha t reveal a spin torque ferromagnetic\nresonance (ST-FMR). We predict that ST-FMR experiments wil l reveal valuable information on the\ncurrent-induced magnetization dynamics of magnetic insul ators and AC spin Hall effect.\nFerrimagnetic insulators such as yttrium iron garnet\n(YIG) with high critical temperatures and very low mag-\nnetization damping have been known for decades to be\nchoice materials for in optical, microwave or data stor-\nage technologies [1]. Near-dissipationless propagation of\nspin waves make YIG wires and circuits interesting for\nlow power data transmission and logic devices. A cru-\ncial breakthrough was the discovery that the magneti-\nzation in YIG can be excited electrically by Pt con-\ntacts [2], thereby creating an interface between elec-\ntronic/spintronic and magnonic circuits. However, the\ngeneration of coherent spin waves by the current-induced\nspin orbit torques in Pt is a strongly non-linear process\nand the low critical threshold currents found by the ex-\nperiments [2] cannot yet be explained by theory [3]. Here\nwe suggest and model a simpler method to get to grips\nwith the important current-magnetization interaction in\nthe YIG|Pt system without a problematic threshold, viz.\nby employing the recently discovered magnetoresistance\nofYIG|PtbilayersorSpinHallMagnetoresistance(SMR)\n[4, 5] to detect current-inducedspin torqueferromagnetic\nresonance (ST-FMR).\nThe SMR is the dependence of the electrical resistance\nofthenormalmetalonthe magnetizationangleofaprox-\nimity insulator and is caused by a concerted action of\nthe Spin Hall Effect (SHE) [ ?] and its inverse (ISHE).\nAn alternative mechanism of the SMR phenomenology\nin terms of an equilibrium proximity magnetization close\nto the YIG interface has been proposed [5]. However,\nthis interpretation has been challenged by experiments\n[4, 7]. Moreover, while experiments of many groups are\ndescribedquantitativelywell bythe SMR model with one\nset of parameters [8–12], we are not aware of a transport\ntheory that explains the observed magnetoresistance in\nterms of a monolayer-order magnetic Pt.\nCurrent-induced spin torque ferromagnetic resonance\n(ST-FMR) has been demonstrated [13–15] in bilayer thin\nfilms made from metallic ferromagnets (FM) and non-\nmagnetic metals (N). In these experiments the SHE\ntransforms an in-plane alternating current (AC) into an\noscillating transverse spin current. The resultant spin\ntransfer resonates with the magnetization at the FMR\nfrequency. The effects induced simultaneously by the\nOersted field can be distinguished by a different sym-metry of the resonance on detuning. The magnetiza-\ntion dynamics leads to a time dependence of the bilayer\nresistance by the anisotropic magnetoresistance (AMR).\nMixing the applied current and the oscillating resistance\ngenerates a DC voltage that is referred to as spin torque\ndiode effect [16, 17].\nThe longitudinal spin Seebeck effect was found to be\nfrequency independent up to 30 MHz [18]. The DC ISHE\ninduced by spin pumping has been observed by many\ngroups, but detection of the AC spin Hall effect [23] has\nonly recently been reported in metallic structures [19–\n21] as well as in Pt |YIG under parametric microwave\nexcitation[22]. ADCvoltagecanbegeneratedinPt |YIG\nunder FMR conditions by rectification of the AC spin\nHall effect by means of the SMR, but this signal was\nfound to be swamped by the DC spin Hall effect [24].\nA study of the spin Hall impedance concludes that the\nmaterial constants of Pt |YIG bilayers do not depend on\nfrequency up to 4 GHz [25].\nIn this paper we suggest to combine the principles\nsketched above to realize ST-FMR for bilayers of a ferro-\nor ferrimagnetic insulator (FI) such as YIG and a normal\nmetal with spin orbit interaction (N) such as platinum\n[26] (see Fig. 1). We derive the magnetization dynam-\nics and DC voltages generated by the SMR-induced spin\ntorque diode effect as a function of the external mag-\nnetic field. Our theory should help to better understand\nthe elusive current-induced magnetization dynamics of\nferromagnetic insulators which should pave the way for\nlow-power devices based on magnetic insulators [1].\nThe spin current through an F |N interface is governed\nby the complex spin-mixing conductance G↑↓[27]. The\nprediction of a large Re G↑↓for interfaces between YIG\nand simple metals by first-principle calculations [28] has\nbeen confirmed by recent experiments [29, 30]. The spin\ntransport in N (spin Hall system) can be treated by\nspin-diffusiontheorywithquantummechanicalboundary\nconditions at the interface to the insulating ferromagnet\n[5, 31]. The AC current with frequency ωa= 2πfain-\nduces a spin accumulation distribution µs(z,t) in N that\nobeys the spin-diffusion equation\n∂tµs=D∂2\nzµs−µs\nτsf, (1)2\nFIG. 1. Schematic set-up to observe the SMR mediated spin\ntorque diode effect. The light blue rectangle is anormal meta l\n(N) film with a finite spin Hall angle, while F is a ferromag-\nnetic insulator. The F |N bilayer film is patterned into a strip\nwith width wand length h. A Bias-Tee allows detection of a\nDC voltage under an AC bias.\nwhereDis the charge diffusion constant and τsfspin-flip\nrelaxation time in N. In position-frequency space the so-\nlution for the spatiotemporal dependence of the spin ac-\ncumulation reads µs(z,ω) =Ae−κ(ω)z+Beκ(ω)z, where\nκ2(ω) = (1 + iωτsf)/λ2,λ=√Dτsfis the spin-diffusion\nlength, and the constant column vectors AandBare\ndetermined by the boundary conditions for the spin cur-\nrent density in the z-direction Js,z(z), where Js,z/|Js,z|\nis the spin polarization vector, which is continuous at the\ninterface to the ferromagnet at z= 0 and vanishes at the\nvacuum interface at z=dN. For planar interfaces\nJs,z(z,ω) =−θSHJc(ω)ˆ y−σ∂zµs(z,ω)\n2e,(2)\nwhereθSHis the spin Hall angle, σthe electri-\ncal conductivity, and Jc(ω) = 2πJ0\ncδ(ωa−ω) the\ncurrents not accounting for spin-orbit interaction.\nJs,z(dN,ω) = 0 and Js,z(0,ω) =/integraltext∞\n−∞Js,z(0,t)e−iωtdt,\nwhereJs,z(0,t) =JT\ns+JP\ns=J(F)\nswith\nJT\ns=Gr\neˆM×/parenleftBig\nˆM×µs(0)/parenrightBig\n+Gi\neˆM×µs(0),(3)\nJP\ns=/planckover2pi1\ne/parenleftBig\nGrˆM×∂tˆM+Gi∂tˆM/parenrightBig\n, (4)\nwhereˆMis the unit vector along the FI magnetization\nandG↑↓=Gr+iGithe complex spin-mixing interface\nconductance per unit area of the FI |N interface. The\nimaginary part Gican be interpreted as an effective ex-\nchange field acting on the spin accumulation, which is\nusually much smaller than the real part. A positive J(F)\ns\n[10, 28] corresponds here to up-spins flowing from FI into\nN. For Pt(Ta) ωaτPt(Ta)\nsf= 1(15)×10−3at the FMR fre-\nquencyfa= 15.5GHz with τPt(Ta)\nsf= 0.01(0.15)ps, in-\ndicating that the condition ωaτPt(Ta)\nsf≪1 is fulfilled forthese metals [23]. In this limit the frequency dependence\nof the spin diffusion length may be disregarded such that\nµs(z,t)\n→ −ˆ yµs0(t)sinh2z−dN\n2λ\nsinhdN\n2λ+J(F)\ns2eλ\nσcoshz−dN\nλ\nsinhdN\nλ,(5)\nJ(F)\ns=µs0(t)\ne/bracketleftBig\nˆM×/parenleftBig\nˆM׈ y/parenrightBig\nRe+ˆM׈ yIm/bracketrightBig\nT\n+/planckover2pi1\ne/bracketleftBig/parenleftBig\nˆM×∂tˆM/parenrightBig\nRe+∂tˆMIm/bracketrightBig\nT, (6)\nwhereµs0(t) = (2 eλ/σ)θSHJ0\nc(t)tanh[dN/(2λ)]\nwithJ0\nc(t) = J0\ncRe(eiωat) and T=\nσG↑↓//bracketleftbig\nσ+2λG↑↓coth(dN/λ)/bracketrightbig\n. The ISHE drives a\ncharge current in the x-yplane by the diffusion spin\ncurrent along z. The total charge current density reads\nJc(z,t) =J0\nc(t)ˆ x+σθSH/parenleftbigg\n∇×µs(z,t)\n2e/parenrightbigg\n.(7)\nThe averaged current density over the film thickness is\nJc,x(t) =d−1\nN/integraltextdN\n0Jc,x(z,t)dz=JSMR(t)+JSP(t) with\nJSMR(t) =J0\nc(t)/bracketleftbigg\n1−∆ρ0\nρ−∆ρ1\nρ/parenleftBig\n1−ˆM2\ny/parenrightBig/bracketrightbigg\n,(8)\nJSP(t) =JP\nrω−1\na/parenleftBig\nˆM×∂tˆM/parenrightBig\ny+JP\niω−1\na∂tˆMy,(9)\nJP\nr(i)=/planckover2pi1ωa\n2edNρθSHRe(Im)η,\nwhereJSMR(t) andJSP(t) are SMR rectification and\nspin pumping-induced charge currents, ρ=σ−1is\nthe resistivity of the bulk normal metal layer and\nwe recognize the conventional DC SMR with ∆ ρ0=\n−ρθ2\nSH(2λ/dN)tanh(dN/2λ) and ∆ ρ1=−∆ρ0Reη/2,\nwhere\nη=2λρG↑↓tanhdN\n2λ\n1+2λρG↑↓cothdN\nλ, (10)\nare effective resistivities that do not depend on frequency\n[5].\nST-FMRexperimentsemploytheACimpedanceofthe\noscillating transverse spin Hall current caused by the in-\nduced magnetization dynamics that is described by the\nLandau-Lifshitz-Gilbert (LLG) equation, including the\ntransverse spin current Eq. (6),\n∂tˆM=−γˆM×Heff+α0ˆM×∂tˆM+γ/planckover2pi1J(F)\ns\n2eMsdF,(11)\nwhereHeff=Hex+Hdywith an external magnetic field\nHexand the sum of the AC current-induced Oersted field\nand the (thin film limit of) the dynamic demagnetization\nHdy=Hac(t)+Hd(t) =/parenleftbig\n0,Haceiωat,−4πMz(t)/parenrightbig\n.γ,α0,\nMsanddFarethegyromagneticratio,theGilbert damp-\ning constant of the isolated film, the saturation magneti-\nzation, and the thickness of the FI film, respectively.3\nWe henceforth disregard the very low in-plane magne-\ntocrystalline anisotropy field of Hk∼3Oe reported [8].\nThe external magnetic field Hexis applied at a polar an-\ngleθin thex-yplane. It is convenient to consider the\nmagnetization dynamics in the XYZ-coordinate system\n(Fig.1)in whichthemagnetizationisstabilizedalongthe\nX-axis by a sufficiently strong external magnetic field.\nDenoting the transformation matrix as R(θ), the mag-\nnetization MR(t) =R(θ)M(t) precesses around the X-\naxis, where MR(t) =M0\nR+mR(t)≈(Ms,mY(t),mZ(t))\nas shown in Fig. 1. M0\nRandmR(t) are the static\nand the dynamic components of the magnetization, re-\nspectively. The LLG equation in the XYZ-system then\nbecomes β∂tMR=−γMR×Heff,R+αˆMR×∂tMR\nwhere the effective magnetic field in the XYZ-system\nisHeff,R=HXˆX+HYeiωatˆY+/parenleftbig\nHZeiωat−4πmZ(t)/parenrightbigˆZ\nwithHX=Hex,HY= (Hac+Hi)cosθandHZ=\nHrcosθwith\nHr(i)=/planckover2pi1\n2eMsdFθSHJ0\ncRe(Im)η, (12)\na modulated damping α=α0+ ∆αand g-factor β=\n1−∆βwith ∆α(∆β) =γ/planckover2pi12/(2e2MsdF)ReT(ImT).For a small-angle precession around the equilib-\nrium direction M0\nR,mR(t) = (0,δmYeiωat,δmZeiωat)\n(Re[δmY] Re[δmZ]≪Ms). Disregarding higher orders\ninδmY(Z)in theR-transformed LLG equation we ar-\nrive at the (Kittel) relation between AC current fre-\nquency and resonant magnetic field HF=−2πMs+/radicalbig\n(2πMs)2+(ωa/γ)2.\nA DC voltage is generated by two different mecha-\nnisms, viz. the time-dependent oscillations of the SMR\nin N (spin torque diode effect) and the ISHE generated\nby spin pumping. This is quite analogous to electrically\ndetected FMR in which the magnetization is driven by\nmicrowavesin cavitiesorcoplanarwaveguides. In metal-\nlic bilayers, the spin pumping signal due to the ISHE can\nbe separated from effects of the magnetoresistance of the\nmetallic ferromagnet by sample design and angular de-\npendences [32, 33]. Here we focus on the current-induced\nmagnetization dynamics that induces down-converted\nDC and second harmonic components in the normal\nmetal. Indicating time-average by /angb∇acketleft···/angb∇acket∇ighttthe open-circuit\nDC voltage is VDC=hρ/angb∇acketleftJc,x(t)/angb∇acket∇ightt=VSMR+VSP,where\nVX=hρ/angb∇acketleftJX(t)/angb∇acket∇ightt. The SMR rectification and spin pump-\ning induced DC voltage are\nVSMR=−h∆ρ1J0\nc\n4FS(Hex)\n∆/bracketleftbigg\nC(Hr+αHac)+C+HacHex−HF\n∆/bracketrightbigg\ncosθsin2θ, (13)\nVSP=hρJP\nr\n4FS(Hex)\n∆C/bracketleftbigg\nC−H2\nr+αHrHac\n∆+C+H2\nac−αHrHac\n∆/bracketrightbigg\ncosθsin2θ, (14)\nwhereC= ˜ωa//radicalbig\n1+ ˜ω2aandC±= 1±1//radicalbig\n1+ ˜ω2awith\n˜ωa=ωa/(2πMsγ),FS(Hex) = ∆2/[(Hex−HF)2+∆2],\n∆ =αωa/γthe line width, Hac= 2πJ0\ncdN/cthe Oersted\nfield from the AC current determined by Amp` ere’s Law\n(in the limit of an extended film), and cspeed of light.\nUsing the material parametersfor YIG [2] and Pt [30, 33]\nshown in Tables I and II we compute the DC voltages in\nEqs. (13) and (14). The calculated VSMRis plotted in\nFig. 2 as a function of an external magnetic field and\nfor different dF, resolved in terms of the contributions to\nthe FMR caused by the spin transfer torque (symmetric)\nand the Oersted magnetic field (asymmetric). In Fig. 3\nweshowthetotalDCvoltagewithbothspintorquediode\nand spin pumping contributions. The DC voltagein F |Pt\nbilayersdependsmoresensitivelyon dFforF = YIGthan\nF = Py/CoFeB because spin pumping is more important\nwhen the Gilbert damping is small. ST-FMR measure-\nments are carried out at relatively high current density,\nso Joule heating in Pt may cause observable effects, the\nmost notable being the spin Seebeck effect (SSE), which\nadds a constant background DC voltage to the SMR rec-\ntification signal [34].\nTheST-FMR spectrainFig.3areenhancedforthickerTABLE I. Material parameters for the FI layer.\nγ[T−1s−1]Ms[Am−1] α0\naYIG 1.76 ×10111.56×1056.7×10−5\naReference 2.\nTABLE II. Material parameters for the N layer.\nGr[Ω−1m−2]ρ[µΩcm] λ[nm] θSH\nPta3.8×1014 a41b1.4b0.12\naReference 30,bReference 33.\nF layers, but these are dominated by the Oersted field\nactuation. These less-interesting contributions can be\neliminatedinatri-layerstructureasinFig.4inwhichthe\nmagnetic insulator is sandwiched by two normal metal\nfilms with the same electric impedance. The second film\nN2 should be Cu or another metal with negligible spin-\norbit interaction and thereby contributions to the ST-\nFMR, the quality of the YIG |N2 interface is therefore\nless of an issue. In Fig. 4 we plot pure ST-FMR signals\nobtained by setting Hac= 0 in Eqs. (13) and (14), which4\n-1.2-0.8-0.4 0  0.4 0.8\n 2.2  2.3  2.4  2.5  2.6 dF=4nm\n=10nm\n=40nmVSMR  ( µV) \nHex  (kOe) YIG(dFnm)|Pt(6nm)\n-0.8-0.6-0.4-0.2 0  0.2\n 2  2.2  2.4  2.6  2.8 Total result Spin torque FMR \nOersted field FMR YIG(4nm)|Pt(6nm)VSMR  ( µV) \nHex  (kOe) (b)(a)\nθ=45rHex \ndF (nm)FI Hex \nFIG. 2. (a) The ferromagnet thickness dependence of calcu-\nlated SMR rectificied voltage for YIG |Pt atfa= 9GHz with\ncurrent density J0\nc= 1010A/m2and F(N) layer length and\nwidthh=w= 30µm andθ= 45◦. (b)dF(N)= 4(6) nm.\n-0.5 0  0.5 1.5 2.5 3.5\n 2.2  2.3  2.4  2.5  2.6 \nHex  (kOe) VDC  ( µV) \ndF=4nm \n=10nm \n=40nm \nYIG(dFnm)|Pt(6nm)-0.8  0  0.6 \n 2  2.2  2.4  2.6  2.8 \nTotal result SMR (V SMR )\nISHE (V SP )dF=4nm VDC  ( µV) \nHex  (kOe) \nFIG. 3. Dependence of the ST-FMR spectra on dFat\nfa= 9GHz and θ= 45◦. Inset: Contributions by SMR\nrectification and spin pumping for dF= 4nm.-0.25-0.2-0.15-0.1-0.05 0 \n 2.2  2.3  2.4  2.5  2.6 \nHex  (kOe) VDC  ( µV) \ndF=4nm \n=10nm \n=40nm \nYIG(dFnm)|Pt(6nm)\nFIG. 4. ST-FMR spectra dependence on dFin a trilayer set-\nup to observe the spin torque induced DC voltages without\nartifacts of the Oersted field. ( fa= 9GHz and θ= 45◦)\nmay now be observed also for thick magnetic layers.\nIn summary, we predict observable AC current-driven\nST-FMR in bilayer systems consisting of a ferromagnetic\ninsulatorsuchasYIG and anormalmetal with spin-orbit\ninteraction such as Pt. Our main results are the DC volt-\nages caused by an AC current as a function of in-plane\nexternal magnetic field and film thickness of a magnetic\ninsulator. The DC voltages generated in YIG |Pt bilay-\ners depend sensitively on the ferromagnet layer thick-\nness because of the small bulk Gilbert damping. The\npredictions can be tested experimentally by ST-FMR-\nlike experiments with a magnetic insulator that would\nyieldimportantinsightsintothenatureoftheconduction\nelectron spin-magnon exchange interaction and current-\ninduced spin wave excitations at the interface of metals\nand magnetic insulators.\nThis work was supported by KAKENHI (Grants-in-\nAid for Scientific Research) Nos. 22540346, 25247056,\n25220910, and 268063, FOM (Stichting voor Funda-\nmenteel Onderzoek der Materie), the ICC-IMR, the\nEU-RTN Spinicur, EU-FET grant InSpin 612759, and\nDFGPriorityProgramme1538“Spin-CaloricTransport”\n(Grant No. BA 2954/1).\n[1] Recent Advances in Magnetic Insulators - From Spin-\ntronics to Microwave Applications, edited by M. Wu andA. Hoffmann, Solid State Physics 64(Academic Press,5\n2013).\n[2] Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida,\nM. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K.\nTakanashi, S. Maekawa, and E. 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Goennenwein, Sign of\ninverse spin Hall voltages generated by ferromagnetic\nresonance and temperature gradients in yttrium iron\ngarnet|platinum bilayers, arXiv:1404.3490."    },    {        "title": "1701.01462v2.Magnetic_control_of_Goos_Hanchen_shifts_in_a_yttrium_iron_garnet_film.pdf",        "content": "1 \n Magnetic control of Goos-Hänchen shi fts in a yttrium-iron-garnet film  \nWenjing Yu,1 Hua Sun,1,3 & Lei Gao1,2,* \n1College of Physics, Optoelectronics and Energy of Soochow University, Collaborative Innovation Center of \nSuzhou Nano Science and Technology, Soochow University, Suzhou 215006, China.  \n2Jiangsu Key Laboratory of Thin Films, Soochow University, Suzhou 215006, China. \n3hsun@suda.edu.cn    \n*leigao@suda.edu.cn \n \nAbstract:  \nWe investigate the Goos-Hänchen (GH) shifts reflected and transmitted by a \nyttrium-iron-garnet (YIG) film for both normal and oblique incidence. It is found that \nthe nonreciprocity effect of the MO material does not only result in a nonvanishing reflected shift at normal incidence, but also  leads to a slab-thickness-independent term \nwhich breaks the symmetry between the reflected and transmitted shifts at oblique \nincidence. The asymptotic behaviors of the normal-incidence reflected shift are \nobtained in the vicinity of two characteristic frequencies corresponding to a minimum reflectivity and a total reflection, respectively. Moreover, the coexistence of two types of negative-reflected-shift (NRS) at oblique incidence is discussed. We show that the \nreversal of the shifts from positive to negative values can be realized by tuning the \nmagnitude of applied magnetic field, the frequency of incident wave and the slab thickness as well as the incident angle. In addition, we further investigate two special cases for practical purposes: the reflected shift with a total reflection and the \ntransmitted shift with a total transmission. Numerical simulations are also performed \nto verify our analytical results. \n \nIntroduction:\n  \nThe GH effect refers to the lateral shift of an incident beam of finite width upon \nreflection from an interface which was first studied by Goos and Hänchen1,2 and \ntheoretically explained by Artmann in terms of the stationary-phase approach in the \nlate 1940s3. Since then, such effect has been very important with development of the \nlaser beams and integrated optics4 and has significant impact on applications as well \nas for investigations of the fundamental problems in physics. And the studies have \nbeen extended from a simple dielectric interface to more complex structures or exotic materials such as metal-dielectric nanocomposites\n5,6, epsilon-near-zero metamaterials7, \ngraphene8-10, PT-symmetric medium11, topological insulator12 etc.  \nThe GH shift by magneto-optical (MO) materials13-18 is obtained by making use \nof ferromagnetic resonances of natural magnetic materials. Similar relation between  \nGH effects and intrinsic resonances was also reported in nonmagnetic dielectric, such as GH shifts arising from phonon resonances in crystal quartz\n19. But what was found \nin MO materials is of particular interest because of the nonreciprocity in scattering 2 \n coefficients originated from the broken time reversal symmetry20. As a result, a lateral \nshift for reflection will occur at the interface between the vacuum and an magnetic material arranged in the V oigt geometry even at normal incidence\n14,15, with both sign \nand magnitude controlled by the applied magnetic field. And the \npolarization-dependence of the GH shift by MO materials makes it possible to separate the incident radiation into beams of different polarizations\n21. However, the \ndetails of the magnetic effects on GH shift are stilled obscure. Most studies only \ndiscussed the effects of a semi-infinite antiferromagnetic material —MnF 2 at low \ntemperature ( T=4.2K), with a dispersion quite different from that of conventional MO \nmaterials adopted in applications. The ro le of material properties and geometric \nfactors (such as finite slab thickness, incident angles etc.) remains unclarified in the \nmagnetic control of GH shifts with MO materials. \nHence we are motivated to perform a theoretical investigation of the GH shifts \nreflected and transmitted by a MO slab made of yttrium-iron-garnet (YIG). As a ferrite well known for its high MO efficiency and low damping\n22-27, YIG has been \nextensively studied and broadly adopted in microwave28-30 and magneto-optics \ntechnologies31-33. The recent realization of one -way waveguides based on YIG \nphotonic crystals sparks even more interest of the application of this traditional MO \nmaterial in the field of subwavelength optics34,35. It was also shown that hyperbolic \ndispersion and negative refraction initia lly investigated in antiferromagnetic \nmaterials36 can be extended to and realized in conventional ferrites37. But the GH-shift \neffects due to a surface/slab of YIG have not been studied.  \nIn this paper we present a theoretical anal ysis of the lateral shifts of both the \nreflected beam and the transmitted beam due to a magnetized YIG slab in the V oigt \ngeometry. It is shown that the nonreciprocity effect caused by the MO material does not only result in a nonvanishing reflected shift at normal incidence, but also leads to a slab-thickness-independent term which br eaks the symmetry between the reflected \nand transmitted shifts at oblique incidence. The asymptotic behaviors of the \nnormal-incidence reflected shift are obtained in the vicinity of two characteristic frequencies (\nrω and cω) corresponding to a minimum reflectivity and a total \nreflection, respectively. And the coexistence of two types of negative-reflected-shift \n(NRS) at oblique incidence is discussed. We also investigate two special cases for \npractical purposes: the reflected shift with a total reflection and the transmitted shift \nwith a total transmission. Analytical expressions of the shifts in these cases are obtained approximately, which is in good agreement with the results from numerical \ncalculations. \n 3 \n \n \nFIG. 1. Schematic diagram of the structur e in the presence of an external field 0h. The incident \nplane wave is polarized along the z-direction and propagates along k\n.  \n \nResults \nGeneral formulas. Consider a YIG film of thickness d surrounded by a \nnon-magnetic background medium of (11,εμ) as shown in Fig. 1. For simplicity we \nset the background medium in Region 1&3 as the vacuum. The magnetic permeability of YIG magnetized along the z\n-axis is of the tensor form \n                             0\n0\n00 1ri\niri\niμμ\nμμ μ−\n=                          (1) \nwhere the digonal and off-diagonal permeabi lities follow the typical dispersion of \nferrites in the microwave region  \n               0\n22\n0()1()m\nri\niωω α ωμωα ω ω+=++−                     (2a) \n                            22\n0()m\niiωωμωα ω ω=+−                        (2b)  \nHere ω is the frequency of the incident light, 0ω, mω are the magnetic resonance \nfrequencies given by \n                            002h ωπ γ= ,                          ( 3 a )  \n                            2ms m ωπ γ=                          ( 3 b )  \nwith 0h and sm denoting the applied magnetic field and the saturated \nmagnetization, respectively. The material parameters of YIG are chosen as:  \n32.8 10 GHz/Oeγ−=× , 1800Gausssm= , 14.5ε=35. The damping factor α is \nquite small and neglected in the following analytic derivations and calculations. However, later in the numerical simulations, we have considered the influence of 4 \n the realistic damping of YIG material. \nTo find the GH shifts due to such a YIG slab, we start by considering an \ns-polarized plane wave of angular frequency ω incident from Region 1 at an angle \nθ. Then the x component of the wave vectors in layers 1, 3 and 2 are given by \n                              1/22\n2\n13 2,xx ykk kcω =−  ,                    ( 4 a )  \n                             1/22\n2\n2 2 xe f f ykkcωεμ =−  ,                   ( 4 b )  \nwith  \n0sinykk θ =                              (5) \nHere ε is the dielectric constant of YIG and effμ is its effective permeability \ngiven by14 \n                        22() /eff r i rμ μμμ=− ,                        ( 6 )  \nand 0 / kcω=  is the wave number of the incident radiation in the background \nvacuum. Note that effμ is only used to calculate the “effective” wave vector k 2x in \nthe MO slab, not to replace the slab by an isotropic one.  Then the electric fields and \nthe magnetic fields in layers 1, 2 and 3 can be expressed as \n                      11\n11\n0()\n1y xxik y i t ik x ik xAe Be e\niω\nωμ− −=+\n=∇ ×1z\n11Ee\nHE                     ( 7 )  \n                      22\n22\n02()\n1y xxik y i t ik x ik xAe Be e\niω\nωμ μ− −=+\n=∇ ×2z\n22Ee\nHE                     (8) \n                     33() ()\n33\n0()\n1y xxik y i t i k xd i k xdAe Be e\niω\nωμ− −− −=+\n=∇ ×3z\n33Ee\nHE             ( 9 )  \nBased on the boundary conditions (1 )\nllz l zxxEE+== , (1 )\nlly l yxxHH+== (1, 2l= ), \nwe obtain the reflection and transmission coefficients as \n                  \n22*\n2\n22(2 sin )\n|| ||xxx\nik d ik dAB i k drAe Be−=−                   ( 1 0 a )  5 \n                   \n2212\n224\n|| ||xxeff x x\nik d ik dkktAe Beμ\n−−=−                   ( 1 0 b )                   \nwith 12 12 ,e f fx x y e f fx x y A k k igk B k k igkμμ=− − =+ +  and ir gμμ=  is the MO V oigt \nconstant of YIG.  \nWhen an electromagnetic beam of finite width illuminates the slab at an incident \nangle 0θ, the lateral shifts of the reflected and transmitted beams can be obtained by \nthe stationary phase method3 \n                         \n0\ny yr\nr\nykkdddkϕ\n==−                          ( 1 1 a )  \n                         \n0\ny yt\nt\nykkdddkϕ\n==−                              (11b) \nwhere 00sino\nykk θ =  and rϕ, tϕ are the phase angles of the reflection and \ntransmission coefficients for plane waves, respectively. Note here the lateral shift of \nthe transmitted beam is measured in the same way as that of the reflected beam38. \nFor a transparent YIG slab, effμ and 2xk are both real when the weak \nabsorption of YIG is neglected (i.e. the damping factor α is assumed to be zero). \nThen the reflected shift derived from Eq. (10)-(11) includes two parts: \n                  \n00(1) (2)\ny yy yr\nyykk kkddddk dkϕϕ\n===− −                       (12) \nwhere ()(1) *Arg AB ϕ=  while (2)ϕ is the phase angle of the complex variable \n               \n22 221\n|| ||xxik d ik dAe Beξ−=−                 ( 1 3 )  \nThe transmitted shift is only determined by the yk-dependence of (2)ϕ: \n                     \n0(2)\ny yt\nykkdddkϕ\n==−                              (14) \nWhen 0 g=, we have 12 0rx x A kk Aμ=− ≡ , 12 0rx x B kk Bμ=+ ≡ . The results of \nthe lateral shifts are reduced to the case of a nonmagneto slab as investigated in Ref. [38]. The first term in Eq. (12) will vanish since \n0A and 0B are both real and \nsymmetric reflected and transmitted shifts will appear. Based on the formulas Eq. \n(12)-(14), we will discuss the behaviors of the shifts at normal incidence and at \noblique incidence, respectively, for a MO slab with 0g≠, in the following sections. \n 6 \n Normal incidence.  When 0g≠, a θ-dependent imaginary part is added to A or \nB so that \n                00 ,ey ey AAi g k B Bi g k=− =+                  ( 1 5 )  \nhere 0eA and 0eB are real parameters for an “effective” slab where the MO \npermeability tensor is replaced by the magnetic-field-controlled scalar effμ. Since \n22 2 2 22 2 2 22\n00 00 ,2ee ee y A BA BA BA B g k−= − += + + , the shift term from (2)ϕ \nis expected to behave like that of the effective slab when the incident angle \napproaches zero and finally vanishes at normal incidence. \nThe first term of Eq. (12) is independent of the slab thickness and contributes a \nnon-vanishing reflected shift at normal incidence: \n                ()()(1)\n0\n0n\nyr\ny effkg dddkλ ϕ\nπμ ε==− =\n−                  (16) \nBy combining Eq. (16) with Eq. (2a) and (2b), we obtain the dependence of ()n\nrd on \nfrequency and magnetic field in the form \n              ()()0\n22 2(, )1nm\nr\nmdHλω ωωπε ω β ω=−−                 (17) \nwith  \n            1/2\n2 21\n11HHεβεε−=+ −−−                   ( 1 8 )  \nHere, 00\nmshHmω\nω≡=  is a dimensionless magnetic field reduced by the saturated \nmagnetization of the MO slab. In vicinity of the discontinuity point cmω βω≡  (This \ndiscontinuity in the frequency spectrum occurs exactly at the reflection minimum, \ncorresponding to effμ ε=)15, the abrupt transition of ()n\nrd from negative to positive \ncan be approximated by \n                  ()()0\n21n\nrdλ\nπβεη≈−                        (19) \nwhere c\ncωωηω−≡  describes a small deviation from cω. Note that the expression of \n(1)ϕ is identical to that by a semi-infinite MO material in Ref. [14,15]. So Eq. \n(16)-(19) are also applied to the case of d→∞ , i.e. a semi-infinite YIG interface. 7 \n \n \nFIG. 2. Calculated normal inciden ce (a) GH shift of reflected field /rdλ and (b) reflectivity as a \nfunction of frequency (express as /2ωπ ). The red, blue and black curves correspond to \n02580,2680,2780Oe h= , respectively. Circles: approximat ed results from Eq. (19); Lines: \nnumerical results. \nFig. 2(a) shows the approximated frequency dependence of ()n\nrd based on Eq. \n(19) for 02580Oe h= , 2680Oe  and 2780Oe  (circles). The numerical results (lines) \ndirectly from Eq. (10) and (11) are disp layed simultaneously for comparison and good \nagreement is found even for moderate deviation from the discontinuity point. Since \nβ increases monotonically with H, cω is red-shifted when the applied magnetic \nfield 0h is decreased, accompanied by the enhancement of ()n\nrd around cω. For a \nlower field 01000Oeh= , we have ()0.0136~n\nrdη, which is larger by 1-2 orders of \nmagnitude than the result for MnF 2 at the same applied magnetic field as reported in \nRef. [14,15]. \nFor practical purposes, a sufficiently large reflectivity is necessary for the \napplication of reflected shift. Fig. 2( b) shows a typical frequency spectrum of \nreflectivity of a YIG slab (0 0.3m, 2680Oedh== ), where 2r is quite small around \ncω but rises rapidly when the frequency approaches the sharp edge of a platform of \n21 r=. The rapid oscillation of reflectivity is a typical interference pattern of a slab \nof finite thickness, which is not exhibited in the spectrum of rd in Fig. 2(a) since the \nreflected shift is independent of slab thickness. The total-reflection platform at \nr ff> occurs when the wave vector 2xk in YIG becomes imaginary, which means a \nnegative effμ in the cases of normal incidence ( 0yk=). According to the dispersion \nrelation of iμ and rμ, it is easy to find \n                      (1 )rm HH ωω=+                         (20) 8 \n and the frequency dependence of effμ and g can be expressed as \n                   ()\n() ()22222\n22 22\n0(1 )r\nr\neff\nrHHωωωω\nμ\nωω ωω−−+=\n−−                 (21a)  \n                     221\n(1 )r\nrg\nHHωω\nωω=− +                   (21b) \nNote that at r ωω= , both g and effμ go infinite, but their ratio has a finite \nvalue \n                      1\nreffg H\nH\nωωμ\n==+                        (22) \nSubstituting this in to Eq. (16), we obtain the reflected shift at rω \n                    ()0()1n\nrrHdHλωωπ==+                   (23) \nThis result tells us the largest ()n\nrd achievable when 21 r=, which increases \nmonotonically with the reduced magnetic field H up to a strong-field limit: 0λ\nπ. \n \nOblique incidence.  When the incident beam is at a certain angle θ, the reflected \nshift rd and the transmitted shift td caused by a YIG-slab of thickness d can be \nexpressed as \n                     (, ) () (, )rdd D dθθ θ =+ Λ                 (24a) \n                     (, ) (, )tdd dθθ =Λ                        (24b) \nwhere the thickness-independent part ()Dθ is according to the first term in Eq. (12), \ngiven by \n                   () [ () () ]Dg F Fθθ θ+− =+                    (25) \nwith  \n            2\n12 21\n12\n22 2\n12() ()\n()()y\neff x x eff x x\nxx\neff x x ykkk kkkkFkk g kμμ\nθμ±±+ ±\n=±+        (26) \nThe expression of (),dθΛ  can be obtained from Eq. (13) and (14) as 9 \n                   ()2,1( )y\nydG dkdGkθΛ=+                    (27) \nHere we have introduced a function \n                22\n2 22() t a n ( )yxABGk k d\nAB+=\n−               (28) \nwhich can be rewritten as \n                0 () () 1 ()ye y yGk G k M k   =+                (29) \nwhere 0()eyGk  is the result of ()yGk  for a slab of scalar permeability effμ while \n()yMk gives the correction term caused by the tensor form of the slab permeability: \n                22\n00\n02 22\n00() t a n ( )ee\ney x\neeABGk k dAB+=−            (30a)  \n            2 2\n22 2 2\n0()1( )y\ny\neff eff eff yk gMkkk μμ μ ε=++ −         (30b) \nNote that 22 2(1 )m\neffmg\nHωω\nμ ωω=+−, hence the condition ()2\n2 11effg\nμ<<+ holds \nfor most frequencies not close to rω in the transparent region r ωω<, and (),dθΛ  \ncan be well approximated by the shifts (),re te edd d θ == Λ  due to an effective \nnon-MO slab of ( ε,effμ) for the same incident angle θ and slab-thickness d38. 10 \n \n \nFIG. 3. (a1) Reflectivity, (b1) GH shift of reflected field /rdλ and (c1) GH shift of transmitted \nfield /tdλ as functions of the slab thickness (expressed as /dλ) and the incident angle θ for \n02780Oe h= , 9.5GHzf= . (2), (3) are the same as (1) but for 9.7GHzf=  and 9.736GHz , \nrespectively.  \nThe competition between ()Dθ and (),dθΛ  leads to the coexistence of two \ntypes of NRS at certain frequencies. Fig. 3 and Fig. 4 illustrate the variance of 2r, \n/rdλ and /tdλ with the incident angle θ and the slab thickness d. The magnetic \nfield 0h is set to be 2780Oe, at which the characteristic frequencies are given by \n9.749GHzcf=  and 9.991GHzrf= . To one’s interest, both reflectivity and the shifts \nshow the periodicity with the change of slab-thickness (shown in Fig. 3). Two NRS regions are revealed in the sign-patterns of the lateral shifts for 9.5GHz f= , \n9.7GHz  and 9.736GHz  in Fig. 3b and 3c, where region A extends from 0θ= to \nA θθ= with only slight thickness dependence while region B for B θθ> shows a \nperiodic positive-to-negative transition of rd (and td as well) with thickness \nvarying.  11 \n \n \nFIG. 4. (a), (c) GH shift of transmitted field /tdλ and (b), (d) GH shift of reflected field /rdλ \nvs the incident angle θ at two certain slab thicknesses: (a, b) 3.03d λ = , (c, d) 3.047d λ =  for \nboth the YIG slab and the corresponding effective slab. The incidence frequency is \n9.736GHzf= . \nIn Fig. 4(a) and (c), the curves of td vs θ at a certain slab thickness for \n9.736GHzf=  are presented for both the YIG slab and the corresponding effective \nslab. It is clearly seen that (),dθΛ  can be well approximated by (),eff dθ Λ , which \naccounts for the transition of td with thickness at larger incident angles. For the \nreflected shift rd (Fig. 4(b) and (d)), ()Dθ dominates the NRS region at smaller \nangles and makes a non-negligible correction to the NRS in region B, breaking the \nsymmetry between rd and td which is an important feature of GH shifts due to a \nnon-MO slab38. \n \nTwo special cases.  Asymptotic behaviors of the GH shifts in two special cases of \nparticular interest for applications can be obtained from the general formulas Eq. \n(24)-(30). The first case is at r ωω=, where total reflection occurs and the reflected \nshift is only determined by ()Dθ even at oblique incidence. By expanding the \nfunction in terms of 0/s i nykkκθ≡=  and keeping terms up to the second order, we \nhave 12 \n        ()22\n2\n33\n022 11\n22ab g ab g gDka a a aθκ++ −−\n+− + −   −−=+ + +  \n        (31) \nwith eff eff an μ±=±  and  1\neff\neffbnμ±=± . \nSince 1effH gHμ=+ at rω, the asymptotic behavior of rd is given by \n                () 2 H+21+1rn\nrrddHωωκ==+                     (32) \nwhere ()n\nrd is the reflected shift in Eq. (23) at normal incidence. The calculated \nresults from Eq. (32) are illustrated in Fig. 5a in comparison with the numerical \nresults for 01000Oeh= , 2000Oe  and 3000Oe . \nThe second case is the transmitted shift accompanied by a 100%  transmittivity \nwhen the slab thickness satisfies 2 ()xkd m m π=∈ Ζ . According to Eq. (24b), the \ntransmitted shift can be written as \n                       21y\ntdd kdφ\nφ=+                            (33) \nwith \n                222 2\n00\n2 22\n002tan( )ee y\nx\neeAB g kkdABφ++=−                (34) \nwhen 2xkd m π= , we have 0φ= and \n               222 2\n00\n22\n20 02ye e y\nyx e ekA B g k d\ndk k A Bφ ++ =−−                 (35) \nAlso keeping the first two terms in the expression of td, we obtain the \nasymptotic behavior of td in this case \n                  2(1 )tdD κ χκ =+                            (36) \nwhere the coefficients are given by \n                  0\n22eff\neff effmDnμ ε λ\nμε+=                            (37) \nand \n               22 2\n22 231\n2eff eff\neff eff effng\nnnμχμ++ −=−+                     (38) \nThe transmitted shift will vanish at r ωω= , because of the divergence of effμ at 13 \n this frequency, and then rise with frequency decreasing. Fig. 5b illustrates the \nfrequency-dependence of td at a certain incident angle ( 30 , 45ooθ= ) for \n03000Oeh=  when the slab thickness satisfies the total transmission condition. Good \nagreement is found between the approximated td in Eq. (36) and the numerical \nresults. \n \nFIG. 5. (a) GH shift of reflected field /rdλ vs sin[ ]θ for 01000,3000Oeh=  at r ff=. (b) \nGH shift of transmitted field /tdλ vs the frequency at two certain incident angles ( 30 , 45ooθ= ) \nfor 03000Oeh= , 2 10 /x dk π= . The solid lines indicate the numerical results and the square \nsymbol lines correspond to the asymptotic behaviors calculated from Eq. (32) and (36).  \n \nNumerical simulations \nTo verify the above theoretical analysis, we performed a numerical simulation of \na YIG slab illuminated by a Gaussian incident beam with the well-known finite-element analysis software COMSOL Multiphysics. The center of the incident \nbeam arrived at the upper interface of the slab is located at the point (0,0)  and the \nhalf-width of the beam is \n7.5λ. The GH shifts can be directly obtained by comparing \nthe field distributions of the incident beam  and the reflected/transmitted beam at the \nrelevant interfaces.    \nNote that the damping of YIG has been neglected in the analytic expressions. In \nour simulations, a more prac tical dispersion of YIG perm eability will be adopted \nwhere the damping factor is set to be 2dH αγ ω =× , with 30Oe dH=35. The low \ndamping (~10-4) implies that no significant absorption effects will occur except for \nfrequencies near ferromagnetic resonance 0 m fHf= . According to Eqs. (17), (18) and \n(20), the two characteristic frequencies for nonreciprocal GH shifts, cf and rf, will \nnot be close to 0f unless the field 0h is in the strong-field limit 0 s hm>> .  \nAt normal incidence the analytical results predict that nonvanishing reflected 14 \n shift occurs in both the transparent region (r ff<) and the opaque region (r ff>) as \nshown in Fig.2. The simulation results of the field distribution along the incident interface for both the incident beam and the reflected beam are given in Fig. 6. The \nparameters are chosen to be the same as those for the points A and B in Fig. 2b, namely h\n0=2680Oe, d=0.3m, 9.01GHzf=  (point A) or 9.72GHzf= (point B). \nThe cases with (black solid lines) an d without (blue soli d lines) damping are \nboth investigated. Table 1 gives the reflected shifts given by analytic expressions, simulations without damp ing and simulations with damping. \nIt is \nshown that the damping has no significant effect on the shift, and the analytic \npredictions is in good agreement with the numerical results. \n \nFig. 6. The COMSOL simulation results for reflected shifts at normal incidence. (a) The \ndistribution of electric field amplit ude along the incident interface for \n9.01GHzf= ,02680Oe h= and 0.3md=  (corresponding to point A in Fig.2b ) ; (b) The \ndistribution of electric field amplit ude along the incident interface for \n9.72GHzf=02680Oeh= and 0.3md=  (point B in Fig.2b ).  The red lines indicate the \nanalytical shift of each case. \n \nTable 1: Comparisons between analytical and simulation results \n Analytical predictions Simulations without \ndamping Simulations \nwith damping \nNormal incidence f=9.01GHz \n0.130rd λ =−  0.134rd λ =−  0.134rd λ =−  \nf=9.72GHz 0.229rd λ =  0.221rd λ =  0.221rd λ =  \nOblique \nincidence Total \nreflection 0.443rd λ =  0.454rd λ =  0.454rd λ =  \nTotal transmission 0.241td λ =  0.245td λ =  0.245td λ =  \n \nAt oblique incidence both reflected and transmitted shifts may be observed at 15 \n certain conditions. Fig.7 gives the simulated results when the incident angle is 45o \nand the external magnetic field 0h is 3000Oe . The frequency and the slab thickness \nare chosen to satisfy the conditions for total reflection (r=ff, Fig. 7a and 7c) and total \ntransmission (2xkd m π= , Fig. 7b and 7d), respectively, since these cases are \nespecially interesting for practical applications. Again both the cases with and without \ndamping are investigated and compared with the analytical results as listed in Table 1. \nTrivial damping effects and good agreement between the analytical and simulation results are found, similar to those at normal incidence. \n \n \nFIG. 7. Numerical simulations of GH shifts when the Gaussian beam is incident from air. (a) The \nfield pattern for 03000Oeh= , r ff= with an incident angle of 45o. (b) The field pattern for \n03000Oeh= , 7GHzf=  and 2 2/x dk π=  with an incident angle of 45o. (c), (d) The distributions \nof field amplitudes along y direct ion near the interface between YI G and air, based on numerical \nresults in (a) and (b). The red lines indi cate the analytical shift of each case.  \n \nConclusions  \nIn this paper, we mainly investigate the lateral shifts of a TE wave both reflected and \ntransmitted from a YIG slab theoretically. It  is shown that the nonreciprocity effect \ncaused by the MO material will result in a nonvanishing reflected shift at normal incidence. In the case of oblique incidence, this effect also leads to a \nslab-thickness-independent term of \nrd which breaks the symmetry between the \nreflected and transmitted shifts which is an  important feature of GH shifts due to a 16 \n non-MO slab. The asymptotic behaviors of the normal-incidence reflected shift are \nobtained in the vicinity of two characteristic frequencies (rω and cω) corresponding \nto a minimum reflectivity and a total reflection, respectively. And the coexistence of \ntwo types of negative-reflected-shift (NRS ) at oblique incidence is discussed. \nNumerical results show that the reversal of the sign of GH shifts can be realized by tuning the magnitude of external magnetic field \n0h, adjusting the incident wave \nfrequency f or changing the thickness d as well as the incident angle θ. We also \ninvestigate two special cases for practical purposes: the reflected shift with a total \nreflection and the transmitted shift with a tota l transmission. Analytical expressions of \nthe shifts in these two cases are obtaine d approximately, which are in good agreement \nwith the results from numerical calculations. \nThough nonreciprocal re flected shifts were also reported in antiferromagnetic \nMnF 216,17, our YIG-based study confirms the possibility of experimental \ndemonstration of these effects in conventional ferrites at room temperatures. And the systematic analysis of both the reflected and the transmitted shifts due to a YIG slab \noffers a deeper insight into the role of magnetic field in tuning the shift sign, \nmagnitude and types (reflected or transmitted).     \n \nMethods \nTheory and simulations.  The numerical simulation results shown in Fig. 6 and Fig. 7 were \nobtained using the finite elemen t solver COMSOL Multiphysics. The scattering bou ndaries were \nset for four sides. Based on the numerical simulati on, the curves of field amplitude in Fig. 6 were \nobtained by performing the line plot along y axis from 4λ−  to 4λ. Due to the interference \neffect, the field amplitudes are oscillating along x di rection. The line plot is located at the first \npeak close to the interface between air and YIG.  Meanwhile, we zoom in the line plot of zE \nenough to get the distance betw een its symmetric axis and y=0, which indicates the lateral shift \nrd. The numerical results in Fig. 7 were obtained by the same technique. \n \nReferences  \n1. Goos, F. & Hänchen, H. Ein neuer und fundamentaler Versuch zur Totalreflexion. Ann. \nPhys , 436, 333–346 (1947). \n2. Goos, F. & Hänchen, H. A new measurement of the displacement of the beam at total \nreflection. Ann. Phys , 6, 251 (1949). \n3. Artmann, K. Berechnung der Seitenversetzung des totalreflektierten Strahles. Annalen der \nPhysik , 437, 87–102 (1948). \n4. 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Experimental observation of a \ngiant Goos–Hänchen shift in graphene using a beam splitter scanning method. Opt. Lett , 39, \n5574–5577 (2014). \n10. Jellal, A., Wang, Y ., Zahidi, Y . & Mekkaou i, M. Zero, positive and negative quantum \nGoos–Hänchen shifts in graphene barrier with vertical magnetic field. Physica E: \nLow-dimensional Systems and Nanostructures , 68, 53–58 (2015). \n11. Chuang, Y . L. & Lee, R. K. Giant Goos-Hänchen shift using PT symmetry. Phys. Rev. A  92, \n013815 (2015). \n12. Kuai, J. & Da, H. X. Electrical and proximity-magnetic effects induced quantum \nGoos–Hänchen shift on the surf ace of topological insulator. Journal of Magnetism and \nMagnetic Materials , 354, 355–358 (2014). \n13. Gruszecki, P., Romero-Vivas, J., Dadoenkova, Y . S., Dadoenkova, N. N., Lyubchanskii, I. L. \n& Krawczyk, M. Goos-Hänchen effect and bend ing of spin wave beams in thin magnetic \nfilms. Appl. Phys. Lett.  105, 242406 (2014). \n14. Lima, F., Dumelow, T., Da Costa, J. A. P. & Albuquerque, E. L. Lateral shift of far infrared \nradiation on normal incidence reflection off an antiferromagnet. Europhysics Letters , 83, \n17003 (2008). \n15. Lima, F., Dumelow, T., Albuquerque, E. L. & da Costa, J. A. P. Power flow associated with \nthe Goos-Hänchen shift of a normally inci dent electromagnetic beam reflected off an \nantiferromagnet. Phys. Rev. B  79, 155124 (2009). \n16. Lima, F., Dumelow, T., Albuquerque, E. L. & da  Costa, J. A. P. Nonreciprocity in the \nGoos-Hänchen shift on oblique incide nce reflection off antiferromagnets. J. Opt. Soc. Am. \nB, 28, 306–313 (2011). \n17. Macêdo, R., Stamps, R. L. & Dumelow, T. Spin  canting induced  nonreciprocal Goos-Hänchen \nshifts. Opt. Express , 22, 28467–28478 (2014). \n18. Macêdo, R. & Dumelow, T. Tunable all-angle negative refraction using \nantiferromagnets. Phys. Rev. B  89, 035135 (2014). \n19. Macedo R & Dumelow T. Beam shifts on reflection of electromagnetic radiation off \nanisotropic crystals at optic phonon frequencies J. Opt . 15, 014013 (2013). \n20. Dumelow, T., Camley, R. E., Abraha, Kamsul. & Tilley, D. R. Nonreciprocal phase behavior \nin reflection of electromagnetic waves from magnetic materials. Phys. Rev. B  58, 897–908 \n(1998). \n21. Tang, T. T., Qin, J., Xie, J. L., Deng, L. J. & Bi, L. Magneto-optical Goos-Hänchen effect in a \nprism-waveguide coupling structure. Opt. Express , 22, 27042–27055 (2014). \n22. Geller, S. & Gilleo, M. A. Structure and ferrimagnetism of yttrium and rare-earth–iron \ngarnets. Acta Crystallographica , 10, 239–239 (1957). \n23. Cherepanov, V ., Kolokolov, I. & L'vov, V . Th e saga of YIG: spectra, thermodynamics, \ninteraction and relaxation of magnons in a complex magnet. Phys. Rep . 229, 81–144 (1993). 18 \n 24. Cooper, R. W., Crossley, W. A., Page, J. L. & Pearson, R. F. Faraday rotation in YIG and \nTbIG . J. Appl. Phys . 39, 565–567 (1968). \n25. Krishnan, R., Le Gall, H. & Vien, T. K. Optical and magnetooptical properties of epitaxial \nYIG films. Phys. stat. sol. (a) , 17, K65–K68 (1973). \n26. Wettling, W. Magnetooptical properties of YIG measured on a continuously working \nspectrometer. Appl. Phys . 6, 367–372 (1975). \n27. Demokritov, S. O., Hillebrands, B. & Slavin, A. N. Brillouin light scattering studies of \nconfined spin waves: linear  and nonlinear confinement. Phys. Rep . 348, 441–489 (2001). \n28. Rodrigue, G. P. A generation of microwave ferrite devices. Proceedings of the IEEE , 76, \n121–137 (1988). \n29. Schloemann, E. F. Circulators for microwave and millimeter-wave integrated \ncircuits. Proceedings of the IEEE , 76, 188–200 (1988). \n30. Stancil, D. D. & Prabhakar, A. Spin Waves: Theory and Applications (2009). \n31. Kahl, S. & Grishin, A. M. Enhanced Faraday rotation in all-garnet magneto-optical photonic \ncrystal. Appl. Phys. Lett . 84, 1438–1440 (2004). \n32. Tomita, S., Kato, T., Tsunashima, S., Iwata, S., Fujii, M. & Hayashi, S. Magneto-optical Kerr \neffects of yttrium-iron garnet thin films incorporating gold nanoparticles. Phys. Rev. Lett . 96, \n167402 (2006). \n33. Uchida, H., Masuda, Y ., Fujikawa, R., Baryshev, A. V . & Inoue, M. Large enhancement of \nFaraday rotation by localized surface plasmon resonance in Au nanoparticles embedded in Bi: \nYIG film. Journal of Magnetism and Magnetic Materials , 321, 843–845 (2009). \n34. Wang, Z., Chong, Y . D., Joannopoulos, J. D. & Solja čić, M. Reflection-free one-way edge \nmodes in a gyromagnetic photonic crystal. Phys. Rev. Lett . 100, 013905 (2008). \n35. Lian, J., Fu, J. X., Gan, L. & Li, Z. Y . Robust and disorder-immune magnetically tunable \none-way waveguides in a gyromagnetic photonic crystal. Phys. Rev. B  85, 125108 (2012). \n36. Macˆedo, R., Dumelow, T., & Stamps, R. L. Tunable Focusing in Natural Hyperbolic \nMagnetic Media. ACS Photonics  3, 1670 (2016). \n37. Lan, C., Bi, K., Zhou, J., & Li, B. Experimental demonstration of hyperbolic property in \nconventional material-ferrite. Appl. Phys. Lett . 107, 211112 (2015). \n38. Li, C. F. Negative  Lateral Shift of a Light Beam Transm itted through a Di electric Slab and \nInteraction of Boundary Effects. Phys. Rev. Lett.  91, 133903 (2003). \n \nAcknowledgments  \nThis work was supported by the National Natural Science Foundation of China (Grant \nNo. 11374223), the National Science of Jiangsu Province (Grant No. BK20161210), the Qing Lan project, “333” project (Gra nt No. BRA2015353), and PAPD of Jiangsu \nHigher Education Institutions. \n \nAuthor Contributions \nL.G. conceived the idea. W.Y . performed mo st theoretical and numerical calculations. \nW.Y . and H.S. analyzed the data. All authors joined discussion extensively and revised the manuscript before the submission. \n \nAdditional Information \nCompeting financial interests:  The authors declare no competing financial interests. "    },    {        "title": "1409.5762v1.Perpendicularly_Biased_YIG_Tuners_for_the_Fermilab_Recycler_52_809_MHz_Cavities.pdf",        "content": "PERPENDICULARLY BIASED YIG TUNERS FOR THE FERMILAB \nRECYCLER 52.809 MHZ RF CAVITIES*  \nR. Madrak#, V . Kashikhin, A. Makarov, D. Wildman,  FNAL, Batavia, IL 60510, USA \nAbstract \nFor NOvA and future experiments requiring high \nintensity proton beams, Fermilab is in the process of \nupgrading the existing accelerator complex for increased \nproton production. One such improvement is to reduce the \nMain Injector cycle time, by performing slip stacking, \npreviously done in the Main Injector, in the now \nrepurposed Recycler Ring. Recycler slip stacking requires \nnew tuneable RF cavities, discussed separately in these \nproceedings. These are quarter wave cavities resonant at \n52.809 MHz with a 10 kHz tuning range. The 10 kHz \nrange is achieved by use of a tuner which has an electrical \nlength of approximately one half wavelength at 52.809 \nMHz. The tuner is constructed from 3⅛ ″ diameter rigid \ncoaxial line, with 5 inches of its length containing \nperpendicularly biased, Al doped Yttrium Iron Garnet \n(YIG). The tuner design, measurements, and high power \ntest results are presented.  \nINTRODUCTION \nTwo new quarter wave copper cavities have been \nconstructed and installed in the Fermilab Recycler Ring \nfor the purpose of slip stacking protons to increase beam  \n \n \n \nFigure 1: Tuner adjustable section with the inner \nconductor removed from the outer conductor. \n  \npower. These have been described in detail in [1]. They \nare to operate at 52.809 MHz with a tuning range of 10 \nkHz. Tuning by 1260 Hz is required for slip stacking \nitself, and additional tuning is needed to compensate for \ntemperature changes. The cavity maximum peak gap \nvoltage is 150 kV with a shunt impedance of 75 kΩ and Q \n~ 6000. The cavities will be pulsed for a maximum of 0.8 \ns every 1.33 s (the Main Injector cycle time).   \nThe cavities are tuned using perpendicularly biased \ngarnet in a coaxial line. This has been done previously \nand is described in [2] and [3].  The original concept, \nthough not for cavity tuning in particular, is described in \n[4]. \nDESIGN \nIn one of the ports located 3 ″ (on center) from the \ncavity shorted end, a fast tuner is loopbcoupled to the \ncavity. The coupling is through a 2.5 ″ diameter coaxial \nceramic window. The tuner loop area of 2.5 in2 was \nadjusted to give a nominal coupling impedance of 50 Ω. \nThe tuner is slightly less than a halfbwavelength long and \nis made up of  standard EIA 3⅛ ″ diameter, 50 Ω rigid \ncopper transmission line, in addition to an “adjustable \nsection” which is partially loaded with a 5 ″ long piece of \nAl doped Yttrium Iron Garnet (YIG). This section is \nimmersed in a variable solenoidal magnetic field which \nprovides a perpendicular bias for cavity tuning. A photo \nof the adjustable section, with the center conductor \nassembly removed from the outer conductor, is shown in \nFigure 1. The tuner line is shorted at the end opposite the \ncavity. \n The garnet (TCI Ceramics type ALb400) has a \nsaturation magnetization (4πM s) of 400 gauss. Its OD and \nID are 3.0″ and 0.65″ respectively. The center conductor \nused in the 15.75 ″ long adjustable section, which does not \nhave the dimensions of standard 3⅛ ″ coax line, is shrink \nfit into the garnet. Over the 9 ″ length closest to the short, \nthe outer conductor thickness has been reduced from the \nstandard 3⅛ ″ coax line wall thickness to 0.020 ″. Both this \n9″ section and the bottom copper shorting plate have a \n0.0197″ wide slot machined through the copper to reduce \neddy current effects. \nFor heat removal, the outer conductor is water cooled \nand the adjustable section is filled with a dielectric fluid \n(Diala AX). \nThe adjustable section had previously been designed \nfor use as a fast phase shifter in vector modulators \noperating at 325 MHz [5].  \nThe entire tuner, attached to the cavity and installed in \nthe Recycler, is shown in Figure 2. \n \n ___________________________________________  \n*Operated by Fermi Research Alliance, LLC under Contract No. DE b\nAC02b07CH11359 with the United States Department of Energy. \n#madrak@fnal.gov \nFERMILAB-CONF-13-363-APCSolenoid \nThe 112 turn solenoid (with TCI Ceramics G4 ferrite \nflux return) was designed to work at relatively high DC \ncurrent (80A), but also at high frequency (~2 kHz) \nAC current. While the AC response is not absolutely \nrequired, the ability to tune at ~2 kHz will be an \nadvantage. Obtaining the desired DC and AC response \nnecessitated the use of Litz Cable of sufficient \ncross section (1/0 AWG) with 270 copper strands of 24 \nAWG wire insulated from each other. The total strand \ncross section reduces the losses in the DC regime, and the \ninsulation between strands minimizes the eddy currents in \nthe cable (for the AC regime). A cooling tube was wound \non the outer surface of the solenoid as a bifilar coil in \norder to eliminate the effect of magnetic fields caused by \neddy currents induced in it. A DC magnetic field \nsimulation predicted the resulting minimum and \nmaximum magnetic fields in the tuner garnet of 0.058 T \nand 0.081 T, for 8000 solenoid amperebturns. \nThe solenoid current is supplied by a Copley Controls \nModel 266 amplifier which has a nominal bandwidth of  \n \n \nFigure 2: Tuner attached to the cavity, installed in the \nRecycler. The tuner is made up several 3⅛ ″ coax sections: \nan elbow, directional coupler, several straight sections, \nand finally the YIG loaded adjustable section (here \nobscured by the solenoid) which is shown in Figure 1. \n5 kHz. The tuner response is currently limited by the \nsolenoid inductance of 1.9 mH and the 60V max/80 A \nmax DC supply for the Copley amplifier.  \nTUNER PERFORMANCE AND \nMEASUREMENTS \nThe tuner is characterized apart from the cavity by the  \nmeasurements plotted in Figure 3. This shows network \nanalyzer S 11 amplitude and phase measurements as a \nfunction of DC solenoid bias. Large phase shifts  \n \nFigure 3: Twobway phase shift in tuner 15.75 ″ adjustable \nsection at 52.809 MHz (left ybaxis) and Loss (right yb\naxis) as a function of solenoid bias. \n \ncorrespond to a large frequency change (tuning) in the \ncavity/tuner system. Operation must be above ~35A, \nbelow which the garnet becomes lossy.  \nThe next set of measurements quantifies the \noperation of the cavity/tuner system as a whole. \nFigure 4 (wide range) and Figure 5 (operation range) \nshow the frequency (left ybaxis) and the Q of the \ncavity/tuner system (right ybaxis) as a function of \nsolenoid bias current. The bias at which we observe \na large change in frequency and a minimum in Q \ncorresponds to the case where the tuner is one half \nwavelength long.  \n \n \nFigure 4: Frequency and Q of the cavity/tuner system as a \nfunction of solenoid bias current, for a wide range of \ntuner biases. \n \nAn additional restriction on tuner range is due to \nthe peak voltage that can be sustained in the tuner \nwithout sparking. Figure 6 shows low level \nmeasurements of frequency and the corresponding tuner \nvoltage. In this case, the cavity was powered with a \nrelatively small signal. Cavity peak accelerating gap \nvoltage and tuner peak voltage were measured \nsimultaneously. The values were then scaled to cavity gap  \n \n \nFigure 5: Frequency and Q of the cavity/tuner system as a \nfunction of solenoid bias current, for solenoid biases in \nthe range that the tuner is operated ( > 35 A). \n \n \n \n Figure 6: Tuner voltage and frequency of cavity/tuner \nsystem as a function of tuner bias. The maximum peak \nvoltage for the tuner 3⅛ ″ line is also shown. (These \nmeasurements were taken from the cavity field probe and \nthe directional coupler which is part of the tuner.)  \n \nvoltages of 100 and 150 kV. Operating with a tuner \nvoltage above the peak voltage limit for 3⅛ ″ coaxial line \nof 13.3 kV will cause arcing and must be avoided. Not \nonly is arcing in the coax line undesirable, but this can \nalso lead to high voltages near the garnet which could be \ndestructive. \nMeasurements shown indicate I ≥ 32 A and I ≥ 40 A to \nrespect the 13.3 kV limit, for cavity voltages of 100 kV \nand 150 kV. With no other adjustments, and taking into \naccount the 35 A limit imposed due to losses in the \ngarnet, the tuning ranges are 11.7 kHz and 7.5 kHz for \ncavity peak voltages of 100 kV and 150 kV.  TUNER SPARK PROTECTION \nTo avoid destruction of the garnet, several methods of \nprotection are in place to shut off RF power to the cavity \nin the event of a tuner spark. \nThe tuner line contains a 60 dB directional coupler with \nboth forward and reverse power taps. The first tuner \nprotection system shuts off the RF drive if a change of \nmore than 3dB/ µs is detected in the reverse power. \nIn the second protection system, a phase detector \nmonitors the phase difference between forward and \nreverse power. If the phase changes more quickly than \nthat due to normal tuning, the RF drive is again shut off. \nThis is accomplished using a capacitive differentiator and \na comparator. \nSTATUS AND HIGH POWER TESTING \nBefore being installed in the Recycler, both cavities \nwere high power tested in a shielded “cavity test cave”. \nThe cavity and tuner systems have been operated \nsuccessfully at high power in the Recycler. \nACKNOWLEDGMENTS \nWe would like to acknowledge the huge efforts of the \nAccelerator Division Mechanical Support and RF Groups, \nand everyone involved in this project. In addition, we \nwould like to thank all of those in Technical Division \ninvolved in the design and construction of the solenoids. \nREFERENCES \n[1] R. Madrak and D. Wildman, “Design and High \nPower Testing of 52.809 MHz RF Cavities for Slip \nStacking in the Fermilab Recycler”, WEPMA013, \nthese proceedings.  \n[2] S. M. Hanna et.al, “YIG Tuners for RF Cavities”, \nIEEE Transactions on Magnetics, Vol 28, No. 5, p. \n3210, Sept 1992. \n[3] R. L. Porier, “Perpendicular Biased Ferrite –Tuned \nCavities”, PAC’93, Washington, D. C., May 1993, p. \n753, (1993). \n[4] A. S. Boxer , S. Hershenov and E. F. Landry  \"A \nHighbPower Coaxial Ferrite Phase Shifter\",  IRE \nTrans. Microwave Theory and Techniques \n(Correspondence) ,  vol. MTTb9,  pp.577, 1961. \n[5] R. Madrak and D. Wildman, “High Power 325 MHz \nVector Modulators for the Fermilab High Intensity \nNeutrino Source (HINS)”, LINAC’08, Victoria, B.C.,  \nOct 2008, THP088. \n \n"    },    {        "title": "1607.02312v2.Magnon_Polarons_in_the_Spin_Seebeck_Effect.pdf",        "content": "arXiv:1607.02312v2  [cond-mat.mtrl-sci]  27 Oct 2016Magnon Polarons in the Spin Seebeck Effect\nTakashi Kikkawa,1,2,∗Ka Shen,3Benedetta Flebus,4Rembert A. Duine,4,5Ken-ichi\nUchida,1,6,7,†Zhiyong Qiu,2,8Gerrit E. W. Bauer,1,2,3,7and Eiji Saitoh1,2,7,8,9\n1Institute for Materials Research, Tohoku University, Send ai 980-8577, Japan\n2WPI Advanced Institute for Materials Research, Tohoku Univ ersity, Sendai 980-8577, Japan\n3Kavli Institute of NanoScience, Delft University of Techno logy, Lorentzweg 1, 2628 CJ Delft, The Netherlands\n4Institute for Theoretical Physics and Center for Extreme Ma tter and Emergent Phenomena,\nUtrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Neth erlands\n5Department of Applied Physics, Eindhoven University of Tec hnology,\nP.O. Box 513, 5600 MB Eindhoven, The Netherlands\n6PRESTO, Japan Science and Technology Agency, Saitama 332-0 012, Japan\n7Center for Spintronics Research Network, Tohoku Universit y, Sendai 980-8577, Japan\n8Spin Quantum Rectification Project, ERATO, Japan Science an d Technology Agency, Sendai 980-8577, Japan\n9Advanced Science Research Center, Japan Atomic Energy Agen cy, Tokai 319-1195, Japan\n(Dated: October 28, 2016)\nSharp structures in the magnetic field-dependent spin Seebe ck effect (SSE) voltages of\nPt/Y3Fe5O12at low temperatures are attributed to the magnon-phonon int eraction. Experimental\nresults are well reproduced by a Boltzmann theory that inclu des magnetoelastic coupling. The SSE\nanomalies coincide with magnetic fields tuned to the thresho ld of magnon-polaron formation. The\neffect gives insight into the relative quality of the lattice and magnetization dynamics.\nPACS numbers: 85.75.-d, 72.25.-b, 75.80.+q, 72.25.Mk\nThe spin Seebeck effect (SSE) [1–19] refers to the gen-\neration of a spin current ( Js) as a result of a tempera-\nture gradient ( ∇T) in magnetic materials. It is well es-\ntablished for magnetic insulators with metallic contacts,\nat which a magnon flow is converted into a conduction-\nelectron spin current by the interfacial exchange interac-\ntion [20] and detected as a transverse electric voltage via\nthe inverse spin Hall effect (ISHE) [21–27] [see Fig. 1(a)].\nThe SSE provides a sensitive probe for spin correlations\nin magnetic materials [8, 9, 12–15].\nThe ferrimagnetic insulator yttrium-iron-garnet\nY3Fe5O12(YIG) is ideal for SSE measurements [19],\nexhibiting a long magnon-propagation length [28–30],\nhigh Curie temperature ( ∼560 K) [31], and high\nresistivity owing to a large band gap ( ∼2.9 eV) [32].\nThe magnon and phonon dispersion relations in YIG\nare well known [33–38]. The magnon dispersion in the\nrelevant regime reads\nωk=/radicalbig\nDexk2+γµ0H/radicalBig\nDexk2+γµ0H+γµ0Mssin2θ,\n(1)\nwhereω,k,θ,γ, andµ0Ms, are the angular frequency,\nwave vector kwith length k, angleθwith the exter-\nnal magnetic field H(of magnitude H), gyromagnetic\nratio, and saturation magnetization, respectively [33–\n36]. The exchange stiffness coefficient Dexas well as the\ntransverse-acoustic (TA) and longitudinal-acoustic (LA)\nsound velocities for YIG are summarized in Table I and\nthe dispersion relations are plotted in Fig. 1(b).\nIn this Letter, we report the observation of a resonant\nenhancement of the SSE. The experimental results are\nwell reproduced by a theory for the thermally induced\nTA phonon magnon \nLA phonon magnetization V\nLTLVLW\nEISHE\n∇T x\nzy\nHJs(a)\n(b) magnon polaron \nμ0H = 1.0 T magnon \nLA phonon \nTA phonon \nk (10 8 m-1 )ω/2 π (THz) \n10 0 51.0\n0.50.56 0.580.0350 0.0355 \n0.2730 0.2750 ω/2 π (THz) \n4.44 4.46 \nk (10 8 m-1 )(c)\n(d)k1\nk2\nFIG. 1: (a) The longitudinal SSE in the Pt/YIG/GGG\nsample, where EISHEdenotes the electric field induced by\nthe ISHE. The close-up of the upper (lower) right shows a\nschematic illustration of a propagating magnon and TA (LA)\nphonon. (b) Magnon [Eq. (1) with µ0Ms= 0.2439 T,\nµ0H= 1.0 T, and θ=π/2], TA-phonon ( ω=c⊥k), and\nLA-phonon ( ω=c||k) dispersion relations for the parameters\nin Table I. (c),(d) Magnon polarons at the (anti)crossings b e-\ntween the magnon and TA-phonon branches at (c) lower and\n(d) higher wave numbers, where k/bardblˆx(θ=π/2 andφ= 0)\nandH/bardblˆz.\nmagnon flow in which the magnetoelastic interaction is2\nT = 50 K\n5 -5 10 -10 005.0\n-5.0 V (μ V)\nμ0H (T)∆T = 1.73 K\n-5.2 -4.8 5.2 \n4.8 \n10 -10 0wave vectorfrequency \n0 0 0TA \nphonon\nmagnonpoint \ntouch \nwave vector wave vector\n-5 0 5-5.005.0V (μ V)T = 50 K(b) (d) \n00.2 \n-0.2 5.0 5.2 \n4.8 \n4.6 V (μ V)\n2.0 2.5 3.0 ∆T = 1.73 K\n∆T = 0 K\nμ0H (T)(a)\nT = 50 Kat 0.1 Tat HTA \n∆T = 1.73 K(c)\n1.0 2.0 02.04.06.0\n∆T (K)V (μ V)\n(h) \n00.2 \n-0.2 5.0 5.2 \n4.8 \n4.6 V (μ V)\n9.0 9.5 10.0 ∆T = 1.73 K\n∆T = 0 K\nμ0H (T)T = 50 Kpoint \ntouch \nTA \nphonon magnon LA \nphonon\nwave vectorfrequency \n0 0 0wave vector wave vector(e)\n1.0 2.0 02.0 4.0 6.0 \n∆T (K)V (μ V)(g) \nat HLA H = HTA H < HTA H > HTA H = HLA H < HLA H > HLA \n(f)\nμ0H (T)2.6 T 9.3 T\nFIG. 2: (a) Magnon and TA-phonon dispersion relations for YI G when H < H TA,H=HTA, andH > H TA. (b)V(H) of\nthe Pt/YIG/GGG sample for ∆ T= 1.73 K at T= 50 K for |µ0H|<6.0 T. (c) V(∆T) of the Pt/YIG/GGG sample at\nµ0H= 0.1 T and µ0HTA. (d) Magnified view of V(H) around HTA. (e) Magnon, TA-phonon, and LA-phonon dispersion\nrelations for YIG when H < H LA,H=HLA, andH > H LA. (f)V(H) of the Pt/YIG/GGG sample for ∆ T= 1.73 K at\nT= 50 K for |µ0H|<10.5 T. The inset to (f) is a magnified view of V(H) for 4.6µV<|V|<5.3µV. (g)V(∆T) of the\nPt/YIG/GGG sample at H=HLA. (h) Magnified view of V(H) around HLA. TheVpeaks at HTAandHLAare marked by\nblue and red triangles, respectively.\ntaken into account. We interpret the experiments as\nevidence for a strong magnon-phonon coupling at the\ncrossings between the magnon and phonon dispersion\ncurves, i.e., the formationofhybridized excitations called\nmagnon polarons [40, 41].\nThe sample is a 5-nm-thick Pt film sputtered on the\n(111) surface of a 4- µm-thick single-crystalline YIG film\ngrown on a single-crystalline Gd 3Ga5O12(GGG) (111)\nsubstrate by liquid phase epitaxy [42]. The sample was\nthen cut into a rectangular shape with LV= 4.0 mm\n(length),LW= 2.0 mm (width), and LT= 0.5 mm\n(thickness). SSE measurements were carried out in a\nlongitudinal configuration [1, 19] [see Fig. 1(a)], where\nthe temperature gradient ∇Tis applied normal to the\ninterfaces by sandwiching the sample between two sap-\nphire plates, on top of the Pt layer (at the bottom of the\nGGG substrate) stabilized to TH(TL) with a tempera-\nture difference ∆ T=TH−TL(>0). ∆Twas measured\nwith two calibrated Cernox thermometers. A uniform\nmagnetic field H=Hˆ zwas applied by a superconducting\nTABLE I: Parameters for the magnon and phonon dispersion\nrelations of YIG [34–39].\nSymbol Value Unit\nExchange stiffness Dex7.7×10−6m2/s\nTA-phonon sound velocity c⊥3.9×103m/s\nLA-phonon sound velocity c||7.2×103m/ssolenoidmagnet. Wemeasuredthedcelectricvoltagedif-\nferenceVbetween the ends of the Pt layer with a highly\nresolved field scan, i.e., at intervals of 15 mT and waiting\nfor∼30 sec after each step.\nFigure 2(b) shows the measured V(H) of the Pt/YIG\nsample atT= 50 K. A clear signal appears by applying\nthe temperature difference ∆ Tand its sign is reversed\nwhen reversing the magnetization. The magnitude of V\natµ0H= 0.1 T is proportional to ∆ T[see Fig. 2(c)].\nThese results confirm that Vis generated by the SSE\n[19].\nOwing to the high resolution of H, we were able to\nresolve a fine peak structure at µ0H∼2.6 T that is\nfully reproducible. A magnified view of the V-Hcurve\nis shown in Fig. 2(d), where the anomaly is marked by\na blue triangle. Since the structures scale with ∆ T[see\nFigs. 2(c) and 2(d)], they must stem from the SSE.\nThe peak appears for the field HTAat which accord-\ning to the parameters in Table I the magnon dispersion\ncurve touches the TA-phonon dispersion curve. By in-\ncreasingH, the magnon dispersion shifts toward high\nfrequencies due to the Zeeman interaction ( ∝γµ0H),\nwhile the phonon dispersion does not move. At µ0H= 0,\nthe magnonbranchintersectsthe TA-phononcurvetwice\n[see Fig. 2(a)]. With increasing H, the TA-phonon\nbranch becomes tangential to the magnon dispersion at\nµ0H= 2.6 T and detaches at higher fields [see Fig. 2(a)].\nIf the anomaly is indeed linked to the “touch” condition,\nthere should be another peak associated with the LA-\nphonon branch. Based on the parameters in Table I,\nwe evaluated the magnon −LA-phonon touch condition3\natµ0HLA∼9.3 T. We then upgraded the equipment\nwith a stronger magnet and subsequently investigated\nthe high-field dependence of the SSE.\nFigure2(f)showsthedependence V(H)ofthePt/YIG\nsample atT= 50 K, measured between µ0H=±10.5 T.\nIndeed, another peak appeared at µ0HLA∼9.3 T pre-\ncisely at the estimated field value at which the LA-\nphonon branch touches the magnon dispersion [see Fig.\n2(e)], sharing the characteristic features of the SSE; i.e.,\nit appears only when ∆ T∝negationslash= 0 and exhibits a linear-∆ T\ndependence [see Figs. 2(g) and 2(h)]. For µ0H >9.3 T\ntheV-Hcurves remain smooth.\nWe carried out systematic measurements of the tem-\nperature dependence of the SSE enhancement at HTA\nandHLA. Figure 3(c) shows the normalized SSE volt-\nageS≡(V/∆T)(LT/LV) as a function of Hfor various\naverage sample temperature Tavg[≡(TH+TL)/2]. The\namplitudeoftheSSEsignalmonotonicallydecreaseswith\ndecreasingTin the present temperature range [8, 9] [see\nFig. 3(b)]. Importantly, the two peaks in SatHTAand\nHLAexhibitdifferent Tdependences[seeFigs. 3(c), 3(d),\nand 3(e)]. The peak shape at HTAbecomes more promi-\nnent with decreasing Tand it is the most outstanding at\nthe lowestT. On the other hand, the Speak atHLAis\nsuppressed below ∼10 K and it is almost indistinguish-\nable at the lowest T. This different Tdependence can\nbe attributed to the different energy scale of the branch\ncrossing point for H=HTAandH=HLA. The fre-\nquency of the magnon −LA-phonon intersection point is\n0.53 THz = 26 K ( ≡TMLA), and it is more than 3 times\nlarger than that of the magnon −TA-phonon intersection\npoint (0.16 THz). Therefore, for T < T MLA, the exci-\ntation of magnons with energy around the magnon −LA-\nphonon intersection point is rapidly suppressed, which\nleads to the disappearance of the Speak atHLAat the\nlowestT.\nThe clear peak structures at low temperatures allow\nus to unravel the behavior of the SSE around HTAin\ndetail. Increasing Hfrom small values, Sincreases up to\na maximum value at H=HTA, as shown in Fig. 3(d)\n(Tavg= 3.46 K). For fields slightly larger than HTA, S\ndrops steeply to a value below the initial one. The SSE\nintensityS(i), wherei(= 0,1,2) represents the number\nof crossing points between the magnon and (TA-)phonon\nbranch curves [see also Fig. 2(a)], can be ordered as\nS(1)>S(2)>S(0) and could be a measure of the num-\nber of magnon polarons.\nThe SSE is generated in three steps: (i) the tem-\nperature gradient excites magnetization dynamics that\n(ii) at the interface to the metal becomes a particle\nspin current and (iii) is converted to a transverse volt-\nage by the ISHE. The latter two steps depend only\nweakly on the magnetic field. For thick enough sam-\nples, the observed anomalies in the SSE originate from\nthe thermally excited spin current in the bulk of the fer-\nromagnet. The importance of the magnetoelastic cou--0.40 -0.20 00.20 0.40 \n-0.20 -0.10 00.10 0.20 34.38 K\n-0.10 00.10 16.24 K\n-0.10 00.10 13.22 K\n-0.0500.0510.48 K\n-0.04-0.0200.020.046.30 K\n-10 0 10-0.0200.023.46 K0.30 0.32\n0.30 0.320.34\n0.160.18\n0.180.20 \n0.050.060.07\n0.060.070.08\n0.040.050.06\n0.050.060.07\n0.030.04\n0.040.05\n0.020.03\n0.020.030.04\n2.0 2.5 3.0 00.010.02\n9.0 9.5 10.0 00.010.02Tavg  = 49.61 K \nμ0H (T) 5 -5 \nμ0H (T) μ0H (T) S (μ V/K )(c) (d) (e) (a) \n0 20 40 600.10.20.30.4S (μ V/K )at 0.1 T\nat HTA \nat HLA (b) \nTavg  (K) 0.5\nLA \nTA magnon\npoint \ntouch point \ntouch \n0 0 0 0ω/2 π\nk k k kHTA  < H < HLA H = HLA H = HTA H < HTA \nFIG. 3: (a) Magnon, TA-phonon, and LA-phonon dispersion\nrelations for YIG when H < H TA,H=HTA,HTA< H <\nHLA, andH=HLA. (b)Tavgdependence of the normalized\nSSE voltage SatH= 0.1 T,HTA, andHLA. (c)S(H) of the\nPt/YIG/GGG sample for various values of Tavgin the range\nof|H|<10.5 T. (d),(e) A blowup of S(H) around (d) HTA\nand (e)HLA.\npling (MEC) for spin transport in magnetic insulators\nhas been established by spatiotemporally resolvedpump-\nand-probe optical spectroscopy [41, 43]. Here we de-\nvelop a semiclassical model for the SSE in the strongly\ncoupled magnon-phonon transport regime [40, 41, 44–\n46]. Our model Hamiltonian consists of magnon ( Hmag),\nphonon ( Hel), and magnetoelastic coupling ( Hmec)\nterms. In second-quantized form Hmag=/summationtext\nkAka†\nkak+\n(Bk/2)(a†\nka†\n−k+a−kak),Hel=/summationtext\nk,µ/planckover2pi1ωµk/parenleftBig\nc†\nµkcµk+1\n2/parenrightBig\n,\nandHmec=/planckover2pi1nB⊥(γ/planckover2pi1\n4Msρ)1/2/summationtext\nk,µkω−1/2\nkµe−iφak(cµ−k+\nc†\nµk)×(−iδµ1cos2θ+iδµ2cosθ−δµ3sin2θ) +h.c..\nIn spherical coordinates the wave vector k=\nk(sinθcosφ,sinθsinφ,cosθ),Ak//planckover2pi1=Dexk2+γµ0H+\n(γµ0Mssin2θ)/2, andBk//planckover2pi1= (γµ0Mssin2θ)/2. Here,4\na†\nk(c†\nµk) andak(cµk) are magnon (phonon) creation and\nannihilation operators, respectively. B⊥is the magne-\ntoelastic couplingconstant, ρis the averagemassdensity,\nn= 1/a3\n0isthe numberdensity ofspins, and a0is the lat-\ntice constant. The magnondispersionfrom Hmagis given\nby Eq. (1), while the phonon dispersions are ωµk=cµk\nwithµ= 1,2 for the two transverse modes and µ= 3\nfor the longitudinal one. δµiinHmecrepresents the Kro-\nneckerdelta. Bydiagonalizing Hmag+Hel+Hmec[47], we\nobtain the dispersion relationof the i-th magnon-polaron\nbranch/planckover2pi1Ωikand the correspondingamplitude |ψik∝angb∇acket∇ight. The\nmagnon-polaron dispersions for θ=π/2 andφ= 0 are\nillustrated in Figs. 1(c) and 1(d), with a magnetic field\nµ0H= 1.0 T andB⊥/(2π) = 1988 GHz [38].\nWe assume diffuse transport that at low tempera-\ntures is limited by elastic magnon and phonon impu-\nrity scattering [45]. We employ the Hamiltonian Himp=/summationtext\nµ/summationtext\nk,k′c†\nµkvph\nk,k′cµk′+/summationtext\nk,k′a†\nkvmag\nk,k′ak′, where, assum-\nings-wave scattering, vph\nk,k′=vphandvmag\nk,k′=vmagde-\nnote the phonon and magnon impurity scattering poten-\ntials, respectively. We compute the spin current driven\nby a temperature gradient [6, 16] and thereby the SSE\nin the relaxation-time approximation of the linearized\nBoltzmann equation. The linear-response steady-state\nspin current Js(r) =−ζ· ∇Tis governed by the SSE\ntensorζ:\nζαβ=/integraldisplayd3k\n(2π)3/summationdisplay\niWs\nikτik(∂kαΩik)(∂kβΩik)∂Tf(0)\nik|T=T(r).\n(2)\nHereWs\nik=|∝angb∇acketleft0|ak|ψik∝angb∇acket∇ight|2is the intensity of the i-\nth magnon-polaron and τikis the relaxation time to-\nwards the equilibrium (Planck) distribution function\nf(0)\nik(r) = (exp( /planckover2pi1Ωik/(kBT(r)))−1)−1. The relax-\nation time τikof thei-th magnon-polaron reads τ−1\nik=\n(2π//planckover2pi1)/summationtext\njk′|∝angb∇acketleftψjk′|Himp|ψik∝angb∇acket∇ight|2δ(/planckover2pi1Ωik−/planckover2pi1Ωjk′). The\nstrong-coupling(weak scattering) approachis valid when\nτ−1\nik1,2≪∆Ω, where ∆Ω is the energy gap at the anti-\ncrossing points k1,2. We disregard the Gilbert damping\nthat is very small in YIG.\nFrom the experiments we infer the scattering param-\neters|vmag|2= 10−5s−2[28] and/vextendsingle/vextendsinglevmag/vph/vextendsingle/vextendsingle= 10,\ni.e., the magnons are more strongly scattered than the\nphonons. The computed longitudinal spin Seebeck coeffi-\ncient(SSC) ζxx[Eq.(2)]isplottedinFig.4(a). Switching\non the magnetoelastic coupling increases the SSC espe-\ncially at the “touching” magnetic fields HTAandHLA.\nAt these points the group velocity of the magnon is iden-\ntical to the sound velocity. Nevertheless, spin transport\ncan be strongly modified when the ratio/vextendsingle/vextendsinglevmag/vph/vextendsingle/vextendsingledif-\nfers from unity. The SSC can be enhanced or suppressed\ncompared to its purely magnonic value. A high acous-\ntic quality as implied by/vextendsingle/vextendsinglevmag/vph/vextendsingle/vextendsingle= 10 is beneficial for\nspintransportandenhancesthe SSCbyhybridization, as\nillustrated by Fig. 4(a). When magnon and phonon scat-2 2.5 30.290.3149.61 K\n9 9.5 100.30 0.32\n2 2.5 300.20.4\n9 9.5 1000.20.42 2.5 300.20.4\n9 9.5 1000.4\n2 2.5 30.040.0613.22 K\n9 9.5 100.050.07\n2 2.5 300.20.4\n9 9.5 1000.20.42 2.5 300.023.46 K\n9 9.5 1000.023.46 K\n0 5 10 0510 50 K\n50 K\n13 K \n13 K \n3.5 K\n3.5 K0.2S (μ V/K )\nζxx  ( 10 22  m-1 s-1 K-1 )T = 13 K \nμ0H (T) (a) (b) (c) \n13.22 K49.61 K\nμ0H (T) μ0H (T) Experiment Calculation ζxx  ( 10 22  m-1 s-1 K-1 )\n×10 HTA \nHLA with MEC\n|v mag /v ph | = 10\nwithout MECwith MEC\n|v mag /v ph | = 1\nEnhancement\nFIG. 4: (a) Calculated SSC ζxxatT= 13 K as a function\nofH,with (red solid curve and blue circles) and without (red\ndashed curve) magnetoelastic coupling (MEC). The red solid\ncurve and the blue circles are computed for ratios of the scat -\ntering potentials of/vextendsingle/vextendsinglevmag/vph/vextendsingle/vextendsingle= 10 and/vextendsingle/vextendsinglevmag/vph/vextendsingle/vextendsingle= 1, re-\nspectively. The bluedashed curveis ablowup ofthedifferenc e\nbetween the red solid and dashed curves. (b) Experimental\nSand (c) theoretical ζxxafter subtraction of the zero MEC\nresults.\ntering potentials would be the same, i.e.,/vextendsingle/vextendsinglevmag/vph/vextendsingle/vextendsingle= 1,\nthe anomalies vanish identically [see the blue circles in\nFig. 4(a)]. The difference between the calculations with\nandwithout MECagreesverywell with the peakfeatures\non top of the smooth background as observed in the ex-\nperiments, see Figs. 4(b) and 4(c). We can rationalize\nthe result by the presence of a magnetic disorder that\nscatters magnons but not phonons.\nFinally, we address the SSE background signal. The\noveralldecrease of the calculated ζxxis not related to the\nphonons, but reflects the field-induced freeze-out of the\nmagnons (that is suppressed in thin magnetic films [8]).\nIntheexperiments, ontheotherhand, theglobal Sbelow\n∼30 K clearly increases with increasing H[Fig. 3(c)].\nWetentativelyattributethisdiscrepancytoanadditional\nspin current caused by the paramagnetic GGG substrate\nthat, when transmitted through the YIG layer, causes\nan additional voltage. Wu et al.[7] found a paramag-\nnetic SSE signal in a Pt/GGG sample proportional to\nthe induced magnetization ( ∼a Brillouin function for\nspin 7/2) [7]. Indeed, the increase of Sin the present\nPt/YIG/GGG sample is of the same order as the para-\nmagnetic SSE in a Pt/GGG sample [8].\nIn conclusion, we observed two anomalous peak struc-\ntures in the magnetic field dependence of the spin See-\nbeck effect (SSE) in Pt/Y 3Fe5O12(YIG) that appear at5\nthe onset of magnon-polaron formation. The experimen-\ntal results are well reproduced by a calculation in which\nmagnons and phonons are allowed to hybridize. Our re-\nsults show that the SSE can probe not only magnon dy-\nnamics but also phonon dynamics. The magnitude and\nshape of the anomalies contain unique information about\nthe sample disorder, depending sensitively onthe relative\nscattering strengths of the magnons and phonons.\nThe authors thank S. Daimon, J. Lustikova, L. J. Cor-\nnelissen, and B. J. van Wees for valuable discussions.\nThis work was supported by PRESTO “Phase Inter-\nfaces for Highly Efficient Energy Utilization” from JST,\nJapan,Grant-in-AidforScientificResearchonInnovative\nArea “Nano Spin Conversion Science” (No. 26103005,\n26103006), Grant-in-Aid for Scientific Research (A) (No.\n15H02012, 25247056) and (S) (No. 25220910) from\nMEXT, Japan, NEC Corporation, The Noguchi Insti-\ntute, the Dutch FOM Foundation, EU-FET Grant In-\nSpin 612759, and DFG Priority Programme 1538 “Spin-\nCaloric Transport” (BA 2954/2). T.K. is supported by\nJSPS through a research fellowship for young scientists\n(No. 15J08026).\n∗Electronic address: t.kikkawa@imr.tohoku.ac.jp\n†Presentaddress: National Institutefor Materials Science ,\nTsukuba 305-0047, Japan.\n[1] K. Uchida, H. Adachi, T. Ota, H. Nakayama, S.\nMaekawa, and E. Saitoh, Observation of longitudinal\nspin-Seebeck effect in magnetic insulators, Appl. Phys.\nLett.97, 172505 (2010).\n[2] J. Xiao, G. E. W. Bauer, K. 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Colpa, Diagonalization of the quadratic boson\nHamiltonian, Physica A 93, 327 (1978)."    },    {        "title": "1807.00610v1.Picosecond_acoustic_excitation_driven_ultrafast_magnetization_dynamics_in_dielectric_Bi_substituted_yttrium_iron_garnet.pdf",        "content": "1 \n  Picosecond acoustic excitation driven  ultrafast magnetization dynamics in \ndielectric Bi-substituted yttrium iron garnet  \n \nMarwan Deb1*, Elena Popova2, Michel Hehn1, Niels Keller2, Stéphane Mangin1, Gregory \nMalinowski1 \n \n1Institut Jean Lamour (IJL), CNRS UMR 7198, Université de Lorraine, 54506 Vandœuvre-\nlès-Nancy, France. \n2Groupe d’Etude de la Matière Condensée (GEMaC ), CNRS UMR 8635, Université de Paris-\nSaclay, 78035 Versailles, France. \n \nAbstract: \nUsing femtosecond optical pulses, we have investigated the ultrafast magnetization dynamics \ninduced in a dielectric film of  bismuth-substituted yttrium iron garnet (Bi-YIG) buried below \na thick Cu/Pt metallic bilayer. We show that exciting the sample from Pt surface launches an \nacoustic strain pulse propagating into the garnet  film. We discovered that this strain pulse \ninduces a coherent magnetization precession in the Bi-YIG at the frequency of the \nferromagnetic resonance. The observed phenomena can be explain by strain-induced changes \nof magnetocristalline anisotropy via the inve rse magnetostriction effect. These findings open \nnew perspectives toward the control of the ma gnetization in magnetic garnets embedded in \ncomplex heterostructure devices.   \n \nPACS numbers: 75.78.Jp, 75.50.Gg, 72.80.Sk, 78.20.Ls. \n  \n \n \n \n \n \n \n*Corresponding author: marwan.deb@univ-lorraine.fr 2 \n I. Introduction: \nThe experimental discovery of the subpicosecond  demagnetization of Ni films following the \nexcitation by 60 femtosecond optical pulse [1] has opened a new and rapidly growing \nresearch field of modern magnetism called femtomagnetism [2,3]. An ultimate goal of this field is to control the magnetization at the fastest possible speed and in the most efficient way. \nTo this end, intense researches are being ca rried out to investigate laser-induced ultrafast \nmagnetic process and understanding the fundamental mechanism behind their excitation. It \nwas demonstrated that ultrashort optical pulses can trigger in magnetic materials various \nimportant magnetic processes, including magnetic phase transition [4-6], magnetization \nswitching [7,8], as well as a coherent spin precession [9,10]. In metallic materials, thermal \neffects resulting from the energy absorbed by th e medium play a crucial role in laser-induced \nmagnetic phenomena [11]. On the other hand, the possibility of controlling the spin \nprecession in dielectric with light was demo nstrated via non-thermal mechanisms like the \ninverse Faraday effect [10,12].  \nRecently, the field of femtomagnetism started investigating alternative ways to control the \nmagnetization using other ultrashort stimuli in cluding hot-electron pulses [13-15], Terahertz \npulses [16,17], as well as acoustic pulses [18-20].  This is due to two main reasons. The first is \nto improve the fundamental understanding in highly debated issues related to ultrafast magnetic phenomena induced by optical pulses like the ultrafast demagnetization [13,21] and \nreversal [14,15]. The second is to discover a versatile tool that allows non-thermal and \nultrafast control and reversal of the magnetization in metal, semiconductor, and dielectric \nfilms and heterostructures. In particular, for the latter reason, the use of acoustic pulse can \noffers at least two advantages. Firstly, acoustic pulses can produce a high mechanical stress of \nseveral GPa [22]. As a result, a large modifi cation of lattice parameter occurs and the \nmagnetization can be therefore non-thermally ch anged via the inverse magnetostriction effect \n[18,19]. Secondly, acoustic pulses have a larg e propagation distance of several millimeters \nwith low energy dissipation [23,24]. Consequently, they provide opportunity for non-thermal \nmanipulation of spins in films deeply embedded in opaque hetorestructure devices. In 2010, \nthe control of the magnetization dynamics by a picosecond acoustic pulse was demonstrated \nby Scherbakov et al. in GaMnAS ferromagnetic  semiconductor [18]. Later, acoustically-\ninduced magnetization dynamic was extended to ferromagnetic metals [19,25,26]. An important question in this context is the possibi lity to take advantage of picosecond acoustic \npulse to trigger a magnetization dynamics in magnetic dielectrics.  3 \n In this paper we present the results of an experimental study exploring the laser induced \nultrafast magnetization dynamics in bismuth-substituted yttrium iron (Bi-YIG) garnets buried \nbelow a thick Cu/Pt metallic bilayer. By exciting the Pt/Cu/Bi-YIG trilayers from Pt surface, \nwe find out that an acoustic strain pulse is generated and propagated into the garnet film. We demonstrate that the strain pulse induces a co herent magnetization precession at the frequency \nof ferromagnetic resonance. The obtained resu lts are explained by strain-induced change of \nmagnetocrystalline anisotropy via the invers e magnetostriction effect. In addition, we \ndemonstrate that we can control the magnetization precession amplitude by tuning the \namplitude of the acoustic pulse.          \nThe paper is organized as follows. First, we describe in Sec. II the experimental methods and \nthe static magnetic and magneto-optical properties of the sample. Then, we present and \ndiscuss in Sec. III the experimental results  of the time-resolved  magneto-optical and \nreflectivity measurements as a function of the external magnetic field and the laser energy \ndensity. Finally, we summarize in Sec. IV our findings.    \nII. SAMPLE PROPERTIES AND EXPERIMENTAL METHODS: \n \nBismuth-substituted yttrium iron garnet (Bi\nxY1-xFe5O12, Bi-YIG) materials have the cubic Ia-\n3d space group, which is characterized by thr ee different crystallographic sites (octahedral \n16a, tetrahedral 24d, and dodecahedral 24c) formed by the oxygen atoms [27]. The non-\nmagnetic Bi and Y atoms occupy the dodecahedral 24c sites, while the magnetic Fe atoms are \ndistributed between the octahedral 16a and tetr ahedral 24d sites. This two Fe sublattices are \nnonequivalent and coupled by a strong antiferroma gnetic superexchange interaction, leading \nto a ferrimagnetic state with high Curie temperature (T C > 550 K). These materials have \nattracted a great deal of attention due to their fascinating proprieties at room temperature, such \nas the good transparency in the infrared and visible spectrum (gap ~ 2.5 eV) and the very \nlarge magneto-optical (MO) Faraday effect (~ 104 deg/cm at 2.4 eV) [28], which make them \nsuitable for MO recoding media and non-reciprocal MO devices [27,29]. Beside the \ntechnological importance of the large MO effects, they have also been used as an efficient tool to study fundamental science related to magnetism in Bi-YIG such as the spin dependent \nband structure [30] as well as light-induced ultrafast magnetization dynamics and switching \n[31-36].  \nThe experiments were performed on 140-nm-thick film of Bi\n2Y1Fe5O12, grown by pulsed \nlaser deposition onto a gadolinium gallium garnet (Gd 3Ga5O12, GGG) (100) substrate. The 4 \n structural properties were characterized in-situ  by reflection high-energy electron diffraction \n(RHEED) and ellipsometry, and ex-situ by X -ray diffraction and transmission electronic \nmicroscopy. The film is single phase and epitaxial with atomically sharp interface. The \nmagnetic and magneto-optical (MO) properties of  the film were investigated using a custom \ndesigned broad band MO spectrometer based on 90°-polarization modulation technique. \nDetails on the experimental setup are described in Ref [32,37]. Figures 1(a) and 1(b) show the \nspectral dependency of the rotation ( ΘF, ΘK) and ellipticity ( εF, εK) Faraday and Kerr spectra \nmeasured at 300 K in polar configuration with  saturating external magnetic field applied \nperpendicular to the film plane. ΘF is negative above 487 nm with a minimum at 520 nm and \npositive between 487 nm and 353 nm with a maximum at 390 nm, whereas εF shows  two pics \ncentered at 474 nm and 357 nm. For the MO Kerr signals, the largest absolute amplitudes occur in the vicinity of the optical band gap: Θ\nK reaches -1.6 deg near 510 nm and εK reaches -\n1.3 deg near 540 nm. We note that the Faraday and Kerr spectra show a good agreement with \nprevious study of MO properties in Bi-YIG [28,37,38]. From a fundamental point of view, \nthey are well described by the crystal field energy levels of Fe3+ ions in tetrahedral and \noctahedral symmetries, which acquire a large enhancement in the spin-orbit splitting due to their hybridization with Bi-6s orbital [37,39,40]. This phenomenon is proposed to be at the \norigin of the increase of MO proprieties in Bi-YIG with increasing Bi content. Let us also \nmention that the positions of the peaks of Θ\nF nicely fit into recent results showing the \nevolution of the energy level transition associated with the tetrahedral and octahedral iron \nsites as a function of Bi content in Bi xY3-xFe5O12 ( 0.5 ≤ xBi  ≤ 3 ) [37], which confirm the \nbismuth concentration in our samples. The normalized polar and longitudinal Kerr hysteresis \nloops of the garnet film are shown in Figs . 1(c) and 1(d) respectively. The normalized \nremanence (M r/Ms) is of 0.05 and 0.45 for the polar and longitudinal configuration \nrespectively. The very weak polar remanence show that the easy axis of the magnetization is \nin the film plane.                   \nIn order to explore the effect of an ultrashor t strain pulse on the magnetization dynamics in \niron garnet, a thick Cu(100)/Pt(5) nonmagnetic metallic bilayer was deposited by dc \nmagnetron sputtering on top of the garnet film (s ee Fig. 1(e)). The numbers in parentheses are \nin nanometers and represent the thickness d of the layer. Let us mention that each metallic \nlayer plays a crucial role in our artificial structur e. The Pt top layer is important due to its high \nelectron-phonon coupling constant (109 1016 W.m-3.K-1) [41] and absorption at 800 nm [13], \nwhich enables the generation of coherent acoustic phonon with quite high frequency [42,43]. 5 \n The Cu layer is important due to its high hot-electron life time. Indeed, a thick Cu layer \nallows protecting the magnetic Bi-YIG film from the direct laser excitation, while the hot-\nelectron can travel balistically for Cu thic kness up to few hundreds of nanometers [13,14]. \nThe arrival of the hot-electron at the back side of Cu modifies its reflectivity and can be therefore used as a mark of the zero time delay that defines the onset of the pump excitation \n[44].    \nThe time-resolved MO and reflectivity measurem ents were performed at 300 K with the all-\noptical pump-probe configuration sketched in Fig. 1(c). Briefly, we have employed a \nfemtosecond laser pulse issued from an amplifie d Ti-Sapphire laser system operating at a 5 \nkHz repetition rate and delivering 35 fs pulses at  800 nm to generate the pump and the probe \nbeams. The pump beam is kept at the fundamental of the amplifier at 800 nm and excites the \nsample at normal incidence from the Pt side, while the probe beam is frequency doubled to \n400 nm using a barium boron oxide crystal and incident with a small angle of 6° onto the \nGGG substrate. Both beams are linearly polarized and focused onto the sample in spot diameters of ~260 µm for the pump and ~60 µm for the probe. The probe wavelength is well \nbelow the optical absorption edge of the GGG [45], which allows the probe to penetrate the \nsubstrate and reach the Bi-YIG layer. After in teraction with the Bi-YIG, the reflected probe \npulses allow measuring the differential changes of the MO polar Kerr rotation ΔΘ\nK (t) and \nreflectivity ΔR (t) induced by the acoustic pulse as a function of the time delay t between the \npump and probe pulses using a synchronization de tection scheme. The external magnetic field \nHext is applied perpendicular to the plane of the film.                     \nIII. RESULTS AND DISCCUSSION: \n Figure 2(a) shows the time resolved MO Kerr effect (TR-MOKE) measurement of the \ndynamics induced by a laser energy density of 11.3 mJ cm\n-2 for H ext = 3.3 kG. We note that \nthe TR-MOKE signal changes its sign when the direction of H ext is reversed. In addition, the \nzero time delay corresponds to the arrivals of hot -electron pulse to the back side of Cu layer, \nas revealed by the TR-MOKE signals measured in areas with and without the Pt/Cu bilayers. \nOn the other hand, a strong peak in the TR-MOKE signal appears at t = 40 ps, which is very \nclose to the time t = d Cu/VCu + d Bi-YIG /VBi-YIG  ≈ 41 ps  required for an acoustic pulse to cross \nthe Cu and Bi-YIG layers, where d and V are the thickness and longitudinal sound velocity \ncharacterizing Cu and Bi-YIG and their values are d Cu = 100 nm, d Bi-YIG = 140 nm, V Cu = 4730 \nm/s [13] and V Bi-YIG ≈ 6700 m/s [46]. The changes observed in ΔΘ K (t) signal in the time 6 \n delay between 0 and 40 ps have a nonmagnetic origin. This phenomenon is the same as \nreported in Ref [47] for semiconductor as it sh ows the same characteristic behaviors: (i) The \nvariation of its amplitude with the magnetic field is the same as the static MO response of the \nsample, i.e, it saturates for H ext higher than the saturating field H sat= 2.5 kOe (see Fig. 1 (c) \nand inset of Fig. 2 (a)) and changes sign when the direction of H ext is reversed (ii) Its \namplitude monotonously increases with the pump energy density. This phenomenon is due to \nthe modulation of the reflectivity signal by hot-electron pulse and the strain pulse which \naffects differently the right ( σ+) and left ( σ-) helicity of light [47]. Since the Kerr rotation can \nbe considered as the phase di fference between the reflected σ+ and σ- helicity, the different \neffect induced in σ+ and σ- is observed in the ΔΘ K(t). From a theoretical point of view, it can \nbe reproduced within the thin-film multilayer reflectivity model based on the transfer matrix method [47,48]. Such a full theoretical study goes however beyond the scope of the present \npaper. Interestingly, after the acoustic pulse leaves the Bi-YIG layer, two resonance modes \nare clearly revealed by the oscillations shown in the ΔΘ\nK (t) signal with the frequencies of 6.4 \nand 63.7 GHz, as seen in the Fourier transform sp ectrum displayed in the inset of Fig. 2(a). As \ndemonstrated hereafter, the first mode is the ferromagnetic resonance mode (f fmr) observed via \nacoustic pulse induced changes of magnetocristalline anisotropy, whereas the second mode \n(facous) results from the modulation of the MO effe ct by the propagation of the acoustic pulse \nin the GGG substrate.           \n \nThe obtained results suggest that an acoustic strain excitation is at the origin of the observed resonance modes. The existence of such a strain  pulse travelling through the sample has been \nexperimentally confirmed by measuring the pump-induced changes in the reflectivity signal \nΔR(t)/R [Fig. 2(b)], which shows oscillations for the time delay higher than 40 ps. Such \noscillations were attributed to the so-called Brillouin oscillations, which are due to the \ninterference between the probe beam reflected at the Bi-YIG interfaces and secondary beams reflected by the strain pulse propagating in the GGG substrate. The frequency associated with \nthe Brillouin oscillations is given by  ݂\nൌ2 ܸ ீீீ√݊ଶെs i nଶߠ⁄ߣ [49], where λ = 400 nm, \nVGGG = 6400 m/s [50], n ≈ 2 [45], and θ = 6° are, respectively the probe wavelength, the \nlongitudinal sound velocity in GGG, the refractive index at the probe wavelength and the \nincidence angle of the probe beam. The calculated value of f B = 63.9 GHz, which is in good \nagreement with the frequency characterizing the oscillations observed in ΔR(t)/R signal [see \ninset of Fig. 2(b)]. \n 7 \n The comparison between the results obtained from ΔΘ K(t)/ΘKmax and ΔR(t)/R measurements \nallows us to conclude that the mode f acous observed in the TR-MOKE  is related to the \npropagation of the acoustic pulse in the GGG substrate, since it has the same frequency as the \nBrillouin oscillations. This result clearly demonstrate that an acoustic pulse can induce in a MO medium a structure with a complex refractive index that allows the modulation of the \nMO effects at a frequency determined by the sound velocity as mentioned by Subkhangulov \net al [51]. On the other hand, the frequency of 6.4 GHz associated with the low-frequency \nmode is in the range of the ferromagnetic reso nance (FMR) frequencies in  Bi-YIG. In order to \nconfirm the magnetic origin of this mode, we investigated the effect of the external magnetic \nfield on the TR-MOKE. ΔΘ\nK(t) measured at selected external magnetic field are displayed in \nFig. 3(a). The frequency and amplitude of the low-frequency mode are clearly influenced by \nthe external magnetic field. To further highlight the behavior of the modes, the field \ndependence of the oscillations frequency and ampl itude as determined by Fourier analyses are \nshown in Figs. 3(b) and 3(c). The variation of the oscillations frequency of the low-frequency \nmode can be described by Kittel formula adapted to the case of our experimental \nconfiguration [52]:  \nωൌ γ  ൫ H\n௫௧െH ൯     ሺ1ሻ  \n \nwhere ω is the angular precession frequency, γ the gyromagnetic ratio, H ext the external \nmagnetic field, and the effective field H eff is defined as (4 πMs - H u - H c) where H u and H c are \nthe uniaxial and cubic anisotropy fields, respectively. The adjustment of Fig. 3(b) with Eq (1) \nusing γ = 2.8 GHz/kG yields H eff = 1.22 kG. This behavior of the low-frequency mode clearly \nindicates that is associated with FMR. We also note that we have found that the initial precession amplitude for the low-frequency mode  has maximum near the saturating field H\nsat \n= 2.5 kG (see Fig 1(c) and Fig (3c)), which is also in a qualitative agreement with the typical \nbehavior obtained for the FMR mode when H ext is applied along a hard magnetization axis as \nin our experimental configuration [53,54] . On the other hand, the frequency of f acous is \nindependent on the magnetic field strength (Fig. 3(c)). As discussed above, this is the \nexpected behavior for the acoustically-induced modulation of the MO effects. Furthermore, \nwe show that the initial oscillations amplitude of f acous as a function of field has the same \nbehavior as the MO response of our sample. Such dependence can be related to the \ncharacteristic behavior of the non-magnetic contribution observed in ΔΘ K (t) for t between 0 \nand 40 ps which modulate the TR-MOKE signal.  8 \n Let’s us now focus on the mechanism behind the excitation of the ferromagnetic resonance in \nour sample. It results from an ultrafast non-thermal modification of the magnetocristalline \nanisotropy induced by the acoustic strain puls es via the inverse magnetostriction effect, as \ninitially shown by Scherbakov et al in GaMnAs [18]. The excitation of the FMR mode for a magnetization already aligned along H\next, i.e higher than H sat = 2.5 kG (see Fig. 1(c) and Fig. \n4), substantiate this interpretation. Indeed, a decrease of the magnetocristalline parameters \ninduced by heating effects does not allow ex citing the FMR mode when  the magnetization is \naligned along H ext [25]. Let us also mention that the non-thermal mechanisms based on the \nCotton-Mouton effect [55] and photo-induced magnetic anisotropy [12,32] usually used to \ninduce in magnetic garnet a magnetization preces sion with a linearly polarized light can be \nexcluded in our case. This is due to the very weak transmitted light (less than 0.1%) from the thick Pt/Cu bilayer to the Bi-YIG layer. Moreover, we have investigated using the same \nconfiguration the ultrafast magnetization dynami cs induced by direct light excitation (not \nshown). No magnetization precession has been observed. This results further highlights the \nimportance of the approach based on acoustic strain pulse for generating spin wave via the \ninverse magnetostriction effect.       \nTo further investigate the two resonance mode s, we performed TR-MOKE measurements as a \nfunction of the laser energy density E\npump. Figure 4(a) shows the TR-MOKE signals measured \nat selected E pump for H ext= 3.3 kG. The pump energy densit y dependence of the oscillations \nfrequency and amplitude extracted using Fourier an alysis are presented in Figs. 4(b) and 4(c). \nThe frequency of both modes is independent of the E pump. The behavior of f fmr is similar to the \none obtained by non-thermal effects induced magnetization precession [32,34]. This is in \nagreement with our interpretation based on stra in-induced changes of magnetic anisotropy via \nthe inverse magnetostriction effect. Indeed, in the case of thermally induced spin precession a \ndependence of the frequency on E pump is usually observed [56,57]. On the other hand, the \nbehavior of f acous as a function of E pump is also in agreement with the prediction that the \nfrequency of acoustically-induced modulation of the MO effects is mainly defined by the \nspeed of sound. Moreover, our experiments show that the oscillations amplitude of the two \nresonance modes increases linearly with the laser energy density within the probed range. \nThis means that the amplitude of the spin prece ssion is proportional to the amplitude of the \nstrain pulse. Therefore, using an engineered st ructure that allows injecting a higher amplitude \nstrain pulse into Bi-YIG can be used for further improving the magnetization precession amplitude or inducing a magnetizati on switching in magnetic garnet. 9 \n VI. CONCLUSION: \n \nWe have studied the laser-induced ultrafast magnetization in a dielectric film of bismuth-\nsubstituted yttrium iron garnet buried below a thick Pt/Cu bilayer. It is found that exciting the \nsample from Pt surface launches coherent stain pulses that propagate into the garnet film.  We \ndemonstrate that this acoustic pulse modifies th e magnetocristalline anisotropy via the inverse \nmagnetostriction effect. 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Lai, Appl Phys Lett 101, 222402 (2012). \n 12 \n 400 600 800-40-200204060\n400 600 800-1.5-1.0-0.50.00.51.01.5\n-4 -2 0 2 4-1.0-0.50.00.51.0\n-4 -2 0 2 4-1.0-0.50.00.51.0ΘF  εFΘF , εF (deg/μm)\nWavelength (nm)(a) (b) ΘK  εK\n ΘK , εK (deg/ μm)\nWavelength (nm)\n  H⊥(c)ΘK/ΘKsat\nMagnetic field (kG) H//(d)\nΘK/ΘKsat\nMagnetic field (kG)\n(e)\nPump\n(800nm)\nTo the \npolarization\nbridge\nPt (5 nm)\nBi2Y1Fe5O12Probe\n(400nm)Cu (100 nm)strain pulse  \n    \n  \n         \n \n \n \n \n \n \n \nFigure 1: Static room temperature magn eto-optical and magnetic properties of Bi 2Y1Fe3O15 \nthin film and the pump-probe experimental conf iguration. (a, b) Magneto-optical Faraday (a) \nand Kerr (b) polar spectra measured over a broad range of wavelength. The filled and open \nsymbols represent, respectively, the rotation ( ΘF, ΘK) and ellipticity ( εF, εK). (c,d) Normalized \nmagneto-optical hysteresis loops measured in polar (c) and longitudinal configuration. (e) \nSketch of the time resolved experimental co nfiguration that allows studying the ultrafast \nmagnetization dynamics induced by acoustic pulse.    \n \n 13 \n 0 100 200 300 400 500 600-0.03-0.02-0.010.000.010.020.030.040.0502 0 4 0 6 0 8 0 1 0 00.00.51.01.5\n0 50 100 150 200 250-0.75-0.50-0.250.000.250.50\n02 0 4 0 6 0 8 0 1 0 00.00.51.01.502468 1 00.000.010.020.03ΔΘΚ/ΘΚsat  \nTime (ps)(a)\n  FFT ( arb.unit)\nFrequency (GHz)ffmr= 6.4 GHz\nfacous= 63.7 GHz\n(b)ΔR/R x10-2 \nTime (ps)\nFFT ( arb.unit)\nFrequency (GHz)fB= 63.7 GHzΔΘΚ/ΘΚsat \nMagnetic Field (kG)t = 20 ps \n \n \n \n \n \n \n \n             \n   \n \n          \nFigure 2: Dynamics of spin and reflectivity in the Bi 2Y1Fe5O12/GGG(100) buried below a \nthick Pt/Cu bilayers. (a, b) ΔΘ K/ΘKsat and ΔR/R induced by a laser energy density of 11.3 mJ \ncm-2 for H ext= 3.3 kG.  Inset (a): the Fourier transform spectrum of the ΔΘ K/ΘKsat data for the \ntime delay t ≥ 50 ps and the ΔΘ K/ΘKsat measured at the time delay t = 20 ps as a function of  \nHext.   Inset (b): the Fourier transform spectrum of the ΔR/R data for the time delay t ≥ 50 ps . \nThe solid lines in (a) are guides to the eyes. 14 \n 0 100 200 300 400 500 6000.00.10.2\n02468 1 0020406080100\n02468 1 00.000.050.109.8 kG1.6 kG\n2.5 kG\n3.3 kG\n4.9 kG\n6.6 kG\n Time (ps)\n ΔΘΚ/ΘΚsat(a) Ferromagnetic modeAcoustic\n   mode\n(b)  Acoustic mode\n FMR mode Frequency (GHz)\nMagnetic Field (kG) FMR mode(c) Amplitude (arb.unit)\nMagnetic Field (kG)0.000.020.040.06 Acoustic mode\nAmplitude (arb.unit) \n \n         \n \n \n     \n \n \n \nFigure 3: Magnetic field dependence of the spin dynamics. (a) ΔΘ K/ΘKsat as a function of the \nmagnetic field. (b, c) Field dependence of th e precession frequencies (b) and amplitudes (c) \nassociated with acoustic and FMR resonance modes. All measurements are obtained for a pump energy density of 11.3 mJ cm\n-2. The dashed lines in (a) and solid lines in (c) are guides \nto the eyes. In (b) the solid line describing the FMR mode is a fit with the Kittel formula, \nwhereas the solid line for the acoustic mode is a guide to the eyes.  \n \n 15 \n 0 100 200 300 400 500 6000.00.10.20.3\n05 1 0 1 50.000.050.100.15\n05 1 0 1 505106070809015.1 mJ cm-2\n13.2 mJ cm-2\n11.3 mJ cm-2\n9.4 mJ cm-2\n7.6 mJ cm-2\n5.6 mJ cm-2\n4.9 mJ cm-2\n3.4 mJ cm-2\n1.9 mJ cm-2ΔΘΚ/ΘΚsat\n Time (ps)(a)\n(c)  Acoustic mode\n  FMR mode \n Amplitude (arb.unit)\nEpump(mJ.cm-2) Acoustic mode\n FMR mode Frequency (GHz)\nEpump (mJ.cm-2)(b) \n    \n \n          \n \n  \n \nFigure 4: Laser energy density dependence of the spin dynamics. (a) ΔΘ K/ΘKsat as a function \nof the laser energy density. (b, c) Variation of the precession frequencies (a) and amplitudes \n(b) associated with acoustic and FMR resonance modes as a function of the laser energy \ndensity. All measurements are obtained for H ext = 3.3 kG. The solid lines in (b) and (c) are \nguides to the eyes.  \n \n "    },    {        "title": "2305.05889v1.Proposal_for_optomagnonic_teleportation_and_entanglement_swapping.pdf",        "content": "Proposal for optomagnonic teleportation and entanglement swapping\nZhi-Yuan Fan,1Xuan Zuo,1Hang Qian,1and Jie Li1,\u0003\n1Interdisciplinary Center of Quantum Information, State Key Laboratory of Modern Optical Instrumentation,\nand Zhejiang Province Key Laboratory of Quantum Technology and Device,\nSchool of Physics, Zhejiang University, Hangzhou 310027, China\n(Dated: May 11, 2023)\nA protocol for realizing discrete-variable quantum teleportation in an optomagnonic system is provided. Us-\ning optical pulses, an arbitrary photonic qubit state encoded in orthogonal polarizations is transferred onto the\njoint state of a pair of magnonic oscillators in two macroscopic yttrium-iron-garnet (YIG) spheres that are placed\nin an optical interferometer. We further show that optomagnonic entanglement swapping can be realized in an\nextended dual-interferometer configuration with a joint Bell-state detection. Consequently, magnon Bell states\nare prepared. We analyze the e \u000bect of the residual thermal occupation of the magnon modes on the fidelity in\nboth the teleportation and entanglement swapping protocols.\nI. INTRODUCTION\nAs an indispensable building block of quantum informa-\ntion science, quantum teleportation refers to the process of\ntransferring an unknown quantum state at one location onto\nanother quantum system some distance away. Ever since it\nwas first proposed by Bennett et al. [1], quantum teleporta-\ntion has been successfully realized in various physical systems\nover the past few decades. These include photons [2–5], nu-\nclear spins [6], trapped ions [7, 8], atomic ensembles [9, 10],\nsolid-state systems [11], high-frequency vibration phonons\n[12], and optomechanical systems [13], among many others.\nThese successful demonstrations lay the foundation for re-\nalizing many other quantum protocols, such as quantum re-\npeaters [14], fault-tolerant quantum computation [15], etc.\nHere we provide an optomagnonic quantum teleportation\nprotocol which can transfer an arbitrary photonic qubit state\nto a dual-rail encoding magnonic system of two yttrium-iron-\ngarnet (YIG) spheres. We adopt an optical interferometer con-\nfiguration, of which each arm contains a YIG sphere sup-\nporting an optomagnonic system. We use the Stokes type\nscattering of the magnon-induced Brillouin light scattering\n(BLS) to create an optomagnonic EPR state. A subsequent\nBell-state measurement onto the input photonic qubit and the\noutput Stokes photon from the interferometer enables such a\nphoton-to-magnon quantum state transfer. The magnon state\ncan be read out by activating the anti-Stokes scattering real-\nizing the optomagnonic state-swap operation, from which the\nmagnonic qubit state is retrieved onto the anti-Stokes photon.\nWe further propose an optomagnonic entanglement swapping\nprotocol based on an extended dual-interferometer configura-\ntion, which realizes the transfer of the optomagnonic entan-\nglement to a dual-rail encoding two-qubit magnonic system\ninvolving four YIG spheres and thus prepares a magnonic Bell\nstate.\nIn what follows, we first introduce the basic optomagnonic\ninteractions in Section II, and describe our two protocols re-\nspectively in Section III and Section IV. We then analyze the\n\u0003jieli007@zju.edu.cne\u000bect of the magnon thermal excitations on the fidelity of the\ntwo protocols in Section V and finally conclude in Section VI.\nII. OPTOMAGNONIC INTERACTION\nIn the past decade, hybrid magnonic systems based on\ncollective spin excitations in ferrimagnetic materials, such\nas YIG, have gained significant attention due to their excel-\nlent ability to coherently couple with a variety of physical\nsystems, including microwave photons [16–18], optical pho-\ntons [19–23], vibration phonons [24–28] and superconduct-\ning qubits [29–33]. The emerging field of hybrid quantum\nmagnonics provides a platform not only for studying strong\ninteractions between light and matter, but also for developing\nnovel quantum technologies to be applied in quantum infor-\nmation processing, quantum sensing and quantum networks\n[34–37]. In particular, the coupling between magnons and\noptical photons, namely the optomagnonic interaction [19–\n23], is indispensable for building a magnonic quantum net-\nwork [37], where the transmission of the information between\nremote quantum nodes is realized by optical photons. Such\nan optomagnonic interaction has been exploited in many pro-\nposals to cool magnons [38], prepare magnon Fock [39], cat\n[40], path-entangled [41] states and magnon-photon entan-\ngled states [37, 42], and realize magnon laser [43, 44], fre-\nquency combs [45], photon blockade [46], polarization-state\nengineering [47], etc.\nThe typical optomagnomic system, as depicted in Figure\n1(a), consists of a YIG sphere, which supports both a magne-\ntostatic mode (i.e., the magnon mode) and optical whispering\ngallery modes (WGMs). The WGM resonator is near the sur-\nface of the YIG sphere and the input optical field is evanes-\ncently coupled to the WGM, e.g., via a tapered fiber [19, 20]\nor prism [21, 22]. The optomagnonic interaction in such a sys-\ntem is embodied by the magnon-induced BLS, where the pho-\ntons of a WGM are scattered by lower-frequency magnons,\ntypically in GHz [19–22], giving rise to sideband photons with\ntheir frequency shifted by the magnon frequency and their\npolarization changed. When the frequency of the scattered\nphotons matches another WGM, namely the triple resonance\ncondition, the optomagnonic scattering probability is maxi-arXiv:2305.05889v1  [quant-ph]  10 May 20232\nFIG. 1: (a) A typical optomagnonic system: a YIG sphere supports a magnon mode and two optical WGMs. (b) Mode frequencies corre-\nsponding to the optomagnonic Stokes scattering, which leads to the PDC interaction. (c) Mode frequencies associated with the optomagnonic\nanti-Stokes scattering, which yields the state-swap (BS) interaction.\nmized. This can be easily achieved by tuning the magnon\nfrequency via changing the strength of the bias magnetic field\n(B0). Typically, there are two types of magnon-induced BLSs,\ni.e., the Stokes and anti-Stokes scatterings, corresponding to\noptomagnonic parametric down conversion (PDC) and state-\nswap (beam-splitter, BS) interactions, respectively [37, 41].\nIn the optomagnonic scattering, the angular momenta of the\nWGM photons and magnons are conserved, which leads to\na selection rule and prominent asymmetry in the Stokes and\nanti-Stokes scattering strengths [48–51]. This allows us to se-\nlect a particular type of the optomagnonic interactions (either\nPDC or BS) on demand.\nThe Hamiltonian accounting for the optomagnonic interac-\ntion in such a three-mode system reads\nH=~=!vaya+!hbyb+!mmym\n+g0(aybm+abymy)+Hdri=~;(1)\nwhere j=a;b(jy) and m(my) are the annihilation (creation)\noperators of the two WGMs and the magnon mode, respec-\ntively, and!k(k=v;h;m) correspond to their resonance fre-\nquencies, satisfying the relations j!v\u0000!hj=!mand!m\u001c\n!v;h. Here the subscripts vandhrepresent two orthogonal\npolarizations of the two WGMs, i.e., the transverse-electric\n(TE) mode and the transverse-magnetic (TM) mode. The op-\ntomagnonic interaction is a three-wave process and g0is the\ncorresponding single-photon coupling strength. The last term\ndenotes the driving Hamiltonian Hdri=i~Ej(jye\u0000i!0t\u0000jei!0t),\nwhere Ej=p\u0014jPj=(~!0) is the coupling strength between the\nWGM j(with decay rate \u0014j) and the optical drive field with\nfrequency!0and power Pj. To enhance the optomagnonic\ncoupling strength, a strong drive field is used to resonantly\npump one of the WGMs, i.e., !0=!vor!h, depending on\nwhich type of the optomagnonic interactions (either PDC or\nBS) is desired.\nFor the case where the TE WGM ( a) is pumped, i.e.,\n!0=!vand\u0001b\u0011!h\u0000!0=\u0000!m, cf. Figure 1(b), the\nstrongly driven WGM acan be treated classically as a num-\nber\u000b\u0011 hai[37], and the e \u000bective optomagnonic Hamilto-\nnian then becomes the PDC form, HS\nint=~g(bm+bymy), with\ng=g0\u000bbeing the e \u000bective optomagnonic coupling strength.\nThis interaction corresponds to the Stokes scattering, where\nthe TE-WGM photons convert into lower-frequency sidebandphotons (resonant with the TM WGM b) by creating magnon\nexcitations. Such a PDC interaction can be used to gener-\nate optomagnonic entangled states [37, 42]. Specifically, the\nWGM band magnon mode mare prepared in a two-mode\nsqueezed state (unnormalized)\nj ib;m=j00ib;m+p\nPj11ib;m+O(P); (2)\nwhere Pis the probability for a single Stokes scattering event\nto occur andO(P) denotes the higher-excitation terms, of\nwhich the probabilities are equal to or smaller than P2. The\nscattering probability increases with the power of the drive\nfield, and when the power is su \u000eciently weak, the scatter-\ning probability P\u001c1. In this weak-coupling limit, the\nprobability of generating two-magnon (photon) state j2im(b)\nand higher-excitation states jnim(b)(n>2) is negligibly small.\nSuch a low scattering probability of creating an entangled pair\nof single excitations was adopted in Ref. [41], which suggests\nan optomagnonic variant of the Duan-Lukin-Cirac-Zoller pro-\ntocol [52]. It was also used in cavity optomechanical ex-\nperiments for creating entangled states of single photons and\nphonons [53, 54].\nSimilarly, when the TM WGM ( b) is resonantly pumped,\ni.e.,!0=!hand\u0001a\u0011!v\u0000!0=!m, cf. Figure 1(c), the anti-\nStokes scattering is activated and the e \u000bective optomagnonic\nHamiltonian becomes the BS type, HAS\nint=~g(aym+amy), with\nthe e\u000bective coupling g=g0\fand\f\u0011hbi. This interaction\nrealizes the state-swap operation between the magnon mode\nand the WGM a, accompanied with the process that TM-\nWGM photons convert into higher-frequency anti-Stokes pho-\ntons (resonant with the TE WGM a) by eliminating magnon\nexcitations. As will be shown later, this interaction is used to\nread out the magnon state.\nIII. OPTOMAGNONIC QUANTUM TELEPORTATION\nWe now proceed to describe our optomagnonic teleporta-\ntion protocol, which is able to transfer a photonic qubit state\n(in polarization encoding) onto a magnonic system consisting\nof two optomagnonic devices placed in two arms of an opti-\ncal interferometer, cf. Figure 2. The two magnon oscillators\nare subject to a simultaneous excitation using a weak pulse3\nFIG. 2: Sketch of the optomagnonic quantum teleportation protocol. It consists of three steps: preparation of the optomagnonic EPR state,\nBell-state measurement and readout of the magnonic state. In the EPR-state preparation setup, an optical interferometer configuration is\nadopted and its each arm contains a YIG sphere which supports an optomagnonic system of a magnon mode and two optical WGMs. See text\nfor detailed descriptions.\nthat drives the TE WGM to activate the optomagnonic Stokes\nscattering. After a Bell-state measurement of the input photon\nwith the Stokes photon from the optomagnonic devices, the\ninput photonic qubit state is then transferred onto the dual-rail\nencoding magnonic system.\nFor simplicity, we assume at most single excitations in\nthe optomagnonic devices. This is the case of using a weak\npulse, where the probability of creating higher-excitation\nstatesjnim(b)(n\u00152) in the Stokes scattering is negligible.\nSince the vacuum component of the state (2) will not trigger\nany coincidences in the Bell-state detection, leading to un-\nsuccessful trials for the teleportation, the protocol can be de-\nscribed using a simplified model [55], where a TE-polarized\nsingle-photon pulse is sent onto a 50 /50 BS to drive the op-\ntomagnonic devices in the interferometer. After the BS, an\noptical path-entangled state,1p\n2(j01iAB+j10iAB), is generated\nin the two outputs (i.e., the upper path A and lower path B).\nIn each path, the pulse resonantly drives the TE WGM of the\nYIG sphere to activate the Stokes scattering. By selecting tri-\nals with successful scattering events, the PDC interaction pre-\npares an optomagnonic Bell state in the form of\nj\u001e+ibm=1p\n2(jHibjLim+jVibjUim); (3)\nwherejLim(jUim) denotes the generated single magnon is at\nthe lower path B (the upper path A), and jHib(jVib) representsthe accompanied Stokes photon is in the horizontal (vertical)\npolarization. The Stokes photon, with equal probability in\npath A or B, then couples to the nanofiber or the prism coupler\nand enters the first polarizing beam splitter (PBS1) before go-\ning to the next stage of the Bell-state detection. Note that the\npolarization of the Stokes photon in the upper path is changed\nfrom HtoVafter passing through a half-wave plate (HWP).\nWe remark that the unsuccessful scattering events leave the\nsingle photons remaining in the vertical polarization. These\nphotons eventually enter the other output of the PBS1 and thus\nhave no impact on the subsequent Bell-state measurement. We\nalso assume that the magnon modes are initially in their quan-\ntum ground state, which is the case for GHz magnons at a low\ntemperature, e.g., of 10 mK. Nevertheless, in Section V we\nshall discuss the e \u000bect of the residual thermal occupation on\nthe fidelity of the teleportation.\nThe input photonic qubit state to be teleported is an ar-\nbitrary superposition of two polarization modes, i.e., j\u001fic=\n\u000bjHic+\fjVic, with the complex coe \u000ecients\u000band\fsatis-\nfyingj\u000bj2+j\fj2=1. Such an arbitrary qubit state can be\nconstructed on the surface of the polarization Poincare sphere\nby using a HWP and a quarter-wave plate (QWP). This in-\nput photon, being resonant with the TM WGM, goes into one\ninput of the PBS2 and meanwhile the output of the PBS1 en-\nters the other input port. Thereby, the joint state before the4\nBell-state measurement is as follows:\nj\u001e+ibm\nj\u001fic=1p\n2\u0000\u000bjHibjHicjLim+\fjHibjVicjLim\n+\u000bjVibjHicjUim+\fjVibjVicjUim\u0001:(4)\nThe Bell-state measurement is performed onto the input pho-\ntonic qubit ( c) and the output Stokes photon ( b) from the in-\nterferometer, and the detection setup consists of a HWP, a\nPBS and two single-photon detectors in each output of the\nPBS2, cf. Figure 2. A coincidence measurement projects\nthe optical modes onto the polarized Bell states j\u001e\u0006ibc=\n1p\n2(jHibjHic\u0006jVibjVic). The fast axis of the HWP is set at\n22:5\u000e, which acts as a Hadamard operation on the polarization\nof the photons passing through PBS2. The Bell states j\u001e\u0006ibc\ncorrespond to di \u000berent coincidence measurements, as can be\nseen from the following\nj\u001e+ibcPBS2 & HWP\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000!1p\n2\u0010\nay\n3;hay\n4;h+ay\n3;vay\n4;v\u0011\njvaci;\nj\u001e\u0000ibcPBS2 & HWP\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000!1p\n2\u0010\nay\n3;hay\n4;v+ay\n3;vay\n4;h\u0011\njvaci;(5)\nwhere the operator ay\ni;h(ay\ni;v) denotes the detection of a single\nhorizontally (vertically)-polarized photon at one of the two\noutputs of the PBS i(i=3;4), andjvacimeans the vacuum\nstate. Note that, in addition to the Bell states j\u001e\u0006ibc, the other\ntwo types of Bell states j \u0006ibc=1p\n2(jHibjVic\u0006jVibjHic) can\nbe realized by using photon-number-resolving detectors, and\nwe disregard these cases for simplicity.\nThe measurement that projects the optical modes bandc\nonto the Bell state j\u001e+ibcprojects the magnonic system onto\nthe state\nj\u001f0im=\u000bjLim+\fjUim; (6)\nwhich indicates the successful teleportation of the input pho-\ntonic qubit statej\u001ficto a dual-rail encoding magnonic system,\nand corresponds to the ideal quantum teleportation without\nrequiring additional correction operations in the readout step.\nOn the other hand, the measurement associated with the Bell\nstatej\u001e\u0000ibcprojects the magnonic system onto the state\nj\u001f00im=\u000bjLim\u0000\fjUim: (7)\nIt has a\u0019-phase di \u000berence with respect to the input optical\nstatej\u001fic, which can be corrected in the readout step using a\nfeed-forward operation by applying a phase shift in the optical\ninterferometer [13, 55].\nTo verify the successful teleportation, we need to retrieve\nthe teleported magnonic qubit state. To achieve this, we ex-\nploit the optomagnonic state-swap (BS) interaction, as intro-\nduced in Section II, where a TM-polarized weak pulse is sent\nto the interferometer to activate the anti-Stokes scattering and\nthe magnonic state is then transferred to the anti-Stokes pho-\nton. Due to the orthogonal polarization with respect to the\nStokes photon (produced in the Stokes scattering), the anti-\nStokes photon leaves from the other output of the PBS1, and\nenters the state-readout setup (cf. Figure 2) [13], from which\nthe magnonic qubit state is retrieved.IV . OPTOMAGNONIC ENTANGLEMENT SWAPPING\nQuantum entanglement plays an essential role in all the\nquantum-teleportation related protocols. Naturally, entangled\nstates can be obtained in directly coupled systems, such as\nthe EPR state produced in the optomagnonic Stokes scatter-\ning. For two systems that have no direct interaction, quantum\nmechanics also manifests its unique capabilities to establish\nquantum entanglement between them, one of which is referred\nto the entanglement swapping [1, 3, 56]. In this section, we\nshow that the entanglement swapping protocol allows us to\nprepare the magnon modes in space-separated YIG spheres\ninto an entangled Bell state.\nThe detailed entanglement swapping protocol is shown in\nFigure 3, which consists of two optical interferometer setups\nused in Section III, a joint Bell-state detection and the as-\nsociated state-readout devices. Similarly to the teleportation\nprotocol, in each interferometer setup containing two YIG\nspheres, a TE-polarized single-photon pulse is sent to activate\nthe Stokes scattering, which prepares an optomagnonic Bell\nentangled statej\u001e+ibm, as in Equation (3). Therefore, the over-\nall state of the two interferometer setups before performing a\njoint Bell-state measurement on the scattered Stokes photons\nreads\nj\titotal=j\u001e+ib1m1\nj\u001e+ib2m2\n=1\n2\u0000jHib1jLim1+jVib1jUim1\u0001\n\u0002\u0000jHib2jLim2+jVib2jUim2\u0001;(8)\nwhere the subscripts 1 and 2 are used to distinguish the two\ninterferometers, and the notation is the same as in Equation\n(3). The above state can be rewritten in the basis of Bell states\nas\nj\titotal=1\n2\u0010\nj +ib1b2j +im1m2+j \u0000ib1b2j \u0000im1m2\n+j\u001e+ib1b2j\u001e+im1m2+j\u001e\u0000ib1b2j\u001e\u0000im1m2\u0011\n;(9)\nwherej\u001e\u0006ib1b2andj \u0006ib1b2are the four Bell states of the\nStokes photons in the two interferometers, which take the\nsame form asj\u001e\u0006ibcandj \u0006ibcin Section III.j\u001e\u0006im1m2and\nj \u0006im1m2are the four Bell states of a pair of dual-rail encod-\ning magnonic systems, defined as j\u001e\u0006im1m2=1p\n2(jLim1jLim2\u0006\njUim1jUim2) andj \u0006im1m2=1p\n2(jLim1jUim2\u0006jUim1jLim2).\nFrom Equation (9), it is clear to see that a Bell-state mea-\nsurement on the Stokes photons from the outputs of the two\ninterferometers projects the magnonic systems onto a cor-\nresponding Bell state. Specifically, a coincidence measure-\nment corresponding to the optical Bell state j\u001e+ib1b2(j\u001e\u0000ib1b2)\nprojects the magnonic systems onto the Bell state j\u001e+im1m2\n(j\u001e\u0000im1m2). This implies that the magnonic systems establish\nthe same form of entanglement as the corresponding optical\nBell state. Similarly as in the teleportation protocol, we disre-\ngard the other two types of the Bell-state measurements asso-\nciated withj \u0006ib1b2.\nThe entanglement swapping can also be understood in the\nframework of quantum teleportation in the sense that it trans-5\nFIG. 3: Sketch of the optomagnonic entanglement swapping protocol. It is based on a dual-interferometer configuration, combined with a joint\nBell-state detection and magnonic state readout devices.\nfers an optomagnonic quantum correlation (instead of a pho-\ntonic qubit statej\u001fic) to the magnonic systems. This can be\nseen by comparing Eqs. (4) and (8). This further reflects the\nversatility of the quantum teleportation protocol, which can\ntransfer not only a qubit state but also multi-qubit states, e.g.,\nquantum correlations. Replacing the optomagnonic system\nwith an optomechanical system [54] in one of the interferom-\neters also allows us to prepare a hybrid magnon-phonon Bell\nstate, which may find potential applications in hybrid quantum\nnetworks [37].\nV . EFFECT OF MAGNONIC THERMAL EXCITATIONS\nIn the teleportation and entanglement swapping protocols,\nwe neglect the dissipation of the magnon modes. This is the\ncase of using fast optical pulses [53] such that the magnon dis-\nsipation can be assumed to be negligible during an experimen-\ntal run. We also assume that the magnon modes are initially\nprepared in their quantum ground state. This is a good approx-\nimation for the magnon modes at \u0018GHz frequency [19–23]\nand at low temperature of, e.g., 10 mK. However, the optical\npulses may heat the magnon modes due to the optical absorp-\ntion of the YIG, causing the magnon modes to be at a thermal\nstate. To include this heating e \u000bect in practical situations, we\nassume that the magnon modes are initially prepared in a ther-\nmal state\u001ath=(1\u0000s)P1\nn=0snjnihnj, with s=¯n0=(¯n0+1) anda thermal occupation ¯ n0\u001c1. Note that the frequencies of the\nmagnon modes are assumed to be (nearly) identical and thus\nthey have equal thermal occupation, i.e., they are in the same\nthermal state. For a small ¯ n0<0:2,s<0:17,s2<0:03 and\ns3<0:005, and the total probability of higher-excitation terms\njni(n>2) is less than 0 :5%. Thus we can safely approximate\n\u001ath'(1\u0000s)\u0010\nj0ih0j+sj1ih1j+s2j2ih2j\u0011\n. The density matrix\nof the two magnon modes (in path A and B) in the teleporta-\ntion scheme is then\n\u001am=\u001aA\n\u001aB'(1\u0000s)22X\nnA;nB=0snA+nBjnAnBihnAnBj;(10)\nwhich is a probabilistic mixture of nine pure states jnAnBi\n(nA;nB=0;1;2). This mixed initial state eventually leads\nto the following teleported magnonic state after the Bell-state\nmeasurement associated with j\u001e+ibc:\n\u001a0\nm=(1\u0000s)22X\nnA;nB=0snA+nBj'nAnBih'nAnBj; (11)\nwherej'nAnBi=\u000bjnA;nB+1iAB+\fjnA+1;nBiABis the tele-\nported magnonic state corresponding to the pure inital state\njnAnBiin Equation (10). Clearly, j'00i=\u000bj01iAB+\fj10iAB\u0011\n\u000bjLim+\fjUimcorresponds to the ideal initial state j00iABcon-\nsidered in Section III. For the Bell-state detection related to6\nFIG. 4: Fidelity in the teleportation (solid) and entanglement swap-\nping (dashed) protocol versus the thermal occupation ¯ n0of the\nmagnon modes.\nj\u001e\u0000ibc, we obtain the same \u001a0\nmas in Equation (11) but with \f\nreplaced by\u0000\finj'nAnBi. All other states in \u001a0\nmare orthogo-\nnal to the desired state j'00i, and result in a reduction of the\nteleportation fidelity, which is\nF1=h'00j\u001a0\nmj'00i=1=\u00001+s+s2\u00012: (12)\nThe solid line in Figure 4 clearly shows a declining fidelity\nversus the thermal occupation ¯ n0. For a genuine quantum tele-\nportation with the fidelity F1>2=3 [57], a small ¯ n0\u0014\u00180.2 is\nrequired. This is similar to the finding in the optomechanical\nteleportation [55].\nBy contrast, in the entanglement swapping protocol, four\nmagnon modes (in path A, B, C and D, respectively) are in-\nvolved, cf. Figure 3. Following the same approach, we obtain\nthe final joint state of the magnonic systems, after the Bell-\nstate measurement associated with j\u001e\u0006ib1b2, given by\n\u001a0\nm1m2=(1\u0000s)42X\nnA(B)=02X\nnC(D)=0snA+nBsnC+nDj'm1m2ih'm1m2j;(13)\nwhere A,B,Cand Dare used to distinguish the\nmagnon modes via their path information, and j'm1m2i=\n1p\n2\u0010\njnA;nB+1;nC;nD+1im1m2\u0006jnA+1;nB;nC+1;nDim1m2\u0011\nis the teleported magnonic state corresponding to the pure\nstatejnAnBnCnDiin the mixed initial state \u001am1m2=\u001am1\n\u001am2,\ncf. Equation (10). For the ideal case of the initial ground\nstate considered in Section IV, we obtain the desired statesj'm1m2i=1p\n2\u0010\nj0101im1m2\u0006j1010im1m2\u0011\n\u0011j\u001e\u0006im1m2after the\nentanglement swapping. The other additional terms in \u001a0\nm1m2are related to the residual thermal excitations in the magnonic\ninitial state, which are unwanted and reduce the fidelity in the\nentanglement swapping protocol. The corresponding fidelity\nis given by\nF2=m2m1h\u001e\u0006j\u001a0\nm1m2j\u001e\u0006im1m2=1=\u00001+s+s2\u00014: (14)\nThe dashed line in Figure 4 shows a decreasing fidelity F2as\nthe thermal occupation ¯ n0increases. It reduces more rapidly\nwith respect to the fidelity F1in the teleportation protocol be-\ncause of the relation F2=F2\n1, as seen from Equations (12)\nand (14).\nVI. CONCLUSION\nWe present two protocols for realizing optomagnonic quan-\ntum teleportation and entanglement swapping, respectively,\nadopting YIG spheres and an optical interferometer configu-\nration. The optomagnonic Stokes and anti-Stokes scatterings\nare the essential elements for preparing optomagnonic EPR\nstates and optically reading out the magnonic states. A Bell-\nstate detection enables the transfer of an arbitrary photonic\nqubit state to a dual-rail encoding magnonic system in the for-\nmer protocol, and the transfer of the optomagnonic entangle-\nment to the magnon modes in a dual-interferometer configu-\nration which are prepared in a Bell state in the latter protocol.\nWe further discuss the e \u000bect of the residual thermal excita-\ntions on the fidelity in both the protocols. Our work suggests\nthat optomagnonic systems could become a new platform for\nrealizing quantum teleportation and entanglement swapping\nwhere quantum superposition and entangled states of macro-\nscopic objects (e.g., YIG spheres with diameter of hundreds\nof microns [19–22]) could be generated. 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J. van Wees1\n1Physics of Nanodevices, Zernike Institute for Advanced Materials,\nUniversity of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands.\n2Kavli Institute of NanoScience, Delft University of Technology, Delft, The Netherlands\n3Institute for Materials Research and WPI-AIMR, Tohoku University, Sendai, Japan\n4Universit\u0013 e de Bretagne Occidentale, Laboratoire de Magn\u0013 etisme de Bretagne CNRS, 6 Avenue Le Gorgeu, 29285 Brest, France.\n(Dated: May 25, 2022)\nWe report the observation of the spin Peltier e\u000bect (SPE) in the ferrimagnetic insulator Yttrium\nIron Garnet (YIG), i.e. a heat current generated by a spin current \rowing through a Platinum\n(Pt)jYIG interface. The e\u000bect can be explained by the spin torque that transforms the spin current in\nthe Pt into a magnon current in the YIG. Via magnon-phonon interactions the magnetic \ructuations\nmodulate the phonon temperature that is detected by a thermopile close to the interface. By \fnite-\nelement modelling we verify the reciprocity between the spin Peltier and spin Seebeck e\u000bect. The\nobserved strong coupling between thermal magnons and phonons in YIG is attractive for nanoscale\ncooling techniques.\nThe discovery of the spin Seebeck e\u000bect (SSE) in\nYIGjPt bilayers [1] opened up a new research direction in\nthe \feld of spin caloritronics. In the SSE a temperature\ndi\u000berence between the magnons in the magnetic insulator\nand the electrons in the metal contact leads to thermal\npumping of a spin current [2{4]. In a suitable metal such\nas Pt this spin current is transformed into an observable\ntransverse voltage by the inverse spin Hall e\u000bect [5]. Nu-\nmerical simulations of the phonon, magnon and electron\ntemperatures show good agreement with experiments [6].\nIn this Letter we report the observation of the spin Peltier\ne\u000bect (SPE), which is the Onsager reciprocal [7] of the\nSSE.\nThe SPE is the generation of a magnon heat current\nin the magnetic insulator by a spin current through the\ninterface with the metal contact. The latter can be gen-\nerated by a charge current in the Pt \flm that by the spin\nHall e\u000bect generates a transverse spin current normal to\nthe interface. The spin Peltier heat current generates\na temperature di\u000berence between magnons and phonons\nin the YIG that when relaxing leads to a change in the\nlattice temperature. We con\frm this scenario experi-\nmentally by picking up such temperature changes via\nproximity thermocouples. According to our modelling\nthe experimental results are consistent with Onsager reci-\nprocity between the SPE and the SSE, which we measure\nseparately (see supplementary IV). Our results con\frm\nrecent indications for a strong magnon-phonon interac-\ntion in YIG at room temperature [6, 8, 9].\nA charge current through a Pt strip generates a trans-\nverse spin current induced by the spin Hall e\u000bect that\nleads to a spin accumulation Vsat the boundaries. At\nthe interface to YIG the spin current is absorbed as a\nspin transfer torque proportional to the spin mixing con-\nductance [10, 11], as depicted in Fig. 1(a). When the\nmagnetic moment of the spin accumulation ( \u0016s) at the\nPtjYIG interface is parallel (antiparallel) to the average\nmagnetization direction, the spin torque transfers mag-netic momentum and energy from the electrons in the\nPt to the magnons in the YIG (or vice versa). Magnons\nare thereby annihilated (excited) (see Fig. 1(b)) lead-\ning to cooling (heating) of the magnetic order parame-\nter (see Fig. 1(c)). Since thermal magnons equilibrate\nwith the lattice by magnon-phonon scattering, the non-\nequilibrium magnons a\u000bect the lattice temperature (see\nFig. 1(b) and (c)) depending on the magnetization di-\nrection.\nIn the SSE [2] the spin current density ( Js) pumped\nfrom the YIG into the non magnetic metal is proportional\nto the temperature di\u000berence between the magnons and\nelectrons at the interface ( Tm-e=Tm\u0000Te) and the in-\nterface spin Seebeck coe\u000ecient LS,Js=LSTm-e. In or-\nder to arrive at a symmetric linear response matrix that\nre\rects Onsager symmetry, the sum of the products of\ncurrents and driving forces should be proportional to the\ndissipation [12], leading to (see supplementary I)\n\u0012Js\nQm-e\u0013\n=\u0012gSLST\nLST{I\nST\u0013\u0012Vs=2\nTm-e=T\u0013\n(1)\nHere we used the Onsager Kelvin relation \u0005 S=SST=\nLS=(gST);where the spin Seebeck SS= (dVs=2dT)Js=0\nand spin Peltier \u0005 S= (dQm-e=dJs)@Tm-e=0coe\u000ecients\nhave been de\fned. gSis the average spin conductance\nper unit area when spin accumulation and magnetiza-\ntion are collinear, i.e. the Vsat the YIGjPt interface is\neither parallel or antiparallel to the average YIG magne-\ntization.gS\u00190:16grat room temperature [13], where\ngris the real part of the spin-mixing conductance per\nunit area. {I\nSis the magnetic contribution to the inter-\nface heat conductance per unit area [6]. The SPE heat\ncurrent density we set out to discover is therefore\nQm\u0000e=LSTVs\n2: (2)\nThe devices designed for observing the SPE are fab-\nricated on top of a 200 nm thick single-crystal (111)arXiv:1311.4772v1  [cond-mat.mes-hall]  19 Nov 20132\nJc\nM\nTpTm\nM/uni03BCsTeT\nT0\nTpTm\nMT\nTe\n(a) (b) (c)\nT0/uni03BCs\nFIG. 1. (color online). Schematic \fgure of the spin Peltier e\u000bect at a Pt jYIG interface. (a) A charge current through the Pt\ncreates a transverse spin current induced by the spin Hall e\u000bect that generates a spin accumulation Vsat the boundaries. (b)\nWhen the spin magnetic moment \u0016sis antiparallel to M the spin torque transfers angular momentum and energy from the\nelectrons in the Pt to the magnons in the YIG thereby cooling the electrons and heating the magnons, e\u000bectively raising the\nmagnon temperature Tmwith respect to the electron temperature Te. (c) When \u0016sis parallel to M the spin torque transfers\nangular momentum and energy from the magnons in the YIG to the electrons in the Pt thereby cooling the magnons, e\u000bectively\nlowering Tmwith respect to Te.\nY3Fe5O12(YIG) \flm grown on a 500 µm thick (111)\nGd3Ga5O12(GGG) substrate by liquid-phase-epitaxy.\nTwo temperature sensors are fabricated in close prox-\nimity to the PtjYIG interface. The optical microscope\nimage in Fig. 2(b) shows the 20 \u0002200µm2and 5 nm\nthick Pt injector \flm. The thermopile sensors consist\nof \fve 40 nm thick Pt-Constantan (Ni 45Cu55) thermo-\ncouples in series that are very sensitive because of the\nlarge di\u000berence in the Seebeck coe\u000ecient of these met-\nals. In the thermopile on the right of the Pt injector the\nPtjNi45Cu55order is reversed for additional cross check\nmeasurements. The two thermopiles and the Pt injector\nare connected to 5 j100 nm thick titanium jgold contacts,\nproviding good thermal anchoring and electrical contact\nto bonding pads 30 µm away. All structures are patterned\nby electron beam lithography. The Pt injector and the\nNi45Cu55are deposited by DC sputtering while electron\nbeam evaporation has been used to make the Au contacts\nand Pt thermocouple components.\nAn AC current is sent through the Pt injector, from I+\nto I\u0000(Fig. 2(b)), to create Vs. The voltage over the ther-\nmopile (V+and V\u0000) is simultaneously recorded. Using\na standard lock-in detection technique the \frst harmonic\nresponse (V/I) is extracted from the measured voltage.\nA low excitation frequency of 17 Hz was used to ensure\na thermal steady-state condition. All measurements are\ncarried out at room temperature.\nIn Fig. 2(a) the \frst harmonic voltage over the ther-\nmopile is shown as a function of an applied in-plane\nmagnetic \feld ( B) for a root-mean-square current of 3\nmA through the Pt injector. A clear switch is observed\njust after the applied \feld becomes positive, in line with\nthe magnetization reversal of YIG at very small coer-\ncive \felds [14]. The signal switches back to its originalvalue when reversing the \feld with a small hysteresis.\nWe measure a SPE signal of 33 nV on top of a back-\nground voltage of 0 :463µV. We observe linear scaling\nof the SPE signal for currents between 1 and 4 mA in\nthe Pt ( IPt injector )(see Fig. 2(c)). Results for four dif-\nferent samples (from two di\u000berent batches) match the\nsignal presented here within 15 %. The measurements\nwere repeated with Brotated 90\u000e. No SPE signal was\nobserved in this con\fguration while the background re-\nmained the same (see supplementary II), which con\frms\nour interpretation.\nIn order to obtain quantitative information we carry\nout 3D \fnite element modelling of our devices [15]. As\ndiscussed above, the SPE heat current ( Qm-e) \rows be-\ntween the electron and magnon systems through the\nPtjYIG interface. Qm-eis calculated using Eq. (2) and\nVs=\u0012Jc\u0001\u0011\u0001tanh\u0012t\n2\u0015\u0013\n(3)\nwhere\u0012is the spin Hall angle, tthe Pt \flm thickness,\nJcthe charge current density through the Pt injector,\n\u001athe Pt resistivity, \u0015the spin-\rip di\u000busion length and\n\u0011= 2\u0015\u001a\u0001[1 +gS\u001a\u0015coth\u0000t\n\u0015\u0001\n]\u00001a back\row correction\nfactor. The heat charge current densities in Pt are mod-\nelled by a three reservoir model of thermalized phonons,\nmagnons and electrons at temperatures Tph,TmandTe,\nrespectively [6]. In linear response the charge ( Jc) and\nheat (Q) current densities in the bulk of the materials\nare related to their driving forces, i.e. gradients of ( V,\nTph,TmandTe) as~Qx=\u0014x~rTxand\n\u0012~Jc\n~Qe\u0013\n=\u0000\u0012\u001b \u001bS\n\u001bST \u0014e\u0013\u0012~rV\n~rTe\u0013\n(4)\nwherexisphorm,\u001bis the electrical conductivity,3\n−10 −5 0 5 100.450.460.470.48Vthermopile (/uni03BCV)\nB (mT)IPt injector= 3mA\n1 2 34 010203040∆Vspin Peltier (nV)\nIPt injector (mA)\nPt\nNiCuNiCu\nPtV+\nV-I+\nI-B(a)\n(b) (c)\nFIG. 2. (color online). (a) First harmonic voltage across\nthe thermopile as a function of applied magnetic \feld. The\ndi\u000berence between the voltage at positive and negative \felds\nis the spin Peltier signal. (b) Optical microscope picture of\nthe device. (c) The spin Peltier signal (\u0001 Vspin Peltier ) as a\nfunction of the charge current through the Pt injector.\nSthe Seebeck coe\u000ecient and \u0014ph,\u0014mand\u0014eare the\nphonon, magnon and electron thermal conductivities, re-\nspectively. The interaction between the magnon and\nphonon subsystems in YIG and between the phonon and\nelectron subsystems in Pt are taken into account by us-\ning thermal relaxation lengths, \u0015m\u0000phand\u0015e\u0000ph;respec-\ntively (see supplementary III),\nr2Tm-ph=Tm-ph\n\u00152\nm-phandr2Te-ph=Te-ph\n\u00152\ne-ph:(5)\nThe phonon interface heat conductance ( \u0014I\nph) and heat\nexchange between magnons and electrons across the in-\nterface ( {I\nS) are treated as boundary conditions [6] (see\nSupplementary III).\nThis model is evaluated for the material parameters\nlisted in table I. Additionally, we adopt gr= 7\u00021014\n\n\u00001m\u00002[16],\u0012= 0:11 [6],\u0015= 1:5 nm [6] and LS=\n7:24\u0002109A/(m2K) [2, 6]. The magnon heat conductivity\nof YIG (\u0014m) at room temperature is not well known so\nwe used a \u0014mof 10\u00002and 10\u00003W/(mK) in order to\ncover the range of estimated values [6, 17]. For Pt jYIG\na\u0014I\nphof 2:78\u0002108W/(m2K) obtained from the acoustic\nmismatch model was adopted [6]. Since this model tends\nto overestimate the heat conductance [18], we also usedTABLE I. Material parameters used in the model. Both \u001band\nS are measured in separate devices [19] except for \u001bof the Pt\ninjector, which is extracted from the SPE devices directly.\n\u0014ph\n\u0014eis adopted from Ref. 6 and the total \u0014=\u0014ph+\u0014eis\ncalculated using \u0014=\u001b\n\u001bbulk\u0014bulk.\n\u001b S\u0014ph \u0014e\n(S/m) ( µV/K) (W/(m\u0001K)) (W/(m\u0001K))\nYIG - - 6 -\nGGG - - 8 -\nAu 2.7\u00011071.7 1 179\nPt injector 3.5 \u0001106-5 3 23\nPt thermocouple 4.2 \u0001106-5 4 28\nNi45Cu55 1\u0001106-30 1 9\n2\u0002108W/(m2K). In \fgure 3(a) the results are shown as a\nfunction of \u0015m-ph. The semi transparent blue horizontal\nbar indicates the range of measured SPE signals that are\nbest \ftted by a \u0015m-phof 0.1 to 0.2 nm for the ranges of\n\u0014mand\u0014I\nphdiscussed above.\nSSE samples were fabricated and simulated by the\nsame model and parameters used above (see Supplemen-\ntary IV). In Fig. 3(b) the results are plotted and best\n\ftted by\u0015m-phbetween 0.2 and 0.5 nm, which is con-\nsistent with the values found for the SPE, as is indeed\nrequired by Onsager reciprocity. This implies that our\nmodel captures the essential physics of the interacting\nelectron, magnon and phonon systems.\nThe observed SPE signal in Fig. 2(a) corresponds to\na phonon temperature di\u000berence of 0.25 mK at the ther-\nmopile, which according to the model is 39 % of the\nphonon temperature di\u000berence directly at the Pt jYIG\ninterface. By engineering devices in which the phonon\nheat loss through the substrate is minimized by thinner\nor etched YIG \flms could therefore signi\fcantly enhance\nthe measured signal. Altering the Pt injector coupling to\nthe heat sink or placing the thermocouple on top of the\nPt injector might also help.\nThe\u0015m-phfound here is smaller than the one adopted\nby Ref. 6 (\u00196nm) by roughly an order of magnitude.\nActually Schreier et al.'s simulations might agree bet-\nter with their measurements for smaller values as well.\n\u0015m-phextracted from Fig. 3 is quite sensitive to small\nvariations in the modelling, which implies a large uncer-\ntainty. Nevertheless even when accepting a large error\nbar from 0.1 to 6 nm for \u0015m\u0000phwe may conclude that\nthermal magnons and phonons interact strongly [8].\nThe background signal in the SPE data is a factor 20\nhigher than we would expect from conventional charge\nPeltier heating and cooling at the Au jPt injector inter-\nfaces. Reference measurements on the second thermopile\non the other side of the Pt injector excludes charge cur-\nrent leakage to the thermopile. For an identical con\fgu-\nration, V+on the same side as I+, we \fnd an opposite\nsign of the measured voltage, as expected for a thermal4\nspin Seebeck signal (/uni03BCV)spin Peltier signal (nV)\nλm–ph (nm) λm–ph (nm) (a) (b)\nκm= 1e-3   and   κI\nph = 2e8κm= 1e-2   and   κI\nph = 2.79e8κm= 1e-2   and   κI\nph = 2e8\nκm= 1e-3   and   κI\nph = 2.79e8κm= 1e-3   and   κI\nph = 2e8κm= 1e-2   and   κI\nph = 2.79e8κm= 1e-2   and   κI\nph = 2e8\nκm= 1e-3   and   κI\nph = 2.79e8\n0 1 2 3 4 504080120\n0 1 2 3 4 501020304050\nFIG. 3. (color online). The modeled SPE (a) and SSE (b) signal versus \u0015m-phfor two di\u000berent values of \u0014m(W/(m K)) and\ntwo di\u000berent values of \u0014I\nph(W/(m2K)). The semi transparent blue bar indicates the range of measured SPE and SSE e\u000bect\nsignals.\nsignal since the Pt jNi45Cu55thermopile sequence is in-\nverted. A current leak would not change sign and can\ntherefore be excluded. The background in the second\nharmonic voltage is likely to be caused by the thermo-\nvoltage across the thermopile due to Joule heating in the\nPt injector, since its value agrees within 17% with the\nmodeled one. Additional measurements of frequency de-\npendent properties (see Supplementary V) rule out pick-\nups due to capacitive or inductive couplings.\nWe checked that the sign of the experimentally ob-\nserved SPE and SSE signals obey reciprocity. Further-\nmore the voltage measured across the Pt detector in a\nRF spin pumping measurement matches the sign of the\nSSE voltage for the same geometry when heating the YIG\nrelative to the Pt, as previously reported [20, 21]. How-\never, the absolute sign of these three e\u000bects is still under\ninvestigation.\nIn conclusion, we report experimental proof that a\nspin accumulation at a Pt jYIG interface induces heat ex-\nchange between electrons and magnons on both sides.\nUsing thermal modelling to knit the theory of inter-\nface transport to the observables we demonstrate that\nthe SPE is the Onsager reciprocal of the SSE and con-\n\frm a strong interaction between thermal magnons and\nphonons in YIG, as reported earlier [8]. We hope that\nthese results can contribute to a better understanding\nof coupling between thermomagnetic and thermoelectric\nproperties. Our proof of principle opens new strategies\nfor nanoscale cooling applications.\nWe would like to acknowledge B. Wolfs, M. de Roosz\nand J. G. Holstein for technical assistance. This work\nis part of the research program of the Foundation for\nFundamental Research on Matter (FOM) and supported\nby NanoLab NL, Marie Curie ITN Spinicur, DFG Pri-\nority Programme 1538 \"Spin-Caloric Transport\", Grant-in-Aid for Scienti\fc Research A (Kakenhi) 25247056 and\nthe Zernike Institute for Advanced Materials.\n\u0003J.Flipse@rug.nl\n[1] K. Uchida, J. Xiao, H. Adachi, J. I. Ohe, S. Takahashi,\nJ. Ieda, T. Ota, Y. Kajiwara, H. Umezawa, H. Kawai,\nG. E. W. Bauer, S. Maekawa, and E. Saitoh, Nature\nMaterials 9, 894 (2010).\n[2] J. Xiao, G. E. W. Bauer, K. Uchida, E. Saitoh,\nS. Maekawa, Phys. Rev. B 81, 214418 (2010).\n[3] H. Adachi, J. I. Ohe, S. Takahashi, S. Maekawa, Phys.\nRev. B 83, 094410 (2011).\n[4] S. Ho\u000bman, K. Sato, Y. Tserkovnyak, Phys. Rev. B 88,\n064408 (2013).\n[5] A. Ho\u000bmann, IEEE Trans. Magnetics 49, 5172 (2013).\n[6] M. Schreier, A. Kamra, M. Weiler, J. Xiao, G. E.\nW. Bauer, R. Gross, S. T. B. Goennenwein, Phys. Rev.\nB88, 094410 (2013).\n[7] L. Onsager, Phys. Rev. 37, 405426 (1931).\n[8] M. Agrawal, V. I. Vasyuchka, A. A. Serga, A.\nD. Karenowska, G. A. Melkov, B. Hillebrands, Phys.\nRev. Lett. 111, 107204 (2013).\n[9] N. Roschewsky, M. Schreier, A. Kamra, F. Schade,\nK. Ganzhorn, S. Meyer, H. Huebl, S. Gepr ags, R. Gross,\nS. T. B. Goennenwein, arXiv 1309.3986v1 (2013).\n[10] A. Brataas, G. E. W. Bauer, P. J. Kelly, Phys. Rep.\n427, 157 (2006).\n[11] Z. Wang, Y. Sun, M. Wu, V. Tiberkevich, A. Slavin,\nPhys. Rev. Lett. 107, 146602 (2011).\n[12] H. B. Callen, Phys. Rev. 73, 1349 (1948).\n[13] H. J. Jiao, J. Xiao, G. E. W. Bauer, in preparation.\n[14] N. Vlietstra, J. Shan, V. Castel, B. J. van Wees, J. Ben\nYoussef, Phys. Rev. B 87, 184421 (2013).\n[15] COMSOL Multiphysics®4.2a.\n[16] N. Vlietstra, J. Shan, V. Castel, J. Ben Youssef, G. E.\nW. Bauer, B. J. van Wees, Appl. Phys. Lett. 103, 032401\n(2013).5\n[17] R. L. Douglass, Phys. Rev. 129, 1132 (1963).\n[18] E. T. Swartz, R. O. Pohl, Rev. Mod. Phys. 61, 605\n(1989).\n[19] F. L. Bakker, J. Flipse, B. J. van Wees, J. Appl. Phys.\n111, 084306 (2012).\n[20] C. W. Sandweg, Y. Kajiwara, A. V. Chumak, A.\nA. Serga, V. I. Vasyuchka, M. B. Jung\reisch, E. Saitoh,\nB. Hillebrands, Phys. Rev. Lett. 106, 216601 (2011).\n[21] M. Weiler, M. Althammer, M. Schreier, J. Lotze,\nM. Pernpeintner, S. Meyer, H. Huebl, R. Gross,\nA. Kamra, J. Xiao, Y. T. Chen, H. J. Jiao, G. E.\nW. Bauer, S. T. B. Goennenwein, Phys. Rev. Lett.111, 176601 (2013).\n[22] W. Wang, D. G. Cahill, Phys. Rev. Lett. 109, 175503\n(2012).\n[23] K. Uchida, H. Adachi, T. Ota, H. Nakayama,\nS. Maekawa, E. Saitoh, Appl. Phys. Lett. 97, 172505\n(2010).\n[24] M. Weiler, M. Althammer, F. D. Czeschka, H. Huebl,\nM. S. Wagner, M. Opel, I. Imort, G. Reiss, A. Thomas,\nR. Gross, S. T. B. Goennenwein, Phys. Rev. Lett. 108,\n106602 (2012).6\nSUPPLEMENTARY INFORMATION\nI. Onsager reciprocity for the spin Seebeck and spin Peltier e\u000bect\nThe linear response matrix of thermoelectrics re\rects Onsager reciprocity when the sum of the products of currents\ntimes driving forces equals the dissipation [12]. When IcandQare the charge and heat currents driven by voltage\nand temperature di\u000berences \u0001 Vand \u0001T,_F=Ic\u0001V+Q\u0001T=T equals the dissipation and we obtain the symmetric\nresponse matrix [12]:\n\u0012\nIc\nQ\u0013\n=\u0000\u0012\nG GST\nGST KT\u0013\u0012\n\u0001V\n\u0001T=T\u0013\n(6)\nwhereGis the electrical conductance, Sthe Seebeck coe\u000ecient, and Kthe heat conductance. Here the Onsager-\nThomson relation for the Peltier coe\u000ecient \u0005 = SThas already been implemented.\nIn the case of the spin Seebeck e\u000bect (SSE) and the spin Peltier e\u000bect (SPE) for magnetic insulators, the spin\naccumulation at the interface drives a spin current. To ensure reciprocity, we have to compute the Joule heating\ncaused by the spin currents:\n_F=I\"\u0001V\"+I#\u0001V#=G\"\u0001V2\n\"+G#\u0001V2\n# (7)\nwhere the subscripts denote the up and down spin contribution. Comparing this with the product of Iswith \u0001Vs:\nIs\u0001Vs= (G\"\u0001V\"\u0000G#\u0001V#)(\u0001V\"\u0000\u0001V#) = 2(G\"\u0001V2\n\"+G#\u0001V2\n#) (8)\nwe conclude that \u0001 Vs=2 is the proper driving force. The linear response relations for spin and heat current densities\nat the interface between a normal metal and a ferromagnet then reads (Eq. (1) of the main text, where all variables\nand parameters are introduced):\n\u0012Js\nQm\u0000e\u0013\n=\u0012gSLST\nLST{I\nST\u0013\u0012Vs=2\nTm-e=T\u0013\n(9)\nand we omitted the charge sector because we are dealing with a ferromagnetic insulator.\nII. Measurement with B rotated 90\u000e\nWe repeated the measurements in the main text after rotating the magnetic \feld by 90\u000e(see Fig. 4). The\nmagnetization direction is here parallel to the current in the Pt \flm such that the current-induced spin accumulation\nis normal to the magnetization. The background voltage and noise level remain unmodi\fed; we do not detect a\nheating or cooling of the ferromagnet. This con\frms our interpretation of the experiments in the main text. The spin\ntorque normal to the magnetization is not expected to a\u000bect the magnon temperature and the SPE should vanish, as\nobserved.\nIII. The 3D \fnite element model\nWe adopt the three reservoir model of thermalized electron, magnon, and phonon systems. Charges are transported\nby the electron system only, while heat currents \row in all subsystems. We take into account spin angular momentum\ncurrents in the electron and magnon system, but disregard the phonon angular momentum current. The bulk charge\nand heat currents in linear response are given by Eq. (4) in the main text. The charge and energy conservation\nrelations read:\n~r\u00010\nBBBBBBBBBB@~Jc\n~Qph\n~Qm\n~Qe1\nCCCCCCCCCCA=0\nBBBBBBBBBB@0\n(1\u0000P2\nm-ph)(\u0014m+\u0014ph)\n4\u00152\nm-ph(Tm\u0000Tph) +(1\u0000P2\ne-ph)(\u0014e+\u0014ph)\n4\u00152\ne-ph(Te\u0000Tph)\n\u0000(1\u0000P2\nm-ph)(\u0014m+\u0014ph)\n4\u00152\nm-ph(Tm\u0000Tph)\nJ2\nc\n\u001b\u0000(1\u0000P2\ne-ph)(\u0014e+\u0014ph)\n4\u00152\ne-ph(Te\u0000Tph)1\nCCCCCCCCCCA(10)7\n−10 −5 0 5 100.450.460.47\n Pt\nNiCuNiCu\nPtV+\nV-I+\nI-BIPt injector= 3mA\nB (mT)Vthermopile (/uni03BCV)\nFIG. 4. (a) First harmonic voltage across the thermopile as a function of applied magnetic \feld parallel to the Pt injector. (b)\nOptical microscope picture of the measured device and measurement geometry.\nwithPm\u0000ph= (\u0014m\u0000\u0014ph)=(\u0014m+\u0014ph),Pe\u0000ph= (\u0014e\u0000\u0014ph)=(\u0014e+\u0014ph) andJ2\nc=\u001baccounts for Joule heating. The\nother terms describe the heat exchange between phonons, magnons and electrons, as indicated by the subscripts.\nAll parameters have been de\fned in the main text, except for the thermal relaxation lengths \u0015m\u0000phand\u0015e\u0000ph:The\nformer is used as an adjustable parameter while the latter \u0015e\u0000ph=p\n\u0014e=gis calculated from the electron-phonon\ncoupling parameter ( g) for Au and Pt [22]. This leads to a \u0015Pt\ne\u0000phof 4.5 nm and a \u0015Au\ne\u0000phof 80 nm at room temperature.\nThe boundary conditions at the metal jYIG interfaces are governed by the phonon ( \u0014I\nph) and magnetic ( {I\nS) interface\nheat conductances. We adopt the values for \u0014I\nphfrom Ref. 6, while that for Ni 45Cu55jYIG is assumed to be equal to\nthat of PtjYIG. The interface magnetic heat conductance is calculated using [6]\n{I\nS=h\ne2kBT\n~\u0016BkBgr\n\u0019MsV2m\u0012\n1 +gS\u001a\u0015coth\u0012t\n\u0015\u0013\u0013\u00001\n(11)\nwhereV2mis the (spin pumping) magnetic coherence volume. The currents through the interface read\n0\nBBBBBBBBBB@JI\nc\nQI\nph\nQI\nm\nQI\ne1\nCCCCCCCCCCA=0\nBBBBBBBBBB@0\n\u0014I\nph(Ta\nph\u0000Tb\nph)\n\u0000{I\nS(Tm\u0000Te)\u0000LSTVs\n2\n{I\nS(Tm\u0000Te) +LSTVs\n21\nCCCCCCCCCCA(12)\nwhereTPt/YIG\nphare the phonon temperatures on the Pt/YIG side of the interface. QI\nmandQI\neare the interface\nmagnetic heat currents. The \frst terms on the r.h.s represent the heat current driven by temperature di\u000berences,\nwhile the second term is the heat current associated with the magnon injection by an applied Vs(see Eq. (2) in the\nmain text) that is responsible for the SPE.\nEq. (4) in the main text is solved for the SPE and SSE con\fgurations, taking into account energy conservation\n(Eq. (10)) and boundary conditions (Eq. (12)). The results are plotted in Figs. 3 (a) and (b) of the main text.\nIV. Spin Seebeck e\u000bect\nTo verify reciprocity we fabricated samples nominally identical to the Pt jYIG heterostructures used for the SPE\nexperiments in order to measure the longitudinal SSE [23, 24]. Fig. 5(b) gives a picture of such a device consisting8\n−20−15 −10 −5 05101520−30−20−100102030VPt detector (/uni03BCV)\nB (mT)Iheater= 5mA\nBI+\nI-V+\nV-\n100 /uni03BCm(a) (b)\nFIG. 5. (a) Second harmonic voltage across the Pt detector as a function of applied magnetic \feld. (b) Optical microscope\npicture of the measured device and the measurement geometry used.\nof a 5 nm thick sputtered Pt detector (250 x 10 \u0016m2) on top of a GGG jYIG substrate as used for the study of the\nSPE e\u000bect. The Pt detector is covered with a \u001870 nm aluminium oxide (Al 2O3) layer that electrically isolates the\nPt detector from a 40 nm Pt \flm heater evaporated on top. Both the detector and heater are contacted by a 100 nm\nthick Au layer to large bonding pads.\nBy Joule heating, a charge current through the heater creates a thermal gradient over the Pt jYIG interface. The\nhot electrons transfer energy to the cold magnons, which is associated with a spin current in Pt that is converted to\nan observable charge current by the inverse spin Hall e\u000bect. This is the SSE.\nIn Fig. 5(a) the second harmonic voltage versus magnetic \feld is a measure of the Joule heating. A clear SSE signal\nis observed, which changes sign when the magnetization is reversed, as expected. The signal scales quadratically with\nthe current and for B parallel to the Pt detector no SSE signal is detected, which con\frms that the voltage is due to\nthe inverse spin Hall e\u000bect.\nThe model discussed in the main text and the previous section can be applied to this measurement geometry to\n\fnd theTm\u0000eat the interface and the transverse voltage over the Pt detector [6]:\nVSSE= 2\u0001gr\r~kB\n2\u0019MsV2mTm\u0000e\u00014\u0019\ne\u0012\u001al\u0001\u0015\nttanh\u0000t\n2\u0015\u0001\n1 +gS\u001a\u0015coth (t=\u0015)(13)\nwherelis the length of the Pt detector. The results obtained from the SSE modeling are shown in Fig. 3(b) of the\nmain text.\n2 4 6 8 10 12 14 16 180.290.30.310.32\n0Vthermocouple X-component (/uni03BCV)\nMeasurement frequency (Hz)2 4 6 8 10 12 14 16 18 0481216Vthermocouple Y-component (/uni03BCV)\nMeasurement frequency (Hz) (a) (b)IPt injector= 2 mA IPt injector= 2 mA\nFIG. 6. (a) First harmonic voltage across the thermopile in phase with the current ( X-component). (b) First harmonic voltage\nacross the thermopile out-of-phase with the current ( Y-component).9\n−10 −5 0 5 100.70.720.74Vthermopile (/uni03BCV)\nB (mT)IPt injector= 4 mA\nFIG. 7. First harmonic voltage across the thermopile as a function of applied magnetic \feld for a 4 mA current through the\nPt injector with frequency of 3 Hz.\nV. Frequency dependent measurements\nWe measured the voltage as function of frequency in order to exclude any capacitive or inductive coupling between\nthe Pt injector and the thermopile (see Fig. 6). The in-phase voltage ( x-component) slightly decreases with frequency,\nleading to a frequency-dependent voltage of around 10% at our usual measurement frequency of 17 Hz (Fig. 6(a)).\nThe out-of-phase voltage ( Y-component) linearly depends on frequency and vanishes for zero frequency, as expected.\nIn Fig. 7 the \frst harmonic voltage across the thermopile is shown for a 4 mA current through the Pt injector\nwith a frequency of 3 Hz. A clear SPE signal is observed with the same magnitude as for the measurements at 17\nHz . From these results we can safely conclude that the measured signal is not a\u000bected by any spurious capacitive or\ninductive signals. Furthermore at these low frequencies the thermal time constants are much shorter than the period\nof the measurement modulation."    },    {        "title": "2405.08519v1.Spin_Hall_magnetoresistance_in_Pt__Ga_Mn_N_devices.pdf",        "content": "   \n1 Spin Hall magnetoresistance in  Pt/(Ga, Mn)N devices \nJ. Aaron Mendoza- Rodarte1,2*, Katarzyna Gas3,4, Manuel Herrera- Zaldívar2, Detlef \nHommel5,  Maciej Sawicki3,6, and Marcos H. D. Guimarães1* \n1Zernike Institute for Advanced Materials, University of Groningen, 9747 AG Groningen, The Netherlands  \n2Centro de Nanociencias y Nanotecnología-Universidad Nacional Autónoma de México, Ensenada, 22800-Baja \nCalifornia, México \n3Institute of Physi cs, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL -02668 Warsaw, Poland \n4Center for Science and Innovation in Spintronics, Tohoku University, Katahira 2-1 -1, Aoba-ku, Sendai 980-8577, \nJapan \n5Lukasiewicz Research Network - PORT Polish Center for Techn ology Development, Stabłowicka 147, Wrocław, \nPoland  \n6Research Institute of Electrical Communication, Tohoku University, Katahira 2-1-1, Aoba-ku, Sendai 980-8577, \nJapan \n \nE-mail: j.a.mendoza.rodarte@rug.nl , m.h.guimaraes@rug.nl  \n \nDiluted magnetic semiconductors (DMS) have attracted significant attention for their potential \nin spintronic applications. Particularly, magnetically -doped GaN is highly attractive due to its \nhigh relevance for the CMOS industry and the possibility of deve loping advanced spintronic \ndevices which are fully compatible with the current industrial procedures . Despite this interest, \nthere remains a need to  investigate the spintronic  parameters that characterize interfaces within \nthese systems. Here , we perform s pin Hall magnetoresistance (SMR) measurements to evaluate \nthe spin transfer at a Pt/(Ga,Mn)N interface. We determine the transparency of the interface \nthrough the estimation of  the real part of the spin mixing conductance  finding 𝐺𝐺𝑟𝑟 = 2.6 ⋅1014 \nΩ-1 m-2, comparable to state- of-the-art yttrium iron garnet (YIG)/Pt interfaces . Moreover, the \nmagnetic ordering probed by SMR above the (Ga,Mn)N  Curie temperature  TC provides  a \nbroader temperature range for the efficient generation and detection of spin currents, relaxing \nthe conditions for this material to be applied in new spintronic devices .  \n     \n2 Recent advances in the manipulation of electron charge and spin have highlighted \nspintronics as a key area of research1,2. This interest stems from the potential of spintronic \ndevices to address the limitations of traditional charged -based devices, such as high energy \nconsumption3. Among various materials, diluted magnetic semiconductors  (DMS) , those \nsystems where a fraction of cations are  substituted by magnetic elements,  stand out as \npromising candidates for spintronic applications due to their unique integration of \nsemiconductor and magnetic properties4. Particularly , Mn-doped compounds  have received \nsignificant attention from the rese arch community since the formation of a ferromagnetic order \nat technologically relevant temperatures was predicted5 and successfully realized6. Among \nthese compounds, single -phase Mn -doped GaN epitax ial layers stand out for their insulating \nand short -range ferromagnetic character at low temperatures . It presents a rich playground to \nexplore magnetic interactions of magnetic ions in a semiconductor  lattice. A brief discussion \non the ferromagnetic mechanism s in (Ga,Mn)N  is included in the  supplementary material. The \ncombination of these factors, coupled with GaN well-established roles in optoelectronics7, \nhigh- frequency8,9, and power electronic s10 could provide substantial technological benefits, \nparticularly in the development of  new GaN- based spintronic devices11. In fact apart from \nexploitation of the magnetoelectric e ffect in (Ga,Mn)N  so far 12, the study of GaN -based DMS \nhas been focused on optimizing growth  conditions13 and magnetic properties14,15, while spin \ntransport dynamics received comparatively less attention16–20. It is therefore timely and \nimportant to exploit spintronic and magnonic21 properties of this insulating ferromagnetic \nmaterial in an aim to foster the development of all -nitride low power information  processing \nand dissipationless communication means based on spin waves  propagation.  \n \nOf particular importance for new nonvolatile magnetic data storage applications, t he field of \nspin-orbit torques (SOTs)  focuse s on manipulating the magnetization in thin film \nheterostructures via electric currents , providing a promising way to manipulate magnetization \nat the nanoscale22. This technology relies on transferring  spin angular momentum from a \nnormal metal into a ferromagnet, using spin- charge interconversion effects to exert  a torque on \nthe magnetization22. Significant a dvancement s in SOT have catalyzed  the development of non-\nvolatile memory devices23,24. However, the success of these devices depends  on efficient \ngeneration, transport, and detection of spin currents. While material research with high spin -\norbit coupling25 or orbital angular momentum effects26 addresses spin current generation,    \n3 transport issues are managed by identifying materials with high interface spin transparency, \nindicated by substantial spin mixing conductance values ( G↑↓). Spin Hall magnetoresistance \n(SMR) has emerged  as a key technique  for assessing  spin transport properties at heterostructure  \ninterface s through electric al methods27. Despite these advances, there is still a significant gap \nin understanding spin transport properties in heterostructures that utilize transition metal -doped \nGaN, which is crucial for SOT applications.  \nIn this study, we investigate spin Hall magnetoresistan ce (SMR) within a Pt/(Ga ,Mn)N \nheterostructure. By performing angle -dependent magnetoresistance measurements, we \ndetermine  the spin mixing conductance  (G↑↓), a critical parameter that dictat es the transport of \nspin information through the interface between the two materials . Additionally, we provide \nanother assessment of the Curie temperature of  the magnetic layer solely through electrical \nmeans, via temperature- dependent measurements. Our results  provide  valuable insights for the \ndevelopment of devices that  incorporate  GaN- based DM S for advanced spintronic applications.  \nTo characterize the Pt/ (Ga,Mn)N interface via SMR,  we fabricated  a 6 nm -thick Pt Hall bar  \non a  100 nm -thick single -phase epitaxy Ga0.922Mn 0.078N film. The film was grown using \nplasma- assisted  molecular beam epitaxy (MBE)  on a 3 μm-thick GaN(0001) template , which \nwas deposited on c -oriented 2- inch sapphire substrates. Detailed methodologies for  the growth \nand magnetic and crystallographic characterization of these films are discussed  in detail in \nreference 14. The Hall bar, measuring 25 μm in wide and 200 μm in length, was fabricated by \nconventional lithography processes . Prior to the deposition of  Pt through Ar+ plasma d.c. \nsputtering, the (Ga ,Mn)N surface was cleaned  using an Ar+ mild etching process, detailed in \nsupplementary material.  Electrical measurements were performed by rotating a n external \nmagnetic  field within the plane of the device . We utilized  conventional  lock-in technique s for \nthe measurements, with a bias current ( 𝐼𝐼𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏) of less than  0.7 mA at a frequency  of 187.77 Hz . \nThis setup allowed us  to measure the transverse  resistance ( Rxy), which is  perpendicular to the \ncurrent path . This layout of the electrical interconnects is illustrated in Fig. 1( a).    \n4   \nFIG. 1. (a) Schematic s of the devices and electrical measurement configuration, illustrating the definition of the \nrotating external magnetic field  angle  α. (b) Angle dependence of the SMR signal in the transverse geometry as a \nfunction of α. The measurements were conducted in the temperature range of 4.7 to 50 K using an external \nmagnetic field of 600 mT. A baseline has been removed so that the relative changes in resistivity are zero at α=0 \ndegrees.  (c-d) Schematic representation of SMR. (c) Low -resistance configuration where 𝑀𝑀��⃗||𝜎𝜎⃗ . (d) High -\nresistance configuration where  𝑀𝑀��⃗⊥𝜎𝜎⃗. \nThe SMR measurements offer insights  into the magnetic ordering at the surface of the \n(Ga,Mn)N thin film  at low temperatures. Fig. 1( b) illustrates  the dependence of the magnetic \nfield angle (α) on the relative changes in the transverse resistivity,  ∆𝜌𝜌𝑥𝑥𝑥𝑥/𝜌𝜌. Here, ∆𝜌𝜌𝑥𝑥𝑥𝑥/𝜌𝜌 is \ndefined as (𝑅𝑅𝑥𝑥𝑥𝑥−𝑅𝑅0,𝑥𝑥𝑥𝑥)/(𝑅𝑅0,𝑥𝑥𝑥𝑥/4.8) where 𝑅𝑅𝑥𝑥𝑥𝑥 is the transverse resistance, and  𝑅𝑅0,𝑥𝑥𝑥𝑥 and \n𝑅𝑅0,𝑥𝑥𝑥𝑥 are the transverse and longitudinal resistance s at zero magnetic field,  respectively,  and \n4.8 is the  geometrical factor of the Hall bar (length /width ). The observed changes in resistivity \ncan be explain ed as follows:  when a charge current ( Jc) flows  through Pt, the  spin Hall effect  \n(SHE ) induces a transverse spin current ( Js) towards the (Ga ,Mn)N/Pt interface. At this \ninterface, spin accumulation interacts with the local magnetic moments of (Ga,Mn)N. If the \nspin polarization 𝜎𝜎⃗ is parallel to the magnetization direction 𝑀𝑀��⃗ of the ferromagnet ( 𝑀𝑀��⃗||𝜎𝜎⃗), the \nelectrons' angular momentum is reflected, resulting in a low -resistance configuration due to the \ninverse spin Hall effect  (ISHE ) [see Fig. 1( c)]. Conversely, when 𝑀𝑀��⃗  is perpendicular to 𝜎𝜎⃗ \n(𝑀𝑀��⃗⊥𝜎𝜎⃗) [refer to Fig. 1( d)], the electrons' angular momentum is absorbed, leading to a high -\nresistance configuration . In transverse SMR  measurements,  the interaction of the generated \n(a) (b)\n(c) (d)\n4.7 K\n15 K\n50 K   \n5 spin current with the (Ga ,Mn) N magnetization 𝑀𝑀��⃗ at the interface results in a reorientation of \nthe outgoing spin current polarization 𝜎𝜎⃗. At angles α = 45 and 135°, this leads  to positive and \nnegative transverse voltage, respectively28. Our experimental observations align with this \ntheory , showing maximum resistance at α  = 45° and minimum at α = 135°, similar to findings \nin YIG/Pt systems29. The dependence of the resistivity in the transverse geometry ( 𝜌𝜌𝑇𝑇) with \nrespect to the magnetization components along the different directions is given by27: \n                     𝜌𝜌𝑇𝑇=∆𝜌𝜌1𝑚𝑚𝑥𝑥𝑚𝑚𝑥𝑥+∆𝜌𝜌2𝑚𝑚𝑧𝑧+∆𝜌𝜌𝐻𝐻𝑏𝑏𝐻𝐻𝐻𝐻𝐵𝐵𝑧𝑧,                (1) \nwhere ∆𝜌𝜌1 and ∆𝜌𝜌2 represents relative  changes in resistivity and 𝑚𝑚𝑥𝑥, 𝑚𝑚𝑥𝑥, and 𝑚𝑚𝑧𝑧 are the \ncomponents (unit vectors) of magnetization in the 𝑥𝑥�-, 𝑦𝑦�- and 𝑧𝑧̂-direction, respectively. The \nangular dependenc ies are defined by the components  𝑚𝑚𝑥𝑥 , 𝑚𝑚𝑥𝑥 , and 𝑚𝑚𝑧𝑧 , where 𝑚𝑚𝑥𝑥=\ncos(𝛼𝛼)cos(𝛽𝛽) , 𝑚𝑚𝑥��=sin(𝛼𝛼)cos(𝛽𝛽) , and 𝑚𝑚𝑧𝑧=sin(𝛽𝛽) , with 𝛽𝛽  representing  the azimuthal \nangle – i.e. the angle with respect to the out -of-plane (OOP) direction. Using the se definitions, \nwe can express the change in the transverse resistivity as follows27: \n                ∆𝜌𝜌𝑥𝑥𝑥𝑥/𝜌𝜌=∆𝜌𝜌𝑥𝑥𝑥𝑥,11\n2sin(2𝛼𝛼)−∆𝜌𝜌𝑥𝑥𝑥𝑥,2sin (𝛽𝛽),                (2) \nwhere ∆𝜌𝜌𝑥𝑥𝑥𝑥,1 and ∆𝜌𝜌𝑥𝑥𝑥𝑥,2 represent  the amplitudes of the relative change in resistivity.26) This \nequation effectively describes our  observed signals and fit s our measurements well. \nAdditionally, t he term ∆𝜌𝜌𝐻𝐻𝑏𝑏𝐻𝐻𝐻𝐻𝐵𝐵𝑧𝑧 account s for the ordinary Hall effect oc curring in Pt  when  a \nmagnetic field is applied i n the 𝑧𝑧̂-direction . The sin (𝛽𝛽) component observed in our results \nlikely results from  OOP component due  to misalignment  of the Hall bar  with respect to the in -\nplane field . At 4.7 K, the values for the amplitudes ∆𝜌𝜌𝑥𝑥𝑥𝑥,1 and ∆𝜌𝜌𝑥𝑥𝑥𝑥,2 are 7.2⋅10-6 and 1.8⋅10-\n6, respectively , indicating  a clear  dominance  of the in- plane component . \nUsing  SMR  measurements, we investigate  the ferromagnetic -paramagnetic transition of \n(Ga,Mn)N solely through electrical methods . To this end, we perform a temperature -dependent  \nmeasurement ranging from 4.7 to 290 K. Fig. 2 displays  the behavior of magnitudes  ∆𝜌𝜌𝑥𝑥𝑥𝑥,1 \nand ∆𝜌𝜌𝑥𝑥𝑥𝑥,2 derived from Eq. (1) across different temperatures. As the temperature increased  \nfrom 4.7 to 10 K, ∆𝜌𝜌𝑥𝑥𝑥𝑥,1 fluctuate d between  7.2⋅10-6 and 7.3⋅ 10-6 with the maximum value  at \n10 K . Above 10 K, a decrease in  ∆𝜌𝜌𝑥𝑥𝑥𝑥,1  was noted, dropping to 6.81⋅ 10-6 at 15 K . This \nreduction in the  SMR signal suggests  a TC between  10 to 15 K , aligning  well with the TC of \n13.0 ± 0.3 K established  by SQUID magnetometry14. The signal diminished above 50 K , and \ndata fitting bec ame unreliable due to a  significant decrease in the signal -to-noise ratio  (SNR).    \n6 Nevertheless, it is important  to note that substantial  SMR signal s have been observe d from 15 \nto 50 K , a range dominated by the  paramagnetic phase. This magnetic ordering is similar to \nthat seen in other Curie -like paramagnetic insulator s such as  Gd3Ga5O12 (GGG)30,31. In such \nmaterials , the SMR arises from the interaction  between  the conduction -electron spins in Pt and \nthe paramagnetic spins S within GGG via the interface exchange interaction, which applies \ntorque on S.  Regarding the  fitted OOP component, its  temperature dependence displayed a \nmonotonic behavior, with ∆𝜌𝜌𝑥𝑥𝑥𝑥,2 values between 1.6 and 1.9 ⋅ 10-6. This points towards  an \nordinary Hall contribution in our transverse measurement s, which can be  attributed to  a sample \nmisalignment.  \n \nFIG. 2. Temperature dependence of the relative changes of the resistivity of ∆𝜌𝜌𝑥𝑥𝑦𝑦,1 and ∆𝜌𝜌𝑥𝑥𝑦𝑦,2. The green shadow \nhighlight s the decrease of the SMR signal observed above  10 K. All the amplitudes were  extracted from Eq. (1) \nusing  measurements performed at 600 mT.  \nThe magnetic field depend ence of ∆𝜌𝜌𝑥𝑥𝑥𝑥,1  and ∆𝜌𝜌𝑥𝑥𝑥𝑥,2  demonstrates quadratic and linear \nbehaviors , respectively , as shown in Fig. 3a and 3b. We can attibute the former dependency to \nthe percolating character of ferromagnetism in our material,  indicating that only at T  = 0 all the \nspins present in the sample are ferromagnetically coupled. On increasing T  some of the spins \nstart decoupling such that at its equilibrium T C only about 20% of spins remain in the infinite \ncluster32. The remaining spins reside in finite  ferromagnetic  clusters of different sizes. These \nmagnetic granules are characterized by their own magnetic moments and are responsible for \nthe glassy characteristics like blocking ( i.e. as in the material being far from thermal \n   \n7 equilibrium). Such material can therefore mimic a ferromagnet well above its equilibrium T C \nwhen it is probed on sufficiently short time scales, as during SMR experiment s. Additionally, \nat 600 mT we do not observe a saturation state, a f inding corroborated by SQUID \nmagnetometry M-H curves, which indicate that even at µ0H = 7 T at T  = 2 K  the system still \nexhibits a positive slope (as indicated in the supplementary material Fig. S2). The coercive \nfield (H c) values from SQUID magnetometry for similar samples suggest an increase of the \naverage magnetization  above 200 mT, aligning with the clear SMR signal observed at 240 mT \nin our results (Fig. 3a). Additionally , the linear dependence seen in  ∆𝜌𝜌𝑥𝑥𝑥𝑥,2  aligns  with the \nexpected behavior  of the ordinary Hall component. To confirm that our measurements are \nwithin  the linear regime, we conducted bias current  (𝐼𝐼𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏) dependenc e tests. Fig. 3c shows the \ntransverse voltage (𝑉𝑉𝑥𝑥𝑥𝑥) as a function of α , and Fig. 3d displays the 𝐼𝐼𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 dependency of the \nrelative change in transverse voltage ∆𝑉𝑉𝑥𝑥𝑥𝑥, where a data point represents the amplitude of the \nangle -dependent voltage, obtained by a fit to 𝑉𝑉𝑥𝑥𝑥𝑥=∆𝑉𝑉𝑥𝑥𝑥𝑥,11\n2sin(2𝛼𝛼)−∆𝑉𝑉𝑥𝑥𝑥𝑥,2sin (𝛽𝛽). Linear \nfits applied to  both the in-plane (∆𝑉𝑉𝑥𝑥𝑥𝑥,1) and out-of-plane (∆𝑉𝑉𝑥𝑥𝑥𝑥,2) components confirm the \nlinear behavior , with  no evident heating effects. \n \nFIG. 3. (a) Relative change of the transverse resistivity  ρxy as a function of angle α at different values of magnetic \nfield B and for bias current  𝐼𝐼𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 =1.0 mA . (b) M agnetic field dependence  of ∆𝜌𝜌𝑥𝑥𝑥𝑥,1  and ∆𝜌𝜌𝑥𝑥𝑥𝑥,2 . (c) Angle \ndependence  of 𝑉𝑉𝑥𝑥𝑥𝑥 at different 𝐼𝐼𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏. (d) Current dependence of ∆𝑉𝑉𝑥𝑥𝑥𝑥. All the measurements were performed at \na fixed temperature of 4.7 K.  A baseline has been removed so that the relative changes in resistivity are zero at \nα=0 degrees.   \n4.7 K4.7 K (a)\n(c) (d)(b)\n600 mT\n480 mT\n360 mT\n0 mT\n1.0 mA\n0.7 mA\n0.3 mA   \n8 The transfer of spin angular momentum at the  interface of the heterostructure  is quantified  \nby the spin mixing conductance, G↑↓, which we estimate from our experimental results. From  \nEq. (1), we can express ∆𝜌𝜌1 and ∆𝜌𝜌2 as follows27: \n                  ∆𝜌𝜌1\n𝜌𝜌=𝜃𝜃𝑆𝑆𝐻𝐻2𝜆𝜆\n𝑑𝑑𝑁𝑁Re�2𝜆𝜆G↑↓tanh2𝑑𝑑𝑁𝑁\n𝜆𝜆\n𝜎𝜎+2𝜆𝜆G↑↓coth𝑑𝑑𝑁𝑁\n𝜆𝜆� ,                (3) \n                  ∆𝜌𝜌2\n𝜌𝜌=−𝜃𝜃𝑆𝑆𝐻𝐻2𝜆𝜆\n𝑑𝑑𝑁𝑁Im�2𝜆𝜆G↑↓tanh2𝑑𝑑𝑁𝑁\n𝜆𝜆\n𝜎𝜎+2𝜆𝜆G↑↓coth𝑑𝑑𝑁𝑁\n𝜆𝜆� .               (4) \nHere,  𝜃𝜃𝑆𝑆𝐻𝐻 represents the spin Hall angle,  λ the spin relaxation length, 𝑑𝑑𝑁𝑁 the thickness, and \n𝜎𝜎 the conductivity of Pt. G↑↓ denotes  the relaxation of the spin current polarization component \ntransverse  to the magnetization at the Pt/( Ga,Mn)N interface33 and is defined as the sum of its \nreal and imaginary part s (G↑↓=𝐺𝐺𝑟𝑟+𝑖𝑖𝐺𝐺𝑏𝑏). In our calculations  the imaginary part is neglected  \ndue to its significantly smaller contribution compared to the real part34. \nThe Pt conductivity 𝜎𝜎=1/𝜌𝜌  is calculated from the resistivity measured  by a four-probe  \nmethod on the Pt Hall bar, yielding a value of 6.3⋅ 106 Ω-1 m-1, consistent with literature  values  \n35–37. Assuming constant values of  𝜃𝜃𝑆𝑆𝐻𝐻 = 0.08, 𝜆𝜆 = 1.1 ± 0.3 nm 38,39, taking 𝑑𝑑𝑁𝑁 = 6 nm , and \nusing our SMR data at 4 .7 K and B  = 600 mT , we calculate  the real part of the spin mixing \nconductance , which under G↑↓≈𝐺𝐺𝑟𝑟 condition attains a value of 2.6 ⋅1014 Ω-1 m-2. This value is \ncomparable to those reported for other  heavy metal/magnetic insulator interfaces such as \nYIG/Pt38.  \nOur study shows that t he Pt/(Ga ,Mn)N interface exhibits  spintronic properties comparable \nto state -of-the-art systems , such as YIG/Pt interfaces39, as evidenced by similar spin mixing \nconductance  values . This indicates the strong potential of magnetically- doped GaN for \nspintronic and magnonic applications suc h as magnonic transistors40 and SOT devices41. \nFurthermore , the magnetic ordering probe d by SMR in (Ga,Mn)N above its equilibrium  Curie \ntemperature  extends  sizably the temperature  range for efficient generating  and detecting spin \ncurrents  in this material. This broadens the operational parameters for its application  in novel  \nspintronic devices with higher operating temperatures . Our findings contribute significantly to \nunderstanding spin transport in (Ga ,Mn)N -based devices and set the stag e for exploring various \ndevice architectures for new spintronic applications  based on the nitride family . \n \n    \n9 SUPPLEMENTARY MATERIAL \nSee supplementary material  [url] for a brief discussion of the ferromagnetic mechanism in \n(Ga,Mn)N insulating systems, a detailed description of Pt/(Ga,Mn)N device fabrication , and \nSQUID magnetometry measurements , which includes Ref.12,14,32,42– 60.  \n \nACKNOWLEDGMENTS  \nWe acknowledge M. Cosset -Chéneau and J. J. L. van Rijn for  helpful discussions  and critically  \nreading the manuscript, and J. Holstein, H. de Vries, F.H. van der Velde, H. Adema, and A. \nJoshua  for technical support . This work was supported by the Dutch Research Council (NWO \n– OCENW.XL21.XL21.058), the Zernike Institute for Advanced Materials, the research \nprogram “Materials for the Quantum Age” (QuMat, registration number 024.005.006), which \nis part of the Gravitation program financed by the Dutch Ministry of Education, Culture and \nScience (OCW),  the National Science Centre (Poland) through project  OPUS (DEC-\n2018/31/B/ST3/03438), and the European Union (ERC, 2D -OPTOSPIN, 101076932, and \n2DMAGSPIN, 101053054). Views and opinions expressed are however those of the author(s) \nonly and do not necessarily reflect those of the European Union or the European Research \nCouncil. Neither the European Union nor the granting authority can be held responsible for \nthem. JAMR is grateful to CONAH CYT for a graduate research fellowship (no. CVU 655591) . \nThe device fabrication and characterization were performed using Ze rnike NanoLabNL \nfacilities.  \n \nAUTHOR DECLARATIONS  \nConflict of Interest  \nThe authors have no conflicts to disclose.  \nAuthor Contributions \nJ. Aaron Mendoza- Rodarte: Conceptualization (equal) ; Methodology (lead); Validation  \n(lead); Formal analysis  (lead); Investigation (lead); Data Curation  (lead); Writing – Original \nDraft  (lead); Visualization  (equal) . Katarzyna Gas: Resources (supporting) ; Writing – Review \n& Editing  (supporting ). Manuel Herrera -Zaldívar: Supervision (supporting ); Writing – \nReview & Editin g (supporting ). Detlef Hommel: Resources (supporting ); Writing – Review \n& Editing  (supporting ). Maciej Sawicki: Resources (supporting) ; Writing – Review & Editing     \n10 (supporting). Marcos H. D. Guimarães : Supervision (lead) ; Project administration (lead); \nConceptualization (equal) ; Visualization  (equal) ; Resources  (lead); Writing – Review & \nEditing  (lead); Funding acquisition (lead). \n \nDATA AVAILABILITY  \nThe data that support the findings of this study are available from the corresponding author \nupon reasonable request.  \n \nREFERENCES  \n1 C. Song, R. Zhang, L. Liao, Y. Zhou, X. Zhou, R. Chen, Y. You, X. Chen, and F. Pan, “Spin -\norbit torques: Materials, mechanisms, performances, and potential applications,” Prog Mater \nSci 118, 100761 (2021).  \n2 B. Dieny, I.L.  Prejbeanu, K. Garello, P. Gambardella, P. Freitas, R. Lehndorff, W. Raberg, U. \nEbels, S.O. Demokritov, J. Akerman, A. Deac, P. Pirro, C. Adelmann, A. Anane, A. V. Chumak, \nA. Hirohata, S. Mangin, S.O. Valenzuela, M.C. Onbaşlı, M. d’Aquino, G. Prenat, G. Finocchio, \nL. Lopez -Diaz, R. Chantrell, O. Chubykalo -Fesenko, and P. 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Gosk, M. Kaminska, A. Twardowski, M. Bockowski, I. \nGrzegory, and S. Porowski, “Optical and magnetic propertie s of Mn in bulk GaN,” Phys Rev B \n69, 115210 (2004).  \n50 J. Gosk, M. Zajac, A. Wolos, M. Kaminska, A. Twardowski, I. Grzegory, M. Bockowski, and \nS. Porowski, “Magnetic anisotropy of bulk GaN:Mn single crystals codoped with Mg acceptors,” \nPhys Rev B 71, 94432 (2005).  \n51 W. Stefanowicz, D. Sztenkiel, B. Faina, A. Grois, M. Rovezzi, T. Devillers, F. d’Acapito, A. \nNavarro -Quezada, T. Li, R. Jakieła, M. Sawicki, T. Dietl, and A. Bonanni, “Structural and \nparamagnetic properties of dilute Ga 1-xMn xN”, Phys Rev B 81, 235210 (2010).  \n52 I.L. Guy, S. Muensit, and E.M. Goldys, “Extensional piezoelectric coefficients of gallium \nnitride and aluminum nitride,” Appl Phys Lett 75, 4133– 4135 (1999).  \n53 J. Blinowski, P. Kacman , and J.A. Majewski, “Ferromagnetic superexchange in Cr -based \ndiluted magnetic semiconductors,” Phys Rev B 53, 9524– 9527 (1996).  \n54 M. Sawicki, T. Devillers, S. Gałęski, C. Simserides, S. Dobkowska, B. Faina, A. Grois, A. \nNavarro -Quezada, K.N. Trohidou, J.A. Majewski, T. Dietl, and A. Bonanni, “ Origin of low-\ntemperature magnetic ordering in Ga 1-xMn xN” Phys Rev B 85, 205204 (2012).  \n55 C. Śliwa, and T. Dietl, “ Electron -hole contribution to the apparent s-d exchange interaction \nin III- V dilute magnetic semicon ductors,” Phys Rev B 78, 165205 (2008).  \n56 A. Bonanni, T. Dietl, and H. Ohno, “Dilute Magnetic Materials,” in Handbook of Magnetism \nand Magnetic Materials , edited by J.M.D. Coey and S.S.P. Parkin, (Springer International \nPublishing, Cham, 2021), pp. 923– 978. \n57 G. Kunert, S. Dobkowska, T. Li, H. Reuther, C. Kruse, S. Figge, R. Jakiela, A. Bonanni, J. \nGrenzer, W. Stefanowicz, J. von Borany, M. Sawicki, T. Dietl, and D. Hommel, “ Ga1−xMn xN \nepitaxial films with high magnetization,” Appl Phys Lett 101, 022413 (2012).  \n58 D. Sztenkiel, K. Gas, J.Z. Domagala, D. Hommel, and M. Sawicki, “Crystal field model \nsimulations of magnetic response of pairs, triplets and quartets of Mn3+ ions in GaN,” New J \nPhys 22, 123016 (2020).  \n59 D. Sztenkiel, “Spin orbital reorientatio n transitions induced by magnetic field,” J Magn \nMagn Mater 572, 170644 (2023).  \n60 K. Gas, and M. Sawicki, “In situ compensation method for high -precision and high -\nsensitivity integral magnetometry,” Meas Sci Technol 30, 085003 (2019).  \n  Supplementary  material  \nSpin Hall magnetoresistance in Pt/(Ga, Mn)N devices  \nJ. Aaron Mendoza -Rodarte1,2*, Katarzyna Gas3,4, Manuel Herrera -Zaldívar2, Detlef Hommel5,  \nMaciej Sawicki3,6, and Marcos H. D. Guimarães1* \n1Zernike Institute for Advanced Materials, University of Groningen, 9747 AG Groningen, The \nNetherlands  \n2Centro de Nanociencias y Nanotecnología- Universidad Nacional Autónoma de \nMéxico, Ensenada, 22800- Baja  California, México \n3Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL -02668 Warsaw, \nPoland \n4Center for Science and Innovation in Spintronics, Tohoku University, Katahira 2- 1-1, \nAoba- ku, Sendai 980- 8577, Japan \n5Lukasiewicz Research Network - PORT Polis h Center fo r Technology Development, \nStabłowicka 147, Wrocł aw, Poland \n6Research Institute of Electrical Communication, Tohoku University, Katahira 2 -1-1, \nAoba- ku, Sendai  980- 8577, Japan \n \nShort -range superexchange mechanism in (Ga,Mn)N insulating systems  \nMn-doped GaN systems are currently identified as d iluted ferro magnetic semiconductors  (DFS), \nand remain as one of the most prominent members of this family, despite the ferromagnetic \ncoupling in (Ga,Mn)N is not mediated by itinerant carriers (holes). This stems from the fact that, \nimportantly to this study, the Mn acceptor derived holes are so strongly localized (mostly due to \np-d hybridization) on Mn acceptor s1 that in the absence of other doping or numerous charged point \ndefects Mn -doped GaN remains insulating up to the highest (currently) available concentrations \nin epitaxy2–6. At these instances, Mn assumes a neutral  acceptor oxidation state , Mn+3 (d4 \nconfi guration)  characterized by spin and orbital momentum quantum numbers S = 2 and L  = 2, \nrespectively7, and a strong single ion uniaxial magnetic anisotropy with respect to the wurtzite c -\naxis of GaN8–10. The latter proved to be controllable by an externally applied electrical field11, via \nthe inverse piezoelectric effect in GaN host lattice along the c  axis12.    \n As a consequence of the strongly insulating character of GaN enriched with Mn, the carrier -\nmediated coupling is precluded in (Ga,Mn) N. The observed low -temperature ferromagnetism is \nbased on a short -range superexchange  scenario2,13,14, which, below the nearest -neighbor \npercolation threshold,  earns  its effectiveness from  the ferromagnetic (FM) sign of the spin -spin \nexchange constants up to the nearest 30th neighbors14,15, and the ferromagnetism is established in \npercolation fashion16,17. \n \nDetails of the Pt/(Ga ,Mn)N devices f abrication  \nOur devices were fabricated on  a Ga 1-xMn xN (100 nm)  single -phase epitaxial film . More details \nof the growth and characterization of the Ga 1-xMn xN films can be found in Ref.18. We patterned 25 \nμm-wide , 200 μ m-long Hall bars using a 950K polymethyl methacrylate  (PMMA) positive resi st, \nwith a solid content of 4% dissolved in ethyl  lactate . To prevent charging effects during the electron \nbeam exposure process, we spin -coated Electra -92 on top of the PMMA, which serves as a water -\nsoluble conductive polymer. The device patterning was conducted using a Raith e -line 150 \nelectron -beam lithography system with an acceleration voltage of 10 kV . For the development \nprocess, the Electra 92 layer is removed by immersing the sample in deionized water . Then, to \ndevelop the PMMA, the sample is immersed in a 1:3 mixture of  isopropyl  alcohol (IPA) and \nmethylisobutylketon (MIBK). Next, a Pt (6 nm)  layer is deposited using a  sputtering machine  with \na base pressure better than 1× 10-7 mbar, using a n Ar plasma with a working pressure of 4 ×10-3 \nmbar with a power 50 W, resulting in  a rate of  0.36 nm/s . Prior to deposition, the sample undergoes  \nan Ar+ mild etching step with a power of 200 W to remove organic residues and a potential top \noxide layer from the  (Ga ,Mn)N exposed surface, resulting in a clean and high- quality interface.  A \nTi (5 nm)/ Au (55 nm) thin layer was deposited to ensure electrical contact via an electron beam \nevaporation system . The lift- off process is carried out by immersing the sample in acetone at 45 °C , \nrinsing it with IPA, and blowing it dry with nitrogen gas. Finally, the devices are bonded to a 44-\npin chip carrier.  \nThe measurements are performed using a flow cryostat (Oxford Instruments) mounted between \nthe poles of a room -temperature electromagnet with the magnetic field aligned to the in -plane \ndirection with respect to the sample. The current is applied using a home -made current source, \nreferenced by the output of a lock- in amplifier and the voltage is pre -amplified by a home -made voltage pre -amplifier before being sent to the lock -in amplifier.  We applie d currents below 1 mA \nat low frequencies (<200 Hz) to avoid heating and high- frequency effects.  \n \n \nFigure  S1. Reflection high -energy electron dff raction, RHEED, pattern along [112�0] azimuth observed after the \ngrowth and cooling the sample down below 200° C. \n \nDuring the  growth the surface quali ty of the layer has been investigated in situ  by refle ction high -\nenergy electron di ffraction (RHEED). The post growth [112 �0] RHEED patterns  collected after the \ngrowth (below 200°C) is  shown in Fig. S 1. The streaky and  sharp RHEED pat tern with clearly \nvisible Kikuchi lines indicates that the(Ga,Mn)N surface is smooth and relatively free of oxides or \nother contaminants.  SQUID magnetometry measurements  \nThe Curie temperature ( TC) of the magnetic layer  used for the Pt/(Ga ,Mn)N devices was \npreviously assesd by five different methods by SQUID magnetometry , converging to the value  of \nTC = 13.0  ± 0.3 K. More details can be found in Ref.18 where the magnetic layer for the devices \nshown in the main text  is labelled  as S605. The magnetization curves , M-H, for this sample are \nshown in Figure S2. We highlight the fact that the M-H curves presented in Figure S2 are \nrepresentative of single -phase  (Ga,Mn)N epitaxial layers18–20. Importantly, we do not register any \nnonlinear magnetization at the weak field region above TC, which  confirm s the single magnetic \nphase of the (Ga,Mn)N layer  investigated here . Below TC ≅ 13.0 K M responces swiftly  to the \napplied field already at very weak fields, yet it does not reach the complete saturation  even at µ 0H \n= 7 T at T  = 2 K. This behaviour highlights the fact that the M in (Ga,Mn)N is largely controlled \nby the single ion properties of the Mn3+ ions (de scribed in the frame of the crystal field model) and \nthat the orbital contribution to M exhibits a much weaker dependence on H  than the spin \ncomponent does20,21. \n0 2 4 6 8050100150 Magnetization   ( kA/m ) \nMagnetic Field  ( T )\n  \nGa0.922Mn0.078N\n300 K50 K25 K2 K\n \nFigure S2. Magnetization , M, curves of the particular sample investigated in the main part of the letter at four selected \ntemperatures. The magnetic field is applied in the plane of the sample (i.e. perpendicularly to the wurtzite c  axis) , \nwhich is the easy orientation for M in this material  8–10. The absolute  value  of M  specific to the (Ga,Mn)N epilayer is \nestablished upon the technique allowing in situ  compensation of the unwanted signal of the substrate (sapphire) \ndescribed  in Ref. 22.  \n \nREFERENCES  \n1 T. Dietl, F. Matsukura , and H. Ohno, “ Ferromagnetism of magnetic semiconductors: Zhang- Rice \nlimit, ” Phys Rev B 66, 33203 (2002). \n2 A. Bonanni, M. Sawicki, T. Devillers, W. Stefanowicz, B. Faina, T. Li, T.E. Winkler, D. Sztenkiel, \nA. Navarro -Quezada, M. Rovezzi, R. Jakieła, A. G rois, M. Wegscheider, W. Jantsch, J. Suffczyński, \nF. D ’Acapito, A. Meingast, G. Kothleitner, and T. Dietl, “Experimental probing of exchange \ninteractions between localized spins in the dilute magnetic insulator (Ga,Mn)N, ” Phys Rev B 84, \n35206 (2011). 3 T. Yamamoto, H. Sazawa, N. Nishikawa, M. Kiuchi, T. Ide, M. Shimizu, T. Inoue, and M. Hata, \n“Reduction in Buffer Leakage Current with Mn- Doped GaN Buffer Layer Grown by Metal \nOrganic Chemical Vapor Deposition, ” Jpn J Appl Phys 52, 08JN12 (2013). \n4 R. Adhikari , W. Stefanowicz, B. Faina, G. Capuzzo, M. Sawicki, T. Dietl, and A. Bonanni, “ Upper \nbound for the s -d exchange integral in n- (Ga,Mn)N:Si from magnetotransport studies ”, Phys Rev \nB 91, 205204 (2015).  \n5 L. Janicki, G. Kunert, M. Sawicki, E. Piskorska -Hommel , K. Gas, R. Jakiela, D. Hommel, and \nR. Kudrawiec, “ Fermi level and bands offsets determination in insulating (Ga,Mn)N/GaN \nstructures, ” Sci Rep 7, 41877 (2017). \n6 K. Kalbarczyk, K. Dybko, K. Gas, D. Sztenkiel, M. Foltyn, M. Majewicz, P. Nowicki, E. \nŁusakow ska, D. Hommel, and M. Sawicki, “ Electrical characteristics of vertical -geometry \nSchottky junction to magnetic insulator (Ga,Mn)N heteroepitaxially grown on sapphire,” J Alloys \nCompd 804, 415 (2019).  \n7 J. Kreissl, W. Ulrici, M. El- Metoui, A.- M. Vasson, A. Vasson, and A. Gavaix, “ Neutral manganese \nacceptor in GaP: An electron -paramagnetic- resonance study, ” Phys Rev B 54, 10508 (1996).  \n8 A. Wolos, M. Palczewska, M. Zajac, J. Gosk, M. Kaminska, A. Twardowski, M. Bockowski, I. \nGrzegory, and S. Porowski, “ Optica l and magnetic properties of Mn in bulk GaN,” Phys Rev B 69, \n115210 (2004). \n9 J. Gosk, M. Zajac, A. Wolos, M. Kaminska, A. Twardowski, I. Grzegory, M. Bockowski, and S. \nPorowski, “Magnetic anisotropy of bulk GaN:Mn single crystals codoped with Mg acceptors ,” \nPhys Rev B 71, 94432 (2005). \n10 W. Stefanowicz, D. Sztenkiel, B. Faina, A. Grois, M. Rovezzi, T. Devillers, F. d’ Acapito, A. \nNavarro -Quezada, T. Li, R. Jakieła, M. Sawicki, T. Dietl, and A. Bonanni, “Structural and \nparamagnetic properties of dilute Ga 1-xMn xN”, Phys Rev B 81, 235210 (2010). \n11 D. Sztenkiel, M. Foltyn, G.P. Mazur, R. Adhikari, K. Kosiel, K. Gas, M. Zgirski, R. Kruszka, R. \nJakiela, T. Li, A. Piotrowska, A. Bonanni, M. Sawicki, and T. Dietl, “ Stretching magnetism with \nan electric field in a nitride semiconductor,” Nat Commun 7, 13232 (2016). \n12 I.L. Guy, S. Muensit, and E.M. Goldys, “ Extensional piezoelectric coefficients of gallium nitride \nand aluminum nitride, ” Appl Phys Lett 75, 4133 (1999).  \n13 J. Blinowski, P. Kacman, and J.A. Majewski, “ Ferromagnetic superexchange in Cr -based diluted \nmagnetic semiconductors, ” Phys Rev B 53, 9524 (1996).  14 M. Sawicki, T. Devillers, S. Gałęski, C. Simserides, S. Dobkowska, B. Faina, A. Grois, A. \nNavarro -Quezada, K.N. Trohidou, J.A. Ma jewski, T. Dietl, and A. Bonanni, “ Origin of low -\ntemperature magnetic ordering in Ga 1-xMn xN,” Phys Rev B 85, 205204 (2012). \n15 C. Śliwa, and T. Dietl, “Electron -hole contribution to the apparent s -d exchange interaction in \nIII-V dilute magnetic semiconduct ors,” Phys Rev B 78, 165205 (2008).  \n16 I.Ya. Korenblit, E.F. Shender, and B.I. Shklovsky, “ Percolation approach to the phase transition \nin very dilute ferromagnetic alloys, ” Phys Lett A 46, 275–276 (1973).  \n17 A. Bonanni, T. Dietl, and H. Ohno, “ Dilute Magn etic Materials, ” in Handbook of Magnetism \nand Magnetic Materials , edited by J.M.D. Coey and S.S.P. Parkin, (Springer International \nPublishing, Cham, 2021), pp. 923–978. \n18 K. Gas, J.Z. Domagala, R. Jakiela, G. Kunert, P. Dluzewski, E. Piskorska -Hommel, W. \nPaszkowicz, D. Sztenkiel, M.J. Winiarski, D. Kowalska, R. Szukiewicz, T. Baraniecki, A. \nMiszczuk, D. Hommel, and M. Sawicki, “ Impact of substrate temperature on magnetic properties \nof plasma -assisted molecular beam epitaxy grown (Ga,Mn)N,” J Alloys Compd 747, 946 (2018).  \n19 G. Kunert, S. Dobkowska, T. Li, H. Reuther, C. Kruse, S. Figge, R. Jakiela, A. Bonanni, J. \nGrenzer, W. Stefanowicz, J. von Borany, M. Sawicki, T. Dietl, and D. Hommel, “ Ga1−xMn xN \nepitaxial films with high magnetization, ” Appl Phys Lett 101, 022413 (2012). \n20 D. Sztenkiel, K. Gas, J.Z. Domagala, D. Hommel, and M. Sawicki, “ Crystal field model \nsimulations of magnetic response of pairs, triplets and quartets of Mn3+ ions in GaN,” New J Phys \n22, 123016 (2020). \n21 D. Sztenkiel , “Spin orbital reorientation transitions induced by magnetic field,” J Magn Magn \nMater 572, 170644 (2023). \n22 K. Gas, and M. Sawicki, “ In situ compensation method for high -precision and high -sensitivity \nintegral magnetometry, ” Meas Sci Technol 30, 085003 (2019). \n  "    },    {        "title": "2005.08539v1.Local_spin_Seebeck_imaging_with_scanning_thermal_probe.pdf",        "content": "Local spin Seebeck imaging with scanning thermal probe\nAlessandro Sola,\u0003Vittorio Basso, and Massimo Pasquale\nIstituto Nazionale di Ricerca Metrologica, Strada delle Cacce 91, 10135, Torino, Italy\nCarsten Dubs\nINNOVENT e.V., Technologieentwicklung, Pr ussingstrasse. 27B, 07745 Jena, Germany\nCraig Barton and Olga Kazakowa\nNational Physical Laboratory, Teddington TW11 0LW, United Kingdom\nIn this work we present the results of an experiment to locally resolve the spin Seebeck e\u000bect in\na high-quality Pt/YIG sample. We achieve this by employing a locally heated scanning thermal\nprobe to generate a highly local non-equilibrium spin current. To support our experimental results,\nwe also present a model based on the non-equilibrium thermodynamic approach which is in a good\nagreement with experimental \fndings. To further corroborate our results, we index the locally\nresolved spin Seebeck e\u000bect with that of the local magnetisation texture by MFM and correlate\ncorresponding regions. We hypothesise that this technique allows imaging of magnetisation textures\nwithin the magnon di\u000busion length and hence characterisation of spin caloritronic materials at the\nnanoscale.\nINTRODUCTION\nThe visualisation of domain structure in magnetism\nand magnetic materials is paramount in aiding the under-\nstanding at the fundamental level and subsequent utilisa-\ntion of such materials in real world applications. Hence,\na signi\fcant e\u000bort is dedicated to the development of a\nvariety of techniques which are suitable to study mag-\nnetic materials and magnetic domains. Magnetic force\nmicroscopy (MFM) [1, 2] is a scanning probe technique\ncapable of sensing the force gradients induced by stray\n\felds over the surface of a magnetic material at the\nnanoscale. Scanning electron microscopy with polariza-\ntion analysis (SEMPA) [3] can also be used to visualize\nmagnetic domains by monitoring the spin polarization of\nthe secondary electrons interacting with the stray \feld\nof a sample under test. In order to visualize the local\nmagnetization M(r) (or the \rux density B(r)) instead of\nthe stray \feld H(r), it is possible to use imaging tech-\nniques such as magneto-optic e\u000bects [4] or Lorentz mi-\ncroscopy [5]. Electrons can also be used as a probe for\nthe imaging of magnetic domains in electron holography\n[6]. The above techniques typically involve the character-\nization of magnetic properties con\fned to the surface or\nthin samples; the investigation of the domain structure\nin bulk materials, however, requires more complex exper-\niments that involves neutron scattering [7{10] or X-ray\nspectroscopy [11{13].\nThe interaction between heat and non-equilibrium spin\ncurrents in magnetic materials represents an alternative\napproach to image magnetic domains. Analogous to stan-\ndard thermoelectric e\u000bects, this interaction has been de-\nscribed as the thermal generation of driving power forelectron spin, i.e. the spin Seebeck e\u000bect (SSE) [14]. This\ne\u000bect involves pure spin currents which are produced by\ndriving the system out of equilibrium through thermal\ngradients; by de\fnition, they carry zero net charge and\ndepend on the local magnetization of the material. The\nlength scale lMthat governs the SSE has been demon-\nstrated to be of the order of micrometers [15{17]. The\nSSE has been observed in magnetic insulator garnet fer-\nrites [18] such as the ferrimagnetic yttrium iron garnet\nY3Fe5O12(YIG), or other ferrites [19{21]. A typical spin\nSeebeck device can be formed by creating a bilayer of a\nmagnetic material and a thin metallic \flm with high spin-\norbit coupling like platinum or tungsten both of which\nare paramagnetic heavy metals. This second layer acts\nas the spin detector thanks to the inverse spin Hall ef-\nfect (ISHE) that enables a spin-charge conversion at the\ninterface [22]. The most useful con\fguration is the lon-\ngitudinal spin Seebeck e\u000bect (LSSE) [23{25] that corre-\nsponds to the generation of a spin current parallel to\nthe temperature gradient both of which are orthogonal\nto the local magnetisation direction which is in the sam-\nple plane. The majority of experiments typically refer to\nspin Seebeck samples that are uniformly heated over the\nwhole sample and are in a uniform magnetic state, i.e. at\nor near saturation. In these conditions, it is possible to\ncompare the spin Seebeck characteristics to the magneti-\nzation loop of the sample; however it is not possible to re-\nsolve the contributions to the spin Seebeck signal coming\nfrom regions where the orientation of the magnetization\ndi\u000bers from the average, i.e. in a multidomain state. To\nresolve magnetic domains, it is necessary to go beyond\nthe experimental con\fguration previously described in\nfavour of a locally injected heat current, as described inarXiv:2005.08539v1  [cond-mat.mes-hall]  18 May 20202\n  \nPt    YIG\njM\njq\nV\nPt\njM\njq\nIqYIG\nPt\nVYIG\nPtYIGThermal probe\nIq\nxyz\na) b)xz z\ny xyzjM\njq\nV\nTop viewCross \nsection2r0Lz\ntPttYIG\nFigure 1. Longitudinal spin Seebeck measurement con\fgurations. (a) Standard con\fguration with uniform heating (the red\narrows at the Pt/YIG interface represent the hot side; the cold side is at the right-hand side of the YIG slab). (b) A local\nheating from the Pt side generates a magnetic moment current density jMin spherical symmetry. On the right side of the\npanel, the projections of the experimental scheme on the x-z and y-z planes are represented: the local heat current iqfrom the\nthermal probe spreads over a circle whose dimensions ( r0) limit the space resolution. For Pt/YIG sample that we investigated\nthe dimensions are: tPt= 5 nm,tYIG= 0:5 mm,LPt\nz= 150\u0016m.\nFigure 1. The dependence of a local spin Seebeck signal\non the local magnetization has been observed by scan-\nning a laser beam on a Pt/YIG structure [26] and in a\ntime-resolved con\fguration [27{29]. By taking account\nof the experimental geometry the same set-up can be\nadopted for the measurement of the time resolved anoma-\nlous Nernst e\u000bect [26, 30]. In this work we present for the\n\frst-time local measurements of the SSE using a thermal\nAFM probe as local source of heat and non-equilibrium\nspin currents. We employed a high-quality bulk YIG sin-\ngle crystal with a Pt strip lithographically de\fned onto\nthe surface as our spin detector. We demonstrate that\nthe measured e\u000bect is unambiguously the local spin See-\nbeck through a series of tests; by varying the heating\npower and the vector of the externally applied magnetic\n\feld. We interpret and support our observations with a\nthermodynamic description of the generation of the local\nmagnetic moment current. This model describes quanti-\ntatively the geometry of local heat current as a circular\nheat source below the thermal probe whose size is larger\nthan the cross-section of the tip due to the non-zero ther-\nmal conductivity at the Pt/YIG interface. The diameter\nof the heat source that we observed by employing our\nmodel is\u00182:8\u0016m.\nSPIN SEEBECK EFFECT BY UNIFORM AND\nLOCAL HEATING\nFigure 1 shows two variations of a spin Seebeck exper-\niment, both demonstrating that the measured signal due\nto the inverse spin Hall e\u000bect (represented by the yellow\nwires in Figure 1) corresponds to the open circuit voltage\nacross the Pt \flm. The \frst one (Figure 1 (a)), representsthe classical experimental geometry reported by the ma-\njority of experimental results. This provides a uniform\ndistribution of the thermally-generated spin current as\nconsequence of the heat current across the whole surface\nof the sample. The proportionality between the voltage\ngradientryVSSEgenerated along the Pt \flm due to the\nSSE and the heat current Iqpassing through the sample\nalong the cross-section Ais described by the following\nexpression:\nryVSSE\nIq\u0014A=\u0012SH\u0010\u0016B\ne\u0011vYIGlYIG\u000fYIG\nvp1\ntPt(1)\nwhere\u0016B=eis the ratio between the Bohr magneton and\nthe elementary charge, \u0014is the thermal conductivity of\nYIG. The other parameters inside Eq. 1 are the spin Hall\nangle\u0012SH, the absolute thermomagnetic power coe\u000ecient\n\u000fYIG, the magnon di\u000busion length lYIGand the thick-\nness of the Pt \flm tPt. The intrinsic magnetic moment\nconductance of YIG is represented by vYIG=lYIG=\u001cYIG\nwhere\u001cYIGis the magnon mean scattering time. The pa-\nrametervprepresents the magnetic moment conductance\nper unit surface area of the Pt/YIG bilayer. This quan-\ntity depends on the intrinsic conductances of YIG and Pt,\non the ratio between the thickness and the magnon dif-\nfusion length, for each layer. We derived the expression\nofvpfrom a thermodynamic description of the magnetic\nmoment currents generation [31]. It is important to note\nthat the geometry shown in Figure 1 and modelled by\nEq. 1 does not allow to resolve the spatial distribution of\nthe underlying magnetic structure, since the spin Seebeck\nvoltage results from an averaged contribution of regions\nwith di\u000berent magnetization. Because of this, the exper-\niments performed in this geometry are usually conducted3\n  \nvoltmeterPt\nYIGScanning probe\na) b)\nc)μ0H\nμV\nd)\ne)80 μm\n80 μm80 μm\n80 μm80 μm\n80 μm80 μm\n80 μm\ny\nxz\nFigure 2. Experimental set-up for the local SSE measurements. (a) Macroscopic schematic representation of the Pt/YIG sample\nwith the nanovoltmeter connected at the edges of the Pt strip. The subsequent images (b-e) are examples of measurements\nobtained from a 80 \u0016m\u000280\u0016m area of the Pt \flm with an applied magnetic \feld of \u00188 mT. (b) MFM image of the same\narea, (c) spin Seebeck voltage originating from the scanning thermal probe, (d) AFM image that is obtained both from MFM\nand local spin Seebeck map, (e) 3D image obtained by overlapping of MFM and spin Seebeck map, taking into account the\ninformation on their mutual shift provided by the AFM images.\nat magnetic saturation or follow the magnetization hys-\nteresis loop.\nHowever, in the second experimental setup (Figure 1\n(b)), which shows the e\u000bect of the heated AFM probe\nin contact with the Pt surface, we can selectively gener-\nate the heat current injected through a point of the Pt\nsurface and propagating with spherical symmetry in the\nvolume. In this arrangement, the thermally generated\nspin current is assigned to a locally limited SSE, which\nis generated at the point of thermal contact, represented\nby the red dot in Figure 1 (b). As in the standard con\fg-\nuration the spin Seebeck voltage depends on the average\nmagnetization, in the con\fguration with local heating the\ne\u000bect scales with the magnetization of a region whose size\nis determined by the locally heated volume, allowing re-\nsolved measurements of the domain distribution within\nthe sample. To describe this experimental con\fguration,\nwe took into account the local magnetization mas a unit\nvector. This leads to an expression of the spin Seebeck\nvoltage, which originates from a circle with radius r0on\nthe top of the sample, into which a heat current Iqis\ninjected. In this way it is possible to rewrite Eq. 1 as\nfollows:\nryVSSE\nIq2\u0019r2\n0k=\u0012SH\u0010\u0016B\ne\u0011vYIGlYIG\u000fYIG\nvp1\ntPt(2)\nwhere the dependence on the radius of the heated region\nr0(red circle in Figure 1 (b)) is highlighted. However, the\nvalue ofVSSEin Eq. 2, and in particular the voltage dif-\nference \u0001VSSEis the one that would be measured at the\ntwo ends of the heated region, limited by r0. Since thisquantity is not accessible by the experiment, it is neces-\nsary to rescale the value of \u0001 VSSE, taking into account\nthe lateral dimensions of the whole Pt \flm that works\nas the spin-voltage detector. Here we use an approxi-\nmation criterion and consider the Pt \flm as an electric\ncircuit formed by a voltage source ( VSSE), two resistances\nin series whose sum is proportional to ( Ly\u00002r0)=2r0and\none resistance in parallel whose value is proportional to\nLy=(Lz\u00002r0). Assuming that 2 r0\u001cLyand 2r0\u001cLz,\nwe can represent the experimental values of the spin See-\nbeck voltage as a rescaled value of \u0001 VSSE\n\u0001VSSE,exp'\u0001VSSE2r0\nLz(3)\nThe expression of \u0001 VSSE,exp can be used to interpret the\nexperimental data of the local SSE. By using the param-\neters included in Eq. 2 to determine the value of VSSE\nat the right of Eq. 3, it is possible to obtain a value of\nr0that can be considered as the space resolution of the\nspin Seebeck imaging.\nEXPERIMENTAL METHODS\nThe measurements of the local SSE were performed\non a high quality YIG single crystal plate ( Ly= 4:95\nmm,Lz= 3:91 mm and thickness tY IG = 0:545 mm).\nBoth crystal surfaces were polished to be optically \rat\n(Rq= 0:4 nm obtained by AFM). A 150 \u0016m wide Pt strip\nwas sputtered onto one side of the YIG crystal along the y\ndirection, where the z direction de\fnes the stripe width.4\nThe thickness of the Pt \flm tY IG was chosen equal to\nbe approximately 5 nm in accordance with [32, 33]. The\nmeasurement technique for the uniform heat current, i.\ne. the classical con\fguration, has been described else-\nwhere [34] for the same sample. The local heat current\nwas generated using a nano thermal analysis probe (Nan-\noTa probe); this consists of a micropatterned AFM can-\ntilever probe that allows a current to be driven around\nthe cantilever resulting in Joule heating that propagates\nto the tip apex. We approximated the temperature of the\nprobe using a series of polymer test samples with known\nglass transition temperatures. We also expected an o\u000b-\nset in our estimated temperatures due to the respectively\nhigher thermal conductivity of the Pt used in our studies\nas compared to the test polymers. With this procedure\nit was possible to obtain an approximate relationship be-\ntween the probe power voltage and the temperature dif-\nference applied to the sample in this geometry. Moreover,\nwe observed a stable spin Seebeck signal when reverting\nthe heating voltage thus verifying absence of electric in-\nterference from the nanoTA cantilever. This setup al-\nlowed us to compare a local domain map obtained by\nMFM with a local SSE map and correlate the two data\nsets. The enlarged area in Figure 2 (a) represents one\nof the speci\fc regions investigated in the experiments.\nFrom the data in Figure 2 (b,c) it is possible to spatial-\nity map the SSE voltage at the location of the thermal\nprobe in contact with the Pt \flm. This makes it possible\nto correlate SSE and MFM and draw qualitative conclu-\nsions. Figures 2 (b) and (c) show a stray \feld gradient\nat the YIG surface obtained by MFM (Figure 2 (b)) and\na SSE map obtained by nanoTA (Figure 2 (c)). The ex-\nperiment was structured as follows: \frst we performed a\nset of nanoTA measurements with a saturating magnetic\n\feld in both directions ( \u0006Ms), achieved using a small\nneodymium magnet adjacent to the sample. This corre-\nsponds to a standard SSE experiment. The second set of\nmeasurements was performed at a magnetic \feld where a\ndomain structure could be observed using MFM. Keep-\ning the applied \feld constant, the nanoTA measurements\nwere performed at di\u000berent heating power levels of the\nthermal probe and at several points of the sample sur-\nface including the transition from the Pt strip to the bare\nYIG surface.\nRESULTS AND DISCUSSION\nLocal spin Seebeck e\u000bect at magnetic saturation\nWe \frst investigated the local SSE of the Pt/YIG bi-\nlayer structure at magnetic saturation. This experiment\nwas performed with the \feld aligned within the plane of\nthe sample and perpendicular to the long axis of the Pt\nstrip (Figure 2 (a)). The experimental data points in\nFigure 3 show the SSE data for each realized tempera-\nVmeas (μV)\nΔT (K)Figure 3. Measured voltage as consequence of the local heat-\ning of the sample at magnetic saturation: spin and ordinary\nSeebeck voltage as function of the temperature di\u000berences\n(red point). Fitting lines after the compensation of the ordi-\nnary Seebeck component (blue dashed lines). The direction\nof the saturating magnetic \feld is shown as a sketch in each\npanel.\nture di\u000berences \u0001T. Here, we averaged the signal that\noriginated from the local heating of a 40 \u0016m\u000220\u0016m\narea of the Pt surface and the experimental uncertainty\nwas evaluated from the standard deviation of these data\nsets. The dependency between the total measured volt-\nage, including the spin Seebeck contribution and \u0001T can\nbe described by the following relation:\nVmeas=\u0006CSSE\u0001T+COSE\u0001T+V0 (4)\nwhereVmeas corresponds to the average voltage recorded\nas a consequence of scanning a given area and \u0001T is the\ndi\u000berence between the probe temperature and the room\ntemperature. The voltage Vmeas contains the following\ncontributions: the \frst one ( CSSE\u0001T) corresponds to the\nspin Seebeck voltage VSSE,exp of Eq. 3; the sign of the\nspin Seebeck coe\u000ecient \u0006CSSEdepends on the direction\nof the magnetisation. The second contribution is the or-\ndinary Seebeck e\u000bect ( COSE); for our work, this is a spu-\nrious component that does not depend on the magnetic\ncon\fguration of the sample. The COSEcomponent de-\nrives from the contact between the di\u000berent metals used\nto electrically connect the sample (silver paint for bond-\ning the platinum strip at the edges of the thin \flm). This\nis due to a small transverse heat loss through the elec-\ntric contacts associated to small geometric asymmetry.\nSuch an artefact can a\u000bect the interpretation of spin-\ncaloritronic measurements and has been previously de-\nscribed for both the spin Seebeck and spin Peltier e\u000bect\n[34] data. The last contribution of Eq. 4 is an o\u000bset volt-\nageV0that originated from the circuit resistance. The\nvoltage due to the Seebeck e\u000bects, both spin and ordi-\nnary e\u000bects, can be plotted as function of \u0001T, according\nto Eq. 4. From the di\u000berence between the absolute values5\nof the slopes obtained by the linear \fts shown in Figure\n3 (red lines), a clear distinction can be made between\nthe ordinary Seebeck and the spin Seebeck components.\nAfter the compensation of the ordinary Seebeck compo-\nnent, we present the spin Seebeck data by blue dashed\nlines in Figure 3, whose coe\u000ecient is CSSE= 2\u000110\u00008\nVK\u00001.\nLocal spin Seebeck e\u000bect at intermediate\nmagnetization obtained with \u00188mT\nIn the second step, we repeated the measurements with\na lower applied magnetic \feld in order to induce a re-\nduction in the magnetostatic energy of the sample and\nintroduce a domain structure where we could distinguish\nseveral magnetization areas from the MFM images. We\napplied the magnetic \feld at lowered level by distancing\nthe neodymium magnet from the sample and we mea-\nsured its value by positioning an Hall probe in place of\nthe sample. The measured value for the applied magnetic\n\feld was approximately 8 mT. An example of two MFM\nimages of the sample at magnetic saturation and with an\napplied \feld of\u00188 mT obtained from the same area is\nrepresented in Figure 4.\n10 \u0001m\na) b)10 \u0001m\nFigure 4. MFM micrograph of the same area of the Pt/YIG\nsurface at magnetic saturation (left panel) and at \u00188 mT\n(right panel).\nHaving applied\u00188 mT, we scanned the thermal probe\nover a 80\u0016m\u000280\u0016m in the same locations as that\nof the MFM, at di\u000berent values of \u0001T in order to anal-\nyse the voltage that arises form the local SSE. We used\n\fve values of heating power on the thermal probe that\nled to the corresponding temperature di\u000berences \u0001T at\neach area investigated. With these data sets we were\nable to determine the SSE dependence on heating power\nand extract the SSE voltage for each pixel of the data\nset. First we removed the ordinary Seebeck component,\nas described for the sample at magnetic saturation; this\nprocedure gives the spin Seebeck coe\u000ecient CSSEand the\nmeasurement o\u000bset V0, represented by the slopes and the\nintercepts of the blue lines in Figure 3. Here we extract\nthe values of these two parameters for the sample at in-\na)  spin Seebeck coefficient CSSE (nV/K)\nb)           offset voltage V0             ( V)\nc)      reduced magnetization12\n8\n4\n0\n0.4\n0.2\n0\n-0.2\n-0.4\n1.0\n0.8\n0.6\n0.4\n0.2\n0.0Figure 5. spin Seebeck maps of one 80 \u0016m\u000280\u0016m area\nof the Pt/YIG sample surface obtained with a local heating\ngenerated by 0.5, 1, 1.5, 1.8 and 2.2 V on the heater. (a)\nSpin Seebeck coe\u000ecients measured at each pixel of the map.\n(b) Map of the spurious o\u000bset voltage component and (c)\nmap of the ratios between the local spin Seebeck voltage and\nthe corresponding quantity at magnetic saturation; the colour\nscale in (c) represents the magnetization as percentage of the\nmagnetic saturation.\ntermediate magnetization by performing a linear \ft of the\nvoltage values ( Vmeas) represented by each pixel as func-\ntion of the temperature di\u000berence. We can now build two\nmaps that represent the parameters of these linear \fts:\nthe map of the spin Seebeck coe\u000ecients CSSE(Figure 5\n(a)) and the map of intercepts V0(Figure 5 (b)). Since\nwe have already measured the upper limit of the spin\nSeebeck signal i.e. the value at magnetic saturation, we6\nFigure 6. Local spin Seebeck voltage maps (right columns) obtained at \u00188 mT with 1.5 V at the heater, shown together with\nthe MFM micrographs (left columns). The \fve pairs of maps correspond to the areas of the sample represented in the scheme,\nwhere the edge of the Pt \flm is shown, together with the direction of the applied magnetic \feld.\ncan use this value to normalize our results with \u00188 mT\nwhich is represented as a percentage of the maximum sig-\nnal at saturation. Figure 5 (c) shows an example of this\nprocedure recorded at the maximum temperature di\u000ber-\nence; the colorbar quanti\fes the level of magnetization\nbetween zero and the values at saturation, represented\nas the range [0\u00001]. From the data in Figure 5 (a), we\nfound that the ratio between the spin Seebeck values and\nthe temperature di\u000berence (i.e. the spin Seebeck coe\u000e-\ncient) has some non-zero positive values in some regions\nand decreases to zero in the round central region of the\nmap. By adapting the de\fnition of the spin Seebeck coef-\n\fcient that is formulated for the saturated sample to the\ncase of intermediate magnetization, we can deduce what\nfollows. In some areas the magnetization was oriented to\nprovide a larger spin Seebeck signal with respect to the\nareas where we observed a lower signal, reasonably due\nto a change in the orientation of the magnetization. On\nthe contrary, the map of the intercepts in Figure 5 (b)\nis considerably \ratter, indicating that the electric o\u000bset\nwas rather steady over the investigated area.\nComparison between MFM and local spin Seebeck\nimages\nThe third step of this experiment involved scanning\ndi\u000berent regions of the sample, at a \fxed heating volt-\nage, with the same applied magnetic \feld as previously\nused (\u00188 mT). The motivation is a qualitative compari-\nson of the spin Seebeck maps with MFM micrographs.We tested the hypothesis that the contrast of the lo-\ncal spin Seebeck map is related to the local magnetiza-\ntion through the local spin Seebeck signal. Scanning of\nlarge areas allowed for overlapping neighbouring data sets\nand while recorded them accordingly; the corresponding\nMFM micrographs and the local spin Seebeck voltage\nmaps are reported in Figure 6. The signal of the local\nspin Seebeck maps was processed using the Gwyddion\ndata analysis software [35]. First we focused on the pres-\nence of some sharp voltage spikes; these have been manu-\nally corrected with an interpolation of the error-free pix-\nels surrounding the spike. The second data process con-\ncerned the artefacts that usually appears because of the\nline by line acquisition. We corrected the voltage shift\nthat rise between neighbouring horizontal lines by mini-\nmizing the median of height di\u000berences, between vertical\nneighbouring pixels [36]. From Figure 6 we observe that\nthe MFM micrographs can provide a great variety of de-\ntails due to the high resolution, whereas the local SSE\nmaps have a lower spatial resolution. Furthermore, we\nhave to keep in mind that the underlying physical phe-\nnomena are di\u000berent for both experiments: the MFM sig-\nnal depends on the stray \felds emanating from the sam-\nple surface whereas the local SSE depends on the mag-\nnetization of the sample and the integrated e\u000bect over\nhot spot provided by the probe. Nevertheless, there is a\nclear correspondence between regions and features in the\nMFM micrographs and the regions of local minima of the\nlocal spin Seebeck maps. To make the correlation more\nvisible, we traced the contour lines extracted by the spin7\nSeebeck maps above the MFM micrographs in Figure 6.\nFor the three images on the left, we selected the con-\ntour lines corresponding to 0 :6\u0016V on the spin Seebeck\nmap, while for the two images on the right we selected\ndi\u000berent values since we did observe a drift on the o\u000bset\nvoltage. For this reason, the two areas labelled by the\ngreen and light-blue frames appear on a slightly di\u000berent\nvoltage scale, but nevertheless it is possible to distinguish\nthe shape of the local minimum in the expected position,\naccording to the 20 \u0016m shift between the two areas. The\ntwo micrographs that refer to the area labelled by the\nred frame in Figure 6 represent a measurement across\nthe edge of the Pt strip, cutting the frame horizontally.\nIn the lower half of the SSE micrograph (the region not\ncovered by the Pt), the spin Seebeck voltage decreases\nin agreement with the local generation of a spin current\nfrom the bare YIG that is not detected by the Pt \flm.\nFinally, it is possible to hypothesise the level of the lo-\ncal magnetization of the sample. This was achieved by\naveraging the values reported in the local spin Seebeck\nmaps of Figure 6 and then normalising these values with\nthe spin Seebeck coe\u000ecient according to the procedure\nadopted for Figure 5 (c). By considering this approach,\nwe obtained a value of the average magnetization cor-\nresponding to 0 :4 times the magnetic saturation. This\nvalue is in agreement with the measurement performed\nwith the Hall probe (8 mT), knowing that over 20 mT the\nsample saturates (see supplementary informations of ref.\n[34]). This value tells us that the sample was far from\nthe magnetic saturation but above the level of magneti-\nzation at which the magnetic-\feld dependence of the SSE\ndeviates from the bulk magnetization curve [37, 38], as\nreported by other experiments performed on bulk sam-\nples [19, 20, 23, 39{41].\nSpatial resolution of the spin Seebeck image\nFinally, we comment on the resolution of the local SSE\nmeasurement technique. We consider the radius of the\nhot spot (r0in in Eq. 2) as a more realistic limit to the\nresolution, compared to the cross-section of the probe.\nStarting from the thermodynamic description of the SSE,\nwe have derived the expression of the spin Seebeck volt-\nage di\u000berence as function of the heat current injected by\nthe heated AFM probe. We also highlighted the need to\nrescale this expression according to the geometry of the\nexperiment, as represented by Eq. 3. By replacing the\nexpression of the spin Seebeck voltage \u0001 VSSE(Eq. 2) in-\nside the expression of the rescaled signal we could derive\nthe following representation of the experimental results:\n\u0001VSSE,exp\n\u0001T=4r0\nLz\u0012SH\u0010\u0016B\ne\u0011vYIGlYIG\u000fYIG\nvp1\ntPt(5)\nwhere the parameters that represent the properties of\nYIG have been chosen according to the semi-in\fnite ap-proximation where the thickness of the YIG is larger than\nthat of the magnon di\u000busion length and Lz= 150\u0016m is\nthe width of the Pt strip. In Eq. 5 the heat current Iq\nthat appears in Eq. 2 has been written as the tempera-\nture di\u000berence \u0001 T, by knowing the thermal conductivity\n\u0014and the geometrical constrains on the YIG layer. By\nusing the experimental data from a previous spin Seebeck\nstudy on the same sample [34], we can use the following\nexperimental value for the YIG.\nvYIGlYIG\u000fYIG\nvp=\u00004:6\u000210\u000010(V/K)(m/s)\u00001(6)\nThe spin Hall angle \u0012SH=\u00000:1 refers to the current\nof magnetic moments which has the opposite sign with\nrespect to a de\fnition of the spin Hall angle based on\nthe spin current [34]. By replacing the value of the spin\nSeebeck coe\u000ecient obtained ( CSSE= 2\u000110\u00008VK\u00001) in\nplace of \u0001VSSE,exp=\u0001Tin Eq. 5, we obtained a value of\nr0= 1:4\u0016m. This value of r0is in reasonable agreement\nwith the lateral point-spread function of a heating laser\npresented by Bartell et. al. [28] with a FWHM of the hot\nspot of 0:606\u0016m, obtained with an optical laser power\nof 0:6 mW.\nCONCLUSIONS\nIn summary, we have employed a scanning probe mi-\ncroscopy technique to locally inject heat currents in a\nPt/YIG bilayer structure. We observed a spatially-\nresolved voltage response dependent on the location of\nthe heated probe and the local magnetisation state which\nwe unambiguously attribute to the SSE. This allows for\nthe \frst time to obtain locally resolved spin Seebeck mea-\nsurements, which we map spatially and compare quali-\ntatively with MFM micrographs obtained on the same\nscanned regions. We have discussed the measured signals\nusing a thermodynamic description in spherical coordi-\nnates. Furthermore, we have derived the spatial resolu-\ntion of the local spin Seebeck measurements which is of\nthe order of few micrometres. Local spin Seebeck imag-\ning represents an innovative tool for the investigation of\nnovel spin caloritronic materials. In particular, it pro-\nvides a signi\fcant step forward for the analysis of bulk\nmagnetic structures, compared to surface characteriza-\ntion techniques, as the signal originates from the bulk\nat a distance that can be considered equivalent to the\nmagnon di\u000busion length. Additionally, this experimental\ntechnique allows us to image the magnetisation structure\nof samples where tip-sample interaction could result in ir-\nreversible changes of the sample state during the imaging\nprocess, thus providing a non-perturbative imaging tool.\nMoreover, this technique could pave the way to new con-\ncepts of scanning probe microscopy, inspired by spintron-\nics and spin-caloritronics. For example, the development\nof V-shaped Pt probes as a point-contact ISHE detector,8\nor the scanning thermal microscopy (SThM) as a probe\nfor spatial magnetic imaging using the spin Peltier e\u000bect.\nThe authors thank the EMRP Joint Research Projects\n15SIB06 NanoMag for \fnancial support. 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Physical Review B , 92(6):064413,\n2015."    },    {        "title": "2203.10566v2.Deposition_temperature_dependence_of_thermo_spin_and_magneto_thermoelectric_conversion_in_Co__2_MnGa_films_on_Y__3_Fe__5_O___12___and_Gd__3_Ga__5_O___12__.pdf",        "content": "1 Deposition  temperature dependence of thermo -spin and magneto -\nthermoelectric conversion in Co 2MnGa films on Y 3Fe5O12 and Gd 3Ga 5O12 \n \nHayato Mizuno1,a), Rajkumar Modak1, Takamasa Hirai1, Atsushi Takahagi2, Yuya Sakuraba1, \nRyo Iguchi1, Ken -ichi Uchida1,3,4,a)  \n \nAFFILIATIONS  \n1National Institute for Materials Science, Tsukuba 305 -0047, Japan  \n2Department of Mechanical Systems Engineering, Nagoya University, Nagoya 464 -8601, Japan  \n3Institute for  Materials Research, Tohoku University, Sendai 980 -8577, Japan  \n4Center for Spintronics Research Network, Tohoku University, Sendai 980 -8577, Japan  \na)Author to whom correspondence should be addressed: MIZUNO.Hayato@nims.go.jp and \nUCHIDA.Kenichi@nims.go.jp  \n \nABSTRACT   \nWe have characterized Co2MnGa (CMG) Heusler alloy films grown on Y3Fe5O12 (YIG) and \nGd3Ga5O12 (GGG) substrates at different deposition  temperatures  and investigated thermo -spin \nand magneto -thermoelectric conversion properties by means of a lock-in thermography \ntechnique . X-ray diffraction, magnetization, and electrical transport measurements show that \nthe deposition at high substrate  temperatures  induces the crystallized  structure s of CMG  while \nthe resistivity  of the CMG  films on YIG (GGG) p repared at and above 500 °C (550 °C) becomes \ntoo high to measure the thermo -spin and magneto -thermoelectric effects due to large roughness , \nhighlighting the difficulty of fabricating  highly ordered continuous CMG films on garnet \nstructures . Our lock -in thermography measurements show that the deposition at high substrate  \ntemperatures  results in an increase  in the current -induced temperature change for CMG/GGG  \nand a decrease in  that for CMG/YIG. The former  indicates  the enhancement of the anom alous \nEttingshausen effect in CMG  through  crystallization . The latter can be explained by the \nsuperposition of the anomalous Ettingshausen effect and  the spin Peltier  effect  induced by  the \npositive (negative) charge -to-spin conversion for the amorphous (crystal lized) CMG  films . \nThese results provide a hint to construct spin -caloritronic devices based on Heusler alloys.  \n  2  Magnetic Heusler alloys have been widely studied as promising spintronic and spin -\ncaloritronic materials because they may show half me tallic1,2 or topological band structure .3 \nFor instance,  Co2MnGa (CMG) with the L21 ordered phase is known to exhibit the large \nanomalous Hall and anomalous Nernst effects due to its topological nature .4,5,6,7 Recent studies \nshow that the CMG films with the L21 and B2 ordered phases also exhibit the large charge -spin \ncurrent conversion efficiency .8,9 This finding suggests that CMG would be useful  for detecting \nand driving the thermo -spin effects such as the spin Seebeck10,11 and spin Peltier effects \n(SPE s).12,13,14 However, the spin -caloritronic effects in CMG have not been investigated except \nfor the anomalous Nernst effect so far.  \n The SPE refers to the conversion of a spin current into a heat current in \nmetal/ magnetic -material  junction system s. In combination with  the charge -to-spin current \nconversion  in the metal layer , e.g., the spin Hall effect  (SHE) ,15 the SPE is served as transverse \nthermoelectric conversion as follows. When  the charge current is applied  to the metal layer, it \ninduces  a transverse conduction electron  spin current  via the SHE  and the conduction electron \nspin current  is transformed into a magnon  spin current in  the attached magnetic layer  via the \nspin-mixing conductance.  The magnon spin current is  eventually converted to a heat current  \nand resultant temperature change via the SPE. Here , the directions of the input charge current \nand output heat current are perpendicular to each other . As shown in Fig s. 1(a) and 1(b), in th e \nin-plane magnetized configuration, the symmetry of the charge -to-heat current conversion due \nto the SHE -driven SPE is the same as that due to the anomalous Ettingshausen effect  (AEE) ,16 \nwhich is the Onsager reciprocal of the anomalous Nernst effect , in conductive ferromagnets . \nHowever, the SPE can be separated from the AEE by using a ferrimagnetic insulator, e.g., \nY3Fe5O12 (YIG),  as the magnetic layer.  Although the SPE ha s mainly been investigated in \nparamagnetic -metal  (e.g., Pt) /YIG junction systems  so far, the SPE -induced temperature change \ncan appear also in ferromagnetic -metal/YIG junction systems because ferromagnetic metals \nexhibit the charge -to-spin conversion  [see Fig. 1 (c)].17,18,19 The quantitative separation of the \nSPE from the AEE in the ferromagnetic -metal/YIG systems and the improvement of transverse \nthermoelectric conversion performance by the hybrid action of the SPE and AEE are important \nissues in spin caloritronics.  \nIn this study, to clarify the potential of CMG as a spin -caloritroni c material,  we have \ngrown CMG films on YIG and Gd 3Ga5O12 (GGG) substrates at different deposition  \ntemperatures  and characterized the SPE and AEE  by means of the lock -in thermography \ntechnique13,20,21 as well as the structural, magnetic, and electrical transport properties . Here, \nGGG is employed as a reference substrate, which has the close lattice constant to and the same 3 crystal structure as YIG. Since GGG is a paramagnetic insulator , it allows us to measure  the \nAEE  contribution  in CMG , which is expected to be large because of the large  anomalous  Nernst \neffect of  CMG ,4,6,7 free from the SPE contribution . This systematic data set will provide a hint \nfor future studies on spin-conversion and spin -caloritronic phenomena  in ferromagnetic Heusler \nalloys , leading to the development of  the spintronic thermal management .22 \nThe sample systems used in this study are the 20-nm-thick CMG films deposited on single -\ncrystalline YIG (111) and GGG (111) substrates  by DC magnetron sputtering. We chose 20 nm \nbecause it is the reported minimum thickness of the ordered phases  of CMG .23 The YIG \nsubstrate consists of a 26-μm-thick YIG layer  grown on a 0.5-mm-thick single -crystalline  GGG \n(111) substrate by a liquid  phase epitaxy method .13 The lattice constant of CMG (YIG and \nGGG) is 5.8 (12.4) Å.4,24 The YIG surface is mechanically polished with Al slurry . The \ndimension  of the YIG and GGG  substrates is 5  × 10 mm2. The CMG films were sputtered from \na Co 41.2Mn 27.5Ga31.3 source with a base pressure of  < 2.0×10−6Pa, working pressure of 0.4 Pa \nin Ar atmosphere , sputtering power  of 50 W, and the deposition rate of 0.06 nm/s. The substrate \ntemperature ( 𝑇sub) during the deposition varied from room temperature  to 550  ℃. Before \ndeposition, the  YIG and GGG  substrates were heated  at 600 ℃ for 20 min  to achieve a clean \nsurface as previous studies performed for MgO substrate s25,26 and then down to 𝑇sub in the \nsputtering chamber. To avoid oxidation, the CMG layers were covered b y the 2-nm-thick Al \ncapping layer s without breaking the vacuum , deposited  at room temperature . The composition \nof the CMG films was  determined to be Co 45.8Mn 26.2Ga28.0 by x-ray fluorescence analysis  (note \nthat the composition of the sputtering source is intentionally tuned to obtain the optimized \ncomposition of the deposited films7). As a reference sample, we deposited the 20-nm-thick Pt \nfilm on the same YIG substrate at room temperature . The structural properties  of the CMG films \nwere characterized  by out -of-plane x -ray diffraction . The magnetic properties were measured \nusing a vibrating sample magnetometer. For electrical transport measurements, the films were  \npatterned into Hall bars using photo lithography and Ar -ion milling process es. The surface \nmorphology of the films was evaluated by atomic force microscopy.  The SPE and AEE were \nmeasured by means of the lock-in thermography method by patt erning the CMG and Pt films \ninto a U shape , the line width of which is 0.2 mm  with the same proce ss as the Hall bars . The \nprocedures of the lock -in thermography measurements are the same as those shown in Ref. 13 . \nAll measurements were performed at room temperature . \n Figures 2 (a) and 2(b)  show the x-ray diffraction patterns of the CMG/YIG and \nCMG/GGG  systems, respectively . No clear peak of CMG appeared in the film s deposited at \nroom temperature , suggesting the ir amorphous structure . In contrast, the weak peaks of the 4 CMG (220) lattice plane were observed for the high 𝑇sub deposited  films. We confirme d that \nthe 220 fundamental diffraction peak showed a ring shape in 2D scan , as shown in the inset of \nFig. 2(b), indicating the polycrystalline structure with most fundamental A2 structure , where \nthe A2 phase indicate s the completely disorder ed structure among Co, Mn, and Ga . Although \nthe presences of the B2 (partially disordered structure between Mn and Ga  while Co atoms \noccupy regular sites ) and L21 (fully ordered structure) phases can be recognized from 002 and \n111 superlattice peaks,  respe ctively,  these peaks were too weak  to be detected within the margin \nof experimental errors  (note that the 002 peak was too weak even in the in -plane x -ray \ndiffraction measurements) . Therefore, the x-ray diffraction results  observed here  suggest the \npresence of polycrystalline structure s but do not identify the ordered  phase s in the high 𝑇sub \ndeposited  CMG films on YIG and GGG.  \nFigure 2(c) shows the in -plane magnetic field H dependence of the magnetization M \nof the CMG films on the GGG s ubstrates . We found that  M saturate s for 𝜇0H > 0.04 T, where \n𝜇0 is the vacuum permeability . The saturation magnetization 𝑀s of the high 𝑇sub deposited  \nfilms was observed to be larger than that of the as -deposited ( 𝑇sub = room temperature ) film.  \nThe as -deposited film shows 𝜇0𝑀s ~ 0.14 T, which is much smaller than 𝜇0𝑀s of the ordered \nCMG but in agreement with the previous reports .27,28 The 𝜇0𝑀s values for our high 𝑇sub \ndeposited  CMG films were  estimated to be ~ 0.45 T, which is smaller than the reported value s \nof the A2 phase ( ~ 0.54 T )27 and B2 ordered phase ( ~ 0.63 T )9,28 of CMG films. Figure 2(d) \nsummarizes  the measured  and reference  𝑀s value s.8,9,2 7,28,29 This result suggests that our high \n𝑇sub deposited  CMG films consist of the mixture of the crystal lized and amorphous structures.  \n We focus on the  effect of 𝑇sub on electrical transport properties of the CMG films. \nFigure 2( e) shows the longitudinal electrical resistivity  𝜌𝑥𝑥 as a function of 𝑇sub. The 𝜌𝑥𝑥 \nvalues of the CMG/YIG systems are larger than those of the  CMG/GGG systems and their 𝑇sub \ndependence  is different from each other. T he resistivity  of the CMG  films on the YIG (GGG) \nsubstrates  prepared at and above 500 ℃ (550 ℃) was too high to apply a charge current  (note \nthat we prepared  four CMG/YIG sample s at 𝑇sub = 450 ℃ and two of them  exhibit no \nelectrical conduction  while the remaining samples exhibit almost the same value of the \nelectric al conductivity ). Figure 2(f) shows the out -of-plane H dependence of  the transverse \nresistivity 𝜌𝑥𝑦 for the CMG/GGG system s. The 𝜌𝑥𝑦 values show the H-odd dependence and \nits magnitude almost saturate s at ~ 0.7 T for the high 𝑇sub deposited  films. By extrapolating \nthe 𝜌𝑥𝑦 data above 1.0 T to zero field, the anomalous Hall resistivity for our high 𝑇sub \ndeposited  films was estimated to be ~ 13 μΩcm, which is comparable to  the previous ly report ed \nvalues  for B2- and L21-ordered CMG films .28,29 The large anomalous Hall resistivity impl ies 5 that the high 𝑇sub deposited  CMG/GGG films partially include the ordered phases.  In order to \nclarify  the reason for  no electric al conduction for the high 𝑇sub deposited  films , we \ninvestigated the surface morp hology of the samples by atomic force microscopy. We confirmed \nthat the average roughness 𝑅a of the CMG/YIG and CMG/GGG systems monotonically \nincreases with increasing 𝑇sub and 𝑅a of CMG/YIG is larger than that of CMG/GGG at each \n𝑇sub [Figs. 2(g) and 2(h)].  The large 𝑅a values for the samples prepared at  high 𝑇sub result  \nin the discontinuity  of the CMG layers . Although the lattice constant and coefficient of thermal \nexpansion of YIG and GGG are comparable ,24,30,31 the growth of CMG is sensitive  to the \nsurface energy , which determines the wettability of the initial growth of the CMG film ; it is \nmore difficult to obtain ordered CMG films with good electrical conduction when the YIG \nsubstrates are used than when the GGG substrates are used .  \n Next, we show the thermo -spin and thermoelectric conversion properties for the \nCMG/YIG and CMG/GGG systems with finite electrical conduction . During the lock -in \nthermography measurements, t he square -wave -modulated AC charge  current 𝐽c with the \nfrequency f = 5 Hz and zero DC offset and in -plane H were applied to the U-shaped CMG films , \nwhich is a standard condition for measuring the SPE and AEE .13,16 By extract ing the first-\nharmonic response of the current -induced temperature change , we can obtai n lock -in amplitude \nA and phase 𝜙 images  respectively showing the amplitude and sign of the thermo -spin and/or \nthermo electric  conversion . To extract the pure SPE and AEE signals , we performed the lock -in \nthermography measurements with applying positive and negative H and calculated the \ntemperature modulation with the H-odd dependence :32,33 𝐴odd𝑒−𝑖𝜙𝑜𝑑𝑑=(𝐴(+𝐻)𝑒−𝑖𝜙(+𝐻)−\n𝐴(−𝐻)𝑒−𝑖𝜙(−𝐻))/2, where the background due to the H-independent Peltier effect is excluded . \nThe 𝐴odd and 𝜙odd image s for the  as-deposited CMG/YIG film  are shown in Figs. 3( a) and \n3(b), respectively,  where 𝐽c = 3.0 mA and 𝜇0|𝐻| = 0.32 T. We found  that the clear  current -\ninduced temperature change appeared on the areas L and R of  the CMG layer , where 𝐉c⊥ 𝐇, \nwhile no signal appea red in the area with 𝐉c∣∣𝐇. As shown in Fig. 3( b), 𝜙odd in L (R) is 0° \n(180°), indicat ing that the sign of the temperature modulation is reversed by reversing the 𝐉c \ndirection . Figure 3( c) [Fig. 3(d)] show s the increases  of 𝐴odd in proportion to 𝐽c for all the \nCMG/YIG (CMG/GGG ) systems for various values of 𝑇sub. Figures 3( e) and 3( f) show the \n𝜇0|𝐻| dependence of 𝐴odd in the CMG/YIG and CMG/GGG systems , respectively.  The \nmagnitude of 𝐴odd is almost constant for 𝜇0|𝐻|  > 0.1 T, where M of the YIG and CMG layers \nis aligned along the H direction . These results  are consistent with th e features  of the temperature \nchange induced by the SPE and AEE . We confirmed that all the samples exhibit the current -\ninduced temperature modulation with the same sign.  6 In Fig. 4, w e summarize the amplitude of the temperature modulation per unit charge \ncurrent density, 𝐴odd/𝑗c, where 𝑗c=𝐽c/(𝑡𝑤) with t (w) being the thickness ( line width) of the \nU-shaped CMG films . We also estimated 𝐴odd/𝑗c for the  conventional Pt/YIG system  to be \n3.7 × 10-13 Km2A-1, which is in good agreement with the previous reports13,34 [see the star data \npoint in Fig. 4(a)] . We found that, for the CMG/YIG systems, 𝐴odd/𝑗c in the as -deposited \nsample  is 2.6 times larger than that in Pt/YIG and its magnitude is decrease d in the  high 𝑇sub \ndeposited samples [ Fig. 4(a)] . In contrast, for the CMG/GGG systems, 𝐴odd/𝑗c is enhanced in \nthe high 𝑇sub deposited sample s compared with the as -deposited sample [Fig. 4(b)]. For the \nas-deposited ( high 𝑇sub deposited ) samples, 𝐴odd/𝑗c in the CMG/YIG systems is larger \n(smaller) than that in the CMG/GGG systems. The striking contrast in the current -induced \ntemperature change between the CMG/YIG and CMG/GGG systems cannot be explained only \nby the AEE in the CMG layers, suggesting the coexistence of the SPE in the CMG/YIG systems.  \nFinally, we discuss the origin of t he difference of 𝐴odd/𝑗c between the room \ntemperature and high 𝑇sub deposited films.  In the CMG/GGG systems, t he SPE does not \ncontribute to the temperature modulation signals  because GGG does not transport a spin current \nat room temperature .35 Therefore, the  signals for the CMG/GGG systems are due purely to the \nAEE in CMG. Since  the deposition at high 𝑇sub induces the crystal lized structures of CMG , \nthe increase in  𝐴odd/𝑗c between the CMG/GGG systems  deposited at and above room \ntemperature [Fig. 4(b)]  indicates  the e nhancement of the AEE in CMG through crystallization . \nWe also note that  the magnitude of 𝐴odd/𝑗c is almost constant within the margin of \nexperimental errors  in the high 𝑇sub deposited  CMG/GGG systems , suggesting that the degree \nof ordering could be unchanged in the  𝑇sub range , as suggested by almost constant 𝑀s and \n𝜌𝑥𝑦 [Figs. 2(c) and 2(f) ]. A similar trend has been reported in Ref. 36, where higher ordered \nphase exhibits a larger anomalous Nernst effect in a ferromagnetic Heusler alloy. Although the \nbehaviors of the AEE in the CMG/YIG systems are expected to be similar to those in the \nCMG/GGG systems,  the 𝐴odd/𝑗c signal in CMG/YIG  is much larger than that in CMG/GGG \nin the as -deposited case and  is decre ased in the  high 𝑇sub deposited films [Fig. 4(a)] . This \nbehavior  can be explained  by assuming that  the SPE due to the charge -to-spin current \nconversion in the CMG layers in the as -deposited ( high 𝑇sub deposited ) CMG/YIG systems \nmakes an additive (subtractive) contribution to the AEE. This assumption indicate s that the  spin \nHall angle of the amorphous ( crystal lized) CMG films is positive (negative) because the sign \nof the temperature modulation observed here is the same as that for the Pt/ YIG systems with \nthe positive spin Hall angle of Pt . Our scenario is consistent with the previous studies,  which \nclarified the negative  spin Hall angle  of the B2-L21-mixed and B2-ordered CMG films .8,9 7 Although the SHE -driven SPE and AEE contributions in our crystal lized CMG /YIG systems  \ncancel each other  out, the replac ement  of YIG by other ferrimagnets with the magnetization \ncompensation can tune the sign of the SHE -driven SPE , enabling the enhanced thermoelectric \nconversion based on the hybrid action of the SPE and AEE .37,38 The quantitative estimation of \nthe spin Hall angle of the amorphous and crystal lized CMG films on YIG remains to be \nperformed. Nevertheless, our experiments reveal the fundamental behaviors of the SHE -driven \nSPE in the CMG/YIG systems a nd provide a possibility of tuning the sign and magnitude of \nthe spin Hall angle of Heusler alloys through their crystallization . \n In conclusion , we investigated the SPE and AEE in the CMG/YIG and CMG/GGG \nsystems with the CMG films sputtered at different deposition  temperatures . The CMG films \ngrown on YIG at 500 and 550 °C and on GGG at 550 °C exhibit no electrical conduction  due \nto the large roughness . These results indicate the difficulty of fabricating highly ordered \ncontinuous CMG films on garnet subst rates . Further  optimization of substrate and deposition \nconditions  remains future work . Our lock-in thermography measurement s show  that the \ntemperature change induced by the AEE in CMG is enhanced by the deposition at high 𝑇sub \nthrough crystallization  and that the SPE induces the additive (subtractive) temperature change  \ndue to the positive (negative) spin Hall angle of  the amorphous ( crystal lized) CMG films. The \nsystematic data set reported here provide s a guideline for future studies of spin  caloritro nics \nusing magnetic Heusler alloy s. \n \nThe authors thank H. Nagano for valuable discussions and M. Isomura , N. Kojima, and R. \nTateishi  for technical supports. This work was supported by the CREST \"Creation of Innovative \nCore Technologies for Nano -enabled Thermal Management\" (No. JPMJCR17I1) from JST, \nJapan; Grant -in-Aid for Scientific Research (S) (No. 18H05246) from JSPS KAKENHI, Japan; \nand the NEC Corporation.  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Schematics of (a) the SPE in the Pt/YIG system, (b) the AEE in the CMG/GGG system, and \n(c) the hybrid thermoelectric conversion based on the SPE and AEE in the CMG/YIG system. \n𝐌YIG (CMG ), 𝐉c(s), and 𝛁𝑇SPE (AEE ) repre sent the magnetization of YIG (CMG), charge current (spatial \ndirection of the spin current), and  temperature gradient  induced by  the SPE (AEE), respectively.   \n \n  \n11 FIG. 2.  (a) and  (b) X -ray diffraction patterns of the CMG/YIG and CMG/GGG systems prepared at \nvarious values of the substrate temperature 𝑇sub. RT represents room temperature. The peaks \nhighlighted in red arise from CMG (220).  The peaks indicated by * arise from the YIG  and GGG \nsubstrates. The inset shows the 2D scan for the CMG/GGG system prepared at 𝑇sub = 500 ℃. (c) In -\nplane H dependence of the magnetization M of the CMG/GGG systems prepared at various values of \n𝑇sub. The paramagnetism contribut ion from the GGG subs trates was  subtracted by linear fitting of the \nraw data in the field range from 𝜇0|𝐻| = 0.1 to 0.2 T, where 𝜇0 is the vacuum permeability. (d)  𝑇sub \n12 dependence of the saturation magnetization 𝑀s of the CMG  films on the GGG  substrates. The areas \nhighlighted in red show the reference values of the amorpho us27,28, A227, B29,28,29, and  B2-L21-\nmixed8,28,29 structures of CMG films. (e ) 𝑇sub dependence of the longitudinal electrical resistivity \n𝜌𝑥𝑥. (f) Out-of-plane H dependence of the transverse resistivity 𝜌𝑥𝑦 of the CMG  films on the GGG  \nsubstrates. (g ) Atomic force microscope images of the CMG/YIG systems  prepared at room \ntemperature and 450 ℃. (h) 𝑇sub dependence of the surface roughness 𝑅a of the CMG/YIG and \nCMG/GGG systems.  \n  13 FIG. 3. (a) and (b) 𝐴odd and 𝜙odd images for the as -deposited CMG/YIG system  at 𝐽c = 3.0 mA \nand in-plane field  𝜇0|𝐻| = 0.32 T. 𝐴odd (𝜙odd) denotes the lock -in amplitude (phase ) with the H-\nodd dependence. 𝐽c denot es the square -wave amplitude of the AC charge current applied to the  U-\nshaped CMG layer. (c) and (d) 𝐽c dependence of 𝐴odd for the CMG/YIG and CMG/GGG systems \nat 𝜇0|𝐻| = 0.32 T. The solid lines are obtained from the linear fitting. (e) and (f) H dependence  of \n𝐴odd for the CMG/YIG and CMG/GGG systems at 𝐽c = 3.0 mA.  The 𝐴odd data in (c) -(f) are \nextracted from the area R i n (a).   \n14 FIG. 4 . 𝐴odd/𝑗c for the (a) CMG/YIG  and (b) CMG/GGG systems for various values of 𝑇sub. 𝑗c \ndenotes  the charge current density in the CMG layer. The 𝐴odd/𝑗c data are  extracted from the slope \nof the linear fitting  of the data  in Fig s. 3(c) and 3(d). The star data point indicates the 𝐴odd/𝑗c signal \nfor the Pt/YIG system.  \n"    },    {        "title": "1410.0551v2.Simultaneous_detection_of_the_spin_Hall_magnetoresistance_and_the_spin_Seebeck_effect_in_Platinum_and_Tantalum_on_Yttrium_Iron_Garnet.pdf",        "content": "Simultaneous detection of the spin-Hall magnetoresistance and the spin-Seebeck e\u000bect\nin Platinum and Tantalum on Yttrium Iron Garnet\nN. Vlietstra, J. Shan, and B. J. van Wees\nPhysics of Nanodevices, Zernike Institute for Advanced Materials,\nUniversity of Groningen, Groningen, The Netherlands\nM. Isasa\nCIC nanoGUNE, 20018 Donostia-San Sebastian, Basque Country, Spain\nF. Casanova\nCIC nanoGUNE, 20018 Donostia-San Sebastian, Basque Country, Spain and\nIKERBASQUE, 48011 Bilbao, Basque Country, Spain\nJ. Ben Youssef\nLaboratoire de Magn\u0013 etisme de Bretagne, CNRS, Universit\u0013 e de Bretagne Occidentale, Brest, France\n(Dated: September 25, 2018)\nThe spin-Seebeck e\u000bect (SSE) in platinum (Pt) and tantalum (Ta) on yttrium iron garnet (YIG)\nhas been investigated by both externally heating the sample (using an on-chip Pt heater on top of\nthe device) as well as by current-induced heating. For SSE measurements, external heating is the\nmost common method to obtain clear signals. Here we show that also by current-induced heating\nit is possible to directly observe the SSE, separate from the also present spin-Hall magnetoresis-\ntance (SMR) signal, by using a lock-in detection technique. Using this measurement technique, the\npresence of additional 2ndorder signals at low applied magnetic \felds and high heating currents is\nrevealed. These signals are caused by current-induced magnetic \felds (Oersted \felds) generated by\nthe used AC-current, resulting in dynamic SMR signals.\nPACS numbers: 72.25.Mk, 72.80.Sk, 75.70.Tj, 75.76.+j\nI. INTRODUCTION\nFor the investigation of pure spin transport phenom-\nena, yttrium iron garnet (YIG) is shown to be a very\nsuitable candidate. YIG is a ferrimagnetic insulating\nmaterial having a low magnetization damping as well\nas a very low coercive \feld. In combination with a\nhigh spin-orbit coupling material such as platinum (Pt),\nmany di\u000berent experiments have been performed, show-\ning spin-pumping1{4, spin transport5,6and spin-wave\nmanipulation7{9as well as the recently discovered spin-\nHall magnetoresistance (SMR).10{15\nRecently also experiments were performed showing\nthe spin-Seebeck e\u000bect16{19(SSE) as well as the spin-\nPeltier e\u000bect20in YIG/Pt systems. The SSE is ob-\nserved when a temperature gradient is present over a\nferromagnetic/non-magnetic interface. In a YIG/Pt sys-\ntem, this temperature gradient causes the creation of\nthermal magnons, resulting in transfer of angular mo-\nmentum at the YIG/Pt interface, generating a pure spin-\ncurrent into the Pt.19This spin-current can then be de-\ntected electrically via the inverse spin-Hall e\u000bect (ISHE).\nSo far, most experiments on the SSE are performed using\nexternal heating sources to create a temperature gradi-\nent over the device. Interestingly, Schreier et al.21showed\nthat a clear SSE signal can also be extracted from more\neasily performed current-induced heating experiments.\nIn their experiments a temperature gradient is created\nby sending a charge current through the detection strip.A disadvantage of their measurement method is the pres-\nence of a much larger signal originated from the SMR,\nwhich should be subtracted to reveal the SSE signal.\nIn this paper we investigate both the SSE and SMR\nin a YIG-based device, showing the possibility to simul-\ntaneously, but separately, detect the SSE and SMR by\nusing a lock-in detection technique. Whereas Schreier\net al. only performed their measurements applying high\nmagnetic \felds, fully saturating the magnetization of the\nYIG, we show that when lowering the applied magnetic\n\feld, dynamic behavior of the magnetization of the YIG\ncan be picked up as additional 2ndorder signal. Only by\nusing a lock-in detection technique these signals can be\nseparately detected and analyzed. Having platinum (Pt)\nor tantalum (Ta) as detection layer, we investigate the\nevolution of the SSE and the SMR signal as a function\nof the magnitude and the direction of the applied \feld,\nfocusing especially on their low-\feld behavior.\nThe \frst experiments described in this paper show SSE\nmeasurements where a temperature gradient is generated\nby externally heating the sample using a second Pt strip\non top of the device. By using devices consisting of both\nPt and Ta on YIG, we con\frm the opposite sign of the\nspin-Hall angle for Ta and Pt.22,23In the secondly shown\nexperiments, the samples are heated by current-induced\nheating through the metal detection strip, such that both\nthe SSE and SMR are present. Additionally detected 2nd\nharmonic signals for low applied \felds and high heating\ncurrents are discussed and ascribed to dynamic behaviorarXiv:1410.0551v2  [cond-mat.mes-hall]  27 Nov 20142\nof the magnetization of the YIG, caused by the applied\nAC-current. Finally, we derive a dynamic SMR term,\nwhich is used to explain the observed features.\nThe same kind of experiments could as well be used\nfor detection of spin-transfer torque e\u000bects on the mag-\nnetization of the YIG, like the generation of spin-torque\nferromagnetic resonance as formulated by Chiba et al.24.\nHowever, as will be shown in this paper, when apply-\ning low magnetic \felds and high currents, the detected\nmagnetization behavior is dominated by current-induced\nmagnetic \felds (like the Oersted \feld). So, to be able to\ndetect any e\u000bect of the spin-transfer torque, its contribu-\ntion should be increased, for example by decreasing the\nYIG thickness.\nII. SAMPLE CHARACTERISTICS\nFor the experiments shown in this paper, two Hall-bar\nshaped devices have been used, one consisting of a 5nm-\nthick Pt layer and the other of a 10nm-thick Ta layer.\nThe Hall-bars have a length of 500 \u0016m and a width of\n50\u0016m, with side contacts of 10 \u0016m width. Both Hall-bars\nare deposited on top of a 4x4mm2YIG sample, by dc\nsputtering.\nThe used sample consists of a 200nm thick layer of\nYIG, grown by liquid phase epitaxy on a single crystal\n(111)Gd 3Ga4O12(GGG) substrate. The YIG magnetiza-\ntion shows isotropic behavior of the magnetization in the\n\flm plane, with a low coercive \feld of only 0.06mT.4,12\nFor external heating experiments, a Ti/Pt bar of\n5/40nm thick is deposited on top of both Hall-bars, sep-\narated from the main channel by a 80nm-thick insulating\nAl2O3layer. The size of the heater is 400x25 \u0016m2. Fi-\nnally, both Hall-bars and Pt heaters are contacted by\nthick Ti/Au pads [5/150nm]. All structures are pat-\nterned using electron-beam lithography. Before each fab-\nrication step the sample has been cleaned by rinsing it\nin acetone, no further surface treatment has been carried\nout. A microscope image of the device is shown in Fig.\n1(a).\nIII. MEASUREMENT METHODS\nTo observe the SSE, two measurement methods have\nbeen investigated. At \frst, to generate a clear SSE sig-\nnal, a temperature gradient is created using an external\nheating source to heat one side of the sample. In our\ncase, we have a Ti/Pt strip on top of the Hall-bar, elec-\ntrically insulated from the detection channel, which can\nbe used as an external heater. By sending a large cur-\nrent (up to 10mA) through the heater, the strip will be\nheated by Joule heating. Hereby, a temperature gradient\nwill be formed over the YIG/Pt(Ta) stack, giving rise to\nthe SSE.\nA second method to generate the SSE, is the genera-\ntion of a temperature gradient by current-induced heat-\n(b)\n(c)I-B\nV-\nI+V+α0(a)\nxy\n-20 -10 0 10 20-10-8-6-4-20246810 \n \n Pt [5nm]\n Ta [10nm]V2  [µV]\nMagnetic Field [mT]Iheater= 10mA\n       α0 = -90o\n-180 -90 0 90 180-10-8-6-4-20246810\nIheater= 10mA\nB = 50mT\n Pt [5nm]\n Ta [10nm]V2  [µV]\nα0 [ο]FIG. 1. (a) Microscope image of the device structure, con-\nsisting of a Pt or Ta Hall-bar detector (bottom layer) and a\nPt heater (top layer), separated by an insulating Al 2O3layer.\nTi/Au pads are used for contacting the device. For external\nheating experiments, the device is contacted as marked. The\napplied \feld direction is given by \u000b0, as de\fned in the \fg-\nure. (b) 2ndharmonic voltage signal generated by the SSE\nfor a \fxed magnetic \feld direction of \u000b0= 90\u000eand (c) angu-\nlar dependence of the SSE signal applying a magnetic \feld of\n50mT, in both Pt and Ta.\ning through the detection strip. In this case a charge cur-\nrent is sent through the Hall-bar itself, which also leads\nto Joule heating, resulting in a temperature gradient over\nthe YIG/Pt(Ta) stack. As the Hall-bar is directly in con-\ntact with the YIG, also the SMR will be present when\nusing this heating method.\nTo separately detect the SSE and the SMR signals, a\nlock-in detection technique is used. Using up to three\nStanford SR-830 Lock-in ampli\fers, the 1st, 2ndand\nhigher harmonic voltage responses of the system are sep-\narately measured. As SMR scales linearly with the ap-\nplied current, its contribution will be picked up as a 1st\nharmonic signal. Similarly, the SSE scales quadratically\nwith current, so its contribution will be detected as a 2nd\nharmonic signal. For lock-in detection an AC-current is\nused with a frequency of 17Hz. The magnitude of the\napplied AC-currents is de\fned by their rms values.\nEvaluating the working mechanism of the lock-in de-\ntection technique in more detail (see appendix), shows\nthat in order to obtain the linear response signal of the\nsystem, both the measured 1stharmonic signal as well as\nthe 3rdharmonic signal have to be taken into account.\nIncluding both harmonic signals following the analysis\nexplained in the appendix, we \fnd the shown 1storder\nresponse. Note that the measured 2ndharmonic signal is\ndirectly plotted, without any corrections.\nIn all experiments, an external magnetic \feld is applied\nto de\fne the direction of the magnetization M of the YIG.\nThe direction of the applied \feld is de\fned by \u000b0, which\nis the in-plane angle between the current direction (along3\n(a) (d) (c)\n(b)\nI-B\nV-I+V+\nα0\n-6-4-2024642444648\n-6 -4 -2 0 2 4 6-15-10-5051015\n-180-90 0 9018030405060\n-180 -90 0 90 180-10-50510Pt [5nm]\nα0= 0o Iac = 4mA\n Iac = 8mA\n Magnetic Field [mT]\n  R1 [mΩ] \n Iac = 4mA\n Iac = 8mAPt [5nm]\nα0= 0oV2 [µV]\nMagnetic Field [mT]Pt [5nm]\nIac= 8mAB[mT]  0.9  1.8  100\nα0 [ο]\n  R1 [mΩ] \nPt [5nm]\nIac= 8mA\n  V2 [µV]\nα0[ο]B [mT]\n 0.9\n 1.8\n 2.8\n 8.5\n100\nFIG. 2. Current-induced heating experiments on the YIG/Pt sample. (a) Magnetic \feld dependence of the 1storder resistance\nfor\u000b0= 0\u000e, showing the SMR signal for applied AC-currents of 4mA and 8mA. (b) Angular dependence of the 1storder SMR\nsignal for I ac=8mA, for applied magnetic \felds of 0.9mT, 1.8mT and 100mT. (c) and (d) show the corresponding 2ndharmonic\nvoltage signals, respectively. For applied magnetic \felds above 10mT, the 2ndharmonic response only shows the SSE signal.\nFor low applied magnetic \felds an additional signal is observed on top of the SSE signal. The black symbols in both \fgures\nare a guide for the eye. They show equal measurement conditions comparing the results shown in both \fgures. The inset of\n(c) shows the used measurement con\fguration.\nx) and the applied \feld direction, as it is marked in Fig.\n1(a). Not only experiments at high saturation magnetic\n\felds are performed, also the low \feld behavior is investi-\ngated. The applied magnetic \feld strength was measured\nby a LakeShore Gaussmeter (model 421) using a trans-\nverse Hall probe, to correct the set magnetic \feld for any\npresent remnant \feld. All measurements are carried out\nat room temperature.\nIV. RESULTS AND DISCUSSION\nA. Spin-Seebeck e\u000bect by external heating\nFor the external heating experiment an AC-current is\nsent through the top Pt strip as marked in Fig. 1(a).\nBy measuring the 2ndharmonic voltage signals along the\nHall-bar, the SSE is detected via the ISHE in Pt and\nTa. Fig. 1(b) shows the typical SSE signals for both\nYIG/Pt and YIG/Ta samples for an applied \feld per-\npendicular to the longitudinal direction of the Hall-bar\n(\u000b0= 90\u000e). Changing the sign of B (and thus M) changes\nthe sign of the signal, as the spin-polarization direction\nof the pumped spin-current is reversed. Due to the low\ncoercive \feld of YIG almost no hysteresis is observed for\nthe reversed \feld sweep. For the YIG/Pt and YIG/Ta\nsample opposite magnetic \feld dependence is observed,\nproving the opposite sign of the spin-Hall angle for Pt\nversus Ta.\nAs the spin-polarization direction of the generated\nspin-current is dependent on the direction of the YIG\nmagnetization, the SSE/ISHE voltage shows a sine\nshaped angular dependence with a period of 360\u000e. By\nrotating the sample in a constant applied magnetic \feldof 50mT, this angular dependence is detected as is shown\nin Fig. 1(c). Also here the e\u000bect of the opposite sign of\nthe spin-Hall angle, for Ta compared to Pt, is clearly\nvisible.\nFrom Fig. 1 it is observed that the SSE signal for\nthe YIG/Ta sample is almost a factor 10 smaller than\nfor the YIG/Pt sample ( VSSE;Pt=VSSE;Ta =\u00009:8). To\ncompare, we calculate the expected ratio from the the-\noretical description of the SSE voltage, as reported by\nSchreier et al.17:\nVSSE=CYIG\u0001\u0001TmeGr\u0002SH\u001al\u0011\u0015\nttanh\u0012t\n2\u0015\u0013\n(1)\nwhereCYIG contains all parameters describing proper-\nties of YIG, including some physical constants (de\fned\nin ref.17), soCYIGis constant for both the YIG/Pt and\nthe YIG/Ta sample. \u0001 Tmeis the temperature di\u000berence\nbetween the magnons and electrons at the YIG/metal\ninterface.\u001a,\u0015,tandlare the resistivity, spin-di\u000busion\nlength, thickness of the normal metal layer (Pt/Ta) and\nthe distance between the voltage contacts, respectively.\n\u0011is the back\row correction factor, de\fned as\n\u0011=\u0014\n1 +Gr\u001a\u0015coth\u0012t\n\u0015\u0013\u0015\u00001\n(2)\nPreviously, in ref.14, we have determined the real part\nof the spin-mixing conductance at the YIG/Pt inter-\nface (Gr= 4:4\u00021014\n\u00001m\u00002), the spin-Hall angle\n(\u0002SH;Pt = 0:08) and the spin-di\u000busion length ( \u0015Pt=\n1:2nm) of Pt. For the YIG/Ta sample we take the mag-\nnitude of these system parameters as reported by Hahn\net al.15(Gr= 2\u00021013\n\u00001m\u00002, \u0002SH;Ta =\u00000:02 and\n\u0015Ta= 1:8nm). As a check, we also used these parameter-\nvalues to calculate the 1storder SMR signals for Ta, and4\nfound good agreement with the measured signals (not\nshown).\nTo get an estimate for VSSE;Pt=VSSE;Ta we assume\n\u0001Tmeto be constant for both samples. By inserting the\nvalues of the mentioned parameters, the dimensions of\nthe Hall-bars and the measured resistivity of the Pt and\nTa layers (\u001aPt= 3:4\u000210\u00007\nm and\u001aTa= 3:5\u000210\u00006\nm,\nrespectively), we \fnd VSSE;Pt=VSSE;Ta =\u000010:6, which\nis close to the experimentally observed ratio.\nB. Current-induced spin-Seebeck e\u000bect\nThe second method used to detect the SSE is by\ncurrent-induced heating through the metal detection\nstrip itself, as recently was reported by Schreier et al.21.\nIn this section we show that we can achieve more directly\nsimilar results, by using a lock-in detection technique. By\nthis technique, the SSE signals can directly be detected\nas a 2ndharmonic signal, fully separated from the SMR\nsignal, which shows up in the 1stharmonic response. Fur-\nthermore, the lock-in detection technique enables us to\nreveal and investigate additional signals appearing when\napplying low magnetic \felds.\nThe inset of Fig. 2(c) shows a microscope image of\nthe sample, marking the position of the current and volt-\nage probes for the current-induced heating experiments.\nThe magnetic \feld direction is again de\fned by \u000b0. This\nmeasurement con\fguration is similar to the method used\nto detect transverse SMR12,14and therefore we expect\nto observe SMR in the 1storder signal, as is shown in\nFigs. 2(a) and (b). In Fig. 2(b) it is observed that down\nto very low applied magnetic \felds, the average magne-\ntization direction of the YIG nicely follows the applied\n\feld direction, resulting in the sin(2 \u000b0) angular depen-\ndence of the SMR.11Only for the lowest applied \feld of\n0.9mT a small deviation of the signal around \u000b0=\u000690\u000e\nis observed, showing this \feld strength is not su\u000ecient\nto assume M being (on average) fully along the applied\n\feld direction.\nSimilar to the external heating experiment, the SSE\nsignal shows up in the 2ndharmonic signal. Figs. 2(c)\nand (d) show the magnetic \feld dependence and angular\ndependence of the detected 2ndharmonic signal, respec-\ntively. Comparing the shape of the 2ndharmonic data\nof the external heating experiments (Fig. 1(b)) to the\ncurrent-induced heating experiments (Fig. 2(c)), an en-\nhanced signal is observed in Fig. 2(c) for \felds of a few\nmT. This additional signal cannot be explained by the\nangular dependence of the SSE, neither by the rotation\nof M in the plane towards B (by which the 1stharmonic\nSMR peaks in Fig. 2(a) are explained12).\nThe angular dependence of this additional signal, as\npresented in Fig. 2(d), shows that besides an increased\namplitude of the SSE signal (black symbols in Figs. 2(c)\nand (d)), also at \u000b0=\u000690\u000eadditional peaks appear for\nlow applied \felds. By increasing the applied magnetic\n\feld, all extra signals disappear, leaving the expected\n(c)(a)\n(b)-180 -90 0 90 180-10-50510\n0102030405060-10-50510Iac (mA)\n 1\n 2\n 4\n 6\n 8Pt [5nm]\nB = 0.9mT\n  V2 (µV)\nα0(deg)\nPt [5nm]\n  V2 (µV)\nI2\nac (mA2)50mT (α0=0o)\n0.9mT (α0=0o)\n0.9mT (α0=90o)\n-180-90 0 90180-10-50510Iac (mA)  1  4  8\n              2  6  \nPt [5nm]\nB = 50mT\n  V2 (µV)\nα0(deg)FIG. 3. AC-current dependence of the 2ndharmonic volt-\nage for an applied \feld of (a) 0.9mT and (b) 50mT. For\nthe shown experiments, the transverse current-induced heat-\ning measurement con\fguration has been used. The verti-\ncal dashed lines mark the data plotted in (c), which shows\nthe AC-current dependence of the magnitude of the signal\nat\u000b0= 0\u000efor B=0.9mT (red dots) and B=50mT (black\nsquares) and the average magnitude of the peaks (peak to\npeak) around \u000b0=\u000690\u000efor B=0.9mT (blue triangles). The\ndashed lines are a guide for the eye.\nSSE signal showing a 360\u000eperiodic angular dependence.\nTo further characterize the additionally observed fea-\ntures at low applied magnetic \felds, also their AC-\ncurrent dependence has been measured and these results\nare shown in Fig. 3. It can be seen that the current\ndependence is very similar to the shown magnetic \feld\ndependence, giving maximal additional signals for low\napplied \felds and high applied AC-currents. From Figs.\n3(a) and (b), the magnitude of the signal at \u000b0= 0\u000eis\nextracted and plotted separately in Fig. 3(c). As can\nbe seen from this \fgure, for both the applied magnetic\n\feld of 0.9mT (Fig. 3(a)) and 50mT (Fig. 3(b)), the\namplitude of the signal quadratically scales with the ap-\nplied AC-current. The magnitude of the peaks around\n\u000b0=\u000690\u000e, plotted in blue in Fig. 3(c), increases faster\nthan quadratically, pointing to the presence of higher or-\nder e\u000bects.\nTo fully exclude the SSE being the origin of the ad-\nditionally detected signals, the current-induced heat-\ning measurements were repeated on the YIG/Ta sam-\nple. Results of those measurements are shown in Fig.\n4. The applied current in those experiments is only5\n1.9mA, limited by the high resistance of the Ta bar\n(\u001aTa= 3:5\u000210\u00006\nm). In both Fig. 4(a) and (b)\nit can be seen that the high-\feld signal nicely changes\nsign compared to the YIG/Pt data, as predicted for the\nSSE/ISHE, because of the opposite sign of the spin-Hall\nangle of Ta compared to Pt. Contrary, the low-\feld peak\nin the magnetic \feld sweep (Fig. 4(a)) keeps the same\nsign as in the YIG/Pt sample (Fig. 2(c)), showing the\nSSE cannot be the origin of this phenomenon. Further-\nmore, this result also excludes the observed feature being\noriginated by any other e\u000bect linearly related to the spin-\nHall angle of a material. So, possible deviations of M\ncaused by spin-transfer torque, due to a spin-current cre-\nated via the SHE, cannot directly be used to explain the\nobserved features. Note that the SMR signal depends\nquadratically on the spin-Hall angle,10,14which makes\nany e\u000bect related to the SMR a likely candidate for ex-\nplaining the observed features.\nSummarizing, the current-induced heating experi-\nments show that when applying a su\u000eciently high mag-\nnetic \feld ( >10mT), the SMR and SSE can be simul-\ntaneously, but separately, detected using an AC-current\ncombined with a lock-in detection technique. By this\nmethod the SSE can thus be very easily and directly de-\ntected, without being interfered with the SMR signal.\nFurthermore, it is observed that for low magnetic \felds,\nand/or high heating currents, additional signals appear\non top of the SSE. The origin of these additional sig-\nnals might be related to the SMR-e\u000bect, which will be\ndiscussed in more detail in the next section.\nC. Dynamic Spin-Hall Magnetoresistance\nDuring the measurements, large AC-currents are sent\nthrough the Hall-bar structure, which can generate mag-\nnetic \felds. One source of these magnetic \felds will\nbe Oersted \felds (B oe) generated around the Hall-bar,\nperpendicular to the current direction. As the Hall-bar\n(b) (a)\n-6-4-2 0 2 4 6-0.050.000.050.100.150.200.25\n-180 -90 0 90 180-0.050.000.050.100.150.200.25  V2 (µV)\nMagnetic Field [mT]Ta [10nm]\nIac= 1.9mA\nα0= 0o\nV2 (µV)\n 50\n 0.9Ta [10nm]\nIac= 1.9mA\n  \nα0(deg) B (mT)\nFIG. 4. 2ndharmonic response of the transverse voltage for\nthe YIG/Ta sample. (a) Magnetic \feld sweep for \u000b0= 0\u000e\nand (b) angular dependence for two di\u000berent applied mag-\nnetic \felds (0.9mT and 50mT). The SSE signal (=signal at\nhigh \feld) has opposite sign compared to the YIG/Pt sam-\nple, whereas the low-\feld behavior is similar for both samples.\nThe black symbols in the \fgures point out the equal measure-\nment conditions comparing both \fgures.structure is very thin compared to its lateral dimen-\nsions, the Oersted \feld above/below the center of the\nbar can be estimated using the in\fnite plane approxima-\ntion:Boe=\u00160I\n2w, where\u00160is the permeability in vacuum,\nIis the applied current and wis the width of the Pt bar.\nNote that the generated \feld in this case is independent\nof the distance from the plane, so the full thickness of the\nYIG below the Hall-bar will be exposed to this \feld.25\nFor example, for an applied current of 8mA, an Oersted\n\feld of 0.1mT will be generated, which is signi\fcant com-\npared to an applied magnetic \feld of 0.9mT. Therefore,\nfor low applied \felds, the magnetization direction of the\nYIG will also be a\u000bected by the generated Oersted \feld.\nAs we are dealing with AC-currents, the generated\nOersted \feld (and any other current-induced magnetic\n\feld) will continuously change sign, which might cause\nM to oscillate around the applied \feld direction. In\nthis case,\u000b(de\fning the direction of M) is current-\ndependent,26giving rise to dynamic equations for both\nthe SMR and the SSE. The current-dependent behavior\nof the SMR signal is derived starting from the equation\nfor transverse SMR10,14\nVT;SMR =IRT;SMR = \u0001R1Imxmy+ \u0001R2Imz(3)\nwhere \u0001R1and \u0001R2are resistance changes dependent\non the spin-di\u000busion length, spin-Hall angle and spin-\nmixing conductance of the system, as de\fned in refs.10,14.\nmx;y;z are the components of M pointing in respectively\nthe x-, y-, and z-direction (where z is the out-of-plane\ndirection). mxandmycan be expressed as sin( \u000b) and\ncos(\u000b), respectively. As the applied magnetic \feld is in-\nplane, as well as the generated Oersted \feld, combined\nwith the large demagnetization \feld of YIG for out-of-\nplane directions, mzwill be small and therefore neglected\nin further derivations.\nSmall oscillations of M, due to the presence of AC-\ncurrent generated magnetic \felds, result in a current-\ndependent SMR signal, which can be expressed in \frst\norder as\nVT;SMR (I)\u0019VT;SMR (\u000b0) +IdVT;SMR\ndI\f\f\f\f\n\u000b0(4)\nwhere\u000b0gives the equilibrium direction of M around\nwhich it is oscillating (assuming it to be equal to the\napplied \feld direction, as concluded from the measured\n1stharmonic SMR response). Calculating the derivative\nin Eq.(4), using Eq.(3), neglecting mzand keeping in\nmind that\u000bis dependent on I, gives\ndVT;SMR\ndI\f\f\f\f\n\u000b0= \u0001R1sin(\u000b0) cos(\u000b0)+\u0001R1Icos(2\u000b0)d\u000b\ndI\n(5)\nThe \frst term on the right side of Eq.(5) describes the\n1storder response (linear with I), showing the expected\ntransverse SMR behavior (Eq.(3)). The second term is a\n2ndorder response ( R2;SMR ) and will therefore show up\nin addition to the expected SSE signal.d\u000b\ndIis the term6\nwhich includes the deviation of M due to current-induced\nmagnetic \felds, and its magnitude is dependent on both\n\u000b0and the magnitude of the total magnetic \feld (applied\n\feld, coercive \feld and the current-induced \felds). For\nlarge applied magnetic \felds, the current-induced mag-\nnetic \felds will have a negligible e\u000bect on M, sod\u000b\ndIgoes\nto zero, leaving only the SSE signal in the 2ndharmonic\nsignal (as is observed). Note that also in the described\nexternal heating experiments Oersted \felds are gener-\nated, in\ruencing M, but there the dynamic SMR signal\nwill not be detected, as the SMR itself is not present.\nTo \fnd an expression ford\u000b\ndI, \frst the direction of M\n(given by\u000b) is de\fned, taking into account the Oersted\n\felds (causing \u0001 \u000b):\n\u000b=\u000b0+ \u0001\u000b=\u000b0+ atan\u0012Boe\nBex\u0013\ncos(\u000b0) (6)\nwhereBexis the applied magnetic \feld and atan(Boe\nBex)\nis the maximum deviation of M from \u000b0, which is the\ncase for\u000b0= 0\u000e(Bexperpendicular to Boe, neglecting\nany other \feld contributions). From Eq.(6) nowd\u000b\ndI(=\nd\u000b\ndBoedBoe\ndI) can be derived, \fnding\nd\u000b\ndI=\u00160\n2wBex\nB2ex+B2oecos(\u000b0) (7)\nSubstituting the derived equation ford\u000b\ndIin Eq.(5), we\ncalculate the expected 2ndharmonic SMR signal due to\ndynamic behavior of M, caused by the current-induced\nOersted \feld as:\nR2;SMR = \u0001R1\u00160\n2wBex\nB2ex+B2oecos(2\u000b0) cos(\u000b0) (8)\nAdditional to R2;SMR , also the SSE will be present as\na 2ndharmonic signal, showing cos( \u000b0) behavior with an\namplitude independent from the applied magnetic \feld\nstrength. The amplitude of the SSE signal can be derived\nfrom the high-\feld measurements shown in Fig. 3(b) and\nFig. 4(b).\nFigures 5(a) and (b) show the total calculated 2ndhar-\nmonic voltage signal for the YIG/Pt sample, taking into\naccount both the SSE, extracted from the measurements,\nand the dynamic SMR term, given by Eq.(8) (where\nV2;SMR = (I2=p\n2)R2;SMR ). For the calculation of \u0001 R1\nsystem parameters from ref.14are used. Both the cal-\nculated current dependence (Fig. 5(a)) as well as the\ncalculated magnetic \feld dependence (Fig. 5(b)) show\nsimilar features as the measurements, only its magnitude\nand exact shape do not fully coincide. Following the ex-\nplanation of the lock-in detection method as described in\nthe appendix, we \fnd that these discrepancies are mainly\ncaused by the presence of a non-negligible 4thharmonic\nsignal (and possibly even higher harmonics). When de-\ntermining the 2ndorder response of the system, taking\ninto account the measured 2ndand 4thharmonic signals,\nthe peaks observed around \u000b0=\u000690\u000eget wider and\n(a) (b)\n(c)-180 -90 0 90 180-20-1001020\nIac [mA]\n 1\n 2\n 4\n 6\n 8Pt [5nm]\nB = 0.9mT\n  V2 [µV]\nα0[ο]-180 -90 0 90 180-20-1001020\nPt [5nm]\nIac = 8mA\n  V2 [µV]\nB [mT]\n 0.9\n 1.8\n 2.8\n 8.5\n 100\nα0[ο]\n-180 -90 0 90 180-0.10-0.050.000.050.10Ta [10nm]\nIac = 1.9mA V2 [µV]B [mT]\n 0.9\n 50\nα0[ο]FIG. 5. (a) Current dependence and (b) magnetic \feld de-\npendence of the calculated 2ndharmonic response, including\nthe dynamic SMR (from Eq.(8)) and the SSE for the YIG/Pt\nsystem. (c) Calculated 2ndharmonic response for the YIG/Ta\nsample using a scaling factor ford\u000b\ndIof 0.5. For all calculations\nthe SSE signal is extracted from the measurements.\nsmoother, more closely reproducing the calculated sig-\nnals as presented in Fig. 5.\nThe amplitude of the overall calculated signal is\nslightly larger than the measurements, up to a factor 1.6\n(even after taking into account the 4thharmonic signal).\nOne reason for this discrepancy might be that, for the cal-\nculations, only the applied magnetic \feld and the gener-\nated Oersted \feld are taken into account, neglecting any\nother present \felds. For example, the coercive \feld of the\nYIG is assumed to be absent, as well as the e\u000bect of spin-\ntorque and the presence of non-uniform magnetic \felds.\nThese additional \felds will in\ruence the amplitude of the\noscillations of M, giving a di\u000berent value ford\u000b\ndIthan the\nassumed deviation from only BoeandBex. Furthermore,\nthe assumed perfect cos( \u000b0) behavior ofd\u000b\ndImight also\nbe disturbed by the presence of these other \felds. Sec-\nondly, in the calculations it is assumed that M is fully\naligned with the total magnetic \feld, which might not\nalways be the case, as we investigate the system apply-\ning magnetic \felds approaching the coercive \feld of the\nYIG. Therefore, the de\fnition of \u000b0can be slightly o\u000b\nfrom the assumed ideal case. Full characterization of the\nmagnetization dynamics of the system and the magnetic\n\feld distribution would be needed to be able to give a\nmore complete theoretical analysis of the observed fea-\ntures.\nThe same calculations have been repeated for the\nYIG/Ta system. The system parameters needed to calcu-\nlate \u0001R1are taken from ref.15, as given in section IV A.\nAs a check, these system parameters \frstly were used to\ncalculate the 1storder SMR signal, \fnding good agree-\nment with the measured signals (not shown). For the\ncalculated 2ndharmonic signal again it is found that the\namplitude ofd\u000b\ndIhas to be lowered to be able to reproduce\nthe measured behavior. When lowering the calculatedd\u000b\ndI7\nby a chosen scaling factor of 0.5, results as shown in Fig.\n5(c) are obtained. Comparing the calculated angular de-\npendence to the measurement as shown in Fig. 4(b),\ngood agreement is found in the observed behavior. Note\nthat for the YIG/Ta system the contribution of the 4th\nharmonic term is much less pronounced, as the applied\ncurrent is only 1.9mA (compared to 8mA for the YIG/Pt\nexperiments).\nFollowing the derivation of the dynamic SMR as ex-\nplained above, also dynamic SSE signals can be expected.\nAs the SSE is a 2ndorder e\u000bect, any dynamic SSE signals\nare expected to appear as a 3rdorder signal. Measure-\nments of the 3rdharmonic signal indeed show addition-\nally appearing signals at low applied magnetic \felds, but\nthese additional signals are one order of magnitude too\nlarge to be explained by the derived possible dynamic\nSSE signals. This shows that other higher harmonic ef-\nfects are present, which makes it at this moment impos-\nsible to exclusively extract any contribution of possibly\npresent dynamic SSE signals.\nConcluding this section, the dynamic SMR is a good\ncandidate for explaining the observed low-\feld 2ndhar-\nmonic behavior. For both the YIG/Pt and YIG/Ta sam-\nple, the features observed in the experiments can be well\nreproduced by the dynamic SMR model. However, one\nhas to keep in mind that more non-linear e\u000bects are\npresent, such that higher harmonic signals need to be\ntaken into account. Furthermore, the derived model is\nnot su\u000ecient to fully be able to reproduce the measured\ndata. Further analysis of the magnetization dynamics in\nthe YIG at low applied \felds and high applied currents\nis necessary to be able to derive a more complete model.V. SUMMARY\nWe have shown the detection of the SSE in YIG/Pt and\nYIG/Ta samples by both external heating and current-\ninduced heating. The external heating experiments di-\nrectly show the SSE and clearly show the e\u000bect of the\nopposite spin-Hall angle for Ta compared to Pt. For\nthe current-induced measurements, besides the SSE, the\nSMR is also present. By using a lock-in detection tech-\nnique we are able to simultaneously, but separately, mea-\nsure the SSE and SMR signals. Investigation of the low-\n\feld behavior of the SMR and SSE, reveals an additional\n2ndharmonic signal. This additional signal is explained\nby the presence of a dynamic SMR signal, caused by al-\nternating Oersted \felds. Calculations show reproducibil-\nity of the observed 2ndharmonic features, however fur-\nther analysis of the magnetization dynamics in the YIG\nis needed to derive a more complete model of the system\nbehavior.\nACKNOWLEDGEMENTS\nWe would like to acknowledge M. de Roosz, H. Adema\nand J. G. Holstein for technical assistance. This work is\nsupported by NanoNextNL, a micro and nanotechnology\nconsortium of the Government of the Netherlands and\n130 partners, by the Marie Curie Actions (Grant 256470-\nITAMOSCINOM), the Basque Government (PhD fellow-\nship BFI-2011-106), by NanoLab NL and by the Zernike\nInstitute for Advanced Materials (Dieptestrategie pro-\ngram).\n1C. W. Sandweg, Y. Kajiwara, A. V. Chumak, A. A. Serga,\nV. I. Vasyuchka, M. B. Jung\reisch, E. Saitoh, and B. Hille-\nbrands, Phys. Rev. Lett. 106, 216601 (2011).\n2C. Hahn, G. de Loubens, M. Viret, O. Klein, V. V. Naletov,\nand J. Ben Youssef, Phys. Rev. Lett. 111, 217204 (2013).\n3K. Harii, T. An, Y. Kajiwara, K. Ando, H. 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Kimura, and Y. Otani, Phys. Rev. B 83,\n174405 (2011).\n24T. Chiba, G. E. W. Bauer, and S. Takahashi, Phys. Rev.\nApplied 2, 034003 (2014).\n25This approximation was checked by modeling the system\nusing COMSOL Multiphysics, showing indeed a nearly\nconstant Oersted \feld at least several hundreds of nanome-\nters above/below the plane.\n26K. Garello, I. M. Miron, C. O. Avci, F. Freimuth,\nY. Mokrousov, S. Blugel, S. Au\u000bret, O. Boulle, G. Gaudin,\nand P. Gambardella, Nature Nanotechnology 8, 587\n(2013).\nAPPENDIX\n1. Lock-in detection\nAll measurements shown in the main text are per-\nformed using a lock-in detection technique. By this tech-\nnique, 1st, 2ndand higher order responses of a system on\nan applied AC-current can be determined. In general,\nany generated voltage can be written as the sum of 1st,\n2ndand higher order current-responses as:\nV(t) =R1I(t) +R2I2(t) +R3I3(t) +R4I4(t) +:::(9)\nwhere Rnis the nthorder response of the measured sys-\ntem to an applied current I(t). By applying an AC-\ncurrentI(t) =p\n2I0sin(!t), with angular frequency !\nand rms value I0, a lock-in ampli\fer can be used to de-\ntect individual harmonic voltage responses of the inves-\ntigated system, making use of the orthogonality of sinu-\nsoidal functions. To extract the separate harmonic re-\nsponses, the output signal and the reference input signal\n(a sine wave function) are multiplied and integrated over\na set time. When both signals have di\u000berent frequen-\ncies, the integration over many periods will result in zero\nsignal, whereas integration of two sine wave functions\nwith the same frequency and no phase shift will result\nin a non-zero signal. Besides being able to separately\nextract the di\u000berent harmonic responses of the system,\nthe lock-in detection technique also reduces the noise inthe signal, compared to dc voltage measurements, as the\nmeasurement is only sensitive to a very narrow frequency\nspectrum.\nThe detected n-th harmonic signal of a lock-in ampli-\n\fer at a set phase \u001eis de\fned as\nVn(t) =p\n2\nTtZ\nt\u0000Tsin(n!s+\u001e)Vin(s)ds (10)\nBy evaluating Eq.(10) for a given input voltage V in, one\ncan obtain the di\u000berent harmonic voltage signals that can\nbe measured by the lock-in ampli\fer ( Vn). Assuming a\nvoltage response up till the 4thorder, the following lock-\nin voltages are calculated:\nV1=R1I0+3\n2R3I3\n0 for\u001e= 0\u000e(11)\nV2=1p\n2(R2I2\n0+ 2R4I4\n0) for\u001e=\u000090\u000e(12)\nV3=\u00001\n2R3I3\n0 for\u001e= 0\u000e(13)\nV4=\u00001\n2p\n2R4I4\n0 for\u001e=\u000090\u000e(14)\nSo, using di\u000berent lock-in ampli\fers to measure the 1st,\n2nd, 3rdand 4thharmonic voltage responses, Rncan\nbe deduced from Eqs.(11)-(14). To detect the 2ndand\n4thharmonic response, the phase of the lock-in ampli\fer\nshould be set to \u001e=\u000090\u000e.\nNote that V1(V2) does not purely scale linearly\n(quadratically) with I0. A 3rd(4th) order current depen-\ndence is also present in the measured voltage response.\nThus, to obtain the 1storder response ( R1) of the mea-\nsured system, not only the measured 1stharmonic signal\n(V1) has to be taken into account, also the 3rdharmonic\nsignal (V3) has to be included:\nR1=1\nI0(V1+ 3V3) (15)\nSimilarly, the 2ndorder response is calculated as\nR2=p\n2\nI2\n0(V2+ 4V4) (16)\nFor the current-induced SSE and SMR measurements\ndescribed in the main text, Fig. 6 shows the e\u000bect of in-\ncluding the higher harmonic responses of the system (up\nto the 4thharmonic), compared to assuming them to be\nnegligible. For this comparison Eq.(15) and Eq.(16) are\nused. In Fig. 6(a) and (c) V3andV4are assumed to be\nzero, whereas in Fig. 6(b) and (d) the measured 3rdand\n4thharmonic response have been taken into account, re-\nspectively. From these \fgures it can be concluded that in9\n(a) (b)\n(c)\n-180 -90 0 90 180-0.3-0.2-0.10.00.10.20.3\n-90 0 90 18030354045505560\n  R2 [V/A2]Assuming V4= 0 Including measured V4\n  Assuming V3= 0\n  R1 [mΩ]Including measured V3\n  \nB[mT]\n0.9 \n1.8 \n100\nPt [5nm]\nIac = 8mA\nα0[ο] α0[ο](d)\nFIG. 6. Evaluation of Eq.(15) for a selected set of measure-\nments: (a) assuming V3= 0 and (b) including the measured\nvalues ofV3. Similarly for Eq.(16): (c) assuming V4= 0 and\n(d) including the measured values of V4.our low-\feld measurements the higher harmonic signals\nare not negligibly small, and thus should be accounted\nfor.\nIn Fig. 2(a) and (b) of the main text, the 1storder\nresponse of the system is plotted, taking into account\nthe measured 3rdharmonic voltage response as derived\nin Eq.(15) and shown in Fig. 6(b). For the other \fg-\nures, regarding the 2ndorder response of the system, the\nmeasured lock-in voltage is directly plotted ( V2). To \fnd\nthe 2ndorder response of the system Eq.(16) should be\nevaluated, including both V2andV4. Taking this cor-\nrection into account, the observed features of V2slightly\nchange, as is shown in Fig. 6(d), more closely following\nthe expected behavior from the calculated dynamic SMR\nsignal."    },    {        "title": "2308.06896v2.Investigation_of_Phonon_Lifetimes_and_Magnon_Phonon_Coupling_in_YIG_GGG_Hybrid_Magnonic_Systems_in_the_Diffraction_Limited_Regime.pdf",        "content": "Investigation of Phonon Lifetimes and Magnon-Phonon Coupling in YIG/GGG\nHybrid Magnonic Systems in the Diffraction Limited Regime\nManoj Settipalli\nAnn and H.J. Smead Aerospace Engineering Sciences,\nUniversity of Colorado Boulder, Boulder, Colorado 80303, USA\nXufeng Zhang\nElectrical and Computer Engineering, Northeastern University, Boston, MA 02115\nSanghamitra Neogi∗\nAnn and H.J. Smead Aerospace Engineering Sciences,\nUniversity of Colorado Boulder, Boulder, CO 80303, USA\nQuantum memories facilitate the storage and retrieval of quantum information for on-chip and\nlong-distance quantum communications. Thus, they play a critical role in quantum information\nprocessing and have diverse applications ranging from aerospace to medical imaging fields. Bulk\nacoustic wave (BAW) phonons are one of the most attractive candidates for quantum memories\nbecause of their long lifetime and high operating frequency. In this work, we establish a modeling\napproach that can be broadly used to design hybrid magnonic high-overtone bulk acoustic wave\nresonator (HBAR) structures for high-density, long-lasting quantum memories and efficient quantum\ntransduction devices. We illustrate the approach by investigating a hybrid magnonic system, where\nBAW phonons are excited in a gadolinium iron garnet (GGG) thick film via coupling with magnons\nin a patterned yttrium iron garnet (YIG) thin film. We present theoretical and numerical analyses of\nthe diffraction-limited BAW phonon lifetimes, modeshapes, and their coupling strengths to magnons\nin planar and confocal YIG/GGG HBAR structures. We utilize Fourier beam propagation and\nHankel transform eigenvalue problem methods and discuss the effectiveness of the two methods to\npredict the HBAR phonons. We discuss strategies to improve the phonon lifetimes, since increased\nlifetimes have direct implications on the storage times of quantum states for quantum memory\napplications. We find that ultra-high, diffraction-limited, cooperativities and phonon lifetimes on\nthe order of ∼105and∼10 milliseconds, respectively, could be achieved using a CHBAR structure\nwith 10 µm lateral YIG dimension. Additionally, the confocal HBAR structure will offer more\nthan 100-fold improvement of integration density. A high integration density of on-chip memory\nor transduction centers is naturally desired for high-density memory or transduction devices. Our\nresults will have direct applicability for devices operating in the cryogenic or milliKelvin regimes. For\nexample, our approach and analyses can be applied to design HBAR devices that could effectively\ncouple with superconducting qubit systems.\nINTRODUCTION\nHybrid quantum systems that integrate diverse plat-\nforms, constitute a rapidly emerging research field due to\ntheir ability to synergistically address the limitations of\nindividual platforms. Among various systems explored,\nhybrid magnonic systems received significant attention in\nrecent years due to the unique advantages they offer [1, 2].\nIn these systems, quantized spin waves (magnons) co-\nherently couple with other information carriers, such as\nphotons and phonons. The coupling presents avenues\nfor both fundamental investigations and practical imple-\nmentations [3–11]. These systems often leverage mag-\nnetic materials with high spin density, such as yittrium\niron garnet (YIG). Such materials enable strong coupling\nof magnons with microwave photons trapped in a cav-\nity, with coupling strengths surpassing their respective\ndissipation rates [3–7]. Several devices have been in-\nvestigated that benefit from the strong magnon-photon\ncoupling, with applications ranging from microwave-to-optical transduction [12, 13] to quantum magnonics [14–\n16], dark matter detection [17, 18] and others [19–24].\nBesides magnon-photon coupling, magnon-phonon\ncoupling enables another important class of hybrid\nmagnonic devices which has attracted attention for co-\nherent and quantum information processing applica-\ntions [11, 25–28]. Phonons are particularly attractive\ndue to their long lifetime compared to other infor-\nmation carriers [29–32]. Magnon-phonon coupling has\nshown success for classical information processing appli-\ncations [33–45]. The coupling of magnons with mechani-\ncal phonons in YIG spheres has been utilized for promis-\ning quantum information processing (QIP) applications,\nsuch as magnon-phonon entanglement [46], nonreciprocal\nphonon propagation [47, 48], and magnon squeezing [49].\nSeparately, recent studies used high-overtone bulk acous-\ntic wave resonator (HBAR) phonons, that operate in the\nGHz regime, for coupling with superconducting qubits\nand demonstrated their remarkable potential for quan-\ntum acoustodynamics [50–52]. The HBAR phonons also\nexist in hybrid magnonic devices and can enable reso-arXiv:2308.06896v2  [cond-mat.mes-hall]  29 Nov 20232\nnant magnon-phonon coupling via the magnetoelastic ef-\nfect [53]. These phonons are commonly supported by\nstructures composed of YIG thin-films [54–57] grown on a\nthick gadolinium gallium garnet (GGG) substrate, which\ncan host a series of HBAR resonances. A recent study\ndemonstrated a triply resonant photon-magnon-phonon\nsystem using a YIG/GGG hybrid magnonic HBAR de-\nvice [58]. Such resonant coupling can enable long-lived\nmultimode phononic quantum memories and other quan-\ntum information processing and transduction applica-\ntions. In this article, we investigate a class of planar and\nconfocal HBAR YIG/GGG hybrid magnonic structures\nfor quantum transduction applications.\nThe performance figure of merit for magnon-phonon\ntransduction can be characterized using cooperativity,\nC= 4g2\nmb/κmκb, where gmb,κm, and κbare magnon-\nphonon coupling strength, magnon dissipation rate, and\nphonon dissipation rate, respectively. The lifetimes of\nmagnons and phonons are given by τm= 1/κmand\nτb= 1/κb, respectively. In YIG/GGG hybrid magnonic\nHBAR devices, the phonon lifetimes usually surpass the\nmagnon lifetime thanks to the excellent mechanical prop-\nerties of the garnets. Past studies reported the magnon\nand phonon lifetime in YIG/GGG HBAR systems to be\nτm= 0.07µs and τb∼0.25µs, respectively [57, 58]. The\nsystem lifetime is primarily determined by the phonon\nlifetimes. It is desirable to further improve the phonon\nlifetimes and consequentially, the system lifetimes, for\nquantum memories and other information processing ap-\nplications. However, a complete understanding about the\nloss mechanisms that limit the phonon lifetime in gar-\nnet devices is not yet available. The phonon lifetime\ncould be limited by material and diffraction losses de-\npending on the device geometry or the operating condi-\ntions. At room temperature, the phonon lifetime is pri-\nmarily limited by acoustic attenuation due to phonon-\nphonon interactions [59, 60]. The acoustic attenuation\neffects, α∝1/τb, follows a T4temperature dependence.\nDue to the T4behavior, these effects are less significant\nat low temperatures, such as millikelvin regime where\nmany qubit systems operate. Ideally, τbcan increase by\nmultiple orders of magnitude at cryogenic temperatures.\nHowever, the diffraction losses play an important role\nin determining the phonon lifetime at low-temperature\noperating conditions, where the material losses are sup-\npressed. In this study, we establish a computational\napproach for analyzing the diffraction losses in HBAR\nstructures and use it to investigate the effects of diffrac-\ntion losses on the HBAR phonons of hybrid YIG/GGG\nmagnonic structures. We focus on the diffraction effects\nbecause they can be analyzed computationally, while ex-\nperimental characterization is needed to investigate the\nacoustic attenuation effects. Note that the knowledge of\nmaterial losses in YIG/GGG material systems is limited.\nThere is a strong need for experimental characterization\nof acoustic attenuation in different magnetic materialsacross temperature and operating frequencies, to unlock\ntheir full potential.\nThe integration density of the hybrid magnonic HBAR\nstructures is an important design aspect for developing\nhigh-density phononic quantum memories. The inte-\ngration density is measured by the number of memory\nor transduction components that could be incorporated\ninto a single chip, in addition to other on-chip circuitry.\nThe YIG film represents the transduction component of\nthe YIG/GGG HBAR structures of interest. Thus, the\nsmaller the lateral dimension of the YIG film, the greater\nthe number of transduction components in a single chip\nof given dimension and the integration density. We de-\nfine the integration density as Di=A0\nd\nAd. Here, Adis the\nlateral area of the YIG film for planar HBAR structures,\nand the cross-sectional area of the confocal dome surface\nfor CHBAR structures in our study, respectively. A0\ndrep-\nresents a reference value for the device. We consider the\nsmallest known lateral area reported for YIG/GGG de-\nvices in literature to be the reference, A0\nd= 0.8×0.9\nmm2= 0.72 mm2[58]. The integration density of the\ndevice can be increased by reducing the lateral dimen-\nsion of the YIG films. However, the reduced aperture\nwill lead to high-diffraction and affect phonon lifetime.\nIn addition, a smaller aperture will reduce the overlap\nbetween the magnon and the phonon modes since they\nonly overlap in the YIG film. A reduced magnon-phonon\noverlap could lead to a weaker magnon-phonon coupling\nstrength. Consequently, the cooperativity will decrease\nas a result of the reduced lifetime and coupling strength.\nIt is therefore imperative to develop a strategy to in-\ncrease the integration density without inducing excessive\ndiffraction losses. Note that we only focus on the lateral\nintegration density while keeping the thickness fixed at\n527.2µm for all structures investigated in this work. The\nsame thickness allows us to keep the free spectral range\nof phonons fixed for all our analysis.\nIn this study, we investigate a set of planar and\nconfocal YIG/GGG HBAR structures. We investigate\nthe phonon lifetime and the magnon-phonon coupling\nstrength in these structures, and identify strategies to\nimprove their performance and integration density. We\nassume a magnon lifetime of 0 .07µs [58] in all struc-\ntures considered in this study. The phonon lifetime τb\nor the quality factor ( Q=ωτb) is of particular inter-\nest due to its direct relevance to the quantum mem-\nory storage time. Henceforth, we drop the subscript b\nfrom τb, unless otherwise needed to distinguish it from\nmagnon and photon lifetimes. Without losing general-\nity, we only consider the fundamental magnon mode, i.e.,\nthe Kittel mode, for simplicity. The Kittel mode has a\nwell-defined analytical function for circular discs. How-\never, such analytical description is still lacking for the\nHBAR phonon modes. Several methods have been im-\nplemented to model HBAR phonons and their lifetimes.3\nFor example, phonon lifetime in planar HBARs was cal-\nculated by decomposing the initial beam in a Bessel func-\ntion basis and calculating the overlap of the reflected\nbeam with the initial profile after each round trip [61].\nHowever, the predicted lifetime was smaller than the ex-\nperimental observations because the effects of the lat-\neral confinement induced by the transducer are not in-\ncluded. Finite element analysis (FEA) is another popular\nmethod to study HBAR phonons, using the COMSOL\nMultiphysics [62] software. A recent study estimated\nthe diffraction loss in an epitaxial planar HBAR struc-\nture by measuring the power received at the opposite\nend to that of the actuator [63]. Another study used\nFEA to predict the modeshapes of a confocal HBAR for\nqubit coupling applications [64]. They considered phonon\nmodes at low-overtones or long wavelengths, hence it was\npossible to perform three-dimensional (3D) FEA sim-\nulations for this system. However, as we will discuss\nlater, it becomes intractable to simulate bulk acoustic\nwaves with overtone numbers n∼3000 using FEA since\nsampling required for these small wavelength modes is\non the order of ∼billion nodes. Moreover, all these\nstudies discussed longitudinal HBAR phonons whereas\nin hybrid magnonic devices shear phonons exhibit much\nstronger coupling with magnon modes [57, 58, 65]. The\nFEA analysis becomes particularly expensive for shear\nphonons since they are not axi-symmetric. One cannot\nleverage the axi-symmetric two-dimensional (2D) FEA\nmodelling available in COMSOL Multiphysics for these\nsystems. Due to these reasons, we explore other ap-\nproaches to model shear phonon modes. We consider\nthe Fourier beam propagation method (FBPM) [51, 61]\nwhich follows a Fox-Li-like [66] iterative approach to ob-\ntain the phonon modes and works with a plane-wave basis\nset. Although past studies used FBPM for longitudinal\nphonon modes, we illustrate that it can effectively model\nthe shear phonon modes of interest. We provide a de-\ntailed description of this method and discuss an adaptive\nalgorithm that allows to overcome some of the challenges\nof the standard FBPM method. Additionally, we con-\nsider another method that uses Hankel transform (HT),\nwhich is a Fourier transfrom for axi-symmetric systems,\nand works with a Bessel function basis. Thus far, HT\nmethod has only been implemented for Fabry-Perot op-\ntical cavities [67, 68]. We adapt this approach to sim-\nulate YIG/GGG HBAR structures by leveraging their\naxi-symmetry and isotropic material properties.\nMETHODS\nHBAR Configurations\nFigure 1 shows the two representative HBAR struc-\ntures investigated in this work. The structure in Fig-\nure 1(a) consists of a thick GGG film joined with a YIGthin film at the bottom. We refer to this structure as\nthe planar HBAR structure since it has planar top and\nbottom surfaces. Figure 1(b) is a focusing HBAR struc-\nture that includes a GGG dome structure at the top and\na planar bottom surface. We refer to this structure as\nthe confocal HBAR (CHBAR) structure. The thickness\nof the GGG film for all configurations considered in this\nstudy, is tGGG = 527 µm, as shown in Fig. 1. All YIG\nfilms are circular films or discs with thickness, tYIG= 200\nnm, and radius of cross-section, RYIG. The total HBAR\ndevice thickness is tHBAR =tGGG+tYIG= 527 .2µm. We\nconsider planar and confocal HBAR structures with sev-\neralRYIG’s ranging from 10 µm to 200 µm. The width W\nof the structure is 1200 µm unless otherwise mentioned.\nThe radius of cross-section of the dome, Rcross, and its\nradius of curvature, Rcurv, vary for different structures\nconsidered. We investigate 8 planar HBARs with varying\nRYIG, 18 CHBARs with varying Rcurvand fixed Rcross,\nand 10 CHBARs with varying Rcrossand fixed Rcurv.\nFIG. 1. Representative HBAR configurations: (a) Pla-\nnar HBAR structure consisting of a GGG thick film and a\nYIG thin film. (b) Confocal HBAR structure with a top dome\nand a planar bottom surface. xaxis is along the out-of-plane\ndirection while yandzaxes point along the lateral and the\nnormal direction of the structures, respectively. Origin of the\ncoordinate axes is at the center of the bottom YIG surface.\nMagnon Modes\nThe YIG thin film hosts magnons, generated by a\nstatic, B0, and an oscillating RF magnetic field, BRF.\nB0is along the zaxis, while BRFacts in the YIG plane.\nThe magnetic fields, B0andBRF, generate forward vol-\nume magnetostatic modes in the YIG film, that precess\nin the x−yplane. The resulting dynamic magnetization\nis given by mYIG(x, y) =m0(x, y)(cos( ωt)i+ sin( ωt)j).\nHere, m0(x, y) represents the magnon modeshape and\nωis the frequency of precession. We practice the fol-\nlowing convention throughout the article, bold-font sym-\nbols represent vector fields and corresponding normal-\nfont symbols represent scalar values, respectively. We\nexpect un-pinned spin waves at the top and bottom YIG\nfilm surfaces because of the following reason. The ex-\nchange interaction effects are expected to dominate at a4\nthickness on the order of 200 nm and below. As a conse-\nquence, the pinning effect is expected to disappear below\na critical width on the order of 200 nm [69]. However,\nwe expect full-pinning of the magnetic spin waves at the\nlateral boundaries of the YIG discs of radii ranging from\n10 micrometers ( µm) to 200 µm. This is because the mag-\nnetic dipolar interaction effects dominate at length scales\non the order of micrometers, over the exchange interac-\ntions [69]. Taking these into account, we assume that\nmYIGis constant throughout the YIG film thickness.\nWe describe the modeshape of the pinned spin waves,\nm0(x, y), using truncated Bessel functions [70, 71]:\nm0(x, y) =(\nJ0\u0010\nR\nRYIGζj\u0011\n,ifR≤RYIG\n0, otherwise(1)\nwhere radial coordinate, R=p\nx2+y2andJ0is the 0th\norder bessel function. The zeroes of ζjwith j= 0,1,2, ...,\ncorrespond to the fundamental and higher-order magnon\nmodes respectively. In this article, we consider the fun-\ndamental magnon mode or the Kittel mode, whose am-\nplitude is represented by the function, J0(R\nRYIGζ0). The\nmodeshape is constant along the z-direction.\nPhonon Modes\nThe forward volume magnon modes, mYIG(x, y), gen-\nerate precessing shear deformations in the YIG region.\nThe dynamic shear strain results in a circularly po-\nlarized chiral phonon traveling wave in the GGG re-\ngion. The helicity of the chiral phonon is determined\nby the precession direction of the Kittel mode. The\nshear displacements can be expressed as uYIG(x, y) =\nu0(x, y)(cos( ωt)i+sin( ωt)j), where u0(x, y) is the phonon\nmodeshape and ωis the precession frequency. In the fol-\nlowing, we refer to the shear displacements, uYIG(x, y),\nasu0(x, y). The traveling wave undergoes reflections\nat the top GGG surface and the reflected waves inter-\nfere with the forward traveling wave. When the forward\nand the reflected helical propagating waves interfere, they\nform rotating standing shear wave modes at specific fre-\nquencies. While the traveling chiral phonons are helical,\nthe standing waves are not helical. The top GGG surface\ndoes not induce a πphase shift to the reflected wave,\nunlike circularly polarized light reflecting off a mirror.\nAs a result, we obtain standing shear wave modes with\nzero net helicity. In this article, we use (1) Fourier beam\npropagation and (2) HT eigenvalue problem approaches\nto analyze the shear phonon modes in the chosen HBAR\nconfigurations.(1) Phonon Modeshape Analysis: Fourier Beam Propagation\nThe Fourier beam propagation method (FBPM), also\nknown as the angular spectrum method, predicts the field\ndisplacements or profiles of propagating waves [51, 72].\nThe advantage of FBPM lies in its simplicity and the\nability to predict the field profiles at any target distances\nfrom the source without needing to calculate the behavior\nat intermediate distances. The FBPM has been widely\nused to analyze beam propagation in the field of op-\ntics [72]. The mathematical formulation of FBPM for\nacoustic waves is well established [51]. Recently, it has\nbeen used to study phonons in planar [61] and CHBAR\nstructures [51], adapting an iterative method similar to\nthe Fox-Li approach [66]. Here, we implement a reformu-\nlated iterative approach in which the propagation is cal-\nculated using projector and propagator operators. The\nreformulation allows us to achieve a seven-fold speed up\nof computation time. Our approach is particularly ad-\nvantageous for isotropic systems such as YIG and GGG.\nWe assume an initial shear displacement field with\nmodeshape\nu(1)\n0(x, y) =(\nJ0\u0010\nR\nRYIGζ0\u0011\n,ifR≤RYIG\n0. otherwise(2)\nThe iterative procedure begins with this initial input field\nand the superscript represents iteration index ( i= 1).\nThe subscript in u(1)\n0(x, y) refers to the fact that the dis-\nplacement is computed at z= 0. Note that the lateral\nphonon modeshape is the same as the magnon Kittel\nmodeshape, as shown in Eq. 1. In FBPM, the input\nbeams for initial and the subsequent iterations are de-\ncomposed into plane-waves. The field displacements at\nintermediate distances is then obtained by multiplying\nphase factors to the decomposed beam. The Fourier de-\ncomposition of the input beam is given by:\n˜u(i)\n0(kx, ky) = FFT[ u(i)\n0(x, y)]. (3)\nHere, iis the iteration index and ˜u(i)\n0is defined on a\nN×N(kx, ky) grid which is the Fourier conjugate of the\nspatial N×N(x, y) grid. We calculate the projections\nof˜u(i)\n0(kx, ky) along the different polarization directions,\nm= 1,2,3, using the projector, Pm, and obtain the am-\nplitudes, A(i)\nm, of the shear ( m= 1,2) and the longitudi-\nnal (m= 3) modes:\nA(i)\nm=Pm·˜u(i)\n0, (4a)\nwithPm=Σcyc(−1)|sgn(j−m)|ˆdj(ˆdm·ˆdj−ˆdk·ˆdl)\n1 + 2Π cyc(ˆdj·ˆdk)−Σcyc(ˆdj·ˆdk)2.\n(4b)\nHere, ˆdmis the polarization vector for the plane-waves\npropagating along ( kx, ky, kz,m), (j, k, l ) refers to the5\nCartesian directions, ( j, k, l ) = (1 ,2,3) and Σ cycand Π cyc\nare cyclic sum and product operators, respectively, that\ncycle through variables j,k, and l. We use the amplitudes\nA(i)\nm(kx, ky), to obtain the displacement field of the propa-\ngated beam, starting from the input beam, u(i)\n0(x, y). For\na wave originating in the YIG thin film, the propagation\ndistance to reach the upper GGG surface of the HBAR\nstructures is tHBAR , as shown in Fig. 1. The displace-\nment field of the propagated beam at the upper GGG\nsurface is given by\nu(i)\ntHBAR(x, y) = IFFT[Σ mA(i)\nmGm], (5a)\nwith Gm(kx, ky) =ˆdm(kx, ky)eikz,m(kx,ky)tHBAR.(5b)\nHere, Gm(kx, ky) is the propagator for plane-waves trav-\neling along ( kx, ky, kz,m) with polarization m. Typically,\nkz,mcan be derived from their respective slowness sur-\nfaces [51] for each polarization and values of ( kx, ky).\nHowever, the functional relation can be simplified to\nkz,m=q\nω2\nv2m−k2x−k2y, for isotropic dispersions. Here, ω\nis the frequency of the initial wave and vmis the velocity\nof phonons with polarization m. We use the simplified re-\nlationship and the material properties of YIG and GGG,\nshown in Table. I, to compute kz,mfor the propagat-\ning waves in our isotropic YIG/GGG HBAR structures.\nSince the properties of YIG and GGG are similar, we\nuse GGG values for both YIG and GGG, for simplicity.\nThis approximation can be further justified by consid-\nering that our structures are composed of GGG thick\nfilms with ultrathin YIG films bonded to it. Note that\nwe compute kz,m,Am(Eq. 4), and utHBAR(x, y) (Eq. 5)\nonly for the first iteration, for a given ω. This aspect re-\nsults in the seven-fold computational speed-up mentioned\nearlier. We choose the longitudinal polarization ˆd3to be\nTABLE I. Material properties of YIG and GGG.\nYoung’s modulus Poisson’s ratio Density\nE (Pa) ν ρ (kg/m3)\nYIG [73] 0.2 ×10120.29 5170\nGGG [74] 0.222 ×10120.28 7080\nalong the unit vector kand the shear polarizations, ˆd1,2,\nto be two mutually perpendicular unit vectors perpendic-\nular to k. For the isotropic systems investigated in this\narticle, dm’s are mutually perpendicular and Amreduces\ntoAm=ˆdm·˜uat each ( kx, ky).\nThe propagated plane-waves are periodic in the direc-\ntion transverse to the beam propagation direction. When\nthe diffracted waves reach the transverse boundaries of\nthe computational domain, they introduce undesired re-\nflections. To avoid these reflections, one needs to con-\nsider a sufficiently wide computational domain that can\ncontain the waves even after multiple reflections. How-\never, such large domains can significantly increase com-\nputational costs. An alternative approach is to intro-duce absorbing boundary regions that completely atten-\nuate any waves that enter these regions [61]. In this\nstudy, we implement absorbing boundaries of thickness\nWAbs= 50×λ, where λis the wavelength of the input\nfield. The blue shaded regions in Fig. 1 show the ab-\nsorbing boundaries. To simulate the effect of absorbing\nboundaries, we multiply u(i)\ntHBAR(Eq. 5) with a reflection\noperator RtHBAR, defined as\nRtHBAR(x, y) =(\n1,ifR≤Weff/2\n0.otherwise(6)\nWeff=W−2WAbsis the effective width of the sim-\nulation window without the absorbing boundaries. We\npropagate the attenuated reflected wave further through\na distance tHBAR , We implement the beam propagation\nby following the approach outlined in Eqs. 3 - 5b, to\ncomplete a full round trip. After every round trip, we\nmultiply the resulting complex displacement field by an\nadditional phase R0,mfor each polarization m:\nR0,m(x, y) =\n\nei2kz0,mtYIG,ifRYIG≤R≤Weff/2\n1, ifR≤RYIG\n0. otherwise\n(7)\nwhere kz0,m=kz,m(kx= 0, ky= 0). The phase fac-\ntor is introduced due to the finite width of the YIG film\ncompared to the GGG width, W. However, it is worth\nmentioning that this approximation is more appropriate\nfor low-diffraction cases. We used this for all cases con-\nsidered to simplify the analysis. We use the resulting\ndisplacement field as the new input field for the next it-\neration. We repeat this process for Nround trips. At\nthe end of Nround trips, we calculate the complex sum\nof the displacement fields, U0(x, y) = ΣN\ni=1u(i)\n0(x, y) at\nz= 0. Using the interference sum, U0(x, y), we restart\nthe iterative process with U0(x, y) as the initial beam of\nthe next restart, u(1)\n0(x, y) =U0(x, y). Such a restart\nprocess using interference sum as an input ensures fast\nconvergence to the desired mode. The other mode com-\nponents are attenuated in the interference sum due to\ndestructive interference. We continue this process until\nthe varation of the modeshape is within a chosen toler-\nance. We obtain a converged standing shear wave with\ndisplacement field, u0(x, y), as a final outcome.\nIn addition to the displacement profiles, we are inter-\nested in identifying the frequencies of the shear modes.\nThese modes could couple with the Kittel magnon modes\ngenerated in the YIG thin-film (Eq. 1), in a rotating coor-\ndinate system. The frequency overtones for plane-waves\ntraveling along z-direction in the HBAR structures are\nexpected to be ωm,n= 2π×nvm\n2tHBAR, with n= 1,2,3, ...∞.\nHere, tHBAR is the thickness of the structure, velocity\nof wave is vmandmrepresents polarization. The over-\ntones are separated byvm\n2tHBAR, known as the free spectral6\nrange (FSR). However, unlike plane-waves, the diffract-\ning waves traveling in the zdirection do not have well-\ndefined kz,m’s leading to Gouy phase effects [75]. The\ndiffraction results in the shift of resonance frequencies\nfrom the monochromatic plane-wave overtones. To iden-\ntify the resonance frequencies, we select beam of fre-\nquencies from a chosen frequency window, propagate the\nbeam in the structure, and calculate the intensities of\nthe interference sums ( I=R\nARe[U0(x, y)]2dA) atz= 0\nafter Nround trips. The frequencies for which the in-\ntensities are maximum are the resonance frequencies of\nthe shear waves in the HBAR structure. We focus on\nthe shear modes with m= 2, however, we could have as\nwell chosen m= 1 as the dominant shear polarization\nin the rotating coordinate system. We sweep through a\nfrequency window of ω0±5×FSR using 50 steps. Here,\nthe frequency of interest, ω0= 2π×9.825 GHz, cor-\nresponds to the frequency of the 2960thovertone of a\nstanding plane-wave in the HBAR structure. We choose\nthis overtone to match with the structure investigated\nin a previous article [58]. We consider a YIG/GGG\nHBAR structure with tYIG= 200 nm, RYIG= 200 µm,\ntGGG = 527 µm, and Lx,GGG =Ly,GGG = 1200 µm, for\nthis analysis.\nFIG. 2. Resonant modes in planar HBAR structures:\n(a) Multimode phonons, represented by the overtones of the\nfundamental mode, and their free spectral range (FSR). (b)\nIsometric view of the y-component of U0(x, y), U 0y(x, y). (c).\nTop view of U 0y(x, y) showing localization effect caused by\nthe YIG film. (d) Weighted deviation (WD) between mode\nprofiles before first and second restarts.\nFigure 2(a) shows the intensities of different propa-\ngated beams with frequencies within the frequency win-\ndow, a frequency window, ω0±5×FSR. The central in-\ntensity peak corresponds to ω0= 2π×2960v2\n2tHBAR, the exact\nvalue of the 2960thplane-wave overtone. The peaks form\nat resonance frequencies separated by FSR = 3 .32 MHz\naround the frequency, ω0/2π∼9.825 GHz, indicatingthe presence of multiple shear wave overtones. In Figs. 2\n(b) and (c), we show different views of the y-component\nof the normalized resonant modeshape, Re[ U0y(x, y)],\nat frequencies near ω0. We obtain these displacement\nfields after 800 round trips. This simulation included one\nrestart after 400 round trips. We perform multiple tests\nand check that this number is large enough such that the\nGouy phase effects are not significant on the interference\nsums. We obtain the normalized displacement fields be-\nfore the 1st and the 2nd restarts, U1\n0(x, y) and U2\n0(x, y),\nrespectively, and compute the weighted deviation (WD)\nbetween them: |(U1\n0(x, y)−U2\n0(x, y))|/|Σx,yU1\n0(x, y)|.\nWe monitor the WD between restarts to check for conver-\ngence. The color scale of Fig. 2 (d) shows that the WD\nis much less than 10−7, indicating that the Figs. 2 (b)\nand (c) represent displacement fields converged within\nsufficient numerical tolerance. Some fringes remain in\nthe modeshapes that are possible artifacts of our numer-\nical analysis. These are the high frequency components\nresulting from the sampling of the sharp phase change\ninduced by the YIG film. They are significantly lower in\nvalues, however, could be further reduced by either using\na low-pass filter or a smoother phase change factor than\nthe function, R0,m(x, y) (Eq. 7), used in this work.\nTo obtain the converged modes and identify the true\nresonant frequency near an intensity peak of interest,\nwe further narrow the frequency sweeping range with\nfiner sampling ( ∼1 kHz). We set the frequency to be\nωHBAR\n0 =ω0+δω, with δω= 2π×2.157 kHz and restart\nthe iterative process with Eq. 2 as the input beam, to\nidentify the true modal frequency near ω0. If such nar-\nrowing is not done, the modeshape can change signifi-\ncantly after each restart due to the Gouy phase effects\nthe beam incurs due to diffraction [75]. These effects\nresults in the detuning of overtones with respect to the\nplane-wave overtones. Figure 3 shows the real and imagi-\nnary parts of the complex sums U(1)\n0y(x, y) and U(2)\n0y(x, y)\ncomputed before first and second restart (after 400 and\n800 round trips), respectively. We calculate the complex\nsums at ω0without performing the narrowed frequency\nsweeping discussed above. Both the real and imaginary\nparts of U(1)\n0yand U(2)\n0yshow significant variations between\nthe restarts. These results establish that the δωcorrec-\ntion is necessary to obtain the converged modes.\nHowever, the gradual narrowing of the frequency\nsweeping is a tedious process to identify the necessary\nfine sampling. We need to restart the iterative process\nmultiple times especially for high-diffraction cases. The\nresults shown in Fig. 2, represent propagating waves in a\nlow-diffraction regime with Fresnel number, NF= 213.\nHowever, this work also considers wave propagation in\nHBAR structures with RYIGas low as 10 µm correspond-\ning to a very low Fresnel number, NF= 1.06, indicating\na high-diffraction regime. We investigate these structures\nto discuss how a reduced actuator lateral area affects7\nFIG. 3. Lateral mode profiles obtained with FBPM\nshowing Gouy phase effects: The real (solid) and imag-\ninary (dashed) components of the y-component of the inter-\nference sum before the first U(1)\n0yand second restarts U(2)\n0yat\nω0. The components deviate significantly between restarts.\n.\nthe phonon modes and their lifetimes. The reduced area\npromises to increase the integration density of the HBAR\ndevices towards high-density memory applications. The\nhigh-diffraction regime of these structures introduces sig-\nnificant Gouy phase effects on the propagating waves and\nresults in detuning of frequencies. It becomes challeng-\ning to identify a resonant frequency by merely narrowing\nthe frequency sweeping range. We implement an adap-\ntive FBPM algorithm to circumvent this challenge. Note\nthat we implement absorbing boundaries in this study to\navoid undesired boundary reflections. The phonon mode-\nshapes can be well predicted using this implementation.\nHowever, their lifetimes can be dependent on the shape\nand size of the boundaries. We leave the extensive inves-\ntigation of the effect of boundary conditions on phonon\nlifetimes for a future investigation.\nPhonon Modeshape: Adaptive Fourier Beam Propagation\nWe formulate the FBPM approach described above as\nan eigenvalue problem:\nRTu 0(x, y) = Λu0(x, y) (8)\nwhere RTis the round trip operator that includes all the\noperations described in the previous section (Eqs. 3- 7)\nand Λ is the eigenvalue corresponding to the eigenvec-\ntoru0. It is possible to solve the eigenvalue problem\nfor 2D problems (e.g., if HBARs are represented by pla-\nnar 2D structures), however, it cannot be directly solved\nfor 3D structures of our interest. We develop an iter-\native method to obtain the resonant mode that satis-\nfies Eq. 8 with a real Λ. Λ is real for a standing wavemode. A complex Λ applies a global phase to the input\nmodeshape after a round trip, and the phase prevents\nthe waves to interfere constructively over multiple round\ntrips. In the adaptive FBPM (a-FBPM) algorithm, we\niteratively adjust the frequencies based on the phase dif-\nference incurred over a round trip to arrive at the desired\nstanding wave modes. We outline the steps of the itera-\ntive method below.\n1. Start with a input beam, u(i)\n0(x, y), for the ith\nround trip. The initial input beam ( i= 1) has\nwavelength, λ(1)=2t\nnand frequency ω(1)= 2π×\nnv2\n2t. Here, nis the overtone number, t=tHBAR\nandv2is the velocity of shear phonons. The wave-\nlength and frequency are estimated based on the\nnthovertone for an open-ended column. The ini-\ntal modeshape is chosen to be a truncated Bessel\nfunction as shown in Eq. 2.\n2. Calculate u(i+1)\n0(x, y) =RTu(i)\n0(x, y).\n3. Estimate the real eigenvalue from, Λ(i)=q\nI(i+1)\nI(i), where I(i)=R\n|u(i)\n0|2dxdy andI(i+1)=\nR\n|u(i+1)\n0|2dxdy .\n4. Calculate the residual, R(i)=||(u(i+1)\n0(x, y)−\nΛ(i)u(i)\n0(x, y))||/||Λ(i)u(i)\n0(x, y)||.\n5. If R(i)≤tolerance (tol), end simulation and output\nfinal eigenvalue and modeshape of resonant modes,\nelse continue to the next step. We choose tol =\n10−6for all a-FBPM calculations of this study.\n6. If R(i)>tol, estimate the global phase factor in-\ntroduced by the round trip operation RT,θ(i)=\nArg(u(i)\n0(0,0))−Arg(u(i+1)\n0(0,0)).\n7. Set ω(i+1)=ω(i)+v2θ(i)\n2tHBARand λ(i+1)=\n2πnv2/ω(i+1). The steps 6 and 7 makes the al-\ngorithm adaptive.\n8. Repeat the process until step 5 is satisfied or the\nmaximum number of Nround trips is reached,\nat which point we set u(1)\n0(x, y) =U0(x, y) =\nΣN\nn=1u(i)\n0(x, y), and restart the iterative procedure.\nFigure 4 illustrates the effectiveness of the a-FBPM al-\ngorithm in predicting the resonant modes of the HBAR\nstructures. We show the results for two planar HBAR\nstructures with RYIG= 200 µm (red) and 10 µm (blue),\nrespectively. The residual, R, decays as a function of\nthe number of steps, as we approach the resonant mode.\nThe step-like features of the residual decay occur after\nevery NRround trips when we compute the interference\nsums and use it as input for the next restart of the iter-\native procedure. The jumps occur because some errors8\nFIG. 4. Obtaining resonant modes using the adaptive\nFBPM algorithm: Residual (R) of modeshapes decays as a\nfunction of number of round trips. Results are shown for two\nplanar HBAR structures with RYIG= 200 µm (red, left and\nbottom axes) and 10 µm (blue, right and top axes). Circles\nwith arrows point to corresponding axes for each plot.\nget canceled due to destructive interference when an in-\nterference sum is computed. In some cases, R can have a\nmomentary rise due to accumulation of errors from pre-\nvious round trips, however, the overall trend continues\nto decrease ultimately reaching convergence. We calcu-\nlate the complex interference sum after every NRround\ntrips. We choose NR= 400 and 40 for the HBARs with\nRYIG= 200 µm and 10 µm, respectively. We choose a\nsmaller number of round trips, NR, for the 10 µm case,\nsince the HBAR with RYIG= 10 µm represents a high-\ndiffraction structure and the modeshape decays rapidly\nfor this case. As shown in Fig. 4, it takes a total of\n∼650 and ∼1200 round trips (including restarts) to ob-\ntain converged resonant modes for the 10 µm and the 200\nµm case, respectively. We continue the iterative process\ntill R <10−8to show stability of the resonant mode even\nafter tolerance is reached.\n(2) Phonon Modeshape Analysis: Hankel Transform\nEigenvalue Problem\nOur a-FBPM approach avoids the frequency sweeping\nprocess and mostly overcomes the slow convergence is-\nsues of the standard FBPM approach. However, it is\nstill challenging to obtain converged phonon modes for\nthe confocal HBAR structures of interest, using the a-\nFBPM approach. Here, we discuss an alternate method\nthat uses the Hankel transform (HT) approach, the axi-\nsymmetric equivalent of the Fourier transform method.\nThe HT approach has been widely used in the field of\noptics, e.g., Fabry-Perot cavities [67]. However, it hasnot been applied for acoustics problems, to the best of\nour knowledge. The HT method allows us to obtain con-\nverged phonon modes with significantly reduced compu-\ntational cost. The approach leverages the axi-symmetry\nof the problem to reduce the 3D problem to a 2D one. For\nexample, the 2D problem can be obtained by represent-\ning HBARs as planar 2D structures. The axi-symmetry\nconditions are applicable for the YIG/GGG HBAR struc-\ntures due to their isotropic material properties and the\ncircular cross-section of the YIG film. Note that the\nscalar field describing the dominant shear-component of\nu0is expected to obey axi-symmetry, however, the vector\nfield of the shear acoustic phonon modes does not obey\naxi-symmetry. Here, we provide a brief description of the\nHT method. We encourage the interested reader to find\na detailed description of the method elsewhere [67, 68].\nWe cast the HBAR structures shown in Fig. 1 into axi-\nsymmetric representations about the z-axis. We trans-\nform the x−ycoordinates to the radial coordinate r,\nwith limit 0 ≤r≤W/2. We write the Hankel eigenvalue\nproblem as RThku(r) = Λ u(r), where u(r) represents the\naxisymmetric scalar phonon modeshape, Λ is the corre-\nsponding eigenvalue and RThkis the round trip operator.\nSince the dimensionality of the problem is reduced from\n3D to 2D, the Hankel eigenvalue problem can be solved\nto obtain the phonon modes directly. We discretize ras\nrj=W/2\u0012ζj\nζN\u0013\n(9)\nwhere j= 1,2,3, .., N andζj(j= 1,2,3, .., N ) are the\nroots of the J1Bessel function. The round trip operator\nfor the HT approach RThkis given by\nRThk=R0PRtHBARP (10)\nwhere P, the N×Npropagator matrix, is defined as:\nP= (H+)−1˜GH+, (11a)\nH+\nij=W2\n2ζ2\nNJ0(ζjζj/ζN)\nζ2\nNJ2\n0(ζj), (11b)\n˜Gij= exp \n−i2ζ2\nj\nW2!\nδjj. (11c)\nHere, H+is the HT whose matrix inverse is taken to\nbe an approximation of the inverse HT, and ˜Gis the\nGreen’s function obtained from the Fourier transform of\nthe Fresnel propagator in the paraxial approximation.\nRtHBAR, andR0are the N×Ndiagonal reflection matrix\noperators at z=tHBAR andz= 0, respectively and the\ndiagonal elements are defined as\nR0,jj=R0,2(rj), (12a)\nRtHBAR ,jj=RtHBAR ,2(rj). (12b)\nThis method, unlike the FBPM/a-FBPM approaches,\ndoes not require a Fox-Li-like iterative process and can9\nalso predict higher-order phonon modes. Although this\napproach has a limited applicability due to its axi-\nsymmetric constraints, it makes it possible to obtain the\nphonon modes and lifetimes of HBAR structures at a\nreduced cost. The expedited analysis allowed us to ana-\nlyze a large class of HBAR structures, as we show in the\nResults and Discussion section.\nDiffraction-Limited Phonon Lifetime\nWe estimate the diffraction-limited phonon lifetimes\nusing three methods: ( A) Eigenvalue method, ( B) Expo-\nnential curve fitting method, and ( C) Clipping method.\n(A) Eigenvalue method: In this method, we obtain the\neigenvalues using the adaptive FBPM simulation and use\nthem to estimate the phonon lifetime. The elastic en-\nergy contained in a acoustic beam with modeshape uis\ngiven by E∝R\n|u|2dxdy . We only integrate over the\ndxdy element since the modeshape largely remains un-\naltered throughout the thickness. We assume that the\nelastic energy of the acoustic beam decays exponentially\nwith the propagation time. For a cavity phonon mode\nwith a total initial elastic energy Etot, we represent the\nelastic energy left in the cavity after one round trip as\nEin=Etote−tRT/τ. Here, tRT(= 2tHBAR /v2) is the time\ntaken by shear waves to complete a round trip and τis\nthe lifetime of the phonon mode. On the other hand, the\nratio of EintoEtotis given by Ein/Etot= Λ2. Here, Λ\nis the real eigenvalue obtained using the a-FBPM simu-\nlations. Using this relation, we can obtain the lifetime of\nthe phonon mode as\nτ=−2tHBAR\nv2ln(Λ2). (13)\nSince these computations are performed on a finite x−y\nmesh grid, Λ, and consequently, τcan be sensitive to the\nmesh density. Hence, it is important to select an optimal\ngrid to obtain the converged values of Λ. In Fig. 5, we\nshow the variation of Λ with mesh density Nx(=Ny)\nfor both the low-diffraction ( RYIG= 200 µm) and the\nhigh-diffraction ( RYIG= 10µm) cases. We find that the\nvariation of Λ is much less than 1% when Nx≥1024.\nConsequentially, we choose Nx=Ny= 1024 to compute\nphonon modes and lifetimes for all cases considered here.\n(B) Exponential curve fitting method: In this method,\nwe calculate the phonon mode lifetime by evaluating the\ndiffraction loss of the beam as it travels in the HBAR\nstructure. We estimate the diffraction loss from the over-\nlap of the propagated mode profile with the initial input\nprofile, after each round trip. We estimate the overlap\nusing the following function:\nI(t) =|⟨u0(t)|u0(0)⟩|2\n|⟨u0(0)|u0(0)⟩|2. (14)\nFIG. 5. Eigenvalues of two planar HBAR structures\nwith RYIG= 200 µm (red) and 10µm (blue): Variation of\nΛ for meshes with density ranging from 128 ×128 to 2400 ×\n2400. The width Wis kept fixed at 1200 µm. The circles with\narrows point to the axes corresponding to each plot.\nHere, tis an integral multiple of the time taken for each\nround trip, tRT. We allow the initial beam to travel for\nmultiple round trips until I(t)<0.1 is reached or the\ntime is t >3 ms, whichever is reached first. In Fig. 6, we\nshow the decay of the overlap function, I(t), for beams\ntraveling in a HBAR structure with RYIG= 200 µm. The\nfinite width of the YIG film (transducer) induces a lo-\ncalizing effect on the propagating beam. The localizing\neffect can be modeled by introducing a phase factor. The\nblue and the red lines shown in Fig. 6 correspond to the\ntwo cases when we obtained the overlap function with\nor without the consideration of the phase factor, respec-\ntively. The two different decays highlight the effect of the\nfinite width of the transducer on the propagating beam.\nWe find that I(t) decays at a much slower rate when the\nYIG-induced phase factor is considered, compared to the\ncase without the phase factor. We fit the two decays with\nexponential functions and obtain the phonon lifetimes as\nτwophase= 0.1 ms and τwphase= 14.1 ms, respectively.\nThe remarkable two-orders of magnitude difference be-\ntween these two predicted lifetimes highlights the im-\nportance of including the transducer or actuator phase\neffects for such an analysis. Subsequently, we included\nthe phase effects in all our analyses to obtain the phonon\nlifetimes. We find that the lifetime predicted with the\nphase effects, τwphase= 14 .1 ms, closely matches with\nτof 15.5 ms, predicted using the eigenvalue method dis-\ncussed earlier. A past study used the exponential curve\nfitting approach to estimate the lifetime of the longitudi-\nnal HBAR phonons in an AlN/Sapphire structure [61].\nHowever, they treated the phonons as superpositions of\nBessel functions and did not account for the localization\neffects induced by the AlN transducer.10\nFIG. 6. Decay of the overlap function, I(t), for prop-\nagating beams in a HBAR structure with RYIG =\n200µm. Solid lines indicate I(t) while dashed lines indicate\ntheir respective exponential fits. The finite width of the YIG\nfilm induces a localizing effect on the beam, modeled with\na phase factor. I(t) decays at a much slower rate when the\nphase effect is included (blue, top and right axes) compared\nto the no-phase-effect case (red, bottom and left axes).\n(C) Clipping method: In this method, we obtain the\nphonon lifetime using a similar expression as used for\nthe eigenvalue method, Eq. 13: τ=−2tHBAR\nv2ln(Ein/Etot). Here,\nEtotis the total initial elastic energy for a cavity phonon\nmode and Einis the elastic energy left in the cavity after\none round trip. We calculate Ein, contained in a volume,\nVin, of a cylinder with radius of the YIG disc and the\nthickness of the HBAR structure, Ein∝R\nVin|u|2dxdy .\nWe consider a converged modeshape, obtained with a-\nFBPM or HT method. This method results in the lowest\nestimate of the phonon lifetimes since it assumes that all\nenergy outside the cylinder of interest is lost after a round\ntrip. Our approach is similar to the clipping method used\nfor Fabry-Perot cavities [67, 68] and HBAR resonators,\nexcited opto-mechanically using photon beams [76]. In\nthese optical systems, the clipping method is more appli-\ncable since the input photon beam spans the entire x−y\nplane and is clipped by the localizing finite cross-section\nmirrors or domes. In our system, the input acoustic beam\nis already laterally confined so this method may have lim-\nited applicability. We still include the discussion here as\na consistency check since the method results in the lower\nbound for the predicted lifetimes.Magnon-Phonon Coupling\nThe magnon-phonon coupling strength, gmb, is given\nby\ngmb=Bp\nAνQη\f\f\f\fZ\nVYIG\u0014du∗\nx\ndzmx+du∗\ny\ndzmy\u0015\ndr\f\f\f\f.(15)\nHere, B= 7×105J/m3, is the magnetoelastic constant\nof YIG, ( mx, my) are the xandycomponents of the\nmagnetization, m=mYIG, (u∗\nx, u∗\ny) and (du∗\nx\ndz,du∗\ny\ndz) rep-\nresent the complex conjugates of the xandycompo-\nnents of displacement uand their corresponding shear\nstrains, respectively. The integration domain is the vol-\nume of the YIG film, VYIG, since the magnon modes\nreside in YIG. The termp\nAνQηcorresponds to the\nmagnon and phonon normalizing constants. We normal-\nize the magnon modes as\nMs\nγZ\nVYIGm∗\nν(r)·(k×mν′(r))dr=−iAνδν,ν′.(16)\nHere, µ0Ms= 0.175T is the saturation magnetization,\nγ= 2π×28.5 GHz/T is the gyromagnetic ratio, kis\nthe unit vector along the z axis, νis the magnon mode\nindex, and Aνis the magnon normalization constant, re-\nspectively. Similarly, we normalize the HBAR phonon\nmodes as\n2ωηρZ\nVHBARu∗\nη(r)·uη′(r)dr=Qηδη,η′. (17)\nHere, ρ= 7080 kg/m3is the GGG material density, η\nis the phonon mode index, and ωηis the corresponding\nphonon frequency, respectively. VHBAR is the volume of\nthe HBAR structure. Qηis the phonon normalization\nconstant and is of the same dimensionality as Aν.\nNote that if we assume zero diffraction, the acoustic\nenergy is fully contained in the volume Vin, the volume\nof the cylinder with radius of the YIG disc and the thick-\nness of the HBAR structure. Such a scenario results in a\ncomplete overlap of lateral mode profiles of the magnon\nand the phonon modes. We obtain the zero diffrac-\ntion limit of the magnon-phonon coupling strength to\nbeg0\nmb/2π= 1.13 MHz. This value is independent of the\nwidth of the YIG film. Our result compares well with a\nprevious experimental result of 1 MHz [57], and is slightly\nhigher than another experimental result of 0 .75 MHz [58].\nHowever, we consider the regime where diffraction causes\nthe spread of acoustic energy outside Vin, which can im-\npact both τandgmbof phonon modes in HBAR struc-\ntures. We present the diffraction-limited HBAR phonon\nlifetimes and the magnon-phonon coupling strengths in\nthe Results and Discussion section.11\nRESULTS AND DISCUSSION\nWe investigate hybrid magnonic HBAR structures that\ncan support the development of high-density phononic\nquantum memories. We can include a greater number of\ntransduction components in a single chip of given dimen-\nsion by reducing the radius of the YIG film. However, the\nreduced aperture increases the diffraction of the acous-\ntic waves propagating into the GGG region. Here, we\ndiscuss the diffraction-limited phonon lifetimes and the\nmagnon-phonon coupling in the HBAR structures.\nDiffraction-Limited Phonon Lifetime\nFigure 7 (a) shows the variation of phonon lifetimes\nin planar HBAR structures as we decrease the YIG ra-\ndius. We compute the lifetimes using the a-FBPM ap-\nproach and following the eigenvalue method (red line),\nthe exponential fitting (blue and purple lines), and the\nclipping methods (green line), respectively. In addition\nto the a-FBPM predictions, we show predictions from\nHT method in Fig. 7 using circles, with colors match-\ning to their corresponding a-FBPM predictions. Note\nthat we use |Λ|instead of Λ in Eq. 13 to calculate τ\nusing the HT method. All methods predict that the life-\ntimes decrease with decreasing RYIGdue to increased\ndiffraction, as expected. The eigenvalue method results\nin the highest lifetime estimates, among the three meth-\nods considered. We obtain the highest lifetime to be\nτ= 15.5ms ( Q=ωτ= 9.57×108) for HBAR structure\nwith RYIG= 200 µm. The lowest lifetime predicted by\nthe eigenvalue method, is τ= 10.5µs (Q= 6.48×105),\ncorresponding to HBAR structure with RYIG= 10 µm,\nrespectively. These results show that the lifetimes are\nreduced by more than three orders of magnitude in the\nhigh-diffraction regime. The lifetimes predicted by the\neigenvalue method (red) closely match with those pre-\ndicted by the exponential fitting method (blue), for the\nlow-to-medium diffraction cases, when the phase effects\nare included. This result can be explained through the\nfollowing argument. The input beam can be thought\nof as a superposition of the eigenmodes of the HBAR\ncavity. Once the initial input beam undergoes multiple\nround trips, only the fundamental mode is likely to sur-\nvive, while the rest of the mode components decay at a\nfaster rate. When we operate at the overtone frequency\nof one of the fundamental modes, the fundamental mode\nbecomes the dominant mode. It can be argued that this\ndominant mode is the same as the fundamental eigen-\nmode found by solving the eigenvalue equation, Eq. 8.\nTherefore, the decay of the overlap function of the input\nfield (exponential fitting method) is expected to have a\nsimilar character to that of the decay of the dominant\neigenmode (eigenvalue method). As a result, we obtain\nsimilar lifetimes using the two methods.However, in the high diffraction case ( RYIG= 10µm),\nwe observe 35-fold reduced lifetimes predicted by the ex-\nponential fit compared to that of the eigenvalue method.\nDue to high-diffraction, significant amount of the energy\nof the input acoustic beam spreads out laterally during\nthe first few round trips, resulting in a rapid initial de-\ncay of I(t). In the exponential fit method, we obtain\nthe lifetime from the exponential fit of I(t), starting at\nt= 0. The loss of acoustic overlap during the initial\ntransient phase results in lower τpredictions. We expect\nthat the two lifetime predictions will be closer if the ex-\nponential fit is obtained after a steady state is reached.\nNote that we obtain the exponential fitting predictions\n(blue) by including the localizing phase effects due to the\nYIG film. We also show the lifetimes predicted by this\nmethod, when we do not consider the phase effects due\nto the YIG film (purple). We find that the τwophase’s are\nconsistently lower than the τwphase’s. The difference be-\ntween the two predictions decreases monotonously from\n135 to 1.35 fold, as we decrease RYIGfrom = 200 µm to\n= 10µm, respectively. This is expected since the local-\nizing effect of the YIG film diminishes as we approach\nRYIG→0. Past studies often ignored the effect of aper-\nture width on the HBAR phonons, however, our results\n(Fig. 6 and Fig. 7) establish that such effects needs to be\nconsidered to make reliable predictions.\nFigure 7 (a) also shows the lifetime predictions from\nthe clipping method (green), which assumes that the en-\nergy outside the cylindrical volume of interest, Vin, is\ncompletely lost after a round trip. We find the life-\ntimes to be more than an order of magnitude lower than\nthose predicted from the eigenvalue method, for all cases\nconsidered. We obtain the lowest lifetime in the high-\ndiffraction regime to be τ= 0.311µs (Q= 1.92×104).\nInterestingly, this prediction is still marginally greater\nthan the experimentally observed values of 0.25 µs and\n1.45×104, respectively [58], for a device operating in the\nlow-diffraction regime. The reason for this discrepancy\nis that the experimental values correspond to HBAR de-\nvices operating at room temperatures. In this limit, the\nperformance of HBAR structures is limited by the ma-\nterial attenuation effects [59, 60]. In our study, we ig-\nnore the material attenuation effects and only discuss\nthe diffraction-limited performance. Our study will cor-\nrespond to devices operating in the cryogenic or poten-\ntially milli Kelvin regimes. For example, our results will\nhave high applicability for the HBAR devices that could\neffectively couple with superconducting qubit systems,\netc., which typically operate in the milli Kelvin regime.\nWe discuss here the uncertainties associated with the\nnumerical prediction of the HBAR phonon lifetimes using\ndifferent methods. In the eigenvalue method, we obtain\nthe lifetimes from the ratio between the HBAR thickness\n(tHBAR ) and a function of the phonon velocity and the\neigenvalue, Λ, as shown in Eq. 13. Following Eq. 13, an12\nFIG. 7. Diffraction-limited properties of phonon modes of planar HBAR structures: (a) Phonon lifetimes, τ,\ncalculated from the eigenvalue (red), exponential fitting (blue, purple), and clipping methods (green). (b) Ratio between\nmagnon-phonon coupling in HBAR structures in various diffraction regimes and that in the zero-diffraction limit, gmb/g0\nmb\n(red). Reduction of gmbas we decrease RYIG, is connected to the spread of modeshape in the high-diffraction regime. Increasing\namount of acoustic energy spreads into the GGG region, causing reduced overlap between phonon and magnon modes in the\nYIG region. The ratio of acoustic energy spreading out to the total energy, Eout/Etot(black), increases in the high-diffraction\nregime. (c) Magnon-phonon cooperativity C.RYIGis varied keeping all other geometric factors fixed. Solid lines correspond\nto a-FBPM predictions, while the circles of the same color correspond to the respective HT predictions.\nerror propagation relation can be written as\n\f\f\f\fdτ\nτ\f\f\f\f=\f\f\f\fv2τ\ntHBAR×dΛ\nΛ\f\f\f\f∝\f\f\f\fτ×dΛ\nΛ\f\f\f\f. (18)\nThe uncertainty\f\fdτ\nτ\f\fincreases monotonously with τ\nwhen other factors are fixed. It can be deduced from\nEq. 18 that to predict a lifetime within x% of variabil-\nity, Λ has to be predicted to be withintHBAR x\nv2τ% of vari-\nability. Here, the order of magnitude of the prefactor\nv2\ntHBAR∼107. The prefactor remains fixed since we keep\nthe material and the HBAR thickness unaltered. This\nimplies that we have to be increasingly more stringent\nwith our Λ convergence criteria for high lifetime calcu-\nlations. Figure 8 shows the conditions for desired ac-\ncuracy (relative tolerance) needed for Λ predictions for\nhigh- τ(high- Q) systems. When τ= 0.1µs and the de-\nsired accuracy is set to 10%, Λ must be predicted within\n15% accuracy, which is attainable following our numer-\nical procedure. The accuracy requirement increases fast\nwhen the lifetime values are much higher. For example,\nwhen τ= 1ms, and the desired accuracy is set to 10%\nofτvalue, Λ must be predicted within ∼10−3% of accu-\nracy. For our RYIG= 200 µm HBAR structure, we con-\ntinue the simulation till we obtain\f\fdτ\nτ\f\f= 7.9×10−4with\f\fdΛ\nΛ\f\f= 4.7×10−7, between the last two restarts. On the\nother hand for the HBAR with RYIG= 10µm, we con-\ntinue till\f\fdτ\nτ\f\f= 3.9×10−7with\f\fdΛ\nΛ\f\f= 3.8×10−8. How-\never, achieving such high accuracy requires high compu-\ntational expense associated with simulating a large num-\nber of iterations. We implement and use an alternative\nHT approach to expedite the numerical analysis.\nFIG. 8. Relative tolerance of Λat various τrequired to\npredict τwith 10% accuracy.\nMagnon-Phonon Coupling in Diffraction-Limited Regime\nWe now turn to discuss the effect of diffraction on\nthe magnon-phonon coupling strength, gmb. We esti-\nmate the effect of diffraction by computing the ratio be-\ntween the coupling strength in the planar HBAR struc-\ntures, gmb, and that in the zero-diffraction limit, g0\nmb.\nIn Fig. 7(b)(red), we show the variation of the ratio,\ngmb/g0\nmb, with decreasing YIG radius. As expected,\nThe ratio is equal to 1 for low-diffraction cases with\nRYIG≳100µm. As we reduce RYIGto 40 µm,gmbonly\nchanges by 5% from the g0\nmblimit. Note that even though\nthis case typically corresponds to a strong-diffraction\nregime, the effect of diffraction on gmbis minimal. This13\nis because the YIG disc helps to localize the beam and\nthus, preserve the magnon-phonon overlap and the cou-\npling strength. However, as we decrease RYIG≤40µm,\nwe observe a sharp decline of gmb. The percentage of the\nratio, gmb/g0\nmb, drops to 77 .4% and 55 .7% for HBAR with\nRYIG= 20µm and 10 µm, respectively. The decrease of\ngmbcan be explained in the following way. In the zero\ndiffraction limit, the acoustic beam is completely local-\nized in the volume Vin, of a cylinder with radius equal\ntoRYIGand the thickness of the HBAR structure. This\nresults in a complete overlap of lateral mode profiles of\nmagnon and phonon modes. Also, the acoustic energy of\na propagating beam is completely contained in the vol-\numeVin,Etot=Ein. However, as we decrease RYIG, the\nmodeshape spreads out beyond Vin, due to diffraction.\nAs a result, there is reduced overlap between the phonon\nand the magnon mode profiles in the YIG region. Cor-\nrespondingly, some acoustic energy also leaks out of Vin\nand spreads into the GGG region. We refer to the en-\nergy of the beam lying outside VinasEout. Figure 7(b)\n(black) shows that the ratio Eout/Etotincreases in the\nhigh-diffraction regime, with decreasing RYIG.\nWe combine the magnon-phonon coupling strength\n(Fig. 7(b)) and the phonon lifetimes (Fig. 7(a)) to de-\ntermine the performance figure of merit of the HBAR\nstructures, defined by the magnon-phonon cooperativity,\nC= 4g2\nmb/κmκb= 4g2\nmbτmτb. We show the variation\nofCwith RYIGin Fig. 7(c). We assume the magnon\nlifetime to be τm= 0.07µs [57, 58]. We compute the\ncooperativity values using τbpredicted from eigenvalue\nmethod (red line), exponential fitting (blue and purple\nlines), and clipping methods (green line), respectively.\nWe obtain a monotonically decreasing diffraction-limited\nCranging from 21 .9×104to 46.4, using τfrom the eigen-\nvalue method, as we decrease RYIGfrom 200 µm to 10 µm.\nOn the other hand, Cis in the range between 1 .2×104\nand 1 .4, predicted using τfrom the clipping method. As\ncan be noted from Fig. 7, τ,gmb, and Cpredicted by the\nHT method are in an excellent agreement with a-FBPM\npredictions, for all RYIGcases considered. The close\nmatch validates our implementation of the HT method.\nAs we have discussed earlier, a-FBPM has broader appli-\ncability compared to HT method, and can be applied for\nsystems with anisotropic material properties and trans-\nducer shapes. However, we find that it is an expensive\nand uncertain process to converge the residual since the\nrates of convergence for different HBAR structures are\nslow and variable. Also, one needs to select the num-\nber of round trips before restarting the simulation using\na trial-and-error approach. In comparison, HT method\nis computationally inexpensive to make predictions. Ad-\nditionally, it can be applied to the YIG/GGG HBAR\nsystems since they are isotropic with YIG transducer be-\ning a circular thin film disc. Henceforth, we only present\npredictions from the HT method in this article.\nNote that our calculations predict high cooperativityfor even the HBAR with RYIG= 10 µm. This implies\nthat if material acoustic attenuation effects are elimi-\nnated (e.g. by operating in the milli Kelvin regime), one\ncan achieve high cooperativity for planar HBAR struc-\ntures. However, the phonon lifetime of the HBAR struc-\nture with RYIG= 10 µm is limited to 10 .5µs (0.31µs),\nas predicted by the eigenvalue (clipping) methods and\nthe maximum magnon-phonon coupling strength is only\n55.7% of g0\nmb. We explore design approaches to further\nimprove these performance parameters.\nEnhancing integration density and Performance in\nDiffraction-Limited Regime\nFIG. 9. Variation of magnon-phonon coupling,\ngmb/g0\nmb, with varying waist of fundamental phonon\nmode: gmb/g0\nmbis maximum when w0/RYIG∼0.65.\nHere, we illustrate that the use of focusing dome-like\nsurface structures could significantly improve the perfor-\nmance of HBAR structures. Past studies demonstrated\nthat the confocal HBAR structures could achieve Q-\nfactors on the order of 107, while resulting in 103-fold\nreduction in device volumes [76]. However, such struc-\ntures have not been explored for hybrid magnomechan-\nical systems. We show that both phonon lifetimes and\nmagnon-phonon coupling can be improved by employ-\ning confocal geometries, leading to improved performance\nof hybrid magnomechanical systems. We first identify\nthe optimal domeshape that will result in the maximum\noverlap between the CHBAR phonon and the magnon\nmodeshapes, and consequently, the highest gmb. We ob-\ntain theoretical estimates of the overlap and identify the\ndome shape where the overlap is maximum. We assume\nthat the phonon modeshapes in CHBAR structures can\nbe described by Laguerre-Gaussian (LG) functions. The\nfundamental mode, LG 00, can be assumed to be of the14\nFIG. 10. Performance of CHBAR structures with varied radius of curvature, Rcurv, of the dome-shape: (a)\nPhonon lifetimes, τ, calculated from the eigenvalue (red) and clipping methods (green), following the HT method, (b) Ratio\nbetween magnon-phonon coupling in CHBAR structures and that in the zero-diffraction limit, gmb/g0\nmband (c) Magnon-phonon\ncooperativity, C.Rcurvis varied keeping RYIG= 10µm and Rcross= 60µm fixed. Solid lines represent corresponding values for\na planar HBAR structure with RYIG= 10µm, while the circles of the same color correspond to the CHBAR values.\nform: uy∝e−R2/w2\n0, due to the axi-symmetry of our sys-\ntem. Here, Ris the radial coordinatep\nx2+y2.w0is the\nwaist of the Gaussian beam at z= 0 (see Fig. 1), that can\nbe varied by changing tHBAR ,ω, or the dome radius of\ncurvature, Rcurv. We normalize the modeshapes accord-\ning to Eqs. 16, 17 and calculate the coupling strengths\nusing Eq. 15. Figure 9 shows the computed gmb/g0\nmbra-\ntio as a function of the beam waist, w0. We find that\nthe peak of gmb/g0\nmboccurs at w0/RYIG∼0.65.gmb\ndecreases sharply if w0≲0.65RYIG, however, decreases\nslowly if w0is increased beyond the optimal value. In-\nterestingly, we find that w0/RYIGis a constant and does\nnot depend on RYIG, for all the CHBAR structures con-\nsidered in this article. We use the w0value to obtain an\ninitial estimate of the radius of curvature, Rcurv, of the\ndome [77]:\nRcurv=1\nRe[1/(q0+tHBAR )], (19a)\nwith q0=iπw2\n0\nλ. (19b)\nWe obtain the analytical estimate to be Rcurv/tHBAR∼1.5\nthat corresponds to w0/RYIG∼0.65 and the peak of\ngmb/g0\nmbas shown in Fig. 9. Note that Rcurvcan be sim-\nilarly estimated for various device thicknesses ( tHBAR ),\nwavelengths ( λ) and waists by appropriately accounting\nfor these variables shown in Eq. 19. This Rcurvestimate\nresults in maximum gmb/g0\nmb, however, this analysis does\nnot consider the effects of the dome shape on phonon life-\ntimes and cooperativities. Also, we ignore the effects of\nthe lateral extent of the HBAR structure, the localizing\neffects of the YIG film, and assume phonon modeshapes\nto be perfectly Gaussian.\nTo obtain a Rcurvestimate that improves the overall\nperformance of these CHBAR structures, we carry out\na systematic numerical analysis using the HT method.We modify the reflection operator at the upper GGG\nsurface to include the effects of the lateral extents of the\ndome and the device, the attenuation window, and the\nlocalization effects of the YIG film. We include the phase\ninduced by the dome surface in Eq. 6 and the radial form\nof Eq. 11c and write the modified reflection opertor as\nRtCHBAR ,m(x, y) =\n\neikz0,mr2/Rcurv,ifr≤Rcross\n1, ifRcross< r≤Weff\n2\n0. otherwise\n(20)\nWe vary Rcurv/tHBAR across the analytical estimate,\n∼1.5, and identify Rcurvthat results in the optimal per-\nformance of the CHBAR structures. We show the effect\nof the focusing dome on the phonon lifetime, magnon-\nphonon coupling and cooperativity of the CHBAR struc-\nture in Fig. 10(a), (b) and (c), respectively. We show\nthe predictions from the eigenvalue method and the clip-\nping method using red and green circles, respectively.\nWe also show the corresponding performance parame-\nters of the planar HBAR structure with red and green\nhorizontal lines, respectively. For the clipping method,\nwe consider the volume, Vin, defined by the dome’s lat-\neral cross-sectional area and the CHBAR thickness. We\nfixRYIG= 10µm and the dome x−ycross-section ra-\ndius, Rcross = 60 µm, for this analysis. As can be seen\nfrom Fig. 10(a), the phonon lifetimes are maximum at\nRcurv/tHBAR = 1.09. This value is less than the ana-\nlytical prediction for maximum gmb,Rcurv/tHBAR∼1.5.\nHowever, the peak value, τ∼218ms (red circles), is ∼500\ntimes larger than the value at Rcurv/tHBAR∼1.5, justi-\nfying the necessity of performing the systematic anal-\nysis. The τpeak value for the CHBAR structure is\n∼1.7×104(red circles) ( ∼7.3×104, green circles) times\ngreater than the corresponding planar HBAR τpredic-\ntion, shown with red (green) horizontal lines, respec-15\nFIG. 11. Performance of CHBAR structures with varied radius of cross-section, Rcross, of the dome-shape: (a)\nPhonon lifetimes, τ, calculated from the eigenvalue (red) and clipping methods (green), following the HT method, (b) Ratio\nbetween magnon-phonon coupling in HBAR structures and that in the zero-diffraction limit, gmb/g0\nmband (c) Magnon-phonon\ncooperativity, C.Rcross is varied keeping RYIG= 10µm and Rcurv/tHBAR = 1.09 fixed. Solid lines represent corresponding\nvalues for a planar HBAR structure with RYIG= 10µm, while the circles of the same color correspond to the CHBAR values.\ntively. The peak τpredicted by the clipping method\nis∼22.7ms, lower than those predicted by the eigenvalue\nmethod, as expected. We note that τdecrease sharply for\nRcurv/tHBAR <1. This corresponds to the situation when\nthe center of curvature of the dome is inside the HBAR\nstructure, resulting in a negative ‘ g−parameter’ [76],\nwhere g= 1−tHBAR\nRcurv. It is shown that one cannot obtain\nreal and finite Gaussian beam solutions for structures\nwith a negative g−parameter [76]. We find that this is\nthe case for the modes corresponding to Rcurv/tHBAR <1\nin our case, that is, they are lossy modes which do not\nhave well-defined Gaussian characters.\nFigure 10(b) shows that the coupling, gmb/g0\nmb, is ap-\nproximately 0.99 at Rcurv/tHBAR∼1.4, a value close to\nthe analytical estimate of ∼1.5. These results show that\nreducing the phonon mode volume by means of focus-\ning does not necessarily improve gmbbeyond the max-\nimum achievable value, g0\nmb, which represents the case\nwhen there is complete lateral overlap of phonon and\nmagnon modes. We use τandgmbto calculate the figure\nof merit, C, and show the results in Fig. 10(c). The peak\nCvalue is predicted to be 2 .5×106(2.6×105), using\nthe eigenvalue (clipping) method. The Cvalues reach\npeak at Rcurv/tHBAR = 1.09, where τis maximum, as\nwe have noted from Fig. 10(a). The τandCvalues of\nthe CHBAR structure are several orders of magnitude\nimproved compared to its planar counterpart and the\ncoupling reaches up to 90% of g0\nmb. This result high-\nlights that τis the dominating factor that controls the\nperformance of CHBAR structures, since gmbis always\nlimited to the maximum value, g0\nmb. Note that we only\nvary Rcurvfor this analysis and keep the radius of the\ndome cross-section, Rcross, fixed at 60 µm. Due to the\nfixed cross-sectional area of the dome, the lateral inte-\ngration density of these devices, Di=A0\nd\nAd, is limited to\n∼64. Here, Adis the cross-sectional area of the confocaldome surface and A0\nd= 0.72 mm2[58].\nTo further improve the integration density, we analyze\nCHBAR structures with reduced dome cross-sectional\narea. We keep Rcurv/tHBAR = 1.09 and RYIG= 10µm\nfixed for this analysis. Figure 11 (a), (b) and (c) show\nthe variation of τ,gmb, and Cwith Rcross/RYIG, re-\nspectively. As we decrease Rcross/RYIGfrom 6 to 1, τ\nis reduced by more than 4 orders of magnitude from\n∼218 ms (22.7 ms) to 4 .83µs (0.134µs), as predicted by\nthe eigenvalue (clipping) method. gmbshows a sharp\ndecline for Rcross/RYIG<3 reaching the lowest value of\ngmb/g0\nmb∼0.32 at Rcross/RYIG= 1, however, remains\nmostly unchanged for Rcross/RYIG≥3. The coopera-\ntivity follows a similar trend to that of τ, as shown in\nFig. 11(c). Using the predictions from the eigenvalue\n(clipping) method, we obtain that Cis decreased by more\nthan 5 orders of magnitude from 2 .5×106(2.6×105) to 7.2\n(0.19) as we decrease Rcross/RYIG. Additionally, τ,gmb,\nandCof these CHBAR structures is reduced below those\nof the planar HBAR values, shown with solid lines, if\nRcross/RYIG<2. Considering all the different aspects, we\nargue that the optimal geometry for the CHBAR struc-\ntures is represented by the condition Rcross/RYIG= 4.5.\nOur analysis using the HT eigenvalue (clipping) method\npredicts that the diffraction-limited lifetime of τ=144.7\nms (11.1 ms) and a integration density of Di= 113 could\nbe achieved at a cooperativity C= 1.64×106(1.25×105)\nfor the CHBAR structure with Rcurv/tHBAR = 1.09,\nRYIG= 10µm and Rcross/RYIG= 4.5. The results pre-\nsented in Figs. 10 and 11 provide a proof-of-concept\nthat including a focusing dome improves the performance\nof the hybrid magnonic HBAR structures for quantum\nmemory and transduction applications.16\nCONCLUSION AND OUTLOOK\nIn this article, we establish a modeling approach that\ncan be broadly used to design hybrid magnonic HBAR\nstructures for high-density, long-lasting quantum memo-\nries and efficient quantum transduction devices. We il-\nlustrate the approach by discussing the magnon-phonon\ntransduction properties of hybrid magnonic YIG/GGG\nHBAR structures. We present analytical and numer-\nical analyses of the bulk acoustic wave phonon mode\nlifetimes, and the magnon-phonon coupling strengths in\nplanar and confocal YIG/GGG HBAR structures. We\ndiscuss strategies to improve the phonon mode lifetimes\nof these structures, since increased lifetimes have di-\nrect implications on the storage times of quantum states\nfor quantum memory applications. Additionally, high\nintegration density of on-chip memory or transduction\ncenters is naturally desired for high-density memory or\ntransduction devices. In our structures, the transduction\ncenters are represented by the YIG films and thus, inte-\ngration density can be increased by reducing the lateral\ndimension of these films. However, the reduced aperture\nresults in high-diffraction that affects the performance\nof these devices. We analyze the diffraction-affected\nshear wave phonon modes in the HBAR structures us-\ning two different methods: (1) Fourier beam propagation\n(FBPM) and (2) Hankel transform (HT) eigenvalue prob-\nlem method. The FBPM method has been widely used\nto analyze beam propagation in the field of optics, and\nmore recently, to study HBAR phonons in planar and\nconfocal HBAR (CHBAR) structures. We find that the\nFBPM method often requires us to analyze large and\nunpredictable number of round trips to obtain converged\nphonon modes. We implement an adaptive FBPM (a-\nFBPM) approach that mostly overcomes the slow con-\nvergence issues. However, we find that a-FBPM still suf-\nfers from convergence challenges when applied to con-\nfocal HBAR structures. To circumvent the challenges,\nwe implement the HT method that allows us to ob-\ntain converged phonon modes with significantly reduced\ncomputational cost. The HT method leverages the axi-\nsymmetry of the problem and the isotropic material prop-\nerties of the YIG/GGG HBAR structures to reduce the\n3D problem to a 2D one and expedite the analysis. The\nHT method has been mostly used in the field of optics,\ne.g., Fabry-Perot cavities, however, it has not been ap-\nplied for acoustics analysis, to the best of our knowledge.\nOur study provides key insights into the diffraction-\nlimited performance of the YIG/GGG HBAR structures.\nWe predict the diffraction-limited τto be on the order\nof milliseconds, for a planar HBAR structure with lateral\nYIG dimension, RYIG= 200 µm. A recent study reported\nthat the performance of a YIG/GGG HBAR structure\nat room temperature is limited by the phonon lifetime at\n0.25µs [58]. The previously studied structure had largerYIG lateral area than our structures and thus, the diffrac-\ntion effects were less dominant. The phonon lifetime\nin HBAR structures could be limited by both material\nand diffraction losses depending on the device geometry\nand/or the operating conditions. At room temperature,\nthe phonon lifetime is primarily limited by acoustic atten-\nuation effects. However, these effects are less significant\nat low temperatures while the diffraction losses play an\nimportant role. Therefore, our results will have high ap-\nplicability for devices operating in the cryogenic or po-\ntentially milliKelvin (mK) regimes. For example, our\napproach and analyses can be applied to design HBAR\ndevices that could effectively couple with superconduct-\ning qubit systems. We acknowledge that the analysis\nof the material losses is necessary to obtain a complete\nunderstanding of the performance of the YIG/GGG sys-\ntems. This may include an understanding of the different\nattenuation effects (e.g., Akhiezer and Landau-Rumer)\nat various temperatures and frequencies of interest, and\nstrategies to overcome them.\nAssuming that the material-limited lifetimes could be\nincreased to ∼0.1 ms at mK temperatures, we find that\nthe planar HBAR structures are not affected by diffrac-\ntion effects even at RYIG= 50 µm,. This structure al-\nready offers a significant 50-fold improved integration\ndensity over the reference structure [58]. The integra-\ntion density can be further improved by scaling down\nthe YIG film lateral area. However, the reduced aperture\nwill affect the phonon lifetime and the magnon-phonon\ncoupling strength, resulting in a decrease of the coop-\nerativity, the performance figure of merit for magnon-\nphonon transduction. It is therefore imperative to de-\nvelop a strategy that increases the integration density of\nHBAR structures without affecting the performance. Ad-\nditionally, the performance of planar HBAR structures\nwas found to be highly sensitive to the parallel nature\nof the surfaces. To address both these aspects, we in-\nvestigate confocal YIG/GGG HBAR structures with top\nfocusing domes and a planar bottom surfaces. Use of\nfocusing dome structures has been proposed to signifi-\ncantly improve the phonon lifetime and integration den-\nsity of these systems. Furthermore, the dome structures\neliminate the necessity of maintaining perfectly parallel\nsurfaces of planar HBAR structures. We first theoreti-\ncally estimate the shape of the dome structure for which\nthe magnon-phonon coupling strength, gmb, is at its max-\nimum value. We then perform a rigorous numerical anal-\nysis and obtain a refined set of shape parameters of the\ndome for which both τandCare optimal, and gmbis\nclose to its peak value. Overall, we find that ultra-high,\ndiffraction-limited, cooperativities and phonon lifetimes\non the order of ∼105and∼10 ms, respectively, could be\nachieved using a CHBAR structure with RYIG= 10µm.\nIn addition to enhanced τandC, the confocal HBAR\nstructure will offer more than 100-fold improvement of\nintegration density.17\nIn this work, we discuss the diffraction-limited perfor-\nmance induced by the lateral area of the YIG film and\nthe dome structure. It will be interesting to apply the\ninsights presented in the article and explore the applica-\nbility of YIG/GGG HBAR structures for quantum trans-\nduction applications, as future work. For example, these\nHBAR structures could be used for coupling with other\nquantum information carriers such as superconducting\nqubits, which operate in the microwave frequency regime\nand at mK temperatures. However, to optimize the per-\nformance of such devices, a further comprehensive anal-\nysis of different geometric and physical parameters will\nbe necessary. For example, we keep the thickness fixed\nat 527 .2µm for all YIG/GGG HBAR structures inves-\ntigated in this work. The same thickness allows us to\nkeep the free spectral range of phonons fixed for all our\nanalysis. The different geometric parameters, such as the\noverall thickness, the thickness of the YIG and the GGG\nregions, geometrical misalignment, and imperfections, or,\nthe physical parameters, such as the photon and magnon\nlifetimes and their coupling strengths, could also play\nimportant role in the overall device performance. We as-\nsume that these parameters remain invariant in our anal-\nysis. A comprehensive optimization analysis can be per-\nformed using the current state-of-the-art machine learn-\ning techniques, which is a promising research direction for\nthe future. Additionally, we only discuss the coupling be-\ntween the fundamental magnon mode and the fundamen-\ntal phonon modes of the planar and the confocal HBAR\nstructures. It will be interesting to consider the effect of\nhigher-order mode couplings in the device performance.\nThe higher-order mode couplings can be particularly rel-\nevant here since these structures host long-lasting phonon\nmodes with broadband nature. Other interesting direc-\ntions to explore are the anharmonic phonon interactions\nat the surface and interfaces, and the coupling of bulk\nacoustic waves and the surface acoustic waves. Our study\nis concerned with coherent states, and it will be impor-\ntant to explore how the insights provided here translate\nto systems involving non-classical states, e.g. squeezed\nstates, cat states, and Fock states.\nACKNOWLEDGEMENTS\nThis work was partially supported by funding from\nthe Quantum Explorations in Science & Technology\n(QuEST) grant provided by the University of Colorado\nBoulder Research & Innovation Office in partnership\nwith the College of Engineering and Applied Science,\nthe College of Arts and Sciences, JILA, and the Na-\ntional Institute of Standards and Technology (NIST).\nWe acknowledge the computing resources provided the\nRMACC Summit supercomputer, which is supported by\nthe National Science Foundation (awards ACI-1532235\nand ACI-1532236), the University of Colorado Boulder,and Colorado State University. 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Here we demonstrate an analogous spin effect near the N ´eel tem-\nperatureTN=296 K of the antiferromagnetic insulator Cr 2O3. Using a yttrium iron garnet\nYIG/Cr 2O3/Pt trilayer, we injected a spin current from the YIG into the Cr 2O3layer, and\ncollected via the inverse spin Hall effect the signal transmitted in the heavy metal Pt. We\nobserved a change by two orders of magnitude in the transmitted spin current within 14 K\nof the N ´eel temperature. This transition between spin conducting and nonconducting states\ncould be also modulated by a magnetic field in isothermal conditions. This effect, that we\nterm spin colossal magnetoresistance (SCMR), has the potential to simplify the design of\nfundamental spintronics components, for instance enabling the realization of spin current\nswitches or spin-current based memories.\nSpin current is a flow of spin angular momentum sharing many analogues with electric cur-\nrents. Spin currents can be carried not only by migrating electrons, but also by magnetic quasi-\nparticles such as magnons or spin waves3, 4. These magnetic excitations are of particular interest in\nthe spintronics community because they allow a spin current to flow in electrical insulators where\ncharge currents cannot5–9. Thus, insulator spintronics may provide a route towards low power de-\nvices where spin currents carry signals and encode information. However, it is not straightforward\nto create all spintronic components which are analogous to their electronic counterparts10. For ex-\nample, a basic spin-current on-off switch has not been demonstrated in insulator spintronics. The\ndifficulty lies in the lack of a simple mechanism to directly gate the carrier density in magnetic\n2insulators. Nevertheless, some clues may be obtained from colossal magnetoresistance (CMR) oc-\ncurring in materials which exhibit metal-insulator transitions, where a large modulation of charge\nconductivity can be induced by a magnetic field1, 2(Fig.1 1a). Spin conductivity may be tunable in\nsystems with a spin conducting-nonconducting transition (Fig.1 1b).\nIn this Letter, we report such a ‘conductor-nonconductor transition’ for spin currents in the\nuniaxial antiferromagnetic insulator Cr 2O311–13. We found that Cr 2O3does not conduct spin cur-\nrents below the N ´eel temperature, but abruptly becomes a good spin conductor above this tempera-\nture. Furthermore, in the vicinity of the transition, the spin-current transmission can be modulated\nby up to 500% with a 2:5T magnetic field: spin colossal magnetoresistance (SCMR). The active\ntransport element is spin angular momentum rather than electrical charge.\nThe spin-current transmission in Cr 2O3was studied by using a trilayer device, sandwiching a\nCr2O3thin film between a magnetic insulator yttrium iron garnet (YIG) and a heavy metal Pt layer\n(Fig. 2a). Here, YIG serves as a spin-current source. By using a temperature gradient, rT, along\nthe out-of-plane direction z, the spin Seebeck effect (SSE)6, 14generates a spin accumulation at\nthe interface of YIG/Cr 2O3, which drives a spin current ( Jin\ns) into the Cr 2O3layer. Spin currents,\ntransmitted through the Cr 2O3layer to the Pt interface ( Jout\ns), are converted into a measurable\nvoltage via the inverse spin Hall effect (ISHE)15.\nWe define the spin-current transmissivity Ts=jJout\nsj=jJin\nsj, describing the relative amount of\nspin-current incident on the YIG/Cr 2O3interface which is transmitted to the Cr 2O3/Pt interface.\nWe ignore the effect of the spin accumulation at the YIG/Cr 2O3interface on Jin;out\ns, which will a\n3posteriori be justified as the data are well explained solely by the intrinsic properties of the Cr 2O3.\nThe spin current entering the Cr 2O3,Jin\ns, flows alongrTand is polarized along M, where Mis\nthe magnetization of the YIG layer which can be easily manipulated by the external magnetic field\nH. The ISHE voltage measured in the Pt layer is thus\nVSSE/Jout\ns=Ts\u0012\nJin\ns\u0002M\njMj\u0013\n\u0001^x; (1)\nwhere ^xis the unit vector along the xaxis.Jin\nscan be roughly estimated by the SSE signal in the\nYIG/Pt device in which the Cr 2O3thickness is zero and the SSE signal in YIG/Cr 2O3/Pt devices\nyields Jout\ns16, 17. Therefore, the spin-current transmissivity Tscan be estimated from Jin\nsandJout\ns\n(Eq.1).\nFigures 2c and 2d show the field dependence of the measured voltage Vfor a YIG/Pt bilayer\nand the YIG/Cr 2O3/Pt trilayer. In both samples—with and without a Cr 2O3layer—the sign of V\nreverses with the sign of H, and the shape of the V-Hcurves agree with the M-H(hysteresis)\ncurve of the YIG film18–20. This confirms that the measured voltage Vin the YIG/Cr 2O3/Pt trilayer\ndevice is induced by the thermal spin currents generated from the YIG.\nFirst, we show a steep conductor-nonconductor transition for spin currents in Cr 2O3. Figure\n2e shows the temperature dependence of the SSE voltage VSSE(H= 0:1T) for the YIG/Cr 2O3/Pt\ntrilayer device. Surprisingly, the voltage exhibits an abrupt change of more than 100\u0002around 290\nK. Above this temperature, a voltage with a peak of VSSE\u0019500 nV appears at T= 296 K. When\nT < 282 K,VSSEis close to the noise floor \u00195 nV (Fig. 2e). By contrast, in the YIG/Pt bilayer\ndevice,VSSEvaries little across the same temperature range (Fig. 2f)21, indicating Jin\nsis nearly\n4constant. This equivalently means that the spin-current transmissivity Tsof Cr 2O3changes more\nthan100\u0002around 290 K, which is calculated according to Eq.1 and plotted in the supplementary\nFig. 2c.\nWe attribute the abrupt change of VSSEin the YIG/Cr 2O3/Pt device to the change in the spin-\ncurrent transmissivity Tsof the Cr 2O3layer, marking the transition of the Cr 2O3layer from a spin\nconductor to a spin nonconductor at T=296 K. This critical temperature coincides with the N ´eel\ntemperature of the Cr 2O3thin film22, 23, and we associate the change in spin-current transmissivity\nwith the onset of magnetic order. We found a similar spin conductor-nonconductor transition\nin a spin pumping measurement for devices with the same YIG/Cr 2O3/Pt structure as shown in\nFig. 2g (also refer to supplementary Note 1), demonstrating that the spin conductor-nonconductor\ntransition in Cr 2O3does not depend on the method of spin current generation. We also ruled\nout magnetic interface effects between the exchanged coupled YIG and Cr 2O3(such as exchange\nbias or spin reorientation transitions) causing the large change of Ts. Using a control sample with\na 5-nm Cu layer (a nonmagnetic metal but good spin conductor) inserted between the YIG and\nCr2O3layers, we observed results similar to that in the YIG/Cr 2O3/Pt trilayer (Fig. 2h, also refer\nto supplementary Note 2). By measuring the VSSEfor a Cr 2O3/Pt bilayer, we also confirmed that\nVSSEcomes from spin current generated in the YIG and transmitted through the Cr 2O3, rather than\nby spin current originating within Cr 2O3(Fig. 2h, also refer to supplementary Note 2).\nHaving established the spin conductor-nonconductor transition, we show that the spin-current\ntransmissivity of Cr 2O3has an anisotropic response to magnetic fields in the critical region of the\n5magnetic transition. The spin-current transmissivity of Cr 2O3depends not only on the magnitude\nbut also on the direction of the magnetic field. By using the SSE and ISHE as sources and probes\nof spin currents, within the critical region we measured the dependence of VSSEon the magnetic\nfield magnitudejHjand angle\u0012in thez-yplane as illustrated in Fig. 3a.\nFigure 3b shows the dependence of VSSEon the angle \u0012at different magnetic field magni-\ntudes. AtT=296 K (in the spin conducting regime), VSSEshows a sinusoidal change with respect\nto\u0012, the same as the relative angle between the YIG magnetization MandJin\nsas expected from\nEq. (1). The magnitude of VSSEchanges only slightly from jHj=0.5 T to 2.5 T. Similar behaviour\nis observed for T > 296 K, indicating that Tsdepends only weakly on \u0012orHin the spin conductor\nregime. However, at T < 296 K,VSSE(\u0012)starts to deviate from this dependence. As the tem-\nperature decreases further, the character of VSSE(\u0012) changes completely. The maximum amplitude\nofVSSEno longer resides at \u0012=\u000690\u000ebut peaks four times through the rotation ( \u0000180\u000e,180\u000e).Ts\nalso becomes strongly dependent on jHj. Thus,Ts(\u0012,H) depends on both \u0012andjHjin the critical\nregion.\nFigure 3c shows the temperature dependence of VSSE(jHj)at\u0012=20\u000e, where thejHjde-\npendence is the most pronounced. The temperature dependence of VSSEis qualitatively similar\nfor all field strengths, featuring a sharp transition between the spin nonconductor and conductor\nregimes. However, the transition edge of VSSEshifts to lower temperatures for stronger magnetic\nfields. TakingjHj= 0:5 Tas a reference,\u0018500% modulation of VSSEis achieved with a 2:5T\nfield (Fig. 3d).\n6Above the N ´eel temperature, the paramagnetic moments of Cr 2O3follow the external mag-\nnetic field and spin current is carried by correlations of the paramagnetic moments as has been\nreported previously9, 16, 24.\nBelow the N ´eel temperature—in the ordered antiferromagnetic phase—the propagation of\nthe spin current is in principle determined by the thermal population of magnons, the magnon\nmean free path and the magnon gap. However, the magnon gap is approximately 10 K12, therefore\nthis description by itself cannot lead to the sharp transition observed at the N ´eel point. In other\nwords, the nonconducting regime cannot be caused by magnon freezing. Rather, it is caused by\nthe anisotropic transmissivity of the antiferromagnet in combination with the device geometry.\nOnly the spin component which is parallel (or antiparallel) to the N ´eel vector can be carried\nby magnons25. Below the N ´eel temperature, due to the strong uniaxial anisotropy, N ´eel vector of\nCr2O3is pinned to the easy axis (out of plane in this work). When the YIG magnetization is in\nthe plane of the film, the spins are polarized perpendicularly to the Cr 2O3N´eel vector and the spin\ncurrent cannot be transmitted into the Cr 2O3. When the YIG magnetization is out of the plane,\nthe spin current can be transmitted but the device geometry prohibits the generation of an ISHE\nvoltage. Furthermore, the strength of the anisotropy in Cr 2O3is almost independent of temperature,\ncollapsing to zero only very close to the N ´eel temperature12. Therefore, the Cr 2O3is strongly\naligned perpendicular to the plane for almost the entire temperature range and no spin current can\nbe transmitted. This small temperature window where the anisotropy decreases corresponds with\nthe increase in ISHE voltage.\n7In the region just below the N ´eel temperature, where the anisotropy is reducing, the trans-\nmissivity can be manipulated with the applied field. The enhanced susceptibility and reduced\nanisotropy in this small temperature window allows the N ´eel vector to be slightly rotated, giving a\nfinitey-component (in the plane) on to which the spin current is projected12, 13. The field induced\nN´eel vector and magnetization y-components of the antiferromagnet (Fig. 3e) are\nLAF\ny\u0019\u0000MsH2\n2HexchHanisin 2\u0012;MAF\ny\u0019MsH\nHexchsin\u0012; (2)\nrespectively. Msis the saturation magnetization of the antiferromagnetic sublattices, Hexchis the\nexchange field between the sublattices, and Haniis the uniaxial anisotropy field. The equation is\nbased on a zero-temperature theory but by allowing the temperature dependence of HexchandHani,\nit appears to be a good approximation even up to the N ´eel temperature. At temperatures much\nbelow the N ´eel temperature, the field required to manipulate the N ´eel vector is approximately\nH\u00196T12. But, when Hanidrops in the transition window, much smaller fields (smaller than the\nspin flop field) can manipulate the N ´eel vector.\nUnder the assumption that spin transport is possible only for angular momentum along the\nN´eel vector and in the linear dynamics regime, we phenomenologically obtain the angular depen-\ndence of the ISHE voltage as\nV(\u0012) =\u0000aLAF\nycos\u0012+bMAF\ny (3)\nwhereaandbare phenomenological parameters. Equations 2 and 3 qualitatively reproduce our\nexperimental results for the \u0012dependence (Fig. 3b) of the voltage (see supplementary Note 4 for\ndetails).\n8In summary, we report the occurrence of the spin conductor-nonconductor transition and\nthe field induced modulation of spin-current transmissivity in Cr 2O3, which is reminiscent of the\nCMR in electronics. We attribute this ‘colossal’ modulation of spin current to the combination of\nthe anisotropic spin current transmission of the antiferromagnet and the device geometry, which is\ncorrelated to the N ´eel vector and anisotropy of Cr 2O3. The SCMR may also be observed in other\nantiferromagnetic materials in which the N ´eel vector responds to magnetic fields. It may therefore\nbe possible to create devices which switch between the spin insulating and conducting states—but\nin response to a completely different stimulus. For example switching the antiferromagnet between\nperpendicular states electrically26.\n1. Ramirez, A. 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Science 351, 587–590 (2016).\nAcknowledgements This work was supported by JST-ERATO ‘Spin Quantum Rectification’, JST-PRESTO\n‘Phase Interfaces for Highly Efficient Energy Utilization’, Grant-in-Aid for Scientific Research on Innova-\ntive Area, ‘Nano Spin Conversion Science’ (26103005 and 26103006), Grant-in-Aid for Scientific Research\n(S) (25220910), Grant-in-Aid for Scientific Research (A) (25247056 and 15H02012), Grant-in-Aid for Chal-\nlenging Exploratory Research (26600067), Grant-in-Aid for Research Activity Start-up (25889003), and\nWorld Premier International Research Center Initiative (WPI), all from MEXT, Japan, ZQ acknowledges\nsupport from ”the Fundamental Research Funds for the Central Universities (DUT17RC(3)073)”. DH ac-\nknowledges support from Grant-in-Aid for young scientists (B) (JP17K14331), JB acknowledges supports\nfrom the Graduate Program in Spintronics, Tohoku University, and Grand-in-Aid for Young Scientists (B)\n(17K14102). KY and OG acknowledge support from Humboldt Foundation and EU ERC Advanced Grant\nNo. 268066. KY acknowledges the Transregional Collaborative Research Center (SFB/TRR) 173 SPIN+X\nand DAAD project ”MaHoJeRo”. O.G. acknowledge the EU FET Open RIA Grant no. 766566 and the\nDFG (project SHARP 397322108).\nAuthor contributions Z.Q. and D.H. designed the experiment, Z.Q. fabricated the samples and collected\nall of the data. Z.Q., D.H., J.B. and K.Y . analyzed the data. J.B., K.Y . and O.G. contribute theoretical\ndiscussions. E.S. supervised this study. All the authors discussed the results and prepared the manuscript.\nCompeting Interests The authors declare that they have no competing financial interests.\n12Correspondence Correspondence and requests for materials should be addressed to D.H. (email: dazhi.hou@imr.tohoku.ac.jp).\nMethods\nPreparation of YIG/Cr 2O3/Pt samples. We grew a 3 \u0016m-thick single-crystalline YIG film on a\n(111) Gd 3Ga5O12wafer by liquid phase epitaxy method at 1203 K in PbO-B 2O3based flux. We\ncut a single wafer into 1.5 mm \u00023 mm in size. A 12 nm-thick Cr 2O3film is grown on top of the\nYIG film by pulsed laser deposition at 673 K and subsequently annealed at 1073 K for 30 min to\nobtain continuous films and improve the crystallinity. 10 nm-thick Pt films were then grown on the\ntop of the Cr 2O3by RF magnetron sputtering.\nSample characterization. Crystallographic characterization for the samples was carried out by X-\nray diffractometry and transmission electron microscopy (TEM). The obtained TEM image shows\nthat the YIG film is of a single-crystal structure, and the easy axis ( c-axis) of the hexagonal Cr 2O3\ngrown on the top of the YIG is along the out-of-plane direction z(Fig. 2b).\nSpin Seebeck experimental set-up. We performed spin Seebeck measurements in a vector mag-\nnet system (Oxford instruments inc). We set the samples on a wave guide and heated the Pt layer\nby using a pulsed microwave27(8 GHz, 1 W), which creates a temperature gradient as shown in\nFig. 2a. We measured the voltage signal between the ends of the Pt layer using a lock-in amplifier.\nData availability\nThe data that support the findings of this study are available from the authors on reasonable\n13Figure 1 Concept of spin colossal magnetorresistance\nSpin current\nSpin currentSpin conductorSpin nonconductor\nSpin\nconductor-nonconductor\ntransitionMetal-insulator\ntransition\nCharge current Charge current \nCharge conductorCharge insulator\nH H SCMR\nSpin colossal magnetoresistance CMRColossal Magnetoresistance a b\nFigure 1: Concept of spin colossal magnetoresistance. a . A schematic illustration of colossal\nmagnetoresistance (CMR). CMR is a property of some materials in which their electrical resistance\nchanges steeply in the presence of a magnetic field, typically due to the strong coupling between a\nsteep metal-insulator transition and a magnetic phase transition. b. A schematic illustration of spin\ncolossal magnetoresistance (SCMR): spin current transmissivity changes steeply due to the change\nin symmetry (in this paper due to a magnetic phase transition). The spin current transmissivity is\nalso modulated by an applied magnetic field.14Figure 2 Spin insulator-metal transition in Cr2O3\nTN\nT (K)100 200 300 V (nV)\n-0.1 0.1 0\nH (T)296 KT=320 K\n280 KPtVSSE\nCr2O3\nY3Fe5O12\nT=300 K\n288 Kc V (nV)\nY3Fe5O12Cr2O3Pt\nc\nad\n00\n0\n0\n0400\n-400\n400\n-400\n400\n-400400\n-4002k\n-2k\n VSSE@0.1 T (nV)a\nfe VSSE@0.1 T (nV)0\n02k4k200400-0.1 0.1 0H (T)\nVSSEJsY3Fe5O12/Pt\nY3Fe5O12/Cr2O3/Pt\nY3Fe5O12/PtY3Fe5O12/Cr2O3/Pt\ne↑– e↓– out\ninJs\n∇T\nH \nb\n VISHE (nV)2000\n1000\n0g Y3Fe5O12/Cr2O3/Pt\ndCr2O3 =12 nm\nYIG/Cr2O3/Pt\nCr2O3/PtVSSE (nV)400\n200\n0\nT (K)350 250 300h\nYIG/Cu/Cr2O3/PtdCr2O3 =\n7.5 nm\n12.0 nm\n24.0 nm\n36.0 nm xz\nyFigure 2: Spin conductor-nonconductor transition in Cr 2O3. a. A schematic illustration shows\nthe concept of the spin-current transmissivity measurement of a YIG/Cr 2O3/Pt trilayer device. A\ntemperature gradient, rT, is along the zdirection while an external magnetic field ( H) is along\ntheydirection. The magnetic insulator YIG is used as a spin source to inject spin currents Jin\ns\ninto the Cr 2O3based on the spin Seebeck effect, and transmitted spin currents Jout\nsthrough the\nCr2O3are detected as voltage signals in the Pt layer via the inverse spin Hall effect. b. A cross\nsectional TEM image of the YIG/Cr 2O3/Pt trilayer device used in this work. The scale bar of the\nimage is 5 nm. The easy axis cof the Cr 2O3is in the out-of-plane direction zof the film as inserted\naxis shows. c. The external magnetic field Hdependencies of the voltage signal Vmeasured in\na YIG/Pt bilayer device at 300 K. d. The external magnetic field Hdependencies of the voltage\nsignalVmeasured in the YIG/Cr 2O3/Pt trilayer device at various temperatures. e. The temperature\ndependence of the spin Seebeck voltage VSSEatH=0.1 T for the YIG/Cr 2O3/Pt trilayer device. f.\nThe temperature dependence of the spin Seebeck voltage VSSEatH=0.1 T for a YIG/Pt bilayer\ndevice. g. The temperature dependence of spin pumping signals VISHE for YIG/Cr 2O3/Pt trilayer\ndevices with various values of the Cr 2O3layer thickness dCr2O3.h. The temperature dependence\nof the spin Seebeck voltage VSSEatH=0.1 T for Y 3Fe5O12/Cu/Cr 2O3/Pt and Cr 2O3/Pt devices.\nThe measurement errors are smaller than the size of data points in all figures.15a\nb\nFigure 3 Spin colossal magnetoresistance in  Cr2O3.H \n∇T θV \nxyz\n-4040T=278 K0\n-80\n080T=284 K0\n-250250\n-90 90T=290 KVSSE@H (nV)\nθ (°)e\n02.5 T\nH=0.5 T0\n-400400T=296 K\nH\nLAFz\ny\nφθ\nMAFMA\nMB\ny y yH H\nHz z z\nθ=0°LAFLAF LAF\nθ=90° 0°<θ<90°Cr2O3\nRatio(    )@H (%)c\nd100200 VSSE@H (nV)\n300 250\nT (K)02.5 T\n2.1 T\n1.7 T\n1.3 T\n0.9T5002.5 Tθ=20°\nH=0.5 T0H=0.5 TFigure 3: Spin colossal magnetoresistance in Cr 2O3. a. A schematic illustration of the out-of-\nplane spin Seebeck set-up for the YIG/Cr 2O3/Pt trilayer device. A temperature gradient, rT, is\nalong thezdirection, and an external magnetic field, H, is applied in the y-zplane.\u0012is the angle\nbetweenrTandH.b.\u0012dependencies of VSSEat different temperatures for the YIG/Cr 2O3/Pt\ntrilayer device with various values of H. Here,VSSErefers to the spin Seebeck voltage signal\ndetected from the Pt layer. The solid curve is a fitting result using Eq. (3). The noise level\nof the voltage measurement is about 5 nV , which is smaller than the size of data points in most\nfigures. c. Temperature dependence of VSSEfor the YIG/Cr 2O3/Pt trilayer device at different\nexternal magnetic fields Hat\u0012=20\u000e. The solid lines are guides to the eye. d. TheTschange\nratioRatio (Ts)@Hat an applied external magnetic field Has functions of temperature. Here,\nRatio (Ts)@H= (VSSE@ H\u0000VSSE@0 :5T)=VSSE@0 :5T.Tsrefers to the spin-current transmissivity in\nthe Cr 2O3layer. The solid lines are guides to the eye. e. A schematic illustration of the relation\nbetween a N ´eel vector, LAF, the induced magnetization MAF, and the external magnetic field H.\nHere,\u0012refers to the angle between Hand thezaxis;\u001erefers to the angle between LAFand the\neasy axiscof Cr 2O3, which is along the zaxis in our sample. MAandMBare the magnetization\nvectors of the two sublattices of Cr 2O3. The lower panel shows the relation between a N ´eel vector,\nLAF, the induced magnetization MAF, and the external magnetic field Hat different values of \u0012.16request, see author contributions for specific data sets.\n27. Vasyuchka, V . I. Microwave-induced spin currents in ferromagnetic-insulator—normal-metal\nbilayer system. Applied Physics Letters 105, 092404 (2014).\n17"    },    {        "title": "2307.09846v1.Enhanced_bipartite_entanglement_and_Gaussian_quantum_steering_of_squeezed_magnon_modes.pdf",        "content": "Enhanced bipartite entanglement and Gaussian quantum steering of squeezed magnon\nmodes\nShaik Ahmed,1M. Amazioug,2Jia-Xin Peng,3and S. K. Singha4,†\n1School of Technology, Woxsen University, Hyderabad, Telangana -502345, India\n2LPTHE-Department of Physics, Faculty of sciences, Ibn Zohr University, Agadir, Morocco\n3State Key Laboratory of Precision Spectroscopy,\nQuantum Institute for Light and Atoms, Department of Physics,\nEast China Normal University, Shanghai 200062, China\n4Graphene and Advanced 2D Materials Research Group, Sunway University, Malaysia\n(Dated: July 20, 2023)\nWe theoretically investigate a scheme to entangle two squeezed magnon modes in a double cavity-\nmagnon system, where both cavities are driven by a two-mode squeezed vacuum microwave field.\nEach cavity contains an optical parametric amplifier as well as a macroscopic yttrium iron garnet\n(YIG) sphere placed near the maximum bias magnetic fields such that this leads to the excitation of\nthe relevant magnon mode and its coupling with the corresponding cavity mode. We have obtained\noptimal parameter regimes for achieving the strong magnon-magnon entanglement and also studied\nthe effectiveness of this scheme towards the mismatch of both the cavity-magnon couplings and\ndecay parameters. We have also explored the entanglement transfer efficiency including Gaussian\nquantum steering in our proposed system.\nI. INTRODUCTION\nQuantum entanglement [1] and Gaussian quantum steering [2, 3] are two major important resources in the field\nof quantum computing [4], quantum cryptography [5] and quantum teleportation [6] including quantum information\nprocessing [7]. Many microscopic as well as macroscopic quantum systems have been proposed over the past decades for\nthe study of quantum entanglement and other nonclassical quantum correlations in superconducting qubits [8], atomic\nensembles [9], cavity optomechanics [10–16] and cavity magnomechanical (CMM) systems [17–22] which paves the\nway for advancements in present era of quantum technology. In CMM systems, the magnons defined as the collective\nexcitation of a large number of spins in ferrimagnetic materials play very important role in the study of light-matter\ninteractions due to their tunability, low damping, high spin density [23, 24] as well the strong coupling with the\nmicrowave photons [25–28]. Moreover, other important mascroscopic quantum phenomena such as magnon-induced\neffects [30], tunable magnomechanically induced transparency and absorption [31, 32], slow light [33], four-wave mixing\n[34], squeezed states [35–38], nonclassical quantum correlations [39, 40], microwave-to-optical carrier conversion [41]\nincluding quantum sensing [42, 43] also successfully investigated in cavity magnomechanical systems.\nTo quantify the bipartite entanglement between the magnon and microwave photon in CMM systems, we use a\nvery well-known witness of bipartite entanglement defined as the logarithmic negativity [44]. A recent theoretical\nwork given in [45] explored the logarithmic negativity between two magnon modes where the optimal conditions for\nachieving the strong magnon-magnon entanglement involve the resonant coupling between the microwave cavity with\nboth the magnon modes whereas in case of two microwave cavities given in [46–48], it is found that the detuning\nof the cavity and magnon mode significantly affects the bipartite entanglement. These research works also found\nthe presence of both one-way and two-way Gaussian quantum steering. So, all these studies demonstrate that the\nbipartite entanglement and quantum steering in CMM systems can be significantly controlled through the various\nphysical parameters. All these recent progress also broaden our understanding of quantum correlations and facilitate\nto further explore the possibility of secure quantum protocols in such kind of macroscopic quantum systems.\nMotivated by these works, we study the quantum correlations and Gaussian quantum steering between two squeezed\nmagnon modes of two yttrium iron garnet (YIG) spheres in a system of two spatially separated microwave cavities.\nEach cavity also contains an optical parametric amplifier (OPA) as well as a macroscopic YIG sphere placed near\nthe maximum bias magnetic fields such that this leads to the excitation of the corresponding magnon mode and its\ncoupling with the cavity mode. In addition, both the cavities are simultaneously driven by a two mode squeezed\nvacuum microwave field in our proposed system [49]. In this present work, we found the generation of a considerable\nbipartite entanglement between the two magnon modes with gradually increasing squeezing parameter and the mean\naCorresponding Author\n†singhshailendra3@gmail.comarXiv:2307.09846v1  [quant-ph]  19 Jul 20232\nthermal magnon number. Moreover, it can be seen clearly from our work that not all entangled states allow for\nquantum steering whereas any state that can be steered must necessarily be entangled.\nThis paper is organized as follows: In Section II, we introduce model Hamiltonian and also evaluate corresponding\nquantum Langevin equations including its solutions (QLEs). In Section III, we discuss in details about mathematical\nformulation of bipartite entanglement and Gaussian quantum steering between two magnon modes. Numerical Results\nand related discussions are given in Section IV whereas we conclude our results in Section V.\nII. THE MODEL HAMILTONIAN\nOur proposed system shown in Fig. 1 consists of two microwave cavities and two magnon modes in two YIG\nspheres, which are respectively placed inside the cavities near the maximum magnetic fields of the cavity modes and\nsimultaneously in uniform bias magnetic fields such that it causes both the magnon modes to strongly couple with\nrespective cavity modes [49, 50]. Each cavity, contains a degenerate optical parametric amplifier (OPA) to produce\nsqueezed light [51]. The Hamiltonian of the system in a rotating frame with frequency ωjcan be written as\nH=X\nj=1,2n\nℏ∆cjc†\njcj+ℏ∆mjm†\njmj+ℏgj\u0000\ncjm†\nj+c†\njmj\u0001\n+icℏλj(eicθc†2\nj−e−icθc2\nj) +icℏµj(eicνm†2\nj−e−icνm2\nj)o\n,(1)\nwhere ∆ cj=ωcj−ωj,∆mj=ωmj−ωj,cj(c†\nj),mj(m†\nj) are the annihilation (creation) operators of the jthcavity and\nmagnon modes, respectively, and we have\u0002\nO,O+\u0003\n= 1 (O=cj, mj).ωcj(ωmj) is the resonance frequency of the jth\ncavity mode (magnon mode). The frequency of the magnon mode ωmjis determined by the external bias magnetic\nfieldHjand the gyromagnetic ratio βviaωmj=βHj, and thus can be flexibly adjusted, and gjis the coupling rate\nbetween the jthcavity and magnon modes. The parameter λjandθjrepresents respectively the nonlinear gain of the\nOPA and the phase of the driving field. with µjbeing the squeezing parameter and νbeing the phase of jthsqueezing\nmode. The magnon squeezing can be achieved by transferring squeezing from a squeezed-vacuum microwave field\n[34], or by the intrinsic nonlinearity of the magnetostriction (the so-called ponderomotive-like squeezing) [52], or by\nthe anisotropy of the ferromagnet [53, 54], etc.\nFIG. 1. Schematic diagram of a double cavity-magnon system where both the cavities are driven by a two-mode squeezed\nvacuum microwave field. Two YIG spheres are respectively placed inside the microcavities near the maximum magnetic fields\nof the cavity modes and simultaneously in uniform bias magnetic fields.\nIn the frame rotating at the frequency ωj, i.e., the frequency of the jthmode of the input two-mode squeezed field,\nthe QLEs of this model Hamiltonian are given by\n˙cj=−(κcj+i∆cj)aj−igjmj+ 2λjeiθc†\nj+p\n2κcjcin\nj, (2)\n˙mj=−(κmj+i∆mj)mj−igjcj+ 2µjeicνm†\nj+p\n2κmjmin\nj,\nwhere κcj(κmj) is the decay rate of the jth cavity mode (magnon mode), ∆ cj=ωcj−ωj,∆mj=ωmj−ωj, and cin\nj\n(min\nj) is the input noise operator for the jthcavity mode (magnon mode). The two cavity input noise operators cin\njare\nquantum correlated due to the injection of the two-mode squeezed field, and have the following non-zero correlations3\nin time domain cin\njandcin†\njare given by [64]\n⟨cin\nj(t)cin†\nj(t′)⟩= (N+ 1)δ(t−t′) (3)\n⟨cin†\nj(t)cin\nj(t′)⟩=Nδ(t−t′) (4)\n⟨cin\nj(t)cin\nj′(t′)⟩=Me−icωM(t+t′)δ(t−t′) ; j̸=j′(5)\n⟨cin†\nj(t)cin†\nj′(t′)⟩=MeicωM(t+t′)δ(t−t′) ; j̸=j′(6)\n. Here N= sinh2r,M= sinh rcoshrandris the squeezing parameter of the two-mode squeezed vacuum field\nwhereas the magnon input noise operators min\njare of zero mean and correlated as follows\n⟨min\nj(t)min†\nj(t′)⟩= (Nmj+ 1)δ(t−t′) (7)\n⟨min†\nj(t)min\nj(t′)⟩=Nmjδ(t−t′) (8)\nwhere Nmj=h\nexp\u0010ℏωmj\nkBT\u0011\n−1i−1\nis the equilibrium mean thermal magnon number of the jthmode, with Tthe\nenvironmental temperature and kBthe Boltzmann constant.\nUsing the linearisation of quantum Langevin equations, the fluctuations of the system are written as\nδ˙cj=−(κcj+i∆cj)δcj−igjδmj+ 2λjeiθδc†\nj+p\n2κcjcin\nj, (9)\nδ˙mj=−(κmj+i∆mj)δmj−igjδcj+ 2µjeicνδm†\nj+p\n2κmjmin\nj.\nTo get the explicit expression of the degree of freedom of optical and magnon modes, we consider the EPR-type\nquadrature fluctuations operators corresponding to the two subsystems defined as δQj= (δcj+δc†\nj)/√\n2, δPj=\ni(δc†\nj−δcj)/√\n2, δqj= (δmj+δm†\nj)/√\n2, δpj=i(δm†\nj−δmj)/√\n2 (we have similar definition for input noises Qin\nj, Pin\nj\nandqin\nj, pin\nj) [55–60],\nThe above QLEs can be simplified as\nδ˙Qj=−κcjδQj+ ∆ cjδPj+gjδpj+ 2λcos(θ)δqj+ 2λsin(θ)δpj+p\n2κcjQin\nj, (10)\nδ˙Pj=−κcjδPj−∆cjδQj−gjδqj−2λcos(θ)δpj+ 2λsin(θ)δqj+p\n2κcjPin\nj,\nδ˙qj=−κmjδqj+ ∆ mjδpj+gjδPj+ 2µcos(ν)δQj+ 2µsin(ν)δPj+p\n2κmjqin\nj,\nδ˙pj=−κmjδpj−∆mjδpj−gjδQj−2µcos(ν)δPj+ 2µsin(ν)δQj+p\n2κmjpin\nj.\nEquation (10) take the following compact matrix form\n˙V(t) =AV(t) +χ(t), (11)\nHere V(t) = [δQ1, δP1, δQ 2, δP2, δq1,δp1, δq2, δp2]T,Ais the drift matrix\nA=\u0012\nA1A3\nA3A2\u0013\n(12)\nwhere\nA1=\n−κc1+ 2λcos(θ) ∆ c1+ 2λsin(θ) 0 0\n−∆c1+ 2λsin(θ)−κc1−2λcos(θ) 0 0\n0 0 −κc2+ 2λcos(θ) ∆ c2+ 2λsin(θ)\n0 0 −∆c2+ 2λsin(θ)−κc2−2λcos(θ)\n (13)4\nand\nA2=\n−κm1+ 2µcos(ν) ∆ m1+ 2µsin(ν) 0 0\n−∆m1+ 2µsin(ν)−κm1−2µcos(ν) 0 0\n0 0 −κm2+ 2µcos(ν) ∆ m2+ 2µsin(ν)\n0 0 −∆m2+ 2µsin(ν)−κm2−2µcos(ν)\n (14)\nand\nA3=\n0g10 0\n−g10 0 0\n0 0 0 g2\n0 0 −g20\n (15)\nandχ(t) = [√2κc1Qin\n1,√2κc1Pin\n1,√2κc2Qin\n2,√2κc2Pin\n2,√2κm1qin\n1,√2κm1pin\n1,√2κm2qin\n2,√2κm2pin\n2]T. The system is\nstable when eigenvalues of the drift matrix A(12) have negative real parts. This corresponds to the so-called Routh-\nHurwitz criterion [61]. The steady state of the system, which is completely characterized by an 8 ×8 covariance matrix\n(CM) Σ, defined as Σ ij(t) =⟨Vi(t)Vj(t′) +Vj(t′)Vi(t)⟩/2. the solution of Σ can be obtained by directly solving the\nLyapunov equation [62]\nAΣ + Σ AT=−D, (16)\nwhere Dis the diffuse matrix defined by Dijδ(t−t′) =⟨χi(t)χj(t′) +χj(t′)χi(t)⟩/2, given by\nD=\nκ′0√κc1κc2M 0 0 0 0 0\n0 κ′0 −√κc1κc2M0 0 0 0√κc1κc2M 0 κ′′0 0 0 0 0\n0 −√κc1κc2M 0 κ′′0 0 0 0\n0 0 0 0 γ′0 0 0\n0 0 0 0 0 γ′0 0\n0 0 0 0 0 0 γ′′0\n0 0 0 0 0 0 0 γ′′\n(17)\nwhere κ′=κc1\u0000\nN+1\n2\u0001\n,κ′′=κc2\u0000\nN+1\n2\u0001\n,γ′=κm1\u0000\n2Nm1+ 1\u0001\nandγ′′=κm2\u0000\n2Nm2+ 1\u0001\n.\nThe covariance matrix Σ (mm)associated with the two magnon modes is given by\nΣ(mm)=\u0012X Z\nZTY\u0013\n(18)\nThe 2 ×2 sub-matrices XandYin Eq. (18) describe the autocorrelations of the two magnon modes and 2 ×2\nsub-matrix Zin Eq. (18) denotes the cross-correlations of the two magnon modes.\nIII. QUANTUM CORRELATIONS\nA. Quantum entanglement\nThe logarithmic negativity Emis a measure or witness of entanglement in bipartite continuous-variable (CV)\nsystems [63]. Mathematically, it can be expressed as:\nEm= max[0 ,−log(2ψ−)] (19)\nwith ψ−being the smallest symplectic eigenvalue of partial transposed covariance matrix Σ (mm)of two magnon modes\nψ−=s\nΓ−pΓ2−4 det Σ (mm)\n2(20)\nwhere the symbol Γ is written as Γ = det X+ detY −detZ. The two magnon modes are entangled if the condition\nψ−<1/2 (i.e. when Em>0) is satisfied.5\nB. Gaussian quantum steering\nQuantum steering is the process of acquiring information about an unmeasurable quantum system by measuring\na single quantum system. Gaussian quantum steering is the asymmetric property between two entangled observers\n(the two mechanical modes), Alice ( A: magnon M1) and Bob ( B: magnon M2). Besides, they are used to quantify\nhow much the two magnon modes are steerable. We use the covariance matrix Σ (mm)of the two magnon modes, the\nGaussian steering A→BandB→Awritten as [12 ?]\nSA→B=SB→A= max\u0014\n0,1\n2ln\u0012det(X)\n4 det Σ (mm)\u0013\u0015\n; (21)\nThere are two possibilities of steerability between AandB: (i) no-way steering if SA→B=SB→A= 0 i.e. Alice can’t\nsteer Bob and vice versa also impossible even if they are not separable, and (ii) two-way steering if SA→B=SB→A>0,\ni.e. Alice can steer Bob and vice versa. Indeed, a non-separable state is not always a steerable state, while a steerable\nstate is always not separable.\nIV. RESULTS AND DISCUSSION\nIn this section, we will discuss the steady state quantum correlations of two magnon modes under various parameters\nregime reported experimentally [49, 64] and given as ωc1/2π= 10 GHz, κc/2π= 5 MHz, κm=κc/5,g1=g2= 5κc,\nθ=πandν= 0.9π. For simplicity, we consider that Nm1=Nm2=Nm,κc1=κc2=κcandκm1=κm2=κm.\nAdditionally, each YIG sphere used in our study has a diameter of 0.5 mm. These spheres are specifically chosen for\ntheir size and contain more than 1017spins.\nFIG. 2. Plot of the logarithmic negativity Embetween two magnon modes versus (a) ∆ c1and ∆ m1, with ∆ c2= ∆m2= 0; (b)\n∆c2and ∆ m2, with ∆ c1= ∆m1= 0,r= 1,λ= 0.2κc,µ= 0.2κcandT= 100 mK. See text for the other parameters.\nIn Fig. 2, we have plotted the logarithmic negativity Emof subsystem magnon-magnon with varying ∆ cjand ∆ mj\n(j= 1,2) whereas all other parameters are fixed. It can be seen in Figs. 2(a) and (b) that when ∆ cj= ∆ mj= 0\n(j= 1,2), the entanglement between two magnon modes is optimal. This observation can be attributed to the\nresonant transfer of quantum correlations from the input fields to the two magnon modes, facilitated by the linear\ncavity-magnon coupling.\nFig. 3(a) shows that the bipartite entanglement Emincreases with the squeezing parameter rof the two-mode\ninput squeezed vacuum field and decreases with the temperature T. The effect of the temperature is due to the\ninfluence of the decoherence phenomenon [65]. Furthermore, it has been observed that the logarithmic negativity Em\nreaches its maximum value when the value of rfalls within the range of (1-1.5) whereas when the parameter rgoes\nto zero both the magnon modes remain separable ( Em= 0) as illustrated in figure 3(a). This shows the dependence\nof the bipartite entanglement of the two magnon modes on the squeezing parameter r. In Fig. 3(b), we plot Emas\na function of squeezing parameter randg2/g1. We found here the generation of the bipartite entanglement between\nthe two magnon modes with a gradual increase in squeezing parameter reven for a wide range of mismatch of the\ntwo coupling strengths g2andg1.6\nFIG. 3. (a) Plot of the logarithmic negativity Emm between the two magnon modes vs the temperature T(K) and the magnon\nsqueezing parameter µwith ∆ c1,2= ∆m1,2= 0,T= 100 mK, λ= 0.2κcandµ= 0.2κc. (b) Plot of the logarithmic negativity\nEmm between the two magnon modes vs g2/g1androf the two-mode squeezed vacuum field with g1= 5κc, ∆c1,2= ∆m1,2= 0,\nT= 100 mK, λ= 0.2κcandµ= 0.2κc.\nFIG. 4. Plots of the steering SA→B,SB→Alogarithmic negativity Emof the two magnon modes versus the equilibrium mean\nthermal magnon number Nmfor various values of the coupling λandµ, with ∆ c1,2= ∆m1,2= 0,r= 1 and T= 100 mK.\nIn Fig.(4), we plot the Gaussian steering SA→B,SB→Aand logarithmic negativity Emfor the subsystem magnon-\nmagnon as a function of the equilibrium mean thermal magnon number Nmfor various values of the parameters λ\nandµwhereas the other parameters are fixed. It can be seen that SA→B,SB→Aand entanglement Emhave the\nsame evolution behavior. This figure studies the effect of Nm(temperature T) and the parameters λandµon the\nbipartite entanglement and quantum steerings. Due to decoherence phenomena both the quantities i.e. bipartite\nentanglement and quantum steering decrease very quickly with increasing Nm. Moreover, when we enhance λandµ\nthe magnon-magnon entanglement as well as two-way quantum steering become finite for a wide range of temperature\nT(the equilibrium mean thermal magnon number Nm). Moreover, as depicted in Fig.(4) the entangled state is\nnot always a steerable state but a steerable state must be entangled, i.e. when SA→B=SB→A>0 which leads to\nEm>0 and hence is the witnesses of the existence of Gaussian two-way steering. This means that both magnon\nmodes are entangled as well as are steerable from AtoBand from BtoA. However, we get no-way steering when\nSA→B=SB→A= 0 and Em>0 and so the measure of Gaussian steering always remains bounded by the bipartite\nentanglement Em.\nIn Fig. 5, we have plotted the logarithmic negativity Emof two magnon modes with κcand ∆ m1for a fixed value\nof all other parameters. It can be seen that the entanglement between the two magnon modes is maximum when\n∆m1= 0 and κc= 3×107Hz. However, the bipartite entanglement Emdecreases with decreasing decay rate κcand\nincreasing detuning ∆ m1.7\nFIG. 5. Plot of the logarithmic negativity Embetween the two magnon modes vs κcand ∆ m1with ∆ c1= ∆ c2= ∆ m2= 0,\nT= 50 mK, r= 1.5,g1= 5κc,λ= 0.2κcandµ= 0.2κc.\nV. CONCLUSIONS\nWe have theoretically investigated a scheme for the generation of the bipartite entanglement and Gaussian quantum\nsteering in a double microwave cavity-magnon hybrid system where a two-mode squeezed microwave vacuum field is\nalso transferred simultaneously into both cavities. We have obtained optimal parameter regimes for achieving the\nstrong magnon-magnon entanglement and also explored the effectiveness of the scheme towards the mismatch of two\ncavity magnon couplings including the entanglement transfer efficiency. 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Rev. A 70, 022318 (2004).\n[64] M. Yu, S. Y. Zhu, and J. Li, Journal of Physics B: Atomic, Molecular and Optical Physics, 53(6), 065402 (2020).\n[65] W. H. Zurek, Rev. Mod. Phys. 75, 715 (2003)."    },    {        "title": "1603.09201v1.Low_damping_transmission_of_spin_waves_through_YIG_Pt_based_layered_structures_for_spin_orbit_torque_applications.pdf",        "content": "arXiv:1603.09201v1  [cond-mat.mes-hall]  30 Mar 2016Low-damping transmission of spin waves through YIG/Pt-bas ed layered\nstructures for spin-orbit-torque applications\nDmytro A. Bozhko,1,2Alexander A. Serga,1,a)Milan Agrawal,1Burkard Hillebrands,1and Mikhail P. Kostylev3\n1)Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universit¨ at Kaiserslautern,\n67663 Kaiserslautern, Germany\n2)Graduate School Materials Science in Mainz, Kaiserslauter n 67663, Germany\n3)School of Physics, M013, University of Western Australia, C rawley, WA 6009, Australia\n(Dated: 12 June 2018)\nWe show that in YIG-Pt bi-layers, which are widely used in experiments on the spin transfer torque and spin\nHall effects, the spin-wave amplitude significantly decreases in comp arison to a single YIG film due to the\nexcitation of microwave eddy currents in a Pt coat. By introducing a novel excitation geometry, where the Pt\nlayer faces the ground plane of a microstrip line structure, we supp ressed the excitation of the eddy currents\nin the Pt layer and, thus, achieved a large increase in the transmissio n of the Damon-Eshbach surface spin\nwave. At the same time, no visible influence of an external dc curren t applied to the Pt layer on the spin-wave\namplitude in the YIG-Pt bi-layer was observed in our experiments with YIG films of micrometer thickness.\nMagnonics addresses the transfer and processing of\ninformation using spin waves (SW) and their quanta,\nmagnons.1One of the main challenges for this field is the\nfinite lifetime ofmagnonseven in alow-dampingmaterial\nsuch as single crystal yttrium-iron-garnet(YIG).2A pos-\nsible way to overcome this issue is the compensation of\nSW damping via transfer of a spin torque to a magnetic\nmedium by a spin-polarized electron current generated\nin an adjacent non-magnetic metal layer with high spin-\norbit interaction. The decrease in the damping of spin\nwaves propagating in a YIG film covered with a current\nconducting Pt film has already been reported.3,4How-\never, the initial spin-wave attenuation in such a struc-\nture was unacceptably high for practical applications\n(≃35dB in Ref.3). Therefore, it is an important task to\ndetermine the origin of the excessive spin-wave damping\nin YIG-Pt bi-layers and find a way to reduce this harm-\nful effect. The latter will only be possible if this damp-\ning does not correspond to the spin-pumping effect – the\nprocess of a back transfer of a spin torque from magnons\nexited in a magnetic medium to a non-magnetic layer.\nHere, we show that in a Pt covered YIG film the SW\namplitude decreases in comparison to a single YIG film\ndue to the resistive losses attributed to the microwave\neddy currents induced in a thin Pt layer by a propagat-\ning spin wave. By placing near the Pt layer of a highly\nconducting metal plate, we suppressed the excitation of\nthe eddy currents in platinum and, thus, achieved a large\nincrease (up to 1000 times) in the transmission of the\nDamon-Eshbach surface spin wave.\nThe experiments were performed using 2mm wide and\n15mm long SW waveguidespreparedfrom a 6.7 µm thick\nYIG film grown by liquid phase epitaxy on a 500 µm\nthick gallium-gadolinium garnet (GGG) substrate. A\nPt layer of 10nm thickness was deposited on top of the\nYIG surface. Spin waves were emitted and received by\na)Electronic mail: serga@physik.uni-kl.de\nFIG. 1. Sketch of the experimental setup. (a) The conven-\ntional SW excitation geometry – a YIG-Pt bi-layer is placed\nnear the microstrip antennas. (b) The inverted SW excitatio n\ngeometry – the YIG-Pt bi-layer is close to the ground plate\nof the microstrip structure.\ntwo short-circuited gold-wire antennas of 50 µm diame-\nter placed 7.6mm apart from each other (see Fig. 1).\nSpin-wave propagation was studied in a frequency band\nfrom 6.2 to 6.5GHz using a vector network analyzer\n(Anritsu MS46322A). A relatively small level of the ap-\nplied microwave power of 0.5mW was chosen to pre-\nvent the development of non-linear effects for the prop-\nagating spin waves. An external bias magnetic field\nH0= 1600Oe was applied in the YIG film plane per-\npendicularly to the SW propagation direction. Thus, the\ngeometry of the Damon-Eshbach (DE) surface spin wave\nwas implemented.2\nThe reference experiment was performed in a conven-\ntional excitation geometry (as in Ref. 3), when a YIG\nfilm lays directly under the antennas (see Fig. 1(a) and\ninsets in Fig. 2). In the case of a bare YIG film – i.e.\nwithout any Pt coating (Fig. 2(a)), the DE spin wave,\nwhich propagates near the YIG surface touching the an-\ntennas (see the sketch of spin-wave distribution over the\nYIG film thickness in inset in Fig. 2(a)), possesses rel-2\nFIG. 2. Conventional excitation geometry. A YIG film is\npositioned directly under the antennas. The localization o f\na Damon-Eshbach wave depends on the direction of the bias\nmagnetic field H0(see insets). Dotted/blue lines correspond\nto the bias magnetic field pointing into the drawing plane.\nSolid/red lines correspond to the bias magnetic field pointi ng\nout of the drawing plane. (a) Transmission ( S12) charac-\nteristic of the bare YIG waveguide. (b) Transmission ( S12)\ncharacteristic of the YIG waveguide covered with 10nm Pt\nlayer. (c) Excitation efficiency by the input antenna ( S22)\nfor the bare YIG waveguide. (d) Excitation efficiency by\nthe input antenna ( S22) for the YIG waveguide covered with\n10nm Pt layer.\natively low minimal transmission losses of around 10dB\nat 6.42GHz. The DE wave localized at the opposite film\nsurface shows one order of magnitude weaker transmis-\nsion. This well known effect relates to the nonreciprocal\nexcitation of a DE wave, whose efficiency depends on the\nrelative orientation of the bias magnetic field H0and the\nSWpropagationdirection.2,5,6In spite ofthe pronounced\ndifference in the transmissioncharacteristics,the efficien-\ncies of SW excitation, which are inverselyproportionalto\nthe microwave reflection from the input antenna S22, are\nidentical for two opposite orientations of the bias mag-\nnetic field (see Fig. 2(c)). That is because in both cases\nthe input antenna radiates spin waves in two opposite\ndirections but only one of the emitted waves reaches the\noutput antenna.\nIn the case of the Pt-covered YIG film, the excita-\ntion efficiency remains almost the same as for the bare\nYIG waveguide (compare Fig. 2(d) to Fig. 2(c)) but the\ntransmissionofthe DE wavelocalizednearthe Ptlayeris\nstrongly suppressed (see the dashed curve in Fig. 2(b)).\nThis suppression can be associated either with eddy cur-\nrents excited in the high-resistive ( RPt= 218Ohm) Pt\nlayer by stray fields of the propagating spin wave or with\nthe spin-pumping effect at the YIG-Pt interface.\nFIG. 3. Inverted excitation geometry. The YIG waveguide is\npositioned directly on the copper ground plate. Dotted/blu e\nlines correspond to the bias magnetic field pointing into the\ndrawing plane. Solid/red lines correspond to the bias mag-\nneticfieldpointingoutofthe drawingplane. (a)Transmissi on\n(S12) characteristics of the bare YIG waveguide. (b) Trans-\nmission ( S12) characteristics of the YIG waveguide covered\nwith 10nm Pt layer. (c) Excitation efficiency of the input\nantenna ( S22) for the bare YIG waveguide. (d) Excitation\nefficiency of the input antenna ( S22) for the YIG waveguide\ncovered with a 10nm Pt layer. Insets demonstrate schematic\nrepresentations of the surface spin wave localizations for dif-\nferent directions of a bias magnetic field.\nAt the second stage of our studies, in order to dis-\ntinguish between these two damping mechanisms, the\nYIG film was facing the ground plate (see Fig. 1(b) and\nFig. 3). It is expected that if the Pt film is placed in\ndirect contact with the bulk copper plate, than the eddy\ncurrents will mostly flow in the low resistive ground\nplate. Hence, no additional Ohmic losses due to currents\nin Pt are expected. At the same time, the spin pumping\neffect, which can potentially contribute to the excessive\nSW damping, should remain unchanged. The SW\nexcitation efficiencies shown in Fig. 3(c) and Fig. 3(d)\nfor the bare and Pt-covered YIG waveguides are close to\nthose obtained in the conventional excitation geometry\n(compare with Figs. 2(c) and 2(d)). At the same time,\none can see from Figures 3(a) and 3(b) that the wave,\nwhich propagates close to the Pt layer, experiences now\nno excessive damping and shows the same transmission\nlosses as the wave in the bare YIG film. This fact\nexplicitly confirms our assumption about the significant\ncontribution of the SW induced eddy currents to the\nSW damping in Pt-YIG bi-layers.\nIt should be noted that in the proposed excitation\ngeometry spin waves are excited by microwave currents\nflowing in the ground plate rather than through the3\nFIG. 4. Inverted excitation geometry. The YIG waveguide\nis positioned near the copper ground plate. An additional\n80µm thick dielectric polypropylen (PP) spacer layer is intro-\nduced to electrically decouple the Pt layer from ground. Dot -\nted/blue lines correspond to the bias magnetic field pointin g\ninto the drawing plane. Solid/red lines correspond to the bi as\nmagnetic field pointing out of the drawing plane. (a) Trans-\nmission ( S12) characteristics of the bare YIG waveguide. (b)\nTransmission ( S12) characteristics of the YIG waveguide cov-\neredwith 10nmPtlayer. (c)Excitation efficiencyoftheinput\nantenna ( S22) for the bare YIG waveguide. (d) Excitation ef-\nficiency of the input antenna ( S22) for the YIG waveguide\ncovered with a 10nm Pt layer. Insets demonstrate schematic\nrepresentations of the surface spin wave localizations for dif-\nferent directions of a bias magnetic field.\nmicrostrip antenna, which is separated from the YIG\nfilm by the thick GGG layer. It decreases the excitation\nefficiency of short-wavelength spin waves because the\nmicrowave currents are localized more weakly in the\nextended ground plate than in the narrow microstrip.\nHowever, the change in the electrodynamic boundary\nconditions due to placing a highly conducting layer near\nthe surface of the YIG film7makes the DE dispersion\nrelation steeper, i.e., increases the eigen-frequencies of\nthe excited long-wavelength spin waves. Moreover, the\nrelated increase in the SW group velocity decreases\nthe SW propagation time and consequently the SW\ntransmission losses. As a result, the SW transmission\nfrequency band remains rather wide.\nThe described geometry enables the surface spin waves\nto propagate rather long distances through the Pt-\ncovered YIG-film waveguides. However, no application\nof a dc electric current to such a structure is possible be-\ncause of electric contact between the Pt layer and the Cu\nground plate. In order to settle the issue, a 80 µm thick\npolypropylene spacerwas placed between the sample and\nthe ground plate (see insets in Figs. 4(a) and 4(b)). Inthis case, the minimal transmission losses (Figs. 4(a) and\n4(b)) remain the same as in the previous measurements\n(see Figs. 2(a), 3(a) and 3(b)), but the transmission\nbandwidth turns to be significantly narrower. The rea-\nsons for this narrowing are obvious. A microwave mag-\nnetic field induced by microwave currents flowing in the\nground plate is becoming more uniform with increase in\nthe distance to the ground plate. This leads to a reduc-\ntion of the excited SW wavenumbers and, alongside with\nthe vanishing of the effect of metallization, to a decrease\nin the frequency bandwidth. However, the coupling be-\ntween the Pt layer and the ground plate appears to be\nhigh enough to shunt the Pt film and reduce, thus, the\neddy currents related SW losses.\nAt the final stage, we used the circuit shown in inset\nin Fig. 4(b) in order to check the ability to control the\nSW damping by a dc electric current applied to the Pt\nlayer. To avoid spurious heating effects the experiment\nwas performed in the pulsed regime: overlapping input\nmicrowave pulse (20ns) and dc current pulse (350ns)\nwere applied with 1kHz repetition rate. It is remarkable\nthat in the studied structure the best SW transmission\ncorresponds to the case when the DE wave is localized\nat the YIG-Pt interface. In the absence of the externally\napplied dc electric current, such localization results in\nrather high dc voltages induced in the Pt layer by the\ncombined action of the spin pumping and inverse spin\nHall effects. However, no influence of both positive and\nnegative electric currents Idc=±370mA on the SW am-\nplitude was measured in the entire SW frequency band.\nIn conclusion, we have demonstrated that by using a\nnovel spin-wave circuitry, where a Pt-covered YIG-film\nwaveguide faces the ground plane of a microstrip line\nstructure, it is possible to reduce the excitation of SW-\nrelated eddy currents in the Pt layer and achieve, thus,\nin a YIG-Pt bi-layer the same transmission losses for the\nDamon-Eshbach surface spin wave as in a bare YIG film.\nAt the same time, no influence of a dc electric current\napplied to the Pt layer on the amplitude of SW pulses\npropagating in the relatively thick YIG waveguide of mi-\ncrometer thickness was observed in our experiments.\nFinancial support by EU-FET InSpin 612759 and by\nthe Australian Research Council is gratefully acknowl-\nedged.\n1A.V. Chumak, V.I. Vasyuchka, A.A. Serga, and B. Hillebrands ,\nNature Phys. 11, 453 (2015).\n2A. A. Serga, A. V. Chumak, and B. Hillebrands, J. Phys. D: Appl .\nPhys.43, 264002 (2010).\n3Z. Wang, Y. Sun, M. Wu, V. Tiberkevich, and A. Slavin, Phys.\nRev. Lett. 107, 146602 (2011).\n4E. Padr´ on-Hern´ andez, A. Azevedo, and S. M. Rezende, Appl.\nPhys. Lett. 99, 192511 (2011).\n5T. Schneider, A. A. Serga, T. Neumann, B. Hillebrands, and M.\nKostylev, Phys. Rev. B 77, 214411 (2008).\n6V. E. Demidov, M.P. Kostylev, K. Rott, P. Krzysteczko, G. Rei ss,\nand S. O. Demokritov, Appl. Phys. Lett. 95, 112509 (2009).\n7J. P. Parekh, K. W. Chang, and H. S. Tuan, Circuits, Systems\nand Signal Processing 4, 9 (1985)."    },    {        "title": "1810.11841v1.Magnetic_Resonance_in_Defect_Spins_mediated_by_Spin_Waves.pdf",        "content": "Magnetic Resonance in Defect Spins mediated by Spin Waves\nClara M uhlherr, V. O. Shkolnikov, and Guido Burkard\nDepartment of Physics, University of Konstanz, D-78464 Konstanz, Germany\nIn search of two level quantum systems that implement a qubit, the nitrogen-vacancy (NV) center\nin diamond has been intensively studied for years. Despite favorable properties such as remarkable\ndefect spin coherence times, the addressability of NV centers raises some technical issues. The\ncoupling of a single NV center to an external driving \feld is limited to short distances, since an\ne\u000ecient coupling requires the NV to be separated by only a few microns away from the source. As\na way to overcome this problem, an enhancement of coherent coupling between NV centers and a\nmicrowave \feld has recently been experimentally demonstrated using spin waves propagating in an\nadjacent yttrium iron garnet (YIG) \flm1. In this paper we analyze the optically detected magnetic\nresonance spectra that arise when an NV center is placed on top of a YIG \flm for a geometry\nsimilar to the one in the experiment. We analytically calculate the oscillating magnetic \feld of the\nspin wave on top of the YIG surface to determine the coupling of spin waves to the NV center. We\ncompare this coupling to the case when the spin waves are absent and the NV center is driven only\nwith the antenna \feld and show that the calculated coupling enhancement is dramatic and agrees\nwell with the one obtained in the recent experiment.\nI. INTRODUCTION\nThe negatively charged nitrogen vacancy (NV) cen-\nter in diamond is an optically active point defect with a\nground state spin triplet, lying deep in the bandgap of\ndiamond2,3. Its bright optical transition and the exis-\ntence of intersystem crossing provides a good mechanism\nfor initialization and read out of the spin state of the\ncenter4. The ground state of the NV center is sensitive\nto magnetic and electric \felds, as well as to strain5{7, and\nthus can be used as a nanosensor to detect them8{10. This\nmakes the NV center extremely interesting for metrol-\nogy. Apart from that, the long spin coherence time of\nthe defect makes it interesting for quantum information\npurposes. Its state can e\u000eciently be manipulated with\noscillating magnetic \felds, that cause transitions between\nthe levels of the spin triplet. In most of the experiments\nthis oscillating \feld was generated by an antenna placed\nin the vicinity of the center, which raises the issue of ad-\ndressability for many NV centers when the antenna can\nno longer be placed close to each of them. Recently an ex-\nperiment was reported in which the NV center was placed\non top of a ferromagnetic material (YIG) that can host\npropagating spin waves. Using an antenna as a source of\nthe spin waves in this material, one can couple distinct\nNV centers to it just like dialog partners are connected\nby a signal line. In this work we theoretically treat the\ncoupling of the spin waves to the NV centers for a spe-\ncial geometry of the device described below. We provide\nan analytical expression for the spin wave \feld and de-\ntermine the coupling enhancement that it produces with\nrespect to the \feld of the antenna only, when spin waves\nare absent.\nII. SETUP\nWe calculate the e\u000bect of spin wave excitation inside\nferromagnetic thin \flms as used for example in the ex-periment presented by Andrich et al.1. The vicinity of a\nspin wave hosting material to a quantum system driven\nby external microwaves provides an additional compo-\nnent of the driving \feld. In that way, spin wave excita-\ntion inside a ferromagnetic material a\u000bects the coupling\nbetween the driving \feld and the given quantum system.\nIn the experimental realization by Andrich et al.1the\nquantum system consisted of defect spins inside a col-\nlection of nanodiamonds patterned on the surface of the\nmagnetic material as depicted in Fig. 1( a). A single nan-\nodiamond, which is selected with the laser focus, hosts an\nensemble of\u0018500 NV centers with an isotropic electron\ng-factor of g\u001923. The nanodiamond is embedded in a\npolydimethylsiloxane (PDMS) \flm, which is on the top\nof a layered structure of YIG and gadolinium gallium gar-\nnet (GGG). Since YIG is a ferromagnetic material with\nultra-low spin wave damping, it is perfect for the usage\nas the spin wave medium. An ac current \rowing through\na microstrip line (MSL) grown on the surface of the YIG\ngenerates a microwave driving \feld, penetrating through\nthe di\u000berent materials and denoted as the antenna \feld\nin Fig. 1(a). Due to the vicinity of the NV centers to the\nYIG surface, the defect spins are not only sensitive to the\nantenna \feld, but also to the stray \feld originating from\nthe magnetic YIG \flm. For a theoretical description of\nthe system we use the coordinate system and the de\fning\nparameters sketched in Fig. 1( b). Choosing the YIG \flm\nto have a thickness dand to lie in the xy-plane, the MSL\norientation can be set as the y-axis and the MSL posi-\ntion marks the origin of the coordinate system. Further,\nthe MSL has a width wand the probed nanodiamond\nis located in the xz-plane atxNV. Based on realistic\ndimensions of experimental systems, some assumptions\nconcerning the boundary conditions of the system are\nreasonable. In Ref. 1, the YIG \flm is only 3 :08µm thick,\nwhich is very thin compared to its dimensions of about\n10 mm along the x- andy-axis1. Thus, the \flm can be as-\nsumed to be in\fnite in the xy-plane. The GGG substrate\nand the PDMS layer are a few hundreds of µm thick, soarXiv:1810.11841v1  [cond-mat.mes-hall]  28 Oct 20182\nFigure 1. ( a) Electron spin resonance in NV spins driven by spin waves demonstrated by Andrich et al.1. An array of\nnanodiamonds is patterned inside a PDMS \flm layered on top of a YIG thin \flm. From a distant MSL grown on the YIG the\nantenna \feld propagates through the PDMS and couples to the NV centers. Inside the YIG the microwave excitation by the\nMSL leads to spin waves propagating in the plane. Due to the given dimensions, the theoretical treatment can be reduced to\na two-dimensional coordinate system as shown in ( b), which is a cross section of the setup in ( a). The layered structure lies\nin thexy-plane and the center of the MSL of width wmarks the origin of the coordinate system and is oriented along the\ny-direction.\nthat they are treated as in\fnite in the z-direction and\nthe only boundary conditions to be ful\flled are those at\nthe YIG interfaces, where both surrounding layers in re-\ngions I and III are approximated as non-magnetic. This\nassumption is justi\fed due to their magnetic permeabil-\nities being isotropic and close to 11112. The MSL has a\nwidth ofw= 5µm, making its height of about 200 nm\nnegligible. Overall, due to the spatial expansion of the\nsystem along the y-direction and the invariance of the\nsystem under y-translation, we assume that all \felds are\nindependent of yand the problem is treated in two di-\nmensions.\nIII. DRIVEN SPIN WAVES\nIn order to calculate the spin wave spectrum of a fer-\nromagnetic thin \flm and the resulting \feld amplitude,\nwe start from Maxwell's equations (MEs). If there is an\nexternal magnetic H-\feld, a magnetization \feld Mis\nbuilt up in the magnetic \flm. In general, the external\nH-\feld can be decomposed into a static part H0and\na time-dependent component h(t) originating from the\nmicrowave antenna \feld. Consequently, the magnetiza-\ntion inside the material depends on the driving frequency\n!and will also have a time-dependent component m(t).\nThe relation between both time-varying components is\ngiven by the constitutive equation m=Xh, where ma-\nterial properties and the geometry of the system enter\nvia the susceptibility tensor X. In case of a strong static\nbias \feldH0along they-direction, the magnetic \flm\nis tangentially magnetized and the static component of\nmagnetization M0saturates. Under these conditions, X\ntakes the form of the Polder susceptibility\nX=0\n@\u001f0\u0000i\u0014\n0 1 0\ni\u00140\u001f1\nA; (1)with the frequency-dependent entries \u001f=!!M=(!2\n0\u0000!2)\nand\u0014=!0!M=(!2\n0\u0000!2)13. The parameters !0=\r\u00160H0\nand!M=\r\u00160MSaccount for the characteristics of the\nmaterial in an external \feld H0. Here,\randMSde-\nnote the gyromagnetic ratio and the saturation magneti-\nzation of the \flm and \u00160denotes the vacuum permeabil-\nity. Using the Polder tensor (1) and assuming the electric\npermittivity of the materials to be 1 for simplicity, the\nsingle components of the four MEs in two dimensions\nform a system of eight coupled di\u000berential equations for\nthe magnetic and electric \feld components Hi(x;z) and\nEi(x;z) (i=x;y;z ) in each region I-III in Fig. 1( b).\nSince the MSL is located right at the PDMS-YIG inter-\nface and is assumed to be in\fnitesimally thin, the \row-\ning current is non-zero only at the boundary between the\ntwo upper layers I and II, whereas inside the bulk regions\nthere are no free currents j\rowing, and thereby, no ad-\nditional source terms. Referring to that, the \felds inside\nthe bulk regions I-III are obtained by solving the system\nof homogeneous MEs with j= 0 separately and the MSL\ncurrent is included by matching the boundary conditions\natz= 0 andz=\u0000dafterwards.\nPerforming a one dimensional Fourier transform of the\nx-coordinate yields a system of ordinary di\u000berential equa-\ntions in the kxz-space, where only one equation actually\nhas to be solved,\n@2\nzHx(kx;z) +a2Hx(kx;z) = 0; (2)\nwherea2= ((1 +\u001f)2\u0000\u00142)k2\n0=(1 +\u001f)\u0000k2\nxandk0=\n!=c. The other non-zero \feld components Hz(kx;z)\nandEy(kx;z) can be expressed in terms of the solution\nHx(kx;z) of (2) and its derivative @zHx(kx;z),\nHz(kx;z) =i\u0014k2\n0Hx(kx;z)\u0000ikx@zHx(kx;z)\nk2x\u0000(1 +\u001f)k2\n0; (3)\nEy(kx;z) =!\nkx(i\u0014Hx(kx;z) + (1 +\u001f)Hz(kx;z):(4)\nSince the PDMS as well as the GGG layer are assumed\nto be in\fnite in positive and negative z-direction, there3\nare no incoming waves in these regions, which could be\ncaused by re\rections at any surfaces. Thus, the ansatz\nHI\nx(kx;z) =C1eiaPDMSz;\nHII\nx(kx;z) =C2eiaYIGz+C3e\u0000iaYIGz;\nHIII\nx(kx;z) =C4e\u0000iaPDMSz;(5)\nis chosen, where the superscripts I-III refer to the regions\nandaicorresponds to ain (2) in the corresponding ma-\nteriali. In order to obtain the actual amplitude of the\nmagnetic \feld, the coe\u000ecients C1-C4have to be derived,\nso there is need to include existing boundary conditions,\nwhich arise at the two interfaces. In the absence of any\nsurface currents, the parallel component of the H-\feld\nand the orthogonal component of the B-\feld are continu-\nous at an interface. But since this is only the case for the\nlower interface at z=\u0000d, the upper boundary condition\nfor the parallel H-\feld component has to be considered\nmore carefully. The \feld component Hxis not continu-\nous atz= 0, where the step between both regions equals\nthe current density at the boundary. Hence, the proper\nboundary condition at the I-II interface is\nHI\nx(kx;0)\u0000HII\nx(kx;0) =j(kx;y) (6)\nin thekxz-space. The current density function j(x;z)\ndescribes the total current I0\rowing through the in-\fnitesimally thin MSL of width w, what can be ex-\npressed as j(x;z) =I0=w#(w=2\u0000x)#(x+w=2)\u000e(z)\nand the corresponding Fourier transform is j(kx;z) =\nj0sin(kxw=2)=kxwithj0= (2=\u0019)1=2I0=w. Combining\nthis discontinuity of the parallel H-\feld at the upper in-\nterface with the known continuity condition at the lower\ninterface and the continuity of the orthogonal B-\feld at\nboth interfaces provides four boundary conditions in to-\ntal. Inserting the ansatz (5) \fnally leads to a system\nof linear equations determining the coe\u000ecients C1\u0000C4.\nAs long as the interacting quantum system is located\nabove the magnetic thin \flm, we deal require the solu-\ntion for positive z-values, so that only the \feld ampli-\ntude in region I is playing a role for any coupling pro-\ncesses. Hence, it is su\u000ecient to concentrate only on the\ncoe\u000ecientC1. The system of equations can be further\nsimpli\fed by introducing approximations relying on the\nrealistic experimental values1. A typical saturation mag-\nnetizationMSof the ferromagnetic material is of the or-\nder of 103G and is achieved in an external magnetic \feld\nB0of\u0018102G. Hence, the assumption !2\n0\u001c!2is jus-\nti\fed for microwave excitation frequencies !in the GHz\nrange and susceptibility parameters \u001fand\u0014of the order\nof 10\u00001and 100. Further, the wave vectors of microwaves\nof aboutk0\u0019101m\u00001in vacuum are much smaller\nthan the experimental wave vectors kx\u0019105m\u00001, i. e.\nk2\n0\u001ck2\nx. Under these conditions, the parameter ain (2)\ncan be approximated independently from the material as\naPDMS\u0019aGGG\u0019aYIG\u0019ikx. Hence, the result for C1\nonly depends on kx,\nC1(kx)\u0019j0sin(kxw=2)\nkx!M(!0+!M+!)\u0000e2kxd(!0+!M\u0000!)(2!0+!M+ 2!)\n!2\nM+e2kxd(4!2\u0000(2!0+!M)2): (7)\nThe full solution for the magnetic \feld Hin the region\nI is given by\n\u0012HI\nx\nHI\nz\u0013\n\u0019\u0012\n1\ni\u0013\nC1(kx)e\u0000kxz\u0011\u0012\n1\ni\u0013\nHI(kx;z)=p\n2;(8)\nwhere the amplitude function HI(kx;z) is introduced.\nFrom equation (8) as a function of kxandz, the ex-\ncited spin wave modes inside the ferromagnetic \flm, i.\ne. the spin wave resonance condition, can be derived by\ndetermining the zeroes of the denominator. This yields\nthe dispersion relation\n!(kx) =1\n2q\n(2!0\u0000!M)2\u0000!2\nMe\u00002kxd; (9)\nthat matches exactly the so called Damon-Eshbach sur-\nface waves (DESW), which were calculated for ferromag-\nnetic thin \flms in the absence of any surface currents14.Hence, the solution of the modi\fed system with non-\nzero current is peaked around the DESW modes, which\nis reasonable against the background of made approxima-\ntions. Assuming large wave numbers kx, i. e.kx\u001dk0,\nthe current function j(kx;z)\u0018sin(kxw=2)=kxdecreases\nin amplitude, so that the result for the resonance con-\ndition is identical to the DESW modes within the given\napproximation.\nAn important feature of the calculated spin wave\nmodes is that for large kxthe frequency in (9) saturates\nto a value that depends on the external magnetic \feld\nB0. Thus, at a \fxed magnetic \feld, spin wave excita-\ntions are limited to a \fxed range of frequencies between\nthe limiting cases kx= 0 andkx!1 ,\np\n!0(!0\u0000!M)\u0014!\u0014!0\u0000!M=2: (10)\nThe lower bound at kx= 0 corresponds to the so called\nuniform precession mode, where all the spins inside the4\nFrequency (GHz)\nFrequency (GHz)\nFrequency\nFigure 2. Magnetic \feld amplitude depending on the microwave frequency !at \fxed external \feld B0= 50 G. The \feld\namplitudes are normalized to the applied current per unit length I0=w. (a) The wide range plot shows the step behavior at\nthe upper bound !max\nswof spin wave excitation. ( b) Magni\fcation of the range around the step frequency indicated by the black\nrectangle in ( a), which resolves a sharp decrease in the amplitude at e!sw. (c) Further magni\fcation shows a second amplitude\ndip at e!\u0003\nsw, below!max\nsw= 2:73 GHz.\nmaterial precess in phase, so that there is an oscillating\nmagnetization, but no spatial propagation. Even in a\nstrong external magnetic \feld of 200 G as used, for ex-\nample, in the reported experiment1, the uniform preces-\nsion mode of the considered YIG \flm oscillates at about\n1:6 GHz, which is far too small to stimulate magnetic\ndipole transitions in the spin triplet of the NV center. In\ncontrast, the upper limit !max\nsw=!0\u0000!M=2 is reached\nfor the same magnetic \feld at \u00183:1 GHz, which lies, for\nexample, in the range of NV resonances, as will be im-\nportant later.\nAfter deriving the driven spin wave modes, we are in-\nterested in the real space solution of the magnetic \feld, to\n\fnally model the interaction with an NV ensemble and to\ngive a quantitative expectation of the coupling strength.\nTherefore, the magnetic \feld HIin (8) has to be Fourier\ntransformed back into real space using the explicit form\nof the coe\u000ecient C1(kx) in (7). The \feld can be written\nas\nHI(x;z) =\u0012\n1\ni\u0013\nHI(x;z)=p\n2 (11)with the Fourier transform of the amplitude function\nHI(x;z) =1p\n2\u0019Z1\n\u00001HI(kx;z)eikxxdkx: (12)\nIn order to calculate this integral, the denominator in (7)\nis expanded to \frst order in kxaround its zero k0\nx, which\ncorresponds to the resonant wave number de\fned by (9),\nk0\nx=1\n2dln\u0012\n\u0000!2\nM\n4!2\u0000(2!0+!M)2\u0013\n: (13)\nThe remaining integral can be analytically evaluated\nby assuming that before the current in the MSL was\nswitched on there were no spin waves. This imposes the\nrule how to go around the pole in the integral above. Ap-\nplying the Sokhotski-Plemelj theorem in case of a real line\nintegral and restricting ourselves to the far \feld regime,\nwhere the condition x\u001d(kx\u0000k0\nx)\u00001holds, the amplitude\nfunctionHI(x;z) in real space is\nHI(x;z)\u0019ip\n2I0\nwsin(k0\nxw=2)\nk0xd\u0014!M(!0+!M+!)\n4!2\u0000(2!0+!M)2e\u00002k0\nxd+!0+!M\u0000!\n2!0+!M\u00002!\u0015\ne\u0000k0\nxzeik0\nxx; (14)\nwhich is one of our main results. Based on this com-\nplex \feld amplitude, the magnetic \feld above the ferro-\nmagnetic \flm is known explicitly. Note, however, that\nthe calculated solution only holds for frequencies within\nthe range (10). The assumptions, which were made in\norder to solve the integral above, are not valid outside\nthis frequency range, where the existence of the solution\n(14) at frequencies above !max\nswis not given. Hence, the\ngiven solution (14) is only valid forp\n!0(!0\u0000!M)\u0014\n!\u0014!0\u0000!M=2 and is set to 0 otherwise, which corre-\nsponds to the absence of spin waves. Within this de\fni-tion range, the absolute value of the \feld amplitude in\n(14) at \fxed magnetic \feld depends on the excitation fre-\nquency as plotted in Fig. 2. The wide range plot in Fig.\n2(a) shows the expected behavior. As discussed before,\nan upper bound at !max\nswcuts o\u000b the spin wave driving\nregime. Above this limit, the amplitude rapidly drops to\na level at least six orders of magnitude smaller. Below\nthis limit, the amplitude increases until the resonance\nfrequency!swfor DESW modes in (9) is reached, which\nbecomes apparent as a high peak just below the cut-o\u000b\nfrequency. Although the increase of jHIjfrom low fre-5\nFigure 3. (a) Possible orientation of NV centers inside the diamond structure. Depending on the alignment of the connection\nline between a substitutional nitrogen atom (green) and the neighboring vacancy (red), the NV-axis (red) can be oriented in\nthe depicted four directions. ( b) The crystal structure of diamond leads to four di\u000berent angles \u0012Bibetween the magnetic \feld\nand the possible NV-axes ^ ci. (c) Resonance frequencies of the magnetic dipole transitions within the ground state triplet of an\nensemble of NV centers embedded in single crystal diamond. Each of the four possible NV center orientations ^ cicontributes\ntwo resonances at !\u0006indicated by the di\u000berent line styles. The resonances depend on the alignment of the magnetic \feld. For\nthe left plot in ( c) the magnetic \feld is \u0012B= 50\u000eat an angle 'B= 30\u000ewith respect to one of the crystal axes and on the right\nthe chosen parameters are \u0012B= 15\u000e;'B= 50\u000e.\nquencies towards !max\nswin Fig. 2(a) seems to be smooth\nat the \frst sight, the magni\fcations in ( b) and (c) high-\nlight a substructure of amplitude dips very close to the\nmaximum. The occurring amplitude dips correspond to\nthe minima of the amplitude function in (14) induced by\nthe zeros of the factor sin( k0\nxw=2)=k0\nx. The position of the\namplitude dip at e!\u0003\nswin (c) di\u000bers by only a few hundreds\nof Hz from the resonance maximum and, therefore, might\nnot be resolved in usual resonance experiments. In con-\ntrast, the second dip in ( b) is more incisive and occurs at\na frequency a few MHz from the maximum. This second\ndecrease in the \feld amplitude should be experimentally\nresolvable as discussed later in section V.\nIV. MAGNETICALLY DRIVEN NV SPINS\nThe spin wave propagation through the magnetic thin\n\flm can increase the interaction between the external\n\feld and a quantum system. As a consequence, the driv-\ning of magnetic dipole transitions of NV centers in di-\namond can be performed more e\u000eciently. In order to\nmodel the coupling strength and to simulate the result-\ning transition spectrum, we have to distinguish whether\na single NV center or an NV ensemble in single crystal\ndiamond is probed.\nA. Single NV\nThe ground state of the NV center is a spin triplet,\nwhich is split into the mS= 0 andmS=\u00061 sublevels.\nThis ground state energy splitting Dof 2:87 GHz3origi-\nnates from spin-spin interaction and the Hamiltonian of\nthe spin triplet is\n^H0=D^S2\nz+g\u0016BB0\u0001S; (15)where the second term accounts for the Zeeman split-\nting with an external magnetic \feld B0= (Bx;By;Bz).\nThe spin vector S= (^Sx;^Sy;^Sz) contains the ( S= 1)\nspin operators ^Si, which are de\fned in the basis of the\nmS= 0;\u00001;1 spin projections corresponding to the ^Sz\neigenstatesfj0i;j\u0000iandj+ig. In cases where the mag-\nnetic \feld is not aligned with the NV-axis ^ c, the Hamil-\ntonian (15) can be written as\n^H0=D^S2\nz+g\u0016BB0(sin\u0012B^Sx+ cos\u0012B^Sz); (16)\nwhere the relative angle \u0012Bquanti\fes the orientation of\nthe external magnetic \feld in the frame of the NV cen-\nter. Since the Hamiltonian (16) depends explicitly on\n\u0012B, the according eigenenergies and, consequently, the\nfrequencies !\u0006\n\u0012Bof the dipole transitions j0i$j\u0006i vary\nfor di\u000berent orientations of the magnetic \feld with re-\nspect to the NV-axis. Hence, the transition spectrum\nof a single NV center consists of two frequency branches\n!\u0006\n\u0012B, whose behavior with respect to varying bias \feld B0\nis determined by the given value of \u0012B.\nB. NV ensemble in single crystal diamond\nIn case of an NV ensemble every single of the numer-\nous contained NV centers contributes two branches to the\ntransition spectrum. If the single NV centers were ran-\ndomly oriented, all branches would combine to a blurred\nspectrum. But due to the particular symmetry of the\ndiamond crystal, the actual transition spectrum of an\nNV ensemble inside a single crystal consists of single\nbranches. For a single crystal diamond, the lattice has a\n\fxed orientation in the lab frame, but inside the diamond\nstructure the actual orientation of an NV center is not\ncontrollable during fabrication. Thus, the NV axis can6\nbe aligned along the four crystal directions shown in Fig.\n3(a), which are at a tetrahedral angle of \u0012t= 109:5\u000eto\neach other. A single nanodiamond contains NV centers\nwith orientations equally distributed over these four di-\nrections. Accordingly, for a \fxed B0-\feld orientation, the\nmagnetic \feld is aligned at a di\u000berent angle with respect\nto each of these four directions. Consequently, there are\nnot numerous overlaying resonance branches, but each of\nthe four possible NV-axis orientations gives rise to two\nresonances and in total there are eight resonances for an\nNV ensemble.\nIn order to describe the expected magnetic resonances\nwith a formula, the NV center orientation is expressed\nin terms of the relative angles \u0012Bibetween the pos-\nsible NV-axis and the magnetic \feld as shown in the\nFig. 3(b). Choosing one of the NV-axes, ^ c0, to de-\n\fne thez-direction and one of the carbon atoms to\nlie on thex-axis, the normalized magnetic \feld vector\nand the four NV-axes ^ ciare parameterized in spherical\ncoordinates as ^B= (sin\u0012Bcos'B;sin\u0012Bcos'B;cos\u0012B)\nand ^ci= (sin\u0012icos'i;sin\u0012icos'i;cos\u0012i) with\u00120= 0,\n\u00121;2;3=\u0012t,'0;1= 0 and'2;3= 2\u0019=3;4\u0019=3. The in-\ntroduced coordinate system is thus \fxed to the crystal\nlattice. Based on these vectors, the angles \u0012Bibetween\nthe single NV-axes and the magnetic \feld in Fig. 3 are\nrelated to the corresponding scalar product as\ncos\u0012Bi=^B\u0001^ci= sin\u0012Bsin\u0012icos('B\u0000'i)+cos\u0012Bcos\u0012i:\n(17)\nInserting the angle \u0012Biobtained from this expression into\nthe Hamiltonian for non-aligned magnetic \feld (16) and\ncalculating the transition frequencies between the corre-\nsponding eigenenergies yields the spectra depending on\nthe polar as well as the azimuth angle of the magnetic\n\feld as plotted with respect to the strength of the applied\nmagnetic \feld in Fig. 3( c). A comparison of the two plots\ncorresponding to di\u000berent parameter sets ( \u0012B;'B) high-\nlights the sensitivity of the resonance spectrum of single\ncrystal diamond hosting NV centers at various orienta-\ntions to rotations of the crystal in a magnetic \feld. In\nthat way, information about the spatial orientations can\nbe extracted by exploiting the existing relation.\nFrom experimental data of the eight di\u000berent magnetic\nresonance frequencies in a nanodiamond of known crystal\norientation, the angles \u0012Band'Bcan be calculated or,\nin turn, from a known magnetic \feld alignment, the crys-\ntal directions of a nanodiamond can be derived. Hence,\nthe resonance experiment can be used to sense a probe\nmagnetic \feld resolving its direction on the one hand and\nto measure the crystal structure of a nanodiamond indi-\nrectly on the other hand.15,16\nC. Spin dynamics\nIn order to model the quantum mechanical coupling of\nan NV ensemble to the calculated spin wave \feld (14),\nthe corresponding Bloch equations are solved. Therefore,we use a \fve level scheme similar to17, which includes the\nNV energy levels relevant for the processes in an ODMR\nexperiment. The NV is pumped by a continuous laser,\nwhich excites the system from its ground state3A2triplet\nfj0i;j\u00061igto the excited state3E tripletfje0i;j\u0006e1ig.\nFrom the excited triplet states, the system partially de-\ncays optically back to the ground state manifold and the\nintensity of the emitted \ruorescent photons is detected\nin ODMR experiments.\nWithout external microwave drive and in the presence\nof optical excitation the \ruorescence would just have a\nconstant intensity and the system would be polarized in\nthems= 0. due to the inter-system crossing channel via\nthe excited singlet1A1statejsi. The induced resonance\nmicrowave transitions within the ground state manifold\npump the system out of ms= 0 state and in the pres-\nence of optical excitation lead to a change of \ruorescence\nintensity. The intensity in fact drops because now the\nsystem spends more time in the jsistate, which is not\noptically active at the same photon frequency.\nThe dynamics of this mechanism are described by the\ntime-dependent \fve-dimensional density matrix \u001a(t) of\nthe system written in the basis of fj0i;j1i;je0i;je1i;jsig.\nThereby, only the driven mS= 1 triplet sublevels are\nregarded, since we neglect transitions from the singlet to\nthe spin state that is not driven by the microwave \feld.\nWhen the transition into the mS=\u00001 state is driven, we\nuse the replacement j1i!j\u0000 1i. The density matrix is\ndetermined by solving the Master equation in Lindblad\nform,\n@t\u001a=i[\u001a;^H] +X\n\u0016>0\u0014\nL\u0016\u001aLy\n\u0016\u00001\n2Ly\n\u0016L\u0016\u001a\u00001\n2\u001aLy\n\u0016L\u0016\u0015\n;\n(18)\nwhere the operators L\u0016describe a non-unitary time evo-\nlution due to dissipative interactions between the sys-\ntem and the environment, which is given by the pho-\nton bath of the pumping laser and the emitted \ruo-\nrescence. For the special case of the \fve level system,\nthe Hamiltonian ^Hfor the closed system in (18) is de-\ncomposed into the static part and the interaction part,\n^H=^H0+^HI. The static Hamiltonian ^H0is diagonal\ncontaining the eigenenergies \"iof the \fve states jiiwith\ni= 0;1;e0;e1;s. Writing the classical microwave \feld as\nBmw(t) =Bmwcos(!t), the interaction Hamiltonian is\ngiven by\n^HI=\nR\n2\u0000\nei!t+e\u0000i!t\u0001\n(j1ih0j+j0ih1j); (19)\nwith the Rabi frequency \n R=\rh0jBmw\u0001Sj1idenot-\ning the coupling strength due to magnetic dipole inter-\naction between the driving \feld and the ground state\ntriplet. Here, the spin vector S= 1=2\u001bcan be repre-\nsented by the Pauli matrices \u001b= (\u001bx;\u001by;\u001bz). The op-\neratorsL\u0016in (18) represent the various dissipation pro-\ncesses as well as the continuous pumping, which excites\nthe system from the ground state manifold to the excited\nstate manifold at pumping rate \u0000 p, where the mSquan-7\ntum number is left unchanged, i. e. L0\np= \u00001=2\npje0ih0j\nandL1\np= \u00001=2\npje1ih1j. The inverse process, the direct\ndecay back into the ground states, happens at rate \u0000 0\nand the corresponding operators are L0\n0= \u00001=2\n0j0ihe0j\nandL1\n0= \u00001=2\n0j1ihe1j. For the inter-system crossing the\ncoupling between the lower level je0iandjsiis much\nsmaller than the coupling between the highest level je1i\nandjsi, therefore, the latter is neglected. Hence, the\ncorresponding operator is Les= \u00001=2\nesjsihe1j. The second\ninter-system crossing from jsito the ground state triplet\nj0iandj1iis modeled to have the same probabilities for\nending up in the \fnal states j0iandj1iwith the operators\nL0\nsg= (\u0000sg=2)1=2j0ihsjandL1\nsg= (\u0000sg=2)1=2j1ihsj. Fur-\nther, theT1-decay at rate \u0000 1= 1=T1from themS=\u00061\nandmS= 0 ground states into the equilibrium, which is\nassumed to lie at equally populated states is included.\nWe also treat the transverse relaxation including de-\nphasing inside the ground state spin triplet at rate \u0000 2.\nThe corresponding operators can be written in terms of\nspin operators as L1= (\u0000 1=2)1=2(j0ih1j+j1ih0j) and\nL2= (\u0000 2=2)1=2^\u001bz. Inserting these operators and the\nHamiltonians ^H0and ^HIinto the Master equation (18)\nyields the time derivative of the density matrix \u001ain the\nform of 15 coupled \frst order di\u000berential equations with\ntime-dependent coe\u000ecients. Since some of these equa-\ntions are redundant, a system of only seven di\u000berential\nequations actually has to be solved, where the rotating\nwave approximation is used. In the special case of con-\ntinuous pumping a single measurement of the ODMR\ncontrast at \fxed driving frequency !is detected over a\ntime long enough to allow the system to evolve into a\ndynamical equilibrium. Hence, the solution of interest is\na steady where the time derivative in (18) can be set to\nzero. Finally, the matrix elements \u001aijare obtained by\nsolving the remaining system of linear equations.\nD. Optical detection\nThe ODMR intensity Ican be calculated from the ob-\ntained solution for the density matrix \u001a. It is given by\nthe number of emitted photons, when the excited states\nje0iandje1idecay optically to the ground state triplet.\nDue to the existence of the inter-system crossing channel,\nonly a part of the population in je1icontributes to the in-\ntensity. In detail, only the fraction \u0000 0=(\u00000+\u0000es) remains\nwithin the spin triplet channel. Treating the detected in-\ntensity to be proportional to these e\u000bective probabilities,\nit can be written as\nI/\u001ae0e0+\u00000\n\u00000+ \u0000es\u001ae1e1: (20)\nFor the steady state solutions, the normalized intensity\ndepending on the detuning \u000e=\"1\u0000!turns out to be\nLorentzian,\nI(\u000e) =\u000e2\n\u000e2+e\r2; (21)wheree\rdescribes the width of a resonance centered at\n\u000e= 0, which is simpli\fed assuming the following. First,\nthe decay rate from the excited state je1iinto the triplet\nground state and the inter-system crossing channel are\nof the same order, i. e. \u0000 es\u0019\u00000. Second, the ground\nstate triplet is not completely depopulated during the\nexperiment, which requires a pumping rate much lower\nthan the decay rate, \u0000 p\u001c\u00000, and theT1-decay being\nmuch slower than the leakage via the inter-system cross-\ning channel, \u0000 1\u001c\u0000sg.17Under these conditions, the\nparametere\rtakes the compact form\ne\r=s\n\u0000\n2\u0000e\u000b\n2\u00012+4\u0000e\u000b\n2(1 + \u0000p=(4\u0000sg))\n\u00001+ \u0000p=4\n2\nR; (22)\nwith the e\u000bective transverse relaxation rate \u0000e\u000b\n2= \u0000 2+\n\u0000p+\u00001=2. So far, we assumed degenerate mS=\u00061 sub-\nlevels. But since the NV ensemble in the considered setup\nis placed in a static bias magnetic \feld, these sublevels\nare split by the corresponding Zeeman energy. Thus, if\nthe NV ensemble is driven with a microwave \feld at fre-\nquency!, each of the transitions j0i$j +iandj0i$j\u0000i\nwill be excited with di\u000berent detunings \u000e\u0006=!\u0006\u0000!.\nIn addition, the resonance spectrum of a single nanodia-\nmond is a combination of the resonances originating from\nfour di\u000berent alignments of the NV-axes as discussed for\nan NV ensemble. To calculate the ODMR spectrum of an\nNV ensemble inside a nanodiamond with eight occurring\nresonances, we assume the eight expected resonances be-\ning far o\u000b-resonant with each other. Hence, at \u000ei\n+= 0 the\ndetuning from the second resonance \u000ei\n\u0000is large enough\nand single transitions are excited separately. Therefore,\nthe contributions I(\u000ei\n\u0006) in (21) from each of the eight\nresonances are summed up to the intensity of ODMR\n\ruorescence,\nI(!) =X\nj=\u00064X\ni=1(!i\nj\u0000!)2\n(!i\nj\u0000!)2+\r2: (23)\nDue to the large magnetic \feld amplitudes of the driving\n\feld in the experiment, the line width e\rin (22) is dom-\ninated by the second term, which means that the lines\ncan be assumed to be purely power broadened17. In the\ncase of low pumping power, the condition \u0000 p=4\u001c\u0000sgis\nful\flled ande\rcan be further approximated by\ne\r\u0019s\n4 \u0000e\u000b\n2\n\u00001+ \u0000p=4\nR: (24)\nFor a plot of the ODMR intensity I(!) in dependence\nof the bias \feld strength, we assume realistic values for\nthe relaxation times T1= 1=\u00001andT2= 1=\u00002of the NV\ncenter to lie in the range of ms18andµs19and a pumping\nrate \u0000p\u0019106s\u00001, so that the line width is approximately\n2\nR.\nBased on the intensity (23) with the simpli\fed line\nwidth (24) and using the spin wave \feld amplitude (14)8\nFigure 4. ( a) and (b): ODMR intensity Idepending on the magnetic \feld and the driving frequency. The used parameters\ncorrespond to a nanodiamond at xNV= 5µm away from the antenna and a driving power of 0 :5µW. For the magnetic \feld\norientation the arbitrary values \u0012B= 14\u000eand'B= 7\u000eare chosen. In ( b), a zoom of the black boxed area in ( a) is shown,\nwhich highlights the substructure of the resonances right at the cut-o\u000b frequency !max\nsw.\ncalculated in section III, a theoretical ODMR spectrum\nis plotted in Fig. 4. In the frequency regime above the\ncut-o\u000b of spin wave excitation, i. e. ! > !max\nsw, we use\na microwave driving \feld amplitude corresponding to a\npure antenna \feld Hantwithout spin wave excitation in-\nside the YIG \flm.\nThe parameters xNVandPentering the function for\nHsw/ant are chosen to be equal to the experimental spec-\ntrum in Ref. 1 (Supplementary), which are given by\nthe distance between the nanodiamond and the antenna\nxNV\u00195µm and a driving power of P= 0:5µW. The\nangle of the bias \feld is chosen arbitrary to be \u0012B= 14\u000e\nand'B= 7\u000e. The zoom of two resonances within the\nblack boxed area in ( a) right at the maximal spin wave\nexcitation frequency is given in Fig. 4( b). The more the\ndriving frequency abroaches the cut-o\u000b frequency !max\nsw,\nthe stronger the resonances are broadened, which results\nfrom the maximum of the amplitude function HIat the\nupper frequency limit. In addition, the high resolution\nin Fig. 4(b) also shows a sharp peak in the intensity ap-\nparent as a bright line parallel to the border between the\ndriving regimes. The reason for this intensity increase\nis the substructure of the amplitude function in the spin\nwave driving regime and it corresponds to the \frst dip in\nFig. 2 at frequency e!sw. Between this peak and the cut-\no\u000b frequency, the spin wave \feld takes a maximal value\nright before it drops to zero. This becomes apparent by\nthe dark line in Fig. 4( b) crossing the resonance branches\nfrom the bottem left to the top right and a similar dark\nline occurs in experimental ODMR spectra1. This con-\nsistency between the experiment and the theory plot sug-\ngests, that the amplitude function (14) describes the \feld\namplitude in the spin wave regime at a satisfying level of\naccuracy.V. ENHANCEMENT\nDue to the presence of spin waves inside the magnetic\n\flm, the magnetic \feld coupling to an NV ensemble is\nenhanced compared to the amplitude of a pure antenna\n\feld. Moreover, the spin wave \feld has the great ad-\nvantage of higher decay length, which becomes appar-\nent when inspecting the explicit dependencies of the spin\nwave \feld in (14). The lack of x-dependence of the \feld\namplitude indicates a very important di\u000berence from a\npure antenna driving \feld. The microwave \feld of the\nantenna decays as 1 =x, so that the coupling strongly de-\npends on the position of the interacting NV center. In\na system of defect spins in a quantum system addressed\nor read out with an oscillating magnetic \feld, informa-\ntion can be transferred over a distance, which increases\nwith the length over which the \feld amplitude is con-\nserved. Hence, the x-independent spin wave \feld ampli-\ntude in (14) appears to lead to a constant coupling over\na large distance. Realistically, the coupling decays due\nto the neglected exchange energy terms in the suscepti-\nbility tensor. Regarding the corresponding terms would\ngive rise to a magnetic amplitude damping in (14) along\nthex-coordinate. But YIG has a particularly low damp-\ning parameter and the excited spin waves propagate at\nlowk, thus justifying the made approximation. Along the\nz-direction, ~C1decays exponentially with the distance to\nthe surface, i.e. the spin wave modes are con\fned to the\nsurface of the thin \flm. This guarantees a highly local-\nized \feld, so that the e\u000bect of spin wave enhancement is\nonly present in the vicinity to the magnetic \flm.\nUsing the obtained result for the spin wave \feld al-\nlows us to investigate the enhancement of the coupling\nstrength \n Rwith respect to the case without spin wave\nexcitation. The enhancement factor is de\fned as \u0011=\n\nsw\nR=\nant\nR, where \nsw\nRdenotes the Rabi frequency in the9\nFigure 5. (a) Plot of the resonance frequencies of an NV center (black) depending on the bias \feld. The NV-axis is oriented\nin thexy-plane at an angle of 78\u000e. The black dashed line represents the step between spin wave and antenna regime. The\nintersections of the driving frequency (blue/red dashed) with one of the branches indicates possible measurement settings.\n(b) Dependence of the e\u000bective enhancement \u0011e\u000bon thex-component of the coupling NV ensemble. The theoretical model\ncorresponds to the red line and the data points represent the experimental results of Andrich et al. presented in Ref. 1.\ncase of spin wave excitation and \nant\nRis the same quan-\ntity for the pure antenna driving without any back action\noriginating from magnetic material. More speci\fcally we\ncalculate the antenna \feld without regarding the stray\n\feld of the YIG, which is reasonable in the regime with-\nout spin wave excitation, since the thin \flm is located in\nan approximately homogeneous magnetic \feld and due\nto its small height the in\ruence on the antenna \feld be-\ncomes negligible. Both quantities, the spin wave and the\nantenna \feld, depend on the angle between the respec-\ntive \feld and the NV center, which is non-zero in the\ngeneral case.\nIn a coordinate system S0withz0-axis parallel to\nthe NV-axis ^ cand the antenna \feld lying in the x0z0-\nplane, the magnetic \felds of antenna and spin waves\nare parametrized as Bant=Bant(sin\u0012ant;0;cos\u0012ant)\nandBsw=Bsw(sin\u0012swcos'sw;sin\u0012swsin'sw;cos\u0012sw),\nwhere\u0012ant/sw denotes the angle with respect to the ^ c-\naxis and the spin wave \feld is at the angle 'swwith\nrespect to the x0z0-plane. The Rabi frequency scales\nwith the o\u000b-diagonal terms in Bsw/ant\u0001S, so that \nsw\nR/\nBswsin\u0012swexp(\u0000i'sw) and analogously for the antenna\ndriving \feld. Inserting this into the enhancement ratio\nyields\n\u0011=Hsw\nHantsin\u0012sw\nsin\u0012ante\u0000i'sw: (25)\nSince the phase factor e\u0000i'swdoes not a\u000bect the absolute\nvalue of this enhancement, it is omitted in the following.\nFurthermore, we want to adapt our result to the exper-\nimental method described in1, where the resonance be-\ntween the same levels is measured twice in the di\u000berent\ndriving regimes, the antenna driving and the spin wave\ndriving regime. To measure both regimes at the samefrequency, a resonance has to lie once above the maximal\nexcitation frequency of spin waves and once below, so\nthat the di\u000berent regimes can be tuned by the bias mag-\nnetic \feld. For the realization of such a measurement,\nthe orientation of the bias \feld has to be non-collinear\nto the NV-axis, whereby the states j0iandj\u0006ibecome\ncoupled and the energy splitting does not depend linearly\non the magnetic \feld. Thus, the resonance branches are\ntilted as plotted in Fig. 5( a). The plot shows the NV\nresonances in the case of the NV-axis being aligned in\nthexy-plane at an angle of \u0012B= 78\u000ewith respect to\nthe bias magnetic \feld. In Fig. 5( a) only six resonances\nare visible, because the magnetic \feld encloses the same\nangle with two of the possible crystal directions lead-\ning to these resonances being two-fold degenerate. An\nexcitation frequency of 2 :86 GHz is on resonance with a\nbias \feldB0antin the antenna driving regime, which is\nmarked by the blue dot in Fig. 5( a). A second resonance\n(red dot) occurs in the regime of spin wave excitation at\nB0sw. For an optimal resolution of the detection in both\nregimes, the antenna current, i. e. the driving power\nPis varied. The antenna \feld decays rapidly with the\ndistance of the NV center from the antenna, so that a\ndriving power of the order of mW is required to obtain\na clear signal. If the same driving power was used in\nthe spin wave driving regime, the resonance would be\nsigni\fcantly power broadened and hence, a lower power\nat several µW is chosen. Hence, the enhancement ratio\n\u0011has to be corrected by the impact of varying driving\npower. Both, the antenna \feld amplitude Hantand the\nspin wave \feld amplitude Hswdepend linearly on the\ndriving current I0= (2P=Z)1=2, which is an e\u000bective ac\ncurrent through a wire with impedance Z. Finally, the\ne\u000bective enhancement ratio is obtained by normalizing\nthe \felds with respect to the driving power Pant/sw ,10\n\u0011e\u000b=r\nPant\nPswHsw\nHantsin\u0012sw\nsin\u0012ant= 4p\n2\u0019r\nPant\nPswsin(k0\nxw=2)\nk0x\u0010\n!M(!0+!M+!)\n4!2\u0000(2!0+!M)2e\u00002k0\nxd+!0+!M\u0000!\n2!0+!M\u00002!\u0011\ne\u0000k0\nxz\nln\u0010\u0000w\n2\u0000x\u00012=\u0000w\n2+x\u00012\u0011sin\u0012sw\nsin\u0012ant:(26)\nWith this expression, we are able to make a theoret-\nical prediction for the enhancement for an NV center\nlocated at xNV. Since the far \feld approximation was\nmade, the condition x\u001dk\u00001has to be ful\flled and for\nwave numbers of the excited spin waves1of the order\nof\u0018105m\u00001, the expression (26) applies for distances\nabove\u001810µm. For example, the resonances of a nan-\nodiamond at xNV= 20 µm driven by a \feld at 2 :862 GHz\noccur in the antenna regime at 15 G and in the spin wave\ndriving regime at 145 G. Using these parameters, the\nenhancement factor of the coupling between the defect\nspin and the driving \feld is \u0011e\u000b\u0019117, which is close to\nthe experimentally measured enhancement \u0011e\u000b\u0019100 de-\ntected via Rabi experiments1. Moreover, when distance\nxin expression (26) is varied, the enhancement factor \u0011e\u000b\nbehaves as plotted in Fig. 5( b), along with the experi-\nmental data of Andrich et al.1. The theoretical prediction\nmatches the experimental behavior, although the exper-\nimental data are slightly shifted to lower enhancement\nvalues with respect to the predicted theory curve. Nev-\nertheless, the linear x-dependence of \u0011e\u000bnot only shows\ngood agreement with the trend of experimental data, but\nalso an achievable enhancement factor of more than 400\nat distances above 70 \u0016m which underscores the great ad-\nvantage of spin wave excitation inside the magnetic \flm.\nFor low magnetic damping the spin waves do not decay\nas fast as the microwave \feld leading to a large enhance-\nment far away from the antenna.\nVI. CONCLUSION\nWe have presented a theoretical model that provides a\ncomplete description of real experimental hybrid systems\nconsisting of driven spin waves coupled to NV spins. All\nresults are obtained analytically and especially the de-\nscription of spin wave modes follows a fundamental ap-\nproach from the basics of Maxwell's equations. In that\nway, we derive an analytical solution for the magnetic\n\feld amplitude, which allows for a theoretical calcula-\ntion of the coupling to an NV ensemble. Therefore, we\nregard every possible spatial orientation of an NV center\nin a real nano-diamond, thus obtaining the full spectrum\ncapturing all eight resonances. Our theoretical intensity\nplots show good agreement with existing experimental\ndata1, what clearly highlights the suitability of the de-\nveloped model. The existence of the substructure and\nan upper frequency bound limiting the spin wave driv-\ning regime are important features, which follow from the\ntheoretical calculations. They provide an accurate but at\nthe same time very applicable method of tuning the drive\nof spin rotations of an NV center, where the resonant cou-pling below the cut-o\u000b frequency in the spin wave driving\nregime are calculated to be much stronger with respect\nto the case of pure antenna driving. This di\u000berence in\nthe coupling strength is a powerful tool to speed up qubit\noperations in the \feld of quantum computing. The de-\nrived enhancement factor \u0011e\u000bis strongly dependent on\nthe distance between the source of the microwave \feld\nand the NV ensemble. Accordingly, the magnetic \feld\nat the NV spin can be enhanced by two orders of magni-\ntude and the predicted behavior for longer distances is in\ngood agreement with recent experiments1indicating the\nstrength of our model as well as the applicability of the\nrelated assumptions and approximations. The presented\nsolutions are obtained using the far \feld approximation,\nwhich holds for distances above \u001810µm. This regime is\nof particular interest for technical realizations, since it is\na general problem to couple a sensitive quantum system\nto a microwave antenna across distances of more than a\nfewµm in view of the rapidly decaying antenna \feld. In\ncontrast, we show that the spin wave \feld is very stable\nacross the distance enabling a coupling to NV centers far\naway from the antenna by using a magnetic thin \flm.\nIf the enhancing e\u000bect over a long distance is of partic-\nular interest the made approximations are reasonable and\nthe model provides good results. But in case of a mod-\ni\fed system operating at a di\u000berent conditions, possibly\nthe limits of our theoretical model could be reached. If\na nanodiamond is positioned very close to the antenna,\nthere is need of a solution outside the far \feld regime,\nwhich is applicable for short distances. An exemplary\nreal system would be an array of nanodiamonds, where\nthe antenna is not grown besides but in between the ar-\nray, so that the NV centers are partly in the near \feld\nand partly in the far \feld regime. Strictly speaking, this\nwould require a mid \feld solution as well.\nFurther unconsidered e\u000bects, which had to be taken\ninto account in order to optimize the presented model,\nare additional boundary e\u000bects caused by the sourround-\ning material being non-perfect magnetic vaccum or spin\nwave re\rections in case of the \flm having \fnite expan-\nsion in the xy-plane. A re\fnement of the model con-\nsidering those aspects would lead to important insights\nwith respect to new applications of a spin wave mediated\ncoupling. Since the spectrum of the spin waves is mainly\ndetermined by the shape of the ferromagnetic sample,\nthe driven spin wave modes could be adapted by shap-\ning, for example, a ferromagnetic sphere or a disk. 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Lett. 108, 197601 (2012).\n19P. L. Stanwix, L. M. Pham, J. R. Maze, D. Le Sage, T. K.\nYeung, P. Cappellaro, P. R. Hemmer, A. Yacoby, M. D.\nLukin, and R. L. Walsworth, Phys. Rev. B 82, 201201\n(2010).\n20K. Kiseki, S. Yakata, and T. Kimura, Applied Physics\nLetters 101, 212404 (2012)."    },    {        "title": "1509.04018v1.Spectral_characteristics_of_time_resolved_magnonic_spin_Seebeck_effect.pdf",        "content": "arXiv:1509.04018v1  [cond-mat.mtrl-sci]  14 Sep 2015Spectral characteristics of time resolved magnonic spin Se ebeck effect\nS. R. Etesami, L. Chotorlishvili, and J. Berakdar\nInstitut f¨ ur Physik, Martin-Luther-Universit¨ at Halle- Wittenberg, 06099 Halle,\nGermany\n(Dated: 15 July 2021)\nSpin Seebeck effect (SSE) holds promise for new spintronic devices w ith low-energy consumption. The un-\nderlying physics, essential for a further progress, is yet to be fu lly clarified. This study of the time resolved\nlongitudinal SSE in the magnetic insulator yttrium iron garnet (YIG) c oncludes that a substantial contri-\nbution to the spin current stems from small wave-vector subther mal exchange magnons. Our finding is in\nline with the recent experiment by S. R. Boona and J. P. Heremans, P hys. Rev. B 90, 064421 (2014).\nTechnically, the spin-current dynamics is treated based on the Lan dau-Lifshitz-Gilbert (LLG) equation also\nincluding magnons back-action on thermal bath, while the formation of the time dependent thermal gradient\nis described self-consistently via the heat equation coupled to the m agnetization dynamics.\nIntroduction. The spin counterpart of the Seebeck\neffect, the spin-Seebeck effect (SSE) refers to the emer-\ngence of a spin current upon applying a thermal bias.\nAside fundamental interest, SSE is of relevance for a\nvariety of applications, including spin-dependent ther-\nmoelectric devices1. Since its first observation2, SSE\nhasbeenstudied extensivelybothexperimentally3–15and\ntheoretically16–23and for a variety of systems. In partic-\nular, SSE in ferromagnetic (FM) insulators4hints on a\nmagnonic origin of the spin current5. Magnonic SSE has\nsome advantages with respect to charge-carrier-related\nspin current in that magnon spin current propagatesover\na length scale of up to a millimeter24while conduction-\nbased spin current is usually much smaller due to spin-\ndependent scattering. The spectral characteristics of the\nmagnons contributing to the magnonic SSE are still un-\nder debate. In a recent spatially resolved experiment6on\nthe magnetic insulator yttrium iron garnet (YIG), the\nmeasured magnon temperature Tmwas related to the\nshort wavelength exchange part of the magnon spectrum\nω/parenleftbig/vectork/parenrightbig\n. An important observation is that a non-vanishing\nspin current emerges even for equal magnon and phonon\ntemperatures Tm=Tph. Note that the standard narra-\ntive attributes the emergence of the magnonic spin cur-\nrent to the difference between magnon and phonon tem-\nperatures. A theoretical explanation to the observed fact\nwas given in terms of the long-wavelength dipolar part\nof the magnon spectrum25. If long-wavelength dipolar\nmagnons are weakly coupled to the phonons their life\ntime is largerand the magnon temperature deviates from\nthe phonon temperature. Thus, dipolar magnons might\ncontribute to SSE. This explanation though comprehen-\nsible, does not exclude the contribution of the long-\nwavelength exchange subthermal magnons in the SSE.\nAs was shown in a recent theoretical paper17constitu-\ntive issuein the formationofthe magnonicSSE is not the\ndifferencebetweenmagnonandphonontemperaturesbut\nrather the nonuniform magnon temperature profile lead-\ning to a nonzero exchange spin torque and the magnon\naccumulation effect that drives the magnonic SSE. In\nthe present paper we study the contribution of the long-\nwavelength (small wave-vector /vectork) exchange subthermalmagnons to the magnonic SSE. We find that contrary\nto the short-wavelength exchange thermal magnons, the\nlong-wavelength exchange subthermal magnons do con-\ntribute substantially to the SSE. Our predictions are\nconsistent with the recent experimental report by S. R.\nBoona and J. P. Heremans8. Magnon modes at thermal\nenergies in YIG were found not responsible for the spin\nSeebeck effect. Subthermal magnons, i.e., those at ener-\ngies below about 30 ±10 [K], were found important for\nthe spin transport in YIG at all temperatures. To assess\nthe partial contribution to SSE of magnons with differ-\nent frequencies ω/parenleftbig/vectork/parenrightbig\nwe analyze the time resolved SSE\nadopting a low-pass filter, as done experimentally9. In\ntime resolved SSE experiments the laser modulation fre-\nquency is the relevant control parameter. If it is smaller\ncompared to the cutoff frequency of the filter then the\nfilter does not affect the spin current, otherwise the low-\npass filter cuts the spin current (external cutoff). Any\ncutoff observed for modulation frequencies smaller than\ncutoff frequency of the filter is thus intrinsic. In our sim-\nulation we implemented this approach, which aside from\nmimicking the experimental situation bears some advan-\ntages against a discrete Fourier transform to obtain the\nspectral dependence of the spin current, as detailed in\nthe supplementary materials26.\nWe note, that while our study is limited to thermally\ninduced magnonic transport in FM insulators, it can\nbe in principle extended to include contributions from\nthermally activated carriers in metals or semiconductors.\nThe method would rely however on further reasonable\ninputs such as the details of the spin torque current and\nthe rescaled exchange interaction parameters.\nTheoretical background . In a FM insulator, the\nlow-lying excitations are spin waves describable by\nLandau-Lifshitz-Gilbert (LLG) equation (for a compar-\nison an implementation based on the Landau-Lifshitz-\nMiyasaki-Seki scheme has also been performed with full\ndetails and results being included in the Supplementary2\nMaterials26)\n∂\n∂t/vectorM(/vector r,t) =−γ/vectorM(/vector r,t)×/bracketleftbigg\nH0ˆz+2A\nM2\nS∇2/vectorM(/vector r,t)/bracketrightbigg\n+α\nMS/vectorM(/vector r,t)×∂\n∂t/vectorM(/vector r,t),(1)\nwhere/vectorM(/vector r,t) stands for the magnetization vector, γ\nis the gyromagnetic ratio, H0is the external mag-\nnetic field, Ais the exchange stiffness, MSis the\nsaturation magnetization and αrefers to the Gilbert\ndamping constant. The LLG equation in the linear\nlimit has the following solution Mx(/vector r,t) +iMy(/vector r,t)∝\nexp/parenleftBig\ni/vectork·/vector r+iωkt/parenrightBig\nexp(−αωkt). The spin wave disper-\nsion relation ωk=γ/parenleftBig\nH0+2A\nMSk2/parenrightBig\nand the damping\nwhich depends on the wave vector αωk,5indicates that\nshort wavevector /vectork(long wavelength)exchangemagnons\nare less damped. Thus magnons have different relaxation\ntimes. Assuming that the magnon-phonon scattering is\nthe main source of damping, the magnon-phonon relax-\nation (thermalization) time reads\nτk\nmp=/bracketleftbigg\nαγ/parenleftbigg\nH0+2A\nMSk2/parenrightbigg/bracketrightbigg−1\n,k= 2nπ/d,n= 0,±1,···\n(2)\nwheredis the length of the FM insulator. Let us as-\nsume that the left edge of the FM insulator is heated\nup periodically with the modulation frequency ωmod(see\nFig. 1). A time periodic thermal bias is meant to mimic\nqualitatively the action of laser pulses (in real exper-\niment the laser electromagnetic energy is absorbed by\nthe sample only partly, however. More details are found\nin the Supplementary Materials26). The inherent ther-\nmal loses relevant to the experiment are treated self-\nconsistently by adopting an additional source term in\nthe heat equation (see Eq. (SM-3) in the Supplemen-\ntary materials26). Thus, the temperature profile imple-\nmented in the LLG equation in our case is calculated\nself-consistently. For unraveling the role of magnons\nwith different frequencies we follow the idea of using a\nlow-pass filter, i.e., a filter with an extrinsic cutoff fre-\nquencyωcwhich detruncates the spin current if the mod-\nulation frequency of the laser pulses exceeds the extrin-\nsic cutoff frequency of the filter ωmod> ωc. The ratio\nbetween ”true” I0and measured I/parenleftbig\nωmod/parenrightbig\nspin current\nreadsI/parenleftbig\nωmod/parenrightbig\n=I0//radicalBig\n1+/parenleftbig\nωmod/ωc/parenrightbig2. Ifωmod< ωc,\nthe measured spin current is not altered by the filter.\nThe decay in the spin current in this case is ascribed\nto the intrinsic cutoff frequencies, which in turn are re-\nlatedtothemagnon-phonon-relaxation-time(seeEq.(2))\nΩk\nmp= 2π/τk\nmp. In this way different internal cutoff fre-\nquencies can be observed.\nModel and simulations. We model a ferromagnetic\ninsulator via a chain of FM cells arranged along the ˆ x\naxis (Fig. 1). The total energy density of the system of\nFIG. 1. Schematic of the contribution of the spin waves with\ndifferent wave vectors kito the spin current in SSE. The\ntemperature of the left edge of the system is varied in time:\nT0(t) =T00S(ωmodt), where S(ωmodt) is a rectangular pulse\nwith a modulation frequency ωmodand levels 0 and 1. Tnand\nMnrepresent the temperature and magnetization in each in-\ndividual cell, respectively, and are calculated self-cons istently\nvia the heat and LLG equations (Eqs. (4) and (6)).\nNcells reads:\ne=−H0N/summationdisplay\nn=1Mz\nn−2A\na2M2\nSN/summationdisplay\nn=1/vectorMn·/vectorMn+1,(3)\nwhereais the size of the cell, /vectorMnis the magnetiza-\ntion vector of nthcubic cell and Ais the exchange stiff-\nness. WeuseEq.(3)tomodelYttrium-iron-garnet(YIG)\nwhichhasbeenemployedextensivelyinSSEexperiments.\nThe effective magnetic field acting on the nthcell reads\n/vectorHeff\nn(t) =−∂e\n∂/vectorMn=H0ˆz+2A\na2M2\nS/parenleftBig\n/vectorMn+1+/vectorMn−1/parenrightBig\n.\nA Gaussian-white noise /vector ηn(t) contribution to the\neffective magnetic field (with a correlation function\n∝angb∇acketleftηni(t)ηmj(t+ ∆t)∝angb∇acket∇ight=2αKBTn(t)\nγMSa3δnmδijδ(∆t)) accounts\nfor thermal fluctuations. Here, ∝angb∇acketleft···∝angb∇acket∇ightmeans average over\ndifferent realization of the noise, nandmare cell num-\nbers and iandjarethe Cartesiancomponents. The time\nand site-dependent temperature Tn(t) obeys the follow-\ning heat equation\nd\ndtTn(t) =κ\nρCTn+1(t)−2Tn(t)+Tn+1(t)\na2,(4)\nwith the initial and the boundary conditions\nTn(t= 0) = 0; n= 0,···N+1\nT0(t) =T00S(ωmodt),TN+1(t) = 0.(5)\nκisthe phononicthermalconductivity, ρisthemassden-\nsity,Cis the phonon heat capacity, T00is the tempera-\nture applied on the left edge and Sis a seriesof rectangu-\nlar laser pulses with the modulation frequency ωmod(see\nFig. 1). For solving the heat equation we implemented\na Forward-Time Central-Space (FTCS) scheme27. The\nhierarchy of the relaxation times for phonons τph=/bracketleftBig\nκ\nρC/parenleftbigπ\nNa/parenrightbig2/bracketrightBig−1\n≈10[ns] (the number and size of cells\nN= 50,a= 10[nm] and phonon thermal conductivity\nκ= 6[W.m−1.K−1]) and magnons τmp=1\n2αω0≈103[ns]3\n0 10 20 30 40 50012345\nTime/LBracket1ns/RBracket1Spincurrent1011/MultiΠly/LBracket1/HBars/Minus1/RBracket1Ωc/LBracket1GHz/RBracket1\n/Infinity\n2Π/MultiΠly10/Minus0.0\n2Π/MultiΠly10/Minus0.6\n2Π/MultiΠly10/Minus1.0\n2Π/MultiΠly10/Minus1.4\nFIG. 2. Spin current at the middle of a chain of 50 FM cells\nversus time and for different extrinsic cutoff frequencies ( ωc).\nWe choose a= 10[nm], H0= 0.057[T],T00= 10[K], α= 0.1.\nThe system is heated up periodically with the modulation\nfrequencies of ωmod= 2π×10−1.0[GHz] and the spin current\nis statistically averaged over 1000 realization of the nois e. For\nthe blue curve no cutoff frequency is implemented on the spin\ncurrent ( ωc=∞) but for the red, green, orange and black\ncurves the cutoff frequencies are ωc= 2π×100.0[GHz],ωc=\n2π×10−0.6[GHz],ωc= 2π×10−1.0[GHz] and ωc= 2π×\n10−1.4[GHz], respectively.\n(ferromagnetic resonance frequency and phenomenologi-\ncal damping constant ω0= 10 [Ghz], α≈10−4) allows\nan adiabatic decoupling which amounts, to a first order,\nto plug the obtained phonon temperature profile directly\nin the LLG equation and study so the magnetization dy-\nnamics self-consistently.\nThe magnetization dynamics is governed by a set of\ncoupled LLG equations\n∂\n∂t/vectorMn(t) =\n−γ\n1+α2/vectorMn(t)×/bracketleftbigg\n/vectorHeff\nn(t)+α\nMS/vectorMn(t)×/vectorHeff\nn(t)/bracketrightbigg\n.\n(6)\nFor the numerical integration of the coupled stochastic\ndifferential equations we utilize the Heun’s method28,29.\nThe spin current tensor is calculated using the following\nTABLE I. Magnon-phonon relaxation times and the cor-\nresponding frequencies according to Eq. (2) for N= 50,\na= 10[nm], H0= 0.057[T] and α= 0.1.\nn 012345\n|k|= 2πn/Na[108m−1]0.000.130.250.380.500.63\nτk\nmp[ns] 1.0000.3030.0980.0460.0260.017\nΩk\nmp/2π[GHz] 1.03.310.221.637.758.4Ωc/LBracket1Hz/RBracket1/Slash12Π\n6.3/MultiΠly107\n1.0/MultiΠly108\n1.6/MultiΠly108\n2.5/MultiΠly108\n4.0/MultiΠly108\n6.3/MultiΠly108\n1.0/MultiΠly109\n1.6/MultiΠly109\n2.5/MultiΠly109\n4.0/MultiΠly109\n6.3/MultiΠly109\n1.0/MultiΠly1010\n1.6/MultiΠly1010\n2.5/MultiΠly1010\n4.0/MultiΠly1010\n/VertEllipsis2\n1.0/MultiΠly1012\n1071081091010101110120.60.650.70.750.80.850.90.951\nΩmod/LBracket1Hz/RBracket1/Slash12ΠNormalized spincurrent\nFIG. 3. Normalized spin current (Iωmod\nI2π×107[Hz]) at the middle\nof a chain of 50 FM cells versus the modulation frequency for\ndifferent extrinsic cutoff frequencies ( ωc) with the parameters\na= 10[nm], H0= 0.057[T],T00= 10[K], α= 0.1. The spin\ncurrent is statistically averaged over 1000 noise realizat ions.\nForωc<2π×109[Hz] the cascades follow the extrinsic cutoff\nfrequencies which are characteristic of Low-Pass filter. Ho w-\never, for ωc≥2π×109[Hz] the cascades occur earlier than\nthe corresponding extrinsic cutoff which are a sign of inhere nt\nintrinsic cutoff in the system. The arrows show the magnon-\nphonon frequencies (Ωk\nmp/2πin TABLE I) for different wave\nvectors evaluated theoretically (Eq. (2)) and coinciding w ith\nthe appearance of intrinsic cutoff frequencies (cascades) i n the\ncurves.\nformula: Iα\nn=−2Aa\nM2\nS/summationtextn\nm=1∝angb∇acketleftMβ\nm(Mγ\nm−1+Mγ\nm+1)∝angb∇acket∇ightεαβγ\nwhereα=x,y,zdefines the spin components of the\ntensor, while nstands for the cite number, εαβγis the\nLevi-Civita antisymmetric tensor and ∝angb∇acketleft···∝angb∇acket∇ightmeans aver-\naging over the different realization of the noise5,17,19,22.\nIn our model, because of the particular geometry of the\nsystem (1D chain aligned along ˆ xaxis) the only non-zero\nelement of the spin current tensor is Iz\nn17. For the out-\nput signal we implemented a recursive low-pass filter27\nIout(t) =ωc∆t\n1+ωc∆tIin(t) +1\n1+ωc∆tIout(t−∆t). Here Iin\nandIoutare the spin currents before and after filtering\nprocedure and ωcis the extrinsic cutoff frequency of the\nfilter (see Supplementary Materials26).\nResults and discussion. Fig. 2 shows the spin See-\nbeck current as a function of time for different extrinsic\ncutoff frequencies ωc. To each cutoff frequency a cer-\ntain color is attributed. The filter with the extrinsic cut-\noff frequency ωccuts the spin current if the modulation\nfrequency of the laser pulses ωmodexceeds the extrinsic\ncutoffωmod> ωc. Thus, the larger the extrinsic cutoff of4\nthe filter ωc, the larger is the spin current. This is what\nwe see in Fig. 2. For small extrinsic cutoff frequencies\n(see Fig. 3) ωc<2π×109[Hz] the spin current is detrun-\ncated extrinsically at modulation frequencies ωmod=ωc\nsmaller than the first intrinsic cutoff frequency of the\nsystem Ωk=0\nmp= 2π/τk=0\nmp. Therefore, no inherent cutoff is\nobserved in this case. However, for an elevated extrinsic\ncutoffωcwe observe a cascade of the inherent intrinsic\ncutoff at the frequencies Ωk\nmp= 2π/τk\nmp< ωc. All these\nintrinsic cutoff frequencies are in the subthermal regime\nof the magnon spectrum k < kmax≈108[m−1]6,8(TA-\nBLE I). To be confident while heating up the system the\nthermal magnons are also activated, the corresponding\ndispersion relation to our parameters is shown in Fig. 4.\nAs can be seen, magnons with a broad range of frequen-\ncies, beyond subthermal regime are also activated. The\nsubthermal regime of the spectrum is shown with a small\ngreen frame6,8.\nΓ/LParen1H0/Plus2A\nMS2/LParen11/MinusCos/LParen1ka/RParen1/RParen1\na2/RParen1\n/Minus3/Minus2/Minus1 0 1 2 30123456\nk/LBracket1108m/Minus1/RBracket1Ω/LBracket11012Hz/RBracket1\nSubthermal regime\nFIG. 4. Spin-wave dispersion relation: The yellow back-\nground shows the absolute value of the discrete fourier tran s-\nformation of mx+imybased on micromagnetic simulations33\nfor achain of 50FM cells underalinear temperature gradient .\nT1= 0,TN= 10 [K], a= 10[nm], H0= 0.057[T]. For small\nk, the dispersion relation reduces to ωk=γ/parenleftBig\nH0+2A\nMSk2/parenrightBig\n.\nThe small green frame shows the subthermal regime of the\nspectrum.\nTo ensure that our findings are not an artefact of a\nparticular choice of parameters/model we performed the\ncalculations for various temperature gradients, different\napplied external magnetic fields, and varying lengths of\nthe chain (Fig. 5). In all these cases the intrinsic cut-\noff frequencies follow the corresponding magnon-phonon\nfrequencies (Eq. (2)). Furthermore, the use of a white\nnoise for a swift time-dependent heating might be ques-\ntioned. Therefore, we implemented the Landau-Lifshitz-\nMiyasaki-Seki scheme31to account for the back-action\nof the magnon subsystem to the surrounding phonon\nbath (see Supplementary Materials26) and arrived basi-\ncallyatthesameconclusionthatsmall-wavevectors /vectorkex-\nchange subthermal magnons contribute substantially totheformationofthermallyactivatedspin current. Hence,\nour finings are expected to be of some generalities for\nmagnon-driven SSE and associated devices, for the em-\nployed schemes are quite ubiquitous, had proven to be\nreliable for finite temperature spin dynamics, and the\nemployed system parameters are generic.\nAcknowledgements. We thank K. Zakeri Lori, Y.-J.\nChen and A. Sukhov for valuable discussions.\nΩc/LBracket1Hz/RBracket1/Slash12Π\n6.3/MultiΠly107\n1.0/MultiΠly108\n1.6/MultiΠly108\n2.5/MultiΠly108\n4.0/MultiΠly108\n6.3/MultiΠly108\n1.0/MultiΠly109\n/VertEllipsis2\n1.0/MultiΠly1012\n1071081091010101110120.80.850.90.9511.05Normalized spincurrent/LParen1a/RParen1\nN/Equal50\na/Equal10/LBracket1nm/RBracket1\nH0/Equal0.057/LBracket1T/RBracket1\nT00/Equal40/LBracket1K/RBracket1\nR/Equal2000\nΩc/LBracket1Hz/RBracket1/Slash12Π\n6.3/MultiΠly107\n1.0/MultiΠly108\n1.6/MultiΠly108\n2.5/MultiΠly108\n4.0/MultiΠly108\n6.3/MultiΠly108\n1.0/MultiΠly109\n/VertEllipsis2\n1.0/MultiΠly1012\n1071081091010101110120.80.850.90.9511.05Normalized spincurrent/LParen1b/RParen1\nN/Equal50\na/Equal10/LBracket1nm/RBracket1\nH0/Equal0.57/LBracket1T/RBracket1\nT00/Equal10/LBracket1K/RBracket1\nR/Equal1000\nΩc/LBracket1Hz/RBracket1/Slash12Π\n6.3/MultiΠly107\n1.0/MultiΠly108\n1.6/MultiΠly108\n2.5/MultiΠly108\n4.0/MultiΠly108\n6.3/MultiΠly108\n1.0/MultiΠly109\n/VertEllipsis2\n1.0/MultiΠly1012\n1071081091010101110120.80.850.90.9511.05\nΩmod/LBracket1Hz/RBracket1/Slash12ΠNormalized spincurrent/LParen1c/RParen1\nN/Equal100\na/Equal10/LBracket1nm/RBracket1\nH0/Equal0.057/LBracket1T/RBracket1\nT00/Equal20/LBracket1K/RBracket1\nR/Equal1500\nFIG. 5. 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Kuanr4,* \n1School of Physical Sciences, Jawaharlal Nehru University, New Delhi, INDIA 110067  \n2Department of Physics, Ramjas College, University of Delhi, New Delhi, INDIA 1100 07 \n3Departme nt of Physics, Morgan State University, Baltimore, MD, USA 21251  \n4Special Centre for Nanosciences, Jawaharlal Nehru Univ ersity, New Delhi, INDIA 110067  \n5Department of Physics, Indian Institute of Science Education and Research, Pune, INDIA \n411007  \n \n \nAbstract  \nFerromagnetic resonance  (FMR)  based spin pumping is a versatile tool to quantify the spin mixing \nconductance and spin to charge conversion  (S2CC)  efficiency of ferromagnet/normal metal \n(FM/NM) heterostructure.  The spin mixing conductance of FM/NM interfac e can also be tuned by  \nthe crystal orientation symmetry of epitaxial FM. In this work, we study the S2CC in epitaxial \nBismuth  substituted Yttrium Iron Garnet  (Bi0.1Y2.9Fe5O12) thin films Bi-YIG ( 100 nm ) interface d \nwith heavy  metal platinum ( Pt (8 nm )) depo sited by pulsed laser deposition process on differen t \ncrystal orientation Gd3Ga5O12 (GGG)  substrates  i.e. [100] and [111] .  The crystal structure and \nsurface roughness characterized  by X -Ray diffraction and  atomic force microscopy  measurements \nestablish epitaxial Bi -YIG[100], Bi -YIG[111]  orientations  and atomically flat surfaces  \nrespectively . The S2CC quantification has been realized by two complimentary techniques , (i) \nFMR -based spin pumping  and inverse spin Hall effect (ISHE) at GHz frequency and (ii) \ntemperature dependent spin Seebeck measurements.  FMR -ISHE results  demonstrate that the [111] \noriented Bi-YIG/Pt sample shows significantly higher values of spin mixing conductance \n((2.31±0.23) 1018 m-2) and spin Hall angle (0.01±0.001)  as compared to the [10 0] oriented Bi -\nYIG/Pt . A longitudinal spin Seebeck measurement reveals that the [111] oriented sample has higher \nspin Seebeck coefficient (106.40±10 nV  mm-1 K-1). Th is anisotropic nature of spin mixing 2 \n conductance  and spin Seebeck coefficient  in [111] and [100] orientation has been discussed using \nthe magnetic environment elongation along the surface normal or parallel to the growth direction. \nOur results aid in understanding  the role of crystal orientation symmetry in S2CC based spintronics \ndevices.  \n \nKeyw ords: Bi-YIG thin films, lattice mismatch, Pulsed Laser Deposition, ferromagnetic \nresonance, Gilbert damping, inhomogeneous broadening  \n \n*Corresponding  authors:  ganeshgurjar4991@gmail.com, spatnaik@mail.jnu.ac.in , \nbijoykuanr@mail.jnu.ac.in  \n \n \n1. Introduction  \n \nSpin pumping is defined as the transfer of electron spins from  a ferromagnet (FM) to a \nnormal metal (NM) in an FM/NM junction  under magnetic precession [1–3]. In general, a time \nvarying magnetization pumps a pure spin current into a NM contact  and this idea of spin pumping \nhas been realized by numerous experimental techniques in which excitation sources are varied from \nGHz to THz time scales [4–6]. This is also observed with temperature gradien t dependent spin \nSeebeck effect [7–9]. The efficiency of spin pumping relates to spin angular momentum \ntransfer [10] and spin mixing conductance [11,12] . The nature of this efficiency is strongly affected \nby interface characteristics of NM in contact with either ferromagnetic metal (FM) [13] or \nferrimagnetic insulator (FMI) [14]. A great deal of theoretical and experimental research has been \nfocused on how to actuate, detect, and regulate the magnetization and spin currents in these \nsystems [15]. One of the most important magnetic materials for spin pum ping is Bismuth substituted \nYttrium Iron Garnet ( Bi-YIG, BixY3-xFe5O12). This is primarily due to its long spin -wave \npropagation length  and low Gilbert damping [16–22]. Several experim ental studies on spin 3 \n pumping have been reported using the Bi-YIG/Pt system [8,23 –26] because of  high spin orbit \ncoupling  (SOC)  in Pt.  \nThe recent theoretical and experimental works on this subject predict that the spin pumping \nefficiency strongly depends on the interface cut and orientation of NM and FM structures [27–31]. \nThis anisotropy has been explained on the basis of local magnetic moment of magnetic ions and \ngeneration of crystal field symmetry due to broken rotational symmetry of magnetic atoms [27]. \nMotivated by the se results, we investigate the role of crystal field symmetry in Bi-YIG/Pt \nmultilayers. Here we discuss the tuning of spin pumping by controlling the interface cut dependent \ngrowth of 100 nm Bi-YIG film on Gd 3Ga5O12 (GGG) substrates. Our Bi-YIG/Pt het erostructures \nhave been deposited using pulse d laser deposition (PLD) technique where GGG  [100] and GGG  \n[111] substrates are used to change  the interface cut orientation.  We utilize the broadband \nferromagnetic resonance  (FMR)  technique to realize the spin pumpi ng based on the enhancement \nof Gilbert damping  and spin to charge conversion (S2CC) using inverse spin Hall effect (ISHE) . \nThe spin mixing conductanc e and spin Hall angle vary significantly  with [111] and [100] \norientations . To further corroborate our find ings, we have used the complementary technique based \non spin Seebeck effect where the variation of spin Seebeck voltage due to interface cut orientation \nhas been observed.           \n \n2. Experimental Technique  \n \nThe Bi-YIG thin film s of thickness 100 nm were g rown on [100] and [111] -oriented GGG \nsubstrates using PLD technique. Bare grown samples, GGG [100]/Bi-YIG and GGG [111]/Bi-YIG \nwere labeled as Bi -YIG [100] and Bi -YIG [111] respectively . After  that, a platinum thin film of 8 4 \n nm thickness  with deposition ra te of 0.8Å/sec  was successfully deposited in -situ on bare Bi -YIG \nusing PLD wi th optimized growth parameters. Platinum deposited samples, GGG [100]/Bi-YIG/Pt, \nand GGG [111]/Bi-YIG/Pt were labeled as Bi -YIG[100]/Pt and Bi -YIG[111]/Pt, respectively . The \nsize of  grown samples is 2 mm   5 mm. Before deposition, GGG substrates were appropriately \ncleaned us ing acetone and isopropanol. All thin films were deposited at a base vacuum of 2×10-7 \nmbar. We used a 248 nm KrF excimer laser with a 10 Hz pulse rate to ablate the material from the \ntarget. During deposition, Oxygen pressure, target to substrate distance, and substrate temperature \nwere all maintained at 0.15 mbar, 5.0 cm, and 825 oC, respectively. The growth rate of deposited \nBi-YIG films was 6 nm/min. The as -grown films were in -situ annealed for 2 hours at 825 oC in the  \npresence of oxygen (0.15 mbar). After deposition of Bi -YIG film the ta rget to substrate distance \nwas changed to 3.0 cm and substrate temperature 100 oC , for deposition of the Pt film,  Thickness \nwas measured using atomic force microscopy (AFM) (WITec  GmbH, Germany). FMR -based spin \npumping measurements have been carried out in a vector network a nalyzer  spectrometer, where \nsamples are placed in a flip -chip arrangement on a microstrip line  with DC ma gnetic field applied \nperpendicular to the high -frequency magnetic field (h RF) onto the film plane. The dc voltage \ngenerated due to ISHE was measured with the help of Keithley 2182 Nanovoltmet er. For spin \nSeebeck effect measurements, standard longitudinal s pin Seebeck effect (LSSE)  geometry was \nused with a temperature gradient formed between the Pt layer and the GGG substrate, and the \nvoltage (perpendicular to an applied magnetic field and temperature gradient) was measured using \na Keithley 2182 Nanovoltmete r. \n \n \n 5 \n  \n \n3. Results and Discussion  \n \n3.1 Structural, surface morphology and static magnetization study  \n \nBi-YIG crystallizes in cubic structure with the Ia -3d space group. Figure 1 (a) -(d) shows the XRD \npattern of Bi -YIG and Bi -YIG/Pt bilayers on [100] and [111] cut GGG substrates. XRD patterns  \nshown in Fig. 1 (a) and 1 (b) indicat e the single -crystalline growth of Bi -YIG and Bi -YIG/Pt thin \nfilms . Figures 1 (c) and 1 (d) show the  evidence for successful growth of Pt layer on top of bare  \nBi-YIG films of  [100] and [111] orientations. The growth of Pt film is nano -crystalline  in nature.  \nThe lattice constant, lattice mismatch (with respect to substrate), and lattice volume obtained from \nXRD data are listed in Table 1.  The cubic structure  lattice constant 𝑎 is calculated using the formula  \n𝑎=𝜆√ℎ2+𝑘2+𝑙2\n2𝑠𝑖𝑛𝑠𝑖𝑛 𝜃                                                         (1) \n Where , 𝜆 is the wavelength of Cu -Kα radiation, 𝜃 is the diffraction angle, and [h, k, l] are the \nMiller indic es of the corresponding XRD peaks.   \nThe lattice mismatch (𝛥𝑎\n𝑎) is calculated using the equation  \n𝛥𝑎\n𝑎=(𝑎𝑓𝑖𝑙𝑚 − 𝑎𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 )\n𝑎𝑓𝑖𝑙𝑚×100               (2) \nHere, 𝑎𝑓𝑖𝑙𝑚 and 𝑎𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒  indicate the latt ice constants of the film and substrate, respectively. The \ncalculated value of lat tice constant are tabulated in T able 1 which are consiste nt with prior \nfindings [16,21,32,33] . The l attice constant slightly increases in the case of [111] as compared to \n[100] because the distribution of Bi3+ at the dodecahedral site is dep endent on t he substrate 6 \n orientation [21,34,35]  which leads to a comparatively  larger lattice mismatch (or strain). Observed \nvalues of lattice mismatch are close to  what has been reported earlier [36,37]. The lattice mismatch \nof the film plays an important role , a smaller lattice mismatch value can reduce the damping \nconstant of the film [38,39] .  \n Furthermore, t he surface roughness plays an important role in magnetization dynamics due \nto the generation of two magnon scattering in films with higher roughness. Figure 1 (e) -(h) shows \nroom temperature AFM images with root mean square (RMS) roughness. We have observed RMS \nroughness around 0.35 nm or less for all grown Bi -YIG films which is comparable to previously \nreported YIG films [18,40] . We have observed that there RMS roughness remains unchanged for \nall grown films. Moreover,  rough ness would be more affected by changes in growth factor s than \nby substrate orientation [18,21] .  \nVSM magnetization measurements were performed at 300 K with an applied magnetic field \nparallel to the film plane (in -plane). The GGG substrate's paramagnetic co ntribution was properly \nsubtracted . Magnetization plots of Bi -YIG thin films for [100] and [111] orientations are shown in \nFigure 1(i).   The schematic of an applied magnetic field dire ction parallel to the film plane is shown \nin the inset of Fig. 1(i). Measu red saturation magnetization ( 0MS) values are 165.60 ± 20.10 mT \nGauss for Bi -YIG [100] and 196.20 ± 19.10 mT for Bi -YIG [111]. Saturation magnetization error \nbars are related to s ample volume unc ertainty. The observed value of 0MS of as grown [100] and \n[111]-oriented Bi -YIG films are consistent with previous reports [33,41 –43].  Bismuth doping \ngenerally affects the magnetic and mechanical propertie s of YIG , However,  the dependence of \nsaturation magnetization on interface cut is minimal because the magnetization of YIG is induced \nvia a super -exchange interaction at the d and a site (in Wycoff notation)  between non -equivalent \nFe3+ ions. Along [111], there is more contribution of Bi3+ ions co mpared to [100] orientation [21], 7 \n and Bismuth l ocated at the dodecahedral site dis tort the tetrahedral and octahedral Fe3+ ions[22]. \nFigure 1 (j) shows the crystal orientation dependent atomic model of YIG with Yttrium (Y) (red), \nFe ‘d site’ (green) and Fe ‘a site’ (blue) atomic position. The inte rface cut dependent magnetic \nenvironment is highly anisotropic in [111] and [100] orientation whi ch is clearly depicted in Fig . \n1(j). Pt interface with these different magnetic environments may ch ange its spin conversion  \nefficiency [44] which will be discussed in  next sections .  \n \nTable 1:  Lattice consta nt and roughness obtained from XRD and AFM.  \nS. No.  Sample  Lattice \nconstant (Å)  Lattice \nMismatch (%)  Lattice \nvolume  (Å3) Roughness  \n(nm)  \n1. Bi-YIG [100] 12.461  0.53 1935.15  0.36 \n2. Bi-YIG [111] 12.472  0.64 1939.80  0.33 \n3. Bi-YIG/Pt [100] 12.402  0.39 1907.81  0.29 \n4. Bi-YIG/Pt [111] 12.455  0.80 1932.31  0.34 \n \n \n \n \n3.2 Ferromagnetic resonance (FMR) study  \n \nFMR measurements  are performed  at room temperature . Figure 2 (a) and (b)  show the FM R \nabsorption spectra of [100] and [111] -oriented thin films, respectively. When Pt is deposited on \ntop of Bi -YIG, the observed FMR absorption spectra  shift right in case of [100] and left in case of 8 \n [111] orientation with respect to bare Bi -YIG film. This  shows that Pt deposition moderately affects \nthe effective saturation magnetization in Bi -YIG[111] due t o its [111] texture . The FMR data at 10 \nGHz with its Lorentzian fit is shown in the inset of Fig. 2 (a) and 2 (b), and the schematic depicts \nthe directi on of the applied DC magnetic field in the film plane. Inset in Fig. 2 (a) and (b) shows \nthe enhancement of FMR linewidth in Bi -YIG/Pt samples as compared to bare Bi -YIG samples. \nThe corresponding FMR linewidth enhancement is ~0.24 mT  in [100] and ~ 0.50 mT in [111] \noriented films. Th is enhancement in linewidth results from spin pumping .  The coupling that \ntransfers angular momentum from Bi -YIG to the metal contributes to the damping due to \nprecess ion of Bi-YIG magnetization  that leads to increasing linew idth. To calculate the increment \nin Gilbert damping due to Pt deposition, we have measured the FMR data at different resonance \nfrequencies ( 𝑓= 2 GHz -12 GHz) with dc magnetic field applied parallel to film plane. The \nLorentzian fitting was used to fit the FMR data and calculate the FMR linewidth (∆H) and \nresonance magnetic field (H rf) or different frequencies. From the fitting of Kittel's in -plane equation \nprovided by Eq. 3[45], the gyromagnetic ratio (γ) and effective magnetization field ( 0𝑀𝑒𝑓𝑓) were \ncalculated.  \n𝑓=𝛾\n2𝜋0√(𝐻𝑟)(𝐻𝑟+𝑀𝑒𝑓𝑓)     (3), \nWhere , 0𝑀𝑒𝑓𝑓=0(𝑀𝑠−𝐻𝑎𝑛𝑖) is the effective field with anisotropy field 𝐻𝑎𝑛𝑖=2𝐾1\n𝑀𝑠. Similarly, \nGilbert damping parameter (α) and inho mogeneous broadening (∆H 0) linewidth were calculated \nfrom the fitting of the Landau –Lifshitz –Gilbert equation (LLG) provided by equation 4[45], \n𝛥𝐻(𝑓)=𝛥𝐻 0+4𝜋𝛼\n𝛾𝑓                                     (4) \n \nFigures 2 (c) and 2 (d) show Kittel and LLG fitted graphs for the [111] orientation, respectively, \nwith an inset showing Kittel and LLG fitted graphs for  the [100] orientation.  9 \n  \nTable 2 lists the calculated parameters obtained from the FMR study. The measured Gilbert \ndamping (α) is consistent with thin films used in spin -wave propagation studies [21,24,43] . For both \norientations [100] and [111], the value of α increases  when Pt is deposited on top of Bi -YIG films \n(Table 2).  The value of α increases significantly in (111) orientation .  However, in the case of pure \nYIG it is reported that Gilbert damping in [100] is more as compared to [111][28]. In our case \nhigher α obtained in [111] may be ascribed qualitatively to the presence of Bi3+ ions, which \nadditional induce spin -orbit coupling (SOC) [16,46,47]  that causes local di stortion of Fe3+ so as to \naffect the magnetic properties as compared with pure YIG[22]. The higher lattice mismatch (strain) \nin (111) could also be the reason for increased electron scattering leading to higher damping [48]. \nFigure 1 (j) elucidates how more  Fe atoms (i.e., more magnetic environment ) and also more yttrium \nsites are available for the  distribution of Bi3+ ions along [111] orientation .  This is the cause of \nhigher lattice mismatch in Bi -YIG [111]. These findings  account for the higher Gilbert da mping \nand 0𝑀𝑒𝑓𝑓 values in the [111] orientation. In conclusion, Bi -YIG/Pt with an orientation of [100] \nhas the lowest damping factor. We note that t hese are the essential desirable factors used in \nspintronic devices that need longer propagation lengths for spin waves.  \n \nTable 2:  Damping and linewidth parameters obtained from FMR  \nS. No.  Sample  α (10-4) ΔH 0 (Oe)  0Meff (Oe)  \n1. Bi-YIG [100] (2.97±0.11)  25.99±0.21  2124.89±8.80  \n2. Bi-YIG [111] (3.62±0.17)  26.69±0.33  2241.70±12.77  \n3. Bi-YIG/Pt [100] (3.74±0.14)  27.33±0.29  2030.12±40.19  10 \n 4. Bi-YIG/Pt [111] (6.11±0.47)  25.19±0.92  2582.26±75.13  \n \n3.3 Inverse spin Hall effect (ISHE) study using FMR based spin pumping and spin \nSeebeck effect  \n \nThe induced crystal field anisotropy due to different interfaces  strongly affects the tuning of \nspin mixing conductance because the magnetic Fe3+ ions can result in different spin currents \naccording to its placement in [111] and [100] orientation. It is argued that the crystal field is \nelongated along the surface norma l in [111] orientation and parallel to the surface in [100] \norientation [27]. This anisotropic nature of the crystal field gives the highest spin mixing \nconductance in [111] direction and considerably lower in [100] orientation. At Bi -YIG/Pt  interfaces \nwher e the spin pumping or the transfer of spin angular momentum occurs, the spin mixing \nconductance ( g) plays a significant role in transfer of spin angular momentum. Spin pumping -\ninduced damping is measured from the enhancement in the Gilbert damping con stant (e.g. due to \nPt layer on top of Bi -YIG film ( 𝛼𝐵𝑖−𝑌𝐼𝐺 /𝑃𝑡−𝛼𝐵𝑖−𝑌𝐼𝐺)).  This is required for calculating the spin \nmixing conductance at the Bi -YIG/Pt interface. Table 2 shows the Gilbert damping values of bare \nBi-YIG and Pt deposited Bi-YIG samples. The value of g as (0.73±0.07)  1018 m-2 for [100] and \n(2.31±0.23)  1018 m-2 for [111] oriented samples are calculated using equation 5 [14]. The observed \nspin mixing conductance values are consistent with previous reports [14,24,49,50]  and shows that \nit depends on crystal orientation. A larger spin mixing conductance in [111] oriented samples  occur  \nthat indicates that [111] grown samples have  a greater spin -injection efficiency than [100] oriented \nsamples.  This is a consequence of [111] oriented sample having a greater Gilbert damping constant \nenhancement owing to the Pt layer.  11 \n g=µ0𝑀𝑠𝑡𝐹\n𝑔𝐵(𝛼𝐵𝑖−𝑌𝐼𝐺 /𝑃𝑡−𝛼𝐵𝑖−𝑌𝐼𝐺)   (5) \nWhere µ0𝑀𝑠 and 𝑡𝐹 are the magnetization and thickness of the Bi -YIG thin film, 𝐵is the Bohr \nmagneton, 𝑔=𝛾ħ\n𝐵 is the g -factor of Bi -YIG[22] and 𝛼𝐵𝑖−𝑌𝐼𝐺 /𝑃𝑡−𝛼𝐵𝑖−𝑌𝐼𝐺 is the enhancement in \nthe Gilbert damping constant due to Pt layer on top of Bi -YIG film.  \n  \n ISHE measurements  are carried out at room temperature using FMR technique. In this \nscheme, a  dc magnetic field (H) is applied in the film plane and  perpendicular to the sample length. \nISHE voltage (V ISHE) is measured along the length of the Pt layer. A resonant rf field ( hrf), is also \nsuperimposed to cause the uniform precession of the Bi -YIG magnetization. When excited \nprecession occurs at the inter face, angular momentum is transferred to the conduction electrons in \nPt, resulting in a pure spin current (J S) in Pt. As a result of ISHE, the spin current travels across the \nPt perpendicular to the plane and is transformed into charge current ( JC).  As a consequence of this, \na voltage difference develops between the two ends of the Pt layer that is measured using a Keithley \nnanovoltmeter. Figure 3 (a) shows the measured dc voltage (V dc) at frequency 10 GHz, and figure \n3 (b) shows its Lorentzian fit ting[45] with symmetric and anti -symmetric contributions. Figure 3 \n(c) shows  the symmetric contribution of the voltage (V ISHE) extracted from the measured dc voltage \n(Vdc). The i nset shows the schematic setup for FMR spin pumping for measurement of ISHE \nvoltage.   \n The ISHE voltages [14] given by equation 6  depends on several material parameters  \n𝑉𝐼𝑆𝐻𝐸 =−𝑒𝑆𝐻\n𝜎𝑁𝑡𝑁+𝜎𝐹𝑡𝐹𝜆𝑆𝐷𝑡𝑎𝑛ℎ (𝑡𝑁\n2𝜆𝑆𝐷) 𝑔𝑓𝐿𝑃 (𝛾ℎ𝑟𝑓\n2𝛼𝜔)2   (6) \nWhere e is the electron charge, 𝑆𝐻 is the spin Hall angle, 𝜎𝑁(𝜎𝐹) and 𝑡𝑁(𝑡𝐹) denotes the \nconductivity and thickness of the NM (FM) thin -film respectively, 𝜆𝑆𝐷=7.3 𝑛𝑚 is the spin 12 \n diffusion length in Pt[16], 𝑔 denotes interfacial spin mixing conductance, 𝜔=2𝜋𝑓 is the \nresonance frequency, L is the length of the sample, and ℎ𝑟𝑓=0.17 𝑂𝑒 in our FMR cavity at power \n= +15 dBm. The ellipticity of the magnetization precession gives rise to the factor  𝑃[13,14] . \n𝑃=2𝜔[𝛾µ0𝑀𝑠+√(𝛾µ0𝑀𝑠)2+4𝜔2]\n(𝛾µ0𝑀𝑠)2+4𝜔2 =1.21      (7) \nValues of 𝑔 and 𝑆𝐻 are calculated using equations (5) -(7). Calculated values of 𝑆𝐻 for [100] \norientation is (0.73±0.07)  10-2 and for [111] orientation is (1.01±0.10)  10-2. Figure 3 (d) shows \nthe comparative results of 𝑔 and 𝑆𝐻 for [100] and [111] oriented samples. Consequently, the \nspin current density 𝐽𝑠 can be calculated using [14], \n𝐽𝑠=𝜎𝑁𝑡𝑁+𝜎𝐹𝑡𝐹\n𝑆𝐻𝜆𝑆𝐷𝑡𝑎𝑛 ℎ (𝑡𝑁\n2𝜆𝑆𝐷) 𝑉𝐼𝑆𝐻𝐸\n𝐿       (8) \n𝐽𝑠 is calculated to be 0.59×107 A-m-2 for [100] and 0.70×107 A-m-2 for [111] orientation. 𝑔, 𝑆𝐻 \nand 𝐽𝑠 was significant ly larger in the case of [111] orientation, indicating the enhanced spin orbit \ncontribution  and spin transparency in [111] oriented Bi -YIG/Pt bilayer structure, resulting in \nmaximum V ISHE signal ~76.30 μV .  For the [100] oriented Bi -YIG/Pt film it was foun d to be ~46.31 \nμV. Our obtained transport parameters are in agreement with the reported  literature  values [14,51 –\n58] in which  order of  spin mixing conductance values varies  from  1016 m-2 to 1018 m-2.  Similar ly \norder of spin Hall angle [57] and spin current density are also in agreement  with published \ndata[27,29 –31].  It show s the effect of crystal orientation on spin mixing conductance . Jia et al. \n[31] observed no crystal orientation dependent g in the pure  YIG/Ag system . Cahaya et a l.[27], \nreported  that asymmetry of spin mixing conductance depends not only on the density of exposed \nmoments and but also on local point symmetry.  The pumped spin -current anisotropy is dependent \non the quadrupole moment, which is dependent on the orbital occup ancy of interface magnetic \natoms.  For half -filled shells  ions (Fe3+) quadrupole moment is zero. It is necessary to break 13 \n symmetry, such as via strain or at an interface, to obtain a finite quadrupole moment. From the \ntheoretical explanation given by [27], we conclude that our crystal orientation dependent findings \nmay be due to rot ational symmetry break ing in Bi -YIG thin films owing to substrate lattice \nmismatch.  As discussed already, a l arger lattice mismatch in [111] is observed compared to [100]. \nFurthermor e, we have us ed Bi -doped YIG system, where Bi3+ affects the magnetic properties of \nYIG[22] and also distribution of Bi3+ ions among dodecahedral sites depends on substrate \norientation [21,59 –61]. Figure 1 (j) shows the interface cut dependent magnetic environment in \n[100] and [111] orientation. The interaction of Pt with these varied magnetic environments may \naffect its spin conversion efficiency. Hence, the enhancement in spin pumping parameters depends \non the orienta tion of crystal growth [27,29 –31], which supports our experim ental findings of ISHE \nresults.  \n \n3.4 Spin Seebeck  effect  \n \nThe integration of magnetic insulators with Pt promises to usher in  several application s of \nspin current technology to thermoelectric devices.  In this regard, we discuss spin Seebeck study \non the s ame films that were  studied for invers e Spin Hall Effect previously. We note that t he doping \nof Bi softens the crystal and increases the growth induced anisotropy [22]. The effective SOC for \nFe3+ ions is increased by mixing 6p orbitals of Bi ions with O -2p orbitals. Energy transmission \nbetween spin and phonon systems is also enhanced  by SOC. Thus  during spin thermoelectric \ngeneration, maximum energy transfer mechanism for the heat simulation of magnetization \nprecession  will result in larger spin currents in Bi -YIG films [23]. The spin Seebeck effec t \nmeasurements  were performed in longitudinal spin Seebeck effect (LSSE)  geom etry. Figure 4  (a) \nshows the schematic setup for LSSE measurements. Figu re 4 (b), 4 (c), and 4 (d) show  the 14 \n comparative results of longitudinal spin Seebeck voltage (V LSSE) as a f unction of temperature (20 \nK to 300 K), applied magnetic field (up to ±0.2 Tesla) and temperature gradient  (upto 11 K)  \nrespectively. At room temperature, the o bserved spin Seebeck coefficient is 106.40±10 nV  mm-1 \nK-1 for [111] and 89.41±0.3 nV  mm-1 K-1 for [100] oriented sample . Because these voltages are \nobtained across various effective resistances, we represent these voltages in their normalized form, \ni.e. 𝑉𝐿𝑆𝑆𝐸 =𝑉𝑑𝑐/𝑅𝛥𝑇𝐿 , where 𝑉𝑑𝑐 is the observed transverse voltage, R and L are the \ncorresponding resistance and distance between contact probes, 𝛥𝑇 is the temperature  gradient \nacross the sample . The Spin  Seebeck effect study indicates the significantly higher values of spin \nSeebeck coefficient observed in the case of the [111]. Thes e findings from spin Seebeck \nmeasurements also provide credence to the higher ISHE  voltage generated in the [111] orientation \nthat was seen in FMR spin pumping studies.  \nAccording to these  experimentally observed ISHE  results, the spin mixing conductance of  \nBi-YIG/Pt bilayers depends on crystal cut and orientation [27]. We find that t he spin pumping \nefficiency of the [111] oriented sample is substantially greater than that of the [100] orientated \nsample.  \n \n4. Conclusion  \n \nWe have successfully grown  the Bi -YIG(100 nm )/Pt(8 nm) bilayer using pulsed laser deposition \ntechnique on top of GGG substrates having [100] and [111] orientation s. AFM and XRD \ncharacterizations revealed that the deposited thin films are phase pure and have smooth surfaces. \nFMR based spin pumping re sults confirm that there are significantly higher values of spin mixing \nconductance (~ three times) and spin Hall angle in the case of the [111] orientated sample compared \nto [100] oriented sample .  This  indicates that significantly higher spin current can transfer to the  Pt 15 \n from Bi -YIG [111]. Consequently, the [111] oriented sample has a higher spin pumping efficiency \nthan the [100] oriented sample . From a longitudinal spin Seebeck measurement  we note that the \n[111] orientated sample had a higher (~20 %) s pin Seebeck coefficient.  In conclusion, t he results \nof inverse spin Hall effect and spin Seebeck experiments show that the fundamental parameters of \nspin pumping can be tuned effectively by substrate ’s orientation.  \n \n \n \nAcknowledg ements  \nThis work is support ed by the MHRD -IMPRINT grant, DST (SERB, AMT, and PURSE -II) grant \nof Govt. of India. Ganesh Gurjar acknowledges CSIR,  New Delhi for financial support. We \nacknowledge AIRF, JNU for access of PPMS facility.  \n \nConflict of interest statement  \nThe authors declare that they have no competing financial interests or personal relationships that \ncould have appeared to influence the work reported in this paper.  \n \n \nData Availability statement  \nThe data that support the findings of this study are available from the corresponding author upon \nreasonable request.  \n \n \n 16 \n  \nReferences  \n[1]  Tserkovnyak Y, Brataas A and Bauer G E W 2002 Enhanced Gilbert Damping in T hin \nFerromagnetic Films Phys. Rev. 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Phys.  58 454–9 \n \n \n \n 23 \n Figure Captions  \n \nFigure 1:  XRD patterns of GGG/Bi -YIG/Pt structure, Bi -YIG XRD pattern shown (a) with [100] \nand (b) [111] orientations, (c) and (d) shows the XRD pattern of Pt film with respec t to bare Bi -\nYIG films on [100] and [111] orientations, (e) -(h) AFM images of Bi -YIG and Bi -YIG/Pt films in \n[100] and [111] orientations, (i)  VSM plot of Bi -YIG film in [100] and [111] orientations (one inset \nshows the schematic of applied magnetic fi eld direction in the film plane, second inset shows the \nschematic of sample grown structure) , (j) shows the crystal orientation dependent atomic model of \nYIG with Yttrium (Y) (red), Fe ‘d site’ (green) and Fe ‘a site’ (blue) atomic position.  \n \nFigure 2: FMR absorption spectra of Bi -YIG and Bi -YIG/Pt films with (a) [100] and (b) [111] \norientation. Inset shows FMR data at 10 GHz with its Lorentzian fit and enhancement in FMR \nlinewidth in Pt deposited sample. The schematic shows the direction of the applied DC  magnetic \nfield in the film plane. (c) and (d) show frequency -dependent FMR field (H r) data fitted with Kittel \nequation and frequency -dependent FMR linewidth (ΔH) data fitted with LLG equation [111] \norientation, respectively, with an inset showing Kittel a nd LLG fitted graphs for the [100] \norientation.  \n \nFigure 3: (a) show the measured dc voltage (V dc) at frequency 10 GHz, (b) show its Lorentzian \nfitting with symmetric and anti -symmetric contributions,  (c) symmetric contribution of the voltage \n(VISHE) extrac ted from the measured dc voltage (V dc) (schematic setup of ISHE measurements), (d) \nshows the comparative results of spin mixing conductance ( 𝑔) and spin Hall angle ( 𝑆𝐻) for [100] \nand [111] oriented samples.  \n  24 \n  \nFigure 4: (a) Shows the schematic setup of spin Seebeck measurements in LSSE geometry, \ncomparative results of longitudinal spin Seebeck voltage (V LSSE) as a function of (a) amb ient \ntemperature, (b) applied magnetic field, (c) temperature gradient, for [100] and [111] oriented \nsamples.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 25 \n Figure 1:  \n \n \n \n \n \n \n \n \n \n26 \n  \nFigure 2:  \n \n \n \n \n \n \n \n \n \n \n \n27 \n  \n \nFigure 3:  \n \n \n \n \n \n \n \n \n \n28 \n  \n \nFigure 4:  \n \n \n"    },    {        "title": "1905.00663v1.Anomalous_spin_Hall_angle_of_a_metallic_ferromagnet_determined_by_a_multiterminal_spin_injection_detection_device.pdf",        "content": "Anomalous spin Hall angle of a metallic ferromagnet determined by a\nmultiterminal spin injection/detection device\nT. Wimmer,1, 2,a)B. Coester,3S. Gepr ags,1R. Gross,1, 2, 4, 5S. T. B. Goennenwein,6H. Huebl,1, 2, 4, 5and\nM. Althammer1, 2,b)\n1)Walther-Mei\u0019ner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching,\nGermany\n2)Physik-Department, Technische Universit at M unchen, 85748 Garching, Germany\n3)School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link,\nSingapore 637371, Singapore\n4)Nanosystems Initiative Munich (NIM), Schellingstra\u0019e 4, 80799 M unchen, Germany\n5)Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, D-80799 M unchen,\nGermany\n6)Institut f ur Festk orper- und Materialphysik and W urzburg-Dresden Cluster of Excellence ct.qmat,\nTechnische Universit at Dresden, 01062 Dresden, Germany\n(Dated: July 5, 2021)\nWe report on the determination of the anomalous spin Hall angle in the ferromagnetic metal alloy cobalt-iron\n(Co25Fe75, CoFe). This is accomplished by measuring the spin injection/detection e\u000eciency in a multiterminal\ndevice with nanowires of platinum (Pt) and CoFe deposited onto the magnetic insulator yttrium iron garnet\n(Y3Fe5O12, YIG). Applying a spin-resistor model to our multiterminal spin transport data, we determine\nthe magnon conductivity in YIG, the spin conductance at the YIG/CoFe interface and \fnally the anomalous\nspin Hall angle of CoFe as a function of its spin di\u000busion length in a single device. Our experiments clearly\nreveal a negative anomalous spin Hall angle of the ferromagnetic metal CoFe, but a vanishing ordinary spin\nHall angle. This is in contrast to the results reported in Refs. 1 and 2 for the ferromagnetic metals Co and\npermalloy.\nThe spin Hall e\u000bect (SHE) is at the origin of a\nplethora of transport e\u000bects relevant for spintronics ap-\nplications1{9. While the charge to spin current conversion\ne\u000eciency is conveniently expressed in terms of the phe-\nnomenological spin Hall angle \u0002 SH, its microscopic origin\nis the spin-orbit interaction causing spin-selective scatter-\ning of charge carriers7,10. Many ferromagnetic metals ex-\nhibit a strong spin-orbit coupling, which manifests itself\nin various electrical transport e\u000bects, among them the\nanomalous Hall e\u000bect (AHE)11. The AHE hinges on the\nsame physical principles as the SHE3,7. While the trans-\nverse charge current arising in the AHE has been studied\nfor more than a century, the pure spin current part has\nonly very recently received broad attention1,2,12,13.\nRecent developments in magnetotransport experi-\nments with incoherent magnons (the quantized excita-\ntions of the magnetization)14{17o\u000ber a suitable plat-\nform for the investigation of the SHE and the anoma-\nlous spin Hall e\u000bect (ASHE) in ferromagnets1,2,18{20.\nIn these experiments, a spin current is injected into an\nadjacent magnetic insulator via the SHE. More speci\f-\nvally, a DC charge current in a metallic electrode gener-\nates a spin accumulation at the interface between the\nmetal and the magnet, which in turn induces a non-\nequilibrium magnon accumulation in the magnet. The\nnon-equilibrium magnons di\u000buse in the magnet, and are\ndetected in a second, electrically separate metallic elec-\na)tobias.wimmer@wmi.badw.de\nb)matthias.althammer@wmi.badw.de\nPt2\nH\nφyz\nxPt1\nCoFe\nYIG\nPt3d\nPt\nd\nCoFeM\nYIGMCoFe\nd\nCoFe+\n-+\n-\n+\n- +\n-Vdet\nVdetVdetIqFigure 1. Schematic depiction of the device, the electrical\nconnection scheme and the coordinate system. A charge cur-\nrentIqis fed through the Pt2 electrode, resulting in a spin\ncurrent injection into YIG via the SHE. The lateral di\u000bu-\nsion of the magnon spin current is electrically detected at the\nPt electrodes ('Pt1' and 'Pt3') and the ferromagnetic metal\nelectrode ('CoFe') as the detector voltage Vdet. The center-to-\ncenter distances between each of the Pt electrodes is constant,\nsuch thatdPt= 2dCoFe.\ntrode as a voltage signal via the inverse SHE. In a recent\nwork, Das et al. reported spin injection and detection in\nYIG via the AHE11using Py electrodes2. They found a\nmagnetic \feld enhanced injection/detection e\u000eciency of\nthe permalloy (Py) electrodes due to the gradual increase\nof the e\u000bective anomalous spin Hall angle \u0002 ASHin Py.\nIn this Letter, we report on the determination of\n\u0002ASHof the ferromagnetic metal alloy Co 25Fe75(CoFe)21\nvia all-electrical magnon transport measurements in thearXiv:1905.00663v1  [cond-mat.mes-hall]  2 May 20192\nPt1 Pt2 Pt3CoFe\nIcVdet\nPt1 Pt2 Pt3CoFe\nIcVdet\nPt1 Pt2 Pt3CoFe\nIcVdet(a) (b) (c)\nFigure 2. Resistance Rdetmeasured using di\u000berent detector\nelectrodes. (a) The signal at the Pt1 detector is taken as a\nreference measurement, with which we can characterize the\nmagnon transport in the YIG layer. (b) The Pt3 detector\nsignal is somewhat smaller than the Pt1 signal owing to the\n\fnite absorption of the magnon current by the CoFe elec-\ntrode in between Pt2 and Pt3. (c) The Rdetassociated with\nthe detector voltage recorded at the CoFe electrode shows a\nsign reversal, indicating that the anomalous spin Hall angle\nin CoFe is negative.\nmagnetic insulator YIG. For this purpose, we utilize a\nmultiterminal structure with four metallic electrodes {\none made of CoFe and three made of Pt { deposited onto\na YIG thin \flm (see Fig. 1).\nOur device consists of a 1 µm thick, commercially avail-\nabe YIG \flm grown on a GGG (Gd 3Ga5O12) substrate\nvia liquid phase epitaxy. Both the Pt and the CoFe elec-\ntrodes were deposited by DC sputtering and patterned\nvia electron beam lithography and lift-o\u000b9. The CoFe\nelectrode was additionally capped with a 2 :5 nm thick Al\nlayer to prevent oxidation. In a further step, Al leads\nand bondpads were deposited to connect the device elec-\ntrically. Each electrode has a width of w= 500 nm and a\nthickness of tPt=tCoFe = 7 nm. The lengths of the strips\narelPt1=lPt3= 148 µm for the outer electrodes and\nlPt2=lCoFe = 162 µm for the inner ones. As indicated in\nFig. 1, the center-to-center distances between the metal\nstrips aredPt= 1:6µm anddCoFe = 0:8µm (cf. Fig. 1).\nFor the injection of magnons, we apply a charge current\nIq= 0:5 mA to the Pt2 electrode (the injector) and de-\ntect the magnon transport signal as the detector voltage\nVdetat the Pt1, Pt3 and CoFe electrodes (see Fig. 1). In\norder to distinguish between electrically (via the SHE)\nand thermally (via Joule heating) injected magnons, we\nutilize the current reversal method15,16. Here, we focus\non the magnon transport via the electrical SHE-induced\nspin current injection. All measurements are conducted\nin a superconducting magnet cryostat at a constant tem-\nperature of T= 280 K22. In order to compare between\ndi\u000berent detector signals, we de\fne a normalized signal\namplitude as Rdet= (Vdet=Iq)\u0001(Ainj=Adet), which ac-\ncounts for the di\u000berent interface areas Ainj(Adet) of the\ninjector (detectors)16.\nTo characterize the magnon transport in our device,\nwe measure Rdetas a function of the magnetic \feld ori-entation'for various in-plane \feld magnitudes \u00160H.\nCorresponding data are shown in Fig. 2 (a)-(c) for three\ndi\u000berent external magnetic \felds. Firstly, panel (a)\nshows the reference measurement using Pt1 as a detector.\nIn accordance with Refs.14,15,23, we observe a sin2(')-\ndependence of Rdetwith reduced amplitudes for increas-\ning external magnetic \feld strengths. Secondly, panel (b)\nshowsRdetrecorded across Pt3. Since the separation of\nthe Pt1 and Pt3 strip to the injector strip Pt2 are the\nsame (cf. 1), one would expect the same signal magni-\ntude. However, the Rdetmodulation recorded across the\nPt3 strip is signi\fcantly smaller, which we attribute to a\npartial absorption of the magnon spin current in the CoFe\nelectrode located in between the Pt2 and Pt3 electrodes.\nFinally, panel (c) shows Rdetmeasured at the CoFe elec-\ntrode. Interestingly, the polarity of the detected voltage\nis inverted. Since all strips were contacted with identical\npolarity in the experiments (see Fig. 1), we conclude that\nthe anomalous spin Hall angle \u0002CoFe\nASH in CoFe is negative\ncompared to the positive spin Hall angle \u0002Pt\nSHin Pt10,24.\nThis is in agreement with the negative spin Hall angles\nreported for both Co and Fe12. Unlike the magnetic \feld\nsuppression observed for the Pt detector strips, we \fnd a\nsigni\fcant enhancement of Rdetfor increasing magnetic\n\felds up to \u00160H= 2 T for the CoFe detector. We at-\ntribute this to the \feld-induced increase of \u0002CoFe\nASH, quali-\ntatively similar to the results reported in Ref.2. For larger\nmagnetic \felds ( \u00160H= 7 T), however, we observe a sup-\npression of the magnon transport signal. Since the CoFe\nmagnetization MCoFe saturates around \u00160H= 2 T21, we\nattribute this \feld suppression to the YIG magnon sys-\ntem in analogy to the situation observed for the Pt de-\ntectors14. Interestingly, we observe a distinct asymmetry\nin the magnitudes of the signal for strong magnetic \felds\n\u00160H > 2 T, which is discussed in more detail in the Sup-\nplementary Information (SI)25.\nIn Fig. 3 (a), we plot the anisotropic magnetoresistance\n(AMR) of the CoFe electrode by measuring its longitudi-\nnal resistance Rlongas a function of the magnetic \feld\nstrength (the sweep direction is indicated by arrows).\nThe blue (green) colored lines correspond to the \feld\ndirection pointing perpendicular (parallel) to the strip\nlength, while dark (light) colored lines correspond to the\nup (down) sweep of the magnetic \feld strength (trace\nand retrace, respectively). Obviously, we observe a clear\nAMR with a maximum (minimum) in resistance for par-\nallel (perpendicular) \feld alignment with respect to the\nstrip length for a coercive \feld of approximately \u000618 mT.\nAdditionally, we \fnd a second peak at a characteristic\n\feld of roughly \u000611 mT. This feature corresponds well\nto the switching \feld observed for the longitudinal resis-\ntance of the Pt2 electrode, which is shown in Fig. 3 (b)\n(switching \feld indicated by gray dashed lines). There-\nfore, the peaks in the resistance of the CoFe strip around\n11 mT can be attributed to a spin Hall magnetoresistance\n(SMR)6contribution related to the magnetization rever-\nsal in the YIG \flm. Most importantly, these magnetore-\nsistance measurements show that exchange coupling be-3\ntween the two ferromagnetic layers is not relevant, since\nno exchange bias e\u000bect can be observed (which would\nlead to a shift of the hysteresis curves along the mag-\nnetic \feld axis). Figure 3 (c), (d) show the detector sig-\nnalsRdetas a function of the magnetic \feld strength\nmeasured at the Pt1 and CoFe electrodes, respectively.\nFor the reference detector (Pt1), we \fnd a continuous\nsuppression of the magnon transport signal Rdetwith in-\ncreasing magnetic \feld strength when the \feld is oriented\nperpendicular to the strips (blue data points in Fig. 3\n(c)). For a parallel alignment of the \feld and the strips\n(green data points), the signal vanishes. This response\nis quantitatively consistent with the \feld-orientation de-\npendent data (Fig. 2). Figure 3 (d) shows the magnetic\n\feld dependence of Rdetwhen the CoFe electrode is used\nas the detector. Here, Rdetis zero for\u00160H= 0 for both\n\feld orientations. When the \feld is oriented prependic-\nular to the strips, Rdetrapidly increases and reaches its\nmaximum at around \u00160H\u00192 T. Since the injection and\ndetection e\u000eciency of the magnons is maximized when\nthe magnetizations MYIGandMCoFe are aligned per-\npendicular to the electrodes2, the maximum of Rdetis\nexpected when the magnetization MCoFe is fully satu-\nrated perpendicularly to the strips overcoming the shape\nanisotropy at around 2 T2. For larger magnetic \felds, we\nagain observe a \feld-induced suppression of the signal,\nwhich was already discussed for the orientation depen-\ndent measurements in Fig. 2 and originates from the \feld\ndependence of the magnon transport in YIG. Following\nRef.2, the contributions of \u0002 SHand \u0002 ASHcan be sepa-\nrated by identifying the magnon transport signal at the\nswitching \feld and the saturation \feld of the CoFe de-\ntector, respectively. It is, however, evident that the CoFe\ndetector signal in Fig. 3 (d) becomes zero for small mag-\nnetic \felds, suggesting that there is no contribution from\na pure SHE in CoFe, in contrast to the results reported\nfor Py2. We note, however, that this observation strongly\ndepends on whether or not the CoFe is in a multidomain\nstate, since a net magnetization MCoFe perpendicular to\nthe electrode could be counterbalanced by a positive SHE\ncontribution. Clearly, we can verify again the asymmetry\nfeature when the CoFe electrode is used as a detector25.\nUtilizing our multiterminal magnon transport device,\nwe are able to extract the anomalous spin Hall angle\n\u0002CoFe\nASH. To this end, we model the spin transport in our\ndevice by employing the spin-resistor circuit model pro-\nposed in Ref.26. This approach is valid as long as the\ndistancedbetween the considered electrodes is smaller\nthan the characteristic magnon di\u000busion length \u0015min\nour YIG \flm. We verify that dPt= 2dCoFe< \u0015 m\u0019\n6µm for a comparable YIG \flm (see SI). The equiva-\nlent spin-resistor circuit diagram for the Pt2-Pt1 con-\ntact pair and the three-terminal Pt2-CoFe-Pt3 contacts\nare shown in Fig. 4 (a) and (b), respectively. Here,\nthe individual resistors are described by three di\u000ber-\nent resistances: \frstly, Rs\ni=\u0015i\u001ai=[liwtanh(ti=\u0015i)] is\nthe spin resistance of electrode i(withi= Pt1, CoFe,\nPt3) with\u0015ithe spin di\u000busion length and \u001aithe elec-\nPt1 Pt2 Pt3CoFe\nIc\n(a) (c)\n(b)\n (d)VdetVdetFigure 3. Longitudinal resistance Rlongand magnon trans-\nport signal Rdetmeasured as a function of the magnetic \feld\nstrength for \feld directions pointing perpendicular (blue) and\nparallel (green) to the strip length. Dark and light colored\nlines correspond to up- and down-sweep curves, respectively.\n(a)Rlongmeasured on the CoFe electrode, showing the AMR\nwith a switching \feld of \u00160H=\u000618 mT. Additionally, a\nsecond switching at lower \felds \u00160H=\u000611 mT (indicated by\ngray dashed lines) is observed, which corresponds to the Rlong\nchange measured on the Pt2 electrode (SMR) in (b). Here,\nthe green curves correspond to the left vertical axis, while the\nblue lines refer to the right axis. (c), (d) show Rdetas a func-\ntion of the magnetic \feld strength measured at the Pt1 and\nCoFe detector, respectively.\ntrical resistivity. Furthermore, ti,liandwdenote the\nthickness, length and width of electrode i. Secondly,\nRs\nint;i= 1=(giliw) is the interface spin resistance, with gi\nthe interface spin conductance of electrode iand lastly\nRs\nYIG=dCoFe=(\u001bmlPt2tYIG) is the YIG spin resistance\nfor a distance of dCoFe with\u001bmthe magnon conductivity\nandtYIGthe thickness of the YIG \flm. For both circuits\nshown in Fig. 4 (a) and (b), the \"spin battery\" of the net-\nwork is characterized by the injected spin chemical po-\ntential\u0016Pt2\ns;inj= 2\u0002Pt\nSHIc\u0015Pt[RPt2=lPt2] tanh (tPt=(2\u0015Pt))27\nat the YIG/Pt2 interface. Here, \u0015Ptis the spin di\u000bu-\nsion length of Pt, \u0002Pt\nSHis the spin Hall angle of Pt and\nRPt2is the electrical resistance of the Pt2 electrode. The\nspin chemical potential \"drop\" across each detector iis\ngiven via the measured detector voltages Vi\ndetas\u0016i\ns;det=\n2ti=(\u0002i\n(A)SHli)\u0000\n1 + [cosh(ti=\u0015i)\u00001]\u00001\u0001\nVi\ndet26. For each\ndetectori, we can then calculate the spin transfer e\u000e-\nciency as\n\u0011i\ns=\u0016i\ns;det\n\u0016Pt2\ns;inj: (1)\nApplying Kirchho\u000b's laws to the spin-resistor network\nshown in Fig. 4 (a), we obtain the spin transfer e\u000eciency4\nRint, CoFes\nRPt2sRint, Pt2sRYIGsRYIGsRCoFes\nRint, Pt3sRPt3s/uni03BCs, detPt3/uni03BCs, detCoFe\n/uni03BCs, injPt2RPt1sRint, Pt1sRYIGsRYIGsRint, Pt2sRPt2s/uni03BCs, detPt1\n/uni03BCs, injPt2(a)\n(b)\n(c) (d)\nPt1 Pt2 Pt3CoFe\nIc\nPt1 Pt2 Pt3CoFe\nIc\n10 mT0.1 T0.3 T0.5 T7 T\nFigure 4. Equivalent spin-resistor network for the Pt2-Pt1\ncontact pair (a) and the Pt2-CoFe-Pt3 contact con\fgura-\ntion (b). (c) Experimentally determined absolute value of the\nanomalous spin Hall angle \u0002CoFe\nASH as a function of the spin\ndi\u000busion length \u0015CoFe for various external magnetic \felds.\nHere, the spin conductance of the YIG/CoFe interface was\nset to a constant value gCoFe = 4\u00021010S=m. (d) Anomalous\nspin Hall angle of CoFe as a function of the applied magnetic\n\felds, assuming a spin di\u000busion length \u0015CoFe = 6 nm.\nof the Pt1 detector as\n\u0011Pt1\ns=Rs\nPt1\nRs\nPt2+Rs\nPt1+Rs\nint;Pt2+Rs\nint;Pt1+ 2Rs\nYIG;(2)\nwhile analyzing the circuit shown in Fig. 4 (b), we \fnd\n\u0011Pt3\ns=Rs\nPt3\u0010\nRs\ntot(1 +\u0010);\n\u0011CoFe\ns =Rs\nCoFe\nRs\ntot(1 +\u0010);(3a)\n(3b)\nfor the Pt3 and CoFe detectors. Here, \u0010= [Rs\nint;CoFe +\nRs\nCoFe]=[Rs\nYIG+Rs\nint;Pt3+Rs\nPt3] andRs\ntotis the total re-\nsistance of the spin-resistor network of Fig. 4 (b).\nOn the basis of this model, we now calculate \u001bm,gCoFe\nof the YIG/CoFe interface and \fnally \u0002CoFe\nASHof CoFe. We\nobtain\u001bmby equating Eqs. (1) and (2) for i= Pt125.\nNote, that the spin conductance gPtfor the YIG/Pt in-\nterfaces was independently determined via longitudinalSMR measurements25. We extract \u001bmfor the di\u000berent\nmagnetic \feld values and \fnd \u001bm= 3:2\u0002104S=m for\nan external magnetic \feld of \u00160H= 0:1 T. In a next\nstep, we extract gCoFe. Since our experiment does not al-\nlow to determine the spin di\u000busion length \u0015CoFe of CoFe,\nwe determine gCoFe as a function of \u0015CoFe from Eqs. (1)\nand (3a) for i= Pt3. Substituting the values extracted\nfor\u001bmfor each of the magnetic \felds measured, we \fnd\nthatgCoFe only varies by \u00180:05 % when changing \u0015CoFe\nfrom 0 nm to 10 nm. Moreover, we \fnd gCoFe to vary\nbetween approximately 2 \u00021010S=m for\u00160H= 7 T and\n4\u00021010S=m for\u00160H= 0:5 T. Since the spin conduc-\ntance is not expected to depend on the applied magnetic\n\feld, we adopt a constant value of gCoFe = 4\u00021010S=m\nin the following25. Then, we can extract \u0002CoFe\nASH as a func-\ntion of\u0015CoFefrom Eqs. (1) and (3b) for i= CoFe. The re-\nsult is shown in Fig.4 (c) for di\u000berent magnetic \felds. Ob-\nviously, \u0002CoFe\nASH saturates as a function of \u0015CoFe at around\n\u00187 nm, which corresponds to the CoFe electrode thick-\nnesstCoFe. This is reasonable, since (experimentally) we\ndo not expect any change of \u0002CoFe\nASH for\u0015CoFe> tCoFe.\nFinally, we estimate the \feld dependence of \u0002CoFe\nASH by as-\nsuming\u0015CoFe = 6 nm28. Note, that the value of \u0015CoFe\nonly a\u000bects the quantitative values for \u0002CoFe\nASH, but the\nqualitative \feld dependence remains the same. Plotting\n\u0002CoFe\nASH as a function of magnetic \feld in Fig. 4 (d), we\n\fnd that \u0002CoFe\nASH rapidly increases with increasing mag-\nnetic \feld (as MCoFe saturates) and reaches its maximum\nvalue for about 2 T-3 T at \u00185 %. For permalloy, a spin\nHall angle of 2 % was reported12. Clearly, the \feld de-\npendence of \u0002CoFe\nASH in our experiment is determined by\nthe magnetization MCoFe aligning perpendicularly to the\nCoFe strip length (i.e. along the magnetic hard axis). As\ndetailed in the SI, however, the application of a Stoner-\nWohlfarth model with uniaxial shape anisotropy29does\nnot reproduce the observed \feld dependence well, sug-\ngesting that the CoFe electrode is in a multidomain state\nfor small magnetic \felds.\nIn conclusion, we demonstrated the determination of\nthe anomalous spin Hall angle \u0002CoFe\nASHof the ferromagnetic\nmetal Co 25Fe75employing a multiterminal spin injec-\ntion/detection device. Using both paramagnetic Pt and\nferromagnetic CoFe electrodes on the ferrimagnetic insu-\nlator YIG, we were able to determine the magnon con-\nductivity of YIG, the spin conductance of the YIG/CoFe\ninterface and \fnally the anomalous spin Hall angle of\nCoFe on a single device. We based our analysis on a\nspin-resistor model26and found that the pure SHE con-\ntribution in CoFe is negligible, which is in contrast to the\n\fnite SHE contribution reported for Py2. The anomalous\nspin Hall angle of CoFe was found to increase strongly\nby saturating MCoFe with an applied magnetic \feld and\nshows a saturation value of \u00185 % for magnetic \felds of\n\u00160H&2 T.\nThis work is funded by the Deutsche Forschungsge-\nmeinschaft (DFG, German Research Foundation) under\nGermanys Excellence Strategy { EXC-2111 { 390814868\nand project AL2110/2-1.5\nREFERENCES\n1D. Tian, Y. Li, D. Qu, S. Y. Huang, X. Jin, and C. L. Chien,\n\\Manipulation of pure spin current in ferromagnetic metals in-\ndependent of magnetization,\" Phys. Rev. B 94, 020403 (2016).\n2K. S. Das, W. Y. Schoemaker, B. J. van Wees, and I. J. 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Schoen, D. Thonig, M. L. Schneider, T. J. Silva, H. T.\nNembach, O. Eriksson, O. Karis, and J. M. Shaw, \\Ultra-low\nmagnetic damping of a metallic ferromagnet,\" Nature Physics\n12, 839{842 (2016).\n22This speci\fc temperature was used due to better temperature\nstability in our cryostat.\n23L. J. Cornelissen and B. J. van Wees, \\Magnetic \feld dependence\nof the magnon spin di\u000busion length in the magnetic insulator\nyttrium iron garnet,\" Physical Review B 93, 020403 (2016).\n24M. Weiler, M. Althammer, M. Schreier, J. Lotze, M. Pernpeint-\nner, S. Meyer, H. Huebl, R. Gross, A. Kamra, J. Xiao, Y.-T.\nChen, H. Jiao, G. E. W. Bauer, and S. T. B. Goennenwein,\n\\Experimental test of the spin mixing interface conductivity con-\ncept,\" Physical Review Letters 111, 176601 (2013).\n25See Supplemental Information, where we present additional mea-\nsurements of the spin Hall magnetoresistance of the YIG/Pt in-\nterface, the magnon di\u000busion length and magnon conductivity of\nthe YIG \flm, a more detailed study of the spin conductance of\nthe YIG/CoFe interface, a study on the in\ruence of the shape\nanisotropy of the CoFe electrode and a more in depth investi-\ngation of the asymmetry feature observed in magnon tranport\nsignals with the CoFe electrodes.\n26L. J. Cornelissen, K. J. H. Peters, G. E. W. Bauer, R. A. Duine,\nand B. J. van Wees, \\Magnon spin transport driven by the\nmagnon chemical potential in a magnetic insulator,\" Physical\nReview B 94, 014412 (2016).\n27Y.-T. Chen, S. Takahashi, H. Nakayama, M. Althammer, S. T. B.\nGoennenwein, E. Saitoh, and G. E. W. 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Hammel, and F. Y. Yang\nOhio State University, Columbus, Ohio 43210, USA\nD. C. Ralph\nCornell University, Ithaca, New York 14853, USA and\nKavli Institute at Cornell for Nanoscale Science, Ithaca, New York, 14853, USA\n(Dated: December 7, 2016)\nWe report measurements of the frequency and temperature dependence of ferro-\nmagnetic resonance (FMR) for a 15-nm-thick yttrium iron garnet (YIG) \flm grown\nby o\u000b-axis sputtering. Although the FMR linewidth is narrow at room tempera-\nture (corresponding to a damping coe\u000ecient \u000b= (9.0\u00060.2)\u000210\u00004), comparable\nto previous results for high-quality YIG \flms of similar thickness, the linewidth in-\ncreases strongly at low temperatures, by a factor of almost 30. This increase cannot\nbe explained as due to two-magnon scattering from defects at the sample interfaces.\nWe argue that the increased low-temperature linewidth is due to impurity relaxation\nmechanisms that have been investigated previously in bulk YIG samples. We suggest\nthat the low-temperature linewidth is a useful \fgure of merit to guide the optimiza-\ntion of thin-\flm growth protocols because it is a particularly sensitive indicator of\nimpurities.arXiv:1612.01954v1  [cond-mat.mes-hall]  6 Dec 20162\nYttrium iron garnet (Y 3Fe5O12, YIG) thin \flms are of considerable interest for appli-\ncations in spintronics and magnonics, since YIG can have one of the lowest damping co-\ne\u000ecients of any magnetic material at room temperature.1High-quality \flms of YIG and\nrelated garnets with thicknesses on the 10's of nm scale and below can be grown by pulsed-\nlaser deposition (PLD),2{9o\u000b-axis sputtering,10{14and molecular-beam epitaxy (MBE).15\nFerromagnetic resonance (FMR) measurements at room temperature for \flms grown by all\nof these techniques show that the FMR linewidth increases with decreasing \flm thickness,\nand the frequency dependence of the linewidth has a nonlinear functional form for very thin\n\flms.2,15,16This behavior has been attributed to two-magnon scattering at the \flm interfaces\nthat becomes increasingly dominant as the \flm thickness decreases.17,18\nWe are interested in extending the use of ultra-thin YIG \flms to cryogenic temperatures,\nfor example so that we can use scannning SQUID microscopy19to study the manipulation\nof YIG devices by spin-orbit torques.20,21In the course of this work we have found that even\napparently high-quality YIG \flms, which possess small FMR linewidths at room tempera-\nture, can have linewidths that increase dramatically with decreasing temperature. The 15\nnm YIG \flm featured in this paper has a linewidth that increases by a factor of 28 as the\ntemperature is lowered from room temperature to 25 K. The linewidth of this thin YIG \flm\nalso shows an increasingly nonlinear frequency dependence as the temperature is lowered.\nWe argue that these strong temperature dependencies cannot be explained by two-magnon\nscattering from the YIG interfaces. Instead, we suggest that the increased linewidth at low\ntemperature is due to magnetic damping associated with impurity mechanisms that have\nbeen studied previously in bulk YIG samples.1,22{24\nWe grow our YIG \flms by o\u000b-axis sputtering10,11,14on a (111)-oriented gadolinium gallium\ngarnet (GGG, Gd 3Ga5O12) substrate (see details in supplementary material). We measure\nthe FMR response using a broadband coplanar waveguide with simultaneous \feld and power\nmodulation.15The waveguide is installed in a continuous-\row He cryostat for temperature-\ndependent studies. Figure 1(a) and (b) show room temperature FMR results at 3 and\n13 GHz respectively, for a 15 nm \flm as deposited ( i.e., without post-annealing). The\nresonances correspond well to derivatives of individual Lorentzians, to which we \ft to extract\nthe linewidth and resonance \feld. In Fig. 1(c) we plot the Lorentzian full-width at half\nmaximum (FWHM) linewidth \u0001 Hversus frequency. The slope of this curve corresponds\nto an e\u000bective Gilbert damping parameter \u000b= (9.0\u00060.2)\u000210\u00004. This agrees well with3\nprevious measurements of a 14.0 nm YIG \flm grown by o\u000b-axis sputtering,12which had a\ndamping parameter \u000b= (11.6\u00060.7)\u000210\u00004, and is within the range of measurements on\nPLD \flms of similar thickness.2\nFigure 2 shows how the in-plane FMR spectra of the same YIG \flm vary as a function\nof temperature. With decreasing temperature the data show a very large increase in the\nlinewidth \u0001 H, a shift in the resonance \feld, and a reduction in the amplitude of the signal,\nvisible in the normalized curves as a reduction of the signal-to-noise ratio. The reduction\nin signal amplitude is consistent with the linewidth increase, given that the amplitude is\nexpected25to scale with (\u0001 H)\u00002. Below roughly 37 K, the resonances become so broad\nthat they are no longer distinguishable using the coplanar waveguide system. This strong\ntemperature dependence is similar to results reported by Shigematsu et al. ,26but it is not\nuniversal in ultra-thin YIG \flms: e.g., Haidar et al. have observed in YIG \flms grown by\nPLD a damping coe\u000ecient that decreased by approximately a factor of two upon decreasing\nTfrom room temperature to 8 K.16\nBy analyzing similar FMR resonances obtained at di\u000berent values of microwave frequency,\nwe can extract both the frequency and temperature dependencies of \u0001 H(Fig. 3). The\nfrequency dependence at room temperature has an approximately linear dependence, similar\nto previous studies of high-quality YIG thin \flms in this thickness range.2,11,27As a function\nof decreasing temperature not only does the overall magnitude of the linewidth grow by a\nlarge factor, but at the same time there are strong deviations from linearity in the frequency\ndependence. These nonlinearities are qualitatively similar to what one might expect from\ntwo-magnon scattering from defects at the interfaces of the YIG \flm, but as we will argue\nbelow this mechanism cannot explain the very strong variations with temperature. We will\ninstead argue that these changes can be accounted for by impurity relaxation within the\nYIG \flm.\nWe can obtain greater sensitivity in the FMR experiments, and thereby extend our study\nto temperatures lower than 37 K, by performing measurements in an X-band cavity. This\ncomes at the cost of operating at \fxed frequency (9.4 GHz). We perform background subtrac-\ntion using in- and out-of-plane measurements in the cavity, as described in the supplementary\nmaterial. Figure 4 shows the Tdependence of the FMR linewidth in these cavity measure-\nments, with a comparison to the broadband coplanar waveguide results. (The waveguide\nvalues are interpolated from measurements at 9 and 10 GHz.) We \fnd excellent quantitative4\nagreement between the two types of measurements. The cavity measurements reveal that\n\u0001Hhas a maximum near 25 K, with a clear decrease at lower temperatures. The maximum\nlinewidth is 28 times larger than the room temperature result at this frequency.\nIn order to evaluate possible mechanisms for these very strong changes in linewidth with\ntemperature, we must \frst characterize how the magnetic anisotropy in the YIG \flm varies\nwith temperature. We do this based on the measured FMR resonance \felds, \ftting to the\nKittel equation for a magnetic thin \flm with an in-plane magnetic \feld25\nf=j\rj\n2\u0019q\nHk\nr(Hk\nr+ 4\u0019M e\u000b): (1)\nHere\ris the gyromagnetic ratio, Hk\nris the in-plane resonance \feld for a given \fxed frequency\nf, and 4\u0019M e\u000bparameterizes the shape anisotropy and any additional contributions to the\nperpendicular magnetic anisotropy. We obtain good \fts (see Fig. 5(a)) with no additional\nin-plane anisotropy contribution. In Eq. (1) we do not include a renormalization shift in\nthe resonance frequency that can result from two-magnon scattering because this is small\non the scale important to our analysis.17,18We also neglect a small shift in resonance \feld\nthat can arise from a static dipole interaction between the YIG and the paramagnetic GGG\nsubstrate28,29because this is also small, less than a 1% shift for temperatures above 15 K\n(see the supplementary material). The values of 4 \u0019M e\u000bwe obtain from the \fts to Eq. (1)\nat di\u000berent temperatures are shown in Fig. 5(b). We \fnd that 4 \u0019M e\u000bis signi\fcantly larger\nthan the simple shape anisotropy generated by the YIG saturation magnetization, 4 \u0019M s\n(determined from vibrating sample magnetometry (VSM) measurements presented in the\nsupplementary material), indicating the presence of a positive uniaxial anisotropy, Hs=\n4\u0019M e\u000b\u00004\u0019M s, favoring an in-plane magnetization. We have con\frmed the value of Hs\nand the form of its temperature dependence using FMR measurements with an out-of-plane\nmagnetic \feld. Figure 5(a) shows the frequency dependence of the resonance position with an\nout-of-plane \feld at room temperature, and Fig. 5(b) shows the extracted value of 4 \u0019M e\u000b\nas a function of temperature from both waveguide and cavity FMR measurements. The\nlarge value of Hsis greater than expected from surface anisotropy15or magneto-crystalline\nanisotropy25of cubic YIG alone, so we tentatively ascribe the result to a growth-induced\nanisotropy, such as caused by tetragonal distortion. This is consistent with predictions and\nobservations in YIG \flms grown by PLD,4where the anisotropy is highly dependent on\nthe growth conditions. The temperature dependence of Hsthat we obtain is qualitatively5\nconsistent with the spin \ructuation model,30,31which predicts Hs(T)/[Ms(T)]2.\nGiven this characterization of Hs(T), we can now evaluate whether two-magnon scattering\nfrom surface defects, a mechanism that is expected to be active for ultra-thin YIG \flms at\nroom temperature,2,15is capable of explaining the large increase in the linewidth \u0001 Hthat we\nobserve at low temperature. This e\u000bect causes a linewidth that is nonlinear with frequency\nf, following the form32\n\u0001H2M= \u0000(T) sin\u00001sp\n!2+ (!0=2)2\u0000!0=2p\n!2+ (!0=2)2+!0=2; (2)\nwhere!= 2\u0019fand!0=\r4\u0019M e\u000b(T). The temperature dependence in this equation is\ndominated by the scattering coe\u000ecient \u0000( T), whose expected temperature dependence17is\n\u0000(T)/[Hs(T)]2. Given our determination of Hs(T) above (using Me\u000bfromHk\nr\fts as a\nworst-case scenario), the temperature dependence expected from the two-magnon scattering\nmechanism is illustrated by the red line in Fig. 4. This mechanism can explain at most a\nfactor of 4 increase in the linewidth as the temperature is reduced from 300 to 0 K, far less\nthan the factor of 28 that we observe. It also is incapable of explaining the peak in \u0001 Hwe\nmeasure near 25 K. Similar conclusions follow if one assumes30,31thatHs(T)/[Ms(T)]2,\ntogether with our VSM measurements of Ms(T).\nAn alternative mechanism that can account for a much stronger temperature dependence\nfor \u0001His impurity relaxation, for example due to rare earth or Fe2+impurities in the\nYIG \flm. Researchers in the 1960's produced a rich body of literature which shows that\nthe linewidth in bulk YIG samples can increase dramatically at low temperatures when\nimpurity relaxation is active.1,22{24,33The frequency and temperature dependence of \u0001 H\nin our samples can be explained well using a model of slowly-relaxing impurities.1,24,34The\ncontribution to the linewidth from this mechanism is expected to have the form\n\u0001HSR=A(T)!\u001c\n1 + (!\u001c)2; (3)\nwhereA(T) is a frequency-independent prefactor and \u001cis a temperature-dependent time\nconstant. The lines in Fig. 3 are \fts assuming that the linewidths are governed by this\nfunctional form plus a linear-in-frequency temperature-independent background contribution\nequal to the room-temperature dependence (Fig. 1(c)). The \ft parameters are shown in the\nsupplemental material. The maximum near 25 K in the temperature dependence of \u0001 H6\n(Fig. 4) is very similar to previous measurements in bulk YIG,1and corresponds within the\nslowly-relaxing impurity model to the condition !\u001c\u00191.\nIn conclusion, even when a YIG \flm has a narrow linewidth at room temperaure indi-\ncating an apparently high-quality \flm, the linewidth can still increase dramatically at low\ntemperature, by well over an order of magnitude. This is generally undesirable. For exam-\nple, this will make manipulation of YIG \flms by anti-damping spin-transfer torques much\nless e\u000ecient at low temperature, and may block it entirely for practical purposes. Based on\nmeasurements of the temperature and frequency dependence of the e\u000bect, we suggest that\nthe increased low-temperature linewidth is due to slowly relaxing impurities, perhaps rare\nearth or Fe2+impurities introduced during growth.1Given the high degree of sensitivity of\nthe low-temperature linewidth to these impurities, we suggest that the low- Tlinewidth can\nserve as a useful \fgure of merit for optimizing growth protocols for ultra-thin YIG \flms.\nSUPPLEMENTARY MATERIAL\nSee the supplementary material for detailed information on the YIG growth by o\u000b-axis\nsputtering, saturation magnetization and substrate susceptibility, broadband and cavity\nmeasurement systems, resonance \feld shift from substrate dipolar \felds, thickness depen-\ndence of the linewidths, and \ftting parameters extracted from the frequency dependence of\nthe linewidths.\nACKNOWLEDGMENTS\nWe acknowledge G. E. Rowlands and S. Shi for their help in building a low-temperature\ncryostat, M. S. Weathers for her help with X-ray re\rectivity measurements, and B. Dzikovski\nfor his help with the low-temperature cavity measurements preformed at ACERT (National\nBiomedical Center for Advanced ESR Technology). This research was supported by the\nNational Science Foundation (DMR-1406333 and DMR-1507274), NSF/MRSEC program\nthrough the Center for Emergent Materials (DMR-1420451), the NSF/MRSEC program\nthrough the Cornell Center for Materials Research (DMR-1120296), the NIH/NIGMS pro-\ngram through ACERT (P41GM103521), the US Department of Energy (DOE) (DE-FG02-\n03ER46054), and the US Department of Defense - Intelligence Advanced Research Projects7\nActivity (IARPA) through the U.S. Army Research O\u000ece (W911NF-14-C-0089). The con-\ntent of the paper does not necessarily re\rect the position or the policy of the Government,\nand no o\u000ecial endorsement should be inferred. 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Normalized ferromagnetic resonance spectra at (a) 3 GHz and (b) 13 GHz for a 15 nm YIG\n\flm at room temperature, with an in-plane applied magnetic \feld. (c) The frequency dependence\nof the linewidth corresponds to an e\u000bective Gilbert damping constant \u000b= (9.0\u00060.2)\u000210\u00004.11\nFIG. 2. Normalized ferromagnetic resonance spectra at 3 GHz with an in-plane applied magnetic\n\feld for the YIG \flm at di\u000berent temperatures. Di\u000berent normalization factors are used for data at\ndi\u000berent temperatures; the actual amplitude of the resonances decreases strongly with decreasing\ntemperature, as re\rected in the decreasing signal-to-noise ratio. With decreasing temperature, we\nobserve a large increase in the resonance linewidth.12\nFIG. 3. Linewidths (Lorentzian FWHM) from the in-plane FMR spectra measured at di\u000berent\ntemperatures. Solid lines are \fts to the sum of the frequency dependence expected from a slowly-\nrelaxing impurity mechanism in addition to the room temperature linear behavior. The dashed\nline for 37 K is a guide to the eye.13\nFIG. 4. FMR linewidth at 9.4 GHz as measured by two techniques: (open black triangles) cavity\nmeasurements and (blue squares) coplanar waveguide measurements. We observe a peak near 25 K,\nwhere \u0001His 28 times larger than at room temperature. The solid red line indicates temperature\ndependence expected from two-magnon scattering; this dependence is too weak to explain the\nvariation in \u0001 H.14\n(a) (b)\nFIG. 5. FMR resonance \feld as a function of frequency for (squares) an out-of-plane applied mag-\nnetic \feld at room temperature and (circles) and in-plane applied \felds at various temperatures.\nSolid lines are \fts to the Kittel equation. (b) Temperature dependence of the e\u000bective magnetiza-\ntion, determined from the Kittel \fts for (black circles) in-plane and (open triangles) out-of-plane\napplied magnetic \felds. The open triangles below 50 K are from cavity measurements. The red line\nis 4\u0019times the saturation magnetization, from a \ft to VSM measurements (see the supplementary\nmaterial). The e\u000bective magnetization re\rected in the magnetic anisotropy is signi\fcantly greater\nthan the saturation magnetization."    },    {        "title": "1707.06509v2.Bistability_of_Cavity_Magnon_Polaritons.pdf",        "content": "Bistability of Cavity Magnon Polaritons\nYi-Pu Wang,1Guo-Qiang Zhang,1Dengke Zhang,1,\u0003Tie-Fu Li,2, 1,yC.-M. Hu,3and J. Q. You1,z\n1Quantum Physics and Quantum Information Division,\nBeijing Computational Science Research Center, Beijing 100193, China\n2Institute of Microelectronics, Tsinghua National Laboratory of Information\nScience and Technology, Tsinghua University, Beijing 100084, China\n3Department of Physics and Astronomy, University of Manitoba, Winnipeg R3T 2N2, Canada\n(Dated: October 8, 2018)\nWe report the first observation of the magnon-polariton bistability in a cavity magnonics system consisting of\ncavity photons strongly interacting with the magnons in a small yttrium iron garnet (YIG) sphere. The bistable\nbehaviors are emerged as sharp frequency switchings of the cavity magnon-polaritons (CMPs) and related to the\ntransition between states with large and small number of polaritons. In our experiment, we align, respectively,\nthe [100] and [110] crystallographic axes of the YIG sphere parallel to the static magnetic field and find very\ndi\u000berent bistable behaviors (e.g., clockwise and counter-clockwise hysteresis loops) in these two cases. The\nexperimental results are well fitted and explained as being due to the Kerr nonlinearity with either positive or\nnegative coe \u000ecient. Moreover, when the magnetic field is tuned away from the anticrossing point of CMPs,\nwe observe simultaneous bistability of both magnons and cavity photons by applying a drive field on the lower\nbranch.\nPACS numbers: 75.30.Ds, 71.36. +c, 42.65.Pc\nBoth quantum information processing [1] and future quan-\ntum internet [2] inevitably need e \u000ecient quantum informa-\ntion transfers among di \u000berent physical systems. Hybrid quan-\ntum systems may provide hopeful solutions to these prob-\nlems [3, 4]. Recently, cavity magnonics has attracted much\ninterest (see, e.g., [5–12]), which involves cavity photons\nstrongly or ultrastrongly [13] interacting with collective spin\nexcitations in a millimeter-scale yttrium iron garnet (YIG)\ncrystal. This hybrid system gives rise to new quasiparticles\ncalled cavity magnon-polaritons (CMPs) [14, 15]. Benefiting\nfrom low damping rates of both magnons and cavity photons,\nthis hybrid system is also expected to become a building block\nof the future quantum information network. Now, a versatile\nquantum information processing platform based on the coher-\nent couplings among magnons, cavity microwave photons [5–\n7], optical photons [16–19], phonons [20] and superconduct-\ning qubits [21, 22] is being established, with the strong cou-\npling between magnons and cavity photons being the core of\nthe hybrid quantum system.\nFor the cavity magnonics system, in addition to the hy-\nbridization between magnons and cavity photons, nonlinear\ne\u000bect can also play an important role. Originating from the\nmagnetocrystalline anisotropy in the YIG [23, 24], this non-\nlinearity is related to the Kerr e \u000bect of magnons and the\nnonlinearity-induced frequency shift has been demonstrated\nin the dispersive regime at cryogenic temperature [25]. In this\nLetter, we observe the CMP bistability in a cavity magnonics\nsystem. In a wide range of parameters, the frequency of the\nCMPs is found to jump up or down sharply at the switching\npoints where the CMPs transition from one state to another.\nTo clearly demonstrate the nonlinear e \u000bect, we implement the\nexperiment by aligning the [100] and [110] crystallographic\naxes of the YIG sphere parallel to the externally applied static\nmagnetic field, respectively. The measured results show very\ndi\u000berent features in these two cases, such as the blue- or red-shift of the frequency of the CMPs and the emergence of the\nclockwise or counter-clockwise hysteresis loop related to the\nCMP bistability. Also, we theoretically fit the experimental\nresults well and explain the very di \u000berent phenomena of the\nCMPs as being due to the Kerr nonlinearity with either a pos-\nitive ornegative coe\u000ecient.\nTo our knowledge, this work is the first convincing obser-\nvation of the bistability in CMPs. As an important nonlin-\near phenomenon, bistability is not only of fundamental in-\nterest in studying dissipative quantum systems [26, 27], but\nalso has potential applications in switches [28, 29] and mem-\nories [30, 31]. Our cavity magnonics system o \u000bers a new\nplatform to explore these applications. Also, the employed\nKittle-mode magnons have distinct merits, such as the tun-\nability with magnetic field and the maintenance of good quan-\ntum coherence even at room temperature [7, 10]. These ad-\nvantages may bring new possibilities in exploring nonlinear\nproperties of the system. Indeed, we also observe the simul-\ntaneous bistability of both magnons and cavity photons at a\nvery o \u000b-resonance point by applying a drive field on the lower\nbranch, where the optical bistability is achieved via the mag-\nnetic bistability. This observation shows that the CMPs can\nserve as a bridge /transducer between optical and magnetic\nbistabilities, which paves a new way for using one e \u000bect to\ninduce and control the other.\nThe experimental setup for our hybrid system is schemat-\nically shown in Fig. 1(a). We use a three-dimensional (3D)\nrectangular cavity made of oxygen-free copper with inner di-\nmensions 44 :0\u000222:0\u00026:0 mm3and have a small YIG sphere\nof diameter 1 mm glued on an inner wall of the cavity at a\nmagnetic-field antinode of the cavity mode TE 102. The cavity\nhas three ports, with ports 1 and 2 as the input and output ports\nfor measuring transmission spectrum and also with a specially\ndesigned port (i.e., port 3) in the vicinity of the YIG sphere for\nconveniently loading a microwave drive field via a loop an-arXiv:1707.06509v2  [quant-ph]  6 Feb 20182\nFIG. 1. (a) Schematic diagram of the experimental setup. The 3D\ncavity with a small YIG sphere embedded is placed in the static\nmagnetic field B0generated by an electromagnet. Ports 1 and 2\nof the cavity, connected to a vector network analyzer (VNA), are\nused for transmission spectroscopy and port 3 connected to a mi-\ncrowave (MW) source is for driving the YIG sphere. At the bottom,\nthe magnetic-field distribution of the TE 102mode is magnified for\nclarity. The small YIG sphere is placed beside a circular-loop an-\ntenna and located at the magnetic-field antinode of the TE 102mode.\n(b) Transmission spectrum of the CMPs measured versus the magnet\ncoil current (i.e., the static magnetic field) and the frequency of the\nprobe field. Two vertical dashed lines indicate, respectively, the res-\nonance and the very o \u000b-resonance points at which we show the CMP\nbistability.\ntenna. Here we focus on the Kittel mode which is a spatially\nuniform mode of the ferromagnetic spin waves [32]. To re-\nduce the disturbance from other magnetostatic modes [33, 34],\nthe small YIG sphere is placed at the uniform field of the an-\ntenna. The whole cavity with the YIG sphere embedded is\nplaced in a static magnetic field B0created by a high-precision\ntunable electromagnet at room temperature. This bias mag-\nnetic field, the magnetic component of the microwave drive\nfield, and the magnetic field of the TE 102mode are nearly per-\npendicular to one another at the site of the small YIG sphere.\nWhen the frequency of the Kittel-mode magnons is tuned\nin resonance with the microwave photons of the cavity mode\nTE102, anticrossing of energy levels occurs owing to the strong\ncoupling between magnons and cavity photons, which gives\nrise to two branches of CMPs [see Fig. 1(b)]. The magnon-\nphoton coupling strength is found to be gm=2\u0019=41 MHz\nfrom the energy splitting at the resonance (anticrossing) point.\nThe fitted cavity-mode linewidth is \u0014=2\u0019\u0011(\u00141+\u00142+\u00143+\n\u0014int)=2\u0019=3:8 MHz, where \u0014i(i=1;2;3) is the decay rate\nof the cavity due to the ith port and \u0014intis the intrinsic loss\nof the cavity. Also, the Kittel-mode linewidth is found to be\n\rm=2\u0019=17:5 MHz. It is clear that the system is in the strong-\ncoupling regime because gm>\u0014;\r m.\nHere we demonstrate the nonlinear e \u000bect in the cavity\nmagnonics system by driving the YIG sphere with a mi-\ncrowave field via the loop antenna. We first focus on the\ncase with the magnons in resonance with the cavity photons,\nwhere a CMP is the maximal superposition of a magnon and\n051015200\n51015200\n5010015020025030035005101520051015200\n5101520-\n60-40-2002 0406005101520/s948LP/2π = -14.1 MHz \nScan forward \nScan backward-\n - - - Theory(a)/s948\nLP/2π = -12.1 MHzPolariton frequency shift ΔLP/2π (MHz)/s948\nLP/2π = -10.1 MHzD\nrive power Pd (mW)(b)P\nd = 25 dBm \nScan forward \nScan backward-\n - - - TheoryP\nd = 23 dBmP\nd = 21 dBmF\nrequency detuning δLP/2π (MHz)FIG. 2. (a) \u0001LP=2\u0019versus Pdwhen\u000eLP=2\u0019=\u000014:1,\u000012:1, and\n\u000010:1 MHz. (b) \u0001LP=2\u0019versus\u000eLP=2\u0019when Pd=25, 23, and\n21 dBm. The black circle (blue triangle) dots are the forward\n(backward)-scanning results. The red dashed curves are theoreti-\ncal results obtained using Eq. (2), with \rLP=2\u0019=(\rm+\u0014)=4\u0019=\n10:65 MHz. In (a), the characteristic constant c=(2\u0019)3is fitted to be\n3:15, 3:55, and 3:82 MHz3/mW for curves from top to bottom. In (b),\nc=(2\u0019)3is fitted to be 1 :85, 2:52, and 3:22 MHz3/mW for curves from\ntop to bottom. The [100] crystallographic axis of the YIG sphere is\naligned parallel to the static magnetic field B0and the magnons are\nin resonance with the cavity mode TE 102.\na cavity photon. Also, the YIG sphere is aligned to have\nits [100] crystallographic axis parallel to the static magnetic\nfield B0. In Fig. 2(a), we measure the frequency shift \u0001LP\nof the lower -branch CMPs versus the drive power Pdfor dif-\nferent values of the drive-field frequency detuning (see [35]\nfor the measurement method). Here the angular frequency\n!LPof the lower -branch CMPs is tuned to be at the anticross-\ning point Ain Fig. 1(b) and the drive-field frequency detuning\n\u000eLP\u0011!LP\u0000!dis relative to the lower -branch CMPs, where\n!dis the angular frequency of the drive field. At the reso-\nnance point with !m=!c(where!mis the frequency of the\nKittel-mode magnons and !cis the frequency of the cavity\nmode TE 102), a hysteresis loop is clearly seen at \u000eLP<0,\nrevealing the emergence of the CMP bistability in the cavity\nmagnonics system. This hysteresis loop is counter-clockwise\nwhen considering the increasing and decreasing directions of\nthe drive power. Moreover, its area reduces when decreasing\nthe frequency detuning j\u000eLPj. In Fig. 2(b), we measure the fre-\nquency shift \u0001LPof the same lower -branch CMPs versus the\nfrequency detuning \u000eLPfor di \u000berent values of Pd. When the\nincreasing and decreasing directions of \u000eLPare considered, a\ncounter-clockwise hysteresis loop is also clearly shown, and\nits area decreases when reducing Pd.\nFor a small YIG sphere driven by a microwave field with\nfrequency!d, when its [100] crystallographic axis is aligned\nparallel to the static magnetic field, the cavity magnonics sys-\ntem has the Hamiltonian (setting ~=1) [25]\nH=!caya+!mbyb+Kbybbyb+gm(ayb+aby)3\n+\n d(bye\u0000i!dt+bei!dt); (1)\nwhere ay(a) is the creation (annihilation) operator of the cavity\nphotons at frequency !c,by(b) is the creation (annihilation)\noperator of the Kittel-mode magnons at frequency !m, and\n\ndis the drive-field strength. As shown in [25], the Kerr term\nKbybbyb, with a positive coe \u000ecient K=\u00160Kan\r2=(M2Vm),\nis intrinsically due to the magnetocrystalline anisotropy in the\nYIG material. Here \u00160is the magnetic permeability of free\nspace, Kanis the first-order anisotropy constant, \r=g\u0016B=~is\nthe gyromagnetic ratio (with gbeing the g-factor and \u0016Bthe\nBohr magneton), Mis the saturation magnetization, and Vm\nis the volume of the YIG sphere. Note that the Kerr e \u000bect is\nstrengthened when reducing Vm.\nWe consider the case with j!LP\u0000!dj\u001cj!UP\u0000!dj, where\n!UPis angular frequency of the upper -branch CMPs. In such\na case, the drive field applied to the YIG sphere generates po-\nlaritons in the lower branch much more than the polaritons in\nthe upper branch. Using a quantum Langevin approach, we\nobtain a cubic equation for the CMP frequency shift \u0001LP[35]\n\u0014\n(\u0001LP+\u000eLP)2+\u0012\rLP\n2\u00132\u0015\n\u0001LP\u0000cPd=0; (2)\nwhere\rLPis the damping rate of the lower -branch CMP and\ncis a coe \u000ecient characterizing the coupling strength between\nthe drive field and the lower -branch CMPs. This cubic equa-\ntion provides a steady-state solution for the frequency shift of\nthelower -branch CMPs as a function of both the drive-field\nfrequency detuning \u000eLPand the drive power Pd. Under ap-\npropriate conditions, it has three solutions, two of them stable\nand the additional one unstable. This corresponds to the bista-\nbility of the system. In Fig. 2, we also show the theoretical\nresults (dashed curves) obtained using Eq. (2), which fit the\nexperimental results very well. This verifies the experimental\nobservation of the CMP bistability in our cavity magnonics\nsystem. The two stable solutions of \u0001LPin Eq. (2) correspond\ntotwostates of the system with large andsmall number of po-\nlaritons in the lower branch [35]. Thus, in Fig. 2, each sharp\nswitching of the frequency shift \u0001LPis related to the transition\nbetween these two states.\nIn the resonance case with !m=!c, we further imple-\nment experiment by aligning the [110] crystallographic axis\nof the YIG sphere parallel to the static magnetic field B0. In\nFig. 3(a), we measure the frequency shift \u0001LPof the lower -\nbranch CMPs versus the drive power Pdfor di \u000berent values\nof the drive-field frequency detuning \u000eLPrelative to the lower -\nbranch CMPs. In sharp contrast to Fig. 2(a), bistability of the\nlower -branch CMPs is now observed at \u000eLP>0. The area\nof the hysteresis loop also decreases when reducing \u000eLP, but\nthe hysteresis loop becomes clockwise. Also, we present in\nFig. 3(b) the frequency shift \u0001LPversus the frequency detun-\ning\u000eLPfor di \u000berent values of Pd. The hysteresis loop is also\ncounter-clockwise, similar to that in Fig. 2(b).\nAs shown in [35], when the YIG sphere is aligned with its\n[110] crystallographic axis parallel to the static magnetic field\nB0, the Hamiltonian of the cavity magnonics system takes the\n-20-15-10-50-\n20-15-10-500\n50100150200250300350-20-15-10-50-30-20-100-\n30-20-100-\n60-40-2002 04060-30-20-100/s948LP/2π = 17.2 MHz \nScan forward \nScan backward-\n - - -  Theory(a)/s948\nLP/2π = 13.2 MHzPolariton frequency shift ΔLP/2π (MHz)/s948\nLP/2π = 9.2 MHzD\nrive power Pd (mW)Pd = 25 dBm \nScan forward \nScan backward-\n - - - Theory(b)P\nd = 23 dBmF\nrequency detuning /s948LP/2π (MHz)Pd = 21 dBmFIG. 3. (a) \u0001LP=2\u0019versus Pdwhen\u000eLP=2\u0019=17:2, 13:2, and\n9:2 MHz. (b) \u0001LP=2\u0019versus\u000eLP=2\u0019when Pd=25, 23, and 21 dBm.\nThe black circle (blue triangle) dots are the forward (backward)-\nscanning results. The red dashed curves are theoretical results ob-\ntained using Eq. (2), with \rLP=2\u0019=10:65 MHz. In (a), c=(2\u0019)3is\nfitted to be\u00004,\u00003:5, and\u00003:1 MHz3/mW for curves from top to bot-\ntom. In (b), c=(2\u0019)3is fitted to be\u00003:01,\u00003:46, and\u00003:6 MHz3/mW\nfor curves from top to bottom. The [110] crystallographic axis of the\nYIG sphere is aligned parallel to B0and the magnons are in reso-\nnance with TE 102.\nsame form as in Eq. (1), but the coe \u000ecient of the Kerr term\nbecomes negative, K=\u000013\u00160Kan\r2=(16M2Vm). The corre-\nsponding theoretical results (dashed curves) obtained using\nEq. (2) are shown in Fig. 3, which are also in good agree-\nment with the experimental results. Moreover, Figs. 2 and 3\nshow that the CMPs have blue(red)-shift in frequency when\nthe [100] ([110]) crystallographic axis of the YIG sphere is\naligned along the direction of the static magnetic field. These\nare due to the positive and negative Kerr coe \u000ecients K’s in\nthe two di \u000berent cases [35].\nFinally, we demonstrate the Kerr e \u000bect of magnons at an\no\u000b-resonance point much away from !m=!c, where the\nmagnet coil current is 4 :8 A [see the left dashed line in\nFig. 1(b)]. At this point, the frequency of the lower -branch\nCMPs is about 9 :91 GHz and the frequency of the upper -\nbranch CMPs is about 10 :08 GHz, as indicated by points B\nandCin Fig. 1(b), respectively. Also, the [100] crystallo-\ngraphic axis of the YIG sphere is aligned parallel to the static\nmagnetic field B0. In Fig. 4(a), we present the frequency shift\n\u0001LPof the lower -branch CMPs versus the drive power Pdfor\ndi\u000berent values of the drive-field frequency detuning \u000eLPrela-\ntive to the lower -branch CMPs. Moreover, the frequency shift\n\u0001LPversus the frequency detuning \u000eLPis shown in Fig. 4(b)\nfor di \u000berent values of Pd. As in Fig. 2, we also see counter-\nclockwise hysteresis loops. Because the magnon is a domi-\nnating component of the lower -branch CMP at this very o \u000b-\nresonance point, nearly the bistability of magnons is actually\nobserved here, directly owing to the magnon Kerr e \u000bect in the\nYIG sphere. The theoretical results (dashed curves) obtained4\n020400\n20400\n2040040800\n40800\n40800\n120\n120\n501001502002503003500120240\n24-\n100- 500 5 01 00024/s948LP/2π = -31.8 MHz \nScan forward \nScan backward-\n - - - Theory(a)/s948\nLP/2π = -27.8 MHzPolariton frequency shift ΔLP/2π (MHz)/s948\nLP/2π = -23.8 MHz(b) \nScan forward \nScan backward-\n - - - TheoryPd = 25 dBmP\nd = 23 dBmP\nd = 21 dBm \nScan forward \nScan backward-\n - - -  Theory(c)/s948LP/2π = -27.8 MHzPolariton frequency shift ΔUP/2π (MHz)/s948\nLP/2π = -23.8 MHz/s948LP/2π = -31.8 MHzD\nrive power Pd (mW)Pd = 25 dBm(d) \nScan forward \nScan backward-\n - - - TheoryP\nd = 23 dBmP\nd = 21 dBmF\nrequency detuning /s948LP/2π (MHz)\nFIG. 4. (a) \u0001LP=2\u0019versus Pdwhen\u000eLP=2\u0019=\u000031:8,\u000027:8, and\n\u000023:8 MHz. (b) \u0001LP=2\u0019versus\u000eLP=2\u0019when Pd=25, 23, and\n21 dBm. The black circle (green triangle) dots are the forward\n(backward)-scanning results. The red dashed curves are theoreti-\ncal results obtained using Eq. (2), with \rLP=2\u0019=16:8 MHz. In\n(a),c=(2\u0019)3is fitted to be 25 :2, 25:0, and 25:3 MHz3/mW for curves\nfrom top to bottom. In (b), c=(2\u0019)3is fitted to be 17 :0, 19:0, and\n22:7 MHz3/mW for curves from top to bottom. (c) and (d) \u0001UP=2\u0019\nmeasured under the same conditions as in (a) and (b), respectively.\nThe black circle (oragne triangle) dots are the forward (backward)-\nscanning results. The blue dashed curves (except for the parts around\nsmall dips) are also obtained using Eq. (2), with the fitted ratio\n\u0018\u0011\u0001UP=\u0001LPfor curves from top to bottom being 0 :060, 0:062, and\n0:061 in (c) and 0 :062, 0:063, and 0:061 in (d). The [100] crystal-\nlographic axis of the YIG sphere is aligned parallel to B0and the\nmagnons are far o \u000bresonance with TE 102. In addition, the small dips\nin, e.g., (c) and (d) can be fitted by introducing a magnetostatic mode\nwith the negative Kerr coe \u000ecient [35].\nusing Eq. (2) also agree well with the experimental results.\nUnder the conditions same as in Figs. 4(a) and 4(b), we fur-\nther show the frequency shift \u0001UPof the upper -branch CMPs\nversus the drive power Pd[Fig. 4(c)], as well as the frequency\nshift \u0001UPof the upper -branch CMPs versus the drive-field\nfrequency detuning \u000eLPrelative to the lower -branch CMPs\n[Fig. 4(d)]. Here, as in Figs. 4(a) and 4(b), the drive field\nis still applied on the lower branch. Now the cavity photon\nis a dominating component of the upper -branch CMP at this\nvery o \u000b-resonance point, so it is nearly the bistability of cavity\nphotons that is observed in Figs. 4(c) and 4(d). As one knows,\nboth optical [40, 41] and magnetic [42] bistabilities are inter-\nesting phenomena of nonlinear systems. Here we observe the\nsimultaneous bistability of both magnons and cavity photonsat a very o \u000b-resonance point by applying the drive field only\non the lower branch, where the optical bistability is achieved\nvia the magnetic bistability. Also, the experimental results in\nFigs. 4(c) and 4(d) fit well with the theoretical results (dashed\ncurves), where the fitted ratio \u0018\u0011\u0001UP=\u0001LPis close to the\ntheoretical value 0 :065 [35]. In addition, some small dips are\nobserved in, e.g., Figs. 4(c) and 4(d). By fitting with these\ndips, we attribute them to other polaritons stemming from the\ncoupling between cavity photons and a magnetostatic mode\nwith the negative Kerr coe \u000ecient [35].\nIn conclusion, we have demonstrated the bistable behaviors\nof the CMPs using a hybrid system consisting of microwave\ncavity photons strongly interacting with the magnons in a YIG\nsphere. We find that the switching points where the CMPs\ntransition from one state to another depend on the drive power\nand the drive-field frequency detuning. When implementing\nthe experiment, we align, respectively, the [100] and [110]\ncrystallographic axes of the YIG sphere parallel to the static\nmagnetic field and find very di \u000berent bistable behaviors in\nthese two cases. We theoretically fit the experimental results\nwell and put these di \u000berent bistable behaviors down to the\nKerr nonlinearity with either a positive or negative coe \u000ecient.\nThe cavity magnonics system possesses the merits of tun-\nability and compatibility with other quantum systems, in\nwhich the magnons can be tuned by an external magnetic field\nand the magnon-photon coupling can be tuned by moving the\nYIG sphere inside the cavity. With regard to the bistabil-\nity of the CMPs, the area of the hysteresis loop can be eas-\nily controlled by a lower drive power and a tunable drive-\nfield frequency. This may bring potential applications in re-\nalizing low-energy switching devices. Also, we observe si-\nmultaneous bistability of both magnons and cavity photons\nby tuning the magnetic field. It paves a new path to con-\ntrol the conversion between magnetic and optical bistabili-\nties. Moreover, our study provides possible routes to ex-\nplore other nonlinear e \u000bects of the system such as the creation\nof a frequency comb for frequency conversion [43] and the\nchaos of CMPs. With the available CMP bistability, the cavity\nmagnonics system can also provide a new platform to under-\nstand the dissipative phase transition and the related critical\nphenomena [26, 44, 45]. These open up di \u000berent directions\nfor future studies.\nThis work was supported by the National Key Re-\nsearch and Development Program of China (Grant\nNo. 2016YFA0301200), the NSFC (grant No. 11774022), the\nMOST 973 Program of China (Grant No. 2014CB848700),\nand the NSAF (Grant No. U1330201 and No. U1530401).\nC.M.H. was supported by the NSFC (Grant No. 11429401).\nY .-P. W and G.-Q. Z contribute equally to this work.\n\u0003Present address: Department of Engineering, University of\nCambridge, Cambridge CB3 0FA, United Kingdom\nylitf@tsinghua.edu.cn5\nzjqyou@csrc.ac.cn\n[1] M. Wallquist, K. Hammerer, P. Rabl, M. Lukin, and P. Zoller,\nHybrid quantum devices and quantum engineering, Phys. Scr.\nT137 , 014001 (2009).\n[2] H. J. Kimble, The quantum internet, Nature (London) 453, 1023\n(2008).\n[3] Z.-L. Xiang, S. Ashhab, J. Q. You, and F. Nori, Hybrid quantum\ncircuits: Superconducting circuits interacting with other quan-\ntum systems, Rev. Mod. Phys. 85, 623 (2013).\n[4] G. 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PMR calculations were\nperformed for yttrium iron garnet (YIG), cobalt ferrite (CFO), and two europium chalcogenides EuO\nand EuS. We \fnd a signi\fcant PMR (up to 100%) values de\fned as a relative change of graphene\nconductance with respect to parallel and antiparallel alignment of two proximity induced magnetic\nregions within graphene. Namely, for high Curie temperature (Tc) CFO and YIG insulators which\nare particularly important for applications, we obtain 22% and 77% at room temperature, respec-\ntively. For low Tc chalcogenides, EuO and EuS, the PMR is 100% in both cases. Furthermore, the\nPMR is robust with respect to system dimensions and edge type termination. Our \fndings show\nthat it is possible to induce spin polarized currents in graphene with no direct injection through\nmagnetic materials.\nI. INTRODUCTION\nGraphene is a two-dimensional (2D) material1,2that\nhas attracted a lot of interest in view of its unique\nphysical properties and applications potential in di-\nverse \felds such as electronics, spintronics and quantum\ncomputing3{5. Due to its weak spin orbit coupling6{15\ngraphene possesses a long spin relaxation time and\nlengths even at room temperature16. While these charac-\nteristics o\u000ber an optimal platform for spin manipulation,\nit remains however a challenge to achieve robust spin po-\nlarization e\u000eciently at room temperature.\nSeveral methods have been proposed in order to\nintroduce ferromagnetic order on graphene, among\nwhich functionalization with adatoms17, addition of\ndefects18,19, and by means of proximity e\u000bect via an ad-\njacent ferromagnet20{25. The latter approach attracted\na lot of interest using magnetic insulators (MI) as a sub-\nstrate to induce exchange splitting in graphene. When\na material is placed on top of a magnetic insulator, it\ncan acquire proximity induced spin polarization and ex-\nchange splitting20resulting from the hybridization be-\ntweenpzorbitals with those of the neighboring magnetic\ninsulator. For practical purposes, the implementation in\nspintronic devices of this kind of materials could lead\nto lower power consumption since no current injection\nacross adjacent ferromagnet (FM) is required as in case\nof traditional spin injection techniques. Experimentally,\nthe existence of proximity exchange splitting via mag-\nnetic insulator in graphene have been demonstrated with\nexchange \felds up to 100 T using the coupling between\ngraphene and EuS23. For yittrium irog garnet/graphene\n(YIG/Gr) based system, using non-local spin transport\nmeasurements, Leutenantsmeyer et al.24demonstrated\nexchange \feld strength of 0.2 T. Another possibility of\ninducing exchange splitting in graphene using FM metal,\nby separating them by alternative 2D material such as\nhexa-boron nitride (hBN), was also proposed theoreti-\ncally21.\nRecent studies have suggested the creation ofgraphene-based devices where EuO-graphene junction\ncan act as a spin \flter and spin valve simultaneously by\ngating the system26. It was also demonstrated27that a\ndouble EuO barrier on top of a graphene strip can exhibit\nnegative di\u000berential resistance making this system a spin\nselective diode. However, the drawback of using EuO\nis its low Curie temperature and the predicted strong\nelectron doping20. It was proposed therefore using high\nCurie temperature materials such as YIG or cobalt fer-\nrite (CFO)28. Indeed, a large change in the resistance\nof a graphene-based spintronic device has been reported\nrecently where the heavy doping induced by YIG could\nbe treated by gating29.\nIn this Letter we demonstrate the existence of Proxim-\nity Magnetoresistance (PMR) e\u000bect in graphene for four\ndi\u000berent magnetic insulators (MI), YIG, CFO, europium\noxide (EuO) and europium sul\fde (EuS). Using ab ini-\ntio parameters reported in Ref. [ 28], we show that for\nYIG and CFO based lateral graphene-based devices with\narmchair edges, PMR values could reach 77% and 22%\nat room temperature (RT), respectively. With chalco-\ngenides, EuS and EuO, PMR values can reach 100% at\n16 K and 70 K, respectively. In addition, we demon-\nstrate the robustness of this e\u000bect with respect to sys-\ntem dimensions and edge type termination. Further-\nmore, our calculations with spin-orbit coupling (SOC)\nincluded does not signi\fcantly a\u000bect the PMR. These\n\fndings will stimulate experimental investigations of the\nproposed phenomenon PMR and development of other\nproximity e\u000bect based spintronic devices.\nII. METHODOLOGY\nIn order to calculate conductances and PMR, we em-\nployed the tight-binding approach with scattering matrix\nformalism conveniently implemented within the KWANT\npackage30. The system modeled is shown in Fig. 1 and\ncomprises two identical proximity induced magnetic re-\ngions of width Wand length Lresulting from insula-\ntors with magnetizations M1andM2, separated by aarXiv:1906.04469v1  [cond-mat.mes-hall]  11 Jun 20192\nFIG. 1. (color online) Lateral spintronic device comprising\ntwo magnetic insulators on top of a graphene sheet. The\nmagnetic graphene regions have a length L, widthWand are\nseparated by a distance d.distancedof nonmagnetic region of graphene sheet with\narmchair edges. Both magnetic graphene regions are sep-\narated from the leads L1andL2by a small pure graphene\nregion. In order to take into account the magnetism aris-\ning in graphene from the proximity e\u000bects induced by\nthe MI's, in the Hamiltonian are used the parameters\nobtained for di\u000berent MI's in Ref. [ 28]. It is important\nto note that the magnetic regions do not a\u000bect the linear\ndispersion of graphene bands, except breaking the valley\nand electron-hole symmetry resulting in spin-dependent\nband splitting and doping. The discretized Hamiltonian\nfor the magnetic graphene regions can be expressed as:\nH=X\ni\u001bX\nltl\u001bcy\n(i+l)1\u001bci0\u001b+h:c:+X\ni\u001b\u001b01X\n\u0016=0[\u000e+ (\u00001)\u0016\u0001\u000e]cy\ni\u0016\u001b[~ m:~ \u001b]ci\u0016\u001b0+X\ni\u001b1X\n\u0016=0[ED+ (\u00001)\u0016\u0001s]cy\ni\u0016\u001bci\u0016\u001b (1)\nwherecy\ni\u0016\u001b(cy\ni\u0016\u001b) creates (annihilates) an electron of type\n\u0016= 0 for A sites and \u0016= 1 for B sites on the unit\ncelliwith spin\u001b=\"(#) for up (down) electrons. ~ m\nand~ \u001brespectively represent a unit vector that points\nin the direction of the magnetization and the vector of\nPauli matrices, so that ~ m:~ \u001b =mx\u001bx+my\u001by+mz\u001bz.\nThe anisotropic hopping tl\u001bconnects unit cells ito their\nnearest neighbor cells i+l. Parameters \u000e, \u0001\u000e, \u0001sare\nde\fned via exchange spin-splittings \u000ee(\u000eh) of the elec-\ntrons (holes) and spin-dependent band gaps \u0001 \u001bde\fned\nin Ref. [ 28]. EDindicates the Dirac cone position with\nrespect to the Fermi level. The Hamiltonian for the whole\ndevice is obtained by making aforementioned parameters\nspatially dependent.\nTo obtain hopping parameters of Hamiltonian (1), we\n\ftted tight-binding bands to those obtained from \frst\nprinciples calculations in Ref. [ 28]. The results of the\n\ftting procedure in case of graphene magnetized by YIG,\nCFO, EuS and EuO are shown in Fig. 2(a), (b), (c) and\n(d), respectively. The corresponding hopping parameters\nare given in Table I. As one can see, the graphene bands\nobtained with tight-binding Hamiltonian given by Eq. 1\nare in good agreement with those obtained using Density\nFunctional Theory (DFT) con\frming suitability of our\nmodel for transport calculations. Of note, due to the\npresence of super\fcial tension at the interface between\nCFO and graphene, hopping parameters in this case are\nanisotropic as they depend on direction to the nearest\nneighbor as speci\fed in the inset of Fig. 2(b).\nThe conductance for parallel and antiparallel con\fgu-\nrations of magnetizations M1andM2in the linear re-\nsponse regime is then obtained according to:\nGP(AP)=e\nhX\n\u001bZ\nT\u001b\nP(AP)\u0012\u0000@f\n@E\u0013\ndE; (2)TABLE I. hopping parameters used in equation 1 for each\nmagnetic insulator considered.\nMaterial Hopping di-\nrectionspin up (eV) spin down (eV)\nYIG t 3.6 3.8\nCFOt1 1.38 1.44\nt2 1:41e\u0000i0:011:48e\u0000i0:01\nt3 1:36e\u0000i0:021:44e\u0000i0:02\nEuS t 4.5 4.8\nEuO t 4.9 4.3\nwhereT\u001b\nP(AP)indicates spin-dependent transmission\nprobability for parallel(antiparallel) magnetizations con-\n\fgurations and f= 1=(e(E\u0000\u0016)=kBT+ 1) represents the\nFermi-Dirac distribution with \u0016andTbeing electro-\nchemical potential (Fermi level) and temperature, respec-\ntively. It is important to note that temperature smearing\nhas been taken into account using the Curie temperature\nof each MI.\nThe PMR amplitude has been de\fned according to fol-\nlowing expression:\nPMR =\u0012GP\u0000GAP\nGP+GAP\u0013\n\u0002100%; (3)\nIn order to determine the impact of the system dimen-\nsions on the PMR, several calculations were carried out\nfor di\u000berent lengths, widths and separations of the mag-\nnetic regions. Furthermore, we checked the robustness of\nPMR on edge type termination by calculating the PMR\nfor systems with zigzag, armchair and rough edges. The\nlatter were created by removing atoms and bounds ran-\ndomly and deleting the dangling atoms at the new edges.3\nFIG. 2. (color online) Band structure obtained using tight-binding Hamiltonian de\fned by Eq. (1) (solid lines) \ftted to the\nband structure from DFT spin majority (green open circles) and spin minority (black \flled circles) data for the cases with (a)\nYIG, (b) CFO, (c) EuS and (d) EuO from Ref. [ 28]. The inset in (b) shows the anisotropic hoppings reported in Table I\n.\nIII. RESULTS\nIn Fig. 3 we present the PMR curves for lateral device\nstructures based on YIG, CFO, EuS and EuO on top of a\ngraphene sheet with armchair edges. Taking into account\nCurie temperatures for these materials, the curves were\nsmeared out using 16 K (70 K) for EuS (EuO), and 300\nK for YIG and CFO cases. For system with YIG we\nfound a maximum PMR value of 77% while for CFO the\nvalue obtained was 22%. In case of chacolgenides EuS\nand EuO used, the maximum PMR values reach 100%.\nAmong the materials studied, YIG represents the most\nsuitable candidate for lateral spintronic applications due\nto both high Curie temperature and considerably large\nPMR value.\nIn order to elucidate the underlying physics behind\nthese PMR results, let us analyze details of the con-\nFIG. 3. (color online) Proximity magnetoresistance de\fned\nby Eq. 3 as a function of energy in respect to the Fermi\nlevel for YIG (blue circles), CFO(red squares), EuS(black dia-\nmonds) and EuO(green triangles) using temperature smeared\nconductances at T=300 K, 300 K, 16 K and 70 K, respec-\ntively. System dimensions are L= 49:2 nm,W= 39:6 nm\nandd= 1:5 nm.ductance behaviour. In Fig. 4(a)-(b) we reproduce the\ngraphene bands in proximity of YIG and corresponding\ntransmission probabilities resolved in spin for P and AP\ncon\fgurations at T= 0 K for a system with dimensions\nL= 49:2 nm,W= 39:6 nm andd= 1:5 nm. One can\nsee that for energies between -0.88 eV and -0.78 eV there\nis no majority spin states present and the only contri-\nbution to transmission T#\nPis from minority spin channel\n(Fig. 4(b), red solid line). In other words, the situation\nwithin this energy range is half-metallic giving rise to\nmaximum PMR values of 100% using \\pessimistic\" def-\ninition given by Eq. (3). The similar situation is for en-\nergy ranges between -0.72 eV and -0.75 eV but this time\nthe only contribution T\"\nPis from majority spin channel\n(Fig. 4(b), red dashed line). One should point out here\nthat the conduction pro\fle here is due combining both\nmagnetic and nonmagnetic regions into one scattering\nregion. The conductance of a pure graphene nanoribbon\nsheet represents quantized steps due to transverse con-\n\fnement with no conductivity at zero energy depending\non its edges. Inducing magnetism within graphene sheet\nleads to symmetry breaking with the shift of exchange\nsplitted gaps in the vicinity of Dirac cone region below\nthe Fermi level. This leads to characteristic conductance\npro\fle with two minima at around -0.8 eV and 0 eV (not\nshown here) due to the Dirac cone regions of magne-\ntized and the pure graphene. The corresponding con-\nductances for the parallel ( GP) and for the antiparallel\n(GAP) magnetic con\fgurations at T= 300 K are shown\nin Fig 4(c). Interestingly, even at room temperature the\nPMR for YIG based structure preserves a very high value\nof about 77% as already pointed above, a behavior that\nis very encouraging for future experiments on PMR. As\na guide to the eye with dashed lines we highlight the en-\nergy value where the PMR has a maximum in Fig. 4.\nSince the edges may strongly in\ruence the aforemen-\ntioned properties of the system, we next explore the\nrobustness of PMR against di\u000berent edge types of the\ngraphene channel of the proposed device. It is well\nknown that electric \feld can trigger half-metallicity in\nzigzag nanoribbons due to the antiferromagnetic inter-\naction of the edges31. On the other hand, graphene4\nFIG. 4. (color online) (a) Band structure reproduced using the DFT parameters from Ref. [ 28] for graphene in proximity of\nYIG. (b) Transmission probabilities for majority (dash lines) and minority (solid) spin channel for parallel (red) and antiparallel\n(blue) magnetization con\fgurations at T= 0 K for a system with dimensions L= 49:2 nm,W= 39:6 nm andd= 1:5 nm.\n(c) Resulting conductance for parallel (red circles) and antiparallel (blue squares) magnetization con\fgurations at 300 K. (d)\nPMR for device with armchair (blue circles), rough (red squares) and zigzag (black triangles) edge termination of graphene.\nPMR pro\fles as a function of (e) L, (f)Wand (g)d. (h) Dependence of PMR for the energy outlined by dashed line in (e),\n(f) and (g) as a function of L(black circles), W(red squares) and d(blue triangles). The green square highlights the region\nwhere PMR becomes independent of system dimensions.\nnanoribbons with armchair edges can display insulating\nor metallic behaviour depending on graphene nanoribbon\n(GNR) width32,33. Armchair and zigzag edges are par-\nticular cases and the most symmetric edge directions in\ngraphene. But one can cut GNR at intermediate angu-\nlar direction between these two limiting cases giving rise\nto an intermediate direction characterized by a chirality\nangle\u001234. Graphene band structure is highly dependent\non\u0012. When the angle is increased, the length of the edge\nstates localized at the Fermi level decrease and eventu-\nally disappear in the limiting case when \u0012= 30\u000e, i. e.\nwhen acquires armchair edge. In the laboratory condi-\ntions, graphene sheets are \fnite and have imperfections\nthat in\ruence their transport properties. For defects at\nthe edges, it has been demonstrated that rough edges\ncan diminish the conductance of a graphene nanoribbon\nas was shown in Ref. [ 35] or may exhibit a nonzero spin\nconductance as reported in Ref. [ 36].\nIn order to demonstrate the robustness of PMR with\nrespect to the edge type, we thus performed calculations\nwith the same system setup (Fig. 1) but this time for\nvarious edge terminations. The resulting PMR behavior\nfor the cases with armchair, rough edges and zigzag are\nshown in Fig. 4(d). The former have been modeled by\ncreating extended vacancies distributed randomly. It is\nclear that the maximum PMR value does not present a\nsigni\fcant variation maintaining for all cases PMR val-\nues around 75%. With this results in hand we can claim\nthat the PMR is indeed robust with respect to edge ter-mination type.\nAs a next step, we checked the dependence of the PMR\non di\u000berent system dimensions, i.e. the length of the\nmagnetic region L, system width Wand the separation\nbetween the magnetic regions d. The corresponding de-\npendences are presented respectively in Fig. 4(e),(f) and\n(g). One can see that for all energy ranges the PMR ratio\nhas a tendency to increase as a function of Lapproaching\nlimiting value of 77% at energies around -0.81 eV indi-\ncated by a dashed line Fig. 4(e). As for dependence of\nthe PMR as a function of GNR width W, clear oscilla-\ntions due to quantum well states formation are present\nwith a tendency to vanish as system widens (Fig. 4(f)).\nOn a contrary, the PMR shows almost constant behav-\nior as a function of separation between the magnets d\n(Fig. 4(g)) due to the fact that transport is in ballistic\nregime. For convenience, we summarize all these depen-\ndencies in Fig. 4(h) at energy -0.81 eV as a function of\nL,Wandd. One can clearly see that the PMR saturates\nas system dimensions are increased. At the same time, it\nshows the oscillations in the PMR for small Was well as\nthe invariance of the PMR with respect to d. For large\ndimensions highlighted by the green box in Fig. 4(h), we\ncan claim that the PMR is indeed robust, and the maxi-\nmum PMR value would be eventually limited only by the\nmagnitude of the spin di\u000busion length in the system.\nFinally, we consider the impact of spin-orbit coupling\non the PMR. Despite weak SOC within graphene, the\nproximity of adjacent materials can induce the interfacial5\nFIG. 5. (color online) PMR dependencies for three values of\nRashba spin-orbit interaction parameter \u0015Rde\fned by Eq. (4)\nfor YIG-based system with armchair edges and of dimensions\nL= 49:2 nm,W= 39:6 nm andd= 1:5 nm. The dashed\nline is a guide to the eye that shows the maximum value when\n\u0015R= 0 eV.\nRashba SOC7. Rashba type SOC is included into our\ntight-binding approach adding the following term:\nHSO=i\u0015RX\ni\u001b\u001b0X\nlcy\n(i+l)1\u001b[\u001bx\n\u001b\u001b0dx\nl\u0000\u001by\n\u001b\u001b0dy\nl]ci0\u001b0+h:c:\n(4)\nwhere the vector ~dl= (dx\nl;dy\nl) connects the two nearest\nneighbours, \u0015Rindicates the SOC strength. The values\nof\u0015Rare generally lie in the range between 1-10 meV\n(see, for instance, in Ref. [ 37]). Keeping in mind this in-\nformation, we present in in Fig. 5 the PMR dependences\nfor three values of spin-orbit interaction. One can see\nthat increasing the strength of SOC \u0015Rlower the PMR.\nThis behavior is expected and could be attributed to thefact that spin-orbit interaction mixes the spin channels.\nThese dependencies allows us to conclude that PMR is\nquite robust also against SOC and even in the worst sce-\nnario remains of the order of 50 % (cf. black triangles\nand blue circles in Fig. 5).\nIV. CONCLUSIONS\nIn this paper we introduced the proximity induced\nmagnetoresistance phenomenon in graphene based lateral\nsystem comprising regions with proximity induced mag-\nnetism by four di\u000berent magnetic insulators. For YIG\nand CFO based devices we found PMR ratios of 77% and\n22% at room temperature, respectively. For chalcogenide\nbased systems, i.e. with EuS and EuO, we found PMR\nvalues of 100% for both at 16 K and 70 K, respectively.\nVery importantly, it is demonstrated that the PMR is\nrobust with respect to system dimensions and edge type\ntermination. 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Ben Youssef3, and B.\nJ. van Wees1\n1Physics of Nanodevices, Zernike Institute for Advanced Materials, University of Groningen, Ni-\njenborgh 4, 9747 AG Groningen, The Netherlands\n2WPI-AIMR &Institute for Materials Research &CSRN, Tohoku University, Sendai 980-8577,\nJapan\n3Lab-STICC, CNRS- UMR 6285, Université de Bretagne Occidentale, 6 Avenue Le Gorgeu,\n29238 Brest Cedex 3, France\n*e-mail: x.wei@rug.nl\n†Current address: Leibniz Institute for Solid State and Materials Research, IFW, 01069 Dresden,\nGermany\nConductivities are key material parameters that govern various types of transport (elec-\ntronic charge, spin, heat etc.) driven by thermodynamic forces. Magnons, the elementary ex-\ncitations of the magnetic order, flow under the gradient of a magnon chemical potential1–3in\nproportion to a magnon (spin) conductivity σm. The magnetic insulator yttrium iron garnet\n(YIG) is the material of choice for efficient magnon spin transport. Here we report an unex-\npected giant σmin record-thin YIG films with thicknesses down to 3.7 nm when the number\n1arXiv:2112.15165v3  [cond-mat.mes-hall]  12 Jan 2022of occupied two-dimensional (2D) subbands is reduced from a large number to a few, which\ncorresponds to a transition from 3D to 2D magnon transport. We extract a 2D spin con-\nductivity (≈1S)at room temperature, comparable to the (electronic) spin conductivity of\nthe high-mobility two-dimensional electron gas in GaAs quantum wells at millikelvin tem-\nperatures4. Such high conductivities offer unique opportunities to develop low-dissipation\nmagnon-based spintronic devices.\nThe spin current density in metals is the difference of the up- and down-spin charge current\ndensities measured in A/m2,which is driven by a gradient of the spin chemical potential (often\ncalled spin accumulation) ∇µs. The spin conductivity σ, defined asjs=j↑−j↓=σs∇µs/e, can\nbe expressed in electrical units as S/m. In a magnetic insulator where charge currents are absent,\neach magnon carries angular momentum /planckover2pi1,which is equivalent to the spin current in metals carried\nby a pair of spin-up ( +/planckover2pi1/2) and spin-down (−/planckover2pi1/2) electrons that flow in opposite directions. A\nmagnon current jmcan be defined as its number current times electron charge e. In magnetic\ninsulator-based spintronic devices, magnon spin currents are injected, detected, and modulated by\nmicrowave striplines or electric contacts made from a heavy metal for charge-spin conversion5–9.\nThe corresponding spin conductivity, magnon conductivity σm, is the current density divided by\nthe gradient of the magnon chemical potential. The unit of the magnon conductivity in jm=\nσm∇µm/e, whereµmis the magnon chemical potential, is then the same as that of electrons in\na metal1. The value of σm= 4×105S/m in a 210 nm thick YIG film at room temperature6\ncorresponds to the electronic conductivity of bad metals.\n2The high magnetic and acoustic quality of magnetic insulators make them the ideal material\nfor all-magnon logical circuits and magnon-based quantum information10, 11. An example of recent\nprogress in magnon-based computing is an integrated magnonic half-adder based on 350 nm wide\nwave guides make from 85-nm-thick YIG films12. However, these devices operate with coherent\nmagnons (∼GHz ) excited by narrow microwave striplines which can not be integrated into an all-\nelectrical circuit. Therefore, it is attractive to inject magnons electrically13, but those are mainly\nthermal (∼Thz) and scatter much stronger at phonons. Also, scalability to smaller structure sizes,\nessential for future high-performance processing units, requires micro and nanofabrication in all\ndimensions. The first step is the growth of films of a few or even a single unit cell. Previously,\nmagnon transport was reported in transistor structures on films down to about 10 nm, which shows\nthat ultrathin films can maintain high quality and display intriguing non-linear magnon effects14, 15.\nHowever, the scattering by surface roughness is expected to be increase in even thinner films16.\nThis could be an obstacle for magnon spin transport in ultrathin YIG films that hinders observation\nof a transition from three dimensional magnons to two dimensional magnon gas when the thermal\nwavelength λthermal = 2π/radicalbig\n/planckover2pi1γD/(kBT)(∼3 nm at room temperature) approaches the thickness\nof the films tYIG, where /planckover2pi1is the reduced Planck constant, γis the gyromagnetic ratio, Dis spin\nwave stiffness and kBis Boltzmann constant.\nHere we report measurements of the magnon conductivity of YIG films with thicknesses\ndown to 3.7 nm. Much to our surprise, the magnon transport turns out to be strongly enhanced\nin the ultrathin regime. We report a drastical increase in magnon conductivity of up to σm=\n1.6×108S/m at room temperature that even exceeds the electronic spin conductivity of high-\n3purity copper. This increase is intimately connected to the small number of occupied subbands and\napparent domination by the lowest subband in our films. These results can importantly boost the\nperformance of magnon-based information technology10, 17.\nWe employ a non-local configuration6(Figure 1a) of two Pt thin film strips with length L\nat a distance don top of YIG films grown on gallium gadolinium garnet (GGG) by liquid phase\nepitaxy. An electric charge current Ithrough the injector generates a transverse spin current due\nto the spin Hall effect18, resulting in a spin accumulation µsin Pt at the interface to YIG. The\ninjector-conversion coefficient ηinj=µs/(eI)depends on the properties and dimensions of the Pt\nstrip as explained in the Section I of the supplementary information (SI Section I). The effective\ninterface spin conductance results from the exchange interaction across the interface and produces\na magnon chemical potential µmon the YIG side of the interface that acts as a magnon source ,\nwhereµm≈µssince the interface spin resistance can be ignored (see SI Section III). The detector\nelectrode is a magnon drain that absorbs magnons and converts them into a spin current jsentering\nthe Pt detector electrode. The inverse spin Hall effect generates a detector voltage Vnlwith detector-\nconversion coefficient ηdet=Vnl/jdet\ns. By reciprocity, ηinj=ηdetwhen injector and detector\ncontacts have the same properties (see SI Section I for details). Since the signal scales with L,\na normalized non-local resistance can be defined as Rnl=Vnl/(IL). The magnon conductance\nfollows from the measured non-local resistance\nGm=1\nηinjηdetVnl\nI=RnlL\nηinjηdet. (1)\nThe magnon conductivity σmas a function of the thickness tYIGof the YIG films in Figure 1 then\n4follows from the magnon spin conductance\nσm=Gmd\ntYIGL. (2)\nFor the films with tYIGmuch smaller than the magnon relaxation length λmas well as the lateral\ndevice dimension, µmcan be considered constant in the zdirection . Therefore, we use following\nequation to describe magnon diffusion6, 19\nRnl=σmtYIGηinjηdet\nλmcschd\nλm→\n\nσmtYIGηinjηdet\nd\n2σmtYIGηinjηdet\nλmexp(−d\nλm)ford/lessmuchλm\nd/greatermuchλm. (3)\nWhen the spacing dis smaller than λm, it is the Ohmic regime in which the magnons are con-\nserved,Rnl(Gm)∼d−1.Otherwise, the signal decays exponentially as a function of distance due\nto magnon relaxation.\nWe measure Rnlat room temperature as a function of an external in-plane magnetic field\nHexwith|Hex|= 50 mT, which we rotate in the plane (Figure 1a). We modulate the AC current\nIby a low frequency (18 Hz) and detect the first/second harmonic signal Vnl(ω)/Vnl(2ω)by lock-\nin amplifiers (see Methods). Vnl(2ω)depends on the spin Seebeck generation and diffusion of\nmagnons under an inhomogeneous temperature profile, which renders interpretation difficult20, 21\n(see SI Section V). Therefore, we focus on Vnl(ω)that follows the formula\nR1ω\nnl(α) =R1ω\nnlcos2α+R1ω\n0, (4)\nwhereR1ω\n0is an offset resistance (see Methods) and αis the angle of Hexwith thex-axis. In\nFigure. 2, the the angle-dependent measurements in various thickness YIG films show that R1ω\nnl\nbecomes four times larger when the film is over fifty times thinner from 210 nm to 3.7 nm. We also\n5observe a strongly increased non-local signal in ultrathin films in Figure 3 as a function of contact\nseparation for a wide range of tYIGincluding results on ultrathin YIG films for 400 nm wide Pt\nstrips and for thicker films tYIG≥210nm6, 22. Figure 4a emphasizes the dramatic enhancement of\nR1ω\nnlfor the thinnest films down to tYIG= 3.7nm and fixed d= 2.5µm, which can be attributed to\nthetYIGdependence of σmbecauseλm>2.5µm for all thicknesses (see SI Section IV for details).\nR1ω\nnlincreases with decreasing thickness and saturates for both the thinnest and thickest films.\nA finite-element model1can simulate the depth ( z) dependence of µmwhentYIG>λm(see\nSI Section I for details). This leads to a limiting σm→3×104S/m in Figure 4b for thicker films,\nwhich represents the bulk value. The simulated Rnlvalues ford= 2.5µm in Figure 4c have been\nfitted toR1ω\nnlin Figure 4a by conductivities that are strongly enhanced in the regime tYIG< λm.\nFortYIG= 3.7nm, the magnon conductivity σm= 1.6×108S/m is four orders of magnitude\nlarger compared to the bulk value, exceeding the electronic conductivity of pure metals such as\ncopper with σe= 6×107S/m23. The observed saturation at tYIG→0appears to reflect an\nincreased role of surface roughness scattering that we do not model explicitly.\nA magnon conductivity that diverges for tYIG→0likeσm∼σ2D\nmt−1\nYIGsimply suggests two-\ndimensional transport. In Figure 4c, it shows that σ2D\nmsaturates for tYIG<10nm, i.e. higher 2D\nsubbands do not contribute significantly even though they are still populated (see below). Extrapo-\nlation to zero thickness leads to σ2D\nm≈1S. This value at room temperature is comparable to that of\nthe high-mobility two-dimensional electron gas at millikelvin temperatures, which is σ2D\ne≈1.4S\nin GaAs quantum wells4.\n6The magnons propagate in the plane with wave vector kand form perpendicular standing\nspin waves (PSSW) in zdirection labeled by an integer n. The exchange interaction scales like\n∼k2and dominates the magnon dispersion εnkat thermal energies ( ≈kBT) with small magne-\ntodipolar corrections. A magnon with energy εnk=/planckover2pi1γD/parenleftbig\nk2+ (nπ/t YIG)2/parenrightbig\ncontributes to the\nconduction proportional to its thermal occupation Nnk= 1/{exp [εnk/(kBT)]−1}. For YIG\nγ/2π= 28 GHz/T and the spin wave stiffness24D= 5×10−17Tm2. The highest occupied\nsubbandndefined as\nn= int/parenleftBigg\ntYIG\nπ/radicalBigg\nkBT\n/planckover2pi1γD/parenrightBigg\n(5)\natεn0<kBTas a function of thickness25, where int(x) is the greatest integer no more than x. For\ntYIG= 3.7nm, only three approximately 2D subbands are occupied at room temperature.\nThe simplest model for the magnon conductivity in ν(=2, 3) dimensions follows from the\nBoltzmann equation with a constant relaxation time τ\nσ(ν)\nm=e2τ(ν)\nkBT/integraldisplaydk\n(2π)ν/parenleftbigg∂εk\n/planckover2pi1∂kz/parenrightbigg2eεk/(kBT)\n(eεk/(kBT)−1)2(6)\nwhereεk=/planckover2pi1γDk2. Magnetic freeze-out experiments show that the contributions from the low-\nfrequency magnons ( ∼GHz ) is significant even at room temperature, presumably reflecting low\nmobilities of thermal exchange magnons26–28. This can be represented by a high momentum cut-\noffK∞∼1/nmat magnon frequencies ε∞//planckover2pi1∼THz . In the high temperature limit kBT/greatermuchεk\nthe conductivities do not depend on γD:\nσ(3)\nm=2e2kBTτ(3)\n3/planckover2pi12π2K∞, (7)\nσ(2)\nm=e2kBTτ(2)\nπ/planckover2pi12logK∞\nK0, (8)\n7whereK0is a low momentum cutoff by the magnon gap of ε0//planckover2pi1∼GHz . By equating these\nequations with the experimental results σ(3)\nm≈3×104S/m and the present σ(2)\nm≈1S and using\nthe scattering times as adjustable parameters, we arrive at τ(3)≈40fs andτ(2)≈0.1ns. The short\nscattering time in three dimensions can be explained by highly efficient magnon-phonon scattering\nat room temperature1. While the high-momentum cut-off plays an important role in 3 dimensions,\nthe near independence of σ(2)\nmemphasizes the importance of the near band-gap excitations for\ntransport in two dimensions. Coherent magnons excited at GHz frequencies can propagate over\ncm’s in spite of their small group velocity because they scatter only weakly at phonons29. Their\ncontribution has a much larger effect on transport in ultrathin films than in the bulk, which is\nconsistent with the magnetic field and temperature dependence reported in the SI. The estimated\nscattering time of τ(2)= 0.1ns may be limited by the film roughness scattering. The precise\nmechanism can be elucidated only by more extensive experimental and theoretical studies of the\ntemperature and field dependence.\nWhile magnon-based devices do not suffer from Joule heating, magnon transport is not\ndissipationless6, 30even for transport on length scales shorter than the magnon relaxation length\nwhere magnons are conserved. The observed giant magnon conductivity is therefore excellent\nnews, implying low dissipation from magnon-phonon scattering even at room temperature. Ultra-\nthin films can therefore be driven with relative ease into the non-linear regime in e.g. magnon spin\ntransistors14, 15, facilitating electrically-induced magnon Bose-Einstein condensation and magnon\nspin superfluidity31–33. The robustness of the transport of the magnetic order for thin films of close\nto the monolayer thickness should allow magnon transport in nanostructures such as constrictions,\n8wires and dots with feature sizes of a few nanometer without loss of magnetic functionality.\nMethods\nFabrication The YIG films are grown on Gd 3Ga5O12(GGG) substrates by liquid-phase epitaxy\n(LPE) at the Université de Bretagne Occidentale in Brest, France, with thicknesses from 3.7 nm\nto 53 nm The effective magnetization Meff(Hk−4πM s) and the magnetic relaxation (intrinsic\ndamping parameter αand extrinsic inhomogenous linewidth ∆Hin) are determined by broadband\nferromagnetic resonance (FMR) in the frequency range 2-40 GHz (see SI Section IV). The device\npatterns are written by three e-beam lithography steps, each followed by a standard deposition\nand lift-off procedure. The first step produces a Ti/Au marker pattern, used to align the subse-\nquent steps. The second step defines the platinum injector and detector strips, as deposited by dc\nsputtering in an Ar+ plasma at an argon pressure with thickness ˜8nm for all devices. The third\nstep defines 5/75 nm Ti/Au leads and bonding pads, deposited by e-beam evaporation. Devices\nhave an injector/detector length L= 30/25µm and the strip widths Ware 400 nm for series A\nand 100 nm for series B. The experimental results in main text are obtained from series A. The\ndistance-dependent non-local resistances for series B can be found in SI Section III.\nMeasurements All measurements were carried out by means of three SR830 lock-in amplifiers\nusing excitation frequency of 18 Hz. The lock-in amplifiers are set up to measure the first and sec-\nond harmonic responses of the sample. Current was sent to the sample using a custom built current\nsource, galvanically isolated from the rest of the measurement equipment. V oltage measurements\n9were made using a custom-built pre-amplifier (gain 103) and amplified further using the lock-in\nsystems. The typical excitation currents applied to the samples are 200 µA (RMS) for series A and\n20µm for series B. The in-plane coercive field of the YIG Bcis below 10 mT for all YIG samples,\nand we apply an external field to orient the magnetization using a physical property measurement\nsystem (PPMS). The samples are mounted on a rotatable sample holder with stepper motor. All\nexperimental data in the main text have been collected at 300 K (room temperature) at an applied\nmagnetic field of 50 mT.\nSimulations Our finite-element model implements the magnon diffusion equation in insulators\nin order to simulate transport of electrically injected magnons. We carried out the simulations\nby COMSOL MULTIPHYSICS (version 5.4) software package with technical details in the SI\nSection I.\nAcknowledgements\nWe acknowledge the helpful discussion with J. Shan and T. Yu. We acknowledge the technical\nsupport from J. G. Holstein, H. de Vries, H. Adema, T. Schouten and A. Joshua. This work is\npart of the research programme \"Skyrmionics\" with project number 170, which is financed by the\nDutch Research Council (NWO). The support by NanoLab NL and the Spinoza Prize awarded in\n2016 to B. J. van Wees by NWO is also gratefully acknowledged. G.B. was supported by JSPS\nKakenhi Grant 19H00645.\n10Author contributions\nB.J.v.W. and X.W. conceived the experiments. X.W. designed and carried out the experiments,\nwith help from O.A.S. J.B.Y . supplied the YIG samples used in the fabrication of devices. X.W.,\nO.A.S., C.H.S.L., G.E.W.B. and B.J.v.W. were involved in the analysis. X.W. wrote the paper with\nO.A.S., G.E.W.B. and B.J.v.W. All authors commented on the manuscript.\nReferences\n1. Cornelissen, L. J., Peters, K. J. H., Bauer, G. E. W., Duine, R. A. & van Wees, B. J. 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Effect of the magnon dispersion\non the longitudinal spin Seebeck effect in yttrium iron garnets. Phys. Rev. B 92, 054436 (2015).\n28. Jamison, J. S. et al. Long lifetime of thermally excited magnons in bulk yttrium iron garnet.\nPhys. Rev. B 100, 134402 (2019).\n29. Streib, S., Vidal-Silva, N., Shen, K. & Bauer, G. E. W. Magnon-phonon interactions in mag-\nnetic insulators. Phys. Rev. B 99, 184442 (2019).\n30. Man, H. et al. Direct observation of magnon-phonon coupling in yttrium iron garnet. Phys.\nRev. B 96, 100406 (2017).\n31. Bender, S. A., Duine, R. A. & Tserkovnyak, Y . Electronic Pumping of Quasiequilibrium\nBose-Einstein-Condensed Magnons. Phys. Rev. Lett. 108, 246601 (2012).\n32. Demokritov, S. O. et al. Bose–Einstein condensation of quasi-equilibrium magnons at room\ntemperature under pumping. Nature 443, 430–433 (2006).\n1433. Divinskiy, B. et al. Evidence for spin current driven Bose-Einstein condensation of magnons.\nNature Communications 12, 6541 (2021).\n15Figure 1: Device layout : a)Schematic representation of the experimental geometry. Two Pt strips\ndeposited on top of YIG serve as magnon detector and injector via the direct and inverse spin\nhall effect. A low-frequency ac current with rms value of Iacthrough the left Pt strip injects\nmagnons. The center-to-center distance of the injector and the detector is dand the length of the\ninjector/detector is L. A spin accumulation µsis formed at Pt|YIG interface due to the SHE when\na charge current passes through the injector and excites magnon non-equilibrium underneath the\ninjector. The diffusive magnons are absorbed at the drain, which induce a spin current density\njs. Using a lock-in technique, the first harmonic voltage is measured simultaneously by the right\nPt strip, i.e. a magnon detector. αis the angle of external magnetic field Hex.b) SEM image\nof the geometry. The parallel vertical lines are the platinum injector and detector, and they are\ncontacted by gold leads. Current and voltage connections are indicated schematically. The scale\nbar represents 2 µm.\n16Figure 2: Dependence of non-local resistance on magnetization direction on ultrathin YIG films,\nfor short center-to-center distances between the injector and the detector. The offset R1ω\n0in Eq.4\nhas been subtracted. This shows R1ω\nnlincreases with decreasing thickness. Comparing with the\nreference from Cornelissen et al.6,R1ω\nnlsignificantly increases in ultrathin films.\n17Figure 3: Non-local resistance as a function of injector-detector distance of the samples of series A\nandtYIGvarying from 3.7 nm to 50000 nm. The width of injector/detector is 400 nm. The results\nfortYIG≥210 nm are adopted from Cornelissen et al.6and Shan et al.22.\n18Figure 4: a)The non-local resistance R1ω\nnlatd= 2.5µm as a function of tYIG. The results for\ntYIG≥210nm are adopted from Cornelissen et al.6and Shan et al.22.b)Thickness dependence\nof the magnon conductivity σmobtained by the best fit for different distances with statistical er-\nror bars. c)Thickness dependence of the 2-dimension spin conductance σ(2)\nmand the non-local\nresistanceRnlfrom the simulation, values based on the best fit for the magnon conductivity. The\nsaturation at tYIG→0indicates that the film approaches the two-dimensional regime in ultrathin\nlimit. The obtained error bar here means the range of the best fitting results for the non-local\nresistance we can get from the FEM simulations (see SI Section II).\n19Giant magnon spin conductivity approaching the two-dimensional transport regime in\nultrathin yttrium iron garnet films\nX-Y. Wei, O. Alves Santos, C.H. Sumba Lusero, G. E. W. Bauer, J. Ben Youssef, and B. J. van Wees\nI. FINITE-ELEMENT MODELLING\nWe measure magnon transport non-locally by monitoring the voltage in a Pt detector as a function of current a Pt\ninjector. The electrical spin accumulation µs(expressed in energy), which is generated by a current Iin the injector\nat the YIG interface reads [S1]\nµs=2eIθPtls\nσetwtanht\n2ls, (S1)\nwheree,t,w,θPt,lsandσsare the electronic charge, the Pt film thickness, width, spin Hall angle, spin relaxation\nlength, and conductivity of the Pt contact. With parameters in Table I the charge current to spin accumulation\nconversion coefficient ηinj=µs/slash.left(eI)=0.05 Ω whenw=400 nm and 0 .21 Ω whenw=100 nm.\nNM1 Injector NM2 Detector\nInterface layer\nYIG\n𝝁𝒔 𝒋𝒛𝒙𝒔xz\ndw w\n𝐻𝑒𝑥 𝑀𝑡\n𝑡𝑖𝑛𝑡\n𝑡𝑌𝐼𝐺a.\n𝜇𝑚𝜇Vb. znm\nxnm\nFIG. S1. a)The sample configuration and its dimensions in the model simulations. Eq. S1 and Eq. S4 describe the spin\naccumulation µsat the interface, used as input of the model, and the voltage build up in the detector, respectively. b)The\nmagnon chemical potential profile for a 100 nm thick YIG film with a charge current I=20µA and 100 nm wide injector/detector\ncontacts and parameters in Table. I. Magnon absorption by the detector electrode causes the dark region close to the detector.\nWe calculate the diffusive magnon spin transport in YIG films numerically by finite-element method (FEM) [S2]\nin the configuration of Figure I a) using the COMSOL MULTIPHYSICS (version 5.4) software package. Since the\nlength of the strips is much longer than the electrode separation, we use a 2D model, where the parameters depend\non x and z only.\nOhm’s Law for magnon current density jmin YIG is\njm=−σm∇µm/slash.lefte. (S2)\nwhere ∇=ˆx∂x+ˆz∂zandσmis the magnon spin conductivity. The local magnon current is proportional to ∇µm. The\ndiffusion equation\n∇2µm=µm\nl2m, (S3)\ngoverns the magnon chemical potential µm,wherelmis the magnon relaxation length. We use the zero-current\nboundary condition (∇⋅n)µm=0 for the bottom as well as top surface that is not covered by Pt, where nis the\nsurface normal.arXiv:2112.15165v3  [cond-mat.mes-hall]  12 Jan 20222\nTABLE I. Parameters used to obtain the results in Figure I b) and Figure S2 b) and c).\nParameter Symbol Value\nMagnon spin conductivity σm 5×105S/m (for 210 nm thickness YIG [S2])\nMagnon spin relaxation length lm 5×10−6m\nEffective spin mixing conductance Geff\ns 2×1012S/m2\nPt spin Hall angle θPt 0.11\nPt conductivity σe 2×106S/m\nPt spin relaxation length ls 1.5×10−9m\nThe spin accumulation in the detector generates an magnon transport driving force proportional to ∂µs/slash.left∂zthat\nleads to a measurable voltage by the integral over the cross-section A=wt\nVnl=θPt\n2A/integral.disp\nA∂µs\n∂zdA. (S4)\nThe non-local resistance Rnl=Vnl/slash.left(IL)(in unit of Ω/m) can be compared with the experimental results. The detector\nefficiency is the same as that of the injector, ηdet=ηinj.\nThe conversion of spin accumulation in Pt into magnons in YIG is governed by an effective spin conductance Geff\ns,\nwhich is a certain fraction, ≈0.06, of the spin mixing conductance [S3, S4]. It can be modelled by a hypothetical\nspacer between Pt and YIG with conductivity σint\ns=Geff\nstint, wheretint=1 nm is the spacer layer thickness.\nFigure I b) shows the magnon chemical potential profile µmin a 100 nm thick YIG under a charge current of\nI=20µA. Injector and detector with w=100 nm/t=8 nm have a center-to-center distance of 300 nm. Table I lists the\nparameters used to produce Figure Ib). The Pt detector strip acts as a spin sink, absorbing magnons and producing\na dark YIG region at the bottom of the detector in Fig. Ib).\nII. YIG FILM THICKNESS DEPENDENCE\nWe measured the non-local resistance in many samples in order to obtain the YIG film thickness and contact-\ndistance dependence. Figure S2 a) shows the experimental results for the non-local resistance for 400 nm wide\ninjector/detector, with center to center distance d=2.5µm, and for different YIG thicknesses tYIG. We distinguish\ntwo regimes: when tYIG>lm,R1ω\nnlsaturates at a small value that does not depend on tYIGanymore. On the other\nhand,R1ω\nnlincreases with decreasing YIG thickness when tYIG<lm, wherelmis the magnon relaxation length.\nFigure S2 b) shows that the non-local resistance calculated with a constant magnon spin conductivity decreases\nwhen the YIG film become thinner . Figure S2 c) shows the results for selected das a function of tYIGthat in contrast\nto experiments, show a completely different thickness dependence.\n024681012141618200.010.11101001000R10nl (R/m)\nDistance (/m)YIG thickness (nm) 5.9      1077 15.8 2321 40 5000 75 10772 210 23208 500  50000\n0.0010.010.11101000100200300400500600R11nl (3/m)\nYIG thickness (nm)Distance: 0.5 3m 0.8  m 1.5 µm 3.0 µm 12.0 µmb.a.c.Experimental dataSimulationSimulation\nFIG. S2. a)YIG thickness dependence of the measured non-local signal R1ω\nnlfor a injector/detector distance of d=2.5µm.b)\nFinite element model results with fixed magnon conductivity σm=5×105S/m obtained previously for tYIG=210 nm [S2]. c)\nR1ω\nnlthickness dependency for different distances, for a fixed value of magnon spin conductivity. This shows an opposite trend\nin comparison with a).\nIn Figure S2 c), R1ω\nnlsaturates above tYIG>lmbecause the magnon current distribution does not change anymore\nwhen increasing tYIG. WhentYIG<lm, on the other hand, the non-local resistance vanishes when tYIG=0, clearly is3\nopposite to the experimental trend. We have to conclude that σmdramatically increases the thin-film limit.\nWe now use σmas a single adjustable parameters that fits the experimental results. The numerical simulations are\nsummarized in Figure S3 a) to j). The values of spin relaxation length used in the simulations, shown in Table II, were\nobtained from the experimental measurements at large distances, where the exponential decay is dominant[S5, S6].\n0.010.11101001000\nR1w\nnl (W/m) Data\n 8.0E3  S/m\n 1.4E4  S/m\n 2.5E4  S/m\n 4.5E4  S/m\n 8.0E4  S/mYIG thickness = 2700 nm\n0 2 4 6 8 10 12 14 16 18 20\n0.010.11101001000\nR1w\nnl (W/m)Distance ( mm)\n Data\n 7.0E7  S/m\n 1.2E8  S/m\n 2.2E8  S/m\n 3.9E8  S/m\n 7.0E8  S/mYIG thickness = 5.9 nm\n0.010.11101001000YIG thickness = 53 nmR1w\nnl (W/m)\n Data\n 2.0E6  S/m\n 3.6E6  S/m\n 6.3E6  S/m\n 1.1E7  S/m\n 2.0E7  S/m\n0.010.11101001000YIG thickness = 210 nm\nR1w\nnl (W/m)\n Data\n 1.0E5  S/m\n 1.8E5  S/m\n 3.2E5  S/m\n 5.6E5  S/m\n 1.0E6  S/m\n0.010.11101001000R1w\nnl (W/m) Data\n 1.0E4  S/m\n 1.8E4  S/m\n 3.2E4  S/m\n 5.6E4  S/m\n 1.0E5  S/mYIG thickness = 1500 nm\n0 2 4 6 8 10 12 14 16 18 200.010.11101001000\nR1w\nnl (W/m) Data\n 5.0E3  S/m\n 7.9E3  S/m\n 1.6E4  S/m\n 3.2E4  S/m\n 5.0E4  S/m\nDistance ( mm)YIG thickness = 50000 nm\n0 2 4 6 8 10 12 14 16 18 200.010.11101001000R1w\nnl (W/m)\nDistance ( mm) Data\n 7.0E3  S/m\n 1.1E4  S/m\n 2.2E4  S/m\n 4.4E4  S/m\n 7.0E4  S/mYIG thickness = 12000 nm\n0 2 4 6 8 10 12 14 16 18 20\n0.010.11101001000Distance ( mm)R1w\nnl (W/m)\n Data\n 5.6E7  S/m\n 9.9E7  S/m\n 1.8E8  S/m\n 3.1E8  S/m\n 5.6E8  S/mYIG thickness = 3.7 nm\n0.010.11101001000\nR1w\nnl (W/m)\n Data\n 8.0E6  S/m\n 1.4E7  S/m\n 2.5E7  S/m\n 4.5E7  S/m\n 8.0E7  S/mYIG thickness = 15.8 nm\n0.010.11101001000\n Data\n 7.0E7  S/m\n 1.2E8  S/m\n 2.2E8  S/m\n 3.9E8  S/m\n 7.0E8  S/mR1w\nnl (W/m)YIG thickness = 7.9 nma. b.\nc. d.\ne. f.\ng. h.\ni. j.\nFIG. S3. Comparison between experimental data (black circles) and the two-dimensional diffusive model obtained for different\nvalues of the magnon spin conductivity σm, (lines brown to green). a)toj)present the distance dependence of the non-local\nresistance for different YIG thickness, from 3.7 nm up to 50 µm, indicated on the top of each figure. Based on that comparison\nwe obtain the value of the magnon spin conductivity presented in Figure 5 of the main text.\nEach black circle in Figure S3 is an independent measurement and susceptible to variations in the individual sample\nparameters. It is clear from Figure S3 that the experiments can be well fitted by a σmthat depends on tYIG.4\nTABLE II. Values of the magnon spin relaxation length adopted in the simulations.\nYIG thickness [nm] Spin relaxation length [ µm]\n5.9 / 7.9 / 15.8 3.5\n53 6.0\n210 9.2\n1500 4.0\n2700 3.5\n12000 / 50000 4.0\nIII. PT /divides.alt0YIG INTERFACE SPIN RESISTANCE\nThe low interface magnon resistance resulting from the large effective spin mixing conductance makes the Pt strips\nact as a good spin source/sink. Figure S4 shows the calculated values of non-local resistance for wide range of effective\nspin conductance, Geff\ns. A significant change in the non-local resistance occurs only when we suppress Geff\nsto below\n2×1012S/slash.leftm2. This value is lower than reported by Kohno et. al., 8 .8×1012S/slash.leftm2[S7] (after local annealing). In\nFigure S4, a higher Geff\nsdoes not significantly change the calculated non-local resistance, which represents that the\ngiant increase of the non-local resistance we observed is not due to the increase of Geff\ns. Figure S5 shows an equivalent\ncircuit model omitting magnon relaxation. The spin current passing through the circuit is dominated by the spin\nconductance of YIG since Rs\nint<Rs\nYIG. Therefore, Rs\nintcan be disregarded and the magnon transport measured by\nRnlis determined by the magnon spin conductivity of YIG rather than the YIG /divides.alt0Pt interface conductance.\n0 2 4 6 8 10 12 14 16 18 2011010010007.9 nm YIG\n1.0E8 S/mNonlocal resistance ( W/m)\nDistance ( mm)Gseff(S/m2):\n 2.0E11\n 4.3E11\n 9.3E11\n 2.0E12\n 4.3E12\n 9.3E12\n 2.0E13\n0 2 4 6 8 10 12 14 16 18 20101001000Nonlocal resistance ( W/m)\nDistance ( mm)Gseff(S/m2):\n 1.0E11\n 2.0E11\n 4.6E11\n 1.0E12\n 2.0E12\n 4.6E12\n 1.0E13210 nm YIG\n4.5E5 S/ma) b)\nFIG. S4. Calculated non-local resistance for a range of values of the effective spin conductance Geff\nsof a 400 nm wide injec-\ntor/detector. a)tYIG=210 nm, and b)tYIG=7.9 nm YIG. In both cases, a significant change in the magnon transport only\noccurs forGeff\ns<2×1012S/slash.leftm2.\nFIG. S5. This equivalent circuit model includes the relevant electronic and magnonic spin resistances and it is valid for short\ndistance (within the magnon relaxation length). µsis the spin accumulation induced by the spin Hall effect. Rs\nPtis the\nspin resistance of the Pt strip (injector/detector). Rs\nintis the interface spin resistance of Pt /divides.alt0YIG.jsis the electronic spin\ncurrent injected into the detector, and Rs\nYIGis the spin resistance of YIG, which is the parameter of interest obtained from the\ntwo-dimensional model.\nIn an additional experiments we placed a third Pt strip between the injector and detector contacts. By absorbing\nmagnons, the middle strip suppresses the non-local signal by a factor of 5 when d=2µm andtYIG=7.9 nm, and by a\nfactor of 2 for d=4µm andtYIG=53 nm. From this, we obtain a best fit with Geff\ns≈2.5×1012S/slash.leftm2fortYIG=7.9 nm,5\nandGeff\ns≈1.0×1012S/slash.leftm2fortYIG=53 nm. We therefore adopted Geff\ns=2×1012S/slash.leftm2as the standard value for the\neffective spin conductance in the other simulations.\nWe confirm the high Geff\nsby comparing results for different widths of the injector/detector contacts. The non-local\nresistance from the Series B devices in Figure S6 with width w=100 nm, is roughly 10 times higher than that of Series\nA devices with w=400 nm, which is mainly due to the higher values of the injector/detector conversion efficiency,\nEq. S1. The simulations show that when Geff\nsis large, magnons are injected/absorbed predominantly in/by only part\nof the interfaces, i.e., the magnon chemical potential in the YIG film covered by the Pt strip decays according to an\nabsorption length Labsorb=/radical.alt1\nσmtYIG/slash.leftGeffsas sketched in Figure S7 b) in terms of the interface chemical potential\nµmand spin current, /uni20D7Js, in the Pt detector. Figure S7 a) shows the calculated resistance for Geff\ns=2×1012S/m2,\nσm=1×108S/m,tYIG=3.7 nm and therefore Labsorb=430 nm that agrees with observations. This confirms our\nvalue forGeff\nseven for the thinnest films.\nFIG. S6. Non-local resistance as a function of injector-detector separation distance for series B devices. The thickness of YIG\nfilms is 3.7 nm and 210 nm. The non-local resistance of 3.7 nm thickness YIG is more than ten times larger than that in 210\nnm thickness YIG at same center-to-center distance. The lines are the simulation results based on a 2D-FEM model. The\nmagnon conductivities σmhere are 1.0 ×108S/m for 3.7 nm, 1.45 ×108S/m for 5.6 nm and 4.5 ×105S/m for 210 nm thickness\nYIG.\n𝜇𝑚𝐽𝑠\n100 150 200 250 300 350 40005001000150020002500300035004000450050005500Nonlocal resistance ( W/m)\nInjector/detector width (nm) tYIG = 3.7 nm\n         d = 1 ma. b.\nmagnons\nFIG. S7. a)Simulation for the non-local resistance with different injector/detector width on 3.7 nm thickness YIG. The\ncenter-to-center distance between the injector and the detector is 1 µm. The simulations in Figure S7 are in agreement with\nthe experimental results in Figure S6. b)Schematic illustration of the decay of the magnon chemical potential decay and spin\ncurrent over the Pt width.6\nIV. MAGNETIC PROPERTIES OF THE YIG FILMS\nTable. III summarizes the magnetic properties of the YIG films are determined by broadband ferromagnetic\nresonance in the range of 2-40 GHz. The magnon spin relaxation lengths obtained by fitting Eq. 3 in the main text,\nare shown in Figure. S8.\nTABLE III. Magnetic properties of the YIG films by FMR characterization.\nYIG thickness Gilbert damping Effective field Inhomogenous linewidth\ntYIG(nm) α(10−4)Heff=Hk−4πMs(Oe) ∆ Hinh(Oe)\n3.7 5 ±2.1 1720 17\n5.6 not available 1900 not available\n5.9 10 ±2.4 1950 43\n7.9 6 .3±4.6 1930 53\n15.8 0 .6±0.5 1960 7.9\n53 1 .0±0.2 1810 1\nFIG. S8. Magnon spin relaxation length and Gilbert damping parameter as a function of YIG film thickness. The thickness of\nYIG films ranges from 3.7 nm to 50000 nm (shown on a logarithmic scale). The results from 210 nm to 50000 nm thickness\nsamples are adopted from Cornelissen et al.[S5] and Shan et al.[S6]. The Gilbert damping parameter of the YIG films is also\npresented here. Note that for ultrathin YIG (around 10 nm), the magnon spin relaxation lengths are similar to each other but\nthe Gilbert damping parameters are quite different.7\nV. THE SECOND HARMONIC NON-LOCAL RESISTANCE\nThe second harmonic signal in the non-local resistance is a measure of the thermal spin generation in the magnetic\nfilm by the spin Seebeck effect. Figures. S9 a) shows results for this R2ω\nnl(α) as a function of the direction of in-plane\nmagnetic field. R2ω\nnlcan be extracted by\nR2ω\nnl(α)=1/slash.left2R2ω\nnlcosα+R2ω\n0, (S5)\nwhereαis the in-plane angle of Hexwith thex-axis.R2ω\n0is an offset signal, possibly by an unintended conventional\nSeebeck voltage in the detector[S8]. We plot the observed amplitude R2ω\nnlas a function of contact separation in\nFigures. S9 b) for tYIG=5.9 nm, 7.9 nm, 15.9 nm (series A with 400 nm wide injector/detector strips). R2ω\nnlin\nultrathin YIG is significantly reduced compared to that in thicker YIG films[S5, S6, S9], because the generation of\nthermal magnons by SSE due to the vertical temperature gradient become less effective. Figures. S9 b) show neither\na simple Ohmic or exponential decay as a function of d, which indicates that the generation and transport of magnons\nis complex and requires detailed modelling of the temperature profile. Just like the magnon conductivity, the spin\nSeebeck coefficient should also depend on tYIG. A reliable extraction of both parameters as a function of thickness\nfrom the second harmonic signals appears impossible at this this time.\nFIG. S9. The second harmonic non-local signals of ultrathin YIG films. a)The angle-dependence for tYIG=7.9 nm and\ncenter-to-center distance of injector and detector d=1.5µm.b)Amplitude of the non-local signals as a function of dfor the\nsamples from series A.\nVI. MAGNETIC FIELD DEPENDENCE OF THE NON-LOCAL SIGNAL IN ULTRATHIN FILMS\nWe measured the non-local transport in the series A devices with tYIG=5.9 nm (Figure. S10) and tYIG=7.9 nm\n(Figure. S11) as a function of the strength of a magnetic field along x(in-plane, normal to the contacts). We use Eq.\n3 from the main text to extract the field-dependence of the magnon conductivity and relaxation length, as shown in\nFigure. S12 and Figure. S13, where we use ηcs=ηsc=0.05 Ω from Section I.\nWhile the magnon spin relaxation length only slightly decreases, the increasing field suppresses the magnon spin\nconductivity stronger for the thinner samples. This is in line with the dominance of the lowest magnon subband in\nthe thinnest samples, which is more susceptible to magnetic freeze-out of the thermal magnon population.8\nFIG. S10. The non-local resistance R1ω\nnlas a function of magnetic field and contact distance of the A series sample with\ntYIG=5.9 nm.\nFIG. S11. The non-local resistance as function of magnetic field for tYIG=7.9 nm.9\nFIG. S12.λmandσmas a function of field in a film with tYIG=5.9 nm. a)The magnon relaxation length λmdecreases slightly\nwith increasing magnetic field. b)The magnetic field strongly suppresses the magnon conductivity σm..\nFIG. S13. a)λmandb)σm, as a function of field in a film with tYIG=7.9 nm.\nVII. TEMPERATURE DEPENDENCE OF THE NON-LOCAL SIGNAL IN ULTRATHIN FILMS\nWe measured the temperature dependence of the non-local signal on series A samples with tYIG=3.7 nm\n(Figure. S14), 5 .9 nm (Figure. S15) and 7 .9 nm (Figure. S16) and a magnetic field of 50 mT. R1ω\nnlfortYIG=3.7 nm at\nlow temperatures is more than five times larger for d=1.3µm than for d=2µm. Since the decrease of Geff\nsdue to the\ntemperature is equivalent for both devices. The large change of the ratio of R1ω\nnlford=1.3µm andd=2µm suggests\nthatR1ω\nnlis dominated by the magnon spin conductivity of YIG and not the Pt /divides.alt0YIG interface spin conductance, which\nsupports the conclusions from Section III.\nResults of the fit by Eq. 3 from the main text are shown in Figure. S17 and Figure. S18. The magnon relaxation\nlength only slightly changes with temperature, in contrast to that σmdecreases strongly with temperature. In thick\nYIG films,Rnl(T)decreases slowly at high temperatures and faster at low temperatures [S10, S11], but the decrease\nat low temperatures is much more pronounced for ultrathin thickness YIG. Also this result is consistent with the\ndominant role of the lowest magnon subband in thermal transport in the thinnest films, since the two-dimensional\nmagnon gas is more susceptible to the freeze out of carriers at low temperatures.10\nFIG. S14. The non-local resistance R1ω\nnlas a function of temperature for different contact distances on a YIG film with\ntYIG=3.7 nm.\nFIG. S15. Non-local resistance as a function of temperature for tYIG=5.9 nm.11\nFIG. S16. Non-local resistance as a function of temperature for tYIG=7.9 nm.\nFIG. S17. a)λmandb)σmas a function of temperature for tYIG=5.9 nm.12\nFIG. S18. a)λmandb)σmas a function of temperature for tYIG=7.9 nm.\nVIII. HIGHEST OCCUPIED EXCHANGE MODE OF YIG AT ROOM TEMPERATURE\nFigure. S19 shows the highest occupied exchange perpendicular standing spin waves (PSSW) 2D subbands nfor\ndifferent thickness YIG films at room temperature. The number of the occupied subbands is n+1. It is reduced from\nover 104to a few when the thickness of the films is down to 3.7 nm.\nFIG. S19. Thickness dependency of the number of occupied PSSW modes defined as the highest occupied subband nat 300 K.\n[S1] Sinova, J., Valenzuela, S. O., Wunderlich, J., Back, C. H. & Jungwirth, T. Spin Hall effects. Rev. Mod. Phys. 87,\n1213–1260 (2015).\n[S2] Cornelissen, L. J., Peters, K. J. H., Bauer, G. E. W., Duine, R. A. & van Wees, B. J. Magnon spin transport driven by\nthe magnon chemical potential in a magnetic insulator. Phys. Rev. B 94, 014412 (2016).\n[S3] Weiler, M. et al. Experimental Test of the Spin Mixing Interface Conductivity Concept. Phys. Rev. Lett. 111, 176601\n(2013).13\n[S4] Qiu, Z. et al. Spin mixing conductance at a well-controlled platinum/yttrium iron garnet interface. Applied Physics\nLetters 103, 092404 (2013).\n[S5] Cornelissen, L. J., Liu, J., Duine, R. A., Ben Youssef, J. & van Wees, B. J. Long-distance transport of magnon spin\ninformation in a magnetic insulator at room temperature. Nature Physics 11, 1022–1026 (2015).\n[S6] Shan, J. et al. Influence of yttrium iron garnet thickness and heater opacity on the nonlocal transport of electrically and\nthermally excited magnons. Phys. Rev. B 94, 174437 (2016).\n[S7] Kohno, R. et al. Enhancement of YIG /divides.alt0Pt spin conductance by local Joule annealing. Applied Physics Letters 118, 032404\n(2021).\n[S8] Sierra, J. F. et al. Thermoelectric spin voltage in graphene. Nature Nanotechnology 13, 107–111 (2018).\n[S9] Shan, J. et al. Criteria for accurate determination of the magnon relaxation length from the nonlocal spin Seebeck effect.\nPhys. Rev. B 96, 184427 (2017).\n[S10] Cornelissen, L. J., Shan, J. & van Wees, B. J. Temperature dependence of the magnon spin diffusion length and magnon\nspin conductivity in the magnetic insulator yttrium iron garnet. Phys. Rev. B 94, 180402 (2016).\n[S11] Gomez-Perez, J. M., V´ elez, S., Hueso, L. E. & Casanova, F. Differences in the magnon diffusion length for electrically\nand thermally driven magnon currents in Y 3Fe5O12.Phys. Rev. B 101, 184420 (2020)."    },    {        "title": "1405.1929v1.Enhancement_of_Spin_Pumping_in___mathrm_Y_3Fe_5O__12__Pt_Ni__81_Fe__19____Trilayer_Film.pdf",        "content": "arXiv:1405.1929v1  [cond-mat.mtrl-sci]  8 May 2014Enhancement of Spin Pumping in Y3Fe5O12/Pt/Ni81Fe19Trilayer\nFilm\nDaichi Hirobe,1,∗Ryo Iguchi,1Kazuya Ando,1,2,3Yuki Shiomi,1and Eiji Saitoh1,4,5,6\n1Institute for Materials Research, Tohoku University, Send ai 980-8577, Japan\n2Department of Applied Physics and Physico-Informatics,\nKeio University, Yokohama 223-8522, Japan\n3PRESTO, Japan Science and Technology Agency, Saitama 332-0 012, Japan\n4WPI Advanced Institute for Materials Research,\nTohoku University, Sendai 980-8577, Japan\n5CREST, Japan Science and Technology Agency, Chiyoda, Tokyo 102-0075, Japan\n6The Advanced Science Research Center,\nJapan Atomic Energy Agency, Tokai 319-1195, Japan\n(Dated: September 12, 2021)\nAbstract\nWe study spin pumping in a Y 3Fe5O12(YIG)/Pt/Ni 81Fe19(Py) trilayer film by means of the\ninverse spin Hall effect (ISHE). When the ferromagnets are not excited simultaneously by a mi-\ncrowave, ISHE-induced voltage is of the opposite sign at eac h ferromagnetic resonance (FMR). The\nopposite sign is consistent with spin pumping of bilayer film s. On the other hand, the voltage is of\nthe same sign at each FMR when both the ferromagnets are excit ed simultaneously. Futhermore,\nthe voltage greatly increases in magnitude. The observed vo ltage is unconventional; neither its\nsign nor magnitude can be expected from spin pumping of bilay er films. Control experiments show\nthat the unconventional voltage is dominantly induced by sp in pumping at the Py/Pt interface.\nInteraction between YIG and Py layers is a possible origin of the unconventional voltage.\n1I. INTRODUCTION\nSpintronics has attracted attention because it is expected to rea lize new magnetic mem-\nories and computing devices by use of electronic spins.1A necessity of spintronics is to\nunderstand the nature of a spin current, a flow of spin angular mom entum. The concept of\na spin current stems from spin-polarized transport in ferromagne tic metals. Ferromagnetic\nmetals conduct a spin-dependent electric current, a charge curr ent accompanied by a spin\ncurrent. This current induces interesting phenomena such as gian t magnetoresistance2and\ncurrent-induced magnetization reversal3–5.\nThe complete separation into spin and charge currents leads to the concept of a purespin\ncurrent, a flow of electronic spins with nocharge currents. Spin pumping6–8is a versatile\nmethod for generating a pure spin current. This method uses magn etization precession in a\nferromagnet to inject a spin current into an adjacent metal.\nFerromagnet (F)/normal metal (NM) bilayer films have offered the mselves as the bench-\nmark for the validity of spin pumping. Experimental and theoretical studies have con-\ncentrated on this structure and contributed to a solid understan ding of spin pumping.\nSpin-current injection enhances magnetization relaxation in F and b roadens ferromagnetic-\nresonance(FMR)linewidth. TheinjectioniselectricallydetectableinN Maswell; spin-orbit\ninteraction in NM can convert an injected spin current into a charge current. This conver-\nsion is known as the inverse spin Hall effect9–14(ISHE) and induces an electric field EISHE\nthat satisfies the relation10\nEISHE∝Js×σ. (1)\nHere,Jsis the spatial direction of a spin current and σis the spin-polarization vector.\nSpin pumping in F/NM/F trilayer films will be of further interest since ne w collective\nexcitation can be induced by magnetic couplings between the ferrom agnets. In fact, such\nexcitation was verified by FMR measurements.7,16This paper presents an experimental\ninvestigation on spin pumping in a Y 3Fe5O12(YIG)/Pt/Ni81Fe19(Py) trilayer film by means\nof ISHE. DC voltage is found to be enhanced when both the ferroma gnets are excited\nsimultaneously. The enhanced voltage is unconventional in the sens e that it cannot be\nexplained byspin-pumping theoryofF/NMbilayer films. Control expe riments demonstrates\nthat the unconventional voltage comes from spin pumping at the Py /Pt interface. We\ndiscuss the origin of the unconventional voltage and narrow down c ontributing factors on\n2an experimental basis.\nII. METHOD\nFigure 1(a) shows a schematic illustration of a YIG /Pt/Py trilayer film. The YIG (Pt)\nlayer was 1 mm wide, 3 mm long and 50 nm (5 nm) thick. The Py layer was 1 m m wide,\n2 mm long and 10 nm thick. We grew the YIG layer on a Gd 3Ga5O12(GGG) substrate by\na metal organic decomposition method15; we then sputtered the Pt layer on the YIG layer\nand finally deposited the Py layer by the electron-beam evaporation in a high vacuum. We\nattached two electrodes to the ends of the Pt layer to measure DC voltage.\nThe film was placed at the center of a TE 011microwave cavity at which a microwave\nmagnetic field (electric field) maximizes (minimizes) itself. A frequency of the microwave\nwas 9.44 GHz. We applied a static magnetic field perpendicular to the mic rowave magnetic\nfield. Figure 1(b) defines the out-of-plane magnetic-field angle θHand the magnetization\nangleθM,YIG(θM,Py) of YIG (Py). Spin pumping injects a spin current into the Pt layer,\nleaving the spin-polarization vector σparallel to the precession axis on temporal average\n[see Fig. 1(c)].10ISHE in the Pt layer converts the injected spin current into DC volta ge.\nDC voltage and an FMR spectrum were measured at room temperatu re with changing\nstatic-field magnitude or direction.\nIII. RESULTS AND DISCUSSION\nFigure 2(a) shows spectra of the microwave absorption signal at θH=±90 deg. (i.e.,\nwith in-plane fields). We see FMR of Py (YIG) near µ0H= 120 (300) mT. Correlating\nwith FMR, DC voltage appears as shown in Fig. 2(b). Remember the dir ection of relevant\nvectors: Js∝ba∇dblˆy,σ∝ba∇dblˆz, andEISHE∝ba∇dblˆx[see Fig. 1(c)]. Here, ˆx,ˆy, andˆzrepresent unit\nvectors in each direction. The voltage spectra are consistent with Eq. (1). At θH= 90\ndeg., voltage is negative at FMR of Py while it is positive at FMR of YIG. Th is opposite\npolarity is due to the opposite directions of spin-current injection ( Js∝ba∇dbl ±ˆy,σ∝ba∇dblˆz, and thus\nEISHE∝ba∇dbl ±ˆx). Moreover, when the field direction is reversed, the voltage sign r everses at each\nFMR. This reversal is due to the opposite directions of spin polarizat ion (Js∝ba∇dblˆy,σ∝ba∇dbl ±ˆz,\nand thus EISHE∝ba∇dbl ±ˆx). These results all support that the voltage is dominated by ISHE.\n3The replacement of Pt with Cu reveals irrelevance of galvanomagnet ic effects such as the\nanomalous Hall effect and the planar Hall effect18. The replacement greatly reduces voltage.\nAs a result, the voltage with Cu cannot reproduce the voltage with P t, even considering a\nconductance difference between Pt and Cu. This reduction is consis tent with the fact that\nISHE is negligibly small in Cu owing to weak spin-orbit interaction.\nFigure 2(c) shows the θHdependence of a resonance field HFMR. The result is consistent\nwith previous studies of bilayer films.17In terms of HFMR, we observed little difference\nbetween the trilayer film and its corresponding YIG/Pt and Py/Pt bila yer films. Different\nsaturation magnetization between YIG and Py results in the differen tθHdependence of\nHFMRvia a demagnetization field along the y-axis [see Fig. 1(b)].17This enables us to\ncontrol magnetization dynamics in each layer. At θH=±90 deg, resonance fields are\ndifferent between YIG and Py. Thus, while one magnetic layer is on res onance with large\nprecessional amplitude, theother isoff resonance withsmall prece ssional amplitude. Around\nθH=±20 deg., on the other hand, resonance fields are nearly identical: ma gnetization\nprecesses simultaneously in the two ferromagnets.\nFigure 3(a) shows the magnetic-field dependence of voltage at var ious values of θH. We\nfirst confine ourselves to voltage at FMR of Py. Voltage monotonica lly decreases in mag-\nnitude as θHapproaches zero and changes its sign across θH= 0 deg.. This behavior is\nconsistent with Eq. (1) because EISHE∝Js×σ∝sinθM,Pyholds [see Fig. 1(b)]. To fit the\nvoltage spectra near FMR of Py, we used a sum of symmetric and asy mmetric Lorentzian\nfunctions10\nV(H) =VSYM(∆H/2)2\n(H−HFMR)2+(∆H/2)2+VASYM2∆H/2(H−HFMR)\n(H−HFMR)2+(∆H/2)2.(2)\nHere, ∆His the full width at half maximum and VSYM(VASYM) is the magnitude of a\nsymmetric (asymmetric) component of voltage. VSYMcorresponds to ISHE and VASYMto\nthe anomalous Hall effect in this study. Fig. 3(b) bottom shows that theθHdependence of\nVSYMisroughlyrepresented byasinecurveandrevealsnodifferencebet weentheYIG/Pt/Py\ntrilayer film and a Py/Pt bilayer film.\nWe next focus on voltage at FMR of YIG for θH>0 deg. Fig. 3(a) shows that as θH\ndecreases from90deg., voltagemagnitude decreases andalmost v anishes at 30 deg.. Notable\nis that voltage of the opposite sign reappears at 20 deg.. This rever sal of sign cannot be\n4explained by spin pumping in a YIG/Pt bilayer film. Application of Eq. (1) t o the bilayer\nfilm yields voltage of a constant sign for θH>0 deg. because EISHE∝Js×σ∝sinθM,YIG\nholds. An unconventional feature is not limited to the sign of voltage : the upper inset in\nFig. 3(a) clearly shows enhancement of voltage near θH= 18 deg, where the resonance\nfields of YIG and Py are equal. Anomalies in sign and magnitude are summ arized in Fig.\n3(b) and (c). At this stage, we cannot conclude that the unconve ntional voltage originates\nin spin pumping at the YIG/Pt interface. We also need to consider spin pumping at the\nPy/Pt interface; YIG and Py can affect mutual dynamics, or spin pu mping via magnetic\ninteraction.\nWe note that the observed unconventional voltage cannot be att ributed to heating effects\ndue to microwave absorption. The microwave absorption at FMR hea ts the magnetic layer\nand may cause a heat current flowing from the magnetic layer to the Pt layer. Eventually\nDC voltage is induced by thermoelectric effects such as the ordinary Nernst effect, the\nanomalous Nernst effect and the longitudinal spin Seebeck effect19(LSSE). The DC voltage\nis proportional to H×∇T(ordinary Nernst effect) and M×∇T(anomalous Nernst effect\nand LSSE), where Tdenotes temperature. The magnitude of ∇Tis proportional to the\nmicrowave absorption at FMR IFMR; thus the DC voltage is proportional to |H|IFMRsinθH\n(ordinary Nernst effect) and |M|IFMRsinθM(anomalous Nernst effect and LSSE). Since\nIFMRdecreases as θHapproaches zero, the heating effects cannot reproduce the obs ervedθH\ndependence of voltage.\nVoltage anomalies appearing when HFMR,YIG=HFMR,Pysuggest that magnetization dy-\nnamics, or spin pumping is affected by a magnetic coupling between the ferromagnets. One\nof the possibilities is a dynamic exchange coupling16, a coupling via spin currents mediated\nby conduction electrons in an intermediate NM layer. This coupling per sists on the spin-\ndiffusion length scale in the NM layer and may exist in the YIG/Pt/Py trila yer film, because\nPt has a spin diffusion length of around 10 nm. To examine a dynamic exc hange coupling,\nwe prepared a YIG/SiO 2(20 nm thick)/Pt (5 nm thick)/Py (10 nm thick) quadrilayer film.\nAlthough a spin current is interrupted across the YIG/Pt interfac e, voltage anomalies per-\nsist near θH=±20 deg.. Moreover, the quadrilayer film reveals the similar θHdependence\nofVSYMat FMR of YIG as observed in the trilayer film [see Fig. 4(a)]. The persis tence and\nthe angular dependence both suggest that a dynamic exchange co upling is irrelevant to the\ntrilayer film.\n5The voltage anomalies in the YIG/SiO 2/Pt/Py quadrilayer film suggest that unconven-\ntional voltage originates in spin pumping at the Py/Pt interface. To r einforce this argument,\nwe prepared a YIG/Pt (5 nm thick)/SiO 2(20 nm thick)/Py (10 nm thick) quadrilayer film,\nwhere the SiO 2layer suppresses a dynamic exchange coupling by interrupting a spin cur-\nrent across the Py/Pt interface. The unconventional voltage is f ound to disappear [see Fig.\n4(b)]. This result further proves that unconventional voltage or iginates in spin pumping at\nthe Py/Pt interface.\nWe need to explain at least the following four features of the unconv entional voltage:\n(feature 1) the unconventional voltage persists even when the f erromagnets are separated by\nupto about20 nm; (feature 2)thevoltage originatesinspin pumping atthePy/Pt interface;\n(feature 3) the voltage appears when both the ferromagnets ar e excited simultaneously;\n(feature 4) the voltage takes extrema at θHwhere YIG and Py are equal in HFMR.\nThe long-range nature (feature 1) suggests that a dipolar couplin g may be relevant to the\nunconventional voltage. As shown below, however, this coupling alo ne is unlikely to yield\nour results consistently. Magnetization dynamics in the YIG layer yie lds a time-dependent\ndipole field20,21. A sum of the time-dependent dipole field and a microwave field act on\nmagnetization in the Py layer. The increased driving field may enhance the magnetization\ndynamics in the Py layer. Note that YIG and Py are greatly different in volume, the\nsaturation magnetization, and the damping constant. These differ ences may enhance spin\npumping of Py while having little influence on that of YIG. On the other h and, a static\ndipole coupling should lead to a resonance-field shift in the YIG/Pt/Py trilayer film. This\nshift will become prominent especially when YIG and Py are excited simu ltaneously.22At\nleast we did not observe resonance-field shifts greater than an er ror range of 10 mT. It is\nunreasonable to infer that a dynamic dipole field enhances magnetiza tion dynamics while a\nstatic counterpart has little influence on resonance fields. The unc onventional voltage will\nneed contributing factors other than a dipolar coupling.\nIV. CONCLUSION\nWe investigated spin pumping in a YIG/Pt/Py trilayer film by means of IS HE. Un-\nconventional voltage appears when both the ferromagnets are e xcited simultaneously by a\nmicrowave. The voltage is attributed to spin pumping at the Py/Pt int erface. The voltage\n6persists when a dynamic exchange coupling is suppressed. Several features are pointed out\nto characterize the unconventional voltage. The voltage persist s with the ferromagnets sep-\narated by up to 20 nm (long-range feature). The voltage magnitud e increases when YIG\nand Py are equal in HFMR(dynamic feature). Considering these features, a time-depende nt\ndipolar coupling was proposed to explain the origin of the unconventio nal voltage. However,\nour experiments show that a static dipolar coupling does not shift re sonance fields. This\nresult appears inconsistent with the proposed mechanism. The orig in of the unconventional\nvoltage requires further elucidation.\nACKNOWLEDGEMENTS\nThis work was supported by CREST-JST ”Creaction of Nanosystem s with Novel Func-\ntions through Process Integration”, a Grant-in-Aid for Scientific Research (A) (24244051)\nfrom MEXT, Japan, The Murata Science Foundation, The Funding Pr ogram for Next Gen-\neration World-Leading Researchers, The Asahi Glass Foundation, The Noguchi Institute,\nand The Mitsubishi Foundation.\n7V\nHmicrowave\nPy \nPt \nYIG\nGGG\nPy \nYIGPt θM, Py \nθM, YIGθH\nH\nσPy Js,Py \nEISHE,Py EISHE Py \nPt \nYIGMPy \nMYIGy\nx\nzy\nz\nσYIGJs,YIG EISHE,YIG σσYIGYIGJISHE ,YIG ,YIG MPy \nMYIG(a)\n(b)\n(c)\nVV+V\n㸫V\nFIG. 1: (a) A schematic illustration of the YIG/Pt/Py trilay er film.His a static magnetic field.\n(b) Definitions of the magnetic-field angle θHand the magnetization angle θM.His a static\nmagnetic field and Mis magnetization. (c) A schematic illustration of spin pump ing at the Py/Pt\n(left) and YIG/Pt interfaces (right). Mdenotes magnetization; Jsthe spatial direction of a spin\ncurrent; σthe polarization vector; EISHEan ISHE-induced electric field.\n8θH = -90 deg.\nθH = 90 deg.VH\nVH(a) (c)\n1200\n1000\n800\n600\n400\n200\n0 \n0\n  MM\n YIG, YIG/Pt/Py\n YIG, YIG/Pt\n  Py, YIG/Pt/Py\n Py, Py/Pt\nθH  (deg.)(mT) FMRHµ0\n90 -90 \n-30-20-1001020\nV  (µV)\n350 300 250 200 150 100\nµ0H (mT)dI/dH  (a.u.)\n-2-1012\nV  (µV)\n310 300 290 280\nµ0H (mT) 90°\n -90°  = -90°, YIG/Pt/Py\n   = 90°, YIG/Pt/Py   = 90°, YIG/Cu/PyθH\nθH\nθH\n00\nT)µ0HFMR, Py\nµ0HFMR, YIG\n(b)\nFIG. 2: (a) The in-plane field Hdependence of the microwave abosrption signal d I/dH.HFMRis\na resonance field. (b) The in-plane field Hdependence of DC voltage Vfor the YIG/Pt/Py trilayer\nfilm (red and blue circles) and the YIG/Cu/Py trilayer film (bl ack circles). The inset shows the\nHdependence of Vnear FMR of YIG. (c) The field-angle θHdependence of the resonance field\nHFMR. The filled circles represent data for the YIG/Pt/Py trilaye r film; the open triangles for the\nYIG/Pt and Py/Pt bilayer films. Data are blue for YIG and red fo r Py.\n9V䢢䢢(µV)\n1200 1000800 600 400 200\nµ䢲H䢢(mT)V䢢䢢(µV)\n-200-1000100200\nµ䢲(H-H䢢FMR,  YIG䢢) (mT) 24°\n 22°\n 20°\n 19°\n 18°\n 16°\n 14° 20 ( µV)\n-100 -50 0 50 100 \nµ䢲(H-H FMR, YIG 䢢) (mT)  䢢= 19° 䢢20䢢(µV)䢢20䢢(µV)θΗ   = 90 °\nθΗ   = 70 °\nθΗ  䢢= 50°\nθΗ   = 30 °\nθΗ   = 20 °\nθΗ  䢢= 10°\nθΗ   = 0°\nθΗ   = -10 °\nθΗ  䢢= -20 °\nθΗ  䢢= -30 °\nθΗ  䢢= -50 °\nθΗ   = -70 °\nθΗ   = -90 ° θHV䢢䢢(µV)(a)\n(d)θH\nθH\nθH\nθH\nθH\nθH\nθH\nθH\nθH\nθH\nθH\nθH\nθHFMR of Py FMR of YIG\n-1.001.0\n-50 0 50800\n400\n0µ0H FMR (mT)\n-4-2024\nVSYM/|VSYM, -90 deg.| YIG/Pt/Py\nat FMR of YIG YIG/Pt\nat FMR of Py YIG/Pt/Py Py/PtțH FMR = H FMR, Py  - H FMR, YIG\nVSYM/|VSYM, -90 deg.|\n θH(deg.)ț(b)\n(c)\nFIG. 3: (a) The field Hdependence of DC voltage Vfor the YIG/Pt/Py trilayer film at magnetic-\nfield angles θHbetween −90 and 90 deg. The blue (red) dashed line represents the reson ance field\nof YIG (Py). Orange (black) circles indicate unconventiona l (conventional) voltage at FMR of\nYIG. The upper inset shows the Hdependence of Vnear FMR of YIG. The lower inset shows\nexperimental data (open circles) and fitting (dashed lines) atθH= 19 deg. Red and blue dashed\nlines are a symmetric Lorenzian function and the green dashe d line is an asymmetric Lorenzian\nfunction. (b) The magnetic-field angle θHdependence of the resonance-field difference δHFMR.\nδHFMRis defined as δHFMR=HFMR,Py−HFMR,YIG. Orange circles indicate where YIG and\nPy are equal in HFMR. (c) The magnetic-field angle θHdependence of the symmetric component\nof DC voltage VSYMat FMR of YIG. VSYM/|VSYM,−90deg|is the normalized DC voltage. Circles\nrepresentdatafortheYIG/Pt/Pytrilayer film; triangles fo rtheYIG/Ptbilayer film. Orangecircles\nindicate where YIG and Py are equal in HFMR. (d) The magnetic-field angle θHdependence of the\nsymmetric component of DC voltage VSYMat FMR of Py. VSYM/|VSYM,−90deg|is the normalized\nDC voltage. Circles represent data for the YIG/Pt/Py trilay er film; triangles for the YIG/Pt\nbilayer film.\n10-0.4-0.200.20.4\nV  (µV)\n-20 -10 0 10 20\nµ0(H-H FMR, YIG  ) (mT)-10-50510\nV  (µV) at FMR of YIG\nat FMR of Py VSYM/VSYM, MAX \nVSYM/VSYM, MAX (a)\n(b)\n-1-0.5  00.51\n-50 0 50-1-0.5 00.511.5\n YIG/Pt/Py (reference)  YIG/SiO 2/Pt/Py\n YIG/Pt/SiO 2/Py\n YIG/Pt/SiO 2/Py YIG/SiO 2/Pt/Py\n YIG/Pt/Py (reference)  YIG/SiO 2/Pt/Py  YIG /Pt/Py  YIG/Pt/ /Py SiO 2\nJs\n \n YIG /Pt/Py YIG/Pt/ /Py SiO 2\n YIG/SiO 2/Pt/Py+\n+-\n-\n  θH(deg.)Py \nPt \nYIGSiO 2Py Py \nPt \nYIG YIGPt SiO 2Js\nJsJs\n = H FMR, Py H FMR, YIG\nFIG. 4: (a) The magnetic-field angle θHdependence of the symmetric component of DC voltage\nVSYMat FMR of YIG. The black circles represent data for the YIG/Si O2/Pt/Py quadrilayer film;\nthe dark brown circles for the YIG/Pt/SiO 2/Py quadrilayer films; the sky blue circles for the\nYIG/Pt/Py trilayer film. The inset shows the magnetic field Hdependence of VSYMnear FMR of\nYIG.θHis such that YIG and Py are equal in the resonance field HFMR. The red (blue) arrow\nstresses a positive (negative) sign of VSYM. (b) The magnetic-field angle θHdependence of the\nsymmetric component of DC voltage VSYMat FMR of Py. The black circles represent data for the\nYIG/SiO 2/Pt/Py quadrilayer film; the dark brown circles for the YIG/P t/SiO2/Py quadrilayer\nfilms; the light pink circles for the YIG/Pt/Py trilayer film.\n11∗Electronic address: daichi.kinken@imr.tohoku.ac.jp\n1Zutic, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004).\n2A. Fert, Rev. Mod. Phys. 80, 1517 (2008).\n3J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n4L. Berger, Phys. Rev. B 54, 9353 (1996).\n5S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J . Schoelkopf, R. A. Buhrman,\nand D. C. Ralph, Nature 425, 380 (2003).\n6S. Mizukami, Y. Ando, and T. Miyazaki, Phys. Rev. B 66, 104413 (2002).\n7Y. 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Woltersdorf, A. Brataas, R. Urban, and G. E. W. Bauer, Phys.\nRev. Lett. 90, 187601 (2003).\n17K. Ando, S. Takahashi, J. Ieda, Y. Kajiwara, H. Nakayama, T. Y oshino, K. Harii, Y, Fujikawa,\nM. Matsuo, S. Maekawa, and E. Saitoh, J. Appl. Phys. 109, 103913 (2011).\n18L. Chen, F. Matsukura, and H. Ohno, Nat. Commun. 4, 2055 (2013)\n19K. Uchida, H. Adachi, T. Ota, H. Nakayama, S. Maekawa, and E. S aitoh, Appl. Phys. Lett. 97,\n12172505 (2010).\n20O. Dmytriiev, T. Meitzler, E. Bankowski, A. Slavin, and V. Ti berkevich, J. Phys. Condens.\nMatter22, 136001 (2010).\n21S. Schafer, N. Pachauri, C. K. A. Mewes, T. Mewes, C. Kaiser, Q . Leng, and M. Pakala, Appl.\nPhys. Lett. 100, 032402 (2012).\n22B. Heinrich, Z. Celinski, J. F. Cochran, W. B. Muir, J. Rudd, Q . M. Zhong, A. S. Arrott, K.\nMyrtle, and J. Kirschner, Phys. Rev. Lett. 64, 673 (1990)\n13"    },    {        "title": "1909.00085v1.Amplitude_and_Phase_Noise_of_Magnons.pdf",        "content": "Amplitude and Phase Noise of Magnons, UC Riverside (2019)  \nAmplitude and Phase Noise of Magnons  \nSergey Rumyantsev1,2, Michael Balinskiy1, Fariborz Kargar1, Alexander Khitun1 and \nAlexander A. Balandin1 \n1Nano -Device Laboratory (NDL) , Department of Electrical and Computer Engineering, University of \nCalifornia, Riverside, California, USA  \n2Center for Terahertz Research and Applications (CENTERA), Institute of High Press ure Physics, Polish \nAcademy of Sciences, Warsaw, Poland  \n \nAbstract — The low -frequency amplitude and phase noise \nspectra of magnetization waves , i.e. magnons , was \nmeasured in the y ttrium iron garnet (YIG) waveguide s. \nThis type of noise , which originates from the fluctuations \nof the physical properties of the YIG  crystal s, has to be \ntaken into account in the design of YIG -based RF \ngenerators and magnonic devices for data processing, \nsensing and imaging applications. It was found that the \namplitude noise level of magnons depends strongly on the \npower level, increas ing sharply at the on -set of nonlinear \ndissipation. The noise spectra of both the amplitude and \nphase noise have the Lorentzian shape with the \ncharacteristic frequencies below 100 Hz.  \nKeywords  — magnon, magnetization wave, phase noise, \namplitude noise, ran dom telegraph signal  noise,  RTS \nI. INTRODUCTION  \nThe majority of devices for in formation processing and  \nsensing applications are based on the charge transfer in \ndifferent media, e.g. semiconductors, metals, or vacuum . \nRecently , a completely different approach –  termed magnonics \n– received significant attention . It is based on manipulation  of \nthe spin currents carried by the magnetization waves –  \nmagnons  – in electrical insulators  [1-7]. Spin current s in \ninsulators avoid Ohmic losses  and, therefore,  Joule heatin g. A \nnumber of new devices based on magnon propagation  have \nalready been proposed and demonstrated for data processing, \nsensing and imaging applications  [8-10]. The operation \nfrequency of magnonic devices  ranges from the low GHz to \nTHz frequenc ies. The key  material for th ese devices is yttrium \niron garnet (YIG). Despite the strong interest to magnonic \ndevices, their low -frequency noise characteristics remained \nlargely unexplored  [11]. \n YIG has also been used for filters and resonators, operating \nat frequenc ies up to 26 GHz [12-15]. Noise properties of RF \ngenerators based on YIG spheres  and delay lines were studied  \nin details [16-20]. The noise of a generator depends on many \nfactors , and the generator scheme may include several noise \nsources. The YIG crystal is only one of them.  However, to the \nbest of our knowledge , the noise properties YIG material have not been rigorously addressed yet. We have recently reported \non the low -frequency amplitude noise in YIG waveguides [ 11]. \nThe low -frequency noise is a ubiqui tous phenomenon, present \nin all kinds of electronic materials and devices [ 21-33]. The \nlow-frequency noise in magnonic devices is also an important \nmetric, which deserves much more attention. In this paper , we \nreport the results of the measurements of the low -frequency  \nnoise of magnons in a YIG waveguide, focusing on the phase  \nnoise.   \nII. EXPERIMENTAL DETAILS  \nThe YIG -film of 9.6 μm thickness and 1.5  mm × 13. 5 mm in \ndimensions was grown on the g adolinium gallium garnet  \n(GGG,  Gd 3Ga5O12) substrate by  the liquid phase epitaxy. The \nTi/Au antenna e for spin wave (SW) e xcitation and detection \nwere fabricated on the sur face of YIG -film waveguide  (see \ninset in Figure 1) . The devices were placed in a magnetic field \ncreated by the permanent neodymium magnet. Depending on \nthe orientation of the magnetic field , the spin waveguide \nstructure supports either the magneto -static s urface spin waves \n(MSSW s) or backward volume magneto -static spin wave s \n(BVMS Ws). The M SSW can propagate either on the top \nsurface of the YIG waveguide (surface waves)  or at the \ninterface between the YIG waveguide and GGG substrate \n(interface waves) . The st rength of the magnetic field \ncorresponded to the f erromagnetic resonance (FMR) \nfrequency of about 5 GHz. Antenna 1 or 3 were used to excite \nspin waves and antenna 2 was used as a receiver (see Fig ure \n1).   \nIn order to confirm the generation and propagation  of \nmagnon current through the electrically insulating waveguide \nwe measured the S -parameters of the waveguide as a function \nof frequency and magnetic field. The measured S –  parameters \nwere compared with known dispersion laws for BVMSW and \nMSSW. Good agre ement with the theory confirmed the type \nof propagating magnons, and allowed for tuning the  fp-H space \nparameters for the magnon noise studies. These data also \nconfirmed that the signal is not a result of direct \nelectromagnetic coupling between antennas.  Amplitude and Phase Noise of Magnons, UC Riverside (2019)  \n \nPropagating in the waveguide, magnon current acquires \nvariations in the amplitude and phase due to the fluctuations of \nthe physical p roperties of the YIG thin film.  In order to \nmeasure these fluctuations, the commercial Schottky diode \n(33330B Keysight Techn ologies Inc. ) was connected to the \nantenna 2 (see Fig ure 1). The DC detected signal from the \ndiode was amplified by the low noise amplifier and analyzed \nby the FFT spectrum analyzer.  \nIII. RESULTS AND DISCUSSIO NS \nFourier transform of the signal from the diode y ields the \nspectrum of the amplitude fluctuations . The amplitude noise \nof the magnons propagating along the interface between YIG \nwaveguide and GGG substrate was the highest, and the lowest \nwas the noise of the volume magnons. The high noise level of \nthe interface magnons was explained by the YIG/GGG \ninterface roughness, resulting in stronger fluctuations of the \nmaterial parameters that govern magnon current propagation.  \nThe dependence of noise on power for interface and surfaces \nmagnons reveals one or more maxima. The positions of the \nnoise peaks correspond to the change of the slope of S 21 \ndependence on the power.  \nThe spectra of the amplitude fluctuations had the shape of \nthe Lorentzian,  SV~1/(1+f2/fc2) with the characteristic corner \nfrequency fc<100- 1000H z. In the time domain, the magnon \nnoise revealed itself as a random telegraph signal (RTS) noise. \nVery small changes in the input power of ~0.1dB  led to the \nsignificant changes in the RTS noise and it spectrum. RTS \nnoise is well known in electronic devices  and charge density \nmaterials and devices [ 34-38]. Observation of RTS noise in \nthe large magnon waveguides can be explained by the \nindividual discrete macro events which contribute to both the \nnoise and magnon dissipation process es [11].  \nFluctuations of the speed of the magnon wave lead to the \nfluctuations of the phase of the wave. In order to measure the \nphase noise, the delay line itself with the three antennas , as \nshown in Fig ure 1, was used as a phase detector. For this \npurpose , the output of the exter nal generator was split and fed \nto antennas 1 and 3, correspondingly (see Fig ure 1). The line \nfor the antenna 3 included an attenuator and a phase shifter. \nUsing the attenuator , the power on the output antenna 2 was \nadjusted to be equal when powered separa tely in each of two \nconfigurations. The r esulting power on antenna 3 was equal to \n~2 dBm. These two signals are merged at the plane, which \ncorresponds to the antenna 3 (see Fig ure 1). Due to the \ninterference, when both antennas 1 and 3 are powered, the \npower on  the output is a function of the phase difference of \nthese two signals. This phase difference is defined by the \ndistance between antennas 2 and 3 and by the external phase \nshifter. Fluctuations of the magnon wave speed contribute to \nthe phase differen ce, and are converted to the voltage \nfluctuations detected at the antenna 2. The s ymbols in Fig ure 2 \nshow the DC voltage on the detector, which is proportional to \nthe output power at the antenna 2 as a function of the phase \nshift adjusted by the external p hase shifter. The s olid line is a \nfit with A×cos2(Ψ/2) function ( A is a fitting parameter). The derivative, R, of this function is a conversion coefficient , \nwhich determines the spectral noise density of the voltage \nfluctuations: Sv=SΨ×R2 (SΨ is the spectral noise density of the \nphase fluctuations).  \n \nFig.1. Schematic view of the YIG waveguide with three antennae . \n0 1 2 3 40.05.0x10-41.0x10-31.5x10-32.0x10-3Detector output voltage, V\nΨ, radAcos2(Ψ/2)\n \nFig. 2. DC voltage on the detector as a function of the phase shift adjusted by \nthe phase shifter. The s ymbols show experimental data while the solid line is a \nfit with A ×cos2(Ψ/2) function.  \nFigure 3 shows the spectral noise density S v at f=10 Hz as a \nfunction of the phase difference. The b lue symbols and line \nshow the measured spectral nose density at f=10 Hz; the red \nsymbols and dashed line show the level of the backgro und \nnoise. As one can see , the noise is minimal at the phase \ndifferences Ψ≈0 and Ψ≈π. The noise at its maximum, i.e., at \nΨ≈π/2 is more that an order of magnitude higher. The \nconversion coefficient, R , has its maximum at this phase. \nTherefore, we can conclu de that this amplitude noise is \npredominately due to the conversion of the phase noise.  \nFigure 4 shows the calculated phase noise S Ψ = S v/R2. As \nseen, within the measurement accuracy, the phase noise is \nindependent on Ψ,  as expected. Although RTS noise was not \nfound in the phase fluctuations, the noise spectra of the pha se \nnoise within the frequency range 10  Hz < f < 1 kHz also had \nthe form of the Lorentzian with the characteristic frequency \nwithin 10  Hz – 100 Hz (see Figure 5 ). With the excitation \npower  of ~2 dBm on one of the antennas , the phase noise at \nthe frequency of 10  Hz was measured to be around - 68 dB/Hz.  Amplitude and Phase Noise of Magnons, UC Riverside (2019)  \n \nWe attributed the measured phase noise to the \nmagnetization wave phase velocity fluctuations. The velocity \nfluctuations can be estimated as S v/V2=SΨ/Ψint2, where Ψint is \nthe phase difference gained by the spin wave between the \nantennas, which can be measured by the vector network \nanalyzer. Ψint was measured by method of substitution used \n“unwrapped phase” format of VNA. Total phase margin at \noperation frequency was first measured with our device, Ψ DUT, \nand then with electrically short SMA connector which \nsubstituted our device, Ψ conn. Interested value Ψ int was \ncalculated as difference Ψ DUT - Ψconn. We found Ψint=93 rad, \nwhich yield s Sv/V2≈ 2×1011 Hz-1 at the frequency of the \nanalysis f=10 Hz. This value of the velocity fluctuations can \nbe used to estimate the phase fluctuations in the waveguides of \nan arbitrary length.  \n0 2 410-1610-1510-1410-13Noise S v, V2/Hz\nΨ, radbackground\nFig. 3. Spectral noise density Sv at f=10 Hz as a function of the phase \ndifference. The d ashed line and symbols show the background noise.  \n \n0 1 2 3 4-120-110-100-90-80-70-60-50-40Phase noise  dB/Hz\nΨ, degree\nFig. 4. Phase noise at f=10  Hz as a function of the phase shift. 100101102103-100-90-80-70-60phase noise, dB/Hz\nFrequency f, Hz\nFig. 5. Phase noise spectrum of magnons in YIG wavegu ide. \n \nIV. CONCLUSIONS  \nIn conclusion, the low -frequency noise of magnons, \npropagating as magneto -static surface spin waves , was \nmeasured in YIG waveguides at the frequencies f<1 kHz. The \nnoise spectra had  the Lorentzian shape w ith the characteristic \nfrequencies  below 100 Hz. At these frequencies , the noise of \nmagnons set s the limit for  data processing, sensing and \nimaging applications . 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Horn, “Low frequency fluctuations in solids: 1/f noise,” \nRevi ew of modern Physics, vol. 53, pp.497 -516, 1981.  \n[24] N.V. Dyakonova, M.E. Levinshtein, S.L . Rumyantsev, \"Nature \nof the bulk 1/f noise in GaAs and Si ( review )\", Soviet Physics \nSemiconductors, vol.25, pp. 1241 -1265, 1991.  \n[25] A. Balandin, “L ow-frequency 1/ f noise in graphene devices,”  Nature \nNanotechnology,  vol.8, pp.549–555, 2013.  \n[26] J. Renteria, R. Samnakay, S. L. Rumyantsev, C. Jiang, P. Goli, M. S. \nShur, and A. A. Balandin, “Low -frequency 1/f noise in MoS 2 transistors: \nRelative contributions of the cha nnel and contacts,”  Appl. Phys. Lett., \nvol. 104, p . 153104,  2014.  \n[27] R. Samnakay, C. Jiang, S. L. Rumyantsev, M. S. Shur, and A. A. \nBalandin, Selective chemical vapor sensing with few -layer MoS 2 thin-\nfilm transistors: Comparison with graphene devices, Appl.  Phys. Lett., \nvol. 106, p. 023115, 2015.  [28] S. L. Rumyantsev ,C. Jiang, R.  Samnakay,M. S. Shur, A. A. 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Balandin, “ Unique features of the g eneration –recombination noise in \nquasi -one-dimensional van der Waals nanoribbons, ” Nanoscale, vol. 10, \npp.19749 –19756 , 2018.  \n[33] R. Salgado, A. Mohammadzadeh, F. Kargar, A. Geremew, Chun -Yu \nHuang, M. A. Bloodgood, S. Rumyantsev, T. T. Salguero, and A. A. \nBalandin, “Low-frequency noise spectroscopy of charge -density -wave \nphase transitions in vertical quasi- 2D 1T -TaS 2 devices, ” Appl . Phys . \nExpress vol.12, p. 037001 , 2019 . \n[34] Z. Çelik -Butler, P. Vasina, and N. V. Amarasinghe, “A method for \nlocating the position of oxide traps responsible for random telegrap h \nsignals in submicron MOSFET’s ,” IEEE Trans. Electron Devices  vol. \n47, pp. 646 –648, 2000.  \n[35] M. J. Kirton, and M. J. Uren, “Noise in solid -state microstructures: a \nnew per spective on individual defects, interface states and l ow-\nfrequency (1/f) noise,”  Adv. Phys , vol.  38, pp. 376–468, 1989. \n[36] I. Bloom, A. C. Marley, and M. B. Weissman, “Nonequilibrium \ndynamics of discrete fluctuators in charge -density waves in NbSe 3,” \nPhys. Rev. Lett. , vol. 71, pp.4385 –4388 , 1993.  \n[37] I. Bloom, A. C. Marley, and M. B. Weissman, “Discrete fluctuators and \nbroadband noise in the charge -density wave in NbSe 3,” Phys. Rev. B , \nvol. 50, pp. 5081– 5088 , 1994.  \n[38] G. Liu, S. Rumyantsev , M. A. Bloo dgood , T. T. Salguero , and  A. A. \nBalandin, “Low-Frequency Current Fluctuations and Sliding of the \nCharge Density Waves in Two -Dimensional Materials, ” Nano \nLetters , vol. 18 , pp. 3630 -3636 , 2018.  \n "    },    {        "title": "2008.13061v3.Exploring_a_quantum_information_relevant_magnonic_material__Ultralow_damping_at_low_temperature_in_the_organic_ferrimagnet_V_TCNE_x.pdf",        "content": " \n 1  \nExploring a quantum-information-relevant magnonic material: ultralow \ndamping at low temperature in the organic ferrimagnet V[TCNE] x \n \nH. Yusuf*1, M. Chilcote*1,2, D. R. Candido3, S. Kurfman1, D. S. Cormode1, Y. Lu1, M. E.  \nFlatté3, E. Johnston-Halperin1 \n \n1Department of Physics, The Ohio State University, Columbus, Ohio 43210 \n \n2School of Applied and Engineering Physics, Cornell University, Ithaca, New York 14853 \n \n3Department of Physics and Astronomy, University of Iowa, Iowa City, Iowa, 52242 \n \n* These authors contributed equally to this work. \n \nAbstract:  Quantum information science and engineering requires novel low-loss magnetic \nmaterials for magnon-based quantum-coherent operations. The search  for low-loss \nmagnetic materials, traditionally driven by applications in microwave electronics near \nroom-temperature, has gained additional constraints from the need to operate at cryogenic \ntemperatures for many applications in quantum information science and technology. \nWhereas yttrium iron garnet (YIG) has been the material of choice for decades, the \nemergence of molecule-based materials with robust magnetism and ultra-low damping has \nopened new avenues for exploration. Specifically, thin-films of vanadium \ntetracyanoethylene (V[TCNE] x) can be patterned into the multiple, connected structures \nneeded for hybrid quantum elements and have shown room-temperature Gilbert damping \n(α = 4 × 10-5) that rivals the intrinsic (bulk) damping otherwise seen only in highly-polished \nYIG spheres (far more challenging to integrate into arrays). Here, we present a \ncomprehensive and systematic study of the low-temperature magnetization dynamics for \nV[TCNE] x thin films, with implications for their application in quantum systems. These \nstudies reveal a temperature-driven, strain-dependent magnetic anisotropy that \ncompensates the thin-film shape anisotropy, and the recovery of a magnetic resonance \nlinewidth at 5 K that is comparable to room-temperature values (roughly 2 G at 9.4 GHz). \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 2 We can account for the se variations of the V[TCNE] x linewidth within the context of \nscattering from very dilute paramagnetic impurities, and anticipate additional linewidth \nnarrowing as the temperature is further reduced.  \n \nThe search for low-loss magnetic materials dates to the early days of radio and \nmicrowave electronics [1–3], and the study of elementary excitations, or magnons, in these \nmagnetically-ordered materials has proven to be a rich area of research for both \nfundamental physics and their potential technological applications. More recently, interest \nin these low-loss systems has expanded to include applications in the field of quantum \ninformation technology such as quantum sensing and quantum transduction [4 –7], wherein \nlow-temperature operation allows for the freeze-out of thermal excitations and access to \nthe single-quantum regime . In this regime the field of quantum magnonics utilizes hybrid \narchitectures for coupling magnons to other quantum degrees of freedom, such as \nmicrowave photons, with the aim of extending their functionality in the quantum limit \n[8,9]. It has been demonstrated that magnons can be resonantly excited over a wide range \nof microwave frequencies, allowing for precise control of qubit states mediated by coherent \nexchange via cavity-mode photon excitations [4 ,7]. Magnons also exhibit the potential to \ncoherently couple localized spin-qubits with high cooperativity [10] . However, while \nmagnons exist in a wide range of materials, the same delocalized electrons that are most \noften responsible for stabilizing ferromagnetic order also contribute to electron-magnon \nscattering [11], leading to substantial losses in most metallic ferromagnets. As a result, the \nstudy of low-dissipati on magnon dynamics for quantum applications has focused on \ninsulating ferromagnets and ferrimagnets, with yttrium iron garnet (YIG) and its close \nrelatives holding pride of place as the benchmark low-loss materials for more than 50 \nyears  [ 4,12–14]. As a result, despite these longstanding and emerging needs, applications \nare still constrained by the materials limitations of YIG; namely the need for growth or \nannealing at high temperatures (typically 800° C)  [15–17] and the resulting difficulty in \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 3 integrating and patterning YIG thin-films with other microwave electronic structures and \ndevices.   \nIn this context, the emergence of the molecule-based ferrimagnet vanadium \ntetracyanoethylene (V[TCNE] x) has dramatically expanded the playing field for low-loss \nmagnets. Despite what one might expect from its molecular building blocks, V[TCNE] x \nhas a magnetic ordering temperature of over 600 K and shows sharp hysteresis at room-\ntemperature  [ 18–20]. Moreover, its dynamic properties are exceptional, showing ultra-\nnarrow ferromagnetic resonance (FMR) linewidth (typically ~ 1 – 1.5 G at 9 .4 GHz) with \na Gilbert damping parameter,  of 4 × 10-5 for thin-films [ 18,21]. As a comparison, the \nbest YIG thin-films typically show  = 6.5 × 10-5  [22] and a value of 4 × 10-5 is competitive \nwith the intrinsic  damping of bulk YIG  = 3 × 10-5  [15,23]. From an applications \nperspective, V[TCNE] x has been shown to deposit on a wide variety of substrates without \ncompromising material quality  [24–26], facile encapsulation allows for direct integration \nwith pre-patterned microwave structures for operation under ambient conditions [ 27], and \nrecent work has demonstrated patterning at length scales down to 10 m without increased \ndamping  [21]. However, while these properties clearly establish the potential of \nV[TCNE] x for new applications in traditional microwave electronics, very little is known \nabout its low-temperature magnetization dynamics and therefore its potential for \napplications in quantum information science and engineering (QISE ). \nHere we present a detailed study of the low-temperature magnetic resonance of \nV[TCNE] x films. We identify two regimes. In the high-temperature regime, extending from \n300 K down to 9 K, we observe a monotonic shift in the resonance frequency consistent \nwith a temperature-dependent strain. This strain results in a crystal-field anisotropy that \nincreases with decreasing temperature with a magnitude of at least 140 Oe and the same \nsymmetry, but opposite sign, to the shape anisotropy of the thin-film.  In addition, we \nobserve an increase in linewidth consistent with magnon scattering from paramagnetic \nimpurities similar to what has been observed in YIG  [23,28,29], but with an amplitude 3 \ntimes smaller ( i.e. an increase in linewidth by 9 times in V[TCNE] x as compared to 28 \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 4 times in YIG  [23, 30]). In the low-temperature regime, starting at 9 K and extending to 5 \nK, we observe a discontinuous change in both anisotropy and linewidth: the anisotropy \nabruptly reverts to the room-temperature symmetry (in-plane easy-axis) and the linewidth \napproaches room-temperature values (2.58 G) at 5 K. This linewidth variation can be \nexplained using a model for scattering between magnons and paramagnetic impurities that \ntakes into account the finite spin-lifetime of the impurity spins  [23,31]. At high \ntemperatures (above 100 K) the spin-lifetime is sufficiently short that changes in \ntemperature do not lead to significant changes in scattering rate, and at low-temperatures \n(below 9 K) the spin-lifetime becomes long with respect to the spin-magnon scattering \ntime, resulting in a saturation of the excited state. At intermediate temperatures (from 9 K \nto 100 K) this spin-magnon scattering dominates relaxation due to the increase of the \nground state impurity population, which results in a local maximum in the linewidth that \nis 9 times larger than the room-temperature value. These results are extremely promising \nfor low-temperature applications of V[TCNE] x magnonics, promising low-temperature \nmagnon resonators with unprecedented low-loss that can be integrated on-chip into \nmicrowave electronic circuits and devices [20,21]. \nFor this study, thin-films of V[TCNE] x are deposited on sapphire (Al 2O3 (0001)) \nsubstrates using chemical vapor deposition (CVD) growth process consistent with prior \nreports [18,19].  Briefly, argon gas transfers the two precursors tetracyanoethylene (TCNE) \nand vanadium hexacarbonyl (V(CO) 6) into the reaction zone of a custom-built CVD reactor \n(Fig. 1(a)) where V[TCNE] x is deposited onto polished sapphire substrates. The system is \ntemperature controlled to maintain the TCNE, V(CO) 6 and the reaction zones at 65° C, 10° \nC and 50° C respectively . After growth the sample is mounted on a custom, microwave-\ncompatible sample holder and sealed using a septa cap in an electron paramagnetic \nresonance (EPR) grade quartz tube in an argon environment. When the sample is not being \nmeasured, it is stored in a - 35° C freezer housed in an argon glovebox and is stable for over \none month [ 27].  \nFerromagnetic resonance (FMR) measurements are performed using a Bruker EMX \nPlus X-band EPR spectrometer at temperatures ranging from 300 K down to 5 K. The \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 5 microwave frequency of the spectrometer is tuned between 9 and 10 GHz for optimal \nmicrowave cavity performance before the measurement, and then the frequency is fixed \nwhile the DC field is swept during data collection.  Figure 1(b) shows a representative \nroom-temperature FMR measurement of a typical V[TCNE] x thin-film with the external \nmagnetic field applied in the plane of the sample. The resonance feature is consistent with \npreviously reported high-quality V[TCNE] x thin-film growth, showing a peak- to-peak \nlinewidth of 1.5 G at 9.4 GHz  [18,19].  \nComparing this data to FMR measurements at temperatures of 80 K and 40 K \n(Figure 1(c)) shows an increase in the resonance field of over 40 G (roughly half of the \nsaturation magnetization, 4𝜋𝑀 𝑠) as the temperature decreases. Since the applied \nmicrowave frequency is held constant at 9.4 GHz , this shift must arise from fields internal \nto the V[TCNE] x film, i.e. magnetic anisotropy fields. Note that since the value of the DC \napplied field varies between 3350 G and 3450 G, well above 4𝜋𝑀 𝑠, changes in the \nmagnetization of the film are not expected to contribute to this field shift.  In a similar \nfashion, changes in the shape-dependent anisotropy fields can be ruled out, leaving only \nchanges to the crystal-field anisotropy as a potential source of this phenomenon. Crystal-\nfield anisotropy originates from the interaction of a material’s mean exchange field and the \nangular momenta of neighboring atoms (ions) in the material , indicating that there is a \ntemperature dependence to the local atomic environment within the V[TCNE] x films , e.g. \ndue to a temperature-dependent strain within the film.  \nIn order to more comprehensively map out this phenomenon angle dependent FMR \nmeasurements are performed to quantitatively track changes in the magnetic anisotropy at \ntemperatures of 300 K, 80 K, and 40 K (Fig. 2). Variation of the magnetic resonance field \nas a function of the angle between the applied field and the princip al axes of the film can \nbe modeled by considering the free energy of the magnetic system with anisotropic \ncontributions. If we consider the case of a uniaxial anisotropy with the hard-axis \nperpendicular to the easy-axis, and where the magnetization is parallel to the external field \n(i.e. external field is much larger than the saturation magnetization, as is the case here) the \ntotal magnetostatic energy is as follows  [ 32]: \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 6   \n                                              𝐸 = −𝑴 · 𝑯 + 2𝜋 (𝑴 · 𝒏 )𝟐− 𝐾 (𝑴 · 𝒖 𝑀 ⁄)2                          (1) \n \nwhere M is the magnetization, H is the applied magnetic field, n is the unit vector parallel \nto the normal of the magnetic sample , u is the unit vector parallel to the easy-axis and K is \nan anisotropy constant. For the case of in-plane uniaxial anisotropy, this simplifies to \n \n           𝐸 = − 𝑀𝐻 (sin𝜙sin2𝜃 + cos2𝜃)+ 2𝜋𝑀2cos2𝜃 − 𝐾 sin2𝜃sin𝜙2               (2) \n    \nwhere 𝜃 is the angle between M and the sample normal and 𝜙  \nis the azimuthal angle. Minimizing the magnetostatic energy with respect to 𝜃 , one will \nfind that the easy-axis orientation occurs when 𝜃 = 2 𝑛𝜋±  𝜋\n2, where n is an integer . Using \nthis simple symmetry analysis, we can see that the data in Fig. 2 indicates that the easy-\naxis lies in-plane at a temperature of 300 K (i.e. the resonance field is smallest when the \napplied magnetic field lies in-plane) and out- of-plane at a temperature of 40 K (i.e. the \nresonance field is smallest when the applied magnetic field is out- of-plane). In this context, \nthe lack of variation in resonance field at 80 K indicates a nearly isotropic magnetic \nresponse. This switch in magnetic easy-axis from in-plane to out- of-plane further supports \nthe proposition that there is an additional temperature-dependent crystal-field contribution \nto the magnetic anisotropy.  \nIn previous studies, templated growth of V[TCNE] x resulting in nanowire \nmorphologies induced an additional in-plane magnetic anisotropy with easy-axis \nperpendicular to the long-axis of the nanowires, strongly suggesting the presence of a \nstrain -dependent contribution to the crystal-field anisotropy  [ 33]. In the thin-films studied \nhere, such a strain-dependent crystal-field effect would be expected to generate anisotropy \nparallel to the surface normal, i.e. in the out- of-plane direction. The anisotropy field would \nthen be parallel to the expected shape anisotropy from a thin-film, though not necessarily \nwith the same sign. As a result, if there is a difference in the coefficient of thermal \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 7 expansion between the V[TCNE] x film and the sapphire substrate then the temperature \ndependence of magnetic anisotropy can potentially be understood as a proxy for a \ntemperature dependence of strain in the thin- film; such variations in strain leads to changes \nin the local atomic structure, leading to the observed changes in magnetic anisotropy. We \nnote that while the coefficient of thermal expansion for V[TCNE] x has not yet been \nmeasured, the value for sapphire is 5.4 ppm/K and typical values for molecular-based solids \ncan range somewhere between 28 –500 ppm/K [ 34]. Assuming no strain at room-\ntemperature, this would then imply a compressive strain between 0.67% to 15% at the \nsapphire –V[TCNE] x interface at 5 K , leading to an out- of-plane distortion whose symmetry \nis consistent with the observed anisotropy.    \nA schematic describing how these two anisotropy fields would be expected to \ninteract as a function of temperature can be found in Fig. 3(a). At a temperature of 300 K \n(Fig. 3(a), upper panels), the orientation of the easy-axis is determined by the shape \nanisotropy, resulting in an in-plane easy-axis for thin-films. But at a temperature of 40 K \n(Fig. 3(a) lower panels), there is an additional crystal-field anisotropy, 𝐻⊥, proposed that \ndominates the shape anisotropy, reorienting the easy-axis to be out-of-plane. This \nsymmetry analysis also explains the lack of orientation dependence at a temperature of 80 \nK, which is apparently the temperature at which the strain-driven crystal-field anisotropy \nperfectly cancels out the shape anisotropy. We note that similar phenomenology is also \nobserved in vanadium methyl tricyanoethylenecarboxylate (V[MeTCEC] x) thin-films (see \nsupplementary materials), indicating that this temperature- and strain-dependent \nanisotropy is a general property of this class of metal-ligand ferrimagnets. \nThe fact that the shape and proposed crystal-field anisotropies have the same \nsymmetry make it challenging to distinguish between the two; therefore, an effective field \nis defined as 𝐻𝑒𝑓𝑓= 4𝜋𝑀 𝑒𝑓𝑓= 4𝜋𝑀 𝑠− 𝐻 ⊥, where 𝑀𝑠 is the saturation magnetization and \n𝐻⊥is the crystal-field anisotropy. Figure 2 shows the effects of this net anisotropy field in \nthe form of resonance field shifts and a change in the easy-axis orientation . Quantitatively \nextracting the magnitude and direction of this anisotropy field provides detailed insight \ninto the role of crystal-field anisotropy in tuning the magnetic response of V[TCNE] x thin-\nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 8 films. To this end, each scan is fit to the sum of the derivatives of absorption and dispersion \nfrom a Lorentzian function to extract the resonance frequency and linewidth (experimental \ndata are obtained using a modulated-field technique that yields the derivative of the \nexpected Lorentzian resonance lineshape). For scans showing an out- of-plane easy-axis a \nsingle derivative sum provides good agreement with the data, while for scans showing in-\nplane easy-axis more complex structure is observed requiring the addition of up to three \nderivative sums. In the results discussed below we focus on the behavior of the primary , \ni.e. largest amplitude, peak (a full description of the fitting and resulting phenomenology \ncan be found in the supplemental material).  \nFigure 3(b) shows the extracted resonance field plotted against sample rotation \nangle for the high-and low-temperature data shown in Fig. 2, 300 K and 40 K, respectively . \nTaking into account a uniaxial out- of-plane anisotropy defined by 𝐻𝑒𝑓𝑓, as described \nabove, the angular dependence for in-plane to out- of-plane rotation of a thin-film sample \nis given by  [19,35,36]: \n \n𝜔\n𝛾 =  √(𝐻 − 𝐻 𝑒𝑓𝑓cos2𝜃)(𝐻 − 𝐻 𝑒𝑓𝑓cos 2𝜃)\n=  √(𝐻 − (4𝜋𝑀 𝑠− 𝐻 ⊥) cos2𝜃)(𝐻 − (4𝜋𝑀 𝑠− 𝐻 ⊥)cos 2𝜃)  \n                                        \nwhere  is the resonance frequency and  is the gyromagnetic ratio. As a result, the \nphenomenology of the data presented in Fig. 2 can be understood as an 𝐻𝑒𝑓𝑓 that is positive \nat 300 K and negative at 40 K, as 𝐻⊥ increases with decreasing temperature, consistent \nwith the mechanism for anisotropy switching described in Fig. 3(a). This qualitative \nunderstanding can be made quantitative by fitting the data in Fig. 2 using Eq. (3) to extract \n𝐻𝑒𝑓𝑓= 4𝜋𝑀 𝑒𝑓𝑓 of 91.2 G  1.6 G and -22.8 G  0.4 G, respectively. \nFigure 3 (c) shows this 𝐻𝑒𝑓𝑓 plotted against temperature over the temperature range \nfrom 300 K to 5 K, extracted from angular dependencies such as the measurements \npresented in Fig. 2 . It should be noted that each anisotropy point in Figure 3(c) represents (3) \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 9 a fit to a complete angular dependence such as the data shown in Figure 3(b).The effective \nfield mak es a smooth transition through zero from positive (in-plane) to negative (out- of-\nplane) at a temperature of roughly 80 K . This behavior is qualitatively consistent with the \nphenomenological model presented above and reveals a magnitude of the variation in 𝐻𝑒𝑓𝑓, \nfrom +91 .2 G ± 1.6 G at 300 K to - 45.2 G ± 1.1 G at 10 K, that is roughly 150% of the \nroom-temperature value.  \nNotably, this more comprehensive study also reveals new phenomenology at the \nlowest temperature of 5 K, where the anisotropy abruptly shifts back to in-plane with a \nvalue of +26.2 G ± 0.6 G (roughly 25% of the room-temperature value). This behavior \nreproduces across all samples measured and is quantitatively reproduced upon temperature \ncycling of individual films. The abruptness of this change is distinct from the broad and \nmonotonic behavior observed for temperatures greater than 9 K . The origin of this abrupt \nchange is unclear, but there are two potential explanations consistent with this \nphenomenology. First, it is possible that the increase in strain results in an abrupt relaxation \nthrough the creation of structural defects. This explanation would require some level of \nself-healing upon warming in order to explain the reproducibility of the transition. Given \nthe lack of long-range structural order in V[TCNE] x films as-grown [ 37] it is possible that \nany residual structural defects do not contribute to additional magnetic loss (damping). \nSecond, it is possible that there exist paramagnetic spins in the system that magnetically \norder at temperatures below 9 K. If such spins were preferentially located in an interface \nlayer, for example, their ordering could create an exchange bias that would then pull the \neasy-axis back to an in-plane orientation. \nThe temperature dependence of the linewidth of the magnetic resonance provides \nan additional avenue for evaluating these potential explanations. Figure 4 shows the \nlinewidth for the in-plane magnetic resonance from 300 K to 5 K, with additional data to \nmore clearly resolve the sharp change between 5 K and 9 K. The linewidth data presented \nin Figure 4 is extracted from a single (in-plane applied magnetic field orientation) scan. As \na result, the initial dataset underlying Figure 3 was supplemented by a second temperature \ndependent scan at fixed angle in Figure 4. This data reveals a monotonic increase in \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 10 linewidth with decreasing temperature from 300 K down to 9 K followed by a dramatic \ndecrease in linewidth between 9 K and 5 K, coincident with the abrupt change in magnetic \nanisotropy. We note that in studies of YIG thin-films broadly similar phenomenology is \nobserved, though with a maximum in linewidth that is both higher amplitude (roughly 28 \ntimes the room-temperature value) and at higher temperature (typically 25 K) than is \nobserved here [23,30]. Prior work [23,28 ] has explained this behavior using a model of \nmagnon scattering from paramagnetic defect spins (also referred to as two-level \nfluctuators, TLF) wherein the scattering cross-section at high temperature increases with \ndecreasing temperature as the thermal polarization of the spins increases. This \nphenomenology competes with magnon-pumping of the paramagnetic spins into their \nexcited state, a process that saturates as the spin-lifetime of the defects becomes long \nrelative to the spin-magnon scattering time. The competition between these two processes \nyields a local maximum in the damping (linewidth) that depends on the temperature \ndependent spin lifetime, ts, the energy separation between majority and minority spin states, \nℏωeg, and the difference between that energy splitting and the uniform magnon energy,  \n(ℏω - ℏωeg ). \nIn this model, the linewidth expression is proportional to the square of the exchange \ninteraction energy between V[TCNE] x atoms and the impurity level ( ℏωint)2 ~ (ℏωeg)2, a \nline-shape factor accounting for the finite spin lifetime, 1/𝑡 𝑠/(ℏ2/𝑡𝑠2 + (ℏω - ℏωeg )2ts2), and \nthe ratio between the ground and excited impurity states for fast impurity relaxation , \ntanh(ℏω/2k BT)  [23, 28], \n \nΔ𝐻 =  𝑆\n𝛾 𝑁𝑖𝑚𝑝\n𝑁 (ℏ𝜔 𝑖𝑛𝑡)2 1/𝑡 𝑠 \nℏ2/𝑡𝑠2+ (ℏ𝜔 − ℏ𝜔 𝑒𝑔)2 𝑡𝑠2tanh (1\n2 ℏ𝜔\n𝑘𝐵𝑇) + 𝐻 𝑂 \n \nwhere Nimp/N is the ratio between number of impurit ies and number of V[TCNE] x atoms, \nand S is the averaged V[TCNE] x spin per site , 𝛾 is the gyromagnetic ratio and 𝐻𝑂 is a \nconstant offset due to other relaxation mechanisms. In addition, we assume spin lifetime ts (4) \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 11 = t∞ 𝑒𝐸𝑏𝑘𝐵𝑇⁄ [31, 38, 39] where t∞ is the spin lifetime limit at very high temperatures, and \nEb is a phenomenological activation energy. Figure 4 includes a fit of Eq. (4) to the \nexperimental linewidth (orange line) that yields for S ~ 1 and ωint ~ ωeg the parameters: \nωegt∞ = 0.98, Eb = 1meV, ω egNimp/N = 36.5GHz and 𝐻𝑜= 1 G . Interestingly, if we assume \na reasonable value for ℏωeg of 1.3 meV, a value of Nimp/N = 0.1 follows, thus indicating \nthat V[TCNE] x is an exceptional low-loss magnetic material even if we assume an impurity \nconcentration as high as 10%. This observation is consistent with the hypothesis of \ninsensitivity to structural defects discussed above. \nHowever, it is important to note that the peak in linewidth coincides with the abrupt \nreversion in anisotropy from an out- of-plane easy-axis to an in-plane easy-axis. This \nchange in magnetic anisotropy has the potential to have a substantial impact on spin-\nmagnon scattering efficiency. For example, this change will result in a shift of the energy \nof the magnon bands (see Eq. 1), and if this change involves a commensurate change in the \nstrain there will also be a modification to the spin-orbit coupling and exchange parameters \nat the paramagnetic defects. It should be noted that although this reentrant anisotropy is an \nintriguing feature, the fits to our model for TLFs in Figure 4 are able to reproduce our \nlinewidth data without reference to this effect. As a result, we interpret this fit as an upper \nbound on Eb. This is represented by the additional fits shown in Figure S7 within the \nSupplemental Material wherein we assume a lower temperature for the nominal peak in \nlinewidth occurring due to spin-magnon scattering that is experimentally preempted by the \nchange in magnetic anisotropy. These alternate fits agree with experimental observations \nat temperatures above 9 K, and therefore must be considered as possible mechanisms. \nMoreover, if the residual paramagnetic spins are ordered at temperatures below 9 K, one \nwould require a large amount of energy (>> ℏω) to populate their excited states, which is \nunlikely to happen. Hence, magnetic ordering of the paramagnetic spins would also \nenhance the suppression of spin-magnon scattering, resulting in the sharp linewidth \nsuppression for T < 9 K.  \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 12 When considering the expected behavior as the temperature is further reduced below \n5 K, as would be the case for many applications in QISE, it is useful to consider recent \nmilliKelvin-range measurements of YIG films [40]. That work confirms the expected \ncontinued narrowing down to 500 mK followed by a modest increase from 500 mK down \nto 20 mK, for an overall line narrowing of roughly a factor of 2. The model of scattering \nfrom TLFs described above is consistent with this result in YIG if one supposes a second \npopulation of TLFs that are dipole coupled to the magnons rather than exchange coupled , \nfor example dilute magnetic impurities in the substrate or environment. We note that \nextending this model into V[TCNE] x requires taking into account: i) the substantial \ndifference in structure and chemistry between V[TCNE] x and YIG, and ii) the fact that Ms. \nin V[TCNE] x is roughly 20 times smaller than in YIG. The former consideration indicates \nthat the presence of these dipole coupled TLFs need not correlate between the two systems, \nwhile the latter predicts that any relaxation associated with their presence should be \nreduced by a factor of 20 from Ref. [40]. As a result, the overall factor of 2 decrease in \nlinewidth observed in YIG between temperatures of 5 K and 20 mK should be taken as an \nextremely conservative lower bound on the performance of V[TCNE] x. Given that the \nlinewidth in V[TCNE] x at 5 K is already on par with its room temperature value, these \nresults firmly establish the suitability for this material for applications in quantum \nmagnonics and related aspects of QISE. \nIn conclusion, this work presents the first systematic study of the magnetization \ndynamics of V[TCNE] x at low temperatures. A strong variation in resonance frequency and \nanisotropy with temperature is observed , and attributed to a temperature-dependent strain \narising from the mismatch in thermal expansion coefficients between V[TCNE] x films and \ntheir sapphire substrates. The resonance linewidth of these films is found to increase with \ndecreasing temperature up to a maximum value of 15 G (roughly 9 times the room-\ntemperature value) and is well fit by a model based on magnon scattering from \nparamagnetic defect spins. At 5 K the magnetic anisotropy reverts to in-plane, coinciding \nwith a nearly complete recovery of the resonance linewidth to room-temperature values; \nquantitative modeling suggests the linewidth behavior arises from scattering from \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 13 paramagnetic defect spins that is suppressed at very low-temperature. This suppression of \nspin-magnon scattering is expected to strengthen as temperature is further decreased into \nthe milli-Kelvin range due to freeze-out of thermal magnons and phonons, providing a \ncompelling case for the utility of V[TCNE] x for low-temperature microwave applications, \nsuch as those emerging in the field of quantum information science and technology. \n \nAcknowledgements:  The authors would like to thank A. Franson for providing a \nsoftware suite for fitting FMR spectra as well as general fitting assistance, and G. Fuchs \nfor fruitful discussions . The work presented in the main text, both experiment and theory, \nwas primarily supported by the U.S. Department of Energy, Office of Basic Energy \nSciences, under Award Number DE-SC0019250. S. Kurfman was supported by NSF \nEFMA-1741666 and grew V[TCNE] x calibration samples used for preliminary \nmeasurements not explicitly included in this paper . Work on V[MeTCEC] x presented in \nthe supplementary material was performed by M. Chilcote and Y. Lu with the support of \nNSF Grant No. DMR- 1741666. \n \nData availability statement:  See supplementary material at URL will be inserted by AIP \nPublishing for datasets pertaining to temperature-dependent anisotropy of V[MeTCEC] x, \nmethod for extracting linewidth of V[TCNE] x from FMR scans and additional fits to \nexperimental data highlighting temperature dependence of V[TCNE] x linewidth. \n \nReferences:  \n \n[1] A. Raveendran, M. T. Sebastian, and S. Raman, “Applications of Microwave \nMaterials: A Review” J. Electron. Mater. 48, 2601 (2019). \n[2] Ü. Özgür, Y. Alivov, and H. Morkoç, “Microwave ferrites, part 1: Fundamental \nproperties” J. Mater. Sci. Mater. Electron. 20, 789 (2009). \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 14 [3] J. M. Silveyra, E. Ferrara, D. L. Huber, and T. C. 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Khartsev, and A. M. Grishin, “Submicron Y3Fe5O12 \nfilm magnetostatic wave band pass filters” J. Appl. Phys. 105, (2009). \n[18] M. Harberts, Y. Lu, H. Yu, A. J. Epstein, and E. Johnston- Halperin, “Chemical \nVapor Deposition of an Organic Magnet, Vanadium Tetracyanoethylene” J. Vis. \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 16 Exp. (2015). \n[19] H. Yu, M. Harberts, R. Adur, Y. Lu, P. C. Hammel, E. Johnston-Halperin, and A. \nJ. Ep stein, “Ultra -narrow ferromagnetic resonance in organic-based thin films \ngrown via low temperature chemical vapor deposition” Appl. Phys. Lett. 105, \n012407 (2014). \n[20] N. Zhu, X. Zhang, I. H. Froning, M. E. Flatté, E. Johnston-Halperin, and H. X. \nTang, “L ow loss spin wave resonances in organic-based ferrimagnet vanadium \ntetracyanoethylene thin films” Appl. 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M iller, “A \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 17 Room-Temperature Molecular/Organic- Based Magnet” Science (80 -. ). 252, 1415 \nLP (1991). \n[26] K. I. Pokhodnya, A. J. Epstein, and J. S. Miller, “Thin -Film V[TCNE]x Magnets” \nAdv. Mater. 12, 410 (2000). \n[27] I. H. Froning, M. Harberts, Y. Lu, H. Yu, A. J. Epstein, and E. Johnston-Halperin, \n“Thin -film encapsulation of the air-sensitive organic-based ferrimagnet vanadium \ntetracyanoethylene” Appl. Phys. Lett. 106, (2015). \n[28] P. E. Seiden, “Ferrimagnetic resonance relaxation in rare -earth iron garnets” Phys. \nRev. 133, A728 (1964). \n[29] A. M. Clogston, “Relaxation Phenomena in Ferrites” Bell Syst. Tech. J. 34, 739 \n(1955). \n[30] C. L. Jermain, S. V. Aradhya, N. D. Reynolds, R. A. Buhrman, J. T. Brangham, M. \nR. Page, P. C. Hammel, F. Y. Yang, and D. C. Ralph, “Increased low -temperature \ndamping in yttrium iron garnet thin films” Phys. Rev. B 95, 174411 (2017). \n[31] W. A. Yager, J. K . Galt, and F. R. Merritt, “Ferromagnetic resonance in two nickel -\niron ferrites” Phys. Rev. 99, 1203 (1955). \n[32] H. Puszkarski and M. Kasperski, On the Interpretation of the Angular Dependence \nof the Main FMR/SWR Line in Ferromagnetic Thin Films  (2012). \n[33] M. Chilcote, M. Harberts, B. Fuhrmann, K. Lehmann, Y. Lu, A. Franson, H. Yu, \nN. Zhu, H. Tang, G. Schmidt, and E. Johnston- Halperin, “Spin -wave confinement \nand coupling in organic- based magnetic nanostructures” APL Mater. 7, (2019).    \n[34]   Y. Mei, P. J. Diemer, M. R. Niazi, R. K. Hallani. K. Jarolimek, C. S. Day, C. \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 18 Risko, J. E. Anthony, A. Amassian and O. D. Jurchescu, “Crossover from band -\nlike to thermally activated charge transport in organic transistors due to strain-\ninduced traps” P NAS 114, 33 (2017). \n[35] H. Suhl, “Ferromagnetic Resonance in Nickel Ferrite Between One and Two \nKilomegacycles” Phys. Rev. 97, 555 (1955). \n[36] J. Smit and H. G. Beljers., “Ferromagnetic resonance absorption in BaFe 12O19” \nPhilips Res. Rep. 10, 113 (1955). \n[37] M. Chilcote, Y. Lu, and E. Johnston-Halperin, Organic-Based Magnetically \nOrdered Films  (World Scientific, 2018). \n[38]   J. K. Galt and E. G. Spencer, “Loss Mechanism in Spinel Ferrites ” Phys. Rev. 127, \n1572, 1962. \n[39]   H. Maier-Flaig , S. Klingler, C. Dubs, O. Surzhenko, R. Gross, M. Weiler, H. \n          Huebl, and S. T. B. Goennenwein , “Temperature dependent damping of yttrium \n          iron garnet spheres ” Phys. Rev. B 95, 214423 (2017). \n[40]   S. Kosen, A. F. van Loo, D. A Bozhko, L. Mihalceanu, R. Gross, and A. D. \n          Karenowska , “Microwave magnon damping in YIG films at millikelvin \n          temperatures ” APL Mater. 7, 101120  (2019 ). \n \n \n \n \n \n \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 19 Figure Legends:  \n \nFigure 1  \n \n(a) Schematic (planar view) of the CVD growth system; (b) FMR scan of V[TCNE] x \nthin film at 300 K with the applied magnetic field applied in the plane (IP) of the \nsample with 𝜃 = 90𝑜 and resonance frequency of 9.4 GHz. ΔH pp denotes the peak-\nto-peak linewidth measured as the difference between the positive and negative peak \npositions; (c) FMR line scans for in-plane field orientation at 300 K, 80 K and 40 K \nwith 𝜃 = 90𝑜 and resonance frequency of 9.4 GHz. \n \nFigure 2  \n \nAngle-dependent FMR spectra at temperatures of 300 K, 80 K and 40 K at different \nfield orientations with respect to the sample normal. Nominally the sample is rotated \nfrom  𝜃 = − 10𝑜 to  𝜃 =  100𝑜 in increments of 10𝑜, where 𝜃 =  90𝑜 and 𝜃 =  0𝑜 \nare in-plane and out- of-plane field orientations respectively. Angle corrections have \nbeen taken into account (through fitting with Eq. ( 3)) to reflect the actual rotation \nangles, denoted by the black arrows to the right of each of the temperature-labeled \npanels. \n \nFigure 3  \n \n(a) Schematic of the changes in anisotropy at 300 K and 40 K. 𝑯𝒂𝒑𝒑 denotes the \nexternal magnetic field, 𝑯𝒅𝒆𝒎𝒂𝒈  represents the demagnetizing field of the \nV[TCNE] x film and 𝑯𝒄𝒓𝒚𝒔𝒕𝒂𝒍  is the crystal-field anisotropy. It should be noted that \na finite thin-film has a (negligibly) small demagnetization field when the external \nfiled is applied in the plane since this is not a truly infinite film; (b) Resonance field \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 20 at different field orientations plotted against sample rotation angles for 300 K (open \ncircles)  and 40 K (filled circles) and fits to Eq. (3) (dashed and solid line, \nrespectively) to extract the effective field 𝑯𝒆𝒇𝒇; (c) 𝑯𝒆𝒇𝒇 plotted against temperature \nranging from 300K – 5K. The inset shows the FMR lineshapes at 300 K and 5 K; \nfitting the data to extract the linewidth at FWHM gives 1.63 G and 2. 58 G \nrespectively , this shows that the two linewidths are indeed comparable with the \nlinewidth at 5 K only about 1.66 times larger than the room-temperature value . For \nboth (b) and (c), experimental errors are smaller than the point size.   \n \nFigure 4  \n \nV[TCNE] x linewidth as a function of temperature (black points) and \ncorresponding curve fit (orange line) using Eq. (4). \n \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 21  \n \n \n \n \n \n \n \n \n \n  \n  \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n     Figure 1 H. Yusuf  et al. \n  \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 22  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 2 H. Yusuf  et al. \n \n \n \n \n \n \n \n \n \n \n \n \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 23  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 3 H. Yusuf  et al. \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 24  \n \n \n \n \n \n \n \nFigure 4 H. Yusuf  et al. \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 25  \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 1 Supplementary Materials for “ Exploring  a quantum -information -relevant  \nmagnonic  material:  ultralow  damping  at low temperature  in the organic  \nferrimagnet  V[TCNE] x” \nH. Yusuf *1, M. Chilcote *1,2, D. R. Candido3, S. Kurfman1, D. S. Cormode1, Y. Lu1, M. E.  \nFlatté3, E. Johnston -Halperin1 \n \n1Department of Physics, The Ohio State University, Columbus, Ohio 43210  \n \n2School of Applied and Engineering Physics, Cornell University, Ithaca, New York 14853  \n \n3Department of Physics and Astronomy, University of Iowa, Iowa City, Iowa, 52242  \n \n* These authors contributed equally to this work.  \n \n1. Temperature -dependent anisotropy of V [MeTCEC ]x \nHere, we investigate the magnetic properties of  vanadium methyl tricyanoethylene  \ncarboxylate  V[MeTCEC ]x thin-films using temperature -dependent cavity ferromagnetic \nresonance (FMR). The MeTCEC ligand is similar to the TCNE described in the main text, \nand these results demonstrate that strain -dependent anisotropy is a general feature of this \nclass of metal -ligand materials. Figure  S1a shows the molecular structures of both the \nTCNE molecule and the MeTCEC molecule discussed below.  Figure  S1b shows \ntemperature -dependent magnetization  data for zero field -cooled (ZFC; open black squares) \nand field -cooled (FC; open red circles) measurements , and electron transport data (filled \nblack squares) collected for V[MeTCEC] x thin-films on the same temperature axis. Notice \nthat the maximum in the ZFC magnetization curve – sometimes referred to as the blocking \ntemperature[13,14] – corresponds to the rapid rise observed in the resistance data. This \nchange in electronic and magnetization properties has been associated with carrier freeze  \nout and a magnetic phase transition in related materials such as magnatites, but in light of  2 the results presented in the main text we note \nthat a structural transition associated with \nincreased strain in the films may also play a role \nin these measurements.  \nV[MeTCEC] x samples are deposited on \nAl2O3(0001) substrates using a previously \nreported synthesis and chemical vapor \ndeposition (CVD) growth process.[4,16] During \nthe deposition, argon gas carries the two \nprecursors, MeTCEC and V(CO) 6, into the \nreaction zone where V [MeTCEC ]x is deposited \nonto one or more substrates. The system \nemploys three independently temperature -\ncontrolled regions for the MeTCEC , V(CO) 6, \nand reaction zone with typical setpoints of \n55 °C, 10 °C, and 50  °C, respectively and with \ntypical flow rates for each precursor of 50 \nsccm. Sample growth, manipulation, and \nhandling is p erformed in an argon glovebox \n(O2 < 1.0 ppm; H 2O < 1.0 ppm).   \nAfter growth, samples are mounted onto custom microwave compatible sample \nholders in the appropriate orientation, protected from undesired rotation, and flame -sealed \nin evacuated electron paramagnetic resonance (EPR) grade quartz tubes without exposure \nto air. When not being measured, the sealed samples are stored in a -55 °C freezer and are \nfound to be stable for weeks.  \nFigure  S2 shows four ferromagnetic resonance (FMR) spectra of V[MeTCEC] x \noriented both in  plane (90 °; see inset to Fig.  3c) and out of plane (0 °) at 140  K and 80  K. \nThe FMR response of magnetic materials is sensitive to the local field environment of the \nFigure S1 (a) The molecular structures of \ntetracyanoethylene (TCNE) and \ntricyanoethylenecarboxylate (MeTCEC). (b) \nMagnetization vs. temperature curves for zero field -\ncooled (ZFC; open black squares) and zero field -\ncooled (FC; open red circles) measurements. On the \nsame  temperature axis, resistance vs. temperature \ndata is shown for a V[MeTCEC]x thin -film (filled \nblack squares). The corresponding dependent -axis is \nshown on the right axis. Note the maximum in the \nmagnetization data corresponds to the rapid rise \nobserved in  the resistance data.   3 sample and therefore allows for sensitive characterization of the anisotropy fields in \nV[MeTCEC] x. FMR measurements are performed using a Bruker electron paramagnetic \nresonance spectrometer setup for X -band measurements with 200 µW of applied \nmicrowave power and fitted with an Oxford Instruments ESR900 cryostat insert. The \ncryostat is cooled by flowing liquid nitrogen and operates at tempe ratures ranging from \n80 K to 300  K with better than 50  mK stability during FMR measurements. In standard \noperation, the microwave frequency of the spectrometer is tuned between 9 and 10  GHz \nfor optimal microwave cavity performance before the measurement, a nd then the frequency \nis fixed while the DC field is swe pt during the measurement.   \nFigure  S2a shows FMR spectra collected at 140  K for the magnetic field applied in  \nplane (𝜃 = 90°) and out  of plane ( 𝜃 = 0°). Consistent with prior FMR measurements of \norganic -based magnetic  materials,[6,16,17] the center field associated with th e resonant \nfeature in the in -plane spectrum is at a lower field than that of the out -of-plane spectrum, \nand therefore the easy magnetization axis is oriented in the plane of the film. This easy -\naxis orientation is the expected outcome resulting from the sh ape anisotropy present in \nthin-film samples. Figure  S2b also shows FMR spectra collected with the magnetic field \napplied in  plane ( 𝜃 = 90° ) and out of plane (𝜃 = 0°).  However, this data is collected at 80  K, \nfurther below the maximum in the V [MeTCEC ]x ZFC magnetization curve than the data \nshown in Fig . S2a. Surprisingly, the center field of the dominant resonance feature in the \nin-plane spectrum is at a higher field than that of the out -of-plane spectrum. This behavior \nseems to indicate that th e sample has an easy  axis oriented out of the plane of the sample; \nthe spectra show signs of a switch in the magnetic easy  axis from in  plane to out  of plane \nas it is cooled from 140  K to 80  K.  4 To investigate this behavior in greater \ndetail, angular -dependent data is collected in 10 ° \nincrements as the ap plied field is rotated from in  \nplane ( 𝜃 = 90° ) to out of plane (𝜃 = 0°)  of the \nsample. The data is shown in Figs . 3a and b for \n140 K and 80  K respectively. A gray dashed line \nis overlaid on the data to serve as a guide to the \neye. The field shifts shown in Figs . S3a and S3b \nare consistent with those shown in Fig . 2 above. \nFigure  S3c shows the center fields extracted from \nthe two -angle series, emphasizing the magnitude \nof the change in the anisotropy.  \nThis switch in the magnetic easy  axis from \nin plane to out  of plane present in the data \nsuggests the presence of an additional \ncontribution to the anisotropy beyond simply \nshape anisotropy. Previously, given the isotropic \nin-plane response of thin  films at room  \ntemperature, additional contributions to  the anisotropy had been excluded. However, the \nresults here warrant the inclusion of an additional term 𝐻#, which is responsible for \ninducing perpendicular anisotropy in thin  films. This phenomenology is consistent with the \nmeasurements of V[TCNE] x thin films presented in the main text. Following that \ndevelopment,  the angular  dependence of the FMR response for in  plane to out of plane \nrotation of a thin -film sample can therefore be described by,[17–19] \n𝜔\n𝛾='(𝐻−4𝜋𝑀-..cos2𝜃)\t(𝐻−4𝜋𝑀-..\tcos2𝜃)\t\n='(𝐻−(4𝜋𝑀6−𝐻7)cos2𝜃)\t(𝐻−(4𝜋𝑀6−𝐻7)\tcos2𝜃), (1) \nFigure S2 (a) Single FMR line scans at 140  K for \na sample oriented in -plane (90°) and out of plane \n(0°) with respect to the externally applied magnetic \nfield. (b) Single FMR line scans at 80  K for a \nsample oriented in -plane (90°) and out of plane \n(0°) with respect to the externally applied magnetic \nfield.   5 where ω is the resonance fequency, γ is the gyromagnetic ratio, 𝐻 is the applied field, and \n𝜃 is the polar angle of the magnetization.  The FMR resonance fields are more than an order \nof magnitude larger than the typical saturation field for V[TCNE] x, and therefore we have \nassumed that the magnetization is effectively p arallel to the applied magnetic field (i.e. \n𝜙≈𝜙> and 𝜃≈𝜃? where 𝜃, 𝜙 and 𝜃?, 𝜙? are the polar and azimuthal angles of the \nmagnetization 𝑀 and the applied bias field 𝐻, respectively). Also, note that the in -plane \nFMR response remains isotropi c, with the 𝜙 dependence dropping out:  \n𝜔\n𝛾='𝐻\t(𝐻+4𝜋𝑀-..)\t\n='𝐻\t(𝐻+(4𝜋𝑀6−𝐻7)) \nThe data in Fig.  S3c are then fit according to the dispersion relation in Eq.  1. The fits \nare shown as solid and dashed lines in Fig.  3c. The 140  K data yields an 𝐻eff=4π𝑀eff= \n15.4 Oe ± 0.1 Oe while fi tting to the 80  K data result in an 𝐻eff value of  -28.3 Oe ± 1.0 Oe. \nThe negative value of 𝐻eff for the 80  K data means that 𝐻#>4π𝑀S and that the film has \nperpendicular magnetic anisotropy. Note that the magnetic energy landscape, and therefore \nthe angular dependence contained in Eq.  1, does not allow for the easy  magnetization axis \nto take on an intermediate vector between in  plane or  out of plane for this set of anisotropy \nfields. This result also implies that prior measurements of the anisotropy of thin  films are \nin fact measuring 4π𝑀eff rather than the bare 4π𝑀S as previously assumed.[17,20] However,  (2) \n  (𝜃=90°).   6 as with previous studies of uniform thin  films, \nit is challenging to disentangle this form of \nanisotropy from 4π𝑀S, leading us to use the \nmore general 𝐻eff=4π𝑀eff. Temperature -\ndependent FMR studies combined with careful \nDC magnetization measurements provide a \npromising avenue to decoupling the two \nanisotropy fields.  \nIn comparing the data shown in Fig.  S3a \nand b, also note that at lower temperatures, the \nresonance response becomes markedly multi -\nmodal and appears to broaden. To investigate \nthis behavior in greater detail, FMR data is \ncollected over a range of temperatures with the \napplied field oriented in the plane of the \nsample. The dat a is shown in Fig.  S4a. Note \nthe clear shift of the resonant features towards \nhigher field at lower temperatures as the in -\nplane orientation, which is the geometry being \nmeasured in this data set, changes from the \neasy magnetization axis to the hard \nmagnet ization axis.  \nThe effective magnetization, 4π𝑀eff, \nextracted from the data shown in Fig.  S3 \ncontains contributions from a perpendicular \nmagnetic  anisotropy energy. This 𝐻# does not \narise from shape anisotropy in thin  films and  \nFigure S3 (a) Shows FMR spectra as the sample is \nrotated from in -plane ( 𝜃 = 90°) to out of plane \n(𝜃 = 0°) with respect to the externally applied \nmagnetic field at 140  K. (b)  Shows the FMR \nspectra as the sample is rotated from in -plane \n(𝜃 = 90°) to out of plane ( 𝜃 = 0°) with respect to \nthe externally applied magnetic field at 80  K. (c) \nShows the extracted center fields from the angular \nseries shown in (a) and (b) with fits shown as solid \nand dashed lines. The inset shows the coordinate \nsystem with respect to the sam ple geometry.  0.1 Oe \n1.1 Oe  7   \n  \n  \n  \nmust instead come  from a crystal -field anisotropy wherein the local exchange vector \nacquires some anisotropy due to some combination of lattice symmetry and strain. Given \nthe large differences in the coefficients of thermal expansion for organic and in organic \nmaterials (often varying by an order of magnitude or more) , stain due to differential thermal \nexpansion at the interface between the substrate and organic -based materials is likely \ncreating an anisotropic strain field in the magnetic material. As the sample temperature is \nlowered, this strain field increases until 𝐻# becomes larger in magnitude than 4π𝑀S and \n4π𝑀eff takes on a negative value. The result is a magnet with an easy -axis out of plane as \nshown in Fig.  S3b.  \nWe note that qualitatively similar results were obtained for vanadium ethyl \ntricyanoethylene carb oxylate ( V[ETCEC ]x). V[ETCEC ]x is a third member of this class of \nmetal ligand ferrimagnets[23,24], supporting the thesis that strain -dependent anisotropy is a \ncommon feature of this class of materials.  \n2. Method for extracting linewidth from FMR scans  \nThe FMR scans are obtained through phase -sensitive detection, where in addition to the \nstatic DC magnet ic field the sample sees a sinusoidally modulated field component that is  Figure S4 (a) Shows FMR spectra of V[MeTCEC ]x sample mounted  in-plane \n(𝜃 = 90°) with respect to the externally applied magnetic field as a function of \ntemperature . (b)  Shows the extracted peak -to-peak linewidths from the \ntemperature -dependent spectra shown in (a)  \n 8 varied  at the same frequency as the amplitude modulation  of the microwaves reflected from \nthe cavity. If there is an EPR signal , that signal is converted into a sine wave whose \namplitude is proportional to the  derivative of  the signal  (change in microwave power \nrelative to field modulation)  and appears as the first derivative of a Lorentzian function . In \naddition, it should be noted that some FMR scans show multi  peaks (for examp le, the 300 \nK scans shown in Figure 2 of the main  text) and a possible reason for that could be \ninhomogeneous strain. As discussed in our main text, strain in our films is induced by \ndifference in thermal expansion coefficients between V[TCNE] x and the substrate. Given \nthat we have taken no special precautions to prevent it, we believe it is likely that this strain \nwill be inhomogeneous, resulting in regions of our sample with differing magnetic \nanisotropy, and therefore the potential for additi onal peaks in FMR spectra. It has been \nreported that strain -induced distortions can alter the local electronic and crystal -field \nenvironment by changing the orbital occupancy, tilt angle between neighboring spins[25] or \nmagnetocrystalline anisotropy[26,27], for instance, leading to local changes in magnetic \nanisotropy which result in the appearance of additional resonance peaks . \n Since the asymmetry of the FMR lineshape and the multi -peaks need to be \naccounted for , scans are  not simply fit by the derivative of a symmetric Lorentzian . In \nphase -sensitive measurements the microwave electric field generates oscillati ng electric \ncurrents in the sample ; the oscillating magnetization due to the microwave magnetic field \nresults in oscilla ting angles between the current flow and magnetization, leading to local \nlattice distortions  which may c ause the observed asymmetry in signal lineshape  due to \ninhomogeneous broadening[28,29]. Another possible source of this asymmetry could be the \nresult of high cavity loading[30] and the resulting phase error introduced by the automatic \nfrequency controller of the EPR spectrometer when the sample is resonantly excited. This \nwarrants the inclusion of a dispersion or antisymmetric term that takes int o account this \nasymmetry, therefore the FMR scans are fitted to the sum of the derivative of an absorption \n(symmetric  term) and dispersion (antisymmetric term ) from a Lorentzian . The derivative s \nhave the following form : (3) (4)  9 absorption derivative =\t−32\t√3\t𝐴\t𝐹𝑊𝐻 𝑀K(𝐵−𝐵M)\n9\t[FWHM \t2+4(𝐵−𝐵M)\t2]\t2 \n \ndispersion derivative =\t−4\t𝐷\t𝐹𝑊𝐻𝑀 \t(𝐵−𝐵M)\nFWHM \t2+4(𝐵−𝐵M)\t2 \nwhere FWHM  is the full -width at half -max, A is the height of the absorption derivative, D \nis the height of the dispersion derivative, 𝐵M is the location of the resonance (center) field  \nand B is the amplitude of the magnetic field that is being swept at each data point. \nTherefore, the resulting li ne shape depends on the relative contributions of these two terms .  \nFor scans with an out -of-plane easy  \naxis fitting with a single  derivative sum \nprovides good agreement with the data  \n(Figure S5). But for scans with in -plane \neasy axis, due to the appearance of a \nmodest satellite peak , obtaining a good fit \nto the data requires addition of up to three \nderivative sum s. For FMR scans in the \nrange 9 K  –  80 K (out -of-plane easy  axis \nbetween 9 K – 100 K and negligible \nanisotropy at 80 K) the date is well fit with a single derivate sum . On the other hand,  fits \nfor scans in the high temperatures between 120 K – 300 K (in -plane easy  axis) give good \nagreement with data when  two derivative sums  are used , a few requiring up to three \nderivative sums (Figure  S6b).  However, FMR scans at 5 K (in-plane easy  axis) and 6 K \n(out-of-plane easy  axis) mimic the high temperature fits by requiring two derivative sums.  \nFor the purposes of this study, which explores the fundamental FMR mode, in scans \nshowing multiple peaks we focus on the contribution fr om the peak that persists to low Figure S5 Shows a single derivative fit to the \nFMR data collected at 2 2 K \n 10 temperatures.  If we plot the individual Lorentzian \ncomponents of the FMR fit, we find that the first \ncomponent YL1 (component with the highest \noverall peak -to-peak magnitude) is present in all \nthe temperatures being consid ered in the range 300 \n– 5 K. Therefore, the linewidth date plotted against \ntemperatures in Figure 4 of the main text is the \nlinewidth at f ull width  half max (FWHM) of YL1.  \nIn Figure S6a it can be seen that fitting the \nFMR scan at 300 K with a single derivative sum \ndoes not provide a great fit to the data. However , \nfrom  Figure S6b it becomes clear that fitting the \nsame data with the superposition of three \nderivative sums or components (each with their \ndistinct A, D and FWHM ) gives a decent  fit. In \nFigure S6c the amplitude of each individual \ncomponent is plotted against magnetic field sweep \nrange to provide a visual understanding of how \neach component contributes to the overall FMR \nline shape.  \n3. Temperature dependent linewidth  \nThe V [TCNE ]x linewidth dependence on \ntemperature can be well explained from the \ninteraction between magnons and defects or Figure S6 (a) Shows a single derivative fit to the \nFMR data collected at 300 K. (b) Shows FMR \nscan at 300 K fitted to superposition of three \nLorentzian derivative  sums. (c) Amplitude of \neach sum or component plotted against magnetic \nfield sweep range.  YL1, YL2 and Y L3 are the \nfirst, second and third components respectively.  \n 11 impurities in V [TCNE ]x.  The \ndefects or impurities are \nconsidered to be a two -level spin \nsystem s. These experience spin -\nflip transitions excited by the \nannihilation of a uniform -magnon \nmode  [31,32]. This process \nintroduces a finite magnon \nlifetime, which in turn leads to the \nlinewidth expression Eq. (4)  in the \nmain  text. In Fig. S7, we use four \ndiffere nt parameter sets to fit the \nhigh temperature experimental \ndata using Eq. (4). All the different \nsets yield a good fitting for T> 9 K, \nalthough the smaller the E b, the \nsmaller the nominal peak in \nlinewidth. As discussed in the main text, this imposes an upp er bound on E b ~ 1meV.   \nHowever, it is important to note that the peak in linewidth coincides with the abrupt \nreversion in anisotropy from an out -of-plane easy  axis to an in -plane easy  axis. This change \nin magnetic anisotropy has the potential to have a substantial impact on spin -magnon \nscattering efficiency. For example, this change will result in a shift of the energy of the \nmagnon bands (see Eq. 1  in main text ), and if this change involves a commensurate change \nin the strain there will also be a modifi cation to the spin -orbit coupling and exchange \nparameters at the paramagnetic defects. This is represented by the fits shown in Fig. S 7 \nwherein we assume a lower temperature for the nominal peak in linewidth occurring due \nto spin -magnon scattering that is experimentally preempted  by the change in magnetic \nanisotropy.  As a result, we interpret th e fit in Fig. 4 of the main text as an upper bound on \nFigure S7 (a) (b), (c) and (d) show V [TCNE ]x linewidth as a \nfunction of temperature and the corresponding fit curves using \nfitting parameters of Eq. (4)  \n 12 Eb. The alternate fits presented in Fig. S7 agree with experimental observations at \ntemperatures above 9 K, and therefore must be considered as possible mechanisms. \nMoreover, if the residual paramagnetic spins are ordered at temperatures below 9 K, one \nwould require a large amount of energy (>>  ℏω) to populate their excited states, which is \nunlikely to happen. 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Rev.  1964 , 133, A728.  \n \n \n "    },    {        "title": "2007.15299v1.Coherent_multi_mode_conversion_from_microwave_to_optical_wave_via_a_magnon_cavity_hybrid_system.pdf",        "content": "arXiv:2007.15299v1  [quant-ph]  30 Jul 2020Coherent multi-mode conversion from microwave to optical w ave via a magnon-cavity\nhybrid system\nYong Sup Ihn, Su-Yong Lee, Dongkyu Kim, Sin Hyuk Yim, and Zaeill Kim∗\nQuantum Physics Technology Directorate, Agency for Defens e Development, Daejeon 34186, Korea\n(Dated: July 31, 2020)\nCoherent conversion from microwave to optical wave opens ne w research avenues towards long\ndistant quantum network covering quantum communication, c omputing, and sensing out of the\nlaboratory. Especially multi-mode enabled system is essen tial for practical applications. Here we\nexperimentally demonstrate coherent multi-mode conversi on from the microwave to optical wave via\ncollective spin excitation in a single crystal yttrium iron garnet (YIG, Y 3Fe5O12) which is strongly\ncoupled to a microwave cavity mode in a three-dimensional re ctangular cavity. Expanding collective\nspin excitation mode of our magnon-cavity hybrid system fro m Kittel to multi magnetostatic modes,\nwe verify that the size of YIG sphere predominantly plays a cr ucial role for the microwave-to-optical\nmulti-mode conversion efficiency at resonant conditions. We also find that the coupling strength\nbetween multi magnetostatic modes and a cavity mode is manip ulated by the position of a YIG\ninside the cavity. It is expected to be valuable for designin g a magnon-hybrid system that can be\nused for coherent conversion between microwave and optical photons.\nI. INTRODUCTION\nStrong coupling induced by resonant light-matter in-\nteraction can give rise to coherent information trans-\nfer between distinct physical systems in quantum and\nclassical information processing1–3. The coherent trans-\nfer of quantum state is a key role in realizing large-\nscale quantum optical networks and long distance quan-\ntum sensing and imaging4–9since it allows quantum in-\nformation to be exchanged between different systems\nthat operate at different energy scales. A platform for\ntransferring multi-mode states will be attractive to the\npractical application of quantum-enhance metrology and\ncommunication10,11. After the first demonstration of op-\ntical frequency conversion12, the photon frequency con-\nversion has been implemented with crystals in optical\ndomain, and with superconducting circuits in the mi-\ncrowave domain13,14. Since it has great potential in\nrealizing large-scale quantum optical networks with su-\nperconducting qubits, recently, the microwave to optical\nfield conversionhas been intensivelyattracted and exper-\nimentally demonstrated by using intermediate systems,\nsuch as optomechanical systems15–20, electro-optical\nsystems21, atomic ensembles22–25, and magnons26. So\nfar, the maximum microwave-to-optical (MO) conver-\nsion efficiency has been demonstrated in a nanomechan-\nical resonator system employing a nano-membrane that\nis combined with an optical cavity while it is coupled\nto a superconducting microwave resonator. Its conver-\nsion efficiency reached 47 % at low temperature16,17. A\nferromagnetic material, an yttrium iron garnet (YIG),\nin a microwave cavity offers strong interaction between\nmagnon and microwave cavity modes at both low and\nroom temperatures since YIG has a Curie temperature\nof 560K and a net spin density of 2 .1×1022µBcm−3(µB;\nBohr magneton) that is few orders of magnitude higher\nthan a net spin density of 1016−1018µBcm−3in para-\nmagnetic materials. High Verdet constant with sharp\nlinewidth ofelectronspinresonancein microwavedomainalso makes YIG noticeable in Faraday effect27,28. Re-\ncently, YIG based materials have been studied on a novel\nconcept for ultrafast magneto-optic polarization modula-\ntion with frequencies up to THz orders29,30. Longer spin\nexcitationtime thanparamagneticspinsystemisanother\nadvantage of YIG and its hybrid system31,32. In this hy-\nbrid system, the MO conversion is achieved through the\nFaraday effect and Purcell effect. The magnetization os-\ncillation induced and amplified by the Purcell effect of\na microwave cavity mode creates the sidebands to the\nincidental optical wave, resulting in coherent conversion\nbetween microwave and optical wave. So far, the MO\nconversion via YIG-cavity system has been focused only\nfor the Kittel mode.\nIn this paper, we report coherent multi-mode conver-\nsion from the microwave to optical wave fields, which is\nbased on a hybrid system consisting of a sphere of YIG\nand a three-dimensional rectangular microwave cavity.\nWe experimentally demonstrate and verify that the size\nof YIG is a dominant factor of the coherent multi-mode\nconversionefficiency. Theconversionefficiencyistheoret-\nically derived by using the interaction Hamiltonian with\nthe ferromagnetic-resonance (FRM or Kittel mode, KM)\nandahighermagnetostaticmode(MSM) andexperimen-\ntally characterized by normal-mode splitting, coupling\nstrengths of the ferromagnetic-resonance, and magneto-\nstatic modes. For the near-uniform microwave cavity\nfield, all the spins in the YIG sphere precess in phase\nthat is called Kittel mode, and therefore the whole YIG\nsphere can be treated as a giant spin33,34. As the YIG\nsphere size increases, the microwave cavity field can no\nlonger be treated as a uniform field, leading to the higher\norder magnon modes which are observed35–37. By posi-\ntioning a YIG off the uniform microwave cavity field re-\ngion, we can manipulate the coupling strength between\nmulti magnetostatic modes and a cavity. As a result,\nit is shown that KM and MSM manifested in each YIG\nsample are successfully transferred through the conver-\nsion process from microwave field to optical wave field.2\n\u000bD\f\n\u000bE\f\nFIG. 1. (a) Schematic of hybrid system for coherent cover-\nsion from microwave to light, where an YIG sphere is inserted\ninto a rectangular microwave cavity. (b) Magnon-cavity hy-\nbrid model. In the hybrid system, magnon mode ˆ smand a\nmicrowave cavity mode ˆ aare strongly coupled with a coupling\nstrength gm. Here,κiandγmpresent internal cavity loss and\nintrinsic loss for magnon modes, respectively. A microwave\nfield mode ˆ aiis coupled to a microwave cavity mode at a rate\nκe, while a traveling optical wave field mode ˆbiis coupled to\nthe magnon mode at a rate δm. Through this process, mi-\ncrowave field is converted to the traveling optical wave field\nwith a conversion efficiency ηt.\nII. THEORETICAL CONVERSION MODEL\nFig. 1(a) shows the main part of our hybrid system, a\nYIG sphere and a 3D rectangular microwave cavity. Due\nto the magnetic and optical properties, a highly polished\nYIG sphere can serve as an excellent magnon resonator\nat microwave frequencies. A 3D rectangular microwave\ncavity intrinsically maintains a low damping rate com-\npatible with one of magnon mode at room temperature\nand enhances the coupling rate between a magnon mode\nand a cavity mode through the Purcell effect. Then, the\nlinearly polarized light travels through a YIG perpen-\ndicular to the magnetization direction along the static\nmagnetic field. Finally the Faraday effect creates the op-\ntical sidebands, or polarization oscillations of the light\ninduced by the magnetization oscillations26,38,39. Fig.\n1(b) describes the schematic diagram for coherent con-version from microwave photons to optical photons. The\ntotal Hamiltonian( ˆHt) describing the conversion process\nincluding the KM and a higher MSM can be given by\nˆHt=ˆHc+ˆHs+ˆHo,\nˆHc=−i√2κe/bracketleftBig\nˆa†ˆai(t)−ˆa†\ni(t)ˆa/bracketrightBig\n,\nˆHs=ωcˆa†ˆa+ωKˆs†\nKˆsK+ωMˆs†\nMˆsM+gK/parenleftBig\nˆa†ˆsK+ˆaˆs†\nK/parenrightBig\n+gM/parenleftBig\nˆa†ˆsM+ˆaˆs†\nM/parenrightBig\n,\nˆHo=−i/radicalbig\n2δK/parenleftBig\nˆsK+ ˆs†\nK/parenrightBig/bracketleftBig\nˆbi(t)eiΩ0t−ˆb†\ni(t)e−iΩ0t/bracketrightBig\n−i/radicalbig\n2δM/parenleftBig\nˆsM+ ˆs†\nM/parenrightBig/bracketleftBig\nˆbi(t)eiΩ0t−ˆb†\ni(t)e−iΩ0t/bracketrightBig\n,\n(1)\nwhere/planckover2pi1= 1. The subscripts KandMstand for the\nKM and a higher MSM, respectively. The MO coversion\nproceeds with the following steps in the Hamiltonians:\nˆHc→ˆHs→ˆHo.ˆHcdescribes the interaction Hamil-\ntonian between an itinerant microwave mode ˆ aiand the\nmicrowave cavity mode ˆ awith external coupling rate κe,\nwhich results from the rotating-wave approximation. ˆHs\nincluding the system Hamiltonian describes the interac-\ntion Hamiltonian between cavity and magnon modes. gK\nandgMrepresent the magnon-microwave photon cou-\npling strengths for KM and MSM, respectively. Here, we\nnotethat gKandgMincludetheoverlappingcoefficient ξ,\nwhich is related to the space variation effect between the\nmagnetic field of the cavity mode and magnon modes32.\nˆa†(ˆa) is the creation (annihilation) operator for the mi-\ncrowavephotonattheangularfrequency ωc. ˆs†\nK(ˆsK)and\nˆs†\nM(ˆsM) represent the collective spin excitations for KM\nand MSM at angular frequency ωKandωM, respectively\n(see Appendix A). The number of spins in a YIG sphere\ncan contribute to both KM and MSM. ˆHodescribes the\ninteraction Hamiltonian between magnon modes ˆ sand a\ntraveling optical photon mode ˆbiwith angular frequency\nΩ0.δKandδMrefer to the optical photon-magnon cou-\npling rate for KM and MSM, respectively. Since it in-\ncludes both Stokes and anti-Stokes processes that are in-\nvolved in the MO conversion, we leaves the Hamiltonian\nˆHowithout the rotating-wave approximation. The MO\nconversion indicates that the itinerant microwave pho-\nton mode ˆ aiis converted to the traveling optical photon\nmodeˆbo.\nIn our experiment, the strongly coupled magnons and\ncavitymicrowavephotonmodecanbedeterminedbynor-\nmal mode splittings in transmission spectra which are\nmeasured from the input and output channels (see Fig.\n2(c)). According to the input-output relation40, equa-\ntions of motions for a cavity mode and magnon modes\ncan be obtained from quantum Langevin equation (see\nAppendix A). As a result, the transmission for multi-\nmagnon modes can be given by\nS21(ω) =−i2κe\nω−ωc+iκt−/summationtext\nmg2mχm,(2)3\nwhere\nχm(ω) = [ω−ωm+iγm]−1. (3)\nHere,κt=κe+κiis the total loss rate which includes\nboth external coupling rate κeand internal loss of the\ncavityκi.\nFor the conversion process from the microwave to op-\ntical wave, the conversion coefficients with amplification\nfactorβmfor the KM and a MSM are obtained as (see\nAppendix A)\nS31,K(ω) =−2/radicalbig\nβKδKκegKχKχc/parenleftbig\n1+g2\nMχMχcTM/parenrightbig\n1−g2\nKχKχc(1+g2\nMχMχcTM),\n(4)\nand\nS31,M(ω) =−2/radicalbig\nβMδMκegMχMχc/parenleftbig\n1+g2\nKχKχcTK/parenrightbig\n1−g2\nMχMχc(1+g2\nKχKχcTK),\n(5)\nwhere\nχc(ω) = [ω−ωc+iκt]−1,\nTm(ω) =/bracketleftbig\n1−g2\nmχmχc/bracketrightbig−1.(6)\nTherefore, the MO conversion efficiency including both\nKM and a MSM modes can be given by\nηt(ω) =|S31,K(ω)|2+|S31,M(ω)|2.(7)\nHere, at the resonant condition ω−ωc=ω−ωK=\nω−ωM= 0, the two-mode conversion efficiency can be\nrepresented in terms of cooperativities for CK=g2\nK\nκtγK\nandCM=g2\nM\nκtγM,\nηt=4κe\n(1+CK+CM)2/bracketleftbigg\nδKβKC2\nK\ng2\nK+δMβMC2\nM\ng2\nM/bracketrightbigg\n.(8)\nIn this work, the two-mode conversion efficiency given\nin Eq.(8) is well matched to the experimental results. If\nwe expand the interaction Hamiltonian to possess higher\norder terms, the multi-mode MO conversion efficiency\ncan be obtained by\nηt=4κe\n(1+/summationtext\nmCm)2/summationdisplay\nm/bracketleftbigg\nδmβmC2\nm\ng2m/bracketrightbigg\n,(9)\nwheremis a mode index for each MSM. Since so far the\nMO conversion for multi-modes has not been reported\nin magnon-cavity system, our theoretical result can be\napplied to multi-mode conversionbasedon ferromagnetic\nmaterial-hybrid systems.\nIII. EXPERIMENTS AND RESULTS\nAs a ferromagnetic material, we use commercial YIG\nspheres of diameter 0.45, 0.75, and 1 mm from Fer-\nrisphere and Microsphere. A 3D rectangular cavity\n\u000bD\f\n\u000bE\f;+)\u0003URJGTGU \n\u000bF\f\n\u0014\u0015\u0016\n[\\\n][\\\n]\u0013\u0011\u0017\u0018\u0003PP \u0013\u0011\u001a\u0018\u0003PP \u0014\u0011\u0013\u0003PP\nFIG. 2. (a) Numerical simulation of the magnetic field distri -\nbution of the fundamental mode TE 101inside the microwave\ncavity with the volume of 20 ×20×4 mm−3. An YIG sphere\nmade by Ferrisphere Inc.41is mounted at the field maximum\nof the fundamental mode. (b) Transmission power (left y-\naxis) and phase (right y-axis) without a YIG sphere through\nthe cavity as a function of the microwave frequency. ωc/2π=\n10.632 GHz, κi/2π= 0.6 MHz, κe/2π= 2.1 MHz. Experi-\nmental data (black solid-circles and solid-diamonds), and the-\noretical results (redand bluecurves). (c) Experimental se t-up\nfor the microwave to optical light conversion. To examine th e\nhybrid system, the transmission data are taken by a vector\nnetwork analyzer. For the conversion measurement, a 1550-\nnm cw laser with z-polarization is injected into a YIG sphere\nwhose beam waist size is about 120 µm. The polarization of\nthe light is oscillated by the magnetization oscillations, which\nis induced by the microwave driving field fed into the channel\n1. Finally, the polarization oscillations of light is detec ted\nby a fast photodiode and amplified by a microwave amplifier\nwith 30 dB amplification along the channel 3. HWP refers to\nthe half-waveplate.\nis made of oxygen-free copper with the volume Vcof\n20×20×4 mm3and the fundamental mode TE 101is\nused for magnetic-dipole coupling. Fig. 2(a) shows the\nmicrowavemagnetic field distribution of the fundamental\nmode TE 101at the resonant frequency ωc/2πof 10.598\nGHz, simulated by COMSOL Multiphysics/circleR. The YIG4\n\u000bD\f \u000bF\f \u000bE\f\n\u000bG\f \u000bH\f \u000bI\f%\u0003 \u0003\u0013\u0011\u0016\u001a\u001c\u001a\u00037\n%\u0003 \u0003\u0013\u0011\u0016\u001a\u001c\u001a\u00037%\u0003 \u0003\u0013\u0011\u0016\u001a\u001c\u001a\u00037\n%\u0003 \u0003\u0013\u0011\u0016\u001a\u001c\u001a\u00037%\u0003 \u0003\u0013\u0011\u0016\u001b\u0013\u0017\u00037\n%\u0003 \u0003\u0013\u0011\u0016\u001b\u0013\u0017\u00037%\u0003 \u0003\u0013\u0011\u0016\u001b\u0013\u0017\u00037\n%\u0003 \u0003\u0013\u0011\u0016\u001b\u0013\u0017\u00037%\u0003 \u0003\u0013\u0011\u0016\u001b\u0014\u0014\u00037\n%\u0003 \u0003\u0013\u0011\u0016\u001b\u0014\u0014\u00037%\u0003 \u0003\u0013\u0011\u0016\u001b\u0014\u0014\u00037\n%\u0003 \u0003\u0013\u0011\u0016\u001b\u0014\u0014\u00037%\u0003 \u0003\u0013\u0011\u0016\u001b\u0014\u001b\u00037\n%\u0003 \u0003\u0013\u0011\u0016\u001b\u0014\u001b\u00037%\u0003 \u0003\u0013\u0011\u0016\u001b\u0014\u001b\u00037\n%\u0003 \u0003\u0013\u0011\u0016\u001b\u0014\u001b\u00037\nFIG. 3. (a) Measured microwave transmission spectrum, |S21(ω)|2of the 0.45-mm YIG-cavity hybrid system as a function of\nthe microwave frequency and the static magnetic field. The ho rizontal and diagonal dashed lines (yellow) show the freque ncy\nof the fundamental mode TE 101and the Kittel mode frequency, respectively. The white-das hed line describes the dispersion\nof the resonance frequency obtained by diagonalizing ˆHsas given in Eq.(1). (b) Cross sections of |S21(ω)|2at static magnetic\nfields corresponding to B= 0.3797, 0.3804, 0.3811, and 0.3818 T. Solid lines are theor etical curves given by Eq.(2) for the\ndata (solid dots). The individual data sets are vertically o ffset for clarity. (c) The phase S21(ω) with theoretical hand-fits at\nthe static magnetic fields. (d) Measured MO conversion spect rum,|S31,K(ω)|2of the same system. Here, the MO conversion\nspectrum is obtained from the raw data which is amplified by a m icrowave amplifier (30 dB) and detected by a fast photodiode.\n(e) Cross sections of |S31,K(ω)|2at static magnetic fields corresponding to B= 0.3797, 0.3804, 0.3811, and 0.3818 T. Solid\nlines are theoretical curves given by Eq.(7) for the data (so lid dots). (f) The phase S31,K(ω) with theoretical hand-fits at the\nstatic magnetic fields. The individual data sets are vertica lly offset for clarity.\nspheremounted onthe aluminarodalongthe crystalaxis\n/angbracketleft110/angbracketrightis placednearthe maximumofthe magneticfieldin\norder to get the largest coupling strength and the unifor-\nmity ofthe field as shownin Fig. 2(a). Fig. 2(b) presents\nmeasured transmission magnitude and phase without a\nYIGspherethroughthecavity. Asaresult, thefrequency\nof TE 101mode (ωc/2π) is 10.632 GHz, and the exter-\nnal cavity loss rate ( κe/2π) and the internal cavity loss\n(κi/2π) are 2.1 MHz and 0.6 MHz, respectively.\nIn order to manipulate the magnetic field, a set of\nneodymium-iron-boron magnets applies a static mag-\nnetic field of around 380 mT to the YIG sphere. The\nmagnetic components of the microwave field perpendic-\nular to the bias field induce the spin flip, and excite the\nmagnon mode in YIG. The magnetic circuit consists of a\nset of permanent magnets and a pair of Helmholtz coils\nwith 800 turns of wires for each. The cavity is placed at\nthe center of a pair of Helmholtz coils, so a static mag-\nnetic field along the z-axis is applied to the crystal axis\n/angbracketleft100/angbracketrightof the YIG sphere across the cavity. Helmholtz\ncoils driven by a bipolar current supplier combine with\nthe permanent magnetsand tunes the resonancefrequen-\ncies of magnon modes. The magnetic field measured bya flux gate sensor (3MTS) provides the field-to-current\nconversion rate of dB/dI= 70 Gauss /A. Fig. 2(c) shows\nthe experimental set-up for the microwave to light con-\nversion. We use temperature controlled butterfly diode\nlaser to deliver 1550-nm cw input power of 5 mW before\nthe YIG and get the transmission of 80 %. We also use\nsome of optics and microwave components such as po-\nlarizer and HWP to define the linear polarization, lens\nto focus the laser into the YIG, fast photo detector to\nreceive the transmitted laser with optical side band, low\nnoise microwave amplifiers with 30 dB amplification and\nisolators to increase the signal, and a vector network an-\nalyzer by probing the transmission through the hybrid\nsystem.\nFig. 3(a) shows the measured microwave transmis-\nsion spectrum, |S21(ω)|2, of the hybrid system with the\nYIG diameter of 0.45 mm as a function of the frequency\nand the static magnetic field. A normal-mode splitting\nis clearly observed, and the avoided crossing manifests\nstrong coupling between the Kittel mode and the mi-\ncrowave cavity mode. As the magnetic field is swept,\nthe Kittel mode approaches the cavity mode up to the\ndegeneracy point. The horizontal dashed line shows the5\nfundamental mode frequency of the cavity and the di-\nagonal dashed line presents the Kittel mode frequency,\nf11=ω11/2π=µ0γHo/2π(see Appendix B). White-\ndashed lines are the dispersion curves of the resonance\nfrequency, ω±=ωc+ωK\n2±1\n2/radicalbig\n4g2\nK+(ωc−ωK)2, which\nare obtained from the diagonalization of the interaction\nHamiltonian ˆHsin Eq.(1) without the last term. In or-\nder to quantify the coupling strength and the damping\nrate of ferromagnetic resonace frequency, the experimen-\ntal data at some of magnetic fields are hand-fitted into\nthe theoretical transmission coefficient S21(ω) given in\nEq.(2) (see Fig. 3(b)). As a result, the total cavity\nlinewidth ( κt/2π), the coupling strength ( gK/2π), and\nthe Kittel mode linewidth ( γK/2π) are determined as\n2.7, 28.6, and 2.3 MHz, respectively. Here, the coupling\nstrength gKcan be represented by gK=gB√\n2Ns=\nξγ\n2/radicalBig\n/planckover2pi1ωcµ0\nVc√\n2Ns, whereγis the electron gyromagnetic\nratio,ωcis the cavity resonance frequency, Vcis the vol-\nume of the cavity of 20 ×20×4 mm3, ands= 5/2 is the\nspin number in YIG26.gBis the coupling strength of a\nsingle Bohr magneton to the cavity which can be calcu-\nlated as 0.325 Hz for TE 101mode in our system. If we\nassumethatallthespinsintheYIGsphereareprecessing\nin phase, the coupling strength gKof the Kittel mode to\nthe cavity mode is proportional to the square root of the\nnumberofnetspins NK42. Inthiscase,almostallofspins\ncontribute to the KM. The coupling strength gM/2πfor\nanMSM islessthan 1.0MHz, sothisterm canbe ignored\nhere. The coefficient ξ≤1 indicates the spatial overlap\nand polarization matching conditions between the mi-\ncrowave and the magnon modes32. From the extracted\nvalue of gK, we can deduce the number of net spins of\nNK= 1.51×1017. The fact which gKis much largerthan\nκtandγK, indicates that the system is in the strong\ncoupling regime even at the room temperature. With\nthe experimental parameters, we obtain a cooperativity\nofCK=g2\nK/κtγK= 132, indicating how well collective\nspins in the YIG sphere couple to the microwave cavity\nmode43,44. Fig. 3(c) shows cross-sectional experimental\ndataandtheoreticalcurvesforthe phasesof S21(ω). The\nphase values of S21(ω) range from −π/2 toπ/2 that are\ntwo times less than the phase values of S31(ω) as shown\nin Fig. 3(f). In Ref. [26], the phase of a reflection spec-\ntrumS11(ω) shows the same feature as that of S31(ω)\nexcept the scale factor of 2. In this work, since all ex-\nperimental data are based on the S21(ω) transmission\nspectra, the phase of reflection spectrum S11(ω) is given\nin Appendix C. Here, Fig. 7 shows the similar feature to\nthe phase of S31(ω). Fig. 3(c) and 3(f) also show that\nthe phase of S21(ω) is clearly converted to the phase of\nS31(ω). Therefore, the conveyance of the phase clearly\nexhibits the coherent conversionfrom microwaveto light.\nFig. 3(d) shows the measured power of the MO con-\nversioncoefficient, |S31,K(ω)|2, ofthehybridsystemwith\nthe same YIG. The conversion spectrum |S31,K(ω)|2is\nalmost similar to |S21(ω)|2, which implies that the mi-\ncrowave field is coherently converted to the optical wavefield. Fig. 3(e) shows the cross sectional experimental\ndata at some of magnetic fields in Fig. 3(d) that are\nhand-fitted into the theoretical transmission coefficient\n|S31,K(ω)|2given in Eq.(7). As a result, gK/2πand\nγK/2πare 28.5 MHz and 2.4 MHz, respectively that are\nquite close to the result of |S21(ω)|2. In orderto evaluate\nthe MO conversion efficiency, we first estimate the opti-\ncal photon-magnon coupling rate δKwhich is given by\nδK=G2\nKl2\n16VmnKP0\n/planckover2pi1Ω026. Withl= 0.45 mm being the length\nof the YIG sample, nK= 3.16×1027m−3andVm=\n4\n3π×0.2253mm3being the spin density and the spatial\nvolume of the magnetostatic mode, V= 3.8 radians/cm\nat 1.55µm49that result in GK= 4V/nK= 4.81×10−25\nm2, andP0//planckover2pi1Ω0= 1.17×1017Hz forP0= 15 mW,\nwe have δK/2π= 0.0036 Hz. Therefore, under the near\nresonant conditions at ω−ωc=ω−ωK= 0, the total\nconversion efficiency ηt=|S31,K|2can be approximated\nin terms of the coupling strength gK,\nηt=/parenleftbigg2√δKκeCK\ngK(1+CK)/parenrightbigg2\n. (10)\nWith all obtained parameters for the YIG sphere with\n0.45 mm diameter, we attain ηt= 8.45×10−11. This low\nconversion efficiency results from the small light-magnon\ncoupling rate. Actually, the maximum conversion effi-\nciency is occured at particular detunings from the cavity\nresonance and the Kittel mode frequency26. However,\nin this experiment, we are interested in the multi-mode\nconversion efficiency at the degenerate point.\nIn orderto examinethe YIG sizedependence ofsystem\nparameters,wealsomeasuredthetransmissionspectraof\nYIG spheres with diameters of 0.75 and 1 mm, as shown\nin Fig. 4. Fig. 4(a) and 4(e) show transmission magni-\ntude,|S21(ω)|2, measured for YIG diameter of 0.75 and\n1.0 mm, respectively. As the size of the YIG sphere in-\ncreases, the larger number of spins in a bigger sphere\ncan contribute the interaction with the microwave cavity\nmode, that makes the normal-mode splitting wider. As\na result, we obtain the coupling strengths of gK/2π=\n67.3 and 91 MHz for 0.75 and 1.0-mm YIG spheres so\nthat the cooperativity CKfor the Kittel mode reaches\nup to about 3.5 ×103as shown in Table I. In addition to\nlarger coupling strengths, another avoided level crossing\nfeature is observed because of the strong coupling cor-\nresponding to the nonuniform MSMs which can be also\ncoupled to the cavity mode. The coupling strengths of\nMSM are gM/2π= 4 and 12 MHz, and the decay rates\nareγM/2π= 1.1 and 0.9 MHz for YIG spheres with\n0.75 and 1.0 mm diameter, respectively. Based on the\nfitting parameters, 2D spectra of |S21(ω)|2for each case\nare simulated in Fig. 4(b) and (f). Here, the red-dashed\nline describes the nonuniform MSM which is identified\nby magnetostatic theory35,37. In general, the relation be-\ntween MSM frequencies and the external magnetic field\nis linear for i−|j|= 0 or 1 as mentioned in Appendix B.\nWhen the YIG sphere is subjected to an oscillating mag-\nnetic field at ωijand a strong coupling regime is reached6\n\u000bD\f \u000bE\f \u000bF\f\n\u000bH\f \u000bI\f \u000bJ\f\u000bG\f\n\u000bK\f\nFIG. 4. (a) Measured |S21(ω)|2of the 0.75-mm YIG-cavity hybrid system as a function of the m icrowave frequency and the\nstatic magnetic field. The horizontal and diagonal dashed li nes show the frequency of the fundamental mode TE 101and the\nKittel mode frequency, respectively. The white-dashed lin e describes the dispersion of the resonance frequency obtai ned by\ndiagonalizing ˆHsas given in Eq.(1). (b) The simulated spectrum of |S21(ω)|2based on Eq.(2). For FMR or the Kittel mode,\ngK/2πandγK/2πare 67.3 and 1.1 MHz, respectively, and for MSM, gM/2πandγM/2πare 4 and 1.5 MHz, respectively.\nThe red-dashed line refers to the (2,0) mode. (c) Measured MO conversion spectrum, ηtof the 0.75-mm YIG-cavity hybrid\nsystem. The measured spectrum is based on the raw data which i s amplified by a microwave amplifier (30 dB) and detected by\na fast photodiode. (d) The simulated spectrum of ηtof the 0.75-mm YIG-cavity hybrid system based on Eq.(7). (e) Measured\n|S21(ω)|2through the 1.0-mm YIG-cavity hybrid system. (f) The simula ted spectrum of |S21(ω)|2based on Eq.(2). For FMR\nor the Kittel mode, gK/2πandγK/2πare 91 and 0.95 MHz, respectively, and for MSM, gM/2πandγM/2πare 12 and 0.9\nMHz, respectively. The red-dashed line refers to the (2,0) m ode as given in Eq.(11). (g) Measured ηtof the 1.0-mm YIG-cavity\nhybrid system. The measured spectrum is based on the raw data which is amplified by a microwave amplifier (30 dB) and\ndetected by a fast photodiode. (h) The simulated spectrum of ηtof the 1.0-mm YIG-cavity hybrid system based on Eq.(7).\natHo, avoidedlevelcrossingsappearatthe regionswhere\ntheresonancefrequenciesoftwosubsystemsarematched,\nthat make it possible to distinguish a MSM with iandj\nassociated to an avoided level crossing45–48. In our case,\nadditional avoided level crossing is placed at the (2,0)\nmode which can be given by37\nω20=µ0γMs/radicalBigg/parenleftbiggHo\nMs−1\n3/parenrightbigg/parenleftbiggHo\nMs+7\n15/parenrightbigg\n.(11)\nFig. 4(c) and (g) show the MO conversion spectra, ηt,\nmeasured for the YIG diameter of 0.75 and 1.0 mm, re-\nspectively. These MO conversion spectra present raw\ndata which are amplified and detected by using a mi-\ncrowave amplifier and a fast photodiode. One can find\nthe same avoided level crossing features, as shown in Fig.\n4(a) and 4(e), which clearly demonstrates the coherent\nconversion from microwave to optical photons. Based\non the fitting parameters, 2D spectra of ηtfor each case\nare simulated in Fig. 4(d) and 4(h). As a result, when\nwe take into account both KM and MSM contributions,\nthe total conversion efficiency ηtare 5.12×10−12and\n3.46×10−12for 0.75 and 1.0 mm YIG spheres, respec-\ntively. We summarize system parameters for each YIGsphere in Table I that are obtained from the two-mode\nMO conversion process.\nTo examine the size dependence of a YIG sphere for\nMOconversionefficiency, wefirstevaluatethevolumede-\npendence ofparametersusedforMLconversionefficiency\nat resonant conditions as shown in Fig. 5. According to\nthe Ref. [30], it demonstrated that the coupling strength\ngKof the Kittel mode is proportional to the square root\nof the volume (or the number of spins) of YIG spheres.\ngKis linear-fitted to f(x) = 130 .97x, where xis the\nsquare root of volume V1/2. For the higher MSM, gM\nis not proportional to the linear function, but rather the\nquardratic function which is f(x) = 22.19x2(Fig. 5(a)).\nThismight be due to the factthat the spin excitationsin-\nduced by non-uniform field do not linearly contribute to\na higher mode. According to the relation of Cm=g2\nm\nκtγm,\nthe cooperativity CKis fitted to f(x) = 6569 .43x2and\nforCM,f(x) = 228.79x4is used.\nInaddition, δKandδMalsohavethedependenceofthe\nnumber of spins. δKis fitted to f(x) = 0.693+0.625/x\nandδMis fitted to f(x) = 0.139/x2as shown in Fig.\n5(b). By using these fitting values of system parameters,\nwe obtain the theoretical fit curve for the MO conversion7\nTABLE I. System parameters for two-mode MO conversion\nParameter 0.45-mm dia. 0.75-mm dia. 1.0-mm dia.\ngK[MHz] 2 π×28.6 2 π×67.3 2 π×91.0\nγK[MHz] 2 π×2.3 2 π×1.1 2 π×0.95\ngM[MHz] <2π×1.0 2 π×4.0 2 π×12.0\nγM[MHz] >2π×2.0 2 π×1.5 2 π×0.9\nCK 132 1373 3487\nCM 0.19 3.6 64\nNK 1.51×10178.36×10171.53×1018\nNM <1.84×10142.95×10152.66×1016\nVm[m3] 4.77 ×10−112.21×10−105.24×10−10\nnK[m−3] 3.16 ×10273.79×10272.92×1027\nnM[m−3]<3.87×10241.34×10255.07×1025\nGK[m2] 4.81 ×10−254.02×10−255.21×10−25\nGM[m2]>\n3.93×10−221.14×10−222.99×10−23\nδK[mHz] 2 π×3.61 2 π×1.80 2 π×1.75\nδM[Hz] 2 π×2.95 2 π×0.512 2 π×0.101\nηt 8.45×10−115.12×10−123.46×10−12\nSubscripts KandMdenote the Kittel mode and MSM, re-\nspectively.\n\u000bD\f\n\u000bE\f\n\u0013\u0011\u0017\u0018\u0003PP\n\u0013\u0011\u001a\u0018\u0003PP\n\u0014\u0011\u0013\u0003PP\u0013\u0011\u0017\u0018\u0003PP\u0013\u0011\u001a\u0018\u0003PP\u0014\u0011\u0013\u0003PP\n[\u0014\u0013\u0010\u0014\u0014[\u0014\u0013\u0016\n/ScriptT\n/ScriptT\nFIG. 5. (a) Extracted values (solid circles) and fit curves\nfor coupling strengths and cooperativities for the KM and\nMSM as a function of the square root of the YIG volume.\n(b) Extracted values (solid circles) and fit curves for optic al\nphoton-magnon coupling rates for the KM and MSM and the\ntotal MO conversion efficiency as a function of the square root\nof the YIG volume.efficiency based on Eq.(8) as presented in Fig 5(b). As a\nYIG size increases, the total MO conversion efficiency at\nthe resonant condition decreases since the increments of\ncoupling strength and cooperativity lead to the drop in\nthe MO conversion efficiency as given by Eq.(8). In our\nsystem, the conversion efficiency at the resonant condi-\ntion is limitted to 10−11order. This mainly comes from\nthe small coupling rate δKandδMbetween the opti-\ncal photons and magnons although it depends on the\nexperimental conditions such as the quality of the sam-\nple and proper alignment. Therefore, we need to im-\nprove the coupling rate δmto enhance coherent quantum\nconversionefficiency between microwaveand optical pho-\ntons. There are several suggestions as mentioned in ref.\n[26]. One possible way was to use the optical whispering\ngallery modes (WGMs) of an YIG sphere50,51. No one\nhasachievedasignificantimprovement,however,suppos-\nedly due to the small overlap between the Kittel mode\nand WGMs. Other suggestions are to utilize a magnetic\nmaterial with a large Verdet constant such as CrBr 3and\niron garnet based on rare-earth atoms52–55.\nIV. DISCUSSION\nWe clearly observe not only the YIG size dependence\nof the MO conversion but also the coupling strength be-\ntween the multi-magnetostatic mode and a cavity. But\nnote that the multi-mode MO conversionfeatures arenot\nprominent compared to the single-mode MO conversion\nsince the most of spins are involved in the KM mode\nthat makes gMandCMmuch lower than the values of\ngKandCK. In order to make the dominant contribution\nof spins to higher MSM, we carefully position a 1.0 mm-\nYIG sphere off the uniform microwave mode region, so\nthat a non-uniform MSM also apprearsat the degenerate\npoint as shown in Fig. 6(a). In this configuration, the\nanti-crossingsduetothehigherMSMbecomelargersince\nthe number of spins contributing to the higher mode in-\ncreases. Fig. 6(b) shows the simultion result of |S21(ω)|2\nbased on Eq.(2). As a result, gK/2πandgM/2πare 83.4\nand 25 MHz, respectively, which are few orders larger\nthan decay rates of γK/2π=1.1 and γM/2π=0.5 MHz,\nthat indicates the strong couplings between the cavity\nmode and the KM and MSM. Fig. 6(c) presents the 2D\nspectrum of ηt. Based on the measured data, ηtis sim-\nulated by using Eq.(7) as shown in Fig. 6(d). We find\nout that the theoretical model is well matched with the\nexperimental results. Here, we ignore higher modes in\nthe tail of the spectrum because their contributions are\nvery small in the MO conversion efficiency. The total\nmulti-mode MO conversion efficiency is 1.02 ×10−11at\nthe resonant condition. To date, adjustable MO conver-\nsion for multi-modes has not been reported in magnon-\ncavity system.8\n\u000bD\f \u000bE\f\n\u000bF\f \u000bG\f\nFIG. 6. (a) Measured |S21(ω)|2through the 1.0-mm YIG-cavity hybrid system while YIG posit ion adjusted to enhance the\nMSMs. (b) The simulated spectrum of |S21(ω)|2from Eq.(2). (c) Measured ηtof the same system. The measured spectrum\nis based on the raw data which is amplified by a microwave ampli fier (30 dB) and detected by a fast photodiode. (d)The\nsimulated spectrum of ηtfrom Eq.(7).\nV. CONCLUSION\nWe have experimentally demonstrated coherent multi-\nmode conversion from microwave to optical fields via a\nYIG sphere in a rectangular microwave cavity. A large\nnumber of spins in ferromagnetic materials easily couple\nthe collective excitation to cavity photons, that makes it\npossible to hybridize the microwave photon modes and\nmagnetostatic modes. A traveling optical field is coupled\nto a microwavefield through this hybrid system. We first\nobserved YIG size dependence of conversion efficiency by\nmeasuring the normal-mode splitting between the mag-\nnetostatic modes and the microwavecavitymodes, where\nthe coupling strength is in the order of magnitude larger\nthan the decay rates. Based on our multi-mode MO con-\nversion model, we analyzed all the system parameters\nwith experimental data, confirming that the theoretical\nmodel is consistent with the experimental results. The\ntotal multi-mode conversion efficiency at the resonant\ncondition reaches 1.02 ×10−11for 1.0 mm-YIG sphere.\nWe also evaluate the multi-mode MO conversion effi-\nciency by manipulating position of the YIG sphere in-\nside the cavity. These sharp and adjustable multi-mode\nconversion shows the possibility of coherent conversionof multi-mode quantum states while keeping coherence\ntime. This work will also provide optimal design condi-\ntions of a cavity magnon-microwave photon system that\ncan be used for coherent conversion between microwave\nand optical photons.\nACKNOWLEDGMENTS\nThis work was supported by a grant to Quantum Fre-\nquency Conversion Project funded by Defense Acquisi-\ntion Program Administration and Agency for Defense\nDevelopment. We would like to thank prof. Changsuk\nNoh for his valuable comment on the quantum input-\noutput relation and also thank Jinwon Yoo, prof. Won-\nmin Son, and prof. Suyeon Cho for productive discussion\non the hybrid system.9\nAppendix A: The interaction Hamiltonian for the\nmulti-mode microwave-to-optical wave conversion\nThe Hamiltonian for the magnon-cavity system can be\nwritten as\nHs=ωcˆa†ˆa+/summationdisplay\nm=K,M/bracketleftBig\ngµBBm\nzˆSm\nz+gm(ˆaˆSm\n++ˆa†ˆSm\n−)/bracketrightBig\n,\n(A1)\nwhereωcis the angular frequency of the cavity mode\nTE101, ˆa†(ˆa) is the microwave photon creation (anihila-\ntion) operator, and m=K,Mdenotes the Kittel mode\n(KM) and magneto static modes (MSM), respectively. g\nis the electron g-factor, µBis the Bohr magneton, and\nBm\nzis the effective magnetic field affected by the magnon\nmodes of the YIG sphere. The exchange interaction be-\ntween electron spins can be ignored because of the long-\nwavelength descrete modes of spins in the YIG sphere.\nSince the frequency of the corresponding magnon mode\nis different from each other, the Hamiltonian for each\nmagnon mode can be written seperately. Here, ˆSmis\nthe collective spin operator for magnon modes which is\ngiven by ( ˆSm\nx,ˆSm\ny,ˆSm\nz). These collective spin operators\ncan be expressed in terms of the bosonic operators ˆ s†\nm,\nˆsmby using the Holstein-Primakoff transformation56,57:\nˆSm\n+=ˆSm\nx+iˆSm\ny= ˆs†\nm/radicalBig\n2Sm−ˆs†\nmˆsm,ˆSm\n−=ˆSm\nx−iˆSm\ny=\n(/radicalBig\n2Sm−ˆs†\nmˆsm)ˆsm, andˆSm\nz= ˆs†\nmˆsm−2Sm, whereSm\nis the total spin number of the corresponding magnon\nmode. For the low-lying excitations/angbracketleftbig\nˆs†\nmˆsm/angbracketrightbig\n≪2Sm, the\nHamiltonian Hscan be obtained as\nHs=ωcˆa†ˆa+/summationdisplay\nm=K,M/bracketleftbig\nωmˆs†\nmˆsm+gm(ˆaˆs†\nm+ˆa†ˆsm)/bracketrightbig\n,\n(A2)\nwhereωm=gµBBm\nzis the angular frequency of the\ncorresponding magnon mode and gm=gB√\n2Sm=\nξγ\n2/radicalBig\n/planckover2pi1ωcµ0\nVc√\n2Sm. Here,γis the electron gyromagnetic\nratio,ωcis the cavity resonance frequency, Vcis the vol-\nume of the cavity, gBis the coupling strength of a single\nBohr magneton to the cavity for TE 101mode, and ξis\nthe spatial overlapping coefficient which is relevant to\nthe spatial variation effect. In the Kittel mode, since the\nmagnetic dipiole coupling between the spins engenders a\nuniform demagnetization field parallel to the magnetiza-\ntion in a sphere, the demagnetizing field plays no role\nin the magnetization dynamics for the Kittel mode. For\nthe non-uniform profile for MSM, the variation in space\nplays a crucial role not only in the frequency calculation\nbut also in the coupling with the exciting field as well as\nthe light.\nTherefore, the interaction Hamiltonian of the multi-\nmode MO conversion can be given by Eq.(1) which con-\nsists of the magnon, microwave photon, optical photon,\nand their interactions. According to the input-output\nrelation40, equations of motions for a cavity mode and\nmagnon modes can be obtained from quantum Langevinequation. For the cavity mode ˆ a,\n˙ˆa(t) =−i[ˆa,ˆHs]−κtˆa(t)−√\n2κeˆai(t)\n=−iωcˆa(t)−i(gKˆsK(t)+gMˆsM(t))−κtˆa(t)\n−√2κeˆai(t),\n(A3)\nwhere the total loss rate κt=κe+κiincludes both exter-\nnal coupling and internal losses of the cavity. The cavity\nmode ˆacan be given by\nˆa(t) =χc(ω)/bracketleftbig\ngKˆsK(t)+gMˆsM(t)−i√\n2κeˆai(t)/bracketrightbig\n,(A4)\nwhere\nχc(ω) = [ω−ωc+iκt]−1. (A5)\nSincemagnonsdonotcoupledirectlytothecavity,noad-\nditional input and output magnons are involved. There-\nfore, the equation of motion of ˆ sKcan be given by\n˙ˆsK(t) =−i[ˆsK,ˆHs]−γKˆsK(t)\n=−iωKˆsK(t)−igKˆa(t)−γKˆsK(t)\nˆsK(t) =χK(ω)gKˆa(t),(A6)\nwhere\nχK(ω) = [ω−ωK+iγK]−1. (A7)\nIn the same manner, ˆ sMhas the similar form which is\nˆsM(t) =χM(ω)gMˆa(t), (A8)\nwhere\nχM(ω) = [ω−ωM+iγM]−1. (A9)\nSubstituting Eq. (A6) and Eq. (A8) into Eq. (A4) and\napplying the Fourier transform of the cavity mode ˆ a, we\ncan derive\nˆa(ω) =−i√\n2κeT−1ˆai(ω), (A10)\nwhere\nT=ω−ωc+iκt−(g2\nKχK+g2\nMχM).(A11)\nIn our experiment, we obtain the transmission spectrum\nwhich can be determined by measuring the output port\n2 from the input port 1. For no input in port 2 and the\nsame external coupling rate at both ports, the boundary\ncondition becomes ˆ ao,2(ω) =√2κeˆa(ω) that results in\nthe transmission,\nS21=ˆao,2\nˆai,1=−i2κeT−1. (A12)\nFor the transmission for multi modes, Eq. (A12) can be\nextended to T=ω−ωc+iκt−/summationtext\nmg2\nmχm.10\nIn the conversion process from microwave to optical\nwave, we can obtain the equation of motions for magnon\nmodes which are given by\n˙ˆsK(t) =−i[ˆsK,ˆHs]−γKˆsK(t)\n−/radicalbig\n2δK/parenleftBig\nˆbi(t)eiΩ0t−ˆb†\ni(t)e−iΩ0t/parenrightBig\n˙ˆsM(t) =−i[ˆsM,ˆHs]−γMˆsM(t)\n−/radicalbig\n2δM/parenleftBig\nˆbi(t)eiΩ0t−ˆb†\ni(t)e−iΩ0t/parenrightBig\n.(A13)\nAs a result, magnon modes ˆ sKand ˆsMare written as\nˆsK(t) =χK/bracketleftBig\ngKˆa(t)−i/radicalbig\n2δKˆbi(t)/bracketrightBig\nˆsM(t) =χM/bracketleftBig\ngMˆa(t)−i/radicalbig\n2δMˆbi(t)/bracketrightBig\n.(A14)\nAfter substituting Eq. (A4) into Eq. (A14) and applying\nthe Fourier transform, we can obtain magnon modes for\nKM and MSM which are given by\nˆsK(ω) =gKgMχKχcTKˆsM(ω)−i√\n2κegKχKχcTKˆai(ω)\n−i/radicalbig\n2δKχKTKˆbi(ω)\nˆsM(ω) =gKgMχMχcTMˆsK(ω)−i√2κegMχMχcTMˆai(ω)\n−i/radicalBig\n2δ′\nMχMTMˆbi(ω),\n(A15)\nwhere\nTK(ω) =/bracketleftbig\n1−g2\nKχKχc/bracketrightbig−1\nTM(ω) =/bracketleftbig\n1−g2\nMχMχc/bracketrightbig−1.(A16)\nIf we substitute ˆ sK(ˆsM) into ˆsM(ˆsK) in Eq. (A15), we\ncan obtain the MO conversion coefficients for KM and\nMSM. For the KM, by consideringthe Stokes (Ω = Ω 0−\nω)andanti-Stokes(Ω = Ω 0+ω)processesandthebound-\nary conditions ˆb†\no(Ω0−ω) =ˆb†\ni(Ω0−ω)+√2δKˆcK(ω) and\nˆbo(Ω0+ω) =ˆbi(Ω0+ω)+√2δKˆcK(ω)26, the conversion\ncoefficient for the KM is obtained as\nS31,K(ω) =√βK\n2i/parenleftBigg/angbracketleftBiggˆb†\no(Ω0−ω)\nˆai(ω)/angbracketrightBigg\n+/angbracketleftBiggˆbo(Ω0+ω)\nˆai(ω)/angbracketrightBigg/parenrightBigg\n=−2/radicalbig\nβKδKκegKχKχc/parenleftbig\n1+g2\nMχMχcTM/parenrightbig\n1−g2\nKχKχc(1+g2\nMχMχcTM),\n(A17)\nwhereβKis the amplification factor. Here, we point out\nthat, if we take into account the MO conversion of only\nthe KM ( gM= 0), Eq. (A17) becomes the same result as\nthe single-mode MO conversion coefficient in Ref. [26].\nIn the same manner, we can induce the MO conversion\ncoefficient for a MSM by using similar boundary condi-\ntions and amplification factor of βMthat is given by\nS31,M(ω) =−2/radicalbig\nβMδMκegMχMχc/parenleftbig\n1+g2\nKχKχcTK/parenrightbig\n1−g2\nMχMχc(1+g2\nKχKχcTK).\n(A18)Therefore,theconversionefficiencyforthetwo-modeMO\nconversion for the KM and a MSM can be obtained as\nηt(ω) =|S31,K(ω)|2+|S31,M(ω)|2.(A19)\nAppendix B: Magnetostatic modes in a\nferromagnetic sphere\nMagnons are spin excitations describing small pertur-\nbations to the magnetization of a ferromagnetic system.\nA small oscillating magnetic field in the plane perpendic-\nular to the bias field can lead the alignment of spins to\ndeviate slightly from the bias direction. The bias field\nexerts a torque on misaligned spins, and then the spins\nbegin precessing around it. L.R. Walker first considered\nthe relationshipbetweenthe resonancefrequencyand the\ninternal static field of a ferromagnetic spheroid35,36. He\nassumed that the microwave magnetic fields in spheroids\nsatisfy the magnetostatic approximations. The allowed\nresonant frequencies of MSMs in a sphere inserted in\na microwave cavity can be derived from the character-\nistic equation in terms of associated Legendre function\nPj\ni(ξ0)37.\ni+1+ξ0Pj\ni′(ξ0)\nPj\ni(ξ0)±jχ2= 0, (B1)\nwhereξ2\n0= 1 +1\nχ1,χ1=γ2MsHi\nγ2H2\ni−f2,χ2=γMsf\nγ2H2\ni−f2,\nHi=H0−Ms\n3, andPj\ni′(ξ0) =dPj\ni(ξ0)\ndξ0. Here, Hiand\nHoareinternalandexternalmagneticfields, respectively.\nµ0Ms= 0.178 T (at 298 K)58,59is the saturation magne-\ntization, µ0is the vacuum permeability,γ\n2π= 28 GHz/T\nis the gyromagnetic ratio, and fis the frequency. iand\njare mode indices that i≥1 is an integer and jis also\nan integer obeying −i≤j≤i.\nFor a single mode solution, it is labelled with ( i,j).\nFor MSMs with i− |j|= 0 or 1, the relations between\nthe resonant frequencies and the external magnetic field\ncan be given by37,60\nωij\nµ0=γHo+/parenleftbiggj\n2j+1−1\n3/parenrightbigg\nγMs(i=j),(B2)\nωij\nµ0=γHo+/parenleftbiggj\n2j+3−1\n3/parenrightbigg\nγMs(i=j+1).(B3)\nHere, the (1 ,1) FMR mode, known as the Kittel mode, is\nthe lowest mode in which all spins precess in phase which\ngives a frequency given by ω11=µ0γHo.\nAppendix C: Microwave reflection spectrum\nFigure 7(a) and (b) show the measured reflection spec-\ntrum|S11(ω)|2and the phase S11(ω) of the hybrid sys-\ntem with the 0.45 mm-dia YIG as a function of the mi-\ncrowavefrequency. 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Here we demon-\nstrate optical whispering gallery modes of a YIG sphere interrogated by a silicon nitride photonic\nwaveguide, with quality factors approaching 106in the telecom c-band after surface treatments.\nMoreover, in contrast to conventional Faraday setup, this implementation allows input photon po-\nlarized colinearly to the magnetization to be scattered to a sideband mode of orthogonal polarization.\nThis Brillouin scattering process is enhanced through triply resonant magnon, pump and signal pho-\nton modes - all of whispering gallery nature - within an \\optomagnonic cavity\". Our results show\nthe potential use of magnons for mediating microwave-to-optical carrier conversion.\nHybrid magnonic systems have been emerging recently\nas an important approach towards coherent informa-\ntion processing1{9. The building block of such systems,\nmagnon, is the quantized magnetization excitation in\nmagnetic materials10,11. Its great tunability and long life-\ntime make magnon an ideal information carrier. Partic-\nularly, in magnetic insulator yttrium iron garnet (YIG),\nmagnons interact with microwave photons through mag-\nnetic dipole interaction, which can reach the strong and\neven ultrastrong coupling regime thanks to the large spin\ndensity in YIG4{6. Besides, the magnon can also couple\nwith the elastic wave12,13and optical light14,15, it is of\ngreat potential as an information transducer that medi-\nates inter-conversion among microwave photon, optical\nphoton and acoustic phonon. Long desired functions,\nsuch as microwave-to-optical conversion, can be realized\non such a versatile platform.\nMagneto-optical (MO) e\u000bects such as Faraday e\u000bect\nhave been long discovered and utilized in discrete op-\ntical device applications16{18. Based on such e\u000bects,\nmagnons can coherently interact with optical photons.\nOn the one hand, magnon can be generated by optical\npumps19{22. On the other hand, optical photons can be\nused to probe magnon through Brillouin light scattering\n(BLS)15,23. However, in previous studies the typical ge-\nometries are all thin \flm or bulk samples inside which\nthe optical photon interacts with magnon very weakly,\nusually only through a single pass. For high e\u000ecient\nmagnon-photon interaction, it is desirable to obtain triple\nresonance condition of high quality ( Q) factor modes, i.e.,\nthe magnon, the input and the output optical photons are\nsimultaneously on resonance.\nIn this Letter, we demonstrate the magnon-photon in-\nteraction in a high Qoptomagnonic cavity which simul-\ntaneously supports whispering gallery modes (WGMs) of\noptical and magnon resonances. With high-precision fab-\nrication and careful surface treatment, the widely used\nYIG sphere structure, which is inherently an excellent\nmagnonic resonator, exhibits high optical Qfactors in\nour measurements. YIG has a high refractive index (2.2\nin the telecom c-band), which poses a challenge for e\u000e-\ncient light coupling with silica \fber tapers. By employingan integrated silicon nitride optical waveguide with an ef-\nfective index (around 2.0) matching that of YIG, we can\ne\u000eciently couple to both the TM and TE optical reso-\nnances in the YIG sphere. By driving the system with\nan optical pump, the magnons excited by an input mi-\ncrowave signal can be converted into optical signal of a\ndi\u000berent color. Our demonstration shows the great po-\ntential of YIG sphere as a platform to bridge the gap\nbetween magnon and optical photons, paving the way\ntowards using magnon as a transducer for coherent in-\nformation processing between distinct carriers.\nIn magnetized YIG material, magnons are the collec-\ntive excitations of spin states of Fe3+ions24. The cre-\nation or annihilation of a single magnon corresponds to\nthe ground spin \rip. At the microwave frequencies, the\nmagnon state can be manipulated straightforwardly by\nmagnetic dipolar transition using the oscillating mag-\nnetic \felds of microwave photons. While at the optical\nfrequencies, the magnon manipulation becomes di\u000ecult\nbecause the magnetic transition is negligible whilst di-\nrect spin-\rip by electric dipole transition is forbidden25.\nAlternatively, the optical photons can modify the ground\nstate spin through a two-photon transition by means of\nan orbital transition and spin-orbit interaction (the MO\ne\u000bect)14. Such a process has been previously studied us-\ning conventional Faraday setups26, in which light prop-\nagates parallel to the magnetic \feld and interacts with\nmagnon in a single pass. It is natural to consider shaping\nthe YIG into an optical cavity to boost up the magnon-\nphoton interaction as light passes the magnetic material\nmultiple times. Therefore, we propose to use a whisper-\ning gallery resonator made by YIG to provide enhanced\nmagnon-photon coupling in a triply resonant con\fgura-\ntion, and the mechanism is explained in the following\ndiscussions.\nThe optomagnonic cavity we used in our experiments\nis a single crystal YIG sphere. Due to the spherical\nsymmetry, lights are con\fned in the sphere by total in-\nternal re\rection and form WGMs. Each optical WGM\nis characterized by three mode numbers ( q;l;n ), which\ncorrespond to the radial, angular and azimuthal order\n(n=\u0000l;:::;l ), respectively27,28. Moreover, the WGMsarXiv:1510.03545v1  [cond-mat.mes-hall]  13 Oct 20152\nTM\nTE\nωωmSignal Pumpx\nyzσ+\nπ\nσ+\nnm−1nmπ σ+ π(b)(a)\n(c)\nFIG. 1. (a) Schematic illustration of the magnon-photon in-\nteraction. The YIG sphere is biased by a magnetic \feld along\nzdirection, while the WGMs propagate along the perimeter\nin thex-yplane. The TM input light excites the \u0019WGM\nin the YIG sphere, which is scattered by magnon into \u001b+\npolarized photon and then converts to the TE output in the\nwaveguide. (b) Energy level diagram of the magnon-photon\ninteraction. (c) Triple resonance condition for the enhanced\nmagnon-photon interaction process in the optomagnonic res-\nonator.\nare also characterized by their polarization, i.e., the di-\nrection of their electric \feld distribution. Conventional\nFaraday setups require the bias magnetic \feld to be par-\nallel to the direction of light propagation. However, for\nWGMs light propagates along the circumference of the\nsphere [Fig. 1(a)], therefore the bias magnetic \feld should\nbe inx-yplane. Due to the geometry symmetry, the MO\ne\u000bect vanishes for such a Faraday con\fguration. At a\n\frst glance, the MO e\u000bect also vanishes for bias mag-\nnetic \feld along zdirection, since it requires circular\npolarization in respect to\u0000 !Hwhile WGMs are linearly\npolarized, either parallel (TE) or perpendicular (TM) to\nthezdirection. However, thanks to the \feld gradient\nat the dielectric interface29{31, there are non-zero optical\nelectric \felds along the propagation direction for the TE\npolarized WGMs. As a result, the electric \feld rotates\nwithin thex-yplane and forms a cycloid trajectory, sim-\nilar to the elliptically polarized light propagating in free\nspace. Therefore, the TE WGMs possess partial circular\npolarization ( \u001b+) and can have magnetic response via\nFaraday e\u000bect, as schematically illustrated in Fig. 1(a).\nNote that the pump light can propagate either clockwise\n(CW) or counterclockwise (CCW), with di\u000berent conser-\nvation conditions accordingly, as will be shown below.\nSimilar to the optical WGMs, the magnon modes in\nYIG sphere can also be characterized by three mode\nnumbers (qm;lm;nm)32. For the uniform magnon mode\nH\nz\nxy\nOutput fiberIntput fiber\nBefore A/g332er\nFibersWaveguideMagnetChip\nAntenna\nMagnet\n(c) (e)(a)\n(d)(b)FIG. 2. (a) and (b) Schematic and optical image of the ex-\nperimental assembly of our optomagnonic device, respectively.\n(c) Scanning electron microscope image of the polished YIG\nsphere. The scale bar is 100 \u0016m. (d) The surface of the YIG\nsphere before and after our surface treatment process. Scale\nbars are 1\u0016m. The sub-micrometer particles vanish after the\nsurface treatment. (e) Optical image of the silicon nitride\ncoupling waveguide chip with glued \fbers on the two sides.\nThe chip and the \fbers are attached to a piece of glass holder\nfor mechanical support and reducing long-term drift.\nwith all the spins precessing in phase, the corresponding\nmode numbers are (1 ;1;1). The microscopic mechanism\nof the magnon-photon interaction is intrinsically a three-\nwave process, as schematically illustrated by Fig. 1(b).\nDue to the spin angular momentum conservation, every\ntime when the magnon number increases by 1 it indi-\ncates that the electron spin increases by 1, which corre-\nsponds to a two-photon transition in the form of \u001b+!\u0019\n(CCW) or \u0019!\u001b\u0000(CW). As a result, there would be\nonly one optical sideband generated for a given pump-\ning light direction. The mesoscopic model of the MO\ne\u000bect is represented by the permittivity tensor \"ij=\n\"0(\"r\u000eij\u0000if\u000fijkMk)24, where\"0is the vacuum permittiv-\nity,\"ris the relative permittivity of YIG, \u000eijand\u000fijkare\nKronecker and Levi-Civita symbols, fis the Faraday co-\ne\u000ecient,Mkis the magnetization, and i,j,kcorrespond\ntox,y,zdirection, respectively. When the energy is con-\nserved for the two-photon and magnon transitions that\n!1\u0000!2=!m, the coupling strength between two opti-\ncal modes is g=\u0001\n\u0001\"ij(\u0000 !x)E\u0003\n1;i(\u0000 !x)E2;j(\u0000 !x)d\u0000 !x3, where\nEp;i(\u0000 !x) (p= 1;2) is the normalized \feld of optical WGM\u0001\n\"ii(\u0000 !x)jEp;i(\u0000 !x)j2d\u0000 !x3=!p, and magnon induced per-\nmittivity change is \u0001 \"ij(\u0000 !x) =\u0000if\"0\u000fijkMk(\u0000 !x). As the\n\feld distributions are in the form of ein\u001ein the spherical\ncoordinate along the azimuthal direction, gis non-zero\nonly for the conservation of orbit angular momentum\nn1\u0000n2=nm. Therefore, when the energy, spin and\norbit angular momentum conservation relations, i.e., the\ntriple resonance condition [Fig. 1(c)] and selection rule\nfor our optomagnonic resonator, are simultaneously sat-\nis\fed, the coupling strength gcan be greatly enhanced.\nThe schematic and optical images of the experiment\nassembly of out optomagnonic cavity integrated with3\nphotonic and microwave circuits are shown in Figs. 2(a)\nand (b), respectively. A 300- \u0016m-diameter single crys-\ntal YIG sphere [Fig. 2(c)] is glued to a 125- \u0016m-diameter\nsupporting silica \fber. Although YIG spheres have been\nwidely used as magnon resonators, their potential as op-\ntical high-QWGM microresonators has been overlooked.\nIn fact, the low absorption loss of YIG in the infrared\nwavelengths (0.13 dB/cm)33can lead to Qfactors as\nhigh as 3\u0002106. Nonetheless, the surface defects and\ncontamination of commercial YIG sphere products in-\nduce strong scattering losses, limiting the highest achiev-\nableQfactor in our experiment. A major contribution\nof the surface contamination is the residual of the sub-\nmicrometer aluminum oxide polishing grit used in the\nYIG sphere production process, which is very di\u000ecult to\nremove using conventional cleaning procedures. By com-\nbining a mechanical polishing procedure (using silicon ox-\nide slurry) and a follow-up chemical cleaning procedure\n(using bu\u000bered oxide etch), we e\u000eciently removed these\ncontamination and obtained very clean sphere surface\n[Fig. 2(d)]. To excite the high- QWGMs, conventional ta-\npered silica \fber (refractive index 1.44) approach cannot\nachieve high e\u000eciency because of the index mismatch34.\nTherefore, we integrate the YIG sphere with a silicon\nnitride optical circuit [Fig. 2(e)], whose waveguide mode\nindex matches that of YIG. The chip is glued to silica\noptical \fbers using UV curable epoxy after careful align-\nment, which provides high e\u000eciency and stable transmis-\nsion. Another coplanar loop antenna circuit is placed in\nvicinity of the YIG sphere to convert microwave signal to\nmagnon. In our experiments, the YIG sphere is always\nbiased by an external magnetic \feld along the support-\ning \fber (z) direction according to the spin conservation\ncondition discussed above.\nBefore studying the magnon-photon interaction, we\n\frst characterize the optical and magnon modes. The\nre\rection microwave spectrum at H= 1840 Oe is plot-\nted in Fig. 3(a), showing multiple dips that correspond\nto magnon modes (to observe high order modes, the YIG\nsphere is placed at the non-uniform \felds of the antenna).\nIn the zoomed-in spectrum of Fig. 3(b), the loaded Qfac-\ntor of the fundamental magnon mode (1 ;1;1) is 1230. In\nthe following magnon-photon interaction measurement,\nthe YIG sphere is placed at the uniform microwave \felds\nof the antenna output such that only the (1 ;1;1) mode\nis excited. The optical transmission spectra are plotted\nin Fig. 3(c), where TE/TM polarized light in the waveg-\nuide are used to probe \u001b+/\u0019polarized WGMs in the YIG\nsphere, receptively. Groups of optical resonances show up\nin the spectra, exhibiting large extinction ratio (beyond\n10 dB) for both polarizations, which con\frms the e\u000e-\ncient coupling between the silicon nitride waveguide and\nthe WGMs. The measured free spectral ranges for both\nthe\u001b+(1.0765 nm) and \u0019(1.1068 nm) polarization agree\nwith the prediction (1.1580 nm) for WGMs. Thanks to\nour surface treatments, very high optical Qfactors are\nachieved:Q\u001b+= 0:593\u0002106andQ\u0019= 0:763\u0002106\n[Figs. 3(d) and (e)].\nMicrowave refl. (dB)\nFrequency (GHz)\nMicrowave refl. (dB)\n5.16 5.17-25-20-15\nf (GHz)Q = 1230\n1546 1548 1550 1552 1554-45-40-35-30-25-20-15Op/g415cal transmission (dBm)\nWavelength (nm)σ+\nπ\nTrans. (dBm) Trans. (dBm)σ+\nQ = 593000\n1542.355 1542.357-31-30-29\nWavelength (nm)\nπ\nQ = 763000\n1540.828 1540.830-33.5-33.0\nWavelength (nm)5.1 5.2 5.3 5.4 5.5-30-25-20-15(a) (b)\n(c) (d)\n(e)FIG. 3. (a) Magnon resonances measured on a 300- \u0016m-\ndiameter YIG sphere biased at 1840 Oe. (b) The zoomed-\nin spectrum of the fundamental magnon mode. (c) Optical\nWGMs for both polarizations ( \u001b+and\u0019) measured on the\nsame YIG sphere using the silicon nitride coupling waveguide.\nLarge extinction ratio and the periodic mode distribution is\nevident. (d) and (e) are the zoomed-in spectrum for the two\npolarizations, respectively.\nTo measure the interaction between optical photon\nand magnon, the YIG sphere is biased at H= 2410\nOe, corresponding to a magnon resonance frequency of\n!m=2\u0019= 6:75 GHz. The optomagnonic resonator is\npumped by a TM polarized laser beam with 1 mW power,\nand the magnons are excited by an on-resonance mi-\ncrowave signal. The laser wavelength is scanned to search\nfor the optical modes that satisfy the energy, spin and\nangular momenta conservation conditions. During the\nsearching process, lock-in technique is adopted to im-\nprove the converted light signal to noise ratio. When the\nconservation conditions are satis\fed, the output light is\nsent to a high resolution spectrometer for further analy-\nsis. It is worth noting that the density of optical WGMs\nis very large, as there are mode degeneracy in the polar\ndirection and high order modes in the radial direction.\nAs a result, the conservation conditions can be satis\fed\naccidentally, similar to the Brillouin scattering in micro-\nsphere optomechanical cavities35,36. A typical spectrum\nof converted photons as a function of the sweeping pump\nlaser wavelength is shown in Fig. 4(a), where the passive\ntransmission spectrum of the pump light is also shown ac-\ncordingly. The correspondence between the resonances\nfor pump light and the peaks of magnon-photon con-\nversion implies the triple resonance enhancement in our\noptomagnonic cavity. The dependence of the converted\nphotons on the microwave resonance [Fig. 4(b)] also con-\n\frms the participation of magnon in the inelastic light4\n1539.0 1539.2 1539.40.000.050.101E-41E-30.01Opt. signal (a.u.)\nWavelength (nm)Pump trans. (mW)\n-6 -4 -2 0 2 4 61E-91E-81E-71E-61E-81E-71E-61E-51E-41E-3Op/g415cal power (mW)\nFrequency (GHz)\n6.67 6.68 6.690.00.51.01.5-20-18-16Opt. signal (a.u.)\nFrequency (GHz)MW refl. (dB)\n-6.9 -6.8 -6.7 -6.6 -6.5 -6.4 -6.351015 Power (nW)\nFrequency (GHz) 245 mW\n 255 mW 270 mW 280 mW 305 mW 320 mW\n250 300 350101520Power (nW)\nP        (mW)MWTM\nTE(a) (c)\n(b) (d)\nFIG. 4. (a) Optical pump transmission and the generated op-\ntical signal as a function of pump laser wavelength. The corre-\nspondence of the generated optical signal peak and the optical\npump resonance dip indicates the satisfaction of the conser-\nvation conditions. (b) Microwave re\rection and the generated\noptical signal as a function of the microwave frequency. (c)\nOptical spectrum of the device output when the triple reso-\nnance condition is satis\fed. The TM and TE components of\nthe output light are separated by a polarization beam splitter.\nThe TM component corresponds to the direct transmission of\nthe pump light, while the TE component contains the scat-\ntered sideband. (d) Power dependence of the sideband on\nthe input microwave power PMW. Inset: extracted sideband\npower as a function of the input microwave power.\nscattering process.\nThe detailed spectrum for one selected tripe-resonance\ncondition is plotted in Fig. 4(c), where the optomagnonic\nresonator is pumped by a TM light at 1534.599 nm. A\npolarization beam splitter is used to separate the two\npolarizations. The TM component of the output light\nshows a single peak as it only contains the transmitted\npump light. On the contrary, the TE component shows\ntwo peaks: a strong peak which corresponds to the trans-mitted pump light that is not completely \fltered out,\nand a weak sideband which corresponds to the gener-\nated photons. Therefore we do have orthogonal polariza-\ntions for the generated signal and the pump light, which\nagrees well with our theory model. The linewidths of the\nmeasured pump and sideband signal are not the physical\nlinewidth of the light but instead only represent the \f-\nnite resolution (67 MHz) of the \flter in the spectrometer.\nThe centers of the pump peak and sideband di\u000ber from\neach other by 6 :75 GHz, matching the input magnon fre-\nquency. The sideband appears only on one side of the\npump as a result of the conservation conditions, as ex-\nplained in above analysis. We measured the converted\nlight at various microwave input powers, which clearly\nshows a linear power dependence [Fig. 4(d)], indicating\na linear magnon to photon conversion. The \ftted raw\npower (system) conversion e\u000eciency is 5 \u000210\u00008. Consid-\nering the imperfect resonance coupling and in-line inser-\ntion losses for both the optical and microwave circuits,\nthe internal power conversion e\u000eciency is about 5 \u000210\u00003.\nThe conversion e\u000eciency can be improved by using YIG\nspheres of smaller size and smoother surface. Further ge-\nometry optimization, such as using YIG microdisk whose\nmodal volume is orders of magnitude smaller, in combina-\ntion with Faraday e\u000bect enhancement via doping, could\nlead to much improved conversion e\u000eciencies.\nIn conclusion, we have demonstrated an excellent op-\ntomagnonic resonator that is made by a highly polished\nYIG sphere. Utilizing an integrated optical chip for\nhigh e\u000eciency optical coupling, high- Qoptical WGMs\nare observed in addition to magnon resonances in the\nYIG sphere after our careful surface treatment. When\nthe triple resonance condition and angular momentum\nconservation condition are satis\fed, the magnon is con-\nverted to optical photon with internal power e\u000eciency\nof about 0:5%. This e\u000eciency can be further improved\nby doping or geometry optimization. Our \fndings show\nthat YIG sphere is a promising platform for designing\ncomplex hybrid systems, which holds great potential to\nrealize information inter-conversion among magnon, mi-\ncrowave photon, and optical photons.\nThe authors thank Liang Jiang and Michael Flatt\u0013 e\nfor fruitful discussions, and funding support from\nDARPA/MTO MESO program (N66001-11-1-4114), a\nUS Army Research O\u000ece grant (W911NF-14-1-0563),\nan AFOSR MURI grant (FA9550-15-1-0029), and the\nPackard Foundation.\n\u0003corresponding email: hong.tang@yale.edu\n1A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hille-\nbrands, Nature Phys. 11, 453 (2015).\n2Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Ya-\nmazaki, K. 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Bahl,\nNature Phys. 11, 275 (2015)."    },    {        "title": "1806.04764v1.Resonant_spin_wave_excitation_in_magnetoplasmonic_bilayers_by_short_laser_pulses.pdf",        "content": "arXiv:1806.04764v1  [cond-mat.mes-hall]  12 Jun 2018Resonant spin wave excitation in magnetoplasmonic bilayer s by short laser pulses\nStanislav Kolodny, Dmitry Yudin, and Ivan Iorsh\nITMO University, Saint Petersburg 197101, Russia\n(Dated: June 14, 2018)\nIn magnetically ordered solids a static magnetic field can be generated by virtue of the transverse\nmagneto-optical Kerr effect (TMOKE). Moreover, the latter w as shown to be dramatically enhanced\ndue to the optical excitation of surface plasmons in nanostr uctures with relatively small optical\nlosses. In this paper we suggest a new method of resonant opti cal excitations in a prototypical\nbilayer composed of noble metal (Au) with grating and a ferro magnet thin film of yttrium iron\ngarnet (YIG) via frequency comb. Based on magnetization dyn amics simulations we show that for\nthe frequency comb with the parameters, chosen in resonant w ith spin-wave excitations of YIG,\nTMOKE is drastically enhanced, hinting towards possible te chnological applications in the optical\ncontrol of spintronics systems.\nI. INTRODUCTION\nIdeas and concepts developed in photonics have been\nrecently applied in related fields, in particular in physics\nof collinear magnets. The rapidly advancing area of\nmagnonics represents photonics with spin waves, or, col-\nlective propagating magnetization excitations in mag-\nnetic materials. Thanks to their unique linear and non-\nlinear properties [1–5] spin waves can be successfully em-\nployed in next generation spintronics devices and quan-\ntum computing. In the meantime, further implication\nin technology requires excitation of spin waves at short\nlength- and time-scales. This can be achieved by the\noptical excitation of magnetic system with femtosecond\nlaser pulses [6–12]. Despite a variety of ways of opti-\ncal generation of spin waves, the nonthermal magneto-\noptical processes such as inverse Faraday and inverse\nKerr are of particular importance, since they are charac-\nterized by high time resolution and do not require heat-\ning of the sample. The essence of the effect is that the\nexternal optical electric field Einduces magnetic field\nh∝E∗×Evia the nonlinear processes such as e.g.\nstimulated Raman scattering [13]. Thus induced mag-\nnetic field drives spin waves in a magnetic structure. Re-\ncently, it was anticipated that in inhomogeneous media\nthe expression E∗×Emay be non-zero even for the lin-\nearlypolarizedwave. Inplanargeometry,thiseffecttakes\nplaceforTMpolarizationonly, andtheinduced magnetic\nfield is perpendicular to the plane of incidence, realizing\nthus the inverse transverse magneto-optical Kerr effect\n(TMOKE) [14]. Among structures which allow inverse\nTMOKE are magnetoplasmonic systems [15], since they\nfacilitate electric field localization and an effective chiral-\nityσ∝ |E∗×E|/|E|2of surface plasmon polariton can\napproach a unity.\nIt is clear that in continuous wave regime the induced\nmagneticfield histime-independent, whereasunderfem-\ntosecond laser pulses this field follows the intensity pro-\nfile. Therefore, a much higher efficiency of spin wave\ngeneration can be achieved in the case of resonant exci-\ntation, when the intensity profile is periodic with a fre-\nquency close to that of ferromagnetic resonance, which\nis typically of the order of 1 GHz. Recently, it has beendemonstratedexperimentally[16]thatusingclockedlaser\nexcitation can substantially enhance the efficiency of the\nspin wave generation at the bottom of magnon band,\nfollowed by their diffusion. However, for the direct im-\nplication in data processing resonant excitation of spin\nwaves with fixed both frequency and wavevector (and\ngroup velocity, as a result) is of vital importance.\nIn this paper we propose a combination of a magne-\ntoplasmonic structure and clocked laser excitation for\nthe resonant excitation of spin waves. We first design\nthe structure and numerically model the distribution of\nelectromagnetic field under the clocked laser excitation.\nUsing the data we evaluate the induced magnetic field\nand study magnetization dynamics based on numerical\nsolution of Landau-Lifshitz-Gilbert (LLG) equation [17].\nWe provide an estimate for a realistic structure, where a\nresonant excitation of spin waves can be achieved with\nefficiency enhancement of at least two orders of magni-\ntude as compared to that in the free magnetic film under\npulsed excitation.\nII. MODEL SYSTEM\nAmong various magnetic materials of collinear order-\ning yttrium iron garnet (YIG) is characterized by the\nbest magneto-elecctrical and magneto-optical properties,\nwhich stand out this material for microwave applications\n[18]. YIG belongs to the class of magnetically ordered\nstructures with cubic symmetry [19, 20] and rather low\nloss [21], and is successfully used in a broad range of\nspintronics applications. We therefore consider a hybrid\nnanostructure consisting of a thin film of YIG and Au\ndiscrete grating on the top (geometry of the system is\nschematicallydepicted inFig.1aundershortlaserpulses.\nTo study magnetization dynamics induced by laser\npulses we are to numerically solve LLG equation\n∂M\n∂t=−γM×Heff+αM×∂M\n∂t,(1)\nwhereγis the gyromagnetic ratio of an electron ( γ= 28\nGHz/T). In general LLG equation describes precession\nof the magnetization, M, around the effective magnetic2\na) b)\nxyzkH0,M0\nFIG. 1. (a) A schematic representation of the bilayer under c onsideration: a thin film of ferromagnetic YIG with the magne ti-\nzationM0aligned along the zaxis is covered by a layer of gold with grating. The system is p laced to the external magnetic\nfieldH0directed normal to the interface; and is irradiated with the laser pulses of intensity Iand duration t, so that the\ninduced magnetic field hin YIG is perpendicular to the plain of incidence ( xOz). The latter is associated with surface plas-\nmon polaritons emerging right at the interface between two m etals. (b) The magnon dispersion relation in YIG obtained by\nnumerical solution of Landau-Lifshitz-Gilbert equation.\nfield,−γHeff=∂H/∂M, created by delectrons of a fer-\nromagnet, whereas the relaxation rate to the direction of\nthe field is associated with the Gilbert damping param-\neter,α. We assume the thin layer of YIG is initially po-\nlarized along the zaxisM0=M0ez, as shown in Fig. 1a,\nand is placed to the external field H0. Being irradiated\nwith laserpulses results in the emergenceofthe magnetic\nfieldhdirected perpendicular to the plain of incidence\nvia TMOKE mechanism. The latter gives rise for the\nmagnetization direction to slightly deviate from collinear\nordering, and thus to spin waves excitation. To get spin\nwave spectrum we linearize Eq. (1) with respect to in-\nplane components of the magnetization m= (mx,my),\ni.e., we plug M=M0ez+mxex+myeyinto (1) on con-\ndition that |mx|,|my| ≪M0. We derive after straight\nforward algebra,\n∂m\n∂t=−γM0×Heff+α\nM0/parenleftbigg\nM0×∂m\n∂t/parenrightbigg\n.(2)\nThe properties of YIG can be well approximated using\nthe microscopic Hamiltonian which includes Heisenberg\nexchangeinteraction, magnetocrystalline anisotropy, and\nZeeman coupling. Thus, the effective field can be repre-\nsented as follows:\n−γHeff=h+H0+Hk+αik∂2m\n∂xi∂xk−ˆNm,(3)\nwhere we assume the summation over repeated indexes;\nˆNis the tensor of magnetocrystalline anisotropy, ˆNm=\nNijmj. In Eq. (3) the exchange tensor αik=Aδikis\nreduced to a scalar owing to the cubic lattice symmetryof YIG with A=3·10−16m2, whileHkcorresponds\nto the effective anisotropy magnetic field [17]. Typically,\nthe effective anisotropy field Hkis material-specific and\nequals 4.6 kA/m for YIG. In the following, we assume\nM0andHkare parallel to H0(see Fig.1a), thus being\nnormal to Au–YIG interface. We put H0= 1 T, and set\nM0to be equal to the saturation magnetization of YIG,\ni.e.,Ms= 140 kA/m. It is therefore clear that with no\nhpresent in the system the vector product M0×Heff\nis zero, producing no magnetization precession.\nIII. RESULTS AND DISCUSSION\nWe put the Gilbert damping parameter α=3.2·10−4,\nalthough it can be decreased as shown in Ref. [22].\nFourier transforming the equation of motion (2) to\nm(ω,k) =G(ω,k)h(ω,k) makes it possible to evaluate\nthe spin wave spectrum from G−1(ω,k) = 0. Thus ob-\ntained the magnon dispersion relation for the YIG thin\nfilm in the presence of his shown in Fig. 1b. One can\nclearly observe that magnons are positioned in 4-7 GHz\nrange for k-vector ranging between 1 and 100 µm−1),\nand are characterized by a very narrow dispersion line\n(about 40 MHz) because of low Gilbert damping. There-\nfore, the direct optical excitation of magnons is hardly to\nbe achieved.\nTo bind optical waves (in the form of plasmons, for\nexample) with spin waves we propose to use a gold grat-\ning that amplifies the electric field and are to determine\nthe field distribution due to strong field localization via3\n6\n5.5\n44.5\n3.5\n3\n0 10 20 30 40\n|k|, 1f, GHza)\nc)b)\nd)\n01|Ez/hmax|\n01|Ex/hmax|\n01|hy/hmax|\n01|hy/hmax|\nFIG. 2. Distribution of electric field components Ex(a) andEz(b) inside the magnetoplasmonic bilayer, normalized by the\nmaximum value of electric field. The induced magnetic field h∝E∗×Eis strongly localized at the Au-YIG interface (c),\nnormalized by the maximum value of the field. Dispersion of h, created by the train of femtosecond optical pulses with\nrepetition rate 4.5 GHz (d). Noteworthy, in these calculati ons we impose the periodic boundary conditions along xandyaxes.\nexcitation of plasmonic resonances [23, 24]. To be more\nspecific, we suppose: the thickness of Au layer dis 100\nnm, periodicity ais100nm, the width wand theheight h\nof array’s element are 50 nm. The latter allows us to ex-\ncite plasmons on the Au-YIG interface with the k-vector\nto be 107m−1.To estimate the distribution of magnetic\nfield inside the hybrid nanostructure we impose the pe-\nriodic boundaries in xandydirections. We further pro-\nceed with eigenmode simulation in CST Microwave Stu-\ndio: the electric field distribution clearly reveals that ex-\ncited field of plasmons possesses two phase-shifted com-\nponentsoftheelectricfield(see2a-b)insidetheYIGfilm.\nThus, it induces time-independent magnetic polarization\nvia the TMOKE mechanism h∝E∗×E[14].\nThe time-independent magnetic field hserves as a\nsourceofspin wavesinaccordancewithEq.(3). Tointro-\nduce time-dependence we propose to excite plasmons in\nthe nanostructure by a train of Gaussian pulses with the\nrepetition rate chosen in resonance with the frequency\nof magnons in the YIG thin film. We assume the dura-tion of each pulse is 40 ps that is shorter in comparison\nwith the repetition rate. As we discussed earlier, the Au\ngrating allows us to generate electromagnetic field with\na fixed value of the wavevector k= 107m−1, thus, in full\naccordance with the magnon dispersion of YIG shown\nin Fig. 1a the resonance frequency of the spin waves in\nthe magnetoplasmonic bilayer under consideration is 4.5\nGHz. To be more realistic we put the finite size of hy-\nbrid nanostructure limited by 1 µm. Such a value of\nrepetition rate is difficult to achieve using only common\nmethodsofGaussianpulsestraingeneration. Meanwhile,\nusing a frequency comb technique facilitates overcoming\nthis problembygeneratingthe Gaussianpulseswith high\nrepetition rate up to THz range [25, 26]. The dispersion\nofinducedmagneticfieldconsistsofadiscretesetofspots\nas shown in Fig.2d, the latter happens due to the peri-\nodicity of the structure and an excitation properties of\nfrequency combs. Therefore, by solving the equation of\nmotion with thus obtained hwe can get the spin waves\ndispersion m(ω,k) as well as to recover their time and4\n|m|, x10-3a.u.\n08\nFIG. 3. Magnon dispersion relation in hybrid Au-YIG nanostr ucture. White dashed curve represents the dispersion line o f spin\nwaves in YIG, crossing points of dashed lines represents the dispersion of h.\nspace dependence by performing inverse discrete Fourier\ntransformation. The resulting dispersion is depicted in\nFig. 3. As it was expected there is only one spot on co-\nincidence of dispersion diagrams of excitation magnetic\nfield and spin waves in YIG (hot spots of magnetic field\ndiagram are represented by crossing of straight dashed\npink lines and dispersion line of magnons in YIG is rep-\nresented by white dashed curve). The dependence of\n2 4 6 8 10\nRepetition frequency, GHz00.0020.0040.0060.0080.01|m|, a.u.\n-0.01 -0.005 0 0.005 0.01-0.01-0.00500.0050.01 a) b)\nmx\n, \u0000 \u0001 \u0002 \u0003\nm\ny\n\u0004\u0005\u0006\u0007\bFIG. 4. (a) Spin wave magnitude in the YIG thin film vs.\nrepetition rate of femstosecond laser pulses. (b) Precessi on\nof/vector maround equilibrium point in time for different excitation\nfrequency: brown curve – resonant excitation frequency 4.5\nGhz, blue curve – 4.48GHz, yellow curve – 4.52 GHz.\n|mon the repetition rate of femtosecond laser pulses\nfor fixed values of kin range 1-10 GHz for maximum\nvalue of excitation magnetic field h/hmaxis presented in\nFig. 4a, which clearly manifests the formation of a reso-\nnance mode. The resonance mode appears at 4.5 GHz.Also there are additional equidistant peaks are placed on\nlower frequencies. The main and additional peaks are\nvery narrow because of low damping of spin waves in\nYIG which is described by Gilbert damping parameter\nas it was mentioned before. In addition, we study hodo-\ngraph of the magnetization at different frequencies: In\nparticular, results for 50 Gaussian pulses are shown in\nFig. 4b.\nIV. CONCLUSIONS\nIn the current studies we considered a typical magne-\ntoplasmonic structure, that represents the bilayer of a\nthin film YIG covered by a layer of gold with grating and\nexplicitly showedthat TMOKEemergingin such a struc-\nturemaybedramaticallyenhancedusingfrequencycomb\ntechnique. In fact, for the repetition rate chosen to be\nin resonant with the ferromagnetic resonanceof the mag-\nnetic layer the direct numerical solution to LLG equation\nreveals a desired behavior. 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Diddams,\nMicroresonator-based optical frequency combs , Science\n332, 555 (2011)."    },    {        "title": "1604.08659v2.Spin_Seebeck_effect_through_antiferromagnetic_NiO.pdf",        "content": "1 \n Spin Seebeck effect through antiferromagnetic  NiO \nArati Prakash1, Jack Brangham1, Fengyuan Yang1, Joseph P. Heremans2,1,3 \n1The Ohio State University Dep artment  of Physics, Columbus, Ohio 43210  \n2The Ohio State University Dep artment  of Mechanical Engineering, Columbus, Ohio 43210  \n3The Ohio State University Dep artment  of Materials Science and Engineering, Columbus, Ohio 43210  \n \nAbstract  \nWe report temperature -dependent  spin-Seebeck measurements on  Pt/YIG bilayers and  \nPt/NiO/YIG trilayers, where YIG ( Yttrium iron garnet , Y3Fe5O12) is an insulating ferrimagnet  \nand NiO is an antiferromagnet at low temperature s.  The  thickness of the NiO layer is varied \nfrom 0 to 10  nm.  In the Pt/YIG bilayers , the temperature gradient  applied to the YIG stimulates \ndynamic spin injection into the Pt, which generates an inverse spin Hall voltage in the Pt.   The \npresence of a NiO layer damp ens the spin injection exponent ially with a decay length of 20.6 \nnm at 180 K .  The decay length increases with temperature and shows a maximum of 5.5 0.8 nm \nat 360 K.   The temperature dependence of the  amplitude of the  spin-Seebeck signal  without NiO \nshows a broad maximum  of 6.50.5 V/K at  20 K.  In the presence of NiO, the maximum shifts \nsharply to higher temperatures, likely correlated to the increase in decay length.   This implies \nthat NiO is mo st transparent to magnon propagation near the paramagnet -antiferromagnet  \ntransition . We do not see the enhancement in spin current driven into Pt reported in other papers \nwhen 1 -2 nm NiO layers are sandwiched between Pt and YIG.  \nIntroduction  \nIn the spin -Seebeck effect (SSE, longitudinal configuration) , 1 a heat current stimulates \nspin propagation across an interface between a ferro magnetic  (FM)  material and a normal metal  \n(NM) , where it is converted into a transverse electric field by the inverse spin Hall effect  2 \n (ISHE) .2  The most commonly stu died materia l system is the Pt/YIG  system.  Recently,3,4,5 SSE \nsignals have been reported in systems  where the magnetic material is an antiferromagnet (AFM) .  \nIn principle,6 in the absence of a net spin polarization, an AFM cannot produce a spin -Seebeck \nsignal.  A net  spin polarization can b e produced by bringing the AFM above the spin -flop \ntransition  by an applied magnetic field , resulting in recent experimental reports of SSE signals in \nCr2O33 and MnF 2,4 as well as a theory for it.7  The effect on MnF 2 is particularly large, reaching \nover 40 μV/K  at high field and low temperature  (T).4 However, an uncertainty remains about the \nexact values of the temperature gradient applied  to the AFM, and thus the absolute val ues of the \nspin-Seebeck coefficients .  The effect on Cr 2O3 was much smaller3 and more in line with values  \nhitherto  observed on the Pt/ YIG system.  \nWe report here on SSE signal s measured  on a thin  AFM layer grown on a FM, the \nPt/NiO/YIG  trilayer structure, which was re cently reported to exhibit  antiferromagnetic magnon \ntransport .8  In that study,  microwave -driven ferromagnetic resonance  (FMR)  in the YIG  FM \nlayer was used to produce a pumped spin current t hat flowed through the  NiO AFM layer and \nreached  the Pt layer , where it produced an ISHE electric field. Monitoring the amplitude of this \nfield as a function  of NiO layers thickness (varied from 0 to 10 0 nm) gave information about the \npropagation  of a dynamic net magnetic moment through the AFM.  The results showed t hat \nmagnon ic transport through NiO layers thicker than 4 nm was attenuated exponentially with a \ndecay length of about 10  1.2 nm at room temperature .  However, the insertion of a  2 nm thin \nNiO layer between YIG and Pt  enhanced  the spin currents driven into Pt.  A subsequent  theory9 \nascribes the origin of this effect to diffusion of thermal antif erromagnetic magnons in the NiO.  \nIf that is so, one would not expect to see the enhancement in the SSE measurements, and this is \nwhy the present experimental st udy was undertaken.  We note that SSE results on the 3 \n Pt/NiO/YIG system were recently published5 after we started this study, in which an \nenhancement  is reported .  In contrast, we do not see  the enhancement and we discuss the possible \nsources for that difference.  \nIn previous studies,10 the temperature dependence of the SSE in the YIG/Pt system \nwithout any NiO shows a broad maximum, attributed to the properties of the magnon dispersion.  \nWe report the same behavior here.  This maximum was explained10 by the fact that t hermally \ndriven magnons have a spectral distribution that covers  almost the entire Bri llouin zone at room \ntemperature, unlike FMR -driven  magnons at ~10 GHz near the zo ne center .  The magnon \ndispersion11 of YIG includes several optical -like branches that have practically no group \nvelocity, and are not expected to contribute much to the SSE signal, but it also includes quasi -\nacoustic branches.   The non -monoton ic temperature dependence then arises as follows.   At the \ntemperature s below the maximum , the SSE signal  is limited by the thermal magnon population ; \nit increases with increasing temperature because  higher energy magnons get excited , and in \nlarger numbers.  I n this temperature range, most magnons have rather similar group velocity \n(only the magnons with energies near about 10 GHz have near -zero group velocity).   At \ntemperatures a bove the maximum, the effects of the magnon inelastic mean free path12 and \ndiffusio n length13 become more important, and the SSE now decreases with increasing \ntemperature.  It was  further  concluded from the magnetic -field dependence of the SSE \nmeasurements10,14 at high field that  the magnon modes at energies below 40 K contribute the \nmost to the SSE effect in the Pt/YIG system.   The present study includes the temperature \ndependence of the SSE  in YIG/Pt, which shows the same behavior as in Ref. [ 10], and to this we \nadd temperature -dependent data of the SSE  through the AFM, which was not included in the \nFMR study.8   4 \n  \nExperiment  \nYIG films were grown epitaxially  on single crystal Gd3Ga5O12 (GGG) <111>  substrates \nby ultrahigh vacuum off -axis sputtering.15,16,17  The YIG was grown at a substrate  temperature of  \n750C while being rotated at 10 /s for optimal sample uniformity at a growth rate of 0.51  \nnm/min. The  thickness of YIG films grown  on GGG  substrates was  250 nm, as it has been  \nshown18 that the intensity of the SSE signal in YIG films grown on GGG increases with \nincreasing YIG thickness, but saturates for thicknesses of about 250 nm. NiO and Pt layers were \nthen deposited at room temperature o n these YIG films also by off -axis sputtering. A series of \nPt/NiO/YIG  samples with length 10  mm and width 5  mm were de posited  with 6  nm thick Pt and \n0-10 nm thick NiO. In all cases , the Pt deposit ion was done in-situ with the NiO growth.  A \nsummary of the samples studied is given in Table I.   \nSample  TM (K) \nPt(6nm) /YIG (250nm)  20 \nPt(6nm) /NiO(1nm)/YIG  120 \nPt(6nm) /NiO(2nm)/YIG  220 \nPt(6nm)/ /NiO(5nm)/YIG  260 \nPt(6nm)/ /NiO(10nm)/YIG  320 \nTable I p roperties of the samples studied.  \nThe SSE measurements wer e conducted as in Ref. [ 10], except for the way the \ntemperature gradient was calculated . High temperature measurements were  conducted in a liquid \nnitrogen flow cryostat from 80 to 420  K. Samples were  mounted using Ap iezon H -grease ( T > 5 \n 300 K) or N -grease ( T < 300 K) on electrically insulating cubic boron nitride  (c-BN) heat-\nspreading pads with  high in-plane  thermal conductivity.  Three r esistive heaters (120 Ohm) were  \napplied  to the c -BN pads  and connected electrically in series , ensuring uniform heating . To \nswitch the SSE signal, we swept the field between  50 mT.  Low temperature measurements \nfrom 2 to 300  K were  conducted  on the Pt/YIG  sample  in a Physical Property Measurement \nSystem (PPMS) by Quantum Design . All experimental parameters were  controlled as much  as \npossible to be identical between the two systems.   The voltage across the Pt layer s was measured \nwith a Keithl ey 2182 nano voltmeter .  Electric fields were calculated from raw traces  (an example \nis shown in Fig. 1 a for a Pt (6nm) /YIG (250nm)/GGG  bilayer ) of the voltage across the Pt strip, \nVy, by averaging the values at -50 mT and + 50 mT, then dividing by the length Ly of the Pt layer  \nto give the induced ISHE field  (\ny\ny\nyVEL  in V/m ).  \nUnlike in Ref [ 10], temperature gradients  \nT  are calculated  from the measured  heater \npower  output  Q per unit cross -sectional area of YIG (in units of W/ m2) and the measured values \nfor bulk thermal conductivity of the YIG12 (\nYIG ) at each temperature :\nYIGQT . This \nprocedure assumes that there is negligible loss of heat between the heater  and the heat sink, so \nthat all the heat goes though the sample as a uniform flux.  Heat spreaders assured uniformity  \nand an adiabatic sam ple mount minimized heat losses. The high thermal conduct ance of the \nsample, due to the high thermal conductivity of GGG and the favorable geometry, combined \nwith the extensive use of radiation shielding and of 25  m copper or manganin voltage and \nheater wires, ensure that heat losses are minimal.  The procedure also  assumes that the thermal \nconductivity of the 250 nm YIG film is the same as that of bulk YIG.  In the absence of reliable \ndata on the temperature dependence of the thermal conductivity of 250 nm -tick YIG films, this  6 \n hypothesis  is not as easily justified.  The error may affect the absolute values reported for the \nspin-Seebeck coefficient Sy, but the relative effect of adding NiO layers between the YIG and the \nPt is much less sensitive to this hypothesis , since the thickness of the YIG layer is held constant .  \nThe values of Sy obtained following this procedure in different cryostats and in different runs are \nsuperimposed on each other in Fig. 1(b).  The spread in the data, of the order of 30% at worst, \ngives a quantitative estimate of the accuracy with which we can make sample to sample \ncomparisons, as we need to do to study the effect of NiO thickness.  We developed this process, \ninstead of using the metho d in Ref [ 10], because it  is not sensitive to thermal contact resistances \nbetween films and substrate or between the different layers, while direct thermometry is.   To test \nthat, Quantum Design thermometry  was mounted on the hot- and cold -side c -BN pads , but the \nresulting measurements were affected by thermal contact resistances between sample and heat \nsink, and proved to be less reproducible than the procedure used here.  The values for the spin -\nSeebeck coefficient  \ny\nyEST  (in units of nV/K) , are reported as the ratios of electric field to \ntemperature gradient, with the geometrical factors of the samples divided out.  The procedure is \nconsistent with our previous measurements ,10 enabling  a quantitative comparison of all results.  \nResults  \nSSE signals are observed on all samples at room temperature . We begin with the cas e of \nthe Pt/YIG sample  (Table  1).  We point out  that the  observed spin -Seebeck coefficients  (Fig. 1 b) \nin the Pt (6nm) /YIG(250 nm) bilayer on GGG  are one to two  order s of magnitude larger than  our \nown previously re ported signals  for Pt on YIG single crystal ,10 but an important  reason is simply \nthe difference in the method used to calculate T.  The maximum SSE signal is 6.50.5 V/K \nnear 20 K.  These results  demonstrate the critical role of the Pt/YIG interface : the pristine \ninterface between Pt and epitaxial YIG film allows  the conduction electrons in the Pt directly 7 \n interact with the localized YIG magnetization in the absence of interfacial defects, which is highly \ndesired for spin transfer acr oss the interface.  In contrast , for our previous10 YIG bulk crystal with \na polished surface  (even for epi -ready YIG single crystals) , it is expected that there is defect layer \nat the YIG surface .  Given that our  previous FMR spin pumping study19 shows that a nonmagnetic \ninsulating barrier of only 1 nm thickness at the interface can reduce the I SHE signal by a factor of \n~250, it is reaso nable to expect that any defect layer will strongly affect the results.  \nFigure 2 shows t he SSE of the  Pt/NiO/YIG  trilayer systems  as a function of temperature.  \nThe SSE signal is hard to disce rn from  the noise fl oor of 50 nV at some lo w-temperature cutoff \npoint, varying from 100 to 180 K depending  on sample thickness: Fig. 2 reports only data above \nthis cut -off temperature  for each sample .  The temperature  dependence of SSE shows a maximum  \nat a temperature  TM that decreases monotonically with the NiO t hickness, and is summarized in \nTable 1.   TM is not necessarily the N éel temperature TN of NiO  films , because two competing \nmechanisms are at work.  On the one hand, it is known that TN in free NiO films is lower  than that \nof bulk NiO and has some systematic variation with film thickness .20 On the other hand, it was \nshown21 by neutron diffraction on CoO/Fe 3O4 films in the same thickness range as those studied \nhere that the ordering temperature of the CoO films is enhanced for small CoO  thickness es.  \nThere is clearly an interaction between the Ni spins  in NiO and the Fe spins in YIG by an \nexchange coupling or biasing effect.   The positive slope of dSy/dT below TM indicates that the \npropagation of magnons though the NiO film improves as the thermal  fluctuation  in the AFM -\nordered NiO  is increased with increasing temperature.  The negative slope of dSy/dT above TM in \nthe Pt/NiO/YIG trilayers simply mirrors the negative slope behavior of the Pt/YIG system in the \nabsence of NiO .  8 \n Figure 3a shows how the SSE signal  decay s exponentially  (i.e.\n0exp( / )yS S th L  where \nth is the NiO thickness, S0 is a prefactor, and L is an attenuation length)  with increasing NiO \nthickness  at 420, 360, 300, 240 and 180 K.  We fit the attenuation length  L to the data and report  \nit as a function of temperature in Fig. 3b.  The decay continues to be observed down to 100  K, but \nthe number of samples on which the SSE is measurable has decreased.   This behavior  and the \nattenuation length  at 300 K  is within about 40%  of that observed8 with FMR driven spin flu x. At \nfirst sight this is surprising, if we consider that the modes excited in FMR -driven spin flux are \nnear the Brillo uin zone center and have a much larger spatial extent than the high energy modes \nexited in thermal spin fluxes.   Recall that  in FMR spin pumping of Pt/NiO/YIG trilayers, the \nexcitation is uniform coherent precession of the YIG magne tization (or, zero -k magnons), while \nfor thermally -driven SSE measurements of the same structure, the excitation is thermal magnons \nthat have finite k and are diffusive.  Therefore the ratio between the NiO film thickness and the \nmagnon wavelength would be  expected to be very different for FMR spin pumping and SSE ; \nthus,  the decay length would be expected to follow th e same  argument as well.  The similar \nbehaviors between the two experiments indicates that not all thermal magnons participate in the \nSSE, but only those at small k-values .  This same conclusion was arrived at during the \ninterpretation of the freeze -out of the SSE effect at high magnetic field.10,14  The attenuation \nlength decreases with decreasing temperature below 300 K, and seems to have a broad maximum \nnear 360 K, although the decrease above 360 K is within experimental error bars .  This hints that \ntwo separate mechanisms are at work here, one related to thermal fluctuations of the magnons, \nand one to the coupling  between Ni spins in NiO and Fe spins in the underlying YIG .  \nUnlike what was observed with FMR -pumped magnons ,8 the SSE measurement of \nPt/NiO/YIG for the 2 nm NiO reported here does not show an enhancement. This also contrast s 9 \n with the recently reported SSE measurements on similar structures using polycrystalline YIG \nsubstrates ,5 where an enhancement is observed  similar to our earlier report of FMR spin pumping .  \nWe believe these two SSE measurements do not contradict each other because of the difference in  \nmeasurement techniques and  sample characteristics .  In this work w e used 250 -nm YIG epitaxial \nfilms and in Ref. 5 , polished polycrystalline substrates  were used .  We also note that i n Ref. 5, the \nmaximum magnitude of Sy for Pt(3 nm)/NiO(2 nm)/YIG(bulk) is about 1.1 V/K at ~260 K, \nwhile  in Fig. 2 of this work, the maximum Sy for Pt(6 nm)/NiO(2 nm)/YIG(250 nm)/GGG is 0.35 \nV/K at 200 – 300 K.  Considering that the resistance of a 3 nm Pt layer is typically a few times \nhigher than that of a 6 nm Pt layer, these results of the same measurements on two similar \nsamples are consistent.   We believe that the AF M magnons and spin fluctuations in NiO are \nlikely responsible for the spin current enhancement through NiO in Ref. 5 , similar to our earlier \nFMR spin pumping results .8 The absence of enhancement in this work is likely due to the \ndifferences in YIG samples (epitaxial films vs. polycrystalline bulk) and interfacial \ncharacteristics, as well as fact that our Pt/YIG sample alrea dy exhibit a large SSE signal.    \nIn summary,  we report the transmission of thermal ly-driven  magnons excited in the YIG \nfilms through a NiO layer in its antiferromagetic and its para magnetic state, with an attenuation \nlength ranging between 2 and 5.5 nm at temperatures from 180 to 420 K .  From this we conclude \nthat the thermally -driven magnons that give rise to the SSE must be limited to the low -energy part \nof the magnon spectrum, consistent with previous work on the high -field suppression of the SSE.  \nAcknow ledgement  \nThis work was primarily  supported by the Army Research Office ( ARO ) MURI W911NF -\n14-1-0016  and the U.S. Department of Energy (DOE), Office  of Science, Basic Energy Sciences, 10 \n under Grants No. DE -SC0001304.   It is partially supported by the Center for Emergent Materials, \nan NSF MRSEC under grant DMR -1420451 . \n \nFigure captions  \nFigure 1 . (a) A  raw trace of  the spin -Seebeck voltage  Vy versus applied field at 300 K for an \napplied heater power of 0.495 W and (b) spin S eebeck Coefficient as a function of temperature \nfor a Pt(6 nm)/YIG(250 nm)  bilayer deposited on GGG <111>.  In (b), t he results of several \ndifferent runs in different instruments are superimposed on each other, illustrating the \nreproducibility of the results, which is of the order of 30%.  A maximum Spin Seebeck \nCoefficient of ~6.50.5  V/K is observed  on the YIG film w ithin a broad peak around 20 K.  \nFigure 2. Temperature dependence of the SSE signal on the Pt/ YIG bilayer  is compared to SSE \nsignals from the YIG/NiO/Pt trilayers , with the NiO film thickness as noted . A decrease in SSE \nsignal is observed with increas ing NiO film thickness. A maximum  at a temperature TM is \nobserve d for each of the YIG/NiO/Pt trilayers . TM  increases with increasing NiO thickness.  \nFigure 3. (a) Magnitude of the SSE signal at 420, 360, 300, 240, and 180  K plotted as a function \nof NiO thickness. The experimental values are the data points.  The lines through them are fitted \nexponential decay functions, with a characteristic  attenuation  length scale given as a function of \ntemperature in (b).  \n  11 \n Figure 1  \n  \n12 \n Figure 2  \n  \n13 \n Figure 3  \n  \n14 \n References:   \n1 K. Uchida, H. Adachi, T. Ota, H. Nakayama, S. Maekawa, and E. Saitoh, Appl. Phys. Lett. 97, \n172505 (2010).  \n2 S. R. Boona, R. C. Myers and J. P. Heremans , Spin Caloritronics, Energy Environ. Sci. 7, 885-\n910 (2014) . \n3 S. Seki, T. Ideue, M. Kubota, Y. Kozuka, R. Takagi, M. Nakamura, Y. Kaneko, M. Kawasaki, \nand Y. Tokura, Phys. Rev. Lett. 115, 266601 (2015) . \n4 S. M. Wu, W . Zhang, K. C. Amit KC, P . Borisov, J . E. Pearson, J. S . Jiang, D . Lederman, A . \nHoffmann, and A . Bhattacharya, Phys. Rev. Lett. 116, 097204 (2016) . \n5 W. W. Lin, K. Chen , S. F. Zhang , and C. -L. Chien, Phys. Rev. Lett. 116, 186601 (2016).  \n6 Y. Ohnuma, H. Adachi, E. Saitoh, and S.  Maekawa, Phys. Rev. B 87, 014423 (2013).  \n7 S. M. 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L. Wang, F. Y. Yang, and P. C. \nHammel, Phys. Rev. B   90, 140407(R) (2014).  \n20 J. Nogues and I. K. Schuller, J. Magn. Magn. Mater. 192, 203 (1999) ; A. Baruth and S. \nAdenwalla, Phys. Rev. B 78, 174407 (2008) . \n21 P. J. van der Zaag, Y. Ijiri, J. A. Borchers, L. F. Feiner, R. M. Wolf, J. M. Gaines, R. W. \nErwin, and M. A. Verheijen, Phys. Rev. Lett.  84, 6102 (2000) . "    },    {        "title": "2110.04506v1.Single_shot_imaging_of_ultrafast_all_optical_magnetization_dynamics_with_a_spatio_temporal_resolution.pdf",        "content": "1 \n Single-shot imaging of ultrafast all-optical magnet ization dynamics  \nwith a spatio-temporal resolution \nT. Zalewski and A. Stupakiewicz \nFaculty of Physics, University of Bialystok, 15-245 Bialystok,  Poland \nAbstract.  We present a laboratory system for single-shot magneto-opti cal (MO) imaging of ultrafast \nmagnetization dynamics with MO Faraday’s rotation sensitivity of 4 mdeg/ µm. We create a stack of MO images \nrepeatedly employing a single pair of a pump and defocused prob e pulses to induce and visualize MO changes in \nthe sample. Both laser beams are independently wavelength-tunabl e allowing for a flexible, resonant adjustable \ntwo-color pump and probe scheme. To increase the MO contrast the pr obe beam is spatially filtered through a \n50 µm aperture. We performed the all-optical switching experime nt in Co-doped yttrium iron garnet films \n(YIG:Co) to demonstrate the capability of the presented metho d. We determine the spatial-temporal distribution \nof the effective field of photo-induced anisotropy driving th e all-optical switching of the magnetization in YIG:Co \nfilm without an external magnetic field. Moreover, using this  imaging method, we tracked the process of the laser-\ninduced magnetization precession.  \nI. INTRODUCTION \nThe development of femtosecond lasers and time-resolved methods provide d the unprecedented \ncapability for the all-optical coherent control of spins on ultrashort ti me scales 1. From an application \nperspective, the main challenge for obtaining an electronic-competiti ve solution is the integration of \nphotonics and spintronic devices with triggering of magnetic order by a n ultrafast all-optical laser \nstimulus 2. The modern magneto-optical methods using pump-probe geometry are attra ctive tools for \nresearch in ultrafast magnetism, offering insight into the time-re solved dynamics of the light-induced \nprocesses on ultrashort timescales. The exceptionally sharp time resolution down to femtoseconds offers \naccess to information inaccessible in the slow, static, or quasi-st atic regime experiments. At the same \ntime, the spatial resolution of the magneto-optical microscopy 3 allows imaging of features as small as \ncouple hundred nanometers, up to the light diffraction limit.  \nThe visualization of the ultrafast dynamics of photo-induced processes i n physics is critical for \nunderstanding the main interactions. The main advantage of such a techn ique is compounding both the \nspatial, and temporal behavior of the phenomena at the same time, givi ng three-dimensional sets of \ninformation. Image, as a 2D representation, gives briefly qualitati ve information, which by appropriate \nscaling of intensity level can be quantitive. Imaging methods, whi ch are based on pump and probe \ntechniques are well-known for almost 40 years where the first process es were recorded on photographic \nfilms.4 Later, a huge amount of different variations of this general ide a were used very widely, mainly \nin chemistry and biology.5,6 However, the pump-and probe methods are inapplicable for all phenomena. \nNonrepetable or difficult to reproduce ultrafast processes, such as irrev ersible chemical reactions, laser-\ninduced damage, or scattering on a living tissue require so-called single-shot imaging. In this context, \nthe “single-shot” term refers to capturing the whole event without repeating it. The whole measurement \nsequence is performed in real-time with only one laser pulse.7 In contradiction, it has to be noticed that \nin this article the discussed method is a repeatable pump and probe  technique and the term single-shot \nrefers to the magnetization switching by a single laser pulse.  \nRecently, different fundamental mechanisms which are responsible for the ul trafast laser-induced \nmagnetization dynamics have been revealed. In the case of the heat load in magnetic materials, they can \nbe separated into thermal and non-thermal. The thermal effect has been observ ed in metals base on the \nultrafast demagnetization during heating close to Curie point.8–10  The non-thermal effects are caused by \nthe inverse Faraday effect 11  and the photo-magnetic effect.12 ,13  Since the discovery of these effects, rapid \ndevelopment, and increased interest in the possibility of magnetiz ation switching by laser pulses in a \nlarge class of materials occurred,14  and time-resolved imaging was a prominent tool for examining the \nspatio-temporal behavior of the magnetization. Two main approaches for time-resolved magneto-optical \nimaging can be noted. It can be realized via scanning a focused bea m through the sample 15–18  or via \ndirect imaging on a CCD camera 19,20  which is also realized in this work. The method is widely known \nand it was used for examining plenty of different materials such a s GdFeCo metallic alloys,18,20–22  the 2 \n rare-earth orthoferrites DyFeO 323  and  HoFeO 3,24  Co/Pt ferromagnetic multilayers,25  garnets.16,26  \nMoreover, the understanding of ultrafast magnetization dynamics was achi eved through the \ndevelopment of time-resolved imaging methods using femtosecond sources of X-ray radiation 27  and \nfree-electron lasers.28  Recently, it was discovered that without any external magneti c field reversible \nphoto-magnetic switching in Co-doped yttrium iron garnet films (YIG:Co) can  be obtained only by a \nsingle laser pulse.29  Such switching mechanism is based on the light-induced effectiv e field of photo-\ninduced magnetic anisotropy, which lifetime is about 20 ps, and it is sufficient to trigger angle precession \nof the magnetization. The photo-magnetic effect is the most intri guing due to the non-dissipative \nmechanism of all-optical switching of magnetization with the l owest heat load and fastest time \nswitching. No long-timescale recovery from excessive heat load give s a high repetition rate limit of 20 \nGHz for a cold photomagnetic recording.30   \nUltimately, imaging with only one ultrashort probe pulse from a lase r source is marked with a large \nnumber of technical difficulties. These are among others: nonuniformity  of the illuminating light, \ntemporal stability of the laser pulse train, separation between pump and probe signal, impact of the \ndiffraction, and mechanical stability. Therefore, to sufficiently inve stigate the dynamic of such a fast \nprocess one has to develop a reliable and repeatable imaging system . For YIG:Co films such \nmeasurement has been also presented in 29 . However, obtained image contrast in performed \nmeasurements was marked as insufficient, concealing the desired i nformation in a spatially averaged, \nblurry picture. Detailed spatial analysis and determination of the spat ial distribution of the effective field \nof the photo-induced anisotropy were not performed.  \nHere, we developed a spatially improved time-resolved system for s ingle-shot imaging of ultrafast laser-\ninduced spin precession and switching. We used a two-color indepen dently tuned laser pump and probe \npulses with duration <40 fs at a wide spectral range of 290 −2570 nm. Moreover, the two separate \nmechanical delay lines for both pump and probe beams allowed to ensure full  flexibility and \nindependence in settings desired time shift resolution <8 fs. We ap plied spatial filtering to the unfocused \nprobe laser beam which allows to cleanse the beam of imperfections ge nerated by defects on the sample \nand the effect of the interference. We demonstrated time-resolved la ser-induced dynamics of \nmagnetization precession and permanent switching with a high sensit ivity of magneto-optical rotation \nin YIG:Co thin films. Moreover, we determined the spatial distributi on of the field of photo-induced \nanisotropy driving the magnetization switching.  \nThis paper is organized as follows. In Sec. II, we describe the e xperimental details of the set-up for \nsingle-shot time-resolved imaging, the magnetic structure of the sa mple, and the image processing. In \nSec. III, we demonstrate the imaging of all-optical photo-magnetic s witching and precession in YIG:Co \nfilm with temporal and spatial resolutions. The conclusions are in Sec. IV. \nII. EXPERIMENTAL DETAILS  \nThe set-up used for the magnetization dynamics imaging is presen ted in FIG. 1. We employed the time-\nresolved technique of magneto-optical polarized microscopy using exc itation and detection with a single \nfemtosecond pump and probe pulses, respectively. A femtosecond laser s ystem is one-box integrated \nTi:Sapphire amplifier (Coherent Astrella). Its output pulse energy is  8 mJ  with the 1 kHz repetition rate \nand pulse duration not longer than 35 fs on 800 nm central wavelength. Next, the laser beam is divide d \ninto two branches the probe and the pump beam. The magneto-optical imagi ng in ferrimagnetic \ndielectrics requires a two-color scheme between pump and probe due t o the spectrally-selective \nexcitation of magnetic order and optimization of magneto-optical  rotation for a probing. Two optical \nparametric amplifiers (Light conversion TOPAS-Prime) and optica l frequency mixers (Light conversion \nNirUVis) were applied to each branch with input pulse energy of 3.5 m J. The wavelength of both pump \nand probe pulses can be independently tuned in the 290 −2570 nm range.  \nAfterward, laser pulses are temporarily shifted by ∆t defined by the mec hanical time-delay line up to \n4.4 ns. The probe beam passes through a pinhole PH with a high damage thres hold and 50 µm diameter. \nNext, the beam goes through the polarizer P as a half-wave plat e and the adjustable lens with a focus \ndistance of 10 mm. It stays on sample S slightly out of focus to illuminate a suffici ently large area. 3 \n  \nFIG. 1. Magneto-optical setup scheme for the two colors ti me-resolved single-shot imaging. BS – beam splitter, OPAs –  optical \nparametric amplifiers, PH – pinhole, SH1- mechanical sh utter, FM – flip mirror, L – adjustable lens, S – sampl e, C – coil, F- \ncolor filter, A – analyzer, P – polarizers, DG – delay gene rator. \nTo retrieve magnetic contrast from images the princ iple of the magneto-optical polarizing microscope \nhas been used. The light propagating through the sa mple is affected by the magneto-optical Faraday \neffect. It induces rotation of the plane of polarizat ion proportional to the magnetization component \nprojected on the direction of the light propagation .3 In used Faraday (transmission) polarizing \nmicroscope geometry, linearly polarized probe pulse propagates through the sample and it is collected \nby the objective O. Afterward, the pulse passes thr ough the analyzer A. The mutual rotation between \npolariser and analyzer defines magneto-optical contra st. The CCD camera (Princeton Instruments \nProEm 1024) working in the full-frame mode is used to  record and digitize 16-bit images with resolution \n1024×1024 pixels. To suppressed thermal noise and u nwanted charge accumulation, the built-in \ncontinuous cleaning of the array procedure was used  between each frame. It removes any charge from \nthe array until a trigger pulse is detected and sto ps as soon as the frame collection begins. The spot  size \nof the probe laser beam localized in the sample surf ace was 350 µm in diameter which is significantly \ngreater than the pump beam spot size of 130 µm in diameter. For adjusting the probe spot size focu sing \nlens L ( f = 100 mm) was used. It makes the beam divergent aft er its focal point resizing the illuminated \narea. To eliminate residual pump light the spectral  bandwidth filter F was placed before the CCD camera . \nThe pump pulse train is suppressed by the synchroni zed mechanical shutter SH1. It can release a single \npulse to excite the sample. For safety reasons addit ional mechanical shutter, not marked in the figure \nwas added to the probe branch. The recorded magneti c domain can be removed by an external in-plane \nmagnetic field H produced by coil C.  \nMoreover, for static imaging and magnetic domain st ructure observation, the sample also was \nilluminated by the white LED light let in by a flip mirror FM in normal incidence. To obtain high-quality  \nimages of magnetic domains the magneto-optical contr ast was optimized by setting the analyzer to \nrespect of polarizer axis in the position which allo ws visualizing magnetic domains structure. It was \ndone on a long time scale with an LED light source polarized in the same plane as the probe pulse. Next , \nthe intensity of the illuminating probe laser pulse s, its uniformity, and angle of incidence were bala nced \nto minimalize the total impact of the imaging aberra tions, diffraction on surface defects, and created \ninterface fringes concealing the image.  \nA. Setup synchronization \nTo automatize and ensure repeatability and stability  of the experiment, the following measurement \nprocedure which consists of four integral steps was  applied. Firstly, the software sets the mechanical  \n4 \n delay line (Newport DL325) to the desired distance,  which defines a time delay between the probe and \npump pulses. The minimum incremental motion with gu aranteed bidirectional repeatability of ±0.15 µm \ncorresponds to about 8 fs resolution, which is shor ter than a laser pulse duration. Secondly, the softw are \nautomatically sends a predefined 100 ms duration vo ltage pulse to the erasing coil using a power suppl y \n(Kepco BOP 72-6 DL). It creates the magnetic field wit h an amplitude of H=100 Oe used to restore the \ninitial state of the sample's magnetization. Next, w hile the pump beam is still suppressed by the \nmechanical shutter, only the probe beam illuminates  the sample. The image recorded in this step acts a s \nthe background image. Ensuring a new background ima ge for every measurement sequence helps to \nprevent the impact of long-time laser beam instabili ties. Finally, the mechanical shutter opens, and bot h \nbeams – pump and probe illuminate the sample during  the single shot of 35 fs creating the time-resolved  \nimage. The whole sequence is consistently repeated for every delay value ∆t. \nThe electrical signal which acts as a trigger is de fined by the laser’s system regenerative amplifier \nAstrella. Its rising edge is synchronized with the op tical pulse generation (see Fig. 2a). To perform \na single-shot imaging experiment in the above sequen ce, one has to be able to pick only one laser pulse  \nfrom the pulse train on demand. Plenty of different solutions, mainly dependent on the laser repetition  \nrate, can be applied to solve this issue. The conce ptually most simplistic approach is to use the \nmechanical shutter. Such devices have a large apertu re size which, does not affect the beam parameters. \nHowever, due to the large inertia of these devices they can be applied only for relatively small (tens  \nof Hz) repetition rates. For higher repetition rates , electro-optic or acousto-optic modulators have to b e \nused. Shutter based on this type of device ensures perfect timing, but they introduce significant loss es \nin light transmission. Moreover in our case, the pu lse selection from both pump and probe pulse trains  \nhas to be independent. \nSignificantly longer open-close time cycle (<35 ms) o f used mechanical shutter (Thorlabs SHB05T) and \nseveral millisecond CCD sensor data acquisition spe ed makes it necessary to decrease the laser repetit ion \nrate. Here, as presented in Fig. 2a,  the laser repetition rate was reduced by the amplif ier’s built-in Pockels \ncell divider by 40 times, from 1 kHz to 25 Hz. Moreov er, to provide a proper pulses synchronization the \ndigital time delay generator DG (Stanford Research Sy stems DG645) was applied. It is capable to pick \nnot more than one trigger pulse per second, definin g the base system operation frequency at 1 Hz (Fig. \n2b). This gives a sufficiently long time window for  other components and ensures a single pulse \nselectivity. Because the opening and closing cycle of the mechanical shutter takes about 10 ms the fir st \npulse is always suppressed. Thus, DG always chooses  second, subsequent pulse (Fig. 2c). The \nintroduced time shift from the triggering pulse to the sequent optical pulse ensures enough time for \nopening the mechanical shutter (Fig. 2d). The CCD ca mera is triggered right after the mechanical \nshutter. Its exposure time is set to 1 ms, but it is  not essential in this type of experiment. The came ra \nrequires additional dozen milliseconds for data acqu isition.  \n \nFIG. 2. The timing synchronization scheme. a) The pump and probe pulses are formed from an electronically triggered  \namplifier's pulse. This signal acts as the synchronizi ng edge. b) DG holds the trigger for 1 s c) DG omits the first  pulse and \nsynchronizes the shutter and CCD camera with subsequen t one d) The mechanical shutter open-close time e) The CCD camera \nexposure and time required for data required. The delay be tween pump and probe is unnoticeable in the presented scale. \n5 \n We note that our femtosecond laser system is also capable of worki ng in a single-shot or burst regime. \nThe built-in Pockels cell can be steered directly, deterministi cally creating the required pulse train. \nHowever, every such event disturbs the amplifier’s cavity thermal equilibrium. We observed that in long \nterm, it negatively affects the generated pulse train stabili ty. Using external modulators makes it possible \nto stabilize the laser cavity. Even for the decreased operation fre quency, the pulse train energy fluctuates \nless than in burst or single-shot regimes. The whole system is unifie d and controlled by the PC using the \nLabView software. It ensures flexibility and gives the possibility of  integrating the system with various \nother components such as coils, step motors, modulators, or heaters, hence des igning many variants of \nexperiments. \nB. Magnetic states in a garnet film \nThe investigated sample is a 7.5 µm-thick film of Co-doped yttrium iron garnet 31 . The sample was grown \nby a liquid phase epitaxy method on a gadolinium gallium garnet substr ate with 4° miscut. The material \nis optically transparent in NIR 32  with a relatively low static Faraday rotation of about 0.4 ° at \na wavelength of 650 nm 33 . The garnet has a cubic lattice with two antiferromagnetically coupled spin \nsublattices of Fe 3+  in both tetrahedral and octahedral sites. In both of the sublattic es, cobalt dopants \nreplace the Fe 3+  with Co 2+ and Co 3+ ions.34  The addition of cobalt ions introduces strong \nmagnetocrystalline anisotropy and large Gilbert damping α = 0.2.35  The negative cubic magnetic \nanisotropy gives 4 easy magnetization axes to be close to the  cube diagonals of [111]-type directions \n(see Fig. 3a) The miscut was implemented to partially break the degenera cy between the magnetization \nstates at room temperature and with no applied field and makes it e asier to magneto-optically distinguish \nthe domain structure 29 . The orientations of easy magnetization axes are close to the cube diagonals \n[111] −type directions. Here, we focus on the simplest case switching  between large domains M- and \nM+, state [11-1] and [1-11] respectively.  \n \nFIG. 3. (a) Easy magnetization axes in YIG:Co film (b ) and image of remanent magnetic domain structure in  the YIG:Co \nobtained at zero magnetic fields. The sample was illum inated by the LED with improved contrast by image proce ssing. Two \nmagnetic states are visible: large domain (1) M- along  [11-1], and small domain (5) along [111] directions.  By applying the \n100 Oe external field with 1 s duration in the direc tion along [110] the small domain pattern can be remo ved. On certain sample \nlocalization, only monodomain (single magnetic phase “ 1”) ROI can be selected. The size of the image is 380x 380 µm 2. \n6 \n C. Imaging and data processing \nImaging of the sample illuminated by the probe ultrashort laser pulse is not sufficiently efficient to \ndirectly visualize the magnetic domains with low magneto-optical contrast in the YIG:Co. To \nqualitatively improve the images a spatial filter was introduc ed into the optical setup. Spatial filter \npinholes are useful components for maintaining high beam quality in hi gh-energy pulsed laser systems 36 . \nSuch a filter was used to reduce the high-frequency noise in the pro be pulses beam profiles (see Fig. 4a). \nAlso, the spatial shift of illumination connected with the delay line  movement is minimized. To couple \nand decouple the beam through the pinhole two lenses with f = 40 mm we re used. The imaging beam is \nnot a plane wave, therefore the light profile is not uniform. However, the presence of the magnetic \ndomain structure can be easily distinguished on the images and on the  profile of the cross-section (see \nFig. 4b). \n \nFIG. 4. The magneto-optical images in YIG:Co film wh ich is illuminated with a single probe pulse. The dist ance profiles \ncorrespond to the yellow cross-section line marked in t he images. a) initial image was taken without a pinho le. Nonuniformity \nof the incident wavefront creates high-frequency noise  concealing the magnetic structure character. b) ima ge obtained using a \npinhole. As in FIG. 3 revealed domain structure corre sponds to the large (1) M- and small (5) M+ magnetic p hases \nschematically marked as stripes on the profile. Images a re represented in a normalized 8-bit grayscale. The i mages size is \n380×380 µm2.  \nTo determine the character of pump-induced changes from the recorded 2D ima ges, one has to separate \nthe contribution of the magneto-optical Faraday effect from pure optical effects. Any local light intensity \nchanges such as probe beam instabilities or noise caused by the pump absorption in the s ample have an \nimpact on the recorded image. Here, the visible difference comes from the rotation of the polarization \nplane of the probe beam, which passed through the sample. While propagating throug h the sample, the \nprobe beam interacts with the magnetic domain structure. Different o rientation of the magnetization \ncomponent of magnetic domains results in the different Faraday rota tion of the polarization plane, which \nis detected on the camera. The position of the analyzer defines the magneto-optic c ontrast. \n \n7 \n  \nFIG. 5. a) The single-shot imaging of YIG:Co at monod omain state with magnetization (1) M- along the [11- 1] direction before \nexcitation, illuminated only by a single probe pulse wi th λ = 650 nm. b) The image recorded after a single p ump pulse with λ \n= 1300 nm for ∆t= 185 ps. The magnetization of the switched domain was along [1-11] direction and corresponds to a state \n(8). c) The differential image (B-A). The pump pola rization was E|| [100]. The slightly elongated shape of the switched do main \nis related to both the beam shape and the miscut angle  of the sample and schematically marked (red dashed lin e) on a profile.  \nThe images of the domain structure can be visualized directly using only a single probe  pulse. It is also \nvalid to the dynamic situation within pump and probe pulses. However, for con trast improvement of \nthese images and to suppress the impact of the probe’s wavefront nonuniform ity differential images \nwere created. These images were obtained by subtracting image B recorded after single pump pulse \nirradiation and background initial image A (see Fig. 5). \nThe most distinct advantage of imaging is that it reveals  the spatial information of the domain, \nconcerning its shape, structure, and size. The time-resolved addition to imaging  allows determining the \ntime scale of pump-induced spatial changes. Therefore, creating repe atable series of images, as was \npreviously proposed in the measurement sequence, is an easy way to re trieve the information concerning \nthe time and localization of pump-induced changes. Such an image stack  contains information about the \nrelative time delay between images, which is retrieved from the delay line d istance.  \n8 \n Spatial information can be obtained directly from a stack of image s, but several factors need to be \nconsidered. Firstly, to compare images obtained with non-constant illumi nation conditions, one has to \nuniformize their intensity levels. In order not to spoil the informat ion concerning the single image \nintensity levels, the contrast-enhancing procedure should be applied to all images in the same manner. \nTherefore, the histogram of the whole stack, instead of individual image s histograms, was used to rescale \nthe intensity levels of each image. Here, float 32-bit representat ions of differential, images were rescaled \nto 0-256 range. Secondly, one has to be sure that the background intensity lev el is flat. For every image, \nit has to be at the same, uniform level. We note that without i t, differential images were spatially \nincomparable. In this case, we used additional post-processing algorithms.37   \nII. RESULTS \nWe used a setup for single-shot time-resolved magneto-optical imagi ng to determine the photo-magnetic \nswitching in YIG:Co film. The experimental data of photo-magnetic  switching retrieved from the \ndifferential image stack were represented in a three-dimensional manner, highlighting the mutual \ntemporal and spatial behavior of the change of magnetization orientat ion as shown in FIG. 6. The \nmagneto-optical differential images for selected ∆t are shown on t he top panel in FIG. 6. The contrast \nin these images is due to the magneto-optical Faraday effect, wh ich is proportional to the perpendicular \nmagnetization component. Change of intensity within the laser pump spot on t hese images allows \ntracking the evolution of the photo-magnetic switching. To visual ize the spatio-temporal redistribution \nof magnetization the 3D map was created from the intensity profiles of  150 differential images which \nwere recorded using a single probe pulse for different ∆t. The intensi ty profile for every image, \nproportional to the time-resolved Faraday rotation, was used data for the 3D map. To improve the signal-\nto-noise ratio, instead of only one profile cross-section line, we chose  the averaged profile from a \nrectangular selection of 10 µm width as shown by yellow on the image in Fig. 6. The color code  on the \nmap corresponds to the out-of-plane magnetization component. In this map, we obs erve the red area of \nphoto-magnetic switching in YIG:Co between M- and M+ states which correspond to magnetization \nstates (1) and (8) in FIG. 3a.  \n \nFIG. 6. Three-dimensional representation of magnetiz ation dynamics under single pump pulse with a fluence o f 75 mJ/cm 2 in \nYIG:Co film. A negative time corresponds to the probe  pulse illuminating the sample before the pump pulse. The characteristic \nblink in overlap position defines precisely ∆t=0 ps time. The top panel shows the differential imag es of magnetic domain \nstructure obtained at various ∆t. The color on the diagram corresponds to the angle of time-resolved Faraday rotation which is \nproportional to the change of the out-of-plane magn etization component. The deep blue color is an initi al magnetization state \n(1) M-. The green color corresponds to the magnetizat ion aligned in-plane. The red color shows the area sw itched to state (8) \nM+.  \n9 \n The mechanism of presented photo-magnetic switching in YIG:Co film is triggered via the precession \nof the net magnetization.29  In this case, the pump light optically excites one of the possible  and the most \neffective electronic transitions for Co ions being in the tetrahedra l sites.38  Such excitation is responsible \nfor the change of magnetic anisotropy in a garnet. For an initial stat e, before pumping the intrinsic \nmagnetic anisotropy is predominantly cubic with a small uniaxia l contribution. The strength of the \neffective field of photo-induced anisotropy is comparable to the intrin sic one.13  It can lower the energy \nbarrier allowing the magnetization to switch to the second state. Be cause of the large damping, the \nlifetime of the photo-induced anisotropy is low and it decays at about 60 ps at room tempera ture which \ncorresponds to the quarter of the laser-induced precession.  \nThe size of a switched domain can be determined by the spot of the pump beam. Here, it is given by the \naveraged laser optical power and total pump spot size on the sample, de termined by the system focusing. \nHowever, because the incident beam has a Gaussian shape, locall y the light intensity differs. It is highest \non the beam axis, and it decreases away from the axis. Therefore, by selecting and ana lyzing a selected \nregion of interest (ROI) one can distinguish the amplitude of photo-induced f ield-related temporal \nchanges in FIG. 7a. Improved sensitivity of the set-up allowed to vis ualize and distinguish four \ncharacteristic regions with different behavior of the magnetizati on vector. The mean value from the \nselected ROIs is subjected to further analysis. Chosen ROIs are marked on the image with appropriate \ncolors. All of them are circles with the same size of about 10 µm diameter and are shifted from the image \ncenter by given ∆ d. Choosing a larger region instead of only individual pixels can reduce noise impact.  \nFor the ∆ d > 60 µm ROI corresponds to the region which is unaffected by the pump puls e so it acts as \na background. No change of magnetization out of plane component appears here.  \n \nFIG. 7 a) The temporal behavior of the out-of-plane  magnetization component for the different characte ristic regions of interest. \nb) The time-resolved precession around the state M- in the long-time domain for ∆ d = 40 µm. The signal obtained by the single-\nshot imaging is compared to the regular pump-probe dyn amic with the same experimental condition of the lase r pulses. \n10 \n The background is used as the image stack reference. It has to be fl at to ensure that all images taken in \nthe measurement series are comparable. For smaller ∆ d = 40 µm the incident pump intensity is still \nbelow the switching threshold. However, its intensity is high enough t o trigger the precessional \nmovement of magnetization. With ∆ d = 20 µm and lower the incident pump intensity is high enough to \nobtain the switching of magnetization from state M- to state M+. After it, the out-of-plane component \nof magnetization is reversed. Presented behavior applies also to the averaged situat ion – ROI including \nwhole switched spot. In the center of the spot ∆ d = 0 µm the local pump intensity is highest. It triggers \nthe magnetization switching from one state to another. Previously, in reference 29  due to the insufficient \nmagneto-optical contrast, this effect was hidden. Here, because of the greater imaging sensitivity, we \ncan perform spatial analysis in detail. Thus, for a sufficiently la rge field of photo-induced anisotropy \nswitching from the state (1) to (8) occurs. Next, after about 60 ps, when the magnetization was switched, \nwe observed damped precession around the direction of the switched magneti zation state. The amplitude \nof this precession corresponds to the FMR mode was small and the magne tization remains at the \nswitched state. This effect was predicted 38  but it was not observed experimentally. \nWe compared precession for ∆ d = 40 µm measured through presented single-shot imaging with the \nsignal obtained through highly sensitive pump-probe measurement (see Fig  7b). By that, we determined \nthe time-resolved Faraday rotation sensitivity of our method as 30 mde g for the 7.5 µm garnets which \ncan be scaled to about 4 mdeg/µm. The data presented as the pump-probe signal was obtained within \nthe same setup but with a typical detection pump-probe scheme.39  The laser system worked with a \nrepetition rate of 1 kHz. The probe beam was modulated by a chopper to 500 Hz and it  was focused on \nthe sample to 50 µm in diameter. Next, the probe was directed to the ha lf-wave plate and the Wollaston \nprism which separates it into two linearly polarized beams with orthogonal polarization. These beams \nare directed to the two separate branches of an auto-balanced photodiode , which monitors the change of \ntheir intensities corresponding to the Faraday rotation. The frequency modula tion alternating the pump \nand probe signal was applied to use lock-in amplifier (Zurich Instruments  MFLI) based detection. \nMoreover, to improve the signal-to-noise ratio boxcar integrator (Stanford Re search Sr 250) was \napplied.  \nIII. CONCLUSIONS \nWe developed and implemented an automated setup for the high-contrast time-resolved single-shot \nimaging of magnetization dynamic. By using two OPAs we obtained independent temporal and spectral \ntunability of two beams. The possibility of selecting the central w avelength from the wide spectrum for \nthe pump beam is a key for inducing different resonant transitions responsi ble for the magnetization \nswitching, especially in dielectrics. The spectral tuning of the  probe beam is necessary to find the \nwavelength assuring an optimal balance between high magneto-optic al contrast and low absorption in \nthe sample. Using two delay lines made it possible to set a pump d elay with an unchanging probe delay \nand vice versa. By applying the pinhole into the probe beam we limit ed the interference noise improving \nthe imaging quality. We observed the switching in single-domain structure and spatially analyzed its \ndynamics. Moreover, right next to the switching with the high sensi tivity of 4 mdeg/ µm, we \ndistinguished regions in which precession of the magnetization appears. The propos ed imaging method \nis characterized by full flexibility and very high sensitivity. It may be further use for examining the \nspatio-temporal behavior of more sophisticated, multi-state magneti c structures, as well as even finding \ncompletely new switching mechanisms in different materials. \nAcknowledgments.  The authors thank D. Afanasiev and A.V. Kimel for the fruitful dis cussion. This \nwork has been funded by the Foundation for Polish Science POIR.04.04.00-00-413C/17-00. \nREFERENCES \n1 A. Kirilyuk, A. V. Kimel, and T. Rasing, Rev. Mod. Phys. 82 , 2731 (2010). \n2 D. Sander, S.O. Valenzuela, D. Makarov, C.H. Marrows, E.E. Fullert on, P. Fischer, J. McCord, P. Vavassori, S. \nMangin, P. Pirro, B. Hillebrands, A.D. Kent, T. Jungwirt h, O. Gutfleisch, C.G. Kim, and A. Berger, J. Phys. D.  \nAppl. Phys. 50 , 363001 (2017). \n3 J. McCord, J. Phys. D. Appl. Phys. 48 , (2015). \n4 M.C. Downer, R.L. Fork, and C. V. Shank, J. Opt. Soc. Am. B 2, 595 (1985). 11 \n 5 D. Davydova, A. de la Cadena, D. Akimov, and B. Dietzek, Las er Photonics Rev. 10 , 62 (2016). \n6 M.C. 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"    },    {        "title": "1504.00895v1.The_effect_of_the_magnon_dispersion_on_the_longitudinal_spin_Seebeck_effect_in_yttrium_iron_garnets__YIG_.pdf",        "content": "1 \n The effect of the magnon dispersion on the longitudi nal spin Seebeck effect in yttrium iron garnets \n(YIG)  \nHyungyu Jin1, Stephen R. Boona1, Zihao Yang2, Roberto C. Myers2,3,4, and Joseph P. Heremans1,3,4 \n1. Department of Mechanical and Aerospace Engine ering, The Ohio State University, Columbus, \nOH, 43210 \n2. Department of Electrical and Computer Engineer ing, The Ohio State University, Columbus, OH, \n43210 \n3. Department of Materials Science and Engineerin g, The Ohio State University, Columbus, OH, \n43210 \n4. Department of Physics, The Ohio State University, Columbus, OH, 43210 \n \n \nABSTRACT \n We study the temperature dependence of the long itudinal spin-Seebeck effect (LSSE) in a yttrium \niron garnet Y 3Fe5O12 (YIG) / Pt system for samples of different  thicknesses. In this system, the thermal \nspin torque is magnon-driven.  The LSSE signal peak s at a specific temperature that depends on the YIG \nsample thickness.  We also observe freeze-out of the LSSE signal at high magnetic fields, which we \nattribute to the opening of an energy gap in the ma gnon dispersion.  We observe partial freeze-out of the \nLSSE signal even at room temperature, where kBT is much larger than the gap. This suggests that a subset \nof the magnon population with an energy below kBTC (TC ∼ 40 K) contribute disp roportionately to the \nLSSE; at temperatures below TC, we label these magnons subthermal magnons .  The T-dependence of the \nLSSE at temperatures below the maximum is interprete d in terms of a new empirical model that ascribes \nmost of the temperature dependence to that of the thermally driven magnon flux. \n  2 \n INTRODUCTION \n The spin Seebeck effect (SSE), first reported in 20081, is the experimental manifestation of \nthermal spin transfer torque.  The SSE2,3 can be understood as the proce ss by which a spin current is \ngenerated by a temperature gradient ∇T in one material, where it is carried by either magnons or spin-\npolarized electrons, and this spin current then cro sses the interface into an adjacent normal metal (NM). \nDue to strong spin-orbit interactions  in the NM, this injected spin current generates an electric field EISHE \nvia the inverse spin-Hall effect (ISHE). Since this electric field is related to ∇T, it is possible in the \nregime of linear relations to define a spin-Seebeck coefficient SSSE= EISHE / ∇T.  The effect manifests \nitself in two geometries, the tran sverse spin-Seebeck effect (TSSE)1,4,5,6,7,8 and the longitudinal spin-\nSeebeck effect (LSSE)9,10,11, in which the direction of the heat fl ux and of the spin flux are collinear and \nnormal to the direction of the magneti zation, respectively.  In the latter geometry, a spin Peltier effect has \nbeen observed12 to be the Onsager reciprocal of the LSSE.  The potential issue of  the admixture of a \nNernst effect into the LSSE signal13 has also been addressed.14,15,16  The LSSE is well understood17 at this \npoint; however, it is only applied to ferromagnetic insulators such as yttrium iron garnet Y 3Fe5O12 (YIG), \nsince the geometry of the measurem ent means that LSSE signals are indistinguishable from classical \nNernst signals if ferromagnetic conductors are used.  The general theory for L SSE is given in Ref [17], \nand the effect in the YIG/Pt system  is attributed to magnon transport.18 , 19 , 20 , 21  To explain this \nschematically, we can say that b ecause there is a magnon contribution κM to the thermal conductivity,22,23 \na magnon heat flux jQM = -κM ∇T exists in the presence of a temper ature gradient, and thus there must \nalso exist a flux of magnons, i.e ., a spin or magnetization flux jM.  In the simplest possible approximation, \nby treating the magnons as an ideal gas and ignoring th e mode and frequency dependence of effects, one \ncould even write that jM = jQM (gL μB)/(k BT) = - κM ∇T (gL μB)/(k BT).  Here, gL is the Landé factor and μB 3 \n the Bohr magneton.  This thermally dr iven magnetization flux then cro sses the interface through a process \ngoverned by a characteristic paramete r known as the spin-mixing conductance g↑↓, which is the ratio that \nrelates the spin current generated in a material to  the energy of the driving spin injection process.24  The \ndirectional spin polarization v ector is set by the magnetization M along the applied field  H (as given in \nFig. 1(b)).  This overall spin polariz ation gives rise in the Pt metal to  an inverse spin-H all field given by \nEISHE = D ISHE (jQM × M), which is measured as a voltage VISHE = | EISHE| L, where where DISHE is the , and \nL the length of the Pt strip.   \nThe dependence of LSSE on YIG thickness  has been studied in thin films,25 along with the behavior at \nlow18 and high26 temperatures and at high fields.16  Particularly interesting is that near the Curie \ntemperature, a specific temperature dependence has been observed that does no t reflect that of the \nmagnetization.  Here, we present temperature-depende nt LSSE data at cryogenic temperatures on both a \nbulk single crystal and a 4 μm thick film, and we show that they  are different.  This difference is \ninterpreted in terms of the magnon  thermal mean free path.  We also extend the low temperature \nmeasurements to high magnetic fields, where we see that the LSSE signal is decreased at high field, \nwhich we interpret in terms of the magnon gap openi ng due to Zeeman splitting of the magnon dispersion.  \nThis suppression is also observed at 300 K, where th e thermal energy is much larger than the magnon \nenergy gap we open in our experiments. The fact that th ere still is a significant suppression of the LSSE at \nthese temperatures and fields is taken as proof that low energy magnon modes appear to constitute a \ndisproportionately large perc entage of the LSSE signal at all temper atures.  This leads us to divide the \nmagnon population into two categories: thermal magnons , which we define as those magnons at all \nenergies up to T × kB, where T is the temperature of the experiment, and subthermal magnons,  which we \ndefine as those at energies below kBTC.  We define the cutoff temperature to be TC ∼ 40 K on the basis of 4 \n the amount of freeze-out measured at room temper ature, and note that the experimental magnon \ndispersion27 at energies of kBTC starts deviating significantly from a simple quadratic function of the \nwavevector.  Obviously, the distinction between subthermal and thermal magnons only holds at T>TC, and \nthat below TC both populations are the same.  The concept of subthermal magnons is inspired by the same \nconcept used for phonons in the context of electron-phonon drag in semiconductors.28  In phonon drag, \nthe only phonons that can interact with the electrons are thos e with propagation vector limited to 2 kF, \nwhere kF is the wavevector of the electrons at the Fermi surface.  In semiconductors where kF is much \nsmaller than the size of the Brillouin zone, and at high temperatures where the phonons of energy kBT \nhave a much larger k-vector, this cutoff means that only those “subthermal” phonons with smaller k-\nvectors can participate in drag.  Through our LSSE data we present here, we offer evidence that \nsubthermal magnons in YIG interact more strongly w ith the electrons in Pt, similarly to phonon drag, \nsuch that LSSE is driven not equally by all modes of the magnon population at th ermal equilibrium at any \ngiven temperature, but instead is biased  toward low-energy subthermal magnons.   \nSince this manuscript was written, a manuscript has appeared on arXiv29 which, for all intents and \npurposes, agrees very well with the da ta and conclusions we put forth here.  In addition, we propose here \nan empirical model for the temperat ure dependence of the LSSE at low temperature and in thin films.  \nThe model takes into account the ch ange of the magnon dispersion, from quadratic at the zone edge to \nkinear (“pseudo-acoutic ” as defined by Plant27) at higher energies; this mode l explains the fact that the \nobserved temperature dependence in thin films be low 100 K does not actually follow a power law of T, \nbut varies slowly from a T1 law at the lowest temperatures, wher e the dispersion of the thermal magnons \nis mostly quadratic, to a T2 law at higher temperatures.   \n 5 \n  \nEXPERIMENTAL \nTwo different samples (A and B) were measured  in this study.  Sample A consists of a 4 μm thick \nsingle-crystalline YIG film  grown on a 0.5 mm thick gadolinium ga llium garnet (GGG) substrate with a \n10 nm thick Pt layer deposited on top of the YIG.  Sample B is a 0.5 mm thick single-crystalline YIG slab \npurchased from Princeton Scientific, Inc, upon which a 10 nm thick Pt layer was also deposited on one \npolished wide surface of the YIG slab using e-beam evaporation after careful cleaning.  Dimensions ( Lx × \nLy × Lz, Fig. 1(b)) of the two samples are 2 mm × 6 mm × ~0.5 mm (sample A) and 5 mm × 10 mm × 0.5 \nmm (sample B).   \nFor LSSE measurements, both samples were mounted  in the same way as follows.  First, two \nvoltage leads were attached on two edges of the sample separated by Ly.  Fine copper wires with 0.001 in. \ndiameter were used for the voltage leads to minimi ze heat losses through the leads.  Point contacts \nbetween the voltage leads and the sample were made  using silver epoxy, and an effort was made to \nminimize the size of the contacts.  The sample was sandwiched between tw o rectangular cubic boron \nnitride (c-BN) pads to promote uniform heat flux across the sample, and Apiezon N grease was used \nbetween the sample and c-BN pads to provi de uniform thermal contacts.  Three 120 Ω resistive heaters \nconnected in series were attached on the top of the upper c-BN pad by silver epoxy.  A Cernox \nthermometer was attached to each c-BN pad usi ng Lakeshore VGE-7031 varnish.  The whole sample \nblock was then pushed onto the samp le platform wherein the Apiezon N grease was used as a thermal \ncontact.  The sample block and the platform were wr apped together by an insulated wire in order to \nmechanically secure the heterostructure and provide  a solid thermal contact between them.  Fig. 1(a) \nshows a picture of sample A mount ed in the way described above. 6 \n The LSSE measurements were performed usi ng a liquid helium cryostat for sample A and a \nPhysical Properties Measurement Syst em (PPMS) for sample B.  Sample B was also measured in the \nliquid helium cryostat for a cross-check, and the data obtained from the two different instruments showed \ngood agreement in terms of the temperature dependence of LSSE signal.  We observed that the magnitude \nof the signal can be slightly different for different measurement sets, possibly because of an error in the \nmeasured temperature difference induced by varying thermal contact resistances between the sample and the c-BN pads.  Nevertheless, we confirmed that th e temperature dependence itself is not affected by the \nsmall variability of the magnitude.  The measurement configuration is shown in Fig. 1(b).  First, the \nCernox thermometers were calibrated as a function of temperature from room temperature down to 2 K.  \nResistances of the thermometers were measured at di fferent temperatures using an AC bridge, and those \nvalues were interpolated to extr act actual temperatures during LSSE measurements.  LSSE measurements \nwere made at different temperatures, and ~1h st abilization time was given after changing the base \ntemperature to ensure thermal equilibrium between the sample and the system.  After a temperature \ngradient \nzT∇  was applied across the sample, additional 15 ~ 30 min wait time was used.  The transverse \nvoltage Vy was measured using a Keithley 2182A nanovoltmeter while the magnetic field Hx was \ncontinuously swept between a same Hx value in both polarities and in both directions (+ Hx → -Hx and - Hx \n→ +Hx).  For low field measurement, Hx was swept between ± 800 Oe at 4 Oe/s, and for high field \nmeasurement, between ± 70 kOe at 30 Oe/s.  The temperature difference zTΔ was separately measured \nfrom the attached thermometers using the same heater power as that used for the Vy measurement.  The \nsample chamber was kept under a high vacuum (<10-6 torr) and covered by the gold-plated radiation \nshield to minimize convective and ra diative heat losses, respectively. \n 7 \n RESULTS AND DISCUSSION \n Raw traces of the measured transverse voltage Vy as a function of applied field Hx for both \nsamples are shown in Fig. 2.  Around room temperatur e, both samples show clear LSSE signals under an \napplied temperature gradient ∇Tz, wherein Vy switches sign with the magnetiz ation in the hysteresis loops \n(Fig 2(a), (d)).  The ma gnitude of the switch, ∆Vy, is defined as the differe nce between the saturation \nvoltages.7 The signals become noisier at low temperatures (Fig. 2(b), (e)) due to the smaller magnitude of \nboth the voltage signals and the temperature difference s across the samples.  The raw traces of sample B \nexhibit an additional loop at low Hx values, and the size of this loop is almost independent of temperature.  \nWhile some of the previous studies reported a si milar repeatable trend in their LSSE signals,9,30 no \nexplicit description has been gi ven thus far to explain these obs ervations.  Based on the lack of \ntemperature dependence of the loop si ze, we may tentatively attribute the additional loop to the presence \nof magnetic domains in the bulk sample .  Fig. 2(c) and 2(f) show that ∆Vy varies linearly with ∆Tz for \nboth samples, so that a spin Seebeck coefficient 2 yy z z yy zSE T L VL T≡∇ = Δ Δ //  can be defined. \n The same LSSE measurement was extended up to Hx = ± 70 kOe for sample B.  At Tavg = 296 K, a \ngradual reduction of Vy / ∆Tz is observed as Hx increases (Fig. 3(a)).  This behavior around room \ntemperature is similar to what has been reported by Kikkawa et al.16 also on Pt/YIG.  The authors \nattributed the decrease of the LSSE  signals to the suppression of magnon excitation caused by the magnon \ngap opening under an applied H.  At low energies (B Tk ω=h  < 30 K), the magnon dispersion relation \nfor YIG can be expressed reasonably well using th e classic Heisenberg ferromagnet model for spin \nwaves,31 which is given as32 \n 22\nkB L gBD a k ωμ=+h   (1), 8 \n where h is Planck’s constant, ωk is the angular frequency of th e magnon mode with wave vector k, μB is \nthe Bohr magneton, gL is the Landé factor ( gL = 2.046 for YIG33), D is a spin-wave stiffness parameter, \nand a is the size of the unit cell.  Applying a non-zero external H creates a forbidden zone in the \ndispersion (Eq. (1)) as well as in  the density of states of magnons due to the Zeeman effect,31 thus \nfreezing out magnons with energy BLgB ωμ<h .  Since thermally excited magnons are believed to induce \nthe SSE in ferromagnetic insulators18,19,20,21,25, their suppression by an external H is expected to reduce the \nmeasured ISHE voltage.  Fig. 3(b)  shows that the suppression of Vy / ∆Tz in applied H is more pronounced \nat Tavg = 10 K.  If we define Vy,max as the maximum value of the ISHE voltage obtained at low H, the Vy / \n∆Tz at Hx = ± 70 kOe ( Vy,70kOe  / ∆Tz) is less than 50 % of Vy,max / ∆Tz.  Surprisingly, ab out 20% suppression \nis observed at all te mperatures up to Tavg = 296 K.  We compared the magnitude of Vy,max / ∆Tz in these \nhigh H measurements (Fig. 3(c) and (d)) with that in the independently measured low H results on sample \nB (Fig. 2(d)-(f)), and found that they  agree well with each other.   \n The significant suppression of the LSSE signal at high H at Tavg = 10 K hints at which magnons \nare mainly responsible for the LSSE in Pt/YIG, which is further revealed by inve stigating the temperature \ndependence of the suppression  effect.  We can define ∆Sy = Sy,max – Sy,70kOe , where \n2 yz y y zSL VL T ≡Δ Δ ,max ,max /  and 70 70 2 yk O e z yk O e y zSL VL T ≡Δ Δ ,, / . Fig. 4(a) shows the temperature \ndependence of ∆Sy / Sy,max derived from the high H measurements (Fig. 3) on sample B.  Above Tavg = 30 \nK, the ∆Sy / Sy,max is almost constant at ~ 0.2 (inset in Fig. 4(a)), indicating that about 20% of the LSSE \nsignal is suppressed at Hx = ± 70 kOe.  The ∆Sy / Sy,max increases gradually below 30 K and then shows a \ndrastic increase below 15 K reachi ng ~ 0.9 at about 7 K, suggesting th at the LSSE almost disappears at \nthis temperature and field.  For YIG, at Hx = ± 70 kOe, a gap of / 9.6 KBL BgH kμ = opens in the \ndispersion relation given by Eq. (1) as shown in Fig. 4(b) .  Therefore, it is like ly that most magnons are 9 \n suppressed below 8 K in Hx = ± 70 kOe and so is the LSSE, which is consistent with the experimental \nresult in Fig. 4(a).  But, as had already been pointed out in Ref [23], magnon suppression cannot be \ncomplete in Hx = ± 70 kOe fields at temperatures above 10 K.  Some amount of field suppression of the \nLSSE is measured at all temperatures, while not in specific heat or thermal conductivity.23 Here, we \nverified experimentally on another piece of Sample A that the thermal conductivity of YIG is independent \nof magnetic field at 300 K to within less than 1%, th e experimental accuracy of that measurement.  We \nthus conclude that low-energy ma gnons (i.e. those of energy below ~ kB × TC) must play a more \nprominent role in producing the LSSE.  We estimate the value of the critical temperature TC ∼ 40 K using \nMaxwell-Boltzmann statistics by setting ,max 0.2 / exp( / ) 1yy B L B CSS g B k T μ =Δ = − , but note that there is \nconsiderable uncertainty in the pr ecise energy scale of the relevant  modes, since we expect all magnon \nmodes to contribute to thermal spin transport and L SSE, just some to contribut e more than others.  In \nparticular, the inflexion point observed in the low- temperature behavior of ,max/yySSΔ at 15 K in Fig. 4(a) \nseems to single this temperature out as well. Similar to the case of phonons,28,34 we call these low-energy \n(E < kB TC), long-wavelength magnons subthermal magnons  to distinguish them from the thermal \nmagnons with relatively higher energies and short wa velengths that populate the magnon spectrum in \nthermal equilibrium at T > TC .  \n We now discuss the temperature and thickness de pendence of the LSSE in Pt/YIG.  Fig. 5 shows \nthe temperature dependence of Sy for sample A and sample B between 10 K and 300 K.  The Sy of sample \nA has a maximum at 180 K. The Sy of sample B, meanwhile, has a maximum at 70 K and a ~ T 1.19 \n(dashed line) dependence at low temperatur e.  The longitudinal thermal conductivity κxxx was measured \nindependently, but is not shown because th e data reproduce thos e of Slack & Oliver35 quite exactly. The \ntemperature dependence of the resistivity of the Pt film on sample B was measured to change from 0.35 10 \n μΩ m below 5 K to 0.52 μΩ m at room temperature.  The te mperature dependence of both these \nproperties is quite different from that of the observed LSSE signal, which suggests that there is likely no \nsignificant correlation between them.  In the absence of information about the temperature dependence of \nthe spin Hall angle in Pt and of the spin mixing c onductance at the YIG/Pt inte rface, we concentrate on \nthe temperature-dependence of the thermally driven magnon flux. \n Therefore, we assume that the temperature depe ndence of the LSSE signal arises primarily from \nthat of the thermally driven magnon flux that reach es the interface. This in terpretation, supported by the \nresults of Ref. [26], allows us to neglect the T-dependence of the ISHE, the spin-Hall angle in the Pt, and \nthe spin-mixing conductance across the interface. In general, since the magnetization flux at the YIG/Pt \ninterface is in YIG driven by a temperatur e gradient, we can describe this flux by \n ## T ML B L Bj gj g L Tμμ== ∇      ( 2 ) ,  \nwhere j# is the magnon number flux, and L#T is the effective kinetic coefficient that relates this number \nflux to the accelerating force, the temperature gradient .  The number flux is also related to the magnon \nheat flux jQ, which is itself related to the magnon thermal conductivity κM by Fourier’s law QMj T κ=− ∇ .  \nUsing the Boltzman diffusion equation in the relaxation time ( τ) approximation, both kinetic coefficients \ncan be written for each magnon mode as an in tegral over the magnon frequencies (or energy E):36 \n 2\n#T\n0\n2\n0()()\n1()()3MAX\nMAXE\nG\nE\nMGfLv E E d ET\nfEv E E dETτ\nκτ∂⎛⎞= ⎜⎟∂⎝⎠\n∂⎛⎞= ⎜⎟∂⎝⎠∫\n∫D\nD     (3). 11 \n Here D(E) is the magnon density of states and f is the Bose-Einstein statisti cal distribution function, so \nthat \n()21x\nxfxe\nTT e∂=∂ − where B xEkT≡  is the reduced energy.   \nAt very low temperature where the magnon dispersion is purely quadratic27 and given by Eq. (1), \nthe group velocity and density of states in  the absence of magnetic field become: \n()21 / 2\n1/2\n3/2 222\n1()4GvD a E\nEE\nDaπ=\n=h\nD       ( 4 ) .  \nAn experimental estimate of th e magnon thermal mean free path Ml is given by Ref. [23] for T > \n2 K.  In the ballistic regime where the ma gnon mean free path is limited by sample size, lM is temperature-\nindependent and () /M G Evτ =l .  Consequently, we can derive the kinetic coefficient of the magnon \nnumber flux as well as the thermal conductivity: \n()\n()22\n#T 2 22\n0\n3 3\n2\n2 22\n031\n2 1\n1\n3 1x\nMB\nx\nx\nMB\nMxkx eLT d xDa e\nkx eTd xDa eπ\nκπ∞\n∞=\n−\n=\n−∫\n∫l\nh\nl\nh     ( 5 ) .  \nOne recognizes in Eq. (5) the familiar magnon thermal conductivity T2 scaling law. In that case, Eq. (5) \npredicts that the thermally dr iven magnon flux, and thus the LSSE signal, should scale as T1 (recall jM = - \nκM ∇T (gL μB)/(k BT)).  At T < 2 K, the T2 law for κM  is observed,22 but not23 above 2 K, for two reasons.  \nFirst, because the measurements are taken on bulk samples, the ballistic regime is not reached until T < \n2K and there is a dependence of lM on T.  The second reason is the non-parabolicity of the magnon 12 \n dispersions. Eq. (1) already breaks  down at energies below 10 K: in deed Ref [27] shows that the \ntemperature dependence of the saturation ma gnetization curves departs from the Bloch T3/2law already \nbelow 10K.  \nAt higher energies (> 8 me V) and temperatures, the magnon dispersion becomes linear27, and the \nthermal conductivity in the ballisti c regime is expected to follow a T3 law, with L#T ∝ T2.  In terms of \ngroup velocity, it increases with energy when the dispersion is quadratic, but saturates at a maximum \nvalue vGS when the dispersion becomes linear.  These feat ures can be captured by  fitting the measured \n“pseudo-acoustic” magnon disp ersion curves in YIG27 (and ignoring the pseudo-opt ical modes) with a \nphenomenological model for the magnon dispersion writ ten by analogy with the electron dispersion in \nnarrow gap semiconductors such as PbTe37 as:  \n22\n0() ( 1 )EE ED a kEγ ≡+ =      ( 6 ) ,  \nwhere E0 parametrizes the saturation value vGS of the group velocity vG, as shown below.38  The group \nvelocity and DOS are given by: \n \n()2\n3/2 22\n02\n'\n' 1()4\n'1 2GDav\nE\nDa\nE d\ndE Eγ\nγ\nγγ\nπ\nγγ=\n=\n≡= +h\nD      (7). \nThe value of E0 is derived from vGS by taking the limit 2\n0lim ( )GS E GDEavv→∞ ==h.  Ref [27] reports for \nD =  46 K × kB, vGS = 1.8 × 104 m s-1 along the (0,0,l) axes, and E0 = 270 K × kB.  13 \n The integrals in Eq. (3) can be fully evaluated and expre ssed in terms of Ml:  \n ()\n()#T 2 2\n0\n2\n2 2\n01()3 1\n1()3 1x\nMGxB\nx\nMM GxBeLE v E d EkT e\neEvE d EkT eκ∞\n∞=\n−\n=\n−∫\n∫l\nlD\nD    (8). \nIn the ballistic regime with a constant Ml: \n()\n()22\n#T 2 22\n0 0\n33\n2\n2 22\n0 03112 1\n113 1x\nMB B\nx\nx\nMB B\nMxkk T x eLT x d xDa E e\nkk T x eTx d xDa E eπ\nκπ∞\n∞⎛⎞=+ ⎜⎟\n− ⎝⎠\n⎛⎞=+ ⎜⎟\n− ⎝⎠∫\n∫l\nh\nl\nh   (9). \nFrom Eqs (5) and (9), a temperature-dependen ce of the LSSE signal intermediate between T2 and T1 is \nexpected, which is shown in Fig. 5 and appears to desc ribe the results on the thin-film sample (sample A) \nbelow the maximum quite well.   \n We can now discuss the difference in the temperature dependence of Sy between sample A and \nsample B in Fig. 5 by considering the length scal e of subthermal magnons in terms of the sample \nthickness.  Based on the suppression of LSSE at high fields, we can st ate that most of the LSSE signal \nobserved at low temperatures comes from magnons at energies below kBTC. We can then extrapolate the \nmean free path data of Ref [23] in to the range of 10 K to 40 K, and we make the reasonable assumption \nthat these subthermal  magnons will have Mlof a few μm or above within this temperature range.  Since \nSample A is only 4 μm thick, we expect the subthermal magnons in this sample to be in the ballistic \nregime with a constant Ml = 4 μm, and therefore Eq. (9) to hold at low temperatures. This assumption of 14 \n ballistic magnon transport in sample A is expected  to break down once the temperature becomes high \nenough that Ml becomes limited by the increased prevalence of magnon-phonon interactions..   \nIn contrast to this situation,  the subthermal magnons in the bulk sample B are expected to \npropagate diffusively at all temperatures due to the much larger YIG thickness (500 μm), since the \nmagnon mean free path observed at 2K in Ref. [23] was only 100 μm.  As in sample A, however, the \nmagnon mean free path in sample B is also a f unction of temperature w ith a negative exponent,23 which \nresults in a slower T dependence of the LSSE in sample A, as observed in Fig. 5.  We attribute this \ndifference to the fact that there is likely no purely ballistic magnon tr ansport in sample B, and thus \ninelastic magnon scattering starts to  affect the subthermal magnons starting from even the lowest \ntemperatures examined in this study.  Furthermore, we can explain the shift in  peak temperature between \nsample A and sample B by considering that magnons in sample B (the bulk crysta l) are sensitive to any \nmechanism that scatters magnons ov er a length scale shorter than 4 μm, whereas those in sample A (the 4 \nμm film) are not. In other words, bounda ry scattering persists as more important than diffuse inelastic \nmagnon-phonon or magnon-magnon39 scattering for magnons up to TC ∼ 40 K in sample A, whereas the \ninelastic processes are already significant above 45 K in sample B.  As a result, the maximum in Sy in \nsample B appears at 70 K, a much lo wer temperature than in sample A ( Tavg = 180 K).   \nIt is noted that the Sy of sample A is larger than that of sample B above 80 K and becomes smaller \nbelow 80 K because of the faster T dependence.  Kehlberger et al.25 demonstrated that the LSSE signal \nincreases as the thickness of YIG incr eases up to 100 nm and st arts to saturate for larger thicknesses at \nroom temperature, while Rezende et al.18 observed no difference in LSSE between 8 μm thick and 1 mm \nthick YIG samples under the same ∇Tz.  Both data are consistent with  each other in the sense that 8 μm is \nlikely to already be larger than the mean free path of subthermal magnons at room temperature (~ 100 15 \n nm), which can be estimated from the data in Ref. [2 3].  The same argument does not apply to our result, \nhowever, as the magnitude of Sy is quite different in sample A and sa mple B at room temperature.  Indeed, \nwe find that the magnitude of Sy can be a quite complex problem b ecause the magnitude of the voltage \ninduced by the LSSE is sensitive to the Pt/YIG interface condition.40,41,42  Furthermore, because of the \ndifferent origins of sample A and sample B in our st udy, we cannot exclude the pos sibility that the Pt/YIG \ninterface condition varies between th e two samples, which may at least partially explain the observed \ndifference in the absolute magnitude of Sy.  \n \nCONCLUSIONS \nIn summary, we present magnetic field and temper ature dependent data of  the longitudinal Spin-\nSeebeck effect on the YIG/Pt system on a 4 μm thick film and on a bulk YIG.  From the freeze-out of the \nLSSE signal in high-field data at different temper atures below 300 K, we conclude that subthermal \nmagnons (defined as those with en ergies below a critical energy kBTC with TC of the order of 40 K) must \nplay a more important role in the LSSE than most magnons present at thermal equilibrium at T > TC.  We \nalso attribute the temperature-depe ndence of the LSSE predominantly to  that of the thermally-driven \nmagnon flux, for which we develop a new model that takes into account the departure of the magnon \ndispersion relations from purely para bolic to a relation that more closely resembles that measured by \nneutron diffraction.  Finally, we argue that it is this  non-parabolic dispersion that  is the reason behind the \nobserved decrease in LSSE signal with decreasing te mperature below the maximum, which deviates from \nthe expected T1\n law for parabolic magnons when thei r mean free path is constant.  \n 16 \n ACKNOWLEDGMENTS \nThe work is supported as part of the ARO MU RI under award number W911N F-14-1-0016, US AFOSR \nMURI under award number FA9550-10-1-0533 (HJ), NSF grants CBET-1133589 and DMR 1420451, \nand NSF MRSEC (SB).  The thin-film sample was kindl y supplied by Professors E.  Saitoh and K. Uchida, \nTohoku University, Sendai, Japan, whom we also acknowledge for useful conversations. 17 \n Figure Captions \nFIG. 1.  (Color online)  Experime ntal setup for LSSE measurement.  (a) Experimental setup on a liquid \nhelium cryostat used for both sample A and B.  (b) Schematic illustration of the measurement geometry \nfor sample A.  The same geometry applies to sample B except that the GGG substrate is absent in sample B.  The sample dimensions  are not drawn to scale. \n \nFIG. 2.  (Color online)  Lo w field LSSE measurement.  (a) – (c) Transverse voltage V\ny as a function of \nmagnetic field Hx (a,b), and ∆Vy versus applied temperature difference ∆Tz (c) for sample A.  (d) – (f) Vy \nas a function of Hx (d,e), and ∆Vy versus ∆Tz (f) for sample B. \n \nFIG. 3.  (Color online)  High field L SSE measurement on sample B.  (a), (b) Vy / ∆Tz as a function of Hx \nmeasured up to Hx = ± 70 kOe (“high field”) at Tavg = 296 K (a) and at Tavg = 10 K (b).  (c), (d) \nMagnification of low field ( Hx = ± 3 kOe) results at Tavg = 296 K (c) and at Tavg = 10 K (d). \n \nFIG. 4.  (Color online)  (a) T dependence of ∆Sy / Sy,max for sample B.  Here, ∆Sy / Sy,max ≡ (Sy,max – Sy,70kOe) \n/ Sy,max with Sy,max and Sy,70kOe defined as in the text.  ∆Sy / Sy,max = 1 indicates the fully suppressed Sy at H = \n± 70 kOe (i.e. Sy,70kOe = 0).  The inset shows the same data extended to 300 K in log scale.  Tavg denotes \nthe average sample temperature.  (b) Schematic di agram of magnon dispersion of  YIG below 0.6 THz (30 \nK) per Ref [27].  The red arrow indicates opening of a magnon gap equivalent to ~ 9 K when H = ± 70 \nkOe is applied.  The red dashed lines mark the te mperature (~ 15 K) below which significant increase in \n∆Sy / Sy,max starts to occur. 18 \n  \nFIG. 5. (Color online)  T dependence of the spin Seebeck coefficient Sy for the 4 μm film sample A (green \ndiamond) and the bulk sample B (orange circle).  Th e dashed lines represent a power law for the bulk \nsample, and values calculated from the model (Eq. 9) in the text for the thin-film sample, renormalized to go through the data.  19 \n Figure 1 (one column) \n \n \n \n \n  \nH∇T\nPt\nGGG \nsubstratez\ny\nx\nLxLzLy\nYIG\nheater\nthermometerYIGc-BNa b20 \n Figure 2 (two columns) \n \n \n  -400 -200 0 200 400\nHx (Oe)-3-113Vy (μV)\nΔTz = 2.5KTavg = 302.5KΔVy\n-400 -200 0 200 400\nHx (Oe)-0.3-0.10.10.3Vy (μV)\nΔTz = 0.9KTavg = 36K\n-800 -400 0 400 800\nHx (Oe)-0.8-0.400.40.8Vy (μV)\nΔTz = 1.1KTavg = 296.4KΔVy\n-800 -400 0 400 800\nHx (Oe)-150-5050150Vy (nV)\nΔTz = 0.36KTavg = 13.2K01234\nΔTz (K)0246ΔVy (μV)Tavg = 302 ± 1K\n00 . 8 1 . 6\nΔTz (K)00.8ΔVy (μV)41 ± 1K\n0 0.4 0.8 1.2\nΔTz (K)00.40.81.21.6ΔVy (μV)\nTavg = 296.4 K00 . 4 0 . 8\nΔTz (K)00.81.6ΔVy (μV) 33 ± 0.1Kac\nd fb\ne21 \n Figure 3 (one column) \n \n \n  -40 0 40\nHx (kOe)-0.800.8Vy / ΔTz (μV K-1) Tavg = 296  K\n-2 0 2\nHx (kOe)-0.800.8Vy / ΔTz (μV K-1)-40 0 40\nHx (kOe)-0.400.4Vy / ΔTz (μV K-1) Tavg = 10  K\n-2 0 2\nHx (kOe)-0.400.4Vy / ΔTz (μV K-1)a\nbc\ndHigh field High field\nLow field Low field\na\ncb\nd22 \n Figure 4 (one column) \n \n \n  a\nb\n Excitation frequency (THz)\nT(K)k\n300.6\n00\n15~ 9 K-π/2\n+π/20 1 02 03 04 0\nTavg (K)00.51ΔSy / Sy,max  \n1 10 100 1000\nTavg (K)00.51ΔSy / Sy,max23 \n Figure 5 (one column) \n \n   \n24 \n References  \n                                                            \n1 K. 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L’vov,4and Burkard Hillebrands1,∗\n1Fachbereich Physik and Landesforschungszentrum OPTIMAS,\nTechnische Universit¨ at Kaiserslautern, 67663 Kaisersla utern, Germany\n2Graduate School Materials Science in Mainz, 67663 Kaisersl autern, Germany\n3Faculty of Radiophysics, Electronics and Computer Systems ,\nTaras Shevchenko National University of Kyiv, Kyiv 01601, U kraine\n4Department of Chemical Physics, Weizmann Institute of Scie nce, Rehovot 76100, Israel\nRecently E. Sonin commented[1] on our preprint “Supercurre nt in a room temperature Bose-\nEinstein magnon condensate” [2, 3], arguing that our “claim of detection of spin supercurrent is\npremature and has not been sufficiently supported by presente d experimental results and their theo-\nretical interpretation.” We consider the appearance of thi s Comment as a sign of significant interest\ninto the problem of supercurrents in Bose-Einstein magnon c ondensates. Here, we explicitly address\nE. Sonin’s comments and show that our interpretation of our e xperimental results as a detection of\na magnon supercurrent is fully supported not only by the expe rimental results themselves, but also\nby independent theoretical analysis [4].\nRecently E. Sonin submitted a short Comment on\nour preprint “ Supercurrent in a room temperature Bose-\nEinstein magnon condensate ”[2, 3], to arXiv[1], which\nwe have addressed in a previous Note[5]. Here we dis-\ncuss the four main statements of Ref.[1] in more details.\nStatement I. “Observation of spin current in Yttrium-\nIron-Garnet (YIG) is hardly possible” .\nThis statement is based on two criteria for the\nexistence of a steady supercurrent, formulated in\nRef.[6] in terms directly applicable for easy-plane\n(anti)ferromagnets, where the minimum of the magnon\nfrequency ω(k) corresponds to k= 0 and BEC is noth-\ning else but the homogeneous precession of the magne-\ntization. The situation in tangentially magnetized YIG\nfilms is quite different: the dispersion relation ω(k) has\ntwo minima at non-zero wavevector values ±k0(with\nk0≃5·104cm−1) and the BEC spate is a magnon\nstate with a wavelength of λ≃10−4cm, which at least\nten times smaller than the hot-spot radius. In this case\nwe have no problem with the anisotropy in the complex\nplaneoftheBECphase(seee.g. Ref.[4])andthemagnon\nphase variation across the hot spot exceeds 2 πat least\nten times.\nStatement II. “The authors did not consider a more\ncommon scenario of spin diffusion, which could probably\nsuccessfully compete with their present interpretation. I t\nis worth noting that spin diffusion would be more effective\nin the condensed state than in a non-condensed gas of\nmagnons” .\nWe present below evidence that spin diffusion cannot\nbe observed in our experiments and theoretical argu-\nments that the spin diffusion is much smaller than the\nspin supercurrent.\nExperimental arguments. We explained in page 4 of\nour preprint, that “the laser heating process decreases\nlocally the saturation magnetization via thermal excita-tion of high-energy magnons with terahertz frequencies,\nbut practically does not increase the local population of\nlow-energy magnons. As a result, the number of thermal\nmagnons in the low-energy spectral region remains neg-\nligibly small in comparison with the number of magnons\noriginating from the parametric pumping process and\ncannot visibly affect the BEC dynamics”. Moreover,\nwe observed a critical value of the bottom-magnon\npopulations Ncr(see horizontal dashed line in Fig. 5)\nthat separates two regions with different spin dynamics;\nin the upper region the only reasonable explanation\nfor the observed spin dynamics is the existance of a\nsupercurrent of magnons within the BEC, because a\ndiffusive spin current is not a threshold effect, and,\nthus, cannot lead to the appearance of Ncr. Thus, we\nconclude: The above described experimental facts exclude\nthe possibility of a significant contribution of the spin\ndiffusion current to the observed spin dynamics .\nTheoretical arguments. To support the above conclu-\nsion theoretically, we could have chosen to present an\nanalytical estimation of the relative roles of the magnon\nsupercurrent and a normal diffusion current. Such an\nestimate has been discussed in a recent PRL paper by\nC.Sun, T. Nattermann and V.Pokrovsky [4] with the\nfollowing result: “Thus, we expect that, in realistic cir-\ncumstances, the spin superfluid current (in YIG films)\nequal to 1022÷1023cm−2s−1is larger than the normal\ncurrent by (3÷5)decimal orders ”. Due to experimen-\ntal limitations regarding the apparatus calibration, we\ncannot give the absolute value of the spin supercurrent,\nbut we believe that for the purpose of our discussion the\nrelative estimate, obtained in Ref.[4], is sufficient.\nStatement III. ”The authors investigated not station-\nary, but time dependent spin currents.”\nIndeed, our experiments are dealing with time-\ndependent decay processes with a characteristic decay of\ntime about (3 ÷4)·10−7s. This time is at least 10 times2\nlonger than the characteristic time of the four-magnon\ninteraction, responsible for the creation of the magnon\nBEC state. Therefore we believe that our BEC can be\nconsidered as being quasi-stationary, at least in first-\norder approximation. However we agree with E. Sonin\nthat an estimate of the effect of the non-stationarity is\nimportant in a consistent theory of the magnon BEC\nphenomenon. We hope to address this problem during\nongoing research.\nStatement IV. ”The agreement (in our Ref.[2, 3]) looks\nvery good indeed, probably due to using a fitting parame-\nter.”\nWe believe that such a good agreement stems from the\nfact, that our model reflects well the basic physics of the\nobserved phenomena. If there exists an alternative quan-\ntitative theoretical interpretation of the experimental re-\nsults, we would be happy to study it to compare, which\nmodel describes our experimental observations better.\nConclusion. We consider E. Sonin’s Comment[1] as\na sign of a wide interest to the subject and thank him\nfor the constructive discussions. We believe that in this\nNote we clarifiedthe differences in the physicalsituations\ndiscussed in Ref.[1, 2, 5]. We are confident that our ex-\nperimental results confirm the existence of a magnon su-\npercurrent in a room-temperature magnon Bose-Einstein\ncondensate.\n∗hilleb@physik.uni-kl.de\n[1] E.B. Sonin, Comment on supercurrent in a room\ntemperature Bose-Einstein magnon condensate,\narXiv:1607.04720.\n[2] D. A. Bozhko, A. A. Serga, P. Clausen, V. I. Vasyuchka,F. Heussner, G. A. Melkov, A. Pomyalov, V. S. L’vov,\nand B. Hillebrands, Supercurrent in a room temperature\nBose-Einstein magnon condensate, arXiv:1503.00482.\n[3] D. A. Bozhko, A. A. Serga, P. Clausen, V. I. Vasyuchka,\nF. Heussner, G. A. Melkov, A. Pomyalov, V. S. L’vov,\nand B. Hillebrands, Supercurrent in a room temperature\nBose-Einstein magnon condensate, Nature Physics, 2016,\nDOI: 10.1038/NPHYS3838.\n[4] C. Sun, T. Nattermann, and V. Pokrovsky, Unconven-\ntional superfluidity in yttrium iron garnet films, Phys.\nRev. Lett. 116, 257205 (2016).\n[5] D. A. Bozhko, A. A. Serga, P. Clausen, V. I.\nVasyuchka, F. Heussner, G. A. Melkov, A. Pomyalov,\nV. S. L’vov, and B. Hillebrands, On supercurrents in\nBose-Einstein magnon condensates in YIG ferrimagnet,\narXiv:1608.01813v1\n[6] E. B. Sonin, Adv. Phys. 59, 181 (2010).\n[7] R. E. Troncoso and ´A. S. N´ uez, Josephson effects in a\nBose-Einstein condensate of magnons. Ann. Phys. 346,\n182–194 (2014).\n[8] K. Nakata, K. A. van Hoogdalem, P. Simon, and D. Loss,\nJosephson and persistent spin currents in Bose-Einstein\ncondensates ofmagnons. Phys.Rev.B 90, 144419 (2014).\n[9] K. Nakata, P. Simon, and D. Loss, Magnon transport\nthrough microwave pumping. Phys. Rev. B 92, 014422\n(2015).\n[10] P. Nowik-Boltyk, O. Dzyapko, V.E. Demidov, N.G.\nBerloff, and S. O. Demokritov, Spatially non-uniform\nground state and quantized vortices in a two-component\nBose-Einstein condensate of magnons. Sci. Rep. 2, 482\n(2012).\n[11] S. Autti et al.Self-trapping of magnon Bose-Einstein\ncondensates in the ground state and on excited levels:\nfrom harmonic to boxconfinement. Phys. Rev. Lett. 108,\n145303 (2012).\n[12] F. Li, W.M. Saslow, andV.L.Pokrovsky, Phase diagram\nfor magnon condensate in yttrium iron garnet film. Sci.\nRep.3, 1372 (2013)."    },    {        "title": "1610.05760v1.Spin_transport_in_antiferromagnetic_NiO_and_magnetoresistance_in_Y__3_Fe__5_O___12___NiO_Pt_structures.pdf",        "content": "arXiv:1610.05760v1  [cond-mat.mtrl-sci]  18 Oct 2016Spin transport in antiferromagnetic NiO and magnetoresist ance in\nY3Fe5O12/NiO/Pt structures\nYu-Ming Hung,1Christian Hahn,1Houchen Chang,2Mingzhong Wu,2Hendrik Ohldag,3and Andrew D. Kent1\n1)Department of Physics, New York University, New York, New Yo rk 10003\n2)Department of Physics, Colorado State University, Fort Col lins, Colorado 80523\n3)Stanford Synchrotron Radiation Lightsource, Menlo Park, C alifornia 94025\n(Dated: 11 August 2018)\nWe have studied spin transport and magnetoresistance in yttrium ir on garnet (YIG)/NiO/Pt trilayers with\nvaried NiO thickness. To characterize the spin transport through NiO we excite ferromagnetic resonance\nin YIG with a microwave frequency magnetic field and detect the volta ge associated with the inverse spin-\nHall effect (ISHE) in the Pt layer. The ISHE signal is found to decay e xponentially with the NiO thickness\nwith a characteristic decay length of 3.9 nm. This is contrasted with t he magnetoresistance in these same\nstructures. The symmetry of the magnetoresistive response is c onsistent with spin-Hall magnetoresistance\n(SMR). However, in contrast to the ISHE response, as the NiO thic kness increases the SMR signal goes\ntowards zero abruptly at a NiO thickness of ≃4 nm, highlighting the different length scales associated with\nthe spin-transport in NiO and SMR in such trilayers.\nI. INTRODUCTION\nAntiferromagnets have attracted a great deal of at-\ntention in spintronics because of their unique proper-\nties, such as a low magnetic susceptibility and ter-\nahertz spin dynamics.1,2In metallic antiferromagnet-\nbased spintronics, electrical switching of the antiferro-\nmagnetic domains in CuMnAs was sucesfully performed3\nand spin pumping studies have demonstrated spin in-\njection into IrMn.4,5In oxide spintronics, ferromag-\nnet (FM)/antiferromagnet (AFM)/heavy metal trilayer\nstructures have been used to study spin-transport in in-\nsulating antiferromagnets.6–13Experimental techniques\ninclude microwave-field-induced magnetization preces-\nsion in the FM to generate a spin-current in the AFM\nand the inverse spin Hall effect (ISHE) in the heavy\nmetal to convert the spin-current transmitted through\nthe AFM into a voltage.6–8,14Such experimental studies\nhave shown that NiO can be an efficient spin-conductor.\nA theoretical model has been proposed that explains\nthese experimental results by spin-currents conducted by\nevanescent spin-waves in NiO.15,16In similar structures,\nYIG/NiO/Pt, the spin-Hall magnetoresistance (SMR)\nhas been measured as a function of temperature. Re-\nsults suggest that NiO can suppress the magnetic prox-\nimity effect measured in a 3 nm thick Pt layer and were\nfurther interpreted to indicate spin-transport between Pt\nand YIG through NiO layers.14\nThese two types of experiments have not been con-\nducted on the same samples as a function of the NiO\nthickness. Here, we investigate spin transport through\nNiO layers with varied thickness at room temperature\nby exciting ferromagnetic resonance (FMR) in YIG and\ndetecting the voltage signal across the Pt film associated\nwith the ISHE. The SMR as a function of the NiO thick-\nness was measured in the same samples. A comparisonof\nthese results shows that the ISHE signal decreasesmono-\ntonically as a function of NiO thickness, while the SMR\nvanishes abruptly at a thickness of about 4 nm.II. EXPERIMENTAL METHODS\nYIG(20 nm)/NiO/Pt(5 nm) trilayers were deposited\non gadolinium gallium garnet (Gd 3Ga5O12, GGG) sub-\nstrates. YIG with a /angbracketleft111/angbracketrightorientation was deposited\nby magnetron sputtering.17,18The YIG film was subse-\nquently exposed to ambient conditions. Ar+ion clean-\ning with a discharge voltage of 800 V was used prior to\ndepositing NiO by radio frequency (rf) magnetron sput-\ntering. A 5 nm thick Pt layer was deposited in-situ by\nelectron beam evaporation. Samples with 0 (i.e. without\nNiO), 2, 4, 6, and 10 nm thick NiO layers were prepared\nand studied.\nFor FMR studies, we mount the sample on a Cu\nstripline with a width of 500 µm, similar to the experi-\nmentalgeometryin Ref.19. We applyamicrowavesignal\nat 3.85 GHz and measure the inverse spin Hall voltage\nVISHEacross the Pt film as a function of the applied field\nusingalock-intechnique. Themicrowavesignalisturned\non and off at a few kHz and the lock-in response at this\nfrequency is measured. Figure 1(a) shows a schematic of\nthe sample and applied field geometry, while Fig. 1(b)\nshows a schematic of the magnetoresistance experiments.\nMagnetoresistance measurements of YIG/NiO/Pt trilay-\ners were performed in a 4-wire configuration, i.e. with\nseparate voltage and current contacts in a line. We ap-\nply an external magnetic field of 0.2 T to align the mag-\nnetization of YIG with the field. The samples are then\nrotated around three different axes to obtain the magne-\ntoresistance as a function of angle. All the experiments\npresented in this paper were conducted at room temper-\nature.\nIII. RESULTS\nFirst we present measurements of the ISHE voltage\nin YIG(20 nm)/Pt(5 nm), thin films that do not have\na NiO layer. Fig. 2(a) shows the ISHE voltage as a2\n(a) \n(b) YIG NiO Pt \n  V \n \n \nYIG NiO Pt \n \n   \n   \nFIG. 1. Schematics of (a) inverse spin Hall voltage measure-\nment by microwave-field-induced magnetization precession in\nYIG(20 nm)/NiO(t nm)/Pt(5 nm) and (b) Pt resistance mea-\nsurement as a function of angle of applied field ( α,β, andγ)\nwith respect to three different axes of the same sample.\nfunction of the applied field. A peak in the ISHE voltage\nsignalisseenattheresonancefieldoftheYIG.(TheFMR\nabsorptiondataisnotshown.) Thereisalsoaminorpeak\nwhich is most likely due to inhomogneity in our extend\nfilm samples. As previously reported in Ref. 19–21, the\nmeasured ISHE voltage VISHEis an odd function of the\nmagneticfield whichisasignaturethat the effect isbased\non spin pumping into Pt and not a thermoelectric signal.\nIn YIG/NiO/Pt trilayers the VISHEas a function of NiO\nthickness is shown in Fig. 2(b). We find slightly different\nresonance fields on different YIG samples, which may be\nassociated with small variation in their magnetization.\nWe checked that the shift in resonance field is symmetric\nin both field directions indicating that it is not caused by\nexchange bias introduced in the YIG by NiO. We extract\nthe peak values of VISHEand plot it as a function of NiO\nthickness in Fig. 4. The VISHEsignal in YIG/NiO/Pt\nindicates that exciting FMR in YIG is able to produce\nspin-currents in NiO and spin-injection into Pt.\nWe now discuss the magnetoresistance measurements\nof YIG/Pt and YIG/NiO/Pt. Magnetoresistance as a\nfunction of angle of applied field with respect to three\ndifferent axes of the sample were measured. In these\nexperiments, the applied field µ0H= 0.2 T is fixed. Fig-\nure3 showsthe magnetoresistanceofYIG(20 nm)/NiO(2\nnm)/Pt(5 nm) as a function of angle α(rotation in the\nx-y plane) in Fig. 3(a), β(rotation in the x-z plane) in\nFig. 3(b), and γ(rotation in the y-z plane) in Fig. 3(c).\nThe insets in Fig. 3 show the experimental geometry and\ndefinitions of angles, α,β, andγ.\nWe measured the magnetoresistance with a resolution\nof∼10−6in all three geometries. If we assume SMR and\nanisotropic magnetoresistance (AMR) as possible contri-\nbutions to the measured magnetoresistance, the angle\ndependent measurements in Fig. 3(a), 3(b), and 3(c)69 71 73 75 77 -4 -3 -2 -1 0\nµ0H (mT) V ( µV) \n  \n0 nm \n2 nm \n4 nm \n6 nm \n10 nm (b) \nt -76 -73 -70 -4 -2 024\nµ0H (mT) V ( µV) \n70 73 76 \nµ0H (mT) (a) \nFIG. 2. Inverse spin Hall voltage VISHEmeasured as a func-\ntion of external field in (a) YIG(20 nm)/Pt(5 nm) and (b)\nYIG(20 nm)/NiO(t nm)/Pt(5 nm) with a 3.85 GHz rf exci-\ntation.\nshould represent contributions of SMR and AMR, SMR\nonly, and AMR only, respectively. The signal observed\nin Fig. 3(c) is of the order of the measurement noise.\nGiven the measurement noise is ∼10−6, the peak value\nof the signal is at least a factor of 10 smaller than the\nSMR signal. This is consistent with the results reported\nin Ref. 14, where it was concluded based on no AMR\nresponse that there are no induced magnetic moments\nin Pt with a NiO interlayer. However, we also find neg-\nligible signal when rotating in the y-z plane to sweep\nthe angle γfor the YIG(20 nm)/Pt(5 nm) sample, which\ndoes not have the NiO interlayer. The AMR signal in-\ndicative of magnetic proximity effect found in Ref. 14\nwas measured in a sample with 3 nm Pt thickness as\nopposed to the thicker films used here and in Refs. 19,\n22, where also no AMR was reported. If magnetic prox-\nimity effect is present in our samples, its contribution\nto the magnetoresistance is negligible compared to the\ncontribution of SMR. Therefore, the resistance varia-\ntions in Figs. 3(a) and 3(b) can be attributed to the\nSMR, which describes how the Pt resistance reflects the\nitinerant electron-spin interactions at the NiO/Pt inter-\nface. The similar magnitude (∆ RMax/R0≃4×10−5)\nin Figs. 3(a) and 3(b) also suggest we only have a SMR\nsignal. We observe a periodic response with a period\nof 180 degrees in Figs. 3(a)(b) which can be described\nwithR−R0= ∆R= ∆RMaxsin2θ.θis the angle be-\ntween the magnetization Min YIG and the spin polar-\nization from the spin Hall effect in Pt. We compute the\nequilibrium magnetization direction as a function of the\napplied field direction, αandβ, based on a macrospin\nmodel considering the demagnetization field and exter-\nnal field. The results are α=θin Fig. 3(a) and\nβ=θ+ arcsin(sin(2 θ)Ms/2H) in Fig. 3(b). The sharp3\n012345\n012345ΔR/R 0 ( ×10 -5 ) \n-180 -90 0 90 180 012345\nα,β,γ (deg.) (a) \n(c) \n/g2185 \n /g2868   \n(b) \n/g2868 /g2185  \n  \n/g2868 /g2185   \n \nFIG. 3. Magnetoresistance of YIG(20 nm)/NiO(2 nm)/Pt(5\nnm) as a function of the field angle rotated in the (a) x-y\nplane an angle α, (b) x-z plane an angle β, and (c) y-z plane\nan angle γ. The dashed line is the expected SMR response\nbased on a macrospin model.\npeaks in the SMR signal at = +/-90 degree seen in Fig.\n3(b) are due to the difference between the angles of the\napplied field ( β) and the YIG magnetization ( θ). The\nmagnetization of YIG was determined by ferromagnetic\nresonance characterization to be µ0Ms= 0.176 T. As\nseen in the formofdashedlines in Fig. 3, plots of∆ R/R0\nversus field angle ( α,β, andγ) fit the experimental data\nwell.\nThe results in Fig. 3 show that the direction of YIG\nmagnetization can affect the resistance measured in Pt.\nThis suggests that the YIG magnetization rotates the\nspins in the NiO which changes the spin-scattering and\naccumulation at the NiO/Pt interface and thus the re-\nsistance of the Pt film. We repeat the same experiment\nfor YIG/Pt and YIG/NiO/Pt with t = 4 and 6 nm to\nextract the maximum value ∆ RMax/R0in Fig. 3(b) and\nplot it as a function of NiO thickness in Fig. 4.\nWe now compare the two experiments as a func-\ntion of NiO thickness. We show the ISHE voltage of\nYIG/NiO/Pt as a function of NiO thickness and plot it\nusing black circles on the left hand axis in Fig. 4. We fit\ntheVISHEdata points with Ae−t/λand obtain a decay\nlength of 3.9 nm. The blue squares plotted with refer-\nence to the right axis in Fig. 4 represent the SMR signal\nof YIG/NiO/Pt as a function of NiO thickness. We see\nfromthe comparisoninFig. 4howboththeISHEvoltage\nand SMR decrease monotonically with NiO thickness. In\ncontrast to the ISHE signal, the SMR shows an abrupt\ndecrease at a thickness of 4 nm. This points to a dif-\nference in how the critical length scales are determined\nin the two effects. In both cases spin flip-scattering of\nelectrons in Pt on the NiO interface is involved. How-\never,theexponentialdecreaseassociatedwithdiffusionof0246810 012345\nNiO thickness (nm) Max V( µV) \n  \n012345\nΔR/R 0(×10 -5 )VISHE \nVISHE =Ae (-t/λ)\nSMR \nFIG. 4. Black circles are the peak values of the ISHE voltage\nextracted from Fig. 2(b), while blue squares are the maxi-\nmum values of SMR ∆ RMax/R0extracted from Fig. 3(b) for\ndifferent NiO thicknesses. Black dashed line is a fit to A e−t/λ.\nBlue dashed line is only a guide to the eyes.\nmagnons in NiO15,16is not reproduced in the SMR data.\nOn the contrary the insertion of a 2 nm NiO layer barely\nreduces the SMR amplitude. We can thus speculate that\nthe source of the SMR, collinearity or non-collinearity\nbetween magnetic moments at the Pt/NiO interface and\nthe electron spins, is abrubtly decoupled from the YIG\nmagnetization above a certain NiO thickness.\nIV. CONCLUSION\nInconclusion,wehavereportedISHEvoltageandSMR\nmeasurements in YIG/NiO/Pt layered structures. From\nthe dependence on the NiO thickness, we found a mono-\ntonic decreaseof the VISHEand SMR signal with the NiO\nthickness. The sharp decrease of the SMR signal at t =\n4 nm and the exponential decrease of the VISHEsignal\nsuggests that the length scales of the phenomena are dis-\ntinct. The abrupt decrease of the SMR signal may reflect\nthe blocking of the NiO moments but this remains to be\ntested.\nACKNOWLEDGMENTS\nThe growth of the YIG films at Colorado State Uni-\nversity was supported by the SHINES, an Energy Fron-\ntier Research Center funded by the U.S. Department of\nEnergy, Office of Science, Basic Energy Sciences under\nAward SC0012670. The research at NYU was sponsored\nbytheInstituteforNanoelectronicsDiscoveryandExplo-\nration (INDEX), a funded center of Nanoelectronics Re-\nsearch Initiative (NRI), a Semiconductor Research Cor-\nporation (SRC) program sponsored by NERC and NIST.\n1T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich,\nNature Nanotechnology 11, 231 (2016).\n2V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono, and\nY. Tserkovnyak, arXiv:1606.04284 (2016).4\n3P. Wadley, B. Howells, J. elezn, C. Andrews, V. Hills, R. P.\nCampion, V. Novk, K. Olejnk, F. Maccherozzi, S. S. Dhesi, S. Y .\nMartin, T. Wagner, J. Wunderlich, F. Freimuth, Y. 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Saitoh,\nPhysical Review Letters 110, 206601 (2013)."    },    {        "title": "2202.02834v1.Enhancing_Perpendicular_Magnetic_Anisotropy_in_Garnet_Ferrimagnet_by_Interfacing_with_Few_Layer_WTe2.pdf",        "content": "   \n \n1 \n Enhancing Perpendicular Magnetic Anisotropy in Garnet Ferrimagnet \nby Interfacing with Few -Layer WTe 2 \nGuanzhong Wu1*, Dongying Wang1, Nishchhal Verma1, Rahul Rao2, Yang Cheng1, Side Guo1, \nGuixin Cao1, Kenji Watanabe3, Takashi Taniguchi4, Chun Ning Lau1, Fengyuan Yang1,  \nMohit Randeria1, Marc Bockrath1, and P. Chris Hammel1 \n1.Department of Physics, The Ohio State University, Columbus, OH, 43210, USA  \n2.Materials and Manufacturing Directorate, Air Force Research Laboratory, Wright -Patterson Air \nForce Base, Dayton, OH, 45433, USA  \n3. Research Center for Functional Materials,  National Institute for Materials Science, 1 -1 Namiki, \nTsukuba 305 -0044, Japan  \n4. International Center for Materials Nanoarchitectonics,  Nationa l Institute for Materials Science,  1-1 \nNamiki, Tsukuba 305 -0044, Japan  \n \n \n \n \n \n \n \n \n \n \n \n*wu.2314@osu.edu     \n \n2 \n Abstract:  \nEngineering magnetic anisotropy in a ferro - or ferrimagnetic (FM) thin film is crucial in \nspintronic device. One way to modify the magnetic anisotropy is through the surface of the FM \nthin film. Here, we report the emergence of a perpendicular magnetic anisotropy (PMA) induced \nby interfacial interactions in a heterostructure comprised of a garnet ferrimagnet, Y 3Fe5O12 \n(YIG), and the low -symmetry, high spin orbit coupling (SOC) transition metal dichalcogenide, \nWTe 2.  At the same time, we also observed an enhancement in Gilbert damping in the WTe 2 \ncovered YIG area. Both the magnitude of interface -induced PMA and the Gilb ert damping \nenhancement have no observable WTe 2 thickness dependence down to single quadruple -layer, \nindicating that the interfacial interaction plays a critical role. The ability of WTe 2 to enhance the \nPMA in FM thin film, combined with its previously rep orted capability to generate out -of-plane \ndamping like spin torque, makes it desirable for magnetic memory applications.   \n \nKey words:  perpendicular magnetic anisotropy, magnetic resonance force microscope, transition metal \ndichalcogenides, ferrimagnetic i nsulator      \n \n3 \n Perpendicular magnetic anisotropy  (PMA)  in a ferromagnetic thin film is of great interest in \nspintronics research and application s. Ferromagnetic nano -element s with PMA overcome  their shape \nanisotropy , greatly ease the memory cell size reduction and improves  memory retention . These \nexceptional properties, improving the performance  of magnetic devices , make PMA highly desirable for \nmagnetic memory  application s.  PMA becomes even more important in the recent development of solid \nstate magnetic random -access memory (MRAM) since it allows MRAM to have lower switching current \nand faster switching speed compare d to in-plane magnetized materials 1, 2.  \nMagnetic storage devices generally rely on metallic magnetic material s due to their robust \nelectrical response . Interfacial magnetic anisotropy plays a critical role in generat ing PMA in metallic \nferromagnet s. When interfacing with a nonmagnetic material (NM), electron orbital angular momentum \nof the magnetic ions at the ferromagnet surface will be modified, in some cases  enabling strong covalent \nbonding, resulting in distinct magnetic properties compare d to the single layer 3-6. However, spintronics \ndevices made of metallic magnetic materials are inherently energy consumptive due to resistive losses. \nRecently, complex oxide ferro - or ferrimagnet  insulator s (FMI)  have  attracted substantial  interest  due to \ntheir ability to transport spin excitation s with low dissipation  7. Inducing PMA in FMIs naturally \nbecomes an important topic both for scientific and technologic al reasons . Several successful route s to \nachiev ing PMA in FMIs ha ve been reported using bulk intrinsic anisotropy 8 or lattice strain 9-12. But in \nmost experiments,  the sign of the resulting interfacial anisotropy in FMI/NM heterostructures is such as \nto enhance  the easy-plane anisotropy  13-15. Only one recent experiment has shown the possibility of \ngenerating interfacial PMA, and this was attributed to topological surface states  16. Nevertheless, t hese \nresults demonstrate  the possibility of controlling magnetic anisotropy through interfac ial interaction s in    \n \n4 \n FMI/NM heterostructures . Here, we report a study on YIG/WTe 2/hBN heterostructures, which shows  \nthat when interfacing with a low symmetry nonmagnetic  van der Waals  material , WTe 2, an additional  \ninterfac e-induced  PMA  (iPMA) term  emerges  in the magnetic anisotropy  of the YIG thin film . The \nabsence of topological surface states at room temperature  in WTe 2 17, 18 forces us to seek an explanation \nfor our observation of enhanced PMA that is distinct from that proposed for top ological insulator/YIG \nbilayers 16. We therefore turn to an analysis of the broken symmetries in WTe 2. We point out that low \nsymmetry WTe 2 has recently shown the capability of generating both in -plane and out -of-plane spin \npolarization in charge -spin conversion experiments 19-22. It also enables field-free switching of PMA \nmagnet ic material , which ease s the application of PMA material s in MRAM application s 23-25.   \nFerrimagnetic insulator  YIG is of significant research interest in spintronics  due to its \nexceptionally  low Gilbert damping  26, which describes the  relaxation rate  of magnetization precession . \nAnd 1T’-WTe 2 is a semi -metallic transition metal dichalcogenide (TMD) layered material with strong \nSOC  27, 28. The crystal structure of 1T’-WTe 2 lacks twofold rotational symmetry about the c -axis (Fig. \n1a). The only symmetry in the  WTe 2 crystal lattice  ab plane is the mirror symmetry about  the bc plane  \n29. This u nique symmetry breaking allows  out-of-plane damping -like torque  to be generated 30, 31, \nenabling efficient switching of the  out-of-plane magnetization  of the adjacent magnetic material  24.  \nA 20nm thick YIG thin film used in our experiment is epitaxially grown on (111) -oriented  \nGd3Ga5O12 (GGG) substrate  by off -axis sputtering  32. WTe 2 flakes are then mechanically exfoliated \nfrom a flux-grown crystal, and dry transferred on to the clean top YIG surface without touching any \nother substances. This whole process is carried out in an Ar -filled glove box with <0.1 ppm of H 2O and    \n \n5 \n O2 to protect the flakes from degradation and  ensure  the clean liness  of the YIG/WTe 2 interface. We \nemploy hexagonal boron nitride ( hBN ) encapsulation to protect the WTe 2 flakes from oxidation after \nbeing removed from the glove box. We make two samples  and focus on the  data taken from sample 1 in \nthe main text. The raw data taken from sample 2 can be found  in Supporting Information  Fig. S 2.  \n \nFig. 1 Crystal structure of WTe 2 and sample schematic . a) Crystal lattice structure of WTe 2 viewed  \nfrom the top along  the c-axis and looking from the side along the a-axis. The black dashed box in the \nside view indicates a monolayer of WTe 2. b) Schematic of the ferromagnetic resonance force \nmicroscope. RF excitation is generated by a stripline underneath the sample , where the hBN \nencapsulation is not shown . The  region of  localized mode is shown as a yellow dot adjacent to the  WTe 2 \nflake, and the probe magnetic moment is shown as a yellow arrow on the particle. The cantilever \noscillation is detected by a fiber laser interferometer.  \nFigure 2a shows an optical image  of the sample 1. Due to the small  lateral size of the exfoliate d \nWTe 2 and hBN  flake s having  length  scale s of 10 μm, we use a home -built ferromagnetic resonance \nforce microscope (FMRFM) to measure  the local ferromagnetic resonance (FMR) signal. FMRFM is a \nsensitive technique to detect  the local magnetic properties with high spatial and spectral resolution  33. In \nour FMRFM, the external magnetic field 𝐻⃗⃗ ext is aligned perpendicular to  the sample plane. The \ncantilever tip holds a  high coercivity SmCo 5 magnetic particle , whose  moment  is magnetized in the  \n   \n \n6 \n direction opposite to 𝐻⃗⃗ ext to create a magnetic field well . The field well supports a set of  localized \nstanding spin wave modes (LMs). During the measurement, we excite spin precession uniformly by a \nstripline underneath the sample at a fixed RF frequency  (2 GHz)  and sweep the magnetic field. The \nresonance of each LM  generate s a stray field, whic h can then  be detected by the SmCo 5 magnetic \nparticle attached on the cantilever through their magnetic dipole -dipole interaction (Fig. 1b). During the \nmeasurement, we keep the probe -to-sample separation around 4 μm. The operation  of FMRFM is \ndescribed in detail in Ref s. 34-36. For reference, w e separate  a region of  YIG that does not contain  \nWTe 2/hBN heterostructures and measur e its Gilbert damping using  broadband FMR. To eliminate two -\nmagnon scattering, w e perform broadband FMR in the out -of-plane  field geometry. The FMR linewidth \nas a function of frequency measured on bare YIG (sample 1) shows a linear dependence (Fig. 2b), from \nwhich we can extract the Gilbert damping  of bare YIG  𝛼YIG=1.05×10−3. We also confirm that the \nWTe 2 used in the experiment is indeed  the 1T’ phase  through polarized Raman measurements. The \npolarization angle dependence of the Raman peak at 212 cm-1 (spectrum is shown in Fig. S4) exhibits \nminimum intensity when the excitation laser polarization is along the crystallographic a axis of WTe 2 37 \nas shown by the polar plot in Fig. 2c and Raman intensity plot in Fig. 2d .  \nWe find t he position of the YIG/WTe 2/hBN heterostructure with the assist ance of magnetic \nalignment markers  (Fig. 2a) . Figure 2e shows t wo raw FMRFM scans taken in the region  of YIG/hBN \nand YIG/WTe 2/hBN , indicated by the blue and the red dot in Fig. 2a, respectively , which  reveals  the \nchange in FMRFM spectra  at two different location s. Here we focus on the 𝑛=1 LM because it has the \nmode radius of around 1 μm and gives the highest spatial resolution. Higher order modes  have \nincreasing  mode radius and therefore, detect  less local ized magnetic  properties.  This is the reason why    \n \n7 \n the quasi -uniform mode at ~  3325 Oe does not show obvious change in resonance field or signal \namplitude.  We further take a  line scan across the edge of WTe 2 flake (Fig. 2 f) to resolve the spatial \nevolution of FMRFM spectra . The line scan in Fig. 2f (along the dashed line shown in Fig. 2a) shows \nthree main features : first, the magnitude of the LM resonance signal is reduced in the YIG/WTe 2/hBN \nregion compare d to the YIG/hBN region; second, the LM resonance field for all LMs is decreased by \n~40 Oe in the YIG/WTe 2/hBN region; third , the LMs  show complex  splitting and crossing when the \nprobe is close to the boundary (−5 μm<𝑋<10 μm).  \n \nFig. 2 FMRFM and Raman measurement data . a) An optical micrograph of the YIG/WTe 2/hBN \nheterostructure under study . WTe 2 crystal a and b axis are labeled.  b) Broadband FMR measurement of \nthe frequency -dependent linewidth of the YIG thin film. The measurement is done on the same piece of \nYIG used to make sample shown in Fig. 1b. c) Polar plot of the 21 2 cm-1 peak  Raman  intensity. Angle \ndenote s the relative angle between the measurement laser polarization and the WTe 2 a axis. d) 2D \nintensity plot showing Raman peak intensities versus polarization angle . e) FMRFM spectra, one over \nthe YIG/hBN  region (blue line)  and the second over the YIG/WTe 2/hBN  region (red line); these \nlocations are  indicated by the  blue and red dot s in Fig. 2a respectively . f) Color plot of field -dependence \n   \n \n8 \n FMRFM scans as a function of position along the trace indicated by the black dashed line in Fig. 2a. A \nconstant background is subtracted to show only the signal from the several LM resonance s. \nIn the following, we will explain the origin of the three observed effects using spin pumping and \nmagnetic anisotropy. The first effect , i.e. signal reduction in  the YIG/WTe 2/hBN area relative to the \nYIG/hBN  area, is the result of enhanced relaxation due to spin pumping from YIG to WTe 2 38. The 𝑛=\n1 LM resonance signal amplitude ∆𝐴 is inversely proportional to the square of Gilbert damping , 𝛼2. We \ndetermine  the Gilbert damping  constant  𝛼 for YIG/WTe 2/hBN  using 𝛼YIG/WTe2/hBN=\n𝛼YIG/hBN×√∆𝐴YIG/hBN∆𝐴YIG/WTe2/hBN ⁄  (see Ref. 39), where 𝛼YIG/hBN is assumed to be the same as \n𝛼YIG=1.05×10−3 due to  the low SOC and insulating character of hBN . The second  effect is  the \ndecrease of 𝑛=1 LM resonance field 𝐻r,1 by ~40 Oe . And the third effect is  splitting and crossing  of \ncomplex modes  in the region −5 μm<𝑋<10 μm. The second and the third effects  are due  to an \nabrupt change of uniaxial anisotropy across the boundary  separating the YIG/WTe 2/hBN and YIG/hBN \nregions  15. Here, the uniaxial anisotropy refers to the magnetic free energy  depends on the angle between \nmagnetization and sample normal  ℱu=−𝐾u𝒎z2, where 𝒎z is the component  of magnetization unit \nvector in the direction normal to sample plane  and 𝐾u is the uniaxial anisotropy constant specific to \nsample  and depends on the total interaction in the sample . When 𝐾u is positive, ℱu is called to be of \nPMA type, on the other hand, if 𝐾u is negative, ℱu is called to be of easy -plane  type.  This uniaxial \nanisotropy will lead to an effective uniaxial magnetic field 𝑯u=−𝜕ℱu𝜕𝑴⁄ , where 𝑴 is the \nmagnetization . And therefore, a change in 𝐾u can modify the resonance field in a FMR measurement . In \nFMRFM spatial mapping, a n abrupt change in 𝐾u spatially could disturb the LM and lead to mode \nsplitting and crossing as described in Ref. 15. Moreover, i n striking contrast to the previously studied    \n \n9 \n YIG/Au interface 15, which result s in a 32 Oe increase of 𝐻r,1 due to  the enhanced easy -plane \nanisotropy, the observed decrease of 𝐻r,1 indicates that the WTe 2 overlayer induces  an iPMA  in YIG. \nWe note that the  magnitude of the shift in  𝐻r,1 is comparable to the easy -plane anisotropy induced by a \nheavy metal 15, 40 or the iPMA generated by topological surface state 16 on garnet ferrimagnetic material .  \nIn order to probe  the global effect of  a WTe 2 overlayer on YIG, we spatial ly map 𝐻r,1 using  the \n𝑛=1 LM. Figure 3a present s an optical image of WTe 2 flakes on a Si/SiO 2 (285nm ) substrate , where \ndifferent c olors of WTe 2 flakes indicat e different WTe 2 thickness es. Figure 3b and 3 c show spatial  maps \nof magnetic properties  in the region enclosed by the black dashed rectangle  in Fig. 3a .  We acquire the \nmaps using the procedure described in Ref. 39, i.e., simultaneously  measuring spatial variation of  the \nmagnetic anisotropy and Gilbert damping using the  𝑛=1 LM resonance field 𝐻r,1 and signal amplitude  \n∆𝐴. The  entire  WTe 2-covered area show s uniformly lower ed 𝐻r,1 and increased Gilbert damping relative \nto the area without WTe 2. In Fig. 3c, despite the not great signal to noise ratio in damping imaging, the re \nis a clear  Gilbert damping enhancement in WTe 2-covered area . The averaged Gilbert damping of YIG in \nWTe 2-covered area is 𝛼̅YIG/WTe2/hBN≈1.30×10−3, about 24% higher than  𝛼YIG. We note that due to \nthe slight relative tilting of the scan plane and the sample plane, there is a color shift in Fig. 3b  that \nmight conceal the contrast difference in different WTe 2 thickness region . Therefore, to study the WTe 2 \nthickness dependence, we will show fine line scans across edge s of flakes  having  different WTe 2 \nthickness es.     \n \n10 \n  \nFig. 3 Two -dimensional FMRFM scan resolving the spatial variation of magnetic anisotropy and \nGilbert damping. a ) Optical micrograph showing the color contrast of different thickness WTe 2 flake s \n(ranging from 4.7 nm to 44.8 nm) on Si/SiO 2(300 nm). Black dashed box outlines the FMRFM scanned \narea for 2D mapping. b) 2D map of the 𝑛=1 LM resonance field . The d ashed line s labeled 1 -4 \ncorrespond to the  four line-scans shown in Fig. S1a-S1d. c) 2D mapping of the Gilbert damping \nextracted from  the 𝑛=1 LM resonance peak amplitude.  \n  \n   \n \n11 \n Next, we want to understand what gives rise to the PMA in WTe 2/YIG. We rule out the effect \ninduced by a modification of the gyromagnetic ratio by showing the resonance field shift across the \nWTe 2 edge does not depend on RF excitation frequency (See Fig. S3). We also exclude a strain induced \neffect given the absence of an epitaxial relation and the weakness of the van der Waals interaction \nbetween YIG and WTe 2. We further note that we can ignore the role of topological surface states  16 in \nour analysis; they are not relevant for our room temperature experiment since WTe 2 is a topological \nWeyl semimetal only below 100 K  17, 18. \nWe show how an analysis based on symmetry and the nature of the interfacial SOC , generalizing \nthe theory in Ref. 41, gives insight into the PMA observed in our experiment. This will also help us \nunderstand why the easy-axis anisotropy we observe in WTe 2/YIG is so different from the results of \nRef. 13, 15  on YIG interfaces with a dozen different metallic and semiconducting materials, all of which \nexhibit interface -induced easy-plane  anisotropy, as is predicted by theory  41.  \nYIG is a ferrimagnetic Mott insulator, with two inequivalent Fe -sites coupled via \nantiferromagnetic (AFM) superexchange interactions. We focus on how interfacial SOC impacts AFM \nsuperexchange in YIG and show that it leads to a very specific form of the mag netic anisotropy that is \ngoverned by the direction of the effective B-field (see Supporting Information for a details).  \nBefore turning to WTe 2/YIG, it is useful to first consider the simpler case when the only broken \nsymmetry is the mirror plane defined b y the interface.  The abrupt change in lattice potential then results \nin an effective electric field that points normal to the interface, which in turn leads to an effective \nmagnetic field in the rest frame of the electron that couples to its spin. Since t he E-field points normal to    \n \n12 \n the interfacial plane in which the electron moves, the resulting B-field arising from SOC lies within  the \ninterfacial plane. As we show in the SI, this leads to a SOC -induced correction to AFM superexchange \nthat necessarily lead s to an easy-plane  anisotropy.  \nIn the case of WTe 2/YIG, however, when there are additional broken symmetries. Not only does \nthe interface break inversion  symmetry , but the crystal structure of WTe 2 itself breaks in -plane inversion  \nsymmetry . The electric field is now no longer normal to the interface, and the effective B-field arising \nfrom SOC necessarily has an out -of-plane component, as shown in Fig S5b in SI. Thus, we see why the \nlower symmetry of WTe 2/YIG can naturally result in an easy-axis or perpendicular magnetic anisotropy  \n(PMA);  see Supporting Information for details.  \nWe note that the lack of two -fold rotational symmetry in the ab plane in WTe 2 that plays a \ncritical role in our understanding of PMA in WTe 2/YIG, has also been pointed  out be crucial for the out -\nof-plane damping -like torque in WTe 2/Permalloy30. We note, however, that the out -of-plane damping -\nlike torque necessarily involves current flow in WTe 2, while the PMA is an equilibrium property of the \nsystem independent of current flow.  \nWe further demonstrate the interfacial  origin of the observed effect by studying the influence of \nWTe 2 thickness.  We show four  line-scans , labeled in Fig. 3b, across the edges of WTe 2 with different \nthickness es, ranging from 4. 7 nm to 44.8 nm . From these four line -scans, we extract  the 𝑛=1 LM \nresonance field 𝐻r,1 and the 𝑛=1 LM resonance signal amplitude  ∆𝐴. Figures S1a-d in the Supporting \nInformation  show the evolution of 𝐻r,1 and ∆𝐴 along the traces labeled correspondingly . The thickness \nof WTe 2 at each measurement location is later measured using atomic force microscop y. From these    \n \n13 \n line-scans , we choose the region s where the probe is far away from the edge of WTe 2 so that the \nmagnetic propert ies are uniform,  to obtain  spatial average s of 𝐻r,1 and ∆𝐴, which are denoted  \n𝐻̅r,1,YIG/hBN and ∆𝐴̅̅̅̅YIG/hBN in the YIG/hBN region , and 𝐻̅r,1,YIG/WTe2/hBN and ∆𝐴̅̅̅̅YIG/WTe2/hBN in the \nYIG/WTe 2/hBN region , respectively . We further extract the 𝑛=1 LM resonance field difference \nbetween two regions using ∆𝐻r,1=𝐻̅r,1,YIG/hBN−𝐻̅r,1,YIG/WTe2/hBN, as well as the Gilbert damping \ndifference using ∆𝛼=𝛼YIG×(√∆𝐴̅̅̅̅YIG/hBN ∆𝐴̅̅̅̅YIG/WTe2/hBN ⁄ −1) as a function of the WTe 2 \nthickness . We note  that the hBN overlayer does not change the Gilbert damping in YIG . The \nsummarized results containing the data from both sample 1 and sample 2  are shown in Fig s. 4a and 4 b. \nRaw data from sample 2 can be found in Supporting Information  Fig. S 2. The thinnest WTe 2 acquired in \nthe experiment is 3.2nm from sample 2, which is approximately the thickness of  a quadruple -layer \nWTe 2.  \nFigures 4a and 4 b indicate  that both ∆𝐻r,1 and ∆𝛼 have  almost no WTe 2 thickness  dependence . \nThere is a  small sample -to-sample variation possibly  due to different YIG/WTe 2 interfacial quality . The \nchange of 𝑛=1 LM resonance field,  ∆𝐻r,1, is as large as ~38 Oe even when the WTe 2 thickness \napproaches the quadruple -layer thickness . This indicates that the modification of magnetic anisotropy is \ndue to the YIG/WTe 2 interfac ial interaction , with no bulk contribution. For the increase of Gilbert \ndamping ∆𝛼, no obvious thickness dependence is observed when comparing the data from the same \nsample. In sample 2, the Gilbert damping enhancement  due to the quadruple -layer WTe 2 has almost the \nsame value as the 50 nm thick WTe 2 flake, indicating that no thickness dependence of spin pumping  can \nbe resolved from our measurement. There are two possible interpretations of these results . First, if the    \n \n14 \n spin current injected into WTe 2 is mainly relaxed due to spin relaxation in the bulk, then the \nexperimental result is a demonstration of ultra -short spin diffusion length along the c axis38, smaller or \ncomparable to the thinnest WTe 2 flake (3.2 nm), employed in this experiment . It is much smaller than \nthe 8nm spin diffusio n length  in the  in-plane direction measured using inverse spin Hall effect  22. Note \nthat due to the chang e in mo bility and the metal -insulator transition  in few layer WTe 2 when its \nthickness reduces  42, the spin diffusion length approximated here could be inaccurate . Alternatively , it is \npossible that the spin relaxation is primarily due to the interfacial SOC induced by inversion symmetry \nbreaking at the interface and in the WTe 2 crystal lattice. In this case, the Gilbert damping enhancement \nwill have no WTe 2 thickness dependence.     \n \n15 \n  \nFig. 4  WTe 2 thickness dependence  of resonance field and damping enhancement . a) 𝐻r,1 in the \nYIG/hBN and YIG/WTe 2/hBN regions  are averaged  respectively to get 𝐻̅r,1,YIG and 𝐻̅r,1,YIG/WTe2, and \n∆𝐻r,1=𝐻̅r,1,YIG−𝐻̅r,1,YIG/WTe2. b) ∆𝛼 as a function of WTe 2 thickness , and  ∆𝛼=𝛼YIG/WTe2−𝛼YIG \nwhere 𝛼YIG is the Gilbert damping of bare YIG measured using broadband FMR for each sample , and \n𝛼YIG/WTe2=𝛼YIG×(√∆𝐴̅̅̅̅YIG/hBN ∆𝐴̅̅̅̅YIG/WTe2/hBN ⁄ ).   \nIn conclusion, we have shown that the YIG/WTe 2 interface plays a critical  role in both interfacial \nmagnetic anisotropy and spin relaxation , making WTe 2 a promising material  in magnetic memory \n   \n \n16 \n application s. Combining the iPMA  created by WTe 2 with the out-of-plane spin orbit torque  generated by \nflowing a charge current along  the a axis of WTe 2, one can possibly achieve field -free switching of a \nPMA magnetic cell  for magnetic memory application s. It will improv e the  scalability , reduc e the  power \nconsumption  and increas e operation speed  of magnetic solid -state devices . Our result reveals new \npossibilities in selecting materials and designing spintronic devices. For example, one can consider other \nmaterials with low lattice symmetry and strong SOC to induce larger PMA type interfacial ani sotropy in \nFMIs. To achieve a fully PMA material, one could utilize thinner FMIs to magnify the role of iPMA. \nMoreover, interfacial SOC also plays an important role in  generat ing topologically protected magnetic \ntextures in the FMIs  43. These findings will motivate further research to reveal the fundamental  physics \narising at the  interface between FMIs and nonmagnetic materials.  \n \nData availability:  \nThe data generated by the present study are available from the corresponding author on request.  \nSupporting Information:  \nA description of raw data on WTe 2 thickness dependence, a FMRFM measurement on a second sample, \na FMRFM measurement at different RF frequency, a description of polarized Raman measurement \nresult, and a detailed illustration of impact of broken  mirror reflection symmetries on the magnetic \nanisotropy.  \nAckno wledgements:     \n \n17 \n This work was primarily supported by the Center for Emergent Materials: an NSF MRSEC under award \nnumber DMR -2011876 (GW, NV , YC, SG, FY , MR and PCH) . KW and TT acknowledge support from \nthe Elemental Strategy Initiative conducted by the MEXT, Japan (Grant Number JPMXP0112101001) \nand JSPS KAKENHI (Grant Numbers 19H05790, 20H00354 and 21H05233).  DW, GC, CNL, and MB \nare supported by NSF under award DMR -2004801.  We gr atefully acknowledge N. Trivedi for insightful \ndiscussions. Fabrication and some characterization were performed in the Ohio State University \nNanoSystems Laboratory.  \n     \n \n18 \n Methods:  \nSample fabrication  \nOur YIG/WTe 2/hBN heterostructure was prepared by means of dry transfer and stacking 44. hBN crystals \nwere mechanically exfoliated under ambient conditions onto SiO 2/Si substrates (285 nm thick SiO 2). 20-\n40 nm thick hBN flakes were identified under an optical microscope and used for the capping lay er for \nthe stack. The hBN was picked up using a polymer -based dry transfer technique and then moved into an \nAr-filled glove box with oxygen and water level below 0.1 ppm. Flux -grown WTe 2 crystals 45 were \nexfoliated inside the glove box and flakes with different thicknesses were optically identified and \nquickly picked up with the capping hBN layer then transferred to the YIG substrate. Finally, we removed \nthe fully encapsulated sample from the glove b ox and performed the e -beam lithography and \nmetallization (Ni/Au) step for alignment in our ferromagnetic resonance force microscope (FMRFM).  \n \nPolarized Raman measurement  \nPolarized Raman spectra from the WTe 2 sample were collected using 633 nm excitation w avelength in \nan inVia Renishaw Raman microscope.  The sample was loaded onto the microscope stage and \npositioned in such a way that the long edge of the flake was aligned parallel to the laser polarization ( θ = \n0°). In this configuration, the incident illu mination is polarized vertically coming out of the laser and is \naligned with the long axis of the WTe 2 flake. The polarization of the incident laser was rotated from 0 to \n360° by 10° increments using a polarization rotator, while an analyzer was set to onl y allow vertically \npolarized light to enter the spectrometer.  Raman spectra were collected at each polarization for 3 \nacquisitions with a 20 s time per acquisition.  The laser power was set to 0.5 mW at the sample to avoid \nany damage by heating.  Followin g spectral collection, the (baseline corrected) integrated intensities \nunder each peak were calculated to make the contour plots and polar plots in Fig. 2c and 2d.  \n \nFMRFM measurement and signal fitting  \nOur FMRFM perform s local ly measures  FMR at room temper ature in vacuum. The cantilever has \nnatural frequency of ~18 KHz, spring constant of 0.2 N/m and Q factor of ~20000, resulting in force \ndetection sensitivity of  10-15 N/Hz1/2. The SmCo 5 magnetic particle attached on the cantilever has a \nmagnetic moment of ~4 nemu.  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B 2015,  92, (4), 041104.  \n \n     \n \n23 \n Enhancing  Perpendicular Magnetic Anisotropy in Garnet  Ferrimagnet \nby Interfacing with Few-Layer  WTe 2 \nGuanzhong Wu1*, Dongying Wang1, Nish chhal  Verma1, Rahul Rao2, Yang Cheng1, Side Guo1, \nGuixin Cao1, Kenji Watanabe3, Takashi Taniguchi4, Chun  Ning Lau1, Fengyuan Yang1,  \nMohit Randeria1, Marc Bockrath1, and P. Chris Hammel1 \n1.Department of Physics, The Ohio State University, Columbus, OH, 43210, USA  \n2.Materials and Manufacturing Directorate, Air Force Research Laboratory, Wright -Patterson Air \nForce Base, Dayton, OH, 45433, USA  \n3. Research Center for Functional Materials,  National Institute for Materials Science, 1 -1 Namiki, \nTsukuba 305 -0044, Japan  \n4. International Center for Materials Nanoarchitectonics , National Institute for Materials Science,  1-1 \nNamiki, Tsukuba 305 -0044, Japan  \n*wu.2314@osu.edu  \n     \n \n24 \n FMRFM  line-scan across edge of WTe 2 with different thickness  \n \nFig. S1 a-d, FMRFM line-scans along the traces 1~4 indicated in Fig. 3b respectively.  The gray shaded \narea in four figures are outlining the location of  WTe2 flake . 𝐻r,1 and ∆𝐴 at each position are derived by \nfitting the 𝑛=1 LM to a Lorentzian line shape. The thickness of WTe2 flake at each location are \nmeasured by atomic force microscope.  \n  \n   \n \n25 \n FMRFM measurement on Sample 2  \n \nFig. S 2 FMRFM measurement on sample 2. a, The optical picture of the YIG/WTe 2/hBN \nheterostructure. b, 2D mapping of the 𝑛=1 LM resonance field in the black dash line circled area.  c, \n2D mapping of the Gilbert damping extracted from 𝑛=1 LM resonance peak amplitude in the black \ndash line circled area. d, FMRFM line scan along the trace indicated by the solid black line in Fig. S 2a. \nA constant background is subtracted to show only the signal from the LMs resonance. e, Fine scan \nzoomed in o n the quadruple layer WTe 2 stripe area  \n  \n   \n \n26 \n FMRFMR measurement at 4 GHz  \n \nFig. S 3 FMRFM measurement across WTe 2 edge at 4 GHz. FMRFM line scan is measured at 4 GHz \nacross the WTe 2 flake edge. The shift of the resonance field 𝐻r,1 is 36 Oe, similar to the 𝐻r,1 shift \nmeasured at 2 GHz. This result excludes the possibility that the resonance field shift arises from \nmodification of the gyromagnetic ratio.  \n \n  \n   \n \n27 \n Polarized Raman measurement  \n \nFig. S 4 Polarized Raman measurement. As shown by the red curve, the Raman  spectrum taken on \nGGG/YIG/WTe 2/hBN heterostructure contains more peaks than WTe 2. The Raman spectrum taken in \nthe GGG/YIG/hBN area identifies the peaks arising from the  substrate GGG/YIG or top hBN \nencapsulation layer. By subtracting the Raman spectrum in the GGG/YIG/hBN area, the Raman spectra \nfrom WTe 2 layer are extracted and plotted in Fig. 2d. The black dash line are the markers indicating the \nRaman peaks of WTe 2  \n   \n \n28 \n Impact of  broken mirror reflection  symmetr ies on the  magnetic anisotropy  \nWe describe theo retical constraints on the interface -induced  magnetic anisotropy in the WTe 2/YIG \nbilayer. We first show that symmetry arguments alone do not provide strong constraints on the anisotropy \ntensor, given that we are dealing with an interface between two crystalline materials at an arbitrary \norientation with respect to each other . We then present qualitative arguments, based on the interfacia l spin -\norbit coupling, that give insight into the magnetic anisotropy in WTe 2/YIG. This helps us understand why \nthe easy-axis anisotropy that we observe  in WTe 2/YIG differ s from the results  of Lee et al. [1] on YIG \ninterfaces with a dozen different metallic and semiconducting materials , all of which exhibit interface -\ninduced easy-plane anisotropy  as predicted  by theory  [2]. \nOn general grounds, the  anisotropy (free) energy can be written as   \nℱ𝑎𝑛𝑖𝑠= ∑ 𝐾𝑎𝑏𝑎,𝑏 𝑚𝑎 𝑚𝑏,           (S1) \nwhere a and b take on values x,y,z. We focus here on the leading term, quadratic in the magnetization, and \nignore higher order anisotropy terms like (mx4+my4+mz4) or (mx2my2+my2mz2+mz 2mx2). The form of \n 𝐾𝑎𝑏=  𝐾𝑏𝑎 is constrained by symmetry. Let us consider three cases , going from the most symmetric to \nthe least .  \nCase I: The only broken symmetry is interfacial inversion (z →  - z), which  is relevant for the \nexperiments of Ref. [1]. The magnetization is an axial vector (or pseudovector) that transforms under  \nrotation s like a vector but is unchanged  under inversion . Thus (𝑚𝑥 ,𝑚𝑦 ,𝑚𝑧)→ (𝑚𝑥 ,− 𝑚𝑦 ,−𝑚𝑧) under \nreflection in  a mirror plane with normal 𝑥̂. Using reflection symmetry in mirror planes normal  to 𝑥̂ and to    \n \n29 \n 𝑦̂ , we can see that all off -diagonal components of  𝐾𝑎𝑏  vanish. Further, f our-fold rotational symmetry \nabout the 𝑧̂ axis shows that  𝐾𝑥𝑥=  𝐾𝑦𝑦. Using mx2+my2+mz2=1, we write  𝐾𝑥𝑥(m𝐱2+m𝐲2) in terms \nof m𝑧2, and d efining 𝐾𝑢=  (𝐾𝑥𝑥−  𝐾𝑧𝑧), we obtain  \nℱ𝑎𝑛𝑖𝑠= −  𝐾𝑢 m𝑧2.         (S2) \nThis symmetry analysis only constrains the form  of the anisotropy energy, but not the sign of  𝐾𝑢. We will \ngive below a simple microscopic argument [2] that shows that   𝐾𝑢<0  (easy plane)  for Case I.  \nCase II: In addition to broken interfacial inversion (z → - z), let u s also break reflection symmetry \nin the plane normal to 𝑥̂. This would be the case if the crystalline axes of WTe 2 were aligned with YIG.  \nThis also breaks four-fold rotational symmetry about 𝑧̂, so that   𝐾𝑥𝑥 ≠ 𝐾𝑦𝑦. However, we  can still use \nreflection symmetry in the plane normal to 𝑦̂ to conclude that  𝐾𝑥𝑦=  𝐾𝑦𝑧=0. Thus  we find that  \n      K = (𝐾𝑥𝑥0 𝐾𝑥𝑧\n0 𝐾𝑦𝑦0\n 𝐾𝑥𝑧0 𝐾𝑧𝑧)                                                               (S3) \nCase III: When the crystalline axes of WTe 2 are not aligned with YIG, which is the experimentally \nrelevant case, all mirror reflection and rotation symmetries are broken. Then there are no symmetr y \nconstrain ts on   𝐾𝑎𝑏 and all six components of this symmetric tensor are in general non -zero.      \nLet us now see how, despite the lack of general symmetry -based constrai nts, we can still get some \nqualitative insight about the form of the anisotropy from simple microscopic considerations informed by \nsymmetry. YIG is a ferrimagnetic Mott insulator, with two inequivalent Fe -sites coupled via    \n \n30 \n antiferromagnetic (AFM) superexch ange interactions. We thus focus on how interfacial spin -orbit \ncoupling (SOC) impacts AFM superexchange.   \nThe broken symmetry at the interface leads to an electric field ℇ=−𝛁𝑉(𝒓), whose direction will \nbe discussed in detail below for three cases. This in turn produces a magnetic field in the rest frame of the \nelectron  which underlies SOC. As the electron moves along  𝐫̂ij from site i to j, it experiences an  SOC field  \nin the direction 𝐝̂ij  which is determined by  ℇ ×𝐫̂ij . The SOC Hamiltonian  is thus given by \n−𝑖𝜆∑𝑐𝐢𝛼†(𝐝̂ij∙𝝈𝛼𝛽)𝑐𝐣𝛃 𝛼𝛽 . Including the effect of this term in addition to the usual hopping t and Hubbard \nU in the standard strong coupling expansion calculation leads to the Hamiltonian  \n                 ℋex=J∑𝐒i∙𝐒j <𝐢,𝐣>+D∑𝐝̂ij∙𝐒i×𝐒j <𝐢,𝐣> +K0∑(𝐝̂ij∙𝐒i)(𝐝̂ij∙𝐒j). <𝐢,𝐣>                  (S4) \nHere  the spin  𝐒i   at site i is coupled to its neighbors via  the AFM superexchange  𝐽 ~𝑡2\n𝑈  and the \nDzyaloshinskii -Moriya interaction (DMI)  𝐷 ~𝑡𝜆\n𝑈. The K0 term will be the focus of our attention belo w as \nit leads  to magnetic anisotropy.  We note that the general form of ℋex is in fact substantially independent \n[2] of the microscopic mechanism and very similar results are obtained not only for superexchange but \nalso for Zener double exchange and RKKY interactions.  \nCase I: Let us again return to the simplest case with broken interfacial inversion (z → - z). This \nleads to an electric field ℇ=−𝛁𝑉(𝒓)  along ẑ , the normal to the interface.  The SOC magnetic field \ndirection is then given by  𝐝̂ij= ẑ ×𝐫̂ij ; see Figure S4(a). This is the well -known Rashba SOC at \ninterfaces . We note in passing that 𝐝̂ij is antis ymmetric under the interchange of i and j, and thus  leads to \na DMI term where 𝐒i×𝐒j is also antisymmetric.     \n \n31 \n  \n \nFig. S 5 Symmetry based selection of magneto -crystalline anisotropy. Interfacial SOC originates from \nan effective Electric field ℇ=−𝛁𝑉(𝒓) whose direction is determined by the broken mirror planes in the \nsystem. This electric field leads to  spin-orbit coupling (SOC), with the 𝐝̂ij= ℇ̂ ×𝐫̂ij, the direction of the \nSOC magnetic  field. Note that the direction of the electron hop 𝐫̂ij lies in the xy plane of the interface. As \nshown in the text 𝐝̂ij controls the interface -induced magnetic anisotropy. (a) When only surface inversion \nis broken, 𝐝𝐢𝐣 is constrained to lie in the int erface and interfacial SOC leads to easy -plane anisotropy. (b) \nIf there are other broken mirror planes, the 𝐝𝐢𝐣 must lie outside the interfacial plane. This can lead to a \nperpendicular magnetic anisotropy in systems like YIG/WTe 2 bilayers.  \n \nWe see that in Case I,  𝐝̂ij lies in the plane  of the interface, and the third term in eq. (S 4) then takes \nthe form K0∑(S𝐢𝑥𝑆𝐢+𝑦𝑥+S𝐢𝑦𝑆𝐢+𝑥𝑦) 𝐢  for a square lattice . To make the connection with  magnetic anisotropy, \nwe look at a continuum approximation with a s lowly  varying magnetization 𝐦(𝐫). We make a Taylor \nexpan sion of  𝐒r in terms of its value at 𝒓, denoted by  𝐦(𝐫), and its spatial derivatives . The exchange and \nDMI terms involve gradients of 𝐦(𝐫), but we focus here on local terms that  do not involve derivatives  to \n   \n \n32 \n understand the magnetic anisotropy . The leading term is + K0(m𝐱2+m𝐲2)  which can be rewritten as \n– K0 m𝑧2 using the fact that mx2+my2+mz2=1 at each 𝒓. Thus, we may identify K0 with the anisotropy  \nK𝑢 defined in eq. (S2).  \nThe microscopic analysis leads to the result K0= − 𝜆2\n𝑈   < 0 and this explains the easy-plane \nanisotropy  arising Rashba SOC at the interface . The easy-plane nature of the anisotropy  is in fact a general \nfeature of various microscopic models as emphasized  in Ref. [2]. We note however that these author s use d \nthe opposite sign convention for anisotropies  from the one we use here . The easy plane vs. easy -axis \ncharacter is , of course, independent of sign conventions.  The FMR experiments of Ref. [1] have seen the \ninterface -induced easy-plane anisotropy  predicted by the theory in a YIG interfaces with several  metallic \nand semiconducting materials . \nThe key difference between the YIG/WTe 2 bilayer studied here and systems  studied earlier [1] is \nthat WTe 2 has a broken mirror plane  (the ac plane ) as shown in  Fig. 1(a)  of the paper . We now  look at the \neffect of this lower symmetry on the microscopic analysis.  \nCase II: Let us break reflection symmetry in the plane normal to 𝑥̂ in addition to broken interfacial \ninversion. We choose x̂ parallel to the b axis, ŷ parallel to a, and ẑ parallel to c. Reflection symmetry in \nthe ŷ mirror plane constrains the  electric field ℇ =−𝛁𝑉(𝒓) to lie in the xz plane, at an angle 𝜃 from the \nz-axis as shown in Fig. S 5(b). Thus  \n𝐝ij=(sin𝜃𝑥̂+cos𝜃𝑧̂)×𝐫̂ij     (S5)    \n \n33 \n where 𝐫̂ij is a vector in the interface  (xy plane ) and 0≤𝜃≤𝜋. Using eq. (S5), we may rewrite  the last \nterm in the  Hamiltonian (S4) as \n K0 sin2𝜃∑(S𝐢𝑧𝑆𝐢+𝑦𝑧)\n𝐢+K0 cos2𝜃∑(S𝐢𝑥𝑆𝐢+𝑦𝑥+S𝐢𝑦𝑆𝐢+𝑥𝑦)\n𝐢\n−K0sin𝜃cos𝜃∑(S𝐢𝑧𝑆𝐢+𝑦𝑥+S𝐢𝑥𝑆𝐢+𝑦𝑧)\n𝐢  \nAs before, we  make  a continuum approximation with a smoothly varying 𝐦(𝐫) and focus only on the \nlocal terms, without gradients, to obtain the magnetic anisotropy . We find that the leading order \ncontribution to anisotropy is −K0cos2𝜃m𝑧2+K0sin2𝜃 mzmx. This analysis correctly captures the non -\nzero K𝑥𝑧 expected on general grounds; see eq. (S3). We did not include here , for simplicity, the effects of \nbroken four -fold rotation that would have led to  𝐾𝑥𝑥 ≠ 𝐾𝑦𝑦. \nCase III: When we lose all mirror symmetries, the case relevant to the YIG/WTe 2 experiment, the \nelectric field ℇ =−𝛁𝑉(𝒓) will point in a general direction specified by 0≤𝜃≤𝜋 and 0≤𝜑≤2𝜋, and \nthere will be no symmetry constraints on the anisotropy tensor  𝐾𝑎𝑏. \n Let us conclude by highlighting the  key qualitative difference  between Case I  on the one hand and \nCases II and III  on the other . In Case I, the only broken symmetry is interfaci al inversion (z → - z). Then \nsymmetry constrains the  𝐝̂ij, the direction of the SOC B-field,  to lie in the plane  of the interface  and this \nleads to easy -plane anisotropy as described above. In Cases II and III, there are other additional broken \nmirror planes, and this leads to the 𝐝̂ij  vector being pulled out of the plane of the interface. This \nimmediately leads to the possibility of an easy -axis like character to the anisotropy, although in the general \ncase one has a non -trivial anisotropy tensor  𝐾𝑎𝑏.     \n \n34 \n Reference  \n[1] Lee, A. J.; Ahmed, A. S.; McCullian, B. A.; Guo, S. D.; Zhu, M. L.; Yu, S. S.; Woodward, P. M.; \nHwang, J.; Hammel, P. C.; Yang, F. Y . Interfacial Rashba -Effect-Induced Anisotropy in Nonmagnetic -\nMaterial -Ferrimagnetic -Insulator Bilayers. Phys. Rev. Lett. 2020,  124, (25), 257202.  \n[2] Banerjee, S.; Rowland, J.; Erten, O.; Randeria, M. Enhanced Stability of Skyrmions in Two -\nDimensional Chiral Magnets with Rashba Spin -Orbit Coupling. Physical Review X 2014,  4, (3), 031045.  \n \n "    },    {        "title": "2001.08940v1.Axion_search_with_a_quantum_limited_ferromagnetic_haloscope.pdf",        "content": "Axion search with a quantum-limited ferromagnetic haloscope\nN. Crescini,1, 2,\u0003D. Alesini,3C. Braggio,2, 4G. Carugno,2, 4D. D'Agostino,5\nD. Di Gioacchino,3P. Falferi,6U. Gambardella,5C. Gatti,3G. Iannone,5\nC. Ligi,3A. Lombardi,1A. Ortolan,1R. Pengo,1G. Ruoso,1,yand L. Ta\u000barello7\n(QUAX Collaboration)\n1INFN-Laboratori Nazionali di Legnaro, Viale dell'Universit\u0012 a 2, 35020 Legnaro (PD), Italy\n2Dipartimento di Fisica e Astronomia, Via Marzolo 8, 35131 Padova, Italy\n3INFN-Laboratori Nazionali di Frascati, Via Enrico Fermi 40, 00044 Roma, Italy\n4INFN-Sezione di Padova, Via Marzolo 8, 35131 Padova, Italy\n5INFN-Sezione di Napoli, Via Cinthia, 80126 Napoli, Italy and Dipartimento di Fisica,\nVia Giovanni Paolo II 132, 84084 Fisciano (SA), Italy\n6IFN-CNR, Fondazione Bruno Kessler, and INFN-TIFPA, Via alla Cascata 56, 38123 Povo (TN), Italy\n7INFN-Sezione di Padova, Via Marzolo 8, , 35131 Padova, Italy\n(Dated: January 27, 2020)\nA ferromagnetic axion haloscope searches for Dark Matter in the form of axions by exploiting\ntheir interaction with electronic spins. It is composed of an axion-to-electromagnetic \feld transducer\ncoupled to a sensitive rf detector. The former is a photon-magnon hybrid system, and the latter\nis based on a quantum-limited Josephson parametric ampli\fer. The hybrid system consists of ten\n2.1 mm diameter YIG spheres coupled to a single microwave cavity mode by means of a static\nmagnetic \feld. Our setup is the most sensitive rf spin-magnetometer ever realized. The minimum\ndetectable \feld is 5 :5\u000210\u000019T with 9 h integration time, corresponding to a limit on the axion-\nelectron coupling constant gaee\u00141:7\u000210\u000011at 95% CL. The scienti\fc run of our haloscope resulted\nin the best limit on DM-axions to electron coupling constant in a frequency span of about 120 MHz,\ncorresponding to the axion mass range 42 :4-43:1\u0016eV. This is also the \frst apparatus to perform an\naxion mass scanning by changing the static magnetic \feld.\nThe axion is a beyond the Standard Model (BSM) hy-\npothetical particle, \frst introduced in the seventies as a\nconsequence of the strong CP problem of quantum chro-\nmodynamics (QCD) [1{3]. Present experimental e\u000borts\nare directed towards \\invisible\" axions, described by the\nKSVZ [4, 5] and DFSZ [6, 7] models, which are extremely\nlight and weakly coupled to the Standard Model parti-\ncles. Axions can be produced in the early Universe by\ndi\u000berent mechanisms [8{11], and may be the main con-\nstituents of galactic Dark Matter (DM) halos. Astro-\nphysical and cosmological constraints [12, 13], as well as\nlattice QCD calculations of the DM density [14, 15], pro-\nvide a preferred axion mass window around tens of \u0016eV.\nNon-baryonic DM is where cosmology meets particle\nphysics, and axions are among the most interesting and\nchallenging BSM particles to detect. Their experimental\nsearch can be carried out with Earth-based instruments\nimmersed in the Milky Way's halo, which are therefore\ncalled \\haloscopes\" [16]. Nowadays, haloscopes rely on\nthe inverse Primako\u000b e\u000bect to detect axion-induced ex-\ncess photons inside a microwave cavity in a static mag-\nnetic \feld. Primako\u000b haloscopes allowed to exclude ax-\nions with masses mabetween 1.91 and 3.69 \u0016eV [17{19],\nand, together with helioscopes [20], are the only exper-\niments which reached the QCD-axion parameter space.\nThe last years saw a \rourishing of new ideas to search for\naxions and axion-like-particles (ALPs) [21{33]. Among\nthese, the QUAX experiment [34, 35] searches for DM ax-\nions through their coupling with the spin of the electron.This experiment aims to implement the idea of Ref. [36]\nas follows.\nThe axion-electron interaction is described by the La-\ngrangian\nLae=gaee\n2me@\u0016a\u0000\u0016 e\r\u0016\r5 e\u0001\n; (1)\nwheregaeeis the axion-electron interaction constant, ais\nthe axion \feld,  eandmeare the electron wavefunction\nand mass, and \r\u0016and\r5are Dirac matrices. This vertex\ndescribes an axion-induced \rip of an electron spin, which\nthen decays back to the ground state emitting a photon.\nSinceva, the relative speed between Earth and the DM\nhalo, is small, we may use the non-relativistic limit of\nEuler-Lagrange equations and recast the interaction term\nLae'\u00002\u0016B\u001b\u0001\u0010gaee\n2e\u0011\nra\u0011\u00002\u0016B\u001b\u0001Ba: (2)\nHere\u00002\u0016B\u001bandeare the spin and charge of the elec-\ntron,\u0016Bis the Bohr magneton, and Bais de\fned as\nthe axion e\u000bective magnetic \feld. As ra/va[36], the\nnon-zero value of varesults in Ba6= 0.\nIf accounting for the whole DM, the numeric axion den-\nsity isna'8\u00021012(42\u0016eV=ma) cm\u00003. Forva'10\u00003c,\nwherecis the speed of light, the de Broglie wave-\nlength and coherence time of the galactic axion \feld are\n\u0015ra= 25 (42\u0016eV=ma) m, and\u001cra= 85 (ma=42\u0016eV)\u0016s\n[34, 35]. The e\u000bective \feld frequency is proportional to\nthe axion mass, !a=2\u0019= 10 (ma=42\u0016eV) GHz, while itsarXiv:2001.08940v1  [hep-ex]  24 Jan 20202\namplitude depends on the properties of the DM halo and\nof the axion model,\nBa=gaee\n2er\nna~\nmacmava'4\u000210\u000023\u0010ma\n42\u0016eV\u0011\nT;(3)\nwhere~is the reduced Planck constant. These features\nallows for the driving of a coherent interaction between\nBaand the homogeneous magnetization of a macroscopic\nsample. The sample is immersed in a static magnetic\n\feldB0to couple the axion \feld to the Kittel mode of\nuniform precession of the magnetization. The interaction\nyields a conversion rate of axions to magnons which can\nbe measured by searching for oscillations in the sample's\nmagnetization. Due to the angle between B0andBa, the\nresulting signal undergoes a full daily modulation [37].\nThe maximum axion-deposited power is related to Eq. (3)\nand to the characteristics of the receiver, namely number\nof spinsNsand system relaxation time \u001cs\nPa=\re\u0016BNs!aB2\na\u001cs; (4)\nwhere\reis the electron gyromagnetic ratio.\nTo detect this signal we devised a suitable receiver. As\nit measures the magnetization of a sample, it is con\fg-\nured as a spin-magnetometer used as an axion haloscope.\nThe device consists of an axion \feld transducer and of\nan rf detection chain.\nAt high frequencies and in free \feld, the electron spin\nresonance linewidth is dominated by radiation damping,\nwhich limits \u001cs[38{40]. To avoid this issue, the mate-\nrial is placed in a microwave cavity. If the frequency\nof the Kittel mode !m=\reB0is close to the cavity\nmode frequency !c, the two resonances hybridize and\nthe single mode splits into two, following an anticrossing\ncurve [41, 42]. The B0-dependent hybrid modes frequen-\ncies are!1and!2and the cavity-material coupling is\ngcm= min(!2\u0000!1). Ifgcmis larger than the hybrid\nmode linewidths \r1;2'(\rc+\rm)=2, where\rcand\rm\nare the cavity and material dissipations, the system is\nin the strong coupling regime. To increase Pa,Nsand\n\u001csmust be large, so a suitable sample has a high spin\ndensity and a narrow linewidth. The best material iden-\nti\fed so far is Yttrium Iron Garnet (YIG), with roughly\n2\u00021022spins=cm3and 1 MHz linewidth [43].\nIn the apparatus that we operated at the Labora-\ntori Nazionali di Legnaro of INFN, the TM110 mode\nof a cylindrical copper cavity is coupled to ten 2.1 mm-\ndiameter spheres of YIG. The spherical shape is needed\nto avoid geometrical demagnetization. We devised an on-\nsite grinding and polishing procedure to obtain narrow\nlinewidth spherical samples starting from large single-\ncrystals of YIG. The spheres are placed on the axis of\nthe cavity, where the rf magnetic \feld is uniform.\nSeveral room temperature tests were performed to de-\nsign the YIG holder: a 4 mm inner diameter fused silica\npipe, containing 10 stacked PTFE cups, each one largeenough to host a free rotating YIG. Free rotation permits\nthe spheres' easy axis self-alignment to the external mag-\nnetic \feld, while a separation of 3 mm prevents sphere-\nsphere interaction. The pipe is \flled with 1 bar of helium\nand anchored to the cavity for thermalization. The cavity\nand pipe are placed inside the internal vacuum chamber\n(IVC) of a dilution refrigerator, with a base temperature\naround 90 mK. Outside the IVC, in a liquid helium bath,\na superconducting magnet provides the static \feld with\nan inhomogeneity below 7 ppm over all the spheres.\nFIG. 1: Measured (left) and modeled (right) transmis-\nsion functions of the HS. The right plot is the function\nfcdmn(!;! m), based on the second quantization of coupled\nharmonic oscillators, while the left one is a SO-to-Readout\n(see Fig. 2) transmission measurement with the JPA o\u000b, per-\nformed at 90 mK. Color scales are in arbitrary units (brighter\ncolors corresponds to higher amplitudes). The dashed line in\nthe left plot identi\fes the hybrid mode frequencies !1, where\nwe performed measurements.\nThe resulting hybrid system (HS) has been studied\nby collecting a B0vs frequency transmission plot, re-\nported in Fig. 1 (left). The measured plot is not a usual\nanticrossing curve. In our system the cavity frequency\n!c=2\u0019= 10:7 GHz and the expected coupling is of the\norder of 600 MHz, thus !2gets close and couples to a\nhigher order mode of the cavity. This hybrid mode fur-\nther splits into others, making the two oscillators de-\nscription unsuitable. Other disturbances are related to\nresidual sphere-sphere interaction and to non-identical\nspheres. To model the HS, we write an hamiltonian based\non two cavity modes, candd, and two magnetic modes,\nmandn\nHcdmn =0\nBB@!c\u0000i\rc\n20gcmgcn\n0!d\u0000i\rd\n2gdmgdn\ngcmgdm!m\u0000i\rm\n2gmn\ngcngdngmn!n\u0000i\rn\n21\nCCA;\n(5)\nwhereg,!and\rindicate their couplings, resonant fre-\nquencies and dissipations, respectively. Fig. 1 (right)\nshows the function fcdmn(!;!m) = det\u0000\n!I4\u0000Hcdmn\u0001\n,\nwhose maxima identify the resonance frequencies of the3\nHS. By comparing the two plots of Fig. 1, one can see that\nthe model appropriately describes the system, allowing\nus to extract the linewidths, frequencies and couplings of\nthe modes through a \ft. The typical measured values are\n\r1'1:9 MHz and gcm'638 MHz, yielding \u001cs'84 ns\nandNs'1:0\u00021021spins, respectively. Remarkably, the\nmode!1is not altered by other modes, thus we will use\nit to search for axion-induced signals. For a \fxed B0the\nlinewidth of the hybrid mode is the haloscope sensitive\nband. By changing B0, we can perform a frequency scan\nalong the dashed line of Fig. 1.\nMagnetmw-cavityYIGD1\nD2-20SO\n-20 -10 -30Aux\nA1A2Readout-20 -30Pump\nJPAsp i90 mK4 K\nTT\nCalibration\nFIG. 2: Schematics of the apparatus. The cavity is reported\nin orange, the ten YIG spheres are in black, and the blue\nshaded region is permeated by a uniform magnetic \feld. The\ncryogenic and room temperature HEMT ampli\fers are A1 and\nA2, respectively, and the JPA ports are the signal (s), idler (i)\nand pump (p). Superconducting cables are brown, the red-\ncircledTs are the thermometers, SO is a source oscillator, and\nattenuators are shown with their reduction factor in dB. As\ninset, we show the calibration of the system gain and noise\ntemperature, obtained by injecting signals in the SO line. The\npower injected in the HS is given in terms of an e\u000bective\ntemperature proportional to Acal. The errors are within the\nsymbol dimension. See text for further details.\nThe electronic schematics, shown in Fig. 2, consists in\nfour rf lines used to characterize, calibrate and operate\nthe haloscope. The HS output power is collected by a\ndipole antenna (D1), connected to a manipulator by a\nthin steel wire and a system of pulleys to change its cou-\npling. The source oscillator (SO) line is connected to aweakly coupled antenna (D2) and used to inject signals\ninto the HS, the Pump line goes to a Josephson paramet-\nric ampli\fer (JPA), the Readout line ampli\fes the power\ncollected by D1, and Aux is an auxiliary line. The Read-\nout line is connected to an heterodyne as described in\n[35], where an ADC samples the down-converted power\nwhich is then stored for analysis. The JPA is a quan-\ntum limited ampli\fer, with resonance frequency of about\n10 GHz resulting in a noise temperature of 0.5 K. Its gain\nis close to 20 dB in a band of order 10 MHz, and its work-\ning frequency can be tuned thanks to a small supercon-\nducting coil [44]. Excluding some mode crossings, hybrid\nmode and JPA frequencies overlap between 10.2 GHz and\n10.4 GHz, and allow us to scan the corresponding axion\nmass range.\nThe procedure to calibrate all the lines of the setup is:\n(i) the transmittivity of the Aux-Readout path KARis\nmeasured by decoupling D1 or by detuning !1; (ii) for\nthe Aux-SO and SO-Readout paths, KASandKSRare\nobtained by critically coupling D1 to the mode !1. The\ntransmittivity of the SO line is KSO'p\nKSRKAS=KAR.\nIf a signal of power Ainis injected in the SO line,\nthe fraction of this power getting into the HS results\nAcal=AinKSO. SinceAcalis a calibrated signal, it can\nbe used to measure gain and noise temperature of the\nReadout line. From this measurement we obtain a sys-\ntem noise temperature Tn= 1:0 K, and a gain of 70.4 dB\nfrom D1 to Readout (see Fig. 2). In our setup, the cou-\npling of D1 can be varied of 8 dB, thus we estimate a\ncalibration uncertainty of 16%. We measured the JPA\ngain, the HEMTs noise temperature, and the cavity tem-\nperature to get the noise budget detailed in Tab. I. The\n0.12 K di\u000berence may be due to unaccounted losses, or\nnon-precise temperature control.\nSource Estimated\nQuantum noise 0.50 K\nThermal noise 0.12 K\nHEMTs noise 0.25 K\nExpected total 0.87 K\nMeasured total Tn0.99 K\nTABLE I: Noise budget of the apparatus. The measured noise\nis compatible with the estimated one.\nTo double check the accuracy of the result, we measure\nthe thermal noise of the HS. The noise di\u000berence for !1\non and o\u000b the JPA resonance (dark blue and light blue)\ngives the noise added by the hybrid mode (orange curve),\nas shown in Fig. 3. The excess noise is compatible with\na temperature of the HS \u001810 mK higher than the one\nof the nearest load, which is realistic. Similar results are\nobtained by changing the D1 antenna coupling for a \fxed\nB0.\nThe axion search consisted in \ffty-six runs, each one4\nFIG. 3: Thermal noise of the HS. The blue curves are the\npower measured at the Readout with !1in the JPA band-\nwidth (dark blue) and out of it (light blue). The di\u000berence\nbetween the two is the HS noise (reported in orange).\nwith \fxedB0. For every run a transmission measurement\nof the hybrid system is used to set !1, to critically couple\nD1 to it, and to measure \r1. The frequency stability of\n!1resulted well below the linewidth within an interval\nof several hours, allowing long integration times. Data\nare stored with the ADC over a 2 MHz band around !1\nfor subsequent analysis. We FFT the data with a 100 Hz\nresolution bandwidth to identify and remove biased bins\nand disturbances in the down-converted spectra. To esti-\nmate the sensitivity to the axion \feld, we rebin the FFTs\nwith a resolution RBW '5 kHz, which at this frequency\ngives the best SNR for the axionic signal [36]. The spec-\ntra are \ftted to a degree \fve polynomial to extract the\nresiduals, whose standard deviation is the sensitivity of\nthe apparatus. We veri\fed that the analysis procedure\nexcludes unwanted bins while preserving the signal and\nSNR by adding a fake axion signal to real data.\nOur data were collected in July 2019 in a total run\ntime of 74 h. The average run length was \u00181 h, and each\none was performed during the maximum of the daily-\nmodulated axionic signal. The measured \ructuations\nare compatible with the estimated noise in every run,\nand we detected no statistically signi\fcant signal consis-\ntent with the DM axion \feld. The minimum measured\n\ructuation is \u001bP= 5:1\u000210\u000024W, for the longest in-\ntegration time t= 9 h, where the Dicke prediction is\nkBTnp\nRBW=t= 4:8\u000210\u000024W. In terms of rf mag-\nnetic \feld, this result corresponds to \u001bB= 5:5\u000210\u000019T,\nwhich, to our knowledge, is a record one for an rf spin-\nmagnetometer. The absence of fast rf bursts in the data\nis veri\fed by using a 1 ms time resolution waterfall spec-\ntrogram.\nEven if the minimum \feld detectable by the haloscope\nis much larger than Ba, these measurements can still be a\nprobe for ALPs, which may also constitute the totality of\nDM [46]. The 95% CL upper limit on the axion electroncoupling constant is\ngaee<e\n\u0019mavas\nkac\u00022\u001bP\n2\u0016B\renaNs\u001cs'1:7\u000210\u000011:(6)\nThe transduction coe\u000ecient of the axionic signal kacwas\ncalculated with a model similar to the one of Eq. (5) [47].\nIt essentially depends on !1and, in our bandwidth, re-\nsults 0:5<kac<1:0. The overall exclusion plot obtained\nwith the ferromagnetic haloscope is given in Fig. 4. All\nthe experimental parameters used to extract the limits\nfrom Eq. (6) are measured within every run, making the\nmeasurement highly self-consistent.\nThese results improved the best previous limits [35]\nby roughly a factor 30 in gaeeand 50 in bandwidth. The\nimprovement over the previous prototype is due to an in-\ncreased material volume, to an almost quantum-limited\nnoise temperature, and to longer integration times. No\naxion-mass scan was performed by previous experiments\nof this kind, and we now demonstrate that it is feasi-\nble to tune a hybrid resonance over hundreds of MHz to\nsearch for axion-deposited power. Our prototype scanned\na range of axion masses of about 0.7 \u0016eV with a \feld vari-\nation of 7 mT, drastically simplifying the tuning of the\nhaloscope.\nIn conclusion, we designed and developed a quantum-\nlimited rf spin-magnetometer used as an axion haloscope.\nThe instrument implements an axion-to-rf transducer,\ni. e. an hybrid system which embeds one of the largest\nquantity of magnetic material to date, and a detection\nelectronics based on a quantum-limited JPA. The oper-\nation of this instrument led to an axion search over a\nspan of 0:7\u0016eV around 42 :7\u0016eV, with a maximum sen-\nsitivity togaeeof 1:7\u000210\u000011. This, to our knowledge,\nis the best reported limit on the coupling of DM axions\nto electrons, and corresponds to a 1- \u001b\feld sensitivity of\n5:5\u000210\u000019T, which is a record one. No showstoppers\nhave been found so far, and hence a further upscale of\nthe system can be foreseen. A superconducting cavity\nwith a higher quality factor was already developed and\ntested [48]. It was not employed in this work since the\nYIG linewidth does not match the superconducting cav-\nity one, and the improvement on the setup would have\nbeen negligible. With this prototype we reached the rf\nsensitivity limit of linear ampli\fers [49]. To further im-\nprove the present setup one needs to rely on bolometers\nor single photon/magnon counters [50]. Such devices are\ncurrently being studied by a number of groups, as they\n\fnd important applications in the \feld of quantum infor-\nmation [51{54].\nWe are grateful to E. Berto, A. Benato and M. Rebes-\nchini, who did the mechanical work, F. Calaon and M.\nTessaro who helped with the electronics and cryogenics,\nand to F. Stivanello for the chemical treatments. We\nthank G. Galet and L. Castellani for the development of\nthe magnet power supply, and M. Zago who realized the5\nFIG. 4: Exclusion plot at 95% CL on the axion-electron coupling obtained with the present prototype (excluded region reported\nin blue and error in light blue), and overview of other searches for the axion-electron interaction. The other results are from\n[35] (orange) and [45] (green), while the DFSZ axion line is at about gaee'10\u000015. The inset is a detailed view of the reported\nresult.\ntechnical drawings of the system. We deeply acknowledge\nthe Cryogenic Service of the Laboratori Nazionali di Leg-\nnaro, for providing us large quantities of liquid helium on\ndemand.\n\u0003Electronic address: nicolo.crescini@phd.unipd.it\nyElectronic address: ruoso@lnl.infn.it\n[1] R. D. Peccei and Helen R. Quinn. CP conservation in the\npresence of pseudoparticles. Phys. Rev. Lett. , 38:1440{\n1443, Jun 1977.\n[2] Steven Weinberg. A new light boson? Phys. Rev. Lett. ,\n40(4):223, 1978.\n[3] Frank Wilczek. 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Bauer2, 1\n1Kavli Institute of NanoScience, Delft University of Technology, 2628 CJ Delft, The Netherlands\n2Institute for Materials Research & WPI-AIMR & CSRN, Tohoku University, Sendai 980-8577, Japan\n(Dated: April 10, 2018)\nInelastic scattering of light by spin waves generates an energy \row between the light and magne-\ntization \felds, a process that can be enhanced and controlled by concentrating the light in magneto-\noptical resonators. Here, we model the cooling of a sphere made of a magnetic insulator, such as\nyttrium iron garnet (YIG), using a monochromatic laser source. When the magnon lifetimes are\nmuch larger than the optical ones, we can treat the latter as a Markovian bath for magnons. The\nsteady-state magnons are canonically distributed with a temperature that is controlled by the light\nintensity. We predict that such a cooling process can signi\fcantly reduce the temperature of the\nmagnetic order within current technology.\nA great achievement of modern physics is the Doppler\ncooling of trapped atoms by optical lasers [1, 2] down\nto temperatures of micro-Kelvin [3]. Subsequently, even\nmacroscopic mechanical objects, such as membranes and\ncantilevers, have been cooled to their quantum me-\nchanical ground state [4{8] by blue shifting the stim-\nulated emission using an optical cavity [4, 5]. `Cav-\nity optomechanics' is a vibrant \feld that achieved suc-\ncessful Heisenberg uncertainty-limited mechanical mea-\nsurements, the generation of entangled light-mechanical\nstates, and ultra-sensitive gravitational wave detection\n[8]. An optical cryocooler based on solid state samples [9]\ncan be superior due to its compactness and lack of mov-\ning components [10]. Optical cooling has been demon-\nstrated for glass [9, 11] and envisioned for semiconductors\n[10, 12, 13].\nAn analogous cooling of a magnet would generate in-\nteresting opportunities. Magnetization couples to mi-\ncrowaves [14{18], electric currents [18{20], mechanical\nmotion [17, 21{23], and, indeed, light [24]. Spin waves\nare the elementary excitations of the ferromagnetic or-\nder, which are quantized as bosonic magnons. Similar\nto phonons, magnons may be considered non-interacting\nup to relatively high temperatures and are Planck-\ndistributed at thermal equilibrium. However, there are\nimportant di\u000berences as well: Magnons have mass and\nchirality [25, 26], both of which are tunable by an ex-\nternal static magnetic \feld. Their long wavelength dis-\npersion in thin \flms is highly anisotropic, with minima\nin certain directions that can collect the Bose-Einstein\ncondensate of magnons [27{29]. Magnons can be used\nas quantum transducers between microwaves and optical\nlight [30] or between superconducting and \rying qubits\n[31].\nMotivated by the potential of a ferromagnet as a ver-\nsatile quantum interface at low temperatures, we discuss\nhere the potential of optical cooling of magnons. The\nmagnon-photon interaction gives rise to inelastic Bril-\nlouin light scattering (BLS) [32], which is a well estab-\nlished tool to study magnon dispersion and dynamics\n[24, 33, 34]. Recently, several groups carried out BLSexperiments on spheres made of ferrimagnetic insulator\nyttrium iron garnet (YIG) [35{40], which has a very\nhigh magnetic quality factor\u0000\n\u0018105\u0001\n[41{43] and sup-\nports ferromagnetic-like magnons with long coherence\ntimes (\u0018\u0016s) [31, 44, 45]. YIG spheres are commercially\navailable for microwave applications, but are also good\ninfrared light cavities due to their large refractive index\nand high optical quality [46, 47], making them good op-\ntomagnonic resonators [35{39, 48{51]. Via proximity op-\ntical \fbers or prisms, external laser light can e\u000eciently\nexcite `whispering gallery modes' (WGMs), i.e. the op-\ntical modes circulating in extremal orbits of dielectric\nspheroids [52, 53].\nThe BLS experiments on YIG spheres discovered a\nlarge asymmetry in the red- (Stokes) and blue-shifted\n(anti-Stokes) sidebands [36{39] due to selective resonant\nenhancement of the scattering cross section [38, 50, 51].\nThe asymmetry can be controlled by the polarization\nand wave vector of the light. When more photons are\nscattered into the blue than the red-shifted sidebands,\nlight e\u000bectively extracts energy from the magnons. Op-\ntomagnonic scattering is enhanced for a `triple resonance\ncondition' [38, 54{57] by tuning both the input and the\nscattered photon frequency to the optical resonances of\nthe cavity. In contrast, optomechanical cooling [4, 5, 8]\nrequires detuning the input laser from a cavity resonance\nwith correspondingly reduced scattering and cooling rate.\nIn this manuscript, we predict that modern technology\nand materials can signi\fcantly reduce the temperature of\nthe magnetic order, showing the potential to manipulate\nmagnons using light.\nWe derive below rate equations for photons and\nmagnons to estimate the steady-state magnon number\nthat can be reached as a function of material and device\nparameters. We consider a spherical magnetic insulator\nwith high index of refraction that is transparent at the\ninput light frequency (Fig. 1) and magnetization perpen-\ndicular to the WGM orbits that are excited by proximity\ncoupling to an external laser. We single out two groups of\nmagnon modes that couple preferentially to the WGMs\n[50]. The small angular momentum (including the Kit-arXiv:1804.02683v1  [cond-mat.mes-hall]  8 Apr 20182\nAin AoutWinMLWRMSWT\nHappInput\nScattered\nMagnon\nFIG. 1. Optomagnonic cooling setup: A ferromagnetic sphere\nin contact with an optical waveguide. A magnetic \feld Happ\n(into the paper) is applied to saturate the magnetization. In-\nput light with amplitude Ainis evanescently coupled to a\nWGMWin. We focus on anti-Stokes scattering by two types\nof magnons that are characterized by their angular momen-\ntum [50]. A small angular momentum magnon MSmaintains\nthe direction of WGMs, converting WintoWT.Wincan be\nre\rected into WRby absorbing a large angular momentum\nmagnonML. Theoretically, both the cases can be treated in\nthe same formalism.\ntel) magnons, MSin Fig. 1, and large angular momen-\ntum magnons, the chiral Damon-Eshbach (DE) modes\nML. The theory presented below is valid for both types\nof magnons.\nWe can understand the basic physics by the minimal\nmodel sketched in Fig. 2. We focus on a single incident\nWGMWpwith index pand frequency !p. It is occupied\nby [8]\nnp=4Kp\n(\u0014p+Kp)2Pin\n~!p(1)\nphotons, with \u0014pbeing the intrinsic linewidth, Kpthe\nleakage rate into the proximity coupler, and Pinthe input\nlight power. An optically active magnon M[with either\nsmall or large angular momentum] is annihilated Wp+\nM!Wcor created Wp!Wh+Mby BLS, where\nWcandWhare blue and red-shifted sideband WGMs,\nrespectively.\nWe \frst derive a simple semi-classical rate equation for\nthe non-equilibrium steady-state magnon number, n(sc)\nm\n[the superscript distinguishes the estimate from nmas\nmore rigorously derived below]. The thermal bath ab-\nsorbs and injects magnons at rates \u0014mn(sc)\nm(nth+ 1) and\n\u0014mnth\u0010\nn(sc)\nm+ 1\u0011\nrespectively, where \u0014mis the inversemagnon lifetime,\nnth=\u0014\nexp\u0012~!m\nkBT\u0013\n\u00001\u0015\u00001\n(2)\nits equilibrium thermal occupation, !mthe magnon fre-\nquency and Tthe ambient temperature. The optical\ncooling rate is R0\ncnpn(sc)\nm, whereR0\ncis the anti-Stokes\nscattering rate of one Wp-photon by one M-magnon and\nwe assumed that there are no photons in Wc. The lat-\nter is justi\fed because of small optomagnonic couplings\ncompared to WGM dissipation rates, \u00182\u0019\u00020:1\u00001 GHz\n[36{38] while R0\ncnpn(sc)\nmis at most\u0018\u0014m\u00182\u0019\u00021 MHz.\nIn the absence of dissipation, Fermi's golden rule gives\nR0\nc= 2\u0019jgcj2\u000e(!p+!m\u0000!c), where ~gcis the opto-\nmagnonic coupling and f!p;!c;!mgare the frequencies\noffWp;Wc;Mg;respectively. When Wchas a \fnite life-\ntime, the\u000e-function is broadened into a Lorentzian, giv-\ning\nR0\nc=jgcj2(\u0014c+Kc)\n(!p+!m\u0000!c)2+ (\u0014c+Kc)2=4; (3)\nwhere\u0014cis its intrinsic linewidth, and Kcis its leak-\nage rate into the proximity coupler. Similarly, the opti-\ncal heating rate is R0\nhnp\u0010\nn(sc)\nm+ 1\u0011\n;whereR0\nhis given\nby Eq. (3) with gc;!c;\u0014c!gh;!h;\u0014hand!m!\u0000!m.\nIn deriving R0\nc;h, we ignore the magnon linewidth since\n\u0014m\u001c\u0014c;\u0014h[38, 50]. In the steady state the cooling and\nheating rates are equal, leading to the estimate\nn(sc)\nm=\u0014mnth+R0\nhnp\n\u0014m+ (R0c\u0000R0\nh)np: (4)\nThis agrees with the result from the more precise theory\ndiscussed below, thus capturing the essential processes\ncorrectly (a posteriori). However, the rate equation can-\nnot access noise properties beyond the magnon number\nthat are important for thermodynamic applications. Fur-\nther, it does not di\u000berentiate between a coherent preces-\nsion of the magnetization and the thermal magnon cloud\nwith the same number of magnons.\nIn order to model the cooling process more rigorously,\nwe proceed from a model Hamiltonian for a system with\nthree photon and one magnon mode. In the Hamiltonian\n^HS=^H0+^Hom[50]\n^H0=~!p^ay\np^ap+~!c^ay\nc^ac+~!h^ay\nh^ah+~!m^my^m;(5)\nand ^axand ^mare the annihilation operators for photon\nWxwithx2fp;c;hgand magnon M. The optomagnonic\ncoupling in the rotating wave approximation reads [50]\n^Hom=~gc^ap^ay\nc^m+~gh^ap^ay\nh^my+ h:c:; (6)\nwheregcandghare the scattering amplitudes and h :c:is\nthe Hermitian conjugate.3\nWpκp\nWc\nκcWh\nκhMκm\nKp\nKh KcR0\nc R0\nh\nFIG. 2. Light-induced cooling of a magnon, M. A proxim-\nity \fber or prism is coupled to the WGMs Wxwith a cou-\npling constant Kx, excitingWpwhile collecting the scattered\nWcandWh. The photons are inelastically scattered by the\nmagnonWp+M!WcandWp!Wh+Mat single particle\nratesR0\ncandR0\nhrespectively, derived in the text. All modes\nare coupled to their respective thermal baths by leakage rates\n\u0014x. When\u0014cis much larger than the corresponding scatter-\ning rate, the bath associated with Wccan become an e\u000ecient\nchannel for dissipation of the magnons in M.\nIn the rotating frame of the \\envelope\" operators\n^Wx(t)4= ^ax(t)ei!xtand ^M(t)4= ^m(t)ei!mtthe (Heisen-\nberg) equation of motion for ^Mbecomes [4, 58]\n_^M=\u0000igh^Wp^Wy\nhei\u000eht\u0000ig\u0003\nc^Wy\np^Wce\u0000i\u000ect\u0000\u0014m\n2^M\u0000p\u0014m^bm;\n(7)\nwhere\u000eh=!h+!m\u0000!pand\u000ec=!c\u0000!m\u0000!pare the\ndetunings from the scattering resonances. ^bm(t) is the\nstochastic magnetic \feld generated by the interaction of\nMwith phonons [59] and/or other magnons [60], whose\nprecise form depends on the microscopic details of the\ninteraction [61].\nWe assume that the correlators of ^bmobey the\n\ructuation-dissipation theorem for thermal equilibrium\n[62, 63]. When \u0014m\u001ckBT=~, which is satis\fed for \u0014m\u0018\n2\u0019\u00021 MHz [36{38] and T\u001d50\u0016K, the (narrow band\n\fltered) noise is e\u000bectively white and generates a canon-\nical Gibbs distribution of the magnons in steady state\n[58]. Their statistics areD\n^bm(t)E\n= 0,D\n^by\nm(t0)^bm(t)E\n=\nnth\u000e(t\u0000t0) andD\n^bm(t0)^by\nm(t)E\n= (nth+ 1)\u000e(t\u0000t0), where\nnthis de\fned in Eq. (2).\nFor weak scattering relative to the input power, we can\nignore any back-action on ^Wpsuch that its dynamics is\ngoverned only by the proximity coupling. When Wpis ina coherent state,D\n^Wp(t)E\n=pnpandD\n^Wy\np(t0)^Wp(t)E\n=\nnp, wherenpis given by Eq. (1).\nThe photons in Wcare generated by ^Homand dissi-\npated into their thermal bath, with Heisenberg equation\nof motion [4, 58]\nd^Wc\ndt=\u0000igc^Wp^Mei\u000ect\u0000\u0014c+Kc\n2^Wc\u0000p\u0014c^bc\u0000p\nKc^Ac;\n(8)\nwhere ^bcand ^Acare noise operators. The physical ori-\ngins of ^bcand \fnite lifetime \u0014\u00001\ncare scattering by im-\npurities, surface roughness, and lattice vibrations. Kcis\nthe leakage rate of Wcinto the proximity coupler and ^Ac\nis the vacuum noise from the latter into Wc. The noise\nsources are white for su\u000eciently small \u0014c.D\n^Xc(t)E\n= 0,D\n^Xy\nc(t0)^Xc(t)E\n= 0 andD\n^Xc(t0)^Xy\nc(t)E\n=\u000e(t\u0000t0) where\nX2f^bc;^Acg, because the thermal occupation of photons\nat infrared and visible frequencies is negligibly small at\nroom temperature e\u0000~!c=(kBT)\u00190.\nThe solution to Eq. (8) is ^Wc(t) =^Wc;th(t) +^Wc;om(t).\nThe thermal contribution is,\n^Wc;th=Zt\n0e\u0000(\u0014c+Kc)(t\u0000\u001c)=2h\n\u0000p\u0014c^bc(\u001c)\u0000p\nKc^Ac(\u001c)i\nd\u001c\n(9)\nwhere the origin of time is arbitrary. For t;t0! 1 ,\nwe get the equilibrium statisticsD\n^Wy\nc;th(t0)^Wc;th(t)E\n= 0\nand\nD\n^Wc;th(t0)^Wy\nc;th(t)E\n= exp\u0014\n\u0000(\u0014c+Kc)jt\u0000t0j\n2\u0015\n;(10)\nindependent of the optomagnonic coupling. ^Wc;omcan\nbe simpli\fed by the adiabaticity of the magnetization\ndynamics that follows from \u0014m\u001c\u0014c. When ^Mis treated\nas a slowly varying constant\n^Wc;om(t)\u0019\u0000igc^M(t)Zt\n0e\u0000(\u0014c+Kc)(t\u0000\u001c)=2^Wp(\u001c)ei\u000ec\u001cd\u001c:\n(11)\n^Wh(t) is obtained by the substitution c!hand ^M!\n^Myin Eqs. (9-11).\nWe can now rewrite Eq. (7) as\nd^M\ndt=\u0000\u0010\u0014m\n2^M+p\u0014m^bm\u0011\n+^Oc+^Oh: (12)\nwith cooling and heating operators that re\rect the light\nscattering processes in Fig. 2:\n^Oc=^Nc+i^\u0006c^M; (13)\n^Oh=\u0000^Ny\nh+i^\u0006y\nh^M: (14)\nFocusing on the cooling process, we distinguish the\nself-energy,\n^\u0006c=ijgcj2Zt\n0e(i\u000ec+(\u0014c+Kc)=2)(\u001c\u0000t)^Wy\np(t)^Wp(\u001c)d\u001c;(15)4\nfrom the noise operator,\n^Nc(t) =\u0000ig\u0003\nc^Wy\np(t)^Wc;th(t)e\u0000i\u000ect: (16)\nIn the weak-coupling regime we may adopt a mean-\feld\napproximation by replacing ^\u0006cby its average,\nD\n^\u0006cE\n=\u0000\u0016!c+i\u0016\u0014c\n24=jgcj2np\n\u000ec\u0000i(\u0014c+Kc)=2; (17)\nwhere \u0016!cis the (reactive) shift of the magnon resonance\nand \u0016\u0014cthe optical contribution to the magnon linewidth.\nThe noise ^Nccan be interpreted as the vacuum \ruc-\ntuations of Wcentering the magnon subsystem via the\noptomagnonic interaction. ^Nchas a very short corre-\nlation time\u0018(\u0014c+Kc)\u00001[see Eq. (10)] compared to\nmagnon dynamics \u0018\u0014\u00001\nm, and thus can be treated as a\nwhite noise source withD\n^Nc(t)E\n= 0,D\n^Ny\nc(t)^Nc(t0)E\n= 0\n, andD\n^Nc(t0)^Ny\nc(t)E\n\u0019Vc\u000e(t\u0000t0). By integrating over\ntime and using the correlation functions of ^Wpand ^Wc;th\nVc=4jgcj2np(\u0014c+Kc)\n4\u000e2c+ (\u0014c+Kc)2= \u0016\u0014c; (18)\nde\fned in Eq. (17). \u0016 \u0014c=\u0014mat resonance \u000ec= 0 is the\ncooperativity between the magnons and Wc-photons due\nto the coupling mediated by Wp-photons.\nAnalogous results hold for ^Oh, with substitutions c!\nhin Eqs. (15)-(18). We arrive at\nd^M\ndt\u0019\u0000i(\u0016!c+ \u0016!h)^M\u0000\u0014tot\n2^M\u0000p\u0014tot^btot;(19)\nwhere\u0014tot=\u0014m+ \u0016\u0014c\u0000\u0016\u0014handp\u0014tot^btot=p\u0014m^bm\u0000\n^Nc+^Ny\nh. The \ructuations of the total noise follow from\nEq. (18)\nD\n^by\ntot(t0)^btot(t)E\n\u0019nm\u000e(t\u0000t0); (20)\nD\n^btot(t0)^by\ntot(t)E\n\u0019(nm+ 1)\u000e(t\u0000t0); (21)\nwhere\nnm=\u0014mnth+ \u0016\u0014h\n\u0014m+ \u0016\u0014c\u0000\u0016\u0014h: (22)\nEq. (19) is equivalent to the equation of motion for\nmagnons in equilibrium after the substitutions !m!\n!m+ \u0016!c+ \u0016!h,\u0014m!\u0014tot, andnth!nm. It implies that\nthe magnons in the non-equilibrium steady state are still\ncanonically distributed with density matrix\n^\u001ane= exp\u0012\u0000~!m^nm\nkBTne\u0013\u0012\nTr\u0014\nexp\u0012\u0000~!m^nm\nkBTne\u0013\u0015\u0013\u00001\n(23)\nwhere the number operator ^ nm= ^my^mand the non-\nequilibrium magnon temperature Tneis implicitly de\fned\nby Eq. (22) and\nnm=\u0014\nexp\u0012~!m\nkBTne\u0013\n\u00001\u0015\u00001\n: (24)We getD\n^MxE\n=D\n^MyE\n= 0, which implies that light\nscattering does not induce a coherent magnon precession,\nin contrast to a resonant ac magnetic \feld. nmis the av-\nerage number of magnons that can be larger or smaller\nthan the equilibrium value nth. The result is consistent\nwithn(sc)\nm[see Eq. (4)] because \u0016 \u0014c;h=R0\nc;hnpas expected\nfrom Fermi's golden rule. The canonical distribution im-\nplies that the steady-state magnon entropy is maximized\nfor the given number of magnons, nm.\nWhen \u0016\u0014h\u0000\u0016\u0014c> \u0014m;i.e. when heating by the laser\novercomes the intrinsic magnon damping, the system be-\ncomes unstable, leading to runaway magnon generation\nand self-oscillations [49, 64, 65]. The instability is reg-\nularized by magnon-magnon scattering, not included in\nour theory.\nHere we focus on the cooling scenario in which \u0016 \u0014h\u001c\u0016\u0014c\n[50]. Magnon cooling can be monitored by the inten-\nsity of the blue-shifted sideband. Using the input-output\nformalism [58, 66] the scattered light amplitude in the\nrotating frame is\n^Aout(t) =\u0000p\nKc^Wc(t): (25)\nIt can be converted into the output power by Pout=\n~!cD\n^Ay\nout(t)^Aout(t)E\n, which is independent of time in\nsteady state. With impedance matching, \u0014p;c=Kp;c,\nand at the triple resonance, \u000ec= 0,\nPout\nPin=jgcj2\n\u0014c\u0014p\u0014mnth\n\u0014m+ 2jgcj2np=\u0014c/1\n1 +Pin=Ps;(26)\nde\fning the saturation power\nPs4=~!p\u0014p\u0014c\u0014m\n2jgcj2: (27)\nTo leading order Pout/Pin[37, 50], but saturates when\nthe magnon number becomes small, which is an exper-\nimental evidence for magnon cooling. Psis the input\npower that halves the number of magnons.\nFor a YIG sphere with parameters !c\u0019!p= 2\u0019\u0002\n300 THz (free space wavelength 1 \u0016m), an optical Q-factor\n!p=(2\u0014p) =!c=(2\u0014c) = 106, [37], magnon linewidth\n\u0014m= 2\u0019\u00021 MHz, and optomagnonic coupling gc=\n2\u0019\u000210 Hz [50], we get Ps= 140 W. Trying to match this\nwithPinis not useful since laser-induced lattice heating\n[10] will overwhelm the cooling e\u000bect. However, Pscan be\nsigni\fcantly reduced by large magnon-photon coupling.\nDoping YIG with bismuth can increase gctenfold [47],\nbringingPsdown to\u00181 W. The spatial overlap between\nthe magnons and photons [50] can be engineered in ellip-\nsoidal or nanostructured magnets [67] which can increase\ngcfurther by an order of magnitude, giving Ps\u001810mW.\nFor an ambient temperature T= 1 K and magnon fre-\nquency!m= 2\u0019\u000210 GHz, the thermal magnon number\nnth= 1:62. 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Schmidt, ArXiv e-prints (2018),\narXiv:1802.03176 [cond-mat.mes-hall]."    },    {        "title": "1804.11231v1.Single_Nitrogen_vacancy_center_quantum_memory_for_a_superconducting_flux_qubit_mediated_by_a_ferromagnet.pdf",        "content": "Single-Nitrogen-vacancy-center quantum memory for a superconducting \rux qubit\nmediated by a ferromagnet\nYen-Yu Lai,1, 2Guin-Dar Lin,1, 2Jason Twamley,3and Hsi-Sheng Goan1, 2,\u0003\n1Department of Physics and Center for Theoretical Physics,\nNational Taiwan University, Taipei 10617, Taiwan\n2Center for Quantum Science and Engineering, National Taiwan University, Taipei 10617, Taiwan\n3Centre for Engineered Quantum Systems, Department of Physics and Astronomy, Macquarie University, NSW 2109, Australia\n(Dated: May 1, 2018)\nWe propose a quantum memory scheme to transfer and store the quantum state of a supercon-\nducting \rux qubit (FQ) into the electron spin of a single nitrogen-vacancy (NV) center in diamond\nvia yttrium iron garnet (YIG), a ferromagnet. Unlike an ensemble of NV centers, the YIG mod-\nerator can enhance the e\u000bective FQ-NV-center coupling strength without introducing additional\nappreciable decoherence. We derive the e\u000bective interaction between the FQ and the NV center by\ntracing out the degrees of freedom of the collective mode of the YIG spins. We demonstrate the\ntransfer, storage, and retrieval procedures, taking into account the e\u000bects of spontaneous decay and\npure dephasing. Using realistic experimental parameters for the FQ, NV center and YIG, we \fnd\nthat a combined transfer, storage, and retrieval \fdelity higher than 0.9, with a long storage time\nof 10 ms, can be achieved. This hybrid system not only acts as a promising quantum memory, but\nalso provides an example of enhanced coupling between various systems through collective degrees\nof freedom.\nPACS numbers: 03.65.Yz, 42.50.Dv, 03.67.-a, 03.65.Ta\nI. INTRODUCTION\nSuperconducting qubits and related circuit-QED de-\nvices [1, 2] with excellent scalability, parametric tunabil-\nity, and strong coupling with external \felds are proving\nto be a powerful platform for quantum information pro-\ncessing. However, they su\u000ber from decoherence due to\ninevitable interactions with their surrounding environ-\nments. In a complex quantum protocol, superconducting\nqubits may experience frequent idles times when they are\nnot involved in active quantum gates. During this idle\ntime, to prevent the decoherence of their quantum in-\nformation, one can transfer their quantum state to an\nadjacent quantum memory for better protection.\nA hybrid system that takes advantage of the fast oper-\nation of superconducting qubits and long coherence times\nof a suitable quantum memory may yield good coherence\npreservation if the state transfer between them is quick\nenough, i.e., faster than the decoherence time of either\nsystem. The spin of a nitrogen-vacancy (NV) center in\ndiamond, which has a relatively long coherence time even\nat room temperature [3], can be a candidate for such a\nquantum memory. This low decoherence rate also means\nthat the NV center normally only couples weakly to a\nsuperconducting qubit. Such a weak coupling leads to a\nslow state transfer and coherence loss can be signi\fcant.\nEnsembles of NV centers [4{8] may make the coupling\nstronger, but at the added cost of increased decoherence\ncaused by internal spin-spin interactions, degrading the\n\fdelity of the quantum memory.\n\u0003goan@phys.ntu.edu.twIn this paper, we propose a scheme to transfer quantum\nstates faithfully between a superconducting \rux qubit\nand a single-NV-center spin via the ferromagnetic ma-\nterial yttrium iron garnet (YIG) [9, 10]. YIG has been\nproposed as a mediator for classical magnetic \felds to\nenhance the sensitivity of a NV magnetometer to achieve\nsingle nuclear spin detection [11]. In addition, the large\nnumber of spins in YIG with strong exchange interac-\ntion leads to collective-excitation modes with narrow\nlinewidths at low temperature [12, 13]. These collective\nmodes are known as quasiparticles or magnons [14], and\nhave been shown to be capable of coupling to di\u000berent\nkinds of quantum systems, such as superconducting mi-\ncrowave cavity modes [13, 15, 16]. Magnons in YIG have\nalso been proposed as a mediator of coherent coupling\nbetween two distant spins (e.g., two spatially distant NV-\ncenter spins) [17]. A CNOT gate between two single-NV-\ncenter spins separated by a distance of about 1 \u0016m with\noperation times of the order of a few tens of nanoseconds\nhas been demonstrated [17]. This shows that a relatively\nstrong coherent coupling between a single-NV-center spin\nand YIG magnons is feasible. On the other hand, a\n\rux qubit (FQ) can display strong coherent coupling to\nan ensemble of NV centers exhibiting a collective cou-\npling of\u001870 MHz [8]. However the spin density of YIG\n(\u001a\u00184:2\u00021021cm3) [13] is almost three orders of magni-\ntude larger than typical NV ensembles ( \u001a\u00185\u00021018cm3)\n[8]. This suggests that the coupling between a \rux qubit\nand a small YIG sample may be similar or even stronger\nthan between a \rux qubit and a NV ensemble. In this\npaper, we show that we can achieve a substantially large\ne\u000bective coupling between a single-NV-center spin and a\nFQ by using the magnons in a small nearby YIG sample\nas a mediator without appreciably sacri\fcing the transferarXiv:1804.11231v1  [quant-ph]  30 Apr 20182\nand storage \fdelity of the quantum state.\nWhen the size of the YIG is small enough, the Kit-\ntel mode (KM) of the YIG sample [13, 15] is gapped\nfrom the higher-energy modes and thus plays an impor-\ntant role in a low-temperature and low-excitation regime.\nIn our scheme, we \fnd that the e\u000bective coupling and\nthe spatial separation between the FQ and NV required\nto attain these coupling strengths via the YIG can be\nsigni\fcantly enhanced. The coupling attained using our\nproposal is of the order of several tenths of MHz, while\nthe spatial separation can be increased to a few tenths\nof\u0016m. This represents an enhancement in the coupling\nstrength of 3{7 times over the direct FQ-NV coupling.\nMore interestingly, it also represents a substantial en-\nhancement in spatial separation required between the FQ\nand NV. For comparison, a direct coupling scheme [18]\n\fnds a coupling strength of \u0018100 kHz, but requires a mi-\nnuscule spatial separation of 20 nm. To achieve larger\ndirect coupling strength, strengths comparable to those\nfound using our scheme would require even tinier spatial\nseparations, which may be physically unrealistic. In con-\ntrast, in our proposal we are able to expand the spatial\nseparation to a few tenths of a \u0016m scale, which is 10{20\ntimes larger than the separation required to attain simi-\nlar coupling strengths via direct FQ-NV coupling. Thus\nour scheme can provide signi\fcant couplings over a sep-\naration, which is technically far easier to engineer. The\nquantum state transfer time with the coupling strength\nfound in our scheme is considerably smaller than the de-\ncoherence time of the FQ so that fast and faithful transfer\ncan be realized without su\u000bering signi\fcant decoherence.\nThe paper is arranged as follows. In Sec. II, we derive\nthe e\u000bective Hamiltonian and coupling strength between\na FQ and a NV-center spin from a FQ-YIG-NV-center\nhybrid system. In Sec. III, we introduce a protocol for\nthe transfer and storage of the quantum state. After that,\nsimulations of the protocol are presented and discussed in\nSec. IV, taking major decoherence e\u000bects into consider-\nation. Finally, a short conclusion is given in Sec. V. All\nthe details of derivations of equations and calculations\nare presented in Appendices A and B.\nII. MODEL\nThe hybrid system in our proposal is schematically il-\nlustrated in Fig. 1 and contains three parts: the FQ, YIG,\nand a single-NV center. The noninteracting Hamiltonian\ndescribing the individual systems [with ( ~= 1)] can be\nwritten as\nHs=HF+HY+HN; (1)where\nHF=1\n2!F\u001b(z)\nF; (2)\nHY=\u0000JX\nhr;r0iSr\u0001Sr0+\reBX\nrS(z)\nr; (3)\nHN=\u0001ZS\u0010\nS(z)\nN\u00112\n\u0000\reBS(z)\nN: (4)\nHere, the FQ is regarded as a typical two-level system de-\nscribed by the Hamiltonian HFin Eq. (2), with !Fthe\ntransition frequency of the FQ and \u001b(z)the Pauli matrix\n(for details, see Appendix A). The Hamiltonian of the\nYIG in an external \feld along the zaxis is given by HY\nin Eq. (3), where Srare the operators of the spin located\nat positionrin the YIG. The parameter Jis the exchange\ncoupling between the spins inside the YIG. We consider\nthe application of B=BL+\u000eB, an external magnetic\n\feld along the zaxis and which is felt by the YIG and the\nNV center (see Fig. 1 and 2). Here, BLis a local magnetic\n\feld generated by a micromagnet [19] without disturbing\nthe FQ (for details, see Sec. IV), and \u000eBis the tuneable\nmagnetic \feld whose value is set below the critical \feld of\nthe FQ. The tuneable dc magnetic \feld could be gener-\nated by a coil. The ground triplet states of the NV center\nis described by the Hamiltonian HNin Eq. (4), where\nS(z)\nNis thezcomponent of the spin-1 operators of the\nsingle-NV center, \u0001 ZS= 2:87 GHz is the zero-\feld split-\nting of the ground triplets, and \re=\u00001:76\u00021011rad s\u00001\nT\u00001is the gyromagnetic ratio of electron spin. To pro-\nceed further, let us simplify the Hamiltonians a little bit.\nThere are two single-photon transitions between j0Niand\nj\u00061Niin the ground triplet states of the NV-center spin,\nand their energy gaps are !(\u00061)= \u0001 ZS\u0007\reB. We use,\nfor instance, the NV-center spin states j0Niandj\u00001Nias\nour storage qubit basis state. In the state transfer stage,\nthis transition frequency is tuned to be resonant with the\nFQ transition frequency, while the j0Nitoj1Nitransi-\ntion is largely detuned (see Fig. 2). Further, as shown\nlater, the e\u000bective coupling between the FQ and the NV-\ncenter storage qubit can be switched on and o\u000b (or very\nsmall) by varying the external B\feld. Here, for the sake\nof deriving the e\u000bective Hamiltonian, we \frst treat the\nNV-center spin as a two-level storage qubit by ignoring\nthe far-detuned transition. We will take into account the\ne\u000bect of the existence of the far-detuned NV j1Nistate\nwhen we run numerical simulations for the dynamics of\nquantum state transfer and storage processes. We then\ntransform the YIG Hamiltonian from a Heisenberg model\nto a magnon form with the Holstein-Primako\u000b transfor-\nmation [17, 20, 21] and harmonic approximation. As a\nresult, the NV and YIG Hamiltonians can be rewritten\nas [17, 21]\nHN'1\n2!N\u001b(z)\nN; (5)\nHY'X\nk!kay\nkak; (6)3\nFigure 1. Schematic illustration of the proposed quantum\nmemory. The FQ is regarded as a two-level system depending\non the sign (direction) of its persistent current Ip, and the KM\nin YIG couples to both FQ and the single-NV-center spin with\nstrengthsgFYandgY N, respectively. The external magnetic\n\feld felt by the YIG and the NV center is B=BL+\u000eB\n(see Fig. 2), where BLis a local magnetic \feld generated by a\nmicromagnet without disturbing the FQ, and \u000eBis a tuneable\nmagnetic \feld, generated by, e.g., a coil.\nwhere!N=!(\u00001), and!k=sJa2k2+\reBare the fre-\nquencies of the NV storage qubit and magnon mode k,\nrespectively, ay\nk(ak) is the creation (annihilation) opera-\ntor of magnon mode k,sis the maximum eigenvalue of\nthe spin operator S(z)\nr, andais the lattice constant of\nthe YIG. For a small-sized YIG sample, the boundary\nconditions at the surface are of great importance and the\nmagnon modes become gapped.\nThe FQ and NV interact indirectly via YIG. The cou-\npling Hamiltonian thus has two parts: the FQ-YIG cou-\npling and the YIG-NV coupling. The current in the loop\nof the FQ generates a magnetic \feld which interacts with\nthe spins in the YIG. The NV electron spin also cou-\nples to these spins by dipole-dipole interaction. Under\nthe rotating-wave approximation (RWA), the coupling\nHamiltonian in terms of the YIG collective-excitation\nmodes (derived in detail in Appendix A) reads\nHc=HFY+HYN; (7)\nHFY'\u0000X\nk\u0010\ngFY(k)\u001b(+)\nFak+H:C:\u0011\n; (8)\nHYN'\u0000X\nk\u0010\ngYN(k)ay\nk\u001b(\u0000)\nN+H:C:\u0011\n; (9)\nwhere\ngFY(k) =\u00160\n2\u0019\reIpr\n2s\nNNX\nrFe\u0000ik\u0001a\nrF; (10)\ngYN(k) =\u0000\u00160\n4\u0019\r2\ne~r\n2s\nNNX\nrN\u00123cos2\u0012rN\u00001\nr3\nN\u0013\neik\u0001a;\n(11)\nare the coupling strength of the FQ and NV with magnon\nmodekin the YIG, respectively. Here, \u00160is the vacuum\npermeability, Ipis the persistent current of the FQ, rF(rN) is the distance between a spin in the YIG and the\nFQ (the NV spin), Nis the number of the spins in the\nYIG, and\u0012rNis the angle between the vector connecting\nthe NV and the spin in YIG, and the direction of the\nexternal magnetic \feld. Under the condition that the\nHeisenberg interaction inside the YIG is much greater\nthan the coupling between a qubit and any single spin\nin the YIG, the qubit then e\u000bectively interacts with the\ncollective mode of all the spins in the YIG. The more\nspins in the YIG following this condition, the stronger\nthe coupling.\nIf one chooses a YIG sphere with a submicrometer di-\nameter, then the energy levels of the YIG magnon modes\nare gapped, largely due to its small size. We consider\nonly the simplest mode of the YIG, i.e., the KM [13, 15],\nwith frequency !Kwhich is far from both the frequen-\ncies of the FQ and NV-center storage qubit. Thus the\nKM is in a virtual coupling regime with the FQ and NV-\ncenter storage qubit. To account for the overall e\u000bect,\nwe use the Schrie\u000ber-Wol\u000b transformation (SWT) to de-\nrive the e\u000bective Hamiltonian up to the second order in\nthe coupling strengths with the YIG by averaging out\nthe far-o\u000b-resonance degrees of freedoms of the YIG. We\nthen obtain (with detailed derivation shown in Appendix\nB) an e\u000bective Hamiltonian between the FQ and the NV-\ncenter qubit as\nHe\u000b'1\n2!F;e\u000b\u001b(z)\nF+1\n2!N;e\u000b\u001b(z)\nN\n+gFN;e\u000b\u0010\n\u001b(+)\nF\u001b(\u0000)\nN+H:C:\u0011\n; (12)\nwhere\n!F;e\u000b=!F+\u000eF; (13)\n!N;e\u000b=!N+\u000eN; (14)\nare the e\u000bective frequencies of the FQ and the NV-center\nstorage qubit with frequency shifts \u000eF=g2\nFY(!K)\n!F\u0000!Kand\n\u000eN=g2\nY N(!K)\n(!N+\u000eY N)\u0000!Kinduced by the KM of the YIG, re-\nspectively, and\ngFN;e\u000b(!K) =1\n2gFY(!K)gYN(!K)\n\u0002\u00141\n!F\u0000!K+1\n(!N+\u000eYN)\u0000!K\u0015\n(15)\nis the e\u000bective coupling strength between the FQ and\nthe NV-center qubit. Here, !Kdenotes the frequency of\nthe KM. Note that to have a substantially large e\u000bective\ncoupling, the detuning ( !F\u0000!K) and detuning ( !N+\n\u000eYN\u0000!K) appearing in the denominator of Eq. (15)\nshould not be too large. On the other hand, to keep\nthe KM in a regime of virtual coupling with the FQ and\nNV-center storage qubit, they should not be too small.\nBy varying the external magnetic \feld, we can control\nthe values of ( !F\u0000!K) and (!N\u0000!K). In particular,\nthe change in !K, due to the variation of the magnetic4\nFigure 2. Energy-level diagram of the FQ (left) and the\nground triplet states of the NV-center spin (right). The NV-\ncenter spin states j\u00061Niare degenerate (dashed line) and have\na zero-\feld splitting \u0001 ZSand an energy shift \u000eNinduced by\nthe indirect coupling scheme via YIG relative to state j0Ni\n(see red arrow). By tuning the magnetic \feld on the NV-\ncenter spin to the value of B=Bres, one can control the tran-\nsition betweenj0Niandj\u00001Nito be resonant with the FQ\n(see blue arrow) or to be o\u000b-resonant at B=Bo\u000b. The tran-\nsition betweenj0Niandj1Niis always set to be o\u000b-resonant\nwith the FQ, i.e., is set to be a disconnected channel.\n\feld, is opposite to the change of the energy di\u000berence\nbetweenj0Niandj\u00001Ni[in contrast to the same energy\nchange betweenj0Niandj1Niresulting in no change in\n(!(+1)\u0000!K)]. This makesj0Niandj\u00001Nia better choice\nof storage qubit basis states.\nIII. QUANTUM MEMORY\nNext we will use the derived e\u000bective Hamiltonian to\ninvestigate the dynamics and the \fdelity of the proposed\nquantum memory scheme. There are two stages that we\nneed to consider: state transfer stage and state storage\nstage.\nTo better assess and calculate the \fdelity of our\nscheme, we take all the lowest triplet states of the NV\nspin into account. In this case, the FQ couples to two\ntransitions in these triplets separately and the e\u000bective\nHamiltonian, given by Eq. (12), becomes\nHe\u000b=1\n2!F;e\u000b\u001b(z)\nF+X\nj=\u00061!N;(j);e\u000b(B)jjNihjNj\n+g(+1)h\n\u001b(+)\nFS(\u0000)\nN;(+1)+H:C:i\n+g(\u00001)h\n\u001b(+)\nFS(\u0000)\nN;(\u00001)+H:C:i\n; (16)\nHere in the spin-1 Hilbert space of the NV center, the\ne\u000bective NV spin frequencies are !N;(\u00061);e\u000b(B) = \u0001 ZS\u0007\n\reB+\u000eN;(\u00061), where \u0001 ZSis the zero-\feld splitting and\n\u000eN;(\u00061)is the frequency shift. The e\u000bective couplingstrengths between the FQ and the NV spin transitions\nare denoted as g(\u00061), corresponding to Eq. (15) with\n!N!!N;(\u00061);e\u000b(B). The subscripts ( \u00061) in the ex-\npression (and in the following), stand for the transitions\nbetweenj0Niandj\u00061Ni, respectively. The operators\nS(\u0006)\nN;(\u00061)with superscript\u0006denote the raising and lower-\ning operators, respectively.\nNow we move to the interaction picture through\nthe unitary transformation U= exp(\u0000itH0;e\u000b), where\nH0;e\u000b=1\n2!F;e\u000b\u001b(z)\nF+P\nj=\u00061!N;(j);e\u000b(Bres)jjNihjNjis\nthe \frst two terms of the e\u000bective Hamiltonian given by\nEq. (16) with magnetic \feld B=Bres, whereBresis the\nmagnetic \feld strength applied to the NV-center spin\nwhen the transition between j0Niandj\u00001Nimatches\nthe energy gap of the FQ (see Fig. 2). Then the e\u000bective\ninteraction Hamiltonian Hintbecomes\nHint=HN;int+HFN;int; (17)\nHN;int=X\nj=\u00061\u000eB;(j)jjNihjNj; (18)\nHFN;int=g(\u00001)h\n\u001b(+)\nFS(\u0000)\nN;(\u00001)+H:C:i\n;\n+g(+1)h\n\u001b(+)\nFS(\u0000)\nN;(+1)e2it\reBres+H:C:i\n;(19)\nand\n\u000eB;(\u00061)=\u0007\re(B\u0000Bres): (20)\nWhen\u000eB;\u00001is tuned to zero, i.e., B=Bres, theg(\u00001)cou-\npling terms start to transfer the quantum state from the\nFQ to the NV-center spin and the fast oscillating com-\nponents in the g(+1)terms can be e\u000bectively neglected.\nWe initially prepare the NV in the ground state,\nj N(0)i=j0Ni. Suppose that the FQ is in a general\nstate characterized by angles \u0012and\u001e. Then the joint\nstate is\nj (0)i=\u0000\ncos\u0012j1Fi+ei\u001esin\u0012j0Fi\u0001\n\nj0Ni\n= cos\u0012j1F;0Ni+ei\u001esin\u0012j0F;0Ni: (21)\nAfter a transfer time t=\u0019=(2g(\u00001)), the target state in\nthe interaction picture becomes\nj (t)i=\u0000icos\u0012j0F;\u00001Ni+ei\u001esin\u0012j0F;0Ni\n=j0Fi\n\u0000\n\u0000icos\u0012j\u00001Ni+ei\u001esin\u0012j0Ni\u0001\n:(22)\nOnce the state has been transferred to the NV-center\nspin, we turn o\u000b the coupling e\u000bectively by enlarging\nthe mismatch of the frequencies between the FQ and the\nNV-center storage qubit. The quantum state can thus\nbe stored for better coherence with dephasing time char-\nacterized by the NV-center spin's T2time. To retrieve\nthe state from the NV-center storage qubit to the FQ,\nwe tune to the NV-FQ resonance again. After a time\nt1=\u0019=(2g(\u00001)), the original state is restored in the FQ\ndegrees of freedom\nj (tf)i=\u0000e\u0000i\u001escos\u0012j1F;0Ni+ei\u001esin\u0012j0F;0Ni;(23)5\nwith an additional phase \u001esthat comes from the coherent\nevolution during the storage time t2and since this is\nknown it can be corrected.\nIV. RESULTS AND DISCUSSION\nWe present the numerical results together with discus-\nsions to verify our scheme here. Before proceeding with\nour numerical calculations for the \fdelity performance,\nwe \frst describe the system parameters used. It is as-\nsumed that the YIG is a sphere of radius about 45 nm\nand contains about 106spins. A local magnetic \feld\nBLis generated by placing a micromagnet [19] of size\n0:2\u00020:2\u00020:2\u0016m3with a uniform perpendicular magne-\ntization of about a hundred Gauss at a vertical distance\n\u001825{50 nm from the YIG. With this local magnetic \feld,\nthe frequency of the KM can reach GHz levels; further-\nmore, by varying an external magnetic \feld the frequency\ndi\u000berence (!F\u0000!K) can achieve a typical value of about\n170 MHz. Since the direction of BLis parallel to the\nplane of the FQ, the FQ is insensitive to BL. Further-\nmore, because the transverse distance from the micro-\nmagnet boundary at which BLdrops to 0, is smaller than\n0.1\u0016m [19], ifrFis considerably larger than the sum of\nthis transverse distance and the distance from the YIG\nto the left boundary of the micromagnet (see Fig. 1),\nwe can realize local magnetic \feld control for the YIG\nwithout disturbing the FQ. Therefore, by choosing a rel-\natively large value of rF\u00180.25\u0016m and using a FQ with\nIp=500 nA and a diamond with a single NV spin at a\ndistancerN\u001860 nm from the YIG, we can estimate the\ne\u000bective coupling strength gFN;e\u000bto be about 350{700\nkHz according to Eqs. (A12), (A14), and (B14). The fre-\nquency shifts of both the FQ and NV-center spin due to\nthe YIG coupling [see Eqs. (B8) and (B9)] are at about\nhundreds kHz, and are thus rather small and negligible in\ncomparison with their own frequencies in the GHz range.\nFollowing the exposition of the system parameters, we\nnow continue to show the numerical results and we plot\nthe dynamics of the transfer process in Fig. 3. We choose\nthe case where the e\u000bective coupling strength is 700 kHz\nand where the initial state is\f\f\b1=2\u000b\n=p\n1=2j1F;0Ni+p\n1=2j0F;0Nisince this state is, in a more realistic situa-\ntion considered later, in\ruenced the most by the dephas-\ning e\u000bect. During the transfer process (whose duration\nis 0.36\u0016s),j1F;0Niis transferred to j0F;\u00001Ni, while\nj0F;0Niis left unchanged. The population of the other\nstates remains zero, except that the j0F;1Nistate has a\nsmall probability ( \u001810\u00007) as shown in Fig. 3(b). This\nsmall probability is due to the detuning between the tran-\nsition frequency from j0Nitoj1Niand the frequency of\nthe FQ, and one can reduce this probability further by\nmaking the detuning larger.\nTo simulate the state transfer and storage processes\nin a more realistic setting, we use the master equation\n[22], which takes into account both spontaneous decay\nFigure 3. (a) Dynamics of the probabilities of the basis states\nof the quantum memory during a state transfer process for the\ninitial state\f\f\b1=2\u000b\n=p\n1=2j1F;0Ni+p\n1=2j0F;0Ni, and cou-\npling strength 700 kHz. During the transfer process, j1F;0Ni\nis transferred toj0F;\u00001Ni, whilej0F;0Niis unchanged. (b)\nDue to the detuning between the j0Nitoj1Nitransition and\nthe FQ transition frequency, state j0F;1Nihas a negligible\nprobability (\u001810\u00007) during the process.\nand pure dephasing of the FQ and the NV-center spin,\nd\u001a\ndt=\u0000i[He\u000b;\u001a]\n+\r(s)\nF\n2L[\u001b(\u0000)\nF] +\r(p)\nF\n2L[j1Fih1Fj]\n+X\nj=\u00061\r(s)\nN;(j)\n2L[S(\u0000)\nN;(j)] +\r(p)\nN;(j)\n2L[jjNihjNj];(24)\nwhereL[O] = 2O\u001aOy\u0000\u001aOyO\u0000OyO\u001a, is the Lindblad\nsuperoperator, and \r(s)\nqand\r(p)\nqare the spontaneous de-\ncay and the dephasing rates, respectively, of the species q.\nNote that they are related to the relaxation time T1and\ndecoherence time T2asT1=1=\r(s)andT2=2=(\r(s)+2\r(p)),\nrespectively [22]. It has been reported recently that both\nthe intrinsic T\u0003\n1andT\u0003\n2of FQ's could be about 10 \u0016s at 33\nmK [23] and the value of the intrinsic T\u0003\n2of an NV-center\nspin could be about 90 \u0016s at room temperature [24]. Fur-\nthermore, it has been shown that by applying dynamical\ndecoupling pulse sequences, T1of ensembles of NV spins\ncould be more than 10 sec and T2could be about 0.6 sec\nat 77 K [25]. In the simulations, the relevant decoherence\ntimes in the state transfer stage are the intrinsic T\u0003\n1and\nT\u0003\n2times of the FQ and the NV-center spin. In the state\nstorage stage, however, the NV-center spin is e\u000bectively\ndecoupled from the YIG and FQ due to the large detun-\ning and thus dynamical decoupling pulse sequences can\nbe applied to protect the NV from decoherence to main-\ntain the transferred state. Consequently, we can use the\nT1andT2values of the NV-center spin measured using\ndynamical decoupling [25] to estimate the \fdelity in the\nstate storage stage. Furthermore, the decoherence times6\nof a NV center solely due to a coupling to a ferromagnet\nYIG have been estimated in Ref. [17], and theses times\ndepend sensitively on the ratio of the magnon excitation\ngap to the YIG temperature. It has been shown that for\na magnon gap of 100 \u0016eV and a temperature of 0.1 K,\nthese times are typically much larger than the (intrinsic)\ndecoherence times of the NV [17]. In other words, the\ninduced decoherence solely due to coupling to the YIG\nfor temperatures smaller than the magnon excitation gap\nis nondetrimental [17]. In our scheme, all of the compo-\nnents (the FQ, YIG, and NV) of the qubit and quantum\nmemory are at the same low temperature as that of the\nFQ. Moreover, because of the small size of the YIG and\nthe applied magnetic \feld, this ratio of the magnon ex-\ncitation gap to the temperature in our scheme is even\nbigger than that used for estimation in Ref. [17]. As a\nresult, the e\u000bect of the induced decoherence solely due to\ncoupling to the YIG will be neglected in our simulations.\nThe e\u000bect of the linewidth of the YIG nanosphere can\nalso be neglected. The linewidths of the KM of a single\nYIG sphere with submillimeter size have been measured\nto be about 1{2 MHz [12, 13]. Although linewidth mea-\nsurement on a single YIG nanosphere is, to our knowl-\nedge, not available so far, a close case of YIG nanodisks\nwith thickness about 20 nm and diameter ranging from\n\u0018300{700 nm has been reported in Ref. [26]. There, the\nlinewidth of the uniform mode, i.e., the lowest-energy\nferromagnetic resonance mode of a nanodisk, was mea-\nsured to be about 7 MHz for a nanodisk with diame-\nter 700 nm at frequency 8.2 GHz. However, since this\nmeasurement was performed at room temperature and\nthe nanodisks with di\u000berent diameters are arranged in a\nrow with 3 \u0016m spacing (i.e., not completely a single-disk\nmeasurement), one may expect that the linewidth could\nbe narrower if the measurements were performed for a\nactual single nanodisk at a low temperature of tens of\nmK and at the frequency down to the value of \u00182 GHz\nas in our proposal. Furthermore, the linewidths of the\nnanodisks do not change much with the diameter [27],\nat least within the range investigated in Ref. [26]. One\nmay thus expect that the linewidth of a YIG nanosphere\nwithout surface defects at low temperature could be sim-\nilar or at most at a few MHz level, which is still much\nsmaller than the detuning ( \u0015170 MHz) between the YIG\nnanosphere and other quantum systems in our proposal.\nConsequently, the e\u000bect of the linewidth of the KM of the\nYIG nanosphere does not appreciably a\u000bect the virtual\nexcitation or virtual coupling picture in our proposal and\nthus is neglected in the subsequent calculations after the\ndegrees of freedoms of the YIG are traced out.\nWe then take the values of the decoherence and relax-\nation times at higher temperatures [24, 25] to make a\nconservative evaluation of the performance of our quan-\ntum memory scheme through the \fdelity of the state\nF=p\nh\tj\u001aj\ti; (25)\nwherej\tiis the target state and \u001ais the actual system\ndensity matrix. We can transform the Hamiltonian to therotating frame to obtain Hintas in Eqs. (17){(19). Since\nduring the storage stage the system is tuned to be o\u000b-\nresonant, i.e., \u000eB;(\u00001)\u001dg(\u00001), the total system approx-\nimately undergoes free evolution during this stage. The\n\fdelities of the quantum state memory for initial states\nj\b1iand\f\f\b1=2\u000b\nare shown in Table I, in which results\nthat make use of more conservative values for T\u0003\n2= 20\u0016s\nfor the NV center are also presented. The initial states\nj\b1iand\f\f\b1=2\u000b\nare chosen because they are in\ruenced\nthe most by the spontaneous decay and dephasing e\u000bect,\nrespectively. We have also simulated for di\u000berent ini-\ntial states ofj\b0i=j0F;0Ni,\f\f\b1=3\u000b\n=p\n2=3j1F;0Ni+p\n1=3j0F;0Ni,\f\f\b1=4\u000b\n=p\n3=4j1F;0Ni+p\n1=4j0F;0Ni,\nand\f\f\b1=5\u000b\n=p\n4=5j1F;0Ni+p\n1=5j0F;0Ni, and the re-\nsult shows thatj\b1ihas the worst \fdelity. This is be-\ncause during the transfer stage, the main factor caus-\ning in\fdelity is the decoherence of the FQ, and T\u0003\n1and\nT\u0003\n2of the FQ is about the same in our case so that the\nspontaneous decay rate is larger than the dephasing rate.\nFurthermore, in the transfer stage, switches take place\nbetweenj\b1i=j1F;0Niandj0F;\u00001Ni, while the state\nj\b0i=j0F;0Niis unchanged. When the portion of j\b1i\nin a general initial state decays into j0F;0Ni, the state\ntransfer process of that portion will stop and will cause\nin\fdelity. This results in the initial state j\b1ibeing the\nworst possible case for the parameters we used. Never-\ntheless, if the e\u000bective coupling is stronger through the\nuse of a YIG moderator containing more spins or if the\nFQ possesses a longer coherence time [28], the \fdelity\ncan be appreciably enhanced.\nIn our scheme, the state transfer interaction can be\ne\u000bectively turned on and o\u000b depending on whether or\nnot the NV storage qubit is resonant with the FQ. We\nthus can attempt to consider engineering a near-perfect\nstep function of the external magnetic \feld \u000eBfrom 0\nG (o\u000b) to 80 G (on). We choose the maximum mag-\nnetic \feld strength such that it is still lower than the\ncritical \feld of the FQ (the critical \feld is 100 G for\nFQ made of aluminum; could be higher if made of other\nsuperconductor)[29]. However, due to technical limits on\nthe charging and discharging times of circuits, the ramp-\ning of the magnetic \feld cannot be instantaneous and\nwe take this rise time into account. We assume a vari-\nation of the magnetic \feld over 200 G in 10 ns, similar\nto what has been reported in experiments [30]. In our\nsimulation, \u000eBis switched from 0 to 80 G with either a\nlinear or exponential ramping over a duration of 4 and\n10 ns (see Fig. 4). The results shown in Table II indi-\ncate that the linear ramping has slightly lower \fdelity\nthan the exponential ramping. This is due to the fact\nthat the linear ramping makes the system stay in the\nnear-resonance regime longer and thus subject to FQ de-\ncoherence longer. Shorter rise times of the magnetic \feld\ncan also improve the \fdelity or, alternatively, one can\n\fne tune the transfer time to correct the rise-time and\nfall-time e\u000bects.7\nTable I. Fidelity at di\u000berent steps for the initial states of j\b1i=j1F;0Niand\f\f\b1=2\u000b\n=p\n1=2j1F;0Ni+p\n1=2j0F;0Niwith the\nstorage time 10 ms.\nInitial state g(eff)\nF\u0000N T\u0003\n2 F(FQ!NV) F(Storage) F(NV!FQ)\n\f\f\b1=2\u000b700 kHz90\u0016s 0.9689 0.9598 0.9318\n20\u0016s 0.9627 0.9548 0.9218\n350 kHz90\u0016s 0.9421 0.9363 0.8880\n20\u0016s 0.9307 0.9270 0.8709\nj\b1i700 kHz90\u0016s 0.9317 0.9284 0.8653\n20\u0016s 0.9268 0.9239 0.8562\n350 kHz90\u0016s 0.8695 0.8668 0.7537\n20\u0016s 0.8608 0.8581 0.7386\nTable II. Fidelity F=p\nh tj\u001aj tiafter the transfer process with di\u000berent lengths and types of the rise times for the initial\nstates ofj\b1i=j1F;0Niand\f\f\b1=2\u000b\n=p\n1=2j1F;0Ni+p\n1=2j0F;0Ni.\nState g(e\u000b)\nF\u0000N T\u0003\n2 rise-time func. F(4 ns) F(10 ns)\n\f\f\b1=2\u000b700 kHz90\u0016sexponential 0.9677 0.9647\nlinear 0.9672 0.9639\n20\u0016sexponential 0.9613 0.9581\nlinear 0.9608 0.9573\n350 kHz90\u0016sexponential 0.9412 0.9395\nlinear 0.9410 0.9392\n20\u0016sexponential 0.9296 0.9278\nlinear 0.9295 0.9275\nj\b1i700 kHz90\u0016sexponential 0.9294 0.9241\nlinear 0.9288 0.9228\n20\u0016sexponential 0.9246 0.9193\nlinear 0.9240 0.9180\n350 kHz90\u0016sexponential 0.8678 0.8648\nlinear 0.8677 0.8646\n20\u0016sexponential 0.8591 0.8561\nlinear 0.8590 0.8559\nFigure 4. Temporal variation of the magnetic \feld with linear\nor exponential ramping from o\u000b-resonance (0 G) to resonance\n(80 G) in 4 ns at the beginning, and using an inverse ramping\nat the end of the state transfer stage. Only the rise-time\nand fall-time regimes are shown and the storage stage, during\nwhich the magnetic \feld is \fxed, is not shown.V. CONCLUSION\nWe have demonstrated how to couple a superconduct-\ning FQ with a single electron spin of a NV-center spin via\na collective KM in a ferromagnetic material, YIG. This\nscheme enhances the e\u000bective coupling between the FQ\nand the NV-center spin, allowing the single-NV-center\nspin to interact with the FQ at a longer spatial distance.\nThis provides greater \rexibility for the design of hybrid\nquantum systems. We have proposed a protocol for quan-\ntum state memory and presented a quantitative analysis\nof the state transfer, taking into consideration the possi-\nble decay channels and imperfect technical issues. This\nYIG architecture can be used not only as a quantum\nmemory but also as a quantum transducer that couples\na single-NV-center spin with other kinds of qubits or with\na magnetic \feld.\nACKNOWLEDGMENTS\nG.D.L. and H.S.G. acknowledge support from the\nthe Ministry of Science and Technology of Taiwan un-8\nder Grants No. MOST 105-2112-M-002-015-MY3 and\nNo. MOST 106-2112-M-002-013-MY3, from the Na-\ntional Taiwan University under Grant No. NTU-CCP-\n106R891703, and from the thematic group program of the\nNational Center for Theoretical Sciences, Taiwan. J.T.\nacknowledges support from the Center of Excellence in\nEngineered Quantum Systems.\nAppendix A: Coupling with magnons\nHere we describe how the magnons in YIG couple to\nthe FQ or the single-NV spin, and give a detailed deriva-\ntion of their coupling strengths. The Hamiltonian of a\nFQ can be written as HF=1\n2!F\u001b(z)\nF+1\n2\u000f\u001b(x)\nF;where!F\nis the energy of the tunnel splitting, \u000f= 2Ip(\b\u0000\b0=2) is\nthe energy bias, Ipis the persistent current of the FQ, \b 0\nis the \ruxon, and \b is the external \rux. If \b is tuned to\nthe optimal point with \b = \b 0=2 so that\u000f= 0, then one\nhasHF=1\n2!F\u001b(z)\nF. As discussed in the main text, the\nFQ-YIG coupling comes from the Zeeman-like interac-\ntion of the spins in the YIG experienced in the magnetic\n\feldBF(r) produced by the persistent current of the FQ\n(see Fig. 1). Since the persistent current carried by the\nside wire of the FQ loop near the YIG is along the z\naxis, the magnetic \feld BF(r) =\u0000\u00160\n2\u0019r\u0001\nIp\u001b(x)\nFgenerated\nby this persistent current is in the xaxis of the YIG [18].\nThen the coupling Hamiltonian reads\nHFY=\u0000X\nr\reBF(rF)\u0001Sr\n=\u0000X\nr\u0012\u00160\reIp\n2\u0019rF\u0013\n\u001b(x)\nFS(x)\nr; (A1)\nwhererFis the distance between a spin in the YIG and\nthe FQ. The YIG-NV coupling Hamiltonian through the\ndipole-dipole interaction is written as\nHYN=\u0000\u00160\n4\u0019\r2\neX\nrN3 (Sr\u0001^rN) (\u001bN\u0001^rN)\u0000(Sr\u0001\u001bN)\n2r3\nN:\n(A2)\nDue to the external magnetic \feld Bapplied in the z\ndirection, which makes the frequencies !Nand\reBmuch\nlarger than the dipole-dipole coupling strength, we can\napply the secular approximation to rewrite Eqs. (A2) as\nHYN=\u0000\u00160\n8\u0019\r2\ne~X\nr\u00123cos2\u0012rN\u00001\nr3\nN\u0013h\n3S(z)\nr\u001b(z)\nN\u0000Sr\u0001\u001bNi\n;\n(A3)\nwhererNis the distance between a spin in the YIG and\nthe NV spin, and \u0012rNis the angle between the vector,\nwhich connects the NV and the spin in YIG, and the\ndirection of the external magnetic \feld. We can rewritethis Hamiltonian as HYN'H0\nYN+H(z)\nYN, where\nH0\nYN=X\nrN\u0012\u00160\r2\ne~\n4\u0019\u0013\u00123cos2\u0012rN\u00001\nr3\nN\u0013\u0010\nS(+)\nr\u001b(\u0000)\nN+S(\u0000)\nr\u001b(+)\nN\u0011\n;\n(A4)\nH(z)\nYN=\u0000X\nrN\u0012\u00160\r2\ne~\n4\u0019\u0013\u00123cos2\u0012rN\u00001\nr3\nN\u0013\nS(z)\nr\u001b(z)\nN:\n(A5)\nMagnons are low-energy spin-wave excitations, which\nare used to describe the collective behavior of the spins\nin YIG. Since the system of the quantum memory is at a\ntemperature much lower than the Curie temperature of\nYIG,Tc=559 K, and the YIG is in an o\u000b-resonant cou-\npling regime to the FQ and NV-center spin, one expects\nthe excitation number to be very small. In this low-\ntemperature and low-excitation regime, it is convenient\nto use the Holstein-Primako\u000b transformation [17, 20],\nS(z)\nr=\u0000s+ay\nrar\u0019\u0000s; (A6)\nS(\u0000)\nr=p\n2sr\n1\u0000nr\n2sar\u0019p\n2sar; (A7)\nS(+)\nr=\u0010\nS(\u0000)\nr\u0011y\n; (A8)\nto transform the spin operators in the YIG into bosonic\noperators, where each operator is associated with a par-\nticle coordinate. Using the creation and annihilation op-\nerators of the magnon modes in the wave-vector repre-\nsentation,\nay\nk=1p\nNX\nre\u0000ik\u0001ray\nr; (A9)\nak=1p\nNX\nreik\u0001rar; (A10)\none can then rewrite the Hamiltonian of YIG as Eq. (6).\nSince the coupling strength in HFYis far smaller\nthan!Fand!Y, it is valid to use the RWA and\nEqs. (A6){(A10) to rewrite the FQ-YIG coupling Hamil-\ntonian, given by Eq. (A1), as\nH0\nFY=\u0000X\nr\u0012\u00160\reIp\n2\u0019rF\u0013\u0010\n\u001b(+)\nFS(\u0000)\nr+H:C:\u0011\n\u0019\u0000X\nkh\ngFY(k)\u001b(+)\nFak+H:C:i\n; (A11)\nwith the coupling strength\ngFY(k) =\u00160\n2\u0019\reIpr\n2s\nNNX\nrFe\u0000ik\u0001a\nrF: (A12)\nSimilarly, the YIG-NV coupling reads\nH0\nYN'\u0000X\nkh\ngYN(k)ay\nk\u001b(\u0000)\nN+H:C:i\n(A13)9\nwhere\ngYN(k) =\u0000\u00160\n4\u0019\r2\ne~r\n2s\nNNX\nrN\u00123cos2\u0012rN\u00001\nr3\nN\u0013\neik\u0001a;\n(A14)\nis the coupling strength between the YIG and NV-center\nspin. Using Eq. (A6), one can also rewrite Eq. (A5) as\nH(z)\nYN'\u000eYN\u001b(z)\nN; (A15)\nwhere\u000eYN=P\nrN\u00160\n4\u0019\r2\ne~\u00103cos2\u0012rN\u00001\nr3\nN\u0011\nsis the induced\nenergy shift to the NV-center storage qubit due to the\ncoupling with the YIG.\nAppendix B: Derivation of the e\u000bective Hamiltonian\nby the Schri\u000ber-Wol\u000b transformation\nHere we describe here the procedure to derive the e\u000bec-\ntive Hamiltonian between the the FQ and the NV-center\nspin. Following Schri\u000ber and Wol\u000b's approach [31{33],\nwe can make a canonical transformation e\u0000\u0011on our orig-\ninal Hamiltonian H=H0+Hc, whereH0=Hs+H(z)\nYN\nwithHsde\fned in Eq. (1), H(z)\nYNde\fned in Eq. (A15),\nandHcde\fned in Eq. (7). The Hamiltonian after the\ntransformation reads\n~H=e\u0011He\u0000\u0011\n=H+ [\u0011;H] +1\n2![\u0011;[\u0011;H]] +\u0001\u0001\u0001: (B1)\nBy choosing proper operator \u0011satisfying [H0;\u0011] =Hc,\none has\n~H'H0+1\n2[\u0011;Hc]: (B2)\nIn our case,\n\u0011= lim\n\u0015!0\u0014\n\u0000i\u00021\n0Hc(t)e\u0000\u0015tdt\u0015\n; (B3)\nand\n~H'H0\u0000lim\n\u0015!0i\n2\u00021\n0[Hc(t);Hc]e\u0000\u0015tdt; (B4)whereHc(t) =eiH0tHce\u0000iH0t. Since the size of the YIG\nis small, the KM of YIG [13, 15] is gapped from the\nhigher-energy modes. Thus, in a low-temperature and\nvirtual-excitation regime of our scheme, we consider only\nthe KM of the YIG to mediate the e\u000bective coupling\nstrength between the FQ and the NV-center spin. To\nobtain the e\u000bective Hamiltonian between the FQ and\nthe NV-center spin without considering the detailed dy-\nnamics of the YIG, we trace out the degrees of freedoms\nof the YIG , i.e., He\u000b=h~HiY, withD\nay\nKaKE\nY=nK\nbeing the mean occupation number of the KM (similar\nto a mean-\feld approximation). This is a good approx-\nimation as the YIG is in a low-temperature and low-\nexcitation regime. Since Hc=H0\nFY+H0\nYN, we can\nrewrite Eq. (B4) by categorizing the terms of the com-\nmutators after a trace over the YIG degrees of freedom\ninto two types,\nHe\u000b\u0011H0\u0000(\u000eHs+\u000eHc) (B5)\nThe \frst type in Eq. (B5) reads\n\u000eHs= lim\n\u0015!0i\n2\u00021\n0h[H0\nFY(t);H0\nFY] + [H0\nYN(t);H0\nYN]iYe\u0000\u0015tdt\n(B6)\n=1\n2\u000eF\u001b(z)\nF+1\n2\u000eN\u001b(z)\nN; (B7)\nwhere\n\u000eF=g2\nFY(!K)\u00121\n!F\u0000!K\u0013\n; (B8)\n\u000eN=g2\nYN(!K)\u00141\n(!F+\u000eYN)\u0000!K\u0015\n; (B9)\nwith!Kdenoting the frequency of the KM, are the en-\nergy shifts of the qubits of the FQ and NV systems, re-\nspectively. We demonstrate how to derive Eq. (B7) from\nEq. (B6) by calculating the term containing \u000eFexplicitly,\nand then the other term containing \u000eNcan be obtained\nin a similar way. The commutator in the \frst term of\nEq. (B6) considering only the KM is10\nh[H0\nFY(t);H0\nFY]iY=g2\nFYD\n[\u001b(+)\nF(t)aK(t);\u001b(\u0000)\nFay\nK] + [\u001b(\u0000)\nF(t)ay\nK(t);\u001b(+)\nFaK]E\nY\n=g2\nFYD\nei(!F\u0000!K)th\n\u001b(\u0000)\nF\u001b(+)\nF+\u001b(z)\nF\u0010\nay\nKaK+ 1\u0011i\n+e\u0000i(!F\u0000!K)th\n\u0000\u001b(\u0000)\nF\u001b(+)\nF\u0000\u001b(z)\nF\u0010\nay\nKaK+ 1\u0011iE\nY\n=g2\nFYn\nei(!F\u0000!K)th\n\u001b(\u0000)\nF\u001b(+)\nF+\u001b(z)\nF(nK+ 1)i\n+e\u0000i(!F\u0000!K)th\n\u0000\u001b(\u0000)\nF\u001b(+)\nF\u0000\u001b(z)\nF(nK+ 1)io\n: (B10)\nThen, integrating it over time as in Eq. (B6), one obtains\nlim\n\u0015!0i\n2\u00021\n0h[H0\nFY(t);H0\nFY]iYe\u0000\u0015tdt=i\n2g2\nFYlim\n\u0015!0\u00021\n0n\nei(!F\u0000!K)th\n\u001b(\u0000)\nF\u001b(+)\nF+\u001b(z)\nF(nK+ 1)i\n+e\u0000i(!F\u0000!K)th\n\u0000\u001b(\u0000)\nF\u001b(+)\nF\u0000\u001b(z)\nF(nK+ 1)io\ne\u0000\u0015tdt\n=g2\nFYlim\n\u0015!0\u00121\n!F\u0000!K+i\u0015\u0013\u00141\n2(2nK+ 1)\u001b(z)\nF+1\n2IF\u0015\n=g2\nFY\u00121\n!F\u0000!K\u0013\u00141\n2(2nK+ 1)\u001b(z)\nF+1\n2IF\u0015\n'1\n2g2\nFY\u00121\n!F\u0000!K\u0013\n\u001b(z)\nF; (B11)\nwhere the constant energy term containing the identity operator IFcan be ignored, and nK!0 since the system is\noperated in a virtual-excitation regime. Equation (B11) is the \frst term of Eq. (B7). Similarly, the second term of\nEq. (B6) can be calculated and yields the term containing \u000eNin Eq. (B7).\nThe other type \u000eHcin Eq. (B5) represents the e\u000bective coupling between the FQ and NV-center spin:\n\u000eHc= lim\n\u0015!0i\n2\u00021\n0h[H0\nFY(t);H0\nYN] + [H0\nYN(t);H0\nFY]iYe\u0000\u0015tdt (B12)\n=gFN;e\u000b\u0010\n\u001b(+)\nF\u001b(\u0000)\nN+H:C:\u0011\n; (B13)\nwith\ngFN;e\u000b=1\n2gFY(K)gYN(K)\u00141\n!F\u0000!K+1\n(!N+\u000eYN)\u0000!K\u0015\n: (B14)\nWe now show how to obtain Eq. (B13) from Eq. (B12). Following the same approach as in Eqs. (B10) and (B11), the\n\frst term of the commutator in Eq. (B12) reads\nlim\n\u0015!0i\n2\u00021\n0h[H0\nFY(t);H0\nYN]iYe\u0000\u0015tdt= lim\n\u0015!0i\n2\u00021\n0gFYgYND\n[\u001b(+)\nF(t)aK(t);ay\nK\u001b(\u0000)\nN] + [\u001b(\u0000)\nF(t)ay\nK(t);aK\u001b(+)\nN]E\nYe\u0000\u0015tdt\n=i\n2gFYgYNlim\n\u0015!0\u00021\n0h\nei(!F\u0000!K)t\u0010\n\u001b(+)\nF\u001b(\u0000)\nN\u0011\n+e\u0000i(!F\u0000!K)t\u0010\n\u0000\u001b(\u0000)\nF\u001b(+)\nN\u0011i\ne\u0000\u0015tdt\n=gFYgYNlim\n\u0015!0\u00141\n2\u00121\n!F\u0000!K+i\u0015\u0013\n\u001b(+)\nF\u001b(\u0000)\nN+1\n2\u00121\n!F\u0000!K\u0000i\u0015\u0013\n\u001b(\u0000)\nF\u001b(+)\nN\u0015\n=1\n2gFYgYN\u00121\n!F\u0000!K\u0013\u0010\n\u001b(+)\nF\u001b(\u0000)\nN+\u001b(\u0000)\nF\u001b(+)\nN\u0011\n: (B15)\nThe second term of the commutator in Eq. (B12) can be evaluated in a similar way and combining with Eq. (B15)\ngive the results of Eqs. (B13) and (B14). Combining all these results, one arrives at the e\u000bective Hamiltonian of\nEq. (12).\n[1] J. Clarke and F. K. 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Loss, Annals of\nPhysics 326, 2793 (2011)."    },    {        "title": "2107.09363v1.Spin_wave_dispersion_measurement_by_variable_gap_propagating_spin_wave_spectroscopy.pdf",        "content": "Spin-wave dispersion measurement by variable-gap propagating\nspin-wave spectroscopy\nMarek Va\u0014 natka,1,\u0003Krzysztof Szulc,2Ond\u0014 rej Wojewoda,1Carsten Dubs,3Andrii Chumak,4\nMaciej Krawczyk,2Oleksandr V. Dobrovolskiy,4Jaros law W. K los,2and Michal Urb\u0013 anek1, 5,y\n1CEITEC BUT, Brno University of Technology, 612 00 Brno, Czech Republic\n2ISQI, Faculty of Physics, Adam Mickiewicz University, Pozna\u0013 n, Poland\n3INNOVENT e.V. Technologieentwicklung, 07745 Jena, Germany\n4Faculty of Physics, University of Vienna, A-1090 Wien, Austria\n5Institute of Physical Engineering, Brno University of Technology, 616 69 Brno, Czech Republic\n(Dated: July 21, 2021)\nMagnonics is seen nowadays as a candidate technology for energy-e\u000ecient data processing in\nclassical and quantum systems. Pronounced nonlinearity, anisotropy of dispersion relations and\nphase degree of freedom of spin waves require advanced methodology for probing spin waves at room\nas well as at mK temperatures. Yet, the use of the established optical techniques like Brillouin light\nscattering (BLS) or magneto optical Kerr e\u000bect (MOKE) at ultra-low temperatures is forbiddingly\ncomplicated. By contrast, microwave spectroscopy can be used at all temperatures but is usually\nlacking spatial and wavenumber resolution. Here, we develop a variable-gap propagating spin-wave\nspectroscopy (VG-PSWS) method for the deduction of the dispersion relation of spin waves in wide\nfrequency and wavenumber range. The method is based on the phase-resolved analysis of the spin-\nwave transmission between two antennas with variable spacing, in conjunction with theoretical data\ntreatment. We validate the method for the in-plane magnetized CoFeB and YIG thin \flms in k?B\nandkkBgeometries by deducing the full set of material and spin-wave parameters, including\nspin-wave dispersion, hybridization of the fundamental mode with the higher-order perpendicular\nstanding spin-wave modes and surface spin pinning. The compatibility of microwaves with low\ntemperatures makes this approach attractive for cryogenic magnonics at the nanoscale.\nI. INTRODUCTION\nProperties of magnetic materials are of high interest\ndue to several application concepts regarding, e.g., mem-\nories, sensors, microwave devices, or logic devices [1].\nIn the emerging \feld of magnonics, which utilizes spin-\nwaves for data transport and processing, the essential\nsystem characteristic is the spin-wave dispersion relation.\nIt provides a connection between the k-space and fre-\nquency space, and it also dictates other properties like\nthe group velocity vgand decay length \u0003. In thin \flms\n(approx. below 30 nm), only the fundamental mode is\nobserved when the experimentally accessible frequency\nrange is limited to few GHz. In contrast, the spin-wave\ndispersion of thicker \flms can be rather complex as mul-\ntiple perpendicularly standing modes may appear in the\nspectrum, exhibiting frequency crossing and hybridiza-\ntion [2]. The spin-wave dispersion measurement is typi-\ncally done by k-resolved [2, 3] or phase-resolved [4, 5] Bril-\nlouin Light Scattering (BLS). Current interest in quan-\ntum computing, quantum magnonics [6] and in super-\nconductor/ferromagnet hybrid systems[7, 8], rises the\nneed for material characterization at ultra-low tempera-\ntures, where optical access is typically extremely compli-\ncated. All electrical measurements are usually preferable\nin these applications.\n\u0003marek.vanatka@live.com\nymichal.urbanek@ceitec.vutbr.czThe spin-wave dispersion measurement is also possible\nusing the propagating spin-wave spectroscopy (PSWS).\nThe PSWS is a technique which uses a vector network\nanalyzer (VNA) connected to a pair of microwave an-\ntennas (e.g., striplines or coplanar waveguides) by mi-\ncrowave probes [9, 10]. The two antennas (i.e., spin-\nwave transmitter and receiver) have a gap between them\nover which the spin-waves propagate, as shown in Fig.\n1(a). Transmitting antenna is powered by the VNA's\nmicrowave source and the receiving antenna serves as an\ninduction pick-up detected by the VNA's second port.\nThe antenna type determines the excitation properties\nand can be adjusted to the experiment. Main three types\nof antennas used in experiments are striplines (rectangu-\nlar wires), U-shaped ground-signal (GS) antennas, and\ncoplanar waveguide (CPW) antennas. Schematic geom-\netry of all three antenna types is shown in Fig. 1 (b).\nStriplines provide a continuous spectrum where the max-\nimum excited k-vector is limited by the stripline width.\nCiubotaru et al. [11] showed scalability of the antennas,\nwhere 125 nm wide striplines provided a wide continu-\nousk-vector band. Good alternatives are GS antennas,\ne.g. [9, 12, 13], or coplanar waveguides (CPW), e.g. [14{\n17], both providing a \fltering capability for the k-vector\nspectrum allowing only speci\fc ranges to exist. PSWS\ncan be used on both nanostructured materials (stripes)\n[11{15, 18, 19] as well as layers [17, 20]. It was previously\nshown that the PSWS signal can be negligibly di\u000berent\nfor continuous layers and wide stripes [17]. PSWS signals\ncan also be modeled [15, 17, 21].\nIn previous reports, the spin-wave dispersion was ex-arXiv:2107.09363v1  [cond-mat.mes-hall]  20 Jul 20212\ntracted from PSWS spectra measured on yttrium iron\ngarnet (YIG) using the CPW excitation. As the CPWs\nexcitation spectrum exhibits distinct peaks in k-space, it\nallows extracting one point in spin-wave dispersion for\neach peak. The central k-vector of each peak is then as-\nsigned to a frequency from either the envelope of the S21\nsweep [18, 22] or by \ftting the S21spectrum [17]. This\napproach is limited to only several extracted points, and\nit is not easily transferable to metallic materials because\nof the low signal amplitude (compared to YIG) caused\nby large damping, making it impossible to use more than\ntwo peaks from the CPW antenna's excitation spectrum.\nHere, we show that the spin-wave dispersion measure-\nment using VNA is possible with a high level of details\ndetermined by the VNA frequency step.\nII. MEASUREMENT SETUP AND SAMPLE\nPREPARATION\nOur setup uses Rohde & Schwarz ZVA50 VNA and\nGGB industries microwave probes to establish a con-\nnection to the two antennas lithographically fabricated\non top of CoFeB and YIG thin \flms. The lithogra-\nphy process consisted of e-beam patterning using PMMA\nresist, e-beam evaporation of Ti 5 nm/Cu 85 nm/Au\n10 nm multilayer, and lift-o\u000b. The CoFeB \flms (nom-\ninal thicknesses 30 nm and 100 nm) were magnetron-\nsputtered from Co 40Fe40B20(at. %) target on (100) GaAs\nsubstrate with 5 nm Ta bu\u000ber layer. The (111) YIG \flms\nwere grown by liquid phase epitaxy on top of a 500 \u0016m,\nthick (111) gadolinium gallium garnet substrate [23, 24].\nThe samples were placed in a gap of a rotatable electro-\nmagnet allowing to apply an in-plane magnetic \feld up\nto 400 mT in an arbitrary direction with respect to the\nspin-wave propagation direction. The VNA was set us-\ning a calibration substrate supplied with the microwave\nprobes. A power sweep was performed before measur-\ning each type of the sample-antenna combination to \fnd\na suitable power level that avoids nonlinear phenomena\n[25] and maintains a su\u000ecient signal-to-noise ratio.\nThe VNA controls and analyses the electric signals of\ntransmitting and receiving antennas both in the domain\nof amplitude and phase, therefore, it can measure the\nphase acquired by the spectral components of spin wave\nwhile it propagates between antennas. Our analysis is\nbased on the S21transmission parameter. The trans-\nmitted spin-wave signal is modi\fed by a nonmagnetic\nbackground Sbackground\n21 that is always present in the ex-\nperiment due to direct electromagnetic crosstalk between\nthe antennas. This background is constant for di\u000berent\nvalues of static magnetic \feld, and therefore it is possible\nto evaluate it as the median over all measured magnetic\n\felds. The subtracted signal \u0001 S21is then calculated as:\n\u0001S21=S21\u0000Sbackground\n21 .\nFig. 1 (c,d) shows the \u0001 S21signal, measured at 0 dBm\npower output for a 30 nm CoFeB thin \flm, over the gap\n1.8\u0016m in the k?Bgeometry (magnetostatic surfacewaves), and with 500 nm wide striplines used as excita-\ntion and detection antennas. This geometry is known to\nbe nonreciprocal with an exponential distribution of the\ndynamic magnetization along the layer's thickness due to\nthe surface localization of the mode [26, 27]. The higher\nsignal amplitude in the + Bpart of the spectrum is caused\nby both the stronger excitation and by the larger induc-\ntion pick up from spin-waves propagating at the nearer\nsurface. To achieve the best result, we can focus on + B\npart of the spectrum (or alternatively on the \u0000Bpart\nand use the reverse transmission parameter S21). Fig.\ngap\nkPort1Port2GND\nGNDGND\nGNDtransmi\u0000ng an tenna\nreceiving an tennavariable(a)\nGND\nGNDGND\nGNDGND\nGNDGND\nGNDGNDGND\nVNA port 1 VNA port 2\nVNA\nport 2VNA\nport 1VNA\nport 1\nVNA\nport 2striplines: GS antennas: CPW antennas(b) antenna types\nk\nk k\nRe(         ) Im(         )\nFIG. 1. (a) Schematics of the PSWS experiment using a\npair of stripline antennas. (b) Schematic geometry of the\nthree most commonly used antenna types. The outcomes of\nthe PSWS measurement: (c) real and (d) imaginary part of\n\u0001S21(B;f) for 30 nm thick CoFeB layer 30 nm in k?B\ngeometry and the gap width of 1 :8\u0016m. The measurement\nshows the nonreciprocity of spin-wave propagation for the k?\nBgeometry which is re\rected by the larger signal in + B\felds\nthan in\u0000B\felds. The plots of (e) show real and imaginary\nparts of \u0001S21(f), and (f) the unwrapped phase of \u0001 S21(f),\nat \fxed \feld B= 20 mT.3\n1(d) shows the plot of real and imaginary parts of \u0001 S21\nmeasured at 20 mT. The corresponding phase, which was\nunwrapped, is shown in Fig. 1(e). The phase rises from\nthe ferromagnetic resonance (FMR) frequency up until\nit reaches the antenna's excitation limits. Beyond this\npoint, the signal loses its coherency due to insu\u000ecient\nsignal-to-noise ratio, and therefore the phase stops evolv-\ning. The slope of the phase depends on the gap size { it\nchanges more rapidly for wider gaps.\nIn the next step, we repeat the measurement on multi-\nple instances of identical antenna structures (with vary-\ning gap width) prepared on the same 30 nm CoFeB thin\n\flm sample. The phases measured over 11 gap widths\nare shown in Fig. 2(a). The phases are on the same level\nbefore the frequency reaches FMR, and then they start\nto rise. The lowest phase corresponds to the smallest\ngap width and the uppermost to the largest measured\ngap width. Then we project the measured phases into a\nphase { gap width plot [selected frequencies are plotted\nin Fig. 2(b)]. In this projection, the phase shows a lin-\near dependence (for a coherent plane wave) that can be\n\ftted; the slope equals to the k-vector at the given fre-\nquency. Now we can plot the extracted k-vectors against\ntheir frequencies, showing the resulting dispersion rela-\ntion in Fig. 2(c). To con\frm the result, we remeasured\nthe same sample using phase-resolved BLS, and we found\na very good agreement. The comparison of the dispersion\nrelations measured by both techniques (VNA { red, BLS\n{ blue) is shown in Fig. 2(c), and the comparison of the\nmeasured phase evolution at the frequency of 11.5 GHz\nis shown in Fig. 2(d).\nIII. MEASURED SPIN-WAVE DISPERSION\nRELATIONS\nFig. 3(a) shows dispersion relations of the same 30 nm\nCoFeB thin \flm measured in k?Bgeometry in mag-\nnetic \felds ranging from 20 mT to 380 mT with a step of\n60 mT. The sample was also measured in kkBgeom-\netry, but the measured signal was insu\u000ecient to recon-\nstruct the dispersion relation. For all measured \felds,\nit was possible to evaluate the dispersion for k-vectors\nranging from 0 up to 8 rad/ µm. The upper limit is given\nby the excitation e\u000eciency [15] of the used antenna (see\nblue lines in Fig. 3). By \ftting the measured dispersions\nusing the Kalinikos-Slavin model [28], we were able to\nobtain material parameters of the measured thin \flms\n(see black lines in Fig. 3 for the model \fts and the \fgure\ncaption for the \ftting results).\nIn addition to the 30 nm CoFeB thin \flm, we also mea-\nsured 100 nm thick CoFeB [Fig. 3(b)] and 100 nm thick\nYIG [Fig. 3(c)] \flms to further explore the possibili-\nties of the presented technique. The measurements were\nperformed using di\u000berent antenna types (stripline, GS,\nCPW) to see their in\ruence on the quality of the obtained\ndispersions, and they were also evaluated for multiple\nmagnetic \felds in k?Band in kkBgeometries. The\n5 10 15\nf (GHz)04 :8 :12 :Arg( \"S21) (rad)\n7GHz\n8GHz\n9GHz\n10GHz\n11GHz\n12GHz\n13GHz\n14GHz\n15GHzB=20mT\n2 4\ngap size (µm)04 :8 :Arg( \"S21) (rad)\n7GHz8GHz9GHz10GHz11GHz12GHz13GHz14GHz15GHz\n0 5\nk (rad/µm)51015f (GHz) VNA\nBLS\n1 2 3\ngap size (µm)02 :4 :Phase (rad)f=11.5GHz\nVNA\nBLS(a) (b)\n(c) (d)FIG. 2. Extraction of the dispersion relation from the out-\ncomes of PSWS experiment on 30 nm CoFeB thin \flm at\nB= 20 mT in k?Bgeometry. (a) The unwrapped \u0001 S21\nphases measured over gap widths ranging from 0 :9\u0016m (the\nlowest line) to 2 :9\u0016m (the steepest line) with the step of\n200 nm. (b) The representative \fts of the phase at selected\nfrequencies versus gap size where the slope of the \ft yields the\ndesiredk-vector at that frequency. (c) The dispersion relation\nextracted for all frequencies within the range with su\u000ecient\nPSWS signal (red line). The dispersion obtained by phase-\nresolved BLS is plotted in the blue points for comparison. (d)\nThe compared outcomes of the phase measurements for VNA\n(red circles) and phase-resolved BLS (blue squares) at single\nfrequencyf= 11:5 GHz.\n100 nm thick CoFeB \flm was measured in the \felds rang-\ning from 20 to 380 mT with a step of 60 mT at a power\nof 0 dBm. In Fig. 3(b) we show the dispersion measured\nusing 500 nm wide stripline antenna for k?Bgeometry\nand with 500 nm coplanar waveguide (signal and ground\nline widths, as well as signal-to-ground gap, were 500 nm)\nink?Bgeometry. We were able to obtain a dispersion\ninkkBgeometry also using a 500 nm wide stripline\nantenna but with substantially worse quality due to its\nlower excitation e\u000eciency. The 100 nm thick YIG \flm\nwas measured in k?BandkkBgeometries in the\n\felds ranging from 20 to 200 mT with a step of 20 mT\nat a power of\u000030 dBm. In this case, it was possible to\nobtain a dispersion in both geometries using all types of\nantennas. Fig. 3(c) shows data acquired by GS antennas\nwith gap widths from 1.0 to 3.4 \u0016m with 400 nm step.\nAs in the case of the coplanar waveguide, the dimensions\nof the GS antennas were 500 nm (signal and ground line\nwidths and signal-to-ground gap width). The dispersions\nare plotted for magnetic \felds from 20 to 200 mT with a\nstep of 20 mT and the power output was set to \u000030 dBm.\nIn the dispersion measured on 100 nm thick CoFeB\n\flm, we further identi\fed visible hybridizations at the\ncrossings' positions between the n= 0 fundamental spin-\nwave mode and n= 1;2;3 higher-order perpendicularly4\n0 5 10\nk (rad/µm)5101520253035f (GHz)\nJexc (normalized)\n-10 -5 0 5 10\nk (rad/µm)10203040f (GHz)\nJexc (normalized)\n-10 -5 0 5 10\nk (rad/µm)2468f (GHz)\nJexc (normalized)(a) (b) (c)\n20 mT\n20 mT20 mT380 mT\n260 mT\n140 mT 380 mT\n260 mT\n140 mT140 mT\n100 mT180 mT\n60 mTP=0 dBm P=0 dBm P=-30 dBm\nFIG. 3. The dispersion relations measured using the PSWS technique (red points) compared to the analytic dispersions\nfrom Kalinikos-Slavin model (black lines) for (a) 30 nm thick CoFeB (b) 100 nm thick CoFeB, (c) 100 nm thick YIG layers.\nFor clarity, the data for k?Bare plotted in the domain of positive wave numbers kand the kkBdata are presented\nwith negative k. The samples were excited by stripline, CPW, and GS antennas with all lateral wire widths dimensions of\n500 nm. The blue lines show calculated (and normalized) wave number excitation spectra of the antennas, calculated form\nthe spatial distributions of microwave \feld (the solid, dashed and dotted lines correspond to the spectra of stripline, GS, and\nCPW antenna, respectively). For Kalinikos-Slavin model we \ftted the following values of parameters: (a) Ms= 1230 kA/m\n(\u00160Ms= 1:54 T),\r=2\u0019= 30:5 GHz/T,t= 29:0 nm; (b)Ms= 1310 kA/m ( \u00160Ms= 1:64 T),\r=2\u0019= 30:1 GHz/T; (c) Ms= 133\nkA/m (\u00160Ms= 0:166 T),\r=2\u0019= 28:3 GHz/T, where tstands for thickness of the layer. In \ftting procedure, we \fxed the\nexchange sti\u000bness to Aex= 15 pJ/m3for CoFeB \flms and Aex= 3:6 pJ/m3for the YIG \flm.\nstanding spin-wave modes (see Fig. 4). The hybridiza-\ntions are present in the measured dispersion relation as\ngap openings, with a portion of the measured data evolv-\ning towards smaller k-vectors [see Fig. 4(d) for the most\nprominent example]. This part of the dispersion has no\nphysical meaning and is only a result of the data process-\ning described earlier in the text. The hybridizations are\nalso directly visible as dips in the magnitude Abs(\u0001 S21)\nof the transmission spectra [see Fig. 4(b)].\nIV. NUMERICAL MODELING\nTo investigate numerically the hybridization between\nthe fundamental mode and perpendicularly standing\nmodes, we solve Landau-Lifshitz equation (LLE) using\nthe \fnite-element method (FEM) in the frequency do-\nmain. The LLE was linearized and implemented in FEM\nsolver (COMSOL Multiphysics) as a set of di\u000berential\nequations for in-plane and the out-of-plane components\nof magnetization mkandm?:\ni!mk=j\rj\u00160[(H0\u00002Aex\n\u00160Ms\u0001)m?+Ms@x?'];(1)\ni!m?=j\rj\u00160[(H0\u00002Aex\n\u00160Ms\u0001)mk+Ms@xk'];(2)\ntogether with the equation for the scalar magnetostatic\npotential'derived from Maxwell equations in magneto-\nstatic approximation [29, 30]:\n\u0001'\u0000@xkmk\u0000@x?m?= 0; (3)\nwherexkandx?denote the directions of the correspond-\ning dynamic components of magnetization.In contrast to the analytical model of Kalinikos and\nSlavin, our numerical calculations truly reproduce the\nspin dispersion relation in the crossover dipolar-exchange\nregime [2], including the hybridizations between the fun-\ndamental spin-wave mode and perpendicularly standing\nspin-wave modes. In our model, we also consider the\npresence of surface anisotropy Ks. The anisotropy is ex-\npected to be di\u000berent on the bottom and the top faces\nof the magnetic layer interfaced with di\u000berent materi-\nals. The surface anisotropy and its asymmetry (between\nthe top and bottom face) is responsible for the spin-wave\npinning and the strength of the hybridization between\nfundamental and perpendicularly standing modes. The\npinning is implemented in the boundary conditions [31]\n@x?mk= 0; (4)\nAex@x?m?\u0000Ksm?= 0: (5)\nThe numerical model (as well as the analytical model\nby Kalinikos and Slavin) is characterized by a few ma-\nterial parameters. We \fxed the value of exchange sti\u000b-\nness toAex= 14 pJ/m2and layer's thickness to nom-\ninal valued= 100 nm. The values of saturation mag-\nnetization Ms= 1275 kA/m and gyromagnetic ratio\nj\rj=2\u0019= 30:8 GHz/T were selected to \ft the ferromag-\nnetic resonance frequency (i.e., the frequency of funda-\nmental mode at k= 0) and the slope of the dispersion\nfor fundamental mode. The surface anisotropy Ksde-\ntermines the spin-wave pinning and therefore is impor-\ntant both for the quantization (and the frequencies) of\nthe perpendicularly standing modes and the strength of\nthe hybridization between and perpendicularly standing\nmodes. To obtain the proper values of the frequencies5\nof perpendicularly standing modes, we did not need to\nreduce the thickness below the nominal value 100 nm but\nwe had to introduce the non-zero Ksinstead, which seems\nto be a more realistic approach. The experimental data\nshow that the hybridizations of fundamental mode with\nthe perpendicularly standing mode are observed both for\nthe perpendicularly standing modes quantized with even\n(n= 2) and odd number of nodes ( n= 1;3) across the\nlayer. The e\u000bect for n= 1;3 is hardly visible in disper-\nsion relation [Fig. 4(c,e)] but is quite distinctive in the\nmeasurement of the transmission amplitude [Fig. 4(b)].\nThese results indicate that the asymmetric pinning (and\ndi\u000berent values of surface anisotropy on both faces of the\nCoFeB layer) must be considered to obtain less symmet-\nric pro\fles of spin-wave modes across the thickness of the\nlayer, which in turn, gives the non-zero cross-sections be-\ntween fundamental mode and perpendicularly standing\nmodes of odd numbers of nodes. There is always some\ndegree of freedom for choosing the values of Kson both\nfaces, therefore, we decided to consider the simplest case\nwhereKs= 0 and 1600 \u0016J/m2at the top (interface with\nvacuum) and bottom (interface with GaAs). With these\nvalues, we succeeded in the induction of the hybridiza-\ntion of fundamental mode ( n= 0) with the \frst ( n= 1)\nand third (n= 3) perpendicularly standing modes, char-\nacterized by the width comparable to the widths of the\ndeeps of the transmission \u0001 S12[orange stripes in Fig.\n4(a,b)]. It is worth noting that in the case of symmetric\npinning (we took Ks= 700 \u0016J/m2on both faces), we do\nnot observe the hybridization with the \frst ( n= 1) and\nthird (n= 3) perpendicularly standing modes [see blue\ndashed lines in Fig. 4(c,e)].\nV. ADDITIONAL DATA ANALYSIS AND\nDISCUSSION\nFrom the measured dispersion relations, it is possi-\nble to extract also other parameters important for spin-\nwave propagation. The group velocity vgcan be cal-\nculated as the numerical derivative of the dispersion\nvg= 2\u0019df=dkand the lifetime \u001ccan be obtained as the\nnumerical derivative of \feld-dependent dispersion [29]:\n\u001c= (\u000b!@!\n@!B)\u00001= (\u000b4\u00192f\n\r@f\n@B)\u00001, where!B=\rBand\u000b\nis a damping constant.\nThe decay length \u000ecan be then calculated by multiply-\ning the group velocity and the lifetime \u000e=vg\u001c. In Fig.\n5(a), we plot the decay length obtained by the above-\ndescribed procedure (orange line) and compare it with\nthe decay length obtained using the more traditional ap-\nproach, i.e., by \ftting the exponential decay of the mag-\nnitude of \u0001 S21signal (blue line). Here, the decay length\n\u000ewas obtained by \ftting [see Fig. 5(b) for representa-\ntive \fts] the formula Abs(\u0001 S21) =Ie\u0000g=\u000ein logarithmic\nform ln[Abs(\u0001 S21)] = ln(I)\u00001\n\u000eg, wheregis the gap\ndistance, and Iis a free parameter proportional to sig-\nnal strength. In Fig. 5(c), we plot \feld-dependent life-\ntime evaluated using the decay length obtained from the\n0 0.5\nk (rad/µm)21222324f (GHz)\n11.5 22.5\nk (rad/µm)242526f (GHz)\n5 6\nk (rad/µm)282930f (GHz)-3\nAbs(         ) (x10)0 2 4 6 8\nk (rad/µm)202530f (GHz)\n0 5(a)\n(e) (d) (c)(b)\nB=260 mTn=1n=2n=3\nn=0\n01-detail 02-detail 03-detailexperiment mod. asym. pinning mod. sym. pinningFIG. 4. (a) Measured dispersion relation (red points) for\n100 nm thick CoFeB layer at 260 nm exhibiting hybridized\nspin-wave modes, visible mainly at the n= 02 crossing. The\ndashed line represents dispersion simulation with asymmet-\nrical surface pinning conditions with parameters KS= 0\nand 1600 \u0016J/m2on top and bottom face of the layer, re-\nspectively. (b) Corresponding magnitude of the \u0001 S21trans-\nmission parameter exhibiting dips at the crossings' positions\n([Abs(\u0001S21)] data shown for the gap width of 1.8 \u0016m). (c-e)\ndetails of the 01, 02, and 03 crossings, respectively. Dashed\nblue lines correspond to the case of symmetric pinning ( KS=\n700\u0016J/m2on both faces).\nderivative of \feld-dependent dispersion ( @!=@!B, orange\ncircles) and from \ftting the exponential decay of the mag-\nnitude of \u0001 S21signal (blue diamonds). Both approaches\ngive roughly the same results for both the decay length\nand the lifetime, but the decay lengths obtained using the\n@!=@!Bapproach agree better with the analytical model\n[see Fig. 5(a), green line]. The lower quality of the data\nobtained from \ftting the exponential decay of the mag-\nnitude of \u0001 S21signal is caused by the limitation of our\nexperimental arrangement. Due to the \fnite length of the\nexcitation antenna, spin-wave caustics can form [32]. An\nexample of such caustics measured by BLS microscopy is\nshown in Fig. 5(e). The phase in the caustic beam is spa-\ntially incoherent [33] and the focusing e\u000bect [34] causes\nmodulation of the spin-wave intensity along the propa-\ngation direction. The associated adverse e\u000bects can be\navoided by measuring at propagation distances that are\nsu\u000eciently small with respect to the stripline length. In\nour experiments, we used the stripline length of 10 \u0016m,\nand the maximum measured propagation distance was\n2.9\u0016m. For phase-resolved measurement such as evalu-\nation of dispersions, the short propagation distance was\nsu\u000ecient. On the other hand, the change in signal at\n2.9\u0016m propagation distance was not su\u000ecient to obtain\na fully reliable \ft of the exponential decay of the \u0001 S21\nmagnitude and at longer propagation distances the mea-\nsurements were distorted by caustics.\nWe also analyzed how the number of measured propa-6\ngation distances (gaps) a\u000bected the reliability of the ob-\ntained dispersion. We took the data measured for 11 gap\nwidths in total and then used di\u000berent combinations of a\nreduced number of gap widths (starting from 2 up to 10)\nto evaluate the dispersion. The dispersion obtained from\nthe reduced number of gap widths was then compared to\nthe analytical \ft Kalinikos-Slavin model [28] of the dis-\npersion obtained from \ftting of the complete set of 11\nmeasured points. As shown in Fig. 5(d), the mean fre-\nquency di\u000berence from the reference dispersion can rise\nto the maximum of 400 MHz when using a combination\nof just two gap widths. This maximum quickly decreases\nto 200 MHz for \fve gap widths and then stays around\n100 MHz for combinations of six and more gaps.\nAs can be seen from the presented data, the variable-\ngap approach allows the reconstruction of the spin-wave\ndispersion relation, including detailed features that might\nbe used for analysis of the spin-wave system. In order to\nobtain the highest possible data quality, we need to fabri-\ncate multiple pairs of antennas where the gap width and\nthe step in the gap width (an increase of the gap between\ntwo pairs of antennas) must be optimized for each exper-\niment. Note that this approach does not require precise\nknowledge of the excitation spot location (phase origin)\nbecause it does not a\u000bect the \ft's slope. It is only essen-\ntial to know the relative di\u000berences between the propa-\ngation distances, i.e., the gap-width step, which can be\neasily measured (and precisely fabricated using e-beam\nlithography).\nThe antenna design is sample-speci\fc and needs to\nbe tailored to ful\fl experimental requirements. The an-\ntenna's type and geometry must be able to excite the\nexpectedk-vector range with su\u000ecient e\u000eciency.\nThe gap widths need to be in the optimum range con-\nsidering the caustics formation and the spin-wave decay\nlength for the given material and geometry. For example,\nfor CoFeB 30 nm in kkBgeometry at k= 5 rad/ \u0016m,\nwe calculated decay lengths of 0.03 \u0016m and 0.45 \u0016m for\nmagnetic \felds of 20 and 200 mT, respectively. Here, the\nquality of the measured data, even for the smallest fabri-\ncated gap distance of 1 \u0016m, was not su\u000ecient to evaluate\ndispersion; thus, this geometry is not presented in Fig.\n3(a).\nBefore \ftting the data as plotted in Fig. 2(b), the\nphase needs to be correctly unwrapped. In the case of\nstripline antennas with a continuous excitation spectrum,\nit is possible to achieve correct unwrap even when the\nphases for neighboring gap widths at the same frequency\nare higher than \u0019rad by unwrapping the phase in the fre-\nquency spectrum [Fig. 2(a)]. It is because the frequency\nstep size of the VNA is usually small and the phase shift\nof the neighboring points is always smaller than \u0019rad.\nOn the other hand, the safest approach is to unwrap the\nphases when plotted against the gap size [Fig. 2(b)]. In\nthis case, the phase change of neighboring points must be\nsmaller than \u0019rad. This is necessary for antenna types\nwith discrete excitation spectra (i.e., CPW, GS, ladders\n[42] or meanders [43]).\nmeas. error\nerror rangef=14 GHz\nf=16 GHz\nf=18 GHzB=100 mT\nk=2 r ad/µm\nB\nBLS intensity (counts)VNA signalGND\nGNDstripline 1\nstripline 2(e)FIG. 5. The determination of decay length and lifetime\nfrom VG-PSWS measurement of 30 nm thick CoFeB \flm (a-\nc,e) and the impact of the number of gaps on the quality\nof the result (d). (a) Blue line shows the spin-wave decay\nlengths obtained directly from exponential \fts, orange line\nshows the decay length calculated by multiplication of the\nlife time and the group velocity and the green line is calcu-\nlated from the Kalinikos-Slavin model. (b) Representative\nexponential \fts of the \u0001 S21magnitude which were used to\nobtain the spin-wave propagation length. (c) Spin-wave life\ntime extracted from the derivative of \feld-dependent disper-\nsion using\u000b= 0:075 (orange circles), calculated by dividing\nthe propagation length (from \u0001 S21magnitude) by the numer-\nically calculated group velocity (blue diamonds) and calcu-\nlated from the Kalinikos-Slavin model (green line). (d) Mean\nfrequency di\u000berence of the dispersion obtained from the re-\nduced number of gap widths compared to the analytical \ft of\nthe dispersion obtained from the complete set of 11 measured\npoints. Blue points represent di\u000berent combinations of gaps\nand the red area represents the di\u000berence spread. (e) Spin-\nwave caustics caused by the \fnite length of the antenna, mea-\nsured on 100 nm CoFeB \flm by BLS with exciation frequency\nof 11 GHz in the external magnetic \feld of 26 mT. Caustics\ncan destroy the phase coherence of propagating spin-waves.\nNote that all presented data were collected at propagation\ndistances (gap widths) between 0.9 \u0016m to 2.9 \u0016m to avoid the\nnegative e\u000bects of the caustics formation.7\nTABLE I. Comparison of experimental techniques used for measurement of spin-wave dispersion relations.\nLateral Resolution Resolution Phase max k Experimental\nresolution in k(rad/ \u0016m) inf(Hz) extraction (rad/ \u0016m) geometries\nin (nm) (MS, BV, FV)\nVG-PSWS [this work] - <0:1* 1 yes 10*** MS, BV, (FV possible)\nPSWS [17, 21] - 2*** 1 no 10*** MS, BV, FV\nMOKE [35, 36] 500 <0:1* 1\u0001108** yes 10 BV, MS\nConventional BLS [2, 37] 10,000 0.5 1 \u0001108no 23.6 BV, MS\nPhase-resolved BLS [4, 38, 39] 250 <0:1* 1\u0001108** yes 7 BV, MS\nSTXM [40, 41] 20 <0:1* 1\u0001108** yes 30 BV, MS\nMS - magnetostatic surface spin waves, BV - backward volume spin waves, FV - forward volume spin waves\n* Thek-resolution can be tailored to the magnetic system under study by an appropriate design of the experiment geometry.\n**The resolution in fin these techniques is limited by the acquisition time.\n*** Excitation limited.\nThe step in the gap width de\fnes the maximum k-\nvector for which the dispersion can be measured. The\nphase change of the spin wave along the step distance\nhas to be smaller than \u0019rad except for certain cases\ndiscussed later. On the other hand, small k-vectors may\nhave a phase change that is too small over a short step\nin the gap width. If we require an accurate \ftting of\nsmallk-vectors, the step in the gap width should be large\nenough to provide su\u000ecient phase change (we suggest at\nleast 1 rad) while respecting the decay length.\nVI. METHOD COMPARISON\nIn Table 1, we compare the variable gap propagat-\ning spin-wave spectroscopy with other experimental tech-\nniques used to obtain spin-wave dispersion relations. To\nget detailed dispersion, one needs to have high resolu-\ntion in both the k-vector and the frequency. Optical and\nX-ray techniques have low frequency resolution, which\nis typically limited to hundreds of MHz. In the case of\nconventional BLS, the resolution is determined directly\nby the Fabry-Perot interferometer [44]. The optical tech-\nniques using microwave excitation, i.e., Magneto-Optical\nKerr E\u000bect microscopy (MOKE), Phase-resolved BLS\nand Scanning Transmission X-ray Microscopy (STXM),\nare limited by slow signal acquisition. It does not allow\ncapturing the required span of frequencies with a high\nresolution in a reasonable time. The propagating spin-\nwave spectroscopy technique, which uses known positions\nof the excitation peaks of the CPW antenna, can cap-\nture the data with a high resolution in frequency. How-\never, thek-resolution is limited with the \fnite widths of\npeaks in the excitation spectra of the used CPW anten-\nnas which in turn a\u000bect also the frequency resolution.\nBig advantage of the VG-PSWS technique is that it\ndoes not require direct optical access to the sample, mak-\ning the method very suitable for, e.g., experiments at\nultra-low temperatures. In addition, compared to other\ntechniques for spin-wave dispersion relation measurement\nVG-PSWS can achieve high resolution in frequency andk-vector at the same time. This combination allows cap-\nturing smooth dispersion curves over the span of all ac-\ncessiblek-vectors and fast acquisition times allow repeat-\ning the experiment in multiple magnetic \feld strengths\nand orientations. The full set of \feld-dependent disper-\nsion curves measured for both k?Band in kkBge-\nometries present a robust 4-dimensional ( f,k,B, angle)\ndataset that can be further evaluated, and all essential\nmaterial and spin-wave parameters can be extracted from\nit.\nA disadvantage of this method is the need for a set of\nantennas with multiple gap widths on top of the sample.\nThe other techniques can obtain the dispersion either\nwithout any antenna (conventional BLS), with one exci-\ntation antenna (MOKE, Phase-resolved BLS, STXM) or\nwith a pair of excitation and detection antennas (PSWS).\nHowever, the need for multiple antennas may be over-\ncome in the future by using freestanding positionable an-\ntennas [45].\nVII. CONCLUSION\nIn conclusion, we presented a new method of extrac-\ntion of high-quality spin-wave dispersion relation from\npropagating spin-wave spectroscopy (PSWS) measure-\nments performed over several propagation distances. We\ndemonstrated this technique on CoFeB and YIG thin\n\flms measured in k?BandkkBgeometries. The re-\nsults on CoFeB thin \flm were veri\fed by phase-resolved\nBLS measurement showing good agreement. When\ncompared with the phase-resolved BLS, the VNA-based\nmethod provides more frequency measurement points in\na shorter acquisition time. Fine detail measurement ca-\npability was demonstrated on the measurement and anal-\nysis of hybridized modes acquired on 100 nm thick CoFeB\nthin \flm, revealing asymmetric surface pinning and the\nvalues of pinning parameters on both interfaces of the\nmagnetic layer. The all electric nature of this method\nmakes it very suitable for characterization of cryogenic\nand quantum magnonics systems and materials.8\nACKNOWLEDGMENTS\nThe work was supported by MEYS CR (project\nCZ.02.2.69/0.0/0.0/19 073/0016948). CzechNanoLab\nproject LM2018110 is gratefully acknowledged for the\n\fnancial support of the measurements and sample\nfabrication at CEITEC Nano Research Infrastruc-\nture. O.W. was supported by Brno PhD talentscholarship. J.W.K. and M.K. acknowledge the sup-\nport of the National Science Centre - Poland for\nthe projects UMO-2020/37/B/ST3/03936 and UMO-\n2020/39/O/ST5/02110. O.V.D. acknowledges the Aus-\ntrian Science Fund (FWF) for support through Grant No.\nI 4889 (CurviMag). C.D. gratefully acknowledges \fnan-\ncial support from the Deutsche Forschungsgemeinschaft\n(DFG, German Research Foundation) - 271741898.\n[1] B. Dieny, I. L. Prejbeanu, K. Garello, P. Gambardella,\nP. Freitas, R. Lehndor\u000b, W. Raberg, U. Ebels, S. O.\nDemokritov, J. Akerman, A. Deac, P. Pirro, C. Adel-\nmann, A. Anane, A. V. Chumak, A. Hirohata, S. Man-\ngin, S. O. Valenzuela, M. C. Onba\u0018 sl\u0010, M. d'Aquino,\nG. Prenat, G. Finocchio, L. Lopez-Diaz, R. Chantrell,\nO. Chubykalo-Fesenko, and P. 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By adjusting an out-of-plane magnetic\nfield, we demonstrate linear center frequency tuning for a 4th-\norder filter from 4.5 GHz to 10.1 GHz while retaining a fractional\nbandwidth of 0.3%, an insertion loss of 6.94 dB, and a -35dB\nrejection. We characterize the filter nonlinearity in the passband\nand stopband with IIP3 measurements of -4.85 dBm and 25.84\ndBm, respectively. When integrated with a tunable magnetic field,\nthis device is an octave tunable narrowband channel-select filter.\nIndex Terms —Micromachining, magnetostatic wave (MSW),\nyttrium iron garnet (YIG), tunable bandpass filter, edge-coupled\nI. I NTRODUCTION\nCOUPLED micro-electromechanical resonators with high\nquality factors (Q-factor) and miniaturized footprints\nhave been an attractive technology for integration in wireless\ncommunication systems as narrowband channel-select filters.\nWang et al [1] has demonstrated high-order micromechan-\nical bandpass filters using one-dimensional (1D) arrays of\nmechanically-coupled resonators. However, for higher-order\nfilters, long mechanical coupling beams become impractical\ndue to size constraints [2] while sensitivity due to fabrica-\ntion variation causes increased insertion loss and passband\ndistortion [3]. The high sensitivity of weak electrostatic or\nmechanical edge-coupled resonators due to structural asym-\nmetries has been leveraged for sensitive parametric mass\nsensing applications in [2], but prohibits their use in filters. 2D\nmicroresonator arrays have shown some success by utilizing\nweak coupling in one dimension to achieve pass band shape\nand strong coupling in the other dimension to reduce effects\nof fabrication variation, but suffer from high insertion loss\nManuscript received on XX XX, 2023; revised on XX XX, 2024; accepted\non XX XX, 2024. This research was developed with funding from the Air\nForce Research Laboratory (AFRL) and the Defense Advanced Research\nProjects Agency (DARPA). The views, opinions and/or findings expressed are\nthose of the authors and should not be interpreted as representing the official\nviews or policies of the Department of Defense or the U.S. Government.\nThis manuscript is approved for public release; distribution A: distribution\nunlimited. ( Corresponding authors: Connor Devitt, Renyuan Wang )\nR.W. invented the device concept, and developed baseline model and device\ndesign. C.D. performed simulations on fabrication variations and total filter\nloss, chip fabrication and characterization, as well as data analysis. Manuscript\nwas prepared by C.D. with inputs from R.W., S.T., and S.A.B.\nConnor Devitt (e-mail: devitt@purdue.edu), Sudhanshu Tiwari (e-mail:\ntiwari40@purdue.edu), and Sunil A. Bhave (e-mail: bhave@purdue.edu) are\nwith the OxideMEMS Lab, Elmore Family School of Electrical and Computer\nEngineering, Purdue University, West Lafayette, IN 47907 USA.\nRenyuan Wang is with FAST Labs, BAE Systems, Inc., Nashua, NH 03060\nUSA (e-mail: renyuan.wang@baesystems.com).\n(a)\n(b)\nFig. 1. (a)Chip microphotograph of multiple MSW bandpass filters and 1-\nport resonators fabricated on a YIG on GGG chip using YIG micromachining\ntechnology [6]. A 4-pole filter (on the left) and a 2-pole filter (on the right) are\nhighlighted in red. A 1-port resonator is highlighted in orange. (b)Rendering\nof a 4-pole bandpass filter featuring four YIG resonators with gold electrodes\nconformally deposited over the etched YIG. Magnetic bias is oriented out-of-\nplane along the z-axis.\n[3]. Coupled resonator arrays utilizing magnetostatic waves\n(MSW) have the potential to overcome the weak coupling\nand extremely narrow bandwidths achieved by electrostatically\ncoupled micromechanical resonators [4], [5] while introducing\na degree of tunability.\nThe magnetostatic wave resonance can be tuned more\nthan an octave in frequency using a static magnetic field\nensuring that the filter size does not scale to sub-micrometer\ndimensions at high-frequencies. Yttrium iron garnet (YIG)\nis the most widely used material for MSW devices due to\nits low Gilbert damping ( α= 2.8×10−4for a 100 nm\nfilm [7]) and experimentally demonstrated Q-factors exceeding\n3000 [8], [9]. In state of the art YIG sphere filters [10]–arXiv:2312.10583v1  [physics.app-ph]  17 Dec 20232\n[12], polished YIG resonators are attached to a thermally\nconductive rod and manually aligned to non-planar inductive\nloops acting as transducers. The assembled sphere and loop\nstructures are then coupled through transmission lines similar\nto the coupling beams in [1] to synthesize a filter. Planar YIG\nresonators can magnetically couple if they are fabricated in\nclose proximity (Fig. 1b), analogous to the electrically coupled\nmechanical resonators in [2], [13] or coupled electromagnetic\nresonators. Magnetically coupled YIG filters can be fabricated\nat scale using micromachining techniques on films grown on\na gadolinium gallium garnet (GGG) substrate, allowing for\nminiaturization and eliminating the need for polishing and\nmeticulous manual alignment.\nII. B ANDPASS FILTER DESIGN\nThe bandpass filter shown in Fig. 1bconsists of a number of\nclosely-spaced rectangular YIG magnetostatic forward volume\nwave (MSFVW) resonators with shorted 300 nm -thick gold in-\nductive transducers conformally deposited over the outermost\nresonators. With an out-of-plane DC magnetic bias, the RF\nmagnetic field from the transducers excite MSFVW modes in\nthe YIG mesa. Forward volume waves are a family of highly\ndispersive modes in a thin film whose lowest order mode is\ndescribed by the dispersion relation [14]:\nω2=ω0\u0014\nω0+ωm\u0012\n1−1−e−kmnt\nkmnt\u0013\u0015\n, (1)\nwhere ωm=µ0γmMs,ω0=µ0γmHeff\nDC,tis the film\nthickness, γmis the gyromagnetic ratio, µ0is the permeability\nof free space, kmnis the wave vector, Msis the saturation\nmagnetization, and Heff\nDCis the effective DC magnetic field.\nConsidering the limits as kmn→0andkmn→ ∞ in the\ndispersion relation, ωis restricted within the range [14]:\nω0≤ω≤p\nω0(ω0+ωm), (2)\ndenoted as the spin wave manifold. When the planar dimen-\nsions of the thin YIG film are bounded, the magnetostatic\nwaves reflects off the edges forming a standing waves with\nwave vectors approximately given by [15]–[19]:\nkmn=r\u0010πm\nl\u00112\n+\u0010πn\nw\u00112\n, m, n = 1,2,3. . . (3)\nwhere landware the length and width of the cavity respec-\ntively. These MSFVW resonances can be further understood\nthrough an analogy to Lamb waves in a piezoelectric plate\n[20]–[23]. An oscillating electric field perturbs the polarization\nof the piezoelectric film generating a stress field and exciting\nan acoustic wave. In the ferrimagnetic film, an oscillating\nmagnetic field, conversely, perturbs the static magnetization\nleading to a precession of spins. Both the piezoelectric and\nmagnetostatic cavities support a discrete number modes whose\nwave vectors depend on the cavity dimensions. Nonlinear\ndispersion relates the wave vectors to the resonant frequencies\nfor both MSFVW and Lamb waves. For MSFVW, this leads\nto irregularly spaced modes which all reside in the spin wave\nmanifold.Unique to MSW, the applied out-of-plane magnetic field\nshifts the MSFVW dispersion relation in frequency where the\ntuning rate for the fundamental mode is given by\n∂ω\n∂Heff\nDC=µ0γm2ω0+ωm\u0010\n1−1−e−kmnt\nkmnt\u0011\n2r\nω0h\nω0+ωm\u0010\n1−1−e−kmnt\nkmnt\u0011i,(4)\nwhich simplifies to µ0γm= 2.8 MHz /Oe(for YIG) when\nkmnt≪1. The magnetostatic scalar potential, ψ, decays\nexponentially outside the YIG mesa [14], [24] so the MSFVW\nresonance in one YIG mesa may couple to adjacent mesas\nif there is sufficient overlap in their scalar potentials [25].\nConsequently, the spacing of adjacent resonators and their ver-\ntical sidewall profile are critical to control the inter-resonator\ncoupling strength. For the 4-pole filter in Fig. 1b, the resonator\nspacings are s1= 10 µm,s2= 15 µm, and s3= 10 µm. Each\nhas length l= 500 µm. The outermost resonators have a width\nw1,4= 70 µmwhile the inner resonators are slightly narrower\natw2,3= 67 µmwhich was found to marginally improve\ninsertion loss and higher order width mode suppression based\non finite element simulation.\nSince each resonator length is much shorter than the elec-\ntromagnetic wavelength over the tuning range, the 4-pole filter\ncan be modeled with the lumped element circuit show in\nFig.2. A Butterworth-Van Dyke circuit [26], [27] is typically\nused to model an acoustic resonance where the mechanical\nmode is described by a series R-L-C tank circuit and the\ntransducers introduce a shunt plate capacitance. In the case\nof magnetic resonators, the transducer introduces a parasitic\nseries inductance and the MSFVW is modeled using a parallel\nR-L-C tank circuit instead. L0andR0represent the parasitic\ninductance and resistivity of the gold electrodes. Rm,Cm, and\nLmrepresent the MSFVW resonance of each YIG mesa. Mnm\nis the inter-resonator coupling between adjacent YIG mesas\nwhile MIOrepresents input/output inductive coupling of the\ngold electrodes setting the out-of-band rejection level. Similar\nto a mechanical coupling coefficient, an effective coupling\nfrom the electrical to magnetostatic domain can be defined\nM12 M23 M34MIOL0 R0 Rm1\nLm1\nCm1\nRm2\nLm2\nCm2\nRm3\nLm3\nCm3\nRm4\nLm4\nCm4 L0 R0\nFig. 2. Lumped circuit model of an edge-coupled 4-pole MSW bandpass\nfilter with electrically short transducers.3\nFig. 3. Measured frequency response of the 1-port resonator highlighted in\nFig.1aat3962 Oe showing a Q= 2206 andk2\neff= 1.53% .\nby (5) where fpandfsrepresents the magnetic resonance\nand anti-resonance respectively.\nk2\neff=π\n2\u0012fp\nfs\u0013\ncot\u0012π\n2fp\nfs\u0013\n(5)\nk2\neff is a function of the resonator design determined by\nthe ratio of LmtoL0. Similar to an acoustic filter, k2\neff\nsets a bound on the maximum achievable filter bandwidth\nand impacts the passband ripple. The representative 1-port\nresonator (highlighted in Fig. 1a) exhibits a measured effective\ncoupling and quality-factor of k2\neff= 1.53% andQ= 2206 at\n3962 Oe with frequency response shown in Fig. 3. From the\nmeasured resonator impedance response, R0,L0,Rm,Cm,\nandLmcan all be extracted through separate fittings near the\nmagnetostatic resonance and outside the spin wave manifold\nwhere the resonator behaves as an inductor. Using the same\nresonator parameters, the measured filter response excluding\nspurious modes can be fit to the lumped model in Fig. 2\nby tuning MIO,M12, and M23under the assumption that\neach resonator has the same resonant frequency and the filter\nis symmetric. Fig. 8bshows the frequency response of the\nlumped model fitted to the measured S21 for a 4-pole filter\nalong with the extracted model parameters.\nIII. F ABRICATION\nThe fabrication process for the MSW bandpass filter is\noutlined in Fig 4where steps (a)-(c) are adapted from [6].\nA thick photoresist mask (SPR220-7.0) is patterned onto a\n3µmYIG film grown via liquid phase epitaxially (LPE) on a\n500µmGGG substrate. The YIG film is etched through at a\nrate of 36 nm /min using an optimized ion milling recipe for\nvertical sidewalls with sufficient intermittent cooling to prevent\nburning of the photoresist. Optimized lithography for the thick\nSPR photoresist is crucial to the filter’s final performance since\nthe inter-resonator coupling factors, Mij, are sensitive to the\nphysical separation of the etched YIG. With this process, we\nare able to fully etch resonator spacings as narrow as 3µm\nsetting the maximum achievable Mij≤1.37% based on finite\nelement simulation. After etching, the resist is removed and\nFig. 4. Fabrication process of the bandpass filter: (a)3µmliquid phase\nepitaxy (LPE) YIG film grown on 500µmGGG substrate. (b)7.8µmthick\nphotoresist (SPR220-7.0) patterned on YIG film as an etch mask. (c)3µm\nion mill etch of YIG film at a rate of 36 nm /min.(d)Photoresist mask is\nremoved and etched YIG is soaked in phosphoric acid at 80◦Cfor20 min .(e)\nBi-layer photoresist (SPR220-7.0 and LOR 3B) mask for a liftoff is patterned\nonto the etched sample. (f)10 nm Ti and 300 nm Au is deposited using\nglancing angle e-beam evaporation followed by a liftoff process.\nFig. 5. SEM of the MSFVW bandpass filter showing vertically etched YIG\nresonators with an inter-resonator spacing of 10.5µmand conformal gold\nelectrodes over the edges of etched YIG.\nthe sample is soaked in phosphoric acid at 80◦Cto remove\nredeposited material. For the gold electrodes, a SPR220-7.0\nphotoresist mask with a liftoff resist (LOR 3B) bi-layer is\npatterned for a liftoff process. Using a glancing angle e-beam\nevaporation, 300 nm of gold and a 10 nm titanium adhesion\nlayer is conformally deposited over the etched YIG resonators.\nFinally, the sample is soaked in Remover PG overnight to\ncomplete the liftoff process. Fig. 5shows an SEM of a\nfabricated filter illustrating the vertically etched YIG and\nconformal gold electrodes. Due to the combination of bi-layer\nliftoff and angled metal deposition, the gold electrodes are\nlarger than designed and show a tapered thickness.4\nFig. 6. Measured 4-pole MSW bandpass filter frequency response at different\nout-of-plane magnetic biases from 3205 Oe to5303 Oe .\nIV. E XPERIMENTAL RESULTS\nFilter s-parameters are measured using an Agilent PNA-L\nN5230A network analyzer with a pair of ground-signal (GS)\nprobes from 3205 Oe to5303 Oe corresponding to a center\nfrequency tuning over 5.6 GHz as shown in Fig. 6. A single-\npole electromagnet powered by a constant current source\nprovides the required out-of-plane magnetic bias (pictured in\nFig.7) and a single-axis Gauss meter is used to map the source\ncurrent to the applied field. Prior to each measurement, the\ndevice under test is aligned to the center of the electromagnet’s\npole to ensure field uniformity. Fig. 8shows the frequency\nresponse near the passband for a 2-pole and 4-pole filter at\n3864 Oe . Outside of the spin wave manifold, no magnetostatic\nwaves may propagate so the filter behaves as two coupled\ninductors. Consequently, the out-of-band rejection is governed\nby the inductive coupling strength between the input and\noutput electrodes. For the 2-pole filter with an electrode\nspacing of 67µm, the rejection is −25 dB while for the 4-\npole filer with a spacing of 229µm, the rejection is −35 dB .\nAt3864 Oe , the 2-pole filter shows an insertion loss (IL) of\n−3.55 dB and a 3 dB bandwidth of 57.0 MHz while the 4-pole\nfilter has an IL of 6.94 dB and a 3 dB bandwidth of 17.0 MHz .\nFrom Fig. 8aand8b, higher-order magnetostatic spurious\nmodes are visible to the right of the passband with a frequency\nseparation of 29 MHz −40 MHz . As described in [15], the\ncurrent distribution along the transducer length can excite\nFig. 7. Experimental setup showing the fabricated filter chip resting on the\npole of an electromagnet, two GS probes connected to one device under test,\nand an optical microscope used for probe landing and device alignment.\n(a)\n(b)\nFig. 8. Frequency responses for the (a)2-pole and (b)4-pole filters high-\nlighted in Fig. 1near the passband at 3864 Oe .(b)also shows a comparison\nof the fitted lumped element model in Fig. 2with measured S21. The fitted\ncircuit assumes all four resonators are identical and excludes the spurious pass\nbands caused by higher order MSFVW modes.\neither even or odd ordered modes. Based on the resonator\ndimensions and electrically short transducer, the frequency\nspacing of these spurs agree well with odd ordered length\nmodes. Fig. 9shows the linear center frequency tuning at a\nrate of 2.7 MHz /Oeand the 3 dB -bandwidth over the applied\nbias for the 4-pole filter. With a well-calibrated bias field,\nthe extrapolated center frequency tuning line should intersect\n−ωmat0 Oe . However, the Gauss meter used for the field\nFig. 9. Measured 3 dB bandwidth and center frequency of the 4-pole MSFVW\nfilter showing a tuning rate of 2.7 MHz /Oe.5\ncalibration is thicker than the fabricated chip, so the reported\nbias is underestimated by approximately 219 Oe . The filter’s\ntuning was also measured using a neodymium permanent\nmagnet mounted on a 3-axis stage to precisely calibrate the\nfield accounting for any thickness difference between the chip\nand sensor. In this setup, the extrapolated 0 Oe intersection\nis atωm=µ0γm·1751 Oe which agrees well the saturation\nmagnetization of LPE YIG.\nConsidering the total loss ( 1− |S11|2− |S21|2) for the 2-\npole filter biased at 3652 Oe and measured far away from all\nmagnetostatic resonances, an average of 43% of the input\npower is dissipated in the thin 300 nm gold transducers,\nradiated, or absorbed by the YIG on GGG substrate. Based\non finite element simulations, the loss is primarily attributed\nto the resistance of the gold transducers. A second sample\nwas fabricated with 3µmelectroplated gold to reduce resistive\nlosses. Fig. 10compares the measured insertion loss and total\nloss for the same 2-pole filter with different gold thicknesses.\nAs expected, the mean out-of-band loss shows significant\nimprovement from 43% to only 12%. The average loss within\nthe 3dB bandwidth exhibits a slight improvement dependent\n(a)\n(b)\nFig. 10. (a)Measured S21 and (b)total loss for 2-pole filters with 300 nm\nand3µmthick gold transducers biased at 3652 Oe and3660 Oe respectively.\nFrequency is plotted relative to the center frequency to account for the slight\ndifference in bias strength.\nFig. 11. Two-tone IIP3 measurement in the passband of a 4-pole filter at\n3652 Oe bias\nTABLE I\nUPPER INPUT TONE FREQUENCIES FOR IIP3 MEASUREMENTS\nBias Field Stopband Low Passband Stopband High\n3652 Oe 5.465 GHz 5.799 GHz 6.055 GHz\nTABLE II\nSUMMARY OF 4-POLE FILTER IIP3\nBias Field Stopband Low Passband Stopband High\n3652 Oe ≥37.95 dBm −4.85 dBm 25.84 dBm\non bias, ranging from 0.5% to14.5%. The insertion loss\nimprovement reflects the change in mean in-band loss with\na maximum improvement from 4.43 dB to2.92 dB around\n3660 Oe .\nThe linearity of the MSW bandpass filter is evaluated by\nmeasuring the input referred 3rdorder intercept point (IIP3) in\nthe passband as well as the stopband both below and above the\npassband at a bias of 3652 Oe . The nonlinearity measurements\nare performed using two Keysight E8257D signal generators\nwith a frequency separation of ∆f= 15 MHz . The higher of\nthe two tone frequencies for each region are listed in Table\nI. A wideband power divider combines the two tones while\nan Agilent PXA spectrum analyzer measures the resultant\nspectrum. A two-stage calibration is performed to remove all\ncable and system losses at every tone frequency and input\npower level. Far away from the passband, the filter is expected\nto be linear and no intermodulation products were observed.\nBased on the maximum output power of the signal generators\nand noise floor of the spectrum analyzer, the lower stopband\nIIP3 is estimated to be greater than 37.95 dBm at3652 Oe .\nThe passband shows the greatest nonlinearity with an IIP3 of\n−4.75 dBm at3652 Oe as shown in Fig. 11. The measured\nIIP3 in each frequency region is summarized in Table II.\nV. C ONCLUSION\nIn this paper, we have demonstrated a novel edge-coupled\nhighly-tunable magnetostatic bandpass filter using state-of-\nthe-art micromachining fabrication techniques. The designed\n2-pole and 4-pole filters have been tuned over an octave6\nTABLE III\nPERFORMANCE COMPARISON WITH OTHER TUNABLE BANDPASS FILTERS\nReference Frequency\nTuning\n(GHz)Insertion\nLoss\n(dB)Bandwidth\n(MHz)Rejection\n(dB)\nThis work (2-pole) 4.5-10.1 <6 29-39 >25\nThis work (4-pole) 4.5-10.1 <11 11-17 >35\nYIG Sphere [10] 2-8 <5 20 >50\nMagnetostatic Sur-\nface Wave [28]3.4-11.1 <5.1 18-25 >25\nRF MEMS Tun-\nable Filter [29]6.5-10 <5.6 306-539 >50\nEvanescent-Mode\nCavity [30], [31]3-6.2 <4 15-25 >50\nfrom 4.5 GHz to10.1 GHz showing a consistent passband\nshape with performance comparable to other state-of-the-art\nfrequency tunable bandpass filters as summarized in Table\nIII. We have also characterized the linearity of the filter in\nthree distinct frequency regions. Our micromachining process\nenables precise control over the YIG mesa shape and spacings\nto synthesize miniaturized MSW channel-select filters analo-\ngous to electromagnetic cavity filter design.\nDATA AVAILABILITY\nThe code and data used to produce the plots within this work\nwill be released on the repository Zenodo upon publication.\nACKNOWLEDGMENTS\nChip fabrication was performed at the Birck Nanotechnol-\nogy Center at Purdue. Resonator and filter measurements and\ncharacterization were performed at Seng-Liang Wang Hall at\nPurdue. The Purdue authors would like to thank Dave Lubelski\nfor assistance with the glancing angle metal deposition and\nYiyang Feng for discussions on the fabrication recipes.\nREFERENCES\n[1] K. Wang and C.-C. Nguyen, “High-order medium frequency microme-\nchanical electronic filters,” Journal of Microelectromechanical systems ,\nvol. 8, no. 4, pp. 534–556, 1999.\n[2] P. Thiruvenkatanathan, J. Yan, J. 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Wartenberg et al. , “Modified butterworth-\nvan dyke circuit for fbar resonators and automated measurement system,”\nin2000 IEEE ultrasonics symposium. proceedings. an international\nsymposium (Cat. No. 00CH37121) , vol. 1. IEEE, 2000, pp. 863–868.\n[27] R. Aigner, “Mems in rf filter applications: Thin-film bulk acoustic wave\ntechnology,” Sensors Update , vol. 12, no. 1, pp. 175–210, 2003.\n[28] X. Du, M. H. Idjadi, Y . Ding et al. , “Frequency tunable magnetostatic\nwave filters with zero static power magnetic biasing circuitry.” [Online].\nAvailable: http://arxiv.org/abs/2308.009077\n[29] K. Entesari and G. Rebeiz, “A differential 4-bit 6.5-10-GHz RF\nMEMS tunable filter,” IEEE Transactions on Microwave Theory\nand Techniques , vol. 53, no. 3, pp. 1103–1110. [Online]. Available:\nhttp://ieeexplore.ieee.org/document/1406317/\n[30] H. Joshi, “Multi band RF bandpass filter design,” ISBN: 978-1-\n124-15604-0 Publication Title: ProQuest Dissertations and Theses.\n[Online]. 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Marzolo 8, I-35131 Padova, Italy\n2INFN, Laboratori Nazionali di Legnaro, Viale dell'Universit\u0012 a 2, I-35020 Legnaro, Italy\nWe demonstrate an all-optical method for manipulating the magnetization in a 1{mm YIG\n(yttrium-iron-garnet) sphere placed in a \u00180:17 T uniform magnetic \feld. An harmonic of the\nfrequency comb delivered by a multi-GHz infrared laser source is tuned to the Larmor frequency\nof the YIG sphere to drive magnetization oscillations, which in turn give rise to a radiation \feld\nused to thoroughly investigate the phenomenon. The radiation damping issue that occurs at high\nfrequency and in the presence of highly magnetizated materials, has been overcome by exploiting\nmagnon-photon strong coupling regime in microwave cavities. Our \fndings demonstrate an e\u000bective\ntechnique for ultrafast control of the magnetization vector in optomagnetic materials via polariza-\ntion rotation and intensity modulation of an incident laser beam. We eventually get a second-order\nsusceptibility value of \u001810\u00007cm2/MW for single crystal YIG.\nIntroduction. |Nonthermal control of spins by short\nlaser pulses is one of the preferable means to achieve ul-\ntrafast control of the magnetization in magnetic materi-\nals [see 1, and references therein], representing a break-\nthrough in potential applications ranging from high den-\nsity magnetic data storage [2], spintronics [3], to quantum\ninformation processing [4, 5]. One of the most interesting\nopto-magnetic mechanisms that allows for coherent con-\ntrol of the magnetization in materials is the Inverse Fara-\nday (IF) e\u000bect, a Raman-like coherent scattering process\nthat entails the generation of a magnetic excitation (i.e.\nmagnon) in a medium undergoing the action of high{\nintensity optical pulses. As it does not require absorption\nand takes place on a femtosecond time scale, it stands\nout as a promising mechanism to control the magnetiza-\ntion at high speeds. This principle has been successfully\napplied in dysprosium orthoferrite (DyFeO 3) with iso-\nlated femtosecond laser pulses, that act as magnetic \feld\npulses of a 0.3 T amplitude [6]. By way of the IF e\u000bect,\nvector control of magnetization in another antiferromag-\nnetic crystal was also demonstrated by varying the delay\nbetween pairs of polarization-twisted ultrashort optical\npulses [7]. In this work we introduce a new approach in\nopto{magnetism based on multi-gigahertz repetition rate\nlasers with optical carrier f0[8]. The power spectrum of\nsuch mode-locked laser sources, as detected by ultrafast\nphotodiodes, is a frequency comb that consists of several\nharmonicsnfr, wherefris the repetition rate and nis a\nsmall number.\nTheir gaussian envelope is determined by the optical\npulse temporal pro\fle [9]. For example, our 4.6 GHz\npassively mode-locked oscillator delivers \u001810 ps-duration\npulses that give rise to a frequency comb up to 100 GHz,\nand the \frst three harmonics have approximately the\nsame amplitude. In principle, any harmonic of the comb\ncan coherently drive the magnetization in the steady\nstate through the process described in the present work,\nprovided it is tuned to electron spin resonances (ESR) of\nthe magnetized material.\nWe study the spin dynamics in a hybridized system\nwhich consists of two strongly coupled oscillators, i.e. amicrowave cavity mode and a magnetostatic mode re-\nlated to ferromagnetic resonance (FMR) with uniform\nprecession [10] of a single crystal yttrium-iron garnet\nY3Fe5O12sphere. Magnetic garnets [11, 12] represent\nthe ideal materials for such investigations for several rea-\nsons, including the possibility to realize large magneto-\noptical e\u000bects due to their strong spin-orbit coupling and\nintrinsically low magnetic damping [12{14].\nWe succeed to optically drive the precession of the\nspins electro-dynamically coupled to the cavity photons\nwith the \frst harmonic of the train of pulses at frtuned\nto one of the hybrid system's resonant frequencies. The\nprocess gives rise to a microwave \feld that is measured\nwith a loop antenna critically coupled to the cavity mode.\nIn this way, we have identi\fed a new observable for\nthe spin precession to explore opto-magnetic phenom-\nena. So far, experiments in this \feld were performed\nwith pump-probe apparatus based on femtosecond lasers\n[1, 7, 12, 15].\nAs it is well known, at very high values of frequency\nand magnetization, the energy radiated from oscillating\nmagnetization through magnetic dipole radiation can be\nan issue for the dynamic control of the magnetization.\nFor instance, in a polarized 1{mm YIG sphere with lin-\near susceptibility \u001f\u001830, the onset of radiation damping\noccurs at\u001810 GHz [16]. However, radiation damping\nmechanism can be conveniently suppressed in the mi-\ncrowave cavity-QED and strong coupling regimes [16, 17],\nas we detail in the following for hybridized systems.\nMoreover, under remarkable conditions of hybridization,\nwe get the control of relevant experimental parameters\nsuch as the number of spins, rf absorbed power, and the\ninvolved relaxation times. System hybridization is how-\never not essential to observe the phenomenon described\nin the present work. In fact, we succeed in controlling the\nmagnetization also in free space, but under experimental\nconditions that do not allow for accurate modeling.\nHybridized system characterization. | Strong interac-\ntion between light (i.e. photons stored in a cavity) and\nmagnetized materials has been accomplished in several\nexperiments that are paving the way toward the develop-arXiv:1609.08147v1  [cond-mat.mes-hall]  24 Sep 20162\nment of quantum information technologies [18{20]. Hy-\nbridization is commonly investigated by measuring the\nmicrowave{cavity transmission spectrum as a function of\nthe static magnetic \feld, as summarized in Fig. 1 for our\nexperimental setup, even though, very recently, some au-\nthors have reported electric detection via spin pumping\n[21]. A YIG sphere made by Ferrisphere Inc. with a ra-\n(a)(b)\nzx\nFIG. 1. Hybridizing magnons and microwave photons. (a)\nCavity transmission spectrum measured as a function of the\nstatic magnetic \feld at room temperature. (b) Simulated\nmagnetic{\feld distribution of the TE 102cavity mode. A\nstatic magnetic \feld B extis applied normal to the xzplane,\nand the microwave magnetic \feld at the YIG sphere (in black\nand not to scale in the representation) position is perpendic-\nular to the static magnetic \feld. The color map represents\nthe amplitude of the cavity magnetic \feld normalized to its\nmaximum value.\ndius of 1 mm is mounted at the center of a 3D rectangular\nmicrowave cavity with dimensions 98 \u000242:5\u000212:6 mm3.\nThe cavity made of oxygen free copper has the TE 102\nmode frequency !c=2\u0019'4:67 GHz, and its internal cav-\nity loss\u0014int. This cavity has two ports characterized by\nthe coupling coe\u000ecients \u00141and\u00142to the considered cav-\nity mode. The sphere is glued to an alumina (aluminum{\noxide) rod that identi\fes the crystal axis [110], perpen-\ndicular to the static magnetic \feld Bext(yaxis) and par-\nallel to the TE 102microwave magnetic \feld lines lying\non thexzplane. Due to the strong coupling between the\ncavity mode and FMR mode an avoided crossing occurs\nwhen their resonant frequencies match. As derived in\nthe input{output theory context [17, 20], when the static\nmagnetic \feld is tuned to drive the magnons in resonance\nwith the cavity mode TE 102, the measured transmission\ncoe\u000ecient can be written as\nS21(!) =p\u00141\u00142\ni(!\u0000!c)\u0000\u0014c\n2+jgmj2\ni(!\u0000!FMR)\u0000\rm=2;(1)\nwhere!FMR and\rmare the frequency and linewidth of\nthe FMR mode, \u0014c=2\u0019= (\u00141+\u00142+\u0014int)=2\u0019is the total\ncavity linewidth, and gmis the coupling strength of the\nFMR mode to the cavity mode. The latter parameter is\nproportional to the square root of the number of precess-\ning spinsNs, i.e.gm=g0pNs, whereg0=\rep\n\u00160~!c=Vcis the coupling strength of a single spin to the cavity\nmode, with \re= 2\u0019\u000228 GHz/T electron gyromagnetic\nratio,\u00160permeability of vacuum and Vcis cavity volume.\nAs discussed in the seminal paper of Bloembergen and\nPound [16], the poles at the anti{crossing point !FMR =\n!c\u0011!0are given by\np\u0006=i\u0010\n!0\u0006p\njgmj2\u0000[(\u0014c\u0000\rm)=4]2\u0011\n\u00001\n2\u0012kc\n2+\rm\n2\u0013\n;(2)\nand their imaginary and real parts represent the frequen-\ncies\n!\u0006=!0\u0006p\njgmj2\u0000[(\u0014c\u0000\rm)=4]2\nand the linewidths \r\u0006= 1=2(kc+\rm) of the hybridized\nmodes, respectively. From eq. 2 hybridization clearly oc-\ncurs only ifjgmj2\u0000[(\u0014c+\rm)=4]2>0 and, as a con-\nsequence, hybridized modes have the same decay time,\nindependent of the sample or cavity volume and cou-\npling strengths, i.e. \u0016 \u001c\u0011\u001c\u0006= (2=\u001cc+ 2=\u001c2)\u00001, where\n\u001cc= 2=\u0014cand\u001c2are the loaded cavity decay time and the\nspin{spin relaxation time, respectively. In the absence of\nhybridization, the term under square root in eq. 2 is neg-\native and thus the poles have the same frequency !0,\nwith two relaxation times \u001ccand\u001c\u0003= (1=\u001cr+ 1=\u001c2)\u00001,\nthat correspond to the damping of the cavity mode and\nmagnetization mode in the presence of radiation damping\n\u001cr=\u0014c=2=jgmj2.\nIn our experimental apparatus for B ext'0:17 T we\nachieve a strong coupling regime with gm=2\u0019= 57 MHz,\nthus the involved precessing spins are Ns\u00181020. Along\nwith the mode frequencies f+= 4:7247 GHz and f\u0000=\n4:6677 GHz, the \ft of the measured S21coe\u000ecient to eq. 1\ngives the mode decay times \u0016 \u001c'65 ns of the hybridized\nsystem, compatible with the value of \u001c2provided by the\nmanufacturer and the measured \u001cc.\nPhotoinduced magnetization. | Once the hybrid sys-\ntem has been characterized, the experimental appara-\ntus illustrated in Fig. 2 is used to investigate the opto{\nmagnetic phenomenon. The 7.2 ps{duration, 1.55 \u0016m{\nwavelength laser pulses are obtained at the idler output\nof an optical parametric oscillator (OPO), synchronously\npumped by the second harmonic of a MOPA (master os-\ncillator power ampli\fer) laser system that has been de-\nscribed elsewhere [23]. It is important to note that we\nare exploiting a non absorptive mechanism as the opti-\ncal wavelength is within the YIG transparency window\n(1:5\u00005\u0016m). The beam waist at the YIG position is\n1:28 mm, and the average intensity of the incident pulses\nis 2:4 MW/cm2, obtained within <1\u0016s{duration macro{\npulses.\nIn Fig. 3 we demonstrate the all{optical coherent con-\ntrol of the magnon{photon mode at f\u0000= 4:67 GHz by\nemploying a train of laser pulses with repetition rate fr\ntuned tof\u0000. The rise and decay time of the microwave\npulse registered at the oscilloscope agrees with the mode\ndecay time \u0016 \u001c= 65 ns we get through the S 12measure-\nments within experimental errors. The duration of the3\nMOPA LASERSH\nOPODMHS\nCGFF532 nm1064 nm1064 nm809 nm1554 nmMWM\nN\nS\nLCC\nB[...]500 nsUPD\u0000/2\nYIG\noscilloscope\nxyA\nFIG. 2. Schematic representation of the experimental ar-\nrangement. The 1064 nm{wavelength macro{pulse delivered\nby a MOPA laser is frequency{doubled (SH) to synchronously\npump an optical parametric oscillator (OPO). The laser rep-\netition rate, macro{pulse uniformity and energy are moni-\ntored at an InGaAs ultrafast photodiode (UPD), a coaxial\nwaveguide device WM [22] and bolometer B respectively. The\n809 nm OPO output beam intensity pro\fle is adjusted at a\ndigital laser camera LC. To make sure that only emission at\n1550 nm impinges on the YIG sphere, several optical \flters are\ninserted in the beam path. CGF is a 610 nm longpass colored\nglass \flter, F transmits \u0015>1500 nm and M is a 1064 nm, high\nre\rectivity dielectric mirror. HS (harmonic separator) is a di-\nelectric mirror that transmits 1064 nm wavelength and has a\nhigh re\rectivity for 532 nm whereas DM is a 1000 nm{cuto\u000b\nwavelength dichroic mirror. The microwave \feld generated\nduring the magnetization precession is detected by means of\nan antenna critically coupled to the TE 102mode and con-\nnected through a short transmission line to a 39 dB{gain am-\npli\fcation stage A. The ampli\fed signal is \fnally registered\nat a 20 GHz sampling oscilloscope.\noptical excitation is set to a value of te'0:5\u0016s>\u0016\u001cal-\nlowing us to control the system in its steady state. This\ndi\u000bers from previous studies in opto{magnetism which\nwere focused on the transient optical control of the mag-\nnetization via single femtosecond laser pulses [see 1, and\nreferences therein]. Moreover, the YIG magnetization\nprecesses in phase with the laser pulses, as demonstrated\nby juxtaposition in Fig. 3 (c) of the signal generated in the\nmicrowave cavity and the output of the laser macro{pulse\nmonitor WM, i.e. a coaxial waveguide hosting a nonlin-\near crystal in which microwaves are generated through\noptical recti\fcation [22]. Another important signature\nof the coherent precession of the magnetization is also\nshown in Fig. 3 (d), where the amplitude of the Fourier\ntransform of the microwave signal is plotted for di\u000berent\nvalues of the laser repetition rate fr. The data are \ftted\nto a lorentzian curve that takes into account the convolu-\ntion between the optical excitation and the pro\fle of the\nhybridized mode at 4 :6711 GHz. As shown in Fig. 3 (b),\nthe spectral component f+is also excited but with a\nmuch smaller strength. These results, combined with the\n-0.1-0.0500.050.1\n-0.04-0.0200.020.04\n0 200 400 600 800cavity - hybrid system\nmacropulse monitor VC (V)VM (V)\ntime (ns)(a)\n-140-120-100-80-60-40-20\n4.554.64.654.74.754.8A (a.u.)frequency (GHz)(b)\n390390.5391391.5392-0.1-0.0500.050.1\n-0.04-0.0200.020.04\ntime (ns)VC (V)VM (V)(c)\n0246810\n4.644.654.664.674.684.69A (a. u.)frequency (GHz)(d)FIG. 3. (Color online) Optically-driven spin precession in the\ntime and frequency domain. (a) Oscilloscope traces displaying\nboth the ampli\fed signal V Cdetected in the microwave cav-\nity hosting the YIG sphere (blue) and the output V Mof the\nlaser macro{pulse monitor (red). (b) Fourier transform am-\nplitude spectrum of the microwave signals displayed in (a).\nThe logarithmic scale is used for the vertical axis. (c) 2 ns{\nduration zoom out of (a) showing the magnetization precess-\ning synchronously with the laser pulses. (d) Tuning the laser\nrepetition rate to the hybridized frequency f\u0000.\n00.20.40.60.811.2\n0π/2π3π/22πAmplitude (a.u.)θ (rad)(a)\n020406080\n00.511.522.53Vrms (mV)I (MW/cm2)(b)\nFIG. 4. (a) Amplitude of the microwave signal in the cavity as\na function of the laser polarization angle. (b) Magnetization\ndependence on the laser intensity.\nassessment of stationary precession of the macroscopic\nmagnetization, unambiguously show that each macro{\npulse acts as an e\u000bective microwave \feld on the ensemble\nof strongly correlated spins of the FMR mode.\nDiscussion. | To con\frm the nonthermal origin of the\nlaser{induced magnetization precession and de\fnitely at-\ntribute the observed opto{magnetic phenomenon to the\nIF e\u000bect, we investigated the dependence of the mi-\ncrowave signal amplitude on the laser polarization [1]\nand the results are reported in Fig. 4 (a). Owing to the\nstrong anisotropy of the magnetic susceptibility of YIG,\nthe time{dependent magnetization vector Mis not ex-\npected to be parallel to the wave vector kas is the case for\nIF in isotropic medium and circularly polarized light. Ac-4\ntually the observed magnetization precession is induced\nby a second{order process conveniently described by a\nthird{rank, axial time{odd tensor \u001f(2)\nijk, provided that k\nis orthogonal to the [1 1 0] crystal direction d[7]. As the\nreference system axes xandycoincide with kandddi-\nrections, the photoinduced magnetization lies in the yz\nplane and reads\nMi=Z\nd!\u001f(2)\nijkE?\nj(!)Ek(!); (3)\nwherei= 2;3, andE2(!) =E(!) cos'andE3(!) =\nE(!) sin'; hereE(!) is the Fourier transform of the\nlaser electric \feld and 'is the polarization angle of the\nincident light with respect to the yaxis. Due to well{\nknown symmetries of second{order susceptibility [1], the\nnon{vanishing components of \u001fijkare\u001f233=\u0000\u001f222=\n\u001f332=\u001f323\u0011\u0004(!) and equation 3 gives the components\nMz=Z\nd!\u0004(!)jE(!)j2cos 2' (4)\nMy=Z\nd!\u0004(!)jE(!)j2sin 2': (5)\nAround the hybridized mode frequencies !\u0006, the real and\nimaginary part of the complex susceptibility \u0004( !) can be\napproximated by absorption \u0004( !)00and dispersion \u0004( !)0\ncomponents of magnetization [24]. In particular, at work-\ning frequency !\u0000we have only absorption with no disper-\nsion, hence the susceptivity \u0004( !\u0000) = \u0004 0!\u0000\u0016\u001c=2 becomes\nreal, and does not a\u000bect the magnetization direction.\nThus the ful\fllment of resonant condition also allow us\nto simplify the geometric description of the photoinduced\nmagnetization vector. Indeed, to explain the 4{fold pe-\nriodicity of the plot displayed in Fig. 4 (a) we only need\nto realize that the cavity selects the Mz/cos 2'compo-\nnent via its geometric projection on TE 102mode (i.e. the\nz direction as shown in Fig. 1), and that the critically\ncoupled antenna cannot distinguish between parallel and\nantiparallel orientation of Mz. Therefore the detected\nmagnetization signal must be proportional to jcos 2'j, as\ncon\frmed by the \ft to the data in Fig. 4 (a). Figure 4 (b)\nshows instead the linearity of the measured spin oscilla-\ntions amplitude as a function of the pump laser intensity,\nin agreement with eq. 3 as well.\nThe strength of the e\u000bective microwave \feld Be\u000bthat\ndrives theMzprecession can be estimated thanks to the\npeculiar dynamics of hybridization. In general, the ab-\nsorbed power in stationary conditions by a magnetized\nsample [24] is given by\nPa=Vs\u001c\n\u0000B\u0001dM\ndt\u001d\n;\nwhereh\u0001idenotes the time average over one period.\nMoreover, at resonance and for a critically coupled in-\nductive loop, the measured power in the microwave cav-\nity isPa=2. In our experimental conditions, the absorbedpower by the YIG crystal at the frequency !\u0000\nPa=Vs\u00040!2\n\u0000\u001cB2\ne\u000b\n\u00160(6)\nis written in terms of quantities that are measured or \ft-\nted to the data, so that the second{order susceptivity can\nbe readily estimated through \u0004 0=Pa\u00160=(Vs!2\n\u0000\u001cB2\ne\u000b),\nwhereBe\u000brepresents the laser induced e\u000bective mag-\nnetic \feld. Due to 1 =fdependence of the power spec-\ntrum generated by downconversion of the picosecond fre-\nquency comb, the infrared optical \feld average amplitude\nBl=p\n\u00160I=c= 10 mT, at fo'190 THz optical fre-\nquency, is suppressed to Be\u000b= 2:5\u000210\u00005BI= 0.25\u0016T\natf\u0000\u00184:7 GHz. With Pa= 3 nW estimated from the\nplots reported in Fig. 3, we eventually get \u0004 0\u001810\u00007\ncm2/MW.\nIn summary, our experimental and theoretical ap-\nproach provides a purely optical, \rexible technique to\nmanipulate the magnetization vector in YIG via polar-\nization rotation and intensity modulation of the incident\nlaser beam. Remarkably, the maximum control speed of\nthis process is only limited by the bandwidth of currently\navailable electro-optic devices. Unlike the ingenious op-\ntical method described in reference [7], here the mode-\nlocked pulses impinging on the magnetized material allow\nfor operation of the system in the steady state, opening a\npath on the ultrafast laser control of hybridized magnon-\nphoton systems. It is worth mentioning that commer-\ncially available compact ultrafast oscillators with 200 pJ-\nenergy output pulses [25] may foster applications of the\npresent method in the opto{magnetism \feld.\nThe authors thank D. Budker and V. S. Zapasskii for\ncarefully reading the manuscript and for useful discus-\nsions. C. B. and M. G. acknowledge partial \fnancial sup-\nport of the University of Padova under Progetto di Ate-\nneo (reference grant number CPDA135499/13). Techni-\ncal support by E. Berto is gratefully acknowledged.\n\u0003Electronic address: caterina.braggio@unipd.it\nyElectronic address: antonello.ortolan@lnl.infn.it\n[1] A. Kirilyuk, A. V. Kimel, and T. Rasing, Rev. 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Pokrovsky1;2\n1Department of Physics, Texas A&M University, College Station, Texas77843-4242 and\n2Landau Institute for Theoretical Physics, Chernogolovka, Moscow District,142432, Russia\n(Dated: May 26, 2022)\nRecently, magnons, which are quasiparticles describing the collective motion of spins, were found\nto undergo Bose-Einstein condensation (BEC) at room temperature in \flms of Yttrium Iron Garnet\n(YIG). Unlike other quasiparticle BEC systems, this system has a spectrum with two degenerate\nminima, which makes it possible for the system to have two condensates in momentum space.\nRecent Brillouin Light scattering studies for a microwave-pumped YIG \flm of thickness d= 5\u0016m\nand \feldH= 1 kOe \fnd a low-contrast interference pattern at the characteristic wavevector Qof the\nmagnon energy minimum. In this report, we show that this modulation pattern can be quantitatively\nexplained as due to non-symmetric but coherent Bose-Einstein condensation of magnons into the\ntwo energy minima. Our theory predicts a transition from a high-contrast symmetric phase to a\nlow-contrast non-symmetric phase on varying the dandH, and a new type of collective oscillations.\nPACS numbers: 75.10.-b, 75.60, 75.70, 75.85\nBose-Einstein condensation (BEC), one of the\nmost intriguing macroscopic quantum phenomena,\nhas been observed in equilibrium systems of Bose\natoms, like4He [1, 2],87Rb [3] and23Na [4]. Recent\nexperiments have extended the concept of BEC to\nnon-equilibrium systems consisting of photons [5]\nand of quasiparticles, such as excitons [6], polari-\ntons [7{9] and magnons [10, 11]. Among these,\nBEC of magnons in \flms of Yttrium Iron Garnet\n(YIG), discovered by the group of Demokritov [11{\n17], is distinguished from other quasiparticle BEC\nsystems by its room temperature transition and\ntwo-dimensional anisotropic properties. In par-\nticular, the spin-wave energy spectrum of a YIG\n\flm shows two energetically degenerate minima.\nTherefore it is possible that the system may have\ntwo condensates in momentum space [18]. An ex-\nperiment by Nowik-Boltyk et al. [17] indeed shows\na low-contrast spatial modulation pattern, indicat-\ning that there is interference between the two con-\ndensates. Current theories [19{24] do not describe\nthe appearance of coherence or the distribution of\nthe two condensates.\nThis report points out that a complete descrip-\ntion of BEC in microwave-pumped YIG \flms must\naccount for the 4th order interactions, includ-\ning previously neglected magnon-non-conserving\nterms originating in the dipolar interactions. The\ntheory explains not only the appearance of coher-\nence but also quantitatively explains the low con-trast of the experimentally observed interference\npattern. Moreover, the theory predicts that, on\nincreasing the \flm thickness dfrom a small value,\nthere is a transition from a high-contrast symmet-\nric phase S for d<dc, with equal numbers of con-\ndensed magnons \flling the two minimum states, to\na low-contrast coherent non-symmetric phase NS\nford > dc, with di\u000berent numbers of condensed\nmagnons \flling the two minimum states. In com-\nparatively thin \flms ( d<0:2\u0016m) the same transi-\ntion can be driven by an external magnetic \feld H.\nFord > d\u0003, whered\u0003is another critical thickness\n(d\u0003> dc), the sum of phases of the two conden-\nsates changes from \u0019to 0; ford=d\u0003the system\nis in a completely non-symmetric phase with only\none condensate, for which there is no interference.\nIn the experiment of Ref.[17] the \flm thickness d\nexceededd\u0003. We suggest that the phase transitions\nmay be identi\fed by measuring the contrast of the\nspatial interference pattern for various dandH.\nWe also predict a new type of collective magnetic\noscillation in this system and discuss the possibil-\nity of domain walls in non-symmetric phases.\nResults\nPhase Diagram.| We consider a YIG \flm of\nthicknessdwith in-plane magnetic \feld H(see in-\nset of Fig. 1). The 4-th order interaction of con-arXiv:1302.6128v1  [cond-mat.quant-gas]  25 Feb 20132\ndensate amplitudes reads [25{27]:\n^V4=A[cy\nQcy\nQcQcQ+cy\n\u0000Qcy\n\u0000Qc\u0000Qc\u0000Q]\n+2Bcy\nQcy\n\u0000Qc\u0000QcQ\n+C[cy\nQcQcQc\u0000Q+cy\n\u0000Qc\u0000Qc\u0000QcQ+h:c:]:(1)\nHerec\u0006Qandcy\n\u0006Qare the annihilation and cre-\nation operators for magnons in the two conden-\nsates located at the two energy minima (0 ;\u0006Q) in\nthe 2-D momentum space (see. Fig.1). The coe\u000e-\nFIG. 1. The magnon spectrum in the kzdirection for\nd= 5\u0016m andH= 1 kOe. The inset is a schematic\ndiagram of YIG \flm.\ncients in Eq.(1) are:\nA=\u0000~!M\n4SN[(\u000b1\u0000\u000b3)FQ\u00002\u000b2(1\u0000F2Q)]\n\u0000DQ2\n2SN[\u000b1\u00004\u000b2];\nB=~!M\n2SN[(\u000b1\u0000\u000b2)(1\u0000F2Q)\u0000(\u000b1\u0000\u000b3)FQ]\n+DQ2\nSN[\u000b1\u00002\u000b2];\nC=~!M\n8SN[(3\u000b1\u000020\n3\u000b3+ 3\u000b2)FQ\n+16\n3\u000b3(1\u0000F2Q)] +DQ2\nSN\u000b3; (2)\nwith\u000b1=u4+ 4u2v2+v4,\u000b2= 2u2v2and\u000b3=\n3uv(u2+v2). Here,uandvare the coe\u000ecients of\nBogoliubov transformation (see the Methods sec-\ntion for details). S= 14:3 is the e\u000bective spin, Nis\nthe total number of spins in the \flm, Mis the mag-\nnetization and ~!M=\r4\u0019M with gyromagneticratio\r= 1:2\u000210\u00005eV=kOe.Dis proportional\nto the exchange constant and Fk= (1\u0000e\u0000kd)=kd.\nSimilar results for the coe\u000ecients AandBwere\nobtained in Ref.[19]. The coe\u000ecient C, which vi-\nolates magnon number conservation, has not been\nconsidered previously. Below we show that Cis the\nonly source of coherence between the two conden-\nsates. The three coe\u000ecients A,BandC, whose\nvalues as functions of Hare shown in Fig.2 for\ntwo typical values of d, determine the distribu-\ntion of condensed magnons in the two degenerate\nminima. Ref.[19] assumed a symmetric phase with\ncondensed magnons in both minima having equal\namplitudes and equal phases. Later, Ref.[20] as-\nsumed \flling of only one minimum. More recently\nRef.[24] considered Josephson-like oscillations by\nstarting from two condensates with equal numbers\nof magnons but di\u000berent phases. Our theory pre-\ndicts coherent condensates and the ratio of their\namplitudes with no additional assumptions.\nIn terms of condensate numbers N\u0006Qand\nphases\u001e\u0006, the condensate amplitudes are c\u0006Q=p\nN\u0006Qei\u001e\u0006. Substituting them into eq.(1) we \fnd:\nV4=A(N2\nQ+N2\n\u0000Q) + 2BNQN\u0000Q\n+2Ccos \b(N3\n2\nQN1\n2\n\u0000Q+N1\n2\nQN3\n2\n\u0000Q):(3)\nClearly the dipole energy depends on the total\nphase \b = \u001e++\u001e\u0000. To minimize this energy,\nforC > 0 we have \b = \u0019and forC < 0 we have\n\b = 0. Fig.2 shows that the sign of Cchanges for\ndi\u000berentdandH, which indicates a transition of\n\b between 0 and \u0019. For both C > 0 andC < 0\nthe dipole energy establishes a coherence between\nthe two condensate amplitudes. In contrast to a\nJosephson-like interaction, the sum rather than the\ndi\u000berence of the two condensate phases is \fxed.\nSince the total number of condensed magnons\nNc=NQ+N\u0000Qis uniquely determined by the\npumping (see Methods), the energy has only a sin-\ngle free variable, the so far unspeci\fed di\u000berence\n\u000e=NQ\u0000N\u0000Q. In terms of Ncand\u000ethe conden-\nsate energy eq.(3) is:\nV4=1\n2h\n(A+B)N2\nc\u0000(B\u0000A)\u000e2\n\u00002jCjNcp\nN2c\u0000\u000e2i\n: (4)\nThe ground state of the condensates depends on\nthe criterion parameter \u0001, de\fned as\n\u0001\u0011A\u0000B+jCj: (5)3\nFIG. 2. Interaction coe\u000ecients A,BandC(in units of\nmK/N, with Nthe total number of spins in the \flm)\nas a function of magnetic \feld Hfor \flm thickness (a)\nd= 1:0\u0016m and (b)d= 0:1\u0016m.\nWhen \u0001>0,\u000e= 0 minimizes the energy, so\nthe two minima have equal numbers of condensed\nmagnons. This is the symmetric phase, with\nNQ=N\u0000Q. When \u0001 <0, the minimum shifts\nto\u000e2\nN2c= 1\u0000C2\n(B\u0000A)2. This is the non-symmetric\nphase (S), with NQ6=N\u0000Q. The transition from\nsymmetric to non-symmetric phase (NS) at \u0001 = 0\nis of the second order. There is no metastable state\nof these phases. At C= 0 one \fnds \u000e=\u0006Nc,\nwhich corresponds to a completely non-symmetric\nphase with only one condensate. The ground state\nof the non-symmetric phase is doubly-degenerate,\ncorresponding to the two possible signs for \u000e. Fig.3\nshows that for a \flm thickness of about 0 :05\u0016m,\nthe symmetric phase is energy favorable up to\nH= 1:2 T. Ford= 0:08\u0016m, on increasing Hto\nabout 0:6 kOe, there is a transition from symmet-\nric to non-symmetric phase. For the larger thick-\nnessesd= 0:1\u0016m andd= 1\u0016m, the ground state\nis non-symmetric for H > 0:3 kOe.\nFig.4 gives the phase diagram in ( d;H) space.\nIt has three di\u000berent regions, separated by two\ncritical transition lines, dc(H) andd\u0003(H), corre-\nsponding to \u0001 = 0 and C= 0, respectively. For\nd= 0:13\u00000:16\u0016the system possesses re-entrant\nbehavior (NS \b = \u0019, to NS \b = 0, to NS \b = \u0019)\nasHincreases. As shown below, measurement of\nthe contrast, or modulation depth [17], of the spa-\ntial interference pattern permits identi\fcation of\nthe di\u000berent condensate phases.\nFIG. 3. Transition criterion \u0001 from non-symmetric to\nsymmetric phase, \u0001 (in units of mK/N), as a function\nof magnetic \feld Hfor di\u000berent thicknesses d.\nFIG. 4. The phase diagram for di\u000berent values of thick-\nnessdand magnetic \feld H.\nZero Sound.| In two-condensate states the\nrelative phase \u000e\u001e=\u001e+\u0000\u001e\u0000is a Goldstone mode.\nIts oscillation, coupled with the oscillation of the\nnumber density \u000en=nQ\u0000n\u0000Qrepresents a new\ntype of collective excitation, which we call zero\nsound (as in Landau's Fermi liquid, this mode\nis driven by the self-consistent \feld rather than\ncollisions). Solving a properly modi\fed Gross-\nPitaevskii equation (see Methods), we \fnd its spec-\ntrum. In the symmetric phase its dispersion rela-\ntion is:\n!=r\n~2k4\n4m2+Nc\u0001k2\nm: (6)4\nThe magnon e\u000bective mass is of the order of the\nelectron mass. The density of condensed magnons\nnc=Nc=Vis about 1018cm\u00003and \u0001\u001910\nmK=N. The sound speed for small kin this case\nisv0s=p\nNc\u0001=m, which is about 100 m =s. Near\nthe transition point \u0001 = 0, the velocity of this zero\nsound goes to zero. For the non-symmetric case,\nthe spectrum is:\n!=r\n~2k4\n4m2\u0014+Nc(B\u0000A)(\u0014\u00001)k2\nm;(7)\nwhere\u0014\u0011(B\u0000A)2\nC2. In the experiment of Ref. [17],\n\u0014\u0018104andB\u0000A= 8:4 mK=N, from which\nwe estimate a sound speed of 3 \u0002103m=s. The\ndispersions of zero sound for symmetric and non-\nsymmetric cases are shown in Fig.5. Note that\nthe range of applicability of the linear approxima-\ntion decreases signi\fcantly for small C, where one\nof the condensate densities is small and the phase\n\ructuations grow.\nFIG. 5. Dispersion of zero sound as a function of\nwave vector in the direction of external magnetic \feld\nfor symmetric and non-symmetric cases, respectively.\nFor the non-symmetric case, we choose H= 1 kOe and\nd= 5\u0016m.\nDomain Wall.| Since the ground state of\nthe non-symmetric phase is doubly degenerate, it\ncan consist of domains with di\u000berent signs of \u000e\nseparated by domain walls. The width wof such a\ndomain wall is of the order ofq\n~2\n2mNcj\u0001j. For the\ndata of experiment [17] we estimate that w\u001910\n\u0016m, and a domain wall energy per unit area of\nabout 10\u00009J/m2.Discussion\nThe ground state wave function \t( z) gener-\nally is a superposition of two condensate ampli-\ntudes \t(z) = (cQeiQz+c\u0000Qe\u0000iQz)=p\nV, where\nc\u0006Q=p\nN\u0006Qei\u001e\u0006andVis the volume of the\n\flm. The spatial structure of \t( z) can be mea-\nsured by Brillouin Light Scattering (BLS), with\nintensity proportional to the condensate density\nj\tj2=nQ+n\u0000Q+2pnQn\u0000Qcos(2Qz+\u001e+\u0000\u001e\u0000).\nIn their recent experiment, Nowik-Boltyk el al\n[17] observed the interference pattern associated\nwith the ground state. They found that the con-\ntrast of this periodic spatial modulation is far be-\nlow 100%; of the order 3%. The present theory\ncan quantitatively explain this result. In their ex-\nperiment, Ref.[17] employ d= 5:1\u0016m andH= 1\nkOe. Then eq.(2) for A,BandCgivesA=\u00000:168\nmK=N,B= 8:218 mK=NandC=\u00000:203 mK=N,\nso \u0001<0. This corresponds to the non-symmetric\nphase, where for spontaneous symmetry-breaking\nwith\u000e=NQ\u0000N\u0000Q>0 the ratio of the num-\nbers of magnons in the two condensates isN\u0000Q\nNQ\u0019\nC2\n4(B\u0000A)2. The contrast is \f=j\tj2\nmax\u0000j\tj2\nmin\nj\tj2max+j\tj2\nmin. Since\nC\u001cBandN\u0000Q\u001cNQ, we have\f\u00192q\nN\u0000Q\nNQ\u0019\njCj\njB\u0000Aj. For the above values of A,BandC,\f\nis of order 2 :4%, in good agreement with experi-\nment. The smallness of C(andA) relative to B\nis associated with the large parameter d=l, where\nl=q\nD\n\u0019\rMis an intrinsic length scale of the sys-\ntem andl\u001810\u00006m. In terms of this parameter,\njCj\nB\u0018(l\nd)2=3.\nIn the experiment of [17] the contrast \freaches a\nsaturation value at a comparatively small pumping\npower, corresponding to the appearance of BEC.\nThis agrees with our expression for \f, which de-\npends only on \flm thickness dand magnetic \feld\nH. By varying dandH, the contrast can be\nchanged. Speci\fcally, in the symmetric phase,\n\f= 1; in the non-symmetric phase, \f < 1; and\nin the completely non-symmetric case with only\none condensate ( d=d\u0003),\f= 0. Therefore, mea-\nsurement of the contrast for di\u000berent values of d\nandHcan give complete information about the\nphase diagram of the system, for comparison with\nthe present theory.\nFig.6 plots C, \u0001 and\fas functions of the \flm\nthicknessdat \fxed magnetic \feld H= 1 kOe. For5\nsmalldthe system is in the high-contrast sym-\nmetric state. On increasing dabovedc= 0:07\n\u0016m, the sign of \u0001 changes, corresponding to the\ntransition from the symmetric to the low-contrast\nnon-symmetric phase. As dfurther increases, to\nd\u0003= 0:17\u0016m,Cchanges sign, and the total phase\n\b changes from \u0019to 0. Only at this point d\u0003does\nthe zero-contrast phase (with only one condensate)\nappear. Correspondingly, a characteristic cusp in\nthe contrast \fappears near d\u0003.\nFIG. 6. (a) Phase transition criterion \u0001 and inter-\naction coe\u000ecient Cas functions of thickness dfor\n\fxed magnetic \feld H= 1 kOe. (b) The contrast\n\f=j\tj2\nmax\u0000j\tj2\nmin\nj\tj2max+j\tj2\nminas a function of thickness dfor\nH= 1 kOe. S and NS denote symmetric and non-\nsymmetric phase, respectively.\nTo conclude, we have calculated the 4-th order\nmagnon-magnon interactions in the condensate of\na \flm of YIG, including magnon-non-conserving\nterms that are responsible for the coherence\nof two condensates. sFor u\u000eciently thin YIG\n\flms (d < 0:1\u0016) we predict a phase transition\nfrom symmetric to non-symmetric phase when\nthe magnetic \feld exceeds the modest value\nof 0:2 kOe. We also predict that within the\nnon-symmetric phase there is a thickness d\u0003(H)\nwhere the modulation in the observed interference\npattern should totally disappear.\nMethods\nMagnon Spectrum.| In a YIG \flm with an\nin-plane external magnetic \feld H, the magnon\ndispersion has been studied extensively [28{30]. At\nlow energies, YIG can be described as a Heisenbergferromagnet with large e\u000bective-spin S= 14:3 [19,\n24] on a cubic lattice. The Hamiltonian consists of\nthree parts:\nH=\u0000JX\nhi;jiSi\u0001Sj+HD\u0000\rHX\niSz\ni;(8)\nthe nearest neighbor exchange energy, the dipolar\ninteraction and the Zeeman energy. We take yto\nbe perpendicular to the \flm and the magnetic \feld\nto be in the plane along z. It is convenient to char-\nacterize the exchange interaction by the constant\nD= 2JSa2= 0:24 eV \u0017A2. The dipolar interaction\ncan be calculated using the method indicated in\nRefs.[20, 30]. The competition between the dipo-\nlar interaction and exchange interaction leads to\na magnon spectrum !kwith minima located at\nthe two points in 2D wave-vector space given by\nk= (0;\u0006Q) (i.e. along z), with an energy gap \u0001 0.\nFor \flm thickness d= 5\u0016m and magnetic \feld\nH= 1 kOe, we \fnd that Q= 7:5\u0002104cm\u00001and\n\u00010= 2:7 GHz. In the experiment of [17] Qwas\nfound to be about 3 :5\u0002104cm\u00001, i.e. about half\nthe predicted value. The reason for this may be\nassociated with a rather shallow energy minimum\nas a function of wavevector. In such a situation\nsmall corrections to our approximate formula can\nhave a large e\u000bect on the value of Q. The lowest\nband of the magnon spectrum can be calculated\nusing the Holstein-Primako\u000b transformation [32],\nwhich expresses the spin operator Sin terms of\nboson operators aanday.\nTo second order in aanday, the Hamiltonian\neq.(8) is:\nH0=X\nkh\nAkay\nkak+1\n2Bkaka\u0000k+1\n2B\u0003\nkay\nkay\n\u0000ki\n;(9)\nwith\nAk=\rH0+Dk2+\r2\u0019M(1\u0000Fk) sin2\u0012+\r2\u0019MFk\nBk=\r2\u0019M(1\u0000Fk) sin2\u0012\u0000\r2\u0019MFk (10)\nwhereFk\u0011(1\u0000e\u0000kd)=kdandMis the magneti-\nzation of the material (4 \u0019M= 1:76 kG). Here, \u0012\nis the angle between the 2D wave vector kand\nthe magnetic \feld direction ( z).H0of eq.(9)\nis diagonalized by the Bogoliubov transformation\nak=ukck+vkcy\n\u0000kwithuk= (Ak+~!k\n2~!k)1=2and\nvk=sgn(Bk)(Ak\u0000~!k\n2~!k)1=2, leading to the magnon\nspectrum:\n~!k= (A2\nk\u0000jBkj2)1=2: (11)6\nFig.1 gives the magnon spectrum along kzfor typ-\nical values of thickness dand magnetic \feld H.\nNumber of condensed magnons Nc=\nNQ+N\u0000Q.|Experimentally, the spin lattice\nrelaxation time is of order 1 \u0016s, whereas the\nmagnon-magnon thermalization time is of order\n100 ns; the magnons are long-lived enough to equi-\nlibrate before decaying, thus making BEC possible\n[11]. After the thermalization time the pumped\nmagnons go to a quasi-equilibrium state with a\nnon-zero chemical potential \u0016. The number of\npumped magnons Np=N(T;\u0016)\u0000N(T;0), where\nN(T;\u0016) =VP\nk1\ne(!k\u0000\u0016)=T\u00001, is determined by the\npumping power and the magnon lifetime. \u0016is a\nmonotonically increasing function of Npbut can-\nnot exceed the energy gap \u0001 0. Therefore, on fur-\nther pumping \u0016= \u0001 0and some of the pumped\nmagnons fall into the condensate. The equation\nNpc=N(T;\u00010)\u0000N(T;0) thus de\fnes the crit-\nical line for condensation. Since \u0001 0\u001cTand\nNp\u001cN(T;0) this equation can be satis\fed at\na rather high temperature. The total number of\ncondensed particles is [11, 31]\nNc=Np\u0000N(T;\u00010)+N(T;0) =N(T;\u0016)\u0000N(T;\u00010):\n(12)\nIn exactly 2D systems BEC formally does not ex-\nist since in the continuum approximation the sum\ninN(T;\u0016) diverges. However, for strong enough\npumping the chemical potential approaches ex-\nponentially close to the energy gap: \u0001 0\u0000\u0016\u0019\n\u00010exp(\u0000Np=N0), whereN0=VTm= ~2. For\nNp=N0>ln(T=\u00010) all pumped magnons occupy\nonly one or two states \u0006Q.\nEq.(12) determines only the total number of par-\nticles in the condensate. The distribution of the\ncondensate particles between the two minima re-\nmains undetermined in the quadratic approxima-\ntion. To resolve this issue we have shown that it is\nnecessary to consider the fourth order terms in the\nHolstein-Primako\u000b expansion of the exchange and\ndipolar energy. Observe that terms of third order\noccur in this expansion of the dipolar interaction,\nbut since the total momentum must be zero, such\nterms vanish for the condensate momenta (0 ;\u0006Q).\nZero Sound.| We now provide details about\ncalculating the zero sound spectrum. We consider\nsmall deviations from the static symmetric solution\nnQ=n\u0000Q=nc=2,\u001e+=\u0019\u0000\u001e\u0000= 0, so that\nn\u0006Q=nc=2 +\u000en\u0006with\u000en+=\u0000\u000en\u0000=\u000en=2 and\u000e\u001e+=\u0000\u000e\u001e\u0000=\u000e\u001e=2. Then\nE=Z\ndr\u0010~2\n2m(jr\t+j2+jr\t\u0000j2))\n+AV(j\t+j4+j\t\u0000j4+ 2BVj\t+j2j\t\u0000j2\n+CV(\t+\t\u0000+ \t\u0003\n+\t\u0003\n\u0000)(j\t+j2+j\t\u0000j2)\u0011\n;\nOn linearizing, the energy reads:\nE=Z\ndr\u0010~2\n4mncjr\u000enj2+~2nc\n4mjr\u000e\u001ej2+\u0001V\n2\u000en2\u0011\n:\nUsing the commutation relation [ \u000e\u001e;\u000en ] =i, and\nthe equation of motion i~\u000e_\u001e= [\u000e\u001e;H ], we obtain:\n~@\u000e\u001e\n@t=\u0000~2\n2mncr2\u000en+ \u0001V\u000en; (13)\n~@\u000en\n@t=~2\n2mncr2\u000e\u001e: (14)\nTaking Fourier transforms of the above two equa-\ntions in coordinate and time, one arrives at the\ndispersion relations in Eq.(6).\n[1] Kapitza, P. Viscosity of Liquid Helium Below the\n\u0015-point. 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Ginzburg-Landau Model of\nBose-Einstein Condensation of Magnons. Phys.\nRev. B 81, 024418 (2010).\n[24] Troncoso, R. E. & Nunez, A. S. Dynamics and\nSpontaneous Coherence of Magnons in Ferromag-\nnetic Thin Films. J. Phys.: Condens. Matter 24,\n036006 (2012).\n[25] Cherepanov, V., Kolokolov, I. & L'vov, V. The\nSaga of YIG: Spectra, Thermodynamics, Inter-\naction and Relaxation of Magnons in a Complex\nMagnet. Phys. Rep. 229, 81-144 (1993).\n[26] Slavin, A. N. & Rojdestvensky, I. V. \\Bright\"\nand \\Dark\" Spin-Wave Envelope Solitons in\nTangentially Magnetized Yttrium-Iron Garnet\nFilms. IEEE Transactions on Magnetics 30, 37-\n45 (1994).\n[27] Krivosik, P. & Patton, C. E. Hamiltonian Formu-\nlation of Nonlinear Spin-Wave Dynamics: Theory\nand Applications. Phys. Rev. B 82, 184428 (2010).\n[28] Damon, R. W. & Eshbach, J. R. Magnetostatic\nModes of a Ferromagnet Slab. J. Phys. Chem.\nSolids 19, 308-320 (1961).\n[29] Sparks, M. Ferromagnetic Resonance in Thin\nFilms. I. Theory of Normal-Mode Frequencies.\nPhys. Rev. B 1, 3831-3856 (1970).\n[30] Kalinikos, B. A. Excitation of Propagating Spin\nWaves in Ferromagnetic Films. IEEE Proc., Part\nH: Microwaves, Opt. Antennas ,127, 4-10 (1980).\n[31] Bunkov, Y. M. & Volovik, G. E. Spin Super\ruidity\nand Magnon BEC. arXiv: 1003.4889v1.\n[32] Madelung, O. Introduction to Solid-State Theory\n(Springer, Berlin, 2008).\nAcknowledgements\nThis work was partly supported by the grant of\nDOE DE-FG02-06ER46278. We are thankful to H.\nSchulte attracting our attention to the experimen-\ntal work [17] and to S. M. Rezende, I.S. Tupitsyn\nand V. E. Demidov for useful discussion.\nAuthor Contributions\nAll authors contributed to the theoretical analysis\nand the preparation of the manuscript.\nAdditional Information\nThere are no competing \fnancial interests."    },    {        "title": "2311.17821v2.Nonlinear_erasing_of_propagating_spin_wave_pulses_in_thin_film_Ga_YIG.pdf",        "content": "Nonlinear erasing of propagating spin-wave pulses in thin-film Ga:YIG\nD. Breitbach,1,a)M. Bechberger,1,a)B. Heinz,1A. Hamadeh,1J. Maskill,1K. O. Levchenko,2B. Lägel,1C.\nDubs,3Q. Wang,4R. Verba,5and P. Pirro1\n1)Fachbereich Physik and Landesforschungszentrum OPTIMAS,\nRheinland-Pfälzische Technische Universität Kaiserslautern-Landau, D-67663 Kaiserslautern, Germany\n2)Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Wien, Austria\n3)INNOVENT e.V. Technologieentwicklung, D-07745 Jena, Germany\n4)School of Physics, Huazhong University of Science and Technology, 430074 Wuhan, China\n5)Institute of Magnetism, Kyiv 03142, Ukraine\n(*Electronic mail: dbreitba@rptu.de)\n(Dated: 2 February 2024)\nNonlinear phenomena are key for magnon-based information processing, but the nonlinear interaction between two\nspin-wave signals requires their spatio-temporal overlap which can be challenging for directional processing devices.\nOur study focuses on a gallium-substituted yttrium iron garnet film, which exhibits an exchange-dominated dispersion\nrelation and thus provides a particularly broad range of group velocities compared to pure YIG. Using time- and space-\nresolved Brillouin light scattering spectroscopy, we demonstrate the excitation of time-separated spin-wave pulses at\ndifferent frequencies from the same source, where the delayed pulse catches up with the previously excited pulse and\noutruns it due to its higher group velocity. By varying the excitation power of the faster pulse, the outcome can be finely\ntuned from a linear superposition to a nonlinear interaction of both pulses, resulting in a full attenuation of the slower\npulse. Therefore, our findings demonstrate the all-magnonic erasing process of a propagating magnonic signal, which\nenables the realization of complex temporal logic operations with potential application, e.g., in inhibitory neuromorphic\nfunctionalities.\nThe last decade has seen an increasing interest in alterna-\ntive computing schemes such as neuromorphic computing1,2,\nparticularly in the context of the growing computational de-\nmands and the physical limitations of current semiconductor\ntechnologies3. Contemporary research has focused consider-\nable attention on wave-based logic concepts, which offer addi-\ntional degrees of freedom, a non-Boolean data carrier, and ef-\nfects such as interference4,5. In this context, spin-wave based\ninformation processing stands out6–8, not least because spin\nwaves exhibit efficient intrinsic nonlinear effects, fundamen-\ntal to any logic operation. This feature in conjunction with\ntheir low power footprint makes spin waves an efficient data\ncarrier for future information processing schemes such as neu-\nromorphic computing9,10.\nTo take advantage of nonlinear effects in the spin-wave do-\nmain requires the spatio-temporal overlap, i.e., the superposi-\ntion, of spin-wave signals, which can be challenging for direc-\ntional processing with propagating spin-waves. Existing spin-\nwave logic typically makes use of structural design to achieve\nsuperposition, e.g., by using waveguides11–14or delay struc-\ntures such as ring resonators15to recouple previous signals.\nHowever, spin waves typically show a pronounced dispersion\nand therefore offer intrinsic means to achieve signal superpo-\nsition in a more elegant way. Here, we study the superpo-\nsition and interaction of separate spin-wave signals emitted\nfrom the same source with a time delay, by exploiting their\ndifference in group velocity, i.e. their dispersive properties.\nFor this purpose, the ultralow spin-wave damping material yt-\ntrium iron garnet16–18(YIG) is well suited since it provides\nsufficiently long propagation distances as well as functional\na)D. Breitbach and M. Bechberger contributed equally to this work.nonlinear properties. Moreover, it features a positive nonlin-\near frequency shift when magnetized out-of-plane, a property\nthat facilitates powerful logic functionalities19. However, a\ndrawback of this system is the limited range of group veloc-\nities in the wavevector regime accessible via standard means\nof RF excitation. Furthermore, operation in the out-of-plane\nmagnetization configuration poses a challenge for the optical\ndetection of the spin-wave signals.\nHere we circumvent these limitations by employing a\nGallium-substituted YIG20–25film instead. This material\nexhibits a nearly compensated magnetic configuration, re-\nsulting in a low saturation magnetization of MS=20.2mT,\nwhich leads to an almost vanishing dipolar contribution to\nthe spin-wave dispersion. More importantly, Ga:YIG shows\na large perpendicular magnetic anisotropy (PMA) and an ex-\nchange length which is several times larger compared to pure\nYIG25,26. Therefore, the spin-wave dispersion in this mate-\nrial is exchange-dominated even at low wavevectors and of\nalmost isotropic, quadratic shape, which is analogous to the\ndispersion of a non-relativistic particle in quantum mechanics.\nThus, the group velocity has a linear wavevector dependence\nand covers a broad range with fine tunability compared to pure\nYIG. This allows for the excitation of spin-wave signals with\na particularly broad difference in group velocities. Further-\nmore, due to the PMA, the material shows a positive nonlin-\near frequency shift coefficient27when magnetized in the in-\nplane configuration, easily accessible by optical means such\nas Brillouin light scattering spectroscopy (BLS). In this work,\nwe utilize these unique features to demonstrate the superpo-\nsition of propagating spin-wave signals which are excited by\nthe same source but with a time delay. Furthermore, we study\ntheir nonlinear interaction with regard to the functionality of\nspin-wave erasing - a process where one temporal signal an-\nnihilates another.arXiv:2311.17821v2  [cond-mat.mtrl-sci]  1 Feb 20242\n25 µm\n267 nm485 nmCPW\nFocal\nspot\nDetector\nLaserTandem \nFabry-Pérot \ninterferometer\nλ = 457 nmBLS time-\ntraces\n0 50 100 150 200BLS intensity (a. u.)\nTime t (ns) \nH H\nFIG. 1. Colorized SEM micrograph of the CPW antenna under\ninvestigation, placed on top of the Ga:YIG film and a schematic of\nthe applied time-resolved BLS microscope. In the experiment, an\nin-plane bias magnetic field of µ0Happ≈86mT is applied either par-\nallel or perpendicular to the scan direction to magnetize the Ga:YIG\nin-plane. Timetraces of an exemplary BLS measurement of a propa-\ngating spin-wave pulse excited at f=1.4GHz are shown to illustrate\nthe measurement process.\nFigure 1 shows the investigated structure and the experi-\nmental setup. The magnonic material under study is an 59 nm\nthin Ga:YIG film, grown by liquid phase epitaxy (LPE)16,17\nadjusted for the deposition of sub-100-nm, single crystalline\nY3Fe5 –xGaxO12films. Its material properties have been thor-\noughly studied in a previous work25. It is characterized\nby a strong PMA of µ0Hu= (94.1±0.5)mT, an exchange\nlength of λex=91.9nm, which is substantially larger com-\npared to pure YIG and the low Gilbert-damping parameter of\nα= (6.1±0.6)×10−4. Unless stated otherwise, for all pre-\nsented results an external in-plane magnetic field of µ0H⊥=\n(86±2)mT is applied to ensure that the magnetization of the\nfilm is aligned in the film plane (see supplementary material,\nSec. I).\nOn top of the film, a coplanar waveguide (CPW) antenna\nis structured for spin-wave excitation, composed of Ti/Au\n(10 nm/70 nm). The antenna is fed by a microwave setup con-\nsisting of two microwave sources connected to two individual\nmicrowave switches which are separately triggered by a pulse\ngenerator. The two circuits are then merged by a Wilkinson\ntype power combiner such that the backward isolation is en-\nsured. Using this setup, short microwave pulses of approxi-\nmately τrf≈15ns are generated to excite a propagating spin-\nwave packet with a certain carrier frequency. The accessi-\nble frequencies are defined by the excitation efficiency of the\nCPW antenna, which is a function of the wavevector kof the\nWavevector k (rad/µm)Group velocity v (µm/ns)G\nFrequency f (GHz)\n0 5 10 15 201.01.52.02.53.0\nDE 88.54  mT BV 85.76  mT\nMeasurement DE\nMeasurement BV\n0.000.250.500.751.001.25\nWavevector k (rad/µm)\n-1\nWavevector k (rad  µm)Wavevector k (rad/µm)\nWavevector k (rad/µm)-1\nGroup velocity v (µm ns) G\n-1\nGroup velocity v (µm ns ) G-1\nGroup velocity v (µm ns ) G\n-1\nGroup velocity v (µm ns) GFrequency f (GHz)\n0\n00\n05\n55\n510\n1010\n1015\n1515\n1520\n2020\n201.01.52.02.53.0\nExperiment\nExperimentCalculation\n0.000.000.00\n0.000.25 0.250.25\n0.250.50 0.500.50\n0.500.750.750.75\n0.751.00\n1.001.00\n1.001.25\n1.251.25\n1.25H \nH H ExperimentH H \nH Frequency f (GHz)Frequency f (GHz)\nFrequency f (GHz)(a)(a)\n(a)\n(b)(b)\n(b)\n1.5\n1.01.5\n2.02.0\n2.52.5\n3.03.0\n1.52.02.53.0\nCalculationH \nExperimentH H \nCalculationCalculationH H \nH FIG. 2. (a)Group velocity as a function of the spin-wave wavevec-\ntork, theoretical curve (black) and experimental data (points). Each\ndatapoint is extracted from a 1D spatial BLS scan, respectively. The\npulses were excited using an excitation pulse length of τRF≈10ns\nand varying microwave powers to account for the excitation effi-\nciency of the CPW (see supplementary material, Sec. III). The\ndatapoints were converted from excitation frequency to the corre-\nsponding wavevector using the dispersion relation shown in panel\n(b), calculated according to Ref25. The external field values are\nµ0H⊥=88.54mT and µ0H∥=85.76mT.\nexcited spin waves (see supplementary material, Sec. II). The\nexperimental investigation of the spin-wave propagation is\nstudied by applying time- and space-resolved BLS28, focusing\na laser beam of λ=457nm with a laser power of P=3.0mW\non the sample. After the spin-wave pulse is excited, it propa-\ngates away from the CPW antenna. As schematically depicted\nin Fig. 1, time-resolved BLS measurements are performed at\ndifferent distances to the antenna, allowing to track the spin-\nwave pulse in time and space and to extract its group velocity.\nThe resulting group velocities vgare shown in Fig. 2 (a)\nas a function of the excited spin-wave wavevector k. The\ndata was collected under two field configurations, k⊥M\nandk∥M. In addition, the dispersion relation of both cases\nis depicted in Fig. 2 (b). It can be seen from these two\nconfigurations at approximately the same field values Happ\nthat the spin-wave properties are highly isotropic, confirming\nthe exchange-dominated character of this system. Figure 2\nalso shows the quadratic shape of the dispersion relation\nω(k). This results in the group velocity vg=∂ω\n∂kbeing almost\nlinearly dependent on the spin-wave wavevector k, which is in\ngood agreement with the obtained data. Due to this property,\nthe system exhibits a particularly large range of group\nvelocities in the low-wavevector regime compared to pure\nYIG, ranging from approximately vg=0 atk=0 to above\nvg=1µmns−1atk=15radµm−1. This property allows for\nan interesting experiment: spin-wave pulses with strongly\ndifferent group velocities excited from the same source could\nstill come into superposition even when excited at different3\nf = 2200 MHz B050100\n0.040.060.080.10.20.40.60.81BLS intensity (arb. u.)\n050100 Time t (ns)\nf+fA  B\n0 5 10 15 20 25050100\nPosition x (µm) f = 1110 MHz A\n050100\nf A\nf BTime t (ns) Time t (ns) Time t (ns)(a)\n(b)\n(c)\n(d)510152025Position x (µm)\n510152025Position x (µm)\n510152025Position x (µm)0.040.060.080.10.20.40.60.81BLS intensity (arb. u.)\nf = 2.20  GHz B\nf+fA  Bf = 1.1 1 GHz A\n25\nf+fA  B(a)\n(b)\n(c)\n(d)\n0510152025\n0 25 50 75 100 125Position x (µm)\nTime t (ns)v= 0.30  µm/ns g\nv= 0.96  µm/ns g\nfBLSI   BLSfBLSI   BLSfBLSI   BLSfBLSI   BLS\nFIG. 3. Time-resolved BLS measurements of two short pulses Aand\nBexcited at different frequencies fA=1.11GHz and fB=2.2GHz\nand with a time delay of ∆t0≈32ns. (a)Only excitation of pulse\nA.(b)Only excitation of pulse B.(c)Combined excitation of pulses\nAandBwith same parameters as in previous panels. All color plots\nhave the same color scale and normalization. (d)Same as (c)but only\nBLS frequencies around fAare shown (marked in red in the inset).\nParameters: PA=0dBm and PB=15dBm.\ntimes. Due to the high isotropy of the system, the following\ninvestigations are carried out in k⊥Mconfiguration.\nFigure 3 (a)shows the time- and space-resolved measure-\nment of a spin-wave pulse excited close to the band bot-\ntom at fA=1.11GHz, which corresponds to a low wavevec-\ntor of kA≈(5±1)radµm−1. On the other hand, Fig. 3 (b)\ndepicts the measurement of a spin-wave pulse that was ex-\ncited with a frequency of fB=2.2GHz and a wavevector of\nkB≈(15.0±0.3)radµm−1. It is noted that while the fre-\nquency fAis required to be close to the band bottom, the ex-\nact choice of fBis not crucial to the effect under study, but\nrather an experimental trade-off between a high group veloc-\nity and a feasible excitation efficiency of the CPW antenna\n(see supplementary material, Sec. IV). Close to the CPW\nantenna, the pulses have a duration (FWHM) of τA=17ns\nandτB=14ns. The pulse timings and their respective width\n(FWHM) are marked on the color plot as extracted by fitting\nremaining \n         intensity          Pulse A\n15 Off\n 2 dBm\n 8 dBmPower P\n-5 0 5 100.00.20.40.60.81.0\nPower P of pulse B (dBm)B Level without fast pulse\nLevel without any pulse\n0 50 100 1500.00.51.0\nBLS intensity (arb. u.) \nTime t (ns) BLS intensity of pulse A\n(arb. u.)B(a) (b)\nfBLSI   BLSFIG. 4. (a)BLS intensity of slow pulse A, extracted for BLS fre-\nquencies around fA, as a function of the applied microwave power\nPBof fast pulse B(see also supplementary material, Sec. IV). (b)\nTimetraces of pulse Aextracted for BLS frequencies around fAfor\ndifferent microwave powers PB. Parameters: PA=0dBm, position:\nx=20.5µm.\nthe time-dependent data with Gaussian functions. In the fol-\nlowing, these two pulses will be referred to as pulse AandB,\nrespectively.\nIt is evident from this measurement that pulse AandBsig-\nnificantly differ in their group velocity. While pulse Awith\nvg=0.30µmns−1is much slower, pulse Bpropagates with\nvg=0.96µmns−1, which is also the main reason for their dif-\nferent decay behavior (see supplementary material, Sec. V).\nIn the following, both pulses are combined in a single ex-\nperiment while all other parameters are kept constant. The re-\nsult is shown in Fig. 3 (c)and demonstrates that, even with a\ntime delay of ∆t0=32ns in their excitation, pulse Bcatches up\nwith pulse Aapproximately 14 ns after its excitation, leading\nto a crossing point. However, pulse Avanishes when being\npassed by the faster pulse B, indicating a direct interaction be-\ntween both pulses. This becomes evident in Fig. 3 (d), which\nshows the combined measurement of both pulses ( fA+fB),\nbut extracted only for BLS frequencies around fA. The fast\npulse Bprovides a moving spatial barrier for spin waves in the\nlow frequency range and erases the slow pulse Aincluding its\npulse tail, which would otherwise reach the furthest point of\nthe measurement (see also supplementary material, Sec. VI).\nThis means that spin-wave signals generated at different times\nbut from the same source can still be brought to superposition\nand even nonlinearly interact.\nTo study the nonlinear interaction between the pulses, the\nremaining intensity of pulse Aat position x=20.5µm is\nconsidered as a function of the applied microwave power PB\nof the fast pulse B, see Fig. 4 (a)and(b). As PBincreases,\nthe intensity of the slow pulse decreases from being almost\nunaffected to almost full attenuation. While pulse Aretains\naround 90% of its original intensity for PB=−5dBm, it is\nreduced to only 10% at PB=13dBm. Varying the excitation\npower of the pulse Btherefore enables either an almost linear\nsuperposition of both pulses or a nearly complete erasure of\nthe slower pulse A. The results of Figs. 3 and 4 are confirmed\nby micromagnetic simulations (see supplementary material,\nSec. VII).4\n0.51.01.52.02.53.0\nTime t (ns) BLS frequency f (GHz)BLS\n0.000.050.100.150.30\nBLS intensity (arb. u.)\n0 25 50 75(c)\n>0.35(b)\nPower P (dBm) cw−20 −15 −10 −5 0 50.81.01.21.41.61.8\n \n Minimum spin-wave frequencyBLS frequency f  (GHz)BLS\n f = 1.1 1 GHz A c = 0B\n c = 0.1 1 = c B crit\n c = 0.15 > cB crit\nf = 1.1 1 GHz A(a)\n0 2 4 6 8 100.81.01.21.41.61.8Frequency f (GHz)\nWavevector k (rad/µm)xBLS\nx(cw)fB\nfA P = 15.0 dBmB\n0.25\n0.20\nFIG. 5. Effect of the nonlinear frequency cross-shift of the mode kB=15radµm−1(not shown) on the linear spin-wave dispersion. (a)\nCalculated spin-wave dispersion relation for the linear case (red) and cases with increased amplitude cBat critical (blue) and overcritical\nlevel (green). Calculated for µ0Heff=88.5mT which includes effects of crystal anisotropy to match the experiment. (b)Measured minimal\nfrequency of the thermal spin-wave signal (BLS) as a function of the applied, continuous microwave power PcwatfB=2.2GHz. Measured at\nposition x(cw)=4.0µm. (c)Time-resolved BLS measurement of pulses AandBbefore the pulse crossing at the power levels PA=0dBm and\nPB=15dBm, measured at position x=4.0µm.\nTo investigate the physical process behind this mechanism,\nthe effect of the nonlinear frequency shift on the linear dis-\npersion is considered. The four-magnon interaction between\ntwo waves with wavevectors kandk′results, in particular, in\na total shift of the spin-wave frequency of mode k29,\n˜ωk=ωk+Tk|ck|2+2Tkk′|ck′|2, (1)\nwhere ωk=2πfkis the linear spin-wave frequency, ckand\nck′are the spin-wave amplitudes of the two modes, Tkis the\nself nonlinear frequency shift coefficient and Tkk′is the cross\nnonlinear frequency shift. Calculations with the Hamiltonian\nformalism30result in a positive cross-shift of TkAkB/(2π)≈\n3.6GHz, as expected for an in-plane magnetized film with\nhigh PMA. This contribution is crucial for the comprehension\nof the experimental results – pulse Bshifts the entire disper-\nsion to higher frequencies, in particular, the bottom of the dis-\npersion, see Fig. 5 (a). The dispersion relations shown are\ncalculated according to Eq. (1) using different spin wave am-\nplitudes cB:fk=f(k,ck→0,cB). The dispersion in the ab-\nsence of pulse B, i.e. the linear dispersion, is shown in red.\nIn the presence of pulse B, it is increasingly shifted to higher\nfrequencies for an increasing amplitude cB. When a critical\nvalue of cB= 0.11 is reached, corresponding to about 9 de-\ngree magnetization precession angle, the minimal frequency\nof the dispersion relation is shifted up to fA=1.1 GHz. For\nhigher amplitudes cB, this low-frequency mode no longer ex-\nists and pulse Abecomes nonresonant. Hence, as long as the\nfrequency is conserved, this prevents further propagation of\nthe slow pulse A. This mechanism occurs when the fast pulse\npasses the slower pulse with a spin-wave amplitude cB>ccrit,\nwhich results in the cut-off of pulse A. Note that according to\nEq. (1) this mechanism does not depend on the phase of pulse\nB, but only on its intensity |cB|2.\nDue to the inaccessibility of absolute spin-wave amplitudes ck\nby means of BLS spectroscopy, only qualitative comparisons\ncan be made between the experiment and this calculation.\nThis can be achieved using a continuous microwave excita-tion at the frequency fBwith increasing excitation amplitude.\nThis allows to investigate the effect of the nonlinear shift on\nthe thermal spin-wave population, i.e. incoherent, thermally\nexcited spin waves that can be detected by BLS at room tem-\nperature. Fig. 5 (b)shows the measured minimal frequency\nof the thermal spin-wave spectrum as a function of the con-\ntinuous excitation power. The minimum frequency increases\ntofAat a power of Pcw≈0dBm and increases even further\nfor higher amplitudes. This experiment demonstrates that us-\ning experimentally feasible spin-wave amplitudes, the linear\nspin-wave frequency can be shifted by several 100MHz.\nSince the previous experiments were done using pulsed RF\nexcitation, the power PBcannot be directly compared to the\ncontinuous power levels in Fig. 5 (b). Fig. 5 (c)shows a\nfrequency- and time-resolved BLS spectrum of the pulses A\nandBfor an excitation power of PB= 15 dBm before the pulse\ncrossing. At this power level, pulse Bcan almost completely\nerase pulse A, as was shown in Fig. 4. It is visible that the\nthermal spin-wave spectrum shifts to frequencies higher than\nfAin the presence of pulse B, which prevents further propa-\ngation of the slow pulse. This interpretation is in agreement\nwith the power-dependent attenuation in Fig. 4, as the nonlin-\near frequency shift is proportional to the spin-wave intensity,\nwhich increases with the applied microwave power (see Fig.\n5(b)). This mechanism occurs when the fast pulse passes the\nslower one, which results in the cut-off of pulse A. Effectively,\nthe faster pulse is an eraser for a spin-wave signal that was sent\nearlier.\nIt remains an open question, however, how the energy of\npulse Ais dissipated or redistributed during the erasing pro-\ncess. Potential channels for this could include reflection or\neven scattering perpendicular to the propagation direction, or\nscattering to the whole spin-wave spectrum as the mode be-\ncomes nonresonant. In our current study, no clear signature of\nsuch energy transfer could be found. The difficulty to unravel\nthe precise mechanisms lies not only in the two-dimensional\nextension of the investigated film, but also the wavevector de-\npendent and -limited BLS sensitivity, which can effectively5\nobscure energy scattering to different frequency channels. It\nis noted that in Fig. 5 (c), spin-waves at intermediate frequen-\ncies occur during the presence of pulse B. However, these are\ncaused by the intense pulse Band are not related to an energy\ntransfer of pulse A, as they also occur in absence of pulse\nA. The clarification of this question requires further experi-\nments and extensive simulations as well as studying a confined\nwaveguide.\nIn conclusion, we demonstrated that temporally separated\nspin-wave signals excited by the same source can be brought\ninto direct superposition by exploiting a systems intrinsic dis-\npersive properties. We achieved this by employing Ga:YIG,\na low-damping material with an exchange-dominated disper-\nsion relation and a high range of excitable group velocities\nwith fine tunability. Utilizing a considerable difference in\ngroup velocity, two temporally separated spin-wave pulses\ncan be excited such that the second pulse catches up with\nthe first, bringing them into direct superposition. As a\nproof-of-principle for the interaction between the pulses,\nwe have demonstrated the tunable nonlinear erasing of the\nslow spin-wave pulse by the faster pulse. This is mediated\nby a positive nonlinear frequency shift induced by the fast\npulse, which locally shifts the spin-wave dispersion to higher\nfrequencies. This renders the slow pulse nonresonant, hence\nhindering its further propagation. Our work demonstrates\nthe direct nonlinear interaction of signals from separated\ntemporal inputs which is key to the realization of complex\ntemporal logic operations and is essential for recurrent\nneuromorphic computing functionalities such as the fading\nmemory. Furthermore, the ability to erase previous informa-\ntion is potentially applicable as an inhibitory component in\nneuromorphic computing.\nSee the supplementary material for a additional data on\nthe in-plane magnetization of the Ga:YIG film, on the exci-\ntation efficiency of the CPW antenna and for details on the\nselected power levels for the group velocity measurements.\nFurthermore, the supplementary material contains details on\nthe choice of excitation frequencies, analysis of the spin-wave\ndecay and additional time-domain data on the erasing process.\nLastly, comparative micromagnetic simulations are shown.\nACKNOWLEDGMENTS\nThis research was funded by the European Research Coun-\ncil within the Starting Grant No. 101042439 \"CoSpiN\", by\nthe Deutsche Forschungsgemeinschaft (DFG, German Re-\nsearch Foundation) within the Transregional Collaborative\nResearch Center—TRR 173–268565370 “Spin + X” (project\nB01) and the project 271741898. The authors acknowledge\nsupport by the Max Planck Graduate Center with the Johannes\nGutenberg-Universität Mainz (MPGC). R.V . acknowledges\nsupport by MES of Ukraine (project 0124U000270). K. L. ac-\nknowledges the Austrian Science Fund FWF for the support\nthrough Grant ESP 526-N \"TopMag\". Q. W. acknowledges\nsupport from the National Key Research and Development\nProgram of China (Grant Nos. 2023YFA1406600) and Na-\ntional Natural Science Foundation of China, the startup grantof Huazhong University of Science and Technology (Grants\nNo. 3034012104).\nAUTHOR DECLARATIONS\nConflict of Interest\nThe authors declare no competing interests.\nDATA AVAILABILITY\nThe data that support the findings of this study are available\nfrom the corresponding author upon reasonable request.\nREFERENCES\n1D. Markovi ´c, A. Mizrahi, D. Querlioz, and J. Grollier, Nat. Rev. Phys. 2,\n499 (2020).\n2J. Torrejon, M. Riou, F. A. Araujo, S. Tsunegi, G. Khalsa, D. Querlioz,\nP. Bortolotti, V . Cros, K. 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B 82, 184428 (2010)."    },    {        "title": "2112.07264v1.Robust_perpendicular_magnetic_anisotropy_in_Ce_substituted_yttrium_iron_garnet_epitaxial_thin_films.pdf",        "content": "arXiv:2112.07264v1  [cond-mat.mtrl-sci]  14 Dec 2021Robust perpendicular magnetic anisotropy in Ce substitute d yttrium iron\ngarnet epitaxial thin films\nManik Kuila,1Archna Sagdeo,2,3Lanuakum A Longchar,4R.J.Choudhary,1S.Srinath,4and V.Raghavendra\nReddy1,a)\n1)UGC-DAE Consortium for Scientific Research, University Cam pus, Khandwa Road, Indore 452001,\nIndia.\n2)Synchrotrons Utilization Section, RRCAT, Indore India.\n3)Homi Bhabha National Institute, Training School Complex, A nushakti Nagar, Mumbai 400094,\nIndia.\n4)School of Physics, University of Hyderabad, Hyderabad-500 046, India\nCerium substituted yttrium iron garnet (Ce:YIG) epitaxial thin films a re prepared on gadolinium gallium\ngarnet (GGG) substrate with pulsed laser deposition (PLD). It is ob servedthat the films grown on GGG(111)\nsubstrate exhibit perpendicular magnetic anisotropy (PMA) as com pared to films grown on GGG(100) sub-\nstrate. The developed PMA is confirmed from magneto-optical Ker r effect, bulk magnetization and ferromag-\nnetic resonance measurements. Further, the magnetic bubble do mains are observed in the films exhibiting\nPMA. The observations are explained in terms of the growth directio n of Ce:YIG films and the interplay of\nvarious magnetic anisotropy terms. The observed PMA is found to b e tunable with thickness of the film and\na remarkable temperature stability of the PMA is observed in all the s tudied films of Ce:YIG deposited on\nGGG(111) substrate.\nKeywords: Garnet thin films, epitaxial thin films, perpendicular magn etic anisotropy, MOKE.\nI. INTRODUCTION\nCerium substitute yttrium iron garnet (Ce:YIG) is a\nwidelyknownmaterialbecauseofitsimportantmagneto-\noptical (MO) properties over a wide range of wavelength\nspectrum (infrared to near UV-visible)1–4. The strong\nMO activity and high transparency usually promotes\nCe:YIG as a promising candidate for non-reciprocal de-\nvice applications2,5–7. Therefore, most of the studies\nwere being focused for improving MO properties, such\nas Faraday and Kerr rotation in polycrystalline and/or\nepitaxial Ce:YIG films8,9. However, another interesting\naspect in these kind of films is the development of per-\npendicular magnetic anisotropy (PMA). The fact that\nthe garnet films are insulating in nature and exhibit low\nmagnetic losses, the development of garnet films with\nPMA has opened immense applications in the fields of\nspintronics, magneonic, spin caloritronics etc10–14. For\nexample, in pure and substituted YIG pure spin currents\nare observeddue to the inverse spin Hall effect in a heavy\nmetal over layer15, an electrical control of magnetization\nhas also been realized in Tm 3Fe5O12(TmIG) with ad-\njacent heavy metal layer due to spin-orbit-torque (SOT)\ninduced effect16etc.\nRecently we have studied epitaxial Ce:YIG films in re-\nflection mode with longitudinal magneto-optical Kerr ef-\nfect (MOKE)andshownthatonecantune theMOactiv-\nity by preparing the films in different O2partial pressure\n(OPP) during deposition and also by varying the thick-\nness (deposition time) at a given OPP17. However, we\ncould not observe PMA in these films. In fact, there\na)Electronic mail: varimalla@yahoo.com; vrreddy@csr.res. inseems to be only one report dealing with the PMA in\nCe:YIG films in literature, wherein the PMA is reported\nto be stable only at temperatures below 150 K18. In\nview of this, in the present work we have undertaken the\ndevelopmentofPMAin Ce:YIG films byfurther optimiz-\ningthe depositionconditionswith pulsed laserdeposition\n(PLD), especially the substrate temperature during de-\nposition and the substrate orientation. Moreover, it is\nobserved that the PMA is stable over a wide temper-\nature range ( ≤400 K) enabling the possibility of room\ntemperature based applications of these materials.\nII. EXPERIMENTAL DETAILS\nPulsed laser deposition (PLD) technique is used to ab-\nlate stoichiometric Ce 1Y2Fe5O12(Ce:YIG) target, the\nsametargetthathasbeenusedinourearlierworkdealing\nwith the Ce:YIG films prepared at different OPP17. The\nfilms are grownin optimized O 2partial pressure of about\n110mTorrandfixedsubstratetemperatureof780oC.The\nother parameters i.e., laser influence, repetition rate and\ntarget to substrate distance are 2.5 J/cm2, 6 Hz, 5 cm,\nrespectively. All these conditions are kept same for ev-\nery deposition except deposition time to vary the thick-\nness. Three films of about 10, 30 and 65 nm thickness\nare prepared on Gd 3Ga5O12(GGG) substrates. For ev-\nery deposition, both the (100) and (111) oriented GGG\nsubstrates are mounted side by side, so that the films are\ndepositedinsameconditions. Thecrystallinestructureof\nthefilmsareanalyzedusingx-raydiffraction(XRD) mea-\nsurements carried out at angle dispersive X-ray diffrac-\ntion (ADXRD) beamline (BL-12) of Indus-2, RRCAT In-\ndia. The x-ray reflectivity (XRR) and reciprocal space\nmapping (RSM) are carried out using Bruker D8 diffrac-2\nTABLE I. Notations of the prepared films and the obtained\nstructural parameters. For example, Ce:YIG10-111 stands f or\nthefilm ofabout 10nmthickness anddeposited on GGG(111)\nsubstrate. All the films are prepared at the same conditions\nexcept the deposition time i.e., film thickness. ’t’ is the th ick-\nness (±0.1 nm), ��is the roughness ( ±0.1 nm). aoutis the\nout-of-plane lattice parameter ( ±0.0001 nm) obtained from\nFig. 4.\nNotation Substrate t ρ a out\n(nm) (nm) (nm)\nCe:YIG10-111 GGG(111) 10.0 0.5 1.2383\nCe:YIG30-111 GGG(111) 30.0 0.9 1.2364\nCe:YIG65-111 GGG(111) 65.7 1.0 1.2350\nCe:YIG10-100 GGG(100) 9.2 0.5 -\nCe:YIG30-100 GGG(100) 29.9 0.9 -\nCe:YIG65-100 GGG(100) 64.0 0.2 -\ntometer equipped with LynxEye detecotr, Goebel mirror\nandEuleriancradle. Furthersurfaceroughnessaredeter-\nmined byatomicforcemicroscopy(AFM) measurements.\nDomain structure and magnetic hysteresis loops of the\nfilms are simultaneously captured with Kerr microscopy\nsystem of M/s Evico magnetics, Germany equipped with\nwhite light LED source. Room temperature ferromag-\nnetic resonance (FMR) measurements are carried out us-\ningJEOL-FA200ElectronSpin ResonanceSpectrometer.\nFMR spectra have been recorded as a function of polar\nangle (θH)19.\n/s48/s46/s48/s53 /s48/s46/s49/s48 /s48/s46/s49/s53 /s48/s46/s50/s48 /s48/s46/s50/s53 /s48/s46/s51/s48 /s48/s46/s48/s53 /s48/s46/s49/s48 /s48/s46/s49/s53 /s48/s46/s50/s48 /s48/s46/s50/s53 /s48/s46/s51/s48/s32/s67/s101/s58/s89/s73/s71/s49/s48/s45/s49/s49/s49/s82/s101/s102/s108/s101/s99/s116/s105/s118/s105/s116/s121/s32/s40/s65/s114/s98/s46/s32/s85/s110/s105/s116/s115/s46/s41\n/s113\n/s122/s45/s49\n/s32/s40/s65/s45/s49\n/s41/s32/s67/s101/s58/s89/s73/s71/s51/s48/s45/s49/s49/s49/s32/s67/s101/s58/s89/s73/s71/s54/s53/s45/s49/s49/s49\n/s40/s98/s41/s32/s67/s101/s58/s89/s73/s71/s54/s53/s45/s49/s48/s48\n/s40/s97/s41\n/s32/s67/s101/s58/s89/s73/s71/s51/s48/s45/s49/s48/s48\n/s32/s67/s101/s58/s89/s73/s71/s49/s48/s45/s49/s48/s48/s82/s101/s102/s108/s101/s99/s116/s105/s118/s105/s116/s121/s32/s40/s65/s114/s98/s46/s32/s85/s110/s105/s116/s115/s46/s41\n/s113\n/s122/s45/s49\n/s32/s40/s65/s45/s49\n/s41\nFIG. 1. X-ray reflectivity (XRR) data of the indicated films.\nSymbols represent the experimental data points and the soli d\nline is thebest fittothedata. Obtainedparameters are shown\nin Table-1.\nIII. RESULTS\nFig. 1 show XRR patterns of Ce:YIG films deposited\non GGG(111) and GGG(100) substrates. Fitting to the\nexperimental data using Parratt formalism gives the pa-\nrameter such as thickness, roughness and the obtained\nparameters are shown in table-I. Further, the surface\n/s40/s99/s41/s40/s97/s41 /s40/s98/s41\nFIG. 2. Atomic force microscopy (AFM) images of (a)\nCe:YIG10-111 (b) Ce:YIG66-111 films.\nmorphology of the films is studied with atomic force mi-\ncroscopy and Fig. 2 shows the AFM images 10 and 66\nnm thick films deposited on GGG(111) substrate. The\ndata indicate a smooth surface with rmsroughness less\nthan 1 nm, corroborating the XRR data.\nSince the lattice parameter of the Ce:YIG films de-\nposited at higher OPP and substrate temperature is ex-\npected to be very close to that of GGG substrate17, the\nfilm peaks are merged with that of substrate in the 2 θ-\nωscans (not shown) of all the films measured using lab\nbased x-ray diffractometer. However no other reflections\nare observed in the 2 θ-ωscans indicating that there are\nno secondary phases present for all the samples. This is\nconsistent with our earlier work in which the film Bragg\npeaks are observed to shift close to GGG substrate peak\nat 110 mTorr OPP17. Further, reciprocal space map-\nping (RSM) data across symmetric and asymmetric re-\nflections are carried out and the representative data of\nCe:YIG66-111 film across (444) and (642) reflections is\nshown in Fig. 3. Since the present RSM measurements\nare not done in high-resolution mode with lab based x-\nray diffractometer (as a result one can see the substrate\nα2reflection also) the film and substrate reciprocal lat-\ntice points are overlapping with each other, even though\nthe data confirms that the films are epitaxial in nature.\nHowever, for samples deposited on GGG(111) substrates\n(which exhibit PMA as discussed below) the 2 θ-ωscans\nare carried out using synchrotron radiation and the data\nis shown in Fig. 4.\n/s45/s48/s46/s48/s53 /s48/s46/s48/s48 /s48/s46/s48/s53/s53/s46/s53/s54/s53/s46/s53/s56/s53/s46/s54/s48/s53/s46/s54/s50/s53/s46/s54/s52/s53/s46/s54/s54/s40/s52/s52/s52/s41/s113\n/s122/s32/s47/s47/s91/s49/s49 /s49 /s93/s32/s40/s110/s109/s45/s49\n/s41\n/s113\n/s120 /s32/s47/s47/s32/s91/s49/s48 /s49 /s93/s32/s40/s110/s109/s45/s49\n/s41/s48/s46/s49/s48/s48/s48/s46/s51/s48/s50/s48/s46/s57/s49/s49/s50/s46/s55/s53/s56/s46/s51/s48/s50/s53/s46/s49/s55/s53/s46/s54/s50/s50/s56/s54/s56/s57/s50/s46/s48/s56/s69/s43/s48/s51/s54/s46/s50/s56/s69/s43/s48/s51/s49/s46/s57/s48/s69/s43/s48/s52/s53/s46/s55/s50/s69/s43/s48/s52\n/s50/s46/s50/s52 /s50/s46/s50/s56 /s50/s46/s51/s50 /s50/s46/s51/s54/s53/s46/s53/s54/s53/s46/s53/s56/s53/s46/s54/s48/s53/s46/s54/s50/s53/s46/s54/s52/s53/s46/s54/s54/s40/s54/s52/s50/s41/s113\n/s122/s32/s47/s47/s32/s91/s49/s49/s49/s93/s32/s40/s110/s109/s45/s49\n/s41\n/s113\n/s120 /s32/s47/s47/s32/s91/s49/s48 /s49 /s93/s32/s40/s110/s109/s45/s49\n/s41/s48/s46/s49/s48/s48/s48/s48/s46/s50/s56/s53/s49/s48/s46/s56/s49/s50/s54/s50/s46/s51/s49/s55/s54/s46/s54/s48/s52/s49/s56/s46/s56/s51/s53/s51/s46/s54/s55/s49/s53/s51/s46/s48/s52/s51/s54/s46/s49/s49/s50/s52/s51/s51/s53/s52/s52/s49/s46/s48/s49/s48/s69/s43/s48/s52/s50/s46/s56/s56/s48/s69/s43/s48/s52\nFIG. 3. Reciprocal space mapping (RSM) data of Ce:YIG66-\n111measuredacrosstheindicatedreflections. Thetwointen se\npeaks (red color) are substrate α1andα2reflections.3\n/s51/s46/s52 /s51/s46/s54 /s53/s46/s54 /s53/s46/s56 /s54/s46/s56 /s55/s46/s48 /s55/s46/s50/s40/s56/s56/s48/s41/s40/s52/s52/s52/s41\n/s40/s56/s56/s56/s41/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s65/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s113/s32/s40/s65/s45/s49\n/s41/s32/s67/s101/s58/s89/s73/s71/s54/s53/s45/s49/s49/s49\n/s32/s67/s101/s58/s89/s73/s71/s51/s48/s45/s49/s49/s49\n/s32/s67/s101/s58/s89/s73/s71/s49/s48/s45/s49/s49/s49\nFIG. 4. 2 θ-ωscans of the Ce:YIG films deposited on GGG(111) substrate mea sured with synchrotron radiation. The data is\nmeasured across symmetric (444), (888) reflections & asymme tric (880) reflection. The small vertical lines are to indica te the\nfilm peak.\nFig. 4 shows the XRD measured around symmetric\n(444), (888) & asymmetric (880) reflections of all the\nfilms deposited on GGG(111) substrate. All the films\nexhibit pronounced Laue oscillations which clearly indi-\ncate that the films are highly smooth, uniform and single\ncrystalline in nature. It may be noted that the thickness\nof the films calculated from Laue oscillations match ex-\ncellently with the XRR data. The out-of-plane (OOP)\nlattice parameter ( aout) is estimated from the (888) re-\nflectiondataandistabulatedintable-1. Usuallyonemay\nexpect that the film peak shifts to higher (lower) angles\nwith respect to the substrate, below a critical thickness\nof the film, due to the expected in-plane tensile (com-\npressive) epitaxial strains in the case of lattice mismatch\nbetween substrate and film20. However, it is observed\nthat for 10 nm thick sample, the film peak is overlapping\nwith that of substrate, which could be due to the ex-\ncellent lattice matching between the film and substrate.\nWith further increase in thickness, the OOP lattice pa-\nrameter is observed to decrease indicating the OOP lat-\ntice compression, unlike most of the cases in which films\nare expected to relax as thickness increases. The plau-\nsible reason for this could be either due to changes in\nthe relative concentration of Ce3+, Ce4+and associated\nissues related to charge neutralization in the system with\nthickness17,21.\nFig. 5 show the MOKE data of all the films along with\nthe bulk magnetization data measured using SQUID-\nVSM for few selected samples. MOKE measurements\nare carried out in two geometries viz., longitudinal (L-\nMOKE) in which the applied magnetic field is in the film\nplane and polar (P-MOKE) in which the applied field\nis out-of-plane of the sample. P-MOKE measurements\nare sensitive to out-of-plane component of the magneti-zation and the L-MOKEis expected to be sensitive to in-\nplanecomponentofmagnetization22. However,itmaybe\nnoted that L-MOKE geometry can also have a sensitiv-\nity of out-of-plane magnetization component in addition\nto the in-plane component due to the oblique incidence\nof light22. Perusal of Fig. 5, for all the films deposited\non GGG(100) substrate the observed L-MOKE data is\ntypical of films exhibiting in-plane magnetization.\nHowever, for the films deposited on GGG(111) sub-\nstrates, an anomalous kind of L-MOKE loops are ob-\nservedas shown in Fig. 5. The data indicates two switch-\ning fields (Fig. 5(d),(e),(f)). As one can see that at mid-\ndle portion of the L-MOKE loops, a clear switching of\nmagnetization with finite H Cis observed in addition to\nthe saturation at sufficiently high magnetic fields as in-\ndicated by vertical dashed line in Fig. 5(d),(e),(f), the\nmagnetization saturates along the field direction. The\ncentral switching part could be due to the presence of\nconsiderable OOP magnetization component or domains\nwith non-zero OOP field components during application\nof in-plane external magnetic field. In view of this, P-\nMOKE measurements are also carried out on films de-\nposited on GGG(111) substrate. Fig. 5 shows the room\ntemperature P-MOKE hysteresis loops. It is very inter-\nesting to note that the observed squareness (ratio of re-\nmanence/saturation, M R/MS) is close to unity for all the\nthreefilmsintheP-MOKEdata. Inordertoconfirmthis,\nbulk magnetization measurementswith SQUID-VSM are\nalsocarriedout on these three films with the applied field\nalong the out-of-plane direction. The SQUID-VSM data\nis observed to match excellently with P-MOKE data ex-\ncept small variation in the coercivity (H C). Therefore,\nboth P-MOKE and SQUID-VSM data clearly suggest\nthat at room temperature the easy magnetization axis4\n/s40/s103/s41/s75/s101/s114/s114/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s117/s110/s105/s116/s115/s41\n/s75/s101/s114/s114/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s117/s110/s105/s116/s115/s41/s67/s101/s58/s89/s73/s71/s47/s71/s71/s71/s40/s49/s49/s49/s41\n/s54/s53/s32/s110/s109 /s51/s48/s32/s110/s109\n/s70/s105/s101/s108/s100/s32/s40/s109/s84/s41/s49/s48/s32/s110/s109/s40/s105/s41\n/s45/s54/s48/s45/s52/s48/s45/s50/s48/s48/s50/s48/s52/s48/s54/s48\n/s45/s52/s48/s45/s50/s48/s48/s50/s48/s52/s48\n/s45/s50/s48/s45/s49/s48/s48/s49/s48/s50/s48\n/s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s32/s40/s101/s109/s117/s47/s99/s109/s51\n/s41/s40/s97/s41/s75/s101/s114/s114/s32/s114/s111/s116/s97/s116/s105/s111/s110/s32/s40/s65/s114/s98/s46/s32/s85/s110/s105/s116/s115/s41/s54/s53/s32/s110/s109\n/s40/s104/s41/s40/s98/s41\n/s70/s105/s101/s108/s100/s32/s40/s109/s84/s41/s51/s48/s32/s110/s109/s67/s101/s58/s89/s73/s71/s47/s71/s71/s71/s40/s49/s48/s48/s41\n/s40/s99/s41\n/s49/s48/s32/s110/s109\nFIG. 5. (a)-(c) L-MOKE loops of Ce:YIG films deposited\non GGG(100) substrate. (d)-(f) L-MOKE loops of Ce:YIG\nfilms deposited on GGG(111) substrate. The vertical dotted\nlines of L-MOKE loops is to indicate the in-plane saturation\nfield (H Sat). (g)-(i) P-MOKE (black color) and SQUID-VSM\n(blue color) loops of Ce:YIG films deposited on GGG(111)\nsubstrate. Thickness of the films is indicated in the respect ive\nframe.\nis found to be along out-of-plane direction for the films\ndeposited on GGG(111) substrate.\nDevelopment of PMA is further substantiated with\nmagnetic microstructure of the films by capturing Kerr\nimages. It may be noted that when the preferred (easy)\ndirection of magnetization is perpendicular to the plane\nof the film, usually magnetic bubble domains are ex-\npected to form23. Bubble domains are cylindrical mag-\nnetic domains that may occur in a thin plate of a mag-\nnetic material exhibiting PMA. Fig. 6 shows room tem-\nperature Kerr images of Ce:YIG10-111 and Ce:YIG66-\n111 films captured in P-MOKE geometry. The samples\nare initially saturated by applying a sufficient negative\nfield and the images are captured at the indicated pos-\nitive field, which is close to H Cand one can clearly see\nthe bubble domain formation confirming the presence of\nPMA in the Ce:YIG films deposited on GGG(111) sub-\nstrate. In order to further confirm the PMA, the FMR\nmeasurements are carried out on two films of each that\nare deposited on GGG(100) and GGG(111) substrates\nand the results are discussed as following.\nFig. 7 shows the FMR derivative spectra of the indi-\ncated films measured asa function ofpolar angle, θH, be-\ntween the normal of the film plane and the applied field.\nFMR signal for 10 nm thick films was weak and hence\nis not shown. The resonance field values ( Hr) extracted\nfrom the FMR data as a function of θHis also shown\nFIG. 6. Room temperature Kerr micrograph of the indicated\nfilms captured in P-MOKE geometry. Scale bar is same for\nboth the images./circledottext\nand/circlemultiplytext\ndenote out-of-plane (OOP) and\nin to the plane respectively.\n/s32/s100/s97/s116/s97\n/s32/s70/s105/s116\n/s68 /s72\n/s80/s80/s61/s49/s49/s46/s48/s32/s109/s84\nFIG. 7. (a),(b) Ferromagnetic resonance (FMR) data of the\nindicated films. Inset of (b) shows the fitting of one represen -\ntative FMR data to extract resonance field (H r) and typical\nline width (∆ Hpp). IP and OOP denote in-plane and out-of-\nplane configuration respectively. (c),(d) show the variati on of\nHrwith OOP angle ( θH), the solid line is guide to eye only.\nin Fig. 7. One can clearly see from the data that for\nthefilmsdepositedonGGG(111)substratetheresonance\nfield (H r) values are the smallest in OOP configuration\n(θH=0◦) and the largest in IP configuration ( θH=90◦)\nwhereas for films deposited on GGG(100) substrate, the\ntrend is opposite. This indicates that easy axis magneti-\nzation is out of plane (i.e., PMA) for the films deposited\nonGGG(111)substrateandit isin-planeforthe filmsde-\nposited on GGG(100) substrate. The observed peak to\npeak lines width (∆ Hpp) forstudied samples arefound to\nbe larger as compared to the pure YIG films24–26. This5\nindicate in the studied Ce:YIG samples have higher mag-\nnetic damping as also expected due to lattice dilation\nand different environments of the Fe cations due to the\nCe doping9,27. It is also to be mentioned that spin orbit\ncouplingpresent in the rareearthiron garnetgivesrise to\nlarger magnetic damping than pure YIG. However, the\nmagnetic damping in our Ce:YIG samples is found to be\ncomparable or lesser than TmIG and other rare earth\ngarnet those are showing PMA28,29.\nThe above observations clearly indicate that the\nPMA is developed in films that are deposited on\nGGG(111) substrates, whereas the films deposited on\nGGG(100) substrate exhibit only in-plane magnetiza-\ntion. One can understand this by considering various\nmagnetic anisotropy terms and their origin such as in-\ntrinsic magneto-crystalline anisotropy (K MC), magneto-\nelastic (K ME), shape anisotropy (K Shape) and induced\nanisotropy as discussed in the following.\nIn general in YIG and most of the other RIG systems,\nbecause of negative cubic anisotropy constant (K 1), easy\naxis of magnetization is always along /angbracketleft111/angbracketrightdirection30,\nwhich is an inherent magneto-crystalline anisotropy fac-\ntor (K MC). In addition to this one can also under-\nstand the easy axis of magnetization as /angbracketleft111/angbracketrightin terms\nof magneto-striction coefficient ( λ) i.e., magneto-elastic\nanisotropy (K ME). Depending upon λ, either compres-\nsiveortensilestraindue tolattice mismatchbetween film\nand substrate can produce large uniaxial magneto-elastic\nanisotropy to overcome shape anisotropy for inducing\nPMA20,31,32. For example, considering YIG, the λvalues\nare given by λ111=-2.73×10−6andλ100=-1.25×10−6,\nwhich are usually small and negative30. However, larger\nin magnitude of λ111thanλ100make/angbracketleft111/angbracketrightdirection as a\nfavorableeasy axis than /angbracketleft100/angbracketright, when a compressivestress\nis applied along the respective direction. Along these\nlines, it may be noted that from our XRD data, the pres-\nence of in-plane tensile strain is argued for higher thick-\nness films which require a negative λvalue to generate\nPMA in the present samples. However, it is reported in\nliterature that the Ce substitution in YIG makes λfrom\nnegativeto positivewith increasingCe concentration33,34\nindicating that in addition to this magneto-elasticcontri-\nbution there could be other contribution for the observed\nTABLE II. M sis the saturation magnetization obtained from\nSQUID measurements. H Kand K Udenote out-of-plane\n(OOP) effective anisotropy field and anisotropy energy den-\nsity, respectively. K MEis the magneto elastic anisotropy\ndensity. H C(PMA)is the coercivity from the P-MOKE data\n(Fig. 5)\n,\nSample 4 πMSHK KUKMEHC(PMA)\n(Gauss) (Oe) ( kJ/m3) (kJ/m3) (mT)\nCe:YIG10-111 200 ±50 4200±200 3.4 - 0.8\nCe:YIG30-111 500 ±25 2500 ±50 5.0 0.50 3.6\nCe:YIG65-111 630 ±12 3130 ±40 7.8 0.82 14.5PMA in the present work. For this we estimated quan-\ntitatively magneto-elastic anisotropy (K ME) as discussed\nin the following.\nThe effective PMA field (H K) values can be calcu-\nlated using equation H Sat= HK-4πMS, where H Satis IP\nsaturation field which is obtained from L-MOKE loops\nand M Sis the saturation magnetization deduced from\nSQUID-VSM M-H data (Fig. 5). The anisotropy K Uis\nrelated through the equation K U= (HKMS)/224. The\nobserved value of K Uof the three studied PMA films are\ntabulated in the table-II.\nThe K MEis calculated for the Ce:YIG30-111 and\nCe:YIG65-111 films using equation K ME=(9/4)λ111C44\n(π/2-β), whereC44isthe shearstiffness constantand βis\nthe shear distortion angle. The β35can be deduced from\nIP and OOP lattice parameter values as obtained from\nx-ray diffraction data (Fig. 4), where the IP lattice pa-\nrameter of the film is considered to be equivalent to that\nof substrate due to coherent growth. The observed βare\nfound to be 90◦(i.e., no distortion), 90.06◦and 90.1◦\nfor Ce:YIG10-111,Ce:YIG30-111 and Ce:YIG65-111, re-\nspectively. Substitution of thus obtained β,λ111=-2.73\n×10−6andC44=76.6 GPa results in K MEas 0.50±0.02,\nand 0.82±0.02,kJ/m3for Ce:YIG30-111 and CeYIG65-\n111 films, respectively. We considered λ111andC44val-\nues of YIG24,36in the above calculation mainly due to\nthe unavailability of these values for Ce:YIG samples,\nbut this calculation is expected to give an upper limit for\nthe values of values of K ME.\n/s45/s50/s48 /s45/s49/s48 /s48 /s49/s48 /s50/s48/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s45/s56 /s45/s52 /s48 /s52 /s56 /s45/s52 /s45/s50 /s48 /s50 /s52 /s45/s52 /s45/s50 /s48 /s50 /s52\n/s45/s50/s48 /s45/s49/s48 /s48 /s49/s48 /s50/s48/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s45/s49/s48 /s48 /s49/s48 /s45/s56 /s45/s52 /s48 /s52 /s56 /s45/s50 /s45/s49 /s48 /s49 /s50\n/s45/s56/s48 /s45/s52/s48 /s48 /s52/s48 /s56/s48/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s45/s54/s48 /s45/s51/s48 /s48 /s51/s48 /s54/s48 /s45/s51/s48 /s45/s49/s53 /s48 /s49/s53 /s51/s48/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s45/s51/s48 /s45/s49/s53 /s48 /s49/s53 /s51/s48/s50/s48/s48/s75/s50/s52/s48/s75 /s51/s52/s48/s75 /s52/s48/s48/s75/s75/s101/s114/s114/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s70/s105/s101/s108/s100/s40/s109/s84/s41\n/s70/s105/s101/s108/s100/s40/s109/s84/s41/s70/s105/s101/s108/s100/s40/s109/s84/s41\n/s70/s105/s101/s108/s100/s40/s109/s84/s41\nFIG. 8. Temperature dependent P-MOKE data of Ce:YIG\nfilms deposited on GGG(111) substrate (top) 10 nm thickness\n(middle) 30 nm thickness (bottom) 66 nm thickness at the\nindicated temperatures.\nClearly, experimentally observed OPP anisotropy K U\n(ref table-II) is one order of magnitude large than what\nis obtained from respective K ME. Even the magneto-6\nelastic field H ME, defined as H ME=2KME/MSis not even\n50% of the demagnetizing fields (4 πMS) of the respec-\ntive films and therefore K MEcannot be solely respon-\nsible for the observed PMA. An extra anisotropy term\nis therefore required to be introduced, which is often\ncalled growth induced anisotropy term (K GROWTH ) as\nreported by Soumah et al., in Bi substituted YIG films10.\nIn view of the small value of K MC, KUis attributed to\nbe sum of two contributions i.e., K MEand K GROWTH\nfor Ce:YIG30-111 and Ce:YIG30-111 films whereas in\ncase of Ce:YIG10-111 film, due to absence of strain, only\nKGROWTH term is expected to be responsible for the ob-\nserved PMA. However the exact microscopic origin of\nKGROWTH and its dependence on factors such as compo-\nsition of the film, thickness, orientation etc., is beyond\nthe scope of the present work.\n/s32/s67/s101/s58/s89/s73/s71/s51/s48/s45/s49/s49/s49\n/s32/s67/s101/s58/s89/s73/s71/s54/s53/s45/s49/s49/s49\nFIG. 9. Temperature variation of coercivity (H C) obtained\nfrom Fig. 8 of all the Ce:YIG films deposited on GGG(111)\nsubstrate.\nFIG. 10. Kerr micrograph of Ce:YIG films at the indicated\ntemperatures captured in P-MOKE geometry. Scale bar is\nsame for all the images.\nThe stability of the developed PMA in Ce:YIG films\ndeposited on GGG(111) is further studied with temper-ature. Temperature dependent P-MOKE measurements\nare carried out over a temperature range of 80 to 460 K\nand the data for all the three films is shown in Fig. 8. It\nis observed that the easy axis of magnetization remain\nperpendicular to the film plane irrespective of film thick-\nnesses much above the room temperature as shown in\nFig. 8. The apparent curved shape of the loops is due\nto the Faraday contribution of the objective lens of the\nmicroscope at higher fields. This is a remarkable stabil-\nity of the PMA in these kind of films as very few reports\ncould show this kind of robust stability of PMA in garnet\nepitaxial thin films. For example, reorientationfrom out-\nof-plane to in-plane easy axis at about 173 K is reported\nin similar Ce:YIG epitaxial films, prepared at different\ngrowth conditions18.\nThecoercivity(H C)isestimatedfromthedataofFig.8\nforallthe filmsandthe variationofH Cisshownin Fig.9.\nThe increase of H Cwith decreasing temperature is ob-\nserved for all the films as expected. The temperature\ndependent Kerr images are captured near H Cduring the\nhigh temperature P-MOKE measurements and the data\nis shown in Fig. 10. For Ce:YIG65-111 film, the bubble\nnucleationpointand the sizeofthe the bubble isfound to\nbe almost same irrespective of temperature. However for\nlower thickness film (Ce:YIG10-111) the size of magnetic\nbubble is found to increase with temperature.\nIV. CONCLUSIONS\nIn conclusion, the present work reports the stabiliza-\ntion of perpendicular magnetic anisotropy (PMA) in\ncerium substituted yttrium iron garnet (Ce:YIG) epitax-\nial thin films on GGG substrate by suitably optimizing\nthe deposition conditions with pulsed laser deposition.\nThe developed PMA is confirmed from bulk and surface\nsensitive magnetic measurements, which is further cor-\nroborated by ferromagnetic resonance and magnetic mi-\ncrostructure. The observed stability of PMA over a wide\nrange of temperatures makes the studied Ce:YIG films\nsuitable for many practical applications. The cerium\nsubstitution in YIG seems to be a better choice for the\ninsulating garnet films as the possibility of enhancing\nmagneto-optical activity17and stabilization of PMA are\ndemonstrated with suitably optimized deposition condi-\ntions.\nV. ACKNOWLEDGMENTS\nDr.R.Venkatesh and Mr. Mohan Gangrade are\nthanked for the AFM measurements. 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Non ferromagnetic  large spin -orbit coupling heavy metal (NM) / ferromagnet (FM) \nhetero structures offer interesting elements of response  to this issue , by granting the  manipulation of \nthe FM magnetization by the NM spin Hall effect (SHE) generated spin current . Additional \nfunctionalities , such as the electric field control of the spin current generation , can be offered using \nmultifunctional ferromagnets. We  have studied  the spin current transfer processes  between Pt and \nthe multifunctional  magnetoelectric Ga 0.6Fe1.4O3 (GFO).  In particular,  via angular dependent \nmagnetotransport measurements , we  were able to differentiate between magnetic proximity effect \n(MPE) -induced anisotropic magnetoresistance (AMR) and spin Hall magnetoresistance (SMR). Our \nanalysis shows that SMR  is the dominant pheno menon at all temperatures and is the only one to be \nconsidered near room temperature, wi th a magnitude comparable to those  observed in Pd/YIG or \nPt/YIG heterostructures. These results indicate that magnetoelectric GFO  thin films show promises for \nachieving  an electric -field control of the spin current generation in  NM/FM oxide -based \nheter ostructures . \nABSTRACT GRAPHIC  \n  \n \nKEYWORDS: magnetic oxides thin films, gallium ferrite, platinum, spintronics, spin Hall \nmagnetoresistance, magnetic proximity effects  \n1. INTRODUCTION  \nNon ferromagnetic heavy metal (NM) / ferromagnet  (FM) hetero structures are currently \nlargely investigated to study viable routes to  exploit the fascinating interplay between heat, \ncharge and spin transport .1–3 In particular , recent developments point at a  suitable  low-power \nmanipulation of magnetization in spintronic devices  through  spin-orbit torques (SOT)  effects , \nwhich ground on the intrinsic spin-orbit coupling.  Indeed , it has  been shown th at the spin \ncurrent generated upon the application of a charge current in the heavy metal nonmagnetic  \nmaterial via the spin Hall effect (SHE) can manipulate the magnetic moments in the adjacent \nferromagnetic  (FM) layer  through SOT .4–10 This has triggered the development of a new \nMagnetic Random Access Memory (MRAM) -based technology, the spin -orbit torque MRAM \n(SOT -MRAM)8, and the possibility to integrate SHE -based spin -valves in standard CMOS -like \nnetworks has already been demonstrated via complementary spintronic logic (CSL) design \nmethodology .11  \nThe interest for NM/FM heterostructures has recently moved from all -metal  systems  to \nsystems in which the FM is an insulating oxide (FMI), and display the interesting phenomenon \nof spin Hall magnetoresistance (SMR) .12–14 The most emblematic NM/FMI system is based on \ngarnet ferrite, with Pt/YIG (Y 3Fe5O12).3,15 –22 However,  spinel fe rrites have also been considered  \nso far , with  systems such as  Pt/NiFe 2O4,3,18 Pt/Fe3O418 and Pt/CoFe 2O4.23–25 Recently, a study \non Pt/Bi0.9La0.1FeO 3 bilayers26 made a step towards the exciting challenge of the electric field \ncontrol of the spin Hall effect . This, by granting the control of the generation of spin current \nin engineered NM/FM -based devices, adds extra functionalities to the realization of future \nspintronics devices .  \nHere we propose the use of another oxide, the  multifunctional  magnetoelectric gallium ferrite \nGa2-xFexO3 (GFO x, 0.8 ≤x≤1.4),27,28,29 with a view to use its magnetoelectric character in the \nfuture for an electric field control of the spin current generation  from NM/FM \nheterostructures.  Ga2-xFexO3 crystallizes in the polar orthorhombic Pna2 1 (equivalently Pc2 1n) \nspace group (S.G.#33) ,30,31 different from the usual perovskite structure adopted by most of \nthe other magnetoelectric compounds, with a = 0.5086(2) nm, b = 0.8765(2) nm, and c = \n0.9422(2) nm for x = 1.4 .32 The material is polar with a polarization of ca. 25  µC/cm2,33,34 and \nferrimagnetic with a  Curie temperature increasing with x  and reaching values above room \ntemperature for x=1.3 .28,35,36 Taking into consi deration the lattice matching possibilities, t hin films of GFO x (00l) can be epitaxially deposited on various substrates such as yttrium stabilized \nzirconia  (001) (YSZ) ,37,38 YSZ (111),38 Pt (111) buffered YSZ (111)37 or strontium titanate SrTiO 3 \n(111)  (STO ).38,39 For symmetry reasons, while the (100) YSZ substrates allow the growth of six \nin-plane variants, the YSZ (111), Pt (111) buffered YSZ (111) and STO (111) ones will only allow \nthree , the lowest number observed until now. GFO1.4 films  were shown to be ferroelectric at \nroom temperature ,39 to have  a magnetic Curie  temperature  of ca. 370 K , and a saturation \nmagnetization of about 100 emu/cm3 at room temperature .37,40  \n \n2. MATERIAL S ELABORATION AND CHARACTERIZATION - EXPERIMENTAL DETAILS  \nHere we show results obtained from Pt/GFO1.4 heterostructures which, for convenience, will \nbe simply indicated as Pt/GFO. These  heterostructures were deposited onto SrTiO 3 (111) (STO) \nsubstrates  (Furuuchi Chemical  Corporation, Japan, with rms -roughness lower than 0.15 nm) , \nby pulsed laser deposition  using a KrF excimer laser (= 248 nm ) with a fluence of 4 J/cm² . The \nchoice of STO as the substrate was dictated by the will to have the lowest numbe r of in -plane \nvariants , and we have recently demonstrated the possibility to have a layer -by-layer growth \nof highly crystalline GFO  on this substrate.40 The growth of such atomically flat GFO films is a \nprerequisite to high quality Pt/GFO interfaces. The GFO layer (ca. 30 nm thick) was deposited \nfirst, at 900°C, by ablati ng a sintered stoichiometric Ga 0.6Fe1.4O3 ceramic target at a repetition \nrate of 2 Hz of in a 0.1 mbar O 2 pressure, using an already optimized procedure desc ribed \nelsewhere .40 The overall composition of the film has been assessed by energy dispersive X-ray \nspectroscopy (EDX) coupled to a scanning electron microscopy technique (JEOL 6700 F). The \nanalysis was performed at 5 keV, ensuring a large surface sensitivity of the EDX sign al. The Pt \ndeposition wa s performed at a repetition rate of 10 Hz, under the system base pressure of \n2 .10-8 mbar, and at room temperature to avoid any interdiffusion between the metal and \noxide layers . The Pt thickness chosen  here for generating spin curr ents was of 5 nm . This \nthickness has been evidenced by other works as optimum f or SHE -induced spin currents from \nits associated  spin diffusion length (λ sd), spin memory loss (SML) at the interface41 and spin \nHall torque efficiency per applied electric field unit .42 The surface of the STO//GFO/Pt \nheterostructure, observed by atomic force microscopy (AFM) with a Bruker ICON microscope \noperated in tapping mode, has a low rms  roughness  of 0.3 nm (Figure 1a). The roughness of a \nGFO layer of similar thickness (32 nm) deposited alone  on a STO (111) substrate , without any Pt layer on top, had already been previously characterized, and is also  of about 0.3 nm.40 \nObservations of the heterostructure cross section by transmission elect ron microscope  (TEM)  \n(JEOL 2100 F) allowed confirming the low roughness of the Pt/GFO interface . A composition \nprofile was measured across the interface by energy dispersive X -ray spectroscopy (EDS) and \nshowed that interdiffusion  is limited in this system and restricted to a less than 2 nm wide \nzone  (Figure 1b).  X-ray diffraction was performed on the heterostructure with a Rigaku Smart \nLab diffractometer equipped with a rotating anode (9 kW) and a monochromated copper \nradiation (1.54056 Å). Both GFO oxide a nd Pt meta l layers of the heterostructure  are well \ncrystallised (Figure 1c). GFO is oriented along its [001] direction and Pt, along its [ 111] \ndirection, on the STO (111) . A layer -by-layer growth together with a smooth Pt/GFO interface \nare demonstrated by clear Laue oscillations for both GFO and Pt around their 004 and 111 \nreflections, respectively. Reflectivity measurements allowed us to determine th at th e precise \nthicknesses of the GFO and Pt layers  are 36 and 5 nm, respe ctively. The magnetic properties \nof the heterostructure were studied with a superconducting quantum interference device \nvibrating sample magnetometer (SQUID VSM MPMS 3, Quantum Design). Temperature \ndependent measurements of the magnetization performed in f ield cooled and zero field \ncooled modes (not shown here) indicate a Curie temperature of 364 K. The sample indeed still \nshows a ferromagnetic behaviour at room temperature with a saturation magnetization of \n100 emu/cm3, as expected for this Fe/Ga ratio.37 A highly anisotropic behaviour is evidenced \nfrom the in -plane and out -of-plane magnetization loops measurements (Figure 1d). The \nmagnetization lies preferentially in -plane, in perfect agreement wit h the fact that the [100] \nand [001]  crystallographic  axes are, respectively, the easy and hard magnetic directions for \nGFO,28 and are observed from X -ray characterizations to lie, respectively, in - and out -of-plane.   \nFigure 1. (a) TEM cross view of the STO//GFO(3 6 nm)/Pt(5 nm) heterostructure with (insert) the AFM image of \nits top surface indicating a rms roughness of 0.3 nm, (b) EDX mapping at the Pt/GFO interface, (c) X -ray \ndiffractogram of the heterostructure in the θ -2θ mode, with a focus on the GFO 004 peak sh owing the Laue \noscillations observed for both the GFO 004 and Pt layers 111 reflections, (d) Magnetization hysteresis loops of \nthe heterostructure measured in both parallel and perpendicular modes at 300 K and (insert) 10 K.\n \n \n3. MAGNETO TRANSPORT STUDY – RESULTS AND DISCUSSION  \nThe Pt layer was  patterned, using standard optical lith ography, into double Hall bars to \nminimize the electrical resistance associated to the metallic contacts, with a longitudinal \nlength L=38 µm, a width w=10 µm, and a thickness  t=5 nm (Figure 2). The current is injected \nin the Pt bar along the STO[0 -11] () direction, that is , perpendicularly to the [100] easy \nmagnetization direction of GFO. The plane of the films will be referred to as  xy, and the normal \nto the film  will be named z. Hz (Hx, Hy) refer s to the magnetic field applied along the z (x, y) \ndirection. The electric current density Jc is applied along  x, and t he longitudinal (transversal) \nresistivity ρxx (ρxy) is calculated  from  the measured voltage  Vxx (Vxy).  \n  \nFigure 2. Schematics of the lithographed double Hall bars on STO//GFO(3 6 nm)/Pt(5 nm) heterostructures with \nindications of length (L), width (w) and thickness (t) . \n \n \nMagnetotransport in heavy metal ( NM) / insulating ferromagnet (FM I) heterostructures may \ninvolve  contributions from two main origins: (1) the magnetic proximity effect  (MPE) , and (2) \nthe spin Hall related  effects  with the spin Hall magnetoresistance (SMR) . Figure 3 offers a \nschematic representation of the microscopic mec hanisms behind each  contribution and for \ntwo different geometries of measurement, i.e. longitudinal and transverse.  \nGoing to details, MPE  implies the existence of some induced interfacial magnetism in the non -\nmagnetic Pt  layer . As a result, anisotropic magnetoresistance (AMR) and anomalous Hall \neffects43 (AHE), ch aracteristic of metallic FM, can play a role.  \nInstead, SMR contribution does not necessitate the existence of any MPE as already put \nforward by Nakayama et al.14 SMR  effects  arise as a combination of the spin Hall effect (SHE) \nand the inverse spin Hall effect (ISHE). When a charge current flows longitudinally in the \nnonmagnetic NM, a spin current is produced along the film normal direction by SHE. The spin \npolarization 𝜎⃗  of this spi n current is perpendicular to both charge and spin current densities, \n𝐽𝑒⃗⃗⃗⃗  and 𝐽𝑠⃗⃗⃗, respectively, in agreement with 𝐽𝑠⃗⃗⃗=𝜃𝑆𝐻(𝜎⃗ ×  𝐽𝑒⃗⃗⃗⃗ ), where 𝜃𝑆𝐻 is the spin Hall \nangle .44,45 The spin current can either be reflected or absorbed by the adjacent FM layer \ndepending on whether 𝜎⃗ is parallel or perpendicular to the  magnetization direction of the  FM \nlayer, respectively .18 The reflected spin current will produce an additional charge current \nthrough the inverse spin Hall effect (ISHE) which will lead to a decrease of the longitudinal \nresistivity. The resistivity of the NM layer will therefore strongly depend upon the orienta tion \nof the FM magnetization.  \n \nIf one neglects contributions from other phenomena such as  the topological Hall effect46 (THE) \nor unidirectional magnetoresistance47,48 (UMR), very unlikely in this collinear -spins s ystem , \nthe longitudinal ( 𝜌𝑥𝑥) and transvers e (𝜌𝑥𝑦) resist ivities  of the studied heterostructure as a \nfunction of an external magnetic field can be described by the following equations :13,18  \n \n𝜌𝑥𝑥=𝜌0+ ∆𝜌𝑀𝑃𝐸  𝐴𝑀𝑅 ⋅ 𝑚𝑥2 + Δ��||𝑆𝑀𝑅  ⋅ 𝑚𝑦2 (1) \n \n𝜌𝑥𝑦=(∆𝜌𝑀𝑃𝐸  𝑃𝐻𝐸−Δ𝜌||𝑆𝑀𝑅)⋅𝑚𝑥⋅𝑚𝑦+ (Δ𝜌𝑀𝑃𝐸  𝐴𝐻𝐸 +Δ𝜌⊥ 𝑆𝑀𝑅) ⋅ 𝑚𝑧  (2) \n \nwhere 𝑚𝑥, 𝑚𝑦, and 𝑚𝑧 are the magnetization unit vector components in the x, y, and z \ndirections, respectively , 𝜌0 is the resistivity of platinum in the absence of a magnetic field, \n∆𝜌𝑀𝑃𝐸  𝐴𝑀𝑅 is the anisotropic magneto -resistivity due to the MPE induced anisotropic \nmagnetoresistance (MPE AMR), and Δ𝜌𝑀𝑃𝐸  𝑃𝐻𝐸  (𝐴𝐻𝐸 ) is the planar ( anomalous ) Hall resistivity \ndue to the MPE induced planar ( anomalous ) Hall effect in the NM. The AMR and AHE  \ncontributions, which depend on the orientation of the MPE -induced magnetic Pt , arise due to \nextrinsic spin -flip scattering mechanism and /or intrinsic mechanism depending on the band \nstructure of Pt. In addition, one has to consider the SMR related phenome na. Δ𝜌||𝑆𝑀𝑅 \nrepresents the SMR effect and Δ𝜌⊥ 𝑆𝑀𝑅 is a Hall -effect -type resistivity that can be relevant in \nsystems where the imaginary part of the spin mixing  conductance is important, but it is usually \nsmaller than Δ𝜌||𝑆𝑀𝑅.18 \n \n  \n  \nFigure 3. Schematics of the various physical phenomena which have to be considered for magnetotransport in a  \nPt/GFO hete rostructure , for both longitudinal , (a) and (b), and transverse , (c) and (d), modes involving  the \nmagnetic proximity effect s (MPE)  with anisotropic magnetoresistance (AMR), anomalous Hall effect (AHE) and \nplanar Hall effect (PHE), and the s pin Hall induced effects (SHE).\n \n \nLongitudinal magnetoresistance  \nThe longitudinal magnetoresistance (MR) is defined  as 𝜌𝑥𝑥 (𝐻) − 𝜌𝑥𝑥 (𝐻=7𝑇)\n𝜌𝑥𝑥 (𝐻=7𝑇) = ∆𝜌𝑥𝑥\n𝜌𝑥𝑥(𝐻). The field \ndependence of the MR ( for a field applied along the z direction) is plotted in Figure 4a for \nvarious temperatures between 20 and 300 K . These curves are a clear evidence  of the \npresence of non-zero longitudinal magnetoresistance  (MR) . The MR decreases with increasing \ntemperature, goes to zero at approximately 120 K, after what it  keeps a negligible value  \n(Figure 4b). This behavior of MR cannot be attributed to weak localization  (WL)  and weak anti -\nlocalization (WAL) mechanisms. If one cannot completely exclude the existence of a 2DEG at \nthe STO/GFO interface, it will however not be possible to observe any of its transport \nproperties in the Pt layer, where t he electric al contacts are made,  because of the insulating \ncharacter of the GFO layer between the STO and Pt layers. Moreover, the temperatures at \nwhich WL and WAL are at play are usually much lower than 120 K, since such quantum effects \nrequire  that the coherence of  the wave functions is kept. Possible origins behind the \ntemperature evolution of the  MR will be discussed later . The MR curves presented in Figure \n4a do not saturate even at magnetic fields beyond the magnetic saturation of the \nheterostructure observed by  SQUID (Figure 1d). This is probably caused by some independent \nmoments at the interface.49 It can be related to the highly anisotropic nature of magnetism in \nthe GFO films, which induces high anisotropy for the induced magnetic Pt as well, through a \nmagnetic proximity effect , and  will inhibit a  saturation point even at high temperatures. The \nmeasured Pt resistivity value of 23 µ .cm at room temperature is in good agreement with \nvalues reported for Pt thin films deposited either by sputtering or molecular beam epitaxy.50,51 \nIts linear decrease with temperature is also in agreement with previous studies.52  \n \n \n \nFigure 4. Longitudinal magnetoresistance  (as defined in the main text) . (a) Measurements at various  \ntemperatures , and (b) Temperature dependence of the longitudinal magnetoresistance estimated for H z= 0 T.\n \n \nTransverse magnetoresistance  \nThe magnetic field dependence of the transverse Hall resistivity 𝜌𝑥𝑦 measured at various \ntemperatures between 20 and 300 K  (the field is applied along the z direction) , is presented \nin Figure 5a , after correction from both the ordinary magneto -resistance (OMR) contribution \ndue to the variation of the temperature and the ordinary Hall resistance (OHR) due t o the \nLorentz force applied onto the carriers . For all temperatures, this corrected transverse \nresistivity 𝜌𝑥𝑦−𝑐𝑜𝑟𝑟 behaves as an odd function (opposed signs for opposed Hz fields). In  the \nlight of Equation (2), this means that it results from the c ontribution of the second term, \n(Δ𝜌𝑀𝑃𝐸  𝐴𝐻𝐸 +Δ𝜌⊥ 𝑆𝑀𝑅). The first one, associated to in -plane projections of magnetization, is \nnot expected to reverse with reversing Hz fields.  \nOne can observe a sign reversal of the 𝜌𝑥𝑦−𝑐𝑜𝑟𝑟 signal with temperature on Figure 5a. A more \nprecise estimation of the temperature at which this sign reversal operates can be done by \nplotting the temperature dependence of the values measured for 𝜌𝑥𝑦−𝑐𝑜𝑟𝑟 at 7 T (Figure 5b). \nThe inversion tempera ture is of about 120 K, which is the same as the one at which the \nlongitudinal resistivity  𝜌𝑥𝑥 goes to zero (Figure 4b).  A sign inversion of the transverse resistivity  \nhas already been observed for Pt/YIG systems and assigned to AHE-induced by MPE  in th e \nPt.15,19,53 Similar temperature variations of the AHE were also observed in the absence of any \nFM layer, in ion -gated platinum thin films and analogously attributed to some induced \nferromagnetic ordering on the Pt surface.16,54 The exp lanation given by Zhou et al.53 for such \na sign inversion is that for paramagnetic Pt, both the density of states (DOS) and curvatur e \nnear the Fermi surface importantly change with temperature. They indeed observed, for \nPt/YIG heterostructures, a sign inversion of the ordinary Hall coefficient R0 of Pt with \ntemperature, indicating a change of the carriers type. The raw resistivity curv es we measured , \nbefore  OMR and OHR corrections,  are shown in SI, Figure S1. The ordinary Hall coefficient of \nPt, R 0, determined as the slope of the linear part of the measurements  at high magnetic fields , \nis always  negative and does not vary significantly with temperature.  It allows determining a \nfree electron density in Pt of 48 .1028 /m3 , in good agreement with values expected for a \nmetal55 and already reported for other Pt thin films.15,16 The absence of any sign inversion of \nR0 in our case could originate from the fact that we have used a thicker Pt layer (5 nm) than \nthe one used by Zhou et al. (1 nm).53 Shimizu et al. ,16 who have used 3.5 nm thick  Pt layers , \nalso observe a constant negative R 0. The modification of the DOS and curvature near the Fermi \nsurface, induced by a MPE, is indeed expected to happen only at the interface between the \nNM and FM materials, and could be  masked in our case, for which the bulk signature \npredominates . We cannot therefore be fully conclusive on the incidence of MPE -induced AHE \non the temperature behaviour of transverse resistivity.  \nIn order to go further in the understanding of our system, we have sought to disentangle the \nMPE -based and SMR contributions  in the longitudinal measurements .  \n \n  \nFigure 5. Transverse Hall resistivity measurements with (a) Transverse resistivity  corrected from  both the \nordinary magnetoresistance and the linear contribution of the ordinary Hall effect (for a selection of \ntemperatures) , (b) Temperature dependence of the transverse resistivity at Hz = 7 T .\n \n \nAngular dependence of the longitudinal resistivity  \n \nOne possible way to distinguish between the MPE -based AMR and SMR contributions is \nthrough the insertion of a nonmagnetic metal such as Cu  or of an antiferromagnetic oxide \nsuch as NiO, which might eliminate the possibility of MPE effects , leaving only SMR effects to \nplay a role . However, this introd uces additional interfacial issues  and important modifications \nof the SMR , depending upon the interlayer thickness,  have been reported in both \nPt/Cu/Co/Pt56 and Pt/NiO/YIG57 heterostructures .  \nAn a lternative way to separate AMR and SMR contributions is by perfor ming  angle -depend ent \nlongitudinal measurements with th e magnetic field in either the xz or the yz planes, while  the \ncurrent density Jc and the measured resistivity ρxx are in the x direction49 as schematized in \nFigure 6 a. This can  be understood from Equation 1: a rotation of the magnetic field within the \nyz (xz) plane will only have  an effect on the SMR  (AMR) and no effect on the AMR  (SMR ) which  \nonly depends on Mx (My). As depicted in Figure 3, w hile the AMR results from the fact that M \nis parallel to the current direction, the SMR  originates from the fact that M is parallel to the \nspins of the electrons of the SHE-generated spin current, which prevents the spins to be \nabsorbed by the FM , and causes them to be reflected into Pt .14 \nWe measured ρxx at H = 7 T while rotating the sample in the yz (xz) plane to study the  () \nangle dependence of ρxx, at various temperatures. The orientation of the magnetic field is \ndescribe d through the α (within the xz plane) and  (within the yz plane) angles. For the \nexperiments, the α and  angles were limited to a 90° rotation, and conventionally, the z \ndirection was chosen as the 90° angle. Both 𝜌(𝛽)−𝜌𝑧\n𝜌𝑧, and 𝜌(𝛼)−𝜌𝑧\n𝜌𝑧 calculated quantities are \nshown in Figure 6b for various temperatures.  The -dependent measurements show that the \nresistivity value decreases when going from Hz to Hy at all temperatures, with a 180° periodic \noscillation. On the other side, t he α measurem ents also show that the resistivity value \ndecreases when going from Hz to Hx but with a smaller  change in resistivity, and no \nunambiguous periodic oscillation. By fitting the   measurements with cos2𝛽 (SI, Figure S 2), the \n𝜌𝑦 −𝜌𝑧\n𝜌𝑧  SMR  values can be  extracted. The  measurements  could  not unambiguously be fitted \nwith cos2𝛼, and the  𝜌𝑥 −𝜌𝑧\n𝜌𝑧  MPE AMR values are extracted from the 𝜌(𝛼)−𝜌𝑧\n𝜌𝑧 measurements at  \n = 0°. This procedure is comforted by the fact that the values of  𝜌𝑦 −𝜌𝑧\n𝜌𝑧  (SMR ), extracted from \nthe  measurements , have the  same values as 𝜌(𝛽)−𝜌𝑧\n𝜌𝑧 for  = 0°. Hence, assuming that the  \nmeasurements are periodic  as well , we extracted the  𝜌𝑥 −𝜌𝑧\n𝜌𝑧 (AMR) values from  𝜌(𝛼)−𝜌𝑧\n𝜌𝑧 \nmeasurements at  = 0°. \nThe SMR and AMR values obtained as explained  are plotted in Figure 6c. Both contributions \nglobally decrease with increasing temperatures. SMR shows a minimum at about 120 K, which \nis also the temperature at which AMR goes to zero. For all temperatures, the SMR  contribution \ndominates over  the AMR one, and it is the only one present above 120 K. The predominance \nof the SMR mechanism over the entire temperature range has also been observed for Pt/YIG \nand Pd/YIG samples. The SMR value measured for the Pt/GFO heterostru ctures is about 2 .10-4 \nat 300 K and 4.5 .10-4 at 20 K. These values are similar  to what is observed in Pd/YIG \nheterostructures, and only slightly smaller than the ones observed in Pt/YIG (4 .10-4 at 300 K \nand 6 .10-4 at 20 K).49 If the AMR contribution we observe  is very similar to the one reported \nin other  works19,49,58,59: same positive sign, similar amplitude of ca. 10-4, and a decrease with \nincreasing temperature wh ich leaves it practica lly insignificant after ab out 1 00 K, we however \nhighlight some differences stemming from the temperature dependence of the SMR \ncontribution.  Studies performed under a high magnetic field, such as the present one of \nPt/GFO performed at  7 T or the study of Pt/YIG under 100 kOe,49 show V -shaped SMR curves, \nwith first a decrease and then an increase with the increasing temperature, the position of the \nminimum varying between 20 and 120 K from one system to another.  The measurements  performed in lower magnetic fields (10 kOe)53,59 globally show an inversed tendency, with first \nan increase and then a decrease, with a maximum at about 100 K.  \n \n \nFigure 6. Identification of the S MR and AMR contributions to the longitudinal effects, (a) Geometries of the \nmeasurements, (b) Angular measurements at various temperatures (selection of temperatures) in both \ngeometries, (c) Temperature dependence of SMR an d AMR as deduced from the ρ(β)−𝜌𝑧\n𝜌𝑧  and ρ(α)−𝜌𝑧\n𝜌𝑧 values \nmeasured at zero degree angle for β and α, respectively.\n \n \nThe magnetic state of the magnetic material is probably to be considered  to explain such  \ndifferences. Recently, a theoretical study  has shown that the orbital hybridization of the \nmagnetic material plays a role in the magnetoresistance, most probably in relation with spin -\norbit coupling.60 The minimum in SMR  we observe here  could  thus be put in  perspective with \nthe modifi cation of the spin -orbit coupling observed in bare GFO thin films near 120 K via our \nXMCD study .61  \nIn fact, since both SMR and MPE -induced m agnetoresistive  contributions stem from the \ninteraction between the charge current flowing in the Pt layer and the magnetic properties of \nthe FM, the modification of the spin-orbit coupling at 120  K for GFO could contribute to  the \nobserved temperature vari ation s of both longitudinal and transverse magnetoresistance  \nmeasurements in the  90-140 K temperature range (Figures 4b and 5b ).  \n \n4. CONCLUSION  \nThe multifunctional  magnetoelectric GFO oxide has successfully  been introduced in FM I/NM \n(with Pt as the NM) heterostructures  of high crystalline quality. The Pt/ GFO  interface  is sharp  \nand t his makes the heterostructure suitable for spin currents transparency. The interactions \nbetween the spin Hall current from Pt and the GFO magnetic orientation have been eviden ced \nby magnetotransport measurements. SMR  has been shown to be the dominant phenomenon  \nat all temperatures , and it is the only one to be considered near room temperature, wi th a \nmagnitude comparable to those  observed in the classically studied Pd/YIG or Pt/YIG \nheterostructures.  This study therefore validates the use of GFO as a multifunctional \nmagnetoelectric material in NM/ FMI heterostructures with a view to control their spin current \ngeneration  by an electric field.  \n \nSUPPORTING INFORMATION  \nTransverse Hall resistivity measurements, Fitting of the longitudinal angular dependent \nmeasurements  \n \nAUTHOR INFORMATION  \nCorresponding Authors  \n*E-mail: juan -carlos.rojas -sanchez@univ -lorraine.fr  (J.-C.R.-S.) \n*E-mail: viart@unistra.fr  (N.V.)  \n \nACKNOWLEDGEMENTS  \nThis work was funded by the French National Research Agency (ANR) through the ANR -18-\nCECE24 -0008 -01 ‘ANR MISSION’ and, within the Interdisciplinary Thematic Institute QMat, as \npart of the ITI 2021 2028 program of the University of Strasbourg, CNRS and Inserm, it was \nsupported by IdEx Unistra (ANR 10 IDEX 0002), and by SFRI STRAT’US project (ANR 20 SFRI \n0012) and ANR -11-LABX -0058_NIE and ANR -17-EURE -0024 under the framework of the French Investments for the Future Program.  The authors wish to thank D. Troadec (IEMN, Lille, \nFrance) and A. -M. Blanchenet (UMET, Lille, France) for the preparation of the TEM FIB \nlamellae, as well as the XRD , MEB -CRO, and TEM platforms of the IPCMS. We acknowledge \npartial support from the French PIA project “Lorraine Université d’Excellence ”, reference  ANR -\n15IDEX -04-LUE. Devices in the present study were patterned at MiNaLor clean -room platform \nwhich is parti ally supported by FEDER and Grand Est Region through the RaNGE project.  \n \n \nREFERENCES  \n(1)  Slachter, A.; Bakker, F. L.; Adam, J. -P.; van Wees, B. J. Thermally Driven Spin Injection from a \nFerromagnet into a Non -Magnetic Metal. Nat. Phys.  2010 , 6 (11), 879 –882. \nhttps://doi.org/10.1038/nphys1767.  \n(2)  Ando, K.; Takahashi, S.; Ieda, J.; Kajiwara, Y .; Nakayama, H.; Yoshino, T.; Harii, K.; Fujikawa, Y.; \nMatsuo, M.; Maekawa, S.; Saitoh, E. Inverse Spin -Hall Effect Induced by Spin Pumping in \nMetallic System. J. Appl. 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Manuscript under \npreparation . \n \n \n "    },    {        "title": "1912.12663v1.Synchronized_excitation_of_magnetization_dynamics_via_spin_waves_in_Bi_YIG_thin_film_by_slot_line_waveguide.pdf",        "content": " 1 Synchronized excitation of magnetization dynamics via spin waves in Bi-YIG thin film by slot line waveguide  Tetsunori Koda1, Sho Muroga2 and Yasushi Endo3,4,5  1 General Education Division, National Institute of Technology, Oshima College, Suo-Oshima 742-2193, Japan 2 Department of Mathematical Science and Electrical-Electronic-Computer Engineering、Graduate School of Engineering Science, Akita University, Akita 010-8502, Japan 3 Department of Electrical Engineering、Graduate School of Engineering, Tohoku University, Sendai 980-8579, Japan 4 Center for Spintronics Research Network, Tohoku University, Sendai 980-8577, JAPAN 5 Center for Science and Innovation in Spintronics (CSIS), Organization for Advanced Studies, Tohoku University, Sendai 980-8577, JAPAN     2 ABSTRACT  We have studied magnetization dynamics of single Bi-YIG thin films by means of the high frequency power response induced by a slot line waveguide. Multiple absorption peaks that correspond to excitement states in magnetization dynamics appeared without the ferromagnetic resonance (FMR) condition. The peaks were strongly influenced by a waveguide line width and a distance between the lines. Micromagnetics simulation reveals that each line induces a local magnetization dynamics oscillation and generates spin waves. The spin wave that propagates from one of the lines interferences with the other side of local magnetization dynamics oscillation around the other line, resulting in an amplification of the oscillation when they are in synchronization with each other. This amplification occurs at both sides of the lines by the interference. Thus, the possible mechanism of the excitation in the magnetization dynamics oscillation is the synchronization of mutual magnetization dynamics oscillation via spin waves. This technique resonantly excites the local magnetization dynamics without the FMR condition, which is applicable as a highly coherent spin waves source.     3 When a magnetic field is applied to a ferromagnetic material, the magnetization precesses around the axis of final state direction. The precession motion of magnetization, magnetization dynamics, is a basic phenomenon of magnetism. Especially, near the ferromagnetic resonance (FMR) conditions, the precession motion is effectively excited. Since the precession frequency is of the order of GHz, this phenomenon is a strong interest to both fundamental physics and an engineering points of view. For example, a spin torque nano oscillator shows promising properties for microwave emission [1-3] and reservoir computing [4-5]. At near FMR condition, a response of magnetization precession motion shows strong nonlinearity with respect to the change of an applied magnetic field applicable for highly sensitive magnetic sensors. We have reported that the change in magnetization dynamics was sensitive enough to detect a small change in magnetic field and is capable of being used for biomagnetic field measurements [6]. There are several ways to induce the magnetization dynamics oscillation. Typically, a coplanar waveguide (CPW) consisted of one signal line and two ground lines is used as a high frequency waveguide. A strong magnetic field induces local magnetization dynamics at around the signal line, which has been widely used to study the magnetization dynamics of magnetic films and sub micro magnetic elements [7-10].  The design of the waveguide is a key factor because it controls magnetic field dependence of magnetization dynamics in ferromagnetic materials. In our previous work, the magnetic field dependence of reflected RF power in Bi doped YIG (Bi-YIG) single crystal thin films was measured  4 using an asymmetrical CPW, and multiple peaks corresponding to the excitation of magnetization dynamics were observed [6]. Those height of peak profile were strongly influenced by the width of each line and the distance between the lines. At a critical condition, the peak height changed drastically. This change could help detecting a small change in magnetic field. However, the study has not completely explained the origin of the multiple peaks. In this paper, we report magnetization dynamics in a Bi-YIG single crystal thin film induced by a slot line waveguide to understand the origin of the multiple peaks. The slot line waveguide is a representative planar waveguide including two lines with the same width. We systematically changed the width of the lines and the distance between the lines and measured the high frequency power response. The results showed that the interference between local magnetization dynamics oscillation and spin waves generated around each line was a crucial rule for the excitation of magnetization dynamics of the system. Bi-YIG (111) single crystal thin films with a thickness of about 10 µm were epitaxially grown on gadolinium gallium garnet (GGG) (111) single crystal substrates using liquid phase epitaxy technique. To evaluate the high-frequency response of the samples, Cu slot line waveguides with a thickness of 500 nm were directly fabricated on the Bi-YIG film using a combination of photolithography, Ar ion milling and sputtering techniques. The width and distance between the lines were changed and the length of the sample’s signal line(s) and ground lines was fixed at 300 µm (Fig.1 (a)). Since all the waveguides were fabricated on a single Bi-YIG thin film, the magnetic property was constant. When magnetization dynamics was resonantly excited, the input power was absorbed for the use of the  5 precession motion of magnetization, resulting in the significant reduction of the reflected signal strength. Magnetization dynamics was evaluated by measuring the reflected power from the slot line waveguide with the vector network analyzer (Keysight Model N5230A) while sweeping the magnetic field parallel to the sample plane. The input signal strength was set at 0 dBm for all the measurements and the frequency was fixed at a frequency between 6.0 and 9.0 GHz for each measurement. It should be noted that the short length of lines help high-frequency input signals propagating along the waveguide, although the waveguide impedance was different from its measurement setup impedance. The numerical simulations based on micromagnetics were performed using object oriented micromagnetic framework (OOMMF) [11]. Figure 2 (a) shows the reflected high frequency signal strength of various input frequency according to the distance between the lines. Multiple peaks were clearly detected in all conditions. The peaks shifted towards higher magnetic field with the increase of frequency. This indicates that the peaks are associated with magnetic phenomenon. For the samples with the same line width, the number of multiple peaks increased and shifted towards higher magnetic field with the increase of the distance between the lines. Furthermore, one of the peaks shifted higher magnetic field with increasing the width for the samples with the same line’s distance as shown in Fig.2 (b).  To understand the experimental results and the origin of the peaks for slot lines waveguides, we carried out the micromagnetics simulation on the RF line’s number dependence of magnetization dynamics. The simulation models are shown in Fig. 3 (a). Those names are one-line and two-line models. The dimensions used in the simulation were 200 µm long (l), 320 µm wide (w), and 10 µm  6 thick (z), and one or two lines were arranged. The cell sizes were set at l=50 nm, w=200 µm and z=1 µm. The length of the lines was fixed at 200 µm, and the width and the distance (d) were varied. The Gilbert damping factor  a of the Bi-YIG thin film was set at 0.001. A small RF magnetic field (0.5 Oe) was applied perpendicular locally at the position of the lines with the frequency at 8.0 GHz. A DC magnetic field was applied parallel to the length direction to the film. The simulation results are shown in Fig 3 (b). Only one peak associated with FMR is observed for the one-line model. The FMR conditions are determined by the magnetization properties and line width. The broad FMR peak is due to the inhomogeneous demagnetizing field caused by the limitation of simulation size. On the other hand, multiple peaks were identified for the two-line models. When the distance between the lines are identical, the peak shifted toward higher magnetic field with the increase of line widths. The magnetic field where magnetization reaches its peak shifts with the increase of the distance between the lines. Those results faithfully reproduce the experimental results. The difference of the magnetic fields for the appearance of the peaks between the experimental and the simulation results is caused by the difference of the internal magnetic field owing to the limitation of the simulation size. In the experiments, the slot line waveguides with the length of 300 µm were directly fabricated on the Bi-YIG thin film with the substrate size of 11 mm ´ 11 mm. We simulated the magnetization orientation distribution for x-direction for one-line model as shown in Fig. 3 (c). The DC magnetic field was set at 3.95 kOe, which was the condition for the appearance of the peak of the two-line model in Fig.3 (b). This shows that the line induces local magnetization dynamics. The oscillation of localized magnetization dynamics propagates perpendicular to the line direction via exchange or dipole  7 interaction, and it generates magneto static surface spin waves (MSSW) [12]. This result reveals that the lines also work as spin wave sources. Figure 3 (c) also shows the combined wave of one-line and the 40 µm shifted waves as the green and red lines, respectively. The geometric condition of the combined wave is the same as that of the two-line model. Figure 3 (d) shows the comparison between the combined wave and the spin wave for the two-line model under the same simulation condition. The amplitude of the spin wave for the two-line model is larger than that for the combined wave. This indicates that the local magnetization dynamics oscillation is more amplified in the two-line model. For the two-line model, a creation of spin waves from the lines propagates toward both sides of the lines, and the spin waves reach the other side of the line. The wavelength of MSSW is described as [13-14] 𝑓\"##$=&(𝑓(+𝑓\"2⁄)-−(𝑓\"2⁄)-exp(−2𝑘𝑑4),       (1) where fH=gH0 and fM=4pgMS and d0 is film thickness. Here, g =2.8 MHz Oe-1 is the electron gyromagnetic ratio, k is the wavenumber of spin wave, H0 is the effective internal field, MS=140 emu/cc is the saturation magnetization of YIG [12]. The wave length increases with the increase in the applied magnetic field. Fig. 4 shows the experimental results for the magnetic fields at the magnetization peaks to the calculated dispersion curve of the equation (1) for several input frequencies. The peaks appeared when the wave length of the spin wave was almost the same as the distance of the lines. This result indicates that the spin wave generation is potentially relevant with the excitement of magnetization dynamics.  If there is an interaction between the local magnetization dynamics and the spin waves, the condition  8 of the interference should be influenced by the phase difference of those dynamics. The phase difference can be set by the input RF magnetic fields condition of the lines. The simulation results are shown in Fig.5. The peaks shifted with the change of the phase difference. This strongly indicates the existence of the interaction between the local magnetization dynamics and the spin waves.  The possible mechanism of the excitation of magnetization dynamics is as follows. Each local magnetization dynamics under each line generates the spin wave. This spin waves propagate each other and interact with the other side of local magnetization dynamics. When the phase between the local magnetization dynamics and the spin wave coincides each other, the local magnetization dynamics is amplified resonantly. The amplified magnetization dynamics emits the spin wave with large amplitude. The enhanced spin waves propagate and amplify the local magnetization dynamics under the other side of the line. Thus, the local magnetization dynamics under the both side of lines are resonantly excited via the spin waves. This is a typical mutual synchronization phenomenon and have been reported in different systems [15-20]. The reason why we did not detect the FMR peak in our system was that the wave length of spin waves at FMR condition was much longer than the distance between the lines. Thus, the local magnetization dynamics directly influenced each other via the spin waves. We found that the phase difference of the slot line waveguide was about 180° based on numerical electromagnetic analysis. This phase difference effectively suppressed the excitation of the magnetization dynamics for both lines.  Our experimental results show that the magnetization dynamics can be controlled by using multiple sources of local magnetization dynamics. This finding gives the artificial control of the emission  9 condition of highly coherent spin waves without using the FMR, which is attractive for the applications that use the interference of spin waves such as logic gates devices [21-23] and magnonic crystals [24-26]. In conclusion, we studied magnetization dynamics of single Bi-YIG thin films by measuring the high frequency power response induced by the slot lines waveguide. Multiple peaks corresponding to the excitement of magnetization dynamics appeared. We found that the peaks were strongly influenced by the width and the distance of lines. Micromagnetics simulation successfully reproduced the experimental results and revealed that the spin waves generated from the local magnetization dynamics at each line propagated and interfered with the other side of the local magnetization. At a critical condition, this interference resonantly excites the both side of magnetization dynamics owing to the synchronization via spin waves. This finding gives a way for the excitement of the local magnetization dynamics without the FMR condition, which is applicable as a highly coherent spin waves source.  ACKNOWLEDGMENT We thank GRANOPT Co,Ltd. for their support of Bi-YIG single crystal thin films. This work was supported by JSPS KAKENHI Grant Number JP18K14114.  References [1] S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J. Schoelkopf, R. A. Buhrman, and  10 D. C. Ralph, Nature 425, 380 (2003). [2] W. H. Rippard, M. R. Pufall, S. Kaka, S. E. Russek, and T. J. Silva, Phys. Rev. Lett. 92, 027201 (2004). [3] D. Grollier, D. Querlioz, and M. D. Stiles, Proc. of the IEEE 104, 2024 (2016). [4] S. Tsunegi, T. Taniguchi, K. Nakajima, S. Miwa, K.Yakushiji, A. Fukushima, S. Yuasa, and H. Kubota, Appl. Phys. Lett. 114, 164101 (2019). [5] T. Kanao, H. Suto, K. Mizushima, H. Goto, T. Tanamoto, and T. Nagasawa, Phys. Rev. Appl. 12, 024052 (2019). [6] T. Koda, S. Muroga, Y . Endo, IEEE Trans. Magn. 55, 4002604 (2019). [7] G. Counil, P. Crozat, T. Devolder, C. Chappert, S. Zoll, and R. Fournel, IEEE Trans. Magn. 42, 3321 (2006). [8] I. Neudecker, G. Woltersdorf, B. Heinrich, T. Okuno, G. Gub- biotti, and C. Back, J. Magn. Magn. Mater. 307, 148 (2006).  [9] C. Bilzer, T. Devolder, P. Crozat, C. Chappert, S. Cardoso, and P . Freitas, J. Appl. Phys. 101, 074505 (2007). [10] I. Harward, T. O’Keevan, A. Hutchison, V . Zagorodnii, and Z. Celinski, Rev. Sci. Instr. 82, 095115 (2011). [11] OOMMF User’s Guide, Version 1.0, M. J. Donahue, D. G. Porter, Interagency Report NISTIR 6376, National Institute of Standards and Technology, Gaithersburg, MD, 1999. http://math.nist.gov/oommf  11 [12] D. D. Stancil, and A. Prabhakar, Spin Waves: Theory and Applications (NewYork, Springer, 2009). [13] D. D. Stancil, Theory of MagnetostaticWaves (NewYork, Springer, 1993) [14] B. A. Kalinikos, and A. N. Slavin, J. Phys. C: Solid State. Phys. 19, 7013 (1986). [15] S. Kaka, M. R. Pufall, W. H. Rippard, T. J. Silva, S. E. Russek, and J. A. Katine, Nature 437, 389 (2005). [17] F. B. Mancoff, N. D. Rizzo, B. N. Engel, and S. Tehrani, Nature 437, 393–395 (2005). [18] V . E. Demidov, H. Ulrichs, S. V . Gurevich, S. O. Demokritov, V . S. Tiberkevich, A. N. Slavin, A. Zholud and S. Urazhdin, Nat. Commun. 5, 3179 (2014). [19] A. Houshang, E. Iacocca, P. Dürrenfeld, S. R. Sani, J. Åkerman and R. K. Dumas, Nat. Nanotechnol. 11, 280–286 (2016). [20] A. A. Awad, P. Dürrenfeld, A. Houshang, M. Dvornik, E. Iacocca, R. K. Dumas and J. Åkerman, Nat. Phys. 13, 292 (2017). [21] Chumak, A. V., Serga, A. A. & Hillebrands, B., Nature Commun. 5, 4700 (2014). [22] K. Vogt, F.Y. Fradin, J.E. Pearson, T. Sebastian, S.D. Bader, B. Hillebrands, A. Hoffmann and H. Schultheiss, Nat. Commun. 5, 3727 (2014). [23] T. Schneidera, A. A. Serga, B. Leven, and B. Hillebrands, Appl. Phys. Lett. 92, 022505 (2008). [24] G Gubbiotti, S Tacchi, M Madami, G Carlotti, A O Adeyeye, and M Kostylev, J. Phys. D 43, 264003 (2010). [25] Y . V . Gulyaev, S. A. Nikitov, L. V. Zhivotovskii, A. A. Klimov, P. Tailhades, L. Presmanes, C.  12 Bonningue, C. S. Tsai, S. L. Vysotskii, and Y. A. Filimonov, JETP Lett. 77, 567–570 (2003). [26] M. Krawczyk, and D. Grundler, J. Phys. Cond. Matter 26, 123202 (2014).    13    \n  Fig. 1 Schematic view of measurement set up for magnetization dynamics using a slot line waveguide.    \n 14  \n  Fig. 2 (a) Applied magnetic field dependence of the power of reflected high frequency wave with various frequency of input power. The distance between lines and the width of the line were systematically changed. (b) Slot line waveguide’s width dependence of the reflected power measured at 8.0 GHz.   \n 15  \n \n \n 16 Fig. 3 (a) Schematic view of micromagnetics simulation models. The number of local areas applied RF magnetic field was set at one or two. The dimensions used in the simulation were 200 µm long, 320 µm wide, and 10 µm thick. and one or two lines were arranged. The length of the lines was fixed at 200 µm, and the width (w) and the distance (d) were varied.  (b) Micromagnetic simulation results for the line’s number dependence of magnetization dynamics. W and D stand for the width of lines and the distance between lines, respectively.  (c) Magnetization orientation distribution for x-direction for one-line model at 3.95kOe. Y-axis indicates the z component of normalized magnetization at each position. The green line shows the 40 µm shifted curve of the blue line. The red line shows the combined curve of the blue and green curves. (d) Comparison for simulation results of the spin waves for two-lines model and the combined curve shown in Fig.3 (c).    17  Fig. 4 Lines’ distance dependence of magnetic field for the appearance of magnetization dynamics. The solid lines show the calculated dispersion curves using the equation (1).       \n  Fig. 5 Magnetization dynamics of the two-line model. The input RF magnetic fields have various range of phase difference.   \n"    },    {        "title": "1604.03272v2.The_effect_of_inserted_NiO_layer_on_spin_Hall_magnetoresistance_in_Pt_NiO_YIG_heterostructures.pdf",        "content": "arXiv:1604.03272v2  [cond-mat.mtrl-sci]  22 Apr 2016The effect of inserted NiO layer on spin-Hall magnetoresista nce in Pt/NiO/YIG\nheterostrucures\nT. Shang,1,∗H. L. Yang,1Q. F. Zhan,1,†Z. H. Zuo,1Y. L. Xie,1L. P. Liu,1S. L.\nZhang,1Y. Zhang,1H. H. Li,1B. M. Wang,1Y. H. Wu,2S. Zhang,3,‡and Run-Wei Li1,§\n1Key Laboratory of Magnetic Materials and Devices &Zhejiang Province\nKey Laboratory of Magnetic Materials and Application Techn ology,\nNingbo Institute of Material Technology and Engineering,\nChinese Academy of Sciences, Ningbo, Zhejiang 315201, China\n2Department of Electrical and Computer Engineering,\nNational University of Singapore, 4 Engineering Drive 3 117 583, Singapore\n3Department of Physics, University of Arizona, Tucson, Ariz ona 85721, USA\n(Dated: August 25, 2021)\nWe investigate the spin-current transport through antifer romagnetic insulator (AFMI) by means\nof the spin-Hall magnetoressitance (SMR) over a wide temper ature range in Pt/NiO/Y 3Fe5O12\n(Pt/NiO/YIG) heterostructures. By inserting the AFMI NiO l ayer, the SMR dramatically decreases\nbydecreasing thetemperature down totheantiferromagneti cally ordered state ofNiO, which implies\nthat the AFM order prevents rather than promotes the spin-cu rrent transport. On the other hand,\nthe magnetic proximity effect (MPE) on induced Pt moments by Y IG, which entangles with the\nspin-Hall effect (SHE) in Pt, can be efficiently screened, and p ure SMR can be derived by insertion\nof NiO. The dual roles of the NiO insertion including efficient ly blocking the MPE and transporting\nthe spin current from Pt to YIG are outstanding compared with other antiferromagnetic (AFM)\nmetal or nonmagnetic metal (NM).\nSpin current, the motion of spin angular moment, has\nattracted intense interest due to the prospects of low\nconsumption spintronic devices1,2. Several experimen-\ntal techniques have been developed to generate and ma-\nnipulate spin current, e.g., spin pumping3–5, spin See-\nbeck effect6–8, and also spin-Hall effect (SHE)9–11. Re-\ncently, the generation and propagation of spin current\nin antiferromagnets (AFMs) including insulators or met-\nals have been extensively investigated by various tech-\nniques12–22. Especially, the thermally injected or dy-\nnamically pumped spin current from ferromagnetic(FM)\nYIG layer can flow into the NiO or CoO AFM insulator\n(AFMI) layer and reach the Pt or Ta nonmagnetic metal\n(NM) layer where it can be converted into charge current\nby means of inverse spin-Hall effect (ISHE)18–22. By in-\nserting thin AFMI layer, the ISHE voltage is largely en-\nhancedandexhibitsnonmonotonictemperatureorAFMI\nthickness dependence, which reaches a maximum near\nthe N´ eel temperature of AFMI or with the AFMI thick-\nness of∼1-2 nm, respectively18–22.\nSeveral theoretical models have been proposed for the\npropagation of injected spin current through AFMs in\nNM/AFMs/FM heterostructures23–26. These models de-\nscribe the spin-current transport and its enhancement\nby assuming that the AFMs are ordered at room tem-\nperatures, while the AFM ordering temperatures of thin\nAFMs are well below the room temperature23–26. How\nthe spin current interacts with AFM order is still beyond\nunderstood. In most of these experimental or theoretical\ninvestigations, the spin currentcarried by spin waves, are\nproducedin YIG layerand flowinto theAFMI layer18–26.\nWhile there is another type of spin current, which is car-\nried by conduction electrons via SHE in strong spin-orbit\ncoupling (SOC) NM9–11. However, the investigationsof spin-current transport from SOC NM into AFMI are\nrare, which can help us to further understand the mech-\nanism of spin-current propagation in AFMs. Herein, we\ninvestigated the spin-current transport in Pt/NiO/YIG\nheterostructures by injecting the spin current from top\nPt layer. Our results show that when approaching the\nantiferromagnetically ordered state of NiO, the spin-\nHall magnetoresistance (SMR) amplitude is largely sup-\npressed, implying that the AFM order prevents rather\nthan promotes the spin-current transport.\nThe Pt/NiO/YIG heterostructures were prepared in\na combined ultra-high vacuum (10−9Torr) pulse laser\ndeposition (PLD) and sputter system. The high-quality\nYIG films were epitaxially deposited on (111)-orientated\nGd3Ga5O12(GGG) substrates via PLD technique. The\nthin NiO films were deposited on YIG films after the\ngrowth of YIG film. The top Pt layers were sputtered in\nanin situprocess. In this study, the thicknesses of YIG\nand Pt films are fixed at 60 nm and 3 nm, respectively,\nwhile the NiO thickness ranges from 0 to 8 nm. Figure\n1(a) plot representative room-temperature x-ray diffrac-\ntion (XRD) patterns for epitaxial YIG/GGG film near\nthe (444) reflections. Clear Laue oscillations indicate the\nflatness and uniformity of the epitaxial films. As shown\nin the insets of Fig. 1(b), no indication of impurities or\nmisorientation was detected in the range of 10 to 80 de-\ngree. The atomic force microscope surface topography\nof Pt/NiO(3)/YIG heterostructure over an area of 3 µm\n×3µm in Fig. 1(c) reveals a root-mean-square surface\nroughness of 0.14 nm (The number in the brackets rep-\nresents the thickness of NiO layer in nm unit), indicating\natomically flat of prepared films, with the other films\nshowing similar surface topology. A representative ferro-\nmagnetic resonance (FMR) derivative absorption spec-/s48/s46/s54/s110/s109 \n/s48/s46/s48/s110/s109 \n/s50/s46/s53 /s109 /s32\n/s50/s46/s48\n/s49/s46/s53\n/s49/s46/s48\n/s48/s46/s53\n/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53 /s109 /s32/s32/s80/s116/s47/s78/s105/s79/s47/s89/s73/s71/s40/s99/s41\n/s50/s48/s54/s48 /s50/s48/s56/s48 /s50/s49/s48/s48 /s50/s49/s50/s48 /s50/s49/s52/s48 /s50/s49/s54/s48/s100/s73\n/s70/s77/s82/s47/s100/s72/s32/s40/s97/s46/s117/s46/s41\n/s70/s105/s101/s108/s100/s32/s40/s79/s101/s41/s72 /s32/s126/s32/s56/s79/s101/s40/s100/s41/s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48/s40/s52/s52/s52/s41\n/s32/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s46/s117/s46/s41\n/s40/s100/s101/s103/s114/s101/s101 /s41/s40/s50/s50/s50/s41/s40/s98/s41\n/s52/s57/s46/s53 /s53/s48/s46/s48 /s53/s48/s46/s53 /s53/s49/s46/s48 /s53/s49/s46/s53 /s53/s50/s46/s48 /s53/s50/s46/s53/s40/s97/s41\n/s71/s71/s71\n/s32/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s46/s117/s46/s41\n/s40/s100/s101/s103/s114/s101/s101 /s41/s89 /s73/s71\nFIG. 1. (Color online) (a) A representative 2 θ-ωXRD pat-\nterns for YIG/GGG film near the (444) reflections. (b) The\nXRD patterns from 20 to 80 degree. (c) Three dimensional\nplot of the atomic force microscope surface topography for\nPt/NiO(3)/YIG heterostructure over an area of 3 µm×3\nµm. (d) A representative FMR derivative absorption spec-\ntrum of YIG film with.\nFIG. 2. Anisotropic magnetoresistance at various tempera-\ntureswiththemagnetic fieldvariedwithin xy(top-rowpanel),\nxz(middle-row panel), and yz(bottom-row panel) planes.\nThe right panel shows the schematic plots of longitudinal an d\ntransverse resistance measurements. The magnetic fields ar e\napplied with angles θxy,θxz, andθyzrelative to the y-,z-,\nandz-axes. The electric current is applied along the x-axis.\nThe AMR for Pt/YIG are shown in (a)-(c), while the results\nof Pt/NiO(1)/YIG are shown in (d)-(f).\ntrum of YIG film (60 nm) shown in Fig. 1(d) exhibits\na line width ∆H of 8 Oe, which was measured at radio\nfrequency 9.39 GHz and power 0.1 mW with an in-plane\nmagnetic field at room temperature. All the above prop-\nerties indicate the excellent quality of prepared films.\nThe recent proposed SMR built on the combined SHE\nand ISHE was applied to investigate the spin-current\ntransport from SOC NM into AFMI in Pt/NiO/YIGheterostructures15,27–30. As shown in the right panel of\nFig. 2, all the films were patterned into Hall-bar ge-\nometry by using mask to measure the longitudinal and\ntransverse Hall resistance. The anisotropic magnetore-\nsistance (AMR) were measured in a magnetic field of\n20 kOe for both Pt/NiO/YIG and Pt/NiO/MgO het-\nerostructures. The absence of AMR in Pt/NiO/Mgo\nimplies that the NiO moments are robust against such\nmagnetic field. Figures 2(a)-(c) plots the AMR for\nPt/YIG at various temperatures, while the results of\nPt/NiO(1)/YIG are shown in Figs. 2(d)-(f). Both the\nPt/YIG and Pt/NiO(1)/YIG heterostructures demon-\nstrate clear SMR, with the amplitudes reaching 6.1 ×\n10−4and 4.5×10−4at room temperature, respectively\n[see Figs. 2(c) and (f)], implying that the spin current\ngenerated by SHE in Pt can transport through the NiO\nto interact with the YIG. Apparently, the spin current\ncan pass through NiO from both sides, i.e., Pt →NiO→\nYIG or YIG →NiO→Pt18,19,22. For Pt/YIG, as shown\nin Fig. 2(b), the magnetic proximity effect (MPE) in-\nduced conventional AMR (CAMR) always coexists with\nSMR and its maximum amplitude of 2.2 ×10−4is com-\nparable to the SMR. However, for Pt/NiO(1)/YIG, there\nis no clear CAMR even down to lowest temperature [see\nFig. 2(e)], and the observed AMR are almost attributed\nto the SMR. As shown in Figs. 2(d)(f), the θxyandθyz\nscansexhibit similarbehaviors, with theiramplitudes be-\ningalmostidenticaltoeachother. TheinsertedNiOlayer\nbetween Pt and YIG efficiently suppresses the MPE at\nthe interface, which also can be revealed by anomalous-\nHall resistance (AHR). For instance, the AHR at 70 kOe\nfor Pt/NiO(1)/YIG is 4.6 mΩ at 5 K, which is 6 times\nsmaller than the Pt/YIG (27.6 mΩ). As further increas-\ning the NiO thickness, the AHR is negligible for tNiO≥\n3 nm. Compared to the Pt/IrMn/YIG, Pt/Cu/YIG, or\nPt/Au/YIG heterostructures, only 26% of the SMR is\nlost by insertion of NiO, while over 80% of the SMR is\nsuppressed by insertion of IrMn, Cu, or Au30–32. Thus,\nthe dual roles of the inserted NiO are outstanding, which\nefficiently blocks the MPE and transports the spin cur-\nrent.\nAll the AMR amplitudes are summarized in Fig. 3\nas a function of temperature. For Pt/YIG, the SMR\nexhibits nonmonotonic temperature dependence and ac-\nquires its maximum value of 6.9 ×10−4around 150 K.\nWhile for θxyscan, where both CAMR and SMR con-\ntribute the total AMR, the amplitude is almost identi-\ncal to the difference between SMR and CAMR [see star\nsymbols in Fig. 3(a)], i.e., |∆ρ|/ρ0(θxy) =|∆ρ|/ρ0(θyz)\n-|∆ρ|/ρ0(θxz), indicating that the MPE induced CAMR\nat Pt/YIG interface competes with the SMR. However,\nafter inserting NiO, the SMR exhibits significantly dif-\nferent behaviors. As shown in Fig. 3(b), the SMR\ncaptures the AMR for θxyscan due to the absence of\nMPE, whose amplitude is almost identical to the θyz\nscan, i.e., |∆ρ|/ρ0(θxy) =|∆ρ|/ρ0(θyz). Similar results\nwere also reported previously in MPE-free Rh/YIG bi-\nlayers33. Moreover, the SMR undergoes a sharp decrease\n2/s48/s51/s48/s54/s48/s57/s48\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s50/s48/s52/s48/s47\n/s48/s40/s49/s48/s45/s53\n/s41\n/s120 /s121/s32\n/s121/s122/s45/s32\n/s120 /s122/s32\n/s120 /s122/s32\n/s121/s122/s40/s97/s41/s32/s80/s116/s47/s89/s73/s71\n/s84/s32/s40/s75/s41/s40/s98/s41/s32/s80/s116/s47/s78/s105/s79/s47/s89/s73/s71/s47\n/s48/s40/s49/s48/s45/s53\n/s41\n/s84\n/s98\nFIG. 3. Temperature dependence of the AMR amplitudes for\n(a) Pt/YIG and (b) Pt/NiO(1)/YIG heterostructures. The\ncubic, circle, and triangle symbols stand for θxy,θxz, andθyz\nscans, respectively. The star symbols represent the differe nce\nbetween SMR and CAMR ( θyz-θxz). The solid lines are\nguides to the eyes.\nas decreasing the temperature and changes its sign be-\nlow 70 K [see Figs. 2(d)(f)], with its amplitude being\ntemperature independent for T <70 K. According to\nmagnetization results, the appearance of the exchange\nbias field around the blocking temperature Tb∼70 K in\nPt/NiO(1)/YIG suggests that the moments in NiO layer\nbecome antiferromagnetically ordered when approaching\nthis temperature(see detailsin Fig. 4). Since the signre-\nversal happens simultaneously around the blocking tem-\nperature, it is likely associated with the interactions be-\ntweenthe antiferromgneticallyalignedNiO moments and\nspin current. For Pt/NiO(1)/YIG, the SMR amplitude\nis almost 10 times smaller for T < T bthan the room-\ntemperature value. Most of the spin current are blocked\nby NiO beforereachingthe NiO/YIG interfacewhen NiO\nmoments are ordered.\nSince the AFM transition temperature for very thin\nAFMs is expected to be well below the room tempera-\nture, we also measured the field dependence of magne-\ntization for Pt/NiO/YIG at various temperatures, from\nwhich the AFM transition temperature can be roughly\ntrackedby blockingtemperature from exchangebias field\nHE. As shownin Fig. 4(a), thenormalizedmagnetichys-\nteresis loops M/Msfor Pt/NiO(1)/YIG at various tem-\nperatures are presented, with the other heterostructures\nshowing similar behaviors. The derived HEas a function\nof temperature are summarized in Fig. 4(b), from which\ntheTbare approximately determined to be around 70 K,\n110 K, and 150 K for Pt/NiO(1)/YIG, Pt/NiO(3)/YIG,\nand Pt/NiO(5)/YIG, respectively. Similar blocking tem-\nperatures were previously reported in Co/NiO/(Co/Pt)\nheterostructures34. Moreover, as shown in the inset of/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s50/s48/s48/s45/s50/s48/s45/s52/s48/s45/s54/s48/s45/s56/s48/s45/s49/s48/s48\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s49/s48/s50/s48/s51/s48/s49/s110/s109/s78/s105/s79\n/s51/s110/s109\n/s53/s110/s109/s72\n/s69/s32/s40/s79/s101/s41\n/s84/s32/s40/s75/s41/s40/s98/s41\n/s72\n/s99/s32/s40/s79/s101/s41\n/s84/s32/s40/s75/s41\n/s45/s49/s48/s48 /s45/s53/s48 /s48 /s53/s48 /s49/s48/s48/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s77/s47/s77\n/s83\n/s70/s105/s101/s108/s100/s32/s40/s79/s101/s41/s50/s75\n/s53\n/s49/s48\n/s49/s53\n/s50/s48\n/s50/s53\n/s51/s48\n/s51/s53\n/s52/s48\n/s52/s53\n/s53/s48\n/s49/s48/s48\n/s51/s48/s48/s80/s116/s47/s78/s105/s79/s40/s49/s110/s109/s41/s47/s89 /s73/s71/s40/s97/s41\nFIG. 4. (a) Field dependence of normalized magnetization\nM/Msfor Pt/NiO(1)/YIG at various temperatures. (b) The\nin-plane exchange bias field HEfor Pt/NiO/YIG with differ-\nent NiO thicknesses versus temperature. The arrows indicat e\nthe AFM block temperatures Tb. The inset plots the coerciv-\nity field HCfor Pt/NiO(1)/YIG.\nFig. 4(b), the coercivity HCof Pt/Ni(1)/YIG also ex-\nhibitsastep-likeincreasenearthe Tbduetotheenhanced\nmagnetic exchangecoupling between NiO and YIG layer.\nDifferent from the spin pumping or spin Seebeck tech-\nniques, where the spin current are produced at the\nAFM/FM interface by the precessing magnetization or\nthermal gradient in the YIG layer and flow through the\nAFMI layer18–22, the direction of spin-current transport\nis reversal in our experiments. According to the SMR\nresults, the spin current generated in Pt via SHE can\ntransport through the NiO to interact with the YIG.\nHowever, the SMR amplitude is strongly suppressed be-\nlow the blocking temperature of NiO. Based on the SMR\nmodel, the reflected spin current interact with NiO mo-\nments again before converting into charge current in Pt,\nwhich is more complicated than spin pumping or spin\nSeebeckprocess. Severaltheoreticalmodelsincludingthe\ncoherent magnetization dynamics and incoherent ther-\nmal magnons have been proposed to explain the spin-\ncurrent transport in AFMI and its enhancement by in-\nsertion of AFMI23–26. In the magnetic dynamics model,\nthe processed magnetization in YIG layer exerts a torque\non the AFM moments and promotes the magnetization\ndynamics propagation to the AFM/FM interface23–25.\nThe other model describe the spin-current transport by\nconsidering the diffusion of incoherent thermal magnons\nwhich is caused by the accumulation of magnons at the\nAFM/FM interface26. According to these theoretical\ncalculations, the coupling of the spin excitations at the\nNiO/YIG interface is much larger than at Pt/NiO or at\nPt/YIG, and the spin mixing conductance of AFM/FM\ninterface also depends on the AFMI layer thickness and\nlinearlyscaleswiththeenhancementofspincurrent,both\nof which can explains the initial increase of spin current\ntransport by increasing the AFMI thickness22–26. In our\ncase, the spin current is generated via SHE in Pt layer\nand is independent of YIG/NiO interface. However, the\n3enhanced coupling of spin excitation at the NiO/YIG in-\nterface can efficiently enhance the spin pumping or spin\nSeebeck signals23–26. So far, all these theoretical models\nassume the AFMI are ordered at room temperature23–26,\nwhere all the simulations have been done, but the order-\ning temperatures of thin AFMs are well below the room\ntemperature. Further theoretical models which include\nthe temperature degree of freedom are highly desirable\nto understand the spin-current transport in AFMs.\nIn summary, we investigated the spin-current trans-\nport properties by means of SMR in Pt/NiO/YIG het-\nerostructures over a wide temperature range. Different\nfrom Pt/YIG, the SMR in Pt/NiO/YIG is significantly\nsuppressedwhen the temperatureapproachesthe antifer-\nromagnetically ordered state of NiO and becomes tem-perature independent below the blocking temperature.\nHowever, the dual roles of the NiO insertion includ-\ning efficiently screening the MPE at Pt/YIG interface\nand transporting the spin current from Pt to YIG lay-\ners are outstanding. These experimental results extend\nthe knowledge for the further theoretical investigations\non the spin-current propagation in AFMs.\nWethankthehighmagneticfieldlaboratoryofChinese\nAcademy of Sciences for the FMR measurements. 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It is shown that t he non -linear magnetic susceptibility of the yttrium iron garnet \n(YIG) ferrite , on account of its narrow line -width Larmor precessional resonance , make it an ideal material for \nthe creation of entangled photons . These can be measured using  a homodyne interferometer , as the much larger \nnumber  of thermally generated photons  associated with  ambient temperature emission  can be screened out . The \nproposed architecture may enable YIG quantum technology -based  sensors to be developed , mimicking in the \nmillimeter -wave band the large number of quantum optical experiments in the near -infrared and visible regions \nwhich had been made possible by use of  the nonlinear beta barium borate ferroelectric , an analogue of YIG . It is \nillustrated here how the YIG parametric amplifier can reproduc e quantum optical Type I and Type II wave \ninteractions, which can be used to create entangled photons in the millimeter -wave band.  It is estimated that \nwhen h alf a cubic centimeter  of YIG crystal is placed in a magnetic field of a few Tesla and pumped with 5 Watt s \nof millimeter -wave radiatio n, approximately 0.5x1012 entangled millimeter -wave photon  pairs per second  are \ngenerated by the spin -wave interaction . This means an integration time of only a few tens of seconds is needed \nfor a successful Bell Test.  A successful demonstration of th is will lead to novel architectures of entanglement -\nbased  quantum technology room temperature sensors, re -envisioning YIG as a modern quantum material.  \n__________________________________________________________________________________  \n1. Introduction   \nThe Einstein -Poldolsky -Rosen (EPR) paper [1] from  1935 \nprovoked a  dialogue questioning  the nature of reality  at the quantum \nlevel , which  stimulat ed hugely the development of quantum theory . \nThe ensuing discussions around this are likely to extend well into the \n21st century , as they are sitting right at the heart of emerging quantum \ncomputers  and sensors .   \nJohn Bell in 1964 conceived a  simple test on the nature of local \nreality  [2] that centered on evaluating an  inequality. If the inequality \nis obeyed , future events in a system are completely determined by past \nhistory, this information being passed on  in postulated hidden \nvariables. The term hidden is used as these variables are not part of \nquantum theory. If the inequality is violated , any local hidden \nvariables are  not dictating future events , and the state of a system is \ndetermined only by its measurement.  \nJohn Clauser  experimentally embodied [3] the Bell Test by using \npolariz ed optical photon s; similar tests followed and these types of test \nare now  referred to as Bell Tests. The maturing technologies of \nnonlinear dielectric crystals, lasers and photo -diodes then enabled the \nfirst demonstrations of a Bell Test  that violated  the Bell inequality  by \n40 standard deviations  [4]. With the exceptions of a few loopholes, \nthis demonstrate d that at the quantum level , reality is defined by a \nmeasurement.  Since then  optical Bell Tests have closed many of the \nloopholes  [5] and reduc ed signal integration times from 200 hours in \nthe 19 70’s [6], to a few seconds with today’s  technology . This \ntechnology  in the optical band is now available commercially for \nundergraduate teaching laborator ies [7].  \nThe initial Bell Tests in the optical band were optimal, as detection \nof the weak entangled photon  flux went unhindered by spurious \nsignals from thermally generated photons. For operation in the \nmillimeter wave band the problem of the thermal photons has been \nsuccessfully comb ated by  cryogenically cool ing the sources and \nreceivers [8], [9].  \nFor operation at ambient temperature in the millimeter wave band, \na novel Bell Test is herein proposed. Such a system  would be  cheaper  \nand lead to easier  experimentation and in-field system deployments . \nThe thermal photon problem is circumvented by using a yttrium iron \ngarnet (YIG) ferrite parametric amplifier source of entangled photons \nwith a phase link to a homodyne interferometer performing  coherent \nintegration  [10]. A key point about this test is that the pump which \ngenerates the entangled photons is used to coherently integrate the \nentangled photon signature in quadrature space.  It is important to \nrecognize that the special phase r elationship between the pump, signal \nand idler is a fundamental enabler of th e coherent integration in the \nproposed Bell Test.  \n \n* Corresponding Author.  \n     E-mail address: n.salmon@mmu.ac.uk (N.A. Salmon).  \n1 ORCid: 0000 -0003 -4786 -7130.  \n2 ORCid: 0000 -0002 -1250 -9432. The proposed homodyne system  generat es non-degenerate \nentangled photons  over a quasi -continuous spectrum , and then  \nmeasure s their phase s and amplitude s, as proposed  in [11] for the \noptical band. This type of measurement is referred to as one of \ncontinuous -variable  (CV) . The other type of entanglement \nmeasurement is that of the discrete -variable  (DV) , where only two \nenergy states are involved. Successful Bell Test s (invalidating the Bell \ninequality) were first demonstrated using discrete -variable \nmeasurements.  Successful continuous -variable Bell Test s came later \n[12].  \nThe method section in this  paper analyses the numbers of \nmillimeter wave thermal photons  produced  at ambient temperature \nand explains how entangled photons could  be generated using \nnonlinear magnetic materials such as a YIG ferrite. The results section \ndetermines the flux of entang led photon pairs in relation to practicable \nlevels of parametric pump power. The discussion section describes the \ntypes of millimeter wave circuits that could be used to induce  Type I \nand Type II interactions and how these may be used to generate \nentangled  photons and the Bell States. Opp ortunities to close possible \nBell Test loopholes are also discussed.    \n2. Method  \nSemiclassical c onsiderations are now made as to  the numbers of \nmillimeter wave thermal photons that are generated at ambient \ntemperature and how entangled millimeter wave photons may be \ngenerated using nonlinear susceptible materials, in particular YIG \nferrites.   \n2.1. Thermally generated photons in the Rayleigh -Jeans and quantum \nsensing regimes  \nThe ratio of p hoton to thermal  energy ( hf/kT) makes a clear \ndistinction between the Rayleigh -Jeans regime ( hf/kT < 1 ) and the \nquantum regime ( hf/kT >1), where f is the photon frequency, T is \nambient temperature, h is Planck’s constant and k is Boltzmann’s \nconstant. At the ambient temperature (290 K) the photon energy \nbecomes  equal to the thermal energy ( hf=kT ) at 6 THz; below this \nfrequency is the Rayleigh -Jean’s regime and above  this is the quantum \nregime.  \nBeing either in the R ayleigh -Jeans  regime or the quantum  regime \nhas a major effect on the mean number of thermally generated photons \n𝑛̅ which occupy a mode in the sensor.  The mean  number of photons is \ngiven  [13] by the Bose -Einstein distribution function  \n 𝑛̅=1\n𝑒ℎ𝑓𝑘𝑇⁄−1 (1) \nwhich at ambient temperature in the optical band (~5x1014 Hz) \nbecom es exp ( -hf/kT) , of the order of  2x10-22 and in the microwave \nband (~10 GHz) becomes  kT/hf , of the order of 60 4. This indicates the 2 \n heart of the problem : In the millimeter wave band, the number of \nthermally generated photons is high and likely to swamp the relatively \nlow fluxes of entangled photons.  \nBecause of the large numbers of thermally generated photons at \nambi ent temperature the photon counting techniques used in the \noptical band are completely unsuitable in this frequency band. The \nabove figure of 604 photons indicates there are these numbers of \nphotons per unit bandwidth. This indicates if the classical radia tion \nfield were sampled at the Nyquist rate there would be this number of \nphotons in a single sample. If there were one o r more entangled \nphotons  also there, they would be completely swamped by the thermal \nphoton flux. For this reason,  the technique propos ed here uses radio \nreceivers to measuring in quadrature space , with a phase link to the \nparametric amplifier pump  to enabl e coherent integration .  \n2.2. Generation of entangled photons in a nonlinear medium  \nPair production is a  process whereby an incident photon vanishes \nand two photons appear in its place, whose energies sum to that of the \noriginal photon . It is a quantum process happening across the \nelectromagnetic spectrum , from microwave s to gamma ray s. The \nincident photon must interact with matter  for the process to proceed, \nas this provides energy levels with which the electromagnetic wave \ncan interact and generate entangled photons . A Feynman diagram for \nthe process is illustrated in Fig. 1.    \n \nFig. 1. Feynman diagram for SPDC pair production, where a pump (P) photon \nvanishes and a signal (S) and idler (I) photon appear in its place.   \nIn quantum optics entangled photons are  generated  when the \nelectric field E from a laser  beam  is sufficiently intense to force the \ndielectric susceptibility  into a nonlinear region. The dielectric \nsusceptibility E is defined by Eq.  (2) where  P is the material \npolarization  [14].  \n𝑷=𝜀0(𝝌(1)E𝑬+𝝌(2)E:𝑬2+𝝌(3)E∷𝑬3+⋯+𝝌(n)E:…:𝑬n) (2) \nAt low intensities only the linear susceptibility 𝜒(1)𝐸 is active, but \nat higher fields t he nth order nonlinear susceptibilities  𝜒(𝑛≥2)𝐸 become \nactive , this happening more readily in non-centrosymmetric crystal s. \nThe unit cells of these crystals can have permanent dipole electric \nmoments  and these may spontaneously align to form a  domain.  These \ntypes of materials are referred to as pyro - or ferro -electric  and are used \nto generate visible entangled photons . The most commonly used \nmaterial for this a t present is beta barium borate (BBO).  \nFerromagnetic material s are the magnetic equivalent of \nferroelectric ones, as magnetic moments in the unit cells \nspontaneously align  to form a domain . Upon the application of a \nmagnetizing  intensity  H, a magnetization  M results, the relationship \nbeing highly nonlinear and hysteretic, and described by  \n𝐌=𝛘(1)𝑀𝐇+𝛘(2)𝑀:𝐇2+𝛘(3)𝑀∷𝐇3+⋯+𝛘(𝑛)𝑀:…:𝐇𝑛 (3) \nwhere 𝜒(1)𝑀 is the linear magnetic  susceptibility and  𝜒(𝑛≥2)𝑀 are the \nhigher order nonlinear susceptibilities  [15]. These materials are not \ncurrently recognized  as being potential sources of entangled photons.   \nEntangled photon generation takes place by a process known as \nspontaneous parametric fluorescence or spontaneous parametric \ndown -conversion (SPDC). It is the second order (n=2) nonlinear \nsusceptibility which is responsible fo r this three -wave interaction that \ncreates a pair of entangled photons from a pump photon [16], the first \nexperimental proof of this being presented in [17].  \nThe semi -classical explanation for this process is that the nonlinear \nmedium generates a photon that has a difference frequency , this being \ndefined as the difference between the pump photon and a vacuum \nphoton from the quantum zero -point field energy. For the difference \nfrequenc y photons to be generated it is necessary for the second order \nnonlinear susceptibility 𝜒(2)𝐸 to be relatively large, typically in the region of a few pm/Volt for many ferro electric s when excited by laser \nradiation.   \n2.3. Energy and momentum conservation in entangled photon \ngeneration  \nConservation of energy dictates that th e vacuum photon must have \nan energy less than that of the pump photon. The energy from the \npump photon then passes to that of the signal and idler photons. Given \nthe Einstei n energy relation  𝐸=ℏ𝜔, conservation of energy relates \nthe frequencies of the pump P, signal S and idler I as  \n𝜔𝑃=𝜔𝑆+𝜔𝐼. (4) \nFrom photon ballistics , the process must also satisfy the \nconservation of momentum. Given the de  Broglie momentum \nrelation  𝐩=ℏ𝐤, where k is the direction propagation vector, \nconservation of momentum relates the propagation vectors of the \npump  kP, signal kS and idler KI as,  \n𝐤P=𝐤S+𝐤I, (5) \nand in the  vector triangle of Fig. 2, the magnitude of the propagation \nwave vector is related to the wavelength  of the photon as | k|=2/.  \n \nFig. 2. The momentum ( k) conservation vector triangle is shown for entangled \nphoton pair creation, where  is the angle between the pump and signal \npropagation vector.  \nSince the magnitude of the wave vector is related to its phase  𝜙, \nthrough the wave description  𝑒𝑥𝑝[𝑗(𝐤.𝒓− 𝜔𝑡 + 𝜙)], the \nconservation of momentum indicates the phases of the pump, signal \nand idler at the point of pair creation are related as   \nϕ𝑃=ϕ𝑆+ϕ𝐼. (6) \nThis phase relationship enables coherent integration to recover the \nsignature of the signal and idler photons in the radio Bell Test from \nthe high level of thermal noise characteristic of the Rayleigh -Jeans \nregion.  Without this basic and very fundamental phase relationship, \nno coherent integration would be poss ible. \n2.4. Phase matching to maximize the flux of entangled photon pairs  \nMost materials display properties of normal dispersion , where \nrefractive index rises with frequency. The refractive index usually \nvaries with the direction of propagation and it is often not possible to \nsatisfy completely the conservation of momentum of Eq. (5) depicted \nin Fig. 2. In these  cases,  there is a mismatch in the wave -vector k and \nthis is quantified as  \n∆𝐤=𝐤𝑃−𝐤𝑆−𝐤𝐼 . (7) \nSince the magnitude of the k-vector can be written 𝜔𝐧𝑐⁄, where n \nis the refractive index in the direction of propagation , the above k -\nvector mismatch can be written as   \nΔ𝐤=(𝜔𝑃𝐧𝑃−𝜔𝑆𝐧𝑆−𝜔𝐼𝐧𝐼)1𝑐⁄ . (8) \nSince matching the wave vectors is associated with the \nwavelengths of the pump, signal and idler radiation, achieving \nmomentum conservation can be achieve d by tailoring the refractive \nindex and angles of propagation. This is referred to as phase matching. \nIn cases where phase matching is achieved at an angle =0, it is \nreferred to as collinear phase matching . The oth er cases , where   0, \nare referred to as non -collinear phase matching.  \nIn the 1980’s attention to e ffective phase matching increased th e \nefficiency of entangled photon generation by several orders of \nmagnitude . As a result, f or a single crystal the probability of detecting \nan entangled pair per pump photon is now in the region of 3x10-11 [18]. \nPeriodic poling in lithium niobate has increased th is probability to \n4x10-6 per pump photon [19]. Other methods of generating entangled \nphotons in the visible band have been SPDC in an AlGaAs \nsemiconductor waveguide [20]. In the microwave band operating at \ncryogenic temperatures an entangled photon power of -110 dBm has \nbeen  generated by pumping a Josephson junction (acting as a \nkS, S \nkP, P \nkI, I \nTime  \nkP \nkI \nkS \n 3 \n parametric amplifier) with -75.6 dBm, indicating efficiency of the \norder of -34.4 dB (3.6x10-4) [9].   \nTwo relatively efficient interactions for generating entangled \nphotons  are referred to as  Type I and Type II [14]. In a Type I \ninteraction each photon of the entangled pair has the same (linear) \npolarization , which  is orthogonal to the pump polarization. This is \nreferred to as an ‘e -oo’ or an ‘o -ee’ type interaction, where ‘o’ refers \nto the ordinary or o -mode polarization (E -vector perpendicular to the \ncrystal incidence plane) and ‘e’ ref ers to the extraordinary or e -mode \npolarization (E -vector parallel to the crystal incidence plane) [14], the \ncrystal incidence plane being defined as that which contains the wave \npropagation vector k and the crystal optic axis.  In the designation ‘e -\noo’, the first letter ‘e’ refers to the polarization of the pump and the \nother two letters refer to the polarizations of the signal and idler \nphotons.  \nIn a Type II interaction,  the polarizations of each photon of a pair \nare orthogonal to one another, with one of these polarizations being \nidentical to that of the pump polarization , these interactions designated \nas ‘e-oe’, ‘e -eo’, ‘o -oe’ or ‘o -eo’.  \n2.5. Ferrite s as a nonlinear medium for entangled photon generation  \nThe potential of ferrites as a nonlinear medium for the generation \nof entangled photons can be  appreciated from  the nonlinear \ndependen ce of the magnetization M on the static magnetizing intensity \nH shown in Fig. 3. Ferrites , a well understood class of materials [21], \n[22], can have their  constitutive magnetic properties (eg. coercivity \nHC, remanence M R, saturation magnetization M S, susceptibility ) \nfinely tuned by appropriate doping and prepared in single crystal or \nthin film form. This is illustrated in Fig. 3 by the hysteresis loops of \nHo doped YIG, described by Y 3-xHo xFe5O12, where x=0 and 1.5  [23], \nthe curves being  modelled heuristically using a modified form of the \nLangevin function L(x), where 𝑀=𝑀𝑠(coth 𝜇0𝑎(𝐻−𝐻𝑐)−1/\n𝜇0𝑎(𝐻−𝐻𝑐)) and a is a fitting parameter , from data presented in [23].    \n \nFig. 3. The nonlinear magnetic susceptibility (M/H) evident from the hysteresis \nloop shown here is typical of a pure and Ho doped (x=1.5) YIG ferrite  \nillustrating the high field (1T) saturation magnetization (M S x=1.5  0.804 T (i.e. \n640 kA/m)) (remanent magnetization (M R x=1.5 =561 A/m) and coercivity (H C \nx=1.5 = 0.013 T).  \nAt the atomic level in ferrites , the Heisenberg uncertainty principle \nprevents th e net atomic or domain magnetic moment (related to \nangular momentum) aligning with the magnetizing intensity vector H, \nas angular momentum and its direction don’t commute. However, the \ntorque to align induces a precessional motion of the magnetic moment \naround the field direction at the (classical) Larmor precession \nfrequency [24] of  \nwhere  \nis the electron spin gyro magnetic ratio and e and me are the electron \ncharge and mass respectively, and 𝑔𝑒 is the electron Landé g -factor, \nhaving a value ~2.002319 . The Larmor precession frequency is \nequivalent to the frequency of the photons absorbed in the quantum \nmechanical representation as magnetic moments flip in an applied field during resonant absorpti on between spin up and spin down states  \n[25].  \nAs the conduction bands in ferrites contain no free electrons the ir \nresistivity is high , typically 107 m, so the  material is transparent to \nmicrowaves and millimeter waves. This means the ferrite interaction \nof radiation is governed b y the complex refractive index  given  [26] by  \nwhere r (=r‘- jr”) and r (=r‘ - jr”)) are the complex relative \npermittivity and permeability of the material, these being related to the \nfirst order (linear) susceptibility as r  = 1 + 𝜒(1)𝐸. For ferrites the \ncomplex relative permittivity ( r‘, jr”) is [27] isotropic and takes a \nscalar value, typically (12 -22, 0.05).  The permeability of the ferrite is \ndictated by the Polder Tensor [24] and the variation of this with \nfrequency over the millimeter wave band determines largely how \nthese materials behave and have been exploited as devices in a large \ncommercial market.  \nThe ferrite response to electromagnetic  radiation  falls into two \northogo nal propagation modes, one couple d strongly to the magnetic \ndipole s and the other weakly coupled , the strength being dependent on \nthe angle of propagation in the medium . For longitudinal propagation \n(along the static magnetic field lines)  the strongly coupled mode is \nright -hand circularly (RHC) polarized and the weakly coupled mode \nis left -hand circularly (LHC) polarized. For transverse propagation (at \nright angles to the static field lines) the strongly couple d mode  is the \ne-mode (the H -field of the wave being perpendicular to the static \nmagnetic field direction ), with the weakly coupled mode being the o -\nmode  (having the wave H-field parallel to the static field direction ). \nFor the strong mode the ferromagnetic resonance shifts from 0=L \nfor longitudinal propagation to (0(0+M)) where M=-SM, for \ntransverse propagation. At intermediate propagation angles the strong \nand weak modes are elliptically polarized.   \nGarnets are a class of ferrites of particular interest as they have \nvery low damping and hence little crystal lattice absor ption of the \nwave energy . This means the ferromagnetic resonance line is very \nnarrow and in yttrium iron garnet (YIG) it has a width [28] of ~1 MH z, \nwhich gives i t a highly non-linear magnetic susceptibility . \nA plot of the complex refractive index is shown in Fig. 4 for \ntransverse propagation  in an infinite solid of YIG for a Larmor \nprecession frequency (associated with the internal field H) o f 15 GHz \nand a  Larmor precession frequency (associated with the internal \nmagnetization M/2 ) of 6.9 GHz . The dielectric constant (’) is \nassumed to be 14.7, the dielectric loss tan ( ”/’) to be 0.0002 [29] and \nthe damping factor  to be (~1/Q) of 0.00007 (typical for YIG) [30]. \nFor the strongly couple d mode there is a (negative permeability) non-\npropagating region extending from the resonant frequency to  the cut -\noff frequency, 0+M, above which th e permeability becomes \npositive.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n -800-4000400800\n-0.4 -0.2 0 0.2 0.4M (kA/m)\nB(T)= 0HMS\nMR\n0HY1.5Ho1.5Fe5O12\nY3Fe5O12\n𝜔𝐿=−𝛾𝑆𝐻 ,  (9) \n𝛾𝑆=−𝑔𝑒𝜇0𝑒\n2𝑚𝑒 .     (10) 𝑛=√𝜀𝑟𝜇𝑟 , (11) 4 \n      \nFig. 4. The real (a) and imaginary (b) parts of the refractive index for the strong and weak transverse propagation modes in YIG.     \nIn general the refractive index is a function of the propagation \nangle in the YIG and its quantitative behavior is well known from the \nsolution to Eq. (11), as illustrated  to good effect  in [24]. From this \nequation two orthogonal elliptical polarization modes arise (which \nbecome linearly (circularly) polarized for propagation across (along) \nthe magnetic field lines. This behavior is similar to the  angular \nvariatio n in the  refractive index  for the (extra)ordinary mode in \nelectro -optic (ferroelectric) crystals , as can be appreciated in the \nrefractive index polar plots in [14]. There fore there is an opportunity \nto use the angular variation of refractive index in YIG for phase \nmatching  (by minimizing the momentum mismatch k from Eq. (8)) \nto maximiz e the  intensity of the entangled photons  in accordance with \nEq. (17). It may also be possible to increase the nonlinear interaction \nlength in YIG (whilst retaining good phase matching), to increase the \nefficiency of entangled photon generation , by periodically poling YIG \n(PPYIG), as is done in periodically polled lithium niob ate (PPLN)  \n[31].  \n3. Results  \nGiven the above mechanism for creating entangled photons, \nestimations are now made of the likely fluxes of entangled photons \nfrom YIG in the millimeter -wave band  using moderate pump powers \nthat are readily available. This will determine the practicability of the \nproposed Bell Test in this spectral band and therefore the likelihood \nthat the system could be used as the basis for a sensor.   \n3.1. The second order nonlinear magnetic susceptibility of YIG  \nOptimization of the second order nonlinear susceptibility of \nferrites has led to these devices being used as frequency doublers and \nparametric amplifiers  [15]. The second order nonlinear susceptibility \nof a frequency doubling ferrite is given  [32] as \nwhere M0 is the YIG static magnetization , L is the Larmor angular \nprecession frequency associated with the externally applied static field \nH0 (given by -SH0) and H is the ferromagnetic resonance linewidth \nin A/m.  It can be seen  from this equation  that a higher second order \nnonlinear susceptibility results from a narrow resonance width. Small \nresonance widths of the order of 0.36 Oersted (~28 A/m) can be \nachieved using a YIG crystal [32]. For a garnet with a static \nmagne tization of 238 kA (~0.3 Tesla) pumped at  a frequency of 20 \nGHz  (to resonate with the externally applied field)  the second order \nnonlinear susceptibility from  Eq. (12) is ~0.015 m/A.  \nFor frequency doubling a nonlinear response is achieved  [32] by \npumping in the transverse mode , with the B -field of the pump \nradiation being perpendicular to the static field. The frequency \ndoubled component is also taken off in the transverse propagation \nconfiguration, with the B -field parallel to the static field.  \nFor parametric amplification in YIG a number of modes are [15], \n[33] possible . These are analogous to the different types of optical \nentangled photon creation: just how these are excited is explained in \nthe following section on the circuits for millimeter wave entangled \nphoton creation.  The mode with the highest gain is the magneto -static \nmode  (or the spin-wave or three -magnon mode ) [34]. Here a linearly \npolarized pump wave at twice the res onance frequency propagates \ntransversely across the static magnetic field lines with its wave H -field \nparallel to the static field direction, almost an exact reverse process of \nthe above frequency doubling mechanism . This system creates \nparametric gain in ferrites at pump power levels of 0.5 W, generating \nsignal and idler photons that are linearly polarized with their H -fields perpendicular to the static field directions. Considering the direction \nof magnetization in a ferrite to be analogous to the optic a xis in a \nferroelectric , the mode is the magnetic equivalent of the Type I \ninteraction in ferroelectrics .  \nA further parametric amplification mode is the electromagnetic \nmode. Here a linearly polarized pump wave at the ferromagnetic \nresonance frequency propagates transversely across the static \nmagnetic field lines with its H -field perpendicular to the static fi eld \ndirection [35]. Signal and idler photons are created having orthogonal \nlinear polarizations, one having the H -field perpendicular and the \nother with the H -field parallel to the static  field direction. This mode \nis the magnet ic equivalent of the Type II entangled photon creation in \nferroelectrics .  \nMiller’s rule [36] is an empirical relationship from the field of \nnonlinear optics for non -centrosymmetric dielectrics stat ing that the \nratio of the second order nonlinear susceptibility to the product of the \nfirst order susceptibilities at the three frequencies (𝜔1+𝜔2,𝜔1,𝜔2) \nof a three -wave interaction, namely  \nis constant .  \nBoyd shows  [31] that the mathematical form of Miller’s rule is a \nconsequence of the anharmonic  motion of the electron  in the E -field \nof a passing electromagnetic wave, this anharmonicity being a direct \nresult of the local crystal E -field in the non -centrosymmetric material. \nHowever, there is only finite second order nonlinear electrical \nsusceptibility  (2)E, if the crystal is non -centrosymmetric. This suggests \nthat if YIG has a finite magnetic susceptibility (2)M, it should also \nfollow Mill er’s rule.   \nAssuming Miller’s rule is followed for the magnetic \nsusceptibilities, this suggests  that the nonlinear susceptibility  of Eq. \n(12) is very close to that for difference  frequency generation , provided \nthe first order susceptibilities are similar. The value from Eq. (12) of \n0.0015 m/A may therefore be used in the estimation of the nonlinear \nsusceptibility for the difference frequency generation in spontaneous \nparametric down -conversion.   \n3.2. The power of entangled photons  \nIn the process of entangled photon generation,  the pump photon \ninteracts with a vacuum photon to annihilate itself and create a pair of \nentangled photons. A high pump power parametrically amplifies  [37] \nthe vacuum photons. If the field-gain of the nonlinear medium is \ndesignated   and its length l, the spectral radiance of the entangled \nphoton flux (in W/m2/sr/(radians /s) is given [38] by  \n𝐼𝜔Ω(𝐤)=𝐼𝜔Ω𝑉𝐴𝑉𝑠𝑖𝑛ℎ2[(𝛾2−|Δ𝐤|2/4)1/2𝑙]\n(1−|Δ𝐤|2/4𝛾2),  (14) \nwhere k is the wavevector  (phase) mismatch  of Eq.  (8), and 𝐼𝜔Ω𝑉𝐴𝑉 is \nthe spectral radiance of the vacuum photon fluctuations, given by  \n𝐼𝜔Ω𝑉𝐴𝑉=ℏ𝜔3𝑛𝑆2\n8𝜋3𝑐2 , (15) \nwhere nS is the medium refractive index.  The vacuum photon \nfluctuations correspond to a single  photon in each mode.  The field -\ngain from  [38] is given as  \n𝜒(2)𝑀=𝛾𝑆𝑀0\n𝜔𝐿Δ𝐻 , (12) 𝜒(2)𝐸(𝜔1+𝜔2,𝜔1,𝜔2)\n𝜒(1)𝐸(𝜔1+𝜔2)𝜒(1)𝐸(𝜔1)𝜒(1)𝐸(𝜔2) (13) \n𝛾𝐸(𝑚−1)=(8𝜔𝑆𝜔𝐼𝐼𝑃𝜇0\n𝑐𝑛𝑃)1\n2\n𝜋𝜒(2)𝐸  , (16) (a) (b) 5 \n where 𝑛𝑃 is the medium refractive index for the pump radiation, which \nhas an intensity (or irradiance) in W/m2 given as 𝐼𝑃. Taking a pump \npower of 1 04 W/m2 (1 W/cm2), a nonlinear electrical susceptibility of \n5x10-12 m/V and a refractive index of 2.2, the field gain of the medium \nis ~ 1.2x10-5 per meter  for a signal and idler frequency of 10 GHz . For \na low gain medium in which  << k, Eq. (14) becomes  \nand if perfect phase matching is satisfied the entangled photon spectral \nradiance in the  nonlinear dielectric medium becomes   \nThis indicates that for the above pump power the spectral \nirradiance of the entangled photon flux would be 6.88 x 10-32 \nWatts/sr/m2/(radians/s)  for an interaction length of 10 cm. If it is \nassumed the bandwidth is 5 GHz and emission is measured over  \nsteradians , the entangled power flux is 6.79 x 10-25 Watts over an area \nof one square centimeter . \nMaking a similar analysis  to [38] for entangled photon generation \nfrom a magnetic material the field -gain is given by  \nwhere  𝜒(2)𝑀 is the 2nd order nonlinear magnetic susceptibility  and 𝑍𝑃 \nis the medium impedance to the pump radiation  𝑍𝑃=√𝜇𝑟𝜇0𝜀𝑟𝜀0 ⁄ . \nPumping YIG possessing  a nonlinear magnetic susceptibility of 0.015 \nm/A and a refractive index of 3.8 with a pump power of 104 W/m2 (1 \nW/cm2) at 20 GHz  results in a field gain from Eq. (19) of 630 per \nmeter . This is a considerably higher gain than that of a nonlinear \ndielectric. Assuming that the conditions of phase matching are \nsatisfied (ie k =0), then the entangled photon spectral radiance in the \nnonlinear magnetic medium from Eq. (14) becomes  \nGiven the estimated magnetic nonlinear susceptibility of 0.015 \nm/A from above  and a refractive index of 3.8  for YIG, the entangled \nphoton spectral irradiance becomes  1.80x10-19 Watts/sr/m2/(radians/s) \nfor an interaction length of 3 mm  at signal and idler frequencies of 10 \nGHz . Assuming a bandwidth of 10 GHz and that emission is collected \nover  steradians , the entangled photon flux f or a pump power of  \n5 W/cm2 is 3.56  10-12 Watts/cm2, which is approximately 0.5  \n1012 photons per second from an area of 1  cm2. This is also \nconsiderably higher than the above counterpart for a nonlinear \ndielectric medium.    YIG ferrites have many potential advantages when it comes to the \ngeneration of entangled states. YIG, as a bulk crystal, regular prism or \nthin film, can be immersed in an applied static bias field, H 0, close  to \nthe resonance radio frequency field, mounted in either a waveguide \n[39] or a very high quality factor (Q~5000) resonant cavity [40]. The \nhigh Q-factor raises  the magnetizing intensity of the applied radiation, \nincreasing  the probability of generating entangled photons.  \n4. Discussion  \nGiven the results on the  fluxes of millimeter  wave entangled \nphotons, a discussion is opened here on architectures for their \ngeneration and a homodyne receiver system for their detection.   \n4.1. Circuits for Type I & Type II ferrite source s of entangled photons  \nGiven that the  cross -sectional area of the YIG crystal  is envisaged \nto be in the region of 1 cm2, the pump beam would need to be either \nfocused from a waveguide or transmitted direct to the sample using a \ntransmission line. Because the sample cross -sectional  dimension is of \nthe order of the wavelength of the signal and idler photons, these will \nemerge from the YIG crystal into a wide angled cone  due to \ndiffraction . This cone would constitute  a single mode of the receiver \nsystem  so signal and idler may be captured by the receiver regardless \nof energy.  This represents an essential difference between what is \nproposed here and the quantum o ptics experiments in entanglement.    \nA circuit using the magneto -static mode of a parametric amplifier  \nto create entangled photons  is illustrated in Fig. 5. This  exploits the \nType I interaction  and is similar to  that in [41], but differs in that it \nuses a r esonant circuit . The system  comprises a pump generating \nlinearly polarized  radiation orientated at an angle of 45, which splits \ninto two orthogonal horizontally and vertically polarized  beams using \na vertically wired grid polarizer  [42]. The horizontally polarized  (E-\nfield) beam enters the YIG-1 sample with the radio -frequency H -field \nparallel to the static H -field in the ferrite. Signal and idler photons \ngenerated with  H-fields perpendicular to the static H -field exit via a \nreflection from the second vertically  wired grid. The vertically \npolarized  (E-field) beam enters the YIG-2 sample with the H -field \nperpendicular to the static H -field stimulating signal and idler photons \nwhich exit by transmission through the second wire grid.  \nThe unused horizontal and vertically polarized  pump radiation exit \nvia the second wire grid polarizer  with a relative phase (epoch) angle \ndefined by the difference in phase delay s between  the two arms of the \nsystem. An adjustable wave -plate then provides sufficient delay \nbetween these polarizations  to generate linear polarization  orientated \nat -45. This maximizes  the amount of radiation that reenter s the \ncavity by reflection from  the 45 -angled polarizer . The polarization  \nrotator is then adjusted to bring the orientation around to +45 , in order \nthat equal fluxes  of horizontal and vertical polarizations  can be \npresented to the two YIG ferrite crystals. The entangled photons exit \nthrough  the vertical wire grid polarizing beam splitter  and are then \nseparated in to two channels by the following beam splitter so they may \nenter the homodyne interferometer of Fig. 8.  \n \nFig. 5. Type I interaction  in a resonant circuit using the magneto -static parametric amplifier mode in a YIG ferrite pumping at twice the ferromagnetic resonance \nfrequency generat es signal and idler photons about the ferromagnetic resonance frequency  0 and the entangled state (|V S>1|Vi>1+ej|HS>2|Hi>2)/2. The orientation s \nof linear polarizations  are indicated as +45, -45, H and V .  \nThe beam splitter of Fig. 5 from  which the signal and idler emerge \ncan be realized in practice in several different forms. 1) It may be a \nnon-polarizing 50/50 beam splitter , similar to [43]. 2) It may be a \nhigh/low pass filter, passing photons of frequency > 𝜔𝑃/2 into one \nchannel and those <  𝜔𝑃/2 into the other channel. 3) It may be a \npolarizing grid orientated at 45 . The latter two forms  would however \nmake a preselection of photons on the basis of their momentum  or \npolarization and thereby may remove one of the degrees of \nentanglement. However, since signal and idler are initially entangled in both momentum and polarization, removing one of these may still \nenable a Bell Test to be constructed.   \nA circuit using the electromagnetic mode of a parametric \namplifier, which exploits the Type II interaction  for the creation of \nentangled photons , is shown in Fig. 6. Pumping the YIG at the \nferromagnetic resonance transversely with the H -field perpendicular \nto the static magnetic field direction , the signal and idler can be taken \noff transversely, one with the wave H -field perpendicular to the static \nfield and the other parallel to the static field. The signal and idler are 𝐼𝜔Ω(k)=𝐼𝜔Ω𝑉𝐴𝑉𝛾2𝑙2𝑠𝑖𝑛𝑐2[(|Δ𝐤|)𝑙/2] (17) \n𝐼𝜔Ω(𝐤)=ℏ𝜇0𝜔𝑆4𝜔𝐼𝐼𝑃𝑛𝑆2𝜒(2)𝐸2𝑙2\n𝜋𝑐3𝑛𝑃 . (18) \n𝛾𝑀(𝑚−1)=(8𝜔𝑆𝜔𝐼𝐼𝑃𝜇0\n𝑐𝑛𝑃)1\n2𝜋𝜒(2)𝑀\n𝑍𝑃 , (19) \n𝐼𝜔Ω(𝐤)=ℏ𝜔3𝑛𝑆2\n8𝜋3𝑐2 sinh2(𝛾𝑙) . (20) \nHP \nUnused pump having polarization epoch angle  \nVertical wire \ngrid polarizing \nbeam splitter  \n|H s >2 ,|H i>2 \n VP \n|Vs>1 ,|V i>1 \n+45 \n Linear  \n +45 \nYIG-1 \nYIG-2 \n45 angled wire \ngrid polarizer  \n \nVertical wire \ngrid polariser  \nWave -plate  \n(variable)  \nPump P =20 \n-45 \nPolarization \nrotator  \nBeam \nsplitter  \nOutputs  \n|Vs>1,|H s>2 \n|Vi>1,|H i>2 6 \n taken off using a dichroic mirror fabricated  from a mesh grid  [42]. The \nunused radiation gets reflected by the mesh filter and as the pump is \nhorizontally polarized  it passes through a half -wave plate to become \nvertically polarized , enabling  it to re-enter the resonance cavity by \nreflection off the gird. The polarization  rotator rotate s the plane of polarization  so that horizontally polarized  radiation is incident on the \nYIG.  \nOn exit via the dichroic mirror the entangled photons pass through \na beam splitter , similar to that in the Type I system above .      \n \nFig. 6. Type II interaction in a resonant circuit using the electromagnet ic mode parametric amplifier in YIG pumping at the ferromagnetic resonance frequency \ngenerating signal and idler photons about half the ferromagnetic resonance frequency , creating the entangled state (|Hs>1|Vi>1+ej|Vs>2|Hi>2)/2. The orientations of \nthe linear polarizations  throughout the system are indicated as H and V.  \nA non-resonant circuit to stimulate  Type II interactions is adopted \nfrom the Sagnac interferometer of [44] and depicted in Fig. 7. Here  \nlinearly polarized pump radiation orientated at 45  encounters a \npolarizing beam -splitter constructed from a vertical grid. A horizontal \npolarization component of the pump passes through the grid and is \nconverted to vertical polarization by the half wave -plate, before \nstimulating the YIG to generate entangled photons in mode 1 . A \nvertical polarization component is reflected from the grid and \nstimulates the YIG crystal directly from the other direction  to create \nentangled photons in mode 2 . The Type II interactions create \northogonal signal  and idler photons in these two modes which exit via the polarizing beam -splitter and the dichroic mirror. The dichroic \nmirror, manufactured from mesh grids  [42], is designed to pass the \npump radiation , but reflect the lower frequency signal and idler \nradiations. The outputs from the circuit pass directly into the channels \nof the homodyne interferometer of Fig. 8. The advantage of this \narchitecture is that the photons remain entangled in both momentum \nand polarization. The unused pump radiation from the Sagnac system \nexits in the direction of the pump, but  this could re -enter the system \nby a reconfiguration using addit ional  resonant circuit components , not \nshown here .   \n \nFig. 7. A circuit based on a Sagnac interferometer [44] using a Type II interaction of the electromagnetic mode in YIG to create the entangled state ( |Vi>1|Hs>1+ \nej|Hs>2|Vs>2)2. \n4.2. The homodyne interferometer  Bell Test   \nA single -channel Bell Test architecture, the type initially proposed \nin [3], but exploiting the attributes of millimeter wave receivers for an \nambient temperature Bell Test is shown in Fig. 8. It creates  the entangled photons by pumping a nonlinear YIG crystal , in the circuits \nof Fig. 5 (for entangled photons created by a Type I interaction ) and \nFig. 6 or Fig. 7 (for entangled photons created by a  Type II \ninteraction ), and then separates the pair  into two channels .  \n|Hs>1, |Vi>1 \n|Vs>2, |H i>2 \nUnused pump horizontal polarization H P \nHP \n HP \n Linear  \nHP \nDichroic Mirror \n(mesh grid filter)  \nYIG \nVertical grid  \nwire polari zer \n \nHalf-wave \nplate  \nHP \nPump P =0 \nVP \n<P \n~P \nPolarisation \nrotator  \nBeam \nsplitter  \nOutputs  \n|H s>1, |V s>2  \n|Vi>1|Hi>2 \nVP \n|H s>2, |V i>2 \n Polarizing \nbeam -splitter \n(vertical grid)  \n|H s>1, |V i>1 \nOutputs  \nVP \nHP \nYIG \nDichroic mirror \n(mesh grid filter)  \n45 linear  \nPump P =0 \nVP \nHalf wave -\nplate \nVP \n|H s>1, |V s>2 \n|Vs>2, |H i>2 \n|Vi>1, |HS>2 7 \n  \nFig. 8. The homodyne interferometer single -channel Bell Test is proposed to overcome the noise generated by thermal photons when operating in the Rayleigh -Jeans \nregime ( hf<kT ) [10], where 2 E is the angular frequency difference between the signal and idler photons and 2 SI is the assumed angular bandwidth of the signal \nand idler photons.  The box labelled ‘entangled photon state creation’ represents  any of the circuits from Fig. 5, Fig. 6 or Fig. 7. \nWave -plates are included in the circuit of Fig. 8 so that all Bell \nStates can be created , as indicated in [45]. Since the system is an \ninterferometer it is sensitive to phase, so adjustments to the phase can \nbe made by the variable phase shifter in one of the arms of the \ninterferometer. The effects of birefringence in components of the \nsystem could potentially lea d to a reduced level of entanglement, but \nthese may be compensated for by extra components in the circuity of \nFig. 8. Since  signal and idler photons need to arrive at more or less the \nsame time at the receivers , the transit time of photons could be \nbalanced by physical  movement or by insertion of low-loss refractive \nindex material into one of the arms of the interferometer , to increase \nits effective optical path length . The above adjustments would be \nmade to maximize the measured entanglement signature, similar to \nthat in the optical band described in  [46].   \nThe radiation in each channel is amplified to drive mixer -1 into its  \nnonlinear regime. The mixer output contains the multiplication of \nsignal and idler fields together with relatively large amount s of \nthermal noise on a 20 GHz carrier frequency . The phases  of the signal \nand idler are summed by this mixing process. The output from mixer -\n1 is then mixed with the original pump radiation at 20 GHz in mixer -\n2, thereby shifting the combined signal and idler signature down to \nbaseband.   \nThe mixer -2 output is lowpass filtered and sampled in a sine and \ncosine receiver to record a complex amplitude , which is then \ncoherently integrated  to recover the stationary complex value for the \nsignature of the signal and idler photon pairs  from the noise . The sine \nand cosine receiver is also referred to by some as an I,Q or a quadrature \nreceiver. It  integrate s to a stationary complex value because the p hase \nnoise n average s to zero, leaving t he phase relation of Eq. (6) to \nprevail.  The square of this stationary value is equivalent to the number \nof coincidence count s, usually designated as N, in the optical Bell Test , \nas indicated by Eq. (21).   \nThe circuit in Fig. 8 is the radio receiver equivalent of a single \nchannel Bell Test. The Bell Test test -statistic, usually designated as S, \nis formed by the usual combination of measurements of N at the Bell \nTest angles [3]. Measurements made at the Bell Test angles maximize \nthe difference between the classical and quantum results, this giving \nthe highest chance of demonstrating violation of the Bell inequality.  \nA twin -channel Bell Test architecture, also proposed in [3], but \nsuitable for ambient temperature operation in the millimeter wave \nband , is constructed b y replacing the polarization  rotators with \nrotating polarizing  beam splitter s, which are then  followed by two \nfurther receiver channels . The twin -channel Bell Test  architecture  has \nsmall er systematic uncertainties than the single -channel  test. The \nsingle -channel Bell Test was initially proposed as a more practicable instrument, but with more sophi sticat ed technology the twin -channel \nmeasurement with its superior performance is now the preferred  \noption , forming the basis for almost all Bell Tests today . As with the \nsingle -channel test, measurements are made at the Bell Test angle s. In \nthis case four  effective coincidence counts are formed  from Eq. (21), \nwhich are then used to determine the so -called quantum correlation, \nusually designated by E for each of the Bell Test angles, from which \nthe Bell Test test -statistic S is then formed  as detailed in  [12] and [47].     \n4.3. System s ignal -to-noise ratio s and integration time s     \nOne of the main sources of noise  comes from the amplifiers and \nthe thermal ( black body ) radiation entering the receiver from the \nambient temperature surroundings. This noise  power  in each channel , \nreferred to the amplifier input is just FkT 0B, where F is the amplifier \nnoise factor, T0 is ambient temperature ~290 K and B is the bandwidth \nof the system. For an amplifier noise factor of 2.5 and a 10 GHz \nbandwidth the noise is ~ 71 pW.  \nThe entangled photon power at the input to the  radio frequency  \namplifier is  𝑃𝑆/𝐿, where L is the loss in the entangled photon flux \narising from stimulated emission and absorption. Since the entangled \nphoton flux from the YIG has been estimate d at 3.56  10-12 Watts  and \nall front -end components in front of the waveguide horn can be \ndesigned to be low loss , it is reasonable to a ssume  L would be at the \nmost  a factor of two , which gives the entangled power flux at the \namplifier input as 1.78  10-12 Watt s, for 5 Watt s of pump power  over \none square centimeter . It is therefore also assumed that the loss of \nsignal due to absorption and s timulated emission would still permit \nsufficient information from the signal and idler to be coherently \nintegrate d for the Bell Test .  \nAs the YIG parametric amplification will also amplify thermally \ngenerated photons which have random phases, these will constitute \nextra noise in the system, having a power level   𝑛𝑃𝑆/𝐿 where 𝑛 is the \nmean photon number of Eq.(1). Since this is estimated at  604 at 10 \nGHz, the power flux from parametrically amplified thermal photons \nis 604  1.78  10-12 = 1.08  10-9 Watt s.  \nSummarizing  the above , the s ignal -to-noise on each of the signal \nand idler arms of the interferometer on the input to mixer -1 is given \nby  \nwhich has a numerical value of 1.55  10-3. Given that mixer one \nmultiplies the signal and idler with their independent noises together, \nthe signal -to-noise ratio output from mixer -1 is  \n 𝑁=𝐴2⇐|𝐴𝑒𝑗(𝜙𝑆+𝜙𝐼)|2⇐|<𝐴𝑛𝑒𝑗(𝜙𝑆+𝜙𝐼+𝜙𝑛)>|2 (21) \n𝑆𝑁𝑅1=𝑃𝑠/𝐿\n𝑛𝑃𝑆/𝐿+𝐹𝑘𝑇0𝐵  , (22) \nSum frequency \nsection P SI \n \nSignal  \n\nS = \nP/2±\nE \n \n  P \n \nP \nMixer -1 \n(taking sum \nfrequencies)  \nBand -pass filter:  \n \nP/2 (\nE +\nSI) \nAmplifier  \nAmplifier  \nPhase shifter \n(variable)  \nPump \noscillator  \nAttenuator \n(variable)  \n PSI \nBand -pass \nfilter  \nBaseband section  \nDC SI \n \nS \nSI  \n \nP \n \nBand -pass filter  \n \nP/2 (\nE +\nSI) \n \nI \nSI  \nIdler    \n \nI = \nP /2 ∓\nE \n \nWave -plate  \nRotatable \npolariser  \nrotat or \nRotatable \npolariser  \nLow-pass filter \n (DC to SI) \nADC sampling clock, sample \nfrequency  SI   \nA \nD \nRadio frequency (RF) section: DC to P \nIntegration \nfollowed by \ncomplex square  \nWave -plate  \nPolarisation \nselector  \nPolarisation \nselector  \nEntangled photon \nstate creation  \nN=||2 \n  \nMixer -2  8 \n which has  a numerical value of 2.41  10-6. \nThe final stage of the receiver mixes the combined and signal and \nidler signatures  (together with noise)  down to a base -band frequency \nby mixing with the pump radiation  in mixer -2. Coherent integration of \nthe output from mixer -2 brings an  improvement in the signal -to-noise \nratio proportional to the root of the number of samples. Sampling at \nthe Nyquist rate of 2B , the signal -to-noise ratio output after an \nintegration time of t INT is \nIt can be seen from this that the highest signal -to-noise ratios will \nbe achieved by using  the largest possible radiation bandwidth. \nInserting numerical values into this equation gives a signal -to-noise \nratio of unity  for an integration time of 8.59 s. Measurement of this \nsignature at each of the Bell Test angles will therefore take a few tens \nof seconds.  \nAs the parametric gain is increased , more entangled photons will \nbe created and with a sufficiently high gain the parametrically \namplified thermal photon fluctuations may dominate the noise \ngenerated by the RF amplifier. In this case 𝑛𝑃𝑆/𝐿≫𝐹𝑘𝑇0𝐵 and Eq. \n(24) simplifies to  \nTransposing the last equation indicates the integration time to achieve \na given signal -to-noise ratio is  \nThis indicates there would be benefits in moving up in frequency, \nso bandwidth B could be increased and the mean numbers of thermal \nphotons would be reduced. Doubling the frequency of the pump from \n20 GHz to 40 GHz  (also requiring the doubling of H 0) and increasing \nthe bandwidth from 10 GHz to 20 GHz, would shorten the integration \ntime by a factor o f 32.  \n4.4. Closing Bell Test loopholes  \nThe two main loopholes in Bell Tests are the ‘detector efficiency \nloophole’ and the ‘locality loophole’, the former being associated with \nthe fact that a photon counting detector does not ensure sufficient \nquantum efficiency and the latter associated with a local \ncommunication link between the two re ceivers which could  corrupt \nthe result.  \nFor the case of the detection efficiency loophole this should be less \nof an issue with radio receivers, as information is more easily captured \nwith homodyne quadrature receivers operating in the continuous -\nvariable mode  [12]. Noise is present due to the ambient thermal \nphotons, which is removed by coherent integration, but the essential \npoint is that the information about  the arrival of each photon of an \nentangled pair is captured.    \nFor the case of the locality loophole s, strategies to remove these \nwould be the following. With the envisaged integration times being a \nmatter of seconds and receivers placed several meters apart,  it might \nbe argued that there would be sufficient time for millimeter wave \ncommunication between receivers , potentially leading to corruption of \ndata. However, this loophole would be closed by randomly changing \nthe plane of polari zation  on a timescale sufficiently rapidly that \nmeasurement at one polarization is shorter than the time taken for \nsignals to  pass between the receivers and the source , as is done in the \noptical Bell Tests [48]. Rapidly switching the polarization can be done \nusing phase shifters and waveplates constructed from pin or \nmicrowave varactor diodes, as these devices switch on nanosecond \ntimescales. This switching would need to be done in synchronization \nwith the data acquisition. This is entirely possible as the sample times \nrequired to satis fy the Nyquist criterion associated with GHz \nbandwidths is sub -nanosecond, a sampling capability readily available \nnow.  \nIn the case  of the  locality loophole associated with the link between \nthe two receiver channels at the site of mixer -1 (before the sampling ) \nthis may be removed by sampling separately in each of the channels \nbefore mixer -1. The action of mixer -1 would then be performed digitally on uncorrupted samples, before a digitally downshifting \naction of mixer -2. In such a configuration it might be argue d there \nmay be a communication link via the sampling clock suppl ied to the \ntwo samplers. This link would be removed by having free -running \nsampling clocks at the sites of the samplers, which are only \nperiodically up -dated to keep them in synchronization.   \nIt is not being claimed here that this system offers a completely \nloophole free Bell Test, only that it offers a new potential direction for \nresearch into the phenomenon of entanglement , with attention paid to \nhow loopholes may be closed .  Furthermore, th e approach here is only \nsemiclassical and that  a full quantum operator evaluation is necessary \nto identify potential quantum effects that may thwart the function of \nthe proposed system. This would constitute a n essential  next phase of \nthe work.   \n5. Conclusion  \nThe proposed combination of a YIG parametric amplifier with a \nhomodyne interferometer enables the measurement of a flux of \nentangled photons in the presence of a much larger flux of thermally \ngenerated photons. Calcula tions indicate that approximately 5 Watt s \nof millimeter -wave radiation pumping 0.3 cm3 of YIG will generate \n~0.5  1012 entangled photon pairs per second. The proposed \narchitecture enables an ambient temperature Bell Test to be made \nwhich would require only tens of seconds of integration time. It is \nencouraging with this architecture of Bell Test that there are good \nopport unities to close the most severe loopholes of detector efficiency \nand locality. A successful demonstration of this test will lead to novel \narchitectures of entanglement -based  quantum technology system s, \npotentially for applications in computers  and millime ter wave \ncommunication systems and sensors .   \nData Availability  \nThe datasets generated during and/or analyzed during the current \nstudy are available from the corresponding author on reasonable \nrequest.  \nAcknowledgements  \nThe authors are grateful for the partial financial  funding  of this \nwork from the T ERANET  Seedcord funding scheme provided by the \nUK EPSRC , through the School of Electronic and Electrical \nEngineering, at the University of Leeds , managed by Professor John \nCunningham and Professor Martyn Chamberlain. The insightful \ncomments and critique provided by the referees has been a n enormous \nhelp in the refinement of the paper.   \nAuthor contributions  \nN.A.S. made the calculations of the entangled photon fluxes and \nthe signal -to-noise ratios in the receiver systems. S.R.H. provided the \ndetails on the YIG ferrites , their hysteresis  and use in resonance \ncircuits.  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Tapster, “Experimental violation of Bell's \ninequ ality based on phase and momentum,” Physical Review Letters, \nvol. 64, no. 21, p. 2495, 1990.  \n[44]  Fedrizzi, A.; Herbst, T.; Poppe, A.; Jennewein, T.; Zeilinger, A.;, “A \nwavelength -tunable fiber -coupled source of narrowband entangled \nphotons,” Optics Express, vol. 15, no. 23, pp. 15377 -15386, 2007.  \n[45]  Agarwal, G.S., Quantum Optics, New York: Cambridge University \nPress, 2013.  \n[46]  Kwiat, P.G.; Mattle, K.; Weinfurter, H.; Zeilinger, A., “New High -\nIntensity Source of Polarization -Entangled Photon Pairs,” Physical \nReivew Letters, vol. 75, no. 24, pp. 433 7-4342, 1995.  \n[47]  Lipson, A.; Lipson, S.G.; Lipson, H., Optical Physics, Fourth Edition, \nCambridge: Cambridge University Press, 2011.  \n[48]  Aspect, A.; Dalibard, J.; Roger, G., “Experimental test of Bell's \ninequalities using time -varying analysers,” Physical Review Letters, vol. \n49, pp. 1804 -1807, 1982.  \n \n "    },    {        "title": "1911.03708v1.The_magnonic_superfluid_droplet_at_room_temperature.pdf",        "content": "arXiv:1911.03708v1  [cond-mat.soft]  9 Nov 2019The magnonic superfluid droplet at room temperature\nYu. M. Bunkov(a),∗A.Farhutdinov(b), A. N. Kuzmichev(a), T. R. Safin(a,b),\nP. M. Vetoshko(a), V. I. Belotelov(a), and M. S. Tagirov(b,c)\naRussian Quantum Center, Skolkovo, 143025 Moscow, Russia\nbKazan Federal University, 420008, Kazan, Russia\ncInstitute of Applied Research of Tatarstan Academy of Scien ces, 420111, Kazan, Russia\n(Dated: November 12, 2019)\nWe declare the observation of spin superfluid state in Yttriu m Iron Garnet (YIG) at room temper-\nature. It is similar to a Homogeneous Precessing State (HPD) , observed earlier in antiferromagnetic\nsuperfluid3He-B. The formation of this state explains bythe repulsive i nteraction between magnons,\nwhich is required as a prior condition for the spin superfluid ity. It establishes an energy gap, which\nstabilizes the long range superfluid transport of magnetiza tion and determines the Ginzburg-Landau\ncoherence length. This discovery paves a way to many quantum applications of supermagnonics at\nroom temperature, such as magnetic Josephson effect, long di stance spin transport, Q-bit, quantum\nlogics, magnetic sensors and others.\nPACS numbers: 67.57.Fg, 05.30.Jp, 11.00.Lm\nKeywords: Supermagnonics, spin supercurrent, magnon BEC, YIG film\nThe coherent quantum state of matter - the super-\nfluid state was discovered in 1938 by P. L. Kapitza in\nliquid4He at temperature of 2 K [1]. The first theoreti-\ncal explanation of this state was done by F. London. He\nsuggested that the superfluidity could have some connec-\ntion with Bose-Einstein condensation (BEC) [2]. In BEC\ntheory by Einstein the coherent state of noninteracting\nparticles was considered [3]. Recently it was successfully\napplied for the weakly interacted deluted gas of atoms at\nextremely low temperatures [4, 5]. But it can not be di-\nrectly applied for the system with a strong inter-particle\ninteraction, like4He liquid. In 1941 L. D. Landau sug-\ngested that superfluidity can be understood in terms of\natomic states, modified by interaction. The phenomeno-\nlogical theory by L. D. Landau successfully explained the\nsuperfluid properties [6]. The depletion of condensate in\n4He is very strong: in the limit of zero temperature only\nabout 10% of particles occupy the state with zero mo-\nmentum. Nevertheless, BEC still remains the key mech-\nanism for the phenomenon of superfluidity in liquid4He:\ndue to BEC the whole liquid (100% of4He atoms) forms\na coherent quantum state at T= 0 and participates in\nthe non-dissipative superfluid flow. Nowadays the super-\nfluid state is well known as a quantum state, governed by\na single wave function. Indeed, the superfluid state may\nexist without BEC, like in BerezinskiiKosterlitzThouless\ntransition in 2D materials. And opposite, BEC may ex-\nist, but does not lead to a superfluid transport. It takes\nplace when there is no energy gap and kinetic energy de-\nstroys the coherent state. In this case the critical current\nis equal to zero.\nThe magnetically ordered materials are described by\nthe ground state and a gas of bosonic excitations which\nare represented by magnons. At a condition of ther-\nmal equilibrium the density of magnons is always be-\nlow the critical density, required for the Bose condensa-\ntion. Indeed, the density of magnons can be increasedup to about Avagadro number by a magnetic resonance\nmethods. Usually the spin-spin interaction time is much\nshorter than the spin-lattice relaxation time and magnon\ngas may exist at a quasi equilibrium state. The magnon\ngasisa veryinterestingobjecttostudy the Bose-Einstein\ncondensation (BEC) and formation of a spin superfluid\nstates. Its properties are strongy depends from the type\nof spin-orbit interactions, which result in attractive or\nrepulsive interaction between magnons. At the first case\nthe formation of magnon BEC should be unstable. In-\ndeed, the recent experiments with magnon BEC forma-\ntion shows a very interesting new dynamic effects at\nthis case. [7, 8]. Contrary, at the repulsive interac-\ntion, the magnon BEC should be stable [9]. Further-\nmore, at higher density of magnons the collective spin\nmodes emerge. It is described by a common wave func-\ntion and shows the properties comparable with the su-\nperfluid component of4He liquid. It exhibit the long dis-\ntance spin supercurrent transport which is characterized\nby a Ginzburg-Lanadau coherence length.\nThe first magnonic superfluid state was discovered in\n1984 by A. S. Borovik-Romanov and Yu. M. Bunkov ex-\nperimental group and described theoretically by I. Fomin\nas a new state of magnetically ordered matter [10, 11]. It\nwas observed as the spontaneously self-organized phase-\ncoherent precession of spins in an antiferromagnetic su-\nperfluid3He-B. This state fulfills all criteria of coher-\nence, suggested later by Snoke [12]. It is radically differ-\nent from the conventional ordered states in magnets. It\nemergesonthebackgroundoftheorderedmagneticstate,\nandcanbedescribedintermsofthecondensationofmag-\nnetic excitations to a superfluid coherent quantum state\n[13–15]. The spin superfluid state exhibit many phe-\nnomena analogues to other superfluid states, including\nlongdistancespin supercurrentin the channelandphase-\nslippageatacriticalspinsupercurrent[16–18], Josephson\nspin-current effect [19, 20] as well as spin-current vortices2\n[21,22]. TheGoldstonecollectiveexcitationsofthis state\n(the analog of the second sound in4He), were also ob-\nserved[23,24]. Thestate, whichisveryneartoamagnon\nBEC state, was observed in a conditions of spatial trap.\nIn this case the BEC signal may ring for about tenth of\nminutes at a frequency of 1 MHz [25, 26]. This new state\ncan be described in a terms of a self trapping model,\nsuggested earlier for the formation of elementary parti-\ncles [27, 28]. The spin superfluid state was also observed\nin3He-A in the conditions, when spin-orbit interaction\nwas changed from attractive to repulsive [29–31]. It was\nalso resently observed in other superfluid state -3He-P\n[32]. The discovery of spin superfluidity in3He has been\nrecognized in 2004 by the Low Temperature community\nby a prestigious F. London prize [33, 34].\nIt is very important to note, that there is no any\nprincipal difference between the magnon gas in super-\nfluid3He and in solid magnetically ordered materials.\nThe magnetic ordering is a quantum phenomenon. The\nmodel of spatially fixed magnetic moments, usually ap-\nplied at some theoretical considerations, describes only\nlimited number of observations. For example, it fails\nto reproduce the temperature dependens of magnetiza-\ntion. The correct consideration follows from Holstein-\nPrimakoff transformation model [35] in which magnons -\nthe quanta of magnetic existations are not localized and\nmay flow as a magnon liquid. That is why the nature of\nsuperflow of magnetization in superfluid3He is similar to\none for solid magnetic materials. The difference is only\nthevalueofGilbertdamping, whichinsuperfluid3Hecan\nbe as small as 10−8while in solid magnetic materials it is\nabout 10−5in the best caseof the magnetic dielectric fer-\nrimagnets represented by yttrium iron-garnets. Indeed,\nthepropertiesofmagnonflow, magnonBECandmagnon\nsupercurrent in antiferromagnetic superfluid3He and in\nYIG film are very similar. The first experimental obser-\nvations of the spin superfluid phenomenon in a YIG film\nat room temperature are presented in this article.\nThe superfluid state is characterized by the off-\ndiagonal long-range order (ODLRO) [36]. In superfluid\n4He and in the coherent atomic systems the operators\nof creation and annihilations of atoms with momentum\np= 0 have the time-dependent vacuum expectation\nvalue. For the creation operator\n/angb∇acketleftˆa0/angb∇acket∇ight=N1/2\n0eiµt+iα, (1)\nwhereN0is the number of particles in the coherent state\nandµandαarechemical potentialand the phaseofwave\nfunction, respectively.\nThe analogy between spin precession and ODLRO in\nsuperfluids is seen if one compares the operator of cre-\nation of particle ˆ a+\n0with the operator ˆS+of creation of\nspin projecton on axis z, whose expectation value is:\n/angbracketleftBig\nˆS+/angbracketrightBig\n=Sx+iSy=/radicalBig\nS −ˆSzeiωt+iα.(2)This analogy suggests that in the coherent spin pre-\ncession the role of particle number Nis played by the\nprojection of total spin on the direction of the external\nmagnetic field Sz[14, 37]. The role of chemical potential\nis taken by the precession frequency. It is important to\nnote, that the Zeeman energy should be incorporated to\nthe chemical potential, since it may vary in space in the\ncase of magnetic field inhomogeneity.\nThe hydrodynamic equations for the magnon super-\nfluid are the Hamilton equations for the canonically con-\njugated variables Nandα:\n˙α=δF\nδN,˙N=−δF\nδα, (3)\nwhereFis the Ginzburg-Landau free energy functional\nof the system [38]. This functional in a frame, rotating\nat a frequency ωhas the conventional form\nF−µN=/integraldisplay\nd3r/braceleftBig|∇Ψ|2\n2m+/bracketleftbig\nωL(r)−ω/bracketrightbig\n|Ψ|2+Fso(|Ψ|2)/bracerightBig\n.\n(4)\nHereωL(r) =γH(r) is the localLarmorfrequency, which\nplays the role of external potential U(r) in atomic con-\ndensates. The last term Fso(|Ψ|2) contains nonlinear-\nity which comes from the spin-orbit interaction. It is\nanalogous to the 4-th order term in the atomic BEC,\nwhich describes the interaction between the atoms. The\nspin-orbit interaction provides the effective interaction\nbetween magnons, which can be attractive or repulsive.\nThe spin-orbitinteractioncontainsquadraticand quartic\nterms in |Ψ|.\nFso(|Ψ|2) =a|Ψ|2+b|Ψ|4+.... (5)\nWhile the quadratic term modifies the potential Uin the\nGinzburg-Landau free energy, the quartic term simulates\nthe interaction between magnons.\nWe are able to rewrite Eq. (4) in the next form [14]:\nF −µN=/integraldisplay\nd3r/braceleftBig|∇Ψ|2\n2m+/bracketleftbig\nω0(r)−ω/bracketrightbig\n|Ψ|2+b|Ψ|4)/bracerightBig\n,\n(6)\nwhereω0(r) =ωL(r)+a(r) isthe frequencyofprecession\nata limit ofsmall excitation. Hereweregropeterms with\n|Ψ|2and|Ψ|4. The magnon number density Nis related\nto the deflection angle βvia\nN=|Ψ|2=M(1−cosβ) (7)\nand the frequency of magnetization precession is\nωS(r) =ω0(r)+2b(1−cos(β(r))), (8)\nwhereMis the magnetization.\nThe gradientsof αexcites the spin supercurrent, which\ntransports the longitudinal magnetization [38]:\nJ=N∇α , (9)3\nThis long distance spin supercurrent was measured di-\nrectly in the experiments, described in [16–18]. It was\nconfirmed that the critical current corresponds to the\ncritical phase gradient, which is the inverse value of the\nGinzburg-Landau coherence length ξGL:\n∇αc= 1/ξGL=/radicalbig\nω0(ωS−ω0)/cSW,(10)\nwherecSWis a spin wave velocity. It is determined by\nthe competition between the energy of repulsive interac-\ntion (third term in Eq. 6) and kinetic energyofflow(first\nterm in Eq. 6). Please note, that at the attractive inter-\naction the coefficient bis negative and spin supercurrents\nunstable.\nThe remarkable consequence of the spin superfluid-\nity is the formation of the homogeneous precession do-\nmain (HPD) even in a strongly inhomogeneous magnetic\nfield. It is formed because the gradient of magnetic field\nleads to a gradient of phase of magnetization precession\nwhich excites the spin supercurrent. The latter trans-\nports magnons to the direction of smaller field. The\nprecession frequency increases untill the gradient of pre-\ncession vanishes. Finally the equilibrium state with the\nprecession frequency ωS(r) =constis established. Any\nperturbation of this state excites the spin supercurrent\nwhich restore the coherent precession. It was shown that\nthe HPDstate appearsspontaneouslyat somedelayafter\nthe pulseofmagneticresonanceexcitation. This statera-\ndiates an extremely long living induction signal [10, 11].\nIt can persist for several minutes due to the very slow re-\nlaxation(evaporation)ofquasiparticles[28,39]. Recently\nthis type of the signal was considered as a time-crystal\n[40]. It is important to note that the HPD state is the\neigen state of the ensemble of excited magnons. At relax-\nation the number of magnons decreases, the frequency of\nHPD decreases but the magnons remain in the coherent\nstate.\nThe other distinctive feature of the HPD state is its\npermanence. In the case of the atomic BEC the state\ndisappears because of the atom evaporation. In the case\nof HPD it is possible to replenish the losses(evaporation)\nof quasiparticles by excitation of new quasiparticles. It\nwas shown experimentally [29, 41, 42], that a weak RF\npumping at a frequency ωSstabilizes chemical poten-\ntial of magnon gas and keeps its density constant. The\ncorresponding phase difference between the magnetiza-\ntion precession and the RF field appears automatically\nto compensate the energy losses.\nIn this article we describe the first observation of HPD\nstate in an out-of-plane magnetized YIG film. The ex-\nperiments were performed on a YIG films of 6 and 1 µm\nthicknessinashapeofadisks0.5and0.3mmindiameter\n(see Methods).\nThe magnetization precession in the out-of-plane mag-\nnetized YIG film has dynamic properties very similar to\nones in a superfluid3He-B. The repulsive interaction be-\ntween magnons leads to the dynamical frequency shift ofthe precession at its deflection on angle β[43] described\nby equation:\nω−ω0=γ4πMS(1−cosβ), (11)\nwhereωis magnonfrequency of the homogeneouspreces-\nsion (k=0) at the deflection angle β,ω0is magnon fre-\nquency at the limit of small excitation, γis gyromagnetic\nratio,and MSissaturationmagnetization. It meansthat\nthe local frequency grows with increasing magnon den-\nsity. As a result the resonance field decreases if we excite\nthe system at a constant frequency.\nWe haveinvestigatedthe FMR adsorptionsignals from\nthe YIG film at frequency 9.26 GHz and different RF\npower of excitation. The signals are shown in Fig. 1. At\nfirst sight it looks like a well known signals of the non-\nlinear (bi-stable) resonance, first described by Anderson\net al. [44]. However this theory does not correspond well\nto the experimental results at relatively high excitation\n[45]. The reason for this discrepancy is that the Ander-\nson’s theory doesn’t take into account the spin super-\ncurrents, which play a very important role at a high an-\ngles of the magnetization deflection, when the density of\nmagnonssurpasethe conditionsofmagnonBose-Einstein\ncondensation, which is about 2-3◦for YIG film [46].\nAt a relatively small excitation power of 0.05, 0.1 and\n0.4 mW (see Fig. 1B) the amplitude of absorption sig-\nnalgrowsproportionallyto the RF field. This propertyis\ncorrespondto an excitationofthe magnongas. At higher\nexcitation the “capture” of a signal takes place. The sig-\nnals start to follow the field change (see curve c in Fig.\n1B). This non-linear behavior starts at an angle of de-\nflection about 2-3◦, when the field shift of the resonance\nbecame bigger then the broadening of the resonance line.\nThis angle of deflection is corresponds to the theoreti-\ncal approximation for magnon BEC formation [46]. At\nhigher excitation the signal follows the magnetic field up\nto its change by 80 Oe, as follows from the curve d at\nthe Fig. 1A. This field shift corresponds to a frequency\nshift of about 320 MHz and the angle of magnetization\ndeflection (see Eq. 7) by about 20◦!\nAt some critical field change the signal disintegrate.\nThis critical field shift (CFS) stronglydepends on the ex-\nciting power. If we sweep the field up the signal restore\nat a some field, which is quite near to the field of disinte-\ngration (see dotted line d in Fig. 1A). The observed be-\nhavior of the signal amplitude shows that the signals are\ngeneratedbya statewith coherentprecessionofmagneti-\nzation, which appears due to a spatial magnons redistri-\nbution by the spin supercurrent. Its properties are wery\nsimilar to one in antiferromagnetic superfluid3He, and\nparticularlytothe signalsin3He-Ain aerogel[29, 30, 47].\nAccording to a theoretical considerationfor the out-of-\nplane magnetized yttrium iron garnet film the density of\nexcited magnons should reach the conditions of BEC at\ntheangleofmagnetizationdeflectionabout β= 2,5◦[46].4\n5040 5060 5080 5100 5120 5140 51600.000.010.020.030.040.050.06\n5140 5145 5150 51550.0000.0010.002d\nc\nb Amplitude, arb. u. 10.0 mW (a)\n 20.0 mW (b)\n 40.0 mW (c)\n 80.0 mW (d)\n  \nH0, Oea(A) (B)\nde\nc\nb\na 0.05 mW (a)\n   0.1 mW (b)\n   0.4 mW (c)\n   0.8 mW (d)\n   1.0 mW (e)\n Amplitude, arb.u. \nH0, Oe\nFIG. 1: Amplitudes of the absorption signals at a frequency\nof 9.26 GHz for different RF pumping powers at a magnetic\nfield sweep down. The curves a, b and c in Fig. B corre-\nsponds to a linear magnetic resonance, when the amplitude\nof the signal is proportional to the RF field. The curve c cor-\nrespond to conditions, when the magnon BEC should forms.\nCurves at higher excitation correspond to the conditions to\na non-linear behavior of signals when the magnon superflow\nplays an important role. It redistribute the magnetization in\nthe sample and provide the stability of coherent precession .\nCurve d in Fig A corresponds to ahighest energy of excitation\nat this experiment. The field shift of the signal about 80 Oe\nwas achieved. It corresponds to an angle of precessing mag-\nnetization deflection of about 20◦. The curve d, which shown\nby dotted line corresponds to a signal we have observed at a\nsweep field up. It shows that the signal of non-linear mag-\nnetic resonance restores at a small hysteresis. (The power o f\n100 mW in these experiments corresponds roughly to the RF\nmagnetic field of 0.1 Oe.)\nThis angle well corresponds to a begining of signals non-\nlinearity (see Fig. 1). At an excitation power higher then\n0.4 mW for the conditions of our experiments the density\nof magnons increasesand strong repulsive interaction be-\ntween magnons modifies the spectrum of magnons. The\nspin dynamics can not be described anymoreby the BEC\ntheory of weekly interacted magnons. The formation of\ncoherent macroscopic states and long distance spin su-\npercurrent should be considered.\nIn Fig. 2 the profile of effective magnetic field in the\n500��m in diameter disk-shaped YIG film is shown. It\nwas calculated by OOMMF micromagnetic software [48].\nThe effective field has a minimum at the disk center due\nto the demagnetization field. One may suggest that at a\nfield sweep down the resonance conditions appear first at\na central part. Then the magnetization deflects and the\nfrequencyfollowstothe resonanceconditionsatasmaller\nfield untill the signal breaks down. But why the signal\nrestore at a field sweep up with a very small hysteresis.\nLet us consider the case, when the resonanse conditions\nappears only near the edge of the sample. If we apply\nthe RF field at the frequency corresponding to the ferro-\nmagnetic resonancenear the edge (at R = 200 µm in Fig.\n2) the spin waves will be excited at this region (see the\nright inset Fig. 2 red dot line). If the excitation power\nN \nFIG. 2: Variation of the local effective magnetic field from th e\ncenter of the sample (red line) and the frequency shift due to\nthemagnonchemical potential (blueline), whichcompensat es\nthe inhomogeneity of magnetic field. The inset to the right\nshows schematically the spatial distribution of magnetiza tion\ndeflection (sin β) at small excitation (point red line), at a\nhigher excitation, when magnons start to superflow to lower\nfield parts of the disk (dashed blue line) and the HPD state\n(solid black line). The inset to the left side shows the HPD\ndroplet of coherent magnons.\nwill be increased the deflection angle also increases and\nmay surpasses the critical BEC angle. The gradient of\nphase of precession and spin supercurrent will appears\ndue to the gradient of effective magnetic field. It will\ntransport magnons in the direction of smaller field, to\nthe central part of the sample (Fig. 2 dashed blue line).\nFinally, the angle of magnetization deflection in the cen-\ntralpartincreasesuptoavalue, whichcorrespondstothe\nprecession frequency equal to RF field (black line). The\nspatial distribution of dynamic frequency shift is shown\nin Fig. 2 by a solid blue line. This shift compensates\nthe inhomogeneity of effective magnetic field and all the\nmagnetizationprecessesatthefrequencyofRFfield. The\nHPD state is the eigenstate in inhomogeneous magnetic\nfield for a given number of magnons. This scenario was\ncarefully investigated in the experiments with superfluid\n3He and MnCO 3. [41, 49].\nTheHPDstatehasaformofdroplet, artisticallyshown\nin the left inset in Fig. 2. Deflected magnetization pre-\ncesses coherently and homogeneously at a frequency:\nω=γH(r)+∆ω(r) =const, (12)\nwhereH(r) is the effective magnetic field at a point r.\nFurthermore, the spin supercurrent compensates the in-\nhomogeneity of magnetization relaxation by redistribu-\ntion of magnons density. This state exists permanently\nin the case of RF pumping at the frequency of the HPD\nstate, which compensates the evaporation of magnons.\nThe superfluid state is supported by a permanent\npumping of the RF magnetic field, HRF, which is trans-\nverse to the applied constant external magnetic field5\nH0. The RF field prescribes the precession frequency,\nω=ωRF, and thus fixes the chemical potential µ=ω.\nIn the precession frame, where both the RF field and the\ndeflected magnetization Mare constant, the interaction\nenergy term is\nFRF=−HRF·M=−HRFM⊥cos(α−αRF),(13)\nwhereHRFandαRFare the amplitude and the phase of\nthe RF field. This term softly breaks the U(1)-symmetry\nand serves as a source of the mass of Nambu-Goldstone\nmode [50]. The phase difference between the condensate\nand RF field, ( α−αRF), is determined by the energy\nlosses due to magnetic relaxation, which is compensated\nby the pumping power of the RF field:\nW=ωMHRFsinβsin(α−αRF).(14)\nThe phase difference is automatically adjusted to com-\npensate the losses. If dissipation is small, the phase shift\nis small ( α−αRF≪1) and can be neglected. The ne-\nglected (α−αRF)2term leads to the nonzero mass of the\nGoldstone boson – quantum of the second sound waves\n(phonons) in the magnonic superfluid [50]. The signal\nbreaks down at the moment, when the RF power is not\nenough to compensate the magnons dissipation. Since\nthe pumping (14) is proportional to sin βsin(α−αRF),\na critical tilting angle βc, at which the pumping cannot\ncompensate the losses, increases with increasing HRF.\nThe breaks down occurs when the phase shift ( α−αRF)\nreaches 90◦. At this moment the adsorption signal cor-\nresponds to a transverse magnetization of the sample.\nIn Fig. 3 it is shown by points. The theoretical curve,\nshowninFig. 3bysolidline, iscalculatedfortheassump-\ntion that the HPD droplet fills up all the region, where\nmagnetic field is less than ωRF/γ. The theoretical curve\nshows a good agreement with the experimental points.\nThe similar results were obtained in other samples.\nLet us compare the experimental results with the the-\nory of non-linear resonance [44]. This theoretical ap-\nproach is based on the properties of a single non-linear\noscillator. It supposes that at increase of the excitation\npower, the angle of deflection increases which leads to a\nfrequency shift. This theory describes well the non-linear\nresonance in ferromagnets at a relatively small angles of\ndeflection but strongy disagrees with the experimental\nresults at higher angles [45]. A more sophisticated the-\nory of autoresonance was suggested in [51], where a singe\nnon-linear oscillator is also considered. This model is not\napplicable to real magnetically ordered materials since\nthe spin system consists of many oscillators with differ-\nent ground frequency due to inhomogeneity of magnetic\nfield. If we try to modify the theory for a number of the\nnon-interacting oscillators with different ground frequen-\ncies we should conclude that the break down points are\ndifferent for different oscillators. As a result the exper-\nimental curve of signal break down should be broaden/s53/s48/s52/s48 /s53/s48/s54/s48 /s53/s48/s56/s48 /s53/s49/s48/s48 /s53/s49/s50/s48 /s53/s49/s52/s48 /s53/s49/s54/s48/s48/s46/s48/s48/s48/s46/s48/s49/s48/s46/s48/s50/s48/s46/s48/s51/s48/s46/s48/s52/s48/s46/s48/s53/s48/s46/s48/s54\n/s53/s48/s52/s48 /s53/s48/s56/s48 /s53/s49/s50/s48 /s53/s49/s54/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52\n/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s32/s115/s105/s110/s32/s115/s105/s110\n/s32/s83/s83/s44/s32/s109/s109/s50\n/s72\n/s48/s44/s32/s79/s101\n/s32/s32/s65/s98/s115/s111/s114/s112/s116/s105/s111/s110/s44/s32/s97/s114/s98/s46/s32/s117/s46\n/s72\n/s48/s44/s32/s79/s101\nFIG. 3: The absorption signal amplitude at the moment\nof breaks down of the HPD state (points). The theoretical\ncurve (red solid line) was calculated for the signal from the\nmagnonic droplet at these conditions. The inset shows the\nmagnetization deflection (sin β) in the center of the magnon\ndroplet and the film surface area, occupied by the droplet (S) .\non the magnetic field. Indeed, experimentally the break\ndown appears very abruptly. The even more important\nargument against this theory is the recovery of signal\nwith a very small hysteresis at a sweep magnetic field\nup. The non-liner oscillator can not be excited by out of\nresonance excitation. It is clearly shown in Fig. 3 of Ref.\n[51]. It may be excited at a sweep field up only near the\nresonance, at a field shift of a homogeneous broadening\nof the line, which is about 2 Oe. Experimentally the ex-\ncitation takes place at uncomparable big shift of 80 Oe.\nThis can be explained only by a resonance excition of\nmagnons at the edge and its redistribution throw all the\nsample by spin supercurrent. The comparison between\nthe model of non-linear oscillator and HPD have been\ninvestigated in Ref. [52].\nIt is very important to note that the spin supercur-\nrent exists only at the conditions of repulsive interaction\nbetween magnons. In the case of attractive interaction\nthe superfluid critical velocity is equal to zero and the\nsuperfluid state is unstable [53].\nFinally, we have shown the big importance of spin su-\npercurrenttransportto explain the propertiesofthe non-\nlinear magnetic resonance. We have demonstrated the\nformation of a macroscopic region with the coherent pre-\ncession of magnetization in an inhomogeneous effective\nmagnetic field in the out-of-plane magnetized YIG film.\nThis state is the first permanent superfluid state of con-\ndensed matter demonstrated at room temperature. It is\nan idealplatform for developmentofmicrowavemagnetic\ntechnologies, which have already resulted in the creation\nof the magnon transistor and the first magnon logic gate\n[54, 55]. The YIG films can be used as the basis for6\nnew solid-state quantum measurement and information\nprocessing technologies including cavity-based QED, op-\ntomagnonics, and optomechanics [56]. A chain of YIG\nsamples with excited HPD droplets may be considered\nas a Q-bits, interacting through Josephson junctions for\naquantumcomputer. TheformationofthemagnonBEC\nin YIG and observation of the spin supercurrent, like in\n3He, should lead to the development of a new branch of\nmodern magnetism - supermagnonics.\nMethods\nAll samples were prepared from yttrium iron gar-\nnet films grown by liquid phase epitaxy on 500\nµm thick GGG substrates with the (111) crystallo-\ngraphic orientation [57]. To reduce the effect of cu-\nbic magnetic anisotropy, we used scandium substituted\nLu1.5Y1.5Fe4.4Sc0.6O12iron garnet films; the introduc-\ntion of lutetium ions was necessary to match the param-\neters of the substrate and film crystal gratings. It is\nknown that the introduction of scandium ions in such\nan amount reduces the field of cubic anisotropy by more\nthan an order of magnitude [58]. In addition, the used\nlutetium and scandium ions practically do not contribute\nto additional relaxation in the YIG. The samples were\nprepared in the form of a disk with diameters of 500 and\n300µm and a thickness of 6 µm. The disk was made\nby photolithography. To avoid magnetic pinning on the\nsurface the sample was etched in a hot phosphoric acid\n[59]. As a result, the edges of the disk have a slope of 45\ndegrees and had a smooth surface.\nThe CW FMR experiments were performed on Varian\nE-12X-bandEPRspectrometeratthe roomtemperature\nand the frequency 9.26 GHz. The RF field was oriented\ninplaneofthesamples. Theamplitudeandthefrequency\nof magnetic field modulation were 0.05 Oe and 100 kHz,\nrespectively. This frequency is much lower than the es-\ntimated frequency of the second sound of the magnon\nBEC (The Goldston mode). 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