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Nonlocal magnon transconductance in extended magnetic insulating films. II: two-fluid behavior. R. Kohno,1K. An,1E. Clot,1V. V. Naletov,1N. Thiery,1L. Vila,1R. Schlitz,2N. Beaulieu,3J. Ben Youssef,3A. Anane,4 V. Cros,4H. Merbouche,4T. Hauet,5V. E. Demidov,6S. O. Demokritov,6G. de Loubens,7and O. Klein1,∗ 1Université Grenoble Alpes, CEA, CNRS, Grenoble INP, Spintec, 38054 Grenoble, France 2Department of Materials, ETH Zürich, 8093 Zürich, Switzerland 3LabSTICC, CNRS, Université de Bretagne Occidentale, 29238 Brest, France 4Unité Mixte de Physique CNRS, Thales, Univ. Paris-Sud, Université Paris Saclay, 91767 Palaiseau, France 5Université de Lorraine, CNRS Institut Jean Lamour, 54000 Nancy, France 6Department of Physics, University of Muenster, 48149 Muenster, Germany 7SPEC, CEA-Saclay, CNRS, Université Paris-Saclay, 91191 Gif-sur-Yvette, France (Dated: June 13, 2023) Thisreviewpresentsacomprehensivestudyofthespatialdispersionofpropagatingmagnonselectricallyemit- ted in extended yttrium-iron garnet (YIG) films by the spin transfer effects across a YIG |Pt interface. Our goal is to provide a generic framework to describe the magnon transconductance inside magnetic films. We experi- mentallyelucidatetherelevantspectralcontributionsbystudyingthelateraldecayofthemagnonsignal. While mostoftheinjectedmagnonsdonotreachthecollector,thepropagatingmagnonscanbesplitintotwo-fluids: i) alargefractionofhigh-energymagnonscarryingenergyofabout 𝑘𝐵𝑇0,where𝑇0isthelatticetemperature,with acharacteristicdecaylengthinthesub-micrometerrange,and ii)asmallfractionoflow-energymagnons,which are particles carrying energy of about ℏ𝜔𝐾, where𝜔𝐾∕(2𝜋)is the Kittel frequency, with a characteristic decay lengthinthemicrometerrange. Takingadvantageoftheirdifferentphysicalproperties,thelow-energymagnons can become the dominant fluid i)at large spin transfer rates for the bias causing the emission of magnons, ii)at large distance from the emitter, iii)at small film thickness, or iv)for reduced band mismatch between the YIG belowtheemitterandthebulkduetovariationofthemagnonconcentration. Thisbroaderpicturecomplements part I [1], which focuses solely on the nonlinear transport properties of low-energy magnons. I. INTRODUCTION Nonlocal devices, such as the geometry shown in Fig. 1, consisting of two lateral circuits deposited on an extended magnetic insulating film have recently attracted much atten- tion as novel electronic devices exploiting the spin degree of freedom[2–6]. As emphasized in part I, one of their origi- nal features is to behave as a spin diode at large currents[1]. These devices rely on the spin transfer effect (STE) to elec- trically modulate the magnon population in a magnetic thin film. The process alters the amplitude of thermally activated spin fluctuations by transferring quanta of 𝛾ℏbetween an ad- jacentmetallicelectrodeandthemagneticthinfilmviaastim- ulated emission process. In unconfined geometries, a wide energy range of eigenmodes is available to carry the exter- nalflowofangularmomentum,spanningafrequencywindow fromGHztoTHz,asschematicallyshowninFig.1(c),which shows the lower branch of the spin wave dispersion over the Brillouin zone [7–9]. At high-energy the curve flattens out at about 30 meV, which corresponds to the thermal energy, 𝐸𝑇≈𝑘𝐵𝑇0, at ambient temperature, while at low-energy it shows a gap, 𝐸𝑔≈ℏ𝜔𝐾≈ 30𝜇eV, around the Kittel fre- quency𝜔𝐾∕(2𝜋)[10]. Betweenthesetwoextremes,thespec- tralidentificationoftherelevanteigenmodesinvolvedinnon- local spin transport has remained mostly elusive. In this review, we propose a simple analytical framework toaccountforthemagnontransconductanceinextendedmag- netic insulating films. We find that the observed behavior can ∗Corresponding author: oklein@cea.frbe well approximated by a two-fluid model, which simplifies thespectralviewasemanatingfromtwoindependenttypesof magnons placed at either end of the magnon manifold. On the one hand, we have magnons at thermal energies, to be re- ferred to as high-energy magnons[4], whose distribution fol- lowsthetemperatureofthelattice. Ontheotherhand,wehave magnonsatthebottomofthebandneartheKittelfrequency,to be referred to as low-energy magnons, whose electrical mod- ulationathighpoweristhefocusofpartI[1]. Theresponseof these two magnon populations to external stimuli is very dif- ferent. The high-energy thermal magnons, being particles of high wavevector, are mostly insensitive to any changes in the external conditions of the sample such as shape, anisotropy andmagneticfield,beinginsteaddefinedbythespin-waveex- change stiffness and the large k-value of the magnon[11, 12]. In contrast, low-energy magnons, sensitive to magnetostatic interaction, depend sensitively on the extrinsic conditions of thesample. Itturnsoutthatnonlocaldevicesprovideaunique meanstostudyeachofthesetwo-fluidsindependentlybycom- paringthedifferencesintransportbehaviorasafunctionofthe separation,𝑑, between the two circuits, thus benefiting from the spatial filtering associated with the fact that each of these two components decays very differently as a function of dis- tance, as schematically shown in Fig. 1(b). The paper is organized as follows. After this introduction, inthesecondsectionwereviewthemainfeaturesthatsupport the two-fluid separation. In the third section, we describe the analytical framework of a two-fluid model and, in particular, theexpectedsignatureinthetransportmeasurement. Thispart builds on the knowledge gained in part I[1] about the nonlin- ear behavior of the low-energy magnon. To facilitate quickarXiv:2210.08283v2 [cond-mat.mes-hall] 11 Jun 20232 FIG. 1. Lateral geometry used for measuring the magnon transcon- ductanceinextendedmagneticinsulatingfilms. (a)Scanningelectron microscopeimageofa4-terminalcircuit(scalebaris5 𝜇m),whose4 polesareconnectedtotwoparallelwires,Pt1andPt2(showninpink), deposited on top of a continuous YIG thin film. A continuous elec- tric current, 𝐼1, injected in Pt1(emitter) produces an electric mod- ulation of the magnon population by the spin transfer effect (STE). Thismodulationisconsequentlydetectedlaterallybythespinpump- ing voltage−𝑅2𝐼2through a second electrode Pt2(collector) placed atadistance 𝑑fromtheemitter. Wedefinethemagnontransmission ratioT𝑠=𝐼2∕𝐼1andthetransconductance T𝑠∕𝑅1. Panel(b)isasec- tional view showing the spatial decay of propagating magnons. (c) Schematic representation of the spin-wave dispersion over the Bril- louin zone. We consider the spin transport properties as originat- ing from two independent fluids located at either end of the disper- sioncurve. Eachofthetwo-fluidshasadifferentcharacteristicdecay length,𝜆𝑇and𝜆𝐾respectively, as shown in (b). reading of either manuscript, we point out that a summary of the highlights is provided after each introduction and, in both papers, the figures are organized into a self-explanatory sto- ryboard, summarized by a short sentence at the beginning of each caption. In the fourth section we will show the experi- mentalevidencethatsupportssuchapictureandfinallyinthe fifth section we will conclude our work by emphasizing the important results and opening to future perspectives. II. KEY FINDINGS The purpose of this review is to present the experimental evidence supporting the separation of the magnon transcon- ductanceintotwocomponents. Thisisachievedbymeasuring the transmission coefficient T𝑠≡𝐼2∕𝐼1of magnons emitted and collected via the spin Hall effect between two parallel Pt wires, Pt1and Pt2, respectively. It is shown that a two-fluid model, where T𝑠=T𝑇+T𝐾is the independent sum of a high-energy and a low-energy magnon contribution, providesasimplifiedcommonframeworkthatcapturesalltheobserved behaviorinnonlocaldeviceswithdifferentinter-electrodesep- aration,differentcurrentbias,differentappliedmagneticfield, differentfilmthicknessormagneticcomposition,anddifferent substrate temperature. Makingaquantitativeanalysisofthetransmissionratio,we findthatmostoftheinjectedspinsremainlocalizedunderthe emitterorpropagateinthewrongdirection(theestimatedfrac- tion is about 2/3), making these materials intrinsically poor magnonconductors. Theremainingpropagatingmagnonsfall into two distinct categories: First, a large fraction carried by high-energy magnons, which follow a diffusive transport be- havior with a characteristic decay length, 𝜆𝑇, in the submi- cron range[13, 14]; and second, a small fraction carried by low-energy magnons,which are responsible forthe asymmet- ric transport behavior [1], and which follow a ballistic trans- port with a characteristic decay length, 𝜆𝐾, in the micrometer range. The different decay behaviors are directly observable experimentally in the change of the nonlinear spin transport behavior with separation, 𝑑. We also carefully study the collapse of the magnon trans- mission ratio with increasing temperature of the emitter, 𝑇1, as it approaches the Curie temperature, 𝑇𝑐. Here, the num- berof spin-polarizedsitesunder theelectrodebecomes ofthe same order as the spin flux coming from the external Pt elec- trode. Thetransitiontothisregimeofmagnetizationreduction leadstoasharpdecreaseinthemagnontransmissionratio. We report signs of interaction between the low-energy and high- energy parts of the liquid in this highly diffusive regime[15– 17]. In addition, the collapse seemsto actually occur well be- forereaching 𝑇𝑐,suggestingthatthetotalnumberofmagnons issignificantlyunderestimatedcomparedtothatinferredfrom the single temperature value of the lattice below the emit- ter. Alternatively, this discrepancy could indicate a rotation of the equilibrium magnetization under the emitter[18, 19]. Since the discrepancy actually becomes more pronounced as the magnetic film gets thinner, this suggests that the culprit is the amount of low-energy magnons. III. ANALYTICAL FRAMEWORK A. Low-energy magnons WerecallthefindinginpartI[1],thatthetransconductance by low-energy magnons in open geometries can be described by the analytical expression: T𝐾∝𝑀1 𝑀2⋅𝑘𝐵𝑇1 ℏ𝜔𝐾⋅𝑒𝜔𝐾 th⋅1 1−(𝐼1∕th)2,(1) where𝑒istheelectroncharge,while 𝑀1and𝑀2arethemag- netizationvaluesundertheemitterandcollector,respectively. The threshold current, th, is the solution of a transcenden- tal equation obtained by combining Eqs. (4), (6) and (7) in Ref. [1]. In our model, its nonlinear behavior is determined solely by two parameters th,0and𝑛sat, which are related to the nominal value of the transmission coefficient at low cur- rentandthesaturationthresholdexpressedinnormalizedunits3 FIG. 2. Current bias characteristic of the magnon transconductance depending on the spectral nature of propagating magnons. Panels (a) and (b) compare the predicted electrical variation of T𝑠for low- energymagnons(seeEq.(1),leftpanel)andforhigh-energymagnons (see Eq. (3), right panel), respectively, when 𝐻𝑥<0. Panel (f) shows the associated variation of 𝑇1=𝑇0+𝜅𝑅𝐼2 1, the lattice tem- perature below the emitter. The current span exceeds 𝐼c, the current bias, whichraises 𝑇1to𝑇𝑐, theCurietemperature. Panels (c)and (d) show the behavior when T𝑠is renormalized by 𝑇1. Panel (e) shows thetwo-fluidfittingfunction: theindependentsumofthelow-energy and high-energy magnon contributions with their respective weights Σ𝑇andΣ𝐾. The inset (g) shows the temperature dependence of the magnetization 𝑀𝑇as measured by vibrating sample magnetometry (cf. Fig. S1), and the solid line is a fit with the analytical expression 𝑀𝑇≈𝑀0√ 1−(𝑇∕𝑇𝑐)3∕2, with𝜇0𝑀0=0.21T and𝑇𝑐=550K. of nonlinear effects, respectively. All information about these feedback effects can be found in Ref. [1]. AsemphasizedindetailinpartI,oneofthepitfallsofnon- local devices is that the emitter electrode cannot be made im- mune to Joule heating due to poor thermalization in the 2D geometry. This leads to a significant increase of the tempera- ture under the emitter with current 𝐼1, which we model by 𝑇1||𝐼2 1=𝑇0+𝜅𝑅𝐼2 1. (2) In our notation, 𝑇0is the substrate temperature at no current and𝜅is the temperature coefficient of resistance for Pt. It is the coefficient that determines the temperature rise per de- posited joule power (see Fig. S1 in Appendix). We addition- allydefine𝐼cthecurrentrequiredtoreachtheCurietempera- ture,𝑇𝑐=𝑇0+𝜅𝑅𝐼2 c[see Fig. 2(f)]. This variation has pro- found consequences both on the level of thermal fluctuations ofthelow-energymagnonsandonthenumberofhigh-energy magnons. In particular, the variation of 𝑇1with𝐼1expressed by Eq. (2) enters into the variation of T𝐾with𝐼1expressed by Eq. (1). The resulting variation of the magnon population as a function of 𝐼1is shown in Fig. 2(a). To account for thevariationof𝑇1producedbyJouleheating,whichexpressesthe influence of a varying background of thermal fluctuations on the STE, we plot T𝐾∕𝑇1in Fig. 2(c). This renormalization is equivalent to looking at the nonlinear behavior from the per- spective of a thermalized background. The resulting shape of the curve as a function of 𝐼1is greatly simplified. In the re- verse bias, marked by the symbol ◂representing the magnon absorption regime, the normalized transconductance is con- stant up to𝐼c. In contrast, in the forward bias, denoted by the symbol▸, which represents the magnon emission regime, a peak appears. This asymmetric peak is called the spin diode effect in part I[1]. The advantage of the 𝑇1normalization of themagnontransmissionratioisthatitmakesthepeakachar- acteristic feature of the spin diode effect. B. High-energy magnons Wenowassumethatthenumberofhigh-energymagnonsis approximatelyequaltothetotalnumberofmagnons,whichis thedifference 𝑀1−𝑀0,where𝑀0isthespontaneousmagne- tization at𝑇=0K and𝑀1is the spontaneous magnetization at𝑇=𝑇1, the temperature of the emitter [20]. We thus an- alytically express the contribution of high-energy magnons to the magnon transconductance by the equation: T𝑇∝𝑀1 𝑀2⋅𝑀0−𝑀1 𝑀0, (3) wheretheprefactor 𝑀1representstheamountofmagneticpo- larizationavailableundertheemitter. Wenotethattheanalyt- ical form expressed by Eq. (3) has been previously proposed to describe spin transmission in paramagnetic materials[21]. As shown in the inset Fig. 2(g), we find that the tempera- ture dependence of 𝑀1is well described by the analytical 𝑀1≈𝑀0√ 1−(𝑇1∕𝑇𝑐)3∕2. The resulting number of ther- mally excited magnons contributing to the nonlocal transport is shown in Fig. 2(b). Repeating the same analysis developed in Fig. 2(c), a more revealing behavior is obtained by renor- malizing T𝑇with𝑇1and the result is shown in Fig. 2(d). In thiscase,thecurrentdependenceof T𝑇∕𝑇1on𝐼1isaconstant function up to 𝐼c. C. Two-Fluid Model An advantage specific to nonlocal transport measurements isthatthepropagationdistance, 𝑑,providesapowerfulmeans to spectrally distinguish different types of magnons, each of which has its characteristic decay length 𝜆𝑘along the𝑥-axis [13,22]. Inthefollowingwewillexaminetheexpectationfor the different extrema of the dispersion curve. For the high-energy magnons, the spin wave spectrum can simply be approximated as 𝜔𝑘=𝜔𝑀𝜆2 ex𝑘2, where𝜔𝑀= 𝛾𝜇0𝑀𝑠= 2𝜋×4.48GHz and𝜆ex≈ 15nm is the exchange length[23]. High-energy magnons at room temperature ( 𝑇0= 300K) have the frequency 𝜔𝑇=𝑘𝐵𝑇0∕ℏ= 2𝜋×6.25THz,4 FIG.3. Dispersioncharacteristicoflow-energymagnons. (a)Disper- sion curves at the bottom of the magnon manifold of a 19 nm thick YIG film for two values of 𝜃𝑘= 0◦(𝑘∥𝑀) and90◦(𝑘⟂𝑀), theanglebetweenthewavevectorandtheappliedmagneticfield. We mark with dots the Kittel mode ( 𝐸𝐾, black dot), the lowest energy mode(𝐸𝑔,bluedot),andthemodedegeneratetotheKittelmodewith the highest wavevector ( 𝐸𝐾, orange dot). The curve is computed for YIG𝐴thin films. (b) Characteristic decay length calculated from the dispersioncurve,assumingthatthemagnonsfollowthephenomeno- logical LLG equation with 𝛼LLG=4⋅10−4. which corresponds to a wavevector 𝑘𝑇=2.5nm−1. It is seri- ously questionable whether the estimate for 𝜆𝑇from the phe- nomenologicalLandau-Lifshitz-Gilbert(LLG)modelisappli- cable to such short-wavelength magnons. Practically i)the Gilbert damping is expected to be increased in the THz range [23].ii)the group velocity is reduced towards the edge of the Brillouin zone [7, 24], and iii)the LLG model does not con- sider the reduction of the characteristic propagation distance due to diffusion processes. Furthermore, YIG is a ferrimag- net, higher (antiferromagnetic) spin wave branches contribute significantlytothemagnontransport[7–9]. Webelievethatthe most reliable estimates have been obtained experimentally by studying the spatial decay of the spin Seebeck signal[13, 25] and have found 𝜆𝑇≈0.3𝜇m. In contrast to its high-energy counterpart, the LLG frame- work should provide a good basis for calculating the propa- gation distance of long-wavelength dipolar spin waves. This interaction gives an anisotropic character to the group veloc- ity of these spin waves. In Fig 3(a) we plot the dispersion curve of a magnon propagating either along the 𝑥-axis (or- angeline)oralongthe 𝑦-axis(blueline). Inthefollowing,we will focus our attention on the branch 𝜃𝑘= 0◦(orange line), which corresponds to the magnon propagating in the normal direction of the Pt wires. As emphasized in part I[1], there are 3 remarkable positions on the curve, each marked by a colored dot on Fig. 3. The energy minimum, 𝐸𝑔(blue dot), does not contribute to the transport because its group veloc- ity is zero. The longest wavelength spin waves correspond to the Kittel mode, 𝐸𝐾(black dot). The damping rate, tak- ing into account the ellipticity of the spin waves, is given by Γ𝐾=𝛼LLG(𝜔𝐻+𝜔𝑀∕2), where𝜔𝐻=𝛾𝐻0[26]. The ve- locity is equal to 𝑣𝐾=𝜕𝑘𝜔=𝜔𝐻𝜔𝑀𝑡YIG∕(4𝜔𝐾), where𝜔𝐾 is the Kittel frequency and 𝑡YIGis the YIG thickness. The re- sulting decay length of the spin transport carried by 𝑘→0 magnons is𝜆𝐾=𝑣𝐾∕(2Γ𝐾)≈2.5𝜇m for𝑡YIG=19nm. Aspointed out in part I[1], the mode that seems to be most rel- evant for long-range magnon transport in nonlocal devices is probably𝐸𝐾, the degenerate mode with the Kittel frequency and the shortest wavelength. This mode is marked by an or- ange dot in Fig. 3. For our 𝑡YIG= 19nm film, it turns out that its group velocity is of the same order as that of the Kit- tel mode, giving a similar decay distance. We will show later that this estimate is quite close to the experimental value. We note, however, that the value of the decay distance at 𝐸𝐾in- creaseswithincreasingfilmthicknesstobecomeindependent of𝑡YIGfor thicknesses above 200 nm. The saturation value is 𝜆𝐾≈20𝜇m, assuming 𝛼LLG=4⋅10−4. Since𝜆𝐾≈ 10×𝜆𝑇, changing𝑑allows tuning from spin transport governed by high-energy magnons to spin transport governed by low-energy magnons. One should also add that thecurrentintensity, 𝐼1,alsoprovidesameanstotunetheratio between the two-fluid as discussed in Ref. [1]. Learningfromtheaboveconsiderations,wecannowputall the contributions together to propose an analytical fit of the data with the two-fluid function: T𝑠=Σ𝑇,0exp−𝑑∕𝜆𝑇T𝑇 T𝑇,𝐼1→0+Σ𝐾,0exp−𝑑∕𝜆𝐾T𝐾 T𝐾,𝐼1→0, (4) combining two independent magnon contributions: one at thermal energy and the second at magnetostatic energy. We assumeherethatbothmagnonfluidsfollowanexponentialde- cay. To ease the notation, we shall refer below at underlined quantity,e.g.T𝑇≡T𝑇∕T𝑇,𝐼1→0, as the normalized quantity by the low current value. We define Σ𝐾||𝑑=Σ𝐾,0exp−𝑑∕𝜆𝐾 andΣ𝑇||𝑑= Σ𝑇,0exp−𝑑∕𝜆𝑇, where the index 0represents the extrapolated value at the emitter position ( 𝑑= 0): see Fig. 1(b). Thus the parameter Σ𝐾∕(Σ𝐾+Σ𝑇)||𝑑represents the variation with distance of the proportion of low-energy propagating magnons over the total number of propagating magnons. An exemplary fit for 𝑑= 0and identical high- energy and low-energy contributions is shown in Fig. 2(e). ItshouldbeemphasizedthatthemodelproposedbyEq.(4), which assigns a fixed decay rate to each magnon category, is certainlytoosimplistic. Forexample,oneshouldkeepinmind thatif𝑀1→0duetoJouleheating,thiscouldhaveprofound consequenceson 𝜆𝑇bychangingthestiffnessoftheexchange constant. This has already been discussed in the context of spin propagation in paramagnetic materials[21]. We will re- turntothisissue belowinthecontextof ourdiscussionofthe discrepancy in the values of 𝑇𝑐extracted from the transport data. IV. EXPERIMENTS In this section we present the experimental evidence sup- porting the two-fluid picture shown above. We focus on the evolutionofspintransportwithcurrent,distance,appliedmag- netic field, substrate temperature and effective magnetization, 𝑀eff. This will allow us to test the validity of our model.5 FIG. 4. Dependence of the collected voltage on an external mag- netic field. Comparison of the nonlocal voltage V2= (𝑉2,⟂−𝑉2,∥) between (a) a short-range device ( 𝑑=1.0𝜇m) and (b) a long-range device (𝑑= 2.3𝜇m). The panels show the zoom at the maximum and minimum of the normalized values. We interpret the detection of a finite susceptibility, 𝜕𝐻𝑥V2<0, as an indication of a magnon transmission ratio by low-energy magnons. In contrast, a constant behavior,𝜕𝐻𝑥V2≈ 0, is indicative of a magnon transmission ratio by high-energy magnons. Finite susceptibility is uniquely observed in the long-range regime when 𝐼1⋅𝐻𝑥<0,i)when the number of low-energymagnonsisincreasedbyinjectingacurrentintheforward direction,and ii)whenthecontributionoftherapidlydecayinghigh- energy magnons becomes a minority. The data are collected on the YIG𝐶thinfilmdrivenbyalargecurrentamplitudeof ±𝐼1=2.0mA. Thenormalizationvalueof V2arerespectively 10.04𝜇Vand8.66𝜇V in panel (a) and (b). A. Magnetic susceptibility of the magnon transmission ratio We begin this section by first presenting some key experi- mentalevidencesupportingthetwo-fluidpicture. Aschematic of the 4-terminal device is shown in Fig. 1(a). It circulates pure spin currents between two parallel electrodes subject to thespinHalleffect[27]: inourcasetwoPtstrips 𝐿Pt=30𝜇m long,𝑤Pt= 0.3𝜇m wide and 𝑡Pt= 7nm thick. The experi- ment is performed here at room temperature, 𝑇0=300K, on a 56 nm thick (YIG𝐶) garnet thin film whose physical prop- erties are summarized in Table 1 of Ref. [1]. While injecting anelectriccurrent 𝐼1intoPt1,wemeasureavoltage 𝑉2across Pt2,whoseresistanceis 𝑅2. Tosubtractallnon-magneticcon- tributions, we define the spin signal V2= (𝑉2,⟂−𝑉2,∥)as thevoltagedifferencebetweenthenormalandparallelconfig- uration of the magnetic field with respect to the direction of the electric current. In practice, the measurement is obtained simply by recording the change in voltage as an in-plane ex- ternal magnetic field, 𝐻0, is rotated along the 𝑥and𝑦direc- tions,respectively[theCartesianframeisdefinedinFig.1(a)]. Fig.4showsthevariationof V2asafunctionof 𝐻𝑥foralarge amplitude of |𝐼1|=2.0 mA, which corresponds to a current density of1⋅1012A/m2. To reduce the influence of Joule heating and also thermal activation of the electrical carriers in YIG[28, 29], we use a pulse method with a 10% duty cy- cle throughout this study to measure the nonlocal voltage[4].In the measurements, the current is injected into the device only during 10 ms pulses with a 10% duty cycle. In Fig. 4 we comparethemagneticfieldsensitivityofthe(normalized)spin transport at two values of the center-to-center distance 𝑑be- tweenemitterandcollectorforpositiveandnegativepolarities of the current. In total, this leads to 4 possible configurations for the pair ( 𝐼1,𝐻𝑥), each labeled by the symbols ◔, ◔,◔,◔ to match the notation of Fig. 5. There, vertical displacement ofthemarkerdissociatesscansofopposite 𝐻𝑥-polarity,while horizontaldisplacementofthemarkerdissociatesscansofop- posite𝐼1-polarity. Looking at Fig. 4, we recover the expected inversion symmetry while enhancement of the spin current is clearlyvisiblewhen 𝐼1⋅𝐻𝑥<0. Thesignalseemstodepend onthemagneticfieldonlyforlargerdistancesand 𝐼1⋅𝐻𝑥<0 (forward bias). Considering that the two Pt wires are both 𝑤= 0.3𝜇m wide, this corresponds to an edge to edge sep- aration𝑠=𝑑−𝑤. In one case the distance is 𝑠≈ (2𝜆𝑇), in the other case 𝑠≈ 4⋅(2𝜆𝑇), where2𝜆𝑇≈ 0.6𝜇m is the esti- mated amplitude decay length of the magnons at thermal en- ergy. Itwillbeshownbelowthatundertheemitterthenumber ofhigh-energymagnonsfarexceedsthenumberoflow-energy magnons. Assuming an exponential decay of the high-energy magnons, one expects in (a) an attenuation of their contribu- tionby50%,whilein(b)itisreducedbyalmost99%. Wethus arriveatasituationwhereat 𝑑=0.5𝜇mthemagnontransport is dominated by the behavior of high-energy magnons, while at𝑑= 2.3𝜇m the magnon transport is dominated by the be- havioroflow-energymagnons(seebelow). InFig.4weassign thefinitesusceptibility 𝜕𝐻𝑥V2<0asanindicationofmagnon transmission through low-energy magnons. Since the energy ofthesemagnonsaswellasthethresholdofdampingcompen- sation depend sensitively on the magnetic field[30, 31], the low-energy magnons are significantly affected by the ampli- tude of the magnetic field, 𝐻𝑥[4, 32, 33]. Such a field de- pendenceisexplainedinEq.(5)ofRef.[1]. Whatisobserved here is that near the peak bias, 𝐼pk≈ 2.2mA (see definition in part I), the device becomes particularly sensitive to a shift ofth. In our case, the external magnetic field shifts thby shifting the Kittel frequency, 𝜔𝐾=𝛾𝜇0√ 𝐻0(𝐻0+𝑀𝑠). In contrast, the constant behavior, 𝜕𝐻𝑥V2≈ 0, is indicative of a magnon transmission ratio by high-energy magnons: because of their short wavelength, their energy is of the order of the exchange energy, and thus independent of the magnetic field strength[34]. Since these 2 plots are measured with exactly the same current bias, and the only parameter changed is 𝑑, it showsthatfilteringbetweenhighandlow-energymagnonscan be achieved by simply changing the separation between emit- terandcollector. Italsodirectlysuggestsadoubleexponential decay, as will be discussed later in Fig. 8. B. Spectral signature in nonlocal measurement. Fig. 5 compares the variation of V2as a function of emitter current𝐼1for two different emitter-collector separations. The maximum current injected into the device is about 2.5 mA, corresponding to a current density of 1.2⋅1012A/m2. The polarity bias for the pair ( 𝐼1,𝐻𝑥) is represented by the sym-6 FIG.5. Measurementofthecollectedelectricalcurrent, 𝐼2,asafunc- tionoftheemittercurrent, 𝐼1. Wecomparethetransportcharacteris- ticsbetweentwononlocaldevices: onewithashortemitter-collector distance in the submicron range ( 𝑑= 0.5𝜇m, left column) and the other with a long distance of a few microns ( 𝑑= 2.3𝜇m, right col- umn). Thefirstrow(a)and(b)shows V2at𝑇0=300Kasafunction of𝐼1, the injected current, for both positive and negative polarity of 𝐻𝑥, the applied magnetic field. In our symbol notation, the marker position indicates the quadrant in the plot pattern. The raw signal V2= −𝑅2𝐼2+V2is decomposed into an electric signal, 𝐼2, and a thermal background signal, V2, as shown in the third row (e,f) and the second row (c,d), respectively. The background, V2, represents the background magnon currents along the thermal gradients. The measurementsareperformedonYIGCthinfilms. Thedataaretaken at𝐻0=0.2T. bols◔, ◔,◔,◔, in replication of the 4-curve pattern. We recover in Fig. 5(a,b) the expected inversion symmetry with V◔ 2≈ −V ◔ 2andV ◔ 2≈ −V◔ 2, while the enhancement of the spin current is visible when 𝐼1⋅𝐻𝑥<0, representing the forward regime. As explained in part I [1], the raw sig- nalV2=V2−𝑅2𝐼2can be decomposed into i)V2|||𝐼2 1a ther- malsignalproducedbytheSpinSeebeckEffect(SSE),which is always odd/even with 𝐻𝑥or𝐼1and shown in panels (c,d), andii)−𝑅2𝐼2||𝐼1, an electrical signal produced by the spin transfer effect (STE), which is in the linear regime even/odd withthepolarityof 𝐻𝑥or𝐼1,respectively,andshowninpan- els (e,f) [35]. This decomposition is obtained by assuming that in reverse bias V◔ 2= −V ◔ 2+𝑅2T𝑠||𝐼1→0T𝑇⋅𝐼1and V ◔ 2= −V◔ 2+𝑅2T𝑠||𝐼1→0T𝑇⋅𝐼1, which evaluates thenumber of absorbed magnons as a linear deviation from the number of thermally excited low-energy magnons, assuming C2continuityofthemagnontransmissionratioacrosstheori- gin. We recall that in our notation T𝑇≡T𝑇∕T𝑇||𝐼1→0. We then construct V ◔ 2=V◔ 2andV◔ 2=V ◔ 2by enforcing that thesignalgeneratedbyJouleheatingisexactlyevenin 𝐼1. We observe that in the short range ( 𝑑= 0.5𝜇m), we get V◓ 2≈ (V ◔ 2+V◔ 2)∕2and𝐼◓ 2=sign(𝐼1)(V ◔ 2−V◔ 2)∕(2𝑅2), which istheexpectedsignatureforasymmetricmagnonsignal. This equality is not satisfied in the long range ( 𝑑= 2.3𝜇m) for V2due to the asymmetry of the signal between forward and reverse bias as explained in part I. The consistency of this data manipulation is confirmed below in Fig. 6(a) and (b) by showing a small asymmetric enhancement of 𝐼2at high𝐼1 by low-energy magnons at short distances and a pronounced enhancement at long distances as discussed in Ref. [1]. The factthatamorepronouncedenhancementisobservedatlarge distances is further evidence for the spatial filtering of high- energy magnons. It is worth noting that one can reach a situation where −𝑅2𝐼2= 0without necessarily having V2vanish as well, as shown in Fig. 5(e) and (f) at 𝐼1= 2.5mA. This is explained bytheformationoflateraltemperaturegradients[36]. Inother words, the observation of 𝑀𝑇=0is a local problem, mostly affecting the region below the emitter. It does not imply that 𝑀=0throughout the thin film. As a next step, we will show how to distinguish the con- tributions of high-energy and low-energy magnons using the analytical model in Fig. 6. Starting from Fig. 5(e,f), we will removetheinfluenceofthespuriouscontributionontheelec- trical spin transport signal. First, we normalize the signal by theemittercurrenttoobtainthemagnontransmissionratioco- efficient T𝑠=𝐼2∕𝐼1as shown in Fig. 6(a,b). For small sepa- ration, we observe that T𝑠shows a quadratic behavior that is symmetricincurrentandconsequentlyweassociateitwiththe device temperature. In contrast, the device with large separa- tion shows an asymmetric enhancement due to the spin diode effect [1]. The influence of the increase of the emitter tem- perature𝑇1due to the Joule heating of 𝐼1can be removed by normalizing with 𝑇1∕𝑇0. This normalization removes the symmetric enhancement of the magnon transmission ratio as reported in previous studies[3, 29, 37, 38], where the justi- fication will be discussed later in Fig. 7 [39]. The obtained traces are shown in Fig. 6(c,d) and can be compared with the theoretical expectation given by Eq. (4), which is graphically summarized in Fig. 2(e). The solid lines are fit curves with ourmodelrepresentingthesumofthecontributionfromlow- energymagnonsandthebackgroundcontributionsfromhigh- energymagnons,withtheparametersofthefitgiveninTableI. The dashed line and the gray shaded area represent the latter Σ𝑇Δ𝑛𝑇. Fromthefitswecanobtaintheratio Σ𝑇∕(Σ𝑇+Σ𝐾)for thetwomagnonfluids,wherethecontributionofhigh-energy magnons decreases from 95% at 0.5 𝜇m to 50% at 2.3 𝜇m, in accordance with the spatial filtering proposed above. To illustrate Eq. (3) experimentally, we repeated the mea- surement for different values of the substrate temperature 𝑇07 FIG. 6. Dependence of the magnon transmission ratio on the sepa- ration between the electrodes. Starting from the extraction of 𝐼2in Fig. 5, the first row compares the variation of the ratio T𝑠=𝐼2∕𝐼1 betweenshort-range(leftcolumn)andlong-range(rightcolumn)de- vices. In the short range, the behavior shows a symmetrical signal of the magnon transmission ratio with respect to the current polar- ity𝐼1, while in the long range, the behavior is asymmetrical. We interpret the difference to be due to two different types of magnons: dominantly high-energy magnons in the short range and dominantly low-energymagnonsinthelongrange. Toeliminatenonlineardistor- tionscausedbyJouleheating, T𝑠isrenormalizedby 𝑇1||𝐼2 1,theemit- ter temperature variation produced by Joule heating (see text). The solidlinesarefittedwithEq.(4),wheretheshadedregionshowsthe background contribution from high-energy magnons Σ𝑇T𝑇, where Σ𝑇∕(Σ𝑇+Σ𝐾)||𝑑represent their relative weight at this distance. In (c) this ratio is about 0.95, while in (d) it drops to about 0.5. at small separation. Fig. 7(a) shows the experimental result for five different values of 𝑇0when𝐼1varies on the same [−2.5,2.5]mA span. Note that the data are plotted as a func- tion of𝑇1=𝑇0+𝜅𝐴𝑅Pt𝐼2 1, the emitter temperature. The ra- tionale for this transformation of the abscissa is apparent in Fig. 7(b) and (c), which show that the nonlinear current de- pendence of both the SSE and STE signals originates from the enhancement of 𝑇1. In particular, Fig. 7(c) shows the rise of the SSE signal V2as a function of 𝐼1for different values of𝑇0. We find that all curves almost overlap on the same parabola, suggesting an identical thermal gradient of the Pt1 electrode through 𝐼1independently of 𝑇0, with a small devia- tionforsmaller 𝑇0duetothedecreaseof 𝑅Pt. Inaddition,Fig. 7(d) shows(T𝑠)−1≡(T𝑠∕T𝑠|𝐼1→0)−1, the inverse transmis- sion ratio of the spin current generated by the STE normal- ized by its low current value[40]. The data from the different curvesoverlapand,similartotheSSE,showaparabolicevolu- tion(seedottedline). Thissuggeststhattheprimarysourceof the symmetric nonlinearity between 𝐼2and𝐼1is simply Joule heating. It therefore justifies the transformation of the current abscissa𝐼1into a temperature scale 𝑇1in Fig. 7(a). Focusing now on the remarkable features of Fig. 7(a), one could no-tice that the low current data taken at 𝑇0= 300K fall on a straight line intercepting the origin, as predicted by Eq. (1), which is𝐼2∕𝐼1∝𝑇1. Another notable feature, as previously reported[3, 38], is that the transmission ratio reaches a maxi- mum at high temperature. To support this picture with experimental data, we have plotted in the inset of Fig. 7 the behavior of 𝑀𝑇(𝑀0−𝑀𝑇) suggestedbyEq.(3). Thisshouldrepresentthemagnontrans- mission ratio by the high-energy fraction, i.e. the number of available high-energy magnons multiplied by the amount of spin polarization available in the film. We find that the ob- served variation of T𝑠with𝑇1follows the expected behavior derivedfromthesingletemperaturevariationofthetotalmag- netization shown in the inset Fig. 7(b). This provides exper- imental evidence that the short range behavior is dominated by high-energy magnons and that the density change follows the analytical expression in Eq. (3). Furthermore, it is con- firmed that the drop in the magnon transmission ratio above 440Kisassociatedwithadropinthesaturationmagnetization as one approaches 𝑇𝑐, precisely where high-energy magnons reachtheirmaximumoccupancy. Thedropsuggeststhathigh- energy magnons actually prevent STE spin transport. This is thenonlineardeviationexpectedforadiffusivegas: thehigher thenumberofparticles,themorethetransportisinhibited(see alsoRef.[1]). Whatitshowshereisthatthemagnontranscon- ductance is dominated by high-energy magnons around the emitter. This confirms the initial finding of Cornelissen et al.[2] whodrew thisconclusion based onthe similarityof the characteristic decay of SSE and STE as a function of 𝑑. C. Double decay of the magnon transmission ratio 1. Thin films with anisotropic demagnetizing effect Having established that the spin current is carried by the two-fluidsandthatthefitallowstoextracttherespectivecon- tributions of high and low-energy magnons, we took a series ofexperimentaldataof T𝑠⋅𝑇0∕𝑇1withdifferentseparations 𝑑 rangingfrom 0.5𝜇mto6.3𝜇m. TheresultsareshowninFig.8. We see directly in Fig. 8 that the decay length of the magnon transmission ratio at small 𝐼1is much shorter than the decay lengthofthemagnontransmissionratioatlarge 𝐼1(spindiode regime). Thisshowsexperimentallythateachofthetwo-fluids has a different decay length with 𝜆𝑇≪ 𝜆𝐾. These are ad- justed by varying Σ𝐾∕(Σ𝐾+Σ𝑇)||𝑑while keeping the other parametersinEq.(1-3). Thefitsareshownasthesolidlinein Fig. 8(a,c). The fit parameters are set according to the values given in Table I. Bymeansoftheanalysis,weobtainedtheamplitudeandthe fractionofhigh-energyvs. low-energymagnonsasafunction of𝑑,whicharesummarizedinpanel(b)andextractthetwode- caylengths𝜆𝐾=1.5𝜇mand𝜆𝑇=0.4𝜇m,respectively. This confirms the short-range nature of the high-energy magnons and the much longer range of the low-energy magnons. We note that since the shortest decay length is of the same order of magnitude as the spatial resolution of standard nanolithog- raphy techniques, the regime of magnon conservation could8 FIG. 7. Dependence of the magnon transmission ratio on the sub- strate temperature, 𝑇0. Short-range measurement ( 𝑑= 0.5𝜇m) of nonlocal spin transport in YIGA. (a) Variation of T𝑠as the emitter current𝐼1isvaried inthe range [−2.5,2.5]mAat differentvalues of the substrate temperature 𝑇0. The data are plotted as a function of 𝑇1=𝑇0+𝜅𝐴𝑅Pt𝐼2 1, the emitter temperature. The resulting temper- ature dependence of T𝑠observed in (a) corresponds to the variation of𝑀𝑇(𝑀0−𝑀𝑇)shown in inset (b), where 𝑀𝑇is the temperature dependence of the saturation magnetization. The dots are the exper- imental points, while the blue solid line is the expected behavior as- suming𝑀𝑇≈𝑀0√ 1−(𝑇∕𝑇𝑐)3∕2. Thistransformationissupported by the observation in (c) and (d) that both the SSE signal V2and the normalized inverse transmission ratio T𝑠vs.𝐼1scale on the same parabolicbehavior(dashedline),suggestingthattherelevantbiaspa- rameter is𝑇1. probably never be achieved in lateral devices. Note that there is the discrepancy that the vanishing of 𝐼2occurs slightly be- fore𝑇𝑐. Wewillshowthatthisoccurssystematicallyonallour samples (see subsection 3). The same analysis applied to the YIGfilmwithlargerthickness(panels(c,d))revealsanidenti- calbehaviorofthehigh-energymagnons,whereasthedecayof thelow-energymagnonsisslightlyslowerwith 𝜆𝐾=1.9𝜇m. We do not see an obvious increase in the transmission ra- tio in thinner films (YIG𝐴), although Eq. (5) of Ref. [1] pre- dicts inverse proportionality as previously observed experi- mentally[41],whichcanbeattributedtothedifferenceinma- terial quality. Nevertheless, an interesting feature observed whencomparingFig.8(a)and(c)isthattheratiooflow-energy magnons to high-energy magnons increases with decreasing filmthickness. Thiscanbeattributedtoanincreaseinthecut- offwavevector,wherethemagnonsbehavetwo-dimensionally, andthusthespectralrange,wherethedensityofstateremains constant, which favors the exposure of the increasing occu- pancyoflow-energymagnons. Thelongerdecaylengthinthe thickerfilmisalsoconsistentwiththelongerpropagationdis- tanceexpectedforballisticlow-energymagnons,whoseprop- agation range is determined by the film thickness. However, theenhancementisnotproportionaltothethickness,suggest- ing that some other undefined process is also involved in this decay. We emphasize that the shape of the decay observed in Fig. 8(b) and (d) corresponds to a double exponential decay with two different decay lengths in unprocessed data. This reinterprets the double decay behavior reported in previous FIG.8. Doubleexponentialspatialdecayofthemagnontransmission ratio. (a,c) Current dependence of the magnon transmission ratio for (a) the 19 nm thick YIG𝐴and (c) the 56 nm thick YIG𝐶thin films. The solid lines are a fit by Eq. (4), where the only variable parame- ter is the value of Σ𝐾∕(Σ𝐾+Σ𝑇)||𝑑. For the YIG𝐶sample, we have added in panel (c) the variation of the spin magnetoresistance (right axis), which corresponds to the conductivity at 𝑑=0. Spatial decay ofthemagnontransmissionratiofor(b)YIG𝐴and(d)YIG𝐶,respec- tively. In both cases, the decay of high-energy magnons follows an exponentialdecaywithcharacteristiclength 𝜆𝑇≈0.5±0.1𝜇m. The decayoflow-energymagnons,ontheotherhand,followsanexponen- tialdecaywithcharacteristiclength 𝜆𝐾=1.5𝜇mforthethinnerfilm (b)andanexponentialdecaywithcharacteristiclength 𝜆𝐾=1.9𝜇m for the thicker film (d). nonlocaltransportmeasurements[2,22,41,42]. Theinterpre- tationpresentedinthisworkisdifferentfromtheoneproposed byCornelissen etal.,whereitwasrelatedtotheboundarycon- dition of the diffusion problem[2]. We note that while chang- ing the current bias 𝐼1can affect the ratio between the two- fluids, it does not change the decay length, as shown by the purple lines in panel (b,d). This is consistent with the notion that the bias affects the mode occupation of the transported magnons but not their character. The obtained decay lengths areinroughagreementwiththeexpecteddecaylengthofthese two populations as discussed in Sect.III. Note also that the high- and low-energy magnon length scales appear to be sim- ilar to the energy and spin relaxation length scales observed in the Spin Seebeck effect as proposed by A. Prakash et al.9 TABLE I. Fitting parameters by Eq. (4). 𝑡YIG(nm) 𝑛sat𝑇⋆ 𝑐(K) th,0(mA) 𝜆𝐾(𝜇m)𝜆𝑇(𝜇m)Σ𝐿→𝑅 𝐾,0Σ𝐿→𝑅 𝑇,0 YIG𝐴 19 4 495 8 1.5 0.4 5 % 37% (Bi-)YIG𝐵 25 11 480 3 3.8 0.5 4 % 39% YIG𝐶 56 4 515 8 1.9 0.6 3 % 15% YIG𝐷 65 4 545 8 [43],thecorrelationbetweenthelengthscalesisacomplexis- suethatwarrantsamorerigoroustheoreticalinvestigation(see also conclusion below). We note that our value of 𝜆𝐾appears to be dependent on thicknessandanisotropy(seeTableI).Thiscontradictsthebe- havior observed for thicker films ( 𝑡YIG>200nm), where the value was reported to be independent of film thickness[41]. The latter observation may be consistent with the assignment of the dominant low-energy propagating magnons to the 𝐸𝐾mode(orangedotinFig.3). Webelievethatthegroupvelocity thereis weaklydependent on 𝑡YIG, atleastfor thickfilms(see discussion above). We should emphasize here that our report does not cover the same dynamic range as those reported in thickerfilms,duetothelowersignal-to-noiseratio. Itispossi- ble that a third exponential decay could appear at much lower signallevels. Apossibleexplanationforthelongrangebehav- iorcouldbethattheangularmomentumiscarriedbycircularly polarized phonons, which have been found to have very long characteristic decay lengths in the GHz range[44, 45]. Finally, it is useful to quantify the spin current emitted by the STE, as shown in panel (b,d). Renormalizing the trans- missionratiocoefficient T𝑠bytheproductofthespintransfer efficiency at both the emitter and collector interfaces, 𝜖1⋅𝜖2 (see Table 1 of Ref. [1]), we observe that only 10% of the generated magnons reach a collector placed at 𝑑= 0.2𝜇m away. This percentage increases to 15% by extrapolating the decay to𝑑=0, which is the proportion of itinerantmagnons among the total generated, and there are about an order of magnitude(×14)morehigh-energymagnonsthanlow-energy magnons below the emitter. Taking into account the fact that magnons can escape from both sides of the emitter, while we monitoronlyoneside,wecanestimatethat70%ofthegener- ated magnons remain localized. This localization is the con- sequence of three combined effects, which mainly affect the low-energy magnons: i)STE primarily favors an increase in densityatthebottomofthemagnonmanifold,whichhaszero group velocity ii)STE, as an interfacial process, efficiently couples to surface magnetostatic modes [46], The nonlinear frequency shift associated with the demagnetizing field [47] produces a band mismatch at high power between the region belowtheemitterandtheoutside,whichpreventsthepropaga- tionofmagnons(seepartI[1]). Thespatiallocalizationcould be induced either by the thermal profile of the Joule heating [13] or by the self-digging ball modes [18, 48, 49]. This ra- tional concerns mainly the magnons whose wavelengths are shorter than the width of the Pt electrode. Another confirmation is the variation of the ratio between low-energy magnons and high-energy magnons with the uni- axial anisotropy. When the latter compensates the out-of-plane depolarization field, we observe a suppression of the low-energy magnon confinement, and the transmitted signal atlargedistances(10 𝜇m)fullyreplicatesthevariationoflow- energy magnons under the emitter. 2. Thin films with isotropically compensated demagnetizing effect In this section, we will clarify the influence of self- localization on the saturation threshold 𝑛satthat we intro- duce in our analytical model. For this purpose, we have re- peated the experiment on a Bi-YIG𝐵sample. This mate- rial has a uniaxial anisotropy corresponding to the saturation magnetization (see Table 1 in Ref. [1]). As a consequence, the Kittel frequency follows the paramagnetic proportional- ity relation𝜔𝐾=𝛾𝐻0(similar to the response of a sphere), where the value of 𝜔𝐾is independent of 𝑀𝑇and the cone angle of precession, and therefore exhibits a vanishing non- linear frequency shift[32, 47, 50] (see further discussion in Ref.[1]). Werefertothisasanisotropicallycompensatedma- terial. We emphasize, however, that although the nonlinear frequency shift is zero, the system is still subject to satura- tion effects[10]. Compensation of the out-of-plane demagne- tization factor eliminates only the ellipticity of the trajectory caused by the finite thickness, but not the self-depolarization effect of the magnons on themselves. The latter depends on the angle between the propagation direction and the equilib- rium magnetization direction and is the origin of the magnon manifold broadening. As shown in Fig. 9(a), the nonlinear behavior of T𝑠ob- served in the Bi-YIG𝐵sample is qualitatively similar to that of YIG𝐶. Quantitatively, however, the magnitude of the spin diode effect is more pronounced in the former case. This is especially noticeable at long distances. Comparing Fig. 9(b) (𝑑=0.70𝜇m)withFig.9(c)( 𝑑=10.3𝜇m),fortheformerthe conductivitycanonlybeincreasedbyafactorof3withrespect to its initial value, while for the latter it can be increased by a factorof15. Thisisagainduetothefilteringoutoftheback- groundofhigh-energymagnons: inthecaseoflargedistances, the contribution of low-energy magnons is more pronounced. Recalling that in YIG𝐶the conductivity was enhanced by a factor of 7 by low-energy magnons (see 𝑑 >4.3𝜇m data in Fig. 8(c) or Fig. 7 of Ref. [1]), here a larger fitting parameter of𝑛sat=11isusedinBi-YIG𝐵while𝑛sat=4isusedinYIG𝐶, indicating a larger threshold for saturation. This is consistent withthesuppressionofthenonlinearfrequencyshiftaffecting thelongwavelengthspinwaveintheYIG𝐶sample. Thisresult suggeststhatremovingtheselfnonlinearityonthelongwave- length magnons improves the ability to generate more propa-10 FIG. 9. Two-fluid behavior in thin films with isotropically compen- sated demagnetization effect ( 𝑀eff= 0). (a) Variation of the spin diode signal T𝑠measured in BiYIG𝐵for different emitter-collector separations𝑑. The main panel (a) shows the normalized magnon transmission ratio as a function of 𝑇1, while the right panels show the corresponding current dependence for (b) 𝑑= 0.7𝜇mand (c) 𝑑=10.3𝜇m. ThesolidlinesarefitsbyEq.(4),withtheonlyvariable parameters,Σ𝐾andΣ𝑇, representing the fraction of low and high- energymagnons. (d)Spatialdecayofthetwo-fluidmodelseparating the contributions of high-energy and low-energy magnons. The ob- served decay can be explained by a short decay 𝜆𝑇≈0.5𝜇m of the high-energymagnoncontribution( 𝑘𝐵𝑇,blackline)andalongdecay 𝜆𝐾≈4.0𝜇mofthelow-energymagnoncontribution( ℏ𝜔𝐾,magenta andbluelines). Thedataat 𝐼1=1.3mAshowthedecaybehaviorin thecondensedregime. (e)Magneticfielddependenceofthenormal- ized magnon transmission ratio at different currents. gating magnons. It can also be understood as the removal of the self-digging process under the emitter in pure YIG sam- ples. The fit parameters are listed in Table I. Note that the discrepancybetween 𝑇𝑐and𝑇⋆ 𝑐,whichmarksthedropof T𝑠, isevenmorepronouncedinthissystem. Thedropoccurs70K below𝑇𝑐. We will return to this point in the last subsection. InFig.9(d)weplotthespatialdecayof T𝑠renormalizedby 𝜖2, obtained from fits with Eq. (4) in percent for high-energy magnonsinblack,low-energymagnonsat 𝐼1=0.4mA(𝜇𝑚≪ 𝐸𝑔) in blue, and 𝐼1=1.3mA (𝜇𝑚≈𝐸𝑔) in purple. The two decay lengths are 𝜆𝑇≈ 0.4𝜇m for high-energy magnons, in agreement with the results in YIG, and a much larger value of𝜆𝐾= 4𝜇m for low-energy magnons. The latter value is similar to the decay length of low-energy magnons observed FIG.10. Dependenceof 𝑇⋆ 𝑐onthethicknessofYIGfilms. Compar- ison of nonlocal devices with approximately the same ratio of high- energy magnons to low-energy magnons at 𝐼1→0. We observe an increasein𝑇𝑐−𝑇⋆ 𝑐withdecreasingfilmthickness,suggestinganin- creasing influence of low-energy magnons at high power 𝐼1→𝐼𝑐 with decreasing film thickness. by BLS in these films[50]. Moreover, it is in good agreement with the estimate made in Sect. III. Forthesakeofcompleteness,weplotthemagneticfieldde- pendencefordifferent 𝐼1inFig.9(e). Thedecreaseofthesig- nal at zero field is due to the residual out-of-plane anisotropy, which forces the magnetization to be along the film normal, resulting in no STE applied by Pt. The magnon transmission ratiobecomesmaximumnear0.05T,whichisthesaturationof the effective magnetization for BiYIG. The field dependence at a larger field than 0.05 T becomes significant for the cur- rent values near the appearance of the peak in (a) at 𝐼1=1.3 mA, where the conductivity of low-energy magnons reaches the highest. As noted in a previous study[4], the fact that we see a dependence with magnetic fields is direct evidence that wearedealingherewithlow-energymagnons. Heretheextra sensitivityof T𝑠tochangesin thnear𝐼pk,asdiscussedabove in the context of describing the behavior of Fig. 4, is clearly illustrated here with the BiYIG sample.11 3. Discrepancy between 𝑇𝑐and𝑇⋆ 𝑐 Finally, we discuss the disappearance of the magnon trans- mission ratio already at 𝑇⋆ 𝑐far below the experimentally de- termined𝑇𝑐(see Fig. S1). We note in Fig. 8 and Fig. 9 that all curves collapse at the same value independent of 𝑑. This clearly points to a problem that only concerns the region be- low the emitter, since there is a lateral temperature gradient. To this end, we summarize the normalized magnon transmis- sionratioforYIGsamplesasafunctionofemittertemperature 𝑇1inFig.10withdifferentthicknesses. Toavoidanyinfluence of thermal gradients, we have chosen devices whose spacing 𝑑leads to a similar ratio between Δ𝑛𝑇andΔ𝑛𝐾. This re- quires𝑑toincreasewithincreasingfilmthickness,suggesting a decreasing contribution of low-energy magnons. We spec- ulate that the collapse can be caused either by the onset of strong electron-magnon scattering as the YIG film becomes conducting[28, 29], or by a reversal of the equilibrium mag- netization below the emitter, which becomes aligned with the injectedspindirection[18,19]. Inthelattercase,themagneti- zation below the emitter and collector are opposite, suppress- ing any spin transport. This process is consistent with the as- sumption that a large fraction of the injected spins remain lo- calized. Thisprocessisalsoconsistentwiththedecreaseof V2 observedatlarge 𝐼1,wherenowtheelectriccurrentdecreases theeffectivetemperatureofthespinsystem(decreasefluctua- tions) despite the fact that 𝐼1⋅𝐻𝑥<0. Weexaminetheothercluesthatsupportthispicture. Ifone comparesthediscrepancybetween 𝑇⋆ 𝑐and𝑇𝑐betweenthedif- ferent samples, one can clearly see on the data in Fig. 10 that the discrepancy increases with decreasing film thickness, as expected for an increased surface effect of STE and reduced volume of polarized spins. Another indication is the fact that the largest discrepancy is observed on films with large uniax- ialanisotropy,asshowninFig.9(a). Thisisinagreementwith theobservationmadeonnano-devicesontheswitchingofthe magnetizationdirectionbythespinHalleffect[51]. Neverthe- less,thediscrepancydoesnotseemtoscalesimplywith 𝑡YIGin our observation, suggesting that there may be additional phe- nomena at play that are responsible for the vanishing magnon transmissionratioathightemperaturewhilethesystemisstill initsferromagneticphase(seealsothediscussionofFig.5of Ref. [1]). We have tentatively calculated 𝐼𝑓, the critical current re- quired to flip the magnetization. We call 𝑛sat=𝑉𝑀1∕(𝛾ℏ) thetotalnumberofspinsthatremainpolarizedundertheemit- ter. We compare this to the number of injected spins within the spin-lattice relaxation time, which is 𝐼𝑓𝜖∕(2𝑒𝛼LLG𝜔𝐾). Equalizing the two quantities, we find that 𝐼𝑓= 2.5mA for YIG𝐴samples. According to the upper scale of Fig. 8, 𝑇𝑐is reached when 𝐼= 2.7mA. Using Fig. S1, we can calculate thetemperaturedifferenceproducedbyJouleheatingbetween thesetwovalues,andtheresultisabout65K.Thisisveryclose to the shift of 50 K observed experimentally on this sample. Whilethereareindicationsthatashiftoccurs,andthenum- ber roughly matches the expected numbers, the above para- graphisstillratherspeculativeatthisstage,andadirectproof isstillmissing. Forthesakeofcompleteness,itisworthmen-tioningthattheremaybealternativeexplanations. Onepossi- bilityisadecreaseof 𝑇𝑐intheregionbelowthePt. Theorigin ofsuchaneffectcouldbeinterdiffusionofPtatomsinsidethe YIG at the interface. More thorough systematic studies will be required to clarify this point. V. CONCLUSION Through these two consecutive reviews, we present a com- prehensive picture of magnon transport in extended magnetic insulating films, covering a wide range of current and mag- netic field bias, substrate temperature, as well as nonlocal ge- ometrieswithvaryingpropagationdistance. Thepictureofthe two-fluid model expressed in this part II, complemented by a picture of the nonlinear behavior of the low-energy magnon expressed in part I, is formulated analytically and it is sup- ported by a series of different experiments that include non- local transport on different thicknesses YIG thin films with different garnet composition, different interfacial efficiency, as well as different thermalization. While providing a com- prehensive study of these materials, our model accounts for almost all the experimental observations within this common framework. What the analytical model allows to do is: i) to describe the expected signal in the linear regime [Eq. (6) in part I] ii) to fit the nonlocal transport data well on the whole cur- rentrangeandfordifferentseparationbetweentheelec- trodes using very few parameters ( th,0,𝑛sat,𝑇⋆ 𝑐,𝜆𝑇, 𝜆𝐾,Σ𝑇andΣ𝐾) iii) to incorporate all relevant physical effects: effect of Joule heating on 𝑀1, divergent form of magnon- magnon relaxation. What it doesn’t do, but could be important: i) to take into account the propagation properties (propa- gationangle,groupvelocity,modeselectionbytheelec- trodegeometry,spatialvariationofthesepropertiesdue tothetemperaturegradient)ofthemagnonsexcitedun- dertheemittertoknowhowtheycontributetothesignal under the collector. ii) to take into account nonlinear magnon localization ef- fects under the emitter (for YIG in particular). iii) to take into account the effects of high power (change in temperature or change in low energy magnon occu- pancy) on damping, exchange constant (and thus group velocity), pumping, and detection efficiency. Thefactthatthesepointsarenotdirectlyconsideredandthat the fits are excellent means that these effects are effectively used in the other components of the model. In particular, Eq. (6) of the relaxation in part I is very general and can absorb many different physical effects, hence the effectiveness of the model.12 In this paper, we assume that low-energy magnons propa- gating in the ballistic regime lead to a magnon transconduc- tancethatfollowsanexponentialspatialdecayinthinfilmge- ometries. This argument follows from the experimental find- ingthatinallBLSexperimentsmonitoringthelow-energypart ofthemagnonmanifold,theamplitudeofthesignalfollowsan exponentialdecay. Nevertheless,thetransportbehaviorinthe cleanlimit,wherethemagnonmeanfreepathislargerthanthe sampleboundary,isinitselfaveryinterestinglineofresearch. Another open question concerns the premature collapse of thesignalat𝑇⋆ 𝑐. Wehavetentativelyexplainedthisasapoten- tialswitchingofthemagnetizationdirectionbelowtheemitter. However, direct evidence for such a process remains elusive. We think that spin transport in materials with low magnetiza- tionorclosetotheparamagneticphasearebothveryinterest- ing topics. Finally, we summarize the main result of our two-fluid model, which separates the low-energy magnons from the high-energy ones. This allows us to propose an alternative explanation for the measured variation of the magnon trans- mission ratio with distance, due to a double exponential de- cay. Each of the fluids has its own transport characteristics, which are expressed by two different propagation lengths. A decay length in the submicron range is assigned to the high- energy magnon and a decay length above the micron range is assigned to the low-energy magnon. This explanation im- plies that even in the short-range regime, the magnon number isnotaconservedquantity,andthusanyanalogytoelectronic transport should take this rapid decay into account. Despite the fact that the model includes several parameters, there are still open questions. The similarity of the decay of SSE and STE currents with 𝑑must be reconciled with our results. A possible reason is that low-energy magnons participate in the SSE transport in the long range[52]. Although the amount of quanta carried is clearly 𝐸𝑇∕𝐸𝐾∼ 103against the latter, we shouldkeepinmindthatwearedealingwithatinysignal. The roleofacousticphonons[44,45]inthisprocessisstillunclear. Recentexperimentshaveshownthattheyarestronglycoupled to low-energy magnons and also benefit from a very low de- cay length. Of particular interest is the contribution of circu- larly polarized acoustic phonons, which have been shown to be strongly coupled to long-wavelength spin waves while al- lowing angular momentum transfer over large distances. ACKNOWLEDGMENTS This work was partially supported by the French Grants ANR-18-CE24-0021 Maestro and ANR-21-CE24-0031 Harmony; the EU-project H2020-2020-FETOPEN k- NET-899646; the EU-project HORIZON-EIC-2021- PATHFINDEROPEN PALANTIRI-101046630. K.A. acknowledges support from the National Research Founda- tion of Korea (NRF) grant (No. 2021R1C1C201226911) fundedbytheKoreangovernment(MSIT).Thisworkwasalso supported in part by the Deutsche ForschungsGemeinschaft (Project number 416727653). FIG. S1. Characterization of garnet thin films. The left column (a,b,c) shows the variation of the Pt resistance as a function of the injected current for YIG𝐴, (Bi-)YIG𝐵and YIG𝐶without and with Al coating, respectively (see Table 1 of Ref. [1]). The right ordinate allows to convert the current bias into a temperature increase in the range [300,600] K due to Joule heating. The upper abscissa gives the corresponding current density in Pt. The right column (d,e,f) showsthecorrespondingvariationofthesaturationmagnetizationin the [300,600] K range. VI. ANNEX A. Sample characterisation The4magneticgarnetfilms(seeTableI)usedinthisstudy havebeengrownby2differentmethods: liquidphaseepitaxy inthecaseofYIG𝐴,𝐶,𝐷andpulsedlaserdepositioninthecase of (Bi-)YIG𝐵. Their macroscopic magnetic properties have beencharacterizedusingacommercialvibratingsamplemag- netometer,wherethesampletemperaturecanbecontrolledby a flow of argon gas from room temperature to 1200K. Curves of magnetization versus temperature in the range of 300K to 600KareshowninFig.S1(d-f). Theyhighlightthevalueofthe Curietemperature( 𝑇𝑐)foreachsamplesummarizedinTable1 in Ref. [1]. Similarly, the Pt metal for the middle electrode was deposited by 2 different techniques: e-beam evaporation13 in the case of YIG𝐴and YIG𝐶and sputtering in the case of (Bi-)YIG𝐵. In this work we convert the Joule heating associated with the circulation of an electric current 𝐼1in the emitter into a temperature increase, which we plot on the abscissa of Fig. 7, Fig.10andFig.6,Fig.7ofRef.[1]. Thisisdonebycalibrating 𝑅Pt||𝐼1: the variation of the resistance Pt1with the injectedelectric current 𝐼1. We introduce the calibration factor 𝜅𝐴,𝐵or𝐶=𝜅Pt𝑅Pt∕𝑅0−1 𝑅Pt𝐼2 1, (5) for the conversion coefficient, with 𝑅0≡𝑅Pt||𝐼1=0= 𝜌Pt𝐿Pt∕(𝑤1𝑡Pt)is the nominal value of the Pt wire resistance andthecoefficient 𝜅Pt=𝑅Pt∕𝜕𝑇𝑅Ptisobtainedbymonitoring the variation of the Pt resistance at low current vs. substrate temperature. The obtained values of 𝜅Ptand𝜌Ptare given in Table 1 in Ref. [1]. Fig. S1(a-c) shows the R-I curves with correspondingtemperatureconsideringJouleheatingforeach sample. [1] R. Kohno, N. Thiery, E. Clot, R. Schlitz, K. An, V. V. Nale- tov,L.Vila,J.BenYoussef,H.Merbouche,V.Cros,N.Anane, T.Hauet,V.E.Demidov,S.O.Demokritov,G.deLoubens,and O. Klein, Physical Review B. 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2022-10-15

This review presents a comprehensive study of the spatial dispersion of propagating magnons electrically emitted in extended yttrium-iron garnet (YIG) films by the spin transfer effects across a YIG$\vert$Pt interface. Our goal is to provide a generic framework to describe the magnon transconductance inside magnetic films. We experimentally elucidate the relevant spectral contributions by studying the lateral decay of the magnon signal. While most of the injected magnons do not reach the collector, the propagating magnons can be split into two-fluids: \textit{i)} a large fraction of high-energy magnons carrying energy of about $k_B T_0$, where $T_0$ is the lattice temperature, with a characteristic decay length in the sub-micrometer range, and \textit{ii)} a small fraction of low-energy magnons, which are particles carrying energy of about $\hbar \omega_K$, where $\omega_K/(2 \pi)$ is the Kittel frequency, with a characteristic decay length in the micrometer range. Taking advantage of their different physical properties, the low-energy magnons can become the dominant fluid \textit{i)} at large spin transfer rates for the bias causing the emission of magnons, \textit{ii)} at large distance from the emitter, \textit{iii)} at small film thickness, or \textit{iv)} for reduced band mismatch between the YIG below the emitter and the bulk due to variation of the magnon concentration. This broader picture complements part I \cite{kohno_SD}, which focuses solely on the nonlinear transport properties of low-energy magnons.

Non-local magnon transconductance in extended magnetic insulating films.\\Part II: two-fluid behavior

2210.08283v2

Non-equilibrium thermodynamics of the spin Seebeck and spin Peltier eects Vittorio Basso, Elena Ferraro, Alessandro Magni, Alessandro Sola, Michaela Kuepferling and Massimo Pasquale Istituto Nazionale di Ricerca Metrologica, Strada delle Cacce 91, 10135 Torino, Italy (Dated: October 4, 2018) We study the problem of magnetization and heat currents and their associated thermodynamic forces in a magnetic system by focusing on the magnetization transport in ferromagnetic insulators like YIG. The resulting theory is applied to the longitudinal spin Seebeck and spin Peltier eects. By focusing on the specic geometry with one YIG layer and one Pt layer, we obtain the optimal conditions for generating large magnetization currents into Pt or large temperature eects in YIG. The theoretical predictions are compared with experiments from the literature permitting to derive the values of the thermomagnetic coecients of YIG: the magnetization diusion length lM0:4m and the absolute thermomagnetic power coecient M102TK1. PACS numbers: 75.76.+j, 85.75.-d, 05.70.Ln I. INTRODUCTION The recent discovery of the longitudinal spin Seebeck eect in ferromagnetic insulators has raised a renewed in- terest in the non equilibrium thermodynamics of spin or magnetization currents1. Experiments have shown that a temperature gradient applied across an electrically in- sulating magnetic material is able to inject a spin current into an adjacent metal, where the spin polarization is re- vealed by means of the inverse spin Hall eect (ISHE)2,3. Typical experiments have been performed by using ferri- magnets, like the yttrium iron garnet (Y 3Fe5O12, YIG) as insulating magnetic material and Pt or other noble metals, as conductors2,3. In analogy to thermoelectrics, the reciprocal of the spin Seebeck eect has been called spin Peltier eect4. This reciprocal eect has been re- cently observed by using the spin Hall eect of Pt as spin current injector and observing the thermal eects on YIG4. All these experiments show that the magneti- zation current can propagate along dierent media using dierent type of carriers. While spin currents in metals are associated to the unbalance in the spin polarization of conduction electrons, in magnetic insulators the mag- netization transport is due to spin waves or magnons5. Spin Seebeck and spin Peltier experiments reveal that the magnetization current carried by magnons in the mag- netic insulator can be transformed into a spin current carried by electrons and viceversa. The mechanism of this conversion is seen as the interfacial s-d coupling be- tween the localized magnetic moment of the ferromagnet (which is often due to d shell electrons) and the con- duction electrons of the metal (which are often s shell electrons)6{8. The thermodynamics of thermo-magneto-electric ef- fects, i.e. spin caloritronics, has been already developed for metals by adding the spin degree of freedom to the thermo-electricity theory9,10. However, spin caloritronics cannot be directly applied to electrical insulating mag- netic materials like YIG. Therefore it is necessary to develop a more general theory which could be applied to both conductors and insulators. The formulations of the problem present in the literature often focus onthe microscopic origin without paying much attention to the formal thermodynamic theory that is expected as a result. Refs.5,11{14describe the non equilibrium magnon distribution through an eective magnon tem- perature dierent from the lattice temperature. How- ever from an experimental point of view in Ref.15it was observed a close correspondence between the spatial de- pendencies of the exchange magnon and phonon temper- atures. The Boltzmann approach for magnon transport was used in Ref.8,16{19, combined by a YIG/Pt interface coupling7,8. Within these approaches the spin accumu- lation and the magnon accumulation take the role of an eective force able to drive the magnetization current. The use of dierent quantities between the two sides of a junction requires therefore the introduction of a spin convertance to account for the magnon current induced by spin accumulation and the spin current created by magnon accumulation20. The aim of the present paper is to dene the macro- scopic non-equilibrium thermodynamics picture for the problems related to magnetization currents that could be used independently of the specic magnetic moment carrier. To this aim we start from the results of the ther- modynamic theory of Johnson and Silsbee9. The main dierence with respect to the classical theories of the thermoelectric eects is that the magnetization current densityjMis not continuous. The magnetic moment can both ow through a magnetization current but also can be locally absorbed and generated by sinks and sources. Here, by limiting the analysis to the scalar case, we state the simplest possible continuity equation for the magne- tization. As a result we nd that the potential for the magnetization current is the dierence H=HHeqbe- tween the magnetic eld Hand the equilibrium eld Heq. The gradient of the potential rHis the thermodynamic force to be associated to the magnetization current. With this denition it is then possible to state the con- stitutive equations for the joint magnetization and heat transport and to identify the absolute thermomagnetic power coecient Mrelating the gradient of the poten- tial of the magnetization current 0rHwith the tem- perature gradient rT, in analogy with thermoelectricity.arXiv:1512.08890v2 [cond-mat.mtrl-sci] 5 May 20162 The same coecient also determines the spin Peltier heat currentMTjMwhen the system is subjected to a mag- netization current. In the present work we apply the previous arguments to describe the generation of a magnetization current by the spin Seebeck eect and the heat transport caused by the spin Peltier eect. To this end we have to complement the constitutive equations for the thermo-magnetic active material (YIG) with the equations for the spin Hall active layer (Pt). Once the equations for the two materials are written by using the same thermodynamic formalism, one can apply the theory to solve specic problems of magne- tization current traversing dierent layers. The diusion length for the magnetization current lM= (0MM)1=2 is related to intrinsic properties of each material: the magnetization conductivity Mand the time constant M, describing how fast the system is able to absorb the magnetic moment in excess. We are also able to show that the passage of the magnetization current from one layer to the other is governed by the ratio between lM=M of the two layers. By focusing on the specic geometry with one YIG layer and one Pt layer, we obtain the optimal conditions for generating large magnetization currents into Pt in the case of the spin Seebeck eect and for generating large heat current in YIG in the case of spin Peltier eects. In both cases we nd that ecient injection is obtained when the thickness of the injecting layer is larger than the critical thickness lMas recently experiments conrm21. We nally determine the values of the thermomagnetic coecients of YIG by comparing the theory to recent experiments4,22. The paper is organized as follows. In Section II we rst discuss the thermodynamic properties of an out-of- equilibrium but spatially uniform magnetic system23and on that basis we introduce, for non spatially uniform sys- tem, the currents and the thermodynamic forces in anal- ogy with the non equilibrium thermodynamics of ther- moelectric eects24. In Section III we set the constitu- tive equations for the magnetization and heat transport in both an insulating ferrimagnet and a metal with the spin Hall eect. Section IV is devoted to the solutions of the magnetization current problem. In Section V we focus on the specic longitudinal spin Seebeck geometry and on the spin Peltier eect. Finally some conclusive remarks are drawn in Section VI. II. THERMODYNAMICS OF MAGNETIZATION CURRENTS A. Thermodynamics of uniform magnetic systems We consider a magnetic system that can be described by a scalar magnetization M. Suitable systems can be ferromagnetic or ferrimagnetic materials where an easy axis is present, due for example to an anisotropic crys- tal structure, along which all the vector quantities arelying. We take spatially uniform quantities and all ex- tensive quantities as volume densities. The derivative of the internal energy density u(s;M) with respect to the magnetization at constant entropy density s, gives the equilibrium state equation Heq=1 0@u @M s(1) where0is the magnetic permeability of vacuum. In equilibrium the magnetic eld His equal to the state equationH=Heq(M;s). WhenHis dierent from its equilibrium value Heqthe system state will try to reach the equilibrium by the action of dissipative processes. In a generic out-of-equilibrium situation the variation of in- ternal energy must take into account that dissipative pro- cesses correspond to an entropy production. The energy balance then reads du=Tds+0HdMTsdt (2) whereT=@u=@s is the temperature, 0HdM is the in- nitesimal work done on the system, sis the entropy production rate, which has to be a denite positive term andtis time. When approaching equilibrium, the mag- netizationMwill change until the equilibrium condition H=Heq(M) will be reached. The typical situation is sketched in Fig.1 showing two processes connecting the equilibrium states (1) and (2). The equilibrium path (solid line) corresponds to the slow variation of eld H fromH1toH2through the equilibrium state equation H=Heq(M). The out-of-equilibrium path (dashed line) passes through the out-of-equilibrium state (10) and cor- responds to the sudden variation of the eld from H1to H2and to the subsequent time relaxation. As the initial and nal states are always equilibrium states, the nal internal energy variation must be the same for any pro- cess. This is obtained by assuming that the part of the work going into the internal energy is always the equi- librium one. Then, by inserting du=0HeqdM(from Eq.(1)) into Eq.(2) at constant entropy ( ds= 0) we nd the expression for the entropy production rate s=0HHeq TdM dt: (3) As expected, the entropy production rate is the product of a generalized force, or anity, represented by the term 0(HHeq)=T, times a generalized ux, or velocity, rep- resented by dM=dt24. If the distance from equilibrium is not too large one can consider the linear system approx- imation and assume the velocity to be proportional to the anity. It is appropriate to describe this fact by in- troducing a typical time constant Mfor the process by dening dM dt=HHeq M; (4)3 where the temperature Tand0appearing in the gener- alized force, have been incorporated into the denition of the time constant. Eq.(4) provides a kinetic equation for the magnetization describing the time relaxation from a generic out-of-equilibrium state by showing that the ve- locitydM=dt depends on the distance from equilibrium HHeq(see Fig.1). H MHeq(M)H-HeqH2M1H1(1)(1')Δu(2)M2 FIG. 1. Equilibrium path (solid line) and out-of-equilibrium path (dashed line) connecting the equilibrium states (1) and (2) in the HversusMdiagram of a magnetic material. Heq(M) is the equilibrium state equation at constant en- tropy. (10) is an out-of-equilibrium state obtained by the sudden change of the eld from H1toH2. In the relaxation path from (10) to (2) the work is 0HdM , the internal en- ergy change is du=0HeqdMand the entropy production is Tsdt=0(HHeq)dM. The relaxation equation is Eq.(4) The interesting physics behind Eq.(4) is that it also expresses the non conservation of the magnetic moment with the presence of sources and sinks, although the total angular momentum for an isolated system is conserved. As a matter of fact in the solid state there is a huge reservoir of angular momentum available (electrons, nu- clei, etc) and only a very small part of it is associated to the magnetic moment. As a result, the magnetization can be easily varied by exchanging angular momentum with the reservoir constituted by non magnetic degrees of freedoms. With this in mind, the physical meaning of Eq.(4) is to express how fast the angular momentum from the magnetization subsystem can be exchanged with the reservoir. Finally, as it happens in many problems involving a non conserved magnetization, also the internal energy is a non conserved quantity. To avoid the problem, we pass to the enthalpy potential ue=u0HM which contains the magnetic eld Hand the entropy sas independent variables. Dealing with out-of-equilibrium processes, the potentialueis also a non equilibrium one which depends on the magnetization Mas an internal variable. From Eq.(2), the enthalpy variation is due=Tds0MdH0(HHeq)dM (5) where we have used the denition of the entropy produc- tion of Eq.(3). The expression for the variation of the en- thalpy potential (5), together with the kinetic equation(4), constitutes the out-of-equilibrium thermodynamics of the system23and can be employed to build up the thermodynamics of uxes and forces. B. Thermodynamics of uxes and forces We now pass from the out-of-equilibrium thermody- namics of a spatially uniform magnetic system to the problem of having a non uniform situation involving cur- rents of the extensive variables, entropy and magnetiza- tion, and the associated thermodynamic forces24. Both the extensive and intensive variable are now allowed to vary as a function of space coordinates r. In the case of extensive variables the volume densities are intended as moving averages over a small volume
Varound the point r. As the magnetization is a non conserved quan- tity, we need to explicitly express the fact that any mag- netization change dMis in part drawn from the reser- voir of angular momentum, which is external to the ther- modynamic system, and in part exchanged between the surrounding regions of the thermodynamic system itself, giving rise to a current of magnetic moment jM. The sources and sinks of the magnetic moment are exactly those described in the previous Section by Eq.(4), then we can immediately write a continuity equation for the magnetization by extending Eq.(4), obtaining @M @t+r
jM=HHeq(M) M: (6) Next, as it is usually done in the non equilibrium theory of uxes and forces24, we use Eq.(5) to pass to the entropy representation by writing the entropy variation ds=1 Tdue+0M TdH+0(HHeq) TdM: (7) As we aim to dene the entropy current as a function of the other currents, we have to look at the previous equa- tion in search for the variations of the extensive variables. Eq.(7) contains the variation of the enthalpy dueand the magnetization dMwhich both have associated currents, while the variation of the magnetic eld dHdoes not corresponds to any current and has not to be taken into account in the denition of the entropy current. Then we dene js=1 Tjue+0(HHeq) TjM (8) where jsis the entropy current and jueis the enthalpy current which obeys the following continuity equation @ue @t+r
jue=0M@H @t; (9)4 from which one notices that the enthalpy is conserved if the eldHis constant in time. The continuity equation for the magnetization is Eq.(6) and nally the entropy obeys the continuity equation @s @t+r
js=s: (10) As it is done in the classical treatment24,25, one expresses the entropy production rate sin terms of a sum of prod- ucts of each current times its thermodynamic force. By using Eqs.(7)-(9) into Eq.(10) and introducing the heat current as jq=Tjs, after a few passages one obtains s=r1 T
jq+FM
jM+1 M0(HHeq)2 T(11) where we have dened the thermodynamic force associ- ated to the magnetization current FM=1 T0r(HHeq): (12) In Eq.(11) we see the products of the heat current jq times its forcer(1=T), of the magnetization current jMtimes its forceFMand the last term which is ex- actly the entropy production associated with the out-of- equilibrium homogeneous processes and not to the uxes. The last term can be also recognized as entropy produc- tion of Eq.(3) where the anity is 0(HHeq)=Tand the magnetization change dM=dt is (HHeq)=Mas given by Eq.(4). As a main result we have found that the gradient of the distance from equilibrium Eq.(12) is the generalized force associated with the magnetization current jM. For simplicity we dene H=HHeqto specify the dis- tance from equilibrium and we observe that the driving force of the magnetization current appears as soon as the system is brought out-of-equilibrium. In that case the system may nd more eective to draw magnetiza- tion from the surroundings rather than from the local spin reservoir. The strength of this eect is given by a further parameter, the magnetization conductivity M, which establishes the relationship between the magneti- zation current jMand the gradient of H jM=M0rH: (13) Hcan be dierent from zero in stationary situation ev- ery time the material experiences the accumulation of magnetization (i.e. spin accumulation in the case of metallic conductors). We have to notice that even if H has the units of a magnetic eld, it is not a magnetic eld in the sense of the Maxwell equations of electro- magnetism. Its status is analogous to the exchange eld or the anisotropy eld of ferromagnets whose origins is inthe quantum mechanics of the solid. Hrepresents the thermodynamic reaction of the system for nding itself in an out-of-equilibrium situation. In the following we refer toHas the potential for the magnetization current. III. CONSTITUTIVE EQUATIONS Having dened the potential Hassociated with the magnetization current, we are ready to write the consti- tutive equations for the two materials of interest for the spin Seebeck and spin Peltier eects: a magnetic insu- lating material with a spin Seebeck eect and a metallic conductor with the spin Hall eect. A. Thermomagnetic eects in magnetic insulators In analogy with the thermoelectric eects24, we can write the constitutive equation for the joint transport of magnetization and heat by using the potential associated with the magnetization current derived in the previous Section. The general case which includes the presence of electric current is reported in Appendix A. Here we limit to insulators and we take currents and forces in one dimension (rx=@=@x ). The equations for the thermo- magnetic eect reads jM=M0rxHMMrxT (14) jq=MMT0rxH(+2 MMT)rxT (15) whereMis the spin conductivity, Mis absolute ther- momagnetic power coecient, jqis the heat current den- sity andis the thermal conductivity under zero mag- netization current. Since the magnetization is not con- served, the magnetization current is not continuous and we have always to add the continuity equation (6). In non-equilibrium stationary states we always ask the con- dition@M=@t = 0 to be true, so Eq.(6) becomes rxjM=H M: (16) 1. Uniform temperature gradient If we disregard for the moment the heat currents, the solution of magnetization current problems will cor- respond to nd solutions to the system composed by Eqs.(14) and (16). Under a uniform temperature gra- dient, whererTis a constant, the second term at the right hand side of Eq.(14) is just a magnetization current density source jMS=MMrxT. Then the solution of jM=jMS+M0rxH(17)5 together with Eq.(16), considering constant coecients, leads to a dierential equation for the potential l2 Mr2 xH=H(18) where lM= (0MM)1=2(19) is a material dependent diusion length. The dierential equation (18) has general solutions in the form H(x) =H exp(x=lM) +H +exp(x=lM) (20) whereH andH +are coecients to be determined on the base of the boundary conditions. By looking at Eqs.(14) and (16) we have that if the conduction process is present in dierent materials, the solution is made by taking Eq.(20) for each material and nally joining the solutiosn by requesting the continuity in both jMand H. 2. Adiabatic conditions When the temperature is not externally controlled, we have to formulate the thermal problem by writing the heat diusion equation. To this aim we need to write the continuity equations for the entropy. In stationary conditions Eq.(10) becomes rxjs=swhere the term at the left hand side is written by using js=jq=Tand Eq.(15) rewritten as jq=MTjMrxT (21) while the term at the right hand side is given by Eq.(11). After a few passages, we obtain rxjq=0rxHjM+0(H)2 M(22) where the terms at the right hand side are due to the en- ergy dissipation of the magnetization current and to the local damping, respectively. Both terms are quadratic in the force and the potential, therefore if we assume small currents and forces we are allowed to neglect them in a rst approximation. In this case we obtain the condi- tionrxjq= 0 which, in one dimension, corresponds to a constant heat ux traversing the material. Moreover we choose here to study the adiabatic condition corre- sponding to jq= 0 in which the two terms at the right hand side of Eq.(21), the spin Peltier term MTjMand the heat conduction caused by the temperature prole T(x), counterbalance each other, giving no net heat ow through the layer. The prole T(x) will be stable if tem- perature of the thermal baths at the boundaries of thematerial are let free to adapt at the temperatures of the two ends. By using the adiabatic condition jq= 0 in Eq.(15) we immediately obtain rxT=1 ^M0rxH(23) where ^Mis the thermomagnetic power coecient in adi- abatic conditions 1 ^M=1 MM +M(24) andM=2 MMT. From Eq.(23) we see that the tem- perature prole depends on the prole of the potential H. This last one is determined by inserting Eq.(23) into Eq.(14). We have nally jM= ^M0rxH(25) that has to be solved with the continuity equation (16) giving again the diusion equation (18) of the previous section. However now the diusion length is the adiabatic value ^lM= (0^MM)1=2where ^M=M +M(26) is the conductivity for the magnetization current in adi- abatic conditions. B. Spin Hall eect in non-magnetic metals The spin Hall eect is due to the spin orbit interac- tion for conduction electrons. This eect is particularly relevant for noble metals with high atomic number. Be- cause of the spin orbit interaction, a spin polarized elec- tric current is de ected by an angle which is called the spin Hall angle SH. To include spin Hall eects into the theory of Section III A one should rst extend the equa- tions for the thermo-magnetic eects to the presence of an additional electric current. This is straightforward and the formal result is reported in Appendix A. How- ever to state the equation for the spin Hall eect, the equations must be further extended for two dimensional ow. The complete constitutive equations are character- ized by six force variables, namely: the derivative along xandyof the three driving forces for magnetic, elec- tric and heat currents. Here we simplify the problem by just disregarding the thermal eects. For our nal aims this is a reasonable approximation, since the contribu- tion arising from the thermomagnetic coecients of Pt is smaller than the other contributions involved in the full matrix of the thermo-magneto-electric eects26. The general constitutive equations for the joint electric and magnetic transport are reported in Appendix B. Here we6 analyze in more detail the case of a non magnetic conduc- tor with negligible Hall eect. We select the conditions in which the electric current jeis always along y, and the magnetization current jMalongx. We have then the equations for the spin Hall and the inverse spin Hall ef- fects from Eqs.(B5) and (B6). By converting to magnetic units one obtains jey=0ryVe+0SHB e 0rxH(27) jMx=0SHB e ryVe+M0rxH(28) whereM=0(B=e)2is the conductivity for the mag- netization current, 0is the electric conductivity, Veis the electric potential, eis the elementary charge and B is the Bohr magneton. The equations contain the spin Hall eects in the non diagonal terms which couples dif- ferent directions and dierent currents. It is worthwhile to notice that the eects are fully described by the spin Hall angle SHwhich for metals is a denite negative quantity. 1. Spin Hall eect In the spin Hall eect a magnetization current is gen- erated in the parallel direction xbecause of an electric current in the perpendicular one y. By eliminatingryVe by Eq.(27) and Eq.(28) we nd that the magnetization current is related to the electric current density by jMx=SHB e jey+0 M0rxH(29) where0 M=M(1 +2 SH). If the electric current density is uniform, the spin Hall eect corresponds to a mag- netization current source jMS=(B=e)SHjey. The prole of the magnetization current jMxwhich is actu- ally traversing the layer also depends on the boundary conditions posed by the adjacent layers. Then, to nd the prolejMx(x), Eq.(29) must be solved together with the continuity equation (16) giving a dierential equation for the driving potential H(x) which has the same from of Eq.(18) but with lM= (00 MM)1=2. 2. Inverse spin Hall eect In the conguration corresponding to the inverse spin Hall eect one has a magnetization current in the parallel direction which generates an electric eect perpendicular to it. The electric equation in the ydirection is jey=0 0ryVe+SHe B jMx (30)where0 0=0(1 +2 SH). The magnetization current traversing the layer is not constant and it will be given by the solution of Eq.(29) if the electric current jeyis constrained or by the solution of Eq.(28) if the electric potentialryVeis constrained. In both cases the constitu- tive equation must be solved together with the continuity equation (16), giving again the dierential equation (18). IV. SOLUTIONS OF THE MAGNETIZATION CURRENT PROBLEM A. Single active material For an active material both the spin Seebeck eects and the spin Hall eect results in a magnetization cur- rent source and the prole of the magnetization current will be due to the boundary conditions. In presence of boundaries blocking the ow of the magnetization cur- rent, the magnetic moments accumulate giving rise to the potential H. The magnetization current close to a boundary is therefore absorbed by the materials itself as the potential His also the driving force for the non con- servation of the magnetic moment (Eq.(6)). As it was shown in the previous Section, both spin Seebeck and spin Hall eects are characterized by constitutive equa- tions that have the same functional form. Then we can work out the solution for the prole of the magnetization current independently of the specic eect and consider- ing boundary conditions only. The specic solution will correspond to use as the current source jMSthe expres- sion derived from the spin Seebeck Eq.(14) or to the spin Hall Eq.(29). We initially consider a single material with generic boundary conditions. The solution of the magne- tization current problem with several layers will then be obtained by applying appropriate boundary conditions and joining the solutions for dierent layers. We take a material from x=d1tox=d2with a uniform source of magnetization current jMS. Starting from the formal solution Eq.(20), we derive the magnetization current by Eq.(17) and we x arbitrary values of the current at both boundaries, i.e. jM(d1) andjM(d2). The expression for the current is jM(x) =jMS(jM(d1)jMS)sinh((xd2)=lM) sinh(t=lM)+ + (jM(d2)jMS)sinh((xd1)=lM) sinh(t=lM) (31) and for the potential is H(x) =(jM(d1)jMS)1 (lM=M)cosh((xd2)=lM) sinh(t=lM)+ + (jM(d2)jMS)1 (lM=M)cosh((xd1)=lM) sinh(t=lM); (32)7 wheret=d2d1. Figs.2 and 3 shows the proles of the magnetization current and the eective eld along the material for dierent thicknesses t=lM. The spin accu- mulation close to the boundaries generates, as a reaction, an eective eld which counteracts the eect considered (e.g. the spin Seebeck eect) in order to let the current to go to zero at the interface. FIG. 2. Magnetization current proles for a single active ma- terial. Curves are Eq.(31) with d1=t=2 andd2=t=2, boundary conditions xed to zero ( jM(t=2) =jM(t=2) = 0) and show dierent thicknesses t=lM. The curves are normal- ized tojMS. FIG. 3. Magnetization potential prole Hfor a single active material. Curves are Eq.(32) with d1=t=2 andd2=t=2, boundary conditions xed to zero ( jM(t=2) =jM(t=2) = 0) and show dierent thicknesses t=lM(same as Fig.2). The curves are normalized to H 0=jMS=(lM=M). B. Injection of a magnetization current We consider the injection of a magnetization current from an active material which is acting as current gen- erator, or current injector, into a passive material which is acting as a conductor. It is known that the quality of the interface plays an important role in the injection of the spin currents27. In Ref.27the condition of thePt/YIG interface was intentionally modied by creating a thin amorphous YIG layer varying from 1 to 14 nm and it was shown that the spin Seebeck eect is depressed as the thickness of the amorphous layer increases. The max- imum value is obtained with a fully crystalline interface and the typical decay length of the eect with thickness is 2:3 nm. In the present theory this kind of interlayer inter- face can be taken into account by introducing a third ef- fective layer, with degraded properties, between the two. In the present paper we consider ideal interfaces between injector and conductor which is appropriate for spin See- beck experiments characterized by crystalline interfaces. To analyze the injection of a magnetization current, we simplify the notation by dropping the Msubscript and employing subscripts describing the role of the material: (1) for the injector and (2) for the conductor. The mag- netization current source is that of the active material (1) and is denoted jMS. The connection between the two me- dia is set at x= 0. The boundary conditions for the mag- netization current is j1(0) =j2(0) =j0and the bound- ary condition for the potential is H 1(0) =H 2(0) =H 0. Appendix C reports the formal solutions in the case in which each layer has nite width. These solutions will be employed in the comparison with real experiments per- formed in bilayers. Here we discuss how the eciency of the injections is determined by intrinsic parameters. To this aim we take the solutions of Appendix C in the limit of semi innite width and we obtain j1(x) =jMS(jMSj0) exp(x=l1) (33) and j2(x) =j0exp(x=l2) (34) for the currents and H 1(x) =j0jMS (l1=1)exp(x=l1) (35) and H 2(x) =j0 (l2=2)exp(x=l2): (36) By setting the boundary condition at the interface be- tween the two media H 1(0) =H 2(0) we nd the value of the current at the interface j0=jMS 1 +r12(37) wherer12= (l1=1)=(l2=2). Ifr121 the current is eciently injected, while if r121 the magnetization current is not transmitted into the conductor. In terms of intrinsic parameters we have8 r12=r1 22 1: (38) So a junction with an ecient injection from (1) to (2) should have a conductor (2) with a magnetization con- ductivity much larger than the injector 21and a time constant much smaller 21. V. SPIN SEEBECK AND SPIN PELTIER EFFECTS In this Section we apply the theory previously devel- oped to the spin Seebeck and spin Peltier eects. A. Spin Seebeck eect The spin Seebeck eect consists in a magnetization current generated by a temperature gradient across a ferromagnetic material. We study the longitudinal spin Seebeck eect (LSSE) where the magnetization current and the temperature gradient are along the same direc- tion. We consider experiments in which the active layer is YIG, the injector, labeled as (1) and the sensor layer is Pt, the conductor, labeled as (2). The geometry of the experiment is schematically shown in Fig.4. The YIG injector has thickness t1=tY IGwhile the Pt conductor has thickness t2=tPt. The interface is set at x= 0. xyzme-jeyHYIGPt jMxt1=tYIGt2=tPtcoldhotM FIG. 4. Geometry of the longitudinal spin Seebeck eect. The temperature gradient is applied along x, the mag- netic eld is along z, the electric eects (ISHE voltage) are measured along y. We consider a constant tempera- ture gradientrxT, therefore the magnetization current source of YIG is jMS=Y IGY IGrxTgiven by the equations of Section III A. The solutions of the magneti- zation current problem are Eqs.(C1) and (C2) reported in Appendix C and the magnetization current at the in- terface is given by Eq.(C5) in which l1=lY IG,1=Y IG andl2=lPt,2=Pt. As the thickness of the Pt layer is generally of the same order of the spin diusion length (tPtlPt10 nm), we can approximate Eq.(C2) forthe case of t2l2and nd that the prole of the mag- netization current is, at a good approximation, a linear decay from j0at the interface x= 0 to zero at the bor- derx=t2. The average magnetization current in the Pt layer is therefore hjMxix=j0=2 wherej0is the magne- tization current injected at the interface. If the experi- ments are performed by measuring the ISHE voltage, by taking Eq.(30) with jey= 0, we obtain the relation be- tween the magnetizations current along xand the electric potential along y. We assume the relation to be valid for the average values along xover the thickness t2. The average potential is then hryVeix=SH ee B hjMxix: (39) wheree, corresponding to 0 0in Eq.(30), is the electric conductivity of Pt. The current injected at the interface j0can therefore be estimated by the gradient of the ISHE voltageryVISHE =hryVeix, j0= 2e SHB e ryVISHE: (40) In experiments, the spin Seebeck coecient is determined asSLSSE =ryVISHE=rxT. The magnetization current at the interface can be calculated by Eq.(40) where the spin Hall angle is evaluated as SH=0:1 from Ref.28. In turn, the relation between the spin Seebeck current jMSandj0at the interface, given by Eq.(C5), will de- pend on the intrinsic parameters of both layers and their thickness. Once the current jMSis calculated, one can estimate the spin Seebeck coecient as Y IG=1 Y IGjMS rT : (41) In Pt the magnetization diusion length is known to be lPt= 7:3 nm28. The spin conductivity can be estimated by assuming that in a normal metal the scattering acts independently of the spin29. Then, by converting the electrical conductivity of Pt e= 6:4
106 1m1, into the conductivity for the magnetization current, we obtain 0Pt= 2:6
108m2s1. The time constant is nally calculated and results Pt=l2 Pt=(0Pt)'2
109s. In YIG the estimations of the magnetization diu- sion length present in literature, range from micron to millimeter30{32for the transverse experiment (in which current and magnetization are parallel) to much lower value (i.e.<1m)33for the longitudinal eect (in which current and magnetization are perpendicular). From Ref.3the LSSE coecient measured on 1 mm of YIG, SLSSE'4
107VK1, results to be larger than the one measured on a 4 m sample22SLSSE'2:8
107VK1, but of the same order of magnitude. Therefore we can guess thatlY IGis of the same order of magnitude of the thinner sample (4 m) in order to allow for an ecient9 injection in both cases. In a more recent study, the de- pendence of the spin Seebeck eect on the thickness of YIG was investigated21. It has been reported that the typical diusion length is below lY IG= 1:5m. We set in the following lY IG= 1m. For the evaluation of the absolute thermomagnetic power coecient Y IG we use the result of Ref.22where the thermal conditions were properly taken into account. These experiments were performed by using a YIG layer of 4 m and a Pt layer of 10 nm. By using the LSSE coecient estimated at the sat- uration magnetization of YIG we obtain j0=(rxT)' 2
103As1K1m22. The only missing intrinsic param- eter is the magnetization conductivity of the YIG, Y IG. To have an order of magnitude we suppose a reasonable injection from YIG into Pt (i.e 50%, with j0= 0:5jMS). Then we set r12= 1, i.e. l1=1=l2=2. By using the resulting value for the magnetization conductivity of YIG0Y IG4
107m2s1, we nally obtain an order of magnitude for the absolute thermomagnetic power coecient as Y IG102TK1. In analogy with the thermoelectric eects where the absolute thermoelec- tric power coecient is compared to the classical value e=kB=e'86
106VK1, the value found here can be compared with the ratio kB=B'1:49 TK117. Fur- thermore, as the experiments show that ryVISHE and thereforejMS, changes sign when the magnetization of the YIG layer is inverted, this means that Y IGchanges sign when inverting the magnetization M. The value re- ported before corresponds to the absolute value when the magnetization of YIG is at saturation. B. Spin Peltier eect In the spin Peltier experiments a magnetization cur- rent is generated by the spin Hall eect in a Pt layer, labeled as (1) and is injected into a YIG layer, labeled as (2). The injection of the magnetization current into the YIG, generates thermal eects. The geometry of the experiment is schematically shown in Fig.5. xyzme-jeyHYIGPt jMxt2=tYIGt1=tPtMcoldhot FIG. 5. Geometry of the spin Peltier eect. The interface is set at x= 0, the electric current isalongy, the magnetization current is along xand the magnetic eld is along z. The magnetization current source is now jMS=SH(B=e)jeygiven by the spin Hall eect in Pt discussed in Section III B. When the magnetization current diuses inside YIG, it also gen- erates a heat current because of the spin Peltier eect described in Section III A 2. The solution of the magne- tization conduction problem is mathematically identical to the spin Seebeck one, but with the role of YIG and Pt inverted. For this reason we have employed label (1) for the injector, which is now Pt, and label (2) for the conductor which is now YIG. The solutions of the mag- netization current problem are again Eqs.(C1) and (C2) reported in Appendix C and the magnetization current at the interface is given by Eq.(C5). With respect to the previous spin Seebeck case, the diusion length of YIG is the adiabatic value ^lY IG = (0^Y IGY IG)1=2. In the spin Peltier experiment the temperature prole in YIG is given by the integration of Eq.(23) T(x)T(0) =1 ^Y IG0(H 2(x)H 2(0)) (42) whereH 2(x) is given by Eq.(C4). The result is shown in Fig.6. !!"#!"$!"%!"&'!!"'!"#!"(!"$!") *+,#
-+
-./)$(#0#+,#1' x/tYIGlYIG/tYIG=0.10.5125 FIG. 6. Temperature prole of YIG for the spin Peltier eect. Curves are
T=T(x)T(0) from Eq.(42) normalized to
TSH=0H SH=^"Y IG andH SH=jMS=(lY IG=Y IG). The parameters are r12= 1; lPt=tPt= 0:1. By looking at the magnetization current prole (Fig.7), we see, as in the spin Seebeck experiment, that in order to have a good eciency, the thickness of each layer should be larger than its diusion length ( t1>l1andt2>l2) to permit to the magnetization current to develop. More- over the eciency of the injection is regulated by the ratio of intrinsic parameters r12= (l1=1)=(l2=2), where (1) is the injector Pt and (2) is the conductor YIG. Again the magnetization current at the interface is large if the ratior12is small. However it should be noticed that given the two materials in the junction (i.e. Pt,YIG) we have that rPt!Y IG = 1=rY IG!Pt. So, the value rPt!Y IG =rY IG!Pt'1 is the value which permits10 relatively ecient injection both from Pt into YIG and from YIG into Pt. Finally from the temperature prole Fig.6 obtained in adiabatic conditions we can reach information about the coecient of the absolute thermomagnetic power in adi- abatic conditions ^ "Y IG. The prole T(x) is normalized to the temperature
TSHwhich gives the typical scale of the eect
TSH=1 ^"Y IG0H SH (43) whereH SH=jMS=(lY IG=Y IG). From the litera- ture the thermal conductivity of YIG is = 6 W K1m1. From Section V A, "Y IG'102TK1and the parameter Y IG'102W K1m1 34. Moreover the potential H SHis related to the spin Hall current jMS=(B=e)SHjeyinjected from Pt. Using the val- ues from34lY IG=Y IG = 3 ms1andSH=0:1 we are able to give an order of magnitude estimate of the temperature change, obtaining
TSH=jey= 4
1013K A1m2. Experimental values are taken from Ref.4, where in correspondence to an electric current density of 3
1010 A m2in Pt, the temperature dierence measured by a thermocouple in YIG was 2 :5
104K, considering that the Joule heating of the electric current in Pt was already subtracted. The parameter
TSHresults 1:2
102K which is of the correct order of magnitude. Consequently by usingt1=tPt= 5 nm and t2=tY IG = 0:2m in Eqs.(C4) and (42), we nd an adiabatic tempera- ture change of T(tY IG)T(0)'2:5
104K with lY IG = 0:4m. This value renes the upper limit of 1m which was found in Section V A, however the phe- nomenology of the spin Peltier eect in YIG seems coher- ent with the absolute thermomagnetic power coecient derived previously. VI. CONCLUSIONS In this paper the problem of magnetization and heat currents is investigated through a non equilibrium ther- modynamics approach. Based on the constitutive equa- tions of a ferromagnetic insulator and a spin Hall active material we are able to solve the problem of the pro- les of the magnetization current and of the potential in the geometry of the longitudinal spin Seebeck and of the spin Peltier eects. By focusing on the specic ge- ometry with one YIG layer and one Pt layer, we obtain the optimal conditions for generating large magnetization currents into Pt in the case of the spin Seebeck eect and for generating large heat current in YIG in the case of spin Peltier eects. In both cases we nd that ecient injection is obtained when the thickness of the injecting layer is larger than the diusion length lM. The the- ory predictions are compared with experiments and this permits to determine the values of the thermomagneticcoecients of YIG: the magnetization diusion length lM0:4m and the absolute thermomagnetic power coecientM102TK1. ACKNOWLEDGMENTS This work has been carried out within the Joint Re- search Project EXL04 (SpinCal), funded by the Eu- ropean Metrology Research Programme. The EMRP is jointly funded by the EMRP participating countries within EURAMET and the European Union. Appendix A: Constitutive equations of the thermo-magneto-electric eects The equations for the thermo-magneto-electric eects relates the current densities of the electric charge je, the magnetic moment jM, and the heat jq, with the gradients of the electric potential Ve, the magnetization potential H, and the temperature T. In one dimension ( rx= @=@x ) the equations are je=erxVe+0rxHeerxT (A1) jM=rxVe+M0rxHMMrxT (A2) jq=eeTrxVe+MMT0rxHisorxT: (A3) whereeis the electrical conductivity, eis the abso- lute thermoelectric power coecient, represents the magneto-electric conductivity, Mis the magnetic con- ductivity,Mis the absolute thermomagnetic power coef- cient andisois the thermal conductivity with rxVe= 0 andrxH= 0. By dening the heat current as jq=Tjswe obtain from Eqs.(10) and (11) T@s @t+rxjq=0rxHjM+0(H)2 MrxVeje:(A4) By solving the previous equation together with the con- stitutive equation (A3), one can obtain the generalized heat diusion equations. Appendix B: Magnetization current carried by electrons We consider the specic case of metals in which the magnetic and electric current are due to the same type of carriers (electrons or holes) with dierent spin. The theory can be equivalently formulated in terms of mag- netic moment (up or down). One subdivides the particle currentjn=jn++jninto the sum of moment up jn+ and moment down jn. The electric current is je=qjn whereqis the charge of the carrier, while the magneti- zation current is B(jn+jn), whereBis the Bohr11 magneton. As it is somehow customary to dene a mag- netization current jmmeasured in the same units of the electric currents, we have then jm= (e=q)(je+je) whereeis the elementary charge. Electrons, moving in the opposite direction of the charge current, with a mag- netic moment up will give a negative jm, while holes with moment up, will give a positive jm. One is allowed to as- sume dierent conductivities among the two sub-bands as a function of the gradients of the potentials rVerel- ative to each sub-band. The equations are je+=+rVe+mixrVe (B1) je=mixrVe+rVe (B2) where one has to introduce both the individual channel conductivities +andand the spin mixing conduc- tivitymix. One obtains  je jm =0 1 +   1 rVe rVm (B3) withVe=Ve++VeandVm= (e=q)(Ve+Ve) where0= (++)=2 is the electric conductivity, =mix=0is the spin mixing coecient ( 1) and = (+)=(20) represents the spin unbalance of the conductivities. Vmis a potential for the current jm with the same units of Ve. The electric conductivity is e=0(1+) and the conductivity for the magnetization currentjmism=0(1). It is often the case that the spin mixing conductivity is very small (i.e. = 0 into Eq.(B3)) because the spin ip events are much more rare than the normal scattering conserving the spin, so m=e. This leads to the Mott's two current model. In that case the spin unbalance coecient is a number between 1 and -1. The previous equations form also the basis to describe the Hall and the spin Hall eects. We need to extend the equations for the magneto-electric eects to two di- mensions. We consider the case in which the spin mixing conductivity is zero and e=m=0. The equations read 0 B@jex jey jmx jmy1 CA=00 B@1HSH H 1SH SH 1H SH  H 11 CA0 B@rxVe ryVe rxVm ryVm1 CA (B4) whereHis the Hall angle and SHis the spin Hall an- gle. It is important to notice that the Hall angle depends on the magnetic eld while the spin Hall angle is a con- stant that is determined by the spin orbit interaction for conduction electrons. We analyze in more detail a non magnetic conductor with= 0 for which the Hall angle is negligible H= 0. Furthermore we select conditions in which the electric current is always along yand the magnetic current alongx. We have nally the equations for the spin Hall and the inverse spin Hall eects jey=0=ryVeSHrxVm (B5) jmx=0=SHryVerxVm: (B6) To convert to magnetic units of Section III B one simply uses rVm=B e 0rH(B7) and jm=e B jM: (B8) Appendix C: One junction Let us consider a bilayer of two materials: the injector (1) fromx=t1tox= 0 which contains a magneti- zation current source jMSand the conductor (2) from x= 0 tox=t2. The connection between the two me- dia is put at x= 0 and the boundary conditions on the magnetization current are: j1(t1) = 0,j2(t2) = 0 and j1(0) =j2(0) =j0. The solutions for the magnetization currents, where only the injector (1) is an active material, are j1(x) =jMS+jMSsinh(x=l1) sinh(t1=l1)+ + (j0jMS)[sinh(x=l1) coth(t1=l1) + cosh(x=l1)] (C1) and j2(x) =j0[sinh(x=l2) coth(t2=l2)cosh(x=l2)] (C2) and for the potentials H 1(x) =jMS (l1=1)cosh(x=l1) sinh(t1=l1)+ +j0jMS (l1=1)[cosh(x=l1) coth(t1=l1) + sinh(x=l1)] (C3) and H 2(x) =j0 (l2=2)[cosh(x=l2) coth(t2=l2)sinh(x=l2)]: (C4) By setting the boundary condition at the interface be- tween the two media H 1(0) =H 2(0) we nd the value of the current at the interface12 j0=jMScosh(t1=l1)1 cosh(t1=l1) +r12sinh(t1=l1) coth(t2=l2)(C5) wherer12= (l1=1)=(l2=2). Figs.7 and 8 shows the pro- les of the magnetization current and the eective eld along the material for dierent values of t1=l1. FIG. 7. Magnetization current proles for a bilayer showing the passage (injection) of a magnetization current generated in medium (1), of nite thickness t1=l1, to the semi innite conductor medium (2). Curves are from Eqs.(C1), (C2). The parameters are r12= 1; l2=l1= 2. The curves show the ef- fect of dierent thicknesses t1=l1of layer (1) on the injected current. FIG. 8. Magnetization potential proles Hfor the bilayer of Fig.7. Curves are from Eqs.(C3), (C4) and normalized to H 0=jMS=(l2=2). Parameters are the same of Fig.7 1K. Uchida, H. Adachi, T. Ota, H. Nakayama, S. Maekawa, and E. Saitoh, Appl. Phys. Lett. 97, 172505 (2010). 2G. Siegel, M. C. Prestgard, S. Teng, and A. Tiwari, Sci. Rep.4, 4429 (2014). 3K. Uchida, T. Kikkawa, A. Miura, J. Shiomi, and E. Saitoh, Phys. Rev. X 4, 041023 (2014). 4J. Flipse, F. K. Dejene, D. Wagenaar, G. E. W. Bauer, J. B. Youssef, and B. J. V. Wees, Physical Review Letters 113, 027601 (2014). 5H. Adachi, K. Uchida, E. Saitoh, and S. Maekawa, Rep. Prog. 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2015-12-30

We study the problem of magnetization and heat currents and their associated thermodynamic forces in a magnetic system by focusing on the magnetization transport in ferromagnetic insulators like YIG. The resulting theory is applied to the longitudinal spin Seebeck and the spin Peltier effects. By focusing on the specific geometry with one YIG layer and one Pt layer, we obtain the optimal conditions for generating large magnetization currents into Pt or large temperature effects in YIG. The theoretical predictions are compared with experiments from the literature permitting to derive the values of the thermomagnetic coefficients of YIG: the magnetization diffusion length $l_M \sim 0.4 \, \mu$m and the absolute thermomagnetic power coefficient $\epsilon_M \sim 10^{-2}$ TK$^{-1}$.

Non-equilibrium thermodynamics of the spin Seebeck and spin Peltier effects

1512.08890v2

1 Structural and Magnetic Study of Metallo -Organic YIG Powder Using 2 -ethylhexanoate Carboxylate Based P recursors S. Hosseinzadeha, P.Elahib, M. Behboudnia*, a, M.H. Sheikhic, S.M. Mohseni*, d aDepartment of Physics, Urmia University of Technology, Urmia, Iran bDepar tment of Material Science, The U niversity of Utah, Utah, U.S. cDepartment of Communications and Electronics, School of Electrical and Computer Engineering, Shiraz, Iran dFaculty of Physics, S hahid Beheshti University , Evin, 19839, Tehran, Iran Abstract The crystallization and magnetic behavior of yttrium iron garnet (YIG) prepared by metallo - organic decomposition (MOD) method are discussed . The chemistry and physics related to synthesis of iron and yttrium carboxylates based on 2 -ethylhexanoic acid (2EHA) are studied , since no literature was found which elucidate s synthesi s of metallo -organic precursor of YIG in spite of the literatures of doped YIG samples such as Bi -YIG. Typically, the metal carboxylates used in preparation of ceramic oxide materials are 2 -ethylhexanoate (2EH) solvents. Herein , the synthesis , thermal behavior and solubility of yttrium and iron 2EH used in synthesis of YIG powder by MOD are reported . The crystallization and magnetic parameters , including saturation magnetization and coercivity of these samples , smoothly change as a function of the annealing temperature. It is observed that high sintering temperature of 1300 to 1400 °C promotes the diffraction peaks of YIG , therefore, we can conclude that the formation of YIG in MOD method increases the crystallization temperature . The maximum value of saturation magnetization and minimum value of coercivity and remanence are observed fo r the sample sintered at 1200°C which are 13.7 emu/g, 10.38 Oe and 1.5 emu/g , respectively . This study cites the drawbacks in chemical synthesis of metallo -organic based YIG production. 2 Keywords YIG; MOD ; metallo -organic precursor ; crystallization ; magnetic particles *Corresponding author. E -mail Address: m-mohseni@sbu.ac.ir , majidmohseni@gmail.com (Seyed Majid Mohseni) . mbehboudnia@gmai l.com (Mahdi Behboudnia) 1. Introduction Magnetic t hin film of Yttrium Iron Garnet (Y 3Fe5O12 – YIG) has found great attention in emerging spintronic devices. Such a thin film with low Gilbert damping constant is suitable for magnonics and beside its insulator behavior ( insulating nature) , it gains a great deal of attention in generation of spin current. With the advent of spintronics, the demand for synthesis and processing of YIG films has surged forward greatly. In general, t he characteri stics and surface morphology of the thin films strongly depends on deposition techniques. Pulsed laser deposition (PLD) has emerged as a preferred technique to deposit complex oxide thin films, heterostructures, and superlattices with high quality in comparison with the other deposition techniques such as e -beam evaporation and sputtering [4, 5]. Contrarily , chemica l solution deposition (C SD) techni ques are cost effective synthesis process es in which the precursor solution is deposited by variety of relatively simple techniques such as spin or dip coating followed by post treatments of drying and anneal ing in case needed . More r ecently , metallo -organic decomposition (MOD) recognized as a chemical technique has been growing due to its extreme ly good molecular level composition controllability in the fabrication process of garnet thin films . Kirihara et.al [6] reported generation of spin -current - driven thermoelectric conversion by using B i-YIG thin layer prepared by MOD. There are also 3 further literature s [7-9] which report deposition of doped YIG and YIP (yttrium iron perovskite, YFeO 3) by MOD technique using metal carboxylate precursors in organic solvents. In majority of the publications , the effect of the annealing temperature and duration of doped YIG powders has been investigated on the physical and magnetic properties . It is shown that at higher temperature s, the observed physical and magnetic properties of garnet are similar to that of solid state reaction techniques [10]. The MOD procedure introduced by Azevedo et. al is used for thin film deposition [11-14]. Their procedure consist s of preparation of metal -organic carboxylates, which exhibit high compatibility to organic solvents , result in better final thin film properties . They succeeded in preparing polycrystalline Gd 2BiFe 5O12 and (DyBi) 3(Fe, Ga) 5O12 thin films on glass substrates. In their work , large carboxylate compounds of 2 -ethylhexanoate (2EH) precursors were synthesized, and then a stoichiometric ratio of precursors was dissolved in xylene followed by spin coating of the solution on preferred substrate. Typically, the metal carboxylates used in the preparation of ceramic oxide materials are 2EH salts which are dispersed in aromatic solvents [10, 12, 13, 15, 16 ]. Metal 2EH have found wide application such as metal –organic precursors , catalysts for ring opening poly merizations and drier agent in paint industries [17]. Realization of the MOD advantages require s in-depth understanding of the precursor solution chemistry such as precursor species, solute concentration, and solvent system and its relation to the final material properties. A detailed study of the fundamental properties of 2EH yttrium 2-ethylhexanoate (Y-2EH) and iron- 2ethylhexanoate (Fe-2EH) precursors used in synthesis of YIG thin films has not been reported yet. The primary objective of this work is to investigate the fundamental properties of yttrium 2- ethylhexanoate (Y-2EH) and iron-2-ethylhexanoate (Fe-2EH) precursor s needed to synthesis 4 YIG thin films, such as chemi cal properties (including solvent/solute concentration ratios, solution structure and internal bonding) a nd on process optimization methodolo gies in order to obtain optimum YIG properties . 2. Materials and methods 2.1.Reagents Iron(III) nitrate nonahydrate (Fe(NO 3)3, Yttrium nitrate hexahydrate (Y(NO 3)3), Y 2O3, 2EHA , Xylene, Toluene, n -Hexane, n -octane, benzene e, THF, Isopropanol, propioni c acid, and glacial acid acetic are of analytical reagent grade. 2.2.Synthesis of Yttrium and Iron carboxylates A metal carboxylate compound is defined as the central metal atoms linked to ligands through a hetero –atom s [14]. 2.2.1 Fe-2EH : The iron carboxylates were prepared by double decomposition from ammonium soaps obeying the following reactions [18] NH 4OH + C7H15COOH 1 C7H15COONH 4 + H 2O Fe (NO 3)3 + 3 C7H15COONH 4 Fe (C7H15COO) 3 + 3 (NH 4) NO 3 The first reaction involves the preparation of the ammoni um soap of 2-ethylhexanoic acid (2EHA) . Then the soap from second reaction was mixed with the aqueous solution of Fe(NO 3)3. After stirring for 10 minutes , the solution was separate d into Fe(RCOO) 3 and (NH 4) NO 3. In a funnel separator, a solvent (xylene ) was added to this solution and the carboxylate which dissolves in xylene was separated and filtered by 0.22 μ -filter and was kept away until xylene evaporated under fume hood to reach a reddish brown powder of Fe-2EH. By decomposing a 5 certain amount of the Fe-2EH into the metal oxide and weighing the oxide, the equivalent amount of iron oxide in Fe -2EH was determined. 2.2.2 Y-2EH : The Y-2EH is prepared according to the following reaction [19] Y2O3 + 6 C7H15COOH 1 2 (C7H15COO) 3Y + 3 H 2O Yttrium oxide ( Y2O3) powder was added gradually to 2EHA while gently stirring and it was kept at 100°C till all liquid has been evaporated. The product was stirred with an excess of toluene for 24h at room temperatu re. Thereupon , it was filtered and was kept under fume hood until the toluene was evaporated and a white solid is achieved . This white so lid is Y-2EH. By decomposing a known amount of the Y -2EH into the metal oxide and weighing the oxide, the equivalent amount of Y2O3 in Y-2EH was determined. 2.3.Synthesis of YIG powder Metallo -organic YIG was prepared using Y-2EH and Fe- 2EH in stoichiometric ratio of 3:5. Firstly, The Y - 2EH and Fe -2EH were dissolved into toluene and xylene , respectively. Secondly, the solutions were mixed together with respect to the weight percentage of each carboxylate to achieve the desired stoichiometry. The process was followed by adding glacial acetic acid un til a homogeneous solution was reached with no precipitation . The powders of MOD solution were prepared by drying for 72 hours at 150 °C and then they were grinded in a mortar . Figure 1 illustrate s the schematic diagram of metallo -organic decomposition of YIG. 6 Figure 1: Schematic diagram for the metallo -organic decomposition process of YIG . 2.4.Characterization of samples To investigate the pyrolysis and crystallization process of the YIG prepared by the MOD method, a thermo -gravimetric -differential thermal analysis (TG A-DTG -DTA ) (model mettle Toledo C1600 analyzer) was carried out from controlled room temperature to 1400 °C in an air atmosphere with the heating rate of 10°C/min. The crystalline structure of samples was characterized using X -ray diffractometer (STOE -STADI) with Cu Kα (λ = 0.154 nm) radiation. Room temperature magnetization measurements were performed using vibrating sample magnetomete r (Meghnatis Daghigh Kavir Co.) 3. Results and discussions As previously reported by Beckel [20] and Neagu [21], the solvent influences the boiling point of the solution and determines the speed of evaporation during heating of the droplets which affects 7 the film roughness. The solvent also impacts the maximum metal carboxylate solubility and spreading behavior of the droplets. The deposition temperature is primarily influenced by the solvent and additives . These organic solvents and additives help in gelation and polymerizations, and modif ication o f the solution properties [21-24], such as viscosity, solubility of metal carboxylate and spreading of the droplets. The propionic acid and mixture of ethanol and 2EHA were used as additives as mentioned in literatures beyond the number. Besides, glacial acetic acid (GAA) were used as additive. This study reveals that GAA improves YIG formation and decreases YIG crystallization temperature . Figure 2 demonstrates the TGA -DTG and DTA curves of the MOD precursor . It show s thermal decomposition proceeds via six-step process es in an air atmosphere . The broad endothermic peak from room temperature to about 200 ° C with a total weight loss of about 3% corresponds to the evaporation of residual solvents including xylene with boiling point (bp) of 138 °C and glacial acetic acid with boiling point of 118 °C [25]. Three exothermic peaks from 200 to about 480 °C with total weight loss of 51.5 % are ascribe d to the volatilization of 2 EHA (bp~228 °C ) and the pyrolysis of possible metal -organic compounds such as Fe-2EH are expected [25, 26] . The three exothermic peaks from 480 to 820 °C represent removal of the three 2 -EH groups from the Y- 2EH molecule to form Y2O3 with total weight loss of 20.7% [27]. The following two exothermic peaks at temperature range of 820 to 920 °C with a weight loss of 2.6% correspond to crystallization of YIG and YIP. The weight loss curve then approaches plateau from temperature range 920 to 1300 °C. The two exothermic peaks observed within this temperature range are attributed to the conversion of YIP to YIG and crystallization of YIG. This crystallization temperature is higher than other chemical solution decomp osition methods [28, 29] . On the other hand, the main advantage of the carboxylate -based -routines is the comparably low crystallization 8 temperature. This is due to the educt molecules that are mixed at the molecular level. Thus, the diffusion paths of metal -and-oxygen -ions are sho rt compared to classical powder -based syntheses of ceramic bulk materials [18, 30] . However, The formation of YIG in the MOD method increases the crystallization temperature which is mentioned previously by Lee et al. [13]. The above results suggest s that the 2 EHA may not serve as an excellent ligan d for yttrium precursor, since the decomposition of the organics for Y -2EH occurs at temperatures higher than 500°C as reported in literatures [27]. Figure 2: TGA -DTG (a), DTA (b) characteristics for YIG powder Figure 3 shows the XRD patterns of YIG particles annealed at 1000 -1400 °C for 2hrs. The XRD pattern of the YIG particles calcined at 1000 °C is associated with the formation of YIG with already formed yttrium oxide (Y 2O3) around 2θ=29° and hematite (α -Fe2O3) around 2θ=33° as the major phases in addition to the traces of YIP around 2θ=48, 33° . As the solid state reaction method suggested, the crystallization process could be described by the following equation s, Y2O3+Fe 2O3 2YFeO 3 at 800 -1200 °C 3YFeO 3+Fe 2O3 Y3Fe5O12 at 1000 -1300 °C 9 which indicates that the Y 2O3 phase would first transform into YIP phase at low temperatures and then convert s to YIG phase by combining with α-Fe2O3 at higher temperatures . The color of the particles calcined at 1000 °C temperature was reddish brown which is due to the presen ce of α-Fe2O3 and YIP as the major phases. When the YIG particles were calcined at 1100 °C and 1200 °C, the color converted to brownish green , indicating the conversion of YIP to YIG phase. For the sample calcined at these two temperatures , the diffraction peak around 2θ=33° became stronger , the Y 2O3 and YIP peaks became weaker, but still there is excess amount of α -Fe2O3 phase. As the annealing temperature is increased to 1300 & 1400 °C there is no change in diffraction peaks of YIG and α -Fe2O3 and the peak related to the α -Fe2O3, located near 2θ= 33°, still remain s the same and the intensity of garnet peaks are not increased as sintering temperature is risen. The remained α-Fe2O3 phase can be explained due to the insufficient amount of Y-2EH during the reaction because of precipitation of Y -precursor. The metal carboxylat e must be soluble and stable at room temperature in the solution, if not precipitation will occur and lead to inhomogeneity repartition of cations in the obtained gel. The successful application of the MOD process significantly depends on the metallo -organic compounds used as precursors for a variety of elements. The ideal compounds should satisfy some requirements such as high solubility in a common solvent. The solutions of individual metallo -organic compounds should mix in the appropriate ratio to give the desired stoichiometry for final formation. The main conclusion is that there is no theoretical database for selecting the optimum solvent suitable for MOD process . In order to explore the interactions between solute and solvent system , the polarity of the solvent and solute should be considered due to evaluation of the effect of dipole -dipole interactions . 10 Generally, to have a good solubility , the polarity of solute and solvent should be close to each other. The longer chain acid like 2EHA can be solved in low polar solvents (eg. Xylene, alcohol etc.). In case of unknown solubility parameter of a compound, a successful approach is to first try a non -polar solvent which has low solubility parameter. If the approach is not successful, then a moderately polar solvent with intermediate solubility paramet er should be tried. In order to find an adequate solubility in the desired solvent s which were compatible with each other, we tested some solvent s recommended by literatures for yttrium and iron 2EH. Therefore, we tested xylene, toluene, benzene, n -hexane, THF for both Fe-2EH and Y-2EH and found that the homogeneity of toluene and xylene are the best for yttrium and iron , respectively. However , the solubility and homogeneity of Y-2EH tends to be much less than that of the Fe-2EH. The Y- 2EH showed precipitation and was not as homogenous as Fe-2EH. As reported by Ishibashi et al. [12], the Y -2EH cann ot dissolve in the solvent introduced by Azevedo et al.[14]. As a result, we suggest that synthesis of Y-2EH is not an easy and homogenous synthesis approach . 11 20 30 40 50 60 70 80 O GGG OGGGG GH G1400 C 1300 C 1200 C 1100 C Intensity (a.u. ) 2 Theta (degree)1000 C YG H OG GG Figure 3. XRD pattern of the YIG powder annealed at 1000 -1400 °C. Assignment of diffraction peaks are indicated as following: G: YIG, O: Y IP, H: α -Fe2O3, Y: Y2O3 Figure 4 (a) show s the sintering temperature dependence of magnetization . Parameters such as MS, H c and M r are shown in Figure 4 (b,c). The MS of the powders sintered at 1000 -1400 °C were 9 to 13 emu/g, and a maximum value of 13.7 emu/g was observed for the powder sintered at 1200°C . From the XRD results, we observed that the intensity of garnet phases around 2θ=32, 45° are strongest at 1000 -1400° C, which can be deduced that the magnetic behavior of sintered powders was strongly determined by the garnet phases due to the weak ferromagnetic properties of the α-Fe2O3 and Y IP phases. The magnetic result s is similar to the magnetic result s report ed by Lee et al. [10]. In the range of 1000 -1400°C, the H c and Mr decrease whereas the MS show s a rise. The decrease in Mr and H c versus the increase in MS are explained due to an increase in the particle size [31]. 12 -10 -8 -6 -4 -2 0 2 4 6 8 10-15-10-5051015 -0.1 0.0 0.1-0.10.00.1M (emu/g) H (kOe) 1000C 1100 C 1200 C 1300 C 1400 C M (emu/g) H (kOe) 1000 1100 1200 1300 140091011121314 sintering temperature ( C)Ms (emu/g) 1020304050607080 Hc (Oe) 1000 1100 1200 1300 14001.41.61.82.02.22.42.62.83.03.2 sintering temperature ( C)Mr (emu/g) 91011121314 Ms (emu/g)A Figure 4. (a) Room temperature magnetization hysteresis loops of powders sintered at 1000 -1400 °C, (b) Variation of MS, Hc, and (c) Mr of YIG powders as a f unction of sintering temperature 4-Conclusion : Metallo -organic precursors of yttrium and iron metal -carboxylates were synthesized and the chemistry and physics related to various fabrication steps were investigated. The metallo -organic (a) (b) (c) 13 compound in work can be dissolved in proper solvent s such as toluene and xylene with the GAA used as an additi ve, to achieve the desired stoichiometry for preparing the YIG powder. Crystallization and magnetic behavior of the YIG was studied. It is observed that the Y-2EH show s precipitation and is not as stable as Fe-2EH and also Y-2EH is not homogenously synthesized . Our results can be valuable to revive useful materials for chemical solution processing of YIG family thin films . Acknowledgments S.M. Mohseni acknowledges support from Iran Scie nce Elites Federation (ISEF), Iran Nanotechnology Initiative Council (INIC) and Iran’s National Elites Foundation (INEF) References 1. Kajiwara, Y., et al., Transmission of electrical signals by spin -wave interconversion in a magnetic insulator. Nature, 2010. 464(728 6): p. 262. 2. Garello, K., et al., Symmetry and magnitude of spin –orbit torques in ferromagnetic heterostructures. Nature nanotechnology, 2013. 8(8): p. 587. 3. Silva, T.J. and W.H. Rippard, Developments in nano -oscillators based upon spin -transfer point -contact devices. Journal of Magnetism and Magnetic Materials, 2008. 320(7): p. 1260 -1271. 4. Hegde, M., Epitaxial oxide thin films by pulsed laser deposition: Retrospect and prospect. Journal of Chemical Sciences, 2001. 113(5-6): p. 445 -458. 5. Christen, H .M. and G. Eres, Recent advances in pulsed -laser deposition of complex oxides. Journal of Physics: Condensed Matter, 2008. 20(26): p. 264005. 6. Kirihara, A., et al., Spin -current -driven thermoelectric coating. Nat Mater, 2012. 11(8): p. 686 -689. 7. Asada, H., et al., Longitudinal Spin Seebeck Effect in Bi -substituted Neodymium Iron Garnet on Gadolinium Gallium Garnet Substrate Prepared by MOD Method. Physics Procedia, 2015. 75: p. 932 -938. 8. Galstyan, O., et al., Influence of bismuth substitution on yttri um orthoferrite thin films preparation by the MOD method. Journal of Magnetism and Magnetic Materials, 2016. 397: p. 310 -314. 9. Yamamoto, A., et al., Evaluation of Correlation Between Orientation of Y 3 Fe 5 O 12 (YIG) Thin Film and Spin Seebeck Effect. IEEE Transactions on Magnetics, 2017. 53(11): p. 1 -4. 10. Lee, H., et al., Magnetic and FTIR studies of Bi x Y 3− x Fe 5 O 12 (x= 0, 1, 2) powders prepared by the metal organic decomposition method. Journal of Alloys and Compounds, 2011. 509(39): p. 9434 -9440. 11. Ishibashi, T., et al., (Re, Bi) 3 (Fe, Ga) 5 O 12 (Re. Journal of Crystal Growth, 2005. 275(1): p. e2427 -e2431. 12. Ishibashi, T., et al., Characterization of epitaxial (Y, Bi) 3 (Fe, Ga) 5 O 12 thin films grown by metal -organic decomposition met hod. Journal of applied physics, 2005. 97(1): p. 013516. 13. Lee, H., et al., Preparation of bismuth substituted yttrium iron garnet powder and thin film by the metal -organic decomposition method. Journal of Crystal Growth, 2011. 329(1): p. 27 -32. 14. Azev edo, A., et al., Deposition of garnet thin films by metallo -organic decomposition (MOD). IEEE Transactions on Magnetics, 1994. 30(6): p. 4416 -4418. 15. Ishibashi, T., et al., Magneto -optical Indicator Garnet Films Grown by Metal -organic Decomposition Metho d. Journal of the Magnetics Society of Japan, 2008. 32(2_2): p. 150 -153. 16. Ishibashi, T., et al. Magneto -optical properties of Bi -substituted yttrium iron garnet films by metal -organic decomposition method . in Journal of Physics: Conference Series . 2010. IOP Publishing. 17. Mishra, S., S. Daniele, and L.G. Hubert -Pfalzgraf, Metal 2 -ethylhexanoates and related compounds as useful precursors in materials science. Chemical Society Reviews, 2007. 36(11): p. 1770 -1787. 14 18. Vest, R.W., Electronic Films From Met allo-Organic Precursors. Ceramic Films and Coatings, edited by JB Wachtman and Richard A. Haber (Noyes Publications, Westwood, NJ, 1993), 1993: p. 303 -347. 19. Teng, K. and P. Wu, Metallo -organic decomposition for superconductive YBa/sub 2/Cu/sub 3/O/sub 7 -x/film. IEEE Transactions on Components, Hybrids, and Manufacturing Technology, 1989. 12(1): p. 96 -98. 20. Beckel, D., et al., Spray pyrolysis of La 0.6 Sr 0.4 Co 0.2 Fe 0.8 O 3 -δ thin film cathodes. Journal of electroceramics, 2006. 16(3): p. 221 -228. 21. Neagu, R., et al., Initial stages in zirconia coatings using ESD. Chemistry of materials, 2005. 17(4): p. 902 -910. 22. Messing, G.L., S.C. Zhang, and G.V. Jayanthi, Ceramic powder synthesis by spray pyrolysis. 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Nguyet, D.T.T., et al., Temperature -dependent magnetic properties of yttrium iron garnet nanoparticles prepared b y citrate sol –gel. Journal of Alloys and Compounds, 2012. 541: p. 18 -22. 29. Vajargah, S.H., H.M. Hosseini, and Z. Nemati, Preparation and characterization of yttrium iron garnet (YIG) nanocrystalline powders by auto -combustion of nitrate -citrate gel. Jour nal of Alloys and Compounds, 2007. 430(1): p. 339-343. 30. Schneller, T. and D. Griesche, Carboxylate Based Precursor Systems , in Chemical Solution Deposition of Functional Oxide Thin Films . 2013, Springer. p. 29 -49. 31. Moreno, E., et al., Preparation of narrow size distribution superparamagnetic γ -Fe2O3 nanoparticles in a sol− gel transparent SiO2 matrix. Langmuir, 2002. 18(12): p. 4972 -4978.

2018-11-29

The crystallization and magnetic behavior of yttrium iron garnet (YIG) prepared by metallo-organic decomposition (MOD) method are discussed. The chemistry and physics related to synthesis of iron and yttrium carboxylates based on 2-ethylhexanoic acid (2EHA) are studied, since no literature was found which elucidates synthesis of metallo-organic precursor of YIG in spite of the literatures of doped YIG samples such as Bi-YIG. Typically, the metal carboxylates used in preparation of ceramic oxide materials are 2-ethylhexanoate (2EH) solvents. Herein, the synthesis, thermal behavior and solubility of yttrium and iron 2EH used in synthesis of YIG powder by MOD are reported. The crystallization and magnetic parameters, including saturation magnetization and coercivity of these samples, smoothly change as a function of the annealing temperature. It is observed that high sintering temperature of 1300 to 1400 {\deg}C promotes the diffraction peaks of YIG, therefore, we can conclude that the formation of YIG in MOD method increases the crystallization temperature. The maximum value of saturation magnetization and minimum value of coercivity and remanence are observed for the sample sintered at 1200{\deg}C which are 13.7 emu/g, 10.38 Oe and 1.5 emu/g, respectively. This study cites the drawbacks in chemical synthesis of metallo-organic based YIG production.

Structural and Magnetic Study of Metallo-Organic YIG Powder Using 2-ethylhexanoate Carboxylate Based Precursors

1811.12514v1

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 explore a magnetooptomechanical system consisting of a single magnetic microparticle optically levitated within the core of a helically twisted single-ring hollow-core photonic crystal fibre. We use newly-developed magnetic particles that have a core of antiferromagnetic yttrium-ortho-ferrite (YFeO3) and a shell of ferrimagnetic YIG (Y3Fe5O12) approximately 50 nm thick. Using a 632.8 nm probe beam, we observe optical-torque-induced rotation of the particle and rotation of the magnetization vector in presence of an external static magnetic field. This one-of-a-kind platform opens a path to novel investigations of optomagnetic physics with levitated magnetic particles. The optical tweezer technique has revolutionized our ability to trap and manipulate mesoscopic particles. [1–3] Magnetic fields provide an additional tool for manipulating tweezered magnetic particles. [4] Strong trapping of micro/nanoparticles in free-space requires a tightly-focused laser beam, whose depth of focus is limited by the Rayleigh length; to maintain a linear trap over long distances, the trapping beam must be both tightly focused and diffraction-free, which are conflicting requirements. [5] Hollow-core photonic crystal fibre (HC-PCF) uniquely provides a means of achieving a linear trap [5–8], light being confined to the hollow core either by a photonic bandgap [9] or by anti-resonant reflection. [10] The hollow core also provides a protected environment that can be adjusted through the addition of gases or liquids, or by evacuation. [11,12] This is an asset in studies of the effects of particle birefringence on the optical forces [13] and in more applied experiments such as living cell delivery [14] or thermal sensing using doped particles. [15] In this paper we report optical trapping of magnetic microparticles and investigate their response to an external magnetic field. We focus on particles formed from yttrium-iron-garnet (YIG), which is a dielectric with a strong magnetic response. [16] Although it is transparent in the infrared, its relatively high refractive index (2.2 at 1064 nm) makes it challenging to trap optically. [17,18] The typical core diameter of HC-PCF is a few tens of µm, so that µm-sized particles can be conveniently accommodated. [5,7] Although it is possible to trap smaller (~100 nm-scale) particles, the overlap between light and particle is very small, making experiments difficult. Various techniques have been used to synthesize nm-scale YIG particles, such as co-precipitation, [19] sol-gel, [20] micro-emulsion, [21] microwave irradiation, [22] and traditional solid–state reaction methods. [23] Although current synthesis techniques permit a degree of control of particle size, they do not yet allow control of particle shape and surface roughness, which is irregular and unpredictable. This prevents the formation of high-Q internal optical resonances which are needed to enhance the weak photon-magnon coupling. [4] The particles used in the experiments have a hybrid core-shell structure, the shell being a layer of cubic ferrimagnetic YIG (Y3Fe5O12) approximately 50 nm thick, and the core a sphere of biaxial antiferromagnetic yttrium-ortho-ferrite (YFeO3). They are propelled into a chiral [24] single-ring HC-PCF [25,26] using a dual-beam trapping scheme [7]. Since the hybrid particles are on average optically biaxial, they experience a torque when subject to a linearly polarized light beam, causing them to align along the electric field of the light. Subsequent application of a static external magnetic field results in anisotropic changes in magnetic permeability that in turn cause anisotropic changes in complex refractive index (the Voigt effect [27–29]) that are probed using HeNe laser light at 632.8 nm. Particle synthesis was carried out in two stages (Fig. 1a and methods in SM). First, the yttrium and iron molecular precursors were solubilized in N, N-dimethylformamide (DMF) and after the addition of the nonionic surfactant sorbitan monooleate- Span 80, the solution was aged under solvothermal conditions at 200°C for 6 hours. The surfactant Span 80 is well known to form micelles in polar solvents and has been used to synthesize porous alumina microparticles with interconnected pores, [30] TiO2 microspheres, [31] gold nanoparticles, [32] BaTiO3, [33] and to control the growth of CaCO3. [34] The solid obtained was then isolated, washed, dried, resulting in amorphous spherical particles. In a second step the particles were calcined in air at temperatures between 700°C and 1000°C for 8 hours (Fig. 1a). The final particles were characterized by powder x-ray diffraction (XRD), scanning and transmission electron microscopy (SEM and TEM), high resolution-TEM, selected area electron diffraction (SAED) and energy-dispersive X-ray spectroscopy (EDS). In Fig. 1b powder diffraction of the particles calcined at 1000°C reveals the co-presence of three phases: orthorhombic YFeO3 (ICSD 43260), cubic YIG (ICSD:14342), and minor quantities of cubic phase Y2O3 (ICSD:33648). It was observed that by increasing the calcination temperature, the proportion of the YIG phase increases (see Methods in SM). SEM characterization of particles isolated at 1000°C showed that the majority of them had a spherical shape with median diameter x50,0 of 1.33±0.5 µm (Fig. 1b, c). Individual particles appear to have a rough and porous surface (Fig. 1b, c). We analysed particles calcined at 700°C by spin-coating them on to a standard silicon wafer and sputtering them with 200 nm thick gold layer. We then cut them open using a Ga focused ion beam (FIB) in a Zeiss NVision 40. The FIB probe was set to 30 kV-80 pA. FIB-cut SEM images of the cut-open particles reveal a complex morphology (Fig. 1d). Their structure is egg-like, consisting of an external shell and a porous interior core. The composition of the different parts of the particles were analysed using EDX in conjunction with SAED and HR-TEM, revealing that the thin shell is made of YIG and the core contains YFeO3 (Fig. 1g-n). There is no specific crystal orientation relationship between the shell and core. In the optical experiments we trapped particles calcined at 700°C in a conventional dual-beam trap (Fig. 2). Light from a continuous-wave ytterbium fibre laser (Keopsys CYFL-KILO) delivering 3W at 1064 nm was used for the trapping beams. The light was split at a polarizing beam splitter (PBS) and coupled into the LP!"-like mode at both ends of a 7-cm-long chiral “single-ring” HC-PCF with core diameter 44 μm. This fibre has weak circular birefringence 𝐵#~10$% and is optically active, i.e., the electric field of a linearly polarized signal rotates slowly with distance while remaining linearly polarized, travelling around the equator of the Poincaré sphere. [24] Over the 7-cm length of the fibre this rotation is Fig. 1: Fabrication and characterization of magnetic YIG micron-sized particles. (a) Schematic of the synthesis of the particles. (b,c) SEM of the fabricated particles calcined at 1000°C and 700°C respectively, (d) FIB-cut of two particles calcined at 700°C revealing a consistent core-shell structure. (e) Powder x-ray diffraction (XRD) for particles fabricated at 1000°C revealing co-presence of three different crystalline materials. (f) HR-TEM image of a single particle calcined at 700°C. (g) High-resolution zoom of the shell region in the HR-TEM image and (h) the corresponding fast Fourier transform (FFT) revealing the cubic phase of YIG. (i) SAED pattern of the shell demonstrating cubic phase of YIG, (j) SAED of the core showing orthorhombic phase of YFeO3. however very small so can be neglected. Precise preservation of linear polarization state is essential since we wish to probe small magnetically-induced anisotropic changes in complex dielectric constant. The fibre was mounted in a V-groove inside a custom-built low-pressure chamber with a transparent acrylic lid so as to allow access to light side-scattered by the particle as it is propelled along the fibre. Fig. 2: Schematic of the experimental setup. HWP: half-wave plate, PBS: polarizing beam splitter, DM: dichroic mirror, NF: notch filter at 1064 nm, BP: 1 nm bandpass filter centred at 632.8 nm. The distance between the end of the magnet and the fibre is 𝑑. Inset: Scanning electron micrograph showing the cross-section of the single-ring HC-PCF. The core diameter of the fiber is 44 μm and the outer diameter ~270 μm. The motion sensor is a quadrant photodiode. Particles were launched using the aerosol method. [7] The particles are first dispersed in a 50-50 mixture of a span-80 and water. A medical nebulizer was used to produce small aerosol droplets which were then delivered through an inlet placed above the fibre input face until one of the particles was trapped in front of the fibre. Once trapped at the entrance of the HC-PCF, a particle could be held there long-term and propelled into the core by momentarily lowering the power of the counter-propagating trapping beam. The trapping success rate with these particles was close to 100%. The motion of the bound particle along the fibre was imaged using a high-speed camera (Mikrotron EoSens mini2) placed above the chamber and a quadrant detector (Thorlabs PDQ30C) connected to an oscilloscope (PicoScope 3406B). A magnet was mounted on a translation stage so as to allow the magnetic field strength to be varied. Fig. 3: (a) Snapshot of an optically trapped magnetic particle inside the core of a HC-PCF captured with a high-speed camera. (b) Spectrum of the damped mechanical motion of the bound magnetic particle at a pressure of 2 mbar. The laser power is 3 W. (c) Measured spectral linewidth (FWHM) as a function of the environment pressure at a fixed laser power of 3 W. (d) Measured central frequency as a function of laser power at 6 mbar pressure. The red line is the theoretical prediction according to Eq. (1). First, we tested the limits of the levitated system by evacuating the chamber with a particle already trapped in the fibre core, as shown in Fig. 3a. At pressures below ~1 mbar the particle escaped from the optical trap and was lost, which we attribute to the onset of the ballistic regime caused by the increased molecular mean-free path. [35] In the transverse direction the trapped particle behaves like a damped mechanical oscillator, driven by Brownian motion. [12] From time-domain data recorded with the position-sensitive quadrant detector we extracted the Lorentzian spectrum of the particle motion (Fig. 3b). As the gas pressure decreases, the viscosity falls, and the linewidth narrows. The relationship between the spectral linewidth Γ and the air pressure 𝑝 follows the Knudsen relation: Γ=𝛾𝑚=12𝜋𝑅𝑚𝜂!21+𝐾&(𝑝)6𝛽"+𝛽'𝑒$(!/*"(,)9:; (1) where 𝛾 is the damping coefficient caused by viscosity, 2𝑅 is a characteristic length (the particle diameter), and m is the particle mass (densities of YIG and YFeO3 are 5.11 and 5.47 g/cm3 respectively) and 𝐾&(𝑝)= 66×10$.(𝑝𝑅)⁄ is the Knudsen number, 𝜂!=18.1 µPa×s is the viscosity of air at atmospheric pressure and 𝛽"=1.231, 𝛽'=0.469 and 𝛽/=1.178 are dimensionless constants. [7] Figure 3(c) plots the measured spectral linewidth as a function of pressure, and the red line is a fit to Eq. (1). The trap stiffness, which governs the resonant frequency, is controlled by the trapping laser power 𝑃. At a fixed pressure (6 mbar in the experiment) the resonant frequency increases linearly with the laser power, as expected (Fig. 3d). Crystalline YFeO3 is orthorhombic and biaxial, displaying optical birefringence, which means that the linearly polarized trapping beam can be used as an optical spanner, [13] permitting measurements to be made as a function of particle orientation. The shell of the particles is formed from YIG, which is cubic and isotropic, becoming optically biaxial when a magnetic field is applied parallel to the (110) crystallographic plane (for details refer to SM). Both crystals are strongly absorbing at 632.8 nm, offering a simple means of probing magnetically induced changes in complex refractive index by monitoring the power and polarization state of the transmitted probe light [27,29,36]. The on-axis magnetic flux of the NdFeB permanent magnet (N35, cross-section 4×4 mm) used in the experiments is plotted in Fig. 6 as a function of the distance from the magnet’s end-face. The magnet was placed with its N-S axis oriented perpendicular to the fibre axis and centred on the trapped particle (Fig. 1), and a motorized translation stage was used to vary the distance 𝑑 between the magnet and the particle. Probe light was provided by a linearly polarized HeNe laser (2 mW, 632.8 nm). The transmitted trapping beam light was filtered out using a combination of dichroic mirror, 1 nm bandpass filter centred at 632.8 nm, and 2 nm notch filter centred at 1064 nm (Fig. 1). In the experiments, both the power and the polarization state of the probe beam was monitored. Fig. 4: Transmitted probe power as the particle is rotated inside the HC-PCF and subject to a constant magnetic field of 29 mT. The red dotted curve is a heuristic fit to Eq. (3). The opto-magnetic response was first investigated by applying a constant magnetic field (B = 29 mT) and rotating the particle by rotating the trapping beam polarization [13]. The transmitted probe power was directly monitored using both a polarimeter and a lock-in amplifier. Figure 4 plots the transmitted probe power as a function of the orientation of the trapping electric field 𝜃, where 𝜃=0° when the trapping and probe beams are co-polarized. For each value of 𝜃 we made repeated measurements of the transmitted power and evaluated the mean (blue dot) and standard deviation (error bar). Over 180° the data shows two distinct peaks, which we attribute to the complex-valued biaxial refractive index of the particle. In the case of pure YIG crystal, assuming its magnetization vector points in the (110) plane, the imaginary part of the dielectric susceptibility can be written in the form [27,28]: Δ𝜒0(𝜃)=𝑀1'𝑛2Q𝑔33+Δ𝑔16(3+2cos2𝜃+3cos4𝜃)U (2) where 𝜃 is angle between the magnetization vector 𝑀VV⃗ and [001] crystal axis (see SM for detail), 𝑛2 is the wavelength-dependent real part of the refractive index of YIG, 𝑀4 is its saturation magnetization, Δ𝑔=𝑔""−𝑔"'−2𝑔33, and 𝑔"",𝑔"' and 𝑔33 are complex numbers representing the non-vanishing tensor elements of the dielectric tensor induced by a magnetic field and causing biaxial linear birefringence and (in the visible) dichroism. Since in our case the particle is a complex hybrid of YIG and YFeO3, the system cannot be so easily modelled. Moreover, when a new particle is launched into the trap, the initial orientations of its optical axes as well as the magnetisation axis are unknown. However, a heuristic fit to the power measurement can be made using a similar function with an added phase shift of 𝜓: 𝑃(𝜃)∝a [1+bcos2(𝜃−𝜓)+ccos4(𝜃−𝜓)] (3) where the phase 𝜓=−50° is added to the angle 𝜃, which in our experiment is the angle between probe and pump beam polarization. We note that 𝑎 represents the average power of our dataset which is 70.8 µW. The other coefficients are respectively 𝑏=2.12×10$/ and 𝑐=4.1×10$/. The Eq. (3) qualitatively fits to the data, as seen in the red dashed curve in Fig. 4. The individual magnetooptomechanic response of each particle is slightly different due to its initial orientation, though they all follow the same general trend, exhibiting two maxima (Fig. 4). The first peak occurs at 𝜃≃50° (Fig. 4), which is in reasonable agreement with the values for pure YIG (50°) and YFeO3 (45°) (see SM for more details). [37] Fig. 5: Transmitted power (blue axis) as the magnet is moved inwards towards the particle. The angle between the orientation of the linearly polarized probe beam and the applied external field B is 19° in (a) and 45° in (b). The red dashed line is a heuristic fitting using Eq. (3) Next, we kept the particle stationary and moved the magnet inwards, keeping the angle between the magnetic field and the probe beam polarization (𝐸,2567) fixed to either 19° or 45°(Fig. 5 inset). In both cases, the magnetic field is orthogonal to the polarization of the trapping beam (Fig. 2). At 𝑑=23 mm, the magnetic field 𝐵V⃗ at the particle is very weak and the magnetization vector 𝑀VV⃗ is unaffected. As the magnet approaches the particle, 𝑀VV⃗ gradually rotates until it aligns almost parallel to 𝐵V⃗ at 𝑑=3 mm. At present, we cannot distinguish between the physical rotation of the particle from the rotation of only the magnetization vector, however they will give rise to same experimental result. Figure 5(a) shows the variation of the transmitted probe beam power with respect to 𝑑 when the angle between magnetic field and the probe beam polarisation is 19° and Fig. 5(b) shows the variation when this angle is 45° as shown in the insets of the figures. The angle 𝜃 by which magnetization vector rotates depends inversely on 𝑑 allowing us to heuristically fit the experimental data with 𝑃(𝜃). For fitting of the data in Fig. 5(a), the added phase shift 𝜓 to 𝑃(𝜃) in Eq. (3) is 𝜓=90° and 𝑎=96 µW, 𝑏=5.73×10$/ and 𝑐=0.098. For Fig. 5(b), the added phase shift is 𝜓= −22.5° and the parameters 𝑎=98 µW, 𝑏=1.53×10$/ and 𝑐=3.98×10$/ respectively. These plots show a behaviour similar to that in Fig. 4, from which we deduce that the imaginary part of the dielectric susceptibility is being probed as a function of the rotation angle 𝜃 and distance d. We also observed that once the external magnetic field was strong enough to align the magnetization vector parallel to itself, the transmitted probe power and polarisation state no longer responded to changes in magnetic field strength (see SM). A similar response could be recovered by moving the particle along the fibre out of the magnetic field and then returning it. In summary, spheroidal µm-sized magnetic particles with a ~50 nm shell of YIG and a core of FeO3 were synthesized. The relative proportion of the two materials could be adjusted by running the calcination process at different temperatures. The particles could be reproducibly trapped long-term in the evacuated HC-PCF core. Measurements with a 632.8 nm probe beam and a single µm-diameter particle reveal detectible changes in the transmitted power and the polarization state. The system is suitable for a wide variety of different applications, such as remote magnetic field sensing [36], interactions between waveguide modes, and the study of rotational degrees of freedom and spin waves in optomechanically cooled resonators [38]. The reported results suggest new possibilities for experiments in particle-based magneto-optomechanical physics., including cooling down to the single quantum regime [38], possibly at room temperature. Methods The particles were fabricated in two steps. At first, an yttrium molecular precursor (Y(NO3)3·6H2O) (1.15 g, 3 mmol) and an iron molecular precursor (Fe(acac)3) (1.8 g, 5.10 mmol) were solubilized at room temperature in 50 mL N,N-dimethylformamide (DMF) and the surfactant sorbitan monooleate - Span 80 was added during stirring. The as-obtained solution was transferred in a Teflon liner and aged in a stainless-steel autoclave at 200°C for 6 hours. Upon cooling to room temperature, toluene was added to the reaction mixture to induce precipitation. The solid was isolated by centrifugation and washed by three redispersion and centrifugation cycles. Finally, it was dried in air at 60°C for 20 hours. The product at this stage comprised amorphous spherical particles. The second step of the fabrication procedure comprised a calcination process in air at temperature between 700 and 1000 °C for 8 hours. This second part leads to an amorphous to crystalline phase transition during which the particles become magnetic. The magnetic field was produced by eight 4×4×3 mm/ NdFeB N35 permanent magnets placed in a row. The on-axis magnetic flux density was measured with a Gaussmeter as a function of d, the distance from the end-face of the magnet (Fig. 6). Fig. 6: Measured magnetic flux density B (mT) of the NdFeB magnet as a function of distance, along the N-S axis, from the one of the poles. References 1. A. 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N. Kolpakova, and Y. M. Yakovlev, "Magnetic birefringence of light in iron garnets," Sov. Phys.-JETP 33, 1175–1182 (1971). 28. J. Ferre and G. A. Gehring, "Linear optical birefringence of magnetic crystals," Rep. Prog. Phys. 47, 513 (1984). 29. W. Wettling, "Magnetooptical properties of YIG measured on a continuously working spectrometer," Appl. Phys. 6, 367–372 (1975). 30. H. Yang, Y. Xie, G. Hao, W. Cai, and X. Guo, "Preparation of porous alumina microspheres via an oil-in-water emulsion method accompanied by a sol–gel process," New J. Chem. 40, 589–595 (2016). 31. Z. Li, M. Kawashita, and M. Doi, "Sol–gel synthesis and characterization of magnetic TiO2 microspheres," J. Ceram. Soc. Jpn 118, 467–473 (2010). 32. C.-L. Chiang, "Controlled Growth of Gold Nanoparticles in Aerosol-OT/Sorbitan Monooleate/Isooctane Mixed Reverse Micelles," J. Colloid Interface Sci. 230, 60–66 (2000). 33. R. Savo, A. Morandi, J. S. Müller, F. Kaufmann, F. Timpu, M. Reig Escalé, M. Zanini, L. Isa, and R. 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2024-04-24

We explore a magnetooptomechanical system consisting of a single magnetic microparticle optically levitated within the core of a helically twisted single-ring hollow-core photonic crystal fibre. We use newly-developed magnetic particles that have a core of antiferromagnetic yttrium-ortho-ferrite (YFeO3) and a shell of ferrimagnetic YIG (Y3Fe5O12) approximately 50 nm thick. Using a 632.8 nm probe beam, we observe optical-torque-induced rotation of the particle and rotation of the magnetization vector in presence of an external static magnetic field. This one-of-a-kind platform opens a path to novel investigations of optomagnetic physics with levitated magnetic particles.

Optomagnetic forces on YIG/YFeO3 microspheres levitated in chiral hollow-core photonic crystal fibre

2404.16182v1

1 Effect of magnons on interfacial phonon drag in YIG/metal systems Arati Prakash1, Jack Brangham1, Sarah J. Watzman2, Fengyuan Yang1, Joseph P. Heremans1,2,3 1 Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA 2 Department of Mechanical Engineering, The Ohio State University, Columbus, Ohio 43210, USA 3 Department of Materials Science and Engineering, The Ohio State University, Columbus, Ohio 43210, USA Abstract We examine substrate -to-film interfacial phonon dra g on typical spin Seebeck heterostructures, in particular studying the effect of ferromagnetic magnons on the ph onon -electron drag dynamics at the interface . We investigate with high precision the effect of magnons in the Pt|YIG heterostructure by designing a magnon drag thermocouple; a hybrid sample with both a Pt|YIG film and Pt|GGG interf ace accessible isothermally via a 6 nm Pt film patterned in a rectangular U shape with one arm on the 250 nm YIG film and the other on GGG. We measure t he voltage between the isothermal ends of the U, while applying a temperature gradient parallel to the arms and perpendicular to the bottom connection. With a uniform applied temperature gradient, the Pt acts as a differential thermocouple. We conduct temperature -dependent longitudinal thermopower measurements on this sample. Results show that the YIG interface actually decreases the thermopower of the film, implying that magnons impede phonon drag. We repeat the experiment using metals with lo w spin Hall angles, Ag and Al, in place of Pt. We find that the phonon drag peak in thermopower is killed in samples where the metallic interface is with YIG. We also investigate magneto -thermopower and YIG film thickness dependence. These measurements confirm our findings that magnons impede the phonon -electron drag interaction at the metallic interface in these heterostructures. 2 Introduction In this study, we focus on the longitudinal thermopower , αxxx, of a typical spin Seebeck heterostructure, i.e. P t|YIG . In this configuration, a temperature bias is applied in the direction parallel to the direction that voltage is measured on the metal, i.e. along the direction of the interface. In comparison to an SSE measurement, in these studies we will essentia lly turn the sample on its side relative to the applied heat flux (see Figure 1). Specifically, we are interested in studying the influence of magnons on the thermopower of the Pt|YIG heterostructure. We s peculate that magnons at the Pt|YIG interface coul d exert a drag-like force on the electrons in the Pt, either directly or via an interaction mediated by phonons in YIG, and we endeavor to measure and discern this effect. In fact, when we aim to elucidate the parameter space of the SSE, which arises from magnon -phonon interactions, the question of drag between magnons, phonons and electrons in the Pt|YIG heterostructure is highly pertinent and is motivated by several contemporary studies, both experimental and theoretical. Nonlocal drag (i.e. interfacial, between substrate and thin film) has been studied both in theory and experiment.1,2 Ref. [ 1] examines nonlocal phonon -electron drag between an insulating sapphire substrate and a Bi 2Te3 thin film , show ing that the temperature dependent thermopower of the Bi 2Te3 follows that of the t hermal conductivity of the sapphire substrate, which implies that electronic transport in the film is enhanced by phononic transport in the sapphire substrate (phonon -electron drag). This demonstrates that substrate -to-thin-film phonon -electron drag can occur even when the two layers have dissimilar crystal structure. From this study, we would expect that there could be strong phonon -electron drag affecting the thermopower on either Pt|GGG or 3 Pt|YIG films. Furthermore, Ref. [ 2] predicts nonlocal magnon -magnon drag in a FM bilayer, arising from dipolar interactions across the interface. Local drag effects (i.e. bulk, not interfacial) have also been studied: local magnon -electron drag (MED), or the advective transport of electrons dragged by magnons, has been examined in bulk metallic ferromagnets, where a thermal gradient drives magnons and electrons along with phonons.3 While all metals contain drag and diffusive contributions to their thermopower, it was shown that the MED contribution actually dominated the thermopower in ferromagnetic metals (Fe, Co). The findings of thermopowers enhanced by magnon drag in Ref. [ 3] and of phonon drag dominating the thermopower even in dissimilar substrate -to-thin film heterostructures in Ref. [ 1], combined with the theoretical predictions in Ref. [ 2] that magnons can also participate in such nonlocal, interfacial drag effects, together inspire a look at magnon effects on the interfacial phonon drag on the Pt|YIG heterostructure. Instead of an out of plane temperature gradient, which drive s a nonloca l spin flux across the interface, we apply an in plane temperature gradient longitudinally as the driving force and examine magnon transport along this direction. We probe these dynamics by measurements of the longitudinal thermopower on the Pt film of Pt|YIG and similar heterostructures . As per Ref. [ 1], one would expect this thermopower as measured on the thin film to follow the thermal conductivi ty of the substrate (YIG or GGG) due to the interfacial phonon -electron drag. However, the question of interest here is what is the effect of magnons on this phonon electron drag? To study this, we compare the thermopower of Pt films grown on ferrimagneti c YIG to that grown on paramagnetic GGG , where the only difference in substrate dynamics can be attributed purely to magnons in the YIG. In order t o isolate the hypothetical drag contribution from the magnons in YIG into the adjacent Pt film, we design a thermocouple device using a hybrid sample 4 with half Pt|GGG and half Pt|YIG(250nm)|GGG (see Figure 2) . With a uniform applied temperature gradient, t he Pt acts as a differential thermocouple. The effective voltage at the isothermal ends of the Pt provides a direct measure of the difference in thermopower of the two systems, which we attribute to magnon dynamics in YIG and their interactions at the Pt|Y IG interface. The effective voltage at the isothermal ends of the Pt provides a direct measure of the difference in thermopower of the two systems, which we attribute to magnon dynamics in YIG and their interactions with phonons and electrons at the Pt|YIG interface. Since the Pt|YIG (or Pt|YIG|GGG) heterostructure is the typical system used to examine spin thermal effects , we primarily focus on that system here. However, the large spin orbit coupling in Pt (which is precisely what makes it advantageous as the ISHE layer for SSE devices) could raise the question of contributions to the thermopower from SHE signals, especially in the presence of a magnetic field. To isolate any contribution from spin Hall effects, we repeat the experiment on similar heterost ructures where the Pt metal is replaced by metals with rather low spin Hall angles. To examine magnonic effects on phonon electron drag (phonons in the YIG, electrons in the m etallic thin film), we choose metals with relatively simple, clean Fermi surfaces as far as possible. We choose p-type Ag and n -type Al . When examining these metals in their thin film form to see how they interact with a substrate, we consider previous knowledge regarding the thermopower of these materials. Temperature dependent thermo power data for these bulk metals were measured decades back .4,5 Although n -type, with a negative diffusion thermopower of -5 μV/K at 300 K, Pt exhibits a sign change in its thermopower around 200 K, with a positive phonon drag thermopower peak at 6 μV/K.4 In contrast to this, Ag has a consistently positive (p -type) thermopower with a phonon drag peak near 1 μV/K and Al has a consistently negative ( n-type) 5 thermopower with a phonon drag peak near -2 μV/K.5 These values inform our interpretation of data in the context of relative strength of t he thermopower measure d here. Experiment In order to circumvent the influence of sample to sample variability on our measurements, we devise a hybrid h eterostructure on which we can make a differential measurement α Pt|YIG vs. αPt|GGG on one sample in situ . We name this hybrid heterostructure the magnon drag thermocouple (MDT). The MDT consists of 3 layers: a GGG substrate, a YIG film (250 nm) grown on half of the substrate with a gradually stepped edge at the longitudinal center fold of the sample, and a P t film (6 nm) deposited across the entire structure then patterned into a squared -U shape with four corners (A, B, C, and D). In addition to the MDT for Pt, we construct an identical MDT for Ag (10 nm) and Al (6 nm). A list of MDT samples created can be fo und in Table 1. All samples were measured for steady state, zero field thermopower, αxxx, in the static - heat-sink configuration using a Quantum Design PPMS as in Ref. [ 3], with thermometry and gold plated copper leads attached to the back face of each substrate. Voltage probes to measure thermopower were attached via small (~25 μm) Ag epoxy contacts placed di rectly on the thin film. With a temperature gradient applied longitudinally, the two ar ms (AB and DC) comprise the therm ocouple on which a differential voltage (Vad) can be measured . The hybrid heterostructure acts effectively as a thermocouple for the Pt interface: because both sides of the bottom bar (B and C) are isothermal with an applied longitudinal temperature g radient, any voltage measured across AD would be due to a difference in the voltage drop across arm AB vs arm DC, i.e., a difference in the i nterfacial thermopower depending on whether or not YIG is present. 6 One can reasonably ask the question of the influence of the bottom bar of the U on the signal. Depending on the direction of the applied magnetic field, this would correspond to a Nernst - like configuration or transverse spin Seebeck effect (see Figure 3 ). Upon the addition of an applied magnetic field, such a point becomes relevant. This question is simply addressed by a measurement across the bar, revealing little -to-none signal on the V bc channel, a result which could also have been predicted noting that the Nernst effect is ten times smaller (often 1 in 2000) than the Seebeck effect in metals.6 Sample Film Deposition Pt(6 nm)|half YIG(250nm)|GGG U Al(6 nm)|half YIG(250 nm)|GGG U Ag(6 nm)|half YIG(250 nm)|GGG U Pt(6 nm)|GGG (control) U (no half -YIG) Table 1. Directory of magnon drag thermocouple (MDT) sample s. Results In order to characterize our P t films, we measure temperature dependence of the resistivity using the standard AC Transport option in the PPMS. We measure in zero field and at 7 Tesla applied magnetic field; results show no anomalies and the resistance behaves as expected (see Figure 4) . We also measure the thermal conductivity of every substrate used in t his study, in situ with the Seebeck measurements. An example of thermal conductivi ty of GGG is shown in Figure 5 . These measurements help to keep track of sample quality and check for induced defects as the study goes 7 on. As is characteristic for phonon th ermal conductivity, there is a low temperature drop off, where the density of carriers decreases as T3. The high temperature drop is attributed to intrinsic phonon - phonon Umklapp scattering, and t he low temperature drop is attributed to phonon -boundary scattering.7 The intermediate temperature range is where phonon transport is maximum and where the phonon drag peak in the thermopowers is expected to be maximal. To test of the validity of the MDT, we measure a control sample of a Pt U deposited on a GGG su bstrate, with no half -YIG film. Although in principle there should clearly be no signal on Vad of such a sample, this measurement demonstrates the validity of the assumption of the U as a reliable thermocouple in practice. Measurements confirm there is no spurious signal from the bottom bar, and that V ad is isovoltaic in the absence of YIG. The low temperature measurements from this control reveal the baseline noise level of the experiment, on the order of 1 µV/K below 6 K. Seebeck measurements from the MDT in differential mode (V ad) are shown in Figure 6 . The data show a temperature dependence roughly following that of the thermal conductivity of the substrate GGG and a positive peak near 8 μV/K. It is worth noting carefully that measurements of the MDT in differential mode actually give the thermopower of Pt on YIG subtracted from the thermopower of Pt on GGG, considering the cold side to be the positive voltage terminal, as is conventional in Seebeck measurements. This means that the positive signal on the Pt MDT in Figure 6 implies that the magnons lower the thermopower of the Pt on YIG relative to the Pt on GGG. This result is surprising, as it implies that magnons at the interface may actually be suppressing the drag effects across the Pt interface. Next, we examine the differential thermopower of the Ag and Al MDTs. Two observations are immediately evident : 1) the thermopower magnitudes of each of the films exceed t hose of their 8 bulk counterparts and 2) t heir temperature dependence roughly follows the rmal conductivity of substrate, implying phonon electron drag, substrate to film. A positive low temperature peak is around 4 μV/K on the Ag sample, and a negative thermopower with a peak is around -14 μV/K in the Al. Given that Ag is p type and Al is n ty pe, we find that the polarity of the effect matches that of the sign of the carrier in the metal. This verifies that the measured thermopower voltage is not related to spin Hall physics, but rather electron drag by phonons. Thus, t hese results are consiste nt with the results on the Pt sample; the YIG yields a lower signal than the GGG interface, implying that the magnons interfere with phonon drag. As a follow up to this observation, we conduct magnetic field -dependent measurements of the the rmopower on th e Pt and Ag MDT s, as shown in Figure 7 . Applied magnetic fields are expected to suppress or “freeze out” magnon dynamics,8 with a more pronounced effect at low temperatures.9 At lowest temperatures, where one would expect the effect of the magnetic field to be strongest (6 to 9 K) the interfacial thermopower itself is difficult to resolve, so that a field dependent study is difficult to obtain. At moderately low temperatures w here signal is strongest, near the phonon drag peak in thermopower, this magnetic field effect is measurable. As the magnetic field is increased from 0 to 9 Tesla at 10 K, we observe a decrease in thermopower on both Pt and Ag films . A decrease in thermopo wer implies a decrease in signa l as magnon dynamics are suppressed at large magnetic fields so that both YIG and GGG exert th e same amount of drag on the Pt. T his effect is more pronounced at low temperatures, but where signal is still well resolvable from the noise floor, which becomes difficult below around 7 K. By contrast, we also measure the magneto -Seebeck coefficient of a Pt|YIG (250) sample. Here, we observe an increase in thermopower as magnetic field is increased from 0 to 9 Tesla below 30 K, as shown in Figure 8. An increase in thermopower implies a recovery of signal as magnon dynamics are suppressed 9 out at large magnetic fields; this result is consistent with the results in Refs. [ 8,9] and differential measurements from the Pt MDT , supporting the conclusion that magnons interfere with phonon drag in these heterostructures . Having isolated that there is a magnonic impedance to the phonon drag effect, we explore the length scale of this effect. In particular, as we decrease YIG thickness, there should be less magnons available, so t hat the Pt|YIG thermopower should increase and ultimately for very thin YIG, match that of Pt|GGG. We now measure a series of Pt|YIG samples with YIG o f varied thickness (bulk, 1 μm, 250 nm, 100 nm, 40 nm). The bulk sample behaves much like the 250 nm YIG sample. At 100 nm, the signal increases, and at 40 nm, the phonon drag peak in the Pt|YIG thermopower is nearly equivalent to that of the Pt|GGG. In the 1 μm samples, the phonon drag peak is killed altogether, but the diffusion thermopower (high temperatur es) equilibrates for all samples above around 100 K. The implications of these data are summarized as follows. Substrate -to-thin film phonon electron drag is observed on Pt|YIG and Pt|GGG with equal magnitudes at high temperatures. The phonon drag peak in thermopower is significantly attenuated in the metal when YIG is present, so that this attenuation is attributed to magnons in the YIG. The thicker the YIG film, the larger the magnon scattering volume, which effectively acts as a barrier for the phonons which otherwise would drag electrons in the neighboring conducting film. At the smallest YIG thicknesses, we recover results very similar in magnitude to the signals on GGG. This length scale dependence on YIG thickness complements the identification of th e magnon energy relaxation length from Ref. [10]. In fact the magnon -to-phonon energy relaxation, or a difference between magnon and phonon temperatures, could reasonably affect scattering rates between magnons and phonons and consequently interrupt the pho non-electron drag effects that occur in the absence of magnons. 10 With an applied magnetic field at temperatures below 30 K , we observe a recovery of the phonon -electron drag as magnon dynamics are partially suppressed. The effect manifests as an increase or recovery in signal on the isolated YIG system, and a decrease in differential signal on the MDT. These observations are consistent with the magnetic field -dependent “freeze out” of magnon dynamics established in Refs. [ 8,9]: as magnons are suppressed with high magnetic fields, the Pt|YIG interface behaves more closely like the Pt| GGG. This series of measurements support our conclusion that magnons in fact inte rfere with the phonon -electron drag interaction at the metallic interface in these heterostructures. Further work should focus on developing a quantitative theoretical model for such an effect, accounting for scattering rates of magnons with phonons and el ectrons in the YIG and at the Pt interface. Acknowledgements Funding for this work was provided by the OSU Center for Emergent Materials, an NSF MRSEC, Grant DMR -1420451 and the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Grant No. DE-SC0001304. Figures 11 Figure 1. Schematic of longitudinal thermopower measurement, where voltage is measured in the direction of the temperature bias, along the direction of the interface on Pt|YIG heterostructure. Figure 2 . Schematic of magnon drag thermocouple with measured voltage in differential mode (Vad) (left ), and photograph of actual Pt|YIG(/GGG) U shaped sample (right ). Figure 3. The direction of the applied magnetic field yields either a Nernst configuration (blue) or a transverse spin Seebeck configuration (red) on the bottom bar of the MDT. 12 Figure 4. Temperature dependent resistivity of Pt on GGG in 7 T field (orange) and in zero -field (purple) shows no anomalous features and serves as an experimental check of the Pt film. Figure 5. Temperature dependent thermal conductivity of bulk single crystal GGG substrate . 13 Figure 6. Temperature dependent thermopower of Pt, Ag, and Al magnon drag thermocouples. Figure 7. Magnetic field dependence of thermopower at various temperatures indicated on the graphs in the Pt and Ag magnon drag thermocouples in differential mode (left). The relative signal can be seen as a relative decrease from the zero field value (right). 14 Figure 8. Magnetic field dependence of thermopower, at various base temperatures indicated on the graphs for the Pt|YIG interface (left). The strength of this effect can be seen as a relative increase from zero -field thermopowe r (right). Figure 9. Temperature dependence of interfacial Pt|YIG thermopower for various YIG thicknesses as shown on the graph. 15 1 G.Wang , L. Endicott, H. Chi, P. Lost’ak, and C. Uher, Phys. Rev. Lett. 111, 046803 (2013). 2 T. Liu, G. Vignale, M. E. Flatte, Phys. Rev. Lett. 116, 237202 (2016). 3 S.J. Watzman, R.A. Duine, Y. Tserkovnyak, H. Jin, A. Prakash, Y. Zheng, and J. P. Heremans, Phys. Rev. B. 94, 144407 (2016). 4 R. P. Huebner, Phys. Rev. 140, 5A (1965). 5 R. J. Gripshover, J. B. VanZytveld, and J. Bass, Phys. Rev. 163, 3 (1967). 6 S. R. Boona, R. C. Myers and J. P. Heremans, Energy Environ. Sci. 7, 885-910 (2014). 7 J. M. Ziman, Principles of the Theory of Solids, Cambridge University Press (1972). 8 T. Kikkawa, K. Uchida, S. Daimon, Z. Qiu, Y. Shiomi, and E. Saitoh, Phys. Rev. B. 92, (6), 064413 (2015). 9 H. Jin, S. R. Boona, Z. Yang, R. C. Myers, and J. P. Heremans, Phys. Rev. B . 92, 054436 (2015). 10 A. Prakash, B. Flebus, J. Brangham, F. Yang, Y. Tserkovnyak, and J.P. Heremans, Phys. Rev. B. 97, (2), 020408(R ) 2018.

2018-04-19

We examine substrate-to-film interfacial phonon drag on typical spin Seebeck heterostructures, in particular studying the effect of ferromagnetic magnons on the phonon-electron drag dynamics at the interface. We investigate with high precision the effect of magnons in the Pt|YIG heterostructure by designing a magnon drag thermocouple; a hybrid sample with both a Pt|YIG film and Pt|GGG interface accessible isothermally via a 6 nm Pt film patterned in a rectangular U shape with one arm on the 250 nm YIG film and the other on GGG. We measure the voltage between the isothermal ends of the U, while applying a temperature gradient parallel to the arms and perpendicular to the bottom connection. With a uniform applied temperature gradient, the Pt acts as a differential thermocouple. We conduct temperature-dependent longitudinal thermopower measurements on this sample. Results show that the YIG interface actually decreases the thermopower of the film, implying that magnons impede phonon drag. We repeat the experiment using metals with low spin Hall angles, Ag and Al, in place of Pt. We find that the phonon drag peak in thermopower is killed in samples where the metallic interface is with YIG. We also investigate magneto-thermopower and YIG film thickness dependence. These measurements confirm our findings that magnons impede the phonon-electron drag interaction at the metallic interface in these heterostructures.

Effect of magnons on interfacial phonon drag in YIG/metal systems

1804.07023v1

Magnetic coupling in Y 3Fe5O12/Gd 3Fe5O12heterostructures S. Becker,1,Z. Ren,1, 2, 3, †F. Fuhrmann,1A. Ross,4, 1S. Lord,1, 2, 5 S. Ding,1, 2, 6R. Wu,1J. Yang,6J. Miao,3M. Kläui,1, 2, 7and G. Jakob1, 2, ‡ 1Institute of Physics, Johannes Gutenberg-University Mainz, Staudingerweg 7, 55128 Mainz, Germany 2Graduate School of Excellence “Materials Science in Mainz” (MAINZ), Staudingerweg 9, 55128 Mainz 3School of Materials Science and Engineering, University of Science and Technology Beijing, Beijing 100083, China 4Unité Mixte de Physique CNRS, Thales, Université Paris-Saclay, 91767 Palaiseau, France 5Department of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom 6State Key Laboratory for Mesoscopic Physics, School of Physics, Peking University, Beijing 100871, China 7Center for Quantum Spintronics, Norwegian University of Science and Technology, 7491 Trondheim, Norway (Dated: April 21, 2021) Ferrimagnetic Y 3Fe5O12(YIG) is the prototypical material for studying magnonic properties due to its ex- ceptionally low damping. By substituting the yttrium with other rare earth elements that have a net magnetic moment, we can introduce an additional spin degree of freedom. Here, we study the magnetic coupling in epitax- ial Y 3Fe5O12/Gd 3Fe5O12(YIG/GIG) heterostructures grown by pulsed laser deposition. From bulk sensitive magnetometry and surface sensitive spin Seebeck effect (SSE) and spin Hall magnetoresistance (SMR) mea- surements, we determine the alignment of the heterostructure magnetization through temperature and external magnetic field. The ferromagnetic coupling between the Fe sublattices of YIG and GIG dominates the overall behavior of the heterostructures. Due to the temperature dependent gadolinium moment, a magnetic compen- sation point of the total bilayer system can be identified. This compensation point shifts to lower temperatures with increasing thickness of YIG due the parallel alignment of the iron moments. We show that we can control the magnetic properties of the heterostructures by tuning the thickness of the individual layers, opening up a large playground for magnonic devices based on coupled magnetic insulators. These devices could potentially control the magnon transport analogously to electron transport in giant magnetoresistive devices. I. INTRODUCTION A major challenge in information technology is solving the issue of Joule heating due to charge currents. One approach is to move away from electron-based to magnon-mediated in- formation transport and processing [1]. This requires the de- velopments of new logic devices, such as magnon valves that allow for the manipulation of spin currents [2]. Material can- didates require an insulating character, magnetic ordering and low magnetic damping. One promising candidate is ferrimag- netic Y 3Fe5O12(YIG = yttrium iron garnet), which is a well- known material in magnetism as it shows ultra-low magnetic damping and low magnetic anisotropy. The net ferrimagnetic moment originates from an antiparallel alignment of Fe3+mo- ments on different crystallographic sites. Each minimal unit cell consists of 12 trivalent Fe3+ions that are tetrahedrally coordinated with oxygen atoms ( dsites) and 8 trivalent Fe3+ ions that are octahedrally coordinated ( asites). The dominant coupling is antiferromagnetic between iron atoms on minor- ityaand majority dsites. By substituting Y3+with Gd3+, an additional moment appears aligned antiparallel to the d-site Fe atoms [3, 4]. Due to the strong temperature dependence of the Gd net magnetic moment, Gd 3Fe5O12(GIG) shows a magnetization compensation at temperature T295 K [5]. For low power information processing, magnons (the quanta of magnetic excitation) are exciting candidates. The magnon spectra of YIG has been the subject of both experi- svenbecker@uni-mainz.de †zengyaoren@163.com; S.B. and Z.R. contributed equally to this work ‡jakob@uni-mainz.demental and theoretical investigation [6, 7]. In heterostructures, magnon-magnon coupling and magnetic coupling play a deci- sive role for the propagation of magnons [2, 8–13]. Here, we investigate the coupling between two different iron based gar- nets YIG and GIG as candidates for an all insulator magnon spin valve. Both YIG and GIG are important ferrimagnetic insulators that can be grown epitaxially on isostructural but paramagnetic Gd 3Ga5O12(GGG) substrates. Recently, mag- netic coupling of YIG to an ultrathin GIG layer was reported [13, 14], where GIG is formed by an interdiffusion process while preparing YIG on GGG. In our work, we realize con- trolled growth of epitaxial YIG/GIG heterostructures on GGG substrates, where the individual layers have sufficient mag- netic moment to be detected by magnetometry. The bilay- ers show high crystalline quality and a magnetic compensa- tion of the entire bilayer, which shifts to lower temperature with increasing thickness of YIG, indicating interlayer mag- netic coupling of YIG and GIG. The alignment of the mag- netization of bilayers with temperature and field can be char- acterized by SQUID magnetometry, which measures the sum signal of all the layers. To identify the magnetization direc- tion, we utilize the surface sensitive spin Hall magnetoresis- tance (SMR), which has proven to be a suitable tool to inves- tigate the magnetic properties of magnetic insulators [15–18]. We further conduct spin Seebeck (SSE) measurements, which have previously been used to investigate pure GIG samples [18–23]. SSE measurements show dominating sensitivity to the top layer of the heterostructure. Our results demonstrate that the respective Fe sublattices of YIG and GIG are ferro- magnetically coupled across the interface. This allows for an unprecedented tunability of the magnetic properties of the bi- layer system by choosing the relative thickness of the YIG andarXiv:2104.09592v1 [cond-mat.mtrl-sci] 19 Apr 20212 GIG layers. II. EXPERIMENTAL DETAILS Y3Fe5O12(YIG) and Gd 3Fe5O12(GIG) are deposited on (001)-oriented Gd 3Ga5O12(GGG) substrates by pulsed laser deposition (PLD) in an ultrahigh vacuum chamber with a base pressure lower than 2 108mbar. For ablation, a KrF ex- cimer laser (248 nm wavelength) with a nominal energy of 130 mJ per pulse is used at a pulse frequency of 10 Hz. The films are grown under a stable atmosphere of 0.026 mbar of O2at 475C substrate temperature. After deposition, the films are subsequently cooled down to room temperature at a rate of 25 K/min. The crystalline structure of the films was deter- mined by x-ray diffraction (XRD). The magnetic moment was measured by using a superconducting quantum interference device magnetometer (SQUID, Quantum Design MPMS II). For spin Seebeck effect (SSE) measurements, the samples are covered with a continuous layer of 4 nm Pt deposited by mag- neton sputtering. The measurements are performed in the lon- gitudinal geometry, where the heat gradient is perpendicular to the sample surface [19]. For spin Hall magnetoresistance (SMR) measurements, a 4 nm thick Pt bar of around 0.3 mm width is defined on the surface along the crystallographic axes. The Pt is deposited ex-situ using magneton sputtering in an Ar atmosphere through a shadow mask. III. RESULTS Fig. 1(a) and 1(b) present the XRD patterns of GGG/YIG, GGG/GIG, and GGG/YIG/GIG bilayer films measured with the scattering vector normal to the (001) oriented cubic sub- strate. As the films grow coherently on the substrate surface, theaandbaxes are equally strained and the (004) peaks and (008) peaks, indicating the length of the c-axis, are evident in the corresponding high resolution XRD patterns, respec- tively. Near the respective (004) (Fig. 1(a)) and (008) (Fig. 1(b)) diffraction peaks, the XRD patterns show Laue oscilla- tions, indicating a smooth surface and interface. While the YIG reflex is partly shadowed by the substrate peak, we can clearly identify the reflex of the GIG top layer. From the GIG peak position, we determine the out-of-plane lattice param- eter to around c=1:26 nm, independent of the thickness of the underlying YIG layer, indicating strained growth for every sample. The absence of a structural alteration of the top GIG layer indicates that the YIG interlayer does not influence the GIG growth. It is therefore expected, that the magnetic prop- erties of the GIG layer are similarly unaffected. The rocking curve of each reflex is around Dw=0:04further showing well aligned unit cells for every bilayer. Together, these XRD measurements indicate the high-quality growth of YIG/GIG heterostructures. To characterize the magnetic properties of the heterostruc- tures, the magnetization vs field ( mH) dependence was measured with a magnetic field applied within the sample plane. The mHof YIG/GIG for magnetic fields up to （a） （b） 2Θ (deg) 2Θ (deg) log(intensity) (arb. units) log(intensity) (arb. units)FIG. 1. Out-of-plane 2 Q=wmeasurements of YIG, GIG and YIG/GIG bilayer films near the (004) peaks (a) and (008) peaks (b) shown in a logarithmic intensity scale. The thicknesses of the indi- vidual layers are detailed in nanometers. 50 mT are obtained by measuring the GGG/YIG/GIG sample and subtracting a linear fit to compensate for the paramag- netic contribution of the substrate. Fig. 2 (a) shows the mH curves for single layer GIG and YIG measured at a temper- ature of T=100 K. Note that the magnetic moment of the YIG layer of around 0 :6107Am2corresponds to a mag- netization of 133 kA/m, which is similar to other YIG thin films and bulk samples [24], confirming the high quality of the thin films. Since heterostructures of varying thickness are investigated, in the following we will focus on the total mag- netic moment of the layers rather than on the magnetization. Fig. 2(b-d) show the mHcurves measured at the same tem- perature for YIG/GIG heterostructures of varying YIG thick- ness. All samples generally show sharp switching features, however, the YIG(36 nm)/GIG(30 nm) (Fig. 2 (d)) shows seg- mented switching features in this magnetic field range. The curve displays an ‘inner hysteresis’ and additionally a larger hard axis loop. Secondly, the total magnetization of the bi- layers at m0H=20 mT decreases with increasing YIG layer thickness. This behavior indicates that the net moments of YIG and GIG are antiparallel in the low-field region. In order to further understand this behavior, the tempera- ture dependence of the mHcurves was measured. The net magnetic moments mof the YIG/GIG samples at 50 mT were obtained. As shown in the top of Fig. 3 (a), the mag- netic moment of the pure YIG layer is only weakly dependent on temperature, while the pure GIG sample is ferrimagnetic with a compensation temperature ( Tcomp ;G) of 280 K, which is close to the bulk value (295 K [5]). The GIG magnetiza- tion is strongly temperature dependent due to the Gd moment increasing towards lower temperatures. Above Tcomp ;G, the direction of the magnetization of the GIG sample is given by the direction of the d-Fe moments, while below Tcomp ;G, the magnetization direction is given by the direction of the c-Gd moments. Note that the antiferromagnetic coupling between thea-Fe and d-Fe sublattices within one layer cannot be bro- ken by magnetic fields accessible in our labs. We therefore3 @100 K（a） （b） （c） （d） ‘inner hysteresis’ FIG. 2. SQUID measurements ( mH-loops) for different YIG/GIG bilayers measured at a temperature of 100 K with a maximum applied field of 50 mT. simplify the description of the magnetic properties by intro- ducing one magnetic Fe lattice for YIG and GIG as the sum of minority a-Fe and majority d-Fe, respectively. Having established the compensation temperature of pure GIG, we observe that when grown in a bilayer with YIG, the compensation temperature shifts to lower temperatures with increasing YIG thickness. We indicate in Fig. 3 three regions of the bilayer, corresponding to: above the compensation of the pure GIG film (grey, zone I), temperatures below the com- pensation of the bilayer (blue, zone III), and an intermedi- ate region (orange, zone II). The shifting of the compensa- tion temperature of the bilayer Tcomp ;Bindicates that there is a coupling between the YIG and GIG layer that is compen- sating for the change in the Gd orientation that occurs. The magnitude of the magnetic moments at a magnetic field of 20 mT at different temperatures is summarized in Fig. 3 (b). At 20 K, the total magnetic moment decreases with increas- ing YIG thickness, similar to the 100 K measurements shown in Fig. 2). At 300 K, the total magnetization increases with increasing thickness of YIG. Taking into account the rever- sal of the magnetization direction of the magnetic sublattices in GIG, this indicates an interlayer ferromagnetic coupling in the whole temperature range of the Fe sublattices of YIG and GIG, i.e. considering the Fe moments and Fe-O bonds only, the bilayer has a coherent magnetic structure at low fields at all temperatures. In order to support our claim of a coherent magnetization structure of the Fe sublattices, we perform a simple simula- tion taking into account the temperature dependence of the Gd and Fe magnetic moments [4]. We model the net Fe mag- netic moment as m(Fe) =jm(d-Fe)m(a-Fe)j, so the magnetic moment of GIG is equal to jm(Fe)m(Gd)j. Above Tcomp ;G, the modulus m(Gd) = m(Fe)m(GIG), below Tcomp ;G,m(Gd) =m(GIG) +m(Fe). Since the total magnetic moment of the Fe ions sublattices is the same in YIG and GIG, the m(Fe) can be seen as m(YIG). The magnetic moment of Gd in GIG can be extracted from the magnetization of individual YIG 1E-61E-51E-4 1E-61E-51E-4 1E-61E-51E-4 20 60 100 140 180 220 260 3001E-61E-51E-4 YIG(18 nm) GIG(30 nm) YIG/GIG (9/30) -3 2 YIG/GIG (18/30) YIG/GIG (36/30) T (K) （a） （b） III II I Tcomp,B Tcomp,G20 K 300 K Tcomp,B Temperature (K)FIG. 3. (a) MTcurves of the YIG, GIG and YIG/GIG films in low magnetic field measured by mH-loops at different temperatures. The shaded regions indicate T>Tcomp ;G(zone I), Tcomp ;G>T> Tcomp ;B(zone II), T<Tcomp ;B(zone III) for the respective samples. (b) The dependence of the magnetic moment m(300 K), m(20 K) and the bilayer compensation temperature on the YIG thickness. (18 nm) and GIG (30 nm) layers shown in Fig. 3(a) top panel. The magnetization of YIG is only weakly temperature de- pendent, so we can model its magnetic moment as constant m(Fe,18 nm) = 5 :4108Am2. We note that the YIG mag- netization should follow T3=2from the Curie temperature TC to 0 K, however, we stay far below TC. The m(Gd) can be ex- tracted from the temperature dependent magnetization curves and the data points are fitted by a third order polynomial used for the simulations. More precise modeling of m(Gd) can be done in principle using a self consistent molecular field acting on the Gd spin moments as input for the Brillouin function [25]. This would require knowledge of the detailed tempera- ture dependence of the Fe sublattice magnetization. However, the exact behavior of the sublattice magnetization is out of scope of this work and the parameters from the bulk cannot be transferred to the thin films that usually have somewhat re- duced Curie temperatures. Therefore, we describe m(Gd) only phenomenologically in the temperature range of our measure- ments to facilitate the analysis. The temperature dependencies of the magnetic moments of Gd3+, Fe3+and simulated GIG are plotted in Fig. 4(a). We model three different cases for the interlayer cou- pling of YIG/GIG bilayers: the respective magnetic moments m(Fe,GIG) of GIG and m(Fe,YIG) of YIG are ferromagnetically coupled, antiferromagnetically coupled and without coupling. For the ferromagnetic Fe-Fe cou- pling case, the overall magnetic moment of the system is given as m=jm(Fe,GIG) +m(Fe,YIG)m(Gd,GIG)j; For the antiferromagnetic Fe-Fe coupling, m= jm(Fe,GIG)m(Fe,YIG)m(Gd,GIG)j; Without coupling m=jm(Fe,GIG)m(Gd,GIG)j+m(Fe,YIG). The respective results are plotted in Fig. 4. As shown in Fig. 4(b), for ferromagnetic coupling, Tcomp ;Bis decreasing with increasing thickness of YIG. In addition, the total magnetic moment increases with the thickness of YIG at 300 K and the total magnetic moment decreases with the thickness of YIG at4 Gd3+Fe3+ Gd3+Fe3+Gd3+Fe3+（a） （b） （c） （d）Exp. Sim. FIG. 4. (a) Modeled mTcurve of the m(Gd) and m(Fe). Modeled mTcurves of GIG and YIG/GIG bilayers, where m(Fe,GIG) of GIG and m(Fe,YIG) of YIG are (b) ferromagnetically coupled, (c) antiferromagnetically coupled and (d) without coupling. The insets show the magnetic sublattice alignment at low magnetic fields. The blue arrow indicate the presence of coupling. 20 K. These features are consistent with the experimental results and support our claim that the layers are indeed ferromagnetically coupled with respect to iron atoms. For an assumed antiferromagnetic Fe-Fe coupling at the YIG-GIG interface (Fig. 4 (c)), the bilayers do not possess compensa- tion temperatures below 300 K for the simulated thicknesses. Without coupling between the YIG and GIG magnetization (Fig. 4 (d)), the bilayers never have a compensation point, but only a total magnetization minimum at Tcomp ;G. The above discussion was on magnetization measurements at low magnetic field and we used simulations based on the individual materials to compare to the measured SQUID data. We have seen that these investigations show a ferromagnetic Fe-Fe coupling across the YIG-GIG interface. The double switching in the YIG(36nm)/GIG(30nm) sample (Fig. 2(d)) indicates that a sufficiently large magnetic field can break the interlayer coupling. To investigate this further, we per- form mHmeasurements exploiting larger magnetic fields at various temperatures. These mHof YIG/GIG can be obtained via measuring the entire GGG/YIG/GIG sample and then subtracting the paramagnetic contribution of GGG de- termined in a separate measurement. As shown in Figs. 5 (a)-(c), we observe double switching over a large tempera- ture range. Further increase of the external field leads to a saturation of the overall magnetic moment. To describe the curves, we introduce Amp 2 as the maximum amplitude of the hysteresis. Here, the Fe-Fe interlayer coupling is broken and the net moments of YIG and GIG layers align parallel so thatm(YIG) +m(GIG) =Amp 2. Approaching the compensa- tion temperature of the GIG layer Tcomp ;G, the signal from the GIG layer becomes too weak to perform this type of analysis with respect to the signal resolution of the SQUID and in the presence of the strong substrate background. Therefore, we 20 60 100 140 180 220 260 300012345 T(K)Amp2Amp1 µ0H0III II I YIGGIG -1,0 -0,5 0,0 0,5 1,0-4-2024 160 K -7 2 µ0H (T)-1,0 -0,5 0,0 0,5 1,0-8-4048 -7 2 µ0H (T)20 K YIG/GIG (36/ 30) -1,0 -0,5 0,0 0,5 1,0-4-2024 120 K -7 2 µ0H (T)(a) (b) (c)(d) -7 2 FIG. 5. mHcurves of a YIG(36)/GIG(30) sample measured at a temperature (a) 20 K (b) 120 K and (b) 160 K in high magnetic field. (d) mTcurve of YIG(36)/GIG(30) with parallel state and antiparallel state and the alignment of the YIG and GIG layer in high and low magnetic fields in different temperature regions evaluate Amp 2 only up to T = 180 K. The ‘inner hysteresis’, as already seen in Fig. 2 (d) and in Fig. 5 (a), which indicates a switching at low magnetic fields has the amplitude Amp 1. In this low-field region, the Fe-Fe coupling is not broken and the magnetic moments of the whole bilayer stack reverses by reversing the small magnetic field keeping the relative orien- tation of the Fe and Gd moments intact. The amplitude of this hysteresis is given by the difference of the net magnetic moments of the individual layers jm(YIG)m(GIG)j=Amp 1. The preservation of the relative orientation of the individual moments implies that the orientation of the net magnetic mo- ments of the YIG and GIG layers changes going from zone II to zone III in Fig. 3. In zone II, where the Gd moment is smaller than the net Fe moment of the coupled layers, the YIG layer aligns parallel to the field while the GIG layer aligns an- tiparallel to the field. If the magnitude of the Gd moment overcomes that of the net bilayer Fe moment in zone III, GIG aligns parallel to the field and YIG antiparallel. This is illus- trated in Fig. 5(d). The relative alignment of Fe (orange) and Gd (red) moments is depicted. In the high temperature region above the compensation temperature of the GIG layer (grey sketched zone I in Fig 5(d)) the magnetization of GIG is dom- inated by the net iron moment. Here the magnetization of the YIG layer and GIG layer are always parallel to the external field at low and high field. We have seen in SQUID measurements that the magnetiza- tion of the YIG and GIG layers aligns differently depending on the temperature and relative thickness. Tuning the thick- ness of the respective layers allows for choosing the magnetic properties of the heterostructures. The larger the YIG:GIG ra- tio, the lower the bilayer compensation temperature Tcomp ;B. We can thereby functionalize the coupled layers and build de- vices that have defined relative orientation of the magnetic5 -50 0 50 100 (deg)-1-0.500.51RL/RL(0)10-3 0.1 T 0.5 T 2.0 T -50 0 50 100 (deg)-4-3-2-101RL/RL(0)10-4 0.1 T 0.5 T 2.0 T -2 0 2-2.5-2-1.5-1-0.50RL/RL(0)10-4 30 K 50 K -2 0 2 µ0H (T) µ0H (T)-4-20246RL/RL(0)10-4 100 K 200 K(a) (b) (c) (d) HI x y30 K 200 Kzone III zone II I FIG. 6. Uniaxial SMR measurements at a GGG/YIG(36)/GIG(30)/Pt sample at various temperatures (a-b) with the field applied in the sample plane perpendicular to the current as depicted in the inset of (a). ADMR for two different temperatures (c-d) where the field is rotated within the sample plane as illustrated in the inset of (c). moments. However, SQUID measurements do not distinguish which layer switches (top or bottom). In order to disentan- gle this, we conduct surface-sensitive methods to probe the top surface layer individually to then, in combination with SQUID, identify which layer switches at which field. A suitable tool to investigate the magnetic properties of the top layer is spin Hall magnetoresistance (SMR) [15, 26]. The spin accumulation in a heavy metal in close contact with a ferrimagnetic insulator interacts with the magnetic order pa- rameter only at the very interface. The resistance of a Pt bar defined on the surface of a GGG/YIG(36)/GIG(30) sample is modulated by DRLµ(1m2 y)[26], where myis the magne- tization component of the GIG layer in plane perpendicular to the Pt bar. We normalize the change of resistance to the zero-field value. We perform uniaxial measurements with the magnetic field applied in-plane perpendicular to the Pt bar. The uniaxial field measurements at 30 K and 50 K describe a sharp drop of resistance at low magnetic fields followed by negligible further resistance changes, which is shown in Fig. 6 (a). The sharp decrease of resistance is likely due to the annihilation of differently aligned domains, going from a multidomain state to a monodomain state when the GIG net magnetic moment aligns with the field. In the absence of an external field, the sample symmetry allows for domains with magnetization aligned along the Pt bar. At temperatures above Tcomp ;B, double switching is observed in the SMR data with increasing magnetic field (see Fig. 6 (b)). This indicates the rotation of the GIG magnetic moment after the interfacial coupling between YIG and GIG is broken. The continuous increase of resistance in the switching process indicates thatthe top layer switching is in fact not an abrupt process, but a gradual rotation since only the alignment of the GIG magne- tization away from the y-direction can increase the resistance due to SMR. We confirm this by performing rotation mea- surements, where the field rotates within the sample plane (a-plane) from around 55to 125, relative to the current direction, and back. The experiment is performed at 30 K, which is in zone III (Fig. 6 (c)) and at 200 K, which is in zone II (Fig. 6 (d)). The longitudinal resistance is given as relative to the resistance value at 0° (along the Pt bar). At low temperatures, where no top-layer switching is observed in the uniaxial SMR measurements, we also do not observe any changes in the phase of the angular-dependent magnetore- sistance (ADMR) for magnetic fields of different magnitude. However, increasing the temperature to 200 K, the ADMR be- comes strongly field-dependent. At low magnetic field and at high magnetic field, the phase of the ADMR is close to 0, indicating the alignment of the GIG magnetization axis with the external magnetic field. At 0.5 T, however, in the region of the GIG switching, a phase shift of almost 90is observed, indicating the perpendicular alignment of GIG magnetization to the magnetic field in the switching region. A complementing tool to investigate the magnetic proper- ties of the bilayer system is the longitudinal spin Seebeck ef- fect (SSE). The SSE can be used to investigate the magnetic properties close to Tcomp ;Gat which SQUID measurements cannot resolve the magnetization of the GIG layer. For the SSE measurement, the sample is exposed to an out-of-plane temperature gradient by sandwiching it between a tempera- ture sensor and a resistive heater (see inset of Fig. 7 (a)). The temperature gradient is estimated by monitoring the resistance of sensor and heater. The external magnetic field is applied in the sample plane, perpendicular to the temperature gradient. Measuring the thermal excitation of spin waves in such a bi- layer can potentially lead to superposing spin currents origi- nating from YIG and GIG, measured as a voltage VISHE via the inverse spin Hall effect (ISHE) in a heavy metal top layer [22, 27]. The SSE of YIG(36)/GIG(30)/Pt(4) was measured at different temperatures. As shown in Fig. 7, a hysteretic volt- age signal VISHE is obtained by sweeping the magnetic field. At the lowest temperature of 35 K, we observe a negative am- plitude of the SSE as shown in Fig. 7(a). Going to 60 K, the sign of the measured voltage changes its sign to positive (Fig. 7(b)). For 110 K, the shape of the SSE signal fundamentally changes as seen in Fig. 7(c). At low magnetic fields, a nega- tive switching is observed before the signal changes sign again at around 40 mT. At 306 K, only one switching event is visible at low magnetic fields, having a negative sign (Fig. 7(d)). These features emphasize the SSE signal to have it’s ori- gin in the GIG layer. We investigate these features in the fol- lowing, by taking into account the magnetic properties of the bilayer system determined by SQUID and the complex behav- ior of SSE signal measured at pure GIG samples [22, 23]. For temperatures below the compensation temperature of the bi- layer Tcomp ;B, we can compare the SSE measurement with the mHloop. In the SQUID measurement at 60 K (zone III), a switching of YIG is expected at approximately 30 mT. How- ever, we do not observe this feature at 30 mT in SSE mea-6 FIG. 7. VISHEHloops of a GGG/YIG(36)/GIG(30)/Pt sample at various temperatures (a) 35 K, (b) 60 K, (c) 110 K and (d) 306 K. For (d) a larger field range is displayed. VISHE is normalized by the estimated temperature gradient and a constant offset of the signal is subtracted. surements. Thus, a possible contribution to the spin current originating from YIG seems to be damped in the GIG layer in this sample. At temperatures above Tcomp ;Band below Tcomp ;G (zone II), a segmented switching is observed. This is expected for the GIG sensitive SSE measurement, as the GIG layer switches at higher fields in this temperature range (thus the second step in the field sweep). The sign change of the SSE signal VISHE at low temperatures also suggests the GIG layer to be the main spin current source. For even lower temper- atures ( T=35 K), the SSE signal undergoes a sign change, which can be explained by the change of spectral weight and temperature dependence of occupied magnon modes in GIG [22, 23]. For pure GIG, another sign change is observed at compensation temperature Tcomp ;G, because of the reversal of the sublattice magnetization for a constant external applied magnetic field [22]. As the SSE signal follows the GIG mag- netization, we can investigate the orientation of the GIG at higher temperatures. When changing the temperature from below to above Tcomp ;G, there is no sign change observed in the YIG/GIG bilayer. For a coupling of YIG and GIG via the net moment of each film, a reversal of the sign of VISHE would be expected at Tcomp ;G, which is not supported by our data. This is in line with our claim, that the Fe sublattice moments of YIG and GIG dominate the coupling, thus when changing the temperature across Tcomp ;Gthe GIG orientation stays the same and no reversal of the sign of VISHE is observed. IV . DISCUSSION Together, our measurements indicate that we have robust Fe-Fe exchange coupling between defined YIG and GIG lay- ers. While SQUID gives a basis of the interpretation of the magnetic behavior of the heterostructures, both SSE and SMR selectively show the behavior of the top GIG layer only. Our high quality thin films access a phase space of samples witharbitrarily aligned magnetization of the bilayers, depending of the relative thicknesses. Up to now, the exchange cou- pling between YIG and GIG has only been described as an interface effect in GGG/YIG samples [12–14]. The artificial stacks grown here give much better control over the bilayer properties and allow for fine tuning of these by choosing the relative thickness of YIG and GIG. We identify three temper- ature regions, where the samples have fundamentally differ- ent responses to an external magnetic field. Above the GIG compensation point Tcomp ;G, the coupled net iron moments are always parallel and show in the direction of the external mag- netic field. Reducing the temperature below Tcomp ;G, we still observe ferromagnetic Fe-Fe coupling across the YIG/GIG in- terface at low magnetic fields. Increasing the field, we break this coupling across the interface and the Gd moment of the GIG layer orients with the external field. Uniaxial SMR mea- surements show the switching of the top GIG layer and rota- tion measurements indicate a continuous rotation of the mag- netic moments in the reorientation regime. Decreasing the temperature further, we observe a second compensation point, where the net GIG layer moment equals the YIG layer mo- ment. We label this bilayer compensation point Tcomp ;B. We show that Tcomp ;Bcan be tuned by choosing the relative thick- ness of the YIG and GIG layers. This allows for devices with controllable magnetization direction. Below Tcomp ;B, we ob- serve again a double switching in SQUID, indicating the re- orientation of one of the layers. Both SMR and SSE measure- ments do not show this switching, indicating that the bottom YIG moment rotates, leaving the top GIG moment aligned with the field. SSE measurements thereby prove to be sen- sitive to the top GIG layer only, since the switching of the bottom YIG spin current source is not observed in our mea- surements. Moreover, SSE measurements show a change of sign at certain temperature, which is not associated with the direction of the net magnetic moment, but with the magnon population as observed in pure GIG layers [22, 23]. Recently, it was reported that magnon hybridization may occur in YIG- GIG heterostructures leading to a reduction of the temperature of the sign change. We note that here, the change of sign occurs at around 35 K for a YIG(36 nm)/GIG(30 nm) sam- ple. SSE measurements performed at pure GIG layers have shown the sign change to occur at around 44 K to 72 K [22], depending on several parameters like magnetic compensation point, heavy metal material and GIG-heavy metal interface quality. The here observed sign change temperature of 35 K is remarkably low compared to these reports, which might in- dicate magnon hybridization. However, a direct comparison with literature values is difficult due to the strong dependence on the surface quality of the samples. A simultaneous depo- sition of the HM layer excludes a different GIG/HM interface for different samples and will be target of futures studies. For the interlayer coupling strength between YIG and GIG, we assume that the coupling energy is equal to the gain of Zeeman energy to align both YIG and GIG magnetization in zones II and III. We calculate the energy at the low temper- ature state of sample YIG(36 nm)/GIG(30 nm). From Fig. 5 (a) we extract the saturation field at 20 K, which is the field at which Amp 2 is reached as m0H=0:1 T. Rotating7 m(YIG) from antiparallel to parallel leads to a gain in Zee- man energy by 2
m(YIG )
H.m(YIG) is given by m(YIG)= 1 2(Amp 2Amp 1)1:5107Am2. Taking into account the interface area of 0.25 cm2, we estimate an effective inter- face coupling energy of 0.0012 J/m2. With 8 Fe atoms (d or a site) on the surface of a (001) oriented unit cell this relates to 1.4 meV per Fe atom at the interface. Comparing this to the dominant exchange parameter J1in YIG, which is found to be 6.8 meV [7] and J1in GIG of 4.0 meV [14], we observe a qualitative agreement, but the interlayer coupling strength ap- pears to be smaller than the direct exchange coupling in pure YIG and pure GIG but of the same order of magnitude as the theoretical limits. For interfacial coupling of YIG-GIG at the YIG/GGG interface, a coupling strength of 0.14 meV was dis- cussed by Gomez et al. based on the c-Gd antiferromagnetic interaction Jcdto d-Fe [14]. Our value is significantly larger than this. This can be understood by the fact that they in- vestigate an ultra-thin layer formed by interdiffusion with the substrate. In our case, however, the Jcdinteraction energy is not a limiting factor as it needs to be integrated over the vol- ume/thickness of the Gd layer and the resulting energy is then much larger than the interfacial exchange coupling between YIG and GIG layers that is limited by Fe-Fe interactions at the interface. In reality, the competition between different exchange interactions can easily lead to more complicated spin structures than in our fully collinear toy model discussed above and we see hints for this in the continuous rotation of the magnetization observed in SMR measurements. Also in our samples we have the YIG-GGG interface at the substrate and therefore the coupling effects induced by intermixing with the substrate should be present. However, as this interface is ultrathin and contributes very little to the total sample magne- tization it can be neglected to first order in our analysis and it was only observed at very low temperatures by Gomez at al. [14]. Traces of this intermixing GIG at the substrate interface might be seen in the low amplitude SQUID curves shown in Fig. 2, where the pure YIG as well as the bilayers with a thin YIG layer appear slightly exchange-biased. In spite of the limitations of the simplified toy model for the analysis we can safely state that the largest interfacial coupling dominating the properties of our bilayers stems from the upper YIG/GIG in- terface.V . CONCLUSION YIG/GIG bilayers were fabricated by PLD and the mag- netic coupling of the samples was investigated by SQUID, SSE and SMR. It is found that the YIG/GIG bilayers show ferrimagnetic features. The compensation temperature of the bilayer system shifts to lower temperatures with increasing thickness of the YIG layer, which originates from the effec- tive ferromagnetic coupling of the iron magnetic sublattices at the YIG-GIG interface, i.e. in the epitaxial bilayer, the iron atoms spin subsystem on respective d-Fe and a-Fe sites is co- herent over the boundary of the two materials in zero mag- netic field. Below the compensation temperature of the GIG layer, an external magnetic field can break the coupling, lead- ing to a parallel magnetization of YIG and GIG layers at high field. At low magnetic fields, the orientation of the YIG and GIG magnetic moment depends on the temperature (Gd mo- ment). SMR and SSE measurements reveal the behavior of the top GIG layer, from which the temperature and field de- pendence of alignment of YIG and GIG can be obtained. Our results demonstrate that the magnetic coupling in insulating YIG/GIG heterostructures can be manipulated analogously to that of metallic spin valves and open the pathway to manipu- late the magnon transport. ACKNOWLEDGMENTS The authors gratefully acknowledge funding by Deutsche Forschungsgemeinschaft (DFG, German Research Founda- tion) Project No. 358671374. This work was supported by the Max Planck Graduate Center with the Johannes Guten- berg–Universität Mainz (MPGC) as well the Graduate School of Excellence Materials Science in Mainz (GSC266). This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) Spin+X (A01+B02) TRR 173 - 268565370. This work is partially supported by the National Key R&D Program of China (Grant No. 2018YFB0704100), the National Science Foundation of China (Grants No. 11974042, No. 51731003, No. 51927802, No. 51971023, No. 11574027, and No. 61674013). Sally Lord gratefully acknowledges the DAAD RISE Germany Scholarship. 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2021-04-19

Ferrimagnetic Y$_3$Fe$_5$O$_{12}$ (YIG) is the prototypical material for studying magnonic properties due to its exceptionally low damping. By substituting the yttrium with other rare earth elements that have a net magnetic moment, we can introduce an additional spin degree of freedom. Here, we study the magnetic coupling in epitaxial Y$_3$Fe$_5$O$_{12}$/Gd$_3$Fe$_5$O$_{12}$ (YIG/GIG) heterostructures grown by pulsed laser deposition. From bulk sensitive magnetometry and surface sensitive spin Seebeck effect (SSE) and spin Hall magnetoresistance (SMR) measurements, we determine the alignment of the heterostructure magnetization through temperature and external magnetic field. The ferromagnetic coupling between the Fe sublattices of YIG and GIG dominates the overall behavior of the heterostructures. Due to the temperature dependent gadolinium moment, a magnetic compensation point of the total bilayer system can be identified. This compensation point shifts to lower temperatures with increasing thickness of YIG due the parallel alignment of the iron moments. We show that we can control the magnetic properties of the heterostructures by tuning the thickness of the individual layers, opening up a large playground for magnonic devices based on coupled magnetic insulators. These devices could potentially control the magnon transport analogously to electron transport in giant magnetoresistive devices.

Magnetic coupling in Y$_3$Fe$_5$O$_{12}$/Gd$_3$Fe$_5$O$_{12}$ heterostructures

2104.09592v1

Spin waves in coupled YIG/Co heterostructures Stefan Klingler,1, 2,∗Vivek Amin,3, 4Stephan Gepr¨ ags,1, 2Kathrin Ganzhorn,1, 2Hannes Maier-Flaig,1, 2Matthias Althammer,1, 2Hans Huebl,1, 2, 5Rudolf Gross,1, 2, 5Robert D. McMichael,3Mark D. Stiles,3Sebastian T.B. Goennenwein,6, 7and Mathias Weiler1, 2 1Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany 2Physik-Department, Technische Universit¨ at M¨ unchen, 85748 Garching, Germany 3Center for Nanoscale Science and Technology, National Institute of Standards and Technology, Gaithersburg, Maryland 20899-6202, USA 4Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, MD 20742 5Nanosystems Initiative Munich, 80799 Munich, Germany 6Institut f¨ ur Festk¨ orperphysik, Technische Universit¨ at Dresden, 01062 Dresden, Germany 7Center for Transport and Devices of Emergent Materials, Technische Universit¨ at Dresden, 01062 Dresden, Germany We investigate yttrium iron garnet (YIG)/cobalt (Co) heterostructures using broadband ferro- magnetic resonance (FMR). We observe an efficient excitation of perpendicular standing spin waves (PSSWs) in the YIG layer when the resonance frequencies of the YIG PSSWs and the Co FMR line coincide. Avoided crossings of YIG PSSWs and the Co FMR line are found and modeled using mutual spin pumping and exchange torques. The excitation of PSSWs is suppressed by a thin aluminum oxide (AlOx) interlayer but persists with a copper (Cu) interlayer, in agreement with the proposed model. In magnonics, information is encoded into the electron spin-angular momentum instead of the electron charge used in conventional CMOS technology [ 1–10]. Magnon- ics based on exchange spin waves is particularly appeal- ing, due to isotropic spin-wave propagation with small wavelengths and large group velocities [ 5]. With its long magnon propagation length, yttrium iron garnet (YIG) is especially interesting for this application. However, an excitation of exchange spin waves by microwave magnetic fields requires nanolithographically defined microwave an- tennas [ 11] that have poor efficiency due to high ohmic losses and impedance mismatch. Here, we show that exchange spin waves can be ex- cited by interfacial spin torques (ST) in YIG/Co het- erostructures. These STs couple the YIG and Co mag- netization dynamics by microwave frequency spin cur- rents. Phenomenological modeling of the coupling reveals a combined action of exchange, damping-like and field-like torques that are localized at the YIG/Co interface. This is in contrast to the previously observed purely damping-like ST in all-metallic multilayers [12]. We study the magnetization dynamics of YIG/Co thin film heterostructures by broadband ferromagnetic res- onance (FMR) spectroscopy. From our FMR data we find an efficient excitation of perpendicular standing spin waves (PSSWs) in the YIG when the YIG PSSW reso- nance frequency is close to the Co FMR line. We ob- serve about 40 different PSSWs with wavelengths down toλPSSW≈50 nm. Clear evidence for the coupling is provided by avoided crossings and corresponding characteristic changes of the ∗stefan.klingler@wmi.badw.delinewidths of the YIG PSSW and the Co FMR line. This coupling and the excitation of PSSWs is also observed when a copper (Cu) layer separates the YIG and the Co films. However, the insertion of an insulating AlOx interlayer completely suppresses the excitation of YIG PSSWs. This allows us to exclude dipolar coupling as the origin of the PSSW excitation and is in agreement with the mediation of the coupling by spin currents. Our data are well described by a modified Landau-Lifshitz-Gilbert equation for the Co layer, which includes direct exchange torques and spin torques from mutual spin pumping at the YIG/Co interface. Simulations of our coupled systems reveal the strong influence of spin currents on the coupling of the different layers. We investigate a set of four YIG/Co samples, which are YIG/Co(50), YIG/Co(35), YIG/Cu(5)/Co(50) and YIG/AlOx(1.5)/Co(50), where the numbers in brackets denote the layer thicknesses in nanometers. The YIG thicknessd2is = 1µm for all samples. The FMR mea- surements are performed at room temperature using a coplanar waveguide (CPW) with a center conductor width ofw= 300µm. The CPW is connected to the two ports of a vector network analyzer (VNA) and we measure the complexS21parameter as a function of frequency fand external magnetic field H(for details of the sample prepa- ration and the FMR setup see Supplemental Material S1 and S2 [13]). Fig. 1 (a) shows the background-corrected field- derivative [ 14] of the VNA transmission spectra |∂DS21/∂H|for the YIG/Co(50) sample as a function ofHandfas explained in S3 [ 13] and we clearly observe two major modes. The low frequency mode corresponds to the YIG FMR line, whereas the high-frequency mode corresponds to the Co FMR line. Within the broad Co FMR line, we find several narrow resonances, of whicharXiv:1712.02561v1 [cond-mat.mes-hall] 7 Dec 20172 YIG/Cu(5 nm)/Co(50 nm) YIG/Co(50 nm) YIG/AlOx(1.5 nm)/Co(50 nm) 5 10 15 20 25 5 10 15 20 25 5 10 15 20 250.1 00.20.30.4 1 0.5 0µ0H (T) f(GHz) f (GHz) f (GHz)(c) (b) (a) |∂DS21/∂H| (arb.u.) YIG Co Co PSSW exchange mode FIG. 1. (Color online) Field-derivative of the Vector Network Analyzer (VNA) transmission spectra for three different samples as a function of magnetic field and frequency. All samples show two modes corresponding to the YIG (low-frequency mode) and Co (high-frequency mode) FMR lines. The color scale is individually normalized to arbitrary values. (a) The YIG/Co(50) sample additionally reveals YIG PSSWs and pronounced avoided crossings of the modes for small frequencies. (b) The YIG/Cu(5)/Co(50) sample also shows the YIG PSSWs, but the frequency splittings of the modes are much smaller than in (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 Co films are magnetically uncoupled. the dispersion is parallel to the YIG FMR. These lines are attributed to the excitation and detection of YIG PSSWs with wavelengths down to 50 nm (for details see Fig. S5 [ 13]). We find avoided crossings between these YIG PSSWs and the Co FMR line (inset), where the frequency splitting geff/2π≤200 MHz (see S4 [ 13] for details). This is a clear indication that the YIG and Co modes are coupled to each other. Furthermore, an addi- tional low-frequency mode with lower intensity is observed in Fig. 1 (a). This line is attributed to an exchange-spring mode of the coupled YIG/Co system. A qualitatively sim- ilar transmission spectrum is observed for the YIG/Co(35) sample (for details see Fig. S6 [ 13]). Furthermore, we observe the first Co PSSW at around f= 22 GHz and µ0H= 0.1 T for samples with a 50 nm thick Co layer. Fig. 1 (b) shows|∂DS21/∂H|for the YIG/Cu(5)/Co(50) sample as a function of Handf. Again, we observe the YIG FMR, YIG PSSWs and the Co FMR lines. However, the frequency splitting between the modes (in- set) is much smaller in comparison to the YIG/Co(50) sample,geff/2π≤40 MHz. This strongly indicates that the coupling efficiency is reduced in comparison to Fig. 1 (a). We attribute this mainly to the suppression of the static exchange coupling by insertion of the Cu layer. This is also in agreement with the vanishing of the exchange mode. Fig. 1 (c) displays |∂DS21/∂H|for the YIG/AlOx(1.5)/Co(50) sample as a function of Hand f. No YIG PSSWs are observed within the Co FMR line (inset Fig. 1 (c)). This provides strong evidence that the insertion of the thin AlOx layer suppresses the coupling between the YIG and Co magnetization dynamics. An analysis of the Co FMR linewidth (for details see S7 [ 13]) also demonstrates that the AlOx layer eliminates anycoupling between the YIG and Co layers. From Fig. 1, we conclude that any magneto-dynamic coupling is sup- pressed by insertion of a thin insulator between the two magnetic layers. This provides strong evidence against a magnetostatic coupling by stray fields, and is in agree- ment with a dynamic coupling mediated by spin currents, which can pass through the Cu layer, but are blocked by the AlOx barrier. Fig. 2 shows the magnetic hysteresis loops of the YIG/Co samples recorded by Superconducting Quantum Interference Device (SQUID) magnetometry. The hys- teresis loop of the YIG/Co(50) sample (solid blue line in Fig. 2) exhibits a sharp switching at the YIG coercive field of about 0.1 mT. However, no sharp switching of the Co layer is visible but a smooth increase of the measured magnetic moment until the bilayer magnetization is satu- rated. This can be explained by a direct, static exchange coupling between YIG and Co magnetizations (inset), as known from exchange springs [ 15,16]. The form of the hysteresis loop suggests an antiferromagnetic coupling, as comparably large magnetic fields are required to force a parallel alignment of the layers. However, without a detailed examination of the remnant state, we cannot rule out any ferromagnetic coupling. By inserting a Cu or AlOx layer between the YIG and the Co (dash-dotted and dashed lines in Fig. 2) we find a sharp switching at the Co coercive field µ0Hc≈1 mT. This switching is in agree- ment with the behavior expected for statically uncoupled magnetic layers [ 17]. However, we still observe a dynamic coupling in Fig. 1 (b) in the YIG/Cu(5)/Co(50) sample. Since we expect no static exchange coupling between Co and YIG in this sample, this observation requires a differ- ent mechanism as the origin of the excitation of the YIG3 PSSWs. We model the data of Fig. 1 with a modified Landau- Lifshitz-Gilbert approach, which includes finite mode cou- pling between the YIG and the Co magnetizations at the YIG/Co interface at z=d2. We model the Co mag- netization M1as a macrospin, which is fixed primarily along they-direction with small transverse parts and the YIG magnetization M2(z) as a vector that depends on the distance zfrom the YIG/Co interface (for detailed calculations see S8, S9 [ 13]). In the limit that the trans- verse parts are small, the equation of motion for the Co macrospin then reads: ˙M1=−γ1ˆy×/bracketleftbigg −µ0HM1−α1 γ1˙M1−µ0Ms,1M1,zˆz −J d1Ms,1(M1−M2(d2))−µ0h/bracketrightbigg −γ1 d1Ms,1/bracketleftbig (τF−τDˆy×)(˙M1−˙M2(d2))/bracketrightbig . (1) Here,α1is the Gilbert damping parameter for Co, γ1and Ms,1its gyromagnetic ratio and saturation magnetization, respectively, ˆzis the unit vector in z-direction, and d1is the thickness of the Co layer. The magnetic driving field from the CPW is denoted by h. In our model, his as- sumed to be spatially uniform, to reflect the experimental situation where the CPW center conductor width is much larger than either the YIG or Co thickness. The exchange coupling constant between the YIG and the Co is given byJ. The torques due to spin currents pumped from normalized magnetization1 -1-0.50.5 0 -4 4 2 2- 0YIG/C o(50) YIG/C u(5)/C o(50) YIG/A lOx/Co(50) YIGCo YIGCuCo µ0H (mT)1 3 -1 -3 FIG. 2. Magnetization of YIG/Co(50) (solid), YIG/Cu(5)/Co(50) (dash-dotted) and YIG/AlOx(1.5)/Co(50) (dashed) normalized to the magnetization at µ0H= 4 mT. The magnetic hysteresis loops of YIG/Co show an enhance- ment of the Co coercive field as well as a rather smooth switching. The samples with a Cu or AlOx interlayer reveal a sharp switching of the magnetization at the Co coercive field µ0Hc≈1 mT. The inset shows a possible static magnetization distribution in a exchange coupled (left) and an uncoupled (right) heterostructure.one layer and absorbed in the other have field-like τFand damping-like τDcomponents. The YIG magnetization direction at the YIG/Co interface is given by M2(d2). The YIG magnetization obeys two boundary conditions. First, the total torque at the YIG/Co interface at z=d2 has to vanish: 0 =2Aˆy×∂zM2(z)|z=d2−Jˆy×(M1−M2(d2)) + (~/e)(τF−τDˆy×)/parenleftbig˙M1−˙M2(d2)/parenrightbig .(2) Here,Ais the exchange constant of YIG. Second, we assume an uncoupled boundary condition at the YIG/substrate interface 0 = 2Aˆy×∂zM2(z)|z=0, (3) where the torque vanishes as well. The Co susceptibility χ1is then derived using the ansatz for the transverse YIG magnetization m2(z,t) = (m2,x(z,t),m2,z(z,t)): m2(z,t) = Re/bracketleftbig c+m2+cos(kz) exp(−iωt) c−m2−cos(κz) exp(−iωt)/bracketrightbig .(4) Here,m2±are the complex eigenvectors of the uncoupled transverse YIG magnetization, c±are complex coefficients, ω= 2πfis the angular frequency, kandκare complex wavevectors of the undisturbed YIG films. The transverse Co magnetization follows a simple elliptical precession: m1= Re [ m1,0exp(−iωt)] (5) where m1= (m1,x,m1,z), and m1,0≈(m1,0,x,m1,0,z) is a complex precession amplitude. After finding the complex coefficients c±, the Co susceptibility χ1can be obtained from Eq. (1). Fig. 3 (a-c) show the simulated microwave signal |∂DS21/∂H| ∝ |∂χ1/∂H|(for details see S3, S9 [ 13]). For all simulations we take the same material parame- ters, namely µ0Ms,1= 1.91 T,A= 3.76 pJ/m,α1= 7.7×10−3,α2= 7.2×10−4,γ1= 28.7 GHz/T and γ2= 27.07 GHz/T, as extracted in S4, S5, S7 [ 13]. The thicknesses are d1= 50 nm and d2= 1µm. For the YIG saturation magnetization we take the literature value µ0Ms,2= 0.18 T [ 18]. In Fig. 3 (a) we show the simula- tions for the YIG/Co(50) sample using τF= 30 A s/m2, τD= 15 A s/m2andJ=−400µJ/m2. The interfacial exchange constant J < 0 models an antiferromagnetic coupling as suggested by the SQUID measurements. The sign of the damping-like torque is required to be positive, as it depends on the real part of the spin mixing con- ductance of the interface. The simulation reproduces all salient features observed in the experiment, in particu- lar the appearance of the YIG PSSWs and their avoided crossing with the Co FMR line. Note that the simulations do not reproduce the YIG FMR, as we only simulate the Co susceptibility. However, we can obtain a similar color plot for a ferromagnetic coupling and a negative field-like torque (see for example S6, S10 [ 13]). The com- bination of exchange torques with the field-like torques4 τF = 30 As/m2, τD = 15 As/m2 J = 0τF =0, τD = 0, J = 0 5 10 15 20 25 5 10 15 20 251 0.5 0 f (GHz) f (GHz)(c) (b) (a)τF =30 As/m2, τD = 15 As/m2 J = -400 µJ/m2 5 10 15 20 25µ0H (T) f(GHz)0.1 00.20.30.4 |∂DS21/∂H| (arb.u.) FIG. 3. (Color online) Calculated |∂DS21/∂H|of the simulated transmission spectra. Simulation of the (a) YIG/Co(50) sample, (b) YIG/Cu(5)/Co(50) sample, (c) YIG/AlOx(1.5)/Co(50) sample. at the FM1/vextendsingle/vextendsingleFM2interface complicates the analysis of the total coupling because both torques affect the coupling in very similar ways. Hence, the signs of the field-like torque and the exchange torque cannot be determined unambiguously for the YIG/Co(50) sample. In Fig. 3 (b) we show the simulations for the YIG/Cu(5)/Co(50) sample. Here, τFandτdare un- changed compared to the values used for the simulation of the YIG/Co(50) sample, but we set J= 0, as no static cou- pling was observed for YIG/Cu(5)/Co(50) in the SQUID measurements. The simulation is in excellent agreement with the corresponding measurement shown in Fig. 1 (b). The elimination of the static exchange coupling results in a strong reduction of the coupling between the YIG and Co magnetization dynamics. However, the Cu layer is transparent to spin currents mediating the field-like and damping-like torques, as the spin-diffusion length of Cu is much larger than its thickness [ 19]. We note that a finite field-like torque is necessary to observe the excitation of the PSSWs for vanishing exchange coupling J. Further- more, the field-like torque is required to be positive to model the intensity asymmetry in the mode branches of the YIG/Cu(5)/Co(50) sample (cf. Fig. S10 [13]). In Fig. 3 (c) we use τF=τD=J= 0, which reproduces the experimental observation for the YIG/AlOx/Co(50) sample. Importantly, no YIG PSSWs are observed in either the experiment or the simulation for this case. In summary, the simulations are in excellent qualitative agreement with the experimental observation of spin dy- namics in the coupled YIG/Co heterostructures. We attribute small quantitative discrepancies between the simulation and the experiment to the fact that we do not take any inhomogeneous linewidth and two-magnon scattering into account, which is, however, present in our system (see S7 [ 13] for details). This results in an under- estimated linewidth of the Co FMR line, in particular for small frequencies. As |∂DS21/∂H|is inversely propor-tional to the linewidths, this causes small quantitative deviations of the simulations and the experimental data. Furthermore, the exchange modes in Fig. 1 (a) are not found in the simulations. We attribute this to the fact that the simulations only represent the Co susceptibil- ity. However, as shown in Fig. S10 [ 13], similar exchange modes can also be found in the Co susceptibility from our simulations. In conclusion, we investigated the dynamic magnetiza- tion coupling in YIG/Co heterostructures using broad- band ferromagnetic resonance spectroscopy. We find ex- change dominated PSSWs in the YIG, excited by spin currents from the Co layer, and static interfacial exchange coupling of YIG and Co magnetizations. An efficient excitation of YIG PSSWs, even with a homogeneous ex- ternal magnetic driving field, is found in YIG/Co(35), YIG/Co(50) and YIG/Cu(5)/Co(50) samples, but is sup- pressed completely in YIG/AlOx(1.5)/Co(50). We model our observations with a modified Landau-Lifshitz-Gilbert equation, which takes field-like and damping-like torques as well as direct exchange coupling into account. Our findings pave the way for magnonic devices which operate in the exchange spin-wave regime. Such devices allow for utilization of the isotropic spin-wave dispersion relations in 2D magnonic structures. An excitation of short-wavelength spin waves by an interfacial spin torque does not require any microstructuring of excitation an- tennas but is in operation in simple magnetic bilayers. Remarkably, this spin torque scheme allows for the cou- pling of spin dynamics in a ferrimagnetic insulator to that in a ferromagnetic metal. The coupling is qualitatively dif- ferent to that found for all-metallic heterostructures [ 12]. Furthermore, the excitation of magnetization dynamics by interfacial torques should allow for efficient manipula- tion of microscopic magnetic textures, such as magnetic skyrmions. Financial support from the DFG via SPP 1538“Spin5 Caloric Transport” (project GO 944/4 and GR 1132/18) is gratefully acknowledged. S.K. would like to thank Meike M¨ uller for fruitful discussions. V.A. acknowledges support under the Cooperative Research Agreement between theUniversity of Maryland and the National Institute of Standards and Technology, Center for Nanoscale Science and Technology, Award 70NANB14H209, through the University of Maryland. [1]A. Khitun, M. Bao, and K. Wang, IEEE Transactions on Magnetics 44, 2141 (2008). [2]A. A. Serga, A. V. Chumak, and B. Hillebrands, Journal of Physics D: Applied Physics 43, 264002 (2010). [3]A. V. Chumak, A. A. Serga, and B. Hillebrands, Nature Communications 5, 4700 (2014). [4]S. Klingler, P. Pirro, T. Br¨ acher, B. Leven, B. Hillebrands, and A. V. Chumak, Applied Physics Letters 105, 152410 (2014). [5]S. Klingler, P. Pirro, T. Br¨ acher, B. Leven, B. Hillebrands, and A. V. Chumak, Applied Physics Letters 106, 212406 (2015). [6]K. Ganzhorn, S. Klingler, T. Wimmer, S. Gepr¨ ags, R. Gross, H. Huebl, and S. T. B. 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2017-12-07

We investigate yttrium iron garnet (YIG)/cobalt (Co) heterostructures using broadband ferromagnetic resonance (FMR). We observe an efficient excitation of perpendicular standing spin waves (PSSWs) in the YIG layer when the resonance frequencies of the YIG PSSWs and the Co FMR line coincide. Avoided crossings of YIG PSSWs and the Co FMR line are found and modeled using mutual spin pumping and exchange torques. The excitation of PSSWs is suppressed by a thin aluminum oxide (AlOx) interlayer but persists with a copper (Cu) interlayer, in agreement with the proposed model.

Spin waves in coupled YIG/Co heterostructures

1712.02561v1

1 Enhanced spin-orbit coupling in a heavy metal via molecular coupling S. Alotibi1, B.J. Hickey1, G. Teobaldi2,3,4,5, M. Ali1, J. Barker1, E. Poli2, D.D. O’Regan6,7, Q. Ramasse1,8, G. Burnell,1 J. Patchett,9 C. Ciccarelli,9 M. Alyami,1 T. Moorsom1 and O. Cespedes1* 1School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom. 2Scientific Computing Department, Science and Technology Facilities Council, Didcot OX11 0QX, United Kingdom 3Beijing Computational Science Research Center, 100193 Beijing, China 4Stephenson Institute for Renewable Energy, Department of Chemistry, University of Liverpool, L69 3BX Liverpool, United Kingdom 5School of Chemistry, University of Southampton, Highfield, SO17 1BJ Southampton, United Kingdom 6School of Physics, Trinity College Dublin, The University of Dublin, Dublin 2, Ireland 7Centre for Research on Adaptive Nanostructures and Nanodevices (CRANN) and the SFI Advanced Materials and Bio-Engineering Research Centre (AMBER), Dublin 2, Ireland 8SuperSTEM, SciTech Daresbury Science and Innovation Campus, Keckwick Lane, Daresbury WA4 4AD, United Kingdom 9Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge CB3 0HE, United Kingdom 5d metals are used in electronic architectures because of their high spin-orbit coupling (SOC) leading to efficient spin ↔ electric conversion and strong magnetic interactions. When C 60 is grown on a metal, the electronic structure is altered due to hybridisation and charge transfer. The spin Hall magnetoresistance for Pt/C 60 and Ta/C 60 at room temperature are up to a factor 6 higher than for the pristine metals, with the spin Hall angle increased by 20-60%. At low fields of 1-30 mT, there is an anisotropic magnetoresistance, increased up to 700% at room temperature by C 60. This is correlated with non-collinear Density Functional Theory simulations showing changes in the acquired magnetic moment of transport electrons via SOC. Given the dielectric properties of molecules, this opens the possibility of gating the effective SOC of metals, with applications for spin transfer torque memories and pure spin current dynamic circuits. * o.cespedes@leeds.ac.uk 2 The spin-orbit interaction is perhaps the most crucial mechanism in the design of magnetic structures and metal device physics. It determines the magnetocrystalline anisotropy, is key to the propagation and electrical conversion of spin currents, determines the magnitude of interfacial mechanisms such as the Dzyaloshinskii–Moriya interaction and guides new paths of research, such as the generation of Majorana fermions and energy band engineering of topological insulators[1-5]. The SOC also controls the efficiency of spin - charge conversion in the spin Hall, spin torque and spin Seebeck effects. All of these are key to reducing the power consumption and energy dissipation of computing and electronic devices, an issue that is quickly coming to the forefront of technology. However, currently we can only tune the SOC by static means, such as doping, preventing the design of architectures where spin, charge and magnetic interactions can be reversibly modified to enhance device performance or to acquire new functionalities. The Spin Hall magnetoresistance (SHMR) can quantify the SOC in thin (~nm) heavy metal layers deposited on a magnetic insulator such as the yttrium iron garnet Y 3Fe5O12 (YIG) [6-8]. When an electric current Jc flows in the metal, the spin Hall effect (SHE) induces a perpendicular spin current Js, with the spin polarization s parallel to the film surface. If the YIG magnetization M is parallel to s, Js cannot flow into the magnet and a spin accumulation forms. The resistance is the same as a bare Pt wire. When M is not parallel to s, the transverse component exerts a torque on the YIG magnetic moments, injecting spin current into the magnet. This opens a channel of dissipation for the spin current, reducing the inverse SHE contribution to Jc so that the resistance of Pt appears to have increased [5,9]. The largest dissipation takes place when M is perpendicular to s and the maximum SHMR should occur. The SHMR is measured by rotating the angle in Fig. 1a, with the applied H field (and therefore M) always orthogonal to the electrical current, but varying from in-plane to out-of-3 plane, and therefore from parallel (R min) to perpendicular to the spin polarization (R max) [5,10,11]. The ratio of the spin to charge current is known as the spin Hall angle: 𝜃ௌு=|𝑱𝒔|/|𝑱𝑪| [12-14]. 𝛩ௌு has technological relevance, as it is correlated with the torque exerted on ferromagnets in spin transfer torque memories [15]. A larger SHMR and therefore increased 𝜃ௌு can result in lower power or smaller switching currents for such devices. By tuning the SOC in conventional magnetic insulator/metal structures with a molecular layer, we can also differentiate spin transport effects based on their physical origins [16,17]. At metallo-molecular interfaces, the electronic and magnetic properties of both materials change due to charge transfer and hybridisation [18-21]. This can lead to the emergence of spin ordering and spin filtering [22-25], or change the magnetic anisotropy [21,26,27]. Even though composed of light carbon, fullerenes with large curvature can produce a large spin-electrical conversion [2,28-30]. Here, we study the effect of metal/C 60 interfaces on the SHMR and anisotropic magnetoresistance (AMR) of YIG/Pt and YIG/Ta. We aim to: investigate the mechanisms behind spin orbit scattering at hybrid metal/C 60 interfaces, maximise technologically-relevant parameters, and open new paths of research towards tunable SOC. Using shadow mask deposition, we grew two metal wires simultaneously on the same YIG substrate and, without breaking vacuum, covered one wire with 50 nm of C 60 –modifying the density of states (DOS) and transport properties of the metal. According to our density functional theory (DFT) calculations for Pt/C 60, 0.18-0.24electrons per C 60 molecule are transferred [31], and the first molecular layer is metallised. This reduces the electron surface scattering, improving the residual resistance ratio (RRR) –Figs. 1b and SI [31]. Our Ta wires have a resistivity (~1-2 ·m) and a negative 4 temperature coefficient (~-500 ·10-6 K-1), consistent with a sputtered -Ta phase [32]. Opposite to Pt, C60 increases the resistivity of Ta –see Fig. 1b. The change in resistivity as the magnetic field is rotated is fitted to a cosଶ(𝛽) function, and the amplitude is taken to be the SHMR [31]. The SHMR saturates once the applied field saturates the magnetisation out-of-plane, at 0.1-0.15 T for a YIG film 170 nm thick at 290 K, and no higher than 0.5 T for any measured condition. However, above this field range, other contributions such as Koehler MR, localisation and the Hanle effect can result in significant linear and parabolic contributions to the MR that would artificially enhance the SHMR ratio and SH (Fig 1c)[31,33]. For a YIG/Pt(2nm) sample, the C 60 layer increases the MR due to spin accumulation by about a factor 3, but reduces the polynomial contributions because of the increased effective (conducting) thickness of the Pt/C 60 bilayer. In YIG/Ta(4nm), where C 60 increases the resistance rather than reducing it, both the SHMR up to 0.15 T and the polynomial MR at higher fields are enhanced (Fig. 1d). 5 FIG. 1 (a) Schematic of the experiment. There are three possible orientations of the magnetic field (H) w.r.t. the electrical current and the YIG film. To measure only the SHMR without AMR effects, we rotate H from perpendicular to transverse (change in ). (b) Typical resistivity of thin Pt (3 nm) and Ta (4 nm) wires on YIG. With C 60 on top, the Pt resistivity is about 40% lower, the RRR factor increases and the upturn at low T is absent. With Ta, we observe the opposite effect, an increase in the resistivity with the molecular interface. Inset: Resistance with different applied fields as a function of the angle . The data is fitted to a cos2( function. We take the amplitude at the lowest field of 0.5 T, when the YIG substrate is saturated but the polynomial contributions are small, as the SHMR value. (c) MR in a Pt wire with H perpendicular . The spin Hall contribution to the MR at 300 K reaches a maximum at ~0.1-0.15 T, where the YIG film is saturated out of plane. (d) MR in a Ta wire with Hperpendicular at 75 K. The maximum in the spin Hall contribution at this temperature is reached at ~0.2 T. 6 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 molecular overlayer in Fig. 2b. The temperature dependence of the SHMR reproduces observations in RF-sputtered YIG/sputtered Pt wires [33]. For Pt grown by evaporation on thicker, liquid epitaxy or pulsed laser deposition YIG, the SHMR has a gentler drop at high temperatures. This is attributed to a smaller temperature dependence of the spin diffusion length [34,35], which could be due to a different resistivity of Pt and different magnetic behaviour of YIG films depending on the growth method. It is possible that the larger SHMR observed in metallo-molecular wires could be due to a change in the spin mixing conductance ( G↑↓) induced by C 60 [6,36]. However, G↑↓ is related to the spin transparency of the YIG/Pt interface, where the effect of the molecular interface is small [31]. Also, we do not observe an increase in the ferromagnetic resonant damping , proportional to G↑↓, of YIG/Pt with C 60 interfaces (Fig. 2c) [31,37]. Furthermore, the SHMR versus temperature results cannot be fitted by changing G↑↓ without also changing 𝛩ௌு [31]. Fig. 2d shows the 𝛩ௌு values taking G↑↓=4×1014 -1m-2 [8]; see the SI for other fitting values [34,38]. For Pt wires of ≤5 nm, there is an increase in 𝛩ௌு with C 60. This effect disappears for thick wires (> 10nm), where the molecular interface does not significantly change the spin Hall angle. A similar molecular enhancement of the SHMR and 𝛩ௌு is observed for Ta wires [31]. Molecules may affect the Rashba effect and spin texture of the metal, leading to changes in the effective SOC of the hybrid wire [39-41]. In our simulations, we consider the perpendicular dipole formed due to charge transfer at the Pt/C 60 interface and its associated potential step breaking symmetry [42]. However, this dipole is maximum at 2.5 nm, where the experiments show a local minimum. Our calculations point rather towards a mechanism mediated by the magnetic moment acquired by the transport electrons, resulting in spin-dependent charge flow [31]. 7 FIG. 2 (a) SHMR for Pt and equivalent Pt/C 60 wires of different thicknesses on GGG/YIG(170 nm) films. (b) SHMR Ratios between Pt/C 60 and Pt. The maximum effect of the molecular layer (factor 4 to 7 change) take place for thin films (1.5 nm) at low temperatures or thick films (5 nm) at room temperature. (c) The magnetic resonance damping is not increased by the C 60 interface; here a comparison of YIG/Pt/Al/C 60 and YIG/Pt/C 60 shows similar or even higher damping values for the decoupled Pt/Al/C 60 sample. (d) For wires ≤5 nm, SH obtained from the SHMR data fits is significantly higher with the molecular overlayer. Inset: Top view of the optimized C 60/Pt(111)-(2√3x2√3)R30° interface DFT model. The C 60 molecules are adsorbed on top of one Pt-vacancy. The black polygon marks the in-plane periodicity of the system. Pt: silver, C: cyan. (e) DFT simulations of the electrical current-induced, in-plane magnetic moments (| mxy|) and experimental SHMR, normalized to the largest calculated (| mxy|) or measured value (SHMR) as a function of the Pt thickness. 8 Non-collinear band structure calculations enable analysis of the atom Projected (energy- dependent) Magnetization Density (PMD) for different Pt and Pt/C 60 film thicknesses. In all cases, we find the PMD for the in-plane (x,y) magnetic moment components ( mx,y) to be larger than for the out of plane one ( mz). It is also possible to observe an enhancement of the PMD oscillation magnitudes due to the adsorption of C 60. The effect becomes smaller as the Pt thickness increases from 1.1 nm to 2.5 nm and 3.9 nm, correlated with the SHMR values in Pt/C 60 (Fig. 2e). The differences in PMD between the C 60/Pt and Pt systems document the role of the Pt/C 60 interfacial re-hybridization, and ensuing changes in the electronic structure, for enhancing SOC-related anisotropies and spin transport in Pt-based systems. The fabrication of YIG films can lead to elemental diffusion and defects that change the magnetic properties of the ferrimagnet and the interpretation of transport measurements [43]. Figs. 3a-b show atomic-resolution aberration corrected cross-sectional scanning transmission electron microscopy (STEM) images and electron energy loss spectroscopy (EELS) chemical maps. It is possible to observe, in addition to a certain level of surface roughness of the YIG film, an area close to the YIG surface and below the sputtered Pt wire into which some Pt metal may have diffused and formed a low density of nm-sized clusters (see also Fig. S4 in [31]). This diffusion can affect the magnetization and anisotropy direction at the surface of the YIG layer, originating the minor loops we observe in the perpendicular field direction in some YIG films [31,43]. For Pt grown on YIG, an additional change in resistance is observed at low magnetic fields <5-20 mT when the direction of an applied magnetic field is changed with respect to the electrical current. The origin of this AMR is controversial. It has been attributed to a proximity-induced magnetization of Pt, which is close to the Stoner criterion, but it is also claimed that there is no evidence for this 9 induced magnetization [16,17]. The same effect is also seen in YIG/Ta. This low field AMR (LF-AMR) is characterized by the presence of peaks, positive or negative depending on the field direction, resembling the AMR observed in magnetic films with domain wall scattering [44,45]. Due to the SOC, in most magnetic materials domain walls reduce the resistance for in-plane fields, and increase it for out of plane fields. This domain wall AMR peaks at the coercive field Hc of the magnet, for the greatest magnetic disorder and domain wall density. In YIG/Pt, the position of out-of-plane LF-AMR peaks coincides with the coercivity of the perpendicular minor YIG loops (Fig. 3c and [31]), which could point to a YIG surface layer with an out-of-plane easy axis. We find that the LF-AMR has the same shape and peak position with or without a molecular overlayer. However, the magnitude of the LF-AMR is larger when C 60 is present. This molecular effect is stronger for the perpendicular configuration (Fig. 3d), which may be due a larger perpendicular magnetic anisotropy induced by C 60, as reported for Co [21]. A larger LF-AMR is also observed in YIG/Ta when C 60 is deposited on top [40]. For YIG films grown on YAG substrates, the in-plane coercivity is increased by 1-2 orders of magnitude, and the LF-AMR peaks appear at higher fields, supporting the correlation between the AMR in Pt and the surface YIG magnetisation (Figs. S5-S7 in [31]). 10 FIG. 3 (a) Cross-sectional high angle annular dark field (HAADF) image of the YIG/Pt interface obtained using a scanning transmission electron microscope (see methods for details). (b) Elemental chemical analysis of the interface using EELS: the relative intensity maps of the Y, Fe and Pt ionization edges are presented with a simultaneously acquired HAADF image of the region, indicated by a white box in (a). Bright clusters immediately below the YIG surface, indicated by white arrows in the Pt map and the overview HAADF image, contain a higher Pt concentration and may be due to Pt diffusion into the YIG. (c) Low field MR and minor hysteresis loop with the field in the perpendicular orientation at 200 K. The full loop uncorrected and other examples can be found in [31]. (d) Room temperature LF-AMR comparison between YIG/Pt and YIG/Pt/C 60. The curves are qualitatively the same, but the magnitude of the effect is enhanced by the molecules. 11 The LF-AMR peak position (coercivity of the YIG surface) and peak width (saturation field of the YIG surface), increase as the temperature is lowered (Figs. 4a-b). Typically, the AMR of YIG/Pt measured at high fields is reported to vanish above 100-150 K. If measuring at 3 T, where quantum localisation and other effects are strong, we observe this same decay with temperature. However, the LF-AMR can be observed up to room temperature. C 60 not only increases the LF-AMR value, but it also makes it less temperature dependent, so that the LF-AMR ratio can be up to 700% higher for Pt/C60 at 290 K. This supports our suggestion from DFT simulations of a mechanism based on C 60- induced re-hybridization enhancing the magnetic moment acquired by transport electrons via SOC (Fig. 4c). The LF-AMR depends on the Pt thickness, 𝑡, as (𝑡−𝑥)ିଵ (Fig. 4d). We identify the value of 𝑥, approximately 1 nm, as the magnetised Pt region contributing to the AMR. This relationship is not affected by the C 60 layer, although the magnitude is uniformly higher with molecules. 12 FIG. 4 (a) Perpendicular LF-AMR for GGG/YIG(170)/Pt(2)/C 60(50). (b) As the sample is cooled, the perpendicular LF-AMR peak position and width are increased in steps, rather than monotonic fashion. (c) Temperature dependence of the maximum LF-AMR, calculated as the change in resistance from the peak in the perpendicular orientation to the minima in the longitudinal. There is a faster temperature drop in the MR values for Pt when compared with Pt/C 60. This may be due to the acquired magnetic moment in Pt/C 60 leading to a more stable induced magnetisation up to higher temperatures. (d) The LF-AMR for Pt and Pt/C 60 can be fitted to a (𝑡−𝑥)ିଵ function, where 𝑡 is the Pt wire thickness and x is a constant of 1 nm that we identify with the magnetically active Pt region. 13 Our results show that molecular overlayers can enhance the spin orbit coupling of heavy metals, as observed in SHMR and AMR measurements. Additionally, the molecular layers aid in distinguishing the origin of spin scattering mechanisms, such as the coupling with YIG surface magnetisation and a LF-AMR measurable at high temperatures. The enhancement of the effective SOC with molecular interfaces has a wide range of applications, e.g. to reduce the current densities in spin transfer torque memories. Given the dependence on surface hybridisation and charge transfer, the effect could be controlled via an applied electrical potential. This is an important development, as nearly all other methods to alter the spin-orbit coupling of a material are static. The inverse SHE can be modified by gating with ionic liquids, but changes to the SOC are undetermined and the electrical conversion may only be quenched [46]. Materials can be doped during fabrication to increase the spin-orbit effect, but that becomes fixed in a circuit, i.e. static. Using UHV grown nanoscale molecular films that can be gated offers a dynamic response – the transport properties of an active circuit, e.g. to control the direction and magnitude of pure spin currents. ACKNOWLEDGMENTS This work was supported by Science Foundation Ireland [19/EPSRC/3605] and the Engineering and Physical Sciences Research Council (EPSRC) UK through grants EP/S030263, EP/K036408, EP/M000923, EP/I004483 and EP/S031081. This work made use of the ARCHER (via the UKCP Consortium, EPSRC UK EP/P022189/1 and EP/P022189/2), UK Materials and Molecular Modelling Hub (EPSRC UK EP/P020194/1) and STFC Scientific Computing Department's SCARF High-Performance Computing facilities. J.B. acknowledges support from the Royal Society through a University Research Fellowship. Electron microscopy work was carried out at SuperSTEM, the National Research Facility for Advanced Electron Microscopy supported by EPSRC. S.A. acknowledges support from Prince Sattam bin Abdulaziz University. 14 [1] F. Hellman et al., Reviews of Modern Physics 89, Unsp 025006 (2017). [2] L. E. Hueso et al., Nature 445, 410 (2007). [3] J. Liu, F. C. Zhang, and K. T. Law, Physical Review B 88, 064509 (2013). [4] W. J. Shi, J. W. Liu, Y. Xu, S. J. Xiong, J. Wu, and W. H. Duan, Physical Review B 92, 205118 (2015). [5] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, Reviews of Modern Physics 87, 1213 (2015). [6] Y. T. 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2019-12-05

Heavy metals are key to spintronics because of their high spin-orbit coupling (SOC) leading to efficient spin conversion and strong magnetic interactions. When C60 is deposited on Pt, the molecular interface is metallised and the spin Hall angle in YIG/Pt increased, leading to an enhancement of up to 600% in the spin Hall magnetoresistance and 700% for the anisotropic magnetoresistance. This correlates with Density Functional Theory simulations showing changes of 0.46 eV/C60 in the SOC of Pt. This effect opens the possibility of gating the molecular hybridisation and SOC of metals.

Enhanced spin-orbit coupling in a heavy metal via molecular coupling

1912.02712v2

arXiv:1412.2367v2 [cond-mat.mes-hall] 12 Dec 2014Magnonic band gaps in YIG based magnonic crystals: array of g rooves versus array of metallic stripes V. D. Bessonov,1,2M. Mruczkiewicz,3R. Gieniusz,2U. Guzowska,2A. Maziewski,2A. I. Stognij,4and M. Krawczyk3∗ 1Faculty of Physics, University of Bia/suppress lystok, Bia/suppress lystok, Poland, 2Institute of Metal Physics, Ural Division of Russian Academ y of Science, Yekaterinburg, Russia, 3Faculty of Physics, Adam Mickiewicz University in Poznan, U multowska 85, Pozna´ n, Poland. 4Scientific-Practical Materials Research Center at Nationa l Academy of Sciences of Belarus, Minsk, Belarus (Dated: June 10, 2021) Abstract The magnonic band gaps of the two types of planar one-dimensi onal magnonic crystals comprised of the periodic array of the metallic stripes on yttrium iron garnet (YIG) film and YIG film with an array of grooves was analyzed experimentally and theoretically. In such periodic magnetic structures th e propagating magnetostatic surface spin waves were excite d and detected by microstripe transducers with vector network an alyzer and by Brillouin light scattering spectroscopy. Pro perties of the magnonic band gaps were explained with the help of the fi nite element calculations. The important influence of the nonreciprocal properties of the spin wave dispersion induc ed by metallic stripes on the magnonic band gap width and its dependence on the external magnetic field has been shown. The usefulness of both types of the magnonic crystals for potent ial applications and possibility for miniaturization are disc ussed. PACS numbers: 75.75.+a,76.50.+g,75.30.Ds,75.50.Bb I. INTRODUCTION The spatial periodicity determines the rule of conser- vation of the the quasi-momentum for excitations in arti- ficial crystals, similar to the conservation of momentum in homogeneous material. In the frequency domain this periodicity causes the formation of the pass bands and band gaps, i.e., frequency regions in which there are no available excitation states and the wave propagation is prohibited. Magnetic structures with artificial transla- tional symmetry are investigated to design new materials with properties that otherwise do not exist in nature, so called metamaterials. In particular, artificial ferromag- netic materials with periodicity comparable to the wave- length of spin waves (SWs), known as magnonic crystals (MCs),1–3have recently focused attention of the physics community. The typical example of the exploitation of MCs is control of the propagation and scattering of SWs. Thefirstexperimentalstudy ofthemagnetostaticSWsin ferromagnetic thin film with periodic surface was made by Sykes et al. already in 1976.4Nowadays, the number of studies about MCs has surged and continues to grow at a fast pace due to interesting physics and potential new applications.5–8 In this study we present the complementary experi- mental and theoretical investigation of SWs in two types of MCs having the same period in dependence on the external magnetic field amplitude. The first type is a system of periodically arranged grooves etched in the yttrium iron garnet (YIG) film, the second is an uni- form YIG crystal with placed atop metallic stripes. Both types of structures have already been studied.9–14The first one was proposed as a SW waveguide and exper- imentally tested as delay line or filter for microwave applications4,15,16and more recently as a basic element of the purely magnonic transistor17or microwave phase(a) 80 μm 150 μm GGGYIG10 μm 1.5μm10μmGGGYIG1μmAuSample A (b) Sample B yx zUnit cell Unit cellH0 FIG. 1. Geometry of the two MCs investigated in the paper. (a) Sample A, 1D MC created by the array of grooves etched in the YIG film on the GGG substrate. The thickness of the film is 10 µm, grooves width and depth is 80 µm and 1.5 µm, respectively. (b) Sample B, 1D MC formed on the basis of homogeneous YIG film of 10 µm thickness by deposition of the array of Au stripes. The stripe width is 80 µm and thickness 1 µm. The same lattice constant 150 µm is kept in both samples. shifter.18Thestructuresofthesecondtypewerealsocon- sidered as delay lines and filters16,19but recently have also been proposed as room temperature magnetic field sensors.20–22 We perform comparative study of these two types of MCs magnetically saturated by the external magnetic field along grooves and metallic stripes. For measure- ments we use the passive delay line with a network ana- lyzer and Brillouin light scattering (BLS) measurements. The SW dynamic in these MCs is modeled with finite element method (FEM) in the frequency domain. Ac- quired information from calculations is used to explain experimental data and to discus properties of magnonic band gap formation in these two kinds of MCs and their usefulness for recently proposed applications. 1The paper is composed as follows, in Sec. II we briefly describe the experimental methods used in the study: measurements of the transmission of the SWs with mi- crowave transducers and BLS measurements. In Sec. III we introduce FEM used for calculation of the magnonic band structure. In the next section, Sec. IV we discuss the results obtained in two types of MCs. The paper is ended with Sec. V where the summary of the paper is presented. II. EXPERIMENTS The fabrication process of the artificial periodic struc- tures with characteristic dimensions in deep nanoscale is very hard to control and so far mainly theoretical stud- ies are available in this scale.23,24Especially it concerns the quality of edges and the interfaces between adjacent materials which makes up MC and can significantly in- fluence magnonic band structure.25–27However, in larger scalewithaperiod startingfromhundrednm the fabrica- tion technology is already well established for thin ferro- magnetic metallic films.28,29The YIG films of the thick- ness of tens nm have been fabricated only recently and the quality of these films increases systematically. The damping comparable to the value in thick YIG (of or- der smaller than in the best metallic ferromagnetic film) has already been achieved.30,31However, the YIG thin films with patterning in nanoscale is not yet investigated. Here, we study dielectric YIG films structurized in larger scale where edge properties have minor influence on the SW dynamics. 10 µm thick YIG films were epitaxially grown on gallium gadolinium garnet (GGG) substrates in a (111) crystallographic plane and serve to fabricate one-dimensional (1D) MCs. The MCs used in our exper- iment had been produced in the form of the waveguide of 3.5 mm width and 50 mm length with (i) an array of parallel grooves chemically etched [sample A, Fig. 1(a)] and (ii) an array of Au microstripes placed on the top of the film [sample B, Fig. 1(b)]. The grooves and Au microstripes were perpendicularly oriented with respect to the SW propagation direction and include nine lines of 80µm width which are spaced 70 µm from each other, so that the period ais 150µm. The external magnetic field H0is applied along the grooves and microstripes in order to form conditions for propagation of the magnetostatic surface spin wave (MSSW), alsocalledasDamon-Eshbachwave. Thiswave has asymmetric distribution of the SW amplitude across the film, which depends on the direction of the magnetic field with respect to the direction of the wave vector, and this asymmetry increases with increasing wavenumber. By putting metal on the ferromagnetic film the nonre- ciprocal dispersion relation of MSSW is induced.32 The MSSW were excited and detected in garnet film waveguide using two 30 µm wide microstripe transduc- ers connected with the microwave vector network ana- lyzer (VNA), one placed in the front and another one be-hind the periodic structure. An external magnetic field (µ0H0= 0.1 T) was strong enough to saturate the sam- ples. Microwave power of 1 mW used to the input trans- ducer was sufficiently small in order to avoid any non- linear effects. A VNA was used to measure amplitude- frequency characteristics collected for the second trans- ducer. The transmission spectra of SWs measured in the reference sample, i.e., a thin YIG film, is shown in Fig. 2. The transmission of the microwave signal above 10 dB is in the band from 4.46 to 5.14 GHz. 4 4 . 4 8 . 5 2 .-70-50-30-10 Frequency (GHz)Transmission (dB) FIG. 2. Transmission spectra of MSSW in the reference sam- ple, i.e., uniform YIG film of 10 µm thickness in the external magnetic field 0.1 T. In BLS measurements the SWs were excited with the single 30 µm wide microwave transducer located in front of the array of Au microstripes (sample B). The SWs scattered on the line of waveguide were detected by space-resolved BLS spectroscopy in the forward scatter- ing configuration.33The probe laser beam was scanned across the sample (in the areas between Au stripes) and the BLS intensity, which is proportional to the square of the dynamic magnetization amplitude, was recorded at various points. This technique allows for a two- dimensional (2D) mapping of the spatial distribution of the SW amplitude with step sizes of 0.02 mm. III. THEORETICAL MODELING In our calculations we assume the stripes and grooves have infinite length (i.e., they are infinite along zaxis). This is reasonable assumption taking into account that length is around 23 times longer than the period of the structure. The structures under investigation remain in a magnetically saturated state along the z-axis due to the static external magnetic field pointing in the same direction. To obtain insight into the formation of the magnonic band structure and opening magnonic band gaps the nu- merical calculations of the dispersion relation were per- formed. For SWs from the GHz frequency range, due to 10µm thickness of YIG film and a small value of the exchange constant in YIG, the exchange interactions can 2be safely neglected. In order to calculate the SW disper- sion relation in magnetostatic approximation we solved the wave equation for the electric field vector E:34 ∇×/parenleftbigg1 ˆµr(r)∇×E/parenrightbigg −ω2√ǫ0µ0/parenleftbigg ǫ0−iσ ωǫ0/parenrightbigg E= 0,(1) whereω= 2πf,fisaSWfrequency, µ0andǫ0denotethe vacuum permeability and permittivity, respectively, and σisconductivity, different fromzeroonlyin sampleB.To describe the dynamics of the magnetization components in the plane perpendicular to the external magnetic field, it is sufficient to solve Eq. (1) for the zcomponent of the electric field vector Ewhich depends solely on xandy coordinates: Ez(x,y).35 The permeability tensor ˆ µ(r) in Eq. (1) can be ob- tained from Landau-Lifshitz (LL) equation.32The as- sumption that the magnetization is in the equilibrium configuration allows us to use the linear approximation in SW calculations, which implies small deviations of the magnetization vector M(r,t) from its equilibrium orien- tation. Thus, for the MCs saturated along z-axis the magnetization vector can split into the static and dy- namic parts: M(r,t) =Mzˆz+m(r,t), and we can ne- glect all nonlinear terms with respect to dynamical com- ponents of the magnetization vector m(r,t) in the equa- tion of motion defined below. Since |m(r,t)| ≪Mz, we can assume also Mz≈MS, whereMSis the saturation magnetization. We consider only monochromatic SWs propagating along the direction of periodicity, thus we can write m(r,t) =m(x,y)exp(iωt). Under these as- sumtions the dynamics of the magnetization vector m(r) with negligible damping is described by stationary LL equation: iωm(r) =γµ0(MSˆz+m(r))×Heff(r),(2) whereγis the gyromagnetic ratio (we assume γ= 176 rad GHz/T) and Heffdenotes the effective magnetic field acting on the magnetization. The effective magnetic field is in general a sum of several components, here we will considertwoterms, the static externalmagneticfield and the dynamic magnetostatic field: Heff(r,t) =H0ˆz+hms(r,t). (3) The permeability tensor ˆ µ(r) in Eq. (1) obtained from the linearized damping-free LL Eq. (2) for ferromagnetic material takes following form: ˆµr= µxxiµxy0 −iµyxµyy0 0 0 1 , (4) where µxx=γµ0H0(γµ0H0+γµ0MS)−ω2 (γµ0H0)2−ω2,(5) µxy=γµ0MSω (γµ0H0)2−ω2, (6) µyx=µxy, µyy=µxx, (7)in non-magnetic areaspermeability is an identity matrix. Equation (1) with the permeability tensor defined in Eq. (4) in the periodic structure has solutionswhich shall fulfill Bloch theorem: Ez(x,y) =E′ z(x,y)eiky·y, (8) whereE′ z(x,y) is a periodic function of y:E′ z(x,y) = E′ z(x,y+a).kyis a wave vector component along yand ais a lattice constant. Due to considering SW propaga- tion along ydirection only, we assume ky≡k. Eq. (1) together with Eq. (8) can be written in the weak form and the eigenvalue problem can be generated, with the eigenvalues being frequencies of SWs or in the inverse eigenproblem with the wavenumbers as eigenvalues. The former eigenproblem is used to obtain magnonic band structure, the later to calculate the complex wavenumber of SW inside the magnonic band gaps. This eigenequa- tion is supplemented with the Dirichlet boundary condi- tions at the borders of the computational area placed far from the ferromagnetic film along xaxis (bold dashed lines in Fig. 1). In FEM the equations are solved on a discrete mesh in the two-dimensional real space [in the plane ( x,y)] lim- ited due to Bloch equation to the single unit cell (marked by the gray box in Fig. 1). In this paper we use one of the realizations of FEM developed in the commercial software COMSOL Multiphysics ver. 4.2. This method has already been used in calculations of magnonic band structure in thin 1D MCs, and their results have been validated by comparing with micromagnetic simulations and experimental data.36–38The detailed description of FEM in its application to calculation of the SW spectra in MCs can be found in Refs. [38 and 39]. IncalculationswehavetakennominalvaluesoftheMC dimensions and the saturation magnetization of YIG as MS= 0.14×106A/m. The conductivity of the metal is assumed as σ= 6×107S/m, which is a tabular value for Au. IV. RESULTS AND DISCUSSION In Fig. 3(a) and (b) we present the results of the SW transmission measurements with the use of microstripe lines in the external magnetic field 0.1 T for sample A and sample B, respectively. We can see a clear evidence of three (centered at 4.81, 4.97 and 5.05 GHz) and two magnonic band gaps (at 4.88 and 5.05 GHz) in sample A and B, respectively. The transmission band in both samples is approximately the same as in the reference sample (Fig. 2), however at high frequencies in sample B a large decrease of the transmission magnitude is ob- served. Thus, in MC with metallic stripes the second magnonic band gap [marked with blue square in Fig. 3 (b)] is already at the part of the low transmission. The estimation of its position and width will be loaded with additional errors and some ambiguity, thus in further in- vestigations we will not consider this band gap. 34 75 . 4 85 .w(a) (b) Sample A Sample B -70 -70-50 -50-30 -30-10 -10 Transmission (dB) Transmission (dB) 4 4 . 4 8 . 5 2 . Frequency (GHz)4 4 . 4 8 . 5 2 . Frequency (GHz) FIG. 3. Transmission spectra of SWs in (a) sample A and (b) in sample B measured with microstripe lines in external magnetic field 0.1 T. The magnonic band gaps are marked by solid symbols: in sample A there are three gaps, in sample B there are two gaps, however the second gap is at the part of low transmission and will not be considered in the paper. In the inset of the figure (a) the enlargement of the spectra around of the first band gap is shown, the width of this gap isw. The calculated magnonic band structures are pre- sented in Fig. 4 with blue dashed and red solid lines for sample A and B, respectively. For MC with grooves the dispersion relation is symmetric and magnonic band gaps are opened at the Brillouin zone (BZ) border (first and third gap) and in the BZ center (the second gap). The frequencies of gaps obtained in calculations agree well with the gaps found in transmission measurements [Fig. 3 (a)]. For sample B, the magnonic band struc- ture is nonreciprocal, i.e., f(k)/negationslash=f(−k).35,40Moreover, the first band has large slope (larger than for sample A), especially in + kdirection has significantly increased group velocity. These effects are results of conducting properties of the Au stripes, which cause fast evanescent of the dynamic magnetic field generated by oscillating magnetization in the areas occupied by metallic stripes. Due to this nonreciprocity in the dispersion relation the magnonic band gap opens inside the BZ and it is an in- direct band gap. Also for sample B we have found good agreement between calculations and measured data. In the measured data shown in Fig. 3 there is visi- ble difference between the width and depth of the first band gap in sample A and B. In order to estimate the depth ofthe gap fromcalculations we need to solvean in- verse eigenproblem, i.e., to fix the frequency as a param- eter and search for a complex wavenumber as an eigen- value. In Fig. 5 the calculated imaginary part of the wavenumber (Im[ k]) as a function of frequency around the first band gap is presented. In figures (a), (b) the external magnetic field was set on 0.1 T, in figures (c), (d) it was enlarged to the value 0.15 T. For sample A [Fig. 5(a) and (c)] the Im[ k] has zero value outside of the gap since the Gilbert damping is neglected in the calcula- tions. However, forsampleB [Fig. 5(b) and(d)] the func- tion Im[k](f) is nonzero outside of the gap, it is because the metal stripes induce attenuation of SWs. Outside of the gap regions in sample B the Im[ k] increases with the frequency and this behavior is observed in the transmis--1 1 04.64.85.0 Wavevector ( / ) k a/c112Frequency (GHz) FIG. 4. Magnonic band structure in the first Brillouin zone calculated for sample A (blue dashed line) and sample B (red solid line) with magnetic field µ0H0= 0.1 T. Magnonic band structure is symmetric and asymmetric with respect of the Brillouin zone center (marked by vertical black dashed line ) in sample A and B, respectively. sion spectra as decrease of the signal at large frequencies (still in the transmission band of the reference sample) in sample B [Fig. 3(b)]. It is observed that the maximal value of Im[ k] in the first band gap is significantly larger for sample B than A (0.113 and 0.062, respectively for 0.1 T magnetic field). Because an inverse of Im[ k] describes the decaying length of SWs, it correlates with the magnonic band gap depth in the transmission measurements. Indeed this finds con- firmation in the experimental data, where the minimal transmission magnitude in the band gap is -18 dB at 4.81 GHz and -39 dB at 4.89 GHz in sample A and B, respectively. This significant suppression of the trans- mission of SW signal in the first magnonic band gap in sample B is confirmed also in BLS measurements pre- sented in Fig. 6, where two excitation frequencies were set to (a) 4.64 GHz and (b) 4.89 GHz. These frequencies were chosen to visualize the SW propagation at frequen- cies from the band and from the band gap, respectively. In both cases the decrease of the SW amplitude with in- creasing the distance form the transducer is found, how- ever in the band gap this decrease is more pronounced. Nevertheless, some signal is still observed at the end of MC for frequencies from the band gap. We suppose that this is due to limited number of Au stripes used in the experiment and direct excitation of SWs from the trans- ducer. There is also another difference between function Im[k](f) for both samples. This is an asymmetry be- tween the bottom and top part of the gap in sample B, while in sample A the function Im[ k](f) is almost sym- metric with respect to the magnonic band gap center. To have some measure of this asymmetry we have cal- culated the derivatives ∂Im[k]/∂fat the points where Im[k] is half of its maximum value, i.e., at points a-d marked in Fig. 5(a) and (b). For the sample A these val- ues are: 2 .23×10−4s/m and −2.28×10−4s/m (points 4Sample A . .b a 4.77 4.78 4.790 00.04 0.040.08 0.080.12 0.12Sample BSample B . .d c 4.87 4.89 4.91 4.93 6.34 6.35 6.36Sample A(a) (b) (c) (d)00.040.080.12 Im[ ] ( / )k a/c112 00.040.080.12 Im[ ] ( / )k a/c112Im[ ] ( / )k a/c112 Im[ ] ( / )k a/c112f(GHz) f(GHz)f(GHz) f(GHz)6.46 6.42 6.44 FIG. 5. The imaginary part of the wavenamber around the first band gap in sample A (a), (c) and in sample B (b), (d) for the two values of the magnetic field µ0H0= 0.1 T (a), (b) and 0.15 T (c), (d). The calculation were done for the inverse eigenproblem with FEM. The letters a-d indicate points wher e the value of Im[ k] takes a half of its maximum. These points may indicate the borders of the band gap extracted from the transmission measurements. a and b, respectively) and for sample B: 1 .50×10−4s/m and−1.97×10−4s/m (respectively points c and d).41 We attribute this difference in Im[ k] between MCs to the different group velocities of SWs around the gaps, i.e., the symmetric and asymmetric dispersion curves of the first (and second) band near the edge of the band gap for the sample A and B, respectively (Fig. 4). We point out that this asymmetry in Im[ k] might appear as asymmet- ric slope in the transmission spectrum (Fig. 3) and it can be of some importance for applications in magnetic field sensors and magnonic transistors.17,21 0.30.91.52.1 0 00.6 0.6 1.2 1.2y(mm) z(mm) z(mm)(a) = 4.64 GHzf (b) = 4.89 GHzf minmax H0k Magnonic crystal FIG. 6. Maps of the SW intensity acquired with BLS from sample B at two frequencies (a) 4.64 GHz and (b) 4.89 GHz related to the transmission band and the band gap. The mi- crostripe transducer aligned along zaxis used to excite SWs is located below presented area. Finally, we study magnonic band gap widths in depen- dence on amplitude of the external magnetic field. The results are presented in Fig. 7(a) and (b) for sample A and B, respectively. In this figure there are points (full dots and squares) extracted from the transmission mea-(a) (b) Sample A Sample B 0.08 0.08 0.12 0.12 0.16 0.1651525 Width of gap (MHz)w Width of gap (MHz)w Magnetic f T ield ( )/c1090H0 Magnetic f T ield ( )/c1090H020304050 FIG. 7. Width of the magnonic band gap as a function of the external magnetic field (a) in sample A and (b) in sam- ple B. The experimental data are marked by full dots and squares, while the results of calculations are shown with so lid and dashed lines for the first and second band gap, respec- tively. The horizontal lines at some selected values of H0show errors of the measured magnonic band gap width. surements and lines (red-solid and blue-dashed) obtained from FEM calculations (first and second band gap, re- spectively). Overall, we have found good agreement be- tween theory and measurements, the calculation results arealwaysin the rangeofthe experimentalerrorsmarked in figuresby solid verticallines. The decreaseofthe band gapwidthwithincreaseofthemagneticfieldweattribute to the decrease of the band width (i.e., decrease of the group velocity) of the MSSW.42The steeper decrease of the band gap width is observed for the MC with metal- lic stripes. These dependencies find also reflection in the values of Im[ k] shown in Fig. 5, where the Im[ k] drops down by 23% in sample B, while for sample A the Im[ k] remains almost the same with the increase of the mag- netic field by 0.05 T. This different dependencies for sample A and B are related to the larger sensitivity of the group velocity of MSSW on changes of the magnetic field in metallized film then in unmetallized, but also to the nonreciprocal magnonic band structure and the presence of the indi- rect band gap in the case of sample B. The sensitiv- ity of the group velocity around the gap might be es- timated analytically. In Fig. 8 the analytical dispersion relation of MSSW in 10 µm thick YIG film with metal overlayeris presented in the empty lattice model (ELM). The periodicity was taken the same as a periodicity of the samples. The crossing point between dispersions of the MSSW propagating in opposing directions, + k(with maximum of the amplitude close to the metal) and −k (with amplitude on the opposite surface, in the figure this dispersion is shifted by the reciprocal lattice vector 2π/a) indicates the Bragg condition, i.e., the condition for opening magnonic band gap. It means that the Bragg condition takes wavenumbers from the first and second BZ for waves propagating into positive (+ k) and nega- tive (−k) direction of the wavevector, respectively.39,43 The group velocities were calculated at crossing points at field values 0.1 T and 0.15 T for structures with and without metal overlayer. Based on these values, we have found that the ratio of group velocity changes with the field is almost twice higher in structure with metal layer 5than without this. Although, the change of the dispersion slope around Bragg condition is larger for + kwave, the magnitude of the wavevector |+k|is smaller than | −k|and small change of the group velocity of −kwave might also have impact on changes in the band gap position. In our case, the increase of the magnetic field from 0.1 T to 0.15 T results also in the shift of the Bragg condition towards BZ center (see Fig. 8). However, in general, the posi- tion of the Bragg condition might shift towards center or edge of the BZ with the increase of the field. To which direction will shift the Bragg condition is determined by both groupvelocitychangeand wavenumberdifferenceof MSSW propagating in opposite directions. We note also, that for both samples, the width of the second band gap is less sensitive to the magnetic field amplitude than the width of the first gap. 1.0 0.05.06.07.0 Wavevector ( / ) k a/c112Frequency (GHz) 0.2 0.6 0.4 0.85.56.5 /c1090 0H=0.10 T/c1090 0H=0.15 T +k -k FIG. 8. Analytical estimation of the Bragg condition for magnonic band gap opening in 10 µm thick metalized YIG film with periodicity a= 150µm at two values of the ex- ternal magnetic field: 0.1 T and 0.15 T. Dispersion relation of the propagating MSSW waves with maximum of the am- plitude close to the metal ( k+) and on opposite side ( k−) of the film are marked with solid and dashed lines, respectively . The dispersion of k−wave is shifted by the reciprocal lattice vector 2π/afrom its original position. The Bragg condition is fulfilled at the cross-section of the k+andk−lines. It is also sample B which has a wider band gap then sample A in the considered magnetic field values [see Fig. 7(a) and (b)]. It was already shown that cover- ing bi-component MC or ferromagnetic film with lat- tice of grooves by a homogeneous metallic overlayer shall increase the band gap width of the MSSW due to in- creased group velocity of MSSW propagating along met- alized surface.43,44However, the influence of metal with finite conductivity depends on the wavenumber(and film thickness), and disappears for large k.35Thus, band gap widthwilldependonthewavenumberatwhichtheBragg condition is fulfilled, i.e., will depend on the lattice con- stant. For sufficiently large k(small period) the influence ofmetal disappearsand band gapswill not form.43In the homogeneous YIG film of 10 µm thickness the influence of the homogeneous Au overlayer on the dispersion rela- tion of MSSW disappears for k≈1.57×106rad m−1, i.e., for the MC with a period a= 2µm at the BZ border the effect of metallization will be absent. The influence of metal will disappear also when the separation between metallic stripes and YIG will be introduced, howeverthiscan be avoided by proper fabrication technique. In sample A an influence of the corrugation shall pre- serve also for small a, thus it is expected that for small lattice constant the band gap in sample A will be wider than for sample B. However, we note also that the band gap width depends also on the grooves depth in sample A, thus there is an additional parameter to be taken into account. In the caseofsmallsurfaceperturbations(small ratioofthe grovedepth to the film thickness) the coupled mode theory shows that the width ofthe gap and the gap depth (maximal Im[ k] in the gap) are proportionalto the perturbation.15Nevertheless, the structure with larger groves has not been found very promissing for magnonic band gaps applications so far, because suppressed trans- mission in the bands due to excitations of the standing spin waves.11,12,45This can change when the very thin YIG samples will be used for MCs, then the frequency of standing exchange SW modes will moved to high fre- quencies. However, the fabrication regularmodulation of the film thickness in deep nanoscale remain challenging task. V. CONCLUSIONS In summary, the SW spectrum of the two planar 1D MCs comprised of a periodic array of etched grooves in YIG film and an array of metallic stripes on homoge- neous YIG film has been fabricated and studied experi- mentally and theoretically. The properties of propagat- ing SWs and magnonic band gaps were in focus of our investigations. The two different kinds of MCs elaborate the fundamental differences in the magnonic band spec- trum, and also their band gap properties are different. To study of band gap widths and depths we used numer- ical method, which is based on FEM in the frequency. We obtained these values according with the measured data. We haveshownthat MC formed bymetallic stripes posses wider magnonic band gap with larger depth than the second MC. Moreover, the fabrication of arrays of metallic stripes is much more feasible than etching of grooves in the dielectric slab. However, the influence of the metal overlayer on the band gaps of magnetostatic wavesis limited to relatively small wavenumbersand this limits the miniaturizing prospective for these MCs. In contrast the MC based on the lattice of grooves does not have such limit, nevertheless its band width and depth is limiting by the excitation of the standing exchange SWs. In both types of MCs, the magnonic band gap width de- creases with increasing external magnetic field, we have identified mechanisms responsible for these changes. The results obtained here should have impact on the applications of MCs, because we have shown the influ- ence of different types of periodicity on the magnonic band gaps. These properties shell be especially impor- tant for magnonic devices, like magnetic field sensors21 or full magnonic transistors17which functionality were already experimentally demonstrated. In these applica- 6tionsthetailoringofthemagnonicbandgapwidth, depth and their edges is crucial to make magnonic devices com- petitive with existing technologies.ACKNOWLEDGMENTS The researchleadingto these resultshasreceivedfund- ing from Polish National Science Centre project no. 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The magnonic band gaps of the two types of planar one-dimensional magnonic crystals comprised of the periodic array of the metallic stripes on yttrium iron garnet (YIG) film and YIG film with an array of grooves was analyzed experimentally and theoretically. In such periodic magnetic structures the propagating magnetostatic surface spin waves were excited and detected by microstripe transducers with vector network analyzer and by Brillouin light scattering spectroscopy. Properties of the magnonic band gaps were explained with the help of the finite element calculations. The important influence of the nonreciprocal properties of the spin wave dispersion induced by metallic stripes on the magnonic band gap width and its dependence on the external magnetic field has been shown. The usefulness of both types of the magnonic crystals for potential applications and possibility for miniaturization are discussed.

Magnonic band gaps in YIG based magnonic crystals: array of grooves versus array of metallic stripes

1412.2367v2

Science Advances Manuscript Template Page 1 of 12 Observation of spin-orbit magnetoresistance in metallic thin films on magnetic insulators Lifan Zhou,1,† Hongkang Song,2,3,† Kai Liu,3 Zhongzhi Luan,1 Peng Wang,1 Lei Sun,1 Shengwei Jiang,1 Hongjun Xiang,3,4 Yanbin Chen,1,4 Jun Du,1,4 Haifeng Ding,1,4 Ke Xia,2 Jiang Xiao,3,4,5,* and Di Wu1,4,* 1National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, P. R. China. 2Department of Physics, Beijing Normal University, Beijing 100875, P. R. China. 3Department of Physics and State Key Laboratory of Surface Physics, Fudan University, Shanghai 200433, P. R. China. 4Collaborative Innovation Center of Advanced Microstructures, Nanjing 21 0093, P. R. China. 5Institute for Nanoelectronics Devices and Quantum Computing, Fudan University, Shanghai 200433, P. R. China. † These authors contributed equally to this work. *Corresponding author: xiaojiang@fudan.edu.cn, dwu@nju.edu.cn. Abstract A magnetoresistance effect induced by the Rashba spin -orbit interaction was predicted, but not yet observed, in bilayers consisting of normal metal and ferromagnetic insulator. Here, we present an experimental observation of this new type of spin-orbit magnetoresistance (SOMR) effect in a bilayer structure Cu[Pt]/Y 3Fe5O12 (YIG), where the Cu/YIG interface is decorated with nanosize Pt islands. This new MR is apparently not caused by the bulk spin-orbit interaction because of the negligible spin-orbit interaction in Cu and the discontinuity of the Pt islands. This SOMR disappears when the Pt islands are absent or located away from the Cu/YIG interface, therefore we can unambiguously ascribe it to the Rashba spin-orbit interaction at the interface enhanced by the Pt decoration. The numerical Boltzmann simulations are consistent with the experimental SOMR results in the angular dependence of magnetic field and the Cu thickness dependence. Our finding demonstrates the realization of the spin manipulation by interface engineering. Introduction Relativistic 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 formation of skyrmions (5, 6). In SHE, a pure spin current transverse to an electric current can be generated in conductors with str ong SOI, such as Pt, Ta etc (7, 8). The inverse SHE (ISHE) is generally used to detect the spin current electrically by converting a pure spi n current into a charge current (9, 10). It was recently discovered that the interplay of the SHE and 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 Hall magnetoresistance (SMR), in which the resistance of the NM layer depends on the direction of the FI magnetization M (11-13). SMR has been observed in several NM/FI systems and even in metallic bilayers (14-17). However, it has been argued that SMR may originate from the magnetic moment in the NM layer induced by the magnetic proximity effect (MPE) (18). These two mechanisms were proposed to be distinguished by the an gular dependent MR measurements (11, 13). Very recently, another type of MR, the Hanle MR (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 external magnetic field H, rather than that of M in SMR. Within the framework of SMR, because of the negligible SOI in Cu (20), one would not expect any MR effect in a Cu/FI bilayer. Recently, Grigoryan et al. predicted a new type of MR effect in the NM/FI systems when a Rashba type SOI is present at the interface between NM and FI ( 21). This new spin-orbit MR (SOMR) works even with light metals such as Cu or Al with negligible bulk SOI, provided that the Rashba SOI is present at the NM/FI interface. Because of the identical angular dependence on M direction for SOMR and SMR, however, it is difficult to distinguish SOMR from SMR in systems like Pt/Y 3Fe5O12 (YIG), where both SOMR and SMR are present in principle. In this work, we report the first observation of SOMR in a Cu/YIG bilayer, where the Rashba SOI at Cu/YIG interface is enhanc ed by an ultrathin Pt layer (< 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 be the origin of the observed MR and the Pt-decoration of the Cu/YIG interface is crucial for 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 of SOMR is clearly different from the non-monotonous dependence of SMR (13, 18). Both the angular- and Cu-thickness- dependence of the observed MR are in good agreement with our Boltzmann simulations based on the SOMR mechanism. In addition, the MR shows two maxima as the Pt layer thickness increases, in sharp contrast with that of SMR (13, 22). Result s and discussions Sample morphology and structure The YIG films used in this study are 10 nm thick, unless otherwise stated, grown by pulsed laser deposition (PLD) on Gd 3Ga5O12 (GGG) (111) substrates. The surface morphology of the YIG films was characterized by atomic force microscopy (AFM), as shown in Fig. 1A. The film is fairly smooth with the root-mean-square (rms) roughness of 0.127 nm and the peak-to-valley fluctuation of 0.776 nm. The 0.4-nm-thick Pt layer, thinner than the peak-to- valley value of the YIG film, deposited on YIG by magnetron sputtering forms the nanosize islands with the rms roughness of ~ 0.733 nm, shown in Fig. 1B. This discontinuous Pt layer is non-conductive with the resistance over the upper limit of a multimeter. The surface roughness is reduced after the deposition of Cu onto Pt, as shown in Fig. S1. Figure 1C presents the cross-section high-resolution transmission electron microscope (HRTEM) image of the Au(3)/Cu(4)[Pt(0.4)]/YIG films, where the numbers are the thicknesses in the unit of nanometer. The YIG film is clearly single-crystalline and smooth. The lattice constant of the YIG film is determined to be 1.2234 nm, to be compared to 1.2366 nm for the bulk YIG. A clear interface is observed between the metallic films and the YIG film. The metallic films are polycrystalline. Field-dependent magnetization and transport measurements In this work, all the measurements were performed at room temperatu re. The YIG film is almost isotropic in the film plane with the coercivity of about 0.4 Oe, shown in Fig. 2A. Due to the large paramagnetic background of the GGG substrate, it is difficult to measure the magnetization of a thin YIG/GGG film in the out -of-plane geometry. We measured a 400 - nm-thick YIG/GGG(111) film instead. As shown in Fig. 2B, the magnetization is saturated at ~ 1800 Oe. The saturation magnetization M s of our YIG film is determined to be 164.5 emu/cc measured by ferromagnetic resonance (FMR ) (see the Supplementary Materials ). In comparison, M s of bulk YIG is 140 emu/cc. Figures 2C and 2D present the resistivity r as a function of H for Cu(2)[Pt(0.4)]/YIG(10) sample. 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 the sample plane and perpendicular to the current direction ( y-axis), and iii) perpendicular to the sample plane (z-axis), respectively. The MR effects are clearly present in all measurements. For H along x- and y-directions, r shows two peaks around the coercive fields of YIG. For H along z-direction, r shows a minimum at H = 0 and remains almost a constant value above the saturation field. These features indicate that the MR effects are intimately correlated with M, meaning that the observed MR effects are not HMR. Angular dependent MR measurements To further study the anisotropy of the MR effects in Cu[Pt]/YIG, we performed the angular dependent MR measurements. Figure 3A shows Dr/r of Cu(3)[Pt(0.4)]/YIG(10) sample with rotation of H in the xy- (a-scan), yz- (b-scan) and xz- (g-scan) planes, where a, b and g are the angles between H and x-, z- and z-directions, respectively, as defined in the inset of Fig. 3A. The applied magnetic field strength ( H = 1.5 T) is large enough to align M with H. The MR effect is clearly anisotropic. The MR ratio, defined as Dr/r = [r(angle) - r(angle = 90o)]/r(angle = 90o)], in a- and b-scans is about 0.012%, comparable to the SMR ratio in Pt/YIG (see Fig. S3) (11, 13, 23). Next, we investigated the origin of the observed MR effect. Considering that Pt on YIG may 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 a negligible MPE (25). The MR effect of 0.002% still appears as shown in Fig. 3B, comparable to the SMR ratio in Au/YIG (see Fig. S 4), ruling out MPE as the origin of the observed MR. Furthermore, the MR ratios of the Cu(3)[Pt(0.4)]/YIG(10) sample in a- and b-scans are comparable and almost one order of magnitude larger than that in g-scan. This is different from AMR of a ferromagnetic metal, where the MR ratio in a- and g-scans is much larger than that in b-scan (11, 14, 24). Therefore, the MPE-induced AMR can be ruled out. In fact, the behaviors of the MR angular dependence follow the SMR scenario well (11, 13-15, 17). However, with several control experiments, we can unambiguously exclude SMR as the explanation for our observations. First, the observed MR amplitude cannot be explained by SMR. In our samples, the 0.4 - nm-thick ultrathin Pt layer is non -conductive and the conductivity of bulk Pt is about one order of magnitude smaller than that of bulk Cu, meaning that the current mainly passes through the Cu layer. We prepared a 3-nm-thick single layer Cu on YIG without interface decoration and performed the MR angular dependent measurement in a-scan. MR is not observed, as shown in Fig. 3B, evidencing that the Pt-decorated interface is indispensable. A conductive 0.4-nm-thick Pt layer is not available experimentally. Considering that a small fraction of current may flow in the Pt islands, there is a possibility of the occurrence of SMR from the Pt islands. According to the reported SMR results in Pt/YIG bilayers, the SMR ratio in Pt/YIG decreases rapidly with decreasing Pt thickness when the Pt th ickness is less than about 3 nm (13, 18). The SMR ratio of Pt(0.4)/YIG is extrapolated to be well below 0.01% from the previously reported Dr/r versus Pt thickness data (13, 18). Considering the pronounced 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 ratio is as large as ~ 0.012% in Cu(3)[Pt(0.4)]/YIG (see Fig. 3A). Therefore, the SMR mechanism cannot explain our observations. Second, the potential enhancement of SMR caused by intermixing or alloying between a strong SOI material and a weak SOI material can be excluded (17, 26, 27). For this purpose, we prepared two types of control samples with the 0.4 -nm-thick Pt layer either on top of or inserted inside the Cu layer: [Pt(0.4)]Cu(3)/YIG and Cu(1)[Pt(0.4)]Cu(3)/YIG. Since both samples are fabricated under the same condition as the Cu[Pt]/YIG samples, the intermixing Science Advances Manuscript Template Page 4 of 12 of Pt and Cu should be similar. The MR vanishes in the [Pt(0.4)]Cu(3)/YIG and Cu(1)[Pt(0.4)]Cu(3)/YIG samples, shown in Fig. 3B. These results rule out the Pt-Cu alloying induced SMR. Thus, we conclude that the observed MR effect is not SMR. Cu-thickness dependent transport measurements To identify the physical origin of the observed unusual MR, we carried out the Cu -thickness dependent measurements. Figure 4A presents the angular dependent MR measurements of Cu(t Cu)[Pt(0.4)]/YIG in a-scans for various Cu thickness (t Cu). Obviously, the MR ratio steadily decreases with increasing tCu, highlighting the importance of the Pt-decorated Cu/YIG interface. This monotonous NM-thickness dependence of this MR is in sharp difference with the non-monotonous behavior of SMR, which peaks at ~ 3 nm for Pt/YIG (13, 18). The Cu-thickness dependence of r and the MR ratio extracted from Fig. 4A are shown in Fig. 4B. For very thin Cu film (t Cu £ 5 nm), r dramatically increases with decreasing t Cu, indicating that r is dominated by the interface/surface scatterings. Besides SMR, there is another type MR predicted recently possessing the same angular dependence as we found (see Fig. 3A) (21). It originates from the Rashba SOI at the interface of a NM/FI bilayer. By comparing the samples of Cu[Pt]/YIG, [Pt]Cu/YIG and Cu[Pt]Cu/YIG, one can see that only the Cu[Pt]/YIG samples exhibit a significant MR (see Fig. 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 interface. First principles calculations and Boltzmann simulations In order to prove that the Pt-decoration can indeed induce Rashba SOI at the Cu/YIG interface, we carried out first principles band structure calculations based on i) a Cu ultra - thin film of 14 monolayers, ii) the same Cu film as i) but covered by Au on surfaces on both sides, iii) the same Cu film as i) but covered by Pt on both surfaces, iv) the Pt layer inside the Cu film. By comparing these four different scenarios, we can see that there is no clear Rashba effect in the bare Cu film and the one covered by Au. A strong Rashba effect appears only for Pt on the Cu film surface (the details of calculations are given in the Supplementary Materials). For a quantitative analysis, we employ a Boltzmann formalism to calculate the charge and spin transport in a NM/FI bilayer structure. We solve the following spin-dependent Boltzmann equation in the NM layer: () ( )( )00 ,0 ,FS0, , ,()() () ( ) xyzfR ef d P fa aa aa aa a d¢¢ ¢==-¶¢ ¢¢ ×- × + ¶åòr,kvk E vk r k ,k k k,k r,k r， (1) where fa=0,x,y,z(r,k) is the four-component distribution function denoting the charge/spin occupation at position r and wavevector k. The interface at FI z = z+ contains a Rashba type SOI described by the Hamiltonian: ( )( )RHz z hd=× ´ -+ σzp! !!, where h is the strength of the Rashba SOI, z! is the normal direction of the interface, p! is the momentum operator. HR gives rise to an anomalous velocity localized at the interface. The Boltzmann equation is solved by discretizing the spherical Fermi surface of Cu and the real space in z direction of Cu film. With the full distribution function, we calculate all charge/spin transport properties, including the longitudinal and transverse conductivities. This method extends the earlier Boltzmann method developed for current-perpendicular-to-plane structure like spin valves to current-in-plane structure like NM/FI bilayers (28-31) by taking into account the surface roughness and Rashba SOI at the interface. The detai ls of the simulations are given in the Supplementary Materials . Science Advances Manuscript Template Page 5 of 12 In the numerical Boltzmann calculation, ther e are only two fitting parameters, the surface roughness and the Rashba coupling constant. All other parameters are either given by the experiment (such as the film thickness) or can be determined otherwise (such the bulk relaxation time). By employing the quantum description of rough surface (32-34), we are able to fit the thickness dependence of r in the ultra-thin Cu film to a reasonably good precision 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 magnetization angular dependence of r, in a good agreement with the experimental results (see Fig. 3A), from which we obtain the SOMR ratio. The calculated SOMR ratio is shown in Fig. 4B, which shows monotonic decreasing behavior as a function of Cu film thickness, consistent with our experiment results but very different from the non -monotonic behavior observed in SMR (13, 18). Pt-thickness dependent MR measurements Finally, to further differentiate SOMR from SMR, we carried out the Pt thickness tPt dependent measurements. To reduce the sample fluctuation, we fabricated the YIG films successively under the same condition. Figure 4A shows the angular dependent MR measurements of Cu(3)/Pt(t Pt)/YIG in a-scans with H = 2000 Oe. The MR ratio extracted from Fig. 4A exhibits non-monotonous behavior with increasing tPt as shown in Fig. 4C. Two separate regimes can be identified: 1) the SOMR regime for tPt < 1 nm and 2) the conventional SMR regime for tPt > 2.2 nm (see Fig. 4C). For tPt < ~0.6 nm, r and Dr/r increase with increasing tPt because the Pt islands not only introduce the interface scattering but also enhance the Rashba SOI. For ~0.6 nm < tPt < ~1 nm, the Pt islands start to form a complete layer, leading to the reduction of the interface roughness and the rapid decrease of r as seen in Fig. 4C. The MR ratio continues to increase in this region because of the enhanced Rashba SOI with increasing Pt coverage on YIG. For ~1 nm < tPt < ~2 nm, r is smaller than the resistivity of Cu/YIG, suggesting that the interface scattering has minor contribution to r. Since SOMR is caused by the interface scattering, the MR ratio rapidly drops in this region. A sizable SMR ratio only appears when tPt > 2 nm in Pt/YIG (13, 18, 22). Therefore, around tPt ~ 2 nm, both SOMR and SMR are small, resulting in a minimum in MR. In the SMR regime, the SMR ratio exhibits a maximum, as expected for SMR (13, 18, 22). This result demonstrates the differences between the SOMR and the SMR. A theoretical calculation shows that a rough interface can enhance SHE ( 34). To understand the role of the roughness to the SOMR, we fabricated a control sample of Cu(3)[Ag(0.7)]/YIG. The rms roughness of Ag(0.7)/YIG is 0.797 nm, as shown in Fig. S9(A), similar as that of Pt(0.4 nm)/YIG. Owing to the weak SOI in Ag, the Rashba SOI in Cu[Ag]/YIG is expected to be weak. We do not observe any MR effect down to 5´10-6 in Cu(3)[Ag(0.7)]/YIG, shown in Fig. S9(B). This result means that the rough surface alone cannot cause the SOMR. Conclusions In conclusion, we report the first observation of the SOMR effect predicted recently ( 21) at room 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 dependence of SOMR is similar to that of SMR, but all other features are different, such as the 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 interfaces. Our finding demonstrates the possibility of realizing spin manipulation by interface decoration. Science Advances Manuscript Template Page 6 of 12 Materials and Methods The single crystalline YIG films were epitaxially grown on GGG (111) substrates by PLD technique using a KrF excimer laser with wavelength of 248 nm. The PLD system was operated at a laser repetition rate of 4 Hz and an energy density of 10 J/cm2. The distance between the substrate and the target is 50 mm. Before films deposition, the chamber was evacuated to a base pressure of 1 × 10−7 torr. The YIG films were deposited at ~ 730 oC in an oxygen pressure of 0.05 Torr. The growth of the YIG films was monitored by in situ reflection high-energy electron diffraction (RHEED). The structure was further examined by X-ray diffraction and HRTEM. The magnetic properties of all YIG films were characterized using a vibration sample magnetometer (VSM). Then we used magnetron sputtering to deposit polycrystalline metallic films onto the YIG films via dc sputtering at room temperature with a shadow mask to define 0.3 -mm-wide and 3-mm-long Hall bars. The deposition rate was calibrated by X -ray reflectivity. After the metallic film deposition, the samples were immediately mounted and transferred into a vacuum chamber for the transport measurements to minimize the metal oxidation. The resistance was measured by a Keithley 2002 multimeter in a four-probe mode. For the angular dependent MR measurements with the magnetic field less than 5000 Oe, the resistance was monitored as the 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) equipped with a rotatory sample holder. H2: Supplementary Materials section S1. AFM images of Cu(t Cu)[Pt (0.4)]/YIG(10)/GGG(111) section S2. Magnetic properties of the YIG films section S3. SMR in Pt/YIG section S4. SMR and AFM image of Au/YIG section S5. First principles calculations section S6. Boltzmann simulations fig. S1. AFM images of Cu(t Cu)[Pt(0.4)]/YIG(10)/GGG (111). fig. S2. FMR of the YIG films. fig. S3. SMR in Pt/YIG. fig. S4. SMR and AFM image of Au/YIG. fig. S5. The band structures of Cu, Au/Cu/Au and Pt/Cu/Pt. fig. S6. The spin textures of outer band and inner band. fig. S7. 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Mat. 9, 014105 (2008). Acknowledgments: Funding: L.Z. and D.W. are supported by National Key R&D Program of China (2017YFA0303202), NSF of China (11674159, 51471086 and 11727808), National Basic Research Program of China (2013CB922103). H.S. and J.X. acknowledge the support by NSF of China (11474065 and 11722430) and National Key R&D Program of China (2016YFA0300702). Author contributions: J.X. and D.W. designed and supervised the project. 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 simulations under supervision of J.X. K.L. performed the first principles calculations under supervision 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 on the manuscript. All authors discussed the results and reviewed the manuscript. Competing interests: The authors declare no competing financial interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors. Science Advances Manuscript Template Page 10 of 12 Figures and Tables A B C Fig. 1. Sample characterization. (A) AFM image of YIG(10)/GGG, the rms roughness is 0.127 nm. (B) AFM image of Pt(0.4)/YIG(10)/GGG, the rms roughness is 0.733 nm. (C) HRTEM image of Au(3)/Cu(4)[Pt(0.4)]/YIG heterostructure where Au is used to prevent the oxidation. A B C D Fig. 2. Field-dependent magnetization and transport measurements. Magnetic hysteresis loops of (A) YIG(10)/GGG with field in-plane and (B) YIG(400)/GGG with field out-of-plane. r measured on the Cu(2)[Pt(0.4)]/YIG(10)/GGG sample for H applied (C) along x-axis, y-axis and (D) z-axis, respectively. Science Advances Manuscript Template Page 11 of 12 Fig. 3. Angular dependent MR measurements. (A) Angular dependent MR measurements in the xy, yz, and xz planes for Cu(3)[Pt(0.4)]/YIG. The solid lines are the Boltzmann simulation results. (B) Angular dependent MR measurements in the xy plane for several control samples. Science Advances Manuscript Template Page 12 of 12 Fig. 4. Cu- and Pt-thickness dependent transport measurements. Angular dependent MR measurements in the xy plane for (A) Cu(t Cu)[Pt(0.4)]/YIG samples and Cu(3)/Pt(t Pt)/YIG samples. (B) Cu thickness dependence of the MR ratio and r, respectively, for Cu(t Cu)[Pt(0.4)]/YIG. The solid lines are the Boltzmann simulation results. (C) The Pt layer thickness dependence of the MR ratio and r, respectively, for Cu(3)/Pt(t Pt)/YIG. The solid lines are guide to the eyes. 1 Supplementary Materials for Observation of spin -orbit magnetoresistance in metallic thin films on magnetic insulators Lifan Zhou, Hongkang Song, Kai Liu, Zhongzhi Luan, Peng Wang, Lei Sun, Shengwei Jiang, Hongjun Xiang, Yanbin Chen, Jun Du, Haifeng Ding, Ke Xia, Jiang Xiao , and Di Wu This PDF file includes: • section S1. AFM images of Cu(t Cu)[Pt (0.4)]/YIG(10)/GGG(111) • section S2. Magnetic properties of the YIG films • section S3. Spin Hall magnetoresistance in Pt/YIG • section S4. SMR and AFM image of Au/YIG • section S5. First principles calculations • section S6. Boltzmann simulations • fig. S1. AFM images of Cu(t Cu)[Pt(0.4)]/YIG(10)/GGG (111). • fig. S2. FMR of the YIG films. • fig. S3. Spin Hall magnetoresistance in Pt/YIG. • fig. S4. SMR and AFM image of Au/YIG. • fig. S5. The band structures of Cu, Au/Cu/Au and Pt/Cu/Pt. • fig. S6. The spin textures of outer band and inner band. • fig. S7. The Rashba splitting in Cu/Pt/Cu. • fig. S8. Specular and diffusive interface scattering in the NM/FI bilayer. • fig. S9. AFM image of Ag(0.7)/YIG and MR of Cu(3)[Ag(0.7)]/YIG. • References (35–52) 2 section S1. AFM images of Cu( tCu)[Pt (0.4)]/YIG(10)/GGG(111) The discontinuous 0.4-nm-thick Pt layer is insulating. The surface morphology gets smoother after the deposition of Cu onto Pt. The rms roughness of Cu(t Cu)[Pt(0.4)]/YIG(10)/GGG(111) increases with increasing tCu, shown in Fig. S1. For the thin Cu film, resistivity r increases with decreasing Cu thickness tCu, indicating that r is dominated by the interface/surface scatterings. section S2. Magnetic properties of the YIG films The saturation magnetization Ms is measured by ferromagnetic resonance (FMR) with an in-plane magnetic field and in an X-band microwave cavity operated at a frequency of f = 9.7798 GHz. Fig. S2A shows the FMR absorption derivative spectrum of the YIG (10 nm) film measured at room temperature . The resonance field Hr is at 2608.8 Oe. Ms is determined to be 164.5 emu/cc by using the Kittel formula : (4rr S fH HMgp=+ where g is the gyromagnetic ratio. In comparison, M s of bulk YIG is 140 emu/cc. section S3. Spin Hall magnetoresistance in Pt/YIG Pt/YIG is a typical system with the SMR. We fabricated a sample of Pt(3.4)/YIG to compare with the magnetoresistance of Cu[Pt]/YIG. Fig. S3 presents the angular dependent magnetoresistance of our Pt(3.4)/YIG sample measured in b-scan at room temperature. The applied magnetic field H = 1.5 T is much larger than the demagnetization field to align the magnetization along H. We determined the SMR ratio of about 4.5´10-4, comparable to previous reports (11, 14). section S4. SMR and AFM image of Au /YIG It would be better to compare the MR ratio of the Cu[Au]/YIG sample with Au/YIG sample. Fig. S4A shows the angular dependent magnetoresistance of the Au(6)/YIG sample 3 measured in a-scan at room temperature. A SMR ratio of ~1.2´10-5 is observed, consistent with the previous reports (17). The SMR ratio in Au/YIG should be larger for the optimized Au thickness. The surface of the Au film is smoother than that of the Pt film, shown in Fig. S4B. Therefore, the observed magnetoresistance ratio of about 2´10-5 in Cu(3)[Au(0.4)]/YIG is reasonable. section S5. First principles calculations The calculations are performed within density -functional theory (DFT) using the projector augmented wave (PAW) method (35) encoded in the Vienna ab initio simulation pa ckage (VASP) (36, 37). The exchange-correlation potential is treated in the generalize d-gradient approximation (GGA) (38). The plane-wave cutoff energy is set to be 400 eV. For geometry optimization, all the internal coordinates are relaxed until the Hellmann - Feynman forces are less than 1meV/Å and SOI is not included. For the band structure calculation, the SOI is included. We build three models. The first model is a pure Cu ultra-thin film of 14 monolayers. The second (third) one is the same film covered by Au (Pt) at surfaces on both sides to keep the inversion symmetry of the whole system. The thickness of the bare Cu film, the vacuum layer and the surface lattice constant are 27 Å, 20 Å and 2.56 Å, respectively. The band structures of the three models are s hown in Fig. S5. There is no obvious Rashba splitting in the bare Cu film and in the film covered by Au, as shown in Fig. S5A and S5B, respectively. While in the Cu film covered by Pt, there is a Rashba splitting, and the splitted bands are highlighted by green bold lines, as shown in Fig. S5C. Near the Gamma 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 environment, characteristics of the k-linear Rashba effect. To further confirm the nature of the Rashba splitting in Fig. S5C, we calculate the spin textures of outer band and inner band around -0.35 eV and -0.10 eV iso-energy surface, respectively, shown in Fig. S6. The 4 inverse rotation of spin orientations of outer band and inner ba nd is characteristic of a pure Rashba splitting. This indicates that Pt can indeed induce a strong Rashba effect at Cu surfaces. In Fig. S7, we show that when Pt is placed inside Cu film away from the surface, the Rashba splitting decreases significantly, and vanishes when Pt is in the middle of the film. This is consistent with our experimental data that when Pt is placed inside Cu, the SOMR disappear. section S6. Boltzmann simulations Based on the Boltzmann method developed for CPP (current-perpendicular-to-plane) structure like spin valves (29-31, 39, 40), we made modifications for the CIP (current-in- plane) structure like the bilayer systems used in SMR/HMR/SOMR. 6.1 Basic formalism of Boltzmann calculation We use four-component distribution function ()0, , ,xyz fa= r,k to denote the charge/spin occupation at position r and wavevector k: f0 is the electric charge distribution and ( ) ,,xyzfff=f is the pure spin (no net charge) distribution. Thus , the majority/minority spin distribution 0 ff±=± f, where ± denotes the majority/minority spin along ˆ±f direction. These four-component distribution function satisfies the generalized spin - dependent Boltzmann equations (31, 39-41), () ( )( )00 ,0 ,FS0, , ,()() () ( ) xyzfR ef d P fa aa aa aa a d¢¢ ¢==-¶¢ ¢¢ ×- × + ¶åòr,kvk E vk r k ,k k k,k r,k r， (S1) where E is the applied external electric field and 0m=vk /! is the velocity in free electron model, and the right-hand side is the scattering-out and scattering-in collision terms and FSd ¢ òkdenotes the integral over Fermi surface. ( ),FS() , Rd Paa a a¢ ¢¢¢ =åòkk k k is the total relaxation rate for ( ) far,k . And ( ),Paa ¢ ¢k,k describes the ¢®kk scattering 5 probability from charge/spin -a¢ to the charge/spin-α, e.g ., ( )0,0P ¢k,k is the scattering probability for electric charge, ( ),xxP ¢k,k is the spin-conserved scattering probability for spin-x, ( ),yxP ¢k,k is the scattering probability with spin flip spin -x → spin-y, ( )0,xP ¢k,k is the scattering probability with spin Hall effect converting pure spin -x to charge, and ( ),0xP ¢k,k is the scattering probability with ISHE converting charge to pure spin-x. In the case of normal metal like Cu without bulk SHE, ( ) ()1 ,F S ,/ PAaa aadt- ¢¢ ¢¢= k,k k in the relaxation time approximation, where AFS is the area of Fermi surface and () t ¢k is the spin-conserved relaxation time for electrons at ¢k.() ()( ) (),0 sf 1/ 1 / Raa td t +- k= k k，where ()sftk is the spin-flip relaxation time. All scatterings are assumed to be elastic, i.e., ¢=kk . We study an NM/FI bilayer structure, as shown in Fig. S8, whose interfaces/surfaces are in x-y plane and locate at /2 zd±=± . The boundary condition for the upper interface at zz+= is given by a surface scattering matrix S+ that connects the impinging distribution function ( 0zk>) and the reflected distribution function ('0zk>): ( ) ()( ),FS,, 0 , , 0zz fz k d S f z kaa a a+ ¢¢ ++¢¢ ¢ ¢ >= < òkk k ,k k . (S2) We regard the interface/surface scattering as specular when ( ) ( )( )( )()(),, xx yy zz z z Sk k k k kk k kaa aadd d d+ ¢¢ ¢¢ ¢ ¢ ¢=- - +Q Q - k,k , (S3) and as diffusive when ( ) ( )()()1 ,, FS zz SAk kaa aadd+- ¢¢ ¢¢ ¢=- Q Q - k,k k k , (S4) where the factor ,aad¢ means that the surface scattering is spin-conserving. Similar boundary condition can be written down at zz-=. Due to conservation of charge, we have the following identity 6 ( ) ( )0,0 0,0FS FS1 dS d S±±¢¢ ¢ òòkk , k = k k , k = . (S5) Since spin is generally not conserved, there is no constraint on the spin related boundary scattering matrix. Once the distribution function has been found by solving Eq. (S1), all transport properties can be calculated accordingly: charge/spin accumulation: ()()3 3FS() 2defaaµ p=-òkr r,k, (S6a) charge current density ( 0a=):() ()()() ()3 00 0 3FS,, 2 xyzdje v f v fbb b aa a p =éù=- + êú ëûå òkrk r,k r ,k,(S6b) spin current density ( ,,xyza= ):() ()()() ()3 00 3FS2dje v f v fbb b aa apéù =- +ëû òkr k r,k r,k ,(S6c) where aµ is the charge accumulation when 0a= and spin-a accumulation when ,,xyza= , 0jb is the charge current flowing in b-direction, jb a and is the spin-α current flowing in b direction when ,,xyza= . The two contributions in 0j and ja are due to the fact that different spins may have different velocities, e.g., the majority/minority spin- a has velocity 0 a¢±vv , where a¢v is the anomalous velocity due to the spin-orbit coupling [see Eq. (S10) below]. In the bilayer structure in Fig. S7 with translational invariance in x-y plane, () ()jj zbb aa=r , and the film conductivities can be calculated from the current density as ()1 z zdzj zEbb aas+ -=ò. (S7) For electric field applied in x direction, the longitudinal conductivity is 0xs, and the transverse Hall conductivity is 0ys. To carry out the Boltzmann calculation numerically, we discretize the Fermi surface in k-space and the real space (in the out-of-plane z-direction only) simultaneously: 7 {}{}1 1,z kn n iji jz= =k . The Boltzmann equation then becomes a set of linear equations, which is solved by matrix inversion. 6.2 Surface roughness For metallic thin films, the rough surface becomes an important or even dominate factor on the transport. There are various models in dealing with a rough surface, including the Fuchs-Sondheimer model (42, 43), the Mayadas-Shatzkes model (44), and the Namba model (45). All these models are phenomenological, and work only in certain circumstances (46-51). To deal with ultrathin films, we adopt the quantum description of a rough surface as developed in Ref. 32-34, in which the relaxation rate becomes channel dependent: ()2 00 ,111 1 1 nn mnn tt t t t tº= + = +¢¢ k with 2 2214 3F cES nd ta=¢ !, (S8) where 0t is the bulk impurity relaxation time and nt¢ is the channel-dependent surface relaxation time, and 23 '13' / 1cn c nSn n ==»å . Here a is the lattice constant and δ parameterizes the magnitude of the surface roughness. In Eq. (S8), the relaxation due to the surface roughness is built into the Boltzmann equation via total relaxation rate as the bulk impurity scattering, rather than a simple surface scattering. The reason for this is the following: for ultrathin film, one cannot view the electron as a classical point particle bouncing back and forth between the two surfaces, and only feels the surface as the electron hits the surface. Instead, the electronic wave function spreads out in the thickness direction and is in contact with the surface all the time, thus the rough surface becomes a ‘bulk’ effect and is felt constantly by the electron and causes scattering from ¢k to k. 6.3 Interfacial Rashba spin-orbit interaction When considering the interfacial Rashba spin-orbit interaction at the NM/FI interface, we assume a Rashba Hamiltonian of the following form (21) 8 ( )( )RHz z hd=× ´ -+ σzp! !!, (S9) where ˆp is the momentum operator, ( ) ˆ ˆ ˆ ˆ ,,xyzsssσ= are the Pauli matrices, η is the strength of the Rashba SOI, and ˆz is the normal direction of the interface. Eq. (S9) gives rise to a spin-dependent anomalous velocity at the interface: (21, 52) [ ] ( )( ) ( )( )ivH z z z z mb aa a a bb bsh d sh d++¢=- = - ´ » - ´Rr, σzm z!! ! ! ! ", (S10) where operator ˆσ is approximated by the magnetization direction m at the top surface in contact with FI. vb a¢ is the anomalous velocity in b direction for spin- a. Therefore, the velocity used in the Boltzmann equation Eq. (S1) is modified with replacement 0, 0 0, 0 , 'aa g a g ddd®+ vv v at the top surface at zz+=. The anomalous velocity also contributes to the evaluation of charge/spin currents in Eq. (S6). Such an anomalous velocity would not change the drift term in the Boltzmann equation (first term in Eq. (S1)) in the bilayer system studied in this paper. The reason is that the anomalous velocity in Eq. (S10) is in-plane and perpendicular to ˆz, while the spatial dependence of fa is in the ˆz direction, therefore the dot product in the drift term with vanishes identically. 6.4 Boltzmann simulation results With the numerical Boltzmann method described above, we are ready to calculate the longitudinal conductivity and the transverse Hall conductivity as function of magneti zation orientation, NM film thickness, and temperature. We adopt the conventional magnetic field scanning scheme as show in Fi g. 3A of main text. We calculate both the longitudinal and transverse resistivity as function of the magnetization angel in the a-, b-, g-scans as in Fig. 3A of main text. Then MR is calculated as ()() o90MR with = , , .rq rqq abgr-= (S11) 9 where r is the average value of ρ(θ), and o90r is the resistivity value at 90o. Fig. 3A of main text shows the angular dependent of ρ and MR for a Cu film of thickness t = 3 nm, where δ is chosen to match the average resistivity. It is seen that all three angular dependences are in agreement with the experimental data. The small oscillation in the experimental data for the γ-scan might be caused by the weak anisotropic MR, which is not included in our simulation. It should note that these angular dependence of r and MR ratio in SOMR are identical to that in the SMR. Therefore, it is impossible to tell SOMR from SMR from this angular dependence. The more interesting part is the NM film thickness dependence. As we know for sure that r must decrease monotonically as increasing t because of the reducing surface scattering. This is exactly what has been observed experimentally and calculated using the Boltzmann method, as shown in Fig. 4C of main text. In the Boltzmann simulation, we have chosen τ0 as the bulk Cu relaxation time at room temperature. And δ is a varying fitting parameter to account for the surface scattering. Using δ as the only fitting parameter (δ = 6), we find that the longitudinal resistivity can be fitted with a s atisfactory level (Fig. 4C), considering the large error bar and extremely thin film. We also note that the Rashba coupling strength has little effect on the longitudinal resistivity as expected. It is well-known that the magnitude of SMR depends on the NM film thickness in a non- monotonic fashion, i.e., there is a peak when the film thickness is comparable to the spin diffusion length of NM. However, the SOMR observed in this work shows a monotonic decreasing behavior as increasing NM thickness. We should note that Cu has very long spin diffusion length, much longer than the film thickness. Such monotonic MR ratio can also be fitted using the Boltzmann simulation with only one fit ting parameter, i.e., the Rashba coupling constant η (η = 0.174). Similar to the SMR effect, the SOMR effect also depends quadratically on the spin-orbit coupling strength, therefore 2MR hµ . 10 fig. S1. AFM images of Cu(t Cu)[Pt(0.4)]/YIG(10)/GGG (111). 11 fig. S2. FMR of the YIG films. FMR absorption derivative spectrum of YIG(10 nm)/GGG(111) film with field in sample plane measured at room temperature. fig. S3. Spin Hall magnetoresistance in Pt/YIG. The angular dependent magnetoresistance of our Pt(3.4)/YIG sample measured in b-scan at room temperature with the magnetic field of 1.5 T. 12 fig. S4. SMR and AFM image of Au/YIG. (A) The angular dependent MR of the Au(6)/YIG sample measured in a-scan with H = 2000 Oe at room temperature. (B) AFM image of Au(0.4)/YIG. The rms roughness is 0.135 nm. fig. S5. The band structures of Cu, Au/Cu/Au and Pt/Cu/Pt. The band structures of (A) the Cu ultra-thin film of 14 monolayers, (B) the same film covered by Au, (C) the same film covered by Pt. The bands marked by green bold lines indicate a R ashba splitting. 13 fig. S6. The spin textures of outer band and inner band. Spin texture of (A) the outer band around -0.35 eV iso-energy surface and (B) the inner band around -0.10 eV iso-energy surface. The outer band and inner band are hig hlighted by green bold lines in Fig. S5C. fig. S7. The Rashba splitting in Cu/Pt/Cu. The band structures of (A) the Cu ultra-thin film of 14 monolayers with Pt located 4 monolayers away from the surface, and it s hows a weak Rashba splitting; (B) the same film with Pt located in the middle, and there is no Rashba splitting. 14 fig. S8. Specular and diffusive interface scattering in the NM/FI bilayer. (A) (B) fig. S9. AFM image of Ag(0.7)/YIG and MR of Cu(3)[Ag(0.7)]/YIG. (A) AFM image of Ag(0.7)/YIG. The rms roughness is 0.797 nm. (B) The angular dependent MR of the Cu(3)[Ag(0.7)]/YIG sample measured in a-scan with H = 2000 Oe at room temperature.

2017-12-09

A magnetoresistance effect induced by the Rashba spin-orbit interaction was predicted, but not yet observed, in bilayers consisting of normal metal and ferromagnetic insulator. Here, we present an experimental observation of this new type of spin-orbit magnetoresistance (SOMR) effect in a bilayer structure Cu[Pt]/Y3Fe5O12 (YIG), where the Cu/YIG interface is decorated with nanosize Pt islands. This new MR is apparently not caused by the bulk spin-orbit interaction because of the negligible spin-orbit interaction in Cu and the discontinuity of the Pt islands. This SOMR disappears when the Pt islands are absent or located away from the Cu/YIG interface, therefore we can unambiguously ascribe it to the Rashba spin-orbit interaction at the interface enhanced by the Pt decoration. The numerical Boltzmann simulations are consistent with the experimental SOMR results in the angular dependence of magnetic field and the Cu thickness dependence. Our finding demonstrates the realization of the spin manipulation by interface engineering.

Observation of spin-orbit magnetoresistance in metallic thin films on magnetic insulators

1712.03322v1

Complex THz and DC inverse spin Hall effect in YIG/Cu 1−xIrx bilayers across a wide concentration range Joel Cramer,1, 2Tom Seifert,3Alexander Kronenberg,1Felix Fuhrmann,1 Gerhard Jakob,1Martin Jourdan,1Tobias Kampfrath,3, 4and Mathias Kl¨ aui1, 2,∗ 1Institute of Physics, Johannes Gutenberg-University Mainz, 55099 Mainz, Germany 2Graduate School of Excellence Materials Science in Mainz, 55128 Mainz, Germany 3Department of Physical Chemistry, Fritz Haber Institute of the Max Planck Society, 14195 Berlin, Germany 4Department of Physics, Freie Universit¨ at Berlin, 14195 Berlin, Germany (Dated: September 7, 2017) Abstract We measure the inverse spin Hall effect of Cu 1−xIrxthin films on yttrium iron garnet over a wide range of Ir concentrations (0 .056x60.7). Spin currents are triggered through the spin Seebeck effect, either by a DC temperature gradient or by ultrafast optical heating of the metal layer. The spin Hall current is detected by, respectively, electrical contacts or measurement of the emitted THz radiation. With both approaches, we reveal the same Ir concentration dependence that follows a novel complex, non-monotonous behavior as compared to previous studies. For small Ir concentrations a signal minimum is observed, while a pronounced maximum appears near the equiatomic composition. We identify this behavior as originating from the interplay of different spin Hall mechanisms as well as a concentration-dependent variation of the integrated spin current density in Cu 1−xIrx. The coinciding results obtained for DC and ultrafast stimuli show that the studied material allows for efficient spin-to-charge conversion even on ultrafast timescales, thus enabling a transfer of established spintronic measurement schemes into the terahertz regime. 1arXiv:1709.01890v1 [cond-mat.mtrl-sci] 6 Sep 2017INTRODUCTION Spin currents are a promising ingredient for the implementation of next-generation, energy-efficient spintronic applications. Instead of exploiting the electronic charge, transfer as well as processing of information is mediated by spin angular momentum. Crucial steps towards the realization of spintronic devices are the efficient generation, manipulation and detection of spin currents at highest speeds possible. Here, the spin Hall effect (SHE) and its inverse (ISHE) are in the focus of current research [1] as they allow for an interconversion of spin and charge currents in heavy metals with strong spin-orbit interaction (SOI). The efficiency of this conversion is quantified by the spin Hall angle θSH. In general, the SHE has intrinsic as well as extrinsic spin-dependent contributions. The intrinsic SHE results from a momentum-space Berry phase effect and can, amongst others, be observed in 4 dand 5dtransition metals [1–3]. The extrinsic SHE, on the other hand, is a consequence of skew and side-jump scattering off impurities or defects [4]. It occurs in (dilute) alloys of normal metals with strong SOI impurity scatterers [5–8], but can also be prominent in pure metals in the superclean regime [9]. As a consequence, the type of employed metals and the alloy composition are handles to adjust and maximize the SHE. Remarkably, it was recently shown that the SHE in alloys of two heavy metals (e.g. AuPt) can even exceed the SHE observed for the single alloy partners [10]. Pioneering work within this research field covered the extrinsic SHE by skew scattering in copper-iridium alloys [5]. However, previously the iridium concentration was limited to 12 % effective doping of Cu with dilute Ir. The evolution of the SHE in the alloy regime for large concentration thus remains an open question and the achievable maximum by an optimized alloying strategy is unknown. The potential of a metal for spintronic applications (i.e. θSH) can be quantified by inject- ing a spin current and measuring the resulting charge response. This can be accomplished by, for instance, coherent spin pumping through ferromagnetic resonance [11–13] or the spin Seebeck effect (SSE) [14, 15]. The SSE describes the generation of a magnon spin cur- rent along a temperature gradient within a magnetic material. Typically, such experiments involve a heterostructure composed of a magnetic insulator, such as yttrium iron garnet (YIG), and the ISHE metal under study [see Fig. 1(a)]. A DC temperature gradient in the YIG bulk is induced by heating the sample from one side. On the femtosecond timescale, 2however, a temperature difference and thus a spin current across the YIG-metal interface can be induced by heating the metal layer with an optical laser pulse [Fig. 1(b)] [16–19]. This interfacial SSE has been shown to dominate the spin current in the metal on timescales below∼300 ns [16]. For ultrafast laser excitation, the resulting sub-picosecond ISHE current leads to the emission of electromagnetic pulses at frequencies extending into the terahertz (THz) range, which can be detected by optical means [20]. Therefore, femtosecond laser excitation offers the remarkable benefit of contact-free measurements of the ISHE current without any need of micro-structuring the sample. The all-optical generation as well as detection of ultra- fast electron spin currents [20, 21] is a key requirement for transferring spintronic concepts into the THz range [22]. So far, however, characterization of the ISHE was conducted by experiments including DC spin current signals as, for instance, the bulk SSE [Fig. 1(a)]. For the use in ultrafast applications, it thus remains to be shown whether alloying yields the same notable changes of the spin-to-charge conversion efficiency in THz interfacial SSE experiments [Fig. 1(b)] and whether alloys can provide an efficient spin-to-charge conversion even at the ultrafast timescale. In this work, we study the compositional dependence of the ISHE in YIG/Cu 1−xIrx bilayers over a wide concentration range (0 .056x60.7), exceeding the dilute doping phase investigated in previous studies [5]. The ISHE response of Cu 1−xIrxis measured as a function of x, for which both DC bulk and THz interfacial SSE are employed. Eventually, we compare the spin-to-charge conversion efficiency in the two highly distinct regimes of DC and terahertz dynamics across a wide alloying range. EXPERIMENT The YIG samples used for this study are of 870 nm thickness, grown epitaxially on (111)-oriented Gd 3Ga5O12(GGG) substrates by liquid-phase-epitaxy. After cleaving the GGG/YIG into samples of dimension 2 .5 mm×10 mm×0.5 mm, Cu 1−xIrxthin films (thick- nessdCuIr= 4 nm) of varying composition ( x= 0.05,0.1,0.2,0.3,0.5 and 0.7) are deposited by multi-source magnetron sputtering. To prevent oxidation of the metal film, a 3 nm Al capping layer is deposited, which, when exposed to air, forms an AlO xprotection layer. For the contact-free ultrafast SSE measurements, patterning of the Cu 1−xIrxfilms into defined 3(a) (b)Figure 1. (a) Scheme of the setup used for DC SSE measurements. The out-of-plane temperature gradient is generated by two copper blocks set to individual temperatures T1andT2. An external magnetic field is applied in the sample plane. The resulting thermovoltage Vtis recorded by a nanovoltmeter. (b) Scheme of the contact-free ultrafast SSE/ISHE THz emission approach. The in-plane magnetized sample is illuminated by a femtosecond laser pulse, inducing a step-like temperature gradient across the YIG/Cu 1−xIrxinterface. The SSE-induced THz spin current in the CuIr layer is subsequently converted into a sub-picosecond in-plane charge current by the ISHE, thereby leading to the emission of a THz electromagnetic pulse into the optical far-field. nanostructures is not necessary. In the case of DC SSE measurements, the unpatterned film is contacted for the detection of the thermal voltage. The DC SSE measurements are performed at room temperature in the conventional longitudinal configuration [15]. While an external magnetic field is applied in the sample plane, two copper blocks, which can be set to individual temperatures, generate a static out-of-plane temperature gradient, see Fig. 1(a). This thermal perturbation results in a magnonic spin current in the YIG layer [23], thereby transferring angular momentum into the Cu 1−xIrx. A spin accumulation builds up, diffuses as a pure spin current and is eventually converted into a transverse charge current by means of the ISHE, yielding a measurable voltage signal. The spin current and consequently the thermal voltage change sign when the 4YIG magnetization is reversed. The SSE voltage VSSEis defined as the difference between the voltage signals obtained for positive and negative magnetic field divided by 2. Since VSSEis the result of the continuous conversion of a steady spin current, it can, applying the notation of conventional electronics, be considered as a DC signal. For the THz SSE measurements, the same in-plane magnetized YIG/Cu 1−xIrxsamples are illuminated at room temperature by femtosecond laser pulses (energy of 2 .5 nJ, duration of 10 fs, center wavelength of 800 nm corresponding to a photon energy of 1 .55 eV, repetition rate of 80 MHz) of a Ti:sapphire laser oscillator. Owing to its large bandgap of 2 .6 eV [24], YIG is transparent for these laser pulses. They are, however, partially (about 50 %) absorbed by the electrons of the Cu 1−xIrxlayer. The spatially step-like temperature gradient across the YIG/metal interface leads to an ultrafast spin current in the metal layer polarized parallel to the sample magnetization [19]. Subsequently, this spin current is converted into a transverse sub-picosecond charge current through the ISHE, resulting in the emission of a THz electromagnetic pulse into the optical far-field. The THz electric field is sampled using a standard electrooptical detection scheme employing a 1 mm thick ZnTe detection crystal [25]. The magnetic response of the system is quantified by the root mean square (RMS) of half the THz signal difference Sfor positive and negative magnetic fields. RESULTS Figure 2(a)-(f) shows DC SSE hysteresis loops measured for YIG/Cu 1−xIrx/AlOx multi- layers with varying Ir concentration x. The temperature difference between sample top and bottom is fixed to ∆ T= 10 K with a base temperature of T= 288.15 K. In the Cu-rich phase, we observe an increase of the thermal voltage signal with increasing x, exhibiting a maximum at x= 0.3. Interestingly, upon further increasing the Ir content VSSEreduces again. This behavior is easily visible in Fig. 2(g), in which the SSE coefficient VSSE/∆Tis plotted as a function of x. The measured concentration dependence shows that VSSE/∆T exhibits a clear maximum in the range from x= 0.3 to 0.5. Thus, as a first key result the maximum spin Hall effect is obtained for the previously neglected alloying regime beyond the dilute doping. For comparison, the resistivity σ−1of the metal film is also shown in Fig. 2(g). We see that the resistivity of the Cu 1−xIrxlayer follows a similar trend as the DC SSE signal. 50.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 x0.20.40.60.81.0 246810 −10−50510 −10−50510VSSE(µV) −10 0 10 µ0H(mT)−10−50510 −10 0 10 µ0H(mT)(a) (b) (c)(d) (e) (f)x = 0.05 x = 0.1 x = 0.2x = 0.3 x = 0.5 x = 0.75.0 µV 5.7 µV 8.1 µV18.1 µV 15.7 µV 9.1 µVVSSE /ΔT µVK−1) )(g) VSSE/∆T 10−7Ωm) ) σ−1σ−1Figure 2. (a)-(f) Measured DC SSE voltage in YIG/Cu 1−xIrx/AlOx stacks for different Ir con- centrations xin ascending order. The temperature difference between sample top and bottom is fixed to ∆T= 10 K. (g) SSE coefficient VSSE/∆T(red squares) and resistivity σ−1(blue circles) as a function of Ir concentration x. Typical THz emission signals from the YIG/Cu 1−xIrx/AlOx samples are depicted in Figs. 3(a)-(f). The THz transients were low-pass filtered in the frequency domain with a Gaussian centered at zero frequency and a full width at half maximum of 20 THz. The RMS of the THz signal odd in sample magnetization is plotted in Fig. 3(g) as a function of x. After an initial signal drop in the Cu-rich phase, the THz signal increases with increasing Ir concentration, indicating a signal maximum in the range between x= 0.3 and 0.5. Further increase of the Ir content leads to a second reduction of the THz signal strength. DISCUSSION In the following, a direct comparison of the signals obtained from the DC and the ultrafast THz measurements is established. To begin with, the emitted THz electric field right behind the sample is described by a generalized Ohm’s law, which in the thin-film limit (film is much 6(a) −4−2024 −4−2024THz signal (arb. u. ) −1.0 −0.5 0.0 0.5 t (ps)−4−2024 −1.0 −0.5 0.0 0.5(b) (c)(d) (e) (f)x = 0.05 x = 0.1 x = 0.2x = 0.3 x = 0.5 x = 0.7 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.60.70.80.91.01.1 x(g)THz signal strength (norm. ) t (ps)Figure 3. (a)-(f) Signal waveforms (odd in the sample magnetization) of the THz pulses emitted from YIG/Cu 1−xIrx/AlOx stacks for different Ir concentrations xin ascending order. (g) THz signal strength (RMS) as a function of Ir concentration x. thinner than the wavelength and attenuation length of the THz wave in the sample) is in the frequency domain given by [21] ˜E(ω)∝θSHZ(ω)/integraldisplayd 0dzjs(z,ω), (1) whereωis the angular frequency. The spin-current density js(z,ω) is integrated over the full thickness dof the metal film. The total impedance Z(ω) can be understood as the impedance of an equivalent parallel circuit comprising the metal film (Cu 1−xIrx) and the surrounding substrate (GGG/YIG) and air half-spaces, 1 Z(ω)=n1(ω) +n2(ω) Z0+G(ω). (2) Here,n1andn2≈1 are the refractive indices of substrate and air, respectively, Z0= 377 Ω is the vacuum impedance, and G(ω) is the THz sheet conductance of the Cu 1−xIrx films. Considering the Drude model and a velocity relaxation rate of 28 THz for pure Cu at room temperature as lower boundary [26], the values of G(ω) vary only slightly over the 7detected frequency range from 1 to 5 THz (as given by the ZnTe detector crystal). Therefore, the frequency dependence of the conductance can be neglected, i.e. G(ω)≈G(ω= 0). Importantly, the metal-film conductance ( G≈8×10−3Ω−1) is much smaller than the shunt conductance ([ n1(ω) +n2(ω)]/Z0≈4×10−2Ω−1) for the investigated metal film thickness (d= 4 nm) and can be thus neglected. Therefore, the Ir-concentration influences the THz emission strength only directly through the ISHE-induced in-plane charge current flowing inside the NM layer. The measured DC SSE voltage, on the other hand, is given by an analogous expression related to the underlying in-plane charge current by the standard Ohm’s law, VSSE ∆T∝θSHR/integraldisplayd 0dzjs(z). (3) Here,Ris the Ohmic resistance of the metal layer between the electrodes, which is inversely proportional to the metal resistivity σ, andjs(z) is the DC spin current density. Therefore, in contrast to the THz data, the impact of alloying on VSSEthroughσ−1is significant. For a direct comparison with the THz measurements, we thus contrast the RMS of the THz signal waveform with the DC SSE current density jSSE=VSSE·σ/∆T. In Fig. 4, the respective amplitudes are plotted as a function of the Ir concentration. Remarkably, DC and THz SSE/ISHE measurements exhibit the very same concentration dependence. This agreement suggests that the ISHE retains its functionality from DC up to THz frequencies, which vindicates the findings and interpretations of previous experiments [21]. Small discrepancies may originate from a varying optical absorptance of the near- infrared pump light, which is, however, expected to depend monotonically on xand to only vary by a few percent [21]. Furthermore, as discussed below, these findings imply that for DC and THz spin currents comparable concentration dependences of spin-relaxation lengths may be expected. To discuss the concentration dependence of the DC and THz SSE signals (Fig. 4), we consider Eqs. (1) and (3). According to these relationships, the THz signal and the SSE voltage normalized by the metal resistivity result from a competition of (i) the spin Hall angleθSHand (ii) the integrated spin-current density/integraltextd 0dzjs(z,ω). At first, we consider the local spin signal minimum at small, increasing Ir concentration x(dilute regime) that appears for both jSSEand the THz signal. In fact, with regard to (i)θSHone would expect the opposite behavior as for the dilute regime the skew scattering 8THz signal strength (norm. ) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 x0.080.100.120.14 0.60.70.80.91.01.1107µV(KΩm)−1) ) jSSE/ΔTjSSE/∆T THzFigure 4. Ir concentration dependence of the thermal DC spin current (red squares) and the RMS of the THz signal (green diamonds). mechanism has been predicted [4] and experimentally shown [5] to yield the dominant ISHE contribution. With increasing SOI scattering center density ( ρimp∝σ−1), a linear increase of the spin signal should appear. In this work, this trend is observed for VSSE[Fig. 2(g)]. The significantly deviating signal shapes of jSSEand the THz signal, however, suggest that the converted in-plane charge current is notably governed by additional effects. An explana- tion can be given by (ii), considering a spatial variation of the spin current density that, as we discuss below, can be influenced by both electron momentum- and spin-relaxation. The initial electron momenta and spin information of a directional spin current become random- ized over length scales characterized by the mean free path /lscriptand the spin diffusion length λsd, yielding a reduction of the spin current density. For spin-relaxation, the integrated spin current density is given by [27]: /integraldisplaydCuIr 0dzjs(z)∝λsdtanh/parenleftBiggdCuIr 2λsd/parenrightBigg j0 s (4) withdCuIrbeing the thickness of the Cu 1-xIrxlayer. According to Niimi et al. [5] the spin- diffusion length λsddecreases exponentially from λsd≈30 nm forx= 0.01 toλsd≈5 nm for x= 0.12. This exponential decay implies that the integrated spin current density is nearly constant for both small and large x, but undergoes a significant decline in the concentration region where λsd≈dCuIr. This effect possibly explains the observed reduction of the signal amplitude from x= 0.05 tox= 0.2. Furthermore, we interpret the fact that for DC and THz SSE signals similar trends are observed as an indication of similar concentration dependences of λsdin the distinct DC and THz regimes. This appears reasonable when considering that spin-dependent scattering rates are of the same order of magnitude as the momentum scattering [28] (e.g. Γmom. Cu = 1/36 fs≈28 THz [26]) and thus above the 9experimentally covered bandwidth. In addition to spin-relaxation, the integrated spin current density is influenced by mo- mentum scattering. As shown in Fig. 2, alloying introduces impurities and lattice defects in the dilute phase, such that enhanced momentum scattering rates occur. Assuming that the latter increase more rapidly than θSH, the appearance of the previously unexpected local minimum near x≈0.2 can be thus explained. We now focus on the subsequent increase of the spin signal at higher x(concentrated phase). It can be explained by a further increase of extrinsic ISHE as well as intrinsic ISHE contributions, as pure Ir itself exhibits a sizeable intrinsic spin Hall effect [2, 3]. A quantitative explanation of the intrinsic ISHE, however, requires knowledge of the electron band structure (obtainable by algorithms based on the tight-binding model [2] or the density functional theory [29]), which is beyond the scope of this work. The decrease of jSSEand the THz Signal at x= 0.7 may then be ascribed to an increase of atomic order and thus a decrease of the extrinsic ISHE. In conclusion, we compare the spin-to-charge conversion of steady state and THz spin currents in copper-iridium alloys as a function of the iridium concentration. We find a clear maximum of the spin Hall effect for alloys of around 40 % Ir concentration, far beyond the previously probed dilute doping regime. While the detected DC spin Seebeck voltage exhibits a concentration dependence different from the raw THz signal, very good qualitative agreement between the DC spin Seebeck current and the THz emission signal is observed, which is well understood within our model for THz emission. Ultimately, our results show that tuning the spin Hall effect by alloying delivers an unexpected, complex concentration dependence that is equal for spin-to-charge conversion at DC and THz frequencies and allows us to conclude that the large spin Hall effect in CuIr can be used for spintronic applications on ultrafast timescales. ACKNOWLEDGMENTS This work was supported by Deutsche Forschungsgemeinschaft (DFG) (SPP 1538 “Spin Caloric Transport”, SFB/TRR 173 ”SPIN+X”), the Graduate School of Excellence Materi- als Science in Mainz (DFG/GSC 266), and the EU projects IFOX, NMP3-LA-2012246102, INSPIN FP7-ICT-2013-X 612759, TERAMAG H2020 681917. 10∗Klaeui@uni-mainz.de [1] J. Sinova, S. O. Valenzuela, J. 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2017-09-06

We measure the inverse spin Hall effect of Cu$_{1-x}$Ir$_{x}$ thin films on yttrium iron garnet over a wide range of Ir concentrations ($0.05 \leqslant x \leqslant 0.7$). Spin currents are triggered through the spin Seebeck effect, either by a DC temperature gradient or by ultrafast optical heating of the metal layer. The spin Hall current is detected by, respectively, electrical contacts or measurement of the emitted THz radiation. With both approaches, we reveal the same Ir concentration dependence that follows a novel complex, non-monotonous behavior as compared to previous studies. For small Ir concentrations a signal minimum is observed, while a pronounced maximum appears near the equiatomic composition. We identify this behavior as originating from the interplay of different spin Hall mechanisms as well as a concentration-dependent variation of the integrated spin current density in Cu$_{1-x}$Ir$_{x}$. The coinciding results obtained for DC and ultrafast stimuli show that the studied material allows for efficient spin-to-charge conversion even on ultrafast timescales, thus enabling a transfer of established spintronic measurement schemes into the terahertz regime.

Complex THz and DC inverse spin Hall effect in YIG/Cu$_{1-x}$Ir$_{x}$ bilayers across a wide concentration range

1709.01890v1

arXiv:1607.03409v1 [cond-mat.mes-hall] 12 Jul 2016Effect of Quantum Tunneling on Spin Hall Magnetoresistance Seulgi Ok,1Wei Chen,1Manfred Sigrist,1and Dirk Manske2 1Institut f¨ ur Theoretische Physik, ETH-Z¨ urich, CH-8093 Z ¨ urich, Switzerland 2Max-Planck-Institut f ¨ur Festk¨orperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, G ermany (Dated: September 5, 2018) We present a formalism that simultaneously incorporates th e effect of quantum tunnelingand spin diffusion on spin Hall magnetoresistance observed in normal metal/ferromagnetic insulator bilayers (such as Pt/Y 3Fe5O12) and normal metal/ferromagnetic metal bilayers (such as Pt /Co), in which the angle of magnetization influences the magnetoresistanc e of the normal metal. In the normal metal side the spin diffusion is known to affect the landscape o f the spin accumulation caused by spin Hall effect and subsequently the magnetoresistance, wh ile on the ferromagnet side the quantum tunneling effect is detrimental to the interface spin curren t which also affects the spin accumulation. The influence of generic material properties such as spin diff usion length, layer thickness, interface coupling, and insulating gap can be quantified in a unified man ner, and experiments that reveal the quantum feature of the magnetoresistance are suggested. PACS numbers: 75.76.+j, 75.47.-m, 85.75.-d, 73.40.Gk I. INTRODUCTION The electrical control of magnetization dynamics has been a central issue in the field of spintronics1,2, owing to its possible applications in magnetic memory devices with low power consumption. A particularly promis- ing mechanism for the electrical control is to utilize the spin Hall effect3–6(SHE) in a normal metal (NM), such as Pt or Ta, to convert an electric current into a spin current, and subsequently to magnetization dynamics in an adjacent magnet via mechanisms such as spin- transfer torque7,8(STT). In reverse, the inverse spin Hall effect9,10(ISHE) can convert the spin current gen- erated by certain means, for instance spin pumping11,12, into an electric signal. A particularly intriguing phe- nomenon that involves both SHE and ISHE is the spin Hall magnetoresistance13–22(SMR), in which a charge current in an NM causes a spin accumulation at the edge of the sample due to SHE, yielding a finite spin current at the interface to a ferromagnet. Through ISHE, the spin current gives an electromotive force along the orig- inal charge current, effectively changing the magnetore- sistance of the NM. The two major ingredients that determine SMR are the spin diffusion25in the NM and the spin current at the NM/ferromagnet interface. The spin diffusion part has been addressed in detail by Chen et al.for the NM/ferromagnetic insulator (NM/FMI) bilayer, such as Pt/Y3Fe5O12(Pt/YIG), and FMI/NM/FMI trilayer22. This approach solves the spin diffusion equation in the presence of SHE and ISHE in a self-consistent manner, where the spin current at the NM/FMI interface serves as a boundary condition. However, the interface spin current remains an external parameter for which exper- imental or numerical input is needed23,24. On the other hand, a quantum tunneling formalism has emerged re- cently as an inexpensive tool to calculate the interface spin current from various material properties such as the insulating gap of the FMI and the interface s−dcoupling26. The quantum tunneling theory also success- fully explains27the reduced spin pumping spin current when an additional oxide layer is inserted between NM andFMI28. It is thenoffundamental importanceto com- bine the spin diffusion approach with the quantum tun- neling formalism for the interface spin current to give a complete theoretical description of the SMR, in particu- lar to quantify how various material properties influence the SMR. In this article we provide a minimal formalism that bridges the quantum tunneling formalism to the spin dif- fusion approach. We focus on the SMR in NM/FMI bilayer realized in Pt/YIG, and the NM/ferromagnetic metal (NM/FMM) bilayer realized in Pt/Co and Ta/Co14. The spin diffusion in the NM is assumed to be described by the same formalism of Chen et al.22, whereas the interface spin current is calculated from the quantum tunneling formalism26,27. In the NM/FMM bi- layer, we consider an FMM that has long spin diffusion length and a small thickness, such that the spin diffu- sion effect is negligible and the spin transport is predom- inately of quantum origin26. This is presumably ade- quate for the case of ultrathin Co films29, but not for materials with very short spin diffusion length such as permalloy30,31. Within thisformalism,theeffectofmate- rial properties including spin diffusion length of the NM, interfaces−dcoupling, insulating gap of the FMI, and the thickness of each layer can all be treated on equal footing. In particular, we reveal the signature of quan- tum interference in SMR in NM/FMM bilayer, and dis- cuss the situation in which it can be observed. The structure of the article is arranged in the follow- ing manner. In Sec. II, we detail the quantum tunneling formalism for the interface spin current in the NM/FMI bilayer, and how it is adopted into the spin diffusion ap- proachthat describestheNM. Section III generalizesthis recipe to the NM/FMM bilayer, and discuss the observ- ability ofthe predicted signatureof quantum interference in SMR. Section IV gives the concluding remark.2 II. NM/FMI BILAYER A. Interface spin current We start with the quantum tunneling formalism that calculates the interface spin current in the NM/FMI bi- layer, which later serves as the boundary condition for the spin diffusion equation that determines SMR. The quantum tunneling formalism describes the NM/FMI bi- layer shown in Fig. 1 (a) by the Hamiltonian HN=p2 2m−µσ x(−lN≤x<0), (1) HFI=p2 2m+V0+ΓS·σ(0≤x≤lFI),(2) whereµσ x=±µx·ˆz/2 is the spin voltage of σ={↑,↓} produced by an in-plane charge current Jc yˆy,ǫFis the Fermi energy, V0−ǫFis the insulating gap, and S= S(sinθcosϕ,sinθsinϕ,cosθ) is the magnetization. We choose Γ<0 such that the magnetization has the ten- dency to align with the conduction electron spin σ. The wave function near the interface is ψN= (Aeik0↑x+Be−ik0↑x)/parenleftbigg 1 0/parenrightbigg +Ce−ik0↓/parenleftbigg 0 1/parenrightbigg ,(3) ψFI= (Deq+x+Ee−q+x)/parenleftbigg e−iϕ/2cosθ 2 eiϕ/2sinθ 2/parenrightbigg +(Feq−x+Ge−q−x)/parenleftbigg −e−iϕ/2sinθ 2 eiϕ/2cosθ 2/parenrightbigg ,(4) wherek0σ=/radicalbig 2m(ǫF+µσ 0)//planckover2pi1andq±=/radicalbig 2m(V0±ΓS−ǫF)//planckover2pi1. The amplitudes B∼E are solved in terms of the incident amplitude Aby matching wave functions and their first derivative at the interface. The x<−lNandx>lFIregions are assumed to be vacuum or insulating oxides that correspond to infinite potentials such that the wave functions vanish there for simplicity. We identify the incident flux with |A|2=NF|µ0|/a3whereNFis the density of states per a3witha= 2π/kF=h/√2mǫFthe Fermi wave length. The spin current inside the FMI at position xis calcu- lated from the evanescent wave function jx=/planckover2pi1 4im/bracketleftbig ψ∗ FIσ(∂xψFI)−(∂xψ∗ FI)σψFI/bracketrightbig .(5) Angularmomentum conservation8,26dictatesthat thein- terface spin current to be equal to the STT exerts on the magnetization j0−jlFI=j0=τ a2 =ΓSNF /planckover2pi1/bracketleftBig GrˆS×/parenleftBig ˆS×µ0/parenrightBig +GiˆS×µ0/bracketrightBig ,(6) which defines the field-like Giand dampling-like Grspin mixing conductance that in turn can be calculated fromthe interface spin current26 ΓSNF /planckover2pi1Gr=2jx 0cosϕ |µ0|sin2θ+2jy 0sinϕ |µ0|sin2θ=−jz 0 |µ0|sin2θ, ΓSNF /planckover2pi1Gi=jx 0sinϕ |µ0|sinθ−jy 0cosϕ |µ0|sinθ. (7) A straight forward calculation yields Gr,i=−4 a3|γθ|2/parenleftbiggq+cothq+lFI−q−cothq−lFI q2 +−q2 −/parenrightbigg ×(Im,Re)/parenleftBig n∗ ↓+n↓−/parenrightBig , (8) whereσx,yisx,ycomponent of Pauli matrix, and nσ±=k0σ (k0σ+iq±cothq±lFI), γθ=n↓+ n↑+cos2θ 2+n↓− n↑−sin2θ 2. (9) Equation (8) describes the spin mixing conductance in STT, as well as that in spin pumping since the Onsager relation32is satisfied in this approach26. BothGrand Gihave very weak dependence (at most few percent) on the angle of magnetization θthroughγθ, which may be considered as higher order contributions26. In the numerical calculation below we set θ= 0.3πwithout loss of generality. Numerical results of the spin mixing conductance Gr,i are shown in Fig. 1, plotted as a function the FMI thick- nesslFIand at different strength of the interface s−d coupling ΓS/ǫF. BothGrandGiincrease with lFIini- tially and then saturate to a constant as expected, since they originatefrom the quantum tunneling ofconduction electrons that only penetrate into the FMI over a very short distance. At a FMI thickness small compared to Fermi wave length lFI≪a, we found that Gr∝l6 FIand Gi∝l3 FI, therefore the damping-like to field-like ratio is|Gr/Gi| ≪1. In most of the parameter space, the torque is dominated by field-like component |Gr/Gi|<1 throughout the whole range of lFI. Only when the mag- nitude ofs−dcoupling is large compared to the insu- lating gap ( V0−ǫF)/ǫFis the torque dominated by the damping-like component |Gr/Gi|>1, consistent with that found previously26and also in accordance with the result from first principle calculation23. The magnitude ofGr,igenerally increases with the s−dcoupling, yet more dramatically for Gr. Note that GrandGido not depend on the NM thickness in this quantum tunneling approach. B. SMR We adopt the spin diffusion approach of Chen et al.22 to address the effect of the interface spin current in Eq. (6) on SMR, which is briefly summarized below. The3 FIG. 1: (color online) (a) Schematics of the bilayer con- sists of an NM with thickness lNand an FMI with thickness lFI. (b) The spin mixing conductance Gr,iversus the FMI thickness lFI, at different values of interface s−dcoupling strength −ΓS/ǫF. The insulating gap strength is fixed at (V0−ǫF)/ǫF= 1.5. The absolute units for Gr,iise2//planckover2pi1a2 which is about 1014∼1015Ω−1m−2depending on the Fermi wave length a. spin diffusion approachis based on the following assump- tions for the spin transport in the NM: (1) The spin cur- rent in NM consists of two parts, one from the spatial gradient of spin voltage and the other the bare spin cur- rent caused directly by SHE, jx=−σc 4e2∂xµx+θSHσcEy 2eˆz, (10) whereθSHis spin Hall angle, σcis the conductivity of NM,Eyis applied external electric in ydirection, and −eis electron charge. (2) The spin voltage obeys the spin diffusion equation ∇2µx=µx/λ2, whereλis the spin diffusion length. (3) Spin current vanishes at the edge of NM ( x=−lN), which serves as one boundary condition. (4) The spin current at the NM/FMI interface isdescribed byEq.(6), whichservesasanotherboundary condition. The self-consistent solution satisfying (1) ∼(4) is22 jx·ˆx jSH=βxsinθ/bracketleftBig cosθcosϕRe/parenleftbig/tildewideG/parenrightbig +sinϕIm/parenleftbig/tildewideG/parenrightbig/bracketrightBig , jx·ˆy jSH=βxsinθ/bracketleftBig cosθsinϕRe/parenleftbig/tildewideG/parenrightbig −cosϕIm/parenleftbig/tildewideG/parenrightbig/bracketrightBig , jx·ˆz jSH= 1−cosh(2x+lN 2λ) cosh(lN 2λ)−βxsin2θRe/parenleftbig/tildewideG/parenrightbig ,(11) where βx=sinh(x+lN λ) sinh(lN λ)tanh(lN 2λ), /tildewideG=αGc 1−αGccoth(lN λ), α=4ΓSNFe2λ /planckover2pi1σc, (12) andjSH=θSHσcEy/2eis the bare spin current. Here α <0 is a negative parameter (because we assume theinterfaces−dcoupling Γ<0) that bridges our tunneling formalism to the spin diffusion equation, and Gc=Gr+ iGiis the complex spin mixing conductance. Through ISHE, the spin currents in Eq. (11) is con- verted back to a chargecurrent in the longitudinal (along ˆy) and transverse (along ˆz) direction ∆jc long(x) =−2eθSH/parenleftBig jx−θSHσcEy 2eˆz/parenrightBig ·ˆz,(13) ∆jc trans(x) = 2eθSH/parenleftBig jx−θSHσcEy 2eˆz/parenrightBig ·ˆy.(14) TheconductivityaveragedovertheNMlayerthenfollows σlong=σ+1 lNEy/integraldisplay0 −lNdx∆jc long(x),(15) σtrans=1 lNEy/integraldisplay0 −lNdx∆jc trans(x). (16) Usingθ2 SH∼0.01≪1, the longitudinal and transverse component of SMR read ρlong=σ−1 long≈ρ+∆ρ0+sin2θ∆ρ1, ρtrans=−σtrans/σ2 long ≈cosθsinθsinϕ∆ρ1−sinθcosϕ∆ρ2,(17) where ∆ρ0/ρ=−θ2 SH2λ lNtanh/parenleftbigglN 2λ/parenrightbigg , ∆ρ1/ρ=−θ2 SHλ lNtanh2/parenleftbigglN 2λ/parenrightbigg Re/parenleftbig/tildewideG/parenrightbig , ∆ρ2/ρ=θ2 SHλ lNtanh2/parenleftbigglN 2λ/parenrightbigg Im/parenleftbig/tildewideG/parenrightbig .(18) Clearly the FMI thickness lFIaffects ∆ρ1and ∆ρ2only through /tildewideG=/tildewideG(lFI). To perform numerical calculation of Eq. (18), we make the following assumption on the parameter αin Eq. (12) that connects the quantum tunneling formalism with the spin diffusion equation. Firstly, αcontains the density of state per a3at the Fermi surface, which is assumed to be the inverse of Fermi energy NF= 1/ǫF. The com- bined parameter Γ SNF= ΓS/ǫFtherefore represents the strength of s−dcoupling. Other parameters that influenceαare the spin diffusion length assumed to be λ≈10nm, the conductivity of the NM film taken to be σc≈5×106Ω−1m−1, and Fermi wave length assumed to be roughly equal to the lattice constant a≈0.4nm, all of which are the typical values for commonly used materials such as Pt. These lead to the dimensionless parameter αGc≈10×(ΓS/ǫF)×/parenleftbig Gc/(e2//planckover2pi1a2)/parenrightbig in Eq. (12) be- ing expressed in terms of the relative strength of s−d coupling and the spin mixing conductance divided by its unit. Inwhatfollows, weexaminethe effectofFMI thick- ness, NM thickness, insulating gap, and interface s−d coupling on SMR. On the contrary, the spin Hall angle,4 FIG. 2: (color online) The longitudinal ∆ ρ1/ρand transverse −∆ρ2/ρcomponent of SMR in the NM/FMI bilayer, plotted against the FMI thickness in units of Fermi wave length lFI/aand NM thickness in units of the spin diffusion length lN/λ, at various strength of s−dcoupling −ΓS/ǫFand the insulating gap ( V0−ǫF)/ǫF. Note that the color scale of each plot is different. spindiffusionlength, andconductivityaretreatedascon- stants, although in reality they may also depend on the layer thickness or on each other in such thin films33. ThenumericalresultofSMRisshowninFig.2, plotted as a function of the FMI thickness lFIand NM thickness lNat several values of insulating gap ( V0−ǫF)/ǫFand s−dcoupling ΓS/ǫF. As a function of the FMI thick- nesslFI, both the longitudinal ∆ ρ1/ρand the transverse ∆ρ2/ρcomponent initially increase and then saturate at aroundlFI/a∼2, which is expected since conduction electrons only tunnel into the FMI over a short depth, so the interface spin current saturates once the FMI is thicker than this tunneling depth. The insulating gap (V0−ǫF)/ǫFobviouslyaffects the tunneling depth, and is particularly influential on the magnitude of longitudinal ∆ρ1/ρ, as can be seen by comparing plots with different (V0−ǫF)/ǫFin Fig. 2. The magnitude of ∆ ρ1/ρalso generally increases with the s−dcoupling ΓS/ǫF, while the transverse component ∆ ρ2/ρat large ΓS/ǫFdisplays a nonmonotonic dependence on the FMI thickness. On the other hand, as a function of NM thickness lN, both SMR components increase and peak at around lN/λ∼1 and then decrease monotonically for large lN. This can be understood because both ∆ ρ1and ∆ρ2are interfaceeffects that become less significant compared to bulk re- sistivityρwhen NM thickness increases, and the spin voltage is known to be maximal when the NM thickness is comparable to the spin diffusion length25lN/λ∼1. III. NM/FMM BILAYER A. Interface spin current and SMR We proceed to address the SMR in the NM/FMM bi- layer, with the assumption that the FMM film is much thinner than its spin diffusion length lFM≪λsuch that quantum tunneling is the dominant mechanism for spin transport in the FMM, while the spin diffusion inside the FMM can be ignored. The calculation of the spin cur- rent at the NM/FMM interface starts with the model schematically shown in Fig. 3 (a). The NM and FMM occupy−lN≤x<0 and 0≤x≤lFM, respectively. The NM region is described by Eqs. (1) and (3), while the FMM layer is described by HFM=p2/2m+ΓS·σand5 FIG. 3: (color online) (a) Schematics of an NM/FMM bi- layer with finite thickness. (b) The ratio of spin mixing con- ductance (c) Grand (d)−Giin this system, plotted against the thickness lFMof the FMM and s−dcoupling −ΓS/ǫF, in units of e2/planckover2pi1/a2whereais the Fermi wave length. the wave function ψFM= (Deik+x+F−ik+x)/parenleftbigg e−iϕ/2cosθ 2 eiϕ/2sinθ 2/parenrightbigg +(Eeik−x+Ge−ik−x)/parenleftbigg −e−iϕ/2sinθ 2 eiϕ/2cosθ 2/parenrightbigg ,(19) wherek±=/radicalbig 2m(ǫF∓ΓS)//planckover2pi1. The wave functions out- side of the bilayer in x > l FMandx <−lNare as- sumed to vanish for simplicity. The coefficients A∼I are again determined by matching wave functions and their first derivative at the interface. The interface spin current and the spin mixing conductance are calculated from Eqs. (5) to (7), with replacing ψFItoψFMandlFI tolFM, resulting in Gr,i=1 a3|γ′ θ|2(Im,Re)/bracketleftBigg Z∗ ↓−+Z↓++ ×/parenleftBig u+−−u++−u−−+u−+/parenrightBig/bracketrightBigg ,(20) where uαβ=iei(αk++βk−)lFM αk++βk−, Wσαβ=k0σ+βkα 2k0σ, Zσαβ=Wσαβe−ikαlFM−WσαβeikαlFM, γ′ θ=Z↑++Z↓−+cos2θ 2+Z↓++Z↑−+sin2θ 2,(21)withβ=−β. Apart from a change in magnitude, the pattern of the spin mixing conductance as a function of s−dcoupling and the FMM thickness shown in Fig. 3 is almost indistinguishable from that reported in Fig. 2 of Ref. 26, which shows clear signals of quantum inter- ference with respect to both s−dcoupling and FMM thickness. This similarity is expected, since the only dif- ference between the formalism here and in Ref. 26 is the insulating gap V0−ǫFof the substrate or vacuum in the x > lFMregion in Fig. 3 (a), which is assumed to be infinite here for simplicity but finite in Ref. 26. The in- sulating gap is spin degenerate and essentially does not influence the spin transport. FIG. 4: (color online)The longitudinal ∆ ρ1/ρandtransverse ∆ρ2/ρcomponent of SMR in the NM/FMM bilayer, plotted against the FMM thickness in units of Fermi wave length lFM/aand NM thickness in units of the spin diffusion length lN/λ, at various strength of s−dcoupling −ΓS/ǫF.6 To get SMR, we use Eq. (17) ∼(18) while taking the Gc=Gr+iGiobtainedfromEq.(20). Theresultsforthe longitudinal ∆ ρ1/ρand transverse ∆ ρ2/ρcomponent of SMR as functions of FMM thickness lFMand NM thick- nesslNare shown in Fig. 4, for several values of s−d coupling ΓS/ǫF. As a function of NM thickness, both components reach a maximal at around the spin diffu- sion length lN/λ∼1 and then decrease monotonically, similartothatreportedinFig.2forNM/FMIbilayerand is due to the spin diffusion effect explained in Sec. IIB. On the other hand, asa function of FMM thickness, both components show clear modulations with an average pe- riodicity that decreases with increasing s−dcoupling, a trend similar to that of GrandGishown in Fig. 3 and is attributed to the quantum interference of spin transport. Intuitively, a larger s−dcoupling renders a faster preces- sion of conduction electron spin when it travelsinside the FMM, hence more modulations appear for a given FMM thickness. The transverse component of SMR is found to be generally one order of magnitude smaller than the longitudinal component. B. To observe the predicted oscillation in SMR The experimental detection of the oscillation of SMR with respect to FMM thickness lFMshown in Fig. 4 would be a direct proof of our approach. In a typical NM/FMI set up, however, there are other sources that contribute to the total resistance measured in experi- ments, therefore it is important to investigate whether there is a situation in which the predicted oscillation of SMR can manifest. To explore this possibility, we use a three-resistor model to characterize the total longitu- dinal resistance14, which contains the resistor that rep- resents the NM layer ( N), the FMM layer ( F), and the interface layer ( I) connected in parallel, each denoted by Ri=R0 i+δRiwithi={N,I,F}. HereR0 iis the contri- bution to the longitudinal resistance in layer ithat does not depend on the angle of the magnetization, and δRi is the part that depends on the angle which is generally much smaller δRi≪R0 i. Expanding the total longitudi- nal resistance to leading order in δRiyields Rtot≈R0 tot+/parenleftbiggR0 IR0 F B/parenrightbigg2 δRN +/parenleftbiggR0 NR0 F B/parenrightbigg2 δRI+/parenleftbiggR0 NR0 I B/parenrightbigg2 δRF, R0 tot=R0 NR0 IR0 F B, B=R0 NR0 I+R0 IR0 F+R0 NR0 F. (22) Eachresistanceisassumed to satisfythe usualrelation to the sample size/braceleftbig R0 i,δRi/bracerightbig =/braceleftbig ρ0 i,δρi/bracerightbig ×L/lih, whereL andhare the length and the width of the sample, respec- tively,ρ0 iandδρiare the corresponding resistivity, and li is the thickness of layer i. The thickness of the interfacelIis assumed to be intrinsically constant, in contrast to lNandlFthat can be varied experimentally14. The per- centage change of the total resistance due to the angle of the magnetization is Rtot−R0 tot R0 tot≈/parenleftbiggρ0 Iρ0 F lIlFC/parenrightbiggδρN ρ0 N+/parenleftbiggρ0 Nρ0 F lNlFC/parenrightbiggδρI ρ0 I +/parenleftbiggρ0 Nρ0 I lNlIC/parenrightbiggδρF ρ0 F, C=ρ0 Nρ0 I lNlI+ρ0 Nρ0 F lNlF+ρ0 Fρ0 I lFlI.(23) Note that the ρ0 iρ0 j/liljCfactors are monotonic functions of the layer thickness {lN,lI,lF}, and are independent from the angle of the magnetization. The contribution to the angular dependent part of RF comes from the anisotropic magnetoresistance (AMR) which takes the form34,35δρF∝(jc·ˆm)2∝(my)2 since the in-plane charge current jcruns along ˆyas shown in Fig. 3 (a), and we denote ˆm=S/S= (sinθcosϕ,sinθsinϕ,cosθ) as the unit vector along the direction of the magnetization. In addition, Zhang et al.35showed that the interface resistance has a quadratic dependence on both myandmz, a result of surface spin- orbit scattering. On the other hand, the SMR in the NM has the angular dependence22described by Eq. (17). These considerations lead to the parametrization of re- sistivity by ρ0 F+δρF=ρ0 F+∆ρb F(my)2, ρ0 I+δρI=ρ0 I+∆ρs I,y(my)2+∆ρs I,z(mz)2, ρ0 N+δρN= (ρ+∆ρ0)+∆ρ1/bracketleftbig (mx)2+(my)2/bracketrightbig . (24) Combinig this with Eq. (23) motivates us to propose the following experiment that should isolate the effect of lon- gitudinal SMR represented by δρN. From Eq. (24), we seethatδρFandδρIvanishifthe magnetizationdoesnot have an in-plane component, i.e., my=mz= 0, while δρNremains finite as long as the out-of-plane component is nonzeromx∝ne}ationslash= 0. Thus we propose to fix the magne- tization of the FMM film to be out-of-plane mx∝ne}ationslash= 0, in which case the percentage change of total longitudinal resistance as a function of FMM thickness takes the form Rtot−R0 tot R0 tot≈l1 lF+l1+l2×∆ρ1 ρ+∆ρ0(mx)2, (formy=mz= 0) (25) whereρ, ∆ρ0, and ∆ρ1are those in Eqs. (17) and (18), l1=lNρ0 F/ρ0 Nandl2=lIρ0 F/ρ0 Iare two length scales that can be treated as fitting parameters in experiments. Equation (25) indicates that, for the case of only out- of-plane magnetization, the percentage change of magne- toresistance decays with the FMM thickness lFdue to thel1/(lF+l1+l2) factor, but also oscillates with lFdue to the ∆ρ1/(ρ+ ∆ρ0)≈∆ρ1/ρfactor as quantified in7 Eq. (18) and shown in Fig. 4. Thus varying FMM thick- ness while keeping its magnetization out-of-plane may be a proper set up to observe the predicted oscillation of longitudinal SMR, provided the FMM thickness remains thinner than its spin relaxation length lF≪λ. Finally, we remark that the convention of labeling coordinate in SMR or STT experiments is that the charge current is defined to be along ˆ xand the direction normal to the film is along ˆ z. Therefore the coordinate in our tun- neling formalism ( x,y,z) corresponds to ( z,x,y) in the experimental convention. IV. CONCLUSION In summary, the quantum tunneling formalism for the interface spin current is incorporated into the spin diffu- sionapproachtostudytheeffectofvariousmaterialprop- erties on SMR, in particular the effect of layer thickness, insulating gap, and interface s−dcoupling. The advan- tage of combining the quantum and diffusive approachis that the effects of all these material properties can be treated on equal footing. For the NM/FMI case, we re- veal an SMR that saturates at large FMI thickness since the conduction electrons only tunnels into the FMI over a short distance, whereas the longitudinal and transverse SMR display different dependence on the insulating gap and interface s−dcoupling. For the NM/FMM case, we predict that SMR may display a pattern of oscillation as increasing FMM thickness due to quantum interference, and propose an experiment to observe it by using fixed out-of-plane magnetization to isolate SMR from other contributions. 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2016-07-12

We present a formalism that simultaneously incorporates the effect of quantum tunneling and spin diffusion on spin Hall magnetoresistance observed in normal metal/ferromagnetic insulator bilayers (such as Pt/YIG) and normal metal/ferromagnetic metal bilayers (such as Pt/Co), in which the angle of magnetization influences the magnetoresistance of the normal metal. In the normal metal side the spin diffusion is known to affect the landscape of the spin accumulation caused by spin Hall effect and subsequently the magnetoresistance, while on the ferromagnet side the quantum tunneling effect is detrimental to the interface spin current which also affects the spin accumulation. The influence of generic material properties such as spin diffusion length, layer thickness, interface coupling, and insulating gap can be quantified in a unified manner, and experiments that reveal the quantum feature of the magnetoresistance are suggested.

Effect of Quantum Tunneling on Spin Hall Magnetoresistance

1607.03409v1

Cavity-mediated superconductor–ferromagnetic insulator coupling Andreas T. G. Janssønn, Henning G. Hugdal,Arne Brataas, and Sol H. Jacobsen Center for Quantum Spintronics, Department of Physics, NTNU Norwegian University of Science and Technology, NO-7491 Trondheim, Norway A recent proof of concept showed that cavity photons can mediate superconducting (SC) signatures to a ferromagnetic insulator (FI) over a macroscopic distance [Phys. Rev. B, 102, 180506(R) (2020)]. In contrast with conventional proximity systems, this facilitates long-distance FI–SC coupling, local subjection to different drives and temperatures, and studies of their mutual interactions without proximal disruption of their orders. Here we derive a microscopic theory for these interactions, with an emphasis on the leading effect on the FI, namely, an induced anisotropy field. In an arbitrary practical example, we find an anisotropy field of 14–16µT, which is expected to yield an experimentally appreciable tilt of the FI spins for low-coercivity FIs such as Bi- YIG. We discuss the implications and potential applications of such a system in the context of superconducting spintronics. I. INTRODUCTION Enabling low-dissipation charge and spin transport, super- conducting spintronics presents a pathway to reducing en- ergy costs of data processing, and provides fertile ground for exploring new fundamental physics [1–3]. Conventionally, superconducting and spintronic systems are coupled by the proximity effect, with properties of adjacent materials trans- ported across an interface. The superconducting coherence length thus limits the extent to which superconducting prop- erties can be harnessed in proximity systems, to a range of nm–m near interfaces [4–8]. By contrast, cavity-coupled systems offer mediation across macroscopic distances [9–13]. They also offer interaction strengths that relate inversely to the cavity volume [14, 15], which is routinely utilized experimentally to achieve strong coupling in e.g. GHz–THz cavity set-ups [16–20]. Further- more, research on the coupling of magnets and cavity photons shows that the effective interaction strengths scale with the number of spins involved [9, 20–28], which has been utilized experimentally to achieve effective coupling strengths far ex- ceeding losses [11, 13, 20, 26–36]. Theoretically, a number of methods have been employed to extract mediated effects in cavity-coupled systems. This includes, but is not limited to, classical modelling for cou- pling two ferromagnets [37], and a ferromagnet to a super- conductor [10]; application of Jaynes–Cummings-like models for coupling a ferromagnet and a qubit [12, 13, 32, 38], and two ferromagnets [39]; perturbative diagonalization by the Schrieffer–Wolff transformation for coupling a ferro- and an- tiferromagnet [9, 21, 28], and a normal metal to itself [14, 20]; and perturbative evolution of the density matrix, as well as perturbative diagonalization by the non-equilibrium Keldysh path integral formalism, for coupling a mesoscopic circuit to a cavity [40]. In this paper, we will employ the Matsubara path inte- gral formalism [41–45] to derive a microscopic theory for the cavity-mediated coupling of a ferromagnetic insulator (FI) with a singlet s-wave superconductor (SC). In particular, we Corresponding author: henning.g.hugdal@ntnu.no FIG. 1. Illustration of the set-up. A thin ferromagnetic insulator and thin superconductor are placed spaced apart inside a rectangular, electromagnetic cavity. The FI is subjected to an aligning external magnetic field Bext. The cavity is short along the zdirection, and long along the perpendicular xydirections, causing cavity modes to separate into a band-like structure. The FI and the SC are respectively placed in regions of maximum magnetic ( z=Lz) and electric ( z= Lz=2) cavity field of the `z= 1modes, as defined in Sec. II B 1 and illustrated above by the colored field cross-section on the right wall. consider the Zeeman coupling to the FI, and the paramag- netic coupling to the SC. We show that with this approach, we may exactly integrate out the net mediated effect by the cavity photons. This is in contrast to the Schrieffer–Wolff ap- proach, which would limit the integrating-out of the cavity to off-resonant regimes [21]. For instance, a pairing term analo- gous to the one found via the Schrieffer–Wolff transformation in Ref. [14] also appears in our calculations, without the lim- itation to an off-resonant regime. Furthermore, unlike many preceding works which single out the coupling to the uniform mode of the magnet [9, 10, 13, 22, 29, 30], we retain the influ- ence of a range of modes in our model. Their non-negligible influence when the magnet exceeds a certain size relative to the cavity, has been emphasized by both experimentalists [30] and theorists [22]. The Matsubara path integral approach was very recently ap- plied to construct a general effective theory of cavity-coupled material systems of identical particles [45], highlighting some of the same advantages of this approach as above. By contrast, we consider the cavity-mediated coupling of lattices of two distinct classes of quasiparticles, specifically magnons and SC quasiparticles. By a careful choice of cavity dimensions and the place- ment of subsystems, we couple the insulator to the momen-arXiv:2209.09308v2 [cond-mat.mes-hall] 31 Jan 20232 tum degrees of freedom of the superconductor. In this case, the cavity acts as an effective spin–orbit coupling. Here, we emphasize the leading effect of the superconductor on the in- sulator, namely, the induction of an anisotropy field. In an arbitrary, practical example, we achieve a field of 14–16µT, which is expected to yield an experimentally appreciable tilt of the FI spins for an insulator of sufficiently low coercivity such as Bi-YIG. Since the cavity facilitates coupling across unconventionally long distances, it enables the FI and SC to be held at different temperatures, be subjected separately to external drives, and have them interact without the same mu- tual disruption of their orders associated with the proximity effect [2, 10], such as the breaking of Cooper pairs by mag- netic fields from the FI. In practical applications, our system may be used to bridge superconducting and other spintronic circuitry. The article is organised as follows. In Section II A we present the set-up: A cavity with an FI and SC film placed at magnetic and electric antinodes as shown in Fig. 1, with no overlap in the xyplane. In Section II B we cover theoret- ical preliminaries: The quantized gauge field, the magnon- basis Hamiltonian for the insulator, and the Bogoliubov quasiparticle-basis Hamiltonian for the superconductor. The system Hamiltonian is subsequently constructed. In Sec. II C– II E, we construct an effective magnon theory using the path integral formalism. Here we exactly integrate out the cav- ity, and perturbatively the superconductor. In Section III, we extract from the effective theory the leading effect of the su- perconductor on the insulator, namely, the induced anisotropy field. In a practical example, we calculate this field numer- ically, and find here an induced field on the order of µ Tin magnitude. Finally, in Section IV, we give concluding re- marks, discussing the results and their significance, and an outlook. In the appendices, we affirm the mathematical con- sistency of the effective theory with an alternative derivation, explore a variation of the set-up with the SC placed at the opposite magnetic antinode, and elaborate on the interpreta- tion of certain quantities in the effective action as an effective anisotropy field. II. THEORY A. Set-up Our set-up is illustrated in Fig. 1. We place two thin layers, one of a ferromagnetic insulator (FI) and one of a supercon- ductor (SC), spaced apart inside a rectangular electromagnetic cavity. The dimensions of the cavity are Lx;LyLz, with Lzon them–mm scale, and Lx;Lyon the cm scale. The aspect ratios render photons more easily excited in the xydi- rections. The FI is placed at the upper magnetic antinode of the`z= 1 modes (cf. Sec. II B 1), and the SC at the corre- sponding electric antinode, as illustrated in Fig. 1. Because the layers are thin in comparison to Lz, the local spatial vari- ation of the modes in the zdirection is negligible, i.e., the modes are treated as uniform in the zdirection. The FI is locally subjected to an aligning and perpendicularuniform, external magnetostatic field, which vanishes across the SC. This was achieved experimentally with external coils and magnetic shielding in Tabuchi et al. [13]. Furthermore, the SC is subjected to an in-plane supercurrent. This may be realized by passing a direct current (DC) through small elec- tric wires, entering the cavity via small holes in the walls and connecting along the sides of the SC, similarly to Ref. [46]. Provided the wires and holes are sufficiently small, their influ- ence on the cavity modes are negligible. Provided the sample width does not exceed the Pearl length 2=dSC[46–48], the leading effect of the DC is to induce an equilibrium supercur- rent with a Cooper pair center-of-mass momentum 2P, with the magnitude of Pdetermined by the current. Here is the effective magnetic penetration depth, and dSCis the sample depth. For Nb thin films, we expect the Pearl length crite- rion to be met at widths of up to 0:1 mm for adSCdown to 1 nm [49]. B. Hamiltonian In the following, we deduce a Hamiltonian HH FI+Hcav 0+HSC: (1) for the system illustrated in Fig. 1. We begin by quantizing the cavity gauge field, and introducing the cavity Hamiltonian Hcav 0. Following this, we deduce a Hamiltonian HFIfor the FI in the magnon basis, including the Zeeman coupling to the cavity. Finally, we deduce a Hamiltonian HSCfor the SC in the quasiparticle basis, including the paramagnetic coupling to the cavity. 1. Cavity gauge field We begin by presenting the expression for the quantized cavity gauge field Acav[15]. Starting from the Fourier de- composition of the classical vector potential, we impose the transverse gauge and quantize the field. We employ reflect- ing boundary conditions at the cavity walls in the zdirection, and periodic boundary conditions at the comparatively distant walls in thexydirections. The gauge field is thus AcavX Q&s ~ 2!Q(aQ& uQ&+ay Q& u Q&): (2) Above, Q(Qx;Qy;Qz)(2`x=Lx;2`y=Ly;`z=Lz)(3) are the momenta of each photonic mode, with `x;`y= 0;1;2;::: and`z= 0;1;2;:::. The discretization of Qz differs from that of QxandQydue to the different bound- ary conditions in the transverse and longitudinal directions. Furthermore, &= 1;2labels polarization directions, is the permittivity of the material filling the cavity, and !Q=cjQj (4)3 is the cavity dispersion relation, with cthe speed of light. ay Q& andaQ&are photon creation and annihilation operators, satis- fying [aQ&;ay Q0&0] =QQ0&&0; (5) where the factors on the right-hand side are Kronecker delta functions. Lastly, the mode functions  uQ&X D^eDOQ &DuQD (6) encapsulate the spatial modulation of the modes. Here, ^eDis the unit vector in the D=x;y;z direction.OQ &Dare elements of a matrix that rotates the original xyz basis of unit vectors to a new basis labeled 123, with the 3direction aligned with Q(see Fig. 2): 0 @^eQ 1 ^eQ 2 ^eQ 31 A=OQ0 @^ex ^ey ^ez1 A; (7) OQ0 @coscos'cossin'sin sin' cos' 0 sincos'sinsin'cos1 A: (8) Here=Qand'='Qare the polar and azimuthal angles illustrated in Fig. 2. OQoriginates from the implementation of the transverse gauge, which amounts to neglecting the lon- gitudinal 3component of the gauge field. Finally, uQDare the mode functions in the xyzbasis, given by uQx=uQy=r 2 VeiQxx+iQyyisinQzz; (9) uQz=r 2 VeiQxx+iQyycosQzz; (10) whereVis the volume of the cavity [50]. Our set-up facilitates coupling to the `z= 1 band of cav- ity modes, as the FI and SC are placed in field maxima as illustrated in Fig. 1. We will only consider variations of the in-plane part qof the general momenta Q, defined via Qq+^ez=Lz: (11) For this reason we will use the subscript qfor functions of Q where thezcomponent is locked to the `z= 1mode, e.g. !q!Q Q=q+^ez=Lz=cs Lz2 +q2: (12) The cavity itself contributes to the system Hamiltonian with the term Hcav 0X q&~!qay q&aq&; (13) where we have disregarded the zero-point energy, since it does not influence our results. FIG. 2. Illustration of the 123coordinate system. Qis the photon momentum vector, and qis its component in the xyplane.(single line) is the polar, and '(double line) the azimuthal angle associated withQin relation to the xyz basis. The 123 axes results from a rotation of the xyzaxes by an angle about theyaxis, followed by a rotation by an angle 'about the original zaxis. In the illustration, the 1axis points somewhat outwards and downwards, the 2axis points somewhat inwards and is confined to the original xyplane, and the 3axis aligns with Q. 2. Ferromagnetic insulator The Hamiltonian of the FI in the cavity is HFIH ex+Hext+HFIcav; (14) with HexJX hi;jiSi
Sj; (15a) HextgB ~BextX iSiz; (15b) HFIcavgB ~X iSi
Bcav(ri): (15c) The first term is the exchange interaction: J > 0is the ex- change interaction strength for a ferromagnetic insulator, Si is the spin at lattice site i, and only nearest neighbor interac- tions are taken into account, as indicated by the angle brack- ets. The next two terms are Zeeman couplings: gis the gyro- magnetic ratio, Bis the Bohr magneton, Bextis a strong (i.e.jBextj  j Bcavj) and uniform external magnetostatic field aligning the spins in the zdirection, and Bcav(ri)is the magnetic component of the cavity field at lattice site i. The corresponding position vector is ri. It is convenient to transition from the spin basis fSix;Siy;Sizgto a bosonic magnon basis fi;y ig. This is achieved with the Holstein–Primakoff transformation [51], which is covered in detail in Refs. [21, 52]. Each FI lattice site carries spin S. The aligning field Bext regulates the excitation energy of magnons (cf. Eq. (21)), hence a sufficiently strong field implies few magnons per lat- tice site, i.e. hy iii2S: (16)4 We can therefore Taylor-expand the Holstein–Primakoff transformation, leading to the relations Siz=~(Sy ii); (17) Sid~p 2S 2(di+ dy i); (18) whered=x;yandfx;yg=f1;ig. Now, upon Fourier-decomposing the magnon operators ri1pNFIX kkeik
ri; (19) we obtain the conventional expression for Hex+Hextin the magnon basis [52]: Hex+HextHFI 0X k~ky kk; (20) where we have introduced the magnon dispersion relation k2~JNS 11 NX eik
! +gB ~Bext:(21) Above,NFIis the total number of FI lattice points, N= 6is the number of nearest-neighbor lattice sites on a cubic lattice (neglecting edges and corners), and = aFI^ex;aFI^ey;aFI^ezare nearest-neighbor lattice vectors. The magnon momenta are k(2mFI x=lFI x;2mFI y=lFI y;0)(kx;ky;0); (22) wheremFI d=j NFI d1 2k ;:::;NFI d1j NFI d1 2k covers the first Brillouin zone (1BZ), with NFI dthe number of FI lat- tice points in direction d, andb
cthe floor function. Here we neglect thekzcomponent; only the kz= 0 modes enter our calculations due to the thinness of the FI film (cf. Eq. (26)). Note that the set of magnon momenta generally does not over- lap with that of photon momenta in Eq. (3). Observe further- more that the magnon energies (21) can easily be regulated experimentally by adjusting Bext. Proceeding to the interaction term, we deduce the magnetic cavity field Bcav(ri)across the FI, which is the curl of the gauge field at zLz: Bcav(ri) FI=rAcav(ri) FI =X qdi2 dqd^edsinqs ~ !qVeiq
ricoszi Lz(aq1+ay q1): (23) Above, d“inverts”dsuch that x=yandy=x, andzi is thezposition of lattice site i. Note that the photon mo- mentum component qdenters the sum with an inverted lower index. Observe that only the 1direction enters the expression, because AcavatzLzpoints purely along the zdirection.The2direction is by definition locked to the xyplane, and does therefore not contribute at zLz. Inserting Eqs. (17)–(19) and (23) into Eq. (15c), we find HFIcavX kdX q&gkq d(dk+ dy k)(aq1+ay q1); (24) and hence a complete FI Hamiltonian HFIHFI 0+HFIcav. Above, we defined the coupling strength gkq dgBqdi2 dsinqs S~NFI 2!qVDFI kqeiq
rFI 0: (25) DFI kqquantifies the degree of overlap between magnonic and photonic modes. An analogous quantity appears in the cavity– SC coupling in Sec. II B 3, so we define it via the general ex- pression DM lMqei(lMq)
rM 0 NMX i2Mei(lMq)
ri coszi Lz; M = FM sinzi Lz; M = SC lM;z0Y dsinc NM dmM d NM d`daM Ld :(26) HereM=fFI;SCgis a material index, lMrepresents ei- ther a magnon or an SC quasiparticle momentum, rM 0is the center position of lattice Mrelative to the origin, and the photon momentum numbers `d=`x;`ywere defined under Eq. (3). The latter, along with other SC quantities, are defined in Sec. II B 3. The sum over iis taken over either FI or SC lattice points, as indicated by M, and the last equality holds forNM d1. DFI kqreduces to a Kronecker delta kqonly whenLd= ld=aFINFI d, i.e. when the FI and the cavity share in- plane dimensions [53]. At the other end of the scale, when the FI becomes infinitely small, DFI kqreduces tok0, im- plying all cavity modes couple exclusively to the uniform magnon mode, which is often assumed in cavity implemen- tations [9, 10, 13, 29]. We assume this uniform coupling only in thezdirection, hence the factor lM;z0in Eq. (26) (thus kz= 0); the condition is that dM=2Lz1, withdMthe thickness of film M[54]. 3. Superconductor The SC Hamiltonian is HSC=Hsing+HBCS+Hpara; (27) with HsingX ppcy pcp0; (28a)5 HBCSX p
pcy p+P;"cy p+P;#+
 pcp+P;#cp+P;" ; (28b) HparaX dX jjd(rj)Adrj+Id+rj 2 ; (28c) Hsingis the single-particle energy, where pis the lattice- dependent electron dispersion, and cpandcy pare fermionic operators for an electron of lattice momentum pand spin. The momenta are discretized as p(2mSC x=lSC x;2mSC y=lSC y;2mSC z=lSC z)(px;py;pz); (29) wheremSC dandmSC zare defined analogously to mFI d(see be- low Eq. (22)), covering the 1BZ of the SC with NSC d(NSC z) the number of SC lattice points in direction d(z). HBCS is the BCS pairing term, with
pthe pairing po- tential. The leading order effect of applying an in-plane DC across the SC is to shift the center of the SC pairing poten- tial from p=0top=P, where 2Pis the generally finite center-of-mass momentum of the Cooper pairs [46, 55, 56]. The maximum value of Pis limited by the critical current of the superconductor. Hparais the paramagnetic coupling. jd(rj)is thedcompo- nent of the discretized electric current operator at lattice site j with the position vector rj, and is defined as [14] jd(rj)iaSCet ~X (cy j+Id;cjcy jcj+Id;): (30) Thezcomponentjzdoes not contribute to our Hamiltonian because the cavity gauge field is in-plane at zLz=2. Above, aSCis the lattice constant, eis the electric charge, tis the lattice hopping parameter, and cjandcy jare real-space fermionic operators for electrons with spin at lattice site j. They relate to cpandcy pvia cj=1pNSCX pcpeip
rj; (31) withNSCthe total number of SC lattice points. Further- more,Idrepresents a unit step in the ddirection with re- spect to lattice labels. For instance, if j= (1;1), then j+Ix= (1 + 1;1) = (2;1). Inserting Eqs. (2), (30) and (31) into Eq. (28c) yields Hpara=X pp0X q&gqpp0 &(aq&+ay q&)cy pcp0:(32) Here, we have introduced the coupling strength gqpp0 &aSCet ~s ~ !qVDSC pp0;qeiq
rSC 0
X d ei(pq=2)
dei(p0+q=2)
d Oq &d;(33) where daSC^edare in-plane primitive lattice vectors. DSC pp0;qis defined in Eq. (26), quantifying the degree of over- lap between two electron modes and a photon mode. It re- duces topp0;qonly when the cavity and the SC share in- plane dimensions, as is the case in Ref. [14]. As we move onto the imaginary-time (Matsubara) path in- tegral formalism in the next sections, it becomes convenient to eliminate creation–creation and annihilation–annihilation fermionic operator products. To this end, we absorb the BCS term (28b) into the diagonal term (28a) by a straight-forward diagonalization: Hsing+HBCS =X pcp+P;" cy p+P;#yp+P
p
 pp+Pcp+P;" cy p+P;# =X p p0 p1y Ep00 0Ep1 p0 p1 : (34) Here we introduced the Bogoliubov (SC) quasiparticle basis f pm; y pmg, withm= 0;1and dispersion relations Epm=1 2 p+Pp+P + (1)mq (p+P+p+P)2+ 4j
pj2 :(35) The elements upandvpof the basis transformation matrix are defined through [48] cp+P;"u p p0+vp p1; cy p+P;#v p p0+up p1: (36) Inserting the above into Eq. (34), one finds the relations
 pvp up=1 2[(Ep0Ep1)(p+P+p+P)]; (37a) jvpj2= 1jupj2=1 2 1p+P+p+P Ep0Ep1 ;(37b) which determine upandvp. RecastingHparain terms of this basis yields Hpara=X pp0X q&X mm0gqpp0 &mm0(aq&+ay q&) y pm p0m0;(38) where the coupling strength is now gqpp0 &mm0 gq;p+P;p0+P &upu p0+gq;pP;p0P &vpv p0gq;p+P;p0+P &upvp0gq;pP;p0P &vpup0 gq;pP;p0P &u pv p0+gq;p+P;p0+P &v pu p0gq;pP;p0P &u pup0+gq;p+P;p0+P &v pvp0! mm0: (39)6 This concludes the derivation of the terms entering the sys- tem Hamiltonian in terms of the various (quasi)particle bases. We now turn our focus to the construction of an effective FI theory. C. Imaginary time path integral formalism We now seek to extract the influence of the SC on the FI, in particular the anisotropy field induced across the FI. Diagonal- izing the Hamiltonian directly, as was done in Eq. (34), would in this case be very challenging, as it couples many more modes, and furthermore contains trilinear operator products. Since the external drives ( Bextand the DC) only give rise to equilibrium phenomena in our system, the Matsubara path in- tegral formalism of evaluating thermal correlation functions is valid [41]. This translates the evaluation into a path integral problem, which is very convenient for our purposes. The path integral approach facilitates aggregation of the influences of specific subsystems into effective actions, without explicit di- agonalization. On this note, for comparison, Cottet et al. [40] analyze a scenario in which the non-equilibrium Keldysh path integral formalism is used to analyze the net influence of a QED circuit on a cavity. The starting point is the imaginary time action SSFI 0+Scav 0+SSC 0+SFIcav int +ScavSC int =Z dX ky k~@k+X q&ay q&~@aq& +X pm y pm~@ pm+H : (40) is a temperature parameter treated as imaginary time, which relates to the thermal equilibrium density matrix exp(H=~), with~=kBTthe inverse temperature Tin units of time, andHthe system Hamiltonian. The dependence of the field operators on temperature ( ) is implied. In formu- lating the path integral, the magnon, photon and Bogoliubov quasiparticle operators have been replaced by eigenvalues of the respective coherent states [41]; i.e. the bosonic operators have been replaced by complex numbers, and the fermionic operators by Graßmann numbers. The magnons, photons and Bogoliubov quasiparticles are furthermore taken to be func- tions of[41]. The integral over is taken over the interval (0;]. Note that we assume the gap to be fixed to the bulk mean field value, and therefore do not include a gap action or integration in the partition function. We now replace the integral over by an infinite sum over discrete frequencies by a Fourier transform of the magnon, photon and Bogoliubov quasiparticle operators with respect to. The conjugate Fourier parameters are Matsubara fre- quencies: n=2n (41)for bosons, and !n=(2n+ 1) (42) for fermions, with n2Z. The transforms read k=1pX m m;kei m; (43a) aq&=1pX na n;q&ei n; (43b) pm=1pX !n !n;pmei!n: (43c) To avoid clutter, we introduce the 4-vectors k( m;k); (44a) q( n;q); (44b) p(!n;p); (44c) and the generally complex energies ~ki~ m+~k; (45a) ~!qi~ n+~!q; (45b) Epmi~!n+Epm: (45c) The actions in (40) then become SFI 0=X k~ky kk; (46a) Scav 0=X q&~!qay q&aq&; (46b) SSC 0=X pmEpm y pm pm; (46c) SFIcav int =X kdX q&gkq d&(dk+ dy k)(aq&+ay q&); (46d) ScavSC int =1pX q&X pmX p0m0gqpp0 &mm0(aq&+ay q&) y pm p0m0; (46e) where we introduced the coupling functions gkq d&gkq d&1 m; n; (47) gqpp0 &mm0gqpp0 &mm0!n0;!n n: (48) We additionally introduced a redundant Kronecker delta func- tion&1to the coupling (47), which will facilitate the gather- ing of interaction terms in Eq. (51). We will use the notation gandg for the magnitudes of the FI–cavity and cavity–SC coupling, respectively. We are now equipped to construct effective actions by in- tegrating out the photonic and fermionic degrees of freedom,7 to which end we will consider the imaginary-time partition function [41, 43] Zhvac;t=1jvac;t=1i =Z D[;y]Z D[a;ay]Z D[ ; y]eS=~; (49) where e.g. Z D[ ; y]Y pmZ D[ pm; y pm] (50) is to be understood as the path integrals over every Bogoliubov quasiparticle mode. D. Integrating out the cavity photons The order in which we integrate out the cavity and the SC is inconsequential. We will begin with the cavity, which can be integrated out exactly. We show that interchanging the order of integrations leads to identical results in Appendix A. We gather the interactions between the cavity and FI and SC, Scav int=X q;&[Jq&aq&+Jq&ay q&]; (51) where we have defined Jq&=X ksgkq d&(dk+ dy k) +1pX pp0X mm0gqpp0 &mm0 y pm p0m0: (52) These interaction terms are illustrated by the diagrams in the top panel of Fig. 3. Integrating out the cavity modes [41], we therefore get the effective action Se=X q&Jq&Jq& ~!q: (53) Inserting the expression for Jq&we get three different terms, Se=SFI 1+SSC 1+Sint, shown diagrammatically in the bottom panel of Fig. 3. The first term, SFI 1=X qkk0X &dd0gkq d&gk0q d0& ~!q (dk+ dy k)(d0k0+ d0y k0); (54) is a renormalization of the magnon theory due to interactions with the cavity, resulting in a non-diagonal theory. The second term, SSC 1=1 X qpp0 oo0X &mm0 nn0gqpp0 &mm0gqoo0 &nn0 ~!q y pm p0m0 y on o0n0; (55) Scav int:agη η + agγ γγ SFM 1 :Gcav SSC 1 :Gcav Sint :GcavFIG. 3. Feynman diagrams [57] of the bare cavity coupling to the FI and SC, and the resulting terms in the FI and SC effective actions after integrating out the cavity photons, where Gcavis the photon propagator. is an interaction term coupling four quasiparticles, similar to the term found in Ref. [14] for a normal metal coupled to a cavity, leading to superconducting correlations. Note that un- like the pairing term found in Ref. [14] via the Schrieffer– Wolff transformation, the term above is not limited to an off- resonant regime. In principle it could also lead to renormaliza- tion of the quasiparticle spectrum and lifetime. Since we are here concerned with the effects of the cavity and SC on the FI, we will neglect this term as it only leads to higher order corrections. Finally, we have the cavity-mediated magnon-quasiparticle coupling, Sint=1pX kpp0X dmm0Vkpp0 dmm0(dk+ dy k) y pm p0m0; (56) where we have defined the effective FI-SC interaction Vkpp0 dmm0=X q&gkq d&gqpp0 &mm01 ~!q+1 ~!q : (57) This term is generally nonzero, and we therefore see that the cavity photons lead to a coupling between the FI and SC, po- tentially over macroscopic distances. This means that the FI and SC will have a mutual influence on each other, possibly leading to experimentally observable changes in the two mate- rials. We therefore integrate out the Bogoliubov quasiparticles and calculate the effective FI theory below. We reiterate that the interaction is exact at this point, not a result of a perturba- tive expansion.8 E. Integrating out the SC quasiparticles — effective FI theory The full effective SC action comprises the sum SSC 0+SSC 1+ Sint. The second term is second order in g , but does not contain FI operators, and will therefore only have an indirect effect on the effective FI action. In a perturbation expansion of the effective FI action, the term SSC 1will therefore contribute higher order correction terms compared to Sint. We therefore neglect this term in the following, leading to the SC action SSCX pp0X mm0 y pm(G1)pp0 mm0 p0m0; (58) where we have defined G1=G1 0+ , with (G1 0)pp0 mm0=Epmpp0mm0; (59) pp0 mm0=1pX kdVkpp0 dmm0(dk+ dy k): (60) Integrating out the SC quasiparticles results in the effective FI action [41] SFI=SFI 0+SFI 1~Tr ln(G1=~): (61) The Green’s function matrix G1contains magnon fields, and will be treated perturbatively in order to draw out the lowest order terms in the effective FI theory. We expand the loga- rithm to second order in the FI–SC interaction, ln G1 ~ ln G1 0 ~ +G01 2G0G0; (62) whereG0is the inverse of G1 0. This expansion is valid when jG0j 1, meaningjgg =~!qEpmj 1, where we use shorthand notation for the couplings gandg between cav- ity photons and and fields respectively. The first term in Eq. (62) does not contain magnonic fields, and therefore does not contribute to the FI effective action [58]. The third term contains bilinear terms in magnonic fields, and gives a correc- tion to the magnon dispersion of order j[gg =~!q]2=Epmj, a factor ofj(g )2=~!qEpmjsmaller than the corrections con- tained inSFI 1, and will therefore also be neglected. Keeping only the second term, and using the fact that G0is diagonal in both quasiparticle type mand momentum p, we therefore get the effective FI action to leading order, SFI=X k~ky kkgBX kdhk d
r S 2(dk+ dy k) +X kk0dd0Qkk0 dd0(dk+ dy k)(d0k0+ d0y k0); (63) where we have defined the anisotropy field due to the coupling to the superconductor, hk d=~ gBr2 SX pmVkpp dmm Epm; (64)and a function Qkk0 dd0 X q&gkq d&gk0q d0& ~!q: (65) describing the cavity-mediated self-interaction in the ferro- magnetic insulator. III. RESULTS The main result of our work is the effective magnon ac- tion (63). The interaction with the cavity and the SC gives rise to linear and bilinear correction terms to the diagonal magnon theory, corresponding to an induced anisotropy field and cor- rections to the magnon spectra. To extract a specific quantity, we consider the leading order effect of coupling the FI to the SC via the cavity, namely the linear magnon term. Physically this can be understood as a contribution from an additional magnetic field trying to reori- ent the FI in a direction other than along the zaxis. We can see this explicitly if we Fourier transform the linear magnon term back to real space and imaginary time, SFI lin=gB ~Z dX riX dhd(ri;)Sid(); (66) where we have used the definition of the in-plane spin compo- nents in Eq. (18), and defined the real space anisotropy field components due to the interaction with the superconductor hd(ri;) =1pNFIX khk deik
ri: (67) Above, we introduced the 4-vector ri(;ri): (68) In order for the anisotropy field components to be real, we re- quirehk d= (hk d). Inserting the expressions for Epmand Vkpp dmmfrom Eqs. (45c) and (57) into Eq. (64), and performing the sum over the Matsubara frequencies [41], we get the fol- lowing expression for the Fourier transposed anisotropy field components, hk d=p NFI m0X q;d04aSCet ~!2qVLzqdqd0 jQj22 deiq
(rFI 0rSC 0) DFI k;qDSC 0;qeiqd0aSC=2Pd0; (69) where the dependence on the supercurrent comes in through the factor Pd=X p sin[(pd+Pd)aSC]jupj2 + sin[(pdPd)aSC]jvpj2 tanhEp0 2~: (70) Notice that the field is finite only for zero Matsubara fre- quency, meaning that it is time-independent (magnetostatic).9 It is possible to show that hk d= (hk d)by letting q!qin the sum in Eq. (69), and using DFI k;q= (DFI k;q),DSC 0;q= (DSC 0;q)from the definition in Eq. (26). Observe that in the case of no DC (i.e. P=0), the summand in Eq. (70) is odd inp, and the sum therefore zero, i.e., Pd= 0 ifPd= 0. Hence there is no anisotropy field induced across the FI in the absence of a supercurrent. This stresses the necessity of breaking the inversion symmetry of the SC in order to induce an influence on the FI. A. Special case: small FM FIG. 4. Illustration of the set-up used in the example given in Sec. III A. A small, square FI and SC are placed spaced apart in the yandzdirections inside a comparatively large cavity. Only a small portion of the cavity length in yis utilized as the contributions by the various mediating cavity modes add constructively only over short distances. The FI and SC are nevertheless separated by hundreds of µm, 2–5 orders larger than typical effectual lengths in proximity systems. The anisotropy field (67) generally gives rise to compli- cated, local reorientation of the FI spins. However, there are special cases in which it takes on a simple form. In partic- ular, assume the FI to be very small relative to the cavity, i.e.`xlFI x;`ylFI yLx;Ly. In this case, the FI sum (26) be- comes highly localized around k=0for the relevant ranges of`xand`y, which are limited by the other factors DSC 0qand (!qjQj)2found in Eq. (69). We may therefore set k=0. For a specified set of material parameters and dimensions, the validity is confirmed numerically. In this case, Eq. (67) thus reduces to hd=h0 dpNFI; (71) representing a uniform anisotropy field across the FI. In this limit we can simplify the expression for the anisotropy field components, hd=X q;d02aSCet ~!2qVLz2 dDFI 0;qDSC 0;qPd0qdqd0 jQj2  cosqxLsep xcosqyLsep ysinqxLsep xsinqyLsep y ; (72) where we have assumed eiqd0aSC=21, which is a good ap- proximation as long as the cavity dimensions far exceed thelattice constant and only low jqjcontribute to the sum, and used the fact that DM 0;q[Eq. (26)] is an even function in q. We have also defined the separation length Lsep d= (rFI 0rSC 0)
^ed. Assuming a finite separation between the FI and SC only in one direction, the last term in the above equation vanishes, making every remaining factor even in qd, except the product qdqd0ford6=d0. The sum over qtherefore picks out terms such that d=d0. In order to get a finite hdwe must, there- fore, have Pd6= 0, i.e., the supercurrent momentum must be finite in the direction d. Hence, in the case that the separation between the FI and SC is finite in only one direction, applying a supercurrent in the xdirection can only induce an anisotropy field in theydirection, and vice versa. We consider the specific case of a small, square FI and SC displaced along yandz(Fig. 4). In Fig. 5 we show numer- ically how the effective anisotropy field varies with the su- percurrent momentum in this special case, using Nb and YIG as material choices for the FI ( lFI x=lFI y= 10 µm) and SC (lSC x=lSC y= 50 µm,dSC= 10 nm ) films, respectively; see Table I. We use Python with the NumPy andMatplotlib li- braries for the numerics. We furthermore use the interpolation formula [59]
= 1:76kBTc0tanh(1:74p Tc0=T1) (73) for the superconducting gap, and a simple cubic tight-binding electron dispersion. With the FI and SC center points sepa- rated by 140µmin theydirection (meaning they are separated edge-to-edge by 115µmin-plane), we find an anisotropy field with a magnitude of .14µT(Fig. 5a). If the constraint on separating the FI and SC in-plane is eased, the maximum mag- nitude increases to 16µTin our specific example (Fig. 5b). We discuss the latter case in the concluding remarks. Two factors determine the inhomogeneous distribution of the responses seen in Fig. 5. First, the anisotropy field is nearly linear in the components Pdof the supercurrent mo- mentum, which is seen by expanding the anisotropy field (see Eq. (70)) around PdaSC= 0(note thatPcaSC0:001). This generally makes the response stronger for larger jPj, which is as expected, since it relies on breaking the p- inversion symmetry. This dependency is evident in Fig. 5. Second, the factor eiq
(rFI 0rSC 0)renders the anisotropy field very sensitive to the separation of the FI and SC center points in the in-plane directions. This factor expresses that cavity modes associated with a range of different in-plane momenta q(i.e., spatial oscillations) with a coherent amplitude at no in-plane separation ( rFI 0rSC 0= 0), become increasingly decoherent with increasing separation. Eventually, this deco- herence causes states in the SC to contribute oppositely, hence destructively, to the effective anisotropy field. The destructive addition at finite separation is limited by the range of low- q cavity modes that contribute to the mediated interaction un- til the coupling is suppressed by the factor DFI 0qDSC 0q=!2 qQ2, which in turn is determined by the dimensions of the three subsystems. For sufficiently small separations (determined by the contributing range of q), this oscillation is mild, and can be used to change the polarity of the anisotropy field without extinguishing the response. This is why the polarity of the response component hxchanges between Figs. 5a and 5b.10 It is furthermore clear by inspection of Eq. (70) that the main contributions to the anistotropy field come from states near the Fermi surface. Series-expanding the expression inP, most terms are seen to cancel due the odd symme- try in pthat was remarked below Eq. (70). The strongest asymmetry caused by Pis seen to originate from the factor sin [(pd0+Pd0)aSC]jupj2+ sin [(pd0Pd0)aSC]jvpj2in the summand, due to the step-like nature of jupj2andjvpj2near the Fermi surface. This is as expected, since we consider in- teractions involving the scattering of SC quasiparticles, hence the low-energy events are concentrated near the Fermi surface. (a) (b) FIG. 5. The magnitude and direction (arrows) of the effective anisotropy field [Eq. (72)] at T= 1 K as a function of the super- current momentum P, for the simple case of a small FI ( lFI x=lFI y= 10µm) relative to the cavity ( Lx=Ly= 10 cm ,Lz= 0:1 mm ). The SC dimensions are lSC x=lSC y= 50 µm, with a depth of dSC= 10 nm . The FI and SC center points are separated by (a)Lsep y= 140 µmand (b) nothing (placed directly over each other). Observe the change in both the strength and direction of the anisotropy field. The plots were produced using Python with the NumPy andMatplotlib libraries.TABLE I. Table of numerical parameter values. YIG (FI) Nb (SC) aFI 1:240 nm [60] aSC 0:330 nm [61] Tc0 6 K[49] t 0:35 eVa Pc 3:1107m1b EF 5:32 eVc[61] aBased on the tight-binding expression t=~2=2ma2 SC[14], with mthe effective electron mass. bBased onPc=jcm=~ens[46], with an estimated critical currentjc= 4 MA=cm2[62], and a superfluid density ns= m= 0e22[48] with a penetration depth = 200 nm [49]. cFermi energy for Nb. Does not appear explicitly in Eq. (72), but is used in the electron dispersion. IV . CONCLUDING REMARKS In this paper, we have calculated the cavity-mediated cou- pling between an FI and an SC by exactly integrating out the cavity photons. The main result is the effective FI action (63), in which linear and bilinear magnon terms appear in addition to the diagonal terms. These respectively correspond to an induced anisotropy field, and corrections to the magnon spec- tra. In contrast to conventional proximity systems, the cavity- mediation allows for relatively long-distance interactions be- tween the FI and the SC, without destructive effects on order parameters associated with proximity systems, such as pair- breaking magnetic fields. The separation furthermore facili- tates subjection of the FI and the SC to separate drives and temperatures. In contrast to common perturbative approaches to cavity-mediated interactions involving the Schrieffer–Wolff transformation [9, 14, 21] or Jaynes–Cummings-like mod- els [12, 13, 39], the path-integral approach allows for an ex- act integrating-out of the cavity, without limitations to off- resonant regimes. This carries the additional advantage of allowing for magnon–photon hybridization; that is, we are not theoretically limited to regimes of weak FI–cavity Zeeman coupling. We furthermore take into account that the finite and different FI, cavity and SC dimensions enable interactions be- tween large ranges of particle modes, which is neglected in various preceding works [9, 10, 13, 14, 22, 29, 30], although its importance has been emphasized by both experimentalists [30] and theorists [22]. In an arbitrary practical example, we estimate numerically the effective anisotropy field induced by leading-order inter- actions across a small YIG film (FI) ( lFI x=lFI y= 10 µm) due to mediated interactions with an Nb film (SC) ( lSC x= lSC y= 50 µm,dSC= 10 nm ). We find it is .14µT, medi- ated across 130µmedge-to-edge accounting for both in-plane and out-of-plane separation, inside a 10 cm10 cm0:1 mm cavity (Fig. 5a). With out-of-plane coercivities in nm-thin Bi- doped YIG films reportedly as low as 300µT[63], this result is expected to yield an experimentally appreciable tilt in the FI spins. The separation is 2–5 orders of magnitude greater than the typical length scales of influence in proximity systems, and facilitates local subjection to different drives and temper- atures. The main contributions from the SC originate from a11 narrow vicinity of the Fermi surface determined by the Cooper pair center-of-mass momentum 2P. The response is very sen- sitive to the in-plane separation of the FI and SC center points due to the spatial decoherence of the mediating cavity modes over distances, which in turn depends on the dimensions of the FI, cavity and SC. For this reason, the in-plane separation of FI and SC was much smaller than the cavity width. In Appendix B we have included the calculation of the anisotropy field when placing the SC at the magnetic antin- ode atz= 0. Since the vector potential is purely out of plane in this case, the paramagnetic coupling is zero, and we there- fore couple the cavity to the SC via the Zeeman coupling. As shown in the appendix, this results in a much weaker cou- pling and therefore much smaller anisotropy field. This can be understood by comparing the effective fields the SC cou- ples to in the two cases. The strength of the Zeeman cou- pling is proportional to qA, which for the lowest cavity modes gives a field strength proportional to jAj=L. How- ever, for the paramagnetic coupling, the effective field is pro- portional to p
A. In both cases, the main contribution to the anisotropy field originates from a narrow vicinity of the Fermi level, the extent of which is determined by the magni- tude of the symmetry-breaking supercurrent (electric antin- ode) or applied field (magnetic antinode). Thus, we have a paramagnetic coupling proportional to pFjAj, wherepF is the Fermi momentum. A Fermi energy of 5:32 eV gives pF1010m11=Lfor cavities with lengths in the mm tocmrange. Together with the fact that the contributing com- ponents of Aare larger for low jqjat the electric antinode compared with the magnetic antinode, the difference in length scales leads to a much larger paramagnetic coupling between cavity and SC compared to the Zeeman coupling, resulting in a much larger effective FI–SC coupling and anisotropy field. One important constraint in our model that can potentially be eased, is that the FI and the SC cannot overlap in-plane. In this case, we found a stronger response (cf. Fig. 5b). This was assumed in order to enable the FI to be subjected to the align- ing magnetostatic field Bextwithout affecting the SC, analo- gously to the experimental set-up in Refs. [12, 13]. Combined with the eventually destructive contributions of various cav- ity modes over finite in-plane distances that limited us to us- ing only a fraction of the cavity width in our example, this leads to significant constraints on the dimensions and rela- tive placements of the FI and SC. However, Ref. [64] reports out-of-plane critical fields of nm-thin Nb films of roughly 1– 4 T, while Ref. [63] reports out-of-plane coercivities in nm- thin Bi-doped YIG films of roughly 3104T. An aligning field can therefore be many orders of magnitude smaller than the SC critical field with appropriate material choices. One would then expect the effect of Bexton the SC to be negli- gible. However, we have not considered here the subsequent effect of the SC on the spatial distribution of Bext, which was taken to be uniform across the FI. Moreover, the Pearl length criterion, which greatly lim- its SC dimensions, can potentially be disregarded if the odd psymmetry of the anisotropy field (64) is broken by other means than a supercurrent. A candidate for this is taking into account spin–orbit coupling on the SC and subjecting it to aweak (non-pair breaking) magnetostatic field. Furthermore, in our set-up, we have considered coupling to the quasiparticle excitations of the SC. This has partly been motivated by the prospect of using the FI to probe detailed spin and momentum information about the SC gap, which would require an extension of our present model. Another in- teresting avenue to explore is coupling directly to the gap by considering fluctuations from its mean-field value. This has been explored for an FI–SC bilayer, where the Higgs mode of the SC couples linearly to a spin exchange field [65]. This has a significant impact on the SC spin susceptibility in a bilayer set-up. Despite coupling to the quasiparticles, we find that the anisotropy field magnitude nearly constant at low tempera- tures, and rapidly decreases to zero near the critical tempera- ture. This can be understood from the fact that the symmetry- breaking supercurrent momentum enters the system Hamilto- nian via the gap (cf. Eq. (28b)). Hence, when the gap van- ishes, so does the quantity that breaks the symmetry. On the other hand, for temperatures substantially below Tc0, the gap varies little with temperature; the anisotropy field becomes close to constant, with a magnitude depending on the momen- tum associated with the inversion symmetry-breaking current P. In the normal state, the DC through the SC induces a sur- rounding magnetostatic field, by the Biot–Savart law. This differs from the response in the superconducting state by in- stead being appreciable above Tc0, and by its spatial distribu- tion; for instance, the magnetostatic field cannot reverse the field direction as observed between Fig. 5a and 5b. Lastly, it is seen from Eq. (64) that the SC quasiparticle modes uniformly affect the anisotropy field in our current set- up, as the sum over fermion momenta pcan be factored out from the sum over photon momenta q. This limits the reso- lution of SC features in the anisotropy field, and by extension the FI. However, to higher order in the calculations, the quan- tityGqq0 &&0defined in Eq. (A6) enters, with sums over fermion momenta pandp0that are inseparable from the cavity mo- menta qandq0. This quantity is a candidate for extracting more features of the SC via the FI. ACKNOWLEDGMENTS We acknowledge funding via the “Outstanding Academic Fellows” programme at NTNU, the Research Council of Nor- way Grant number 302315, as well as through its Centres of Excellence funding scheme, project number 262633, “QuS- pin”. Appendix A: Integrating out the SC first The order in which we integrate out the cavity and the SC is inconsequential. We show this here by integrating out the SC first, starting from the partition function (49).12 We introduce the interaction matrix Gwith elements Gpp0 mm01pX q&gqpp0 &mm0(aq&+ay q&); (A1) and furthermore the diagonal matrix Ewith elements Epp0 mm0Epmpp0mm0: (A2) Hence the action involving the SC can be written as SSC 0+ScavSC int =X pmX p0m0(E+G)pp0 mm0 y pm p0m0:(A3) The part of the partition function (49) which depends on the SC is a Gaussian integral, and can now be written as [41] ZSCZ D[ ; y] exp2 41 ~X pmX p0m0(E+G)pp0 mm0 y pm p0m03 5 exp Tr E1GE1GE1G=2 : (A4) In the last line, we neglected a factor exp Tr ln (E=~)that is constant with respect to the integration variables, and ex- panded another logarithm to second order in jE1Gj. Hence, integrating out the SC to second order in the cav–SC coupling yields an effective action Scav 1~Tr E1GE1GE1G=2 =~pX q&X pmgqpp &mm Epm(aq&+ay q&) +X q&X q0&0Gqq0 &&0(aq&+ay q&)(aq0&0+ay q0&0);(A5) where we introduced the coefficient Gqq0 &&0~ 2X pmX p0m0gqpp0 &mm0gq0p0p &0m0m EpmEp0m0: (A6)We now proceed to isolate the photonic terms and integrate out the cavity, i.e., we will perform the integral ZcavZ D[a;ay]eScav=~; (A7) where the effective cavity action is ScavScav 0+Scav 1+SFIcav int: (A8) To this end, we introduce the current operator Jq&X kdGkq d&(dk+ dy k) +sq&; (A9) and perform a shift of integration variables aq&!aq&+Jq&=~!q; (A10a) ay q&!ay q&+Jq&=~!q: (A10b) The quantities Gkq d&(to be distinguished from Gqq0 &&0) andsq& are coefficients of linear photon terms to be determined. We now require that the shifts (A10a)–(A10b) absorb the explicit linear photon terms in the action (A8), leaving only bilinear and constant terms in the shifted variables. This leads to self-consistency equations for Gkq d&andsq&. However, to second order injE1Gj, it can be shown that only the lowest- order expressions for Gkq d&andsq&affect the anisotropy field to be extracted at the end, cf. Sec. III. These are Gkq d&=gkq d&; (A11) sq&=~pX pmgqpp &mm Epm: (A12) Hence, the action (A8) can be written as Scav=Scav bil+Scav con (A13) where Scav bilX q&~!qay q&aq&+X q&X q0&0Gqq0 &&0(aq&+ay q&)(aq0&0+ay q0&0);(A14) Scav conX q&Jq&Jq& ~!q+X q&X q0&0Gqq0 &&0Jq&Jq0&01 ~!q+1 ~!q1 ~!q0+1 ~!q0 : (A15) Scav bilcontains all bilinear terms with respect to the shifted vari- ables, andScav conall constant terms. Returning to the integral (A7), by Eq. (A13), we now have Zcav=Z D[a;ay]eScav=~=eScav con=~Z D[a;ay]eScav bil=~: (A16)The integrand is now independent of magnons, and therefore inconsequential to the physics of the ferromagnetic insulator. We can therefore neglect the integral, leaving only the expo- nential prefactor. We are thus left with an effective FI partition13 function ZFIZ D[;y]eSFI=~; (A17) where the effective FI action is SFISFI 0+Scav con: (A18)Neglecting magnon-independent terms, SFIreads, after some rewriting, SFI=X k~ky kk+X kdX k0d0Qkk0 dd0(dk+ dy k)(d0k0+ d0y k0)gBX kdhk d
r S 2(dk+ dy k): (A19) Above, we introduced Qkk0 dd0X q&2 4gkq d&gk0q d0& ~!q+X q0&0Gqq0 &&01 ~!q+1 ~!q1 ~!q0+1 ~!q0 gkq d&gk0q0 d0&03 5; (A20) hk d=~ gBr2 SX pmVkpp dmm Epm; (A21) which to leading order in the paramagnetic coupling are indeed the same as Eqs. (64) and (65). Appendix B: SC at magnetic antinode FIG. 6. Illustration of the set-up with the SC placed at the mag- netic antinode. The SC is subjected to an aligning external in-plane magnetic field BSC ext. This set-up is otherwise identical to the one illustrated in Fig. 1. To compare our results for the FI-SC coupling with the SC placed at the electric antinode, we examine what happens when we place the superconductor at a magnetic maximum at z0, cf. Fig. 6. In this case the vector potential Apoints purely in the zdirection, and therefore does not couple to the SC via the paramagnetic coupling term used above. We there- fore couple the SC to the cavity via the Zeeman coupling, and calculate the resulting anisotropy field across the FI. For the setup considered in the main text, it was necessary to break the inversion symmetry to get a finite anisotropy field, achieved, for instance, by applying a DC current. For the present setup, it is necessary to break the in-plane spin rotation symmetry, which can be achieved by applying an in-plane magnetic field to the SC. This becomes evident when considering the cou- pling between the cavity and SC. Placing the SC at z0, thecavity magnetic field is purely in-plane, pointing in the oppo- site direction to the field at z=Lz[Eq. (23)], resulting in a coupling term, SZeeman =X qpp0X 0gqpp0 0(aq1+ay q1)cy pcp00; (B1) with interaction matrix gqpp0 0= n;!n!0n s ~2 B !qVDSC pp0;qeiq
rSC 0isinq(q)0
^ez: (B2) This interaction alone would lead to a SC-cavity coupling that is off-diagonal in quasiparticle basis. The anisotropy field, corresponding to the diagram for Sintin Fig. 3 with connected quasiparticle lines will therefore be exactly zero unless one breaks the spin-rotation symmetry by an in-plane magneto- static field BSC ext. The latter can for example be experimen- tally realized using external coils, as suggested for Bext. In that case the quasiparticle bands are spin-split, resulting in the SC term SSC 0=X pn(i~!n+Epn) y pn pn; (B3) with the four quasiparticle bands Epn= (1)bn=2cEp+ (1)nH; (B4) withEp=q 2p+j
pj2,n2[0;1;2;3]andH=jBBSC extj. The bands are independent of in-plane direction of the field14 BSC ext, with the directional dependence entering through the coupling between the quasiparticles and the cavity photons, SSCcav int =1 2pX qppX nn0gqpp0 nn0(aq1+ay q1) y pn p0n0;(B5)where we have defined the interaction matrix in the Bogoli- ubov quasiparticle basis gqpp0 nn0=1 2gqpp0 "#ei [uy pup0+vpvy p0][z+iy] [uy pvp0vpuy p0][0x] [vy pup0upvy p0][0+x] [vy pvp0+upuy p0][ziy]! nn0 1 2gqpp0 #"ei [uy pup0+vpvy p0][ziy] [uy pvp0vpuy p0][0+x] [vy pup0upvy p0][0x] [vy pvp0+upuy p0][z+iy]! nn0; (B6) where0is the 22identity matrix, and is the angle of the in-plane field relative to the xaxis. We have also defined the functions up=eips 1 2 1 +p Ep ; (B7a) vp=eips 1 2 1p Ep ; (B7b) which satisfyjupj2+jv2 pj= 1. Here 2pis the phase of the order parameter. Following the same procedure of integrating out the cav- ity photons and quasiparticles in the SC, we get an expres- sion identical to Eq. (63), with the only change coming in the anisotropy field, which is now defined as hk d~p2SgBX pnVkpp dnn Epn; (B8)with Vkpp0 dnn0=X qgkq d1gqpp0 nn01 ~!q+1 ~!q : (B9) The additional factor of 1=2in the definition of hk dis due to the field integral resulting in the Pfaffian of the antisymmetrized Green’s function in this case, which is the square root of the determinant [66]. The reason for this is the necessity of an ex- panded Nambu spinor, which contains both creation and anni- hilation operators of both types of quasiparticles when includ- ing an in-plane field [67]. Inserting Eqs. (B6) and (B9) into Eq. (B8) and performing the sum over fermionic Matsubara frequencies [41], we get hk d=p m0p 2SgBX qpgkq d ~!q[gqpp "#ei+gqpp #"ei]  tanh(Ep+H) 2~tanh(EpH) 2~ ;(B10) where we have used the fact that !qis even in q. Here it is clear that the anisotropy field is exactly zero when the in-plane field is zero, since the last two terms exactly cancel in that case. Moreover, since the anisotropy field is independent of the frequency m, we define the time-independent anisotropy fieldhk d=P mhk dei m=p. Inserting the expressions forgkq dandgqpp 0from Eqs. (25) and (B2) we get hk d=BpNFI VX qeiq
(rFI 0rSC 0)DFI kqDSC 0qsin2q !2qqd2 d[qycosqxsin]X p tanh(Ep+H) 2~tanh(EpH) 2~ : (B11) We focus on the anisotropy field averaged across the FI, hhdi=P ihd(ri;)=NFI=P iP khk deik
ri=N3=2 FI=h0 d=pNFI (cf. Eq. (67)), rewrite the first sum such that it becomes dimensionless, and transform the second sum into an integral using a15 free electron gas dispersion k=~2p2=2m. Assuming cavity dimensions Lx=Ly=Land ans-wave gap, we get hhdi=BVSC(m
0)3=2 p 22~3c2VX qeiq
(rFI 0rSC 0)DFI 0qDSC 0q`d2 d[`ycos`xsin][`2 x+`2 y]  `2x+`2y+ L 2Lz22 max=
0Z =
0dxr x+
0 tanh1:764Tcp x2+j
=
0j2+H=
0 2Ttanh1:764Tcp x2+j
=
0j2H=
0 2T : (B12) HereVSCand
0are the volume and zero temperature gap of the superconductor, respectively, and mthe electron mass. `x and`yare integer indexes corresponding to cavity momen- tumq. From the above expression we expect terms even in`dto dominate, resulting in the anisotropy field and ex- pectation values of the in-plane spin components to have a dependence given by hk x hSixi / cosandhk y hSiyi / sin. This is in good agreement with numer- ical solutions of Eq. (B12) in an arbitrary practical exam- ple, as shown in Fig. 7. The results were obtained using the Python libraries NumPy andMatplotlib , and sub-package scipy.integrate . Notice, however, that the magnitude of the anisotropy field is very small, on the order of 109T. This is several orders of magnitude smaller than the previously considered setup, and we do not expect this to be a measurable effect. Here we have neglected the effect of an in-plane finite separation between the SC and FI by placing them directly above each other. A finite separation would further reduce the anisotropy field. At zero temperature the two hyperbolic tangent functions in Eq. (B12) are always equal to one, as long as H <
0. Since the field must be below the critical field Hc0=
0=p 2 in the superconducting state, the two terms in the integral al- ways cancel exactly at zero temperature. On the other hand, in the case of temperatures just above the critical tempera- ture,T&Tc, and;max> H , we get the analytical re- sult4Hp=
3=2 0for the integral, assuming that the main contribution to the integral comes from energies close to the Fermi level. Hence we expect the anisotropy field to in- crease from zero to the normal state value as temperature in- creases towards Tc, and thathhdiincreases linearly with ap- plied field in the normal state. This is found to be in good agreement with numerical results, see the inset in Fig. 7 for jHj> Hc. In the numerical calculations we have assumed ;max
0, and that the gap’s dependence on temper- ature and applied field is described by Eq. (73) multiplied withp 1(H=Hc)2[59, 68], and the critical field depends on temperature as Hc=Hc0[1(T=Tc0)2][48], where Tc0 is the critical temperature for zero field. Below the critical temperature and field, the field-dependence of the anisotropy field is more complicated due to the additional effect of re- ducing the superconducting gap, see inset in Fig. 7. The dif- ference in temperature and applied field-dependence of the anisotropy field between the normal and superconducting state -10 -5 0 5 10 Hx[T]-10-50510Hy[T] 0.000.250.500.751.001.251.50 |⟨h⟩|[T]×10−9 1010−9HcFIG. 7. Absolute value (contour plot) and direction (arrows) of the averaged anisotropy field as a function of applied field strength and direction. The anisotropy field points opposite the applied field over the SC, following a cosandsindependence for the xandycom- ponent respectively. The inset shows the absolute value of the in- plane projection as a function of the field strength. The temperature is set toT= 0:5Tc0. The cavity dimensions are Lx=Ly=L= 10 cm andLz= 1 mm , and the FI and SC have sides of length 0:001Lin thexandydirections, and are placed at the center of the cavity. The thickness of the SC is dSC= 10 nm . could therefore in principle be a way of detecting the onset of superconductivity without directly probing the supercon- ductor, though the anisotropy field calculated in this arbitrary example is too small to be detectable. Appendix C: Linear terms as an anisotropy field In this appendix, we take a closer look at the interpretation of the linear magnon terms as interactions with an effective anisotropy field. Consider an FI in an inhomogeneous applied field, H=JX hi;jiSi
SjX iHi
Si: (C1) Above, Hi= (Hx i;Hy i;Hz)is the inhomogeneous external field, withHzassumed homogeneous and much larger than16 Hx i;Hy i. We therefore assume ordering in the zdirection when performing the Holstein–Primakoff transformation, re- sulting in the Fourier-transformed Hamiltonian H=E0+X kh ~ky kkhky kh kki : (C2) Here~kis the dispersion defined in Eq. (21), the classical ground state energy is E0=~SNFI[J~SN+Hz]; (C3) and the momentum-dependent in-plane magnetic energy hk=r S 2NFI~X i(Hx i+iHy i)eik
ri: (C4) Since the applied field has in-plane components, the zdi- rection is not the exact ordering direction in the ground state, leading to a non-diagonal Hamiltonian with linear terms. To get rid of these terms, we translate the fields according to k!k+tk; y k!y k+t k;(C5) and require that linear terms cancel. Translating the fields leads to the Hamiltonian H!E0+X kn ~ky kk+ [~ktkhk]y k + [~kt kh k]k+~kt ktkhkt kh ktko ; (C6)and we therefore require tk=hk ~k: (C7) The resulting diagonal Hamiltonian is H=E0+X k[~ky kk~kt ktk]: (C8) The last term in the above equation results in a renormal- ization of the classical ground state, E0!E0X k~kt ktk =E0X i;j;kS~2(Hx i+iHy i)(Hx jiHy j)2eik
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2022-09-19

A recent proof of concept showed that cavity photons can mediate superconducting (SC) signatures to a ferromagnetic insulator (FI) over a macroscopic distance [Phys. Rev. B, 102, 180506(R) (2020)]. In contrast with conventional proximity systems, this facilitates long-distance FI$\unicode{x2013}$SC coupling, local subjection to different drives and temperatures, and studies of their mutual interactions without proximal disruption of their orders. Here we derive a microscopic theory for these interactions, with an emphasis on the leading effect on the FI, namely, an induced anisotropy field. In an arbitrary practical example, we find an anisotropy field of $14 \unicode{x2013} 16$ $\mu$T, which is expected to yield an experimentally appreciable tilt of the FI spins for low-coercivity FIs such as Bi-YIG. We discuss the implications and potential applications of such a system in the context of superconducting spintronics.

Cavity-mediated superconductor$\unicode{x2013}$ferromagnetic insulator coupling

2209.09308v2

Coherent long-range transfer of angular momentum between magnon Kittel modes by phonons K. An,1A.N. Litvinenko,1R. Kohno,1A.A. Fuad,1V. V. Naletov,1, 2L. Vila,1 U. Ebels,1G. de Loubens,3H. Hurdequint,3N. Beaulieu,4J. Ben Youssef,4N. Vukadinovic,5G.E.W. Bauer,6A. N. Slavin,7V. S. Tiberkevich,7and O. Klein1, 1Univ. Grenoble Alpes, CEA, CNRS, Grenoble INP, Spintec, 38054 Grenoble, France 2Institute of Physics, Kazan Federal University, Kazan 420008, Russian Federation 3SPEC, CEA-Saclay, CNRS, Universit e Paris-Saclay, 91191 Gif-sur-Yvette, France 4LabSTICC, CNRS, Universit e de Bretagne Occidentale, 29238 Brest, France 5Dassault Aviation, Saint-Cloud 92552, France 6Institute for Materials Research and WPI-AIMR and CSRN, Tohoku University, Sendai 980-8577, Japan 7Department of Physics, Oakland University, Michigan 48309, USA (Dated: March 13, 2020) Abstract We report ferromagnetic resonance in the normal conguration of an electrically insulating mag- netic bi-layer consisting of two yttrium iron garnet (YIG) lms epitaxially grown on both sides of a 0.5 mm thick non-magnetic gadolinium gallium garnet (GGG) slab. An interference pattern is observed and it is explained as the strong coupling of the magnetization dynamics of the two YIG layers either in-phase or out-of-phase by the standing transverse sound waves, which are excited through the magneto-elastic interaction. This coherent mediation of angular momentum by circu- larly polarized phonons through a non-magnetic material over macroscopic distances can be useful for future information technologies. 1arXiv:1905.12523v3 [cond-mat.mes-hall] 12 Mar 2020The renewed interest in using acoustic oscillators as coherent signal transducers [1{3] stems from the extreme nesse of acoustic signal transmission lines. The low sound atten- uation factor abenets the interconversion process into other wave forms (with damping s) as measured by the cooperativity, C= 2=(2as) [4, 5], leading to strong coupling as dened by C>1 even when the coupling strength is small. Here we present experimen- tal evidence for coherent long-distance transport of angular momentum via the coupling to circularly polarized sound waves that exceeds previous benchmarks set by magnon diusion [6{8] by orders of magnitude. The material of choice for magnonics is yttrium iron garnet (YIG) with the lowest mag- netic damping reported so far [9, 10]. The ultrasonic attenuation coecient in garnets is also exceptional, i.e.up to an order of magnitude lower than that in single crystalline quartz [11, 12]. Spin-waves (magnons) hybridize with lattice vibrations (phonons) by the magnetic anisotropy and strain dependence of the magneto-crystalline energy [13{18]. Although often weak in absolute terms, the magneto-elasticity leads to new hybrid quasiparticles (\magnon polarons") when spin-wave (SW) and acoustic-wave (AW) dispersions (anti)cross [19{21]. This coupling has been exploited in the past to produce microwave acoustic transducers [22, 23], parametric acoustic oscillators [24] or nonreciprocal acoustic wave rotation [25, 26]. Recent studies have identied their benecial eects on spin transport in thin YIG lms by pump-and-probe Kerr microscopy [27, 28] and in the spin Seebeck eect [29]. The adiabatic conversion between magnons and phonons in magnetic eld gradients proves their strong coupling in YIG [30]. But phonons excited by magnetization dynamics can also transfer their angular momen- tum into an adjacent non-magnetic dielectrics [32, 33]. When the latter acts as a phonon sink, the \phonon pumping" increases the magnetic damping [34]. The substrate of choice for YIG is single crystal gadolinium gallium garnet (GGG) which in itself has very long phonon mean-free path [35, 36] and small impedance mismatch with YIG [37], raising the hope of a phonon-mediated dynamic exchange of coherence through a non-magnetic insulating layer [34]. Here we report ferromagnetic resonance experiments (FMR) of a \dielectric spin-valve" stack consisting of half a millimeter thick single-crystal GGG slab coated on both sides by thin YIG lms. We demonstrate that GGG can be an excellent conductor of phononic angular momentum currents allowing the coherent coupling between the two magnets over 2FIG. 1. (Color online) a) Schematic and picture of the ferromagnetic resonance (FMR) setup. A butter y shaped stripline resonator [31] with 0.3 mm wide constriction is in contact with the bottom layer of the YIG1( d= 200 nm)jGGG(s= 0:5 mm)jYIG2(d= 200 nm) \dielectric spin-valve" stack. The microwave antenna can be tuned in or out of its fundamental resonance (5 :11 GHz) as shown in the re ectivity spectrum b). c) Schematic of the coupling between the top (red) and bottom (blue) YIG layers by the exchange of coherent phonons: the magnetic precession m+generates a circular shear deformation u+of the lattice that can be tuned into a coherent motion of all elds. Constructive/destructive interference between the dynamics of the two YIG layers occurs for even/odd mode numbers ncausing d) a contrast
in the absorbed microwave power ( Pabs) between tones separated by half a phonon wavelength. e) Density plot of the spectral modulation of Pabsproduced by Eq.(1) when magnetic bi-layers are strongly coupled to coherent phonon modes. The orange/green dots indicate the spectral position of the even/odd acoustic resonances. millimeter distance. Figure 1a illustrates the experimental setup in which an inductive antenna monitors the coherent part of the magnetization dynamics. The spectroscopic signature of the dynamic coupling between the two YIG layers is a resonant contrast pattern as a function of microwave frequency (see intensity modulation along FMR2 in Figure 1e). Before turning to the experimental details, we sketch a simple phenomenological model that captures the dynamics of the elds as described by the continuum model for magneto- elasticity with proper boundary conditions [34]. The perpendicular dynamics of a trilayer 3with in-plane translational symmetry can be mapped on three coupled harmonic oscillators, viz. the Kittel modes of the two magnetic layers mi=1;2and then-th mechanical mode, un, in the dielectric, which obey the coupled set of equations (!s!1+js)m+ 1= 1u+ n=2 +1h+(1a) (!s!2+js)m+ 2= 2u+ n=2 +2h+(1b) (!s!n+ja)u+ n= 1m+ 1=2 + 2m+ 2=2 (1c) Here!n=(2) =v=n, wherevis the AW velocity and n=2 = (2d+s)=nis a half wavelength that ts into the total sample thickness 2 d+s, withnbeing an integer (mode number). The dynamic quantities m+ i= (mx+jmy)iare circularly polarized magnetic complex amplitudes (jbeing the imaginary unit) precessing anti-clockwise around the equilibrium magnetization at Kittel resonance frequencies !16=!2. In our notation s=aare the magnetic/acoustic relaxation rates [38] and the constants iandiare the magneto-elastic interaction and inductive coupling to the antenna, respectively. Coherence eects between m1andm2can be monitored by the power Pabs=iIm(h?mi) as a function of the microwave frequency !sof the driving eld with circular amplitude h+[39]. Note that Eq.(1) holds when the characteristic AW decay length exceeds the lm thickness (see below). The acoustic modes with odd and even symmetry couple with opposite signs, i.e. 2= (1)n 1(see Figure 1c), which aects the dynamics as sketched in Figure 1d. When nis odd (even), the top layer returns (absorbs) the power from the electromagnetic eld, because the phonon amplitude is out-of(in) phase with the direct excitation, corresponding to destructive (constructive) interference. In other words, the phonons pumped by the dynamics of the layer 1 are re ected vs. absorbed by layer 2. According to Eq.(1), a contrast
should emerge between tones separated by half a wavelength. This is illustrated in Figure 1e by plotting the calculated modulation of the magnetic absorption when two Kittel modes with slightly dierent resonance frequencies and dierent inductive coupling to the antenna interact via strong coupling to coherent phonons (see below values in Table.(I)). In the gure the eect is more visible around the resonance of the layer with weaker coupling 2< 1to the antenna (FMR2, red dashed line) since, according to the model, the amplitude of the contrast is proportional to the amplitude ratio of the microwave magnetic elds felt by the two YIG layers:
/1=2. We employ here a stripline with width (0 :3 mm) that couples strongly to the lower layer YIG1, while still allowing to monitor the FMR absorption of YIG2.[40] 4FIG. 2. (Color online)a) Microwave absorption spectra of a YIG(200 nm) jGGG(0.5 mm) crystal, revealing a periodic modulation of the intensity interpreted as the avoided-crossing between the FMR mode (see blue arrow) at !1= 0(H0M1), and thenthstanding (shear) AW nresonances across the total thickness (horizontal dash lines in orange and green) at !n=nv= (d+s) . The right panels (b,c,d) show the intensity modulation for 3 dierent cuts (blue, magenta and red) along the gyromagnetic ratio ( i.e.parallel to the resonance condition). The solid lines in the 4 panels are ts by the oscillator model (cf. Eq.(1) with t values in Table.(I)). Figure 1a is a picture of the bow-tie =2-resonator (with re ectivity spectrum shown in Figure 1b) with which we perform spectroscopy around 5 GHz. The later fullls the \half- wave condition" of the phonon relative to the YIG thickness that maximizes the phonon pumping [34]. The sample was grown by liquid phase epitaxy, i.e. by immersing a GGG monocrystal substrate with thickness s= 0:5 mm and orientation (111) into molten YIG. The concomitant growth leads to nominally identical YIG layers, with thickness d= 200 nm on both sides of the GGG. The Gilbert damping parameter 9105, measured as the slope of the frequency dependence of the line width, is evidence for the high crystal quality. All experiments have been carried out at room temperature and on the same sample. Because of that, the results shall be presented in inverse chronological order. Having removed YIG2 by mechanical polishing, we rst concentrate on the dynamic behavior of a single magnetic layer. Figure 2a shows the FMR absorption of YIG1 jGGG bilayer [41{44] around 5 :56 GHz i.e.for a detuned antenna having weak inductive coupling. 5These spectra are acquired in the perpendicular conguration, where the magnetic precession is circular, by magnetizing the sample with a suciently strong external magnetic eld, H0, applied along the normal of the lms. Figure 2a provides a detailed view of the ne structure within the FMR absorption that is obtained when one sweeps the eld/frequency in tiny steps of 0.01 mT/0.1 MHz, respectively. The FMR mode (see arrow) follows the Kittel equation !1 0(H0M1) [45], with =(2) = 28:5 GHz/T, the gyromagnetic ratio and 0M1= 0:1720 T, the saturation magne- tization, but its intensity vs. frequency is periodically modulated [42, 46] which we explain by the hybridization with the comb of standing shear AWs described by Eq.(1) truncated to one magnetic layer. We ascribe the periodicity of 3.50 MHz in the signal of Figures 2 to the equidistant splitting of standing phonon modes governed by the transverse sound velocity of GGG along (111) ofv= 3:53103m/s [42, 46, 47] via v=(2d+ 2s)3:53 MHz [48]. This value thus separates two phononic tones, which dier by half a wavelength. At 5.5 GHz, the intercept between the transverse AW and SW dispersion relations occurs at 2 = n=!s=v105cm1, which corresponds to a phonon wavelength of about n700 nm with index number n1400. The modulation is strong evidence for the high acoustic quality that allows elastic waves to propagate coherently with a decay length exceeding twice the lm thickness, i.e. 1 mm. For later reference we point out that the absorption is the same for odd and even phonon modes, whose eigen-values are indicated here by green and orange dots. In Figures 2bcd we focus on the line shapes at detunings parallel to the FMR resonance as a function of eld and frequency indicated by the blue, magenta, and red cuts in Fig- ure 2a. The amplitude of the main resonance (blue line) in Figure 2b dips and the lines broaden at the phonon frequencies [42, 46]. The minima transform via dispersive-looking signal (magenta in 2ac) into peaks (red 2ad) once suciently far from the Kittel resonance as expected from the complex impedance of two detuned resonant circuits, illustrating a constant phase between miandunalong these cuts. The miare circularly polarized elds rotating in the gyromagnetic direction, that interact only with acoustic waves unwith the same polarity, as implemented in Eq. (1) [30]. The observed line shapes can be used to extract the lifetime parameters in Eq. (1). We rst concentrate on the observed 0 :7 MHz full line width of the acoustic resonances in Figure 2d. Far from the Kittel condition, the absorbed power is governed by the sound 6attenuation. According to Eq. (1), the absorbed power at large detuning reduces to Pabs/ ((!s!n)2+2 a)1. The AW decay rate a=(2) = 0:35 MHz is obtained as the half line width of the acoustic resonance, leading to a characteristic decay length =v=a2 mm for AW excited around 5.5 GHz. The acoustic amplitude therefore decays by 20% over the half millimeter lm thickness. The sound amplitude in both magnetic layers are therefore roughly the same, as assumed in Eq.(1). This gure is consistent with the measured ultrasonic attenuation in GGG: 0.70 dB/ s at 1GHz [36, 49], i.e., a lifetime of about 0.5 s at 5GHz. The SW lifetime 1 =sfollows from the broadening of the absorbed power at the Kittel condition which contains a constant inhomogeneous contribution and a frequency-dependent viscous damping term. When plotted as function of frequency, the former is the extrapola- tion of the line widths to zero frequency, in our case 5.7 MHz (or 0.2 mT). On the other hand, the Gilbert phenomenology (see above) of the homogeneous broadening s=!scor- responds to a s=(2) = 0:50 MHz at 5.5 GHz. The dominantly inhomogeneous broadening is here caused by thickness variations, a spatially dependent magnetic anisotropy, but also by the inhomogeneous microwave eld. Conspicuous features in Figure 2a are the clearly resolved avoided-crossing of SW and AW dispersion relations, which prove the strong coupling between two oscillators. Fitting by hand the dispersions of two coupled oscillators through the data points (white lines), we extract a gap of =(2) = 1 MHz and a large cooperativity C3. From the overlap integral between a standing shear AW conned in a layer of thickness sand the Kittel mode conned in a layer of thickness d, one can derive the analytical expression for the magneto-elastic coupling strength [42, 50]: =Bp 2r !sM1sd 1cos!sd v (2) where [35]B= (B2+ 2B1)=3 = 7105J/m3, withB1andB2being the magneto-elastic coupling constants for a cubic crystal, and = 5:1 g/cm3is the mass density of YIG. From Eq.(2) we infer that coherent SW excited around !s=(2)5:5 GHz have a dynamic coupling to shear AW of the order of =(2) = 1:5 MHz, close to the value extracted from the experiments. The material parameters extracted for our YIG jGGG are summarized in Table (I). Nu- merical solutions of Eq. (1) using these values are shown as solid lines in Figure 2bcd. The agreement with the data is excellent, conrming the validity of the model and parameters. 7TABLE I. Material parameters used in the oscillator model (all values are expressed in units of 2106rad/s). !1!2 !n+1!n s a 40 3.50 1.0 0.50 0.35 FIG. 3. (Color online) FMR spectroscopy of the YIG1 jGGGjYIG2 trilayer. Panel a) is a transpar- ent overlay of magnetic eld sweeps for frequencies in the interval 5 :1010:008 GHz by 0.1 MHz steps. Dark lines reveal two acoustic resonances marked by orange and green dots. Panel b) and c) show the frequency modulation of the FMR amplitude for respectively the bottom YIG1 layer and the top YIG2 layer, in which a contrast
appears between neighboring acoustic resonances. The solid lines show the modulation predicted by Eq.(1). The other needed parameter for solving Eq.(1) in the general case is the attenuation ratio 2=17 deducted from a factor of 50 decreased power when ipping the single YIG layer sample upside down on the antenna. The layer is then separated 0.5 mm from the antenna, and the observed reduction agrees with numerical simulations using electromagnetic eld solvers. We turn now our attention to the magnetic sandwich in which YIG1 touches the antenna and the nominally identical YIG2 is 0.5 mm away, where a slight dierence in uniaxial anisotropy causes separate resonance frequencies. Since we want to detect also the resonance 8of the top layer, we have to compensate for the decrease in inductive coupling by tuning the source frequency to the antenna resonance at 5.11 GHz (see Figure 1b). This enhances the signal by the quality factor Q30 of the cavity at the cost of an increased radiative damping of the bottom layer signal [51]. Figure 3a is a transparent overlay of eld sweeps for frequency steps of 0.1 MHz in the interval 5:1010:008 GHz. We attribute the two peaks separated by 1.4 mT (or 40 MHz) to the bottom and top YIG Kittel resonances, the later shifted due to a slight dierence in eective magnetization 0M2=0M1+ 0:0014 T. Note that the detuning between the two Kittel modes is large compared to the strength of the magneto-elastic coupling . In Figure 3b and Figure 3c we compare the measured modulation of the resonance amplitude for respectively the bottom YIG1 layer and top YIG2 layers. This corresponds to performing 2 cuts at the resonance condition FMR1 and FMR2 in the same fashion as Figure 2b. The top YIG2 signal is modulated with a period of 7.00 MHz (Figure 3c) with a contrast
between even and odd modes. This agrees with the prediction of Eq.(1) (see solid lines) due to constructive/destructive couplings mediated by even/odd phonon modes, the modulation period of the absorbed power doubles along the resonance of the top layer (FMR2), when compared to the case of a single YIG layer (Figure 2). Figure 3b illustrates also that the strong coupling 1to the antenna hinders clear observation of this modulation in the bottom YIG1 layer resonance. Nevertheless, the anticipated sign change of
(by the inverted phase ofunrelative tom2in Eq.(1)) between FMR1 and FMR2 remains observable. We now address the acoustic resonances revealed by the dark lines in Figure 3a for odd/even indices labeled by green/orange circles in the wings. The phonon line with even index (orange marker) progressively disappears when approaching the YIG2 Kittel resonance from the low eld (left side) of the resonance, while the opposite behavior is observed for the odd index feature (green marker), which disappears when approaching the YIG2 Kittel resonance from the high eld (right side). This behavior agrees with the model in Figure 1e. The contrast in the acoustic resonance intensity mirrors the contrast of the amplitude of the FMR resonance. Figure 4a shows the observed FMR absorption spectrum around 5.11 GHz measured at xed eldH0= 0:3453 T. We enhance the ne structure in Figure 4b by subtracting the FMR envelope and progressively amplifying the weak signals in the wings. The orange/green color code emphasizes the constructive/destructive interference of the even/odd acoustic 9FIG. 4. (Color online) a) Frequency sweep at xed eld performed on the magnetic bi-layer. The ne regular modulation within the FMR envelop is ascribed to the excitation of acoustic shear waves resonances. The acoustic pattern is enhanced in panel b) by subtracting the FMR envelop emphasizing the constructive/destructive interferences of the even/odd acoustic resonances in the vicinity of the YIG2 FMR mode. Panel c) shows on a logarithmic scale the predicted modulation using the experimental parameters of Table.(I). Panel d) shows on a linear scale the corresponding power absorbed by the top magnetic layer only. resonances in the top-layer signal. This feature can be explained by Eq. (1), as shown by the calculated curves in Figure 4cd. The acoustic modes change character from even to odd (or vice versa) across the FMR frequency, which is caused by the associated phase shift by 180of the acoustic drive, again explaining the experiments. The absorption by the YIG2 top layer in Figure 4d may even become negative so the phonon current from YIG1 drives the magnetization in YIG2. This establishes both angular momentum and power transfer of microwave radiation via phonons. In summary, we report interferences between the Kittel resonances of two ferromagnets over macroscopic distance through the exchange of circularly polarized coherent shear waves propagating in a nonmagnetic dielectric. We show that magnets are a source and detector for phononic angular momentum currents and that these currents provide a coupling, analogous to the dynamic coupling in metallic spin valves [52], but with an insulating spacer, over much larger distances, and in the ballistic/coherent rather than diuse/dissipative regime. This 10should lead to the creation of a dynamical gap between collective states when the two Kittel resonances are tuned within the strength of the magneto-elastic coupling. Our ndings might have implications on the non-local spin transport experiments [53], in which phonons provide a parallel channel for the transport of angular momentum. While the present experiments are carried out at room temperature and interpreted classically, the high acoustic quality of phonon transport and the strong coupling to the magnetic order in insulators may be useful for quantum communication. This work was supported in part by the Grants No.18-CE24-0021 from the ANR of France, No. EFMA-1641989 and No. ECCS-1708982 from the NSF of the USA, by the Oakland University Foundation, the NWO and Grants-in-Aid of the Japan Society of the Promotion of Science (Grant 19H006450). V.V.N. acknowledges support from UGA through the invited Prof. program and from the Russian Competitive Growth of KFU. We would like to thank Simon Streib for illuminating discussions. Corresponding author: oklein@cea.fr [1] A. Bienfait, K. J. Satzinger, Y. P. Zhong, H.-S. Chang, M.-H. Chou, C. R. Conner, E. Dumur, J. Grebel, G. A. Peairs, R. G. Povey, and A. N. Cleland, Science 364, 368 (2019). [2] B. A. Moores, L. R. Sletten, J. J. Viennot, and K. Lehnert, Physical Review Letters 120 (2018), 10.1103/physrevlett.120.227701. [3] Y. Tsaturyan, A. Barg, E. S. Polzik, and A. Schliesser, Nature Nanotechnology 12, 776 (2017). [4] S. Al-Sumaidae, M. H. Bitarafan, C. A. Potts, J. P. Davis, and R. G. DeCorby, Optics Express 26, 11201 (2018). [5] N. Spethmann, J. Kohler, S. Schreppler, L. Buchmann, and D. M. Stamper-Kurn, Nature Physics 12, 27 (2015). [6] L. J. Cornelissen, J. Liu, R. A. Duine, J. B. Youssef, and B. J. van Wees, Nat Phys 11, 1022 (2015). [7] K. Oyanagi, S. Takahashi, L. J. Cornelissen, J. Shan, S. Daimon, T. Kikkawa, G. E. W. Bauer, B. J. van Wees, and E. Saitoh, Nature Communications 10(2019), 10.1038/s41467- 019-12749-7. 11[8] R. Lebrun, A. Ross, S. A. Bender, A. Qaiumzadeh, L. Baldrati, J. Cramer, A. Brataas, R. A. Duine, and M. Kl aui, Nature 561, 222 (2018). [9] E. G. Spencer, R. C. LeCraw, and A. M. Clogston, Phys. Rev. Lett. 3, 32 (1959). [10] V. Cherepanov, I. Kolokolov, and V. L'vov, Physics Reports 229, 81 (1993). [11] R. LeCraw and R. Comstock, in Physical Acoustics vol. 3 (Elsevier, 1965) pp. 127{199. [12] E. G. Spencer, R. T. Denton, and R. P. Chambers, Physical Review 125, 1950 (1962). [13] C. Kittel, Physical Review 110, 836 (1958). [14] H. B ommel and K. Dransfeld, Physical Review Letters 3, 83 (1959). [15] R. Damon and H. van de Vaart, Proceedings of the IEEE 53, 348 (1965). [16] M. Seavey, Proceedings of the IEEE 53, 1387 (1965). [17] L. Dreher, M. Weiler, M. Pernpeintner, H. Huebl, R. Gross, M. S. Brandt, and S. T. B. Goennenwein, Physical Review B 86(2012), 10.1103/physrevb.86.134415. [18] X. Zhang, C.-L. Zou, L. 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2019-05-29

We report ferromagnetic resonance in the normal configuration of an electrically insulating magnetic bilayer consisting of two yttrium iron garnet (YIG) films epitaxially grown on both sides of a 0.5-mm-thick nonmagnetic gadolinium gallium garnet (GGG) slab. An interference pattern is observed and it is explained as the strong coupling of the magnetization dynamics of the two YIG layers either in phase or out of phase by the standing transverse sound waves, which are excited through a magnetoelastic interaction. This coherent mediation of angular momentum by circularly polarized phonons through a nonmagnetic material over macroscopic distances can be useful for future information technologies.

Coherent long-range transfer of angular momentum between magnon Kittel modes by phonons

1905.12523v3

First-principles study of magnon-phonon interactions in gadolinium iron garnet Lian-Wei Wang,1, 2, 3Li-Shan Xie,1Peng-Xiang Xu,3and Ke Xia1, 2, 3,  1The Center for Advanced Quantum Studies and Department of Physics, Beijing Normal University, Beijing, 100875, China 2Shenzhen Institute for Quantum Science and Engineering, and Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China 3Center for Quantum Computing, Peng Cheng Laboratory, Shenzhen 518005, China (Dated: December 24, 2019) We obtained the spin-wave spectrum based on a first-principles method of exchange constants, calculated the phonon spectrum by the first-principles phonon calculation method, and extracted the broadening of the magnon spectrum,
!, induced by magnon-phonon interactions in gadolinium iron garnet (GdIG). Using the obtained exchange constants, we reproduce the experimental Curie temperature and the compensation temperature from spin models using Metropolis Monte Carlo (MC) simulations. In the lower-frequency regime, the fitted positions of the magnon-phonon dispersion crossing points are consistent with the inelastic neutron scattering experiment. We found that the
!and magnon wave vector khave a similar relationship in YIG. The broadening of the acoustic spin-wave branch is proportional to k2, while that of the YIG-like acoustic branch and the optical branch are a constant. At a specific k, the magnon-phonon thermalization time of mpare approximately 109s,1013s, and 1014s for acoustic branch, YIG-like acoustic branch, and optical branch, respectively. This research provides specific and effective information for developing a clear understanding of the spin-wave mediated spin Seebeck effect and complements the lack of lattice dynamics calculations of GdIG. I. INTRODUCTION The collinear multi-sublattice compensated ferrimagnetic insulator gadolinium iron garnet ( Gd3Fe5O12, GdIG) has the same crystalline structure as YIG1–4, only if yttrium is re- placed by the magnetic rare-earth element, gadolinium.5–9In comparison with YIG10,11, GdIG also has a low Gilbert damp- ing constant of nearly 103,12but has three sublattices, where the 12 Gd sublattice moments (dodecahedrals) are ferromag- netically coupled to the 8 Fe moments (octahedrals) and an- tiferromagnetically coupled to the 12 Fe moments (tetrahe- drals),4–9so that GdIG has more complex spin-wave modes than YIG, which have been obtained by first-principles study of exchange interactions, indicating that the accurate calcula- tion method can improve and compensate for the abnormality in the spin-wave spectrum caused by exchange constants.4,13 GdIG has high compensation temperatures T comp =286- 295 K,14–17which is close to room temperature. Recently, the heterostructures consisting of YIG18–21and heavy metals (FMI/NM) have been frequently used to study the spin See- beck effect (SSE)22–24and spin Hall magnetoresistance effect (SMR).25–27. Similar to YIG, GdIG has been frequently used to study the SSE in FMI/NM heterostructures.22–24SSE ex- periments have shown two sign changes of the current signal upon decreasing temperature.28,29One can be explained by the inversion of the sublattice magnetizations at T comp , where the net magnetization vanishes and the other can be attributed to the contributions of Ferrimagnetic resonance mode ( -mode) and a gapped optical magnon mode ( -mode).28–30The SMR experiments shows that GdIG has a canted configuration31and a sign change of SMR signal32at around T comp . Unlike in SSE,28,29the sign change of SMR is decided by the orientation of the sublattice magnetic moments associated with exchange interaction.32Thus, these experiments28,32indicate that mul- tiple magnetic sublattices in a magnetically ordered system have different individual contribution and highlight the im- portance of the multiple spin-wave modes determined by ex-change interactions. However, the microscopic mechanisms responsible for these spin current associated effect are still under investigation. A major question is whether the high- frequency magnons play an important role in the SSE, and the fitting exchange parameters used in the literature through lim- ited experimental data7,33,34are always physically credible. In addition to the pivotal magnon-driven18,30,35,36effect, phonon-drag37,38effect plays non-negligible roles in the SSE through magnon-phonon interactions,39–41which play an im- portant role in YIG based spin transport phenomena.22,30,39–42 Thus, the understanding of the scattering process of magnon- phonon interactions is important and meaningful. In fact, the magnon-phonon thermalization (or spin-lattice relaxation) time,mp,39,43,44is an important parameter used to describe the magnon-phonon interactions and calculate magnon dif- fusion length30,39. We have extracted the mp(109s) from the broadening of magnon spectrum quantitatively13, in good agreement with reported data39,44,45, however, the value is three orders of magnitude lower than the reported mp 106s30,43,46,47. For the spin-wave spectrum and phonon spec- trum to aid our understanding of the magnon-phonon scatter- ing mechanism, the temperature-dependent magnon spectrum and lattice dynamic properties of GdIG have still not been completely determined. Here, we investigate these charac- teristics of GdIG based on the operable and effective method used in YIG.4,13 To computationally reveal the microscopic origin of SSE in these hybrid nanostructures, the magnon spectrum, phonon spectrum and magnon-phonon coupling dominant effect in GdIG will also be investigated step by step. First, we use density functional theory (DFT) technology to study the elec- tronic structure and exchange constants, and using Metropolis Monte Carlo (MC) simulations, we obtain the Curie temper- ature (TC) and compensation temperature ( Tcomp ). Second, we obtain the spin-wave spectrum using numerical methods combined with exchange constants. Then, the phonon spec- trum is studied using first-principles calculations, allowing usarXiv:1912.10432v1 [cond-mat.mtrl-sci] 22 Dec 20192 to extract intersecting points of magnon branch and acoustic phonon branch. In the end, we study the temperature depen- dence of spin moment, exchange constants, and magnon spec- trum, and calculated broadening of the spin-wave spectrum of GdIG is used to extract the magnon-phonon thermalization time. II. COMPUTATIONAL DETAILS AND RESULTS In this study, we investigate GdIG, which belongs to the cubic centrosymmetric space group, No. 230 Ia3d.6,7The cubic cell contains eight formula units, as shown in Fig. 1, where rare-earth gadolinium ions occupy the 24c Wyckoff sites (green dodecahedrals), the FeOand FeToccupy the 16a sites (blue octahedrals) and 24d sites (yellow tetrahedrals), re- spectively, and the O ions occupy the 96h sites (red balls). The atomic sites from the experimental structural parameters (TABLE I)6–8are used in the study. /s40/s98/s41/s40/s97/s41 /s70/s101/s79 /s71/s100 /s70/s101/s84/s79 /s74 /s100/s99/s74 /s97/s99 /s74 /s97/s97/s74 /s100/s100 /s74 /s97/s100 Figure 1. (a) 1/8 of the GdIG unit cell. The dodecahedrally co- ordinated Gd ions (green) occupy the 24c Wyckoff sites, the octa- hedrally coordinated FeOions (blue) occupy the 16a sites, and the tetrahedrally coordinated FeTions (yellow) occupy the 24d sites. (b) The dashed lines denote the nearest-neighbor (NN) exchange interac- tions. Subscripts aa,dd,ad,acanddcstand for FeO-FeO,FeT-FeT, FeO-FeT,FeO-GdandFeT-Gd interactions, respectively. To calculate the electronic structure and total energy of GdIG, we use DFT, as implemented in the Vienna ab initio simulation package (V ASP).48,49The electronic structure is described by the generalized gradient approximation (GGA) of the exchange correlation functional. Projector augmented wave pseudopotentials50are used. By using a 500 eV plane-Table I. Atomic positions in the GdIG unit cell. The lattice constant isa= 12:465Å. Wyckoff Position x y z FeO16a 0.0000 0.0000 0.0000 FeT24d 0.3750 0.0000 0.2500 Gd 24c 0.1250 0.0000 0.2500 O 96h 0.9731 0.0550 0.1478 -1.00-0.75-0.50-0.250.000.250.500.751.00- 1.00-0.75-0.50-0.251.61.822.22.42.62.833.23.4- 1.00-0.75-0.50-0.251.61.822.22.42.62.833.23.4 0.0(a)( b)( c) 0.0 E-Ef (eV) Majority spin Minority spin ΓP H (U-J)Fe = 5.7 eVGGA+U( U-J)Gd = 6.3 eV E-Ef (eV) ΓP H E-Ef (eV) ΓP H GGA+U( U-J)Fe = 5.7 eVG GA Figure 2. The energy band structure of the GdIG ground state under different calculation conditions. (a) GGA calculation results. (b) GGA + U , thedorbital of the Featom plusU, where theUJ value is 5:7eV . (c) GGA + U , thedorbital of the Featom plusU, where theUJvalue is 5:7eV; theforbital of the Gd atom plus U, where theUJvalue is 6:3eV . The green lines represent 0 eV . wave cutoff and a 666Monkhorst-Pack k-point mesh we obtain results that are well converged. A. Electronic structure The calculated energy band structures of the ferrimagnetic ground-state structure, are shown in Fig. 2. The apparent band gap indicates the properties of the insulator. The total moment (including Fe, Gd and O ions) per formula unit is consistently 16B, which is consistent with experimental data9,51. The Fe and Gd sublattice contribute the majority of the spin moments within the unit cell. In the DFT-GGA calculation, the spin moments of the Fe ions are 3.69Bfor FeOand 3.63B for FeT, which are lightly larger than the computational data9, but spin moments for the Gd and O ions are the similar values 6.85Band 0.08Brespectively, and the electronic band gap is 0.55 eV , as shown in Fig. 2(a). And just like we did in YIG,4because DFT is not good at predicting the energy gap of insulators, DFT-GGA+ Ucalculations with UJford orbital of Fe in the range of 2:2–5:7eV andUJforforbital of Gd in the range of 0:3–6:3eV are conducted to determined the Hubbard Uand Hund’s Jparameters. The variation in3 /s50 /s51 /s52 /s53 /s54/s51/s46/s55/s52/s46/s48/s54/s46/s56/s55/s46/s48/s55/s46/s50 /s48 /s50 /s52 /s54/s52/s46/s48/s52/s46/s49/s52/s46/s50/s54/s46/s56/s55/s46/s48/s55/s46/s50 /s83/s40/s70/s101/s79 /s41 /s32/s32/s32/s83/s40/s70/s101/s84 /s41 /s32 /s83/s40/s71/s100/s41 /s32/s32/s83/s112/s105/s110/s32/s109/s111/s109/s101/s110/s116/s40 /s41 /s40/s85/s45/s74 /s41 /s70/s101/s32/s40/s101/s86/s41/s40/s85/s45/s74 /s41 /s71/s100/s61/s32/s51/s46/s51/s101/s86/s40/s97/s41/s40/s98/s41 /s32/s32/s83/s112/s105/s110/s32/s109/s111/s109/s101/s110/s116/s32/s40 /s41 /s40/s85/s45/s74 /s41 /s71/s100/s32/s40/s101/s86/s41/s83/s40/s70/s101/s79 /s41 /s32/s32/s32/s83/s40/s70/s101/s84 /s41 /s32 /s83/s40/s71/s100/s41/s40/s85/s45/s74 /s41 /s70/s101/s32/s61/s52/s46/s55/s101/s86 Figure 3. Variation in the spin moments of Fe and Gd ions accord- ing to different GGA+U calculations of the forbital of Gdandd orbital of FeplusU. (a) Variation in the fixed (UJ)Gd= 3:3 eV calculations and (b) Variation in the fixed (UJ)Gd= 4:7eV calculations. the spin magnetic moments of different atoms under different conditions is shown in Fig. 3. As shown in Figs. 2(b) and (c) and in Fig. 3, the electronic energy gap and the spin moments slightly increase with UJ. For the GGA+U calculations(Fig. 2(b)), when the UJvalue for the Fe atom is constant, the band structure of the GdIG near the Fermi energy is similar to that of the YIG. When the Gd atom have (UJ)Gd= 6:3eV , the energy band of Gd moves up, as shown in Fig. 2(c). For the largest values of UJ, the spin moments of FeO, FeT, and Gd are4.26B, 4.18Band7.05B, respectively, and the electric band gap is approximately 2.08 eV . Even for the largest values of UJ, the moments are much smaller than expected for the pure Fe3+, electronic spin S= 3=2state [s=gp S(S+ 1) = 5:916B]and for the pure Gd3+, electronic spin S= 7=2 state [s=gp S(S+ 1) = 7:937B]. Compared with the electronic structure calculation for YIG, the results of the spin moments of Fe and the energy gap have been found to be sim- ilar.4 B. Exchange constants To obtain the five independent nearest-neighbor(NN) ex- change constants, Jaa,Jdd,Jad,JacandJdccovering the inter- and intra-sublattice interactions, as shown in Fig. 1(b). In TABLE II, we map ten different collinear spin configura- tions(SCs) a-j on the Heisenberg model without external mag- netic field energy or anisotropic energy. The calculation de- tails can be found in Ref. 4 In the NN model, with Eac=JacSaScandEdc= JdcSdSc,Eaa,Edd, andEadare just as the work in Ref. 4, whereSa,Sd, andScare the +=directions of the FeO, FeT and Gd ions, the total energies, Etotof the Heisenberg model are determined as listed in TABLE II. Here Ecalare the calcu- lated total energies for fixed (UJ)Fe= 3:4eV and different (UJ)Gdvalues relative to the ground state of SC (a). When all or part of the magnetic moment directions of Gd atoms are flipped at (UJ)Gd= 0 eV , SC (e), (g), and (j) have lower total energies than SC (a), which is in contrast to the experi- /s50 /s52 /s54/s48/s46/s48/s48/s46/s50/s48/s46/s52/s50/s51/s52 /s50 /s52 /s54/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50 /s48 /s50 /s52 /s54/s48/s46/s48/s48/s46/s49/s48/s46/s50/s51/s46/s49/s51/s46/s50/s51/s46/s51 /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 /s40/s85 /s32/s45/s32 /s74 /s41 /s70/s101/s32/s40/s101/s86/s41/s32/s74 /s97/s97 /s32/s74 /s100/s100 /s32/s74 /s97/s100 /s69/s110/s101/s114/s103/s121/s32/s40/s109/s101/s86/s41 /s40/s85 /s32/s45/s32 /s74 /s41 /s70/s101/s32/s40/s101/s86/s41/s32/s74 /s97/s99 /s32/s74 /s100/s99/s32/s32/s69/s110/s101/s114/s103/s121/s32/s40/s109/s101/s86/s41 /s40/s85 /s32/s45/s32 /s74 /s41 /s71/s100/s32/s40/s101/s86/s41/s32/s74 /s97/s97 /s32/s74 /s100/s100 /s32/s74 /s97/s100 /s69/s110/s101/s114/s103/s121/s32/s40/s109/s101/s86/s41 /s40/s85 /s32/s45/s32 /s74 /s41 /s71/s100/s32/s40/s101/s86/s41/s32/s74 /s97/s99 /s32/s74 /s100/s99/s40/s97/s41/s40/s98/s41 /s40/s99/s41 /s40/s100/s41Figure 4. (a) and (b) Five different exchange interactions change with different values of (UJ)Fefor fixed (UJ)Gd= 3:3eV . (c) and (d) Five different exchange interactions change with different values of(UJ)Gdfor fixed (UJ)Fe= 3:4eV. The negative value of the exchange constants indicates that the magnetic moment tends to be aligned in the same direction. mental result, and the energy differences between these three SCs and SC (a) decrease as (UJ)Gdincreases. Furthermore, in SC (g), the static magnetic moment of the Gd sublattice is 0B, and the total magnetic moment of GdIG unit molecular formula is 5 B, indicating that it is necessary to add U for Gd ions. Through the differences of
Ebetween the Ecal andEtot, we also find that
Eincrease when (UJ)Gdis too small or too large. For (UJ)Gd= 3:3eV , the maximum j
Ejis 0.63 %, which is acceptable. The exchange constants shown in Fig. 4 are obtained by the least-squares of six linear equations using the SCs a-g listed in TABLE II. SCs h-j are selected to check whether the results are reasonable. The exchange constants Jaa,JddandJadare positive (antiferromagnetic), whereas the exchange constants JacandJdcdepend on the value of UJ. In Figs. 4(a) and (b),Jaa,JddandJaddecrease as (UJ)Feincreases when (UJ)Gdis kept constant 3.3 eV , which is similar to the situation for YIG.4The values of Jadare approximately 4 % and 2 % lower compared with the ones for YIG when (UJ)Feis 4.7 eV and 5.7 eV , respectively. JacandJdc with different signs decrease slightly as (UJ)Feincreases and forjJdcj>jJacj. In Fig. 4(c) and (d), when (UJ)Feis kept constant at 3.4 eV , JaaandJddmaintain almost the same values, whereas Jadincrease slightly as (UJ)Gdincreases. JacandJdcdecrease to zero and then change their signs as (UJ)Gdincreases. Among all the results, Jadis one order of magnitude larger than the other interactions, whereas Jaais approximately half of Jddand the absolute value of Jacis al- ways smaller than that of Jdc. Thus, the strong inter-sublattice exchange interaction, Jad, dominates the other smaller ener- gies and helps maintain the ferrimagnetic ground state of the4 Table II. Comparison of the calculation of the total energies for different SCs in the NN models. The b j are obtained by changing the magnetization directions of part of the magnetic ions based on the ferrimagnetic ground state SC a. Etotis the total energy fitting formula. Ecalis the total energy (in units of meV) calculated via ab initio with different (UJ)Gdat fixed (UJ)Fe= 3:4(in units of eV).
Eis the difference between EtolandEcal.Ecalof SC a is denoted as zero. SC EtotEcal
E 0.0 1.3 3.3 5.3 0.0 1.3 3.3 5.3 aE0+ 32Eaa+ 24Edd+ 48Ead+ 48Eac+ 24Edc 0.00 0.00 0.00 0.00 0.01 0.000.010.02 bE0+ 32Eaa+ 24Edd48Ead48Eac+ 24Edc3957.97 5036.60 5145.78 5236.59 0.00 0.00 0.010.03 c E0+ 32Eaa24Edd48Eac 1729.27 2661.94 2582.27 2506.26 0.01 0.000.010.03 d E032Eaa+ 24Edd+ 24Edc 1364.90 2399.29 2454.38 2500.18 0.01 0.000.010.03 eE0+ 32Eaa+ 24Edd+ 48Ead48Eac24Edc187.34 599.72 331.46 88.54 0.01 0.000.010.02 f E0+ 32Eaa24Edd 1770.30 2662.68 2532.24 2412.53 0.00 0.00 0.010.02 g E0+ 32Eaa+ 24Edd+ 48Ead589.27 299.52 165.63 43.94 0.96 0.34 0.09 0.31 h E032Eaa+ 24Edd 1807.52 2700.43 2570.38 2450.69 0.630.540.31 0.00 i E0+ 32Eaa24Edd+ 48Eac 1813.35 2664.33 2482.58 2319.38 2.020.900.380.60 jE0+ 32Eaa+ 24Edd+ 48Ead32Eac16Edc327.56 496.21 274.47 71.62 6.56 3.56 1.74 2.14 bulk.7,52,53Moreover, with a change in the (UJ)Gdvalue, JacandJdcmay change signs, which implies that it is possi- ble to change the direction of the Gd atomic magnetic moment in the ground state. A comparison of our exchange constants with those found in prior studies is provided in TABLE III. We find that dif- ferent methods provide different exchange constants. Us- ing limited experimental data, neither the magnetization fit- ting7nor the molecular field approximation52,53can effec- tively determine whether the interaction between the inter- and the intra-sublattice is ferromagnetic or antiferromagnetic coupling. Although our calculated value is smaller than the value provided in the TABLE III and the obtained exchange constants between Fe atoms in GdIG are smaller than those in YIG4, the relative size relationship is Jad> Jdd> Jaa, Jad> Jdc> Jac. Here, we can well determine the type of exchange constants between sublattices, and use the ex- change constants to obtain a reasonable experimental Curie temperature and compensation temperature. Therefore, the first-principles method of exchange constants4,13undoubtedly provides an effective way to calculate the interaction parame- ters in GdIG. C. Magnetization, Curie temperature, and compensation temperature To obtain the temperature dependence of the magnetiza- tion, Curie temperature ( TC), and compensation temperature (Tcomp ), we use the spin models by Metropolis MC simula- tions on a 323232 supercell with a unit cell containing 32 spins under periodic boundary conditions. The computa- tional details can be found in Ref. 4. The results are shown in Fig. 5. With the parameters of (UJ)Gd= 3:3eV and (U J)Fe= 4:7eV , the temperature dependence of magnetiza- tion,Ma,Md, andMcof FeO, FeT, and Gd, respectively, and the total magnetization ( M=Ma+Mb+Mc) of a for- /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 /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 /s32/s32/s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s32/s40 /s41 /s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s32/s84/s111/s116/s97/s108 /s32/s70/s101/s79 /s32/s70/s101/s84 /s32/s71/s100/s84 /s99/s111/s109/s112/s32/s61/s32/s51/s49/s48/s32/s75/s40/s97/s41 /s40/s98/s41 /s32/s32/s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110 /s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s40/s85/s45/s74 /s41 /s71/s100/s32/s32 /s32/s32/s48/s32/s101/s86 /s32/s32/s50/s46/s51/s32/s101/s86 /s32/s32/s51/s46/s51/s32/s101/s86 /s32/s32/s52/s46/s51/s32/s101/s86 /s32/s32/s82/s101/s102Figure 5. (a) Temperature dependence of magnetization of FeO, FeT, and Gd and the total magnetization of a formula unit with exchange constants fitted to the ab initio energies for (UJ)Gd= 3:3eV and(UJ)Fe= 4:7eV . The black arrow represents the position of the compensation temperature, Tcomp = 310 K (b) The absolute value of the total magnetization is jMj=jM(T= 0 K)jfor different (UJ)Gdat different temperatures. The reference curves (green lines) are calculated using the exchange constants in Ref. 7.5 Table III. Exchange constants taken from the literature and our study. In calculation, (UJ)Gd= 3:3eV . The unit for the interaction coefficient is meV . (UJ)Fe= 4:7eV is used for comparing with the result in YIG.4 JaaJddJadJacJdc Method Reference 0.78 0.78 3.94 0.22 0.87 Magnetization fit Ref.7 1.051.47 3.140.11 0.58 Molecular field approximation Ref.52 0.56 1.04 2.59 0.05 0.16 Molecular field approximation Ref.53 0.081 0.137 2.487 0.032 0.157 Ab initio GGA+U ( (UJ)Fe= 4.7eV) This paper 0.103 0.185 3.018 0.035 0.170 Ab initio GGA+U ( (UJ)Fe= 3.7eV) This paper mula unit are determined, as shown in Fig. 5(a). The crossing point of the total magnetization curve (black) and the hori- zontal dash line shows that Tcomp (310 K) and TC(550 K), which are in good agreement with the experimental values of 290 K16,17and 560 K5,54, respectively. Through the Fig. 5(a), we can find that the change of the total spin moments of FeO-sublattice and FeT-sublattice is considerably flat. How- ever, the total spin moment of Gd-sublattice rapidly declines with increasing temperature until approximately 200 K; As the temperature continued to increase, owing to competition between the Gd and Fe magnetic moments, the total spin mo- ments undergoes a transition dominated by Gd to Fe. The direction of the total magnetization changes from Gd (FeO) to FeTand the value first decreases and then increases; then, Tcomp emerges16,17, wherein one sign change of SSE signal appears.28With a further increase in temperature, the decreas- ing trend of Gd-sublattice spin moments slows down. Ad- ditionally, the decreasing trend of the spin moments of FeO- sublattice and FeT-sublattice becomes steeper, and the total magnetic moment slowly increases and then decreases to 0 B at transition temperature TC. The temperature dependence of the magnetization of GdIG is similar to that reported in the literature.28 As shown in Fig. 5(b), we determine the absolute values of the total magnetization, jMj, normalized by its value at zero temperature,jM(T= 0 K)j, for different (UJ)Gd at the fixed (UJ)Fe= 3:4eV . There is no Tcomp with (UJ)Gd= 0 eV , whereas the calculated Tcomp decreases as(UJ)Gdincreases.Tcomp of the reference curve(green line) calculated using the exchange constants in Ref. 7 is also approximately 310 K. Compared with Fig. 4, when the (UJ)Gdvalue is small, JacandJdcin Fig. 4 are positive andJac<Jdc. Thus, the magnetization direction of Gd with FeOand FeTis anti-parallel, the latter is dominant, the ground state corresponds to SC (g) in TABLE II, and half of the Gd has an inverted magnetic moment. With increasing (UJ)Gd, JacandJdcreverse so that Gd tended to be parallel to FeOand anti-parallel to FeT; thus, the ground state corresponds to SC (a) in TABLE II. Under this condition, the Tcomp of the system decreases gradually with increasing (UJ)Gdvalues. There- fore, with appropriate parameters of (UJ)Gd= 3:3eV and (UJ)Fe= 4:7eV , we can reproduce the experimental com- pensation temperature and transition temperature.D. Magnon spectrum Using the exchange interaction obtained under conditions of(UJ)Gd= 3:3eV and (UJ)Fe= 4:7eV , as shown in TABLE III, we obtain the spin-wave spectrum at zero temper- ature, as shown in Fig. 6. The details of the calculation can be found in Ref. 4. In Fig. 6(a), the special ferrimagnetic resonance -mode, lower-frequency optical modes with a slight gap, YIG-like acoustic-mode and lower-frequency optical -mode are marked by red, orange, yellow, and green curves. Other high optical modes are marked by blue curves. To contrast with the lowest optical mode of YIG in Ref. 4, a black dash line indicates the frequency. Fig. 6(b) clearly shows the lower- frequency branches below 0.5 THz. The flat part (orange) has two modes at approximately 425 GHz and two modes at approximately 440GHz, which are dominated by the Gd precessing.7The second derivative of the -mode at the - point is 191041J
m2, which is approximately one quarter of the spin-wave stiffness of YIG, D= 771041J
m2.4 The-mode around 1.4 THz has a similar parabolic branch with the acoustic branch of YIG in which the second deriva- tive of the mode at the point is 621041J
m2. The gap between- and-mode at the point depends on JdcandJac interaction,28and the gap between the second parabolic low- est -mode and the -mode is approximately 5THz, which is consistent with the conditions of the NN model in YIG.4As the temperature increases, the -mode will red-shift to gain a sufficient thermal magnon population below KbT, then the SSE signal changes sign.28,35However, the -mode will also red-shift below 6.25 THz (T =300 K), which may also have some indispensable effects in the SSE. The precession patterns of these special low-frequency modes at are shown in Fig. 7. In Fig. 7(a), for the -mode, the magnetization of FeOis parallel to Gd, anti-parallel to FeT, and near the point, where !andksatisfy the square relationship, which is similar to the acoustic mode of YIG13. For the-mode, the pattern is different from the acoustic branch modes of YIG, the magnetization of 8 FeOions, 12 Gd ions and 12 FeTions have different directions with re- spect to the-mode, and the magnetization of Gd has a small angle (0:12) with the z-axis. For the -mode, the mag- netization of FeOhas different directions with respect to Gd and FeT, and the magnetization of FeOand Gd have small angles (0:11) and (0:16) with respect to the z-axis, re-6 024681012140 .00.10.20.30.40.5ΗΝ ω (ΤΗz) Γ(a)( b) α βγ ω (ΤΗz)N Γ H 1100 011 100 01(b)(c)(d)(e) Figure 6. Spin-wave spectrum at zero temperature in the first Bril- louin zone at (UJ)Gd= 3:3eV and (UJ)Fe= 4:7eV . (a) The entire spin wave spectrum. The black dash line represents the position of the lowest optical branch frequency at 4.8 THz for YIG calculated in the NN model from Ref. 4. The notations (red), (yellow), and (green) mark the three main spin-wave modes, indicating positive, negative and positive polarization, respectively. The orange line marks the two nearly clearance modes at approxi- mately 0.4 THz. These low-frequency optical modes are the Gd mo- ments precession dominant. (b) The partial enlarged details of the low-frequency modes that are red- and orange- marked in (a) around 0.4 THz whereas (b), (c), (d), and (e) mark the modes at approxi- mately 0.4 THz. The directions in the k-space use the standard labels for a bcc reciprocal lattice. spectively. The -mode has different polarizations with the -mode, but the same as -mode. The polarizations of - and -mode switch at Tcomp , which is related to the other sign change at a lower temperature in the SSE.28However, the two modes induce the same sign in the detected SMR signal in a magnetic canted phase of GdIG.26,32. Therefore, spin-wave modes need to be verified in greater detail by experiments in the future. For the 425 GHz case, two patterns with two and three degenerated modes have FeOthat spins lie along the z-axis, while FeTspins precess at small angles, as shown in Fig. 7(b) and (c). For the 440GHz case, for the two patterns with three and three degenerated modes, FeTspins align along the z- axis, and FeOspins precess at small angles or take the op- posite direction as the FeTspins, as shown in Fig. 7(d) and (e). In both cases, Gd spins precess at a larger angle than the Fe spins in the exchange field of the Fe spins.7,28The gap Figure 7. Precession patterns of the low-frequency modes color marked except for blue in Fig. 6 at the -point. (a) The patterns mark the-,-, and -mode. The red arrows represent the differ- ent chiral patterns. (b) and (c) show two patterns at approximately 425 GHz with two and three degenerated modes, respectively. (d) and (e) show two patterns at approximately 440 GHz with three and three degenerated modes, respectively. ( )a, b, c and d denote the different precession angles. The lower optical modes indicate that the Gd moments precess around the exchange field induced by Fe moments. /s48/s49/s50/s51 /s32/s32 /s32/s32 /s32/s74 /s97/s99/s48/s59/s32/s74 /s100/s99 /s32/s74 /s97/s99/s48/s59/s32/s74 /s100/s99 /s32/s74 /s97/s99/s48/s59/s32/s74 /s100/s99 /s32/s74 /s97/s99/s48/s59/s32/s74 /s100/s99/s40 /s122 /s41 /s49/s49/s48 /s32/s78 Figure 8. The spin-wave spectrum is affected by the change in ex- change constants at (UJ)Gd= 3:3eV and (UJ)Gd= 4:7eV . Jaa,Jdd, andJadare unchanged. For black ball curves JacandJdc are both 0 meV; for green star curves, Jacis 0 meV and Jdcis the original value; for blue triangle, Jdcis 0 meV and Jacis the original value; for red triangle curves, JacandJdcare both original values. between these modes at the point and the -mode is ap- proximately 1THz, which is dominated by the interactions of Fe and Gd. To show that the gap is primarily derived from the exchange interaction between the Gd atoms and Fe atoms, we fixJaa,JddandJad, then change JacandJdcto show the change in spectrum along the highly symmetric direction7 /s48 /s53 /s49/s48 /s49/s53 /s50/s48/s48/s53/s49/s48/s49/s53/s50/s48 /s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53 /s48 /s53 /s49/s48 /s49/s53 /s50/s48/s48/s53/s49/s48/s49/s53/s50/s48 /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 /s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s84/s72/s122/s41/s32/s79 /s89 /s70/s101/s79/s70/s101/s84 /s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s84/s72/s122/s41 /s32/s32/s84/s68/s79/s83/s32/s89/s73/s71/s32/s32/s32/s80/s68/s79/s83 /s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s84/s72/s122/s41/s71/s100/s79 /s70/s101/s84 /s70/s101/s79/s40/s100/s41/s40/s99/s41/s40/s98/s41 /s32/s32/s84/s68/s79/s83 /s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s84/s72/s122/s41/s32/s71/s100/s73/s71/s40/s97/s41 Figure 9. First-principles phonon calculation of YIG and GdIG at zero temperature. (a) Projected density of states (PDOS) for YIG. (b) Total density of states (TDOS) for YIG. (c) PDOS for GdIG. (d) TDOS for GdIG. FeOrepresents the octahedral atom sites, and FeT represents the tetrahedral atom positions, the same representation as shown in Fig. 1. (110), as shown in Fig. 8. We find that with the reduction of Jac, the intersection point between the branch with the low- est frequency and the boundary of Brillouin region decreases and the band gap becomes narrower. As Jdcdecreases, the in- tersection point shows a more obvious reduction, the spectral lines near 0:4THz degenerate, and the frequency decreases. WhenJacandJdcsimultaneously decrease and the reduction effect is superimposed, the spectrum near 0:4THz completely disappears. E. Phonon spectrum Different ab initio techniques and methods can be em- ployed to calculate the phonon spectrum.55–59The density functional perturbation theory method is typically used to ob- tain the real-space force constants of GdIG whose DFT+U ground state is used for self-consistent linear-response cal- culations in V ASP as above. Phonon band structures, partial density of states (PDOS), total density of states (TDOS) and the phonon velocity of GdIG are investigated using the force constants via the Phonopy code.56,60We also use the same cal- culation method to obtain the phonon spectrum of YIG, where the calculation parameters come from Ref. 4. In the calculation, we first obtain PDOS and TDOS of GdIG and YIG, as shown in Fig. 9. Figs. 9(a) and (c) show the PDOS for different atoms. Figs. 9(b) and (d) show the TDOS. The clear results for YIG are similar to the results from Ref. 58. The phonon gap in GdIG is approximately 2THz, which is consistent with the phonon spectrum for YIG. There is a dif- ference between YIG and GdIG in the low frequency region (0-5 THz). 051015200 24680 51015200 2468(d)( c)(b) Frequency (THz)Γ H N P ΓYIG Frequency (ΤΗz)( 110)( 001)TALA(a)N Γ H Frequency (THz) ΓH N P ΓGdIG N Γ H ( 110)( 001)TALA Figure 10. (a)Phonon spectrum of YIG along the -H-N-P- high- symmetry lines. (b) Comparison between the phonon and spin-wave spectra along the -N and -H high-symmetry directions in YIG. (c) Phonon spectrum of GdIG along the same high-symmetry lines with (a). (d) Comparison between phonon and spin-wave spectra along the same directions with (b) in GdIG. The longitudinal acoustic (LA) and transverse acoustic (TA) phonons are marked on (b) and (d). The partial spin-wave spectra in (b)(red dots) are obtained from Fig. 5 in Ref. 4, and the representations in (d) are the same as Fig. 6(a), the partial phonon spectra are extracted from (a) and (c), respectively. The phonon spectrum along the path of -H-N-P- in the Brillouin zone of the bcc lattice for YIG and GdIG are shown in Figs. 10(a) and (c), which cover 240phonon branches. The phonon spectrum of YIG is consistent with the results calcu- lated by the finite-displacement method in Ref. 58. We are in- terested in the low-frequency phonon branches, labeled as lon- gitudinal acoustic (LA) and transverse acoustic (TA) phonons in Figs. 10(b) and (d). The frequency of the special branches shows a linear kdependence in the lower frequency region and the TA modes are double degenerate. The slope of the TA(LA) phonon dispersion is presented in TABLE IV. For YIG, the velocity of TA (LA), v= 3:8kms1(6.74 kms1), is consis- tent with experiment results.42For GdIG, the TA (LA) veloc- ityv= 3:3kms1(6.08 kms1), is almost consistent with the transverse (longitudinal) sound velocity found via experi- ments.61,62 In Fig. 10(b) for YIG, the spin-wave acoustic branch (red) taken from the Ref. 4 has an intersecting point at a very lower- energy approximately 1:38(5:13) meV with the TA (LA) phonon branch. These fitting magnon-phonon intersecting points are almost consistent with experiment results.42 In Fig. 10(d) for GdIG, the intersection points are more complicated than for YIG. The LA phonons and TA phonons have only one cross point with the -modes (red) at the point, and not intersection points with the -modes (yel- low). However, they have many more intersection points with flat lower-frequency optical branches (orange) because many multiple degenerated branches stay here as shown in Fig. 6(b),8 Table IV . Comparison between the calculated and reported values of the phonon velocities for YIG and GdIG SystemLA velocity TA velocitySource(105cm/s) (105cm/s) YIG 7.200 3.900 Ref.42 YIG 7.209 3.843 Ref.63 GdIG 6.500 3.390 Ref.62 YIG 6.740 3.800 This paper GdIG 6.080 3.300 This paper such as eight crossing points along the -Hpath and eleven crossing points along the -Npath. We speculate that in the lower-frequency region, low-frequency lattice vibrations (phonons) can couple with magnons and there may be com- plicated and interesting magnon-phonon40,42,64and magnon- magnon65coupling effects. The results are useful for un- derstanding the scattering process of magnon-phonon inter- actions in the SSE. F. Magnon-phonon coupling To investigate the variation of frequency and linewidth in the spin-wave spectrum at room temperature ( T= 300 K), the temperature-induced atomic vibration is considered. The statistical mean square of the displacements, ui, of thei-th atom with its mass, Mi, are determined by the Debye model,13 hjuij2i=9~2 MikBD1 4+T2 2 DZD T 0x ex1dx ;(2.1) wheremGd= 157:25amu,mFe= 55:85amu,mO= 15:99amu and the Debye temperature is D= 655:00K.9 Here, the change in atomic displacement does not cause sig- nificant lattice deformation. The atomic vibration displace- ments modeled in Eq. 2.1 are added to the experimental struc- ture shown in TABLE I. Forty atomic configurations, which are denoted as cf01 cf40, respectively, are used to ob- tain the spin-wave spectrum. We chose the parameters of (UJ)Gd= 3:3eV and (UJ)Fe= 4:7eV for the total energy calculations because they provide reasonable TCand Tcomp . The magnon-phonon relaxation time can be extracted from the broadening spin-wave spectrum. For calculation de- tails, we refer to Ref. 13. The spin moments of Fe and Gd ions for the ferrimagnetic ground-state structure with these 40 configurations are shown in Fig. 11 (a). The average moments of the S(FeO),S(FeT), andS(Gd) ions are marked as black squares, red dots, and blue triangles, respectively. In comparison with the zero tem- perature values (marked as dash lines), the average moments of the Fe ions are lower for all configurations, whereas the ones for Gd ions showed no signigicant difference. The error bars denote the minimum and maximum range for each con- figuration. The spin moments of the Fe ions have a variation /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 /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/s32/s45 /s83 /s40/s70/s101/s79 /s41 /s32/s32 /s83 /s40/s70/s101/s84 /s41 /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 /s99/s102/s52/s48 /s99/s102/s51/s49/s99/s102/s50/s49/s99/s102/s49/s49 /s99/s102/s48/s50/s83/s112/s105/s110/s32/s109/s111/s109/s101/s110/s116/s32/s40 /s66/s41 /s32 /s99/s102/s48/s49 /s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48 /s51/s46/s53/s32/s74 /s97/s97 /s32/s74 /s100/s100 /s32/s74 /s97/s100 /s32/s74 /s97/s99 /s32/s74 /s100/s99/s32/s74 /s105/s106/s32/s40/s109/s101/s86/s41/s99/s102/s50/s48 /s99/s102/s50/s49 /s99/s102/s50/s50/s99/s102/s49/s50/s99/s102/s49/s49 /s99/s102/s49/s48 /s99/s102/s48/s50 /s99/s102/s48/s49 /s99/s102/s52/s48 /s99/s102/s51/s50 /s99/s102/s51/s49/s99/s102/s51/s48/s40/s98/s41/s40/s97/s41Figure 11. Spin moments and exchange constants of 40 atomic con- figurations (cf01-cf40). (a) Spin moments of the ions (in units of B) for these ground-state structures. The S(FeO) (black square), S(FeT) (red dot), andS(Gd) (blue triangle) curves represent differ- ent spin moments. The error bars represent the magnitude of moment change. (b) ab initio calculation of exchange constants (in units of meV) in the NN models as like in TABLE.II. The calculation param- eters for (UJ)Gd= 3:3eV and (UJ)Fe= 4:7eV . The results at zero temperature are indicated by dashed lines. range of approximately 0.1 B, which is much wider than the Gd ions at room temperature. The calculated exchange con- stants for each configuration are shown in Fig. 11(b). The re- sults show that the antiferromagnetic exchange constants, Jad, still dominate. Exchange constants Jaa,Jdd,Jac, andJdcmay change their signs, where Jdchas the largest variation range from0.6 meV to 0.6 meV . The ground state of GdIG is still a ferrimagnetic configuration, in which the moments of the FeO atoms are arranged anti-parallel to the FeTatoms and parallel to the Gd atoms. We can see that magnon-phonon coupling can induce small fluctuation of magnetic moment and vari- ation of exchange constants, so that the broadening of spin- spectrum can be shown. As shown in Fig. 12, at room temperature, spin-wave modes9 024681012140 2468101214 ω (THz) N 110 Γ 001 Hα β γ Γ 111 P(b)( a) Figure 12. (a) Spin-wave spectrum in the first Brillouin zone derived from ab initio calculations of exchange constants with (UJ)Gd= 3:3eV and (UJ)Fe= 4:7eV for 40 atomic configurations. The entire spin-wave spectrum at zero temperature is denoted using gray lines, and the picked modes for ,, are marked in red, yellow, and green. The blue curves with error bars denote the range changes of the spectrum induced by atomic vibration. (b) Spin-wave spectrum in highly symmetric direction 111. The color means the same as in (a). The directions in the k-space have the standard labels for a bcc reciprocal lattice. are plotted as the blue curves with error bars governed by the NN exchange constants in Fig.11(b). At zero temperature, the lowest frequency -mode and two slightly higher frequency parabolic- and -mode are shown by the red, yellow, and green curves, respectively, which is the same as Fig.6. Other modes are marked by gray curves. We can see that the blue curves can superimpose with other modes and show a signifi- cant spread in energy at room temperature. For the -modes, the frequency of different phonon configurations is nearly the same as red curves in different directions, and the spectral line had a slightly larger distribution range at the Brillouin zone boundary. For the -modes, the spectral lines distribute around the yellow curves, and the distribution range increase as thekvalue increases in all directions. Compared with the acoustic branch of YIG,13the spectrum shows a smaller dis- tribution range. For the -modes, the spectral lines also dis- tribute around green curves and disperse larger than the - mode. However, the distribution in all directions decreases then increases with increasing k, which is not the case for the YIG.13. So the spin-wave spectrum using the phonon con- figurations at room temperature shows a noticeable broaden- ing. As shown in Fig. 13, the broadening of the spectrum,
!, for the-,-, and -modes are extracted from 40room- temperature configurations by using the method in Ref. 13. In Fig. 13(a),
!has a strong dependence on the value of k. When the k-value is small,
!in the three high symme- try directions are very close to each other, and have differ- ent trends with increasing k. For the-mode,
!increase slowly with increasing k, but in the (111) direction, there is a slight decrease when the kvalue approaches the Brillouin zone boundary. Compared with the three directions, we find /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 /s48 /s52 /s56 /s49/s50/s48/s49/s50 /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 /s32/s49/s49/s48 /s32/s49/s49/s49/s32/s40/s84/s72/s122/s41 /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 /s32/s40/s84/s72/s122/s41 /s32/s48/s48/s49 /s32/s49/s49/s48 /s32/s49/s49/s49/s32/s40/s84/s72/s122/s41 /s107 /s32/s40 /s47/s97/s41/s40/s97/s41 /s40/s98/s41 /s45/s109/s111/s100/s101 /s45/s109/s111/s100/s101 /s45/s109/s111/s100/s101Figure 13. (a) Calculated broadening of the spin-wave spectrum of GdIG,
!, at room temperature as a function of k. Inset:
!replot- ted as a function of the spin-wave frequency, !. (b)
!(k)replotted on a log-log scale. The black lines indicate a constant
!and a quadratic dependence on kfor the optical branch ( -, - modes) and the acoustic branch ( -mode), as shown in Figs. 6 and 12, respec- tively. the relationship of
!(001)>
!(110)>
!(111) in the region ofk > =a . For the-mode,
!increases with in- creasingk. Upon comparing the three directions, we find the relationship of
!(001)>
!(111)>
!(110) in the re- gion ofk > =a . For the -mode,
!decrease as kin- crease; however, in the (001) direction, there is a small in- crease when kapproaches the Brillouin zone boundary. Upon comparing the three directions, we find the relationship of
!(001)>
!(110)>
!(111) in the region of k >=a . The trend for the three modes can also be obtained from the in- set in Fig. 13(a), where the curves in each direction are almost exactly the same, indicating that the anisotropy plays a negli- gible role in the broadening
!. Combined with Fig.7(a), we find that the - and - modes have the same positive polar- ization direction, and the trend of broadening is consistent as the wave vector changes in different directions. However, the - mode has a negative polarization direction, and the trend is10 different. Using the broadening
!of the spin-wave spectra at room temperature and the uncertainty relationship of
!
mp=~, we calculate the magnon-phonon thermalization time, mp or spin-lattice relaxation time to explore the magnon-phonon interactions. In Fig. 13(b),
!is replotted on a logarith- mic scale for observing the asymptotic behavior in the long- wavelength region, where we find a quadratic dependence onkof
!for the- modes and constants
!= 2:07 meV and
! = 9:14meV for the - and - modes, re- spectively, corresponding to  mp= 3:181013s and  mp= 7:191014s as illustrated by the black solid lines. As shown in Fig. 11, the lattice vibrations can induce fluctu- ations of magnetic moment and the exchange constants. Addi- tionally, the phonon-induced fluctuation of the exchange con- stants has an obvious effect on the magnon spectrum and can induce broadening of the spin-wave spectrum at room temper- ature (as shown in Fig. 12). At the long-wavelength limit ( k !0), the acoustic phonon represents the centroid motion of atoms in the same unit cell, so that the change in atomic dis- placement caused by the temperature has little effect on the lattice; so for the -mode, lattice vibration induced spin-wave broadening is approximately zero. In addition, the decay rate of the spin-wave is found to be proportional to the square of kat the long-wavelength limit, as shown in the hydrodynamic theory for spin-wave.66,67Thusmpis proportional to k2for the acoustic -mode. For - and - modes, as the optical phonon represents the reverse motion of the positive and neg- ative ions in the unit cell, the temperature causes the fluctua- tion of the average displacement of atoms, which can induce a constant spin-wave broadening, so mpis constant for the optical modes. To compare with YIG, we also chose a specific wave vector, k= 5:67105cm1, from Ref. 13 and 45, and values for the
!of three modes are 6:49105THz, 4:86101THz, and1:70THz. We obtain mp= 2:45109s,3:271013s, and9:361014s for the-,-, and -modes, respectively, which are approximately 4:3times, 0:6103times, and 0:1 times the values for the acoustic branch and lowest-frequency optical branch of YIG. As shown in Fig. 10(d), for the YIG- like-mode, the sufficient density of state of the phonons can induce larger magnon-phonon scattering rate in the long- wavelength region so that the magnon-phonon thermalization timempis rather small, which is similar to the case of YIG.13. For the optical -mode, it also has a relatively high frequency, where the phonons have a large density of state so that the magnon-phonon scattering rate is quite large, which can re- sult in a smaller magnon-phonon thermalization time. III. CONCLUSION In conclusion, we investigate the NN exchange interaction coefficient using a more reliable and accurate method, which has been applied to YIG. We obtained the Curie temperature and magnetic compensation temperature that matched the ex- periment well. We found that the spin-wave spectrum ob-tained by numerical methods using the exchange constants can explain the experimental phenomena in SSE well. We reveal the spin-wave precession mode in the low frequency region, which indicates that the acoustic branch -modes and YIG-like optical branch -modes have different chiral charac- teristics, but the same as the lower optical -modes. A first- principles phonon calculation method was used to obtain the phonon spectrum of GdIG and YIG at zero temperature. We reproduce the fitting intersecting point of the spin-wave and phonon branches(LA, TA) that are in good agreement with experiment results in the very low-energy region. We discuss the interaction between magnons and phonons in GdIG by in- troducing temperature-dependent lattice shifts. Three special spin-wave modes ( ,, and ) are found to exhibit differ- ent broadening of the spin-wave spectrum,
!of GdIG. In a small wave vector region, the
!of the- modes have a square relationship with wave vector k (
!k2). For the - modes, the
!are nearly a constant, which is similar to the lower optical branch of YIG.13A higher optical branch -mode also exists below KbT6:25THz at room tempera- ture, which may play an indispensable role in magnon-phonon coupling, and the
!has also a constant relationship with k. At a particular wave vector, the magnon-phonon thermaliza- tion time,mp, for these branches at room temperature is also different from that of YIG. mp109s for the-mode is bigger than the acoustic branch of YIG, the mpof the- and -mode (1013and1014s) are smaller than the acous- tic branch and lower optical branch of YIG, respectively. The magnon-phonon coupling effect may play more central role in higher spin-wave modes compared with lower modes. Additionally, we also do ab initio phonon calculations us- ing the finite-displacement method in the packages V ASP,48,49 ABACUS,68and QUANTUM ESPRESSO(QE) package69 combined with Phononpy56,60to obtain the phonon spectrum of YIG and GdIG, and the results are consistent with those presented in this paper (not shown here). A well-known prob- lem with most of the theories of magnon-phonon coupling is that they do not take into account the magnon-magnon cou- pling or magnon-phonon coupling directly. Thus we aim to develop a set of first-principles calculations in the future to include full interactions to study magnon transport properties and lattice dynamics. ACKNOWLEDGMENTS The authors would like to thank Li-Xin He and Xiao- Hui Liu for helpful discussions about ABACUS phonon cal- culation, and thank Joseph Barker for providing atomistic spin dynamics simulation program to calculate Curie tem- perature. This work was financially supported by National Key Research and Development Program of China (Grant No. 2017YFA0303300 and 2018YFB0407601) and the National Natural Science Foundation of China (Grants No.61774017, No.11734004, and No.21421003). We also acknowledge the National Supercomputer Center in Guangzhou (Tianhe II) for providing the computing resources.11 kexia@bnu.edu.cn 1S. Geller and M. A. Gilleo, Journal of Physics and Chemistry of Solids 3, 30 (1957). 2S. Geller and M. A. Gilleo, Journal of Physics and Chemistry of Solids 9, 235 (1959). 3F. Bertaut, F. Forrat, A. 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2019-12-22

We obtained the spin-wave spectrum based on a first-principles method of exchange constants, calculated the phonon spectrum by the first-principles phonon calculation method, and extracted the broadening of the magnon spectrum, $\Delta \omega$, induced by magnon-phonon interactions in gadolinium iron garnet (GdIG). Using the obtained exchange constants, we reproduce the experimental Curie temperature and the compensation temperature from spin models using Metropolis Monte Carlo (MC) simulations. In the lower-frequency regime, the fitted positions of the magnon-phonon dispersion crossing points are consistent with the inelastic neutron scattering experiment. We found that the $\Delta \omega$ and magnon wave vector $k$ have a similar relationship in YIG. The broadening of the acoustic spin-wave branch is proportional to $k^{2}$, while that of the YIG-like acoustic branch and the optical branch are a constant. At a specific $k$, the magnon-phonon thermalization time of $\tau_{mp}$ are approximately $10^{-9}$~s, $10^{-13}$~s, and $10^{-14}$~s for acoustic branch, YIG-like acoustic branch, and optical branch, respectively. This research provides specific and effective information for developing a clear understanding of the spin-wave mediated spin Seebeck effect and complements the lack of lattice dynamics calculations of GdIG.

First-principles study of magnon-phonon interactions in gadolinium iron garnet

1912.10432v1

Propagation of Coupled Acoustic, Electromagnetic and Spin W aves in Saturated Ferromagnetoelastic S olids Qingguo Xiaa, Jianke Dua,* and Jiashi Yangb, aSmart Materials and Advanced Structures Laboratory , School of Mechanical Engineering and Mechanics, Ningbo University, Ningbo, Zhejiang 315211, China bDepartment of Mechanical and Materials Engineering, University of Nebraska -Lincoln, Lincoln, NE 68588 -0526, USA *E-mail: dujianke@nbu.edu.cn (Jianke Du) E-mail: jyang1@unl.edu (Jiashi Yang) Conflict of interest statement : On behalf of all authors, the corresponding author states that there is no conflict of interest. Data availability statement : The data that supports the f indings of this study are available within the article. Keywords : acoustic; electroma gnetic; ferromagnet ic; photon -phonon -magnon interaction Abstract We study the propagation of plane waves in an unbounded body of a saturated ferromagnetoelastic solid. Tiersten’s equations for small fields superposed on finite initial fields in a saturated ferromagnetoelastic material are employed, with their quasistatic magnetic field extended to dynamic electric and magnetic fields governed by Maxwell’s equations for electromagnetic waves. Dispersion relations of the plane waves are obtained. The cutoff frequenc ies and long -wave approximation of the dispersion cur ves are determined. Results show that acousti c, electromagnetic and magnetic spin waves are coupled in such a material. For YIG which is a cubic crystal without piezoelectric coupling, the acoustic and electromagnetic waves are not directly coupled but they can still interact indirectly through spin waves. 1. Introduction In saturated ferromagnetic solids, the magnetization vector has a fixed magnitude (saturation magnetization) and can change its direction only in a precessional motion. Below the Curie temperature, n eighboring magnetization v ectors align themselves in a certain direction called an easy axis of the material to form a distribution of spontaneous magnetization. A disturbance of the magnetization field propagates as spin waves with various applications [1]. Spin waves can interact wit h acoustic waves through magnetoelastic couplings such as piezoma gnetic and magnetostrictive effects, which is called magnon -phonon interaction for which many references can be found in [2], a recent review article. Obviously, as a motion of magnetic moments, spin waves interact with electromagnetic waves directly as go verned by Maxwell’s equations which is referred to as photon -magnon coupling (see [3] and the references therein). Thus, in deformable ferromagnetic solids, acoustic waves and electromagnetic waves can interact indirectly through spin waves. If the materia l is piezoelectric, acoustic waves and electromagnetic waves are also coupled piezoelectrically . It has been reported recently [4] that the couplings among surface acoustic w aves (SAW), spin waves and electromagnetic waves can be used for making electromag 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 - photon -magnon interaction ). 2. Governing Equations Consider the widely -used Yttrium Iron G arnet (Y3Fe5O12) or YIG. In Gaussian units, the governing equations are [5,6] 0 ,,M ij i i j i jM h u , (1) 0M d , (2) 0M b , (3) 10M M Ct be , (4) 1M M Ct dh , (5) 00 ,1()ML ijk j k lk l k ijk j k iM h a h m H m , (6) where τ is the stress tensor. ρ is the mass density. u is the displacement vector. eM, dM, bM and hM are the Maxwellian electric field, electric displacement, magnetic induction and magnetic field. C is the speed of light in a vacuum. M0 and H0 are the initial m agnetization and initial magnetic field which are static . The initial electric and polarization fields are assumed to be zero. hL is an effective local magnetic field which describes the interaction between the magnetic spin and the lattice [5]. a describe s the exchange interaction between neighboring magnetic spins [5]. m is the incremental magnetization vector . γ is the gyromagnetic ratio which is a negative number. (1) is the linear momentum equation . (2)-(5) are Maxwell’ s equations . (6) is the angular momentu m equation of the magnetic spin . (2) and (3) are essentially implied by (4) and (5). We also have the following relationship s: 0 ,4, 4 4 ,MM i i i MM i i i i j jd e p b h m M u (7) where p is the electric polarization vector. Magne toelectric coupling , if present, is not considered . YIG is a cubic crystal of class (m3m) . Let the spontaneous magnetization M0 (and H0) be along the x3 axis. In this case m3=0 because of the saturation condition MM =(M0)2 which implies that M0·m=0 where M=M0+m and m is small . The constitutive relations are [6] 1 11 11 1,1 12 2,2 12 3,3 2 22 12 1,1 11 2,2 12 3,3 3 33 12 1,1 12 2,2 11 3,3, , ,c u c u c u c u c u c u c u c u c u (8) 0 4 23 44 2,3 3,2 44 2 0 5 31 44 1,3 3,1 44 1 6 12 44 1,2 2,1( ) 2 , ( ) 2 , ( ),c u u b M m c u u b M m c u u (9) ore M M M i i i ip e d e , (10) 0 2 0 1 1 44 1,3 3,1 0 2 0 2 2 44 2,3 3,2 3( ) 2 ( ), ( ) 2 ( ), 0,L L Lh M m b M u u h M m b M u u h (11) 3 0 2 0 1 1 44 1,3 3,1 0 2 0 2 2 44 2,3 3,2 3( ) 2 ( ), ( ) 2 ( ), 0,L L Lh M m b M u u h M m b M u u h (12) 11 ,2ib b iam , (13) where 3 11 2 11 11 2 11 2 12 44 22 11 12 44 5 2 11 2 4 12 4 11 11 7 2 05.172 g/cm , 26.9 10 dyn/cm , 10.77 10 dyn/cm , 7.64 10 dyn/cm , 1.66 10 , 1.66 10 , 3 3.36 10 Oe , 1.87 10 cm , 1.76 10 Oe-cm /dyn-sec, 1750 / 4 G.c cc b b b M (14) YIG is nonpiezo electric and nonpiezomagnetic in its natural state without any fields. Due to the spontaneous magnetization and magnetostriction, it becomes effectively piezomagnetic. 3. Antiplane Motion Consider cubic crystals of class (m3m) such as YIG in Gaussi an units . With the initial magnetization M0 and magnetic field H0 along the x3 axis, for antiplane problems [7] with u1=u2=0 and ∂/∂x3 = 0, the relevant fields are 1 2 3 3 1 2 1 2 3 3 1 2 1 2 3 3 1 2 1 1 1 2 2 2 1 2 3 1 1 1 2 2 2 1 2 30, ( , , ), 0, 0, ( , , ), 0, 0, ( , , ), ( , , ), ( , , ), 0, ( , , ), ( , , ), 0.M M M M M M M M M M M M M M M M M Mu u u u x x t e e e e x x t d d d e x x t b b x x t b b x x t b h h x x t h h x x t h (15) In this case (2) is trivially satisfied. (4) and (5) reduce to 12 3,2 3,1110, 0MM MM bbeeC t C t , (16) 3 2,1 1,21M MM dhhCt . (17) We also have: 0 44 3,11 3,22 44 1,1 2,2 3 0 0 0 3 0 2 0 2 11 2,11 2,22 2 44 3,2 2 1 0 0 0 3 0 2 0 1 11 1,11 1,22 1 44 3,1 1 2 1,1 2,2 1,1 2,2( ) 2 ( ) , 12 ( ) ( ) 2 ( ) , 12 ( ) ( ) 2 ( ) , 4 ( ) 0,M M MMc u u b M m m u M h M m m M m b M u H m m M h M m m M m b M u H m m h h m m (18) where (18) 4 is essentially implied by (16). Then (16), (17) and (18) 1-3 can be written as six equations for u3, 3Me , 1Mh , 2Mh , 1m and 2m . 4 4. Propagation of Plane Waves Let 3 1 1 3 2 1 1 3 1 2 4 1 1 5 1 2 6 1exp[ ( )], exp[ ( )], exp[ ( )], exp[ ( )], exp[ ( )], exp[ ( )].M MMu A i x t e A i x t h A i x t h A i x t m A i x t m A i x t (19) The substitution of (19) into (16), (17) and (18) 1-3 results in a system of six linear and homogeneous equations for A1 through A6. For nontrivial solutions, the determinant of the coefficient matrix has to vanish. This leads to the following equation that determines the dispersion relatio n of the wave: 2 2 2 6 2 2 2 4 44 44 2 2 2 44 44 2 0 2 24 2 0 2 2 2 6 2 2 2 4 44 44 2 2 2 44 44 2 0 2(2 4 ) ( 4 ) (2 4 )() ( 4 )() 2 ( 4 ) ( 4 ) ( 4 ) 2 ( 4 )()C c c P e ce P c P P PM PPM c c P e ce P c P PM 2 2 2 4 2 0 2( 4 ) 0,()PM (20) where 0 0 0 0 2 44 112 , 2 , / , ( )e b M P H M K K M . (21) In the special case of b44=0, the magnetoelastic coupling disappears and (20) reduces to th e product of two factors. One is for uncoupled acoustic waves: 22 44 0 c . (22) The other is for coupled electromagnetic and spin waves : 22 2 2 2 2 2 2 2 2 0 2 2 0 2( )( 4 ) ( 4 ) 0( ) ( )C P P PMM . (23 ) When C→∞, (23 ) reduces to the following dispersion relation for uncoupled spin waves : 2 2 4 2 2 0 2(2 4 ) ( 4 )()P P PM . (24) When α=0 and γ→∞, (23 ) reduces to the following dispersion relation for uncoupled electromagnetic waves 22 2( 4 )CP P . (25) When ε=1 and M0→0, we have P→∞ and (25 ) reduces to ω/ξ=C for electromagnetic waves in a vacuum. When H0=1500 Oe and ε=14, the disper sion relatio ns of the uncoupled waves in (22), (24) and (25) are shown in Fig. 1 in logarithmic scales for both the coordinate and the abscissa. The acoustic 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 straight li ne intersects with the curve at two points. Fig. 1. Uncoupled acoustic, electromagnetic and spin waves. When H0=1500 Oe and ε=14, the dispersion relations of the coupled waves determined by (20) are shown in Fig. 2 . Near B and C there are strong couplings between acoustic and spin waves. Near A and D there are strong couplings between electromagnetic and spin waves. Since the material is nonpiezoelectric, there is no direct coupling between acoustic and electromagnetic waves. Fig. 2 . Dispersio n curves of coupled waves when H0=1500 Oe and ε=14. 6 H0 is an independent parameter. If it is varied a little, its effect s on the d ispersion curves are shown in Fig. 3. The spin waves are sensitive to H0 but the two other waves are not, which is reasonable. Fig. 3. Effects of H0 (in Oe) on dispersion relations of coupled waves. ε=14. The numerical value of ε for YIG in the literat ure ran ges from 14 to 18. The effect of slightly different values of ε on the dispersion curves of the coupled waves is shown in Fig. 4 where ε is denoted by εr which mainly affects the electromagnetic waves as expected. Fig. 4 . Effects of εr=ε on disper sion curves of coupled waves . H0=1500 Oe. 7 5. Conclusion s In saturated ferromagnetoelastic solids such as YIG, acoustic, electromagnetic and magnetic spin waves are coupled. Thus it is possible to manipulate one wave by another or design transducers usin g the couplings of these waves . This offers more possibilities for new devices. At present, the literature on the three -wave coupling of photons, phonons and magnons are limited, with an absence of the mechanics community. Since the equations of elasticity represent a maj or part of the coupled theory for these three waves, mechanics researchers can play an important role in this interdisciplinary area. Acknowledgment This work was supported by the National Natural Science Foundation of China (Nos. 12072167 and 11972199), the Zhejiang Provincial Natural Science Foundation of China ( No. LR12A02001 ), and the K. C. Wong Magana Fund through Ningbo University . References [1] D.D. Stancil , A. Prabhakar , Spin Waves : Theory and Applications , Springer, New York, 2009 [2] D.A. Bozhko, V.I. Vasyuchka, A.V. Chumak, A.A. Serga, Magnon -phonon interactions in magnon spintronics (Revi ew article), Low Temp. Phys., 46, 383 -399, 2020. [3] B. Bhoi, S. -K. Kim, Roadmap for photon -magnon coupling and its applications, Solid State Physics , Volume 71, Chapter 2, 39 -71, Elsevier, 2020. [4] R. Fabiha, J. Lundquist, S. Majumder, E. Topsakal, A. Ba rman , S. Bandyopadhyay , Spin wave electromagnetic nano -antenna enabled by tripartite phonon -magnon -photon coupling, Adv. Sci., 9, 2104644 , 2022 . [5] H.F. Tiersten, Coupled magnetomechanical equation for magnetically saturated insulators, J. Math. Phys., 5 , 1298 -1318, 1964. [6] H.F. Tiersten, Thickness vibrations of saturated m agneto elastic p lates, J. Appl. Phys., 36, 2250 -2259, 1965. [7] Q.G. Xia, J.K. Du, J.S. Yang, Antiplane problems of saturated ferromagnetoelastic solids, Acta Mech., under review.

2023-07-18

We study the propagation of plane waves in an unbounded body of a saturated ferromagnetoelastic solid. The equations by Tiersten for small fields superposed on finite initial fields in a saturated ferromagnetoelastic material are employed, with their quasistatic magnetic field extended to dynamic electric and magnetic fields for electromagnetic waves. Dispersion relations of the plane waves are obtained. The cutoff frequencies and long wave approximation of the dispersion curves are determined. Results show that acoustic, electromagnetic and magnetic spin waves are coupled in such a material. For YIG which is a cubic crystal without piezoelectric coupling, the acoustic and electromagnetic waves are not directly coupled but they can still interact indirectly through spin waves.

Propagation of Coupled Acoustic, Electromagnetic and Spin Waves in Saturated Ferromagnetoelastic Solids

2307.09171v1

1Formation of Bright Solitons from Wave Packets with Repulsive Nonlinearity Zihui Wang ,1 Mikhail Cherkasskii,2 Boris A. Kalinikos ,2 Lincoln D. Carr ,3 and Mingzhong Wu1* 1Department of Physics, Colorado State University, Fort Collins, Colorado 80523 , USA 2St.Petersburg Electrotechnical University, 197376, St.Petersburg, Russia 3Department of Physics, Colorado School of Mines, Golden, Colorado 80401, USA Formation of bright envelope solitons from wave packets with a repulsive nonlinearity was observed for the first time. The e xperiments used surface spin -wave packets in magnetic yttrium iron garnet (YIG) thin film strips. When the wave packet s are narrow and ha ve low power, they undergo self -broadening during the propagation. When the wave packet s are relatively wide or their power is relatively high, they can experience self-narrowing or even evolve into bright soliton s. The experimental results were reproduced by numerical si mulations based on a modified nonlinear Schrödinger equation model . 2Solitons are a universal phenomenon in nature, appearing in systems as diverse as water, optical fibers, electromagnetic transmission lines, deoxyribonucleic acid, and ultra -cold quantum gases .1,2,3,4,5 The formation of solitons from large -amplitude waves can be described by paradigmatic nonlinear equations, one of which is the nonlinear Schrödinger equation (NLSE). In the terms of the NLSE model, two classes of envelope solitons, bright and dark, can be excited in nonlinear media. A bright envelope soliton is a localized excit ation on the envelope of a large -amplitude carrier wave. It typically takes a hyperbolic secant shape and has a constant phase across its width.6 A dark envelope soliton is a dip or null in a large -amplitude wave background. When the dip goes to zero, one has a black soliton. When the amplitude at the dip is nonzero, one has a gray soliton. A dark soliton has a jump in phase at its center. For a black soliton, such a phase jump equals to π. For a gray soliton, the phase jump is between 0 and π. The envelope of a dark soliton can be described by a unique function.3 For a black soliton, this function is typically a hyperbolic tangent function. According to the NLSE model, the formation of a bright soliton from a large -amplitude wave packet is possible in systems with an attractive (or self -focusing) nonlinearity and is prohibited in systems with a repulsive (or defocusing) nonlinearity . The un derlying physics is as follows . The attractive nonlinearity produces a pulse self -narrowing effect ; at a certain power level the self -narrowing can balance the dispersion -induced pulse self-broadening and give rise to the formation of a bright envelope so liton. In contrast, in systems with a repulsive nonlinearity the nonlinearity induces self -broadening of the wave packet, just as the dispersion does , and thereby disables the formation of a bright soliton . Previous experiments show good agreement s with these theoretical predictions: the formation of bright solitons from wave packets has been demonstrated in different systems with an attractive nonlinearity ,3,7 while the self-broadening has been observed for wave packets in systems with a repulsive nonlinearity.8 This letter reports on the first observation of the formation of bright solitons from wave packets with a repulsive nonlinearity. The experiments made use of spin waves traveling along long and narrow magnetic yttrium iron garnet (Y3Fe5O12, YIG)9 thin film strip s. The YIG strips were magnetized by static magnetic field s applied in the ir plane s and perpendicular to the ir length direction s. This film/field configuration supports the propagation of surface spin waves with a repulsive nonlinearity.10,11 To excite a spin wave packet in the YIG strip, a microstrip line transducer was placed on one end of the YIG strip and was fed with a microwave pulse . As the spin wave packet propagates along the YIG strip, it was measured by either a secondary microstrip line or a magneto -dynamic inductive probe located above the YIG strip. When the input microwave pulse is relatively narrow and has relatively low power, one 3observes the broadening of the spin wave packet during its propagation . At certain large input pulse width s and high power level s, however, the spin wave packet undergoes self -narrowing and evolves into a bright envelope soliton. The formation of this soliton is contradictory to the prediction of the standard NLSE model , but was reproduced by numerical simulations with a modified NLSE model that took into account damping and saturable nonlinearity . Figure 1 shows representative data on the formation of bright solitons from surface spin -wave packets. Graph (a) shows the experimental configuration. The YIG film strip was cut from a 5.6-mm-thick (111) YIG wafer grown on a gadolinium gallium garnet substrate. The strip was 30 mm long and 2 mm wide. The magnetic field was set to 910 Oe. The input and output transducers were 50 -mm-wide striplines and were 6.3 mm apart. The input microwave pulses had a carrier frequency of 4.51 GHz. Note that, in Fig. 1 and other figures as well as the discussions below, Pin denotes the nominal microwave pulse power applied t o the input transducer, tin denotes the half-power width of the input microwave pulse, Pout is the power of the output signal, and tout represents the half -power width of the output pulse. In Fig. 1, g raphs (b), (c), (d), (f), and (g) give the power profiles of the output signals measured with different FIG. 1. Propagation of spin -wave packets in a 2.0-mm-wide YIG strip. (a) Experimental setup. (b), (c), (d), (f), and (g) Envelopes of output signals obtained at different input pulse power levels ( Pin) and widths ( tin). (e) Phase ( q) profile for the signal shown in (d). (h) Width of output pulse ( tout) as a function of Pin. (i) Width of output pulse as a function of tin. 4Pin andtin values , as indicated . The circles in (d) shows a fit to the hyperbolic secant squared function.1,3 Graph (e) shows the corresponding phase ( q) profile of the signal shown in (d) . Here , the profile shows the phase relative to a reference continuous wave whose frequency equals to the carrier frequency of the input microwave.6 Graph (h) shows the 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, as indicated. The data in Fig. 1 show three important results. (1) The data in Figs. 1 (b) -(e) and (h) show the change of the output signal with the input power Pin. One can see that the output pulse is broader than the input pulse when Pin=13 mW, as shown in (b), and is significantly narrower when Pin>30 mW , as shown in (c), (d), and (h). This indicates that the spin-wave packet undergoes self -broadening at low power and self -narrowing at relatively high power. (2) The data in Figs. 1 ( d), (f), (g), and ( i) show the change of the output signal with the input pulse width tin. It is evident that 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. These results indicate that the spin-wave packe t experiences strong self -narrowing when it is relatively broad. (3) The pulses shown in (d) and (g) are indeed bright solitons. As shown representatively in (d) and (e), they have a hyperbolic secant shape and a constant phase profile at their centers, which are the two key signatures of bright solitons.1,6 The data from Fig. 1 clearly demonstrate the formation of bright solitons from surface spin -wave packets when the energy of the initial signals (the product of Pin and tin) is beyond a certain level. This result is contradictory to the predictions of the NLS E model. One possible argument is that the width of the YIG strip might play a role in the observed formation of bright solitons. To rule out this possibility, similar measurements were carried out with a n YIG strip that is an order of magnitude narrower. The main data are as fo llows. Figure 2 gives the data measured with a 0.2 -mm-wide YIG strip . This figure is shown in the same format as in Fig. 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 secondary microstrip transducer. The distance between the input transducer and the inductive probe was about 2.6 mm. The magnetic field was set to 1120 Oe. The input microwave pulse had a carrier frequency of 5.07 GHz. The data in Fig. 2 show results very similar to those shown in Fig. 1. Specifically, the low -power, narrow spin- wave packets undergo self -broadening, as shown in (b), (c), (f), and (h); as the power and width are increased to certain levels, the spin-wave packets experience self -narrowing, as shown in (h) and (i), and can also evolve into solitons, as shown in (d), (e), and (g). Therefore, the data in Fig. 2 clearly confirm the results from Fig. 1. This indicate s that the formation of solitons reported here is n ot due to any effects associated with the YIG strip width. Note that the solitons 5shown in Fig. 2 are narrower than those shown in Fig. 1. This difference results mainly from the fact that the spin- wave amplitudes and dispersion properties were different in the two experiments . The spin -wave dispersion differed in the two experiments because the magnetic field s were different and the wave number s of the excited spin -wave mode s were also not the same. Turn now to the spatial formation of solitons from surface spin -wave packets. F igure 3 shows representative data. Graph (a) gives the profile of an inpu t signal. The power and carrier frequency of the input signal were 700 mW and 5.07 GHz, respectively. Graphs (b) -(f) give the corresponding output signals measured with the same experimental configuration as depicted in Fig. 2(a). The signals were measured by placing the inductive probe at different distances (x) from the input transducer, as indicated. The red curves in ( b)-(f) are the corresponding phase profiles. The 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 to that of the input pulse. As the packet propagates to x=2.1 mm, it develops into a soliton, which is not only much FIG. 2. Propagation of spin -wave packets in a 0.2 -mm-wide YIG strip. (a) Experimental setup. (b), (c), (d), (f), and (g) Envelopes of output signals obtained at different input pulse power levels ( Pin) and widths ( tin). (e) Phase ( q) profile for the signal shown in (d). (h) Width of output pulse ( tout) as a function of Pin. (i) Width of output pulse as a function of tin. 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 shown in (c). At x=2.6 mm, the packet has a lower amplitude due to the magnetic damping but still maintains its solitonic nature, as shown in (d). As the packet continues to propagate further, it loses its solitonic properties and undergoes self -broadening, as shown in (e) and (f) , due to significant reduction in amplitude . Note that the phase profiles for all the signals in (b), (e), and (f) are not constant . These results support the above -drawn conclusion, namely, that it is possible to produce a bright soliton from a surface spin -wave packet. The data in Fig. 3 also indicate the other two important results. (1) The development of a soliton takes a certain distance, about 2 mm for the above -cited conditions, due to the fact that the nonlinearity effect needs a certain propagation distance to develop. (2) The soliton exists on ly in a relatively short range, about 1 -2 mm for the above - cited conditions , due to the damping of carrier spin waves. To increase the "life" distance or lifetime of a spin-wave soliton, one can take advantage of parametric pumpin g13 or active feedback9 techniques. As mentioned above, the soliton formation presented here is contradictory to the standard NLSE model. However, it can be reproduced by numerical simulations based on the equation FIG. 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 output signals measured by an inductive probe placed at different distances ( x) from the input transducer. The red curves in ( b) and ( f) are the corresponding phase profiles. 800 850 900 950 10000.00.20.40.60.8Power (W)(a) Input pulse 800 850 900 950 10000.000.020.040.06 (b) x=1.1 mmPower (mW) 800 850 900 950 10000.000.010.020.030.04 (f) x=4.6 mm (e) x=3.6 mm(d) x=2.6 mm (c) x=2.1 mmPower (mW) 800 850 900 950 10000.000.010.02Power (mW) 800 850 900 950 10000.000.010.02Power (mW) Time (ns)800 850 900 950 10000.000.010.02Power (mW) Time (ns) Phase 180 ºPhase 180 º Phase 180 º180 º Phase 180 º Phase 7( )22 4 2102gu u ui v u D N u S u ut x xh¶ ¶ ¶é ù+ + - + + =ê ú¶ ¶ë û ¶ (1) where u is the amplitude of a spin -wave packet, x and t are spatial and temporal coordinates, respectively, vg is the group velocity, h is the damping coefficient, D is the dispersion coefficient, and N and S are the cubic and quintic nonlinearity coefficients, respectively. The quantic nonlinearity term is included because the cubic nonlinearity i s insufficient to capture the experimental observations presented abov e. This additional term is an expansion to the lowest order of saturable nonlinearity. The simulations used the split -step method to solve the derivative terms with respect to x and used the Runge -Kutta method to solve the equation with the rest of the terms.14,15 A high-order Gaussian profile was taken in simulations for the input pulse because it is much closer to the experimental situation than a squared pulse. The use of a square pulse as in the input pulse gave rise to numerical noise due to the discontinuity at the pulse's edges. The use of a fundamental Gaussian function did not onsiderably change the simulation results. It sho uld be noted that both the standard and modified NLSE models are for nonlinear waves in one-dimensional (1D) systems, and previous work had demonstrated the feasibility of using the 1D NLSE models to describe nonlinear spin waves in quasi -1D YIG film strips.16,17 Figure 4 shows representative results obtained for different initial pulse amplitudes ( u0), as indicated. In each panel, the left and right diagrams show the power and phase profiles, res pectively. The simulations were carried out for a 20-mm-long 1D film strip and a total propagation time of 250 ns. The film strip was split into 9182 steps, and the temporal evolution step was set to 0.05 ns. The input pulse was a high -order Gaussian profile with an order number of 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, D=-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 calculated according to the properties of the YIG film, 9 and the S was optimized for the reproduction of the experimental responses. The profiles in Fig. 4 indicate that, at low initial power , the pulse is broader than the initial pulse and has a phase profile which is not constant at the pulse center, as shown in (a) and (b); at relatively high power, however, the pulse is not only significantly narrower than the initial pulse but also has a constant phase across its center portion, as shown in (c). These results agree with the experimental results presented above. The reproduction of the experimental responses with the modified NLSE model indicates the underlying physical 8processes for the formation of bright solitons from surface spin -wave packets. In comparison with the standard NLSE, the additional terms in the modified equation are uh and 4S u u . The term uh accounts for the damping of spin waves in YIG films, while the term 4S u u is needed for the reproduction of the experimental responses. Since the sign of S was opposite to that of N, the term 4S u u played a role opposite to 2N u u and caused nonlinearity saturation. In particular, for the configuration cited for Fig. 4(c) the term 4S u u overwhelm ed the term 2N u u , resulting in a repulsive -to-attractive nonlinearity transition and the formation of a bright soliton. Thus, one can see that the saturable nonlinearity played a critical role in the formation of the bright soliton s from surface spin -wave packets . It should be noted that t he saturable nonlinearity has been known as a critical factor for the formation of solitons in optical fibers.18 In summary, this letter reports the first observation of the formation of bright solitons from surface spin -wave packets propagating in YIG thin films . The formation of such soliton s was observed in YIG film strips with significantly different widths. The spatial evolution of the solitons was measured by placing an inductive probe at different po sitions along the YIG strip. The experimental observation was reproduced by numerical simulations based FIG. 4. Power (left) and phase (right) profiles of spin -wave packets propagati ng in a YIG strip. The profiles were obtained from simulations 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.) Time (ns)160 180 200-180-90090180Phase (degree) Time (ns)(a) Initial pulse amplitude u0=0.0005 150160170 1801902002100.00.20.40.60.81.0Power (a.u.) Time (ns)150 160170180 190200210-180-90090180Phase (degree) Time (ns) 150160170 1801902002100.00.20.40.60.81.0Power (a.u.) Time (ns)150 160170180 190200210-180-90090180(c) Initial pulse amplitude u0=0.071(b) Initial pulse amplitude u0=0.005Phase (degree) Time (ns) 9on a modified NLSE model. The agreement between the experimental and numerical results indicates that the saturable nonlinearity played important role s in the soliton formation. This work was supported in part by U. S. National Science Foundation (DMR -0906489 and ECCS -1231598) and the Russian Foundation for Basic Research . *Corresponding author. E-mail: mwu@lamar.colostate.edu 10 1 M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (SIAM, Philadelphia, 1985). 2 A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Oxford, New York, 1995). 3 M. Remoissenet, Waves Called Solitons: Concepts and Experiments (Springer -Verlag, Berlin, 1999). 4 Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystal s (Academic, New York, 2003). 5 P. G. Kevrekidis, D. J. Frantzeskakis, and R. Carretero -González, Emergent Nonlinear Phenomena in Bose -Einstei n Condensates (Springer -Verlag, Berlin, 2008). 6 J. M. Nash, P. Kabos, R. Staudinger, and C. E. Patton, J. Appl. Phys. 83, 2689 (1998). 7 M. Chen, M. A. Tsankov, J. M. Nash, and C. E. Patton, Phys. Rev. B 49, 12773 (1994) . 8 M. Chen, M. A. Tsankov, J. M. N ash, and C. E. Patton, Phys. Rev. Lett. 70, 1707 (1993) . 9 Mingzhong Wu, “Nonlinear Spin Waves in Magnetic Film Feedback Rings, ” in Solid State Physics Vol. 62, Edited by Robert Camley and Robert Stamps (Academic Press, Burlington, 2011), pp. 163 -224. 10P. Kabos and V. S. Stalmachov, Magnetostatic Waves and Their Applications (Chapman and Hall, London, 1994) . 11D. D. Stancil and A. Prabhakar, Spin Waves – Theory and Applications (Springer, New York, 2009). 12 M. Wu, M. A. Kraemer, M. M. Scott, C. E. Patton, an d B. A. Kalinikos, Phys. Rev. B 70, 054402 (2004) . 13 A. V. Bagada, G. A. Melkov, A. A. Serga, and A. N. Slavin, Phys. Rev. Lett. 79, 2137 (1997) . 14 J. A. C. Weideman and B. M. Herbst, SIAM Journal on Numerical Analysis 23, 485 (1986) . 15 D. Pathria and J. L. Morris, J. Comp. Phys. 87, 108 (1990). 16 H. Y. Zhang, P. Kabos, H. Xia, R. A. Staudinger, P. A. Kolodin, and C. E. Patton, J. Appl. Phys. 84, 3776 (1998). 17 Z. Wang, A . Hagerstrom, J . Q. Anderson, W . Tong, M . Wu, L . D. Carr, R . Eykholt, and B . Kalinikos, Phys. Rev. Lett. 107, 114102 (2011). 18 S. Gatz and J. Herrmann, J. Opt. Soc. Am. B 8, 2296 (1991) .

2015-05-07

Formation of bright envelope solitons from wave packets with a repulsive nonlinearity was observed for the first time. The experiments used surface spin-wave packets in magnetic yttrium iron garnet (YIG) thin film strips. When the wave packets are narrow and have low power, they undergo self-broadening during the propagation. When the wave packets are relatively wide or their power is relatively high, they can experience self-narrowing or even evolve into bright solitons. The experimental results were reproduced by numerical simulations based on a modified nonlinear Schr\"odinger equation model.

Formation of Bright Solitons from Wave Packets with Repulsive Nonlinearity

1505.01882v1

Critical Cavity-Magnon Polariton Mediated Strong Long-Distance Spin-Spin Coupling Miao Tian,1Mingfeng Wang,1Guo-Qiang Zhang,2,Hai-Chao Li,3,yand Wei Xiong1,z 1Department of Physics, Wenzhou University, Zhejiang 325035, China 2School of Physics, Hangzhou Normal University, Hangzhou, Zhejiang 311121, China 3College of Physics and Electronic Science, Hubei Normal University, Huangshi 435002, China (Dated: April 28, 2023) Strong long-distance spin-spin coupling is desperately demanded for solid-state quantum infor- mation processing, but it is still challenged. Here, we propose a hybrid quantum system, consisting of a coplanar waveguide (CPW) resonator weakly coupled to a single nitrogen-vacancy spin in di- amond and a yttrium-iron-garnet (YIG) nanosphere holding Kerr magnons, to realize strong long- distance spin-spin coupling. With a strong driving eld on magnons, the Kerr eect can squeeze magnons, and thus exponentially enhance the coupling between the CPW resonator and the squeezed magnons, which produces two cavity-magnon polaritons, i.e., the high-frequency polariton (HP) and low-frequency polariton (LP). When the enhanced cavity-magnon coupling approaches the critical value, the spin is fully decoupled from the HP, while the coupling between the spin and the LP is sig- nicantly improved. In the dispersive regime, a strong spin-spin coupling is achieved with accessible parameters, and the coupling distance can be up to cm. Our proposal provides a promising way to manipulate remote solid spins and perform quantum information processing in weakly coupled hybrid systems. I. INTRODUCTION Solid spins such as nitrogen-vacancy centers in dia- mond [1], having good tunability [2] and long coher- ence time [3{5], are regarded as promising platforms for quantum information science [14, 15]. However, direct spin-spin coupling is weak due to their small magnetic dipole moments [16{21]. Moreover, the coupling dis- tance is directly determined by their separation. To over- come these, the natural ideal is to look for quantum in- terfaces [6{13] as bridges to couple long-distance spins, forming diverse hybrid quantum systems [14, 15]. Recently, the emerged low-loss magnons (i.e., the quanta of collective spin excitations) in ferromagnetic materials [22{25] have shown great potential in me- diating distant spin-spin coupling [26{31]. For ex- ample, magnons in the Kittle mode of a nanometer- sized yttrium-iron-garnet (YIG) sphere have been used to strongly couple spins with tens of nanometers dis- tance [26{28], via enhancing the local magnetic eld. To further improve the coupling distance between two spins from nanometer to micronmeter, magnons with Kerr eect as quantum interface are proposed [32]. Also, the YIG nanosphere can be used to realize strong spin- photon coupling in a microwave cavity [33]. Besides these, magnons in a bulk material [29, 30] and thin ferro- magnet lm [31] have been suggested to coherently cou- ple remote spins. However, achieved strong coupling is severely limited by the distance between two spins. Motivated by this, we propose a hybrid spin-cavity- magnon system to realize a strong spin-spin coupling zhangguoqiang@hznu.edu.cn yhcl2007@foxmail.com zxiongweiphys@wzu.edu.cnwith coupling distance centimeter . In the proposed sys- tem, the spin in diamond is located at tens of nanometers from the central line of the CPW resonator, and weakly coupled to the CPW resonator. The nanometer-sized YIG sphere supporting Kerr magnons (i.e., magnons with Kerr eect) is employed but weakly coupled to the CPW resonator. Experimentally, strong and tunable magnon Kerr eect, originating from the magnetocrys- talline anisotropy, has been demonstrated [34], giving rise to bi- and multi-stabilities [35{39], nonreciprocity [40], sensitive detection [41], quantum entanglement [42] and quantum phase transition [43, 44]. Under a strong driv- ing eld, this Kerr eect can squeeze magnons, and thus the coupling between magnons and the CPW resonator is exponentially enhanced to the strong coupling regime. The strong magnon-cavity coupling generates two polari- tons, i.e., the high-frequency polariton (HP) and the low- frequency polariton (LP). When the enhanced magnon- cavity coupling strength approaches to the critical value, the LP becomes critical. Then the coupling between the spin and the HP is fully suppressed in the polari- ton representation, while the coupling between the spin and the LP is greatly enhanced. By further consider the case of two spins dispersively coupled to the LP, an in- direct and strong spin-spin coupling can be induced by adiabatically eliminating the degrees of freedom of LP. Moreover, the coupling strength is not limited by the separation between two spins, it is actually determined by the length of the CPW resonator. Experimentally, the centimeter-sized CPW resonator has been fabricated [45]. Therefore, the achieved strong spin-spin coupling can be up tocm. Our proposal privides an alternative path to remotely manipulate solid spin qubits and perform- ing quantum information processing in weakly coupled spin-cavity-magnon systems.arXiv:2304.13553v2 [quant-ph] 27 Apr 20232 /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000018 /uni00000014/uni00000011/uni00000013 /uni00000014/uni00000011/uni00000018 /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 ms=1 ms=0/uni0000000b/uni00000045/uni0000000c /uni00000013/uni00000011/uni00000013/uni00000013 /uni00000013/uni00000011/uni00000014/uni00000018 /uni00000013/uni00000011/uni00000016/uni00000013 /uni00000035/uni00000003/uni0000000b/uni00000050/uni0000000c /uni00000013/uni00000014/uni00000015/uni00000016/uni0000004am/2/uni00000003/uni0000000b/uni00000030/uni0000002b/uni0000005d/uni0000000c /uni0000000b/uni00000046/uni0000000c FIG. 1: (a) Schematic diagram of a hybrid quantum system. The single nitrogen vacancy (NV) center spin, located ddis- tance away from the central line, is weakly coupled to the CPW resonator. The YIG sphere is driven by a microwave eld through a microwave antenna. (b) The level structure of the triplet ground state of the NV center, ms= 0 and ms=1 are selected to form a spin qubit. (c) The cavity- magnon coupling verus the radius Rof the YIG nanosphere. II. MODEL AND HAMILTONIAN We consider a hybrid quantum system consisting of a coplanar waveguide (CPW) resonator weakly coupled to both a single NV spin in diamond and a nanometer-sized YIG sphere, as shown in Fig. 1(a). The spin is fabri- cated far away from the YIG sphere to avoid their direct coupling. In addition, the magnon Kerr eect, stemming from the magnetocrystallographic anisotropy [35, 36], is taken into account. Thus, the total Hamiltonian of the hybrid system can be written as (setting ~= 1), Htot=HNV+HCM+HCS+HK+HD; (1) whereHNV=1 2!NVz, with the transition frequency !NV=DgeBBexbetween the lowest two levels of the triplet ground state of the NV [see Fig. 1(b)], is the free Hamiltonian of the NV spin. Here, D= 22:87 GHz is the zero-eld splitting, ge= 2 is the Land efactor,B is the Bohr magneton, and Bexis the external magnetic eld to lift the near-degenerate stats jms=1i. The second term HCM=!caya+gm aym+amy
(2) represents the Hamiltonian of the coupled magnon-cavtiy susbsytem, where !cis the frequency of the CPW res- onator and gmis the coupling strength [33], nearly propo- tional to the radius Rof the YIG sphere [see Fig. 1(c)].Obviously, strong coupling can be obtained by using micronmeter-sized sphere, which is widely employed in experiments [46{49]. For the nanometer-sized sphere such asR50 nm, we have gm20:2 MHz, which is much smaller than the typical decay rates of the cav- ity (c=21 MHz) [50] and Kittle mode ( m=21 MHz) [51], i.e., gm<  c;m. This indicates that the coupling between the Kittle mode of the nanosphere and the cavity is in the weak coupling regime, consistent with our assumption. The Hamiltonian HCSin Eq. (1) describes the inter- action between the spin qubit and the cavity. With the rotating-wave approximation, HCScan be governed by [19, 20] HCS= +a+ay
; (3) where= 2geBB0;rms(d) [19] is the coupling strength, withB0;rms(d) =0Irms=2d,Irms=p ~!c=2La, andd being the distance between the spin and the center con- ductor of the CPW resonator. To estimate ,!c22 GHz andLa2 nH [51] are chosen. For d5m, 270 Hz, and d50 nm,27 kHz [19], leading to< c. This shows that the coupling between the spin qubit and the cavity is also in the weak coupling regime. Due to this fact, we here assume that the spin qubit is placed close to the central line of the CPW res- onator to obtain a moderate coupling strength, althoght it is still weakly coupled to the cavity. Experimentally, such weak spin-cavity weak couplings can be measured. The Hamiltonian HKin Eq. (1) denotes the magnon Kerr eect, characterizing the coupling among magnons in the YIG sphere and provides the anharmonicity of the magnons, which is given by [36] HK=!mmym+Kmymymm; (4) where!m= B020Kan 2s=M2Vm+0Kan 2=M2Vm is the frequency of the Kittle mode, with the gyromag- netic ration =2=geB=~(Bis the Bohr magneton), the vacuum permeability 0, the rst-order anisotropy constant of the YIG sphere Kan, the amplitude of a bias magnetic eld B0, the saturation magnetization M, and the volume of the YIG sphere Vm.K=0Kan 2=M2Vm is the coecient. Apparently, the Kerr coecient is in- versely proportional to the volume of the YIG sphere, i.e.,K/1=Vm, the Kerr eect can become signicantly important for a YIG nanosphere. For example, when R50 nm,K=2128 Hz, but K=20:05 nHz forR0:5 mm (the usual size of the YIG sphere used in various previous experiments). Obviously, Kis much smaller in the latter case. Because our proposal mainly relies on the Kerr eect, we here use the nanometer-sized YIG sphere to obtain strong Kerr eect. The last term HD= d myei!dt+mei!dt
: (5) in Eq. (1) describes the interaction between the Kittle mode and the driving eld, where dis the Rabi fre- quency and !dis the frequency of the driving eld.3 /uni00000013/uni00000011/uni00000013/uni00000013 /uni00000013/uni00000011/uni00000013/uni00000018 /uni00000013/uni00000011/uni00000014/uni00000013 /uni00000013/uni00000011/uni00000014/uni00000018 /uni00000035/uni00000003/uni0000000b/uni00000050/uni0000000c /uni00000013/uni00000016/uni00000013/uni00000019/uni00000013/uni0000001c/uni00000013/uni0000002a/uni00000012/uni00000015/uni00000003/uni0000000b/uni00000030/uni0000002b/uni0000005d/uni0000000c /uni0000000b/uni00000044/uni0000000crm=0 rm=3 rm=5 /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000015 /uni00000013/uni00000011/uni00000017 /uni00000013/uni00000011/uni00000019 /uni00000013/uni00000011/uni0000001b /uni00000014/uni00000011/uni00000013 G/s /uni00000014 /uni00000013/uni00000014/uni00000015/uni000000162 ±/2 s Gc /uni0000000b/uni00000045/uni0000000c2 + 2 FIG. 2: (a) The coupling strength between the squeezed magnons and the CPW resonator versus the radius of the YIG nanosphere with dierent squeezing parameters rm= 0;3;5. (b) The square of polariton frequencies versus the coupling between the squeezed magnons and the CPW resonator. In the rotating frame with respect to the driving fre- quency (!d), the total Hamiltonian in Eq. (1) becomes Hsys=1 2
NVz+
caya+H0 K+ d my+m
+ +a+ay
+gm aym+amy
;(6) where
NV=!NV!dis the frequency detuning of the spin qubit from the driving eld,
c=!c!dis the fre- quency detuning of the cavity eld from the driving eld, andH0 K=mmym+Kmymymm[32], withm=!m!d being the frequency detuning of the Kittle mode from the driving eld. Due to the large d, the Hamiltonian Hsys in Eq. (6) can be linearized by writting each system op- erator as the expectation value plus its uctuation [52]. By neglecting the higher-order uctuation terms, Eq. (6) is linearize as Hlin=1 2
NVz+
caya+HK + +a+ay
+gm aym+amy
;(7) with HK=
mmym+Ks m2+my;2
; (8) where the eective magnon frquency detuning
m= m+ 4Kjhmij2is induced by the Kerr eect, which has been demonstrated experimentally [35, 36]. The ampli- ed coecient Ks=Khmi2is the eective strength of the two-magnon process, which can give rise to squeeze magnons in the Kittle mode. Aligning the biased mag- netic eld along the crystalline axis [100] or [110] of the YIG sphere [34, 36], Kcan be positive or negative, and we can have Ks>0 orKs<0. There we choose Ks<0 whenK < 0. The linearized Kerr Hamiltonian HKin Eq. (8) describes the two-magnon process, which can give rise to the magnon squeezing. Below we operate the proposed hybrid system in the magnon-squeezing frame by diagnolizing the Hamilto- nianHKwith the Bogoliubov transformation m= mscosh (rm) +my ssinh (rm), whererm=1 4ln
m2Ks
m+2Ksis the squeezing parameter. After diagnolization, HK becomes HKS=
smy sms (9)with
s=p
2m4K2sbeing the frequency of the squeeze magnon, and Eq. (7) is transformed to HS=1 2
NVz+HCMS+ +a+ay
; (10) where HCMS=
caya+
smy sms+G ay+a
my s+ms
(11) is the eective Hamiltonian of the CPW resonator cou- pled to the squeezed magnons, G=1 2gmermis the exponentially enhanced coupling strength between the squeezed magnons and the CPW resonator. Because both the parameters
mandKscan be tuned, so rmcan be very large when
m2Ks, leading to the strong Geven for nanometer-sized YIG sphere [see curves in Fig. 2(a)]. Specically, when rm= 0, i.e., magnons in the Kittle mode is not squeezed, the coupling strength between the CPW resonator and the Kittle mode is un- amplied, giving rise to weak G[see the black curve in Fig. 2(a)]. When magnons in the Kittle mode are squeezed but with moderate squeezing parameters such asrm= 3 andrm= 5, we nd the coupling strength Gcan be signicantly improved for the YIG nanosphere. For example, R50 nm and rm3, we haveG=2= 2 MHz, which is comparable with the decay rates of the CPW resonator ( c) and the Kittle mode ( m). But whenrm5,G=2= 17 MHz, which is much larger than bothcandm. These indicates that indicates that strong coupling between the squeezed magnons and the CPW resonator can be realized by tuning the squeezing parameterrm. In addition, Gcan be further enhanced by using the larger radius of the YIG sphere when rm is xed. Once the strong coupling between the squeezed magnons and the CPW resonator is achieved, the coun- terrotating terms /aymy sandamsin Eq. (11) are related to two-mode squeezing, while rotating terms /aymsand amy sallow quantum state transfer between the squeezed magnons and the CPW resonator. By combining these, polaritons with criticality can be formed, as shown below. III. STRONG COUPLING BETWEEN THE SINGLE NV SPIN AND THE LOW-FREQUENCY POLARITON By further diagnolizing the Hamiltonian HCMS in Eq. (11), two polaritons with eigenfrequencies !2 =1 2
2 c+
2 sq (
2c
2s)2+ 16G2
c
s (12) can be obtained. This is owing to the fact of the achieved strong coupling between the squeezed magnons and the CPW resonator. For convenience, we call two polaritons with frequencies !+and!as the high- and low-frequency polaritons (HP and LP). The diagnolized HCMS reads Hdiag=!+ay +a++!ay a; (13)4 /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000014 /uni00000013/uni00000011/uni00000015 /uni00000037/uni0000004c/uni00000050/uni00000048/uni00000003/uni0000000b/uni00000056/uni0000000c /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 /uni00000033/uni00000052/uni0000004f/uni00000044/uni00000055/uni0000004c/uni00000057/uni00000052/uni00000051 0.0 0.2 0.4 0.6 0.8 1.0 Time (s) 0.00.51.0Occupation(b) Spin Polariton FIG. 3: The occupation of the LP and spin qubit versus the evolution time at G!Gcand
c
s(a) without and (b) with dissipations. The spin decay rate is ?1 kHz and the LP dacay rate is 1 MHz. In both (a) and (b), the spin qubit is initially prepared in the excited state and the LP is in the ground state, and the coupling strength is gr=23:5 MHz. where a=cos 2p
c!h a(
c+!) +ay (
c!)i +sin 2p
c!+h a+(
c+!+) +ay +(
c!+)i :(14) Substituting Eqs. (13) and (14) into Eq. (10), the Hamil- tonian of the coupled spin-CMP can be given by HCMP =1 2
NVz+!+ay +a++!ay a (15) +gr +a+ay  +gcr +ay +a +g0 r +a++ay + +g0 cr +ay ++a+ ; wheregc(cr)=cos(
c!)=2p
c!denote the ef- fective coupling strength between the NV spin and the LP,g0 c(cr)=sin(
c!+)=2p
c!+represent the ef- fective coupling strength between the NV spin and the HP. Obviously, both gc(cr)andg0 c(cr)can be tuned by the driving eld on the Kittle mode of the YIG sphere. The parameteris dened by tan(2 ) = 4Gp
c
s=(
2 c
2 s). To show the behavior of two polaritons with thecoupling strength G, we plot the square of polariton fre- quencies versus the coupling strength Gin Fig. 2(b). Clearly, one can see that !2 +increases with G, but!2 de- creases. When !2 = 0,Greduces to the critical coupling strengthGc, i.e., G=Gc1 2p
c
s; (16) which means !is real forG < G c, while!is imagi- nary (the low-polariton is unstable) for G > G c. When we operate the coupled cavity-magnon subsystem around the critical point (i.e., G!Gc) and
c
sis satised, we havegrgcr!1 2p
c=!,g0 rg0 cr!0. Due to the large
cand the extremely small !,grgcr. These indicate that coupling between the NV spin and the HP is completely decoupled, while the coupling be- tween the NV spin and the LP is signicantly enhanced. By choosing
c= 106!,gr=gcr103are esti- mated. Obviously, three orders of magnitude of the spin- low-polariton coupling is improved. Using d= 50 nm, = 27 kHz is obtained, resulting in gr=2= 3:5 MHz, which is larger than the decay rates of the CPW resonator and the Kittle mode, i.e., gr(cr)>c;m. This suggests that the coupling between the spin and the LP can be in the strong coupling regime. In principle, gr(cr) can be further enhanced by using the larger
cor much smaller!. In the strong coupling regime, the rotating- wave approximation is still valid, and the counterrotating term related to gcrin Eq. (15) can be safely ignored, so Eq. (15) reduces to HJC=1 2
NVz+!ay a+gr +a+ay  ;(17) which is the so-called Janes-Cumming model with the strong coupling, allowing quantum state exchange be- tween the spin and the LP, as demonstrated in Fig. 3(a), where the spin is initially prepared in the excited state and the the LP is in the ground state. When dissipations are included, the dynamics of the system can be described by the master equation, d dt=i[HJC;] +D[a]+ ?D[]; (18) whereD[o]=ooy1 2 oyo+oyo
, and ?is the transversal relaxation rate of the NV spin [53], is the decay rate of the LP. In Fig. 3(b), we use the qutip package in python [54, 55] to numerically simulate the dynamics of the spin and LP governed by Eq. (18). The results show that state exchange between the spin and the LP can be realized in the presence of dissipations such as1 MHz and ?1 kHz [58], althougth the occupation probability decreases with long evolution time.5 0 10 20 30 40 50 Time (s) 0.00.51.0Occupation (a)spin 1 spin 2 Polariton 0 20 40 60 80 100 Time (s) 0.00.51.0Occupation (b)spin 1 spin 2 Polariton FIG. 4: The occupation of two spins and the LP versus the evolution time in the dispersive regime (a) without and (b) with dissipations. The parameters are the same as in Fig. 3. IV. THE EFFECTIVE STRONG COUPLING BETWEEN TWO SINGLE NV SPINS Here, we further consider the case that two identical NV spins are symmetrically placed away from the YIG sphere in the CPW resonator. Thus, two spins interact with the CPW resonator with the same coupling strength . By operating the cavity-magnon subsystem around the critical point, the couplings between two spins and the HP can be fully suppressed, while the couplings be- tween two spins and the LP is greatly enhanced, similar to the single spin case. Therefore, the Hamiltonian of the hybrid system with two identical spins can be eectively described by Tavis-Cumming model, HTC=!ay a+1 2
NV (1) z+(2) z +grh (1) ++(2) + a+ h:c:i : (19) In the dispersive regime, i.e., j
NV!jgr, the LP can be as an interface to induce an indirect coupling be- tween two spins by using the Fr ohlich-Nakajima transfor- mation [56, 57]. By adiabatically eliminating the degrees of freedom of the LP, we can obtain the eective spin-spin Hamiltonian as He=1 2!e (1) z+(2) z +ge (1) +(2) +(1) (2) + ; (20)where!e=
NV+ 2gen+geis the eective tran- sition frequency of the NV spin, depending on the mean occupation number n=hay aiof the LP, ge= g2 r=
NVis the eective spin-spin coupling strength in- duced by the LP. To estimate ge, we assume the dis- tance between the spin and the central line of the CPW resonatord= 50 nm, so gr=2= 3:5 MHz, thus we have ge=2= 12:7 kHz when
NV=2= 960 MHz. Obvi- ously,ge ?1 kHz, i.e., the strong spin-spin cou- pling is achieved. This can be directly demonstrated by simulating the dynamics of the eective system, governed by Eq. (20) or Eq. (19) in the dispersive regime, with the master equation. The simulating results are presented in Fig. 4. One can see that quantum states of two spins can be exchanged each other with [see Fig. 4(a)] and with- out [see Fig. 4(b)] dissipations, while the LP is always in the initial state. Note that the achieved strong spin-spin coupling is not limited by the separation between two spins, it is only determined by the length of the CPW res- onator. Experimentally, the centimeter-sized cavity has been fabricated, so the distance of the strong spin-spin coupling can be improved to centimeter level. Compared to previous proposals of directly coupled spins to a YIG nanosphere [26{28], the distance here is nearly enhanced bysixorders of magnitude. V. CONCLUSIONS In summary, we have proposed a hybrid system con- sisting of a CPW resonator weakly coupled to NV spins and a YIG nanosphere supporting magnons with Kerr eect. With the strong driving eld, the Kerr eect can squeeze magnons, giving rise to exponentially en- hanced strong cavity-magnon coupling, and thus CMPs can be formed. By approaching the cavity-magnon cou- pling strength to the critical value, the spin-LP coupling is greatly enhanced to the strong coupling regime with the accessible parameters, while the coupling between the spins and the HP is fully suppressed. Using the LP as quantum interface in the dispersive regime, strong long- distance spin-spin coupling can be achieved, which allows quantum state exchange between two spins. 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2023-04-26

Strong long-distance spin-spin coupling is desperately demanded for solid-state quantum information processing, but it is still challenged. Here, we propose a hybrid quantum system, consisting of a coplanar waveguide (CPW) resonator weakly coupled to a single nitrogen-vacancy spin in diamond and a yttrium-iron-garnet (YIG) nanosphere holding Kerr magnons, to realize strong long-distance spin-spin coupling. With a strong driving field on magnons, the Kerr effect can squeeze magnons, and thus exponentially enhance the coupling between the CPW resonator and the squeezed magnons, which produces two cavity-magnon polaritons, i.e., the high-frequency polariton (HP) and low-frequency polariton (LP). When the enhanced cavity-magnon coupling approaches to the critical value, the spin is fully decoupled from the HP, while the coupling between the spin and the LP is significantly improved. In the dispersive regime, a strong spin-spin coupling is achieved with accessible parameters, and the coupling distance can be up to $\sim$cm. Our proposal provides a promising way to manipulate remote solid spins and perform quantum information processing in weakly coupled hybrid systems.

Critical Cavity-Magnon Polariton Mediated Strong Long-Distance Spin-Spin Coupling

2304.13553v2

Tunable space-time crystal in room-temperature magnetodielectrics Alexander J. E. Kreil,Halyna Yu. Musiienko-Shmarova, Dmytro A. Bozhko, Sebastian Eggert, Alexander A. Serga, and Burkard Hillebrands Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universit at Kaiserslautern, 67663 Kaiserslautern, Germany Anna Pomyalov and Victor S. L'vov Department of Chemical and Biological Physics, Weizmann Institute of Science, Rehovot 76100, Israel We report the experimental realization of a space-time crystal with tunable periodicity in time and space in the magnon Bose-Einstein Condensate (BEC), formed in a room-temperature Yttrium Iron Garnet (YIG) lm by radio-frequency space-homogeneous magnetic eld. The magnon BEC is pre- pared to have a well dened frequency and non-zero wavevector. We demonstrate how the crystalline \density" as well as the time and space textures of the resulting crystal may be tuned by varying the experimental parameters: external static magnetic eld, temperature, thickness of the YIG lm and power of the radio-frequency eld. The proposed space-time crystals provide a new dimension for exploring dynamical phases of matter and can serve as a model nonlinear Floquet system, that brings in touch the rich elds of classical nonlinear waves, magnonics and periodically driven systems. Spontaneous symmetry breaking is a fundamental concept of physics. A well known example is the breaking of spatial translational symmetry, which leads to a phase transition from uids to solid crystals. By analogy, one can think about a \time crystal" as the result of breaking translational symmetry in time. More generally, one expects the appearance of a \space-time crystal" as a consequence of breaking translational symmetry both in time and in space. If a time crystal exists it should demonstrate time-periodic motion of its ground state [1]. In addition to the time periodicity, the space-time crystals should be periodic in space, similar to an ordinary crystal. It was recently argued that time- and space-time crystals cannot be realized in thermodynamic equilib- rium [2, 3]. This led to a search of space-time symme- try breaking in a wider context, for example in a sys- tem with ux-equilibrium, rather than in the thermo- dynamic equilibrium. Needless to say that oscillatory non-equilibrium states are well known already. One can remember, for example, gravity waves exited on a sea surface under windy conditions, quasi-periodic current instabilities in some semiconductors carrying strong DC electric current (a Gunn eect [4] in A 3B5semiconduc- tors, such as n-type GaAs, used in police radar speed guns), instabilities of an electron beam in plasma. More recent similar example is microwave generation by nano- sized magnetic oscillators driven by a spin-polarized DC electric current [5]. On the other hand new physical phe- nomena can be observed in time-crystals which do not absorb or dissipate energy from external pumping, but instead build up a coherent time-periodic quantum state due to the complexity in the interactions. Possibly a rst nontrivial example, found in Ref. [6], was a periodi- cally driven Floquet quantum disordered system, demon- strating subharmonic behavior. Non trivially, this system does not actually absorb and dissipate any of the exter-nally pumped energy because the disorder in the system makes the energy states isolated from one another, see also Refs. [7, 8]. The possibly most recent observation of a time crystal with very long energy relaxation time com- pared to energy-conserving interaction processes is the Bose-Einstein condensate (BEC) of magnons in a exible trap in super uid3He-Bunder periodic driving by an ex- ternal magnetic eld [9]. Other important requirements for the existence of time crystals include the robustness| the independence of their features to the perturbations of the physical system (e.g. level of disorder) and appear- ance of the soft modes [10{12]. The problems related to space-time crystals [13] are more involved and up to now were explored mostly theoretically [14{18]. In this paper we report the experimental realization of a space-time crystal with tunable periodicity in time and space in the magnon BEC [19{24], formed in a room- temperature Yttrium Iron Garnet (YIG, Y 3Fe5O12) lm. The condensate spontaneously arises as a result of scat- tering of parametrically injected magnons to the bot- tom of their spectrum. The scattered magnons initially form spectrally localized groups, which can be best de- scribed as a time-polycrystal with partial coherence. Af- ter switching o the pumping, we observe an interaction- driven condensation into two coherent spatially extended spin waves|magnon BECs|which are best character- ized by a space-time crystal. We show that this coherent state has the hallmark of non-universal relaxation times, which are much longer than the intrinsic time scales and the crystallization time. We consider the magnon BEC as a model object in studies of Floquet nonlinear wave systems, subject to intensive periodical in time impact. The frequency spectrum of magnons in this system, shown in Fig. 1, has two symmetric minima with non- zero frequency and wave-vectors !min=!(qmin). The possible BEC has accordingly two components with the wave vectorsqminand the frequency !min. The simplestarXiv:1811.05801v1 [cond-mat.quant-gas] 14 Nov 20182 FIG. 1. Magnon spectrum of the rst 48 thickness modes in 5.6-m-thick YIG lm magnetized in plane by a bias magnetic eldH= 1400 Oe, shown for the wavevector qkH(lower part of the spectrum, blue curves) and for q?H(upper part, magenta curves). The red arrow illustrates the magnon injection process by means of parallel parametric pumping. Two orange dots indicate positions of the frequency minimum !min(qmin) occupied by the BECs of magnons. form of their common wave function is a standing wave: C(r;t) =C0cos(qmin
r) exp(i!mint): (1) In this experiment we create a BEC by microwave radia- tion of frequency !p'2
13:6 GHz that can be consid- ered space homogeneous with wave number qp0. The decay instability of this eld with the conservation law !p=)!(q) +!(q) = 2!(q) (2) excites \parametric" magnons with frequency !(q) =!p=2 and wave vectors q. These para- metric magnons further interact mainly via 2 ,2 scattering with the conservation laws: !(q1) +!(q2) =!(q3) +!(q4);q1+q2=q3+q4;(3) that preserves the total number af magnons and their energy. It is known from the theory of weak wave turbulence [25] (see also Ref. [26]) that the scattering process (3) results mostly in a ux of energy towards largeq, which leads to a nonessential accumulation of energy at large q, and to a ux of magnons toward small q. This in turn results in an accumulation of magnons near the bottom of the frequency spectrum !min. The same 2,2 processes (3) lead to eective thermalization of the bottom magnons during some time th.5070 ns and the subsequent creation of the BEC state [24, 27]. The described processes that lead to the creation of BEC, are an experimental manifestation of the space- SwitchPower amplifierYIG filmProbing laser beam and back- scattered light Microwave resonatorHθǁ Super- currents z yx Microwave source Attenuator Pulsed microwave pumpingPulse generatorLaser AOMBeam splitter Fabry-Pérot interferometerPhoto- detector Time-resolved analysis Lens qBECFIG. 2. Experimental set-up. The lower part of the gure shows the microwave circuit, consisting of a microwave source, a switch and an amplier. This circuit drives a microstrip res- onator, which is placed below the in-plane magnetized YIG lm. Light from a solid-state laser ( = 532 nm) is chopped by an acousto-optic modulator (AOM) and guided to the YIG lm. There it is scattered inelastically from magnons, and the frequency-shifted component of the scattered light is selected by the tandem Fabry-P erot interferometer, detected, and an- alyzed in time. time crystal (STC): a system, driven away from ther- modynamic equilibrium by a space-homogeneous, time- periodic (with frequency !p) pumping eld, sponta- neously chooses a space-time periodic state (1) with the frequency!minand non-zero wavevectors qmin. Impor- tantly, the parameters !minandqminare fully deter- mined by intrinsic interactions in the system and are in- dependent of the pumping frequency in a wide range of its values. By varying the strength and direction of the ex- ternal time-independent homogeneous magnetic eld H, the temperature Tand the thickness of the YIG lm, we can change the magnon spectrum !(q) and consequently !minandqminindependently of !p. Note that the life- timeBECof the condensate is much longer than th, en- abling the observation of the magnon BEC state and the study of related eects, such as magnon supercurrent [22] and Nambu-Goldstone modes|the Bogolyubov second sound [28]. All these meet the presently accepted crite- ria of a space-time crystal, i.e the spontaneous symmetry breaking in time and in space, manifested by long-range order and soft modes [17] (in our case the Bogolyubov second sound [28]). The BEC is created from the gaseous incoherent magnons, that accumulate in a relatively narrow fre- quency band
fSTPC near the bottom of the spectrum. To keep in line with the crystal analogy, we will re- fer to this state as a space-time-polycrystal (STPC). In our measurements, the autocorrelation time of these magnons (1 =
fSTPC2 ns, see Fig. 4) signicantly ex- ceeds the wave period 2 =! min0:150:3 ns, similarly3 to the autocorrelation length in polycrystals that spans many unit cell sizes. In our experiments, the magnon BEC in the room- temperature YIG lms was detected by means of pulsed Brillouin light scattering (BLS) spectroscopy. Here the focused laser beam acts both as a probe of the magnon density and as a heating source, which induces a thermal gradient across the probing light spot. The temperature in the spot, and thus the value of thermal gradient, was controlled by the duration of a probing laser pulse. The thermal gradient locally changes the saturation magne- tization and induces a frequency shift between dierent parts of the magnon condensate [29]. Consequently, a phase gradient in the BEC wavefunction is gradually cre- ated and a magnon supercurrent [22, 23], owing out of the hot region of the focal spot is excited. Such a process reduces the number of magnons in the heated area and results in the disappearance of the condensate and in the subsequent disappearance of the supercurrent. The con- ventional relaxation dynamics of the magnons is then re- covered. More details about our experimental techniques one nds in a sketch of the experimental setup shown in Fig. 2, in Ref. [22] and in the supplementary material. We demonstrate here how to change all three param- eters of the STC, Eq. (1): the BEC magnon density jC0j2=NBEC, the frequency !minand the wave number qmin. The STC lifetime can be controlled as well. The most interesting information may be obtained by varying NBEC. We succeeded to change NBECby more than an order of magnitude by tuning the power of the pumping eld. The measured BLS intensity is shown in Fig. 3a as a function of time for selected pumping powers Ppump and two probing laser pulse durations L. The pumping pulse acts during the time interval from 2000 ns to 0 ns. Clearly, a decrease in the pumping power from Ppump = 31 dBm to 19 dBm and a consequent reduction in the number of parametric magnons, which are injected at!(q) =!p=2, leads to a weakening of 2 ,2 magnon scattering and, thus, to an increasing delay in the appearance of these magnons near the bottom of the energy spectrum, as observed by BLS. The density of the bottom magnons, proportional to the intensity of the measured BLS signal, decreases as well (see the yellow shaded area in Fig. 3a, labeled \Polycrystalline phase"). After the pumping pulse is switched o (for t>0 ns), the magnons condense in the energetic minimum of the spectrum, creating the STC. In case of strong heating (L= 80s), this process results in the appearance of a magnon supercurrent, which only involves the con- densed and therefore coherent magnons. This out ow of magnons (blue shaded area in Fig. 3a labeled space- time crystal ) results in a higher decay rate of the magnon density in the laser focal point. This eective decay rate, which is in uenced by the inherent damping of both co- herent and incoherent magnons to the phonon bath and by the supercurrent-related leakage of the magnon BEC, FIG. 3. Transition from the polycrystalline magnon phase to the space-time crystal phase and back. (a) Temporal dynamics of the measured magnon density for a few pumping power values Ppump at dierent temperatures of the probing spot. The bias magnetic eld H= 1400 Oe. The top BLS waveform measured for Ppump = 31 dBm corresponds to the case of the weakly heated YIG sample (duration of the probing laser pulse L= 6s) and, therefore, is not aected by a supercurrent magnon out ow. In all other cases, the non-uniform heating of the YIG sample ( L= 80s) creates a magnon supercurrent owing out from the heated area result- ing in a higher decay rate of the magnons in the BEC phase. This eective decay rate falls with the pumping power. Below a critical magnon density Ncr, characterizing the transition from the space-time crystal phase back to the polycrystalline phase, the decay rate is approximately the same for all cases. (b) The decay times decof the space-time crystal phase (open and solid circles) and polycrystalline phase (squares) as functions of the pumping power Ppump. The space-crystal phase does not exist at pumping powers below 21 dBm. is strongly dependent on the pumping power. This de- pendence stems from the fact, that a lower pumping power leads to a reduced magnon density and therefore to a smaller fraction of BEC magnons. Below a certain threshold density Ncr, when the majority of condensed magnons are own out of the measured region of the BEC, the observed decay rate approaches the same value for all dierent pumping powers. This decay rate corre- sponds to the inherent decay rate of a narrow package of the remaining polycrystalline magnon phase. It is worth noting that the same decay rate is observed4 FIG. 4. BLS intensity (color code) measured for q=qminas a function of the magnon frequency fand of the bias magnetic eldH. Film thickness 5 :6m, pumping power 40 W, pump- ing frequency 14 GHz, pumping pulse duration 1 s, pump- ing period 200 s. The dashed line represents the analytical dependence of the frequency of the spin wave spectrum bot- tom onH:fqmin(H) = H, where the gyromagnetic ratio = 2:8 MHz/Oe. during the entire decay time when heating of the YIG sample can be neglected, and therefore there is no su- percurrent that takes away the coherent magnons. For example, the black top waveform in Fig. 3a was measured at shorter heating times L= 6s. The pumping power is the same as for the red waveform (hot spot, Ppump = 31 dBm), therefore it corresponds to a well-formed BEC. However, it is not possible to distinguish between the BEC and the incoherent magnons via the decay rate mea- surements in this case. The latter fact contradicts a pre- vious interpretation of similar dynamics of the magnon BEC and the incoherent magnon phase in Ref. [30] as be- ing a result of the sensitivity of the BLS technique to the degree of coherence of the scattering magnons. Further- more, two dierent lifetimes of BEC observed at the same pumping power prove our ability to control the lifetime of the magnon BEC by a thermal gradient. Thereby, the density (Fig. 3a) and the lifetime (Fig. 3b) of the magnon space-time crystal are tunable by the parametric pumping power and by the proper adjustment of a spatial temperature prole. The time periodicity 1 =!minof the STC can be easily changed by variation of the bias magnetic eld. Fig- ure 4 shows the BLS intensity from the bottom of the magnon spectrum ( q=qmin) as a function of the fre- quency and the magnetic eld. The color coded inten- sity of the BLS re ects the eciency of the parametric magnon transfer to the bottom of the frequency spec- trum during the pumping pulse [23]. The dependence fqmin(H) is well described by the analytical dependence FIG. 5. Theoretical dependencies of the energy minimum wavelength min(black line) and frequency !min(red line) on the YIG lm thickness for T= 300 K and H= 1400 Oe. !min= 2fqmin(H) = 2 H , where is the gyromag- netic ratio. The spatial periodicity of the STC can be changed in a wide range from'0:5m to'4m by a proper choice of the YIG lm thickness, see Fig. 5. Note that, except for very thin lms, the !minis insensitive to the lm thickness. The tunable magnon space-time crystal, realized by a periodically driven room-temperature YIG lm, rep- resents an example of a nonlinear Floquet system and therefore serves as a bridge between magnonics and clas- sical nonlinear wave physics from one side and the Flo- quet time-crystal description of the periodically driven systems from another. Joining these two perspectives may give birth to a new eld of physical research: \Flo- quet (or periodically driven) nonlinear wave physics". The advantage of a macroscopic system that may be studied at room temperature as compared to small sam- ples at milli-Kelvin temperatures, is obvious. Moreover, strong nonlinearity, non-reciprocity, topology, local ma- nipulation via external electric and magnetic elds and sample patterning, available in a magnonic system, com- bined with tunability and space-, time-, wave-vector- and frequency-resolved measurements using BLS, makes the suggested system a good experimental basis for the newly proposed eld. On the other hand, concepts discussed in the framework of the Floquet systems such as quasi- energy, umklapp scattering, forbidden bands in quasi- momentum space, once applied to magnon space-time crystals, may give new insight into the rich physics of this system, creating new physical ideas and paving a way to new engineering applications. Financial support by the European Research Coun- cil within the Advanced Grant 694709 \SuperMagnon- ics" and by Deutsche Forschungsgemeinschaft (DFG) within the Transregional Collaborative Research Center SFB/TR49 \Condensed Matter Systems with Variable Many-Body Interaction" as well as by the DFG Project INST 248/178-1 is gratefully acknowledged.5 kreil@rhrk.uni-kl.de [1] F. Wilczek, Quantum time crystals , Phys. Rev. Lett. 109, 160401 (2012). [2] P. Bruno, Comment on \Quantum time crystals" , Phys. Rev. Lett. 110, 118901 (2013). [3] H. Watanabe and M. Oshikawa, Absence of quantum time crystals , Phys. Rev. Lett. 114, 251603 (2015). [4] J.B. Gunn, Intabilities of current in III-V semiconduc- tors, IBM J. Res. Dev. 8141{159, (1964). [5] O.R. Sulymenko, O.V. Prokopenko, V.S. Tyberkevych, A.N. Slavin, and A.A. Serga, Bullets and droplets: Two- dimensional spin-wave solitons in modern magnonics , Low Temp. Phys. 44, 602{617 (2018). [6] V. Khemani, A. Lazarides, R. Moessner, and S.L. Sondhi, Phase structure of driven quantum systems , Phys. Rev. Lett. 116, 250401 (2016). [7] J. Zhang, P.W. Hess, A. Kyprianidis, P. Becker, A. Lee, J. Smith, G. Pagano, I.-D. Potirniche, A.C. Potter, A. Vishwanath, N.Y. Yao, and C. Monroe, Observation of a discrete time crystal , Nature 543, 217{220 (2017). [8] S. Choi, J. Choi, R. Landig, G. Kucsko, H. Zhou, J. Isoya, F. Jelezko, S. Onoda, H. Sumiya, V. Khemani, C. von Keyserlingk, N.Y. Yao, E. Demler, and M.D. Lukin, Observation of discrete time-crystalline order in a dis- ordered dipolar many-body system , Nature 543, 221{225 (2017). [9] S. Autti, V.B. Eltsov, and G.E. Volovik, Observation of a time quasicrystal and its transition to a super uid time crystal , Phys. Rev. Letts. 120215301 (2018). [10] C.W. von Keyserlingk, V. Khemani, and S.L. Sondhi, Absolut stability and spatiotemporal long-range order in Floquet systems , Phys. Rev. B 94, 085112 (2016). [11] D.V. Else, B. Bauer, and C. Nayak, Floquet time crystals , Phys. Rev. Lett. 117, 090402 (2016). [12] N.Y. Yao, A.C. Potter, I.-D. Potirniche, and A. Vish- wanath, Discrete time crystals: Rigidity, criticality, and realization , Phys. Rev. Lett. 118, 030401 (2017). [13] J. Smits, L. Liao, H.T.C. Stoof, and P. van der Straten, Observation of a space-time crystal in a super uid quan- tum gas , Phys. Rev. Lett. 121, 185301 (2018). [14] S. Xu and C. Wu, Space-time crystal and space-time group , Phys. Rev. Lett. 120, 096401 (2018). [15] G.N. Borzdov, Dirac electron in a chiral space-time crystal created by counterpropagating circularly polarized plane electromagnetic waves , Phys. Rev. A 96, 042117 (2017). [16] C. Yannouleas and U. Landman, Trial wave functions for ring-trapped ions and neutral atoms: Microscopic de- scription of the quantum space-time crystal , Phys. Rev. A96, 043610 (2017). [17] T. Li, Z.-X. Gong, Z.-Q. Yin, H.T. Quan, X. Yin, P. Zhang, L.-M. Duan, and X. Zhang Space-time crystals of trapped ions , Phys. Rev. Lett. 109, 163001 (2012). [18] Z.-L. Deck-L eger and C. Caloz, Scattering in superlu- minal space-time (ST) modulated electromagnetic crys- tals, 2017 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, 63{64 (2017). [19] S.O. Demokritov, V.E. Demidov, O. Dzyapko, G.A. Melkov, A.A. Serga, B. Hillebrands, and A.N. Slavin, Bose-Einstein condensation of quasi-equilibriummagnons at room temperature under pumping , Nature 443, 430 (2006). [20] P. Nowik-Boltyk, O. Dzyapko, V.E. Demidov, N.G. Berlo, and S.O. Demokritov, Spatially non-uniform ground state and quantized vortices in a two-component Bose-Einstein condensate of magnons , Sci. Rep. 2, 482 (2012). [21] T. Giamarchi, C. R uegg, and O. Tchernyshyov, Bose- Einstein condensation in magnetic insulators , Nat. Phys. 4, 198{204 (2008). [22] D.A. Bozhko, A.A. Serga, P. Clausen, V.I. Vasyuchka, F. Heussner, G.A. Melkov, A. Pomyalov, V.S. L'vov, and B. Hillebrands, Supercurrent in a room-temperature Bose- Einstein magnon condensate , Nat. Phys. 12, 1057{1062 (2016). [23] A.J.E. Kreil, D.A. Bozhko, H.Yu. Musiienko-Shmarova, V.S. L'vov, A. Pomyalov, B. Hillebrands, and A.A. Serga, From kinetic instability to Bose-Einstein condensation and magnon supercurrents , Phys. Rev. Lett. 121077203 (2018). [24] A.A. Serga, V.S. Tiberkevich, C.W. Sandweg, V.I. Vasyuchka, D.A. Bozhko, A.V. Chumak, T. Neumann, B. Obry, G.A. Melkov, A.N. Slavin, and B. Hillebrands, Bose-Einstein condensation in an ultra-hot gas of pumped magnons , Nat. Commun. 5, 3452 (2014). [25] V.E. Zakharov, V.S. L'vov, and G.E. Falkovich, Kol- mogorov Spectra of Turbulence I (Wave Turbulence) (Springer, Berlin, 1992). [26] Yu.M. Bunkov, E.M. Alakshin, R.R. Gazizulin, A.V. Klochkov, V.V. Kuzmin, V.S. L'vov, and M.S. Tagirov, High-Tcspin super uidity in antiferromagnets , Phys. Rev. Lett. 108, 177002 (2012). [27] P. Clausen, D.A. Bozhko, V.I. Vasyuchka, B. Hillebrands, G.A. Melkov, and A.A. Serga, Stimulated thermalization of a parametrically driven magnon gas as a prerequisite for Bose-Einstein magnon condensation , Phys. Rev. B 91, 220402 (2015). [28] D.A. Bozhko, A.J.E. Kreil, H.Yu. Musiienko-Shmarova, A.A. Serga, A. Pomyalov, V.S. L'vov, and B. Hille- brands, Long-distance supercurrent transport in a room- temperature Bose-Einstein magnon condensate , arXiv: 1808.07407 (2018). [29] M. Vogel, A.V. Chumak, E.H. Waller,T. Langner, V.I. Vasyuchka, B. Hillebrands, and G. von Freymann, Op- tically recongurable magnetic materials , Nat. Phys. 11, 487{491 (2015). [30] V.E. Demidov, O. Dzyapko, S.O. Demokritov, G.A. Melkov, and A.N. Slavin, Observation of spontaneous co- herence in Bose-Einstein condensate of magnons , Phys. Rev. Lett. 100, 47205 (2008).6 Supplemental material: Experimental technique Figure 2 provides a sketch of the experimental setup. The sample is placed between the poles of an electromagnet, which creates a homogeneous magnetizing eld Hly- ing in the plane of YIG lm. In order to reach a high enough density of the magnon gas to form a BEC phase, a rather strong microwave pulse with the peak power Pmax= 41 dBm is applied to the half-wave microstrip resonator at a carrier frequency fpump = 13:6 GHz. The resonator creates an Oersted eld q(t)kHin the YIG lm to excite magnons by means of parallel paramet- ric pumping [31, 32]. Additionally, a variable attenua- tor is implemented in the microwave circuit to allow for power reduction of the amplied microwave pulse to a lower value PpumpPmax. The microwave pulse dura- tion is kept constant at p= 2s with a repetition rate offrep= 1 kHz to ensure that microwave heating eects are negligible. Both the detection of the excited magnons and the heating of the YIG lm were made by using the probing laser beam with a power of 30 mW. The beam is chopped by an acusto-optic modulator (AOM) to control the en- ergy input into the YIG sample. A duration of the laser pulse ofL= 80s is used for local heating, while the pulse duration L= 6s is used to avoid the heatingof the probing point. The probing laser pulse is syn- chronized with the microwave pumping and has the same repetition rate frep. The laser pulses of both durations are switched on before the application of the microwave pulse and are switched o 3 s after its end. From the sample backscattered laser light is collected and sent to a Tandem-Fabry-P erot interferometer for frequency and time-of- ight analysis with a frequency and time resolu- tion of approximately 100 MHz and 1 ns. In order to selectively detect only the magnons con- densed in the lowest energy state of the magnonic sys- tem with a wave number qk4:5 radm1(qkkH) and a frequency fmin= 4 GHz (see Fig. 2, the angle of incidence kof the probing laser beam has to be chosen accordingly. The condition to detect a magnon with a specic wavevector qSWlying in a lm plane is qSW= 2qlightsin(#). Therefore the angle of the incident light has been chosen to be k= 12. kreil@rhrk.uni-kl.de [31] A.G. Gurevich and G.A. Melkov, Magnetization Oscilla- tions and Waves (CRC Press, Boca Raton, 1996). [32] V.S. L'vov, Wave Turbulence Under Parametric Excita- tion (Applications to Magnets) (Springer, Berlin, 1994).

2018-11-14

We report the experimental realization of a space-time crystal with tunable periodicity in time and space in the magnon Bose-Einstein Condensate (BEC), formed in a room-temperature Yttrium Iron Garnet (YIG) film by radio-frequency space-homogeneous magnetic field. The magnon BEC is prepared to have a well defined frequency and non-zero wavevector. We demonstrate how the crystalline "density" as well as the time and space textures of the resulting crystal may be tuned by varying the experimental parameters: external static magnetic field, temperature, thickness of the YIG film and power of the radio-frequency field. The proposed space-time crystals provide a new dimension for exploring dynamical phases of matter and can serve as a model nonlinear Floquet system, that brings in touch the rich fields of classical nonlinear waves, magnonics and periodically driven systems.

Tunable space-time crystal in room-temperature magnetodielectrics

1811.05801v1

Compact localised states in magnonic Lieb lattices Grzegorz Centa la and Jaros law W. K los Institute of Spintronics and Quantum Information, Faculty of Physics, Adam Mickiewicz University, Pozna n, Uniwersytetu Pozna nskiego 2, Pozna n 61-614, Poland (Dated: March 26, 2023) Lieb lattice is one of the simplest bipartite lattices where compact localized states (CLS) are observed. This type of localisation is induced by the peculiar topology of the unit cell, where the modes are localized only on one sublattice due to the destructive interference of partial waves. The CLS exist in the absence of defects and are associated with the at bands in the dispersion relation. The Lieb lattices were successfully implemented as optical lattices or photonic crystals. This work demonstrates the possibility of magnonic Lieb lattice realization where the at bands and CLS can be observed in the planar structure of sub-micron in-plane sizes. Using forward volume conguration, we investigated numerically (using the nite element method) the Ga-dopped YIG layer with cylindrical inclusions (without Ga content) arranged in a Lieb lattice of the period 250 nm. We tailored the structure to observe, for the few lowest magnonic bands, the oscillatory and evanescent spin waves in inclusions and matrix, respectively. Such a design reproduces the Lieb lattice of nodes (inclusions) coupled to each other by the matrix with the CLS in at bands. Keywords: at bands, compact localized states, Lieb lattice, spin waves, nite element method I. INTRODUCTION There are many mechanisms leading to wave localiza- tion in systems with long-range order, i.e. in crystals or quasicrystals. The most typical of these require (i) the local introduction of defects, including the defects in the form of surfaces or interfaces [1] (ii) the presence of global disorder [2], (iii) the presence of external elds [3] or (iv) the existence of many-body phenomena [4]. How- ever, since at least the late 1980s, it has been known that localization can occur in unperturbed periodic sys- tems in the absence of elds and many-body eects, and is manifested by the presence of at, i.e., dispersion-free bands in the dispersion relation. The pioneering works are often considered to be the publications of B. Suther- land [5] and E. H. Lieb [6], who found the at bands of zero energy [7] for bipartite lattices with use of the tight- binding model Hamiltonians, where the hoppings occur only between sites of dierent sublattices. The simplest realization of this type of system is regarded as the Lieb lattice [6, 8], where the nodes of one square sublattice, of coordination number z= 4, connect to each other only via nodes with a coordination number z= 2 from other two square sublattices (Fig. 1). In the case of extended Lieb lattices [9, 10], the nodes of z= 2 form chains: dimmers, trimmers, etc.(Fig. 2). An intuitive explana- tion for the presence of the at bands is the internal isolation of excitations located in one of the sublattices. The cancelling of excitations at one sublattice is the re- sult of forming destructive interference and local symme- try within the complex unit cell [11]. When only one of the sublattices is excited, the other sublattice does not mediate the coupling between neighbouring elementary klos@amu.edu.plcells, and the phase dierence between the cells is irrel- evant to the energy (or the frequency) of the eigenmode on the whole lattice - i.e. the Bloch function. Modes of this type are therefore degenerated for dierent wave vector values in innite lattices. We are dealing here with the localization on specic arrangements of struc- ture elements, which are isolated from each other. Such kinds of modes are called compact localized states (CLS) [12{16] and show a certain resistance to the introduc- tion of defects [17, 18]. The at band systems with CLS are the platform for the studies of Anderson localization [19], and unusual properties of electric conductivity [20]. A similar localization is observed in the quasicrystals, where the arrangements of the elements composing the structure are replicated aperiodically and self-similarly throughout the system [21, 22] and the excitation can be localized on such patterns. The CLS in nite Lieb lattices have a form of loops (plaquettes) occupying the majority nodes (z= 2). These states are linearly dependent and do not form a complete basis for the at band. Therefore occupancy gaps need to be lled (for innite lattice) by states occupying only one sublattice of majority nodes, localizes at lines, called noncontractible loop states (NLS) [14, 15, 23]. The topic of Lieb lattices and other periodic struc- tures with compact localization and at bands was re- newed [8] about 10 years ago when physical realizations of synthetic Lieb lattices began to be considered for elec- tronic systems [24, 25], optical lattices [26, 27], supercon- ducting systems [28, 29], in phononics [30] and photonics [14, 31]. In a real system, where the interaction cannot be strictly limited to the nearest elements of the struc- ture, the bands are not perfectly at. Therefore, some authors use the extended denition of the at band to consider the bands that are at only along particular di- rections or in the proximity of high-symmetry Brillouin zone points [32]. In tight-binding models, this eect canarXiv:2303.14843v1 [cond-mat.mes-hall] 26 Mar 20232 be included by considering the hopping to at least next- nearest-neighbours [33, 34]. Similarly, the crossing of the at band by Dirac cones can be transformed into anti- crossing and lead to opening gaps, separating the at band from dispersive bands. This eect can in induced by the introduction of spin-orbit term to tight-binding Hamiltonian (manifested by the introduction of Peierls phase factor to the hopping) or by dimerization of the lattice (by alternative changes of hoppings or site ener- gies) [33{37]. The later scenario can be easily observed in real systems where the position of rods/wells (mimicking the sites of Lieb lattice) and contrast between them can be easily altered [38]. Opening the narrow gap between at band and dispersive bands for Lieb lattice is also fun- damentally interesting because it leads to the appearance of so-called Landau-Zener Bloch oscillations [39]. The isolated and perfectly at bands for Lieb lattices are topologically trivial { their Chern number is equal to zero [40]. For weakly dispersive (i.e. almost at) bands the Chern numbers can be non-zero [41]. However, when the at band is intersected by dispersive bands then it can exhibit the discontinuity of Hilbert{Schmidt distance between eigenmodes corresponding to the wave vectors just before and just after the crossing. Such an eect is called singular band touching [16]. This limiting value of Hilbert{Schmidt distance is bulk invariant, dierent from the Chern number. One of the motivations for the photonic implementa- tion of systems with at, or actually nearly at bands [42], was the desire to reduce the group velocity of light in order to compress light in space, which leads to the concentration of the optical signal and an increase in the light-matter interaction, or the enhancement of non- linear eects. Another, more obvious application is the possibility of realizing delay lines that can buer the sig- nal to adjust the timing of optical signals [43]. A promis- ing alternative to photonic circuits are magnonic sys- tems, which allow signals of much shorter wavelengths to be processed in devices several orders of magnitude smaller [44, 45]. For this reason, it seems natural to seek a magnonic realization of Lieb lattices. In this paper, we propose the realization of such lat- tices based on a magnonic structure in the form of a perpendicularly magnetized magnetic layer with spa- tially modulated material parameters or spatially vary- ing static internal eld. Lieb lattices have been studied also in the context of magnetic properties, mainly due to the possibility of enhancing ferromagnetism in systems of correlated electrons [46], where the occurrence of at bands with zero kinetic energy was used to expose the in- teractions. There are also known single works where the spin waves have been studied in the Heisenberg model in an atomic Lieb lattice, such as the work on the magnon Hall eect [47]. But the comprehensive studies of spin waves in nanostructures that realize magnonic Lieb lat- tices and focus on wave eects in a continuous model have not been carried out so far. In this work, we demonstrate the possibility of realization of magnonic lattices in pla-nar structure based on low spin wave damping material: yttrium iron garnet (YIG) where the iron is partially sub- stituted by gallium (Ga). We present the dispersion rela- tion with a weakly depressive ( at) band exhibiting the compact localized spin waves. The at is almost inter- sected at the Mpoint of the 1stBrillouin zone by highly dispersive bands, similar to Dirac cones. We discuss the spin wave spectra and compact localized modes both for simple and extended Lieb lattices. The introduction is followed by the section describing the model and numerical method we used, which pre- cedes the main section where the results are presented and discussed. The paper is summarized by conclusion and supplemented with additional materials where we showed: (A) the results for extended Lieb-7 lattice, (B) an alternative magnonic Lieb lattice design via shaping the demagnetizing eld, and (C) an attempt of formation magnonic Lieb lattice by dipolarly coupled magnetic na- noelements, (D) discussion of small dierences in the de- magnetizing eld of majority and minority nodes respon- sible for opening a small gap in the Lieb lattice spectrum. II. STRUCTURE Magnonic crystals (MCs) are regarded as promis- ing structures for magnonic-based device applications [45, 48]. In our studies, we consider planar MCs to de- sign the magnonic Lieb lattice, owing to the relative ease of fabrication of such structures and their experimental characterization [49{51]. We proposed realistic systems that mimic the main features of the tight-binding model of Lieb lattice [16, 33]. Investigated MCs consist of yttrium iron garnet doped with gallium (Ga:YIG) matrix and yttrium iron gar- net (YIG) cylindrical inclusions arranged in Lieb lattice Fig. 1. Doping YIG with Gallium is a procedure where magnetic Fe3+ions are replaced by non-magnetic Ga3+ ions. This method not only decreases saturation mag- netizationMSbut, simultaneously, arises uni-axial out- of-plane anisotropy, that ensures the out-of-plane orien- tation of static magnetization in Ga:YIG layer at a rel- atively low external eld applied perpendicularly to the layer. Discussed geometry, i.e. forward volume magne- tostatic spin wave conguration, does not introduce an additional anisotropy in the propagation of spin waves, related to the orientation of static magnetization. The design of the Lieb lattice requires the partial local- ization of spin wave in inclusions, which can be treated as an approximation of the nodes from the tight-binding model. Furthermore, the neighbouring inclusions in the lattice have to be coupled strongly enough to sustain the collective spin wave dynamics, and weakly enough to minimize the coupling between further neighbours. Therefore, the geometrical and material parameters were selected to ensure the occurrence of oscillatory excita- tions in the (YIG) inclusions and exponentially evanes- cent spin waves in the (Ga:YIG) matrix. The size of3 BAa) b) FIG. 1. Basic magnonic Lieb lattice. The planar magnonic structure consists of YIG cylindrical nanoelements embedded within Ga:YIG. Dimensions of the ferromagnetic unit cell are equal to 250x250x59 nm and the unit cell contains three in- clusions of 50 nm diameter. (a) The structure of basic Lieb lattice, and (b) top view of the Lieb lattice unit cell where the node (inclusion) from minority sublattice Aand two nodes (inclusions) from two majority sublattices Bare marked. inclusions was chosen small enough to separate three lowest magnonic bands with almost uniform magnetiza- tion precession inside the inclusion from the bands of higher frequency, where the spin waves are quantised in- side the inclusions. Also, the thickness of the matrix and inclusion was chosen in a way that there are no nodal lines inside the inclusion. The condition which guaran- tee the focussing magnetization dynamics inside the in- clusions is fullled in the frequency range below the ferro- magnetic resonance (FMR) frequency of the out-of-plane magnetized layer made of Ga:YIG (matrix material): fFMR;Ga:YIG =4.95 GHz and above the FMR frequency of out-of-plane magnetized layer made of YIG (inclusions material):fFMR;YIG= 2:42 GHz. These limiting values were obtained using the Kittel formula for out-of-plane magnetised lm: fFMR = 2j0H0+0Hani0MSj, where we used the following values of material param- eters [52] for YIG: gyromagnetic ratio = 177GHz T, magnetization saturation 0MS= 182:4 mT, exchange stiness constant A= 3:68pJ m, (rst order) uni-axial anisotropy eld 0Hani=3:5 mT, and for Ga:YiG: = 179GHz T,0MS= 20:2 mT,A= 1:37pJ m,0Hani= 94:1 mT. Since the greatest impact of the rst order uniaxial anisotropy eld ( 0Hani), we decided to ne- glect higher order terms of uni-axial anisotropy and cubic anisotropy of (Ga:)YIG. Due to the presence of out-of- plane anisotropy and relatively low saturation magneti- zation, we could consider a small external magnetic eld 0H0= 100 mT to reach saturation state. It is worth noticing that without the evanescent spin waves in the ferromagnetic matrix, the appropriate cou- BAa) b)FIG. 2. Extended magnonic Lieb lattice { Lieb - 5. Dimen- sions of the unit cell are 375x375x59 nm and contain 5 inclu- sions of size 50 nm in diameter. Also, we maintain the same separation (distance between centres of neighbouring sites is 125 nm) as for considered basic Lieb lattice { Fig. 1. (a) The structure of Lieb-5 lattice, and (b) top view on Lieb-5 lattice unit cell where the node (inclusion) from minority sublattice Aand four nodes (inclusions) from two majority sublattices Bare marked. pling between inclusions would not be possible. There- fore the realization of the Lieb lattice in form of the array of ferromagnetic nanoelemets embedded in air/vacuum seems to be very challenging { see the exemplary results in Supplementary Information C. We also tested the possibility of other realizations of magnonic Lieb lattices. One solution seemed to be the design of a structure in which the concentration of the spin wave amplitude in the Lieb lattice nodes would be achieved through an appropriately shaped prole of the static demagnetizing eld { Supplementary Information B. However, the obtained results were not as promising as for YIG/Ga:YIG system. In the main part of the manuscript, we present the re- sults for the basic Lieb lattice (showed in Fig. 1) and extended Lieb-5 lattice (showed in Fig. 2), based on YIG/Ga:YIG structures. The further extension of the Lieb lattice may be realized by increasing the number of Bnodes between neighbouring Anodes. Supplementary Information A presents the results for Lieb-7, where for each site (inclusion) from minority sublattice A, we have six nodes (inclusions), grouped in three-element chains, from majority sublattices B. III. METHODS The spin waves spectra and the spatial proles of their eigenmodes were obtained numerically in a semi- classical model, where the dynamics of magnetization4 vector M(r;t) is described by the Landau-Lifshitz equa- tion [53]: dM dt= 0[MHe+ MSM(MHe)]:(1) The symbol He(r;t) denotes eective magnetic eld. In numerical calculations, we neglected the damping term since is small both for YIG and for YIG with Fe substituted partially by Ga (for Ga:YIG = 6:1104 andYIG= 1:3104[52]). The eective magnetic eld Hecontains the following components: the external eld H0, exchange eld Hex, bulk uniaxial anisotropy eld Haniand dipolar eld Hd: He(r;t) =H0^ z+2A 0M2 Sr2M(r;t)+Hani(r)^ zr'(r;t); (2) where the zdirection is normal to the plane of the magnonic crystal. We assume that the sample is satu- rated inzdirection and magnetization vector precesses around this direction. The material parameters ( MS,A, and ) are constant within matrix and inclusions. Using the magnetostatic approximation the dipolar term of the eective magnetic eld may be expressed as a gradient of magnetic scalar potential: Hd(r;t) =r'(r;t) (3) By using the Gauss equation magnetic scalar potential may be associated with magnetisation as follows: r2'(r;t) =r
M(r;t) (4) Spin-wave dynamics is calculated numerically using the nite-element method (FEM). We used the COMSOL Multiphysics [54] to implement the Landau-Lifshitz equa- tion (Eq. 1) and performed FEM computation for the de- ned geometry of magnonic Lieb lattices. The COMSOL Multiphysics is the software used for solving a number of physical problems, since many implemented modules it becomes more and more convenient. Nevertheless, all the equations were implemented in the Mathematics module which contains dierent forms of partial dierential equa- tions. Eq. 1 was solved by using eigenfrequency study, on the other hand, to solve Eq. 4 we used stationary study. To obtain free decay of scalar magnetic potential in the model we applied 5 m of a vacuum above and under- neath the structure. At the bottom and top surface of the model with vacuum, we applied the Dirichlet boundary condition. We use the Bloch theorem for each variable (magnetostatic potential and components of magnetiza- tion vector) at the lateral surfaces of a unit cell. We selected the symmetric unit cell with minority node Ain the centre to generate a symmetric mesh which does not perturb the four-fold symmetry of the system { this ap- proach is of particular importance for the reproduction of the eigenmodes proles in high-symmetry points. In our numerical studies, we used 2D wave vector k=kx^x+ky^y as a parameter for eigenvalue problem which was selectedalong the high symmetry path XM to plot the dispersion relation. We considered the lowest 3, 5 and 7 bands for basic Lieb lattice, Lieb-5 lattice and Lieb-7 lattice, respectively. IV. RESULTS The tight-biding model of the basic Lieb lattice, with hopping restricted to next-neighbours gives three bands in the dispersion relation. The top and bottom bands are symmetric with respect to the second, perfectly at band, and intersect with this dispersionless band at Mpoint of 1stBrillouin zone, with constant slope forming two Dirac cones[8, 27]. In a realistic magnonic system, the spin wave spectrum showing the particle-hole symmetry with a zero energy at band is dicult to reproduce because (i) the dipolarly dominated spin waves, propagating in magnetic lm, experience a signicant reduction of the group velocity with an increase of the wave vector (this tendency is reversed for much larger wave vectors were the exchange interaction starts to dominate) [53], (ii) the dipolar interaction is long-range. The rst eect makes the lowest band wider than the third band, and the latter one { induces the nite width of the second band [33]. We are going to show, that this weakly dispersive band supports the existence of CLS. Therefore, we will still refer to it as at band , which is a common practice for dierent kinds of realization of Lieb lattices in photonics or optical lattices. The results obtained for the basic magnonic Lieb lat- tice, (Fig. 1), are shown in Fig. 3. As we predicted, three lowest bands form a band structure which is similar to the dispersion relation known from the tight-binding model [10]. However, in a considered realistic system there is an innite number of higher bands, not shown in Fig. 3(a). For higher bands, spin waves can propagate in an oscil- latory manner in the matrix hence the system does not mimic the Lieb lattice where the excitations should be associated with the nodes (inclusions) of the lattice. Due to the fourfold symmetry of the system, the dis- persion relation could be inspected along the high sym- metry path XM. Frequencies of the rst three bands are in the range fFMR;YIGfFMR;Ga:YIG . Their total width is about 0:78 GHz. The rst and third band form Dirac cones at Mpoint, separated by a tiny gap15 MHz. The possible mechanism responsible for opening the gap is a small dierence in the demagnetizing eld in the areas of inclusions A(from the minority lat- tice) and inclusions B(from two majority sublattices) { see Supplementary Information D. Inclusions A(B) have four (two) neighbours of type B(A). Although inclusions AandBhave the same size and are made of the same material, the static eld of demagnetization inside them diers slightly due to the dierent neighbourhoods. This eect is equivalent to the dimerization of the Lieb lattice by varying the energy of the nodes in the tight-binding model, which leads to the opening of a gap between Dirac5 a) b)BA FIG. 3. Dispersion relation for the basic magnonic Lieb lat- tice, containing three inclusions in the unit cell: one inclusion Afrom minority sublattice and two inclusions Bfrom ma- jority sublattices (see Fig. 1). (a) The dispersion relation is plotted along the high symmetry path -X-M- (see the inset). The lowest band (blue) and the highest band (red) create Dirac cones almost touching (b) in the M point. The middle band (green) is relatively at in the vicinity of the M point. cones and parabolic attening of them in very close prox- imity to the M-point. It is worth noting that in the in- vestigated system, the gap opens between the rst and second bands, while the second and third bands remain degenerated at point M, with numerical accuracy. The middle band can be described as weakly disper- sive. The band is more at on the XMpath and, in particular, in the vicinity of Mpoint { see Fig. 3(b). The small width of the second band can be attributed to long-range dipolar interactions which govern the mag- netization dynamics in a considered range of sizes and wave vectors. It is known that even the extension of the range of interactions to next-nearest-neighbours in the tight-binding model induces the nite width of the at band for the Lieb lattice. To prove that the second band supports the CLS re- gardless of its nite width, we plotted the proles of spin wave eigenmodes at Mpoint and in its close vicinity. The results are presented in Fig. 4. The proles were shown for innite lattice and are presented in the form of square arrays containing 3x3 unit cells, where the dashed lines BA M1 M2 M3 M1← M2← M3←CLS NLS +- - + - -NLS++ - -FIG. 4. The spin wave proles obtained for the basic magnonic Lieb lattice, composed of three inclusions in the unit cell (see Fig. 1). The modes are presented for each band exactly at M(left column) and in its proximity ( M ) on the path M (right column). In the presented proles, the sat- uration and the colour denote the amplitude and phase of the dynamic, in-plane component of magnetization. The compact localized states (CLS) are presented at the point M for the second band { right column. The CLS do not occupy minority sublattice A. The inclusions B, in which the magnetization dynamics is focused, are quite well isolated from each other. One can easily notice that the lattice is decorated by loops (marked by grey patches) where the phase of the precessing magnetization ips between inclusions (+ and signs). Ex- actly at point M{ left column, we observe the degeneracy of the second and third bands. The spin waves occupy Bin- clusions only in one majority sublattice, i.e. along vertical or horizontal lines, lliping the phase from inclusion to inclusion which gives the pattern characteristic to noncontractible loop states (NLS) - marked by grey stripes. mark their edges. It is visible that the spin waves are concentrated in the cylindrical inclusions, where the am- plitude and phase of precession is quite homogeneous. In calculations, we used the Bloch boundary conditions ap- plied for a single unit cell, which means that at Mpoint the Bloch function is ipped after translation by lattice period, in both principal directions of the lattice and we6 will not see the single closed loops of CLS or lines of NLS. Exactly at Mpoint, all three bands have zero group ve- locity. Therefore, the corresponding modes (left column) are not propagating. The lowest band ( M1) occupy only inclusionsAfrom the minority sublattice where the static demagnetizing eld is slightly lower than inside inclu- sionsB(see Supplementary Information D), which jus- ties its lower frequency and lifting the degeneracy with two higher modes M2andM3of the same frequency. Each of the modes M2andM3occupy only one of two sublattices B, therefore they can be interpreted as NLS. To observe the pattern typical for CLS, we need to move slightly away from Mpoint. The rst and third modes have then the linear dispersion with high group velocity and the second band remains at. We selected the point M shifted from Mpoint toward point by 5% of M distance (right column). We can see that the rst and third modes M 1,M 3occupy now all inclusions and the modeM 2from the at band has a prole typical for CLS, predicted by tight-binding models [8{10, 55, 56]: jmk>= [eiky 2|{z} B;0|{z} A;eikx 2|{z} B] (5) wheremkis the complex amplitude of the Bloch function in the base of unit cell (i.e. on two inclusion Bfrom ma- jority sublattices and one inclusion Afrom minority sub- lattice), k= [kx;ky] is dimensionless wave vector. From (Eq. 5), we can see that (i) CLS do not occupy the mi- nority nodes Aand (ii) close to Mpoint the phases at two nodesB, from dierent majority sublattices, are op- posite. These two features are reproduced for M 2mode in investigated magnonic Lieb lattice. In the prole of this mode, we marked (by a grey patch) the elementary loop of CLS which is easily identied in nite systems. Here, in an innite lattice with Bloch boundary condi- tions, the loops are innitely replicated with phase shift after each translation xandydirection. The lo- calization at the inclusions Band the absence of the spin wave dynamics in inclusions Ais observed regardless of the wave vector. Therefore, the coupling can take place only between the next neighbours (inclusions B), i.e. on larger distances and mostly due to dipolar interactions, that makes the second band not perfectly at. Let's discuss now the presence of at bands and CLS in an extended magnonic Lieb lattice (Lieb-5), contain- ing ve inclusions in the unit cell: one inclusion Aform minority sublattice and four inclusions Bfrom majority sublattices, as it is presented in Fig. 2. In the considered structure, we add two additional inclusions Binto the unit cell in such a way that neighbouring inclusions A are linked by the doublets of inclusions B. The sizes of inclusions, distances between them, the thickness of the layer and the material composition of the structure re- mained the same as for the basic Lieb lattice, discussed earlier (Fig. 1). The dispersion relation obtained for the magnonic Lieb-5 lattice can be found in Fig. 5(a). The proper- ties of the extended Lieb lattices are well described in a) b) BAFIG. 5. Dispersion relation for the extended magnonic Lieb lattice Lieb-5, containing ve inclusions in the unit cell: one inclusion Afrom minority sublattice and four inclusions B from majority sublattices (see Fig. 2). (a) The dispersion relation is plotted along the high symmetry path -X-M- (see the inset). The rst, third and fth bands (dark blue, red and cyan) are strongly dispersive bands, while the second and fourth bands (green and magenta) are less dispersive and related to the presence of CLS. The system does not support the appearance of Dirac cones, even in case when the inter- action is ctitiously limited only to inclusions, according to tight-binding model. (b) The zoomed regions in the vicinity of (in dark green frame) and Mpoints show the essential gaps with relatively low, parabolic-like curvatures for top and bottom bands. the literature [10, 57{59]. The tight-binding model de- scription of Lieb-5 lattices, with information about their dispersion relation and the proles of the eigenmodes, are presented in numerous papers[10, 16, 55]. Therefore, it is possible to compare the obtained results with the theoretical predictions of the tight-binding model. The tight-binding model of Lieb-5 lattice predicts two at bands with CLS: the second (green) and fourth (ma- genta) band in the spectrum. The at bands in the tight- binding model are not intersected by Dirac cones but they are degenerated at and Mpoint with the third band (red). These features are reproduced in investigated magnonic Lieb-5 lattice (Fig. 2). The dispersion relation for this system is presented in the Fig. 5(a). Also, we have marked, with two rectangles (dark green and violet), the vicinities of and Mpoints, where the at bands (the7 Γ3 Γ4 Γ5BA M1 M2 M3 M1← M2← M3← Γ3← Γ4← Γ5←- +-+- ++- -+ -+-++- FIG. 6. The proles obtained for the extended Lieb lattice consisted of 5 inclusions in the unit cell. The modes are presented for bands No. 3-5 in point and its proximity (the rst and second column). In the third and fourth columns, we presented the proles for bands No. 1-3 at Mpoint and its vicinity M . Each prole of eigenmode is presented on a grid composed of 3x3 unit cells - dashed lines mark the edges of unit cells. The scheme of the unit cell is presented in top-left corner. Exactly at (and M) point the bands No. 3 and 4 (No. 2 and 3) are degenerated and proles: 3and 4(M2andM3) have non-standard (for CLS) complementary form { i.e. their combinations 3i4(M2iM3) gives NLS. To obtain proper proles of CLS, where the phase of procession ips around CLS loop, we need to explore the vicinity of ( M) point { see the grey patches for the mode 4(M 2) with + and signs. fourth and second bands) become degenerated with the third, dispersive band { Fig. 5(b). It is easy to notice the essential frequency gaps ( 33 MHz and84 MHz at andMpoints, respectively), which qualitatively corre- sponds to the prediction of the tight-binding model. It is worth noting that although the low dispersion bands (the second and fourth band) are in general not perfectly at. Nevertheless, around the point and Mpoints the bands are attened and the XandXMsections are very at for the fourth and second band, respectively. The spin wave proles of CLS at the high symmetry points: and Mare presented in Fig. 6. Exactly at andM(the rst and third column), we can see the pairs of degenerated mods 3, 4andM2,M3which exhibit features of CLS predicated by the tight-binding model(see the loops of sites on grey patches): (i) modes oc- cupy only the inclusions Bfrom majority sublattices, (ii) doublets of inclusions Bin the loops of CLS have oppo- site (the same) phases at ( M) point. The signicant dierence is that; once we switch one to another B-B doublet, circulating the CLS loop the phase of preces- sion charges by=2 not by 0 or . However, when we make combinations of degenerated modes: 3i4or M2iM3, then we obtain the NLS occupying the hor- izontal or vertical lines, where the precession at exited Binclusion will be in- or out-of-phase. The CLS modes are clearly visible when we move slightly away from the high symmetry point where the degeneracy occurs. In the proximity of and Mpoint, one can see the CLS modes 4andM 2for which the phase of precession8 takes the relative values close to 0 or . The small dis- crepancies, are visible as a slight change in the colours representing the phase, resulting from the fact that we are not exactly in high symmetry points but shifted by 5% on the path M. The extension of the presented analysis to magnonic Lieb - 7 lattice, where the inclusions Aare liked by the chains composed of three inclusions B, is presented in Supplementary Information A. V. CONCLUSIONS We proposed a possible realisation of the magnonic Lieb lattices where the compact localized spin wave modes can be observed in at bands. The presented system qualitatively reproduces the spectral properties and the localization features of the modes, predicted by the tight-binding model and observed for photonic andelectronic counterparts. The magnonic platform for the experimental studies of Lieb lattices seems to be attrac- tive due to the larger exibility in designing magnonic systems and the steering of its magnetic conguration by external biases. The idea of the magnonic Lieb latices allows considering many problems related to dynamics, localization and interactions in at band systems taking the advantage of the magnonic systems: presence and possibility of tailoring of long-range interactions, intrin- sic non-linearity, etc. VI. ACKNOWLEDGEMENTS This work has received funding from National Sci- ence Centre Poland grants UMO-2020/39/O/ST5/02110, UMO-2021/43/I/ST3/00550 and support from the Pol- ish National Agency for Academic Exchange grant BPN/PRE/2022/1/00014/U/00001. [1] S. Davison and M. 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Zavislyak, C. Dubs, O. Surzhenko, B. Hillebrands, A. V. Chumak, and P. Pirro, Fast long-wavelength exchange spin waves in partially compensated Ga:YIG, Appl. Phys. Lett. 120, 102401 (2022). [53] A. Gurevich and G. Melkov, Magnetization Oscillations and Waves (CRC Press., 1996). [54] P. Dechaumphai and S. Sucharitpwatskul, Finite Ele- ment Analysis with COMSOL (Alpha Science Interna- tional, Limited, 2019). [55] D. Leykam, O. Bahat-Treidel, and A. S. Desyatnikov, Pseudospin and nonlinear conical diraction in Lieb lat- tices, Phys. Rev. A 86, 031805 (2012). [56] R. Chen and B. Zhou, Finite size eects on the helical edge states on the Lieb lattice, Chin. Phys. B 25, 067204 (2016). [57] R. G. Dias and J. D. Gouveia, Origami rules for the con- struction of localized eigenstates of the Hubbard model in decorated lattices, Sci. Rep. 5, 16852 (2015). [58] J. Liu, X. Mao, J. Zhong, and R. A. R omer, Localization, phases, and transitions in three-dimensional extended Lieb lattices, Phys. Rev. B 102, 174207 (2020). [59] X. Mao, J. Liu, J. Zhong, and R. A. R omer, Disorder eects in the two-dimensional Lieb lattice and its ex- tensions, Physica E Low Dimens. Syst. Nanostruct. 124, 114340 (2020). [60] P. Graczyk and M. Krawczyk, Coupled-mode theory for the interaction between acoustic waves and spin waves in magnonic-phononic crystals: Propagating magnetoelas- tic waves, Phys. Rev. B 96, 024407 (2017). [61] R. A. Gallardo, A. Banholzer, K. Wagner, M. K orner, K. Lenz, M. Farle, J. Lindner, J. Fassbender, and P. Lan- deros, Splitting of spin-wave modes in thin lms with ar- rays of periodic perturbations: theory and experiment, New J. Phys. 16, 023015 (2014).10 SUPPLEMENTARY INFORMATION A. Double extended Lieb lattice We can generate further extensions of the magonic Lieb lattice by adding more inclusions B, i.e. by introducing additional majority sublattices. We considered here a doubly extended Lieb lattice (Lieb-7) to check to what extent the magnonic system corresponds to the tight- binding model. The mentioned lattice consists of seven nodes; six belong to majority sublattices Band one be- longs to minority sublattice A(Fig. 7). The magnetic parameters were kept as for basic and Lieb-5 lattices, considered in the manuscript. The geometrical param- eters have changed only as a result of the introduction of additional inclusions B. Therefore, the unit cell has increased to the size 500x500 nm. a) b) BA FIG. 7. Doubly extended magnonic Lieb lattice: Lieb- 7. Dimensions of the ferromagnetic unit cell are equal to 500x500x59 nm. The unit cell contains seven inclusions of 50 nm diameter. (a) The structure of extended Lieb lattice, and (b) top view on Lieb-7 lattice unit cell where the node (inclu- sion) from minority sublattice Aand two nodes (inclusions) from two majority sublattices Bare marked. In the case of a doubly extended Lieb lattice (Lieb- 7), we expect (according to the works [10, 59]) to obtain seven bands in the dispersion relation. The tight-binding model predicts that the bands will be symmetric with respect to the fourth band, exhibiting particle-hole sym- metry. However, due to the dipolar interaction, we did not expect such symmetry. Another feature which one may deduce from the tight-binding model is that bands No. 2, 4 and 6 should be at while bands No. 1, 3, 5 and 7 are considered dispersive. Moreover, bands No. 3 and 5 suppose to form a Dirac cone intersecting at band No. 4 at the point. We calculated the dispersion relation for magnonic Lieb-7 lattice (Fig. 8(a)), which share many properties with those characteristic for the tight-binding model [59]: (i) third and fth bands form the Dirac cones which al-most intersect the atter forth band at point; (ii) the third (and fth) band has a parabolic shape at Mpoint where it is degenerated with the second (and six) band which is weakly dispersive. The mentioned regions of dispersion are presented as 3D plots in Fig. 8(b). Also, we are going to discuss shortly the proles of spin wave eigenmodes (including CLS) in these two regions of the dispersion relation, which are presented in Fig. 9. Dirac cones appear at the point for bands No. 3 and 5. At this point, as for the basic magnonic Lieb lat- tice (Fig. 3), there is a very narrow gap of the width 2 MHz. The proles 4and 5(left column in Fig. 9) represent the degenerated states originating from at and dispersive bands. Both of them do not occupy the inclusions Aand are more focused on two inclusions a) b)BA FIG. 8. Dispersion relation for the double extended magnonic Lieb lattice (Lieb-7) containing seven inclusions in the unit cell: one inclusion Afrom minority sublattice and six in- clusions Bfrom majority sublattices (see Fig. 2). (a) The dispersion relation is plotted along the high symmetry path -X-M- (see the inset). The rst, third, fth and seven bands (dark blue, red, cyan and orange) are dispersive, while the second, fourth and sixth bands (green, magenta and dark green bands) are the atter bands, supporting the magnonic CLS. Dirac cones occur at the point and almost interact with the atter fourth band, while at Mpoint, we observe the degeneracy of the dispersive parabolic third (fth) band with a atter second (six) band. (b) The zoomed vicinity of point (dark green frame) and Mpoint (violet frame) regions are presented in 3D.11 M5 M7Γ3 Γ4 Γ5M6BA 3coor YIG inclusions in Ga:YIG Matrix Done reformulation Hani gamma Symmetrized M. not FIG. 9. The proles of eigenmodes were obtained for magnonic Lieb-7. The modes are presented for bands No. 3-5 at point and 5-7 at Mpoint. The modes denoted as 3and 4are degenerated whereas the 5is separated from them by extremely small gap 2 MHz. At Mpoint, we showed the proles for bands No. 5, 6 and 7. The modes M5 andM6are degenerated and separated from M7by essential gap { predicted by the tight-binding model. Barranged in horizontal ( 4) and vertical lines ( 5) { see grey stripes. Therefore, their proles are similar to NLS, where the rst and third inclusion Bin each three-element chain, linking inclusions A, precesses out- of-phase and the second (central) inclusion Bremains unoccupied. At theMpoint, the M5andM6bands are degen- erated. For these bands, the spin waves are localized in all inclusions Band do not occupy inclusions A(see right column of Fig. 6) The rst and third inclusion B in each three-element chain, linking inclusions A, pre- cess in-phase, whereas the second (central) inclusion B precesses out-of-phase with respect to the rst and third one. This pattern of occupation of inclusions and the phase relations between them is similar to one observed for CLS (see grey patches marking the loops of inclu- sions in the left column of Fig. 6), but has one signi- cant dierence. The phase dierence between successivethree-element chains of inclusions B, in the loop, is equal to=2. However, the linear combination of the modes M5iM6produces, similarly to the case of the Lieb-5 lattice, the NLS. To observe the proper proles of CLS or NLS, we need to shift slightly from the high symmetry points and Mto cancel the degeneracy. B. Realization of Lieb lattice by shaping demagnetizing eld We have considered also an alternative realisation method for a magnonic Lieb lattice in a ferromagnetic layer. This approach is based on shaping the internal de- magnetizing eld. The structure under consideration is presented in Fig. 10. It consists of a thin (28.5 nm) and innite CoFeB layer on which a Py antidot lattice (ADL), of 28.5 nm thickness, is deposited. The cylindrical holes in ADL are arranged in shape of the basic Lieb lattice. The size of the unit cell and diameter of holes remains the same as for the basic Lieb lattice proposed in the main part of the manuscript (see Fig. 1). Due to the ab- sence of perpendicular magnetic anisotropy (PMA), we decided to apply a much larger external magnetic eld (H0= 1500 mT) to saturate the ferromagnetic material in an out-of-plane direction. a) b) BA FIG. 10. Basic magnonic Lieb lattice where spin wave exci- tations in the CoFeB layer are shaped by demagnetizing eld from Py antidot lattice. Dimensions of the ferromagnetic unit cell are equal to 250x250x59 nm and contain 3 inclusions of 50 nm diameter. (a) structure of basic Lieb lattice, (b) top view on basic Lieb lattice unit cell and dierentiation to nodes of sublattice AandB. We assumed the same gyromagentic ratio for both ma- terials = 187GHz T, the following values of material parameters for CoFeB [60]: saturation magnetization - MS= 1150kA m, exchange stiness constant - A= 15pJ m. For Py, we used material parameters [61]: saturation magnetization - MS= 796kA m, exchange stiness con- stant -A= 13pJ m.12 a) b)BA FIG. 11. The dispersion relation obtained for basic Lieb lattice formed by demagnetizing eld of antidot lattice (see Fig. 10). (a) The dispersion relation, (b) the 3D plot of dis- persion relation in the region marked with the green frame in (a). Results were obtained for H0= 1500 mT applied out-of-plane. The deposition of the ADL made of Py (material of lowerMS) above the CoFeB layer (material of higher MS) is critical for spin wave localization in CoFeB below the exposed parts (holes) of the ADL. The demagnetiza- tion eld produced on CoFeB/Air interface creates wells partially conning the spin waves. However, this pattern of internal demagnetizing eld becomes smoother with increasing distance from the ADL. The obtained dispersion relation is shown in Fig. 11. It is worth noting that the lowest band is very disper- sive, while the highest band is attened more than in the case of the structure presented in the main part of the manuscript (see Fig. 3). The middle band, which sup- pose to support CLS, varies in extent similar to the third band. For this structure, Dirac cones in the Mpoint cannot be clearly unidentied. C. Lieb lattice formed by YIG inclusions in non-magnetic matrix The periodic arrangement of ferromagnetic cylinders surrounded by nonmagnetic material (e.g. air) seems to be the simplest realization of the Lieb lattice. To refer a) b)FIG. 12. Dispersion relations for basic Lieb lattice. (a) The results obtained for YIG inclusions in Ga:YIG matrix (dashed lines) and YIG inclusions without matrix (solid lines). (b) The zoomed dispersion relation obtained for YIG inclusions without matrix, marked in (a) by the frame. this structure to the bi-component system investigated in the main part of the manuscript, we assumed the same material and geometrical parameters for inclusions as for the structure presented in Fig. 1. The advantage of this system is that the connement of spin waves within the areas of inclusions is ensured for arbitrarily high frequency. We are not limited here by the FMR frequency of the matrix, as it was for bi-component Lieb lattices (Figs. 1, 2). However, the coupling of mag- netization dynamics between the inclusions is here pro- vided solely by the dynamical demagnetizing eld, i.e. the evanescent spin waves do not participate in the cou- pling. Therefore, the interaction between inclusions is much smaller in general, which leads to a signicant nar- rowing of all magnonic bands (Fig. 11). The widths of the second and third band can be even smaller than the gap separating from the rst bands { Fig. 11(b). Such strong modication of the spectrum makes the applicability of the considered system for the studies of magnonic CLS questionable.13 D. Demagnetizing eld in YIG|Ga:YIG Lieb lattice The diculty in designing the magnonic system is not only due to the adjustment of geometrical parameters of the system but also due to the shaping of the internal magnetic eld He. The components of the eective magnetic eld can be divided into long-range and short-range. The realiza- tion of our model is inseparably linked to the long-range dipole interactions through which the coupling between inclusions is possible. This kind of interaction is sensitive to the geometry of the ferromagnetic elements forming the magnonic system. In Lieb lattice, the nodes of minority sublattice Ahave four neighbours and the nodes of majority sublattice B have two. As a result, identical inclusions (in terms of their shapes and material parameters) become dis- tinguishable, because of slightly dierent values of the internal demagnetising eld. This has consequences for the formation of a frequency gap between Dirac cones at pointMin the dispersion relation obtained for the basic Lieb lattice. In the literature, this phenomenon has been described for the tight-binding model and is called node dimerisation of the lattice [37]. In Fig. 13 we have shown the prole of the z-component of the demagnetising eld. For each inclusion through which the cut line passes, we have marked the minimum value of the demagnetising eld. The slightly lower value of internal led for inclusions Ais responsible for a tiny lowering of the frequency for the mode M1(concentrated in inclusions A) respect the degenerated modes M2and M3(conned in inclusions B). a) b)BA FIG. 13. Prole of static demagnetizing eld plotted at cut through (a) Lieb lattice unit cell. (b) The z-component of the demagnetizing eld along the cut line is shown in (a). In the plot, we have marked peaks for the areas of inclusions Aand B. Please note the slightly dierent values of demagnetizing in the centre of AandBinclusion due to dierent the number neighboring of nodes: four for inclusion A, two for inclusion B.

2023-03-26

Lieb lattice is one of the simplest bipartite lattices where compact localized states (CLS) are observed. This type of localisation is induced by the peculiar topology of the unit cell, where the modes are localized only on one sublattice due to the destructive interference of partial waves. The CLS exist in the absence of defects and are associated with the flat bands in the dispersion relation. The Lieb lattices were successfully implemented as optical lattices or photonic crystals. This work demonstrates the possibility of magnonic Lieb lattice realization where the flat bands and CLS can be observed in the planar structure of sub-micron in-plane sizes. Using forward volume configuration, we investigated numerically (using the finite element method) the Ga-dopped YIG layer with cylindrical inclusions (without Ga content) arranged in a Lieb lattice of the period 250 nm. We tailored the structure to observe, for the few lowest magnonic bands, the oscillatory and evanescent spin waves in inclusions and matrix, respectively. Such a design reproduces the Lieb lattice of nodes (inclusions) coupled to each other by the matrix with the CLS in flat bands.

Compact localised states in magnonic Lieb lattices

2303.14843v1

arXiv:1510.09007v1 [cond-mat.mes-hall] 30 Oct 2015Pure spin-Hall magnetoresistance in Rh/Y 3Fe5O12 hybrid T. Shang1,Q.F.Zhan1,*,L.Ma2, H.L.Yang1, Z.H.Zuo1, Y. L.Xie1,H. H.Li1, L.P. Liu1,B. M. Wang1,Y. H.Wu3,S. Zhang4,†,and Run-WeiLi1,‡ 1Key Laboratory of Magnetic Materials andDevices& Zhejiang ProvinceKey Laboratory ofMagnetic Materials and Application Technology,Ningbo Institute ofMaterial T echnology and Engineering, Chinese Academy of Sciences,Ningbo, Zhejiang 315201, P.R.China 2Department of Physics,Tongji University,Shanghai, 20009 2, P.R.China 3Department of Electrical andComputer Engineering, Nation al Universityof Singapore, 4Engineering Drive3 117583, Singapore 4Department of Physics,Universityof Arizona,Tucson,Ariz ona85721, USA *zhanqf@nimte.ac.cn †zhangshu@email.arizona.edu ‡runweili@nimte.ac.cn ABSTRACT We report an investigation of anisotropic magnetoresistan ce (AMR) and anomalous Hall resistance (AHR) of Rh and Pt thin films sputtered on epitaxial Y 3Fe5O12(YIG) ferromagneticinsulator films. For the Pt/YIG hybrid, large spin-Hall magne- toresistance (SMR) along with a sizable conventional aniso tropic magnetoresistance (CAMR) and a nontrivial temperat ure dependenceof AHR were observed in the temperaturerange of 5 -300 K. In contrast, a reduced SMR with negligibleCAMR and AHR was found in Rh/YIG hybrid. Since CAMR and AHR are char acteristics for all ferromagnetic metals, our results suggest that the Pt is likely magnetized by YIG due to the magn etic proximity effect (MPE) while Rh remains free of MPE. ThustheRh/YIGhybridcouldbeanidealmodelsystem toexplo rephysicsanddevicesassociatedwithpurespincurrent. Introduction Thestudiesofmagneticinsulator-basedspintronicshaver ecentlygeneratedgreatinterestduetotheirsegregatedch aracteristic of spin current from charge current.1The interplay between spin and charge transports in nonmagn eticmetal/ferromagnetic insulator(NM/FMI)hybridsgivesrise tovariousinteresti ngphenomena,suchasspin injection,2,3spinpumping,4–6andspin Seebeck.7,8The previous investigations on NM/FMI hybrids, e.g., Pt/Y 3Fe5O12(Pt/YIG), also demonstrated a new-type of magnetoresistance9–13in whichtheresistivityoffilms, ρ,hasanunconventionalangulardependence,namely, ρ=ρ0−Δρ/bracketleftbigˆm·(ˆz׈j)/bracketrightbig2(1) whereˆmandˆjareunitvectorsinthedirectionsofthemagnetizationandt heelectriccurrent,respectively,and ˆzrepresentsthe normal vector perpendicular to the plane of the film; ρ0is the zero-field resistivity. The above angular-dependent resistivity has been named as the spin-Hall magnetoresistance (SMR) in o rder to differentiate from the conventional anisotropic ma g- netoresistance (CAMR) in which ρ=ρ0+Δρ(ˆm·ˆj)2. A theoretical model outlined below has been proposed to exp lain the SMR.Anelectriccurrent( je)inducesaspincurrentduetothespin-Halleffectandintur n,theinducedspincurrent,viainverse spin-Hall effect, generates an electric current whose dire ction is opposite to the original current.14–21Thus , the combined spin-Hallandinversespin-Halleffectsleadtoanaddition alresistanceinbulkmaterialswithspin-orbitcoupling(S OC).How- ever, in an ultra-thin film, the spin current could be either r eflected back at the interface or absorbed at the interface th rough spin transfer torque. In the former case, the total spin curr ent in the metal layer is reduced and thus the additional resi stance is minimized. The spin current reflection is strongest when t he magnetization direction ˆmof the ferromagnetic insulator is parallelto thespinpolarization ˆz׈jofthespincurrent,leadingtothe resistiveminimumasdesc ribedin Eq.(1).14,15 However, the magnetic proximity effect (MPE), in which a non -magnetic metal develops a sizable magnetic moment in the close vicinityof a ferromagneticlayer,maycomplicate the interpretationofthe SMR. Pt is neartheStonerferromag netic instability and could become magnetic when in contact with f erromagnetic materials, as experimentally shown by x-ray magnetic circular dichroism (XMCD), anomalous Hall resist ance (AHR), spin pumping, and first principle calculations o f the Pt/YIG hybrid.22–26In order to separate the MPE form the pure SMR, many attempts h ave been made. By inserting a 1/s53/s46/s48 /s109/s48/s46/s48 /s45/s48/s46/s53/s110/s109/s48/s46/s53/s110/s109 /s40/s99/s41 /s89/s73/s71/s48 /s57/s48 /s49/s56/s48 /s50/s55/s48 /s51/s54/s48 /s32/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s46/s117/s46/s41 /s40/s100/s101/s103/s114/s101/s101/s41/s32/s71/s71/s71/s32 /s32/s89/s73/s71/s40/s98/s41 /s45/s52 /s45/s50 /s48 /s50 /s52/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48 /s32/s32/s77/s47/s77 /s83 /s70/s105/s101/s108/s100/s32/s40/s75/s79/s101/s41/s32/s105/s110/s45/s112/s108/s97/s110/s101 /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 /s32/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s46/s117/s46/s41 /s50 /s40/s100/s101/s103/s114/s101/s101/s41/s71/s71/s71/s32/s40/s52/s52/s52/s41 /s40/s97/s41 Figure1. (Coloronline)(a)A representative2 θ-ωXRD patternsforYIG/GGG filmnearthe(444)peaksofGGGsubst rate andYIGfilm. (b)The ϕ-scanofYIG/GGGfilm. (c)AFM surfacetopographyofa represe ntativeYIGfilm. (d)Thefield dependenceofnormalizedmagnetizationforYIG/GGGfilm mea suredatroomtemperature. Forthein-plane(out-of-plane) magnetization,the magneticfieldisappliedparallel(perp endicular)tothefilm surface. layer of Au or Cu between NM and FMI, the MPE can be effectively screened, but the SMR amplitude is largelysuppressed as well.9,10Furthermore, the insertion of an extra layer would introduc e an additional interface whose quality is not easily accessible. An alternative approach to pursue the pure SMR i s to find proper NM metals in direct contact with YIG, but without the MPE. The Au has a SOC strength comparableto Pt or P d and it is freeof the MPE, but it hasan extremelyweak inverse spin-Hall voltage and SMR.26,27According to the theoretical calculation,28the 4dmetal Rh also possesses a large SOC strengthandspin-Hall conductivity,and a small magnet icsusceptibility,implyinganinsignificantMPE in the Rh me tal whenincontactwithferromagneticmaterials. ThusRh might be anexcellentmaterialforthe pureSMRstudy. Inthisarticle,theanisotropicmagnetoresistance(AMR)a ndAHRofRh/YIGandPt/YIGhybridswereinvestigatedinthe temperaturerangeof5-300K.Indeed,weshowthatthediffer encesinmagneto-transportpropertiesbetweenthesetwohy brids are attributed to the strong (Pt) and weak (Rh) MPE, and thus, Rh/YIG providesan ideal model system for purespin-current investigations. Results Figure1(a)plotsarepresentativeroom-temperature2 θ-ωXRDscanofepitaxialYIG/GGGthinfilmnearthe(444)reflecti ons ofgadoliniumgalliumgarnet(GGG)substrateandYIGfilm. Cl earLaueoscillationsindicatetheflatnessanduniformityo fthe epitaxial YIG film. The epitaxial nature of YIG film was charac terizedby ϕ-scan measurementswith a fixed 2 θvalue at the (642)reflections,asshowninFig. 1(b). Inthisstudy,theth icknessesofYIGandRhorPtfilms,determinedbyfittingthex- ray reflectivity(XRR) spectra,areapproximately50nmand3nm, respectively. TheAFM surfacetopographyofarepresentati ve YIG film in Fig. 1(c) reveals a surface roughnessof 0.15 nm, in dicating atomically flat of the epitaxial YIG film. As shown in Fig. 1(d),thein-planeandout-of-planecoercivitiesof theYIGfilm are <1Oe and60Oe, respectively. Theparamagnetic backgroundoftheGGGsubstratehasbeensubtractedandthem agnetizationisnormalizedtothesaturationmagnetizatio nMs. The out-of-planemagnetization saturates at a field above 2. 2 kOe, which is consistent with previousresults.12,22The above propertiesindicatetheexcellentqualityofourepitaxial YIGfilm. Figures 2(a)-(c) plot the room-temperature AMR for the Rh/Y IG (open symbols) and Pt/YIG (closed symbols) hybrids. As shown in the rightpanels, the Rh/YIG and Pt/YIG hybridsar e patternedinto Hall-bargeometryandthe electriccurrent is applied along the x-axis. The AMR is measuredin a magnetic field of 20 kOe, which i s sufficiently strong to rotate the YIG magnetization in any direction. Here the total AMR is defined asΔρ/ρ0= [ρ(M/bardblI) -ρ(M⊥I)]/ρ0. We note that when the magnetic field scanswithin the xyplane [Fig. 2(a)],boththe CAMR and SMR contributeto the tot al AMR, and it is difficult to separate them from each other; for the xzplane [Fig. 2(b)], the magnetization of YIG is always perpen dicularto the spin 2/7Figure2. Anisotropicmagnetoresistanceforthe Rh/YIG(opensymbol s)andPt/YIG (closedsymbols)hybridswith the magneticfieldscanningwithinthe xy(a),xz(b),andyz(c)planes. TheAMRismeasuredatroomtemperatureina fieldo f µ0H =20kOe. Thesolidlinesthroughthedataarefitsto cos2θwitha 90degreephaseshift. Therightpanelsshowthe schematicplotsoflongitudinalresistance andtransverse Hall resistancemeasurementsandnotationsofdifferentfie ldscans in thepatternedHallbarhybrids. The θxy,θxz,andθyzdenotetheanglesoftheappliedmagneticfieldrelativeto th ey-,z-, andz-axes,respectively. polarizationof the spin currentand the SMR is absent, and th e resistance changesare attributed to the MPE-inducedCAMR . Fortheyzplane[Fig. 2(c)],theelectriccurrentisalwaysperpendic ularto themagnetization,theCAMRiszero,andonlythe SMR survives. According to Eq. (1), the amplitudes of CAMR or SMR (Δρ/ρ0) oscillate as a function of cos2θ, as shown by the solid black lines in Fig. 2. Both the Rh/YIG and Pt/YIG h ybrids display clear SMR at room temperature, with the amplitudes reaching 7.6 ×10−5and 6.1×10−4, respectively [see Fig. 2(c)]. On the other hand, the CAMR al so exists in the 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, theθxzscan shows negligible AMR and the resistivity is almost inde pendent of θxz, indicating the extremely weak MPE at theRh/YIG interfaceincontrasttothesignificanteffectat thePt/YIGinterface. TheMPEat Pt/YIGinterfacewasprevio usly evidencedfromthemeasurementsofXMCD,AHR, andspinpumpi ng.22,24,25 Upondecreasingtemperature,theSMRpersistsdownto5Kinb oththeRh/YIGandPt/YIGhybrids[seeFig. 3]. However, thereisnosizableCAMRintheRh/YIGhybriddowntothelowes ttemperature[seeFig. 3(b)],indicatingtheextremelywea k MPE at the interface even at low temperature. While for the Pt /YIG hybrid, as shown in Fig. 3(e), the amplitude of CAMR is almost independentof temperaturefor T>100K, andthen decreasesby furtherloweringtemperature,w ith theamplitude reaching 1.0 ×10−4at 5 K. The above features are quite different from the Pd/YIG hybrid, where the amplitude of CAMR increases as the temperature decreases, showing a comparab le value to the SMR at 3 K.12The reason for these different behaviors is unclear, and further investigations are neede d. Since the CAMR is negligible in Rh/YIG, the SMR dominates the AMR when the magnetic field is varied within the xyplane. Figures 3(g) and (h) plot the temperature dependence of SMR amplitude for the Rh/YIG and Pt/YIG, respectively. The S MR amplitudes exhibit strong temperature dependence, reachingamaximumvalueof1.1 ×10−4(Rh/YIG)and6.9 ×10−4(Pt/YIG)around150K.Suchnonmonotonictemperature dependenceofSMRamplitudewaspreviouslyreportedin Pt/Y IG hybrid,whichcan bedescribedbya single spin-relaxatio n mechanism.29It is noted that the hybridswith different Rh thicknesses ex hibit similar temperaturedependentcharacteristics with different numerical values compared to the Rh(3 nm)/YI G hybrid shown here. For example, the Rh(5 nm)/YIG hybrid reachesitsmaximumSMRamplitudeof0.8 ×10−4around100K. In order to characterize the MPE at the NM/FMI interface, we a lso carried out the measurements of transverse Hall resistance 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 films, the ordinary-Hall resistance (OHR), which is proport ional to the external field, is subtracted from the measured R xy, 3/7/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s51/s48/s54/s48/s57/s48 /s47 /s48/s40/s49/s48/s45/s53 /s41 /s84/s32/s40/s75/s41/s32/s72/s47/s47/s121/s122/s40/s104/s41/s48/s52/s56/s49/s50 /s45/s50/s48/s50 /s48 /s57/s48 /s49/s56/s48 /s50/s55/s48 /s51/s54/s48/s45/s56/s45/s52/s48/s48/s50/s48/s52/s48/s54/s48 /s45/s50/s48/s45/s49/s48/s48 /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 /s32/s32 /s51/s48/s48/s75 /s50/s53/s48/s32 /s50/s48/s48/s32/s32/s32 /s49/s53/s48/s32/s32 /s49/s48/s48/s32/s32 /s53/s48 /s49/s48/s32/s32/s32/s32 /s53/s82/s104/s47/s89/s73/s71 /s32/s47 /s48/s32/s40/s49/s48/s45/s53 /s41 /s40/s98/s41 /s40/s99/s41 /s32 /s32/s40/s100/s101/s103/s114/s101/s101/s41/s80/s116/s47/s89/s73/s71 /s40/s100/s41 /s40/s101/s41 /s47 /s48/s40/s49/s48/s45/s53 /s41 /s40/s102/s41 /s32/s40/s100/s101/s103/s114/s101/s101/s41 /s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s52/s56/s49/s50 /s72/s47/s47/s120/s121 /s72/s47/s47/s121/s122/s47 /s48/s40/s49/s48/s45/s53 /s41 /s84/s40/s75/s41/s40/s103/s41 Figure3. Anisotropicmagnetoresistanceforthe Rh/YIGhybridat var ioustemperaturesdownto 5K forthe θxy(a),θxz(b), andθyz(c)scans. TheresultsofPt/YIG areshownin(d)-(f). TheAMR ismeasuredina fieldof µ0H= 20kOe. (g)and(h) plotthe temperaturedependenceofSMRamplitudefortheRh/ YIGandPt/YIG hybrids,respectively. Thecubicandtriangl e symbolsstandforthe θxyandθyzscans,respectively. 4/7/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54 /s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s45/s50/s48/s48/s50/s48/s52/s48 /s40/s101/s41/s82 /s65/s72/s82/s32/s40/s109 /s41 /s84/s32/s40/s75/s41 /s82 /s65/s72/s82/s32/s40/s109 /s41 /s84/s32/s40/s75/s41/s40/s102/s41 /s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s49/s48/s50/s48/s51/s48/s45/s56/s45/s52/s48/s52/s56 /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 /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 /s32/s82 /s120/s121/s32/s40/s109 /s41 /s51/s48/s48/s75 /s40/s97/s41 /s40/s99/s41 /s32/s32/s82 /s65/s72/s82/s32/s40/s109 /s41 /s70/s105/s101/s108/s100/s32/s40/s75/s79/s101/s41/s40/s98/s41 /s82 /s120/s121/s32/s40/s109 /s41/s53/s75 /s51/s48/s48/s75 /s40/s100/s41 /s82 /s65/s72/s82/s32/s40/s109 /s41 /s70/s105/s101/s108/s100/s32/s40/s75/s79/s101/s41 Figure4. TransverseHall resistanceR xyfortheRh/YIG (a)andPt/YIG (b)hybridsasafunctionofmagn eticfieldupto70 kOe atdifferenttemperatures. TheanomalousHall resistan ceRAHRfortheRh/YIG(c)andPt/YIG(d)at different temperatures. TheR AHRcanbederivedbysubtractingthelinearbackgroundofOHR. T emperaturedependenceofR AHRfor the Rh/YIG(e)andPt/YIG (f). All R AHRareaveragedby[R AHR(70kOe)-R AHR(-70kOe)]/2. Theerrorbarsarethe results ofsubtractingOHRindifferentfieldranges. i.e.,RAHR=Rxy-ROHR×µ0H,RAHRistheanomalousHallresistance. TheresultingR AHRasafunctionofmagneticfieldfor the Rh/YIG and Pt/YIG hybrids are shown in Figs. 4(c) and (d), respectively. The AHR is proportional to the out-of-plane magnetization,andthusprovidesanotionforMPEattheNM/F MIinterface. Atroomtemperature,fortheRh/YIGhybrid,th e RAHR= 0.57mΩ,which is 22 timessmaller than the Pt/YIG hybrid,implyingt he extremelyweak MPE at Rh/YIG interface, being consistent with the AMR results in Fig. 2(b). We note th at the R AHRof Rh/YIG hybridswith different Rh thicknesses was 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 temperature,respectively. The temperaturedependenceof RAHRfor the Rh/YIG and Pt/YIG hybridsare summarizedin Figs. 4 (e) and (f), respectively. As can be seen, the R AHRexhibits significantly different behaviors: the R AHRroughly decreases on lowering temperature in Rh/YIG. However, in Pt/YIG, the m agnitude of R AHRdecrease with temperature for T>50 K and then it suddenly increases upon further decreasing temp erature. Moreover, the R AHRof Pt/YIG changes sign below 50 K, while it is stays positive for Rh/YIG. Similar non-trivia l AHR were also observed in Pt/LCO hybrids,30but there is no existing quantitativetheory to comparethese results, fur thertheoretical and experimentalinvestigationsare need edto clarify the dominatingmechanisms. Summary In summary, we carried out measurements of angular dependen ce of magnetoresistance and transverse Hall resistance in Rh/YIG and Pt/YIG hybrids. Both hybrids exhibit SMR down to v ery low temperature. The observed AHR and CAMR indicateasignificantMPEatthePt/YIGinterface,whileiti snegligibleattheRh/YIGinterface. Ourfindingssuggestth atthe absenceoftheMPE makestheRh/YIGbilayersystem anidealpl aygroundforpurespin-currentrelatedphenomena. Methods The Rh/YIG and Pt/YIG hybrids were prepared in a combined ult ra-high vacuum (10−9Torr) pulse laser deposition (PLD) and magnetron sputter system. The high quality epitaxial YI G thin films were grown on (111)-orientated single crystalli ne GGG substrate via PLD technique at 750◦C. The thin Rh and Pt films were deposited by magnetron sputter ing at room temperature. All the thin filmswere patternedintoHall-bar geometry. The thicknessand crystalstructurewere charact erized 5/7by using Bruker D8 Discover high-resolution x-ray diffract ometer (HRXRD). The thickness was estimated by using the software package LEPTOS (Bruker AXS). The surface topograp hy and magnetic properties of the films were measured in Bruker Icon atomic force microscope (AFM) and Lakeshore vib rating sample magnetometer (VSM) at room temperature. The measurements of transverse Hall resistance and longitu dinal resistance were carried out in a Quantum Design physic al propertiesmeasurementsystem(PPMS-9T)witha rotationop tionina temperaturerangeof5-300K. Acknowledgments We acknowledgethefruitfuldiscussionswithS.M.Zhou. Thi sworkisfinanciallysupportedbytheNationalNaturalScien ce Foundation of China (Grants No. 11274321, No. 11404349, No. 11174302, No. 51502314, No. 51522105) and the Key Research Program of the Chinese Academy of Sciences (Grant N o. KJZD-EW-M05). S. Zhang was partially supported by the U.S. NationalScienceFoundation(GrantNo. ECCS-14045 42). Author contributions Q. F. Z., S. Z.,and R. W. L. plannedthe experiments. T.S., L. M ., andY. L. X. synthesizedthe hybrids. Structurecharacter i- zation,magneticandtransportmeasurementswereperforme dbyT.S.,H.L.Y.,Z.H.Z.,H.H.L.,andL.P.L.Thedatawere analysed 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 authorsparticipatedin discussionsandapprovedthe submi ttedmanuscript. Additionalinformation Competingfinancialinterests: Theauthorsdeclarenocompetingfinancialinterests. References 1.Wu, M. Z. and Hoffmann, A. Recent Advances in Magnetic Insula tors - From Spintronics to Microwave Applications (AcademicPress , SanDiego,Vol 64,2013). 2.Ohno,Y. et al.Electricalspininjectionina ferromagneticsemiconducto rheterostructure. Nature402,790(1999). 3.Jedema,F. 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2015-10-30

We report an investigation of anisotropic magnetoresistance (AMR) and anomalous Hall resistance (AHR) of Rh and Pt thin films sputtered on epitaxial Y$_3$Fe$_5$O$_{12}$ (YIG) ferromagnetic insulator films. For the Pt/YIG hybrid, large spin-Hall magnetoresistance (SMR) along with a sizable conventional anisotropic magnetoresistance (CAMR) and a nontrivial temperature dependence of AHR were observed in the temperature range of 5-300 K. In contrast, a reduced SMR with negligible CAMR and AHR was found in Rh/YIG hybrid. Since CAMR and AHR are characteristics for all ferromagnetic metals, our results suggest that the Pt is likely magnetized by YIG due to the magnetic proximity effect (MPE) while Rh remains free of MPE. Thus the Rh/YIG hybrid could be an ideal model system to explore physics and devices associated with pure spin current.

Pure spin-Hall magnetoresistance in Rh/Y3Fe5O12 hybrid

1510.09007v1

Design of X-Band Bicontrollable Metasurface Absorber Comprising Graphene Pixels on Copper-Backed YIG Substrate Govindam Sharma1, Akhlesh Lakhtakia2, and Pradip Kumar Jain1 1Department of Electronics and Communication Engineering, National Institute Technology Patna, Patna 800005, India 2Departtment of Engineering Science and Mechanics, The Pennsylvania State University, University Park, PA 16802, USA Abstract The planewave response of a bicontrollable metasurface absorber with graphene-patched pixels was simulated in the X band using commercial software. Each square meta-atom is a 44 array of 16 pixels, some patched with graphene and the others unpatched. The pixels are arranged on a PVC skin which is placed on a copper-backed YIG substrate. Graphene provides electrostatic controllability and YIG provides magnetostatic controllability. Our design delivers absorptance0:9 over a 100-MHz spectral regime in the X band, with 360 MHz kA1m mag- netostatic controllabity rate and 1 MHz V1m electrostatic controllability rate. Notably, electrostatic control viagraphene in the GHz range is novel. Keywords: Bicontrollability, Magnetostatic controllability, Electrostatic controllability, Pixe- lation, Graphene, Yttrium iron garnet, Meta-atom, Metasurface, GHz. 1 Introduction Metasurfaces are thin compared to the operational wavelength, accounting for their popularity in the R&D arena. The use of materials that respond electromagnetically to a stimulus allows controllable metasurfaces to be designed for beam-steering re ectors/lters [1], mirrors/lenses with variable focus [2], and absorbers/lters [3, 4] in a wide spectrum beginning with the microwave frequencies and ending with the visible frequencies. Typically, controllable metasurfaces are designed to operate at high frequencies [5]. Direct scaling [6, 7] of controllable metasurface absorbers from THz frequencies to GHz frequencies is not always feasible, since constitutive parameters are frequency dependent. Generally, at GHz frequencies, metal is used to design the top layer of the metasurface, but using materials such as ferrites [8], graphene [9], and conductive rubber [10] allow control of metasurface absorbers. Huang et al. experimentally demonstrated a magnetostatically controllable (or tunable) X-band absorber containing a ferrite slab, with a 300-MHz controllability range for absorptance A>0:9 [8]. Fallahi et al. design an electrostatically controllable metasurface absorber containing patterned graphene|but only the maximum absorptance Amax, not the maximum-absorptance frequency fmaxA, can be controlled with that design [11]. Yi et al. used shape memory polymers to thermally controlfmaxA2[11:3;13:5] GHz [12]. None of these metasurface absorbers covers the complete X band with absorptance in excess of 0 :9, which is an important requirement for wide use. 1arXiv:2211.03510v1 [physics.app-ph] 1 Nov 2022Bicontrollable X-band metasurface absorbers are desirable for weather radar, police speed radar, and direct broadcast television. With that in mind, Sharma et al. designed a pixellated metasur- face absorber with coarse magnetostatic and ne thermal controllability of fmaxA over the entire X band [13]. The meta-atoms in this design comprise yttrium iron garnet (YIG)-patched pixels and con- ductive rubber (CR)-patched pixels on a metal-backed silicon substrate. Continuing in the same vein, we are now reporting a pixelated metasurface absorber with fmaxA controllable both magnetostatically and electrostatically in the entire X band, while keeping A0:9 over a 100-MHz spectral regime. Each meta-atom is a NrNrarray of pixels, some patched with graphene and the others unpatched. In contrast to numerous designs [14,15], the patches are not metallic. The pixels are arranged on a PVC skin which is placed on top of a copper-backed YIG substrate. Graphene provides electrostatic controllability and YIG provides magnetostatic controllability. Pixel size as well as the conguration of patched pixels were decided by examining the absorptance spectrums of many designs. The plan of this paper is as follows. Section 2 provides information on the metasurface geometry, the relative permeability dyadic of YIG, the surface conductivity of graphene, and theoretical simulations. Numerical results are presented and discussed in Sec. 3. Some remarks in Sec. 4 conclude the paper. An exp(j!t) dependence on time tis implicit, with j=p1,!= 2fas the angular frequency, andfas the linear frequency. The free-space wavenumber is denoted by k0=!p"00= 2= 0, where0is the free-space wavelength, "0is the free-space permittivity, and 0is the free-space permeability. Vectors are denoted by boldface letters; the Cartesian unit vectors are denoted by ^x, ^y, and ^z; and dyadics are double underlined. 2 Materials and Methods 2.1 Device Structure The metasurface extends to innity in all directions in the xyplane, but it is of nite thickness along thezaxis, as depicted in Fig. 1. The metasurface is a biperiodic array of square meta-atoms whose sides are aligned along the xandyaxes. Each meta-atom is of side a. The front surface of the meta-atom is an array of NrNrsquare pixels of side b, each pixel separated from every neighboring pixel by a distance da, so that Nr=a=(b+d). Some of the pixels are patched with graphene, but others are not. Underneath the pixels is a polyvinyl-chloride (PVC) skin of thickness LPVC, a YIG substrate of thickness Lsub, and a copper sheet of thickness Lmserving as a back re ector. We xedNr= 4,LPVC= 0:08 mm,Lsub= 0:2 mm, and Lm= 0:07 mm. In addition, we xed a= 6 mm,b= 1:45 mm, and d= 0:05 mm, after multiple iterations of parameter sweeps. 2.2 YIG The relative permeability dyadic  YIGof YIG depends on the magnitude and direction of the external magnetostatic eld H0. With this eld aligned along the xaxis (i.e, H0=H0^x), we have [16]  YIG=^x^x+yy(^y^y+^z^z) +jyz(^y^z^z^y); (1a) 2Figure 1: Schematics of four meta-atoms: (a) copper-backed YIG substrate; (b) graphene on top of a PVC skin overlaying a copper-backed YIG substrate; (c) the same as (b) but with graphene partitioned as a 4 4 array of graphene patches; and (d) the same as (c) but with only ten pixels patched with graphene. The Cartesian coordinate system is also shown. where yy= 1 +42 0 Ms H0+j4H 2 (0 )2 H0+j4H 22 !2(1b) and yz=4! 0 Ms (0 )2 H0+j4H 22 !2: (1c) In these equations, = 1:761011C kg1is the gyromagnetic ratio, 4H= 1:98 kA m1is the resonance linewidth, and Ms= 0:18 Wb m2is the saturation magnetization. The relative permittivity scalar of YIG is "YIG= 15. Note that H0^xcan be applied by placing the metasurface between two magnets, so long as the lateral extent of the metasurface is in excess of 10 0. 2.3 Graphene Graphene is not aected signicantly by H0^x, because that magnetostatic eld is wholly aligned in the plane containing the carbon atoms [17]. It is, however, aected by the external electrostatic eldE0=E0^zaligned normal to that plane, which can be applied using transparent electrodes signicantly above and below the metasurface. 3The surface conductivity of graphene grcomprises an intraband term and an interband term, the latter being negligibly small compared to the former in the X band [17,18]. Accordingly, gr=jq2 ekBT ~2(!2j1gr) c kBT+ 2 ln 1 + exp c kBT ; (2) whereqe= 1:602 1771019C is the elementary charge, kB= 1:380 6491023J K1is the Boltzmann constant, and ~= 1:054 5721034J s is the reduced Planck constant. All calculations were made for temperature T= 300 K. We xed the momentum relaxation time gr= 0:4 ps after examining values of the maximum absorptance Amaxand the controllability rate @fmaxA=@E 0for gr2[0:01;1] ps. This relaxation time can be controlled by impurity level [3]. The value of the chemical potential cdepends on E0as well as on the d.c. relative permittivity "PVC= 2:7 of PVC [19]. Thus [4,17], "0~22 F qek2 BT2"PVCE0= Li 2 exp c kBT Li2 expc kBT ; (3) whereF= 106m s1[20] is the Fermi speed for graphene and Li () is the polylogarithm function of orderand argument [21]. The Newton{Raphson technique [22] was used to determine cas a function of E0. 2.4 Theoretical Simulations The pixels of the metasurface were taken to be illuminated by a normally incident, linearly polarized plane wave whose electric eld phasor can be written as Einc=^xexp(jk0z); (4) withas its amplitude. As the metasurface is periodic along the xandyaxes, the re ected eld must be written as a doubly innite series of Floquet harmonics [23]. Since a <  0=4 in the entire X band, only specular components of the re ected eld are non-evanescent. Therefore, the re ected electric eld asz!1 may be written as Eref'(xx^x+yx^y) exp(jk0z); (5) wherexx2Cis the co-polarized re ection coecient and yx2Cis the cross-polarized re ection coecient. The transmitted eld in the region beyond the metallic back re ector was negligibly small in magnitude, because Lmis much larger than the penetration depth in copper. Hence, the absorptance was calculated as A= 1(jxxj2+jyxj2): (6) Normal incidence on several congurations of the pixelated-metasurface absorber was simulated using the commercial tool CST Microwave Studio ™2020. Periodic boundary conditions were applied 4along thexandyaxes. The option open was chosen for the zaxis and the planewave condition applied. The meta-atom was partitioned into as many as 10,026 tetrahedrons for each simulation in order to achieve convergent results. The absorptance Awas calculated for f2[8;12] GHz, H02[180;270] kA m1, andE02[0;100] Vm1. 3 Numerical Results We begin by discussing the response of the copper-backed YIG substrate shown in Fig. 1(a). Fig- ure 2(a) shows the computed spectrums of AforH02f180;210;240;270gkA m1, this metasurface being unaected by E0. The maximum-absorptance frequency fmaxA blueshifts as the magneto- static eld H0increases, but the maximum absorptance Amax0:8. Hence, the copper-backed YIG substrate does not satisfy the requirement of Amax2[0:9;1] in any spectral regime within the X band. Figure 2: Absorptance spectrums of (a) the YIG/copper structure of Fig. 1(a) for H02 f180;210;240;270gkA m1and (b) the graphene/PVC/YIG/copper structure of Fig. 1(b) for E02f0;50;100gVm1andH0= 240 kA m1. Covering the YIG substrate on the top, rst by a PVC skin and then by graphene, as in Fig. 1(b), certainly aects the absorptance. Graphene makes this structure susceptible to E0, in addition to the YIG-mediated susceptibility to H0. The spectrums of Aare shown in Fig. 2(b) for E02f0;50;100gVm1andH0= 240 kA m1. Now,Amaxbecomes a decreasing function ofE0, although the controllability of fmaxA byH0(results not shown) is maintained. Therefore, the copper-backed YIG substrate with or without the graphene/PVC bilayer is inadequate as the desired bicontrollable metasurface absorber. For the next set of simulations, we partitioned the graphene in Fig. 1(b) into 16 patches per meta-atom, as shown in Fig. 1(c). The absorptance spectrums in Fig. 3(a) for E02 f0;50;100gVm1andH0= 240 kA m1clearly indicate that pixelation can increase Amaxand make it less susceptible to variations in E0, when compared with the spectrums in Fig. 2(b). The 5Figure 3: Absorptance spectrums of the pixelated metasurface of Fig. 1(c), with all 16 pixels per meta-atom patched with graphene. (a) E02f0;50;100gVm1andH0= 240 kA m1. (b) H02f180;210;240;270gkA m1andE0= 50 Vm1. absorptance spectrums in Fig. 3(b) for H02f180;210;240;270gkA m1andE0= 50 Vm1con- rm the magnetostatic controllability of fmaxA. Finally, we present the absorptance spectrums calculated for the metasurface of Fig. 1(d), which has ten graphene-patched and six unpatched pixels. The specic conguration of unpatched pixels was selected after studying the absorption spectrums for many other congurations. The spectrums in Fig. 4(a) for E02f0;50;100gVm1andH0= 240 kA m1and Fig. 4(b) for H02f180;210;240;270gkA m1andE0= 50 Vm1indicate that a bicontrollable spectral regime with A0:9 andAmax0:99 can be achieved with 360 MHz kA1m magnetostatic control and 1 MHz V1m electrostatic control of fmaxA. Coarse control is possible through H0 and ne control through E0. The bandwidth 4fA0:9of this absorber is about 100 MHz, which is suitable for many X-band applications. Table 1 compares the proposed metasurface absorber with previously reported absorbers. Yuan et al. [24] designed a voltage-controlled metasurface absorber containing varactor diodes, for X-band operation with fmaxA controlled in a 440-MHz range. Huang et al. [8] incorporated a meta-atom with a metal resonator printed on FR4 and axed to a metal-backed ferrite substrate. Their meta- surface absorber has a wider bandwidth than the proposed absorber redhas, but the controllability range is smaller than of the proposed absorber. Sharma et al. [13] reported a meta-atom with a square array of pixels patched with conductive rubber and YIG on a metal-backed silicon substrate. This bicontrollable metasurface has a wider bandwidth with stable maximum absorptance in the entire X band, and ne control is thermal rather than electrostatic as for the proposed absorber. I 6Figure 4: Absorptance spectrums of the pixelated metasurface of Fig. 1(d), with only 10 pixels per meta-atom patched with graphene. (a) E02f0;50;100gVm1andH0= 240 kA m1. (b) H02f180;210;240;270gkA m1andE0= 50 Vm1. 4 Concluding Remarks We conceived, designed, and investigated a electrostatically and magnetostatically controllable metasurface absorber for operation in the entire X band. The meta-atom comprises ten graphene- patched pixels and six unpatched pixels in a 4 4 array on a PVC skin that is axed to a metal- backed YIG substrate. Graphene provides electrostatic controllability and YIG provides magneto- static controllability. Electrostatic control of the maximum-absorptance frequency using graphene- patched pixels in the GHz range is novel. The conguration of graphene-patched and unpatched pixels was optimized to achieve stable maximum absorptance of 0 :99, with pixelation performing better than continuous graphene. According to our simulations, the chosen design delivers absorp- tance0:9 over a 100-MHz band, with 360 MHz kA1m magnetostatic controllabity rate and 1 MHz V1m electrostatic controllability rate. The proposed X-band absorber can be used to improve the performance of radar systems. References [1] Wu PC, Pala RA, Shirmanesh GK, Cheng W-H, Sokhoyan R, Grajower M, Alam MZ, Lee D, Atwater HA. Dynamic beam steering with all-dielectric electro-optic III{V multiple-quantum- well metasurfaces. Nat. Commun. 2019;10(1):3654. [2] Ding P, Li Y, Shao L, Tian X, Wang J, Fan C. Graphene aperture-based metalens for dynamic focusing of terahertz waves. Opt. Exp. 2018;26(21):28038{28050. [3] Kumar P, Lakhtakia A, Jain PK. Tricontrollable pixelated metasurface for stopband for tera- hertz radiation. J. Electromag. Waves Appl. 2020;34(15):2065{2078. 7Table 1: Structure, type of control, controllability range of maximum-absorptance frequency (fmaxA), bandwidth (4fA0:9), and controllability rate of reported metasurface absorbers and the proposed metasurface absorber. Ref. Structure Control fmaxA4fA0:9 Controllability method(s) (GHz) (MHz) rate 8 Metal resonator/FR4/ magnetostatic 9.3{9.7 150 3 MHz kA1m ferrite/metal sheet 24 Metal pads separated by varactor diodes/FR4 electrical 8.25{9.25 400 100 MHz V1 sheet/metal sheet 13 YIG- and CR-patched pixels/ magnetostatic 8{13 200 360 MHz kA1m silicon/metal sheet and thermal and 1 MHz K1 This Graphene pixels/PVC magnetostatic 8{12 100 360 MHz kA1m work skin/YIG/metal sheet and electrostatic and 1 MHz V1m [4] Kumar P, Lakhtakia A, Jain PK. Tricontrollable pixelated metasurface for absorbing terahertz radiation. Appl. Opt. 2019;58(35):9614{9623. [5] He Q, Sun S, Zhou L. Tunable/recongurable metasurfaces: physics and applications. Re- search. 2019;2019:1849272. [6] Sinclair G. Theory of models of electromagnetic systems. Proc. IRE. 1948;36(11):1364{1370. [7] Lakhtakia A. Scaling of elds, sources, and constitutive properties in bianisotropic media. Microw. Opt. Technol. Lett. 1994;7(7):328{330. [8] Huang Y, Wen G, Zhu W, Li J, Si LM, Premaratne M. Experimental demonstration of a magnetically tunable ferrite based metamaterial absorber. Opt. Exp. 2014;22(13):16408{16417. [9] Yi D, Wei XC, Xu YL. Tunable microwave absorber based on patterned graphene. IEEE Trans. Microw. Theory Tech. 2017;65(8):2819{2826. [10] Qiu K, Jin J, Liu Z, Zhang F, Zhang W. A novel thermo-tunable band-stop lter employing a conductive rubber split-ring resonator. Mater. Des. 2017;116:309{315. [11] Fallahi A, Perruisseau-Carrier J. Design of tunable biperiodic graphene metasurfaces. Phys. Rev. B. 2012; 86(19):195408. [12] Yi J, Wei M, Lin M, Zhao X, Zhu L, Chen X, Jiang ZH. Frequency-tunable and magnitude- tunable microwave metasurface absorbers enabled by shape memory polymers. IEEE Trans. Antennas Propagat. 2022;70(8):6804{6812. [13] Sharma G, Kumar P, Lakhtakia A, Jain PK. Pixelated bicontrollable metasurface absorber tunable in complete X band. J. Electromag. Waves Appl. 2022;36(17):2505-2518. 8[14] Mahabadi RK, Goudarzi T, Fleury R, Sohrabpour S, Naghdabadi R. Eects of resonator geometry and substrate stiness on the tunability of a deformable microwave metasurface. AEU Int. J. Electron. Commun. 2022;146:154123. [15] Yousaf A, Murtaza M, Wakeel A, Anjum S. A highly ecient low-prole tetra-band meta- surface absorber for X, Ku, and K band applications. AE U Int. J. Electron. Commun. 2022;154:154329. [16] Pozar DM. Microwave engineering. USA: Wiley;2011. [17] Hanson GW. Dyadic Green's functions for an anisotropic, non-local model of biased graphene. IEEE Trans. Antennas Propagat. 2008;56(3):747{757. [18] Geng M-Y, Liu Z-G, Wu W-J, Chen H, Wu B, Lu W-B. A dynamically tunable microwave absorber based on graphene. IEEE Trans. Antennas Propagat. 2020;68(6):4706{4713. [19] Riddle B, Baker-Jarvis J, Krupka J. Complex permittivity measurements of common plastics over variable temperatures. IEEE Trans. Microw. Theory Tech. 2003;51(3):727{733. [20] Novoselov KS, Geim AK, Morozov SV, Jiang D, Katsnelson MI, Grigorieva I, Dubonos S, Firsov A. Two-dimensional gas of massless Dirac fermions in graphene. Nat. 2005;438(7065):197{200. [21] Cvijovi c D. New integral representations of the polylogarithm function. Proc. R. Soc. London A 2007;463(2080):897{905. [22] Jaluria Y. Computer methods for engineering. Taylor & Francis;1996. [23] Ahmad F, Anderson TH, Civiletti BJ, Monk PB, Lakhtakia A. On optical-absorption peaks in a nonhomogeneous thin-lm solar cell with a two-dimensional periodically corrugated metallic backre ector. J. Nanophoton. 2018;12(1):016017. [24] Yuan H, Zhu BO, Feng Y. A frequency and bandwidth tunable metamaterial absorber in X band. J. Appl. Phys. 2015;117(17):173103. 9

2022-11-01

The planewave response of a bicontrollable metasurface absorber with graphene-patched pixels was simulated in the X band using commercial software. Each square meta-atom is a 4x4 array of 16 pixels, some patched with graphene and the others unpatched. The pixels are arranged on a PVC skin which is placed on a copper-backed YIG substrate. Graphene provides electrostatic controllability and YIG provides magnetostatic controllability. Our design delivers absorptance equal to or in excess of 0.9 over a 100-MHz spectral regime in the X band, with 360 MHz/kA magnetostatic controllabity rate and 1 Hz m/V electrostatic controllability rate. Notably, electrostatic control via graphene in the GHz range is novel.

Design of X-Band Bicontrollable Metasurface Absorber Comprising Graphene Pixels on Copper-Backed YIG Substrate

2211.03510v1

1 UNEXPECTED STRUCTURAL AND MAGNETIC DEPTH DEPENDENCE OF YIG THIN FILMS J.F.K. Cooper, C.J. Kinane, S. Langridge ISIS Neutron and Muon Source, Rutherford Appleton L aboratory, Harwell Campus, Didcot, OX11 0QX M. Ali, B.J. Hickey Condensed Matter group, School of Physics and Astro nomy, E.C. Stoner Laboratory, University of Leeds, LS2 9JT T. Niizeki WPI Advanced Institute for Materials Research, Toho ku University, Sendai 980-8577, Japan K. Uchida National Institute for Materials Science, Tsukuba 3 05-0047, Japan Institute for Materials Research, Tohoku University , Sendai 980-8577, Japan Center for Spintronics Research Network, Tohoku Uni versity, Sendai 980-8577, Japan PRESTO, Japan Science and Technology Agency, Saitam a 332-0012, Japan E. Saitoh WPI Advanced Institute for Materials Research, Toho ku University, Sendai 980-8577, Japan Institute for Materials Research, Tohoku University , Sendai 980-8577, Japan Center for Spintronics Research Network, Tohoku Uni versity, Sendai 980-8577, Japan WPI Advanced Institute for Materials Research, Toho ku University, Sendai 980-8577, Japan Advanced Science Research Center, Japan Atomic Ener gy Agency, Tokai 319-1195, Japan H. Ambaye Neutron Sciences Directorate, Oak Ridge National La boratory, Oak Ridge, Tennessee 37831, USA A. Glavic Laboratory for Neutron Scattering and Imaging, Paul Scherrer Institut, Villigen PSI, Switzerland Neutron Sciences Directorate, Oak Ridge National La boratory, Oak Ridge, Tennessee 37831, USA PACS: 75.70.-I, 75.47.Lx, 68.55.aj 2 Abstract We report measurements on yttrium iron garnet (YIG) thin films grown on both gadolinium gallium garnet (GGG) and yttrium aluminium garnet (YAG) sub strates, with and without thin Pt top layers. We provide three principal results: the observation of an interfacial region at the Pt/YIG interface, we place a limit on the induced magnetism of the Pt layer and confirm the existence of an interfacial layer at the GGG/YIG interface. Polarised neutron r eflectometry (PNR) was used to give depth dependence of both the structure and magnetism of t hese structures. We find that a thin film of YIG on GGG is best described by three distinct layers: an interfacial layer near the GGG, around 5 nm thick and non-magnetic, a magnetic ‘bulk’ phase, an d a non-magnetic and compositionally distinct thin layer near the surface. We theorise that the b ottom layer, which is independent of the film thickness, is caused by Gd diffusion. The top layer is likely to be extremely important in inverse spi n Hall effect measurements, and is most likely Y 2O3 or very similar. Magnetic sensitivity in the PNR t o any induced moment in the Pt is increased by the ex istence of the Y 2O3 layer; any moment is found to be less than 0.02 uB/atom. Introduction Yttrium iron garnet (YIG) has long been known to be a ferrimagnetic insulator and is used widely as a tuneable microwave filter or, when doped with other rare earth elements, for a variety of optical and magneto-optical applications. However, since th e discovery of the spin Seebeck effect in insulators 1,2 , YIG, particularly when grown on gadolinium galliu m garnet (GGG), has become the model system for investigating the physics of the s pin Seebeck effect. The spin Seebeck effect combines two future technologies: the pure spin cur rents of spintronics promise to eliminate Joule heating in computing and many other industries 3–5, whereas energy recovery materials seek to harvest waste heat and movement to reduce energy lo sses, either actively 6–8, or passively from the conventional Seebeck effect 9 or otherwise 10 . The combination of these into a single material provides a great opportunity for energy efficiency and a new generation of future devices. It is therefore extremely important that both the i nterfacial physics of GGG/YIG and the physical system, with all possible imperfections, are well u nderstood. Much work has been done on the theoretical understanding of the spin Seebeck effec t 11–14 , with a general consensus that the effect is magnon driven, with non-equilibrium phonons also pl aying a role. This paper seeks to explore the material science aspect of the GGG/YIG system, and t o understand the effects of interfacial structure on high quality epitaxial films. Thin films of YIG, with different annealing times, were grown on GGG by sputtering and these films were characterised using polarised neutron reflecti vity (PNR) to extract a magnetic depth profile, as well as x-ray reflectivity (XRR) and magnetometry. Films were measured both with and without thin Pt layers on top of the YIG; these layers are conve ntionally used for inverse spin Hall effect (ISHE) measurements, to quantify the strength of the spin Seebeck effect. Additional films were grown on yttrium aluminium garnet (YAG) in order to investig ate the effects of the substrate on the films. 3 Methods Samples were grown in both Leeds and Tohoku Univers ities with common growth methodologies with the exception of the annealing time. The Leeds samples were sputtered in a RF magnetron sputter chamber with a base pressure of 2x10 -8 Torr, with oxygen and argon flow rates of 1.2 and 22.4 sccm respectively. They were deposited onto ei ther GGG or YAG substrates, 1” in diameter. The samples were then removed from the vacuum and annea led in air at 850 oC for 2 hours. They were then sputtered with a thin layer of Pt ~ 27 Å thick , a typical thickness for ISHE measurements. The films grown in Tohoku were prepared according to th e methods detailed in the work by Lustikova et al 15 , where the same annealing temperature was used, bu t for 24 hours instead of 2. A total of eight samples were measured: six from Le eds, four grown on GGG substrates and two on YAG, and two from Tohoku, both on GGG. The Leeds fil ms on GGG were either 300 Å or 800 Å thick with and without a thin Pt layer on top. Both of th e YAG films were 800 Å, and one had a Pt top layer. The Tohoku films were both around 1500 Å thi ck, one with a Pt layer (135 Å thick) on top and one without. PNR measurements record the neutron reflectivity as a function of the neutron’s wave-vector transfer and spin eigenstate. Modelling of the resu ltant data allows the scattering length density (SLD) to be extracted and provides a quantitative d escription of the depth dependent structural and magnetic profile 16 . PNR measurements were taken on the Polref and Offsp ec 17 beamlines at the ISIS neutron and muon source, as well as the Magnetism Reflectometer beam line at the Spallation Neutron Source, in Oak Ridge. X-ray measurements and magnetometry was carr ied out in the R53 Characterisation lab at ISIS. Fitting to both the neutron reflectivity and the x- ray reflectivity data was performed in the GenX fitting package 18 . The Pt cap layers thickness were determined by XR R since the scattering contrast between Pt and YIG or GGG is very good for x-rays and reduced for neutrons, whereas the contrast between YIG and GGG is poor for x-rays and good for neutrons. Magnetometry was performed using a Durham Magneto Optics NanoMOKE3 and showed that a ll of the films presented here had a coercivity of <5 Oe and generally ~1-2 Oe, indicati ng high quality YIG. Results and Discussion Figure 1 presents the polarised neutron reflectivit y and resultant SLD for the YIG layer on GGG and YAG substrates. From the nominal structure of the s puttered samples the neutron reflectivity can be calculated. This simple model does not accurately d escribe the observed data. To describe the GGG system an additional interfacial YIG-like layer was required, see layer (a) in Figure 1. This layer wa s either non-magnetic, or had a very small moment (~< 0.1 µ B /unit cell), and was ~50 Å thick, irrespective of the total YIG film thickness. This layer was not present in the films grown on a YAG substrate. The large roughness of the interface bet ween the GGG and the YIG in the models suggests that a diffusion process created this layer. This l ayer was not formed at the interface with the YAG substrates indicating that this process must be eit her Ga or Gd diffusion. A recent temperature dependent study of this interface also using neutro ns showed that it is Gd 19 . 4 Beyond the initial 50 Å non-magnetic layer, the str uctural and magnetic properties of the sputtered YIG films of differing thickness were very similar and did not have any thickness dependences. This was true of the films grown on both GGG and YAG, mea ning that the effects of the substrate, for these at least, are minimal beyond its ability to d iffuse during annealing. From the magnetic SLD the moment of the bulk YIG (a way from both interfaces) was found to be consistently 3.8(1) µ B / unit cell, this value did not depend on the film thickness. This value compares well with literature values 20 at room temperature, though very slightly higher. The measurements on the YIG films with extended ann ealing times were less conclusive. Models both with and without the non-magnetic layer at the GGG interface gave similar fits to the PNR data. These films were significantly thicker than the fil ms with a 2 hr anneal, ~1500 Å, and as such the sensitivity to the GGG/YIG interface is reduced. As a results it is not possible to conclusively ident ify the presence of an interfacial layer. Since the ann ealing procedures for both sets of films are very similar we can assume that the Gd diffusion will al so be similar, and a common feature of the GGG/YIG interface. In addition to the substrate interface layer, an ad ditional layer was discovered for all of the sample s, labelled as layer (b) in Figure 1. This layer is ar ound 15(5) Å in thickness, with little variation, a cross all samples measured. This layer was distinct from both the Pt and the YIG, as it had a markedly lower scattering length density than either of them . Figure 2 shows datasets for both thin Leeds (300 Å) and thick Tohoku (1500 Å) YIG on GGG, with the b est fit to the data. Since the SLD of YIG and Pt is similar, the low frequency oscillations in both dat asets results from the low SLD layers contrast between the Pt and the YIG. Analysing x-ray reflect ivity curves of the same samples, with and without the Pt cap also require this layer. Examini ng the two scattering length densities (neutrons and x-rays) of the layer involved we can elucidate its composition. Pt alloys would generate a strong x-ray contrast, and the layer would not appear in u ncapped samples and can therefore be ruled out. Both iron and all forms of its oxide have too large an SLD for neutrons so it can be ruled out. This leaves yttrium based compounds: pure Y, Y 2O3, (yttria) and YN (which is possible, though unlike ly, due 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 layer required for a good fit, as shown in Figure 3 . In addition to the matching SLD, we remark that Y3Fe 5O12 has an oxidation state of +3 for Y, which is the s ame as Y 2O3 and both have similar oxygen co-ordination. The Y-O bond length in Y 2O3 is between 2.225 and 2.323 Å 21 , which represents a slight contraction with respect to the bond length in YIG, which is between 2.37 and 2.43 Å 22 . The magnetic signal from the YIG decays across this layer (see Figure 3) and as it is likely that the layer is predominantly yttria (a non-magnetic insul ator), the electrical resistivity at some point in the film becomes that of the Pt. This means that any IS HE effects are likely sampling a lot more of the yttria, than the YIG. Several studies have investig ated the influence of the interface quality 23–25 and have determined that, as might be expected, a high quality interface yields better ISHE results. Qiu et al .24 found that, the ISHE voltage varied from around 3. 5 µV/K for a minimal interface yttria region, to nearly 0 for regions over 7 nm thick. Th ey also found that, at least for samples grown by liquid phase epitaxy, optimisation of after growth annealing could minimise this layer’s formation. This interface has also been investigated by Song et al. 26 using electron microscopy, though they attribute the layer to being oxygen deficient iron (whose magnetic moment is then reduced). 5 Knowledge of the existence and the information abou t the nature of this layer extracted here gives an opportunity to eliminate it in all cases; either by appropriate annealing, or selective etching. As a result of the low SLD of this layer and its co ntrast to the Pt layer, we have gained unusual sensitivity to any induced magnetism in the Pt laye r. Proximity effects in the Pt have been widely studied, with many works extracting the origins of the observed voltages from the iSHE 27–29 . Theoretical studies such as Liang et al. 30 show how a non-magnetic, or reduced magnetic layer would be important in proximity effects in this system. F igure 4 shows a portion of the spin asymmetry (difference in reflectivity of the two spin states normalised by their sum) from GGG/YIG(8000)/Pt(27), at high momentum transfer. Th e spin asymmetry is sensitive to the magnetism, and is, to a first approximation, indepe ndent of the exact structure. The reflectivity can be approximated as the Fourier transform of the lay er structure, so at high Q we are sensitive to thinner layers, e.g. the top Pt layer. The best model line is shown in grey and is clearly a good fit, even by this, more highly processed, measure. In addition to the best fit in Figure 4 (s hown in grey) are two models (blue and red), with the same structural parameters and bulk YIG magneti sm, but with an induced moment added to the Pt of ±0.05 µ B/Pt atom to show how this would affect the fit. Fro m the deviation of these models from the data, we can clearly see that the average magnitude of any induced moment in the Pt is certainly less than 0.05 µ B/Pt atom, and likely less than 0.02 µ B/Pt atom, within a 1σ error bound. This result is in line with previous experiments 31 which specifically tried to measure an induced moment in Pt, albeit with the YIG grown by a differ ent method. It is worth noting here that it may still be possible that a more magnetic sub-region o f the Pt may exist, since PNR cannot probe infinitely thin layers. However, the magnitude of t he total magnetism would still have to remain below our upper bound, e.g. if half was non-magneti c and half was polarised, then it would have to be below 0.04 µ B/Pt atom, etc. Conclusion We have used polarised neutron reflectivity to dete rmine the magnetic depth dependence of yttrium iron garnet thin films grown on gadolinium gallium garnet and yttrium aluminium garnet substrates, with and without a Pt layer on the surf ace. It was found that if the YIG is grown on a GGG substrate there can be a ~50 Å non-magnetic layer a t the substrate interface, this does not depend on the YIG film thickness. This is likely to be cau sed by Gd diffusion during annealing, since this la yer does not appear when the YIG is grown on YAG, this is in line with recent investigations 19 . The effect of growing YIG on YAG, other than the absence of th is interface layer was minimal, with roughnesses and magnetic moments extremely similar to those gro wn on GGG. We also see an additional layer at the YIG/Pt inter face, roughly 15 Å for all samples measured; further investigation and cross referencing with x- ray measurements identifies this layer as Y 2O3. While the existence of this (non-magnetic and usual ly insulating) layer may have large repercussions for the interpretation of ISHE measurements on this model system, knowledge of its existence and composition means it may be possible to eliminate i t. 6 Our measurements also give us unusual sensitivity t o any induced magnetism in the Pt layer, and allow us to give an upper bound on the magnitude of the moment of ±0.02 µ B/Pt atom. Acknowledgements The neutron work in this paper was performed at bot h the Spallation Neutron Source in the Oak Ridge National Laboratory (IPTS-13192), USA, and at the ISIS Pulsed Neutron and Muon Source, which were supported by a beamtime allocation from the Science and Technology Facilities Council (RB1410610 and RB1510146). We would like to thank t he sample environment support staff at both facilities for their help with the experiments. This work is partially supported by PRESTO "Phase I nterfaces for Highly Efficient Energy Utilization" and ERATO "Spin Quantum Rectification Project" from JST, Japan, and by Grant-in-Aid for Scientific Research (A) (No. JP15H02012) and Grant-in-Aid for Scientific Research on Innovative Area "Nano Spin Conversion Science" (No. JP26103005) from JSPS KAKENHI, Japan. References 1 K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahash i, J. Ieda, T. Ota, Y. Kajiwara, H. Umezawa, H. Kaw ai, G.E.W. Bauer, S. Maekawa, and E. Saitoh, Nat. Mater . 9, 894 (2010). 2 K. Uchida, H. Adachi, T. Ota, H. Nakayama, S. Maek awa, and E. Saitoh, Appl. Phys. Lett. 97 , 172505 (2010). 3 I. Žutić and S. Das Sarma, Rev. Mod. Phys. 76 , 323 (2004). 4 D.D. Awschalom and M.E. Flatté, Nat. Phys. 3, 153 (2007). 5 S.A. Wolf, D.D. Awschalom, R.A. Buhrman, J.M. Daug hton, S. von Molnár, M.L. Roukes, A.Y. Chtchelkanova, and D.M. Treger, Science 294 , 1488 (2001). 6 R.J.M. Vullers, R. van Schaijk, I. Doms, C. Van Ho of, and R. Mertens, Solid. State. Electron. 53 , 684 (2009). 7 D. Guyomar, A. Badel, E. Lefeuvre, and C. Richard, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 52 , 584 (2005). 8 S.P. Beeby, M.J. Tudor, and N.M. White, Meas. Sci. Technol. 17 , R175 (2006). 9 T.J. Abraham, D.R. MacFarlane, and J.M. Pringle, E nergy Environ. Sci. 6, 2639 (2013). 10 D.L. Andrews, in Int. Symp. Opt. Sci. Technol. , edited by A. Lakhtakia, G. Dewar, and M.W. McCall (International Society for Optics and Photonics, 20 02), pp. 181–190. 11 J. Xiao, G.E.W. Bauer, K. Uchida, E. Saitoh, and S . Maekawa, Phys. Rev. B 81 , 214418 (2010). 12 H. Adachi, K. Uchida, E. Saitoh, and S. Maekawa, R ep. Prog. Phys. 76 , 36501 (2013). 13 H. Adachi, J. Ohe, S. Takahashi, and S. Maekawa, P hys. Rev. B 83 , 94410 (2011). 7 14 S.M. Rezende, R.L. Rodríguez-Suárez, R.O. Cunha, A .R. Rodrigues, F.L.A. Machado, G.A. Fonseca Guerra, J.C. Lopez Ortiz, and A. Azevedo, Phys. Rev . B 89 , 14416 (2014). 15 J. Lustikova, Y. Shiomi, Z. Qiu, T. Kikkawa, R. Ig uchi, K. Uchida, and E. Saitoh, J. Appl. Phys. 116 , 153902 (2014). 16 B.P. Toperverg and H. Zabel, in Exp. Methods Phys. Sci. , 48th ed. (2015), pp. 339–434. 17 J.R.P. Webster, S. Langridge, R.M. Dalgliesh, and T.R. Charlton, Eur. Phys. J. Plus 126 , 112 (2011). 18 M. Björck and G. Andersson, J. Appl. Crystallogr. 40 , 1174 (2007). 19 A. Mitra, O. Cespedes, Q. Ramasse, M. Ali, S. Marm ion, M. Ward, R.M.D. Brydson, C.J. Kinane, J.F.K. Cooper, S. Langridge, and B.J. Hickey, Sci. Rep. submitted , (2017). 20 A. Bouguerra, G. Fillion, E.K. Hlil, and P. Wolfer s, J. Alloys Compd. 442 , 231 (2007). 21 J. Zhang, F. Paumier, T. Höche, F. Heyroth, F. Syr owatka, R.J. Gaboriaud, and H.S. Leipner, Thin Solid Films 496 , 266 (2006). 22 S. Geller and M.A. Gilleo, J. Phys. Chem. Solids 3, 30 (1957). 23 Z. Qiu, K. Ando, K. Uchida, Y. Kajiwara, R. Takaha shi, H. Nakayama, T. An, Y. Fujikawa, and E. Saitoh, Appl. Phys. Lett. 103 , 92404 (2013). 24 Z. Qiu, D. Hou, K. Uchida, and E. Saitoh, J. Phys. D. Appl. Phys. 48 , 164013 (2015). 25 S. Vélez, A. Bedoya-Pinto, W. Yan, L.E. Hueso, and F. Casanova, (2016). 26 D. Song, L. Ma, S. Zhou, and J. Zhu, Appl. Phys. L ett. 107 , 42401 (2015). 27 S.Y. Huang, X. Fan, D. Qu, Y.P. Chen, W.G. Wang, J . Wu, T.Y. Chen, J.Q. Xiao, and C.L. Chien, Phys. Rev. Lett. 109 , 107204 (2012). 28 T. Kikkawa, K. Uchida, Y. Shiomi, Z. Qiu, D. Hou, D. Tian, H. Nakayama, X.-F. Jin, and E. Saitoh, Phy s. Rev. Lett. 110 , 67207 (2013). 29 T. Kikkawa, K. Uchida, S. Daimon, Y. Shiomi, H. Ad achi, Z. Qiu, D. Hou, X.-F. Jin, S. Maekawa, and E. Saitoh, Phys. Rev. B 88 , 214403 (2013). 30 X. Liang, Y. Zhu, B. Peng, L. Deng, J. Xie, H. Lu, M. Wu, and L. Bi, ACS Appl. Mater. Interfaces 8, 8175 (2016). 31 S. Geprägs, S. Meyer, S. Altmannshofer, M. Opel, F . Wilhelm, A. Rogalev, R. Gross, and S.T.B. Goennenwein, Appl. Phys. Lett. 101 , 262407 (2012). 8 Figure 1 Structural (blue) and magnetic (green) scattering length densities of fitted models for a Y IG/Pt bilayer grown on a GGG substrate (top) and YAG subs trate (bottom). The top shows a thin ~300 Å YIG layer, and the bottom shows a thick ~800 Å YIG lay er, however, the data are representative of all the samples grown on each substrate irrespective of YIG thickness. The GGG substrate has two extra layers: (a), at the substrate interface, and (b), a t the YIG/Pt interface. (a) is non-magnetic at room temperature and around 50 Å thick irrespective of t he thickness of the YIG layer. This layer is not present at the YAG/YIG interface. The nature of lay er (b) discussed later and is absolutely required f or a good fit in all samples with a Pt layer. 9 Figure 2 Polarised Neutron Reflectivity data (points) and mo delled fit (line) of thin (~300 Å ) GGG/YIG/Pt from Leeds, (a), and thicker (~1500 A) Tohoku, (b), samples, both of which have Pt top layers. The low frequency oscill ations (which is the majority of the curve in (a) since it has a thinner Pt layer) are visible due to the low scattering le ngth density layer between the YIG and the Pt and is required in models for al l samples with Pt on to get a reasonable fit. 10 Figure 4 A selected portion of the spin asymmetry of a thick (~800 Å ) YAG/YIG/Pt structure, with a zoom inset. The higher Q range is where we are sensitive to any induced ma gnetism in the Pt layer. The grey curve shows the a symmetry produced by the optimal fit with no induced magneti sm in the Pt. The red and blue curves show the same model with +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, irrespective 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 subtle step in the SLD is required in order to corr ectly model the slow oscillation in the reflectivit y data. By using both the x-ray and neutron scattering length densities we ca n deduce that the top layer in this, and all other samples measured in this study, is extremely likely to be yttria (Y 2O3), whose bulk SLD is indicated by the dotted red li ne.

2017-03-26

We report measurements on yttrium iron garnet (YIG) thin films grown on both gadolinium gallium garnet (GGG) and yttrium aluminium garnet (YAG) substrates, with and without thin Pt top layers. We provide three principal results: the observation of an interfacial region at the Pt/YIG interface, we place a limit on the induced magnetism of the Pt layer and confirm the existence of an interfacial layer at the GGG/YIG interface. Polarised neutron reflectometry (PNR) was used to give depth dependence of both the structure and magnetism of these structures. We find that a thin film of YIG on GGG is best described by three distinct layers: an interfacial layer near the GGG, around 5 nm thick and non-magnetic, a magnetic bulk phase, and a non-magnetic and compositionally distinct thin layer near the surface. We theorise that the bottom layer, which is independent of the film thickness, is caused by Gd diffusion. The top layer is likely to be extremely important in inverse spin Hall effect measurements, and is most likely Y2O3 or very similar. Magnetic sensitivity in the PNR to any induced moment in the Pt is increased by the existence of the Y2O3 layer; any moment is found to be less than 0.02 uB/atom.

Unexpected structural and magnetic depth dependence of YIG thin films

1703.08752v1

Temperature dependence of the eective spin-mixing conductance probed with lateral non-local spin valves K. S. Das,1,a)F. K. Dejene,2B. J. van Wees,1and I. J. Vera-Marun3,b) 1)Physics of Nanodevices, Zernike Institute for Advanced Materials, University of Groningen, 9747 AG Groningen, The Netherlands 2)Department of Physics, Loughborough University, Loughborough LE11 3TU, United Kingdom 3)School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom We report the temperature dependence of the eective spin-mixing conductance between a normal metal (aluminium, Al) and a magnetic insulator (Y 3Fe5O12, YIG). Non-local spin valve devices, using Al as the spin transport channel, were fabricated on top of YIG and SiO 2substrates. By comparing the spin relaxation lengths in the Al channel on the two dierent substrates, we calculate the eective spin-mixing conductance (Gs) to be 3:31012 1m2at 293 K for the Al/YIG interface. A decrease of up to 84% in Gsis observed when the temperature ( T) is decreased from 293 K to 4.2 K, with Gsscaling with ( T=T c)3=2. The real part of the spin-mixing conductance ( Gr5:71013 1m2), calculated from the experimentally obtained Gs, is found to be approximately independent of the temperature. We evidence a hitherto unrecognized underestimation of Grextracted from the modulation of the spin signal by rotating the magnetization direction of YIG with respect to the spin accumulation direction in the Al channel, which is found to be 50 times smaller than the calculated value. The transfer of spin information between a normal metal (NM) and a magnetic insulator (MI) is the crux of electrical injection and detection of spins in the rapidly emerging elds of magnon spintronics1and an- tiferromagnetic spintronics2,3. The spin current ow- ing through the NM/MI interface is governed by the spin-mixing conductance4{7,G"#, which plays a cru- cial role in spin transfer torque8{10, spin pumping11,12, spin Hall magnetoresistance (SMR)13,14and spin See- beck experiments15. In these experiments, the spin- mixing conductance ( G"#=Gr+iGi), composed of a real (Gr) and an imaginary part ( Gi), determines the transfer of spin angular momentum between the spin ac- cumulation ( ~ s) in the NM and the magnetization ( ~M) of the MI in the non-collinear case. However, recent experi- ments on the spin Peltier eect16, spin sinking17and non- local magnon transport in magnetic insulators18,19neces- sitate the transfer of spin angular momentum through the NM/MI interface also in the collinear case (~ sk~M). This is taken into account by the eective spin-mixing conductance ( Gs) concept, according to which the trans- fer of spin angular momentum across the NM/MI inter- face can occur, irrespective of the mutual orientation be- tween~ sand~M, via local thermal uctuations of the equilibrium magnetization (thermal magnons20) in the MI. The spin current density ( ~js) through the NM/MI interface can, therefore, be expressed as17,21,22: ~js=Gr^m(~ s^m) +Gi(~ s^m) +Gs~ s;(1) where, ^mis a unit vector pointing along the direction of~M. WhileGrandGihave been extensively studied a)e-mail: K.S.Das@rug.nl b)e-mail: ivan.veramarun@manchester.ac.ukin spin torque and SMR experiments23{25, direct experi- mental studies on the temperature dependence of Gsare lacking. In this letter, we report the rst systematic study of Gsversus temperature ( T) for a NM/MI interface. For this, we utilize the lateral non-local spin valve (NLSV) geometry, which provides an alternative way to study the spin-mixing conductance using pure spin currents in a NM with low spin-orbit coupling (SOC)17,26,27. A low SOC of the NM in the NLSV technique also ensures that the spin-mixing conductance is not overestimated due to spurious proximity eects in NMs with high SOC or close to the Stoner criterion, such as Pt28{30. We exclu- sively address the temperature dependence of Gsfor the aluminium (Al)/Y 3Fe5O12(YIG) interface, which is ob- tained by comparing the spin relaxation length ( N) in similar Al channels on a magnetic YIG substrate and a non-magnetic SiO 2substrate, as a function of tempera- ture.Gsdecreases by about 84% when the temperature is decreased from 293 K to 4.2 K and scales with ( T=T c)3=2, whereTc= 560 K is the Curie temperature of YIG, con- sistent with theoretical predictions19,31{33. The real part of the spin-mixing conductance ( Gr) is then calculated from the experimentally obtained values of Gsand com- pared with the modulation of the spin signal in rotation experiments, where the magnetization direction of YIG (~M) is rotated with respect to ~ s. The NLSVs with Al spin transport channel were fab- ricated on top of YIG and SiO 2thin lms in multiple steps using electron beam lithography (EBL), electron beam evaporation of the metallic layers and resist lift-o technique, following the procedure described in Ref. 34. The 210 nm thick YIG lm on Gd 3Ga5O12substrate and the 300 nm thick SiO 2lm on Si substrate were obtained commercially from Matesy GmbH and Silicon Quest In- ternational, respectively. Permalloy (Ni 80Fe20, Py) hasarXiv:1812.09766v1 [cond-mat.mes-hall] 23 Dec 20182 (b) (a) (c) AlI L Py1 YIG M xz y Py2V+_ PS Magnons 500 nmIPy V+ _ AlPy xy B θ L FIG. 1. (a)Schematic illustration of the experimental geometry. The spin accumulation ( ~ s), injected into the Al channel by the Py injector, has an additional relaxation pathway into the (insulating) magnetic YIG substrate due to local thermal uctuations of the equilibrium YIG magnetization ( ~M) or thermal magnons. (b)SEM image of a representative NLSV device along with the illustration of the electrical connections for the NLSV measurements. An alternating current ( I) was sourced from the left Py strip (injector) to the left end of the Al channel and the non-local voltage ( VNL) was measured across the right Py strip (detector) with reference to the right end of the Al channel. An external magnetic eld ( B) was swept along the y-axis in the non-local spin valve (NLSV) measurements. In the rotation measurements, Bwas applied at dierent angles ( ) with 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 the YIG substrate (red) and on the SiO 2substrate (black). been used as the ferromagnetic electrodes for injecting and detecting a non-equilibrium spin accumulation in the Al channel. A 3 nm thick Ti underlayer was deposited prior to the evaporation of the 20 nm thick Py electrodes. The Ti underlayer prevents direct injection and detection of spins in the YIG substrate via the anomalous spin Hall eect in Py35,36.In-situ Ar+ion milling for 20 seconds at an Ar gas pressure of 4 105Torr was performed, prior to the evaporation of the 55 nm thick Al chan- nel, ensuring a transparent and clean Py/Al interface. A schematic of the device geometry is depicted in Fig. 1(a) and a scanning electron microscope (SEM) image of a representative device is shown in Fig. 1(b). A low fre- quency (13 Hz) alternating current source ( I) with an r.m.s. amplitude of 400 A was connected between the left Py strip (injector) and the left end of the Al channel. The non-local voltage ( VNL) due to the non-equilibrium spin accumulation in the Al channel was measured be- tween the right Py strip (detector) and the right end of the Al channel using a standard lock-in technique. The measurements were carried out under a low vacuum at- mosphere in a variable temperature insert, placed within a superconducting magnet. In the NLSV measurements, an external magnetic eld (B) was swept along the y-axis and the corresponding non-local resistance ( RNL=VNL=I) was measured. In Fig. 1(c), NLSV measurements for an Al channel length (L) of 300 nm at T= 293 K are shown for two devices, one on YIG (red) and another on SiO 2(black). The spin signal, Rs=RP NLRAP NL, is dened as the dierence in the two distinct states corresponding to the parallel (RP NL) and the anti-parallel ( RAP NL) alignment of the Py electrodes' magnetizations. The Rswas measured as a function of the separation ( L) between the injector andthe detector electrodes for several devices fabricated on YIG and SiO 2substrates, as shown in Fig. 2(a). To de- termine the spin relaxation length ( N) in the Al chan- nels on YIG ( N, YIG ) and SiO 2(N, SiO 2) substrates, the experimental data in Fig. 2(a) were tted with the spin diusion model37for transparent contacts: Rs=42 F (12 F)2RNRF RN2eL=N 1e2L=N; (2) where,Fis the bulk spin polarization of Py, RN= NN=wNtNandRF=FF=wNwFare the spin resis- tances of Al and Py, respectively. N(F),N(F),wN(F) andtNare the spin relaxation length, electrical re- sistivity, width and thickness of Al (Py), respectively. At room temperature, N, YIG = (27630) nm and N, SiO 2= (46820) nm were extracted, with FF= (0:840:05) nm. The NLSV measurements were carried out at dier- ent temperatures, enabling the extraction of N, YIG and N, SiO 2, as shown in Fig. 2(b). From this tempera- ture dependence, it is obvious that N, YIG is lower than N, SiO 2throughout the temperature range of 4.2 K to 293 K. The corresponding electrical conductivities of the Al channel ( N) on the two dierent substrates were also measured by the four-probe technique as a function of T, as shown in Fig. 2(c). The similar values of Nfor the Al channels on both YIG and the SiO 2substrates suggests that there is no signicant dierence in the structure and quality of the Al lms between the two substrates. There- fore, considering the dominant Elliott-Yafet spin relax- ation mechanism in Al38, dierences in the spin relax- ation rate within the Al channels cannot account for the dierence in the eective spin relaxation lengths between the two substrates.3 (a) (b) (c) FIG. 2. (a)The spin signal ( Rs) plotted as a function of the Al channel length ( L) for NLSV devices on YIG (red circles) and SiO2(black square) substrates at 293 K. The solid lines represent the ts to the spin diusion model (Eq. 2). (b)The eective spin relaxation length in the Al channel ( N) extracted at dierent temperatures ( T).Nis smaller on the YIG substrate as compared to the SiO 2substrate. (c)The electrical conductivity ( N) of the Al channels on the YIG and the SiO 2substrates as a function of temperature. The close match between the two conductivities suggests similar quality of the Al lm grown on both substrates. The smaller values of N, YIG as compared to N, SiO 2 suggest that there is an additional spin relaxation mech- anism for the spin accumulation in the Al channel on the magnetic YIG substrate. This is expected via additional spin- ip scattering at the Al/YIG interface, mediated by thermal magnons in YIG and governed by the eective spin-mixing conductance ( Gs). As described in Ref. 17, N, YIG andN, SiO 2are related to Gsas 1 2 N, YIG=1 2 N, SiO 2+1 2r; (3) where,r= 2Gs=(tAlN). Using the extracted values ofNfrom Fig. 2(b) and the measured values of Nfor the devices on YIG from Fig. 2(c), we calculate Gs= 3:31012 1m2at 293 K. At 4.2 K, Gsdecreases by about 84% to 5 :41011 1m2. The temperature dependence of Gsis shown in Fig. 3(a). Since the concept of the eective spin-mixing conductance is based on the thermal uctuation of the magnetization (thermal magnons), Gsis expected to scale as (T=T c)3=2, whereTcis the Curie temperature of the magnetic insulator6,19,31,32. UsingTc= 560 K for YIG, we t the experimental data to C(T=T c)3=2, which is depicted as the solid line in Fig. 3(a). The temperature independent prefactor, C, was found to be 8:61012 1m2. The agreement with the experimental data conrms the expected scaling of Gswith tempera- ture. Note that the deviation from the ( T=T c)3=2scaling at lower temperatures could be in part due to slightly dierent quality of the Al lm on the YIG substrate. Nevertheless, the small dierence of 10% in the elec- trical conductivities of the Al channel on the two dierent substrates at T < 100 K in Fig. 2(c) cannot account for the dierences in N. On the other hand, we note that quantum magnetization uctuations39,40in YIG can also play a role at low T, leading to an enhanced Gs. Next, we investigate the temperature dependence of the real part of the spin-mixing conductance ( Gr). Forthis, we rst calculate Grfrom the experimentally ob- tainedGs, using the following expression19: Gs=3(3=2) 2s3Gr; (4) where(3=2) = 2:6124 is the Riemann zeta function cal- culated at 3 =2,s=S=a3is the spin density with total spinS= 10 in a unit cell of volume a3= 1:896 nm3, and  =p 4Ds=kBTis the thermal de Broglie wavelength for magnons, with Ds= 8:4581040Jm2being the spin wave stiness constant for YIG19,41. The temperature dependence of the calculated Gris shown in Fig. 3(b). Keeping in mind that Eq. 4 is not valid in the limits of T!TcandT!0, we ignore the data points below 100 K. Above this temperature, Gris almost constant at 5:71013 1m2, represented by the dashed line in Fig. 3(b). This is consistent with Ref. 25, where Grwas found to be T-independent. Moreover, the magnitude ofGris comparable with previously reported values for (a) (b) FIG. 3. (a)Temperature dependence of the eective spin-mixing conductance (black symbols). Gsscales with the temperature as ( T=T c)3=2(solid line). (b) The real part of the spin-mixing conductance ( Gr) is calculated from Eq. 4 by using the experimentally obtained values of Gs.Gr (5:71013 1m2) is essentially found to be constant (dashed line) for T >100K.4 (a) (b) (c) FIG. 4. (a)NLSV measurement for a device on the YIG substrate with L= 300 nm at 150 K. (b)Rotation measurement for the same device with B= 20 mT applied at dierent angles ( ) with respect to the y-axis. The black and the red symbols correspond to the average of ten rotation measurements carried out with the magnetization of the Py electrodes in the parallel (P) and the anti-parallel (AP) congurations, respectively. (c)The spin signal ( Rs=RP NLRAP NL) exhibits a periodic modulation of magnitude
Rswhen the angle between the magnetization direction in YIG ( ~M) and the spin accumulation direction in Al (~ s) is changed. The black symbols represent the experimental data at 150 K, while the red line is the numerical modelling result corresponding to Gr= 11012 1m2. Al/YIG17and Pt/YIG19,42interfaces. An alternative approach for extracting Grfrom the NLSVs fabricated on the YIG substrate, is by the rota- tion of the sample with respect to a low magnetic eld in thexy-plane. We have also followed this method, de- scribed in Refs. 17 and 26. In the rotation experiments, the anglebetween the magnetization direction in YIG (~M) and the spin accumulation direction in Al ( ~ s) is changed, which results in the modulation of the spin sig- nal in the Al channel due to the transfer of spin angu- lar momentum across the Al/YIG interface, as described in Eq. 1, dominated by the Grterm. First, the NLSV measurement for a device with L= 300 nm was carried out at 150 K, as shown in Fig. 4(a). In the next step, B= 20 mT was applied in the xy-plane and the sample was rotated, with the magnetization orientations of the Py electrodes set in the parallel (P) or the anti-parallel (AP) conguration. For improving the signal-to-noise ratio, ten measurements were performed for each of the congurations (P and AP). The average of these measure- ments is shown in Fig. 4(b). The spin signal is extracted from Fig. 4(b) and plotted as a function of in Fig. 4(c). Rsexhibits a periodic modulation with the maxima at = 0and minima at =90, consistent with the behaviour predicted in Eq. 1. The modulation in the Rs, dened as(R0 sR90 s) R0 s=
Rs R0 s, was found to be 2 :8%. A similar modulation of 2 :9% was reported in Ref 26 for an NLSV with a Cu channel on YIG with L= 570 nm at the same temperature. Gris estimated from the rotation measurements us- ing 3D nite element modelling, as described in Ref. 17. From the modelled curve for the spin signal modula- tion, shown as the red line in Fig. 4(c), we extract Gr= 11012 1m2. This value is comparable to that reported in Ref. 26, within a factor of 2, for an evaporated Cu channel on YIG. However, this value is more than 50 times smaller than our estimated value from Eq. 4, andalso that reported in Ref. 17 for a sputtered Al chan- nel on YIG. One reason behind the small magnitude of Grextracted from the rotation measurements can be at- tributed to the thin lm deposition technique used. In Ref. 14, it was shown that the SMR signal for a sputtered Pt lm on YIG is about an order of magnitude larger than that for an evaporated Pt lm. Moreover, during the fabrication of our NLSVs, an Ar+ion milling step is carried out prior to the evaporation of the NM chan- nel for ensuring a clean interface between the NM and the ferromagnetic injector and detector electrodes17,26. Consequently, this also leads to the milling of the YIG surface on which the NM is deposited, resulting in the formation of an2 nm thick amorphous YIG layer at the interface43. Since an external magnetic eld of 20 mT is not sucient to completely align the magnetization di- rection within this amorphous layer parallel to the eld direction44, the resulting modulation in the spin signal will be smaller. This might lead to the underestima- tion ofGr. Note that since the eect of Gsdoes not depend on the magnetization orientation of YIG (Eq. 1), the milling does not aect the estimation of Gs. Our observations are consistent with a similarly small value ofGr41012 1m2reported in Ref. 26 for the Cu/YIG interface, where the Cu channel was evaporated following a similar Ar+ion milling step. Using the re- ported values of N= 522 nm (680 nm) on YIG (SiO 2) substrate for the 100 nm thick Cu channel at 150 K in Ref. 26, we extract Gs= 21012 1m2, which is 5 times larger than their reported Grextracted from rota- tion measurements. In summary, we have studied the temperature depen- dence ofGsandGrusing the non-local spin valve tech- nique for the Al/YIG interface. From NLSV measure- ments, we extracted Gsto be 3:31012 1m2at 293 K, which decreases by about 84% at 4.2 K, approximately obeying the ( T=T c)3=2law. While Grremains almost constant with the temperature, the value extracted from5 the modulation of the spin signal (1 1012 1m2) was around 50 times smaller than the calculated value (5:71013 1m2). The lower estimate of Grfrom the rotation experiment can be attributed to the formation of an amorphous YIG layer at the interface due to Ar+ ion milling prior to the evaporation of the Al channel, a consideration missing in the literature so far. ACKNOWLEDGMENTS We acknowledge the technical support from J. G. Hol- stein, H. M. de Roosz, H. Adema, T. Schouten and H. de Vries and thank G. E. W. Bauer and F. Casanova for discussions. We acknowledge the nancial support of the Zernike Institute for Advanced Materials and the Future and Emerging Technologies (FET) programme within the Seventh Framework Programme for Research of the Eu- ropean Commission, under FET-Open Grant No. 618083 (CNTQC). This project is also nanced by the NWO Spinoza prize awarded to Prof. B. J. van Wees by the NWO. 1A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands, Nature Physics 11, 453 (2015). 2T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich, Nature Nanotechnology 11, 231 (2016). 3V. 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2018-12-23

We report the temperature dependence of the effective spin-mixing conductance between a normal metal (aluminium, Al) and a magnetic insulator ($\text{Y}_3\text{Fe}_5\text{O}_{12}$, YIG). Non-local spin valve devices, using Al as the spin transport channel, were fabricated on top of YIG and SiO$_2$ substrates. By comparing the spin relaxation lengths in the Al channel on the two different substrates, we calculate the effective spin-mixing conductance ($G_\text{s}$) to be $3.3\times10^{12}$~$\Omega^{-1}\text{m}^{-2}$ at 293~K for the Al/YIG interface. A decrease of up to 84\% in $G_\text{s}$ is observed when the temperature ($T$) is decreased from 293~K to 4.2~K, with $G_\text{s}$ scaling with $(T/T_\text{c})^{3/2}$. The real part of the spin-mixing conductance ($G_\text{r}\approx 5.7\times10^{13}~ \Omega^{-1}\text{m}^{-2}$), calculated from the experimentally obtained $G_\text{s}$, is found to be approximately independent of the temperature. We evidence a hitherto unrecognized underestimation of $G_\text{r}$ extracted from the modulation of the spin signal by rotating the magnetization direction of YIG with respect to the spin accumulation direction in the Al channel, which is found to be 50 times smaller than the calculated value.

Temperature dependence of the effective spin-mixing conductance probed with lateral non-local spin valves

1812.09766v1

Anomalous Hall eect in YIG jPt bilayers Sibylle Meyer,1, 2,a)Richard Schlitz,1, 2Stephan Gepr ags,1Matthias Opel,1Hans Huebl,1, 3Rudolf Gross,1, 2, 3and Sebastian T. B. Goennenwein1, 3 1)Walther-Meiner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany 2)Physik-Department, Technische Universit at M unchen, 85748 Garching, Germany 3)Nanosystems Initiative Munich, 80799 M unchen, Germany (Dated: 8 April 2015) We measure the ordinary and the anomalous Hall eect in a set of yttrium iron garnetjplatinum (YIGjPt) bilayers via magnetization orientation dependent magne- toresistance experiments. Our data show that the presence of the ferrimagnetic in- sulator YIG leads to an anomalous Hall eect like voltage in Pt, which is sensitive to both Pt thickness and temperature. Interpretation of the experimental ndings in terms of the spin Hall anomalous Hall eect indicates that the imaginary part of the spin mixing conductance Giplays a crucial role in YIG jPt bilayers. In particular, our data suggest a sign change in Gibetween 10 K and 300 K. Additionally, we report a higher order Hall eect contribution, which appears in thin Pt lms on YIG at low temperatures. a)Electronic mail: sibylle.meyer@wmi.badw.de 1arXiv:1501.02574v3 [cond-mat.mtrl-sci] 7 Apr 2015The generation, manipulation and detection of pure spin currents are fascinating chal- lenges in spintronics . In normal metals with large spin orbit interaction, the spin Hall eect (SHE)1{4and its inverse (ISHE)5enable the generation viz. detection of spin currents in the charge transport channel. In this context, the spin Hall angle SHand the spin diusion lengthare key material parameters1,2. Additionally, the spin mixing conductance Gwas proposed as a measure for the number of spin transport channels per unit area across a nor- mal metal (NM)jferromagnet (FM) interface, in analogy to the Landauer-B uttiker picture in ballistic charge transport6,7. Here,G=Gr+{Giis introduced as a complex quantity8{12. The real part Gris linked to an in-plane magnetic eld torque13,14and accessible e.g. from spin pumping experiments5{7,15,16. The imaginary part Giis related to the spin precession and interpreted as a phase shift between the spin current in the NM and the one in the FM.Githus can be either positive or negative7. As suggested recently, the spin Hall mag- netoresistance (SMR)17{19based on the simultaneous action of SHE and ISHE allows for quantifying Gifrom measurements of anomalous Hall-type eects (AHE) in ferromagnetic insulatorjNM hybrids, referred to as spin Hall anomalous Hall eect (SH-AHE)19. Here, we present an experimental study of ordinary and anomalous Hall-type signals ob- served in yttrium iron garnet (Y 3Fe5O12, YIG)jplatinum (Pt) bilayers. We discuss the lm thickness and temperature dependence of the AHE signals in terms of the SH-AHE. While the AHE voltage observed in metallic ferromagnets usually obeys VH/Mn ?withn= 1 and M?the component of the magnetization along the lm normal, we observe a more complex AHE-type response with higher order terms VH/Mn ?at low temperatures in YIG jPt sam- ples with a Pt lm thickness tPt5 nm. The higher order contributions are directly evident in our experiments, since we measure the magneto-transport response as a function of exter- nal magnetic eld orientation, while conventional Hall experiments are typically performed as a function of eld strength in a perpendicular eld arrangement. For comparison, we also study thin Pt lms deposited directly onto diamagnetic substrates. In these samples, we neither nd a temperature dependence of the ordinary Hall-eect (OHE), nor an AHE-type signal, not to speak of higher order AHE contributions. We investigate two types of thin lm structures, YIG jPt bilayers and single Pt thin lms on yttrium aluminum garnet (Y 3Al5O12, YAG) substrates. The YIG jPt bilayers are obtained by growing epitaxial YIG thin lms with a thickness of t60 nm on single crystalline YAG or gadolinium gallium garnet (Gd 3Ga5O12,GGG) substrates using pulsed laser deposition20,21. 2In an in situ process, we then deposit a thin polycrystalline Pt lm onto the YIG via electron beam evaporation. We hereby systematically vary the Pt thickness from sample to sample in the range 1 nmtPt20 nm. In this way, we obtain a series of YIG jPt bilayers with xed YIG thickness, but dierent Pt thicknesses. For reference, we furthermore fabricate a series of YAGjPt bilayers, depositing Pt thin lms with thicknesses 2 nm tPt16 nm directly onto YAG substrates. We employ X-ray re ectometry and X-ray diraction to determine tPt and to conrm the polycrystallinity of the Pt thin lms22. For electrical transport measure- ments, the samples are patterned into Hall bar mesa structures (width w= 80m, contact separation l= 600m)23[c.f. Fig. 1(a)]. We current bias the Hall bars with Iqof up to 500A and measure the transverse (Hall like) voltage Vtranseither as a function of the mag- netic eld orientation (angle dependent magnetoresistance, ADMR21,24) or of the magnetic eld amplitude 0H(eld dependent magnetoresistance, FDMR), for sample temperatures Tbetween 10 K and 300 K. For all FDMR data reported below, the external magnetic eld was applied perpendicular to the sample plane ( 0Hkn, c.f. Fig. 1(a)). For the ADMR measurements, we rotate an external magnetic eld of constant magnitude 1 T 0H7 T in the plane perpendicular to the current direction j23. Here,His dened as the angle be- tween the transverse direction tand the magnetic eld H. In all ADMR experiments, we choose0Hlarger than the anisotropy and the demagnetization elds of the YIG lm. As a result, the YIG magnetization Mis always saturated and oriented along Hin good ap- proximation. The transverse resistivity trans(H;H) =Vtrans(H;H)tPt=Iqof the Pt layer is calculated from the voltage Vtrans(H) along t. Figure 1(b-d) show FDMR measurements carried out at 300 K in YIG jPt bilayers with tPt= 2:0;6:5 and 19:5 nm. Extracting the ordinary Hall coecient OHE=@trans(H)=@(H) from the slope, we obtain OHE(19:5 nm) =25:5p m=T for the thickest Pt layer [see Fig. 1(b)], close to the literature value for bulk Pt25. Additionally, we observe a small superimposed S-like feature around 0H= 0 T, indicating the presence of an AHE like contribution. To quantify this contribution, we extract the full amplitude of the S-shape corresponding to an AHE like contribution AHEfrom linear ts to 0H= 0 T,as indicated in Fig. 1(d). In the sample with tPt= 6:5 nm [Fig. 1(c)], OHEdecreases to23:1p m=T and we nd an increased AHE(tPt= 6:5 nm) = (61)p m. For tPt= 2:0 nm [see Fig. 1(d)], we observe OHE= 7 p m=T, i.e. an inversion of the sign of the OHE. Additionally, we nd AHEequal to (121) p m. The presence of an AHE like behavior in YIG jPt samples 3(a) Iqn t jH Vtrans βH -2 0 2YIG|Pt(6.5nm) µ0H (T)-2 0 2-80-4004080 YIG|Pt(19.5nm)ρtrans(pΩm) µ0H (T)-2 0 2YIG|Pt(2.0nm) µ0H (T)(b) (c) (d) 2αAHE -80-404080 0FIG. 1. (a)Sample and measurement geometry. (b)-(d):Transverse resistivity trans taken from FDMR measurements for YIG jPt bilayers with (b)tPt= 19:5 nm, (c)6:5 nm and (d)2:0 nm, respectively. All data are taken at 300 K. The dashed red lines in panel (d)indicate the extraction ofAHEfrom linear ts to trans(H) extrapolated to 0H= 0 T. coincides with recent reports18,26{30. However, our study of AHEas a function of platinum thickness and temperature in addition reveals a pronounced thickness dependence of AHE fortPt10 nm that will be addressed below [c.f. Fig. 3(b))]. For reference, we also per- formed FDMR measurements on Pt thin lms deposited directly onto diamagnetic YAG substrates. In these samples, we nd a similar thickness dependence of the ordinary Hall- eect (OHE), but no AHE-type signal22. Thus, the sign inversion of the OHE is intimately connected to the Pt thin lm regime18. Complementary to the FDMR experiments, we further investigate transas a function of the magnetic eld orientation (ADMR). In Fig. 2(a) we show ADMR data for a YIG jPt(2:0 nm) hybrid recorded at 10 K. In ADMR experiments, the OHE is expected to depend only on the component H?=Hsin(H), i.e.,(H)/sin(H). However, our experimental data reveals additional higher than linear order contributions of the form Vtrans/Mn ?, with trans/Asin(H) +Bsin3(H) +


. A fast Fourier transformation22of the ADMR data suggests the presence of sinn(H) contributions up to at least n= 522. However, a quantita- tive determination of corresponding higher order coecients is dicult, since the amplitudes of the contributions for n5 are below our experimental resolution of 1 p m. A behavior similar to that shown in Fig. 2(a) is found in all YIG jPt samples with tPt5 nm, but not in plain Pt lms on YAG22. To allow for simple analysis, we use trans=Asin(H) +Bsin3(H) (1) 4in the following. Fits of the ADMR curves measured at dierent eld magnitudes according to Eq. (1) are shown as solid lines in Fig. 2(a). The magnetic eld dependence of the t parameters AandBis shown in Figs. 2(c),(d) for two samples with tPt= 3:1 nm22and tPt= 2:0 nm. We disentangle magnetic eld dependent (OHE like) and "eld independent" (AHE like) contributions to Aby tting the data to A(0H) =AOHE0H+AAHE. As evi- dent from Fig. 2, the OHEandAHEvalues derived from FDMR and ADMR measurements are quantitatively consistent. TheAOHEas a function of tPtis shown in Fig. 3(a). Obviously, AOHEdeviates from the bulk OHE literature value25in YIGjPt bilayers with tPt10 nm and also exhibits a tempera- ture and thickness-dependent sign change for small tPt. A thickness-dependent behavior of the OHE without sign change has also been reported in Ref. 18. However, these authors found an increase of the OHE coecient in the thin lm regime, which could be due to the formation of a thin, non-conductive \dead" Pt layer at the interface as, e.g., reported for NijPt31. In contrast, we attribute the thickness dependence of the OHE in our samples 23456750100150 234567-40-200 YIG|3.1nm PtYIG|2.0nm Pt B(pΩm)(a) (b)A(pΩm) µ0H (T)(c) (d)ρtrans(pΩm) βH -6-4-20246 0°90°180°270°360°-150-100-50050100150 YIG|Pt (2.0nm) 1T 2T 4T 7T µ0H (T)µ0H (T) FIG. 2. (a)ADMR and (b)FDMR data of a YIG jPt sample with tPt= 2:0 nm, taken at 10 K for dierent0H(open symbols). The dashed horizontal lines are intended as guides to the eye, to show that the trans values inferred from FDMR and ADMR are consistent for identical magnetic eld congurations.The ts of Eq. (1) to the data are shown as solid lines. (c)and(d)show the t parameters AandBobtained from Eq. (1) for YIG jPt(3:1 nm) (black) and YIG jPt(2:0 nm) (red) atT= 10 K. Linear ts to the magnetic eld dependence of AandBare shown as solid lines. 5solely to a modication of the Pt properties in the thin lm regime. Further experiments will be required in the future to clarify the origin of the temperature dependence of the OHE in YIGjPt hybrids. The anomalous Hall coecient AAHE, present only in YIG jPt hybrids, i.e., when a magnetic insulator is adjacent to the NM, is depicted in Fig. 3(b). We observe a strong dependence ofAAHEontPtsimilar to the thickness dependent magnetoresistance obtained from lon- gitudinal transport measurements reported earlier21, but with a sign change in AAHEbe- tween 100 K and 10 K. This observation agrees with recent reports of AAHE= 54 p m for YIGjPt(1:8 nm)30andAAHE= 6 p m for YIG jPt(3 nm)29, both taken at 10 K. Our study suggests a maximum in AAHEaroundtPt= 3 nm, compatible with a complete disappearance ofAAHEfortPt!0. This observation however is at odds with the attribution of the AHE in YIGjPt to a proximity MR as postulated in Ref. 29. In this case one would expect a monotonous increase of the AHE signal with decreasing Pt layer thickness, and eventually a saturation when the entire nonmagnetic layer is spin polarized. The absence of a proximity MR in our Hall data is consistent with XMCD data on similar YIG jPt samples20as well as other ferromagnetic insulator jNM hybrids32. However, we want to point out that a magnetic proximity eect has been reported in some YIG jPt samples33,34. We now model our experimental ndings in terms of the SH-AHE theory19 trans=222 SH tPtGitanh2tPt 2
(+ 2GrcothtPt 
)2mn; (2) where=1is the electric conductivity of the Pt layer and mnthe unit vector of the projection of the magnetization orientation monto the direction n(c.f. Fig. 1). To t the nonlinear behavior of AAHE(tPt), we combine this expression with the thickness dependence of the sheet resistivity for thin Pt lms35as discussed in Ref. 23. We use the parameters = 1:5 nm,Gr= 41014 1m2,SH(300 K) = 0 :11 andSH(10 K) = 0 :07 obtained from longitudinal SMR measurements on similar YIG jPt bilayers23. As obvious from the solid lines in Fig. 3(b), Eq. (2) reproduces our thickness dependent AHE data upon using Gi= 11013 1m2for 300 K and Gi=31013 1m2for 10 K. For 300 K, the value forGinicely coincides with earlier reports21as well as theoretical calculations36. In the SH-AHE model, the only parameter allowing to account for the sign change in transas a function of temperature is Gi. In this picture, our AHE data thus indicate a sign change in Gibetween 300 K and 10 K. 6FIG. 3. (a)-(c) Field dependent (OHE-like) and eld independent (AHE-like) Hall coecients AandBproportional to sin ( H) and sin3(H), respectively, plotted versus the Pt thickness for T= 300 K (blue), T= 100 K (red) and T= 10 K (black). The data is obtained from ADMR measurements for YAG jPt (open symbols) and YAG jYIGjPt (full symbols). AOHEdepicted in (a) describes the conventional Hall eect, the olive dashed line corresponds to the literature value for bulk Pt25.(b)Thickness dependence of AAHE. The solid lines show ts to the SH-AHE theory usingGi= 11013 1m2forT= 300 K (blue) and Gi=31013 1m2forT= 10 K (black). Panel (c)shows the thickness dependence of the eld independent coecient BAHEof the sin3(H) term. We nally address the thickness and temperature dependence of the sin3(H) contribution parametrized by B=BAHE+BOHE0H, that cannot straightforwardly be explained in a conventional Hall scenario. As evident from the linear ts in Fig. 2(c), Bis nearly eld independent. A slight eld dependence BOHE1 p m=T might arise due to tting errors caused by neglected higher order terms ( n5). Therefore, we focus our discussion on the eld independent part BAHEin the following. BAHEexhibits a strong temperature and thickness dependence as shown in Fig. 3(c), suggesting a close link to AAHEand therefore the SH-AHE. However, we do not observe a temperature-dependent sign change in BAHE. Expanding the SMR theory19to include higher order contributions of the magnetization directionsmi(i=j;t;n ) in analogy to the procedure established for the AMR of metal- 7lic ferromagnets24,37, sin3(H) terms appear in trans, but with an amplitude proportional to4 SH. Assuming SH(Pt)0:1, this would lead to BAHE=AAHE0:01, which disagrees with our experimental nding BAHE=AAHE0:2. Additionally, we study the in uence of the longitudinal resistivity on AHE. For metallic ferromagnets, one usually considers AHE/M(H) longwith 1238,39. Applying this approach to Vtransof the YIGjPt samples discussed here is not possible: Since the longitudinal resistance is modulated by the SMR with 1=010321,AHE/would imply BAHE=AAHE103. This is in contrast to our experimental ndings. Thus, a dependence of the form AHE/ longcannot account for our experimental observations. Finally, a static magnetic proximity eect26,33,34 also cannot explain BAHE, since the thickness dependence of BAHEshown in Fig. 3 (c) clearly indicates a decrease for tPt2:5 nm. Consequently, within our present knowledge, neither a spin current related phenomenon (SMR, SH-AHE), nor a proximity based eect can explain the origin or the magnitude of this anisotropic higher order anomalous Hall eect. We also would like to point out that the higher order sin3(H) term can be resolved only in ADMR measurements. In conventional FDMR experiments, such higher order contributions cannot be discerned. In summary, we have investigated the anomalous Hall eect in YIG jPt heterostructures for dierent Pt thicknesses, comparing magnetization orientation dependent (ADMR) and magnetic eld magnitude dependent (FDMR) measurements at temperatures between 10 K and 300 K. In Pt thin lms on diamagnetic (YAG) substrates, we observe a Pt thickness dependent ordinary Hall eect (OHE) only. However, in YIG jPt bilayers, an AHE like signal is present in addition. The AHE eect changes sign as a function of temperature and can be modeled using a spin Hall magnetoresistance-type formalism for the transverse transport coecient. However, we need to assume a sign change in the imaginary part of the spin mixing interface conductance to describe the sign change in the anomalous Hall signal ob- served experimentally. Finally, we identify contributions proportional to sin3(H) and higher orders in the ADMR data for YIG jPt. The physical mechanism responsible for this behavior could not be claried within this work and will be subject of further investigations. The observation of higher order contributions to the AHE in angle dependent magnetotransport measurements conrms the usefulness of magnetization orientation dependent experiments. Clearly, magnetotransport measurements as a function of the magnetic eld magnitude only, i.e. for a single magnetic eld orientation (perpendicular eld), as usually performed to study 8Hall eects, are not sucient to access all transverse transport features. We thank T. Brenninger for technical support and M. Schreier for fruitful discussions. Fi- nancial support by the Deutsche Forschungsgemeinschaft via SPP 1538 (project no. GO 944/4) is gratefully acknowledged. References 1M. Dyakonov and V. Perel, \Current-induced spin orientation of electrons in semiconduc- tors," Phys. Lett. A 35, 459{460 (1971). 2J. E. Hirsch, \Spin hall eect," Phys. Rev. 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Chien, \Self-consistent deter- mination of spin hall angles in selected 5 dmetals by thermal spin injection," Phys. Rev. B89, 140407 (2014). 29B. F. Miao, S. Y. Huang, D. Qu, and C. L. Chien, \Physical origins of the new magne- toresistance in pt/yig," Phys. Rev. Lett. 112, 236601 (2014). 30Y. Shiomi, T. Ohtani, S. Iguchi, T. Sasaki, Z. Qiu, H. Nakayama, K. Uchida, and E. Saitoh, \Interface-dependent magnetotransport properties for thin pt lms on ferri- magnetic y3fe5o12," Appl. Phys. Lett. 104, 242406 (2014). 31S.-C. Shin, G. Srinivas, Y.-S. Kim, and M.-G. Kim, \Observation of perpendicular mag- netic anisotropy in ni/pt multilayers at room temperature," Appl. Phys. Lett. 73, 393{395 (1998). 32D. K. Satapathy, M. A. Uribe-Laverde, I. Marozau, V. K. Malik, S. Das, T. Wagner, C. Marcelot, J. Stahn, S. Br uck, A. R uhm, S. Macke, T. Tietze, E. Goering, A. Fra~ n o, J. H. Kim, M. Wu, E. Benckiser, B. Keimer, A. Devishvili, B. P. Toperverg, M. Merz, P. Nagel, S. Schuppler, and C. Bernhard, \Magnetic proximity eect in yba2cu3o7=la2=3ca1=3mno 3 andyba2cu3o7=lamn 3+superlattices," Phys. Rev. Lett. 108, 197201 (2012). 33Y. M. Lu, Y. Choi, C. M. Ortega, M. X. Cheng, J. W. Cai, S. Y. Huang, L. Sun, and C. L. Chien, \Pt magnetic polarization on y3fe5o12 and magnetotransport characteristics," 11Phys. Rev. Lett. 110, 147207 (2013). 34Y. M. Lu, J. W. Cai, S. Y. Huang, D. Qu, B. F. Miao, and C. L. Chien, \Hybrid magnetoresistance in the proximity of a ferromagnet," Phys. Rev. B 87, 220409 (2013). 35G. Fischer, H. Homann, and J. Vancea, \Mean free path and density of conductance electrons in platinum determined by the size eect in extremely thin lms," Phys. Rev. B 22, 6065{6073 (1980). 36X. Jia, K. Liu, K. Xia, and G. E. W. Bauer, \Spin transfer torque on magnetic insulators," Europhys. Lett. 96, 17005 (2011). 37W. Limmer, J. Daeubler, L. Dreher, M. Glunk, W. Schoch, S. Schwaiger, and R. Sauer, \Advanced resistivity model for arbitrary magnetization orientation applied to a series of compressive- to tensile-strained (Ga,Mn)As layers," Phys. Rev. B 77, 205210 (2008). 38R. Karplus and J. M. Luttinger, \Hall eect in ferromagnetics," Phys. Rev. 95, 1154{1160 (1954). 39L. Berger, \Side-jump mechanism for the hall eect of ferromagnets," Phys. Rev. B 2, 4559{4566 (1970). 40O. Panchenko, P. Lutsishin, and Y. G. Ptushinskii, \Galvanomagnetic eects in thin lms of some transition metals," JETP 29, 134 (1969). 12Supplemental Materials: Anomalous Hall eect in YIG jPt bilayers I Reference measurements in YAG jPt bilayers Here, we discuss the reference samples consisting of plain Pt thin lms on single-crystalline diamagnetic Yttrium Aluminum Garnet (YAG). Figures S 4(b),(d),(f) show the character- istic linear behavior for trans(H)/H, i.e., an ordinary Hall eect, without any AHE contribution. Extracting the ordinary Hall coecient OHE =@trans(H)=@(H) from the slope, we obtain OHE(tPt= 2:0 nm) =3:1 p m=T,OHE(tPt= 3:5 nm) =15:9 p m=T, andOHE(tPt= 15:6 nm) =23:1 p m=T with a systematic error of
OHE= 0:1 p m=T. WhileOHEof the thickest Pt lm with tPt= 15:6 nm is consistent with the literature value OHE=24:4 p m=T25, we nd signicantly smaller OHE coecients for the 3 :5 nm and the 2:0 nm thick Pt lm. This behavior in the thin lm regime ( tPt10 nm) agrees with earlier reports40. Measurements of OHE(T) show aTindependent OHEand the absence of any AHE like contribution. Complementary to the FDMR experiments, we investigate transas a function of the magnetic eld orientation (ADMR). As evident from Fig. S 4(c) and (e), we obtain that the OHE depends only on the component H?=Hsin(H), i.e., trans(H)/sin(H). II Magnetization Orientation and Field Magnitude Dependent Measurements for YIGjPt(3:1 nm ) In Fig. S 5(a) we show a set of ADMR data for a YIG jPt (3:1 nm) sample taken at 10 K as a reference to Fig. 2 in the main text. This additional data substantiates the reproducibility of the observation of higher order contributions to trans up to at least n= 3 in a set of ADMR measurements [see Fig. S 5]. Please note that the data shown in Fig. S 5(a) is taken at 10 K, while the FDMR measurements performed on similar samples shown in Fig. 1 were taken at 300 K and thus have a dierent OHE and AHE behavior. In particular, for T= 10 K, we observe an almost vanishing OHE signal in this sample, OHE= 6 p m=T and therefor the sin3(H) contribution becomes prominent even for the 7 T data, which otherwise would be overwhelmed by the sin( H) characteristic of the OHE. The tting parameters A 13(a) Iqn t jH Vtrans βH(c) (d) (e) (f)-80-4004080ρtrans(pΩm) 1T3T YAG|Pt(3.5nm) -2 0 2 0° 180°360°-80-4004080ρtrans(pΩm) µ0H (T) βH YAG|Pt(15.6nm)1T3T -80-4004080ρtrans(pΩm) YAG|Pt(2.0nm)(b) -2 0 2 µ0H (T)FIG. 4. (a)Sample and measurement geometry. (b)Transverse resistivity trans taken from a FDMR measurement for YAG jPt (2:0 nm). (c)trans as a function of Hfor YAGjPt (3:5 nm). (d)Corresponding FDMR data for H= 90.(e, f) : ADMR and FDMR measurements for tPt= 15:6 nm on YAG. The colored, horizontal, dashed lines in panels (c,d) and (e,f) are intended as guides to the eye, to show that the transvalues inferred from FDMR and ADMR are consistent for identical magnetic eld congurations. All data taken at 300 K. 0° 180° 360°-20-1001020 ρtrans (pΩm) 2T 4T 7TYIG|3.1nm Pt 02468100.1110 Frequenc y (1/360° )Amplitude (pΩm) 7T 10K βH (a) (b) FIG. 5. (a)Transverse resistivity trans as a function of Hfor a YIGjPt bilayer with tPt= 3:1 nm, taken at 10 K. (b)Fast Fourier transform (FFT) of the ADMR data taken at T= 10 K with 0H= 7 T for the YIGjPt(3:1 nm) sample shown in (a). The dashed line indicates the experimental noise level of 1 p m. andBobtained from ts of Eq.(1) to the ADMR data shown in S 5(a) for YIG jPt (3:1 nm) are represented by the black data points in Fig. 2(c) and (d) in the main article. For a full 14picture of the temperature dependence of OHE and AHE contributions to the parameters AandB, we refer to Fig. 3 in the main text. III Fast Fourier Transform As shown in Fig. S 5(a) and in Fig. 2(a) in the main text, our magnetization orientation dependent measurements on YIG jPt bilayers reveal additional higher order contributions totrans, such that we can formulate trans/Asin(H) +Bsin3(H) +


. To specify the particular contributions, we perform fast Fourier transformations (FFT) of the ADMR data as exemplarily shown in Fig. S 5(b) for YIG jPt(3:1 nm) taken at 10 K [see Fig. S 5(a)]. For the FFT, we use a rectangular window with amplitude correction. The amplitude spectrum of the FFT for this set of data reveals the presence of sin( nH) contributions up to at least n= 5. Possibly occurring higher order contributions could not be quantied, since the amplitude for the n= 5 contribution is already comparable to our experimental resolution of 1 p m. Please note that the FFT results depicted in S 5(b) are not sign-sensitive and can not straightforwardly be compared to results for AandBobtained from ts using Eq. (2). The FFT algorithm species frequency components proportional to sin( nH), while our approx- imation in Eq. (2) is a power series proportional to sinn(H). However, both expressions represent the same phenomenology and can be transformed into the respective other by fundamental algebra. 15IV Table of Samples A detailed information on the lm thicknesses for both types of thin lm structures used in our study is listed in Tab. S I. The parameter hrepresents the surface roughness of Pt obtained from high-resolution X-ray re ectometry. For YIG jPt bilayers, we determine an averaged surface roughness of h= (0:70:2) nm, while for plain Pt on diamagnetic substrate, we obtain a slightly lower value of h= (0:50:1) nm. However, within the estimated errors, the interface roughnesses of both types of samples are comparable and thus we expect no in uence of the surface roughnesses on our OHE and AHE data. substratetYIG(nm)tPt(nm)h(nm) YAG 34 0.8 0.7 YAG 56 3.1 1.0 YAG 38 1.2 0.9 YAG 63 6.5 0.9 YAG 57 2.0 0.8 GGG 61 11.1 0.6 YAG 49 2.0 0.6 YAG 61 19.5 1.0 YAG 58 2.5 1.1 YAG 0 2.0 0.4 YAG 0 15.6 0.6 YAG 0 3.5 0.5 TABLE I. Substrate material, YIG thickness tYIG, platinum thickness tPtand platinum roughness hfor all samples investigated in this work. References 1M. Dyakonov and V. Perel, \Current-induced spin orientation of electrons in semiconduc- tors," Phys. Lett. A 35, 459{460 (1971). 2J. E. Hirsch, \Spin hall eect," Phys. Rev. Lett. 83, 1834{1837 (1999). 163Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, \Observation of the spin hall eect in semiconductors," Science 306, 1910 {1913 (2004). 4S. O. Valenzuela and M. Tinkham, \Direct electronic measurement of the spin hall eect," Nature 442, 5 (2006). 5E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, \Conversion of spin current into charge current at room temperature: Inverse spin-Hall eect," Appl. Phys. Lett. 88, 182509 (2006). 6Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, \Spin pumping and magnetization dynamics in metallic multilayers," Physical Review B 66, 10 (2002). 7K. Xia, P. Kelly, G. E. W. Bauer, A. Brataas, and I. Turek, \Spin torques in ferromagnetic/normal-metal structures," Phys. Rev. B 65, 220401 (2002). 8D. Huertas Hernando, Y. Nazarov, A. Brataas, and G. E. W. Bauer, \Conductance modu- lation by spin precession in noncollinear ferromagnet normal-metal ferromagnet systems," Phys.l Rev. B 62, 5700 (2000). 9A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, \Finite-element theory of transport in ferromagnetnormal metal systems," Phys. Rev. Lett. 84, 2481{2484 (2000). 10M. D. Stiles and A. Zangwill, \Anatomy of spin-transfer torque," Phys. Rev. B 66, 014407 (2002). 11Z. Wang, Y. Sun, M. Wu, V. Tiberkevich, and A. Slavin, \Control of spin waves in a thin lm ferromagnetic insulator through interfacial spin scattering," Phys. Rev. Lett. 107, 146602 (2011). 12E. Padron-Hernandez, A. Azevedo, and S. M. Rezende, \Amplication of spin waves in yttrium iron garnet lms through the spin hall eect," Appl. Phys. Lett. 99, 192511 (2011). 13D. C. Ralph and M. D. Stiles, \Spin transfer torques," J. MMM 320, 1190{1216 (2008). 14Z. Wang, Y. Sun, Y.-Y. Song, M. Wu, H. Schulthei, J. E. Pearson, and A. Homann, \Electric control of magnetization relaxation in thin lm magnetic insulators," Appl. Phys. Lett. 99, 162511 (2011). 15Y. 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{266 (2010). 16F. D. Czeschka, L. Dreher, M. S. Brandt, M. Weiler, M. Althammer, I. Imort, G. Reiss, A. Thomas, W. Schoch, W. Limmer, H. Huebl, R. Gross, and S. T. B. Goennenwein, 17\Scaling behavior of the spin pumping eect in Ferromagnet-Platinum bilayers," Phys. Rev. Lett. 107, 046601 (2011). 17H. Nakayama, M. Althammer, Y.-T. Chen, K. Uchida, Y. Kajiwara, D. Kikuchi, T. Ohtani, S. Gepr ags, M. Opel, S. Takahashi, R. Gross, G. E. W. Bauer, S. T. B. Goennenwein, and E. Saitoh, \Spin hall magnetoresistance induced by a nonequilibrium proximity eect," Phys. Rev. Lett. 110, 206601 (2013). 18N. Vlietstra, J. Shan, V. Castel, J. Ben Youssef, G. E. W. Bauer, and B. J. v. Wees, \Exchange magnetic eld torques in yig/pt bilayers observed by the spin-hall magnetore- sistance," Appl. Phys. Lett. 103, 032401 (2013). 19Y.-T. Chen, S. Takahashi, H. Nakayama, M. Althammer, S. T. B. Goennenwein, E. Saitoh, and G. E. W. Bauer, \Theory of spin hall magnetoresistance," Phys. Rev. B 87, 144411 (2013). 20S. Gepr ags, S. Meyer, S. Altmannshofer, M. Opel, F. Wilhelm, A. Rogalev, R. Gross, and S. T. B. Goennenwein, \Investigation of induced pt magnetic polarization in pt/y3fe5o12 bilayers," Appl. Phys. Lett. 101, 262407 (2012). 21M. Althammer, S. Meyer, H. Nakayama, M. Schreier, S. Altmannshofer, M. Weiler, H. Huebl, S. Gepr ags, M. Opel, R. Gross, D. Meier, C. Klewe, T. Kuschel, J.-M. Schmal- horst, G. Reiss, L. Shen, A. Gupta, Y.-T. Chen, G. E. W. Bauer, E. Saitoh, and S. T. B. Goennenwein, \Quantitative study of the spin hall magnetoresistance in ferromagnetic insulator/normal metal hybrids," Phys. Rev. B 87, 224401 (2013). 22See supplemental material at [URL will be inserted by publisher] for details. 23S. Meyer, M. Althammer, S. Gepr ags, M. Opel, R. Gross, and S. T. B. Goennenwein, \Temperature dependent spin transport properties of platinum inferred from spin hall magnetoresistance measurements," App. Phys. Lett. 104, 242411 (2014). 24W. Limmer, M. Glunk, J. Daeubler, T. Hummel, W. Schoch, R. Sauer, C. Bihler, H. Huebl, M. S. Brandt, and S. T. B. Goennenwein, \Angle-dependent magnetotransport in cubic and tetragonal ferromagnets: Application to (001)- and (113)A-oriented (Ga,Mn)As," Phys. Rev. B 74, 205205 (2006). 25P. Gehlho, E. Justi, and M. Kohler, \Der hall-eekt von iridium," Z. Naturforschg. 5a, 16{18 (1950). 26S. Y. Huang, X. Fan, D. Qu, Y. P. Chen, W. G. Wang, J. Wu, T. Y. Chen, J. Q. Xiao, and C. L. Chien, \Transport magnetic proximity eects in platinum," Phys. Rev. Lett. 18109, 107204 (2012). 27S. Shimizu, K. S. Takahashi, T. Hatano, M. Kawasaki, Y. Tokura, and Y. Iwasa, \Electri- cally tunable anomalous hall eect in pt thin lms," Phys. Rev. Lett. 111, 216803 (2013). 28D. Qu, S. Y. Huang, B. F. Miao, S. X. Huang, and C. L. Chien, \Self-consistent deter- mination of spin hall angles in selected 5 dmetals by thermal spin injection," Phys. Rev. B89, 140407 (2014). 29B. F. Miao, S. Y. Huang, D. Qu, and C. L. Chien, \Physical origins of the new magne- toresistance in pt/yig," Phys. Rev. Lett. 112, 236601 (2014). 30Y. Shiomi, T. Ohtani, S. Iguchi, T. Sasaki, Z. Qiu, H. Nakayama, K. Uchida, and E. Saitoh, \Interface-dependent magnetotransport properties for thin pt lms on ferri- magnetic y3fe5o12," Appl. Phys. Lett. 104, 242406 (2014). 31S.-C. Shin, G. Srinivas, Y.-S. Kim, and M.-G. Kim, \Observation of perpendicular mag- netic anisotropy in ni/pt multilayers at room temperature," Appl. Phys. Lett. 73, 393{395 (1998). 32D. K. Satapathy, M. A. Uribe-Laverde, I. Marozau, V. K. Malik, S. Das, T. Wagner, C. Marcelot, J. Stahn, S. Br uck, A. R uhm, S. Macke, T. Tietze, E. Goering, A. Fra~ n o, J. H. Kim, M. Wu, E. Benckiser, B. Keimer, A. Devishvili, B. P. Toperverg, M. Merz, P. Nagel, S. Schuppler, and C. Bernhard, \Magnetic proximity eect in yba2cu3o7=la2=3ca1=3mno 3 andyba2cu3o7=lamn 3+superlattices," Phys. Rev. Lett. 108, 197201 (2012). 33Y. M. Lu, Y. Choi, C. M. Ortega, M. X. Cheng, J. W. Cai, S. Y. Huang, L. Sun, and C. L. Chien, \Pt magnetic polarization on y3fe5o12 and magnetotransport characteristics," Phys. Rev. Lett. 110, 147207 (2013). 34Y. M. Lu, J. W. Cai, S. Y. Huang, D. Qu, B. F. Miao, and C. L. Chien, \Hybrid magnetoresistance in the proximity of a ferromagnet," Phys. Rev. B 87, 220409 (2013). 35G. Fischer, H. Homann, and J. Vancea, \Mean free path and density of conductance electrons in platinum determined by the size eect in extremely thin lms," Phys. Rev. B 22, 6065{6073 (1980). 36X. Jia, K. Liu, K. Xia, and G. E. W. Bauer, \Spin transfer torque on magnetic insulators," Europhys. Lett. 96, 17005 (2011). 37W. Limmer, J. Daeubler, L. Dreher, M. Glunk, W. Schoch, S. Schwaiger, and R. Sauer, \Advanced resistivity model for arbitrary magnetization orientation applied to a series of compressive- to tensile-strained (Ga,Mn)As layers," Phys. Rev. B 77, 205210 (2008). 1938R. Karplus and J. M. Luttinger, \Hall eect in ferromagnetics," Phys. Rev. 95, 1154{1160 (1954). 39L. Berger, \Side-jump mechanism for the hall eect of ferromagnets," Phys. Rev. B 2, 4559{4566 (1970). 40O. Panchenko, P. Lutsishin, and Y. G. Ptushinskii, \Galvanomagnetic eects in thin lms of some transition metals," JETP 29, 134 (1969). 20

2015-01-12

We measure the ordinary and the anomalous Hall effect in a set of yttrium iron garnet$|$platinum (YIG$|$Pt) bilayers via magnetization orientation dependent magnetoresistance experiments. Our data show that the presence of the ferrimagnetic insulator YIG leads to an anomalous Hall like signature in Pt, sensitive to both Pt thickness and temperature. Interpretation of the experimental findings in terms of the spin Hall anomalous Hall effect indicates that the imaginary part of the spin mixing interface conductance $G_{\mathrm{i}}$ plays a crucial role in YIG$|$Pt bilayers. In particular, our data suggest a sign change in $G_{\mathrm{i}}$ between $10\,\mathrm{K}$ and $300\,\mathrm{K}$. Additionally, we report a higher order Hall effect, which appears in thin Pt films on YIG at low temperatures.

Anomalous Hall effect in YIG$|$Pt bilayers

1501.02574v3

Nonlocal magnon-polaron transport in yttrium iron garnet L.J. Cornelissen,1,K. Oyanagi,2,T. Kikkawa,2, 3Z. Qiu,3T. Kuschel,1G.E.W. Bauer,1, 2, 3, 4B.J. van Wees,1and E. Saitoh2, 3, 4, 5 1Physics of Nanodevices, Zernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlandsy 2Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 3WPI Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 4Center for Spintronics Research Network, Tohoku University, Sendai 980-8577, Japan 5Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan The spin Seebeck eect (SSE) is observed in magnetic insulator jheavy metal bilayers as an inverse spin Hall eect voltage under a temperature gradient. The SSE can be detected nonlocally as well, viz. in terms of the voltage in a second metallic contact (detector) on the magnetic lm, spatially separated from the rst contact that is used to apply the temperature bias (injector). Magnon-polarons are hybridized lattice and spin waves in magnetic materials, generated by the magnetoelastic interaction. Kikkawa et al. [Phys. Rev. Lett. 117, 207203 (2016)] interpreted a resonant enhancement of the local SSE in yttrium iron garnet (YIG) as a function of the magnetic eld in terms of magnon-polaron formation. Here we report the observation of magnon-polarons in nonlocal magnon spin injection/detection devices for various injector-detector spacings and sample temperatures. Unexpectedly, we nd that the magnon-polaron resonances can suppress rather than enhance the nonlocal SSE. Using nite element modelling we explain our observations as a competition between the SSE and spin diusion in YIG. These results give unprecedented insights into the magnon-phonon interaction in a key magnetic material. When sound travels through a magnet the local dis- tortions of the lattice exert torques on the magnetic order due to the magnetoelastic coupling1. By reci- procity, spin waves in a magnet aect the lattice dy- namics. The coupling between spin and lattice waves (magnons and phonons) has been intensively researched in the last half century2,3. Yttrium iron garnet (YIG) has been a singularly useful material here, because it can be grown with exceptional magnetic and acoustic quality2. Magnons and phonons hybridize at the (anti)crossing of their dispersion relations, a regime that has attracted recent attention4{10. When the quasiparticle lifetime- broadening is smaller than the interaction strength, the strong coupling regime is reached; the resulting fully mixed quasiparticles have been referred to as magnon- polarons6,7. In spite of the long history and ubiquity of the magnon- phonon interaction, it still leads to surprises. Evidence of a sizeable magnetoelastic coupling in YIG was recently found in experiments on spin caloritronic eects, i.e. the spin Peltier11and spin Seebeck eect12,13(SPE and SSE respectively). Recently, Kikkawa et al. showed that the hybridization of magnons and phonons can lead to a res- onant enhancement of the local SSE in YIG9. Bozhko et al. found that this hybridization can play a role in the thermalization of parametrically excited magnons using Brillouin light scattering. They observed an ac- cumulation of magnon-polarons in the spectral region near the anticrossing between the magnon and trans- verse acoustic phonon modes14. However, these previous experiments did not address the transport properties of magnon-polarons. Nonlocal spin injection and detection experiments are of great importance in probing the transport of spin inmetals15, semiconductors16and graphene17. Varying the distance between the spin injection and detection con- tacts allows for the accurate determination of the trans- port properties of the spin information carriers in the channel, such as the spin relaxation length18. Recently, it was shown that this kind of experiments are not limited to (semi)conducting materials, but can also be performed on magnetic insulators19, where the spin information is carried by magnons. Such nonlocal magnon spin trans- port experiments have provided additional insights in the properties of magnons in YIG, for instance by studying the transport as a function of temperature20{23or exter- nal magnetic eld24. Finally, the nonlocal magnon spin injection/detection scheme can play a role in the develop- ment of ecient magnon spintronic devices, for example magnon based logic gates25,26. In this study, we make use of nonlocal magnon spin injection and detection de- vices to investigate the transport of magnon-polarons in YIG. Magnons can be excited magnetically using the oscil- lating magnetic eld generated by a microwave frequency ac current25, or electrically using a dc current in an ad- jacent material with a large spin Hall angle, such as platinum19. Finally, they can be generated thermally by the SSE27{30, in which a thermal gradient in the mag- netic insulator drives a magnon spin current parallel to the induced heat current. The generation of magnons via the SSE can be de- tected in several congurations: First, the heater-induced conguration (hiSSE)31, which consists of a bilayer YIGjheavy metal sample that is subject to external Peltier elements to apply a temperature gradient nor- mal to the plane of the sample. The SSE then gener- ates a voltage across the heavy metal lm (explained inarXiv:1706.04373v1 [cond-mat.mes-hall] 14 Jun 20172 more detail below), which can be recorded. Second, the current-induced conguration (ciSSE)28,32in which the heavy metal detector used to detect the SSE voltage is si- multaneously used as a heater. A current is sent through the heavy metal lm, creating a temperature gradient in the YIG due to Joule heating. Due to this temperature gradient, the SSE generates a voltage across the heavy metal lm, which can again be recorded. Third, the non- local SSE (nlSSE)19,33, in which a current is sent through a narrow heavy metal strip to generate a thermal gradi- ent via Joule heating as well. However, the SSE signal resulting from this thermal gradient is detected in a sec- ond heavy metal strip, located some distance away from the injector. In the nlSSE, the magnons responsible for generating a signal in the detector strip are generated in the injector vicinity and then diuse through the magnetic insula- tor to the detector. The temperature gradient under- neath a detector located several microns to tens of mi- crons from the injector does not contribute signicantly to the measured voltage23,34. In contrast, the hiSSE and ciSSE always have a signicant temperature gradient di- rectly underneath the detector. The hiSSE and ciSSE are therefore local SSE congurations, contrary to the nlSSE which is nonlocal. In all three congurations, the resulting voltage across the heavy metal lm is due to magnons which are ab- sorbed at the YIG jdetector interface, causing spin- ip scattering of conduction electrons and generating a spin current and spin accumulation in the detector. Due to the inverse spin Hall eect35, this spin accumulation is converted into a charge voltage that is measured. At specic values for the external magnetic eld, the phonon dispersion is tangent to that of the magnons and the magnon and phonon modes are strongly coupled over a relatively large region in momentum space. At these resonant magnetic eld values, the eect of the magne- toelastic coupling is at its strongest and magnon-polarons are formed eciently. If the acoustic quality of the YIG lm is better than the magnetic one, magnon-polaron formation leads to an enhancement in the hiSSE signal at the resonant magnetic eld9. This enhancement is at- tributed to an increase in the eective bulk spin Seebeck coecient, which governs the generation of magnon spin current by a temperature gradient in the magnet. This was demonstrated experimentally by measuring the spin Seebeck voltage in the hiSSE conguration9, estab- lishing the role of magnon-polarons in the thermal gen- eration of magnon spin current. Here we make use of the nlSSE conguration to di- rectly probe not only the generation, but also the trans- port of magnon-polarons. We show that in the YIG sam- ples under investigation not only , but also the magnon spin conductivity mis resonantly enhanced by the hy- bridization of magnons and phonons, which leads to sig- natures in the nonlocal magnon spin transport signals clearly distinct from the hiSSE observations. Notably, resonant features in nonlocal transport experiments havevery recently been theoretically predicted by Flebus et al.10, who calculated the in uence of magnon-polarons on the YIG transport parameters such as the magnon spin and heat conductivity and the magnon spin diu- sion length. I. RESULTS A. Sample characteristics Our nonlocal devices consist of multiple narrow, thin platinum strips (typical dimensions are 100 m100 nm10 nm [lwt]) deposited on top of a YIG thin lm and separated from each other by a centre-to-centre distanced. We have performed measurements of non- local devices on YIG lms from Groningen and Sendai, both of which are grown by liquid phase epitaxy on a GGG substrate. The YIG lm thickness is 210 nm (2.5 m) for YIG from Groningen (Sendai). In Sendai, ve batches of devices where investigated (sample S1 to S4) on pieces cut from the same YIG wafer. In Groningen, two batches of devices where investigated (G1 and G2). The platinum strips are contacted using Ti/Au contacts (see Methods for fabrication details) Figure 1a shows an optical microscope image of a typical device, with the electrical connections indicated schematically. The cen- tral strip functions as a magnon injector while the two outer strips are magnon detectors, measuring the nonlo- cal signal at dierent distances from the injector. B. Experimental results A nonlocal signal is generated in the devices by passing a low frequency ac current I(!) (typically w=(2)<20 Hz andIrms= 100A) through the injector. This leads to both thermal and electrical generation of magnons, as outlined above. The voltage that is due to the ther- mally generated magnons is proportional to the excita- tion current squared, and hence can be directly detected in the second harmonic detector response V(2!) (i.e. the voltage measured at twice the excitation frequency). Si- multaneously, the voltage due to electrically generated magnons can be measured in the rst harmonic response V(1!)19. The sample is placed in an external magnetic eldH, under an angle = 90to the injector/detector strips. Figure 1b shows the results of two typical nonlo- cal measurements at dierent distances, in which 0H is varied from3:0 to 3:0 T. Several distinct features can be seen in these results. As the magnetic eld is swept through zero, the YIG magnetization and hence the magnon spin polarization change direction, since a magnon always carries a spin opposite to the majority spin in the magnet. This causes a reversal of the polar- ization of the spin current absorbed by the detector and3 consequently the voltage VnlSSE changes sign. Addition- ally,VnlSSE for short distance d(Figure 1b bottom pan- els) shows an opposite sign compared to VnlSSE for long distance (Figure 1b top panels). This sign-reversal for short distances is a characteristic feature of the nlSSE19 that has so far been observed to depend on both the thickness of the YIG lm tYIG(roughly speaking, at room temperature when d<t YIGthe sign will be opposite to that ford > t YIG33) as well as the sample temperature, where a lower temperature reduces the distance at which the sign-change occurs21,22. The sign for short distances corresponds to the sign one obtains when measuring the SSE in its local cong- urations (hiSSE, indicated schematically in Figure 1c or ciSSE). The results for a hiSSE measurement on sample S3 as a function of Hare shown in Figure 1d, and VhiSSE clearly shows the same sign as VnlSSE for short distance. We will discuss the origin of this sign-change in more de- tail later in this manuscript. The data shown in Figure 1 are from samples with tYIG= 2:5m, hence the dierent signs ford= 2m andd= 6m. Resonant features can be observed in the data for j0Hj=0HTA2:3 T, where the subscript TA sig- nies that these features stems from the hybridization of magnons with phonons in the transverse acoustic mode, rather than the longitudinal acoustic mode (LA) which is expected at larger magnetic elds. The rightmost panels of Figure 1 show a close-up of the data around H=HTA. For smalldthe magnon-phonon hybridization causes a resonant enhancement (the absolute value is increased) ofVnlSSE , while for large da resonant suppression (the absolute value is reduced) occurs. Figure 2 shows the results of a magnetic eld sweep from sample G1 for both electrically generated magnons (rst harmonic) and thermally generated magnons (sec- ond harmonic). A feature at jHj=HTAcan be resolved both in the rst and second harmonic voltage. This sug- gests that magnon-phonon hybridization does not only aect the YIG spin Seebeck coecient, as the rst har- monic signal is generated independent of . It indicates that not only the generation, but also the transport of magnons is aected by the hybridization. In the second harmonic, the signal is clearly suppressed at the resonant magnetic eld. Unfortunately, because the feature in the rst harmonic is barely larger than the noise oor in the measurements (see Fig. 2a and inset), we cannot conclude whether the signal due to electrical magnon generation is enhanced or suppressed at the resonance. Due to the fact that the eect in the rst harmonic is so small, in the remainder of this paper we present a systematic study of the eect in the second harmonic, the nlSSE. The resonant magnetic elds are dierent for the TA and LA modes ( HTAandHLA, respectively). Due to the higher sound velocity in the LA phonon mode, HTA< HLA, and the resonance due to magnons hybridizing with phonons in this mode can also be observed in our nonlo- cal experiments. In the Supplementary Material (section A) we show the results of a magnetic eld scan over anextended eld range, and it can be seen that the res- onance atHLAalso causes a suppression of the nlSSE signal, similar to the HTAresonance. This is compara- ble to the case for the hiSSE conguration, in which the HLAandHTAresonances both show similar behaviour in the sense that they both enhance the hiSSE signal. For the nlSSE case at distances larger than the sign-change distance, both resonances suppress the signal. We now focus on the resonance at HTAin the nlSSE data and carried out nonlocal measurements as a function of magnetic eld for various temperatures and distances. Figure 3a (b) shows the distance (temperature) depen- dent results, obtained from sample S1 (S2). The regions where the sign of the nlSSE equals that of the hiSSE are shaded blue. From Figure 3a the sign-change in VnlSSE can be clearly seen to occur between d= 2 andd= 5 m, as atd= 2m the nlSSE sign is equal to that of the hiSSE for any value of the magnetic eld, whereas for d= 5m it is opposite. Additionally, when comparing theVnlSSEHcurves for 300 K and 100 K in Figure 3b, the eect of the sample temperature on the sign-change is apparent: At 100 K, the nlSSE sign is opposite to that of the hiSSE over the whole curve. Furthermore, Figure 3b demonstrates the in uence of the magnetic eld on the sign change, for instance in the curve for T= 160 K. At low magnetic elds, the nlSSE sign still agrees with the hiSSE sign (inside the blue shaded region), but around j0Hj= 1:5 T the signal changes sign. In addition, Figure 3a shows that the role of the magnon-polaron resonance changes as the nlSSE signal undergoes a sign change. For d2m, magnon-phonon hybridization enhances VnlSSE atH=HTA, whereas for d5mVnlSSE is suppressed at the resonance mag- netic eld. Similarly, from Figure 3b we observe that at temperatures T > 160 K, magnon-phonon hybridization enhances the nlSSE signal at H=HTA, while atT160 K the nlSSE is suppressed at HTA. Since the thermally generated magnon spin current is related to the thermal gradient by jm/rT, a resonant enhancement in  should lead to an enhancement of the nlSSE signal at all distances and temperatures, which is inconsistent with our observations. This is a further indication that not only the generation, but also the transport of magnons is in uenced by magnon-polarons. The temperature dependence of the low-eld ampli- tude of the nlSSE V0 nlSSE and the magnitude of the reso- nanceVTA(dened in Figure 1b) are shown in Figure 4a and 4b respectively. The curve for V0 nlSSE atd= 6m agrees well with an earlier reported temperature depen- dence of the nlSSE at distances which are larger than the lm thickness23, while that at d= 2m qualitatively agrees with earlier reports for distances shorter than the YIG lm thickness21,22. Moreover, from the distance de- pendence of V0 nlSSE we have extracted the magnon spin diusion length mas a function of temperature, which is shown in the Supplementary Material (section B). m(T) obtained from the Sendai YIG approximately agrees with that for Groningen YIG23for temperatures T > 30 K,4 but diers in the low temperature regime. For further discussion we refer to the Supplementary Material of this manuscript. The temperature dependence of VTAis dif- ferent from that of V0 nlSSE , since rst of all no change in sign occurs here even for d= 2m and furthermore a clear minimum appears in the curve around T= 50 K. This indicates that the resonance has a dierent origin than the nlSSE signal itself, i.e. magnon-polarons are aected dierently by temperature than pure magnons. The resonant magnetic eld HTAdecreases with in- creasing temperature, reducing from 0HTA2:5 T at 3 K to0HTA2:2 T at room temperature as shown in Figure 4c. In earlier work by some of us regarding the magnetic eld dependence of the nonlocal magnon transport signal at room temperature, structure in the data at0H= 2:2 T was indeed observed24, but not understood at that time. It is now clear that this struc- ture can be attributed to magnon-phonon hybridization. HTAdepends on the following three parameters9: The YIG saturation magnetization Ms, the spin wave sti- ness constant Dexand the TA-phonon sound velocity cTA.Dexis approximately constant for T < 300 K36 and bothMsandcTAdecrease with temperature. The reduction of HTAas temperature increases from 3 K to 300 K can be explained by accounting for a 7 % decrease ofcTAin the same temperature interval, taking the tem- perature dependence of Msinto consideration37. The results regarding the behaviour of the magnon-polaron resonance qualitatively agree for the Sendai and Gronin- gen YIG (see Supplementary Material (section C) for the temperature dependent results for sample G2). Moreover, we performed measurements of the nlSSE signal as a function of the injector current, and found that the nlSSE scales linearly with the square of the current at high temperatures, as expected. However, at low temperatures ( T < 10 K) and suciently high cur- rents (typically, I > 50A), this linear scaling breaks down (see Supplementary Material (section D)). This could be a consequence of the strong temperature depen- dence of the YIG and GGG heat conductivity at these temperatures38,39. The injector heating causes a small in- crease in the average sample temperature which increases the heat conductivities of the YIG and GGG, thereby driving the system out of the linear regime. However, it might also be related to the bottleneck eect which is ob- served in parametrically excited YIG14. A more detailed investigation is needed in order to establish the origin of the nonlinearity. Finally, we have investigated the ciSSE conguration, meaning that current heating of the Pt injector is used to drive the SSE and the (local) voltage across the injector is measured. The sign of the ciSSE voltage corresponds to that obtained in the hiSSE conguration. However, no resonant features were observed in the ciSSE measure- ments, contrary to the hiSSE and nlSSE congurations. We believe that this is due to the low signal-to-noise ratio in the ciSSE conguration, which could cause the feature to be smaller than the noise level in our ciSSE measure-ments. We refer to the Supplementary Material (section E) for further discussion. C. Modelling The physical picture underlying the thermal genera- tion of magnons has been a subject of debate in the magnon spintronics eld recently. Previous theories ex- plain the SSE as being due to thermal spin pumping, caused by a temperature dierence between magnons in the YIG and electrons in the platinum13,40,41. However, the recent observations of nonlocal magnon spin trans- port and the nlSSE give evidence that not only the inter- face but also the bulk magnet actively contributes and even dominates the spin current generation. At elevated temperatures the energy relaxation should be much more ecient than the spin relaxation, which implies that the magnon chemical potential (and its gradient) is more im- portant as a non-equilibrium parameter than the temper- ature dierence between magnons and phonons. A model for thermal generation of magnon spin currents based on the bulk SSE42which takes into account a non-zero magnon chemical potential has been proposed in order to explain the observations34. This model has been reasonably successful in explain- ing the nonlocal signals (due to both thermal and electri- cal generation) in the long distance limit23,33, yet is not fully consistent with experiments in the short distance limit for thermally generated magnons33. The model is explained in detail in Refs. 33 and 34, and is described concisely in the Methods section of this manuscript. The physical picture captured by the model is explained in Figure 5a and b, where for this study we focus on the thermally generated magnons driving the nlSSE. In Fig- ure 5a a schematic side-view of the YIG jGGG sample with a platinum injector strip on top is shown. A cur- rent is passed through the injector, causing it to heat up to temperature TH. The bottom of the GGG substrate is thermally anchored at T0. As a consequence of Joule heating, a thermal gradient arises in the YIG, driving a magnon current Jm Q==TrTparallel to the heat cur- rent, i.e. radially away from the injector. This reduces the number of magnons in the region directly below the injector (magnon depletion). In Figure 5b the same schematic cross-section is shown, but now the colour coding refers to the magnon chemical potentialm. Directly below the injector contact m is negative due to the magnon depletion in this region (). At the YIGjGGG interface, magnons accumulate since they are driven towards this interface by the SSE but are re ected by the GGG, causing a positive magnon chemical potential +to build up. Note that the  and+regions are not equal in size since part of the magnon depletion is replenished by the injector contact, which acts as a spin sink. Due to the gradient in magnon chemical potential, a diuse magnon spin current Jm dnow arises in the YIG given by Jm d=mrm.5 The combination of these two processes leads to a typ- ical magnon chemical potential prole as shown in Fig- ure 5c, which is obtained from the nite element model (FEM) at room temperature. The sign change from  to+occurs at a distance of roughly dsc= 2:6m from the injector, comparable to the YIG lm thickness. Here we used the eective spin conductance of the PtjYIG interface gsas a free parameter in order to get approximate agreement between the modelled and exper- imentally observed sign-change distance dsc(see Methods for the further details of the model). The value for gsis approximately a factor 30 lower than what we calculated from theory34and used in our previous work23. When usinggs= 9:61012S/m2as in previous work, dsc300 nm which is much shorter than what we observe in the experiments. This discrepancy between the models for electrically and thermally generated magnon transport might indicate that some of the material parameters such as spin or heat conductivity and spin diusion length (for both YIG and platinum) we use are not fully accurate. However, it is also conceivable that the models are not complete and need to be rened further33, for instance by including temperature dierence at material interfaces which are currently neglected. The value of dscdepends mainly on four parameters: The thickness of the YIG lm tYIG, the transparency of the platinumjYIG injector interface, parameterized in the eective spin conductance gs, the magnon spin con- ductivity of the YIG mand nally the magnon spin diusion length m. At high temperatures (i.e. close to room temperature), the thermal conductivities GGG andYIGare similar in magnitude43and aectdsconly weakly, allowing us to focus here on the spin transport. IncreasingtYIGormincreasesdscsince this reduces the spin resistance of the YIG lm, allowing the depleted region to spread further throughout the YIG. However, increasinggsormcauses the opposite eect and re- ducesdscsince this increases the amount of which is absorbed by the injector contact compared to that which relaxes in the YIG. The precise dependency of dscon these parameters is nontrivial but can be explored using our nite element model. Ganzhorn et al. and Zhou et al.in Refs. 21 and 22 observed that dscbecomes smaller with lower temperatures. This indicates that the ratio of the eective spin resistance of YIG to that of the Pt con- tact increases, causing spins to relax preferentially into the contact and thereby reducing the extend of . Flebus et al. developed a Boltzmann transport theory for magnon-polaron spin and heat transport in magnetic insulators10. Here we implement the salient features of magnon-polarons into our nite element model. We ob- serve that when the combination of gs,m,m,tYIG anddis such that the detector is probing the depletion region, i.e. , the magnon-polaron resonance causes enhancement of the nlSSE signal. Conversely, when the detector is probing +the resonance causes a suppres- sion of the signal. This cannot be explained by assuming that the only eect of the magnon-polaron resonance isthe enhancement of , as this would simply increase the thermally driven magnon spin current Jm Qand hence en- hance both and+. To understand this behaviour, we have to account for the enhancement of mby the magnon-polaron resonance as well. A resonant increase in mleads to an increased diu- sive back ow current Jm d, which can lead to a reduction of the magnon spin current reaching the detector at large distances. We model the eect of the magnon-phonon hy- bridization by assuming a eld-dependent magnon spin conductivity m(H) and bulk spin Seebeck coecient (H), which are both enhanced at the resonant eld HTA. Note that the eld-dependence only includes the con- tribution from the magnon-polarons10, and does not in- clude the eect of magnons being frozen out by the mag- netic eld24,44{46since this is not the focus of this study. The parameter values used in the model are given in the Methods section of this paper. The model is used to cal- culate the spin current owing into the detector contact as a function of magnetic eld, from which we calculate the voltage drop over the detector due to the inverse spin Hall eect. We then vary the ratios of enhancement for mand, i.e.f=m(HTA)=0 mandf=(HTA)=0, where0 mand0are the zero eld magnon spin conduc- tivity and spin Seebeck coecient and m(HTA); (HTA) are these parameters at the resonant eld. The ratio of enhancement =f=fis crucial in obtaining agreement between the experimental and modelled data. To change delta, we x f= 1:09 and vary f. The value for f is comparable to the enhancement in calculated from theory for low temperatures10. D. Comparison between model and experiment Figure 6 shows a comparison between the distance de- pendence of V0 nlSSE andVTAobtained from experiments (Fig. 6a) and the nite element model (Fig. 6b and c) at room temperature. In Figure 6a, V0 nlSSE shows a change in sign around d= 4m, whileVTAhas a positive sign over the whole distance range. Fig. 6b shows the model results for V0 nlSSE (red), and the voltage measured at H=HTAfor= 2 (green) and = 0:5 (purple). While the voltage obtained from the model is approximately one order of magnitude lower than in experiments, the qualitative behaviour of the experimental data is repro- duced. In particular, the modelled dscapproximately agrees with the experimentally observed distance. For= 2, the modelled voltage at HTAis always en- hanced with respect to V0 nlSSE (ford < dsc,V(HTA)< V0 nlSSE and ford > dsc,V(HTA)> V0 nlSSE ). This is not consistent with the experiments as it leads to a sign change in VTA, which is dened as VTA=V0 nlSSE V(HTA), as can be seen from Fig. 6c. However, for = 0:5,V(HTA) is enhanced with respect toV0 nlSSE ford < dscbut suppressed for d > dsc. This results in a positive sign for VTAover the full distance range, comparable to the experimental observations. The6 full magnetic eld dependence obtained from the model can be found in the Supplementary Material (section F). As can be seen from the inset in Fig. 6c, = 0:5 results in a decay ofVTAwith distance which is comparable to the experimentally observed VTA(d) (inset Fig. 6a). We tted the data for VTAobtained from both the experiments and the simulations to VTA(d) =Aexpd=`TA, whereAis the amplitude and `TAthe length scale over which VTA decays. From the ts, we obtain `exp TA= 6:31:2m and`sim TA= 10:60:1m at room temperature, where we have tted to the model results for = 0:5. From the simulations, we nd that `TAis in uenced by the value used for , where a smaller leads to a longer `TA. This could indicate that has to be increased slightly to obtain better agreement between `exp TAand`sim TA. Therefore, in order to explain the observations, 0 :5<  < 1, i.e. the relative enhancement due to magnon- phonon hybridization in mhas to be larger than that of .`exp TAis enhanced at low temperatures (see Supplemen- tary Material (section B) for the distance dependence of VTAat low temperatures). This could indicate that  decreases with decreasing temperatures. For further dis- cussion we refer to the Supplementary Material (section B). The model results depend sensitively on gs. A largergs reduces the dscobserved in the model, so that our model no longer qualitatively ts the distance dependence of VnlSSE obtained in experiments. As a consequence, the needed to model the resonant suppression of the signal atHTAfor long distances decreases further, which would imply that the enhancement in mis much stronger than that in. Such a strong enhancement in mshould result in a clear magnon-polaron resonance in the electrically generated magnon spin signal, whereas we observed only a small eect here (see Fig. 2a). This is an indication that our choice of reducing gscompared to our previous work is justied. II. DISCUSSION We report resonant features in the nlSSE as a function of magnetic eld, which we ascribe to the hybridization of magnons and acoustic phonons. They occur at mag- netic elds that obey the \touch" condition at which the magnon frequency and group velocity agree with that of the TA and LA phonons. The signals are enhanced (peaks) for short injector-detector distances and high temperatures, but suppressed (dips) for long distances and/or low temperatures. The temperature dependence of the TA resonance diers from that of the low-eld nlSSE voltage, indicating that dierent physical mechan- sims are involved (this in contrast to the local SSE con- guration). The sign of the nlSSE signal corresponds to that of the signal in the hiSSE conguration for dis- tances below the sign-change distance. In this regime the magnon-polaron feature causes signal enhancement, sim- ilar to the hiSSE conguration. For distances longer thanthe sign-change distance, the nlSSE signal is suppressed at the resonance magnetic eld. These results are consistent with a model in which transport is diuse and carried by strongly coupled magnons and phonons10(magnon-polarons). Theory predicts an enhancement of all transport coecients when the acoustic quality of the crystal is better than the magnetic one. Simulations show that the dip observed in the nlSSE is not caused by deteriorated acoustics, but by a competition between the thermally generated, SSE driven magnon current and the diuse back ow magnon current which are both enhanced at the resonance. More experiments including thermal transport as well as an ex- tension of the Boltzmann treatment presented in Ref. 10 to 2D geometries are necessary to fully come to grips with heat and spin transport in YIG. Additionally, we observed features in the electrically generated magnon spin signal at the resonance magnetic eld. This is further evidence that not only the gener- ation of magnons via the SSE, but additional transport parameters such as the magnon spin conductivity are af- fected by magnon-polarons. The nonlocal measurement scheme provides an excel- lent platform to study magnon transport phenomena and opens up new avenues for studying the magnetoelastic coupling in magnetic insulators. Finally, these results are an important step towards a complete physical picture of magnon transport in magnetic insulators in its many as- pects, which is crucial for developing ecient magnonic devices. III. METHODS Sample fabrication. The YIG lms used in this study were all grown on gadolinium gallium garnet (GGG) substrates by liquid phase epitaxy (LPE) in the [111] direction. The samples from the Sendai group have a thickness of 2.5 m, the samples used in Groningen are 210 nm thick. The Sendai samples were grown in-house, whereas the Groningen samples were obtained commer- cially from Matesy GmbH. In Sendai, ve batches of de- vices where fabricated from the same YIG wafer (S1 to S4). The fabrication method and platinum strip geome- try are the same for all batches, but they were not fabri- cated at the same time, which might lead to variations in for instance the interface quality from batch to batch. In Groningen, two batches of devices were investigated (G1 and G2). The nonlocal devices fabricated in Groningen are dened in three lithography steps: the rst step was used to dene Ti/Au markers on top of the YIG lm via e-beam evaporation, used to align the subsequent steps. In the second step, Pt injector and detector strips were deposited using magnetron sputtering in an Ar+plasma. In the nal step, Ti/Au contacts were deposited by e- beam evaporation. Prior to the contact deposition, a brief Ar+ion beam etching step was performed to remove any polymer residues from the Pt strip contact areas to7 ensure optimal electrical contact to the devices. The non- local devices fabricated in Sendai were dened in a single lithography step. Two parallel Pt strips and contact pads were patterned using e-beam lithography followed by a lift-o process, in which 10-nm-thick Pt was deposited using magnetron sputtering in an Ar+plasma. Measurements. Electrical measurements were car- ried out in Groningen and in Sendai, using a current- biased lock-in detection scheme. A low frequency ac current of angular frequency !(typical frequencies are !=(2)<20 Hz, and the typical amplitude is I= 100 Arms) is sent through the injector strip, and the voltage on the detector strip is measured at both the frequencies !(the rst harmonic response) and 2 !(the second har- monic response). This allows us to separate processes that are linear in the current, which govern the rst harmonic response, from processes that are quadratic in the current which are measured in the second harmonic response19,28,47. The measurements in Sendai were carried out in a Quantum Design Physical Properties Measurement Sys- tem (PPMS), using a superconducting solenoid to apply the external magnetic eld (eld range up to 0H= 10:5 T). The measurements in Groningen were carried out in a cryostat equipped with a Cryogenics Limited variable temperature insert (VTI) and superconducting solenoid (magnetic eld range up to 0H=7:5 T). Electronic measurements in Groningen are carried out us- ing a home built current source and voltage pre-amplier (gain 104) module galvanically isolated from the rest of the measurement electronics, resulting in a noise level of approximately 3 nV r:m:s:at the output of the lockin amplier for a time constant of = 3 s and a lter slope of 24 dB/octave. The electronic measurements in Sendai were carried out by means of an ac and dc cur- rent source (Keithley model 6221) and a lockin amplier using a time constant of = 1 s and a lter slope of 24 dB/octave. The data shown in Figure 1b and Fig- ure 3 is the asymmetric part of the measured voltage with respect to the magnetic eld. The antisymmetriza- tion procedure includes both the forward and backward magnetic eld sweep, and the voltage shown in the g- ures is given by VH+= (Vbackward (H)Vbackward (H))=2 andVH= (Vforward (H)Vforward (H))=2, whereVH+ is the voltage at postive magnetic eld values and VH that at negative magnetic eld values. Simulations. The two-dimensional nite element model is implemented in COMSOL MultiPhysics (v4.4). The linear response relation of heat and spin transport in the bulk of a magnetic insulator reads 2e ~jm jQ = m=T ~=2e  rm rT ; (1) where jmis the magnon spin current, jQthe total (magnon and phonon) heat current, mthe magnon chemical potential, Tthe temperature (assumed to be the same for magnons and phonons by ecient thermal- ization),mthe magnon spin conductivity, the total(magnon and phonon) heat conductivity and the spin Seebeck coecient. We disregard temperature dier- ences arising from the Kapitza resistances at the Pt jYIG or YIGjGGG interfaces. eis the electron charge and ~ the reduced Planck constant. The diusion equations for spin and heat read r2m=m 2m; (2) r2T=j2 c ; (3) wherejcis the charge current density in the injector con- tact,andthe electrical and thermal conductivity and mthe magnon spin diusion length. Eq. (3) represents the Joule heating in the injector that drives the SSE. In the simulations, tYIG= 2:5m andwYIG= 500m are the thickness and width of the YIG lm, on top of a GGG substrate that is 500 m thick.wYIGis much larger than mand nite size eects are absent. The injector has a thickness of tPt= 10 nm and a width ofwPt= 300 nm. The spin and heat currents normal to the YIGjvacuum, Ptjvacuum and GGG jvacuum inter- faces vanish. At the bottom of the GGG substrate the boundary condition T=T0is used, i.e. the bottom of the sample is taken to be thermally anchored to the sam- ple probe. Furthermore, a spin current is not allowed to ow into the GGG. The spin current across the Pt jYIG interface is given by jint m=gs(sm), wheregsis the eective spin conductance of the interface, sis the spin accumulation on the metal side of the interface and m is the magnon chemical potential on the YIG side of the interface. The nonlocal voltage is then found by calculat- ing the average spin current density hjsi owing in the detector, which is then converted to non-local voltage usingVnlSSE =SHLhjsi=, whereSHis the spin Hall angle in platinum and Lis the length of the detector strip. The spin current in the platinum contact relaxes over the characteristic spin relaxation length s. The parameters we use for platinum in the model are SH= 0:11,= 1:9106S/m,s= 1:5 nm and= 26 W/(m K). For YIG, we use m= 3:7105S/m, m= 9:4m which was obtained in our previous work23. Furthermore, we use = 7 W/(m K), based on YIG thermal conductivity data from Ref. 39. For the bulk spin Seebeck coecient at zero eld we use 0= 500 A/m, based on our previous work in which we gave an estimate for at room temperature33. For GGG, the spin conductivity and spin Seebeck coecient are set to zero. For the GGG thermal conductivity we use = 9 W/(m K), based on data from Refs. 38 and 43. Finally, for the eective spin conductance of the interface we use gs= 3:41011S/m2. We note that this is roughly a factor 30 smaller than in our earlier work23. This variation of the interface transparency in dierent experiments indicates the presence of physical processes that are not taken into account in the modeling.8 IV. ACKNOWLEDGEMENTS We thank H. M. de Roosz, J.G. Holstein, H. Adema and T.J. Schouten for technical assistance and R.A. Duine, B. Flebus and K. Shen for discussions. This work is part of the research program of the Nether- lands Organization for Scientic Research (NWO) and supported by NanoLab NL, EU FP7 ICT Grant No. 612759 InSpin, the Zernike Institute for Advanced Mate- rials, Grant-in-Aid for Scientic Research on Innovative Area "Nano Spin Conversion Science" (Nos. JP26103005 and JP26103006), Grant-in-Aid for Scientic Research (A) (No. JP25247056) and (S) (No. JP25220910) from JSPS KAKENHI, Japan, and ERATO "Spin Quantum Rectication Project" (No. JPMJER1402) from JST, Japan. Further support by the DFG priority program Spin Caloric Transport (SPP 1538, KU3271/1-1) is grate-fully acknowledged. K.O. acknowledges support from GP-Spin at Tohoku University. T.Ki. is supported by JSPS through a research fellowship for young scientists (No. JP15J08026). V. AUTHOR CONTRIBUTIONS B.J.v.W., L.J.C., T.Ki. and E.S. conceived the ex- periments. Z.Q. fabricated the Sendai YIG lms. 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Salce, Journal of Mag- netism and Magnetic Materials 27, 315 (1982).39S. R. Boona and J. P. Heremans, Physical Review B 90, 064421 (2014). 40J. Xiao, G. E. W. Bauer, K.-i. Uchida, E. Saitoh, and S. Maekawa, Physical Review B 81, 214418 (2010). 41H. Adachi, K.-i. Uchida, E. Saitoh, and S. Maekawa, Re- ports on Progress in Physics 76, 36501 (2013). 42S. M. Rezende, R. L. Rodr guez-Su arez, R. O. Cunha, A. R. Rodrigues, F. L. A. Machado, G. A. Fonseca Guerra, J. C. Lopez Ortiz, and A. Azevedo, Physical Review B 89, 014416 (2014). 43G. A. Slack and D. W. Oliver, Physical Review B 4, 592 (1971). 44T. Kikkawa, K.-i. Uchida, S. Daimon, Z. Qiu, Y. Shiomi, and E. Saitoh, Physical Review B 92, 064413 (2015). 45H. Jin, S. R. Boona, Z. Yang, R. C. Myers, and J. P. Heremans, Physical Review B 92, 054436 (2015). 46T. Kikkawa, K. I. Uchida, S. Daimon, and E. Saitoh, Journal of the Physical Society of Japan 85, 065003 (2016). 47F. L. Bakker, A. Slachter, J.-P. Adam, and B. J. van Wees, Physical Review Letters 105, 136601 (2010).10 FIG. 1. Experimental geometries and main results. Figure ais an image of a typical device, with schematic current and voltage connections. The three parallel lines are the Pt injector/detector strips, connected by Ti/Au contacts. is the angle between the Pt strips and an applied magnetic eld H(inb-d= 90).bThe nonlocal spin Seebeck (nlSSE) voltage for an injector-detector distance d= 6m (top) and d= 2m (bottom) as a function of 0H. Atj0Hj=j0HTAj2:3 T, a resonant structure is observed that we interpret in terms of magnon-polaron formation (indicated by blue triangles as a guide to the eye). The right column is a close-up of the anomalies for H > 0. The results can be summarized by the voltages V0 nlSSE andVTAas indicated in the lower panels. cSchematic geometry of the local heater-induced hiSSE measurements. Here the temperature gradient rTis applied by external Peltier elements on the top and bottom of the sample. dThe hiSSE voltage measured as a function of magnetic eld. The close-up around the resonance eld (right column) focusses on the magnon-polaron anomaly. All results were obtained at T= 200 K. The results for d= 6,d= 2 andd= 0m were obtained from sample S1, S2, S3, respectively (see Methods for sample details).11 FIG. 2. Nonlocal voltage due to electrically and thermally generated magnons as a function of magnetic eld. Figure ashows the nonlocal voltage generated by magnons that are excited electrically (rst harmonic response to an oscillating current in the injector contact). An anomaly is observed at H=jHTAj(the eld that satises the touching condition for magnons and transverse acoustic phonons). The inset shows a second set of data from the same sample, taken with a higher magnetic eld resolution ( 0
H= 15 mT), sweeping the magnetic eld both in the forward (black) and backward (red) directions. Figure bshows the nlSSE voltage (second harmonic response) for the same device. VnlSSE is suppressed at H=jHTAj. The inset shows the corresponding second harmonic data of the high resolution eld sweep. The results were obtained on sample G1 (thickness 210 nm) with d= 3:5m andI= 150Ar:m:s:, at room temperature. A constant background voltage Vbg= 575 nV was subtracted from the data in Fig. a.12 Temperature dependence for d = 2 μm Distance dependence for T = 300 K a b nonlocal sign = local sign FIG. 3.VnlSSE vs magnetic eld as a function of distance and temperature. Figure ais a plot of VnlSSE vsHfor various injector-detector separations at T= 300 K, while Figure bshowsVnlSSE vsHfor dierent temperatures and d= 2 m. The data in Figs. aandbare from sample S1 and S2, respectively. The magnon-polaron resonance is indicated by the blue arrows. The blue shading in the graphs indicates the region in which the sign of the nlSSE signal agrees with that of the hiSSE. The right column in both aandbshows close-ups of the data around the positive resonance eld (blue triangles). The data in the close-ups has been antisymmetrized with respect to H, i.e.V= (V(+H)V(H))=2. Fig. ashows that when the contacts are close ( d2m), the magnon-polaron resonance enhances VnlSSE , while for long distances VnlSSE is suppressed at the resonance magnetic eld. For very large distances ( d20m), the resonance cannot be observed anymore. Similarly in Fig.b, for temperatures T180 K, the magnon-polaron resonance enhances the nlSSE signal, while for lower temperatures the nlSSE signal is suppressed. The excitation current I= 100Ar:m:s:for all measurements.13 FIG. 4. Temperature dependence of V0 nlSSE,VTAandHTA. adisplays the temperature dependence of the low-eld V0 nlSSE , ford= 2m andd= 6m. For 2m, the signal changes sign around T= 143 K. The blue shading in the graph indicates the regime in which the sign agrees with that of the hiSSE. The temperature dependence of the magnon-polaron resonance VTAis shown in Figure b. Here, no sign change but a minimum around T= 50 K is observed, which is absent in Figure a. Figure cshows the temperature dependence of the resonance eld HTA. Error bars in bandcre ect the peak-to-peak noise in the data used to extract VTAand the step size in the magnetic eld scans ( 0
H= 20 mT), respectively. FIG. 5. Physical concepts underlying the nlSSE signal and simulated magnon chemical potential prole. Figure asketches the eects of Joule heating in the injector, heating it up to temperature TH, which leads to a thermal gradient in the YIG. The bulk SSE generates a magnon current Jm Qantiparallel to the local temperature gradient, spreading into the lm away from the contact. When the spin conductance of the contact is suciently small, this leads to a depletion of magnons below the injector, indicated in Figure bas. When the magnons are re ected at the GGG interface, Jm Qaccumulates magnons at the YIGjGGG interface, shown in Figure bas+. The chemical potential gradient induces a backward and sideward diuse magnon current Jm d. Both processes in Figure aandbare included in the nite element model (FEM). Its results are plotted in Figure cin terms of a typical magnon chemical potential prole. mchanges sign at some distance from the injector, also at the YIG surface, where it can be detected by a second contact. The magnon-polaron resonance enhances both the spin Seebeck coecientand the magnon spin conductivity m. The increased back ow of magnons to the injector causes a suppression of the nonlocal signal at long distances (see Figure 6).14 FIG. 6. Comparison of the experimental and simulated V0 nlSSE andVTA.Figure ashows the distance dependence of V0 nlSSE andVTA(inset) measured at room temperature. The dashed line in the inset is an exponential t to the data. V0 nlSSE changes sign around d= 4m, whileVTAremains positive. Figure bis a plot of the calculated distance dependence of V0 nlSSE at zero magnetic eld (red) and at the resonant eld for = 2 (green) and = 0:5 (purple). Here is a parameter that measures the relative enhancement of the spin Seebeck coecient compared to the magnon spin conductivity, as explained in the main text. The inset shows the signal decay at long distances on a logarithmic scale. Figure cshows the modelled distance dependence of VTAfor various values of on a linear scale (inset for logarithmic scale). = 0:5 results in a positive sign for VTAover the full distance range with a slope that roughly agrees with experiments (cf. insets of Figure aandc). Reducing  further leads to a more gradual slope for VTA. In the simulations, the SSE enhancement is f= 1:09, whilefis varied with .

2017-06-14

The spin Seebeck effect (SSE) is observed in magnetic insulator|heavy metal bilayers as an inverse spin Hall effect voltage under a temperature gradient. The SSE can be detected nonlocally as well, viz. in terms of the voltage in a second metallic contact (detector) on the magnetic film, spatially separated from the first contact that is used to apply the temperature bias (injector). Magnon-polarons are hybridized lattice and spin waves in magnetic materials, generated by the magnetoelastic interaction. Kikkawa et al. [Phys. Rev. Lett. \textbf{117}, 207203 (2016)] interpreted a resonant enhancement of the local SSE in yttrium iron garnet (YIG) as a function of the magnetic field in terms of magnon-polaron formation. Here we report the observation of magnon-polarons in \emph{nonlocal} magnon spin injection/detection devices for various injector-detector spacings and sample temperatures. Unexpectedly, we find that the magnon-polaron resonances can suppress rather than enhance the nonlocal SSE. Using finite element modelling we explain our observations as a competition between the SSE and spin diffusion in YIG. These results give unprecedented insights into the magnon-phonon interaction in a key magnetic material.

Nonlocal magnon-polaron transport in yttrium iron garnet

1706.04373v1

arXiv:1910.04046v1 [cond-mat.mtrl-sci] 9 Oct 2019Magnetic field dependence of the nonlocal spin Seebeck effect in Pt/YIG/Pt systems at low temperatures Koichi Oyanagi,1,a)Takashi Kikkawa,1,2and Eiji Saitoh1,2, 3, 4,5 1)Institute for Materials Research, Tohoku University, Send ai 980-8577, Japan 2)WPI Advanced Institute for Materials Research, Tohoku Univ ersity, Sendai 980-8577, Japan 3)Center for Spintronics Research Network, Tohoku Universit y, Sendai 980-8577, Japan 4)Department of Applied Physics, University of Tokyo, Hongo, Tokyo 113- 8656, Japan 5)Advanced Science Research Center, Japan Atomic Energy Agen cy, Tokai 319-1195, Japan (Dated: 10 October 2019) We report the nonlocal spin Seebeck effect (nlSSE) in a later al configuration of Pt/Y 3Fe5O12(YIG)/Pt systems as a function of the magnetic field B(up to 10 T) at various temperatures T(3 K<T<300 K). The nlSSE voltage decreases with increasing Bin a linear regime with respect to the input power (the applie d charge-current squared I2). The reduction of the nlSSE becomes substantial when the Zeem an energy exceeds thermal energy at low temperatures, which can be interpreted as freeze-out of magnons relevant f or the nlSSE. Furthermore, we found the non-linear power dependence of the nlSSE with increasing Iat low temperatures ( T<20 K), at which the B-induced signal reduction becomes less visible. Our experimental results suggest tha t in the non-linear regime high-energy magnons are over populated than those expected from the thermal energy. We al so estimate the magnon spin diffusion length as functions ofBandT. Spin caloritronics1is an emerging field to study the inter- conversion between spin and heat currents. The spin Seebeck effect (SSE) is one of the fundamental phenomena in this field , referring to the spin-current generation from a heat curren t. The SSE is well studied in a longitudinal configuration2,3, which consists of a heavy metal(HM)/ferromagnet(FM) bi- layer system, typically Pt/Y 3Fe5O12(YIG) junction. When a thermal gradient is applied perpendicular to the interface , a magnon spin current is generated in FM and converted into a conduction-electron spin current in HM via the interfacia l exchange interaction4, which is subsequently detected as a transverse electric voltage via the inverse spin Hall effec t (ISHE)5,6. Recent studies of the longitudinal SSE (LSSE)7–9 suggest that magnon transport in FM plays a key role in SSEs. Nonlocal experiment is a powerful tool to investigate the transport of spin currents in various magnetic insulators10–20. Especially, when a spin current is excited via a thermal grad i- ent, it is called the nonlocal SSE (nlSSE)10,11. A typical non- local device consists of two HM wires on top of a magnetic insulator, which are electrically separated with the dista nce d. In nlSSE measurements, one of the HM wires is used as a heater; the Joule heating of an applied charge current ( I) drives magnon spin currents in the magnetic insulator. Some of the magnons reach the other HM wire and inject a spin current, which is converted into a voltage via the ISHE. By changing the injector-detector separation distance d, we can address the transport property of magnon spin currents. In this paper, we report the high magnetic field ( B) depen- dence of the nlSSE in lateral Pt/YIG/Pt systems at various temperatures from T=300 K to 3 K and up to |B|= 10 T. We observed that the nlSSE signal V2ωdecreases with increasing B, but the feature turns out to depend on the amplitude of the applied I. In a linear regime ( V2ω∝I2), substantial B-induced a)Electronic mail: k.oyanagi@imr.tohoku.ac.jpsuppression of the nlSSE was observed below 10 K, which is consistent with the previous LSSE results in Pt/YIG8,21. In a non-linear regime ( V2ω/negationslash∝I2), however, the nlSSE signal re- mains almost unchanged under high Bat low Ts. By mea- suring the ddependence, we estimate the magnon diffusion length λas functions of TandB. We prepared series of nonlocal Pt/YIG/Pt devices, schemat- ically shown in Fig. 1(a). A 2.5- µm-thick YIG film was grown by liquid phase epitaxy on a Gd 3Ga5O12(111) substrate22. On top of the YIG film, we fabricated two Pt wires using e-beam lithography and the lift-off process19. The dimension of the Pt wires is 200 µm length, 100 nm width, and 10 nm thickness. The Pt wires were deposited by mag- netron sputtering in Ar+atmosphere. We investigated four batches of samples (S1-S4) cut from the same YIG wafer. The ddependence was studied in S1 ( d=5,6,8,13,15, and 20 µm) and S2 ( d=9µm), while the Idependence at lowTs in S3 ( d=2µm) and S4 ( d=8µm). We measured a nonlocal voltage using a lock-in detection technique; we ap - plied an a.c. charge current, I, of 13.423 Hz in frequency to the injector Pt wire and measured a second harmonic nonlocal voltage V2ωacross the detector Pt wire11. First, we confirmed that the obtained nonlocal voltage sat- isfies the features of the nlSSE at room temperature. Figure 1(b) shows typical V2ωas a function of in-plane B(θ=90◦) at 300 K in the d=9µm sample. A clear V2ωappears, whose sign changes with respect to the Bdirection. V2ωdis- appears when Bis applied perpendicular to the plane ( θ=0). This symmetry is consistent with that of the SSE3. We de- fine the low-field amplitude of the voltage signal as VnlSSE= [V2ω(0.18 T)−V2ω(−0.18 T)]/2, at which the magnetization of YIG is fully saturated along B. As shown in Fig. 1(c), VnlSSE is proportional to I2, indicating that VnlSSE appears due to the Joule heating. With increasing B,V2ωgradually de- creases, and at around ±2.2 T, sharp dip structures show up, which are induced by magnon polarons due to magnon −TA-2 BV Js YIG Joule heat I θ(a) VnlSSE ɹ 7 0.010.101.00 0 10 d (µm)5 20 15 T = 300 K I = 100 µAExponential fitExperiment B = 0.18 T(c) V2ωɹ 7 0 B (T)(b) -0.50.5 2 -2 0T = 300 K d = 9.0 µmθ = 0 θ = 90 ° -1 1 3 -3 VnlSSE Pt 0 100 200 I (µA)0.20.4VnlSSE (µV) (d) T = 300 K d = 9.0 µm B = 0.18 TFittingExperiment FIG. 1. (a) A schematic illustration of the nlSSE measuremen t in a lateral Pt/YIG/Pt system. B,θ,I, and Jsdenote the external magnetic field, angle between Band sample surface normal, charge current through the Pt injector, and spin current at the Pt/YIG detec tor inter- face, respectively. An a.c. charge current is applied to the Pt injector, and the second harmonic voltage V2ωis measured across the Pt de- tector. (b) The Bdependence of the nonlocal voltage V2ωin the d= 9µm sample. V2ωatθ=90◦(θ=0) is measured with I=200µA (100 µA).VnlSSE=[V2ω(0.18 T)−V2ω(−0.18 T)]/2 represents the amplitude of the nlSSE. (c) VnlSSE(I)in the d=9µm sample. The solid red line shows a I2fitting to data. (d) Semi logarithmic plot of VnlSSE(d). The red line is fit with VnlSSE=Cexp(−d/λ). The error bars represent the 68% confidence level ( ±s.d.). phonon hybridization16,23,24. By changing the injector-detector separation distance d, we estimate the length scale of the magnon spin current25. As shown in Fig. 1(d), VnlSSE decreases with increasing d. A one-dimensional spin diffusion model11,26describes the de- cay, which reads VnlSSE=Cexp/parenleftbigg −d λ/parenrightbigg , (1) where λis the magnon spin diffusion length and Cis the d- independent constant. We fit Eq. (1) to the ddependence of VnlSSE and obtain λ=6.76±0.16µm at 300 K. Similar values are reported in previous studies in both thin (200 nm)11, and thick (50 µm)27YIG films. Next, we measured the Tdependence of V2ωwith I= 100µA. As shown in Fig. 2(a), at 300 K negative voltages are observed for the d=0.5 and 1.5µm samples, while the positive ones show up for the d=8 and 15 µm samples. With decreasing T, the d=8 and 15 µm samples exhibit a mono- tonic increase of VnlSSE . On the other hand, with decreasing T the negative voltages observed for the d=0.5 and 1.5µm samples at 300 K change their sign at several tens Kelvin. The sign change of VnlSSE with changing dandThas been observed in previous nlSSE experiments and explained as a result of a spatial profile of the magnon chemical potential µmthat governs the sign and amplitude of VnlSSE ; a negatived = 0.5 µm 1.5 µm 8.0 µm 15.0 µm 100 10 T (K)VnlSSE ɹ 7 -10010 -5 5I = 100 µA B = 0.18 T(a) 2 1B (T)0 310 6λ (µm) 8 4 2T = 20 K 50 K100 K 300 KI = 100 µA(b) 10 5 B (T)-5 -10 0V2ωɹ 7 -1 0 -2 12 T = 300 K d = 1.5 µm I = 100 µA(c) 10040 0 T (K)20 60 80 -20 100 10 (d) d = 0.5 µm 1.5 µm 8.0 µm 15.0 µmδɹ % FIG. 2. (a) Semi logarithmic plot of VnlSSE(T)for various dwith I=100µA. (b) λ(B)at various Ts. We obtained λbyCexp(−d/λ) fitting to the ddependence of VnlSSE . (c) V2ω(B)in the d=1.5µm sample with I=100 µA at 300 K. (d) δ(T)for different d.δis defined by Eq. (2). µmcreated beneath the Pt injector exponentially decays apart from the injector and above a certain distance a positive one manifests due to the presence of YIG/GGG interface. The overall µmprofile varies with T14–16,25–27. Furthermore, we found a second sign change for the d=0.5µm sample at 3 K, which is unclear at this moment. We now focus on the magnetic field Bdependent features of V2ω. Figure 2(b) shows the Bdependence of λat various Ts obtained by fitting Eq. (1) to VnlSSE(d). At 300 K, λdecreases with increasing Bby 30 % up to 3 T [from λ=6.8µm at B=0.18 T to 4 µm at 3 T, see blue filled circles in Fig. 2(b)]. A similar field-induced decrease of λhas been observed in the time-resolved LSSE28, nlSSE29, and electrically excited magnon transport experiment29at room temperature. On the other hand, at lower Ts,λwas found to be less sensitive to B [see Fig. 2(b)]. To further investigate the effect of high Bon the nlSSE, we applied larger magnetic fields up to 10 T. Figure 2(c) shows a typical V2ω−Bresult for |B|<10 T in the d=1.5µm sample with I=100 µA at 300 K. High B-induced suppression of V2ωis clearly observed. In Fig. 2(d) we plot the degree of B-induced V2ωsuppression up to 8 T, defined as δ=100×/parenleftbigg 1−V8T 2ω V0.18T 2ω/parenrightbigg (2) as a function of Tfor the d=0.5, 1.5, 8.0, and 15 µm sam- ples. At 300 K, all the samples show the substantial high B- induced V2ωreduction; 65 % <δ<75 % for the d=1.5, 8.0, and 15 µm samples and δ=39 % for the d=0.5µm sam- ple. For the d=8.0 and 15 µm samples, with decreasing T, δgradually decreases in the range of 20 K <T<300 K and slightly increases below 20 K. For the d=0.5 and 1.5 µm samples, more complicated Tdependences were observed, which may be related to the non-monotonic Tresponses of3 (a) 0 50 100 I (µA)2 1S (kΩ/A) 3 d = 2 µm 8 µmT = 3 K B = 0.18 T 9 µmLinear regime 10 5 B (T)02 1S (kΩ/A) 3 I = 100 µA 10 µA 5 µA 3 µA50 µAd = 2.0 µm T = 3 K(c) (b) 2 1S (kΩ/A) 3 20 10 2 T (K)0d = 8.0 µm B = 0.5 TLinear Non-linear (I = 100 µA) 100 40 20 60 80 0 δ (%) T (K) d = 8.0 µm 20 10 2(d) Linear Non-linear LSSE (I = 100 µA) FIG. 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 the 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 sample 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) and 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% confidence level ( ±s.d.). V2ωas shown in Fig. 2(a). The T−δbehavior above 20 K for thed=8.0 and 15 µm samples qualitatively agrees with the previous LSSE result in Pt/YIG-bulk systems8,30. However, below 20 K, the present nlSSE and previous LSSE results are totally different; δof the LSSE becomes more outstanding with decreasing Tand reaches δ∼100% at∼3 K21, much greater than the present nlSSE results. Significantly, we found that the disagreement at low tem- peratures is relevant to the applied current intensity I. So far, the nlSSE experiments were carried out with I=100µA. Be- low 20 K, however, V2ωturned out to deviate from the I2scal- ing in this Irange. To see this, we introduce the normalization factor, S=V2ω I2. (3) IfV2ωis proportional to I2,Skeeps a constant with I, which was indeed confirmed above 20 K for I<100µA. Figure 3(a) shows the Idependence of Sat 3 K at the low Bof 0.18 T for thed=2,8, and 9 µm samples. Stakes almost the same value for I/lessorsimilar5µA [see the gray colored area in Fig. 3(a)], but for I/greaterorsimilar5µA,Sdecreases with increasing I. We refer the former region to the linear regime ( V2ω∝I2), while the lat- ter to the non-linear regime ( V2ω/negationslash∝I2). In Fig. 3(b), we plot theTdependence of Sin the linear and non-linear regimes at B=0.5 T for the d=8µm sample. The difference in Sbe- tween the linear and non-linear regimes becomes significant with decreasing T, and at 3 K Sin the linear regime is about 4 times greater than that in the non-linear regime. Important ly, theBdependence of V2ωandδalso vary between the linear and non-linear regimes. In Fig. 3(c), we show representativ e results on V2ωversus Bwith several Ivalues at 3 K for the d=2.0µm sample. In the linear regime (for I=3µA), clear B-induced V2ωsuppression was observed ( δ=78 %). By in- creasing Iand entering into the non-linear regime, however, theB-induced V2ωreduction becomes less visible and, when I=100µA,V2ωis almost flat against B(δ=−0.1 %). In Fig. 3(d), we summarize the δvalues as a function of Tobtained in the linear (red filled circles) and non-linear (blue filled cir- cles) regimes for the d=8.0µm sample and compare them to the previous LSSE result (gray filled triangles)21. Interest- ingly, the Tdependence of δfor the nlSSE agrees well withthat for the LSSE. The matching of the T−δresults in the low- Trange be- tween the nlSSE in the linear regime and the LSSE indicates that the same mechanism governs the B-induced suppression. In Ref. 8 and 21, the Tdependence of δfor the LSSE at lowTs was well reproduced based on a conventional LSSE theory in which the effect of the Zeeman-gap opening in a magnon dispersion ( ∝gµBB, where gis the g-factor and µBis the Bohr magneton) was taken into account; the competition between thermal occupation of the magnon mode relevant for the LSSE (whose energy is of the order of kBT) and the Zee- man gap ( gµBB) dominates the B-induced LSSE reduction. When kBT≪gµBB(≈10 K at 8 T), magnons cannot be ther- mally excited, leading to the suppression of the LSSE (see Fig. 3(d)). Our results indicate that the same scenario is va lid also for the nlSSE in the linear regime. Finally, we discuss the non-linear feature of the nlSSE. Both the Sandδvalues of the nlSSE in the non-linear regime gradually increase with decreasing T[see Figs. 3(b) and 3(d)]. However, their increasing rates are much smaller than those for the linear regime; both Sandδat 3 K in the non-linear regime are ∼4 times smaller than those at the same Tfor the linear regime and also comparable to those at 12 K for the linear regime. These results suggest that the energy sca le of magnons driving the nlSSE in the non-linear regime at 3 K may be much higher than the thermal energy kBTat 3 K and the Zeeman energy gµBBat 8 T. We note that, in the non-linear regime, the system temperature at least remains un- changed during the measurements, indicting that temperatu re rise due to the Joule heating is negligible. Furthermore, we found that, in the non-linear regime of I=100 µA, the in- tensity of magnon-polaron dips at 3 K at B=2.5 T (9.2 T) is smaller (larger) than that in the linear regime of I=3µA at the same T[see the dip structures marked by blue (red) trian- gles in Fig. 3(c)]. Here, the dip at the low B(high B) originates from the spin currents carried by hybridized magnon −TA- phonon (magnon −LA-phonon) modes with the fixed energy ofEMTA≈6 K ( EMLA≈26 K). The dip intensity should thereby be maximized when the magnon mode at the energy ofEMTA(EMLA) is most significantly occupied under the con- dition of kBT≈6 K (26 K), and apart from this temperature the intensity of magnon-polaron dip decreases. Therefore,4 the small (large) magnon-polaron dip at B=2.5 T (9.2 T) at 3 K in the non-linear regime also indicates the over occu- pation of high-energy magnons than that expected from the thermal energy kBTat 3 K, as with the Sandδresults dis- cussed above. Future work should address the origin of such high-nonequilibrium state realized in this regime. In summary, we systematically investigated the nonlocal spin Seebeck effect (nlSSE) in the lateral Pt/YIG/Pt system s as functions of separation distance ( d), magnetic field ( B), temperature ( T), and excitation current ( I). We found that below 20 K, the nlSSE voltage V2ωdeviates from the conven- tional I2scaling for I/greaterorsimilar5µA. In this non-linear regime, the amplitudes of V2ωandB-induced signal reduction δbecome smaller than those in the linear regime, where V2ω∝I2and I<5µA. In the linear regime, the Tdependence of δof the nlSSE agrees well with that of the longitudinal SSE (LSSE), which can be attributed to the suppression of magnon excita- tion by the Zeeman effect. Our results provide an important clue in unraveling the B-induced suppression of the nlSSE and useful information on the non-linear effect in nonlocal spi n transport at low temperatures. We thank G. E. W. Bauer, B. J. van Wees, L. J. Cornelis- sen, J. Shan, T. Kuschel, F. Casanova, J. M. Gomez-Perez, S. Takahashi, Z. Qiu, Y . Chen, and R. Yahiro for fruitful dis- cussion, and K. Nagase for technical help. This work is a part of the research program of ERATO Spin Quantum Rec- tification Project (No. JPMJER1402) from JST, the Grant- in-Aid for Scientific Research on Innovative Area Nano Spin Conversion Science (No. JP26103005), the Grant-in-Aid for Scientific Research (S) (No. JP19H05600), and Grant-in- Aid for Research Activity Start-up (No. JP19K21031) from JSPS KAKENHI, JSPS Core-to-Core program, the Interna- tional Research Center for New-Concept Spintronics Device s, World Premier International Research Center Initiative (W PI) from MEXT, Japan. K.O. acknowledges support from GP- Spin at Tohoku University. 1G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Nat. Mater. 11, 391 (2012). 2K. Uchida, H. Adachi, T. Ota, H. Nakayama, S. Maekawa, and E. S aitoh, Appl. Phys. Lett. 97, 172505 (2010). 3K. Uchida, M. Ishida, T. Kikkawa, A. Kirihara, T. Murakami, a nd E. Saitoh, J. Phys. Condens. 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2019-10-09

We report the nonlocal spin Seebeck effect (nlSSE) in a lateral configuration of Pt/Y$_3$Fe$_5$O$_{12}$(YIG)/Pt systems as a function of the magnetic field $B$ (up to 10 T) at various temperatures $T$ (3 K < $T$ < 300 K). The nlSSE voltage decreases with increasing $B$ in a linear regime with respect to the input power (the applied charge-current squared $I^2$). The reduction of the nlSSE becomes substantial when the Zeeman energy exceeds thermal energy at low temperatures, which can be interpreted as freeze-out of magnons relevant for the nlSSE. Furthermore, we found the non-linear power dependence of the nlSSE with increasing $I$ at low temperatures ($T$ < 20 K), at which the $B$-induced signal reduction becomes less visible. Our experimental results suggest that in the non-linear regime high-energy magnons are over populated than those expected from the thermal energy. We also estimate the magnon spin diffusion length as functions of $B$ and $T$.

Magnetic field dependence of the nonlocal spin Seebeck effect in Pt/YIG/Pt systems at low temperatures

1910.04046v1

Theory of the magnon Kerr e ect in cavity magnonics Guo-Qiang Zhang,1, 2Yi-Pu Wang,2and J. Q. You2, 1Quantum Physics and Quantum Information Division, Beijing Computational Science Research Center, Beijing 100193, China 2Interdisciplinary Center of Quantum Information and Zhejiang Province Key Laboratory of Quantum Technology and Device, Department of Physics and State Key Laboratory of Modern Optical Instrumentation, Zhejiang University, Hangzhou 310027, China (Dated: March 12, 2019) We develop a theory for the magnon Kerr e ect in a cavity magnonics system, consisting of magnons in a small yttrium iron garnet (YIG) sphere strongly coupled to cavity photons, and use it to study the bistability in this hybrid system. To have a complete picture of the bistability phenomenon, we analyze two di erent cases in driving the cavity magnonics system, i.e., directly pumping the YIG sphere and the cavity, respectively. In both cases, the magnon frequency shifts due to the Kerr e ect exhibit a similar bistable behavior but the corresponding critical powers are di erent. Moreover, we show how the bistability of the system can be demonstrated using the transmission spectrum of the cavity. Our results are valid in a wide parameter regime and generalize the theory of bistability in a cavity magnonics system. I. INTRODUCTION Owing to the fundamental importance and promising ap- plications in quantum information processing, hybrid quan- tum systems consisting of di erent subsystems have recently drawn considerable attention [1, 2]. Among them, the spin ensemble in a single-crystal yttrium iron garnet (YIG) sam- ple coupled to a cavity mode was theoretically proposed [3– 5] and experimentally demonstrated [6–11] in the past few years. In contrast to spin ensembles in dilute paramagnetic systems, e.g., nitrogen-vacancy centers in diamond [12], the ferromagnetic YIG material possesses a higher spin density (2:11022cm3) and essentially is completely polarized below the Curie temperature ( 559 K) [13]. It is found that a strong coupling between the microwave cavity mode and the spin ensemble in a small YIG sample with a low damp- ing rate can be achieved [6–10], which is a challenging task for spin ensembles in paramagnetic materials. In this cav- ity magnonics system, many exotic phenomena, such as cav- ity magnon-polaritons [14–16], magnon Kerr e ect [17–19], bidirectional microwave-optical conversion [20], ultrastrong coupling [21, 22], magnon dark modes [23], cavity spintron- ics [24, 25], optical manipulation of the system [26], synchro- nized spin-photon coupling [27], strong interlayer magnon- magnon coupling [28], cooperative polariton dynamics [29] and non-Hermitian physics [30–32] have been investigated. Moreover, the coupling of magnons to other quantum sys- tems, e.g., the superconducting qubit [33, 34], phonons [35] and optical whispering gallery modes [36–42] was also imple- mented. The cavity magnon polaritons are new quasiparticles re- sulting from the strong coupling of magnons to cavity pho- tons [14–16]. In Ref. [17], the bistability of the cavity magnon polaritons was experimentally demonstrated by directly driv- ing a small YIG sphere placed in a microwave cavity, and the conversion from magnetic to optical bistability was also observed. However, a special case was focused on there by jqyou@zju.edu.cnconsidering the situation that only the lower-branch polari- tons were much generated [17]. In fact, to have a complete picture of the bistability phenomenon, one needs to study the more general case with both lower- and upper-branch polari- tons considerably generated and also consider the coupling between them. Moreover, one can use a drive tone supplied by a microwave source to pump the cavity [43] instead of the YIG sphere and tune the drive-field frequency from on-resonance to far-o -resonance with the magnons. These important issues were not studied in Ref. [17]. In this work, we develop a theory to study the bistability of the cavity magnonics system in a wide parameter regime, which applies to the di erent cases mentioned above. In sect. II, we present a theoretical model to describe the cav- ity magnonics system. This hybrid system consists of a mi- crowave cavity strongly coupled to the magnons in a small YIG sphere which is magnetized by a static magnetic field. In comparison with the model of two strongly-coupled harmonic oscillators [7–9], there is an additional Kerr term of magnons in the Hamiltonian of the system, resulting from the magne- tocrystalline anisotropy in the YIG [44, 45]. In sect. III, we develop the theory for the bistability of the cavity magnonics system. We analyze two di erent cases of driving the hybrid system corresponding to two experimental situations [18, 43], i.e., directly pumping the YIG sphere and the cavity, respec- tively. In both cases, the magnon frequency shifts due to the Kerr e ect exhibit a similar bistable behavior but the corre- sponding critical powers are di erent. Here the positive (neg- ative) Kerr coe cient corresponds to the blue-shift (red-shift) of the magnon frequency. When the cavity and Kittel modes are on-resonance (o -resonance), the critical power for driv- ing the cavity is approximately equal to (much larger than) the critical power for driving the YIG sphere. Finally, in sect. IV, we derive the transmission coe cient of the cavity with the small YIG sphere embedded and show how the bistability of the system can be demonstrated via the transmission spectrum of the cavity. Our results bring the studies of cavity magnonics from the linear to nonlinear regime. Compared with other hybrid sys- tems, the cavity magnonics system owns good tunabilities with, e.g., the magnon frequency, the cavity-magnon interac-arXiv:1903.03754v1 [quant-ph] 9 Mar 20192 tion [31], the drive power, and the drive-field frequency. The easily controllable bistability of the cavity magnonics sys- tem may have promising applications in memories [46, 47], switches [48, 49], and the study of the dissipative phase tran- sition [50, 51]. In the future, more nonlinear phenomena such as auto-oscillations and chaos [52] may be explored by us- ing an even stronger drive field and a smaller YIG sphere to enhance the nonlinearity of the cavity magnonics system. II. THE HAMILTONIAN OF THE SYSTEM As schematically shown in Fig. 1, we study a system con- sisting of a small YIG sphere (with the order of submilimeter or milimeter in size) coupled to a three-dimensional (3D) rect- angular microwave cavity via the magnetic field of the cavity mode. Here we focus on the case in which the YIG sphere is uniformly magnetized to saturation by a bias magnetic field B0=B0ezin the z-direction, where ei,i=x;y;z, are the unit vectors in the rectangular coordinate system. This cor- responds to the Kittel mode of spins in the YIG sphere, i.e., the uniform procession mode with homogeneous magnetiza- tion [18]. In this mode, the Heisenberg-type exchange cou- pling and the dipole-dipole interaction between spins can be neglected since their contributions to the Hamiltonian of the system become constant in the considered long-wavelength limit [53]. For instance, the Heisenberg interaction between any two neighboring spins becomes Ji jsi
sj=Ji js2(i.e., a con- stant) for the Kittel mode, because all spins uniformly precess in phase together. Here Ji jis the exchange coupling strength andsi(sj) is the spin operator of the ith (jth) spin in the YIG sphere with the spin quantum number s=~=2. As given in Appendixes A and B, this hybrid system can be described us- ing a nonlinear Dicke model (setting ~=1) Hs=!caya B0Sz+DxS2 x+DyS2 y+DzS2 z +gs(S++S)(ay+a);(1) where aandayare the annihilation and creation operators of the cavity mode at the frequency !c, =geB=~is the gy- romagnetic ration with the g-factor geand the Bohr magne- tonB,S=P jsj(Sx;Sy;Sz) and SSxiSyare the macrospin operators with the summationP jover all spins in the YIG sphere, and gsdenotes the coupling strength between each single spin and the cavity mode. The nonlinear terms DxS2 x+DyS2 y+DzS2 zin Eq. (1) originate from the magne- tocrystalline anisotropy in the YIG [44, 45] and their coe - cients Direly on the crystallographic axis of the YIG, along which the external magnetic field B0is applied. When the crystallographic axis aligned along B0is [110], the nonlinear coecients Diread (see Appendix A) Dx=30Kan 2 2M2Vm;Dy=90Kan 2 8M2Vm;Dz=0Kan 2 2M2Vm; (2) where0is the vacuum permeability, Kan(>0) is the first- order anisotropy constant of the YIG, Mis the saturation mag- netization, and Vmis the volume of the YIG sample. The YIG YIG Sphere YIG Sphere3D cavity B0hc x yzhcmax minΩdωd ωpεpFIG. 1. Upper panel: schematic diagram of a YIG sphere coupled to a 3D microwave cavity. Lower panel: the simulated magnetic- field distribution of the fundamental mode of the cavity, where the magnetic-field amplitude and direction are indicated by the colors and blue arrows, respectively. The YIG sphere, which is magnetized to saturation by a bias magnetic field B0aligned along the z-direction, is mounted near the cavity wall, where the magnetic field hcof the cavity mode is the strongest and polarized along x-direction to ex- cite the magnon mode in YIG. Either the cavity mode or the magnon mode is driven by a microwave field with frequency !dand Rabi frequency d. A weak probe field with frequency !pand its cou- pling strength "pto the cavity mode is also applied for measuring the transmission spectrum of the cavity. sphere is here required to be in the macroscopic regime to contain a su cient number of spins. Usually, the diameter of the YIG sphere used in the experiment varies from 0.1 mm to 1 mm. Directly pumping the YIG sphere with a microwave field of the frequency !d, the interaction Hamiltonian is (see Ap- pendix B) Hd= s(S++S)(ei!dt+ei!dt); (3) where sis the drive-field Rabi frequency. In the experiment, a drive coil near the YIG sample goes out of the cavity through one port of the cavity connected to a microwave source [18]. Also, a probe field at frequency !pacts on the input port of the cavity, which can be described by the Hamiltonian Hp="p(ay+a)(ei!pt+ei!pt); (4) where"pis the coupling strength between the cavity and the probe field. In the experiment, compared with the drive field, the probe tone is usually extremely weak, and the probe-field frequency!pis tuned to be o resonance with the drive-field frequency!d, so as to avoid interference between them [17]. Now, we can write the total Hamiltonian H=Hs+Hd+Hp3 of the hybrid system in Fig. 1 as H=!caya B0Sz+DxS2 x+DyS2 y+DzS2 z +gs(S++S)(ay+a)+ s(S++S)(ei!dt+ei!dt) +"p(ay+a)(ei!pt+ei!pt): (5) Using the Holstein-Primako transformation [54], S+=p 2Sbybb; S=byp 2Sbyb; Sz=Sbyb;(6) we can convert the macrospin operators to the magnon opera- tors, where by(b) is the magnon creation (annihilation) opera- tor,S=sVmsis the spin quantum number of the macrospin, ands=2:11022cm3is the net spin density of the YIG sphere. Under the condition of low-lying excitations with hbybi=2S1,p 2Sbybcan be expanded, up to the first order of byb=2S, asp 2Sbybp 2S(1byb=4S), so S+p 2S 1byb 4S b; Sp 2S by 1byb 4S :(7) Substituting the expression Sz=Sbybin Eq. (6) and Eq. (7) into Eq. (5), as well as neglecting the constant terms and the fast oscillating terms via the rotating-wave approximation (RWA) [55], we can reduce the total Hamiltonian Hto H=!caya+!mbyb+Kbybbyb+gm 1byb 4S! (ayb+aby) + d 1byb 4S! (byei!dt+bei!dt) +"p(ayei!pt+aei!pt); (8) where !m= B0+130ssKan 2 8M2(9) is the angular frequency of the magnon mode, K=130Kan 2 16M2Vm(10) is the Kerr nonlinear coe cient, gmp 2S gsis the collec- tively enhanced magnon-photon coupling strength and dp 2S sis the Rabi frequency. However, when the crystallographic axis aligned along B0 is [100], the nonlinear coe cients Diin Eq. (2) become (see Appendix A) Dx=Dy=0;Dz=0Kan 2 M2Vm: (11) 0306090120150gm/2π (MHz)(a) 0.0 0.2 0.4 0.6 0.8 1.010-1100101102|K/2π| (nHz) d (mm) [100] [110](b)FIG. 2. (a) The coupling strength gmwith gs=2=39 mHz and (b) the Kerr coe cient K(log scale) as a function of the diameter dof the YIG sphere. The black solid (red dashed) curve in (b) cor- responds to the case with the crystalline axis [100] ([110]) aligned along B0. Other parameters are 0Kan=2480 J=m3,M=196 kA /m, and =2=28 GHz /T. In the RWA, the Hamiltonian Hin Eq. (5) is also converted to the same form as in Eq. (8) using Eq. (7) and the expression Sz=Sbybin Eq. (6), but the magnon frequency is !m= B020ssKan 2 M2; (12) and the Kerr coe cient is K=0Kan 2 M2Vm: (13) It is worth noting that the magnon frequency !mis irrelevant to the volume Vmof the YIG sphere, but the Kerr coe cient is inversely proportional to Vm, i.e., K/V1 m. Thus, the Kerr e ect of magnons can become important for a small YIG sphere. Moreover, the Kerr coe cient becomes positive (neg- ative) when the crystallographic axis [100] ([110]) of the YIG is aligned along the static field B0. In the experiment, instead of using a drive tone supplied by a microwave source to directly pump the YIG sphere, one can also apply a drive field with frequency !ddirectly on the cavity [43]. In this case, the total Hamiltonian of the hybrid system under the RWA is written as H=!caya+!mbyb+Kbybbyb+gm 1byb 4S! (ayb+aby) + d(ayei!dt+aei!dt)+"p(ayei!pt+aei!pt): (14)4 Note that in both cases, we use the same symbols dand!d for simplicity. Here we estimate the collective coupling strength gmand the Kerr coe cient K. As shown in Fig. 2, we plot gm and Kversus the diameter dof the YIG sphere, where we choose the experimentally obtained single-spin coupling strength gs=2=39 mHz [7]. From Fig. 2, it can be seen that when the diameter dis reduced from 1 mm to 0.1 mm (the usual size of the YIG sphere used in experiments), the cou- pling strength gmdecreases one order of magnitude but the Kerr coe cient Kincreases from 0.05 nHz to 100 nHz, i.e., a three orders of magnitude increase. Thus, it is vital to choose a YIG sphere of suitably small size, so as to have strong non- linear e ect of magnons but still maintain the hybrid system in the strong coupling regime. III. THE NONLINEAR EFFECT ON THE HYBRID SYSTEM A. Pump the YIG sphere When directly pumping the YIG sphere with a drive field, considerable magnons are usually generated in the YIG sphere. The magnon number operator bybcan be expressed as a sum of the mean value hbybiand the fluctuation byb, i.e.,byb=hbybi+byb, so bybbyb=(hbybi+byb)(hbybi+byb) =(hbybi)2+2hbybibyb+(byb)2:(15) When a considerable number of magnons are generated in the YIG sphere by the drive field, i.e., hbybihbybi, we can neglect the high-order fluctuation term and have bybbyb(hbybi)2+2hbybibyb =(hbybi)2+2hbybibyb:(16) Under this mean-field approximation (MFA), the Hamiltonian in Eq. (8) can then be written as H=!caya+(!m+2Khbybi)byb + 1hbybi 4S! gm(ayb+aby) + 1hbybi 4S! d(byei!dt+bei!dt) +"p(ayei!pt+aei!pt):(17) Note that the generated magnons may yield an appreciable shift
m=2Khbybito the magnon frequency [17, 18]. How- ever, if the drive field is not too strong, the condition hbybi 2Scan easily be satisfied owing to the very large number of spins in the YIG sphere. Therefore, we can take the approxi- mation 1hbybi=(4S)1 in Eq. (17), and then the Hamilto- nian becomes H=!caya+(!m+
m)byb+gm(ayb+aby) + d(byei!dt+bei!dt)+"p(ayei!pt+aei!pt):(18)With the Heisenberg-Langevin approach [55], we can de- scribe the dynamics of the coupled hybrid system by the fol- lowing quantum Langevin equations: da dt=i(!cic)aigmbi"pei!pt+p 2cain; db dt=i(!m+
mi m)bigmai dei!dt+p 2 mbin; (19) wherec=i+o+intis the decay rate of the cavity mode, withi(o) being the decay rate of the cavity mode due to the input (output) port and intbeing the intrinsic decay rate of the cavity mode, mis the damping rate of the Kittel mode, andainandbinare the input noise operators related to the cavity and Kittel modes, whose mean values are zero, i.e., haini=hbini=0. These input noise operators result from the respective environments of the cavity and Kittel modes, which include both quantum noise and thermal noise. If we write a=hai+aandb=hbi+b, wherehai(hbi) is the expectation value of the operator a(b) anda(b) is the corre- sponding fluctuation, it follows from Eq. (19) that the steady- state valueshaiandhbisatisfy dhai dt=i(!cic)haiigmhbii"pei!pt; dhbi dt=i(!m+
mi m)hbiigmhaii dei!dt:(20) Experimentally, the drive field is much stronger than the probe field, i.e.,"p d, so the probe field can be treated as a perturbation. We assume that the expectation values haiand hbican be written as hai=A0ei!dt+A1ei!pt; hbi=B0ei!dt+B1ei!pt;(21) where the amplitudes A0andB0are the expectation values of operators aandbin the absence of the probe field, and the amplitudes A1andB1result from the perturbation (i.e., probe field). A1andB1are significantly smaller than A0andB0. In this case, the magnon frequency shift
mcan be written as
m=2KjB0j2. At the steady states for both A0andB0 (A1andB1),dA0=dt=0 and dB0=dt=0 (dA1=dt=0 and dB1=dt=0). Then, we have (cic)A0+gmB0=0; (m+
mi m)B0+gmA0+ d=0;(22) and (!c!p)icA1+gmB1+"p=0;(!m+
m!p)i mB1+gmA1=0;(23) wherec(m)!c(m)!dis the frequency detuning of the cavity mode (Kittel mode) relative to the drive field. The first equa- tion in Eq. (22) can be expressed as A0=gmB0=(cic). By inserting this expression of A0into the second equation in Eq. (22), we obtain (0 m+
mi 0 m)B0+ d=0; (24)5 0 100 200 30005101520 (a) ∆=0, K>0 0 100 200 30005101520 (b) ∆=3gm, K>0 0 100 200 300-20-15-10-50 (c) ∆=0, K<0 0 100 200 300-20-15-10-50 (d) ∆=3gm, K<0 ∆/2π (MHz) /2π (MHz) P (mW) P (mW)m ∆m d d FIG. 3. The magnon frequency shift
mversus the drive power Pd for di erent
andK, where
=!c!mis the frequency de- tuning of the cavity from the magnon. (a) Frequency shift
mver- susPdwhen
=0 and K>0. Herem=2=36:2 MHz for the (black) solid curve, m=2=35 MHz for the (red) dashed curve, and m=2=34 MHz for the (blue) dotted curve. (b) Frequency shift
mversus Pdwhen
= 3gmandK>0. Herem=2=9 MHz for the (black) solid curve, m=2=4 MHz for the (red) dashed curve, andm=2=1 MHz for the (blue) dotted curve. (c) Frequency shift
mversus Pdwhen
=0 and K<0. Herem=2=43 MHz for the (black) solid curve, m=2=45 MHz for the (red) dashed curve, andm=2=47 MHz for the (blue) dotted curve. (d) Fre- quency shift
mversus Pdwhen
= 3gmand K<0. Here m=2=15 MHz for the (black) solid curve, m=2=18 MHz for the (red) dashed curve, and m=2=21 MHz for the (blue) dotted curve. The constant is c=(2)3=2 MHz3=mW in both (a) and (b), andc=(2)3=2 MHz3=mW in both (c) and (d). Other parameters aregm=2=40 MHz, and c=2= m=2=2 MHz. where the e ective frequency detuning 0 mand the e ective damping rate 0 mof the Kittel mode are given, respectively, by 0 m=mc; 0 m= m+c; (25) with =g2 m=(2 c+2 c): (26) Using Eq. (24) and its complex conjugate expression, we ob- tain 0 m+
m
2+ 0 m2
mcPd=0; (27) where 2 Kj dj2=cPd, with Pdbeing the drive power and c a coe cient characterizing the coupling strength between the drive field and the Kittel mode. Note that Eq. (27) is a cubic equation for the magnon fre- quency shift
m. Under specific parameter conditions,
mhas two switching points for the bistability, at which there must be dPd=d
m=0, i.e., 3
2 m+40 m
m+0 m2+ 0 m2=0: (28) 9.96 9.98 10.00 10.02 10.0405101520 (a) K>0 ∆m/2π (MHz) ωm/2π (GHz) Pd=80 mW Pd=140 mW Pd=200 mW 9.98 10.00 10.02 10.04 10.06-20-15-10-50 ωm/2π (GHz)∆m/2π (MHz)(b) K<0 Pd=80 mW Pd=140 mW Pd=200 mWFIG. 4. The magnon frequency shift
mversus!mfor di erent val- ues of the drive power Pdin the cases of (a) K>0 and (b) K<0. Here Pd=80 mW for the (black) solid curve, Pd=140 mW for the (red) dashed curve, and Pd=200 mW for the (blue) dotted curve. The constant is c=(2)3=2 MHz3=mW in (a) and c=(2)3= 2 MHz3=mW in (b);!c=2=10 GHz and c=2=35 MHz in both (a) and (b). Other parameters are the same as in Fig. 3(a). According to the root discriminant of the quadratic equation with one unknown, if Eq. (28) has two real roots (correspond- ing to the two switching points), 0 mand 0 mmust satisfy the relation 40 m212 0 m2>0, i.e., 0 m<p 3 0 m;K>0; 0 m>p 3 0 m;K<0:(29) When 40 m212 0 m2=0, Eq. (28) has only one real so- lution and the two switching points coalesce to one point, which means that the bistability disappears. In the case of 40 m212 0 m2=0, the corresponding power Pd, called the critical power, is given by Pm= +()8p 3 0 m3 9c; (30) with cbeing positive (negative) for K>0 (K<0). For 40 m212 0 m2<0, Eq. (28) has no real solution and the magnon frequency shift
mincreases monotonically with the drive power Pd. In Fig. 3(a), the magnon frequency shift
mversus the driv- ing power Pdis plotted for several di erent values of detuning mwhen
=0 and K>0, where
!c!mis the frequency6 detuning of the cavity from the magnon. In a certain parame- ter regime,
mexhibits a bistable behavior. It is clearly shown that the value of the detuning mbetween the Kittel mode and the drive field is crucial for the bistability of
m. Moreover, the frequency shift
mversus the driving power Pdin the case of
=3gmandK>0 is shown in Fig. 3(b) for di erent values ofm. We also see hysteresis loops. In both the on-resonance and o -resonance cases, we further study the relationship be- tween the magnon frequency shift
mand the drive power Pd, as shown in Figs. 3(c) and 3(d), when K<0. We also ob- serve the similar bistability, but the magnon frequency shift is negative because the Kerr coe cient is negative in this case. From the cubic equation in Eq. (27), we can further study the magnon frequency shift
mversus the e ective frequency detuning0 m. In the experiment, 0 mcan be tuned by either sweeping the magnon frequency !m(i.e., the bias magnetic field B0) or sweeping the drive-field frequency !d. Because
mhas similar behaviors when sweeping !mor!d, here we only focus on the magnon frequency shift
mversus!m. Fig- ure 4(a) displays the magnon frequency shift
mversus!mfor dierent values of the drive power Pdwith a fixed !dwhen K>0. With a small drive power,
mdepends nonlinearly on!mbut has no bistable behavior [see the black solid curve in Fig. 4(a)]. When increasing the drive power Pd,
mversus !mshows the bistability and the hysteresis-loop area increases with Pd[see the red dashed curve and the blue dotted curve in Fig. 4(a)]. In the case of K<0, we plot
mversus!min Fig. 4(b). With appropriate parameters, there is also the bista- bility but
mis negative. B. Pump the cavity When a microwave field is applied to directly pump the cav- ity rather than the YIG sphere, linearizing the nonlinear terms via the MFA, the total Hamiltonian in Eq. (14) becomes H=!caya+(!m+
m)byb+gm(ayb+aby) + d(ayei!dt+aei!dt)+"p(ayei!pt+aei!pt);(31) where we have also used the approximation 1 hbybi=(4S) 1. When directly driving the cavity, the dynamics of the cou- pled hybrid system follows the quantum Langevin equations: da dt=i(!cic)aigmbi dei!dti"pei!pt+p 2cain; db dt=i(!m+
mi m)bigma+p 2 mbin: (32) In this case, the evolution equation of the expectation value hai(hbi) is given by dhai dt=i(!cic)haiigmhbii dei!dti"pei!pt; dhbi dt=i(!m+
mi m)hbiigmhai: (33) -400 -200 0 200 400 ∆/2π (MHz) Pm Pc 100101102103Critical power (mW)FIG. 5. The critical powers PmandPc(log scale) versus the detun- ing
when the crystalline axis [100] is aligned along the external magnetic field B0. Other parameters are the same as in Fig. 3(a). Substituting Eq. (21) into Eq. (33), A1andB1also satisfy Eq. (23), but the steady-state equations of A0andB0become (cic)A0+gmB0+ d=0; (m+
mi m)B0+gmA0=0:(34) Eliminating A0in Eq. (34), we have (0 m+
mi 0 m)B0 e=0; (35) where e=gm d=(cic) is the e ective driving strength on the YIG sphere, which depends not only on the Rabi fre- quency dbut also on the coupling strength gmand the fre- quency detuning cbetween the cavity mode and the drive field. From Eq. (35), it is straightforward to obtain a cubic equation for
m, 0 m+
m
2+ 0 m2
mcPd=0; (36) withgiven in Eq. (26). Comparing Eq. (36) with Eq. (27), Pdis the e ective drive power on the YIG sphere. By substi- tuting the drive power Pdin Eq. (27) with the e ective drive powerPd, the bistable condition in Eq. (29) is still valid, but the critical power PcforK>0 (K<0) now becomes PcPm = +()8p 3 0 m3 9c: (37) Also, cis positive (negative) when K>0 (K<0). Because the values of Pm(Pc) are approximately equal for a specific value of
in both cases of aligning the crystalline axes [100] and [110] of the YIG along the external magnetic field B0, we only study the critical powers PmandPcver- sus the detuning
when the axis [100] is aligned along B0 (K>0). As shown in Fig. 5, PmandPcare approximately equal in the near-resonance region jgm=
j>1, but Pcis much larger than Pmin the dispersive regime jgm=
j1. The un- derlying physics is that in the case of j
jgm, the magnon and cavity are nearly decoupled, so directly driving the cavity has weak influence on the magnon subsystem and then it be- comes hard to observe the nonlinear e ect in the hybrid sys- tem. In the experiment, it is di cult to apply an extremely7 (a) 9.909.9510.0010.0510.10ωp/2π (GHz) 9.909.9510.0010.0510.10ωp/2π (GHz)(b) 9.90 9.95 10.00 10.05 ωm/2π (GHz)9.90 9.95 10.00 10.05 ωm/2π (GHz)9.90 9.95 10.00 10.05 10.10 ωm/2π (GHz)(d)(c) (e) (f)|S21(ωp)|2 (dB)-80-400 FIG. 6. Transmission spectrum of the cavity-magnon system versus the probe-field frequency !pand the magnon frequency !mwhen the drive-field frequency is fixed at c=35 MHz. (a) The transmission spectrum when Pd=0. (b) The transmission spectrum when Pd=80 mW andK>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. (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 directions and the switching points of the bistability are indicated, respectively, by the black arrows and the vertical black dashed lines. Here we choosei=2=o=2=0:7 MHz, and other parameters are the same as in Fig. 4. strong microwave field to pump a cavity. Therefore, in the dispersive regime, it is better to directly pump the magnon to observe the nonlinear e ect of the hybrid system. In the case of aligning the crystalline axis [110] along B0(K<0), the above conclusions are still valid. IV . TRANSMISSION SPECTRUM In the experiment, one can probe the bistability via the transmission spectrum of the cavity. In this section, we show the eect of the magnon frequency shift
m(due to the Kerr nonlinearity) on the transmission spectrum of the cavity. From Eq. (23), the amplitude A1of the cavity field due to the probe field reads A1=i"p i(!c!p)+c+ (!p); (38) where (!p)=g2 m i(!m+
m!p)+ m: (39) According to the input-output theory [55], because there is no input field on the output port, the output of the cavity field from the output port is ha(out) pi=p 2ohai=p 2oA0ei!dt+p 2oA1ei!pt;(40)where the first (second) term of the output field ha(out) piis due to the drive (probe) field. The probe field to be input into the cavity via the input port can be written as [55] ha(in) pi= i"pei!pt=p2i. Then, we obtain the transmission coe cient S21(!p) of the cavity at frequency !p, S21(!p)p2oA1i"p=p2i
=2pio i(!c!p)+c+ (!p);(41) where the self-energy (!p), as given in Eq. (39), includes the contribution from the magnon frequency shift
m. Note that the transmission coe cient given in Eq. (41) is valid in both cases of the drive field applied on the YIG sphere and the cavity. Let us consider the case of directly driving the YIG sphere for an example. In Fig. 6, using Eqs. (41) and (27), we plot the transmission spectrum for the cavity magnonics system ver- sus the probe-field frequency !pand the magnon frequency !m(which is related to the bias magnetic field B0) for di er- ent values of the drive power Pdwhen fixing the drive-field frequency!d. The corresponding
mversus!mcan be found in Fig. 4. When the drive field is o , i.e., Pd=0, a pro- nounced avoided crossing of energy levels resulting from the strong coupling between magnons and cavity photons can be observed [see Fig. 6(a)]. Sweeping the magnon frequency !m up and down at Pd=80 mW, we obtain a similar transmis- sion spectrum [Fig. 6(b)] but it looks di erent from Fig. 6(a) at around!m=2=10 GHz, due to the magnon Kerr e ect. We further study the transmission spectrum in the case of K>08 (K<0) in Figs. 6(c) and 6(d) [Figs. 6(e) and 6(f)] when Pd=200 mW. The arrows indicate the sweep directions of the bias magnetic field B0(i.e.,!m) and the vertical dashed lines indicate the switching points of the bistability. Clearly, the transmission spectrum depends on the sweep directions, displaying the bistability of the system. Therefore, one can extract the unique information of the magnon frequency shift
mby measuring the cavity transmission spectrum in the ex- periment. V . DISCUSSIONS AND CONCLUSIONS In our work, the temperature e ect is not explicitly shown. When the frequencies of the cavity mode and the magnon mode are chosen to be a few gigahertz (the usual values of !cand!min the experiment), the numbers of cavity photons and magnons excited by the thermal field are about 1 103 even at the Curie temperature ( 559 K) of the YIG mate- rial [13]. However, when pumping either the YIG sphere or cavity, the pumping field generates magnons and cavity pho- tons up to 11016[17] for observing the bistability in cavity magnonics. Therefore, the approximation of neglecting the temperature e ect is reasonable, and our theoretical predic- tions are valid below the Curie temperature. The bistability of a cavity magnonics system was experi- mentally investigated by directly driving a small YIG sphere coupled to a cavity mode [17] in a special case with only the lower-branch polaritons much generated. However, the theory used in Ref. [17] fails to accurately describe the bistability in the cavity magnonics system when di erent experimental conditions are used (e.g., both lower- and upper-branch po- laritons are considerably generated, the cavity [43] rather than the YIG sphere is directly pumped, and the drive-field fre- quency is swept from on-resonance to far-o -resonance with the magnons). It is the limitation of the theory using the po- lariton basis in Ref. [17], because the coupling between the lower- and upper-branch polaritons is neglected when deriv- ing the equation for bistability. In these more general cases, we can use the theory developed here. In conclusion, we have studied the Kerr-e ect-induced bistability in a cavity magnonics system consisting of a small YIG sphere strongly coupled to a microwave cavity and de- veloped a theory for it which works in a wide regime of the system parameters. We analyze two di erent cases of driv- ing this hybrid system which correspond to the two typical experimental situations [18, 43], i.e., directly pumping the YIG sphere and the cavity, respectively. In both cases, the magnon frequency shifts due to the Kerr e ect exhibit a simi- lar bistable behavior, but the corresponding critical powers are dierent. Specifically, it is shown that directly driving the cav- ity needs a larger critical power than directly driving the YIG sphere when the magnons are o -resonance with the cavity photons. Furthermore, we show how the bistability of the cav- ity magnonics system can be probed using the transmission spectrum of the cavity. Our results provide a complete picture for the bistability phenomenon in the cavity magnonics sys- tem and also generalize the theory of bistability in Ref. [17].ACKNOWLEDGMENTS This work is supported by the National Key Re- search and Development Program of China (Grant No. 2016YFA0301200) and the National Natural Science Foundation of China (Grant Nos. 11774022 and U1530401). Appendix A: The uniformly magnetized YIG sphere As shown in Fig. 1, the YIG sphere used is magnetized to saturation by an externally applied magnetic field B0=B0ez along the z-direction, where ei,i=x;y;z, are the unit vectors along three orthogonal directions. For the magnetized YIG sphere, the internal magnetic field Hinin the YIG sphere is Hin=Hex+Hde+Han; (A1) where the exchange field Hexis caused by the exchange in- teraction, the demagnetization field Hderesults from the mag- netic dipole-dipole interaction, and the anisotropic field Han is induced by the magnetocrystalline anisotropy of the YIG. When Zeeman energy is included, the Hamiltonian of the YIG sphere reads [56] (setting ~=1) Hm=Z VmM
B0d0 2Z VmM
Hind; (A2) where0is the vacuum permeability, Vmis the volume of the YIG sample and M=(Mx;My;Mz) is the magnetization of the YIG sphere. For the uniformly magnetized YIG sphere with a uniform magnetization M, the exchanged field, i.e., the molecular field in Weiss theory, is [44, 45] Hex=M, with the molecu- lar field constant . The induced demagnetizing field is [57] Hde=M=3 for a YIG sphere, but the anisotropic field Han depends on which crystallographic axis of the YIG is aligned along the externally applied static field B0. When the crystal- lographic axis [110] is aligned along B0, the anisotropic field can be written as [58] Han=3KanMx M2ex9KanMy 4M2eyKanMz M2ez; (A3) where we only consider the dominant first-order anisotropy constant Kan(>0) and Mis the saturation magnetization. Then, the Hamiltonian of the YIG sphere in Eq. (A2) takes the form Hm=B0MzVm+0KanVm 8M2(12M2 x+9M2 y+4M2 z);(A4) where a constant term (1 +3)0M2Vm=6, which includes the demagnetization energy and the exchange energy, has been ignored. For the jth spin in the YIG sphere, the magnetic moment ismj sj, where =geB=~is the gyromagnetic ration, geis the g-factor,Bis the Bohr magneton, and sjis the spin operator with the spin quantum number s=1=2. The YIG sphere acting as a macrospin has the magnetization [3, 4] M=P jmj Vm S Vm; (A5)9 where we have introduced the macrospin operator S=P jsj (Sx;Sy;Sz), with the summationP jover all spins in the sphere. Substituting Eq. (A5) into Eq. (A4), we have Hm= B0Sz+DxS2 x+DyS2 y+DzS2 z; (A6) where the nonlinear coe cients are Dx=30Kan 2 2M2Vm;Dy=90Kan 2 8M2Vm;Dz=0Kan 2 2M2Vm:(A7) However, when the crystalline axis [100] is aligned along the bias magnetic field B0, the exchange field and the demag- netization field remain unchanged, but the anisotropic field becomes [58] Han=2KanMz M2ez: (A8) Using the expressions in Eqs. (A2) and (A5), we can write the Hamiltonian Hmin the same form as in Eq. (A6) but the nonlinear coe cients become Dx=Dy=0;Dz=0Kan 2 M2Vm; (A9) where we have omitted the constant demagnetization and ex- change energies. Appendix B: The YIG sphere coupled to a 3D cavity So far, the Hamiltonian Hmof the YIG sphere has been obtained. Then, we derive the Hamiltonian of the cavity magnonics system. The 3D microwave cavity is usually machined from high- conductivity copper to have a high Qfactor. When focusing only on one cavity mode (e.g., the fundamental mode), this 3D resonator can be described by the Hamiltonian Hc=!c aya+1 2 ; (B1)where a(ay) denotes the annihilation (creation) operator of the cavity mode with frequency !c. To achieve a strong coupling between magnons and cav- ity photons, we can place the small YIG sphere near a wall of the cavity (see Fig. 1), where the magnetic field hcof the microwave cavity mode becomes the strongest and is po- larized along the x-direction [8]. Also, the static magnetic field B0is aligned perpendicular to hc. The field hcinduces the spin-flipping and excites the magnon mode. In com- parison with the microwave cavity, the small dimensions of the YIG sphere permit us to regard the cavity field as be- ing nearly uniform around the YIG sample. Thus, we can write hc=h0(ay+a)ex, with h0=p ~!c=(0Vc) being the magnetic-field amplitude and Vcthe volume of the cavity. The interaction Hamiltonian between the YIG sphere and the 3D cavity reads HI=0Z VmM
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2019-03-09

We develop a theory for the magnon Kerr effect in a cavity magnonics system, consisting of magnons in a small yttrium iron garnet (YIG) sphere strongly coupled to cavity photons, and use it to study the bistability in this hybrid system. To have a complete picture of the bistability phenomenon, we analyze two different cases in driving the cavity magnonics system, i.e., directly pumping the YIG sphere and the cavity, respectively. In both cases, the magnon frequency shifts due to the Kerr effect exhibit a similar bistable behavior but the corresponding critical powers are different. Moreover, we show how the bistability of the system can be demonstrated using the transmission spectrum of the cavity. Our results are valid in a wide parameter regime and generalize the theory of bistability in a cavity magnonics system.

Theory of the magnon Kerr effect in cavity magnonics

1903.03754v1

1 Magnetically tunable zero -index metamaterials Yucong Yang1,2, Yueyang Liu3, Jun Qin1,2, Songgang Cai1,2, Jiejun Su1,2, Peiheng Zhou1,2, Longjiang Deng1,2*, Yang Li3* and Lei Bi1,2* 1National Engineering Research Centre of Electromagnetic Radiation Control Materials, University of Electronic Science and Technology of China, Chengdu 610054, China 2State Key Laboratory of Electronic Thin -Films and Integrated Devices, University of Electronic Science and Technology of China, Chengdu 610054, China 3State Key Laboratory of Precision Measurement Technology and Instrument, Department of Precision Instrument, Tsinghua University, Beijing 100084, China Corresponding author denglj@uestc.edu.cn, yli9003@mail.tsinghua.edu.cn , bilei@uestc.edu.cn 2 Abstract Zero -index metamaterials (ZIMs) feature a uniform electromagnetic mode over a large area in arbitrary shapes, enabling many applications including high- transmission supercouplers with arbitrary shapes, direction- independent phase matching for nonlinear opt ics, and collective emission of many quantum emitters. However, most ZIMs reported till date are passive, with no method for the dynamic modulation of their electromagnetic properties. Here, we design and fabricate a magnetically tunable ZIM consisting of yttrium iron garnet (YIG) pillars sandwiched between two copper clad laminates in the microwave regime. By harnessing the Cotton- Mouton effect of YIG, the metamaterial was successfully toggled between gapless and bandgap states, leading to a "phase transition" between a zero -index phase and a single negative phase of the metamaterial. Using an S -shaped ZIM supercoupler, we experimentally demonstrated a tunable supercoupling state with a low intrinsic loss of 0.95 dB and a high extinction ratio of up to 30.63 dB at 9 GHz. Our work enables dynamic modulation of the electromagnetic chara cteristics of ZIMs, enabling various applications in tunable linear, nonlinear, quantum and nonreciprocal electromagnetic devices. Introduction Zero -index materials are materials or composite structures that exhibit an effective refractive index of zero at a given frequency1-4, resulting in an infinite spatial wavelength. This effect can be leveraged to overcome the limitations imposed by the finite spatial wavelength of electromagnetic waves, thereby enabling various novel physical phenomena and applications in linear5-7, nonli near8-10 and quantum electromagnetic systems11,12. Recently, zero-index materials , such as indium tin oxide10,13,14, waveguides at cut-off frequencies15-17, fishnet metamaterials18, and doped ε-near-zero (ENZ) media19, have garnered significant research interest2. However, the aforementioned materials and artificial structures exhibit large ohmic losses because of their metallic components20. In contrast, zero- index metamaterials ( ZIMs) based on all -dielectric photonic crystals exhibit zero ohmic loss, enabling the realization of ZIMs over a large area of arbitrary shapes. A photonic crystal -based ZIM was first realized in the microwave regime based on Al 2O3 pillars embedded in a parallel metal waveguide21. Subsequently, photonic -crystal -based ZIM ranging from acoustic22,23 to photonic regimes24,25 have been reported, demonstrating fascinating physical phenomena and applications , such as supercoup ling1,26, leaky- wave antennas27, cloaking21,28, superradiance12,25, and direction- independent phase matching for nonlinear optics9. Despite this progress, most of the ZIMs reported thus far are passive, with constant post -fabrication electromagnetic properties, limiting their applications in passive devices. Active ZIMs, whose magnetic (µeff) and electric ( εeff) properties can be tuned using external stimuli , may allow the dynamic tuni ng of 3 delicate photonic band structure s, thereby inducing "phase transitions " in metamaterials. In turn, this unique mechanism is expected to enable energy- efficient modulation of electromagnetic wave propagation with low insertion loss, high extinction ratio, and compact device footprint —which are all essential factors in microwave and optical communication applications4,19,29,30. In this study, we design and experimentally investigate a magnetically tunable ZIM consisting of an array of gyromagnetic pillars embedded within a parallel -plate copper waveguide. Under an applied magnetic field of 430 Oe, the photonic band structure of the proposed ZIM changes from a zero -index state to a photonic bandgap state, corresponding to a transition from a “zero -index phase” to a “single negative phase”. Based on this property, we propose a magnetic field-induced on- off switch of the supercoupling state in a n S-shaped ZIM waveguide, resulting in a low intrinsic loss of 0.95 dB and a high extinction ratio of 30.63 dB at 9 GHz. We also demonstrate a magnetic field-controlled switch to effect transitions between a zero -index state and a nontrivial topological boundary state in the magnetic ZIM. These results demonstrate the potential of applying active ZIMs in electromagnetic wave modulators and nonreciprocal devices. Results First, we design ed a Dirac- like cone -based zero -index metamaterial (DCZIM) consisting of a square array of dielectric pillars embedded in a parallel -plate copper waveguide21. At the Γ point, the accidental degeneracy of two linear dispersion bands and a quadratic dispersion band form ed a Dirac -like cone dispersion, corresponding to an impedance -matched zero effective index21. In contrast to conventional passive ZIMs, we achieved active modulation by fabricating a square lattice of pillars constructed using a magnetic dielectric material, yttrium iron garnet (YIG). By applying a magnetic field to the YIG pillars along the direction perpendicular to the wave vector of the transverse magnetic (TM) mode, an effective permeability modulation that is quadratically proportional to the YIG magneti zation was observed owing to the Cotton- Mouton effect ( see Supplementary Information S1 for further details ). In turn, this effect enable d the modulation of the photonic band structure and the effective index of metamaterials. We implemented the proposed design using the structure depicte d in Figure 1 . Figure 1a illustrates the experimentally fabricated metamaterial consisting of a square lattice of gyromagnetic YIG pillars with a 3.53- mm radius and a 17.9- mm lattice constant. The dielectric constant31 and permeability of the YIG material were characterized (see the Supplementary Information S2 for further deta ils). The YIG pillars were placed in a waveguide consisting of two parallel copper clad laminate s separated by 4 mm. A magnetic field was applied to each YIG pillar by placing a neodymium iron boron (NdFeB) permanent magnet under each pillar and behind the copper back plate. A uniform magnetic field along the z-direction was observed, 4 whose intensity reached 430 Oe in the middle of the two copper clad laminates . This was sufficient to saturate the YIG pillars (see Supplementary Information S3 for further details ). Figure 1 . Schematic diagram of the active DCZIM structure. (a) The structure of an active DCZIM based on a gyromagnetic photonic crystal . (b) Schematic diagram of a unit cell. The YIG pillars were placed in a parallel -plate copper -clad waveguide with height , h1 = 4 mm. The thickness of the waveguide plates was h2 = 2 mm. Permanent magnets with 5 -mm diameter and height h3 = 5 mm we re placed in an a crylic matrix underneath the waveguide and we re aligned with the YIG pillars. To characterize the properties of DCZIM, we first calculated and then experimentally measured the photonic band structure in the plane of the YIG array . The gray dotted line in Figure 2a represents the calculated band structure for the TM modes (the electric field is polarized in the z direction) of this photonic crystal , as observed based on a simulat ion using COMSOL MULTIPHYSICS. We observed a clear Dirac- like cone dispersion at 9 GHz . When a magnetic field of +430 Oe was applied along the z direction, the off - diagonal component induced the Cotton- Mouton effect in YIG, changing the frequencies of the three photonic bands forming the Dirac -like cone. Owing to time reversal symmetry (TRS) breaking, t he photonic crystal transitioned from symmetry to symmetry (see Supplementary Information S4 for further details ). As a result, the degeneracy was broken, resulting in two bandgaps, as indicated by the gray dotted lines in the right panel of Figur e 2a. The modes supported by this structure were analyzed by considering those supported by a square array of two-dimensional YIG pillars. As depicted in Figure 2b –d, this structure supported three modes for TM polarization near the 9 GHz frequency—a monopole mode, a transverse magnetic dipole mode, and a longitudinal magnetic dipole mode. When a magnetic field was applied, as illustrated in the right part of the panel, these three modes were displaced to different frequencies (8.95 GHz, 9.52 GHz, 11.04 GHz , respectively ). Because of the broken TRS , all three modes required to be rotated through 180 ° to coincide 5 with each other. We further calculated the Chern number of each photonic band from low frequency to high frequency near the Dirac point to be 0, 1, and -1, respectively. Based on the Chern number of each band, we calculated the Chern numbers of the bandgaps , 1 and 2 (right panel of Figure 2a) , to be and , respectively. Based on the Chern numbers of the bandgaps , 1 and 2, we can determine their topological nature —bandgaps 1 and 2 were topologically trivial and nontrivial, respectively. Figure 2 . Theoretical and experimental demonstration of an active ZIM. ( a). Measured and calculated (grey dots) photonic bands of the active ZIM using Fourier transform field scan (FTFS) of the TM modes corresponding to applied magnetic fields of 0 (left p anel) and 430 Oe (right panel). ( b)–(c). Simulated three- dimensional dispersion surfaces near the Dirac -point frequency, depicting the relationship between the frequency and the wave vectors ( kx and k y) (d). COMSOL -computed R e(Ez) on the cross- section of a ZIM unit cell at the frequencies indicated by dashed arrows , depicting an electric monopole mode , a transverse magnetic dipole mode , and a longitudinal magnetic dipole . The black circles indicate the boundaries of the YIG pillar s. 6 We experimentally characterized this metamaterial using the setup proposed by Zhou et al.31 The sizes of all the samples were designed to be 10 × 10 periods, as illustrated in Figure 1a. The parallel -plate waveguide consisting of two copper clad laminate s separated by 4 mm supported only the fundamental TEM mode in a parallel -plate waveguide below 37.5 GHz. To facilitate the excitation and measurement of electromagnetic fields inside the waveguide, a square array of holes, with a lattice constant of 5 mm , was drilled through the top surface of the copper -clad laminate, enabling the insertion of a probe for field measurement. The thickness of the copper -clad laminate s was taken to be 2 mm to tune the uniform distribution of the magnetic flux. First, we measured the photonic band structure of the metamaterial under an applied magnetic field of 0, as illustrated using the intensity plot in the left panel of Figure 2a. T he band structure of the TM bulk states was obtained by applying two -dimensional discre te Fourier transform (2D-DFT) to the measured complex field distribution over the metamaterial (see Supplementary Information S5 for further details ). The measured photonic band structure exhibited good agreement with the simulation results ( represented by gray dots). Both the measured and computed band structures exhibited a bandgap between 5 GHz and 7 GHz as well as bulk modes between 7 GHz and 12 GHz, indicating a Dirac -like cone dispersion near 9 GHz. The nondegeneracy of the photonic bands was also recorded after applying a 430 Oe magnetic field along z-direction , as depicted in the right panel in Figure 2a. The bandgaps were experimentally measured to be at 8.95–9.60 GHz and 10.24–11.04 GHz, corroborating the simulation results. Figure 3 . Magnetic field -induced phase transition of active ZIM . Real and imaginary parts of the effective permittivity ( εeff) and permeability tensor elements ( µ and κ) (a) under an applied magnetic field of 0, and (b) under an applied magnetic field of 430 Oe in the bandgap frequency range of 8.95–9.60 GHz and (c) the bandgap frequency range of 10.24 –11.04 GHz 7 The magnetic field-induced band structure and transmittance modulation can be regarded to be a phase transition process from a material perspective. This phenomenon can be observed by retrieving the effective permittivity and permeability tensor elements of the metamaterial in the presence and absence of an applied magnetic field, as illustrated in Figure 3. We used the boundary effective medium approach (BEMA) to calculate the effective constitutive parameters , as proposed in a previous study on ZIM32. In Figure 3a –c, εeff denotes the effective permittivity, µ denotes t he diagonal of the effective permeability tensor, and κ denotes the off -diagonal component of the effective permeability tensor , . Figure 3a depicts the results corresponding to an applied magnetic field of 0. The real parts of εeff and µ cross zero simultaneously and linearly at 9 GHz, exhibiting ε-and-µ-near-zero ( EMNZ ) behavior corresponding to the zero- index phase . The imaginary parts of εeff and µ were both close to 0. This low loss was attributed to the small loss tangent (0.0002) of the YIG material in this frequency range . When a magnetic field of 430 Oe was applied, phase transitions occurred from the zero -index phase to the µ- negative (MNG) or the ε−negative (ENG) phase, as depicted in Figure s 3b and 3c , respectively . In the bandgap , 8.95 GHz –9.6 GHz (Figure 3b), εeff is positive , whereas µeff ( )33 is negative, which corresponds to the MNG phase. T he impedance at 9 GHz was tuned from 1.84 to 0.62i after applying the magnetic field , leading to a total reflection of the incident EM wave owing to impedance mismatch. In the bandgap , 10.24 –11.04 GHz (Figure 3c), εeff is negative , whereas µeff is positive, which corresponds to the ENG phase. Corresponding to both frequency ranges , the real parts of εeff and µeff decreased as the frequency increased, exhibiting anomalous dispersion . The incident EM wave was still reflected due to impedance mismatch. Additionally, as depicted in the inset of Figure 3c in the ENG frequency regime, a nontrivial topological boundary state was observed at the metamaterial edge because of the difference between the Chern numbers of the upper and lower structure s. The complex phase transition phenomena discussed above were attributed to the location of the ZIM at the origin of the metamaterial phase diagram, which allowed it to reach all quadrants of the phase diagram via appropriate tuning of the constitutive parameters33,34. Further discussion on the attainment of other phases based on the phase diagram is depicted in Figure s S5–7 of Supplementary Information S6–7. Such unique properties of an active ZIM are indicative of its potential with respect to the modulation of the propagation of EM waves. 8 Figure 4 Structure and characterization of a microwave switch based on active DCZIM. ( a) Photograph of the microwave switch sample. (b) The measured transmissions in the absence and presence of an applied magnetic field of 430 Oe. (c) The real part of the Ez distribution observed at 9 GHz inside the metamaterial (each pillar is indicated in gray) in the absence of a magnetic field. (d) The real part of the Ez distribution observed at 9 GHz inside the metamaterial (each pillar is indicated in gray) in the presence of a 430 Oe magnetic field. To showcase a microwave switch based on the phase transition effect of active ZIM, we fabricated a ZIM waveguide switch by leveraging the contingency of the supercoupling state on the applied magnetic field, as illustrated in Figure 4a. First, a ZIM waveguide co mprising top and bottom metal plates and 100 YIG pillars was fabricated, forming two sharp 90-degree bends , to verify the supercoupling effect experimentally (see Supplementary Information S8 for further details ). The waveguide was coupled to the coaxial line via SMA connectors. Linear taper ed sections were used to sustain the TE 10 mode and induce its gradual evolution into the TM mode of the metamaterial during propagation from the source to the waveguide. P erfect magnetic conductor (PMC) boundary condition s were realized using aluminum alloy walls at a distance of λ /4 from the metamaterial as the lateral boundaries26 (see Supplementary Information S8 for further details). Using the device depicted in 9 Figure 4a, we successfully switch ed between the bulk state ( supercoupling state) and the photonic bandgap state (off state) by applying a ppropriate magnetic field s to the YIG pillars. Figure 4b depicts the measured transmission spectra corresponding to both states. A large transmission contrast was observed over 8.9–9.4 GHz, w hich was consistent with the bandgap frequencies calculated via numerical simulation in Figure 2a. The difference in device transmission under zero and 430 Oe magnetic field s was larger than 30 dB at approximately 9 GHz and t he device insertion loss was 3.75 dB. Considering that the coupling loss induced by the SMA connectors was 2.8 dB, the intrinsic loss of the ZIM waveguide was as low as 0.95 dB, primarily induced by the absorption of YIG materials. We verified the supercoupling behavior in the absence of an applied magnetic field by measuring the Ez component of the electric field at each point of the metamaterial, as illustrated in Figure 4c. A t 9 GHz, the electric field tunnel ed through the metamaterial with almost no phase ch ange in the form of a bulk mode, verifying supercoupling behavior. In contrast, in the absence of YIG pillars in the waveguide, the wave was reflected back to the incident port, as described in Supplementary Information S9. This confirmed that the supercoupling behavior was induced by the DCZIM. As depicted in Figure 4d, in the presence of an applied magnetic field , the electromagnetic wave decayed exponentially in the metamaterial owing to the photonic bandgap 1 depicted in Figure 2a, leading to a high ext inction ratio. We also constructed a switch between the supercoupling state and the topological one -way transmission state by probing at the upper bandgap frequency of 10.6 GH z (see Supplementary Information S10 for further details) . These unique properties are indicative of the potential of active ZIMs in novel active electromagnetic devices. Discussion In this study, we propose d and experimentally operated a magnetically tunable ZIM. The metamaterial was operated by leveraging the Cotton -Mouton effect of the constitutive YIG pillars under applied magnetic fields, which alter the symmetry and bandgap opening of the Dirac -like cone -based ZIM. From a material perspective, the proposed metamaterial exhibited a phase transition from the zero -index phase to a single negative phase, leading to an effective index change from 0 to 0.09i at 9 GHz. Based on this property, we constructed and verified the function of a microwave switch by manipulating the supercoupling effect , thereby reducing the intrinsic loss to 0.95 dB and achieving a high extinction ratio of 30.63 dB at 9 GHz . We believe that this study introduces a new approach to active ZIMs, particularly with respect to the development of efficient active elect romagnetic and nonreciprocal devices. By appropriately engineering the slots in parallel- plate copper waveguide s27, continuous beam steering over the broadside can be achieved by varying the applied magnetic field. Moreover, our design can be extended to the optical regime by embedding YIG pillars in a polymer matrix with gold films cladded24, enabling the dynamic modulation of optical DCZIM. Further, such a magnetically tunable DCZIM can be used to modulate the four -wave 10 mixing process in DCZIM by manipulating the zero -index phase -matching condition9. Additionally , we were able to modulate DCZIM- based large -area single -mode photonic crystal surface- emitting lasers with high output power35. Finally, we also successfully modulate d the extended superradiance by tuning the effective index of the DCZIM, in which many quantum emitters were embedded12. Methods Numerical simulation . The permittivity of the YIG used during numerical simulation was taken from that published by Zhou et al.36 The band structure was calculated using the COMSOL software. The results were obtained by calculating the modes with periodic boundary conditions along the x and y directions. The TM polarization mode was selected by considering the electric field to be an out -of-plane vector. Under magnetic fields, t he DCZIM was simulated to obtain frequency -dependent electromagnetic field profiles. The transitions between the different metamaterial phases in the presence and absence of magnetic field s were simulated by considering the off-diagonal elements o f the YIG permeability tensor. Home -made MATLAB codes based on BEMA and 2D -DFT were used to calculate the effective permittivity, permeability tensor, and photonic band structure. Device Fabrication . The active DCZIM consisted of a square array of YIG pill ars with a radius of 3.53 mm, a height of 4 mm , and a lattice constant of 17.9 mm ( as indicated in Figure 1a). Th e waveguide was formed using two 500 mm × 500 mm copper -clad laminates. During the measurement of the nontrivial boundary state, a copper bar w ith a length of 40 cm and a width of 2 cm was placed at the edge of the metamaterial to form its boundary. The YIG pillars were fabricated from YIG bulk crystals using an ultrahigh- accuracy computer numerical control ( CNC) machine (Mazak V ARIAXIS i -700) with a dimensional accuracy exceeding 0.05 mm. As depicted in Figure 1a, an external magnetic field was introduced to maintain the saturation magnetization of the garnet. To affix the YIG pillars to the copper clad lamin ate substrate tightly, double - sided tape with a radius of 3.53 mm was applied to one side of each pillar. The position of each YIG pillar was precisely defined by placing the other side in an acrylic mold with 10 × 10 holes with a radius of 3.54 mm and a l attice constant of 17.9 mm. The copper clad laminate substrate was pressed onto the side of the YIG pillars with tape, which affixed the YIG pillars tightly and precisely onto the copper clad laminate substrate . The acrylic mold was removed carefully to prevent extrusion and damage to the YIG pillars. Each YIG pillar was properly magnetized by aligning the NdFeB magnet used to apply the magnetic field precisely underneath each YIG pillar . To this end, a 2- mm- thick acrylic sheet was first covered with a double -sided tape to form the substrate. Then, w e fixed a 5- mm-thick acrylic sheet including 10 × 10 11 perforations with a radius of 5 mm and a lattice constant of 17.9 mm onto a 2 -mm-thick acrylic substr ate. Finally, with respect to the designated direction of the magnetic field, we placed the NdFeB magnets into the holes of the 5 -mm- thick acrylic sheet, achieving a good alignment with the array of the YIG pillars (Figure 1b). Characterization Setup . We measure d the photonic band structure, transmission spectra, and near -field distribution of the active metamaterial sample using two dipole antennas as the transmitter and receiver. Both antennas were inserted into the waveguide via holes drilled using an ul trahigh -precision CNC machine and they were connected to a vector network analyzer ( Rohde & Schwarz ZNB 20) to measure the S parameters. Prior to measurement, 3.5 -mm 85052D through- open- short -load calibrations were performed. As a result , the measured S parameters included only the insertion loss of the tapered waveguides and the metamaterial. References 1. Li, Y., Chan, C. T. & Mazur, E. Dirac -like cone -based electromagnetic zero -index metamaterials. Light Sci. Appl. 10 , 203 (2021). 2. Kinsey, N. et al. Near -zero-index materials for photonics. Nat. Rev. Mater. 4, 742- 760 (2019). 3. Liberal, I. & Engheta, N. Near -zero refractive index photonics. Nat. Photonics 11, 149- 158 (2017). 4. Vulis, D. I. et al. Manipulating the flow of light using Dirac -cone zero -index metamaterials. Rep. Prog. Phys. 82, 012001 (2019). 5. Alù, A. et al. 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Unidirectional single -photon generation via matched zero- index metamaterials. Phys. Rev. B 94, 220103 (2016). 12. Mello, O. et al. Extended many- body superradiance in diamond epsilon near -zero metamaterials. Appl. Phys. Lett. 120 (2022). 13. Yang, Y. et al. High -harmonic generation from an epsilon- near-zero material. Nat. Phys. 15, 1022- 1026 (2019). 14. Jia, W. et al. Broadband terahertz wave generation from an epsilon- near-zero material. Light Sci. Appl. 10, 11 (2021). 15. Vesseur, E. J. et al. Experimental verification of n = 0 structures for visible light. Phys. Rev. Lett. 110, 013902 (2013). 16. Zhou, Z. & Li, Y. Effective Epsilon- Near -Zero (ENZ) Antenna Based on Transverse Cutoff Mode. IEEE Trans. Antennas Propag. 67, 2289- 2297 (2019). 12 17. Qin, X. et al. Waveguide effective plasmonics with structure dispersion. Nanophotonics 11, 1659 - 1676 (2022). 18. Yun, S. et al. Low-Loss Impedance- Matched Optical Metamaterials with Zero -Phase Delay. ACS Nano 6, 4475- 4482 (2012). 19. Liberal, I. et al. Photonic doping of epsilon- near-zero media. Science 355, 1058- 1062 (2017). 20. Tang, H. et al. Low -Loss Zero -Index Materials. Nano Lett. 21, 914- 920 (2021). 21. Huang, X. et al. Dirac cones induced by accidental degeneracy in photonic crystals and zero- refrac tive- index materials. Nat. Mater. 10, 582- 586 (2011). 22. Dubois, M. et al. Observation of acoustic Dirac- like cone and double zero refractive index. Nat. Commun. 8, 14871 (2017). 23. Xu, C. et al. Three -Dimensional Acoustic Double -Zero -Index Medium with a Fourfold Degenerate Dirac -like Point. Phys. Rev. Lett. 124, 074501 (2020). 24. Li, Y. et al. On-chip zero -index metamaterials. Nat. Photonics 9, 738- 742 (2015). 25. Moitra, P. et al. Realization of an all -dielectric zero -index optical metamaterial. Nat. P hotonics 7, 791- 795 (2013). 26. Camayd -Muñoz, P. Integrated zero -index metamaterials. PhD thesis thesis, Harvard University, (2016). 27. Memarian, M. & Eleftheriades, G. V. Dirac leaky -wave antennas for continuous beam scanning from photonic crystals. Nat. Commun. 6, 5855 (2015). 28. Chu, H. et al. A hybrid invisibility cloak based on integration of transparent metasurfaces and zero- index materials. Light Sci. Appl. 7, 50 (2018). 29. Engheta, N. Pursuing Near -Zero Respons. Science 340, 286- 287 (2013). 30. Maas, R. et al. Experimental realization of an epsilon- near-zero metamaterial at visible wavelengths. Nat. Photonics 7, 907- 912 (2013). 31. Zhou, P. et al. Observation of Photonic Antichiral Edge States. Phys. Rev. Lett. 125, 263603 (2020). 32. Wang, N. et al. Effective medium theory for a photonic pseudospin- 1/2 system. Phys. Rev. B 102, 094312 (2020). 33. Davoyan, A. R. & Engheta, N. Theory of wave propagation in magnetized near -zero-epsilon metamaterials: evidence for one -way photonic states and magnetically switched transparency and opacity. Phys . Rev. Lett. 111 , 257401 (2013). 34. Smith, D. R., Pendry, J. B. & Wiltshire, M. C. K. Metamaterials and Negative Refractive Index. Science 305 , 788- 792 (2004). 35. Chua, S.- L. et al. Larger -area single -mode photonic crystal surface -emitting lasers enabled by an accidental Dirac point. Opt. Lett. 39, 2072- 2075 (2014). 36. Zhou, P. et al. Photonic amorphous topological insulator. Light Sci. Appl. 9, 133 (2020). Acknowledgements The authors appreciate the discussions with Hengbin Cheng from the Institute of Physics, Chinese Academy of Science , and their assistance . The authors are also grateful for the support received from the Ministry of Science and Technology of the People’s Republic of China (MOST) (Grant No. 2018YFE0109200, 2021YFA1401000, a nd 2021YFB2801600), National Natural Science Foundation of China (NSFC) (Grant Nos. 51972044, 52021001, and 62075114), Sichuan Provincial Science and Technology Department (Grant No. 2019YFH0154), the Fundamental Research Funds for the Central Universities (Grant No. ZYGX2020J005), the Beijing Natural Science Foundation (Grant No. 4212050), 13 and the Zhuhai Industry- University Research Collaboration Project (ZH22017001210108PWC). This work was supported by the Center of High- Performance Computing, Tsinghua Un iversity.

2022-06-09

Zero-index metamaterials (ZIMs) feature a uniform electromagnetic mode over a large area in arbitrary shapes, enabling many applications including high-transmission supercouplers with arbitrary shapes, direction-independent phase matching for nonlinear optics, and collective emission of many quantum emitters. However, most ZIMs reported till date are passive, with no method for the dynamic modulation of their electromagnetic properties. Here, we design and fabricate a magnetically tunable ZIM consisting of yttrium iron garnet (YIG) pillars sandwiched between two copper clad laminates in the microwave regime. By harnessing the Cotton-Mouton effect of YIG, the metamaterial was successfully toggled between gapless and bandgap states, leading to a "phase transition" between a zero-index phase and a single negative phase of the metamaterial. Using an S-shaped ZIM supercoupler, we experimentally demonstrated a tunable supercoupling state with a low intrinsic loss of 0.95 dB and a high extinction ratio of up to 30.63 dB at 9 GHz. Our work enables dynamic modulation of the electromagnetic characteristics of ZIMs, enabling various applications in tunable linear, nonlinear, quantum and nonreciprocal electromagnetic devices.

Magnetically tunable zero-index metamaterials

2206.04237v1

Temperature dependence of the mean magnon collision time in a spin Seebeck device Vittorio Basso, Alessandro Sola, Patrizio Ansalone and Michaela Kuepferling Istituto Nazionale di Ricerca Metrologica, Strada delle Cacce 91, 10135, Torino, Italy Abstract Based on the relaxation time approximation, the mean collision time for magnon scattering c(T) is computed from the experimental spin Seebeck coecient of a bulk YIG / Pt bilayer. The scattering results to be composed by two processes: the low temperature one, with a T1=2dependence, is at- tributed to the scattering by defects and provides a mean free path around 10 m; the high temperature one, depending on T4, is associated to the scat- tering by other magnons. The results are employed to predict the thickness dependence of the spin Seebeck coecient for thin lms. Keywords: spin Seebeck eect, yttrium iron garnet, magnon scattering 1. Introduction A spin Seebeck device is a bilayer composed by a ferromagnetic insulator (i.e. the ferrimagnet YIG) and a metal with a strong spin Hall eect (i.e. Pt). The spin Seebeck eect is obtained by applying a temperature gradient to the YIG that generates a current of magnetic moment. Part of the magnetic moment current from YIG is injected at the interface into the side metallic layer where it is carried by electrons. The use of Pt as a metallic layer per- mits to detect the magnetic moment current because of the inverse spin Hall eect that laterally de ects polarized electrons and creates an electric volt- age in the transverse direction [1]. In the last decade the spin Seebeck eect has attracted attention as an alternative thermoelectric generator and as a source of a magnetic moment current for spintronic devices without involv- ing a charge current [2, 3]. However, to optimize the eect, the underlying physics needs to be appropriately understood. At any nite temperature the Preprint 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 spontaneous value because of the thermal excitation of spin waves (magnons). It is believed that the temperature gradient across the ferromagnet forces the diusion of the thermal magnons in the direction of the gradient, therefore generating a current of magnetic moment [4, 5, 6]. A crucial point is therefore to understand the transport properties of magnons and their scattering [7]. Experiments of the temperature dependence of the spin Seebeck coecient [8, 9] revealed non obvious features: in bulk YIG single crystals a peak is observed at temperatures around 70 K. The peak is shifted at higher tem- peratures and partially suppressed by both decreasing the thickness [8] or decreasing the grain size in polycrystalline materials [10, 11]. The interpre- tation of this eect has stimulated a debate on the possible sources of the scattering of the magnons: impurities, phonons, other magnons and so on [10, 7, 9]. One of the problems is to disentangle the temperature dependence of the magnon scattering processes from the temperature dependence other parameters such as the conductance and the diusion length of Pt. Unlike the classical Seebeck eect of metals, in which the ratio voltage over temperature gives the absolute thermo-electric power coecient, in the spin Seebeck eect the coecient given by the ratio between the voltage gradient over the temperature gradient (in the geometry of Fig.1) depends not only on the absolute thermomagnetic power coecient of the YIG, YIG, and on the spin Hall angle of the platinum, SH, but also on the magnetic moment conductances, the diusion lengths and the thicknesses of the two layers [12]. Even if certain parameters can be obtained by independent ex- periments, the expression of the spin Seebeck coecient still contains at least two free parameters: YIGand the magnetic moment conductivity M;YIGof the ferromagnet. To simplify the problem in this work we set YIG=0:956 T/K as predicted by the Boltzmann approach to the transport of magnons in the relaxation time approximation[13, 14]. Then the magnetic moment conductivity can be easily deduced from the experimental data and the re- laxation time c(T) computed as a function of temperature. c(T) is an estimate of the typical scattering time and gives insights on the evolution of the scattering processes as a function of the temperature. The main result of this paper is to show that the scattering is composed by two processes. At low temperature the active process changes with temperature as T1=2, a dependence that can be attributed to the scattering by defects. At high tem- perature it depends strongly on TasT4a variation that could be associated to the scattering by other magnons. 2xyzme-HYIGPt dYIGdPtcoldhotM-∇xT∇yVFigure 1: Spin Seebeck bilayer composed by a ferromagnetic insulator (i.e. the ferrimagnet YIG) and a metal with a strong spin Hall eect (i.e. Pt). The temperature gradient along xon YIG injects a current of magnetic moment into Pt where it is revealed as a voltage gradient along ybecause of the inverse spin Hall eect. 2. Spin Seebeck eect The spin Seebeck coecient is given by the ratio between the voltage gradient in platinum and the temperature gradient in YIG (see Fig.1) SSSE=ryVe rxT(1) To derive a theoretical expression for the spin Seebeck coecient one has to use a thermodynamic theory describing the transport of the magnetic moment in the ferromagnet caused by the temperature gradient and the transverse electric eect of the inverse spin Hall eect in the metal [15]. For both materials one has also to take into account that the magnetic moment is a non conserved quantity and therefore the transport of the magnetic moment is characterized by a diusion length lMand by a typical time constant M. The two values and their ratio vM=lM=M, a parameter that assumes the meaning of a magnetic moment conductance, are characteristic values of each material (M = Pt, YIG) and can be temperature dependent. From the thermodynamic theory by exploiting the reciprocity of both the intrinsic thermal and magnetic transport in the ferromagnet and the spin Hall eect in the metal [16], one derives the following expression (see appendix A for a derivation) 3SSSE=SHB elYIGvYIG dPtvYIG (2) whereSHis the spin Hall angle of the metal (the angle for the magnetic moment is opposite with respect to the one for the spin i.e. negative for Pt and positive for W and Ta), YIGis the thermomagnetic power coecient of the ferromagnet, lYIGandvYIGare the diusion length and the magnetic moment conductance of the ferromagnet. The product lYIGvYIG=0M;YIG is proportional to the magnetic moment conductivity of the ferromagnet M;YIG.vis an eective conductance of the bilayer and summarizes the eects of the ratio between the thicknesses of the layers and their own intrinsic diusion lengths and contains the sum of the eective conductances. Its expression is 1 v=f(dPt=lPt)f(dYIG=lYIG) vPttanh(dPt=lPt) +vYIGtanh(dYIG=lYIG)(3) wheref(x) = tanh(x) tanh(x=2). in the case of a thick YIG with dYIGlYIG it is simplied as 1 v=f(dPt=lPt) vPttanh(dPt=lPt) +vYIG(4) and if the conductance of the metal is much larger than the one of the ferromagnet, vPttanh(dPt=lPt)vYIG, it is directly given by the properties of the metal v=vPtcoth(dPt=(2lPt)). The aim of this paper is to compute lYIGandvYIGof YIG as a function of temperature by using Eq.(2) and the experimental data for the spin Seebeck coecient. In Eq.(2) many parameters are known from other experiments. The missing one is the intrinsic parameter YIG. In this work we employ the result of the transport theory of magnons which predicts a temperature independent constant YIG=0:956 T/K [14]. We initially simplify the problem by considering a bulk YIG single crystal at constant temperature. For a bulk YIG we have dYIGlYIGand, expecting vPtvYIG, we estimate v'vPtcoth(dPt=(2lPt)). The magnetic moment conductivity of Pt is M;Pt= (B=e)2e;Pt. Then, with e;Pt= 6:4106 1m1we obtain0M;Pt= 2:6108m2s1. As0M;Pt=lPtvPt, with lPt= 7:3 nm [17], we get vPt= 3:5 m/s,Pt= 2:1 ns and nally v'11:2 m/s. The spin Hall angle is taken as SH=0:1 [17]. From recent experiments 4resistivity (Ωm)temperature (K)Pt temperature (K)lM(nm)conductance vPtlenght lPtvM(m/s)Figure 2: Left: standard resistivity of Pt (from Platinum Metals Rev., 28 (4) 164 (1984)) and tting function %(T) =x%L+(1x)%Hwith%L= 6
1014T3,%H= 4:11010(T32), x= [1 + tanh[( T40)=10]]=2. Right: estimated lPt= (0M;PtPt)1=2andvPt=lPt=Pt withPt= 2:1 ns,M;Pt= (B=e)2e;Pt,e;Pt= (%(T) +%0)1with residual resistivity %0= 0:43107 m. at room temperature with a bulk YIG single crystal ( dYIG= 0:5 mm) we foundSSSE=4:8107V K1[16]. Then we compute the only missing parameter, the product lYIGvYIG, resulting 5 :3109m2/s. By estimating the time constant of the YIG as YIG= (0 LM0)1whereis the damping, we getYIG= 1106s with= 2:5105(M0= 1:95105A/m is the spontaneous magnetization of YIG). Therefore, at room temperature one nds a diusion length lYIG'70 nm and a conductance vYIG'0:07 m/s. It must be remarked that the diusion length computed in this way is very similar to the one deduced from the YIG thickness dependence in Ref.[18]. The same calculation can be performed by considering the tempera- ture dependence of the spin Seebeck coecient for the YIG single crystal (dYIG= 1 mm) of Ref.[8] (we take from now on the SSSEas an absolute value without the minus sign given by our choice of the reference system). For platinum, the spin Hall angle is not expected to change signicantly with T[19], therefore we assume SH=0:1, while the temperature dependence of the diusion length and of the conductance of Pt are estimated on the basis of the temperature dependence of the resistivity of Pt (see Fig.2 left) as shown in Fig.2 right. For YIG, YIGis expected from the diusion theory to be temperature independent [14], while the temperature dependence of the time constant YIGis attributed to the temperature dependence of . In Ref.[20] it was found that is approximately linear with T. With these as- sumptions one derives lYIG(T) andvYIG(T) shown in Fig.3 right. Again here 5it is remarkable that, by decreasing the temperature, the diusion length reaches a plateau at around 1 m that matches the low temperature estimate of Ref.[18]. Now, having the temperature dependence of lYIG(T) andvYIG(T) it is possible to employ Eq.(2) with Eq.(3) to compute the spin Seebeck coecient as a function of both YIG thickness and temperature. The result is shown in Fig.4 left. The set of curves fully captures the phenomenology of the experimental curves of Refs.[8] and [9] but the peak starts to decrease at a YIG thickness ( 2m) much smaller than the one seen in experiments (20m). To understand more we have therefore to go into the details of the scattering processes. lYIG(m)temperature (K)τc (s)1/T41/T1/2temperature (K)∇yV/∇xT (V/K) temperature (K) vYIG(m/s)temperature (K)bulk YIG / Pt 1/T3/2 1/T1/2T Figure 3: Top left: spin Seebeck coecient for a bulk (1 mm) YIG from Ref.[8]. Top right: diusion length lYIG(T). Bottom right: magnetic moment conductance vYIG(T). Bottom left: relaxation time c. Points are computed from the data of Ref.[8]. Dashed lines are tted behavior in terms of the powers of temperature marked on the graphs. Full line 1 c=1 c;L+1 c;Hwithc;L= 1:8109(T=Tm)1=2s,c;H= 3:41012(T=Tm)4s. 63. Magnon collision time The previous results have been obtained by xing the thermomagnetic power coecient of the ferromagnet YIGto the constant -0.956 T/K which is the result of the Boltzmann transport theory in the relaxation time ap- proximation [13, 14]. With the relaxation time approximation one assumes that the diusing magnons relax to the equilibrium distribution with a typi- cal time constant c. From the theory the magnetic moment conductivity is given by M;YIG=(2B)2nmc m m(5) where 2Bis the magnetic moment carried by the magnon, m mis its eective massm m=}2=(2D), whereDis the spin wave exchange stiness, cis the relaxation time and nmis the number of magnons. The number of magnons is expected to change signicantly with temperature and nmis given by nm=n 83=2T Tm3=2 (6) wherenis the volume density of localized magnetic moments in the system n= 1=a3,Tm=D=(a2kB) andkBis the Boltzmann constant. ais typical distance between localized magnetic moments and is related to the sponta- neous magnetization by M0=B=a3. For YIG one has D= 8:61040J m2, m m= 2:51028kg (about 280 times the electron mass) and Tm= 480 K [21]. As the temperature dependence of the magnetic moment conductivity 0M;YIG=lYIGvYIGhas been estimated from the spin Seebeck coecient, we can deduce the temperature dependence of the relaxation time c(T) de- scribing the collision processes. The result is shown in Fig.3 bottom left. It appears that the scattering of the magnons is the superposition of two mechanisms. If we apply the Matthiessen's rule 1 c=1 c;L+1 c;H(7) we can separate two dierent time constants. At low temperature the active processc;Lappears to change with temperature as T1=2while at high tem- peraturec;Hdepends strongly on TasT4. The low temperature process can be attributed to the scattering by defects. If we take the velocity of the magnons to vary with temperature as vm=p kBT=mand compute a mean 7free path due to intrinsic defects as =vm
c;Lwe get a temperature inde- pendent value of '10m. We recall that the mean free path was obtained from the bulk data. In the case of thin lms with dYIG<therefore we have to introduce an additional extrinsic time constant c;L;ewhich is thickness dependent as c;L;e=dYIG=vmin order to add the scattering at the inter- face. The curves calculated with this additional scattering mechanism can be seen in Fig.4 right. It must be observed that the plot at the right shows the decrease of the signal at thicknesses much larger with respect to the case of the plot of the left and in better agreement with the experiments. Con- sidering that all the parameters have been derived only from the data of the bulk, the agreement is reasonably good. The high temperature time constant c;Hdecreasing as T4is exactly what is expected for the magnon-magnon scattering process [22]. temperature (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 Figure 4: Spin Seebeck coecient as a function of thickness and temperature. Lines are theoretical predictions. Left: computed by using the scattering corresponding to the bulk for all thicknesses. Right: including a an additional scattering contribution due to interfaces a bulk. In both cases the points corresponds to the data of the spin Seebeck coecient YIG from Ref.[8] (black squares: 1mm, red triangles 10 m, blue stars 1 m, purple diamonds 0.3 m). 4. Conclusions In the paper we have performed a study of the temperature dependence of the spin Seebeck coecient of the YIG/Pt bilayer. We have interpreted the literature data [8, 9] by using the thermodynamic theory of Refs.[15, 14] 8in which the magnon transport occurs by diusion in presence of a gradi- ent of the thermodynamic temperature and in the gradient of the potential for the magnon diusion. The resulting temperature dependence of the re- laxation time describes two sources of scattering: the scattering by defects or interfaces, changing with temperature as T1=2, and mainly active at low temperature and a scattering by other magnons, depending on T4and active at high temperature. The classical literature investigating the contributions of magnons to the thermal conductivity of ferromagnetic insulators, has always considered that the magnon-magnon process was very dicult to identify. This is because it is relevant only at high temperatures, a range in which the contribution of magnons to the transport of heat is irrelevant with respect to the phonon one. The spin Seebeck eect represents an case for the detailed study of magnon-magnon scattering because it reveals the transport of magnetic mo- ment. In reference to the recent literature claiming a magnon-phonon drag eect [23], the results obtained in this study are indicating that the magnon- phonon process as a main source of scattering can be excluded. However it cannot be a priori excluded that, especially in materials with a strong mag- netoelastic interaction, the magnon-phonon processes, by means of a phonon drag eect, could give an enhancement of the eective YIG[23]. From our study it appears that such a mechanism is not necessary to explain the peak of the spin Seebeck signal of YIG. Future works could clarify the limits and possibilities of these two approaches. A possible interesting extension of the present approach to the magnon transport is at high temperatures close to the Curie point. This extension is not straightforward because the spin wave spectrum (especially the long wavelengths) develops over a background saturation magnetization Mswhich is less than the spontaneous M0and such a decrease is due to the spin wave themselves (especially the short wavelengths). Possible advancements of a Boltzmann transport approach are possible in terms of a self consistent ap- proach as suggested by Robert Brout with his random phase approximation technique [24]. Appendix A. Spin Seebeck coecient The spin Seebeck coecient of Eq.(2) is calculated as follows [15]. The transverse voltage gradient in Pt is given by the formula 9ryVe=1 e;PtSHe B <jM>Pt (A.1) where<jM>Ptis the average magnetic moment current in Pt. By solving the diusion equation for the magnetic moment in Pt, <jM>Ptis estimated as a function of the magnetic moment current injected at the interface by the ferromagnet, jM;0, by assuming that the current at the other end of Pt is zero. The result is <jM>Pt=lPt dPttanh(dPt=(2lPt))jM;0 (A.2) Similarly by solving the diusion equation also in the YIG and imposing the boundary conditions between the metal and the ferromagnet, jM;0is computed as a function of the magnetic moment current source jMSdue to the intrinsic spin Seebeck eect in YIG. jM;0is given by the current divider equation jM;0=vPttanh(dPt=lPt) vPttanh(dPt=lPt) +vMtanh(dM=lM)jMS (A.3) where the expressions of the type vMtanh(dM=lM) are the eective conduc- tances (depending on the ratio dM=lM) andvMis the intrinsic magnetic mo- ment conductance which is a property of each material. For a ferromagnet of nite thickness one has that the magnetic moment source is jMS=tanh(dYIG=(2lYIG)) tanh(dYIG=lYIG)YIGM;YIGrxT (A.4) By joining the previous equations one obtains expression (2) with vgiven by Eq.(3). References [1] K. Uchida, H. Adachi, T. Ota, H. Nakayama, S. Maekawa, and E. Saitoh. Observation of longitudinal spin-seebeck eect in magnetic insulators. Appl. Phys. Lett. , 97:172505, 2010. [2] K.I. Uchida, H. Adachi, T. Kikkawa, A. Kirihara, M. Ishida, S. Yorozu, S. Maekawa, and E. Saitoh. Thermoelectric generation based on spin seebeck eects. Proceedings of the IEEE , 104:1946, 2016. 10[3] Xiao-Qin Yu, Zhen-Gang Zhu, Gang Su, and A.-P. Jauho. Spin- caloritronic batteries. Phys. Rev. Applied , 8:054038, Nov 2017. [4] Jiang Xiao, Gerrit E. W. Bauer, Ken-chi Uchida, Eiji Saitoh, and Sadamichi Maekawa. Theory of magnon-driven spin seebeck eect. Phys. Rev. B , 81:214418, Jun 2010. [5] Steven S.-L. Zhang and Shufeng Zhang. Magnon mediated electric current drag across a ferromagnetic insulator layer. Phys. Rev. Lett. , 109:096603, Aug 2012. [6] S. M. Rezende, R. L. Rodriguez-Suarez, R. O. Cunha, A. R. Rodrigues, F. L. A. Machado, G. A. Fonseca Guerra, J. C. Lopez Ortiz, and A. Azevedo. Magnon spin-current theory for the longitudinal spin- seebeck eect. Phys. Rev. B , 89:014416, 2014. [7] Stephen R Boona and Joseph P Heremans. Magnon thermal mean free path in yttrium iron garnet. Physical Review B , 90(6):064421, 2014. [8] Takashi Kikkawa, Ken-ichi Uchida, Shunsuke Daimon, Zhiyong Qiu, Yuki Shiomi, and Eiji Saitoh. Critical suppression of spin seebeck eect by magnetic elds. Phys. Rev. B , 92:064413, Aug 2015. [9] Er-Jia Guo, Joel Cramer, Andreas Kehlberger, Ciaran A. Ferguson, Donald A. MacLaren, Gerhard Jakob, and Mathias Kl aui. In uence of thickness and interface on the low-temperature enhancement of the spin seebeck eect in yig lms. Phys. Rev. X , 6:031012, Jul 2016. [10] K. Uchida, T. Ota, H. Adachi, J. Xiao, T. Nonaka, Y. Kajiwara, G. E. W. Bauer, S. Maekawa, and E. Saitoh. Thermal spin pump- ing and magnon-phonon-mediated spin-seebeck eect. J. Appl. Phys. , 111:103903, 2012. [11] Asuka Miura, Takashi Kikkawa, Ryo Iguchi, Ken ichi Uchida, Eiji Saitoh, and Junichiro Shiomi. Probing length-scale separation of ther- mal and spin currents by nanostructuring yig. ArXiv , page 1704.07568, 2017. [12] Vittorio Basso, Michaela Kuepferling, Alessandro Sola, Patrizio Ansa- lone, and Massimo Pasquale. The spin seebeck and spin peltier reciprocal relation. IEEE Magnetics Letters , 9:1{4, 2018. 11[13] Kouki Nakata, Pascal Simon, and Daniel Loss. Wiedemann-franz law for magnon transport. Phys. Rev. B , 92:134425, Oct 2015. [14] Vittorio Basso, Elena Ferraro, and Marco Piazzi. Thermodynamic trans- port theory of spin waves in ferromagnetic insulators. Phys. Rev. B , 94:144422, Oct 2016. [15] V. Basso, E. Ferraro, A. Sola, A. Magni, M. Kuepferling, and M. Pasquale. Nonequilibrium thermodynamics of the spin seebeck and spin peltier eects. Phys. Rev. B , 93:184421, 2016. [16] Alessandro Sola, Vittorio Basso, M. Kuepferling, Carsten Dubs, and Massimo Pasquale. Experimental proof of the reciprocal relation be- tween spin peltier and spin seebeck eects in a bulkyig/pt bilayer. Sci- entic reports , 9:2047, 2019. [17] H. L. Wang, C. H. Du, Y. Pu, R. Adur, P. C. Hammel, and F. Y.Yang. Scaling of spin Hall angle in 3d, 4d, and 5d metals from Y 3Fe5O12/metal spin pumping. Phys. Rev. Lett. , 112:197201, 2014. [18] A. Kehlberger, U. Ritzmann, D. Hinzke, E.J. Guo, J. Cramer, G. Jakob, M. C. Onbasli, D. H. Kim, C. A. Ross, M. B. Jung eish, B. Hillebrands, U. Nowak, and M. Kl aui. Length scale of the spin seebeck eect. Phys. Rev. Lett. , 115:096602, 2015. [19] Miren Isasa, Estitxu Villamor, Luis E. Hueso, Martin Gradhand, and Felix Casanova. Temperature dependence of spin diusion length and spin Hall angle in Au and Pt. Phys. Rev. B , 91:024402, 2015. [20] Hannes Maier-Flaig, Stefan Klingler, Carsten Dubs, Oleksii Surzhenko, Rudolf Gross, Mathias Weiler, Hans Huebl, and Sebastian TB Goennen- wein. Temperature-dependent magnetic damping of yttrium iron garnet spheres. Physical Review B , 95(21):214423, 2017. [21] D. D. Stancil and A. Prabhakar. Spin Waves. Theory and Applications . Springer, New York, 2009. [22] P Erd os. Low-temperature thermal conductivity of ferromagnetic insu- lators containing impurities. Physical Review , 139(4A):A1249, 1965. 12[23] H. Adachi, K. Uchida, E. Saitoh, J. Ohe, S. Takahashi, and S. Maekawa. Gigantic enhancement of spin Seebeck eect by phonon drag. Applied Physics Letters , 97:252506, 2010. [24] Rober Brout. Statistical mechanics of ferromagnetism. In Rado and Suhl, editors, Magnetism . Academic Press, 1965. 13

2021-01-12

Based on the relaxation time approximation, the mean collision time for magnon scattering $\tau_c(T)$ is computed from the experimental spin Seebeck coefficient of a bulk YIG / Pt bilayer. The scattering results to be composed by two processes: the low temperature one, with a $T^{-1/2}$ dependence, is attributed to the scattering by defects and provides a mean free path around 10 $\mu$m; the high temperature one, depending on $T^{-4}$, is associated to the scattering by other magnons. The results are employed to predict the thickness dependence of the spin Seebeck coefficient for thin films.

Temperature dependence of the mean magnon collision time in a spin Seebeck device

2101.04405v1

Generation of coherent spin-wave modes in Yttrium Iron Garnet microdiscs by spin-orbit torque M. Collet,1X. de Milly,2O. d'Allivy Kelly,1V. V. Naletov,3, 4R. Bernard,1P. Bortolotti,1V. E. Demidov,5S. O. Demokritov,5, 6J. L. Prieto,7M. Mu~ noz,8V. Cros,1A. Anane,1G. de Loubens,2and O. Klein3 1Unit e Mixte de Physique CNRS/Thales and Universit e Paris Sud 11, 1 av. Fresnel, 91767 Palaiseau, France 2Service de Physique de l' Etat Condens e (CNRS UMR 3680), CEA Saclay, 91191 Gif-sur-Yvette, France 3INAC-SPINTEC, CEA/CNRS and Univ. Grenoble Alpes, 38000 Grenoble, France 4Institute of Physics, Kazan Federal University, Kazan 420008, Russian Federation 5Department of Physics, University of Muenster, 48149 Muenster, Germany 6Institute of Metal Physics, Ural Division of RAS, Yekaterinburg 620041, Russian Federation 7Instituto de Sistemas Optoelectr onicos y Microtecnolog a (UPM), Madrid 28040, Spain 8Instituto de Microelectr onica de Madrid (CNM, CSIC), Madrid 28760, Spain (Dated: August 13, 2018) Spin-orbit eects1{4have the potential of radically changing the eld of spintronics by allowing transfer of spin angular momentum to a whole new class of materials. In a seminal letter to Nature5, Kajiwara et al. showed that by depositing Platinum (Pt, a normal metal) on top of a 1.3 m thick Yttrium Iron Garnet (YIG, a magnetic insulator), one could eectively transfer spin angular momentum through the interface between these two dierent materials. The outstanding feature was the detection of auto-oscillation of the YIG when enough dc current was passed in the Pt. This nding has created a great excitement in the community for two reasons: rst, one could control electronically the damping of insulators, which can oer improved properties compared to metals, and here YIG has the lowest damping known in nature; second, the damping compensation could be achieved on very large objects, a particularly relevant point for the eld of magnonics6,7whose aim is to use spin-waves as carriers of information. However, the degree of coherence of the observed auto-oscillations has not been addressed in ref.5. In this work, we emphasize the key role of quasi- degenerate spin-wave modes, which increase the threshold current. This requires to reduce both the thickness and lateral size in order to reach full damping compensation8, and we show clear evidence of coherent spin-orbit torque induced auto-oscillation in micron-sized YIG discs of thickness 20 nm. When spin transfer eects were rst introduced by Slonczweski and Berger in 19969,10, the authors immedi- ately recognized that the striking signature of the trans- fer process would be the emission of microwave radia- tion when the system is pumped out of equilibrium by a dc current. Since the spin transfer torque on the mag- netisation is collinear to the damping torque, there is an instability threshold when the natural damping is fully compensated by the external ow of angular momentum, leading to spin-wave amplication through stimulated emission. Using analogy to light, the eect was called SWASER10, where SW stands for spin-wave. Until 2010, all SWASER devices required a charge current perpen- dicular to the plane to transfer angular momentum be- tween dierent magnetic layers9,10. This implied that the eect was restricted to conducting materials. The situa- tion has radically changed since spin-orbit eects such as the spin Hall eect (SHE)11,12are used to produce spin currents in normal metals. Here a right hand side rule links the de ected direction of the electron and the orien- tation of its spin. This allows the creation of a pure spin current transversely to the charge current, with an e- ciency given by the spin Hall angle  SH. Using a metal with large  SH, such as Pt, a charge current owing in plane generates a pure spin current owing perpendic- ular to the plane, which can eventually be transferredthrough an interface with ferromagnetic metals, result- ing in the coherent emission of spin-waves13, but also with non-metals such as YIG5. The microscopic mechanisms of transfer of angular mo- mentum between a normal metal and a ferromagnetic layer are quite dierent depending on the latter being metallic or not. In the rst case, electrons in each layer have the possibility to penetrate the other one, whereas in the second case the transfer takes place exactly and solely at the interface. It is thus much more sensitive to the imperfection of the interface. Still, a direct experi- mental evidence that spin current can indeed cross such an hybrid interface is through the so-called spin pump- ing eect14: adding a normal metal on top of YIG in- creases its ferromagnetic resonance (FMR) linewidth15, which is due to the new relaxation channel at the inter- face through which angular momentum can escape and get absorbed in the metal. This eect being interfacial, the broadening scales as 1 =tYIG, wheretYIGis the thick- ness of YIG. Even for YIG, whose natural linewidth is only a few Oersted at 10 GHz, it is hardly observable iftYIGexceeds a couple hundreds of nanometers. For these thick lms though, the spin pumping can still be detected through inverse spin Hall eect (ISHE). In a nor- mal metal with strong spin-orbit interaction, the pumped spin current is converted into a transverse charge current.arXiv:1504.01512v1 [cond-mat.mtrl-sci] 7 Apr 20152 This generates a voltage proportional to the length of the sample across the metal, which can easily reach several tens of microvolts in millimeter-sized samples. Since the rst experiment by Kajiwara et al.5, many studies re- ported the ISHE detection of FMR using dierent metals on YIG layers16{18, hereby providing clear evidence of at least partial transparency of the hybrid YIG jmetal inter- faces to spin currents. Due to Onsager relations, these results made the community condent that a spin cur- rent could thus be injected from metals to YIG and lead to the SWASER eect. From the beginning it was anticipated that the key to observe auto-oscillations in non-metals was to reduce the threshold current. The rst venue is of course to choose a material whose natural damping is very low. In this re- spect YIG is the optimal choice. The second thing is to reduce the thickness since the spin-orbit torque (SOT) is an interfacial eect. This triggered an eort in the fabrication of ultra-thin lms of YIG of very high dy- namical quality19,20. For 20 nm thick YIG lms with damping constant as low as = 2:3
104, a striking re- sult was that there were no evidence of auto-oscillations in millimeter-sized samples at the highest dc current pos- sible in the top Pt layer20,i.e., before it evaporates. It is worth mentioning at this point that reducing further the thickness or the damping parameter of such ultra- thin YIG lms21does not help anymore in decreasing the threshold current, as the relevant value of the damping is that of the YIG jPt hybrid, which ends up to be com- pletely dominated by the spin-pumping contribution. But most notably, none of these high-quality ultra- thin YIG lms display a purely homogeneous FMR line. The reason for that is well known. In such extended lms, there are many degenerate modes with the main, uniform FMR mode, which through the process of two- magnon scattering broaden the linewidth22,23. A striking evidence of these degenerate modes can be obtained by parametrically pumping the SW modes. It reveals an uncountable number of modes which are at the same energy as the FMR mode24. Any threshold instabil- ity will be aected by the presence of those modes, as learnt from LASERs where mode competition is known to have a strong in uence on the emission threshold25. Thus, the next natural step was to reduce as well the lateral size in order to lift the degeneracy between modes through connement. The rst microstructures of YIG appeared revealing that the patterning indeed narrowed the linewidth through a decrease of the inhomogeneous part26. The eect is clear in the perpendicular geometry, where magnon-magnon processes are suppressed owing to the fact that the FMR mode lies at the bottom of the SW dispersion relation. However, this is not the case in the parallel geometry where the FMR mode is not the lowest energy SW mode. Even then, we showed that the linewidth in a micron-sized YIG jPt disc could be still tuned thanks to SOT8. In the following, we describe the direct electrical detection of auto-oscillations in similar samples and show that the threshold current is increased FIG. 1. Inductive detection of auto-oscillations in a YIGjPt microdisc. a , Sketch of the sample and measure- ment conguration. The bias eld His oriented transversely to the dc current Idc owing in Pt. The inductive voltage Vy produced in the antenna by the precession of the YIG mag- netisation is amplied and monitored by a spectrum anal- yser. b{e, Power spectral density maps measured at xed jHj= 0:47 kOe and variable Idc. The four quadrants corre- spond to dierent possible polarities of HandIdc. by the presence of quasi-degenerate SW modes. We study magnetic microdiscs with diameter 2 m and 4m which are fabricated based on a hybrid YIG(20 nm)jPt(8 nm) bilayer. The 20 nm thick YIG layer is grown by pulsed laser deposition20and the 8 nm thick Pt layer is sputtered on top of it27. Their physical parameters are summarized in Table I. We stress that the extended YIG lm is characterized by a low Gilbert damping parameter 0= (4:80:5)
104and a remark- ably small inhomogeneous contribution to the linewidth,
H0= 1:10:3 Oe. Each microdisc is connected to electrodes enabling the injection of a dc current Idcin the Pt layer, and a microwave antenna is dened around it to obtain an inductive coupling with the YIG magneti- sation, as shown schematically in Fig. 1a. First, we monitor with a spectrum analyser the voltage produced in the antenna by potential auto-oscillations of the 4m YIG disc as a function of the dc current Idcin- jected in Pt. The in-plane magnetic eld H= 0:47 kOe3 TABLE I. Physical parameters of the Pt and bare YIG layers, and of the hybrid YIG jPt bilayer. Pt tPt(nm) ( .cm) SD(nm)  SH from ref.278 17 :30:6 3 :40:4 0 :0560:010 YIG tYIG(nm) 0 4Ms(G) (107rad.s1.G1) this study 20 (4 :80:5)
104215050 1 :7700:005 YIGjPt tYIGjtPt(nm)  g "#(1018m2) T this study 20j8 (2 :050:1)
1033:60:5 0 :20:05 FIG. 2. ISHE-detected FMR spectroscopy in YIG jPt microdiscs. a , Sketch of the sample and measurement congu- ration. The bias eld His oriented perpendicularly to the Pt electrode and to the excitation eld hrfproduced by the antenna at xed microwave frequency. The dc voltage Vxacross Pt is monitored as a function of the magnetic eld. b, ISHE-detected FMR spectra of the 4 m and 2m YIG(20 nm)jPt(8 nm) discs at 1 GHz and 4 GHz, respectively. c, Dispersion relation of the main FMR mode of the microdiscs. The continuous line is a t to the Kittel law. d, Frequency dependence of the FMR linewidth in the two microdiscs. The vertical bars show the mean squared error of the lorentzian ts. The continuous lines are linear ts to the data. The dashed line shows the homogeneous contribution of the bare YIG. is applied in a transverse direction with respect to Idc, as shown in Fig. 1a. This is the most favorable con- guration to compensate the damping and obtain auto- oscillations in YIG, as spins accumulated at the YIG jPt interface due to SHE in Pt will be collinear to its magneti- sation. Color plots of the inductive signal measured as a function of the relative polarities of HandIdcare pre- sented in Fig. 1b{e. At H < 0, we observe in the power spectral density (PSD) a peak which starts at around 2.95 GHz and 13 mA and then shifts towards lower fre- quency asIdcis increased (Fig. 1b), a clear signature that spin transfer occurs through the YIG jPt interface. The linewidth of the emission peak, lying in the 10{20 MHz range for 13 < I dc<17 mA, also proves the coherent nature of the detected signal. An identical behaviour is observed at H > 0 andIdc<0 (Fig. 1d). In contrast, the PSD remains at in the two other cases (Fig. 1c{d). Therefore, an auto-oscillation signal is detected only if H
Idc<0, in agreement with the expected symmetry of SHE. In order to characterize the ow of angular momen- tum across the YIG jPt interface, we now perform ISHE- detected FMR spectroscopy on our microdiscs. The con- guration of this experiment is similar to the previous case, but now the antenna generates a uniform microwave eldhrfto excite the FMR of YIG while the dc voltage across Pt is monitored at zero current (see Fig. 2a). In other words, we perform the reciprocal experiment of theone presented in Fig. 1. As described in the introduction, a voltageVISHE develops across Pt when the FMR con- ditions are met in YIG. This voltage changes sign as the eld is reversed, which is expected from the symmetry of ISHE, and shown in Fig. 2b, where the FMR spectra of the 4m and 2m microdiscs are respectively detected at 1 GHz and 4 GHz. We also note that for a given eld polarity, the product between VISHE andIdcmust be negative to compensate the damping8, which enables to observe auto-oscillations in Fig. 1. From these ISHE measurements, the dispersion rela- tion and frequency dependence of the full linewidth at half maximum of the main FMR mode can be deter- mined, as shown in Figs. 2c and 2d, respectively. The dispersion relation follows the expected Kittel law. The damping parameters of the 4 m and 2m microdiscs, extracted from linear ts to the data,
H= 2!= +
H0(continuous lines in Fig. 2d, !is the pulsation fre- quency and the gyromagnetic ratio), are found to be similar with an average value of = (2:050:1)
103. The small inhomogeneous contribution to the linewidth observed in both microdiscs,
H0= 1:30:4 Oe and
H0= 0:70:4 Oe, respectively, decreases with the diameter and is attributed to the presence of several un- resolved modes within the resonance line8. In order to emphasize the increase of damping due to Pt, we have reported in Fig. 2d the broadening produced by the ho- mogeneous contribution of the bare YIG using a dashed4 line. The observed increase of damping is due to spin pumping14,15, 0=g"# ~ 4MstYIG; (1) where ~is the reduced Planck constant, Msthe satu- ration magnetisation and g"#the spin-mixing conduc- tance of the YIGjPt interface. This allows us to extract g"#= (3:60:5)
1018m2, which lies in the same window as previously reported values26,28. From the spin-mixing conductance g"#, the spin diusion length SDand the resistivityof the Pt layer, we can also calculate the transparency of the YIG jPt interface to spin current29, T= 0:20:05. The physical parameters extracted for the YIGjPt hybrid bilayer are summarized in the last raw of Table I. To gain further insight about the origin of the auto-oscillation signal, we now monitor how the auto- oscillations of the 4 m disc evolve as the angle be- tween the in-plane bias eld xed to H= 0:47 kOe and the dc current Idcis varied from 30to 150. The re- sults are summarized in Fig. 3. Pannels b{d show the auto-oscillation voltages detected in the antenna ( Vy) and across the Pt electrode ( Vx). At= 90, the auto- oscillation signal is only visible in the Vychannel. At = 60, bothVxandVychannels exhibit the auto- oscillation peak. At = 40, it almost vanishes in Vy, while it slightly increases in Vx. The normalized signals as a function of are plotted in Fig. 3e. The Pt elec- trode and antenna loop being oriented perpendicularly to each other (see Fig. 3a), the ac ux due to the precession of magnetisation picked up by each of them respectively varies as cos and sin(dashed lines in Fig. 3e). More importantly, this study of angle dependence also allows us to extract the threshold current for auto- oscillations as a function of . Asdeviates from the optimal orientation 90, the absolute value of the thresh- old current rapidly increases, see Fig. 3f. In fact, the SOT acting on the oscillating part mof the magneti- sation scales as msm/sin, where sis the spin polarisation of the dc spin current produced by SHE in Pt at the YIGjPt interface. Therefore, the expected thresh- old current scales as 1 =sin, which is plotted as a dashed line in Fig. 3f, in very good agreement with the data. In summary, the results reported in Figs. 1 and 3 un- ambiguously demonstrate that the auto-oscillations ob- served in our hybrid YIG jPt discs result from the ac- tion of SOT produced by Idc. We have also shown that they correspond to the reverse eect of the spin pump- ing mechanism illustrated in Fig. 2d and its detection through ISHE in Fig. 2b. In the last part of this letter, we analyse quantitatively the main features of auto-oscillations, which allows us to determine their nature and to understand the role of quasi-degenerate SW modes in the SOT driven dynamics. For this, we compare the auto-oscillations observed in the 4m and 2m microdiscs. Figs. 4a and 4b re- spectively present the inductive signal Vydetected in the FIG. 3. Auto-oscillations as a function of the angle between the dc current and the bias eld. a , Sketch of the sample and measurement conguration. The bias eld His oriented at an angle with the dc current Idcin the Pt. The precession of the YIG magnetisation induces volt- agesVxin the antenna and Vyacross Pt, which are amplied and monitored by spectrum analysers. b{d,VxandVyat H= 0:47 kOe for three dierent angles .e, Dependence of the normalized signals in both circuits and f, of the thresh- old current for auto-oscillations on . Dashed lines show the expected angular dependences. antenna coupled to these two discs as a function of Idc. The conguration is the same as depicted in Fig. 1a, with a slightly larger bias eld set to H= 0:65 kOe. One can clearly see a peak appearing in the PSD close to 3.6 GHz in both cases, at a threshold current of about 13:5 mA in the 4 m disc and7:4 mA in the 2 m disc. These two values correspond to a similar threshold current density in both samples of (4 :40:2)
1011A.m2, in agreement with our previous study8. As the dc cur- rent is varied towards more negative values, the peaks shift towards lower frequency (Fig. 4c), at a rate which is twice faster in the smaller disc. This frequency shift is mainly due to linear and quadratic contributions in Idcof Oersted eld and Joule heating, respectively8(from the Pt resistance, the maximal temperature increase in both5 FIG. 4. Quantitative analysis of auto-oscillations in YIG jPt microdiscs .a, Inductive voltage Vyproduced by auto- oscillations in the 4 m and b, 2m YIGjPt discs as a function of the dc current Idcin the Pt. The experimental conguration is the same as in Fig. 1a, with the bias eld xed to H= 0:65 kOe. c, Auto-oscillation frequency, d, linewidth and e, integrated power vs.Idc.f, Dependence of the onset frequency and g, of the threshold current on the applied eld in both discs. Expectations taking into account only the homogeneous linewidth or the total linewidth are respectively shown by dashed and continuous lines. samples is estimated to be +40C). At the same time, the signal rst rapidly increases in amplitude, reaches a maximum, and then, more surprisingly, drops until it cannot be detected anymore, as seen in Fig. 4e, which plots the integrated power vs.Idc. The maximum of power measured in the 4 m disc (2.9 fW) is four times larger than the one measured in the 2 m disc (0.7 fW), which is due to the inductive origin of Vy. The lat- ter can be estimated from geometrical considerations, Vy=(!0DtYIGMssin)=2. Here0is the magnetic constant,Dthe diameter of the disc and the angle of uniform precession (the prefactor '0:1 accounts for microwave losses and impedance mismatch in the mea- sured frequency range with our microwave circuit). For the same, the inductive voltage produced by the 4 m disc is thus twice larger than by one produced by the 2m disc, hence the ratio four in power. Moreover, the maximal angle of precession reached by auto-oscillations is found to be about 1in both microdiscs8. Finally, the disappearance of the signal as Idcgets more nega- tive is accompanied by a continuous broadening of the linewidth, which increases from a few MHz to several tens of MHz (Fig. 4d). This rather large auto-oscillation linewidth is also consistent with a small precession angle, i.e., a small stored energy in the YIG oscillator30. By repeating the same analysis as a function of H, we can determine the bias eld dependence of the auto- oscillations in both microdiscs. The onset frequency and threshold current at which auto-oscillations start are plotted in Figs. 4f and 4g, respectively. The onset fre- quency in the 4 m and 2m microdiscs is identical and closely follows the dispersion relation of the main FMR mode plotted as a continuous line. The small redshift to- wards lower frequency, which increases with the applied eld, is ascribed to the Joule heating and Oersted eld in- duced byIdc(the Kittel law in Figs. 2c and 4f is obtained atIdc= 0 mA). We also note that the main FMR mode is the one which couples the most eciently to our induc- tive electrical detection, because it is the most uniform.Hence, we conclude that the detected auto-oscillations are due to the destabilisation of this mode by SOT. In order to reach auto-oscillations, the additional damping term due to SOT has to compensate the natural relaxation rate rin YIG. Given the transparency Tof the YIGjPt interface and the spin Hall angle  SHin Pt, this condition writes: TSH~ 2e tYIGMsIth tPtD=r; (2) wheretPtDis the section of the Pt layer. The homoge- neous contribution to ris given by the Gilbert damping rate, which for the in-plane geometry is: G= (H+ 2Ms): (3) We remind that this expression is obtained by convert- ing the eld linewidth to frequency linewidth through
!=
H(@!=@H ). If only the homogeneous contribu- tion to the linewidth is taken into account, the threshold currentIthis thus expected to depend linearly on H, as shown by the dashed lines plotted in Fig. 4g using Eqs. 2 and 3, and the parameters listed in Table I (the only adjustment made is for the 4 m disc, where the calcu- latedIthhas been reduced by 20% in order to reproduce asymptotically the experimental slope of Ithvs.H). It qualitatively explains the dependence of Ithat large bias eld in both microdiscs, but underestimates its value and fails to reproduce the optimum observed at low bias eld. To understand this behaviour, the nite inhomoge- neous contribution to the linewidth
H0measured in Fig. 2d should be considered as well. In fact, this contri- bution dominates the full linewidth at low bias eld. In that case, the expression of the relaxation rate writes: r= G+
H0 2H+ 2Msp H(H+ 4Ms): (4) The form of the last term in Eq. 4 is due to the Kittel dispersion relation and is responsible for the existence6 of the optimum in IthatH6= 0. Using the value of
H0= 0:7 Oe extracted in Fig. 2d for the 2 m disc in Eq. 4 in combination with Eq. 2, the continuous blue line of Fig. 4g is calculated, in very good agreement with the experimental data. To get such an agreement for the 4m disc,
H0has to be increased by 25% compared to the value determined in Fig. 2d. In this case, both the position of the optimum (observed at H'0:30:5 kOe) and the exact value of Ithare also well reproduced for the 4m disc, as shown by the continuous red line in Fig. 4g. Hence, it turns out that quasi-degenerate SW modes, which are responsible for the inhomogeneous contribu- tion to the linewidth, strongly aect the exact value and detailed dependence vs.HofIth. In fact, it is the to- tal linewidth that truly quanties the losses of a mag- netic device regardless of the nature and number of mi- croscopic mechanisms involved. Even in structures with micron-sized lateral dimensions, there still exist a few quasi-degenerate SW modes as evidenced by the nite
H0observed in Fig. 2d. Due to magnon-magnon scat- tering, they are linearly coupled to the main FMR mode, which as a result has its eective damping increased, along with the threshold current. The presence of these SW modes is also known to play a crucial role in SOT driven dynamics. The strongly non-equilibrium distri- bution of SWs promoted by SOT in combination with nonlinear interactions between modes can lead to mode competition, which might even prevent auto-oscillations to start31. We believe that the observed behaviours of the integrated power (Fig. 4d) and linewidth (Fig. 4e) vs.Idcare reminiscent of the presence of these quasi- degenerate SW modes. A meaningful interpretation of these experimental results is that as the FMR mode starts to auto-oscillate and to grow in amplitude as the dc current is increased above the threshold, its coupling to other SW modes { whose amplitudes also grow due to SOT { becomes larger, which makes the ow of energy out of the FMR mode more ecient. This reduces the inductive signal, as non-uniform SW modes are poorly coupled to our inductive detection scheme. At the same time, it enhances the auto-oscillation linewidth, which re ects this additional nonlinear relaxation channel. The smaller inhomogeneous linewidth in the 2 m disc (Fig. 2d) results in a eld dependence of the threshold current closer to the one expected for the purely homo- geneous case (Fig. 4g). This indicates that reducing further the lateral size of the microstructure will allow to completly lift the quasi-degeneracy between spin-wave modes26, as predicted by micromagnetic simulations, which show that this is obtained for lateral sizes smaller than 1m. This could extend the stability of the auto- oscillation for the FMR mode, and experimental tech- niques capable of detecting SWs in nanostructures8,31 should be used to probe this transition. Very importantly for the eld of magnonics, it was recently shown that this constraint on connement could be relaxed in one dimen-sion such as to produce a propagation stripe32. Other strategies might consist in using specic non-uniform SW modes or to engineer the SW spectrum using topological singularities such as vortices, or bubbles, which could be most relevant to design active magnonics computational circuits. METHODS Samples { Details of the PLD growth of the YIG layer can be found in ref.20. Its dynamical proper- ties have been determined by broadband FMR measure- ments. The transport parameters of the 8 nm thick Pt layer deposited on top by magnetron sputtering have been determined in a previous study27. The YIGjPt mi- crodiscs are dened by e-beam lithography, as well as the Au(80 nm)jTi(20 nm) electrodes { separated by 1 m from each other { which contact them. This electrical circuit is insulated by a 300 nm thick SiO 2layer, and a broadband microwave antenna made of 250 nm thick Au with a 5m wide constriction is dened on top of each disc by optical lithography. Measurements { The samples are mounted between the poles of an electromagnet which can be rotated to vary the angle shown in Fig. 3a. Two 50 matched picoprobes are used to connect to the microwave antenna and to the electrodes which contact the Pt layer. The lat- ter are connected to a dc current source through a bias- tee. To perform ISHE-detected FMR measurements, a microwave synthesizer is connected to the microwave an- tenna, and the output power is turned on and o at a modulation frequency of 9 kHz. The voltage across Pt is measured by a lock-in after a low-noise preamplier (gain 100). For the detection of auto-oscillations, high- frequency low-noise ampliers are used (gain 33 dB to 39 dB, depending on the frequency range). Two spec- trum analysers simultaneously monitor in the frequency domain the voltages VxandVyacross the Pt layer and in the microwave antenna, respectively (Fig. 3a). The resolution bandwidth employed in the measurements is set to 1 MHz. ACKNOWLEDGMENTS We acknowledge E. Jacquet, R. Lebourgeois and A. H. Molpeceres for their contribution to sample growth, and M. Viret and A. Fert for fruitful discussion. This research was partially supported by the ANR Grant Trinidad (ASTRID 2012 program). V. V. N. acknowledges sup- port from the program CMIRA'Pro of the region Rh^ one- Alpes. 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2015-04-07

Spin-orbit effects [1-4] have the potential of radically changing the field of spintronics by allowing transfer of spin angular momentum to a whole new class of materials. In a seminal letter to Nature [5], Kajiwara et al. showed that by depositing Platinum (Pt, a normal metal) on top of a 1.3 $\mu$m thick Yttrium Iron Garnet (YIG, a magnetic insulator), one could effectively transfer spin angular momentum through the interface between these two different materials. The outstanding feature was the detection of auto-oscillation of the YIG when enough dc current was passed in the Pt. This finding has created a great excitement in the community for two reasons: first, one could control electronically the damping of insulators, which can offer improved properties compared to metals, and here YIG has the lowest damping known in nature; second, the damping compensation could be achieved on very large objects, a particularly relevant point for the field of magnonics [6,7] whose aim is to use spin-waves as carriers of information. However, the degree of coherence of the observed auto-oscillations has not been addressed in ref. [5]. In this work, we emphasize the key role of quasi-degenerate spin-wave modes, which increase the threshold current. This requires to reduce both the thickness and lateral size in order to reach full damping compensation [8] , and we show clear evidence of coherent spin-orbit torque induced auto-oscillation in micron-sized YIG discs of thickness 20 nm.

Generation of coherent spin-wave modes in Yttrium Iron Garnet microdiscs by spin-orbit torque

1504.01512v1

Room temperature and low-eld resonant enhancement of spin Seebeck eect in partially compensated magnets R. Ramos,1,T. Hioki,2Y. Hashimoto,1T. Kikkawa,1, 2P. Frey,3A. J. E. Kreil,3V. I. Vasyuchka,3A. A. Serga,3B. Hillebrands,3and E. Saitoh1, 2, 4, 5, 6 1Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 2Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 3Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universit at Kaiserslautern, 67663 Kaiserslautern, Germany 4Department of Applied Physics, The University of Tokyo, Tokyo 113-8656, Japan 5Center for Spintronics Research Network, Tohoku University, Sendai 980-8577, Japan 6Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan (Dated: November 10, 2021) Abstract Resonant enhancement of spin Seebeck eect (SSE) due to phonons was recently discovered in Y 3Fe5O12(YIG). This eect is explained by hybridization between the magnon and phonon dispersions. However, this eect was observed at low temperatures and high magnetic elds, limiting the scope for applications. Here we report observation of phonon-resonant enhancement of SSE at room temperature and low magnetic eld. We observed in Lu 2BiFe 4GaO 12an enhancement 700 % greater than that in a YIG lm and at very low magnetic elds around 101T, almost one order of magnitude lower than that of YIG. The result can be explained by the change in the magnon dispersion induced by magnetic compensation due to the presence of non-magnetic ion substitutions. Our study provides a way to tune the magnon response in a crystal by chemical doping with potential applications for spintronic devices. 1arXiv:1903.09007v1 [cond-mat.mtrl-sci] 21 Mar 2019Heat is an ubiquitous and highly underexploited energy source, with about two-thirds of the used energy being lost as wasted heat1, thus representing an opportunity for ther- moelectric conversion devices2. Recently, a new thermoelectric conversion mechanism has emerged: the spin Seebeck eect (SSE)3,4, driven by thermally induced magnetization dy- namics in magnetic materials, which generates a spin current. The spin current is then injected into an adjacent metal layer, where it is converted into an electric current by the inverse spin Hall eect5,6. Lately, it has been shown that the magnetoelastic coupling can improve the conversion ef- ciency of SSE, as demonstrated by the observation of a resonant enhancement of the SSE voltage in YIG lms7,8. The observed SSE enhancement has been explained by the magnon- phonon hybridization at the crossing points of the magnon and phonon dispersions due to momentum ( k) and energy ( h!) matching, at certain magnetic eld values the dispersions tangentially touch each other and the magnon-phonon hybridization eects are maximised [see Figure 1(a)], resulting in peak structures due to the reinforcement of the magnon lifetime aected by the phonons. However, this eect has only been observed at low temperatures and high magnetic elds7{14. Engineering the magnon dispersion oers the possibility to tune the magnetic eld at which the magnon-phonon hybridization eects are maximised, one possible approach is modica- tion of the magnetic compensation of a ferrimagnet15,16: intuitively, it can be expected that as the magnetic compensation of the system is introduced, the magnon dispersion gradually evolves from a parabolic k-dependence in the non-compensated state (similar to a ferro- magnet) towards a linear dispersion when the system becomes fully compensated (i.e. zero net magnetization, antiferromagnetic case), schematically shown in gure 1(b). Ideally, in the antiferromagnetic state in which spin-wave velocity is same as the sound velocity the magnon-phonon coupling eects can be maximised even further as a consequence of poten- tially larger overlap between the magnon and phonon dispersions, however this is yet to be experimentally observed. Here, in order to investigate the in uence of increased magnetic compensation on the magnon-phonon coupling eects, we use a Lu 2BiFe 4GaO 12(BiGa:LuIG) lm as a model system and study the magnetic eld and temperature dependences of the SSE. This system is a garnet ferrite, similar to YIG: it has two magnetic sublattices, where all the ions carrying spin angular momentum are Fe3+. The ferrimagnetic order originates from the dierent site 2occupation between the two dierent magnetic sublattices, with the 3:2 ratio of tetrahedral (d) to octahedral (a) sites, resulting in non-zero magnetization. Substitution of Fe with Ga reinforces the magnetic compensation of the system due to the preferential occupation of the tetrahedral sites by the Ga ions17, resulting in the reduction of the magnetic moment and ordering temperature (see methods). The use of BiGa:LuIG lm allows systematic evaluation of spin wave dispersion using magneto-optical spectroscopy18,19. I. RESULTS Room temperature resonant enhancement of the SSE. We performed SSE mea- surements in the longitudinal SSE conguration, as schematically depicted in Fig. 2a. The magnetic-eld dependence of the SSE voltage measured at 300 K is shown in Fig. 2b. We observed clear peaks in the voltage at magnetic eld values of 0HTA0.42 T and 0HLA 1.86 T, as shown in the data blown ups in Fig.2c and Fig.2d. These correspond to the hy- bridization of the magnons with the transversal acoustic (TA) and longitudinal acoustic (LA) phonons, respectively. The observation of the clear enhancement of the SSE voltage at room temperature is in a stark contrast to previous observations of the magnon-polaron SSE in YIG and other ferrite systems7,9{11,13. In fact, the observed enhancement in BiGa:LuIG is about 700% greater than that observed in a YIG lm at room temperature (see Fig. 2e). Moreover, the features of the resonant SSE enhancement are already present at very low eld values, as visible in the inset of Fig. 2b, where we can see a larger background SSE signal forH <H TA. Let us consider possible reasons for the observation of clear magnon-polaron SSE peaks at room temperature. The energy transfer between phonon (heat) and spin system via magnon-phonon coupling is resonantly enhanced at the crossing of the magnon and phonon dispersion relations. In the theory of the magnon-polaron SSE8, the peaks in the voltage are understood in terms of increased magnon-phonon hybridization at the magnetic eld values where the magnon and phonon dispersions just tangentially touch each other. At these positions the overlap over momentum space between the magnon and the TA or LA phonon dispersions is maximised. This results in a resonant enhancement of the SSE when the acoustic quality of the crystal is larger than its magnetic quality, with the enhancement proportional to the ratio =jph=magj>1, wheremagandphare the values of magnon 3and phonon lifetime7,8, respectively. In the case of a crystal having a better magnetic than acoustic quality (  <1), dip structures instead of peaks in the SSE voltage are expected8. Therefore, according to the picture above, there is two possible scenarios to explain the larger magnon-polaron SSE peaks observed at room temperature: (1) an increased overlap overk-space at the touching elds ( HTA, LA ) or (2) a larger ratio between the magnetic and acoustic quality of the crystal ( 1). The rst scenario can be discarded in our system, since the eect of larger magnetic compensation increases the curvature of the magnon dispersion (obtained in the next section), which results in a rather reduced overlap between the magnon-phonon dispersions at the touching points (see Supplementary Note 3). Then we must look at the scenario (2): it has been shown that the Bi substitution results in a decrease in the magnon lifetime in YIG20. As a consequence, the larger ratio between the impurity scattering potentials, or , can be expected, and therefore a greater enhancement of the lifetime of magnons by hybridization with the phonons at the touching points, resulting in a greater SSE enhancement. Detailed knowledge of the magnon and phonon lifetimes, mag, ph , is required in order to quantitatively discuss the magnitude of the resonant enhancement of the SSE. We will now centre the rest of our discussion mainly on the magnetic eld dependence of the magnon-polaron peaks and its relation to the dispersion characteristics. Let us now try to understand the magnitude of the magnetic elds required for the observation of the SSE peaks by considering the magnon and phonon dispersions. First, we consider linear TA and LA phonon dispersions !p=cTA, LAk, where the phonon velocities of the sample have been determined using optical spectroscopy18, obtaining: cTA= 2.9103ms1and cLA=6.2103ms1for TA and LA modes, respectively. If we now assume the conventional magnon dispersion for a simple ferromagnet: !m= 0H+Dexk2, as previously used for YIG7, we can never explain the observed results without assuming an exceedingly large spin wave stiness parameter, with a value about four times higher than previously reported for YIG (D ex= 7.7106m2s1)7,21,22and LuIG23. Even if we take into account the eect of Bi-doping, this can only explain a 1.4 times increase of the spin wave stiness magnitude, as shown in previous reports for similar Bi-doping in YIG20. Moreover, if we consider the expression of the spin wave stiness for a ferromagnet Dex/JSa2(hereJcorresponds to the exchange constant, Sthe spin of the magnetic atoms and athe distance between neighboring spins), this estimation would imply an increased magnitude of Jdespite the 4presence of non-magnetic ion substitutions, contrary to expectations. This shows that the magnon dispersion of simple ferromagnets cannot capture the microscopic features of our system, therefore in the following we will focus our attention on the ferrimagnetic ordering: eects of the Ga-doping on the magnetic compensation, and its impact on the magnon dispersion characteristics of our system. Magnon dispersion: theoretical model and experimental determination. We will now discuss the magnon dispersion characteristics of a ferrimagnetic material as a func- tion of the degree of magnetic compensation. As previously explained, the ferrimagnetic order in iron garnet systems typically arises from the dierent Fe3+occupation between the two magnetic sublattices, with the 3 to 2 ratio of tetrahedral (d) to octahedral (a) sites, as shown in Fig. 1b. Therefore, the degree of magnetic compensation can be modied by substitution of the magnetic Fe3+ions in the tetrahedral sites by non-magnetic ones (i.e. Ga). To describe this eect, we consider the conventional Heisenberg Hamiltonian for a ferrimagnetic system15,24, and express it as a function of the occupation numbers in the d and a sites. The expression of the Hamiltonian with the exchange and the Zeeman interaction terms is given below, where we have neglected dipolar and magnetic anisotropy interactions for simplicity: H=Jad h2NaX i;Sa;iSd;i+ 0HNaX iSz a;i +Jad h2NdX j;Sa;j+Sd;j 0HNdX jSz d;j: (1) Here, the upper (lower) part of the Hamiltonian accounts for the octrahedral (tetrahedral) sites, with Na(Nd) magnetic ions per unit volume. Jadis the nearest-neighbour inter- sublattice exchange (positive in the above expression), represents a vector connecting the nearest-neighbour a-d sites, Sx(x = a, d) are the spin operators (for a, d sites), 0His the external magnetic eld, and =gB=his the gyromagnetic ratio, with gthe spectroscopic splitting factor, Bthe Bohr magneton,  hthe reduced Planck constant. Then, using the standard Holstein-Primako approximation24,25, we can obtain the magnon dispersion rela- tion as a function of the ratio of Fe3+ions occupying a and d sites (see Supplementary Note 51 for details of the derivation): !m= 0H+JadS(zda+zad) 2h8 < :  " 2 +4k2a2 3#1=29 = ;;(2) whereS= 5=2 for Fe3+,ais the nearest-neighbor a-d distance, zad(zda) and=Na=N (=Nd=N) correspond to the number of nearest-neighbours and occupation ratio of mag- netic ions in octahedral (tetrahedral) sites, respectively. Where Nis the total number of magnetic Fe3+ions per unit volume. Using the above expression we calculate the magnon dispersion in the two sublattice ferrimagnet: Lu 2Bi[AxFe2x](DyFe3y)O12, where A (D) de- notes the non-magnetic ions in octahedral (tetrahedral) sites with concentrations x(y), the occupation ratio of the a (d) sites can be expressed as a function of x(y) as:= 0:42x 2
(= 0:63y 3
)26. Equation 2 can explain the magnetic eld values of previously observed magnon-polaron SSE in YIG7withx=y= 0 (no substitutions). Comparing our model with magnon-polaron SSE measurements in YIG at room temperature10, we estimated the value of the exchange constant at 300 K, obtaining Jad= (4:30:2)1022J (see Sup- plementary Note 2), which shows reasonable agreement with recently reported values by neutron scattering measurements ( Jad= 4:651022J)27. We now evaluate the eect of the non-magnetic ion substitutions on the magnon spectrum, we can see that as the magnetic compensation of the system increases (larger y) the dis- persion becomes gradually steeper (see Fig. 3a), with higher magnon frequencies for the same wave-vector values. When the system is fully compensated ( x= 0,y= 1) a linear dispersion is obtained, as expected for an antiferromagnetic system. To further test the validity of our model, we also performed wave-vector-resolved Brillouin light scattering (BLS) spectroscopy measurements19,28,29to obtain the spin wave dispersion relation of our system and compare it with our model. As shown in Fig. 3b, the peak frequency in BLS spectra for dierent wavenumbers measured at 0H= 0:18 T can be explained with the calculated magnon dispersion using x= 0:101 andy= 0:909. This com- position is in close agreement with the one estimated by x-ray spectroscopy (see Methods). This result further proves the good agreement between the experiment and the theoretical model. Note that the small deviation of BLS plots at small kregion is due to dipole-dipole interaction, which is not considered in our model. Now we are in a position to look into the magnetic eld dependence of the SSE voltage and 6the conditions for the observation of magnon-polaron in our system. The low magnetic eld value for the observation of the magnon polaron SSE is attributed to an increased magnetic compensation which makes the magnon dispersion steeper compared to the non-doped case. This results in the touching condition for the magnon and phonon dispersions at lower magnetic elds than that of YIG. Figures 3c and 3d show the comparison of the magnon and phonon dispersions with the above estimated composition at the magnetic elds HTA andHLAfor which the SSE peaks are observed. In Fig. 3c, we can distinguish three dierent regions of the SSE response with respect to the magnitude of the magnetic eld: forH < H TAthe SSE has both magnonic and phononic contributions (possibly from the crossing points of the dispersions), at H=HTAthe SSE is resonantly enhanced showing peak structures, due to the increased eect of magnon-phonon hybridization at the touching condition between the dispersions and at H >H TA, a shift of the SSE voltage background signal can be observed, which is likely due to the fact that the magnon and TA-phonon dispersions do not cross at any point of the !kspace and the contribution from the TA phonons is suppressed. The presence of this shift in the SSE voltage indicates that the contribution from magnon-phonon coupling eects can be already sensed at magnetic elds values much lower than that of the touching condition (see inset of Fig. 2b). Temperature dependence. Let us now investigate the temperature dependence of the magnon-polaron SSE. Magnetization measurements as a function of temperature show that the increased magnetic compensation, due to the presence of Ga substitution, results in the reduction of the saturation magnetization and ferrimagnetic ordering temperature ( Tc) in LuIG with Tc= 401.60.2 K (see Fig.4b). Figure 4a shows the result of the magnetic- eld dependent SSE voltages at dierent temperatures, we can see that the magnitude of the SSE decreases close to the transition temperature as previously reported on YIG30, and eventually, at T400 K, the magnon-driven SSE is suppressed, showing only a weak voltage with a paramagnetic-like magnetic eld dependence. The magnon-polaron SSE is gradually weakened as the temperature increases toward the transition temperature. The peaks are strongly suppressed at temperatures well below the transition temperature (the maximum temperature for observation of the peaks is T = 380 K for TA and T = 315 K for LA phonon). Here, we will now focus on the temperature dependence of the magnetic eld at which the magnon-polaron peaks are observed, which is shown in Fig. 4c: we can see that the 7magnitude of the magnetic elds required for the magnon-polaron to appear increase with the temperature. This trend can be understood by the softening of the magnon dispersion upon increasing temperature; in the previously obtained magnon dispersion (Eq. 2), most of the parameters are temperature independent except for the intersite exchange energy Jad. The observed temperature dependence of the magnon polaron magnetic eld can be explained in terms of the magnitude of Jadwhich decreases around the transition temperature, in agreement with the previous observations in YIG and other ferrimagnets31{33. The temperature dependence of Jadestimated from the touching condition between magnon and phonon dispersions is shown in Fig. 4d, the obtained dependence can be understood in terms of a temperature dependent exchange energy, with an expression similar to that used in Ref. 38: Jad=J0(1T5=2), withJ0= (6:410:05)1022J, which is obtained from considering the eect of magnon-magnon interactions in a ferromagnet34. These results possibly suggest that the magnon-magnon interactions might play a role in the magnon- polaron SSE in the temperature region studied here. II. DISCUSSION. In this study, we reported the resonant enhancement of SSE in a partially compensated ferrimagnet. Sharp peaks were observed in SSE voltage at room temperature and low mag- netic elds. The resonant enhancement of SSE is 700 % greater than that observed in YIG lms, atributable to reduced magnon lifetime of BiGa:LuIG in comparison to YIG, which results in larger reinforcement of the magnon lifetimes aected by the phonon system via the hybridization. The observed resonant enhancement of SSE at low magnetic elds is attributable to steeper magnon dispersion caused by the increased magnetic compensation of the system. Our results show the possibility to tune the spin wave dispersion by chemical doping, which allows exploring magnon-phonon coupling eects at dierent regions of the spin-wave spec- trum. The value of the intersite exchange parameter ( Jad) was also estimated with values in reasonable agreement with previous studies. This fact shows the potential of the SSE as a table-top tool to investigate the spin wave dispersion characteristics in comparison to other more expensive and less accessible techniques, such as inelastic neutron scattering9,35. 8III. METHODS A. Sample We used epitaxial Lu 2BiFe 4GaO 12(3m) and YIG(2.5 m) lms grown by liquid phase epitaxy on Gd 3Ga5O12[001] and [111]-oriented substrates, respectively. The composition of the Lu 2BiFe 4GaO 12lm was determined using wavelength dispersive x-ray spectroscopy (WDX). Magnetization measurements show a saturation magnetization 0MS= 0.024 T at 300 K and a transition temperature of TC402 K. A 5 nm Pt layer was deposited at room temperature by (DC) magnetron sputtering in a sputtering system QAM4 from ULVAC, with a base pressure of 105Pa. The sample dimensions for the SSE measurements were Ly= 6 mm,Lx= 2 mm and Lz= 0:5 mm. B. Measurements The SSE measurements were performed in a physical property measurement (PPMS) Dynacool system of Quantum Design, Inc., equipped with a superconducting magnet with elds of up to 9 Tesla. The system allows for temperature dependent measurements from 2 to 400 K. For the SSE measurements the sample is placed between two plates made of AlN (good thermal conductor and electrical insulator): a resistive heater is attached to the upper plate and the lower plate is in direct contact with the thermal link of the cryostat, providing the heat sink. The temperature gradient is generated by applying an electric current to the heater, while the temperature dierence between the upper and lower plate is monitored by two E-type thermocouples connected dierentially. The samples are contacted by Au wire of 25m diammeter. To minimize thermal losses, the wires are thermally anchored to the sample holder. The thermoelectric voltage is monitored with a Keithley 2182A nanovolt- meter. The magnetization measurements were performed using the VSM option of a PPMS system by Quantum Design, Inc, the temperature dependent magnetization measurements were obtained by performing isothermal M-H loops at each temperature and extracting the sat- uration magnetization value for each of the temperatures. The Brillouin light scattering (BLS) measurements were performed using an angle-resolved Brillouin light scattering setup. Brillouin light scattering is an inelastic scattering of light 9due to magnons. As a result, some portion of the scattered light shifts in the frequency equivalent to that of magnon. The scattered light is introduced to multi-pass tandem Fabry- Perot interferometer to determine the frequency shift. Wavenumber resolution was realized by collecting the back-scattered light from the sample by changing the incident angle ( in). As a result of conservation of wavenumber of magnon ( km) and light ( kl), the wavenum- ber of magnon is determined as km= 2klsin(in). All the spectrum was obtained at room temperature. IV. ACKNOWLEDGMENTS We thank J. Barker for fruitful discussions. This work was supported by ERATO \Spin Quantum Rectication Project" (Grant No. JPMJER1402) and Grant-in-Aid for Scientic Research on Innovative Area, \Nano Spin Conversion Science" (Grant No. JP26103005), Grant-in-Aid for Research Activity Start-up (No. JP18H05841) from JSPS KAKENHI, JSPS Core-to-Core program \the International Research Center for New-Concept Spintron- ics Devices" Japan, the NEC Corporation and the Noguchi Institute. V. AUTHOR CONTRIBUTIONS R.R. performed the measurement, analysed the data and developed the theoretical model with 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 the BLS measurements. T.K. performed the SSE measurements in YIG. R.R., Y.H. and E.S. planned and supervised the study. R.R. wrote the manuscript with review and input from T.H., Y.H., T.K. and E.S. All authors discussed the results and commented on the manuscript. ramosr@imr.tohoku.ac.jp 1K. Biswas, J. He, I. D. Blum, C-I. Wu, T. P. Hogan, D. N Seidman, V. P. Dravid, and M. G. Kanatzidis, \High-performance bulk thermoelectrics with all-scale hierarchical architectures," Nature 489, 414 (2012). 102J. He and T. M. Tritt, \Advances in thermoelectric materials research: Looking back and moving forward," Science 357, eaak9997 (2017). 3K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh, \Observation of the spin Seebeck eect," Nature 455, 778 (2008). 4K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi, J. Ieda, T. Ota, Y. Kajiwara, H. Umezawa, H. Kawai, G. E. W. Bauer, S. Maekawa, and E. Saitoh, \Spin Seebeck insulator," Nat. Mater. 9, 894 (2010). 5A. Homann, \Spin Hall Eects in Metals," IEEE Trans. Magn. 49, 5172 (2013). 6J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, \Spin Hall eects," Rev. Mod. 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Commun. 7, 10452 (2016). 17P. Hansen, C.-P. Klages, J. Schuldt, and K. Witter, \Magnetic and magneto-optical properties of bismuth-substituted lutetium iron garnet lms," Phys. Rev. B 31, 5858 (1985). 18Y. Hashimoto, S. Daimon, R. Iguchi, Y. Oikawa, K. Shen, K. Sato, D. Bossini, Y. Tabuchi, T. Satoh, B. Hillebrands, G. E. W. Bauer, T. H. Johansen, A. Kirilyuk, T. Rasing, and E. Saitoh, \All-optical observation and reconstruction of spin wave dispersion," Nat. Comm. 8, 15859 (2017). 19T. Sebastian, K. Schultheiss, B. Obry, B. Hillebrands, and H. Schultheiss, \Micro-focused brillouin light scattering: imaging spin waves at the nanoscale," Frontiers in Physics 3, 35 (2015). 20G. G. Siu, C. M. Lee, and Y. Liu, \Magnons and acoustic phonons in Y 3xBixFe5O12," Phys. Rev. B 64, 094421 (2001). 21C. M. Srivastava and R. Aiyar, \Spin wave stiness constants in some ferrimagnetics," J. Phys. C: Sol. Stat. Phys. 20, 1119 (1987). 22V. Cherepanov, I. Kolokolov, and V. L'vov, \The saga of YIG: Spectra, thermodynamics, interaction and relaxation of magnons in a complex magnet," Phys. Rep. 229, 81 (1993). 23R. Gonano, E. Hunt, and H. Meyer, \Sublattice magnetization in yttrium and lutetium iron garnets," Phys. Rev. 156, 521 (1967). 24M. Sparks, Ferromagnetic-relaxation theory (McGraw-Hill, New York, 1964). 25Stancil D. D. and Prabhakar A., Spin Waves: Theory and Applications (Springer, New York, 2009). 26N. Miura, I. Oguro, and S. Chikazumi, \Computer simulation of temperature and eld depen- dences of sublattice magnetizations and spin- ip transition in gallium-substituted yttrium iron garnet," J. Phys. Soc. Jpn. 45, 1534{1541 (1978). 27S. Shamoto, T. U. Ito, H. Onishi, H. Yamauchi, Y. Inamura, M. Matsuura, M. Akatsu, K. Ko- dama, A. Nakao, T. Moyoshi, K. Munakata, T. Ohhara, M. Nakamura, S. Ohira-Kawamura, 12Y. Nemoto, and K. Shibata, \Neutron scattering study of yttrium iron garnet," Phys. Rev. B 97, 054429 (2018). 28C. W. Sandweg, M. B. Jung eisch, V. I. Vasyuchka, A. A. Serga, P. Clausen, H. Schultheiss, B. Hillebrands, A. Kreisel, and P. Kopietz, \Wide-range wavevector selectivity of magnon gases in brillouin light scattering spectroscopy," Rev. Sci. Instrum. 81, 073902 (2010). 29A. A. Serga, C. W. Sandweg, V. I. Vasyuchka, M. B. Jung eisch, B. Hillebrands, A. Kreisel, P. Kopietz, and M. P. Kostylev, \Brillouin light scattering spectroscopy of parametrically excited dipole-exchange magnons," Phys. Rev. B 86, 134403 (2012). 30K. Uchida, T. Kikkawa, A. Miura, J. Shiomi, and E. Saitoh, \Quantitative temperature de- pendence of longitudinal spin seebeck eect at high temperatures," Phys. Rev. X 4, 041023 (2014). 31R. C. LeCraw and L. R. Walker, \Temperature dependence of the spin-wave spectrum of yttrium iron garnet," J. Appl. Phys. 32, S167 (1961). 32L. Vasiliu-Doloc, J. W. Lynn, A. H. Moudden, A. M. de Leon-Guevara, and A. Revcolevschi, \Structure and spin dynamics of La 0:85Sr0:15MnO 3," Phys. Rev. B 58, 14913 (1998). 33J. Barker and G. E. W. Bauer, \Thermal spin dynamics of yttrium iron garnet," Phys. Rev. Lett. 117, 217201 (2016). 34W. Nolting and A. Ramakanth, Quantum theory of magnetism (Springer-Verlag, Berlin Heidel- berg, 2009). 35A. J. Princep, R. A. Ewings, S. Ward, S. T oth, C. Dubs, D. Prabhakaran, and A. T. Boothroyd, \The full magnon spectrum of yttrium iron garnet," npj Quantum Materials 2, 63 (2017). 36S. Geller, J. A. Cape, G. P. Espinosa, and D. H. Leslie, \Gallium-substituted yttrium iron garnet," Phys. Rev. 148, 522 (1966). 37P. Hansen, P. Rschmann, and W. Tolksdorf, \Saturation magnetization of gallium-substituted yttrium iron garnet," J. Appl. Phys. 45, 2728 (1974). 13kmagnon kphonon x yz(a) (b) H = HTA Magnetic compensation Compensated Non-compensated non-magnetic substitution [ ]a: spin-down ion )d: spin-up ion ( )d: non-magnetic ion (ω ωm ωp k kω ωFIG. 1. Lattice and spin waves in a magnetic material. (a) Schematic representation of a propagating phonon (lattice) and magnon (spin) excitation in a magnetic medium. Magnon- polarons are formed due to magnon-phonon hybridization when the magnon and phonon dispersions tangentially touch each other (i.e. they have coincident wavevector kmagnon =kphonon , and group velocity@! @kjmagnon =@! @kjphonon ), as schematically depicted by the dispersion relation shown in the inset of (a) representing the magnon dispersion (blue curve) and phonon dispersions (dashed line) under an applied magnetic eld with magnitude HTA, in this condition a resonant enhancement of the SSE can be observed. (b) Schematic representation of the eect of magnetic compensa- tion on the magnon dispersion of a ferrimagnetic system: a parabolic dispersion is expected when the system is non-magnetically compensated with non-zero magnetization (black curve), and as the magnetic compensation increases the magnon dispersion gradually evolves towards a linear dependence when the system reaches magnetic compensation (antiferromagnet). This can be ex- perimentally achieved by introduction of non-magnetic ion substitutions into the lattice as shown in the bottom part of (b). Red and blue circles represent magnetic ions in two magnetic sublattices with oppositely oriented spins and the orange circle represents the non-magnetic ion substitution. 14T (a) (Lu 2Bi)Fe 4GaO 12 (b) (c) (d)(d) (e)x zy Pt S0δSFIG. 2. Spin Seebeck eect measurement and magnon-polaron peaks in Ga-doped garnet system (a) Schematic of the SSE and ISHE mechanism and measurement geometry. (b) Magnetic eld dependence of the SSE thermopower measured at T= 300 K in a (Lu 2Bi)Fe 4GaO 12 (BiGa:LuIG) lm (normalized by the sample geometry: Ey=rT=S= (V=
T)(Lz=Ly)). Inset shows detail of the SSE in the 0 <  0H < 0:7 T range, showing that the signal increase due to magnon-phonon hybridization is already present at very small elds. c, d Detail of the SSE signal in the vicinity of magnetic elds 0HTA= 0.42 T and 0HLA= 1.86 T, depicting the resonant enhancement of the SSE voltage, resulting from the magnon and phonon hybridization at the magnetic elds when the magnon dispersion just tangentially touches the transversal acoustic (TA) (c)and the longitudinal acoustic (LA) (d)phonon dispersions, respectively. (e)Comparison of the resonant enhancement of the SSE centred around the peak position ( HTA) for YIG and BiGa:LuIG at 300 K (the enhancement at the peak position is estimated as: S=S 0, whereS0is the extrapolated background SSE coecient at the peak position and S=S(HTA)S0). 15(c) (d)(b) (a) HTA HLA 0.2 T0.6 T 0.42 TFIG. 3. Eect of magnetic compensation on the spin wave dispersion (a) Results of the magnon dispersion calculated using our model for dierent concentrations of non-magnetic ion substitutions ( x,y), illustrating the eect of magnetic compensation in a ferrimagnetic system. A 90 % preferential tetrahedral site occupation by Ga ions is assumed according to previous reports.36,37(b)Comparison between the magnon dispersion measured experimentally by Brillouin light scattering and the theoretical dispersion for a ferrimagnet with non-magnetic ion substitution ofx= 0:101 (octahedral site occupation) and y= 0:909 (tetrahedral site occupation) (theoretical dispersion is vertically shifted to account for demagnetizing and anisotropy eld contributions). (c, d) Magnon and phonon dispersions at magnetic elds of 0H= 0:42 T and0H= 1.86 T depicting the condition for the observation of magnon-polaron SSE enhancement by hybridization of magnon with TA (c)and LA (d)phonons in our system (insets show detail of the measured SSE voltage enhancement at the peak positions). In (c)the magnon dispersion for three dierent eld values, corresponding to the arrows in the inset, is shown. 16(a) (b) (c) (d)0.1FIG. 4. Temperature dependent magnetization and heat-driven spin transport prop- erties of BiGa:LuIG lm (a) Magnetic eld dependence of the measured spin Seebeck voltage at dierent temperatures (b)Saturation magnetization ( 0MS) as a function of temperature. Red line shows tting to 0MS/(TcT), withTc= 401:60:2 K and= 0:3050:007.(c)Temper- ature dependence of the magnetic eld magnitude for the observation of the magnon-polaron SSE, HTA(black squares) and HLA(red circles) (lines interconnecting experimental points are for visual guide). (d)Temperature dependence of the intersite exchange integral ( Jad) estimated from the condition for tangential touching of the magnon and phonon dispersions at dierent temperatures. Red line shows tting to Jad=J0(1T5=2), withJ0= (6:410:05)1022J. 17

2019-03-21

Resonant enhancement of spin Seebeck effect (SSE) due to phonons was recently discovered in Y3Fe5O12 (YIG). This effect is explained by hybridization between the magnon and phonon dispersions. However, this effect was observed at low temperatures and high magnetic fields, limiting the scope for applications. Here we report observation of phonon-resonant enhancement of SSE at room temperature and low magnetic field. We observed in Lu2BiFe4GaO12 and enhancement 700 % greater than that in a YIG film and at very low magnetic fields around 10-1 T, almost one order of magnitude lower than that of YIG. The result can be explained by the change in the magnon dispersion induced by magnetic compensation due to the presence of non-magnetic ion substitutions. Our study provides a way to tune the magnon response in a crystal by chemical doping with potential applications for spintronic devices.

Room temperature and low-field resonant enhancement of spin Seebeck effect in partially compensated magnets

1903.09007v1

Piezoelectric microresonators for sensitive spin detection Cecile Skoryna Kline, Jorge Monroy-Ruz, and Krishna C. Balram∗ Quantum Engineering Technology Labs and Department of Electrical and Electronic Engineering, University of Bristol, Woodland Road, Bristol BS8 1UB, United Kingdom (Dated: May 6, 2024) Piezoelectric microresonators are indispensable in wireless communications, and underpin radio frequency filtering in mobile phones. These devices are usually analyzed in the quasi-(electro)static regime with the magnetic field effectively ignored. On the other hand, at GHz frequencies and especially in piezoelectric devices exploiting strong dimensional confinement of acoustic fields, the surface magnetic fields ( B1) can be significant. This B1field, which oscillates at GHz frequen- cies, but is confined to µm-scale wavelengths provides a natural route to efficiently interface with nanoscale spin systems. We show through scaling arguments that B1∝f2for tightly focused acoustic fields at a given operation frequency f. We demonstrate the existence of these surface magnetic fields in a proof-of-principle experiment by showing excess power absorption at the focus of a sur- face acoustic wave (SAW), when a polished Yttrium-Iron-Garnet (YIG) sphere is positioned in the evanescent field, and the magnon resonance is tuned across the SAW transmission. Finally, we out- line the prospects for sensitive spin detection using small mode volume piezoelectric microresonators, including the feasibility of electrical detection of single spins at cryogenic temperatures. I. INTRODUCTION Piezoelectric microresonators [1, 2] have revolutionized wireless communication by enabling small form-factor, high performance radio frequency (RF) filters that can be compactly packaged into mobile phones. In addition, these devices have had a broad impact on areas ranging from sensing [3] to quantum communication [4]. A piezo- electric material enables conversion of RF electromag- netic fields into acoustic fields, which have wavelengths ≈µm at GHz frequencies, 105smaller than the cm-scale wavelengths of the RF fields. This deeply-subwavelength confinement is the key driver for the majority of applica- tions involving piezoelectric devices. The constitutive relations for a piezoelectric device [5] relate the stress ( ⃗T) induced by an applied electric field (⃗E). While this is strictly true for applied DC fields, the equations are extended to RF fields under the quasi- (electro)static approximation [6], wherein the Poisson equation for electrostatics is substituted for Maxwell’s equations for the electromagnetic field, and is solved along with the elastic wave equation to propagate acous- tic fields in piezoelectric devices. The quasistatic ap- proximation is usually justified because the deeply sub- wavelength confinement provided by the acoustic field ensures that the far-field electromagnetic radiation com- ponent is minimal. By definition, using this approxima- tion forces the magnetic field to be strictly zero. On the other hand, it is known that piezoelectric de- vices radiate electromagnetically [7] and that this radia- tion presents a limit on the achievable mechanical qual- ity factor ( Qm) in piezoelectric resonators [8]. While the far-field radiation efficiency is low for the reasons out- lined above, it can be significantly enhanced in a reso- nant geometry and provides significant size advantages ∗krishna.coimbatorebalram@bristol.ac.ukin the design of very low frequency antennas [9]. In this work, we focus on the near-field (surface) component of this oscillating magnetic field ( B1), and ask if these B1 fields can be exploited for improving the spin detection sensitivity of nanoscale electron spin resonance (ESR) ex- periments [10, 11]. Our aim is to apply ideas from cavity quantum electrodynamics (cQED) [12] to nanoscale spin systems [13, 14], with the key sensitivity enhancement being provided by the vastly reduced mode volume ( Vm) of piezoelectric microresonators in comparison to their electromagnetic counterparts. II. SURFACE CURRENT DENSITY SCALING IN PIEZOELECTRIC DEVICES In a piezoelectric material, the propagating acoustic displacement field is accompanied by a surface polariza- tion ( ρ). The evanescent electric fields ( ⃗E) generated at the material-air boundary curl as depicted in Fig.1(a). The plot shows a finite element method (FEM) simula- tion of a surface acoustic wave (SAW) mode propagating on a Scandium Aluminum Nitride (ScAlN) on Si sub- strate. The accompanying electric fields are shown by an arrow plot. This curling of surface fields is one instance of the universal phenomenon of spin-momentum locking [15] that applies to all evanescent fields (cf. Appendix A for a discussion of the analogy between the surface fields in surface plasmons and surface acoustic waves). The ac- companying surface magnetic field ( B1) orientation can be directly obtained from the ( ⃗E) fields, and in this case would form loops that come periodically into and out of the plane (shown schematically in Appendix A). One can verify the ⃗B1orientation by noting the direction of the surface polarization currents ( J=∂ρ ∂t), plotted in Fig.1(b) and invoking Ampere’s law. For focusing acous- tic field geometries as would be necessary for increasing the local field strength, the field orientation is shown inarXiv:2405.02212v1 [physics.app-ph] 3 May 20242 (a) (b) (c)(d)Air SiScAlN (1 µm) 1 µm750 nm 1 µm 0 + FIG. 1. (a) FEM simulation showing the surface displace- ment of a surface acoustic wave propagating on a ScAlN on Si substrate. The accompanying ≈2.9 GHz electric field is shown using an overlaid arrow plot. As can be seen there exists an evanescent field in the air whose orientation (helic- ity) is determined by spin momentum locking [15], (b) FEM simulation, same as (a) but showing the overlaid oscillating polarization current density using an arrow plot. This os- cillating current density which exists at both the interfaces (air-ScAlN and ScAlN-Si) is responsible for the surface ⃗B1 fields we investigate in this work. (c) Focusing the SAW using curved electrodes in AlN (d) At the focus, the displacement, and corresponding evanescent field and current densities, are enhanced by the focusing ratio. Fig.1(c), with a zoomed-in plot of the focus region shown in Fig.1(d). We would like to note here that while the ⃗Efield lines can be computed using an FEM solver like COMSOL, due to the quasistatic approximation being imposed on the solver, the magnetic field is strictly zero. Therefore, current FEM solvers cannot be used to visu- alize this ( B1) field directly from taking the curl of the ⃗E field. The results presented in Figs.1(a-d) make certain simplifications. We model (Sc)AlN films using aluminum nitride’s material parameters, and in Fig.1(c,d), we simu- late focusing on a thin AlN film to reduce the simulation’s memory constraints. To estimate the scaling of this surface B1field, we therefore start with the oscillating surface current density (J, [A m−2]) instead [3] following prior work on SAW based sensors [16]: J2= 2K2ωk2(ϵc+ϵs)P (1) where K2is the material’s piezoelectric coefficient and represents the fraction of the acoustic wave energy that is stored in the electric field. ωis the wave frequency, k= 2π/λ the wave vector, ϵc,s[F m−1] represent the dielectric constants of the cladding (air) and substrate respectively, and Pis the power density of the acoustic wave expressed in power per unit beam width [W m−1]. We can see that for a tightly focused acoustic beam withfocus width ≈λ,P∝1/λ, and hence B2 1∼J2∝f4, or the surface field B1∝f2. With K2= 0.05, λa= 1µm,f= 3 GHz, ϵc+ϵs≈10, beam width at focus ≈λaand Pa=1 mW, we expect a surface current density at the focus of ≈81.15 MA m−2. The scaling of Jwith ϵcan be visualized directly in Fig.1(b), where the current den- sity at the (Sc)AlN-Si interface is stronger than at the (Sc)AlN-air interface. Assuming the current flows uniformly in a (semi- circular) loop of size ≈λa, one can roughly estimate theB1field as B1≈µ0Jλa/4 = 25 .5µT, with µ0the vacuum permeability. By trapping the acoustic field in wavelength scale microcavities [17], the surface current density and therefore the accompanying B1field can be enhanced by the cavity mechanical quality factor Qm, with Qm≈104feasible in crystalline media at ambi- ent conditions [18]. Alternately, one can also estimate the spin-resonator single photon coupling strength gin a cavity-QED framework [10, 12] with: g=µB ℏr 2µ0ℏω0 Vm(2) where µBis the Bohr magneton, ℏthe reduced Planck constant, µ0vacuum permeability and Vmis the cavity mode volume. Assuming an acoustic cavity mode with dimensions of 5 λ3 aat 3 GHz with λa=1µm, this gives us a coupling strength g≈2π∗14 kHz. We would like to em- phasize that we are interested in the near-field B1here. Due to the alternating signs of the surface current den- sity, the far-field radiation is relatively weak. Such piezoelectric approaches to enhancing spin de- tection sensitivity are in many ways complementary to superconducting cavity approaches which have recently achieved single spin sensitivity [11, 19] at mK temper- atures. Both approaches rely on a Purcell-like [12] en- hancement of the spin detection sensitivity which scales ∝p Qc/Vc, where Qcis the quality factor and Vcis the mode volume of the cavity. The superconducting approaches support high Qfactors ( >105) and small magnetic mode volume ( ≈10−12λ3 RF) [20] by exploiting lumped element inductor-capacitor ( LC) cavity designs, where the B1field is strongly localized in close proximity to the inductor. Piezoelectric resonators, on the other hand provide moderate Qm(≈104), but vastly reduced cavity mode volumes ( ≈10−15λ3 RF) with the additional advantage of enabling experiments in ambient conditions, which makes it feasible to use this technique with chem- ical and biological samples which might deteriorate ap- preciably in cryogenic environments. We would like to distinguish our work from previous results that have studied the interaction between acoustic fields and spin systems (both nanomagnets and magnetic thin films) that primarily exploit the magnetostrictive ef- fect [21–23] or the strain field to perturb spin systems [24–26]. In this work, we instead focus on using piezo- electric devices for generating the oscillating magnetic field ( B1) directly and the methods developed here can be applied in principle to any spin resonance experiment.3 (a) <110> (b) (c) 200 µmSide ViewTop View YIG IDT FIG. 2. (a) Schematic of the experimental setup used to probe the B1. Curved IDTs are used to launch and detect SAWs on a ScAlN on Si substrate. A polished YIG sphere is brought in proximity (non-contact) of the evanescent field at the focus of the SAW and the IDT transmission is monitored as the magnon mode is tuned across the SAW transmission resonance. The magnitude and phase of the transmission is monitored using a vector network analyzer and the magnon mode is tuned by a Bzfield applied using an electromagnet underneath the sample. (b) and (c) show images from the side and top view cameras of the experiment during operation. Given the field generated is on the surface and evanes- cent, physical contact between the sample and the spin system can be completely avoided, as discussed in the experiments below. III. USING SPIN RESONANCE ABSORPTION TO PROBE THE EVANESCENT B1FIELD We perform spin resonance measurements by position- ing a polished Yttrium-Iron-Garnet (YIG) sphere (diam- eter≈150µm) in the evanescent field ( < λa≈1µm) of the SAW devices. YIG spheres support high-Q ( ≈104 at 10 GHz) collective spin wave excitations (magnons), which have been used for a wide range of applications, ranging from tunable filters and low noise oscillators for wireless communication [27] to hybrid quantum trans- duction, wherein quantum states from superconducting qubits are mapped back and forth from the magnon modes [28]. By mounting the YIG sphere on a mov- able three-axis translation stage and positioning it in the evanescent field of the acoustic wave, we can spatially probe the surface B1field by monitoring the SAW trans- mission and looking for changes in the amplitude and phase response of the received SAW signal as the magnon mode frequency is tuned across the SAW resonance. The schematic of our experimental setup is illustrated in Fig.2(a). Fig.2(b,c) show the side and top view of the YIG sphere positioned at the centre of the SAW transmit-receive circuits as captured by the two zoom lenses indi- cated in Fig.2(a). SAWs are launched and detected us- ing a vector network analyzer (VNA) by patterning a set of interdigitated transducers (IDT) [2] on a piezoelectric (c-axis oriented) Sc 0.06Al0.94N film deposited on a [100] oriented silicon substrate. We pattern IDTs with both straight fingers to launch quasi plane waves of sound, and curved fingers [29] to focus the acoustic field down to a beam width of 1-5 λa. The curved IDT devices are designed as confocal transmit-receive pairs [17, 30]. To achieve the highest spin detection sensitivity, focusing devices are necessary as they significantly enhance the lo- cal power density ( Pin equation 1) by the focusing ratio (≈20−50x) and the strongest B1fields are therefore al- ways generated at the focus. The size of the YIG sphere in our experiments presents two constraints. It requires us to separate the focusing IDTs by ≈100−125λa, which magnifies the effect of wave diffraction and the anisotropy of the underlying silicon substrate with the net result be- ing a relatively low acoustic throughput.The second issue, which can be seen from Fig.2(c) is the difficulty of deter- mining the precise acoustic focus location while aligning the YIG sphere for maximum SAW extinction. The magnon frequency can be tuned by applying a static DC magnetic field ( Bz) using an electromagnet, as shown in Fig.2(a). Due to the frequency of opera- tion and the orientation (shown in Fig.2(b)), the YIG sphere is not saturated, and the magnon mode relaxes to lower frequencies with time. Therefore, there is a finite time offset between the setting of the voltage on the elec- tromagnet and the data acquisition on the VNA, which would result in a lower effective Bzdue to relaxation. In principle, one can park the magnon mode on the higher frequency side of the SAW transmission and let the mode relax and down-shift in frequency across the SAW reso- nance, while recording VNA traces as a function of time. In practice, we found this method did not give us the resolution needed to capture the exact crossing between the magnon mode and the SAW. Therefore, we rely on incrementing the voltage in small steps and acquiring the data quickly to minimize the effects of mode relaxation. As the Bzfield is tuned, ferromagnetic resonance (FMR) in the YIG sphere induces an excess absorption which shows up as a reduced transmission (attenuation) at the SAW frequency. This excess absorption is maxi- mized when the frequencies of the SAW and the magnon mode are identical. This is shown in Fig.3. Fig.3(a,c) correspond to datasets taken from two distinct curved IDT devices with the same period but different curva- ture. The colorbar on the 2D plot indicates the mag- nitude of the (normalized) transmitted signal ( S21) as a function of both Bzand frequency. As the plot shows, we observe a variety of magnon modes [31] in our exper- iment, with varying coupling strengths and quality fac- tors. Without mode imaging [31], it is hard to ascertain the mode symmetry from a pure microwave transmission experiment. Here, we focus on the mode that gives us the strongest signal and generically label it as magnon4 (a) (b)MagnonSAW Magnon SAW (c) (d) 0 + FIG. 3. The SAW-magnon interaction is probed by monitoring the VNA transmission ( S21) as a function of Bz. The 2D colorplots in (a,c) plot |S21|as a function of frequency and Bzfor two different curved IDT devices with the same period, but different curvature. The SAW mode frequency is constant with Bz, whereas the magnon mode frequency shifts linearly with Bz. When the magnon mode frequency crosses the SAW transmission resonance, one expects an excess attenuation in the SAW transmission due to magnon mode excitation and dissipation. (b,d) show representative linecuts from (a,c) respectively for fm< fs(red), fm≈fs(black), and fm> fs(blue). The excess attenuation when fm≈fsis especially clear in (b), and the same trend can also be observed, albeit with lower magnitude in (d). The plotted data are time gated with a 40 ns. Uncorrected datasets are available in Appendix B, and the bare IDT reflection and transmission spectra are plotted in Appendix D. in Fig.3(a,c). As the 2D color plots show, the SAW fre- quency stays relatively independent of Bz, whereas the magnon mode linearly tunes to higher frequency with in- creasing Bz. When the magnon mode frequency ( fm) crosses the SAW mode ( fs), one can clearly see an ex- cess absorption which is stronger in Fig.3(a) compared to the dataset in Fig.3(c). One can see this excess ab- sorption effect more clearly by taking 1D cuts through the 2D data corresponding to the cases when fm< fs, fm≈fsandfm> fs. These 1D cuts are shown respec- tively in Fig.3(b) and (d), with the cases colored red, black and blue respectively. The background SAW trans- mission when the magnon mode is way off resonance (fm≪fs, labelled fs) is indicated in magenta. Especially, in Fig.3(b) the excess attenuation as the magnon mode passes through the SAW frequency is clear. While we do observe a similar effect in Fig.3(d), the magnitude is much weaker which we believe is due to a combination of the mode drifting with time and the difficulty of posi- tioning the YIG sphere at the focus, given the geometry shown in Fig.2(c). Our analysis here is complicated by the fact that there is significant electromagnetic crosstalk between the two ports and one needs to distinguish between the localmagnon-SAW interaction occurring on the sample and possible interference effects occurring at the VNA. In particular, the electromagnetic radiation from the probes and the IDT, which act as inefficient antennas [32], can excite the magnon mode and the scattered signal picked up by the receiving port will interfere with the acous- tic transmission. We refer to this second pathway as a nonlocal interaction due to the phase sensitive detection employed by the VNA. In our experiments, the crosstalk signal is larger than the acoustic transmission because of the diffraction effects mentioned above. One can see this in action, by noting that in the case of fm> fsand fm< fs, the magnon signal appears as a transmission peak rather than a dip, which is clear signature of a non- local interference pathway [33]. One way to reduce this background is to exploit the significantly lower speed of sound compared to light and time-gate the VNA trans- mission [34]. The datasets shown in Fig.3 have been time gated with a notch of 40 ns. While this helps to improve the signal to noise ratio (cf. the raw datasets in Appendix B), the high quality factor of the YIG sphere makes the cross talk persist for significantly longer than the electro- magnetic transit time. This residual cross-talk makes it challenging for us to extract the SAW-magnon interac-5 tion strength from the experiments, although the excess local interaction with the SAW through the surface B1 field is clear from Fig.3(a,b). As a control, we repeat the experiment with a straight IDT device and do not ob- serve an excess attenuation at the magnon-SAW crossing (cf. Appendix C). Given that the SAW-magnon mode interaction is ob- served with the YIG sphere not touching the sample, this experiment provides strong preliminary evidence that GHz frequency, localized B1fields can be generated on the surface of piezoelectric devices and can be used to interface with spin systems. While this is encouraging, we would like to emphasize that the experimental modal- ity has a few limitations obvious with hindsight and the results should be interpreted within these constraints. In particular, the size of commercially available YIG spheres limits the sensitivity of the experiment by physically re- quiring the IDTs be separated by >100λa, and making it challenging to determine the focus in real-time. The size of the YIG sphere is also responsible for the mag- nitude of the crosstalk, which makes it challenging to infer the coupling strength and the coupling dynamics from our experiments. In particular, one of the key ef- fects we were hoping to confirm was the frequency de- pendence of the interaction, which should scale ∝f2. Although we made devices with varying SAW frequency, we were unable to quantify this effect. Moving forward, by impedance matching the transducers (to reduce elec- tromagnetic radiation and reflections) and working with high Q YIG samples with dimensions <5µm, one can potentially scan the surface and verify the spatial extent of the B1field both in-plane ( x, y) and in z. Mapping the spatial confinement of the B1field is critical to the spin sensitivity enhancement experiments and this is some- thing we are unable to do with our current setup. Finally, moving to higher acoustic frequencies (and higher Bz) would enable us to saturate the YIG sphere and avoid the relaxation drift with time, which is another source of error in our experiments. In passing, we would like to note that the B1field description provides a compli- mentary route towards understanding the spin rotation effects in nanoparticles interacting with SAWs [35], and interpret the switching of magnetization observed in pre- vious experiments [36]. IV. PROSPECTS FOR SINGLE SPIN ELECTRICAL READOUT As noted in a recent review [10], the problem of improv- ing spin detection sensitivity in electron spin resonance experiments boils down to focusing the magnetic field to deeply sub-wavelength geometries while maintaining high-Q. In effect, piezoelectric microresonators are ideal in that they naturally provide both strong confinement and high Q, and therefore provide a natural complement to traditional electromagnetic approaches [37]. The key issue is whether the magnetic field strengths can be sub-stantial in these devices. As we have shown above us- ing both scaling arguments and proof-of-principle exper- iments, the surface current density has a ∝f2and can be further enhanced by working with stronger piezoelec- tric materials / orientations ( K2 eff>0.2) and designing small mode volume acoustic cavities [17] to exploit the power scaling. With advances in materials and device ge- ometries pushing acoustic device operation to ever-higher frequencies ( >50 GHz) [38], the prospect of acoustics en- abled X-Band ESR is within reach. There is still the open question of how one can efficiently load the near field of these devices efficiently to separate the pure field in- duced effects from strain effects. One possible route could be to employ suspended membranes (similar to [39], but made with an insulator like alumina) in close proximity (<50 nm) to the piezoelectric substrate. We can estimate the minimum spin detection sensitiv- ity, following [40]: Nmin=κ 2gprnw κ2(3) where Nminis the single-shot spin detection sensitiv- ity (per echo), κis the total cavity decay rate given by κ=ω0/QLwith ω0= 2πfbeing the operating frequency andQLthe loaded Q factor of the cavity. We assume the cavity is operated at critical coupling with external cou- pling rate κ2≈κ/2,nis the average number of noise pho- tons, given by n=kBT/ℏω0,pis the spin polarization p=1−e−ℏω0 kBT 1+e−ℏω0 kBT, and the spin resonance width w= 2/T2 with T2the average spin dephasing time. gis the spin cavity coupling rate defined above in eqn.2. For oper- ating frequencies of 10 GHz, temperature 4 K, mode vol- ume 5 µm3andT2≈50 ms, achieving Nmin≈1 requires QL≈105, which is challenging for mechanical systems, but feasible given recent results [18] and the favourable scaling of acoustic dissipation with temperature. In any case, this shows that provided the B1field in piezoelectric microresonators has a spatial extent comparable to that of the acoustic field which requires experimental confir- mation, then detecting individual magnetic nanoparticles at room temperature ( Nspins≥105) is well-within reach using current devices. The exquisite spin detection potential of these res- onators can be understood from a different perspective by treating them as the high frequency analogs of mechani- cal cantilever based spin sensors [41], where the piezoelec- tric effect enables inductive detection. It was noted [42] that the signal to noise ratio for both inductive and me- chanical detection of spin resonance scales ∝p ω0Q/k m, with ω0is the operating frequency, Qthe cavity qual- ity factor and kman effective magnetic spring constant which scales with the cavity mode volume Vmfor in- ductive detection. The sensitivity enhancement there- fore derives from achieving high quality factors in deeply sub-wavelength mode volumes, which gives an effective Purcell enhancement ∝p Q/V mto the sensitivity [12].6 We would like to conclude by noting that while in this work, we have primarily focused on using piezoelectric devices for improving spin detection sensitivity in ESR, the strong field confinement is also of interest in scenarios involving large-scale closely packed efficient spin address- ing [43] without deleterious crosstalk effects, as would be necessary in future spin-based quantum computing. V. ACKNOWLEDGEMENTS We would like to thank John Rarity, Joe Smith, Vivek Tiwari, Hao-Cheng Weng, Alex Clark, Rowan Hoggarth and Edmund Harbord for valuable discussions and sug- gestions. We acknowledge funding support from the UK’s Engineering and Physical Sciences Research Coun- cil (EP/N015126/1, EP/V048856/1) and the European Research Council (ERC-StG SBS 3-5, 758843). Appendix A: Spin Momentum Locking of Evanescent fields in surface acoustic waves and surface plasmons ++ ++++ ++++ ++-- ---- -- MetalAir𝐸 (𝐸𝑥,𝐸𝑦) 𝐻𝑧𝑘Surface plasmon on a metal (THz) -- --++ ++-- ---- --++ ++𝐸 (𝐸𝑥,𝐸𝑦)𝐻𝑧 (b)Surface acoustic wave on a piezoelectric 𝑘 Surface polarization(GHz) xy(a) FIG. 4. (a) Schematic illustration of surface plasmon propa- gating on a metal-air interface. The evanescent electric field and the magnetic field orientation are shown, alongwith the surface charges that terminat the electric field on the metal surface. (b) Illustration of a surface acoustic wave propagat- ing on a piezoelectric material. The evanescent electric field orientation is similar to that in (a), but terminated by bound polarization charges induced in the piezoelectric. By analogy, the orientation of the B1field can be determined, as shown. One can see the universality of spin-momentum lock- ing [15] as applied to all evanescent fields by lookingat two very different surface waves: a surface plas- mon propagating at a metal air interface (at >100 THz) and a surface acoustic wave ( <50 GHz) propagating at a piezoelectric-air interface. The respective cases are shown in Fig.4(a,b). As can be seen in both cases, the evanescent electric fields curl with an orientation deter- mined by spin-momentum locking. The main difference between the two scenarios is the termination of the fields on free charges in the metal and on bound polarization charges in the piezoelectric case. Given that the surface plasmon dispersion relation is traditionally derived by solving for the magnetic field, the B1orientation can be derived by analogy as shown in Fig.4(b). Appendix B: Uncorrected datasets without time-gating (a) (b) 0 + FIG. 5. (a) Uncorrected data sets for the data correspond- ing to Fig.3(a,b). Without the time-gating, the background electromagnetic crosstalk makes it impossible to observe the excess attenuation during the SAW-magnon mode crossing, although even with the raw dataset, a net attenuation (cf. black curve) is clearly visible. Figure 5 plots the raw data sets corresponding to the time-gated datasets plotted in Fig.3(a,b). As discussed in the main text, the presence of the background electro- magnetic crosstalk makes it challenging to infer the SAW- magnon interaction, which can be best seen by comparing the black curves ( fm≈fs) in Fig.3(b) and Fig.5(b). On the other hand, even with the raw datasets, we can clearly observe an overall attenuation as the magnon mode fre- quency comes close to the SAW frequency. This is in contrast to what we observe with the straight IDT de-7 vices, as discussed in the next section. We would like to note that the choice of the notch gate time (40 ns) was not optimized for this analysis and neither was the gate shape. Given the separation between IDTs was 200 µm, and a speed of sound of ≈3750 m /sec, the acoustic wave takes ≈53 ns to reach the receiving IDT. The choice of gate time was made as a rough trade- off between minimizing the background crosstalk signal and ensuring minimal attenuation of the acoustic signal in the transmitted spectrum. We would like to note again that time gating does not fully eliminate the background crosstalk because of the Q factor of the YIG sphere. Appendix C: Straight IDT control results (a) (b) 0 + FIG. 6. (a) 2D colorplot of the time-gated |S21|for a straight IDT transmit-receive device with the YIG sphere positioned in between. The YIG was mounted vertically in this exper- iment to align the 110 axis with z, which shifts the modes to higher frequencies in the 3 .4 GHz range (b) 1D datasets from (a) corresponding to the three different cases ( fm< fs, fm≈fs, and fm> fs), along with the background SAW trans- mission (magenta). The excess attenuation at the magnon- SAW crossing is not observable here. The power dependence of J2in equation 1 can be tested by measuring the relative performance of IDT with curved and straight fingers in inducing magnon absorp- tion. Given the local field intensity in curved devices at the focus is increased by the focusing ratio, the straight IDT devices here serve as a control experiment to sepa- rate the local SAW-magnon interaction from the nonlocal EM crosstalk-SAW interaction occuring in the VNA, dis- cussed in the main text.Experiments identical to that reported in Fig.3 were done with a straight IDT device. The YIG sphere in these experiments was mounted vertically with a view towards saturating the sphere and avoiding the time drift in the magnon modes. This moves the magnon modes to higher frequencies in the 3.4 GHz range, and the IDT period was reduced correspondingly to shift the SAW response to higher frequencies. Fig.6(a) plots the 2D colorplot of the transmitted |S21|as a function of frequency and Bz. To achieve the higher Bz, we use a permanent magnet in combination with our electromagnet. Fig.6(b) shows linecuts from 6(a) that correspond to the three different cases fm< fs,fm≈fsandfm> fs. Here, we don’t ob- serve an excess attenuation as the magnon mode crosses through the SAW resonance, which can be interpreted as a signature of the dependence of the local current density on the local power density. While mounting the YIG sphere vertically makes the magnon mode frequency stable due to field saturation, it makes it very challenging to determine the positioning of the sphere with respect to the beam focus using imaging cameras. In particular, the top view camera, shown in Fig.2(a) can not be used anymore and one has to rely more on the side view camera with associated parallax errors. 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2024-05-03

Piezoelectric microresonators are indispensable in wireless communications, and underpin radio frequency filtering in mobile phones. These devices are usually analyzed in the quasi-(electro)static regime with the magnetic field effectively ignored. On the other hand, at GHz frequencies and especially in piezoelectric devices exploiting strong dimensional confinement of acoustic fields, the surface magnetic fields ($B_{1}$) can be significant. This $B_1$ field, which oscillates at GHz frequencies, but is confined to ${\mu}$m-scale wavelengths provides a natural route to efficiently interface with nanoscale spin systems. We show through scaling arguments that $B_1{\propto}f^2$ for tightly focused acoustic fields at a given operation frequency $f$. We demonstrate the existence of these surface magnetic fields in a proof-of-principle experiment by showing excess power absorption at the focus of a surface acoustic wave (SAW), when a polished Yttrium-Iron-Garnet (YIG) sphere is positioned in the evanescent field, and the magnon resonance is tuned across the SAW transmission. Finally, we outline the prospects for sensitive spin detection using small mode volume piezoelectric microresonators, including the feasibility of electrical detection of single spins at cryogenic temperatures.

Piezoelectric microresonators for sensitive spin detection

2405.02212v1

Is spin super uidity possible in YIG lms? E. B. Sonin Racah Institute of Physics, Hebrew University of Jerusalem, Givat Ram, Jerusalem 91904, Israel (Dated: August 17, 2021) Recently it was suggested that stationary spin supercurrents (spin super uidity) are possible in the magnon condensate observed in yttrium-iron-garnet (YIG) magnetic lms under strong external pumping. Here we analyze this suggestion. From topology of the equilibrium order parameter in YIG one must not expect energetic barriers making spin supercurrents metastable. However some small barriers of dynamical origin are possible nevertheless. The critical phase gradient (analog of the Landau critical velocity in super uids) is proportional to intensity of the coherent spin wave (number of condensed magnons). The conclusion is that although spin super uidity in YIG lms is possible in principle, the published claim of its observation is not justied. The analysis revealed that the widely accepted spin-wave spectrum in YIG lms with magne- tostatic and exchange interaction required revision. This led to revision of non-linear corrections, which determine stability of the magnon condensate with and without spin supercurrents. I. INTRODUCTION Spin super uidity has already been discussed from 70s of the last century1,2. Its investigation continues nowa- days (see recent reviews in Refs. 3 and 4). The inter- est to this phenomenon was revived after emergence of spintronics. Manifestation of spin super uidity is a sta- ble spin supercurrent. Experimental observation of it in magnetically ordered solids would be an essential break- through in the condensed matter physics. A spin super- current is proportional to the gradient of the phase ' (spin rotation angle in a plane) and is not accompanied by dissipation, in contrast to a dissipative spin diusion current proportional to the gradient of spin density. In general the spin current proportional to the gradi- ent of the phase 'is ubiquitous and exists in any spin wave or domain wall, although in these cases variation of the phase 'is small (very small in weak spin waves and not more then on the order in domain walls and in disordered materials). Analogy with mass and charge persistent currents (supercurrents) arises when at long (macroscopical) spatial intervals along streamlines the phase variation is many times larger than 2 . The super- current state is a helical spin structure, but in contrast to equilibrium helical structures is metastable. An elementary process of relaxation of the supercur- rent is phase slip. In this process a vortex with 2 phase variation around it crosses streamlines of the supercur- rent decreasing the total phase variation across stream- lines by 2. Phase slips are suppressed by energetic bar- riers for vortex creation, which disappear when phase gradients reach critical values determined by the Landau criterion. Recently Sun et al.5suggested (see also Ref. 6) that spin super uidity is possible in a coherent magnon con- densate created in yttrium-iron-garnet (YIG) magnetic lms by strong parametric pumping7, and Bozhko et al.8 declared experimental detection of spin supercurrent in a decay of this condensate. Although the experimental evi- dence of spin super uidity was challenged9,10(see discus- sion in the end of the paper) the very idea of spin super- uidity in YIG lms deserves a further analysis. In YIG the equilibrium order parameter in the spin space was not conned to some easy plane analogous to the order parameter complex plane in super uids. The easy-plane order parameter topology providing a barrier stabilizing a supercurrent was considered as a necessary condition for spin super uidity.3,4However, one cannot rule out that metastability of supercurrent states is provided by barriers not connected with topology of the equilibrium order parameter. The goal of the present paper was to in- vestigate this possibility and to determine critical values of possible supercurrents at which metastability is lost. The critical supercurrents are determined from the principle similar to that of the Landau criterion for super- uids: any weak perturbation of the current state always increases the energy, and therefore the current state is metastable. This requires an analysis of nonlinear correc- tions to spin waves in the Landau{Lifshitz{Gilbert (LLG) theory. But calculation of nonlinear corrections is based on knowledge of the wave pattern and the wave spectrum in the linear theory. It was revealed that the commonly accepted and used up to now the linear theory of spin waves in YIG lms5,11{13must be revised. This was done properly taking into account boundary conditions at lm surfaces. For lms now used in experiments on coherent magnon condensation the boundary problem in the pres- ence of exchange and magnetostatic interaction has an accurate analytical solution, which gives the wave pat- tern and the wave spectrum dierent from known before. This is important for the stability analysis of the magnon condensate with and without spin supercurrents. Section II discusses connection of metastability of cur- rent states and topology of the order parameter space, which is a continuum of all degenerate ground states emerging from continuous symmetry (gauge symmetry in super uids, rotational symmetry of the spin space in fer- romagnets). It is also demonstrated how non-equilibrium state of spin precession supported by magnon pumping creates an eective \easy plane" for the order parameter, which allows metastable spin supercurrents. Section III reviews the LLG theory and the dispersion relation of lin-arXiv:1702.06994v2 [cond-mat.other] 10 Apr 20172 ear plane spin waves in YIG bulk. Section IV considers linear spin waves in YIG lm and determines their pat- tern and dispersion relation solving the boundary prob- lem in the presence of the exchange and the magneto- static interaction. The result diers from known before, and the origin of this dierence is discussed. Section V analyzes nonlinear corrections, which determine stabil- ity of the coherent magnon condensate and the distri- bution of magnons between two energy minima in the k space. The quasi-equilibrium approach xing the total number of magnons does not predict stable condensate, in con ict with observation of the coherent condensate in experiments. It was suggested to modify the quasi- equilibrium approach xing magnon numbers condensed in any of two minima, but not only their total number. Finally Sec. VI derives critical gradients in supercurrents from the Landau criterion generalized on spin super u- idity. The last section VII discusses and compares the results of the present work with results of previous inves- tigations. II. TOPOLOGY AND SUPERFLUID SPIN CURRENTS A knowledge on why super uid currents can be metastable is provided by the analysis of topology of the order parameter space. Spin super uidity was suggested by the analogy with the more commonly known mass su- per uidity, and we start from discussion of the latter. At the equilibrium the order parameter of a super uid is a complex wave function = 0ei', where the modulus 0 of the wave function is a positive constant determined by minimization of the energy and the phase 'is a degener- acy parameter since the energy does not depend on 'be- cause of gauge invariance. Any from degenerate ground states in a closed annular channel (torus) maps on some point at a circumference j j= 0in the complex plane , while a current state with the phase change 2 naround the torus maps onto a circumference (Fig. 1a) winding around the circumference ntimes. It is evident that it is impossible to change nkeeping the path on the circum- ferencej j= 0all the time. In the language of topology states with dierent nbelong to dierent classes, and n is atopological charge . Only a phase slip can change it when the path in the complex plane leaves the circumfer- ence. This should cost energy, which is spent on creation of a vortex crossing the cross-section of the torus channel and changing nton1. The state with a vortex in the channel maps on the full circle j j 0. If we consider transport of spin parallel to the axis zthe analog of the phase of the super uid order parameter is the rotation angle of the spin component in the plane xy, which we note also as '. Here we neglect the processes, which break rotational invariance in spin space (analog of gauge invariance in. super uids) and violate the conser- vation law for the total spin. These processes can be of principle importance and were thoroughly investigated.3 ReψImψψa) b)c)d) MzzMzzxyHxyHxyHFIG. 1. Mapping of current states on the order parameter space. a) Mass currents in super uids. The current state in torus maps on a circumference of radius j jon the complex plane . b) Spin currents in an isotropic ferromagnet. The current state in torus maps on an equatorial circumference on the sphere of radius M(top). Continuous shift of mapping on the sphere (middle) reduces it to a point at the northern pole (bottom), which corresponds to the ground state without cur- rents. c) Spin currents in an easy-plane ferromagnet. Easy-plane anisotropy contracts the order parameter space to an equato- rial circumference in the xyplane topologically equivalent to the order parameter space in super uids. d) Spin currents in an isotropic ferromagnet in a magnetic eld parallel to the axis zwith nonequilibrium magnetization Mzsupported by magnon pumping. Spin is conned in the plane parallel to the xyplane but close to the northern pole. This plane is an \easy plane" of dynamical origin. But in the present discussion their eect can be ignored for the sake of simplicity. In isotropic ferromagnets the order parameter space is a sphere of radius equal to the absolute value of the mag- netization vector M(Fig. 1b). All points on this sphere correspond to the same energy of the ground state. Sup- pose we created the spin current state with monotonously varying phase 'in a torus. This state maps on the equatorial circumference on the order parameter sphere. Topology allows to continuously shift the circumference and to reduce it to the point of the northern pole. Dur- ing this process shown in Fig. 1b the path remains on the sphere all the time and therefore no energetic barrier resists to the transformation. Thus metastability of the current state is not expected. In a ferromagnet with easy-plane anisotropy the order parameter space contracts from the sphere to an equa- torial circumference in the xyplane. This makes the order parameter space topologically equivalent to that in super uids (Fig. 1c). Now transformation of the equa-3 torial circumference to the point shown in Fig. 1b costs anisotropy energy. This allows to expect metastable spin currents (supercurrents). They relax to the ground state via phase slips events, in which magnetic vortices cross spin current streamlines. States with vortices maps on a hemisphere of radius Meither above or below the equa- tor. Up to now we considered states close to the equilibrium (ground) state. In a ferromagnet in a magnetic eld the equilibrium magnetization is parallel to the eld. How- ever, by pumping magnons into the sample it is possible to tilt the magnetization with respect to the magnetic eld. This creates a nonstationary state, in which the magnetization precesses around the magnetic eld. Al- though the state is far from the true equilibrium, but it, nevertheless, is a state of minimal energy at xed mag- netizationMz. Because of inevitable spin relaxation the state of uniform precession requires permanent pumping of spin and energy. However, if these processes violat- ing the spin conservation law are weak, one can ignore them and treat the state as a quasi-equilibrium state. The state of uniform precession maps on a circumference parallel to the xyplane, but in contrast to the easy-plane ferromagnet (Fig. 1c) the plane conning the precessing magnetization is much above the equator and not far the northern pole (Fig. 1d). One can consider also a current state, in which the phase (the rotation angle in the xy plane) varies not only in time but also in space with a constant gradient. The current state will be metastable due to the same reason as in an easy-plane ferromagnet: in order to relax via phase slips the magnetization should go away from the circumference on which the state of uniform precession maps, and this increases the energy. Then the plane, in which the magnetization precesses, can be considered as an eective \easy plane" originat- ing not from the equilibrium order parameter topology but created dynamically. Further the concept of dynami- cal easy plane will be applied to YIG magnetic lms with some modications. They take into account that the spin conservation law is not exact due to magnetostatic energy and the precession is not uniform since spin waves in YIG lms have the energy minima at non-zero wave vectors. In contrast to the equilibrium state, stability of the dy- namically supported non-equilibrium state even without current is not for granted and must be checked. In our discussion of topology we assumed that phase gradients were small and ignored the gradient-dependent (kinetic) energy. At growing gradient and gradient- dependent energy, we reach the critical gradient at which barriers making the supercurrent stable vanish. For su- per uids the critical gradient (critical velocity) is deter- mined from the famous Landau criterion. The analo- gous criterion was also known for spin super uidity in easy-plane anti- and ferromagnets.3In the present paper we derive this criterion for possible spin supercurrents in YIG magnetic lms with the easy plane of dynamical origin.III. LANDAU{LIFSHITZ{GILBERT THEORY AND LINEAR SPIN WAVES IN YIG BULK The coherent state of magnons is nothing else but a classical spin wave, and one can use the classical equa- tions of the LLG theory. In the LLG theory the abso- lute value of the magnetization vector Mdoes not vary in space and time, and the classical LLG equations are reduced to two equations for only two independent mag- netization components: _Mx= MzH My;_My= MzH Mx; (1) where is the gyromagnetic ratio and H=M xand H=M yare functional derivatives of the hamiltonian H. The third LLG equation for _Mzis not independent and can be derived from two equations (1). Instead of two real functions MxandMyone can introduce one com- plex function =Mx+iMy. The equation for directly follows from and fully equivalent to the LLG equations (1). By analogy with the theory of super uids they call it the Gross{Pitaevskii equation. There is another form of the equations in the LLG the- ory especially convenient for the analysis of spin trans- port. Magnetization dynamics is described in the terms of two independent variables, Mzand the angle 'of the magnetization rotation around the zaxis: _Mz=H '=@H @'+ri@H @ri'=r
j+Tz;(2) _'=H Mz(3) These are the Hamilton equations for the pair of conju- gated canonical variables \moment{angle" analogous to the conjugated pair \momentum{coordinate". The rst equation is the balance equation for magnetization along the axiszproportional to the zcomponent of spin den- sity, and we introduced the magnetization current jand the torqueTz: j=@H @r'; Tz=@H @': (4) There was decades-long discussion of ambiguity in def- inition of the spin current. Ambiguity emerges because the continuity equation for Mzcontains the torque Tz, which violates the spin conservation law. Indeed, one can add any vector bto the magnetization current jand com- pensate it by adding the divergence r
bto the torque Tz. This does not aect the nal balance. There were numerous attempts to nd a proper denition of the spin current. It was argued3that no denition is more proper than others. But some denition can be more convenient than others, and the convenience criterion may vary from case to case. The choice of denition should not aect4 nal physical results like the choice of gauge in electro- dynamics. YIG is a ferrimagnet with complicated magnetic struc- ture consisting of numerous sublattices.14However at slow degrees of freedom relevant for our analysis one can treat it simply as an isotropic ferromagnet15with the spontaneous magnetization Mdescribed by the hamil- tonian H=Z H
M+DriM
riM 2 dr +Zr
M(r)r
M(r1) 2jrr1jdrdr1: (5) Here the rst term is the Zeeman energy in the mag- netic eld H, the second term /Dis the inhomogeneous exchange energy, and the last one is the magnetostatic (dipolar) energy. Let us consider a spin wave in a YIG bulk propagating in the plane xzin a magnetic eld H parallel to the axis z. In a weak spin wave MzMM2 ? 2M;r
Mr xMx; (6) whereM?=q M2x+M2y, and the linearized equations of motion (1) are _Mx= HM y+ DM (r2 xMy+r2 zMy); _My= HM x DM (r2 xMx+r2 zMx) MrxZrxMx(r1) jrr1jdr1 : (7) The equations look as integro-dierential equations be- cause of the magnetostatic term in the equation for My. But applying the Laplace operator r2to this equation makes this term purely dierential. After exclusion of any of two component MxorMyone receives a dieren- tial equation of the 6th order. For the plane wave with the frequency !and the wave vector k(kx;0;kz) Eqs. (7) become i!M x= My(H+DMk2); i!M y= Mx H+DMk2+4Mk2 x k2 :(8) A solution of linear equations is an elliptically polarized running spin wave, Mx=m0cos(k
r+!t); My=s 1 +4Mk2x (H+DMk2)k2m0sin(k
r+!t);(9) with the wave vector k(kx;0;kz) and the frequency !(k) = s (H+DMk2) H+DMk2+4Mk2x k2 : (10)The energy density in the spin wave mode is E=m2 0 2M H+DMk2+4Mk2 x k2 !(MhMzi) ; (11) wherehMziis the averaged magnetization. In quantum- mechanical description the magnon density nm=E=~! diers from the dierence of MhMzionly by a constant factor. IV. SPIN WAVES IN FILMS, BOUNDARY CONDITIONS A spin wave propagating in the lm of thickness dpar- allel to the plane yz(Fig. 2) must satisfy the boundary conditions at two lm surfaces x=d=2. Neglecting the exchange interaction /Dthe spin wave reduces to a magnetostatic wave investigated in the past by Damon and Eshbach16. The boundary conditions are imposed on the magnetostatic magnetic eld induced by magnetic charges 4r
M(r) and determined from the equation r
(h+ 4M) = 0: (12) The magnetostatic eld is curl-free and is given by h=r ; (r) =Zr
M(r1) jrr1jdr1: (13) At any lm surface the tangential component of the mag- netic eld hand the normal component of the magnetic induction h+ 4Mmust be continuous. For the mag- netostatic mode Mx/m0coskxxeikzzi!tthe magneto- static potential inside the lm is =4Mk x k2sinkxxeikzzi!t: (14) Outside the lm at x>d= 2 there is no magnetic charges and the magnetostatic potential must satisfy the Laplace equation
= 0. Continuity of the tangential compo- nent of the magnetic eld hz=rz at the lm boundary requires continuity of , and atx>d= 2 =4Mk x k2sinkxd 2ekz(d=2x)+ikzzi!t:(15) Hdx yz FIG. 2. The YIG lm of thickness din a magnetic eld H parallel to the axis z.5 Continuity of the normal component of the magnetic in- ductionhx+ 4Mx=rx + 4Mxalso takes place if tankxd 2=kz kx: (16) This equation determines discrete values of kxfor mag- netostatic modes of Damon and Eshbach16. In our case the exchange interaction cannot be ignored, and this imposes additional boundary conditions. One cannot satisfy all boundary conditions by a single plane wave and must consider a superposition of plane waves with the same frequency !and the wave number kzbut with dierent values of kx. The dierential equations are of the 6th order in space. Correspondingly the dispersion relation (10) at xed ! andkzis a characteristic equation of the 6th order with respect tokxbut is tri-quadratic (cubic with respect to k2 x). The roots of the characteristic equation determinekxin the superposition. This approach was used in the past17. The rst root of the cubic equation for k2 xyields a small real kx, which determines the bulk mode with the frequency!. Other two roots can be found analytically if the relevant wave number k=p k2z+k2xis much smaller than 1=ld, whereld=p D= is a small scale determined by the exchange energy. The values k2 of two additional roots of the cubic equation for k2 xare negative and k are imaginary and very large (on the order of 1 =ld): k2 1 D 2H Ms 42+!2 2M2! 1 l2 d 2H Mr 42+H2 M2! :(17) These values correspond to evanescent modes conned to surface layers of rather small width ld. Close to the surface x=d=2 the boundary conditions are satised by a superposition of three modes: Mx/h coskxx+a+ep+(d=2x)+aep(d=2x)i eikzzi!t; My/"r 1 +4Mk2x H+DMk2coskxx+a+s 1 +4M HDMp2 +ep+(d 2x)+as 1 +4M HDMp2 ep(d 2x)# eikzzi!t; (18) wherep=ikare real and positive and aare ampli- tudes of two evanescent modes. The exchange boundary condition for unpinned spins11 arerxMx=rxMy= 0. They are satised if kxsinkxd 2a+p+ap= 0; kxr 1 +4Mk2x H+DMk2sinkxd 2 a+p+s 1 +4M HDMp2 + aps 1 +4M HDMp2 = 0: (19) Repeating derivation of the magnetostatic boundary con- dition done above for magnetostatic modes one obtains k2 x k2coskxd 2+a++a =kxkz k2sinkxd 2+a+kz p++akz p: (20) Equation (19) shows that the amplitudes of evanescent modes are of the order akxsinkxd 2=p. Then their contribution to the magnetostatic boundary condition (20) by a small factor kz=pkzldless than the otherterms and can be ignored. Eventually we return back to the equation (16) for kxobtained for magnetostatic waves of Damon and Eshbach16without eects of ex- change interaction. Thus even though evanescent modes are indispensable for satisfying all boundary conditions they do not aect the shape of the wave in the most of the bulk. At largekzdEq. (16) yields kx==d, and in the bulk the magnetization components Mxcoskxxand Mycoskxxvanish at the lm surfaces. This au- tomatically satises the exchange boundary conditions Mx=My= 0 for pinned spins without adding evanes- cent modes. Ignoring narrow surface layers where evanes- cent modes can be important, the plane wave propagat- ing in the lm plane is Mx=p 2m0cosx dcos(kzz+!t); My=p 2 1 +23M Hk2zd2 m0cosx dsin(kzz+!t)(21) independently from the exchange boundary conditions. The wave frequency is !(kz)  H+DMk2 z+23M k2zd2 : (22) This dispersion relation diers from the spin-wave spec- trum derived for YIG lms by Kalinikos and Slavin11and6 widely used in the past, in particular, in articles address- ing Bose{Einstein condensation and spin super uidity in YIG lms5,12,13. Kalinikos and Slavin11received a dis- persion relation, in which the term 2 M(1ekzd)=kzd replaces the magnetostatic contribution in our dispersion relation (22) (the third term /1=k2 zd2). Instead of solv- ing dierential equations Kalinikos and Slavin11approx- imately solved the integro-dierential equations. They approximated the magnetization distribution in space by a superposition of functions, which do not satisfy dierential equations in the bulk. This is easily seen in the recent simplied derivation of their spectrum by Rezende13. Rezende approximated a spin wave in the lm bulk by a superposition of plane-wave modes with dierent values of kxas in our solution (our axis xcor- respond to the axis yof Rezende and vice versa). But Rezende'skxwere not roots of the characteristic equa- tion of the relevant system of dierential equations. As a result, frequencies of modes in his superposition dier one from another and from the frequency given by the dispersion relation. In particular, two of his modes have valueskx=ikz, for which k2=k2 x+k2 zvanishes and the spectrum (10) of a single plane spin wave gives an innite frequency! Thus Rezende's superposition does not describe a proper monochromatic eigenmode at all. Correspondingly the spectrum of Kalinikos and Slavin following from this superposition is invalid. The spectrum of Kalinikos and Slavin and the spec- trum Eq. (22) are compared in Fig. 3. Quantitate dif- ference between two spectra is not so dramatic. More important is that our analysis predicts an essentially dif- ferent distribution of magnetization across the lm. The component Mxnormal to the lm approaches to zero close to the lm surface (but still outside narrow bound- ary layers, where evanescent modes are important). On the other hand, according to Rezende13, in the approxi- 1520253035400.150.200.250.300.35!!L⇡Mkzd!!L⇡Mkzd12 FIG. 3. Comparison of the linear spin-wave spectrum in a YIG lm calculated by Kalinikos and Slavin11(curve 1) and in the present paper (curve 2). Here !L= His the Larmor frequency.mation of Kalinikos and Slavin11variation of Mxacross the lm is negligible. This is important for evaluation of nonlinear corrections, which determine stability of su- percurrent states investigated further in the paper. Inaccuracy of the theory of Kalinikos and Slavin11has already been noticed by Kreisel et al.18. They calculated numerically the linear spin-wave spectrum in the micro- scopic theory and revealed that the numerically calcu- lated spectrum lies lower than the spectrum of Kalinikos and Slavin as curve 2 in Fig. 3 calculated in the LLG theory. Agreement between the microscopic and macro- scopic LLG theory is not surprizing since all scales rele- vant for our analysis are larger than atomic. V. COHERENT MAGNON CONDENSATE AND ITS STABILITY By strong parametric pumping Demokritov et al.7 were able to create a coherent state of magnons con- densed at states with lowest energies with non-zero wave vectors, which was called a magnon Bose{Einstein con- densate. A condition for emerging of the magnon conden- sate is that magnon-magnon interactions violating the spin conservation law are much weaker than interactions thermalizing the magnon gas. Despite the magnon gas required at least weak pumping for compensation of lost spin (magnons) it was treated as a quasi-equilibrium gas with xed total number of magnons (see below). The energy and the frequency !(kz) given by Eq. (22) have two degenerate minima15at nitekz=k0where magnons can condense (Fig. 4). Here k0=23 Dd21=4 =22 l2 dd21=4 : (23) In the linear theory the distribution of magnons between two condensates is arbitrary and does not aect the total energy (at xed magnetization hMzi, i.e., at xed con- densate magnon density). But non-linear corrections lift this degeneracy.19Let us consider the eect of a non- linear term/M4 ?in the expansion for hMzi: hMzi=MhM2 ?i 2MhM4 ?i 8M3: (24) The energy density of the condensate spin wave as a func- tion ofMhMziis E=H(MhMzi) + DMk2 z+23M k2zd2 MhMzihM4 ?i 8M3 :(25) For the running wave given by Eq. (21) (all magnons con- densate in one minimum) hM4 ?i= 6(MhMzi)2. The sign of the nonlinear correction is negative. This cor- responds to attraction between magnons, and the con- densate is unstable. For the running wave (21) all other nonlinear corrections are smaller and cannot aect this conclusion.7 ! kz k0 k0+K k0K! kz k0 k0+K k0K ! kz k0 k0+K k0K! kz k0 k0+K k0K! kz k0 k0+K k0K! kz k0 k0+K k0K! kz k0 k0+K k0K! kz k0 k0+K k0K FIG. 4. The spin-wave spectrum in a YIG lm. In the ground state the magnon condensate occupies two minima in the k space with kz=k0(large circles). In the current state two parts of the condensate are shifted to k=k0+K(small circles). However, if magnons condense in two minima there is another nonlinear term arising from the magnetostatic energy: Ems=ZrzMz(r)rzMz(r1)) 2jrr1jdrdr1: (26) For the running wave this term is negligible compared to the term considered above, because zvariation of Mzis weak. But the nonlinear magnetostatic term is maximal for the standing wave (two energy minima are equally populated by magnons): Mx= 2m0cosx dcoskzzcos!t; My= 2 1 +23M Hk2zd2 m0cosx dcoskzzsin!t:(27) In the standing wave Mz=Mm2 0 M(1 + cos 2kzz)(1 + cos 2kxx); (28) and Ems=3m4 0 8M2=3(MhMzi)2 2: (29) Now magnon interaction is repulsive. But this does not mean that the standing-wave condensate is absolutely stable, because the interaction energy at xed hMzide- creases when the distribution of magnons between twocondensates becomes more and more asymmetric. Even- tually the condensate spin wave transforms to the run- ning wave in which magnon-magnon interaction is at- tractive and the interaction energy is negative. Thus the magnon condensate cannot be stable! Then in- evitably a question arises why a relatively stable long- living magnon condensate was observed. Instability of the magnon condensate in YIG lms was already re- vealed earlier by Tupitsyn et al.12. In order to explain the paradox that the magnon condensate was observed despite its expected instability, they referred to size ef- fects. Another scenario is also possible. Apparently the quasi-equilibrium approach determining distribution of magnons between two energy minima from the condition of the minimal energy at xed hMzi, i.e., at xed to- tal magnon number, is not satisfactory, and instead the magnon distribution between two minima must be re- ceived from the dynamical balance taking into account spin pumping and spin relaxation. There is no evident reason why pumped magnons prefer to condensate in one minimum rather than in another, and R uckriegel and Kopietz20numerically investigated the dynamical pro- cess of the magnon condensate formation in the LLG theory assuming that the two minima are lled symmet- rically. Malomed et al.21solved numerically the Gross{ Pitaevskii equation with added spin pumping and relax- ation and found that sometimes asymmetric magnon dis- tributions emerge, but only at asymmetric boundary con- ditions. Experimentally Nowik-Boltyk et al.22revealed spatial periodic oscillations of magnon density, which are possible only if magnons condense in the both energy minima. Apparently possible asymmetry of magnon distribu- tion in the process of formation of the magnon conden- sate still deserves further investigations similar to those in Refs. 20 and 21, but it is beyond the scope of this work. Studying stability of current states (the next sec- tion) we shall use a modied quasi-equilibrium approach assuming that dynamical processes (spin pumping and relaxation) x not only the total number of magnons but also distribution of them between two energy minima. We shall focus on a pure standing wave with symmetric magnon distribution in the kspace for which critical gra- dients are higher than for asymmetric distribution. Thus we look for the upper bound for critical gradients. VI. SPIN-SUPERCURRENT STATE AND ITS STABILITY (LANDAU CRITERION) The phase variation in space in the magnon condensate depends on distribution of magnons between two energy minima. In the running wave (21) '= arctanMy Mx=!t+kzz+3M k2zd2sin 2(!t+kzz);(30)8 while in the standing wave '=!t+3M k2zd2sin 2!t: (31)Thus apart from nonessential small periodical oscilla- tions the phase gradient is rz'=kzin the running wave but vanishes in the standing wave. In the standing wave the magnetization (spin) current appears if the wave numbers kzof two condensates dier fromk0(Fig. 4), and neglecting weak ellipticity Mx=m0coskxx[cos(k0z+Kz+!t) + cos(k0zKz!t)] = 2m0coskxxcosk0zcos(Kz+!t); My=m0coskxx[sin(k0z+Kz+!t)sin(k0zKz!t)] = 2m0coskxxcosk0zsin(Kz+!t): (32) Thusrz'=K=kzk0k0. Keeping the mag- netizationhMzixed as before and taking into account the nonlinear magnetostatic term (29) the energy in the spin-current state apart from some constant terms is
E=d2!(k0) dk2zMhMzi (rz')2 2+3(MhMzi)2 2; (33) where d2!(k0) dk2z= M 2D+123 k4 0d2 =163 M k4 0d2: (34) Stability of the spin-current state can be checked fol- lowing the principal idea of the Landau criterion of super uidity3: If weak perturbations of the current state (creation of a quasiparticle in the Landau case) always in- crease energy, the current state is metastable. If there are perturbations decreasing the energy super uid transport with suppressed dissipation is impossible. Let us con- sider slowly varying in space weak perturbations mz= MzhMziandrz'0=rz'K. Quadratic in mzand rz'0terms in expansion of the energy (33) are
E0=d2!(k0) dk2zMhMzi (rz'0)2 2 Kmz rz'0 +3m2 z 2: (35) For stability of the supercurrent the quadratic form in perturbations mzandrz'0must be always positive. This takes place as far as rz'=Kis less than the critical value (rz')cr=vuut3 (MhMzi) d2!(k0) dk2z=r 3(MhMzi) Mk2 0d 4: (36) This corresponds to the critical group magnon velocity vcr=d2!(k0) dk2z(rz')cr=42 M k2 0dr 3(MhMzi) M: (37) Note that applying our course of derivation to super- uid hydrodynamics one obtains exactly the Landau crit- ical velocity equal to the sound velocity (see Sec. 2.1 in Ref. 3).We conclude this section by estimation of the magneti- zation supercurrent jusing the canonical expression (4). Close to the energy minimum the magnetization current along thezaxis is jz=@E @kz=MhMzi d! dkz MhMzi d2!(k0) dk2z(kzk0): (38) At our denition of the current it is proportional to the group velocity d!=dk zof magnons3and therefore van- ishes in the ground state of the condensate both for the running and the standing wave. VII. DISCUSSION AND CONCLUSIONS The derived critical gradient is essentially lower than obtained by Sun et al.5who determined the critical su- percurrent equating the kinetic energy to the high Zee- man energy. Our analysis demonstrates that the Zeeman energy does not aect the stability condition at all. The magnetostatic term (29) stabilizing supercurrents plays the same role as easy-plane anisotropy in easy-plane mag- nets, but the former is of dynamical origin and much smaller than the latter being proportional to the wave intensity (density of condensed magnons). A byproduct of our analysis was revision of the widely accepted spin-wave spectrum in YIG lms, which took into account proper magnetostatic and exchange bound- ary conditions on lm surfaces. This in uenced estima- tions of non-linear corrections to spin waves crucial for metastability of the magnon condensate with and with- out spin supercurrents. Let us make some numerical estimations. Accord- ing to Dzyapko et al.23the magnon density can reach 1018cm3. Assuming that 10 % of magnons are in the coherent state, this corresponds to rather small ratio (MhMzi)=M0:32104. Then Eq. (37) yields for k0= 5:5 104cm1andd= 5 104cm the critical veloc- ityvcrabout 3.6 m/sec (instead of 420 m/sec found by Sunet al.5). In the light of the presented analysis let us discuss the report by Bozhko et al.8on detection of spin supercur-9 rents in observation of a decaying magnon condensate prepared in a YIG magnetic lm by magnon pumping. The major problem with this claim is small total phase variation along streamlines of the supposed current re- alized in the experiment. Bozhko et al. applied a tem- perature gradient to the magnon BEC cloud, which led to a dierence !of the frequency of magnetization pre- cession (phase rotation velocity) across the condensate cloud. This produced a total phase variation '=!t across the BEC cloud growing linearly with time tand generating spin currents. For the maximal != 2550 rad/sec and the maximal life time t= 0:5sec of the con- densate in the experiment of Bozhko et al.8(see their Fig. 5) one can conclude that the total phase variation 'never exceeded about 1/3 of the full 2 rotation. As discussed in introduction, only currents with large num- ber of full 2 rotations along streamlines deserve the title of \supercurrent" manifesting spin super uidity. One might consider it as a purely semantic issue. But calling any current / r'supercurrent demonstrating spin super uidity would reduce spin super uidity to a trivial ubiquitous phenomenon. Currents produced by such small phase variations cannot relax via phase slips and are trivially stable. They emerge in any spin wave or domain wall. Any inhomogeneity produces them, and they must present in the experiment of Bozhko et al.8but in contrast to authors' claim they have nothing to do with the macroscopic phenomenon of super uidity. In summary, spin super uidity in YIG lms is possible in principle, although the recent report on its experimen- tal observation8is not founded. Metastability of spin supercurrents in this material is provided by energetic barriers not of topological but of dynamic origin, which depend on intensity of a nonlinear spin wave describing the coherent magnon condensate. It is worth noting that at growing magnetic eld in YIG lms the orientational phase transition takes place from the state with the total and sublattice magnetiza- tions along the magnetic eld to the state, in which mag- netizations deviate from the magnetic eld direction and have large components in the plane normal to the mag- netic eld.15This is a state with easy-plane anisotropy, for which spin super uidity have been predicted. But this requires magnetic elds 105G, which are orders of magnitude larger than elds nowadays used in exper- iments on magnon condensation. ACKNOWLEDGMENTS The author thanks S.O. Demokritov, V.S. L'vov, V.L. Pokrovsky, and A.A. Serga for useful discussions. 1E. B. Sonin, Zh. Eksp. Teor. Fiz. 74, 2097 (1978), [Sov. Phys.{JETP, 47, 1091{1099 (1978)]. 2E. B. Sonin, Usp. Fiz. Nauk 137, 267 (1982), [Sov. Phys.{ Usp., 25, 409 (1982)]. 3E. B. Sonin, Adv. Phys. 59, 181 (2010). 4H. Chen and A. H. MacDonald, \Spin-super uidity and spin-current mediated non-local transport," ArXiv:1604.02429. 5C. Sun, T. Nattermann, and V. L. Pokrovsky, Phys. Rev. Lett. 116, 257205 (2016). 6C. Sun, T. Nattermann, and V. L. Pokrovsky, J. Phys. D: Appl. Phys. 50, 143002 (2017). 7S. O. Demokritov, V. E. Demidov, O. Dzyapko, G. A. Melkov, A. A. Serga, B. Hillebrands, and A. N. Slavin, Nature 443, 430 (2006). 8D. A. Bozhko, A. A. Serga, P. Clausen, V. I. Vasyuchka, F. Heussner, G. A. Melkov, A. Pomyalov, V. S. L'vov, and B. Hillebrands, Nat. Phys. 12, 1057 (2016). 9E. B. Sonin, \Comment on \Supercurrent in a room temperature Bose{Einstein magnon condensate"," ArXiv:1607.04720. 10D. A. Bozhko, A. A. Serga, P. Clausen, V. I. Vasyuchka, F. Heussner, G. A. Melkov, A. Pomyalov, V. S. L'vov, and B. Hillebrands, \On supercurrents in Bose{Einstein magnon condensates in YIG ferrimagnet,"ArXiv:1608.01813. 11B. A. Kalinikos and A. N. Slavin, J. Phys. C: Solid State Phys. 19, 7013 (1986). 12I. S. Tupitsyn, P. C. E. Stamp, and A. L. Burin, Phys. Rev. Lett. 100, 257202 (2008). 13S. M. Rezende, Phys. Rev. B 79, 174411 (2009). 14V. Cherepanov, I. Kolokolov, and V. L'vov, Phys. Rep. 229, 81 (1993). 15A. G. Gurevich and G. A. Melkov, Magnetization Oscilla- tions and Waves (CRC Press, 1996). 16R. W. Damon and J. R. Eshbach, J. Phys. Chem. Solids 19, 308 (1961). 17T. Wolfram and R. E. DeWames, Prog. Surf. Sci. 2, 233 (1972). 18A. Kreisel, F. Sauli, L. Bartosch, and P. Kopietz, Eur. Phys. J B 71, 59 (2009). 19F. Li, W. M. Saslow, and V. L. Pokrovsky, Sci. Rep. 3, 1372 (2013). 20A. R uckriegel and P. Kopietz, Phys. Rev. Lett. 115, 157203 (2015). 21B. A. Malomed, O. Dzyapko, V. E. Demidov, and S. O. Demokritov, Phys. Rev. B 81, 024418 (2010). 22P. Nowik-Boltyk, O. Dzyapko, V. E. Demidov, N. G. Berlo, and S. O. Demokritov, Sci. Rep. 2, 482 (2012). 23O. Dzyapko, V. E. Demidov, S. O. Demokritov, G. A. Melkov, and A. N. Slavin, New J. Phys. 9, 64 (2007).

2017-02-22

Recently it was suggested that stationary spin supercurrents (spin superfluidity) are possible in the magnon condensate observed in yttrium-iron-garnet (YIG) magnetic films under strong external pumping. Here we analyze this suggestion. From topology of the equilibrium order parameter in YIG one must not expect energetic barriers making spin supercurrents metastable. However some small barriers of dynamical origin are possible nevertheless. The critical phase gradient (analog of the Landau critical velocity in superfluids) is proportional to intensity of the coherent spin wave (number of condensed magnons). The conclusion is that although spin superfluidity in YIG films is possible in principle, the published claim of its observation is not justified. The analysis revealed that the widely accepted spin-wave spectrum in YIG films with magnetostatic and exchange interaction required revision. This led to revision of non-linear corrections, which determine stability of the magnon condensate with and without spin supercurrents.

Is spin superfluidity possible in YIG films?

1702.06994v2

1 I. INTRODUCTION HERE is growing interest in novel co mputational devices able to overcome the limits of the current complimentary - metal -oxide -semiconductor (CMOS) technology and provide further increase of the computational throughput [1]. So far, the majority of the “beyond CMOS” proposals are aimed at the development of new switching technologies [2, 3] with increased scalability and improved power consumption characteristics over the silicon transistor. However , it is difficult to expect that a new switch will outperform CMOS in all figures of merit, and more importantly, will be able to provide multiple generations of improvement as was the case for CMOS [4]. An alternative route to the computational power enhancement is via the development of novel computing devices aimed not to replace but to complement CMOS by special task data processing [5]. Spin wave (magnonic) logic devices are one of the alternative approaches aimed to take the advantages of the wave interference at nanometer scale and utilize phase in addition to amplitude for building logic units for parallel data processing. A spin wave is a collective oscillation of spins in a magnetic lattice, analogous to phonons, the col lective oscillation of the nuclear lattice. The typical propagation speed of spin waves does not exceed 107cm/s, while the attenuation time at room temperature is about a nanosecond in the conducting ferromagnetic materials (e.g. NiFe, CoFe) and may be hu ndreds of nanoseconds in non -conducting materials (e.g. YIG). Such a short attenuation time explains the lack of interest in spin waves as a potential information carrier in the past. The situation has changed drastically as the technology of integrated l ogic circuits has scaled down to the deep sub - micrometer scale, where the short propagation distance of spin waves (e.g. tens of microns at room temperature) is more than sufficient for building logic circuits. At the same time, spin waves have several in herent appealing properties making them promising for building wave -based logic devices. For instance, spin wave propagation can be directed by using magnetic waveguides similar to optical waveguides. The amplitude and the phase of propagating spin waves c an be modulated by an external magnetic field. Spin waves can be generated and detected by electronic components (e.g. multiferroics [6]), which make them suitable for integration with conventional logic circuits. Finally, the coherence length of spin waves at room temperature may exceed tens of micr ons, which allows for the utilization of spin wave interference for logic functionality. It makes spin waves much more prone to scattering than a single electron and resolves one of the most difficult problems of spintronics associated with the necessity to preserve spin orientation while transmitting information between the spin -based units. During the past decade, there have been a growing number of theoretical and experimental works exploring spin wave propagation in a variety of magnetic structure s [7, 8], the possibility of spin wave propagation modulation by an external magnetic field [9, 10], and spin wave interference and diffraction [11-15]. The collected experimental data revealed interesting and unique properties of spin wave F. Gertz1, A. Kozhevnikov2, Y. Filimonov2, D.E. Nikonov3 and A. Khitun1 1)Electrical Engineering Department, University of California - Riverside, Riverside, CA, USA, 92521 2)Kotel’niko v Institute of Radioengineering and Electronics of Russian Academy of Sciences, Saratov Branch, Saratov, Russia, 410019 3)Technology & Manufacturing Group Intel Corp. , 2501 NW 229th Avenue, Hillsboro, OR, USA, 97124 Magnonic Holographic Memory: from Proposal to Device T In this work, we present recent developments in m agnonic holographic memory devices exploiting spin waves for information transfer. The devices comprise a magnetic matrix and spin wave generating/detecting elements placed on the edges of the waveg uides. The matrix consists of a grid of magnetic waveguide s connected via cross junctions. Magnetic memory elements are incorporated within the junction while the read -in and read -out is accomplished by the spin waves propagating through the waveguides. We present experimental data on spin wave propagation throu gh NiFe and YIG magnetic crosses. The obtained experimental data show prominent spin wave signal modulation (up to 20 dB for NiFe and 35 dB for YIG) by the external magnetic field, where both the strength and the direction of the magnetic field define the transport between the cross arms. We also present experimental data on the 2 -bit magnonic holographic memory built on the double cross YIG structure with micro -magnets placed on the top of each cross. It appears possible to recognize the state of each ma gnet via the interference pattern produced by the spin waves with all experiments done at room temperature. Magnonic holographic devices aim to combine the advantages of magnetic data storage with wave -based information transfer. We present estimates on th e spin wave holographic devices performance, including power consumption and functional throughput. According to the estimates, magnonic holographic devices may provide data processing rates higher than 1×1018 bits/cm2/s while consuming 0.15mW. Technologic al challenges and fundamental physical limits of this approach are also discussed. Index Terms — spin waves, holography, logic device. 2 transport (e.g. non -reciprocal spin wave propagation [16]) for building magnonic logic circuits. The first working spin -wave based logic device has been experimentally demonstrated by Kostylev et al in 2005 [17]. The authors used the Mach – Zehnder -type current -controlled spin wave interferometer to demonstrate output voltage modulation as a result of spin wave interference. Later on, exclusive -not-OR (NOR) and not - AND (NAND) gates were experimentally demonstrated utilizing a similar structure [18]. The idea of using Mach – Zehnde r-type spin wave interferometers has been further evolved by proposing a spin wave interferometer with a vertical current -carrying wire [19]. With zero applied current, the spin waves in two branches interfere constructively and propagate through the structure. The waves interfere destructively and do not propagate through the structure if a certain electric current is applied. At some point, these first magnonic logic devices resemble the classical field effect transistor, where the magnetic field produced by the electric current modulates the propagation of the spin wave —an analogue to the electric current. Then, it was proposed to combine spin wave with nano -magnetic logic aimed to combine the advantages of non -volatile data storage in magnetic memory and the enhanced functionality provided by the spin wave buses [20]. The use of spin wave interference makes it possible to realize Majority gates (which can be used as AND or OR gates) and NOT gates with a fewer number of elements than is required for transistor -based circuitry, promising the further reduction of the size of the l ogic gates. There were several experimental works demonstrating three - input spin wave Majority gates [21,22]. Ho wever, the integration of the spin wave buses with nano -magnets in a digital circuit, where the magnetization state of the nano - magnet is controlled by a spin wave has not yet been realized. An alternative approach to spin wave -based logic devices is to build non -Boolean logic gates for special task data processing. The essence of this approach is to maximize the advantage of spin wave interference. Wave -based analog logic circuits are potentially promising for solving problems requiring parallel operatio n on a number of bits at time (i.e. image processing, image recognition). The concept of magnonic holographic memory (MHM) for data storage and special task data processing has been recently proposed [23]. Holographic devices for dat a processing have been extensively developed in optics during the past five decades. The development of spin wave -based devices allows us to implement some of the concepts developed for optical computing to magnetic nanostructures utilizing spin waves instead of optical beams. There are certain technological advantages that make the spin wave approach even more promising than optical computing. First, short operating wavelength (i.e. 100nm and below) of spin wave devices promises a significant increase of the data storage density (~λ2 for 2D and ~λ3 for 3D memory matrixes). Second, even more importantly, is that spin wave bases devices can have voltage as an input and voltage as an output, which makes them compatible with the conventional CMOS circuitry. Th ough spin waves are much slower than photons, magnonic holographic devices may possess a higher memory capacity due to the shorter operational wavelength and can be more suitable for integration with the conventional electronic circuits. In this work, we p resent recent experimental results on magnonic holographic memory and discuss the advantages and potential shortcomings of this approach. The rest of the paper is organized as follows. In Section II, we describe the structure and the principle of operation of magnonic holographic memory. Next, we present experimental data on the first 2 -bit magnonic holographic memory in Section III. The advantages and the challenges of the magnonic holographic devices are discussed in the Sections IV. In Section V, we pres ent the estimates on the practically achievable performance characteristics. II. MATERIAL STRUCTURE AN D THE PRINCIPLE OF O PERATION The schematics of a MHM device are shown in Figure 1(A). The core of the structure is a magnetic matrix consisting of the grid of magnetic waveguides with nano -magnets placed on top of the waveguide junctions. Without loss of generality, we have depicted a 2D mesh of orthogonal magnetic waveguides, though the matrix may be realized as a 3D structure comprising the layers of magnetic waveguides of a different topology (e.g. honeycomb magnetic lattice). The waveguides serve as a media for spin wave propagation – spin wave buses. The buses can be made of a magnetic material such as yttrium iron garnet Y 3Fe2(FeO 4)3 (YIG) or permalloy ( Ni81Fe19) ensuring maximum possible group velocity and minimum attenuation for the propagating spin waves at room temperature. The nano -magnets placed on top of the waveguide junctions act as memory elements holding information encoded in the magnetizatio n state. The nano - magnet can be designed to have two or several thermally stable states of magnetization, where the number of states defines the number of logic bits stored in each junction. The spins of the nano -magnet are coupled to the spins of the junction magnetic wires via the exchange and/or dipole -dipole coupling affecting the phase of the propagation of spin waves. The phase change received by the spin wave depends on the strength and direction of the magnetic field produced by the nano -magnet. At the same time, the spins of the nano -magnet are affected by the local magnetization change caused by the propagating spin waves. The input/output ports are located at the edges of the waveguides. These elements are aimed to convert the input electric si gnals into spin waves, and vice versa, convert the output spin waves into electrical signals. There are several possible options for building such elements by using micro - antennas [24, 25], spin torque oscillators [26], and multiferroic elements [6]. For example, the micro -antenna is a conducting contour placed in the vicinity of the spin wave bus. An electric current passed through the contour generates a magnetic field around the current -carrying wires, which excites spin waves in the magnetic material, and vice versa, a propagating spin wave changes the magnetic flux from the magnetic waveguide and generates an inductive vo ltage in the antenna contour. The advantages and shortcomings of different input/output 3 elements will be discussed later in the text. Spin waves generated by the edge elements are used for information read -in and read -out. The difference among these two modes of operation is in the amplitude of the generated spin waves. In the read -in mode, the elements generate spin waves of a relatively large amplitude, so two or several spin waves coming in -phase to a certain junction produce magnetic field sufficien t for magnetization change within the nano - magnet. In the read -out mode, the amplitude of the generated spin waves is much lower than the threshold value required to overcome the energy barrier between the states of nano - magnets. So, the magnetization of the junction remains constant in the read -out mode. The details of the read -in and read-out processes are presented in Ref. [23]. The formation of the hologram occurs in the following way. The incident spin wave beam is produced by the number of spin wave generating elements (e.g. by the elements on the left side of the matrix as illustrated in Figure 1(B)). All the elements are biased by the same RF generator exciting spin waves of the same frequency, f, and amplitude, A0, while th e phase of the generated waves are controlled by DC voltages applied individually to each element. Thus, the elements constitute a phased array allowing us to artificially change the angle of illumination by providing a phase shift between the input waves . Propagating through the junction, spin waves accumulate an additional phase shift, Δ, which depends on the strength and the direction of the local magnetic field provided by the nano -magnet, Hm: , (1) where the particular form of the wavenumber k(H) dependence varies for magnetic materials, film dimensions, the mutual direction of wave propagation and the external magnetic field [27]. For example, spin waves propagating perpendicular to the external magnetic field (magnetostatic surface spin wave – MSSW) and spin waves propagating parallel to the direction of the external field (backward volume magnetostatic spin wave – BVMSW) may obtain significantly different phase shifts for the same field strength. The phase shift produced by the external magnetic field variation H in the ferromagnetic film can be expressed as follows [17]: 2 22 2 2)( HMH dl HS (BVMSW), ) 4 () 2 ( 2 22 SS M HHM H dl H (MSSW), (2) where is the phase shift produced by the change of the external magnetic field H, l is the propagation len gth, d is the thickness of the ferromagnetic film, is gyromagnetic ratio, ω=2πf , 4πM s is the saturation magnetization of the ferromagnetic film. The output signal is a result of superposition of all the excited spin waves traveling through the different paths of the matrix. The amplitude of the output spin wave is detected by the voltage generated in the output element (e.g. the inductive voltage produced by the spin waves in the antenna contour). The amplitude of the output voltage is corresponding to t he maximum when all the waves are coming in -phase (constructive interference), and the minimum when the waves cancel each other (destructive interference). The output voltage at each port depends on the magnetic states of the nano -magnets within the matri x and the initial phases of the input spin waves. In order to recognize the internal state of the magnonic memory, the initial phases are varied (e.g. from 0 to ). The ensemble of the output values obtained at the different phase combinations constitute a hologram which uniquely corresponds to the internal structure of the matrix. In general, each of the nanomagnets can have more than 2 thermally stable states, which makes it possible to build a multi -state holographic memory device (i.e. zN possible memory states, where z is the number of stable magnetic states of a single junction and N is the number of junctions in the magnetic matrix). The practically achievable memory capacity depends on many factors including the operational wavelength, coherence length, the strength of nano -magnets coupling with the spin wave buses, and noise immunity. In the next Section, we present experimental data on the operation of the prototype 2 -bit magnonic holographic memory. III. EXPERIMENTAL DATA The set of experiments st arted with the spin wave transport study in a single cross structure, which is the elementary building block for 2D MHM as depicted in Fig.1. Two types of single cross devices made of Y 3Fe2(FeO 4)3 (YIG) and Permalloy (Ni 81Fe19) were fabricated. Both of t hese materials are promising for application in magnonic waveguides due to their high coherence length of spin waves. At the same time, YIG and Permalloy differ significantly in electrical properties (e.g. YIG is an insulator, permalloy is a conductor) and in fabrication method. YIG cross structures were made from single crystal YIG films epitaxially grown on top of Gadolinium Gallium Garnett (Gd 3Ga5O12) substrates using the liquid -phase transition process. After the films were grown, micro -patterning was done by laser ablation using a pulsed infrared laser ( λ≈1.03 μm), with a pulse duration of ~256 ns. The YIG cross junction has the following dimensions: the length of the whole structure is 3mm; the width of the arm in 360µm; thickness is 3.6um. Permalloy crosses were fabricated on top of oxidized silicon wa fers. The wafer was spin coated with a 5214E Photoresist at 4000 rpm and exposed using a Karl Suss Mask Aligner. After development, a permalloy metal film was deposited via Electron -Beam Evaporation with a thickness of 100nm and with an intermediate seed layer of 10 nm of Titanium to increase the adhesion properties of the Permalloy film. Lift -off using acetone completed the process. Permalloy cross junction has the following dimensions: the length of the whole structure is 18um; the width of the arm in 6µm; thickness is 100 nm. Spin waves in YIG and Permalloy structures were excited and detected via micro -antennas that were placed at the edges r mdrHk 0)( 4 of the cross arms. Antennas were fabricated from gold wire and mechanically placed directly at the top of the YIG cross. In the case of permalloy, the conducting cross was insulated with a 100nm layer of SiO 2 deposited via Plasma -Enhanced Chemical Vapor Deposition (PECVD) and gold antennas were fabricated using the same photolithographic and lift -off procedure as with the permalloy cross structures. A Hewlett - Packard 8720A Vector Network Analyzer (VNA) was used to excite/detect spin waves within the structures using RF frequencies. Spin waves were excited by the magnetic field generated by the AC electric curre nt flowing through the antenna(s). The detection of the transmitted spin waves is via the inductive voltage measurements as described in Ref. [28]. Propagat ing spin waves change the magnetic flux from the surface, which produces an inductive voltage in the antenna contour. The VNA allowed the S -Parameters of the system to be measured; showing both the amplitude of the signals as well as the phase of both the transmitted and reflected signals. Samples were tested inside a GMW 3472 -70 Electromagnet system which allowed the biasing magnetic field to be varied from -1000 Oe to +1000 Oe. The schematics of the experimental setup for spin wave transport study in the single cross structures are shown in Figure 2. First, we studied spin wave propagation between the four arms of the permalloy cross -structure as shown in Fig.3(A -B) under different bias magnetic field. The input/output ports are numbered from 1 to 4 sta rting at the 9 O’ clock position and then enumerated sequentially in along a clockwise direction. In order to define the angle between the external magnetic field and the direction of signal propagation, we define the X axis along the line from port 1 to port 3, and the Y axis along the line from port 4 to port 2 propagating as depicted in Fig.2. Spin waves were excited on port 2 (the top of the magnetic cross) and read out from port 4 (the bottom of the cross) (see figure 1). The graph in Fig.3 shows th e change of the amplitude of the transmitted signal as a function of the strength of the external magnetic field directed perpendicular to the propagating spin waves as depicted in the inset to Fig.3. Hereafter, we show the relative change of the amplitude in decibels normalized to some value (e.g. to the maximum value). The normalization is needed as the input power varies significantly for permalloy and YIG structures as well as for the type of experiment. The reference transmission level is taken at 300 Oe, where the S 12 parameter is at its absolute maximum. At small magnetic fields below 100 Oe a very small amplitude signal was observed. At approximately 150 Oe there is a noticeable increase in the amplitude followed by a plateau in the response as th e field is increased to 500 Oe. Also of interest is the response of the signal as a function of the applied magnetic field direction. In Fig.3(D), we present an example of the experimental data showing the influence of the direction of the bias field on spin waves transport from port 2 to port 4. The results demonstrate prominent change in the amplitude of the transmitted signal [18dB] when the field is applied between 20° and 30°. The main observations of these experiments are the following. (i) Spin w ave propagation through the cross junction can be efficiently controlled by the external magnetic field. (ii) Both the amplitude and the direction of the magnetic field can be utilized for spin wave control. We conducted similar experiments on the YIG single cross device as shown in Figure 4. It was observed that prominent signal modulation could be determined by the direction and the strength of the external magnetic field. In Fig.4, there is shown an example of experimental data on the spin wave transport between ports 2 and 1. The maximum transmission between the orthogonal arms occurs when the field is applied at 68°, while the minimum is seen when the field is applied at 0°. The On/Off ratio for the YIG cross reaches 35dB. Of noticeable interest is also the effect of non -reciprocal spin wave propagation. The two curves in Figure 4(D) show signal propagation from port 2 to port 4, and in the opposite direction from port 4 to port 2. The measurements are done at the same bias magnetic field of 998 Oe. There is a difference of about 5dB for the signals propagating in the opposite direction. The effect is observed in a relatively narrow frequency range (e.g. from 5.2GHz to 5.4GHz). The effect of non -reciprocal spin wave propagation may be of some practic al interest for building magnonic diodes, though a more detailed study is required. Concluding on the spin wave transport in the permalloy and YIG single cross structures, prominent signal modulation has been observed in both cases. For the chosen paramete rs, the operation frequency is slightly higher for YIG structure (~5GHz) than for permalloy (~3GHz). The speed of signal propagation is slightly faster in permalloy (3.5×106 cm/s) than in YIG (3.0×106 cm/s). The difference in the spin wave transport can be attributed to the differences between the intrinsic material properties of YIG and Permalloy as well as the difference in the cross dimensions. It is important to note, that in both cases the level of the power consumption was at the microwatt scale (e. g. 0.1µW -1.0µW for permalloy and 0.5µW -5.0µW for YIG) with no feasible effect of micro heating on the spin wave transport. The summary of the experimental findings for permalloy and YIG single cross junctions cab be found in Table I. Next, we carried out experiments on spin wave transport and interference in the double -cross structure made of YIG as shown in Figure 5. The choice of material is mainly due to the larger size of the structure and spin wave detecting antennas, where the larger the area of the detecting contour results in higher the observed output inductive voltage. The multi -port double -cross YIG structure is suitable for the study of spin wave interference. In this study, several coherent spin wave signals were excited by ports 2,3,4 and 5 connected to one port of the VNA. The output is detected at port 6. The phase shifters were employed to vary the phase difference between the ports as shown in Figure 5(B). Figure 6 show the experimental data on the output voltage collected in the frequency range from 5.3GHz to 5.5GHz. The curves of the different color correspond to the different phase shifts between the spin wave generated ports. Phase 1 represents a change in the phase of ports 4 and 6 and Phase 2 represents a change in the phase of ports 3 and 5. Figures 6(B -D) show the slices of 5 data taken at a frequency of 5.385GHz, 5.410GHz and 5.45GHz, respectively. The black markers depict the experimentally obtained data, and the red markers depict the theoretical output for the ideal case of the interfering waves of the same frequency and amplitude. The theoretical data is normalized to have the same maximum value as the experimental data at phase difference zero (constructive interference). Taking l=1.1mm, d=360μm, H=1000 Oe, 4πM s=1750G, and H=20Oe, we estimated possible phase shift by Eqs. (1 -2) to be about π/2, which is in good agreement with the experimental data. This fact implies the dominant role of wave interference in the output signal formation. Discrepancies in the amplitude can be attributed to parasitic noise which raises the base amplitude of the signal to greater than nonzero value even when the phase should be perfectly destructive. Also, it should be noted that Eqs. (1 -2) are derived for sp in wave propagating in a homogeneous magnetic field, while the magnetic field produced by the micromagnets in the experiment may be inhomogeneous across the thickness and in lateral dimensions. The data presented in Figure 7 are collected in the experimen ts where Phase 2 (ports 3 and 5) was changed, while Phase 1 (ports 2 and 4) was kept constant. The ability to independently change the initial phases of the spin waves is equivalent to changing the angle of illumination for building a holograms as illustra ted in Fig.1(B). In Figure 7, we present experimental data showing the holographic image of the double -cross structure without memory elements. The surface is a computer reconstructed 3 -D plot showing the output voltage as a function of Phase 1 and Phase 2. The excitation frequency is 5.40 GHz, the bias magnetic field is 1000 Oe directed from port 1 toward port 6. In this case, antennas on ports 2 and 4 generated spin waves with the initial Phase 1, and antennas on ports 3 and 5 generated spin waves with initial Phase 2. No signal is applied to port 1. The output is detected at port 6. The change of the output inductive voltage is a result of spin wave interference. It has maximum values in the case of the constructive interference (i.e. Phase 1=Phase 2, (0,0) or (π,π)), and shows minimum output signal when the waves are coming out -of-phase ((0,π) or (π,0)). Finally, we conducted experiments to demonstrate the operation of a prototype 2 -bit magnonic holographic memory device. Two micro -magnets made of co balt magnetic film were placed on top of the junctions of the double -cross YIG structure. As mentioned in Section II, these magnets serve as a memory element, where the magnetic state represents logic zeroes and ones. The schematic of the double -cross st ructure with micro -magnets attached are shown in Fi gure 8(A). The length of each magnet is 1.1mm, the width is 360μm and each has a coercivity of 200 -500 Oersted (Oe). For the test experiments, we used four mutual orientations of micro - magnets, where the magnets are oriented parallel to the axis connectin g ports 1 -6, or the axis connecting 2 -4; and two cases when the micro -magnets are oriented in the orthogonal directions. Holographic images were collected for each case. Fig.8 shows the collection of data corresponding to output voltage obtained for differ ent magnetic configurations. The phases of the input elements are the same as in the previous experiment. Markers of different shape and color in the legend of figure 8 represent the direction of the “north” end of the micro magnet. The output from the sam e structure varies significantly for different phase combinations. In some cases, the magnetic states of the magnets can be recognized by just one measurement (e.g. (0,0) phase combination). It is also possible that different magnetic states provide almost the same output (e.g. parallel and orthogonal magnet configurations measured at (π,0) phase combination). The main observation we want to emphasize is the feasibility of parallel read -out and reconstruction of the magnetic state via spin wave interference . As one can see from the data in Figure 8(B), it is possible to distinctly identify the magnetic states by changing the phases of the interfering waves, which is similar to changing the angle of observation in a conventional optical hologram. We would li ke to emphasize that all experiments reported in this Section are done at room temperature. IV. DISCUSSION The obtained experimental data show the practical feasibility of utilizing spin waves for building magnonic holographic logic devices and helps to illust rate the advantages and shortcomings of the spin wave approach. Of these results there are several important observations we wish to highlight. First, spin wave interference patterns produced by multiple interfering waves are recognized for a relatively lo ng distance (more than 3 millimeters between the excitation and detection ports) at room temperature. Despite the initial skepticism [29], coding information into the phase of the spin waves appears to be a robust instrument for information transfer showing a negligible effect to thermal noise and immunity to the structures imperfections. This immunity to the thermal fluctuations can be explain ed by taking into account that the flicker noise level in ferrite structures usually does not exceed -130 dBm [30]. At the same time, spin waves are not sensitive to the structure’s imperfections which have dimensions much shorter than the wavelength. These facts explain the good agreement between the experimental and theo retical data (e.g. as shown in Figure 6). Second, spin wave transport in the magnetic cross junctions is efficiently modulated by an external magnetic field. Spin wave propagation through the cross junction depends on the amplitude as well as the directi on of the external field. This provides a variety of possibilities for building magnetic field - effect logic devices for general and special task data processing. Boolean logic gates such as AND, OR, NOT can be realized in a single cross structure, where an applying external field exceeding some threshold stops/allows spin wave propagation between the selected arms. The ability to modulate spin wave propagation by the direction of the magnetic field is useful for application in non -Boolean logic devices. It is important to note that in all cases the magnitude of the modulating magnetic field is of the order of hundreds of Oersteds, which can be produced by micro - and nano - magnets. 6 Finally, it appears possible to recognize the magnetic state of the magnet pla ced on the top of the cross junction via spin waves, which introduces an alternative mechanism for magnetic memory read out. This property itself may be utilized for improving the performance of conventional magnetic memory devices. However, the fundament al advantage of the magnonic holographic memory is the ability to read -out a number of magnetic bits in parallel though the obtained experimental data demonstrates the parallel read -out of just two magnetic bits. In the rest of this Section, we discuss the fundamental limits and the technological challenges of building multi -bit magnonic holographic devices and present the estimates on the device performance. We start the discussion with the choice of magnetic material for building spin waveguides. Spin wa ve transport in nanometer scale magnetic waveguides has been intensively studied during the past decade [28, 31-33]. There are two materials that have become predominant, permalloy (Ni 81Fe19) and YIG, for spin wave devices prototyping. The coherence length of spin waves in permalloy is about tens of microns at room temperature [28, 31], while the coherence length in a non-conducting YIG exceeds millimeters [34]. The attenuation time for spin waves at room temperature is about a nanosecond in permalloy and a hundreds of nanoseconds in YIG[34]. However, the fabrication of YIG waveguides require a special gadolinium gallium garnet (GGG) substrate. In contrast, a permalloy film can be deposited onto a silicon platform by using the sputtering technique. Though YIG has better properties in terms of the coherence length and a lower attenuation, permalloy is mo re convenient for making magnonic devices on a silicon substrate. There are two major physical mechanisms affecting the amplitude/phase of the spin wave propagating under the junction magnet: (i) interaction with magnetic field produced by the magnet, and (ii) damping due to the presence of the conducting material. The effect of conducting films on spin wave propagation has been studied for MSSWs in the ferrite - metal structures [35, 36]. It was found that the strength of the spin wave dispersion modification is defined by the critical parameter G given as following: (3) where t is the thickness of the conducting film, q is the wave number, and lsk is the skin depth . The presence of a metallic film results in a prominent spin wave dispersion modi fication for G>3, if the is width of the gap between the ferrite and metallic film is less than the wavelength. Spin wave is completely damped in the range 1/3<G<3 due to the excessive absorption by the conducting electrons. The effect vanishes for G<1/3. In our exper iments, we used junction magnets made of cobalt with the thickness of 50nm (bulk electric conductivity σ 1.6107 S/m), which correspond to lsk 1.5 μm. The range of wave numbers q is restricted by the size of the sample L q >π/L>10 cm-1 , and by the width o f the micro -antennas W 30μm, q <π/W<1000 cm-1. In all experiments, the range of the wave numbers is confined within the following range: 10 cm-1<q<1000 cm-1, which, in turn, corresponds to the range for parameter G: 0.002< G<2. It should be noted that Eq. 3 has been derived for an infinite ferrite waveguide, which ignores the effect of space confinement. Taking into account the real dimensions of the YIG substrate 3.6μm, we obtain the minimum boundary for q≈80 cm-1. Thus, we have G<1/3 (negligible effect of spin wave dispersion modification due to the losses) for all experiments. In general, there may be other physical phenomena contributing to the spin wave dispersion modification in a magnetic cross -structure. The development of MHM devices will require a great deal of efforts in the theoretical study and numerical modeling of spin wave transport in magnetic nanostructures. In order to make a multi -bit magnonic holographic devices, the operating wavelength should be scaled down below 100nm [23]. The main challenge with shortening the operating wavelength is associated with the building of nanometer -scale spin wave generating/detecting elements. There are several possible ways of building input/output elements by using micro -antennas [28], spin torque oscillators [26], and multi - ferroic elements [6]. So far, micro -antennas are the most convenient and widely used tool for spin wave excitation and detection in ferromagnetic films [31]. Reducing the size of the antenna will lead to the reduction of the detected inductive voltage. This fact limits the practical application of any types of conducting contours for spin wave detection. The utilization of spin torque oscillators makes it possible to scale down the size of the elementary input/output port to several nanometers [37]. The main challenge for the spin torque o scillators approach is to reduce the current required for spin wave generation. More energetically efficient are the two -phase composite multiferroics comprising piezoelectric and magnetostrictive materials [38]. An electric field applied across the piezoelectric produces stress, which, in turn, affects the magnetization of the magnetoelastic mater ial. The advantage of the multiferroic approach is that the magnetic field required for spin wave excitation is produced via magneto -electric coupling by applying an alternating electric field rather than an electric current. For example, in Ni/PMN - PT synt hetic multiferroic reported in Ref. [39], an electric field of 0.6MV/m has to be applied acro ss the PMN -PT in order to produce 90 degree magnetization rotation in Nickel. Such a relatively low electric field required for magnetization rotation translates in ultra -low power consumption for spin wave excitation [20]. At the same time, the dynamics of the synthetic multiferroics, especially at the nanometer scale, remains mostly unexplored. To benchmark the performance of the magnonic holographic devices, we apply the charge -resistance approach as developed in R ef. [40] The details of the estimates and the key assumpt ions are given in Appendix A. According to the estimates, MHM device consisting of 32 inputs, with a 60nm separation distance between the inputs would consume as low as 150µW of power or 72fJ per computation. At the same time, the functional throughput o f the MHM scales proportional to the number of cells per area/volume and 2 sklqtG 7 exceeds 1.5×1018 bits/cm2/s for a 60nm feature size. It is interesting to note, that holographic logic units can be used for solving certain nondeterministic polynomial time (NP) clas s of problems (i.e. finding the period of the given function). The efficiency of holographic computing with classical waves is somewhere intermediate between digital logic and quantum computing, allowing us to solve a certain class of problems fundamentall y more efficient than general -type processors but without the need for quantum entanglement [41]. Image recognition and processing are among the most promising applications of magnonic holographic device exploiting its ability to process a large number of bits/pixels in parallel within a single core. Ther e are many questions on spin wave transport (e.g. in magnetic crosses), which remain mostly unexplored. For instance, it is not clear the mechanism responsible for spin wave splitting between the orthogonal arms. To the best of our knowledge, there is no t heoretical work explaining the observed spin wave propagation in magnetic crosses. It would be of great interest to study the dynamics of spin wave redirection depending on the geometry of the cross. It may be expected that the amplitude/phase of the redir ected (bended) spin wave depends on the wavelength/size ratio, the material properties of the cross, and magnetic field produced by the nano -magnet. Also, in this work, we attribute the change in the interference pattern to the different phase shifts accum ulated by spin waves propagating under the nano -magnets of different orientation (i.e. Eqs. 1 -2). A real picture may be much more complicated due to the difference in amplitudes, which may arise to the different factors. There is no doubt that the develo pment of magnonic holographic memory devices will take a great deal of efforts including experimental as well as theoretical studies. V. CONCLUSIONS The collected experimental data show rich physical phenomena associated with spin wave propagation in single - and double -cross structures. Prominent signal modulation by the direction, rather than the amplitude of magnetic field and the low effect of thermal noise on spin wave propagation at room temperature are among the many interesting findings presented her e. The effect of spin wave redirection between the cross arms by the external magnetic field may be further exploited for building a variety of logic devices. Besides, spin waves appear to be a robust instrument allowing us to sense the magnetic state of micro -magnets by the change in the interference pattern. Quite surprisingly, it is possible to recognize the unique holographic output for the different orientations of micro -magnets in a relatively long device at room temperature. Overall, the obtained data demonstrates the practical feasibility of building magnonic holographic devices. These holographic devices are aimed not to replace but to complement CMOS in special type data processing such as speech recognition and image processing. According to estimates, scaled magnonic holographic devices may provide more than 1×1018 bits/cm2/s data processing rate while consuming less than 0.2mW of energy. The main technological challenges are associated with the scaling down the operating wavelength and buil ding nanometer scale spin wave generating/detecting elements with spin torque oscillators and multiferroics being among the most promising solutions. At the same time, it is expected that reducing the operating wavelength will make spin waves more sensitiv e to structure imperfections. The development of scalable magnonic holographic devices and their incorporation with conventional electronic devices may pave the road to the next generations of logic devices with functional capabilities far beyond current C MOS. APPENDIX This section contains the details on the power consumption estimates. We assume that the input to each magnetoelectric (ME) cell is an AC voltage with amplitude inV created in a RLC oscillator. Then the power dissipation on resonance is RVPin in22 . (1) The amplitude of the electric field in the piezoelectric of thickness pzt is pz in in tV E / . (2) The amplitude of strain created in the piezoelectr ic of the ME cell is in xx Ed31 , (3) where the piezoelectric coefficient is 31d . Hereafter, it is assumed that spin wave is propagating along the X axis as shown in the inset to Fig2. The stress transferre d to the ferromagnet is xxY , (4) where the Young’s modulus of the piezoelectric is Y . The change in the magnetic anisotropy due to magnetostriction is 23msU , (5) where the magnetostriction coefficient of the ferromagnet is . Then the maximum amplitude of magnetization change is sms xMUM 02 , (6) where the permeability of vacuum is , and the saturation magne tization is sM . This can be expressed via the magnetoelectric coefficient s inx MYd EM31 0 3 . (7) Then the generated dimensionless amplitude of the spin wave can be approximated as follows pzsin sx intMV MMA 0 , (8) The spin waves interact and attenuate as they propagate. If the distance between inputs is L , the number of inputs is N , and the attenuation length is atl , then the propagation distance is 8 NL Ltot (9) and let the minimum amplitude needed to be detected is a quarter of the average output amplitude attot in lL AA exp4min . (10) If the group velocity of the spin waves is sw totcL/ , then the time needed for one holographic imaging is swcNL/ . (11) Assuming that the detection occurs by the inverse of the ME effect, and that its coefficients are the same as for the direct ME effect, we obtain that the connection between the magnetization amplitude and the amplitude of the gene rated electric field MEpz min0 . (12) Thus the minimum output voltage is 0min min min pzpz s pztAMtE V . (13) Combining expressions (10) and (13), we have the following for voltages 002 min exp4 pz atin lNL VV . (14) Then the total driving power for N inputs is 4422 min22exp8 2 c lNL RNV RNVPpz atin tot , (15) where c is the speed of light. For minimum detectability, the output voltage should exceed the Johnson noise voltage by 5X. The spectral density of noise is TRk VB n42 . (16) The required power within the bandwidth B (approximately equal to the ac voltage frequency) is 4422exp 800 c lNLT NBk Ppz atB tot . (17) And the total energy for one imaging is tot totP E . 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Valiant, "Holographic algorithms," Proceedings. 45th Annual IEEE Symposium on Foundations of Computer Science, pp. 306-15, 2004 2004. Figure Captions Figure 1. (A) Schematics of Magnonic Holographic Memory consisting of a 4×4 magnetic matrix and an array of spin wave generating/detecting elements. For simplicity, the matrix is depicted as a two -dimensional grid of magnetic wires with just 4 elements on each side. These wires serve as a media for spin wave propagation. The nano -magnet on the top of the junction is a memory element, where information is encoded into the magnetization state. The spins of the nano -magnet are coupled to the spins of the magnetic wires via the dipole -dipole or exchange interaction. (B) Illustration of the principle of operation. Spin waves are excited by the elements on one or several sides of the matrix (e.g. left side), propagate through the matrix and detected on the other side (e.g. right side) of the structure. All input waves are of the same amplitude and frequency. The initial phases of the input waves are controlled by the generating elements. The output waves are the results of the spin wave interference within the matrix. The amplitude of the output wave depends on the initial phases and the magnetic states of th e junctions. Figure 2. Schematics of the experimental setup for single cross structures testing. The input and the output micro -antennas are connected to the Hewlett -Packard 8720A Vector Network Analyzer (VNA). The VNA generates input RF signal and measur es the S parameters showing the amplitude and the phases of the transmitted and reflected signals. The device under study is placed inside a GMW 3472 -70 Electromagnet system which allows the biasing magnetic field to be varied from -1000 Oe to +1000 Oe. T he in -plane axes X and Y are defined along the lines from port 1 to port 3, and from port 4 to port 2, respectively. Figure 3. (A) Microscope image of Permalloy single cross structure with antennas placed on the edges of the structure. The length of the w hole structure is 18um; the width of the arm in 6µm; thickness is 40 nm. (B) Photo of the packaged device with microwave input/output ports used for connection to VNA. (C) Experimental data showing the relative change of the output signal (inductive volta ge) as a function of the strength of the external magnetic field. The output is normalized to the maximum output detected at 300 Oe. The signal is transmitted from port 2 to port 4, the bias magnetic field is along the X axis as depicted in the inset. The input frequency is 3.16GHz. (D) Experimental data showing the relative change of the output signal amplitude as a function of 10 the direction of the external magnetic field. The output is normalized to the maximum amplitude at zero degrees (parallel to X axis). The signal is transmitted from port 2 to port 4. The measurements are taken at the different angles α of the bias magnetic field of 148Oe, where α is defined as the angle to the X axis as depicted in the inset. Figure 4. Microscope image of YIG s ingle cross structure. The length of the whole structure is 3mm; the width of the arm in 360µm; thickness is 3.6um. (B) Experimental data showing the relative change of the output signal amplitude as a function of the direction of the external magnetic fie ld. The signal is transmitted from port 2 to port 1. (C) Experimental data on the signal transmission from port 4 to port 2 (black curve) and from port 2 to port 4 (red curve). The data show about 5dB difference for the signal propagating in the opposite directions in the frequency range from 5.2GHz to 5.5GHz. Figure 5. (A) Microscope image of YIG double cross structure. The length of the whole structure is 3mm; the width of the arm in 360µm; thickness is 3.6um. (B) Schematics of the double -cross device with six micro -antennas fabricated on the edges under study. (C) Schematics of the experimental setup. The input and the output micro -antennas are connected to the Hewlett -Packard 8720A Vector Network Analyzer (VNA). There is asset of splitters (depicted as S), attenuators (depicted as A), and phase shifters (depicted as Ph) used for the connections with VNA. The device under study is placed inside a GMW 3472 -70 Electromagnet system. Figure 6. Experimental data on the output voltage at port 1 as a result of spin wave interference. The data are collected in the frequency range from 5.3GHz to 5.5GHz. The bias magnetic field is 1000 Oe directed from port 1 toward port 6. The curves of the different color correspond to the different phase shifts between the spin wave generated ports. Phase 1 represents a change in the phase of ports 4 and 6 and Phase 2 represents a change in the phase of ports 3 and 5. (B -D) Slices of data taken at the frequencies of 5.385GHz, 5.410GHz and 5.45GHz, respectively. The black markers depict the experimentally obtained data, and the red markers depict the theoretical data for the ideal case of the interfering waves of the same frequency and amplitude. The theoretical data is normalized to have the same maximum value as the experimental data at phase difference zero (constructive interference). Figure 7. Holographic image of the double -cross structure without memory elements. The cyan surface is a com puter reconstructed 3 -D plot based on the experimental data: output voltage as a function of the phases of the interfering spin waves. The output is detected at port 6. The excitation frequency is 5.40 GHz, the bias magnetic field is 1000 Oe directed from port 1 toward port 6. No signal is applied to port 1. The legend and schematics on the right side explain the phases of the spin waves generated at the six ports. Phase 1 and Phase 2 correspond to the phases generated at the ports 2,4 and 3,5, respectivel y. Figure 8. Collection of experimental data showing the output of the double -cross structure with micro -magnets placed on the top of the junctions. The phase coordinates show the combination 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. 0,)means that the spin waves generated in the ports 2,4 and 3,5 have a - difference in the initial phase. The markers of different shape and color correspond to the different magnetic configurations as il lustrated on the right side. All results are obtained at room temperature. 11 Figure 1 12 Figure 2 13 Figure 3 14 Figure 4 15 Permalloy YIG Cross dimensions L=18µm, w=6µm, d=100nm L=3mm, w=300µm, d=3.8µm Operational Frequency 3GHz -4GHz 5GHz -6GHz SW group velocity 3.5×106 cm/s 3.0×106 cm/s Maximum On/Off ratio 20dB 35dB Power consumption 0.1µW -1µW 0.5µW -5µW Compatibility with Silicon Yes No Table I. Summary on the experimental d ata collected for permalloy and YIG single cross structures 16 Figure 5 17 Figure 6 18 Figure 7 19 Figure 8

2014-11-12

In this work, we present recent developments in magnonic holographic memory devices exploiting spin waves for information transfer. The devices comprise a magnetic matrix and spin wave generating/detecting elements placed on the edges of the waveguides. The matrix consists of a grid of magnetic waveguides connected via cross junctions. Magnetic memory elements are incorporated within the junction while the read-in and read-out is accomplished by the spin waves propagating through the waveguides. We present experimental data on spin wave propagation through NiFe and YIG magnetic crosses. The obtained experimental data show prominent spin wave signal modulation (up to 20 dB for NiFe and 35 dB for YIG) by the external magnetic field, where both the strength and the direction of the magnetic field define the transport between the cross arms. We also present experimental data on the 2-bit magnonic holographic memory built on the double cross YIG structure with micro-magnets placed on the top of each cross. It appears possible to recognize the state of each magnet via the interference pattern produced by the spin waves with all experiments done at room temperature. Magnonic holographic devices aim to combine the advantages of magnetic data storage with wave-based information transfer. We present estimates on the spin wave holographic devices performance, including power consumption and functional throughput. According to the estimates, magnonic holographic devices may provide data processing rates higher than 10^18 bits/cm2/s while consuming 0.15uW. Technological challenges and fundamental physical limits of this approach are also discussed.

Magnonic Holographic Memory: from Proposal to Device

1411.3388v1

Pure spin current transport in gallium doped zinc oxide Matthias Althammer,1, 2, a)Joynarayan Mukherjee,3Stephan Gepr ¨ags,1Sebastian T. B. Goennenwein,1 Matthias Opel,1M.S. Ramachandra Rao,3and Rudolf Gross1, 2, 4 1)Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany 2)Physik-Department, Technische Universit ¨at M¨unchen, 85748 Garching, Germany 3)Department of Physics, Nano Functional Materials Technology Centre and Materials Science Research Centre, Indian Institute of Technology Madras, Chennai, Tamil Nadu 600036, India 4)Nanosystems Initiative Munich (NIM), 80799 M ¨unchen, Germany (Dated: 12 August 2018) We study the flow of a pure spin current through zinc oxide by measuring the spin Hall magnetoresistance (SMR) in thin film trilayer samples consisting of bismuth-substituted yttrium iron garnet (Bi:YIG), gallium-doped zinc oxide (Ga:ZnO), and platinum. We investigate the dependence of the SMR magnitude on the thickness of the Ga:ZnO interlayer and compare to a Bi:YIG/Pt bilayer. We find that the SMR magnitude is reduced by almost one order of magnitude upon inserting a Ga:ZnO interlayer, and continuously decreases with increasing interlayer thickness. Nevertheless, the SMR stays finite even for a 12 nm thick Ga:ZnO interlayer. These results show that a pure spin current indeed can propagate through a several nm-thick degenerately doped zinc oxide layer. We also observe differences in both the temperature and the field dependence of the SMR when comparing tri- and bilayers. Finally, we compare our data to predictions of a model based on spin diffusion. This shows that interface resistances play a crucial role for the SMR magnitude in these trilayer structures. The generation and detection of pure spin currents, i.e. of net flows of angular (spin) momentum without accompany- ing charge currents, has been intensively studied in theory and experiments. Prominent spin current based phenomena are spin pumping1,2, the spin Seebeck effect3–5and the spin Hall magnetoresistance6–10(SMR). The SMR manifests itself in ferromagnetic insulator (FMI) / normal metal (NM) bilayer samples, as a dependence of the electric resistance of the NM on the orientation of the magnetization in the FMI. The mag- nitude of the SMR depends on the size of the spin Hall an- gle in the NM. In a FMI/NM bilayer system, the longitudinal (rlong) and transverse resistivity ( rtrans) of the NM can be writ- ten as6,8,9,11 rlong =r0+r1(mt)2; (1) rtrans=r2mnr1(mjmt)2; (2) where mj,mt, and mnare the projections of the net magnetiza- tion unit vector m=M=Mdetermined by the magnetization orientation in the FMI on the directions jandtparallel and perpendicular to the current direction, respectively, and the film normal n(cf. Fig. 1). The resistivity parameters ride- pend on the material parameters of the hybrid structure. A detailed analysis allows to extract the spin Hall angle qSH;NM and the spin diffusion length lsf;NMof the NM, if the spin mix- ing conductance g"#at the FMI/NM interface is known.9,12 Up to now, most SMR studies were based on FMI/NM bi- layer structures. Only in a few experiments an additional con- ductive interlayer was inserted at the FMI/NM interface to rule out the contribution of a proximity-polarized NM layer.9,12–14 However, trilayer structures also allow to study the transport of pure spin currents in interlayer materials with a negligible a)Electronic mail: matthias.althammer@wmi.badw.despin Hall angle. In particular, the effect of interlayer resistiv- ity and spin diffusion length onto the SMR has not yet been investigated. Moreover, a quantitative comparison between a spin diffusion theory model and experiment is still missing for such trilayer systems. In this letter, we present SMR ex- periments conducted on bismuth-substituted yttrium iron gar- net (Bi 0:3Y2:7Fe5O12, Bi:YIG) / platinum (Pt) bilayers and Bi:YIG / gallium doped zinc oxide (Ga 0:01Zn0:99O, Ga:ZnO) / Pt trilayers. Our results demonstrate pure spin current trans- port across a degenerately doped, several nm-thick ZnO inter- layer, resulting in a sizeable SMR effect. In addition, we ex- ploit the tunability of the resistivity and spin diffusion length in Ga:ZnO with temperature to quantitatively compare our ex- perimental data with the predictions of a model based on spin diffusion. This comparison suggests that interface resistances and their possible spin selectivity may play a crucial role for the SMR in these trilayer structures. The samples have been grown in-situ , without breaking the vacuum, on (111)-oriented yttrium aluminium garnet (YAG) substrates in an ultra high vacuum deposition system. Bi:YIG and Ga:ZnO layers were deposited via laser-molecular beam epitaxy (laser-MBE) from stoichiometric targets in an oxy- gen atmosphere. For Bi:YIG we applied an energy density at the target of ED L=1:5 J=cm2, an oxygen partial pressure of pO2=25mbar, and a substrate temperature of Tsub=450C (see Ref. 9 for more details). For Ga:ZnO we used: ED L= 1 J=cm2,pO2=1mbar,Tsub=400C (See Ref. 15 for more details). Pt was deposited via electron-beam evaporation at a base pressure of 1 108mbar at room temperature. The in-situ deposition process ensures very clean interfaces. From the structural and magnetic characterization of a Bi:YIG(54)/Pt(4) bilayer and a Bi:YIG(54)/Ga:ZnO(8)/Pt(4) trilayer, where the numbers in parentheses give the film thick- nesses in nm, we conclude that the structural and the magnetic properties of the bi- and trilayer sample are nearly identical (see supplemental materials). This suggests that the influencearXiv:1612.07239v1 [cond-mat.mes-hall] 21 Dec 20162 of the Ga:ZnO deposition on the magnetic properties of the Bi:YIG layer is negligible. For electrical transport measurements, the samples have been patterned into Hall bar-shaped mesastructures (80 mm wide, 800 mm long) via photolithography and Ar-ion milling, and then mounted in a superconducting magnet ( m0H7 T) cryostat (5 KT300 K). Resistivity data have been taken using a DC current reversal technique. From the measured longitudinal and transverse voltage we then calculated rlong andrtrans.9,16We performed angle-dependent magnetoresis- tance (ADMR) measurements, where the direction of the ap- plied magnetic field of constant magnitude is rotated within three orthogonal planes as illustrated in the left column of Fig. 1: in the film plane (ip), in the plane perpendicular to thej-direction (oopj), and in the plane perpendicular to the t-direction (oopt). Although the smallest field magnitude is below the full saturation field of Bi:YIG of 2 T, it is sufficient to investigate the main angular dependence. We first discuss the results obtained from ADMR exper- iments at T=10 K and m0H=1 T for our bilayer and trilayer sample, as shown in Fig. 1. For the bilayer we measure the typical fingerprint expected for SMR9: For the ip rotation plane we observe a sin2(a)-dependence of rlong and a sin (a)cos(a)-dependence of rtrans(see Fig. 1(a)), for the oopj rotation plane we observe a sin2(b)-dependence of rlong(see Fig. 1(c)), while we do not observe any sizeable angular-dependence of rlongin the oopt rotation plane (see Fig. 1(e)). Furthermore, rtransonly shows a very small cosine dependence in the oopj and oopt rotation planes, due to the nearly vanishing ordinary Hall coefficient (OHC) of Pt thin films at low temperatures.17We note that the remaining sin2- dependencies, for rlongin oopt and for rtransin oopj and oopt configuration, can be explained by a non-vanishing SMR con- tribution in these rotation planes due to a small tilting ( 3) of the actual rotation plane with respect to the surface normal because of experimental limitations in the sample mounting. For the trilayer sample the angular dependence looks qualita- tively the same (Fig. 1(b,d,f)). As the observed ADMR data reflects the symmetry expected for SMR, we can safely as- sume the SMR as the only cause for the magnetoresistance in the bilayer as well as for the trilayer. Quantitatively, however, there are differences. In the tri- layer, rlongis about 2 times larger than in the bilayer, which can be explained by the one order of magnitude larger resis- tivity of the ZnO layer as compared to the bare Pt layer and the effective average of resistivity from Pt and Ga:ZnO for the trilayer (see Fig. 3(b) for the extracted resisitivity of the ZnO layer). For further quantitative comparison, we simulated the SMR response for rlongandrtrans(red lines in the graphs) by using Eqs. (1) and (2), while assuming that the magnetization orientation is always parallel to the external magnetic field and including the ordinary Hall effect as a contribution pa- rameterized by a field dependence of r2. From the simulation we extract the SMR amplitude SMR =jr1=r0j. For the bi- layer we obtain SMR =4:0104, which agrees nicely with our previous results in YIG/Pt hybrids.9,18For the trilayer we find SMR =2:2105, which is about an order of magni- tude smaller. There are two obvious reasons for the decreasein SMR amplitude upon the insertion of a Ga:ZnO interlayer. First, in the trilayer the Ga:ZnO layer acts as a resistive shunt for the Pt layer thereby reducing the SMR amplitude. Sec- ond, part of the spin current generated in the Pt layer is lost when diffusing across the Ga:ZnO layer. In other words, the Ga:ZnO only acts as a parallel and spin Hall-inactive resistor for the SMR. Moreover, when comparing the transverse resis- tivity for both samples, the amplitude of the ADMR signal is different in all rotation planes for the two different samples (please note that the same scale has been used for rtransin all graphs of Fig. 1). For the ip rotation plane the bilayer has a much larger amplitude than for the trilayer. This is expected due to the larger SMR in the bilayer, which is the only contri- bution to the ADMR for this rotation plane. For the oopj and oopt rotation planes, however, the rtransamplitude is larger for the trilayer. This can be traced back to the contribution of the ordinary Hall effect to the ADMR data: for the bilayer the OHC is rather small17(0:1 mW=T), while for the tri- layer it is an effective average of the Pt layer and the Ga:ZnO interlayer, resulting in a larger OHC. From the field depen- dence of r2for the trilayer, we extract an OHC of 30 m W=T for the Ga:ZnO, corresponding to a carrier concentration of 21021cm3. This value nicely agrees with control measure- ments on blanket Ga:ZnO layers (see supplemental material). To get a deeper insight into the SMR of the trilayer we measured the temperature and field dependence and com- pare it to the results for the bilayer sample. To this end, we conducted ADMR measurements in the ip rotation plane for 5T300 K and m0H=1;3;5;and 7 T and extracted the SMR by fitting the data using Eq.(1). The result is shown in Fig. 2. We first discuss the temperature dependence for the bi- layer sample in Fig. 2(a). The SMR shows a maximum around T=225 K and then decreases with decreasing temperature. This observation nicely agrees with results on YIG/Pt hy- brids18,19and can be attributed to the temperature dependence ofqSH;Pt.18However, the temperature dependence is clearly different for the trilayer sample (Fig. 2(b)). Here, the SMR is only weakly changing with temperature, displays an upturn towards low temperatures and reaches its maximum value for 5 K. We may explain this upturn by two contributions. On the one hand, the spin diffusion length for Ga:ZnO increases with decreasing temperature as it is dominated by the D’yakonov- Perel’ mechanism for spin dephasing.15,20This leads to an in- crease of the SMR and a saturation at low temperatures. On the other hand, due to the laser-MBE deposition of Ga:ZnO an intermixing of Bi:YIG and Ga:ZnO at the interface can occur, which may lead to the formation of isolated paramag- netic moments at the interface between Ga:ZnO and Bi:YIG. These isolated paramagnetic moments align parallel to the ex- ternal magnetic field at low temperatures, which would also result in an increase of SMR with decreasing temperatures. The temperature dependent data of the bilayer also shows a weak plateau at low temperatures, which could be related to isolated paramagnetic moments. However, more detailed in- vestigations of the interface properties will be necessary in the future in order to really unravel the underlying physics. The evolution of the magnetic field dependence of the SMR with temperature for the bilayer and the trilayer sam-3 -90° 0° 90° 180°270°700.26700.27700.28 αρlong(nΩm) -5051015 ρtrans(nΩcm) -90° 0° 90° 180°270°699.79699.80699.81 βρlong(nΩm) -5051015 ρtrans(nΩcm) -90° 0° 90° 180°270°679.59679.60679.61 γρlong(nΩm) -5051015 ρtrans(nΩcm)-90° 0° 90° 180°270°322.4322.5322.6322.7ρlong(nΩm) α-5051015 ρtrans(nΩcm) -90° 0° 90° 180°270°322.4322.5322.6322.7ρlong(nΩm) β-5051015 ρtrans(nΩcm) -90° 0° 90° 180°270°322.4322.5322.6322.7ρlong(nΩm) γ-5051015 ρtrans(nΩcm)h||t γh jtn hjtn α βh jtnh||-t h||j h||t h||-t h||j h||t h||-t h||n h||t h||-t h||n h||-j h||j h||n h||-j h||j h||nip oopj ooptρlong ρtrans ρlong ρtrans ρlong ρtransρlong ρtrans ρlong ρtrans ρlong ρtrans1 T, 10 K 1 T, 10 K 1 T, 10 K1 T, 10 K1 T, 10 K1 T, 10 K(b) (a) (d) (c) (f)(e) FIG. 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 (f)) sample grown on YAG (111) substrates. The data has been recorded at 10 K. The three orthogonal rotation planes for the magnetic field are sketched in the left column. In the plot, black and blue symbols represent the experimental data of the longitudinal rlongand transverse resistivity rtrans, respectively. The red lines are simulations using Eqs. (1),(2). 0 2 4 6 456789µ0H(T)SMR (x10-4)0 100200300 46810T(K)SMR (x10-4) 0 1002003000123SMR (x10-4) T(K)0 2 4 6012SMR (x10-4) µ0H(T)7 T 5 T300 K 3 T 1 T7 T 5 K 300 K5 K(a) (c) (d) (b)5 T 3 T 1 T FIG. 2. Temperature dependence of the SMR signal extracted from ADMR measurements for (a) a Bi:YIG(54)/Pt(4) bilayer and (b) a Bi:YIG(54)/Ga:ZnO(8)/Pt(4) trilayer at magnetic fields of 1 ;3;5;7 T. Magnetic field dependence of SMR from ADMR measurements for the bilayer (c) and the trilayer (d) at 5 K (black squares) and 300 K (red circles). ple is also different. For the bilayer the relative increase of the SMR from 1 T to 7 T is 37% at 5 K and 27% at 300 K(Fig. 2(c)). For the trilayer we obtain 600% at 5 K and 1500% at 300 K(Fig. 2(d)). The much stronger field de- pendence observed for the trilayer sample again might be ex- plained by the presence of isolated paramagnetic moments at the Bi:YIG/Ga:ZnO interface. From our bilayer data it is not directly possible to pin point the origin for the observed fielddependence. Hanle magnetoresistance as proposed by V ´elez et al.21can be ruled out, as this should yield a quadratic field dependence of the SMR, while we observe a linear/square root dependence in our data. Clearly, more detailed studies are necessary to explain the observed field dependence. To analyze the pure spin current transport in Ga:ZnO we first extract the resistivity of Pt ( rPt(T)) from the rlong(T) of the bilayer and then determine the resistivity of Ga:ZnO (rZnO(T)) from the measured rlong(T)of the trilayer by as- suming a parallel conductance model and the same rPt(T)as for the bilayer (see Fig. 3(a) and (b)). For both materials we observe a decrease in resistivity with decreasing temperature, while rZnOis about an order of magnitude larger than rPt. From this we conclude that the Ga:ZnO layer just behaves like a dirty metal due to degenerate doping. In a next step we model the dependence of the SMR on the Ga:ZnO thickness ( dZnO) using the SMR theory approach8 and adding the spin diffusion through the Ga:ZnO to the theory model, while neglecting any interface resistance ef- fects, i.e. assuming continuity of the spin-dependent electro- chemical potential at the Pt/Ga:ZnO interface (see supplemen- tal materials). We note that ADMR measurements on bare Bi:YIG/Ga:ZnO reference samples yield no SMR response within our experimental resolution (for a 8 nm thick Ga:ZnO layer on Bi:YIG, we find SMR 8106for 5 KT 300 K and m0H7 T), such that we assume qSH;ZnO=0. From this model, two parameters define the dZnOdependence of the SMR: rZnOand the spin diffusion length lsf;ZnOin the Ga:ZnO layer. The resistivity rZnOhas a twofold implica- tion. On the one hand, it determines the amount of charge current flowing through Pt and, hence, the amount of spin4 current generated via the spin Hall effect in Pt. On the other hand, it parameterizes the amount of spin current diffusing through Ga:ZnO due to the gradient in the spin-dependent electrochemical potential. To test whether or not this sim- ple model explains our experimental findings, we simulated SMR at 5 K and 300 K and compared it to our experimen- tal data as shown in Fig. 3(c) and (d). We also included ADMR data obtained for Bi:YIG(54)/Ga:ZnO(12)/Pt(9) and Bi:YIG(54)/Ga:ZnO(4)/Pt(7) trilayers, which showed a sim- ilar surface roughness determined from x-ray reflectometry and thus similar interface properties. For better comparison, we renormalized the resistivity data to a Pt thickness of 4 nm using our previous results18. For the simulation we used rPt andrZnOfrom Fig. 3(a) and (b), while we chose temperature- independent values of g"#=11019m2andlsf;Pt=1:5 nm from Refs. 9, 12, and 18. For qSH;Ptwe used 0 :11 at 300 K and 0:07 at 5 K from Ref. 18 as well as 4 nm at 300 K and 12 nm at 5 K for lsf;ZnOfrom Ref. 15. Our very simple model can only reproduce the general trend of the measurements (see Fig. 3(c) and (d)). The differences between simulation and data evident from Fig. 3 most likely originate from the crucial role of a spin dependent interface resistance15at the Pt/Ga:ZnO interface and a possible change of g"#when going from a Bi:YIG/Pt to a Bi:YIG/Ga:ZnO interface. However, to extract these pa- rameters from our measurements, a more systematic study is necessary, which goes beyond the scope of this paper. In summary, we experimentally investigated the flow of a pure spin current through a Ga:ZnO layer by utilizing the SMR in Bi:YIG/Ga:ZnO/Pt trilayer thin film samples. Our results show the possibility to transfer a pure spin current through a degenerately doped ZnO interlayer. Our results also highlight the importance of interface quality for the SMR. Fi- nally, using a spin diffusion model for the SMR response, we achieve reasonable agreement between simulation and exper- 0 5 1010-210-1100101102SMR (x10-4) dZnO(nm)0 5 1010-1100101102SMR (x10-4) dZnO(nm)0100200300300350400ρPt(nΩm) T(K)0 100200300200030004000ρZnO(nΩm) T(K)(a) (c)(b) (d) T=300 K T=5 K 7 T 5 T 3 T 1 T7 T5 T3 T 1 T FIG. 3. Extracted temperature dependence of the resistivity of Pt (a) from the bilayer and the resistivity of Ga:ZnO (b) extracted from the trilayer ADMR data by using the resistivity of Pt extracted from the bilayer. For Pt and ZnO the resistivity increases with increasing temperature. SMR amplitude as a function of Ga:ZnO thickness at 300 K (c) and 5 K(d). The red line is a spin diffusion based simulation of the SMR amplitude.iment. Our results demonstrate how SMR experiments in tri- layers can be used to study pure spin currents in materials with vanishing spin Hall angle qSH;NM. SUPPLEMENTARY MATERIAL See supplementary online material for the structural and magnetic characterization of the bi- and trilayer samples, the carrier concentration of the Ga:ZnO layer, and a more elabo- rate discussion of the SMR trilayer simulation. ACKNOWLEDGMENTS M.S.R. and J.M. would like to thank for funding from De- partment of Science and Technology, New Delhi, that facil- itated the establishment of Nano Functional Materials Tech- nology Centre (Grant: SR NM/NAT/02-2005). J.M. would like to thank UGC for SRF fellowship. We acknowledge fi- nancial support by the German Academic Exchange Service (DAAD) via project no. 57085749. 1K. Ando, Y . Kajiwara, K. Sasage, K. Uchida, and E. Saitoh, IEEE Trans. Magn. 46, 3694 (2010). 2F. D. Czeschka, L. Dreher, M. S. Brandt, M. Weiler, M. Althammer, I.-M. Imort, G. Reiss, A. Thomas, W. Schoch, W. Limmer, H. Huebl, R. Gross, and S. T. B. Goennenwein, Phys. Rev. Lett. 107(2011). 3K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. 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Reiss, L. Shen, A. Gupta, Y .-T. Chen, G. E. W. Bauer, E. Saitoh, and S. T. B. Goennenwein, Phys. Rev. B 87, 224401 (2013). 10N. Vlietstra, J. Shan, V . Castel, B. J. van Wees, and J. B. Youssef, Phys. Rev. B 87(2013). 11K. Ganzhorn, J. Barker, R. Schlitz, B. A. Piot, K. Ollefs, F. Guillou, F. Wil- helm, A. Rogalev, M. Opel, M. Althammer, S. Gepr ¨ags, H. Huebl, R. Gross, G. E. W. Bauer, and S. T. B. Goennenwein, Phys. Rev. B 94(2016). 12M. Weiler, M. Althammer, M. Schreier, J. Lotze, M. Pernpeintner, S. Meyer, H. Huebl, R. Gross, A. Kamra, J. Xiao, Y .-T. Chen, H. Jiao, G. E. W. Bauer, and S. T. B. Goennenwein, Phys. Rev. Lett. 111, 176601 (2013). 13S. Gepr ¨ags, S. Meyer, S. Altmannshofer, M. Opel, F. Wilhelm, A. Rogalev, R. Gross, and S. T. B. Goennenwein, Appl. Phys. Lett. 101, 262407 (2012). 14B. F. Miao, S. Y . Huang, D. Qu, and C. L. Chien, Phys. Rev. Lett. 112 (2014). 15M. Althammer, E.-M. Karrer-Muller, S. T. B. Goennenwein, M. Opel, and R. Gross, Appl. Phys. Lett. 101, 082404 (2012). 16M. Schreier, N. Roschewsky, E. Dobler, S. Meyer, H. Huebl, R. Gross, and S. T. B. Goennenwein, Appl. Phys. Lett. 103, 242404 (2013).5 30°35°40°45°50°101102103104105 -0.4°0.0°0.4°020406080I(cps) 2θI(cps) ∆ω 30°35°40°45°50°101102103104105 -0.4°0.0°0.4°01020304050I(cps) 2θI(cps) ∆ω -6 -3 0 3 6-100-50050100M(kA/m) µ0H(T)-6-3 0 3 6-100-50050100M(kA/m) µ0H(T)0.025° Bi:YIG (444)** * *YAG (444) Bi:YIG (444) YAG (444)0.331° 0.494°0.024°(b) (c) Ms=106 kA/m Ms=108 kA/m(d)(a) T=300 K T=300 K Bi:YIGGa:ZnOPt 54 nm8 nm4 nm Bi:YIGPt 54 nm4 nm FIG. S1. Structural and magnetic properties of the laser-MBE grown Bi:YIG(54)/Pt(4) bilayer and Bi:YIG(54)/Ga:ZnO(8)/Pt(4) trilayer samples grown on YAG (111) substrates. 2 qwscan of bilayer (a) and trilayer (b) samples, the insets of (a) and (b) are the rocking curve of the Bi:YIG (444) reflection, numbers represent the FWHM of the corresponding Gaussian fit. Peaks marked with asterisks are background reflections from the sample holder. In-plane magnetiza- tion versus applied magnetic field curve of bilayer (c) and trilayer (d) samples measured at 300 K. 17S. Meyer, R. Schlitz, S. Gepr ¨ags, M. Opel, H. Huebl, R. Gross, and S. T. B. Goennenwein, Appl. Phys. Lett. 106, 132402 (2015). 18S. Meyer, M. Althammer, S. Gepr ¨ags, M. Opel, R. Gross, and S. T. B. Goennenwein, Appl. Phys. Lett. 104, 242411 (2014). 19A. Aqeel, N. Vlietstra, J. A. Heuver, G. E. W. Bauer, B. Noheda, B. J. van Wees, and T. T. M. Palstra, Phys. Rev. B 92(2015). 20S. Ghosh, V . Sih, W. H. Lau, D. D. Awschalom, S.-Y . Bae, S. Wang, S. Vaidya, and G. Chapline, Appl. Phys. Lett. 86, 232507 (2005). 21S. V´elez, V . N. Golovach, A. Bedoya-Pinto, M. Isasa, E. Sagasta, M. Aba- dia, C. Rogero, L. E. Hueso, F. S. Bergeret, and F. Casanova, Phys. Rev. Lett.116(2016). I. STRUCTURAL AND MAGNETIC CHARACTERIZATION We first compare the results obtained for a Bi:YIG(54)/Pt(4) bilayer and a Bi:YIG(54)/Ga:ZnO(8)/Pt(4) trilayer, where the numbers in parentheses give the film thicknesses in nm. The structural quality of the thin films was analyzed by x-ray diffraction (XRD). The magnetic proper- ties were obtained by superconducting quantum interference device magnetometry. The XRD results presented in Fig. S1 for the bilayer (panel a) as well as for the trilayer (panel b) indicate excellent struc- tural quality. In both samples we only observe reflections that can be assigned to the substrate or the layers itselfthemselves. In the trilayer sample no reflections of the Ga:ZnO layer could be observed, which we attribute to the small thickness of the layer. The XRD rocking curves of the Bi:YIG (444) reflec- tion (insets in Fig. S1(a),(b)) consist in both samples of one rather broad peak superimposed on a second narrow peak. We attribute this to partial strain relaxation in the film due to the large lattice mismatch of 3 %, between YAG and Bi:YIG. Nevertheless, the full width at half maximum (FWHM) ex- tracted from Gaussian fits to the data yield nearly identical 0501001502002503001.51.61.7n(x1021cm-3) T(K)FIG. S2. Carrier concentration of a 8 nm thick Ga:ZnO grown on sapphire. The carrier concentration is independent of temperature indicating the degenerate doping of the Ga:ZnO layer. values of 0 :025for the bilayer and 0 :024for the trilayer sample. However, we obtain different values for the FWHM of the broader peak, 0 :331for the bilayer and 0 :494for the trilayer sample. The broad peak in the Bi:YIG (444) rock- ing curve can be attributed to the strain relaxation layer in the Bi:YIG forming at the YAG substrate interface. Taking this into account, we attribute the change in width to a possible change in strain relaxation by defect formation in the Bi:YIG layer, due to the different thermal treatment by the additional Ga:ZnO deposition in the trilayer. However, further investiga- tions are necessary to confirm this assumption. The magnetization curves recorded at 300 K are shown in Fig. S1(c) for the bilayer and (d) for the trilayer sample, where a diamagnetic background from the substrate has been sub- tracted from the raw data. We extract a saturation magneti- zation Ms=106 kA =m for the bilayer and Ms=108 kA =m for the trilayer sample. Both numbers are smaller than the bulk value?Ms=144 kA =m, which we attribute to defects in our layers. These defects can either be structural defects in the strain relaxation layer present at the YAG substrate in- terface?, effectively reducing the total saturation magnetiza- tion over the whole film thickness, or iron and oxide vacancies present in the whole Bi:YIG film?. The coercive field for both samples is 0 :6 mT. Taken together, the structural and the mag- netic properties of the bi- and trilayer sample are nearly identi- cal. This suggests that the influence of the Ga:ZnO deposition on the magnetic properties of the Bi:YIG layer is negligible. II. ORDINARY HALL EFFECT IN GA:ZNO We used ordinary Hall effect (OHE) measurements on a 8 nm thick Ga:ZnO layer grown via pulsed laser deposition on a (0001)-oriented sapphire substrate under identical depo- sition conditions as for the trilayer structures investigated for the SMR experiments. From the OHE measurements, carried out at various temperatures, we then extracted the n-type car- rier concentration nas a function of temperature, the results are shown in Fig. S2.6 The carrier concentration is independent of temperature with n1:61021cm3. This indicates that the Ga:ZnO is degenerately doped. III. SIMULATION OF THE SMR IN TRILAYER-SYSTEMS For modelling the SMR in trilayer structures we follow the approach outlined in Ref. 8. For the calculations, we use the coordinate system depicted in Fig. S3. Our system consists of two conductive layers (NM1, NM2), where the pure spin current in both layers is carried by the spin angular momen- tum of the charge carriers. In our calculations we assume that only the charge current jq;NM1(z)flowing in NM1 gets con- verted into a spin current js;NM1(z), while there is no spin cur- rent generation via the spin Hall effect in NM2(spin Hall in- active layer). At the NM1/NM2 interface we assume that the spin-dependent electrochemical potentials s;NM1(z),ms;NM2(z) and the spin current flowing across the interface are contin- uous, and thus neglecting any interface resistance contribu- tions. This leads to the following boundary conditions: ms;NM1(dNM2) =ms;NM2(dNM2); js;NM1(dNM2) =js;NM2(dNM2); js;NM1(dNM2+dNM1) =0; js;NM2(0) =1=e[Grm(mms;NM2(0))+Gi(mms;NM2(0))]; where GrandGiare the real and imaginary part of the spin mixing conductance per unit area. We then solve the spin diffusion equation with the above boundary conditions as de- tailed in Ref. 8. The analytical expressions obtained from this procedure are lengthy and therefore not written down in the text here. With these solutions, we then calculate the spin and charge currents as outlined in Ref. 8. Averaging over the film thicknesses and expanding up to the second order in spin Hall angle we then obtain an expression for rlongof the whole tri- layer stack. Using this expression we can then determine the SMR amplitude by determining rlongformkjandmkt. We note that similar calculations can also be carried out for a spin Hall active NM2 layer using the very same approach. Using these result we can then calculate the NM2 thickness dependence of the SMR amplitude for different resisitivity ra- tiosrNM2=rNM1, while using the following values for the re- maining parameters: g"#=11019m2,lsf;NM1=1:5 nm, qSH;NM1=0:11,lsf;NM2=12 nm; The result of these calcu- lations is shown in Fig. S3(b). For rNM2=rNM1=1, the SMR gives the largest values for a finite thickness of NM2. For rNM2=rNM1=0:1, we find a lower SMR amplitude as com- pared to rNM2=rNM1=1 as now a large part of the electri- cal current runs through the spin Hall inactive layer and thus is not contributing to the SMR. For rNM2=rNM1=10 and rNM2=rNM1=100 the SMR gets enhanced for ultrathin NM2 layers, which can effectively understood as an enhancement of the interface resistance at the FMI/NM2 interface which boosts the SMR effect similar to the experiments on match- ing spin mixing conductance in spin pumping experiments as elaborated in Ref. ?. In our experiments we used Ga:ZnO as the spin Hall inac- tive NM2 to vary the spin diffusion length and resistivity of 0 5 10 1510-610-510-410-310-2SMR dNM2(nm)NM2NM1 FMI dNM2dNM1z dNM1+dNM2 dNM2 0(a) (b) ρNM2/ρNM1= 1 10 1000.1FIG. S3. Simulation of the SMR in a trilayer structure with two conductive layers and one FMI layer. (a) Schematic drawing of the coordinate system used for the calculations. (b) Dependence of the SMR on the thickness of the spin Hall inactive interlayer for rNM2=rNM1=0:1 (black line), 1 (red line), 10 (blue line), and 100 (green line). Similar to the conductivity mismatch problem for spin injection into semiconductors, the SMR effect is reduced. the NM2 layer as a function of temperature. Similar effects will occur if the carrier concentration is changed in the ZnO layer, as this will again affect the spin diffusion length and the resistivity of the NM2 layer. 1K. Ando, Y . Kajiwara, K. Sasage, K. Uchida, and E. Saitoh, IEEE Trans. Magn. 46, 3694 (2010). 2F. D. Czeschka, L. Dreher, M. S. Brandt, M. Weiler, M. Althammer, I.-M. Imort, G. Reiss, A. Thomas, W. Schoch, W. Limmer, H. Huebl, R. Gross, and S. T. B. Goennenwein, Phys. Rev. Lett. 107(2011). 3K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh, Nature 455, 778 (2008). 4K. ichi Uchida, H. Adachi, T. Ota, H. Nakayama, S. Maekawa, and E. Saitoh, Appl. Phys. Lett. 97, 172505 (2010). 5M. Weiler, M. Althammer, F. D. Czeschka, H. Huebl, M. S. Wagner, M. Opel, I.-M. Imort, G. Reiss, A. Thomas, R. Gross, and S. T. B. Goen- nenwein, Phys. Rev. Lett. 108(2012). 6H. Nakayama, M. Althammer, Y .-T. Chen, K. Uchida, Y . Kajiwara, D. Kikuchi, T. Ohtani, S. Gepr ¨ags, M. Opel, S. Takahashi, R. Gross, G. E. W. Bauer, S. T. B. Goennenwein, and E. Saitoh, Phys. Rev. Lett. 110, 206601 (2013). 7C. Hahn, G. de Loubens, O. Klein, M. Viret, V . V . Naletov, and J. B. Youssef, Phys. Rev. B 87(2013). 8Y .-T. Chen, S. Takahashi, H. Nakayama, M. Althammer, S. T. B. Goennen- wein, E. Saitoh, and G. E. W. Bauer, Phys. Rev. B 87, 144411 (2013). 9M. Althammer, S. Meyer, H. Nakayama, M. Schreier, S. Altmannshofer, M. Weiler, H. Huebl, S. Gepr ¨ags, M. Opel, R. Gross, D. Meier, C. Klewe, T. Kuschel, J.-M. Schmalhorst, G. Reiss, L. Shen, A. Gupta, Y .-T. Chen, G. E. W. Bauer, E. Saitoh, and S. T. B. Goennenwein, Phys. Rev. B 87, 224401 (2013). 10N. Vlietstra, J. Shan, V . Castel, B. J. van Wees, and J. B. Youssef, Phys. Rev. B 87(2013). 11K. Ganzhorn, J. Barker, R. Schlitz, B. A. Piot, K. Ollefs, F. Guillou, F. Wil- helm, A. Rogalev, M. Opel, M. Althammer, S. Gepr ¨ags, H. Huebl, R. Gross, G. E. W. Bauer, and S. T. B. Goennenwein, Phys. Rev. B 94(2016). 12M. Weiler, M. Althammer, M. Schreier, J. Lotze, M. Pernpeintner, S. Meyer, H. Huebl, R. Gross, A. Kamra, J. Xiao, Y .-T. Chen, H. Jiao, G. E. W. Bauer, and S. T. B. Goennenwein, Phys. Rev. Lett. 111, 176601 (2013). 13S. Gepr ¨ags, S. Meyer, S. Altmannshofer, M. Opel, F. Wilhelm, A. Rogalev, R. Gross, and S. T. B. Goennenwein, Appl. Phys. Lett. 101, 262407 (2012). 14B. F. Miao, S. Y . Huang, D. Qu, and C. L. Chien, Phys. Rev. Lett. 112 (2014). 15M. Althammer, E.-M. Karrer-Muller, S. T. B. Goennenwein, M. Opel, and R. Gross, Appl. Phys. Lett. 101, 082404 (2012). 16M. Schreier, N. Roschewsky, E. Dobler, S. Meyer, H. Huebl, R. Gross, and S. T. B. Goennenwein, Appl. Phys. Lett. 103, 242404 (2013).7 17S. Meyer, R. Schlitz, S. Gepr ¨ags, M. Opel, H. Huebl, R. Gross, and S. T. B. Goennenwein, Appl. Phys. Lett. 106, 132402 (2015). 18S. Meyer, M. Althammer, S. Gepr ¨ags, M. Opel, R. Gross, and S. T. B. Goennenwein, Appl. Phys. Lett. 104, 242411 (2014). 19A. 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). 20S. Ghosh, V . Sih, W. H. Lau, D. D. Awschalom, S.-Y . Bae, S. Wang, S. Vaidya, and G. Chapline, Appl. Phys. Lett. 86, 232507 (2005). 21S. V´elez, V . N. Golovach, A. Bedoya-Pinto, M. Isasa, E. Sagasta, M. Aba- dia, C. Rogero, L. E. Hueso, F. S. Bergeret, and F. Casanova, Phys. Rev. Lett.116(2016).

2016-12-21

We study the flow of a pure spin current through zinc oxide by measuring the spin Hall magnetoresistance (SMR) in thin film trilayer samples consisting of bismuth-substituted yttrium iron garnet (Bi:YIG), gallium-doped zinc oxide (Ga:ZnO), and platinum. We investigate the dependence of the SMR magnitude on the thickness of the Ga:ZnO interlayer and compare to a Bi:YIG/Pt bilayer. We find that the SMR magnitude is reduced by almost one order of magnitude upon inserting a Ga:ZnO interlayer, and continuously decreases with increasing interlayer thickness. Nevertheless, the SMR stays finite even for a $12\;\mathrm{nm}$ thick Ga:ZnO interlayer. These results show that a pure spin current indeed can propagate through a several nm-thick degenerately doped zinc oxide layer. We also observe differences in both the temperature and the field dependence of the SMR when comparing tri- and bilayers. Finally, we compare our data to predictions of a model based on spin diffusion. This shows that interface resistances play a crucial role for the SMR magnitude in these trilayer structures.

Pure spin current transport in gallium doped zinc oxide

1612.07239v1

Quantum transduction of superconducting qubit in electro-optomechanical and electro-optomagnonical system. Roson Nongthombam,Pooja Kumari Gupta,yand Amarendra K. Sarmaz Department of Physics, Indian Institute of Technology Guwahati, Guwahati-781039, India We study the quantum transduction of a superconducting qubit to an optical photon in electro- optomechanical and electro-optomagnonical systems. The electro-optomechanical system comprises a ux-tunable transmon qubit coupled to a suspended mechanical beam, which then couples to an optical cavity. Similarly, in an electro-optomagnonical system, a ux-tunable transmon qubit is coupled to an optical whispering gallery mode via a magnon excitation in a YIG ferromagnetic sphere. In both systems, the transduction process is done in sequence. In the rst sequence, the qubit states are encoded in coherent excitations of phonon/magnon modes through the phonon/magnon- qubit interaction, which is non-demolition in the qubit part. We then measure the phonon/magnon excitations, which reveal the qubit states, by counting the average number of photons in the optical cavities. The measurement of the phonon/magnon excitations can be performed at a regular intervals of time. I. INTRODUCTION `Quantum network' is a rapidly developing area owing to its potential applications in scaling up quantum computers by connecting multiple quantum processors. Recently, much research has been initiated on developing a modular quantum computer based on linking multiple superconducting chips where each chip has a few high- quality qubits. Instead of cramping more qubits onto a single chip, which will result in high error rates and complex hardware, creating a network of modules containing few high-quality qubits on a single chip is better. This modular quantum computing approach has lower error rates and lesser hardware constraints. For connecting the modules, optical bers, which have low propagation loss in a noisy thermal environment, are employed. So, the qubit operations must rst be transferred to the ying optical photons in the optical ber. The transduction of the qubit to the optical photon cannot be achieved directly due to the vast separation of the frequencies between the two (qubit in GHz and optical photon in THz). One way to achieve transduction is by introducing a bosonic system as a mediator that couples both the qubit and the optical photon, forming a hybrid qubit-boson-optical system. In this work, we discuss transduction in two such hybrid systems, namely, electro-optomechanical and electro-optomagnonic systems. The electro- optomechanical system consists of a superconducting microwave circuit coupled to a mechanical resonator which in turn is connected to an optical cavity. In recent years, this hybrid system has been extensively studied experimentally[1{5] and theoretically[6{12] for microwave-to-optical photon transduction. There are several ways of coupling a transmon qubit, formed by n.roson@iitg.ac.in ypooja.kumari@iitg.ac.in zaksarma@iitg.ac.ina superconducting microwave circuit, to a mechanical resonator [13{17]. Here, we consider a ux tunable transmon qubit that is coupled to a suspended mechanical beam [18]. The mechanical beam is then integrated as an end mirror of an optomechanical cavity forming the required hybrid system[10]. The hybrid electro-optomagnonic system consists of a superconducting microwave circuit coupled to a ferromagnetic magnon excitation [19{22], which is coupled to an optical photon [23{27]. This hybrid system is less explored. It is mainly due to the weak coupling between the magnon and the optical photon. However, there has been some progress recently [28]. For example, enhancement of magnon-photon coupling under the triple resonance condition of input photon, magnon and output photon is demonstrated in [25, 29]. By implementing the triple resonant condition, a microwave-to-optical conversion based on multiple magnon mode interaction with the optical photon mode is demonstrated in [30]. Another theoretical study to improve the magnon-optical coupling based on optical optical whispering gallery mode (WGM) coupled to localized vortex magnon mode in a magnetic microdisk is done in [31]. In this work, we construct the hybrid electro- optomagnonical system by merging the scheme proposed in [26], where a ux-tunable transmon qubit is coupled to a magnon mode formed in a m size YIG sphere, and the optomagnonical setup experimentally demonstrated in [29], where an optical WGM interacts with magnon mode in a YIG sphere of radius having few hundred m. One main diculty in realizing this hybrid system is the size gap in the YIG spheres. However, the possibility of reducing the sphere used in the optomagnonic case is pointed out in [29]. Assuming this to be possible, we consider a YIG sphere of few m size radius that couples both the superconducting qubit and the optical WGM present in the sphere. The technique used here for measuring the qubit states from the optical photon is similar to the one employed in [3]. The idea is to rst associate or encodes the qubit states in the magnon/phonon coherent excitationsarXiv:2305.11629v1 [quant-ph] 19 May 20232 and then measure these excitations by counting the average number of the photon in the optical cavity. Although measuring the qubit states by detecting the optical photon count is demonstrated in [3], our scheme exhibits two distinct features: (1) The interaction of the qubit and the magnon/phonon commutes with the intrinsic Hamiltonian of the qubit. In other words, the initial state of the qubit remains the same during the interaction. (2) Due to the coherent and oscillatory evolution of the magnon/phonon and photon states during the interaction, we can perform measurements of the qubit states from the photon count at regular intervals of time. The paper is organized as follows. We describe the two hybrid systems under study in Sec.II. The rst sequence of the transduction process, i.e., encoding the qubit state in the magnon/phonon coherent excitation is studied in Sec.III A. The measurement of the qubit state from the optical photon count is done in Sec.III B. Finally, we conclude by summarizing our work in Sec. IV. II. THE HYBRID SYSTEM We rst consider the hybrid electro-optomechanical system. This hybrid system comprises a ux-tunable transmon (formed by a SQUID loop ( EJ, J)), coupled to a mechanical resonator (realized by suspending one arm of another SQUID loop ( EM, M))[18] which can oscillate out of plane. The suspended mechanical membrane is then integrated as an end mirror of an optical cavity forming an optomechanical cavity [10], as shown in Fig. 1(a). The Hamiltonian of the system is described by ^Heom=^H0+^Htm+^Hom+^Hd (1) where, ^H0=~
c^ay^a+~!m^by^b+~!t^cy^cEc 2^cy^cy^c^c;(2a) ^Htm=~gtm^cy^c(^b+^by); (2b) ^Hom=~gom^ay^a(^b+^by); (2c) ^Hd=~E0 ^a+ ^ay
: (2d) Here, ^a(^ay),^b(^by), and ^c(^cy) are the annihilation (creation) operators of the optical photon, the mechanical phonon and the transmon, respectively. ^H0 is the Hamiltonian of the individual components of the hybrid system in the absence of any interactions. The transmon-mechanical resonator interaction is described by^Htm, and the optomechanical interaction by ^Hom. The strength of the coupling constant gomis generally quite small (1Hz). So, we drive the optomechanical cavity to increase the coupling strength. This drive is included in the Hamiltonian of the hybrid system as ^Hd. The Hamiltonian ^Homis written in the optomechanical drive frame (
c=!c!d, where!cis the cavity C E M ɸ M ɸ J E J B z X ω d ω c Y X Z (a) E J E J C ɸ J B z X Z Y TM TE (b)FIG. 1. (Color online) (a) Schematic diagram of an electro- optomechanical system. On application of an in-plane magnetic eld Bz, the transmon qubit formed by a SQUID loop (EJ;J) is coupled to a mechanical beam suspended at one arm of the loop ( EM;M) [18]. The mechanical resonator is integrated as a movable plate of an optomechanical cavity whose resonance frequency is !c. The optomechanical cavity is shined on by a red-detuned laser light. (b) A YIG ferromagnetic sphere that supports both the magnon excitation and optical WGM is placed near a ux-tunable transmon qubit formed by the loop ( EJ;J) [26]. An optical channel is mounted on top of the YIG sphere. A TM- polarized light given at the input of the channel passes the YIG sphere in a clockwise direction, and a TE-polarized light emerges at the channel output. An in-plane magnetic eld Bz, responsible for the transmon-magnon coupling, is passed through the ferromagnetic sphere. frequency and !dis the drive frequency.). The qubit- mechanical and optomechanical interactions arise from the displacement of the mechanical resonator. On application of an in-plane magnetic eld Bz, as shown in Fig. 1(a), the displacement of the mechanical resonator picks up a ux in the Josephson energy. This motional dependent Josephson energy leads to the transmon- mechanical resonator interaction ^Htm[18]. Note that in addition to the third order non-linear interaction term, ^Htm, higher order non-linear interaction terms are present as demonstrated in [18]. But, here, we have excluded this higher order corrections due to their negligible contribution to the system dynamics. The mechanical motion of the resonator also simultaneously alters the resonator frequency of the optomechanical cavity, which results in the optomechanical interaction ^Hom. We have considered a small SHM (simple harmonic3 motion) displacement of the resonator. We next consider the hybrid electro-optomagnonic system. In this system, a YIG sphere having a diameter in them range is placed near a ux-tunable transmon formed by a symmetric SQUID loop ( EJ,J) [26]. The YIG sphere is then mounted on an optical ber placed just above the plane of the SQUID loop [29], as shown in Fig. 1(b). Just like in the previous system, an in- plane magnetic eld Bzis applied. This eld magnetizes the magnetic sphere along the z-direction. Because of this magnetization, a uniform magnetostatic mode or kittle mode is excited on the YIG sphere, whose magnetic moment produces a stray eld and traverse along the transmon SQUID loop. Subsequently, the stray eld picks up an additional ux in the loop thereby changing the Josephson energy and frequency of the transmon and eventually leading to a transmon-magnon interaction. On the other hand, a TM (transverse magnetic eld) polarized light is pumped at the input of the optical waveguide. When in resonance, this pumped light or signal is conned in the YIG sphere forming WGM (whispering gallery mode) in the clockwise direction. The TM input signal then interacts with the magnons in the magnetic sphere. This interaction has two signicant features. One is that it changes the TM-polarized input signal to a circulating TE WGM and then comes out as a TE-polarized signal at the output. The other feature is that it changes the input and output signal frequencies. The amount of change in the input and output signal frequencies is equivalent to that of the magnon frequency. The above two features are the outcomes of satisfying the triple-resonance condition. The triple resonance condition is experimentally demonstrated in a 100- 300m size ferromagnetic sphere [29]. In our case, the size of the sphere is considered to be around 3 m. Although the size we have considered here is not yet experimentally realized for the triple resonance interaction, the possibility of sizing down the sphere is pointed out in [29]. The eective Hamiltonian of the hybrid electro-optomagnonic system is described by ^H0 eom=^H0 0+^H0 t+^H0 tm+^H0 om+^H0 d (3) where, ^H0 0=~!v^ay v^av+~!h^ay h^ah+~!0 m^my^m; (4a) ^H0 t=~!0 t^cy^cEc 2^cy^cy^c^c; (4b) ^H0 om=~g0 om(^ay h^av^m+ ^ah^my^ay v); (4c) ^H0 tm=~g0 tm^cy^c( ^m+ ^my); (4d) ^H0 d=~Ev ^avei!Lt+ ^ay vei!Lt
: (4e) Here, ^av(^ay v), ^ah(^ay h), ^m( ^my), and ^c(^cy) are the annihilation (creation) operators of the input TM optical photon, the output TE optical photon, the magnon and the transmon, respectively. ^H0 0and ^H0 tare the Hamiltonian of the individual components of the hybrid system in the absence of any interactions. Thetransmon-magnon interaction is described by ^H0 tmand the optomagnonic interaction by ^H0 om. Here, the interaction term ^H0 tmis for the symmetric SQUID loop. ^H0 dis the optical drive of the TM mode. III. TRANSDUCTION Here, we discuss the quantum transduction of qubit states to optical photons via mechanical phonons or YIG sphere magnons. The transduction process is realized in sequence. First, we encode the qubit states to the phonon/magnon excitations, and next, we measure these excitations by counting the average number of photons in the optical cavity. This section is divided into two parts. The rst part discusses the process of encoding the qubit states in the mechanical phonon states, and in the second part, we discuss how the phonon states, and hence the qubit states, are determined from the optical photon number. A. Qubit-phonon/magnon transduction We rst consider the qubit-mechanical interaction in the hybrid electro-optomechanical system and show how qubit states can be encoded to the phonon excitations. The qubit-mechanical coupling rate gtmis much larger than the single-photon optomechanical coupling rate gom. So, if there is no optomechanical cavity drive, we can neglect the optomechanical interaction in the Hamiltonian given by Eq. 2. Now we are left with just the electromechanical part of the hybrid system. ^Hem=^H0+~gtm^cy^c(^b+^by): (5) Here, the coupling constant gtmis dependent on the external ux bias  mas [26], gtm=g0sin(b); (6) where,b=b 0and  0=h 2eis the ux quantum. g0is coupling constant. Next, we enhance the coupling rate by modulating it parametrically by applying a weak ac biasb=accos(!act) (ac<<1) as done in [26]. gtm=g0accos(!act): (7) By substituting this modulated time dependent coupling constant in the Hamiltonian 5, and then transforming the resultant Hamiltonian in the reference frame of the ac drive (U=ei!actbyb), we get ^H0 em=^H0+~g0ac^cy^c(^b+^by)!ac^by^b: (8) Here, we have ignored the fast rotating terms since 2!ac>>g 0ac. To do the qubit transduction, we convert the transmon to a transmon qubit by considering only4 the rst two energy levels. We then let the system evolve under resonant modulation ( !m=!ac). If the qubit is initially in the ground state jgiand the mechanical resonator is in the vacuum state j0bithen after some time t, the qubit will remain in the ground state and the mechanical resonator will change to a coherent state jb=ig0acti. Similarly, if the qubit is initially in the excited state jeiand the resonator in the vacuum state, then the qubit will remain in the excited state, and the mechanical resonator will evolve to another coherent statejb=ig0actiafter some time t. jg;0bi0!jg;b=g0actit je;0bi0!je;b=ig0actit An overall phase term induced from the intrinsic qubit Hamiltonian is not included as it does not contribute to the transduction process. We see that as the system evolves, the mechanical resonator changes from a vacuum state to a coherent state, whereas the qubit state remains as it is. It is because the interaction between the qubit and the mechanical resonator commute with the intrinsic Hamiltonian of the qubit. In other words, the interaction is `non-demolition' in the qubit part. We next consider transduction in the electro- optomagnonic case and analyze how qubit states can be encoded to magnon excitations. Just like in the previous case, we can neglect the optomagnonic part and consider only the electro-magnonic part since the single magnon- photon coupling g0 omis much less than the transmon- magnon coupling g0 tmfor no optical drive. The electro- magnonic part is described by ^H0 em=^H0 0+~g0 tm^cy^c( ^m+ ^my); (9) where, g0 tm=g0 0sin(m)p jcos(m)j; (10) where,m=m 0and  0=h 2eis the ux quantum. g0is coupling constant. Similar to the previous system, here also we enhance the coupling rate by modulating it parametrically by applying a weak ac bias m= 0 accos(!0 act) (0 ac<<1) as done in [26]. g0 tm=g0 00 accos(!0 act): (11) Substituting Eq. 11 in Eq. 9 and then transforming in drive frame ( U=ei!0 actmym) gives ^H0 em=^H0+~g0 00 ac^cy^c( ^m+ ^my)!0 ac^my^m: (12) The fast rotating terms are neglected for 2 !0 ac>>g0 00 ac. We now take the rst two levels of the transmon and allow the system to evolve. For resonance modulation (!0 m=!0 ac), we obtain results similar to that of the electro-mechanical case, i.e.,jg;0mi0!jg;m=ig0 00 actit je;0mi0!je;m=ig0 00 actit Here,jg;0mi0andje;0mi0are the initial states, and jg;m=ig0 00 actandje;m=ig0 00 actitare the nal states of the qubit-magnonic system after time t. An overall phase is not included. So far, we have not included the noise factor while evolving the system. To include the noisy environment, we allow the system to evolve under the Lindblad master equation. For the electro-mechanical system _^em=i[^Hem;^em] + L[ ^z] + L[^] + b(nth+ 1)L[^b] + bnthL[^by]; (13) and for the electro-magnonic system _^0 em=i[^H0 em;^0 em] + L[ ^z] + L[^] + m(n0 th+ 1)L[ ^m] + mn0 thL[ ^my];(14) whereL[^o] = (2^o^^oy^oy^o^^^oy^o)=2. Here, is the decay rate of the transmon qubit, b( m) is the decay rate of phonon (magnon), nth(n0 th) is the thermal phonon (magnon) number, and ^ em(^0 em) is the density operator of the qubit-mechanical (qubit-magnonic) system. To observe the coherent excitations of phonon and magnon in the dissipating environment, we plot the Wigner functions in Fig 2. Here, we observe that the Wigner functions of the phonon and magnon at some time = (3=2)sand for coupling constants g0ac=g0 00 ac= 2 MHz show coherent state prole. The amplitude of the coherent states when the qubit is in the ground state isjb=ig0aci=j3iifor the phonon and jm= g0 00 aci=j3iifor the magnon, as shown in the gure. On the other hand, when the qubit is in the excited state, the coherent amplitudes are jb=ig0aci=j3ii andjm=ig0 00 aci=j3ii. These changes in the amplitude of the coherent states corresponding to the qubit ground and excited states are similar to the ones that are observed in the non-dissipative case. So, in both the dissipative and non-dissipative qubit- mechanical and qubit-magnonic systems, we observe that the ground state of the qubit is encoded or associated with a coherent excitation of both the phonon and magnon and the excited state of the qubit is encoded in another coherent excitation of the same magnon and phonon having amplitudes exactly opposite to that of the excitation associated with qubit ground state. B. Qubit-optical photon transduction We have seen that the state of the qubit can be encoded in the coherent excitations of phonon and magnon. Here, we will complete the qubit transduction sequence by transferring the phonon and magnon states to the optical photon. In the phonon case, this can be achieved through the optomechanical interaction, and in the magnon case,5 FIG. 2. (Color online) Wigner function representation of the coherent states of phonon and magnon. In (a) and (b), the phonon and the magnon excites to the coherent states jb= 3iitandjm= 3iitwhen the qubit is in the ground statejgi. The coherent states of the phonon jb=3iit and the magnonjm=3iitwhen the qubit is in the exited statejeiis shown in (c) and (d), respectively. The coherent states are taken at time t== (3=2)sfor coupling constantsg0ac=g0 00 ac= 2MHz. The other parameters aregamma b=2= 1 Hz,nth= 400,gamma m=2= 0:1 GHz, n0 th= 0:5, =2= 0:1 GHz it can be achieved through the optomagnonic interaction satisfying the triple-resonant condition. First, we consider the optomechanical transfer. We have previously seen from the electro-mechanical interaction that the mechanical resonator can be coherently excited with dierent amplitudes depending on the initial states of the qubit. So, we rst excite the mechanical resonator to coherent states jb=ig0acti and then switch o the interaction g0acby turning o the ux bias ac. The interacting system remaining is then the optomechanical system. ^H=~
c^ay^a+~!m^by^b+~gom^ay^a(^by+^b) +~E0(^ay+ ^a): (15) Sincegom1Hz is very weak, we drive the cavity with an intense laser. Because of this strong drive, we can separate the amplitudes of the mechanical resonator and optical cavity into a semi-classical coherent part ( ;) and a small quantum uctuation ( ^a;^b) around it, i.e., ^a!^a+and^b!^b+. We substitute this separation in Eq.15. By retaining only the interacting term, which is multiplied by the factor (jj103), the Hamiltonian reads ^Hom=~
0^ay^a+~!m^by^b+~Gom(^ay+ ^a)(^by+^b);(16) where
=
c(+)gomandGom=gomjjfor a constant phase preference of alpha. For simplicity we have rewritten ^ato ^aand^bto^b. Note that while writing Eq.16, we have ignored all the constant terms and all the linear terms containing ^ a, ^ay,^band^byare equated to zero [32].The coherent state of the mechanical resonator prepared from the electro-mechanical interaction is in the mechanical frame !m=!ac. So, we transform the Hamiltonian 16 in the mechanical frame. We further transform the system in the cavity detuning frame
. Therefore, for a red-detuned laser drive
= !m, Eq.16 becomes ^Hom=~Gom(^ay^b+^by^a): (17) Here, the fast rotating terms are ignored provided Gom<< 2!m. For studying the state transfer from mechanical phonon to optical photon, we write down the dynamics of average number of photon and phonon in the presence of dissipation. dh^ay^ai dt=i(h^ay^bih^by^ai)Gomh^ay^ai(18a) dh^by^bi dt=i(h^by^aih^ay^bi)Gom bh^by^bi + bnth (18b) dh^by^ai dt=(+ b) 2h^by^aiiGom(h^ay^ai h^by^bi) (18c) By choosing the initial state of the mechanical resonator as the coherent state jb(0)iprepared from the electro- mechanical interaction, the average number of photon in the absence of dissipation is given by h^ay^a(t)i= (g0ac)2(1sin(2Gomt)) (19) when the qubit is in the ground state ( jb(0)i=j+ ig0aci), and h^ay^a(t)i= (g0ac)2(1 +sin(2Gomt)) (20) when the qubit is in the excited state ( jb(0)i=j ig0aci). Here, we have taken the initial state of the cavity photon to be j(0)i=jg0aci. The reason for choosing this particular initial state is discussed in the Appendix A. The evolution of the average photon number is shown in Fig. 3(a). From the gure, we observe that if we measure the average photon number in the cavity at the interval of t==2Gom(starting from t==4Gom) , then we either detect or do not detect the presence of photons depending on the state of the qubit. If we detect photons in the cavity at the interval of t= (2n+ 1)=4Gom, wheren= 0;2;4;:::, then we know that the qubit is in the ground state, and if at the same interval, we do not detect any photons then the qubit is in the excited state. Similarly, if we detect photons in the cavity at the interval of t= (2n+ 1)=4Gom, wheren= 1;3;5;:::, then we know that the qubit is in the excited state, and if at the same interval, we do not detect any photons then the qubit is in the ground state. We have chosen the above particular intervals because the average photon numbers at these intervals are at the maximum separation, and the qubit states can be determined more eciently than the other intervals.6 FIG. 3. (Color online) Evolution of the average number of photonh^ay^aiin the optical cavity. (a) In the absence of dissipation, the oscillatory evolution of h^ay^aikeeps on going. When the qubit is in the ground state, the oscillation is represented by the red colour, and when the qubit is in the ground state, the oscillation is represented by the black colour. The evolution of average photon number in the presence of dissipation is shown in (b) for = 2Gom, (c) and (d) for =Gom. In (a), (b), and (c), =2= 0:01 GHz is used, and in (d),=2= 1 MHz is used. The other common parameters are = 1 Hz and nth= 400 In the presence of dissipation, the oscillatory nature of h^ay^a(t)idecays with time, and in order to know the state of the qubit by counting the photon number we require that the optomechanical coupling rate Gomshould be comparable to the decay rate of the cavity. In Fig. 3(b), we show the decay of cavity photon number for = 2Gom, a moderate coupling strength. At this coupling strength, we are able to make an ecient measurement of qubit states at just two intervals, t= 0:02sand t= 0:075s, before the number of average photon decay to zero. At a coupling strength lower than this, we will not be able to identify the qubit states from the optical photon. We also plot the case when the coupling strength is equal to the decay rate in Fig. 3(c). Here, more oscillations can be seen, and hence more time intervals to measure the qubit states. Furthermore, we can increase the time period for a same number of oscillations by decreasing the decay rate as shown in Fig. 3(d). One could go on and nd out the delity of state transfer of coherent state from the mechanical phonon to the optical photon. However, in our case, it is not necessary since our purpose of determining the qubit state is achieved by simply counting the cavity photon number. Since we are dealing with coherent states, we can quantify how well the measured photon number indicates that the qubit is in a particular state. In Fig. 4, we plot the probability distribution of the coherent state for the = 2Gomcoupling case (Fig. 3(b)) at the measurement time = 0:075s. We see that even when the qubit is in the excited state, there is still some probability of not nding any photons in the FIG. 4. (Color online) Probability distribution of coherent states of the optical photon number in the presence of dissipation. (a) and (b) shows the distribution when the qubit is in the ground and excited state, respectively. The coherent states are measured at time = 0:075s. The average photon number in (a) is 3.4, and 0 in (b) cavity. The dierence in the probability of not nding photons in the cavity when the qubit is in the excited state (Pe= 0:035) and when it is in the ground state (Pg= 0:999) gives the eciency of determining the qubit state,P=PePg= 0:964. This eciency decreases for less average photon number and vice versa. So, we need to repeat the counting measurement several times before concluding the nature of the qubit state. We now move on to the optomagnonic state transfer. Just like in the optomechanical case, we rst excite the magnon to coherent state jm=ig0 om0 omtifor some time t, and then switch o the interaction g0 om0 om by turning o the ux bias 0 om. The remaining optomagnonic system in the drive frame then reads ^H0 om=~v^ay v^av+~h^ay h^ah+~!0 m^my^m+~Ev ^av+ ^ay v
~g0 om(^ay h^av^m+ ^ah^my^ay v); (21) whereh=!h!Landv=!v!L. We write down the dynamics of the system. d^m dt=( m 2+i!0 m) ^m+ig0 om^ah^ay v; (22a) d^av dt=(v 2+iv)^av+ig0 om^ah^my+iEv;(22b) d^ah dt=(h 2+ih)^ah+ig0 om^av^m: (22c) Here, m,vandhare the decay rates of magnon, input TM eld, and output TE eld, respectively. The intrinsic magnon-photon coupling rate g0 omis of the order of 10Hz, which is relatively very weak. We can enhance this coupling strength up to the order of MHz by performing an optical drive ( Ev) to the ferromagnetic sphere. After the drive, we can separate the input eld into semi-classical mean amplitude ( v) and small quantum uctuation around it ( ^av), i.e., ^av!v+^av. Substituting this separation in Eq. 22 and writing the7 quantum and classical parts separately, we have d^m dt=( m 2+i!0 m) ^m+ig0 om(^ah^ay v+ ^ah v);(23a) d^av dt=(v 2+iv)^av+ig0 om(^ah^my+h^my);(23b) d^ah dt=(h 2+ih)^ah+ig0 om^av^m; (23c) and dv dt=(v 2+iv)v+iEv: (24) The linear coupling terms in Eq. 23 are multiplied by a factor ofvor v(jvj103) compared to the non-linear coupling terms. Therefore, we can neglect the non-linear coupling terms and retain only the linear coupling terms. The corresponding linear Hamiltonian reads ^H0 om=~v^ay v^av+~h^ay h^ah+~!0 m^my^m+ ~G0 om(^ay h^m+ ^my^ah); (25) whereG0 om=g0 omjvj.jvjis given by the steady value of Eq. 24. Since the initial coherent state of the magnon prepared from the electro-magnonic interaction is in the magnon frame, we transform the Hamiltonian 25 in the magnon frame. Thus, for a resonant optical drive v= 0, the resultant Hamiltonian of the system in the magnon as well as the output TE eld frame of reference yields ^H0 om=~G0 om(^ay h^m+ ^my^ah); (26) Here, since the interaction satises the triple resonance condition, we have taken h=!0 mor!h=!v+!0 m and ignored the fast-rotating terms provided 2 !0 m>> G0 om. We see that Hamiltonian 26 and 17 are identical. Therefore, the analysis that we have done for determining the qubit states in the optomechanical system is also applicable here. The dissipative and non-dissipative dynamics studied in the optomechanical system and all the plots in Fig. 3 and 4 will be similar. The optomagnonic parameters that produce similar plots in Fig. 3 areG0 om= 0:5h, m= 0:1 Mhz,h= 0:01 GHz andn0 th= 0:5. IV. CONCLUSION In conclusion, we have studied quantum transduction of superconducting ux-tunable transmon qubit in two hybrid systems: electro-optomechanical and electro-optomagnonical system. The realization and advancement of quantum transduction in such hybrid systems are very crucial for the development of quantum network, quantum internet, etc. The transduction is done in two stages. First, we encode the qubit states in the coherent excitations of mechanical phonon or ferromagnetic sphere magnon without disturbing the qubit state (non-demolition interaction) and in the next stage, we identify these excitations by counting the average number of photon in the optomechanical oroptomagnonic WGM cavity. Because of the coherent interaction between the phonon/magnon and the optical photon, the average photon number oscillates with time. The oscillation when the qubit is in the ground state and when in the excited state is exactly opposite. As a result, we can make multiple measurements of the photon number at a regular interval of time and hence know the state of the qubit at each interval. In the presence of dissipation, the optomechanical and optomagnonical coupling strength should be atleast moderately strong in order to perform any measurements before the photon number altogether decays to zero. The required coupling strength in the optomechanical system is extensively studied. But, in the optomagnonic system, the required coupling regime to perform the transduction is not yet explored. However, the possibility of optomagnonic coupling strength going upto 10 MHz is disscued in [29]. One of the ways to reach such coupling magnitude is to reduce the size of the YIG sphere to few m, which is comparable to the size considered in the hybrid system proposed in this work. ACKNOWLEDGEMENT RN gratefully acknowledges support of a research fellowship from CSIR, Govt. of India. Appendix A: Non-dissipative dynamics. The analytical solution of Eq. 18 in the absence of dissipation is given by h^ay^ai=1 2fj0j2+j0j2+ (j0j2j0j2)cos(2Gomt) i( 000 0)sin(2Gomt)g (A1) Here,j0j2=h^ay^ai0andj0j2=h^by^bi0are the initial values of photon and phonon/magnon. The above equation can be further simplied by simply choosing j0j2=j0j2. h^ay^ai=j0j2Im( 00)sin(2Gomt) (A2) The initial coherent amplitudes of phonon/magnon is xed at0=ig0ac. To keep the oscillatory part in Eq. A2, which is necessary for the transduction, we require that the initial coherent amplitude of the cavity photon0should have a non-zero real part. Therefore, we choose0=g0ac. The oscillation of Eq. A2 then becomes h^ay^ai= (g0ac)2f1sin(2Gomt)g; (A3) when the qubit is in the excited state ( 0=ig0ac), and h^ay^ai= (g0ac)2f1 +sin(2Gomt)g; (A4) when the qubit is in the ground state ( 0=ig0ac).8 [1] W. Jiang, C. J. Sarabalis, Y. D. Dahmani, R. N. Patel, F. M. Mayor, T. P. McKenna, R. Van Laer, and A. H. 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2023-05-19

We study the quantum transduction of a superconducting qubit to an optical photon in electro-optomechanical and electro-optomagnonical systems. The electro-optomechanical system comprises a flux-tunable transmon qubit coupled to a suspended mechanical beam, which then couples to an optical cavity. Similarly, in an electro-optomagnonical system, a flux-tunable transmon qubit is coupled to an optical whispering gallery mode via a magnon excitation in a YIG ferromagnetic sphere. In both systems, the transduction process is done in sequence. In the first sequence, the qubit states are encoded in coherent excitations of phonon/magnon modes through the phonon/magnon-qubit interaction, which is non-demolition in the qubit part. We then measure the phonon/magnon excitations, which reveal the qubit states, by counting the average number of photons in the optical cavities. The measurement of the phonon/magnon excitations can be performed at a regular intervals of time.

Quantum transduction of superconducting qubit in electro-optomechanical and electro-optomagnonical system

2305.11629v1

Nano scaled magnon transistor based on stimulated three-magnon splitting Xu Ge1, Roman V erba2, Philipp Pirro3, Andrii V . Chumak4, Qi Wang1,* 1 School of Physics, Huazhong University of Science and Technology, Wuhan, China 2 Institute of Magnetism, Kyiv, Ukraine 3 Fachbereich Physik and Landesforschungszentrum OPTIMAS, Rheinland -Pfälzische Technische Universität Kaiserlautern -Landau, Kaiserslautern, Germany 4 Faculty of Physics, University of Vienna, Vienna, Austria Abstract Magnonics is a rapidly growing field , attracting much attention for its potential applications in data transport and processing. Many individual magnonic devices have been proposed and realized in laboratories . However, an integrated magnonic circuit with several separate magnonic elements has yet not been reported due to the lack of a magnonic amplifier to compensate for transport and processing losses . The magnon transistor reported in [Nat. Commun. 5, 4700, (2014)] could only achieve a gain of 1.8, which is insufficient in many practical cases. Here, we use the stimulated three -magnon splitting phenomenon to numerically propose a concept of magnon transistor in which the energy of the gate magnons at 14.6 GHz is directly pumped into the energy of the source magnons at 4.2 GHz, thus achieving the gain of 9. The st ructure is based on the 100 nm wide YIG nano -waveguides, a directional coupler is used to mix the source and gate magnons, and a dual- band magnon ic crystal is used to filter out the gate and idler magnons at 10.4 GHz frequency. The magnon transistor preser ves the phase of the signal and the design allows integration into a magnon circuit. Spin wave s (magnons) , having low intrinsic losses , high-frequency range (gigahertz to terahertz) , and short wavelengths (down to several nanometers) , are promising candidates for data transport and processing devices [1-5]. In co ntrast to sound and light waves, spin waves exhibit stronger and more diverse intrinsic nonlinear phenomena [6-9], making it easier to construct all - magnonic integrated circuits, in which the magnons are controlled by the magnons themselves without any intermediate conversion to electr ic currents . In the last decade, s everal individual magnonic devices have been proposed such as spin-wave logic gate [ 10,11], magnon transistor [12,13], majority gate [14,15], magnon valve [1 6,17], and direction al coupler [18,19]. However, the realization of an integrated magnonic network [20,21] with several separate magnonic devices is still a challenge . The main reason is that spin wave amplitude decreases after passing through the upper -level device due to magnetic damping and could not reach the nonlinear threshold of the next - level device. The key element to solve this problem is the magnonic amplifier, which compensates the loss and brings the spin -wave amplitude back to the initial state. In recent years, researchers have explor ed effective ways to amplify propagating spin waves. One of the method s is based on the reduction of magnetic damping, which enhances spin-wave signals by spin transfer torque and spin orbit torque generated by a DC current [22,23]. Recent ly, Merbouche et al. [24] reported a true amplification of spin waves based on spin currents. In this *Author to whom correspondence should be addressed: williamqiwang@hust.edu.cn research, the direction of the external field and the composition of the materials were precisely tuned to avoid the occurrence of nonlinear magnon scattering and auto -oscillations. Another common approach is paral lel parametric pumping, in which one microwave photon splits into two magnon s at half the frequency under the conservation of energy and momentum to amplify the propagating spin waves [25-27]. V ery recently, Breitbach et al. [28] proposed a spin -wave amplifier based on rapid cooling, which is a purely thermal effect. The magnon system is brought into a state of local disequilibrium with an excess of magnons. A propagating spin -wave packet reaching this region stimulates the subsequent redistribution process, and is in turn amplified. In modern CMOS electronics, transistors are used to amplify electrical signals. The magnon transistor reported in [1 2] was designed to suppress one magnon flux by another in order to perform all-magnon logic operations. Since the suppression efficiency was very high, it was shown that using a special interferometric scheme, the transistor could also be used for amplifica tion, but with a rather small gain of 1.8. Another magnon transistor concept suitable for direct amplification of the magnon fluxes is need ed. In this letter, we propose a magnon transistor concept based on stimulated three -magnon splitting and validate it using micromagnetic simulations . First, we study three -magnon scattering in a nanoscale straight magnonic waveguide with different external field directions . A pronounced three -magnon scattering is observed, in which one gate (pump) magnon (14.6 GHz) splits into two magnons (10.4 GHz and 4.2 GHz) according to the laws of energy and momentum conservation . It is found that the additional injection of the source magnons at the frequency of 4.2 GHz leads to a drastic enhancement of the splitting of the gate magnons, a phenomenon we refer to as stimulated magnon splitting. Based on this mechanism, we propose a magnon transistor design that employs a directional coupler to mix gate and source magnons. In addition, a dual -band magnon crystal is used to filter out the gate (14.6 GHz) and idler (10.4 GHz) magnons from the transistor drain. The transistor allows the drain magnon density to be 9 times higher than the source magnon density. As shown in Fig. 1 (a), we consider a yttrium iron garnet (YIG) waveguide of length l = 20 m, width w = 100 nm , and thickness t = 50 nm . To investigate the effects of three -magnon splitting, micromagnetic simulations are preformed using Mumax3 [29] with the following parameters of YIG [30]: saturation magnetization Ms = 1.4 105 A/m, exchange constant A = 3.5 10−12 J/m, and damping coefficient = 2 10−4. An external bias magnetic field Bext = 100 mT is applied in the xy plane forming an angle H with the negative x-axis, as illustrated in Fig. 1(a) . The maximum splitting efficiency is obtained at approxi mately H = 60°, which correspond s to M = 45° (the angle between the magnetization direction and the negative x-axis). It is a known feature of the three -magnon interaction in an effectively one -dimensional system [31], which is additionally explained in the supplementary materials. The magnetization direction is not aligned with the direction of the external field due to the non negligible demagnetic field along the width direction in the nanoscale waveguide . In the following studies, the external field is fixed at the optimal value H = 60°. The prorogating spin wave in the waveguides is excited by an alternating field as a sinusoidal function of time: hrf = bsin(2ft)ez with the oscillation amplitude b and the excitation frequency f. The applied field is in the center of waveguide over an area of 30 nm in length [blue area shown in Fig. 1(a)]. To avoid spin wave reflection, the damping coefficient has an exponential increase to 0.5 at both ends of the waveguide [32]. In addition , all the simulations are performed with room temperature T = 300 K. We recorded the time-dependent magnetization data from t = 0 ns to 50 ns across the entire wave guide and then obtained the spin wave dispersion curves by using two-dimensional fast Fourier transform [33,34]. Fig. 1. (a) A sketch of spin-wave excitation system: a 30-nm wide excitation region is used to generate oscillation magnetic field hrf, and the in-plane magnetic field is applied at H = 60° . Dispersion curve of the propagating spin wave s and spin-wave intensity as a function of frequency for the different amplitude s of excitation field are shown in (b) b = 20 mT and (c) b = 90 mT. In the current study , the excitation frequency fG is considered to be 14.6 GHz. The left part of Fig. 1(b) and 1(c) shows the dispersive curves corresponding to different excitation field amplitudes . At b = 20 mT, the dispersive curve exhibits only one weak bright spot [Fig. 1(b) left], and it can also be seen that there is only one weak peak in the spin-wave intensity spectrum , corresponding to the directly pump ed magnon of frequency 14.6 GHz [Fig. 1(b) right]. When the amplitude b is increas ed to 90 mT, two other stronger bright spots at lower frequencies appear in the dispers ion curve [Fig. 1(c) left] . These peaks , observed at 14.6, 10.4, and 4.2 GHz , correspon d to three distinct magnons: the gate magnon fG (kG = 93.5 rad/m) and the scattering idler magnons fI (kI = 73.1 rad/m) and source magnons fS (kS = 20.4 rad/m), as shown in the right of Fig. 1(c). This observed process adheres to the energy -momentum conservation laws, satisfying fG = fI+ fS, kG = kI + kS. This phenomenon is nothing else th an three -magnon splitting [ 31,35,36]. Now we consider stimulated three -magnon scattering, by exciting both the gate and one of the split ting (source) magnons. Namely, we considered the excited field as hrf = bSsin(2fSt)ez + bGsin(2fGt)ez with one more excitation frequency fS = 4.2 GHz, and collected the spin-wave intensity of drain magnons having frequency equal to the frequency of the source magnons fD = fS = 4.2 GHz for the varying amplitude of excitation field bG in the range from 1 mT to 130 mT. As shown in Fig. 2, the spontaneous three -magnon splitting process appears when bG is greater than 30 mT, see black line (rigorously speaking, it is not absolute three -wave instability, but a convective one [3 7, 38]). The red and blue lines correspond to the stimulated process, and were obtained by applying two excitation frequencies of fG = 14.6 GHz and fS= 4.2 GHz . The spin-wave intensity spectra of the red and blue lines are obtained by subtracting the energy of the directly excited magnons at a frequency of 4.2 GHz, and thus show the same ground intensity as the black line. First, we see, that stimulated process takes place at a lower gate power than the spontaneous one and, in fact, does not demonstrate any threshold. This feature is expected for three -wave processes and, for magnon system, in particular, was demonstrated in [3 6] by inspecting idler magnon density . For the same excitation amplitude bG, the red and blue lines show stronger spin-wave intensity in contrast to the black line. This indicates that the existing magnon of frequency fS = 4.2 GHz can stimulate not only the appearance of idler magnons (as shown in [3 6]), but also increases the population of the source magnons itself , and the scattering efficiency depends on the source magnon density. In addition, the stronger density of scattering magnon is achieved when the amplitude bS of fS = 4.2 GHz is increased. This enhancement of the three -magnon splitting by introducing one of the splitting magnons can be used for the amplification of propagation spin waves. FiG. 2. Intensity of the scatter ing spin waves at frequency of fD = 4.2 GHz as a function of the gate excitation field bG, which is proportional to gate magnon density . The black line corresponds to the case when only the gate magnons are directly excited at fG = 14.6 GHz. The red and blue lines depict the case when gate and source spin wave s are simultaneously excited by alternating field at fG = 14.6 GHz and fS = 4.2 GHz for two different densities of source magnons , driven by source excitation field bS = 4 mT (red line) and bS = 10 mT (blue line). Estimation of the amplification efficiency can be done in a way, similar to that used in nonlinear optics [3 7]. The maximum amplification rate can be calculated in the conservative approximation, i.e. neglecting the damping. As derived in the supplementary materials, the maximum amplitude of the drain magnons AD is given by: G S2 2 2 D,ma G x SvA A Av−= , (1) where AS and AG are initial amplitudes of source and gate magnons at fS and fG, respectively, while vS and vG are their group velocities ; amplitudes are defined as standard canonical amplitudes [ 31]. This linear dependence of the gain on the gate power ()22 D,max SAA− ~ bG2 is nicely reproduced in the simulations; only at high gate powers, above bG > 50 mT, the gain saturates due to the influence of other nonlinear effects, in particular, self - and cross - nonlinear frequency shift . We see, that the amplification rate AD/AS is larger for smaller source amplitude AS, since the total power, which can be transferred to the source wave, is limited by the gate magnon power. Also, amplification efficiency increases with the ratio vG/vS. In the case of spin waves in waveg uides, higher -frequency (gate ) magnon almost always has larger group veloc ity than lower -frequency (source ) magnon, improving the amplification mechanism (In our case, t he ratio vG/vS, estimated to be approximately 6.8, is derived from the dispersive curve depicted in Fig. 1(c) ). Of course, damping decreases the amplification efficiency and introduces a dependence on the transferred power on the initial power of source magnons. Detailed consideration of this process lie s beyond the scope of this work . A magnon transistor is design ed as shown in Fig. 3(a) consisting of one magnonic directional coupler [39] and a dual-band magnon ic crystal with different periods [40]. The directional coupler [41] is employed for combining spin wave signals with different frequencies from two separated waveguides into one waveguide via frequency -dependent dipolar coupling strength [19,39]. The directional coupler is carefully design ed with the following geometric dimensions: the length of coupled waveguides ( L1) is 390 nm , the angle between waveguides ( 𝛷) is 10° , the gap between coupled waveguides ( δ) is 10 nm, and the edge -to-edge distance ( d1) is 200 nm , to make sure that gate magnon s can pass through it without significant energy loss, and the source magnon s will be completely guided from the bottom waveguide to the top one as shown by the black and red arrows in Fig. 3(a) . Once the source magnon is coupled into the top waveguide and mixed with the gate magnon, the three -magnon splitting is enhanced and the gate magnon efficiently splits into an idler magnon (10.4 GHz) and a drain magnon with the frequency of 4.2 GHz resulting in an amplification of the source magnon. Figure 3(b) shows the working principle of the magnon transistor . When only the gate magnon ( fG = 14.6 GHz, bG = 40 mT) is applied, the two -dimensional colormap shows the weak magnon density of 4.2 GHz due to the low three magnon splitting efficiency [top of Fig. 3(b) ]. The middle panel of Fig. 3(b) shows the case of only the source magnon is injected from the bottom waveguide. It clearly shows that all the source magnons are guided from the bottom waveguide to top waveguide (transistor’s drain) by the directional coupler as expected . Once both gate and source magnons are simultaneously excited , the drain magnon density at 4.2 GHz is dramatically increased due to the stimulation of the three magnon splitting [botto m of Fig. 3(b) ]. In order to get only the flux of drain magnon s at the output, a specially designed dual -band magnonic crystal , in the form of two in -line connected magnonic crystals of different periodicities, is used to filter out the gate and idler magnon s produced by three magnons splitting [42]. The design principle of magnonic crystals is discussed in detail in supplementary materials. Figure 3(c) shows the magnetization oscillations spectr a extracted from the end of the top waveguide as marked by the red dashed rectangular regions in Fig. 3(a) . It shows two properties: (1) The source magnon has a gain factor of 9. (2) The gate magnon and idler magnon have been efficiently suppressed by the dual-band magnon ic crystal . Importantly, that such a large gain is observed for moderate source magnon power, which is required for logic applications – amplification of nonlinear spin waves is much more nontrivial task than of small -amplitude ones [ 43]. Furthermore, the amplification is not s ensitive to the relative phase between the gate and source magnons (see the supplementary materials). Therefore, the output signal can be directly used to connect the next logic gate without any phase modulations and the proposed magnonic transistor is suitable for further integration. Fig. 3. (a) Schematic of magnon transistor designed with a directional coupler with dual-band magnon ic crystal . (b) The intensity distributions of spin waves at 4.2 GHz and (c) the magnetization oscillation spectrum extracted at the end of the top waveguide marked by red dashed rectangular regions for different cases. In summary, we numerically demonstrate magnon transistor based on the phenomenon of stimulated three -magnon splitting within a nano -scaled waveguide with an in -plane inclined static external field. The three -magnon splitting efficiency of the gate magnons is substantially enhance d by directly introducing one of the signal magnons . A magnon directional coupler is used to mix the gate and source magnons, and a dual -band magnon crystal is used to filter the idler and gate magnons from the drain of the transistor. The transistor has a gain of 9, measured as the ratio of the drain magnon density normalized to the source magnon density. 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Lett. 95, 262508 (2009). [41] A. V . Sadovnikov, S. A. Odintsov, E. N. Beginin, S. E. Sheshukova, Y . P . Sharaevskii, and S.A. Nikitov, Phys. Rev. B 96, 144428 (2017). [42] K.-S. Lee, D. -S. Han, and S. -K. Kim, Phys. Rev. Lett. 102, 127202 (2009). [43] R. Verba, M. Carpentieri, G. Finocchio, V . Tiberkevich, A. Slavin, Appl. Phys. Lett. 112, 042402 (2018). Supplementary Materials for “Nanoscaled magnon transistor based on stimulated three -magnon splitting ” Xu Ge1, Roman V erba2, Philipp Pirro3, Andrii V . Chumak4, Qi Wang1 1 School of Physics, Huazhong University of Science and Technology, Wuhan, China 2 Institute of Magnetism, Kyiv, Ukraine 3 Fachbereich Physik and Landesforschungszentrum OPTIMAS, Rheinland -Pfälzische Technische Universität Kaiserlautern -Landau, Kaiserslautern, Germany 4 Faculty of Physics, University of Vienna, Vienna, Austri a S1. Identification the inclination angle H based on the scattering intensity Figure S1. The scattering spin -wave intensity as function of the angle M the angel between the magnetization direction and negative x-axis (bottom transverse axis ). The three -magnon Hamiltonian, governing considered process, is derived as ( ( )(3) * 12,3 1 2 3 1 2 3 123 c.c.) V c c c= + + − k k k H (S1) where ki = kiex is spin-wave wave vector (1, 2, and 3 denote the three magnons) , directed along the waveguide (x direction). Spin-wave amplitudes ci are defined as usual canonic variables [1], i.e. have an order of dimensionless dynamic magnetization. In our case, magnons 1, 2, and 3 are source, idle and gate magnons, respectively. The three -magnon coefficient V12,3 could be found using scalar [1] or vectorial [2] spin-wave Hamiltonian formalism. General expression is quite cumbersome and is not presented here. However, using a simplified expression for the dynamic demagnetization tensor equal to one of a film, and neglecting minor ellipticity -related term, three -magnon coefficient is derived as ()() ( ) 12,3 1 2cos sin 22M M MV f k t f k t + , (S2) where 𝜔𝑀=𝛾𝜇0𝑀𝑆, with 𝛾 representing the gyromagnetic ratio, 𝜇0 as the vacuum permeability , and 𝑀𝑆 as the saturation magnetization . Additionally , ( ) 1 (1 exp[ | |])/ | |f x x x= − − − is the so - called thin -film function. Consequently, the calculated value of M = 45° corresponds to the maximum strength of three -magnon splitting determined by the above theoretical equation of the scattering coefficient [Eq. ( S2)]. Furthermore , as seen in Fig. S1, we extracted the magnetization direction angle M and the intensity of lower frequency scattering magnons by varying the direction of the in -plane external field from H = 0° to H = 90°. The strong est peak is observed at around M = 45°, which is consistent with the above theoretical analysis , in turn, the corresponding angle H is found to be around 60° (Fig. S1 ), and hence , a tilt angle H = 60° of the external field is utilized to investigate the magnon scattering in the main text. S2 Spin -wave amplification by three -magnon scattering Using well -developed approach from nonlinear optics [3], one can derive equations for spatial - temporal evolution of spin waves envelope amplitudes ai = ai (x, t): ( ) ( ) ( )* 1 1 1 2 3 * 2 2 2 1 3 3 3 3 1 22 2 2, , .tx tx txv a iVa a v a iVa a v a iVa a + + =− + + =− + + =− (S3) Here vi is spin-wave group velocity, i the damping rate, 12,3 VV , and equations are written for the resonant case, i.e. when f3 = f1+ f2. Eq. (S3) does not allow for exact analytical solution. Let’s first look on the amplification conditions. For this, assume the pumping wave amplitude a3 to be constant, signal wave initial value a1 (x = 0) = A1 and absent idler wave a2 (x = 0) = 0. Also, stationary regime ( 0t→ ) is considered. Then, solution of the first two equations of Eq. (S3) is a simple sum of exponents 1 2 1 1 2~xxa c e c e+ , where ( )( )( )2 2 1,2 1 2 2 2 1 2 2 2 1 2 3 1 2 12144 v v v v v v VAvv= − + + + − . (S4) It is clear, that if 3 1 22VA , one of the exponent becomes positive, κ1 > 0, meaning amplification of signal wave. This condition is independent of initial amplitude of the signal spin- wave and is common for any parametric pumping process . To estimate amplification rate, we look on conservative approximation of Eq. ( S3), i.e. neglect the damping. Then, introducing new variables 2/i i j lb Va v v= , j, l i, this system is reduced to standard form: * 1 2 3 * 2 1 3 3 1 2, , .x x xb ib b b ib b b ib b =− =− =− (S5) Exact solution of this system is expressed via elliptic functions and could be found in [3]. Here we note only that this system possesses three integrals of motion (strictly speaking, only two of which are independent), called the Manley -Rowe relations: 22 12( ) ( )b x b x− = const, 22 13( ) ( )b x b x+ = const, and 22 23( ) ( )b x b x+ = const. It is clear, that the powers of signal and idler wave change synchronously – either both increase taking energy from the pumping, either both decrease when the energy flows back to the pumping wave. Also, the power of signal wave cannot be smaller than the the initial power, as the initial power of idler wave is zero. Thus, signal wave is always amplified in the conservative approximation. The maximal power is 222 1 1 3max(0) (0) b b b=+ . Returning to the initial variables, we get the gain 2 22 3 1 1 3max 1va A Av−= , (S6) which is presented in the main text [Eq. (1) ] accounting for the notation of gate, source and drain spin-wave amplitudes. S3. Frequency filtering with nanostrip magnonic crystals Figure S2 . (a) Schematic of magnonic filter: spin waves are generated in the blue area of magnonic waveguide, and filtered out in the dual-band magnonic crystals (labeled as MC 1 and MC 2) region. (b) Intensities of propagating spin wave s at different positions along the waveguide for the indicated frequenc ies of 14.6 , 10.4, and 4.2 GHz . The propagation characteristics of spin waves can be controlled through spatially periodic modulation in magnonic crystals, effectively serving as spin -wave filters due to band gaps [4,5]. As seen in Fig. S2 (a) and Fig. 3 (a) of the main text , we demonstrate the dual-band magnonic filter composed of two parts . One is the 100 -nm wide spin-wave propagation waveguide in the left and the other parts are two magnonic crystals (MC 1 and MC 2) with periodicity P1 = 40 nm and P2 = 50 nm, respectively. Fig. S2 (b) presents the calculated intensities of spin waves propagating along the waveguide . After traveling through MC 1 and MC 2, the spin -wave intensities with frequencies of 14.6 and 10.4 GHz (black and blue lines) are reduced by more than 70% , while spin wave with frequency of 4.2 GHz (red line) can propagate through the two magnonic crystals without additional loss. S4. Dependence of the output spin -wave intensity on the relative phase between the pumping and source magnons For construction of an integrated magnonic circuit, it is hard to accurately control the relative phase between the pumping and source magnons. T o elucidate the relationship between the output spin-wave intensity and the relative phase between the pumping and source magnons, we introduced a variable, denoted as , representing the relative phase between two magnons . Subsequently, we recorded the output spin -wave intensity at 4.2 GHz for various values spanning from 0 ° to 360°. The results , depicted in Fig. S3 , indicate that the output spin -wave intensity exhibits relatively low sensitivity to alterations in the relative phase between the gate and source magnons. The results suggest that the magnonic amplifier based on three -magnon scattering has potential to help to construct an integrated magnonic circuits. Figure S3 The output spin -wave intensity with frequency of 4.2 GHz as function of the varied relative phase between the gate and source magnons Reference: [1] P. Krivosik and C. E. Patton, Phys. Rev. B 82, 184428 (2010). [2] V . Tyberkevych, A. Slavin, P . Artemchuk, and G. Rowlands, ArXiv :2011.13562. [3] N. Bloembergen, Nonlinear optics (Addison -Wesley Pub. Co., 1965). [4] K.-S. Lee, D. -S. Han, and S. -K. Kim, Phys. Rev. Lett. 102, 127202 (2009 ). [5] S.-K. Kim, K. -S. Lee, and D. -S. Han, Appl. Phys. Lett. 95, 082507 (2009).

2023-11-30

Magnonics is a rapidly growing field, attracting much attention for its potential applications in data transport and processing. Many individual magnonic devices have been proposed and realized in laboratories. However, an integrated magnonic circuit with several separate magnonic elements has yet not been reported due to the lack of a magnonic amplifier to compensate for transport and processing losses. The magnon transistor reported in [Nat. Commun. 5, 4700, (2014)] could only achieve a gain of 1.8, which is insufficient in many practical cases. Here, we use the stimulated three-magnon splitting phenomenon to numerically propose a concept of magnon transistor in which the energy of the gate magnons at 14.6 GHz is directly pumped into the energy of the source magnons at 4.2 GHz, thus achieving the gain of 9. The structure is based on the 100 nm wide YIG nano-waveguides, a directional coupler is used to mix the source and gate magnons, and a dual-band magnonic crystal is used to filter out the gate and idler magnons at 10.4 GHz frequency. The magnon transistor preserves the phase of the signal and the design allows integration into a magnon circuit.

Nanoscaled magnon transistor based on stimulated three-magnon splitting

2311.18479v1

arXiv:1301.3266v2 [cond-mat.mes-hall] 8 May 2013Spin-Hall Magnetoresistance in Platinum on Yttrium Iron Ga rnet: Dependence on platinum thickness and in-plane/out-of-plane magnetizat ion N. Vlietstra, J. Shan, V. Castel, and B. J. van Wees University of Groningen, Physics of nanodevices, Zernike In stitute for Advanced Materials, Nijenborgh 4, 9747 AG Groningen, The Netherlands. J. Ben Youssef Universit´ e de Bretagne Occidentale, Laboratoire de Magn´ etisme de Bretagne CNRS, 6 Avenue Le Gorgeu, 29285 Brest, Fra nce. (Dated: September 24, 2018) The occurrence of Spin-Hall Magnetoresistance (SMR) in pla tinum (Pt) on top of yttrium iron garnet (YIG) has been investigated, for both in-plane and ou t-of-plane applied magnetic fields and for different Pt thicknesses [3, 4, 8 and 35nm]. Our experimen ts show that the SMR signal directly depends on the in-plane and out-of-plane magnetization dir ections of the YIG. This confirms the theoretical description, where the SMR occurs due to the int erplay of spin-orbit interaction in the Pt and spin-mixing at the YIG/Pt interface. Additionally, t he sensitivity of the SMR and spin pumping signals on the YIG/Pt interface conditions is shown by comparing two different deposition techniques (e-beam evaporation and dc sputtering). PACS numbers: 72.25.Ba, 72.25.Mk, 75.47.-m, 75.76.+j I. INTRODUCTION Platinum (Pt) is a suitable material to be used as a spin-current to charge-currentconverterdue to its strong spin-orbit coupling.1A spin current injected into a Pt film will generate a transverse charge current by the In- verse Spin-Hall Effect (ISHE), which can then be electri- cally detected. The ISHE has been used to detect for ex- ample spin pumping into Pt from various materials such as permalloy2(Py) and yttrium iron garnet (YIG).3–5 For the opposite effect, to use Pt as a spin current in- jector, a charge current is sent through the Pt, creating a transverse spin accumulation by the Spin-Hall Effect (SHE).6–8 Recently, Weiler et al.9and Huang et al.10observed magnetoresistance (MR) effects in Pt on YIG and re- lated those effects to magnetic proximity. These MR ef- fects havebeen further investigatedby Nakayamaet al.11 and they found and explained a new magnetoresistance, calledSpin-HallMagnetoresistance(SMR).11,12Achange in resistance due to SMR can be explained by a com- bination of the Spin-Hall Effect (SHE) and the Inverse Spin-Hall Effect (ISHE), acting simultaneously. When a charge current /vectorJeis sent through a Pt strip, a trans- verse spin current /vectorJsis generated by the SHE following /vectorJe∝/vector σ×/vectorJs,13–16where/vector σis the polarization direction of the spin current. Part of this created spin current is directed towards the YIG/Pt interface. At this interface the electrons in the Pt will interact with the localized moments in the YIG as is shown in Fig. 1. Depend- ing on the magnetization direction of the YIG, electron spins will be absorbed ( /vectorM⊥/vector σ) or reflected ( /vectorM∝bardbl/vector σ). By changing the direction of the magnetization of the YIG, the polarization direction of the reflected spins, and thus thedirectionoftheadditionalcreatedchargecurrent,can be controlled. A charge current with a component in thedirection perpendicular to /vectorJecan also be created, which generates a transverse voltage. In this paper, the angular dependence of the SMR in Pt on YIG is investigated for different Pt thicknesses (3, 4, 8 and 35nm) and different deposition techniques (e-beam evaporation and dc sputtering), for applied in- plane as well as out-of-plane magnetic field sweeps, re- vealing the full magnetization behaviour of the YIG.17 All measurements are performed at room temperature. The magnitude of the SMR is shown to be dependent on the magnetization direction of the YIG, as well as on the Pt thickness, indicating its relation to the spin diffusion length. Also the used deposition technique is found to be an important factor for the magnitude of the measured signals. MJe MJabsJreflPt Pt YIG YIGa) b) MPt YIG JabsJreflc) Je Je FIG. 1. Schematic drawing explaining the SMR in a YIG/Pt system. (a) When the magnetization /vectorMof YIG is perpen- dicular to the spin polarization /vector σof the spin accumulation created in the Pt by the SHE, the spin accumulation will be absorbed ( /vectorJabs) by the localized moments in the YIG. (b) For /vectorMparallel to /vector σ, the spin accumulation cannot be absorbed, which results in a reflected spin current back into the Pt, where an additional charge current /vectorJreflwill be created by the ISHE. (c) For /vectorMin any other direction, the component of/vector σperpendicular to /vectorMwill be absorbed and the component parallel to /vectorMwill be reflected, resulting in a current /vectorJrefl which is not collinear with the initially applied current /vectorJe.2 II. SAMPLE CHARACTERISTICS Pt Hall bars with thicknesses of 3, 4, 8, and 35nm were deposited on YIG by dc sputtering. Similar Pt Hall bars werealsodepositedonaSi/SiO 2substrate,asareference. FinallyasamplewasfabricatedwherealayerofPt(5nm) was deposited on YIG by e-beam evaporation. Fig. 2(a) shows the dimensions of the Hall bars. The thickness of the deposited Pt layers was measured by atomic force microscopy with an accuracy of ±0.5nm. The used YIG (single-crystal) has a thickness of 200nm and is grown by liquid phase epitaxy on a (111) Gd 3Ga5O12(GGG) substrate. By using a vibrating sample magnetometer, the magnetic field dependence of the magnetization was determined, as shown in Fig. 2(b). The magnetic field dependence shows the same magnetization behaviour for all in-plane directions, indicating isotropic behaviour of the magnetization in the film plane, with a low coercive field of only 0.06mT. To saturate the magnetization of thisYIGsampleintheout-of-planedirection, anexternal magnetic field higher than the saturation field ( µ0Ms= 0.176T)5has to be applied. YIG(111) [200nm] Pt [3, 4, (5), 8, 35 nm] 800µm20 µm 100 µm 100 µma) b) -0.4 -0.2 0 0.2 0.4 -1.0 -0.5 00.5 1.0 -B c M / Ms B [mT] +Bc FIG. 2. (a) Schematics of the used Pt Hall bar geometry. (b) In-plane magnetic field dependence of the magnetization M of the pure single-crystal of YIG. Bcindicates the coercive field of 0.06mT. III. RESULTS AND DISCUSSION A. In-plane magnetic field dependence First, the longitudinal resistance of the Pt strip was measured (using a current I= 100µA) while sweeping an externally applied in-plane magnetic field. For subse- quent measurements the magnetic field was applied for differentin-planeangles α, asdefinedinFig. 3(a). Asthe in-plane magnetization of YIG shows isotropic behaviour with a coercive field Bcof only 0.06mT, its magnetiza- tion will easily align with the applied in-plane magnetic field. It was observed that the measured longitudinal re- sistanceisdependent onthe directionofthe appliedmag- netic field, and thus of the magnetization direction of the YIG, as can be seen in Fig. 3(c) for the YIG/Pt [4nm] sample. For clarity, a background resistance R0of 1007- 1008Ω was subtracted in the plots (the small change in R0between different measurementsoccurreddue to ther- mal drift). A maximum in resistance was observed whenthe magnetic field was applied parallel to the direction of the charge current Je(α= 0◦). The resistance was min- imized for the case where BandJewere perpendicular (α= 90◦). These results are consistent with the SMR as described by Fig. 1 and as observed by Nakayama et al.11. The measured resistivity for the longitudinal configuration can be formulated as11 ρL=ρ0−∆ρmy2(1) whereρ0is a constant resistivity offset, ∆ ρis the mag- nitude of the resistivity change, which can be calculated from the measurements, giving ∆ ρ= 2×10−10Ωm, and myis the component of the magnetization in the y- direction. Thesameexperimentswererepeatedforthetransverse resistance, where the resistance wasmeasured perpendic- ular to the current path as shown in Fig. 3(b). Also in this configuration it was found that the measured resis- tance depends on the direction of the applied in-plane magnetic field, as shown in Fig. 3(d) for the YIG/Pt [4nm] sample. Here a maximum resistance is observed forα= 45◦, and a minimum for α= 135◦. The observed SMR resistivity for the transverse configuration can be formulated as11 ρT= ∆ρmxmy (2) wheremxis the component of the magnetization in thex-direction. From the shown measurements, a ratio ∆RL/∆RT≈7 is found, which is close to the expected ratio of 8 following from equations (1) and (2). For both the longitudinal and the transverse configu- ration, there is a peak and/or dip observed around + Bc for all measurements. This can also be explained by the above described SMR. While sweeping the magnetic field (here from negative to positive B), the magnetization of theYIG willchangedirectionwhenpassing+ Bc(seeFig. 2(b)). Due to its in-plane shape anisotropy, the magne- tization of the YIG will rotate fully in-plane towards B. This rotation of Mresults in a change in measured resis- tance, passing the maximum and/or minimum possible resistance,whichisobservedasapeakand/ordiparound +Bc(when sweeping the field from positive to negative B, a peak/dip will occur at - Bc). Similar features were not observed by Huang et al.10and Nakayama et al.11. They do observe some peaks and dips, but these do not cover the maximum and minimum possible resistances, and thus do not show the full rotation of the magnetiza- tion in the plane. The absence of the full peaks and dips can be explained by different magnetization behaviour of their YIG samples, showing higher coercive fields and switching of the magnetization which is probably domi- nated by non-uniform reversal processes. Theresistancemeasurementsforthe in-planemagnetic fields were repeated for all different samples. A sum- mary of these measurements is shown in Fig. 3(e). Here ∆RLis defined as the difference between the maximum (α= 0◦) and minimum ( α= 90◦) measured longitudinal3 a) b) YIG(111) [200nm] V -B α YIG(111) [200nm] JeV -B α α = 0° α = 45° α = 90° α = 135° α = 180°α = 0° α = 45° α = 90° α = 135° α = 180°d) c) e) 400420440460 400420440460 400420440460 400420440460 -3 -2 -1 0 1 2 3400420440460 RT [mΩ] B [mT] ∆RT∆RL 0 5 10 15 20 25 30 35 40 4502468 SiO2/Pt: Pt Sputtered YIG/Pt: Pt Sputtered Pt Evaporated∆R L / R0 Pt thickness [nm]x 10 - 4-0.4-0.3-0.2-0.10.0 -0.4-0.3-0.2-0.10.0 -0.4-0.3-0.2-0.10.0 -0.4-0.3-0.2-0.10.0 -3 -2 -1 0 1 2 3-0.4-0.3-0.2-0.10.0 R L - R0 [Ω] B [mT]xy Je FIG. 3. Results of the in-plane magnetic field dependence of the resistance of the Pt strip with a thickness of 4nm. Config- uration for (a) longitudinal and (b) transverse resistance mea- surements. (c) and (d) show the measured resistance of the Pt strip while applying an in-plane magnetic field for differe nt anglesα, for thelongitudinal and transverse configuration, re- spectively. R0has a magnitude of 1007-1008Ω. (e) Thickness dependence of the measured magnetoresistance for YIG/Pt and SiO 2/Pt samples. ∆ RLis defined as the maximum differ- ence in longitudinal resistance ( RL(α= 0◦)−RL(α= 90◦)) andR0isRL(α= 0◦). The solid red line is a theoretical fit.11,12 resistance and R0isRL(α= 0◦). The shown thickness dependent measurements are in agreement with data as published by Huang et al.10, though they do not relate their results to SMR. The red line shows a theoretical fit11,12of the SMR signal. The position and width of the peak are mostly determined by the spin relaxationlength λofPt, andthemagnitudeofthesignalbyacombination of the spin-Hall angle θSHand the spin-mixing conduc-tanceG↑↓of the YIG/Pt interface. For the shown fit, λ= 1.5nm,θSH= 0.08,G↑↓= 1.2×1014Ω−1m−2and a thickness dependent electrical conductivity as used in Ref18, were used. When YIG is replaced by SiO 2, the SMR signal to- tally disappears, showing the effect is indeed caused by the magnetic YIG layer. More notable, the e-beam evap- orated Pt layer on YIG did show only a very low SMR signal (≈10−5). This suggests that the spin-mixing con- ductance (which is determined by the interface)19is an important parameter for the occurrence of SMR. B. Out-of-plane magnetic field dependence To further investigate the characteristics of the Pt layer, also the transverse resistance was measured while applying an out-of-plane magnetic field, as shown in Fig. 4(a). The Pt layers on the Si/SiO 2substrate showed linear behaviour with transverse Hall resistances of 1.3, 0.9 and 0.3 ±0.05mΩ for Pt thicknesses of 4, 8 and 35nm, respectively, at B= 300mT. These re- sults, due to the normal Hall effect, are in agreement with the theoretical description RHall=RHB/d, where RH=−0.23×10−10m3/C is the Hall coefficient of Pt20 anddis the Pt thickness. For the YIG/Pt samples, results of the out-of-plane measurements are shown in Fig. 4(b). At fields lower than the saturation field, a large magnetic field depen- dence is observed. The magnitude of this dependence de- creases with Pt thickness and disappears for the thickest Pt layer of 35nm. The occurrence of this magnetic field dependence can be explained by the SMR, using the re- sults of the in-plane measurementsas shownin Fig. 3(d), because for applied fields lower than the saturation field, the magnetization of the YIG will still have an in-plane component. To investigate its effect on the transverse resistance measurements, the direction of the in-plane magnetization in the YIG should be known. To achieve this, the external magnetic field was applied with a small intended deviation φfrom the out-of-plane z-direction towards the - y-direction as defined in Fig. 4(a). This small deviation results in a small in-plane component of the applied field, which controls the magnetization di- rection of the YIG. Using this configuration the sign of the signal due to the SMR can be checked according to Fig. 3(d) by varying the direction of the in-plane com- ponent of the applied magnetic field. Fig. 4(c) shows results applying an external field fixing φ=−1◦for var- ious angles θ, whereθis an additional rotation from the z- towards the x-direction. According to the theory of the SMR and also comparing the results shown in Fig. 3(c), a maximum additional resistance due to SMR is expected for an in-plane magnetic field with α= 45◦, which is the direction of the in-plane component when applying a magnetic field choosing φ=−1◦andθ= 1◦. Similarly, for φ=θ=−1◦, the in-plane component of the field will be α= 135◦, resulting in a minimum addi-4 a) b) c) YIG(111) [200nm] θ-B V Jeϕ d) -300 -200 -100 0 100 200 300-30-20-100102030 ϕ = -1 ο -10 ο -1 οRT [mΩ] B [mT] 1 ο 15 ο 45 ο 90 οθ = YIG/Pt [4nm]z - yx -300 -200 -100 0 100 200 300-10010203040506070θ = 1 ο ϕ = -1 ο YIG/Pt: 3 nm 4 nm 8 nm 35 nmRT [mΩ] B [mT] -300 -200 -100 0 100 200 300-30-20-100102030 θ =-1 ο ϕ = -1 ο RT [mΩ] B [mT]θ =1 ο YIG/Pt [4nm] FIG. 4. Results of the out-of-plane magnetic field dependenc e of the transverse resistance. (a) Configuration for the tran s- verse resistance measurements. φis definedas a rotation from thez- towards the - y-direction, whereas θgives a rotation from the z- towards the x-direction. (b) Magnetic field de- pendence of the transverse resistance for different thickne sses of Pt on top YIG, for φ=−1◦andθ= 1◦. (c) Dependence of the transverse resistance on θ, fixingφ=−1◦, pointing out the effect of the direction of the in-plane component of the ap - plied magnetic field on the observed signal. (d) Theoretical fitsoftheSMRsignal forout-of-planeappliedfieldslowerth an the saturation field, assuming a linear background resistan ce, as shown by the dotted red line. For all shown measurements, a constant background resistance of 10-900mΩ is subtracted . tional resistance. Results as shown in Fig. 4(c) confirm that the sign and magnitude of the magnetic field depen- dence are consistent with the SMR observed for in-plane fields. The shape of the curve can be explained by the dependence of the resistance on the direction of M, as only the component of σparallel to M(σM) will be re- flected. For out-of-plane applied fields, σMis given by σM=σcosβcosα, whereβis the angle by which Mis tilted out of the x/y-plane. Using the Stoner-Wohlfarth Model,21for an applied field in the z-direction, it was derived that β= arcsin( b), where b=B/BsandBsis the saturation field. Assuming that the transverse re- sistivity change due to SMR scales linearly with the in- plane component of σM(σM,in−plane=σMcosβ), this gives (for applied fields close towards the z-direction and φ=θ=±1) ρT=±1 2∆ρ(1−b2) (3) Two fits using this equation are shown in Fig. 4(d). For both fitted curves, an assumed linear background re- sistance,asindicatedbythedottedredline, isalsoadded. Thederivedfitsareingoodagreementwiththemeasured data for applied fields below the saturation field, which confirms the presence of SMR and its dependence on the magnetization direction,12,22,23also for out-of-plane ap- plied fields. Also for the out-of-plane measurements a peak and/or a dip is observed at zero applied field. These peaks anddips have the same origin as those observed for the in- plane measurements, which is the rotation of the mag- netization in the plane towards the new magnetic field direction. For applied magnetic fields above the saturation field no in-plane component of M is left, but still a small mag- neticfielddependenceisobserved. At B= 300mT,trans- verse resistances of 10.1, 5.1, 1.5 and 0.3 ±0.05mΩ were measured for Pt thicknesses of 3, 4, 8 and 35nm, respec- tively. So for thin Pt layers, at applied fields above the saturation field, an increased transverse resistance is ob- served compared to the SiO 2/Pt sample. Possible origins of this difference might be related to the imaginary part of the spin-mixing conductance, or to the (spin-) anoma- lous Hall effect. C. Comparison of e-beam evaporated and dc sputtered Pt Additional to the thickness and angular dependence of the SMR signal, also the difference in signal for two deposition techniques, e-beam evaporation and dc sput- tering was investigated. It was observed that the e-beam evaporated Pt layer did show very low SMR effects com- pared to the sputtered layers. To compare, Fig. 5(a) shows the out-of-plane transverse measurement for both the sputtered [4nm] and evaporated [5nm] Pt layers. The value of the signal at applied fields higher than the satu- ration field is the same, but the additional signal which is described to SMR is lowered by a factor 7. As the evaporated Pt layer showed lower SMR signals compared to the sputtered Pt layers, the effect of using a differentdepositiontechniqueonthe spinpumping/ISHE signalwas also investigated. By using a rf-magnetic field, the magnetization of the YIG was brought into reso- nance. During resonance, a spin current will be pumped into the Pt layer where it will be converted in a charge current by the ISHE. A more detailed description of the used measurement technique can be found in ref.5. Fig. 5(b) shows a measurement of the spin pumping voltage for both e-beam evaporated Pt and dc sputtered Pt on YIG. A rf-frequency and power of 1.4GHz and 10mW, respectively, were used to excite the magnetization pre- cession in the YIG. The same measurement was repeated for different rf-frequencies between 0.6 and 4GHz, all at a power of 10mW (not shown). For all measurements, the spin pumping signalofthe evaporatedPt layerwasfound to be a factor 12 smaller than the signal of the sputtered layer. This change in magnitude of the signal shows the difference of the YIG/Pt interface between both deposi- tion techniques, determining a probable difference in the spin-mixing conductance. As e-beam evaporation is a much softer deposition technique compared to dc sput- tering, the spin-mixingconductance at the YIG/Pt inter- face might be lower in case of evaporation, resulting in less spin pumping.19Also the structure of the Pt layers might be different, resulting in different spin-Hall angles5 and/or different spin diffusion lengths. YIG/Pt: 4 nm (S) 5 nm (E) -0.5 0 0.5 1024681012141618 VISHE [µV] B - Bres [mT] -300 -200 -100 0 100 200 300-5051015202530θ = 1 ο ϕ = -1 ο YIG/Pt: 4 nm (S) 5 nm (E) RT [mΩ] B [mT]a) b) FIG. 5. Comparison of (a) transverse resistance for an out-o f- plane applied magnetic field, and (b) spin pumping/ISHE sig- nal (using an rf-frequency of 1.4GHz with a power of 10mW) for Pt on top of YIG, deposited by e-beam evaporation (E) and dc sputtering (S). IV. SUMMARY In summary, the SMR in Pt layerswith different thick- nesses [3, 4, 8 and 35nm], deposited on top of YIG, was investigated for both in-plane and out-of-plane applied magnetic fields. In-plane magnetic field scans clearly show the presence of SMR for the transverse as well as the longitudinal configuration. Out-of-plane measure- ments present a magnetic field dependence which canalso be assigned to the SMR. The sign and magnitude of the SMR signal are shown to be determined by the magnetization direction of the YIG. Further, thickness dependence experiments show that the SMR signal de- creases in magnitude when increasing the Pt thickness. No SMR signals were observed for SiO 2/Pt samples. For Pt layers deposited by e-beam evaporation, in stead of dc sputtering, the found SMR signals are decreased by a factor 7. Also spin pumping experiments show reduced signals for e-beam evaporated Pt compared to sputtered Pt. The difference in spin pumping signals and SMR sig- nals show the possible importance of the YIG/Pt inter- face, and connectedtothis, the spin-mixingconductance, for this kind of experiments. ACKNOWLEDGEMENTS We would like to acknowledge B. Wolfs, M. de Roosz and J. G. Holstein for technical assistance and prof. dr. ir. G. E. W. Bauer for useful comments regardingthe ex- planation of the measurements. This work is part of the researchprogram(Magnetic Insulator Spintronics) of the Foundation for Fundamental Research on Matter (FOM) andis supported byNanoNextNL, a microandnanotech- nology consortium of the Government of the Netherlands and 130 partners, by NanoLab NL and the Zernike Insti- tute for Advanced Materials. 1K. Ando, Y. Kajiwara, K. Sasage, K. Uchida, and E. Saitoh, IEEE Transactions on Magnetics 46, 3694 (2010). 2E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Applied Physics Letters 88, 182509 (2006). 3K. Ando, S. Takahashi, J. Ieda, Y. Kajiwara, H. Nakayama, T. Yoshino, K. Harii, Y. Fu- jikawa, M. Matsuo, S. Maekawa, and E. Saitoh, Journal of Applied Physics 109, 103913 (2011). 4H. Kurebayashi, O. Dzyapko, V. E. Demidov, D. Fang, A. J. Ferguson, and S. O. Demokritov, Nat. Mater. 10, 660 (2011). 5V. Castel, N. Vlietstra, B. J. van Wees, and J. B. Youssef, Phys. Rev. B 86, 134419 (2012). 6K. Ando, S. Takahashi, K. Harii, K. Sasage, J. Ieda, S. Maekawa, and E. Saitoh, Phys. Rev. Lett. 101, 036601 (2008). 7L. Liu, O. J. Lee, T. J. Gudmundsen, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett. 109, 096602 (2012). 8E. Padr´ on-Hern´ andez, A. Azevedo, and S. M. Rezende, Applied Physics Letters 99, 192511 (2011). 9M. Weiler, M. Althammer, F. D. Czeschka, H. Huebl, M. S. Wagner, M. Opel, I.-M. Imort, G. Reiss, A. Thomas, R. Gross, and S. T. B. Goennenwein, Phys. Rev. Lett. 108, 106602 (2012). 10S. Y. Huang, X. Fan, D. Qu, Y. P. Chen, W. G. Wang, J. Wu, T. Y. Chen, J. Q. Xiao, and C. L. Chien, Phys. Rev. Lett. 109, 107204 (2012).11H. Nakayama, M. Althammer, Y.-T. Chen, K. Uchida, Y. Kajiwara, D. Kikuchi, T. Ohtani, S. Gepr¨ ags, M. Opel, S. Takahashi, R. Gross, G. E. W. Bauer, S. T. B. Goen- nenwein, and E. Saitoh, Phys. Rev. Lett. (in press), ArXiv e-prints (2012), arXiv:1211.0098 [cond-mat.mtrl-sci]. 12Y.-T. Chen, S. Takahashi, H. Nakayama, M. Althammer, S. T. B. Goennenwein, E. Saitoh, and G. E. W. Bauer, Phys. Rev. B 87, 144411 (2013). 13T. Kimura, Y. Otani, T. Sato, S. Takahashi, and S. Maekawa, Phys. Rev. Lett. 98, 156601 (2007). 14H. Nakayama, K. Ando, K. Harii, T. Yoshino, R. Taka- hashi, Y. Kajiwara, K. Uchida, Y. Fujikawa, and E. Saitoh, Phys. Rev. B 85, 144408 (2012). 15A. Azevedo, L. H. Vilela-Le˜ ao, R. L. Rodr´ ıguez- Su´ arez, A. F. Lacerda Santos, and S. M. Rezende, Phys. Rev. B 83, 144402 (2011). 16Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi, S. Maekawa, and E. Saitoh, Nature (London) 464, 262 (2010). 17Nakayama et al.11also investigated the out-of-plane be- havior of the SMR, but only for saturated magnetization directions, which are fully aligned to the applied field. 18V. Castel, N. Vlietstra, J. Ben Youssef, and B. J. van Wees, Applied Physics Letters 101, 132414 (2012). 19C. Burrowes, B. Heinrich, B. Kardasz, E. A. Mon- toya, E. Girt, Y. Sun, Y.-Y. Song, and M. Wu, Applied Physics Letters 100, 092403 (2012).6 20C. M. Hurd, The Hall Effect in Metals and Alloys (Plenum Press, New York, 1972). 21E. Stoner and E. Wohlfarth, IEEE Transactions on Magnetics 27, 3475 (1991). 22M. Althammer, S. Meyer, H. Nakayama, M. Schreier, S. Altmannshofer, M. Weiler, H. Huebl, S. Gepr¨ ags, M. Opel, R. Gross, D. Meier, C. Klewe, T. Kuschel,J.-M. Schmalhorst, G. Reiss, L. Shen, A. Gupta, Y.-T. Chen, G. E. W. Bauer, E. Saitoh, and S. T. B. Goennenwein, ArXiv e-prints (2013), arXiv:1304.6151 [cond-mat.mes-hall]. 23C. Hahn, G. De Loubens, O. Klein, M. Viret, V. V. Naletov, and J. Ben Youssef, ArXiv e-prints (2013), arXiv:1302.4416 [cond-mat.mes-hall].

2013-01-15

The occurrence of Spin-Hall Magnetoresistance (SMR) in platinum (Pt) on top of yttrium iron garnet (YIG) has been investigated, for both in-plane and out-of-plane applied magnetic fields and for different Pt thicknesses [3, 4, 8 and 35nm]. Our experiments show that the SMR signal directly depends on the in-plane and out-of-plane magnetization directions of the YIG. This confirms the theoretical description, where the SMR occurs due to the interplay of spin-orbit interaction in the Pt and spin-mixing at the YIG/Pt interface. Additionally, the sensitivity of the SMR and spin pumping signals on the YIG/Pt interface conditions is shown by comparing two different deposition techniques (e-beam evaporation and dc sputtering).

Spin-Hall Magnetoresistance in Platinum on Yttrium Iron Garnet: Dependence on platinum thickness and in-plane/out-of-plane magnetization

1301.3266v2

1 Imaging Magnetization Structure and Dynamics in Ultrathin YIG/Pt Bilayers with High Sensitivity Using the Time -Resolve d Longitudinal Spin Seebeck Effect Jason M. Bartell1, Colin L. Jermain1, Sriharsha V. Aradhya1, Jack T. Brangham2, Fengyuan Yang2, Daniel C. Ralph1,3, Gregory D. Fuchs1 1Cornell University, Ithaca , NY 14853 , USA 2Department of Physics, The Ohio State University, Columbus , OH 43016, USA 3Kavli Institute at Cornell for N anoscale Science, Ithaca, NY 14853, USA Abstract We demonstrate an instrument for time -resolved magnetic imaging that is highly sensitive to the in-plane magnetization state and dynamics of thin-film bilayers of yttrium iron garnet (Y3Fe5O12,YIG)/Pt: the time -resolved longitudinal s pin Seebeck (TRLSSE) effect microscope. We detect the local , in-plane magnetic orientation within the YIG by focusing a picosecond laser to generate thermally -driven spin current from the YIG into the Pt by the spin Seebeck effect, and then use the inverse spin Hall effect in the Pt to transduce this spin current to an output voltage. To establish the time resolution of TRLSSE , we show that pulsed optical heating of patterned YIG (20 nm)/ Pt(6 nm)/Ru (2 nm) wires generates a magnetization -dependent voltage pulse of less than 100 ps. We demonstrate TRLSSE microscopy to image both static magnetic structure and gigahertz -frequency magnetic resonance dynamics with sub-micron spatial resolution and a sensitivity to magnetic orientation below 0.3 deg/ √𝐻𝑧 in ultrathin YIG. 2 Main Text Ultrathin bilayers of the magnetic insulator YIG interfaced with a heavy , non-magnetic metal (NM) such at Pt are being intensely studied for the development of high -efficiency magnetic memory and logic devices operated by spin -orbit torque [1,2] , for magnon generation and propagation [3–5], and as a model system for understanding spin -current generation by the longit udinal spin Seebeck effect (LSSE) and spin pumping [6–9]. For all of these research areas , it would be useful to have a high -sensitivity and local probe of magnetization dynamics in the YIG layer , especially for the ultrathin films required in many devices . This has proven challenging , and although m agneto -optical techniques such as Brillouin light scattering and the magneto -optical Kerr effect (MOKE) have proven valuable [3,10 –14], they have not enabled direct time -resolved imaging of magnetic precession or direct imag ing of in-plane magnetization of ultra -thin YIG films (20 nm and below) . An alternative approach that enables in -plane imaging of YIG/Pt bilayer devices was demonstrated by Weiler et al. [15]. In that work, the authors use laser heating to image the in -plane magnetic structure of YIG , but not its dynamics . Here we extend the approach into the time domain to perform high sensitivity imaging of the in - plane magnetic orientation (< 0.3°/√𝐻𝑧) with sub -micron spatial resolution and sub -100 ps temporal resolution. Using TR LSSE microscopy we can observe, for example, that the resonance field in ultra -thin YIG films can vary by up to 30 Oe within micron -scale regi ons of a YIG/Pt device . Our results d emonstrate that TR LSSE microscopy is a powerful tool to characterize static and dynamic magnetic properties in ultrathin YIG. The principle behind the TRLSSE microscope , shown schematically in Fig. 1, is the generatio n and detection of a thermally generated local spin current [16]. For the case of YIG/Pt , a local thermal gradient perpendicular to the film plane is generated by laser heating of Pt. The 3 gradient creates a thermally -induced spin current that is proportional to th e local magnetization [17–19]. The spin current that flows into the Pt is detected with the ISHE [20,21] in which spin -orbit coupling leads to a spin-dependent transverse electric field . For this work, the resulting voltage can be described as [17,19] 𝑉𝐿𝑆𝑆𝐸 ∝ − 𝜉𝑆𝐻 𝑆𝐌(𝒙,𝑡) 𝑀𝑠×𝛁𝐓(𝒙,𝑡), where, 𝜉𝑆𝐻 is the spin Hall efficiency, S is the spin -Seebeck coefficient, M is the local magnetization, Ms is the saturation magnetization and 𝛁𝐓 is the thermal gradient . The LSSE has been attributed to both thermal gradien ts across the thickness of YIG and to interfacial temperature differences between YIG and Pt [17–19,22,23] . Our experiment cannot definiti vely distinguish between these two mechanisms . Thus , here we discuss only 𝛁𝐓 as single quantity for simplicity and for consistenc y with our prior work using the anomalous Nernst effect , however, this question requires further study . VLSSE is a read -out of the local magnetization my because the electric field is generated in response to the spatially local z-component of the thermal gradient, ∇𝑇𝑧 (coordinates as defined in Fig. 1) [15,24] . To extend LSSE imaging into the time -domain, we use picosecond laser heating to stroboscopic ally sample magnetization. We have previously shown , in metallic ferromagnets, that picosecond heati ng can be used for stroboscopic magnetic microscopy using the time- resolved anomalous Nernst effect (TRANE) [25]. In TRANE microscopy , the temporal resolution is set by the excitation and decay of a thermal gradient within a single material that both absorbs the heat from the laser pulse and produces a TRANE voltage from internal spin - orbit interactions [26,27] . In the LSSE however, the timescale of spin current generation can depend on both the timescale of the thermal gradient and the timescale of energy transfer between the phonons and magnons. Recent experiments indicate that in the qausi -static regime the magnon -phonon relaxation rate may play a dominant role [28–31]. Using picosecond heating 4 and time -resolved electrical det ection to move beyond the quasi -static regime , we show a TRLSSE in agreement with a recent all -optical experiment [22]. We grew our samples using off-axis sputtering onto (110) -oriented gadollinum gallium garnet (Gd 3Ga5O12, GGG), [32–34] followed by ex situ . deposition of 6 nm of Pt with a 2 nm Ru capping layer. Photolithography and ion milling were used to pattern wires and contacts for wirebonding . We present measurements of a 2 µm × 10 µm wire and a 4 µm × 10 µm wire with DC resistances of 296 Ω and 111 Ω respectively. In this room temperature study , we neglect the potential anomalous Nernst effect of interfacial Pt with induced magnetization [35,36] , and we neglect a poss ible photo -spin voltaic effect [37], neither of which can be distinguished from TRLSSE in presented measurements. Our TRLSSE measurement consists of pulsed laser heating and homodyne electrical detection as shown in Fig. 2a. We use a Ti:Sapphire laser pulse to locally heat the s ample w ith 3 ps pulses of 780 nm light at a repetition rate of 25.5 MHz . The electrical signal produced at the sample is the sum of the LSSE dependent voltage, 𝑉𝐿𝑆𝑆𝐸 (∇𝑇𝑧,𝐌), and a voltage, 𝑉𝐽(Δ𝑇,𝐽), which is generated when a current density J is passing through the local region of Pt with increased resistance due to laser heating [38]. To reject noise and recover the signal of the resulting electrical pulse s, we use a time-domain h omodyne technique in which we mix the VLSSE + VJ pulse train with a synchronized reference pulse train, Vmix, in a broadband (0.1 -12 GHz) electrical mixer. The mixer output is the convolution of the two pulse trains given by [38] 𝑉𝑠𝑖𝑔(𝒙,𝜏)=𝛫∫(𝑉𝐿𝑆𝑆𝐸 (𝛻𝑇𝑧(𝒙,𝑡),𝐌(𝒙,𝑡))+ 𝑉𝐽(𝛥𝑇(𝒙,𝑡),𝐽(𝒙,𝑡)) 𝑉𝑚𝑖𝑥(𝜏−𝑡)𝑑𝑡Γ 0, (1) 5 where x(x,y) is the laser spot position in the sample plane , Γ is the period of the laser pulses, 𝛫 is the transfer coefficient , and 𝜏 is the relative delay. A relative delay of zero corresponds to the maximum of both pulse train s arriving at the mixer simultaneously. We study the timescale of the LSSE signal generated by a picosecond pulse by measurin g Vsig as a function of mixer delay 𝜏. Fig. 2b shows the result of this measurement using a 100 ps mixing pulse reference, Vmix, at a saturating magnetic field, H, perpendicular to the wire at H = +414 Oe and – 414 Oe , respectively. In Figure 2 c we plot the difference between these two voltage traces to reject non -magnetic contributions. We find that the full-width at half -maximum (FWHM) is 100±10 ps, which is followed by electrical oscillations that we attribute to non - idealities in the detection circuit (see the SI for further discussion.) Because the duration of the magnetic component of Vsig is experimentally indistinguishable from the FWHM of Vmix, we conclude that 100 ps is an experimental upper bound for the TR LSSE signal duration . To our knowledge, this is the first direct electrical measurement of picosecond duration LSSE voltages. To calibrate the local change in the Pt temperature , ΔTPt, due to picosecond heating and to quantify the rate of thermal relaxation , we measur e VJ in the presence of a DC current , which uses the local Pt resistivity as an ultra -fast thermometer. Figure 2 d shows VJ as a f unction of mixer delay, VJ(τ) = Vsig(τ, J = 4.2 MA/cm2) – Vsig(τ, J = -4.2 M A/cm2), for applied currents of ±0.5 mA. VJ (τ) is proportional to ΔTpt through VJ, but it is not proportional to either the magnetic state of the sample or ∇𝑇𝑧. We observe that VJ relaxes to zero faster than the laser repetition period , indicating that the sample thermally recovers between pulses. To quantitatively consider the spatiotemporal thermal evolution , we performed a time-domain finite element (TDFE) calculation of focused laser heating in the wire . Additional details are available in the SI, and se e Ref . [25] for a lengthier discussion of the procedure. The comparison of the 6 spatiotemporal profile of the calculation and the known temperature dependence of resistivity enable us to calibrate the spatiotemporal temperature rise due to laser heating. We find that the peak film temperature changes by ~50 K in the platinum and ~ 10 K in the YIG for a laser fluence of 5.8 mJ/cm2, which is the maximum fo r the presented measurements. Note that we assume all laser heating is mediated by optical absorption in Pt because YIG and GGG are transparent at 780 nm [39,40] . The TDFE calculation reveals that , in agreement wi th experiment, ∇𝑇𝑧 across the YIG thickness decays more quickly than the full thermal relaxation of the Pt back to the ambient temperature (e.g. ΔTpt = 0). This difference in timescales between ∇𝑇𝑧 and ΔTpt is important because the magnetic signal in our experiment is sensitive to only ∇𝑇𝑧(𝑡), not ΔTpt (t) of the Pt . The s ub-100 ps spin current lifetime in our experiment is short enough that the TRLSSE is useful for stroboscopic measurements of resonant YIG magnetization dynamics. To confirm this idea, we use TRLSSE microscopy to measure ferromagnetic resonance (FMR) by driving a gigahertz -frequency a.c. current into the Pt, which generates magnetic torques on YIG from both the Oersted magnetic field and from spin currents generate d by the spin Hall effect [41–43]. The current is generated with an arbitrary waveform generator (AWG ) that is phase -locked to the laser repetition rate and coupled to the YIG/Pt device through a circulator (see schematic in Fig. 3a). Synchronizing the a.c. current and the laser repetition rate ensures a constant but controllable phase between the precessing magnetization and the sensing heat pulse for a given driving frequency and magnetic field . In our FMR measurements , we fix 𝜏 = 0 and align the wire axis parallel to the external magnetic field . In this configuration, the TRLSSE signal is stroboscopically sensitive to the magnetic projection my at a particular phase of the magneti c precession about the x-axis. In addition to VLSSE, Vsig contains a contribution from VJ that is 7 proportional to the local a.c. current amplitude and phase [38]. We separate the magnetic VLSSE from the non -magnetic VJ by measuring Vsig with a lock -in amplifier referenced t o a 383 Hz , 7.6 Oe RMS modulation of the external magnetic field. Fig. 3 b shows LSSE FMR spectra as a function of field that is excited using a 0.5 mA a.c. current at 4.1 and 4.9 GHz. In the limit that the modulation magnetic field is small compared to the FMR linewidth, we can interpret the resulting signal Vmod as a derivative signal that contains a linear combination of the real and imaginary parts of the dynamic susceptibility , 𝜒, 𝑉𝑚𝑜𝑑(𝐻)∝𝑑𝜒′ 𝑑𝐻𝑆𝑖𝑛(𝜃)+𝑑𝜒′′ 𝑑𝐻𝐶𝑜𝑠(𝜃). This relation is used to fit the FMR spectra to extract the amplitude, phase , linewidth, and resonant field. For mo re details on fitting see refs [25,38] . To demonstrate that the TRLSSE microscope is a phase -sensitive stroboscope, we rotate d the phase of the microwave current by 180° and re - measure FMR . As expected, inverting the phase of the drive in verts the phase of the FMR lineshape (Fig. 3 c). Next, we quantify the sensitivity of TRLSSE microscopy for our ultra-thin YIG/Pt samples . Figure 4 shows representative LSSE measurements of the YIG magnetization versus magnetic field perpendicular to the wire at several optical powers . In this geometry, t he positive and negative saturation value s of VLSSE quantify the full range of magnetization, +M to –M. The n, using the standard deviation of the noise in the LSSE voltage, 𝜎𝐿𝑆𝑆𝐸, we can quantify the angular sensitivity noise floor assuming small angle magnetic deviations from the wire axis , such as for stroboscopic FMR measurements . The sensitivity is calculated using [25] 𝜃min= 𝜎LSSE sin(𝜃o)(𝑉LSSEmax−𝑉LSSEmin)/2√𝑇𝐶 where TC is the lock-in time constant . We find a sensitivity of 0.3 deg/√Hz for an optical power of 0.6 mW, corresponding to a laser fluence of 5.8 mJ/cm2. It is 8 important to note that the sensitivity is sample dependent through both sample geometry and the impedance match with the detection circuit [25]. The i nterface quality of the sample plays a key role in determining the sensitivity . As spin current diffuses into the platinum , it is subject to loss at the interface. A good indication of interfacial spin transparency is the spin Hall magnetoresistance ( SMR ) [44,45] , which is sensitive to the spin mixing conductance at the interface. For the data presented here, the devices show a SMR of 0.063%, which is the largest value by a factor of 2 from the other devices we patterned . This is consistent with a number of recent SMR reports [44–48], and we expect the high SMR value indicates strong spin transparency at the YIG/Pt interface . We also studied YIG/Pt samples with no measureable SMR which we expect to have a significantly reduced LSSE induced ISHE voltage . We found that the LSSE signal in these devices is approximately an order of magnitude lower for the same laser fluence . Additional details are in the SI. Having placed upper bounds on the time resolution and quantified the sensitivity, next we demonstrate the application of TRLSSE microscopy for imaging of static magnetization . We acquire i mages by scanning the laser focus and making a point -by-point measurement of the TRLSSE voltage and reflect ed light . Figures 5a and 5b show a reflected light image and saturated LSSE image, respectively , for a 4 μm wide YIG/Pt device . In the ref lection image , we see the structure of the wire and the contact pads at both ends . We acquired the TRLSSE image at H =–405 Oe and shifted the background level for clarity of the color scale. N o other image processing was performed . We observe a uniform magnetization state of the YIG/Pt device , as expected from the previously presented ma gnetic hysteresis measurements (Fig. 4 ). When we reduc e the field to near zero ( H = 4 Oe ) and re-imag e the wire (Fig. 5c ), magnetic texture is revealed that indicates non-uniform canting of the device magnetization. To more clearly show 9 the variation in contrast between images , we plot l ine cuts of Figs. 5a -c in Fig. 5d. Despite the inhomogeneous remanence that is evident in Fig. 5c, we were not able to observe domains with oppositely aligned magnetization ; possibly because once a reversal domain is nucleated, the domain wall propagate s without strong pinning . Without a 180o domain wall the spatial resolution of TRLSSE cannot be directly evaluated . Nevertheless , we use the reflected light image and TDFE simulat ions to study the possibility that lateral thermal spreading degrades the res olution. To approximate the lateral point spread function of the laser, we fit a scan of the wire step edge to a Gaussian point spread function . This yields a spot FWHM of 0.606 μm. Calculations of the heating indicate that the thermal gradient does not spread laterally in the Pt, thus we expect that the resolution of the TRLSSE is the same as the diffraction -limited optical resolution in this experiment . We now demonstrate that TRLSSE microscopy has the sensitivity to image dynamic magnetization in the 4 μm YIG/Pt device , which provides quantitative and spatially localized information about dynamical properties of ultrathin YIG materials . As described above , for FMR characterization we orient the external magnetic field parallel to the wire axis a nd drive a 1.1 mA , 4.9 GHz current into the wire. We image dynamical magnetization at a series of magnetic fields near the resonance field, from H = 896 Oe to 1105 Oe, and pl ot a selection of the unprocessed images in Figs. 5e-g. The data show that at H far from resonance (Fig. 5 e) where precession amplitudes are tiny , the TRLSSE signal at the center of the wire is well below the detection noise floor. There is a small , current -induced , non-magnetic signal artifact at the edge s of the wire which we discuss further in the supplementary information. For H near the resonant field, Hres, the device has a strong , position -dependent TRLSSE response . To quantitatively analyze the data, images are correct ed for background offset and sample drift before fitting a 10 resonance field curve for each pixel . We plot a selection of curves from individual pixels i n Fig. 6a. We then construct a spatial map of each fitting parameter : Hres, relative phase, 𝜙, amplitude, A, and linewidth, ΔH, and offset, all of which are shown in Fig. 6b-f. We immediately notice spatial variation in these images that is qualitatively similar to the non-uniform magnetic remanence texture shown in Fig . 5c. Together, these measurements confirm the presence of varying local magnetic anisotropy and quantif y both static and dynamic magnetic properties in each region. The ability to quantitatively relate the spatial variation of static and dynamic properties in ultrathin YIG/Pt devices is a unique capability of our microscope . In conclusion, w e have demonstrated sensitive and high-resolution TRLSSE microscopy of ultrathin YIG/Pt devices that we expect will prove useful for developing spintronic applications . Using picosecond heating, we demonstrate that TRLSSE microscopy is a sub-100 picosecond probe of ultra-thin YIG/Pt device magnetization , both for static magnetic configurations and for dynamical measurements at gigahertz frequencies . We have demonstrated an angular sensitivity of 0.3 °/√𝐻𝑧, which to our knowledge is the most sensitive experimental probe of ultra-thin YIG magnetic orientation reported to date . Acknowledgments We thank J. Kimling and D. G. Cahi ll for helpful comments on an early version of the manuscript, and for providing the interface thermal resistance of YIG/Pt. This research was supported by the U.S. Air Force Office of Scientific Research , under Contract No. FA9550 -14-1- 0243 , and by U.S. National Science Foundation under Grant s No. DMR -1406333 and DMR - 1507274 and through the Cornell Center for Materials Research (CCMR) (DMR -1120296). This work made use of the CCMR Shared Facilities and the Cornell NanoScale Facility, a member of 11 the Nation al Nanotechnology Coordinated Infrastructure, which is supported by the NSF (Grant No. ECCS -1542081) . 12 References [1] B. Behin -Aein, D. Datta, S. Salahuddin, and S. Datta, Proposal for an all -spin logic device with built -in memory, Nat. Nanotechnol. 5, 266 (2010). [2] K. Ganzhorn, S. Klingler, T. Wimmer, S. Geprägs, R. Gross, H. Huebl, and S. T. B. Goennenwein, Magnon -based logic in a multi -terminal YIG/Pt nanostructure, Appl. Phys. Lett. 109, 022405 (2016). [3] C. W. Sandweg, Y. Kajiwara, A. V. Chumak, a. a. Serga, V. I. Vasyuchka, M. B. Jungfleisch, E. Saitoh, and B. 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The heating from the laser creates a temperature gradient, ∇𝑇𝑧. The pulsed heating drives a pulsed magnon flux, Js, from the YIG into the Pt where it is transduced into a pulsed voltage via the ISHE . 17 FIG. 2 (a) Schematic of the LSSE detection circuit used for time -resolved voltage measurements . (b) Time -domain measurement of the LSSE generated voltage in the 2 µm wide wire. The time -varying LSSE signal is measured by electrically mixing the pulsed laser generated voltage with a 100 ps voltage pulse from the AWG. Comparing measurements of the YIG at +414 Oe (filled blue circles) and –414 Oe (open orange circles) shows that the signal depends on the orientation of the magnetic moment. Here d.c. level noise and has been removed. The data was acquired with a lock-in time constant of 500 ms and integration time of 2 s per point . (c) The solid blue circles show the difference between the two curves i n (b), The orange line is a model , normalized by the data amplitude, of the signal determined by numerically convolving the calculated thermal gradient with the measured mixing pulse . (d) Difference signal of the temperature dependent voltage VJ measured using +/ – 0.5 mA and a 600 ps mixing pulse . In (b -d) we report the voltage as detected at the lock-in after passing through the r.f. mixer , not the LSSE signal at the sample itself. 18 FIG. 3 Stroboscopic detection of ferromagnetic resonance a) Schematic of measurement circuit for detection of ma gnetization dynam ics in the 2 µm wide wire . b) TRLSSE detected FMR for 4.1 GHz (blue, closed circles) and 4.9 GHz (orange, open circles) excitation. The solid lines are a fit to the data using a modified Lorentzian. c) Demonstration of stroboscopic FMR det ection in which we measure the response of the YIG driven at phases that differ by 180 degrees. The data was acquired with a lock-in time constant of 1s and integration time of 5 s per point. 19 FIG. 4 Measurement of YIG magnetization with LSSE measuring VLSSE versus external magnetic field for different laser powers and wire widths. For these curves, a DC background was subtracted . The inset shows the wire geometry. We define the signal size to be one -half of the difference in voltage when the magnetization is saturated in opposing directions. The data was acquired with a lock-in time constant of 500 ms and integration time of 2 s per point. 20 FIG. 5 Images of the 4 µm wide YIG/Pt wire (a) Reflected light image of the YIG/Pt wire measured with a photodiode at the same time as the LSSE voltage. (b) Background subtracted LSSE voltage at sa turated magnetization and (c) remnant magnetization at 4 Oe after saturation. 21 (d) Line cuts of the 2D scans. The normalized reflection signal is shown with black squares, blue circles represent the saturated magnetization, and the orange triangles represent the magnetization of the remnant state. Note, that in the line cuts the low field line cut is normalized with respect to the saturation magnetization. The right side of the figure represents the raw images of the 4 μm wire at different fields around the resonance : (e) 896 Oe. (f) 1007 Oe, ( g) 1025 Oe. Images (e -g) share the same color scale. Line cuts of the images are shown in (h) black squares, blue circl es, and orange triangles correspond to the boxed regions of (e), (f), and (g) respectively. For (e -g) the data was acquired with a lock-in time constant of 200 ms and an integration time of 2 s. 22 FIG. 6 Spatial maps of FMR fitting parameters for the 4 µm wide wire. (a) Traces are the pixel values of three points on the sample as a function of magnetic field . b-f) Spatial maps of the FMR fitting parameters made by fitting of the FMR curves at each pixel in the sequence of images measured with LSSE. Before fitting, we correct for image -to-image offset and use a 3x3 pixel moving average to smooth the data. (b) Resonance field, the symbols mark the pixels corresponding to the FMR spectra shown in (a). (c) Resonance amplitude , (d) resonance phase , (e) resonance linewidth (f) offset used in the fit. Imaging Magnetization Structure and Dynamics in Ultrathin YIG/Pt Bilayers with High Sensitivity Using the Time -Resolved L ongitudinal Spin Seebeck Effect Supplemental information Jason M. Bartell1, Colin L. Jermain1, Sriharsha V. Aradhya1, Jack T. Brangham2, Fengyuan Yang2, Daniel C. Ralph1,3, Gregory D. Fuchs1,3 1Cornell University, Ithaca, NY 14853, USA 2Department of Physics, The Ohio State University, Columbus, OH 43016, USA 3Kavli Institute at Cornell for Nanoscale Science, Ithaca, NY 14853, USA Optica l path To heat the YIG/Pt bilayers, we use a Ti:Sapphire laser tuned to 780 nm and pulse durations of 3 ps at 76.5 MHz. An electro -optic modulator referenced to the laser pulses is used to reduce the repetition rate to 25.5 MHz , which allows time for thermal recovery . Next, a photoelastic modulator and a polarizer are used to modulate the optical amplitude at 100 kHz for lock-in detection. The resulting vertically polarized light is focused on the sample with a 0.9 NA objective. A fast -steering mirror with a 4 -f lens pair is used to scan the laser focus across the sample. The light reflected from the sample is detected with a photodiode bridge . FIG. S1. Schematic of TRLSSE microscope. Model of TRLSSE temporal convolution We develo p a model of the detection circuit to clarify the impact of circuit bandwidth and electrical artifacts on the TRLSSE trace s shown in Figs. 2b and 2c . The time domain measurements shown i n Fig. 2 show that the duration of Vsig matches the ~100 ps duration of the mixing pulse. This implies that thermal gradient induced VLSSE must be sufficiently short -lived to sample the mixing pulse , and thus it is suitable for stroboscopic measurement of GHz frequency dynamics. In addition to t he main pulse, we also observe oscillations that can be attributed to non-idealities in the mixing reference pulse produced by the arbitrary waveform gene rator (AWG) and the RF mixer itself . To account for these effects , we develop a phenomenological model of the signal, which we describe as the convolution of the TRLS SE-induced electrical pulse from the sample and the reference pulse from the AWG as a function of relative delay, 𝞽 [1]. We account for bandwidth contributions and the realistic profile of the mixing reference pulse . The model consist s of a 12 GHz low-pass filter leading to the radio frequency and local oscillator inputs of an idealized mixer (Fig. S 2a). The output of the circuit is described by 𝑉𝑠𝑖𝑔(𝑡) ~ ℱ−1[ℱ(𝑉𝑚𝑖𝑥)∗ℱ(𝑉𝐿𝑆𝑆𝐸 )∗𝐿𝑃(𝑓)2] (S1) Where LP(f) is a first -order low -pass filter 𝐿𝑃(𝑓)=1 1+𝑓/𝑓𝑐 for frequency f and cut -off frequency fc = 12 GHz. The Fourier transform ℱ(𝑉) is given by ℱ(𝑉𝛿𝑡 )= 1 √𝑇∑ 𝑉𝛿𝑡 e2 𝜋 𝑖 (𝛿𝑡 −1)(𝑓−1)/𝑇 𝑇 𝛿𝑡 =1 where T = 12.9 ns is the duration of the kernel, 𝛿𝑡 = 2.5 ps is the time step, and f is frequency. In the experiment, the mixing pulse Vmix is generated by an arbitrary wave form generator (AWG) synchronized to the laser repetition rate with a sampling rate of 9.98 GSamples /s. For mixing voltage pulse Vmix we use the output of the AWG measured using a LeCroy SDA 11000 Oscilloscope (Fig. S2b ). To model the signal from the sample, VLSSE, we use the normalized thermal gradient determined from time -domain finite element (TDFE) calculations (further discussion below). In the main text, we use a 100 ps mixing pulse to acquire the data presented in Fig. 2b,c. and a 600 ps mixing pulse to acquire the other data. Figure 1 of the main text shows Vsig calculated via Eq. S1 normalized to the measured data along 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, supports the idea that the oscillations in the data are electrical artifacts , not magnetic oscillations . FIG. S 2. (a) Schematic of circuit model for interpretation of time -domain circuit. The arbitrary waveform generator (AWG) creates a mixing pulse that goes through a 12 GHz low -pass filter before being mixed with the pulse from the sample that has also been sent th rough a 12 GHz low-pass filter. (b) Oscilloscope measurements of the mixing pulses used in the experiments. Determination of temperature change from laser heating Although we know the laser fluence, we do not know the film absorbance for this thin -film limit in which the Pt film is much thinner than the optical skin depth. To determine the temperature change in our experiment we use the following methodology: (1) we numerically calculate the spatiotemporal thermal response to focused laser heating assuming the peak absorbed power is 1 W (an absorbed fluence of 0.7 mJ/cm2). We take the model’s predictions for the spatiotemporal thermal evolution to be correct but the total temperature change amplitude as being unc alibrated. (2) We calibrate and measure VJ, which is equivalent to using the sample resistivity change as a thermometer. (3) We calculate the VJ from our spatiotemporal thermal model calculations and compare it to the measured VJ. We assume there is linear response between the amplitude of the absorbed laser energy and the maximum temperature increase , therefore the ratio of the measured to the calculated values of VJ determines the scale factor of the absorbance . This also scales the temperature inc rease from the model to a value that agrees with our electrical measurement. Additional details have been described previously in the supporting information of Ref. [1]. We base our model on TDFE calculations of thin -film thermal diffusion to determine the spatiotemporal profile of the thermal gradient temperature distri bution. We consi der a GGG/YIG(20 nm)/Pt(6 nm) trilayer with material parameters given by Table S1. The YIG/Pt layer s are modeled as a 2 µm x 10 µm bar to match the measured device. Heat transfer in the structure is calculated using the diffusion equation 𝜌 𝐶𝑝𝛿𝑇(𝐱,𝑡) 𝛿𝑡−𝜅∇2𝑇(𝐱,𝑡)=𝑄(𝐱,𝑡) (S2) with the COMSOL Multiphysics® software package. In Eq. S2 𝜌 is the material density, Cp, is the specific heat, 𝜅, is the thermal conductivity, Q is the heat source, x is the 3D spatial coordinate, and t is time. We assume the YIG/ Pt interfacial thermal conductance is 170 W m- 2 K- 1 [2]. We also assume that laser heating only takes place in the Pt layer because of the negligible optical absorption in the YIG [3] and GGG [4]. Thus, the laser is effectively a radially symmetric heat source , with radius r, in the platinum with a spatial temporal distribution, for positive z, given by, 𝑄(𝐱,𝑡)=𝐸𝑥𝑝 (−𝑧 𝜀)∗(1 2𝜋 𝑑2)∗𝐸𝑥𝑝 (−𝑟2 2 𝑑2)∗𝐸𝑥𝑝 (−(𝑡−𝑡0)2 2 𝑤2), where d = 257 nm is the focused laser spot size (see “determination of optical spot size ” below ), 𝜀 = 12 nm is the skin depth [5,6] , w = 1.27 ps is the laser pulse width for a 3 ps FWHM Gaussian pulse, t0 = 100 ps is the time that the heat source is at the maximum. The heat source is applied every 39.6 ns and the simulation runs from time t = 0 ns to t = 42 ns to capture two pulses. Figure S3 shows the result of the model calculation in the space and time domain s. The z - component of the thermal gradien t within the YIG decays to 1/e in 92 ps and the t emperature difference between the Pt and YIG decays in 91ps, time scales that are experimentally indistinguishable in our measurement and consistent with the time domain measurement shown in 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 support that the TRLSSE signal originates from ∇𝑇𝑧(𝑡) (or indistinguis hably in this work , the temperature difference between YIG and Pt) and that it is localized in time making it suitable for stroboscopic measurements . The model calculation predicts about a 400 K change in the Pt, however , as discussed above, we calculated the amplitude of the laser -induced temperature change without experimental knowledge of the absorbed fluence. Therefore, the true temperature change in the Pt may be scaled up or down to account fo r correct value of the absorbed laser power. To establish the absorbance experimentally, w e compar e the measured values of VJ, which originates from the resistance change of the metal due to laser heating, with a model calculated value of VJ, which is determined from the resistance change expected from our model calculation. Specifically, we calculate VJ using the 3D temperature distribution created from laser heating to determine the sample resistance increase . We use the linear relationship between the resistance and the temperature , 𝑅(𝑇)=𝑅𝑜(1+𝛼 𝑇), with the resistance correction factor α = 1.3 × 10-3 K-1 measured for the Pt films used in our experiment . To compare the calculated value to the experimentally measured VJ, we also determine the electrical circuit transfer function in which we account for the measurement bandwidth and gain (see Ref. [1] for further discussion) . From this analysis we find that our experimentally measured V J is 0.12 times the calculated V J, indicating the peak temperature change in the Pt is 5 0 K, corresponding to a peak absorbed fluence of 0. 09 mJ/cm2, 1.6% of the incident laser energy. The uncertainty in the temperature is estimated to be on the order of 25% based on uncertainties in the circuit calibration . TABLE S1 M aterial parameters used in the TDFE simulations of laser heating aReference [7] bReference [8] cReference [9] Specific Heat, C p (J/kg*K) Density, 𝜌 (kg/m3) Thermal conductivity, 𝜅 (W/m*K) Pt 133a 21500a 71.6a YIG 570b 5170c 6b GGG 400b 7080b 7.94b FIG. S 3. Time -domain fini te element calculations of the temperature and thermal gradient using COMSOL. (a) Time -domain thermal profiles at the YIG/Pt interface calculated with COMSOL assuming an absorbed fluence of 0.7 mJ/cm2 and showing the z -component of thermal gradient in the YIG (orange curve), change in temperature of the Pt (blue curve), and temperature difference between the Pt and the YIG across the interface (black dashed line). The laser turns on at 100 ps in the calculation. (b) Calculated t emperature vs. z -axis positio n showing heating as a function of film depth at the maximum temperature difference (orange curve) and 16 ps later (blue curve) . (c,d) The curves from (a) and (b) scaled by the correction factor. Effect of interface spin transparency The spin Hall magnet oresistance (SMR) is the change in resistance due to spin-dependent transport in a heavy , nonmagnetic metal that shares an interface with a ferromagnet [10]. Thus, for bilayers of the same materials but different spin mixing conductance , measuring SMR provides insight into the efficiency with which spins can cross the interface. The efficiency of interfacial spin transport is important for TR LSSE measurements because in order for the magnetization to be transduced into a voltage, the thermally driven spins must cross the interface. For the data presented in the main text we find a SMR of 0.063%. We compare the signal from this wire with a rel atively strong SMR to the TRLSSE signal from a wire without detectable SMR above the 0.003% noise floor of our lock -in measurement . Both wires were 2 μm x 10 μm with resistances of 296 Ω and 220 Ω for the sample with and without SMR respectivly. The sample without SMR had a thinner YIG film ( 8 nm ), however this is not expected to effect the SMR since SMR is an interfacial effect [11]. Figure S4 shows representative plots of the TRLSSE signal versus field for the different wires at similar laser powers. We find that the sample with SMR has a signal approximately an order of magnitude greater than the sample without. The d ifference is consistent with the model of TRLSSE driving spin current across the YIG/Pt interface. We also note that even though the signal is reduced, it is still measurable in both samples, enabling measurement of YIG magnetization even in systems that c annot be measured electrically. FIG. S 4. TRLSSE signal as a function of applied external field for a sample with 0.063% SMR (blue triangles) and a sample with no measurable SMR (orange squares). The applied laser fluences are 5.4 mJ/cm2 and 6.7 mJ/cm2 for the blue an orange curves respectively. For the data presented here, the laser repetition rate was 76.5 MHz and no amplifier was used between the sample and the RF mixer. Determination of optical spot size We determine the diameter of the illuminated area by modeling a Guassian laser focus and fitting the traces of the image shown in Fig . 5a. Fig. S4 shows a y -axis cross section of the image. The trace shows an approximately flat region on the wire surface a nd a sigmoidal edge due to the convolution of the sharp wire edge with the point -spread function of the laser focus . To fit the reflection signal, I, at the edge, we use the convolution of a Guassian with a step function , 𝐼=1 𝑏 √2 𝜋∫ exp (−(𝑥−𝑎)2 2 𝑏2)∞ −∞ Θ(𝑥−𝑎)𝑑𝑥, (S3) in which 𝑏 determines the Guassian width , a is the center of the peak , and Θ is the step function defined as Θ(𝑥−𝑎)={0 ,𝑥<𝑎 1 ,𝑥≥𝑎. The fit of the data yields b = 0.240 ± 0.007 µm and b = 0.274 ± 0.010 µm for the left and right edges respectively . We take the average to be the optical spot size. We attribute t he difference between the two edges to a slight out-of-plane tilt of the sample leading to asymmetry in the reflection. As a comparison, we fi t a y-axis scan of the TRLSSE signal to Eq. S3. The result gives b = 0.380 ± 0.006 µm and b = 0.381 ± 0.009 µm for the left and right edges respectively. This difference corresponds to a difference of ~1 pixel between the rise-width of the reflection signal and TRLSSE signal . FIG. S 5. Fit of step edge signal for determination of optical spot size. (a) Line cut in y -axis direction of the reflected light image, shown in Fig. 5a, and the TRLSSE image of the static saturated moment, shown in F ig. 5b. (in set) schematic representation of the sample tilt that can lead to the observed anisotropy . Analysis of dynamic TRLSSE images To image the ferromagnetic resonance of YIG in the 4 µm wide wire a series of images was taken at fields ranging from 896 to 1 105 for an applied RF power of 1.1 mA . A selection of unprocessed images is shown in Fig. 5e -g of the main text. Although the signal is quite clear, we account for sample drift and noise, before fitting the FMR curves. We correct for sample drift using autocorrelation to find the image overlap. The kernel for the autocorrelation is a 5 ×12.5 µm region from the center of the reflected light image at H = 896 Oe (the first image in the series). We determine t he drift of subsequent images by finding the distance between the centers of the kernel and the minimum of the autocorrelation. M ost of the sample drift is on the order of a pixel (0.25 µm) with a maximum sample drift of Δy = 0.75 µm and Δx = 0.25 µm. We correct for t he offset by shifting the images and then cropping the borders. The scans cover a large enough area that the cropped region is well away from the wire. After correcting for the sample drift, we remove the background from the vibration edge artifacts by subtract ing the TRLSSE signal of the wire at 896 Oe from the subsequent images. Finally, we reduce random pixel to pixel noise, smoothing the signal with a 3x3 pixel moving average. The 3x3 pixel window is approximately the sampling spot size (see determination of optical spot size). We attribute t he small signal features at the edges of the wires in Fig . 5 of the main text to magnetic field modulation induced relative motion between the microscope objective and the sample . As mentioned in the main text, we separate VJ (which is in princ iple non -magnetic) from VTRLSSE (which is magnetic) by adding a modulation magnetic field (7.6 Oe RMS, ω H = 383 Hz) to the d.c. magnetic field. We then demodulate Vsig with respect to ω H using a lock -in amplifier . Although this procedure is effective for isolating VTRLSSE from VJ when we focus in the center of the wire (away from the wire edge) , the modulation field induces a tiny “wobble” in the laser focus on the sample . When the laser is focused on the sample edge and a current is applied to the sample, the wobble introduces a slight modulation of VJ at ωH because 𝑑𝑉𝐽 𝑑𝐻=(𝑑𝑉𝐽 𝑑𝑦)(𝑑𝑦 𝑑𝐻), where 𝑑𝑦 𝑑𝐻 is due to field-induced mechanical motion and 𝑑𝑉𝐽 𝑑𝑦 is large at the sample edge. We note that these edge signals are independent of external field but that they are sensitive to the current amplitude and phase , both of which are consistent with this interpretation of the artifact . In Fig. S6 we plot both the profile of the externally modulated fi eld signal in the y -direction and the numerical derivative of VJ measured by the lock-in referenced to the 100 kHz laser modulation rate, which demonstrates their correspondence. FIG. S 6. (a) Spatial variation of the TRLSSE in a 4 × 10 μm YIG/Pt wire at 911 Oe. The signal measured by a lock-in amplifier referenced to the frequency of an a.c. magnetic field. (b) Profile of the TRLSSE signal shown in (a) (blue circles) and the derivative of V J from the same area of the wire (orange triangle s). The trace is the average of twenty -six y-axis line scans from along the length of the wire . References [1] J. M. Bartell, D. H. Ngai, Z. Leng, and G. D. Fuchs, Towards a table -top microscope for nanoscale magnetic imaging using picosecond thermal gradients, Nat. Commun. 6, 8460 (2015). [2] J. Kimling and D. G. Cahill (private communication) [3] S. H. Wemple, S. L. Blank, J. A. Seman, and W. A. Biolsi, Optical properties of epitaxial iron garnet thin films, Phys. Rev. B 9, 2134 (1974). [4] D. L. Wood and K. Nassau, Optical properties of gadolinium gallium garnet, Appl. Opt. 29, 3704 (1990). [5] J. H. Weaver, Optical properties of Rh, Pd, Ir, and Pt, Phys. Rev. B 11, 1416 (1975). [6] A. D. Rakić, A. B. Djurišić, J. M. Elazar, and M. L. Majewski, Optical properties of metallic films for vertical -cavity optoelectronic devices, Appl. Opt. 37, 5271 (199 8). [7] E. W. M. Haynes, editor , CRC Handbook of Chemistry and Physics , 97th Editi (CRC Press/Taylor & Francis, Boca Raton, FL., n.d.). [8] A. M. Hofmeister, Thermal diffusivity of garnets at high temperature, Phys. Chem. Miner. 33, 45 (2006). [9] A. E. C lark and R. E. Strakna, Elastic Constants of Single -Crystal YIG, J. Appl. Phys. 32, 1172 (1961). [10] M. Weiler, M. Althammer, M. Schreier, J. Lotze, M. Pernpeintner, S. Meyer, H. Huebl, R. Gross, A. Kamra, J. Xiao, Y. -T. Chen, H. Jiao, G. E. W. Bauer, and S. T. B. Goennenwein, Experimental Test of the Spin Mixing Interface Conductivity Concept, Phys. Rev. Lett. 111, 176601 (2013). [11] N. Vlietstra, J. Shan, V. Castel, B. J. van Wees, and J. Ben Youssef, Spin -Hall magnetoresistance in platinum on yttrium i ron garnet: Dependence on platinum thickness and in -plane/out -of-plane magnetization, Phys. Rev. B 87, 184421 (2013).

2016-12-22

We demonstrate an instrument for time-resolved magnetic imaging that is highly sensitive to the in-plane magnetization state and dynamics of thin-film bilayers of yttrium iron garnet (Y3Fe5O12,YIG)/Pt: the time-resolved longitudinal spin Seebeck (TRLSSE) effect microscope. We detect the local, in-plane magnetic orientation within the YIG by focusing a picosecond laser to generate thermally-driven spin current from the YIG into the Pt by the spin Seebeck effect, and then use the inverse spin Hall effect in the Pt to transduce this spin current to an output voltage. To establish the time resolution of TRLSSE, we show that pulsed optical heating of patterned YIG (20 nm)/Pt(6 nm)/Ru (2 nm) wires generates a magnetization-dependent voltage pulse of less than 100 ps. We demonstrate TRLSSE microscopy to image both static magnetic structure and gigahertz-frequency magnetic resonance dynamics with sub-micron spatial resolution and a sensitivity to magnetic orientation below 0.3$^{\circ}/\sqrt{\text{Hz}}$ in ultrathin YIG.

Imaging Magnetization Structure and Dynamics in Ultrathin YIG/Pt Bilayers with High Sensitivity Using the Time-Resolved Longitudinal Spin Seebeck Effect

1612.07610v2

arXiv:1706.02154v1 [cond-mat.other] 6 Jun 2017Spin Seebeck effect and magnon-magnon drag in Pt/YIG/Pt stru ctures I.I. Lyapilin1,2, M.S. Okorokov1∗ 1Institute of Metal Physics, UD RAS, Ekaterinburg, 620137, R ussia 2Ural Federal University after the first President of Russia B .N. Yeltsin, Yekaterinburg, 620002 Russia (Dated: November 5, 2021) The formation of the two: injected (coherent) and thermally excited, different in energies magnon subsystems and the influence of its interaction with phonons and between on drag effect under spin Seebeck effect conditions in the magnetic insulator part of t he metal/ferromagnetic insulator/metal structure is studied. An approximation of the effective para meters, when each of the interacting subsystems (”injected”, ”thermal” magnons, and phonons) i s characterized by its own effective temperature and drift velocities have been considered. The analysis of the macroscopic momentum balance equations of the systems of interest conducted for d ifferent ratios of the drift velocities of the magnon and phonon currents show that the injected magnon s relaxation on the thermal ones is possible to be dominant over its relaxation on phonons. Th is interaction will be the defining in the forming of the temperature dependence of the spin-wav e current under spin Seebeck effect conditions, and inelastic part of the magnon-magnon intera ction is the dominant spin relaxation mechanism. I. INTRODUCTION The influence of non-equilibrium phonons on kinetic coefficients in electron-phonon or spin-phonon systems has been theoretically studied chiefly by the Boltzmann kinetic equation method or using the formalism of the Kubo response theory. The present work employs the methodofthenon-equilibriumstatisticaloperator(NSO) to analyze how interactions between three flows (two magnon and phonon ones) affect the drag effect and the temperature dependence of the spin Seebeck effect (SSE) within the above model. The concept of magnon spintronics, i.e., the genera- tion, detection and manipulation of pure spin currents in the form of spin wave quanta1, the magnons, has at- tracted growing interest in the recent years2,3. Magnons are quasiparticles representing a low-energy excited state offerromagnets. Aquantized magnonis a boson and car- ries basic spin angular momentum quanta of ¯ h4. Similar to spintronics and electronics, magnonics refers to using magnons for data storage and information processing5. Up to now, magnons in the field of spintronics have been investigated within the content of magnetostatic spin waves which describe the nonuniform spatial and tem- poral distribution of the classical magnetization vector. Recently, a great deal of attention is devoted to the investigation of thermally excited magnons, particularly in studies of the spin Seebeck effect5–9in Pt/YIG/Pt structure. In SSE effect, an applied electric current in one Pt layer accompanies an electron spin current due to the spin Hall effect (SHE)8,9. When the spin cur- rent flows to the boundary between the Pt and the YIG, nonequilibrium spins are accumulated and, consequently, due to the s-d exchange interaction between conduction electrons in normal metal (NM) and magnetic moments in ferromagnetic insulator (FI), magnons are created at the interface10,11. The induced magnons subsequently diffuse in FI to the other interface where the magnon current converts back to an electron spin current in theother NM layer, leading to a charge current due to the inverse spin Hall effect (ISHE)12–14. Thus, the induced electric current in the second NM layer which is electri- cally insulated from the current-flowing NM layer by a FI would elucidate the magnons as spin information car- riers. One of the key advantages of magnon spin currents is their large damping length, which can be several or- dersofmagnitude higherthan the spindiffusion length in conventional spintronic devices based on spin-polarized electron currents15. The propagation of magnons in a magnetic insula- tor is described by two characteristic quantities: mean free path and spin diffusion length that are governed, in turn, by various magnon relaxation mechanisms. A se- ries of experiments determine the range of the diffusion lengths as being quite wide: from 4 µm to 120 µm16–20. To explain so large values of the spin diffusion lengths, the number of papers has put forward several concepts of appearance, along with ”thermal” magnons, of long- wave (”subthermal”) magnons in a magnetic insulator. Under SSE conditions, the former are characterized by short wavelength, the latter are weakly coupled with the lattice17,20. Forinterpretingtheexperimentalresults,the works17,20,21haveadoptedthehypothesisoftheexistence ofthetwomagnonsubsystemswithdifferentenergies. As tothetemperaturedependence oftheSeebeckcoefficient, it is non-monotonic and reaches its maximum within the rangeof50- 100K.And asthe investigationshaveshown, it is affected by strength of a magnetic field, dimensions of the samples, and quality of the interface20,22. To explain the low temperature enhancement23pro- posed the phonon-drag SSE scenario based on a theoret- ical model24–26. Back in 1946, in the context of ther- moelectricity, Gurevich pointed out that thermopower can be generated by nonequilibrium phonons driven by a temperature gradient, which then drag electrons and cause their motions24. It was suggested first by Bailyn27 that the theory of magnon-drag should be analogous to that of phonon-drag24. Following this work the magnon-2 drag component of the thermopower has been calculated by Grannemann and Berger28. In this phenomenologi- cal model, the temperature dependence of the phonon life time is involved, which reaches a maximum at low temperatures. Based on this, they propose a strong in- teraction between the phonons and magnons, which are responsible for the heat transport in the system. The phonons flow along the thermal gradient and interact with thermally excited magnons. These phonons drag the magnons. Thus, the phonon-magnon coupling is sug- gested to explain the observed enhancement of SSE sig- nal at low temperature23,24. The first investigation of the phonon-magnon interaction in magnetic insulators was conducted by Adachi et al., reporting a giant en- hancementofSSEin LaY2Fe5O12atlowtemperatures23. However, the observed transport of magnons over a long distance of up to millimeters in magnetic insulators im- plies a relative weak interaction with phonons and impu- rities, and the measurements of the temperature depen- dent thermal conductance of YIG single crystals show that the phonon contribution to the thermal conductiv- ity reaches its maximum at around 25 K16, which is 50 K lower than the observed peak in the SSE. In28the temperature dependence of the spin Seebeck effect was measured in ( Ga,Mn)As, and the data showed a pro- nounced peak at low temperatures. Here, we study how the formation of two interact- ing magnon subsystems with distinguished energies af- fects the SSE17,19–21. We assume that the first group of magnons is ”thermal” ones subjected to a non-uniform temperature field applied to the magnetic insulator. The energy of such magnons is of the order of the tempera- turekBT. Further, under the SSE, inelastic scattering of spin-polarized electrons of the metal by localized spins located near the interface causes the magnons to inject into the magnetic insulator. The energy of the injected (”coherent”) magnons is of the order of the spin accu- mulation energy ∆ sof conduction electrons of the metal. It can be generated, for example, by the spin Hall effect when passing a direct electric current through the metal (Pt) [1]. Under the SSE conditions, the injection of magnons into the magnetic insulator dominates scattering pro- cesseswithmagnonabsorptionprovidedthattheinequal- ity ∆s> kBTis fulfilled. Thus, it can be said that the magnetic system of the insulator forms another subsys- tem of ”injected”(”coherent”) magnons that are actu- ally responsible for the SSE. As a consequence, in the presence of a non-uniform temperature field, there are threeflowsinsidethemagneticinsulator, namely, phonon and two magnon ones. The evolution of the magnon and phonon subsystems to equilibrium occurs due to the re- laxation of both their energy and their moment. These subsystemstend tobecomebalancedwith differentveloc- ities. Obviously, the interaction between the flows gives rise to the drag effect29–31. It is worth noting that the paper32has already dis- cussed the influence ofmutual draggingbetween phononsand spin excitations on thermal conductivity of a spin system. Once the magnon subsystems are thermalized in energy, the magnon system can be characterized solely by a temperature Tm. As to the moment relaxation pro- cesses, this time is defined, as well as in the event of spin- flip electron scattering, by inelastic magnon scattering. The time relaxation is large enough, which, apparently, provides the existence of the SSE over long distances16. The paper is structured in the following manner. The first part is devoted to the splitting of the ”injected” magnonsflow responsiblefor the spin Seebeck effect from themagnoncurrent. Inthesecondpartthebalanceequa- tions for the magnons and phonons in the approximation of effective parameters when each subsystem is charac- terized by its effective temperature and drift velocity are built and analysed. SPIN CURRENT The density of the spin current Js(r) can be repre- sented as the sum of two terms: collisional ˙ sz (sm)(r) and collisionless Isz(r). The former is controlled by the in- elastic spin-flip electron scattering by localized moments at the interface, the latter is due to the flows of electrons with different spin orientation33 Js(r) =d dtsz(r) =1 i¯h[sz(r),H] =−∇Isz(r)+ ˙sz (sm)(r), Isz(r) =/summationdisplay isz i{pi/m,δ(r−ri)}, ˙sz (sm)(r) =1 i¯h[sz(r),Hsm]. (1) HereHis the Hamiltonian of the system considered, Hsmis the density of the exchange interaction energy between conduction electrons and localized moments at the interface34. Hsm=−J0/summationdisplay j/integraldisplay drs(r)S(Rj)δ(r−Rj),(2) J0is the exchange integral, S(Rj) is the operator of a localized spin with the coordinate Rjat the interface. s(r) is the spin density of the electrons in the metal. Ac- companiedby the creationofmagnons, the inelastic scat- tering of spin-polarized conduction electrons by localized impurity centers dominates other processes. This leads to magnon accumulation ( δN(r) =N(r)−N0(r))33, N(r), N0(r) are the non-equilibrium and equilibrium magnon distribution functions near the interface in the magnetic insulator. The macroscopic spin current can be found by averaging the expression with the non- equilibrium statistic operator ρ(t)35: /an}b∇acketle{tJs(r)/an}b∇acket∇i}htt=−∇ /an}b∇acketle{tIsz(r)/an}b∇acket∇i}htt+/angbracketleftBig ˙sz (ms)(r)/angbracketrightBigt ,(3) where/an}b∇acketle{t.../an}b∇acket∇i}htt=Sp(ρ(t)...).3 Given that [ sz i,s± j] =±s± iδij, we have /angbracketleftBig ˙sz (ms)(r)/angbracketrightBigt = (−J0/2)/summationdisplay j/integraldisplay dr/angbracketleftbig (s+(r)S−(Rj)− −s−(r)S+(Rj))δ(r−Rj)/angbracketrightbigt(4) Restricting ourselves to the linear approximation in the interaction Hsm, we omit the interaction Hsmin the operator ρ(t) in which the averaging is performed. In this case < ... >t=< ... >t e< ... >t m, i.e. the averaging in the electron system and the localized spin system is carried out separately: /angbracketleftbig sα(r)Sβ(Rj)/angbracketrightbigt→ /an}b∇acketle{tsα(r)/an}b∇acket∇i}htt e/angbracketleftbig Sβ(Rj)/angbracketrightbigt m. Thus, we arrive at, /angbracketleftBig ˙sz (ms)(r)/angbracketrightBigt = (−J0/2)/summationdisplay j/integraldisplay dr{/angbracketleftbig s+(r)/angbracketrightbigt e/angbracketleftbig S−(Rj)/angbracketrightbigt m− −/angbracketleftbig s−(r)/angbracketrightbigt e/angbracketleftbig S+(Rj)/angbracketrightbigt m}δ(r−Rj).(5) Let us calculate S±(R). We write down the equations of motion for the transverse components ˙S±(R) = (i¯h)−1[S±(R), Hm+Hk m+Hmp+Hms], whereHm, Hk m, Hmp, Hmsare the energy operators of the magnetic subsystem (Zeeman and kinetic), the magnon-phonon( mp) and exchange( sm) interaction, re- spectively. Computing the commutators, we obtain ˙S±(R) =∓iωmS±(R)−∇IS±(R)+˙S± (mp)(R)+˙S± (ms)(R), ˙A(ik)(R) = (i¯h)−1[A(R),Hik] (6) and the density of the magnon flows at the interface IS±(R) =/summationdisplay jS± j{Pj/M,δ(R−Rj)}(7) Pj,Mare the magnon momentum and the effective magnon mass. The last two summands in the right- hand side of (6)describe the scattering of the magnons by phonons and electrons at the interface. Conducting the averaging, in the stationary case we come to ∓iωm/angbracketleftbig S±(R)/angbracketrightbig m=∇/an}b∇acketle{tIS±(R)/an}b∇acket∇i}htm− −/angbracketleftBig ˙S± (mp)(R)/angbracketrightBig m−/angbracketleftBig ˙S± (ms)(R)/angbracketrightBig m.(8) Further, weinsertthe expression(8) intothe equationfor the spin current/angbracketleftBig ˙sz (ms)(r)/angbracketrightBig and estimate the summands. Thefirsttermintheright-handsideoftheexpression(5), approximatelyproportionalto ∼J0, governsthe magnon flow excited at the interface due to electron scattering by localized moments. The second term is proportional to ∼J0Upand sets forth the magnon scattering by phonons (Upcharacterizesthe intensity of the magnon-phonon in- teraction). Finally, the last term in the right-hand sideof the expression (5) ∼J2 0. Putting that Up≫J0, we leave this term aside. Let us unravel the evolution of the magnetic subsys- tem. ˙Sz(R) = (i¯h)−1[Sz(R), Hm+Hk m+H(mp)+H(ms)] Then we have ˙Sz(R) =−∇ISz(R)+˙Sz (ms)(R)+˙Sz (mp)(R).(9) The first term in the right-hand side of (9) involves the spin-density flow of localized spins (magnons), and the terms˙Sz (mp)(R),˙Sz (ms)(R), described the scattering of the localized spins by phonons at the interface. Thus, the macroscopic spin-wave current realized in the magnetic insulator can be written as /an}b∇acketle{tIS(R)/an}b∇acket∇i}ht=−∇/an}b∇acketle{tISz(R)/an}b∇acket∇i}ht+/angbracketleftBig ˙Sz (ms)(R)/angbracketrightBig m+/angbracketleftBig ˙Sz (mp)(R)/angbracketrightBig m+ +(−J0)/summationdisplay j/integraldisplay dr{/angbracketleftbig s+(r)/angbracketrightbig e/angbracketleftbig S−(Rj)/angbracketrightbig m)− −/angbracketleftbig s−(r)/angbracketrightbig e/angbracketleftbig S+(Rj)/angbracketrightbig m}δ(r−Rj),(10) where /angbracketleftbig S±(R)/angbracketrightbig m=iω−1 m{−∇/an}b∇acketle{tIS±(R)/an}b∇acket∇i}htm−/angbracketleftBig ˙S± (mp)(R)/angbracketrightBig m/bracerightbig .(11) In (11), we have omitted the summand that describes the magnon scattering at the interface ( ∼J2 0). It can be seen from (10), (11) that the magnetic subsystem real- izes two magnon flows. The first is due to a non-uniform temperature perturbation of the magnetic subsystem. It is a flow of ”thermal” magnons. The mean energy of these magnons is of the temperature. The second is ∼J0∇IS±(R) and isbroughtabout bymagnonsinjected into the magnetic subsystem as a result of inelastic scat- tering of conduction electrons by localized moments at the interface. The energy of such magnons is of the or- der of the spin-accumulation energy of the conduction electrons and is equal to ∆ s≫kbT. MACROSCOPIC MOMENTUM BALANCE EQUATIONS The influence of non-equilibrium phonons on kinetic coefficients in electron-phonon or spin-phonon systems has been theoretically studied chiefly by the Boltzmann kinetic equation method or using the formalism of the Kubo response theory32. The present work employs the methodofthenon-equilibriumstatisticaloperator(NSO) for analyzing how interactions between three flows (two magnon and phonon one) affect the drag effect and the temperaturedependence ofthe spinSeebeck effect within the above model. In constructing macroscopic momen- tum balance equations for the system at hand, we should use the Hamiltonian H=HM+HP+HV (12)4 HereHMis the Hamiltonian of the magnetic sys- tem that consists of two magnetic subsystems: of ”injected”(”coherent”) ( Hm1) and ”thermal”( Hm2) magnons and their mutual interaction HM=/integraldisplay dr(/summationdisplay iHmi(r)+Hmimi(r)), i= 1,2 (13) The integration is performed over the volume occupied by the magnetic insulator FI. Hmi(r) is the energy den- sity operator of the (i) magnetic subsystem. Hmimi(r) is the Hamiltonian of the magnon-magnon interaction in- side each the subsystems Suppose the magnon gas to be free: Hmimi=/summationtext kε(k)b+ kbk, ε(k) =P2/(2M) is the sum of the en- ergies of quasi-particles, ferromagnons having a quasi- momentum P= ¯hkwith their effective mass Mand magnetic momentum34).b+ k, bkare the creation and an- nihilationoperatorsforthemagnonswiththewavevector k. Hpis the lattice Hamiltonian Hp=/integraldisplay dr(Hp(r)+Hpp(r)), (14) whereHp(r) is the energy density operator for the phonon subsystem. Hpp(r) is the phonon scattering by non-magnon relaxation mechanisms (scattering by the boundaries of the sample, impurities and defects of the lattice, etc.) HV(r) =Hmimj(r)+Hmip(r)+Hmis(r) (15) is the energy density operator of interaction between the phonons. Hmip(r) is the energydensity operatorofinter- action between the phonon and magnetic (i) subsystems. Hmimj(r) describes the interaction between ”thermal” and ”coherent” (injected) magnons. Hmis(r) is the en- ergy density operator of exchange interaction between conduction electrons and localized magnetic moments at the interface. Under the influence ofa non-uniform temperature field (a temperature gradient) applied to the system, the magnonsandphononsbegintravelling; theirmacroscopic drift affects the propagation of the spin-wave current. Obviously, the drag effects that may arise in the sys- tem considered are governed by both magnon-phonon collision frequencies and phonon relaxation mechanisms by other mechanisms of their scattering. The prob- lem to be solved reduces to constructing and analyz- ing a set of macroscopic momentum balance equations ˙Pi(r) = (i¯h)−1[Pi(r),H] for the magnon ( i= 1,2 ) and phonon ( i=p) subsystems. In writing the Hamiltonian, we have omitted the ex- change interaction between localized spins and conduc- tion electrons at the interface. In doing so, we have put that it is the exchange interaction that is responsible for the magnon injection into the magnetic insulator and makes no significant contribution to the momentum re- laxation of the magnons and phonons.The equations of motion for the magnon and phonon momenta have the form: ∂ ∂tPmi(r) =−∇IPmi(r)+˙P(mi,v)(r),(i/ne}ationslash=j= 1,2) ∂ ∂tPp(r) =−∇IPp(r)+˙P(p,pp)(r)+˙P(p,v)(r),(16) where ˙A(i,v)= (i¯h)−1[Ai,Hv]. The first terms in the right-hand sides of (16) are the flows of appropriate momenta IP(r) =/summationtext i{Pi/M,δ(r− ri)}. The rest of the terms in the right-hand side of these equations describe the relaxation processes: magnon- phonon and magnon-magnon scattering. To derive the macroscopic equations (16) /angbracketleftBig ˙Pi(r)/angbracketrightBigt =Sp{˙Pi(r)ρ(t)},(i=m1,m2,p), the expression for the NSO needs to be sought. Accord- ing to35,36, forρ(t) we have: ρ(t) =ǫ0/integraldisplay −∞dt′eiǫt′eit′Lρq(t+t′), ǫ→+0, ρq(t) =e−S(t), eitLA=e−itH/¯hAeitH/¯h,(17) Hereρq(t) is the quasi-equilibrium statistical operator. The non-equilibrium state of the system considered cor- respondsintermsofaveragedensityvaluestotheentropy operator S(t) =S0+δS(t) = = Φ(t)+/integraldisplay dr{βmi(r,t)[Hmi(r)+Hmimi(r)+ +Hmimj(r)]−βµmi(r,t)Nmi(r)+ +βp(r,t)[Hp(r)+Hpp(r)+Hpmi(r)]− −βmi(r,t)Vmi(r,t)Pmi(r)+βp(r,t)Vp(r,t)Pp(r)}.(18) HereS0is the entropy operator for the equilibrium sys- tem.δS(t) describes the deviation of the system from its equilibrium state. Φ( t) is the Massieu-Plank func- tional.βmi(r,t) are local-equilibrium values of the in- verse temperatures of the magnon ( i= 1,2) and phonon subsystems βp(r,t).µmi(r,t)is a local equilibrium value of the chemical potential of the magnons. N(r) = Nm1(r) +Nm2(r) is the magnon number density oper- ator.Vmi,Vpare the drift velocities of the magnons (i= 1,2) and the phonons respectively. Magnons, as well as phonons are Bose particles; their distribution function is the Bose-Einstein function with a zero chemical potential. However, the situation be- comes quite different if magnons are non-equilibrium. In our case, the non-equilibrium magnon system may be described by introducing the non-equilibrium chemical potential of magnons37–39.5 Forρ(t), in the linear approximation in deviation from equilibrium, we arrive at ρ(t) =ρq(t)−0/integraldisplay −∞dt′eǫt′eit′L1/integraldisplay 0dτ ρτ 0˙S(t+t′)ρ−τ 0ρ0. (19) ˙S(t) =∂S(t)/∂t+(i¯h)−1[S(t),H] is the entropy produc- tion operator. ρ0= exp{−S0}.Thus, the problem boils down to finding the entropy production operator. We write down the equations of motion for the opera- tors involved in the entropy operator. Then, we have ˙Hmi(r) =−∇IHmi(r)+˙H(mi,v)(r) ˙Hp(r) =−∇IHp(r)+˙H(p,pp)(r)+˙H(p,v)(r), ˙Nmi(r) =−∇INmi(r)+˙N(mi,v)(r). (20) The first terms in the right-hand sides of these equations control the flows of appropriate quantities: the energy and number of magnons, phonons, meanwhile, the rest of the terms describe relaxation processes. INm(r) is the density of the magnon flow. Substituting the equations of motion into the entropy production operator, we come to δ˙S(t)=∆/integraldisplay dr{−δβmi(r,t)∇IHmi(r)+ +βµmi(r,t)∇INmi(r)−δβp(r,t)∇IHp(r)+ +βmi(r,t)Vmi(r,t)∇IPmi(r)+βp(r,t)Vp(r,t)∇IPp(r)+ +δβmimj(r,t)˙H(mi,mimj)(r)−βµmi(r,t)˙N(mi,v)(r)− βmi(r,t)Vmi(r,t)˙P(mi,v)(r)−βp(r,t)Vp(r,t)˙P(p,v)(r)},(21) whereδβmimj=βmi−βmj,∆A=A−< A > 0. We integrate by parts the terms containing the flow divergences. Then, we ignore the surface integrals and write down the entropy operator as ˙S(t)=∆/integraldisplay dr{−βINmi(r)∇µmi(r,t)+ +δβmimj(r,t)˙H(mi,mimj)(r)+I∗ mi(r)∇βmi(r,t)+ +I∗ p(r)∇βp(r,t)−βµmi(r,t)˙N(mi,v)(r)− −βVmi(r,t)˙P(mi,v)(r)−βVp(r,t)[˙P(p,pp)+˙P(p,v)(r)]}.(22) Here I∗ mi(r)=[IHmi(r)+IPmi(r)Vmi(r)], I∗ p(r)=[IHp(r)+Vp(r)IPp(r)] and we have taken into account that ∇(βk(r,t)Vk(r,t))∼Vk(t)∇βk(r,t). Before going over to the macroscopic momentum balance equations, we should find a relation between the chemical potential and effective temperature of themagnon subsystem. From the quasi-equilibrium distri- butionρq(t) it follows that δ/an}b∇acketle{tNm1(r)/an}b∇acket∇i}ht=−/integraldisplay dr′{δβm1(r′,t)(Nm1(r),Hm1(r′))− βµm1(r′,t)(Nm1(r),Nm1(r′))−βm1Vm1(r′,t)(Nm1(r),Pm1(r′))}, (23) where δ/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 0dλSp{Aρλ 0∆Bρ1−λ 0}. If one admits that Nm1(r) =constin a non-equilibrium but steady-state case, (23) implies that µm1≃(βm1/β−1)R−(βm1/β)R1 R=(Nm1,Hm1) (Nm1,Nm1)R1=Vm1(Nm1,Pm1) (Nm1,Nm1).(24) Note that as βm1→β, Vm1= 0, the chemical potential of magnons tends to zero: µm→0. MACROSCOPIC EQUATIONS Insertingthe entropyproductionoperator(22) intothe expression for the NSO (19), we average the operator equations (16) for momenta of the subsystems under dis- cussion. Then we have /angbracketleftBig ˙Pmi(r)/angbracketrightBigt = =−0/integraldisplay −∞dt′eǫt′/integraldisplay dr′{β(∇IPmi(r),INmj(r′,t′))∇µmj(r′,¯t)+ +(∇IPmi(r),I∗ mj(r′,t′))∇βmj(r′,¯t)+ +(˙P(mi,v)(r),˙P(mj,v)(r′,t′))βVmj(r′,¯t)}, (25) here¯t≡t+t′. The first summand in the right-hand side of (25) describes the diffusion and drift of magnons due to the gradients of the chemical potential and the tem- perature, the last summand the magnon-magnonscatter- ing processes both inside each of the magnon subsystems and the ”coherent” magnon scattering by the ”thermal” magnons. Analogously, we characterize the relaxation processes in the phonon subsystem: /angbracketleftBig ˙Pp(r)/angbracketrightBigt = =0/integraldisplay −∞dt′eǫt′/integraldisplay dr′{(∇IPp(r),I∗ p(r′,t′))∇βp(r′,¯t)+ −(˙P(p,v)(r),˙P(mi,v)(r′,t′))βVi(r′,¯t)− (˙P(p,v)(r),˙P(p,v)(r′,t′))βVp(r′,¯t)}. (26)6 Given that the chemical potential and the effective tem- perature are related as in (24), we introduce the general diffusion coefficient Dmimj(r,r′,t′) : β(∇IPmi(r),INmj(r′,t′))∇µmj(r′,¯t)+ +(∇IPmi(r),I∗ mj(r′,t′))∇βmj(r′,¯t) = =Dmimj(r,r′,t′)∇µmj(r′,¯t) (27) where βDmimj(r,r′,t′)=(∇IPmi(r),INmj(r′,t′)) + +(∇IPmi(r),I∗ mj(r′,t′))/(R−R1).(28) The above equations represent the temperature gradi- ent as a driving force. Therefore, the entropy operator involves the additional summands such as βi(r,t)Viin- stead of βVi(r,t). Now, revealing explicitly the correlation functions de- scribing the relaxation processes and appearing in the momentum balance equations (25), (26), we have (˙P(mi,v)(r),˙P(mj,v)(r′,t′))= =(˙P(mi,mp)(r),˙P(mi,mp)(r′,t′))+ +(˙P(mi,mimj)(r),˙P(mi,mimj)(r′,t′)),(29) The first summand in the right-hand side of (29) de- scribes the magnon-phonon scattering, the second the magnon-magnon scattering processes both inside each of the magnon subsystems and the injected” magnon scat- tering by the ”thermal” magnons. (˙P(p,v)(r),˙P(p,v)(r′,t′)) = = (˙P(p,pp)(r),˙P(p,pp)(r′, t′))+ +(˙P(p,pm)(r),˙P(p,pm)(r′,t′)).(30) The first term describes the processes of non-magnon re- laxation of phonons; the second one governsthe magnon- phonon scattering. Introducing the notation L(k,v)(r,r′,t′) = (˙P(k,v)(r),˙P(k,v)(r′,t′)),(31) we re-write down the momentum balance equations in a convenient form for further analysis /angbracketleftBig ˙Pm1(r)/angbracketrightBigt = −0/integraldisplay −∞dt′eǫt′/integraldisplay dr′β{Dm1m1(r,r′,t′)∇µm1(r′,¯t) + +L(m1,m1p)(r,r′,t′)δVm1,p(r′,¯t)+ +L(m1,m1m2)(r,r′,t′)δVm1,m2(r′,¯t)},(32) /angbracketleftBig ˙Pm2(r)/angbracketrightBigt = −0/integraldisplay −∞dt′eǫt′/integraldisplay dr′β{Dm2m2(r,r′t′)∇µm2(r′,¯t) + +L(m2,m2p)(r,r′,t′)δVm2,p(r′,¯t)+ +L(m2,m1m2)(r,r′,t′)δVm2,m1(r′,¯t)},(33)/angbracketleftBig ˙Pp(r)/angbracketrightBigt = −0/integraldisplay −∞dt′eǫt′/integraldisplay dr′β{−Dpp(r,r′,t′)∇βp(r′,¯t)+ +L(p,m1p)(r,r′,t′)δVp,m1(r′,¯t)+ +L(p,m2p)(r,r′,t′)δVp,m2(r′,¯t)+ +L(p,pp)(r,r′,t′)Vp(r′,¯t)}.(34) HereδVik=Vi−Vk, Dpp(r,r′,t′) =β(∇IPp(r),I∗ p(r′,t′)). Equations (32) - (34) allow conducting the analysis of how the interaction between the subsystems at hand af- fects the implementation ofthe drageffect. We introduce the average values of the forces induced by the chemical potential and temperature gradients: Fmi(r)=0/integraldisplay −∞dt′eǫt′/integraldisplay dr′Dmimj(r,r′,t′)∇µmj(r′,¯t) Fp(r)=0/integraldisplay −∞dt′eǫt′/integraldisplay dr′Dpp(r,r′,t′)∇βp(r′,¯t). Besides, introduce the inverse times of the magnon and phonon momentum relaxation caused by interac- tion with phonons processes of non-magnon relaxation of phonons. Let us designate them as ω(mp), andω(pp), respectively40,41 ω(γ,v)= (Pγ,Pγ)−10/integraldisplay −∞dt′eǫt′(˙P(γ,v),˙P(γ,v)(t′)), γ=m1,m2,p(35) We restrict ourselves to the discussion of a stationary case. For this purpose, we average the balance equa- tions over time t. To start the analysis, we consider the simplest case when the drift velocities of the magnon sys- tems are equal: Vm1=Vm2≡Vmandβm1=βm2. This actually means that we deal with one magnon and one phonon systems. In addition, we shall assume that the phonon momentum is maintained from the outside by an unchanged. Then, the momentum balance equations appear as Fm=Pmω(m,mp)(Vm−Vp), (36) 0 =Ppω(p,mp)(Vp−Vm)+Ppω(p,pp)Vp.(37) The balance equation for the magnon momentum ac- quires the form Fm=Vmω(p,pp)ω(m,mp) ω(m,mp)+ω(p,pp)Pm. (38)7 wherePm≡(Pm,Pm),Pp≡(Pp,Pp). Finally, the draggingleads to the change in frequency of the magnon- phonon collisions, and the quantity Ω =ω(p,pp)ω(m,mp) ω(m,mp)+ω(p,pp) is the inverse relaxation time of the magnon momentum by non-equilibrium phonons. From the expression (38) it follows that the drag ef- fect has an influence on the magnon-phonon collision fre- quency. The phonon subsystem almost always remains in equilibrium, and the inverse relaxation time is defined by the frequency ω(m,mp)provided that the inequality ω(p,pp)> ω(m,pm)is fulfilled. The latter means that the phonon momentum gained quickly relaxes in the pro- cesses of non-magnon relaxation. If the opposite inequal- ityω(p,pp)< ω(m,pm)holds, the leakage of the phonon momentum occurs slower than the gain momentum rate in the magnon-phonon collisions. In this case, the mech- anism of the non-magnon phonon relaxation mainly con- tributes to the drag effect. In addition, ¯ω(m,mp)≃ω(p,pp)(ω(m,mp)/ω(m,pm))= =ω(p,pp)(Pp/Pm)=Pm0/integraldisplay −∞dteǫt(˙P(p,pp),˙P(p,pp)).(39) Thus, the criterion of realizing the drag effect consists in the requirement ω(m,pm)>ω(p,pp)that coincides with the solution of the kinetic equation41. It is worth emphasiz- ing that to calculate the correlation function in the for- mulafor ω(p,pp), it isnecessaryto knowparticularmecha- nisms of the non-magnon phonon momentum relaxation. Forconsideringthedrageffectstherearetwomechanisms such as the Herring mechanism ω(p,pp)∼(kBT)3and the Simons mechanism ω(p,pp)∼(kBT)4leading to a rather strongtemperaturedependenceoftherelaxationfrequen- cies. Another limiting case corresponds to the situation when the drift velocities of thermal magnons and phonons are equal to Vm2=Vp(βm2=βp). In this case, thermal magnonsandphonons formone subsystem. From balance equations (32), (33) we obtain Fm1=Vm1ω(p,pp)[ω(p,mp)+ω(m,m1m2)] ω(p,mp)+ω(p,pp).(40) From the expression (41) it follows that if ω(p,pp)≫ ω(p,mp)thenF1∼ω(p,mp)andF1∼ω(m,m1m2)if ω(p,mp)≪ω(m,m1m2).If the opposite inequality, when ω(p,pp)≪ω(p,mp)thenF1∼ω(p,pp)[1+ω(m,m1m2)/ω(p,mp)] andF1∼ω(p,pp)ifω(m,m1m2)≪ω(p,mp)andF1∼ ω(p,pp)ω(m,m1m2)/ω(p,mp)whenω(m,m1m2)≫ω(p,mp). Now we look into the drag effect in the event of two magnon and one phonon systems. Then, the momentum balance equations can be written as follows. The set of the equations (36), (37) implies Fm1={ω(m1,mp)+ω(m,m1m2)−ω(m1,mp)ω(m1,mp) Ω}Vm1−−{ω(m1,mp)ω(m2,mp) Ω+ω(m,m1m2)}· ·{Fm2+(ω(m,m1m2)+ω(m1,mp)ω(m2,mp)/Ω)Vm1 ω(m2,mp)+ω(m,m1m2)−ω(m2,mp)ω(m2,mp)/Ω},(41) where Ω = ω(m1,mp)+ω(m2,mp)+ω(p,pp). Let the energy transfer channels from the magnonsub- systems to the phonon subsystem be equal ω(m1,mp)= ω(m2,mp)=ω(m,mp), Vm1=Vm. In this case we have Fm1={ω(m,mp)+ω(m,m1m2)−ω(m,mp)/Ω}Vm− −{ω(m,mp)/Ω+ω(m,m1m2)}× ×{Fm2+(ω(m,m1m2)+ω(m,mp)/Ω)Vm ω(m,mp)+ω(m,m1m2)−ω(m,mp)/Ω}.(42) Here Ω = 2+ ω(p,pp)/ω(m,mp). Ifω(p,pp)≫ω(m,mp), then Fm1={ω(m,mp)+ω(m,m1m2)}Vm− −ω(m,m1m2)·{Fm2+ω(m,m1m2) ω(m,mp)+ω(m,m1m2)}.(43) The expression (43) claims that the spin-wave current ∼F1is determined by the relations between the correla- tion functions ω(m,m1m2)andω(m,mp). As it follows from the expression (43) that if ω(m,m1m2)≪ω(m,mp)then F1∼ω(m,mp). In this case magnon-phonon interaction is the dominant channel of a magnon relaxation. If we have the opposite inequality ω(m,m1m2)≫ω(m,mp)then F1∼ω(m,m1m2). In this case the interaction between ”in- jected” and ”thermal” magnons is the dominant channel of a magnon relaxation. Moreover, the inelastic scatter- ing of the ”injected” magnons by ”thermal” ones can be regarded as scattering by impurity centers whose con- centration is temperature-varied. This interaction will determine the temperature-field behaviour of the spin- wave current under the conditions of the Seebeck spin effect. Because of the existence of two relaxation channels (magnon-phonon and magnon-magnon), the inelastic scattering of the ”injected” magnons by ”thermal” ones may give rise to the bottleneck effect and heating of the ”thermal” magnons. Such a situation emerges if the ”thermal”-magnon subsystem gains energy through the magnon-magnon channel faster than loses it along the magnon-phonon channel, i.e. ω(m,m1m2)≫ω(m,mp). CONCLUSION The formation of the two: ”injected” (coherent) and thermally excited, different in energies magnon subsys- tems and the influence of its interaction with phonons and between on drag effect under spin Seebeck ef- fect conditions in the magnetic insulator part of the metal/ferromagnetic insulator/metal structure is stud- ied. The analysis of the macroscopic momentum bal- ance equations of the systems of interest conducted for different ratios of the drift velocities of the magnon and8 phonon currents show that the injected magnons relax- ation on the thermal ones is possible to be dominant over its relaxation on the phonons. This interaction will be the defining in the forming of the temperature dependence of the spin-wave current under SSE condi- tions, and inelastic part of the magnon-magnon inter- action is the dominant spin relaxation mechanism. The existenceofthe tworelaxationchannels(magnon-phonon and magnon-magnon) in the case of inelastic scattering of the injected magnons on the thermal ones is shown tobe leading to the warming of the letter and to Narrow- neck effect. Such situation could be realized in the case of energy input rate threw the magnon-magnon channel to the ”thermal” magnons domination over its leaking rate threw magnon-phonon mechanism Acknowledgments The given work has been done as the part of the state task on the theme ”Electron” 01201463330 (project 12- T-2-1011) with the support of the Ministry of Education of the Russian Federation (Grant 14.Z50.31.0025) ∗Okorokovmike@gmail.com 1S. Maekawa, S. O. Valenzuela, E. Saitoh, T. Kimura (ed.) Spin Current (Oxford Univ. Press, 2012). 2A. V. Chumak, V. I. Vasyuchka, A. A. Serga and B. Hille- brands, Natur. Phys. 11, 453 (2015). 3V. V. Kruglyak, S. O. Demokritov, D. Grundler, J. Phys. D: Appl. Phys. 43, 264001 (2010). 4C. 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2017-06-06

The formation of the two: injected ("coherent") and "thermally" excited, different in energies magnon subsystems and the influence of its interaction with phonons and between on drag effect under spin Seebeck effect conditions in the magnetic insulator part of the metal/ferromagnetic insulator/metal structure is studied. An approximation of the effective parameters, when each of the interacting subsystems ("injected", "thermal" magnons, and phonons) is characterized by its own effective temperature and drift velocities have been considered. The analysis of the macroscopic momentum balance equations of the systems of interest conducted for different ratios of the drift velocities of the magnon and phonon currents show that the "injected" magnons relaxation on the "thermal" ones is possible to be dominant over its relaxation on phonons. This interaction will be the defining in the forming of the temperature dependence of the spin-wave current under spin Seebeck effect conditions, and inelastic part of the magnon-magnon interaction is the dominant spin relaxation mechanism.

Spin Seebeck effect and magnon-magnon drag in Pt/YIG/Pt structures

1706.02154v1

Microwave magnon damping in YIG lms at millikelvin temperatures S. Kosen,1,A. F. van Loo,1, 2D. A. Bozhko,3, 4, 5L. Mihalceanu,3and A. D. Karenowska1 1Clarendon Laboratory, Department of Physics, University of Oxford, Oxford OX1 3PU, United Kingdom 2Center for Emergent Matter Science, RIKEN, Wako, Saitama 351-0198, Japan 3Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universitaet Kaiserslautern, 67663 Kaiserslautern, Germany 4School of Engineering, University of Glasgow, Glasgow G12 8LT, United Kingdom 5Department of Physics and Energy Science, University of Colorado at Colorado Springs, Colorado Springs CO 80918, USA (Dated: October 28, 2019) Magnon systems used in quantum devices require low damping if coherence is to be maintained. The ferrimagnetic electrical insulator yttrium iron garnet (YIG) has low magnon damping at room temperature and is a strong candidate to host microwave magnon excitations in future quantum devices. Monocrystalline YIG lms are typically grown on gadolinium gallium garnet (GGG) sub- strates. In this work, comparative experiments made on YIG waveguides with and without GGG substrates indicate that the material plays a signicant role in increasing the damping at low tem- peratures. Measurements reveal that damping due to temperature-peak processes is dominant above 1 K. Damping behaviour that we show can be attributed to coupling to two-level uctuators (TLFs) is observed below 1 K. Upon saturating the TLFs in the substrate-free YIG at 20 mK, linewidths of 1:4 MHz are achievable: lower than those measured at room temperature. Microwave magnonic systems have been subject to ex- tensive experimental studies for decades. This work is motivated not only by an interest in their rich basic physics, but also by their potential application as infor- mation carriers in beyond-CMOS electronics [1, 2]. Re- cently, enthusiasm has grown for the study of magnon dynamics at millikelvin (mK) temperatures, the tempera- ture regime in which solid-state microwave quantum sys- tems operate [3{12]. This work oers the possibility to explore the dynamics of microwave magnons in the quan- tum regime and to study novel quantum devices with magnonic components [13{15]. Arguably the most important material in the context of room-temperature experimental magnon dynamics is the ferrimagnetic insulator yttrium iron garnet (Y 3Fe5O12, YIG). Pure monocrystalline YIG has the lowest magnon damping of any known material at room temperature [16] and is produced in the form of bulk crystals and lms. Films suitable for use as waveguides in conjunction with micron-scale antennas are grown by liquid-phase epitaxy to a thickness of between 1 and 10 m on gadolin- ium gallium garnet (Gd 3Ga5O12, GGG) substrates. The use of GGG is motivated by the need for tight lattice matching to assure a high crystal quality. Recently, YIG lms were recognised as promising media for the study of magnon Bose-Einstein condensation and related macro- scopic quantum transport phenomena [17{20]. In the context of quantum measurements and information pro- cessing, YIG lms hold noteworthy promise, however, if they are to be practical, they must be shown to exhibit the same (or better) dissipative properties at mK tem- peratures as they do at room temperature. Magnon linewidths in YIG at mK temperatures have thus far only been characterised in bulk YIG resonators sandoko.kosen@physics.ox.ac.uk B (Top View)(Cross Section)B Sample Spacer Signal line0 -1 Signal (dB) Frequencyf0=4GHzAbsorption Spectrum d w∆f A (b) InputOutput(a) -20dB-20dB (20mK - 9K)100mK4K70K -20dB BPF50Ω SampleBPF LODAQ LPF 50ΩRF LOQ IRF room temperature50ΩFIG. 1. (a) The measurement conguration used to charac- terise the sample's damping. The sample and the microstrip (signal line) are separated from each other by a spacer. When the microwave drive is resonant with the magnons in the sam- ple, a decrease in the transmitted signal is observed. (b) The low-temperature setup and its corresponding data acquisition system at room temperature. (specically, spheres) [4, 6, 7, 10, 21]. Bulk YIG has been shown to retain its low magnon damping at mK temperatures. However, in the case of YIG lms grown on GGG, the story is more complex. GGG is a geomet- rically frustrated magnetic system [22] and it has long been known that at temperatures below 70 K, it exhibits paramagnetic behaviour that has been reported to in- crease damping in lms grown on its surface [23{25]. The behaviour of GGG at mK temperatures is yet to be thoroughly characterised [26{28], but recent results at mK temperatures have suggested that magnon damp- ing in YIG lms grown on GGG is higher than expected if the properties of the YIG system alone are consideredarXiv:1903.02527v3 [cond-mat.mes-hall] 25 Oct 20192 4 5 6 7 Δf (MHz) Δf (MHz) fo (GHz)T = 300K T = 20mK, Pb = -65dBm T = 20mK, Pb = -100dBmSubstrate-free YIG YIG/GGG T = 300K T = 20mK, Pb = -65dBm T = 20mK, Pb = -100dBm1234 371115 FIG. 2. Magnon linewidths (
f) versus resonance frequency (f) for a YIG/GGG lm and a substrate-free YIG lm. Datasets at room temperature (300 K, ) are obtained with an input power of -25 dBm. Datasets at 20 mK are obtained for two input powers Pb=65 dBm (N), and Pc=100 dBm (). Each error bar represents the standard deviation of the linewidth values obtained from repeated measurements. Dashed lines are linear ts, the details of these ts are sum- marised in table I. Note the dierent scaling of the vertical axes of the plots. [9, 11, 29]. In this work we report a comparative set of ex- periments on YIG lms with and without GGG and move toward a more complete understanding of the damping mechanisms involved. We present data from the measurement of two YIG samples: a 11 m-thick lm and a substrate-free 30 m- thick lm. Both samples are grown using liquid phase epitaxy with the surface normal of the YIG lm (and the substrate) parallel to the h111icrystallographic direction. The substrate-free YIG is obtained by mechanically pol- ishing o the GGG until a 30 m-thick pure YIG lm is obtained [25]. The corresponding lateral size of each sample can be found in table I. We measure the damping in both lms using the microstrip-based technique illustrated in g. 1(a) [30]. The sample is positioned above a microstrip and magne- tised by an out-of-plane magnetic eld ( B). Continuous- wave microwave signals transmitted through the mi- crostrip probe the ferromagnetic resonance of the sam- ple. In the room-temperature experiments, the trans- mitted signals are measured by connecting the two ends of the microstrip directly to a commercial network anal- yser. In our low-temperature experiments, the sample is mounted on the mixing-chamber plate of a dilution re- frigerator as shown in g. 1(b), similar to that used in Ref. [11]. A microwave source is used to generate the input microwave signal. At the input line, three 20 dB attenuators are used to ensure an electrical noise tem- perature that is comparable to the temperature of theTABLE I. Comparing results at 300 K and at 20 mK YIG/GGG Substrate-free YIG Size 2 mm3 mm10m 1 mm1 mm30m w=d 1.7 mm / 70 m 0.9 mm / 540 m 300 K 1a= (224)1052a= (8:90:5)105
f;1a= (0:70:4) MHz
f;2a= (0:90:1) MHz 20 mK 1b= (745)1052b= (2:30:7)105 Pb=-65 dBm
f;1b= (1:70:6) MHz
f;2b= (1:10:1) MHz 20 mK 1c= (856)1052c= (9:31:0)105 Pc=-100 dBm
f;1c= (2:60:6) MHz
f;2c= (2:00:1) MHz sample. The output signals then pass through two cir- culators, a bandpass lter, and an amplier, before they are down-converted to a 500 MHz signal at room temper- ature. A DAQ card then digitises the transmitted signal at a 2.5 GHz sampling frequency. Signals are usually av- eraged about 50000 times before being digitally down- converted in order to obtain a signal similar to the one shown in g. 1(a). The magnon linewidth is given by the full-width at half maximum (FWHM) of the Lorentzian t to the transmitted signal. The measurement frequency range is between 3.5 GHz and 7 GHz. The low-frequency limit is imposed by the limited bandwidth of the circu- lators used for our low-temperature measurement setup and the top of the measurement band is determined by the maximum external eld that can be applied by our magnet. The measured damping comprises contributions from the sample and from radiation damping caused by its interaction with the microstrip. In our experiments, ra- diation damping originates from eddy currents excited in the microstrip by the magnetic eld of the magnons [31, 32] and can be decreased by increasing the separation between the sample and the microstrip ( d) at the expense of reducing the measured absorption signal strength ( A). There is therefore a tradeo to be made between be- ing able to measure linewidths very close to the intrinsic linewidth of the sample (thick spacer, negligible radiation damping) and being able to achieve sucient signal-to- noise (SNR) ratio (thin spacer, non-negligible radiation damping). Table I lists the microstrip-sample spacings (d) in our experiments. Since earlier experiments sug- gested that the YIG/GGG linewidth would be higher at low temperature, the YIG/GGG sample is closer to the microstrip to maintain sucient SNR. Within the YIG lm itself, the primary contributions to magnon damping are: intrinsic processes [33, 34], temperature-peak processes [35, 36], two-level uctuator (TLF) processes [4] and two-magnon processes [35, 36]. Intrinsic processes are due to interactions with optical phonons and magnons; they are expected to decrease with reducing temperature. Temperature-peak processes originating from interactions with rare-earth impurities are only signicant at low temperature (above 1 K). TLF processes are due to damping sources that behave as two- level systems; they are dominant below 1 K. Two-magnon processes have their origins in inhomogeneities in the ma-3 0.02 0.1 1 1003691210152025Δf (MHz) Δf (MHz) T (K)YIG/GGG Substrate-free YIGfo = 4GHz fo = 7GHz fo = 4GHz fo= 7GHz FIG. 3. Temperature ( T) dependent magnon linewidths (
f) for both YIG/GGG lm and substrate-free YIG, measured with input power Pc=100 dBm. Note the dierent scaling of the vertical axes of the plots. terial; in our experiments, they are minimized by mag- netising the sample out of plane [37, 38]. Figure 2 compares the magnon linewidth (
f) of each sample at 300 K (room temperature) and at 20 mK as a function of the ferromagnetic resonance frequency ( f). Results at 300 K are obtained by sweeping the input microwave frequency under constant B-eld. Results at 20 mK are obtained by sweeping the B-eld at con- stant input microwave frequency. In the latter case, the linewidths are measured in terms of magnetic eld (
B) and converted to units of frequency (
f) via the rela- tion
f= ( =2)
B, where is the gyromagnetic ra- tio. Note that there is no conversion factor other than =2that is used to translate the low-temperature eld- domain data into the frequency domain. A linear t to
f= 2f+
fgives the characteristic Gilbert damp- ing constant (unitless) and the inhomogeneous broad- ening contribution
f. Table I summarises the results of linear ts to data in g. 2. We rst compare the results at 300 K and 20 mK obtained at relatively high input drive level ( Pb= 65 dBm). The substrate-free YIG shows a measured linewidth decreasing from the room temperature value to approximately 1 :4 MHz at 20 mK. The reduction in damping is as anticipated by existing models that de- scribe the intrinsic damping of YIG [33{35]. The ra- diation damping contribution to the linewidth for the substrate-free YIG is small due to the large spacing from the microstrip ( d= 540 m). The YIG/GGG sample is substantially closer ( d= 70m) to the microstrip and its measured therefore includes a non-negligible radiation damping contribution r. In our raw data, uncorrected for this eect, we mea- sure a damping constant at 20 mK ( 1b) that is 3:4 timeslarger than the room temperature value ( 1a). Following Ref. [31], the radiation damping can be modelled with an equivalent Gilbert damping constant r=CgMs, where Cgdepends on the geometry of the system and Msis the saturation magnetisation of the sample. As both 300 K and 20 mK measurements are performed with iden- tical sample geometry, it is reasonable to expect that the change in ras the temperature is lowered is due to the change in Ms. Therefore, the increase in rbetween 20 mK and 300 K is determined by the ratio of the sat- uration magnetisation, i.e. Ms(20 mK)=Ms(300 K)1:4 [39]. The fact that we see a signicantly larger damp- ing increase ( 1b=1a3:4) in the YIG/GGG and a decrease (2b=2a0:26) in the substrate-free YIG in- dicates that the GGG plays an important role in increas- ing the magnon linewidth of the YIG/GGG sample at 20 mK. The parameters and
fin both samples increase as the input drive level ( Pc) reduces as shown in Table I. This behaviour can be explained by the TLF model upon which we shall elaborate later. Figure 3 shows the temperature dependence of the magnon linewidths for both samples measured at low in- put power ( Pc=100 dBm). For the YIG/GGG results in g. 3, the radiation damping contribution ( r=CgMs [31]) across the examined temperature range can be con- sidered to be an approximately constant vertical shift to each dataset. This is due to the small change (less than 0.07%) inMsof YIG between 20 mK and 9 K [39]. Above 1 K, linewidths of both samples increase as the temperature is increased up to 9 K. In this temperature range, damping is dominated by temperature-peak pro- cesses caused by the presence of rare-earth impurities in the YIG [25, 35, 39{41]. When temperature-peak pro- cesses are dominant, the linewidth of the sample peaks at a characteristic temperature ( Tch) determined by the damping mechanism and the type of impurity. Temperature-peak processes at low temperatures fall into two categories [35, 36]: those associated with (1) rapidly relaxing impurities, and (2) slowly relaxing impu- rities. Rapidly relaxing impurities produce a Gilbert-like damping and a characteristic temperature Tchthat is in- dependent of the magnon resonance frequency f. Slowly relaxing impurities produce a non-Gilbert-like damping and a corresponding characteristic temperature that de- creases as the resonance frequency fis lowered. The behaviour observed in g. 3 at 9 K, with the linewidth for thef= 4 GHz being higher than that at f= 7 GHz, in- dicates that impurities of slowly relaxing type dominate in this temperature range. As the temperature is decreased below 1 K, linewidths for the substrate-free YIG start to increase and eventu- ally saturate as shown in g. 3. This can be explained by the presence of two-level uctuators (TLFs) and has been previously observed in a bulk YIG [4]. In the TLF model, the damping sources are modelled as an ensem- ble of two-level systems with a broad frequency spectrum4 -120 -100 -80 -601234 -120 -100 -80 -600.02 0.1 0.3 1.0 3.012345Δf (MHz) Δf (MHz)T (K) P (dBm) P (dBm)Substrate-free YIG 20mK 300mK 1K20mK 300mK 1KPsat(20mK) Psat(300mK)(a) (b)low drive high drivehigh drive level: Pb = -65dBm low drive level: Pc = -100dBm fo = 4GHz fo = 7GHz Psat(20mK)4GHz 4GHz7GHz 7GHz Substrate-free YIG Substrate-free YIG FIG. 4. Magnon linewidths (
f) in the substrate-free YIG lm for various temperatures ( T) and input powers ( P). (a) Temperature dependent linewidths for two dierent input powers Pb=65 dBm and Pc=100 dBm. (b) Power de- pendent linewidths at 20 mK, 300 mK, and 1 K. The dashed lines are the ts to the data. [42, 43]. The linewidth contribution can be expressed as
fTLF=CTLF!tanh ( ~!=2kBT)p 1 + (P=P sat)(1) whereCTLF is a factor that depends on the TLF and the host material properties. The power-dependent term can be rewritten as P=P sat= 2 r12, where ris the TLF Rabi frequency, 1and2are respectively the TLF longitudinal and transverse relaxation times [44]. At high temperatures ( kBThfTLF), thermal phonons saturate the TLFs and therefore the material behaves as if the TLFs were not present. At low temper- atures (kBThfTLF) and low drive levels ( PPsat), most of the TLFs are unexcited. Under these condi- tions, the TLFs increase the damping of the material by absorbing and re-emitting magnons or microwaves at rates set by their lifetimes, coupling strength, and den- sity. When the drive level is increased past a certain threshold (PPsat), the TLFs are once again saturated and therefore do not contribute to the damping. Evidence for the presence of the TLFs is shown in gs. 2 and 4. The datasets for 20 mK in g. 2 show that the linewidths for both samples are lower when the drive level is higher ( PbvsPc). Figure 4(a) shows a similar be- haviour in the substrate-free YIG. Above 1 K, linewidths for both drive levels are similar: an indication that the relevant TLFs are saturated by thermal phonons. Thedierences in linewidths for the two drive levels begin to appear as the temperature is lowered below 1 K. Figure 4(b) shows the linewidths of the substrate-free YIG as a function of drive level ( P) at three dierent tem- peratures (1 K, 300 mK, and 20 mK). At 1 K, there is no observable power dependence as the relevant TLFs have been saturated by the thermal phonons. At 20 mK and 300 mK, the linewidth increases as the power decreases, saturating at mK temperatures. This is in agreement with the theory previously articulated and the ts shown by dashed lines in g. 4(b). The data are tted using eq. (1) with an additional y-intercept to account for non- TLF linewidth contributions. For thef= 7 GHz dataset in g. 4(b), Psatat 300 mK is clearly larger than at 20 mK. This is in-line with ex- pectations: 1and2are anticipated to decrease as the temperature is increased, leading to a higher Psat(recall thatPsat/1=12) [44{46]. The exact temperature de- pendence of 1 =12is not clear; in previous experiments, a phenomenological model was suggested with the quan- tity 1=12varying from T2toT4[45]. This places the ratioPsat(300 mK)=Psat(20 mK) in the range of 23.5 dB to 47 dB. The tted Psatvalues from our data correspond to a ratio of approximately 22.5 dB, suggestive of a T2 behaviour. It should be noted that the f= 4 GHz,T= 300 mK dataset shows a very weak TLF eect since there are su- cient thermal phonons to saturate the TLFs with central frequencies around 4 GHz; this is not the case for higher frequency datasets taken at the same temperature. A higherPsatis also observed at 300 mK for f= 5 GHz andf= 6 GHz (data not shown). Figure 4(a) shows that the input power Pb=65 dBm used in our experiments is not enough to saturate the relevant TLFs for temperatures between 100 mK and 1 K. The datasets obtained with high drive level ( Pb) in g. 4(a) show that the linewidth dierence f= j
f(f= 7 GHz)
f(f= 4 GHz)jbroadens as the temperature is increased from 100 mK to 300 mK, nar- rowing back as the temperature reaches 1 K. If a higher drive level is used, fis expected to be smaller at tem- peratures between 100 mK and 1 K. In conclusion, the substrate GGG on which typical YIG lms are grown signicantly increases the magnon linewidth at mK temperatures. However, if the substrate is removed, it is possible to obtain YIG linewidths at mK temperatures that are lower than the room-temperature values. Measured linewidths of both YIG/GGG and substrate-free YIG systems above 1 K are consistent with the temperature-peak processes typically observed in YIG containing rare earth impurities. Damping due to the presence of unsaturated two-level uctuators is ob- served in both YIG/GGG and substrate-free YIG lms below 1 K. We observe the TLF saturation power to be higher at higher temperatures in agreement with the ex- isting literature. We further verify that using high drive level reduces the linewidths of the substrate-free YIG lms down to1:4 MHz (f= 3:5 GHz to 7.0 GHz)5 at 20 mK. Looking forward, our measurements suggest that|in the context of the development of magnonic quantum information or measurement systems|it may be worth- while to investigate the possibility of growing YIG lms on substrates other than GGG, or techniques which cir- cumvent the use of a substrate entirely [47{50]. It should be emphasised that the current experimental congura- tion does not allow us to pinpoint the origin of the TLFs; further investigations into TLFs in YIG would be useful in obtaining high-quality YIG magnonic devices that op- erate in the quantum regime. Note added - A pre-print by Prrmann et al. [21] recently reported experiments concerning the eect of two-level uctuators on the linewidth of bulk YIG. This work helpfully complements our investigations into thebehaviour of YIG lms. ACKNOWLEDGMENTS We thank A.A. Serga for helpful discussions, and J.F. Gregg for the use of his room-temperature magnet. Support from the Engineering and Physical Sciences Re- search Council grant EP/K032690/1 (SK, AFvL, ADK), the Deutsche Forschungsgemeinschaft Project No. INST 161/544-3 within SFB/TR 49 (DAB, LM), the Indonesia Endowment Fund for Education (SK), and the Alexan- der von Humboldt Foundation (DAB) is gratefully ac- knowledged. AFvL is an International Research Fellow at JSPS. [1] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands, Nature Physics 11, 453 (2015). [2] A. V. Chumak, A. A. Serga, and B. Hillebrands, Jour- nal of Physics D: Applied Physics 50, 244001 (2017), arXiv:1702.06701. [3] H. Huebl, C. W. Zollitsch, J. Lotze, F. Hocke, M. Greifen- stein, A. Marx, R. Gross, and S. T. B. Goennen- wein, Physical Review Letters 111, 127003 (2013), arXiv:1207.6039. [4] Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Us- ami, and Y. 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2019-03-06

Magnon systems used in quantum devices require low damping if coherence is to be maintained. The ferrimagnetic electrical insulator yttrium iron garnet (YIG) has low magnon damping at room temperature and is a strong candidate to host microwave magnon excitations in future quantum devices. Monocrystalline YIG films are typically grown on gadolinium gallium garnet (GGG) substrates. In this work, comparative experiments made on YIG waveguides with and without GGG substrates indicate that the material plays a significant role in increasing the damping at low temperatures. Measurements reveal that damping due to temperature-peak processes is dominant above 1 K. Damping behaviour that we show can be attributed to coupling to two-level fluctuators (TLFs) is observed below 1 K. Upon saturating the TLFs in the substrate-free YIG at 20 mK, linewidths of 1.4 MHz are achievable: lower than those measured at room temperature.

Microwave magnon damping in YIG films at millikelvin temperatures

1903.02527v3

Nonlocal Anomalous Hall Eect Steven S.-L. Zhang and Giovanni Vignale Department of Physics and Astronomy, University of Missouri, Columbia, MO 65211 (Dated: May 27, 2022) The anomalous Hall eect is deemed to be a unique transport property of ferromagnetic metals, caused by the concerted action of spin polarization and spin-orbit coupling. Nevertheless, recent experiments have shown that the eect also occurs in a nonmagnetic metal (Pt) in contact with a magnetic insulator (yttrium iron garnet (YIG)), even when precautions are taken to ensure there is no induced magnetization in the metal. We propose a theory of this eect based on the combined action of spin-dependent scattering from the magnetic interface and the spin Hall eect in the bulk of the metal. At variance with previous theories, we predict the eect to be of rst order in the spin-orbit coupling, just as the conventional anomalous Hall eect { the only dierence being the spatial separation of the spin orbit interaction and the magnetization. For this reason we name this eect nonlocal anomalous Hall eect and predict that its sign will be determined by the sign of the spin Hall angle in the metal. The AH conductivity that we calculate from our theory is in good agreement with the measured values in Pt/YIG structures. Introduction.The anomalous Hall (AH) eect is the generation of an electric current perpendicular to the electric eld in a ferromagnetic metal [1]. At variance with the ordinary Hall eect, which arises from the ac- tion of a magnetic eld on the orbital motion of the electrons, the AH eect is ascribed to strong spin-orbit coupling in concert with spin-polarized itinerant electrons. The spin orbit coupling plays a cen- tral role in inducing a left-right asymmetry (with respect to the direction of the electric eld) in the scattering of electrons of opposite spins. It is this asymmetry that generates a transverse charge current from a longitudi- nal spin current. The same scattering process generates a pure transverse spin current for systems with spin unpo- larized electrons, which is known as spin Hall eect [2{5]. Based on this picture, the conventional AH eect appears atrst order in spin orbit coupling, no matter which kind of microscopic mechanisms predominates. Recently, an AH signal has also been detected in a Platinum (Pt) layer in direct contact with a YIG layer [6{ 8]. The former is a non-magnetic heavy metal with strong spin orbit coupling and the latter is a well-known ferro- magnetic insulator. In view of the two aforementioned ingredients for the AH eect in ferromagnets, it is puz- zling that an AH current would arise in Pt in the absence of spin polarized conduction electrons. In a rst attempt to solve the puzzle, Huang et. al. [6] showed that the Pt layer in close proximity with YIG acquires ferromagnetic characteristics, which essentially subsumes the novel AH eect under the conventional AH eect for ferromagnetic metals. This explanation ran into diculties when it was found that the AH eect persists in Pt/Cu/YIG tri- layers [9] where the Cu layer is deliberately inserted to eliminate the magnetic proximity eect. An alternative explanation was then proposed [9, 10],based on the physical mechanism depicted in panel (a) of Fig. 1. In this mechanism the applied charge cur- rentjxgenerates, via the spin Hall eect, a spin current Qy zpropagating in the zdirection with spin along the ydirection. When those electrons carrying Qy zare re- ected back from the magnetic interface, spin rotation occurs and gives rise to a spin current of Qx z, which in turn induces a transverse charge current jyvia the in- verse spin Hall eect [11, 12]. Based on this picture, the transverse electric current is of second order in the spin orbit coupling or spin Hall angle, which is qualitatively dierent from a conventional AH current. It is worth mentioning that a t to the experimental data based on this model [7, 10], requires a spin diusion length on the order of 1 nm. Such a short spin diusion length, an order of magnitude smaller than the room-temperature electron mean path of Pt [13], casts doubt on the internal consistency of the spin diusion model. In this paper, we propose a dierent mecha- nism for the AH current observed in hybrid heavy- metal/ferromagnetic-insulator structures. The essential new ingredient is the scattering of electrons from the (rough) metal-insulator interface. Because the insulator is magnetic, the scattering rate is spin-dependent (see Appendix B for a proof). This means that a charge cur- rent owing parallel to the interface is partially converted to a spin current, while a spin current owing parallel to the interface is partially converted to a charge current. The surface-induced conversion of charge to spin current and viceversa conspires with the spin Hall eect in the bulk of the metal to produce the observed AH current. This may happen in two ways: in the rst process, (b1), the charge current jxgenerates, via spin-dependent inter- facial scattering a spin current Qz x, which subsequently gives rise to the transverse spin polarized current jyvia the inverse spin Hall eect; in the second process, (b2),arXiv:1512.04146v1 [cond-mat.mes-hall] 14 Dec 20152 M-𝑸𝑧𝑦 jy -𝑸𝑧𝑥 -jx 𝑥𝑦𝑧 (a) Spin Hall AH eect ( /2 sh) M jy -𝑸𝑥𝑧 -jx (b1) M -𝑸𝑦𝑧 -jx jy (b2) (b) Nonlocal AH eect ( /sh) FIG. 1: Schematics of two dierent mechanisms of the AH eect in heavy-metal (HM)/ferromagnetic-insulator (FMI) bilayers: (a) the spin Hall AH mechanism and (b) nonlocal AH mechanism with two coexisting physical processes depicted separately in panels b1 and b2. The curved arrows represent the trajectories of electrons upon spin orbital scattering and the dotted arrows stand for spin dependent scattering at the magnetic interface. the applied charge current jxrst generates, via spin Hall eect, a transverse spin current Qz y, which is then turned into a spin polarized current jydue to spin dependent in- terfacial scattering. Both physical processes involve spin orbit scattering only once (through the spin Hall eect) and hence the resulting AH current is of rst order in the spin orbit coupling or spin Hall angle. As a matter of fact, this AH eect has the same physical nature as its conventional counterpart in bulk ferromagnets, and diers from the latter only in the spatial separation of the spin orbit interaction and the magnetization: it is for this reason that we name it nonlocal AH eect . Compared to the double spin Hall eect mechanism proposed in Refs. [9, 10], our proposal replaces one of the spin Hall steps, the rst or the second, by a spin- dependent interfacial scattering. This leads to a good quantitative description of the transverse current with- out the need of introducing an exceedingly small spin diusion length, as we will show in details in the remain- der of the paper. In fact, our mechanism survives in the limit of innite spin diusion length, while the double spin Hall eect mechanism vanishes in that limit [10]. In addition, the new mechanism has distinctive features that can be tested experimentally, the most striking one being the sign of the eect, which we predict to track the sign of the bulk spin Hall angle. Linear response theory Let us consider a metal/insulator bilayer as shown in Fig. 1 with an external electric eld applied in the xdirection (i.e., Eext=Eext^x) and with the magnetization of the insula- tor layer pointing in the zdirection, i.e., m=^z. We also assume that both surfaces of the metal are rough, but on the average translational invariance is recovered so that the transport properties are independent of xandy coordinates. The linear response of current densities tospin dependent electric elds can be written as follows j(z) =C0E(z) +CsEk(z) Qk(z) =C0Ek(z) +CsE(z) Q?(z) =C0 rE?(z) +C00 r^zE?(z) (1) where j(z) = (jx;jy) is the in-plane current density (note thatjz= 0 everywhere in the metal layer due to the open boundary conditions), Qk= (Qz x;Qz y) is the in- plane spin-current density (with spin in the zdirection), andQ?= (Qx z;Qy z) is the perpendicular-to-plane spin current density with Qx zandQy zcarrying the xandy components of the spin. The corresponding elds are E= (Ex;Ey),Ek= (Ez x;Ez y) and E?= (Ex z;Ey z). Notice thatCkis dened as the integral operator with kernel ck(z;z0), i.e.,Ckf(z)R dz0ck(z;z0)f(z0). WhileC0is an ordinary in-plane conductivity, Csdescribes the gen- eration of an in-plane spin current from an electric eld in the presence of surface scattering. As we show below, Csis the essential ingredient of our theory, producing an AH current of rst order in the spin Hall angle. On the other hand, C0 randC00 r{ respectively the real and the imaginary part of the spin-mixing conductance [14] { contribute only to second order. In particular, C00 ris the essential ingredient of the spin Hall mechanism of the AH eect [10]. In the presence of the spin-orbit scattering, the driving electric elds E;Ek;E?are self-consistently determined by the internal current densities as follows E=Eext+sh^z(QkQ?) Ek=sh^zj E?=sh^zj (2) wheresh0shwith0being the Drude resistivity andshthe spin Hall angle of the metal layer. Solving3 the system of linear equations (1) and (2), we obtain a general expression for the AH current density up to O 2 sh
jy(z) = shfC0;Csg2 shC0C00 rC0 Eext (3) wheref;grepresents the anticommutator of the two in- tegral operators. Note that with nite Csthe AH eect appears already at the rst order of sh. The two or- derings ofC0andCsin the anticommutator of Eq. (3) correspond to the processes b1 and b2 of Fig. 1. The sec-ond term on the right hand side of Eq. (3) corresponds to the spin Hall AH eect which is of second order in sh and is proportional to the imaginary part of the spin- mixing conductivity kernel. In what follows, we employ the Boltzmann transport theory to explicitly construct the integral kernels C0andCsin the presence of a rough magnetic interface. Boltzmann theory To quantitatively describe the non- local AH eect in a heavy metal thin layer with an external electric eld applied in the xdirection (see Fig. 1(b)), we make use of the spinor Boltzmann equation in the relaxation time approximation [3, 15{17] vz@^f(k;z) @zeEextvx @^f0 @"k! +
[ek^(k;z)] so=^f(k;z)^f(k;z) +2^f(k;z)^ITr^f(k;z) sf(4) where ^f0and ^f(k;z) are 22 matrices repre- sent respectively the equilibrium and nonequilibrium spinor distribution functions, v=d"k=~dkis conduc- tion electron velocity,^f(k;z)(1=4)R d k^f(k;z) is the angular average of the distribution and ^(k;z)(1=4)R d kek^f(k;z) is its dipolar moment, withekthe unit vector of k. Non-spin- ip and spin-slip processes are included, with andsfbeing the momen- tum and spin relaxation times respectively. The addi- tional source term 1 so
[ek^(k;z)], where1 sois the spin-orbit scattering rate, is responsible for the spin Hall eect [18{20]. It is this term that generates the current- dependent elds in Eq. (2). The crucial step in our theory is the description of spin- dependent interfacial scattering via boundary conditions for the distribution function. For the interface (at z= 0) between the heavy metal and the ferromagnetic insulator, we impose the following generalized Fuchs-Sondheimer boundary condition [21], ^f+(k;0) =1 2^s^Ry^f(k;0)^R+1 2 ^I^sD ^f(k;0)E +h:c: (5) whereh:c:represents hermitian conjugate which ensures ^f+to be an hermitian, ^Iis the 22 identity matrix,D ^fE = (2)1R dk^fwithkthek-space azimuthal an- gle, and both ^ sand ^Rare 22 matrices in spin space which are responsible for spin dependent specular re ec- tion and spin rotation of incident electrons. The matrix ^R, satisfying ^Ry^R=^I, is the re ection amplitude matrix which captures the spin rotation of electrons that are specularly re ected from the magnetic interface (Note that we assume such a coherent spin ro- tation does not occur for the diusively scattered elec-trons). The explicit form of ^Rcan be determined by electron wave function matching subject to the following spin-dependent potential barrier ^V(z) = Vb^IJex^z  (z) (6) whereVbis the averaged potential barrier of the insulator, Jexmeasures the spin splitting of the energy barrier, ^ z is thezcomponent of the Pauli spin matrices, and  ( z) is the unit step function. Explicitly, ^Rtakes the following form (see Appendix B for the derivation) ^R=R"+R# 2 ^I+R"R# 2 ^z (7) whereR=(+ikz)=(ikz) withkzthe zcomponent of the electron wave vector, p 2me(VbJex)k2z(we have let ~= 1 for notation convenience) and m ebeing the electron eective mass. The matrix ^ s, on the other hand, is introduced to de- scribe the averaged eects of spin dependent scattering at the magnetic interface due to roughness, impurities, etc. In general, we write [22, 23] ^s=s0 ^I+ps^z (8) wheres0 s"+s#
=2 is the average of the specular re- ection coecients s"ands#for spin-up and spin-down electrons with \up" and \down" dened with respect to m(=^z), andps s"s#
= s"+s#
is their asymme- try. A simple model calculation for the rough interface yields (see Appendix B for the detailed calculation), to the lowest order in Jex=Vb, the specular re ection asym- metryps' 2Jex Vb(1s0) fors0.1. Note that ps4 isnegative , meaning that more spin-down electrons are specularly scattered than spin-up electrons, for the for- mer encounter a higher energy barrier. Also, we notice that a rough magnetic interface is essential for the spin asymmetry of the specular re ection coecients: for an ideally at interface, both s"ands#are exactly equal to one, and no charge/spin conversion can occur. For the outer surface at z=d, we assume, for simplic- ity, that the scattering is diusive, i.e., ^f(k;d) =D ^f+(k;d)E (9) Note that the boundary conditions given by Eqs. (5) and (9) demand that both charge and spin currents ow- ing along the z-direction vanish at the outer (non mag- netic) surface, whereas only the charge current and the z-component of the spin current owing along the z- direction vanish at the magnetic surface. By solving the Boltzmann equation (4) with the boundary conditions given by Eqs. (5) and (9), we have calculated the current densities in the heavy-metal layer. Up to rst order in sh(=so), the Hall current density can be expressed as follows jah y(z) =shEextZd 0dz0 le[cs(z;z0) c0(z0) +c0(z;z0) cs(z0)] (10) whereleis the electron mean free path, the nonlocal in- tegral kernels cs(z;z0) andc0(z;z0) are given by c0(z;z0) =3 4Z1 0d 1
 s0ez+z0 le+ejzz0j le (11) and cs(z;z0) =3 4psZ1 0d 1
s0ez+z0 le (12) with their spatial averages dened as  c0(z)Rd 0dz0 lec0(z;z0) and cs(z)Rd 0dz0 lecs(z;z0). The non- locality of the AH eect, i.e., the spatial separation of the spin-orbit scattering and the magnetization, is clearly re ected in the structure of these integral kernels which depend on the relative distance between the current and eld points as well as the distance of their center of mass coordinate from the interface. Equations (10)-(12) are the main results of this paper. One of the most remarkable features of the nonlocal AH eect is that it appears at the rst order of the spin Hall angle, which is distinctly dierent from the spin Hall AH eect which occurs at the second order. Since psis negative, the directions of the nonlocal AH and the spin Hall AH currents would be the same for positive shbut theopposite for negative sh, as can be seen from Eq. (3). Furthermore, the nonlocal AH is independent of spin dif- fusion and thus is present in both ballistic and diusive 10-210-11001011020.00.51.01.5 s0 = 0.6 s0 = 0.7 s0 = 0.8 s0 = 0.9Ιahy / Ι0 ( 10-5 )d / leFIG. 2: The ratio of total AH current Iah ytoI0(=c0Eextwd) as a function of the thickness of the heavy metal layer for several specular re ection parameters. Other Parameters: sh= 0:05,Jex= 0:01eVandVb= 12eV. regimes, whereas the spin Hall AH eect vanishes as the thickness of the metal layer becomes much smaller than the spin diusion length [10]. The total AH current can be calculated from Eq. (10) by integrating the AH current density over the thickness of the layer, i.e., Iah y(d)wRd 0dzjah y(z) withwbe- ing the width of the metal bar. By doing so, we nd Iah y(d) = 2shEextwRd 0dz0 lecs(z0) c0(z0) where the fac- tor of 2 shows that the two physical processes that we described in Fig. 1b contribute equally to the total AH current. In Fig. 2, we show the thickness dependence of the total AH current for several values of the specular re ection coecient. We nd that Iah ybegins to saturate when the thickness reaches the electron mean free path. Also, we note that the saturation current is smaller for a smoother surface (larger s0), as expected from the above discussions. Experimentally, a most relevant quantity is the ratio of the spatially averaged AH resistivity to the longitudinal resistivity, i.e., ahah xy(d)=xx(d). The AH resistiv- ity can be obtained by inverting the conductivity tensor. Sincepss0sh.101, to a good approximation, we can take ah xy'cah xy=c2 xxwhere cah xyd1Rd 0dzjah y(z)=Eext withjah y(z) given by Eq. (10). In Fig. 3, we show the thickness dependence of ahfor several values of the spec- ular re ection coecient s0. Fordle,ahtends to zero, because  xx(d) increases with decreasing layer thickness. In the opposite limit of dle,ahalso dimin- ishes since the nonlocal AH eect is essentially an inter- face eect, which saturates for thicknesses larger than the electron mean path. By choosing the following parame- ters for a Pt (7 nm)/YIG bilayer at room temperature:5 10-310-210-11001011020.00.51.01.5 AH angle θah ( 10-5 )d / le s0 = 0.6 s0 = 0.7 s0 = 0.8 s0 = 0.9 FIG. 3: The AH angle ah(ah xy(d)=xx(d)) as a function of thickness of the heavy metal layer for several values of the specular re ection coecient. Other Parameters: sh= 0:05, Jex= 0:01eVandVb= 12eV. sh= 0:05 [24],s0= 0:6,Jex= 0:01eV[25],Vb= 12 eVandle= 20nm[13], we estimate the AH angle arising from our mechanism to be about 1 :3105, which is in good agreement with experimental observations [6, 7]. As a nal point, we suggest a crucial verication of our mechanism by contrasting the directions of the Hall cur- rent (or the signs of Hall voltages) of two trilayer struc- tures Pt/Cu/YIG and -Ta/Cu/YIG. Since the spin Hall angles of Pt and -Ta are of opposite signs [26{28] we predict that the Hall current directions in these two tri- layers will be opposite. A Cu layer, thinner than the electron mean free path, may be inserted between the heavy-metal and the magnetic insulator in order to elim- inate the magnetic proximity eect, while the nonlocal AH eect will still be operative. Acknowledgement. It is a pleasure to thank O. Heinonen, S. Zhang, A. Homann, W. Jiang and W. Zhang for various stimulating discussions. One of the author, S. S.-L. Zhang is deeply indebted to O. Heinonen for his hospitality at Argonne National Lab, where part of the work was done. This work was supported by NSF Grants DMR-1406568. Appendix A: Spinor re ection amplitude at a metal/magnetic-insulator interface Consider the following free electron Hamiltonian for a metal/magnetic-insulator interface ^H=^p2 2me+ Vb^IJex^z  (z) (A1)whereVbis the spin-averaged barrier for electrons to go from the metal to the insulator, Jexis the exchange cou- pling which is responsible for the spin-splitting of energy levels in the insulator, and  ( z) is the unit step func- tion. Here we have chosen the spin quantization axis to be parallel to the magnetization m(=^z). For an inci- dent electron (from the metal side, z >0) with its spin pointing in the direction ( ;) with respect to m, we can write the scattering wave function as follows ^ (r) = cos 2 ei=2'"(r)j"i+sin 2 ei=2'#(r)j#i (A2) where the spatial parts of the spinor wave function are '(r) = eikzz+Reikzz
eiq
,z>0 Tezeiq
, z<0(A3) where="(#),RandTare the corresponding re ection and transmission amplitudes, k= (q;kz) and r= (;z) are the wave vector and spatial coordinates respectively, and =p k2 bk2 Jk2zwithkbp 2meVb=~2andkJp 2meJex=~2. By matching the wave functions and their derivatives at z= 0, we nd R=+ikz ikz(A4) and T= 1 +R=2ikz ikz. (A5) Appendix B: Spin dependent specular re ection coecient In this section, we prove that the specular re ection co- ecientsis spin-dependent for a rough metal/magnetic- insulator interface. In the absence of interface roughness, the bilayer can be modeled as a simple spin-dependent step potential as given in Eq. (A1), the corresponding free electron Green's function (setting ~= 1) reads g q(z;z0;E) =m e ikzh eikzjzz0j+Reikz(z+z0)i (B1) wherez <0 andz0<0,kz=p 2meEq2withEthe total kinetic energy and qthe in-plane momentum, and the re ection amplitude for electron with spin is given by Eq. (A4). Now we model a rough interface by a set of randomly- distributed impurities localized at the interface ( z= 0) with-correlated potential Vimp(r) satisfying the follow- ing properties [29{31] hVimp(r)i= 0 (B2) and hVimp(r)Vimp(r0)i= (0)(z)(z0) (B3)6 wherehidenotes the impurity ensemble average and de- scribes the amplitude of the uctuation. Up to rst order in , the impurity-averaged Green's function reads [30] G q(z;z0;E) =g q(z;z0;E) + g q(z;0;E)g q(0;z0;E)N(0;E) (B4) whereN(0;E)Rdq0 (2)2g q0(0;0;E). By placing Eq. (B1) into Eq. (B4), we nd G q(z;z0;E) =m ikzn eikzjzz0j +eikz(z+z0)R 12i N(0;E)kz U b (B5) whereU bVbJexis the spin-dependent barrier. Comparing Eq. (B5) with Eq. (B1), we identify the ef- fective re ection amplitude in the presence of the surface roughness as R=R 12i N(0;E)kz U b (B6) Up toO( ), the re ection coecient is r=jRj2 12 A(0;E)kz U b (B7) with the surface spectral function dened as A(0;E) = 2=mN(0;E). We thus identify the specular re ection coecient as s= 12 A(0;E)kz U b(B8) By placing Eq. (B1) into Eq. (B8) and carrying out the integration over in-plane momentum q, we obtain an ex- plicit expression for s s= 1 2kz(2m eE)3=2 3(U b)2'1 2kzk3 F 3V2 b 1 +2Jex Vb (B9) where we have replaced the total kinetic energy Eby the Fermi energy and kept term up to O(Jex=Vb). Therefore, we have shown that the specular re ection coecient is indeed spin-dependent. We also note that sis in gen- eral dependent on the direction of the incident momen- tum. For brevity, we shall work with an angle-averaged specular re ection coecient, i.e., s=Z d ks(q)=4= 1 k4 F 32V2 b 1 +2Jex Vb (B10)It follows that the spin averaged specular re ection co- ecient as well as the spin symmetry of the specular re ection can be expressed as (up to O(Jex=Vb; )) s0s"+ s# 2= 1
k4 F 32V2 b(B11) and pss"s# s"+ s#'
Jex Vb
2k4 F 32V2 b(B12) Interestingly, we note the pshas a negative sign; in other words, the specular re ection coecient for spin-up elec- trons is smaller than that of the spin-down electrons as the latter encounter a higher barrier. Eliminating the parameter from Eqs. (B11) and (B12), we nd an approximate relation between s0and ps ps'2Jex Vb(1s0) (B13) This relation is valid for a moderately rough interface, i.e.,s0.1. [1] N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Rev. Mod. Phys. ,82, 1539 (2010). [2] J. E. Hirsch, Phys. Rev. Lett. ,83, 1834 (1999). [3] S. Zhang, Phys. Rev. Lett. ,85, 393 (2000). [4] G. Vignale, J. Supercond. Novel Magn. ,23, 3 (2010). [5] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, ArXiv e-prints (2014). [6] S. Y. Huang, X. Fan, D. Qu, Y. P. Chen, W. G. Wang, J. Wu, T. Y. Chen, J. Q. Xiao, and C. L. Chien, Phys. Rev. Lett. ,109, 107204 (2012). [7] M. Althammer, S. Meyer, H. Nakayama, M. Schreier, S. Altmannshofer, M. Weiler, H. Huebl, S. Gepr ags, M. Opel, R. Gross, D. Meier, C. Klewe, T. Kuschel, J.-M. Schmalhorst, G. Reiss, L. Shen, A. Gupta, Y.-T. Chen, G. E. W. Bauer, E. Saitoh, and S. T. B. Goennenwein, Phys. Rev. B ,87, 224401 (2013). [8] B. F. Miao, S. Y. Huang, D. Qu, and C. L. Chien, Phys. Rev. Lett. ,112, 236601 (2014). [9] H. Nakayama, M. Althammer, Y.-T. Chen, K. Uchida, Y. Kajiwara, D. Kikuchi, T. Ohtani, S. Gepr ags, M. Opel, S. Takahashi, R. Gross, G. E. W. Bauer, S. T. B. Goennenwein, and E. Saitoh, Phys. Rev. Lett. ,110, 206601 (2013). [10] Y.-T. Chen, S. Takahashi, H. Nakayama, M. Althammer, S. T. B. Goennenwein, E. Saitoh, and G. E. W. Bauer, Phys. Rev. B ,87, 144411 (2013). [11] E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. Lett. ,88, 182509 (2006). [12] T. Kimura, Y. Otani, T. Sato, S. Takahashi, and S. Maekawa, Phys. Rev. Lett. ,98, 156601 (2007). [13] G. Fischer, H. Homann, and J. Vancea, Phys. Rev. B , 22, 6065 (1980). [14] A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, Phys. Rev. Lett. ,84, 2481 (2000).7 [15] T. Valet and A. Fert, Phys. Rev. B ,48, 7099 (1993). [16] J. Zhang, P. M. Levy, S. Zhang, and V. Antropov, Phys. Rev. Lett. ,93, 256602 (2004). [17] Y. Qi and S. Zhang, Phys. Rev. B ,67, 052407 (2003). [18] E. M. Hankiewicz and G. Vignale, Phys. Rev. B ,73, 115339 (2006). [19] S. K. Lyo and T. Holstein, Phys. Rev. Lett. ,29, 423 (1972). [20] E. G. Mishchenko and B. I. Halperin, Phys. Rev. B ,68, 045317 (2003). [21] S. S.-L. Zhang, G. Vignale, and S. Zhang, Phys. Rev. B , 92, 024412 (2015). [22] R. E. Camley and J. Barna s, Phys. Rev. Lett. ,63, 664 (1989). [23] R. Q. Hood, L. M. Falicov, and D. R. Penn, Phys. Rev. B,49, 368 (1994). [24] L. Liu, R. A. Buhrman, and D. C. Ralph, ArXiv e-prints(2011). [25] 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, Nature ,464, 262 (2010). [26] L. Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, Science ,336, 555 (2012). [27] M. Morota, Y. Niimi, K. Ohnishi, D. H. Wei, T. Tanaka, H. Kontani, T. Kimura, and Y. Otani, Phys. Rev. B , 83, 174405 (2011). [28] C. Hahn, G. de Loubens, O. Klein, M. Viret, V. V. Naletov, and J. Ben Youssef, Phys. Rev. B ,87, 174417 (2013). [29] X.-G. Zhang and W. H. Butler, Phys. Rev. B ,51, 10085 (1995). [30] D. A. Stewart, W. H. Butler, X.-G. Zhang, and V. F. Los, Phys. Rev. B ,68, 014433 (2003). [31] V. F. Los, Phys. Rev. B ,72, 115441 (2005).

2015-12-14

The anomalous Hall effect is deemed to be a unique transport property of ferromagnetic metals, caused by the concerted action of spin polarization and spin-orbit coupling. Nevertheless, recent experiments have shown that the effect also occurs in a nonmagnetic metal (Pt) in contact with a magnetic insulator (yttrium iron garnet (YIG)), even when precautions are taken to ensure there is no induced magnetization in the metal. We propose a theory of this effect based on the combined action of spin-dependent scattering from the magnetic interface and the spin Hall effect in the bulk of the metal. At variance with previous theories, we predict the effect to be of first order in the spin-orbit coupling, just as the conventional anomalous Hall effect -- the only difference being the spatial separation of the spin orbit interaction and the magnetization. For this reason we name this effect \textit{nonlocal anomalous Hall effect} and predict that its sign will be determined by the sign of the spin Hall angle in the metal. The AH conductivity that we calculate from our theory is in good agreement with the measured values in Pt/YIG structures.

Nonlocal Anomalous Hall Effect

1512.04146v1

1 Macroscopic , layered onion shell like magnetic domain structure generated in YIG film using ultrashort , megagauss magnetic pulses Kamalika Nath1, P. C. Mahato1, Moniruzzaman Shaikh2, Kamalesh Jana2, Amit D Lad2, Deep Sarkar2, Rajdeep Sensarma3, G. Ravindra Kumar2#, S. S. Banerjee1* 1Department of Physics, Indian Institute of Technology, Kanpur, Uttar Pradesh, India, 2Department of Nuclear and Atomic Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai, India. 3Department of Theo retical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai, India. Corresponding authors’ email: #grk@tifr.res.in, *satyajit@iitk.ac.in. Abstract: Study of the formation and evolution of large scale, ordered structures is an enduring theme in science. The generation, evolution and control of large sized magnetic domains are intriguing and challenging task s, given the complex nature of competing intera ctions present in any magnetic system. Here, we demonstrate large scale non - coplanar ordering of spins, driven by picosecond, megagauss magnetic pulses derived from a high intensity, femtosecond laser. Our studies on a specially designed Yttrium Iron Garne t (YIG)/dielectric/m etal film sandwich target , show the creation of complex , large, concentric, elliptical shaped magnetic domains which resembl e the layered shell structure of an onion . The largest shell has a major axis of over hundreds of micrometers, in stark contrast to conventional sub micrometer scale polygonal, striped or bubble shaped magnetic domains found in magnetic materials, or the large dumbbell shaped domains produced in magnetic films irradiated with accelerat or based relativistic electron beams. Through micromagnetic simulations , we show that the giant magnetic field pulses create ultrafast terahertz (THz) spin wave s. A snapshot of these fast propagating spin waves is 2 stored as the layered onion shell shaped domains in the YIG film. Typically , information transport via spin waves in magnonic devices occurs in the gigahertz ( GHz) regime, where the devices are susceptible to thermal disturbance s at room temperature. Our intense laser light pulse - YIG sandwich target combination, paves the way for room temperature table - top THz spin wave device s, which operate just above or in the range of the thermal noise floor. This dissipation -less device off ers ultrafast control of spin information over distances of few hundreds of microns. Conventional condensed matter systems display a diverse variety of static magnetization configurations like Bloch or Neel domain walls, magnetic vortices (1,2), stripe domains (3) and skrymions (4). These domains arise out of a competition between different magnetic exchange and magneto static energies (5,6). Over the past few decades, a great deal of effort has gone into using light for p robing different aspects of condensed matter physics (7,8,9,10,11,12). Effects of light interacting with magnetism ha ve been primarily studied with low intensity (I ~ 105-106 Wcm-2) femtosecond (fs) lasers (7,8,9,10) for exploring demagnetization processes occurring on time scales of a few picosec onds (ps) (10,13,14,15). Recent ly, optical coupling of angular momentum of light with spins in magnetizable media has been shown to create micron -sized domains on fs timescales (8,16,17,18). The use of high intensity femtosecond laser pulses (I~ 1014 - 1018 Wcm-2) for such studies has however not been attempted so far. This may be attributed to the apprehension that the enormous energy scale associated with such excitation would overwhelm the spin - spin interaction energy scale and the thermal damage induced by such intense laser pulse would obliterate the possibility of seeing any ordered spin configuration . Indeed, d irect irradiation with such intense pulses typically ablates the material creating a high temperature plasma. However, interaction of an intense fs laser pulse with t he plasma is interesting as it is known to produce giant megagauss (MG) magnetic field pulses of p icosecond 3 duration (19,20,21,22,23). It is therefore worthwhile studying the response of magnetic materials to such intense magnetic pulses . In this paper, with an innovative target design and careful control of experimental conditions, we demonstrate the creation of novel, unusual spin structures created by this magnetic pulse. Here we study the response of Yttrium Iron Garnet (YIG) film subjected to megagauss magnetic field pulse s produced by the interaction of a few hundred petawatt/cm2 intensity , 30 fs laser pulse with a solid target . YIG is a well -known ferrimagnetic insulator film with very low damping , which in recent times has become an attractive material for studying magnon dynamics (24,25,26) (magnons are quasi particles associated with spin waves). Low damping of magnetization dynamics coupled with large magnon diffusion lengths reaching several microns (27), make YIG an important material for applications in magnonics (28,29,30), spin caloritronics (31,32,33) and magnon -based microwave application s (34,35). A careful target design is however, crucial for eliminating the ablative degradation of YIG due to laser induced ionization and subsequent heating. We therefore implement a novel sandwich target geometry of metal film (Al)-dielectric -YIG (Fig. 1 A), where the laser irradiates the top Al layer , leavi ng the lower YI G layer unaffected by the laser induced damage . Magneto -optical microscopy (MOM) of the YIG samples exposed to the laser generated giant magnetic field shows the creation of novel, large, concentric, elliptically shaped magnetic domains exte nding up to a few hundreds of microns from the projected irradiation location . The shape resembles layered shells of an onion. Furthermore, we see that the local magnetic field direction flips up and down periodically across the se elliptical domain structures and its magnitude also has a periodic variation with distance from the center of the irradiation . Micro magnetic simulati on of a YIG film subjected to megagauss field pulse show s the excitation of ripples of spin waves travelling across the low d amping YIG film, a few picoseconds after the pulse . The spin waves cause moments to gradually rotate out of the film plane periodically resulting in the observed behavior of the measured local field. These fast spin waves diffuse up to a few hundred s of microns in the YIG film from the projected laser 4 irradiation site , giving rise to a non-collinear spin configuration , which in turn we propose, gives rise to an additional Dzyloshinskii –Moriya type interaction contribution to the magnetic energy of YIG. This interaction together with pinning effects , results in the spin waves getting stored as the layered onion shell like magnetic domain structure in YIG . Each target (Fig. 1 A) consists of a 16 m thick Al film suspended over a GGG (Gallium Gadolinium Garnet ) substrate in a sandwic h configuration. The lower side of the GGG substrate has a B ismuth doped YIG film grown on it (36,37). We use single pulses of 25 femtosecond (fs) laser (p-polarized , center wavelength 800 nm) having 20 m beam diameter to irradiate identical points at different locations on the Al layer at an angle of incidence of 45 . The laser intensities used are be tween 3 × 1017 to 1 × 1018 Wcm-2 (details of setup in Supplementary section ). The YIG film is isolated from the optical field of the intense laser as well the heating effects it generate s. The sandwich configuration provides a two level protection to the YIG film. Firstly, it eliminate s laser induced ionization of the YIG film and the resulting thermal heat load that could lead to direct damage of the film , since the intense laser pulse ablates the sacrificial Al layer which takes away the se deleterious effects . Secondly, t he magnetic YIG layer has additional shield ing from the heat ing effects provided by the intervening 200 m thick dielectric air gap and the GGG layer present between Al and YIG film layer. The YIG film was devoid of any micron sized magnetic domains prior to irradiation. As late as four days after irradiation, the irradiated region is imaged using a high sensitivity magneto -optic microscope (MOM setup details in Supplementary information and Ref. 23). The magneto -optical intensity is 2 zB, where Bz is the component of local field perpendicular to the surface. The Bz(x,y) distribution is determined from the MOM intensity distribution by suitable calibration (23) (the x and y axes are in the film plane while z axis is perpendicular to the film) . 5 Fig. 1, MOM Images of laser Irradiated YIG films. (A), Schematic shows the incidence of the femtosecond laser pulse on a 16 m thick Al film suspended 100 m above the GGG substrate, on the lower side of which is the YIG film. A dielectric layer of thickness ~ 200 m composed of air and the GGG substrate, exists between the Al and the YIG film. The mark is the estimated projected location of the laser spot on the YIG film . (B,D,F ) MOM images of concentric elliptical magnetic domains generated in YIG film are shown after irradiation at laser intensities (I) of 3 1017 Wcm-2, 6 1017 Wcm-2 and 1018 Wcm-2, respectively. (C) Color coded three -dimensional map of the Bz(x,y) distribution in the elliptical domains of 1( B). The colors represent the magnitude of the Bz values as shown in the color -bar scale. (E) A one-dimensional map of Bz profile ( viz., Bz(r)) measured along the red line in Fig. 1( D), show s periodic oscillations in Bz as one traverses the concentric rings of the elliptical domain pattern . The maximum amplitude of the oscillating Bz(r) is 40 G. (F) Shows two domain patterns generated in YIG wit h laser pulse irradiation of intensity of 1018 Wcm-2. The smaller upper domain structure is more circular than the one below . Also seen is a defect in the YIG film which was present before irradiation. ( G) MOM image of the region same as that in ( F), imaged after 10 days of laser irradiation, with the black defect as the identifier. Here we see layered onion shell like magnetic domain patterns have disappeared. The MOM images in Figs. 1 B, D and F show the creation of magnetic domain patterns recorde d in the YIG film layer , after irradiat ion by single pulses of laser intensities of 3 1017 Wcm-2, 6 1017 Wcm-2 and 1018 Wcm-2, respectively. All images show the formation of concentric black and white elliptical shaped onion shell -like magnetic domains in the YIG film. The overall sizes of the elliptic domain patterns at different laser intensities are comparable and 6 do not scale with laser intensity. The number of concentric rings however increases with increasing laser intensity. The full extent of the concentric domain patterns is at least an order of magnitude larger than the laser beam diameter of 20 m. The peaks and troughs in the color - coded three -dimensional view of Bz (x,y) in Fig. 1C, correspond to periodic modulation of l ocal field Bz across the bright and dark regions of the pattern in Fig. 1 B. A comparison of the concentric domains in Fig. 1 B with Fig. 1 D shows the number of concentric rings increasing with intensity ( I) of the laser. Furthermore, Fig. 1 F shows that at a high intensity of 1018 Wcm-2 there is a large elliptical domain pattern with concentric rings, and an adjacent satellite concentric domain with a relatively lower eccentricity. Fig. 1 G shows a MOM image of the same region of the YIG film a s in Fig. 1 F recorded 10 days after irradiation. Note that image of the defect in the YIG film layer of Fig. 1F is retained in Fig. 1G, however the magneto -optical contrast of the domain patterns ha s diminished such that the patter ns are no longer discerni ble. This disappearance of the pattern is natural ly expected, as the remnant magnetization of the film at room temperature decays out to zero with time. This feature suggests the domain patterns are not permanent and irreversible, i.e. , they are not related to laser induced heating damage. The magnetized regions are generated by the action of the megagauss magnetic pulse created by the laser. The following sections explore the origin of this quasi -stable domain feature observed in YIG fi lm. It is well established in intense laser -solid interaction studies that a high intensity, p - polarized femtosecond laser pulse incident on a target at a non -normal angle sets up electron waves in the generated plasma, which grow to large amplitude before breaking. This breaking unleashes a giant current pulse ( ~mega -ampere) that travels normally into the planar target and the entire process is known as resonance absorption (RA ) (22,38). The current pulse is due to RA generated single or multiple collimated relativistic electron jets (39). These jets generate giant , azimuthal magnetic fields (B), having peak pulse height of few hundreds of Megagauss with 7 typical pulse widths of a few ps (19-21) (Schematic in Fig. 3 C and d etails on RA in supplementary ) Fig. 2. Simulation of YIG films using MuMax . (A) Schematic of the magnetic precession of the in - plane moments of magnetic film due to the megagauss magnetic field (see text for details). (B,C) Simulation of YIG films showing r ipples in the magnetic moment configurations are generated in the YIG film with magnetic pulse s, B0 = 7 megagauss and 35 megagauss, recorded at time ( t) = 0.5 secs (see text for details) . The ripples spread out as a function of increasing time. ( D) Plot of 1tanz xM M across the red solid line in (D) shows periodic modulations of the magnetic moment configuration in the film at t = 0.5 ps. To explore the effect of these RA generated giant magnetic field pulses on YIG films , we model t he temporal evolution of in -plane local magnetization M in the YIG film under the influence of a magnetic field pulse (refer Fig. 2 A) through the Landau -Lifshitz -Gilbert (LLG) equation using the MuMax software (40,41) (see Methods for simulation details) 8 eff eff sdMB M M ( M B )dt M …………………….. eqn. 1 The first term on the right in eqn.1 is the torque on M due to the effective magnetic field ( effB = Applied in plane azimuthal field ( B) + Demagnetizing Field ( dB ) + Anisotropy field ( aniB )), where is the gyromagnetic ratio. The in -plane azimuthal field pulse (see Fig. 2A) is ˆB B where ˆis the azimuthal unit vector in film plane . The second term is a damping term with damping constant . The B pulse first causes Mto flip out of the film plane by an angle 0 due to a torque , BM . For YIG a large B pulse of 7.5 MG (discussed subsequently) gives a 0( ) tB = 10.5 in a pulse duration t ~ 0.5 ps (YIG has = 28 GHzT-1 (42)). The flipped M generates a demagnetization field ( dB 0 ~ M sin zˆ , zˆis to film ) (see Fig. 2A) around which M precesses . At this stage, spin waves are excited in the magnetic material. Simultaneously, damping ( second term in eqn.1) comes into play, leading to the decay of the waves as the precessing M gradually loses energy and falls towards the film plane (see blue dashed trajectory in Fig. 2A), until its energy is just lower than the anisotropy energy barrier ( Ku sin(), - angle between M and aniB ). Note that the anisotropy energy is minimum for parallel ( = 0) or antiparallel ( = ) orientation . Here M performs a damped precess ion around aniB before falling back into the film plane , with M oriented either along or opposite to aniB . Finally , minimizing the free energy leads to domains with in-plane M, where the in-plane M orientation periodically flips by across the domain s. There appears to be only one other group that has studied the influence of ultra-strong , ultrashort magnetic pulses on magnetic films (43,44) and it is therefore interesting to make a comparison with their results . Their studies subject ed films of high Ku (Ku ~ 0.1 MJ m-3) viz., 9 high aniB to (a) magnetic field pulses of the order of a few tens of Tesla and also (b) electric (E) fields ~109 V/m. The se fields were generated by irradiating the film directly with relativistic electron (e-) bunches with 28 and 40 GeV energies at SLAC (43,44). The presence of a strong anisotropy field in the magnetic film material resulted in anisotropic dumbbell like domain pattern (43,44). The in -plane M across adjacent domains in the dumbbell pattern were anti- parallely aligned . A comparison with our work and shows the distinct spatial symmetry of our domain shapes , viz., layered onion shell structure, which is in contrast to the anisotropic dumbbell shaped domain s found in the SLAC studies . Another distinguishin g feature of our domains is t hat - as one moves along a line cutting across the domains the M periodically twist s out of the plane leading in turn to a periodic modulation of local Bz (see Fig. 1E), while in the SL AC study , M always remains in -plane . These contrasting results suggest significant differences in the physical processes leading to the distinct domain shapes seen in our experiments . We would like to suggest that our use of YIG has properties which are completely different from those used in the SLAC study. YIG has a time independent damping constant (), which is nearly two orders of magnitude smaller compared to that of the material used in the SLAC study (see supplementary information) . Furthermore , the domain structures seen in the SLAC study are only explained by considering that for the material s used in their study, the increase exponentially with time until, the spin wave carries away all the energy from the precessing M . For YIG, not only is low but also we do not need to consider any time dependence of to explain the domain feature we observe. Due to low value, for our YIG simulations the damping (second) term in eqn.1 has a relatively small effect (compared to that in the SLAC studies of refs. 43,44) resulting in stable spin waves excited by the field pulse which propagate a cross the YIG film. Also note that for YIG, Ku = 6.1 x 10-4 MJ m-3 is nearly two orders smaller compared to the strong magnetic anisotropy material used at SLAC (43,44), hence the domains formed in YIG are more 10 symmetrical compared to the asymmetrical dumbbell pattern found in the SLAC study . It is also important to mention the differences in the method s used to generate the field pulses in both cases . Our intense laser generated field pulses are essential ly magnetic while both B and E pulses are generated by the relativistic e - bunches in the SLAC experiments (43). The electron beam in the SLAC study traverse s the film and cause s film damage, while such a deleterious effect is completely avoided in our study . Lastly, our B pulse is larger by at least an order of magnitude (19,20) (~ 100 T) compared to that in the SLAC study ( ~ 10 T) (43). For our simulation , the azimuthal magnetic field distribution experienced by the YIG film is approximated using an expression similar to that for the field distribution at positions located away from the relativistic electron bunch (45), viz., 0( , , ) ( ) ()pBB x y t t tr for r> , where , r is the distance from the projected center of laser irradiation on the YIG film and is the diameter of region around irradiation center within which there are the RA generated current jet(s). We use as the diameter of the irradiating laser beam. In the region r < , we use a uniform B = B0. The temporal behavior of the pulse is ( ) 1ptt for 0 t tp = 1 ps and ( ) 0ptt for t > tp. In the above expression , 0 0 3/22 (2 )pneBt , is the field (45) present outside the boundary of radius /2, where n is the number of electrons in the current jet and e is the electron charge . Figures 2 B- C show results of the LLG simulations in YIG with a time independent damping constant , at time ( t) = 0.5 ps after the application of magnetic field pulse with peak field B0 of 7 MG and 35 MG respectively. Note that B0 = 7.5 MG (n = 1.2 x 1012) and 35 MG (n = 5.6 x 1012) correspond to an order of magnitude larger number of electrons in our intense laser pulse generated electron jet compared to those in the relativistic electron bunch at SLAC . The Zeeman energy associated with our giant B pulse in YIG is ~ 14296 MJ m-3 (see supplementary section) , is sufficiently large to completely overwhelm the low magnetic anisotropy energy of YIG ( Ku = 6.10 x 102 J m-11 3). Figures 2B and C show that the B pulse excites circular spin wave ripples in the YIG film around or (center of the pulse). With increasing time, the ripples spread out across the film (see movie ( 46)), until the y reach the film edge which occurs within 1 ns. At long time s (~ 100 ms) , a complex rectangular multi -domain configuration is stabilized in the Y IG film , which are unlike the domains we have recorded in Fig. 1 (see movie at link Ref. 46, and supplementary information section ). We do not observe any dependence of our simulation results on the lateral film dimensions as long as they are larger than the ripple wavelength. The rippling feature seen in the simulations closely resemble s our observed elliptical layer ed onion shell domain structure of Fig. 1. Further similarit y between the simulated features and our experiment is seen i n Fig. 2 D, where we calculate the orientation ( ) of local M in the film w.r.t the film plane viz., 1tanz xM M , measured along the radial direction drawn as a red line in Fig. 2 C. The contrast modulations seen in Figs. 2 B and C clearly correspond to the periodic flipping of z component of M , which is evident from the (r) behavior in Fig. 2 D. The periodic flipping of M , viz., the behavior of (r) in Fig. 2C is similar to the behavior of Bz(r) in Fig. 1 E. Figures 2 B-C show that at 0.5 ps, the diameter of the outer edges of the rippling structure are comparable for B0 = 7.5 MG and 35 MG . The number of rings inside the concentric circular structures increases with B0. These observation s from the simulations match with that of Fig. 1 - while the overall size of the domain is independent of the laser intensity the number of concentric rings (N) increase with intensity (cf. Figs. 1 B and 1 F). The simulations show that increase in N is related to increase in the pulse peak field B0 (cf. Figs. 2B and 2 C). By comparing the number of rings ( N), determined as a function of laser intensity ( I) (experiments , like Figs. 1 B, D and F) and as a function of B0 (determined from simulations like those in Fig. 2), an empirical relation between B0 and I is obtained (details in s upplementary). Our experiments reveal that for I 1017 Wcm-2, no elliptical domain s form . Using the empirical relation, this intensity correspond s to a 12 peak field of B0 ~ 0.1 MG . Our simulations also confirm that there are no discerni ble magnetized ripples excited in YIG below 0.1 MG field. Simulating eqn. 1 using a general form of B with multi -peak s (19-22), viz., 0, , ( , , ) ( ) ()i pi i iBB x y t t tr with i = 1 to 2, we observe in Fig. 3B two well separated rippling structures (after 0.5 ps) , which are similar to the features in Fig. 1 F. By reducing the spacing between the field pulses down to 18 m we observe (Fig. 3C) a single rippling structure produced by the overlap of two spin wave ripples. The resulting ripple is not circular but elliptical in shape . It is these elliptical shapes that we observe in our experiment s (Fig. 1 ). The multi -peaked field structure may result from multiple closely spaced e-jets generated during RA process as shown in the schematic of Fig. 3 A. Fig. 3 . Correlation of simulations with experiments. (A) Schematic of the Resonance Absorption processes in which fs laser hits the Al film to form a plasma plume and interactions with the laser plasma creates multiple electron (e) jets. These e -jets moving at relativistic speeds carrying mega -amps of current generate magnetic field around it (B) which is shown only around one of the e -jet paths for the sake of cla rity. These electron jets give rise to megagauss azimuthal magnetic fields inside the YIG. (B,C) Simulations done on the YIG film using two sharply peaked magnetic field pulses having B0 of 15 MG and 7 MG, separated by 30 m and 18 m. The ripples created in the YIG film are well separated when the separation between the field pulses is significant while the two ripples merge into an elliptical shaped ripple when the pulses are closer to each other. The ripple patterns shown are obtained by stopping the sim ulation at t = 0.5 ps. 13 From our simulations, we determine the phase velocity ( vp) of the spin waves generated by the giant field pulse, as the ratio of the distance covered by the crest of a ripple to the time taken. The vp turns out to be in the range of ~ 107 m s-1 which is four orders of magnitude greater than the typical limit of domain wall velocity (~ 1000 ms-1) reported for YIG films (47). Note that vp is not related to physical motion of domain walls. Using vp and the measured wavelength of the ripple () excited at different energies, we obtain 2()mpkv for our spin wave to be in the range of ~ 30 to 100 THz (depending on the laser intensity) . Some earlier m easurements suggested that spin waves with frequencies m ~ few THz (48,49) can be excited in YIG films . We show that one to two order of magnitude higher THz frequency spin waves can be excited using our intense laser and the novel metal/dielectric/YIG film sandwich target combination. The cl ose match between our simulations and experiments in YIG demonstrate s a unique effect, viz., excitation of fast propagating spin waves excited in the YIG film . Furthermore, t he YIG effectively capture s a snapshot of the propagating spin wave . It is pertinent to ask by what mechanism is the spin wave transform ing into a static magnetic domain pattern ? To get a glimpse into the answer , we det ermine the amount of non -collinear M configuration generated at the domain site i in YIG by the giant field pulse viz., the average value of ||ijMM within a region of 10 m 10 m inside the simulated ripple of magnetic disturbance in Fig. 2. The average value of ||ijMM per unit area associated with each ripple turns out to be ~ 0.9 m-2 compared to zero m-2 prior to the pulse. We propose that the spin waves excited in YIG generate a non collinear twisted moment configuration which leads to an enhanced contribution of magnetic energy associated with the spin configuration in YIG coming from the Dzyloshinskii -Moriya (DM) (5,6) interaction energy , where DM | |j iM M . The addition al DM interaction in YIG excited by the spin wave we believe stabilizes the non -collinear magnetization configuration resulting in the spin wave ripple pattern being frozen into YIG as the elliptical, concentric onion 14 shell -like domains as seen in Fig. 1. It may be recall ed that DM interactions in conventional condensed matter systems are crucial for stabilizing the small chiral magnetic structures like skrymions (4). The ever present pinning in the YIG film would also contribute to stabilizing the M configuration excited by the spin wave. Pinning of the outermost spin waves ripples by microscopic defects in the film is responsible for the irregular shape of the out ermost ring seen in the images of Fig. 1. From the layered onion shell domains recorded in YIG film we show that f emto second laser pulse excites THz spin waves in YIG with diffusion lengths upto hundred s of microns . This result has potential applications for spin wave based devices . In recent times, the generation, control and manipulation of spin waves ha ve emerged as an important research area (50,51) as they can be utilized in dissipation -less transfer of information across a device . In conventional electron transport based devices , increasing electrical resistance with miniaturization significantly increases the dissipation losses , thereby limit ing the processing speed of these device s. However in magnonic devices , spin waves with frequencies, m < 10 GHz , are launched via spin Hall effect or spin torque effect using an electrical current (27,34). The electrical contacts in these magnonic devices for current injection result in unavoidable Joule heating dissipation at the contacts . These devices also need to be operated at low temperatures as the spin waves are susceptible to thermal noise , since for m < 10 GHz the mB kT (=0.025 eV at 300 K). However , the spin waves excited in our metal/dielectric/ YIG sandwich using indirect irradiation of YIG with the intense laser pulses , completely eliminat e the need for using electric currents for exciting the waves and hence limit Joule heating dissipation . Furthermore, the fast propagating spin waves with long diffusion lengths reaching up to hundreds of microns have frequenc ies that can be varie d between 10 to 100 THz by varying the laser intensity . Hence magnonic devices employing our method are suitable f or room temperature operation, as the energy of spin waves excited in YIG by intense laser pulse is far above the thermal noise floor ( as for m~10 to 100 15 THz, mBkT ). We also re-emphasize that our design for launch ing spin wave s has multiple advantages compared to doing the same with an accelerator based relativistic electron beam (43,44). We have a much more compact table top design, less complexity in operation and the strength of the field pulses and pulse durations can be conve niently controlled by varying the laser intensity , the target design and the interaction geometries. Conclusions We have demonstrated novel , macroscopi c, concentric elliptical onion shell -like magnetic domains created in Yttrium Iron Garnet (YIG) film via giant magnetic field pulses of picosecond duration , generated from femtosecond, intense laser pulses . Spin waves excited in the YIG film by the field pulse generate non -collinear spin configurations in the YIG film which gives rise to DM interacti ons which initially w as very weak in the YIG film. These interactions together with pinning in these films cause the spin waves to stabilize into surprisingly long lived elliptical domain patterns. Our innovative metal/dielectric/YIG structure irradiated with intense laser pulse offers a new way to excite ultrafast spin waves with frequency in the few tens to hundred s THz range with large diffusion lengths. These high frequency spin waves are undistu rbed by thermal fluctuations . Hence our tabletop route offers a new way for developing a dissipation less high frequency magnonic devices which are capable of operat ing at room temperature (52). Acknowledgements: SSB would like to acknowledge the funding support from IIT Kanpur and the Department of Science and Technology, Government of India, New Delhi. GRK acknowledges partial support from the Science and Engineering Board (SERB), Government of India, New Delh i through a J C Bose National Fellowship (JCB -2010/037). GRK acknowledges late Predhiman K. Kaw for stimulating discussions. 16 Author Contribution statement: GRK and SB conceived and guided the study. KN prepared the targets. MS, KJ, ADL, KN and DS were involved in the laser irradiation experiments. KN conducted simulations and MOM experiments . SSB, GRK, ADL, KN , interpreted the data and wrote the paper, with the approval of other authors. Methods: Simulations using MuMax: We simulate YIG films with dime nsions 150 × 150 µm2 and thickness 3 µm. The material parameters used in the calculations are as follows: Ms = 1.4 × 105 A/m, exchange constant A = 3.6 × 10−12 J/m, α = 0.0005 and anisotropy constant K = 610 J/m3. The film is discretized cells with grid lengths of 1 µm × 1 µm × 1 µm. At the vertex of each cell a magnetic moment is placed. 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Science 366, 1125 (2019) . 51. J. Han et al., Mutual control of coherent spin waves and magnetic domains in a magnonic device. Science 366, 1121 -1125 (2019). 52. A.S. Sandhu et al. , Real time study of fast electron transport inside dense, hot plasmas, Phys. Rev. E , 73, 036409 (2006) 20 Figure 1 Fig. 1, MOM Images of laser Irradiated YIG films. (A), Schematic shows the incidence of the femtosecond laser pulse on a 16 m thick Al film suspended 100 m above the GGG substrate, on the lower side of which is the YIG film. A dielectric layer of thickness ~ 200 m composed of air and the GGG substrate, exists between the Al and the YIG film. The mark is the estimated projected location of the laser spot on the YIG film . (B,D,F ) MOM images of concentric elliptical magnetic domains generated in YIG film are shown after irradiation at laser intensities (I) of 3 1017 Wcm-2, 6 1017 Wcm-2 and 1018 Wcm-2, respectively. (C) Color coded three -dimensional map of the Bz(x,y) distribution in the elliptical domains of 1( B). The colors represent the magnitude of the Bz values as shown in the color -bar scale. (E) A one-dimensional map of Bz profile ( viz., Bz(r)) measured along the red line in Fig. 1( D), show s periodic oscillations in Bz as one traverses the concentric rings of the elliptical domain pattern . The maximum amplitude of the oscillating Bz(r) is 40 G. (F) Shows two domain patterns generated in YIG with laser pulse irradiation of intensity of 1018 Wcm-2. The smaller upper domain structure is more circular than the one below . Also seen is a defect in the YIG film which was present before irradiation. ( G) MOM image of the region same as that in ( F), imaged after 10 days of laser irradiation, with the black defect as the identifier. Here we see layered onion shell like magnetic domain patterns have disappeared. 21 Figure 2 Fig. 2. Simulation of YIG films using MuMax . (A) Schematic of the magnetic precession of the in - plane moments of magnetic film due to the megagauss magnetic field (see text for details). (B,C) Simulation of YIG films showing r ipples in the magnetic moment configurations are generated in the YIG film with magnetic pulse s, B0 = 7 megagauss and 35 megagauss, recorded at time ( t) = 0.5 secs (see text for details) . The ripples spread out as a function of increasing time. ( D) Plot of 1tanz xM M across the red solid line in (D) shows periodic modulations of the magnetic moment configuration in the film at t = 0.5 ps. 22 Figure 3 Fig. 3 . Correlation of simulations with experiments. (A) Schematic of the Resonance Absorption processes in which fs laser hits the Al film to form a plasma plume and interactions with the laser plasma creates multiple electron (e) jets. These e -jets moving at relativistic speeds carrying mega -amps of current generate magnetic field around it (B) which is shown only around one of the e -jet paths for the sake of clarity. These electron jets give rise to megagauss azimuthal magnetic fields inside the YIG. (B,C) Simulations done on the YIG film using two sharply peaked magnetic field pulses having B0 of 15 MG and 7 MG, separated by 30 m and 18 m. The ripples created in the YIG film are well separated when the separation between the field pulses is significant while the two ripples merge into an elliptical shaped ripple when the pulses are closer to each other. The ripple patterns shown are obtained by stopping the simulation at t = 0.5 ps. 23 Supplementary Materials for Macroscopic , layered onion shell like magnetic domain structure generated in YIG film using ultrashort , megagauss magnetic pulses Kamalika Nath1, P. C. Mahato1, Moniruzzaman Shaikh2, Kamalesh Jana2, Amit D . Lad2, Deep Sarkar2, Rajdeep Sensarma3, G. Ravindra Kumar2#, S. S. Banerjee1* 1Department of Physics, Indian Institute of Technology, Kanpur, Uttar Pradesh, India 2Department of Nuclear and Atomic Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai, India 3Department of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai, India Corresponding authors ’ Email: #grk@tifr.res.in, *satyajit@iitk.ac.in Materi als and Methods Simulations using MuMax We simulate YIG films with dimensions 150 × 150 µm2 and thickness 3 µm. The material parameters used in the calculations are as follows: Ms = 1.4 × 105 A/m, exchange constant A = 3.6 × 10−12 J/m, α = 0.0005 and anisotropy constant K = 610 J/m3. The film is discretized cells with grid lengths of 1 µm × 1 µm × 1 µm. At the vertex of each cell a magnetic moment is placed. We use large grid lengths in our simulation as we are interested to capturing long wavelength magnetic modes which maybe excited on the YIG films by the application of giant magnetic field pulses. W e are not interested in the excitation of small (submicron) wavelength modes. The initial magnetic configuration of YIG in the simulations is in -plane. We have used magnetic field pulse of pulse height varying between 0.07 MG (half -maxima) and 350 MG , pul se width σ = 10 m and pulse time, t = 1 ps (see text for details). 24 Experimental set -up The laser irradiation experiments are performed in experimental chamber with base vacuum of 10-5 mbar. The experimental schematic is shown in Fig. S1 (a). To irradiate a fresh portion of sample every laser shot, the sample is mounted on precise X -Y-Z-θ stage assembly. P-polarized 25 fs, 800 nm laser pulses were focused to 20 µm diameter spot by off -axis parabolic mirror (OAP) on to a sample to create a plasma. The inten sity on the sample is varied from 1017 to 1018 Wcm-2, by changing the laser energy appropriately. The angle of incidence on sample is maintained to 45o to maximize resonance absorption (RA) (discussed later). Figure S1: (a) Schematic of intense, femtosecond laser irradiation set -up. M1, M2: Mirrors, OAP: off -axis parabolic mirror. (b) Schematic of Magneto -optical Imaging set -up. The laser irradiated samples are imaged by high sensitivity magneto -optical imaging technique, which is based on the principle of Faraday effect. Figure S1(b) shows a schematic of the magneto -optical setup where the components refer to: un -polarized Light source (L), Polarizer (P), Beam Splitter (BS), Analyzer (A). In Faraday rotation, the plane of polarization of a linearly 25 polarized light undergoes rotation by an angle in presence of magnetic field in the direction of light propagation. The Faraday rotated angle ( ) is given by, = VBd, where V is the Verdet constant (depends on material and th e wavelength of light), B is the magnetic field and d is the path length in the sample. Therefore, we get information about the magnitude and direction of the local magnetic field from the sample by knowing the degree of rotation . The Faraday rotated intensity distribution is captured in a CCD camera and henceforth calibrated to the corresponding B values. In a MOI, the intensity of the Faraday rotated light I(x,y) is proportional to the local field 2 zB . We calibrate the I(x,y) vers us well calibrated magnetic field by applying known fields to the YIG film. Using this information, we convert the I(x,y) in the MO images of Fig. 1 (main text) into Bz(x,y). The bright and dark contrasts in the MO images represent the local Bz(x,y) fields and hence Mz pointing either out or into the YIG film respectively. The shade in the image corresponds to the magnitude of Bz. Resonance absorption We give a short summary of the generation and effects of magnetic fields as a result of reson ance absorption (RA) in laser -matter interaction. During the RA process, the incoming p-polarized ( E- field in the plane of incidence) high intensity laser pulse impinges on the sample at oblique incidence and generates dense plasma. The preformed plasma cr eated by the laser pre -pulse expands away from the target surface as shown in the schematic (Fig. S2). The laser light propagating into such plasma encounters a density gradient in the plasma, with the density being highest close to the site of irradiation . The electrons of mass me and charge e in the plasma with a density ne oscillates with a frequency p , known as the plasma frequency. If the incident light field has a frequency p then the light is reflected from the plasma. Inside the plasma dome the density increases as one approach the surface of the material. Hence, the femtosecond laser 26 beam is being reflected from the ‘critical surface’ (where the local plasma frequency matches that of the light wave). However, an evanescent wave continues deeper into the plasma. Figure S2: Schematic of Resonance Absorption mechanism during fs laser -matter interaction. up to a region inside the plasma with a critical density. If the frequency of the evanescent light field (ω) matches the plasma frequency ( p ) in the dense region of the plasma then large resonant amplitude oscillations are excited in the plasma. Due to the collapse of the resonant waves in the critical layer of the plasma, energ y is transferred from the plasma wave energy to the electrons in the plasma, thereby generating a collimated jet of hot electrons channeling through the target material at relativistic speeds. Consequently giant, megagauss range azimuthal magnetic field ( viz., fields in the plane of the target material and perpendicular to the electron jet) are generated from the strong mega ampere currents associated with the hot electrons jet. While these magnetic fields ( B) are very large, their typical lifetime is a fe w picoseconds. Thus resonance absorption leads to generation of megagauss magnetic field pulses for picosecond time interval. plasma Hot e-Jetω ωp Magnetic layerInsulator BPoint of collapse of the resonant waves set up inside plasma Azimuthal magnetic fields27 Relation between P and B0 : The log -log plots in Figs. S3(I), (II), show a linear relation between N and P and N with B0. The trends of the experimental data and simulation results of N vs P and B0 behave linearly in the double log plot suggesting N = C1P and N = C2B0. From the same N using the values of P and B0 deduced from panels I and II we plot on a double log scale the linear relationship between B0 and P which is displayed on a log – log plot in Fig. S3(III). From the fit in the panel we obtain an empirical relation and this empirical relation is approximately valid until the ripples reach the edges of the film. Using this empirical relationship, we estimate that our laser intensity P ~ 7 × 1017 W cm-2 corresponds to generating a gigantic field pulse B0 ~ 100 MG experienced by the YIG. We would like to mention that we did not observe formation of any of the elliptical domain structures below a lower threshold of P = 1017 W cm-2, which from backward extrapolation in Fig. S3(III) corresponds to B0 ~ 0.1 MG. This estimate concurs well with our simulations which show no excitation of long wavelength ripples for B0 < 0.07 MG. Figure S3: Panel I shows a plot of N vs P from experiments and panel II shows a plot of N vs B0 obtained from simulations (both on log -log scale). Panel III shows the correlation between P and B0. 28 A comparison of material parameters between YIG and the material studied in Ref. [i] (Ref. [43] in main MS), viz., Co70Fe30 film. Material Parameters YIG Co70Fe30 Magneto -crystalline Anisotropy ( Ku) 6.10 × 102 J.m-3 7.6 × 104 J.m-3 Damping Constant ( α) 0.0005 0.015 A comparison from the above table shows that the Ku and of the Co 70Fe30 films used in the SLAC study [i] is two orders of magnitude larger than those in YIG. YIG has a cubic lattice with lattice parameter of a = 12.37 A, magnetic moment of = 40 B per unit cell [ ii]. The Zeeman energy density of YIG associated with giant magnetic field pulses, B ~ 7.5 MG can be written as, 314296B a MJ.m-3. This Zeeman energy ( Ez), generated either by the relativistic electron bunch at SLAC [43] or via intense FS laser pulses, overwhelm the magnetic anisotropy energy barrier of materials (for YIG Ez ~ 14296 MJ.m-3 >> its magnetic anisotropy energy, Ku = 6.10 x 102 J.m-3). Shape of domains formed in YIG after a long time (~ 100 ms after field pulse) In YIG, the equilibrium (100 msecs after B pulse) domain configuration is rectangular shaped, which minimizes the free energy (Fig. S4). These are the natural equilibrium shapes of the domains in YIG which minimize the free energy of the domains. Note the layered onion shell like magnetic domains in YIG film (Fig. 1 of our main MS) are completely different from these equilibrium rectangular shaped domains. The layered onion shell lik e magnetic domain structure is a snapshot of a rippling spin wave excited in YIG film by the intense fs laser pulse. 29 Figure S4 : Showing the simulated magnetization configuration of YIG, recorded 100 msecs after the magnetic field pulse. [i] S. J. Gamble et al., Electric Field Induced Magnetic Anisotropy in a Ferromagnet. Phys. Rev. Lett. 102, 217201 (2009). [ii] Sahalos , John N, Kyriacou , George A. , Tunable Materials with Applications in Antennas and Microwaves , (Morgan & Claypool Publishers ), 2019.

2020-08-21

Study of the formation and evolution of large scale, ordered structures is an enduring theme in science. The generation, evolution and control of large sized magnetic domains are intriguing and challenging tasks, given the complex nature of competing interactions present in any magnetic system. Here, we demonstrate large scale non-coplanar ordering of spins, driven by picosecond, megagauss magnetic pulses derived from a high intensity, femtosecond laser. Our studies on a specially designed Yttrium Iron Garnet (YIG)/dielectric/metal film sandwich target, show the creation of complex, large, concentric, elliptical shaped magnetic domains which resemble the layered shell structure of an onion. The largest shell has a major axis of over hundreds of micrometers, in stark contrast to conventional sub micrometer scale polygonal, striped or bubble shaped magnetic domains found in magnetic materials, or the large dumbbell shaped domains produced in magnetic films irradiated with accelerator based relativistic electron beams. Through micromagnetic simulations, we show that the giant magnetic field pulses create ultrafast terahertz (THz) spin waves. A snapshot of these fast propagating spin waves is stored as the layered onion shell shaped domains in the YIG film. Typically, information transport via spin waves in magnonic devices occurs in the gigahertz (GHz) regime, where the devices are susceptible to thermal disturbances at room temperature. Our intense laser light pulse - YIG sandwich target combination, paves the way for room temperature table-top THz spin wave devices, which operate just above or in the range of the thermal noise floor. This dissipation-less device offers ultrafast control of spin information over distances of few hundreds of microns.

Macroscopic, layered onion shell like magnetic domain structure generated in YIG film using ultrashort, megagauss magnetic pulses

2008.09473v1

1 Synthetic a ntiferromagnetic coupling between ultra -thin insulating garnets Juan M. Gomez -Perez1, Saül Vélez1, †, Lauren McKenzie -Sell2, Mario Amado2, Javier Herrero -Martín3, Josu López -López1, S. Blanco -Canosa1,4, Luis E. Hueso1,4, Andrey Chuvilin1,4, Jason W. A. Robinson2, Fèlix Casanova1,4, * 1CIC nanoGUNE, 20018 Donostia -San Sebastian, Basque Country, Spain 2Department of Materials Science and Metallurgy, University of Cambridge, 27 Charles Babbage Road, Cambridge CB3 0FS, United Kingdom 3ALBA Synchrotron Light Source, Carrer de la Llum 2 –26, 08290 Cerdanyola del Vallès, Catalonia, Spain 4 IKERBASQUE, Basq ue Foundation for Science, 48013 Bilbao, Basque Country, Spain †Present address: Department of Materials, ETH Zürich, 8093 Zürich, Switzerland. *Email: f.casanova@nanogune.eu Abstract The use of magnetic insulators is attracting a lot of interest due to a rich variety of spin- dependent phenomena with potential applications to spintronic devices . Here w e report ultra - thin yttrium iron garnet (YIG) / gadolinium iron garnet (GdIG) insulating bilayer s on gadolinium iron garnet (GGG) . From spin Hall magnetoresistance (SMR) and X-ray magnetic circular dichroism measurements , we show that the YIG and GdIG magnetically couple antiparallel even in moderate in-plane magnetic fields . The results demonstrate an all- insulating equivalent of a synthetic antiferromagnet in a garnet -based thin film heterostructure and could open new venues for insulators in magnetic devices . As an example, we demonstrate a memory element with orthogonal magnetization switching that can be read by SMR. I. Introduction Spintronics is an emerging field that involves the manipulat ion of not only electron charge but also electron spin, and is seen as a promising alternative to conventional charge -based electronics. The application of magnetic insulators for spintronics is gaining interest because such materials offer advantages over metals such as long spin transmission length s1 and the absence of energy dissipation due to Ohmic losses2. Heavy metal (HM)/ferromagnetic insulator (FMI) heterostructures are an interesting platform where a plethora of novel spintronics phenomena ha s been discover ed, including spin pumping1,3,4, spin Hall magnet oresistance (SMR)4,5, spin Seebeck effect4,6 and many others1,7–15. SMR is based on the interaction between the spin-Hall-induced spin accumulation at the HM layer and the magnetization of the FMI at the HM/FMI interface16. SMR is thus a good candidate to explore the magnetic properties of surfaces17–19 because it is only sensitive to the first atomic planes of the FMI20. The most extensively used FMI in insulating spintronics is y ttrium iron garnet or YIG (Y3Fe5O12), due to its low Gilbert damping, soft ferrimagnetism and negligible magnetic anisotropy1,4–7,10–15,17,18,21 –30. Alternative magnetic insulator s include antiferromagnet s31–33, non-collinear magnets34–36, hexagonal ferrites8, ferrima gnetic spinel s19,37–39 and other ferrimagnetic garnets9,40–45. Downscaling 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 magnetization reversal due to spin-orbit torque 8,9 or for guiding magnon s2. Since a top -down approach to nano fabrication requires the use of thin film materials, there is much effort focused on obtain ing high quality YIG thin films. Standard growth techniques such as liquid phase epitaxy ( LPE) are being pushed towards the 100 nm thickne ss46, but sub-100-nm-thick films still require alternative techniques such as pulsed laser deposition (PLD)47–49 or magnetron sputtering50–53. However, material quality in these cases is not as consistently high as seen in LPE-based YIG . For example, r ecent works report unusual magnetic anisotropy related to Fe3+ vacancie s in PLD -grown YIG47,48, and either exceptionally high magnetization53 or a magnetization suppression52 in sputtered films that could be related to the interface between YIG and the used substrate Gd3Ga5O12 (GGG ). The variety in the results and interpretations that can be found in the literature calls for an in -depth characterization of those thin YIG films . In this paper , we report ultra-thin (13 nm thick) epitaxial YIG on GGG. Structural and compositional analysis by transmis sion electron microscopy (TEM)/ scanning TEM (STEM) reveal a well-defined GdIG interlayer at the YIG/GGG interface. The magnetic properties of the top YIG layer , characterized by SMR ( using Pt as the spin -Hall material ) and X-ray magnetic circular dichroism (XMCD) measurements , are dramatically modified with the YIG magnetization pinned antiparallel to the GdIG one. The results demonstrate the presence of a negative exchange interaction between YIG and GdIG that constitutes a novel insulating synthetic antiferromagnet ic state, with a potential use in insulating spintronic devices54. For instance , we show that the complex interplay between the negative exchange interaction and the demagnetizing fields of the layers induce a memory effect that could be exploited as a device. II. Experimental details Epitaxial YIG (13 nm thick) is grown on (111) oriented GGG by pulse laser deposition (PLD) in an ultra-high vacuum chamber with a base pressure of better than 5×10-7 mbar . Prior to film growth, the GGG is rinsed with deionised water, acetone and is opropyl alcohol and annealed ex situ in a constant flow of O2 at 1000C for 8 hours. The YIG is deposited using KrF excimer laser (248 nm wa velength ) with a nominal energy of 4 50 mJ and fluence of 2.2 W/cm2. The films are grown under a stable atmosphere of 0.12 mbar of O 2 at 750C and fixed frequency of 4 Hz for 20 minutes. An in-situ postannealing at 850C is performed for 2 hours in 0.5 mbar partial pressure of static O 2 and subsequently cooled down to room temperature at a rate of -5 C/min . A 5-nm-thick Pt layer was magnetron -sputtered ex situ (80 W; 3 mtorr of Ar ) and a Hall bar (w idth 450 nm, length 80 m) was patterned by negative e- beam lithography and Ar -ion milling. Unpatterned s amples for TEM/STEM and XMCD were capped with a 2-nm-thick la yer of sputtered Pt. TEM/STEM was performed on a Titan 60 -300 electron microscope (FEI Co., The Netherlands) equipped with EDAX detector (Ametek Inc., USA), high angular annular dark field ( HAADF )-STEM detector and imaging Cs corrector. High resolution TEM (HR -TEM) images were obtained at 300 kV at negative Cs imaging conditions55 so that atoms look bright. Composition profiles were acquired in STEM mode with drift correction utilizing energy dispersive X -ray spectroscopy ( EDX ) signal. Geometrical Phase Analysis (GPA) was performed on HR -TEM images using all strong reflections for noise suppression56. Magnetotransport measurements were performed in a liquid -He cryostat (with a temperature T 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 method57–59. XMCD measurements were performed across t he Fe -L2,3 absorption edges at the BL29 -BOREAS beamline60 of the ALBA Synchrotron Light Source (Barcelona, Spain), using surface -sensitive total electron yield (TEY) detection. III. Results and Discussion III.a. Structural characterization Figure 1 shows the structural and compositional analysis of a Pt/YIG film by TEM/STEM. Figure 1(a) shows a HR-TEM cross -sectional micrograph where t he top layer corresponds to Pt (polycrystalline) , with epitaxial YIG on single crystal GGG beneath . The YIG/GGG interface reveals an extended region with visually different contrast. Comparison of high - resolution contrast in the YIG, interfacial and GGG regions [averaged unit cells are shown in the insets in Fig. 1(a)] show that within the same crystallographic structure there is a variation in distribution of heavy atoms from region to region . To confirm the nature of this middle region , we performed EDX analysis of a spatial distribution of the elements along the out-of-plane direction [ see Fig. 1(b), the scan line is indicated in Fig. 1(a)] . From this analysis , we confirm that the film consists of 2-nm-thick Pt on t he top surface, followed by a 12-nm-thick YIG layer that is Ga-doped. The i nterface between Pt and YIG is assumed to be atomically sharp, thus the inclination of Y curve and declination of Pt curve give the estimation of spatial resolution of composition measurement, which is of the order of 1 nm. At the depth of 1 2 nm, Y concentration decreases to zero, though the slope of declination is lower than at the upper interface, indicating a smooth change of concentration in this case. Gd concentration in the same region increases complementar ily to Y. At the same time the decrease of Fe concentration is delayed by ~3 nm relative to Y, and Ga concentration changes complementar ily to Fe. Thus , it may be concluded that , starting from a depth of 12 nm, Gd gradually (within a range of ~2 nm) substitutes Y in the lattice ; similarly Ga substitutes Fe , but with ~ 3 nm delay in depth . This d elay results in the formation of a 2.8-nm-thick interlayer with a nominal composition corresponding to gadolinium iron garnet (GdIG). The detailed analytical deconvolution of the concentration profiles gives a thickness of the “pure” GdIG as 2.2 nm61. Further insight into the nature of the layers can be obtained from the analysis of the interplanar distances in the direction normal to the surface. This is done by generalized GPA on the base of HR -TEM images56. Variations of the interplanar distance are calculated in terms of strain with respect to the GGG lattice . The obtained strain profile is presented as a black line in Fig. 1(b) , and shows that the region corresponding to GdIG composition is expanded by 1.1% with respect to GGG. This is lower than the 2.3% theoretically expected in epitaxial GdIG on GGG62, which could be explain ed by the presence of an inter-diffusion layer betwee n GGG and GdIG that reduces the strain as compared to a sharp interface . The YIG layer shows an unexpected 0.2% expansion of the lattice (on average) with respect to GGG , in spite of the very similar lattice constant62,63. The out -of-plane expansion of the YIG lattice, which may be attributed to the presence of vacancies47,48, is consistent with the X -ray diffraction measu rements (0.35 -0.6% expansion of the lattice ) in the same deposition batch. This detailed analysis confirms that we have a magnetic garnet bilayer. The presence of a Gd - doped YIG interlayer in YIG/GGG films after similar postannealing treatments has been recently reported51,52, but in ou r case we can confirm a well -defined, 2 .2-nm-thick GdIG layer, and the fact that the YIG layer is Ga -doped. 4 FIG. 1. (a) HR-TEM micrograph of Pt (2 nm)/ YIG (13 nm) (thickness es are nominal) on GGG (111) . Inset: average d unit cells obtained in the different regions shown corresponding to YIG, GdIG and GGG from top to bottom. (b) S patial distribution of the elements extracted along the white arrow in (a) by spatially resolved EDX . The strain , extracted from the HR-TEM image as a v ariation of the interplanar distance with respect to the GGG lattice, is also plotted as a black line. Strain of +0.01 means a lattice expansion by 1% in out -of-plane direction. III.b. Spin Hall magnetoresistance measurements The magnetic properties of th e ultra-thin magnetic garnet bilayer cannot be extracted using standard magnetometry, because the 500-m-thick GGG substrate shows a dominating paramagnetic background that masks the magnetic signal from the bilayer . We performed longitudinal SMR measurements , which only probe the top surface magnetization17–20, and thus the magnetization of the GdIG interlayer at the bottom interface is not expected to influence the SMR signal52. SMR depends on the relative angle of the surface magnetization in the FMI and the spin accumulation in the HM. When the spin accumulation and the magnetization are parallel (perpendicular) the longitudinal resistance state is low (high ). A two-point measurement on the films confirmed that the YIG is insulat ing at room temperature61,64. The patterned Hall bar on the YIG (see section II) corresponds to a Pt/Y IG structure widely measured before5,17,21,22,24,30. Figure 2 shows the longitudinal resistance RL from a 4-point configuration at 2 K vs H applied along the three main axes of the sample [see Fig. 2(a) ]. These field-dependent magnetoresistance (FDMR) curves are expected to show the features of SMR: i) a low resistance when the magnetic field satur ates the magnetization in the y-direction (i.e., parallel to the spin -Hall-induced spin accumulation in Pt) with a peak at low H corresponding to the magnetization reversal of the YIG film; ii) a high resistance value when H saturates the magnetization in the x- or z- direction (i.e., perpendicular to the spin accumulation in Pt) with a dip at low fields due to the magnetization rev ersal . However, t he FDMR curves are very different from the ones observed so far in YIG5,17,21,22. A high H ~ 8 T is needed to saturate the magnetization of the film [see FDMR curve along the y–direction in Fig. 2(a) ], while Y IG is expected to saturate within a few mT in plane22. This result already suggests that the top surface magnetization of the 12 -nm-thick YIG is strongly influenced by the 2.2-nm-thick GdIG at the bottom . Moreover, at relativ ely low H (below ~1.5 T) and at low temperature (below ~100 K , see Ref. [61] for high temperature behavior ) the FDMR curves along the three main axes show unexpected crossings [see Fig. 2(b) ], indicat ing complex magnetic behavior with the magnetization being non -collinear with the applied H. 5 FIG. 2. (a) Longitudinal FDMR measurements at 2 K along the three main axes (sketch indicates the definition of the axes, color code of the magnetic field direction , and the measurement configuration) . (b) Zoom of the FDMR curves at low magnetic field s. Three different zones associate d to the magnetization behavior are indicated (see text for details) . To understand better the magnetic properties and to confirm the non -collinear magnetization behavior of th e bilayer , we performed angular -dependent magnetoresistance (ADMR) measurements in –, – and –planes (see sketches in Fig. 3 ) at 2 K . The AD MR curves have three distinct behaviors depending on the applied H [zones 1 -3 indicated in Fig. 2(b )]. At high H [above ~1.5 T, zone 3, Fig. 3(c) ], we have a sin2 dependence with the angle for – and –planes , and no modulation for the –plane , which is the expected dependence for SMR16–19,38 when the magnetization is saturated and collinear with H. The same angular behavior is expected for Hanle magnetoresistance22 (HMR) , which has a common origin with SMR and is only relevant at ver y large fields . At low H [below ~0.25 T, zone 1, Fig. 3(a)] we still have the sin2 dependence in –plane, but the amplitude is smaller because the bilayer is not saturated (as evidenced in Fig. 2 (a)). However, we have an unusual ADMR for – and – planes . In –plane , the ADMR curve does not follow a sin2 dependence, indicat ing that the magnetization and H are not collinear. When H is perfectly out -of-plane (=0º and 180º ) the magnetization also points ou t-of-plane. As soon as H rotates away from the out -of-plane into the y–direction, the magnetization switches abruptly to the in -plane y-direction (=90º and 270º). This effect can not be simply explained by the demagnetization field due to the strong shape anisotropy expected in the ultra -thin film65,66. As we will s ee below, the presence of the GdIG layer also plays a role in this behavior . Accordingly, the same abrupt switch ing from the out -of-plane (=0º and 180º) into the in -plane x–direction ( =90º and 270º ) when 6 rotating H along the –plane should not give any ADMR modulation; however, the dip in ADMR at =0º and 180º shows that a small net contribution of the magnetization along y exists , probably because the YIG film breaks into domains . FIG. 3. Longitudinal ADMR measurements at 2 K along the three relevant H-rotation planes ( ) for different applied magnetic fields : (a) 0.1 T ( zone 1 ), (b) 0.5 T ( zone 2 ), and (c) 9 T ( zone 3 ). A different background RL0 is subtracted for the ADMR curves at each field. Sketches indicate the definition of the angles, the axes, and the measurement configuration. Dotted line at each sketch corresponds to 0º. At intermediate magnetic fields (0.25 T ≤ H ≤1.5 T , zone 2 , Fig. 3(b)), we can see an extra modulation in the ADMR curves for – and –plane s. In the case of –plane (plane) when H rotates from out -of-plane to in-plane parallel (perpendicular) to the y-direction , we observe a high (low) resistance state, suggest ing that the magnetization and H are not collinear in the plane of the sample . This is confirmed by the ADMR curve for –plane and H = 0.5 T , where the sin2 dependence is maintained, but with a phase shift 0 which can be either ~112º or ~68º and should correspond to the angle between H and the surface magnetization . To study with more detail the behavior of 0, we performed ADMR measurements for different applied magnetic fields (from 20 m T to 2 T) . Figure 4(a) show s how the phase of the ADMR curves shift with increas ing H. Figure 4(b) plots 0 as a function of H, showing a monotonic change between 0º and 180º. Although we cannot , in principle , determine if the phase shift at low fields corresponds to 0º or 180º, we assume that 0 goes from 180º at low fields to 0º at high fields because it is physically more plausible. The three different zones already described can be distinguished in this plot : (i) zone 1, where the surface magnetization is antiparallel to the applied H; (ii) zon e 2, where the surface magnetization has certain angle with H; (iii) zone 3, where the surface magnetization is almost align ed with H. 7 FIG. 4. (a) Longitudinal ADMR measurement s for different applied magnetic fields in the –plane at 5 K. (b) Phase shift ( 0) as a function of magnetic field taken from data in (a). 0 corresponds to the effective angle between the magnetization vector and the applied magnetic field . (c) Hysteresis loop measured by XMCD with the magnetic field applied in plane at 2 K . III.c. X-ray magnetic circular dichroism measurements To confirm this unconventional behavior that suggests that the surface magnetization of YIG opposes a low external field and only aligns parallel under a high enough field (> 1.5 T) , we perform ed XMCD , a technique that extract s information of the magnetization associated to each atomic species. The sample is oriented with its surface forming a grazing angle with respect to the propagation direction of incident x -rays (in -plane configuration ), H is applied parallel to the x -ray beam and TEY detectio n is used, which is sensitive to the surface . We obtained the typical XMCD spectrum for standard YIG at Fe L 2,3 absorption edges61. From these data, and applying the sum rules for XMCD spectra at different H values, we can estimate the magnetization per Fe ion and plot the hysteresis loop (see Fig. 4(c) ). The loop clearly confirms our scenario: a negative net magnetization is measured at low applied H, i.e., the magnetization vector of the YIG surface is aligned antiparallel to H. The net magnetization is reduced with incre asing H because the magnetization vector starts rotating monotonously towards the applied H and, at certain value of H, becomes perpendicular to H, leading to no net magnetization. At higher H, the net magnetization becomes positive while the magnetization vector approaches a collinear configuration with H, finally saturating at very high fields. Note that the saturation magnetization (3-3.5 μ B/Fe) is lower than expected in YIG (5 μ B/Fe), which can be explained by th e presence of Ga substituting Fe along our YIG film67. The behavior of the surface magnetization of YIG observed both via SMR and XMCD can be explained if we consider that YIG is in fact coupled antiparallel to the GdIG interlayer . A hysteresis loop similar to the one in Fig. 4(c) has been recently observed in Ni/Gd layers and attributed to the negative exchange coupling between the transition metal and rare -earth ferromagnets68. YIG has two magnetic sublatti ces [3 tetrahedrally coordinated (“FeD”) and 2 octahedrally coordinated (“FeA”) Fe3+ ions per formula unit ] which are antiferromagnetically coupled, leading to its ferri magnetism , with the magnetization dominated by the FeD sublattice . GdIG has the same iron garnet crystal structure, with a third magnetic sublattice ( 3 dodecahedrally coordinated Gd3+ ions per formula unit ), which is ferro magnetically coupled to the FeA sublattice. The strong variation of the magnetization of the Gd sublattice with temperature makes GdIG a compensated ferrimagnet , with the magnetization dominated by the Gd and FeA sublattices below room temperature43,45,69. We hypothesize that t he perfect epitaxy of the crystal structure at the YIG/GdIG interface (Fig. 1 (a)) would favor the continuity of the FeA and FeD s ublattices across the interface . Such continuity leads to an 8 antiferromagnetic coupling betwe en the net magnetization of the GdIG (dominated by Gd ) and the net magnetization of the YIG (dominated by FeD ). This very same coupling has been deduced from recent magnetooptical spectroscopy51 and polarized neutron reflectivity52 experiments in YIG/GGG interfaces . In our case, t he Gd magnetization at low T is so high that a 2.2-nm-thick GdIG layer can pin the whole 12-nm-thick YIG layer antiparallel to H. When increasing H in plane above ~0.25 T, the surface magnetization of YIG coherently rotates (see Fig. 4(b)) , becoming parallel above ~1.5 T. Note that the behavior of this bilayer is equivalent to that of a synthetic antiferromagnet70, although we are not aware of previous reports of such man-made system with insulating materials. III.d. Memory effect The presence of the negative exchange coupling between YIG and GdIG would also explain the sharp switch ing from the out -of-plane to in -plane magnetization deduced from th e shape of the ADMR in –plane (Fig. 3a and 3b). The strong and opposing demagnetization field s expected from the YIG and GdIG layer s combined with the antiferromagnetic coupling between them would favor the switching of the entire bilayer magnetization to the plane, much sharper than the case of a single YIG layer of similar thickness65. This effect is confirmed with detailed FDMR measurements sweeping at low magnetic fields along z- direction (Fig. 5): the higher resistance state corresponds to the YIG magnetization point ing out-of-plane and the lower resistance state around zero field corresponds to in -plane magnetization. Interestingly, the sw itching has a clear hysteretic behavior , which is probably due to the complex interp lay between antiparallel coupling and the opposing demagnetizing fields of each layer . Switching between two metastable states with orthogonal magnetic configurations can b e used in a memory device which is written with a very low magnetic field and read by longitudinal SMR. This w ould be an advantage with respect to p revious proposal s of a memory device based on magnetic insulators with perpendicular magnetic anisotropy , because they use the transverse SMR to read the magnetization state, which has a resistance change almost three order s of magnitude smaller 8,9. FIG. 5. Longitudinal FD MR measurement (trace and retrace) at 2 K with the magnetic field applied along the z-direction (out -of-plane) . 9 IV. Conclusions We structurally and magnetically characterized ultra-thin epitaxial YIG films on GGG , which reveal an atomically well-defined interlayer of GdIG at the YIG/GGG interface. From SMR and XMCD we demonstrate that t he YIG magnetization opposes moderate external magnetic fields. This unconventional behavior occurs because YIG /GdIG couple magnetically antiparallel , forming the equivalent to a synthetic antiferromagnet, with the exceptional fact of being insulating. Furthermore, we observe a memory effect between orthogonal magnetization orientations, which can be read with an adjacent Pt film via longitudinal SMR measurements. This bilayer system could be further engineered to optimize the functionalities exploited in insulating spintronic devices, such as writing operation s with spin-orbit torque and reading operation s with SMR in insulating magnetic memories8,9, or in envisioned devices where the application of antiferromagnets71 and their synthetic versions54 is advantageous . Acknowledgments The work was supported by the Spanish MINECO under the Maria de Maeztu Units of Excellence Programme (MDM -2016 -0618) and under Project No. MAT2015 -65159 -R and by the Regional Council of Gipuzkoa (Project No. 100/16). J.M.G. -P. thanks the Spanish MINECO for a Ph.D. fellowship (Grant No. BES -2016 -077301) . J.L.-L. thanks the Basque Government for a Ph.D. fellowship (Grant No. PRE-2016 -1-0128 ). J.W.A.R., M.A., and L.M-S. a cknowledge funding from the Royal Society and the EPSRC through the “International network to explore novel superconductivity at advanced oxi de superconductor/magnet inter faces and in nanodevices ” (EP/P026 311/1) and the Programme Grant “ Superspin” (EP/N017242/1). L.M -S. also acknowle dges funding the Winton Trust. J.W.A.R and M.A. also acknowledges support from the MSCA -IFEF -ST Marie Curie Grant 656485 -Spin3. XMCD measurements were performed at BL29 -BOREAS beamline with the collaboration of ALBA staff. References 1. Kajiwara, Y. et al. Transmission of electrical signals by spin -wave interconversion in a magnetic insulator. Nature 464, 262–266 (2010). 2. Chumak, A. V., Vasyuchka, V. I., Serga, A. A. & Hillebrands, B. Magnon spintronics. Nat. Phys. 11, 453–461 (2015). 3. Saitoh, E., Ueda, M., Miyajima, H. & Tatara, G. Conversion of spin current into charge current at room temperature: Inverse spin -Hall effect. Appl. Phys. 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Exchange magnetic field torques in YIG/Pt bilayers observed by the spin-Hall magnetoresistance. Appl. Phys. Lett. 103, 32401 (2013). 31. Hoogeboom, G. R., Aqeel, A., Kusche l, T., Palstra, T. T. M. & Van Wees, B. J. Negative spin Hall magnetoresistance of Pt on the bulk easy -plane antiferromagnet NiO. Appl. Phys. Lett. 111, 52409 (2017). 32. Baldrati, L. et al. Full angular dependence of the spin Hall and ordinary magnetoresistance in epitaxial antiferromagnetic NiO(001)/Pt thin films. arXiv:1709.00910 (2017). 33. Ji, Y. et al. Spin Hall magnetoresistance in an antiferromagnetic magnetoelectric Cr2O3/heavy -metal W heterostructure. Appl. Phys. Lett. 110, 262401 (2017). 34. Aqeel, A. et al. Electrical detection of spiral spin structures in Pt|Cu2OSeO3 heterostructures. Phys. Rev. B 94, 134418 (2016). 35. Aqeel, A. et al. Spin-Hall magnetoresistance and spin Seebeck effect in spin -spiral and paramagnetic phases of multiferroic CoCr2O4 films. Phys. Rev. B 92, 224410 (2015). 36. Jonietz, F. et al. Spin Transfer Torques in MnSi. Science (80 -. ). 330, 1648 –1652 (2011). 37. Valvidares, M. et al. Absence of magnetic proxim ity effects in magnetoresistive Pt/CoF e2 O4 hybrid interfaces. Phys. Rev. B 93, 214415 (2016). 38. Isasa, M. et al. Spin Hall magnetoresistance at Pt/CoFe2O4 interfaces and texture effects. Appl. Phys. Lett. 105, 142402 (2014). 39. Shan, J. et al. Nonloca l magnon spin transport in NiFe 2 O 4 thin films. Appl. Phys. Lett. 110, 132406 (2017). 40. Maier -Flaig, H. et al. Tunable magnon -photon coupling in a compensating ferrimagnet - from weak to strong coupling. Appl. Phys. Lett. 110, 132401 (2017). 41. Cramer , J. et al. Magnon Mode Selective Spin Transport in Compensated Ferrimagnets. Nano Lett. 17, 3334 –3340 (2017). 42. Ganzhorn, K. et al. Non-local magnon transport in the compensated ferrimagnet GdIG. arXiv:1705.02871 (2017). 43. Ganzhorn, K. et al. Spin Hall magnetoresistance in a canted ferrimagnet. Phys. Rev. B 94, 94401 (2016). 44. 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See Supplemental Material at [URL will be inserted by publisher] for a detailed analysis of the concentration profile, electrical characterization, FDMR curves at different temperatures and XMCD data of the samples. 62. The comparison of the lattice consta nt of GGG (12.375 Å, very close to YIG, 12.376 Å) with the one in GdIG (12.471 Å) translates into a change of volume that, due to epitaxial conditions, gives a 2.3% change in strain in the out -of-plane direction. Values extracted from [ref. 6 3]. 63. Geller , S., Espinosa, G. P. & Crandall, P. B. Thermal expansion of yttrium and gadolinium iron, gallium and aluminum garnets. J. Appl. Crystallogr. 2, 86 (1969). 64. Thiery, N. et al. Electrical properties of single crystal Yttrium Iron Garnet ultra -thin films a t high temperatures. arXiv:1709.07207 (2017). 65. Hahn, C. et al. Comparative measurements of inverse spin Hall effects and magnetoresistance in YIG/Pt and YIG/Ta. Phys. Rev. B 87, 1–8 (2013). 66. Wang, P. et al. Spin rectification induced by spin Hall mag netoresistance at room temperature. Appl. Phys. Lett. 109, (2016). 67. Dionne, G. F. Molecular field coefficients of substituted yttrium iron garnets. J. Appl. Phys. 41, 4874 –4881 (1970). 68. Higgs, T. D. C. et al. Magnetic coupling at rare earth ferromagnet/transition metal ferromagnet interfaces: A comprehensive study of Gd/Ni. Sci. Rep. 6, 30092 (2016). 69. Yamagishi, T. et al. Ferrimagnetic order in the mixed garnet (Y1 -xGd x)3Fe5O12. Philos. Mag. 85, 1819 –1833 (2005). 70. Parkin, S. et al. Magnetically engineered spintronic sensors and memory. IEEE 91, 661–679 (2003). 71. Jungwirth, T., Marti, X., Wadley, P. & Wunderlich, J. Antiferromagnetic spintronics. Nat. Nanotechnol. 11, 231–241 (2016). 13 SUPPLEMENTAL MATERIAL S1. Analytical deconvolution of the concentration profiles obtained by TEM Concentration profiles were fitte d by sigmo id functions [Fig. S1(a)]. This fit gives a precise measure of the thickness of YIG layer (the distance between the surface and position of 50% Y concentration) as 12.0 nm, and the thickness of Fe -Gd (measured as the distance between 50% Gd and 50% Fe concentrations) mixing zon e as 2.8 nm. Assuming that Pt and top Y interfaces should be abrupt step functions from the parameters of sigmo id function , we can extract a transfer function of EDX mapping and make an analytical deconvolution of concentration p rofiles at the interfaces. Figure S1(b) shows deconvoluted profiles at the interface region normalized to 0 -1 interval in order to compare their relative sharpness. From these deconvolutions we can estimate the “true” widths of concentrations decay/increase. In terms of 25 -75% inte rval this will be: for Y – 0.7 nm, for Gd – 0.9 nm, for Ga – 0.8 nm, for Fe – 0.5 nm. Deconvoluted profiles also give the estimation of the thickness of “pure” GdIG layer (in terms of >75% of both Gd and Fe), which is 2.2 nm [shown i n Fig. S1(b) ]. FIG. S1. (a) Concentration profiles ( solid lines ) fitted by sigmo id functions (dashed lines) . (b) Deconvoluted fitted profiles in the GdIG layer region. Inte nsities are normalized to 0 -1 interval in order to compare the sharpness of the transitions for differen t elements. S2. Transport properties of the ultra -thin iron garnet bilayer In order to check the electrical behavior of the fabricated film, we applied a voltage between 60-m-long Pt strips (separated by 24 m) and detected the charge current flowing through the YIG/G dIG bilayer , see Fig. S2. The high resistance measured (~1012 ), similar to thick crystalline YIG, confirms that our YIG /GdIG ultra-thin film behaves as an insulator . This control experiment rules out leakage currents through the YIG/GdIG bilayer as the origin of the magnetoresistance effects observed in the longitudinal resistance of the Pt strips. 14 FIG. S2. Current -voltage characteristics of our YIG/GdIG ultrathin film measured between two Pt strips at 300 K . S3. Field -dependent magnetoresistance measurements at high er temperature s We show the FDMR curves obtained at higher temperatures corresponding to the same sample used in the main text . The unexpected crossings shown by the FDMR curves along the three main axes remain up to 100 K. Above this temperature , no signature of SMR is observed, suggesting that the surface of our YIG ultra -thin film is non-magnetic . The features and symmetry of t he observed magnetoresistance correspond to Hanle magnetoresistance , an effect related to SMR occurring solely at the Pt thin film1. FIG. S3. Longitudinal FDMR measurements performed at (a) 30 K, (b) 50 K, (c) 70 K, (d) 100 K, (e) 200 K and ( f) 300 K, along the three different main axes (sketch in ( f) indicates the definition of the axes, colour code of the magnetic field direction, and the measurement configuration). Insets in (a), (b), (c) and (d) are a zoom between -2 T and 2 T. S4. Magnetic characterization , XMCD measurements Figure S4(a) shows the spectra for circular polarization and the corresponding x-ray absorption spectr um (XAS) of the sample YIG (13 nm)/Pt (2 nm) from where we extract ed 15 the XMCD curve shown in Fig. S4(b). This XMDC spectrum is consistent with the Fe L 2,3 edge in thick YIG. We obtained the hysteresis loops by sweeping t he magnetic field between 6 T and –6 T in in-plane and out -of-plane configuration and measuring the difference of the XMCD absorption peak (710.2 eV, Fe L 3 peak ) at 2 K . The in -plane hysteresis loop is shown in the main text, whereas the out -of-plane hysteresis loop is shown in Fig. S4(c), confirming the hard magnetization behavior of our sample due to the strong shape anisotropy . FIG. S4. (a) Absorption spectra for positive (black line) and negative (red line) circular ly polarize d light and X -ray absorption spectrum (blue line) at 2 K and 6 T. (b) XMCD spectrum extracted from the XAS measurements. (c) Hysteresis loop measured by XMCD with the magnetic field applied o ut of plane at 2 K. References 1. Vélez, S. et al. Hanle Magnetoresistance in Thin Metal Films with Strong Spin -Orbit Coupling. Phys. Rev. Lett. 116, 16603 (2016).

2018-03-15

The use of magnetic insulators is attracting a lot of interest due to a rich variety of spin-dependent phenomena with potential applications to spintronic devices. Here we report ultra-thin yttrium iron garnet (YIG) / gadolinium iron garnet (GdIG) insulating bilayers on gadolinium iron garnet (GGG). From spin Hall magnetoresistance (SMR) and X-ray magnetic circular dichroism measurements, we show that the YIG and GdIG magnetically couple antiparallel even in moderate in-plane magnetic fields. The results demonstrate an all-insulating equivalent of a synthetic antiferromagnet in a garnet-based thin film heterostructure and could open new venues for insulators in magnetic devices. As an example, we demonstrate a memory element with orthogonal magnetization switching that can be read by SMR.

Synthetic antiferromagnetic coupling between ultra-thin insulating garnets

1803.05545v1

Unraveling the origin of antiferromagnetic coupling at YIG/permalloy interface Jiangchao Qian,1Yi Li,2Zhihao Jiang,1Robert Busch,1Hsu-Chih Ni,1 Tzu-Hsiang Lo,1Axel Hoffmann,1Andr´ e Schleife,1and Jian-Min Zuo1,∗ 1Materials Research Laboratory and Department of Materials Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA 2Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA (Dated: March 26, 2024) We investigate the structural and electronic origin of antiferromagnetic (AFM) coupling in the Yttrium iron garnet (YIG) and permalloy (Py) bilayer system at the atomic level. Ferromagnetic Resonance (FMR) reveal unique hybrid modes in samples prepared with surface ion milling, in- dicative of antiferromagnetic exchange coupling at the YIG/Py interface. Using atomic resolution scanning transmission electron microscopy (STEM), we found that AFM coupling appears at the YIG/Py interface of the tetrahedral Fe terminated YIG surface formed with ion milling. The EELS measurements suggest that the interfacial AFM coupling is predominantly driven by an oxygen- mediated super-exchange coupling mechanism, which is confirmed by the density functional theory (DFT) calculations to be energetically favorable. Thus, the combined experimental and theoreti- cal results reveal the critical role of interfacial atomic structure in determining the type magnetic coupling in a YIG/ferromagnet heterostructure, and prove that the interfacial structure can be experimentally tuned by surface ion-milling. Yttrium iron garnet (Y 3Fe5O12) is well-known for its low magnetic damping [1–7], making it the material of choice for efficient spin interactions in magnonics [8–11], spin transport [12], cavity spintronics [13–16], and quan- tum information science [17, 18]. Considerable atten- tion has been directed toward building YIG-based thin film heterostructures for spin-based information process- ing by taking advantage of interfacial spin interactions. One example is the YIG/Pt bilayers, with experimental observations of spin pumping [19], nonlocal spin injection [12], and spin Hall magnetoresistance [20], where the in- terlayer exchange coupling has significantly enhanced the spin transmission across the YIG-Pt interface. Recently, another YIG heterostructure with a ferro- magnetic (FM) layer, i.e. YIG/FM bilayer, has aroused increasing interests owing to its potential in hybrid magnonics from the interlayer magnon-magnon coupling [21–24]. The ferromagnetic resonance mode in the FM layer can form strong coupling with the perpendicular standing spin wave modes in YIG due to the interfa- cial exchange interaction, leading to new physical phe- nomena such as coherent spin pumping [24], magneti- cally induced transparency [25], and efficient excitations of short-wavelength spin waves [26]. However, the physi- cal mechanisms underlying the interfacial exchange cou- pling, crucial for coupling spin excitations between the two magnetic systems, remain not fully understood. Par- ticularly, recent works have revealed pronounced antifer- romagnetic coupling across YIG/CoFeB, YIG/Co, and YIG/Py interfaces, where the origin of antiferromag- netic coupling between YIG and FM layers has been a topic of considerable debate. Previous studies, reported either ferromagnetic coupling or antiferromagnetic cou- pling in different prepared YIG/FM bilayers [21, 27, 28]. Specifically, Fan et al. [27], Quaterman et al. [28], andKlingler et al. [21] have posited theories of direct ex- change coupling, while the possibility of super-exchange coupling also remains. These findings hint at the signif- icant role of interfacial structure in determining bilayer coupling mechanisms, yet tuning magnetic interactions through interface engineering remains insufficiently ex- plored. Here we elucidate the interface structure between YIG and Py and their magnetic coupling. By integrat- ing FMR with STEM/EELS and DFT calculations, our study reveals the significant role of surface treatments us- ing ion-milling in enhancing antiferromagnetic coupling through promoting the oxygen mediated super-exchange coupling mechanisms at the YIG/Py interface. In line with the preparation of YIG/Py bilayer in our prior work [24], we first deposited YIG (100 nm) onto two (111)-oriented Gd 3Ga5O12substrates by magnetron sputtering. Then the amorphous YIG films were an- nealed in air at 850 °C for 3 hours, and slowly cooled down to room temperature by 0.5 °C/min, yielding epi- taxial YIG films with light yellow color. To study the formation of antiferromagnetic interfacial exchange cou- pling, we slightly ion milled one YIG film in the sput- tering chamber under Ar environment by applying an RF bias voltage through the substrate holder, where the holder acts as an effective sputtering gun, and trigger Ar+ion bombarding of the substrate surface.The milling rate was 3 nm/mins and the milling process lasted for 1 min 30 s. A Py (10 nm) thin film was subsequently sput- tered on the milled YIG film without breaking the vac- uum, ensuring efficient interfacial exchange coupling. A control YIG/Py bilayer sample was also prepared with- out ion milling the YIG surface. Figure 1a shows the cross-sectional film structure. The quality of the bilayers was checked using X-Ray Diffraction. Clear (444) peaks of YIG and GGG were measured along with Laue os-arXiv:2402.14553v2 [cond-mat.mtrl-sci] 22 Mar 20242 FIG. 1. The YIG/Py bilayer structure and FMR character- ization. (a) STEM image together with the designed bilayer structure. (b) Illustration of the FMR setup: The copla- nar waveguide is located underneath the YIG/Py bilayer thin films. (c)-(d): FMR induced magnetization excitations in YIG/Py bilayer samples on top of the coplanar waveguide. The Line shapes of (c) without ion-milling and (d) with ion- milling samples for the detected resonance mode (n=2) of YIG and the uniform mode (n=0) of Py. cillations, indicating that both samples possess high film quality and maintain epitaxial relationships (Suppl. FIG. 1). The two samples we prepared will be named subse- quently as IM for with ion milling and WoIM for without ion milling. We first conducted a FMR measurement on the two samples we prepared with and without ion milling. Fig- ure 1b details this experimental arrangement using the same setup with the coplanar waveguide beneath the YIG/Py bilayer as in our previous work [24]. The FMR results are depicted in FIG. 1c and 1d, showing FMR- induced magnetization excitations in the YIG/Py bi- layer samples. Notably, two hybrid modes are present in the ion-milled samples, but absent in those without ion- milling. As illustrated in FIG. 1d, these observed hybrid modes in the YIG/Py bilayer system’s spin pumping ex- periment signify antiferromagnetic exchange coupling at the interface. The broader linewidth mode, exhibiting a higher resonance field than the narrower linewidth mode, acts as a key indicator of this antiferromagnetic coupling. This finding is discussed in detail in our previous work [24]. To investigate the structural origin of the magnetic dif- ferences between samples with and without ion-milling,we performed cross-sectional STEM of the YIG/Py bi- layer films using a high-angle annular dark-field detec- tor (HAADF) for Z-contrast (details in Suppl. Notes). Figure 2a shows the STEM-HAADF images of the two samples, providing an atomic-scale inspection of their in- terfacial structural differences. The YIG surface termi- nation in the IM sample, as revealed in FIG. 2a, com- prises a plane of visible Fe and Y atoms, and invisible O atoms, leading to an interface layer that sharply con- nects to the polycrystalline Py. In contrast, the interface in the WoIM sample exhibits roughness and features an amorphous layer approximately 0.7 nm in width (Fig. 2b). The direct adjacency of the termination plane to the Py layer in the IM sample eliminates the observed gap in the WoIM sample. This distinction in interfacial structure between the two samples is further substanti- ated by the normalized intensity line profiles depicted in FIG. 2c. The YIG surface termination in the IM sample is further examined in FIG. 2d, with line profiles marked in FIG. 2a. These profiles demonstrate that the termina- tion plane ends with a combination of Y and Fe contain- ing atomic columns, with Py atoms directly adjoining the crystalline garnet structure in the IM sample. (a) (d)(b) (c) Interfacial layer without surface ion-milling #1 #2With Surface Ion -Milling Without Surface Ion -Milling Interfacial layer with surface ion-milling FIG. 2. Interfacial structure as seen by atomic-resolution STEM-HAADF along the YIG [110] zone axis. STEM- HAADF images for the with ion-milling (a) and without ion- milling (b) samples. The scale bars are both 0.5 nm. (c) The marked horizontal line profiles of HAADF images in (a) and (b) show the different YIG/Py interface width. (d) The line profiles of HAADF intensity across the dashed lines in (a), showing the termination of Y and Fe, with Py atoms adja- cent to YIG. From our STEM-HAADF observations, the garnet structure appears consistent up to the interface, sug-3 gesting a minimal structural modification. Electron en- ergy loss spectroscopy (EELS) analysis was performed in the STEM mode to further examine the chemical sharp- ness of the YIG/Py interface in the IM sample. Figures 3a,b display the oxygen K-edge fine structure, alongside a HAADF survey image (FIG. 3c) with a sampling reso- lution of 0.5 nm. This combination provides insights into both the oxygen content and the nature of its chemical bonding. The oxygen K-edge fine features are observed up to the interface. A drop in the K-edge intensity is ob- served near within the distance of 1 nm to the interface, which can be attributed to the electron probe spreading effect [29]. In addition to the pre-peak for oxygen, the fine features in the range of 535 eV to 540 eV also exhibit changes near the interfaces, indicative of slightly altered yttrium-oxygen (Y-O) bonding. FIG. 3d,e showcase the EELS fine structures for the Fe L 3-edge, coupled with a HAADF survey image (FIG. 3f). The marked peaks of 708.7 eV and 710 .5 eV in FIG. 3d are features for Fe0 and Fe3+respectively. The lower layer consists of Fe0 from Py, while the upper layer contains Fe3+from YIG. The evolution of the oxygen K-edge and Fe L 3edge are further highlighted in FIG. 3b,e, which plot the hyper- spectral EELS data versus the STEM probe position. At the interface, the Fe L 3EELS signal is approximately a composite of Fe0and Fe3+, while O2−is detected up to the interface. The presence of O2−and Fe3+at the interface and slightly beyond, and its sharper transition than the L 3edge of Fe, suggests an oxygen-terminated YIG surface. The fine feature of L 3edge shows the domi- nant peak is still 710 .5 eV with a wider shoulder left.The coexistence of Fe0and Fe3+in the interfacial transition region suggests a small amount of interfacial defects, pos- sibly Fe interstitial atoms inside a rather open YIG struc- ture. The EELS map shows a oxidized YIG surface with remaining garnet structures directly adjacent to the Py layer. To uncover the origin of experimentally observed an- tiferromagnetic coupling at the ion-milled interface of YIG/Py theoretically, we employed density functional theory (DFT) calculations to simulate the YIG/Py in- terfacial configurations. The interface structural models we proposed based on the experimental observations are depicted in FIG. 4a,b. The YIG (111) surface is termi- nated with yttrium, coordinated by 8 oxygen ions, and tetrahedral coordinated Fe, respectively. The superposi- tion of the YIG structural motifs on the magnified atomic resolution image from FIG. 2a shows good agreement be- tween the model and the observed interfacial atomic ar- rangements. The Py atoms, which are not individually resolved in FIG. 4a, appear slightly disordered due to in- teractions with the YIG surface. For simplicity, we have assumed two perfect layers of Py atoms parallel to the YIG surface in our model. Two potential magnetic arrangements can exist at the YIG/Py interface: antiferromagnetic (AFM), where bulk (a) (b) (c) O K Fe L3(d) (e) (f) Fe0Fe3+FIG. 3. With surface ion-milling sample: the EELS near- edge fine structures for (a) O K-edge; and (d) Fe L3-edge, across the YIG/Py interface in the corresponding regions shown in the HAADF images at right, (c) and (f), respec- tively. The spectra are integrated horizontally. The 0 nm po- sition is set at the YIG/Py interfaces. Hyperspectral images for O K-edge (b) and (e) Fe L3-edge shows the fine structure evolution. The scale bars are both 1 nm in (c) and (f). The results here directly evident of the presence of O2−and Fe3+ at the interface. YIG and Py have opposite magnetic moments, and ferro- magnetic with same magnetic moments. Using the struc- ture model in FIG.4b, we performed spin-polarized DFT calculation with different Fe and Ni arrangements in Py (details in Suppl. Notes). The results indicate that the AFM arrangement is energetically more favorable than FM arrangement with an average energy difference of 2.2 meV per atom. However, when we placed the Py layers on the YIG slab with another different surface termina- tion, the simulation results show no preference between FM and AFM interficial coupling (detailed structures of simulation cells of different interface terminations can be seen in FIG. S2 and FIG. S3 in SI). Within the YIG, oc- tahedral Fe and tetrahedral Fe are antiferromagnetically coupled, while the tetrahedral Fe provides the dominant spin. In the AFM arrangement, the spins of tetrahe- dral Fe are also antiferromagnetically coupled with Py atoms. The DFT simulation results thus supports our experimental observation of AFM in the ion-milled sam- ple where the tetrahedral Fe ions interact with Py atoms with the mediation of oxygen ions as highlighted by cir- cles in Fig. 4b. The surface on the YIG without the ion-milling in comparison are rough with a mixture of surface terminations in addition to the amorphous-like layer that breaks the AFM coupling between bulk YIG and Py. Given the coexistence of Fe0, Fe3+, and oxygen at the YIG/Py interface, we propose that oxygen-mediated super-exchange coupling could be the predominant mech-4 anism for the antiferromagnetic interaction between Py and YIG, conceptualized as < Fe0| ↑>−−O−−< Fe3+| ↓>. The substitution of Ni0for Fe0likely does not alter the direction of the magnetic moments. This insight is crucial in our discussion, as it suggests that the antiferromagnetic coupling between Py and YIG is pre- dominantly driven by oxygen-mediated super-exchange coupling, a key finding of this study. Permalloy YIG Fe Y Py(b) (a) O FIG. 4. YIG surface termination and structure model. (a) Zoomed-in HAADF image in FIG. 2a showing the over- lap with YIG motif structures at surface terminations. (b) YIG/Py slab model for Density Functional Theory (DFT) simulation. Highlighted white hexagons represent YIG mo- tifs. Both (a) and (b) are illustrated at [110] projection. The inset illustrates the coupling between Py atom and tetrahe- dral coordinated Fe In conclusion, we have identified interfacial atomic structure as a key factor in determining the type of mag- netic coupling between YIG and Py. AFM coupling, as implied from our FMR measurements, is associated with a sharp YIG/Py interface with tetrahedral Fe as the first magnetic layer on YIG side. DFT calculations quantitatively confirm that AFM coupling is an energeti- cally favorable magnetic state for this particular interface structure. The origin of AFM is explained by oxygen- mediated super-exchange interaction between tetrahedral Fe in YIG and magnetic atoms (Fe and Ni) in Py. Our experiments also point out that such particular interfa- cial structure enhancing AFM coupling can be fabricated with surface ion-milling. Acknowledgement. The work was supported by the U.S. DOE, Office of Science, Basic Energy Sciences, Ma- terials Sciences and Engineering Division, with all parts of the manuscript preparation supported under contract No. DE-SC0022060. 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2024-02-22

We investigate the structural and electronic origin of antiferromagnetic coupling in the Yttrium iron garnet (YIG) and permalloy (Py) bilayer system at the atomic level. Ferromagnetic Resonance (FMR) reveal unique hybrid modes in samples prepared with surface ion milling, indicative of antiferromagnetic exchange coupling at the YIG/Py interface. Using scanning transmission electron microscopy (STEM), we highlight significant interfacial differences introduced by ion-milling. The observations suggests that the antiferromagnetic coupling in YIG/Py bilayers is predominantly driven by an oxygen-mediated super-exchange coupling mechanism on the tetrahedral Fe terminated YIG surface, which is supported by density functional theory (DFT) calculations. This research provides critical insight into the fundamental mechanisms governing the efficiency of coupling in magnetic bilayers and underscores the pivotal role of oxide surface termination in modulating magnetic interfacial dynamics.

Unraveling the origin of antiferromagnetic coupling at YIG/permalloy interface

2402.14553v2

Acoustic excitation and electrical detection of spin waves and spin currents in hypersonic bulk waves resonator with YIG/Pt system N. I. Polzikovaa,*, S. G. Alekseeva, V. A. Luzanovb, A. O. Raevskiyb aKotel'nikov Institute of Radio Engineering and Electronics of Russian Academy of Sciences, Mokhovaya str. 11, build. 7, Moscow, 125009, Russia bFryazino branch Kotel'nikov Institute of Radio Engineering and Electronics of Russian Academy of Sciences Vvedenskiy sq.,1, Fryazino, Moscow Region, 141190, Russia Abstract We report on the self-consisted semi -analytical theory of magnetoelastic excitation and electrical detection of spin waves and spin currents in hypersonic bulk acoustic waves resonator with ZnO - GGG -YIG/Pt layered structure. Electrical detection of acoustically driven spin waves occurs due to spin pumping from YIG to Pt and inverse spin Hall (ISHE ) effect in Pt as well as due to electrical response of ZnO piezotransducer . The frequency -field dependences of the resona tor frequencies and ISHE voltage UISHE are correlated with experimental ones observed previously. Their fitting allows to determine some magnetic and magnetoelastic parameters of YIG. The analysi s of the YIG film thickness influence on UISHE gives the possibility to find the optimal thickness for maximal UISHE value. Key words : magnetoelastic interaction, acoustic spin pumping, bulk acoustic waves, resonator, YIG/Pt, ZnO 1.Introduction In recent years, acoustically driven spin waves (ADSW) are of great interest in connection with the key objectives of next -generation spin -based technologies [1-14]. The piezoelectric generation of ADSW in composite magnetoelastic structures [8 - 14] is promising for use in low power consumption devices free from energy dissipation due to o hmic losses . In particular, acoustic spin pumping - the generation of spin -polarized electron currents from ADSW [7-9] - is promising for microwave spintronics and attracts much attention of the researchers. The acoustic waves (AW) and spin waves (SW) coupling , caused by linear magnetostriction, is most significant under the condition of the phase syn chronism, i.e. at magnetoelastic resonance (MER) . As far as MER frequency generally lies in the gigahertz range [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 with the frequencies up to 20 GHz is the use of a high overtone (n ~ 102 ÷103) bulk acoustic wave resonator (HBAR) [16]. Previously , in [13,14] we demonstrated the piezoelectric excitation of ADSW at 2 GHz by means of HBAR containing ferrimagnetic yttrium iron garnet (YIG) and piezoelectric zinc oxide ( ZnO) films . Quite recently, the resonant acoustic spin pumping in HBAR containing YIG/ Pt system was proposed and implemented in our works [17, 18]. This paper presents a theoretical consideration for acoustic spin pumping in HBAR and detection of the ADSW through the inverse spin Hall effect (ISHE) in Pt. Accounting for the back action of ADSW in YIG on the elastic system in all layers of the structure (in non -magnetic layers through boundary conditions) makes it possible to determine and compare the frequency and magnetic field dependences of HBAR resonance frequ encies , fn , and dc ISHE voltage, UISHE . Comparison of the se theoretical and experimental [18] dependences shows a qualitative agreement and allows us to determine a number of magnetic parameters of the YIG films. The calculation a lso shows that there is the optimal YIG film thickness for acoustic spin pumping efficienc y, which may be an order of magnitude high er than observed previously by means of HBAR . 2. HBAR s tructure In Fig.1 the HBAR structure is shown. It contains a gadolinium -gallium garnet (GGG) substrate 4 and two YIG films 3, 5 on both sides of the substrate [19]. A bulk AW transducer consisting of a piezoelectric ZnO film 1, sandwiched between thin -film Al electrodes 2, is deposited on one side of the YIG -GGG -YIG structure. To excite the bulk AW propagating along the x-axis, the rf voltage Ũ(f) with frequency f is applied across the transducer. A thin Pt strip 6 is attached to the YIG film 5 underneath the acoustic resonator aperture. Below we will use for the layer with the index i =1…6 the notations l(i) for the thickness and xi for the coordinate of the lower surface. The external magnetic field H lies in the plane of the structure along the z-axis and magnetizes YIG films up to uniform saturation magnetization M0|| H. It is assumed that ZnO film with an inclination of piezoelectric c-axis excites shear bulk AW polarized along the z-axis [20]. In YIG layers , this wave drives magnetization dynamics due to the magnetoelastic interaction. The AW and SW interaction results in the shift Δfn(H) = fn(H) - fn(0) of HBAR resonance frequencies in the magnetic field [13, 14]. The resonance frequencies itself correspond to the extrem a in the frequency response of the transducer's electrical impedance. Thus, the SW excitation and detection are performed electrically by the same 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 an electrostatic dc field EISHE along the y-axis (idle circuit) [22]. Fig. 1. HBAR structure: 1 — ZnO film, 2 —Al electrodes, 3, 5 —YIG films, 4 — GGG substrate, 6 — thin film Pt strip with direction a perpendicular to the figure plane . The typical layer thicknesses: l(1) = 3 μm, 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 electrodes 2 has a diameter aa = 170 μm. 3. Theory Further, we assume that all the thicknesses of the layers l(i) are much smaller than transverse dimensions in the plane ( y, z). In this case , in linear approximation , all variables depend on coordinate x and time t as exp[j(k(i)x – ωt)], where 1j , ω=2πf , k(i) is a wave number . For each nonmagnetic layers ( i=1,4) Newton equation of motion for the elastic displacement u= uz along with Hooke’s law lead to the relation ships k(i) = ω/V(i). Here )( )( )(/i i iC V is AW velocity , ρ(i) is the mass density, C(i) is the effect ive elastic modulus account ing for piezoelectric stiffening in layer 1 [23]. Using two roots ± k(i), one can obtain the general solutions for u(i). The normal stress can be represented as T (i)=T(i)zx= C(i)(u(i)/x)+ eIδi1/(jωl(1)C0), where second term exists only in the piezoelectric layer 1. Here e and C0 are the piezoelectric modulus and capacity of the layer , I is the displacement current flowing in the layer [24]. For YIG layers (i=3, 5) from the Newton equation and Landau -Lifshitz equation for the precession of magnetization vector ),,(0Mmmy xM together with Maxwell equations , we obtain the secular equation in the form 0 ) )( (22 2 02 22 2 VkωωωωVkωM H , (1) where we omit the upper indices ( i). Here , ] )( )[( )(2 2 2 2 02 0 M H H k k k , )( )(2 eff2k H kH , eff 4MM , Heff (k2)= H + Dk2 and Meff ≈M0 are the uniform effective magnetic field and magnetizatio n, D is the exchange stiffness, is the gyromagnetic ratio, ) 4/(2 eff2CM b is the dimensionless coupling parameter , and b is the magnetoelastic constant [15, 25, 26 ]. Herei nafter, the system of Gaussian -CGS units is used , but for convenience, the layer thicknesses are given in the SI system : micro - and nanometers . As it is known the crossover of two independent solutions (1) in case of ξ = 0 determine s MER frequency and wave number: ωMER(H)=2πfMER(H) and kMER(H). In case of ξ≠0 the formation of coupled waves and the repulsion of the solutions in the vicinity of ωMER take place . As one can see, for a given real, positive ω, there are three real roots of the secular equation , k2p (ω) (p= 1,2,3 ). Using six roots ± k1,2,3, one can obtain the general solutions for u, mx,y, and normal stress component T =Tzx=C(u/x)+bmx/M0. The solutions obtain ed for all layers should satisfy the elastic and electrodynamic boundary conditions at the interfaces. At the magnetic layers’ interfaces , the additional magnetization boundary conditions for ac magnetization should be taken into account. Here the case of free spins is considered : mx,y/x =0. Magnetic and acoustic losses are taken into account phenomenologically with the help of the following substitutions: )( )( )( i i ii C C and HiH H , where, )(i and H are viscosity factor and ferromagnetic resonance (FMR) line width [23, 27] . Further we use the impedance method for calculating the multilayer resonator structure characteristics [23]. The input electric impedance of piezo transducer may be represented as [28] ZE = IU/~ (1+ ZAW)/(jωC 0), (2) where, ZAW is the function of transducer parameters (material and geometrical) and of the acoustic impedance Z of transducer l oad (layers 2-6). It follows from the impedance continuity condition that the load impedance of layer i-1 is equal to the input impedance of layer i: )/) (/() ()( )( )( in t lxu lxT zi ii ii . Since the influence of 150–200 nm thickness electrodes on the properties of HBAR is negligible, we can assume that )3( inzZ . In the absence of magnetoelastic interaction ( ξ = 0), the load impedance is calculated by the sequential application of the impedance transformation formula [23] ) sin cos /() sin cos ()( )1( in)( )( )( )( )( )1( in)( )( ini i i i i i i i i ijz z jz zz z , ( 3) where φ(i) = k(i)l(i) and )()( )( i i iV z are the phase shift and the material acoustic impedance . For magnetic layers with magnetoelastic interaction , the expression ( 3) is not applicable, because 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 impedance s zin(3,5) the formula analogous to ( 3) with the corresponding substitutions: 2 12 1 )5,3( 2 121 )5,3( 21 )5,3(cos,2sin,~zzzz zzzz zzjz . (4) Here )2/ tg(3 11 sk zp p pp , )2/ ctg(3 12 sk zp p pp , 3 1~ pp , s = l(3,5), and p , p are the coefficients determined in [ 28]. Thus, the relations ( 2) - (4) allow us to describe the HBAR spectrum, and determine (f,H) dependences of fn and resonance Qn factor s. Let us now consider the features of acoustic spin pumping in our structure. The time - averaged spin current polarized along z, n mm j2 r5)/ ( xx s t g , flows from YIG layer 5 into Pt layer 6 [21]. Here rg is the real part of spin mixing conductance, ] /)()( Im[2 0 5 5*Mxmxmy x is the magnetization precession cone angle at YIG/Pt interface x5, and n is the normal to the interface. The ISHE in Pt leads to an electrostatic field ) ( ) (2 SH ISHE zn zj E s , where SH is the spin Hall angle of Pt [22]. For a rectangular Pt strip, the dc voltage between its ends in the direction a is )) (( ) (2 ISHE ISHE azn a E U . ( 5) In (5 ), the constants rg and SH are omitted, since their values are considered to be independent of the field, frequency, and thickness of YIG and Pt . We also omitted the factor resulting from the current density averaging across the Pt thickness. After substitution the general solutions in magnetic layer 5 to magnetic and elastic boundary conditions we obtain qp qpqp qp qppq xxyxNzzxujmm 3 1, 2 14 ) (~)( 5 , (6) where )]2/ (tg)2/ (tg/[)]2/ (tg)2/ ([tg2 2sk sk sk sk Nq p q p qp pq . An explicit view of the amplitude coefficients p and p is given in [2 8] . For allowance of the ADSWs back action on all elastic subsystems , the displacement u(x4) should be expressed via an electrical parameter of the transducer, for example, the voltage U~ applied to the electrodes. The transformation ) sin cos /( ) ( )()( )1( in)( )( )( 1i i i i i i i jz z zxu xu (7) is used to express u(x4) in terms of u(x3). For the transformation of u(x3) to u(x2) via ( 7) the substitution rules ( 4) are needed. Finally, from the equations for the piezoelectric layer 1 [24], we obtain )] 1)( cos sin ( /[)2/ (sin~2)(AW)1( )1( )1( )1( )1( 2 2 Z Z izl Uje xu . (8) Substituting ( 4), (6) - (8) in (5) we can get an analytical expression for UISHE, which is then analyzed numerically. 4. Results and discussions Figures 2 (a),(b) demonstrate the frequency dependences of Re[ k1,2,3(f)] and Im[ k1,2,3(f)] for infinite magnetic media at H= 740 Oe. We attribute k1, k2 to the continuous magneto elastic branches, which for f < fMER are quasi AW and SW , but for f > fMER are quasi SW and AW. The third root, k3(f), is always imaginary and plays a certain role for the satis faction of boundary conditions a s well as the root k2(f) in case f < fFMR= ω0(0)/2π [4, 29, 30]. Fig. 2. Frequency dependences of: real (a) and imaginary (b) parts of wave numbers k1 (blue lines 1), k2 (red lines 2 ), and k3 (green lines 3 ); normalized voltages for structures II (c) and I (d) at fixed magnetic field H = 740 Oe. The elastic and magnetic parameters: (111) oriented YIG – V(3,5) = 3.9 ×105 cm/s, ρ(3,5) = 5.17 g/cm3 [15], b = 4 × 106 erg/cm3, D =4.46 ×10–9 Oe cm2, 4πMeff = 955 G [30], H =0.7Oe; GGG – V(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 thicknesses are given in caption to Fig.1 Figures 2 (c), (d) show the frequency dependence of normalized UISHE(f) at H= 740 Oe for two structures, I and II: with two YIG films (II) (Fig.2(c)) and with only one film 5 (I) (Fig.2(d)) . The parameters of the YIG / Pt interface and the Pt itself are not considered in this case and the voltage was normalized to the maximum for the structure I. As one can see from Fig. 2(c), (d) the dependences for both struc tures, I and II are similar in the form and differs only by the scale. Note that the absolute maximum of UISHE(f) corresponds not to fMER , but to a lower frequency near fFMR. The local maxima frequencies, as it was shown in [28], coincide with HBAR resona nce frequencies. Next, we consider the frequency dependences in a varying magnetic field and compare them with the experimental ones observed previously in [18]. Figure 3(a) shows for the structure II the calculated dependence UISHE(f,H), which is in a good agreement with the experimental result s, shown in Fig. 3 (b). The voltage magnitudes follow the change in the positi on of the resonance frequencies fn(H) - the red lines in Fig. 3 (a) and the red points in Fig. 3 (b). Both calculated and experimental UISHE magnitudes have different behavior above and below the line fMER(H) (line 1), which is a consequence of excitation of SW wave s with basically different wavenumbers . Higher than fMER(H) line , the sho rt SW s with wavenumbers more than 5×104 cm-1 are excited, whereas below the line the wavenumbers of excited modes are essentially smaller (see Fig.2 (a)). Fitting of the theoretical and experimental dependences allows us to evaluate magnetic parameters b, 4πMeff, and D listed in the Fig.2 (a) caption [31 ]. Fig. 3. 3D colour plot UISHE(f,H) for the structure II: (a) – calculated, (b) – experimental adopted from [18] (http://creativecommons.org./licenses/by/4.0/) . The red lines (a) and points (b) correspond to positions of HBAR resonant frequency fn positions. The calculated frequencies fMER(H) and fFMR(H) are shown by line 1 and 2 in (a). For calculation the same parameters as listed in Fig.2 were used. They correspond to the experimental ones. Up to this point, the consideration concerned rather thick YIG films of about 30 μm. But today , much attention is paid to the technology and the study of submicron and nanometer -sized YIG films. In particular, the effective magnetoelastic interaction in such films was observed [32]. Let us consider the effect of YIG thickness l(5) = s on the UISHE for the structure I. Figure 4 shows the calculated UISHE(f0 max, s) dependence for a maximum located at frequency f0 max ≈ closest to fFMR. With the decrease of s from a few tens microns to a micron, the effect increases by an order of magnitude , oscillating with the period ~ 0.65 µ m, which corresponds to the AW half-length. In this case, the local minima correspond to s = (p+1)V/(2f) and maxima – to s = (p+0.5)V/(2f), where p is an integer . At s ~ 3÷2 µm other excitation zones appear at higher frequencies ft max, corresponding to the SW resonance conditions for free spins : k2s ≈ πt, where t = 1, 2, 3, ... A detailed description of the behavior of these high order SW resonance is beyond the scope of this paper. We just note that at certain s the value UISHE (ft max,s) induced by ADSW resonances with t ≤ 3 become s larger than the UISHE(f0 max, s) (as it is shown in the inse rtion). With the decrease s up to 120 nm the frequencies ft max beco me so high that are shifted out of the MER influence region. Finally, at the s ~ 100 nm only one signal UISHE (f0 max, s) remains in the spectrum. With a further thickness decrease, the signal reaches a maximum and then begins to drop. Since in these calculations we did not take into account the additional magnetic damping due to the spin pumping , the dependence on thickness is entirely determined by the method of the SW excitation. It should be noted that at thicknesses s< 200 nm the additional magnetic damping becomes noticeable . Assuming that the FMR linewidth broadening ΔH sp~ rg /s [21, 33] one can obtain a steeper curve for the small thicknesses. Fig. 4. The voltage UISHE (f0 max) dependence on YIG thickness s at the fixed magnetic field H = 740 Oe. The insertion: the spectrum of the signal UISHE (f, s=0.2 μm) with two zones of maximal excitation near f0 max and f1 max . 5. Conclusion A semi -analytic theory for ADSW in hypersonic composite HBAR with ZnO -GGG - YIG/Pt layered structure is developed . The theoretical and experimental dependenc es of the electric voltage UISHE(f, H) in Pt are in good agreement : the significant asymmetry of the UISHE(fn(H)) value in reference to the magnetoelastic resonance line fMER(H) position , experimentally observed previously , manifest s itself also in the theoretical calculations. This asymmetry is due to the SW spectrum governed by nonuniform exchange: near the fFMR the efficiency of quasiuniform SW excitation is higher than the efficiency for the frequencies exceeding fMER. The theory involved takes into account the self -consistent mutual influence of the AW and SW and gives the possibility to evaluate some magnetic parameters of the YIG films including exchange stiffness. The analysis of YIG film thickness inf luence on UISHE at the main frequency f0 max show s that this value reaches a maximum for thicknesses about 100 nm. Also , UISHE maxima due to the high SW resonances at higher frequencies can be detected. So we believe that acoustic spin pumping created by means of HBAR is a sensitive spectroscopic technique for the investigation of magnetic films properties. Acknowledgments This work was partially supported by grants 16 -07-01210 and 17 -07-01498 from the Russian Foundation for Basic Research. REFERENCES [1] A. S. Salasyuk, A. V. Rudkovskaya, A. P. Danilov, B. A. Glavin, S. M. Kukhtaruk, M. Wang, A. W. Rushforth, P. A. Nekludova, S. V. Sokolov, A. A. Elistratov, D. R. Yakovlev, M. Bayer, A. V. Akimov, A. V. Scherbakov, Generation of a localized microwave magnetic field by coherent phonons in a ferromagnetic nanograting, Phys.Rev. B, 97 (2018) 060404(R). https://doi.org/10.1103/PhysRevB.97.060404 [2] P.Graczyk, M. Krawczyk, Coupled -mode theory for the interaction between acoustic waves and spin waves in magnonic -phononic crystals: Propagating magnetoelastic waves , Phys.Rev. B (2017) 024407. https://doi.org/10.1103/PhysRevB.96.024407 [3] A. Barra, A. Mal, G. Carman, A. Sepulveda, Voltage induced mechanical/spin wave propagation over long distances, Appl. Phys. Lett. 110 (2017) , 072401. http://dx.doi.org/10.1063/1.4975828 [4] A. Kamra, H. Keshtgar, P. Yan, G. E. W. Bauer, Coherent elastic excitation of spin waves, Phys.Rev. B , 91 (2015) 104409. https://doi.org/10.1103/PhysRevB.91.104409 [5] A. V. Azovtsev, N. A. Pertsev Coupled magnetic and elastic dynamics generated by a shear wave propagating in ferromagnetic heterostructure Appl. Phys. Lett., 111 (2017) 222403. https://doi.org/10.1063/1.5008572 [6] P. Kuszewski, I. S. Camara, N. Biarrotte, L .Becerra, J. von Bardeleben, W. Savero Torres, A. Lemaître , C .Gourdon, J. -Y. Duquesne, L Thevenard. Resonant magneto -acoustic switching: influence of Rayleigh wave freq uency and wavevector, J. Phys.: Cond. Matt., 30 (2018) 244003. https://doi.org/10.1088/1361 -648X/aac152 [7] K. Uchida, T An., Y. Kajiwara, M. Toda, E. Saitoh. 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Wang, Electric -field-induced spin wave generation using multiferroic magnetoelectric cells, Appl. Phys. Lett., 104 (2014) 082403. https://doi.org/10.1063/1.4865916 [12] G. Yu, H. Zhang, Y. Li, J. Li, D. Zhang, N. Sun, Resonance of magnetization excited by voltage in magnetoelectric heterostructures, Mater. Res. Express, 5 (2018) 045021. https://doi.org/10.1088/2053 -1591/aab91a [13] N. Polzikova, S. Alekseev, I. Kotelyanskii, A. Raevskiy, Yu. Fetisov, Magnetic field tunable acoustic resonator with ferromagnetic -ferroelectric layered st ructure, J. Appl. Phys ., 113 (2013) 17C704. https://doi.org/10.1063/1.4793774 [14] N. I. Polzikova, A. O. Raevskii, A. S. Goremykina, Calculation of the spectral characteristics of an acoustic resonator containing layered multiferroic structure, J. Commun. Technol. Electron., 58 (2013) 87 -94. https://doi.org/ 10.1134/S1064226912120 066 [15] W.Strauss, Magnetoelastic properties of yttrium iron garnet, in W.P. Mason (Ed.), Physical Acoustics, Vol. IV(B), Academic Press, New York, 1968, pp. 211 -268. [16] B. P. Sorokin, G. M. Kvashnin, A. S. Novoselov, V. S. Bormashov, A. V. Golovanov , S. I. Burkov, V. D. Blank, Excitation of hypersonic acoustic waves in diamond -based piezoelectric layered structure on the microwave frequencies up to 20 GHz, Ultrasonics, 78 (2017) 162 –165. https://doi.org/10.1016/j.ultras.2017.01.014 [17] N. I. Polzikova, S. G. Alekseev, I. I. Pyataikin, I. M. Kotelyanskii, V. A. Luzanov, A. P. Orlov, Acoustic spin pumping in magnetoelectric bulk acoustic wave resonator, AIP Advances, 6 (2016) 056306. http://dx.doi.org/10.1063/1.4943765 [18] N. I. Polzikova, S. G. Alekseev, I. I. Pyataikin, V. A. Luzanov, A. O. Raevskiy, V. A. Kotov, Frequency and magnetic field mapping of magnetoelastic spin pumping in high overtone bulk acoustic wave resonator, AIP Advances, 8 (2018) 056128. https://doi.org/10.1063/1.5007685 [19] Note that the presence of the upper YIG film 3 is not necessary. However, we take into consideration this film because such a structure we used in the experiments [18]. [20] V. A. Luzanov, S. G. Alekseev, N. I. Polzikova, Deposition process optimization of zinc oxide films with inclined texture axis, J. Commun. Technol. Electron., 63 (2018) 1076 –1079. DOI: 10.1134/S1064226918090127 [21]Y. Tserkovnyak, A. Brataas, G.E.W. Bauer, Enhanced Gilbert damping in thin ferromagnetic films, Phys. Rev. Lett., 88 (2002) 117601. https://doi.org/10.1103/PhysRevLett.88.117601 [22] E. Saitoh, M. Ueda, H. Miyajima, G. Tatara, Conversion of spin current into charge current at room temperature: inverse spi n-Hall effect, Appl. Phys. Lett., 88 (2006) 182509. 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. [24] D. Royer, E. Dieulesaint, Elastic Waves in Solids II. Generation. Acousto -optic Interaction, Applications, Springer -Verlag, Berlin Heidelberg, 2000. [25] In the geometry involved, the demagnetization field )0,0, 4(d xmh contributes only to the ac component of the effective field. The effect of the YIG crystalline anisotropy is not considered in detail in this work, but it can be taken into account both in Meff and Heff. For (111) YIG films orientation, as in Ref. 18, ani sotropy contributes mainly to the Meff, rather than to the H eff [26]. In this case t he coupling constant b is determined by a linear combination of two cubic constants b2 and b1 with coefficients depending on the field direction in the (111) plane [15]. [26] S. Klingler, A. V. Chumak, T. Mewes, B. Khodadadi, C. Mewes, C. Dubs, O. Surzhenko, B. Hillebrands, and A. Conca, Measurements of the exchange stiffness of YIG films using broadband ferromagnetic resonance techniques Journal of Physics D: Applied Phys ics 48, (2015) 015001 https://doi.org/10.1088/0022 -3727/48/1/015001 [27] A. G. Gurevich and G. A. Melkov, Magnetization Oscillations and Waves (CRC -Press, Boca Raton, 1996), p. 464 . [28] N.I.Polzikova, S.U.Alekseev, V.A.Luzanov, A.O.Raevskiy, Electroacoustic excitation of spin waves and their detection via inverse spin Hall effect, Phys. Solid State, 60 ( 2018) 2211. https://doi.org/10.1134/S1063783418110252 [29] N.M.Salanskii, M.S.Yerukhimov, The Physical Properties and Applications of Magnetic Films, Nauka, Novosibirsk, 1975 (in Russian). [30] H.F.Tiersten, Thickness vibrations of saturated magnetoelastic plates, J. Appl. Phys., 36 (1965) 2250 -2259. https://doi.org/10.1063/1.1714459 [31] Note that the reduced magnetization 4π Meff = 955 G is characteristic for La, Ga -substituted YIG epitaxial films used in the experiment. [32] Yu. V. Khivintsev, V. K. Sakharov, S. L. Vysotskii, Yu. A. Filimonov, A. I. Stognii, S. A. Nikitov, Magnetoelastic waves in submicron yttrium –iron garnet films manufactured by means of ion -beam sputtering onto gadolinium –gallium garnet sub strates, Technical Phys ., 63 (2018) 1029 -1035. https://doi.org/ 10.1134/S1063784218070162 [33] M. B. Jungfleisch, A. V. Chumak, A. Kehlberger, V. Lauer, D. H. Kim, M. C. Onbasli, C. A. Ross, M. Kl äui, and B. Hillebrands Thickness and power dependence of the spin -pumping effect in Y3Fe5O12/Pt heterostructures measured by the inverse spin Hall effect , Phys. Rev. B 91, (2015) 134407 https://do i.org/10.1103/PhysRevB.91.134407

2019-03-28

We report on the self-consisted semi-analytical theory of magnetoelastic excitation and electrical detection of spin waves and spin currents in hypersonic bulk acoustic waves resonator with ZnO-GGG-YIG/Pt layered structure. Electrical detection of acoustically driven spin waves occurs due to spin pumping from YIG to Pt and inverse spin Hall (ISHE) effect in Pt as well as due to electrical response of ZnO piezotransducer. The frequency-field dependences of the resonator frequencies and ISHE voltage $U_{ISHE}$ are correlated with experimental ones observed previously. Their fitting allows to determine some magnetic and magnetoelastic parameters of YIG. The analysis of the YIG film thickness influence on $U_{ISHE}$ gives the possibility to find the optimal thickness for maximal $U_{ISHE}$ value.

Acoustic excitation and electrical detection of spin waves and spin currents in hypersonic bulk waves resonator with YIG/Pt system

1903.12130v2

arXiv:2003.13760v4 [quant-ph] 23 Sep 2020Tunable multiwindow magnomechanically induced transpare ncy, Fano resonances, and slow to fast light conversion Kamran Ullah,∗M. Tahir Naseem,†and¨Ozg¨ ur E. M¨ ustecaplıo˘ glu‡ Department of Physics, Ko¸ c University, Sarıyer, ˙Istanbul, 34450, Turkey We investigate the absorption and transmission properties of a weak probe field under the in- fluence of a strong control field in a cavity magnomechanical s ystem. The system consists of two ferromagnetic-material yttrium iron garnet (YIG) spheres coupled to a single cavity mode. In addi- tion to two magnon-induced transparencies (MITs) that aris e due to magnon-photon interactions, we observe a magnomechanically induced transparency (MMIT ) due to the presence of nonlinear magnon-phonon interaction. We discuss the emergence of Fan o resonances and explain the splitting of a single Fano profile to double and triple Fano profiles due t o additional couplings in the pro- posed system. Moreover, by considering a two-YIG system, th e group delay of the probe field can be enhanced by one order of magnitude as compared with a singl e-YIG magnomechanical system. Furthermore, we show that the group delay depends on the tuna bility of the coupling strength of the first YIG with respect to the coupling frequency of the sec ond YIG, and vice versa. This helps to achieve larger group delays for weak magnon-photon coupl ing strengths. Keywords: Magnon induced transparency; magnomechanical i nduced transparency; Fano resonances; sublu- minal and superluminal effects. I. INTRODUCTION Storing information in different frequency modes of light has attracted much attention due to its critical role inhigh-speed, long-distancequantumcommunicationap- plications [1–3]. The spectral distinction of optical sig- nals eliminates their unintentional coupling to the sta- tionary information or memory nodes in a communica- tion network. For that aim, multiple transparency win- dow Electromagnetically Induced Transparency (EIT) schemes have been considered for multiband quantum memory implementations mainly in the medium ofthree- level cold atoms. Experimental demonstrations of three EIT windows have been reported [4], and extended to seven windows using external fields [5]. Observation of nineEITwindowshasbeenexperimentallydemonstrated quite recently, using an external magnetic field in a va- por cell of Rubidium atoms [6]. A practical question is if such results can be achieved at higher temperatures, for example, for a room temperature multiband quan- tum memory. In recent years, remarkable developments have been achieved to strongly couple spin ensembles to cavity pho- tons, leading to the emerging field of cavity spintron- ics. Quanta of spin waves, magnons, are highly ro- bust against temperature [7–11], and hence significant magnon-photon hybridization and magnetically induced transparency(MIT) havebeen successfullydemonstrated even at room temperature [11]. Tunable slow light and its conversion to fast light based upon room tempera- ture MIT has been theoretically shown recently [12]. Be- ∗Electronic address: kamran@phys.qau.edu.pk †Electronic address: mnaseem16@ku.edu.tr ‡Electronic address: omustecap@ku.edu.trsides, at strong magnon-photon interaction, a wide tun- ability of slow light via applied magnetic field has been shown in [13]. These results demonstrate the promising value of these systems for practical quantum memories [12]. Here we explore how to split such a MIT window into multiple bands for a room temperature multimode quantum memory. Our idea is to exploit the coupling of magnons to thermal vibrations, which is known to yield magnomechanically induced transparency (MMIT) [14], in combination with multiple spin ensembles to achieve multiple bands in MIT. We also discuss the emergence of Fano resonance in the output spectrum and explore the suitable system parameters for its observation. Fano res- onance was first reported in the atomic systems [15], and it emerges due to the quantum interference of different transition amplitudes which give minima in the absorp- tion profile. In later years, it has been discussed in differ- ent physical systems, such as photonic crystal [16], cou- pled microresonators [17], optomechanical system [18]. Recently, Fano-like asymmetric shapes have been exper- imentally reported in a hybrid cavity magnomechanical system [14]. Our model consists of two ferromagnetic insulators, specifically yttrium iron garnets (YIGs), hosting long- lived magnons at room temperature, placed inside a three-dimensional (3D) microwave cavity; we remark that another equivalent embodiment of our model could be to place the YIGs on top of a superconducting co- planar waveguide, which can have further practical sig- nificance being an on-chip device [19]. Specific benefits of YIG as the host of spin ensemble over other systems, such as paramagneticspin ensembles in nitrogen-vacancy centers is due to its high spin density of 2 .1×1022µB cm−3(µB is the Bohr magneton) and high room tem- perature spin polarization below the Curie temperature (559 K). In addition to multimode quantum memories, our results can be directly advantageous for readily in-2 tegrated microwave circuit applications at room temper- ature such as multimode quantum transducers coupling differentsystemsatdifferent frequencies[20], tunable fre- quency quantum sensors [21] or fast light enhanced gyro- scopes [22]. In addition to the magnetic dipole interac- tion between the cavity field and the spin ensemble, we take into account coupling between the magnons and the quanta of YIG lattice vibrations, phonons, arising due to the magnetostrictive force [14]. We only consider the Kittel mode [23] of the ferromagnetic resonance modes of the magnons. Such three-body quantum systems can be of fundamental significance to examine macroscopic quantum phenomena towards thermodynamic limit and quantum to classical transitions [24]. In our model, tunable slow and fast light emerges as a natural consequence of tunable splitting of MIT window. Slow-light propagation at room temperature has been investigated recently in a cavity-magnon sys- tem and the group delays are found to be in the ∼µs range[12]. InasingleYIGmagnomechanicalsystemwith strong magnon-photon coupling strength, slow-light has achieved with a maximum group delay of <0.8 ms [13]. In this paper, we discuss the slow and fast light in a two YIGs magnomechanical system. Further, we exlain the group delay depends on the tunability of the magnon- photon coupling of the first YIG (YIG1) with respect to magnon-photon coupling of the second YIG (YIG2). Thisnotonlyhelpstoachievelargergroupdelaysatweak magnon-photon coupling, but also increase the group de- lay of the transmitted probe field by one order of magni- tude, which is not possible with a single YIG system [13]. The rest of the paper is organized as follows: We de- scribe the model system in Sec. II and present dynam- ical equations with steady-state solutions. The results and discussions for MMIT are presented in the Sec. III. We discuss the emergence and tunability of the multiple Fano resonances in Sec. IV. Next, in Sec. V, we present the transmission of the probe field and discuss the group delays for slow and fast light propagation. Finally, in Sec. VI, we present the conclusion of our work. II. SYSTEM HAMILTONIAN AND THEORY We consider a hybrid cavity magnomechanical sys- tem that consists of two YIG spheres placed inside a microwave cavity, as shown in Fig. 1. A uniform bias magnetic field (z-direction) is applied on each sphere, which excites the magnon modes and these modes are coupled with the cavity field via magnetic dipole inter- action. The excitation of the magnon modes inside the spheres leads to the variation magnetization that results in the deformation of their lattice structures. The mag- netostrictive force causes vibrations of the YIGs which establishes magnon-phonon interaction in these spheres. The single-magnon magnomechanical coupling strength is very weak [14], and it depends on the spheres diam- eters and external bias field directions. Either by con- FIG. 1: (color online) A schematic illustration of a hybrid cavity magnomechanical system. It consists of two ferromag - netic yttrium iron garnet (YIG) spheres placed inside a mi- crowave cavity. A Bias magnetic field is applied in the zdi- rection on each sphere, which excites the magnon modes, and these modes are strongly coupled with the cavity field. The bias magnetic fields activate the magnetostrictive (magnon - phonon) interaction in both YIGs. The single-magnon mag- nomechanical coupling strength is very weak [14], and it de- pends on the spheres diameters and external bias field direc- tions. Either byconsidering a larger YIG1sphere or adjusti ng the direction of the bias field on it, the magnomechanical cou - pling of this sphere can be ignored. Here, we assume the di- rection of the bias field on YIG1 such that the single-magnon magnomechanical interaction becomes very weak and can be ignored [14]. However, the magnomechanical interaction of YIG2 is enhanced by directly driving its magnon mode via a microwave drive (y direction). This microwave drive plays t he role of a control field in our model. Cavity, phonon, magnon modes are labeled as a,b, andmi(i= 1,2), respectively. sidering a larger YIG1 sphere or adjusting the direction of the bias magnetic field on it, the magnomechanical coupling of this sphere can be ignored [24]. Here, we assume the direction of the bias field on YIG1 such that thesingle-magnonmagnomechanicalinteractionbecomes very weak and can be ignored [14]. However, the mag- nomechanicalinteractionofYIG2 is enhancedbydirectly driving its magnon mode via a external microwave drive. This microwave drive plays the role of a control field in our model. In addition, the cavity is driven by a weak probe field. In this work, we consider high quality YIG spheres, each has a 250 µm diameter, and composed of ferric ions Fe+3 of density ρ= 4.22×1027m−3. This causes a total spin S= 5/2ρVm= 7.07×1014, whereVmis the volume ofthe YIG and Sis the collective spin operator which satisfy the algebra; [ Sα,Sβ] =iεαβγSγ. The Hamiltonian of the3 system reads [24] H//planckover2pi1=ωaˆa†ˆa+ωbˆb†ˆb+2/summationdisplay j=1[ωjˆm† jˆmj+gj(ˆm† jˆa+mjˆa†)] +gmbˆm† 2ˆm2(ˆb+ˆb†)+i(Ωdˆm† 2e−iωdt−Ω⋆ dˆm2eiωdt) +i(ˆa†εpe−iωpt−ˆaε⋆ peiωpt) (1) wherea†(a) andb†(b) are the creation (annihilation) op- erators of the cavity and phonon modes, respectively. The resonance frequencies of the cavity, phonon and magnon modes are denoted by ωa,ωbandωj, respec- tively. Moreover, mjis the bosonic operator of the Kit- tle mode of frequency ωjand its coupling strength with the cavity mode is given by gj. The frequency ωjof the magnon mode mjcan be determined by using gy- romagnetic ratio γjand external bias magnetic field Hj i.e.,ωj=γjHjwithγj/2π= 28 GHz. The Rabi fre- quency Ω d=√ 5/4γ√ NB0[23], represents the coupling strength of the drive field of amplitude B0and frequency ωd. Furthermore, in Eq. (1), ωpis the probe field fre- quency having amplitude εpwhich can be expressed as; εp=/radicalbig 2Ppκa//planckover2pi1ωp. Note that in Eq. (1), we have ignored the non-linear termKˆm† jˆm† jˆmjˆmjthat mayarisedue to stronglydriven magnon mode [25, 26]. To ignore this nonlinear term, we must have K|/angbracketleftm2/angbracketright|3≪Ω, and for the system pa- rameters we consider in this work, this condition always satisfies. The Hamiltonian in Eq. (1) is written after applying the rotating-wave approximation in which fast oscillating terms gj(ˆaˆmj+ ˆa†ˆm†) are dropped. This is valid for ωa,ωj≫gj,κa,κmjwhich is the case to be considered in the present work. Where κaandκmjare the decay rates of the cavity and magnon modes, respec- tively. In the frame rotating at the drive frequency ωd, the Hamiltonian of the system is given by H//planckover2pi1=∆aˆa†ˆa+ωbˆb†ˆb+2/summationdisplay j=1[∆mjˆm† jˆmj+gj(ˆm† jˆa+ mjˆa†)]+gmbˆm† 2ˆm2(ˆb+ˆb†)+i(Ωdˆm† 2−Ω⋆ dˆm2)+ i(ˆa†εpe−iδt−ˆaε⋆ peiδt), (2) here, ∆ a=ωa−ωd, ∆mj=ωj−ωd, andδ=ωp−ωd. The quantum Heisenberg-Langevin equations based on the Hamiltonian in Eq. (2) can be written as ˙ˆa=−i∆aˆa−i2/summationdisplay j=1gjˆmj−κaˆa+εpe−iδt+√2κaˆain(t), ˙ˆb=−iωbˆb−igmbˆm† 2ˆm2−κbˆb+√2κbˆbin(t), ˙ˆm1=−i∆m1ˆm1−ig1ˆa−κm1m1+√2κm1ˆmin 1(t), ˙ˆm2=−i∆m2ˆm2−ig2ˆa−κm2m2−igmbˆm2(ˆb+ˆb†) +Ωd+√ 2κm2ˆmin 2(t). (3)Whereκbis the dissipation rate ofthe phonon mode, and ˆbin(t), ˆmin j(t)and ˆain(t)arethevacuuminputnoiseoper- ators which have zero mean values i.e., /angbracketleftˆqin/angbracketright= 0 [27, 28], and (q=a,m,b). The magnon mode m2is strongly driven by a microwave drive that causes a large steady- state amplitude |/angbracketleftm2s/angbracketright| ≫1ofmagnonmode, and due to beam splitter interaction, this leads to the large steady- state amplitude of the cavity mode |/angbracketleftas/angbracketright| ≫1. Conse- quently, we can linearize the quantum Langevin equa- tions around the steady-state values and take only the first-order terms in the fluctuating operator: /angbracketleftˆO/angbracketright= Os+ˆO−e−iδt+ˆO+eiδt[29], here ˆO=a,b,m j. First, we consider the zero-order solution, namely, steady-state solutions which are given by as=−i/summationdisplay 1,2gjmjs κa+i∆a, bs=−igmb|m2s|2 κb+iωb, m1s=−ig1as κm1+i∆m1,m2s=Ωd−ig2as κm2+i˜∆m2, ˜∆m2= ∆m2+gmb(bs+b⋆ s).(4) We assume that the coupling of the external microwave drive on magnon mode m2is much stronger than the amplitude ǫpof the probe field. Under this assumption, the linearizedquantum Langevinequations can be solved by considering the first-order perturbed solutions and ig- noring all higher order terms of ǫp. The solution for the cavity mode is given by a−=εp A′+C′ 1+g2 2 β′+α⋆α′ β⋆β′+A⋆−C⋆ 1+g2 2 β⋆ −1 , (5) where A=κa+i(∆a+δ),B=G2 mbωb ω2 b−δ2+iδκb, C1=g2 1 κm1+i(∆m1+δ),C2=g2 2 κm2+i(˜∆m2+δ), A′=κa+i(∆a−δ),B′=G2 mbωb ω2 b−δ2−iδκb, C′ 1=g2 1 κm1+i(∆m1−δ),C′ 2=g2 2 κm2+i(˜∆m2−δ), α=g2 2B C2+iB,α′=g2 2B′ C′ 2+iB′, β=C2−iC′⋆ 2B C′⋆ 2+iB,β′=C′ 2−iC⋆ 2B′ C⋆ 2+iB′. HereGmb=i√ 2gmbm2sis the effective magnon-phonon coupling. We use the input-output relation for the cavity fieldεout=εin−2κa/angbracketlefta/angbracketright[30], and the amplitude of the output field can be written as4 FIG. 2: (Color online). AbsorptionRe[ εout] profilesare shown against the normalized probe field detuning δ/ωb. (a)g1= gmb= 0,g2/2π= 1.2 MHz and (b) g1= 0,g2/2π= 1.2 MHz, Gmb/2π= 2.0MHz(c) g1/2π=g2/2π= 1.2MHz,Gmb/2π= 2 MHz and (d) g1/2π=g2/2π= 1.2 MHz, Gmb/2π= 3.5 MHz. The other parameters are given in Sec. III. εout=2κaa− εp. (6) The real and imaginaryparts of εoutaccount for in-phase (absorption) and out of phase (dispersion) output field quadratures at probe frequency. III. MMIT WINDOWS PROFILE For the numerical calculation, we use parameters from a recent experiment on a hybrid magnomechanical sys- tem [14], unless stated differently. Frequency of the cav- ity fieldωa/2π= 10GHz, ωb/2π= 10MHz, κb/2π= 100 Hz,ω1,2/2π= 10 GHz, κa/2π= 2.1 MHz, κm1/2π= κm2/2π= 0.1 MHz, g1/2π=g2/2π= 1.5 MHz, Gmb/2π= 3.5 MHz, ∆ a=ωb, ∆mj=ωb,ωd/2π= 10 GHz. We first illustrate the physics behind the multiband transparency by systematically investigating the role of different couplings in the model. Fig. 2 displays the response of the probe field in the absorption spectrum of the output field for different coupling strengths. In Fig. 2(a), we assume the magnon-phonon coupling ( gmb) and magnon mode m1coupling ( g1) with the cavity are absent. Therefore, only magnon mode m2is coupled with the cavity. Under these considerations, we observe a magnon induced transparency (MIT) in which a typi- cal Lorentzian peak of the output spectrum of the simple cavity splits into two peaks with a single dip, as shown in Fig. 2(a). The width of this transparency window can be controlled via microwave driving field power and the magnon-photon coupling g2. On increasing the coupling strength g2the width of the window increases, and vice versa.FIG. 3: (Color online) Dispersion Im[ εout] profiles are shown againstthenormalizedprobedetuning δ/ωb. (a)g1=gmb= 0 andg2/2π= 1.2 MHz and (b) g1= 0,g2/2π= 1.2 MHz, Gmb/2π= 2.0 MHz (c), (d) g1/2π=g2/2π= 1.2 MHz, and (c)Gmb/2π= 2 MHz and (d) Gmb/2π= 3.5 MHz. The other parameters are given in Sec. III. We observe two transparency windows in the absorp- tion as we switch on the magnon-phonon coupling ( gmb) and keeping g1= 0. Due to the non-zero magnetostric- tive interaction, single MIT window in Fig. 2(a) splits into double window shown in Fig. 2(c). The right trans- parency window in Fig. 2(c) is associated with magnon- phonon interaction, and this is so called magnomechani- callyinducedtransparency(MMIT)[14]window. Wecan observe double MIT by removing magnon-phonon cou- plinggmb, and considering non-zero couplings between the magnon modes and the cavity field. Finally, if we consider all three couplings simultane- ously non-zero, then the transparency window splits into three windows consist of four peaks and three dips, this is shown in Fig. 2(c). In this case, one window is as- sociated with the magnomechanical interaction, and the rest of the two are induced by magnon-photon couplings. The width and peaks separation of these windows in- creases and broadens, respectively, at higher values of magnon-phonon coupling Gmb, which can be seen in Fig. 2(d). Moreover, we have a symmetric multi-window transparency profile where the splitting of the peaks oc- curs at side-mode frequencies ωp=ωb±ωd. In Figs. 3(a-d), we plot the dispersion spectrum of the output field versus normalized frequency of the probe field. The single MIT dispersion spectrum in the absence of YIG1 and magnon-phonon coupling gmbis shown in Fig. 3(a). The dispersion spectra for the case of g1= 0, g2/negationslash= 0 and gmb/negationslash= 0 is plotted in the Fig. 3(b). In the presence of all three couplings, the dispersion spectrum of the output field is given in the Figs. 3(c-d). It is clear fromFigs.3(c-d), bytheincreaseintheeffectivemagnon- phononcoupling Gmb, the transparencywindowsbecome wider. We like to point out that the magnomechanically induced amplification (MMIA) of the output field, in5 our system, can be obtained in the blue detuned regime; ∆m2=−ωb. FIG. 4: (Color online) Fano line shapes in the asymmetric absorption Re[ εout] profiles are shown against the normalized probe frequency δ/ωb. (a) ∆ m2= 0.7ωb,g2= 1.5 MHz, g1=gmb= 0, and (b) ∆ m2= 0.7ωb,g1= 0,g2= 1.5 MHz, Gmb= 3.5 MHz. (c) ∆ m1,2= 0.7ωb,g1=g2/2π= 1.5 MHz and Gmb/2π= 3.5MHz, and (d) ∆ m1,2=ωb,g1= g2/2π= 1.5 MHz and Gmb/2π= 3.5 MHz. In all panels, g1=g2/2π= 1.5 MHz,Gmb/2π= 3.5 MHz, and rest of the parameters are give in Sec. III. IV. FANO RESONANCES IN THE OUTPUT FIELD In the following, we discuss the emergence and phys- ical mechanism of the Fano line shapes in the output spectrum. The shape of the Fano resonance is distinctly differentthanthesymmetricresonancecurvesintheEIT, MIT, optomechanically induced transparency (OMIT) and MMIT windows [14, 31]. Fano resonance has ob- servedinthesystemsinwhichEIThasreportedbyasuit- able selection of the system parameters [14, 31–36]. The physical origin of Fano resonance in the systems having optomechanical-likeinteractionshasexplained dueto the presence of non-resonant interactions. For example, in a standard optomechanical system, if the anti-Stokes pro- cess is not resonant with the cavity frequency, asymmet- ric Fano shapes appear in the spectrum [31–33]. In our system, thiscorrespondsto∆ m1/negationslash=ωb, becauseinsteadof a cavity mode, magnon mode m1is coupled with phonon modeviaoptomechanical-likeinteraction. Theasymmet- ric Fano shapes can be seen in Figs. 4(a-c) for differ- ent non-resonant cases, where the absorption spectrum of the output field as a function of normalized detuning δ/ωbis shown. In Fig. 4(a), we consider g1=gmb= 0, and coupling of the magnon mode m2with the cavity is non-zero. Due to the presence of non-resonant process (∆m2= 0.7ωm), the absorption spectrum of the sym- metric MIT (Fig. 2(a)) profile changes into asymmetricFIG. 5: (Color online). The transmission |tp|2spectrum as a function of normalized probe field frequency δ/ωbis shown for different values of g1. (a)g1/2π= 0.5 MHz (b) g1/2π= 0.8 MHz (c) g1/2π= 1.2 MHz (d) g1/2π= 1.5 MHz. In all panels,g2/2π= 1.5 MHz,Gmb/2π= 3.5 MHz and the other parameters are given in Sec. III. window profile, as shown in Fig. 4(a). Such asymmetric MIT band can be related to Fano-like resonance, emerg- ing frequently in optomechanical systems [31–35]. If we remove YIG1 and consider only YIG2 is coupled with the cavity mode, and ∆ m2= 0.7ωm. We observe double Fano resonance in the output spectrum, which is shown in Fig. 4(b). Similarly, in the presence of all three cou- plings and ∆ m1,m2= 0.7ωm, the double Fano resonance goes over to a triple Fano profile, as shown in Fig. 4(c). This is because the cavity field can be build up by three coherent routes provided by the three coupled systems (the magnons, cavity, and phonon modes), and that can interferewith eachother. TheFanoresonancesdisappear when we consider a resonant case ∆ m1= ∆m2=ωb, as shown in Fig. 4(d). V. NUMERICAL RESULTS FOR SLOW AND FAST LIGHTS Here we investigate the transmission and group delay of the output signal, and show the effect of the magnon- photon and magnon-phonon couplings on the transmis- sion spectrum. From Eq. (6), the rescaled transmission field corresponding to the probe field can be expressed as tp=εp−2κaa− εp. (7) In Figs. 5(a-d), we plot the transmission spectrum of the probe field against the scaled detuning δ/ωb, for different values of g1. It is clear from Fig. 5(a), the transmis- sion peak associated with the magnon-photon coupling of YIG1 is smaller than the other two peaks. This is be- cause in Fig. 5(a) g1coupling is weaker than the other6 twointeractions g2andGmbpresentinthesystem. Byin- creasing the coupling strength g1, the peak of the middle transparency profile grows up in height and reaches close to unity, as shown in Figs. 5(b-c). In addition, Fig. 5(d) shows that the width of the transparency window can be increased at higher higher values of the magnon-photon coupling g1. FIG. 6: (Color online). The transmission |tp|2spectrum as a function of normalized probe field frequency δ/ωbis shown. (a)Gmb/2π= 0.5 MHz (b) Gmb/2π= 1.0 MHz. In (c) g2/2π= 0.4 MHz, and (d) g2/2π= 0.8 MHz. The other parameters are same as in Fig. 5. In Figs. 6(a-b), the transmissionspectrum ofthe probe field as a function of dimensionless detuning is shown for different values of Gmb. In Figs. 6(a-b), we consider both g1andg2to be the same in the strong coupling regime. However, the effective coupling ˜ g2=g2αsdepends on the steady-state amplitude of the cavity field αswhich depends on the m2s. Consequently, ˜ g2andGmbare re- lated and it can be seen from Eq. (4). For a smaller value ofGmbin Fig. 6(a), we have two small peaks associated withg2andGmb, in addition, the third-highest peak is associatedwith g1. Forafixedvalueof gmb, ifweincrease Gmb, it increases ˜ g1, and the peaks associated with these twocouplingsbecome morevisible, asshownin Fig. 6(b). Similarly, in Fig. 6(c-d), we observe a similar increase in the height of two peaks associated with g2andGmb, for the variation in g2. The phase φtof the transmitted probe field tpis given by the relation φt= Arg[tp]. The plot of φtas a func- tion of normalized detuning δ/ωbis shown in Fig. 7. In the inset of Fig. 7(a), we consider both g1andgmb are switched off, and only g2is non-zero. This gives a conventional phase of the transmitted field with a sin- gle MIT curve, which appears similar to the standard single OMIT curve [31]. In Fig. 7(b), we switch-off the YIG1 coupling with the field ( g1= 0), and the other two couplings are present ( gmb/negationslash= 0,g2/negationslash= 0), due to which the single transparency window splits into a double win- dow. Ifwekeepallthree couplingsnon-zero,wegettripletransparency window which is shown in Fig. 7(c). FIG. 7: (Color online) The phase φtof the transmitted probe field versus normalized detuning δ/ωbfor different coupling strengths. (a) g1=gmb= 0, (b) g1= 0,g2/2π= 1.5 MHz,Gmb/2π= 4 MHz (c) g1/2π=g2/2π= 1.5 MHz, and Gmb/2π= 4 MHz. Rest of parameters are given in Sec. III. FIG. 8: (Color online) Group delay τgof the output probe field against the amplitude of the magnetic field B0for (a) g1= 0, and (b) g1/2π= 1.5 MHz. The other parameter are g2/2π= 1.5,Gmb/2π= 3.5 MHz,κb/2π= 100 Hz, κm1/2π= κm2/2π= 0.1 MHz,κa/2π= 2.1 MHz and Ω d= 1.2 THz. The transmitted probe field phase is associated with the group delay τgof the output field and it is defined as τg=∂φ(ωp) ∂ωp, (8) which means a more rapid phase dispersion leads to a larger group delays and vice versa. In addition, a nega- tive slope of the phase represents a negative group delay or fast light ( τg<0) whereas, a positive slope of the transmitted field indicates positive group delay or slow light (τg>0). From Fig. 7, we observethat in the regime of the narrowtransparency window, there is a rapid vari- ation in the probe phase, and this rapid phase dispersion can lead to a significant group delay. Fig. 8 shows the group delay τgcan be tuned by the variation of the bias magnetic field B0applied on YIG2. In the absence of YIG1 (Fig. 8(a)), we have a lower slope of Eq. (8), as a result, a maximum group delay of τg= 1 ms is achieved. This group delay can be enhanced by one order of magnitude once second YIG is introduced see Fig. 8(b). The slope of Fig. 8(b) become steeper and the time delay for slow light is increased up to 13.8 ms. This shows the two YIGs system is a good choice to observe a longer group delay in a magnomechanical system while a singleYIG systemcannotdoso. Moreover,thenumerical value of the group delay τgcan be tuned from positive7 (slow light) to negative (fast light) by tuning the magnon fielddetuning∆ m1=ωbto∆m1=−ωb. Here, it isworth mentioning that Fig. 8(b) can be switched into fast light with a maximum group delay in the order of τg≈ −1.4 ms in the presence of both YIGs and we not show in the figure. This negative group delay for the fast light propagation is one order of magnitude greater than a single YIG magnomechanical system [13]. FIG. 9: (Color online) The group delay τgof the transmitted probe field as a function of the driving power Pdfor several values of the magnon-photon couplings. (a) ∆ m1=ωb, (b) ∆m1=−ωb, and the other parameter are same as given in Fig. 8. Finally, we investigate the control of group delay with the external microwave driving power and magnon- photon couplings. For this purpose, in Figs. 9(a-b), we plotτgagainst the driving power for different strengths of the magnon-photon coupling of YIG1 with respect to the coupling frequency of YIG2. Fig. 9(a) shows that the magnitude of the group delay increases with the increase ofg1corresponding to g2, which indicates an enhanced group delay of the transmitted probe field in a two-YIG system. We tuned the coupling strength of YIG1 ( g1) for different values via keeping the coupling strength of YIG2 (g2) constant. This shows increasing the magnon- photon coupling strength increases the group delay of the transmitted probe field and vice versa. This helps us to obtain larger group delays at relatively weak magnon- photoncouplingstrengthswhichisnototherwisepossible with a single YIG magnomechanical system [13]. Similar results can also be obtained by increasing the magnon- photon coupling g2and fixing g1. For the blue detuned regime∆ m1=−ωb, groupdelaybecomes negative. How- ever, the effect of magnon-photon couplings remains the same, as shown in Fig. 9(b). From Fig. 8 and Fig. 9, we see that two YIGs magnomechanical system providesnot only extra tunability, but also drastically enhances the group delays compared to single YIG system studied in Ref. [13]. Our system can be used as a tunable switch, which can be controlled via different system parameters, and our results are comparable with the existing propos- als based on various hybrid quantum systems [37–40]. VI. CONCLUSION We have investigated the transmission and absorption spectrum of a weak probe field under a strong control field in a hybrid magnomechanical system in the mi- crowave regime. Due to the presence of a nonlinear phonon-magnon interaction, we observed magnomechan- ically induced transparency (MMIT), and the photon- magnon interactions lead to magnon induced trans- parency (MIT). We found single MMIT, a result of the single-phonon process, and found two MIT windows in the output probe spectra due to the presence of two magnon-photon interactions. This is demonstrated by plotting the absorption, dispersion, and transmission of theoutput field. WediscussedtheemergenceofFanores- onances in the output field spectrum of the probe field. These asymmetric line shapes appeared due to the pres- enceofanti-Stokesprocessesinthesystem. We examined conditions of slow and fast light propagation in our sys- tem, which can be controlled by different system param- eters. It has shown that in a two YIGs magnomechan- ical system, the tunability of the first coupling strength (YIG1) corresponding to the coupling of the second YIG (YIG2) has an immense effect on the slow and fast light and vice versa. This not only helped to investigate larger group delays at a weak magnon-photon coupling but also enhanced the group delay of the transmitted probe field, which is not possible in a single YIG system. Our re- sults suggest that this system may find its applications to implement multi-band quantum memories [12]. Acknowledgment We acknowledge Prof. M. Cengiz Onbasli for fruit- ful discussions. We also thank the anonymous referees for their valuable comments which greatly improved our manuscript. [1] M. Afzelius, C. Simon, H. de Riedmatten, and N. Gisin, Phys. Rev. A 79, 052329 (2009). [2] N. Sinclair, E. Saglamyurek, H. Mallahzadeh, J. A. Slater, M. George, R. Ricken, M. P. Hedges, D. Oblak, C. Simon, W. Sohler, and W. Tittel, Phys. Rev. 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2020-03-12

We investigate the absorption and transmission properties of a weak probe field under the influence of a strong control field in a hybrid cavity magnomechanical system in the microwave regime. This hybrid system consists of two ferromagnetic material yttrium iron garnet (YIG) spheres strongly coupled to a single cavity mode. In addition to two magnon-induced transparency (MIT) that arise due to strong photon-magnon interactions, we observe a magnomechanically induced transparency (MMIT) due to the presence of nonlinear phonon-magnon interaction. In addition, we discuss the emergence and tunability of the multiple Fano resonances in our system. We find that due to strong photon-magnon coupling the group delay of the probe field can be enhanced significantly. The subluminal or superluminal propagation depends on the frequency of the magnons, which can be easily tuned by an external bias magnetic field. Besides, the group delay of the transmitted field can also be controlled with the control field power.

Tunable multiwindow magnomechanically induced transparency, Fano resonances, and slow-to-fast light conversion

2003.13760v4

Magnon Bose-Einstein condensates: from time crystals and quantum chromodynamics to vortex sensing and cosmology J.T. M¨ akinen,1,a)S. Autti,2,b)and V.B. Eltsov1,c) 1)Low Temperature Laboratory, Department of Applied Physics, Aalto University, POB 15100, FI-00076 AALTO, Finland 2)Department of Physics, Lancaster University, Lancaster LA1 4YB, UK (Dated: 19 December 2023) Under suitable experimental conditions collective spin-wave excitations, magnons, form a Bose-Einstein condensate (BEC) where the spins precess with a globally co- herent phase. Bose-Einstein condensation of magnons has been reported in a few systems, including superfluid phases of3He, solid state systems such as Yttrium- iron-garnet (YIG) films, and cold atomic gases. Among these systems, the super- fluid phases of3He provide a nearly ideal test bench for coherent magnon physics owing to experimentally proven spin superfluidity, the long lifetime of the magnon condensate, and the versatility of the accessible phenomena. We first briefly recap the properties of the different magnon BEC systems, with focus on superfluid3He. The main body of this review summarizes recent advances in application of magnon BEC as a laboratory to study basic physical phenomena connecting to diverse areas from particle physics and cosmology to new phases of condensed matter. This line of research complements the ongoing efforts to utilize magnon BECs as probes and components for potentially room-temperature quantum devices. In conclusion, we provide a roadmap for future directions in the field of applications of magnon BEC to fundamental research. a)jere.makinen@aalto.fi b)s.autti@lancaster.ac.uk c)vladimir.eltsov@aalto.fi 1arXiv:2312.10119v1 [cond-mat.quant-gas] 15 Dec 2023I. INTRODUCTION Spin waves are a general feature of magnetic materials. Their quanta are called magnons, spin-1 quasiparticles that obey bosonic statistics. At sufficiently large number density and low temperature, magnons form a Bose-Einstein condensate (BEC), akin to neutral atoms in superfluid4He or ultracold gases. The magnon BEC is manifested as spontaneous coherence of spin precession across a macroscopic ensemble in both frequency and phase even in the presence of incohering forces resulting from, e.g., magnetic field gradients. Bose-Einstein condensation of magnons (in the form of a homogeneously precessing do- main , HPD) was first observed in 19841in nuclear magnetic resonance (NMR) experiments in the superfluid B phase of3He. The experiments discovered a spontaneous formation of coherent precession of magnetization within a superfluid with antiferromagnetic ground state. In the early experiments, spontaneous coherence of magnons was manifested in pulsed NMR measurements2in a very peculiar manner. At first the amplitude of the free induction decay signal dropped, as expected for a dephasing set of local oscillators in an inhomoge- neous magnetic field. However, shortly after the signal amplitude was restored and, even more surprisingly, the observed ringing time was a few orders of magnitude longer than the initial dephasing time! These observations were explained using the terminology of spin superfluidity that acted as the mechanism establishing spontaneous coherence. Later experiments demonstrated spin supercurrent between two homogeneously precess- ing domains connected by a channel3, and showcased the spin current analog of the DC Josephson effect4. Further signatures of spin superfluidity and macroscopic coherence in- clude the observation of topological objects, spin vortices5, and the collective Goldstone modes, i.e. oscillations of the phase of precession6,7. These modes can be treated as phonons in a time crystal, discussed in Sec. III B. Treating the coherent spin precession as a Bose-Einstein condensate of magnons allows for a simplified description of the system. The condensate wave function amplitude describes the density of magnons and the phase the precession of magnetization. The magnon BEC differs from the atomic BECs in one important respect: magnons are quasparticles and, thus, their number is not conserved. In a thermodynamic equilibrium, magnon chemical potential is always zero, µeq≡0, and thus no equilibrium BEC of magnons can exist. If the lifetime of magnons τNis much larger than the thermalization time τEwithin the 2Pump ...Thermalization BECDecayMagnonFIG. 1. Creation of the magnon BEC. In a typical scheme quanta of spin-wave excitations (magnons) are pumped into a higher energy level by e.g. a radio-frequency pulse. The pumped magnons then thermalize with time constant τE. The magnon decay from the ground state is characterized by the decay time τN. Under sufficiently strong pumping, or if the magnon decay time is much longer than the thermalization time, τN≫τE, a macroscopic number of magnons occupy the ground state of the system, forming a BEC. magnon subsystem, i.e., τN≫τE, or if magnons are continuously pumped into the system, the magnon condensate obtains a nonzero µand forms a lowest-energy-level condensate analogous to a BEC as illustrated in Fig. 1. These conditions are well met e.g. in3He-B, where the thermalization time is a fraction of a second while the lifetime of the free coherent precession and the corresponding magnon BEC can reach tens of minutes at the lowest temperatures8. Magnon condensation in the lowest energy level(s) occurs when the system-specific critical magnon density ncis exceeded. Roughly speaking, nccorresponds to the point when the inter-magnon separation becomes comparable to the thermal de Broglie wavelength λdB, i.e. when n−1/3 c∼λdB. At this point the chemical potential of the magnon system µ asymptotically approaches the ground state energy ϵ0and the ground state population n0=1 e(ϵ0−µ)/kBT−1(1) diverges. Here kBis the Boltzmann constant and Tis temperature. As a result, the ground state becomes populated by a macroscopic number of constituent particles that sponta- 3neously form a macroscopically coherent state that is stable against decohering perturba- tions. Bose-Einstein condensates of non-conserved quasiparticles are ubiquitous in nature; sim- ilar phenomenology is used to describe systems consisting of phonons9, rotons10, photons11, excitons12, exciton-polaritons13, etc. To date BECs consisting of magnons, in particular, have been reported in various superfluid phases of3He14–16, in cold atomic gases17,18, and in a few solid-state systems19,20. Moreover, the antiferromagnetic hematite ( α-Fe2O3) has been put forwards as a promising candidate system for condensation of magnons21,22. We note that while the concept of magnon BEC is also useful for describing the onset of magnetic- field-induced magnetic order in spin-dimer compounds23, such as TlCuCl 3, the excitations in such systems are in thermal equilibrium and therefore the chemical potential is always zero, i.e. µ= 0. While the phenomenology is similar24, such systems are outside the scope of this Perspective. The nature and experimental realization of the magnon BEC varies widely between differ- ent physical systems. In1H, a magnon BEC was created at high magnetic field of multiple tesla by initially preparing a dense cloud of cold gas in a higher-energy low-field-seeking spin state, while magnons are created by pumping a number of atoms to a lower-energy, high-field-seeking state17. The magnons become collective excitations via an effect known as ”identical spin rotation”, where a single spin-flip is carried through multiple atom-atom collisions25–27. In the87RbF= 1 spinor condensate the magnon (quasi-) BEC results from spin-exchange collisions between different internal spin states in the same hyperfine mani- fold at low magnetic fields18. In superfluid phases of3He, magnons are collective excitations stemming from the Nambu-Goldstone modes of the underlying order parameter28. Finally, out of the solid state systems where magnon BEC has been realized, perhaps the most promising systems are thin Yttrium-iron-garnet (YIG) films, where the magnon condensate is created either by continuous radio frequency pumping19or by laser-induced spin currents29 at room temperature in highly non-equilibrium state and, moreover, in the momentum space instead of in real space as for the other examples. Regardless of the system, a magnon BEC can be viewed as the ground state of the relevant subsystem, which is an essential viewpoint in the context of time crystals. For example, if one ignores the weak non-conservation of magnons in3He-B caused by the spin- orbit interaction, the ground state of the subsystem would evolve in time, realizing the time 4crystal as originally suggested by Wilczek30. However, due to conservation of particles the time evolution of such a system is unobservable31–34. A similar scenario is realized in atomic BECs as well as in superconductors (in superconductors the coherent precession is that of Anderson pseudospins), where the evolution of the superfluid phase is unobservable. In principle, for long enough measurement times, the phase should become observable due to proton decay. In some cases, such as in1H and in3He, the absolute value of the phase can be directly monitored in real time as the phase of the precessing magnetization can be measured by oriented pickup coils which provide the necessary loss channel. Such direct measurement of the absolute value of the phase of the macroscopic wave function is rather uncommon and can be exploited for a variety of purposes, as discussed later in this Perspective. The spontaneously formed coherent precession of magnetization has many faces: spin superfluidity, off-diagonal long-range order (ODLRO), the Bose-Einstein condensation of nonequilibrium (pumped) quasiparticles, and time crystal. In Section II, we briefly describe the basic properties of the magnon BEC. Section III focuses on magnon-BEC time crystals. Magnon BECs can be utilized to simulate different processes and objects in particle physics, such as spherical charge solitons (Section IV), the light Higgs particle (Section V) and analogue event horizons (Section VI). Section VII concentrates on probing topological defects using magnon BEC and, finally, Section VIII contains an outlook on future prospects of magnon BECs in various systems. II. COHERENT PRECESSION AND SPIN SUPERFLUIDITY A. Off-diagonal long-range order The phenomenon of Bose-Einstein condensation was originally suggested by Einstein for stable particles with integer spin. Under suitable conditions, the BEC gives rise to macro- scopic phase coherence and superfluidity, first observed in liquid4He. This a consequence of the spontaneous breaking of the global U(1) gauge symmetry related to the conservation of the particle number N, e.g. of4He atoms. As distinct from many other systems with spontaneously broken symmetry, such as crys- tals, liquid crystals, ferro- and antiferromagnets, the order parameter in superfluids and superconductors is manifested in the form known as the off-diagonal longe-range order 50 0.5 1 1.5 2-0.2-0.100.10.2Mx(arb. un.) time (s) ba cpumping pulseFIG. 2. Observing magnon Bose-Einstein condensation: (a) Magnons are pumped to the system with a radio-frequency pulse at zero time, seen as the sharp peak in the data. As illus- trated in the central panel on a colored background, the pumping is followed by dephasing of the precession. If the magnon density is high enough, a BEC emerges after τE, manifest in coherent precession of magnetization Mx+iMy∝D ˆS+E =√ 2S⟨ˆa0⟩=S⊥eiωt. This is picked up by the NMR coils and measured as an oscillating voltage. Magnetic relaxation in superfluid3He is very slow, and the number of magnons Ndecreases with time constant τN(here τN∼10 s), seen as a slow decrease in the signal amplitude, shown in panel ( b). Panel ( c) shows a further zoom-in into the band indicated by the green line. Here, the sinusoidal pick-up signal generated by the preces- sion of magnetization is clearly seen. Data shown in this figure was measured at 0 bar pressure and 131µK temperature35. 6(ODLRO). In bosonic superfluids (such as liquid4He) the manifestation of the ODLRO is that the average values of the creation and annihilation operators for the particle number are nonzero in the superfluid state, i.e., Ψ =D ˆΨE ,Ψ∗=D ˆΨ†E . (2) In conventional (i.e. not superfluid or superconducting) states the creation or annihilation operators have only the off-diagonal matrix elements, such asD N|ˆΨ†|N+ 1E , describing the transitions between states with different number of particles. In the thermodynamic limit N→ ∞ , the states with different numbers of particles in the Bose condensate are not distinguished, and the creation or annihilation operators acquire the nonzero average values. In superconductors and fermionic superfluids such as superfluid3He, the ODLRO is represented by the average value of the product of two creationD a† ka† −kE or two annihilation ⟨aka−k⟩operators, which reflects the Cooper pairing in fermionic systems. In quantum the- ory, states with nonzero values of the creation or annihilation operators are called squeezed coherent states. B. ODLRO and coherent precession The magnetic ODLRO can be represented in terms of magnon condensation, applying the Holstein-Primakoff transformation. The spin operators are expressed in terms of the magnon creation and annihilation operators ˆa0s 1−¯ha† 0a0 2S=ˆS+ √ 2S¯h,s 1−¯ha† 0a0 2Sˆa† 0=ˆS− √ 2S¯h, (3) ˆN= ˆa† 0ˆa0=S − ˆSz ¯h. (4) Eq. (4) relates the number of magnons Nto the deviation of spin Szfrom its equilibrium valueS(equilibrium) z =S=χHV/γ , where χandγare spin susceptibility and gyro-magnetic ratio, respectively. Pumping Nmagnons into the system (e.g. by a RF pulse) reduces the total spin projection by ¯ hN, i.e.Sz=S −¯hN. The ODLRO in magnon BEC is given by: ⟨ˆa0⟩=N1/2eiωt+iα=r 2S ¯hsinβ 2eiωt+iα, (5) where βis the tipping angle of precession. The role of the chemical potential µis played by the global frequency of the coherent precession ω, i.e. µ≡¯hωand the phase of precession 7αplays the role of the phase of the condensate, i.e. Φ ≡α. A typical experimental signal showing an exciting pulse, the formation of the BEC and the slow decay is shown and analysed in Fig. 2. The experimental setup used in this particular experiment is shown in Fig. 3. Note that the analogy with atomic BECs is valid only for the dynamic states of the magnetic subsystem and not e.g. for static magnets with ω= 0. C. Gross-Pitaevskii and Ginzburg-Landau description As for atomic Bose condensates, the magnon BEC is described by the Gross-Pitaevskii equation. The local order parameter is obtained by extension of Eq. (5) to the inhomoge- neous case, ˆ a0→ˆΨ(r, t), and is determined as the vacuum expectation value of the magnon field operator: Ψ(r, t) =D ˆΨ(r, t)E , n=|Ψ|2,N=Z d3r|Ψ|2. (6) where nis the magnon density. If the dissipation and pumping of magnons are ignored, the corresponding Gross- Pitaevskii equation has the conventional form: −i¯h∂Ψ ∂t=δF δΨ∗, (7) where F{Ψ}is the free energy functional forming the effective Hamiltonian of the spin subsystem. In the coherent precession, the global frequency is constant in space and time Ψ(r, t) = Ψ( r)eiωt, (8) and the Gross-Pitaevskii equation transforms into the Ginzburg-Landau equation with ¯ hω= µ: δF δΨ∗−µΨ = 0 . (9) The free energy functional reads F −µN=Z d3r1 2gik∇iΨ∗∇kΨ + ¯h(ωL(r)−ω)|Ψ|2+Fso(|Ψ|2) , (10) where ωLis the local Larmor frequency ωL(r) =γH(r) and gikdescribes rigidity of the magnon system. The spin-orbit interaction energy Fsois a sum of contributions proportional 8NMR excitation and pick upMagnon BEC IminH M3HeB ˆntexture spin orbit energyrz Zeeman energyβ N1N2 ω2 ω1N1>N2 ω1< ω 2FIG. 3. Magnon BEC in magneto-textural trap in superfluid3He. The magnetization M of the condensate is deflected by an angle βfrom the direction of magnetic field Hand precesses coherently around the field direction with the frequency ω. Magnons are confined to the nearly harmonic 3-dimensional trap formed by the spatial variation of the field H(r) via Zeeman energy and by the spatial variation (texture) of the orbital anisotropy vector ˆn(r) of3He-B via spin-orbit interaction energy. The orbital texture is flexible and yields with increasing number of magnons Nin the trap, resulting in lower radial trapping frequency. As a result, the chemical potential, observed as the precession frequency, decreases. This leads to inter-magnon interaction (Sec. II C) and eventually to Q-ball formation (Sec. IV A). to|Ψ|2and|Ψ|4, see e.g. Ref. 36. Thus, the free energy functional can be compared with the conventional Ginzburg-Landau free energy of an atomic BEC: F −µN=Z d3r1 2gik∇iΨ∗∇kΨ + ( U(r)−µ)|Ψ|2+b|Ψ|4 (11) with the external potential U(r) formed by the magnetic field profile and a part of the spin- orbit interaction energy. The fourth order term, which describes the interaction between magnons, originates from the rest of spin-orbit interaction Fso(|Ψ|2). Fig. 3 illustrates the appearance of the inter-magnon interaction via flexible orbital texture in the case of trapped magnon BEC. 9The gradient energy in Eq. (11) is responsible for establishing coherence across the sample. In the London limit, it can be expressed via gradients of the precession phase α Fgrad=1 2gik∇iΨ∗∇kΨ =1 2Kik∇iα∇kα=1 2n m−1
ik∇iα∇kα . (12) A necessary condition for spin superfluidity and phase coherence is that the gradient energy is positively determined. This condition is not universally valid in all systems with magnons, but is applicable, e.g., in3He-B. The spin superfluid currents are then generated by the gradient of the phase. The rigidity tensor Kikcan be further expressed via magnon mass mik(n), which in general depends on the magnon density nand is anisotropic due to applied magnetic field37. III. MAGNON BEC AS A TIME CRYSTAL Originally, time crystals were suggested as class a quantum systems for which time trans- lation symmetry is spontaneously broken in the ground state30. It was quickly pointed out that the original concept cannot be realized and observed in experiments, essentially because that would constitute a perpetual motion machine31–34. That is, if the system is strictly isolated, i.e. when the number of particles Nis conserved, there is no reference frame for detecting the time dependence38. This no-go theorem led researchers to search for spontaneous breaking of the time-translation symmetry on more general grounds, turning to out-of-equilibrium phases of matter (see e.g. reviews 39–41). With this adjustment, feasible candidates of time-crystal systems include those with off-diagonal long range order, such as superfluids42, Bose gases43, and magnon condensates34. Any system with ODLRO can be characterized by two relaxation times34: the lifetime of the corresponding (quasi)particles τNand the thermalization time τEduring which the BEC is formed. If τN≫τE, the system has sufficient time to relax to a minimal energy state with (quasi-)fixed N(i.e. to form the condensate). During the intermediate interval τN≫t≫τE the system has finite µcorresponding to spontaneously-formed uniform precession that can be directly observed as shown in Fig. 2. In3He-B τNcan reach tens of minutes at the lowest temperatures8– this is the closest an experiment has got to a time crystal in equilibrium. Finally, we point out that in the grand unification theory extensions of Standard Model the conservation of the number of atoms is absent due to proton decay44. Therefore, in 10principle, the oscillations of an atomic superfluid in its ground state can be measured, albeit the time scale for the decay is at least in the ∼1036years range44. A. Discrete and continuous magnon time crystals Time crystals are commonly divided into two broad categories based on their symmetry classification. If, relative to the Hamiltonian before the phase transition to the time crystal phase, the system spontaneously breaks the discrete time translation symmetry, it is called a discrete time crystal . Such a system is realized e.g. in parametric pumping scenarios, when the periodicity of the formed time crystal differs from that of the drive. On the other hand, if the spontaneously broken symmetry is the continuous time translation symmetry (the preceding Hamiltonian is not periodic in time), the system is a continuous time crystal . Note that both discrete and continuous time crystals may still possess a discrete time translation symmetry. Both types of time crystals have been observed in magnon BEC experiments35,45–48. Dis- crete time crystals are realized under an applied RF drive, when the frequency of the co- herent spin precession deviates from that of the drive. If the induced precession frequency is incommensurate with the drive, the system obtains the characteristics of a discrete time quasicrystal. On the other hand, if the magnon number decay is sufficiently slow, i.e., τ−1 N≪ω, where ωis the frequency of motion of the time crystal, the coherent precession can be observed for a long time after the drive has been turned off. This is a continuous time crystal, which can generally be formed in magnon BEC materials as τ−1 E< ωby definition. In3He-B continuous time crystals reach life times longer than 107periods35. B. Phonon in a time crystal Spontaneous breaking of continuous time translation symmetry in a regular crystal results in the appearance of the well-known Nambu-Goldstone mode – a phonon. Similarly, the spontaneous breaking of time translation symmetry in a continuous time crystal should lead to a Nambu-Goldstone mode, manifesting itself as an oscillation of the phase of the periodic motion of the time crystal, Fig. 4a. This mode can be called a phonon in the time crystal. In time crystals formed by magnon BECs, the phononic mode is equivalent to the Nambu- 110 0.1 0.2 0.3 0.4 0.5 0.6 Hrf (µT)024681012141618M2 NG (103 Hz2) experiment theory no fit0.43 Tc 0.64 Tc60 80 100 120 Modulation freq. (Hz)-50510Modulation resp. (arb.un.) MNGabsorption dispersion ωNG|k=π La b ccrystal time crystalx t ωωNGMFIG. 4. Phonon in a magnon-BEC time crystal. (a) In a crystal in a ground state, atoms occupy periodic locations in space (empty circles), while phonon excitation results in a periodic shift from these positions (filled circles). A time crystal is manifested by a periodic process (thin line), and a phonon excitation leads to periodic variation of the phase of that process (thick line). (b) In a magnon-BEC time crystal, the periodic process is the precession of magnetization Mat frequency ω. The phonon excitation modulates the precession phase at the frequency ωNG(right). This mode can be excited by applying modulation to the rf drive of the condensate and observed by detecting response in the induction signal from the pick-up coil at the modulation frequency (left). Two standing-wave modes in the sample are visible (marked with the vertical dashed lines). ( c) As the measurements on the panel (b) are done with finite rf excitation of the amplitude Hrf, the phonon becomes a pseudo-Nambu-Goldstone mode with the mass MNG, Eq. (13). Extrapolation to the freely evolving time crystal at zero Hrfshows that the phonon becomes massless, as expected for spontaneous symmetry breaking. The measurements are done at two temperatures at a pressure of 7.1 bar in the polar phase of3He15. 12Goldstone mode related to spin-superfluid phase transition49. It is easier to excite in exper- iments when the spin precession is driven by a small applied rf field Hrfby modulating the phase of the drive, Fig. 4b. In this case the time translation symmetry is already broken explicitly by the drive, and the phonon becomes a pseudo-Nambu-Goldstone mode with the mass (gap) MNG. Its dispersion relation connecting the wave vector kand the frequency ωNGbecomes ω2 NG=M2 NG+c2 NGk2. (13) For the sample size of L, standing-wave resonances can be seen for k=nπ/L , where nis an integer, Fig. 4b, and the mass MNGand the propagation speed cNGcan be determined. According to the theory MNG∝Hrf– experiments in the polar phase of3He demonstrate excellent agreement without fitting parameters15, Fig. 4c. This mode was also observed in time crystals formed by magnon BEC in the B phase of3He6,7. Extrapolation of the mass to the case of a freely evolving time crystal at Hrf= 0 leads to a true massless Nambu-Goldstone mode – a phonon in a time crystal. C. Interacting time crystals Interacting time crystals have been realized in3He-B by creating two continuous time crystals with close natural frequencies close to each other45,46. In a magneto-textural trap such as used in Refs. 35,45,46 the radial trapping potential is provided by the spin-orbit interaction via the spatial order parameter distribution. The magnetic feedback of the magnon number to the order parameter texture means the time crystal frequency (period) is regulated internally, ωB(NB).50The frequency increases as the magnon number slowly decreases (the decay mechanism is not important but details can be found in Refs. 51,52). The second time crystal is created against the edge of the superfluid where such feedback is suppressed. The result is a macroscopic two-level system described by the Hamiltonian H= ¯h ωB[NB(t)]−Ω −Ω ωS (14) where the coupling Ω is determined by the spatial overlap of the time crystals’ wave functions. In this configuration, the two time crystals may interact by exchanging the constituent quasiparticles. The exchange of magnons results in opposite-phase oscillations in the re- 13spective magnon populations of the two time crystals (Fig. 5) which is equivalent to the AC Josephson effect in spin superfluids53. Further two-level quantum mechanics can be accessed by making the precession frequen- cies of two time crystals cross during the experiment using the dependence ωB(NB). The result is magnons moving from the ground state to the excited state of the two-level Hamil- tonian in a Landau-Zener transition, see Fig. 6. Remarkably, these phenomena are directly observable in a single experimental run, including the chemical potentials and absolute phases of the two time crystals, implying that such basic quantum mechanical processes are also technologically accessible for magnonics and related quantum devices. IV. MAGNON BEC AND COSMOLOGY In the context of particle physics and cosmology, the magnon BEC provides a laboratory test bench for otherwise inaccessible or convoluted theoretical concepts. This phenomenology may eventually play a major role in the technological toolbox for magnonics, albeit potential applications cannot yet be predicted. Here we will discuss the analogs between trapped magnons and two cosmological concepts: the Q-ball and the MIT bag model. A. Magnonic Q-ball Self-bound macroscopic objects encountered in everyday life are made of fermionic matter, while bosons mediate interactions between and within them. Compact (self-bound) objects made purely from interacting bosons may, however, be stabilized in relativistic quantum field theory by conservation of an additive quantum number Q54–56. Spherically symmetric non- topological Q-charge solitons are called Q-balls. They generally arise in charge-conserving relativistic scalar field theories. Observing Q-balls in the Universe would have striking consequences beyond support- ing supersymmetric extensions of the Standard Model57,58– they are a candidate for dark matter58–61, may play a role in the baryogenesis62and in formation of boson stars63, and supermassive compact objects in galaxy centers may consist of Q-balls64. Nevertheless, un- ambiguous experimental evidence of Q-balls has so far not been found in cosmology or in high-energy physics. Analogues of Q-balls have been speculated to appear in atomic BECs 14a Ms Mb 1 2 3 4 5 time (s)242628303234frequency (Hz)time (s)b 1 2 3 4 5160180200220240260280chemical potential / h (Hz) csurface (MS)bulk (MB)side band side bandFIG. 5. Josephson (Rabi) population oscillations between two magnon-BEC time crys- tals. (a) We can create two local minima for magnon-BEC time crystals, one in the bulk and one against a free surface of the superfluid. ( b) Both are populated with a pulse at zero time, after which the bulk frequency is slowly changing due to changes in the trap shape as the magnon number slowly decreases. The two levels are coupled, resulting in Josephson population oscillations between them, observed as the side bands above and below the main traces. The side band fre- quency separation (green arrow), shown by the green line in panel c, corresponds to the separation of the main traces (red arrow), shown by the red line in panel c. This ties the population oscillation to the chemical potential difference of the time crystals and thus to Josephson oscillations. The oscillations the bulk population and the surface population are shown to take place with opposite phases in Ref. 45. The Josephson oscillations become Rabi oscillations if the two-level frequencies are brought close to one another as explained in Ref. 46. 15Tunneling through avoided crossingFIG. 6. Landau-Zener tunneling between two magnon-BEC time crystals. One of the two levels is populated at time zero with an exciting pulse (framed out to emphasise the rest of the signal). The chemical potential of this state increases gradually as the magnon number decays, crossing the second level after 3 s. As the avoided crossing is traversed at finite rate (not adiabatically), a part of the magnon population tunnels to the excited level at the avoided crossing46. in elongated harmonic traps65and possibly play a role in the3He A-B transition puzzle66. Additionally, the properties of bright solitons in 1D atomic BECs67and Pekar polarons in ionic crystals68bear similarities with Q-balls. The trapped magnon BEC in3He-B provides a one-to-one implementation of the Q-ball Hamiltonian. The charge Qis the number of magnons and the BEC precession frequency corresponds to the frequency of oscillations of the relativistic field within the Q-ball. Above a critical magnon number, the radial trapping potential for the magnons changes from harmonic to a ”Mexican-hat” potential. The modification is eventually limited by the un- derlying profile of the magnetic field (see Fig. 7). Here, the systems’ Hamiltonian mimics that of the Q-ball. All essential features of Q-balls, including the self-condensation of bosons into a spontaneously formed trap, long lifetime, and propagation in space across macroscopic distances (here several mm) have been demonstrated experimentally as shown in Fig. 7.69 16a b cFIG. 7. Magnon BEC as a self-propagating Q-ball soliton. (a) The magnon BEC is created with a pulse (not shown) with a very large initial magnon number. The time-dependent frequency spectrum of the recorded signal is shown here in such a way that dark corresponds to no magnons and yellow to large magnon population. In the beginning (red dot and panel b) the BEC (orange line and orange blob) is located in a side minimum, where the trapping potential (blue line, blue surface) is modified down to the unyielding trap component controlled by the magnetic field (black line, dark surface). As the magnon number decreases due to slow dissipation, the trapping potential evolves and the BEC gradually moves across several millimeters to the symmetric central position (magenta dot and panel ( c)). Here the BEC is illustrated by the green line and blob.69 B. Magnons-in-a-box – The MIT Bag model The confinement mechanism of quarks in colorless combinations in quantum chromody- namics (QCD) is an open problem. One of the most successful phenomenological models, 17u d da b True vacuumFIG. 8. Magnon BEC as a bosonic analogue to the MIT bag. a In the limit of large magnon density the magnon BEC carves a potential well (white), described by the charge field Φ(r), in the neutral field ζwhich plays the role of the true vacuum. bIn the context of QCD, the quarks carve a potential well (false vacuum, white) in the true QCD vacuum, illustrated here for the charge-neutral neutron consisting of two down-quarks (labeled d) and one up-quark (labeled u). coined MIT bag model70as per the affiliation of its inventors, assumes a step change from zero potential within the confining region to a positive value elsewhere, a cavity surrounded by the QCD vacuum. The cavity is filled with false vacuum, in which the confinement is absent and quarks are free, thus creating the asymptotic freedom of QCD. Outside the cavity there is the true QCD vacuum, which is in the confinement phase, and thus a single quark cannot leave the cavity. Within the cavity quarks occupy single-particle orbitals, and there the zero point energy compensates the pressure from true vacuum. A similar situation is realized for a magnon BEC, if the magnetic maximum applied in the Q-ball experiment discussed in the previous Section is removed. Under these conditions the magnon BEC forms a self-trapping box analogously to the MIT bag model50,71, c.f. Fig. 8. The flexible Cooper pair orbital momentum distribution ˆlplays either the role of the pion field or the role of the non-perturbative gluonic field, depending on the microscopic structure of the confinement phase. Much like quarks, magnons dig a hole in the confining ”vacuum”, pushing the orbital field away due to the repulsive interaction. The main difference from the MIT bag model is that magnons are bosons and may therefore macroscopically occupy the same energy level in the trap, forming a BEC, while in MIT bag model the number of fermions on the same energy level is limited by the Pauli exclusion principle. The bosonic bag becomes equivalent to the 18fermionic bag in the limit of large number of quark flavors due to bosonization of fermions. This phenomenon has been observed in cold gas experiments for SU( N) fermions72. V. LIGHT HIGGS BOSONS: PARTICLE PHYSICS IN MAGNON BEC Both in the Standard Model (SM) of particle physics and in condensed matter physics the spontaneous symmetry breaking during a phase transition gives rise to a variety of collective modes. This is how the Higgs boson arises from the Higgs field in the Standard Model, for example. The gapless phase modes related to the breaking of continuous symmetries are called the Nambu-Goldstone (NG) modes, while the remaining gapped amplitude modes are called the Higgs modes. In superfluid3He, we can make the magnon BEC interact with other collective modes, implementing scenarios that in the Standard Model context may require years of measurements using a major collider facility. The superfluid transition in3He takes place via formation of Cooper pairs in the L= 1, S= 1 channel, for which the corresponding order parameter is a complex 3 ×3 matrix combining spin and orbital degrees of freedom. Thus,3He possesses 18 bosonic degrees of freedom, both massive (amplitude or Higgs modes) and massless (phase or Nambu-Goldstone modes), see Fig. 9. The 14 Higgs modes have masses (energy gaps) of the order of the superfluid gap ∆ /h∼100 MHz, where his the Planck constant, while the four Nambu- Goldstone modes (a sound wave mode and three spin wave modes) are massless at this energy scale. Higgs modes have been investigated for a long time theoretically73–75and experimentally76–78in3He-B as well as s-wave superconductors79,80and ultracold Fermi gases81. At low energies, the superfluid B phase of3He breaks the relative orientational freedom of the spin and orbital spaces, and the resulting order parameter (at zero magnetic field) becomes Aαj= ∆eiφRαj(ˆn, θ), (15) where the rotation matrix Rαjdescribes the relative orientation of the spin and orbital spaces; the spin space is obtained from the orbital space by rotating it by angle θwith respect to vector ˆn. If the spin-orbit interaction is neglected or, equivalently, one considers energy scales of the order of the superfluid gap ∆, the order parameter obtains an additional (”hidden”) symmetry with respect to spin rotation. That is, the energy is degenerate with 19(a) (b) (c) (d)FIG. 9. Collective modes and decay channels. (a) The collective mode spectrum in3He-B contains six separate branches of collective modes. The 14 gapped Higgs modes (orange) are: four degenerate pair-breaking modes with gap 2∆ /h∼100 MHz, five imaginary squashing modes with gapp 12/5∆, and five real squashing modes with gapp 8/5∆. The gapless modes are a sound wave mode (oscillations of Φ, yellow), a longitudinal spin wave mode (oscillations of θ, purple), and two transverse spin wave modes (oscillations of ˆn, green). (b) The longitudinal spin wave mode acquires a gap of Ω L/h∼100 kHz due to spin-orbit interaction and becomes a light Higgs mode. The transverse spin wave modes are split by the Zeeman effect in the presence of a magnetic field into optical and acoustic magnons. The arrows indicate possible decay channels. (c) The spatial extent of the optical magnon BEC in a typical experiment is of the order of a millimeter, and can be used as a probe for quantized vortices. (d) In a container of fixed size R, the spin wave modes form standing wave resonances. respect to ˆnandθ. The spin-orbit interaction lifts the degeneracy with respect to θ, and the minimum energy corresponds to a rotation between the spin and orbital spaces by the Leggett angle θL= arccos( −1/4)≈104◦. Due to this broken symmetry, one of the Nambu-Goldstone modes (the longitudinal spin wave mode) obtains a gap with magnitude equal to the Leggett frequency ΩL/h∼100 kHz. In the B-phase the longitudinal spin wave mode therefore becomes a light Higgs mode. Additionally, the presence of the magnetic field breaks the degeneracy of the transverse spin wave modes, one of which becomes gapped by the Larmor frequency ωL=|γ|H, where γis the gyromagnetic ratio in3He. The gapped transverse spin wave mode is called optical and the gapless mode acoustic. Throughout the manuscript, the term ”magnon BEC” in the context of3He-B refers to a BEC of optical magnons. 20Magnons in the BEC can be converted into other collective modes in the system. For example, the decay of the optical magnons of the BEC into light Higgs quasiparticles has been observed via a parametric decay channel in the absence of vortices82, and via a direct channel in their presence83(see Sec. VII) as illustrated in Fig. 9. The parametric decay channel is directly analoguous to the production of Higgs modes in the Starndard Model. The separation of the Higgs modes in3He-B into the heavy and light Higgs modes poses a question whether such a scenario would be realized in the context of the Standard Model as well. In particular, we note that the observed 125 GeV Higgs mode84–86is relatively light compared to the electroweak energy scale and, additionally, later measurements at higher energies87show another statistically significant resonance-like feature at the electroweak energies of ≈1 TeV related by the authors to possible Higgs pair-production. Entertaining the possibility of a3He-B-like scenario, the observed feature could stem from formation of a ”heavy” Higgs particle; in this case the 125 GeV Higgs boson would correspond to a pseudo-Goldstone (or a ”light” Higgs) boson, whose small mass results from breaking of some hidden symmetry (see e.g. Ref. 88 and references therein). VI. CURVED SPACE-TIME: EVENT HORIZONS The properties of the magnon BECs have also been utilized to study event horizons. In the conducted experiment89, two magnon BECs were confined by container walls and the magnetic field in two separate volumes connected by a narrow channel, Fig. 10. The channel contains a restriction, controlling the relative velocities of the spin supercurrents traveling in the bulk fluid and the spin-precession waves traveling along the surfaces of the magnon BEC. The magnitude and direction of the spin superflow is controlled by the phase difference of the two magnon BECs, both of which are driven continuously by separate phase-locked volt- age generators. The phase difference controls the spin supercurrents, while spin-precession waves are created by applied pulses. For a sufficiently large phase difference, spin-precession waves propagating opposite to the spin superflow are unable to propagate between the two volumes and instead are blocked by the spin superflow. This situation is analogous to a white-hole event horizon. 21HPD HPDSpin superflow Spin- precession wavesSource DetectionFIG. 10. Magnon BECs and event horizon. In the experiment two volumes filled with superfluid3He-B, in which the magnetization precesses uniformly (HPD) are connected by a narrow channel. An imposed phase difference between the precessing HPDs creates a spin supercurrent proportional to the phase difference. For sufficiently large magnitude of the spin superflow, counter- propagating spin-precession waves (surface-wave-like excitations of the HPD) can not propagate between the two volumes, analogously to the white-hole horizon. VII. MAGNON BEC AS A PROBE FOR QUANTIZED VORTICES Magnon BEC has proved itself as a useful probe of topological defects, especially quan- tized vortices. Vortices affect both the precession frequency of the condensate through modification of the trapping potential for magnons90and the relaxation rate of the con- densate through providing additional relaxation channels83,91–93. Trapped magnon BECs provide a way to probe vortex dynamics locally and down to the lowest temperatures94–96, where there are still many open questions related to vortex dynamics, see e.g. Refs. 97,98. The effect of vortex configuration on the textural energy, which determines the radial magnon BEC trapping frequency, may be written as90 Fv=2 5amH2λ ΩZ d3r ωv·ˆl2 ωv, (16) where amis the magnetic anisotropy parameter, Ω is the angular velocity, λis a dimensionless parameter characterizing vortex contribution to textural energy, and ω=1 2⟨∇ × vs⟩is the spatially averaged vorticity. The vortex contribution to the textural energy may be introduced via a dimensionless parameter λ, which is contains the contributions from the orienting effect related to the superflow and the vortex core contribution. While the equilibrium vortex configuration is 22well understood99, the mechanism of dissipation in the zero-temperature limit, in particular, remains an open problem. If the equilibrium vortex configuration is perturbed, i.e. ω:=Ω+ω′, where Ωis the equilibrium vorticity and ω′is a random contribution with ⟨ω′⟩= 0, parameter λis replaced by an effective value λeff=λ1 + (ωv∥/Ω)2−(ωv⊥/Ω)2 p 1 + (ωv∥/Ω)2+ 2(ωv⊥/Ω)2. (17) Here ωv∥andωv⊥are the random contributions along the equilibrium orientation and per- pendicular to it, respectively. This effect has been observed in experiments95by introducing vortex waves via modulation of the angular velocity around the equilibrium value and mon- itoring the precession frequency (i.e. the ground state energy) of the magnon BEC, see Fig. 11. Vortex core contribution can be extracted separately from the measured magnon energy levels by comparing measurements with and without vortices90. Based on numerical 1D calculations using the uniform vortex tilt model from Ref. 100, where all vortices are tilted relative to the equilibrium position by the same angle, the magnon BEC ground state frequency, see Fig 12 a, is found to scale as ∆f≈ −f0sin2θ , (18) where the sensitivity f0∼100 Hz is found to scale linearly with the vortex core size, see Fig. 12 b. Using Eq. (18) one can then extract the average tilt angle of vortices within the volume occupied by the magnon BEC from the measured frequency shift. This method has been utilized for probing transient vortex dynamics95. When quantized vortices penetrate the magnon BEC, like in Fig. 9c, they also contribute to enhanced relaxation of the condensate. Distortion of the superfluid order parameter around vortex cores opens direct non-momentum-conserving conversion channels of optical magnons from the condesate to other spin-wave modes, predominantly light Higgs101, see Fig. 9b. The decay rate of macroscopic (mm-sized) condensate depends on the internal structure of microscopic (100 nm-sized) vortex core. This effect has been used in experiments to distinguish between axially symmetric and asymmetric double-core vortex structures91 and to measure the vortex core size83. 23FIG. 11. Probing vortex dynamics with magnon BEC. a Vortex waves can be excited by applying perturbation in the form of angular modulation (top) on a steady vortex lattice. The vortex configuration is monitored locally with two separate magnon BECs (’Upper BEC’ and ’Lower BEC’), which allows extracting time scales relevant for turbulence buildup. The angular drive results in decreased trapping frequency of the magnon BEC due to the ωv⊥term in Eq.(17). b As expected, the extracted value for λeffdecreases monotonously with increasing drive amplitude. Both data are measured under the same experimental conditions with the same relative drive frequency ω/Ω0, where Ω 0is the mean angular velocity during the drive. Inset shows the definition of the tilt angle θwith respect to the axis of rotation. cSchematic illustration of the vortex (red) array evolution in response to the angular drive and the eventual relaxation after the drive is stopped. VIII. OUTLOOK Bose-Einstein condensates of magnon quasiparticles have been realized experimentally in different systems including solid-state magnetic materials, dilute quantum gases, and super- 2400.20.40.60.81 Vortex tilt angle, sin2100200300400500Radial trap frequency fr, Hz 100 150 200 250 Vortex core size, nm100200300400500Tilt sensitivity f0, Hza b fr = (466 -303, sin2), Hz P = 4.1 barFIG. 12. Magnon BEC as a probe for vortex configuration. a The radial trapping potential for a magnon BEC, originating from the textural configuration in a cylindrical trap, scales with the vortex tilat angle roughly as fr=ωr/2π∝sin2θ, where θis the tilt angle of vortices relative to the axis of rotation. The dashed line is a linear fit to the numerically calculated frequency shift using experimentally determined value for ( λ/Ω)|θ=0at 4.1 bar pressure (vortex core size ∼170 nm). The numerical model assumes uniform vortex tilt. bThe tilt sensitivity f0, calculated using the measured ( λ/Ω)|θ=0at all pressures, is found to scale with roughly linearly with the vortex core size (1 + Fs 1/3)ξ0, where Fs 1is the first symmetric Fermi-liquid parameter and ξ0is the T= 0 coherence length. fluid3He. Spin superfluidity of those condensates and phenomena such as the Josephson effect may be viewed as analogs of superconductivity in the magnetic domain. Supercon- ducting quantum electronics, based on Josephson junctions, is one of the most important platforms for quantum technologies, however requiring dilution refrigerator temperatures of about 10 mK. Coherent magnetic phenomena are generally more robust to temperature than superconductivity with existing room-temperature demonstrations of magnon BEC, spin supercurrents and the magnon Josephson effect19,102,103. Thus one of the strong axes of research on magnon condensates is the development of practically useful (super-)magnonic devices operating at ambient conditions22,104–106. In this Perspective, we have shown that there is another important dimension of magnon BEC applications as a laboratory to study fundamental questions in various areas of physics from Q-balls and Higgs particles to time crystals and quantum turbulence. These fundamental phenomena can and should be utilized in future magnon-based devices. 25Advances outlined in this Perspective have triggered suggestions for further applications of magnon BEC for fundamental research. In particular, magnon BEC has been proposed to be used as a dark matter axion detector107,108via the coupling between axions and coherently precessing spins. The coupling gives rise to a term in the Hamiltonian of the BEC that looks like an effective magnetic field, oscillating with the frequency depending on the axion mass. When the frequency of precession matches the frequency of the axion field, potentially observable glitches in the magnon BEC decay are expected. The change of the frequency of the magnon BEC during decay due to inter-magnon interactions (see an example in Fig. 7a) allows to probe a continuous range of the axion masses. Magnon BECs could also be used as a source and a detector of spin currents in particular to probe composite topological matter at the interface of superfluid3He and graphene109. Atoms of3He do not penetrate through a graphene sheet, but coupling of the spin degrees of freedom of two superfluids on the opposite sides of the sheet immersed in the liquid is nevertheless possible through the excitations of the graphene itself (electronic or ripplons) or via magnetic coupling of the quasiparticles living at the interface between graphene and helium superfluid, including Majorana surface states. As in the original observation of the spin Josephson effect in3He4, two magnon condensates separated by a channel can be maintained at the controlled phase difference of the magnetization precession, which drives a spin current through the channel similar to Fig. 10. In this case, instead of the constriction, one would place a graphene membrane across the channel and find whether the Josephson coupling is nevertheless observable. Even a single magnon BEC placed in contact with the surface of topological superfluid can provide valuable information on the topological surface states. Trapping a condensate with the magnetic field profile, like in Fig. 3, allows to move the BEC around the fluid simply by adjusting the trapping field. Preliminary measurements show that magnon loss from BEC is significantly enhanced when the condensate is brought from bulk to the surface of3He-B sample110. Future experiments should clarify whether this relaxation increase is caused by Majorana surface states111,112. The ability to manipulate Majorana states would come with applications in quantum information processing. Development of optical lattices opened many new areas of physics for probing in cold atom experiments113. So far experiments on magnon BECs were limited to one or two condensates. An exciting development will be to form magnon condensates on a lattice to 26probe solid-state physics in magnetic domain, perhaps utilizing spinor cold gases114on a 2D lattice of elongated trapping tubes18created by superimposing two orthogonal standing waves of attractive lase beams, or in solid-state systems where one may similarly utilize optical beamshaping techniques to directly print a 2D lattice and impose spin currents29. For magnon BEC in superfluid3He, multiple regularly arranged traps can potentially be formed using orbital part of the trapping potential, Fig. 3. Array of the quantized vortices, created by rotation, modulate the orbital degrees of freedom of superfluid3He and at certain conditions may form individual magnon traps around the vortex cores115. The spacing of the lattice sites can then be regulated by rotation velocity. This will change coupling between condensates at individual sites which will allow to observe, for example, a superfluid-Mott insulator transition116but for spin superfluid. Excitations of the magnon BEC, in particular its Nambu-Goldstone mode (phonon of a time crystal) provide ample possibilities to model propagation of particles in curved space in acoustic-metric type experiments117. In such models, effective metric is created by the fluid flow which is externally controlled, while the dispersion relation of the propagating modes is usually nature-given, like gravity waves on water118–120. For magnon BEC, remarkably, the spectrum of Goldstone bosons can typically be adjusted in a wide range by external magnetic field, while the spin flow is formed by the phase of the coherent precession controlled with rf pumping. Thus non-trivial metrics can be realized even without using geometrical flow conditioners (like a channel in Fig. 10) and cases impossible for phonons or ripplons in classical fluids can be achieved. For example, in a magnon BEC in the polar phase of 3He, the propagation speed of the Goldstone boson is controlled by the angle of the static magnetic field with the orbital anisotropy axis of the superfluid. The mode can be brought to a complete halt at a critical angle, and beyond this angle the metric changes signature from Minkowski to Euclidean121. Using magnon BEC one can potentially study instability of quantum vacuum in such signature transition. Magnon BECs make one of the most versatile implementations of time crystals that also comes the closest to the ideal time crystal of all systems in the laboratory. Expanding on the experiments summarized in this Perspective, one may pose fundamental questions such as is it possible to melt a time crystal into a time fluid, is it possible to seed time crystallization122, or how time crystals interact with different types of matter. The time crystal description of magnon BECs also emphasizes potential for quantum magnonics applications: the mag- 27nitude and phase of the wave function of a single magnon-BEC time crystal, or that of a multi-level composite system of time crystals, is directly accessible in experiments, revealing basic quantum mechanical processes such as Landau-Zener transitions and Rabi oscillations in a non-destructive measurement in real time. These can therefore be harnessed unimpeded for also technological applications. Additionally, physics similar to (and beyond!) that outlined in this perspective can per- haps be studied in systems for which the experimental realization is yet to come. One promising system is the superfluid fermionic spin-triplet quantum gas, which could be real- ized by synthetic gauge fields e.g. through Rashba-coupling scenarios123, by tuning into a p-wave Feshbach resonance124, or perhaps by induced interactions125. In the context of the weak-coupling theory, the B-like phase is always expected to be the lowest energy superfluid phase28. Therefore, it is expected that the spin-orbit interaction opens up a gap for one of the Goldstone modes, giving rise to the light Higgs mode. We note that such a scenario allows for unique research directions as the spin-orbit coupling strength is likely controllable e.g. via the Rashba coupling strength, via the amplitude of magnetic field, or via density of the inducing component, depending on the experimental setup. Another interesting research project would be to study the properties of magnon BECs in a (putative) spin-triplet superconductor, such as UTe 2, see e.g. Ref. 126 and references therein. The order parameter of UTe 2may take multiple forms, including the one whose d-vector representation is ˆd(k)∝(0, ky, kz). Such an order parameter corresponds to the B3girreducible representation of the D2hpoint symmetry group in UTe 2and to the so-called planar phase28in the context of3He. In3He the planar phase is predicted to never be the lowest energy phase, as its energy always lies between the B phase and the polar phase28. Due to the presence of the discrete point symmetry group, similar argumentation may not apply in UTe 2, making this a unique possibility to study yet another novel topological phase, including its collective modes such as spin waves and by extension the magnon BEC. The strength of the spin-orbit coupling in UTe 2remains an open question126, but it is expected to be non-zero and quite possibly significant (bare uranium has a large spin-orbit interaction strength). As long as the spin-orbit coupling is non-zero, a gap opens in the longitudinal magnon spectrum which then becomes a light Higgs mode. In principle, the respective order parameter is also unique in that it supports isolated monopoles, i.e. monopoles that do not act as termination points for linear objects such as Dirac monopoles127. 28To conclude, magnon BECs are interesting systems in their own right, as they form analogies to various fields of physics, from time crystals and particle physics to QCD, and provide non-invasive methods for probing the dynamics and structure of quantized vortices. Moreover, magnon BECs hold enormous future potential for accessing novel physics, as replacements for electronic components, or perhaps for detecting dark matter. DATA AVAILABILITY Data sharing is not applicable to this article as no new data were created or analyzed in this study. AUTHOR DECLARATIONS The authors have no conflicts to disclose. ACKNOWLEDGMENTS We thank Grigory Volovik for stimulating discussions. The work was supported by Academy of Finland grant 332964 and by UKRI EPSRC grant EP/W015730/1. REFERENCES 1A. Borovik-Romanov, Y. Bun’kov, V. Dmitriev, and Y. Mukharskii, “Long-lived induc- tion signal in superfluid3He-B,” JETP Letters 40, 1033–1037 (1984). 2A. Borovik-Romanov, Y. Bun’kov, V. Dmitriev, Y. Mukharskii, and K. Flachbart, “Ex- perimental study of separation of magnetization precession in3He-B into two magnetic domains,” Soviet Physics - JETP 61, 1199–1206 (1985). 3A. S. Borovik-Romanov, Y. M. Bunkov, V. V. Dmitriev, Y. M. Mukharskiy, and D. A. Sergatskov, “Investigation of spin supercurrents in3He-B,” Phys. Rev. Lett. 62, 1631– 1634 (1989). 4A. Borovik-Romanov, Y. Bunkov, A. Vaard, V. Dmitriev, V. Makrotsieva, Y. 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2023-12-15

Under suitable experimental conditions collective spin-wave excitations, magnons, form a Bose-Einstein condensate (BEC) where the spins precess with a globally coherent phase. Bose-Einstein condensation of magnons has been reported in a few systems, including superfluid phases of $^3$He, solid state systems such as Yttrium-iron-garnet (YIG) films, and cold atomic gases. Among these systems, the superfluid phases of $^3$He provide a nearly ideal test bench for coherent magnon physics owing to experimentally proven spin superfluidity, the long lifetime of the magnon condensate, and the versatility of the accessible phenomena. We first briefly recap the properties of the different magnon BEC systems, with focus on superfluid $^3$He. The main body of this review summarizes recent advances in application of magnon BEC as a laboratory to study basic physical phenomena connecting to diverse areas from particle physics and cosmology to new phases of condensed matter. This line of research complements the ongoing efforts to utilize magnon BECs as probes and components for potentially room-temperature quantum devices. In conclusion, we provide a roadmap for future directions in the field of applications of magnon BEC to fundamental research.

Magnon Bose-Einstein condensates: from time crystals and quantum chromodynamics to vortex sensing and cosmology

2312.10119v1

First harmonic measurements of the spin Seebeck eect Yizhang Chen,1,Debangsu Roy,1,Egecan Cogulu,1Houchen Chang,2Mingzhong Wu,2and Andrew D. Kent1,y 1Center for Quantum Phenomena, Department of Physics, New York University, New York, New York 10003, USA 2Department of Physics, Colorado State University, Fort Collins, Colorado 80523, USA (Dated: August 24, 2018) We present measurements of the spin Seebeck eect (SSE) by a technique that combines alter- nating currents (AC) and direct currents (DC). The method is applied to a ferrimagnetic insula- tor/heavy metal bilayer, Y 3Fe5O12(YIG)/Pt. Typically, SSE measurements use an AC current to produce an alternating temperature gradient and measure the voltage generated by the inverse spin- Hall eect in the heavy metal at twice the AC frequency. Here we show that when Joule heating is associated with AC and DC bias currents, the SSE response occurs at the frequency of the AC current drive and can be larger than the second harmonic SSE response. We compare the rst and second harmonic responses and show that they are consistent with the SSE. The eld dependence of the voltage response is used to characterize the damping-like and eld-like torques. This method can be used to explore nonlinear thermoelectric eects and spin dynamics induced by temperature gradients. A central theme in spintronics is the interconversion of charge and spin currents [1]. Recently, a focus has been on magnetic insulators where spin transport occurs through spin-wave propagation and spin currents can be generated by either spin injection [2] or by thermal gra- dients [3, 4]. These phenomena can be studied in simple bilayer lms consisting of a ferrimagnetic (FIM) insula- tor, such as Y 3Fe5O12(YIG), and a heavy metal (HM) with large spin-orbit coupling such as Pt. Spin to charge current conversion in such bilayers occurs by the inverse spin-Hall [5{7] and Rashba-Edelstein eects [8, 9]. Spin to charge conversion enables determination of the spin Seebeck eect (SSE)[10{13]. A thermal gradient across the FIM lm produces a spin current into a neigh- boring heavy metal lm, resulting in a transverse charge current or a voltage across the heavy metal lm in an open circuit situation. This leads to a convenient route to characterize the spin transport as well as a means to study the inverse eects, such as the spin torque on the FIM magnetization in response to spin currents associ- ated with charge current ow in the HM. In fact, the SSE also enables detection of the FIM magnetization direction by relatively simple electrical measurements. In this article, we present rst harmonic measurements of the SSE by a technique that combines AC and DC currents in a YIG/Pt bilayer. The temperature gradient is created by Joule heating in a Pt strip and both the linear and nonlinear responses in the longitudinal and transverse voltages are determined as a function of the angle between the AC current and an in-plane external magnetic eld. An analysis of the responses shows that the SSE accounts for the main component of the second harmonic voltage response, corroborating results in the literature [14]. However, when a DC current is present, These two authors contributed equally yElectronic address: andy.kent@nyu.eduseveral new features are observed. First, we detect eld- induced switching of the YIG magnetization in both the rst and second harmonic longitudinal voltage measure- ments. Second, both the rst and second harmonic trans- verse voltages show a step when the magnetization re- verses. Interestingly, the step height in the rst harmonic response has a linear dependence on DC current density and cosine dependence on in-plane eld angle. Of par- ticularly interest is that the presence of a DC current superposed on the AC current enables measurements of the SSE in the rst harmonic response with an increase in signal amplitude relative to the second harmonic signal. The samples we studied consist of a 20 nm thick epi- taxial YIG lm grown on a gadolinium gallium garnet (Gd 3Ga5O12) substrate by RF sputtering [15] and a 5 nm thick Pt lm grown by DC sputtering in separate de- position systems. The YIG lm is transferred in air and Ar+plasma cleaning is performed prior to the deposi- tion of the Pt lm. A Hall bar with a width of 4 m and a length between the voltage contacts of 90 m is fab- ricated using e-beam lithography and ion milling. The current ows in the x-direction and the voltage is mea- sured both along the current direction (V xx) and trans- verse to the current direction (V xy) with separate elec- trical contacts (Fig. 1(a)). Lock-in ampliers are used to measure the rst harmonic and second harmonic voltages with phases 1= 0oand2=90and a time constant of 300 ms. The AC current frequency is 953 Hz and its rms amplitude is indicated in the gures. All the angu- lar dependent data are averaged 50 times to improve the signal-to-noise ratio. The measurements are conducted at room temperature. Figure 1(b) and (c) show the second harmonic longitu- dinalV2! xxand transverse V2! xyvoltage, respectively, as a function of the in-plane angle of a 400 mT magnetic eld, a eld sucient to saturate the magnetization of the YIG layer. To conrm that the second harmonic signal is as- sociated with the SSE, measurements were repeated as a function of the applied magnetic eld magnitude [16, 17].arXiv:1808.07813v1 [cond-mat.mes-hall] 23 Aug 20182 FIG. 1: (a) Measurement setup. ~jcis the charge current density along the x direction. VxxandVxyare voltages mea- sured in the longitudinal and transverse directions, respec- tively, while 'is the angle between the applied eld and the current. (b) Angular dependence of second harmonic longitu- dinal voltage V2! xxat a xed current density of jac= 1:51010 A/m2with an applied eld of 0H = 400 mT. The curve is a t toV2! xx(0) sin('). (c) Angular dependence of second har- monic transverse voltage V2! xyat the same current density, jac= 1:51010A/m2. The curve is a t to V2! xy(0) cos('). (See the Supplementary section [19].) It is important to note that there are contributions to the second harmonic signal from the damping-like (DL) torque, eld-like (FL) torque and Oersted (Oe) elds. By characterizing the eld dependence of the second harmonic response these eects can be separated, particularly at small applied elds at which these torques and Oersted elds intro- duce additional structure in the angular dependence of the second harmonic signal. This is discussed in the sup- plementary section [19], where the relative contributions of SSE, DL, FL and Oe eld torques are determined [16{ 18]. We nd that for an applied eld of 400 mT, the second harmonic signal is dominated by the SSE. Figure 2(a) and (b) show the eld dependence of sec- ond harmonic V2! xy(Fig. 2(a)) and rst harmonic V! xy (Fig. 2(b)) response with the eld applied along the cur- rent direction ( '= 0) at a xed AC current density of 1:51010A/m2as the DC component of the current density is varied, jdc= 0;2:5;5:01010A/m2. The SSE response is expected to change sign when the mag- netization direction reverses, which is evident in Fig. 2(a) in the step change in V2! xynear zero eld at the coerciv- ity of the YIG ( 0Hc'10 mT). The step in voltage in the second harmonic signal is nearly independent of the DC current. Interestingly, the rst harmonic response depends systematically on the DC current. At zero DC current there is virtually no response, only small signal variations near zero eld. However, when the DC current density is non-zero, a clear voltage step is evident near zero eld, with a change in voltage that depends on the FIG. 2: Field dependent measurement of second and rst harmonic transverse voltage with xed AC current jac= 1:5 1010A/m2and varying DC current. (a) Field dependence of V2! xyat'= 0andjdc= 0;2:5;5:01010A/m2. (b) Field dependence of V! xywith'= 0andjdc= 0;2:5;5:01010 A/m2. DC current. The magnitude of the rst harmonic signal is about one order of magnitude larger than the second harmonic signal. In addition, the step in the rst harmonic sig- nal changes sign when the DC current is reversed. Fig- ure 3(a) and (b) show how the steps in voltage depends on DC current. The step in the second harmonic signal
V2! xy(Fig. 3(a)) is slightly modied due to DC current, whereas there is a clear linear relation between the step in the rst harmonic signal
V! xy(Fig. 3(b)) and the DC current. In order to understand this behavior one needs to con- sider Joule heating by the AC and DC current through the Pt. This leads to a power dissipation given by: P= [p 2jaccos(!t) +jdc]2RA2 = [j2 accos(2!t) + 2p 2jacjdccos(!t) +j2 ac+j2 dc]RA2; (1) wherejacis the rms AC current density, Ris the resis- tance of the Pt and Aits cross sectional area, the lm thickness times the width of the current line. The tem- perature gradient rTzis proportional to the power dis- sipation. It follows that the SSE voltage generated has the following form: VISHE/j2 accos(2!t)+2p 2jacjdccos(!t)+j2 ac+j2 dc:(2) There is thus an SSE response at two times the oscillation frequency of the current, the second harmonic, 2 !, as expected, as well as a signal at frequency, !, the rst harmonic. Thus the combination of AC and DC currents provides a technique to measure the SSE voltage as a rst harmonic response. The relative magnitude of the rst and the second harmonic signals is given by V! xy=V2! xy= 2p 2jdc=jac. The rst harmonic response is thus about 2.8 times larger than the second harmonic response when the AC and DC currents are the same. The linear relation between
V! xyandjdcin Fig. 3(b) conrms this model. Further, we experimentally verify the symmetry and magnitude of the SSE rst harmonic response in com- parison to the conventional second harmonic signal. We have performed eld dependent measurements of the3 FIG. 3: Dependence of the second and rst harmonic trans- verse voltage amplitudes on the DC current density with jac xed at 1:51010A/m2. (a) Second harmonic transverse volt- age versus DC current. (b) First harmonic transverse voltage versus DC current. rst harmonic transverse voltage by sweeping the ex- ternal magnetic eld between -400 mT and +400 mT at dierent in-plane angles 'from 0to 360at xed jac= 1:51010A/m2andjdc=5:01010A/m2. Using these results, we have determined
V! xyusing the proce- dure mentioned in the preceding section and plot its vari- ation with'(Fig. 4). The SSE voltage is proportional to the projection of the magnetization on the axis perpen- dicular to the voltage probes. The temperature gradient is along the z-axis, whereas the spin polarization is along the YIG magnetization direction. Therefore the angular dependence of the rst harmonic and second harmonic transverse response are V! xy/2p 2jacjdccos(');V2! xy/ j2 accos('), as seen experimentally. Measurements of
V! xycan be tted well with
V! xy(0) cos', denoted by the solid line in Fig. 4. The second harmonic transverse voltage was measured with varying 'at a xed eld of +400 mT and a xed jac= 1:51010A/m2(Fig. 1(c)). Equation 2 predicts that for jac= 1:51010A/m2and jdc=5:01010A/m2, the relative magnitudes of the rst and second harmonic signals should be 9 :4. We have extracted the maximum
V! xy(0) = 0:1870:053V and
V2! xy(0) = 0:02270:0007V by tting the data in Fig. 4 and Fig. 1(d) respectively. The experimentally obtained ratio of the rst and second harmonic signals is 8:22:7. The experimentally obtained ratio is thus consistent with our simple AC and DC current heating model. The data in Fig. 4 clearly indicates that the rst harmonic response has a much higher signal-to-noise ra- tio than that of the second harmonic voltage (Fig. 1(c)). In summary, we have determined the SSE-produced linear and nonlinear voltage responses in a YIG/Pt bi- layer system. The second harmonic longitudinal voltage has a sine relation with respect to the in-plane eld angle 'when the YIG is saturated. Angular dependence mea-surement of the longitudinal and transverse voltages as a function of the applied eld magnitude enabled estima- tion of the contributions from SSE, DL, FL and Oe eld torques. It was found that the SSE dominates over the other contributions when the applied eld is sucient to saturate the YIG layer. In addition, by applying an AC current with DC bias, we determined that SSE can be measured by a rst harmonic lock-in technique, and can 0 90 180 270 360 φ(o)−0.2−0.10.00.10.2ΔVω xy FIG. 4: Angular dependence of
V! xymeasured with an AC current of 1 :51010A/m2and DC current 5 :01010A/m2. The curve is a t to the data of the form
V! xy(0)cos('). be more sensitive and have higher signal to noise than the conventional second harmonic metthod. This technique can be used to characterize the SSE in ferromagnetic (or ferrimagnetic) and non-magnetic bilayer systems as well as to study nonlinear thermoelectric eects and spin dy- namics induced by temperature gradients. Acknowledgements The instrumentation used in this research was support in part by the Gordon and Betty Moore Foundations EPiQS Initiative through Grant GBMF4838 and in part by the National Science Foundation under award NSF- DMR-1531664. This work was supported partially by the MRSEC Program of the National Science Foundation un- der Award Number DMR-1420073. ADK received sup- port from the National Science Foundation under Grant No. DMR-1610416. At CSU, lm growth was sup- ported by the U.S. National Science Foundation (EFMA- 1641989), and lm characterization was supported by the U.S. Department of Energy, Oce of Science, Basic En- ergy Sciences (DE-SC-0018994). [1] A. Brataas, A. D. Kent, and H. Ohno, Nat Mater. 11, 372 (2012). [2] L. Cornelissen, J. Liu, B. van Wees, and R. Duine, Phys. Rev. Lett. 120, 097702 (2018). [3] K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae,K. Ando, S. Maekawa, and E. Saitoh, Nature 455, 778 (2008). [4] A. D. Avery, M. R. Pufall, and B. L. Zink, Phys. Rev. Lett. 109, 196602 (2012). [5] M. I. Dyakonov and V. I. Perel, JETP Lett. 13, 4674 (1971). [6] J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999). [7] S. Zhang, Phys. Rev. Lett. 85, 393 (2000). [8] J.-C. Rojas-Sanchez, L. Villa, G. Desfonds, S. Gambarelli, J. P. Attane, J. M. De Teresa, C. Magen, and A. Fert, Nat. Commun. 4, 2944 (2013). [9] K. Shen, G. Vignale, and R. Raimondi, Phys. Rev. Lett. 112, 096601 (2014). [10] K. Uchida, H. Adachi, T. Ota, H. Nakayama, S. Maekawa, and E. Saitoh, Appl. Phys. Lett. 97, 172505 (2010). [11] M. Schreier, N. Roschewsky, E. Dobler, S. Meyer, H. Huebl, R. Gross, Sebastian T and B. Goennenwein, Appl. Phys. Lett. 103, 242404 (2013). [12] D. Qu, S. Y. Huang, J. Hu, R. Wu, and C. L. Chien, Phys. Rev. Lett. 110, 067206 (2013). [13] T. Kikkawa, K. Uchida, Y. Shiomi, Z. Qiu, D. Hou, D. Tian, H. Nakayama, X.-F. Jin, and E. Saitoh, Phys. Rev. Lett. 110, 067207 (2013). [14] C. O. Avci, K. Garello, M. Gabureac, A. Ghosh, A. Fuhrer, S. F. Alvarado, and P. Gambardella, Phys. Rev. B90, 224427 (2014). [15] Z. Wang, Y. Sun, M. Wu, V. Tiberkevich, and A. Slavin, Phys. Rev. Lett. 107, 146602 (2011). [16] N. Vlietstra, J. Shan, B. J. van Wees, M. Isasa, F. Casanova, and J. Ben Youssef Phys. Rev. B 90, 174436 (2014). [17] M. Hayashi, J. Kim, M. Yamanouchi, and H. Ohno, Phys. Rev. B 89, 144425 (2014). [18] C. O. Avci, K. Garello, J. Mendil, A. Ghosh, N. Blasakis, M. Gabureac, M. Trassin, M. Fiebig, and P. Gambardella, Appl. Phys. Lett. 107, 192405 (2015). [19] See the supplementary section. I. SUPPLEMENTAL MATERIALS: FIRST HARMONIC MEASUREMENTS OF THE SPIN SEEBECK EFFECT Separation of the Spin Seebeck Eect, anti-damping spin-orbit torque, eld-like torque and Oersted eld contributions In the main text we state that the origin for both the longitudinal and transverse second-harmonic signals for a 400 mT applied eld are due to the spin Seebeck ef- fect arising from current-induced Joule heating in the Pt strip. Here, we estimate the relative contributions to V2! xyandV2! xxfrom SSE, anti-damping spin-orbit torque (AD), eld-like torque (FL) and Oersted eld contribu- tions (Oe). Angular dependence measurements with applied mag- netic eld ranging from 2 mT to 400 mT are used to separate there contributions using the following rela- tions [S1,S2]: V2! xx(') =
V2!;' xx sin('+
') +
V2!;3' xx sin('+
') cos2('+
') +A0(S1)
V2!;' xx is the SSE and AD contributions and
V2!;3' xxis the FL and Oe contributions. A0is the oset of the signal. V2! xy(') =
V2!;' xy cos('+
') +
V2!;3' xy cos('+
') cos[2('+
')] +B0(S2)
V2!;' xy is the SSE and AD contributions and
V2!;3' xy is the FL and Oe contributions. B0is the oset of the signal. We plot
V2!;' xx=xyversus the inverse applied eld, 0H:
V2!;' xx=xy=kxx;' 0H+C' xx=xy(S3) to determine the slope kxx;'and intercept Cxx=xy . We then plot
V2!;3' xx=xyversus the inverse applied eld, 0H:
V2!;3' xx=xy=kxx;3' 0H+D3' xx=xy(S4) to determine the slope kxx;3'and intercept Dxx=xy . Fig. S1 (a) and Fig. S1 (b) show the angular depen- dence ofV2! xxandV2! xyversus applied magnetic eld rang- ing from 2 to 400 mT. When 0H= 400 mT, V2! xxand V2! xyt well to sin( ') and cos('), with negligible 3 '- contributions. For 0Hsmaller than 25 mT, clear 3 '- symmetry can be observed due to the non-negligible FL + Oe eects, indicating that the applied eld torque is com- parable to the AD, FL and Oe torques. By tting V2! xx Fig. S 1: Second harmonic measurements of V2! xxandV2! xy with dierent applied magnetic elds. The AC current den- sityjac= 1:51010A=m2and applied magnetic eld 0H ranges from 2 to 400 mT. Solid lines denotes the ts. (a) Angular dependence of V2! xx, data ts to equation (S1); (b) Angular dependence of V2! xy, data ts to equation (S2); (c) Field dependence of
V2!;' xx and
V2!;3' xx with the inter- ceptsC' xx= 0:59VandD' xx= 3:8103V; (d) Field dependence of
V2!;' xy and
V2!;3' xy with the intercepts D' xy = 22:2103VandD' xx= 2:1103V.5 with Eqn. S1 and V2! xywith Eqn. S2, we can extracted the relative SSE, AD, FL and Oe contributions.
V2!;' xx and
V2!;' xy are denoted as the '-contributions while
V2!;3' xx and
V2!;3' xy indicate the 3 '-contributions. Fig. S1(c) and Fig. S1(d) show the eld dependence of 'and 3'-contributions. The 'contributions are almostindependent of the applied magnetic elds, showing neg- ligible AD. However, the 3 'contributions are inversely proportional to the applied magnetic elds with negligi- ble intercepts. When the applied eld is in the range of 3 mT, the 3 'contributions surpass the 'contributions, showing increasing FL and Oe eects at low elds. [S1] N. Vlietstra, J. Shan, B. J. van Wees, M. Isasa, F. Casanova, and J. Ben Youssef, Phys. Rev. B 90, 17443 [S2] M. Hayashi, J. Kim, M. Yamanouchi, and H. Ohno, Phys. Rev. B 89, 144425 (2014).

2018-08-23

We present measurements of the spin Seebeck effect (SSE) by a technique that combines alternating currents (AC) and direct currents (DC). The method is applied to a ferrimagnetic insulator/heavy metal bilayer, Y$_3$Fe$_5$O$_{12}$(YIG)/Pt. Typically, SSE measurements use an AC current to produce an alternating temperature gradient and measure the voltage generated by the inverse spin-Hall effect in the heavy metal at twice the AC frequency. Here we show that when Joule heating is associated with AC and DC bias currents, the SSE response occurs at the frequency of the AC current drive and can be larger than the second harmonic SSE response. We compare the first and second harmonic responses and show that they are consistent with the SSE. The field dependence of the voltage response is used to characterize the damping-like and field-like torques. This method can be used to explore nonlinear thermoelectric effects and spin dynamics induced by temperature gradients.

First harmonic measurements of the spin Seebeck effect

1808.07813v1

Magnon Kerr e ect in a strongly coupled cavity-magnon system Yi-Pu Wang,1,Guo-Qiang Zhang,1,Dengke Zhang,1Xiao-Qing Luo,1Wei Xiong,1Shuai-Peng Wang,1Tie-Fu Li,2, 1,yC.-M. Hu,3and J. Q. You1,z 1Quantum Physics and Quantum Information Division, Beijing Computational Science Research Center, Beijing, 100193 China 2Institute of Microelectronics, Tsinghua National Laboratory of Information Science and Technology, Tsinghua University, Beijing 100084, China 3Department of Physics and Astronomy, University of Manitoba, Winnipeg R3T 2N2, Canada (Dated: December 8, 2016) We experimentally demonstrate magnon Kerr e ect in a cavity-magnon system, where magnons in a small yttrium iron garnet (YIG) sphere are strongly but dispersively coupled to the photons in a three-dimensional cavity. When the YIG sphere is pumped to generate considerable magnons, the Kerr e ect yields a perceptible shift of the cavity’s central frequency and more appreciable shifts of the magnon modes. We derive an analytical relation between the magnon frequency shift and the drive power for the uniformly magnetized YIG sphere and find that it agrees very well with the experimental results of the Kittel mode. Our study paves the way to explore nonlinear e ects in the cavity-magnon system. PACS numbers: 71.36. +c, 42.50.Pq, 76.50. +g, 75.30.Ds I. INTRODUCTION Hybridizing two or more quantum systems can harness the distinct advantages of di erent systems to implement quan- tum information processors (see, e.g., Ref.[1, 2]). Recently, a cavity-magnon system has attracted considerable attention [3– 9], because of the enhanced coupling between magnons in a yttrium iron garnet (YIG) single crystal and microwave pho- tons in a high-finesse cavity. This hybrid system involves magnon polaritons [10, 11]. Thus, a series of phenomena re- alized in other polariton systems [12, 13], including the Bose- Einstein condensation of exciton polaritons [14, 15] and the optical bistability in semiconductor microcavities [16], can be explored using the magnon polaritons. Based on the strongly coupled cavity-magnon system, coherent interaction between a magnon and a superconducting qubit was realized [17], and magnon dark modes in a magnon gradient memory [18] were utilized to store quantum information. When combined with spin pumping techniques, this cavity-magnon system provides a new platform to explore the physics of spintronics and to design useful functional devices [7, 9]. Potentially acting as a quantum information transducer, microwave-to-optical fre- quency conversion between microwave photons generated by a superconducting circuit and optical photons of a whisper- ing gallery mode supported by a YIG microsphere was also explored [19–22]. Furthermore, coherent phonon-magnon in- teractions relying on the e ect of magnetostrictive deforma- tion in a YIG sphere was demonstrated [23]. Now, a versatile quantum information processing platform based on the coher- ent couplings among magnons, microwave photons, optical photons, phonons, and superconducting qubits is being estab- lished. These authors contributed equally. ylitf@tsinghua.edu.cn zjqyou@csrc.ac.cnIn this paper, we report an experimental demonstration of the magnon Kerr e ect in a strongly coupled cavity-magnon system. The magnons in a small YIG sphere are strongly but dispersively coupled to the microwave photons in a three- dimensional (3D) cavity. When considerable magnons are generated by pumping the YIG sphere, the Kerr e ect gives rise to a shift of the cavity’s central frequency and yields more appreciable shifts of the magnon modes, including the Kit- tel mode [24], which holds homogeneous magnetization, and the magnetostatic (MS) modes [25–27], which have inhomo- geneous magnetization. We derive an analytical relation be- tween the magnon frequency shift and the pumping power for a uniformly magnetized YIG sphere and find that it agrees very well with the experimental results of the Kittel mode. In contrast, the experimental results of MS modes deviate from this relation, which confirms the deviation of the MS modes from homogeneous magnetization. To enhance the magnon Kerr e ect, the pumping field is designed to directly drive the YIG sphere and its coupling to the magnons is strengthened using a loop antenna. Moreover, this pumping field is tuned very o -resonance with the cavity mode to avoid producing any appreciable e ects on the cavity. Our paper is a convinc- ing study of a cavity-magnon system with magnon Kerr e ect and paves the way to experimentally explore nonlinear e ects in this tunable cavity-magnon system. II. EXPERIMENTAL SETUP The experimental setup is diagrammatically shown in Fig. 1. The 3D cavity is made of oxygen-free copper with inner dimensions of 44 :020:06:0 mm3and contains three ports labeled as 1, 2 and 3 (here ports 1 and 2 are used for transmission spectroscopy and port 3 is for loading the drive field). The frequency of the cavity mode TE 102that we use is !c=2=10:1 GHz. A samll YIG sphere of diameter 1 mm is glued on an inner wall of the cavity at the magnetic-field antinode of the TE 102mode (c.f., the magnetic-field intensityarXiv:1609.07891v2 [quant-ph] 7 Dec 20162 B xz y Room Temp. Amplifier HEMT Amplifier Attenuator Isolator Magnetic FieldYIG Sphere Magnet300K 50K 4K 500mK 100mK 22mK B0 YIG Sphere Cavity -23dB 60dB 10dB 10dB 6dB 3dB 10dBmax minSMA Connector Bc dMW VNA FIG. 1. (color online). Schematic of the experiment setup. The 3D cavity is placed in the uniform magnetic field created by a supercon- ducting magnet. Ports 1 and 2 are used for transmission spectroscopy and port 3 is for driving the YIG sphere via a superconducting mi- crowave line with a loop antenna at its end. The total attenuation of the input port is 99dB. The right part shows the magnetic-field distri- bution of the cavity mode TE 102. The bias magnetic field, the drive magnetic field and the magnetic field of the TE 102mode are mutually perpendicular at the site of the YIG sphere, where the magnetic field of the TE 102mode is maximal. distribution of this mode marked by colored shades in Fig. 1). We apply a static magnetic field generated by a superconduct- ing magnet to magnetize the YIG sphere. This bias magnetic field is tunable in the range of 0 to 1 T, so the given frequency of the Kittel mode (i.e., the ferromagnetic resonance mode) ranges from several hundreds of megahertz to 28 GHz. The cavity is placed in a BlueFors LD-400 dilution refrigerator under a cryogenic temperature of 22 mK. The spectroscopic measurement is carried out with a vector network analyzer by probing the transmission of the cavity. A drive tone supplied by a microwave source can directly drive the YIG sphere via a superconducting microwave line going through port 3. More- over, a loop antenna is attached to the end of the superconduct- ing microwave line near the YIG sphere (see Appendix A), so as to strengthen the coupling between the drive field and the YIG sphere. Here the driving magnetic field Bd, the bias mag- netic field B0(which is aligned along the hard magnetization axis [100] of the YIG sphere), and the magnetic field Bcof the TE 102mode are orthogonal to each other at the site of the YIG sphere. Also, a series of attenuators and isolators is used to prevent thermal noise from reaching the sample and the output signal is amplified by two low-noise amplifiers at the stages of 4 K and room temperature, respectively. III. STRONG COUPLING REGIME We first measure the transmission spectrum of the cavity containing the YIG sphere, without applying a drive field on the YIG sphere. The transmission spectrum as a function of the probe microwave frequency and B0are recorded by vector network analyzer [see Fig. 2(a)]. At the point where FIG. 2. (color online). (a) Transmission spectrum for the normal- mode splitting measured as a function of the bias magnetic field and the probe microwave frequency. The large anticrossing indicates strong coupling between the Kittel mode and the cavity mode TE 102. The small splittings are due to the MS modes coupled with the cavity mode. (b) Transmission spectrum at three values of the bias magnetic field. The curves are o set vertically for clarity. the Kittel mode is resonant with the cavity mode TE 102, a distinct anti-crossing of the two modes occurs, indicating strong coupling between them. Some other small splittings are due to the couplings between the cavity mode and the MS modes in the YIG sphere. The coupling strength be- tween the Kittel mode and the cavity mode TE 102is found to be gm=2=42 MHz from the magnon polariton splitting at the resonance point [see the red cure in Fig. 2(b)]. By fitting the measured transmission spectrum [8], the cavity- mode linewidth =2(1+2+int)=2and the Kittel- mode linewidth m=2are determined to be 2 :87 MHz and 24:3 MHz, respectively. Here 1(2) is the loss rate due to the port 1 (2) and intis due to the intrinsic loss of the cavity. The obvious increase in the Kittel mode damping rate compared with the previous work [4, 5] is due to the antenna close to the YIG sphere, which acts as an additional decay channel. Note that all the linewidths throughout the paper are defined as the full width at half maximum (FWHM). Because gm> ; m, the hybrid system falls in the strong-coupling regime with a cooperativity C4g2 m= m=101: IV . RESULTS AND ANALYSIS OF THE DISPERSIVE MEASUREMENT A. Dispersive measurement We tune the static bias magnetic field B0to 346:8 mT, yield- ing about 9.55 GHz for the frequency of the Kittel mode. As shown in Fig. 3(a), we first measure the transmission spec- trum of the cavity (i.e., the black curve) by tuning the fre- quency of the probe field, but without the drive field on the YIG sphere. The measured central frequency of the cavity mode is 10.1035 GHz, which has a frequency shift of about 3 MHz compared with the intrinsic frequency of 10.1003 GHz of the TE 102mode of an empty cavity. This cavity mode has a detuning
=2550 MHz from the Kittel mode. Because
>10gm, the coupled hybrid system is in the dispersive3 (a) Kittel mode-26 -27 -28 9.3 9.4 9.6 9.9 9.8 9.7 9.5Drive power : 1 1 dBm|S 21 | Drive Frequency (GHz)(b)-27 -29 -31 -33 -35|S 21 | 10.1035 GHz 10.1042 GHz 10.101 10.106 10.105 10.104 10.103 10.1020.7MHz MW Drive OFF MW Power : 1 1 dBm, Freq : 9.5915GHz Probe Frequency (GHz) (c) Increasing drive power|S 21 | -26 -27 -28 Drive Frequency (GHz)9.3 9.4 9.6 9.9 9.8 9.7 9.5MS mode 1 MS mode 22 2 2(dB) (dB) (dB) FIG. 3. (color online). (a) Central frequency shift of the cavity mode TE102when the drive field is on (red curve) and o (black curve), respectively. (b) Transmission spectrum of the cavity measured as a function of the drive-field frequency. The blue arrow indicates the response of the Kittel mode, whereas the orange and purple arrows indicate the MS modes 1 and 2, respectively. (c) Transmission spec- trum of the cavity measured as a function of the drive frequency by successively increasing the driving power. The probe field is fixed at 10:1035 GHz in both (b) and (c). regime. We then measure the transmission spectrum of the cavity by both tuning the frequency of the probe field and ap- plying a drive field on the YIG sphere in resonance with the Kittel mode. The measured red curve corresponds to the drive power of 11 dBm. This transmission spectrum has a central frequency of 10 :1042 GHz, with a frequency shift of about 0:7 MHz from the measured central frequency without apply- ing a drive field. Figures 3(b) and 3(c) show the measured transmission spec- tra by tuning the frequency of the drive field, where the fre- quency of the probe field is fixed at the central frequency of 10:1035 GHz of the cavity containing the YIG sphere. Theprobe field power is -129 dBm. The corresponding average cavity probe photon number can be estimated by [28] ¯n=1Pp ~!p[
2p+(=2)2]; (1) where Ppis the probe field power and
p=!p!c. In our ex- periment, it is measured that 1=2=0:70 MHz, i.e., 1=4. Also, the probe field frequency !pis tuned in resonance with the cavity mode TE 102. Then, the average cavity probe photon number is reduced to ¯ n=Pp=(~!p)1. Here the probe tone is chosen extremely weak, so as to avoid producing any ap- preciable e ects on the system. In Fig. 3(b), the power of the drive field is 11 dBm. It can be seen that when the frequency of the drive microwave field is resonant with the Kittel mode, the transmission coe cient has a large decrease at 9 :59 GHz (see the main dip indicated by a blue arrow), caused by the shift of the central frequency of the cavity mode. The dips indicated by orange and purple arrows correspond to two dif- ferent MS modes. In addition, we vary the power of the drive field from -5 to 10 dBm in Fig. 3(c) and observe two interest- ing features, i.e., when increasing the drive power, the main dip becomes deeper successively and it simultaneously shifts rightwards. This reveals that the Kittel mode has a blue shift with the increase in the drive power. The responses of MS modes are similar. B. Origin of the Kerr term For a YIG sphere uniformly magnetized by an external magnetic field along the zdirection, when the magnetization is saturated, the induced internal magnetic field includes the demagnetizing field [29] Hde=M=3 and the anisotropic field [30, 31] Han=(2Kan=M2)Mz, where M(Mx;My;Mz) is the magnetization, Mis the saturation magnetization and Kanis the first-order magnetocrystalline anisotropy constant of the YIG sphere. When both the Zeeman energy and the magnetocrystalline anisotropic energy are included (see Ap- pendix B), the Hamiltonian of the YIG sphere in the magnetic field B0is given by (setting ~=1) Hm= B0Sz0 2Kan M2VmS2 z; (2) where =2=28 GHz /T is the gyromagnetic ratio, 0is the vacuum permeability and Sz=MzVm= is a macrospin operator of the YIG sphere, with Vmbeing the volume of the YIG sample. The macrospin operator Szis related to the magnon operators via the Holstein-Primako transforma- tion [32]: Sz=Sbyb, where by(b) is the magnon creation (annihilation) operator. When including the drive field, the cavity mode, and the in- teraction between the cavity photon and the magnon, the total Hamiltonian of the coupled hybrid system is (see Appendix B) H=!caya+!mbyb+Kbybbyb +gm(ayb+aby)+ d(byei!dt+bei!dt); (3)4 where ˆ ay(ˆa) is the creation (annihilation) operator of the cav- ity photons at frequency !c,Kbybbybrepresents the Kerr ef- fect of magnons owing to the magnetocrystalline anisotropy in the YIG sphere, with K=0Kan 2=(M2Vm), d(i.e., the Rabi frequency) denotes the strength of the drive field, and !dis the drive field frequency. Thus, our experimental setup provides a strongly coupled cavity-magnon system with the magnon Kerr e ect, which is an extension of the cavity-magnon sys- tem without the nonlinear e ect [33]. Note that Kis inversely proportional to Vm, so the Kerr e ect can become important when using a small YIG sphere. C. Cavity and magnon frequency shifts Below we study the case of considerable magnons gener- ated by the drive field. Because the coupled hybrid system is in the dispersive regime, its e ective Hamiltonian can be written as (see Appendix C) He=" !c+g2 m
+2g2 m
2Khbybi# aya +" !mg2 m
+ 12g2 m
2! Khbybi# byb + 0 d(byei!dt+bei!dt);] (4) with the e ective Rabi frequency 0 dgiven by 0 d= 11 2(!c!d)g2 m
+2g2 m
2Khbybi d; (5) where
=!c!m. Due to the coupling between the cav- ity and the YIG sphere, the cavity frequency shifts from the intrinsic cavity mode frequency !cto!c+g2 m=
, with g2 m=
being the dispersive shift. The measured central frequency of 10.1035 GHz corresponds to !c+g2 m=
. When pumping the YIG sphere with a drive field, the magnon number hbybi increases. Then the cavity frequency has an additional blue shift of
c=(2g2 m=
2)Khbybidue to the Kerr e ect. Also, the Kerr e ect yields a blue shift to the magnon frequency,
m=(12g2 m=
2)KhbybiKhbybi. Both cavity frequency shift and magnon frequency shift due to the Kerr e ect have a similar trend depending on hbybi, which is related to the drive power. V . RELATION BETWEEN THE MAGNON FREQUENCY SHIFT AND THE DRIVE POWER In Fig. 4, we extract the Kerr-e ect-induced frequency shifts of the magnon as well as the central frequency shift of the cavity mode at each given drive power P. From Fig. 4(a), it is clear that both the Kittel-mode frequency shift and the cavity central frequency shift indeed have similar behaviors depending on the drive power, as predicted above. We also see that all the frequency shifts exhibit nonlinear dependence on the drive power. As given in Appendix D, we derive an an- alytical relation between the magnon frequency shift
mand 0 2 4 6 8 10010203040MagnonFrequencyShift(MHz) 0.00.40.81.21.6CavityFrequencyShift(MHz)Kittel mode (Left) Cavity mode (Right) 0 2 4 6 8 10010203040 MagnonFrequencyShift(MHz) DrivePower(mW)MS mode 1 MS mode 2 (a) (b)FIG. 4. (color online). (a) Frequency shift of the Kittel mode (blue square) and the central frequency shift of the cavity mode TE 102(red circle) measured at various values of the drive power. The blue fitting curve for the Kittel mode is obtained using Eq. (6). (b) Frequency shifts of MS mode 1 (orange up-triangle) and MS mode 2 (purple down-triangle) measured at various values of the drive power. The corresponding orange and purple fitting curves also are obtained us- ing Eq. (6). The frequency shifts of the Kittel mode, MS mode 1 and MS mode 2 are here referenced from 9.5526, 9.4758, 9.6174GHz, respectively. the drive power Pusing a Langevin equation approach, 
2 m+ m 22
mcP=0; (6) where cis a characteristic parameter reflecting the coupling strength of the drive field with the magnon mode. For the Kittel mode, we have already measured its linewidth m=2= 24:3 MHz. We use Eq. (6) to fit the experimental results of the Kittle mode. As shown in Fig. 4(a), the obtained theoretical (blue) curve fits very well with the experimental data, where c=(2)34:71024kg1m2. For the MS modes, we have two unknown parameters, the MS mode linewidth mand the parameter c. We manage to fit the experimental data in Fig. 4(b) with m=15 MHz and c=(2)31:351024kg1m2for MS mode 1 (orange curve), and with m=30 MHz and c=(2)361024kg1m2for MS mode 2 (purple curve). Note that the theoretical curves do not fit the experimental data of the MS modes so well as those of the Kittel mode, especially in the region around the threshold power [see the region of 1-3 mW in Fig. 4(b)]. In fact, as a collective mode of spins with a zero wavevector,5 the Kittel mode is the uniform precession mode with homo- geneous magnetization, whereas the MS modes are nonuni- form precession modes holding inhomogeneous magnetiza- tion and have a spatial variation comparable to the sample dimensions [26, 27, 31]. The appreciable deviations of the experimental data from the theoretical fitting curves are due to the inhomogeneous magnetization of the MS modes. Note that when the drive power is small, m
m, so Eq. (6) reduces to  m 22
mcP=0; (7) i.e., the magnon frequency shift depends linearly on the drive power in the small drive power limit. When the drive power becomes su ciently large,
m m, and then Eq. (6) reduces to
3 mcP=0: (8) It yields
m=(cP)1=3; (9) i.e., in the large drive power limit, the magnon frequency shift depends linearly on the cubic root of the drive power. These limit results are consistent with the previous work in Ref. [34], where there is a threshold power separating the small and large driving power regions. A dispersive measurement on the cavity transmission was implemented at room temperature in Ref. [35], but the cav- ity’s central frequency shift due to the magnon Kerr e ect was not observed. In Ref. [35], the drive field was applied on the cavity rather than the YIG. This is di erent from our setup in which the YIG sphere is directly pumped by the drive field and the nonlinear e ect of large-amplitude spin waves can be induced [36]. Moreover, choosing a suitable angle be- tween the external magnetic field and the crystalline axis is also important to observe the magnon Kerr e ect because the value of the Kerr coe cient Kstrongly depends on this an- gle [37]. In our case, the bias magnetic field B0is aligned along the hard magnetization axis [100] of the YIG sphere, which gives rise to the largest K. Furthermore, our exper- iment is implemented at a cryogenic temperature where the magnetocrystalline anisotropy constant Kan(so the Kerr coef- ficient K) is several times larger than that at room tempera- ture [3, 31]. These may be the reasons why appreciable Kerr eect was not observed in Ref. [35]. VI. CONCLUSION We have realized a strongly coupled cavity-magnon system with magnon Kerr e ect. By directly pumping the YIG sphere with a drive field, we have demonstrated the Kerr-e ect- induced central frequency shift of the cavity mode as well as the frequency shifts of the Kittel mode and MS modes. An an- alytical relation between the magnon frequency shift and the pumping power for a uniformly magnetized YIG sphere is de- rived, which agrees very well with the experimental results ofthe Kittel mode. In contrast, the experimental results of MS modes deviate from this relation owing to the spatial varia- tions of the MS modes over the sample. We can use this rela- tion to characterize the degrees of deviation of the MS modes from the homogeneous magnetization. Our setup can provide a flexible and tunable platform to further explore nonlinear eects of magnons in the cavity-magnon system. Moreover, this coupled hybrid system involves magnon polaritons. It can be used to explore a series of phenomena realized in other po- lariton systems [12, 13]. ACKNOWLEDGMENTS This work was supported by the National Key Re- search and Development Program of China (Grant No. 2016YFA0301200), the MOST 973 Program of China (Grant No. 2014CB848700), and the NSAF (Grant No. U1330201 and No. U1530401). C.M.H. was supported by the NSFC (Grant No. 11429401). Appendix A: CA VITY USED IN THE EXPERIMENT Figure. 5 shows the three-dimensional (3D) rectangular cavity used in our experiment. It has inner dimensions of 44:020:06:0 mm3and contains three ports. The Port 1 (2) is used for the probe field into (out of) the cavity and the Port 3 is used for inputting the drive field [Fig. 5(a)]. The drive antenna is placed just beside the YIG sphere [Figs. 5(b) and 5(c)] and it is connected to a superconducting microwave line that goes into the cavity via port 3. This makes it e cient to pump the YIG sphere with a drive field. FIG. 5. (a) The 3D cavity, which is made of oxygen-free copper and plated with gold. (b) The drive antenna, which is connected to a superconducting microwave line that goes into the cavity via port 3. (c) The small YIG sphere, which has a diameter of 1 mm and is placed near the antenna and glued on the inner wall of the cavity.6 Appendix B: HAMILTONIAN OF THE COUPLED HYBRID SYSTEM The hybrid system shown in Fig. 5 consists of a small YIG sphere coupled to a 3D rectangular cavity and driven by a microwave field. Its Hamiltonian can be written as (setting ~=1) H=Hc+Hm+Hint+Hd: (B1) Here Hc=!cayais the Hamiltonian of the cavity mode TE 102 used in our experiment, with !canday(a) being the frequency and creation (annihilation) operator of the cavity mode, re- spectively. When Zeeman energy, demagnetization energy and magnetocrystalline anisotropy energy are included, the Hamiltonian of the YIG sphere, which has a volume Vm, can be written as [29] Hm=Z VmM
B0d 0 2Z VmM
(Hde+Han)d(B2) where0is the magnetic permeability of free space, B0= B0ezis the static magnetic field applied in the zdirection which is aligned along the crystalline axis [100] of the YIG sphere in our experiment, Mis the magnetization of the YIG sphere, Hdeis the demagnetizing field induced by the static magnetic field, and Hanis the anisotropic field caused by the magnetocrystalline anisotropy in YIG. For a uni- formly magnetized YIG sphere, the induced demagnetizing field is [29] Hde=M=3, and the anisotropic field is [30] Han=(2Kan=M2)Mz, where only the dominant first-order anisotropy constant Kanis taken into account and Mis the saturation magnetization. Then, the Hamiltonian in Eq. (B2) becomes Hm=B0MzVm+0 6M2Vm+0Kan M2M2 zVm: (B3) The YIG sphere can act as a macrospin S=MVm=  (Sx;Sy;Sz), where =gB=~is the gyromagnetic ration [33], with gbeing the g-factor and Bthe Bohr magneton. With the macrospin operator introduced, the Hamiltonian Hmreads Hm= B0Sz+0Kan 2 M2VmS2 z: (B4) where we have neglected the constant term 0M2Vm=6. The interaction Hamiltonian between the macrospin and the cavity mode is Hint=gs(S++S)(ay+a)2gsSx(ay+a); (B5) where gsdenotes the coupling strength between the macrospin and the cavity mode, and SSxiSyare the raising and lowering operators of the macrospin, respectively. In our experiment, the YIG sphere (i.e., the macrospin) is directly pumped by a drive field with frequency !d. The interaction between the macrospin and the drive field is Hd= s(S++S)(ei!dt+ei!dt)4 sSxcos(!dt);(B6)where scharacterizes the coupling strength of the drive field with the macrospin. The macrospin operators are related to the magnon opera- tors via the Holstein-Primako transformation [32]: S+=p 2Sbyb b; S=byp 2Sbyb ; (B7) Sz=Sbyb; where Sis the total spin number of the macrospin opera- tor and by(b) is the creation (annihilation) operator of the magnon with frequency !m. For the low-lying excitations withhbybi=2S1, one has S+bp 2S, and Sbyp 2S. Then, the Hamiltonian in Eq. (B1) becomes H=!caya+!mbyb+Kbybbyb +gm(a+ay)(b+by) + d(b+by)(ei!dt+ei!dt);(B8) where!m= B020Kan 2S=(M2Vm) is the frequency of the magnon mode, K=0Kan 2=(M2Vm) is a coe cient characterizing the strength of the nonlinear magnon e ect, gm=p 2S gsdenotes the magnon-photon coupling strength, and d=p 2S sdenotes the coupling strength of the drive field with the magnon mode. In the rotating-wave approxima- tion, the Hamiltonian is reduced to H=!caya+!mbyb+Kbybbyb+gm(ayb+aby) + d(byei!dt+bei!dt):(B9) Note that because the YIG sphere contains a very large num- ber of spins, the condition hbybi=2S1 for the low-lying excitations can be easily satisfied [8], even when considerable magnons are generated by the drive field. Appendix C: EFFECTIVE HAMILTONIAN IN THE DISPERSIVE REGIME For convenience of calculations, we first transform the Hamiltonian Hin Eq. (B9) to a rotating reference frame with respect to the frequency of the drive field by the unitary trans- formation R1=exp(i!dayati!dbybt); (C1) i.e., H0=Ry 1HR 1iRy 1@R1 @t =!caya+!mbyb+Kbybbyb+gm(ayb+aby) + d(by+b)(!daya+!dbyb) =caya+mbyb+Kbybbyb+gm(ayb+aby) + d(by+b);(C2)7 withc(m)!c(m)!d. Here the coupled hybrid system is in the strong coupling regime, i.e., gm; m, where( m) is the decay rate of the cavity (magnon) mode. The Hamiltonian (C2) can be divided into two parts, H0=H0+HI, with the free part H0=caya+mbyb+Kbybbyb+ d(by+b); (C3) and the interaction part HI=gm(ayb+aby): (C4)Below we use a Fr ¨ohlich-Nakajima transformation to re- duce the Hamiltonian H0. It needs to find a unitary transfor- mation U=exp(V), where Vis an anti-Hermitian operator Vy=Vand satisfies [ H0;V]+HI=0. Up to the second order, the reduced Hamiltonian is given by H0 e=UyH0UH0+1 2[HI;V]: (C5) We choose V=1(aybaby)+2(aya). Then, [H0;V]+HI=h caya+mbyb+Kbybbyb+ d(by+b);1(aybaby)+2(aya)i +gm(ayb+aby) =1(cm)(ayb+aby)1Kh (2byb+1)ayb+aby(2byb+1)i 1 d(ay+a) +2c(ay+a)+gm(ayb+aby):(C6) In our experiment, we use a drive field to directly pump the YIG sphere, so as to generate considerable magnons. In this case, the mean-field approximation can be applied to the term bybin Eq. (C6). Because hbybi 1, Eq. (C6) can approxi- mately be written as [H0;V]+HIh 1(cm)21Khbybi+gmi (ayb+aby) +(1 d+2c)(ay+a): (C7) Using the relation [ H0;V]+HI=0, we get 1(cm)21Khbybi+gm=0; 1 d+2c=0;(C8) which give 1=gm cm2Khbybi =gm
2Khbybi; 2= d c1= d cgm
2Khbybi;(C9) where
=!c!m. Therefore, Vhas the form V=gm
2Khbybi(aybaby) d cgm
2Khbybi(aya):(C10) Also, we apply the mean-field approximation to the Kerr term in Eq. (C5). Then, the Hamiltonian (C5) becomes H0 eH0+1 2[HI;V]  c+g2 m
+2g2 m
2Khbybi aya+ mg2 m
+ 12g2 m
2! Khbybi byb + 0 d(by+b); (C11) with 0 d= 11 2cg2 m
+2g2 m
2Khbybi d: (C12) Finally, we further rotate the reduced Hamiltonian H0 eus- ing the unitary transformation R2Ry 1=exp(i!dayat+i!dbybt); (C13) which is the inverse transformation of R1in Eq. (C1). The derived Hamiltonian is given by He=Ry 2H0 eR2iRy 2@R2 @t = !c+g2 m
+2g2 m
2Khbybi aya + !mg2 m
+ 12g2 m
2! Khbybi byb + 0 d(byei!dt+bei!dt): (C14) This is the e ective Hamiltonian of the coupled hybrid sys- tem obtained in the dispersive regime [i.e., Eq. (4)]. In our experiment, the drive field is tuned to be in resonance with the magnon mode, !d=!mg2 m
+ 12g2 m
2! Khbybi: (C15)8 Appendix D: RELATION BETWEEN THE MAGNON FREQUENCY SHIFT AND THE DRIVE POWER With Hamiltonian (C2), we can obtain the quantum Langevin equations for the coupled hybrid system, da dt=icaigmb 2a; db dt=imbi(2Kbyb+K)b igmai d m 2b:(D1) Here we write the operator a(b) as a sum of the steady-state value and the fluctuation, i.e., a=A+aandb=B+b. It follows from Eq. (D1) that AandBsatisfy dA dt=icAigmB 2A; dB dt=imBi(2KjBj2+K)B igmAi d m 2B:(D2) From Eq. (C15), we have m!m!dg2 m=
KjBj2, because
gmin the dispersive regime. Also, c!c !d=
+m
. At the steady states for both AandB,dA=dt=0 and dB=dt=0. Then, it follows from Eq. (D2) that i
AigmB 2A=0; i KjBj2+g2 m
 BigmAi d m 2B=0:(D3) Eliminating Ain Eq. (D3), we get  KjBj2+g2 m
g2 m
i(=2)i m 2 B+ d=0: (D4) Because
and m, Eq. (D4) is reduced to  KjBj2i m 2 B+ d=0: (D5) Using Eq. (D5) and its complex conjugate expression, we ob- tain  KjBj22+ m 22 jBj2 2 d=0: (D6) In our experiment, the measured frequency shift of THE magnons is
m= 12g2 m
2 KhbybiKhbybi: (D7) Note thathbybi=jBj2for small fluctuation b, corresponding to the case with considerable magnons generated in the YIG sphere. With
mKjBj2andK 2 d=cP, where Pis the drive power and cis a constant coe cient, Eq. (D6) reads 
2 m+ m 22
mcP=0; (D8) which is the relation between the magnon frequency shift and the drive power [i.e., Eq. (6)]. [1] Z.-L. Xiang, S. Ashhab, J. Q. You, and F. Nori, Rev. Mod. Phys. 85, 623 (2013). [2] G. Kurizki, P. Bertet, Y . Kubo, K. Mølmer, D. Petrosyan, P. Rabl, and J. Schmiedmayer, Proc. Natl. Acad. Sci. U.S.A. 112, 3866 (2015). [3] H. Huebl, C. W. Zollitsch, J. Lotze, F. Hocke, M. Greifenstein, A. Marx, R. Gross, and S. T. B. Goennenwein, Phys. Rev. Lett. 111, 127003 (2013). [4] Y . Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Usami, and Y . Nakamura, Phys. Rev. Lett. 113, 083603 (2014). [5] X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, Phys. Rev. Lett. 113, 156401 (2014). [6] M. Goryachev, W. G. Farr, D. L. Creedon, Y . Fan, M. Kostylev, and M. E. Tobar, Phys. Rev. Appl. 2, 054002 (2014). [7] L. Bai, M. Harder, Y . P. Chen, X. Fan, J. Q. Xiao, and C.-M. Hu, Phys. Rev. Lett. 114, 227201 (2015). [8] D. Zhang, X.-M. Wang, T.-F. Li, X.-Q. Luo, W. Wu, F. Nori, and J. Q. You, npj Quantum Information 1, 15014 (2015). [9] C.-M. Hu, Physics in Canada 72, No. 2 76 (2016).[10] Y . Cao, P. Yan, H. Huebl, S. T. B. Goennenwein, and G. E. W. Bauer, Phys. Rev. B 91, 094423 (2015). [11] S. Kaur, B. M. Yao, J. W. Rao, Y . S. Gui and C.-M. Hu, Appl. Phys. Lett. 109, 032404 (2016). [12] I. Carusotto and C. Ciuti, Rev. Mod. Phys. 85, 299 (2013). [13] A. Kavokin, J. Baumberg, G. Malpuech, and F. Laussy, Micro- cavities (Oxford University Press, Oxford, 2007). [14] J. Kasprzak, M. Richard, S. Kundermann, A. Baas, P. Jeam- brun, J. M. J. Keeling, F. M. Marchetti, M. H. Szymaska, R. Andr, J. L. Staehli, V . Savona, P. B. Littlewood, B. Deveaud, and Le Si Dang, Nature (London) 443, 409 (2006). [15] H. Deng, H. Haug, and Y . Yamamoto, Rev. Mod. Phys. 82, 1489 (2010). [16] A. Baas, J. Ph. Karr, H. Eleuch, and E. Giacobino, Phys. Rev. A69, 023809 (2004). [17] Y . Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Yamazaki, K. Usami, and Y . Nakamura, Science 349, 405 (2015). [18] X. Zhang, C.-L. Zou, N. Zhu, F. Marquardt, L. Jiang, and H. X. Tang, Nat. Commun. 6, 8914 (2015).9 [19] A. Osada, R. Hisatomi, A. Noguchi, Y . Tabuchi, R. Yamazaki, K. Usami, M. Sadgrove, R. Yalla, M. Nomura, and Y . Naka- mura, Phys. Rev. Lett. 116, 223601 (2016). [20] R. Hisatomi, A. Osada, Y . Tabuchi, T. Ishikawa, A. Noguchi, R. Yamazaki, K. Usami, and Y . Nakamura, Phys. Rev. B 93, 174427 (2016). [21] J. A. Haigh, S. Langenfeld, N. J. Lambert, J. J. Baumberg, A. J. Ramsay, A. Nunnenkamp, and A. J. Ferguson, Phys. Rev. A 92, 063845 (2015). [22] X. Zhang, N. Zhu, C.-L. Zou, and H. X. Tang, Phys. Rev. Lett. 117, 123605 (2016). [23] X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, Sci. Adv. 2, e1501286 (2016). [24] C. Kittel, Phys. Rev. 73, 155 (1948). [25] L. R. Walker, Phys. Rev. 105, 390 (1957). [26] L. R. Walker, J. Appl. Phys. 29, 318 (1958). [27] P. C. Fletcher and R. O. Bell, J. Appl. Phys. 30, 687 (1959). [28] M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Rev. Mod. Phys. 86, 1391 (2014).[29] S. Blundell, Magnetism in Condensed Matter (Oxford Univer- sity Press, Oxford, 2001). [30] J. R. Macdonald, Proc. Phys. Soc. A 64, 968 (1951). [31] D. D. Stancil and A. Prabhakar, Spin Waves (Springer, Berlin, 2009). [32] T. Holstein and H. Primako , Phys. Rev. 58, 1098 (1940). [33] O. O. Soykal and M. E. Flatte, Phys. Rev. Lett. 104, 077202 (2010). [34] Y . S. Gui, A. Wirthmann, N. Mecking, and C.-M. Hu, Phys. Rev. B 80, 060402 (2009). [35] J. A. Haigh, N. J. Lambert, A. C. Doherty, and A. J. Ferguson, Phys. Rev. B 91, 104410 (2015). [36] A. F. Cash and D. D. Stancil, IEEE Trans. Mag. 32, 5188 (1996). [37] A. G. Gurevich and G. A. Melkov, Magnetization Oscillations and Waves (CRC, Boca Raton,FL, 1996).

2016-09-26

We experimentally demonstrate magnon Kerr effect in a cavity-magnon system, where magnons in a small yttrium iron garnet (YIG) sphere are strongly but dispersively coupled to the photons in a three-dimensional cavity. When the YIG sphere is pumped to generate considerable magnons, the Kerr effect yields a perceptible shift of the cavity's central frequency and more appreciable shifts of the magnon modes. We derive an analytical relation between the magnon frequency shift and the drive power for the uniformly magnetized YIG sphere and find that it agrees very well with the experimental results of the Kittel mode. Our study paves the way to explore nonlinear effects in the cavity-magnon system.

Magnon Kerr effect in a strongly coupled cavity-magnon system

1609.07891v2

arXiv:2107.06508v3 [cond-mat.mtrl-sci] 13 Sep 2021Sublattice spin reversal and field induced Fe3+spin-canting across the magnetic compensation temperature in Y 1.5Gd1.5Fe5O12rare-earth iron garnet Manik Kuila,1Jose Mardegan,2Akhil Tayal,2Sonia Francoual,2and V.Raghavendra Reddy1,∗ 1UGC-DAE Consortium for Scientific Research, University Cam pus, Khandwa Road, Indore 452001, India. 2Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, 2 2607 Hamburg, Germany. (Dated: September 14, 2021) In the present work Fe3+sublattice spin reversal and Fe3+spin-canting across the magnetic compensation temperature (T Comp) are demonstrated in polycrystalline Y 1.5Gd1.5Fe5O12(YGdIG) by means of in-field57FeM¨ossbauer spectroscopy measurements. Corroborating in-fiel d57Fe M¨ossbauer measurements, both Fe3+& Gd3+sublattice spin reversal has also been manifested with x-ray magnetic circular dichroism (XMCD) measurement in hard x-ray region. Moreover from in-field57FeM¨ossbauer measurements, estimation and analysis of effective internal hyperfine field (Heff), relative intensity of absorption lines in a sextet elucid ated unambiguously the signatures of Fe3+spin reversal, their continuous transition and field induce d spin-canting of Fe3+sublattices across T Comp. Further, Fe K- (Gd L 3-) edge XMCD signal is observed to consist of additional spec - tral features, those are identified from Gd3+(Fe3+) magnetic ordering, enabling us the extraction of both the sublattices (Fe3+& Gd3+) information from a single edge analysis. The evolution of the magnetic moments as a function of temperature for both ma gnetic sublattices extracted either at the Fe K- or Gd L3-edge agree quite well with values that are extracted from bulk magnetization data of YGdIG and YIG (Y 3Fe5O12). These measurements pave new avenues to investigate how the magnetic behavior of such complex system acts across the compensation point. I. INTRODUCTION Rare-earth iron garnets (RIG), R 3Fe5O12, where R is rare-earth (Y, & La-Lu) have become an important class of ferrimagnetic oxide materials finding a signif- icant role in many microwave, bubble memories and magneto-optical device applications [1–4]. Remarkable intriguing magnetic properties and their chemical sta- bility with a large variety of elemental substitutions are one of the prime reasons for exploring these materials by various groups since their discovery [5–11]. Among theseRIGsystems, yttrium irongarnet(Y 3Fe5O12, YIG) received a renewed attention in recent years owing to its low damping, low optical absorption, magneto-optical switching, thermoelectric generation in spin Seebeck in- sulators and other spintronics, magneonic based applica- tion [12–15]. Doping of magnetic light rare-earths (e.g., Ce, Nd andGd) orevenheaviermetals, suchasBiatoms, on the yttrium sites yield a significantly enhanced visi- ble to near-infraredFaraday /Kerrrotation and magneto- optical (MO) figure of merit, without losing their mag- netic insulator character [16–20]. RIG consist of three different magnetic ions viz., two in-equivalent Fe3+ions located at tetrahedral ( d-) and oc- tahedral ( a-) oxygen polyhedron and the third one is the R3+ion situated at dodecahedral ( c) oxygen polyhedron [5]. The d- anda- site Fe3+are always coupled anti- parallel and the resultant moment of Fe3+is also coupled anti-parallel with the R3+. These two magnetic sublat- tices exhibit quite a different temperature dependance and as a result there exists a point in temperature at ∗varimalla@yahoo.com; vrreddy@csr.res.inwhich the resultant Fe3+magnetic moments are equal and opposite to the R3+magnetic moments resulting in zero total magnetization [6], known as magnetic com- pensation temperature (T Comp). Therefore, a thorough understanding of the macroscopic magnetic properties in RIG compounds is achieved from the knowledge of the different sublattices magnetization [21–27]. Usually at low temperatures i.e., T <TCompthe R3+ moments dominate the macroscopic magnetic prop- erties and align along the externally applied mag- netic field (H ext), whereas for intermediate temperature, TComp<T<TC(Curie temperature) it is the resultant Fe3+moments which will dominate and align along the Hext. The net magnetization (M) always align along the Hext. Therefore, the magnetic sub-lattices rearrange ac- cordingly whether the temperature of the system is be- low or above the T Compas shown schematically in Fig- ure. 1. Now, if the R-atom is non-magnetic (like as in YIG systems) or for temperatures above the T Comp, the dominant Fe3+atd- sites decide the resultant magneti- zation direction, whereas for temperatures below T Comp, thed- site Fe3+moments will be in opposite direction to Hext. However, the competition between H extthat always tend to align all the moments parallel to it and the strong anti-ferromagnetic super-exchange interaction between the sublattices result field induced spin-canted phase (between magnetic R3+and resultant Fe3+) close to TComp, which is shown to be a second-order phase transition in literature [21–23]. In this context, the in-field57FeM¨ossbauer spec- troscopy is an ideal method to track the evolution of Fe3+sublattice magnetization, demonstrate their inver- sion acrossT Compand also probe the spin-canting unam- biguously by analyzing the variation of effective internal field (H eff), which is the vector sum of internal hyper-2 FIG. 1. Schematic diagram showing the relation between in- ternal (H int), effective (H eff) hyperfine and externally applied (Hext)fieldsbelowandabovemagneticcompensation (T Comp) temperature. M’s represent the magnetic moments and the corresponding arrows are direction of the moments of the re- spective sublattice. fine field (H int), Hext& the relative line intensities in a given six-line pattern [28, 29]. However, there seems to be no systematic in-field M¨ ossbauer study across T Comp in RIG systems in literature. Most of the M¨ ossbauer lit- erature focused on the estimation of cation distribution in RIG systems, except the work of Stadnik et al., and Seidel et al [30, 31]. Seidel et al., reported that there is no spin-canting in Gd 3Fe5O12across T Compwith zero- field M¨ossbauer measurements[30] and Stadnik et al., re- ported canting of the Fe3+spins in Sc doped Eu 3Fe5O12 with in-field M¨ ossbauer measurements [31]. The in-field57FeM¨ossbauer measurements reported in this paper, are further complimented by x-ray mag- netic circular dichroism (XMCD) measurements. In RIG systems, most of the XMCD investigation is focused on the transition metal (TM) L 2,3(2p→3d) and R- M 4,5 (3d→4f) absorption edges, since in both cases the mag- netism is directly probed via electric-dipole transitions [24, 25]. Recently studies have shown that due to signifi- cant improvement in the 3rd generation synchrotron, the XMCD method is also possible to be carried out with much higher signal to noise ratio even at hard x-rays regime i.e., across the Fe-K and R L-edges [32]. Fur- ther it is shown in the previous literature that Fe K-edge (R L-edge) XMCD signal is strongly influenced by the R-atom (Fe-atom) magnetic ordering in rare-earth ironcompounds such as RFe 2, NdFeB and RIGs. Therefore, by performing XMCD measurements in hard x-ray re- gion across a single edge, one would be able to get the element specific magnetic information from both Fe and R sublattices in these type of compounds [21, 22, 33–35]. The present work reports the temperature dependent in-field57FeM¨ossbauer spectroscopy and XMCD mea- surements on polycrystalline Y 1.5Gd1.5Fe5O12(YGdIG) and Y 3Fe5O12(YIG) samples with the aim of tracing sublattice spin across T Comp, looking for the possible spin-canting across T Compand decomposition of XMCD data into Gd-like and Fe-like spectra from a single edge (i.e., Gd L 3-edge or Fe K-edge) measurement. II. EXPERIMENTAL Polycrystalline Y 1.5Gd1.5Fe5O12and Y 3Fe5O12sam- ples are prepared with conventional solid-state-reaction method starting with high purity ( ≥99.9%) oxide pre- cursors. The structural characterization of the prepared samples is carried out with Brucker D8-Discover x-ray diffraction system equipped with LynxEye detector and Cu-Kαsource.57FeM¨ossbauer measurements are car- riedout in transmissionmode usinga standardPC-based M¨ossbauer spectrometer equipped with a WissEl veloc- ity drive in constant acceleration mode. The velocity calibration of the spectrometer is done with natural iron absorber at room temperature. For the low temperature highmagneticfield M¨ ossbauermeasurements,the sample is placed inside a Janis superconducting magnet and an Hextis appliedparalleltothe γ-rays. Bulkmagnetization measurementsare performed with vibrating sample mag- netometer (VSM). X-ray absorption spectroscopy (XAS) measurements are carried out at beamline P09 at PE- TRA III (DESY) at low temperature at the Gd L 3and Fe K-absorptionedges in transmissiongeometryin which both incident and transmitted x-ray beams are recorded using silicon photo-diodes. The pellets, prepared from sampleswithsuitableamountofboronnitrateadd-mixer, are cooled down by an ARS cryostat with temperature range between 5 to 300 K. XAS /XMCD measurements were performed fast-switching the beam helicity between /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 /s32/s89/s99/s97/s108 /s99 /s32/s89/s111/s98/s115/s45/s89/s99/s97/s108 /s99 /s32/s66/s114/s97/s103/s103/s95/s112/s111/s115/s105/s116/s105/s111/s110/s89/s73/s71/s40/s97/s41 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48/s89/s71/s100/s73/s71 /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 /s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s32/s89 /s71/s100/s73/s71 /s32/s89 /s73/s71 /s84 /s67/s111/s109/s112 FIG. 2. (a) X-ray diffraction data of Y 1.5Gd1.5Fe5O12and Y3Fe5O12(b) M-T data measured in field-cooled protocol with 500 Oe field. The magnetic compensation (T Comp) is clearly seen for YGdIG as indicated by black dash line.3 left and right circular polarization to improve the signal- to-noise ratio. The dichroic signal is revealed when the difference in the absorption spectra for left and right cir- cularly polarized radiation is performed [36]. In order to align the domains and to correct for nonmagnetic arti- facts in the XMCD data, an external magnetic field of approximately 1 T was applied parallel and antiparallel to the incident beam wave vector→ k. III. RESULTS A. Structural and magnetic characterization Figure. 2(a) shows the XRD patterns of the YIG and YGdIG samples and the data confirms the phase purity of the prepared samples. Further, the XRD data is an- alyzed with FullProf Rietveld refinement [37] program considering the Ia-3dspace-group for the estimation of lattice parameters and the obtained lattice parameters are 12.374(1) ˙Aand 12.424(1) ˙Afor YIG and YGdIG, respectively, which also match with the literature [38]. Figure. 2(b) shows the temperature dependent (M-T) magnetizationdataoftheYIG andYGdIG samples. One can clearly see the magnetic compensation temperature (TComp)forYGdIGatabout126Kandnosuchsignature is seen for YIG as expected. Since, the bulk magnetiza- tion data of YIG is considered to be coming only from Fe3+moments this gives a rough idea about the con- tribution of Fe3+sublattice to the total magnetization in YGdIG. So, one can get an estimated contribution of Gd3+sublattice tothe overallmomentin YGdIG bysuit- ably subtracting the bulk magnetization data of YGdIG from YIG data. This method is employed to cross-check the reliability of the XMCD data analysis results as dis- cussed in the later sections. /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 /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 /s40/s100 /s45/s115/s105/s116/s101/s41 /s32/s70/s101/s51/s43 /s40/s97 /s45/s115/s105/s116/s101/s41 /s49/s53/s48/s32/s75/s49/s50/s55/s32/s75/s49/s49/s54/s32/s75/s53/s32/s75 FIG. 3.57FeM¨ossbauer data of Y 1.5Gd1.5Fe5O12at the se- lected temperatures. Black symbols represent the experime n- tal data and the black solid line is the best fit to the data.B.57FeM¨ossbauer spectroscopy results Figure. 3 shows the temperature dependent57Fe M¨ossbauer spectra of YGdIG measured across T Comp. One can clearly see two components corresponding to Fe3+sublattices located at d- anda- sites and the ob- tained hyperfine parameters match with literature val- ues of garnet [30, 40, 41]. Values of H int, area fraction and A 23are shown in Table-1. Further, the area frac- tion of Fe3+atd- anda- sites is found to remain same at all the temperatures and the values of A 23(area ra- tio of second and third lines in a given sextet) is fixed as 2.0 in accordance with the random distribution of θ corresponding to the powder samples measured in zero- field conditions [28]. However, the in-field M¨ ossbauer measurements across T Compwith 5 T external magnetic field applied parallel to the γ-rays, as shown in Figure. 4, FIG. 4. In-field57FeM¨ossbauer data of Y 1.5Gd1.5Fe5O12at the selected temperatures under 5 T magnetic field applied parallel to the γ-rays. Black symbols represent the exper- imental data and the black solid line is the best fit to the data. Blue arrow represents Fe phase1aligned anti-parallel to Hext, orange arrow represents Fe phase2aligned parallel to H ext and the red arrow represents the resultant of these two. H ext is the applied external field which is parallel to the inciden t γ-rays.4 TABLE I. Hyperfine parameters obtained from the analysis of zero field57FeM¨ossbauer spectroscopy data as shown in Figure. 3. δis centre shift, ∆E is quadrupole shift, H intis the internal hyperfine field, A 23is the area ratio of second and third lines of the respective sextet. T (K)δ(mm/s) ∆E(mm/s) H int(T) A 23% Area Fe3+ ±0.01 ±0.02 ±0.1 ±2 Site 5 0.29 0.01 49.5 2.00 58 d 0.57 0.11 56.9 2.00 42 a 116 0.27 0.01 48.5 2.00 58 d 0.56 0.10 56.4 2.00 42 a 125 0.26 -0.01 48.3 2.00 57 d 0.56 0.13 56.4 2.00 43 a 127 0.26 0.02 48.4 2.00 57 d 0.55 0.08 56.3 2.00 43 a 136 0.26 0.01 48.4 2.00 58 d 0.54 0.09 56.3 2.00 42 a 150 0.25 0.01 48.2 2.00 59 d 0.55 0.08 56.3 2.00 41 a TABLE II. Representative hyperfine parameters obtained from the analysis of in-field57FeM¨ossbauer spectroscopy data as shown in Figure. 4. H effis the effective field which is the vector sum of H ext, Hintand demagnetizing fields [39]. A23is the area ratio of second and third lines of the respec- tive sextet. Fe phase1denotes the phase aligned anti-parallel to Hextand Fe phase2denotes the phase aligned parallel to H ext as shown in Figure. 4. T (K) H eff(T) A 23% Area Fe3+Phase ±0.1±0.1 ±2 Site 5 53.2 0.00 58 dFePhase1 54.9 0.00 42 aFePhase1 130 51.6 1.22 25 dFePhase1 53.8 1.22 17 aFePhase1 45.3 1.22 34 dFePhase2 61.1 1.22 24 aFePhase2 150 43.6 0.10 59 dFePhase2 61.1 0.10 41 aFePhase2 reveal a very interesting information regarding the rever- sal of Fe3+moments, spin-canting etc., as elaborated in section-IV. C. Hard x-ray magnetic circular dichroism (XMCD) results Figures. 5 and 6 show the XMCD data measured at the Fe K- and Gd L 3- edges of YGdIG sample at differ- ent temperatures. For the comparison purpose, the Fe K-edge XMCD data of YIG sample is also shown in Fig- ure. 5. As a representative, XAS data collected at the FeK-edge and at the Gd L 3- edges for the YGdIG sample are also shown in Figure. 5 (a) and 6 (a), respectively. A clear reversal of XMCD signals at Fe K- and Gd L 3- edges are observed in YGdIG sample when measured at 5 and 290 K i.e., far below and above the T Comp. Addi- tional features that are only observed in the Fe K-edge spectra of YGdIG in between 7141 and 7154 eV photon energy, could be due to induced signal of Gd magnetic moments. It may be noted that such a clear signal of rare-earth contribution in Fe K-edge XMCD data was not detected previously in RIG systems, even though the opposite case i.e., Fe contribution in rare-earth L-edge XMCD is observed in many RIG systems [22, 42–44]. However, unlike RIG systems, the rare-earth contribu- tion wasobservedat Fe K-edgeXMCD data in rare-earth transition metal intermetallics (RTI), which could be due to the fact that the R(5d) electronic orbitals hybridize with the outermost states of absorbing Fe(3d) in RTI systems whereas it is mediated via oxygen in case of RIG systems [21, 33, 34]. The direct subtraction method from a reference spectrum (YIG, where Y nonmagnetic at c- site) is employed to separate out the Gd contribution at every temperature, as shown in Figure. 5(i)-(p), and is further discussed in the following sections. Gd L3- edge XMCD spectrum can broadly be charac- terized by the structures labeled as P1, P2, and P3 (Fig- ure. 6). The peak P1, in the lower energy region that is eventuallyburied in the profile ofpeak P2 at low temper- atures, can be ascribed to the Fe contribution that be- comes visible at temperatures higher than T Compwhere Gd magnetic moments are weak. Whereas, P2 and P3 mainly originate from the Gd electronic states [42–44]. The Gd L 3- edge XMCD spectra are analyzed with sin- gular value decomposition (SVD) method to extract the Fe3+contribution as discussed in the following section. IV. DISCUSSIONS The in-field57FeM¨ossbauer data (Figure. 4) is ana- lyzed considering two Fe sites at temperatures well below and above T Comp(5, 100 and 150 K) which correspond tod- anda- sites of Fe3+. Representative H effvalues for someofthetemperaturesareshowninTable-II..H extwill be added (subtracted) to the H intofa- site above (be- low) the T Compand it will be vice versa for the d- site as shown schematically in Figure. 1. Hence, the H extfor the two spectral components corresponding to d- anda- sites will be very close to each other below T Comp, whereas significant difference will be observed above T Comp. As a result of this, one would observe well resolved two sex- tets corresponding to d- anda- sites above T Compand a single broad sextet due to the overlapping components below T Compas shown in Figure. 4, specifically at tem- peratures of 150 K and 5 K. Therefore, the obtained H eff values which is the vector sum of H intand H extunam- biguously demonstrates the reversal of Fe3+sublattice across T Compin YGdIG, which can be generalized to all5 /s32/s89/s71/s100/s73/s71/s32/s88/s65/s83 FIG. 5. (a)-(h) Temperature variation of Fe K-edge X-ray mag netic circular dichroism data of Y 1.5Gd1.5Fe5O12(YGdIG) and Y3Fe5O12(YIG) at the indicated temperatures. (i)-(p) shows the enla rged view of region of interest depicting the development of Gd contribution at low temperatures. The Gd contribution is extracted by subtracting the YGdIG data from YIG data and its temperature variation is shown in Figure. 9 /s32/s88/s65 /s83/s32/s100/s97/s116/s97 /s88/s65/s83/s32/s40/s65/s114/s98/s46/s32/s85/s110/s105/s116/s115/s41 FIG. 6. Temperature variation of Gd L 3-edge X-ray magnetic circular dichroism data of Y 1.5Gd1.5Fe5O12. Symbols repre- sent the experimental data, and red lines are reconstructed spectrausingthefirsttwodominating contributionswhich a re shown by shaded areas. Deep green represents Gd- contribu- tion and cyan associated with the Fe- contribution. Labelin g of the peaks and the data fitting carried out using singular value decomposition (SVD) method are discussed in the text. X-ray absorption spectra of the sample measured at 5 K is also shown as a representative data.RIG systems exhibiting magnetic compensation. However, an inspection of the in-field M¨ ossbauer data closeto T Comprevealthe presenceofmorethan twocom- ponents. This is very clearly seen at 127, 130, 136 K and a careful inspection also reveals the presence of these components at 116 and 120 K as shown in Figure. 4. The spectra could be resolved into four sites correspond- ing to four Fe3+sublattices of two ferrimagnetic phases of YGdIG i.e., the phases with resultant Fe3+moment aligned along (Fe phase1) and opposite (Fe phase2) to H ext acrossT Comp. Asmentionedintheprecedingsection, the significant contrast between these two phases in terms of Heffenable the quantitative study of their evolution with temperature. In view of this, the data at these temper- atures is fitted with four sextets and the obtained phase fractions of these two phases (Fe phase1, Fephase2) is plot- ted in Figure. 7. As the temperature is changing, one can see that there is a gradual change in the area fraction of these phases indicating a continuous spin reversal across TComp. In addition to this information, the inspection of in- field M¨ossbauer data close to T Compreveal the presence of considerable intensity for the second and fifth lines (corresponding to ∆m=0 transitions) of a given sextet unlike the data of 5, 100 and 150 K i.e., well below and above T Comp. Quantitatively this is estimated by A 23 parameter for given sextet and the value of A 23is found to be zero at temperatures 5, 100 and 150 K indicat- ing a collinear magnetic structure of Fe3+[28]. How- ever, the in-field M¨ ossbauer data close to T Compis fit- ted keeping A 23parameter as variable and is constrained to be same for all the sites. The obtained variation of6 A23as a function of temperature is also shown in Fig- ure. 7 and it is interesting to note that the magnetic structure deviates from collinear configuration as one ap- proaches T Comp. This is considered to be a signature of spin-cantingasshownrecentlyfromthemagneticcircular dichroism [22] and spin Hall magnetoresistance [45] ex- periments. Considering two-sublattice model (Fe3+and R3+sublattices), simulated magnetic field versus tem- perature (H-T) phase diagram of compensated RIG sys- temsclearlyshowtheregionofcollinear,cantedferrimag- netic and aligned phases [21–23, 45]. The present in-field M¨ossbauer study indicates the presence of such canted phases across T Compunambiguously. Across this region of temperatures, the two Fe phase1and Fe phase2phases are mixed up with forming similar but opposite angle at a given temperature with respect to H ext. As a conse- quence, resultant Fe moment exhibits a continuous ro- tation across this canted region in accordance with the two sublattice model as shown schematically in Fig. 4. It may be emphasized here that even the XMCD data might not be able to distinguish the mixing of these two components as it measures resultant projection of the Fe magnetic moments along the beam direction. Apart from showing a clear reversal of the magnetic signals at the Fe K- and the Gd L 3- edges in YGdIG acorss the T Comp, the XMCD spectra reported in Fig- ure. 5and 6 are employed to extract quantitative infor- /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 /s40/s99/s41/s65 /s50/s51 /s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s32/s32/s70/s101 /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 /s40/s98/s41/s77/s32/s40/s101/s109/s117/s47/s103/s109/s41/s32/s70/s67 /s84 /s67/s111 /s109/s112/s40/s97/s41 /s48/s53/s48/s49/s48/s48 /s32/s70/s101 /s80 /s104 /s97 /s115/s101 /s50 /s37/s32/s65/s114/s101/s97 FIG. 7. (a) M-T data, reproduced from Figure. 2(b). Tem- perature variation of (b) Phase fraction (c) area ratio of se c- ond and third line intensities (A 23) as obtained parameters from the analysis of in-field57FeM¨ossbauer data as shown in Figure. 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 /s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s32/s71/s100/s32/s108/s105/s107/s101 /s32/s70/s101/s32/s108/s105/s107/s101 FIG. 8. Eigen or basis spectra of Gd-like and Fe-like com- ponents calculated by singular value decomposition (SVD) method for the Gd L 3-edge XMCD data set. mation about both the sublattices magnetization from a single absorption edge measurement. Mainly two ap- proachesareusedinliteraturetoextractthespectralcon- tributions of different sublattices from such complicated XMCD data [33, 44]. Subtraction method in which the magnetic dichroic signal from the investigated system is subtracted from a reference sample has been extensively exploited and has provided very reliable interpretation [33].In our present work, XMCD spectra obtained at the Fe K-edge for the YIG sample are considered as reference to estimate the magnetic contribution from the Gd ions at the Fe K-edge. Using this simple methodology, the Gd magnetic contribution can be obsevred via XMCD at the Fe K-edge as shown in Figure. 5(i)-(p). In order to check the reliability of thus obtained Gd3+ contribution, the total area of these features observed at the Fe K post-edge are compared with the temperature variation of Gd3+moment, as calculated from tempera- ture dependent bulk magnetization data of YGdIG and YIG.The results using this approach are summarized in Fig. 9(a). The bulk magnetization data of YIG is consid- ered to have its origin basically from the Fe3+ions since wecanassumethatbothY3+andO2-ionshavenegligible magnetic moments and their hybridization with the Fe3+ ions does not play an important role to the total mag- netism of the system. Therefore, subtracting bulk mag- netization data of YGdIG from the YIG data (Fig. 2), results magnetic signal which is primarily from the Gd3+ ions. The sign of the obtained sublattice magnetization asafunctionoftemperatureischangedaccordingtotheir magnetization direction w.r.t H extacross T Comp(Fig. 9 dashed line). The extracted Gd3+XMCD amplitude (blue dot Fig. 9(a)) from Fe K edge XMCD data (us- ing direct subtraction method) almost follow the trend of Gd3+moment variation as a function of temperature belowthe T Comp. However,aboveT CompGd3+magnetic contribution to Fe K-edge XMCD data is weaker and7 /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 /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 /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 /s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101 /s40/s75/s41/s32/s89 /s71/s100/s73/s71 /s32/s70/s101/s32/s109/s111/s109/s101/s110/s116/s32/s40/s89 /s73/s71/s41 /s32/s71/s100/s32/s109/s111/s109/s101/s110/s116/s32 /s40/s89 /s71/s100/s73/s71/s45/s89 /s73/s71/s41 /s40/s97/s41 /s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s32/s70/s101 /s88 /s77/s67 /s68 /s70/s101 /s88 /s77 /s67 /s68 /s32/s40/s65/s114/s98/s46/s32/s85/s110/s105/s116/s115/s41 /s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54 /s32/s71/s100 /s88 /s77/s67 /s68 /s71/s100 /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 /s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101 /s40/s75/s41/s32/s89 /s71/s100/s73/s71 /s32/s70/s101/s32/s109/s111/s109/s101/s110/s116/s32/s40/s89 /s73/s71/s41 /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 /s51/s32/s101/s100/s103/s101/s32/s88/s77/s67/s68/s32/s115/s112/s101/s99/s116/s114/s97 /s45/s52/s45/s50/s48/s50/s52/s54/s32/s70/s101 /s88 /s77/s67 /s68 /s70/s101 /s88/s77/s67 /s68 /s32/s40/s65/s114/s98/s46/s32/s85/s110/s105/s116/s115/s41 /s45/s52/s48/s45/s50/s48/s48/s50/s48/s52/s48/s54/s48 /s32/s71/s100 /s88 /s77/s67 /s68 /s71/s100 /s88/s77/s67 /s68 /s32/s40/s65/s114/s98/s46/s32/s85/s110/s105/s116/s115/s41 FIG. 9. The two sublattice (Fe3+and Gd3+) magnetization values (proportional to XMCD amplitude) obtained from the (a) Fe K- and (b) Gd L 3- edge XMCD data analysis. The values are shown by symbols. Temperature variation of bulk magnetization data of YGdIG sample (Fig. 2) and the calcu- lated contribution of Fe3+and Gd3+sublattices to the bulk magnetization are shown in both frames. M-T data of YIG is considered as due to Fe3+, except that the sign of the data is changed accordingly across T Comp. hence giving discrepancy to the comparison. We have also compared Fe XMCD value (red symbols in Fig. 9(a), peak height of the XMCD dispersion spectra located at the pre-edge region ( ≈7115 eV) with the temperature variationoftheFe-onlymagnetizationdatafromtheYIG sample and it was observed similar trend (red dashed line) as shown in Fig. 9(a). As mentioned above the direct subtraction method or linear combination fit according to individual magnetic moment value as function of temperature are mostly em- ployed in literature to decompose the XMCD spectra of RIG and RTI systesm [33, 44]. However, recently it is shown that the application of singular value decomposi- tion (SVD) rationalizes previous approaches in a more general framework and simplifies the exploration of mag- netic phase diagramofsuch compounds[22, 35]. One can analyze the shape and amplitude of the hidden compo- nentsin XMCDdatafromcorrelateddataset. Cornellius et al., successfully analyzed and separated Fe contribu- tion from the Er L 2,3edges using SVD method [22]. The same procedure is adopted in the present work also to de-convolute Fe3+contribution from temperature depen-dence Gd L 3edge XMCD data set (Figure. 6). According to SVD theorem any data matrix A(m×n) can be decomposed into three matrices as A=UΣVT whereU(m×m) andV(n×n) are orthogonal matrices and Σ(m×n) is diagonal matrix of singular values. We have used eight XMCD spectra of Gd L 3- edge over the temperature range (5-300 K) to form data matrix Aand MATLAB software is used to perform the SVD on the data matrix Ato find the U, V,Σ. Only the first two eigenvectors or basis which are dominated over the noise levelareused to reconstructthe originaldataas shownin Figure. 8. The reconstructed plot along with separated Fe and Gd like components to the original Gd L 3- edge XMCD data are shown in Figure. 6. It is to be men- tioned that spectral features of Fe contribution from Gd L3- edge XMCD obtained by SVD in the present work match with previous literature[22, 44] and the XMCD signal amplitude (red symbols), as shown in Fig. 9(b), is also following the similar temperature variation as of Fe sublatticemagnetizationdata(reddashedline)fromYIG in terms of its magnitude and direction. The SVD sepa- rated Gd XMCD amplitudes (blue symbols) at indicated temperature and corresponding temperature variation of Gd only sublattice magnetization (blue dashed line) are also shown in the Fig. 9(b). V. CONCLUSIONS In conclusion, in the present work in-field57Fe M¨ossbauer spectroscopy is employed to demonstrate the Fe3+spin reversal and signatures of spin-canting across the magnetic compensation temperature (T Comp) in Y1.5Gd1.5Fe5O12. The M¨ ossbauer data also clearly demonstrate the continuous rotation of Fe3+moment across the T comp, which is nothing but a second order field induced phase transition. Inversion of sublattice spin across the T Compis also confirmed by Fe K- and Gd L3- edge XMCD data. The quantitative estimation of the two sublattice contribution viz., Fe3+and Gd3+ to the net magnetization is separated out from XMCD spectra collected from either Fe K- or Gd L 3- edge using direct subtraction method from reference spectrum and SVD method, respectively i.e., in RIG systems one can probe both Fe3+and Gd3+magnetism from only single edge XMCD measurements in the hard x-ray region. VI. ACKNOWLEDGMENTS We acknowledge DESY (Hamburg, Germany), a mem- ber of the Helmholtz Association HGF, for the provision of experimental facilities. Parts of this research were car- ried out at beamline P09. Beamtime was allocated for proposal I-20180676 within the India@DESY collabora- tion in Photon Science. 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2021-07-14

In the present work $Fe^{3+}$ sublattice spin reversal and $Fe^{3+}$ spin-canting across the magnetic compensation temperature ($T_{Comp}$) are demonstrated in polycrystalline $Y_{1.5}Gd_{1.5}Fe_{5}O_{12}$ (YGdIG) by means of in-field $^{57}Fe$ M$\ddot{o}$ssbauer spectroscopy measurements. Corroborating in-field $^{57}Fe$ M$\ddot{o}$ssbauer measurements, both $Fe^{3+}$ & $Gd^{3+}$ sublattice spin reversal has also been manifested with x-ray magnetic circular dichroism (XMCD) measurement in hard x-ray region. Moreover from in-field $^{57}Fe$ M$\ddot{o}$ssbauer measurements, estimation and analysis of effective internal hyperfine field ($H_{eff}$), relative intensity of absorption lines in a sextet elucidated unambiguously the signatures of $Fe^{3+}$ spin reversal, their continuous transition and field induced spin-canting of $Fe^{3+}$ sublattices across $T_{Comp}$. Further, Fe K- (Gd $L_{3}$-) edge XMCD signal is observed to consist of additional spectral features, those are identified from $Gd^{3+}$ ($Fe^{3+}$) magnetic ordering, enabling us the extraction of both the sublattices ($Fe^{3+}$ & $Gd^{3+}$) information from a single edge analysis. The evolution of the magnetic moments as a function of temperature for both magnetic sublattices extracted either at the Fe K- or Gd $L_3$-edge agree quite well with values that are extracted from bulk magnetization data of YGdIG and YIG ($Y_{3}Fe_{5}O_{12}$). These measurements pave new avenues to investigate how the magnetic behavior of such complex system acts across the compensation point.

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

2107.06508v3

Strongly e xchange -coupled and surface -state -modulated m agnetization dynamics in Bi 2Se3/YIG heterostruct ures Y . T. Fanchiang1, K. H. M. Chen2, C. C. Tseng2, C. C. Chen2, C. K. Cheng1, C. N. Wu2, S. F. Lee3*, M. Hong1*, and J. Kwo2* 1Department of Physics, National Taiwan University, Taipei 1 0617, Taiwan 2Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan 3Institute of Physics, Academia Sinica, Taipei 11529, Ta iwan Abstract We report strong interfacial exchange coupling in Bi 2Se3/yttrium iron garnet (YIG) bilayers manifested as large in-plane interfacial magnetic anisotropy (IMA) and enhancement of damping probed by ferromagnetic resonance (FMR) . The IMA and spin mixing conductance reached a maximum when Bi 2Se3 was around 6 quintuple -layer (QL) thick. The unconventional Bi 2Se3 thickness dependence of the IMA and spin mixing conductance are correlated w ith the evolution of surface band structure of Bi2Se3, indicating that topological surface states play an important role in the magnetization dynamics of YIG. Temp erature - dependen t FMR of Bi2Se3/YIG revealed signatures of magnetic proximity effect of Tc as high as 180 K , and an effective field parallel to the YIG magnetization direction at low temperature . Our study shed s light o n the effects of topological insulators on magnetization dynamics, essential for dev elopment of TI -based spintronic devices. Topological insulato rs (TIs) are emergent quantum materials hosting topologically protected surface states, with dissipationless trans port prohibiting backscattering [1,2] . Strong spin-orbit coupling (SOC) along with time reversal symmetry (TRS) ensures that the electrons in surface states have thei r direction of motion and spin "locked" to each other [1,3,4] . When interfaced with a magnetic layer, the interfacial excha nge coup ling can induce magnetic order in TIs and break TRS [5-8]. The resulting gap opening of the Dirac state is necessa ry to realize novel phenomena such as topological magneto -electric effect [9] and quantum anomalous Hall effect [10,11] . Another appro ach of studying a TI/ferromagnet (FM) system focus es on spin - transfer c haracteristic at the interface and intend s to exploit the helical spin texture of topological surface states (TSSs). Attempts have b een made to estimate the spin -charge conversion (SCC) effici ency, either by using microwave -excited d ynamical method [12-16] (e.g. spin p umping and spin-torque FMR) or thermally induced spin injection [17]. Very large values of SCC ratio have been reported [13,15,16] . Recently, TIs are shown to be excellent sources of spin -orbit torques for efficient magnetization switching [18]. Since the magnetic proximity effect (MPE) an d spin -transfer process rel y on interfacial exchange coupling of TI/FM , understanding the magnetism at the interface has attracted strong interest s in recent years. Several technique s have been adopted to investigate the interfacial magnetic properties, including spin -polarized neutron reflectivity [7,19] , second harmonic generation [20], electrical transport [6,21] , and magneto -optical Kerr effect [6]. All these studies clearly indicate the existence of MPE resulting from exchange coupling and strong spin-orbit interaction in TIs. For device application, however, it is equally important to understand how the interfacial exchange coupling affect s the magnetization dyn amic of FM. For example, TIs can introduce additional magnetic damping and greatly alter the dynamical properties of FM layer, as commonly ob served in FM/heavy metals systems [22-24]. The enhanced damping is visualized as larger linewidth of FMR spectra [22-24]. In TI/FM, t he presence of TSS and, possibly, MPE complicate the system under study, and is still an open question to answer. In this work w e systematically investigate d FMR characteristic of the ferrimagnetic insulator YIG under the influence of the prototypical three -dimensional TI Bi 2Se3 [25]. We choose YI G as the FM layer because of its technological importa nce, with high 𝑇𝑐 ~550 K and extremely low damping coefficient α [26]. When YIG is interfaced with TIs, its good thermal stability minimize s interdiffusion of materials . Through Bi 2Se3 thickness dependence study, w e observed strong modulation of FMR properties , attributed to the TSS of Bi 2Se3. Temperature -dependent study unraveled an effective field parallel to the magnetization direction existing in Bi 2Se3/YIG. Such effective field built up as temperature decreased, which was utilized to demonstrate the zero -applied -field FMR of YIG . Furthermore, we identified the signature of MPE of 𝑇𝑐 as high as 180 K manifested as enhanced spin pumpin g in a fluctuating spin system. Bi2Se3 thin films were grown by molecular beam epitaxy (MBE) [27] on magnetron - sputtered YIG films. The high quality of Bi 2Se3/YIG samples in this work is verified by high resolution X -ray diffraction (XRD) and transmission electron microscope (TEM) [28]. To investigate the magnetic properties of Bi 2Se3/YIG, room -temperature angle - and frequency - dependent FMR measurements were performed independently using a cavity and co -planar waveguide (CPW), respective ly (Figure 1( a) and (b)). For the temperat ure-dependent FMR , the CPW was mounted in a cryogenic probe station (Lake Shore, CPX -HF), which enable samples to be cooled as low as 5 K. The external field is modulated for lock -in detection in all of the measurements. The FMR spectra in Figure 1( c) are compared for single layer YIG(12) and Bi2Se3(25)/YIG(12) bilayer (digits denote thickness in nanometer) , showing a large shift of resonance field ( 𝐻𝑟𝑒𝑠) ~317 Oe after the Bi2Se3 growth plus a markedly broadened peak -to- peak width ∆𝐻 for Bi 2Se3/YIG. Figure 1(d) shows 𝐻𝑟𝑒𝑠 vs applied field angle with respect to 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 directed in the film plane, clear negative 𝐻𝑟𝑒𝑠 shifts induced by Bi 2Se3 were observed at all microwave frequency f in Figure 1 e. The data can be fitted in the scheme of magnetic thin films having uniaxial perpendicular magnetic anisotropy [28]. Since the XRD results show th at the YIG films did not gain additional strain after growing Bi2Se3, the enhanced anisotropy cannot be attributed to changes of magnetocrstalline anisotropy. Defining eff ective demagnetization field 4𝜋𝑀𝑒𝑓𝑓=4𝜋𝑀𝑠−𝐻𝑎𝑛−𝐻𝑖𝑛𝑡, where 4𝜋𝑀𝑠, 𝐻𝑎𝑛 and 𝐻𝑖𝑛𝑡 are the demagnetization field of bare YIG , the magnetocrystalline anisotropy field of YIG, and the anisotropy field induced by Bi 2Se3, we obtain 𝐻𝑖𝑛𝑡 = -926 and -1005 Oe fr om Figure 1 (d) and 1 (e), respectively. The minus sign indicates the additional anisotropy points in the film plane. The above observations suggest ed the presence of IMA in Bi 2Se3/YIG. To ve rify this assumption, we systematically varied the thickness of YIG 𝑑𝑌𝐼𝐺 while fixing the thickness of Bi2Se3. Figure 2( a) presents the 𝑑𝑌𝐼𝐺 dependence of 4𝜋𝑀𝑒𝑓𝑓 for single and bilayer samples. The 4𝜋𝑀𝑒𝑓𝑓 of single layer YIG was independent of 𝑑𝑌𝐼𝐺 varying from 12 to 30 nm. In sharp contrast, 4𝜋𝑀𝑒𝑓𝑓 of Bi 2Se3/YIG became significant ly larger , especially at thinner YIG, which is a feature of an interfacial effect. The f- and 𝜃𝐻-dependent FMR were performed independently to doubly confirm the trends. The IMA can be further characterized by defining the effective anisotropy constant 𝐾𝑒𝑓𝑓= (1/2)4𝜋𝑀𝑒𝑓𝑓𝑀𝑠 =(1/2)(4𝜋𝑀𝑠−𝐻𝑎𝑛)+𝐾𝑖/ 𝑑𝑌𝐼𝐺, with the interfacial anisotropy constant 𝐾𝑖=𝑀𝑠𝐻𝑎𝑛𝑑𝑌𝐼𝐺/2. The 𝐾𝑒𝑓𝑓𝑑𝑌𝐼𝐺 vs 𝑑𝑌𝐼𝐺 data in Figure 2(b) is well fitted by a linear function, indicating that the 𝑑𝑌𝐼𝐺 dependence presented in Figure 2(a) is suitably described by the current form of 𝐾𝑒𝑓𝑓. The intercept obtained by extrapolating the linear function corresponds to 𝐾𝑖=−0.075 erg/cm2. To further investigate the physical origin of the IMA , we next varied the thick ness of Bi2Se3 (𝑑𝐵𝑆) to see how 𝐾𝑖 evolved with 𝑑𝐵𝑆. Figure 2( c) shows the 𝑑𝐵𝑆 dependence of 𝐾𝑖. Starting from 𝑑𝐵𝑆=40 nm sample, the magnitude of 𝐾𝑖 went up as 𝑑𝐵𝑆 decreased. An extremum 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 exhibited a sign change in the interval. The Ki value of 0.014 erg/cm2 at 𝑑𝐵𝑆=3 nm corresponds to weak interfacial perpendicu lar anisotropy. Based on previous investigation on surface band structure of ultrathin Bi2Se3 [29], 𝑑𝐵𝑆=6 nm was identified as the 2D quantum tunneling limit of Bi 2Se3. When 𝑑𝐵𝑆<6 nm, the hybridization of top and bottom TSS developed a ga p of the surface states. Spin -resolved photoemission study later showed that the TSS in this regime exhibited decre ased in-plane spin polarization . The modulated spin texture may lead to the weaker IMA than that in 3D regime [30,31] . We thus divide Figure 2( c) into two regions and correlate the systematic magnetic properties with the surface state band structure. The sharp change of 𝐾𝑖 around 𝑑𝐵𝑆=6 nm strongly suggests that the IMA in Bi2Se3/YIG is of topological origins . The ∆𝐻 broadening in FMR spectra after growing Bi 2Se3 on YIG indicates additional damping arising from spin-transfer from YIG to Bi 2Se3, which is a s ignature of spin pumping effect [24]. The spin pump ing efficiency of an interface can be evaluated by spin mixing conduct ance 𝑔↑↓, using the following relation [24], 𝑔↑↓=4𝜋𝑀𝑠𝑑𝑌𝐼𝐺 𝑔𝜇𝐵(𝛼𝐵𝑆/𝑌𝐼𝐺−𝛼𝑌𝐼𝐺) (1) , where 𝑔, 𝜇𝐵, 𝛼𝐵𝑆/𝑌𝐼𝐺 and 𝛼𝑌𝐼𝐺 are the Landé g factor, Bohr magneton, the damping coefficient of Bi 2Se3/YIG and YIG, respectively. Figure 2( d) displays the dBS dependence of effective spin mixing conductance of Bi 2Se3/YIG . Similar to Ki in Figure 2( c), 𝑔↑↓ had its maximum at 𝑑𝐵𝑆=7 nm with a very large value ~2.2×1015 cm-2, about four times larger than that of our Pt/YIG sample indicated by the red dashed line . The inset shows ∆𝐻 vs 𝑓 data for Bi 2Se3 (7)/YIG(13) and YIG(13) fitted by linear functions. One can clearly see a significant change of slope, from which we determined 𝛼𝐵𝑆/𝑌𝐼𝐺−𝛼𝑌𝐼𝐺 to be 0.014. The large 𝑔↑↓ of Bi 2Se3/YIG implies an efficient spin pumping to an excel lent spin sink of Bi 2Se3. We again 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= 3 nm. The trend in Figure 2( d) is distinct from that of no rmal metal (NM)/FM structures . In NM/FM, the 𝑔↑↓ increase s with increasing NM thickness as a result of vanishing spin backflow in thicker NM [32]. It is worth noting that the conducting bulk of Bi 2Se3 can dissipate the spin-pumping -induced spin accumulation at the interface [12,33] . In this regard, the 𝑑𝐵𝑆= 7 nm sample has the largest weight of surface state contribution to 𝑔↑↓. Such unconventional 𝑑𝐵𝑆 dependence of 𝑔↑↓ suggests that TSS play a dominant role in the damping enhancement . Since the effects of TSS are expected to enhance at low temperature , we next performed temperature -dependent FMR on Bi2Se3/YIG . Two bilayer samples Bi2Se3(25)/YIG(15) and Bi2Se3(16)/YIG(17) , and a single layer YIG(23) were measured for comparison. Figure 3( a) and (b) show the 𝐻𝑟𝑒𝑠 vs 𝑓 data at various T for YIG(23) and Bi 2Se3(25)/YIG( 15). The 𝐻𝑟𝑒𝑠 of both samples show negative shifts at all f with decreasing T. The data of YIG(23) can be reproduced by the Kittel equation with increasing 𝑀𝑠 of YIG at low T. In sharp contrast, Bi2Se3(25)/YIG (15) exhibited negative intercepts at 𝐻𝑟𝑒𝑠, and the intercepts gained its magnitude when the sample was cooled dow n. This behavior of non -zero intercept is common for all of our Bi 2Se3/YIG samples . To account for the behavior, a phenomenological effective field 𝐻𝑒𝑓𝑓 is add ed to the Kittel equation, i.e. 𝑓=𝛾 2𝜋√(𝐻𝑟𝑒𝑠+𝐻𝑒𝑓𝑓)(𝐻𝑟𝑒𝑠+𝐻𝑒𝑓𝑓+4𝜋𝑀𝑒𝑓𝑓). (2) The solid lines in F igure 3( b) generated by the modified Kittel equation fitted the experimental data very well . Figure s 3(c) and (d) presents the T dependence of 𝐻𝑟𝑒𝑠 and ∆𝐻 for the YIG(23) and two Bi2Se3/YIG samples. As we lowered T, all of the samples had decreasing 𝐻𝑟𝑒𝑠, which was viewed as effects of the concurrent increasing 𝑀𝑒𝑓𝑓 and 𝐻𝑒𝑓𝑓 as seen in Figure 3( a) and (b). On the other hand, ∆𝐻 built up with decreasing T. We first examine ∆𝐻 of the YIG(23) single 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 explored in vario us rare -earth iron garnet and was explained by the low T slow -relaxation process via rare -earth elements or Fe2+ impuritie s [34]. For sputtered YIG films, specifically, the increase of ∆𝐻 was less prominent in thicke r YIG, indicating that the dominant impurities locate d near the YIG surface [35]. Distinct from that of YIG(23) , the ∆𝐻 progressive ly increase d for the bilayer samples. We were not able to detect FMR signals with ∆𝐻 beyond 100 Oe due to our instrumental limits. However, one can clearly see that, for Bi2Se3(25)/YIG(15) and Bi 2Se3(16)/YIG(17), the ∆𝐻 broaden ed due to increased spin pumping at first . For Bi 2Se3(25)/YIG(15) , the ∆𝐻 curve gradually leveled off, and intersect ed with that of YIG(23) at T ~40 K . The seeming ly "anti -damping" by Bi 2Se3 at low T may be related to the modification of the YIG surface chemistry during the Bi 2Se3 deposition. Additional analyses are needed to verify t he scenario , which is, however, beyond the scope of this work. In both 𝐻𝑟𝑒𝑠 and ∆𝐻 curves , bump-like feature s located at T = 140 and 180 K (indicated by the arrows) were revealed for Bi 2Se3(25)/YIG(15) and Bi 2Se3(16)/YIG(17), respectively. The bumps are reminiscent of spin pumping in the case of fluctuating magnets , where an enhancement of spin pumping is expected as the spin sink is closed to its magnetic phase transition point [36-38]. In our system, a possibl y newly formed magnetic phase would be the interfacial magnetization driven by the proximity effect , namely, Tc = 140 and 180 K for our Bi2Se3(25)/YIG(15) and Bi 2Se3(16)/YI G(17) , respectively . In fact, the Tc values of our samples are in good agreement with the reported ones of MPE in TI/YIG systems [6,21] . We did not detect anomalous Hall effect in our sample s, which might be o bscured by the bulk conduction of Bi 2Se3 in the magneto -transport measurements . Using Eq. (2 ), we further determine the T dependence of 4𝜋𝑀𝑒𝑓𝑓 and 𝐻𝑒𝑓𝑓 of YIG(23) and the two bilayer s samples, as shown in Figures 3(e) and (f) [28]. The 4𝜋𝑀𝑒𝑓𝑓 of YIG(23) went 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 slightly at low T. With 4𝜋𝑀𝑒𝑓𝑓𝐵𝑆/𝑌𝐼𝐺−4𝜋𝑀𝑒𝑓𝑓𝑌𝐼𝐺≈−𝐻𝑖𝑛𝑡, such T dependence corresponds to decreasing in-plane IMA below 150 K. Note that 150 K is close d to our assumed 𝑇𝑐 from MPE at 140 and 180 K. Calculations of total electronic energy at an TI/FM interface show that perpendicular anisotrop y is in favor [39], which, in our case, may effectively weaken the in - plane IM A. The decreasing in -plane IMA can also be viewed as a result of modified spin texture by MPE. The analyses thus provide another clue suppor ting the existence of MPE in our samples. The 𝐻𝑒𝑓𝑓 of bilayer samples, again, show different T evolution than that of the single layer in Figure 3( f). 𝐻𝑒𝑓𝑓 built up with decreasing T in bilayers while the 𝐻𝑒𝑓𝑓 of the YIG single layer was T-independent and closed to zero. The positive 𝐻𝑒𝑓𝑓 corresponds to an effective field parallel to the magnetization vector M and resembles the exchange bias field. However, we did not observe shifts of magnetization hysteresis loop characteristic of an exchange bias effect [28,40] . The 𝐻𝑒𝑓𝑓 present in FMR m easurement suggests it may come from spin imbalance at the interface [41]. Finally, we demonstrated that the large IMA and 𝐻𝑒𝑓𝑓 in Bi 2Se3/YIG are strong enough to ind uce FMR without an 𝐻𝑒𝑥𝑡, which we term zero-field FMR . Figure 4( a) displays T evolution of FMR first derivative spectra of Bi 2Se3(25)/YIG(15) at 𝑓=3.5 GHz. The spectral shape started to deform when the 𝐻𝑟𝑒𝑠 was approaching zer o. The sudden twists at 𝐻𝑒𝑥𝑡 ~ +30 ( -30) for positive (negative ) field sweep arose from magnetization switching of YIG, and therefore led to hysteric spectra. The two spectra merged at 25 K and then separated again when T was further decreased. Figure 4( b) shows the microwave absorption intensity I spectra with positive field sweep s. We traced the position of I spectrum 𝐻𝑝𝑒𝑎𝑘 using the red dashed line, and found it coincided with zero 𝐻𝑒𝑥𝑡 at the zero -field FMR temperature 𝑇0 ~25 K. Below 25 K, 𝐻𝑝𝑒𝑎𝑘 moved across the origin and one needed to reverse 𝐻𝑒𝑥𝑡 to counte r the internal effective 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 when 𝐻𝑒𝑓𝑓 is finite would the system exhibit non -zero intercepts as we have seen in Figure 3(b). We further calculate 𝑇0 as a function of microwave excitation frequency f (Figure 4 (f)) using Eq. (2) and the extracted 𝐻𝑒𝑓𝑓 of Figure 3(f). We obtain that, with f inite 𝐻𝑒𝑓𝑓 persisting up to room temperature, zero -field FMR can be realized at high T provided f is sufficiently low. However, we emphasize that it’s advantageous for YIG to be microwave - excited above 3 GHz. When 𝑓<3 GHz, parasitic effects such as th ree-magnon splitting [42,43] take place and significantly decrease the microwave absorpti on in YIG. Here, we demonstrate that the strong exchange coupling between Bi 2Se3 and YIG gave rise to zero -field FMR in the feasible high frequency operation regime of YIG. Further improvement of interface quality of Bi 2Se3/YIG is expected to raise 𝐻𝑒𝑓𝑓 and 𝑇0 for room temperature, field free spintron ic application. In summary , we investigate the magnetization dynamics of YIG in the presence of interfacial exchange coupling and TSS of Bi2Se3. The significantly modulated magnetization dynamics at room temperature is shown to be TSS -originated through the Bi 2Se3 thickness dependence study. It can be expected that the strong coupling between Bi 2Se3 and YIG will modify interface band structures an d even the spin texture of TSS. The underlying mechanism of the c oexisting large in -plane IMA and pronouncedly enhanced damping calls for further theoretical modeling and understanding . The temperature -dependent FMR results revealed rich information, inclu ding signatures of MPE and increasing exchange effective field at low temperature . Our study is an important step toward realization of topological spintronics. We would like to thank helpful discussion with Profs. Mingzhong Wu and Hsin Lin. The work is supported by MOST 105 -2112 -M-007-014-MY3 , 105-2112 -M-002-022-MY3 , 106- 2112 -M-002-010-MY3, 102 -2112 -M-002-022-MY3, and 105 -2112 -M-001-031-MY3 of the Ministry of Sc ience and Technology in Taiwan. References [1] M. 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(a), (b) FMR using the cavity and co -planar configuration for angle - and frequency - dependent study, respectively. A dc external field 𝐻𝑒𝑥𝑡 was applied and ℎ𝑟𝑓 denotes the microwave field. (c) FMR spectra of Bi 2Se3(25)/YIG(12) and YIG(12) measured by the cavity. (d), (e) 𝜃𝐻 and f dependence of 𝐻𝑟𝑒𝑠 of Bi 2Se3(25)/YIG(12) and YIG(12), respectively. Figure 2. FIG. 2. (a) The 𝑑𝑌𝐼𝐺 dependence of 4𝜋𝑀𝑒𝑓𝑓 of Bi 2Se3/YIG (solid triangles) and YIG (hollow squares) obtained from 𝜃𝐻 (red) and f (blue) dependent FMR. (b) The 𝐾𝑒𝑓𝑓𝑑𝑌𝐼𝐺 vs 𝑑𝑌𝐼𝐺 plot for determining 𝐾𝑖 using a linear fit. The intercept of the y -axis corresponds to the 𝐾𝑖 value. (c) 𝑑𝐵𝑆 dependence of 𝐾𝑖. The figure is divided into two regions. For 𝑑𝐵𝑆>6 nm, the Dirac cone of TSS is intact, with the Fermi level located in the bulk conduction band. For 𝑑𝐵𝑆<6 nm, a gap and quantum well states form. (d) The 𝑑𝐵𝑆 dependence of spin mixing conductance 𝑔↑↓. The inset shows ∆𝐻 as a function of Bi 2Se3(7)/YIG(13) a nd YIG(13) for calculating 𝑔↑↓. The red dashed line indicates the typical value of 𝑔↑↓ of Pt/YIG. Figure 3. FIG. 3. (a), (b) 𝑓 vs 𝐻𝑟𝑒𝑠 data for various T for YIG(23) and Bi 2Se3(25)/YIG(15), respectively. Solids lines are fitted curves using Eq. (2). (c), (d), (e), and (f) T dependence of 𝐻𝑟𝑒𝑠, ∆𝐻, 4𝜋𝑀𝑒𝑓𝑓 and 𝐻𝑒𝑓𝑓 of the one YIG (23) single layer and two Bi 2Se3/YIG bilayer samples, Bi 2Se3(25)/YIG(15) and Bi 2Se3(16)/YIG(17), respectively. The arrows in (c) and (d) denote the position of the "bumps". Solids line are guide of eyes obtained by properly smoothing the experiment al data. Figure 4. FIG. 4. (a), (b) FMR first derivative and microwave absorption s pectra for variou s T, respectively. The arrows indicate the 𝐻𝑒𝑥𝑡 sweep direction. The dashed line in (b) traces the T evolution of the absorption peak 𝐻𝑝𝑒𝑎𝑘. (c), (d), an d (e) Schematics of the Bi 2Se3/YIG sample when T = 100, 25, and 5 K, respectively. (f) Zero -field FMR temperature 𝑇0 as a function of f. Supplemental materials Growth and F MR characteristics of YIG films The YIG thin films were deposited on (111) -oriented GGG substrates by off -axis sputtering at room temperature. The GGG(111) substrates were first ultrasonically cleaned in order of acetone, ethanol an d DI -water before mounted in a sputtering chamber with the base pressure of 2×10−7 mTorr. For YIG deposition, a 2 -inch YIG target was sputtered with the following conditions: an applied rf power of 75 W , an Ar pressure of 50 mtorr and a growth rate of 0.6 nm/min. The samples were then annealed at 800oC with O2 pressure of 11.5 mtorr for 3 hours. Fig. S1(a) displays the atomic force microscopy (AFM) image of the YIG surface, showing a flat surface with roughness of 0.19 nm. Fig . S1(b) shows the high-angle annular dark -field (HAADF) image of YIG/GGG. The YIG thin film was epitaxially grown on the GGG substrate with excellent crystallinity. No crystal defects were observed at the YIG bulk and YIG/GGG interface. FIG. S1. (a) and (b) AFM surface image and HAADF -STEM image of YIG/GGG. Fig. S2(a) shows the representative FMR data of our YIG film measured by coplanar wav eguide. The FMR spectra exhibit Lorentzian lineshape at all measured frequencies ranging from 3 to 9.76 GHz. To determine the 4𝜋𝑀𝑒𝑓𝑓 and 𝛼, the resonance fields 𝐻𝑟𝑒𝑠 and peak -to-peak widths ∆𝐻 of these spectra were plotted as a function of 𝑓 as shown in Fig. S2(b) and (c), respectively. We obtain the 4𝜋𝑀𝑒𝑓𝑓 of our 23 nm YIG to be 1630 Oe. We note that the 4𝜋𝑀𝑒𝑓𝑓 value is lower th an the reported values of the YIG prepared by either sputtering or pulsed laser deposition [S1- 3]. The difference may come from growth conditions dependent on different systems. For determinati on of 𝛼, the data in Fig. S2(c) is fitted to the following equation, ∆𝐻=∆𝐻0+4𝜋𝑓𝛼 √3𝛾. (S1) Here, ∆𝐻0 and 𝛾 are the inhomogeneous broadening and gyromagnetic ratio. The linear 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 to the Kittel equation (red line) . (c) ∆𝐻 as a function of 𝑓 data. From the linear fit (red line) the 𝛼 value of the sample is obtained. Structural characterizations of Bi 2Se3/YIG heterostructures The YIG/GGG samples were annealed at 450oC in the MBE growth chamber for 30 min prior to Bi 2Se3 growth at 280oC [27]. The base pressure of the system was kept about 2×10−10 Torr. Elemental Bi (7N) and Se (7N) were evaporated from regular effusion cells. As shown in Fig . S3a, streaky reflection high -energy electron diffraction (RHEED) patterns of Bi 2Se3 were observed. Fig . S3b displays the s urface morphology of 7 QL Bi2Se3 taken by atomic force microscopy (AFM). The image shows layer -by- layer growth of Bi 2Se3 with the step heights ~ 1nm, which corresponds to the thickness of 1 QL. The l ayer structure of Bi 2Se3 was also revealed by the HAADF image shown in Fig . S3c. Despite the high quality growth of Bi2Se3, an amorphous interfacial layer of ~ 1 nm formed. The excellent crystallinity of our samples was verified by clear FIG. S3 . (a) RHEED patterns of MBE grown 7 QL Bi 2Se3 on YIG/GGG(111) substrates. (b) AFM image of a 7 QL Bi 2Se3. (c) HAADF -STEM image of Bi 2Se3/YIG/GGG heterostructures. (d) SR-XRD of our Bi2Se3(25)/YIG(12) sample. Clear Pendellösung fringes of YIG and Bi 2Se3 indicates excellent crystallinity. The inset shows the radial scans of YIG before and after Bi2Se3 growth. Pendellösung fringes of the synchrotron radiation x -ray diffraction (SR -XRD) data shown in Fig . S3(d). In particular, the radial scans data of YIG/GGG(22 -4) before and after growing Bi 2Se3 are perfectly matched, indicating the absence of Bi 2Se3-induced strains in YIG t hat might contribute additional magnetic anisotropy [S4]. Analyses of magnetic anisotropy We express the free energy density E of the system as 𝐸=−𝑴∙𝑯+1 2𝑀𝑠(4𝜋𝑀𝑠−𝐻𝑎𝑛−𝐻𝑖𝑛𝑡)cos2𝜃𝑀 (S2) , where 𝑴, 𝑯, 𝑀𝑠, 𝐻𝑎𝑛and 𝜃𝑀 are magnetization vector, applied field vector, saturation magnetization, the anisotropy field induced by Bi 2Se3 and magnetization angle with respect to the surface normal, respectively. We further define the effective demagnetization field 4𝜋𝑀𝑒𝑓𝑓=4𝜋𝑀𝑠−𝐻𝑎𝑛−𝐻𝑖𝑛𝑡, where 𝐻𝑎𝑛 is the magnetocrystalline anisotropy field of sputtered YIG and 𝐻𝑖𝑛𝑡 is the interfacial anisotropy field. We have 𝐻𝑖𝑛𝑡=0 for the YIG single layer by definition . The first term of Eq. ( S2) is the Zeeman energy and the second term accounts for uniaxial out - of-plane anisotropy. Here, we neglect higher order terms that are relatively small for a strain -free cubic system. The 𝐻𝑟𝑒𝑠 can be calculated by minimizing 𝐸 and, at the equilibrium angle of 𝑴, solving the Smit -Beljers equation [S5], (𝜔 𝛾)2 =1 𝑀2sin2𝜃𝑀[𝜕2𝐸 𝜕𝜃𝑀2𝜕2𝐸 𝜕𝜑𝑀2−(𝜕2𝐸 𝜕𝜃𝑀2𝜕𝜑𝑀2)2 ]. (S3) As 𝜃𝐻=𝜋/2, the FMR conditions reduce to the Kittel equation 𝑓= 𝛾 2𝜋√𝐻𝑟𝑒𝑠(𝐻𝑟𝑒𝑠+4𝜋𝑀𝑒𝑓𝑓). We can safely assume that the 𝐻𝑎𝑛 did not change before and after the growth of Bi2Se3 based on Figure S3(d). With this in mind, w e further notice that 𝐾𝑖 can be alternatively expressed as (1/2)(4𝜋𝑀𝑒𝑓𝑓𝑌𝐼𝐺−4𝜋𝑀𝑒𝑓𝑓𝐵𝑆/𝑌𝐼𝐺)𝑀𝑠𝑑𝑌𝐼𝐺 , where 4𝜋𝑀𝑒𝑓𝑓𝑌𝐼𝐺 (4𝜋𝑀𝑒𝑓𝑓𝐵𝑆/𝑌𝐼𝐺) represents the 4π𝑀𝑒𝑓𝑓 of YIG (Bi 2Se3/YIG) for a specific 𝑑𝑌𝐼𝐺. The calculated 𝐾𝑖 using this expression for 𝑑𝐵𝑆 = 25 nm gives an average of - 0.068 erg/cm-2, in good agreement with a 𝐾𝑖 of -0.075 erg/cm2 obtained from the linear fit in Figure 2(b). Temperature -dependent FMR spectra of YIG and Bi 2Se3/YIG Extraction of 𝟒𝝅𝑴𝒆𝒇𝒇 and 𝑯𝒆𝒇𝒇 from temperature dependence of 𝑯𝒓𝒆𝒔 Since the effective damping constant of YIG and Bi 2Se3/YIG increased pronouncedly at low 𝑇, the weakened FMR signal was inevitably accompanied by larger uncertainties of measured 𝐻𝑟𝑒𝑠. The uncertainties are even more serious when FIG. S4 . Temperature -dependent FMR first-derivative spectra of (a) YIG(23) and (b) Bi2Se3(25)/YIG(15). The Δ𝐻 increased with decreasing 𝑇, accompanied by decreas ed 𝑑𝐼/𝑑𝐻𝑒𝑥𝑡 peak magnitude s due to enhanced damping. The 𝐻𝑒𝑥𝑡 scale are fixed to clearly show the pronounce d change s of 𝐻𝑟𝑒𝑠 and Δ𝐻 induced by Bi2Se3. FIG. S5 . (a) 𝐻𝑟𝑒𝑠 vs 𝑇 data of Bi 2Se3(25)/YIG (15) measured at 4 and 4.5 GHz. (b) and (c) Comparison 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 the 𝑓>5 GHz region to the Kittel equation gives large error s of 4𝜋𝑀𝑒𝑓𝑓 and 𝐻𝑒𝑓𝑓, which obscure the temperature dependency of these two quantities. To reduce the uncertainties in the process of extracting 4𝜋𝑀𝑒𝑓𝑓 and 𝐻𝑒𝑓𝑓, we focused on the FMR data for 𝑓=4 and 4.5 GHz , from which we obtained 𝐻𝑟𝑒𝑠 with satisfactory accuracy (Fig . S5(a)). The Kittel equation can be arranged in the following form, 𝐻𝑒𝑓𝑓2+(2𝐻𝑟𝑒𝑠+4𝜋𝑀 𝑒𝑓𝑓)𝐻𝑒𝑓𝑓+(4𝜋𝑀 𝑒𝑓𝑓𝐻𝑟𝑒𝑠+𝐻𝑟𝑒𝑠2−4𝜋2𝑓2 𝛾2)=0. (S3) With 𝛾=1.77×1011 t−1s−1, the two sets of data in Fig . S5a provided us with sufficient information to explicitly solve the second -order equation. Fig . S5b and S5 c compare the results of "fitting" and "solving" the Kittel equation. For 𝑇>150 K, where FMR can be accurately measured up to 7 GHz, the 4𝜋𝑀𝑒𝑓𝑓 and 𝐻𝑒𝑓𝑓 obtained from solving and fitting met hod agree well, de monstrating the reliability of the solving method. Temperature dependence of magnetization hysteresis loop FIG. S6 . Temperature dependence of magnetization hysteresis loop of Bi2Se3(16)/YIG (17) measured by a SQUID magnetometer. The paramagnetic background of the GGG substrate has been subtracted. No shifts of hysteresis loops were observed. References: [S1] O. d'Allivy Kelly, A. Anane, R. Bernard, J. Ben Youssef, C. Hahn, A. H. Molpeceres, C. Carrétéro, E. Jacquet, C. Deranlot, P. Bortolotti, R. Lebourgeois, J. C. Mage, G. de Loubens, O. Klein, V . Cros, and A. Fert, Appl. Phys. Lett. 103, 082408 (2013). [S2] J. C. Gallagher, A. S. Yang, J. T. Brangham, B. D. Esser, S. P. White, M. R. Page, K.-Y . Meng, S. Yu, R. Adur, W. Ruane, S. R. Dunsiger, D. W. McComb, F. Yang, and P. C. Hammel, Appl. Phys. Lett. 109, 072401 (2016). [S3] H. Chang, P. Li, W. Zhang, T. Liu , A. Hoffmann, L. Deng, and M. Wu, IEEE Magn. Lett. 5 (2014). [S4] H. L. Wang, C. H. Du, P. C. Hammel, and F. Yang, Phys. Rev. B 89 (2014). [S5] J. Smit and Beljers, Philips Res. Repts. 10, 113 (1955).

2017-08-02

We report strong interfacial exchange coupling in Bi2Se3/yttrium iron garnet (YIG) bilayers manifested as large in-plane interfacial magnetic anisotropy (IMA) and enhancement of damping probed by ferromagnetic resonance (FMR). The IMA and spin mixing conductance reached a maximum when Bi2Se3 was around 6 quintuple-layer (QL) thick. The unconventional Bi2Se3 thickness dependence of the IMA and spin mixing conductance are correlated with the evolution of surface band structure of Bi2Se3, indicating that topological surface states play an important role in the magnetization dynamics of YIG. Temperature-dependent FMR of Bi2Se3/YIG revealed signatures of magnetic proximity effect of $T_c$ as high as 180 K, and an effective field parallel to the YIG magnetization direction at low temperature. Our study sheds light on the effects of topological insulators on magnetization dynamics, essential for development of TI-based spintronic devices.

Strongly exchange-coupled and surface-state-modulated magnetization dynamics in Bi2Se3/YIG heterostructures

1708.00593v1

Ultrastrong coupling between a microwave resonator and antiferromagnetic resonances of rare earth ion spins Jonathan Everts,1Gavin G. G. King,1Nicholas Lambert,1Sacha Kocsis,2Sven Rogge,2and Jevon J. Longell1 1The Dodd-Walls Centre for Photonic and Quantum Technologies, Department of Physics, University of Otago, Dunedin, New Zealand 2Centre for Quantum Computation and Communication Technology, The University of New South Wales, Sydney, New South Wales 2052, Australia Quantum magnonics is a new and active re- search eld, leveraging the strong collective cou- pling between microwaves and magnetically or- dered spin systems [1]. To date work in quan- tum magnonics has focused on transition met- als and almost entirely on ferromagnetic reso- nances in yttrium iron garnet (YIG) [2, 3]. An- tiferromagnetic systems have gained interest as they produce no stray eld, and are therefore robust to magnetic perturbations and have nar- row, shape independent resonant linewidths [4]. Here we show the rst experimental evidence of ultrastrong-coupling between a microwave cav- ity and collective antiferromagnetic resonances (magnons) in a rare earth crystal. The combina- tion of the unique optical and spin[5, 6] properties of the rare earths and collective antiferromagnetic order paves the way for novel quantum magnonic applications. Superconducting qubits have provided a new set of ca- pabilities in quantum computing, control and measure- ment. In turn, this has generated much interest in \hy- brid quantum systems" [7]. In such approaches the super- conducting qubits capabilities are enhanced by coupling them to another type of system. Amongst the challenges that the hybrid approach tries to address is the long term storage of quantum information by using spins, as well as long distance, room temperature, quantum communi- cation by microwave to optical conversion [8]. Rare earth ions oer exciting possibilities for quantum information because of the long coherence times available for both their optical [9] and optically addressed spin transitions [10]. Quantum memories based on ensembles of rare earth ions have demonstrated large bandwidths [11], high eciencies and very long storage times [10]. Rare earth ions in solids are also being investigated for microwave to optical conversion [12{16]. When using rare earth doped samples for these applications, there is a tradeo when it comes to the concentration of the dopant. Higher dopant concentrations cause a desirable increase in the collective coupling strength to electro- magnetic waves but an undesirable increase in both ho- mogeneous and inhomogeneous linewidths. For optical transitions in low concentration rare earth dopants, this increase is often due to Stark shifts or an increase in crys- tal strain from dopants degrading the crystal quality [17]. For electron spin transitions it is often due to magnetic dipole-dipole interactions [18].A way out of this concentration-linewidth trade o is to instead use fully concentrated rare earth crystals where the rare earth is part of the host crystal. In the case of optical transitions it has been shown that very narrow inhomogeneous linewidths (25 MHz)[6] can be achieved in fully concentrated samples, linewidths comparable with the narrowest seen in doped samples [5, 19]. Here we investigate the coupling between antiferro- magnetic magnon modes in the fully concentrated rare earth crystal gadolinium vanadate (GdVO 4) and a mi- crowave cavity. We show that not only can ultrastrong- coupling be achieved but that the linewidth of the magnon resonances are narrow, an important property for their use in hybrid quantum systems. The magnetic and thermal properties of fully concen- trated rare earth phosphates and vanadates have been investigated before [20, 21], and collective resonances were observed in gadolinium aluminate (GdAlO 4) and GdVO 4. These were observed however, at much higher temperatures and showed much larger linewidths than we observe here. In general antiferromagnetic resonance in rare earth systems is a largely unexplored area. GdVO 4has a tetragonal crystal lattice with space groupD19 4h(I4=amd ). There are four Gd3+ions per unit cell with site symmetry D2d. At room tempera- ture GdVO 4is paramagnetic but upon cooling at zero- magnetic eld to T2.495 K, it orders antiferromagnet- ically as a simple two sublattice system parallel to the crystallographic c-axis [22]. The low ordering tempera- ture is a characteristic of rare earth spins as the exchange interaction between the highly shielded 4 felectrons is weak, and often the magnetic dipole-dipole interaction dominates [23]. The Gd3+ions are in an8S7=2ground state, a predominantly L= 0 state; crystal eld eects are therefore small and the g-factor is approximately isotropic with a value of 2. GdVO 4is iso-structural with the ubiquitous laser host material yttrium vanadate and is a good laser host in its own right. The sample we used was commerically grown and available o the shelf. The antiferromagnetic resonance [24] seen in our sys- tem can be understood by modelling each sublattice by a single spin-1 =2 and using the following interaction Hamil- tonian H=ASz 1Sz 2BgB
(S1+S2): (1) The rst term describes the (predominantly) magnetic dipole-dipole interaction that causes antiferromagneticarXiv:1911.11311v1 [quant-ph] 26 Nov 20192 ordering and the second term describes the interaction of the spins with an applied magnetic eld. We takeSz 1to be our upward pointing spin ( S1 z1 2, S2 z1 2), and apply a static magnetic eld of strength B0along thez-axis. Assuming only small excitations of the spins we make the Holstein-Primako transforma- tions [25] Sz 1=1 2^ay^a (2) Sz 2=1 2+^by^b (3) S+ 1^a (4) S 2^b; (5) where ^aand^bare bosonic annihilation operators for a spin excitation (spin ip) on sublattice 1 and 2 respec- tively. Substituting into Eq.1 and keeping terms up to second-order in creation/annihilation operators leads to the Hamiltonian H= h!0(^ay^a+^by^b) +gBB0(^ay^a^by^b); (6) where!0=A=2h. Thus in our simple model there are two antiferromagnetic resonance modes each involving Larmor precession of just one sublattice. At zero mag- netic eld, the two modes are degenerate at a frequency that is directly related to the strength of the ordering interaction. The frequency of mode ^ a, which sees the applied magnetic eld in addition to the internal eld, increases linearly with applied magnetic eld, where as the frequency of mode ^b, which sees the applied eld op- posing the internal eld, decreases linearly with applied magnetic eld. Antiferromagnetic resonance in GdVO 4has been seen in electron spin resonance experiments, with a predicted low temperature zero-eld frequency of 34 GHz [26]. Using a g-factor of 2 the antiferromagnetic resonance fre- quencies predicted by Eq. 6 are plotted in Fig. 1(a). At 1:2 T the lower antiferromagnetic resonance branch intercepts the zero frequency axis. At this point a transi- tion occurs between the antiferromagnetic phase and the spin- op phase. In the spin op phase, the energy penalty associated with having one of the sublattices anti-aligned with the external eld is relieved by the spins switching from the crystal c-axis to the crystal a-axis, and tilting a small angle towards the external eld. As the ex- ternal eld is increased, increases, eventually returning the crystal to the paramagnetic phase. The magnetic phase in GdVO 4is shown as a function of temperature and magnetic eld in Fig. 1(b). The experimental setup is shown in Fig. 5 the GdVO 4 sample was placed inside a loop-gap resonator and cooled within a dilution refrigerator. Figure 2(a) shows the transmission through the cavity as the cavity is cooled from room temperature to 4 K, with zero applied mag- netic eld. With decreasing temperature the resonance ~FIG. 1. (a) The antiferromagnetic resonance frequencies of GdVO 4as a function of magnetic eld. Beyond the spin- op transition Eq.6 no longer holds, spin resonance theory in the spin- op state is required instead [27]. (b) The magnetic phase diagram of GdVO 4as a function of applied magnetic eld strength and temperature [28]. 50100 150 200 T (K)11.411.611.8f (GHz) -70-60-50S21(dB) 0 1 2 3 B (T)11.111.211.311.411.5f (GHz) -60-50-40-30S21(dB) Abs. Abs. Abs. Abs.T=300 K B=0 T T=4 K B=0 T T=4 K B=3 T T<2.5 K B=0 T f (GHz) 11.25 34 c FIG. 2. (a) The cavity transmission as cooled from room temperature to 4 K. (b) Cavity transmission as a function of applied magnetic eld with a xed temperature of T= 4 K. (c) A sketch showing the relative locations of the cavity (blue) and spin (orange) resonances at dierent elds and temperatures. frequency of the cavity is seen to increase and broaden until the resonance completely disappears at T35 K. This occurs due to the interaction between the cavity and the spins. At 300 K, there is a very broad paramagnetic resonance centered on 0 Hz. Because it is so broad there is very little absorbtion at the cavity frequency. As the temperature decreases, spin lattice relaxation decreases and the spins resonance narrows, leading to more loss at3 FIG. 3. T= 25 mK, transmission through the cavity as a function of magnetic eld and frequency. The red line indi- cates the onset of the spin- op phase. the cavity frequency. Once the temperature reaches 4 K the attenuation has increased to the point that the cavity resonance disappears. Keeping the temperature xed at 4 K an external eld is applied (along the crystals c-axis) to shift the param- agnetic resonance of the ions away from the cavity. In Fig. 2(b) we see that as the magnetic eld increases and the resonance of the ions is moved, the source of loss to the cavity is reduced allowing the cavity resonance to reappear. The cavity resonance also reappears when the sample is cooled past the transition temperature T2.495 K. Be- low the transition temperature the spins become locked together in a long-range order giving a narrow resonant peak at the zero eld antiferromagnetic resonance fre- quency,34 GHz. The cavity is well detuned from this peak and hence resonates as if it were empty. To investigate the coupling between the antiferromag- netic resonant mode and the microwave cavity, the cavity transmission is measured as the lower antiferromagnetic resonant branch (shown in Fig. 1) is pulled through the cavity resonance via sweeping the applied magnetic eld. With the temperature xed at 25 mK Fig. 3(a) shows the cavity transmission as a function of applied magnetic eld. As the lower antiferromagnetic resonant branch passes through the cavity resonance a clear avoided cross- ing is seen centered at 0:9 T. The shape is not sym- metric, there is a signicant bump in the dressed state frequency at1:1 T due to the onset of the spin- op transition. A detailed investigation of this phenomenon will be left to future work. We t to our data (for B < 1:1 T) the eigenvalues of a simple coupling Hamiltonian H=!c^cy^c+!m^my^m+G ^cy^m+ ^my^c
; (7) 0 100 200 300 400 500 600 700 Tmxc (mK)050100150200250γ (MHz) 00.20.40.60.811.21.41.6 G (GHz)FIG. 4. Blue - polariton linewidth measured at a xed fre- quency of 15.6 GHz as a function of the mixing chamber tem- perature, Orange - coupling strength Gobtained from tting the coupling Hamiltonian Eq.(7). The grey points indicate the region where we haven't ruled out the possibility that the mixing chamber and the sample are dierent temperatures. where!c(!m) is the resonant frequency of the cavity mode ^c(magnon mode ^ m) andG=p Ngis the cou- pling strength, proportional to the number of spins N and single ion coupling strength g. From the t we ob- tain a coupling strength of G= 1.72 GHz, which divided by the central frequency of the cavity ( f0= 11:245 GHz) gives a coupling gure of G=f 0= 0.15 putting the system in the ultrastrong coupling regime. Measuring the linewidth of the polariton modes at points far away from the central cavity frequency gives us an estimate for the linewidth of the antiferromag- netic resonant mode. In order to avoid the background frequency dependence of the experimental components (cables/attenuators/amplier) we measure the magnetic linewidth of the polariton. The linewidth measured in Tesla is then converted into the frequency domain via the relation (!) = (B)gB, where (!) and (B) are the polariton linewidths in the frequency and magnetic domains respectively. Fixing the frequency at 15.6 GHz the linewidth of the upper branch was measured as a function of the mixing chamber temperature ( Tmxc), as shown in Fig. 4. As the temperature of the mixing chamber is lowered the magnon linewidth reduces un- til a temperature of Tmxc200 mK where the linewidth remains roughly constant at 35 MHz. The reduction in linewidth with temperature is expected because of a reduction in multimagnon processes [29]. The constant linewidth we see for Tmxc200 mK could be due to some non-magnetic broadening mechanism, however it could also be explained by imperfect thermalisation between the mixing chamber and our sample leaving the sam-4 ple temperature constant at 200 mK despite the lower mixing chamber temperature. The smooth line shown behind the linewidth data points is a t to the equation =A+BT4, where the coecients were calculated as A= 34:8 MHz and B= 8:61010MHz/K4. The coupling strength Gmeasured via tting Eq. (7) is also shown as a function of Tmxcin Fig. 4. The coupling strength follows a similar trend to the linewidth data, re- maining constant at 1:72 GHz up until Tmxc200 mK, after which it starts decreasing proportional to T4. Fit- ting the expression G=ABT4gives coecients A= 1:7 GHz and B= 9:391012GHz/K4. Our results put us in the ultrastrong coupling regime, G=f 0= 0:15>0:1, the region where counter rotating terms in the coupling Hamiltonian start to become sig- nicant. With a lower frequency resonator ( 1:7 GHz) the deep strong coupling regime could be reached. The challenge will be keeping narrow magnon linewidths in spite of this low frequency which will require low spin temperatures to maintain high spin order. Our results have signicant implications for microwave to optical conversion, however GdVO 4is unlikely to be the best material for this. The Gd3+ions have half lled 4forbitals and as a result the lowest energy 4 f4ftran- sition from the ground-state is in the ultraviolet spectrum around 315 nm. Unfortunately the VO3 4ions absorb strongly at this wavelength. Dierent rare earth crys- tals with better optical properties for upconversion will be the subject of future work. Another exciting prospect provided by rare earth ions is the large spectroscopic gfactors available, particularly in erbium and dysprosium where g15 is common. With this the collective coupling, which is linear in g, could be further improved. Our measurements show that narrow collective mag- netic resonances occur in rare earth crystals. In com- parison to rare earth doped samples the concentration to linewidth ratio seen here is orders of magnitude higher. Quantum magnonics using rare earth crystals has an ex- citing future. The remarkable properties of the rare earth 4f4ftransitions, the easy accessibility of antiferomag- netic resonances as well as the large gfactors, open up many possibilities for improvements to hybrid quantum systems.I. METHODS The experimental setup used to investigate the cou- pling between antiferromagnetic modes and a microwave cavity is shown in Fig. 5. A loop-gap microwave res- onator [30], with a central frequency of 11.245 GHz and a Q factor of 1300, is mounted on the coldest stage of a dilution fridge (BlueFors LD-250) and inside the bore of a 3 T superconducting magnet. The magnetic eld was applied along the crystal c-axis, which was identied by observing the crystal through crossed polarisers. Microwaves are coupled into the resonator using two wire antennas attached to SMA connectors. A vector network analyser is used as the microwave signal source and detector. To reduce the electronic noise the input signal is attenuated at various temperature stages of the dilution fridge, while on the output line a cryoamplier is used to improve the signal-to-noise ratio. An attenuator is also added between the amplier and the cavity; this is to suppress heating due to backaction from the input of the amplier. 36+ 22 GdVO4 FIG. 5. Schematic of the experimental setup. [1] D. Lachance-Quirion, Y. Tabuchi, A. Gloppe, K. Usami, and Y. Nakamura, Applied Physics Express 12, 070101 (2019). [2] Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Us- ami, and Y. Nakamura, Physical Review Letters 113, 083603 (2014).[3] M. Goryachev, W. G. Farr, D. L. Creedon, Y. Fan, M. Kostylev, and M. E. Tobar, Phys. Rev. Applied 2, 054002 (2014). [4] H. Y. Yuan and X. R. Wang, Applied Physics Letters 110, 082403 (2017), https://doi.org/10.1063/1.4977083. [5] C. W. Thiel, T. B ottger, and R. L. Cone, Journal of5 Luminescence Selected papers from DPC'10, 131, 353 (2011). [6] R. Ahlefeldt, M. Hush, and M. Sellars, Physical Review Letters 117, 250504 (2016). [7] G. Kurizki, P. Bertet, Y. Kubo, K. Mlmer, D. Pet- rosyan, P. Rabl, and J. Schmiedmayer, Proceedings of the National Academy of Sciences 112, 3866 (2015). [8] N. J. Lambert, A. Rueda, F. Sedlmeir, and H. G. L. Schwefel, arXiv:1906.10255 [physics, physics:quant-ph] (2019), arXiv: 1906.10255. [9] T. B ottger, Y. Sun, C. W. Thiel, and R. L. Cone, Phys- ical Review B 74, 075107 (2006). [10] M. Zhong, M. P. Hedges, R. L. Ahlefeldt, J. G. Bartholomew, S. E. Beavan, S. M. Wittig, J. J. Longdell, and M. J. Sellars, Nature 517, 177 (2015). [11] C. Clausen, I. Usmani, F. Bussi eres, N. Sangouard, M. Afzelius, H. d. Riedmatten, and N. Gisin, Nature 469, 508 (2011). [12] C. O'Brien, N. Lauk, S. Blum, G. Morigi, and M. Fleis- chhauer, Physical Review Letters 113, 063603 (2014). [13] L. A. Williamson, Y.-H. Chen, and J. J. Longdell, Phys- ical Review Letters 113, 203601 (2014). [14] X. Fernandez-Gonzalvo, S. P. Horvath, Y.-H. Chen, and J. J. Longdell, Physical Review A 100, 033807 (2019). [15] S. Welinski, P. J. Woodburn, N. Lauk, R. L. Cone, C. Si- mon, P. Goldner, and C. W. Thiel, Physical Review Letters 122, 247401 (2019). [16] J. R. Everts, M. C. Berrington, R. L. Ahlefeldt, and J. J. Longdell, Physical Review A 99, 063830 (2019). [17] M. J. Sellars, E. Fraval, and J. J. Longdell, Journal of Lu- minescence Proceedings of the 8th International Meeting on Hole Burning, Single Molecule, and Related Spectro- scopies: Science and Applications, 107, 150 (2004). [18] R. S. d. Biasi and A. A. R. Fernandes, Journal of Physics C: Solid State Physics 16, 5481 (1983). [19] R. M. Macfarlane, R. S. Meltzer, and B. Z. Malkin, Physical Review B 58, 5692 (1998).[20] B. Bleaney, Applied Magnetic Resonance 19, 209 (2000). [21] G. J. Bowden, Australian Journal of Physics 51, 201 (1998). [22] J. Cashion, A. Cooke, L. Hoel, D. Martin, and M. Wells, inElements des terres rares , Vol. 2, Centre National de la Recherche Scientique (Paris) ( Editions du Centre Na- tional de la Recherche Scientique, Paris, 1970) pp. 417{ 426. [23] E. Lagendijk, H. W. J. Bl ote, and W. J. Huiskamp, Physica 61, 220 (1972). [24] C. Kittel, Physical Review 82, 565 (1951). [25] T. Holstein and H. Primako, Phys. Rev. 58, 1098 (1940). [26] M. M. Abraham, J. M. Baker, B. Bleaney, J. Z. Pfeer, and M. R. Wells, Journal of Physics: Condensed Matter 4, 5443 (1992). [27] W. Yung-Li and H. B. Callen, Journal of Physics and Chemistry of Solids 25, 1459 (1964). [28] B. Mangum and D. Thornton, AIP Conference Proceed- ings5, 311 (1972). [29] S. M. Rezende and R. M. White, Physical Review B 14, 2939 (1976). [30] J. R. Ball, Y. Yamashiro, H. Sumiya, S. Onoda, T. Ohshima, J. Isoya, D. Konstantinov, and Y. Kubo, Applied Physics Letters 112, 204102 (2018). II. ACKNOWLEDGEMENTS The authors would like to thank Rose Ahlefeldt, Matt Berrington and Matt Sellars for valuable discussions. This work was supported by Army Research Oce (ARO/LPS) (CQTS) grant number W911NF1810011, the Marsden Fund (Contract No. UOO1520) of the Royal Society of New Zealand, the ARC Centre of Excellence for Quantum Computation and Communication Tech- nology (Grant CE170100012) and the Discovery Project (Grant DP150103699).

2019-11-26

Quantum magnonics is a new and active research field, leveraging the strong collective coupling between microwaves and magnetically ordered spin systems. To date work in quantum magnonics has focused on transition metals and almost entirely on ferromagnetic resonances in yttrium iron garnet (YIG). Antiferromagnetic systems have gained interest as they produce no stray field, and are therefore robust to magnetic perturbations and have narrow, shape independent resonant linewidths. Here we show the first experimental evidence of ultrastrong-coupling between a microwave cavity and collective antiferromagnetic resonances (magnons) in a rare earth crystal. The combination of the unique optical and spin properties of the rare earths and collective antiferromagnetic order paves the way for novel quantum magnonic applications.

Ultrastrong coupling between a microwave resonator and antiferromagnetic resonances of rare earth ion spins

1911.11311v1

Spin conductance in extended thin lms of YIG driven from thermal to subthermal magnons regime by large spin-orbit torque N. Thiery,1A. Draveny,1V. V. Naletov,1, 2L. Vila,1J.P. Attan e,1C. Beign e,1G. de Loubens,3M. Viret,3N. Beaulieu,3, 4J. Ben Youssef,4V. E. Demidov,5S. O. Demokritov,5, 6 A. N. Slavin,7V. S. Tiberkevich,7A. Anane,8P. Bortolotti,8V. Cros,8and O. Klein1, 1SPINTEC, CEA-Grenoble, CNRS and Universit e Grenoble Alpes, 38054 Grenoble, France 2Institute of Physics, Kazan Federal University, Kazan 420008, Russian Federation 3SPEC, CEA-Saclay, CNRS, Universit e Paris-Saclay, 91191 Gif-sur-Yvette, France 4LabSTICC, CNRS, Universit e de Bretagne Occidentale, 29238 Brest, France 5Department of Physics, University of Muenster, 48149 Muenster, Germany 6Institute of Metal Physics, Ural Division of RAS, Yekaterinburg 620041, Russian Federation 7Department of Physics, Oakland University, Michigan 48309, USA 8Unit e Mixte de Physique CNRS, Thales, Universit e Paris-Saclay, 91767 Palaiseau, France (Dated: October 26, 2021) We report a study on spin conductance through ultra-thin extended lms of epitaxial Yttrium Iron Garnet (YIG), where spin transport is provided by propagating spin waves, that are generated and detected by direct and inverse spin Hall eects in two Pt wires deposited on top. While at low current the spin conductance is dominated by transport of exchange magnons, at high current, the spin conductance is dominated by magnetostatic magnons, which are low-damping non-equilibrium magnons thermalized near the spectral bottom by magnon-magnon interaction, with consequent a sensitivity to the applied magnetic eld. This picture is supported by microfocus Brillouin Light Scattering spectroscopy. FIG. 1. (Color online) a) Top view of the lateral device. Two vertical Pt wires (grey) are placed at a distance d= 1:2m apart on top of a 18 nm thick YIG lm (scale bar is 5 m). The non-local conductance I-V(injector-detector) is mea- sured using negative and positive current pulses while rotat- ing the magnetic eld H0in-plane by an angle . Panel b) shows the temperature elevation produced in the Pt injector by Joule heating while increasing the pulse amplitude I. The recent demonstrations that spin orbit torques (SOT) allow one to generate and detect pure spin cur- rents [1{8] has triggered a renewed eort to study magnons transport in extended magnetic lms. This topic is currently recognized as one of the important emerging research direction in modern magnetism [9]. This is because, in contrast to spin transfer process in conned geometries ( e.g.nano-pillars or nano-contacts) where usually the uniform magnon mode dominate the dynamics, very little is known about spin transfer in extended geometries, which have continuous spin-wavespectra containing many modes which can take part in the magnon-magnon interactions. A large eort has con- centrated so far on yttrium iron garnet (YIG), a magnetic insulator, which is famous for having the lowest known magnetic damping parameter [10]. From a purely funda- mental point of view, the studies of magnon transport in YIG by means of the direct and inverse spin Hall eects (ISHE) [2, 11{19] are very interesting, as they provide new means to alter eciently the energy distribution of magnons and, potentially, even to trigger condensation [20]. Contrary to the case of magnons excited coherently, e.g.by means of a ferromagnetic resonance or parametric pumping, when the frequencies of the excited magnons are fully determined by the frequencies of the external signals, the excitation of magnons by means of spin trans- fer process lacks frequency selectivity [21], and, therefore, can lead to their excitation in a broad frequency range. This poses a challenge for the identication of the nature of magnons modes excited by SOT. It has been already shown in [22], that it is convenient and useful to introduce the concepts of subthermal (having energy close to the bottom of the spin wave spectrum) and thermal (having energy close to kBT) magnons. On one hand, it has been well established [23, 24] that subthermal magnons can be very eciently thermalized near the spectral bottom (re- gion of so-called magnetostatic waves) by the intensive magnon-magnon interaction, whose decay rate between quasi-degenerate modes increases with power, to reach a quasi-equlibrium state by a non-zero chemical potential [23{25] and an eective temperature [26]. On the other hand, it has been shown, both experimentally and the-arXiv:1702.05226v3 [cond-mat.mtrl-sci] 17 Sep 20172 oretically [26], that the groups of subthermal and ther- mal magnon are eectively decoupled from each other, as under the intensive parametric pumping one can reach a state where the eective temperature of subthermal magnons exceeds the real remperature characterizing the thermal magnons by a factor of 100. Under spin transfer process, whose eciency is known to increase with decreasing magnon frequency, in con- ned geometries with localized spin-current injection ( i.e. when there are no quasi-degenerate modes), it has been shown, that one can reach current induced coherent GHz- frequency magnon dynamics in YIG [16, 17, 27]). In extended thin-lms, the recently discovered non-local magnon transport [28{32] suggests that the magnon transport properties of YIG lms subjected to small SOT are dominated by thermal magnons, whose number over- whelmingly exceeds the number of other modes at any non-zero temperature. The interesting challenge is to elucidate what will happen to this spectrum (in partic- ular the interplay between the thermal and subthermal part [33]) when one applies large SOT to a magnon con- tinuum. We propose herein to measure the room temperature spin conductance of YIG lms when the driving current is varied in a wide range magnitudes [34, 35] creating, rst, a quasi-equilibrium transport regime, and, then, driving the system to a strongly out-of-equilibrium state. To reach this goal the spin current injected in the YIG by SOT shall be increased by more than one order of magni- tude compared to previous works, while simultaneously reducing the lm thickness by also an order of magnitude, using ultra-thin lms of YIG grown by liquid phase epi- taxy (LPE) [14, 36]. A series of lateral devices have been patterned on a 18 nm thick YIG lms. Ferromagnetic res- onance (FMR) characterization of the bare lm are sum- marized in Table 1. On these lms, we have deposited Pt wires, 10 nm thick, 300 nm wide, and 20 m long. The measured resistance of the Pt wire at room temperature isR0= 1:3 k . The lateral device geometry is shown in FIG.1a. One monitors the voltage Valong one wire as a current I ows through a second wire separated by a gap of 1.2 m. Here the Pt wires are connected by 50 nm thick Al electrodes colored in yellow. Since large amount of electrical current needs to ow in the Pt, a pulse method is used to reduce signicantly Joule heat- ing. In the following the current is injected during 10 ms pulses series enclosed in a 10% duty cycle. Temperature sensing is provided by the change of relative resistance of the Pt wire during the pulse. In FIG.1b, we have plotted Pt(RIR0)=R0as a function of the current I, where the coecient Pt= 254 K is specic to Pt. We observe that the pulse method allows to keep the absolute tem- perature of our YIG below 340 K [37] at the maximum current amplitude of 2.5 mA. Avoiding excessive heating of the YIG is crucial because, in a joint review paper [38], it is shown that epitaxial YIG lms grown by LPETABLE I. Summary of the physical properties of the materials used in this study. YIGtYIG(nm) 4M s(G)YIG
H0(Oe) 18 1 :61034:41043.7 PttPt(nm)( .cm) YIGjPtg"#(m2) 10 19.5 2 :410331018 FIG. 2. (Color online) Angular dependence of the non-local voltagesV measured while inverting the polarity of the ap- plied eldH0=2kOe (red/blue) respectively for a) nega- tive and b) positive current pulses I=1:5mA. The mea- sured signal can be decomposed c) and d) in three compo- nents:  (green): the signal sum,
(orange): the signal dif- ference and Vk: the oset; respectively even/odd, odd/even in eld/current, and an independent contribution (dashed). Panel e) shows the current dependence of the amplitude  and
. behave as a large gap semiconductor, with an electrical resistivity that decreases exponentially with increasing temperature following an activated behavior. As shown in Ref.[38], at 340 K, however, the electrical resistivity of YIG remains larger than 106
cm and thus the YIG can still be considered a good insulator ( R> 30 G ) over the current range explored herein. The lateral device is biased by an in-plane magnetic eld,H0set at a variable polar angle , with respect to the perpendicular of the Pt wires. FIG.2a-d displays the results when I= 1:5 mA andH0= 2 kOe [39]. For each value of , 4 measurements V are performed corre- sponding to the 4 combinations of the polarities of H0 andI(the polarity convention is dened in FIG.1a).3 FIG.2a and 2b show the raw data obtained respectively for negative and positive current pulses. Clearly the non- local voltage oscillates around an oset, Vk, dened as the voltage measured at =90. This oset is indepen- dent of the current polarity and its amplitude scales with the temperature elevation of Pt produced by Joule heat- ing (see FIG.2c in [38]). We ascribe it to thermoelectric eects produced by a small temperature dierence at the two PtjAl contacts of the detector circuit [40]. By con- trast, the anisotropic part of the voltage is ascribed to magnons transport. To gain more insight, the data are sorted according to their symmetry with respect to the ( H0;I)-polarity. This is done in FIG.2c and 2d by constructing the signal sum = (V+ +V )=2Vk(green tone) even in eld and the signal dierence
= (V+ V )=2 (orange tone) odd in eld. This separation is exposed in their angular dependences, which follow two dierent behaviors, one in cos2, the other one in cos , respectively. The solid lines in FIG.2c and in 2d are a t by these two functions. Comparing the behavior, we observe that the signal  is odd in current, while the signal
is even in current. As noticed in ref [28], these symmetries of  and
are the hallmark of respectively SOT [17] and spin Seebeck eects [41, 42]. Hereafter, we shall use the t of the whole angular dependence as a mean to extract precisely the amplitude of  and
at = 0. FIG.2e shows their evolution as a function of current. One observes that the correspondence between the sym- metries of  and of
with respect to the polarities of H0andIis respected (within our measurement accuracy) on the whole current range. While
approximately fol- lows the parabolic increase of the Pt temperature (cf. FIG.1a), as expected for thermal eects, the interesting novel feature is the fact that  deviates from a purely linear transport behavior at large I. It is important also to notice that, when the high/low binding posts of the current source and voltmeter are biased in the same ori- entation (cf. FIG.1a), the sign of (
I)<0. This is a signature that the observed non-local voltage is produced by ISHE and not by leakage electrical currents inside the YIG [38], although these eects are only expected to oc- cur at much higher temperatures ( >370 K[43]). While in both scenarii the induced electrical current ows in the same direction in the two parallel Pt wires, for ISHE, the YIG acts as a source and the potential increases along the current direction [44], in contrast, for Ohmic loss, the YIG acts as a load and the potential drops along the current direction ( i.e.(
I)>0 cf. FIG2b in [38]). We should though add that selecting the component of the non-local voltage that is -dependent is another eective mean to eliminate Ohmic contribution, since the later are independent of the in-plane orientation of H0. We have repeated this measurement on other devices either on the same lm or on dierent LPE YIG lms of similar thick- ness. On all the devices, we observe an up-turn of  at FIG. 3. (Color online) Current dependence of the absolute sum signal  averaged over the two current polarities. The dashed line is a linear t of the low current regime and the ar- row marks the onset at which the spin transport deviates from linear behavior. Spin transport excited by SOT can be sepa- rated in two independent channels: a linear contribution (t) taken on by magnons transporting thermal heat and (s), an additional magnons' conduction channel that emerges above Ic. Variation of  as a function of magnetic eld for two dierent current intensities b) above and c) bellow Ic. the same current density. While the sign of (
I) is al- ways negative, this is not the case for the sign for (

I), which depends on the lm quality (the same (

I) sign is observed for all devices on the same lm but could change depending on the lm quality). Further progress on the later issue requires a better understanding of the dierent phenomena contributing to thermal eects (
- signal) and the means to separate them. In the following, we shall concentrate exclusively on the non-linear behav- ior of  which measures the number of magnons created by SOT relatively to the number of magnons annhilated by SOT while being immune to eects caused by Joule heating. Using open dots, we have plot in FIG.3a both the vari- ation ofjjandj+jas a function of the current inten- sity. Both data set show clearly the emergence of a new spin transport channel at large current densities, as ev- idence by the deviation from a purely linear conduction regime. Since both quantities jjfollow the same be- havior on the whole current range, for the sake of simplic- ity we shall call simply  (dark green) their averaged. At low current, the SOT signal follows rst a linear behavior (t)which has been shown to be dominated by thermal magnons transport [28] [45]. We shall dene (s)the ad- ditional conduction contribution, i.e.the deviation from the extrapolated linear behavior. Quite remarkably the enhancement of the conductance due to (s)occurs very gradually. We emphasize that such a low rise is very dierent from the sudden surge of coherent magnons observed at the critical thresold in con- ned geometries [46, 47]. Fitting a straight line through the low current regime,  I2[0;0:9]mA, and the high cur- rent deviation,  I2[1:6;2:3]mA, the intersection provides4 an estimation of the onset current of this new conduction channel,Ic1:5 mA, which corresponds to a current densityJc51011A/m2. This value is very close to the threshold current for damping compensation of coherent modes observed at the same applied eld ( H0= 2 kOe) in micron-sized disks [17] and stripes [48] with similar characteristics. More insight about the nature of the magnons excited aboveIccan be obtained by studying the eld depen- dence of  [49]. The results are shown in FIG.3b and 3c for two values of the current I= 0:4 and 2.5 mA, re- spectively below and above Ic. While in the eld range explored, the signal is almost independent of H0when I <I c, it becomes strongly eld dependent when I >I c. This dierent behaviors are consistent with assigning the spin transport to thermal magnons below Icand mainly to subthermal magnons above Ic. In the former case, the magnons' energy is of the order of the exchange energy, which is much larger than the Zeeman energy, while in the latter case, because of their long wavelength, their energy is of the order of the magnetostatic energy. In consequence,  is expected to increase with decreasing eld at xed I, because of the associated decrease of Ic. The behavior scales well with the reduced quan- tityI=Ic. This is shown by the solid line in FIG.3b, which displays the expected eld dependence of 1 =Ic(H0) [17]) where Ic/(!H+!M=2) (+
H0=(2!K)), where !H= H0and!M= 4 M s, being the gyromagnetic ratio, and!K=p !H(!H+!M) is the Kittel's law. We have used here the amount of inhomogeneous broaden- ing,
H0= 1:5G (probably position dependent), as an adjustable parameter, while the value of the other pa- rameters are those extracted from Table.1. The above interpretation has been checked by preform- ing microfocus Brillouin light scattering ( -BLS) in the sub-thermal energy range. For this measurement, we have used a second series [50] of non-local devices, where the Pt thickness has been reduced to 7 nm (thus compar- ison of the results between the 2 series should be done by juxtaposing data obtained with indentical current densi- ties, cf. upper scale). FIG.4a and 4b show on a logarith- mic scale the spectral distribution of the BLS intensity, J, a) underneath the injector and b) underneath the detec- tor, which are here separated by d= 0:7m. The distri- bution is measured at I=2 mA ( i.e.9:51011A/m2) while the eld is set to H0= +2 kOe and = 0. In both cases, an enhancement of the subthermal magnons population is observed when ( I
H0)<0 (blue), which corresponds to the conguration where the SOT com- pensates the damping (cf convention in FIG1a). The measurement for the opposite case ( I
H0)>0 (red) pro- vides a reference about the out-of-equilibrium state pro- duced by Joule heating. The maximum intensity of the red curve indicates the resonance frequency of the Kittel mode,!K=2at the corresponding temperature. This is because the -BLS response function is centered around FIG. 4. (Color online) Micro-BLS studies of the subthermal magnons spectrum at H0= +2 kOe a) underneath the in- jector and b) underneath the detector 0.7 m away. The left column show the spectral distribution of the BLS intensity J measured at I=2mA. The arrows indicate the Kittel fre- quency. The dierence J(grey area) indicate the spectral distribution of the magnons excited by SOT relative to the ones annihilated. The panel c) plots the current evolution of the integrated intensity Junderneath the injector. the long wavelength magnons. Indeed, the detected sig- nal decreases once the magnon wavelength is smaller than the spot size (approximately 0.4 m: diraction limited). In order to isolate the contribution produced by SOT, we substract the spectral contribution measured at + I to the one measured at I(grey shaded area). This al- lows to cancel out the spectral deformation produced by Joule heating but, as for the -signal, this only measures the enhancement of the magnons created by SOT rela- tive to the magnons annihilated by SOT. One can clearly see on the shaded data that SOT enhances the magnons population in a spectral window between the Kittel fre- quency and the bottom of the magnon manifold. Next, we have plotted in FIG4c how the spectral integration of this dierential signal J=R Jd!varies as a func- tion of the current amplitude underneath the injector. One observes a regime of linear rise at small current, fol- lowed by a growth above Jc51011A/m2in a similar fashion as the one reported in FIG3a. The -BLS exper- iment thus provides a direct evidence that an additional spin conduction channel has indeed emerged in the GHz frequency range (subthermal) at large current when SOT is in the range to compensate the damping. It also shows that the magnons newly created are spread at the bottom of the magnon manifold. In summary, we have shown that while at low values of the spin current the main contribution to the spin conductance comes from thermal magnons, the subther- mal magnons mainly determine the magnon transport at high values of the spin current, comparable to the criti- cal magnitude at which damping compensation of coher- ent magnons takes place. We believe that our current ndings are not only important from the from the funda- mental point of view, but might be also useful for future5 applications. While transport of thermal magnons are dicult to control due to their relatively high energies, the subthermal magnons could be eciently controlled by variation of relatively weak magnetic elds. This research was supported in part by the CEA pro- gram NanoScience (project MAFEYT), by the priority program SPP1538 Spin Caloric Transport (SpinCaT) of the DFG and by the program Megagrant 14.Z50.31.0025 of the Russian ministry of Education and Science.The work at Oakland University was supported by the Grants Nos. EFMA-1641989 and ECCS-1708982 from the NSF of the USA, by the CND, NRI and by DARPA. VVN ac- knowledges fellowship from the emergence strategic pro- gram of UGA, and Russian competitive growth program. We thank G. Zhand, T. van Pham, A. Brenac for their help in the fabrication of the lateral devices. Corresponding author: oklein@cea.fr [1] S. O. Valenzuela and M. Tinkham, Nature 442, 176 (2006). [2] Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. 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Castel, N. Vlietstra, B. J. van Wees, and J. Ben Youssef, Phys. Rev. B 86, 134419 (2012). [37] YIG spontaneous magnetization decreases by about 4G/C. [38] N. Thiery, V. V. Naletov, L. Vila, A. Marty, J.-F. Jacquot, G. de Loubens, M. Viret, A. Anane, V. Cros, J. B. Youssef, V. E. Demidov, S. O. Demokritov, and O. Klein, (to be published) (2017). [39]H0is much higher than the Oersted eld of the current (<50 Oe). [40] Considering the Seebeck coecient of Pt jAl, the 100Voset measured in FIG.1b, corresponds to a tem- perature dierence of less than 0.03C between the top and bottom electrodes. [41] K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi, J. Ieda, T. Ota, Y. Kajiwara, H. Umezawa, H. Kawai, G. E. W. Bauer, S. Maekawa, and E. Saitoh, Nature Mater. 9, 894 (2010).[42] H. Jin, S. R. Boona, Z. Yang, R. C. Myers, and J. P. Heremans, Phys. Rev. B 92, 054436 (2015). [43] This corresponds to injecting in the Pt wire current den- sities>1:01012A
m2. [44] This eect is independent of the sign of the spin Hall angle. [45] The non-local linear resistance between the two Pt wires is (t)=I= 0:019 m . [46] A. Slavin and V. Tiberkevich, IEEE Trans. Magn. 45, 1875 (2009). [47] A. Hamadeh, G. de Loubens, V. V. Naletov, J. Grollier, C. Ulysse, V. Cros, and O. Klein, Phys. Rev. B 85, 140408 (2012). [48] M. Evelt, V. E. Demidov, V. Bessonov, S. O. Demokri- tov, J. L. Prieto, M. Muoz, J. Ben Youssef, V. V. Nale- tov, G. de Loubens, O. Klein, M. Collet, K. Garcia- Hernandez, P. Bortolotti, V. Cros, and A. Anane, Appl. Phys. Lett. 108, 172406 (2016). [49] L. J. Cornelissen and B. J. van Wees, Phys. Rev. B 93, 020403 (2016). [50] This second series has been used to investigate the elec- trical properties of YIG thin lms at high temperature [38].

2017-02-17

We report a study on spin conductance in ultra-thin films of Yttrium Iron Garnet (YIG), where spin transport is provided by propagating spin waves, that are generated and detected by direct and inverse spin Hall effects in two Pt wires deposited on top. While at low current the spin conductance is dominated by transport of thermal magnons, at high current, the spin conductance is dominated by low-damping non-equilibrium magnons thermalized near the spectral bottom by magnon-magnon interaction, with consequent a sensitivity to the applied magnetic field and a longer decay length. This picture is supported by microfocus Brillouin Light Scattering spectroscopy.

Spin conductance of YIG thin films driven from thermal to subthermal magnons regime by large spin-orbit torque

1702.05226v3

arXiv:1702.00846v2 [cond-mat.quant-gas] 11 Dec 2017Magnon Condensation and Spin Superfluidity Yury M. Bunkovaand Vladimir L. Safonovb,c (a)Kazan Federal University, Kremlevskaya 18, 420008 Kazan, R ussia (b)Mag and Bio Dynamics, Inc., Granbury, TX 76049, USA (c)Physical Science Department, Tarrant County College - South Campus, Fort Worth, TX 76119, USA Abstract We consider the Bose-Einstein condensation (BEC) of quasi- equilibrium magnons which leads to spin superfluidity, the coherent quantum transfer of magn etization in magnetic material. The critical conditions for excited magnon density in ferro- an d antiferromagnets, bulk and thin films, are estimated and discussed. It was demonstrated that only t he highly populated region of the spectrum is responsible for the emergence of any BEC. This fin ding substantially simplifies the BEC theoretical analysis and is surely to be used for simulat ions. It is shown that the conditions of magnon BEC in the perpendicular magnetized YIG thin film is fulfillied at small angle, when signals are treated as excited spin waves. We also predictth at the magnon BEC should occur in the antiferromagnetic hematite at room temperature at much low er excited magnon density compared to that of ferromagnetic YIG. Bogoliubov’s theory of Bose-E instein condensate is generalized to the case of multi-particle interactions. The six-magnon re pulsive interaction may beresponsible for the BEC stability in ferro- and antiferromagnets where the f our-magnon interaction is attractive. PACS numbers: 75.45.+j Keywords: Bose-Einstein condensation, magnons, YIG, hematite 1INTRODUCTION Spin deviations from the magnetic order in a magnetic material (ferr omagnet, antifer- romagnet or ferrites) are manifested by spin waves and their quan ta, magnons. Magnons are quasiparticles which represent a very useful quantum theore tical tool to describe various dynamic and thermodynamic processes in magnets in terms of magno n gas. Since magnons have magnetic moments, the external alternating magnetic field ca n excite extra magnons and increase the disorder in the magnetic system. However, in cert ain conditions, the in- crease of magnon density leads to a new state, so-called, magnon c ondensate, in which a macroscopic number of magnons forms a coherent quantum state (see, e.g., [1], [2]). This macroscopic state can significantly change the properties of magn on gas, its dynamics and transport. An example is the phenomenon of quasi-equilibrium Bose- Einstein condensation (BEC) of excited magnons on the bottom of their spectrum as a sing le-particle long-range coherent state of quantum liquid. This state generate an uniform lo ng-lived precession of spins formed by quantum specificity of the magnon gas when the mag non density exceeds certain critical value. The spatial gradients of this state exhibit a s pin superfluidity, the non-potential transport of deflected magnetization. The spin su perfluidity is an extremely interesting phenomenon for both fundamental and applied studies . It should be emphasized that the main paradigm of magnetic dynamics, the Landau-Lifshitz- Gilbert equation, does not contain complete information about the Bose-Einstein condens ate of magnons. BEC is the principal result of quantum statistics and for magnons it can ex ist at room and even higher temperatures. For the first time the existence of quasi-equilibrium Bose condensat e was demonstrated in the experiment with nuclear magnons in the superfluid antiferroma gnetic liquid crystal 3He-B in 1984 [3]. The theoretical explanation of this phenomenon [4] was developed on the basis of global Ginzburg-Landau energy potential. A similar approac h was later developed to explain the atomic BEC [5]. In the experiments with an antiferromag netic3He-B, the following phenomena were observed: a) transport of magnetizatio n by spin supercurrent between two cells with magnon BEC; b) phase-slip processes at the c ritical current; c) spin current Josephson effect; d) spin current vortex formation ; d) Goldstone modes of magnon BEC oscillations. Comprehensive reviews of these studies ca n be found in Refs.[6– 8]. Currently magnon BEC found in different magnetic systems: i) in an tiferromagnetic 2superfluid3He-A [9, 10]; ii) in in-plane magnetized yttrium iron garnet Y 3Fe5O12(YIG) film (with two minima in the magnon spectrum) [11, 12] and in normally ma gnetized YIG film [13]; iii) in antiferromagnets MnCO 3and CsMnF 3with Suhl-Nakamura indirect nuclear spin-spin interaction [14–16]. An explanation of analogy between the atomic and magnon BEC is given in Ref. [17]. A microscopic theory of quasi-equilibrium magnon BEC was developed in Refs.[18]-[21] (”KS theory”). It was predicted that the external strong pump ing of magnons leads to a rapid growth of magnon density and saturation. This state can be c onsidered in terms of weakly non-ideal gas of ”dressed” magnons in a thermodynamic qua si-equilibrium with an effective chemical potential µand effective temperature T. The dressed magnon energy is defined by εk=ε(0) k+δεk, whereε(0) k=/planckover2pi1ωkis the energy spectrum of bare magnons and δεk is the energy shift due to magnon gas nonlinearities. Magnon-magno n scattering processes retainthetotalnumberofdressedmagnonsinthesystemandhold theirdistributionfunction of the form nk=/parenleftbigg expεk−µ kBT−1/parenrightbigg−1 . (1) The instability at µ= minεkin the quasi-equilibrium magnetic system is an analog of BEC phenomenon for the bottom dressed magnons. The distribution (1 ) seems to underlie the phenomenon of spin superfluidity, since it nullifies the integral of fou r-magnon collisions I(4){nk} ∝/integraldisplay d3k1d3k2d3k3|Φ4(k,k1;k2,k3)|2(2) ×[(nk+1)(nk1+1)nk3nk4−nknk1(nk2+1)(nk3+1)] ×δ(εk+εk1−εk2−εk3)∆(k+k1−k2−k3) and thus this energy loss channel vanishes. KS theory qualitatively explained the parallel pumping experiments ([2 2],[23] (YIG at room temperature) and [24] (nuclear magnons in CsMnF 3), where the accumulation of magnons at the bottom of the spin wave spectrum was observed. O ne and a half decade later, purposeful experiment [11] directly demonstrated BEC of quasi equilibrium magnons in the thin film of YIG. Subsequent experimental studies have shown qualitative correspon- dence with the predicted distribution of excited magnons [25] and ag reement with the BEC under noisy pumping [26]. 3In this paper we analyze critical conditions of quasi-equilibrium magno n BEC in ferro- andantiferromagnets, bulkandthinfilms, andevaluatethepossibilit iesoftheirexperimental achievements. BEC OF BOSE PARTICLES Let us first briefly discuss BEC of real bose particles. Their distribu tion is defined by Eq.(1), where εk= (/planckover2pi1k)2/2mis the kinetic energy of particle with the wave vector kand massm. The total number of bosons in the system is N(µ,T) =N=Vs/integraldisplay nkd3k (2π)3, (3) whereVsis the volume of the system. For the critical condition µ= minεk, from (3) follows well-known formula for the BEC critical temperature versus the de nsity of bosons: TBEC=κ0/planckover2pi12 kBm/parenleftbiggN Vs/parenrightbigg2/3 , κ0=2π /bracketleftbig ζ/parenleftbig3 2/parenrightbig/bracketrightbig2/3≃3.31. (4) It is interesting to note that the BEC is formed mainly by bosons with h igh populations when Eq.(1) can be written as nk≃kBT εk−µ. (5) Let us prove it by direct calculation. Substituting the high temperat ure population (5) into Eq.(3) and cutting the upper integral limit by the thermal energy εT≃kBT, one obtains: TBEC≃˜κ0/planckover2pi12 kBm/parenleftbiggN Vs/parenrightbigg2/3 ,˜κ0=π4/3 21/3≃3.65. (6) We see that the only difference between Eqs.(4) and (6) is a slightly diff erent (∼10%) numerical factor. The fact that the high population Eq.(5) is dominant does not mean th at the BEC phenomenon is a classical one. The criterion of classical Maxwell-Bolt zmann statistics exp(µ/kBT)≪1 in this case can be written as (see, e.g., [27]): exp(µ/kBT) =/bracketleftbiggVs N/integraldisplay exp/parenleftbigg −εk kBT/parenrightbiggd3k (2π)3/bracketrightbigg−1 ≪1, (7) or 4N Vsλ3 T=N Vs/parenleftbigg2π/planckover2pi12 mkBT/parenrightbigg3/2 ≪1, (8) whereλTis the thermal de Broglie wavelength. Substituting BEC temperatur e Eq.(6) into (8), we obtain the opposite relation: 2 .26>1, which obviously corresponds to a degenerate bose gas. BEC OF MAGNONS Now let us consider a Bose-Einstein condensation of so-called, “dre ssed” magnons as an instability in the externally pumped quasi-equilibrium magnon gas. The t otal number of magnons N(µ,T)is equal to thenumber of thermal magnons N(0,T) ata giventemperature Tand the number of magnons Npcreated by external pumping. So far as the energy shift of dressed magnons is usually much less than the energy of bare mag nonsδεk≪minε(0) k, for simplicity we can approximate εk≃ε(0) k. BEC in a ferromagnet Consider a ferromagnet with the quadratic spectrum (we neglect d etails of the dipole- dipole interactions): εk=ε0+εex(ak)2. (9) Hereεexis the exchange interaction constant and ais the elementary cell linear size. The quasi-equilibrium BEC will be mainly determined by pumping if the number o f pumped magnons is much greater than the thermal magnon number Np≫N(0,T). In this case we obtain an analog of Eq.(4): TBEC=κ02εex kB/parenleftbigg a3Np Vs/parenrightbigg2/3 , (10) or, TBEC≃˜κ02εex kB/parenleftbigg a3Np Vs/parenrightbigg2/3 (11) in the high-population approximation. 5The above formula, however, does not work for a BEC estimate if Np/lessorsimilarN(0,T). Using a high-population approximation, we write Np=N(µ,T)−N(0,T) ≃Vs/integraldisplayεT ε0/parenleftbiggkBT εk−µ−kBT εk/parenrightbiggk2dk 2π2, (12) and obtain at µ=ε0 Np Vs≃kBTBEC 4πa3ε1/2 0 ε3/2 ex, TBEC≃4πεex kB/parenleftbiggεex ε0/parenrightbigg1/2/parenleftbigg a3Np Vs/parenrightbigg . (13) These formulas coincide with the accuracy of notations with the exa ct calculation given in Ref.[2]. This is one more direct proof that BEC is formed by the high-po pulated part of spectrum. An estimate for YIG, where εexa2//planckover2pi1= 0.092 cm2s−1forε0//planckover2pi1= 2π×2.5 GHz gives TBEC≃2.14×10−17(Np/Vs) cm3K. Thus, we obtain a room-temperature BEC TBEC≃300 K at the pumped magnon density Np/Vs= 1.41×1019cm−3that in order of magnitude corresponds to the experiment [11]. As in the above case of particles, the opposition of high density of ma gnons to their high population makes the classical criterion exp[( µ−ε0)/kBT]≪1 inapplicable to the case of condensation when µ=ε0. As in the previous section, we can write the criterion for Maxwell-Boltzmann statistics exp[(µ−ε0)/kBT] =/bracketleftBigg Vs N(µ,T)/integraldisplay exp/parenleftBigg −εex(ak)2 kBT/parenrightBigg d3k (2π)3/bracketrightBigg−1 ≪1,(14) or N(µ,T) Vsλ3 T=N(µ,T) Vs/parenleftbigg4πεexa2 kBT/parenrightbigg3/2 ≪1. (15) In the case of high temperatures we have N(µ,T)≃N(0,T). Substituting the thermal density N(0,T) Vs=/parenleftbiggkBT 2κ0εexa2/parenrightbigg3/2 (16) 6into (15), we obtain the opposite relation: 2 .61>1. In other words, magnon gas which undergoes Bose-Einstein condensation is always a degenerate bos e gas. Thus the assertion of recent publication [28] “that the experimentally observed conde nsation of magnons in yttrium-iron garnet at room temperature is a purely classical phen omenon” is untenable. BEC in an antiferromagnet Consider now the magnon energy of the form εk=/radicalBig ε2 0+ε2 ex(ak)2. (17) This is typical for magnons in the ”easy-plane” (or, canted) antife rromagnets. Taking into account that k=/radicalbig ε2 k−ε2 0 aεexandkdk=εkdεk ε2exa2, in the high-population approximation, one can write N(µ=ε0,T) Vs≃kBT 2π21 a3ε3ex/integraldisplayεT ε0/radicalbiggε+ε0 ε−ε0εdε. (18) IfNp≫N(0,T), forkBT≫ε0we obtain TBEC≃(2π)2/3εex kB/parenleftbigg a3Np Vs/parenrightbigg1/3 . (19) IfNp/lessorsimilarN(0,T), one can rewrite Eq.(12) in the form: Np Vs≃kBT 2π2ε0 a3ε3ex/integraldisplayεT ε0/radicalbiggε+ε0 ε−ε0dε. (20) After integration, at kBT≫ε0we obtain Np Vs≃(kBTBEC)2 2π2ε0 a3ε3 ex, or, TBEC≃√ 2πεex kB/parenleftbiggεex εo/parenrightbigg1/2/parenleftbigg a3Np Vs/parenrightbigg1/2 . (21) Note that the BEC temperature for antiferromagnet has lower po wer dependence on small parameter a3Np/Vs≪1 and therefore one can expect much lower densities of pumped magnons to achieve condensation. An estimate for α−Fe2O3(hematite), where εexa//planckover2pi1≈ 24×105cm/s forε0//planckover2pi1= 2π×2.5 GHz gives TBEC≈10−6(Np/Vs)1/2cm3/2K. Thus we obtain a room-temperature BEC, T= 300 K at Np/Vs= 0.89×1017cm−3. This estimate is more lower than corresponding estimate for a ferromagnetic YIG. This means that hematite is a very attractive object to observe BEC of magnons experiment ally. 7BEC IN A FERROMAGNETIC FILM Let us now consider an ultra-thin ferromagnetic film. There are two principal cases: 1) external magnetic field His parallel to the the film surface and 2) His perpendicular to this surface. In the first case, the BEC condition µ= min/planckover2pi1ωkgives us two minima at ±kmin∝ne}ationslash= 0. This case was demonstrated experimentally for the YIG film in Ref.[11], where the critical density of pumped magnons was estimated numerically. Later, in Ref .[1] it was considered analytically. Here we focus on the second case with just one energy minimum at k= 0. The magnon spectrum of the perpendicular magnetized ultra-thin f erromagnetic film can be written as [29]: ωk=/braceleftbig [ωH+ωex(ak)2][ωH+ωex(ak)2+ωMf(kτ)]/bracerightbig1/2, (22) whereγ= 2π2.8 MHz/Oe is the gyromagnetic ratio, ωH=γHi,Hi=He−4πMs+H⊥is an effective internal magnetic field, H⊥is the perpendicular anisotropy field, Ms= 139 Oe is the saturation magnetization, ωM= 4πγMs= 2π×4.9 GHz,ωexa2= 2π×1.09×10−2 Hz cm2is the exchange constant, f(kτ) = 1−[1−exp(−kτ)]/kτ,τis the thickness of the film. All numerical parameters are given for YIG. Spin waves are ass umed to be propagated only in the film plane and there is a uniform magnetization along the dept h. The critical density of pumped magnons at temperature Tcan be estimated by the following equation: Np As=N(µ,T) As−N(0,T) As, (23) where the sample volume isreplaced by thefilm area As. The Eq.(23) inthe high-population approximation has the form: Np As≈kBT 2π/integraldisplaykT 0/parenleftbigg1 /planckover2pi1ωk−µ−1 /planckover2pi1ωk/parenrightbigg kdk, (24) wherekTcorresponds to the frequency ωk≃kBT//planckover2pi1. This integral at µ→/planckover2pi1ω0has a logarithmic divergence for an infinitely large film. However, for a finite film we have a magnetostatic mode on the bottom, which can be separated from t he spin-wave spectrum by a small gap ∆ ω. Thus, we can estimate Eq.(24) as Np As≈kBT 4π/planckover2pi1ωexa2ln/parenleftBigω0 ∆ω/parenrightBig . (25) Note that this formula corresponds to the model of ultra-thin film, in which the magnon dynamics is considered in two dimensions. Magnetic excitations acros s the film plane are 8discrete and they interact weakly with magnons in the plane of the film . For this reason they practically do not affect the quasi-equilibrium in the two-dimensio nal system. Formula (25) is convenient for simple estimates, which can be refined by nume rical calculations with accounting for all magnetic degrees of freedom. Critical angle Let us now find a critical angle of the magnetic moment deviation from the equilibrium, which is assumed to correspond to the critical number of excited ma gnons. This angle is defined by the ratio of perpendicular spin component to its longitudin al component tan θ= S⊥/Sz. The perpendicular component is equal to S⊥=/radicalBig S2x+S2y=/radicalbigg S+S−+S−S+ 2(26) ≈√ 2Sa∗a=/radicalbig 2SNp. Substituting Sz≃S, for small angles one obtains θ≈/radicalbigg 2Np S=/radicalbigg 2/planckover2pi1γ MsNp Asτ. (27) For the film thickness τ= 1µm,ω0= 2π×2.5 GHz, ∆ ω= 2π×1 Hz at room temperature T= 300 K we have θfilm≈0.044 (2.5◦). An estimate for a bulk material at the same conditions gives θbulk≈0.061 (3.5◦). Taking into account finite thickness of the film, one can expect the experimental value of the angle will be within θfilm< θ < θ bulk. For comparison, the BEC deflection angle in the antiferromagnetic liquid crystal3He-B was about 10−3[1]. BEC STABILITY The critical density is required but not a sufficient condition for a unif orm magnon BEC. For the BEC stability it is important to consider the interaction betwe en magnons. The Hamiltonian of magnon system described by creation ( b† k) and annihilation ( bk) bose oper- ators can be written in the form: H=/summationdisplay k(εk−µ)b† kbk+H4+H6+H8+... (28) 9where the magnon-magnon interaction terms are H4=1 2/summationdisplay 1,2;3,4Φ4(k1,k2;k3,k4)b† 1b† 2b3b4∆(k1+k2−k3−k4), H6=1 3/summationdisplay 1,2,3;4,5,6Φ6(k1,k2,k3;k4,k5,k6)b† 1b† 2b† 3b4b5b6 ×∆(k1+k2+k3−k4−k5−k6) and so fourth. According to Bogoliubov’s theory (see, e.g.,[2],[30]), we have to single out classical condensate amplitudes with k=0:b† 0=b0=√N0,N0=N−N′. HereN= /summationtext kb† kbkis the total magnon number and N0is the magnon number in the condensate. Assuming that N′/N≪1, we can reduce the Hamiltonian (28) to H=H0+H2, where H0= (ε0−µ)N0+1 2T4(0)N2 0+1 3T60N3 0+... (29) is the condensate energy and H2=/summationdisplay k/negationslash=0/bracketleftbigg Akb† kbk+Bk 2/parenleftBig bkb−k+b† kb† −k/parenrightBig/bracketrightbigg . (30) Here Ak=εk−µ+2T4(k)N0+3T6(k)N2 0+... (31) T4(k) = Φ 4(k,0;k,0), T6(k) = Φ 6(k,0,0;k,0,0)... and Bk=S4(k)N0+S6(k)N2 0+... (32) S4(k) = Φ 4(k,−k;0,0), S6(k) = Φ 6(k,−k,0;0,0,0)... Diagonalizing the quadratic form (30) by the linear canonical transf ormation bk=ukck+ vkc† −k, we find that H2=U2+/summationdisplay k/negationslash=0/tildewideεkd† kdk, (33) where U2=1 2/summationdisplay k/negationslash=0(/tildewideεk−Ak) (34) 10and /tildewideεk= sign(Bk)/parenleftbig A2 k−B2 k/parenrightbig1/2(35) is the spectrum of quasiparticles. From this formula obviously follows the following criterion of the condensate stability: Bk>0, A2 k−B2 k>0. (36) For the theory with four-magnon interactions we obtain S4(k)>0,[2T4(k)]2−[S4(k)]2>0. (37) If this condition does not work (for an attractive interaction, S4(k)<0), the six-magnon interactions which arerepulsive inferro-andantiferromagnets (d ueto specificity ofHolstein- Primakoff representation of spin operators by the bose operator s) can make the system stable: S4(k)+S6(k)N >0, (38) [2T4(k)+3T6(k)N]2−[S4(k)+S6(k)N]2>0. In this case one can expect a sharp appearence of the uniform con densate at N > −S4(k)/S6(k). A detailed analysis will be published elsewhere. DISCUSSION The Bose-Einstein condensate of magnons with k= 0 is a uniform precession of the magnetic moment in the effective magnetic field. How does this preces sion differ from the usual precession of the magnetic moment? It is known that the unif orm precession of a magnetic moment deviated from equilibrium precesses in an effective fi eld and is gradually damped due to losses in the magnetic material and radiation damping. In this case we have an excited coherent state of magnons with k= 0, and the magnons of the entire spectrum are in thermodynamic equilibrium with the chemical potential µ= 0. In the case of BEC, we also have an excited coherent state of magnons with k= 0, but this state arose as a result of a change in the density of magnons and their chemical pote ntial becomes µ=ε0. In this case, the losses of the precessing condensate in the magne tic material disappears, the spin superfluidity arises. There remains only a weak radiation damping [19, 21], which leads 11to a long-lived coherent precession. This long-lived state was first o bserved experimentally in antiferromagnetic3He-B [3]. InthispaperwehavefocusedoncriticalconditionsofBECinferro- andantiferromagnets, bulk and thin films. Recent similar analysis [31] demonstrated a good a greement with experiments for nuclear magnon BEC. Let us list main results. 1. Wehaveshownthatingeneraltheonlyhighlypopulatedregionoft hespectrum(which has been previously considered for particular cases in Refs.[1],[28]) is r esponsible for the for- mation of any BEC. The high population approximation, nk≃kBT/(εk−µ), however, does not mean that we deal with the classical physics, this is an expressio n for the Bose-Einstein distribution if nk/greaterorsimilar1. The opposition of high density of particles or quasiparticles to the ir high population makes the classical criterion of Maxwell-Boltzmann st atistics inapplicable which means that the Bose-Einstein condensation condition µ= minεkalways occurs in the degenerate bose gas. The high population approximation substant ially simplifies analysis of the critical conditions and can simplify magnetic dynamics simulation s of systems with BEC by the use of classical variables instead of operators. So far a s the BEC occurs in the k-space, the most convenient form of simulation is the use of kinetic e quations for nkwith the integrals of magnon-magnon collisions (see, e.g., (2)), sources of pumping and relaxation [19],[12]. 2. We have found that the condition of magnon BEC in one of the most interesting materials, perpendicular magnetized YIG thin film, is fulfilled at a small a ngle, when signals are usually treated as excited spin waves [32]. 3. We have estimated that in hematite, high-temperature antiferr omagnet, the BEC shouldoccur at much lower level ofthemagnonexcitation compared tothat offerromagnetic YIG. We believe that this theoretical prediction can open a new direc tion of purposeful studies for fundamental and applied research. 4. We have generalized the Bogoliubov’s BEC theory to the case of mu lti-particle inter- actions. According to this theory the uniform Bose-Einstein conde nsate is unstable if the four-particle interaction is attractive. This situation sometimes ta kes place in ferro- and antiferromagnets. The account of six-magnon interactions (whic h are repulsive in magnets) can resolve the problem of BEC stability. In principle, there are also f actors of time, the sample size, and relaxation for the BEC instability has to be developed in the system. One more fundamental problem that remains in this field is to connect microscopic and 12phenomenological description of bose system with Bose-Einstein co ndensate. This problem exists for a long time and it’s solution, say, on the base of magnon sys tems, will advance the understanding and progress in all areas where the BEC is manifeste d. The solution of this problem will help understand the spin superfluidity as a property of t he magnetic system that accompanies BEC of quasi-equilibrium magnons. In conclusion, we emphasize that the Bose-Einstein condensation o f quasi-equilibrium magnons is a fundamental law of physics. BEC appears due to quant um statistics of quasi- particles in magneto-ordered systems and can exist at room, or ev en higher temperatures. One of the most intriguing properties of the BEC is a superfluid spin cu rrent, a coherent quantum flow of energy and information. This understanding of mag non BEC in different magnetic materials can be very useful for spin transport and magn onic quantum devices. An interest in the spin currents, the magnetization projection trans fer in magnetic materials, is growing every year. TheauthorswishtothankGrigoryVolovikandanonimousreviewer fo rhelpfulcomments. For Yu. M. B. this work was financially supported by the Russian Scien ce Foundation (grant RSF 16-12-10359). [1] Yu. M. Bunkov, G. E. Volovik, Spin superfluidity and magnon BEC , Chapter IV of the book ”Novel Superfluids”, eds. K. H. Bennemann and J. B. Ketterson (Oxford, University press, 2013) [2] V. L. Safonov, Nonequilibrium Magnons (Wiley-VCH, Weinheim, 2012). [3] A. S. Borovik-Romanov, Yu. M. Bunkov, V. V. Dmitriev, Yu. M. Mukharskiy, JETP Lett. 40 (1984) 1033. [4] I. A. Fomin, JETP Lett. 40 (1984) 1037. [5] L. Pitaevskii and S. Stringari, Bose-Einstein condensation (Clarendon Press, Oxford, 2003) [6] G. E. Volovik, J. Low Temp. Phys. 153 (2008) 266. [7] Yu. M. Bunkov and G. E. Volovik, J. Low Temp. Phys. 150 (200 8) 135; J. Phys.: Condens. Matter 22 (2010) 164210. [8] Yu. M. Bunkov, J. Phys.: Condens. Matter 21 (2009) 164201 . [9] T. Sato , T. Kunimatsu , K. Izumina , A. Matsubara , M. Kubot a , T. Mizusaki and Yu. M. 13Bunkov, Phys. Rev. Lett. 101 (2008) 055301. [10] P. Hunger, Yu. M. Bunkov, E. Collin and H. Godfrin, J. Low Temp. Phys. 158 (2010) 129. [11] S. O. Demokritov, V. E. Demidov, O. Dzyapko, G. A. Melkov , A. A. Serga, B. Hillebrands, and A. N. Slavin, Nature 443 (2006) 430. [12] D. A. Bozhko, A. A. Serga, P. Clausen, V. I. Vasyuchka, F. Heussner, G. A. Melkov, A. Pomyalov, V. S. Lvov and B. Hillebrands, Nature Physics 12 (2 016) 1057. [13] Yu. M. Bunkov, P. M. Vetoshko, I. G. Motygullin, T. R. Safi n, M. S. Tagirov and N. A. Tukmakova, Mag. Res. in Solids 17 (2015) 15205. [14] Yu. M. Bunkov, Physics Uspekhi 53 (2010) 848. [15] Yu. M. Bunkov, E. M. Alakshin, R. R. Gazizulin, A. V. Kloc hkov, V. V. Kuzmin, V. S. L’vov, M. S. Tagirov, Phys. Rev. Lett. 108 (2012) 177002. [16] M. Borich, Yu. M.Bunkov, M. I.Kurkin, A. P. Tankeev, JET P Lett. 105 (2017) 24. [17] Yu. M. Bunkov, J. Low Temp. Phys. 185 (2016) 399. [18] Yu. D. Kalafati and V. L. Safonov, Sov. Phys. JETP 68 (198 9) 1162. [19] Yu. D. Kalafati and V. L. Safonov, JETP Lett. 50 (1989) 14 9; Sov. Phys. JETP 73 (1991) 836. [20] V. L. Safonov. Physica A 188 (1992) 675. [21] Yu. D. Kalafati and V. L. Safonov, J. Magn. Magn. Mater. 1 23 (1993) 184. [22] A. V. Lavrinenko, V. S. L’vov, G. A. Melkov, and V. B. Cher epanov, Sov. Phys. JETP 54 (1981) 542. [23] G. A. Melkov, V. L. Safonov, A. Yu. Taranenko, and S. V. Sh olom, J. Magn. Magn. Mater. 132 (1994) 180. [24] S. A. Govorkov and V. A. Tulin, Sov. Phys. JETP 68 (1989) 8 07. [25] V. E. Demidov, O. Dzyapko, S. O. Demokritov, G. A. Melkov , and A. N. Slavin, Phys. Rev. Lett. 99 (2007) 037205. [26] A. V. Chumak, G. A. Melkov, V. E. Demidov, O. Dzyapko, V. L . Safonov, and S. O. Demokri- tov, Phys. Rev. Lett. 102 (2009) 187205. [27] K. Huang, Introduction to statistical physics (Taylor and Francis, New York, 2001). [28] A. R¨ uckriegel and P. Kopietz, Phys. Rev. Lett. 115 (201 5) 157203. [29] G. N. Kakazei, P. E. Wigen, K. Yu. Guslienko, V. Novosad, A. N. Slavin, V. O. Golub, N. A. Lesnik, and Y. Otani, Appl. Phys. Lett. 83 (2004) 443. 14[30] N. N. Bogolubov and N. N. Bogolubov Jr., Introduction to quantum statistical mechanics (World Scientific Publishing Company, Singapore, 2009)). [31] R. R. Gazizulin, Yu. M. Bunkov, V. L. Safonov, JETP Lette rs 102 (2015) 766. [32] Yu. K. Fetisov, C. E. Patton, and V. T. Synogach, IEEE Tra ns. Magn. 35 (1999) 4511. 15

2017-02-02

We consider the phenomenon of Bose-Einstein condensation of quasi-equilibrium magnons which leads to a spin superfluidity, the coherent quantum transfer of magnetization in magnetic materials. These phenomena are beyond the classical Landau-Lifshitz-Gilbert paradigm. The critical conditions for excited magnon density for ferro- and antiferromagnets, bulk and thin films are estimated and discussed. The BEC should occur in the antiferromagnetic hematite at much lower excited magnon density compared to the ferromagnetic YIG.

Magnon Condensation and Spin Superfluidity

1702.00846v2

Quantum Simulation of the Fermion-Boson Composite Quasi-Particles with a Driven Qubit-Magnon Hybrid Quantum System Yi-Pu Wang,1, 2Guo-Qiang Zhang,1, 2Da Xu,1Tie-Fu Li,3, 4,Shi-Yao Zhu,1J. S. Tsai,5, 6and J. Q. You1,y 1Zhejiang Province Key Laboratory of Quantum Technology and Device, Department of Physics and State Key Laboratory of Modern Optical Instrumentation, Zhejiang University, Hangzhou 310027, China 2Quantum Physics and Quantum Information Division, Beijing Computational Science Research Center, Beijing 100193, China 3Institute of Microelectronics, Tsinghua University, Beijing 100084, China 4Beijing Academy of Quantum Information Sciences, 100193 Beijing, China 5Department of Physic, Tokyo University of Science, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan 6Center for Emergent Matter Science, RIKEN, Wako-shi, Saitama 351-0198, Japan (Dated: April 1, 2019) We experimentally demonstrate strong coupling between the ferromagnetic magnons in a small yttrium-iron-garnet (YIG) sphere and the drive-eld-induced dressed states of a superconducting qubit, which gives rise to the double dressing of the superconducting qubit. The YIG sphere and the superconducting qubit are embedded in a microwave cavity and the eective coupling between them is mediated by the virtual cavity photons. The theoretical results t the experimental observations well in a wide region of the drive-eld power resonantly applied to the superconducting qubit and reveal that the driven qubit-magnon hybrid quantum system can be harnessed to emulate a particle- hole-symmetric pair coupled to a bosonic mode. This hybrid quantum system oers a novel platform for quantum simulation of the composite quasi-particles consisting of fermions and bosons. By exploiting the advantages of dierent components, hybrid quantum systems can provide outstanding archi- tectures for quantum information processing (see, e.g., Refs. [1{4]). Recently, a qubit-magnon hybrid quan- tum system was implemented by strongly coupling both a superconducting qubit and the Kittel mode of magnons in a small yttrium-iron-garnet (YIG) sphere to a three- dimensional (3D) microwave cavity [5, 6], where an ef- fective appreciable qubit-magnon coupling was achieved by exchanging virtual cavity photons between the qubit and magnons. For the Kittel mode in the YIG sphere, the magnons are collective excitations of spins with zero wave number (i.e., in the long-wavelength limit) where all exchange-coupled spins in the sample precess uni- formly [7]. When hybridized with microwave cavity photons [8, 9], optical photons [10], and phonons [11], these magnons can provide a platform to implement var- ious novel phenomena and applications [12{26], including magnon gradient memory [27, 28], cavity spintronics [16, 29], quantum transducer [30{33], parity-time symme- try [34], bistability of cavity magnon-polaritons [35, 36], and the generations of magnon-photon-phonon entan- glement [37] and squeezed magnon-phonon states [38]. Moreover, ultrstrong-coupling regime between magnons and cavity photons have also be shown to be imple- mentable [15, 39, 40], in addition to the strong cou- plings [12{17]. The superconducting qubit as a quantum informa- tion processing unit can be used as the core of a solid- state hybrid quantum system [1, 2]. Coherent dress- ing of the qubit can provide access to a new quantum system with improved properties (e.g., longer coherencetimes [41, 42]) and oer various applications in quantum information (see, e.g., Refs. [43{45]). Quantum mechan- ically, magnons behave similar to other bosons, so their strong coupling to the superconducting qubit can yield a coherent dressing of the qubit, as analogous to the dress- ing of a two-level system via photons [46]. In this Letter, we study a driven qubit-magnon hy- brid quantum system consisting of a 3D transmon qubit and a small YIG sphere embedded in a rectangular 3D microwave cavity. When driven in resonance by a monochromatic microwave eld, the transmon qubit is dressed by the microwave eld. Owing to the qubit- magnon coupling mediated by the virtual cavity photons, the 3D transmon qubit is further dressed by the magnons. From the dispersive readout results, we clearly show these newly-formed doubly dressed states of the superconduct- ing qubit by demonstrating their frequency splittings versus the drive power. The theoretical results t the experimental observations well in a wide region of the drive power and reveal that this driven qubit-magnon hy- brid system can be harnessed to emulate a particle-hole- symmetric pair coupled to a bosonic mode. These doubly dressed states behave like composite quasi-particles con- sisting of fermions and bosons and can be more fermion- like or not, depending on the portion of magnons in the hybridized normal modes. Our work opens a new avenue to quantum simulation of the fermion-boson composite quasi-particles with a hybrid quantum system. Figure 1(a) schematically show the experimental setup of the hybrid quantum system. The rectangular 3D mi- crowave cavity has inner dimensions of 58 326 mm3. It consists of two parts which are made of aluminum andarXiv:1903.12498v1 [quant-ph] 29 Mar 20192 ffMagnon modeTransmon qubit YIG sphere 6.95 6.907.007.05 240 245 250 255 260 FIG. 1. (color online). (a) Schematic of the qubit-magnon hybrid quantum system in a rectangular 3D microwave cavity. A small YIG sphere is placed in the part of the cavity made of oxygen-free copper at the magnetic-eld antinode of the cavity mode TE 102. To mount the transmon qubit, the part of the cavity made of aluminum is further separated into two parts that clamp the qubit at the electric-eld antinode of the cavity mode TE 102. (b) Transmission spectrum of the cavity when the Kittel mode of magnons in the YIG sphere is magnetically tuned to be near resonance with the cavity mode TE 102. oxygen-free copper, respectively. The cavity is placed in a BlueFors LD-400 dilution refrigerator at a base cryogenic temperature of20 mK. At such a low temperature, the aluminum is superconducting and becomes diamag- netic. The 3D transmon qubit [47, 48] mounted in the superconducting aluminum part of the cavity can be pro- tected from the magnetic eld generated by the outside superconducting magnet. In this superconducting qubit, two aluminum pads are attached to the small Joseph- son junction, which, together with the cavity, provide a large shunt capacitor to suppress the charge noise in the qubit, as in the capacitively-shunted ux qubit [49, 50], 2D transmon [51], and Xmon [52]. A small YIG sphere of diameter 1 mm is placed in the copper part of the cavity which is not superconducting at the base temperature of the dilution refrigerator. The static magnetic eld pro- duced by the outside superconducting magnet can go into the copper part of the cavity to adjust the frequency of the Kittel mode in the YIG sphere. The static magnetic eld is aligned along the hard magnetization axis [100] of the YIG sphere and both the 3D transmon qubit and the Kittel mode are strongly coupled to the cavity mode TE102. The frequency of this cavity mode TE 102, whichis measured to be !c=26:99 GHz, is designed to have a large detuning from the transition frequency of the 3D transmon qubit. When the Kittel mode is tuned to be nearly resonant with the qubit by the static magnetic eld (i.e., the cavity mode TE 102is also largely detuned from the Kittel mode), an eective qubit-magnon cou- pling can be achieved via the exchange of virtual cavity photons between the qubit and magnons [5, 6]. Before exhibiting the coupling between the 3D trans- mon qubit and the Kittel mode of magnons, we rst mea- sure the coupling between the Kittel mode and the cavity mode TE 102by tuning the frequency of the Kittel mode in resonance with this cavity mode. Owing to the strong coupling between them, two branch of magnon polaritons occur around the anticrossing point [Fig. 1(b)], with a level splitting86 MHz at the anticrossing point. The transition frequency between ground state jgiand rst excited statejeiof the 3D transmon qubit is measured to be!q=26:49 GHz. Because the qubit is now largely detuned from both the Kittel mode and the cavity mode TE102, its eect on the magnon polaritons can be ne- glected at around the anticrossing point. Therefore, from the measured level splitting at the anticrossing point, the coupling strength between the Kittel mode and the cavity mode TE 102can be obtained as gm43:0 MHz. To achieve an eective coupling between the qubit and the Kittel mode, we then tune the frequency !mof the Kittel mode to be near the transition frequency !qof the qubit. Now, the cavity mode TE 102is largely de- tuned from both the qubit and the Kittel mode, and an eective qubit-magnon coupling can be achieved via the virtual photons of the cavity mode TE 102[5, 6]. At the cryogenic temperature T20 mK,kBT!q;!m, so both the qubit and the Kittel mode stay nearly in their ground statesjgiandj0i, respectively, and a transition fromjeitojgiinduces a transition from the vacuum to single-magnon states (i.e., from j0itoj1i). As in Ref. [5], the spectroscopic measurement is carried out with a vec- tor network analyzer (VNA) by probing the transmis- sion of the cavity mode TE 103. During measurement, the probe eld is applied at the resonant frequency of the cavity mode TE 103. Largely detuned from the cavity mode, the qubit is usually in the ground state. When the qubit is excited, the cavity mode has an observable frequency shift in this dispersive regime and the probed eld transmission thereby changes. Also, a series of at- tenuators and isolators are used to prevent thermal noise from reaching the sample and the signal going out from the output port of the cavity is amplied by two low-noise ampliers at the stages of 4K and room temperature, re- spectively. Moreover, two microwave elds from dierent sources are applied to the input port of the cavity. One is used as an excitation eld and tuned to excite the hy- bridized normal modes of the system, and the other has a xed frequency near or in resonance with the 3D trans- mon qubit and is harnessed to drive the qubit via the3 Excitation field□ |,0□ |,1 □ |,0 0(a) (b) 2302312322332342352366.446.466.486.506.526.546.56 Magneticfield(mT)Excitationfrequency(GHz) -89.00-84.00-79.00-74.00-69.00-64.00S212 (dB) Magnonmode Qubit2gqm/2π> >> FIG. 2. (color online) (a) Energy levels of the qubit-magnon system with only the vacuum and single-magnon states in- volved for the Kittel mode. The coupling between jg;1iand je;0iinduces the vacuum Rabi splitting 0. (b) The vacuum Rabi splitting of the qubit-magnon system measured via the transmission spectrum of the cavity by tuning both the exci- tation eld and the static magnetic eld. The probe eld is applied in resonance with the cavity mode TE 103. large capacitor shunted to the Josephson junction. In Fig. 2(a), we show the energy levels of the qubit- magon hybrid quantum system at the cryogenic temper- ature when the drive eld on the qubit is turned o. The interaction between jg;1iandje;0igives rise to the vacuum Rabi splitting 0. Owing to the exchange of virtual cavity photons between the qubit and the Kit- tel mode, the eective qubit-magnon coupling is [53] gqm=1 2gqgm(1=
q+ 1=
m), where
q!c!q (
m!c!m) is the frequency detuning of the cavity mode TE 102from the qubit (Kittel mode). The eec- tive Hamiltonian of the coupled qubit-magnon system is given by (we set ~= 1) Hqm=!q 2z+!mbyb+gqm(+b+by); (1) wherezandare Pauli operators of the qubit and by (b) is the creation (annihilation) operator of the magnons. When the Kittel mode is tuned to be in resonance with the 3D transmon qubit, gqmis reduced to [5] gqm= gqgm=
, with
q=
m=
. The coupling between the qubit and the Kittel mode is measured in Fig. 2(b) by tuning the static magnetic eld and scanning the fre- quency of the excitation eld. From the vacuum Rabi splitting measured at the anticrossing point ( 0= 2gqm), we obtaingqm=220:1 MHz. Comparing the linewidth of the mode at around 230 and 237 mT, we see that the linewidth becomes broader at a stronger magnetic eld,indicating that some static magnetic eld penetrates into the cavity to aect the quantum coherence of the qubit. Actually, in our experiment, the static magnetic eld was already designed to be parallel to the chip surface of the qubit, so as to reduce its in uence on the qubit as much as possible. With the measured
0:5 GHz, gm42:0 MHz and gqm=220:1 MHz, we can de- duce using gqm=gqgm=
thatgq239 MHz, which reveals that the interaction between the qubit and the cavity mode TE 102is in the strong-coupling regime. When turning on the monochromatic drive eld of fre- quency!dappied to the 3D transmon qubit, the Hamil- tonian of the driven qubit-magnon hybrid system is writ- ten asHd=Hqm+1 4 d(+ei!dt+ei!dt), where d represents the coupling strength between the drive eld and the qubit. In the rotating frame with respect to the drive frequency !d, the Hamiltonian of this driven qubit- magnon system becomes H=1 2qz+1 2 dx+mbyb+ gqm(+b+by), whereq!q!d(m!m!d) is the frequency detuning between the qubit (magnon) and the drive eld. The qubit is now dressed by a clas- sical drive eld and the corresponding Rabi frequency is e d=q 2 d+2q. Due to the coupling between the qubit and magnons, the qubit is then further dressed by the magnons. Therefore, the driven qubit-magnon hybrid system can be described as a doubly dressed qubit . Below we focus on the resonant case with q=m= 0. The Hamiltonian Hbecomes H= d 2x+gqm(+b+by): (2) Even for this simple model, one cannot exactly solve it, because the model of a double dressed qubit is not ex- actly solvable [54]. With the new basis states ji= (jgijei)=p 2 and the mean-eld approximation, the Hamiltonian (2) can be reduced to [53] H=Hp+Hh, with Hp= d 2aya+ ~gqm(ayb+aby); Hh= d 2hyh~gqm(hyb+hby); (3) wherea=hyjih +j, and ~gqm=1 2gqm(A+ 1), with A=hai. Becausefa;ayg= 1 andfa;ag=fay;ayg= 0, these newly-dened operators are fermionic. Here Hhhas the same form as Hp, but its two parameters dand ~gqm change the signs. This means that the Hamiltonian H= Hp+Hhhas the particle-hole symmetry . Thus, we can use the driven qubit-magnon hybrid system to emulate a particle-hole-symmetric pair of fermions coupled to a bosonic mode. The two eigenvalues of the Hamiltonian Hp(de- noted as!4and!2) are=1 2[( d=2)[( d=2)2+ (2~gqmp N+ 1)2]1=2, whereNis the number of magnons excited. The eigenstates correspond to the particle- magnon composite quasi-particle states. The two eigen- values ofHh(denoted as !1and!3) are=, and4 6.45 6.50 6.553dBm(b) 5dBm(c)1dBm(a) 7dBm Excitationfrequency(GHz)(d)11dBm (e) (h)13dBm (g)(f) 15dBm 6.40 6.45 6.50 6.55 6.6017dBm Excitationfrequency(GHz)-86-84-82-80 -86-84-82-80 -86-84-82-80 -86-84-82-80-94-92-90 -94-92-90 -94-92-90 -94-92-9012 3 4 1 2 3 4 1 2 3 4 1 2341 2 3 4 1 4 1 4 1 4S212 (dB)23 FIG. 3. (color online) Dispersive readout of the hybridized normal modes of the driven qubit-magnon system. An ex- citation eld is tuned to excite the hybridized normal modes and a probe eld is applied in resonance with the cavity mode TE103. The power of the microwave eld to drive the super- conducting qubit is tuned to be (a) 1 dBm, (b) 3 dBm, (c) 5 dBm, (d) 7 dBm, (e) 11 dBm, (f) 13 dBm, (g) 15 dBm, and (h) 17 dBm, respectively. the eigenstates correspond to the hole-magnon composite quasi-particle states. By tuning d, these hybridized nor- mal modes can be more fermion-like or not, depending on the portion of magnons in these modes. The frequency splitting between modes 4 and 1 and the frequency split- ting between modes 3 and 2 are !4!1= d 2+h ( d=2)2+ (2~gqmp N+ 1)2i1=2 ; !3!2= d 2+h ( d=2)2+ (2~gqmp N+ 1)2i1=2 :(4) In Fig. 3, we display the transmission spectrum of the driven hybrid system measured in the resonant case of q=m= 0 by successive increasing the power Pdof the drive eld. The spectrum exhibits a clear mirror symmetry owing to the particle-hole symmetry in the Hamiltonian. When increasing Pd, two absorption dips are split into four, with the outer two dips (labelled as 1 and 4) departing away. Accordingly, the inner two dips (labelled as 2 and 3) gradually approach and then merge together. The Rabi frequency dis proportional to the drive-eld amplitude ", so d/pPd. From Eq. (4) it follows that !4!1increases and !3!2decreases when increasing Pd. In particular, for a suciently strong drive eld with d4~gqmp N+ 1,!3!2!0. These behaviors agree with the observations in Fig. 3. For zero d(i.e., when turning o the drive eld), Eq. (4) gives !4!1=!3!2= 2~gqmp N+ 1. In fact, when d= 0, the model in Eq. (2) becomes exactly solvable, with the Rabi splitting N= 2gqmp N+ 1. In this limiting case, we obtain ~ gqm1 2gqm(A+ 1) =gqm, i.e.,A= 1. In Fig. 4, we display the frequency split- tings!4!1and!3!2versus the drive power d, 0.51 .01 .54060801001201401600 .250 .500 .751020304050( b)(ω4−ω1)/2π (GHz)D rive Power (µW)(a)( ω3−ω2)/2π (GHz)D rive Power (µW)FIG. 4. (color online) (a) Fitting the experimental data of the frequency splitting between hybridized normal modes 1 and 4 versus the drive power Pd. (b) Fitting the experimental data of the frequency splitting between hybridized normal modes 2 and 3 versus the drive power Pd. To t the data, we use Eq. (4), where ~ gqm=gqm,N= 0, and d=kpPd, with k= 103 MHz/mW1=2. with the data extracted from Fig. 3. Fitting these data, we use Eq. (4), where d=kpPdand ~gqmis replaced bygqm. WithN= 0, we obtain the only tting pa- rameterk= 103 MHz/mW1=2. Here the attenuation of the drive eld from the microwave source to the input port of the cavity is 45 dB. In the region of a weaker drive eld, the numerical results with N= 0 agree well with the experimental results. This reveals that for a weaker drive eld, the Kittel mode still stays nearly in the ground state and the drive eld does not appreciably aect it. However, when the drive eld becomes strong, the numerical results with N= 0 deviate from the exper- imental observations, i.e., smaller (larger) than the data for!4!1(!3!2). This is what we expect because more magnons may be excited when the drive eld is strengthened to raise the temperature in the cavity. In conclusion, we have convincingly demonstrated strong coupling between the Kittel-mode magnons in a small YIG sphere and the drive-eld-induced dressed states of a 3D transmon qubit. The coupling between the magnons and the dressed states of the superconducting qubit is mediated by the virtual cavity photons. The the- oretical results t the experimental observations well in a wide region of the drive power of the microwave eld res- onantly applied to the transmon qubit. Our work opens up new possibilities to manipulate a hybrid quantum sys- tem composed of several degrees of freedom and paves the way to quantum simulation of composite quasi-particles consisting of fermions and bosons. This work was supported by the National Key Re- search and Development Program of China (Grant No. 2016YFA0301200) and the National Natural Science Foundation of China (Grants No. U1801661 and No.5 11774022). T.-F.L. was partially supported by Science Challenge Project (Grant No. TZ2018003). J.S. 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2019-03-29

We experimentally demonstrate strong coupling between the ferromagnetic magnons in a small yttrium-iron-garnet (YIG) sphere and the drive-field-induced dressed states of a superconducting qubit, which gives rise to the double dressing of the superconducting qubit. The YIG sphere and the superconducting qubit are embedded in a microwave cavity and the effective coupling between them is mediated by the virtual cavity photons. The theoretical results fit the experimental observations well in a wide region of the drive-field power resonantly applied to the superconducting qubit and reveal that the driven qubit-magnon hybrid quantum system can be harnessed to emulate a particle-hole-symmetric pair coupled to a bosonic mode. This hybrid quantum system offers a novel platform for quantum simulation of the composite quasi-particles consisting of fermions and bosons.

Quantum Simulation of the Fermion-Boson Composite Quasi-Particles with a Driven Qubit-Magnon Hybrid Quantum System

1903.12498v1

Field-dependence of magnon decay in yttrium iron garnet thin lms A. L. Chernyshev1 1Department of Physics, University of California, Irvine, California 92697, USA (Dated: September 13, 2021) We discuss threshold eld-dependence of the decay rate of the uniform magnon mode in yttrium iron garnet (YIG) thin lms. We demonstrate that decays must cease to exist in YIG lms of thickness less than 1 m, the lengthscale dened by the exchange length. We show that due to the symmetry of the three-magnon coupling the decay rate is linear in
H= (HcH) in the vicinity of the threshold eld Hcinstead of the step-like k=0/ (
H) expected from the two-dimensional character of magnon excitations in such lms. For thicker lms, the decay rate should exhibit multiple steps due to thresholds for decays into a sequence of the two-dimensional magnon bands. For yet thicker lms, such thresholds merge and crossover to the three-dimensional single-mode behavior: k=0/j
Hj3=2. PACS numbers: 75.10.Jm, 75.40.Gb, 75.30.Ds Introduction. |Extensive experimental and theoretical research on the ferromagnetic insulator, yttrium iron gar- net [Y 3Fe2(FeO 4)3or YIG] was started more than half a century ago has beneted from this material's excep- tional purity, high Curie temperature, and relative sim- plicity of the low-energy magnon spectrum.1Recent dis- covery of the Bose-Einstein condensation of the highly occupied magnon states created by microwave pumping in YIG thin lms has attracted substantial interest.2 Recently, threshold eects due to the so-called three- magnon splitting have been reported in a quasi-two- dimensional (2D) thin lms of YIG under microwave pumping and as a function of external eld.3In this, as well as in the other recent works,4control over the spin current enhancement in layered metal-ferromagnet struc- tures by the three-magnon processes is sought. This calls for a deeper theoretical insight into the decay dynamics of such structures. Fundamentally, given its outstand- ing properties,1YIG may oer a fertile playground in the studies of threshold phenomena, because the decay conditions in it can be varied continuously by both lm thickness and external magnetic eld. Decays. |In this work, we discuss the decay rate of the uniform mode ( k=0 magnon) in an insulating ferromag- netic thin lm as a function of external magnetic eld and lm thickness. In particular, using magnon dispersion in the lowest-mode approximation, we outline the ranges of the eld and lm thickness that favor decays in YIG. Specically, we show that kinematic conditions for three- magnon decays cannot be met for the YIG lms of thick- nessdmin1m or less and for the elds Hmax c600 Oe or higher. The upper limit on the external eld is dened solely by the magnetization of a ferromagnet, in agree- ment with Ref. 3. Less obvious is the existence of the limit on the lm thickness, which is xed by another fun- damental characteristics of a ferromagnet: its exchange length. The physical reason for that limit is the decreas- ing role of the long-range dipolar interactions with the decrease of the lm thickness. The presence of such a limit must be important for the control of relaxation and transfer of the spin current in layered structures, whichrely on the three-magnon processes in YIG.4 We nd that the threshold eld-dependence of the de- cay rate for the k= 0 magnon near the threshold eld Hcis k=0/j
Hj
(
H), where
H= (HcH), because the three-magnon interaction vanishes along the direction of the lm magnetization, which is also precisely thek-direction where magnon band minima are located. This leads to a reduction of the phase space for decays and results in a vanishing decay rate near the threshold, contrary the na ve expectation of the step-like increase of k=0/(
H), when the symmetry of the three-magnon interaction is ignored.5We have supported our considera- tion by explicit calculations of both T= 0 andT= 300K relaxation rate dependencies on the eld for several rep- resentative YIG lm thicknesses. The nite-size quantization in the lm thickness direc- tion leads to the formation of multiple magnon bands.6 We argue that the eld-dependence for the decay rate should exhibit multiple steps linear in jHciHj, corre- sponding to the Hcithresholds for decays into a sequence of magnon bands. For thick lms, these multiple thresh- olds should merge in a continuum, and the decay rate will crossover to a three-dimensional (3D) single-band behav- ior: k=0/j
Hj3=2. These results should be helpful in nding an optimal set of parameters for spin-current enhancement.3 Dispersion, density of states. |Despite its fairly com- plicated crystal structure, at low energies YIG can be described with great success as an eective large-spin Heisenberg ferromagnet on a cubic lattice with nearest- neighbor exchange, long-range dipolar interactions, and negligible spin anisotropy.1,7Thus, at long wavelength, magnon energy in YIG is determined by a competition between three couplings: exchange, dipolar, and Zeeman. For a ferromagnetic crystal of the lm geometry and external eld directed in-plane where it co-aligns with the magnetization direction, as is done most commonly in experiments, the lowest-mode approximation for thearXiv:1208.0831v1 [cond-mat.str-el] 3 Aug 20122 kydkzdEk(GHz)xyzdH||M 012345E 0DoS (arb. units)θ = 0°(b) d = 5 µm 050100150|k|d12345Ek1000 Oe700 Oe300 Oe1 Oe(a)kmkmkmkm(a) FIG. 1. (Color online) (a) 2D plot of magnon dispersion in YIG in the lowest-mode approximation, Eq. (1), for H= 1000 Oe,d= 5m, and inkz>0 sector. Inset: axes conventions relative to the lm and the eld/magnetization direction. (b) 2D magnon density of states (DoS) for several representative eld values, Eis in GHz. magnon energy yields:5,6,8,9 Ek=r h+k2+e
ksin2k h+k2+
k ;(1) where
k=fk
ande
k= (1fk)
andh=H,= JSa2, and
= 4 M are the energy scale parametriza- tions of the external eld, exchange, and dipolar interac- tions, respectively. The form factor fk= (1ejkjd)=jkjd is from the dipolar sums in the direction of the lm thick- nessda, andkis the angle between the ferromagnet's magnetization (directed in-plane, zaxis) and magnon's in-plane 2D wave vector k, see Fig. 1(a). We use =gB whereg= 2 is an eective gfactor and Bis the Bohr magneton. In this work we adhere to the notations and units of Ref. 6, which has provided a detailed micro- scopic spin-wave theory of YIG in 1 =Sapproximation, and we use experimental parameters for YIG, magnetiza- tion 4M= 1750 G, exchange stiness == 5:17
1013 Oe m2, and lattice constant a= 12:376A. In Fig. 1(a), the magnon dispersion from Eq. (1) for a representative eldH= 1000 Oe and lm thickness d= 5m is shown. It should be noted that the dipolar interactions are re- sponsible for the nontrivial structure of magnon band with the minima at nite wave vectors, which allow the decay conditions to occur. Although approximate, Eq. (1) provides a close quantitative description of the lowest 2D magnon energy band,6quantized due to - nite lm thickness din thexdirection [Fig. 1(a), inset]. At small k, dipolar interactions dominate over the ex- change and result in a steep decrease from the energy of the uniform mode, E0=p h(h+
), for k's along the magnetization direction. At larger k, exchange energy dominates, giving Ekh+k2; while at intermediate k, competition between exchange and dipolar terms re- sults in peculiarly shaped minima at km= (0;km), see Fig. 1(a). 0.1 1 10 100 d (µm)0100200300400500600700 H (Oe) 200 400 600 H (Oe)0.511.52 2Emin/E0d = 1 µm d = 2 µm d = 5 µm d = 100 µm HcHcHcdminHcmaxFIG. 2. (Color online) dHdecay diagram for YIG, shaded area is where decays are allowed. Solid (dashed) boundary is the numerical (analytical) solution for the threshold bound- ary, see text. Hmaxanddminare shown. Inset: 2 Emin=E0vs Hfor severald's, upper threshold elds are indicated. The 2D density of magnon states from Eq. (1) is shown in Fig. 1(b) for several representative eld values. Pre- dictably, 2D DoS exhibits a step-like increase at the band minimum, which is followed by an unusual, almost linear increase, reminiscent of the similar behavior for the rela- tivistic dispersion. The latter is due to a nonparabolicity of the long-wavelength magnon dispersion, see Fig. 1(a). \Decay diagram". |For the decays to take place the kinematic conditions must be fullled. For the two- magnon decay (three-magnon splitting) of the uniform mode, the condition to be satised is simply E0= 2Eq. With the microscopic parameters, such as exchange sti- ness and magnetization, xed, some other parameters can be varied to allow or to forbid decays altogether. As is clear from Eq. (1), the external eld increases the energies of both the uniform mode and the minimum, making decays kinematically impossible at some higher eld value.3Another parameter is the lm thickness d, which enters Eq. (1) through the form factor fk. While the manner in which din uences decays is not a priori clear, both trends, vs eld and vs thickness, can be ex- amined numerically. Such an examination is exemplied in Fig. 2, which shows 2 Emin=E0vs eld for several lm thicknesses. Clearly, when the plotted quantity is <1, decays are allowed, and the crossing of 1 corresponds to a threshold eld for decays. Our Fig. 2 gives the complete dH\decay diagram" for YIG with the shaded area showing the parameter space where decays are allowed. Two key results are clear in Fig. 2: (i) the upper threshold eld does not exceed some Hmax ceven for large values of d, and (ii) there exists a lower limit on the lm thickness dmin, below which the decays of the uniform mode may not occur at all. The solid boundary is the numerical solution of Eq. (1) for the energy minimum and the decay conditions. The dashed line is an approximate analytical solution, which turns out to be very precise. The latter also gives us a deeper3 insight into the nature of Hmax canddmindiscussed next. Large-jkmjdapproximation |At large enough dthe wavevector of the magnon energy minimum satises jkmjd1. Then the formfactor fkm1=jkmjdis small, re ecting the reduced role of dipolar interactions in the energy of the km-magnon. One can then show10that both the exchange and the dipolar energies for the km- magnon scale as/d2=3and thus arehfor any reason- able eld. This implies that the energy of the magnon band minimum is Eminh, expected for the uniform mode in the absence of dipolar interactions. Then, the decay threshold equation, 2 Emin=E0, trivially gives hc=
=3, relating saturated value of the decay thresh- old eld to the material's magnetization: Hmax c=4M= 3 (= 583 Oe for YIG in Fig. 2(b)), see also Ref. 3. The physical question is: what parameter of the ferro- magnet denes dmin? One can extend the large- jkmjd approach and nd10that the energies of the dipolar and exchange interactions must be related by jkmjd=
d2=4
1=3. With this, the decay threshold condition 2Emin=E0leads to an algebraic equation in Hcvsd, which can be resolved in a compact form.10It is plot- ted as a dashed line in Fig. 2(b), which coincides almost exactly with the decay boundary obtained from Eq. (1) numerically. From the same equation we nd the mini- mal thickness to be dminCp =
, explicitly related to the exchange length of the ferromagnet, `ex=p =
, al- beit with a large numerical coecient C62:04.10Using parameters for YIG, the exchange length is `ex13:9a and the minimum thickness dmin=1:067m, remarkably close to the numerical result dmin=1:017m. The physical reason for the very existence of dminis the decreasing role of long-range dipolar interactions in the magnon's energy with the decrease of lm thickness, as the relation of dminto exchange length implies. The fact that the decay boundary in dexceeds the exchange length by a large numerical factor is due to the rather stringent requirements of the decay conditions. We emphasize that the provided dHdiagram should apply equally to the other thin-lm ferromagnets. Decay rate. |Transitions that involve changing the number of magnons, such as decays, recombination, or coalescence, originate from the dipolar interactions that couple longitudinal and transverse spin components and therefore do not conserve magnetic1,7as well as mechani- cal angular momentum.3Microscopically, dipolar interac- tions result in anharmonic couplings of magnons, which, for the decay processes of the k= 0 uniform mode into two magnons at qandq, can be written as H(3)=1 2X qV(3) 0;q;q ay qay qa0+ H:c: : (2) The three-magnon coupling in Eq. (2) has an angular de- pendence:V(3) 0;q;q=V0sin 2q, withV0=
=p 2S.1,11,12 This angular dependence is essential since it re ects the symmetry of dipolar coupling of the transverse, Sx(Sy), and longitudinal, Sz, spin components: Vxz/xz=r3 0 100 200 300 400 500 H (Oe)0.00.51.01.52.02.53.0Γk=0 (KHz)d = 10 µm d = 5 µm d = 2 µm Hc Hc Hc1 _ 5 × Γ0FIG. 3. (Color online) The T= 0 decay rate k=0vsHfor d= 2, 5, and 10 m. Dotted line is1 5k=0ford= 10m with the angular-dependence of the three-magnon coupling omitted,V(3) 0;q;q)V0. (Vyz/yz=r3). In particular, it is natural for this cou- pling to vanish for the spin-wave propagating with the momentum qalong the direction of magnetization M[z axis, Fig. 1(a)].11We would like to point out that it is also precisely the direction along which the minima of the magnon band are located.13Therefore, the ampli- tude of the decay of k= 0 magnon into two magnons at the band minima qmandqmis zero. This will lead to a rather spectacular violation of the na ve expectation: while kinematic conditions for the decay of k=0 magnon intoqmare just met at Hc, the corresponding decay amplitude is vanishing. Thus, the decay rate must in- crease gradually from the threshold, not in a jump-like fashion as in the DoS, Fig. 1(b). AtT= 0 only spontaneous magnon decays are allowed.14The three-magnon recombination processes have to obey the same kinematic constraints as the de- cay, having therefore the same threshold conditions. The three-magnon coalescence processes involving the k= 0 mode correspond to the \vertical" transitions, which are forbidden either kinematically as in the single mode case or by the quantum number of the interband transition in the multi-band situation. The four-magnon scattering amplitude from the exchange interaction vanishes iden- tically for the uniform mode, while the remaining four- magnon interactions from the dipole-dipole interaction together with impurity scattering should be providing a background with weak HandTdependence, distinct from the threshold behavior discussed here. With this in mind, using the kinetic approach,1,7which takes into account the balance between decay and recom- bination processes in the relaxation time approximation, the decay rate15of the uniform mode is k=0=X qV(3) 0;q;q2 2nq+ 1 (E02Eq) (3) wherenq= [e~Eq=kbT1]1is the Bose occupation fac- tor. It is clear from Eq. (3) that while the magnitude of4 0 100 200 300 400 500 H (Oe)010203040506070Γk=0 (MHz)d = 10 µm d = 5 µm d = 2 µm HcHc Hc1 _ 4 × Γ0 FIG. 4. (Color online) Same as in Fig. 3, but for T= 300 K. the decay rate at nite Tcan be substantially modied from theT= 0 result by the Bose-occupation factors, the qualitative threshold behavior must remain the same. One can investigate the threshold behavior of Eq. (3) analytically and obtain k=0/jE02EminjforH!Hc.10 Given the proportionality between ( E02Emin) and
H=(HcH) demonstrated in Fig. 2, this yields linear dependence of the decay rate on the eld relative to the threshold: k=0/j
Hj
(
H), see Figs. 3 and 4. Once again, this is the consequence of an eective suppression of the phase space for decays due to the discussed angular dependence of the three-magnon coupling. In Figs. 3 and 4 we show k=0for three dierent lm thicknesses and for T= 0 andT= 300 K, respectively. While the overall scale in the two gures is in a com- pletely dierent frequency range,10the shapes of vs H are qualitatively very similar, especially concerning their threshold behavior vs eld, in agreement with the above analysis. The relative dierence between the curves for dierentd's re ects the smaller phase space for decays in thinner lms. In the same Figs. 3 and 4 we demonstrate a dramatic contrast with the results of Eq. (3) if the angu- lar dependence of the three-magnon coupling in Eq. (2) is neglected,V(3) 0;q;q)V0. The latter results exhibit jumps atHc's and a linear increase after that, similar to the 2D magnon DoS in Fig. 1(b). One should also note that the overall decay rate is also markedly overestimated if the angular dependence of the three-magnon interactionis ignored. In thicker lms, the decay rate will be further modied by the multiple magnon bands that occur due to nite- size quantization in the lm thickness direction.6It will exhibit a sequence of steps linear in jHciHj, whereHci is the threshold eld for decay into an ith band, increas- ing the decay rate every time the corresponding kine- matic conditions are met. Strictly speaking, the angle qinV(3) 0;q;qis between the 3D qvector and magnetiza- tion vector M. In the quasi-2D geometry with the levels quantized in the xdirections,qxhas discrete values and the minimal value of min qimqmin x;i=jqjis zero only for the lowest magnon band. Thus, the thresholds in Hci's will have some small step-like behavior. However, this eect must be negligibly smaller compared to the steps in Figs. 3 and 4 when the angular dependence in V(3)is ignored altogether. Extending our analysis to the limit of thicker lms where multiple bands merge into a single 3D band, using Eq. (3) we obtain10that the decay rate near the threshold eld will crossover to k=0/j
Hj3=2. Conclusions. | In this work, we have discussed the eld-dependence of the decay rate of the uniform magnon mode in YIG thin lms and the eects of lm thickness in it. As a result of our analysis, we have established that the two key characteristics of a ferromagnet, its magneti- zation and exchange length, dene the extent of the dH parameter range that favors decays in thin lm geome- try. Our calculations of the decay rate should provide an important guidance for the experimentalists in designing the optimal conditions for the control of spin current and its relaxation in thin lms. A particularly intriguing sug- gestion to pursue is to study the properties of a thin lm with varying thickness, which may permit decays and, as a consequence, the spin current enhancement in one part of the lm and forbid it in the other. Acknowledgments. |I am deeply indebted to Doug Mills for introducing me to this problem, for many illumi- nating discussions and constant encouragement. I thank Ilya Krivorotov for numerous enlightening conversations, his genuine interest and support, as well as generosity with his time spent on educating me in this eld. I also acknowledge useful conversation with Andreas Kreisel. This work was supported by the U.S. DoE under grant DE-FG02-04ER46174. 1V. Cherepanov, I. Kolokolov, and V. L'vov, Phys. Rep. 229, 81 (1993). 2S. O. Demokritov, V. E. Demidov, O. Dzyapko, G. A. Melkov, A. A. Serga, B. Hillebrands, and A. N. Slavin, Nature 443, 430 (2006); V. E. Demidov, O. Dzyapko, S. O. Demokritov, G. A. Melkov, and A. N. Slavin, Phys. Rev. Lett. 99, 037205 (2007); 100, 047205 (2008). 3H. Kurebayashi, O. Dzyapko, V. E. Demidov, D. Fang, A. J. Ferguson, and S. O. Demokritov, Nature Mater. 10, 660(2011). 4K. Ando and E. Saitoh, Phys. Rev. Lett. 109, 026602 (2012); Z. Wang, Y. Sun, M. Wu, V. Tiberkevich, and A. Slavin, Phys. Rev. Lett. 107, 146602 (2011); H. Schultheiss, X. Janssens, M. van Kampen, F. Ciubotaru, S. J. Hermsdoerfer, B. Obry, A. Laraoui, A. A. Serga, L. Lagae, A. N. Slavin, B. Leven, and B. Hillebrands, Phys. Rev. Lett. 103, 157202 (2009). 5S. M. Rezende, Phys. Rev. B 79, 174411 (2009).5 6A. Kreisel, F. Sauli, L. Bartosch, and P. Kopietz, Eur. Phys. J. B 71, 59 (2009). 7A. G. Gurevich and G. A. Melkov, Magnetization Oscilla- tions and Waves , (CRC Press, Boca Raton, 1996). 8I. S. Tupitsyn, P. C. E. Stamp, and A. L. Burin, Phys. Rev. Lett. 100, 257202 (2008). 9B. A. Kalinikos and A. N. Slavin, J. Phys. C 19, 7013 (1986). 10See supplemental material for details of the calculations of thedHdecay boundary and of the decay rate threshold eld-dependence. 11M. Sparks and C. Kittel, Phys. Rev. Lett. 4, 232, 320 (1960). 12M. Sparks, R. Loudon, and C. Kittel, Phys. Rev. 122, 791 (1961); E. Schl omann, Phys. Rev. 121, 1312 (1961); B. Lemaire, H. Le Gall, and J. L. Dormann, Solid State Commun. 5, 499 (1967); A. V. Syromyatnikov, Phys. Rev. B82, 024432 (2010). 13J. Hick, F. Sauli, A. Kreisel, and P. Kopietz, Eur. Phys. J. B78, 429 (2010). 14M. E. Zhitomirsky and A. L. Chernyshev, arXiv:1205.5278. 15Our k=0corresponds to half-width at half-maximum, i.e., k=0=1 21 k=0. I. SUPPLEMENTAL MATERIAL A. large-jkmjdapproximation Assuming that for large enough dthe magnon en- ergy minimum satises jkmjd1, makes the formfactor fk= (1ejkmjd)=jkmjd1=jkmjdsmall and allows to simplyEkin Eq. (1) to an algebraic expression in reduced energy scales of the dipolar and exchange inter- actions, 
=
=jkmjdand =jkmj2: Ekm=r h+  h+ +
 ; (1) where we also used the fact that the minimum is located along thez-axis, sokm= 0. The minimum condition @Ek=@k= 0 yields: h+ +
4=h

4; (2) which, in the limit of both reduced exchange and dipolar energy scales being small compared to the eld,  ;
h, yields a straightforward relation between the two: 
= 4: (3) This relation gives the eld-independent answer for jkmjd: jkmjd=
d2 41=3 ; (4) which justies the above approximation  ;
hfor large enough dbecause: 
= 4=4
2 d21=3 : (5)Having the dipolar and exchange scales related by Eq. (3), we can substitute it into Eq. (1) for Eminand use it to solve the decay threshold condition 2 Emin=E0 withE0=p h(h+
). This leads to a quadratic equa- tion in the threshold eld hc(=Hc) vs 
(which, in turn, is a function of dby Eq. (5)), that can be resolved in a compact form: hc1;2=
6
 s
6
2 5 12
2;(6) with two solutions being the upper and the lower thresh- old elds. They give the (approximate) threshold bound- ary in Fig. 2(b), shown by the dashed line. From the same solution one can nd the \maximal" 
mthat cor- responds to the lower boundary on d=dminfor which the solution of the threshold condition exists. This is also the point at which hc1=hc2in Eq. (6) and the content of the square-root is zero, solving for which gives 
m=
(dmin) = 6p 15 21!
: (7) Using Eq. (5) we nally obtain: dmin= 221 6p 153=2r
62:04r
:(8) Although the following result can be trivially obtained from the solution for hc1in Eq. (6) taking the d!1 limit ( 
!0), one can derive the large- dlimit forHc1! Hmax cvia an observation that at d!1 energy minimum in Eq. (1) is Eminh, as 
;!0. This reduces the decay condition to a simple one: 2hc=p hc(hc+
); (9) with two solutions: hc2= 0 andhc1=Hmax c=
=3, thus relating the saturated magnetization of the material to the upper limit of the threshold eld for decays. B. threshold eld-dependence of k=0 Since we are interested in the threshold behavior, only spontaneous ( T= 0) decay rate is considered. Near the threshold, decays are happening into the magnons in the vicinity of the magnon band minima qm= (0;qm). Expanding in ( qqm) near the minima gives: k=0=X qV(3) 0;q;q2(E02Eq) (10) =V2 0 2Z dqyZ 0dqzsin22q
E
q2 m ; where
q= (qqm),
E= (E02Emin) andmis the eective magnon mass. Rewriting sin 2 qas 2qyqz=q2 and shifting integration in qzbyqmyields k=0=2V2 0 jqmj2Z q2 ydqyZ dqz
Eq2 m ;(11)6 200 300 400 H (Oe)0123456Γk=0 (arb. units)∆H3/2 Hc0Hc1Hc2 FIG. 5. (Color online) A sketch of k=0vsHfor the case of multiple magnon bands. Dotted lines show linear in jHciHj contributions of each band, solid line with the shading is the total eect, dashed line is the 3D limit /
H3=2. where qz=qzqm,q2= q2 z+q2 y, and in approximating q2q2 mthe smallness of  qz's andqy's in comparisonwithjqmjwas used. It nally leads to: k=0=V2 0m2 jqmj2E02Emin: (12) SincejE02Eminj/jHcHj, this yields k=0/
H. At niteTEmin, the decay rate near the threshold is k=0=2T2 E2 minV2 0m2 jqmj2E02Emin: (13) For the 3D case, when the 2D bands merge into a con- tinuum, the above consideration results in: 3D k=0/V2 0 jqmj2Z q3 ?dq?dqz
Eq2 m ;(14) where q2= q2 z+q2 ?now and possible mass anisotropies, while neglected, are not going to change the result qual- itatively: 3D k=0/V2 0m2 jqmj2E02Emin3=2; (15) and thus, 3D k=0/
H3=2.

2012-08-03

We discuss threshold field-dependence of the decay rate of the uniform magnon mode in yttrium iron garnet (YIG) thin films. We demonstrate that decays must cease to exist in YIG films of thickness less than 1 \mu m, the lengthscale defined by the exchange length. We show that due to the symmetry of the three-magnon coupling the decay rate is linear in \Delta H=(Hc-H) in the vicinity of the threshold field Hc instead of the step-like \Gamma \Theta(\Delta H) expected from the two-dimensional character of magnon excitations in such films. For thicker films, the decay rate should exhibit multiple steps due to thresholds for decays into a sequence of the two-dimensional magnon bands. For yet thicker films, such thresholds merge and crossover to the three-dimensional single-mode behavior: \Gamma |\Delta H|^{3/2}.

Field-dependence of magnon decay in yttrium iron garnet thin films

1208.0831v1

Nonlocal magnon transconductance in extended magnetic insulating films. I: spin diode effect. R. Kohno,1K. An,1E. Clot,1V. V. Naletov,1N. Thiery,1L. Vila,1R. Schlitz,2N. Beaulieu,3J. Ben Youssef,3A. Anane,4 V. Cros,4H. Merbouche,4T. Hauet,5V. E. Demidov,6S. O. Demokritov,6G. de Loubens,7and O. Klein1,∗ 1Université Grenoble Alpes, CEA, CNRS, Grenoble INP, Spintec, 38054 Grenoble, France 2Department of Materials, ETH Zürich, 8093 Zürich, Switzerland 3LabSTICC, CNRS, Université de Bretagne Occidentale, 29238 Brest, France 4Unité Mixte de Physique CNRS, Thales, Univ. Paris-Sud, Université Paris Saclay, 91767 Palaiseau, France 5Université de Lorraine, CNRS Institut Jean Lamour, 54000 Nancy, France 6Department of Physics, University of Muenster, 48149 Muenster, Germany 7SPEC, CEA-Saclay, CNRS, Université Paris-Saclay, 91191 Gif-sur-Yvette, France (Dated: June 13, 2023) This review presents a comprehensive study of the nonlinear transport properties of magnons electrically emitted or absorbed in extended yttrium-iron garnet (YIG) films by the spin transfer effect across a YIG |Pt interface. Ourgoalistoexperimentallyelucidatethepertinentpicturebehindtheasymmetricelectricalvariation of the magnon transconductance, analogous to an electric diode. The feature is rooted in the variation of the density of low-lying spin-wave modes (so-called low-energy magnons) via an electrical shift of the magnon chemical potential. As the intensity of the spin transfer increases in the forward direction (magnon emission regime), the transport properties of low-energy magnons pass through 3 distinct regimes: i)at low currents, wherethespincurrentisalinearfunctionoftheelectriccurrent,thespintransportisballisticanddeterminedby thefilmthickness; ii)foramplitudesoftheorderofthedampingcompensationthreshold,itswitchestoahighly correlated regime, limited by the magnon-magnon scattering process and characterized by a saturation of the magnontransconductance. Herethemainbiascontrollingthemagnondensityisthermalfluctuationsbelowthe emitter.iii)AsthetemperatureundertheemitterapproachestheCurietemperature,scatteringwithhigh-energy magnons starts to dominate, leading to diffusive transport. We find that such a sequence of transport regimes is analogous to the electron hydrodynamic transport in ultrapure media predicted by Radii Gurzhi. This study, restricted to the low-energy part of the magnon manifold, complements part II of this review, which focuses on the full spectrum of propagating magnons. I. INTRODUCTION. Diodesarekeycomponentsintheartofelectronics1. Their distinctive function is to create an asymmetric electrical con- ductance that facilitates transport in the forward direction while blocking it in the reverse direction. This asymmetry is exploited in rectification or clipping devices. Controlling the forward threshold voltage is the basis of the bipolar junction transistor. Afewofthesediodesalsooffertheunusualfeature of negative differential resistance, which is exploited in oscil- lators and active filters2,3. Until recently, it was believed that solid state diodes could only be realized with semiconductor materials. Their band structure allows strong modulation of carrier density by an electrical shift of the chemical potential between the valence and conduction bands. Recent advances in the field of spintronics have shown that it is possible to design new types of diode devices that rely on the transport of the electron’s spin instead of its charge4–7. Inthisnewparadigm,electricalinsulatorsaregood spin conductors by allowing spin to propagate between lo- calized magnetic moments via spin-waves (or their quanta magnons) to carry spin information within the crystal lattice withoutanyJouledissipation8–11. TheabsenceofJouledissi- pationusuallygivesthedielectricmaterialsverylowmagnetic damping, which is associated with a high propensity of the magnons to behave nonlinearly. In ultra-low damping materi- als, a variation of 0.1% from the thermal occupancy is usu- ally sufficient to drive the magnetization dynamics into thenonlinear regime12. This effect could be exploited for spin diodes10,13,spinamplifiers14–16orspinrectifiersofmicrowave signals17–21. Animportantmilestoneinthisnascentoffshootofspintron- ics using electrical insulators, called insulatronics22, was the discoveryoftheinterconversionprocessbetweenthespinand charge degrees of freedom (e.g., by the spin Hall effect23–29), which allows the electrical control of 𝜇𝑀, the magnon chem- ical potential, via the spin transfer effects (STE) from an ad- jacent metal electrode14,30–32. The expected benefit is the re- alization of a new form of solid state spin diode (see Fig. 1) obtainedbyelectricallyshifting 𝜇𝑀relativeto𝐸𝑔,theenergy gap in the magnon band diagram (see Fig. 2). The lever-arm is set here by the magnetic damping parameter, 𝛼LLG30,33–36: the smaller is the damping, the larger is the electrical shift of 𝜇𝑀atagivencurrent. Therefore,theferrimagnetyttrium-iron garnet(YIG),withthelowestmagneticdampingknowninna- ture, is expected to lead to the highest benchmark in asym- metric transport performance37. While the spin diode effect inlaterallyconfinedgeometriesisfairlywellunderstood38–40, its generalization to extended thin films has been largely elusive41. The corresponding difficulty is the collective dy- namics that emerges inside the continuum of magnons when the level splitting between eigenmodes becomes smaller than their linewidth34,42. Given that magnons are bosons, the pre- dicted appearance of new condensed phases at high powers is ofparticular interest. These includea superfluid phasepro- ducedbyaBose-Einsteincondensate(BEC)43–49,whichcouldarXiv:2210.08304v2 [cond-mat.mes-hall] 11 Jun 20232 lead to novel coherent transport phenomena15,50–52. The manuscript is organized as follows. In the following section, we emphasize the highlights of our transport study observed at high currents on lateral devices deposited on ex- tended thin films. In the third section, we present the relevant analytical framework to account for the electrical variation of the magnon transconductance. This framework is based on a two-fluidmodelintroducedinpartII53,whichsplitstheprop- agatingmagnonsintoeitherlow-orhigh-energymagnons. To facilitatequickreadingofeithermanuscript,wepointoutthat a summary of the highlights is provided after each introduc- tion and, in both papers, the figures are organized into a self- explanatorystoryboard,summarizedbyashortsentenceatthe beginning of each caption. In this first part, we focus mainly onthenonlinearpropertiesthatleadtothespindiodeeffect. It willbeshownthatthesearemainlycontrolledbythevariation ofthespin-spinrelaxationratesbetweenlow-energymagnons. In the fourth section, we present the experimental evidence supporting our interpretation. Finally, we conclude the paper by expressing what we believe to be the interesting future di- rections of this field. II. KEY FINDINGS. Themainexperimentalresultpresentedinthisreviewisthat thenonlineargrowthofthemagnonconductivityobservedfor the forward bias (magnon emission regime) is rapidly capped by a saturation threshold in the case of extended geometries. Such behavior can be understood by focusing on the non- linear properties of low-energy magnons, where the limited growth is reminiscent of the saturation effect observed in the ferromagnetic resonance of magnetic films at high power55. There, the amplitude of the uniform precession mode, the so- called Kittel mode, is constrained not to exceed a maximum cone angle by a power dependent magnon-magnon scatter- ing rate, which increases as the mode amplitude approaches saturation. This process involves the increase in parametric coupling via the magnetostatic interaction between pairs of counter-propagatingmagnonmodes,alloscillatingattheKit- tel frequency. This process is also known as the Suhl second- order instability or the four-magnon process. We believe that the same process is at work here between our low-energy magnons. We therefore propose a model in which Γ𝑚, the magnon-magnonrelaxationrate,alsoincreasesstronglyas 𝜇𝑀 approaches𝐸𝑔30,41. This conclusion is drawn from the ex- perimental observation that in extended thin films the regime 𝜇𝑀≥𝐸𝑔is never reached despite the use of very large cur- rent densities. The fact that it is never reached cannot be at- tributedtoathresholdcurrent, 𝐼th,whichwouldnominallybe outsidethecurrentrangeexplored. Infact,thecurrentrangeis largeenoughtoreachtheparamagneticstatebelowtheemitter due to Jouleheating. Assuming that the dampingrate is inde- pendentofthecurrentvalue,thevanishingmagnetizationisin principlesufficienttobring 𝐼thintotheexploredcurrentrange, sincethethresholdcurrentdropstozeroattheCurietempera- ture. Despitethisreductionofthemagnetizationvalue,wedo not detect nonlocally any evidence of full damping compen- FIG. 1. Spin diode effect in a lateral device. (a) Perspective and (b) sectionalviewofthemagnoncircuit: anelectriccurrent, 𝐼1,injected into Pt1emits or absorbs magnons via the spin transfer effect (STE). The change in density is consequently sensed nonlocally in Pt2(col- lector)bythespinpumpingcurrentdefinedasb,−𝐼2=(𝒱2−𝒱2)∕𝑅2, where𝒱2is a background signal generated by magnon migration alongthermalgradients. (c)Deviationfromthenominalthermaloc- cupation of low-energy magnons when 𝐻𝑥<0. We have shaded in blue the deviation in the magnon emission regime corresponding to the forward bias ( ▸) and in red the magnon absorption regime cor- responding to the reverse bias ( ◂). The inset (d) shows the behav- ior expressed as a transmission ratio 𝒯𝑠≡Δ𝐼2∕𝐼1, whereΔ𝐼2= (𝐼▸ 2−𝐼◂ 2)∕2. Asshownin(c),anasymmetricnonlineargrowthof 𝐼2 occursintheforwardbias,correspondingtotheso-calledspindiode effect. However,thisgrowthislimitedby 𝑛sat,aneffectivesaturation threshold due to nonlinear coupling between low-energy modes, as expressed by Eq. (7). The solid and dashed lines show the behavior for two values of the saturation level, 𝑛sat. The colored current scale in (c) distinguishes 3 transport regimes, ➀(green window): a linear regime,𝐼2∝𝐼1atlowcurrents,whichalsoincludesthereversebias; ➁(blue window): a nonlinear asymmetric increase above a forward bias of the order of 𝐼𝑐∕2b; and➂(red window): above 𝐼pk, a col- lapse at𝐼cof the spin conduction as the temperature of the emitter, 𝑇1→𝑇𝑐. When||𝐼1||>||𝐼𝑐||all magnon transport disappears (black window). aThe negative sign in front of 𝐼2is a reminder that the origin of the spin Hall effect is an electromotive force whose polarity is opposite to the ohmic losses54 bAccording to Eq. (4) and Eq. (5), 𝒯−1 𝐾follows a parabola that intersects the abscissa at 𝐼thand the current that reaches a 25% drop10in signal is the landmark of 𝐼th∕2. sation. We therefore conclude that the increase of 𝜇𝑀by the STE is eventually compromised by the concomitant increase ofthedamping,whichcontinuouslypushes 𝐼thtoanunattain- able higher level. The analytical model proposed to describe the asymptotic approach of 𝜇𝑀to𝐸𝑔is governed by a Γ𝑚, whose power dependence is enhanced by a Lorentz factor to limit the growth of mode amplitude below a saturation value.3 Suchananalyticalformwasfirstproposedtodescribethesat- uration effects of dipolarly coupled spin waves56,57. We show that this simple model captures well the high-power transport regime of each of our nonlocal devices, where we have var- ied the aspect ratio of the electrodes, the magnetic properties, andthefilmthickness 𝑡YIG(seeTable.I).Theotheradvantage ofthis simplifiedmodel isthat itreproduces theexperimental behaviorwithaverylimitedsetofeffectivefittingparameters. Complementing the transport results with Brillouin light scattering (BLS) experiments, we show experimentally that the asymmetric transport behavior, i.e., the spin diode effect, is indeed predominantly caused by low-energy magnons with energies around 𝐸𝐾=ℏ𝜔𝐾≈30𝜇eV, where𝜔𝐾∕(2𝜋)is the Kittel frequency58. This picture is further confirmed experi- mentallybytheenhancementofthespindiodeeffectinmate- rials with isotropically compensated demagnetization factors, where the uniaxial perpendicular anisotropy 𝐾𝑢compensates the out-of-plane demagnetization field. Here the vanishing smalleffectivemagnetization 𝑀eff=𝑀1−2𝐾𝑢∕(𝜇0𝑀1)≈0, where𝜇0is the vacuum permeability, reduces the limiting nonlinear magnon-magnon interaction and thus increases the saturation threshold. Nevertheless, even in the case of these thin films, we observe that the growth of the spin diode effect is capped by a saturation threshold, indicating that the rapid growth ofΓ𝑚as𝜇𝑀approaches𝐸𝑔is a generic process. In fact,reducing 𝑀effonlyallowstoreducethestaticcomponent ofthedipolarinteraction,butlow-energymodescanstillinter- actthroughthedynamiccomponentofthedipolarinteraction. Thepicturethatemergesfromourstudyofnonlocalmagnon transport in extended thin films at high power59is thus very different from the frictionless many-body condensate around thelowestenergymode,indicatinga reduction ofitsrelaxation rate. On the contrary, our results indicate an increase in the magnon-magnon relaxation rate, which prohibits the appear- ance of BEC phenomena. In other words, the reported signal suggests coupled dynamics between numerous modes rather than the BEC picture of a single dominant mode43–46. Finally, we report a collapse of the magnon transconduc- tance as the emitter temperature, 𝑇1, approaches the Curie temperature, 𝑇𝑐. In this limit, the population of high-energy magnons becomes significantly larger than the number of po- larized spins, spin conduction by low-lying spin excitations is perturbed by collisions with their high-energy counterpart, leading to a sharp decrease in the transmission ratio. Westudyhowthemagnondensityenhancementvarieswith the physical properties of the nonlocal device itself. By sys- tematically varying the transparency of the emitter-collector interface and the thermal gradient near the emitter, we show that the density of low-energy magnons is dominated by ther- mal fluctuations rather than by the effective damping value, which always remains finitely positive ( >0). The process for the forward polarity (magnon emission regime) leads to the finding that the density of low-energy magnons seems to be determinedbythetemperatureoftheemitter, 𝑇1,withaparti- cledensitythatincreaseswithincreasingtemperature. Thisis in contrast to the usual (ground state) condensation phenom- enathatnormallyoccurasthetemperaturedecreases. Thebe- havior of our system is similar to that of a free electron gasinsideultrapure2Dmaterialssuchasmonolayergrapheneen- capsulated by two layers of hexagonal boron nitride60,61. In the latter, electron transport becomes hydrodynamic at high temperature and can thus be described by the Navier-Stokes equation62. This leads to unusual transport behavior, as first predicted by Radii Gurzhi in the 1960s63. Our results thus suggest that very similar processes are at work in ultra-low damping magnon conductors64–66, the consequence of which is the emergence of a magneto-hydrodynamic regime at high power. III. ANALYTICAL FRAMEWORK. Since the discovery of spin injection into an adjacent mag- netic layer67, there have been extensive efforts to describe the high-power regime of magnets excited by STE from an exter- nal source. For highly confined geometries, such as nanopil- lars, the pertinent picture that emerged is that of a dynam- ical state dominated by a single eigenmode whose damp- ing is reduced or enhanced by the external flow of angu- lar momentum34–36,38. Its peculiarity is the emergence of an auto-oscillation regime, which characterizes the damping compensation threshold, 𝐼th. The selection of the dominant eigenmode is mostly based on the relaxation rate criterion, which favors the lowest possible value36,38. It usually coin- cideswiththeKittelmode,whosecharacteristicistohavethe longestwavelengthorsmallestwavevector(acriterionmostly valid for strong confinement). In pillars based on ultrathin films, the eigenfrequency is roughly determined by the so- called Kittel frequency68, which has a simple analytical form 𝜔𝐾≡𝛾𝜇0√ 𝐻0(𝐻0+𝑀𝑠)for an in-plane magnetized sam- ple, where𝛾is the gyromagnetic ratio. In nanopillars, the degeneracy with other magnons modes is removed by con- finement, preventing them from playing a significant role in the dynamics38,69. An additional advantage of working with nanopillarsisthatthefreemagneticlayerisefficientlythermal- ized with the substrate and any Joule effect produced by high current densities is reduced. In the case of current perpen- dicular to the plane (CPP), the value of the threshold current for damping compensation is set by 𝐼th=2Γ𝐾𝑒N∕𝜖1, where Γ𝐾=𝛼LLG𝜔𝐾is the relaxation rate of the Kittel mode, 𝛼LLG is the damping parameter, N=𝑉1𝑀1∕(𝛾ℏ)is the effective number of spins that can flip within 𝑉1=𝑡YIG⋅𝑤1⋅𝐿Pt, the magnetic volume fed by the spin emitter, and 𝜖1is the overall efficiency of both the spin-to-charge interconversion and the interfacialspintransferprocessthatdependsonspindiffusion length of Pt and spin mixing conductance at the Pt |YIG in- terface [see below Eq. (10)]33,38. While the introduction of a well-defined 𝐼th,asdescribedabove,successfullyallowstode- scribe the high-power regime of STE in confined geometries, wherethemagnonmodesarediscrete,itwillbeshownbelow that the application to extended thin films, where the magnon dispersioniscontinuous,requirestheadditionalconsideration ofcouplingbetweendegeneratemodes. Thisisdonebyintro- ducingapowerdependentdampingrateleadinginthiscaseto a threshold current that increases as 𝐸𝑔−𝜇𝑀(𝐼1)vanishes.4 FIG. 2. Magnon transport in extended magnetic thin films. (a) At the bottom of the magnon manifold, magnons behave as a two- dimensional gas with a step-like density of states, 𝐷(𝐸)(blue). The non-equilibrium magnon distribution, 𝑛(𝐸)(orange), can be modu- latedbyshifting 𝜇𝑀,themagnonchemicalpotential. (b)Inourcase, the shift of𝜇𝑀is produced by an electric current, 𝐼1, injected into Pt1(emitter). (c) Magnon dispersion for different in-plane propaga- tiondirections, 𝜃𝑘. TheplotisshownfortheYIG𝐶filmat300K.The grayshadingemphasizesthedegeneracyweightofthemagnons,i.e., their tendency to excite parametrically degenerate modes. The black dot denotes𝐸𝐾, the energy of the Kittel mode ( 𝑘→0), the blue dot denotes𝐸𝑔,theminimumofthemagnonband,andtheorangedotde- notes𝐸𝐾,themodedegeneratewith 𝐸𝐾withthehighestwavevector. ThesaturationinstabilitydescribedinFig.1(b)producesastrongen- hancement of Γ𝑚, the nonlinear coupling between degenerate modes intheenergyrange 𝐸𝐾±(𝐸𝐾−𝐸𝑔). Thisforces 𝜇𝑀toasymptotically approach𝐸𝑔at large currents, as shown in (b). A. Magnon transport in extended thin films. Extending the zero-dimensional (0D) model38,42,70to either the one-dimensional (1D)71–73or two-dimensional (2D)10,35,74casesystemshasprovenchallenging. Experimen- tally,thisextensionhasbeenmadepossiblebythediscoveryof regenerativespintransfermechanisms,suchastheSpinHallor Rashba effects, which provide STE while allowing the charge current to flow in-plane (CIP). The nonlocal device used to measuremagnontransconductanceisshowninFig.1(a). Ithas two contact electrodes deposited on YIG and subjected to the spin Hall effect (in our case, two Pt strips 𝐿Pt=30𝜇m long, 𝑤Pt=0.3𝜇mwideand𝑡Pt=7nmthick). Thetwostripesare separated by a center-to-center distance, 𝑑. To reach the high power regime, current densities up to 1012A/m2are injected in Pt1. To characterize the transport properties, we propose to fo- cusonthedimensionlesstransmissioncoefficient 𝒯𝑠≡𝐼2∕𝐼1, which corresponds to the ratio of the emitted and collected spin Hall currents circulating in the two Pt strips, Pt1and Pt2. Thequantity 𝒯𝑠∕𝑅1canbelooselyrelatedtothemagnon transconductance75. NotethatsinceSTEisatransverseeffect,thedifferentcross sections of the spin and charge currents must be taken into account,resultingintheSTEefficiency 𝐼CIP=𝐼CPP⋅𝐿Pt∕𝑡Pt. Assuming that all injected spins remain localized below theemitter, one can obtain an estimate for the amplitude of the criticalcurrentthatcompensatesforthe Γ𝐾attenuationforthis confined mode: 𝐼th 𝑒=2Γ𝐾 𝜖1𝑡YIG⋅𝑤1⋅𝑡Pt⋅𝑀1 𝛾ℏ, (1) where𝑒istheelectroncharge. Anumericalcalculationyields 𝐼th≈ 1.25mA for YIG𝐶films (see Table. I). As will be shown below, the corresponding expression for the threshold current in extended thin films ,Ith, differs significantly from the simplistic estimate above. First, the magnon continuum with the absence of discrete energy levels in unconfined ge- ometries allows for nonlinear interaction between degener- ate eignemodes56,76, whose signature is a power dependent magnon-magnonscatteringtime77–80. Second,theJouleheat- ing and the less efficient thermalization of the emitter lead to a significant increase of the emitter temperature 𝑇1||𝐼2 1= 𝑇0+𝜅𝑅𝐼2 1forlargecurrents. Inournotation 𝜅representsthe thermalization efficiency of our Pt stripe. It is the coefficient thatdeterminesthetemperatureriseperdepositedjoulepower (seeFig.S1in53). Heretheriseisproducedrelativeto 𝑇0,the temperature of the substrate. This introduces additional com- plexityduetoasignificantvariationof 𝑀1,themagnetization under the emitter. These difficulties are large spectral shifts of the magnon manifold on the scale of 𝐸𝐾−𝐸𝑔: the rele- vantenergyrangeforlargechangesinthelow-energymagnon densityinthinfilms. Inthefollowingwewillpresentamodel thatincludesalltheseeffectsandallowstomodelthemagnon transmission ratio [see Fig. 1(c)]. In this model, Ithdepends on a low current nominal estimate of Ith,0whose value is to be extracted from the fit to the data. Although the estimate provided by Eq. (1) gives the correct order of magnitude, the fitvalueissystematicallylarger,indicatingthatthevolumeaf- fectedbySTEismuchlargerthanjusttheYIGvolumecovered by the Pt1electrode. B. Spin transfer effect in extended thin films. 1. Magnon chemical potential. On the analytical side, the single mode picture38is no longer relevant and should be replaced by a statistical distri- bution filled by an integration over all possible wavevectors ∫𝑑𝐤∕(2𝜋)314. Theappropriateframeworktodescribetheout- of-equilibriumregimeofthemagnongasistheBose-Einstein statistics [see Fig. 2(a)]: 𝑛(𝜔𝑘)=1 exp[(ℏ𝜔𝑘−𝜇𝑀)∕(𝑘𝐵𝑇1)]−1,(2) where𝜇𝑀is the electrically controlled chemical potential of the magnons whose value follows the expression30: 𝜇𝑀=𝐸𝑔𝐼1∕Ith, (3) where the analytical expression of Ithin extended thin films willbedefinedlaterinEq.(8). Adrasticchangeinthedensity5 oflow-energymagnonsisexpectedwhen 𝜇𝑀approaches𝐸𝑔, indicated by a blue dot in Fig. 2(c)46. This energy level cor- respondstothespin-waveeigenmodewiththelowestpossible energy in the dispersion relation of in-plane magnetized thin films. Itoccursforspin-wavespropagatingalongthemagneti- zationdirectionwithwavevectors 𝑘𝑔≈2𝜋⋅105cm−1,orwave- length𝜆𝑔≈600nm. Suchawavelengthisstillverylargecom- pared to the film thickness, and for all practical purposes we willassumethatthemagnonsbehaveasa2Dfluidinthelow- estspectralpartofthemagnonmanifoldasshowninFig.2(c), and thus the density of states is a step function as shown in Fig. 2(a)81. In addition,in thin films,the gap between 𝐸𝑔and theenergyoftheKittelmode, 𝐸𝐾(blackdot),issmall,sothat for all practical purposes 𝐸𝑔≈𝐸𝐾can be approximated by the analytical expression of ℏ𝜔𝐾. It will be shown below that in extended thin films both Ithand𝐸𝑔depend on𝐼1, the cur- rentbias,fortwodifferentreasons: i)asmentionedabove,the poorthermalizationoftheelectrodesleadstoadecreaseof 𝑀1 withincreasingthermalfluctuationproducedbyJouleheating (∝𝐼2 1), andii)the finite degeneracy of the magnon bands al- lowsnonlinearcouplingbetweeneigenmodesviadipolarcou- pling [see Fig. 2(b)]. As a consequence, Γ𝑀, the magnon- magnonrelaxationratewhichdefines Ith,increasesforhigher densities of low-energy magnons. A drastic simplification can be made by noting that for bo- sonstatisticstherearetwoingredientsthatgiverisetoachange in the magnon occupation described by Eq. (2): one is the chemical potential, 𝜇𝑀; the second is the temperature of the emitter,𝑇1. The former dominantly affects the low-lying spin excitations,whilethelatteraffectsthewholespectrum,mostly weightedbythehigh-energypart. Itisthereforenaturaltosim- plify the problem as a competing two-fluid problem: one of magnetostaticnatureatenergyaround 𝐸𝐾andasecondofex- change nature at energy around 𝐸𝑇∼𝑘𝐵𝑇1. The segregation withinthetwo-fluidsispreciselythefocusoftherelatedwork inpartII53. Thus,todescribethemagnontransconductancein ournonlocaldevices,weproposetosplitthetransmissionratio 𝒯𝑠=𝒯𝐾+𝒯𝑇into two separate components, each account- ingforthecontributionofthelow-energyandthehigh-energy magnon53.2. Transconductance by low-energy magnons. We now concentrate on deriving an analytical expression forthelow-energymagnontransmissionratio 𝒯𝐾inthelinear regime(𝐼1≪𝐼th). StartingfromEq.(2),wederivethe linear variationofthelow-energymagnonsaroundtheKittelenergy, 𝐸𝐾, which is measured by Δ𝑛𝐾=||𝑛𝐾(+𝐼1)−𝑛𝐾(−𝐼1)||∕2, wherethenumberofmagnonsemittedbytheSTE( 𝐼1⋅𝐻𝑥<0) inFig. 1(b)ismeasuredrelativetothenumberofmagnonsab- sorbedwhilereversingthecurrent(ormagnetization)direction (𝐼1⋅𝐻𝑥>0)82. InFig.1(c)thesetwobiasesaresymbolizedby ▸, for the forward bias (magnon emission), and by ◂, for the reverse bias (magnon absorption). The subtraction allows to distinguishelectricallyproducedmagnons( 𝑛(𝜔𝑘)isoddin𝐼1) from magnons produced by pure Joule heating ( 𝑛(𝜔𝑘)is even in𝐼1)8,83–85. Followingtheexpressionof 𝑛(𝜔𝑘)inEq.(2),one obtainsananalyticalexpressionforthevariationofthenumber of low-energy magnons82: Δ𝑛𝐾||𝐼1≈𝑘𝐵𝑇1 ℏ𝜔𝐾𝐼1 Ith1 1−(𝐼1∕Ith)2, (4) whereIthwas introduced in Eq. (3). The application of cur- rent causes a relative decrease of the magnetization accord- ing to the expression Δ𝑀1= Δ𝑛𝐾𝛾ℏ∕𝑉1, where𝑉1is the effective propagation volume of these magnons. These fluc- tuations are then detected nonlocally by the change in the spin pumping signal generated in the collector: Δ𝐼2∕𝑒= 𝜖2𝜔𝐾(𝜎Pt𝑤2∕𝐺0)Δ𝑀2∕𝑀2, where𝜖2is the total interfacial efficiency of spin-to-charge interconversion at the collector, 𝑤2isthewidthofthecollector, 𝐺0=2𝑒2∕ℎisthequantumof theconductance,and 𝑀1and𝑀2arethemagnetizationvalues under the emitter and collector, respectively. Since the latter two values differ when the emitter and collector are at differ- ent temperatures, we have Δ𝑀2=𝜁 𝑀1Δ𝑀1∕(2𝑀2), where 𝜁=e−𝑑∕𝜆𝐾≪1is the attenuation ratio of the spin signal asmagnonspropagatefromtheemittertothecollector,where 𝜆𝐾is the characteristic decay length and the factor 1/2 takes into account that an equal flux of magnons will propagate un- detectedintheoppositedirection. Thus,thetransmissionratio of low-energy magnons 𝒯𝐾||𝐼1≡Δ𝐼2 𝐼1∝𝜖2⋅𝑒𝜔𝐾 𝐼1⋅𝑀1 𝑀2⋅Δ𝑛𝐾||𝐼1(5) finds an analytical expression, which in the linear regime (𝐼1≪Ith) simplifies as: 𝒯𝐾||𝐼1→0∝e−𝑑∕𝜆𝐾⋅𝜖1𝜖2⋅𝑘𝐵𝑇1 𝛼LLGℏ𝜔𝐾⋅𝜎Pt𝑤2 𝐺0⋅𝑀1 𝑀2⋅𝛾ℏ 𝑀1𝑡YIG𝑤1𝑡Pt, (6) where the value of Ith,0has been replaced by the analytical estimation of 𝐼thas expressed by Eq. (1). The above ex- pression predicts an increase in the magnon transmission ra-tio with decreasing film thickness ∝𝑡−1 YIG, consistent with re- centresults86. Thisfollowsdirectlyfromthefactthatthespin pumpingsignal, 𝐼2,isproportionaltochangesinmagnonden-6 sity. Relating this to an external flow of spins from the in- terface decaying at a fixed rate, the result simply translates to theconcentrationbeinginverselyproportionaltothemagnetic film thickness, all else being equal87. This raises the ques- tion of how far the transmission can go when 𝑡−1 YIG→0. It will be shown below that the answer is disappointing because the nonlocal geometry inherently prevents efficient transport oflow-energymagnons. Theothergeometricalinfluencesare confirmed in the appendix [see Fig. S1]. Note that at large currents,Jouleheatingincreasesthetemperatureofthelattice below the emitter, as described in Fig. 3(a). The correspond- inglateraltemperaturegradientsmayalsointroduceadditional dependencies88,89, which are not considered in Eq. (6). Also, wedonotconsiderheretheeffectsofthelocalgradientofthe externalmagneticfieldneartheemittercausedbytheOersted field generated by the current 𝐼190. C. High power regime. Having established the linear response, this section re- viewsthephenomenathataffectthepropagationoflow-energy magnonsinthestronglyout-of-equilibriumregime. Wedistin- guishbetweenthermaleffectsandnonlineareffectswithinthe low-energy magnon gas. 1. Joule heating. As mentioned above, the 2D geometry prevents efficient thermalization of the magnetic material. Therefore, at high current densities, one should expect a significant variation of 𝑇1. We define 𝑇1=𝑇0+𝜅𝑅𝐼2 1, the temperature rise of the emitter caused by joule heating when the substrate is at 𝑇0, and again𝜅is the coefficient that determines the temper- ature rise per deposited joule power. Fig. 3(a) shows the typ- ical parabolic increase of 𝑇1produced by passing high cur- rent densities through a very thin Pt strip of 300 nm width. The collateral damage is a decrease in the saturation mag- netization, which decreases with increasing temperature [see Fig. 3(b)], approximately along the analytical form 𝑀𝑇≈ 𝑀0√ 1−(𝑇∕𝑇𝑐)3∕2, where𝑀0is the saturation magnetiza- tion at𝑇= 0K53,91. These temperature changes induce a large spectral shift of the magnon manifold. The relevant en- ergyscalehereissetbythedifferencebetween 𝐸𝐾−𝐸𝑔,which islessthanhalfaGHzinthesethinfilmsasshowninFig.2(c). Furthermore, the rise can easily exceed 𝑇𝑐, the Curie temper- ature,evenwhentryingtousethepulsemethodasameansto reduce the duty cycle54. We define the current 𝐼cneeded to reach𝑇𝑐=𝑇0+𝜅𝑅𝐼2 c. In this case, we expect 𝑀1=0at𝐼c, as shown by the arrow in Fig3(b). In turn, the decrease of 𝑀1should cause a collapse below 𝐼cof any threshold currents at the emitter site. At this stage we still assume that the magnons remain non-interacting. We define𝐼pk=Ith,0⋅𝑀𝑇pk∕𝑀𝑇082, the conjectured threshold currentbelowtheemitterheatedbytheJouleeffect,while Ith,0 is the nominal threshold current estimated from the magnetic propertiesat 𝐼1=0. Inournotation, 𝑇pkisthetemperatureof FIG. 3. Electrical variation of magnetic properties at high power. (a) Temperature rise at the emitter due to Joule heating, 𝑇1||𝐼2 1. As shown in (b), this results in a reduction of the magnetization below the emitter, 𝑀1||𝑇1. We define 𝐼c, the critical current to reach 𝑇𝑐, the Curie temperature. (c) Variation of the presumed threshold cur- rent for damping compensation. The reduction of 𝑀1causes a col- lapse below 𝐼cof𝐼pk=Ith,0𝑀𝑇pk∕𝑀𝑇0, the expected onset of auto- oscillation for non-interacting magnons. The orange arrow in (c) in- dicates𝐼pkwhen the normal threshold Ith,0=5 mA. Due to para- metric instability, the occupancy of any eigenmode is capped at 𝑛sat byasharpincreasein Γ𝑚,thenonlinearcouplingbetweendegenerate modes, as shown in (d). This translates in (c) as a sharp increase of Ith=Ith,0Γ𝑚∕(𝛼LLG𝜔𝐾). theemitterat 𝐼pk,while𝑇0isitstemperatureat 𝐼1=0. Thepo- sitionof𝐼pkcanbeobtainedgraphicallybylookingattheinter- sectionofthedashedcurve,whichrepresents Ith,0⋅𝑀𝑇1∕𝑀𝑇0and a straight line of slope 1 crossing the origin, shown by thedottedpointsinpanelFig.3(c). Theverticalorangearrow indicates the expected position of the auto-oscillation onset, assuming that the nominal value is Ith,0= 5mA92. It is im- portant to note that the position of 𝐼pkis very weakly depen- dent on the nominal value if Ith,0≫ 𝐼𝑐because of the rapid decrease of𝑀1near𝐼c[see Fig. 3(b)]. We will return to this observationwhendiscussingbelowtherelevantbiastorenor- malize the data. The important conclusion at this stage is that, taking into account the Joule heating, the damping compensation should alwaysbeachievedwithintherange [−𝐼𝑐,𝐼𝑐],regardlessofthe valueof Ith,0. Thus,aslongasthesinglemodepictureremains valid, one should always observe a divergent increase of the number of magnons within the currently explored range. We willshowbelowthatthisisnotthecase,andthattheculpritis the increased magnon-magnon scattering, which prevents the divergent growth of the magnon density. 2. Self-localization effect. We begin this discussion by emphasizing a trivial observa- tionaboutthemagnondispersioncurve,whichconcernsnon- local devices, i.e., magnon transport outside the volume be- low the Pt |YIG interface. Lateral geometries are notappro-7 priatetostudycondensation,whichselectivelyfavorsthelow- estlyingenergymode,suchasBEC.Forin-planemagnetized thin films, the minimum in the dispersion relation, 𝐸𝑔, corre- sponds to a mode with vanishing group velocity, i.e., a non- propagatingmode93: cf. bluedotinFig.2(c). Thismeansthat nonlocal devices are inherently insensitive to changes in the magnon population that occur in a localized mode. This sit- uation is exacerbated when changes in bias or design end up increasingthemagnonconcentrationbecauseofthenonlinear redshiftofthemagnonspectrum. Thispushestheentirespec- tral range of low-lying spin fluctuations at the emitter below the magnon bandgap for outside the emitter. This prevents these magnons from reaching the collector and further pro- motes self-localization90,94,95. The latter is further enhanced by temperature variations caused by Joule heating when large currents are circulated in the emitter. Theseeffectscanbemitigatedbyloweringthenonlinearfre- quencyshift74,96–98. Afirstpossibilityisbytiltingthesample out-of-plane. There is a peculiar angle where the depolarisa- tioneffectvanishes99. Asecondpossibilityistouseamaterial whose uniaxial anisotropy, 𝜇0𝐾𝑢, compensates for the out-of- plane demagnetization factor, 𝜇0𝑀𝑠, leading to a vanishing effective magnetization. Note that full redshift cancellation requirestuningtheuniaxialanisotropytoaprecisionoftheor- der of(𝐸𝐾−𝐸𝑔)∕(𝛾ℏ)(see below). In this case, the Kittel frequencysimplyreducesto 𝜔𝐾=𝛾𝐻0andisindependentof the magnetization amplitude or direction. It has already been notedthatthiseliminatestheself-localizationeffectofthede- polarizationfactorandthuspromotesspinpropagationoutside theemitterregion74. ItwillbeshowninpartIIthatevenifthe redshift is extinguished and the product 𝜖1⋅𝜖2≪1is omit- ted,thetransmissionratioremainswellbelow50%. Thisupper limitisactuallyexpected,consideringthatlessthanhalfofthe magnonspropagateinadirectioncapturedbythecollector. Fi- nally,weaddthatthisdifficultyofefficienttransmissionaffects notonlylow-energymagnons,butalsohigh-energymagnons, which suffer from very short decay lengths (see part II). 3. Lorentz factor enhanced magnon-magnon decay rate. Wenowfocusontheinter-magnonnonlinearitythatoccurs at high power. We are mainly concerned with saturation ef- fects. Thisinstabilityarisesfromthenon-isochronouspreces- sion of the magnetization (inherent to elliptical orbits), which radiates at harmonics of the eigenfrequency, allowing para- metric excitation of other modes100. This problem was first described by Harry Suhl57,101. It is interpreted that the num- ber of magnons that can fill a particular mode is limited by its nonlinear coupling with other magnon modes, by intro- ducingamagnon-magnonrelaxationtimethatdependsonthe mode occupation. The usual requirement is to find a degener- ate eigenmode within the linewidth. Since this effect depends on the level of degeneracy, it becomes dominant in extended thin films due to the increase in mode density. Moreover, the peculiarshapeofthebandstructureofthemagnonsatlowen- ergy levels introduces a discrimination between the different frequencies in terms of the number of degenerate modes. Itturns out that the energy level with the largest number of de- generate modes occurs precisely at the Kittel frequency. This isemphasizedinFig.2(c)byshadingthedegeneracyweightin gray. It should also be noted that among all the modes being degenerate at 𝐸𝐾, the mode with the highest group velocity, whichalsopropagatesalongthenormaldirectiontothewires, isthepoint𝐸𝐾markedbyanorangedotinFig.2(c). Interest- ingly, the wavelength at the orange dot here is of the order of 1∕𝑤1, the lateral size of the Pt1electrode. This suggests that low-energymagnontransconductanceispreferentiallycarried bymagnonsatthisparticularpositioninthedispersioncurve. To describe the nonlinear interaction between magnons we introduce a saturation occupancy Nsat, which marks the max- imum number of magnons that one can put in one mode be- fore decay to degenerate energy levels starts to kick in. Near this threshold, we assume that the damping rate follows the equation102: Γ𝑚||𝐼1=Γ𝐾√ 1−( Δ𝑛𝐾||𝐼1∕Nsat)2, (7) withΓ𝑚−Γ𝐾representing the nonlinear enhancement of the relaxationratecausedbymagnon-magnonscattering. Thede- pendence ofΓ𝑚on𝐼1is plotted in Fig. 3(d). 4. Current threshold in extended thin films. Toaccountfortheincreaseofcorrelationbetweenmagnons discussed above, we replace 𝐼thin Eq. (1) by Ith||𝐼1=Ith,0𝑀𝑇1 𝑀𝑇0Γ𝑚||𝐼1 Γ𝐾. (8) Introducing this new expression of Ithinto Eq. (4) gives a transcendental equation for Ith, whose dependence on 𝐼1is shown in Fig. 3(c). As Δ𝑛𝐾(𝐼1)approaches Nsatby increas- ing𝐼1,Ithrises sharply together with the damping Γ𝑚(𝐼1)as shown in Fig. 3(d) according to Eq. (7-8). The consequence of this increase is that Ithremains unreachable due to a redis- tribution of the injected spin among an increasing number of degenerateeigenmodes. Theconsequenceforthedependence of𝜇𝑀,thechemicalpotentialofthemagnons,on 𝐼1isshown in Fig. 2(b). Near the origin, the linear dependence of 𝜇𝑀on 𝐼1is set by the intrinsic damping parameter. As Joule heat- ing begins to decrease the magnetization 𝑀1below the emit- ter, it shifts the curve upward. In the same way, the decrease in magnetization pushes 𝐸𝑔to a lower energy in accordance with the red-shift nonlinear frequency coefficient. The sharp riseinΓ𝑚(𝐼1)near𝐼cstopstheriseand 𝜇𝑀,whicheventually approaches𝐸𝑔asymptotically at high currents. Fig.1(c)showstheexpectedbehaviorproducedat 𝐼2fortwo valuesof𝑛sat=5(solidline)or10(dashedline). Hereweuse as parameter 𝑛sat=Nsat∕NNLthe value expressed relative to NNL=NΓ𝐾∕𝜔𝑀, which marks the onset when the change in magnetization becomes of the order of the linewidth, with 𝜔𝑀=𝛾𝜇0𝑀1≈2𝜋×4.48GHz58. In these data we assume8 that𝐼𝑐= 2.5mA and𝐼pk= 2.2mA, which is equivalent to assuming that Ith,0=5mA [see Fig. 3(c)]. Depending on the current values, we observe 3 regimes of transport: ➀:𝐼1< 𝐼𝑐∕2, where we have a linear behavior 𝐼2∝𝐼1;➁:𝐼1∈ [𝐼𝑐∕2,𝐼pk], where we have an asymmetric polynomial increase 𝐼2∝ (1−𝐼2 1∕𝐼2 𝑐)−1, the regime for the spindiodeeffectinanextendedfilm; ➂:𝐼1∈[𝐼pk,𝐼𝑐],where we have a drop of 𝐼2∝ (1−𝐼3 1∕𝐼3 𝑐)1∕2. In the following we willuseEq.(4)combinedwithEq.(8)tofitthedata. Notethat theresultisobviouslyinvertedbyreversingthefielddirection (not shown). We conclude this section by emphasizing that the suscepti- bility of low-energy magnons to capture external angular mo- mentum flux in a thermally changing environment predicts a distinctive nonlinear shape for the current dependence of the magnontransmissionratio,whichwillbeconfirmedbyexper- imental data in the following sections. IV. EXPERIMENTS. In this section, we present the experimental evidence sup- portingthephysicalpicturepresentedabove. Wefocusonthe nonlinear and asymmetric transport properties, our so-called spindiodeeffect,andtheextractionoftherelevantparameters that govern it. A. Nonlocal magnon transport. 1. Measurement of the magnon transconductance. The experiment is performed here at room temperature, 𝑇0=300K, on a 56 nm thick (YIG𝐶) garnet thin film whose physical properties are summarized in Table. I. As explained in part II, we deliberately choose a device with a large sep- aration between the two Pt electrodes ( 𝑑= 2.3𝜇m) to al- low the low-energy magnons to dominate the transport prop- erties. Byinjectinganelectriccurrent 𝐼1intoPt1,wemeasure a voltage𝑉2across Pt2, whose resistance is 𝑅2. To subtract all non-magnetic contributions, we define the magnon signal 𝒱2=(𝑉2,⟂−𝑉2,∥)asthevoltagedifferencebetweenthenormal andparallelconfigurationofthemagnetizationwithrespectto the direction of current flow in Pt. Fig. 4(a) shows the mea- suredvariationof 𝒱2foralargespanof 𝐼1. Themaximumcur- rentinjectedintothedeviceisabout2.5mA,correspondingto a currentdensity of 1.2⋅1012A/m2. TheJoule heatingat this intensity is large enough to reach 𝑇𝑐, the Curie temperature. The resulting voltage 𝒱2is shown in Fig. 4 for both positive (𝐻𝑥pointing to+𝑥) and negative ( 𝐻𝑥pointing to−𝑥) polar- ity of the applied field, whose amplitude is 𝜇0𝐻0=0.2T. In Fig.4(a),theexpectedinversionsymmetrywithrespecttothe field polarity has been folded between 𝐻𝑥>0(left ordinate label in pink) and 𝐻𝑥<0(right ordinate label in blue) to di- rectlyemphasizetheSTEinduceddeviationbetweenmagnon emission and absorption. The measured signal decomposes into two contributions 𝒱2= −𝑅2𝐼2||𝐼1+𝒱2|||𝐼2 1: one is𝐼2, FIG. 4. Experimental observation of the spin diode signal. Panel (a) shows the𝐼1dependence of the nonlocal voltage 𝒱2=(𝑉2,⟂−𝑉2,∥): voltage difference between the normal and parallel configuration of the magnetization with respect to the direction of current flow. The dataareshownforbothpositive(leftaxisinpink)andnegative(right axis in cyan) polarity of the applied magnetic field, 𝐻𝑥. The sign of𝒱2is reversed when the field is inverted. The nonlocal voltage 𝒱2= −𝑅2𝐼2+𝒱2is decomposed into an electric signal, 𝐼2, and a thermal signal, 𝒱2(black). Panel (b) shows the variation of 𝐼2at 𝐻𝑥<0in forward and reverse bias. The blue and red shaded areas in panels (a) and (b) highlight respectively the regimes of magnon emissionandabsorptionrespectivelyfortheforwardandreversebias. Panel (c) plots the magnon transmission ratio 𝒯𝑠= Δ𝐼2∕𝐼1, where Δ𝐼2≡(𝐼▸ 2−𝐼◂ 2)∕2. The solid line is a fit with Eq. (4), (7) and (8) using𝑛sat=4andIth,0=6mA. The black shaded region shows the assumedbackgroundcontributiontospinconductionbyhigh-energy magnons,Σ𝑇(cf.53). The data are collected on the YIG𝐶thin film at ambient temperature, 𝑇0=300K, using a device operating in the long-range regime ( 𝑑=2.3𝜇m). theelectricalsignalproducedbytheSTE,andtheotheris 𝒱2, abackgroundvoltageassociatedwithmagnontransportalong thermal gradients. The latter voltage corresponds to the Spin Seebeck Effect (SSE). One expects 𝒱2≈ 0in well thermal- ized devices. We emphasize the minus sign in front of 𝐼2, which accounts for the fact that the spin-charge conversion is anelectromotiveforce,sothecurrentflowsintheoppositedi- rection to the voltage drop. It is thus a reminder that the re- sultingpolarityisoppositetotheohmiclosses54. Inthelinear regime(𝐼1→0),theelectricalsignaliseven/oddwiththepo- larityof𝐻𝑥or𝐼1,whilethethermalsignalisalwaysodd/even with𝐻𝑥or𝐼110. Focusingonthenonlinearbehaviorobservedforthe 𝐻𝑥<0 configuration(bluedata),103,welabeltheforwardbiasas 𝒱▸ 2as the nonlocal voltage for 𝐼1>0and the reverse bias as 𝒱◂ 2as the nonlocal voltage for 𝐼1<0. The cancellation of Joule effects can be obtained simply by calculating the dif-9 TABLE I. Physical properties of the magnetic garnet films (values at 𝑇0=300K). 𝑡YIG(nm)𝜇0𝑀𝑠(T)𝐻𝐾𝑢(T)𝑇𝑐(K)𝛼YIG(×10−4)𝜌Pt(𝜇Ω.cm)𝑡Pt(nm)𝜅Pt(K)𝐺↑↓(×1018m−2)𝜖(×10−3) YIG𝐴 19 0.167 +0.005 545 3.2 27.3 7 480 0.6 2.1 (Bi-)YIG𝐵25 0.147 +0.174 560 4.2 42.0 6 890 2 .4 7.8 YIG𝐶 56 0.178 -0.001 544 2.0 19.5 7 476 ◦:0.64/•:1.9◦:1.3/•:3.6 ference𝒱◂ 2−𝒱▸ 2, which eliminates any contribution that is even in current. The latter quantity represents the number of magnons produced by the STE relative to the number of magnons absorbed by the STE at a given current bias |𝐼1|. The inset (c) of Fig. 4 shows the observed averaged behav- ior,Δ𝐼2≡(𝐼▸ 2−𝐼◂ 2)∕2 = (𝒱◂ 2−𝒱▸ 2)∕(2𝑅2), expressed as a renormalized quantity 𝒯𝑠=Δ𝐼2∕𝐼1, a strictly positive pa- rameter, as suggested by Eq. (6). However, the subtraction operationdoesnotallowtoseparatethebehaviorbetweenthe forward and the backward direction. In fact, the asymmetry of the electrical signal cannot be obtained from the transport data alone. It requires additional input information. This will be provided below by the measurement of the integral inten- sityoftheBLSsignal,whichdirectlymonitorsthelow-energy partofthemagnonspectrum. BasedontheexperimentalBLS observation in Fig. 5(e), we expect the magnon transmission ratiotoshowcontinuityovertheorigin( 𝐼1=0)andtobehave asastepfunctionofamplitudeinthereversebiasupto 𝐼c. This leadstothefollowingexpression 𝒯◂ 𝑠≈𝒯𝑠||𝐼1→0𝑇1∕𝑇0forthe magnontransmittanceinthereversedbiasbelow 𝐼𝑐. Theratio 𝑇1∕𝑇0takesintoaccountthatthenumberofthermallyexcited low-energymagnonsinthereversepolarizationvarieswith 𝐼1 duetoJouleheating. Thisintroducesasecondorderdistortion which will be discussed in more detail in the next section and inpartII.Soweconstruct 𝒱2=−𝒱◂ 2+𝑅2𝒯◂ 𝑠⋅𝐼1inthere- versebiasfromtheoppositemagneticconfiguration. Wethen force𝒱2to be even in current to get the forward bias behav- ior. TheresultisshownasblackdotsinFig.4(a). Wecanthen derive𝐼▸ 2=(𝒱2−𝒱▸ 2)∕𝑅2and𝐼◂ 2=(𝒱2−𝒱◂ 2)∕𝑅2,which is shown in Fig. 4(b). In Fig. 4(b) we observe 3 different transport regimes as predicted above. We have ➀with𝐼1∈ [−2,1]mA: the spin conductance is approximately constant; ➁with𝐼1∈ [1,2.2]mA:thespinconductanceincreasesgraduallyandsat- urates quickly; and ➂with𝐼1∈ [2.2,2.5]mA: the spin con- ductance decreases abruptly to vanish at 𝐼c. Only the regime ➀is anti-symmetric in current. We emphasize at this stage that the sequence of behavior is reminiscent of the 3 regimes predicted in Fig. 1(c). We can also repeat the same analysis to construct the data when 𝐻𝑥>0(see part II). As expected, thepolarityoftheasymmetryreverseswhenthemagnetization direction is changed. In all cases, the forward regime occurs only when𝐼1⋅𝐻𝑥<0, which corresponds to the polarity of the damping compensation. Inthenextsection,wewillconfirmtheabovebehaviorusing BLS spectroscopy. Throughout the rest of the paper, we will alwaysdiscussthedataasshowninFig.4(c)intheformof Δ𝐼2 as a function of |𝐼1|and properly renormalized by 𝐼1and𝑇1 to allow comparison between different devices (see part II53).2. Spin diode effect. We now discuss in more detail the amplitude of the asym- metric rise of the 𝐼2signal in Fig. 4(c). A striking feature of Fig. 4(c) is the limited growth of the spin diode signal, which is capped by a meager factor of 3 rise compared to the value at small currents. This variation is significantly smaller than the changes in cone angles observed at the damping compen- sation threshold in nanopillars, which reach several orders of magnitude. Suchinefficiencyisfurtherconfirmedbyprevious reports aiming at modulating the transport by damping com- pensationwithanadditionalheavymetalelectrodeplacedbe- tween the emitter and the collector, which increases the con- ductionbyafactorof6atmost14–16,98,104. AsshowninFig.2, this cannot be explained by a threshold current larger than the current explored window [−𝐼𝑐,+𝐼𝑐], but rather indicates a strong coupling between the magnons that prevents a large growthofthemagnondensity. Theasymmetriccontributionis wellaccountedforbyEq.(4). Thebestfitisobtainedbyusing 𝑇⋆ 𝑐=515K,Ith,0=6mAand𝑛sat=4andisindicatedinthe plot by the solid line. The low-energy magnon contribution is added to a background indicated by the dashed line. This background accounts for the competing contribution of high- energy thermal magnons to the electrical transport. Its origin and analytical expression can be found in Ref.53. In our fit, it representsa50%additionalcontribution Σ𝑇∕(Σ𝐾+Σ𝑇)=0.5 inEq.(4)ofref.53. Wealsonotethatthevalueof 𝑇⋆ 𝑐usedfor the fit is significantly different from the Curie temperature of thisfilm(seeFig.S1ofref.53). Thispointwillbeinvestigated in more detail in part II53. B. Brillouin Light Scattering. Itisusefulatthisstagetoconfirmspectroscopicallythatthe spindiodeeffectreportedaboveisindeedduetoanasymmet- ricmodulationofthelow-energymagnons. BLSisatechnique of choice for this purpose, since it is specifically designed to monitor spectral shifts in the magnon population at GHz en- ergies. Furthermore, its high sensitivity allows the detection of fluctuations down to the thermal level. While in the past we have performed comparative studies of the transport and BLS behavior on exactly the same device10,54, in this partic- ular case we will introduce in the discussion a thinner YIG𝐴 garnet film, whose physical properties are also given in Table I.Thechangeofsamplesisnotintentionalandispurelyrelated to the chronological context of the experiments. For all prac- tical purposes, the only relevant difference is a change in the valueof𝐼cfrom𝐼𝑐=2.5mAto2.1mAforYIG𝐶andYIG𝐴, respectively,duetodifferencesinPtresistivity. Wehaveveri-10 FIG.5. Experimentalevidencethatthespindiodeeffectisdominated bylow-energymagnons. Panels(a)and(c)showthemicrofocusBril- louin light scattering ( 𝜇-BLS) spectra as a function of 𝐼1performed underthetwoPtelectrodes(seeblackdotontheinset)atPos1(emit- ter) and Pos2 (collector) for 𝐻𝑥<0. Each BLS spectrum is renor- malizedbytheamplitudeoftheKittelpeakat 𝐼1=0. Panels(b)and (d) show two sections at fixed current 𝐼1= ±2mA near𝐼pk. The greenmarkerindicatesthespectralpositionof 𝐸𝐾at𝐼1=2mA.The spectralpositionoftheself-localizedspinfluctuations, 𝐸𝐾1,(seecir- cles)at𝐼1=+2mA(seeorangemarker)occurwellbelow 𝐸𝑔2≈𝐸𝐾2 at Pos2 (see green marker). The shift 𝐸𝐾1−𝐸𝑔2is about 0.7 GHz. Theparamagneticlimit, 𝛾𝐻0∕(2𝜋),isindicatedbytheyellowvertical dotted line at 5.8 GHz. Panel (e) shows the integrated BLS intensity at Pos1 as a function of 𝐼1. The solid blue line is a guide for the eye. Theblackdashedlineshowstheexpectedvariationofthermally activated low-energy magnons produced by Joule heating. In echo with Fig. 4, the blue and red shaded areas highlight the variation of magnonemissionandabsorptionwithrespecttothermalfluctuation. Theinsetpanel(f)showstheevolutionof 𝑀1inducedbyJouleheat- ing. ThebluepointsareinferredfromtheevolutionoftheKittelfre- quency (dashed parabola in panel (a)). The dashed lines are inferred fromtheSQUIDmeasurementsshowninFig.S1ofRef.53. Thedata arecollectedontheYIG𝐴thinfilmat𝑇0=300Kusingadevicewith a distance𝑑=1.0𝜇m between the two Pt electrodes. fiedthatallotherpropertiesdiscussedherearegenerictoboth samples. Sinceweareinterestedinlocalchangesinthemagnonpop- ulation,weusemicro-focusBrillouinlightscattering( 𝜇-BLS) spectroscopy105. Theprobelightwithawavelengthof532nm andapowerof0.1mWisgeneratedbyasingle-frequencylaser with a spectral linewidth of <10MHz. It is focused through the sample substrate into a submicron diffraction-limited spot using a 100×corrected microscope objective with a numeri-cal aperture of 0.85. The scattered light was collected by the samelensandanalyzedbyasix-passFabry-Perotinterferome- ter. Thelateralpositionoftheprobespotwascontrolledusing a custom-designed high-resolution optical microscope. The measured signal (the BLS intensity at a given magnon fre- quency)isproportionaltothespectraldensityofthemagnons at that frequency and at the position of the spot. By moving the focal spot across the film surface, we can obtain informa- tion about the spatial variations of the magnon spectral dis- tribution. Note that only low-energy magnons contribute to theBLSintensity,sincetheBLStechniqueissensitiveonlyto the wavevectors smaller than 2.4⋅105cm−1. However, since the frequency of magnetostatic magnons also depends on the total number of magnons in the sample, the spectral position oftheKittelpeakallowstoderiveinformationaboutthelocal temperature from the BLS data106,107. Fig.5comparesthecurrentmodulationofthespectraloccu- pationbelowtheemitterandcollectorelectrodeslabeledPos1 (𝑥= 0, emitter) and Pos2 ( 𝑥= +𝑑, collector). The position of the spot is indicated by a black circle in the inset images in Fig. 5(a) and (c). To allow a quantitative comparison of the different locations, all curves have been renormalized by the value of the thermal fluctuations at 𝐼1=0, which is assumed tobeconstantthroughoutthethinfilms. Inallthesemeasure- ments, the field is fixed at 𝜇0𝐻𝑥= −0.2T: a value identical to that used in the transport measurement shown in Fig. 4. We start the analysis by concentrating first on panel (a) of Fig. 5, which shows in a density plot the actual variation of themagnonspectraatPos1. ThespectralvariationoftheBLS signal at𝐼1= 0leads experimentally to a peak instead of thepredictedstepfunctionshowninFig.2(a)for2Dsystems. The attenuation of the spectral sensitivity at high frequencies is an experimental artifact related to the extreme focus of the optical beam, which renders the scattered light insensitive to spin-waves whose wavevectors are larger than the inverse of the beam waist. The decrease of the signal above the Kit- tel frequency is thus directly related to the transfer function of the detection scheme, which attenuates short wavelength magnons35,108,109. The black dashed lines in the plots Fig. 5(a) show the shift oftheKittelfrequencyasafunctionof 𝐼1. Atlowcurrent,the shift follows a parabolic behavior as expected for Joule heat- ing. Note that the curvature of the parabola increases as one moves away from the heat source, as shown in Fig. 5(c) taken at Pos2, which we associate with the lateral thermal gradient 𝜕𝑥𝑇1<086,91. ItisalsoworthnotingthatforthesignalatPos1 in Fig. 5(a), the local curvature increases dramatically as one approaches𝐼c,whichweattributetothedecreasein 𝑀1as𝑇1 approaches𝑇𝑐. Interestingly, the effect is more pronounced at 𝐼1>0than at𝐼1<0, suggesting a self-digging effect due to asymmetric excitation of low-energy magnons. Also visible inFig.5(a)istheextinctionoftheBLSintensityintheregion ➂when𝐼1≥2.1mA. Finally, the BLS signal decays at large currents atPos1 (below theemitter), while remainingfinite at Pos2 (below the collector). The density plot in Fig. 5(c) further confirms the enhance- ment / attenuation of the spin fluctuations depending on the polarity of the current. Starting from thermal fluctuations at11 𝐼1= 0, the signal decreases for 𝐼1⋅𝐻𝑥>0and increases for𝐼1⋅𝐻𝑥<0. A more detailed analysis at low current am- plitude(notshown)confirmsalinearvariationin 𝐼1inthere- gion➀. BLS spectra at large currents are shown in Fig. 5(b). They compare the magnon distribution observed in Fig. 5(a) at𝐼1=2mA for both negative (green horizontal dash-dotted cut) and positive (orange horizontal dash-dotted cut) polar- ity of the current. At 𝐼1= −2mA, the maximum ampli- tude in Kittel mode (green vertical marker at 6.7 GHz) has a normalized amplitude below 1, i.e., a lower amplitude than at𝐼1= 0. At𝐼1= +2mA the density plot shows a sig- nificant enhancement of the signal, highlighted by the dashed circles in Fig. 5(a). The corresponding amplitude of the sig- nal [orange dots in panel (b)] shows an enhancement of more thananorderofmagnitude. Thisenhancementcorrespondsto the increase in spin conduction in the region ➁. In addition, the BLS data taken at Pos1 show the self-localization of the asymmetric excitation due to the red shift that arises for large cone angles30,97. It is important to note that the induced shift isalmostaslargeasitcanbe,sincetheKittelfrequencyalmost reachestheparamagneticlimit 𝜔𝐻=𝛾𝐻0,visibleonthepan- els(a)and(c)asalightverticaldotedlineatabout5.8GHz110. Notethattheparamagneticlimitisreachedonlyfor 𝐼1>0and not for𝐼1<0. This suggests that the film is still in its ferro- magnetic phase when the signal disappears at 𝐼𝑐≈ 2.1mA. This feature will be discussed in connection with the discrep- ancy between 𝑇⋆ 𝑐and𝑇𝑐in part II. The excitation pocket cre- ated under the emitter within the circular area also appears as apeakatPos2inFig.5(d),centeredaroundtheorangevertical marker at 6.7 GHz, i.e., well below the position of the Kittel modeatthatposition(greenverticalmarkerat7.4GHz). This suggests that the collector could still probe magnetic fluctua- tionsthatarespectrallybelow(about0.7GHz) 𝐸𝑔,theenergy bandgapatthisposition. Althoughtheamplitudeofthepeakat the orange marker is significantly reduced as one moves from Pos1toPos2,confirmingnumerousindicationsthatthemajor- ityoflow-energymagnonsremainlocalizedbelowtheemitter electrode(seebelow),thesespectralfluctuationsarenotcom- pletelysuppressed. Weinterpretthisapparentcontradictionas a signature that the wavelength of the mode excited under the emitter(𝜆around0.6𝜇m)remainslargecomparedtothewidth oftheemitterwell,andtheratiocorrespondstotheevanescent decaybetween Pos1andPos2. Theissueof spatialdecaywill be investigated in more detail in part II53. To gain further insight, we plot the spectral integration of the BLS signal as a function of 𝐼1in Fig. 5(e). The first key feature directly observed in the plot is the continuity of the signal across the origin for both polarity of the current 𝐼1. This observation is used above to extract the asymmetry of the magnon transmission ratio. It will be shown below that the BLS measurement confirms some other key features ob- served in the magnon transport properties shown in Fig. 4. The second key feature is the limited growth of the intensity at𝐼pk. However, the size of the peak is larger at Pos1 than at Pos2, suggesting the importance of localized magnons under the emitter. The third key feature is the idea that there are 3 transportregimes. Inparticular,weobservethecollapseofthe BLS signal at |𝐼1|>|𝐼𝑐|under the emitter.To emphasize the similarity with transport, in Fig. 5(e) we tentatively plot the evolution with 𝐼1of the number of ther- mally activated low-energy magnons produced by Joule heat- ing with a dashed line. The behavior follows the curve Δ𝑛𝑇1 introduced in part II53. The underlying parabolic increase of this background signal is directly visible in the evolution of themaximumBLSintensityalongthedashedlineinFig.5(e). The blue and red shaded areas show the deviation of the BLS intensity from the nominal thermal occupancy. They indicate the amount of magnons emitted and absorbed by the STE in forward and reverse bias, respectively, and the deviation is analogous to the blue and red shaded areas in Fig. 4(b). For 𝐼1< 𝐼𝑐∕2the curves deviate equally from the dashed curve, indicating that equal amounts of magnons are transferred be- tweenthemetalelectrodeandtheYIGfilminthelinearregime between forward and reverse bias. The curvature at the origin should scale as the variation of 𝑀1with𝐼1. To this end, we plottheevolutionof 𝑀1undertheemitterintheinsetFig.5(f). Thebluepointsarederivedfromthevariationwithcurrentof thespectralpositionoftheKittelfrequency,whichissensitive to temperature. The observed behavior is consistent with the expected evolution of 𝑀1with𝐼1, which is derived from the dependence of 𝑇1on𝐼1as shown in Fig. S1 of Ref.53. The resultisshownasadashedlineandtheagreementensuresthe validity of the evolution with 𝐼1of the thermal background shown in Fig. 5(e). In this data treatment, we do not attempt to correct for the use of different pulse duty cycles, which re- sults in a small discrepancy in the value of 𝐼c. To conclude this section, we interpret the fact that all the distinct transport signatures observed in Fig. 4(b) are present in the BLS intensity shown in Fig. 5(e) as a strong indication thattheyarepredominantlydrivenbythechangeindensityof low-energymagnonsbelowtheemitter. However,weobserve a difference in the scaling of the effect, which we attribute to the self-localization of these low-energy magnons. This will be the subject of the band mismatch below. A more rigorous quantificationofthelocalizationfractionispartoftheanalysis in part II. C. Influence of the lattice temperature. Inthissectionweinvestigatetherelevantparametersthatin- fluencethespindiodeeffect. Tothisend,weproposeinFig.6 tocompare theasymmetricregimewhile independentlyvary- ingeither𝜖,theinterfacialefficiencyofspin-to-chargeconver- sion, or𝜅, the efficiency of device thermalization. It builds on our two recent studies91and111, where more details can be found. The device was patterned on YIG𝐶, which is the same film as the one studied in Fig. 4, but twice as thick as the YIG𝐴film studied in Fig. 5 (see Table. I). In this batch, local annealing111can be used to apparently increase the spin mixing conductance 𝐺↑ ↓of a single electrode. The term spin mixingconductanceshouldbeinterpretedhereasaneffective fittingquantity. AsexplainedbyKohno etal.111,thephysicsat playinthisannealingmechanismdoesnotinvolveachangein intrinsicproperties,butratherachangeinextrinsicproperties via an expansion of the contact area. We denote 𝜖◦and𝜖•as12 FIG. 6. Experimental evidence that the spin diode effect is mainly controlled by thermal fluctuations below the emitter. Spin diode ef- fect as a function of emitter temperature raised by Joule heating, 𝑇1=𝑇0+𝜅𝐶𝑅Pt𝐼2 1, when either (a) 𝜖, the spin-transfer efficiency, or (d)𝜅𝐶, the heat dissipation coefficient, are independently varied. Theupperscaleshowsthecorrespondingcurrentbias, 𝐼1. Thelegend of panel (a) shows with the symbols ◦/•which electrode of the pair (emitter,collector)hasbeensubjectedtolocalannealing111resulting in𝜖•/𝜖◦≈3asshowninTable.I.Panel(d)showsthebehaviorwhen theannealeddeviceiscoveredbyanAlheatsink,whichreducesthe efficiencyoftheJouleheating 𝜅′ 𝐶<𝜅𝐶. Inpanels(a)and(d),allsolid lineshavethesameshapeasinFig.4(b),albeitwithdifferentscaling factors. Therightpanelshowsthecorrespondingcurrentdependence of the SSE voltage 𝒱2in panels (b) and (e) and the normalized in- verse transmission coefficient of the spin current 𝒯−1 𝑠=𝐼1∕𝐼2in panels (c) and (f). The dashed lines are parabolic fits. theSTEefficiencybeforeandafterannealing. Wehaveshown thatapplyinglocalannealingfor60minsat560Kcanreduce 𝜖•∕𝜖◦≈3without changing the Pt resistance, i.e., 𝜅𝐶111. We first investigate changes in the spin transport when al- ternatively the collector and emitter are annealed using the 4 possiblecombinatorialconfigurationspossibleassummarized inthelegendofFig.6(a). WefindinFig.6(c)thatthenormal- ized current ratio 𝒯𝑠−1≡( 𝒯𝑠/ 𝒯𝑠|𝐼1→0)−1 (9) of the 4 curves fall on the same parabola. Note that we have introduced the notation of underlined symbols, which will be referred to in the following as the quantity normalized to thelow current value. The parabolic shape is consistent with the behaviorreportedinFig.7(c)ofpartII53,whichsuggestsade- creaseas1−(𝐼1∕𝐼𝑐)2inthe➁regime. Thisalreadysupports thatthevalueof 𝜖playsverylittleroleintherelativeamountof electrically excited low-energy magnons. If the sample were well thermalized 𝑇1≈𝑇0, the zero point of the parabolic fit shiftssignificantlytohighercurrentvalues,asshownbelowin Fig. 6(f), suggesting that in this case the extrapolated decay 𝒯𝑠−1pointstotheamplitudeof Ithassuggestedbytheinver- sionofEq.(4). Theinterceptvaluedecreasesastheinfluence ofJouleheating,or 𝜅𝐶,increasesandisboundedby 𝐼casdis- cussedinFig.3. Werecallthatlocalannealinghasnotchanged thePtresistanceandthusinFig.6(a-c)theefficiencyofJoule heating𝜅𝐶is identical for all 4 cases. These conclusions are alsoconsistentwiththebehavioroftheSSEsignal 𝒱2shown in Fig.6(b). The 4data sets areexactly divided intotwo pairs ofcurves,whicharescaledbytheratio (𝜖•∕𝜖◦). Thedifference is solely determined by the value of the efficiency coefficient 𝜖2on the collector side. As expected for SSE, changes in the value of𝜖1on the emitter side are irrelevant. We now plot in Fig. 6(a) the variation of the renormal- ized STE transmission coefficient 𝒯𝑠⋅𝑇0∕𝑇1as a function of 𝑇1=𝜅𝐶𝑅Pt𝐼2 1+𝑇0, with𝑇0=300K. The normalization by 𝑇1allows to view the magnon density changes from the point ofviewofawellthermalizeddevice,assuggestedbyEq. (6). Theupperabscissascalecanbeusedasanabacustoconvert 𝑇1 backto𝐼1. Onalldeviceswefindthat 𝒯𝑠⋅𝑇0∕𝑇1increasesnon- linearly above 360 K. Above 460 K the spin transmission de- creaseswithincreasingtemperature,asexplainedinFig.4(b). As expected, we find that the amplitude of 𝒯𝑠⋅𝑇0∕𝑇1scales as the product of 𝜖1⋅𝜖2. This observation confirms that the Pt is weakly coupled to the YIG, i.e., only a small part of the spincurrentcirculatingintheYIGisdetectedbythecollector. If this were not the case, the signal would not depend on the collector efficiency 𝜖2. This is further confirmed by the width dependence of 𝒯𝑠(see Fig. S1). The most interesting feature is the approximate superposi- tionofthe2curves( ◦,•)and( •,◦),whichconsistsininverting emitterandcollectorinadevicewhereoneelectrodeis3times more efficient at emitting magnons. While the superposition is almost perfect at low currents, at high currents the magnon conductance with emitter and collector inverted (orange dots) leads to a significantly larger magnon conductance than the normal configuration (green dots). However, the difference is smallcomparedtothefactorof3producedbyannealing. Thus for all practical purposes, we interpret the data as somewhat confirming that the spin current circulating between the con- tact electrodes is proportional to 𝜖1⋅𝜖2, as Eq. (6) explains it. ThesuperpositionofthetwocurvesinFig.6(a)showsthatthe densityofmagnonsintheYIG(hereitisvariedbyafactorof 3)seemstohavenoeffectontheshapeof 𝒯𝑠⋅𝑇0∕𝑇1,andthe pertinent parameter that determines its behavior remains the emitter temperature 𝑇1. Moreover, if one compares in Fig. 8 of53𝒯𝑠versus𝑇1(as opposed to as versus 𝐼1) taken on two differentYIGthicknesses,itseemsthatthethicknessplaysno roleindeterminingthenonlinearbehaviorbetween 𝐼2and𝐼2 1. To further support this notion, the coefficient 𝜅𝐶can be modified while keeping 𝜖constant. As explained in91, one13 can cover the annealed device with a 105 nm thick Al layer that acts as a heat sink (the heat sink is separated from the Pt electrode by a 20 nm thick protective layer of Si3N4). The in- fluenceoftheheatsinkcanbeseenbycomparingtheabacusin Fig.6(a)andFig.6(d). TheAllayerhasreducedtheefficiency oftheJouleheating 𝜅𝐶by27%. Thechangesaremostvisible in Fig. 6(e), which shows a completely different nature of the SSE signal compared to Fig. 6(b) (different curvature, differ- entpolarity91). However,once 𝐼1isconvertedto 𝑇1,Fig.6(d) scales with Fig. 6(a), even though a larger amount of current 𝐼1is circulating in the former. This leads to the conclusion that the nonlinearity of the magnontransmissionratio 𝒯𝑠⋅𝑇0∕𝑇1producedbySTEseems to be governed mainly by the lattice temperature. This obser- vationaloneseemstocontradictourpreviousconclusionfrom the BLS experiment, which attributes the spin diode effect to low-energy magnons. The apparent discrepancy is explained inFig.3(c). Knowingthatthedensityoflow-energymagnons is also sensitive to temperature, the disappearance of 𝑀1at 𝐼cdue to Joule heating shifts the position of 𝐼pkbelow𝐼c, which then becomes weakly dependent on the room temper- ature value of the threshold current, Ith,0. The process still requires𝜇𝑀toapproach𝐸𝑔byinjectingspins,butonceinthe vicinityandthenonlinearprocessisfullyinplace,thetemper- ature change is dominant. Thus, this artifact is a consequence oftheinabilitytoefficientlythermalizethePtelectroderather thananintrinsicphenomenon. Nevertheless,itemphasizesthe importanceofthermalfluctuationsinnonlinearphenomena. It underlinesthattheparametricthresholdisalsodeterminedby the initial magnon thermal fluctuation in the sample, bearing in mind that all nonlinearities are suppressed at absolute zero temperature. All experimental data in Fig. 6(a) and Fig. 6(d) are fitted with the same curve as shown in Fig. 4(b). The only param- eter that varies is the vertical scaling factor. The fact that all the curves have exactly the same shape also supports the sug- gestionthat𝑇1determinesthenatureof 𝒯𝑠⋅𝑇0∕𝑇1,wherethe shaded region shows the background contribution from high- energythermalmagnons 𝒯𝑇,whereΣ𝑇∕(Σ𝑇+Σ𝐾)≈0.5rep- resents the relative weight at this distance. The Al capping also changes the spatial profile of 𝑇(𝐫), whichdefinestheconfinementpotentialof 𝑀𝑇neartheemit- ter, as shown in Ref.91. Thus, the absence of a large discrep- ancybetweenFig.6(a)andFig.6(d)indicatesthatthedepthof bandshiftproducedbyJouleheatingisthedominantparame- ter, while the spatial extent of the confinement region defined here by𝜕𝑥𝑇is not significant. D. Magnon band mismatch. Inthissectionwewillfurtherelucidatewhichnonlinearef- fects lead to the suppression of spin propagation in the high power regime. FIG. 7. Experimental evidence that the spin diode effect involves a narrow spectral window of magnons around the Kittel energy. The upper panels illustrate the band mismatch in the GHz spectral range between two regions at different temperatures. In panel (a), calcu- latedforYIG𝐶thinfilms,apropagationgapofalmost0.7GHzrises between the low-lying magnon modes, when the temperature differ- enceattains150K.There,thefrequenciesofthelow-energymagnons below the emitter are below the bulk bandgap and thus prevented frompropagatingandinsteadencouragedtoself-localize,limitingthe growthofthenonlocalsignal. Inpanel(b),calculatedfor(Bi)-YIG𝐵, this barrier vanishes when 𝐾𝑢−𝜇0𝑀𝑠→0. There, the low-energy magnonsareallowedtocontributesignificantlytothelong-rangespin transport. Comparison of the propensity of low-energy magnons to travel long distances ( 𝑑= 4.3𝜇m) between (c) YIG𝐶and (d) (Bi- )YIG𝐵. We observe a significant increase in the spin diode signal between(c)YIG𝐶and(d)(Bi-)YIG𝐵. Weexplainthisbythesuppres- sion of self-localization effects produced by the nonlinear frequency shift. 1. Nonlinear frequency shift. We begin this discussion by examining the issue of non- linear frequency shift. We recall that this effect comes from the depolarization factor along the film thickness 𝑁𝑧𝑧= −1, which introduces a correction to the Kittel frequency that de- pendsonthemagnetization, 𝑀𝑠. Thiscoefficientcorresponds to a red shift, i.e., a decrease of 𝑀𝑠causes a red shift of the Kittel mode and thus of 𝐸𝑔. The maximum shift that can be induced is(𝜔𝐾−𝛾𝐻0)∕(2𝜋) ≈ 2GHz, which is the en- ergydifferencebetweentheKittelmodeandtheparamagnetic limit at𝜇0𝐻0= 0.2T. Fig. 5 shows experimental evidence for shifts as large as 0.7 GHz, produced either by Joule heat- ing of the emitter [see the dashed line in Fig. 5(a) and (c)] or by self-localization of the mode (see the dashed circle in Fig. 5). An increase in the precession angle 𝜃also produces a redshiftoftheresonanceduetoadecreaseintheinternalfield,14 which depends on the time-averaged magnetization following 𝑀𝑠cos𝜃77,80. The latter effect is responsible for the foldover of the main resonance. As shown in the inset of Fig. 7(c), such a shift is strong enough to push the entire spectral win- dowoflow-energymagnonsfluctuatingbelowtheemitterun- derneaththeenergygap 𝐸𝑔oftheYIGfilmoutside. Thissug- geststhatspinfluctuationsproducedbySTEcanstillbesensed below the collector despite having energies below the propa- gationbandwindowofmagnonsatthisposition,andtheprop- agation length 𝜆𝐾is the decay length of the evanescent spin- waveoutsidetheenergywell. Thisfavorstheself-localization effect90,94,95as reported in the analysis of Fig. 8 of Ref.53. We want to compare this behavior with the transport prop- erties in (Bi-)YIG𝐵thin films (see material parameters in Ta- ble. I). The peculiarity of sample B is that the YIG sample is doped with Bi. For the right concentration of Bi it is possible to match the uniaxial anisotropy with the saturation magneti- zation (see Table. I), which leads to 𝑀eff≈ 0. This corre- spondstoathinfilmthathasanisotropicallycompensatedde- magnetization effect. In this case, the Kittel mode becomes isochronous and independent of the precession cone angle. The Kittel frequency simply follows the paramagnetic pro- portionality relation 𝜔𝐾=𝛾𝐻0(similar to the response of a sphere). In particular, the value of 𝜔𝐾is independent of 𝑀𝑇and therefore the nonlinear frequency shift is null74,97. This implies that the conduction band of the magnons be- tween emitter and collector remains aligned as schematically shown in Fig. 7(d), which then prevents the self-localization effect of the nonlinear frequency shift. The improvement of themagnontransmissionratioisclearlyvisiblewhencompar- ingFig.7(c)andFig.7(d). Furtherevidenceforthereduction ofself-localizationinthelattercaseistheobservationthatthe variationofthetransmissionratioinFig.7(d)nowmimicsthe observed variation of low-energy magnons under the emitter by BLS, as shown in Fig. 5(e). We will return to this impor- tant point in PartII53while discussing the spatial decay of the magnon transmission ratio. On the quantitative side, we find that the ratio of initial to maximum values is 15.1 for (Bi-)YIG𝐵compared to 7.2 for YIG𝐶). A first important observation is that while the sup- pression of the elliptical precession and the temperature de- pendence of 𝜔𝐾by the uniaxial anisotropy compensating the dipolarfieldfavorsthepopulationoflow-energymagnons,the signal still saturates in the case of (Bi-)YIG𝐵. The solid line in Fig. 7(c) and (d) is a fit with Eq. (4) using 𝑛satas a free pa- rameter. While for the YIG𝐶sample we find that the best fit is obtained by using 𝑛sat=4as explained above, the value of the saturation threshold increases to 𝑛sat= 11in the case of Bi-YIG𝐵. This is also a direct experimental evidence that the contribution of low-energy magnons to the spin diode effect concerns a rather narrow spectral window around the Kittel energy,withanupperlimitwidthofabout1GHz. Thisbroad- eningshouldbecorrelatedwiththeenhancedmagnon-magnon scattering time indicated by Γ𝑚in Fig. 2(c). Another relevant energy scale is the difference 𝐸𝐾−𝐸𝑔, which varies as 𝑡Y1G. Thisimplies thatinultrathin filmsofgarnet thisphenomenon of localization of low-energy magnons is enhanced.2. Saturation of the magnon density. We emphasize, however, that although the nonlinear fre- quency shift is zero in the case of (Bi-)YIG𝐵, the system is stillsubjecttothesaturationeffect. Forexampletheobserved factor15.1isstillsignificantlysmallerthanthechangeincone angle observed in nanopillars at the damping compensation threshold. This implies that the aforementioned saturation ef- fects and the rapid growth of the magnon-magnon relaxation rate is a general phenomenon that is not suppressed by reduc- ing the nonlinear frequency shift. The vanishing nonlinear frequency coefficient only eliminates the ellipticity of the tra- jectoryforthelongwavelengthmagnonswhosewavevectoris smallerthantheinverseofthefilmthickness. Itdoesnotelimi- natetheself-depolarizationeffectofthespin-wave. Thisvalue depends mainly on 𝜃𝑘, the angle between the propagation di- rectionandtheequilibriummagnetizationdirection,thelatter beingtheoriginofthemagnonmanifoldbroadening. Wenote that for the wavevectors around 𝐸𝑔this is the dominant ori- gin of the ellipticity, since the broadening almost reaches the maximum value of 𝛾𝜇0𝑀𝑠, as shown in Fig. 1(d). Thisshowsthatthetuningof 𝑀eff,whichallowstoremove the non-isochronicity of the long wavelength magnons, is re- sponsible for an increase of the saturation threshold, which is found to be almost three times higher. V. CONCLUSION. In this work we draw a comprehensive picture of the role of low-energy magnons in the electrical transconductivity of extended magnetic thin films. While spin conduction at low intensitiesappearstobehavelargelyasexpected,thebehavior at high intensities is markedly different from that observed in highly confined geometries such as nanopillars. The main difference is related to the tendency of the in- jected spin to spread between different degenerate eigen- modes. Thus, there are phenomena related to the two- dimensionality that intrinsically prevent a single mode from dominating the others (as is possible in 0D and 1D45). In a confinedgeometry,theenergygapcreatedbyconfinementbe- tween different eigenmodes protects the main fluctuator from relaxationintoothermodes. Inanextendedthinfilm,thisbar- rier is removed and degeneracies arise, leading to an efficient redistribution of energy between degenerate modes. Even if one mode is pumped more efficiently than the others (e.g. the mode with wavelength 1∕𝑤1), nonlinear saturation phenom- ena quickly come into play, so that the critical current can never be reached. This leads to the magnon-magnon relax- ation rate becoming power dependent, and in particular to a sharpincreaseaboveacertainmodeoccupationthreshold. We use this to paint a picture of a condensate that appears to be- have like a liquid. This picture is supported by a number of different experiments, including nonlocal transport on differ- ent thicknesses of YIG thin films as well as different garnet compositions, Brillouin light spectroscopy, independent vari- ation of the spin mixing conductance or the thermal gradient near the emitter.15 We have shown that doping with Bi improves the nonlo- cal signal. The first reason is that we avoid the nonlinear red shift under the emitter, which produces localization, which is obviously detrimental to the nonlocal geometry. The second reasonisthatthelateraltemperaturegradienthasnoeffecton the magnon spectrum, since 𝑀eff≈ 0regardless of the tem- perature. So the magnons excited under the emittor have no problem propagating to the collector. And the third reason is thatsincetheprecessionisquasi-circular,theparametricexci- tationofothermodesofmagnonsisstronglylimited,allowing 𝑛satto become larger. Wehavealsoshownthattheinabilitytothermalizetheemit- ter electrode plays a crucial role in the decrease of 𝑀1under theemitter. Thisisaverystrongeffectin2Dgeometrywhich, combined with the fact that a single mode cannot be excited and the critical current goes to infinity, means that we reach the Curie temperature before it self-oscillates. Although this reduction in 𝑀1may seem favorable for reaching the critical current (which tends to 0 as 𝑀1→0), the nonlinear effects mentionedabove(couplingbetweenmodes,locationunderthe emitter, etc.) make it inaccessible. WehavenotfoundanydirectsignatureofBECinourtrans- portstudiesonnonlocaldevices. Allofourexperimentaldata point to strong interaction between degenerate modes rather than fluctuation of a single mode. This problem plagues the nonlocalgeometry,whereoneonlyobservesthemagnonprop- agatingoutsidethePtelectrode,which,representsonlyasmall fraction of the total injected spin. In this respect, our work above does not provide answers about what happens directly under the emitter. AlthoughthereisnoBECcondensationoutsidetheareabe- low the emitter, the analogy with the Gurzhi effect in an ul- trapureelectronconductorseemsappropriate. Furtherstudies wouldberequiredtoprovidedirectevidenceforsuchmagneto- hydrodynamicfluidbehaviorathighpower. Theuniquesigna- turewouldbefeaturesthatcanonlybeattributedtotheNavier- Stokes transport equation62. ACKNOWLEDGMENTS This work was partially supported by the French Grants ANR-18-CE24-0021 Maestro and ANR-21-CE24-0031 Harmony; the EU-project H2020-2020-FETOPEN k- NET-899646; the EU-project HORIZON-EIC-2021- PATHFINDEROPEN PALANTIRI-101046630. K.A. acknowledges support from the National Research Founda- tion of Korea (NRF) grant (No. 2021R1C1C201226911) fundedbytheKoreangovernment(MSIT).Thisworkwasalso supported in part by the Deutsche Forschungs Gemeinschaft (Project number 416727653). FIG. S1. Impact of collector width, 𝑤2. The change of 𝑤2occurs while keeping both the emitter width, 𝑤1= 300nm, and the edge to edge distance between the two Pt strips, 𝑠=𝑑−(𝑤1+𝑤2)∕2 = 0.7𝜇m, where𝑑is the center to center distance. (a) shows 𝒯𝑠as a functionof𝐼1for3differentvaluesof 𝑤2and(b)shows 𝒯𝑠at𝐼1=1 mA as a function of 𝑤2. The dashed line is a linear fit through the datapoint. Theinsetisaschematicsideviewofthedevices,defining the various dimensions used throughout the paper. FIG. S2. Impact of emitter width, 𝑤1. The change of 𝑤1occurs while keeping both the collector width, 𝑤2=300nm, and the edge to edge distance between the two Pt strips, 𝑠=𝑑−(𝑤1+𝑤2)∕2 = 0.7𝜇m, constant. (a) shows 𝒯𝑠as a function of the current density 𝐽1=𝐼1∕(𝑤1𝑡Pt)for 2 different values of 𝑤1and (b) shows the 𝒱2 voltage (∝SSE) as a function of 𝑇1. VI. ANNEX A. Sample characterization Allofthemagneticgarnetfilmsusedinthisstudyhavetheir macroscopic magnetic properties fully characterized. Curves of magnetization versus temperature are shown in the ap- pendixof53. ThesameistrueforthePtmetalelectrode,whose resistivity and its dependence on temperature are given in the same appendix. All values are summarized in Table. I. Charge to spin (or vice versa) interconversion is provided16 by the Spin Hall effect. Its efficiency process is described by the relation 𝜖≡𝐺↑ ↓𝜃SHEtanh[𝑡Pt∕(2𝜆Pt)] 𝐺↑ ↓coth(𝑡Pt∕𝜆Pt)+𝜎Pt∕(𝐺0𝜆Pt),(10) where𝜃SHEisthespinHallangle, 𝐺↑ ↓isthespinmixingcon- ductance,𝑡Ptis the thickness of the Pt layer, 𝐺0= 2𝑒2∕ℎis thequantumofconductance,and 𝜆Ptistheinternalspindiffu- sion length. To calculate the value of 𝜖, we assume that 𝜆Pt= 3.8nm42and𝜆Pt𝜃SHE=0.18nm111–113. FromTable.Iwesee thatthevalueof 𝐺↑ ↓≪𝜎Pt∕(𝐺0𝜆Pt)andthusEq.(10)reduces to𝜖≈𝜃SHE𝑇withproportionality 𝑇=𝐺↑ ↓⋅𝐺0𝜆Pt∕𝜎Pt≈0.1 which is maximized when 𝑡Pt≈7nm. Numerical evaluations of𝜖are found in Table. I. The maximum achieved value is 𝜖≈0.08observed in (Bi-)YIG𝐵. NotethatforPt, 𝜃SHE>0. Thismeansthattheinterconver- sion is governed by the right-hand rule. Using the convention of Fig. 1, a positive current (i.e., circulating along −̂ 𝑦) injects spinspolarizedalong +̂ 𝑥,intoanadjacentlayer. Thus,theam- plificationofspinfluctuationsrequiresthat 𝑀𝑠isalignedwith −̂ 𝑥or that𝐻𝑥<0, as indicated in the figure. B. Linear regime Finally, it seems relevant to emphasize that the data pre- sentedaboveprovidesomecross-checkingofthedimensional dependenceofthemagnontransmissionratiowithexternalpa- rameters,assuggestedbyEq.(6). ThedashedlineinFig.7of Ref.53showstheproportionaldependence of 𝒯𝑠on𝑇1. Com- parison of the nonlocal signals using local annealing of the Pt electrode as shown in Fig. 6(a) shows the proportional de- pendence of 𝒯𝑠on𝜖1⋅𝜖2. The Groeningen group has exten- sivelystudiedthethicknessdependenceoftheconductivityof electrically excited magnons (low-energy magnons) and theyobserved the monotonous decrease with increasing thickness, confirming the 1∕𝑡YIGbehavior86. Eq. (6) also predicts the linear and inverse linear relation- ship of the magnon transmission ratio with the width of the collector𝑤2and the emitter 𝑤1. Fig. S1 shows the influence of𝑤2on𝒯𝑠with different 𝑤2as a function of 𝐼1for (a) and of𝑤2for (b). Note that the width of the emitter and the edge to edge distance between the two Pt strips remain constant as 𝑤1=300nmand𝑠=𝑑−(𝑤1+𝑤2)∕2=0.7𝜇m. Thesede- vicesarefabricatedatthesametimeastheYIG𝐶devicesand weverifythatthespinmixingconductance 𝐺↑ ↓foreach𝑤2is relativelysimilar. Takingthevalueatlowcurrent 𝐼1=1.0mA to see the behavior of the linear regime, the proportional de- pendenceon 𝑤2isrevealed. Thisobservationsuggeststhatthe YIG|Pt interface is weakly coupled, i.e., the angular momen- tumtransferbetweenYIGandPtcanbeconsideredasineffec- tive due to the poor transparency at the interface, which leads toonlyasmallfractionofmagnonsbeingabsorbedintothePt electrodes. ThisisconsistentwiththeobservationinFig.6(a) that the magnon emission is proportional to 𝜖1. On the con- trary, Fig. S2 shows the influence of the emitter width 𝑤1on (a)𝒯𝑠asafunctionoftheappliedcurrentdensity 𝐼1∕(𝑤1𝑡Pt). For the sake of completeness, we show in Fig. S2(b) its in- fluence on the 𝒱2voltage as a function of the current den- sity. The behavior shows that the thermalization of Pt1im- proves with decreasing emitter width. Fig. S2(a) qualitatively confirms that the smaller 𝑤1gives greater conduction of low- energymagnons. 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Li, Ap- plied Physics Letters 102, 082401 (2013). 108S.DemokritovandV.Demidov,IEEETransactionsonMagnetics 44, 6 (2008). 109J. Sandercock and W. Wettling, Journal of Applied Physics 50, 7784 (1979). 110The resonance line at 5.72 GHz in Fig. 5(a,c) probably corre- sponds to the paramagnetic response of the substrate. For clarity it has been removed from the plot by background subtraction. 111R. Kohno, N. Thiery, K. An, P. Noël, L. Vila, V. V. Naletov, N. Beaulieu, J. Ben Youssef, G. de Loubens, and O. Klein, Ap- plied Physics Letters 118, 032404 (2021). 112J.-C. Rojas-Sánchez, N. Reyren, P. Laczkowski, W. Savero, J.-P. Attané,C.Deranlot,M.Jamet,J.-M.George,L.Vila, andH.Jaf- frès, Phys. Rev. Lett. 112, 106602 (2014). 113J.C.R.Sánchez,L.Vila,G.Desfonds,S.Gambarelli,J.P.Attané, J.M.D.Teresa,C.Magén, andA.Fert,NatureCommunications 4(2013), 10.1038/ncomms3944. 114N.Beaulieu,N.Kervarec,N.Thiery,O.Klein,V.Naletov,H.Hur- dequint, G. de Loubens, J. Ben Youssef, and N. 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2022-10-15

This review provides a comprehensive study of the nonlinear transport properties of magnons, which are electrically emitted or absorbed inside extended YIG films by spin transfer effects via a YIG$\vert$Pt interface. Our purpose is to experimentally elucidate the pertinent picture behind the asymmetric electrical variation of the magnon transconductance analogous to an electric diode. The feature is rooted in the variation of the density of low-lying spin excitations via an electrical shift of the magnon chemical potential. As the intensity of the spin transfer increases in the forward direction (regime of magnon emission), the transport properties of low-energy magnon go through 3 distinct regimes: \textit{i)} at low currents, where the spin current is a linear function of the electrical current, the spin transport is ballistic and set by the film thickness; \textit{ii)} for amplitudes of the order of the damping compensation threshold, it switches to a highly correlated regime limited by magnon-magnon relaxation process and marked by a saturation of the magnon transconductance. Here the main bias, that controls the magnon density, are thermal fluctuations beneath the emitter. \textit{iii)} As the temperature under the emitter approaches the Curie temperature, scattering with high-energy magnons dominates, leading to diffusive transport. We note that such sequence of transport regimes bears analogy with electron hydrodynamic transport in ultra-pure media predicted by Radii Gurzhi. This study restricted to low energy part of the magnon manifold complements part II of this review\cite{kohno_2F}, which concentrates instead on the whole spectrum of propagating magnons.

Non-local magnon transconductance in extended magnetic insulating films.\\ Part I: spin diode effect

2210.08304v2

Strong to ultra-strong coherent coupling measurements in a YIG/cavity system at room temperature Guillaume Bourcin,1, 2,Jeremy Bourhill,3Vincent Vlaminck,1, 2and Vincent Castel1, 2,y 1IMT Atlantique, Technopole Brest-Iroise, CS 83818, 29238 Brest Cedex 3, France 2Lab-STICC (UMR 6285), CNRS, Technopole Brest-Iroise, CS 83818, 29238 Brest Cedex 3, France 3Quantum Technologies and Dark Matter Labs, Department of Physics, University of Western Australia, 35 Stirling Hwy, 6009 Crawley, Western Australia. (Dated: April 27, 2023) We present an experimental study of the strong to ultra-strong coupling regimes at room tem- perature in frequency-recongurable 3D re-entrant cavities coupled with a YIG slab. The observed coupling rate, dened as the ratio of the coupling strength to the cavity frequency of interest, ranges from 12% to 59%. We show that certain considerations must be taken into account when analyzing the polaritonic branches of a cavity spintronic device where the RF eld is highly focused in the magnetic material. Our observations are in excellent agreement with electromagnetic nite element simulations in the frequency domain. Keywords: cavity spintronics, cavity-spintronics, Yttrium Iron Garnet, ultra-strong coupling I. INTRODUCTION Cavity spintronics is an emerging research eld that investigates light-matter interactions within magnetism, specically the interactions between cavity photons and the quanta of spin waves based on the magnetic dipole interaction { magnons. At the core of cavity spintron- ics are cavity-magnon polaritons (CMPs) which are the associated bosonic quasiparticles, i.e., hybridized cavity- magnon-photon states in the strong coupling regime. cavity spintronics has drawn a growing interest since the rst theoretical prediction in 2010 [1], and then shortly after experimental demonstration of CMPs at both mil- likelvin (mK) temperatures [2, 3] and room temperature (RT) [4]. Cavity spintronics display a broad range of ap- plicability for quantum information systems and RF de- vices such as adjustable sensitive lter [5{7], isolators or circulators [8], gradient memories [9] and for engineering chiral states of electromagnetic radiation [10, 11]. In a cavity{magnon system, when the magnon fre- quency is tuned by an externally applied static magnetic eld towards the cavity resonance frequency, the sys- tem undergoes hybridization (e.g. forms a CMP) with a characteristic anti-crossing signature in the dispersion spectrum. The interaction is quantied by the coupling strengthg=2and by its ratio g=!with the cavity fre- quency!=2. When the coupling g=2is larger than the systems losses, there exist three dierent coupling regimes. These have commonly been referred to as: (i) Strong coupling when g=! < 0:1, (ii) Ultra-Strong Cou- pling (USC) for 0 :1< g=! < 1, and (iii) Deep-Strong Coupling (DSC) for g=!> 1, a regime that still remains largely unexplored. The value of g=!= 0:1 is considered as a threshold between the SC and USC regimes, but guillaume.bourcin@imt-atlantique.fr yvincent.castel@imt-atlantique.frthis is only a historical convention, supposedly indicat- ing the cuto beyond which the coupling rate grepresents a \sizeable fraction" of the system energy and therefore cannot be deemed to be a slowly rotating term in the rotating wave approximation. The USC regime was predicted theoretically in inter- subband cavity polaritons in 2005 [12] and rst observed in 2009 [13] in n-doped GaAs quantum wells embedded in a microcavity, with g=! = 0:11. Since this experi- mental observation, several research groups have exper- imentally achieved the USC regime [14, 15] in dierent systems such as superconducting circuits [16], polaritons [17], and optomechanics [18]. So far, the USC regime in cavity spintronics has been experimentally achieved at low temperature [19{25] and investigated theoretically [26, 27]. Very recently, Golovchanskiy et al. [25] proposed an approach to achieve on-chip USC hybrid magnonic sys- tems reaching g=! = 0:6 and based on superconduct- ing/insulating/ferromagnetic multilayered microstruc- tures operating below 10 K. They highlighted in particu- lar the drastic failure of currently adopted models in the USC regime. Here, we present measurements and simulations of a recongurable hybrid system that allows the study of the transition from the SC to USC regimes at room temper- ature in the 0 :115 GHz frequency range. We utilize a magnetic eld-focusing double-post re-entrant cavity rst described by Goryachev et al. [19]. A set of three dif- ferent resonators (by their dimensions and posts shape) allow us to follow the evolution of the coupling strength through USC regime (starting from the SC/USC limit). With these results, we conrm that it is necessary to add an extra term in the expression of the Ferromagnetic Res- onance (FMR) frequency equation to accurately describe the observed hybridization (measurements and simula- tions) with the commonly used Dicke model [28]. We show that this additional term does not depend on the coupling rate but on the level of connement of the RFarXiv:2209.14643v2 [quant-ph] 26 Apr 20232 magnetic eld in the magnetic material. Moreover, this added term can be negligeable in the SC regime, while it is essential in the USC regime. II. HYBRID SYSTEM DESCRIPTION The hybrid system presented here is made of a com- mercial single crystal of YIG (Yttrium Iron Garnet, Y3Fe5O12) and a modied re-entrant cavity. The YIG is a slab of 3.826.090.61 mm3. The multiple post re-entrant cavity [19] is a unique type of microwave cavity. There are two rst-order reso- nant modes, termed the Dark Mode (DM) and the Bright Mode (BM). Both contain the electric eld of the mode between the top of the post and the lid of the cavity. For the DM (as shown in Fig. 1 (a)), the RF electric elds ( e-elds) focused above the two posts are in-phase, resulting in the circulating RF magnetic elds ( h-elds) destructively interfering in the region between the posts (hence \dark"), whilst the opposite is true for the BM (as shown in Fig. 1 (b)). The advantages of such a cavity are three-fold: rst, the highly localized electric eld results in extremely large frequency sensitivity to any perturbations inside this region (displacement of the containment area or modication of the dielectric mate- rial). Secondly, the physical separation of the electric and magnetic elds permits separate interaction with both magnetic and electrically sensitive devices at dierent locations, potentially simultaneously. Finally, the mag- netic eld focusing between the posts results in extremely strong interactions with any magnetically susceptible ma- terial placed there. /vectorh /vectore /uni0000000b/uni00000044/uni0000000c /vectorh /vectore /uni0000000b/uni00000045/uni0000000c /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 /uni0000005d/vectorH0 /uni00000013/uni00000015/uni00000017/uni00000019/uni0000001bω/2π/uni00000003/uni0000003e/uni0000002a/uni0000002b/uni0000005d/uni00000040 /uni0000000b/uni00000046/uni0000000c /uni00000025/uni00000030 /uni00000027/uni00000030 /uni00000013 /uni00000015/uni00000013 /uni00000017/uni00000013 /uni00000019/uni00000013 /uni0000001b/uni00000013 /uni00000014/uni00000013/uni00000013 d/uni00000003/uni0000003e/uni00000097/uni00000050/uni00000040/uni00000013/uni00000011/uni00000017/uni00000013/uni00000011/uni00000018/uni00000013/uni00000011/uni00000019/uni00000013/uni00000011/uni0000001a/uni00000013/uni00000011/uni0000001bg/ω /uni0000000b/uni00000047/uni0000000c /uni00000013/uni00000011/uni0000001a/uni0000001a/uni00000013/uni00000011/uni0000001a/uni0000001b/uni00000013/uni00000011/uni0000001a/uni0000001c/uni00000013/uni00000011/uni0000001b/uni00000013 η FIG. 1. Re-entrant cavity with electromagnetic simulation overlay where jhj2is displayed for the rst two photonic modes: (a) the DM; and (b) the BM. The interaction between a single cavity mode and the FMR can be described by two coupled harmonic oscilla- tors, for which the Hamiltonian is read as: ^H=^Hc+^Hm+^Hint; (1) where ^Hc=~!^cy^crepresents the photonic mode, ^Hm= ~!m^by^bthe magnon mode, and ^Hintis the Zeeman inter-action [22], which describes the coupling between the two oscillators for this system. ~is the reduced Planck con- stant,!c(m)=2is the cavity (magnon) frequency, and ^ cy and ^c(^byand^b) are the creation and annihilation cavity (magnon) operators, respectively. Following [22], and as demonstrated in appendix A, the physics system is described by the Dicke model reading as: ^H=~=!^cy^c+!b^by^b+g(^cy+ ^c)(^by+^b): (2) An easy way to solve eigenvalues of the Dicke Hamilto- nian is to use the Rotating Wave Approximation (RWA) where the counter-rotating terms, ^ cy^byand ^c^b, are ne- glected. In the case of a system being in the USC regime, it is well known [14, 15] that this approximation no longer describes this system. Using the Hopeld-Bogoliubov transformation allows one to solve for the system eigen- frequencies whilst considering co-rotating, ^ cy^band ^c^by, and counter-rotating terms: !=1p 2r !2+!2mq (!2!2m)2+ 16g2!!m:(3) Moreover, the coupling strength is dened as [29]: g 2= 4r 0S~! Vm=p! 4r glB0~ns; (4) where = 228 GHz.T1is the gyromagnetic ratio for YIG,gl= 2 is the Land e g-factor for an electron spin, 0 is the vacuum permeability, Bis the Bohr magneton, = 5Bis the magnetic moment of the sample, ns= 4:221027m3is the spin density for YIG [29], and  is the lling factor, where =vuuutR Vmh
^xdV2 +R Vmh
^ydV2 VmR Vcjhj2dV: (5) The lling factor describes the proportion of the h-eld (x- and y-axis components), perpendicular to the static magnetic eld ( H-eld), named H0in Fig. 1 compared to the h-eld for all directions inside the entire cavity volumeVc. III. OPTIMIZATION An appropriate optimization of the cavity allows one to maximize the coupling and to obtain a quasi- homogeneous h-eld inside the YIG slab. With the use of Finite Element Modeling (FEM) and following the pro- cedure described by Bourhill et al. [29], we were able to precisely predict and therefore optimize prior to con- struction, the cavity frequency, frequency tuning range,3 and the coupling strength considering equation (4). The optimization of the cavity design was based on the maximization of the lling factor and the h-eld homogeneity at the rst BM inside the YIG slab. For a correct distribution of the RF eld inside the cavity (seen as a Perfect Electric Conductor, PEC), it is necessary to consider the electrical property of the YIG, namely a relative dielectric permittivity of 15. Dynamic mag- netic properties are not useful at this stage and instead of considering the magnetic permeability with the Polder tensor, we consider it as that of vacuum. There exist only three free parameters for the optimiza- tion of the hybrid system, two for the size of the posts, the widthWand the length L, and one for the cavity, the radiusR. The other parameters such as the height of the cavity and the distance between the posts were xed by the constraints imposed by the YIG dimension and the cavity manufacturing accuracy. The optimization step is described in appendix B, and the optimized values are W= 0:6 mm,L= 6 mm, and R= 12 mm. /uni00000013 /uni00000015/uni00000013 /uni00000017/uni00000013 /uni00000019/uni00000013 /uni0000001b/uni00000013 /uni00000014/uni00000013/uni00000013 d/uni00000003/uni0000003e/uni00000097/uni00000050/uni00000040/uni00000013/uni00000011/uni00000017/uni00000013/uni00000011/uni00000018/uni00000013/uni00000011/uni00000019/uni00000013/uni00000011/uni0000001a/uni00000013/uni00000011/uni0000001bg/ω /uni0000000b/uni00000044/uni0000000c /uni00000013/uni00000011/uni00000013 /uni00000017/uni00000011/uni00000013 /uni00000019/uni00000011/uni00000013 /uni0000001b/uni00000011/uni00000013 /uni00000014/uni00000013/uni00000011/uni00000013 /uni00000014/uni00000011/uni0000001a/uni00000014/uni00000011/uni00000014 ω/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 /uni00000036/uni00000026/uni00000038/uni00000036/uni00000026/uni00000027/uni00000036/uni00000026/uni0000000b/uni00000045/uni0000000cη/uni00000003/uni00000020/uni00000003/uni00000014 η/uni00000003/uni00000020/uni00000003/uni00000013/uni00000011/uni0000001a/uni0000001c /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 η /uni00000013 /uni00000015/uni00000018 /uni00000018/uni00000013 /uni0000001a/uni00000018 /uni00000014/uni00000013/uni00000013 d/uni00000003/uni0000003e/uni00000097/uni00000050/uni00000040/uni00000013/uni00000015/uni00000017/uni00000019/uni0000001bω/2π/uni00000003/uni0000003e/uni0000002a/uni0000002b/uni0000003d/uni00000040 /uni00000025/uni00000030 /uni00000027/uni00000030 FIG. 2. (a) Evolution of the ratio g=!(with!=!BM) in blue andin red versus dfor Eigen-Mode simulations (EM). Inset: Evolution of simulated DM and BM frequencies versus d. (b) Evolution of g=!versus the BM frequency for = 1 and= 0:79. The simulated evolution of the two eigenmodes (DM and BM) are shown in the inset of Fig. 2 (a) with respect to the distance dbetween posts and the lid of the cavity, with a range from 1 to 100 m. Decreasing dwill decrease the frequency of the eigenmodes and the frequency dif- ference between the BM and the DM. Fig. 2 (a) shows electromagnetic simulation results for (right y-axis) andg=!(left y-axis) versus dfor a cavity with the optimized dimensions, where !=!BMthe frequency mode of in- terest in our study. is maximized for d= 9m. The variation of over this range of dvalues is only is 2.7%, therefore we may consider it more or less invariant. The tuneability of the distance dplays a role on the g=!ratio as shown in Fig. 2 (a). Indeed, !=2is decreasing with d, andis remaining almost constant. Considering Eq. (4),g=2is a function of and the square root of !. Therefore, the ratio g=!will increase with the inverse of the square root of !from 36.8 to 80.5% as ddecreases from 100 to 1 m. Fig. Fig. 2 (b) illustrates the SC to DSC transition for YIG with the frequency dependence of g=!. The blue dots correspond to the values extracted from EM simu- lation already discussed in Fig. 2 (a) and the solid line dependencies are based on equation (4) for two constant values of, 0.79 (blue) and 1 (green). The magnetic properties of YIG require working in a specic frequency range in order to explore the DSC. For the maximum reachable value of (green line), which corresponds to the entire h-eld perpendicular to H0and fully conned toVm, DSC is possible when the magnons are coupled to a microwave mode below 1.72 GHz [29]. In our case (withclose to 0.79), DSC is achievable but at a smaller resonant frequency (1.07 GHz). Note that the optimized cavity conguration of this work does not allow to reach the DSC due to the presence of the dark mode which con- taminates the low frequency response and the diculty to control distance dlower than 3 m. IV. RESULTS AND DISCUSSION A. Simulation details To compare the experimental results, simulations in the frequency domain (FD), solving for the S21scatter- ing parameter were conducted for dierent values of d from 2 to 100 m. For these simulations, we consid- ered the excitation probes and hence the coupling losses. Losses due to nite conductivity of the cavity walls are also taken into consideration. The static and dynamic magnetic properties of YIG are used to solve the frequency response of the entire system as a function of the applied magnetic eld. The spin dynamics of ferrimagnetic systems can be described by the Landau-Lifshitz-Gilbert (LLG) equation and the frequency dependence of the coupled dynamics can be accurately estimated by using a linear solution of the LLG equation in solving Maxwell's equations. Some con- sideration regarding the shape of the YIG sample must be taken into account. The FMR dispersion for a rela- tively thick slab geometry requires careful consideration. Based on the works of Kittel [30], R. I. Joseph and E. Schl omann [31], the demagnetizing eld expression has been adapted to our non-ellipsoidal sample of YIG (as described in Appendix B). From these results, it is de-4 /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/uni00000040d/uni00000003/uni00000020/uni00000003/uni0000001a/uni00000018/uni00000003/uni00000097/uni00000050 /uni0000000b/uni00000044/uni0000000c g//uni00000003/uni00000020/uni00000003/uni00000013/uni00000011/uni00000016/uni00000019 /uni00000030/uni00000048/uni00000044/uni00000056/uni00000058/uni00000055/uni00000048/uni00000050/uni00000048/uni00000051/uni00000057/uni00000056 /uni00000015/uni00000017/uni00000019/uni0000001b/uni00000014/uni00000013/uni0000000b/uni00000046/uni0000000c g//uni00000003/uni00000020/uni00000003/uni00000013/uni00000011/uni00000017/uni00000019 d/uni00000003/uni00000020/uni00000003/uni00000014/uni00000013/uni00000003/uni00000097/uni00000050 /uni00000015/uni00000017/uni00000019/uni0000001b/uni0000000b/uni00000048/uni0000000c g//uni00000003/uni00000020/uni00000003/uni00000013/uni00000011/uni00000018/uni0000001c d/uni00000003/uni00000020/uni00000003/uni00000016/uni00000003/uni00000097/uni00000050 /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000014 /uni00000013/uni00000011/uni00000015 /uni00000013/uni00000011/uni00000016 /uni00000013/uni00000011/uni00000017 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 /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 /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 /uni00000014/uni00000017/uni00000013 /uni00000014/uni00000015/uni00000013 /uni00000014/uni00000013/uni00000013 /uni0000001b/uni00000013 /uni00000019/uni00000013 /uni00000017/uni00000013 /uni00000015/uni00000013 /uni00000036/uni00000003/uni0000003e/uni00000047/uni00000025/uni00000040 fDM fBM fFMR f± FIG. 3. Transmission spectra versus frequency and H0for (a), (c), (e) measurements at RT, and (b), (d), (f) simulations. Comparison spectra between measurement and simulations are shown for dierent distances das labelled. A t with the Dicke model with a shifted magnon frequency is shown superimposed on (c) and (d) where the FMR frequency ( fFMR =!FMR=2) is shown in black, the DM frequency ( fDM=!DM=2) in red, the BM frequency ( fBM=!BM=2) in orange and the two polariton frequencies ( f=!=2) in white. termined that the demagnetizing eld is signicantly dif- ferent from the thin-lm form, and therefore for accurate simulations proper consideration of this dierence must be taken into consideration. Hence, the eective static magnetic eld in the YIG is dierent from the applied one and read as: Hi=H0Nzz(x;y;z )M0 (6) whereHiis the internal static magnetic eld along the z-axis, and Nzzis the spatially dependent demagnetizing component along the z-axis, and is describe in Eq. C3 in appendix C. B. Experimental set-up To reach the specications described above, an alu- minum cavity with an accuracy of 20 m has been ma- chined. For the applied static magnetic eld, we used an elec- tromagnet where the produced eld is aligned along the z-axis (see Fig. 1), in the direction of the height of the posts. H0aligns all the spin moments along the z- axis and to saturate the macroscopic YIG magnetization. With the shape of the cavity, the h-eld for the BM, con- sidered as the perturbative eld, is only along the x-axis inside the YIG slab between the two posts, as shown in Fig. 1 (b) due to the constructive interference of the twoh-elds around each post. A gaussmeter allows one to measure in situH0magnitudes. S-parameters are mea- sured with a two-port Vector Network Analyzer (VNA), with the magnitude and phase of the scattering parame- ters recorded between 0.1 to 15 GHz with an input power of10 dBm. All measurements are conducted at RT The magnitude of the S21transmission spectra as a function of H0are displayed in Fig. 3 for measurement and simulation with diering sized gaps between the top of the posts and the roof of the cavity. Experimentally, this is varied by using dierent cavity lids which had recesses of diering heights machined into them. C. Results Measurement and simulation results of magnetic spec- troscopy of the cavity magnon system are shown in Fig. 3 as the rst and second row, respectively, for dierent val- ues ofd. Each column represents a comparison between a measurement and a simulation with a distance close to the measured value. The latter can be determined by the unperturbed value of fDM, which acts as a calibration for d. The external magnetic eld was always applied sym- metrically for negative and positive values. This allows to improve the t accuracy on measurements, because we have twice as many data points. All measurements with5 complete frame are shown in appendix F. We can easily distinguish the two hybrid eigenfrequen- ciesf+=!+=2(for the higher branch) and f=!=2 (for the lower branch) from either side of the BM fre- quency. It should be noted that at low H0values the BM is not visible, whilst we can clearly see the DM which is the lowest frequency mode and has a negligible coupling with the magnon mode, hence is constant versus H0. Some minor discrepancies between simulation and ex- periment should be pointed out: ( i) an in ection point on the curvature of the upper CMP in frequency at low H0(observed only in the USC regime) for measurements, appearing neither in simulation nor analytic ts; ( ii) anti- resonances only appearing either in measurement, the horizontal one around 4.3 GHz in Fig. 3 (a) and (e), or in simulation with a S-like shape, around 10, 4, and 2 GHz in respectively Fig. 3 (b), (d) and (f). Let us notice that this anti-resonance does not appear in mea- surements when a cavity mode is overlapping with this transmission dip, as shown for d= 10 µm in Fig. 3 (c) and Fig. 9 (e), and for d= 116 µm in Fig. 9 (a); ( iii) another magnon mode exists near the upper CMP in simulations. It is clear that it is another magnon mode because its H-eld's frequency dependence does not change as dis varied. Dierences given in the two last point could be explained by the fact that the YIG sample is a perfect rectangu- lar prism in simulation whereas the real sample is not. The imperfections of the YIG geometry could result in a weak transmission, which could be not detected in mea- surement. Despite these minor deviations, the agreement between simulations and measurements on the magnon-photon coupling and the resulting CMPs is excellent. In particu- lar, we validated the spatial distribution of the demagne- tizing eld, hence the expression of the FMR for a slab, and the ability of the Maxwell's equations to describe the system. This permits one to conduct a simulation with a magnetic eld larger than experimentally possible in or- der to extract the BM frequency. Indeed, it is impossible to measure the unperturbed BM frequency fBMin the USC regime even when applying a high magnetic eld near to 2 T. D. Model Description In the USC regime, the Tavis-Cummings model be- comes no longer applicable [13, 32], as g=!> 0:1 leads to a failure of the rotating wave approximation as the inter- action term of the Hamiltonian can no longer be assumed to be \slowly rotating" compared to the system terms. The standard model for cavity magnonics is the Dicke model (see Eq. (2)). However, we have noticed that in the coupling regime of our system, even the Dicke model cannot describe observed polariton frequency dispersion for measurements and simulations, as shown in Fig. 7 in appendix D. Another standard model describing light-matter interactions is the Hopeld model [33], similar to the Dicke model with an additional diamagnetic term. This well known model neither t the measured data with the use of the Hopeld model, as shown in appendix E. To remedy these issues, it has been proposed to modify the Dicke model with the addition of a H-eld in the FMR dispersion equation [24]. We also modied the term of the FMR frequency dependence in equation (3) to: !m!!m+
m (7) where
m= 2f
is a frequency shift, which will be further discussed in section IV E. This modied Dicke model was found to t best the experimental and simu- lation spectra, as seen in the white dash lines of Fig. 3 for d= 75m in (a) and (b), d= 10m in (c) and (d) and d= 3m in (e) and (f). Measurement t, shown in Fig. 3 (a), (c), and (e), is achieved with the BM frequency fBM (in orange), the coupling strength g=2, and the added frequencyf
as tting parameters. For simulation t, shown in Fig. 3 (b), (d), and (f), the BM frequency is considered as xed parameter. Indeed, simulations were performed at an articial high H-eld (H0= 10 T), in order to tune the magnon mode many orders of coupling strength away, and clearly distinguish the two photonic modes. An oset far detuned from the BM frequency at a zero H-eld arises for high g=! when the FMR is shifted. When the FMR is not shifted, in the standard Dicke model, no BM frequency detuning at zero H-eld exists for anyg=!value. See appendix G for more details about the frequency detuning at zero H-eld. All values of t parameters, for measurements and sim- ulations, are available in appendix F, and are pooled in Fig. 4. For the measurements (shown in blue), the dis- tancedhas been estimated from the measured DM fre- quency. The tted BM frequencies of the measurements are in good agreement with simulations (shown in black in the inset of Fig. 4). Regarding the coupling strength g=!, we achieve a ratio g=!ranging from 0.35 to 0.59, corresponding to d= 116m tod= 4m, respectively. As mentioned in section III, the values of g=!are dier- ent from the optimization step ones (dotted red curve), mainly due to the dierent estimated frequencies, shown in the inset. Once again, the correlation between tted simulations and measurements for the ratio g=!are also good. This clearly demonstrates the validity of the sim- ulations. E. Discussion We discuss here the physical meaning of the frequency shift in the modied Dicke model. For a deeper un- derstanding of the behavior of this added term, we in- vestigated the transition between the SC and the USC regimes. In order to study a wide range or g=!values, we have used two other cavities with the same YIG sam-6 /uni00000013 /uni00000015/uni00000013 /uni00000017/uni00000013 /uni00000019/uni00000013 /uni0000001b/uni00000013 /uni00000014/uni00000013/uni00000013 /uni00000014/uni00000015/uni00000013 d/uni00000003/uni0000003e/uni00000097/uni00000050/uni00000040/uni00000013/uni00000011/uni00000017/uni00000013/uni00000011/uni00000019/uni00000013/uni00000011/uni0000001b/uni00000014/uni00000011/uni00000013g/ω /uni00000028/uni00000030/uni00000003/uni00000036/uni0000004c/uni00000050/uni00000058/uni0000004f/uni00000044/uni00000057/uni0000004c/uni00000052/uni00000051 /uni00000029/uni00000027/uni00000003/uni00000036/uni0000004c/uni00000050/uni00000058/uni0000004f/uni00000044/uni00000057/uni0000004c/uni00000052/uni00000051/uni00000056 /uni00000030/uni00000048/uni00000044/uni00000056/uni00000058/uni00000055/uni00000048/uni00000050/uni00000048/uni00000051/uni00000057/uni00000056/uni00000025/uni00000030 /uni00000027/uni00000030 /uni00000013 /uni00000015/uni00000013 /uni00000017/uni00000013 /uni00000019/uni00000013 /uni0000001b/uni00000013 /uni00000014/uni00000013/uni00000013 /uni00000014/uni00000015/uni00000013 d/uni00000003/uni0000003e/uni00000050/uni00000040 /uni00000013/uni00000015/uni00000017/uni00000019/uni0000001bω/2π/uni00000003/uni0000003e/uni0000002a/uni0000002b/uni0000005d/uni00000040 FIG. 4.g=! versusdfor tted FD simulations (in black) and for tted measurements (in blue). The simulation trend is plotted in the black, dashed line. Inset: DM and BM fre- quencies versus the distance d, in the black dashed line are shown DM and BM reading values from simulations at ex- tremely high applied H-eld. Eigen-Mode (EM) simulations are shown as the red dashed line. ple. The rst described machined cavity will be named \CAV 01" in the following. This cavity operates in a g=! range from 0.35 to 0.59, as mentioned in the table II in appendix F. The second cavity, \CAV 02", has been 3D printed and has the same shape as CAV 01, but with smaller height posts. This cavity is performing in a certain range of g=!, from 0.28 to 0.32 (see table III in appendix F). The third cavity, \CAV 03'", is also a double re-entrant 3D printed cavity with cylinder posts, adjustable in height. This cavity was used in a previous work [29] to experimentally verify a reworked theory that predicts coupling values from simulations alone. The cavity has radiusRcav= 20 mm and height Hcav= 4.6 mm, whilst the posts have radius Rpost= 2.05 mm and are spaced to 2.7 mm. The operating ratio g=!is lower than the two others cavities and enables to have experimental results at the SC/USC threshold, with g=!comprised between 0.12 and 0.25 (see table IV in appendix F). The operating range in BM frequencies, coupling strengths, and added frequencies for the three cavities are summarized in Table I. TABLE I. Operating range of the cavities CavityfBM[GHz]g=2[GHz]
m=2[GHz] CAV 01 2.80 - 7.65 1.64 - 2.68 2.27 - 2.59 CAV 02 7.63 - 9.79 2.42 - 2.72 1.63 - 1.74 CAV 03 2.35 - 5.53 0.58 - 0.69 0.29 - 0.50 Thanks to the validation of the FD simulations, we were able to simulate the USC CAV 01design for dier- ent dimensions of the YIG slab, while keeping the aspect ratio of the slab constant. Since the demagnetizing com- ponents described in Eq. (C3) are only dependent on this aspect ratio, the FMR remains unchanged. However, stilldecreasing the YIG slab dimensions decreases the lling factor, therefore the coupling strength and g=! from 63 to 5 % with d= 50m. /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000015 /uni00000013/uni00000011/uni00000017 /uni00000013/uni00000011/uni00000019 /uni00000013/uni00000011/uni0000001b g/ω/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013/uni00000014/uni00000011/uni00000018∆m/ω /uni00000036/uni00000026 /uni00000038/uni00000036/uni00000026/uni0000000b/uni00000044/uni0000000c CAVsimu 01 CAVmeas 01 CAVmeas 02 CAVmeas 03 /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000015 /uni00000013/uni00000011/uni00000017 /uni00000013/uni00000011/uni00000019 /uni00000013/uni00000011/uni0000001b /uni00000014/uni00000011/uni00000013 /uni00000014/uni00000011/uni00000015 /uni00000014/uni00000011/uni00000017 g2/2πω/uni00000003/uni0000003e/uni0000002a/uni0000002b/uni0000005d/uni00000040/uni00000013/uni00000014/uni00000015/uni00000016/uni00000017∆m/2π/uni00000003/uni0000003e/uni0000002a/uni0000002b/uni0000005d/uni00000040 ∝η2/uni0000000b/uni00000045/uni0000000c FIG. 5. (a)
m=!versusg=!and (b)
m=2versusg2=2. Shown are the FD simulations on CAV 01in red and measure- ments in blue, in green for CAV 02, and in purple for CAV 03. In (a) and (b), tted values for a reduced CAV 01with an as- pect ratio equal to 0.025 as the black square. The two data points circled in (a) corresponds to the same value of
m=2 in (b). We plotted
m=!versusg=!in Fig. 5 (a) which clearly display a quadratic dependence. For g=!0:1,
m=! is more or less negligible. This description agrees with the commonly situated transition point (shown as the red dotted line) between the SC and USC regimes where all models converge. Our simulations show the need for the
m=2parameter to properly t the data. Fig. 5 (b) shows
m=2versusg2=2!which is proportional to the square of the lling factor, 2. According to this observation and the denition of , we noticed that the more this energy is conned in the YIG, the larger the shift in the magnon frequency will be. In the literature, the parameter is not so often considered or estimated. In ref [29], we had the opportunity to test the model of equations (4) and (5) on multiple published experimental results, and rarely exceeds 0.05 in any of them. As a reminder, and in view of the description in Fig. 2, our system (CAV 01) proposes a of about 0.79. In Fig. 5 is represented by square marker a simulation where dimensions of the cavity and the YIG are reduced by a ratio equal to 0.025 for d= 50m. By decreasing the dimensions of the entire cavity CAV 01by this ratio, the BM frequency is increased to 275 GHz. Then, this cavity operates in the SC regime. However, the propor-7 tion of the h-eld in the YIG remains the same, hence also. In (a), are circled the reduced system performing in the SC regimes, and the unmodied cavity in the USC regime presenting the same value. In (b), it is shown that the frequency shift
m=2is the same for both cav- ities, and for the same value of g2=2!. Considering eq. 4,
m=2depends of the magnetic properties of the YIG and2. It is then important to note that this eect is not bounded to the coupling strength and hence to the cou- pling regime, but instead to the lling factor, something that has never been discussed so far. As physical mechanism of
m=2, the nonlinear op- tical processes having similar behavior, would be a good candidate. Among them, two dierent eects at- tracted our attention: the multi-photon Rabi oscillations [14, 34, 35] for its eective coupling being proportional tog2=!. When the coupling between an articial sin- gle atom and a cavity is in the USC regime, the sys- tem can exchange several photons (and undergo multi- photon Rabi oscillations) instead to a single one (com- monly known as Rabi oscillation); and the self-Kerr, and the cross-Kerr eects [36{40] presenting a frequency shift of the magnon, due to magnetocrystalline anisotropy and magnon-magnon interactions, respectively. V. CONCLUSION In conclusion, we proposed a double re-entrant cav- ity design to achieve USC magnon/photon coupling at microwave frequencies, which was supported by both ex- perimental data and electromagnetic simulations. To the best of our knowledge, this is the only demonstration of USC magnon/photon coupling at room temperature so far. Noteworthily, reaching the USC without cryogenic temperatures is promising for the development of RF ap- plications based on cavity spintronics. We explained the importance of optimizing the lling- factorfor reaching the USC, aside from just the fre-quency of the resonator and the spin density. Impor- tantly, the cavity we proposed is parametrized by the distancedbetween the posts and the lid. We showed that tuning this parameter allowed to continuously go from the regular SC to the USC regime. The ability to study the transition from the SC to USC regime is a sig- nicant step towards understanding the physics of USC magnon/photon coupling. Indeed, we showed that the standard models describ- ing the coupling of a single resonator mode to many dipoles (e.g. the Dicke and Hopeld models) failed to properly decsribe our experimental data. Nevertheless, thanks to the validation of our electromagnetic simula- tions, we showed that a frequency shift in the magnon frequency adequately modelled our data, which we note is fully captured by the classical Maxwell's equations. Furthermore, we showed that this frequency shift only de- pended on the lling-factor , highlighting its importance for hybrid magnon/photon systems. While the physical origin of the magnon's frequency shift is still unknown, we hope that its relation with will motivate further re- search into deriving a proper theoretical model for USC magnon/photon coupling. ACKNOWLEDGMENTS This work is part of the research program supported by the European Union through the European Regional Development Fund (ERDF), by Ministry of Higher Ed- ucation and Research, Brittany and Rennes M etropole, through the CPER Project SOPHIE/STIC & Ondes, by the CPER SpaceTechDroneTech, by Brest M etropole, and the ANR project MagFunc. 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Hopeld. Theory of the contribution of excitons to the complex dielectric constant of crystals. Physical Review , 112(5):1555{1567, December 1958. [34] Anton Frisk Kockum, Adam Miranowicz, Vincenzo Macr , Salvatore Savasta, and Franco Nori. Deterministic quantum nonlinear optics with single atoms and virtual photons. Phys. Rev. A , 95:063849, Jun 2017. [35] Luigi Garziano, Roberto Stassi, Vincenzo Macr , An- ton Frisk Kockum, Salvatore Savasta, and Franco Nori. Multiphoton quantum rabi oscillations in ultrastrong cavity QED. Physical Review A , 92(6), December 2015. [36] Zhi-Bo Yang, Hua Jin, Jing-Wen Jin, Jian-Yu Liu, Hong- Yu Liu, and Rong-Can Yang. Bistability of squeezing and entanglement in cavity magnonics. Physical Review Research , 3(2), May 2021. [37] M. X. Bi, C. J. Dai, Jun-Ling Che, Ming-Liang Hu, and X. H. Yan. Bistability of cavity magnon polaritons be- yond the holstein{primako transformation. Journal of Applied Physics , 130(24):243902, December 2021. [38] Zhi-Bo Yang, Wei-Jiang Wu, Jie Li, Yi-Pu Wang, and J. Q. You. Steady-entangled-state generation via the cross-kerr eect in a ferrimagnetic crystal. Physical Re- view A , 106(1), July 2022.9 [39] Wei-Jiang Wu, Da Xu, Jie Qian, Jie Li, Yi-Pu Wang, and J. Q. You. Observation of magnon cross-kerr eect in cavity magnonics, 2021. [40] GuoQiang Zhang, YiPu Wang, and JianQiang You. Theory of the magnon kerr eect in cavity magnonics. Science China Physics, Mechanics &amp; Astronomy , 62(8), March 2019.[41] F. Rana. Quantum optics lectures ECE5310, Cornell Uni- versity. [42]Spin Waves . Springer US, 2009. [43] Klaus Hepp and Elliott H Lieb. On the superradi- ant phase transition for molecules in a quantized radi- ation eld: the dicke maser model. Annals of Physics , 76(2):360{404, April 1973.10 Appendix A: Physics Description The system under consideration is best described by the Hamiltonian of two coupled harmonic oscillators. The oscillators represent the cavity photonic mode ^Hc=~!^cy^c, and the uniformly precessing Kittel magnon mode ^Hm= ~!m^by^b, where ~is the reduced Planck constant, !=2and!m=2are respectively the cavity and magnon frequencies, and ^cy(^by) and ^c(^b) are the creation and annihilation cavity (magnon) operators. The coupling is then read as a an interaction ^Hint, and the entire system Hamiltonian can then be written as: ^H=^Hc+^Hm+^Hint (A1) The quantization of the Maxwell's equation leads to the expression of the vector potential: ^A(r;t) =X n^qn(t)p0r;nUn(r) (A2) where0the vacuum permittivity, r;nthe relative permittivity experienced by the cavity mode n, ^qn(t) is the temporal term, and Un(r) is the space dependent operator. This expression is generalized for all modes in a cavity. In the following, we will concern ourselves with only a single mode. This potential vector in a cavity is comparable to a simple harmonic oscillator, where radiation modes are dened according to annihilation and creation operators: ^c=1p 2~! !^qi^_q ^cy=1p 2~! !^q+i^_q (A3) Then, the RF magnetic eld ( h-eld) bounded to the cavity mode is[41]: ^h=1 0r ^A=1 0s ~ 2!0r;c ^cy+ ^c
rU (A4) where0is the vacuum permeability, 0the vacuum permittivity, r;cthe relative permittivity experienced by the cavity mode, and Uis the space dependent operator of the potential vector. The component of this eld perpendicular to the sample's magnetization direction will couple to the Kittel magnon mode. For such a uniform precession of the magnetic sample, we introduce the macrospin operator considering the entire sample, as: ^S=Vm ^M (A5) where ^Mis the magnetization operator, Vmthe volume of the magnetic sample, and the gyromagnetic ratio. We consider a saturated magnetization by the use of an applied static magnetic eld ( H-eld) in the z-axis di- rection. It is then useful to introduce spin raising ^S+and lowering ^Soperators. Following the Holstein-Primako transformation[42] and considering low excitation numbers versus the total spin number of the macrospin operator, we obtain: ^S+=^Sx+i^Sy=q 2S^by^b^bp 2S^b ^S=^Sxi^Sy=^byq 2S^by^bp 2S^by Sz=S^by^b(A6) whereS= glBNsis the total spin number of the macrospin, Bis the Bohr magneton, is the magnetic moment of the sample, glis the Land e g-factor, and Ns=nsVmis the number of spins in the sample, with nsthe spin density. The interaction term corresponds in this case to the Zeeman energy: ^Hint=0Z Vm^M
^hdV (A7)11 Substituting ^hand ^Min equation (A7) by their expressions in equations (A4) and (A5), and replacing cartesian macrospin values by raising and lowering ones, with neglecting z-axis terms, we arrive at: ^Hint=~=gx(^c+ ^cy)(^b+^by) +igy(^c+ ^cy)(^b^by) (A8) where the coupling strengths are dened as: gx= 2Vms ~S !r;c0Z Vm(rU)
^xdV gy= 2Vms ~S !r;c0Z Vm(rU)
^ydV(A9) In order to consider the integration of the rUterm in the magnetic sample volume in x- and y-axis, it is needed to rewrite it considering the entire h-eld \seen" by the sample. Therefore, it is convenient to normalize the h-eld against that of the entire cavity. The classical expression for the h-eld energy for a single cavity mode is: E=0 2Z h
hdV=0 20r;cq
qZ (rU)
(rU) dV (A10) whereR Vc(rU)
(rU) =r;c!2 c2[41]. Regarding the ratio of the h-eld energy in the magnetic sample versus the one in the whole cavity, we get: EVm EVc=R Vmh
hdVR Vcjhj2dV=R Vm(rU)
(rU) dV r;c!2=c2(A11) Using the center and right terms of the above equation, deriving numerators and applying the square root, we nally read the innitesimal normalized energy amplitude of the h-eld: rU=pr;c! chqR Vcjhj2dV(A12) Equation (A1), with the use of equation (A8), can be rewritten over a matrix form as: ^H=1 2 ^cy^by^c^b H ^cy^by^c^by+const H=2 64! gx+igy 0gxigy gxigy!mgxigy 0 0gx+igy! gxigy gx+igy 0gx+igy!m3 75(A13) Using Hopeld-Bogolubov transformation [24], the solution of the problem is to nd polariton operators ^ p, ex- pressed as a linear combination of ^ c, ^cy,^b, and ^by. Being bosonic operators, they should obey the Hopeld formulation [33]: h ^p;^Hi =!^p (A14) where!=2are frequency eigenvalues associated with the eigen-operators ^ p. As previously, the Hamiltonian in the polaritonic basis can be rewritten as: ^H=1 2 ^py ^py +^p^p+ M ^py ^py +^p^p+y(A15) In order to respect equation A14, the Hopeld matrix Mhas to be read as: M=Hdiag (1;1;1;1) (A16)12 Solving eigenvalues of the matrix Mleads to: !=1p 2r !2+!2mq (!2!2m)2+ 16g2!!m (A17) where the coupling strength g=q g2x+g2yis dened as: g 2= 4r 0S~! Vm=p! 4r glB0~ns (A18) with the lling factor: =vuuutR Vmh
^xdV2 +R Vmh
^ydV2 VmR Vcjhj2dV(A19) It is important to note that eigenfrequencies are solution of the Dicke model[14, 15]. Finally the Hamiltonian can be rewritten over the easier form: ^H=~=!^cy^c+!b^by^b+g(^cy+ ^c)(^by+^b) (A20) Appendix B: Cavity Optimization Fig. 6 is a representation of the optimization of the lling factor for two of the variable parameters; the width (W) and the length ( L) of the posts, with dchosen equal to 50 µm. The cavity radius ( R) has been chosen at its optimized value. The containment of the hfield inside the YIG is at its maximum when the post dimensions are of the slab dimensions. Hence, the width of the posts has been optimized over a range from 0.1 mm to 2 mm, and their lengths from 4 mm to 8 mm. The radius of the cavity does not have a big impact on . The cavity radius has been optimize over a range from 10 mm to 14 mm. Each contour represents the value of with respect to WandL. The hashed contour delimits the surface where 78:5%. For better feasibility, we choose the largest values of WandL. This leads to an optimal value of for W= 0:6 mm,L= 6 mm, and R= 12 mm. /uni00000017 /uni00000018 /uni00000019 /uni0000001a /uni0000001b 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 /uni00000013/uni00000011/uni00000019/uni00000013/uni0000003a/uni0000004c/uni00000047/uni00000057/uni0000004b/uni00000003/uni0000003e/uni00000050/uni00000050/uni000000400.70 0.72 0.740.76 0.77 0.78 0./uni00000019/uni000000190.680.700.720.740.760.78 η FIG. 6. lling factor function of the width ( W) and the length ( L) of the two posts13 Appendix C: FMR model Using the Landau-Lifshitz equation of the magnetization with the proper approximations leads us to the FMR pulsation for all types of ferromagnet shapes [30]: !0= s!e 2 [(Nxy+Nyx)Ms]2(C1) whereMsis the saturation magnetization, Ni;jis a component of the demagnetizing tensor at the ithcolumn and the jthrow, and!eis the FMR pulsation for an ellipsoidal body read as: !e= q [jHzj+ (NxxNzz)Ms] [jHzj+ (NyyNzz)Ms] (C2) Using the perturbation theory with a small perturbation on the H-eld=Mz Hzat the rst order, it is shown that the demagnetizing components for a rectangular prism are for the diagonal components[31]: N(1) kk=1 48 >< >:cot1[f(xi;xj;xk)] +cot1[f(xi;xj;xk)] + cot1[f(xi;xj;xk)] +cot1[f(xi;xj;xk)] + cot1[f(xi;xj;xk)] +cot1[f(xi;xj;xk)] + cot1[f(xi;xj;xk)] +cot1[f(xi;xj;xk)]9 >= >;(C3) with : f(xi;xj;xk) =q (aixi)2+ (ajxj)2+ (akxk)2(akxk) (aixi) (ajxj)(C4) and for the o-diagonal terms: N(1) ik=1 4logG(rjai;aj;ak)G(rjai;aj;ak)G(rjai;aj;ak)G(rjai;aj;ak) G(rjai;aj;ak)G(rjai;aj;ak)G(rjai;aj;ak)G(rjai;aj;ak) (C5) with: G(rjai;aj;ak) = (ajxj) +q (aixi)2+ (ajxj)2+ (akxk)2(C6) Let us notice that the demagnetizing components are spatially dependent and were averaged to x, y, and z equal to zero for analytical equations. For the YIG dimensions mentioned in the manuscript, the o-diagonal components of the demagnetizing tensor are equal to zero, then the slab as he same FMR frequency as read in Eq. C2. Appendix D: Dicke model 1. Normal phase The Dicke model is the simplest model to describe the magnon-photon interaction. It consider each Hamiltonian of the cavity photonic mode and magnon as well as the interaction Hamiltonian[24]: ^H=!^cy^c+!m^by^b+g(^cy+ ^c)(^by+^b) (D1) From this equation, we can easily solve for the eigenmodes, that is to say the polaritronic modes described in Eq. (A17). This equation is only valid when the ratio g=!is less than 0.5. For a description of a system with a ratio higher than 0.5, it is necessary to use the Dicke superradiant phase [43].14 2. Superradiant phase The superradiant phase is a quantum transition in the Dicke model and represents the displacement of bosonic modes [24]: ^ cy!^ay+pand^by!^dyp, whereandrepresent averaged values of the displaced ground states for the photon and the magnon, respectively. Using Holstein-Primako transformation in the Dicke Hamiltonian, the eigen-frequencies become: !=1p 2r !2+ ~g4!2mq (!2~g4!2m)2+ 4!2!2m (D2) where ~g= 2g !. In this case, Fig. 7 (b) shows tted measurement with the superradiant Dicke model. For this t, we do not need to add a frequency term on the FMR and we found that != 4:75 GHz and g= 2:58 GHz. With comparing BM frequencies, DM frequencies, and gas done in section IV C, the BM frequency and gshould be higher than those obtained from simulation. Because of this mismatch, it seems that the superradiant phase is not reached. /uni00000014/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000018 /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000018 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 /uni00000013/uni00000011/uni0000001a/uni00000018 /uni00000013/uni00000011/uni00000018/uni00000013 /uni00000013/uni00000011/uni00000015/uni00000018 /uni00000013/uni00000011/uni00000013/uni00000013 /uni00000013/uni00000011/uni00000015/uni00000018 /uni00000013/uni00000011/uni00000018/uni00000013 /uni00000013/uni00000011/uni0000001a/uni00000018H0/uni00000003/uni0000003e/uni00000037/uni00000040/uni0000000b/uni00000045/uni0000000cfDM fBM fFMR f± /uni00000014/uni00000017/uni00000013 /uni00000014/uni00000015/uni00000013 /uni00000014/uni00000013/uni00000013 /uni0000001b/uni00000013 /uni00000019/uni00000013 /uni00000017/uni00000013 /uni00000015/uni00000013 /uni00000036/uni00000003/uni0000003e/uni00000047/uni00000025/uni00000040 FIG. 7. Transmission spectra versus the RF frequency and the H-eld. Fitted polariton branches are shown in white. The BM frequency (in orange) and the coupling strength are variables. The FMR is shown in black and the DM in red. Fits were be done with the normal phase of the Dicke model. In (a) the measurement for d= 50m and a t with the Dicke model. In (b) the measurement for d= 4m and a t with the Dicke superradiant phase model. Fig. 7 shows transmission spectra with respect to the frequency and the H-eld. A t has be done with the standard Dicke model. The two eigen-modes of the t shown in dotted white line which are not consistent with the measurement prove the inability to t with the standard Dicke model. Appendix E: Hopeld model 1. Standard The Hopeld model is equivalent to the Dicke one with a supplementary term: the diamagnetic one. Considering a carried particle in a magnetic eld, we redene the impulse of the system[24]: ^p!^pq^A (E1) with ^Athe vector potential and associated to the photonic mode: ^A/(^cy+ ^c) We nally have the Hopeld Hamiltonian of the system: ^H=!^cy^c+!m^by^b+g(^cy+ ^c)(^by+^b) +D(^cy+ ^c)2(E2)15 whereDis the diamagnetic term where the Thomas-Reiche-Kuhn sum rules gives D=g2 !. With using Hopeld-Bogolubav transformation and redening gasgp!m !we have: !=1p 2r !2+!2m+ 4g2q (!2+!2m+ 4g2)24!2!2m (E3) 2. Modied Following ref. [17], where a prefactor dis added before the diamagnetic term in Eq. E4, we tried to t with the modied Hopeld model by varying this prefactor. !=1p 2r !2c+ 4dD!c+!2cq (!2c+ 4dD!c!2m)2+ 16g2!c!m (E4) whereD=g2=!m /uni00000014/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000018 /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000018 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 /uni00000014/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000018 /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000018 /uni00000014/uni00000011/uni00000013H0/uni00000003/uni0000003e/uni00000037/uni00000040/uni0000000b/uni00000045/uni0000000c fFMR fDM fBM f± /uni00000014/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000018 /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000018 /uni00000014/uni00000011/uni00000013H0/uni00000003/uni0000003e/uni00000037/uni00000040/uni0000000b/uni00000046/uni0000000c /uni00000014/uni00000017/uni00000013 /uni00000014/uni00000015/uni00000013 /uni00000014/uni00000013/uni00000013 /uni0000001b/uni00000013 /uni00000019/uni00000013 /uni00000017/uni00000013 /uni00000015/uni00000013 /uni00000036/uni00000003/uni0000003e/uni00000047/uni00000025/uni00000040 FIG. 8. Transmission spectra versus the RF frequency and the H-eld. Fitted polariton branches are shown in white. The BM frequency (in orange) and the coupling strength are variables. The FMR is shown in black and the DM in red. Fits were be done with the modied Hopeld model where the prefactor is: (a) d<1; (b)d= 1; and (c) d>1. Fig. 8 shows the t with the modied Hopeld model when the prefactor is less, equal, or more than 1 in respectively (a), (b), and (c). Let us notice that the standard Hopeld model is for d= 1. Finally the only eect of this prefactor is equivalent to increase (for d<1) or decrease (for d>1) the BM frequency whereas it is needed to have a model which aects the FMR.16 Appendix F: Measurements 1. CAV 01 /uni00000014 /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 /uni00000014/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000018 /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000018 /uni00000014/uni00000011/uni00000013/uni0000000b/uni00000047/uni0000000c /uni00000014/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000018 /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000018 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 /uni00000013/uni00000011/uni00000018 /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000018/uni0000000b/uni00000048/uni0000000c /uni00000014/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000018 /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000018 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 /uni00000013/uni00000011/uni00000018 /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000018H0/uni00000003/uni0000003e/uni00000037/uni00000040/uni0000000b/uni00000049/uni0000000c /uni00000014/uni00000017/uni00000013 /uni00000014/uni00000015/uni00000013 /uni00000014/uni00000013/uni00000013 /uni0000001b/uni00000013 /uni00000019/uni00000013 /uni00000017/uni00000013 /uni00000015/uni00000013 /uni00000036/uni00000003/uni0000003e/uni00000047/uni00000025/uni00000040 fDM fBM fFMR f± FIG. 9. Transmission spectra versus the RF frequency and the H-eld. Fitted polariton branches are shown in white. The BM frequency (in orange) and the coupling strength are variables. The FMR is shown in black and the DM in red. Fitted parameters are shown in Table II.17 TABLE II. Cavity parameters from Fig. 9 Numbering d[m]fDM[GHz]fBM[GHz]g=2[GHz]g=! g2=2![GHz]
m=2[GHz]fgap[GHz] (a) 116 3.75 7.65 2.68 0.35 0.94 2.35 0.58 (b) 75 3.19 7.31 2.62 0.36 0.94 2.29 0.54 (c) 65 3.05 7.16 2.56 0.36 0.92 2.31 0.54 (d) 36 2.40 6.44 2.41 0.37 0.90 2.27 0.64 (e) 10 1.38 4.46 2.03 0.46 0.92 2.39 0.87 (f) 3 0.81 2.80 1.64 0.59 0.96 2.59 1.22 2. CAV 02 /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 fDM fBM fFMR f± /uni0000000b/uni00000046/uni0000000c /uni00000013/uni00000011/uni00000017 /uni00000013/uni00000011/uni00000015 /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000015 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 /uni00000013/uni00000011/uni00000017 /uni00000013/uni00000011/uni00000015 /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000015 /uni00000013/uni00000011/uni00000017H0/uni00000003/uni0000003e/uni00000037/uni00000040/uni0000000b/uni00000047/uni0000000c /uni00000014/uni00000017/uni00000013 /uni00000014/uni00000015/uni00000013 /uni00000014/uni00000013/uni00000013 /uni0000001b/uni00000013 /uni00000019/uni00000013 /uni00000017/uni00000013 /uni00000015/uni00000013 /uni00000036/uni00000003/uni0000003e/uni00000047/uni00000025/uni00000040 FIG. 10. Transmission spectra versus the RF frequency and the H-eld. Fitted polariton branches are shown in white. The BM frequency (in orange) and the coupling strength are variables. The FMR is shown in black and the DM in red. Fitted parameters are shown in Table III. TABLE III. Cavity parameters from Fig. 10 Numbering fDM[GHz]fBM[GHz]g=2[GHz]g=! g2=2![GHz]
m=2[GHz]fgap[GHz] (a) 4.06 9.79 2.72 0.28 0.76 1.63 0.24 (b) 3.26 8.76 2.59 0.30 0.77 1.71 0.30 (c) 3.01 8.32 2.52 0.30 0.76 1.74 0.31 (d) 2.64 7.63 2.42 0.32 0.77 1.69 0.3418 3. CAV 03 /uni00000013/uni00000011/uni00000015 /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 fDM fBM fFMR f± /uni00000013/uni00000011/uni00000015 /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000015/uni00000015/uni00000017/uni00000019/uni0000001b/uni00000014/uni00000013/uni0000000b/uni00000046/uni0000000c /uni00000013/uni00000011/uni00000015 /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 /uni00000013/uni00000011/uni00000014/uni00000013 /uni00000013/uni00000011/uni00000013/uni00000018 /uni00000013/uni00000011/uni00000013/uni00000013 /uni00000013/uni00000011/uni00000013/uni00000018 /uni00000013/uni00000011/uni00000014/uni00000013H0/uni00000003/uni0000003e/uni00000037/uni00000040/uni00000014/uni00000015/uni00000016/uni00000017/uni00000018/uni0000000b/uni00000047/uni0000000c /uni00000014/uni00000017/uni00000013 /uni00000014/uni00000015/uni00000013 /uni00000014/uni00000013/uni00000013 /uni0000001b/uni00000013 /uni00000019/uni00000013 /uni00000017/uni00000013 /uni00000015/uni00000013 /uni00000036/uni00000003/uni0000003e/uni00000047/uni00000025/uni00000040 FIG. 11. Transmission spectra versus the RF frequency and the H-eld. Fitted polariton branches are shown in white. The BM frequency (in orange) and the coupling strength are variables. The FMR is shown in black and the DM in red. Fitted parameters are shown in Table IV. TABLE IV. Cavity parameters from Fig. 11 Numbering fDM[GHz]fBM[GHz]g=2[GHz]g=! g2=2![GHz]
m=2[GHz]fgap[GHz] (a) 3.02 5.53 0.65 0.12 0.08 0.33 0.01 (b) 2.29 4.36 0.69 0.16 0.11 0.29 0.02 (c) 1.44 2.92 0.63 0.22 0.14 0.37 0.02 (d) 1.30 2.35 0.58 0.25 0.14 0.50 0.0519 Appendix G: Gap Study /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 g/ω/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000015/uni00000013/uni00000011/uni00000017/uni00000013/uni00000011/uni00000019∆g/ω /uni0000000b/uni00000044/uni0000000cCAVsimu 01 CAVmeas 03 CAVmeas 02 CAVmeas 01 /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000015 /uni00000013/uni00000011/uni00000017 /uni00000013/uni00000011/uni00000019 /uni00000013/uni00000011/uni0000001b /uni00000014/uni00000011/uni00000013 /uni00000014/uni00000011/uni00000015 /uni00000014/uni00000011/uni00000017 ∆m/ω /uni0000000b/uni00000045/uni0000000c FIG. 12.
g=!versus (a)g=!; (b)
m=!. Shown are the FD simulations on CAV 01in red and measurements in blue, in green for CAV 02, and in purple for CAV 03. Without adding
mto the FMR in the Dicke model, and without applied static magnetic eld, the frequency of the upper polariton is equal to the cavity one. However, when the FMR is shifted, an observable forbidden gap in frequency appears. Considering Fig. 12,
g=!is not observable when g=!is equal or lower to 0.2. For higher g=! values,
g=!is quadratic, as shown in (a). In (b) is shown the evolution of
g=!versus
m=!.

2022-09-29

We present an experimental study of the strong to ultra-strong coupling regimes at room temperature in frequency-reconfigurable 3D re-entrant cavities coupled with a YIG slab. The observed coupling rate, defined as the ratio of the coupling strength to the cavity frequency of interest, ranges from 12% to 59%. We show that certain considerations must be taken into account when analyzing the polaritonic branches of a cavity spintronic device where the RF field is highly focused in the magnetic material. Our observations are in excellent agreement with electromagnetic finite element simulations in the frequency domain.

Strong to ultra-strong coherent coupling measurements in a YIG/cavity system at room temperature

2209.14643v2

Entropy production rate and correlations of cavity magnomechanical system Collins O. Edet,1, 2Muhammad Asjad,3,∗Denys Dutykh,3, 4Norshamsuri Ali,5and Obinna Abah6,† 1Institute of Engineering Mathematics, Universiti Malaysia Perlis, 02600 Arau, Perlis, Malaysia 2Department of Physics, University of Cross River State, Calabar, Nigeria 3Department of Mathematics, Khalifa University, Abu Dhabi 127788, United Arab Emirates 4Causal Dynamics Pty Ltd, Perth, Australia 5Advanced Communication Engineering (ACE) Centre of Excellence, Universiti Malaysia Perlis, 01000 Kangar, Perlis, Malaysia 6School of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon Tyne NE1 7RU, United Kingdom We present the irreversibility generated by a stationary cavity magnomechanical system composed of a yttrium iron garnet (YIG) sphere with a diameter of a few hundred micrometers inside a microwave cavity. In this system, the magnons, i.e., collective spin excitations in the sphere, are coupled to the cavity photon mode via magnetic dipole interaction and to the phonon mode via magnetostrictive force (optomechanical-like). We employ the quantum phase space formulation of the entropy change to evaluate the steady-state entropy production rate and associated quantum correlation in the system. We find that the behavior of the entropy flow between the cavity photon mode and the phonon mode is determined by the magnon-photon coupling and the cavity photon dissipation rate. Interestingly, the entropy production rate can increase/decrease depending on the strength of the magnon-photon coupling and the detuning parameters. We further show that the amount of correlations between the magnon and phonon modes is linked to the irreversibility generated in the system for small magnon-photon coupling. Our results demonstrate the possibility of exploring irreversibility in driven magnon-based hybrid quantum systems and open a promising route for quantum thermal applications. I. INTRODUCTION Hybrid quantum systems play a crucial role in the advance- ment of quantum technologies [ 1–6]. In the last decade, re- markable progress has been made in this field with applications spanning through quantum computing [ 5,7], quantum simu- lation [ 8], quantum communication [ 6], quantum sensing [ 9], and quantum thermodynamics [ 10]. A typical example is the cavity optomechanical system, which combines mechanical degrees of freedom with electromagnetic (EM) cavities [ 11]. Recently, hybrid quantum systems based on magnons, quanta of collective spin excitations in ordered ferrimagnetic materi- als, e.g. yttrium iron garnet (YIG), have attracted considerable attention due to great frequency tunability and very good co- herence [ 4]. Magnons can be coherently coupled to different degrees of freedom, such as phonons via the magnetostrictive force [ 12], microwave photons via the magnetic dipole interac- tion [ 4,13,14], optical photons [ 15,16] and superconducting qubits [ 17–19]. The realization of this cavity magnomechan- ical system, a system of photon-magnon-phonon interaction with YIG spheres interacting with microwave cavities has open up possible applications in the preparation of macroscopic quantum states [ 20], generation of squeezed states [ 21], the generation of high-performance detectors [ 22], entanglement generation [ 23,24], ground state cooling [ 25], and quantum information processing [26–29]. Quantum technological and nanofabrication advancements have motivated the design of microscopic and coherent thermo- dynamic machines – quantum thermal machines [ 10], as well as investigating the interplay between quantum information and thermodynamic processes [ 30]. Optomechanical thermal ∗asjad qau@yahoo.com †obinna.abah@newcastle.ac.ukmachines have been proposed in different configurations [ 31– 37]. The irreversibility (such as, friction, disorder) that in- fluences the machine thermodynamic processes/performance can be quantified by entropy production [ 38,39]. Thus, quan- tifying the degree of irreversible entropy generated in a dy- namic process is useful for the distinctive description of the non-equilibrium processes, and decreasing it, enhances a ther- mal machine efficiency [ 40]. Based on quantum phase-space method, the measure of quantifying the irreversible entropy pro- duction of quantum systems that interact with nonequilibrium reservoirs has been formulated in [ 41–43] and experimentally verified in two distinct setup - an optomechanical system and a driven Bose-Einstein condensate coupled to a high finesse cav- ity [44]. The effect of self-correlation on irreversible entropy production rate in a parametrically driven dissipative system has been investigated [ 45]. It has been recently shown that the presence of nonlinearity via an optical parametric oscillator placed inside the cavity optomechanical system influences the stationary state entropy production rate [ 46]. Moreover, it has been established that the entropy produced in a bipartite quan- tum system is related to the amount of correlations shared by its subsystems [41]. Here, we investigate the generation of irreversibility in a hybrid cavity magnomechanical setup comprising a microwave cavity and a YIG sphere. In this system, magnons are simul- taneously coupled to the phonons of the vibrational sphere via magnetostrictive interaction and to the cavity photons via magnetic-dipole interaction, while there is no direct interaction between the cavity mode and the mechanical mode. We find that the magnon-photon coupling and cavity photon dissipation rate influence the entropy production rate. Furthermore, we demonstrate that the amount of correlations in the cavity mag- nomechanical system deviates from the steady-state entropy production rate for large magnon-photon coupling. The rest of this paper is organized as follows. In Section II,arXiv:2401.16857v1 [quant-ph] 30 Jan 20242 FIG. 1. Diagrammatic representation of a cavity magnomechani- cal system consisting of photon, magnon, and phonon modes. The magnon-photon interaction strength and the magnon-phonon interac- tion of coupling strength are denoted by gamandgmbrespectively. we present the physical model of the cavity magnomechanical system. We derive the linearized Hamiltonian of the system via quantum Langevin equations of motion and standard lin- earization techniques. In Section III, the stationary entropy production rate and quantum correlation quantified by mutual information are presented using the experimentally feasible parameters. Finally, the conclusions are summarized in Section IV. II. CA VITY MAGNOMECHANICAL MODEL We consider a hybrid cavity magnomechanical system, which consists of a microwave cavity and a small sphere (a onemm-diameter, highly-polished YIG sphere is considered in Ref. [ 47,48]). The YIG sphere is positioned close to the maximal microwave magnetic field of the cavity mode, and a variable external magnetic field Hin the z-axis is added to establish the magnon–photon coupling [ 47,49]. The coupling rate can be tailored by adjusting the position of the sphere. The magnons couple to phonons via the magnetostrictive effect. The vibrational modes (phonons) result from the geometric deformation of the YIG sphere because the magnon excita- tion inside the YIG sphere induces a varying magnetization. The magnomechanical coupling can be enhanced by directly driving the magnon mode with a microwave source [ 12]. In addition, the size of the sphere considered is much smaller than the wavelength of the microwave field, such that the in- teraction between cavity microwave photons and phonons can be neglected (i.e. the radiation pressure effect is negligible). Thus, the system has three modes: cavity photon, magnon, and phonon modes, which can be schematically depicted by the equivalent coupled harmonic oscillator model as shown in Fig. 1. The Hamiltonian of the hybrid quantum system under rotating-wave approximation in a frame rotating with the fre- quency ωdof the driving field can be expressed as ( ℏ= 1)[23, 48]: ˆH:= ∆ aˆa†ˆa+ Ω bˆb†ˆb+ ∆ mˆm†ˆm+gam(ˆaˆm†+ ˆa†ˆm) +gmbˆm†ˆm(ˆb+ˆb†) + i(Ω dˆm†−Ω∗ dˆm), (1) where the bosonic annihilation (creation) operators ˆa(ˆa†), ˆb(ˆb†)andˆm( ˆm†)denote, respectively, cavity photon, phonon and maganon modes whose resonant frequencies are taken to beωa,Ωb, and ωmcorrospondingly. The detuning param- eters ∆a: =ωa−ωd, where ωa/2π= 10 GHz [ 12], and ∆m: =ωm−ωd. The uniform magnon mode frequency in the YIG sphere is ωm=γgH, where γg/2π=28 GHz/T is the gyromagnetic ratio, and we set ωmat the Kittel mode frequency [50], which can lead to cavity polaritons by strongly coupling magnon and cavity photons. The parameters gamandgmb are the optomagnon (photon-magnon) and magnomechanical (magnon-phonon) interaction coupling strengths respectively. The last term, (Ωdˆm†−Ω∗ dˆm)in the Hamiltonian describes the external driving of the magnon mode. The Rabi frequency Ωd≡√ 5 4γg√NtB0(assuming low-lying excitations) denotes the coupling strength of the drive magnetic field [ 23]. The amplitude and frequency of the drive field are B0andωd, re- spectively, and the total number of spins Nt=ρ V, where Vis the volume of the sphere and ρ=4.22×1027m−3is the spin density of the YIG. Each of the modes is coupled to an indepen- dent noise reservoir, their energy decay rates are γa,γm, and γb, for photon, magnon and phonon respectively. Experimen- tally, gmbis extremely weak [ 12], but the magnomechanical interaction can be enhanced by driving the magnon mode with a strong microwave field [ 47,49]. The magnon-photon cou- pling rate gamcan be larger than the dissipation rates of the cavity and magnon modes, γaandγm, entering into the strong coupling regime, gam>max{γa, γm}[13, 14]. As a result of strong driving, the Hamiltonian in Eq. (1) can be linearized around the coherent steady-state amplitude: ˆO→Os+ˆO(O∈ {a,b,m}), where Osand the operators ˆO, represent the steady-state amplitudes and quantum fluctu- ations of the corresponding mode. We have the steady-state amplitudes ms=Ωd(i∆a+γa) g2am+ (i∆ m+γm)(i∆ a+γa), (2) andbs=−igmb|ms|2/(i Ωb+γb). Then, the linearized Hamiltonian can be derived as ˆHlin= ∆ aˆa†ˆa+ Ω bˆb†ˆb+˜∆mˆm†ˆm+gam(ˆaˆm†+ ˆa†ˆm) + (G∗ mbˆm+Gmbˆm†)(ˆb+ˆb†), (3) where the enhanced magnon-phonon coupling Gmb=gmbms, and˜∆m= ∆ m−gmb(bs+b∗ s)is the effective magnon de- tuning incorporating the magnetostriction. For the considered parameters, gmb(bs+b∗ s)≪∆m, so we can have, ˜∆m≈∆m. Since the driving field affects ms, we can improve Gmbby adjusting the external driving field Ωd. From the Hamiltonian in Eq. (3), we obtain the quantum3 Langevin equations as; ˙ˆa=−(i∆a+γa) ˆa−igamˆm−p 2γaˆain, ˙ˆm=− i˜∆m+γm ˆm−igamˆa−iGmb(ˆb+ˆb†)−p 2γmˆmin, ˙ˆb=−(iΩb+γb)ˆb−i(Gmbˆm†+G∗ mbˆm)−p 2γbˆbin, (4) where ˆfin∈ {ˆain,ˆmin,ˆbin}are input noise operators for the cavity, magnon and mechanical modes, respectively, which are zero mean and characterized by the follow- ing correlation functions [ 51]:⟨ˆfin(t)ˆf† in(t′)⟩= (Nk+ 1)δ(t−t′)and⟨ˆf† in(t)ˆfin(t′)⟩=Nkδ(t−t′), where Nk= 1/ eℏωk/kBT−1
(k∈{a,m}), are the equilibrium mean thermal photon, and magnon number, respectively, while Nb= 1/ eℏΩb/kBT−1
is the equilibrium mean thermal phonon number, Tis the environmental temperature and kBis the Boltzmann constant . Eq. (4) represents the evolution of the fluctuation incorporating the interplay with the environment via the noise operators [ 25]. The photon number and magnon occupation number are approximately zero, i.e., Na,m≈0, due to the high frequencies of their modes. Since the nature of the noise is Gaussian, all the infor- mation is contained in the first and second-order moments of the operators. In particular, it is convenient to intro- duce the quadratures ˆxandˆyof the photon, magnon, and phonon modes by using the relation ˆO= (ˆxO+ iˆyO)/√ 2 withO∈ {a,m,b}, and elements of the corresponding co- variance matrix are defined as Vij=1 2⟨Ri(∞)Rj(∞) + Rj(∞)Ri(∞)⟩, (i, j∈ {1,2,3,4,5,6}). Here, Ri(t) := [ˆxa,ˆya,ˆxm,ˆym,ˆxb,ˆyb]⊤is the column vector of quadratures and the stationary covariance matrix is obtained by solv- ing the algebraic equation A⊤V+A V +D= 0, where D=diag(γa, γa, γm, γm, γb(2Nb+ 1), γb(2Nb+ 1)) and the drift matrix Ais expressed as: A= −γa∆a 0 gam 0 0 −∆a−γa−gam 0 0 0 0gam−γm˜∆m−Gmb 0 −gam 0−˜∆m−γm 0 0 0 0 0 0 0 Ω b 0 0 0 Gmb−Ωb−γb .(5) The system must be stable for a steady state to exist, to as- certain this, the Routh-Hurwitz criterion [ 52] is employed to characterize the stability of the system. To achieve this, the real part of the spectrum of the drift matrix A, Eq. (5), must be negative, this means that all the eigenvalues of the drift matrix Ahave non-positive real parts. III. ENTROPY PRODUCTION RATE The basic thermodynamics principle asserts the entropy of an open system, (classical or quantum) evolves as: dS dt= Π−Φ, (6)where Π⩾0is the irreversible entropy production rate and Φis the entropy flow from the system to the reservoir. In thermal equilibrium, the steady states are characterized by dS/dt=Π=Φ=0 . However, when the system is connected to multiple reservoirs or being externally driven, it may instead reach a nonequilibrium steady states where dS/dt= 0 but Π=Φ⩾0. In this nonequilibrium steady state case, the system is characterized by the continuous production of entropy, all of which flows to the reservoir. We now move to study the entropy production rate from a multipartite system. In analogy with the bipartite case [ 41], we combine quantum phase-space methods and the Fokker-Planck equation to characterise the irreversible entropy production of quantum systems interacting with reservoirs. In general, the entropy production rate of the quantum system described in Section II is given by (see, Appendix A): Πs:=3X i=12γiV2i−1,2i−1+V2i,2i 2Ni+ 1−1 , (7) where Γi∈ {Γa,Γm,Γb}withΓ∈ {γ,N}. Here, we focus on characterizing the entropy production rates in a cavity magnomechanical system. Without loss of generality, we consider the effective entropy flow between the magnon mode and the mechanical resonator. Thus, the rate of entropy production Πsat a steady-state reads Πs= 2γm(V33+V44−1) + 2 γbV55+V66 2Nb+ 1−1 .(8) We remark, when the system is in the equilibrium state, we haveV11+V22= 1,V33+V44=2Nb+ 1, and hence, Πs≡0. To proceed, we study the entropy production rate in a magnon-phonon-photon system at a steady state in the resolved sideband, where the magnon dissipation rate is comparable to or well below the mechanical resonance frequency (i.e., γm<Ωb). We assume the following parameters close to those employed in the experimental realizations [ 12], as the phonon frequency Ωb/2π=10 MHz, the cavity dissipation rate γa/2π=3MHz, the magnon dissipation rate γm/2π=1MHz, the phonon damping rate γb= 300 Hz, the phonon-magnon coupling gmb/2π≃1Hz, and the temperature T= 10−100 mK (i.e., Nb≃200). In the following analysis, we will uti- lize dimensionless quantities; that is, the quantities will be expressed in units of the phonon frequency, Ωb. In Fig. 2, we present the entropy production rate Πsis plot- ted as a function of the normalized magnon detuning ∆m/Ωb for various values of the photon-magnon coupling gam. In Fig. 2(a) and (b), we consider the dissipation rate of the cavity γa= 10−1Ωbfor distinct occupation number Nb= 10 and Nb= 100 , respectively. It can be seen that in the absence of magnon-photon interaction gam= 0, the entropy production rateΠspeaks at ∆m/Ωb=±1. The two peaks in the rate of entropy production for positive/negative detuning imply the cooling/heating processes behave differently in two distinct regimes of the system. For non-zero gam, the smaller peak smears out with reduced maximum Πs. We observe that for gam=Ωb(gam=2 Ω b), the maximum Πsis in the red (blue)4 -6 -4 -2 0 2 4 60.010.050.100.501510 -6 -4 -2 0 2 4 60.010.10110100 -6 -4 -2 0 2 4 60.010.050.100.501510 -6 -4 -2 0 2 4 60.010.10110100 FIG. 2. Plot of entropy Πs(blue dashed) as a function of normalized magnon detuning ∆m/Ωbfor different values of magnon-photon coupling gam= 0(black curve), gam= Ω b(blue curve) and gam= 2Ωb(red curve), where (a)Nb= 10 (b)Nb= 100 ,(c),Nb= 10 , and(d)Nb= 100 . In(a)&(b):γa= 0.1Ωb, andγa= Ω bin(c)& (d). The other parameters are Gmb= 10−1Ωb,γb= 10−2Ωband γm=Ωb/2. sideband region. The maximum entropy flow between the phonon mode and the effective magnon mode occurs when gam=2Ω b, at∆m=ωb. Fig 2 (c) and (d) show more clearly the effect and interplay between the cavity photon dissipation rate and the magnon-photon coupling. For the case of γb=Ωb, Fig. 2(c) and (d), the value of Πsfurther decreases with a broaden peak as magnon-phonon coupling gamincreases. In the blue-sideband, for gam̸= 0, the value of Πsis always reduced compare to the gam=0scenario. It can be seen that Πscould be enhanced close to the negative detuning region ∆m<0. The effects of thermal fluctuations of the environ- ment on the entropy production rate is shown in Fig. 2. The Fig. 2 shows that increasing the phonon thermal excitation (occupation number) Nb, increases Πsin magnitude (see the magnitude of Πsin Fig. 2(a) and (c) compare to (b) and (d) respectively). The non-uniform behaviour when varying gam for different γais a direct implication of the complex nature of the interaction between the dissipation processes in the cavity magnomechanical system. In Fig. 3, the effects of the cavity photon dissipation rate of the environment on the entropy production rate is explored. We present in Fig.3 (a), the plot of the entropy production rate as a function of magnon-photon coupling with various values of the cavity decay rate, while Fig. 3 (b) shows the entropy production rate as function of cavity decay rate with different values of the magnon-photon coupling. For large thermal exci- tations, Nb=100 , and ∆m=Ωb, the entropy production rates decrease in oscillatory form as the magnon-photon coupling increases. For γa= 10−1Ωb, the maximum entropy produc- tion rate corresponds to the gam= 2Ω b. It can be seen that the entropy production rate can be increase/decrease by tuning the cavity decay value γaand the magnon-photon coupling, seegam≃0.2−3 Ωb. The entropy production rate Πslinearly decreases with respect to the magnon-photon coupling when 0 1 2 3 4 50.010.10110100 0 1 2 3 4 50.10.5151050100FIG. 3. (a)The entropy production rate Πsas a function of magnon- photon coupling gam/Ωbfor different values of cavity photon decay rateγa= 10−1Ωb(black curve), γa= Ω b(blue curve) and γa= 2 Ωb(red curve). (b)Entropy production rate Πsas a function of cavity photon decay rate γa/Ωbfor different values of magnon-photon coupling gam= 10−1Ωb(black curve), gam= Ω b(blue curve) and gam= 2 Ω b(red curve). Other parameters are Nb= 100 ,∆m= Ω b, Gmb=10−1Ωb,γb=10−2Ωbandγm=Ωb/2. gam>3/Ωbbut the value slightly increase when we increase γa. This non-monotonic behaviour (increase/decrease) of the entropy production with respect to the magnon-photon cou- pling can be attributed to the imbalance in populations between the interacting modes induced by the environment. Further- more, Fig. 3 (b) shows that the entropy production rate can increase/decrease when γa< gambut increases slightly after γa=gam. In the region, γa≫gam, the entropy production rate saturates to a quasi-constant value with decreasing value as the magnon-photon gamincreases. This is due to the fact that the individual modes are far from resonance and they are effec- tively decoupled, in such a way that the individually thermalize their own bath. Next, we analyze the behaviour of the entropy production rate with respect to the amount of correlations shared between the magnon and phonon modes. For coupled quantum system, it has been demonstrated that the irreversibility generated by the dissipative system at steady state and the total amount of correlations shared between the subsystems are closely related in small coupling limit as [ 41];I ≈Πs/(2γtot), where Iis the quantum mutual information between the modes at the stationary state and γtotis the sum of the dissipation rates to the local baths. Since the quantum noises are Gaussian, the mutual information between the magnon and phonon modes can be computed as [53]: I(Va:b) =1 2lndet(Va)det(Vb) det(Vab) , (9) where Va(Vb)is the covariance matrix of magnon (mechani- cal) mode. In Fig. 4, we show a comparative plot of the entropy pro- duction rate ΠSto the correlations established by the cavity magnomechanical system, as quantified by the mutual informa- tionI. Considering small thermal phonon excitation, Nb=10 , Fig. 4 (a) shows a good similarity in both ΠsandIcurves for|∆m/Ωb|⩽2. As the magnon-photon coupling increases, both the entropy production rate and mutual information are de- creasing as well as an increase in both quantities deviation, see Fig. 4 (b) and (c). The increase in deviation between ΠsandI for increasing detuning ∆m, clearly demonstrates the interplay5 -4 -2 0 2 40.010.050.100.501510 -4 -2 0 2 40.0010.0050.0100.0500.1000.5001 -4 -2 0 2 40.0010.0050.0100.0500.1000.5001 FIG. 4. Entropy rate Πs(blue curve) and mutual information I(red curve) as a function of normalized magnon detuning ∆m/Ωbfor different 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 , Gmb=10−1Ωb,γb= 10−2Ωbandγm= Ω b/2. among magnon-photon coupling and dissipation rates. This shows that degree of irreversibility induced by the stationary process is directly related to the amount of correlations shared in the system. IV . CONCLUSIONS We have investigated the entropy production rate in a hy- brid magnomechanical system where a microwave cavity mode is coupled to a magnon mode in a YIG sphere, and the latter is simultaneously coupled to a phonon mode via magnetostric- tive force. Specifically, within the quantum phase space for- mulation of the entropy change, we evaluate the steady-state entropy production rate and associated quantum correlation in the hybrid system. We have shown that the entropy flow between the effective magnon mode and the phonon mode is influenced by the magnon-photon coupling and the cavity photon dissipation rate. Our numerical analysis shows non- uniform behavior of irreversibility resulting from the complex and competing nature of the interactions in the cavity mag- nomechanical system. We find that the entropy production rate can be increased/decreased when the cavity decay rate is less than the magnon-photon coupling. Furthermore, we studied the range of validity of the link between irreversibility and the amount of correlations in a mesoscopic quantum system. These results provide insight into the impact of magnonics on the thermodynamics processes of hybrid quantum systems. Finally, we anticipate that our study will open new perspectives for quantum thermodynamics applications, such as realizing thermal machines in the deep quantum regime and quantum thermal transport. ACKNOWLEDGEMENT COE and NA have been supported by LRGS Grant LRGS/1/2020/UM/01/5/2 (9012-00009) provided by the Min- istry of Higher Education of Malaysia (MOHE). MA and DD have been supported by the Khalifa University of Science and Technology under Award No. FSU- 2023-014.Appendix A: Entropy production In this appendix, we provide the explicit expression for the entropy production rate presented in Eq. 7. Considering the Gaussian nature of the quantum system, their states are characterized by the Wigner function of the form W(R) =1 πn√ detσexp −1 2(R−¯R)⊤σ−1(R−¯R) , (A1) where nis the number of the bosonic mode and σis the covariance matrix (CM), whose entries are given by σij= 1 2⟨RiRj+RjRi⟩ − ⟨Ri⟩⟨Rj⟩. The entropy of the Gaussian system is associated to the corresponding Shannon entropy (Wigner entropy) of the Wigner distribution. Following the quantum Langenvin equations for the system dynamics described in Section II, we can recast the Fokker- Planck equation for the Wigner function W(u, t)as a local conservation equation ∂tW=−∂uJ(u, t), (A2) where u= (xa, ya, xm, ym, xb, xb)⊤is a point in phase space, ∂uis the phase-space gradient, and Jis the total probability current vector that reads J(u, t)=AuW(u, t)−1 2D∂uW(u, t). (A3) For the three mode bosonic system model that we consider, the drift matrix Aand the diffusion matrix Dare defined in Section II (Eq. 5). Then, introducing the time-reversal operator E=diag(1,−1,1,−1,1,−1), the dynamical variables can be split according to their symmetry. The drift matrix A can be rewritten by splitting it into the irreversible part Airr which is even under time reversal and the reversible part Arev that is odd [ 41]. They can be evaluated as Airr:=1 2(A+ EAE⊤)andArev:=1 2(A−EAE⊤). The irreversible part is associated with the damping rates, while the reversible part comes from the Hamiltonian part of the dynamics. The drift matrix separation results in splitting of the probability currents asJ(u, t)≡ Jrev(u, t) +Jirr(u, t), where Jrev(u, t) :=ArevuW(u, t), (A4)6 and Jirr(u, t) :=AirruW(u, t)−1 2D∂uW(u, t). (A5) Taking the derivative of the Wigner entropy with respect to time and integrating by parts, we get SW=Z d2α(JirrW)⊤∂uW W . (A6) Equation (A6) can be rewritten in the usual form in terms of the entropy production rate [41]: Π =1 2Z du1 W(u, t)Jirr(u, t)⊤D−1Jirr(u, t), (A7) and entropy flux rate Φ =−1 2Z duJirr(u, t)D−1Airru. (A8) Moreover, for a Gaussian state, ∂uWσ(u, t) = −Wσ(u, t)σ−1(t)u, then the irreversible component of probability current Jirr(u, t) =Wσ(u, t)(Airru+D)σ−1(t)u. Therefore, the explicit integration of Eq. (A7) gives theentropy production rate as [41] Π =1 2tr σ−1D + 2tr[Airr] + 2tr (Airr)⊤D−1Airrσ . 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= 2γm(V33+V44−1) + 2 γbV55+V66 2Nb+ 1−1 , (A12) whereAirr=diag{0,0,−γm,−γm,−γb,−γb}. This is the expression given in the main text. [1]Z.-L. Xiang, S. Ashhab, J. Q. You, and F. Nori, Hybrid quan- tum circuits: Superconducting circuits interacting with other quantum systems, Rev. Mod. Phys. 85, 623 (2013). [2]B. Bauer, D. Wecker, A. J. Millis, M. B. Hastings, and M. Troyer, Hybrid quantum-classical approach to correlated materials, Phys. Rev. X 6, 031045 (2016). [3]J. Li, X. Yang, X. Peng, and C.-P. Sun, Hybrid quantum-classical approach to quantum optimal control, Phys. Rev. Lett. 118, 150503 (2017). [4]D. Lachance-Quirion, Y . Tabuchi, A. Gloppe, K. Usami, and Y . Nakamura, Hybrid quantum systems based on magnonics, Appl. Phys. Express 12, 070101 (2019). [5]A. Clerk, K. Lehnert, P. Bertet, J. Petta, and Y . Nakamura, Hy- brid quantum systems with circuit quantum electrodynamics, Nat. Phys. 16, 257 (2020). [6]A. Callison and N. 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2024-01-30

We present the irreversibility generated by a stationary cavity magnomechanical system composed of a yttrium iron garnet (YIG) sphere with a diameter of a few hundred micrometers inside a microwave cavity. In this system, the magnons, i.e., collective spin excitations in the sphere, are coupled to the cavity photon mode via magnetic dipole interaction and to the phonon mode via magnetostrictive force (optomechanical-like). We employ the quantum phase space formulation of the entropy change to evaluate the steady-state entropy production rate and associated quantum correlation in the system. We find that the behavior of the entropy flow between the cavity photon mode and the phonon mode is determined by the magnon-photon coupling and the cavity photon dissipation rate. Interestingly, the entropy production rate can increase/decrease depending on the strength of the magnon-photon coupling and the detuning parameters. We further show that the amount of correlations between the magnon and phonon modes is linked to the irreversibility generated in the system for small magnon-photon coupling. Our results demonstrate the possibility of exploring irreversibility in driven magnon-based hybrid quantum systems and open a promising route for quantum thermal applications.

Entropy production rate and correlations of cavity magnomechanical system

2401.16857v1

arXiv:1404.2360v2 [cond-mat.mes-hall] 20 Aug 2014Current-induced spin torque resonance of magnetic insulat ors Takahiro Chiba1, Gerrit E. W. Bauer1,2,3, and Saburo Takahashi1 1Institute for Materials Research, Tohoku University, Send ai, Miyagi 980-8577, Japan 2WPI-AIMR, Tohoku University, Sendai, Miyagi 980-8577, Jap an and 3Kavli Institute of NanoScience, Delft University of Techno logy, Lorentzweg 1, 2628 CJ Delft, The Netherlands (Dated: August 10, 2018) We formulate atheory ofthe ACspinHall magnetoresistance ( SMR)in abilayer system consisting of a magnetic insulator such as yttrium iron garnet (YIG) and a heavy metal such as platinum (Pt). We derive expressions for the DC voltage generation based on the drift-diffusion spin model and quantum mechanical boundary condition at the interface tha t reveal a spin torque ferromagnetic resonance (ST-FMR). We predict that ST-FMR experiments wil l reveal valuable information on the current-induced magnetization dynamics of magnetic insul ators and AC spin Hall effect. Ferrimagnetic insulators such as yttrium iron garnet (YIG) with high critical temperatures and very low mag- netization damping have been known for decades to be choice materials for in optical, microwave or data stor- age technologies [1]. Near-dissipationless propagation of spin waves make YIG wires and circuits interesting for low power data transmission and logic devices. A cru- cial breakthrough was the discovery that the magneti- zation in YIG can be excited electrically by Pt con- tacts [2], thereby creating an interface between elec- tronic/spintronic and magnonic circuits. However, the generation of coherent spin waves by the current-induced spin orbit torques in Pt is a strongly non-linear process and the low critical threshold currents found by the ex- periments [2] cannot yet be explained by theory [3]. Here we suggest and model a simpler method to get to grips with the important current-magnetization interaction in the YIG|Pt system without a problematic threshold, viz. by employing the recently discovered magnetoresistance ofYIG|PtbilayersorSpinHallMagnetoresistance(SMR) [4, 5] to detect current-inducedspin torqueferromagnetic resonance (ST-FMR). The SMR is the dependence of the electrical resistance ofthenormalmetalonthe magnetizationangleofaprox- imity insulator and is caused by a concerted action of the Spin Hall Effect (SHE) [ ?] and its inverse (ISHE). An alternative mechanism of the SMR phenomenology in terms of an equilibrium proximity magnetization close to the YIG interface has been proposed [5]. However, this interpretation has been challenged by experiments [4, 7]. Moreover, while experiments of many groups are describedquantitativelywell bythe SMR model with one set of parameters [8–12], we are not aware of a transport theory that explains the observed magnetoresistance in terms of a monolayer-order magnetic Pt. Current-induced spin torque ferromagnetic resonance (ST-FMR) has been demonstrated [13–15] in bilayer thin films made from metallic ferromagnets (FM) and non- magnetic metals (N). In these experiments the SHE transforms an in-plane alternating current (AC) into an oscillating transverse spin current. The resultant spin transfer resonates with the magnetization at the FMR frequency. The effects induced simultaneously by the Oersted field can be distinguished by a different sym-metry of the resonance on detuning. The magnetiza- tion dynamics leads to a time dependence of the bilayer resistance by the anisotropic magnetoresistance (AMR). Mixing the applied current and the oscillating resistance generates a DC voltage that is referred to as spin torque diode effect [16, 17]. The longitudinal spin Seebeck effect was found to be frequency independent up to 30 MHz [18]. The DC ISHE induced by spin pumping has been observed by many groups, but detection of the AC spin Hall effect [23] has only recently been reported in metallic structures [19– 21] as well as in Pt |YIG under parametric microwave excitation[22]. ADCvoltagecanbegeneratedinPt |YIG under FMR conditions by rectification of the AC spin Hall effect by means of the SMR, but this signal was found to be swamped by the DC spin Hall effect [24]. A study of the spin Hall impedance concludes that the material constants of Pt |YIG bilayers do not depend on frequency up to 4 GHz [25]. In this paper we suggest to combine the principles sketched above to realize ST-FMR for bilayers of a ferro- or ferrimagnetic insulator (FI) such as YIG and a normal metal with spin orbit interaction (N) such as platinum [26] (see Fig. 1). We derive the magnetization dynam- ics and DC voltages generated by the SMR-induced spin torque diode effect as a function of the external mag- netic field. Our theory should help to better understand the elusive current-induced magnetization dynamics of ferromagnetic insulators which should pave the way for low-power devices based on magnetic insulators [1]. The spin current through an F |N interface is governed by the complex spin-mixing conductance G↑↓[27]. The prediction of a large Re G↑↓for interfaces between YIG and simple metals by first-principle calculations [28] has been confirmed by recent experiments [29, 30]. The spin transport in N (spin Hall system) can be treated by spin-diffusiontheorywithquantummechanicalboundary conditions at the interface to the insulating ferromagnet [5, 31]. The AC current with frequency ωa= 2πfain- duces a spin accumulation distribution µs(z,t) in N that obeys the spin-diffusion equation ∂tµs=D∂2 zµs−µs τsf, (1)2 FIG. 1. Schematic set-up to observe the SMR mediated spin torque diode effect. The light blue rectangle is anormal meta l (N) film with a finite spin Hall angle, while F is a ferromag- netic insulator. The F |N bilayer film is patterned into a strip with width wand length h. A Bias-Tee allows detection of a DC voltage under an AC bias. whereDis the charge diffusion constant and τsfspin-flip relaxation time in N. In position-frequency space the so- lution for the spatiotemporal dependence of the spin ac- cumulation reads µs(z,ω) =Ae−κ(ω)z+Beκ(ω)z, where κ2(ω) = (1 + iωτsf)/λ2,λ=√Dτsfis the spin-diffusion length, and the constant column vectors AandBare determined by the boundary conditions for the spin cur- rent density in the z-direction Js,z(z), where Js,z/|Js,z| is the spin polarization vector, which is continuous at the interface to the ferromagnet at z= 0 and vanishes at the vacuum interface at z=dN. For planar interfaces Js,z(z,ω) =−θSHJc(ω)ˆ y−σ∂zµs(z,ω) 2e,(2) whereθSHis the spin Hall angle, σthe electri- cal conductivity, and Jc(ω) = 2πJ0 cδ(ωa−ω) the currents not accounting for spin-orbit interaction. Js,z(dN,ω) = 0 and Js,z(0,ω) =/integraltext∞ −∞Js,z(0,t)e−iωtdt, whereJs,z(0,t) =JT s+JP s=J(F) swith JT s=Gr eˆM×/parenleftBig ˆM×µs(0)/parenrightBig +Gi eˆM×µs(0),(3) JP s=/planckover2pi1 e/parenleftBig GrˆM×∂tˆM+Gi∂tˆM/parenrightBig , (4) whereˆMis the unit vector along the FI magnetization andG↑↓=Gr+iGithe complex spin-mixing interface conductance per unit area of the FI |N interface. The imaginary part Gican be interpreted as an effective ex- change field acting on the spin accumulation, which is usually much smaller than the real part. A positive J(F) s [10, 28] corresponds here to up-spins flowing from FI into N. For Pt(Ta) ωaτPt(Ta) sf= 1(15)×10−3at the FMR fre- quencyfa= 15.5GHz with τPt(Ta) sf= 0.01(0.15)ps, in- dicating that the condition ωaτPt(Ta) sf≪1 is fulfilled forthese metals [23]. In this limit the frequency dependence of the spin diffusion length may be disregarded such that µs(z,t) → −ˆ yµs0(t)sinh2z−dN 2λ sinhdN 2λ+J(F) s2eλ σcoshz−dN λ sinhdN λ,(5) J(F) s=µs0(t) e/bracketleftBig ˆM×/parenleftBig ˆM׈ y/parenrightBig Re+ˆM׈ yIm/bracketrightBig T +/planckover2pi1 e/bracketleftBig/parenleftBig ˆM×∂tˆM/parenrightBig Re+∂tˆMIm/bracketrightBig T, (6) whereµs0(t) = (2 eλ/σ)θSHJ0 c(t)tanh[dN/(2λ)] withJ0 c(t) = J0 cRe(eiωat) and T= σG↑↓//bracketleftbig σ+2λG↑↓coth(dN/λ)/bracketrightbig . The ISHE drives a charge current in the x-yplane by the diffusion spin current along z. The total charge current density reads Jc(z,t) =J0 c(t)ˆ x+σθSH/parenleftbigg ∇×µs(z,t) 2e/parenrightbigg .(7) The averaged current density over the film thickness is Jc,x(t) =d−1 N/integraltextdN 0Jc,x(z,t)dz=JSMR(t)+JSP(t) with JSMR(t) =J0 c(t)/bracketleftbigg 1−∆ρ0 ρ−∆ρ1 ρ/parenleftBig 1−ˆM2 y/parenrightBig/bracketrightbigg ,(8) JSP(t) =JP rω−1 a/parenleftBig ˆM×∂tˆM/parenrightBig y+JP iω−1 a∂tˆMy,(9) JP r(i)=/planckover2pi1ωa 2edNρθSHRe(Im)η, whereJSMR(t) andJSP(t) are SMR rectification and spin pumping-induced charge currents, ρ=σ−1is the resistivity of the bulk normal metal layer and we recognize the conventional DC SMR with ∆ ρ0= −ρθ2 SH(2λ/dN)tanh(dN/2λ) and ∆ ρ1=−∆ρ0Reη/2, where η=2λρG↑↓tanhdN 2λ 1+2λρG↑↓cothdN λ, (10) are effective resistivities that do not depend on frequency [5]. ST-FMRexperimentsemploytheACimpedanceofthe oscillating transverse spin Hall current caused by the in- duced magnetization dynamics that is described by the Landau-Lifshitz-Gilbert (LLG) equation, including the transverse spin current Eq. (6), ∂tˆM=−γˆM×Heff+α0ˆM×∂tˆM+γ/planckover2pi1J(F) s 2eMsdF,(11) whereHeff=Hex+Hdywith an external magnetic field Hexand the sum of the AC current-induced Oersted field and the (thin film limit of) the dynamic demagnetization Hdy=Hac(t)+Hd(t) =/parenleftbig 0,Haceiωat,−4πMz(t)/parenrightbig .γ,α0, MsanddFarethegyromagneticratio,theGilbert damp- ing constant of the isolated film, the saturation magneti- zation, and the thickness of the FI film, respectively.3 We henceforth disregard the very low in-plane magne- tocrystalline anisotropy field of Hk∼3Oe reported [8]. The external magnetic field Hexis applied at a polar an- gleθin thex-yplane. It is convenient to consider the magnetization dynamics in the XYZ-coordinate system (Fig.1)in whichthemagnetizationisstabilizedalongthe X-axis by a sufficiently strong external magnetic field. Denoting the transformation matrix as R(θ), the mag- netization MR(t) =R(θ)M(t) precesses around the X- axis, where MR(t) =M0 R+mR(t)≈(Ms,mY(t),mZ(t)) as shown in Fig. 1. M0 RandmR(t) are the static and the dynamic components of the magnetization, re- spectively. The LLG equation in the XYZ-system then becomes β∂tMR=−γMR×Heff,R+αˆMR×∂tMR where the effective magnetic field in the XYZ-system isHeff,R=HXˆX+HYeiωatˆY+/parenleftbig HZeiωat−4πmZ(t)/parenrightbigˆZ withHX=Hex,HY= (Hac+Hi)cosθandHZ= Hrcosθwith Hr(i)=/planckover2pi1 2eMsdFθSHJ0 cRe(Im)η, (12) a modulated damping α=α0+ ∆αand g-factor β= 1−∆βwith ∆α(∆β) =γ/planckover2pi12/(2e2MsdF)ReT(ImT).For a small-angle precession around the equilib- rium direction M0 R,mR(t) = (0,δmYeiωat,δmZeiωat) (Re[δmY] Re[δmZ]≪Ms). Disregarding higher orders inδmY(Z)in theR-transformed LLG equation we ar- rive at the (Kittel) relation between AC current fre- quency and resonant magnetic field HF=−2πMs+/radicalbig (2πMs)2+(ωa/γ)2. A DC voltage is generated by two different mecha- nisms, viz. the time-dependent oscillations of the SMR in N (spin torque diode effect) and the ISHE generated by spin pumping. This is quite analogous to electrically detected FMR in which the magnetization is driven by microwavesin cavitiesorcoplanarwaveguides. In metal- lic bilayers, the spin pumping signal due to the ISHE can be separated from effects of the magnetoresistance of the metallic ferromagnet by sample design and angular de- pendences [32, 33]. Here we focus on the current-induced magnetization dynamics that induces down-converted DC and second harmonic components in the normal metal. Indicating time-average by /angb∇acketleft···/angb∇acket∇ighttthe open-circuit DC voltage is VDC=hρ/angb∇acketleftJc,x(t)/angb∇acket∇ightt=VSMR+VSP,where VX=hρ/angb∇acketleftJX(t)/angb∇acket∇ightt. The SMR rectification and spin pump- ing induced DC voltage are VSMR=−h∆ρ1J0 c 4FS(Hex) ∆/bracketleftbigg C(Hr+αHac)+C+HacHex−HF ∆/bracketrightbigg cosθsin2θ, (13) VSP=hρJP r 4FS(Hex) ∆C/bracketleftbigg C−H2 r+αHrHac ∆+C+H2 ac−αHrHac ∆/bracketrightbigg cosθsin2θ, (14) whereC= ˜ωa//radicalbig 1+ ˜ω2aandC±= 1±1//radicalbig 1+ ˜ω2awith ˜ωa=ωa/(2πMsγ),FS(Hex) = ∆2/[(Hex−HF)2+∆2], ∆ =αωa/γthe line width, Hac= 2πJ0 cdN/cthe Oersted field from the AC current determined by Amp` ere’s Law (in the limit of an extended film), and cspeed of light. Using the material parametersfor YIG [2] and Pt [30, 33] shown in Tables I and II we compute the DC voltages in Eqs. (13) and (14). The calculated VSMRis plotted in Fig. 2 as a function of an external magnetic field and for different dF, resolved in terms of the contributions to the FMR caused by the spin transfer torque (symmetric) and the Oersted magnetic field (asymmetric). In Fig. 3 weshowthetotalDCvoltagewithbothspintorquediode and spin pumping contributions. The DC voltagein F |Pt bilayersdependsmoresensitivelyon dFforF = YIGthan F = Py/CoFeB because spin pumping is more important when the Gilbert damping is small. ST-FMR measure- ments are carried out at relatively high current density, so Joule heating in Pt may cause observable effects, the most notable being the spin Seebeck effect (SSE), which adds a constant background DC voltage to the SMR rec- tification signal [34]. TheST-FMR spectrainFig.3areenhancedforthickerTABLE I. Material parameters for the FI layer. γ[T−1s−1]Ms[Am−1] α0 aYIG 1.76 ×10111.56×1056.7×10−5 aReference 2. TABLE II. Material parameters for the N layer. Gr[Ω−1m−2]ρ[µΩcm] λ[nm] θSH Pta3.8×1014 a41b1.4b0.12 aReference 30,bReference 33. F layers, but these are dominated by the Oersted field actuation. These less-interesting contributions can be eliminatedinatri-layerstructureasinFig.4inwhichthe magnetic insulator is sandwiched by two normal metal films with the same electric impedance. The second film N2 should be Cu or another metal with negligible spin- orbit interaction and thereby contributions to the ST- FMR, the quality of the YIG |N2 interface is therefore less of an issue. In Fig. 4 we plot pure ST-FMR signals obtained by setting Hac= 0 in Eqs. (13) and (14), which4 -1.2-0.8-0.4 0 0.4 0.8 2.2 2.3 2.4 2.5 2.6 dF=4nm =10nm =40nmVSMR ( µV) Hex (kOe) YIG(dFnm)|Pt(6nm) -0.8-0.6-0.4-0.2 0 0.2 2 2.2 2.4 2.6 2.8 Total result Spin torque FMR Oersted field FMR YIG(4nm)|Pt(6nm)VSMR ( µV) Hex (kOe) (b)(a) θ=45rHex dF (nm)FI Hex FIG. 2. (a) The ferromagnet thickness dependence of calcu- lated SMR rectificied voltage for YIG |Pt atfa= 9GHz with current density J0 c= 1010A/m2and F(N) layer length and widthh=w= 30µm andθ= 45◦. (b)dF(N)= 4(6) nm. -0.5 0 0.5 1.5 2.5 3.5 2.2 2.3 2.4 2.5 2.6 Hex (kOe) VDC ( µV) dF=4nm =10nm =40nm YIG(dFnm)|Pt(6nm)-0.8 0 0.6 2 2.2 2.4 2.6 2.8 Total result SMR (V SMR ) ISHE (V SP )dF=4nm VDC ( µV) Hex (kOe) FIG. 3. Dependence of the ST-FMR spectra on dFat fa= 9GHz and θ= 45◦. Inset: Contributions by SMR rectification and spin pumping for dF= 4nm.-0.25-0.2-0.15-0.1-0.05 0 2.2 2.3 2.4 2.5 2.6 Hex (kOe) VDC ( µV) dF=4nm =10nm =40nm YIG(dFnm)|Pt(6nm) FIG. 4. ST-FMR spectra dependence on dFin a trilayer set- up to observe the spin torque induced DC voltages without artifacts of the Oersted field. ( fa= 9GHz and θ= 45◦) may now be observed also for thick magnetic layers. In summary, we predict observable AC current-driven ST-FMR in bilayer systems consisting of a ferromagnetic insulatorsuchasYIG and anormalmetal with spin-orbit interaction such as Pt. Our main results are the DC volt- ages caused by an AC current as a function of in-plane external magnetic field and film thickness of a magnetic insulator. The DC voltages generated in YIG |Pt bilay- ers depend sensitively on the ferromagnet layer thick- ness because of the small bulk Gilbert damping. The predictions can be tested experimentally by ST-FMR- like experiments with a magnetic insulator that would yieldimportantinsightsintothenatureoftheconduction electron spin-magnon exchange interaction and current- induced spin wave excitations at the interface of metals and magnetic insulators. This work was supported by KAKENHI (Grants-in- Aid for Scientific Research) Nos. 22540346, 25247056, 25220910, and 268063, FOM (Stichting voor Funda- menteel Onderzoek der Materie), the ICC-IMR, the EU-RTN Spinicur, EU-FET grant InSpin 612759, and DFGPriorityProgramme1538“Spin-CaloricTransport” (Grant No. BA 2954/1). [1] Recent Advances in Magnetic Insulators - From Spin- tronics to Microwave Applications, edited by M. Wu andA. Hoffmann, Solid State Physics 64(Academic Press,5 2013). [2] 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 mag- netic insulator, Nature 464, 262 (2010). [3] Y. Zhou, H. J. Jiao, Y.-T. Chen, G. E. W. Bauer, and J. Xiao, Current-inducedspin-wave excitation in Pt/YIG bilayer, Phys. Rev. B 88, 184403 (2013). [4] H.Nakayama, M. Althammer, Y.-T. Chen, K. Uchida, Y. Kajiwara, D. Kikuchi, T. Ohtani, S. 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2014-04-09

We formulate a theory of the AC spin Hall magnetoresistance (SMR) in a bilayer system consisting of a magnetic insulator such as yttrium iron garnet (YIG) and a heavy metal such as platinum (Pt). We derive expressions for the DC voltage generation based on the drift-diffusion spin model and quantum mechanical boundary condition at the interface that reveal a spin torque ferromagnetic resonance (ST-FMR). We predict that ST-FMR experiments will reveal valuable information on the current-induced magnetization dynamics of magnetic insulators and AC spin Hall effect.

Current-induced spin torque resonance of magnetic insulators

1404.2360v2

1 Magnetic control of Goos-Hänchen shi fts in a yttrium-iron-garnet film Wenjing Yu,1 Hua Sun,1,3 & Lei Gao1,2,* 1College of Physics, Optoelectronics and Energy of Soochow University, Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, Suzhou 215006, China. 2Jiangsu Key Laboratory of Thin Films, Soochow University, Suzhou 215006, China. 3hsun@suda.edu.cn *leigao@suda.edu.cn Abstract: We investigate the Goos-Hänchen (GH) shifts reflected and transmitted by a yttrium-iron-garnet (YIG) film for both normal and oblique incidence. It is found that the 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 which breaks the symmetry between the reflected and transmitted shifts at oblique incidence. The asymptotic behaviors of the normal-incidence reflected shift are obtained 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 reversal of the shifts from positive to negative values can be realized by tuning the magnitude 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 transmitted shift with a total transmission. Numerical simulations are also performed to verify our analytical results. Introduction: The GH effect refers to the lateral shift of an incident beam of finite width upon reflection from an interface which was first studied by Goos and Hänchen1,2 and theoretically explained by Artmann in terms of the stationary-phase approach in the late 1940s3. Since then, such effect has been very important with development of the laser beams and integrated optics4 and has significant impact on applications as well as for investigations of the fundamental problems in physics. And the studies have been extended from a simple dielectric interface to more complex structures or exotic materials such as metal-dielectric nanocomposites 5,6, epsilon-near-zero metamaterials7, graphene8-10, PT-symmetric medium11, topological insulator12 etc. The GH shift by magneto-optical (MO) materials13-18 is obtained by making use of ferromagnetic resonances of natural magnetic materials. Similar relation between GH effects and intrinsic resonances was also reported in nonmagnetic dielectric, such as GH shifts arising from phonon resonances in crystal quartz 19. But what was found in MO materials is of particular interest because of the nonreciprocity in scattering 2 coefficients originated from the broken time reversal symmetry20. As a result, a lateral shift for reflection will occur at the interface between the vacuum and an magnetic material arranged in the V oigt geometry even at normal incidence 14,15, with both sign and magnitude controlled by the applied magnetic field. And the polarization-dependence of the GH shift by MO materials makes it possible to separate the incident radiation into beams of different polarizations 21. However, the details of the magnetic effects on GH shift are stilled obscure. Most studies only discussed the effects of a semi-infinite antiferromagnetic material —MnF 2 at low temperature ( T=4.2K), with a dispersion quite different from that of conventional MO materials adopted in applications. The ro le of material properties and geometric factors (such as finite slab thickness, incident angles etc.) remains unclarified in the magnetic control of GH shifts with MO materials. Hence we are motivated to perform a theoretical investigation of the GH shifts reflected 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 22-27, YIG has been extensively studied and broadly adopted in microwave28-30 and magneto-optics technologies31-33. The recent realization of one -way waveguides based on YIG photonic crystals sparks even more interest of the application of this traditional MO material in the field of subwavelength optics34,35. It was also shown that hyperbolic dispersion and negative refraction initia lly investigated in antiferromagnetic materials36 can be extended to and realized in conventional ferrites37. But the GH-shift effects due to a surface/slab of YIG have not been studied. In this paper we present a theoretical anal ysis of the lateral shifts of both the reflected beam and the transmitted beam due to a magnetized YIG slab in the V oigt geometry. 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 and transmitted shifts at oblique incidence. The asymptotic behaviors of the normal-incidence reflected shift are obtained in the vicinity of two characteristic frequencies ( rω and cω) corresponding to a minimum reflectivity and a total reflection, respectively. And the coexistence of two types of negative-reflected-shift (NRS) at oblique incidence is discussed. We also investigate two special cases for practical purposes: the reflected shift with a total reflection and the transmitted shift with a total transmission. Analytical expressions of the shifts in these cases are obtained approximately, which is in good agreement with the results from numerical calculations. 3 FIG. 1. Schematic diagram of the structur e in the presence of an external field 0h. The incident plane wave is polarized along the z-direction and propagates along k . Results General formulas. Consider a YIG film of thickness d surrounded by a non-magnetic background medium of (11,εμ) as shown in Fig. 1. For simplicity we set the background medium in Region 1&3 as the vacuum. The magnetic permeability of YIG magnetized along the z -axis is of the tensor form 0 0 00 1ri iri iμμ μμ μ− = (1) where the digonal and off-diagonal permeabi lities follow the typical dispersion of ferrites in the microwave region 0 22 0()1()m ri iωω α ωμωα ω ω+=++− (2a) 22 0()m iiωωμωα ω ω=+− (2b) Here ω is the frequency of the incident light, 0ω, mω are the magnetic resonance frequencies given by 002h ωπ γ= , ( 3 a ) 2ms m ωπ γ= ( 3 b ) with 0h and sm denoting the applied magnetic field and the saturated magnetization, respectively. The material parameters of YIG are chosen as: 32.8 10 GHz/Oeγ−=× , 1800Gausssm= , 14.5ε=35. The damping factor α is quite small and neglected in the following analytic derivations and calculations. However, later in the numerical simulations, we have considered the influence of 4 the realistic damping of YIG material. To find the GH shifts due to such a YIG slab, we start by considering an s-polarized plane wave of angular frequency ω incident from Region 1 at an angle θ. Then the x component of the wave vectors in layers 1, 3 and 2 are given by 1/22 2 13 2,xx ykk kcω =− , ( 4 a ) 1/22 2 2 2 xe f f ykkcωεμ =− , ( 4 b ) with 0sinykk θ = (5) Here ε is the dielectric constant of YIG and effμ is its effective permeability given by14 22() /eff r i rμ μμμ=− , ( 6 ) and 0 / kcω= is the wave number of the incident radiation in the background vacuum. Note that effμ is only used to calculate the “effective” wave vector k 2x in the MO slab, not to replace the slab by an isotropic one. Then the electric fields and the magnetic fields in layers 1, 2 and 3 can be expressed as 11 11 0() 1y xxik y i t ik x ik xAe Be e iω ωμ− −=+ =∇ ×1z 11Ee HE ( 7 ) 22 22 02() 1y xxik y i t ik x ik xAe Be e iω ωμ μ− −=+ =∇ ×2z 22Ee HE (8) 33() () 33 0() 1y xxik y i t i k xd i k xdAe Be e iω ωμ− −− −=+ =∇ ×3z 33Ee HE ( 9 ) Based on the boundary conditions (1 ) llz l zxxEE+== , (1 ) lly l yxxHH+== (1, 2l= ), we obtain the reflection and transmission coefficients as 22* 2 22(2 sin ) || ||xxx ik d ik dAB i k drAe Be−=− ( 1 0 a ) 5 2212 224 || ||xxeff x x ik d ik dkktAe Beμ −−=− ( 1 0 b ) with 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 constant of YIG. When an electromagnetic beam of finite width illuminates the slab at an incident angle 0θ, the lateral shifts of the reflected and transmitted beams can be obtained by the stationary phase method3 0 y yr r ykkdddkϕ ==− ( 1 1 a ) 0 y yt t ykkdddkϕ ==− (11b) where 00sino ykk θ = and rϕ, tϕ are the phase angles of the reflection and transmission coefficients for plane waves, respectively. Note here the lateral shift of the transmitted beam is measured in the same way as that of the reflected beam38. For a transparent YIG slab, effμ and 2xk are both real when the weak absorption of YIG is neglected (i.e. the damping factor α is assumed to be zero). Then the reflected shift derived from Eq. (10)-(11) includes two parts: 00(1) (2) y yy yr yykk kkddddk dkϕϕ ===− − (12) where ()(1) *Arg AB ϕ= while (2)ϕ is the phase angle of the complex variable 22 221 || ||xxik d ik dAe Beξ−=− ( 1 3 ) The transmitted shift is only determined by the yk-dependence of (2)ϕ: 0(2) y yt ykkdddkϕ ==− (14) When 0 g=, we have 12 0rx x A kk Aμ=− ≡ , 12 0rx x B kk Bμ=+ ≡ . The results of the 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 0A and 0B are both real and symmetric reflected and transmitted shifts will appear. Based on the formulas Eq. (12)-(14), we will discuss the behaviors of the shifts at normal incidence and at oblique incidence, respectively, for a MO slab with 0g≠, in the following sections. 6 Normal incidence. When 0g≠, a θ-dependent imaginary part is added to A or B so that 00 ,ey ey AAi g k B Bi g k=− =+ ( 1 5 ) here 0eA and 0eB are real parameters for an “effective” slab where the MO permeability tensor is replaced by the magnetic-field-controlled scalar effμ. Since 22 2 2 22 2 2 22 00 00 ,2ee ee y A BA BA BA B g k−= − += + + , the shift term from (2)ϕ is expected to behave like that of the effective slab when the incident angle approaches zero and finally vanishes at normal incidence. The first term of Eq. (12) is independent of the slab thickness and contributes a non-vanishing reflected shift at normal incidence: ()()(1) 0 0n yr y effkg dddkλ ϕ πμ ε==− = − (16) By combining Eq. (16) with Eq. (2a) and (2b), we obtain the dependence of ()n rd on frequency and magnetic field in the form ()()0 22 2(, )1nm r mdHλω ωωπε ω β ω=−− (17) with 1/2 2 21 11HHεβεε−=+ −−− ( 1 8 ) Here, 00 mshHmω ω≡= is a dimensionless magnetic field reduced by the saturated magnetization of the MO slab. In vicinity of the discontinuity point cmω βω≡ (This discontinuity in the frequency spectrum occurs exactly at the reflection minimum, corresponding to effμ ε=)15, the abrupt transition of ()n rd from negative to positive can be approximated by ()()0 21n rdλ πβεη≈− (19) where c cωωηω−≡ describes a small deviation from cω. Note that the expression of (1)ϕ is identical to that by a semi-infinite MO material in Ref. [14,15]. So Eq. (16)-(19) are also applied to the case of d→∞ , i.e. a semi-infinite YIG interface. 7 FIG. 2. Calculated normal inciden ce (a) GH shift of reflected field /rdλ and (b) reflectivity as a function of frequency (express as /2ωπ ). The red, blue and black curves correspond to 02580,2680,2780Oe h= , respectively. Circles: approximat ed results from Eq. (19); Lines: numerical results. Fig. 2(a) shows the approximated frequency dependence of ()n rd based on Eq. (19) for 02580Oe h= , 2680Oe and 2780Oe (circles). The numerical results (lines) directly from Eq. (10) and (11) are disp layed simultaneously for comparison and good agreement is found even for moderate deviation from the discontinuity point. Since β increases monotonically with H, cω is red-shifted when the applied magnetic field 0h is decreased, accompanied by the enhancement of ()n rd around cω. For a lower field 01000Oeh= , we have ()0.0136~n rdη, which is larger by 1-2 orders of magnitude than the result for MnF 2 at the same applied magnetic field as reported in Ref. [14,15]. For practical purposes, a sufficiently large reflectivity is necessary for the application of reflected shift. Fig. 2( b) shows a typical frequency spectrum of reflectivity of a YIG slab (0 0.3m, 2680Oedh== ), where 2r is quite small around cω but rises rapidly when the frequency approaches the sharp edge of a platform of 21 r=. The rapid oscillation of reflectivity is a typical interference pattern of a slab of finite thickness, which is not exhibited in the spectrum of rd in Fig. 2(a) since the reflected shift is independent of slab thickness. The total-reflection platform at r ff> occurs when the wave vector 2xk in YIG becomes imaginary, which means a negative effμ in the cases of normal incidence ( 0yk=). According to the dispersion relation of iμ and rμ, it is easy to find (1 )rm HH ωω=+ (20) 8 and the frequency dependence of effμ and g can be expressed as () () ()22222 22 22 0(1 )r r eff rHHωωωω μ ωω ωω−−+= −− (21a) 221 (1 )r rg HHωω ωω=− + (21b) Note that at r ωω= , both g and effμ go infinite, but their ratio has a finite value 1 reffg H H ωωμ ==+ (22) Substituting this in to Eq. (16), we obtain the reflected shift at rω ()0()1n rrHdHλωωπ==+ (23) This result tells us the largest ()n rd achievable when 21 r=, which increases monotonically with the reduced magnetic field H up to a strong-field limit: 0λ π. Oblique incidence. When the incident beam is at a certain angle θ, the reflected shift rd and the transmitted shift td caused by a YIG-slab of thickness d can be expressed as (, ) () (, )rdd D dθθ θ =+ Λ (24a) (, ) (, )tdd dθθ =Λ (24b) where the thickness-independent part ()Dθ is according to the first term in Eq. (12), given by () [ () () ]Dg F Fθθ θ+− =+ (25) with 2 12 21 12 22 2 12() () ()()y eff x x eff x x xx eff x x ykkk kkkkFkk g kμμ θμ±±+ ± =±+ (26) The expression of (),dθΛ can be obtained from Eq. (13) and (14) as 9 ()2,1( )y ydG dkdGkθΛ=+ (27) Here we have introduced a function 22 2 22() t a n ( )yxABGk k d AB+= − (28) which can be rewritten as 0 () () 1 ()ye y yGk G k M k =+ (29) where 0()eyGk is the result of ()yGk for a slab of scalar permeability effμ while ()yMk gives the correction term caused by the tensor form of the slab permeability: 22 00 02 22 00() t a n ( )ee ey x eeABGk k dAB+=− (30a) 2 2 22 2 2 0()1( )y y eff eff eff yk gMkkk μμ μ ε=++ − (30b) Note that 22 2(1 )m effmg Hωω μ ωω=+−, hence the condition ()2 2 11effg μ<<+ holds for most frequencies not close to rω in the transparent region r ωω<, and (),dθΛ can be well approximated by the shifts (),re te edd d θ == Λ due to an effective non-MO slab of ( ε,effμ) for the same incident angle θ and slab-thickness d38. 10 FIG. 3. (a1) Reflectivity, (b1) GH shift of reflected field /rdλ and (c1) GH shift of transmitted field /tdλ as functions of the slab thickness (expressed as /dλ) and the incident angle θ for 02780Oe h= , 9.5GHzf= . (2), (3) are the same as (1) but for 9.7GHzf= and 9.736GHz , respectively. The competition between ()Dθ and (),dθΛ leads to the coexistence of two types of NRS at certain frequencies. Fig. 3 and Fig. 4 illustrate the variance of 2r, /rdλ and /tdλ with the incident angle θ and the slab thickness d. The magnetic field 0h is set to be 2780Oe, at which the characteristic frequencies are given by 9.749GHzcf= and 9.991GHzrf= . To one’s interest, both reflectivity and the shifts show 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= , 9.7GHz and 9.736GHz in Fig. 3b and 3c, where region A extends from 0θ= to A θθ= with only slight thickness dependence while region B for B θθ> shows a periodic positive-to-negative transition of rd (and td as well) with thickness varying. 11 FIG. 4. (a), (c) GH shift of transmitted field /tdλ and (b), (d) GH shift of reflected field /rdλ vs the incident angle θ at two certain slab thicknesses: (a, b) 3.03d λ = , (c, d) 3.047d λ = for both the YIG slab and the corresponding effective slab. The incidence frequency is 9.736GHzf= . In Fig. 4(a) and (c), the curves of td vs θ at a certain slab thickness for 9.736GHzf= are presented for both the YIG slab and the corresponding effective slab. It is clearly seen that (),dθΛ can be well approximated by (),eff dθ Λ , which accounts for the transition of td with thickness at larger incident angles. For the reflected shift rd (Fig. 4(b) and (d)), ()Dθ dominates the NRS region at smaller angles and makes a non-negligible correction to the NRS in region B, breaking the symmetry between rd and td which is an important feature of GH shifts due to a non-MO slab38. Two special cases. Asymptotic behaviors of the GH shifts in two special cases of particular interest for applications can be obtained from the general formulas Eq. (24)-(30). The first case is at r ωω=, where total reflection occurs and the reflected shift is only determined by ()Dθ even at oblique incidence. By expanding the function in terms of 0/s i nykkκθ≡= and keeping terms up to the second order, we have 12 ()22 2 33 022 11 22ab g ab g gDka a a aθκ++ −− +− + − −−=+ + + (31) with eff eff an μ±=± and 1 eff effbnμ±=± . Since 1effH gHμ=+ at rω, the asymptotic behavior of rd is given by () 2 H+21+1rn rrddHωωκ==+ (32) where ()n rd is the reflected shift in Eq. (23) at normal incidence. The calculated results from Eq. (32) are illustrated in Fig. 5a in comparison with the numerical results for 01000Oeh= , 2000Oe and 3000Oe . The second case is the transmitted shift accompanied by a 100% transmittivity when the slab thickness satisfies 2 ()xkd m m π=∈ Ζ . According to Eq. (24b), the transmitted shift can be written as 21y tdd kdφ φ=+ (33) with 222 2 00 2 22 002tan( )ee y x eeAB g kkdABφ++=− (34) when 2xkd m π= , we have 0φ= and 222 2 00 22 20 02ye e y yx e ekA B g k d dk k A Bφ ++ =−− (35) Also keeping the first two terms in the expression of td, we obtain the asymptotic behavior of td in this case 2(1 )tdD κ χκ =+ (36) where the coefficients are given by 0 22eff eff effmDnμ ε λ με+= (37) and 22 2 22 231 2eff eff eff eff effng nnμχμ++ −=−+ (38) The transmitted shift will vanish at r ωω= , because of the divergence of effμ at 13 this frequency, and then rise with frequency decreasing. Fig. 5b illustrates the frequency-dependence of td at a certain incident angle ( 30 , 45ooθ= ) for 03000Oeh= when the slab thickness satisfies the total transmission condition. Good agreement is found between the approximated td in Eq. (36) and the numerical results. FIG. 5. (a) GH shift of reflected field /rdλ vs sin[ ]θ for 01000,3000Oeh= at r ff=. (b) GH shift of transmitted field /tdλ vs the frequency at two certain incident angles ( 30 , 45ooθ= ) for 03000Oeh= , 2 10 /x dk π= . The solid lines indicate the numerical results and the square symbol lines correspond to the asymptotic behaviors calculated from Eq. (32) and (36). Numerical simulations To verify the above theoretical analysis, we performed a numerical simulation of a YIG slab illuminated by a Gaussian incident beam with the well-known finite-element analysis software COMSOL Multiphysics. The center of the incident beam arrived at the upper interface of the slab is located at the point (0,0) and the half-width of the beam is 7.5λ. The GH shifts can be directly obtained by comparing the field distributions of the incident beam and the reflected/transmitted beam at the relevant interfaces. Note that the damping of YIG has been neglected in the analytic expressions. In our simulations, a more prac tical dispersion of YIG perm eability will be adopted where the damping factor is set to be 2dH αγ ω =× , with 30Oe dH=35. The low damping (~10-4) implies that no significant absorption effects will occur except for frequencies near ferromagnetic resonance 0 m fHf= . According to Eqs. (17), (18) and (20), the two characteristic frequencies for nonreciprocal GH shifts, cf and rf, will not be close to 0f unless the field 0h is in the strong-field limit 0 s hm>> . At normal incidence the analytical results predict that nonvanishing reflected 14 shift occurs in both the transparent region (r ff<) and the opaque region (r ff>) as shown 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 parameters are chosen to be the same as those for the points A and B in Fig. 2b, namely h 0=2680Oe, d=0.3m, 9.01GHzf= (point A) or 9.72GHzf= (point B). The cases with (black solid lines) an d without (blue soli d lines) damping are both investigated. Table 1 gives the reflected shifts given by analytic expressions, simulations without damp ing and simulations with damping. It is shown that the damping has no significant effect on the shift, and the analytic predictions is in good agreement with the numerical results. Fig. 6. The COMSOL simulation results for reflected shifts at normal incidence. (a) The distribution of electric field amplit ude along the incident interface for 9.01GHzf= ,02680Oe h= and 0.3md= (corresponding to point A in Fig.2b ) ; (b) The distribution of electric field amplit ude along the incident interface for 9.72GHzf=02680Oeh= and 0.3md= (point B in Fig.2b ). The red lines indicate the analytical shift of each case. Table 1: Comparisons between analytical and simulation results Analytical predictions Simulations without damping Simulations with damping Normal incidence f=9.01GHz 0.130rd λ =− 0.134rd λ =− 0.134rd λ =− f=9.72GHz 0.229rd λ = 0.221rd λ = 0.221rd λ = Oblique incidence Total reflection 0.443rd λ = 0.454rd λ = 0.454rd λ = Total transmission 0.241td λ = 0.245td λ = 0.245td λ = At oblique incidence both reflected and transmitted shifts may be observed at 15 certain conditions. Fig.7 gives the simulated results when the incident angle is 45o and the external magnetic field 0h is 3000Oe . The frequency and the slab thickness are chosen to satisfy the conditions for total reflection (r=ff, Fig. 7a and 7c) and total transmission (2xkd m π= , Fig. 7b and 7d), respectively, since these cases are especially interesting for practical applications. Again both the cases with and without damping are investigated and compared with the analytical results as listed in Table 1. Trivial damping effects and good agreement between the analytical and simulation results are found, similar to those at normal incidence. FIG. 7. Numerical simulations of GH shifts when the Gaussian beam is incident from air. (a) The field pattern for 03000Oeh= , r ff= with an incident angle of 45o. (b) The field pattern for 03000Oeh= , 7GHzf= and 2 2/x dk π= with an incident angle of 45o. (c), (d) The distributions of field amplitudes along y direct ion near the interface between YI G and air, based on numerical results in (a) and (b). The red lines indi cate the analytical shift of each case. Conclusions In this paper, we mainly investigate the lateral shifts of a TE wave both reflected and transmitted from a YIG slab theoretically. It is shown that the nonreciprocity effect caused 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 slab-thickness-independent term of rd which breaks the symmetry between the reflected and transmitted shifts which is an important feature of GH shifts due to a 16 non-MO slab. The asymptotic behaviors of the normal-incidence reflected shift are obtained in the vicinity of two characteristic frequencies (rω and cω) corresponding to a minimum reflectivity and a total reflection, respectively. And the coexistence of two types of negative-reflected-shift (NRS ) at oblique incidence is discussed. Numerical results show that the reversal of the sign of GH shifts can be realized by tuning the magnitude of external magnetic field 0h, adjusting the incident wave frequency f or changing the thickness d as well as the incident angle θ. We also investigate two special cases for practical purposes: the reflected shift with a total reflection and the transmitted shift with a tota l transmission. Analytical expressions of the shifts in these two cases are obtaine d approximately, which are in good agreement with the results from numerical calculations. Though nonreciprocal re flected shifts were also reported in antiferromagnetic MnF 216,17, our YIG-based study confirms the possibility of experimental demonstration 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 offers a deeper insight into the role of magnetic field in tuning the shift sign, magnitude and types (reflected or transmitted). Methods Theory and simulations. The numerical simulation results shown in Fig. 6 and Fig. 7 were obtained using the finite elemen t solver COMSOL Multiphysics. The scattering bou ndaries were set for four sides. Based on the numerical simulati on, the curves of field amplitude in Fig. 6 were obtained by performing the line plot along y axis from 4λ− to 4λ. Due to the interference effect, the field amplitudes are oscillating along x di rection. The line plot is located at the first peak close to the interface between air and YIG. Meanwhile, we zoom in the line plot of zE enough to get the distance betw een its symmetric axis and y=0, which indicates the lateral shift rd. The numerical results in Fig. 7 were obtained by the same technique. References 1. 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Tunable Focusing in Natural Hyperbolic Magnetic Media. ACS Photonics 3, 1670 (2016). 37. Lan, C., Bi, K., Zhou, J., & Li, B. Experimental demonstration of hyperbolic property in conventional material-ferrite. Appl. Phys. Lett . 107, 211112 (2015). 38. Li, C. F. Negative Lateral Shift of a Light Beam Transm itted through a Di electric Slab and Interaction of Boundary Effects. Phys. Rev. Lett. 91, 133903 (2003). Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant No. 11374223), the National Science of Jiangsu Province (Grant No. BK20161210), the Qing Lan project, “333” project (Gra nt No. BRA2015353), and PAPD of Jiangsu Higher Education Institutions. Author Contributions L.G. conceived the idea. W.Y . performed mo st theoretical and numerical calculations. W.Y . and H.S. analyzed the data. All authors joined discussion extensively and revised the manuscript before the submission. Additional Information Competing financial interests: The authors declare no competing financial interests.

2017-01-03

We investigate the Goos-Hanchen (G-H) shifts reflected and transmitted by a yttrium-iron-garnet (YIG) film for both normal and oblique incidence. It is found that the 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 which breaks the symmetry between the reflected and transmitted shifts at oblique incidence. The asymptotic behaviors of the normal-incidence reflected shift are obtained 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 reversal of the shifts from positive to negative values can be realized by tuning the magnitude 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 transmitted shift with a total transmission. Numerical simulations are also performed to verify our analytical results.

Magnetic control of Goos-Hanchen shifts in a yttrium-iron-garnet film

1701.01462v2

PERPENDICULARLY BIASED YIG TUNERS FOR THE FERMILAB RECYCLER 52.809 MHZ RF CAVITIES* R. Madrak#, V . Kashikhin, A. Makarov, D. Wildman, FNAL, Batavia, IL 60510, USA Abstract For NOvA and future experiments requiring high intensity proton beams, Fermilab is in the process of upgrading the existing accelerator complex for increased proton production. One such improvement is to reduce the Main Injector cycle time, by performing slip stacking, previously done in the Main Injector, in the now repurposed Recycler Ring. Recycler slip stacking requires new tuneable RF cavities, discussed separately in these proceedings. These are quarter wave cavities resonant at 52.809 MHz with a 10 kHz tuning range. The 10 kHz range is achieved by use of a tuner which has an electrical length of approximately one half wavelength at 52.809 MHz. The tuner is constructed from 3⅛ ″ diameter rigid coaxial line, with 5 inches of its length containing perpendicularly biased, Al doped Yttrium Iron Garnet (YIG). The tuner design, measurements, and high power test results are presented. INTRODUCTION Two new quarter wave copper cavities have been constructed and installed in the Fermilab Recycler Ring for the purpose of slip stacking protons to increase beam Figure 1: Tuner adjustable section with the inner conductor removed from the outer conductor. power. These have been described in detail in [1]. They are to operate at 52.809 MHz with a tuning range of 10 kHz. Tuning by 1260 Hz is required for slip stacking itself, and additional tuning is needed to compensate for temperature changes. The cavity maximum peak gap voltage is 150 kV with a shunt impedance of 75 kΩ and Q ~ 6000. The cavities will be pulsed for a maximum of 0.8 s every 1.33 s (the Main Injector cycle time). The cavities are tuned using perpendicularly biased garnet in a coaxial line. This has been done previously and is described in [2] and [3]. The original concept, though not for cavity tuning in particular, is described in [4]. DESIGN In one of the ports located 3 ″ (on center) from the cavity shorted end, a fast tuner is loopbcoupled to the cavity. The coupling is through a 2.5 ″ diameter coaxial ceramic window. The tuner loop area of 2.5 in2 was adjusted to give a nominal coupling impedance of 50 Ω. The tuner is slightly less than a halfbwavelength long and is made up of standard EIA 3⅛ ″ diameter, 50 Ω rigid copper transmission line, in addition to an “adjustable section” which is partially loaded with a 5 ″ long piece of Al doped Yttrium Iron Garnet (YIG). This section is immersed in a variable solenoidal magnetic field which provides a perpendicular bias for cavity tuning. A photo of the adjustable section, with the center conductor assembly removed from the outer conductor, is shown in Figure 1. The tuner line is shorted at the end opposite the cavity. The garnet (TCI Ceramics type ALb400) has a saturation magnetization (4πM s) of 400 gauss. Its OD and ID are 3.0″ and 0.65″ respectively. The center conductor used in the 15.75 ″ long adjustable section, which does not have the dimensions of standard 3⅛ ″ coax line, is shrink fit into the garnet. Over the 9 ″ length closest to the short, the outer conductor thickness has been reduced from the standard 3⅛ ″ coax line wall thickness to 0.020 ″. Both this 9″ section and the bottom copper shorting plate have a 0.0197″ wide slot machined through the copper to reduce eddy current effects. For heat removal, the outer conductor is water cooled and the adjustable section is filled with a dielectric fluid (Diala AX). The adjustable section had previously been designed for use as a fast phase shifter in vector modulators operating at 325 MHz [5]. The entire tuner, attached to the cavity and installed in the Recycler, is shown in Figure 2. ___________________________________________ *Operated by Fermi Research Alliance, LLC under Contract No. DE b AC02b07CH11359 with the United States Department of Energy. #madrak@fnal.gov FERMILAB-CONF-13-363-APCSolenoid The 112 turn solenoid (with TCI Ceramics G4 ferrite flux return) was designed to work at relatively high DC current (80A), but also at high frequency (~2 kHz) AC current. While the AC response is not absolutely required, the ability to tune at ~2 kHz will be an advantage. Obtaining the desired DC and AC response necessitated the use of Litz Cable of sufficient cross section (1/0 AWG) with 270 copper strands of 24 AWG wire insulated from each other. The total strand cross section reduces the losses in the DC regime, and the insulation between strands minimizes the eddy currents in the cable (for the AC regime). A cooling tube was wound on the outer surface of the solenoid as a bifilar coil in order to eliminate the effect of magnetic fields caused by eddy currents induced in it. A DC magnetic field simulation predicted the resulting minimum and maximum magnetic fields in the tuner garnet of 0.058 T and 0.081 T, for 8000 solenoid amperebturns. The solenoid current is supplied by a Copley Controls Model 266 amplifier which has a nominal bandwidth of Figure 2: Tuner attached to the cavity, installed in the Recycler. The tuner is made up several 3⅛ ″ coax sections: an elbow, directional coupler, several straight sections, and finally the YIG loaded adjustable section (here obscured by the solenoid) which is shown in Figure 1. 5 kHz. The tuner response is currently limited by the solenoid inductance of 1.9 mH and the 60V max/80 A max DC supply for the Copley amplifier. TUNER PERFORMANCE AND MEASUREMENTS The tuner is characterized apart from the cavity by the measurements plotted in Figure 3. This shows network analyzer S 11 amplitude and phase measurements as a function of DC solenoid bias. Large phase shifts Figure 3: Twobway phase shift in tuner 15.75 ″ adjustable section at 52.809 MHz (left ybaxis) and Loss (right yb axis) as a function of solenoid bias. correspond to a large frequency change (tuning) in the cavity/tuner system. Operation must be above ~35A, below which the garnet becomes lossy. The next set of measurements quantifies the operation of the cavity/tuner system as a whole. Figure 4 (wide range) and Figure 5 (operation range) show the frequency (left ybaxis) and the Q of the cavity/tuner system (right ybaxis) as a function of solenoid bias current. The bias at which we observe a large change in frequency and a minimum in Q corresponds to the case where the tuner is one half wavelength long. Figure 4: Frequency and Q of the cavity/tuner system as a function of solenoid bias current, for a wide range of tuner biases. An additional restriction on tuner range is due to the peak voltage that can be sustained in the tuner without sparking. Figure 6 shows low level measurements of frequency and the corresponding tuner voltage. In this case, the cavity was powered with a relatively small signal. Cavity peak accelerating gap voltage and tuner peak voltage were measured simultaneously. The values were then scaled to cavity gap Figure 5: Frequency and Q of the cavity/tuner system as a function of solenoid bias current, for solenoid biases in the range that the tuner is operated ( > 35 A). Figure 6: Tuner voltage and frequency of cavity/tuner system as a function of tuner bias. The maximum peak voltage for the tuner 3⅛ ″ line is also shown. (These measurements were taken from the cavity field probe and the directional coupler which is part of the tuner.) voltages of 100 and 150 kV. Operating with a tuner voltage above the peak voltage limit for 3⅛ ″ coaxial line of 13.3 kV will cause arcing and must be avoided. Not only is arcing in the coax line undesirable, but this can also lead to high voltages near the garnet which could be destructive. Measurements shown indicate I ≥ 32 A and I ≥ 40 A to respect the 13.3 kV limit, for cavity voltages of 100 kV and 150 kV. With no other adjustments, and taking into account the 35 A limit imposed due to losses in the garnet, the tuning ranges are 11.7 kHz and 7.5 kHz for cavity peak voltages of 100 kV and 150 kV. TUNER SPARK PROTECTION To avoid destruction of the garnet, several methods of protection are in place to shut off RF power to the cavity in the event of a tuner spark. The tuner line contains a 60 dB directional coupler with both forward and reverse power taps. The first tuner protection system shuts off the RF drive if a change of more than 3dB/ µs is detected in the reverse power. In the second protection system, a phase detector monitors the phase difference between forward and reverse power. If the phase changes more quickly than that due to normal tuning, the RF drive is again shut off. This is accomplished using a capacitive differentiator and a comparator. STATUS AND HIGH POWER TESTING Before being installed in the Recycler, both cavities were high power tested in a shielded “cavity test cave”. The cavity and tuner systems have been operated successfully at high power in the Recycler. ACKNOWLEDGMENTS We would like to acknowledge the huge efforts of the Accelerator Division Mechanical Support and RF Groups, and everyone involved in this project. In addition, we would like to thank all of those in Technical Division involved in the design and construction of the solenoids. REFERENCES [1] R. Madrak and D. Wildman, “Design and High Power Testing of 52.809 MHz RF Cavities for Slip Stacking in the Fermilab Recycler”, WEPMA013, these proceedings. [2] S. M. Hanna et.al, “YIG Tuners for RF Cavities”, IEEE Transactions on Magnetics, Vol 28, No. 5, p. 3210, Sept 1992. [3] R. L. Porier, “Perpendicular Biased Ferrite –Tuned Cavities”, PAC’93, Washington, D. C., May 1993, p. 753, (1993). [4] A. S. Boxer , S. Hershenov and E. F. Landry "A HighbPower Coaxial Ferrite Phase Shifter", IRE Trans. Microwave Theory and Techniques (Correspondence) , vol. MTTb9, pp.577, 1961. [5] R. Madrak and D. Wildman, “High Power 325 MHz Vector Modulators for the Fermilab High Intensity Neutrino Source (HINS)”, LINAC’08, Victoria, B.C., Oct 2008, THP088.

2014-09-19

For NOvA and future experiments requiring high intensity proton beams, Fermilab is in the process of upgrading the existing accelerator complex for increased proton production. One such improvement is to reduce the Main Injector cycle time, by performing slip stacking, previously done in the Main Injector, in the now repurposed Recycler Ring. Recycler slip stacking requires new tuneable RF cavities, discussed separately in these proceedings. These are quarter wave cavities resonant at 52.809 MHz with a 10 kHz tuning range. The 10 kHz range is achieved by use of a tuner which has an electrical length of approximately one half wavelength at 52.809 MHz. The tuner is constructed from 3 1/8 inch diameter rigid coaxial line, with 5 inches of its length containing perpendicularly biased, Al doped Yttrium Iron Garnet (YIG). The tuner design, measurements, and high power test results are presented.

Perpendicularly Biased YIG Tuners for the Fermilab Recycler 52.809 MHz Cavities

1409.5762v1

arXiv:1607.02312v2 [cond-mat.mtrl-sci] 27 Oct 2016Magnon Polarons in the Spin Seebeck Effect Takashi Kikkawa,1,2,∗Ka Shen,3Benedetta Flebus,4Rembert A. Duine,4,5Ken-ichi Uchida,1,6,7,†Zhiyong Qiu,2,8Gerrit E. W. Bauer,1,2,3,7and Eiji Saitoh1,2,7,8,9 1Institute for Materials Research, Tohoku University, Send ai 980-8577, Japan 2WPI Advanced Institute for Materials Research, Tohoku Univ ersity, Sendai 980-8577, Japan 3Kavli Institute of NanoScience, Delft University of Techno logy, Lorentzweg 1, 2628 CJ Delft, The Netherlands 4Institute for Theoretical Physics and Center for Extreme Ma tter and Emergent Phenomena, Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Neth erlands 5Department of Applied Physics, Eindhoven University of Tec hnology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands 6PRESTO, Japan Science and Technology Agency, Saitama 332-0 012, Japan 7Center for Spintronics Research Network, Tohoku Universit y, Sendai 980-8577, Japan 8Spin Quantum Rectification Project, ERATO, Japan Science an d Technology Agency, Sendai 980-8577, Japan 9Advanced Science Research Center, Japan Atomic Energy Agen cy, Tokai 319-1195, Japan (Dated: October 28, 2016) Sharp structures in the magnetic field-dependent spin Seebe ck effect (SSE) voltages of Pt/Y3Fe5O12at low temperatures are attributed to the magnon-phonon int eraction. Experimental results are well reproduced by a Boltzmann theory that inclu des magnetoelastic coupling. The SSE anomalies coincide with magnetic fields tuned to the thresho ld of magnon-polaron formation. The effect gives insight into the relative quality of the lattice and magnetization dynamics. PACS numbers: 85.75.-d, 72.25.-b, 75.80.+q, 72.25.Mk The spin Seebeck effect (SSE) [1–19] refers to the gen- eration of a spin current ( Js) as a result of a tempera- ture gradient ( ∇T) in magnetic materials. It is well es- tablished for magnetic insulators with metallic contacts, at which a magnon flow is converted into a conduction- electron spin current by the interfacial exchange interac- tion [20] and detected as a transverse electric voltage via the inverse spin Hall effect (ISHE) [21–27] [see Fig. 1(a)]. The SSE provides a sensitive probe for spin correlations in magnetic materials [8, 9, 12–15]. The ferrimagnetic insulator yttrium-iron-garnet Y3Fe5O12(YIG) is ideal for SSE measurements [19], exhibiting a long magnon-propagation length [28–30], high Curie temperature ( ∼560 K) [31], and high resistivity owing to a large band gap ( ∼2.9 eV) [32]. The magnon and phonon dispersion relations in YIG are well known [33–38]. The magnon dispersion in the relevant regime reads ωk=/radicalbig Dexk2+γµ0H/radicalBig Dexk2+γµ0H+γµ0Mssin2θ, (1) whereω,k,θ,γ, andµ0Ms, are the angular frequency, wave vector kwith length k, angleθwith the exter- nal magnetic field H(of magnitude H), gyromagnetic ratio, and saturation magnetization, respectively [33– 36]. The exchange stiffness coefficient Dexas well as the transverse-acoustic (TA) and longitudinal-acoustic (LA) sound velocities for YIG are summarized in Table I and the dispersion relations are plotted in Fig. 1(b). In this Letter, we report the observation of a resonant enhancement of the SSE. The experimental results are well reproduced by a theory for the thermally induced TA phonon magnon LA phonon magnetization V LTLVLW EISHE ∇T x zy HJs(a) (b) magnon polaron μ0H = 1.0 T magnon LA phonon TA phonon k (10 8 m-1 )ω/2 π (THz) 10 0 51.0 0.50.56 0.580.0350 0.0355 0.2730 0.2750 ω/2 π (THz) 4.44 4.46 k (10 8 m-1 )(c) (d)k1 k2 FIG. 1: (a) The longitudinal SSE in the Pt/YIG/GGG sample, where EISHEdenotes the electric field induced by the ISHE. The close-up of the upper (lower) right shows a schematic illustration of a propagating magnon and TA (LA) phonon. (b) Magnon [Eq. (1) with µ0Ms= 0.2439 T, µ0H= 1.0 T, and θ=π/2], TA-phonon ( ω=c⊥k), and LA-phonon ( ω=c||k) dispersion relations for the parameters in Table I. (c),(d) Magnon polarons at the (anti)crossings b e- tween the magnon and TA-phonon branches at (c) lower and (d) higher wave numbers, where k/bardblˆx(θ=π/2 andφ= 0) andH/bardblˆz. magnon flow in which the magnetoelastic interaction is2 T = 50 K 5 -5 10 -10 005.0 -5.0 V (μ V) μ0H (T)∆T = 1.73 K -5.2 -4.8 5.2 4.8 10 -10 0wave vectorfrequency 0 0 0TA phonon magnonpoint touch wave vector wave vector -5 0 5-5.005.0V (μ V)T = 50 K(b) (d) 00.2 -0.2 5.0 5.2 4.8 4.6 V (μ V) 2.0 2.5 3.0 ∆T = 1.73 K ∆T = 0 K μ0H (T)(a) T = 50 Kat 0.1 Tat HTA ∆T = 1.73 K(c) 1.0 2.0 02.04.06.0 ∆T (K)V (μ V) (h) 00.2 -0.2 5.0 5.2 4.8 4.6 V (μ V) 9.0 9.5 10.0 ∆T = 1.73 K ∆T = 0 K μ0H (T)T = 50 Kpoint touch TA phonon magnon LA phonon wave vectorfrequency 0 0 0wave vector wave vector(e) 1.0 2.0 02.0 4.0 6.0 ∆T (K)V (μ V)(g) at HLA H = HTA H < HTA H > HTA H = HLA H < HLA H > HLA (f) μ0H (T)2.6 T 9.3 T FIG. 2: (a) Magnon and TA-phonon dispersion relations for YI G when H < H TA,H=HTA, andH > H TA. (b)V(H) of the 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 µ0H= 0.1 T and µ0HTA. (d) Magnified view of V(H) around HTA. (e) Magnon, TA-phonon, and LA-phonon dispersion relations 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 T= 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 Pt/YIG/GGG sample at H=HLA. (h) Magnified view of V(H) around HLA. TheVpeaks at HTAandHLAare marked by blue and red triangles, respectively. taken into account. We interpret the experiments as evidence for a strong magnon-phonon coupling at the crossings between the magnon and phonon dispersion curves, i.e., the formationofhybridized excitations called magnon polarons [40, 41]. The sample is a 5-nm-thick Pt film sputtered on the (111) surface of a 4- µm-thick single-crystalline YIG film grown on a single-crystalline Gd 3Ga5O12(GGG) (111) substrate by liquid phase epitaxy [42]. The sample was then cut into a rectangular shape with LV= 4.0 mm (length),LW= 2.0 mm (width), and LT= 0.5 mm (thickness). SSE measurements were carried out in a longitudinal configuration [1, 19] [see Fig. 1(a)], where the temperature gradient ∇Tis applied normal to the interfaces by sandwiching the sample between two sap- phire plates, on top of the Pt layer (at the bottom of the GGG substrate) stabilized to TH(TL) with a tempera- ture difference ∆ T=TH−TL(>0). ∆Twas measured with two calibrated Cernox thermometers. A uniform magnetic field H=Hˆ zwas applied by a superconducting TABLE I: Parameters for the magnon and phonon dispersion relations of YIG [34–39]. Symbol Value Unit Exchange stiffness Dex7.7×10−6m2/s TA-phonon sound velocity c⊥3.9×103m/s LA-phonon sound velocity c||7.2×103m/ssolenoidmagnet. Wemeasuredthedcelectricvoltagedif- ferenceVbetween the ends of the Pt layer with a highly resolved field scan, i.e., at intervals of 15 mT and waiting for∼30 sec after each step. Figure 2(b) shows the measured V(H) of the Pt/YIG sample atT= 50 K. A clear signal appears by applying the temperature difference ∆ Tand its sign is reversed when reversing the magnetization. The magnitude of V atµ0H= 0.1 T is proportional to ∆ T[see Fig. 2(c)]. These results confirm that Vis generated by the SSE [19]. Owing to the high resolution of H, we were able to resolve a fine peak structure at µ0H∼2.6 T that is fully reproducible. A magnified view of the V-Hcurve is shown in Fig. 2(d), where the anomaly is marked by a blue triangle. Since the structures scale with ∆ T[see Figs. 2(c) and 2(d)], they must stem from the SSE. The peak appears for the field HTAat which accord- ing to the parameters in Table I the magnon dispersion curve touches the TA-phonon dispersion curve. By in- creasingH, the magnon dispersion shifts toward high frequencies due to the Zeeman interaction ( ∝γµ0H), while the phonon dispersion does not move. At µ0H= 0, the magnonbranchintersectsthe TA-phononcurvetwice [see Fig. 2(a)]. With increasing H, the TA-phonon branch becomes tangential to the magnon dispersion at µ0H= 2.6 T and detaches at higher fields [see Fig. 2(a)]. If the anomaly is indeed linked to the “touch” condition, there should be another peak associated with the LA- phonon branch. Based on the parameters in Table I, we evaluated the magnon −LA-phonon touch condition3 atµ0HLA∼9.3 T. We then upgraded the equipment with a stronger magnet and subsequently investigated the high-field dependence of the SSE. Figure2(f)showsthedependence V(H)ofthePt/YIG sample atT= 50 K, measured between µ0H=±10.5 T. Indeed, another peak appeared at µ0HLA∼9.3 T pre- cisely at the estimated field value at which the LA- phonon branch touches the magnon dispersion [see Fig. 2(e)], sharing the characteristic features of the SSE; i.e., it appears only when ∆ T∝negationslash= 0 and exhibits a linear-∆ T dependence [see Figs. 2(g) and 2(h)]. For µ0H >9.3 T theV-Hcurves remain smooth. We carried out systematic measurements of the tem- perature dependence of the SSE enhancement at HTA andHLA. Figure 3(c) shows the normalized SSE volt- ageS≡(V/∆T)(LT/LV) as a function of Hfor various average sample temperature Tavg[≡(TH+TL)/2]. The amplitudeoftheSSEsignalmonotonicallydecreaseswith decreasingTin the present temperature range [8, 9] [see Fig. 3(b)]. Importantly, the two peaks in SatHTAand HLAexhibitdifferent Tdependences[seeFigs. 3(c), 3(d), and 3(e)]. The peak shape at HTAbecomes more promi- nent with decreasing Tand it is the most outstanding at the lowestT. On the other hand, the Speak atHLAis suppressed below ∼10 K and it is almost indistinguish- able at the lowest T. This different Tdependence can be attributed to the different energy scale of the branch crossing point for H=HTAandH=HLA. The fre- quency of the magnon −LA-phonon intersection point is 0.53 THz = 26 K ( ≡TMLA), and it is more than 3 times larger than that of the magnon −TA-phonon intersection point (0.16 THz). Therefore, for T < T MLA, the exci- tation of magnons with energy around the magnon −LA- phonon intersection point is rapidly suppressed, which leads to the disappearance of the Speak atHLAat the lowestT. The clear peak structures at low temperatures allow us to unravel the behavior of the SSE around HTAin detail. Increasing Hfrom small values, Sincreases up to a maximum value at H=HTA, as shown in Fig. 3(d) (Tavg= 3.46 K). For fields slightly larger than HTA, S drops steeply to a value below the initial one. The SSE intensityS(i), wherei(= 0,1,2) represents the number of crossing points between the magnon and (TA-)phonon branch curves [see also Fig. 2(a)], can be ordered as S(1)>S(2)>S(0) and could be a measure of the num- ber of magnon polarons. The SSE is generated in three steps: (i) the tem- perature gradient excites magnetization dynamics that (ii) at the interface to the metal becomes a particle spin current and (iii) is converted to a transverse volt- age by the ISHE. The latter two steps depend only weakly on the magnetic field. For thick enough sam- ples, the observed anomalies in the SSE originate from the thermally excited spin current in the bulk of the fer- romagnet. The importance of the magnetoelastic cou--0.40 -0.20 00.20 0.40 -0.20 -0.10 00.10 0.20 34.38 K -0.10 00.10 16.24 K -0.10 00.10 13.22 K -0.0500.0510.48 K -0.04-0.0200.020.046.30 K -10 0 10-0.0200.023.46 K0.30 0.32 0.30 0.320.34 0.160.18 0.180.20 0.050.060.07 0.060.070.08 0.040.050.06 0.050.060.07 0.030.04 0.040.05 0.020.03 0.020.030.04 2.0 2.5 3.0 00.010.02 9.0 9.5 10.0 00.010.02Tavg = 49.61 K μ0H (T) 5 -5 μ0H (T) μ0H (T) S (μ V/K )(c) (d) (e) (a) 0 20 40 600.10.20.30.4S (μ V/K )at 0.1 T at HTA at HLA (b) Tavg (K) 0.5 LA TA magnon point touch point touch 0 0 0 0ω/2 π k k k kHTA < H < HLA H = HLA H = HTA H < HTA FIG. 3: (a) Magnon, TA-phonon, and LA-phonon dispersion relations for YIG when H < H TA,H=HTA,HTA< H < HLA, andH=HLA. (b)Tavgdependence of the normalized SSE voltage SatH= 0.1 T,HTA, andHLA. (c)S(H) of the Pt/YIG/GGG sample for various values of Tavgin the range of|H|<10.5 T. (d),(e) A blowup of S(H) around (d) HTA and (e)HLA. pling (MEC) for spin transport in magnetic insulators has been established by spatiotemporally resolvedpump- and-probe optical spectroscopy [41, 43]. Here we de- velop a semiclassical model for the SSE in the strongly coupled magnon-phonon transport regime [40, 41, 44– 46]. Our model Hamiltonian consists of magnon ( Hmag), phonon ( Hel), and magnetoelastic coupling ( Hmec) terms. In second-quantized form Hmag=/summationtext kAka† kak+ (Bk/2)(a† ka† −k+a−kak),Hel=/summationtext k,µ/planckover2pi1ωµk/parenleftBig c† µkcµk+1 2/parenrightBig , andHmec=/planckover2pi1nB⊥(γ/planckover2pi1 4Msρ)1/2/summationtext k,µkω−1/2 kµe−iφak(cµ−k+ c† µk)×(−iδµ1cos2θ+iδµ2cosθ−δµ3sin2θ) +h.c.. In spherical coordinates the wave vector k= k(sinθcosφ,sinθsinφ,cosθ),Ak//planckover2pi1=Dexk2+γµ0H+ (γµ0Mssin2θ)/2, andBk//planckover2pi1= (γµ0Mssin2θ)/2. Here,4 a† k(c† µk) andak(cµk) are magnon (phonon) creation and annihilation operators, respectively. B⊥is the magne- toelastic couplingconstant, ρis the averagemassdensity, n= 1/a3 0isthe numberdensity ofspins, and a0is the lat- tice constant. The magnondispersionfrom Hmagis given by Eq. (1), while the phonon dispersions are ωµk=cµk withµ= 1,2 for the two transverse modes and µ= 3 for the longitudinal one. δµiinHmecrepresents the Kro- neckerdelta. Bydiagonalizing Hmag+Hel+Hmec[47], we obtain the dispersion relationof the i-th magnon-polaron branch/planckover2pi1Ωikand the correspondingamplitude |ψik∝angb∇acket∇ight. The magnon-polaron dispersions for θ=π/2 andφ= 0 are illustrated in Figs. 1(c) and 1(d), with a magnetic field µ0H= 1.0 T andB⊥/(2π) = 1988 GHz [38]. We assume diffuse transport that at low tempera- tures is limited by elastic magnon and phonon impu- rity scattering [45]. We employ the Hamiltonian Himp=/summationtext µ/summationtext k,k′c† µkvph k,k′cµk′+/summationtext k,k′a† kvmag k,k′ak′, where, assum- ings-wave scattering, vph k,k′=vphandvmag k,k′=vmagde- note the phonon and magnon impurity scattering poten- tials, respectively. We compute the spin current driven by a temperature gradient [6, 16] and thereby the SSE in the relaxation-time approximation of the linearized Boltzmann equation. The linear-response steady-state spin current Js(r) =−ζ· ∇Tis governed by the SSE tensorζ: ζαβ=/integraldisplayd3k (2π)3/summationdisplay iWs ikτik(∂kαΩik)(∂kβΩik)∂Tf(0) ik|T=T(r). (2) HereWs ik=|∝angb∇acketleft0|ak|ψik∝angb∇acket∇ight|2is the intensity of the i- th magnon-polaron and τikis the relaxation time to- wards the equilibrium (Planck) distribution function f(0) ik(r) = (exp( /planckover2pi1Ωik/(kBT(r)))−1)−1. The relax- ation time τikof thei-th magnon-polaron reads τ−1 ik= (2π//planckover2pi1)/summationtext jk′|∝angb∇acketleftψjk′|Himp|ψik∝angb∇acket∇ight|2δ(/planckover2pi1Ωik−/planckover2pi1Ωjk′). The strong-coupling(weak scattering) approachis valid when τ−1 ik1,2≪∆Ω, where ∆Ω is the energy gap at the anti- crossing points k1,2. We disregard the Gilbert damping that is very small in YIG. From the experiments we infer the scattering param- eters|vmag|2= 10−5s−2[28] and/vextendsingle/vextendsinglevmag/vph/vextendsingle/vextendsingle= 10, i.e., the magnons are more strongly scattered than the phonons. The computed longitudinal spin Seebeck coeffi- cient(SSC) ζxx[Eq.(2)]isplottedinFig.4(a). Switching on the magnetoelastic coupling increases the SSC espe- cially at the “touching” magnetic fields HTAandHLA. At these points the group velocity of the magnon is iden- tical to the sound velocity. Nevertheless, spin transport can be strongly modified when the ratio/vextendsingle/vextendsinglevmag/vph/vextendsingle/vextendsingledif- fers from unity. The SSC can be enhanced or suppressed compared to its purely magnonic value. A high acous- tic quality as implied by/vextendsingle/vextendsinglevmag/vph/vextendsingle/vextendsingle= 10 is beneficial for spintransportandenhancesthe SSCbyhybridization, as illustrated by Fig. 4(a). When magnon and phonon scat-2 2.5 30.290.3149.61 K 9 9.5 100.30 0.32 2 2.5 300.20.4 9 9.5 1000.20.42 2.5 300.20.4 9 9.5 1000.4 2 2.5 30.040.0613.22 K 9 9.5 100.050.07 2 2.5 300.20.4 9 9.5 1000.20.42 2.5 300.023.46 K 9 9.5 1000.023.46 K 0 5 10 0510 50 K 50 K 13 K 13 K 3.5 K 3.5 K0.2S (μ V/K ) ζxx ( 10 22 m-1 s-1 K-1 )T = 13 K μ0H (T) (a) (b) (c) 13.22 K49.61 K μ0H (T) μ0H (T) Experiment Calculation ζxx ( 10 22 m-1 s-1 K-1 ) ×10 HTA HLA with MEC |v mag /v ph | = 10 without MECwith MEC |v mag /v ph | = 1 Enhancement FIG. 4: (a) Calculated SSC ζxxatT= 13 K as a function ofH,with (red solid curve and blue circles) and without (red dashed curve) magnetoelastic coupling (MEC). The red solid curve and the blue circles are computed for ratios of the scat - tering potentials of/vextendsingle/vextendsinglevmag/vph/vextendsingle/vextendsingle= 10 and/vextendsingle/vextendsinglevmag/vph/vextendsingle/vextendsingle= 1, re- spectively. The bluedashed curveis ablowup ofthedifferenc e between the red solid and dashed curves. (b) Experimental Sand (c) theoretical ζxxafter subtraction of the zero MEC results. tering potentials would be the same, i.e.,/vextendsingle/vextendsinglevmag/vph/vextendsingle/vextendsingle= 1, the anomalies vanish identically [see the blue circles in Fig. 4(a)]. The difference between the calculations with andwithout MECagreesverywell with the peakfeatures on top of the smooth background as observed in the ex- periments, see Figs. 4(b) and 4(c). We can rationalize the result by the presence of a magnetic disorder that scatters magnons but not phonons. Finally, we address the SSE background signal. The overalldecrease of the calculated ζxxis not related to the phonons, but reflects the field-induced freeze-out of the magnons (that is suppressed in thin magnetic films [8]). Intheexperiments, ontheotherhand, theglobal Sbelow ∼30 K clearly increases with increasing H[Fig. 3(c)]. Wetentativelyattributethisdiscrepancytoanadditional spin current caused by the paramagnetic GGG substrate that, when transmitted through the YIG layer, causes an additional voltage. Wu et al.[7] found a paramag- netic SSE signal in a Pt/GGG sample proportional to the induced magnetization ( ∼a Brillouin function for spin 7/2) [7]. Indeed, the increase of Sin the present Pt/YIG/GGG sample is of the same order as the para- magnetic SSE in a Pt/GGG sample [8]. In conclusion, we observed two anomalous peak struc- tures in the magnetic field dependence of the spin See- beck effect (SSE) in Pt/Y 3Fe5O12(YIG) that appear at5 the onset of magnon-polaron formation. The experimen- tal results are well reproduced by a calculation in which magnons and phonons are allowed to hybridize. Our re- sults show that the SSE can probe not only magnon dy- namics but also phonon dynamics. The magnitude and shape of the anomalies contain unique information about the sample disorder, depending sensitively onthe relative scattering strengths of the magnons and phonons. The authors thank S. Daimon, J. Lustikova, L. J. Cor- nelissen, and B. J. van Wees for valuable discussions. This work was supported by PRESTO “Phase Inter- faces for Highly Efficient Energy Utilization” from JST, Japan,Grant-in-AidforScientificResearchonInnovative Area “Nano Spin Conversion Science” (No. 26103005, 26103006), Grant-in-Aid for Scientific Research (A) (No. 15H02012, 25247056) and (S) (No. 25220910) from MEXT, Japan, NEC Corporation, The Noguchi Insti- tute, the Dutch FOM Foundation, EU-FET Grant In- Spin 612759, and DFG Priority Programme 1538 “Spin- Caloric Transport” (BA 2954/2). 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2016-07-08

Sharp structures in magnetic field-dependent spin Seebeck effect (SSE) voltages of Pt/Y$_{3}$Fe$_{5}$O$_{12}$ (YIG) at low temperatures are attributed to the magnon-phonon interaction. Experimental results are well reproduced by a Boltzmann theory that includes the magnetoelastic coupling (MEC). The SSE anomalies coincide with magnetic fields tuned to the threshold of magnon-polaron formation. The effect gives insight into the relative quality of the lattice and magnetization dynamics.

Magnon Polarons in the Spin Seebeck Effect

1607.02312v2

1 Picosecond acoustic excitation driven ultrafast magnetization dynamics in dielectric Bi-substituted yttrium iron garnet Marwan Deb1*, Elena Popova2, Michel Hehn1, Niels Keller2, Stéphane Mangin1, Gregory Malinowski1 1Institut Jean Lamour (IJL), CNRS UMR 7198, Université de Lorraine, 54506 Vandœuvre- lès-Nancy, France. 2Groupe d’Etude de la Matière Condensée (GEMaC ), CNRS UMR 8635, Université de Paris- Saclay, 78035 Versailles, France. Abstract: Using femtosecond optical pulses, we have investigated the ultrafast magnetization dynamics induced in a dielectric film of bismuth-substituted yttrium iron garnet (Bi-YIG) buried below a thick Cu/Pt metallic bilayer. We show that exciting the sample from Pt surface launches an acoustic strain pulse propagating into the garnet film. We discovered that this strain pulse induces a coherent magnetization precession in the Bi-YIG at the frequency of the ferromagnetic resonance. The observed phenomena can be explain by strain-induced changes of magnetocristalline anisotropy via the inve rse magnetostriction effect. These findings open new perspectives toward the control of the ma gnetization in magnetic garnets embedded in complex heterostructure devices. PACS numbers: 75.78.Jp, 75.50.Gg, 72.80.Sk, 78.20.Ls. *Corresponding author: marwan.deb@univ-lorraine.fr 2 I. Introduction: The experimental discovery of the subpicosecond demagnetization of Ni films following the excitation by 60 femtosecond optical pulse [1] has opened a new and rapidly growing research 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. To this end, intense researches are being ca rried out to investigate laser-induced ultrafast magnetic process and understanding the fundamental mechanism behind their excitation. It was demonstrated that ultrashort optical pulses can trigger in magnetic materials various important magnetic processes, including magnetic phase transition [4-6], magnetization switching [7,8], as well as a coherent spin precession [9,10]. In metallic materials, thermal effects resulting from the energy absorbed by th e medium play a crucial role in laser-induced magnetic phenomena [11]. On the other hand, the possibility of controlling the spin precession in dielectric with light was demo nstrated via non-thermal mechanisms like the inverse Faraday effect [10,12]. Recently, the field of femtomagnetism started investigating alternative ways to control the magnetization using other ultrashort stimuli in cluding hot-electron pulses [13-15], Terahertz pulses [16,17], as well as acoustic pulses [18-20]. This is due to two main reasons. The first is to improve the fundamental understanding in highly debated issues related to ultrafast magnetic phenomena induced by optical pulses like the ultrafast demagnetization [13,21] and reversal [14,15]. The second is to discover a versatile tool that allows non-thermal and ultrafast control and reversal of the magnetization in metal, semiconductor, and dielectric films and heterostructures. In particular, for the latter reason, the use of acoustic pulse can offers at least two advantages. Firstly, acoustic pulses can produce a high mechanical stress of several GPa [22]. As a result, a large modifi cation of lattice parameter occurs and the magnetization can be therefore non-thermally ch anged via the inverse magnetostriction effect [18,19]. Secondly, acoustic pulses have a larg e propagation distance of several millimeters with low energy dissipation [23,24]. Consequently, they provide opportunity for non-thermal manipulation of spins in films deeply embedded in opaque hetorestructure devices. In 2010, the control of the magnetization dynamics by a picosecond acoustic pulse was demonstrated by Scherbakov et al. in GaMnAS ferromagnetic semiconductor [18]. Later, acoustically- induced 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 pulse to trigger a magnetization dynamics in magnetic dielectrics. 3 In this paper we present the results of an experimental study exploring the laser induced ultrafast magnetization dynamics in bismuth-substituted yttrium iron (Bi-YIG) garnets buried below a thick Cu/Pt metallic bilayer. By exciting the Pt/Cu/Bi-YIG trilayers from Pt surface, we 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 of ferromagnetic resonance. The obtained resu lts are explained by strain-induced change of magnetocrystalline anisotropy via the invers e magnetostriction effect. In addition, we demonstrate that we can control the magnetization precession amplitude by tuning the amplitude of the acoustic pulse. The paper is organized as follows. First, we describe in Sec. II the experimental methods and the static magnetic and magneto-optical properties of the sample. Then, we present and discuss in Sec. III the experimental results of the time-resolved magneto-optical and reflectivity measurements as a function of the external magnetic field and the laser energy density. Finally, we summarize in Sec. IV our findings. II. SAMPLE PROPERTIES AND EXPERIMENTAL METHODS: Bismuth-substituted yttrium iron garnet (Bi xY1-xFe5O12, Bi-YIG) materials have the cubic Ia- 3d space group, which is characterized by thr ee different crystallographic sites (octahedral 16a, tetrahedral 24d, and dodecahedral 24c) formed by the oxygen atoms [27]. The non- magnetic Bi and Y atoms occupy the dodecahedral 24c sites, while the magnetic Fe atoms are distributed between the octahedral 16a and tetr ahedral 24d sites. This two Fe sublattices are nonequivalent and coupled by a strong antiferroma gnetic superexchange interaction, leading to a ferrimagnetic state with high Curie temperature (T C > 550 K). These materials have attracted a great deal of attention due to their fascinating proprieties at room temperature, such as the good transparency in the infrared and visible spectrum (gap ~ 2.5 eV) and the very large magneto-optical (MO) Faraday effect (~ 104 deg/cm at 2.4 eV) [28], which make them suitable for MO recoding media and non-reciprocal MO devices [27,29]. Beside the technological 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 band structure [30] as well as light-induced ultrafast magnetization dynamics and switching [31-36]. The experiments were performed on 140-nm-thick film of Bi 2Y1Fe5O12, grown by pulsed laser deposition onto a gadolinium gallium garnet (Gd 3Ga5O12, GGG) (100) substrate. The 4 structural properties were characterized in-situ by reflection high-energy electron diffraction (RHEED) and ellipsometry, and ex-situ by X -ray diffraction and transmission electronic microscopy. The film is single phase and epitaxial with atomically sharp interface. The magnetic and magneto-optical (MO) properties of the film were investigated using a custom designed broad band MO spectrometer based on 90°-polarization modulation technique. Details on the experimental setup are described in Ref [32,37]. Figures 1(a) and 1(b) show the spectral dependency of the rotation ( ΘF, ΘK) and ellipticity ( εF, εK) Faraday and Kerr spectra measured at 300 K in polar configuration with saturating external magnetic field applied perpendicular to the film plane. ΘF is negative above 487 nm with a minimum at 520 nm and positive between 487 nm and 353 nm with a maximum at 390 nm, whereas εF shows two pics centered 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 K reaches -1.6 deg near 510 nm and εK reaches - 1.3 deg near 540 nm. We note that the Faraday and Kerr spectra show a good agreement with previous study of MO properties in Bi-YIG [28,37,38]. From a fundamental point of view, they are well described by the crystal field energy levels of Fe3+ ions in tetrahedral and octahedral 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 origin of the increase of MO proprieties in Bi-YIG with increasing Bi content. Let us also mention that the positions of the peaks of Θ F nicely fit into recent results showing the evolution of the energy level transition associated with the tetrahedral and octahedral iron sites as a function of Bi content in Bi xY3-xFe5O12 ( 0.5 ≤ xBi ≤ 3 ) [37], which confirm the bismuth concentration in our samples. The normalized polar and longitudinal Kerr hysteresis loops of the garnet film are shown in Figs . 1(c) and 1(d) respectively. The normalized remanence (M r/Ms) is of 0.05 and 0.45 for the polar and longitudinal configuration respectively. The very weak polar remanence show that the easy axis of the magnetization is in the film plane. In order to explore the effect of an ultrashor t strain pulse on the magnetization dynamics in iron garnet, a thick Cu(100)/Pt(5) nonmagnetic metallic bilayer was deposited by dc magnetron sputtering on top of the garnet film (s ee Fig. 1(e)). The numbers in parentheses are in nanometers and represent the thickness d of the layer. Let us mention that each metallic layer plays a crucial role in our artificial structur e. The Pt top layer is important due to its high electron-phonon coupling constant (109 1016 W.m-3.K-1) [41] and absorption at 800 nm [13], which enables the generation of coherent acoustic phonon with quite high frequency [42,43]. 5 The Cu layer is important due to its high hot-electron life time. Indeed, a thick Cu layer allows protecting the magnetic Bi-YIG film from the direct laser excitation, while the hot- electron can travel balistically for Cu thic kness up to few hundreds of nanometers [13,14]. The 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 [44]. The time-resolved MO and reflectivity measurem ents were performed at 300 K with the all- optical pump-probe configuration sketched in Fig. 1(c). Briefly, we have employed a femtosecond laser pulse issued from an amplifie d Ti-Sapphire laser system operating at a 5 kHz repetition rate and delivering 35 fs pulses at 800 nm to generate the pump and the probe beams. The pump beam is kept at the fundamental of the amplifier at 800 nm and excites the sample at normal incidence from the Pt side, while the probe beam is frequency doubled to 400 nm using a barium boron oxide crystal and incident with a small angle of 6° onto the GGG 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 below the optical absorption edge of the GGG [45], which allows the probe to penetrate the substrate and reach the Bi-YIG layer. After in teraction with the Bi-YIG, the reflected probe pulses allow measuring the differential changes of the MO polar Kerr rotation ΔΘ K (t) and reflectivity ΔR (t) induced by the acoustic pulse as a function of the time delay t between the pump and probe pulses using a synchronization de tection scheme. The external magnetic field Hext is applied perpendicular to the plane of the film. III. RESULTS AND DISCCUSSION: Figure 2(a) shows the time resolved MO Kerr effect (TR-MOKE) measurement of the dynamics induced by a laser energy density of 11.3 mJ cm -2 for H ext = 3.3 kG. We note that the TR-MOKE signal changes its sign when the direction of H ext is reversed. In addition, the zero time delay corresponds to the arrivals of hot -electron pulse to the back side of Cu layer, as revealed by the TR-MOKE signals measured in areas with and without the Pt/Cu bilayers. On the other hand, a strong peak in the TR-MOKE signal appears at t = 40 ps, which is very close to the time t = d Cu/VCu + d Bi-YIG /VBi-YIG ≈ 41 ps required for an acoustic pulse to cross the Cu and Bi-YIG layers, where d and V are the thickness and longitudinal sound velocity characterizing Cu and Bi-YIG and their values are d Cu = 100 nm, d Bi-YIG = 140 nm, V Cu = 4730 m/s [13] and V Bi-YIG ≈ 6700 m/s [46]. The changes observed in ΔΘ K (t) signal in the time 6 delay between 0 and 40 ps have a nonmagnetic origin. This phenomenon is the same as reported in Ref [47] for semiconductor as it sh ows the same characteristic behaviors: (i) The variation of its amplitude with the magnetic field is the same as the static MO response of the sample, i.e, it saturates for H ext higher than the saturating field H sat= 2.5 kOe (see Fig. 1 (c) and inset of Fig. 2 (a)) and changes sign when the direction of H ext is reversed (ii) Its amplitude monotonously increases with the pump energy density. This phenomenon is due to the modulation of the reflectivity signal by hot-electron pulse and the strain pulse which affects differently the right ( σ+) and left ( σ-) helicity of light [47]. Since the Kerr rotation can be considered as the phase di fference between the reflected σ+ and σ- helicity, the different effect induced in σ+ and σ- is observed in the ΔΘ K(t). From a theoretical point of view, it can be 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 paper. Interestingly, after the acoustic pulse leaves the Bi-YIG layer, two resonance modes are clearly revealed by the oscillations shown in the ΔΘ K (t) signal with the frequencies of 6.4 and 63.7 GHz, as seen in the Fourier transform sp ectrum displayed in the inset of Fig. 2(a). As demonstrated hereafter, the first mode is the ferromagnetic resonance mode (f fmr) observed via acoustic pulse induced changes of magnetocristalline anisotropy, whereas the second mode (facous) results from the modulation of the MO effe ct by the propagation of the acoustic pulse in the GGG substrate. The 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 experimentally confirmed by measuring the pump-induced changes in the reflectivity signal ΔR(t)/R [Fig. 2(b)], which shows oscillations for the time delay higher than 40 ps. Such oscillations were attributed to the so-called Brillouin oscillations, which are due to the interference 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 the Brillouin oscillations is given by ݂ ൌ2 ܸ ீீீ√݊ଶെs i nଶߠ⁄ߣ [49], where λ = 400 nm, VGGG = 6400 m/s [50], n ≈ 2 [45], and θ = 6° are, respectively the probe wavelength, the longitudinal sound velocity in GGG, the refractive index at the probe wavelength and the incidence angle of the probe beam. The calculated value of f B = 63.9 GHz, which is in good agreement with the frequency characterizing the oscillations observed in ΔR(t)/R signal [see inset of Fig. 2(b)]. 7 The comparison between the results obtained from ΔΘ K(t)/ΘKmax and ΔR(t)/R measurements allows us to conclude that the mode f acous observed in the TR-MOKE is related to the propagation of the acoustic pulse in the GGG substrate, since it has the same frequency as the Brillouin 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 MO effects at a frequency determined by the sound velocity as mentioned by Subkhangulov et al [51]. On the other hand, the frequency of 6.4 GHz associated with the low-frequency mode is in the range of the ferromagnetic reso nance (FMR) frequencies in Bi-YIG. In order to confirm the magnetic origin of this mode, we investigated the effect of the external magnetic field on the TR-MOKE. ΔΘ K(t) measured at selected external magnetic field are displayed in Fig. 3(a). The frequency and amplitude of the low-frequency mode are clearly influenced by the external magnetic field. To further highlight the behavior of the modes, the field dependence of the oscillations frequency and ampl itude as determined by Fourier analyses are shown in Figs. 3(b) and 3(c). The variation of the oscillations frequency of the low-frequency mode can be described by Kittel formula adapted to the case of our experimental configuration [52]: ωൌ γ ൫ H ௫௧െH ൯ ሺ1ሻ where ω is the angular precession frequency, γ the gyromagnetic ratio, H ext the external magnetic field, and the effective field H eff is defined as (4 πMs - H u - H c) where H u and H c are the uniaxial and cubic anisotropy fields, respectively. The adjustment of Fig. 3(b) with Eq (1) using γ = 2.8 GHz/kG yields H eff = 1.22 kG. This behavior of the low-frequency mode clearly indicates 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 sat = 2.5 kG (see Fig 1(c) and Fig (3c)), which is also in a qualitative agreement with the typical behavior obtained for the FMR mode when H ext is applied along a hard magnetization axis as in our experimental configuration [53,54] . On the other hand, the frequency of f acous is independent on the magnetic field strength (Fig. 3(c)). As discussed above, this is the expected behavior for the acoustically-induced modulation of the MO effects. Furthermore, we show that the initial oscillations amplitude of f acous as a function of field has the same behavior as the MO response of our sample. Such dependence can be related to the characteristic behavior of the non-magnetic contribution observed in ΔΘ K (t) for t between 0 and 40 ps which modulate the TR-MOKE signal. 8 Let’s us now focus on the mechanism behind the excitation of the ferromagnetic resonance in our sample. It results from an ultrafast non-thermal modification of the magnetocristalline anisotropy induced by the acoustic strain puls es via the inverse magnetostriction effect, as initially shown by Scherbakov et al in GaMnAs [18]. The excitation of the FMR mode for a magnetization already aligned along H ext, i.e higher than H sat = 2.5 kG (see Fig. 1(c) and Fig. 4), substantiate this interpretation. Indeed, a decrease of the magnetocristalline parameters induced by heating effects does not allow ex citing the FMR mode when the magnetization is aligned along H ext [25]. Let us also mention that the non-thermal mechanisms based on the Cotton-Mouton effect [55] and photo-induced magnetic anisotropy [12,32] usually used to induce in magnetic garnet a magnetization preces sion with a linearly polarized light can be excluded 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 configuration the ultrafast magnetization dynami cs induced by direct light excitation (not shown). No magnetization precession has been observed. This results further highlights the importance of the approach based on acoustic strain pulse for generating spin wave via the inverse magnetostriction effect. To further investigate the two resonance mode s, we performed TR-MOKE measurements as a function of the laser energy density E pump. Figure 4(a) shows the TR-MOKE signals measured at selected E pump for H ext= 3.3 kG. The pump energy densit y dependence of the oscillations frequency and amplitude extracted using Fourier an alysis are presented in Figs. 4(b) and 4(c). The frequency of both modes is independent of the E pump. The behavior of f fmr is similar to the one obtained by non-thermal effects induced magnetization precession [32,34]. This is in agreement with our interpretation based on stra in-induced changes of magnetic anisotropy via the inverse magnetostriction effect. Indeed, in the case of thermally induced spin precession a dependence of the frequency on E pump is usually observed [56,57]. On the other hand, the behavior of f acous as a function of E pump is also in agreement with the prediction that the frequency of acoustically-induced modulation of the MO effects is mainly defined by the speed of sound. Moreover, our experiments show that the oscillations amplitude of the two resonance modes increases linearly with the laser energy density within the probed range. This means that the amplitude of the spin prece ssion is proportional to the amplitude of the strain pulse. Therefore, using an engineered st ructure that allows injecting a higher amplitude strain pulse into Bi-YIG can be used for further improving the magnetization precession amplitude or inducing a magnetizati on switching in magnetic garnet. 9 VI. CONCLUSION: We have studied the laser-induced ultrafast magnetization in a dielectric film of bismuth- substituted yttrium iron garnet buried below a thick Pt/Cu bilayer. It is found that exciting the sample from Pt surface launches coherent stain pulses that propagate into the garnet film. We demonstrate that this acoustic pulse modifies th e magnetocristalline anisotropy via the inverse magnetostriction effect. This triggers a coherent magnetization precession at the frequency of the ferromagnetic resonance. Importantly, we can control the amplitude of the spin precession by tuning the amplitude of the acoustic strain pulse. Our results highlight the suitability of acoustic strain pulse for ge nerating spin wave in dielectric materials. ACKNOWLEDGMENTS The authors acknowledge M. Bargheer for inte resting discussions. This work was supported by the ANR-NSF Project,ANR-13-IS04-0008 -01, COMAG, ANR- 15-CE24-0009 UMAMI and by the ANR-Labcom Project LSTNM, by the Institut Carnot ICEEL for the project « Optic-switch » and Matelas and by the French PIA project ‘Lorraine Université d’Excellence’, reference ANR-15-IDEX-04- LUE. 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Lai, Appl Phys Lett 101, 222402 (2012). 12 400 600 800-40-200204060 400 600 800-1.5-1.0-0.50.00.51.01.5 -4 -2 0 2 4-1.0-0.50.00.51.0 -4 -2 0 2 4-1.0-0.50.00.51.0ΘF εFΘF , εF (deg/μm) Wavelength (nm)(a) (b) ΘK εK ΘK , εK (deg/ μm) Wavelength (nm) H⊥(c)ΘK/ΘKsat Magnetic field (kG) H//(d) ΘK/ΘKsat Magnetic field (kG) (e) Pump (800nm) To the polarization bridge Pt (5 nm) Bi2Y1Fe5O12Probe (400nm)Cu (100 nm)strain pulse Figure 1: Static room temperature magn eto-optical and magnetic properties of Bi 2Y1Fe3O15 thin film and the pump-probe experimental conf iguration. (a, b) Magneto-optical Faraday (a) and Kerr (b) polar spectra measured over a broad range of wavelength. The filled and open symbols represent, respectively, the rotation ( ΘF, ΘK) and ellipticity ( εF, εK). (c,d) Normalized magneto-optical hysteresis loops measured in polar (c) and longitudinal configuration. (e) Sketch of the time resolved experimental co nfiguration that allows studying the ultrafast magnetization dynamics induced by acoustic pulse. 13 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 0 50 100 150 200 250-0.75-0.50-0.250.000.250.50 02 0 4 0 6 0 8 0 1 0 00.00.51.01.502468 1 00.000.010.020.03ΔΘΚ/ΘΚsat Time (ps)(a) FFT ( arb.unit) Frequency (GHz)ffmr= 6.4 GHz facous= 63.7 GHz (b)ΔR/R x10-2 Time (ps) FFT ( arb.unit) Frequency (GHz)fB= 63.7 GHzΔΘΚ/ΘΚsat Magnetic Field (kG)t = 20 ps Figure 2: Dynamics of spin and reflectivity in the Bi 2Y1Fe5O12/GGG(100) buried below a thick Pt/Cu bilayers. (a, b) ΔΘ K/ΘKsat and ΔR/R induced by a laser energy density of 11.3 mJ cm-2 for H ext= 3.3 kG. Inset (a): the Fourier transform spectrum of the ΔΘ K/ΘKsat data for the time delay t ≥ 50 ps and the ΔΘ K/ΘKsat measured at the time delay t = 20 ps as a function of Hext. Inset (b): the Fourier transform spectrum of the ΔR/R data for the time delay t ≥ 50 ps . The solid lines in (a) are guides to the eyes. 14 0 100 200 300 400 500 6000.00.10.2 02468 1 0020406080100 02468 1 00.000.050.109.8 kG1.6 kG 2.5 kG 3.3 kG 4.9 kG 6.6 kG Time (ps) ΔΘΚ/ΘΚsat(a) Ferromagnetic modeAcoustic mode (b) Acoustic mode FMR mode Frequency (GHz) Magnetic Field (kG) FMR mode(c) Amplitude (arb.unit) Magnetic Field (kG)0.000.020.040.06 Acoustic mode Amplitude (arb.unit) Figure 3: Magnetic field dependence of the spin dynamics. (a) ΔΘ K/ΘKsat as a function of the magnetic field. (b, c) Field dependence of th e precession frequencies (b) and amplitudes (c) associated with acoustic and FMR resonance modes. All measurements are obtained for a pump energy density of 11.3 mJ cm -2. The dashed lines in (a) and solid lines in (c) are guides to the eyes. In (b) the solid line describing the FMR mode is a fit with the Kittel formula, whereas the solid line for the acoustic mode is a guide to the eyes. 15 0 100 200 300 400 500 6000.00.10.20.3 05 1 0 1 50.000.050.100.15 05 1 0 1 505106070809015.1 mJ cm-2 13.2 mJ cm-2 11.3 mJ cm-2 9.4 mJ cm-2 7.6 mJ cm-2 5.6 mJ cm-2 4.9 mJ cm-2 3.4 mJ cm-2 1.9 mJ cm-2ΔΘΚ/ΘΚsat Time (ps)(a) (c) Acoustic mode FMR mode Amplitude (arb.unit) Epump(mJ.cm-2) Acoustic mode FMR mode Frequency (GHz) Epump (mJ.cm-2)(b) Figure 4: Laser energy density dependence of the spin dynamics. (a) ΔΘ K/ΘKsat as a function of the laser energy density. (b, c) Variation of the precession frequencies (a) and amplitudes (b) associated with acoustic and FMR resonance modes as a function of the laser energy density. All measurements are obtained for H ext = 3.3 kG. The solid lines in (b) and (c) are guides to the eyes.

2018-07-02

Using femtosecond optical pulses, we have investigated the ultrafast magnetization dynamics induced in a dielectric film of bismuth-substituted yttrium iron garnet (Bi-YIG) buried below a thick Cu/Pt metallic bilayer. We show that exciting the sample from Pt surface launches an acoustic strain pulse propagating into the garnet film. We discovered that this strain pulse induces a coherent magnetization precession in the Bi-YIG at the frequency of the ferromagnetic resonance. The observed phenomena can be explain by strain-induced changes of magnetocristalline anisotropy via the inverse magnetostriction effect. These findings open new perspectives toward the control of the magnetization in magnetic garnets embedded in complex heterostructure devices.

Picosecond acoustic excitation driven ultrafast magnetization dynamics in dielectric Bi-substituted yttrium iron garnet

1807.00610v1

Proposal for optomagnonic teleportation and entanglement swapping Zhi-Yuan Fan,1Xuan Zuo,1Hang Qian,1and Jie Li1, 1Interdisciplinary Center of Quantum Information, State Key Laboratory of Modern Optical Instrumentation, and Zhejiang Province Key Laboratory of Quantum Technology and Device, School of Physics, Zhejiang University, Hangzhou 310027, China (Dated: May 11, 2023) A protocol for realizing discrete-variable quantum teleportation in an optomagnonic system is provided. Us- ing optical pulses, an arbitrary photonic qubit state encoded in orthogonal polarizations is transferred onto the joint state of a pair of magnonic oscillators in two macroscopic yttrium-iron-garnet (YIG) spheres that are placed in an optical interferometer. We further show that optomagnonic entanglement swapping can be realized in an extended dual-interferometer configuration with a joint Bell-state detection. Consequently, magnon Bell states are prepared. We analyze the e ect of the residual thermal occupation of the magnon modes on the fidelity in both the teleportation and entanglement swapping protocols. I. INTRODUCTION As an indispensable building block of quantum informa- tion science, quantum teleportation refers to the process of transferring an unknown quantum state at one location onto another quantum system some distance away. Ever since it was first proposed by Bennett et al. [1], quantum teleporta- tion has been successfully realized in various physical systems over the past few decades. These include photons [2–5], nu- clear spins [6], trapped ions [7, 8], atomic ensembles [9, 10], solid-state systems [11], high-frequency vibration phonons [12], and optomechanical systems [13], among many others. These successful demonstrations lay the foundation for re- alizing many other quantum protocols, such as quantum re- peaters [14], fault-tolerant quantum computation [15], etc. Here we provide an optomagnonic quantum teleportation protocol which can transfer an arbitrary photonic qubit state to a dual-rail encoding magnonic system of two yttrium-iron- garnet (YIG) spheres. We adopt an optical interferometer con- figuration, of which each arm contains a YIG sphere sup- porting an optomagnonic system. We use the Stokes type scattering of the magnon-induced Brillouin light scattering (BLS) to create an optomagnonic EPR state. A subsequent Bell-state measurement onto the input photonic qubit and the output Stokes photon from the interferometer enables such a photon-to-magnon quantum state transfer. The magnon state can be read out by activating the anti-Stokes scattering real- izing the optomagnonic state-swap operation, from which the magnonic qubit state is retrieved onto the anti-Stokes photon. We further propose an optomagnonic entanglement swapping protocol based on an extended dual-interferometer configura- tion, which realizes the transfer of the optomagnonic entan- glement to a dual-rail encoding two-qubit magnonic system involving four YIG spheres and thus prepares a magnonic Bell state. In what follows, we first introduce the basic optomagnonic interactions in Section II, and describe our two protocols re- spectively in Section III and Section IV. We then analyze the jieli007@zju.edu.cneect of the magnon thermal excitations on the fidelity of the two protocols in Section V and finally conclude in Section VI. II. OPTOMAGNONIC INTERACTION In the past decade, hybrid magnonic systems based on collective spin excitations in ferrimagnetic materials, such as YIG, have gained significant attention due to their excel- lent ability to coherently couple with a variety of physical systems, including microwave photons [16–18], optical pho- tons [19–23], vibration phonons [24–28] and superconduct- ing qubits [29–33]. The emerging field of hybrid quantum magnonics provides a platform not only for studying strong interactions between light and matter, but also for developing novel quantum technologies to be applied in quantum infor- mation processing, quantum sensing and quantum networks [34–37]. In particular, the coupling between magnons and optical photons, namely the optomagnonic interaction [19– 23], is indispensable for building a magnonic quantum net- work [37], where the transmission of the information between remote quantum nodes is realized by optical photons. Such an optomagnonic interaction has been exploited in many pro- posals to cool magnons [38], prepare magnon Fock [39], cat [40], path-entangled [41] states and magnon-photon entan- gled states [37, 42], and realize magnon laser [43, 44], fre- quency combs [45], photon blockade [46], polarization-state engineering [47], etc. The typical optomagnomic system, as depicted in Figure 1(a), consists of a YIG sphere, which supports both a magne- tostatic mode (i.e., the magnon mode) and optical whispering gallery modes (WGMs). The WGM resonator is near the sur- face of the YIG sphere and the input optical field is evanes- cently coupled to the WGM, e.g., via a tapered fiber [19, 20] or prism [21, 22]. The optomagnonic interaction in such a sys- tem is embodied by the magnon-induced BLS, where the pho- tons of a WGM are scattered by lower-frequency magnons, typically in GHz [19–22], giving rise to sideband photons with their frequency shifted by the magnon frequency and their polarization changed. When the frequency of the scattered photons matches another WGM, namely the triple resonance condition, the optomagnonic scattering probability is maxi-arXiv:2305.05889v1 [quant-ph] 10 May 20232 FIG. 1: (a) A typical optomagnonic system: a YIG sphere supports a magnon mode and two optical WGMs. (b) Mode frequencies corre- sponding to the optomagnonic Stokes scattering, which leads to the PDC interaction. (c) Mode frequencies associated with the optomagnonic anti-Stokes scattering, which yields the state-swap (BS) interaction. mized. This can be easily achieved by tuning the magnon frequency via changing the strength of the bias magnetic field (B0). Typically, there are two types of magnon-induced BLSs, i.e., the Stokes and anti-Stokes scatterings, corresponding to optomagnonic parametric down conversion (PDC) and state- swap (beam-splitter, BS) interactions, respectively [37, 41]. In the optomagnonic scattering, the angular momenta of the WGM photons and magnons are conserved, which leads to a selection rule and prominent asymmetry in the Stokes and anti-Stokes scattering strengths [48–51]. This allows us to se- lect a particular type of the optomagnonic interactions (either PDC or BS) on demand. The Hamiltonian accounting for the optomagnonic interac- tion in such a three-mode system reads H=~=!vaya+!hbyb+!mmym +g0(aybm+abymy)+Hdri=~;(1) where j=a;b(jy) and m(my) are the annihilation (creation) operators of the two WGMs and the magnon mode, respec- tively, and!k(k=v;h;m) correspond to their resonance fre- quencies, satisfying the relations j!v!hj=!mand!m !v;h. Here the subscripts vandhrepresent two orthogonal polarizations of the two WGMs, i.e., the transverse-electric (TE) mode and the transverse-magnetic (TM) mode. The op- tomagnonic interaction is a three-wave process and g0is the corresponding single-photon coupling strength. The last term denotes the driving Hamiltonian Hdri=i~Ej(jyei!0tjei!0t), where Ej=pjPj=(~!0) is the coupling strength between the WGM j(with decay rate j) and the optical drive field with frequency!0and power Pj. To enhance the optomagnonic coupling strength, a strong drive field is used to resonantly pump one of the WGMs, i.e., !0=!vor!h, depending on which type of the optomagnonic interactions (either PDC or BS) is desired. For the case where the TE WGM ( a) is pumped, i.e., !0=!vand
b!h!0=!m, cf. Figure 1(b), the strongly driven WGM acan be treated classically as a num- ber hai[37], and the e ective optomagnonic Hamilto- nian then becomes the PDC form, HS int=~g(bm+bymy), with g=g0being the e ective optomagnonic coupling strength. This interaction corresponds to the Stokes scattering, where the TE-WGM photons convert into lower-frequency sidebandphotons (resonant with the TM WGM b) by creating magnon excitations. Such a PDC interaction can be used to gener- ate optomagnonic entangled states [37, 42]. Specifically, the WGM band magnon mode mare prepared in a two-mode squeezed state (unnormalized) j ib;m=j00ib;m+p Pj11ib;m+O(P); (2) where Pis the probability for a single Stokes scattering event to occur andO(P) denotes the higher-excitation terms, of which the probabilities are equal to or smaller than P2. The scattering probability increases with the power of the drive field, and when the power is su ciently weak, the scatter- ing probability P1. In this weak-coupling limit, the probability of generating two-magnon (photon) state j2im(b) and higher-excitation states jnim(b)(n>2) is negligibly small. Such a low scattering probability of creating an entangled pair of single excitations was adopted in Ref. [41], which suggests an optomagnonic variant of the Duan-Lukin-Cirac-Zoller pro- tocol [52]. It was also used in cavity optomechanical ex- periments for creating entangled states of single photons and phonons [53, 54]. Similarly, when the TM WGM ( b) is resonantly pumped, i.e.,!0=!hand
a!v!0=!m, cf. Figure 1(c), the anti- Stokes scattering is activated and the e ective optomagnonic Hamiltonian becomes the BS type, HAS int=~g(aym+amy), with the eective coupling g=g0andhbi. This interaction realizes the state-swap operation between the magnon mode and the WGM a, accompanied with the process that TM- WGM photons convert into higher-frequency anti-Stokes pho- tons (resonant with the TE WGM a) by eliminating magnon excitations. As will be shown later, this interaction is used to read out the magnon state. III. OPTOMAGNONIC QUANTUM TELEPORTATION We now proceed to describe our optomagnonic teleporta- tion protocol, which is able to transfer a photonic qubit state (in polarization encoding) onto a magnonic system consisting of two optomagnonic devices placed in two arms of an opti- cal interferometer, cf. Figure 2. The two magnon oscillators are subject to a simultaneous excitation using a weak pulse3 FIG. 2: Sketch of the optomagnonic quantum teleportation protocol. It consists of three steps: preparation of the optomagnonic EPR state, Bell-state measurement and readout of the magnonic state. In the EPR-state preparation setup, an optical interferometer configuration is adopted and its each arm contains a YIG sphere which supports an optomagnonic system of a magnon mode and two optical WGMs. See text for detailed descriptions. that drives the TE WGM to activate the optomagnonic Stokes scattering. After a Bell-state measurement of the input photon with the Stokes photon from the optomagnonic devices, the input photonic qubit state is then transferred onto the dual-rail encoding magnonic system. For simplicity, we assume at most single excitations in the optomagnonic devices. This is the case of using a weak pulse, where the probability of creating higher-excitation statesjnim(b)(n2) in the Stokes scattering is negligible. Since the vacuum component of the state (2) will not trigger any coincidences in the Bell-state detection, leading to un- successful trials for the teleportation, the protocol can be de- scribed using a simplified model [55], where a TE-polarized single-photon pulse is sent onto a 50 /50 BS to drive the op- tomagnonic devices in the interferometer. After the BS, an optical path-entangled state,1p 2(j01iAB+j10iAB), is generated in the two outputs (i.e., the upper path A and lower path B). In each path, the pulse resonantly drives the TE WGM of the YIG sphere to activate the Stokes scattering. By selecting tri- als with successful scattering events, the PDC interaction pre- pares an optomagnonic Bell state in the form of j+ibm=1p 2(jHibjLim+jVibjUim); (3) wherejLim(jUim) denotes the generated single magnon is at the lower path B (the upper path A), and jHib(jVib) representsthe accompanied Stokes photon is in the horizontal (vertical) polarization. The Stokes photon, with equal probability in path A or B, then couples to the nanofiber or the prism coupler and enters the first polarizing beam splitter (PBS1) before go- ing to the next stage of the Bell-state detection. Note that the polarization of the Stokes photon in the upper path is changed from HtoVafter passing through a half-wave plate (HWP). We remark that the unsuccessful scattering events leave the single photons remaining in the vertical polarization. These photons eventually enter the other output of the PBS1 and thus have no impact on the subsequent Bell-state measurement. We also assume that the magnon modes are initially in their quan- tum ground state, which is the case for GHz magnons at a low temperature, e.g., of 10 mK. Nevertheless, in Section V we shall discuss the e ect of the residual thermal occupation on the fidelity of the teleportation. The input photonic qubit state to be teleported is an ar- bitrary superposition of two polarization modes, i.e., jic= jHic+jVic, with the complex coe cientsandsatis- fyingjj2+jj2=1. Such an arbitrary qubit state can be constructed on the surface of the polarization Poincare sphere by using a HWP and a quarter-wave plate (QWP). This in- put photon, being resonant with the TM WGM, goes into one input of the PBS2 and meanwhile the output of the PBS1 en- ters the other input port. Thereby, the joint state before the4 Bell-state measurement is as follows: j+ibm jic=1p 2jHibjHicjLim+jHibjVicjLim +jVibjHicjUim+jVibjVicjUim
:(4) The Bell-state measurement is performed onto the input pho- tonic qubit ( c) and the output Stokes photon ( b) from the in- terferometer, and the detection setup consists of a HWP, a PBS and two single-photon detectors in each output of the PBS2, cf. Figure 2. A coincidence measurement projects the optical modes onto the polarized Bell states jibc= 1p 2(jHibjHicjVibjVic). The fast axis of the HWP is set at 22:5, which acts as a Hadamard operation on the polarization of the photons passing through PBS2. The Bell states jibc correspond to di erent coincidence measurements, as can be seen from the following j+ibcPBS2 & HWP!1p 2 ay 3;hay 4;h+ay 3;vay 4;v jvaci; jibcPBS2 & HWP!1p 2 ay 3;hay 4;v+ay 3;vay 4;h jvaci;(5) where the operator ay i;h(ay i;v) denotes the detection of a single horizontally (vertically)-polarized photon at one of the two outputs of the PBS i(i=3;4), andjvacimeans the vacuum state. Note that, in addition to the Bell states jibc, the other two types of Bell states j ibc=1p 2(jHibjVicjVibjHic) can be realized by using photon-number-resolving detectors, and we disregard these cases for simplicity. The measurement that projects the optical modes bandc onto the Bell state j+ibcprojects the magnonic system onto the state j0im=jLim+jUim; (6) which indicates the successful teleportation of the input pho- tonic qubit statejicto a dual-rail encoding magnonic system, and corresponds to the ideal quantum teleportation without requiring additional correction operations in the readout step. On the other hand, the measurement associated with the Bell statejibcprojects the magnonic system onto the state j00im=jLimjUim: (7) It has a-phase di erence with respect to the input optical statejic, which can be corrected in the readout step using a feed-forward operation by applying a phase shift in the optical interferometer [13, 55]. To verify the successful teleportation, we need to retrieve the teleported magnonic qubit state. To achieve this, we ex- ploit the optomagnonic state-swap (BS) interaction, as intro- duced in Section II, where a TM-polarized weak pulse is sent to the interferometer to activate the anti-Stokes scattering and the magnonic state is then transferred to the anti-Stokes pho- ton. Due to the orthogonal polarization with respect to the Stokes photon (produced in the Stokes scattering), the anti- Stokes photon leaves from the other output of the PBS1, and enters the state-readout setup (cf. Figure 2) [13], from which the magnonic qubit state is retrieved.IV . OPTOMAGNONIC ENTANGLEMENT SWAPPING Quantum entanglement plays an essential role in all the quantum-teleportation related protocols. Naturally, entangled states can be obtained in directly coupled systems, such as the EPR state produced in the optomagnonic Stokes scatter- ing. For two systems that have no direct interaction, quantum mechanics also manifests its unique capabilities to establish quantum entanglement between them, one of which is referred to the entanglement swapping [1, 3, 56]. In this section, we show that the entanglement swapping protocol allows us to prepare the magnon modes in space-separated YIG spheres into an entangled Bell state. The detailed entanglement swapping protocol is shown in Figure 3, which consists of two optical interferometer setups used in Section III, a joint Bell-state detection and the as- sociated state-readout devices. Similarly to the teleportation protocol, in each interferometer setup containing two YIG spheres, a TE-polarized single-photon pulse is sent to activate the Stokes scattering, which prepares an optomagnonic Bell entangled statej+ibm, as in Equation (3). Therefore, the over- all state of the two interferometer setups before performing a joint Bell-state measurement on the scattered Stokes photons reads j itotal=j+ib1m1 j+ib2m2 =1 2jHib1jLim1+jVib1jUim1
jHib2jLim2+jVib2jUim2
;(8) where the subscripts 1 and 2 are used to distinguish the two interferometers, and the notation is the same as in Equation (3). The above state can be rewritten in the basis of Bell states as j itotal=1 2 j +ib1b2j +im1m2+j ib1b2j im1m2 +j+ib1b2j+im1m2+jib1b2jim1m2 ;(9) wherejib1b2andj ib1b2are the four Bell states of the Stokes photons in the two interferometers, which take the same form asjibcandj ibcin Section III.jim1m2and j im1m2are the four Bell states of a pair of dual-rail encod- ing magnonic systems, defined as jim1m2=1p 2(jLim1jLim2 jUim1jUim2) andj im1m2=1p 2(jLim1jUim2jUim1jLim2). From Equation (9), it is clear to see that a Bell-state mea- surement on the Stokes photons from the outputs of the two interferometers projects the magnonic systems onto a cor- responding Bell state. Specifically, a coincidence measure- ment corresponding to the optical Bell state j+ib1b2(jib1b2) projects the magnonic systems onto the Bell state j+im1m2 (jim1m2). This implies that the magnonic systems establish the same form of entanglement as the corresponding optical Bell state. Similarly as in the teleportation protocol, we disre- gard the other two types of the Bell-state measurements asso- ciated withj ib1b2. The entanglement swapping can also be understood in the framework of quantum teleportation in the sense that it trans-5 FIG. 3: Sketch of the optomagnonic entanglement swapping protocol. It is based on a dual-interferometer configuration, combined with a joint Bell-state detection and magnonic state readout devices. fers an optomagnonic quantum correlation (instead of a pho- tonic qubit statejic) to the magnonic systems. This can be seen by comparing Eqs. (4) and (8). This further reflects the versatility of the quantum teleportation protocol, which can transfer not only a qubit state but also multi-qubit states, e.g., quantum correlations. Replacing the optomagnonic system with an optomechanical system [54] in one of the interferom- eters also allows us to prepare a hybrid magnon-phonon Bell state, which may find potential applications in hybrid quantum networks [37]. V . EFFECT OF MAGNONIC THERMAL EXCITATIONS In the teleportation and entanglement swapping protocols, we neglect the dissipation of the magnon modes. This is the case of using fast optical pulses [53] such that the magnon dis- sipation can be assumed to be negligible during an experimen- tal run. We also assume that the magnon modes are initially prepared in their quantum ground state. This is a good approx- imation for the magnon modes at GHz frequency [19–23] and at low temperature of, e.g., 10 mK. However, the optical pulses may heat the magnon modes due to the optical absorp- tion of the YIG, causing the magnon modes to be at a thermal state. To include this heating e ect in practical situations, we assume that the magnon modes are initially prepared in a ther- mal stateth=(1s)P1 n=0snjnihnj, with s=¯n0=(¯n0+1) anda thermal occupation ¯ n01. Note that the frequencies of the magnon modes are assumed to be (nearly) identical and thus they have equal thermal occupation, i.e., they are in the same thermal state. For a small ¯ n0<0:2,s<0:17,s2<0:03 and s3<0:005, and the total probability of higher-excitation terms jni(n>2) is less than 0 :5%. Thus we can safely approximate th'(1s) j0ih0j+sj1ih1j+s2j2ih2j . The density matrix of the two magnon modes (in path A and B) in the teleporta- tion scheme is then m=A B'(1s)22X nA;nB=0snA+nBjnAnBihnAnBj;(10) which is a probabilistic mixture of nine pure states jnAnBi (nA;nB=0;1;2). This mixed initial state eventually leads to the following teleported magnonic state after the Bell-state measurement associated with j+ibc: 0 m=(1s)22X nA;nB=0snA+nBj'nAnBih'nAnBj; (11) wherej'nAnBi=jnA;nB+1iAB+jnA+1;nBiABis the tele- ported magnonic state corresponding to the pure inital state jnAnBiin Equation (10). Clearly, j'00i=j01iAB+j10iAB jLim+jUimcorresponds to the ideal initial state j00iABcon- sidered in Section III. For the Bell-state detection related to6 FIG. 4: Fidelity in the teleportation (solid) and entanglement swap- ping (dashed) protocol versus the thermal occupation ¯ n0of the magnon modes. jibc, we obtain the same 0 mas in Equation (11) but with  replaced byinj'nAnBi. All other states in 0 mare orthogo- nal to the desired state j'00i, and result in a reduction of the teleportation fidelity, which is F1=h'00j0 mj'00i=1=1+s+s2
2: (12) The solid line in Figure 4 clearly shows a declining fidelity versus the thermal occupation ¯ n0. For a genuine quantum tele- portation with the fidelity F1>2=3 [57], a small ¯ n00.2 is required. This is similar to the finding in the optomechanical teleportation [55]. By contrast, in the entanglement swapping protocol, four magnon modes (in path A, B, C and D, respectively) are in- volved, cf. Figure 3. Following the same approach, we obtain the final joint state of the magnonic systems, after the Bell- state measurement associated with jib1b2, given by 0 m1m2=(1s)42X nA(B)=02X nC(D)=0snA+nBsnC+nDj'm1m2ih'm1m2j;(13) where A,B,Cand Dare used to distinguish the magnon modes via their path information, and j'm1m2i= 1p 2 jnA;nB+1;nC;nD+1im1m2jnA+1;nB;nC+1;nDim1m2 is the teleported magnonic state corresponding to the pure statejnAnBnCnDiin the mixed initial state m1m2=m1 m2, cf. Equation (10). For the ideal case of the initial ground state considered in Section IV, we obtain the desired statesj'm1m2i=1p 2 j0101im1m2j1010im1m2 jim1m2after the entanglement swapping. The other additional terms in 0 m1m2are related to the residual thermal excitations in the magnonic initial state, which are unwanted and reduce the fidelity in the entanglement swapping protocol. The corresponding fidelity is given by F2=m2m1hj0 m1m2jim1m2=1=1+s+s2
4: (14) The dashed line in Figure 4 shows a decreasing fidelity F2as the thermal occupation ¯ n0increases. It reduces more rapidly with respect to the fidelity F1in the teleportation protocol be- cause of the relation F2=F2 1, as seen from Equations (12) and (14). VI. CONCLUSION We present two protocols for realizing optomagnonic quan- tum teleportation and entanglement swapping, respectively, adopting YIG spheres and an optical interferometer configu- ration. The optomagnonic Stokes and anti-Stokes scatterings are the essential elements for preparing optomagnonic EPR states and optically reading out the magnonic states. A Bell- state detection enables the transfer of an arbitrary photonic qubit state to a dual-rail encoding magnonic system in the for- mer protocol, and the transfer of the optomagnonic entangle- ment to the magnon modes in a dual-interferometer configu- ration which are prepared in a Bell state in the latter protocol. 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2023-05-10

A protocol for realizing discrete-variable quantum teleportation in an optomagnonic system is provided. Using optical pulses, an arbitrary photonic qubit state encoded in orthogonal polarizations is transferred onto the joint state of a pair of magnonic oscillators in two macroscopic yttrium-iron-garnet (YIG) spheres that are placed in an optical interferometer. We further show that optomagnonic entanglement swapping can be realized in an extended dual-interferometer configuration with a joint Bell-state detection. Consequently, magnon Bell states are prepared. We analyze the effect of the residual thermal occupation of the magnon modes on the fidelity in both the teleportation and entanglement swapping protocols.

Proposal for optomagnonic teleportation and entanglement swapping

2305.05889v1

Observation of the spin Peltier eect J. Flipse,1,F. K. Dejene,1D. Wagenaar,1G. E. W. Bauer,2, 3J. Ben Youssef,4and B. J. van Wees1 1Physics of Nanodevices, Zernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands. 2Kavli Institute of NanoScience, Delft University of Technology, Delft, The Netherlands 3Institute for Materials Research and WPI-AIMR, Tohoku University, Sendai, Japan 4Universit e de Bretagne Occidentale, Laboratoire de Magn etisme de Bretagne CNRS, 6 Avenue Le Gorgeu, 29285 Brest, France. (Dated: May 25, 2022) We report the observation of the spin Peltier eect (SPE) in the ferrimagnetic insulator Yttrium Iron Garnet (YIG), i.e. a heat current generated by a spin current owing through a Platinum (Pt)jYIG interface. The eect can be explained by the spin torque that transforms the spin current in the Pt into a magnon current in the YIG. Via magnon-phonon interactions the magnetic uctuations modulate the phonon temperature that is detected by a thermopile close to the interface. By nite- element modelling we verify the reciprocity between the spin Peltier and spin Seebeck eect. The observed strong coupling between thermal magnons and phonons in YIG is attractive for nanoscale cooling techniques. The discovery of the spin Seebeck eect (SSE) in YIGjPt bilayers [1] opened up a new research direction in the eld of spin caloritronics. In the SSE a temperature dierence between the magnons in the magnetic insulator and the electrons in the metal contact leads to thermal pumping of a spin current [2{4]. In a suitable metal such as Pt this spin current is transformed into an observable transverse voltage by the inverse spin Hall eect [5]. Nu- merical simulations of the phonon, magnon and electron temperatures show good agreement with experiments [6]. In this Letter we report the observation of the spin Peltier eect (SPE), which is the Onsager reciprocal [7] of the SSE. The SPE is the generation of a magnon heat current in the magnetic insulator by a spin current through the interface with the metal contact. The latter can be gen- erated by a charge current in the Pt lm that by the spin Hall eect generates a transverse spin current normal to the interface. The spin Peltier heat current generates a temperature dierence between magnons and phonons in the YIG that when relaxing leads to a change in the lattice temperature. We conrm this scenario experi- mentally by picking up such temperature changes via proximity thermocouples. According to our modelling the experimental results are consistent with Onsager reci- procity between the SPE and the SSE, which we measure separately (see supplementary IV). Our results conrm recent indications for a strong magnon-phonon interac- tion in YIG at room temperature [6, 8, 9]. A charge current through a Pt strip generates a trans- verse spin current induced by the spin Hall eect that leads to a spin accumulation Vsat the boundaries. At the interface to YIG the spin current is absorbed as a spin transfer torque proportional to the spin mixing con- ductance [10, 11], as depicted in Fig. 1(a). When the magnetic moment of the spin accumulation ( s) at the PtjYIG interface is parallel (antiparallel) to the average magnetization direction, the spin torque transfers mag-netic momentum and energy from the electrons in the Pt to the magnons in the YIG (or vice versa). Magnons are thereby annihilated (excited) (see Fig. 1(b)) lead- ing to cooling (heating) of the magnetic order parame- ter (see Fig. 1(c)). Since thermal magnons equilibrate with the lattice by magnon-phonon scattering, the non- equilibrium magnons aect the lattice temperature (see Fig. 1(b) and (c)) depending on the magnetization di- rection. In the SSE [2] the spin current density ( Js) pumped from the YIG into the non magnetic metal is proportional to the temperature dierence between the magnons and electrons at the interface ( Tm-e=TmTe) and the in- terface spin Seebeck coecient LS,Js=LSTm-e. In or- der to arrive at a symmetric linear response matrix that re ects Onsager symmetry, the sum of the products of currents and driving forces should be proportional to the dissipation [12], leading to (see supplementary I) Js Qm-e =gSLST LST{I STVs=2 Tm-e=T (1) Here we used the Onsager Kelvin relation  S=SST= LS=(gST);where the spin Seebeck SS= (dVs=2dT)Js=0 and spin Peltier  S= (dQm-e=dJs)@Tm-e=0coecients have been dened. gSis the average spin conductance per unit area when spin accumulation and magnetiza- tion are collinear, i.e. the Vsat the YIGjPt interface is either parallel or antiparallel to the average YIG magne- tization.gS0:16grat room temperature [13], where gris the real part of the spin-mixing conductance per unit area. {I Sis the magnetic contribution to the inter- face heat conductance per unit area [6]. The SPE heat current density we set out to discover is therefore Qme=LSTVs 2: (2) The devices designed for observing the SPE are fab- ricated on top of a 200 nm thick single-crystal (111)arXiv:1311.4772v1 [cond-mat.mes-hall] 19 Nov 20132 Jc M TpTm M/uni03BCsTeT T0 TpTm MT Te (a) (b) (c) T0/uni03BCs FIG. 1. (color online). Schematic gure of the spin Peltier eect at a Pt jYIG interface. (a) A charge current through the Pt creates a transverse spin current induced by the spin Hall eect that generates a spin accumulation Vsat the boundaries. (b) When the spin magnetic moment sis antiparallel to M the spin torque transfers angular momentum and energy from the electrons in the Pt to the magnons in the YIG thereby cooling the electrons and heating the magnons, eectively raising the magnon temperature Tmwith respect to the electron temperature Te. (c) When sis parallel to M the spin torque transfers angular momentum and energy from the magnons in the YIG to the electrons in the Pt thereby cooling the magnons, eectively lowering Tmwith respect to Te. Y3Fe5O12(YIG) lm grown on a 500 µm thick (111) Gd3Ga5O12(GGG) substrate by liquid-phase-epitaxy. Two temperature sensors are fabricated in close prox- imity to the PtjYIG interface. The optical microscope image in Fig. 2(b) shows the 20 200µm2and 5 nm thick Pt injector lm. The thermopile sensors consist of ve 40 nm thick Pt-Constantan (Ni 45Cu55) thermo- couples in series that are very sensitive because of the large dierence in the Seebeck coecient of these met- als. In the thermopile on the right of the Pt injector the PtjNi45Cu55order is reversed for additional cross check measurements. The two thermopiles and the Pt injector are connected to 5 j100 nm thick titanium jgold contacts, providing good thermal anchoring and electrical contact to bonding pads 30 µm away. All structures are patterned by electron beam lithography. The Pt injector and the Ni45Cu55are deposited by DC sputtering while electron beam evaporation has been used to make the Au contacts and Pt thermocouple components. An AC current is sent through the Pt injector, from I+ to I(Fig. 2(b)), to create Vs. The voltage over the ther- mopile (V+and V) is simultaneously recorded. Using a standard lock-in detection technique the rst harmonic response (V/I) is extracted from the measured voltage. A low excitation frequency of 17 Hz was used to ensure a thermal steady-state condition. All measurements are carried out at room temperature. In Fig. 2(a) the rst harmonic voltage over the ther- mopile is shown as a function of an applied in-plane magnetic eld ( B) for a root-mean-square current of 3 mA through the Pt injector. A clear switch is observed just after the applied eld becomes positive, in line with the magnetization reversal of YIG at very small coer- cive elds [14]. The signal switches back to its originalvalue when reversing the eld with a small hysteresis. We measure a SPE signal of 33 nV on top of a back- ground voltage of 0 :463µV. We observe linear scaling of the SPE signal for currents between 1 and 4 mA in the Pt ( IPt injector )(see Fig. 2(c)). Results for four dif- ferent samples (from two dierent batches) match the signal presented here within 15 %. The measurements were repeated with Brotated 90. No SPE signal was observed in this conguration while the background re- mained the same (see supplementary II), which conrms our interpretation. In order to obtain quantitative information we carry out 3D nite element modelling of our devices [15]. As discussed above, the SPE heat current ( Qm-e) ows be- tween the electron and magnon systems through the PtjYIG interface. Qm-eis calculated using Eq. (2) and Vs=Jc

tanht 2 (3) whereis the spin Hall angle, tthe Pt lm thickness, Jcthe charge current density through the Pt injector, the Pt resistivity, the spin- ip diusion length and = 2
[1 +gScotht 
]1a back ow correction factor. The heat charge current densities in Pt are mod- elled by a three reservoir model of thermalized phonons, magnons and electrons at temperatures Tph,TmandTe, respectively [6]. In linear response the charge ( Jc) and heat (Q) current densities in the bulk of the materials are related to their driving forces, i.e. gradients of ( V, Tph,TmandTe) as~Qx=x~rTxand ~Jc ~Qe = S ST e~rV ~rTe (4) wherexisphorm,is the electrical conductivity,3 −10 −5 0 5 100.450.460.470.48Vthermopile (/uni03BCV) B (mT)IPt injector= 3mA 1 2 34 010203040∆Vspin Peltier (nV) IPt injector (mA) Pt NiCuNiCu PtV+ V-I+ I-B(a) (b) (c) FIG. 2. (color online). (a) First harmonic voltage across the thermopile as a function of applied magnetic eld. The dierence between the voltage at positive and negative elds is the spin Peltier signal. (b) Optical microscope picture of the device. (c) The spin Peltier signal (
Vspin Peltier ) as a function of the charge current through the Pt injector. Sthe Seebeck coecient and ph,mandeare the phonon, magnon and electron thermal conductivities, re- spectively. The interaction between the magnon and phonon subsystems in YIG and between the phonon and electron subsystems in Pt are taken into account by us- ing thermal relaxation lengths, mphandeph;respec- tively (see supplementary III), r2Tm-ph=Tm-ph 2 m-phandr2Te-ph=Te-ph 2 e-ph:(5) The phonon interface heat conductance ( I ph) and heat exchange between magnons and electrons across the in- terface ( {I S) are treated as boundary conditions [6] (see Supplementary III). This model is evaluated for the material parameters listed in table I. Additionally, we adopt gr= 71014 1m2[16],= 0:11 [6],= 1:5 nm [6] and LS= 7:24109A/(m2K) [2, 6]. The magnon heat conductivity of YIG (m) at room temperature is not well known so we used a mof 102and 103W/(mK) in order to cover the range of estimated values [6, 17]. For Pt jYIG aI phof 2:78108W/(m2K) obtained from the acoustic mismatch model was adopted [6]. Since this model tends to overestimate the heat conductance [18], we also usedTABLE I. Material parameters used in the model. Both and S are measured in separate devices [19] except for of the Pt injector, which is extracted from the SPE devices directly. ph eis adopted from Ref. 6 and the total =ph+eis calculated using = bulkbulk.  Sph e (S/m) ( µV/K) (W/(m
K)) (W/(m
K)) YIG - - 6 - GGG - - 8 - Au 2.7
1071.7 1 179 Pt injector 3.5
106-5 3 23 Pt thermocouple 4.2
106-5 4 28 Ni45Cu55 1
106-30 1 9 2108W/(m2K). In gure 3(a) the results are shown as a function of m-ph. The semi transparent blue horizontal bar indicates the range of measured SPE signals that are best tted by a m-phof 0.1 to 0.2 nm for the ranges of mandI phdiscussed above. SSE samples were fabricated and simulated by the same model and parameters used above (see Supplemen- tary IV). In Fig. 3(b) the results are plotted and best tted bym-phbetween 0.2 and 0.5 nm, which is con- sistent with the values found for the SPE, as is indeed required by Onsager reciprocity. This implies that our model captures the essential physics of the interacting electron, magnon and phonon systems. The observed SPE signal in Fig. 2(a) corresponds to a phonon temperature dierence of 0.25 mK at the ther- mopile, which according to the model is 39 % of the phonon temperature dierence directly at the Pt jYIG interface. By engineering devices in which the phonon heat loss through the substrate is minimized by thinner or etched YIG lms could therefore signicantly enhance the measured signal. Altering the Pt injector coupling to the heat sink or placing the thermocouple on top of the Pt injector might also help. Them-phfound here is smaller than the one adopted by Ref. 6 (6nm) by roughly an order of magnitude. Actually Schreier et al.'s simulations might agree bet- ter with their measurements for smaller values as well. m-phextracted from Fig. 3 is quite sensitive to small variations in the modelling, which implies a large uncer- tainty. Nevertheless even when accepting a large error bar from 0.1 to 6 nm for mphwe may conclude that thermal magnons and phonons interact strongly [8]. The background signal in the SPE data is a factor 20 higher than we would expect from conventional charge Peltier heating and cooling at the Au jPt injector inter- faces. Reference measurements on the second thermopile on the other side of the Pt injector excludes charge cur- rent leakage to the thermopile. For an identical congu- ration, V+on the same side as I+, we nd an opposite sign of the measured voltage, as expected for a thermal4 spin Seebeck signal (/uni03BCV)spin Peltier signal (nV) λm–ph (nm) λm–ph (nm) (a) (b) κm= 1e-3 and κI ph = 2e8κm= 1e-2 and κI ph = 2.79e8κm= 1e-2 and κI ph = 2e8 κm= 1e-3 and κI ph = 2.79e8κm= 1e-3 and κI ph = 2e8κm= 1e-2 and κI ph = 2.79e8κm= 1e-2 and κI ph = 2e8 κm= 1e-3 and κI ph = 2.79e8 0 1 2 3 4 504080120 0 1 2 3 4 501020304050 FIG. 3. (color online). The modeled SPE (a) and SSE (b) signal versus m-phfor two dierent values of m(W/(m K)) and two dierent values of I ph(W/(m2K)). The semi transparent blue bar indicates the range of measured SPE and SSE eect signals. signal since the Pt jNi45Cu55thermopile sequence is in- verted. A current leak would not change sign and can therefore be excluded. The background in the second harmonic voltage is likely to be caused by the thermo- voltage across the thermopile due to Joule heating in the Pt injector, since its value agrees within 17% with the modeled one. Additional measurements of frequency de- pendent properties (see Supplementary V) rule out pick- ups due to capacitive or inductive couplings. We checked that the sign of the experimentally ob- served SPE and SSE signals obey reciprocity. Further- more the voltage measured across the Pt detector in a RF spin pumping measurement matches the sign of the SSE voltage for the same geometry when heating the YIG relative to the Pt, as previously reported [20, 21]. How- ever, the absolute sign of these three eects is still under investigation. In conclusion, we report experimental proof that a spin accumulation at a Pt jYIG interface induces heat ex- change between electrons and magnons on both sides. Using thermal modelling to knit the theory of inter- face transport to the observables we demonstrate that the SPE is the Onsager reciprocal of the SSE and con- rm a strong interaction between thermal magnons and phonons in YIG, as reported earlier [8]. We hope that these results can contribute to a better understanding of coupling between thermomagnetic and thermoelectric properties. Our proof of principle opens new strategies for nanoscale cooling applications. We would like to acknowledge B. Wolfs, M. de Roosz and J. G. Holstein for technical assistance. This work is part of the research program of the Foundation for Fundamental Research on Matter (FOM) and supported by NanoLab NL, Marie Curie ITN Spinicur, DFG Pri- ority Programme 1538 "Spin-Caloric Transport", Grant-in-Aid for Scientic Research A (Kakenhi) 25247056 and the Zernike Institute for Advanced Materials. J.Flipse@rug.nl [1] K. Uchida, J. Xiao, H. Adachi, J. I. Ohe, S. Takahashi, J. Ieda, T. Ota, Y. Kajiwara, H. Umezawa, H. Kawai, G. E. W. Bauer, S. Maekawa, and E. Saitoh, Nature Materials 9, 894 (2010). [2] J. Xiao, G. E. W. Bauer, K. Uchida, E. Saitoh, S. Maekawa, Phys. Rev. B 81, 214418 (2010). [3] H. Adachi, J. I. Ohe, S. Takahashi, S. Maekawa, Phys. Rev. B 83, 094410 (2011). [4] S. Homan, K. Sato, Y. Tserkovnyak, Phys. Rev. B 88, 064408 (2013). [5] A. Homann, IEEE Trans. Magnetics 49, 5172 (2013). [6] M. Schreier, A. Kamra, M. Weiler, J. Xiao, G. E. W. Bauer, R. Gross, S. T. B. Goennenwein, Phys. Rev. B88, 094410 (2013). [7] L. Onsager, Phys. Rev. 37, 405426 (1931). [8] M. Agrawal, V. I. Vasyuchka, A. A. Serga, A. D. Karenowska, G. A. Melkov, B. Hillebrands, Phys. Rev. Lett. 111, 107204 (2013). [9] N. Roschewsky, M. Schreier, A. Kamra, F. Schade, K. Ganzhorn, S. Meyer, H. Huebl, S. Gepr ags, R. Gross, S. T. B. Goennenwein, arXiv 1309.3986v1 (2013). [10] A. Brataas, G. E. W. Bauer, P. J. Kelly, Phys. Rep. 427, 157 (2006). [11] Z. Wang, Y. Sun, M. Wu, V. Tiberkevich, A. Slavin, Phys. Rev. Lett. 107, 146602 (2011). [12] H. B. Callen, Phys. Rev. 73, 1349 (1948). [13] H. J. Jiao, J. Xiao, G. E. W. Bauer, in preparation. [14] N. Vlietstra, J. Shan, V. Castel, B. J. van Wees, J. Ben Youssef, Phys. Rev. B 87, 184421 (2013). [15] COMSOL Multiphysics®4.2a. [16] N. Vlietstra, J. Shan, V. Castel, J. Ben Youssef, G. E. W. Bauer, B. J. van Wees, Appl. Phys. Lett. 103, 032401 (2013).5 [17] R. L. Douglass, Phys. Rev. 129, 1132 (1963). [18] E. T. Swartz, R. O. Pohl, Rev. Mod. Phys. 61, 605 (1989). [19] F. L. Bakker, J. Flipse, B. J. van Wees, J. Appl. Phys. 111, 084306 (2012). [20] C. W. Sandweg, Y. Kajiwara, A. V. Chumak, A. A. Serga, V. I. Vasyuchka, M. B. Jung eisch, E. Saitoh, B. Hillebrands, Phys. Rev. Lett. 106, 216601 (2011). [21] M. Weiler, M. Althammer, M. Schreier, J. Lotze, M. Pernpeintner, S. Meyer, H. Huebl, R. Gross, A. Kamra, J. Xiao, Y. T. Chen, H. J. Jiao, G. E. W. Bauer, S. T. B. Goennenwein, Phys. Rev. Lett.111, 176601 (2013). [22] W. Wang, D. G. Cahill, Phys. Rev. Lett. 109, 175503 (2012). [23] K. Uchida, H. Adachi, T. Ota, H. Nakayama, S. Maekawa, E. Saitoh, Appl. Phys. Lett. 97, 172505 (2010). [24] M. Weiler, M. Althammer, F. D. Czeschka, H. Huebl, M. S. Wagner, M. Opel, I. Imort, G. Reiss, A. Thomas, R. Gross, S. T. B. Goennenwein, Phys. Rev. Lett. 108, 106602 (2012).6 SUPPLEMENTARY INFORMATION I. Onsager reciprocity for the spin Seebeck and spin Peltier eect The linear response matrix of thermoelectrics re ects Onsager reciprocity when the sum of the products of currents times driving forces equals the dissipation [12]. When IcandQare the charge and heat currents driven by voltage and temperature dierences
Vand
T,_F=Ic
V+Q
T=T equals the dissipation and we obtain the symmetric response matrix [12]:  Ic Q = G GST GST KT
V
T=T (6) whereGis the electrical conductance, Sthe Seebeck coecient, and Kthe heat conductance. Here the Onsager- Thomson relation for the Peltier coecient  = SThas already been implemented. In the case of the spin Seebeck eect (SSE) and the spin Peltier eect (SPE) for magnetic insulators, the spin accumulation at the interface drives a spin current. To ensure reciprocity, we have to compute the Joule heating caused by the spin currents: _F=I"
V"+I#
V#=G"
V2 "+G#
V2 # (7) where the subscripts denote the up and down spin contribution. Comparing this with the product of Iswith
Vs: Is
Vs= (G"
V"G#
V#)(
V"
V#) = 2(G"
V2 "+G#
V2 #) (8) we conclude that
Vs=2 is the proper driving force. The linear response relations for spin and heat current densities at the interface between a normal metal and a ferromagnet then reads (Eq. (1) of the main text, where all variables and parameters are introduced): Js Qme =gSLST LST{I STVs=2 Tm-e=T (9) and we omitted the charge sector because we are dealing with a ferromagnetic insulator. II. Measurement with B rotated 90 We repeated the measurements in the main text after rotating the magnetic eld by 90(see Fig. 4). The magnetization direction is here parallel to the current in the Pt lm such that the current-induced spin accumulation is normal to the magnetization. The background voltage and noise level remain unmodied; we do not detect a heating or cooling of the ferromagnet. This conrms our interpretation of the experiments in the main text. The spin torque normal to the magnetization is not expected to aect the magnon temperature and the SPE should vanish, as observed. III. The 3D nite element model We adopt the three reservoir model of thermalized electron, magnon, and phonon systems. Charges are transported by the electron system only, while heat currents ow in all subsystems. We take into account spin angular momentum currents in the electron and magnon system, but disregard the phonon angular momentum current. The bulk charge and heat currents in linear response are given by Eq. (4) in the main text. The charge and energy conservation relations read: ~r
0 BBBBBBBBBB@~Jc ~Qph ~Qm ~Qe1 CCCCCCCCCCA=0 BBBBBBBBBB@0 (1P2 m-ph)(m+ph) 42 m-ph(TmTph) +(1P2 e-ph)(e+ph) 42 e-ph(TeTph) (1P2 m-ph)(m+ph) 42 m-ph(TmTph) J2 c (1P2 e-ph)(e+ph) 42 e-ph(TeTph)1 CCCCCCCCCCA(10)7 −10 −5 0 5 100.450.460.47 Pt NiCuNiCu PtV+ V-I+ I-BIPt injector= 3mA B (mT)Vthermopile (/uni03BCV) FIG. 4. (a) First harmonic voltage across the thermopile as a function of applied magnetic eld parallel to the Pt injector. (b) Optical microscope picture of the measured device and measurement geometry. withPmph= (mph)=(m+ph),Peph= (eph)=(e+ph) andJ2 c=accounts for Joule heating. The other terms describe the heat exchange between phonons, magnons and electrons, as indicated by the subscripts. All parameters have been dened in the main text, except for the thermal relaxation lengths mphandeph:The former is used as an adjustable parameter while the latter eph=p e=gis calculated from the electron-phonon coupling parameter ( g) for Au and Pt [22]. This leads to a Pt ephof 4.5 nm and a Au ephof 80 nm at room temperature. The boundary conditions at the metal jYIG interfaces are governed by the phonon ( I ph) and magnetic ( {I S) interface heat conductances. We adopt the values for I phfrom Ref. 6, while that for Ni 45Cu55jYIG is assumed to be equal to that of PtjYIG. The interface magnetic heat conductance is calculated using [6] {I S=h e2kBT ~BkBgr MsV2m 1 +gScotht 1 (11) whereV2mis the (spin pumping) magnetic coherence volume. The currents through the interface read 0 BBBBBBBBBB@JI c QI ph QI m QI e1 CCCCCCCCCCA=0 BBBBBBBBBB@0 I ph(Ta phTb ph) {I S(TmTe)LSTVs 2 {I S(TmTe) +LSTVs 21 CCCCCCCCCCA(12) whereTPt/YIG phare the phonon temperatures on the Pt/YIG side of the interface. QI mandQI eare the interface magnetic heat currents. The rst terms on the r.h.s represent the heat current driven by temperature dierences, while the second term is the heat current associated with the magnon injection by an applied Vs(see Eq. (2) in the main text) that is responsible for the SPE. Eq. (4) in the main text is solved for the SPE and SSE congurations, taking into account energy conservation (Eq. (10)) and boundary conditions (Eq. (12)). The results are plotted in Figs. 3 (a) and (b) of the main text. IV. Spin Seebeck eect To verify reciprocity we fabricated samples nominally identical to the Pt jYIG heterostructures used for the SPE experiments in order to measure the longitudinal SSE [23, 24]. Fig. 5(b) gives a picture of such a device consisting8 −20−15 −10 −5 05101520−30−20−100102030VPt detector (/uni03BCV) B (mT)Iheater= 5mA BI+ I-V+ V- 100 /uni03BCm(a) (b) FIG. 5. (a) Second harmonic voltage across the Pt detector as a function of applied magnetic eld. (b) Optical microscope picture of the measured device and the measurement geometry used. of a 5 nm thick sputtered Pt detector (250 x 10 m2) on top of a GGG jYIG substrate as used for the study of the SPE eect. The Pt detector is covered with a 70 nm aluminium oxide (Al 2O3) layer that electrically isolates the Pt detector from a 40 nm Pt lm heater evaporated on top. Both the detector and heater are contacted by a 100 nm thick Au layer to large bonding pads. By Joule heating, a charge current through the heater creates a thermal gradient over the Pt jYIG interface. The hot electrons transfer energy to the cold magnons, which is associated with a spin current in Pt that is converted to an observable charge current by the inverse spin Hall eect. This is the SSE. In Fig. 5(a) the second harmonic voltage versus magnetic eld is a measure of the Joule heating. A clear SSE signal is observed, which changes sign when the magnetization is reversed, as expected. The signal scales quadratically with the current and for B parallel to the Pt detector no SSE signal is detected, which conrms that the voltage is due to the inverse spin Hall eect. The model discussed in the main text and the previous section can be applied to this measurement geometry to nd theTmeat the interface and the transverse voltage over the Pt detector [6]: VSSE= 2
gr ~kB 2MsV2mTme
4 el
 ttanht 2
1 +gScoth (t=)(13) wherelis the length of the Pt detector. The results obtained from the SSE modeling are shown in Fig. 3(b) of the main text. 2 4 6 8 10 12 14 16 180.290.30.310.32 0Vthermocouple X-component (/uni03BCV) Measurement frequency (Hz)2 4 6 8 10 12 14 16 18 0481216Vthermocouple Y-component (/uni03BCV) Measurement frequency (Hz) (a) (b)IPt injector= 2 mA IPt injector= 2 mA FIG. 6. (a) First harmonic voltage across the thermopile in phase with the current ( X-component). (b) First harmonic voltage across the thermopile out-of-phase with the current ( Y-component).9 −10 −5 0 5 100.70.720.74Vthermopile (/uni03BCV) B (mT)IPt injector= 4 mA FIG. 7. First harmonic voltage across the thermopile as a function of applied magnetic eld for a 4 mA current through the Pt injector with frequency of 3 Hz. V. Frequency dependent measurements We measured the voltage as function of frequency in order to exclude any capacitive or inductive coupling between the Pt injector and the thermopile (see Fig. 6). The in-phase voltage ( x-component) slightly decreases with frequency, leading to a frequency-dependent voltage of around 10% at our usual measurement frequency of 17 Hz (Fig. 6(a)). The out-of-phase voltage ( Y-component) linearly depends on frequency and vanishes for zero frequency, as expected. In Fig. 7 the rst harmonic voltage across the thermopile is shown for a 4 mA current through the Pt injector with a frequency of 3 Hz. A clear SPE signal is observed with the same magnitude as for the measurements at 17 Hz . From these results we can safely conclude that the measured signal is not aected by any spurious capacitive or inductive signals. Furthermore at these low frequencies the thermal time constants are much shorter than the period of the measurement modulation.

2013-11-19

We report the observation of the spin Peltier effect (SPE) in the ferrimagnetic insulator Yttrium Iron Garnet (YIG), i.e. a heat current generated by a spin current flowing through a Platinum (Pt)|YIG interface. The effect can be explained by the spin torque that transforms the spin current in the Pt into a magnon current in the YIG. Via magnon-phonon interactions the magnetic fluctuations modulate the phonon temperature that is detected by a thermopile close to the interface. By finite-element modelling we verify the reciprocity between the spin Peltier and spin Seebeck effect. The observed strong coupling between thermal magnons and phonons in YIG is attractive for nanoscale cooling techniques.

Observation of the spin Peltier effect

1311.4772v1

1 Spin Hall magnetoresistance in Pt/(Ga, Mn)N devices J. Aaron Mendoza- Rodarte1,2*, Katarzyna Gas3,4, Manuel Herrera- Zaldívar2, Detlef Hommel5, Maciej Sawicki3,6, and Marcos H. D. Guimarães1* 1Zernike Institute for Advanced Materials, University of Groningen, 9747 AG Groningen, The Netherlands 2Centro de Nanociencias y Nanotecnología-Universidad Nacional Autónoma de México, Ensenada, 22800-Baja California, México 3Institute of Physi cs, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL -02668 Warsaw, Poland 4Center for Science and Innovation in Spintronics, Tohoku University, Katahira 2-1 -1, Aoba-ku, Sendai 980-8577, Japan 5Lukasiewicz Research Network - PORT Polish Center for Techn ology Development, Stabłowicka 147, Wrocław, Poland 6Research Institute of Electrical Communication, Tohoku University, Katahira 2-1-1, Aoba-ku, Sendai 980-8577, Japan E-mail: j.a.mendoza.rodarte@rug.nl , m.h.guimaraes@rug.nl Diluted magnetic semiconductors (DMS) have attracted significant attention for their potential in spintronic applications. Particularly, magnetically -doped GaN is highly attractive due to its high relevance for the CMOS industry and the possibility of deve loping advanced spintronic devices which are fully compatible with the current industrial procedures . Despite this interest, there remains a need to investigate the spintronic parameters that characterize interfaces within these systems. Here , we perform s pin Hall magnetoresistance (SMR) measurements to evaluate the spin transfer at a Pt/(Ga,Mn)N interface. We determine the transparency of the interface through the estimation of the real part of the spin mixing conductance finding 𝐺𝐺𝑟𝑟 = 2.6 ⋅1014 Ω-1 m-2, comparable to state- of-the-art yttrium iron garnet (YIG)/Pt interfaces . Moreover, the magnetic ordering probed by SMR above the (Ga,Mn)N Curie temperature TC provides a broader temperature range for the efficient generation and detection of spin currents, relaxing the conditions for this material to be applied in new spintronic devices . 2 Recent advances in the manipulation of electron charge and spin have highlighted spintronics as a key area of research1,2. This interest stems from the potential of spintronic devices to address the limitations of traditional charged -based devices, such as high energy consumption3. Among various materials, diluted magnetic semiconductors (DMS) , those systems where a fraction of cations are substituted by magnetic elements, stand out as promising candidates for spintronic applications due to their unique integration of semiconductor and magnetic properties4. Particularly , Mn-doped compounds have received significant attention from the rese arch community since the formation of a ferromagnetic order at technologically relevant temperatures was predicted5 and successfully realized6. Among these compounds, single -phase Mn -doped GaN epitax ial layers stand out for their insulating and short -range ferromagnetic character at low temperatures . It presents a rich playground to explore magnetic interactions of magnetic ions in a semiconductor lattice. A brief discussion on the ferromagnetic mechanism s in (Ga,Mn)N is included in the supplementary material. The combination of these factors, coupled with GaN well-established roles in optoelectronics7, high- frequency8,9, and power electronic s10 could provide substantial technological benefits, particularly in the development of new GaN- based spintronic devices11. In fact apart from exploitation of the magnetoelectric e ffect in (Ga,Mn)N so far 12, the study of GaN -based DMS has been focused on optimizing growth conditions13 and magnetic properties14,15, while spin transport dynamics received comparatively less attention16–20. It is therefore timely and important to exploit spintronic and magnonic21 properties of this insulating ferromagnetic material in an aim to foster the development of all -nitride low power information processing and dissipationless communication means based on spin waves propagation. Of particular importance for new nonvolatile magnetic data storage applications, t he field of spin-orbit torques (SOTs) focuse s on manipulating the magnetization in thin film heterostructures via electric currents , providing a promising way to manipulate magnetization at the nanoscale22. This technology relies on transferring spin angular momentum from a normal metal into a ferromagnet, using spin- charge interconversion effects to exert a torque on the magnetization22. Significant a dvancement s in SOT have catalyzed the development of non- volatile memory devices23,24. However, the success of these devices depends on efficient generation, transport, and detection of spin currents. While material research with high spin - orbit coupling25 or orbital angular momentum effects26 addresses spin current generation, 3 transport issues are managed by identifying materials with high interface spin transparency, indicated by substantial spin mixing conductance values ( G↑↓). Spin Hall magnetoresistance (SMR) has emerged as a key technique for assessing spin transport properties at heterostructure interface s through electric al methods27. Despite these advances, there is still a significant gap in understanding spin transport properties in heterostructures that utilize transition metal -doped GaN, which is crucial for SOT applications. In this study, we investigate spin Hall magnetoresistan ce (SMR) within a Pt/(Ga ,Mn)N heterostructure. By performing angle -dependent magnetoresistance measurements, we determine the spin mixing conductance (G↑↓), a critical parameter that dictat es the transport of spin information through the interface between the two materials . Additionally, we provide another assessment of the Curie temperature of the magnetic layer solely through electrical means, via temperature- dependent measurements. Our results provide valuable insights for the development of devices that incorporate GaN- based DM S for advanced spintronic applications. To characterize the Pt/ (Ga,Mn)N interface via SMR, we fabricated a 6 nm -thick Pt Hall bar on a 100 nm -thick single -phase epitaxy Ga0.922Mn 0.078N film. The film was grown using plasma- assisted molecular beam epitaxy (MBE) on a 3 μm-thick GaN(0001) template , which was deposited on c -oriented 2- inch sapphire substrates. Detailed methodologies for the growth and magnetic and crystallographic characterization of these films are discussed in detail in reference 14. The Hall bar, measuring 25 μm in wide and 200 μm in length, was fabricated by conventional lithography processes . Prior to the deposition of Pt through Ar+ plasma d.c. sputtering, the (Ga ,Mn)N surface was cleaned using an Ar+ mild etching process, detailed in supplementary material. Electrical measurements were performed by rotating a n external magnetic field within the plane of the device . We utilized conventional lock-in technique s for the measurements, with a bias current ( 𝐼𝐼𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏) of less than 0.7 mA at a frequency of 187.77 Hz . This setup allowed us to measure the transverse resistance ( Rxy), which is perpendicular to the current path . This layout of the electrical interconnects is illustrated in Fig. 1( a). 4 FIG. 1. (a) Schematic s of the devices and electrical measurement configuration, illustrating the definition of the rotating external magnetic field angle α. (b) Angle dependence of the SMR signal in the transverse geometry as a function of α. The measurements were conducted in the temperature range of 4.7 to 50 K using an external magnetic field of 600 mT. A baseline has been removed so that the relative changes in resistivity are zero at α=0 degrees. (c-d) Schematic representation of SMR. (c) Low -resistance configuration where 𝑀𝑀��⃗||𝜎𝜎⃗ . (d) High - resistance configuration where 𝑀𝑀��⃗⊥𝜎𝜎⃗. The SMR measurements offer insights into the magnetic ordering at the surface of the (Ga,Mn)N thin film at low temperatures. Fig. 1( b) illustrates the dependence of the magnetic field angle (α) on the relative changes in the transverse resistivity, ∆𝜌𝜌𝑥𝑥𝑥𝑥/𝜌𝜌. Here, ∆𝜌𝜌𝑥𝑥𝑥𝑥/𝜌𝜌 is defined as (𝑅𝑅𝑥𝑥𝑥𝑥−𝑅𝑅0,𝑥𝑥𝑥𝑥)/(𝑅𝑅0,𝑥𝑥𝑥𝑥/4.8) where 𝑅𝑅𝑥𝑥𝑥𝑥 is the transverse resistance, and 𝑅𝑅0,𝑥𝑥𝑥𝑥 and 𝑅𝑅0,𝑥𝑥𝑥𝑥 are the transverse and longitudinal resistance s at zero magnetic field, respectively, and 4.8 is the geometrical factor of the Hall bar (length /width ). The observed changes in resistivity can be explain ed as follows: when a charge current ( Jc) flows through Pt, the spin Hall effect (SHE ) induces a transverse spin current ( Js) towards the (Ga ,Mn)N/Pt interface. At this interface, spin accumulation interacts with the local magnetic moments of (Ga,Mn)N. If the spin polarization 𝜎𝜎⃗ is parallel to the magnetization direction 𝑀𝑀��⃗ of the ferromagnet ( 𝑀𝑀��⃗||𝜎𝜎⃗), the electrons' angular momentum is reflected, resulting in a low -resistance configuration due to the inverse spin Hall effect (ISHE ) [see Fig. 1( c)]. Conversely, when 𝑀𝑀��⃗ is perpendicular to 𝜎𝜎⃗ (𝑀𝑀��⃗⊥𝜎𝜎⃗) [refer to Fig. 1( d)], the electrons' angular momentum is absorbed, leading to a high - resistance configuration . In transverse SMR measurements, the interaction of the generated (a) (b) (c) (d) 4.7 K 15 K 50 K 5 spin current with the (Ga ,Mn) N magnetization 𝑀𝑀��⃗ at the interface results in a reorientation of the outgoing spin current polarization 𝜎𝜎⃗. At angles α = 45 and 135°, this leads to positive and negative transverse voltage, respectively28. Our experimental observations align with this theory , showing maximum resistance at α = 45° and minimum at α = 135°, similar to findings in YIG/Pt systems29. The dependence of the resistivity in the transverse geometry ( 𝜌𝜌𝑇𝑇) with respect to the magnetization components along the different directions is given by27: 𝜌𝜌𝑇𝑇=∆𝜌𝜌1𝑚𝑚𝑥𝑥𝑚𝑚𝑥𝑥+∆𝜌𝜌2𝑚𝑚𝑧𝑧+∆𝜌𝜌𝐻𝐻𝑏𝑏𝐻𝐻𝐻𝐻𝐵𝐵𝑧𝑧, (1) where ∆𝜌𝜌1 and ∆𝜌𝜌2 represents relative changes in resistivity and 𝑚𝑚𝑥𝑥, 𝑚𝑚𝑥𝑥, and 𝑚𝑚𝑧𝑧 are the components (unit vectors) of magnetization in the 𝑥𝑥�-, 𝑦𝑦�- and 𝑧𝑧̂-direction, respectively. The angular dependenc ies are defined by the components 𝑚𝑚𝑥𝑥 , 𝑚𝑚𝑥𝑥 , and 𝑚𝑚𝑧𝑧 , where 𝑚𝑚𝑥𝑥= cos(𝛼𝛼)cos(𝛽𝛽) , 𝑚𝑚𝑥𝑥=sin(𝛼𝛼)cos(𝛽𝛽) , and 𝑚𝑚𝑧𝑧=sin(𝛽𝛽) , with 𝛽𝛽 representing the azimuthal angle – i.e. the angle with respect to the out -of-plane (OOP) direction. Using the se definitions, we can express the change in the transverse resistivity as follows27: ∆𝜌𝜌𝑥𝑥𝑥𝑥/𝜌𝜌=∆𝜌𝜌𝑥𝑥𝑥𝑥,11 2sin(2𝛼𝛼)−∆𝜌𝜌𝑥𝑥𝑥𝑥,2sin (𝛽𝛽), (2) where ∆𝜌𝜌𝑥𝑥𝑥𝑥,1 and ∆𝜌𝜌𝑥𝑥𝑥𝑥,2 represent the amplitudes of the relative change in resistivity.26) This equation effectively describes our observed signals and fit s our measurements well. Additionally, t he term ∆𝜌𝜌𝐻𝐻𝑏𝑏𝐻𝐻𝐻𝐻𝐵𝐵𝑧𝑧 account s for the ordinary Hall effect oc curring in Pt when a magnetic field is applied i n the 𝑧𝑧̂-direction . The sin (𝛽𝛽) component observed in our results likely results from OOP component due to misalignment of the Hall bar with respect to the in - plane field . At 4.7 K, the values for the amplitudes ∆𝜌𝜌𝑥𝑥𝑥𝑥,1 and ∆𝜌𝜌𝑥𝑥𝑥𝑥,2 are 7.2⋅10-6 and 1.8⋅10- 6, respectively , indicating a clear dominance of the in- plane component . Using SMR measurements, we investigate the ferromagnetic -paramagnetic transition of (Ga,Mn)N solely through electrical methods . To this end, we perform a temperature -dependent measurement ranging from 4.7 to 290 K. Fig. 2 displays the behavior of magnitudes ∆𝜌𝜌𝑥𝑥𝑥𝑥,1 and ∆𝜌𝜌𝑥𝑥𝑥𝑥,2 derived from Eq. (1) across different temperatures. As the temperature increased from 4.7 to 10 K, ∆𝜌𝜌𝑥𝑥𝑥𝑥,1 fluctuate d between 7.2⋅10-6 and 7.3⋅ 10-6 with the maximum value at 10 K . Above 10 K, a decrease in ∆𝜌𝜌𝑥𝑥𝑥𝑥,1 was noted, dropping to 6.81⋅ 10-6 at 15 K . This reduction in the SMR signal suggests a TC between 10 to 15 K , aligning well with the TC of 13.0 ± 0.3 K established by SQUID magnetometry14. The signal diminished above 50 K , and data fitting bec ame unreliable due to a significant decrease in the signal -to-noise ratio (SNR). 6 Nevertheless, it is important to note that substantial SMR signal s have been observe d from 15 to 50 K , a range dominated by the paramagnetic phase. This magnetic ordering is similar to that seen in other Curie -like paramagnetic insulator s such as Gd3Ga5O12 (GGG)30,31. In such materials , the SMR arises from the interaction between the conduction -electron spins in Pt and the paramagnetic spins S within GGG via the interface exchange interaction, which applies torque on S. Regarding the fitted OOP component, its temperature dependence displayed a monotonic behavior, with ∆𝜌𝜌𝑥𝑥𝑥𝑥,2 values between 1.6 and 1.9 ⋅ 10-6. This points towards an ordinary Hall contribution in our transverse measurement s, which can be attributed to a sample misalignment. FIG. 2. Temperature dependence of the relative changes of the resistivity of ∆𝜌𝜌𝑥𝑥𝑦𝑦,1 and ∆𝜌𝜌𝑥𝑥𝑦𝑦,2. The green shadow highlight s the decrease of the SMR signal observed above 10 K. All the amplitudes were extracted from Eq. (1) using measurements performed at 600 mT. The magnetic field depend ence of ∆𝜌𝜌𝑥𝑥𝑥𝑥,1 and ∆𝜌𝜌𝑥𝑥𝑥𝑥,2 demonstrates quadratic and linear behaviors , respectively , as shown in Fig. 3a and 3b. We can attibute the former dependency to the percolating character of ferromagnetism in our material, indicating that only at T = 0 all the spins present in the sample are ferromagnetically coupled. On increasing T some of the spins start decoupling such that at its equilibrium T C only about 20% of spins remain in the infinite cluster32. The remaining spins reside in finite ferromagnetic clusters of different sizes. These magnetic granules are characterized by their own magnetic moments and are responsible for the glassy characteristics like blocking ( i.e. as in the material being far from thermal 7 equilibrium). Such material can therefore mimic a ferromagnet well above its equilibrium T C when it is probed on sufficiently short time scales, as during SMR experiment s. Additionally, at 600 mT we do not observe a saturation state, a f inding corroborated by SQUID magnetometry M-H curves, which indicate that even at µ0H = 7 T at T = 2 K the system still exhibits a positive slope (as indicated in the supplementary material Fig. S2). The coercive field (H c) values from SQUID magnetometry for similar samples suggest an increase of the average magnetization above 200 mT, aligning with the clear SMR signal observed at 240 mT in our results (Fig. 3a). Additionally , the linear dependence seen in ∆𝜌𝜌𝑥𝑥𝑥𝑥,2 aligns with the expected behavior of the ordinary Hall component. To confirm that our measurements are within the linear regime, we conducted bias current (𝐼𝐼𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏) dependenc e tests. Fig. 3c shows the transverse voltage (𝑉𝑉𝑥𝑥𝑥𝑥) as a function of α , and Fig. 3d displays the 𝐼𝐼𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 dependency of the relative change in transverse voltage ∆𝑉𝑉𝑥𝑥𝑥𝑥, where a data point represents the amplitude of the angle -dependent voltage, obtained by a fit to 𝑉𝑉𝑥𝑥𝑥𝑥=∆𝑉𝑉𝑥𝑥𝑥𝑥,11 2sin(2𝛼𝛼)−∆𝑉𝑉𝑥𝑥𝑥𝑥,2sin (𝛽𝛽). Linear fits applied to both the in-plane (∆𝑉𝑉𝑥𝑥𝑥𝑥,1) and out-of-plane (∆𝑉𝑉𝑥𝑥𝑥𝑥,2) components confirm the linear behavior , with no evident heating effects. FIG. 3. (a) Relative change of the transverse resistivity ρxy as a function of angle α at different values of magnetic field B and for bias current 𝐼𝐼𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 =1.0 mA . (b) M agnetic field dependence of ∆𝜌𝜌𝑥𝑥𝑥𝑥,1 and ∆𝜌𝜌𝑥𝑥𝑥𝑥,2 . (c) Angle dependence of 𝑉𝑉𝑥𝑥𝑥𝑥 at different 𝐼𝐼𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏. (d) Current dependence of ∆𝑉𝑉𝑥𝑥𝑥𝑥. All the measurements were performed at a fixed temperature of 4.7 K. A baseline has been removed so that the relative changes in resistivity are zero at α=0 degrees. 4.7 K4.7 K (a) (c) (d)(b) 600 mT 480 mT 360 mT 0 mT 1.0 mA 0.7 mA 0.3 mA 8 The transfer of spin angular momentum at the interface of the heterostructure is quantified by the spin mixing conductance, G↑↓, which we estimate from our experimental results. From Eq. (1), we can express ∆𝜌𝜌1 and ∆𝜌𝜌2 as follows27: ∆𝜌𝜌1 𝜌𝜌=𝜃𝜃𝑆𝑆𝐻𝐻2𝜆𝜆 𝑑𝑑𝑁𝑁Re�2𝜆𝜆G↑↓tanh2𝑑𝑑𝑁𝑁 𝜆𝜆 𝜎𝜎+2𝜆𝜆G↑↓coth𝑑𝑑𝑁𝑁 𝜆𝜆� , (3) ∆𝜌𝜌2 𝜌𝜌=−𝜃𝜃𝑆𝑆𝐻𝐻2𝜆𝜆 𝑑𝑑𝑁𝑁Im�2𝜆𝜆G↑↓tanh2𝑑𝑑𝑁𝑁 𝜆𝜆 𝜎𝜎+2𝜆𝜆G↑↓coth𝑑𝑑𝑁𝑁 𝜆𝜆� . (4) Here, 𝜃𝜃𝑆𝑆𝐻𝐻 represents the spin Hall angle, λ the spin relaxation length, 𝑑𝑑𝑁𝑁 the thickness, and 𝜎𝜎 the conductivity of Pt. G↑↓ denotes the relaxation of the spin current polarization component transverse to the magnetization at the Pt/( Ga,Mn)N interface33 and is defined as the sum of its real and imaginary part s (G↑↓=𝐺𝐺𝑟𝑟+𝑖𝑖𝐺𝐺𝑏𝑏). In our calculations the imaginary part is neglected due to its significantly smaller contribution compared to the real part34. The Pt conductivity 𝜎𝜎=1/𝜌𝜌 is calculated from the resistivity measured by a four-probe method on the Pt Hall bar, yielding a value of 6.3⋅ 106 Ω-1 m-1, consistent with literature values 35–37. Assuming constant values of 𝜃𝜃𝑆𝑆𝐻𝐻 = 0.08, 𝜆𝜆 = 1.1 ± 0.3 nm 38,39, taking 𝑑𝑑𝑁𝑁 = 6 nm , and using our SMR data at 4 .7 K and B = 600 mT , we calculate the real part of the spin mixing conductance , which under G↑↓≈𝐺𝐺𝑟𝑟 condition attains a value of 2.6 ⋅1014 Ω-1 m-2. This value is comparable to those reported for other heavy metal/magnetic insulator interfaces such as YIG/Pt38. Our study shows that t he Pt/(Ga ,Mn)N interface exhibits spintronic properties comparable to state -of-the-art systems , such as YIG/Pt interfaces39, as evidenced by similar spin mixing conductance values . This indicates the strong potential of magnetically- doped GaN for spintronic and magnonic applications suc h as magnonic transistors40 and SOT devices41. Furthermore , the magnetic ordering probe d by SMR in (Ga,Mn)N above its equilibrium Curie temperature extends sizably the temperature range for efficient generating and detecting spin currents in this material. This broadens the operational parameters for its application in novel spintronic devices with higher operating temperatures . Our findings contribute significantly to understanding spin transport in (Ga ,Mn)N -based devices and set the stag e for exploring various device architectures for new spintronic applications based on the nitride family . 9 SUPPLEMENTARY MATERIAL See supplementary material [url] for a brief discussion of the ferromagnetic mechanism in (Ga,Mn)N insulating systems, a detailed description of Pt/(Ga,Mn)N device fabrication , and SQUID magnetometry measurements , which includes Ref.12,14,32,42– 60. ACKNOWLEDGMENTS We acknowledge M. Cosset -Chéneau and J. J. L. van Rijn for helpful discussions and critically reading the manuscript, and J. Holstein, H. de Vries, F.H. van der Velde, H. Adema, and A. Joshua for technical support . This work was supported by the Dutch Research Council (NWO – OCENW.XL21.XL21.058), the Zernike Institute for Advanced Materials, the research program “Materials for the Quantum Age” (QuMat, registration number 024.005.006), which is part of the Gravitation program financed by the Dutch Ministry of Education, Culture and Science (OCW), the National Science Centre (Poland) through project OPUS (DEC- 2018/31/B/ST3/03438), and the European Union (ERC, 2D -OPTOSPIN, 101076932, and 2DMAGSPIN, 101053054). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them. JAMR is grateful to CONAH CYT for a graduate research fellowship (no. CVU 655591) . The device fabrication and characterization were performed using Ze rnike NanoLabNL facilities. AUTHOR DECLARATIONS Conflict of Interest The authors have no conflicts to disclose. Author Contributions J. Aaron Mendoza- Rodarte: Conceptualization (equal) ; Methodology (lead); Validation (lead); Formal analysis (lead); Investigation (lead); Data Curation (lead); Writing – Original Draft (lead); Visualization (equal) . Katarzyna Gas: Resources (supporting) ; Writing – Review & Editing (supporting ). Manuel Herrera -Zaldívar: Supervision (supporting ); Writing – Review & Editin g (supporting ). Detlef Hommel: Resources (supporting ); Writing – Review & Editing (supporting ). Maciej Sawicki: Resources (supporting) ; Writing – Review & Editing 10 (supporting). Marcos H. D. Guimarães : Supervision (lead) ; Project administration (lead); Conceptualization (equal) ; Visualization (equal) ; Resources (lead); Writing – Review & Editing (lead); Funding acquisition (lead). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1 C. Song, R. Zhang, L. Liao, Y. Zhou, X. Zhou, R. Chen, Y. You, X. Chen, and F. Pan, “Spin - orbit torques: Materials, mechanisms, performances, and potential applications,” Prog Mater Sci 118, 100761 (2021). 2 B. Dieny, I.L. Prejbeanu, K. Garello, P. Gambardella, P. Freitas, R. Lehndorff, W. Raberg, U. Ebels, S.O. Demokritov, J. Akerman, A. Deac, P. Pirro, C. Adelmann, A. Anane, A. V. Chumak, A. Hirohata, S. Mangin, S.O. Valenzuela, M.C. Onbaşlı, M. d’Aquino, G. Prenat, G. Finocchio, L. Lopez -Diaz, R. Chantrell, O. Chubykalo -Fesenko, and P. 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Guimarães1* 1Zernike Institute for Advanced Materials, University of Groningen, 9747 AG Groningen, The Netherlands 2Centro de Nanociencias y Nanotecnología- Universidad Nacional Autónoma de México, Ensenada, 22800- Baja California, México 3Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL -02668 Warsaw, Poland 4Center for Science and Innovation in Spintronics, Tohoku University, Katahira 2- 1-1, Aoba- ku, Sendai 980- 8577, Japan 5Lukasiewicz Research Network - PORT Polis h Center fo r Technology Development, Stabłowicka 147, Wrocł aw, Poland 6Research Institute of Electrical Communication, Tohoku University, Katahira 2 -1-1, Aoba- ku, Sendai 980- 8577, Japan Short -range superexchange mechanism in (Ga,Mn)N insulating systems Mn-doped GaN systems are currently identified as d iluted ferro magnetic semiconductors (DFS), and remain as one of the most prominent members of this family, despite the ferromagnetic coupling in (Ga,Mn)N is not mediated by itinerant carriers (holes). This stems from the fact that, importantly to this study, the Mn acceptor derived holes are so strongly localized (mostly due to p-d hybridization) on Mn acceptor s1 that in the absence of other doping or numerous charged point defects Mn -doped GaN remains insulating up to the highest (currently) available concentrations in epitaxy2–6. At these instances, Mn assumes a neutral acceptor oxidation state , Mn+3 (d4 confi guration) characterized by spin and orbital momentum quantum numbers S = 2 and L = 2, respectively7, and a strong single ion uniaxial magnetic anisotropy with respect to the wurtzite c - axis of GaN8–10. The latter proved to be controllable by an externally applied electrical field11, via the inverse piezoelectric effect in GaN host lattice along the c axis12. As a consequence of the strongly insulating character of GaN enriched with Mn, the carrier - mediated coupling is precluded in (Ga,Mn) N. The observed low -temperature ferromagnetism is based on a short -range superexchange scenario2,13,14, which, below the nearest -neighbor percolation threshold, earns its effectiveness from the ferromagnetic (FM) sign of the spin -spin exchange constants up to the nearest 30th neighbors14,15, and the ferromagnetism is established in percolation fashion16,17. Details of the Pt/(Ga ,Mn)N devices f abrication Our devices were fabricated on a Ga 1-xMn xN (100 nm) single -phase epitaxial film . More details of the growth and characterization of the Ga 1-xMn xN films can be found in Ref.18. We patterned 25 μm-wide , 200 μ m-long Hall bars using a 950K polymethyl methacrylate (PMMA) positive resi st, with a solid content of 4% dissolved in ethyl lactate . To prevent charging effects during the electron beam exposure process, we spin -coated Electra -92 on top of the PMMA, which serves as a water - soluble conductive polymer. The device patterning was conducted using a Raith e -line 150 electron -beam lithography system with an acceleration voltage of 10 kV . For the development process, the Electra 92 layer is removed by immersing the sample in deionized water . Then, to develop the PMMA, the sample is immersed in a 1:3 mixture of isopropyl alcohol (IPA) and methylisobutylketon (MIBK). Next, a Pt (6 nm) layer is deposited using a sputtering machine with a base pressure better than 1× 10-7 mbar, using a n Ar plasma with a working pressure of 4 ×10-3 mbar with a power 50 W, resulting in a rate of 0.36 nm/s . Prior to deposition, the sample undergoes an Ar+ mild etching step with a power of 200 W to remove organic residues and a potential top oxide layer from the (Ga ,Mn)N exposed surface, resulting in a clean and high- quality interface. A Ti (5 nm)/ Au (55 nm) thin layer was deposited to ensure electrical contact via an electron beam evaporation system . The lift- off process is carried out by immersing the sample in acetone at 45 °C , rinsing it with IPA, and blowing it dry with nitrogen gas. Finally, the devices are bonded to a 44- pin chip carrier. The measurements are performed using a flow cryostat (Oxford Instruments) mounted between the poles of a room -temperature electromagnet with the magnetic field aligned to the in -plane direction with respect to the sample. The current is applied using a home -made current source, referenced 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 at low frequencies (<200 Hz) to avoid heating and high- frequency effects. Figure S1. Reflection high -energy electron dff raction, RHEED, pattern along [112�0] azimuth observed after the growth and cooling the sample down below 200° C. During the growth the surface quali ty of the layer has been investigated in situ by refle ction high - energy electron di ffraction (RHEED). The post growth [112 �0] RHEED patterns collected after the growth (below 200°C) is shown in Fig. S 1. The streaky and sharp RHEED pat tern with clearly visible Kikuchi lines indicates that the(Ga,Mn)N surface is smooth and relatively free of oxides or other contaminants. SQUID magnetometry measurements The Curie temperature ( TC) of the magnetic layer used for the Pt/(Ga ,Mn)N devices was previously assesd by five different methods by SQUID magnetometry , converging to the value of TC = 13.0 ± 0.3 K. More details can be found in Ref.18 where the magnetic layer for the devices shown in the main text is labelled as S605. The magnetization curves , M-H, for this sample are shown in Figure S2. We highlight the fact that the M-H curves presented in Figure S2 are representative of single -phase (Ga,Mn)N epitaxial layers18–20. Importantly, we do not register any nonlinear magnetization at the weak field region above TC, which confirm s the single magnetic phase of the (Ga,Mn)N layer investigated here . Below TC ≅ 13.0 K M responces swiftly to the applied field already at very weak fields, yet it does not reach the complete saturation even at µ 0H = 7 T at T = 2 K. This behaviour highlights the fact that the M in (Ga,Mn)N is largely controlled by the single ion properties of the Mn3+ ions (de scribed in the frame of the crystal field model) and that the orbital contribution to M exhibits a much weaker dependence on H than the spin component does20,21. 0 2 4 6 8050100150 Magnetization ( kA/m ) Magnetic Field ( T ) Ga0.922Mn0.078N 300 K50 K25 K2 K Figure S2. Magnetization , M, curves of the particular sample investigated in the main part of the letter at four selected temperatures. The magnetic field is applied in the plane of the sample (i.e. perpendicularly to the wurtzite c axis) , which is the easy orientation for M in this material 8–10. The absolute value of M specific to the (Ga,Mn)N epilayer is established upon the technique allowing in situ compensation of the unwanted signal of the substrate (sapphire) described in Ref. 22. REFERENCES 1 T. Dietl, F. Matsukura , and H. Ohno, “ Ferromagnetism of magnetic semiconductors: Zhang- Rice limit, ” Phys Rev B 66, 33203 (2002). 2 A. Bonanni, M. Sawicki, T. Devillers, W. Stefanowicz, B. Faina, T. Li, T.E. Winkler, D. Sztenkiel, A. Navarro -Quezada, M. Rovezzi, R. Jakieła, A. G rois, M. Wegscheider, W. Jantsch, J. Suffczyński, F. D ’Acapito, A. Meingast, G. Kothleitner, and T. Dietl, “Experimental probing of exchange interactions between localized spins in the dilute magnetic insulator (Ga,Mn)N, ” Phys Rev B 84, 35206 (2011). 3 T. Yamamoto, H. Sazawa, N. Nishikawa, M. Kiuchi, T. Ide, M. Shimizu, T. Inoue, and M. Hata, “Reduction in Buffer Leakage Current with Mn- Doped GaN Buffer Layer Grown by Metal Organic Chemical Vapor Deposition, ” Jpn J Appl Phys 52, 08JN12 (2013). 4 R. Adhikari , W. Stefanowicz, B. Faina, G. Capuzzo, M. Sawicki, T. Dietl, and A. Bonanni, “ Upper bound for the s -d exchange integral in n- (Ga,Mn)N:Si from magnetotransport studies ”, Phys Rev B 91, 205204 (2015). 5 L. Janicki, G. Kunert, M. Sawicki, E. Piskorska -Hommel , K. Gas, R. Jakiela, D. Hommel, and R. Kudrawiec, “ Fermi level and bands offsets determination in insulating (Ga,Mn)N/GaN structures, ” Sci Rep 7, 41877 (2017). 6 K. Kalbarczyk, K. Dybko, K. Gas, D. Sztenkiel, M. Foltyn, M. Majewicz, P. Nowicki, E. Łusakow ska, D. Hommel, and M. Sawicki, “ Electrical characteristics of vertical -geometry Schottky junction to magnetic insulator (Ga,Mn)N heteroepitaxially grown on sapphire,” J Alloys Compd 804, 415 (2019). 7 J. Kreissl, W. Ulrici, M. El- Metoui, A.- M. Vasson, A. Vasson, and A. Gavaix, “ Neutral manganese acceptor in GaP: An electron -paramagnetic- resonance study, ” Phys Rev B 54, 10508 (1996). 8 A. Wolos, M. Palczewska, M. Zajac, J. Gosk, M. Kaminska, A. Twardowski, M. Bockowski, I. Grzegory, and S. Porowski, “ Optica l and magnetic properties of Mn in bulk GaN,” Phys Rev B 69, 115210 (2004). 9 J. Gosk, M. Zajac, A. Wolos, M. Kaminska, A. Twardowski, I. Grzegory, M. Bockowski, and S. Porowski, “Magnetic anisotropy of bulk GaN:Mn single crystals codoped with Mg acceptors ,” Phys Rev B 71, 94432 (2005). 10 W. Stefanowicz, D. Sztenkiel, B. Faina, A. Grois, M. Rovezzi, T. Devillers, F. d’ Acapito, A. Navarro -Quezada, T. Li, R. Jakieła, M. Sawicki, T. Dietl, and A. Bonanni, “Structural and paramagnetic properties of dilute Ga 1-xMn xN”, Phys Rev B 81, 235210 (2010). 11 D. Sztenkiel, M. Foltyn, G.P. Mazur, R. Adhikari, K. Kosiel, K. Gas, M. Zgirski, R. Kruszka, R. Jakiela, T. Li, A. Piotrowska, A. Bonanni, M. Sawicki, and T. Dietl, “ Stretching magnetism with an electric field in a nitride semiconductor,” Nat Commun 7, 13232 (2016). 12 I.L. Guy, S. Muensit, and E.M. Goldys, “ Extensional piezoelectric coefficients of gallium nitride and aluminum nitride, ” Appl Phys Lett 75, 4133 (1999). 13 J. Blinowski, P. Kacman, and J.A. Majewski, “ Ferromagnetic superexchange in Cr -based diluted magnetic semiconductors, ” Phys Rev B 53, 9524 (1996). 14 M. Sawicki, T. Devillers, S. Gałęski, C. Simserides, S. Dobkowska, B. Faina, A. Grois, A. Navarro -Quezada, K.N. Trohidou, J.A. Ma jewski, T. Dietl, and A. Bonanni, “ Origin of low - temperature magnetic ordering in Ga 1-xMn xN,” Phys Rev B 85, 205204 (2012). 15 C. Śliwa, and T. Dietl, “Electron -hole contribution to the apparent s -d exchange interaction in III-V dilute magnetic semiconduct ors,” Phys Rev B 78, 165205 (2008). 16 I.Ya. Korenblit, E.F. Shender, and B.I. Shklovsky, “ Percolation approach to the phase transition in very dilute ferromagnetic alloys, ” Phys Lett A 46, 275–276 (1973). 17 A. Bonanni, T. Dietl, and H. Ohno, “ Dilute Magn etic Materials, ” in Handbook of Magnetism and Magnetic Materials , edited by J.M.D. Coey and S.S.P. Parkin, (Springer International Publishing, Cham, 2021), pp. 923–978. 18 K. Gas, J.Z. Domagala, R. Jakiela, G. Kunert, P. Dluzewski, E. Piskorska -Hommel, W. Paszkowicz, D. Sztenkiel, M.J. Winiarski, D. Kowalska, R. Szukiewicz, T. Baraniecki, A. Miszczuk, D. Hommel, and M. Sawicki, “ Impact of substrate temperature on magnetic properties of plasma -assisted molecular beam epitaxy grown (Ga,Mn)N,” J Alloys Compd 747, 946 (2018). 19 G. Kunert, S. Dobkowska, T. Li, H. Reuther, C. Kruse, S. Figge, R. Jakiela, A. Bonanni, J. Grenzer, W. Stefanowicz, J. von Borany, M. Sawicki, T. Dietl, and D. Hommel, “ Ga1−xMn xN epitaxial films with high magnetization, ” Appl Phys Lett 101, 022413 (2012). 20 D. Sztenkiel, K. Gas, J.Z. Domagala, D. Hommel, and M. Sawicki, “ Crystal field model simulations of magnetic response of pairs, triplets and quartets of Mn3+ ions in GaN,” New J Phys 22, 123016 (2020). 21 D. Sztenkiel , “Spin orbital reorientation transitions induced by magnetic field,” J Magn Magn Mater 572, 170644 (2023). 22 K. Gas, and M. Sawicki, “ In situ compensation method for high -precision and high -sensitivity integral magnetometry, ” Meas Sci Technol 30, 085003 (2019).

2024-05-14

Diluted magnetic semiconductors (DMS) have attracted significant attention for their potential in spintronic applications. Particularly, magnetically-doped GaN is highly attractive due to its high relevance for the CMOS industry and the possibility of developing advanced spintronic devices which are fully compatible with the current industrial procedures. Despite this interest, there remains a need to investigate the spintronic parameters that characterize interfaces within these systems. Here, we perform spin Hall magnetoresistance (SMR) measurements to evaluate the spin transfer at a Pt/(Ga,Mn)N interface. We determine the transparency of the interface through the estimation of the real part of the spin mixing conductance finding $G_r = 2.6\times 10^{14} \, \Omega^{-1} m^{-2}$, comparable to state-of-the-art yttrium iron garnet (YIG)/Pt interfaces. Moreover, the magnetic ordering probed by SMR above the (Ga,Mn)N Curie temperature TC provides a broader temperature range for the efficient generation and detection of spin currents, relaxing the conditions for this material to be applied in new spintronic devices.

Spin Hall magnetoresistance in Pt/(Ga,Mn)N devices

2405.08519v1

Local spin Seebeck imaging with scanning thermal probe Alessandro Sola,Vittorio Basso, and Massimo Pasquale Istituto Nazionale di Ricerca Metrologica, Strada delle Cacce 91, 10135, Torino, Italy Carsten Dubs INNOVENT e.V., Technologieentwicklung, Pr ussingstrasse. 27B, 07745 Jena, Germany Craig Barton and Olga Kazakowa National Physical Laboratory, Teddington TW11 0LW, United Kingdom In this work we present the results of an experiment to locally resolve the spin Seebeck eect in a high-quality Pt/YIG sample. We achieve this by employing a locally heated scanning thermal probe to generate a highly local non-equilibrium spin current. To support our experimental results, we also present a model based on the non-equilibrium thermodynamic approach which is in a good agreement with experimental ndings. To further corroborate our results, we index the locally resolved spin Seebeck eect with that of the local magnetisation texture by MFM and correlate corresponding regions. We hypothesise that this technique allows imaging of magnetisation textures within the magnon diusion length and hence characterisation of spin caloritronic materials at the nanoscale. INTRODUCTION The visualisation of domain structure in magnetism and magnetic materials is paramount in aiding the under- standing at the fundamental level and subsequent utilisa- tion of such materials in real world applications. Hence, a signicant eort is dedicated to the development of a variety of techniques which are suitable to study mag- netic materials and magnetic domains. Magnetic force microscopy (MFM) [1, 2] is a scanning probe technique capable of sensing the force gradients induced by stray elds over the surface of a magnetic material at the nanoscale. Scanning electron microscopy with polariza- tion analysis (SEMPA) [3] can also be used to visualize magnetic domains by monitoring the spin polarization of the secondary electrons interacting with the stray eld of a sample under test. In order to visualize the local magnetization M(r) (or the ux density B(r)) instead of the stray eld H(r), it is possible to use imaging tech- niques such as magneto-optic eects [4] or Lorentz mi- croscopy [5]. Electrons can also be used as a probe for the imaging of magnetic domains in electron holography [6]. The above techniques typically involve the character- ization of magnetic properties conned to the surface or thin samples; the investigation of the domain structure in bulk materials, however, requires more complex exper- iments that involves neutron scattering [7{10] or X-ray spectroscopy [11{13]. The interaction between heat and non-equilibrium spin currents in magnetic materials represents an alternative approach to image magnetic domains. Analogous to stan- dard thermoelectric eects, this interaction has been de- scribed as the thermal generation of driving power forelectron spin, i.e. the spin Seebeck eect (SSE) [14]. This eect involves pure spin currents which are produced by driving the system out of equilibrium through thermal gradients; by denition, they carry zero net charge and depend on the local magnetization of the material. The length scale lMthat governs the SSE has been demon- strated to be of the order of micrometers [15{17]. The SSE has been observed in magnetic insulator garnet fer- rites [18] such as the ferrimagnetic yttrium iron garnet Y3Fe5O12(YIG), or other ferrites [19{21]. A typical spin Seebeck device can be formed by creating a bilayer of a magnetic material and a thin metallic lm with high spin- orbit coupling like platinum or tungsten both of which are paramagnetic heavy metals. This second layer acts as the spin detector thanks to the inverse spin Hall ef- fect (ISHE) that enables a spin-charge conversion at the interface [22]. The most useful conguration is the lon- gitudinal spin Seebeck eect (LSSE) [23{25] that corre- sponds to the generation of a spin current parallel to the temperature gradient both of which are orthogonal to the local magnetisation direction which is in the sam- ple plane. The majority of experiments typically refer to spin Seebeck samples that are uniformly heated over the whole sample and are in a uniform magnetic state, i.e. at or near saturation. In these conditions, it is possible to compare the spin Seebeck characteristics to the magneti- zation loop of the sample; however it is not possible to re- solve the contributions to the spin Seebeck signal coming from regions where the orientation of the magnetization diers from the average, i.e. in a multidomain state. To resolve magnetic domains, it is necessary to go beyond the experimental conguration previously described in favour of a locally injected heat current, as described inarXiv:2005.08539v1 [cond-mat.mes-hall] 18 May 20202 Pt YIG jM jq V Pt jM jq IqYIG Pt VYIG PtYIGThermal probe Iq xyz a) b)xz z y xyzjM jq V Top viewCross section2r0Lz tPttYIG Figure 1. Longitudinal spin Seebeck measurement congurations. (a) Standard conguration with uniform heating (the red arrows 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 heating from the Pt side generates a magnetic moment current density jMin spherical symmetry. On the right side of the panel, the projections of the experimental scheme on the x-z and y-z planes are represented: the local heat current iqfrom the thermal probe spreads over a circle whose dimensions ( r0) limit the space resolution. For Pt/YIG sample that we investigated the dimensions are: tPt= 5 nm,tYIG= 0:5 mm,LPt z= 150m. Figure 1. The dependence of a local spin Seebeck signal on the local magnetization has been observed by scan- ning a laser beam on a Pt/YIG structure [26] and in a time-resolved conguration [27{29]. By taking account of the experimental geometry the same set-up can be adopted for the measurement of the time resolved anoma- lous Nernst eect [26, 30]. In this work we present for the rst-time local measurements of the SSE using a thermal AFM probe as local source of heat and non-equilibrium spin currents. We employed a high-quality bulk YIG sin- gle crystal with a Pt strip lithographically dened onto the surface as our spin detector. We demonstrate that the measured eect is unambiguously the local spin See- beck through a series of tests; by varying the heating power and the vector of the externally applied magnetic eld. We interpret and support our observations with a thermodynamic description of the generation of the local magnetic moment current. This model describes quanti- tatively the geometry of local heat current as a circular heat source below the thermal probe whose size is larger than the cross-section of the tip due to the non-zero ther- mal conductivity at the Pt/YIG interface. The diameter of the heat source that we observed by employing our model is2:8m. SPIN SEEBECK EFFECT BY UNIFORM AND LOCAL HEATING Figure 1 shows two variations of a spin Seebeck exper- iment, both demonstrating that the measured signal due to the inverse spin Hall eect (represented by the yellow wires in Figure 1) corresponds to the open circuit voltage across the Pt lm. The rst one (Figure 1 (a)), representsthe classical experimental geometry reported by the ma- jority of experimental results. This provides a uniform distribution of the thermally-generated spin current as consequence of the heat current across the whole surface of the sample. The proportionality between the voltage gradientryVSSEgenerated along the Pt lm due to the SSE and the heat current Iqpassing through the sample along the cross-section Ais described by the following expression: ryVSSE IqA=SHB evYIGlYIGYIG vp1 tPt(1) whereB=eis the ratio between the Bohr magneton and the elementary charge, is the thermal conductivity of YIG. The other parameters inside Eq. 1 are the spin Hall angleSH, the absolute thermomagnetic power coecient YIG, the magnon diusion length lYIGand the thick- ness of the Pt lm tPt. The intrinsic magnetic moment conductance of YIG is represented by vYIG=lYIG=YIG whereYIGis the magnon mean scattering time. The pa- rametervprepresents the magnetic moment conductance per unit surface area of the Pt/YIG bilayer. This quan- tity depends on the intrinsic conductances of YIG and Pt, on the ratio between the thickness and the magnon dif- fusion length, for each layer. We derived the expression ofvpfrom a thermodynamic description of the magnetic moment currents generation [31]. It is important to note that the geometry shown in Figure 1 and modelled by Eq. 1 does not allow to resolve the spatial distribution of the underlying magnetic structure, since the spin Seebeck voltage results from an averaged contribution of regions with dierent magnetization. Because of this, the exper- iments performed in this geometry are usually conducted3 voltmeterPt YIGScanning probe a) b) c)μ0H μV d) e)80 μm 80 μm80 μm 80 μm80 μm 80 μm80 μm 80 μm y xz Figure 2. Experimental set-up for the local SSE measurements. (a) Macroscopic schematic representation of the Pt/YIG sample with the nanovoltmeter connected at the edges of the Pt strip. The subsequent images (b-e) are examples of measurements obtained from a 80 m80m area of the Pt lm with an applied magnetic eld of 8 mT. (b) MFM image of the same area, (c) spin Seebeck voltage originating from the scanning thermal probe, (d) AFM image that is obtained both from MFM and local spin Seebeck map, (e) 3D image obtained by overlapping of MFM and spin Seebeck map, taking into account the information on their mutual shift provided by the AFM images. at magnetic saturation or follow the magnetization hys- teresis loop. However, in the second experimental setup (Figure 1 (b)), which shows the eect of the heated AFM probe in contact with the Pt surface, we can selectively gener- ate the heat current injected through a point of the Pt surface and propagating with spherical symmetry in the volume. In this arrangement, the thermally generated spin current is assigned to a locally limited SSE, which is generated at the point of thermal contact, represented by the red dot in Figure 1 (b). As in the standard cong- uration the spin Seebeck voltage depends on the average magnetization, in the conguration with local heating the eect scales with the magnetization of a region whose size is determined by the locally heated volume, allowing re- solved measurements of the domain distribution within the sample. To describe this experimental conguration, we took into account the local magnetization mas a unit vector. This leads to an expression of the spin Seebeck voltage, which originates from a circle with radius r0on the top of the sample, into which a heat current Iqis injected. In this way it is possible to rewrite Eq. 1 as follows: ryVSSE Iq2r2 0k=SHB evYIGlYIGYIG vp1 tPt(2) where the dependence on the radius of the heated region r0(red circle in Figure 1 (b)) is highlighted. However, the value ofVSSEin Eq. 2, and in particular the voltage dif- ference
VSSEis the one that would be measured at the two ends of the heated region, limited by r0. Since thisquantity is not accessible by the experiment, it is neces- sary to rescale the value of
VSSE, taking into account the lateral dimensions of the whole Pt lm that works as the spin-voltage detector. Here we use an approxi- mation criterion and consider the Pt lm as an electric circuit formed by a voltage source ( VSSE), two resistances in series whose sum is proportional to ( Ly2r0)=2r0and one resistance in parallel whose value is proportional to Ly=(Lz2r0). Assuming that 2 r0Lyand 2r0Lz, we can represent the experimental values of the spin See- beck voltage as a rescaled value of
VSSE
VSSE,exp'
VSSE2r0 Lz(3) The expression of
VSSE,exp can be used to interpret the experimental data of the local SSE. By using the param- eters included in Eq. 2 to determine the value of VSSE at the right of Eq. 3, it is possible to obtain a value of r0that can be considered as the space resolution of the spin Seebeck imaging. EXPERIMENTAL METHODS The measurements of the local SSE were performed on a high quality YIG single crystal plate ( Ly= 4:95 mm,Lz= 3:91 mm and thickness tY IG = 0:545 mm). Both crystal surfaces were polished to be optically at (Rq= 0:4 nm obtained by AFM). A 150 m wide Pt strip was sputtered onto one side of the YIG crystal along the y direction, where the z direction denes the stripe width.4 The thickness of the Pt lm tY IG was chosen equal to be approximately 5 nm in accordance with [32, 33]. The measurement technique for the uniform heat current, i. e. the classical conguration, has been described else- where [34] for the same sample. The local heat current was generated using a nano thermal analysis probe (Nan- oTa probe); this consists of a micropatterned AFM can- tilever probe that allows a current to be driven around the cantilever resulting in Joule heating that propagates to the tip apex. We approximated the temperature of the probe using a series of polymer test samples with known glass transition temperatures. We also expected an o- set in our estimated temperatures due to the respectively higher thermal conductivity of the Pt used in our studies as compared to the test polymers. With this procedure it was possible to obtain an approximate relationship be- tween the probe power voltage and the temperature dif- ference applied to the sample in this geometry. Moreover, we observed a stable spin Seebeck signal when reverting the heating voltage thus verifying absence of electric in- terference from the nanoTA cantilever. This setup al- lowed us to compare a local domain map obtained by MFM with a local SSE map and correlate the two data sets. The enlarged area in Figure 2 (a) represents one of the specic regions investigated in the experiments. From the data in Figure 2 (b,c) it is possible to spatial- ity map the SSE voltage at the location of the thermal probe in contact with the Pt lm. This makes it possible to correlate SSE and MFM and draw qualitative conclu- sions. Figures 2 (b) and (c) show a stray eld gradient at the YIG surface obtained by MFM (Figure 2 (b)) and a SSE map obtained by nanoTA (Figure 2 (c)). The ex- periment was structured as follows: rst we performed a set of nanoTA measurements with a saturating magnetic eld in both directions ( Ms), achieved using a small neodymium magnet adjacent to the sample. This corre- sponds to a standard SSE experiment. The second set of measurements was performed at a magnetic eld where a domain structure could be observed using MFM. Keep- ing the applied eld constant, the nanoTA measurements were performed at dierent heating power levels of the thermal probe and at several points of the sample sur- face including the transition from the Pt strip to the bare YIG surface. RESULTS AND DISCUSSION Local spin Seebeck eect at magnetic saturation We rst investigated the local SSE of the Pt/YIG bi- layer structure at magnetic saturation. This experiment was performed with the eld aligned within the plane of the sample and perpendicular to the long axis of the Pt strip (Figure 2 (a)). The experimental data points in Figure 3 show the SSE data for each realized tempera- Vmeas (μV) ΔT (K)Figure 3. Measured voltage as consequence of the local heat- ing of the sample at magnetic saturation: spin and ordinary Seebeck voltage as function of the temperature dierences (red point). Fitting lines after the compensation of the ordi- nary Seebeck component (blue dashed lines). The direction of the saturating magnetic eld is shown as a sketch in each panel. ture dierences
T. Here, we averaged the signal that originated from the local heating of a 40 m20m area of the Pt surface and the experimental uncertainty was evaluated from the standard deviation of these data sets. The dependency between the total measured volt- age, including the spin Seebeck contribution and
T can be described by the following relation: Vmeas=CSSE
T+COSE
T+V0 (4) whereVmeas corresponds to the average voltage recorded as a consequence of scanning a given area and
T is the dierence between the probe temperature and the room temperature. The voltage Vmeas contains the following contributions: the rst one ( CSSE
T) corresponds to the spin Seebeck voltage VSSE,exp of Eq. 3; the sign of the spin Seebeck coecient CSSEdepends on the direction of the magnetisation. The second contribution is the or- dinary Seebeck eect ( COSE); for our work, this is a spu- rious component that does not depend on the magnetic conguration of the sample. The COSEcomponent de- rives from the contact between the dierent metals used to electrically connect the sample (silver paint for bond- ing the platinum strip at the edges of the thin lm). This is due to a small transverse heat loss through the elec- tric contacts associated to small geometric asymmetry. Such an artefact can aect the interpretation of spin- caloritronic measurements and has been previously de- scribed for both the spin Seebeck and spin Peltier eect [34] data. The last contribution of Eq. 4 is an oset volt- ageV0that originated from the circuit resistance. The voltage due to the Seebeck eects, both spin and ordi- nary eects, can be plotted as function of
T, according to Eq. 4. From the dierence between the absolute values5 of the slopes obtained by the linear ts shown in Figure 3 (red lines), a clear distinction can be made between the ordinary Seebeck and the spin Seebeck components. After the compensation of the ordinary Seebeck compo- nent, we present the spin Seebeck data by blue dashed lines in Figure 3, whose coecient is CSSE= 2
108 VK1. Local spin Seebeck eect at intermediate magnetization obtained with 8mT In the second step, we repeated the measurements with a lower applied magnetic eld in order to induce a re- duction in the magnetostatic energy of the sample and introduce a domain structure where we could distinguish several magnetization areas from the MFM images. We applied the magnetic eld at lowered level by distancing the neodymium magnet from the sample and we mea- sured its value by positioning an Hall probe in place of the sample. The measured value for the applied magnetic eld was approximately 8 mT. An example of two MFM images of the sample at magnetic saturation and with an applied eld of8 mT obtained from the same area is represented in Figure 4. 10
m a) b)10
m Figure 4. MFM micrograph of the same area of the Pt/YIG surface at magnetic saturation (left panel) and at 8 mT (right panel). Having applied8 mT, we scanned the thermal probe over a 80m80m in the same locations as that of the MFM, at dierent values of
T in order to anal- yse the voltage that arises form the local SSE. We used ve values of heating power on the thermal probe that led to the corresponding temperature dierences
T at each area investigated. With these data sets we were able to determine the SSE dependence on heating power and extract the SSE voltage for each pixel of the data set. First we removed the ordinary Seebeck component, as described for the sample at magnetic saturation; this procedure gives the spin Seebeck coecient CSSEand the measurement oset V0, represented by the slopes and the intercepts of the blue lines in Figure 3. Here we extract the values of these two parameters for the sample at in- a) spin Seebeck coefficient CSSE (nV/K) b) offset voltage V0 ( V) c) reduced magnetization12 8 4 0 0.4 0.2 0 -0.2 -0.4 1.0 0.8 0.6 0.4 0.2 0.0Figure 5. spin Seebeck maps of one 80 m80m area of the Pt/YIG sample surface obtained with a local heating generated by 0.5, 1, 1.5, 1.8 and 2.2 V on the heater. (a) Spin Seebeck coecients measured at each pixel of the map. (b) Map of the spurious oset voltage component and (c) map of the ratios between the local spin Seebeck voltage and the corresponding quantity at magnetic saturation; the colour scale in (c) represents the magnetization as percentage of the magnetic saturation. termediate magnetization by performing a linear t of the voltage values ( Vmeas) represented by each pixel as func- tion of the temperature dierence. We can now build two maps that represent the parameters of these linear ts: the map of the spin Seebeck coecients CSSE(Figure 5 (a)) and the map of intercepts V0(Figure 5 (b)). Since we have already measured the upper limit of the spin Seebeck signal i.e. the value at magnetic saturation, we6 Figure 6. Local spin Seebeck voltage maps (right columns) obtained at 8 mT with 1.5 V at the heater, shown together with the MFM micrographs (left columns). The ve pairs of maps correspond to the areas of the sample represented in the scheme, where the edge of the Pt lm is shown, together with the direction of the applied magnetic eld. can use this value to normalize our results with 8 mT which is represented as a percentage of the maximum sig- nal at saturation. Figure 5 (c) shows an example of this procedure recorded at the maximum temperature dier- ence; the colorbar quanties the level of magnetization between zero and the values at saturation, represented as the range [01]. From the data in Figure 5 (a), we found that the ratio between the spin Seebeck values and the temperature dierence (i.e. the spin Seebeck coe- cient) has some non-zero positive values in some regions and decreases to zero in the round central region of the map. By adapting the denition of the spin Seebeck coef- cient that is formulated for the saturated sample to the case of intermediate magnetization, we can deduce what follows. In some areas the magnetization was oriented to provide a larger spin Seebeck signal with respect to the areas where we observed a lower signal, reasonably due to a change in the orientation of the magnetization. On the contrary, the map of the intercepts in Figure 5 (b) is considerably atter, indicating that the electric oset was rather steady over the investigated area. Comparison between MFM and local spin Seebeck images The third step of this experiment involved scanning dierent regions of the sample, at a xed heating volt- age, with the same applied magnetic eld as previously used (8 mT). The motivation is a qualitative compari- son of the spin Seebeck maps with MFM micrographs.We tested the hypothesis that the contrast of the lo- cal spin Seebeck map is related to the local magnetiza- tion through the local spin Seebeck signal. Scanning of large areas allowed for overlapping neighbouring data sets and while recorded them accordingly; the corresponding MFM micrographs and the local spin Seebeck voltage maps are reported in Figure 6. The signal of the local spin Seebeck maps was processed using the Gwyddion data analysis software [35]. First we focused on the pres- ence of some sharp voltage spikes; these have been manu- ally corrected with an interpolation of the error-free pix- els surrounding the spike. The second data process con- cerned the artefacts that usually appears because of the line by line acquisition. We corrected the voltage shift that rise between neighbouring horizontal lines by mini- mizing the median of height dierences, between vertical neighbouring pixels [36]. From Figure 6 we observe that the MFM micrographs can provide a great variety of de- tails due to the high resolution, whereas the local SSE maps have a lower spatial resolution. Furthermore, we have to keep in mind that the underlying physical phe- nomena are dierent for both experiments: the MFM sig- nal depends on the stray elds emanating from the sam- ple surface whereas the local SSE depends on the mag- netization of the sample and the integrated eect over hot spot provided by the probe. Nevertheless, there is a clear correspondence between regions and features in the MFM micrographs and the regions of local minima of the local spin Seebeck maps. To make the correlation more visible, we traced the contour lines extracted by the spin7 Seebeck maps above the MFM micrographs in Figure 6. For the three images on the left, we selected the con- tour lines corresponding to 0 :6V on the spin Seebeck map, while for the two images on the right we selected dierent values since we did observe a drift on the oset voltage. For this reason, the two areas labelled by the green and light-blue frames appear on a slightly dierent voltage scale, but nevertheless it is possible to distinguish the shape of the local minimum in the expected position, according to the 20 m shift between the two areas. The two micrographs that refer to the area labelled by the red frame in Figure 6 represent a measurement across the edge of the Pt strip, cutting the frame horizontally. In the lower half of the SSE micrograph (the region not covered by the Pt), the spin Seebeck voltage decreases in agreement with the local generation of a spin current from the bare YIG that is not detected by the Pt lm. Finally, it is possible to hypothesise the level of the lo- cal magnetization of the sample. This was achieved by averaging the values reported in the local spin Seebeck maps of Figure 6 and then normalising these values with the spin Seebeck coecient according to the procedure adopted for Figure 5 (c). By considering this approach, we obtained a value of the average magnetization cor- responding to 0 :4 times the magnetic saturation. This value is in agreement with the measurement performed with the Hall probe (8 mT), knowing that over 20 mT the sample saturates (see supplementary informations of ref. [34]). This value tells us that the sample was far from the magnetic saturation but above the level of magneti- zation at which the magnetic-eld dependence of the SSE deviates from the bulk magnetization curve [37, 38], as reported by other experiments performed on bulk sam- ples [19, 20, 23, 39{41]. Spatial resolution of the spin Seebeck image Finally, we comment on the resolution of the local SSE measurement technique. We consider the radius of the hot spot (r0in in Eq. 2) as a more realistic limit to the resolution, compared to the cross-section of the probe. Starting from the thermodynamic description of the SSE, we have derived the expression of the spin Seebeck volt- age dierence as function of the heat current injected by the heated AFM probe. We also highlighted the need to rescale this expression according to the geometry of the experiment, as represented by Eq. 3. By replacing the expression of the spin Seebeck voltage
VSSE(Eq. 2) in- side the expression of the rescaled signal we could derive the following representation of the experimental results:
VSSE,exp
T=4r0 LzSHB evYIGlYIGYIG vp1 tPt(5) where the parameters that represent the properties of YIG have been chosen according to the semi-innite ap-proximation where the thickness of the YIG is larger than that of the magnon diusion length and Lz= 150m is the width of the Pt strip. In Eq. 5 the heat current Iq that appears in Eq. 2 has been written as the tempera- ture dierence
T, by knowing the thermal conductivity and the geometrical constrains on the YIG layer. By using the experimental data from a previous spin Seebeck study on the same sample [34], we can use the following experimental value for the YIG. vYIGlYIGYIG vp=4:61010(V/K)(m/s)1(6) The spin Hall angle SH=0:1 refers to the current of magnetic moments which has the opposite sign with respect to a denition of the spin Hall angle based on the spin current [34]. By replacing the value of the spin Seebeck coecient obtained ( CSSE= 2
108VK1) in place of
VSSE,exp=
Tin Eq. 5, we obtained a value of r0= 1:4m. This value of r0is in reasonable agreement with the lateral point-spread function of a heating laser presented by Bartell et. al. [28] with a FWHM of the hot spot of 0:606m, obtained with an optical laser power of 0:6 mW. CONCLUSIONS In summary, we have employed a scanning probe mi- croscopy technique to locally inject heat currents in a Pt/YIG bilayer structure. We observed a spatially- resolved voltage response dependent on the location of the heated probe and the local magnetisation state which we unambiguously attribute to the SSE. This allows for the rst time to obtain locally resolved spin Seebeck mea- surements, which we map spatially and compare quali- tatively with MFM micrographs obtained on the same scanned regions. We have discussed the measured signals using a thermodynamic description in spherical coordi- nates. Furthermore, we have derived the spatial resolu- tion of the local spin Seebeck measurements which is of the order of few micrometres. Local spin Seebeck imag- ing represents an innovative tool for the investigation of novel spin caloritronic materials. In particular, it pro- vides a signicant step forward for the analysis of bulk magnetic structures, compared to surface characteriza- tion techniques, as the signal originates from the bulk at a distance that can be considered equivalent to the magnon diusion length. Additionally, this experimental technique allows us to image the magnetisation structure of samples where tip-sample interaction could result in ir- reversible changes of the sample state during the imaging process, thus providing a non-perturbative imaging tool. Moreover, this technique could pave the way to new con- cepts of scanning probe microscopy, inspired by spintron- ics and spin-caloritronics. For example, the development of V-shaped Pt probes as a point-contact ISHE detector,8 or the scanning thermal microscopy (SThM) as a probe for spatial magnetic imaging using the spin Peltier eect. The authors thank the EMRP Joint Research Projects 15SIB06 NanoMag for nancial support. In particular, A. S. thanks the Researcher Mobility Grant 15SIB06- RMG3. The EMRP is jointly funded by the EMRP participating countries within EURAMET and the Eu- ropean Union. We thank Dr. Vladimir Antonov, Royal Holloway, Uni- versity of London (RHUL) for Pt deposition. A. S. thanks Marco Coisson for useful discussions and preliminary MFM measurements. C. D. thanks R. Meyer and B. Wenzel for technical assistance. a.sola@inrim.it [1] Y. Martin and H. Kumar Wickramasinghe. Magnetic imaging by force microscopy with 1000 A resolution. Ap- plied Physics Letters , 50(20):1455{1457, 1987. [2] J.J. Saenz, N. Garcia, P. Gr utter, E. Meyer, H. Heinzel- mann, R. Wiesendanger, L. Rosenthaler, H.R. Hidber, and H.J. 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Non-equilibrium thermodynamic approach to spin pumping and spin Hall torque eects. Journal of Physics D: Applied Physics , 51(21):214006, 2018. [18] K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi, J. Ieda, T. Ota, Y. Kajiwara, H. Umezawa, H. Kawai, G.E.W. Bauer, S. Maekawa, and E. Saitoh. Spin Seebeck insulator. Nature Materials , 9:894, 2010. [19] K. Uchida, T. Nonaka, T. Ota, and E. Saitoh. Longitu- dinal spin-Seebeck eect in sintered polycrystalline (Mn, Zn) Fe 2O4.Applied Physics Letters , 97(26):262504, 2010. [20] K. Uchida, T. Nonaka, T. Kikkawa, Y. Kajiwara, and E. Saitoh. Longitudinal spin Seebeck eect in various garnet ferrites. Physical Review B , 87(10):104412, 2013. [21] D. Meier, T. Kuschel, L. Shen, A. Gupta, T. Kikkawa, K. Uchida, E. Saitoh, J.M. Schmalhorst, and G. Reiss. Ther- mally driven spin and charge currents in thin NiFe 2O4/Pt lms. Physical Review B , 87(5):054421, 2013. [22] E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara. 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2020-05-18

In this work we present the results of an experiment to locally resolve the spin Seebeck effect in a high-quality Pt/YIG sample. We achieve this by employing a locally heated scanning thermal probe to generate a highly local non-equilibrium spin current. To support our experimental results, we also present a model based on the non-equilibrium thermodynamic approach which is in a good agreement with experimental findings. To further corroborate our results, we index the locally resolved spin Seebeck effect with that of the local magnetisation texture by MFM and correlate corresponding regions. We hypothesise that this technique allows imaging of magnetisation textures within the magnon diffusion length and hence characterisation of spin caloritronic materials at the nanoscale.

Local spin Seebeck imaging with scanning thermal probe

2005.08539v1

1 Deposition temperature dependence of thermo -spin and magneto - thermoelectric conversion in Co 2MnGa films on Y 3Fe5O12 and Gd 3Ga 5O12 Hayato Mizuno1,a), Rajkumar Modak1, Takamasa Hirai1, Atsushi Takahagi2, Yuya Sakuraba1, Ryo Iguchi1, Ken -ichi Uchida1,3,4,a) AFFILIATIONS 1National Institute for Materials Science, Tsukuba 305 -0047, Japan 2Department of Mechanical Systems Engineering, Nagoya University, Nagoya 464 -8601, Japan 3Institute for Materials Research, Tohoku University, Sendai 980 -8577, Japan 4Center for Spintronics Research Network, Tohoku University, Sendai 980 -8577, Japan a)Author to whom correspondence should be addressed: MIZUNO.Hayato@nims.go.jp and UCHIDA.Kenichi@nims.go.jp ABSTRACT We have characterized Co2MnGa (CMG) Heusler alloy films grown on Y3Fe5O12 (YIG) and Gd3Ga5O12 (GGG) substrates at different deposition temperatures and investigated thermo -spin and magneto -thermoelectric conversion properties by means of a lock-in thermography technique . X-ray diffraction, magnetization, and electrical transport measurements show that the deposition at high substrate temperatures induces the crystallized structure s of CMG while the resistivity of the CMG films on YIG (GGG) p repared at and above 500 °C (550 °C) becomes too high to measure the thermo -spin and magneto -thermoelectric effects due to large roughness , highlighting the difficulty of fabricating highly ordered continuous CMG films on garnet structures . Our lock -in thermography measurements show that the deposition at high substrate temperatures results in an increase in the current -induced temperature change for CMG/GGG and a decrease in that for CMG/YIG. The former indicates the enhancement of the anom alous Ettingshausen effect in CMG through crystallization . The latter can be explained by the superposition of the anomalous Ettingshausen effect and the spin Peltier effect induced by the positive (negative) charge -to-spin conversion for the amorphous (crystal lized) CMG films . These results provide a hint to construct spin -caloritronic devices based on Heusler alloys. 2 Magnetic Heusler alloys have been widely studied as promising spintronic and spin - caloritronic materials because they may show half me tallic1,2 or topological band structure .3 For instance, Co2MnGa (CMG) with the L21 ordered phase is known to exhibit the large anomalous Hall and anomalous Nernst effects due to its topological nature .4,5,6,7 Recent studies show that the CMG films with the L21 and B2 ordered phases also exhibit the large charge -spin current conversion efficiency .8,9 This finding suggests that CMG would be useful for detecting and driving the thermo -spin effects such as the spin Seebeck10,11 and spin Peltier effects (SPE s).12,13,14 However, the spin -caloritronic effects in CMG have not been investigated except for the anomalous Nernst effect so far. The SPE refers to the conversion of a spin current into a heat current in metal/ magnetic -material junction system s. In combination with the charge -to-spin current conversion in the metal layer , e.g., the spin Hall effect (SHE) ,15 the SPE is served as transverse thermoelectric conversion as follows. When the charge current is applied to the metal layer, it induces a transverse conduction electron spin current via the SHE and the conduction electron spin current is transformed into a magnon spin current in the attached magnetic layer via the spin-mixing conductance. The magnon spin current is eventually converted to a heat current and resultant temperature change via the SPE. Here , the directions of the input charge current and output heat current are perpendicular to each other . As shown in Fig s. 1(a) and 1(b), in th e in-plane magnetized configuration, the symmetry of the charge -to-heat current conversion due to the SHE -driven SPE is the same as that due to the anomalous Ettingshausen effect (AEE) ,16 which is the Onsager reciprocal of the anomalous Nernst effect , in conductive ferromagnets . However, the SPE can be separated from the AEE by using a ferrimagnetic insulator, e.g., Y3Fe5O12 (YIG), as the magnetic layer. Although the SPE ha s mainly been investigated in paramagnetic -metal (e.g., Pt) /YIG junction systems so far, the SPE -induced temperature change can appear also in ferromagnetic -metal/YIG junction systems because ferromagnetic metals exhibit the charge -to-spin conversion [see Fig. 1 (c)].17,18,19 The quantitative separation of the SPE from the AEE in the ferromagnetic -metal/YIG systems and the improvement of transverse thermoelectric conversion performance by the hybrid action of the SPE and AEE are important issues in spin caloritronics. In this study, to clarify the potential of CMG as a spin -caloritroni c material, we have grown CMG films on YIG and Gd 3Ga5O12 (GGG) substrates at different deposition temperatures and characterized the SPE and AEE by means of the lock -in thermography technique13,20,21 as well as the structural, magnetic, and electrical transport properties . Here, GGG 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 AEE contribution in CMG , which is expected to be large because of the large anomalous Nernst effect of CMG ,4,6,7 free from the SPE contribution . This systematic data set will provide a hint for future studies on spin-conversion and spin -caloritronic phenomena in ferromagnetic Heusler alloys , leading to the development of the spintronic thermal management .22 The sample systems used in this study are the 20-nm-thick CMG films deposited on single - crystalline YIG (111) and GGG (111) substrates by DC magnetron sputtering. We chose 20 nm because it is the reported minimum thickness of the ordered phases of CMG .23 The YIG substrate consists of a 26-μm-thick YIG layer grown on a 0.5-mm-thick single -crystalline GGG (111) substrate by a liquid phase epitaxy method .13 The lattice constant of CMG (YIG and GGG) is 5.8 (12.4) Å.4,24 The YIG surface is mechanically polished with Al slurry . The dimension of the YIG and GGG substrates is 5 × 10 mm2. The CMG films were sputtered from a Co 41.2Mn 27.5Ga31.3 source with a base pressure of < 2.0×10−6Pa, working pressure of 0.4 Pa in Ar atmosphere , sputtering power of 50 W, and the deposition rate of 0.06 nm/s. The substrate temperature ( 𝑇sub) during the deposition varied from room temperature to 550 ℃. Before deposition, the YIG and GGG substrates were heated at 600 ℃ for 20 min to achieve a clean surface as previous studies performed for MgO substrate s25,26 and then down to 𝑇sub in the sputtering chamber. To avoid oxidation, the CMG layers were covered b y the 2-nm-thick Al capping layer s without breaking the vacuum , deposited at room temperature . The composition of the CMG films was determined to be Co 45.8Mn 26.2Ga28.0 by x-ray fluorescence analysis (note that the composition of the sputtering source is intentionally tuned to obtain the optimized composition of the deposited films7). As a reference sample, we deposited the 20-nm-thick Pt film on the same YIG substrate at room temperature . The structural properties of the CMG films were characterized by out -of-plane x -ray diffraction . The magnetic properties were measured using a vibrating sample magnetometer. For electrical transport measurements, the films were patterned into Hall bars using photo lithography and Ar -ion milling process es. The surface morphology of the films was evaluated by atomic force microscopy. The SPE and AEE were measured by means of the lock-in thermography method by patt erning the CMG and Pt films into a U shape , the line width of which is 0.2 mm with the same proce ss as the Hall bars . The procedures of the lock -in thermography measurements are the same as those shown in Ref. 13 . All measurements were performed at room temperature . Figures 2 (a) and 2(b) show the x-ray diffraction patterns of the CMG/YIG and CMG/GGG systems, respectively . No clear peak of CMG appeared in the film s deposited at room 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 the 220 fundamental diffraction peak showed a ring shape in 2D scan , as shown in the inset of Fig. 2(b), indicating the polycrystalline structure with most fundamental A2 structure , where the A2 phase indicate s the completely disorder ed structure among Co, Mn, and Ga . Although the presences of the B2 (partially disordered structure between Mn and Ga while Co atoms occupy regular sites ) and L21 (fully ordered structure) phases can be recognized from 002 and 111 superlattice peaks, respe ctively, these peaks were too weak to be detected within the margin of experimental errors (note that the 002 peak was too weak even in the in -plane x -ray diffraction measurements) . Therefore, the x-ray diffraction results observed here suggest the presence of polycrystalline structure s but do not identify the ordered phase s in the high 𝑇sub deposited CMG films on YIG and GGG. Figure 2(c) shows the in -plane magnetic field H dependence of the magnetization M of the CMG films on the GGG s ubstrates . We found that M saturate s for 𝜇0H > 0.04 T, where 𝜇0 is the vacuum permeability . The saturation magnetization 𝑀s of the high 𝑇sub deposited films was observed to be larger than that of the as -deposited ( 𝑇sub = room temperature ) film. The as -deposited film shows 𝜇0𝑀s ~ 0.14 T, which is much smaller than 𝜇0𝑀s of the ordered CMG but in agreement with the previous reports .27,28 The 𝜇0𝑀s values for our high 𝑇sub deposited CMG films were estimated to be ~ 0.45 T, which is smaller than the reported value s of the A2 phase ( ~ 0.54 T )27 and B2 ordered phase ( ~ 0.63 T )9,28 of CMG films. Figure 2(d) summarizes the measured and reference 𝑀s value s.8,9,2 7,28,29 This result suggests that our high 𝑇sub deposited CMG films consist of the mixture of the crystal lized and amorphous structures. We focus on the effect of 𝑇sub on electrical transport properties of the CMG films. Figure 2( e) shows the longitudinal electrical resistivity 𝜌𝑥𝑥 as a function of 𝑇sub. The 𝜌𝑥𝑥 values of the CMG/YIG systems are larger than those of the CMG/GGG systems and their 𝑇sub dependence is different from each other. T he resistivity of the CMG films on the YIG (GGG) substrates prepared at and above 500 ℃ (550 ℃) was too high to apply a charge current (note that we prepared four CMG/YIG sample s at 𝑇sub = 450 ℃ and two of them exhibit no electrical conduction while the remaining samples exhibit almost the same value of the electric al conductivity ). Figure 2(f) shows the out -of-plane H dependence of the transverse resistivity 𝜌𝑥𝑦 for the CMG/GGG system s. The 𝜌𝑥𝑦 values show the H-odd dependence and its magnitude almost saturate s at ~ 0.7 T for the high 𝑇sub deposited films. By extrapolating the 𝜌𝑥𝑦 data above 1.0 T to zero field, the anomalous Hall resistivity for our high 𝑇sub deposited films was estimated to be ~ 13 μΩcm, which is comparable to the previous ly report ed values 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 clarify the reason for no electric al conduction for the high 𝑇sub deposited films , we investigated the surface morp hology of the samples by atomic force microscopy. We confirmed that the average roughness 𝑅a of the CMG/YIG and CMG/GGG systems monotonically increases with increasing 𝑇sub and 𝑅a of CMG/YIG is larger than that of CMG/GGG at each 𝑇sub [Figs. 2(g) and 2(h)]. The large 𝑅a values for the samples prepared at high 𝑇sub result in the discontinuity of the CMG layers . Although the lattice constant and coefficient of thermal expansion of YIG and GGG are comparable ,24,30,31 the growth of CMG is sensitive to the surface energy , which determines the wettability of the initial growth of the CMG film ; it is more difficult to obtain ordered CMG films with good electrical conduction when the YIG substrates are used than when the GGG substrates are used . Next, we show the thermo -spin and thermoelectric conversion properties for the CMG/YIG and CMG/GGG systems with finite electrical conduction . During the lock -in thermography measurements, t he square -wave -modulated AC charge current 𝐽c with the frequency f = 5 Hz and zero DC offset and in -plane H were applied to the U-shaped CMG films , which is a standard condition for measuring the SPE and AEE .13,16 By extract ing the first- harmonic response of the current -induced temperature change , we can obtai n lock -in amplitude A and phase 𝜙 images respectively showing the amplitude and sign of the thermo -spin and/or thermo electric conversion . To extract the pure SPE and AEE signals , we performed the lock -in thermography measurements with applying positive and negative H and calculated the temperature modulation with the H-odd dependence :32,33 𝐴odd𝑒−𝑖𝜙𝑜𝑑𝑑=(𝐴(+𝐻)𝑒−𝑖𝜙(+𝐻)− 𝐴(−𝐻)𝑒−𝑖𝜙(−𝐻))/2, where the background due to the H-independent Peltier effect is excluded . The 𝐴odd and 𝜙odd image s for the as-deposited CMG/YIG film are shown in Figs. 3( a) and 3(b), respectively, where 𝐽c = 3.0 mA and 𝜇0|𝐻| = 0.32 T. We found that the clear current - induced temperature change appeared on the areas L and R of the CMG layer , where 𝐉c⊥ 𝐇, while no signal appea red in the area with 𝐉c∣∣𝐇. As shown in Fig. 3( b), 𝜙odd in L (R) is 0° (180°), indicat ing that the sign of the temperature modulation is reversed by reversing the 𝐉c direction . Figure 3( c) [Fig. 3(d)] show s the increases of 𝐴odd in proportion to 𝐽c for all the CMG/YIG (CMG/GGG ) systems for various values of 𝑇sub. Figures 3( e) and 3( f) show the 𝜇0|𝐻| dependence of 𝐴odd in the CMG/YIG and CMG/GGG systems , respectively. The magnitude of 𝐴odd is almost constant for 𝜇0|𝐻| > 0.1 T, where M of the YIG and CMG layers is aligned along the H direction . These results are consistent with th e features of the temperature change induced by the SPE and AEE . We confirmed that all the samples exhibit the current - induced temperature modulation with the same sign. 6 In Fig. 4, w e summarize the amplitude of the temperature modulation per unit charge current density, 𝐴odd/𝑗c, where 𝑗c=𝐽c/(𝑡𝑤) with t (w) being the thickness ( line width) of the U-shaped CMG films . We also estimated 𝐴odd/𝑗c for the conventional Pt/YIG system to be 3.7 × 10-13 Km2A-1, which is in good agreement with the previous reports13,34 [see the star data point in Fig. 4(a)] . We found that, for the CMG/YIG systems, 𝐴odd/𝑗c in the as -deposited sample is 2.6 times larger than that in Pt/YIG and its magnitude is decrease d in the high 𝑇sub deposited samples [ Fig. 4(a)] . In contrast, for the CMG/GGG systems, 𝐴odd/𝑗c is enhanced in the high 𝑇sub deposited sample s compared with the as -deposited sample [Fig. 4(b)]. For the as-deposited ( high 𝑇sub deposited ) samples, 𝐴odd/𝑗c in the CMG/YIG systems is larger (smaller) than that in the CMG/GGG systems. The striking contrast in the current -induced temperature change between the CMG/YIG and CMG/GGG systems cannot be explained only by the AEE in the CMG layers, suggesting the coexistence of the SPE in the CMG/YIG systems. Finally, we discuss the origin of t he difference of 𝐴odd/𝑗c between the room temperature and high 𝑇sub deposited films. In the CMG/GGG systems, t he SPE does not contribute to the temperature modulation signals because GGG does not transport a spin current at room temperature .35 Therefore, the signals for the CMG/GGG systems are due purely to the AEE in CMG. Since the deposition at high 𝑇sub induces the crystal lized structures of CMG , the increase in 𝐴odd/𝑗c between the CMG/GGG systems deposited at and above room temperature [Fig. 4(b)] indicates the e nhancement of the AEE in CMG through crystallization . We also note that the magnitude of 𝐴odd/𝑗c is almost constant within the margin of experimental errors in the high 𝑇sub deposited CMG/GGG systems , suggesting that the degree of ordering could be unchanged in the 𝑇sub range , as suggested by almost constant 𝑀s and 𝜌𝑥𝑦 [Figs. 2(c) and 2(f) ]. A similar trend has been reported in Ref. 36, where higher ordered phase exhibits a larger anomalous Nernst effect in a ferromagnetic Heusler alloy. Although the behaviors of the AEE in the CMG/YIG systems are expected to be similar to those in the CMG/GGG systems, the 𝐴odd/𝑗c signal in CMG/YIG is much larger than that in CMG/GGG in the as -deposited case and is decre ased in the high 𝑇sub deposited films [Fig. 4(a)] . This behavior can be explained by assuming that the SPE due to the charge -to-spin current conversion in the CMG layers in the as -deposited ( high 𝑇sub deposited ) CMG/YIG systems makes an additive (subtractive) contribution to the AEE. This assumption indicate s that the spin Hall angle of the amorphous ( crystal lized) CMG films is positive (negative) because the sign of the temperature modulation observed here is the same as that for the Pt/ YIG systems with the positive spin Hall angle of Pt . Our scenario is consistent with the previous studies, which clarified 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 cancel each other out, the replac ement of YIG by other ferrimagnets with the magnetization compensation can tune the sign of the SHE -driven SPE , enabling the enhanced thermoelectric conversion based on the hybrid action of the SPE and AEE .37,38 The quantitative estimation of the spin Hall angle of the amorphous and crystal lized CMG films on YIG remains to be performed. Nevertheless, our experiments reveal the fundamental behaviors of the SHE -driven SPE in the CMG/YIG systems a nd provide a possibility of tuning the sign and magnitude of the spin Hall angle of Heusler alloys through their crystallization . In conclusion , we investigated the SPE and AEE in the CMG/YIG and CMG/GGG systems with the CMG films sputtered at different deposition temperatures . The CMG films grown on YIG at 500 and 550 °C and on GGG at 550 °C exhibit no electrical conduction due to the large roughness . These results indicate the difficulty of fabricating highly ordered continuous CMG films on garnet subst rates . Further optimization of substrate and deposition conditions remains future work . Our lock-in thermography measurement s show that the temperature change induced by the AEE in CMG is enhanced by the deposition at high 𝑇sub through crystallization and that the SPE induces the additive (subtractive) temperature change due to the positive (negative) spin Hall angle of the amorphous ( crystal lized) CMG films. The systematic data set reported here provide s a guideline for future studies of spin caloritro nics using magnetic Heusler alloy s. The authors thank H. Nagano for valuable discussions and M. Isomura , N. Kojima, and R. Tateishi for technical supports. This work was supported by the CREST "Creation of Innovative Core Technologies for Nano -enabled Thermal Management" (No. JPMJCR17I1) from JST, Japan; Grant -in-Aid for Scientific Research (S) (No. 18H05246) from JSPS KAKENHI, Japan; and the NEC Corporation. R.M. is supported by JSPS through the “JSPS Postdoctoral Fellowship for Research in Japan (Stan dard)” (P21064). AUTHOR DECLARATIONS Conflict of Interest The authors have no conflicts to disclose . DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. 8 References 1C. Felser and G. H. Fecher, Spintronics: From Materials to Devices (Springer, Berlin/Heidelberg, 2013). 2K. Elphick, W. Frost, M. Samiepour, T. Kubota, K. Takanashi, H. Sukegawa, S. Mitani and A. Hirohata, Sci. Technol. Adv. Mater. 22, 235 (2021). 3K. Manna, Y. Sun, L. Muechl er, J. Kübler, and C. Felser, Nat. Rev. Mater. 3, 244 (2018). 4A. Sakai, Y. P. Mizuta, A. A. Nugroho, R. Sihombing, T. Koretsune, M. -T. Suzuki, N. Takemori, R. Ishii, D. Nishio -Hamane, R. Arita, P. Goswami, and S. Nakatsuji, Nat. Phys. 14, 1119 (2018). 5I. Belopolski, K. Manna, D. S. Sanchez, G. Q. Chang, B. Ernst, J. X. Yin, S. S. Zhang, T. Cochran, N. Shumiya, H. Zheng, B. Singh, G. Bian, D. Multer, M. Litskevich, X. T. Zhou, S. - M. Huang, B. K. Wang, T. -R. Chang, S. -Y. 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Barker, S. Maekawa, G. E. W. Bauer, E. Saitoh, R. Gross, S. T. B. Goennenwein, and M. Klaui, Nat. Commun. 7, 10452 (2016). 38A. Yagmur, R. Iguchi, S. Geprägs, A. Erb, S. Daimon, E. Saitoh, R. Gross, and K. Uchida, J. Phys. D: Appl. Phys. 51, 194002 (2018). 10 FIG. 1 . Schematics of (a) the SPE in the Pt/YIG system, (b) the AEE in the CMG/GGG system, and (c) the hybrid thermoelectric conversion based on the SPE and AEE in the CMG/YIG system. 𝐌YIG (CMG ), 𝐉c(s), and 𝛁𝑇SPE (AEE ) repre sent the magnetization of YIG (CMG), charge current (spatial direction of the spin current), and temperature gradient induced by the SPE (AEE), respectively. 11 FIG. 2. (a) and (b) X -ray diffraction patterns of the CMG/YIG and CMG/GGG systems prepared at various values of the substrate temperature 𝑇sub. RT represents room temperature. The peaks highlighted in red arise from CMG (220). The peaks indicated by * arise from the YIG and GGG substrates. The inset shows the 2D scan for the CMG/GGG system prepared at 𝑇sub = 500 ℃. (c) In - plane H dependence of the magnetization M of the CMG/GGG systems prepared at various values of 𝑇sub. The paramagnetism contribut ion from the GGG subs trates was subtracted by linear fitting of the raw data in the field range from 𝜇0|𝐻| = 0.1 to 0.2 T, where 𝜇0 is the vacuum permeability. (d) 𝑇sub 12 dependence of the saturation magnetization 𝑀s of the CMG films on the GGG substrates. The areas highlighted in red show the reference values of the amorpho us27,28, A227, B29,28,29, and B2-L21- mixed8,28,29 structures of CMG films. (e ) 𝑇sub dependence of the longitudinal electrical resistivity 𝜌𝑥𝑥. (f) Out-of-plane H dependence of the transverse resistivity 𝜌𝑥𝑦 of the CMG films on the GGG substrates. (g ) Atomic force microscope images of the CMG/YIG systems prepared at room temperature and 450 ℃. (h) 𝑇sub dependence of the surface roughness 𝑅a of the CMG/YIG and CMG/GGG systems. 13 FIG. 3. (a) and (b) 𝐴odd and 𝜙odd images for the as -deposited CMG/YIG system at 𝐽c = 3.0 mA and in-plane field 𝜇0|𝐻| = 0.32 T. 𝐴odd (𝜙odd) denotes the lock -in amplitude (phase ) with the H- odd dependence. 𝐽c denot es the square -wave amplitude of the AC charge current applied to the U- shaped CMG layer. (c) and (d) 𝐽c dependence of 𝐴odd for the CMG/YIG and CMG/GGG systems at 𝜇0|𝐻| = 0.32 T. The solid lines are obtained from the linear fitting. (e) and (f) H dependence of 𝐴odd for the CMG/YIG and CMG/GGG systems at 𝐽c = 3.0 mA. The 𝐴odd data in (c) -(f) are extracted from the area R i n (a). 14 FIG. 4 . 𝐴odd/𝑗c for the (a) CMG/YIG and (b) CMG/GGG systems for various values of 𝑇sub. 𝑗c denotes the charge current density in the CMG layer. The 𝐴odd/𝑗c data are extracted from the slope of the linear fitting of the data in Fig s. 3(c) and 3(d). The star data point indicates the 𝐴odd/𝑗c signal for the Pt/YIG system.

2022-03-20

We have characterized Co$_2$MnGa (CMG) Heusler alloy films grown on Y$_3$Fe$_5$O$_{12}$ (YIG) and Gd$_3$Ga$_5$O$_{12}$ (GGG) substrates at different deposition temperatures and investigated thermo-spin and magneto-thermoelectric conversion properties by means of a lock-in thermography technique. X-ray diffraction, magnetization, and electrical transport measurements show that the deposition at high substrate temperatures induces the crystallized structures of CMG while the resistivity of the CMG films on YIG (GGG) prepared at and above 500 {\deg}C (550 {\deg}C) becomes too high to measure the thermo-spin and magneto-thermoelectric effects due to large roughness, highlighting the difficulty of fabricating highly ordered continuous CMG films on garnet structures. Our lock-in thermography measurements show that the deposition at high substrate temperatures results in an increase in the current-induced temperature change for CMG/GGG and a decrease in that for CMG/YIG. The former indicates the enhancement of the anomalous Ettingshausen effect in CMG through crystallization. The latter can be explained by the superposition of the anomalous Ettingshausen effect and the spin Peltier effect induced by the positive (negative) charge-to-spin conversion for the amorphous (crystallized) CMG films. These results provide a hint to construct spin-caloritronic devices based on Heusler alloys.

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$_{12}$

2203.10566v2

Simultaneous detection of the spin-Hall magnetoresistance and the spin-Seebeck eect in Platinum and Tantalum on Yttrium Iron Garnet N. Vlietstra, J. Shan, and B. J. van Wees Physics of Nanodevices, Zernike Institute for Advanced Materials, University of Groningen, Groningen, The Netherlands M. Isasa CIC nanoGUNE, 20018 Donostia-San Sebastian, Basque Country, Spain F. Casanova CIC nanoGUNE, 20018 Donostia-San Sebastian, Basque Country, Spain and IKERBASQUE, 48011 Bilbao, Basque Country, Spain J. Ben Youssef Laboratoire de Magn etisme de Bretagne, CNRS, Universit e de Bretagne Occidentale, Brest, France (Dated: September 25, 2018) The spin-Seebeck eect (SSE) in platinum (Pt) and tantalum (Ta) on yttrium iron garnet (YIG) has been investigated by both externally heating the sample (using an on-chip Pt heater on top of the device) as well as by current-induced heating. For SSE measurements, external heating is the most common method to obtain clear signals. Here we show that also by current-induced heating it is possible to directly observe the SSE, separate from the also present spin-Hall magnetoresis- tance (SMR) signal, by using a lock-in detection technique. Using this measurement technique, the presence of additional 2ndorder signals at low applied magnetic elds and high heating currents is revealed. These signals are caused by current-induced magnetic elds (Oersted elds) generated by the used AC-current, resulting in dynamic SMR signals. PACS numbers: 72.25.Mk, 72.80.Sk, 75.70.Tj, 75.76.+j I. INTRODUCTION For the investigation of pure spin transport phenom- ena, yttrium iron garnet (YIG) is shown to be a very suitable candidate. YIG is a ferrimagnetic insulating material having a low magnetization damping as well as a very low coercive eld. In combination with a high spin-orbit coupling material such as platinum (Pt), many dierent experiments have been performed, show- ing spin-pumping1{4, spin transport5,6and spin-wave manipulation7{9as well as the recently discovered spin- Hall magnetoresistance (SMR).10{15 Recently also experiments were performed showing the spin-Seebeck eect16{19(SSE) as well as the spin- Peltier eect20in YIG/Pt systems. The SSE is ob- served when a temperature gradient is present over a ferromagnetic/non-magnetic interface. In a YIG/Pt sys- tem, this temperature gradient causes the creation of thermal magnons, resulting in transfer of angular mo- mentum at the YIG/Pt interface, generating a pure spin- current into the Pt.19This spin-current can then be de- tected electrically via the inverse spin-Hall eect (ISHE). So far, most experiments on the SSE are performed using external heating sources to create a temperature gradi- ent over the device. Interestingly, Schreier et al.21showed that a clear SSE signal can also be extracted from more easily performed current-induced heating experiments. In their experiments a temperature gradient is created by sending a charge current through the detection strip.A disadvantage of their measurement method is the pres- ence of a much larger signal originated from the SMR, which should be subtracted to reveal the SSE signal. In this paper we investigate both the SSE and SMR in a YIG-based device, showing the possibility to simul- taneously, but separately, detect the SSE and SMR by using a lock-in detection technique. Whereas Schreier et al. only performed their measurements applying high magnetic elds, fully saturating the magnetization of the YIG, we show that when lowering the applied magnetic eld, dynamic behavior of the magnetization of the YIG can be picked up as additional 2ndorder signal. Only by using a lock-in detection technique these signals can be separately detected and analyzed. Having platinum (Pt) or tantalum (Ta) as detection layer, we investigate the evolution of the SSE and the SMR signal as a function of the magnitude and the direction of the applied eld, focusing especially on their low-eld behavior. The rst experiments described in this paper show SSE measurements where a temperature gradient is generated by externally heating the sample using a second Pt strip on top of the device. By using devices consisting of both Pt and Ta on YIG, we conrm the opposite sign of the spin-Hall angle for Ta and Pt.22,23In the secondly shown experiments, the samples are heated by current-induced heating through the metal detection strip, such that both the SSE and SMR are present. Additionally detected 2nd harmonic signals for low applied elds and high heating currents are discussed and ascribed to dynamic behaviorarXiv:1410.0551v2 [cond-mat.mes-hall] 27 Nov 20142 of the magnetization of the YIG, caused by the applied AC-current. Finally, we derive a dynamic SMR term, which is used to explain the observed features. The same kind of experiments could as well be used for detection of spin-transfer torque eects on the mag- netization of the YIG, like the generation of spin-torque ferromagnetic resonance as formulated by Chiba et al.24. However, as will be shown in this paper, when apply- ing low magnetic elds and high currents, the detected magnetization behavior is dominated by current-induced magnetic elds (like the Oersted eld). So, to be able to detect any eect of the spin-transfer torque, its contribu- tion should be increased, for example by decreasing the YIG thickness. II. SAMPLE CHARACTERISTICS For the experiments shown in this paper, two Hall-bar shaped devices have been used, one consisting of a 5nm- thick Pt layer and the other of a 10nm-thick Ta layer. The Hall-bars have a length of 500 m and a width of 50m, with side contacts of 10 m width. Both Hall-bars are deposited on top of a 4x4mm2YIG sample, by dc sputtering. The used sample consists of a 200nm thick layer of YIG, grown by liquid phase epitaxy on a single crystal (111)Gd 3Ga4O12(GGG) substrate. The YIG magnetiza- tion shows isotropic behavior of the magnetization in the lm plane, with a low coercive eld of only 0.06mT.4,12 For external heating experiments, a Ti/Pt bar of 5/40nm thick is deposited on top of both Hall-bars, sep- arated from the main channel by a 80nm-thick insulating Al2O3layer. The size of the heater is 400x25 m2. Fi- nally, both Hall-bars and Pt heaters are contacted by thick Ti/Au pads [5/150nm]. All structures are pat- terned using electron-beam lithography. Before each fab- rication step the sample has been cleaned by rinsing it in acetone, no further surface treatment has been carried out. A microscope image of the device is shown in Fig. 1(a). III. MEASUREMENT METHODS To observe the SSE, two measurement methods have been investigated. At rst, to generate a clear SSE sig- nal, a temperature gradient is created using an external heating source to heat one side of the sample. In our case, we have a Ti/Pt strip on top of the Hall-bar, elec- trically insulated from the detection channel, which can be used as an external heater. By sending a large cur- rent (up to 10mA) through the heater, the strip will be heated by Joule heating. Hereby, a temperature gradient will be formed over the YIG/Pt(Ta) stack, giving rise to the SSE. A second method to generate the SSE, is the genera- tion of a temperature gradient by current-induced heat- (b) (c)I-B V- I+V+α0(a) xy -20 -10 0 10 20-10-8-6-4-20246810 Pt [5nm] Ta [10nm]V2 [µV] Magnetic Field [mT]Iheater= 10mA α0 = -90o -180 -90 0 90 180-10-8-6-4-20246810 Iheater= 10mA B = 50mT Pt [5nm] Ta [10nm]V2 [µV] α0 [ο]FIG. 1. (a) Microscope image of the device structure, con- sisting of a Pt or Ta Hall-bar detector (bottom layer) and a Pt heater (top layer), separated by an insulating Al 2O3layer. Ti/Au pads are used for contacting the device. For external heating experiments, the device is contacted as marked. The applied eld direction is given by 0, as dened in the g- ure. (b) 2ndharmonic voltage signal generated by the SSE for a xed magnetic eld direction of 0= 90and (c) angu- lar dependence of the SSE signal applying a magnetic eld of 50mT, in both Pt and Ta. ing through the detection strip. In this case a charge cur- rent is sent through the Hall-bar itself, which also leads to Joule heating, resulting in a temperature gradient over the YIG/Pt(Ta) stack. As the Hall-bar is directly in con- tact with the YIG, also the SMR will be present when using this heating method. To separately detect the SSE and the SMR signals, a lock-in detection technique is used. Using up to three Stanford SR-830 Lock-in ampliers, the 1st, 2ndand higher harmonic voltage responses of the system are sep- arately measured. As SMR scales linearly with the ap- plied current, its contribution will be picked up as a 1st harmonic signal. Similarly, the SSE scales quadratically with current, so its contribution will be detected as a 2nd harmonic signal. For lock-in detection an AC-current is used with a frequency of 17Hz. The magnitude of the applied AC-currents is dened by their rms values. Evaluating the working mechanism of the lock-in de- tection technique in more detail (see appendix), shows that in order to obtain the linear response signal of the system, both the measured 1stharmonic signal as well as the 3rdharmonic signal have to be taken into account. Including both harmonic signals following the analysis explained in the appendix, we nd the shown 1storder response. Note that the measured 2ndharmonic signal is directly plotted, without any corrections. In all experiments, an external magnetic eld is applied to dene the direction of the magnetization M of the YIG. The direction of the applied eld is dened by 0, which is the in-plane angle between the current direction (along3 (a) (d) (c) (b) I-B V-I+V+ α0 -6-4-2024642444648 -6 -4 -2 0 2 4 6-15-10-5051015 -180-90 0 9018030405060 -180 -90 0 90 180-10-50510Pt [5nm] α0= 0o Iac = 4mA Iac = 8mA Magnetic Field [mT] R1 [mΩ] Iac = 4mA Iac = 8mAPt [5nm] α0= 0oV2 [µV] Magnetic Field [mT]Pt [5nm] Iac= 8mAB[mT] 0.9 1.8 100 α0 [ο] R1 [mΩ] Pt [5nm] Iac= 8mA V2 [µV] α0[ο]B [mT] 0.9 1.8 2.8 8.5 100 FIG. 2. Current-induced heating experiments on the YIG/Pt sample. (a) Magnetic eld dependence of the 1storder resistance for0= 0, showing the SMR signal for applied AC-currents of 4mA and 8mA. (b) Angular dependence of the 1storder SMR signal for I ac=8mA, for applied magnetic elds of 0.9mT, 1.8mT and 100mT. (c) and (d) show the corresponding 2ndharmonic voltage signals, respectively. For applied magnetic elds above 10mT, the 2ndharmonic response only shows the SSE signal. For low applied magnetic elds an additional signal is observed on top of the SSE signal. The black symbols in both gures are a guide for the eye. They show equal measurement conditions comparing the results shown in both gures. The inset of (c) shows the used measurement conguration. x) and the applied eld direction, as it is marked in Fig. 1(a). Not only experiments at high saturation magnetic elds are performed, also the low eld behavior is investi- gated. The applied magnetic eld strength was measured by a LakeShore Gaussmeter (model 421) using a trans- verse Hall probe, to correct the set magnetic eld for any present remnant eld. All measurements are carried out at room temperature. IV. RESULTS AND DISCUSSION A. Spin-Seebeck eect by external heating For the external heating experiment an AC-current is sent through the top Pt strip as marked in Fig. 1(a). By measuring the 2ndharmonic voltage signals along the Hall-bar, the SSE is detected via the ISHE in Pt and Ta. Fig. 1(b) shows the typical SSE signals for both YIG/Pt and YIG/Ta samples for an applied eld per- pendicular to the longitudinal direction of the Hall-bar (0= 90). Changing the sign of B (and thus M) changes the sign of the signal, as the spin-polarization direction of the pumped spin-current is reversed. Due to the low coercive eld of YIG almost no hysteresis is observed for the reversed eld sweep. For the YIG/Pt and YIG/Ta sample opposite magnetic eld dependence is observed, proving the opposite sign of the spin-Hall angle for Pt versus Ta. As the spin-polarization direction of the generated spin-current is dependent on the direction of the YIG magnetization, the SSE/ISHE voltage shows a sine shaped angular dependence with a period of 360. By rotating the sample in a constant applied magnetic eldof 50mT, this angular dependence is detected as is shown in Fig. 1(c). Also here the eect of the opposite sign of the spin-Hall angle, for Ta compared to Pt, is clearly visible. From Fig. 1 it is observed that the SSE signal for the YIG/Ta sample is almost a factor 10 smaller than for the YIG/Pt sample ( VSSE;Pt=VSSE;Ta =9:8). To compare, we calculate the expected ratio from the the- oretical description of the SSE voltage, as reported by Schreier et al.17: VSSE=CYIG

TmeGrSHl ttanht 2 (1) whereCYIG contains all parameters describing proper- ties of YIG, including some physical constants (dened in ref.17), soCYIGis constant for both the YIG/Pt and the YIG/Ta sample.
Tmeis the temperature dierence between the magnons and electrons at the YIG/metal interface.,,tandlare the resistivity, spin-diusion length, thickness of the normal metal layer (Pt/Ta) and the distance between the voltage contacts, respectively. is the back ow correction factor, dened as = 1 +Grcotht 1 (2) Previously, in ref.14, we have determined the real part of the spin-mixing conductance at the YIG/Pt inter- face (Gr= 4:41014 1m2), the spin-Hall angle (SH;Pt = 0:08) and the spin-diusion length ( Pt= 1:2nm) of Pt. For the YIG/Ta sample we take the mag- nitude of these system parameters as reported by Hahn et al.15(Gr= 21013 1m2, SH;Ta =0:02 and Ta= 1:8nm). As a check, we also used these parameter- values to calculate the 1storder SMR signals for Ta, and4 found good agreement with the measured signals (not shown). To get an estimate for VSSE;Pt=VSSE;Ta we assume
Tmeto be constant for both samples. By inserting the values of the mentioned parameters, the dimensions of the Hall-bars and the measured resistivity of the Pt and Ta layers (Pt= 3:4107 m andTa= 3:5106 m, respectively), we nd VSSE;Pt=VSSE;Ta =10:6, which is close to the experimentally observed ratio. B. Current-induced spin-Seebeck eect The second method used to detect the SSE is by current-induced heating through the metal detection strip itself, as recently was reported by Schreier et al.21. In this section we show that we can achieve more directly similar results, by using a lock-in detection technique. By this technique, the SSE signals can directly be detected as a 2ndharmonic signal, fully separated from the SMR signal, which shows up in the 1stharmonic response. Fur- thermore, the lock-in detection technique enables us to reveal and investigate additional signals appearing when applying low magnetic elds. The inset of Fig. 2(c) shows a microscope image of the sample, marking the position of the current and volt- age probes for the current-induced heating experiments. The magnetic eld direction is again dened by 0. This measurement conguration is similar to the method used to detect transverse SMR12,14and therefore we expect to observe SMR in the 1storder signal, as is shown in Figs. 2(a) and (b). In Fig. 2(b) it is observed that down to very low applied magnetic elds, the average magne- tization direction of the YIG nicely follows the applied eld direction, resulting in the sin(2 0) angular depen- dence of the SMR.11Only for the lowest applied eld of 0.9mT a small deviation of the signal around 0=90 is observed, showing this eld strength is not sucient to assume M being (on average) fully along the applied eld direction. Similar to the external heating experiment, the SSE signal shows up in the 2ndharmonic signal. Figs. 2(c) and (d) show the magnetic eld dependence and angular dependence of the detected 2ndharmonic signal, respec- tively. Comparing the shape of the 2ndharmonic data of the external heating experiments (Fig. 1(b)) to the current-induced heating experiments (Fig. 2(c)), an en- hanced signal is observed in Fig. 2(c) for elds of a few mT. This additional signal cannot be explained by the angular dependence of the SSE, neither by the rotation of M in the plane towards B (by which the 1stharmonic SMR peaks in Fig. 2(a) are explained12). The angular dependence of this additional signal, as presented in Fig. 2(d), shows that besides an increased amplitude of the SSE signal (black symbols in Figs. 2(c) and (d)), also at 0=90additional peaks appear for low applied elds. By increasing the applied magnetic eld, all extra signals disappear, leaving the expected (c)(a) (b)-180 -90 0 90 180-10-50510 0102030405060-10-50510Iac (mA) 1 2 4 6 8Pt [5nm] B = 0.9mT V2 (µV) α0(deg) Pt [5nm] V2 (µV) I2 ac (mA2)50mT (α0=0o) 0.9mT (α0=0o) 0.9mT (α0=90o) -180-90 0 90180-10-50510Iac (mA) 1 4 8 2 6 Pt [5nm] B = 50mT V2 (µV) α0(deg)FIG. 3. AC-current dependence of the 2ndharmonic volt- age for an applied eld of (a) 0.9mT and (b) 50mT. For the shown experiments, the transverse current-induced heat- ing measurement conguration has been used. The verti- cal dashed lines mark the data plotted in (c), which shows the AC-current dependence of the magnitude of the signal at0= 0for B=0.9mT (red dots) and B=50mT (black squares) and the average magnitude of the peaks (peak to peak) around 0=90for B=0.9mT (blue triangles). The dashed lines are a guide for the eye. SSE signal showing a 360periodic angular dependence. To further characterize the additionally observed fea- tures at low applied magnetic elds, also their AC- current dependence has been measured and these results are shown in Fig. 3. It can be seen that the current dependence is very similar to the shown magnetic eld dependence, giving maximal additional signals for low applied elds and high applied AC-currents. From Figs. 3(a) and (b), the magnitude of the signal at 0= 0is extracted and plotted separately in Fig. 3(c). As can be seen from this gure, for both the applied magnetic eld of 0.9mT (Fig. 3(a)) and 50mT (Fig. 3(b)), the amplitude of the signal quadratically scales with the ap- plied AC-current. The magnitude of the peaks around 0=90, plotted in blue in Fig. 3(c), increases faster than quadratically, pointing to the presence of higher or- der eects. To fully exclude the SSE being the origin of the ad- ditionally detected signals, the current-induced heat- ing measurements were repeated on the YIG/Ta sam- ple. Results of those measurements are shown in Fig. 4. The applied current in those experiments is only5 1.9mA, limited by the high resistance of the Ta bar (Ta= 3:5106 m). In both Fig. 4(a) and (b) it can be seen that the high-eld signal nicely changes sign compared to the YIG/Pt data, as predicted for the SSE/ISHE, because of the opposite sign of the spin-Hall angle of Ta compared to Pt. Contrary, the low-eld peak in the magnetic eld sweep (Fig. 4(a)) keeps the same sign as in the YIG/Pt sample (Fig. 2(c)), showing the SSE cannot be the origin of this phenomenon. Further- more, this result also excludes the observed feature being originated by any other eect linearly related to the spin- Hall angle of a material. So, possible deviations of M caused by spin-transfer torque, due to a spin-current cre- ated via the SHE, cannot directly be used to explain the observed features. Note that the SMR signal depends quadratically on the spin-Hall angle,10,14which makes any eect related to the SMR a likely candidate for ex- plaining the observed features. Summarizing, the current-induced heating experi- ments show that when applying a suciently high mag- netic eld ( >10mT), the SMR and SSE can be simul- taneously, but separately, detected using an AC-current combined with a lock-in detection technique. By this method the SSE can thus be very easily and directly de- tected, without being interfered with the SMR signal. Furthermore, it is observed that for low magnetic elds, and/or high heating currents, additional signals appear on top of the SSE. The origin of these additional sig- nals might be related to the SMR-eect, which will be discussed in more detail in the next section. C. Dynamic Spin-Hall Magnetoresistance During the measurements, large AC-currents are sent through the Hall-bar structure, which can generate mag- netic elds. One source of these magnetic elds will be Oersted elds (B oe) generated around the Hall-bar, perpendicular to the current direction. As the Hall-bar (b) (a) -6-4-2 0 2 4 6-0.050.000.050.100.150.200.25 -180 -90 0 90 180-0.050.000.050.100.150.200.25 V2 (µV) Magnetic Field [mT]Ta [10nm] Iac= 1.9mA α0= 0o V2 (µV) 50 0.9Ta [10nm] Iac= 1.9mA α0(deg) B (mT) FIG. 4. 2ndharmonic response of the transverse voltage for the YIG/Ta sample. (a) Magnetic eld sweep for 0= 0 and (b) angular dependence for two dierent applied mag- netic elds (0.9mT and 50mT). The SSE signal (=signal at high eld) has opposite sign compared to the YIG/Pt sam- ple, whereas the low-eld behavior is similar for both samples. The black symbols in the gures point out the equal measure- ment conditions comparing both gures.structure is very thin compared to its lateral dimen- sions, the Oersted eld above/below the center of the bar can be estimated using the innite plane approxima- tion:Boe=0I 2w, where0is the permeability in vacuum, Iis the applied current and wis the width of the Pt bar. Note that the generated eld in this case is independent of the distance from the plane, so the full thickness of the YIG below the Hall-bar will be exposed to this eld.25 For example, for an applied current of 8mA, an Oersted eld of 0.1mT will be generated, which is signicant com- pared to an applied magnetic eld of 0.9mT. Therefore, for low applied elds, the magnetization direction of the YIG will also be aected by the generated Oersted eld. As we are dealing with AC-currents, the generated Oersted eld (and any other current-induced magnetic eld) will continuously change sign, which might cause M to oscillate around the applied eld direction. In this case,(dening the direction of M) is current- dependent,26giving rise to dynamic equations for both the SMR and the SSE. The current-dependent behavior of the SMR signal is derived starting from the equation for transverse SMR10,14 VT;SMR =IRT;SMR =
R1Imxmy+
R2Imz(3) where
R1and
R2are resistance changes dependent on the spin-diusion length, spin-Hall angle and spin- mixing conductance of the system, as dened in refs.10,14. mx;y;z are the components of M pointing in respectively the x-, y-, and z-direction (where z is the out-of-plane direction). mxandmycan be expressed as sin( ) and cos(), respectively. As the applied magnetic eld is in- plane, as well as the generated Oersted eld, combined with the large demagnetization eld of YIG for out-of- plane directions, mzwill be small and therefore neglected in further derivations. Small oscillations of M, due to the presence of AC- current generated magnetic elds, result in a current- dependent SMR signal, which can be expressed in rst order as VT;SMR (I)VT;SMR (0) +IdVT;SMR dI 0(4) where0gives the equilibrium direction of M around which it is oscillating (assuming it to be equal to the applied eld direction, as concluded from the measured 1stharmonic SMR response). Calculating the derivative in Eq.(4), using Eq.(3), neglecting mzand keeping in mind thatis dependent on I, gives dVT;SMR dI 0=
R1sin(0) cos(0)+
R1Icos(20)d dI (5) The rst term on the right side of Eq.(5) describes the 1storder response (linear with I), showing the expected transverse SMR behavior (Eq.(3)). The second term is a 2ndorder response ( R2;SMR ) and will therefore show up in addition to the expected SSE signal.d dIis the term6 which includes the deviation of M due to current-induced magnetic elds, and its magnitude is dependent on both 0and the magnitude of the total magnetic eld (applied eld, coercive eld and the current-induced elds). For large applied magnetic elds, the current-induced mag- netic elds will have a negligible eect on M, sod dIgoes to zero, leaving only the SSE signal in the 2ndharmonic signal (as is observed). Note that also in the described external heating experiments Oersted elds are gener- ated, in uencing M, but there the dynamic SMR signal will not be detected, as the SMR itself is not present. To nd an expression ford dI, rst the direction of M (given by) is dened, taking into account the Oersted elds (causing
): =0+
=0+ atanBoe Bex cos(0) (6) whereBexis the applied magnetic eld and atan(Boe Bex) is the maximum deviation of M from 0, which is the case for0= 0(Bexperpendicular to Boe, neglecting any other eld contributions). From Eq.(6) nowd dI(= d dBoedBoe dI) can be derived, nding d dI=0 2wBex B2ex+B2oecos(0) (7) Substituting the derived equation ford dIin Eq.(5), we calculate the expected 2ndharmonic SMR signal due to dynamic behavior of M, caused by the current-induced Oersted eld as: R2;SMR =
R10 2wBex B2ex+B2oecos(20) cos(0) (8) Additional to R2;SMR , also the SSE will be present as a 2ndharmonic signal, showing cos( 0) behavior with an amplitude independent from the applied magnetic eld strength. The amplitude of the SSE signal can be derived from the high-eld measurements shown in Fig. 3(b) and Fig. 4(b). Figures 5(a) and (b) show the total calculated 2ndhar- monic voltage signal for the YIG/Pt sample, taking into account both the SSE, extracted from the measurements, and the dynamic SMR term, given by Eq.(8) (where V2;SMR = (I2=p 2)R2;SMR ). For the calculation of
R1 system parameters from ref.14are used. Both the cal- culated current dependence (Fig. 5(a)) as well as the calculated magnetic eld dependence (Fig. 5(b)) show similar features as the measurements, only its magnitude and exact shape do not fully coincide. Following the ex- planation of the lock-in detection method as described in the appendix, we nd that these discrepancies are mainly caused by the presence of a non-negligible 4thharmonic signal (and possibly even higher harmonics). When de- termining the 2ndorder response of the system, taking into account the measured 2ndand 4thharmonic signals, the peaks observed around 0=90get wider and (a) (b) (c)-180 -90 0 90 180-20-1001020 Iac [mA] 1 2 4 6 8Pt [5nm] B = 0.9mT V2 [µV] α0[ο]-180 -90 0 90 180-20-1001020 Pt [5nm] Iac = 8mA V2 [µV] B [mT] 0.9 1.8 2.8 8.5 100 α0[ο] -180 -90 0 90 180-0.10-0.050.000.050.10Ta [10nm] Iac = 1.9mA V2 [µV]B [mT] 0.9 50 α0[ο]FIG. 5. (a) Current dependence and (b) magnetic eld de- pendence of the calculated 2ndharmonic response, including the dynamic SMR (from Eq.(8)) and the SSE for the YIG/Pt system. (c) Calculated 2ndharmonic response for the YIG/Ta sample using a scaling factor ford dIof 0.5. For all calculations the SSE signal is extracted from the measurements. smoother, more closely reproducing the calculated sig- nals as presented in Fig. 5. The amplitude of the overall calculated signal is slightly larger than the measurements, up to a factor 1.6 (even after taking into account the 4thharmonic signal). One reason for this discrepancy might be that, for the cal- culations, only the applied magnetic eld and the gener- ated Oersted eld are taken into account, neglecting any other present elds. For example, the coercive eld of the YIG is assumed to be absent, as well as the eect of spin- torque and the presence of non-uniform magnetic elds. These additional elds will in uence the amplitude of the oscillations of M, giving a dierent value ford dIthan the assumed deviation from only BoeandBex. Furthermore, the assumed perfect cos( 0) behavior ofd dImight also be disturbed by the presence of these other elds. Sec- ondly, in the calculations it is assumed that M is fully aligned with the total magnetic eld, which might not always be the case, as we investigate the system apply- ing magnetic elds approaching the coercive eld of the YIG. Therefore, the denition of 0can be slightly o from the assumed ideal case. Full characterization of the magnetization dynamics of the system and the magnetic eld distribution would be needed to be able to give a more complete theoretical analysis of the observed fea- tures. The same calculations have been repeated for the YIG/Ta system. The system parameters needed to calcu- late
R1are taken from ref.15, as given in section IV A. As a check, these system parameters rstly were used to calculate the 1storder SMR signal, nding good agree- ment with the measured signals (not shown). For the calculated 2ndharmonic signal again it is found that the amplitude ofd dIhas to be lowered to be able to reproduce the measured behavior. When lowering the calculatedd dI7 by a chosen scaling factor of 0.5, results as shown in Fig. 5(c) are obtained. Comparing the calculated angular de- pendence to the measurement as shown in Fig. 4(b), good agreement is found in the observed behavior. Note that for the YIG/Ta system the contribution of the 4th harmonic term is much less pronounced, as the applied current is only 1.9mA (compared to 8mA for the YIG/Pt experiments). Following the derivation of the dynamic SMR as ex- plained above, also dynamic SSE signals can be expected. As the SSE is a 2ndorder eect, any dynamic SSE signals are expected to appear as a 3rdorder signal. Measure- ments of the 3rdharmonic signal indeed show addition- ally appearing signals at low applied magnetic elds, but these additional signals are one order of magnitude too large to be explained by the derived possible dynamic SSE signals. This shows that other higher harmonic ef- fects are present, which makes it at this moment impos- sible to exclusively extract any contribution of possibly present dynamic SSE signals. Concluding this section, the dynamic SMR is a good candidate for explaining the observed low-eld 2ndhar- monic behavior. For both the YIG/Pt and YIG/Ta sam- ple, the features observed in the experiments can be well reproduced by the dynamic SMR model. However, one has to keep in mind that more non-linear eects are present, such that higher harmonic signals need to be taken into account. Furthermore, the derived model is not sucient to fully be able to reproduce the measured data. Further analysis of the magnetization dynamics in the YIG at low applied elds and high applied currents is necessary to be able to derive a more complete model.V. SUMMARY We have shown the detection of the SSE in YIG/Pt and YIG/Ta samples by both external heating and current- induced heating. The external heating experiments di- rectly show the SSE and clearly show the eect of the opposite spin-Hall angle for Ta compared to Pt. For the current-induced measurements, besides the SSE, the SMR is also present. By using a lock-in detection tech- nique we are able to simultaneously, but separately, mea- sure the SSE and SMR signals. Investigation of the low- eld behavior of the SMR and SSE, reveals an additional 2ndharmonic signal. This additional signal is explained by the presence of a dynamic SMR signal, caused by al- ternating Oersted elds. Calculations show reproducibil- ity of the observed 2ndharmonic features, however fur- ther analysis of the magnetization dynamics in the YIG is needed to derive a more complete model of the system behavior. ACKNOWLEDGEMENTS We would like to acknowledge M. de Roosz, H. Adema and J. G. Holstein for technical assistance. This work is supported by NanoNextNL, a micro and nanotechnology consortium of the Government of the Netherlands and 130 partners, by the Marie Curie Actions (Grant 256470- ITAMOSCINOM), the Basque Government (PhD fellow- ship BFI-2011-106), by NanoLab NL and by the Zernike Institute for Advanced Materials (Dieptestrategie pro- gram). 1C. W. Sandweg, Y. Kajiwara, A. V. Chumak, A. A. Serga, V. I. Vasyuchka, M. B. Jung eisch, E. Saitoh, and B. Hille- brands, Phys. Rev. Lett. 106, 216601 (2011). 2C. Hahn, G. de Loubens, M. Viret, O. Klein, V. V. Naletov, and J. Ben Youssef, Phys. Rev. Lett. 111, 217204 (2013). 3K. Harii, T. An, Y. Kajiwara, K. Ando, H. Nakayama, T. Yoshino, and E. Saitoh, Journal of Applied Physics 109, 116105 (2011). 4V. Castel, N. Vlietstra, B. J. van Wees, and J. Ben Youssef, Phys. Rev. B 86, 134419 (2012). 5A. V. Chumak, A. A. Serga, M. B. Jung eisch, R. Neb, D. A. Bozhko, V. S. Tiberkevich, and B. Hillebrands, Ap- plied Physics Letters 100, 082405 (2012). 6Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi, S. Maekawa, and E. Saitoh, Nature (Lon- don)464, 262 (2010). 7G. L. da Silva, L. H. Vilela-Leao, S. M. Rezende, and A. Azevedo, Applied Physics Letters 102, 012401 (2013). 8H. Kurebayashi, O. Dzyapko, V. E. Demidov, D. Fang, A. J. Ferguson, and S. O. Demokritov, Nat. Mater. 10, 660 (2011). 9E. Padr on-Hern andez, A. Azevedo, and S. M. Rezende, Phys. Rev. Lett. 107, 197203 (2011).10Y.-T. Chen, S. Takahashi, H. Nakayama, M. Althammer, S. T. B. Goennenwein, E. Saitoh, and G. E. W. Bauer, Phys. Rev. B 87, 144411 (2013). 11M. Althammer, S. Meyer, H. Nakayama, M. Schreier, S. Altmannshofer, M. Weiler, H. Huebl, S. Gepr ags, M. Opel, R. Gross, D. Meier, C. Klewe, T. Kuschel, J.- M. Schmalhorst, G. Reiss, L. Shen, A. Gupta, Y.-T. Chen, G. E. W. Bauer, E. Saitoh, and S. T. B. Goennenwein, Phys. Rev. B 87, 224401 (2013). 12N. Vlietstra, J. Shan, V. Castel, B. J. van Wees, and J. Ben Youssef, Phys. Rev. B 87, 184421 (2013). 13H. Nakayama, M. Althammer, Y.-T. Chen, K. Uchida, Y. Kajiwara, D. Kikuchi, T. Ohtani, S. Gepr ags, M. Opel, S. Takahashi, R. Gross, G. E. W. Bauer, S. T. B. Goen- nenwein, and E. Saitoh, Phys. Rev. Lett. 110, 206601 (2013). 14N. Vlietstra, J. Shan, V. Castel, J. Ben Youssef, G. E. W. Bauer, and B. J. van Wees, Applied Physics Letters 103, 032401 (2013). 15C. Hahn, G. de Loubens, O. Klein, M. Viret, V. V. Naletov, and J. Ben Youssef, Phys. Rev. B 87, 174417 (2013). 16K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi, J. Ieda, T. Ota, Y. Kajiwara, H. Umezawa, H. Kawai, G. E. W. Bauer, S. Maekawa, and E. Saitoh, Nature Ma- terials 9, 894 (2010).8 17M. Schreier, A. Kamra, M. Weiler, J. Xiao, G. E. W. Bauer, R. Gross, and S. T. B. Goennenwein, Phys. Rev. B88, 094410 (2013). 18S. M. Rezende, R. L. Rodr guez-Su arez, R. O. Cunha, A. R. Rodrigues, F. L. A. Machado, G. A. Fonseca Guerra, J. C. Lopez Ortiz, and A. Azevedo, Phys. Rev. B 89, 014416 (2014). 19H. Adachi, K. Uchida, E. Saitoh, and S. Maekawa, Reports on Progress in Physics 76, 036501 (2013). 20J. Flipse, F. K. Dejene, D. Wagenaar, G. E. W. Bauer, J. Ben Youssef, and B. J. van Wees, Phys. Rev. Lett. 113, 027601 (2014). 21M. Schreier, N. Roschewsky, E. Dobler, S. Meyer, H. Huebl, R. Gross, and S. T. B. Goennenwein, Applied Physics Letters 103, 242404 (2013). 22L. Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, Science 336, 555 (2012). 23M. Morota, Y. Niimi, K. Ohnishi, D. H. Wei, T. Tanaka, H. Kontani, T. Kimura, and Y. Otani, Phys. Rev. B 83, 174405 (2011). 24T. Chiba, G. E. W. Bauer, and S. Takahashi, Phys. Rev. Applied 2, 034003 (2014). 25This approximation was checked by modeling the system using COMSOL Multiphysics, showing indeed a nearly constant Oersted eld at least several hundreds of nanome- ters above/below the plane. 26K. Garello, I. M. Miron, C. O. Avci, F. Freimuth, Y. Mokrousov, S. Blugel, S. Auret, O. Boulle, G. Gaudin, and P. Gambardella, Nature Nanotechnology 8, 587 (2013). APPENDIX 1. Lock-in detection All measurements shown in the main text are per- formed using a lock-in detection technique. By this tech- nique, 1st, 2ndand higher order responses of a system on an applied AC-current can be determined. In general, any generated voltage can be written as the sum of 1st, 2ndand higher order current-responses as: V(t) =R1I(t) +R2I2(t) +R3I3(t) +R4I4(t) +:::(9) where Rnis the nthorder response of the measured sys- tem to an applied current I(t). By applying an AC- currentI(t) =p 2I0sin(!t), with angular frequency ! and rms value I0, a lock-in amplier can be used to de- tect individual harmonic voltage responses of the inves- tigated system, making use of the orthogonality of sinu- soidal functions. To extract the separate harmonic re- sponses, the output signal and the reference input signal (a sine wave function) are multiplied and integrated over a set time. When both signals have dierent frequen- cies, the integration over many periods will result in zero signal, whereas integration of two sine wave functions with the same frequency and no phase shift will result in a non-zero signal. Besides being able to separately extract the dierent harmonic responses of the system, the lock-in detection technique also reduces the noise inthe signal, compared to dc voltage measurements, as the measurement is only sensitive to a very narrow frequency spectrum. The detected n-th harmonic signal of a lock-in ampli- er at a set phase is dened as Vn(t) =p 2 TtZ tTsin(n!s+)Vin(s)ds (10) By evaluating Eq.(10) for a given input voltage V in, one can obtain the dierent harmonic voltage signals that can be measured by the lock-in amplier ( Vn). Assuming a voltage response up till the 4thorder, the following lock- in voltages are calculated: V1=R1I0+3 2R3I3 0 for= 0(11) V2=1p 2(R2I2 0+ 2R4I4 0) for=90(12) V3=1 2R3I3 0 for= 0(13) V4=1 2p 2R4I4 0 for=90(14) So, using dierent lock-in ampliers to measure the 1st, 2nd, 3rdand 4thharmonic voltage responses, Rncan be deduced from Eqs.(11)-(14). To detect the 2ndand 4thharmonic response, the phase of the lock-in amplier should be set to =90. Note that V1(V2) does not purely scale linearly (quadratically) with I0. A 3rd(4th) order current depen- dence is also present in the measured voltage response. Thus, to obtain the 1storder response ( R1) of the mea- sured system, not only the measured 1stharmonic signal (V1) has to be taken into account, also the 3rdharmonic signal (V3) has to be included: R1=1 I0(V1+ 3V3) (15) Similarly, the 2ndorder response is calculated as R2=p 2 I2 0(V2+ 4V4) (16) For the current-induced SSE and SMR measurements described in the main text, Fig. 6 shows the eect of in- cluding the higher harmonic responses of the system (up to the 4thharmonic), compared to assuming them to be negligible. For this comparison Eq.(15) and Eq.(16) are used. In Fig. 6(a) and (c) V3andV4are assumed to be zero, whereas in Fig. 6(b) and (d) the measured 3rdand 4thharmonic response have been taken into account, re- spectively. From these gures it can be concluded that in9 (a) (b) (c) -180 -90 0 90 180-0.3-0.2-0.10.00.10.20.3 -90 0 90 18030354045505560 R2 [V/A2]Assuming V4= 0 Including measured V4 Assuming V3= 0 R1 [mΩ]Including measured V3 B[mT] 0.9 1.8 100 Pt [5nm] Iac = 8mA α0[ο] α0[ο](d) FIG. 6. Evaluation of Eq.(15) for a selected set of measure- ments: (a) assuming V3= 0 and (b) including the measured values ofV3. Similarly for Eq.(16): (c) assuming V4= 0 and (d) including the measured values of V4.our low-eld measurements the higher harmonic signals are not negligibly small, and thus should be accounted for. In Fig. 2(a) and (b) of the main text, the 1storder response of the system is plotted, taking into account the measured 3rdharmonic voltage response as derived in Eq.(15) and shown in Fig. 6(b). For the other g- ures, regarding the 2ndorder response of the system, the measured lock-in voltage is directly plotted ( V2). To nd the 2ndorder response of the system Eq.(16) should be evaluated, including both V2andV4. Taking this cor- rection into account, the observed features of V2slightly change, as is shown in Fig. 6(d), more closely following the expected behavior from the calculated dynamic SMR signal.

2014-10-02

The spin-Seebeck effect (SSE) in platinum (Pt) and tantalum (Ta) on yttrium iron garnet (YIG) has been investigated by both externally heating the sample (using an on-chip Pt heater on top of the device) as well as by current-induced heating. For SSE measurements, external heating is the most common method to obtain clear signals. Here we show that also by current-induced heating it is possible to directly observe the SSE, separate from the also present spin-Hall magnetoresistance (SMR) signal, by using a lock-in detection technique. Using this measurement technique, the presence of additional 2nd order signals at low applied magnetic fields and high heating currents is revealed. These signals are caused by current-induced magnetic fields (Oersted fields) generated by the used AC-current, resulting in dynamic SMR signals.

Simultaneous detection of the spin-Hall magnetoresistance and the spin-Seebeck effect in Platinum and Tantalum on Yttrium Iron Garnet

1410.0551v2

Investigation of Phonon Lifetimes and Magnon-Phonon Coupling in YIG/GGG Hybrid Magnonic Systems in the Diffraction Limited Regime Manoj Settipalli Ann and H.J. Smead Aerospace Engineering Sciences, University of Colorado Boulder, Boulder, Colorado 80303, USA Xufeng Zhang Electrical and Computer Engineering, Northeastern University, Boston, MA 02115 Sanghamitra Neogi∗ Ann and H.J. Smead Aerospace Engineering Sciences, University of Colorado Boulder, Boulder, CO 80303, USA Quantum memories facilitate the storage and retrieval of quantum information for on-chip and long-distance quantum communications. Thus, they play a critical role in quantum information processing and have diverse applications ranging from aerospace to medical imaging fields. Bulk acoustic wave (BAW) phonons are one of the most attractive candidates for quantum memories because of their long lifetime and high operating frequency. In this work, we establish a modeling approach that can be broadly used to design hybrid magnonic high-overtone bulk acoustic wave resonator (HBAR) structures for high-density, long-lasting quantum memories and efficient quantum transduction devices. We illustrate the approach by investigating a hybrid magnonic system, where BAW phonons are excited in a gadolinium iron garnet (GGG) thick film via coupling with magnons in a patterned yttrium iron garnet (YIG) thin film. We present theoretical and numerical analyses of the diffraction-limited BAW phonon lifetimes, modeshapes, and their coupling strengths to magnons in planar and confocal YIG/GGG HBAR structures. We utilize Fourier beam propagation and Hankel transform eigenvalue problem methods and discuss the effectiveness of the two methods to predict the HBAR phonons. We discuss strategies to improve the phonon lifetimes, since increased lifetimes have direct implications on the storage times of quantum states for quantum memory applications. We find that ultra-high, diffraction-limited, cooperativities and phonon lifetimes on the order of ∼105and∼10 milliseconds, respectively, could be achieved using a CHBAR structure with 10 µm lateral YIG dimension. Additionally, the confocal HBAR structure will offer more than 100-fold improvement of integration density. A high integration density of on-chip memory or transduction centers is naturally desired for high-density memory or transduction devices. Our results will have direct applicability for devices operating in the cryogenic or milliKelvin regimes. For example, our approach and analyses can be applied to design HBAR devices that could effectively couple with superconducting qubit systems. INTRODUCTION Hybrid quantum systems that integrate diverse plat- forms, constitute a rapidly emerging research field due to their ability to synergistically address the limitations of individual platforms. Among various systems explored, hybrid magnonic systems received significant attention in recent years due to the unique advantages they offer [1, 2]. In these systems, quantized spin waves (magnons) co- herently couple with other information carriers, such as photons and phonons. The coupling presents avenues for both fundamental investigations and practical imple- mentations [3–11]. These systems often leverage mag- netic materials with high spin density, such as yittrium iron garnet (YIG). Such materials enable strong coupling of magnons with microwave photons trapped in a cav- ity, with coupling strengths surpassing their respective dissipation rates [3–7]. Several devices have been in- vestigated that benefit from the strong magnon-photon coupling, with applications ranging from microwave-to-optical transduction [12, 13] to quantum magnonics [14– 16], dark matter detection [17, 18] and others [19–24]. Besides magnon-photon coupling, magnon-phonon coupling enables another important class of hybrid magnonic devices which has attracted attention for co- herent and quantum information processing applica- tions [11, 25–28]. Phonons are particularly attractive due to their long lifetime compared to other infor- mation carriers [29–32]. Magnon-phonon coupling has shown success for classical information processing appli- cations [33–45]. The coupling of magnons with mechani- cal phonons in YIG spheres has been utilized for promis- ing quantum information processing (QIP) applications, such as magnon-phonon entanglement [46], nonreciprocal phonon propagation [47, 48], and magnon squeezing [49]. Separately, recent studies used high-overtone bulk acous- tic wave resonator (HBAR) phonons, that operate in the GHz regime, for coupling with superconducting qubits and demonstrated their remarkable potential for quan- tum acoustodynamics [50–52]. The HBAR phonons also exist in hybrid magnonic devices and can enable reso-arXiv:2308.06896v2 [cond-mat.mes-hall] 29 Nov 20232 nant magnon-phonon coupling via the magnetoelastic ef- fect [53]. These phonons are commonly supported by structures composed of YIG thin-films [54–57] grown on a thick gadolinium gallium garnet (GGG) substrate, which can host a series of HBAR resonances. A recent study demonstrated a triply resonant photon-magnon-phonon system using a YIG/GGG hybrid magnonic HBAR de- vice [58]. Such resonant coupling can enable long-lived multimode phononic quantum memories and other quan- tum information processing and transduction applica- tions. In this article, we investigate a class of planar and confocal HBAR YIG/GGG hybrid magnonic structures for quantum transduction applications. The performance figure of merit for magnon-phonon transduction can be characterized using cooperativity, C= 4g2 mb/κmκb, where gmb,κm, and κbare magnon- phonon coupling strength, magnon dissipation rate, and phonon dissipation rate, respectively. The lifetimes of magnons and phonons are given by τm= 1/κmand τb= 1/κb, respectively. In YIG/GGG hybrid magnonic HBAR devices, the phonon lifetimes usually surpass the magnon lifetime thanks to the excellent mechanical prop- erties of the garnets. Past studies reported the magnon and phonon lifetime in YIG/GGG HBAR systems to be τm= 0.07µs and τb∼0.25µs, respectively [57, 58]. The system lifetime is primarily determined by the phonon lifetimes. It is desirable to further improve the phonon lifetimes and consequentially, the system lifetimes, for quantum memories and other information processing ap- plications. However, a complete understanding about the loss mechanisms that limit the phonon lifetime in gar- net devices is not yet available. The phonon lifetime could be limited by material and diffraction losses de- pending on the device geometry or the operating condi- tions. At room temperature, the phonon lifetime is pri- marily limited by acoustic attenuation due to phonon- phonon interactions [59, 60]. The acoustic attenuation effects, α∝1/τb, follows a T4temperature dependence. Due to the T4behavior, these effects are less significant at low temperatures, such as millikelvin regime where many qubit systems operate. Ideally, τbcan increase by multiple orders of magnitude at cryogenic temperatures. However, the diffraction losses play an important role in determining the phonon lifetime at low-temperature operating conditions, where the material losses are sup- pressed. In this study, we establish a computational approach for analyzing the diffraction losses in HBAR structures and use it to investigate the effects of diffrac- tion losses on the HBAR phonons of hybrid YIG/GGG magnonic structures. We focus on the diffraction effects because they can be analyzed computationally, while ex- perimental characterization is needed to investigate the acoustic attenuation effects. Note that the knowledge of material losses in YIG/GGG material systems is limited. There is a strong need for experimental characterization of acoustic attenuation in different magnetic materialsacross temperature and operating frequencies, to unlock their full potential. The integration density of the hybrid magnonic HBAR structures is an important design aspect for developing high-density phononic quantum memories. The inte- gration density is measured by the number of memory or transduction components that could be incorporated into a single chip, in addition to other on-chip circuitry. The YIG film represents the transduction component of the YIG/GGG HBAR structures of interest. Thus, the smaller the lateral dimension of the YIG film, the greater the number of transduction components in a single chip of given dimension and the integration density. We de- fine the integration density as Di=A0 d Ad. Here, Adis the lateral area of the YIG film for planar HBAR structures, and the cross-sectional area of the confocal dome surface for CHBAR structures in our study, respectively. A0 drep- resents a reference value for the device. We consider the smallest known lateral area reported for YIG/GGG de- vices in literature to be the reference, A0 d= 0.8×0.9 mm2= 0.72 mm2[58]. The integration density of the device can be increased by reducing the lateral dimen- sion of the YIG films. However, the reduced aperture will lead to high-diffraction and affect phonon lifetime. In addition, a smaller aperture will reduce the overlap between the magnon and the phonon modes since they only overlap in the YIG film. A reduced magnon-phonon overlap could lead to a weaker magnon-phonon coupling strength. Consequently, the cooperativity will decrease as a result of the reduced lifetime and coupling strength. It is therefore imperative to develop a strategy to in- crease the integration density without inducing excessive diffraction losses. Note that we only focus on the lateral integration density while keeping the thickness fixed at 527.2µm for all structures investigated in this work. The same thickness allows us to keep the free spectral range of phonons fixed for all our analysis. In this study, we investigate a set of planar and confocal YIG/GGG HBAR structures. We investigate the phonon lifetime and the magnon-phonon coupling strength in these structures, and identify strategies to improve their performance and integration density. We assume a magnon lifetime of 0 .07µs [58] in all struc- tures considered in this study. The phonon lifetime τb or the quality factor ( Q=ωτb) is of particular inter- est due to its direct relevance to the quantum mem- ory storage time. Henceforth, we drop the subscript b from τb, unless otherwise needed to distinguish it from magnon and photon lifetimes. Without losing general- ity, we only consider the fundamental magnon mode, i.e., the Kittel mode, for simplicity. The Kittel mode has a well-defined analytical function for circular discs. How- ever, such analytical description is still lacking for the HBAR phonon modes. Several methods have been im- plemented to model HBAR phonons and their lifetimes.3 For example, phonon lifetime in planar HBARs was cal- culated by decomposing the initial beam in a Bessel func- tion basis and calculating the overlap of the reflected beam with the initial profile after each round trip [61]. However, the predicted lifetime was smaller than the ex- perimental observations because the effects of the lat- eral confinement induced by the transducer are not in- cluded. Finite element analysis (FEA) is another popular method to study HBAR phonons, using the COMSOL Multiphysics [62] software. A recent study estimated the diffraction loss in an epitaxial planar HBAR struc- ture by measuring the power received at the opposite end to that of the actuator [63]. Another study used FEA to predict the modeshapes of a confocal HBAR for qubit coupling applications [64]. They considered phonon modes at low-overtones or long wavelengths, hence it was possible to perform three-dimensional (3D) FEA sim- ulations for this system. However, as we will discuss later, it becomes intractable to simulate bulk acoustic waves with overtone numbers n∼3000 using FEA since sampling required for these small wavelength modes is on the order of ∼billion nodes. Moreover, all these studies discussed longitudinal HBAR phonons whereas in hybrid magnonic devices shear phonons exhibit much stronger coupling with magnon modes [57, 58, 65]. The FEA analysis becomes particularly expensive for shear phonons since they are not axi-symmetric. One cannot leverage the axi-symmetric two-dimensional (2D) FEA modelling available in COMSOL Multiphysics for these systems. Due to these reasons, we explore other ap- proaches to model shear phonon modes. We consider the Fourier beam propagation method (FBPM) [51, 61] which follows a Fox-Li-like [66] iterative approach to ob- tain the phonon modes and works with a plane-wave basis set. Although past studies used FBPM for longitudinal phonon modes, we illustrate that it can effectively model the shear phonon modes of interest. We provide a de- tailed description of this method and discuss an adaptive algorithm that allows to overcome some of the challenges of the standard FBPM method. Additionally, we con- sider another method that uses Hankel transform (HT), which is a Fourier transfrom for axi-symmetric systems, and works with a Bessel function basis. Thus far, HT method has only been implemented for Fabry-Perot op- tical cavities [67, 68]. We adapt this approach to sim- ulate YIG/GGG HBAR structures by leveraging their axi-symmetry and isotropic material properties. METHODS HBAR Configurations Figure 1 shows the two representative HBAR struc- tures investigated in this work. The structure in Fig- ure 1(a) consists of a thick GGG film joined with a YIGthin film at the bottom. We refer to this structure as the planar HBAR structure since it has planar top and bottom surfaces. Figure 1(b) is a focusing HBAR struc- ture that includes a GGG dome structure at the top and a planar bottom surface. We refer to this structure as the confocal HBAR (CHBAR) structure. The thickness of the GGG film for all configurations considered in this study, is tGGG = 527 µm, as shown in Fig. 1. All YIG films are circular films or discs with thickness, tYIG= 200 nm, and radius of cross-section, RYIG. The total HBAR device thickness is tHBAR =tGGG+tYIG= 527 .2µm. We consider planar and confocal HBAR structures with sev- eralRYIG’s ranging from 10 µm to 200 µm. The width W of the structure is 1200 µm unless otherwise mentioned. The radius of cross-section of the dome, Rcross, and its radius of curvature, Rcurv, vary for different structures considered. We investigate 8 planar HBARs with varying RYIG, 18 CHBARs with varying Rcurvand fixed Rcross, and 10 CHBARs with varying Rcrossand fixed Rcurv. FIG. 1. Representative HBAR configurations: (a) Pla- nar HBAR structure consisting of a GGG thick film and a YIG thin film. (b) Confocal HBAR structure with a top dome and a planar bottom surface. xaxis is along the out-of-plane direction while yandzaxes point along the lateral and the normal direction of the structures, respectively. Origin of the coordinate axes is at the center of the bottom YIG surface. Magnon Modes The YIG thin film hosts magnons, generated by a static, B0, and an oscillating RF magnetic field, BRF. B0is along the zaxis, while BRFacts in the YIG plane. The magnetic fields, B0andBRF, generate forward vol- ume magnetostatic modes in the YIG film, that precess in the x−yplane. The resulting dynamic magnetization is given by mYIG(x, y) =m0(x, y)(cos( ωt)i+ sin( ωt)j). Here, m0(x, y) represents the magnon modeshape and ωis the frequency of precession. We practice the fol- lowing convention throughout the article, bold-font sym- bols represent vector fields and corresponding normal- font symbols represent scalar values, respectively. We expect un-pinned spin waves at the top and bottom YIG film surfaces because of the following reason. The ex- change interaction effects are expected to dominate at a4 thickness on the order of 200 nm and below. As a conse- quence, the pinning effect is expected to disappear below a critical width on the order of 200 nm [69]. However, we expect full-pinning of the magnetic spin waves at the lateral boundaries of the YIG discs of radii ranging from 10 micrometers ( µm) to 200 µm. This is because the mag- netic dipolar interaction effects dominate at length scales on the order of micrometers, over the exchange interac- tions [69]. Taking these into account, we assume that mYIGis constant throughout the YIG film thickness. We describe the modeshape of the pinned spin waves, m0(x, y), using truncated Bessel functions [70, 71]: m0(x, y) =( J0 R RYIGζj ,ifR≤RYIG 0, otherwise(1) where radial coordinate, R=p x2+y2andJ0is the 0th order bessel function. The zeroes of ζjwith j= 0,1,2, ..., correspond to the fundamental and higher-order magnon modes respectively. In this article, we consider the fun- damental magnon mode or the Kittel mode, whose am- plitude is represented by the function, J0(R RYIGζ0). The modeshape is constant along the z-direction. Phonon Modes The forward volume magnon modes, mYIG(x, y), gen- erate precessing shear deformations in the YIG region. The dynamic shear strain results in a circularly po- larized chiral phonon traveling wave in the GGG re- gion. The helicity of the chiral phonon is determined by the precession direction of the Kittel mode. The shear displacements can be expressed as uYIG(x, y) = u0(x, y)(cos( ωt)i+sin( ωt)j), where u0(x, y) is the phonon modeshape and ωis the precession frequency. In the fol- lowing, we refer to the shear displacements, uYIG(x, y), asu0(x, y). The traveling wave undergoes reflections at the top GGG surface and the reflected waves inter- fere with the forward traveling wave. When the forward and the reflected helical propagating waves interfere, they form rotating standing shear wave modes at specific fre- quencies. While the traveling chiral phonons are helical, the standing waves are not helical. The top GGG surface does not induce a πphase shift to the reflected wave, unlike circularly polarized light reflecting off a mirror. As a result, we obtain standing shear wave modes with zero net helicity. In this article, we use (1) Fourier beam propagation and (2) HT eigenvalue problem approaches to analyze the shear phonon modes in the chosen HBAR configurations.(1) Phonon Modeshape Analysis: Fourier Beam Propagation The Fourier beam propagation method (FBPM), also known as the angular spectrum method, predicts the field displacements or profiles of propagating waves [51, 72]. The advantage of FBPM lies in its simplicity and the ability to predict the field profiles at any target distances from the source without needing to calculate the behavior at intermediate distances. The FBPM has been widely used to analyze beam propagation in the field of op- tics [72]. The mathematical formulation of FBPM for acoustic waves is well established [51]. Recently, it has been used to study phonons in planar [61] and CHBAR structures [51], adapting an iterative method similar to the Fox-Li approach [66]. Here, we implement a reformu- lated iterative approach in which the propagation is cal- culated using projector and propagator operators. The reformulation allows us to achieve a seven-fold speed up of computation time. Our approach is particularly ad- vantageous for isotropic systems such as YIG and GGG. We assume an initial shear displacement field with modeshape u(1) 0(x, y) =( J0 R RYIGζ0 ,ifR≤RYIG 0. otherwise(2) The iterative procedure begins with this initial input field and the superscript represents iteration index ( i= 1). The subscript in u(1) 0(x, y) refers to the fact that the dis- placement is computed at z= 0. Note that the lateral phonon modeshape is the same as the magnon Kittel modeshape, as shown in Eq. 1. In FBPM, the input beams for initial and the subsequent iterations are de- composed into plane-waves. The field displacements at intermediate distances is then obtained by multiplying phase factors to the decomposed beam. The Fourier de- composition of the input beam is given by: ˜u(i) 0(kx, ky) = FFT[ u(i) 0(x, y)]. (3) Here, iis the iteration index and ˜u(i) 0is defined on a N×N(kx, ky) grid which is the Fourier conjugate of the spatial N×N(x, y) grid. We calculate the projections of˜u(i) 0(kx, ky) along the different polarization directions, m= 1,2,3, using the projector, Pm, and obtain the am- plitudes, A(i) m, of the shear ( m= 1,2) and the longitudi- nal (m= 3) modes: A(i) m=Pm·˜u(i) 0, (4a) withPm=Σcyc(−1)|sgn(j−m)|ˆdj(ˆdm·ˆdj−ˆdk·ˆdl) 1 + 2Π cyc(ˆdj·ˆdk)−Σcyc(ˆdj·ˆdk)2. (4b) Here, ˆdmis the polarization vector for the plane-waves propagating along ( kx, ky, kz,m), (j, k, l ) refers to the5 Cartesian directions, ( j, k, l ) = (1 ,2,3) and Σ cycand Π cyc are cyclic sum and product operators, respectively, that cycle through variables j,k, and l. We use the amplitudes A(i) m(kx, ky), to obtain the displacement field of the propa- gated beam, starting from the input beam, u(i) 0(x, y). For a wave originating in the YIG thin film, the propagation distance to reach the upper GGG surface of the HBAR structures is tHBAR , as shown in Fig. 1. The displace- ment field of the propagated beam at the upper GGG surface is given by u(i) tHBAR(x, y) = IFFT[Σ mA(i) mGm], (5a) with Gm(kx, ky) =ˆdm(kx, ky)eikz,m(kx,ky)tHBAR.(5b) Here, Gm(kx, ky) is the propagator for plane-waves trav- eling along ( kx, ky, kz,m) with polarization m. Typically, kz,mcan be derived from their respective slowness sur- faces [51] for each polarization and values of ( kx, ky). However, the functional relation can be simplified to kz,m=q ω2 v2m−k2x−k2y, for isotropic dispersions. Here, ω is the frequency of the initial wave and vmis the velocity of phonons with polarization m. We use the simplified re- lationship and the material properties of YIG and GGG, shown in Table. I, to compute kz,mfor the propagat- ing waves in our isotropic YIG/GGG HBAR structures. Since the properties of YIG and GGG are similar, we use GGG values for both YIG and GGG, for simplicity. This approximation can be further justified by consid- ering that our structures are composed of GGG thick films with ultrathin YIG films bonded to it. Note that we compute kz,m,Am(Eq. 4), and utHBAR(x, y) (Eq. 5) only for the first iteration, for a given ω. This aspect re- sults in the seven-fold computational speed-up mentioned earlier. We choose the longitudinal polarization ˆd3to be TABLE I. Material properties of YIG and GGG. Young’s modulus Poisson’s ratio Density E (Pa) ν ρ (kg/m3) YIG [73] 0.2 ×10120.29 5170 GGG [74] 0.222 ×10120.28 7080 along the unit vector kand the shear polarizations, ˆd1,2, to be two mutually perpendicular unit vectors perpendic- ular to k. For the isotropic systems investigated in this article, dm’s are mutually perpendicular and Amreduces toAm=ˆdm·˜uat each ( kx, ky). The propagated plane-waves are periodic in the direc- tion transverse to the beam propagation direction. When the diffracted waves reach the transverse boundaries of the computational domain, they introduce undesired re- flections. To avoid these reflections, one needs to con- sider a sufficiently wide computational domain that can contain the waves even after multiple reflections. How- ever, such large domains can significantly increase com- putational costs. An alternative approach is to intro-duce absorbing boundary regions that completely atten- uate any waves that enter these regions [61]. In this study, we implement absorbing boundaries of thickness WAbs= 50×λ, where λis the wavelength of the input field. The blue shaded regions in Fig. 1 show the ab- sorbing boundaries. To simulate the effect of absorbing boundaries, we multiply u(i) tHBAR(Eq. 5) with a reflection operator RtHBAR, defined as RtHBAR(x, y) =( 1,ifR≤Weff/2 0.otherwise(6) Weff=W−2WAbsis the effective width of the sim- ulation window without the absorbing boundaries. We propagate the attenuated reflected wave further through a distance tHBAR , We implement the beam propagation by following the approach outlined in Eqs. 3 - 5b, to complete a full round trip. After every round trip, we multiply the resulting complex displacement field by an additional phase R0,mfor each polarization m: R0,m(x, y) = ei2kz0,mtYIG,ifRYIG≤R≤Weff/2 1, ifR≤RYIG 0. otherwise (7) where kz0,m=kz,m(kx= 0, ky= 0). The phase fac- tor is introduced due to the finite width of the YIG film compared to the GGG width, W. However, it is worth mentioning that this approximation is more appropriate for low-diffraction cases. We used this for all cases con- sidered to simplify the analysis. We use the resulting displacement field as the new input field for the next it- eration. We repeat this process for Nround trips. At the end of Nround trips, we calculate the complex sum of the displacement fields, U0(x, y) = ΣN i=1u(i) 0(x, y) at z= 0. Using the interference sum, U0(x, y), we restart the iterative process with U0(x, y) as the initial beam of the next restart, u(1) 0(x, y) =U0(x, y). Such a restart process using interference sum as an input ensures fast convergence to the desired mode. The other mode com- ponents are attenuated in the interference sum due to destructive interference. We continue this process until the varation of the modeshape is within a chosen toler- ance. We obtain a converged standing shear wave with displacement field, u0(x, y), as a final outcome. In addition to the displacement profiles, we are inter- ested in identifying the frequencies of the shear modes. These modes could couple with the Kittel magnon modes generated in the YIG thin-film (Eq. 1), in a rotating coor- dinate system. The frequency overtones for plane-waves traveling along z-direction in the HBAR structures are expected to be ωm,n= 2π×nvm 2tHBAR, with n= 1,2,3, ...∞. Here, tHBAR is the thickness of the structure, velocity of wave is vmandmrepresents polarization. The over- tones are separated byvm 2tHBAR, known as the free spectral6 range (FSR). However, unlike plane-waves, the diffract- ing waves traveling in the zdirection do not have well- defined kz,m’s leading to Gouy phase effects [75]. The diffraction results in the shift of resonance frequencies from the monochromatic plane-wave overtones. To iden- tify the resonance frequencies, we select beam of fre- quencies from a chosen frequency window, propagate the beam in the structure, and calculate the intensities of the interference sums ( I=R ARe[U0(x, y)]2dA) atz= 0 after Nround trips. The frequencies for which the in- tensities are maximum are the resonance frequencies of the shear waves in the HBAR structure. We focus on the shear modes with m= 2, however, we could have as well chosen m= 1 as the dominant shear polarization in the rotating coordinate system. We sweep through a frequency window of ω0±5×FSR using 50 steps. Here, the frequency of interest, ω0= 2π×9.825 GHz, cor- responds to the frequency of the 2960thovertone of a standing plane-wave in the HBAR structure. We choose this overtone to match with the structure investigated in a previous article [58]. We consider a YIG/GGG HBAR structure with tYIG= 200 nm, RYIG= 200 µm, tGGG = 527 µm, and Lx,GGG =Ly,GGG = 1200 µm, for this analysis. FIG. 2. Resonant modes in planar HBAR structures: (a) Multimode phonons, represented by the overtones of the fundamental mode, and their free spectral range (FSR). (b) Isometric view of the y-component of U0(x, y), U 0y(x, y). (c). Top view of U 0y(x, y) showing localization effect caused by the YIG film. (d) Weighted deviation (WD) between mode profiles before first and second restarts. Figure 2(a) shows the intensities of different propa- gated beams with frequencies within the frequency win- dow, a frequency window, ω0±5×FSR. The central in- tensity peak corresponds to ω0= 2π×2960v2 2tHBAR, the exact value of the 2960thplane-wave overtone. The peaks form at resonance frequencies separated by FSR = 3 .32 MHz around the frequency, ω0/2π∼9.825 GHz, indicatingthe presence of multiple shear wave overtones. In Figs. 2 (b) and (c), we show different views of the y-component of the normalized resonant modeshape, Re[ U0y(x, y)], at frequencies near ω0. We obtain these displacement fields after 800 round trips. This simulation included one restart after 400 round trips. We perform multiple tests and check that this number is large enough such that the Gouy phase effects are not significant on the interference sums. We obtain the normalized displacement fields be- fore the 1st and the 2nd restarts, U1 0(x, y) and U2 0(x, y), respectively, and compute the weighted deviation (WD) between them: |(U1 0(x, y)−U2 0(x, y))|/|Σx,yU1 0(x, y)|. We monitor the WD between restarts to check for conver- gence. The color scale of Fig. 2 (d) shows that the WD is much less than 10−7, indicating that the Figs. 2 (b) and (c) represent displacement fields converged within sufficient numerical tolerance. Some fringes remain in the modeshapes that are possible artifacts of our numer- ical analysis. These are the high frequency components resulting from the sampling of the sharp phase change induced by the YIG film. They are significantly lower in values, however, could be further reduced by either using a low-pass filter or a smoother phase change factor than the function, R0,m(x, y) (Eq. 7), used in this work. To obtain the converged modes and identify the true resonant frequency near an intensity peak of interest, we further narrow the frequency sweeping range with finer sampling ( ∼1 kHz). We set the frequency to be ωHBAR 0 =ω0+δω, with δω= 2π×2.157 kHz and restart the iterative process with Eq. 2 as the input beam, to identify the true modal frequency near ω0. If such nar- rowing is not done, the modeshape can change signifi- cantly after each restart due to the Gouy phase effects the beam incurs due to diffraction [75]. These effects results in the detuning of overtones with respect to the plane-wave overtones. Figure 3 shows the real and imagi- nary parts of the complex sums U(1) 0y(x, y) and U(2) 0y(x, y) computed before first and second restart (after 400 and 800 round trips), respectively. We calculate the complex sums at ω0without performing the narrowed frequency sweeping discussed above. Both the real and imaginary parts of U(1) 0yand U(2) 0yshow significant variations between the restarts. These results establish that the δωcorrec- tion is necessary to obtain the converged modes. However, the gradual narrowing of the frequency sweeping is a tedious process to identify the necessary fine sampling. We need to restart the iterative process multiple times especially for high-diffraction cases. The results shown in Fig. 2, represent propagating waves in a low-diffraction regime with Fresnel number, NF= 213. However, this work also considers wave propagation in HBAR structures with RYIGas low as 10 µm correspond- ing to a very low Fresnel number, NF= 1.06, indicating a high-diffraction regime. We investigate these structures to discuss how a reduced actuator lateral area affects7 FIG. 3. Lateral mode profiles obtained with FBPM showing Gouy phase effects: The real (solid) and imag- inary (dashed) components of the y-component of the inter- ference sum before the first U(1) 0yand second restarts U(2) 0yat ω0. The components deviate significantly between restarts. . the phonon modes and their lifetimes. The reduced area promises to increase the integration density of the HBAR devices towards high-density memory applications. The high-diffraction regime of these structures introduces sig- nificant Gouy phase effects on the propagating waves and results in detuning of frequencies. It becomes challeng- ing to identify a resonant frequency by merely narrowing the frequency sweeping range. We implement an adap- tive FBPM algorithm to circumvent this challenge. Note that we implement absorbing boundaries in this study to avoid undesired boundary reflections. The phonon mode- shapes can be well predicted using this implementation. However, their lifetimes can be dependent on the shape and size of the boundaries. We leave the extensive inves- tigation of the effect of boundary conditions on phonon lifetimes for a future investigation. Phonon Modeshape: Adaptive Fourier Beam Propagation We formulate the FBPM approach described above as an eigenvalue problem: RTu 0(x, y) = Λu0(x, y) (8) where RTis the round trip operator that includes all the operations described in the previous section (Eqs. 3- 7) and Λ is the eigenvalue corresponding to the eigenvec- toru0. It is possible to solve the eigenvalue problem for 2D problems (e.g., if HBARs are represented by pla- nar 2D structures), however, it cannot be directly solved for 3D structures of our interest. We develop an iter- ative method to obtain the resonant mode that satis- fies Eq. 8 with a real Λ. Λ is real for a standing wavemode. A complex Λ applies a global phase to the input modeshape after a round trip, and the phase prevents the waves to interfere constructively over multiple round trips. In the adaptive FBPM (a-FBPM) algorithm, we iteratively adjust the frequencies based on the phase dif- ference incurred over a round trip to arrive at the desired standing wave modes. We outline the steps of the itera- tive method below. 1. Start with a input beam, u(i) 0(x, y), for the ith round trip. The initial input beam ( i= 1) has wavelength, λ(1)=2t nand frequency ω(1)= 2π× nv2 2t. Here, nis the overtone number, t=tHBAR andv2is the velocity of shear phonons. The wave- length and frequency are estimated based on the nthovertone for an open-ended column. The ini- tal modeshape is chosen to be a truncated Bessel function as shown in Eq. 2. 2. Calculate u(i+1) 0(x, y) =RTu(i) 0(x, y). 3. Estimate the real eigenvalue from, Λ(i)=q I(i+1) I(i), where I(i)=R |u(i) 0|2dxdy andI(i+1)= R |u(i+1) 0|2dxdy . 4. Calculate the residual, R(i)=||(u(i+1) 0(x, y)− Λ(i)u(i) 0(x, y))||/||Λ(i)u(i) 0(x, y)||. 5. If R(i)≤tolerance (tol), end simulation and output final eigenvalue and modeshape of resonant modes, else continue to the next step. We choose tol = 10−6for all a-FBPM calculations of this study. 6. If R(i)>tol, estimate the global phase factor in- troduced by the round trip operation RT,θ(i)= Arg(u(i) 0(0,0))−Arg(u(i+1) 0(0,0)). 7. Set ω(i+1)=ω(i)+v2θ(i) 2tHBARand λ(i+1)= 2πnv2/ω(i+1). The steps 6 and 7 makes the al- gorithm adaptive. 8. Repeat the process until step 5 is satisfied or the maximum number of Nround trips is reached, at which point we set u(1) 0(x, y) =U0(x, y) = ΣN n=1u(i) 0(x, y), and restart the iterative procedure. Figure 4 illustrates the effectiveness of the a-FBPM al- gorithm in predicting the resonant modes of the HBAR structures. We show the results for two planar HBAR structures with RYIG= 200 µm (red) and 10 µm (blue), respectively. The residual, R, decays as a function of the number of steps, as we approach the resonant mode. The step-like features of the residual decay occur after every NRround trips when we compute the interference sums and use it as input for the next restart of the iter- ative procedure. The jumps occur because some errors8 FIG. 4. Obtaining resonant modes using the adaptive FBPM algorithm: Residual (R) of modeshapes decays as a function of number of round trips. Results are shown for two planar HBAR structures with RYIG= 200 µm (red, left and bottom axes) and 10 µm (blue, right and top axes). Circles with arrows point to corresponding axes for each plot. get canceled due to destructive interference when an in- terference sum is computed. In some cases, R can have a momentary rise due to accumulation of errors from pre- vious round trips, however, the overall trend continues to decrease ultimately reaching convergence. We calcu- late the complex interference sum after every NRround trips. We choose NR= 400 and 40 for the HBARs with RYIG= 200 µm and 10 µm, respectively. We choose a smaller number of round trips, NR, for the 10 µm case, since the HBAR with RYIG= 10 µm represents a high- diffraction structure and the modeshape decays rapidly for this case. As shown in Fig. 4, it takes a total of ∼650 and ∼1200 round trips (including restarts) to ob- tain converged resonant modes for the 10 µm and the 200 µm case, respectively. We continue the iterative process till R <10−8to show stability of the resonant mode even after tolerance is reached. (2) Phonon Modeshape Analysis: Hankel Transform Eigenvalue Problem Our a-FBPM approach avoids the frequency sweeping process and mostly overcomes the slow convergence is- sues of the standard FBPM approach. However, it is still challenging to obtain converged phonon modes for the confocal HBAR structures of interest, using the a- FBPM approach. Here, we discuss an alternate method that uses the Hankel transform (HT) approach, the axi- symmetric equivalent of the Fourier transform method. The HT approach has been widely used in the field of optics, e.g., Fabry-Perot cavities [67]. However, it hasnot been applied for acoustics problems, to the best of our knowledge. The HT method allows us to obtain con- verged phonon modes with significantly reduced compu- tational cost. The approach leverages the axi-symmetry of the problem to reduce the 3D problem to a 2D one. For example, the 2D problem can be obtained by represent- ing HBARs as planar 2D structures. The axi-symmetry conditions are applicable for the YIG/GGG HBAR struc- tures due to their isotropic material properties and the circular cross-section of the YIG film. Note that the scalar field describing the dominant shear-component of u0is expected to obey axi-symmetry, however, the vector field of the shear acoustic phonon modes does not obey axi-symmetry. Here, we provide a brief description of the HT method. We encourage the interested reader to find a detailed description of the method elsewhere [67, 68]. We cast the HBAR structures shown in Fig. 1 into axi- symmetric representations about the z-axis. We trans- form the x−ycoordinates to the radial coordinate r, with limit 0 ≤r≤W/2. We write the Hankel eigenvalue problem as RThku(r) = Λ u(r), where u(r) represents the axisymmetric scalar phonon modeshape, Λ is the corre- sponding eigenvalue and RThkis the round trip operator. Since the dimensionality of the problem is reduced from 3D to 2D, the Hankel eigenvalue problem can be solved to obtain the phonon modes directly. We discretize ras rj=W/2ζj ζN (9) where j= 1,2,3, .., N andζj(j= 1,2,3, .., N ) are the roots of the J1Bessel function. The round trip operator for the HT approach RThkis given by RThk=R0PRtHBARP (10) where P, the N×Npropagator matrix, is defined as: P= (H+)−1˜GH+, (11a) H+ ij=W2 2ζ2 NJ0(ζjζj/ζN) ζ2 NJ2 0(ζj), (11b) ˜Gij= exp −i2ζ2 j W2! δjj. (11c) Here, H+is the HT whose matrix inverse is taken to be an approximation of the inverse HT, and ˜Gis the Green’s function obtained from the Fourier transform of the Fresnel propagator in the paraxial approximation. RtHBAR, andR0are the N×Ndiagonal reflection matrix operators at z=tHBAR andz= 0, respectively and the diagonal elements are defined as R0,jj=R0,2(rj), (12a) RtHBAR ,jj=RtHBAR ,2(rj). (12b) This method, unlike the FBPM/a-FBPM approaches, does not require a Fox-Li-like iterative process and can9 also predict higher-order phonon modes. Although this approach has a limited applicability due to its axi- symmetric constraints, it makes it possible to obtain the phonon modes and lifetimes of HBAR structures at a reduced cost. The expedited analysis allowed us to ana- lyze a large class of HBAR structures, as we show in the Results and Discussion section. Diffraction-Limited Phonon Lifetime We estimate the diffraction-limited phonon lifetimes using three methods: ( A) Eigenvalue method, ( B) Expo- nential curve fitting method, and ( C) Clipping method. (A) Eigenvalue method: In this method, we obtain the eigenvalues using the adaptive FBPM simulation and use them to estimate the phonon lifetime. The elastic en- ergy contained in a acoustic beam with modeshape uis given by E∝R |u|2dxdy . We only integrate over the dxdy element since the modeshape largely remains un- altered throughout the thickness. We assume that the elastic energy of the acoustic beam decays exponentially with the propagation time. For a cavity phonon mode with a total initial elastic energy Etot, we represent the elastic energy left in the cavity after one round trip as Ein=Etote−tRT/τ. Here, tRT(= 2tHBAR /v2) is the time taken by shear waves to complete a round trip and τis the lifetime of the phonon mode. On the other hand, the ratio of EintoEtotis given by Ein/Etot= Λ2. Here, Λ is the real eigenvalue obtained using the a-FBPM simu- lations. Using this relation, we can obtain the lifetime of the phonon mode as τ=−2tHBAR v2ln(Λ2). (13) Since these computations are performed on a finite x−y mesh grid, Λ, and consequently, τcan be sensitive to the mesh density. Hence, it is important to select an optimal grid to obtain the converged values of Λ. In Fig. 5, we show the variation of Λ with mesh density Nx(=Ny) for both the low-diffraction ( RYIG= 200 µm) and the high-diffraction ( RYIG= 10µm) cases. We find that the variation of Λ is much less than 1% when Nx≥1024. Consequentially, we choose Nx=Ny= 1024 to compute phonon modes and lifetimes for all cases considered here. (B) Exponential curve fitting method: In this method, we calculate the phonon mode lifetime by evaluating the diffraction loss of the beam as it travels in the HBAR structure. We estimate the diffraction loss from the over- lap of the propagated mode profile with the initial input profile, after each round trip. We estimate the overlap using the following function: I(t) =|⟨u0(t)|u0(0)⟩|2 |⟨u0(0)|u0(0)⟩|2. (14) FIG. 5. Eigenvalues of two planar HBAR structures with RYIG= 200 µm (red) and 10µm (blue): Variation of Λ for meshes with density ranging from 128 ×128 to 2400 × 2400. The width Wis kept fixed at 1200 µm. The circles with arrows point to the axes corresponding to each plot. Here, tis an integral multiple of the time taken for each round trip, tRT. We allow the initial beam to travel for multiple round trips until I(t)<0.1 is reached or the time is t >3 ms, whichever is reached first. In Fig. 6, we show the decay of the overlap function, I(t), for beams traveling in a HBAR structure with RYIG= 200 µm. The finite width of the YIG film (transducer) induces a lo- calizing effect on the propagating beam. The localizing effect can be modeled by introducing a phase factor. The blue and the red lines shown in Fig. 6 correspond to the two cases when we obtained the overlap function with or without the consideration of the phase factor, respec- tively. The two different decays highlight the effect of the finite width of the transducer on the propagating beam. We find that I(t) decays at a much slower rate when the YIG-induced phase factor is considered, compared to the case without the phase factor. We fit the two decays with exponential functions and obtain the phonon lifetimes as τwophase= 0.1 ms and τwphase= 14.1 ms, respectively. The remarkable two-orders of magnitude difference be- tween these two predicted lifetimes highlights the im- portance of including the transducer or actuator phase effects for such an analysis. Subsequently, we included the phase effects in all our analyses to obtain the phonon lifetimes. We find that the lifetime predicted with the phase effects, τwphase= 14 .1 ms, closely matches with τof 15.5 ms, predicted using the eigenvalue method dis- cussed earlier. A past study used the exponential curve fitting approach to estimate the lifetime of the longitudi- nal HBAR phonons in an AlN/Sapphire structure [61]. However, they treated the phonons as superpositions of Bessel functions and did not account for the localization effects induced by the AlN transducer.10 FIG. 6. Decay of the overlap function, I(t), for prop- agating beams in a HBAR structure with RYIG = 200µm. Solid lines indicate I(t) while dashed lines indicate their respective exponential fits. The finite width of the YIG film induces a localizing effect on the beam, modeled with a phase factor. I(t) decays at a much slower rate when the phase effect is included (blue, top and right axes) compared to the no-phase-effect case (red, bottom and left axes). (C) Clipping method: In this method, we obtain the phonon lifetime using a similar expression as used for the eigenvalue method, Eq. 13: τ=−2tHBAR v2ln(Ein/Etot). Here, Etotis the total initial elastic energy for a cavity phonon mode and Einis the elastic energy left in the cavity after one round trip. We calculate Ein, contained in a volume, Vin, of a cylinder with radius of the YIG disc and the thickness of the HBAR structure, Ein∝R Vin|u|2dxdy . We consider a converged modeshape, obtained with a- FBPM or HT method. This method results in the lowest estimate of the phonon lifetimes since it assumes that all energy outside the cylinder of interest is lost after a round trip. Our approach is similar to the clipping method used for Fabry-Perot cavities [67, 68] and HBAR resonators, excited opto-mechanically using photon beams [76]. In these optical systems, the clipping method is more appli- cable since the input photon beam spans the entire x−y plane and is clipped by the localizing finite cross-section mirrors or domes. In our system, the input acoustic beam is already laterally confined so this method may have lim- ited applicability. We still include the discussion here as a consistency check since the method results in the lower bound for the predicted lifetimes.Magnon-Phonon Coupling The magnon-phonon coupling strength, gmb, is given by gmb=Bp AνQηZ VYIGdu∗ x dzmx+du∗ y dzmy dr.(15) Here, B= 7×105J/m3, is the magnetoelastic constant of YIG, ( mx, my) are the xandycomponents of the magnetization, m=mYIG, (u∗ x, u∗ y) and (du∗ x dz,du∗ y dz) rep- resent the complex conjugates of the xandycompo- nents of displacement uand their corresponding shear strains, respectively. The integration domain is the vol- ume of the YIG film, VYIG, since the magnon modes reside in YIG. The termp AνQηcorresponds to the magnon and phonon normalizing constants. We normal- ize the magnon modes as Ms γZ VYIGm∗ ν(r)·(k×mν′(r))dr=−iAνδν,ν′.(16) Here, µ0Ms= 0.175T is the saturation magnetization, γ= 2π×28.5 GHz/T is the gyromagnetic ratio, kis the unit vector along the z axis, νis the magnon mode index, and Aνis the magnon normalization constant, re- spectively. Similarly, we normalize the HBAR phonon modes as 2ωηρZ VHBARu∗ η(r)·uη′(r)dr=Qηδη,η′. (17) Here, ρ= 7080 kg/m3is the GGG material density, η is the phonon mode index, and ωηis the corresponding phonon frequency, respectively. VHBAR is the volume of the HBAR structure. Qηis the phonon normalization constant and is of the same dimensionality as Aν. Note that if we assume zero diffraction, the acoustic energy is fully contained in the volume Vin, the volume of the cylinder with radius of the YIG disc and the thick- ness of the HBAR structure. Such a scenario results in a complete overlap of lateral mode profiles of the magnon and the phonon modes. We obtain the zero diffrac- tion limit of the magnon-phonon coupling strength to beg0 mb/2π= 1.13 MHz. This value is independent of the width of the YIG film. Our result compares well with a previous experimental result of 1 MHz [57], and is slightly higher than another experimental result of 0 .75 MHz [58]. However, we consider the regime where diffraction causes the spread of acoustic energy outside Vin, which can im- pact both τandgmbof phonon modes in HBAR struc- tures. We present the diffraction-limited HBAR phonon lifetimes and the magnon-phonon coupling strengths in the Results and Discussion section.11 RESULTS AND DISCUSSION We investigate hybrid magnonic HBAR structures that can support the development of high-density phononic quantum memories. We can include a greater number of transduction components in a single chip of given dimen- sion by reducing the radius of the YIG film. However, the reduced aperture increases the diffraction of the acous- tic waves propagating into the GGG region. Here, we discuss the diffraction-limited phonon lifetimes and the magnon-phonon coupling in the HBAR structures. Diffraction-Limited Phonon Lifetime Figure 7 (a) shows the variation of phonon lifetimes in planar HBAR structures as we decrease the YIG ra- dius. We compute the lifetimes using the a-FBPM ap- proach and following the eigenvalue method (red line), the exponential fitting (blue and purple lines), and the clipping methods (green line), respectively. In addition to the a-FBPM predictions, we show predictions from HT method in Fig. 7 using circles, with colors match- ing to their corresponding a-FBPM predictions. Note that we use |Λ|instead of Λ in Eq. 13 to calculate τ using the HT method. All methods predict that the life- times decrease with decreasing RYIGdue to increased diffraction, as expected. The eigenvalue method results in the highest lifetime estimates, among the three meth- ods considered. We obtain the highest lifetime to be τ= 15.5ms ( Q=ωτ= 9.57×108) for HBAR structure with RYIG= 200 µm. The lowest lifetime predicted by the eigenvalue method, is τ= 10.5µs (Q= 6.48×105), corresponding to HBAR structure with RYIG= 10 µm, respectively. These results show that the lifetimes are reduced by more than three orders of magnitude in the high-diffraction regime. The lifetimes predicted by the eigenvalue method (red) closely match with those pre- dicted by the exponential fitting method (blue), for the low-to-medium diffraction cases, when the phase effects are included. This result can be explained through the following argument. The input beam can be thought of as a superposition of the eigenmodes of the HBAR cavity. Once the initial input beam undergoes multiple round trips, only the fundamental mode is likely to sur- vive, while the rest of the mode components decay at a faster rate. When we operate at the overtone frequency of one of the fundamental modes, the fundamental mode becomes the dominant mode. It can be argued that this dominant mode is the same as the fundamental eigen- mode found by solving the eigenvalue equation, Eq. 8. Therefore, the decay of the overlap function of the input field (exponential fitting method) is expected to have a similar character to that of the decay of the dominant eigenmode (eigenvalue method). As a result, we obtain similar lifetimes using the two methods.However, in the high diffraction case ( RYIG= 10µm), we observe 35-fold reduced lifetimes predicted by the ex- ponential fit compared to that of the eigenvalue method. Due to high-diffraction, significant amount of the energy of the input acoustic beam spreads out laterally during the first few round trips, resulting in a rapid initial de- cay of I(t). In the exponential fit method, we obtain the lifetime from the exponential fit of I(t), starting at t= 0. The loss of acoustic overlap during the initial transient phase results in lower τpredictions. We expect that the two lifetime predictions will be closer if the ex- ponential fit is obtained after a steady state is reached. Note that we obtain the exponential fitting predictions (blue) by including the localizing phase effects due to the YIG film. We also show the lifetimes predicted by this method, when we do not consider the phase effects due to the YIG film (purple). We find that the τwophase’s are consistently lower than the τwphase’s. The difference be- tween the two predictions decreases monotonously from 135 to 1.35 fold, as we decrease RYIGfrom = 200 µm to = 10µm, respectively. This is expected since the local- izing effect of the YIG film diminishes as we approach RYIG→0. Past studies often ignored the effect of aper- ture width on the HBAR phonons, however, our results (Fig. 6 and Fig. 7) establish that such effects needs to be considered to make reliable predictions. Figure 7 (a) also shows the lifetime predictions from the clipping method (green), which assumes that the en- ergy outside the cylindrical volume of interest, Vin, is completely lost after a round trip. We find the life- times to be more than an order of magnitude lower than those predicted from the eigenvalue method, for all cases considered. We obtain the lowest lifetime in the high- diffraction regime to be τ= 0.311µs (Q= 1.92×104). Interestingly, this prediction is still marginally greater than the experimentally observed values of 0.25 µs and 1.45×104, respectively [58], for a device operating in the low-diffraction regime. The reason for this discrepancy is that the experimental values correspond to HBAR de- vices operating at room temperatures. In this limit, the performance of HBAR structures is limited by the ma- terial attenuation effects [59, 60]. In our study, we ig- nore the material attenuation effects and only discuss the diffraction-limited performance. Our study will cor- respond to devices operating in the cryogenic or poten- tially milli Kelvin regimes. For example, our results will have high applicability for the HBAR devices that could effectively couple with superconducting qubit systems, etc., which typically operate in the milli Kelvin regime. We discuss here the uncertainties associated with the numerical prediction of the HBAR phonon lifetimes using different methods. In the eigenvalue method, we obtain the lifetimes from the ratio between the HBAR thickness (tHBAR ) and a function of the phonon velocity and the eigenvalue, Λ, as shown in Eq. 13. Following Eq. 13, an12 FIG. 7. Diffraction-limited properties of phonon modes of planar HBAR structures: (a) Phonon lifetimes, τ, calculated from the eigenvalue (red), exponential fitting (blue, purple), and clipping methods (green). (b) Ratio between magnon-phonon coupling in HBAR structures in various diffraction regimes and that in the zero-diffraction limit, gmb/g0 mb (red). Reduction of gmbas we decrease RYIG, is connected to the spread of modeshape in the high-diffraction regime. Increasing amount of acoustic energy spreads into the GGG region, causing reduced overlap between phonon and magnon modes in the YIG region. The ratio of acoustic energy spreading out to the total energy, Eout/Etot(black), increases in the high-diffraction regime. (c) Magnon-phonon cooperativity C.RYIGis varied keeping all other geometric factors fixed. Solid lines correspond to a-FBPM predictions, while the circles of the same color correspond to the respective HT predictions. error propagation relation can be written as dτ τ=v2τ tHBAR×dΛ Λ∝τ×dΛ Λ. (18) The uncertaintydτ τincreases monotonously with τ when other factors are fixed. It can be deduced from Eq. 18 that to predict a lifetime within x% of variabil- ity, Λ has to be predicted to be withintHBAR x v2τ% of vari- ability. Here, the order of magnitude of the prefactor v2 tHBAR∼107. The prefactor remains fixed since we keep the material and the HBAR thickness unaltered. This implies that we have to be increasingly more stringent with our Λ convergence criteria for high lifetime calcu- lations. Figure 8 shows the conditions for desired ac- curacy (relative tolerance) needed for Λ predictions for high- τ(high- Q) systems. When τ= 0.1µs and the de- sired accuracy is set to 10%, Λ must be predicted within 15% accuracy, which is attainable following our numer- ical procedure. The accuracy requirement increases fast when the lifetime values are much higher. For example, when τ= 1ms, and the desired accuracy is set to 10% ofτvalue, Λ must be predicted within ∼10−3% of accu- racy. For our RYIG= 200 µm HBAR structure, we con- tinue the simulation till we obtaindτ τ= 7.9×10−4withdΛ Λ= 4.7×10−7, between the last two restarts. On the other hand for the HBAR with RYIG= 10µm, we con- tinue tilldτ τ= 3.9×10−7withdΛ Λ= 3.8×10−8. How- ever, achieving such high accuracy requires high compu- tational expense associated with simulating a large num- ber of iterations. We implement and use an alternative HT approach to expedite the numerical analysis. FIG. 8. Relative tolerance of Λat various τrequired to predict τwith 10% accuracy. Magnon-Phonon Coupling in Diffraction-Limited Regime We now turn to discuss the effect of diffraction on the magnon-phonon coupling strength, gmb. We esti- mate the effect of diffraction by computing the ratio be- tween the coupling strength in the planar HBAR struc- tures, gmb, and that in the zero-diffraction limit, g0 mb. In Fig. 7(b)(red), we show the variation of the ratio, gmb/g0 mb, with decreasing YIG radius. As expected, The ratio is equal to 1 for low-diffraction cases with RYIG≳100µm. As we reduce RYIGto 40 µm,gmbonly changes by 5% from the g0 mblimit. Note that even though this case typically corresponds to a strong-diffraction regime, the effect of diffraction on gmbis minimal. This13 is because the YIG disc helps to localize the beam and thus, preserve the magnon-phonon overlap and the cou- pling strength. However, as we decrease RYIG≤40µm, we observe a sharp decline of gmb. The percentage of the ratio, gmb/g0 mb, drops to 77 .4% and 55 .7% for HBAR with RYIG= 20µm and 10 µm, respectively. The decrease of gmbcan be explained in the following way. In the zero diffraction limit, the acoustic beam is completely local- ized in the volume Vin, of a cylinder with radius equal toRYIGand the thickness of the HBAR structure. This results in a complete overlap of lateral mode profiles of magnon and phonon modes. Also, the acoustic energy of a propagating beam is completely contained in the vol- umeVin,Etot=Ein. However, as we decrease RYIG, the modeshape spreads out beyond Vin, due to diffraction. As a result, there is reduced overlap between the phonon and the magnon mode profiles in the YIG region. Cor- respondingly, some acoustic energy also leaks out of Vin and spreads into the GGG region. We refer to the en- ergy of the beam lying outside VinasEout. Figure 7(b) (black) shows that the ratio Eout/Etotincreases in the high-diffraction regime, with decreasing RYIG. We combine the magnon-phonon coupling strength (Fig. 7(b)) and the phonon lifetimes (Fig. 7(a)) to de- termine the performance figure of merit of the HBAR structures, defined by the magnon-phonon cooperativity, C= 4g2 mb/κmκb= 4g2 mbτmτb. We show the variation ofCwith RYIGin Fig. 7(c). We assume the magnon lifetime to be τm= 0.07µs [57, 58]. We compute the cooperativity values using τbpredicted from eigenvalue method (red line), exponential fitting (blue and purple lines), and clipping methods (green line), respectively. We obtain a monotonically decreasing diffraction-limited Cranging from 21 .9×104to 46.4, using τfrom the eigen- value method, as we decrease RYIGfrom 200 µm to 10 µm. On the other hand, Cis in the range between 1 .2×104 and 1 .4, predicted using τfrom the clipping method. As can be noted from Fig. 7, τ,gmb, and Cpredicted by the HT method are in an excellent agreement with a-FBPM predictions, for all RYIGcases considered. The close match validates our implementation of the HT method. As we have discussed earlier, a-FBPM has broader appli- cability compared to HT method, and can be applied for systems with anisotropic material properties and trans- ducer shapes. However, we find that it is an expensive and uncertain process to converge the residual since the rates of convergence for different HBAR structures are slow and variable. Also, one needs to select the num- ber of round trips before restarting the simulation using a trial-and-error approach. In comparison, HT method is computationally inexpensive to make predictions. Ad- ditionally, it can be applied to the YIG/GGG HBAR systems since they are isotropic with YIG transducer be- ing a circular thin film disc. Henceforth, we only present predictions from the HT method in this article. Note that our calculations predict high cooperativityfor even the HBAR with RYIG= 10 µm. This implies that if material acoustic attenuation effects are elimi- nated (e.g. by operating in the milli Kelvin regime), one can achieve high cooperativity for planar HBAR struc- tures. However, the phonon lifetime of the HBAR struc- ture with RYIG= 10 µm is limited to 10 .5µs (0.31µs), as predicted by the eigenvalue (clipping) methods and the maximum magnon-phonon coupling strength is only 55.7% of g0 mb. We explore design approaches to further improve these performance parameters. Enhancing integration density and Performance in Diffraction-Limited Regime FIG. 9. Variation of magnon-phonon coupling, gmb/g0 mb, with varying waist of fundamental phonon mode: gmb/g0 mbis maximum when w0/RYIG∼0.65. Here, we illustrate that the use of focusing dome-like surface structures could significantly improve the perfor- mance of HBAR structures. Past studies demonstrated that the confocal HBAR structures could achieve Q- factors on the order of 107, while resulting in 103-fold reduction in device volumes [76]. However, such struc- tures have not been explored for hybrid magnomechan- ical systems. We show that both phonon lifetimes and magnon-phonon coupling can be improved by employ- ing confocal geometries, leading to improved performance of hybrid magnomechanical systems. We first identify the optimal domeshape that will result in the maximum overlap between the CHBAR phonon and the magnon modeshapes, and consequently, the highest gmb. We ob- tain theoretical estimates of the overlap and identify the dome shape where the overlap is maximum. We assume that the phonon modeshapes in CHBAR structures can be described by Laguerre-Gaussian (LG) functions. The fundamental mode, LG 00, can be assumed to be of the14 FIG. 10. Performance of CHBAR structures with varied radius of curvature, Rcurv, of the dome-shape: (a) Phonon lifetimes, τ, calculated from the eigenvalue (red) and clipping methods (green), following the HT method, (b) Ratio between magnon-phonon coupling in CHBAR structures and that in the zero-diffraction limit, gmb/g0 mband (c) Magnon-phonon cooperativity, C.Rcurvis varied keeping RYIG= 10µm and Rcross= 60µm fixed. Solid lines represent corresponding values for a planar HBAR structure with RYIG= 10µm, while the circles of the same color correspond to the CHBAR values. form: uy∝e−R2/w2 0, due to the axi-symmetry of our sys- tem. Here, Ris the radial coordinatep x2+y2.w0is the waist of the Gaussian beam at z= 0 (see Fig. 1), that can be varied by changing tHBAR ,ω, or the dome radius of curvature, Rcurv. We normalize the modeshapes accord- ing to Eqs. 16, 17 and calculate the coupling strengths using Eq. 15. Figure 9 shows the computed gmb/g0 mbra- tio as a function of the beam waist, w0. We find that the peak of gmb/g0 mboccurs at w0/RYIG∼0.65.gmb decreases sharply if w0≲0.65RYIG, however, decreases slowly if w0is increased beyond the optimal value. In- terestingly, we find that w0/RYIGis a constant and does not depend on RYIG, for all the CHBAR structures con- sidered in this article. We use the w0value to obtain an initial estimate of the radius of curvature, Rcurv, of the dome [77]: Rcurv=1 Re[1/(q0+tHBAR )], (19a) with q0=iπw2 0 λ. (19b) We obtain the analytical estimate to be Rcurv/tHBAR∼1.5 that corresponds to w0/RYIG∼0.65 and the peak of gmb/g0 mbas shown in Fig. 9. Note that Rcurvcan be sim- ilarly estimated for various device thicknesses ( tHBAR ), wavelengths ( λ) and waists by appropriately accounting for these variables shown in Eq. 19. This Rcurvestimate results in maximum gmb/g0 mb, however, this analysis does not consider the effects of the dome shape on phonon life- times and cooperativities. Also, we ignore the effects of the lateral extent of the HBAR structure, the localizing effects of the YIG film, and assume phonon modeshapes to be perfectly Gaussian. To obtain a Rcurvestimate that improves the overall performance of these CHBAR structures, we carry out a systematic numerical analysis using the HT method.We modify the reflection operator at the upper GGG surface to include the effects of the lateral extents of the dome and the device, the attenuation window, and the localization effects of the YIG film. We include the phase induced by the dome surface in Eq. 6 and the radial form of Eq. 11c and write the modified reflection opertor as RtCHBAR ,m(x, y) = eikz0,mr2/Rcurv,ifr≤Rcross 1, ifRcross< r≤Weff 2 0. otherwise (20) We vary Rcurv/tHBAR across the analytical estimate, ∼1.5, and identify Rcurvthat results in the optimal per- formance of the CHBAR structures. We show the effect of the focusing dome on the phonon lifetime, magnon- phonon coupling and cooperativity of the CHBAR struc- ture in Fig. 10(a), (b) and (c), respectively. We show the predictions from the eigenvalue method and the clip- ping method using red and green circles, respectively. We also show the corresponding performance parame- ters of the planar HBAR structure with red and green horizontal lines, respectively. For the clipping method, we consider the volume, Vin, defined by the dome’s lat- eral cross-sectional area and the CHBAR thickness. We fixRYIG= 10µm and the dome x−ycross-section ra- dius, Rcross = 60 µm, for this analysis. As can be seen from Fig. 10(a), the phonon lifetimes are maximum at Rcurv/tHBAR = 1.09. This value is less than the ana- lytical prediction for maximum gmb,Rcurv/tHBAR∼1.5. However, the peak value, τ∼218ms (red circles), is ∼500 times larger than the value at Rcurv/tHBAR∼1.5, justi- fying the necessity of performing the systematic anal- ysis. The τpeak value for the CHBAR structure is ∼1.7×104(red circles) ( ∼7.3×104, green circles) times greater than the corresponding planar HBAR τpredic- tion, shown with red (green) horizontal lines, respec-15 FIG. 11. Performance of CHBAR structures with varied radius of cross-section, Rcross, of the dome-shape: (a) Phonon lifetimes, τ, calculated from the eigenvalue (red) and clipping methods (green), following the HT method, (b) Ratio between magnon-phonon coupling in HBAR structures and that in the zero-diffraction limit, gmb/g0 mband (c) Magnon-phonon cooperativity, C.Rcross is varied keeping RYIG= 10µm and Rcurv/tHBAR = 1.09 fixed. Solid lines represent corresponding values for a planar HBAR structure with RYIG= 10µm, while the circles of the same color correspond to the CHBAR values. tively. The peak τpredicted by the clipping method is∼22.7ms, lower than those predicted by the eigenvalue method, as expected. We note that τdecrease sharply for Rcurv/tHBAR <1. This corresponds to the situation when the center of curvature of the dome is inside the HBAR structure, resulting in a negative ‘ g−parameter’ [76], where g= 1−tHBAR Rcurv. It is shown that one cannot obtain real and finite Gaussian beam solutions for structures with a negative g−parameter [76]. We find that this is the case for the modes corresponding to Rcurv/tHBAR <1 in our case, that is, they are lossy modes which do not have well-defined Gaussian characters. Figure 10(b) shows that the coupling, gmb/g0 mb, is ap- proximately 0.99 at Rcurv/tHBAR∼1.4, a value close to the analytical estimate of ∼1.5. These results show that reducing the phonon mode volume by means of focus- ing does not necessarily improve gmbbeyond the max- imum achievable value, g0 mb, which represents the case when there is complete lateral overlap of phonon and magnon modes. We use τandgmbto calculate the figure of merit, C, and show the results in Fig. 10(c). The peak Cvalue is predicted to be 2 .5×106(2.6×105), using the eigenvalue (clipping) method. The Cvalues reach peak at Rcurv/tHBAR = 1.09, where τis maximum, as we have noted from Fig. 10(a). The τandCvalues of the CHBAR structure are several orders of magnitude improved compared to its planar counterpart and the coupling reaches up to 90% of g0 mb. This result high- lights that τis the dominating factor that controls the performance of CHBAR structures, since gmbis always limited to the maximum value, g0 mb. Note that we only vary Rcurvfor this analysis and keep the radius of the dome cross-section, Rcross, fixed at 60 µm. Due to the fixed cross-sectional area of the dome, the lateral inte- gration density of these devices, Di=A0 d Ad, is limited to ∼64. Here, Adis the cross-sectional area of the confocaldome surface and A0 d= 0.72 mm2[58]. To further improve the integration density, we analyze CHBAR structures with reduced dome cross-sectional area. We keep Rcurv/tHBAR = 1.09 and RYIG= 10µm fixed for this analysis. Figure 11 (a), (b) and (c) show the variation of τ,gmb, and Cwith Rcross/RYIG, re- spectively. As we decrease Rcross/RYIGfrom 6 to 1, τ is reduced by more than 4 orders of magnitude from ∼218 ms (22.7 ms) to 4 .83µs (0.134µs), as predicted by the eigenvalue (clipping) method. gmbshows a sharp decline for Rcross/RYIG<3 reaching the lowest value of gmb/g0 mb∼0.32 at Rcross/RYIG= 1, however, remains mostly unchanged for Rcross/RYIG≥3. The coopera- tivity follows a similar trend to that of τ, as shown in Fig. 11(c). Using the predictions from the eigenvalue (clipping) method, we obtain that Cis decreased by more than 5 orders of magnitude from 2 .5×106(2.6×105) to 7.2 (0.19) as we decrease Rcross/RYIG. Additionally, τ,gmb, andCof these CHBAR structures is reduced below those of the planar HBAR values, shown with solid lines, if Rcross/RYIG<2. Considering all the different aspects, we argue that the optimal geometry for the CHBAR struc- tures is represented by the condition Rcross/RYIG= 4.5. Our analysis using the HT eigenvalue (clipping) method predicts that the diffraction-limited lifetime of τ=144.7 ms (11.1 ms) and a integration density of Di= 113 could be achieved at a cooperativity C= 1.64×106(1.25×105) for the CHBAR structure with Rcurv/tHBAR = 1.09, RYIG= 10µm and Rcross/RYIG= 4.5. The results pre- sented in Figs. 10 and 11 provide a proof-of-concept that including a focusing dome improves the performance of the hybrid magnonic HBAR structures for quantum memory and transduction applications.16 CONCLUSION AND OUTLOOK In this article, we establish a modeling approach that can be broadly used to design hybrid magnonic HBAR structures for high-density, long-lasting quantum memo- ries and efficient quantum transduction devices. We il- lustrate the approach by discussing the magnon-phonon transduction properties of hybrid magnonic YIG/GGG HBAR structures. We present analytical and numer- ical analyses of the bulk acoustic wave phonon mode lifetimes, and the magnon-phonon coupling strengths in planar and confocal YIG/GGG HBAR structures. We discuss strategies to improve the phonon mode lifetimes of these structures, since increased lifetimes have di- rect implications on the storage times of quantum states for quantum memory applications. Additionally, high integration density of on-chip memory or transduction centers is naturally desired for high-density memory or transduction devices. In our structures, the transduction centers are represented by the YIG films and thus, inte- gration density can be increased by reducing the lateral dimension of these films. However, the reduced aperture results in high-diffraction that affects the performance of these devices. We analyze the diffraction-affected shear wave phonon modes in the HBAR structures us- ing two different methods: (1) Fourier beam propagation (FBPM) and (2) Hankel transform (HT) eigenvalue prob- lem method. The FBPM method has been widely used to analyze beam propagation in the field of optics, and more recently, to study HBAR phonons in planar and confocal HBAR (CHBAR) structures. We find that the FBPM method often requires us to analyze large and unpredictable number of round trips to obtain converged phonon modes. We implement an adaptive FBPM (a- FBPM) approach that mostly overcomes the slow con- vergence issues. However, we find that a-FBPM still suf- fers from convergence challenges when applied to con- focal HBAR structures. To circumvent the challenges, we implement the HT method that allows us to ob- tain converged phonon modes with significantly reduced computational cost. The HT method leverages the axi- symmetry of the problem and the isotropic material prop- erties of the YIG/GGG HBAR structures to reduce the 3D problem to a 2D one and expedite the analysis. The HT method has been mostly used in the field of optics, e.g., Fabry-Perot cavities, however, it has not been ap- plied for acoustics analysis, to the best of our knowledge. Our study provides key insights into the diffraction- limited performance of the YIG/GGG HBAR structures. We predict the diffraction-limited τto be on the order of milliseconds, for a planar HBAR structure with lateral YIG dimension, RYIG= 200 µm. A recent study reported that the performance of a YIG/GGG HBAR structure at room temperature is limited by the phonon lifetime at 0.25µs [58]. The previously studied structure had largerYIG lateral area than our structures and thus, the diffrac- tion effects were less dominant. The phonon lifetime in HBAR structures could be limited by both material and diffraction losses depending on the device geometry and/or the operating conditions. At room temperature, the phonon lifetime is primarily limited by acoustic atten- uation effects. However, these effects are less significant at low temperatures while the diffraction losses play an important role. Therefore, our results will have high ap- plicability for devices operating in the cryogenic or po- tentially milliKelvin (mK) regimes. For example, our approach and analyses can be applied to design HBAR devices that could effectively couple with superconduct- ing qubit systems. We acknowledge that the analysis of the material losses is necessary to obtain a complete understanding of the performance of the YIG/GGG sys- tems. This may include an understanding of the different attenuation effects (e.g., Akhiezer and Landau-Rumer) at various temperatures and frequencies of interest, and strategies to overcome them. Assuming that the material-limited lifetimes could be increased to ∼0.1 ms at mK temperatures, we find that the planar HBAR structures are not affected by diffrac- tion effects even at RYIG= 50 µm,. This structure al- ready offers a significant 50-fold improved integration density over the reference structure [58]. The integra- tion density can be further improved by scaling down the YIG film lateral area. However, the reduced aperture will affect the phonon lifetime and the magnon-phonon coupling strength, resulting in a decrease of the coop- erativity, the performance figure of merit for magnon- phonon transduction. It is therefore imperative to de- velop a strategy that increases the integration density of HBAR structures without affecting the performance. Ad- ditionally, the performance of planar HBAR structures was found to be highly sensitive to the parallel nature of the surfaces. To address both these aspects, we in- vestigate confocal YIG/GGG HBAR structures with top focusing domes and a planar bottom surfaces. Use of focusing dome structures has been proposed to signifi- cantly improve the phonon lifetime and integration den- sity of these systems. Furthermore, the dome structures eliminate the necessity of maintaining perfectly parallel surfaces of planar HBAR structures. We first theoreti- cally estimate the shape of the dome structure for which the magnon-phonon coupling strength, gmb, is at its max- imum value. We then perform a rigorous numerical anal- ysis and obtain a refined set of shape parameters of the dome for which both τandCare optimal, and gmbis close to its peak value. Overall, we find that ultra-high, diffraction-limited, cooperativities and phonon lifetimes on the order of ∼105and∼10 ms, respectively, could be achieved using a CHBAR structure with RYIG= 10µm. In addition to enhanced τandC, the confocal HBAR structure will offer more than 100-fold improvement of integration density.17 In this work, we discuss the diffraction-limited perfor- mance induced by the lateral area of the YIG film and the dome structure. It will be interesting to apply the insights presented in the article and explore the applica- bility of YIG/GGG HBAR structures for quantum trans- duction applications, as future work. For example, these HBAR structures could be used for coupling with other quantum information carriers such as superconducting qubits, which operate in the microwave frequency regime and at mK temperatures. However, to optimize the per- formance of such devices, a further comprehensive anal- ysis of different geometric and physical parameters will be necessary. For example, we keep the thickness fixed at 527 .2µm for all YIG/GGG HBAR structures inves- tigated in this work. The same thickness allows us to keep the free spectral range of phonons fixed for all our analysis. The different geometric parameters, such as the overall thickness, the thickness of the YIG and the GGG regions, geometrical misalignment, and imperfections, or, the physical parameters, such as the photon and magnon lifetimes and their coupling strengths, could also play important role in the overall device performance. We as- sume that these parameters remain invariant in our anal- ysis. A comprehensive optimization analysis can be per- formed using the current state-of-the-art machine learn- ing techniques, which is a promising research direction for the future. Additionally, we only discuss the coupling be- tween the fundamental magnon mode and the fundamen- tal phonon modes of the planar and the confocal HBAR structures. It will be interesting to consider the effect of higher-order mode couplings in the device performance. The higher-order mode couplings can be particularly rel- evant here since these structures host long-lasting phonon modes with broadband nature. Other interesting direc- tions to explore are the anharmonic phonon interactions at the surface and interfaces, and the coupling of bulk acoustic waves and the surface acoustic waves. Our study is concerned with coherent states, and it will be impor- tant to explore how the insights provided here translate to systems involving non-classical states, e.g. squeezed states, cat states, and Fock states. ACKNOWLEDGEMENTS This work was partially supported by funding from the Quantum Explorations in Science & Technology (QuEST) grant provided by the University of Colorado Boulder Research & Innovation Office in partnership with the College of Engineering and Applied Science, the College of Arts and Sciences, JILA, and the Na- tional Institute of Standards and Technology (NIST). We acknowledge the computing resources provided the RMACC Summit supercomputer, which is supported by the National Science Foundation (awards ACI-1532235 and ACI-1532236), the University of Colorado Boulder,and Colorado State University. The Summit supercom- puter is a joint effort of the University of Colorado Boul- der and Colorado State University. ∗sanghamitra.neogi@colorado.edu [1] D. Lachance-Quirion, Y. Tabuchi, A. Gloppe, K. Usami, and Y. Nakamura, Hybrid quantum systems based on magnonics, Applied Physics Express 12, 070101 (2019). [2] Y. Li, W. Zhang, V. Tyberkevych, W.-K. Kwok, A. Hoff- mann, and V. Novosad, Hybrid magnonics: Physics, cir- cuits, and applications for coherent information process- ing, Journal of Applied Physics 128(2020). [3] H. Huebl, C. Zollitsch, J. Lotze, F. Hocke, M. Greifen- stein, A. Marx, R. Gross, and S. Goennenwein, High co- operativity in coupled microwave resonator ferrimagnetic insulator hybrids, Phys. Rev. Lett. 111, 127003 (2013). [4] X. 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2023-08-14

Quantum memories facilitate the storage and retrieval of quantum information for on-chip and long-distance quantum communications. Thus, they play a critical role in quantum information processing and have diverse applications ranging from aerospace to medical imaging fields. Bulk acoustic wave (BAW) phonons are one of the most attractive candidates for quantum memories because of their long lifetime and high operating frequency. In this work, we establish a modeling approach that can be broadly used to design hybrid magnonic high-overtone bulk acoustic wave resonator (HBAR) structures for high-density, long-lasting quantum memories and efficient quantum transduction devices. We illustrate the approach by investigating a hybrid magnonic system, where BAW phonons are excited in a gadolinium iron garnet (GGG) thick film via coupling with magnons in a patterned yttrium iron garnet (YIG) thin film. We present theoretical and numerical analyses of the diffraction-limited BAW phonon lifetimes, modeshapes, and their coupling strengths to magnons in planar and confocal YIG/GGG HBAR structures. We utilize Fourier beam propagation and Hankel transform eigenvalue problem methods and discuss the effectiveness of the two methods to predict the HBAR phonons. We discuss strategies to improve the phonon lifetimes, since increased lifetimes have direct implications on the storage times of quantum states for quantum memory applications. We find that ultra-high, diffraction-limited, cooperativities and phonon lifetimes on the order of ~10^5 and ~10 milliseconds, respectively, could be achieved using a CHBAR structure with 10mum lateral YIG dimension. Additionally, the confocal HBAR structure will offer more than 100-fold improvement of integration density. A high integration density of on-chip memory or transduction centers is naturally desired for high-density memory or transduction devices.

Investigation of Phonon Lifetimes and Magnon-Phonon Coupling in YIG/GGG Hybrid Magnonic Systems in the Diffraction Limited Regime

2308.06896v2

Spin colossal magnetoresistance in an antiferromagnetic insulator Zhiyong Qiu1;2, Dazhi Hou3?, Joseph Barker1, Kei Yamamoto1;4;5;6, Olena Gomonay4, Eiji Saitoh1;3;6;7 1Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 2Key Laboratory of Materials Modification by Laser, Ion, and Electron Beams (Ministry of Ed- ucation), School of Materials Science and Engineering, Dalian University of Technology, Dalian 116024, China 3WPI Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 4Institut f ¨ur Physik, Johannes Gutenberg Universit ¨at Mainz, D-55128, Mainz 5Department of Physics and Astronomy, The University of Alabama, AL 35487, USA and Centre of Materials for Information Technology, The University of Alabama, AL 35401, USA 6Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan 7(Present address) Department of applied physics, The University of Tokyo, Tokyo 113-8656, Japan 1arXiv:1804.04516v1 [cond-mat.mtrl-sci] 12 Apr 2018Colossal magnetoresistance (CMR) refers to a large change in electrical conductivity induced by a magnetic field in the vicinity of a metal-insulator transition and has inspired extensive studies for decades1, 2. Here we demonstrate an analogous spin effect near the N ´eel tem- peratureTN=296 K of the antiferromagnetic insulator Cr 2O3. Using a yttrium iron garnet YIG/Cr 2O3/Pt trilayer, we injected a spin current from the YIG into the Cr 2O3layer, and collected via the inverse spin Hall effect the signal transmitted in the heavy metal Pt. We observed a change by two orders of magnitude in the transmitted spin current within 14 K of the N ´eel temperature. This transition between spin conducting and nonconducting states could be also modulated by a magnetic field in isothermal conditions. This effect, that we term spin colossal magnetoresistance (SCMR), has the potential to simplify the design of fundamental spintronics components, for instance enabling the realization of spin current switches or spin-current based memories. Spin current is a flow of spin angular momentum sharing many analogues with electric cur- rents. Spin currents can be carried not only by migrating electrons, but also by magnetic quasi- particles such as magnons or spin waves3, 4. These magnetic excitations are of particular interest in the spintronics community because they allow a spin current to flow in electrical insulators where charge currents cannot5–9. Thus, insulator spintronics may provide a route towards low power de- vices where spin currents carry signals and encode information. However, it is not straightforward to create all spintronic components which are analogous to their electronic counterparts10. For ex- ample, a basic spin-current on-off switch has not been demonstrated in insulator spintronics. The difficulty lies in the lack of a simple mechanism to directly gate the carrier density in magnetic 2insulators. Nevertheless, some clues may be obtained from colossal magnetoresistance (CMR) oc- curring in materials which exhibit metal-insulator transitions, where a large modulation of charge conductivity can be induced by a magnetic field1, 2(Fig.1 1a). Spin conductivity may be tunable in systems with a spin conducting-nonconducting transition (Fig.1 1b). In this Letter, we report such a ‘conductor-nonconductor transition’ for spin currents in the uniaxial antiferromagnetic insulator Cr 2O311–13. We found that Cr 2O3does not conduct spin cur- rents below the N ´eel temperature, but abruptly becomes a good spin conductor above this tempera- ture. Furthermore, in the vicinity of the transition, the spin-current transmission can be modulated by up to 500% with a 2:5T magnetic field: spin colossal magnetoresistance (SCMR). The active transport element is spin angular momentum rather than electrical charge. The spin-current transmission in Cr 2O3was studied by using a trilayer device, sandwiching a Cr2O3thin film between a magnetic insulator yttrium iron garnet (YIG) and a heavy metal Pt layer (Fig. 2a). Here, YIG serves as a spin-current source. By using a temperature gradient, rT, along the out-of-plane direction z, the spin Seebeck effect (SSE)6, 14generates a spin accumulation at the interface of YIG/Cr 2O3, which drives a spin current ( Jin s) into the Cr 2O3layer. Spin currents, transmitted through the Cr 2O3layer to the Pt interface ( Jout s), are converted into a measurable voltage via the inverse spin Hall effect (ISHE)15. We define the spin-current transmissivity Ts=jJout sj=jJin sj, describing the relative amount of spin-current incident on the YIG/Cr 2O3interface which is transmitted to the Cr 2O3/Pt interface. We ignore the effect of the spin accumulation at the YIG/Cr 2O3interface on Jin;out s, which will a 3posteriori be justified as the data are well explained solely by the intrinsic properties of the Cr 2O3. The spin current entering the Cr 2O3,Jin s, flows alongrTand is polarized along M, where Mis the magnetization of the YIG layer which can be easily manipulated by the external magnetic field H. The ISHE voltage measured in the Pt layer is thus VSSE/Jout s=Ts Jin sM jMj
^x; (1) where ^xis the unit vector along the xaxis.Jin scan be roughly estimated by the SSE signal in the YIG/Pt device in which the Cr 2O3thickness is zero and the SSE signal in YIG/Cr 2O3/Pt devices yields Jout s16, 17. Therefore, the spin-current transmissivity Tscan be estimated from Jin sandJout s (Eq.1). Figures 2c and 2d show the field dependence of the measured voltage Vfor a YIG/Pt bilayer and the YIG/Cr 2O3/Pt trilayer. In both samples—with and without a Cr 2O3layer—the sign of V reverses with the sign of H, and the shape of the V-Hcurves agree with the M-H(hysteresis) curve of the YIG film18–20. This confirms that the measured voltage Vin the YIG/Cr 2O3/Pt trilayer device is induced by the thermal spin currents generated from the YIG. First, we show a steep conductor-nonconductor transition for spin currents in Cr 2O3. Figure 2e shows the temperature dependence of the SSE voltage VSSE(H= 0:1T) for the YIG/Cr 2O3/Pt trilayer device. Surprisingly, the voltage exhibits an abrupt change of more than 100around 290 K. Above this temperature, a voltage with a peak of VSSE500 nV appears at T= 296 K. When T < 282 K,VSSEis close to the noise floor 5 nV (Fig. 2e). By contrast, in the YIG/Pt bilayer device,VSSEvaries little across the same temperature range (Fig. 2f)21, indicating Jin sis nearly 4constant. This equivalently means that the spin-current transmissivity Tsof Cr 2O3changes more than100around 290 K, which is calculated according to Eq.1 and plotted in the supplementary Fig. 2c. We attribute the abrupt change of VSSEin the YIG/Cr 2O3/Pt device to the change in the spin- current transmissivity Tsof the Cr 2O3layer, marking the transition of the Cr 2O3layer from a spin conductor to a spin nonconductor at T=296 K. This critical temperature coincides with the N ´eel temperature of the Cr 2O3thin film22, 23, and we associate the change in spin-current transmissivity with the onset of magnetic order. We found a similar spin conductor-nonconductor transition in a spin pumping measurement for devices with the same YIG/Cr 2O3/Pt structure as shown in Fig. 2g (also refer to supplementary Note 1), demonstrating that the spin conductor-nonconductor transition in Cr 2O3does not depend on the method of spin current generation. We also ruled out magnetic interface effects between the exchanged coupled YIG and Cr 2O3(such as exchange bias or spin reorientation transitions) causing the large change of Ts. Using a control sample with a 5-nm Cu layer (a nonmagnetic metal but good spin conductor) inserted between the YIG and Cr2O3layers, we observed results similar to that in the YIG/Cr 2O3/Pt trilayer (Fig. 2h, also refer to supplementary Note 2). By measuring the VSSEfor a Cr 2O3/Pt bilayer, we also confirmed that VSSEcomes from spin current generated in the YIG and transmitted through the Cr 2O3, rather than by spin current originating within Cr 2O3(Fig. 2h, also refer to supplementary Note 2). Having established the spin conductor-nonconductor transition, we show that the spin-current transmissivity of Cr 2O3has an anisotropic response to magnetic fields in the critical region of the 5magnetic transition. The spin-current transmissivity of Cr 2O3depends not only on the magnitude but also on the direction of the magnetic field. By using the SSE and ISHE as sources and probes of spin currents, within the critical region we measured the dependence of VSSEon the magnetic field magnitudejHjand anglein thez-yplane as illustrated in Fig. 3a. Figure 3b shows the dependence of VSSEon the angle at different magnetic field magni- tudes. AtT=296 K (in the spin conducting regime), VSSEshows a sinusoidal change with respect to, the same as the relative angle between the YIG magnetization MandJin sas expected from Eq. (1). The magnitude of VSSEchanges only slightly from jHj=0.5 T to 2.5 T. Similar behaviour is observed for T > 296 K, indicating that Tsdepends only weakly on orHin the spin conductor regime. However, at T < 296 K,VSSE()starts to deviate from this dependence. As the tem- perature decreases further, the character of VSSE() changes completely. The maximum amplitude ofVSSEno longer resides at =90but peaks four times through the rotation ( 180,180).Ts also becomes strongly dependent on jHj. Thus,Ts(,H) depends on both andjHjin the critical region. Figure 3c shows the temperature dependence of VSSE(jHj)at=20, where thejHjde- pendence is the most pronounced. The temperature dependence of VSSEis qualitatively similar for all field strengths, featuring a sharp transition between the spin nonconductor and conductor regimes. However, the transition edge of VSSEshifts to lower temperatures for stronger magnetic fields. TakingjHj= 0:5 Tas a reference,500% modulation of VSSEis achieved with a 2:5T field (Fig. 3d). 6Above the N ´eel temperature, the paramagnetic moments of Cr 2O3follow the external mag- netic field and spin current is carried by correlations of the paramagnetic moments as has been reported previously9, 16, 24. Below the N ´eel temperature—in the ordered antiferromagnetic phase—the propagation of the spin current is in principle determined by the thermal population of magnons, the magnon mean free path and the magnon gap. However, the magnon gap is approximately 10 K12, therefore this description by itself cannot lead to the sharp transition observed at the N ´eel point. In other words, the nonconducting regime cannot be caused by magnon freezing. Rather, it is caused by the anisotropic transmissivity of the antiferromagnet in combination with the device geometry. Only the spin component which is parallel (or antiparallel) to the N ´eel vector can be carried by magnons25. Below the N ´eel temperature, due to the strong uniaxial anisotropy, N ´eel vector of Cr2O3is pinned to the easy axis (out of plane in this work). When the YIG magnetization is in the plane of the film, the spins are polarized perpendicularly to the Cr 2O3N´eel vector and the spin current cannot be transmitted into the Cr 2O3. When the YIG magnetization is out of the plane, the spin current can be transmitted but the device geometry prohibits the generation of an ISHE voltage. Furthermore, the strength of the anisotropy in Cr 2O3is almost independent of temperature, collapsing to zero only very close to the N ´eel temperature12. Therefore, the Cr 2O3is strongly aligned perpendicular to the plane for almost the entire temperature range and no spin current can be transmitted. This small temperature window where the anisotropy decreases corresponds with the increase in ISHE voltage. 7In the region just below the N ´eel temperature, where the anisotropy is reducing, the trans- missivity can be manipulated with the applied field. The enhanced susceptibility and reduced anisotropy in this small temperature window allows the N ´eel vector to be slightly rotated, giving a finitey-component (in the plane) on to which the spin current is projected12, 13. The field induced N´eel vector and magnetization y-components of the antiferromagnet (Fig. 3e) are LAF yMsH2 2HexchHanisin 2;MAF yMsH Hexchsin; (2) respectively. Msis the saturation magnetization of the antiferromagnetic sublattices, Hexchis the exchange field between the sublattices, and Haniis the uniaxial anisotropy field. The equation is based on a zero-temperature theory but by allowing the temperature dependence of HexchandHani, it appears to be a good approximation even up to the N ´eel temperature. At temperatures much below the N ´eel temperature, the field required to manipulate the N ´eel vector is approximately H6T12. But, when Hanidrops in the transition window, much smaller fields (smaller than the spin flop field) can manipulate the N ´eel vector. Under the assumption that spin transport is possible only for angular momentum along the N´eel vector and in the linear dynamics regime, we phenomenologically obtain the angular depen- dence of the ISHE voltage as V() =aLAF ycos+bMAF y (3) whereaandbare phenomenological parameters. Equations 2 and 3 qualitatively reproduce our experimental results for the dependence (Fig. 3b) of the voltage (see supplementary Note 4 for details). 8In summary, we report the occurrence of the spin conductor-nonconductor transition and the field induced modulation of spin-current transmissivity in Cr 2O3, which is reminiscent of the CMR in electronics. We attribute this ‘colossal’ modulation of spin current to the combination of the anisotropic spin current transmission of the antiferromagnet and the device geometry, which is correlated to the N ´eel vector and anisotropy of Cr 2O3. The SCMR may also be observed in other antiferromagnetic materials in which the N ´eel vector responds to magnetic fields. It may therefore be possible to create devices which switch between the spin insulating and conducting states—but in response to a completely different stimulus. For example switching the antiferromagnet between perpendicular states electrically26. 1. Ramirez, A. P. Colossal magnetoresistance. Journal of Physics: Condensed Matter 9, 8171– 8199 (1997). 2. Tokura, Y . Critical features of colossal magnetoresistive manganites. Reports on Progress in Physics 69, 797–851 (2006). 3.ˇZuti´c, I. & Das Sarma, S. Spintronics: Fundamentals and applications. Reviews of Modern Physics 76, 323–410 (2004). 4. Maekawa, S. Concepts in Spin Electronics (Oxford Univ. Press, 2006). 5. Kajiwara, Y . etal. Transmission of electrical signals by spin-wave interconversion in a mag- netic insulator. Nature 464, 262–266 (2010). 6. Uchida, K. etal. Observation of the spin seebeck effect. Nature 455, 778–781 (2008). 97. Takei, S., Halperin, B. I., Yacoby, A. & Tserkovnyak, Y . Superfluid spin transport through antiferromagnetic insulators. Phys. Rev. B 90, 094408 (2014). 8. Cornelissen, L., Liu, J., Duine, R., Ben Youssef, J. & Van Wees, B. Long-distance transport of magnon spin information in a magnetic insulator at room temperature. Nature Physics 11, 1022–1026 (2015). 9. Qiu, Z. etal. Spin-current probe for phase transition in an insulator. Nature communications 7, 12670 (2016). 10. Tserkovnyak, Y . Spintronics: An insulator-based transistor. Nature nanotechnology 8, 706– 707 (2013). 11. Brockhouse, B. N. Antiferromagnetic structure in Cr 2O3.The Journal of Chemical Physics 21, 961–962 (1953). 12. Foner, S. High-field antiferromagnetic resonance in Cr 2O3.Physical Review 130, 183–197 (1963). 13. Nagamiya, T., Yosida, K. & Kubo, R. Antiferromagnetism. Advances in Physics 4, 1–112 (1955). 14. Xiao, J. etal. Theory of magnon-driven spin seebeck effect. Physical Review B 81, 214418 (2010). 15. Saitoh, E., Ueda, M., Miyajima, H. & Tatara, G. Conversion of spin current into charge current at room temperature: Inverse spin-Hall effect. Applied Physics Letters 88, 182509 (2006). 1016. Wang, H., Du, C., Hammel, P. C. & Yang, F. Spin transport in antiferromagnetic insulators mediated by magnetic correlations. Physical Review B 91, 220410(R) (2015). 17. Moriyama, T. etal. Anti-damping spin transfer torque through epitaxial nickel oxide. Applied Physics Letters 106, 162406 (2015). 18. Uchida, K. etal. Spin Seebeck insulator. Nature Materials 9, 894–897 (2010). 19. Uchida, K. etal. Longitudinal spin Seebeck effect: from fundamentals to applications. Journal of physics. Condensed matter : an Institute of Physics journal 26, 343202 (2014). 20. Qiu, Z., Hou, D., Uchida, K. & Saitoh, E. Influence of interface condition on spin-Seebeck effects. Journal of Physics D: Applied Physics 48, 164013 (2015). 21. Kikkawa, T. etal. Critical suppression of spin seebeck effect by magnetic fields. Physical Review B 92, 064413 (2015). 22. Tobia, D., Winkler, E., Zysler, R. D., Granada, M. & Troiani, H. E. Size dependence of the magnetic properties of antiferromagnetic Cr 2O3nanoparticles. Physical Review B 78, 104412 (2008). 23. Pati, S. P. etal. Finite-size scaling effect on N ´eel temperature of antiferromagnetic Cr 2O3 (0001) films in exchange-coupled heterostructures. Physical Review B 94, 224417 (2016). 24. Wu, S. M., Pearson, J. E. & Bhattacharya, A. Paramagnetic spin seebeck effect. Phys. Rev. Lett. 114, 186602 (2015). 1125. Kim, S. K., Tserkovnyak, Y . & Tchernyshyov, O. Propulsion of a domain wall in an antiferro- magnet by magnons. Physical Review B 90, 104406 (2014). 26. Wadley, P. etal. Electrical switching of an antiferromagnet. Science 351, 587–590 (2016). Acknowledgements This work was supported by JST-ERATO ‘Spin Quantum Rectification’, JST-PRESTO ‘Phase Interfaces for Highly Efficient Energy Utilization’, Grant-in-Aid for Scientific Research on Innova- tive Area, ‘Nano Spin Conversion Science’ (26103005 and 26103006), Grant-in-Aid for Scientific Research (S) (25220910), Grant-in-Aid for Scientific Research (A) (25247056 and 15H02012), Grant-in-Aid for Chal- lenging Exploratory Research (26600067), Grant-in-Aid for Research Activity Start-up (25889003), and World Premier International Research Center Initiative (WPI), all from MEXT, Japan, ZQ acknowledges support from ”the Fundamental Research Funds for the Central Universities (DUT17RC(3)073)”. DH ac- knowledges support from Grant-in-Aid for young scientists (B) (JP17K14331), JB acknowledges supports from the Graduate Program in Spintronics, Tohoku University, and Grand-in-Aid for Young Scientists (B) (17K14102). KY and OG acknowledge support from Humboldt Foundation and EU ERC Advanced Grant No. 268066. KY acknowledges the Transregional Collaborative Research Center (SFB/TRR) 173 SPIN+X and DAAD project ”MaHoJeRo”. O.G. acknowledge the EU FET Open RIA Grant no. 766566 and the DFG (project SHARP 397322108). Author contributions Z.Q. and D.H. designed the experiment, Z.Q. fabricated the samples and collected all of the data. Z.Q., D.H., J.B. and K.Y . analyzed the data. J.B., K.Y . and O.G. contribute theoretical discussions. E.S. supervised this study. All the authors discussed the results and prepared the manuscript. Competing Interests The authors declare that they have no competing financial interests. 12Correspondence Correspondence and requests for materials should be addressed to D.H. (email: dazhi.hou@imr.tohoku.ac.jp). Methods Preparation of YIG/Cr 2O3/Pt samples. We grew a 3 m-thick single-crystalline YIG film on a (111) Gd 3Ga5O12wafer by liquid phase epitaxy method at 1203 K in PbO-B 2O3based flux. We cut a single wafer into 1.5 mm 3 mm in size. A 12 nm-thick Cr 2O3film is grown on top of the YIG film by pulsed laser deposition at 673 K and subsequently annealed at 1073 K for 30 min to obtain continuous films and improve the crystallinity. 10 nm-thick Pt films were then grown on the top of the Cr 2O3by RF magnetron sputtering. Sample characterization. Crystallographic characterization for the samples was carried out by X- ray diffractometry and transmission electron microscopy (TEM). The obtained TEM image shows that the YIG film is of a single-crystal structure, and the easy axis ( c-axis) of the hexagonal Cr 2O3 grown on the top of the YIG is along the out-of-plane direction z(Fig. 2b). Spin Seebeck experimental set-up. We performed spin Seebeck measurements in a vector mag- net system (Oxford instruments inc). We set the samples on a wave guide and heated the Pt layer by using a pulsed microwave27(8 GHz, 1 W), which creates a temperature gradient as shown in Fig. 2a. We measured the voltage signal between the ends of the Pt layer using a lock-in amplifier. Data availability The data that support the findings of this study are available from the authors on reasonable 13Figure 1 Concept of spin colossal magnetorresistance Spin current Spin currentSpin conductorSpin nonconductor Spin conductor-nonconductor transitionMetal-insulator transition Charge current Charge current Charge conductorCharge insulator H H SCMR Spin colossal magnetoresistance CMRColossal Magnetoresistance a b Figure 1: Concept of spin colossal magnetoresistance. a . A schematic illustration of colossal magnetoresistance (CMR). CMR is a property of some materials in which their electrical resistance changes steeply in the presence of a magnetic field, typically due to the strong coupling between a steep metal-insulator transition and a magnetic phase transition. b. A schematic illustration of spin colossal magnetoresistance (SCMR): spin current transmissivity changes steeply due to the change in symmetry (in this paper due to a magnetic phase transition). The spin current transmissivity is also modulated by an applied magnetic field.14Figure 2 Spin insulator-metal transition in Cr2O3 TN T (K)100 200 300 V (nV) -0.1 0.1 0 H (T)296 KT=320 K 280 KPtVSSE Cr2O3 Y3Fe5O12 T=300 K 288 Kc V (nV) Y3Fe5O12Cr2O3Pt c ad 00 0 0 0400 -400 400 -400 400 -400400 -4002k -2k VSSE@0.1 T (nV)a fe VSSE@0.1 T (nV)0 02k4k200400-0.1 0.1 0H (T) VSSEJsY3Fe5O12/Pt Y3Fe5O12/Cr2O3/Pt Y3Fe5O12/PtY3Fe5O12/Cr2O3/Pt e↑– e↓– out inJs ∇T H b VISHE (nV)2000 1000 0g Y3Fe5O12/Cr2O3/Pt dCr2O3 =12 nm YIG/Cr2O3/Pt Cr2O3/PtVSSE (nV)400 200 0 T (K)350 250 300h YIG/Cu/Cr2O3/PtdCr2O3 = 7.5 nm 12.0 nm 24.0 nm 36.0 nm xz yFigure 2: Spin conductor-nonconductor transition in Cr 2O3. a. A schematic illustration shows the concept of the spin-current transmissivity measurement of a YIG/Cr 2O3/Pt trilayer device. A temperature gradient, rT, is along the zdirection while an external magnetic field ( H) is along theydirection. The magnetic insulator YIG is used as a spin source to inject spin currents Jin s into the Cr 2O3based on the spin Seebeck effect, and transmitted spin currents Jout sthrough the Cr2O3are detected as voltage signals in the Pt layer via the inverse spin Hall effect. b. A cross sectional TEM image of the YIG/Cr 2O3/Pt trilayer device used in this work. The scale bar of the image is 5 nm. The easy axis cof the Cr 2O3is in the out-of-plane direction zof the film as inserted axis shows. c. The external magnetic field Hdependencies of the voltage signal Vmeasured in a YIG/Pt bilayer device at 300 K. d. The external magnetic field Hdependencies of the voltage signalVmeasured in the YIG/Cr 2O3/Pt trilayer device at various temperatures. e. The temperature dependence of the spin Seebeck voltage VSSEatH=0.1 T for the YIG/Cr 2O3/Pt trilayer device. f. The temperature dependence of the spin Seebeck voltage VSSEatH=0.1 T for a YIG/Pt bilayer device. g. The temperature dependence of spin pumping signals VISHE for YIG/Cr 2O3/Pt trilayer devices with various values of the Cr 2O3layer thickness dCr2O3.h. The temperature dependence of the spin Seebeck voltage VSSEatH=0.1 T for Y 3Fe5O12/Cu/Cr 2O3/Pt and Cr 2O3/Pt devices. The measurement errors are smaller than the size of data points in all figures.15a b Figure 3 Spin colossal magnetoresistance in Cr2O3.H ∇T θV xyz -4040T=278 K0 -80 080T=284 K0 -250250 -90 90T=290 KVSSE@H (nV) θ (°)e 02.5 T H=0.5 T0 -400400T=296 K H LAFz y φθ MAFMA MB y y yH H Hz z z θ=0°LAFLAF LAF θ=90° 0°<θ<90°Cr2O3 Ratio( )@H (%)c d100200 VSSE@H (nV) 300 250 T (K)02.5 T 2.1 T 1.7 T 1.3 T 0.9T5002.5 Tθ=20° H=0.5 T0H=0.5 TFigure 3: Spin colossal magnetoresistance in Cr 2O3. a. A schematic illustration of the out-of- plane spin Seebeck set-up for the YIG/Cr 2O3/Pt trilayer device. A temperature gradient, rT, is along thezdirection, and an external magnetic field, H, is applied in the y-zplane.is the angle betweenrTandH.b.dependencies of VSSEat different temperatures for the YIG/Cr 2O3/Pt trilayer device with various values of H. Here,VSSErefers to the spin Seebeck voltage signal detected from the Pt layer. The solid curve is a fitting result using Eq. (3). The noise level of the voltage measurement is about 5 nV , which is smaller than the size of data points in most figures. c. Temperature dependence of VSSEfor the YIG/Cr 2O3/Pt trilayer device at different external magnetic fields Hat=20. The solid lines are guides to the eye. d. TheTschange ratioRatio (Ts)@Hat an applied external magnetic field Has functions of temperature. Here, Ratio (Ts)@H= (VSSE@ HVSSE@0 :5T)=VSSE@0 :5T.Tsrefers to the spin-current transmissivity in the Cr 2O3layer. The solid lines are guides to the eye. e. A schematic illustration of the relation between a N ´eel vector, LAF, the induced magnetization MAF, and the external magnetic field H. Here,refers to the angle between Hand thezaxis;refers to the angle between LAFand the easy axiscof Cr 2O3, which is along the zaxis in our sample. MAandMBare the magnetization vectors of the two sublattices of Cr 2O3. The lower panel shows the relation between a N ´eel vector, LAF, the induced magnetization MAF, and the external magnetic field Hat different values of .16request, see author contributions for specific data sets. 27. Vasyuchka, V . I. Microwave-induced spin currents in ferromagnetic-insulator—normal-metal bilayer system. Applied Physics Letters 105, 092404 (2014). 17

2018-04-12

Colossal magnetoresistance (CMR) refers to a large change in electrical conductivity induced by a magnetic field in the vicinity of a metal-insulator transition and has inspired extensive studies for decades\cite{Ramirez1997, Tokura2006}. Here we demonstrate an analogous spin effect near the N\'eel temperature $T_{\rm{N}}$=296 K of the antiferromagnetic insulator \CrO. Using a yttrium iron garnet \YIG/\CrO/Pt trilayer, we injected a spin current from the YIG into the \CrO layer, and collected via the inverse spin Hall effect the signal transmitted in the heavy metal Pt. We observed a change by two orders of magnitude in the transmitted spin current within 14 K of the N\'eel temperature. This transition between spin conducting and nonconducting states could be also modulated by a magnetic field in isothermal conditions. This effect, that we term spin colossal magnetoresistance (SCMR), has the potential to simplify the design of fundamental spintronics components, for instance enabling the realization of spin current switches or spin-current based memories.

Spin colossal magnetoresistance in an antiferromagnetic insulator

1804.04516v1

Enhanced bipartite entanglement and Gaussian quantum steering of squeezed magnon modes Shaik Ahmed,1M. Amazioug,2Jia-Xin Peng,3and S. K. Singha4,† 1School of Technology, Woxsen University, Hyderabad, Telangana -502345, India 2LPTHE-Department of Physics, Faculty of sciences, Ibn Zohr University, Agadir, Morocco 3State Key Laboratory of Precision Spectroscopy, Quantum Institute for Light and Atoms, Department of Physics, East China Normal University, Shanghai 200062, China 4Graphene and Advanced 2D Materials Research Group, Sunway University, Malaysia (Dated: July 20, 2023) We theoretically investigate a scheme to entangle two squeezed magnon modes in a double cavity- magnon system, where both cavities are driven by a two-mode squeezed vacuum microwave field. Each cavity contains an optical parametric amplifier as well as a macroscopic yttrium iron garnet (YIG) sphere placed near the maximum bias magnetic fields such that this leads to the excitation of the relevant magnon mode and its coupling with the corresponding cavity mode. We have obtained optimal parameter regimes for achieving the strong magnon-magnon entanglement and also studied the effectiveness of this scheme towards the mismatch of both the cavity-magnon couplings and decay parameters. We have also explored the entanglement transfer efficiency including Gaussian quantum steering in our proposed system. I. INTRODUCTION Quantum entanglement [1] and Gaussian quantum steering [2, 3] are two major important resources in the field of quantum computing [4], quantum cryptography [5] and quantum teleportation [6] including quantum information processing [7]. Many microscopic as well as macroscopic quantum systems have been proposed over the past decades for the study of quantum entanglement and other nonclassical quantum correlations in superconducting qubits [8], atomic ensembles [9], cavity optomechanics [10–16] and cavity magnomechanical (CMM) systems [17–22] which paves the way for advancements in present era of quantum technology. In CMM systems, the magnons defined as the collective excitation of a large number of spins in ferrimagnetic materials play very important role in the study of light-matter interactions due to their tunability, low damping, high spin density [23, 24] as well the strong coupling with the microwave photons [25–28]. Moreover, other important mascroscopic quantum phenomena such as magnon-induced effects [30], tunable magnomechanically induced transparency and absorption [31, 32], slow light [33], four-wave mixing [34], squeezed states [35–38], nonclassical quantum correlations [39, 40], microwave-to-optical carrier conversion [41] including quantum sensing [42, 43] also successfully investigated in cavity magnomechanical systems. To quantify the bipartite entanglement between the magnon and microwave photon in CMM systems, we use a very well-known witness of bipartite entanglement defined as the logarithmic negativity [44]. A recent theoretical work given in [45] explored the logarithmic negativity between two magnon modes where the optimal conditions for achieving the strong magnon-magnon entanglement involve the resonant coupling between the microwave cavity with both the magnon modes whereas in case of two microwave cavities given in [46–48], it is found that the detuning of the cavity and magnon mode significantly affects the bipartite entanglement. These research works also found the presence of both one-way and two-way Gaussian quantum steering. So, all these studies demonstrate that the bipartite entanglement and quantum steering in CMM systems can be significantly controlled through the various physical parameters. All these recent progress also broaden our understanding of quantum correlations and facilitate to further explore the possibility of secure quantum protocols in such kind of macroscopic quantum systems. Motivated by these works, we study the quantum correlations and Gaussian quantum steering between two squeezed magnon modes of two yttrium iron garnet (YIG) spheres in a system of two spatially separated microwave cavities. Each cavity also contains an optical parametric amplifier (OPA) as well as a macroscopic YIG sphere placed near the maximum bias magnetic fields such that this leads to the excitation of the corresponding magnon mode and its coupling with the cavity mode. In addition, both the cavities are simultaneously driven by a two mode squeezed vacuum microwave field in our proposed system [49]. In this present work, we found the generation of a considerable bipartite entanglement between the two magnon modes with gradually increasing squeezing parameter and the mean aCorresponding Author †singhshailendra3@gmail.comarXiv:2307.09846v1 [quant-ph] 19 Jul 20232 thermal magnon number. Moreover, it can be seen clearly from our work that not all entangled states allow for quantum steering whereas any state that can be steered must necessarily be entangled. This paper is organized as follows: In Section II, we introduce model Hamiltonian and also evaluate corresponding quantum Langevin equations including its solutions (QLEs). In Section III, we discuss in details about mathematical formulation of bipartite entanglement and Gaussian quantum steering between two magnon modes. Numerical Results and related discussions are given in Section IV whereas we conclude our results in Section V. II. THE MODEL HAMILTONIAN Our proposed system shown in Fig. 1 consists of two microwave cavities and two magnon modes in two YIG spheres, which are respectively placed inside the cavities near the maximum magnetic fields of the cavity modes and simultaneously in uniform bias magnetic fields such that it causes both the magnon modes to strongly couple with respective cavity modes [49, 50]. Each cavity, contains a degenerate optical parametric amplifier (OPA) to produce squeezed light [51]. The Hamiltonian of the system in a rotating frame with frequency ωjcan be written as H=X j=1,2n ℏ∆cjc† jcj+ℏ∆mjm† jmj+ℏgj cjm† j+c† jmj
+icℏλj(eicθc†2 j−e−icθc2 j) +icℏµj(eicνm†2 j−e−icνm2 j)o ,(1) where ∆ cj=ωcj−ωj,∆mj=ωmj−ωj,cj(c† j),mj(m† j) are the annihilation (creation) operators of the jthcavity and magnon modes, respectively, and we have O,O+ = 1 (O=cj, mj).ωcj(ωmj) is the resonance frequency of the jth cavity mode (magnon mode). The frequency of the magnon mode ωmjis determined by the external bias magnetic fieldHjand the gyromagnetic ratio βviaωmj=βHj, and thus can be flexibly adjusted, and gjis the coupling rate between the jthcavity and magnon modes. The parameter λjandθjrepresents respectively the nonlinear gain of the OPA and the phase of the driving field. with µjbeing the squeezing parameter and νbeing the phase of jthsqueezing mode. The magnon squeezing can be achieved by transferring squeezing from a squeezed-vacuum microwave field [34], or by the intrinsic nonlinearity of the magnetostriction (the so-called ponderomotive-like squeezing) [52], or by the anisotropy of the ferromagnet [53, 54], etc. FIG. 1. Schematic diagram of a double cavity-magnon system where both the cavities are driven by a two-mode squeezed vacuum microwave field. Two YIG spheres are respectively placed inside the microcavities near the maximum magnetic fields of the cavity modes and simultaneously in uniform bias magnetic fields. In the frame rotating at the frequency ωj, i.e., the frequency of the jthmode of the input two-mode squeezed field, the QLEs of this model Hamiltonian are given by ˙cj=−(κcj+i∆cj)aj−igjmj+ 2λjeiθc† j+p 2κcjcin j, (2) ˙mj=−(κmj+i∆mj)mj−igjcj+ 2µjeicνm† j+p 2κmjmin j, where κcj(κmj) is the decay rate of the jth cavity mode (magnon mode), ∆ cj=ωcj−ωj,∆mj=ωmj−ωj, and cin j (min j) is the input noise operator for the jthcavity mode (magnon mode). The two cavity input noise operators cin jare quantum correlated due to the injection of the two-mode squeezed field, and have the following non-zero correlations3 in time domain cin jandcin† jare given by [64] ⟨cin j(t)cin† j(t′)⟩= (N+ 1)δ(t−t′) (3) ⟨cin† j(t)cin j(t′)⟩=Nδ(t−t′) (4) ⟨cin j(t)cin j′(t′)⟩=Me−icωM(t+t′)δ(t−t′) ; j̸=j′(5) ⟨cin† j(t)cin† j′(t′)⟩=MeicωM(t+t′)δ(t−t′) ; j̸=j′(6) . Here N= sinh2r,M= sinh rcoshrandris the squeezing parameter of the two-mode squeezed vacuum field whereas the magnon input noise operators min jare of zero mean and correlated as follows ⟨min j(t)min† j(t′)⟩= (Nmj+ 1)δ(t−t′) (7) ⟨min† j(t)min j(t′)⟩=Nmjδ(t−t′) (8) where Nmj=h expℏωmj kBT −1i−1 is the equilibrium mean thermal magnon number of the jthmode, with Tthe environmental temperature and kBthe Boltzmann constant. Using the linearisation of quantum Langevin equations, the fluctuations of the system are written as δ˙cj=−(κcj+i∆cj)δcj−igjδmj+ 2λjeiθδc† j+p 2κcjcin j, (9) δ˙mj=−(κmj+i∆mj)δmj−igjδcj+ 2µjeicνδm† j+p 2κmjmin j. To get the explicit expression of the degree of freedom of optical and magnon modes, we consider the EPR-type quadrature fluctuations operators corresponding to the two subsystems defined as δQj= (δcj+δc† j)/√ 2, δPj= i(δc† j−δcj)/√ 2, δqj= (δmj+δm† j)/√ 2, δpj=i(δm† j−δmj)/√ 2 (we have similar definition for input noises Qin j, Pin j andqin j, pin j) [55–60], The above QLEs can be simplified as δ˙Qj=−κcjδQj+ ∆ cjδPj+gjδpj+ 2λcos(θ)δqj+ 2λsin(θ)δpj+p 2κcjQin j, (10) δ˙Pj=−κcjδPj−∆cjδQj−gjδqj−2λcos(θ)δpj+ 2λsin(θ)δqj+p 2κcjPin j, δ˙qj=−κmjδqj+ ∆ mjδpj+gjδPj+ 2µcos(ν)δQj+ 2µsin(ν)δPj+p 2κmjqin j, δ˙pj=−κmjδpj−∆mjδpj−gjδQj−2µcos(ν)δPj+ 2µsin(ν)δQj+p 2κmjpin j. Equation (10) take the following compact matrix form ˙V(t) =AV(t) +χ(t), (11) Here V(t) = [δQ1, δP1, δQ 2, δP2, δq1,δp1, δq2, δp2]T,Ais the drift matrix A= A1A3 A3A2 (12) where A1= −κc1+ 2λcos(θ) ∆ c1+ 2λsin(θ) 0 0 −∆c1+ 2λsin(θ)−κc1−2λcos(θ) 0 0 0 0 −κc2+ 2λcos(θ) ∆ c2+ 2λsin(θ) 0 0 −∆c2+ 2λsin(θ)−κc2−2λcos(θ) (13)4 and A2= −κm1+ 2µcos(ν) ∆ m1+ 2µsin(ν) 0 0 −∆m1+ 2µsin(ν)−κm1−2µcos(ν) 0 0 0 0 −κm2+ 2µcos(ν) ∆ m2+ 2µsin(ν) 0 0 −∆m2+ 2µsin(ν)−κm2−2µcos(ν) (14) and A3= 0g10 0 −g10 0 0 0 0 0 g2 0 0 −g20 (15) andχ(t) = [√2κc1Qin 1,√2κc1Pin 1,√2κc2Qin 2,√2κc2Pin 2,√2κm1qin 1,√2κm1pin 1,√2κm2qin 2,√2κm2pin 2]T. The system is stable when eigenvalues of the drift matrix A(12) have negative real parts. This corresponds to the so-called Routh- Hurwitz criterion [61]. The steady state of the system, which is completely characterized by an 8 ×8 covariance matrix (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 Lyapunov equation [62] AΣ + Σ AT=−D, (16) where Dis the diffuse matrix defined by Dijδ(t−t′) =⟨χi(t)χj(t′) +χj(t′)χi(t)⟩/2, given by D= κ′0√κc1κc2M 0 0 0 0 0 0 κ′0 −√κc1κc2M0 0 0 0√κc1κc2M 0 κ′′0 0 0 0 0 0 −√κc1κc2M 0 κ′′0 0 0 0 0 0 0 0 γ′0 0 0 0 0 0 0 0 γ′0 0 0 0 0 0 0 0 γ′′0 0 0 0 0 0 0 0 γ′′ (17) where κ′=κc1 N+1 2
,κ′′=κc2 N+1 2
,γ′=κm1 2Nm1+ 1
andγ′′=κm2 2Nm2+ 1
. The covariance matrix Σ (mm)associated with the two magnon modes is given by Σ(mm)=X Z ZTY (18) The 2 ×2 sub-matrices XandYin Eq. (18) describe the autocorrelations of the two magnon modes and 2 ×2 sub-matrix Zin Eq. (18) denotes the cross-correlations of the two magnon modes. III. QUANTUM CORRELATIONS A. Quantum entanglement The logarithmic negativity Emis a measure or witness of entanglement in bipartite continuous-variable (CV) systems [63]. Mathematically, it can be expressed as: Em= max[0 ,−log(2ψ−)] (19) with ψ−being the smallest symplectic eigenvalue of partial transposed covariance matrix Σ (mm)of two magnon modes ψ−=s Γ−pΓ2−4 det Σ (mm) 2(20) where the symbol Γ is written as Γ = det X+ detY −detZ. The two magnon modes are entangled if the condition ψ−<1/2 (i.e. when Em>0) is satisfied.5 B. Gaussian quantum steering Quantum steering is the process of acquiring information about an unmeasurable quantum system by measuring a single quantum system. Gaussian quantum steering is the asymmetric property between two entangled observers (the two mechanical modes), Alice ( A: magnon M1) and Bob ( B: magnon M2). Besides, they are used to quantify how much the two magnon modes are steerable. We use the covariance matrix Σ (mm)of the two magnon modes, the Gaussian steering A→BandB→Awritten as [12 ?] SA→B=SB→A= max 0,1 2lndet(X) 4 det Σ (mm) ; (21) There are two possibilities of steerability between AandB: (i) no-way steering if SA→B=SB→A= 0 i.e. Alice can’t steer Bob and vice versa also impossible even if they are not separable, and (ii) two-way steering if SA→B=SB→A>0, i.e. Alice can steer Bob and vice versa. Indeed, a non-separable state is not always a steerable state, while a steerable state is always not separable. IV. RESULTS AND DISCUSSION In this section, we will discuss the steady state quantum correlations of two magnon modes under various parameters regime reported experimentally [49, 64] and given as ωc1/2π= 10 GHz, κc/2π= 5 MHz, κm=κc/5,g1=g2= 5κc, θ=πandν= 0.9π. For simplicity, we consider that Nm1=Nm2=Nm,κc1=κc2=κcandκm1=κm2=κm. Additionally, each YIG sphere used in our study has a diameter of 0.5 mm. These spheres are specifically chosen for their size and contain more than 1017spins. FIG. 2. Plot of the logarithmic negativity Embetween two magnon modes versus (a) ∆ c1and ∆ m1, with ∆ c2= ∆m2= 0; (b) ∆c2and ∆ m2, with ∆ c1= ∆m1= 0,r= 1,λ= 0.2κc,µ= 0.2κcandT= 100 mK. See text for the other parameters. In Fig. 2, we have plotted the logarithmic negativity Emof subsystem magnon-magnon with varying ∆ cjand ∆ mj (j= 1,2) whereas all other parameters are fixed. It can be seen in Figs. 2(a) and (b) that when ∆ cj= ∆ mj= 0 (j= 1,2), the entanglement between two magnon modes is optimal. This observation can be attributed to the resonant transfer of quantum correlations from the input fields to the two magnon modes, facilitated by the linear cavity-magnon coupling. Fig. 3(a) shows that the bipartite entanglement Emincreases with the squeezing parameter rof the two-mode input squeezed vacuum field and decreases with the temperature T. The effect of the temperature is due to the influence of the decoherence phenomenon [65]. Furthermore, it has been observed that the logarithmic negativity Em reaches its maximum value when the value of rfalls within the range of (1-1.5) whereas when the parameter rgoes to zero both the magnon modes remain separable ( Em= 0) as illustrated in figure 3(a). This shows the dependence of the bipartite entanglement of the two magnon modes on the squeezing parameter r. In Fig. 3(b), we plot Emas a function of squeezing parameter randg2/g1. We found here the generation of the bipartite entanglement between the two magnon modes with a gradual increase in squeezing parameter reven for a wide range of mismatch of the two coupling strengths g2andg1.6 FIG. 3. (a) Plot of the logarithmic negativity Emm between the two magnon modes vs the temperature T(K) and the magnon squeezing parameter µwith ∆ c1,2= ∆m1,2= 0,T= 100 mK, λ= 0.2κcandµ= 0.2κc. (b) Plot of the logarithmic negativity Emm between the two magnon modes vs g2/g1androf the two-mode squeezed vacuum field with g1= 5κc, ∆c1,2= ∆m1,2= 0, T= 100 mK, λ= 0.2κcandµ= 0.2κc. FIG. 4. Plots of the steering SA→B,SB→Alogarithmic negativity Emof the two magnon modes versus the equilibrium mean thermal magnon number Nmfor various values of the coupling λandµ, with ∆ c1,2= ∆m1,2= 0,r= 1 and T= 100 mK. In Fig.(4), we plot the Gaussian steering SA→B,SB→Aand logarithmic negativity Emfor the subsystem magnon- magnon as a function of the equilibrium mean thermal magnon number Nmfor various values of the parameters λ andµwhereas the other parameters are fixed. It can be seen that SA→B,SB→Aand entanglement Emhave the same evolution behavior. This figure studies the effect of Nm(temperature T) and the parameters λandµon the bipartite entanglement and quantum steerings. Due to decoherence phenomena both the quantities i.e. bipartite entanglement and quantum steering decrease very quickly with increasing Nm. Moreover, when we enhance λandµ the magnon-magnon entanglement as well as two-way quantum steering become finite for a wide range of temperature T(the equilibrium mean thermal magnon number Nm). Moreover, as depicted in Fig.(4) the entangled state is not always a steerable state but a steerable state must be entangled, i.e. when SA→B=SB→A>0 which leads to Em>0 and hence is the witnesses of the existence of Gaussian two-way steering. This means that both magnon modes are entangled as well as are steerable from AtoBand from BtoA. However, we get no-way steering when SA→B=SB→A= 0 and Em>0 and so the measure of Gaussian steering always remains bounded by the bipartite entanglement Em. In Fig. 5, we have plotted the logarithmic negativity Emof two magnon modes with κcand ∆ m1for a fixed value of all other parameters. It can be seen that the entanglement between the two magnon modes is maximum when ∆m1= 0 and κc= 3×107Hz. However, the bipartite entanglement Emdecreases with decreasing decay rate κcand increasing detuning ∆ m1.7 FIG. 5. Plot of the logarithmic negativity Embetween the two magnon modes vs κcand ∆ m1with ∆ c1= ∆ c2= ∆ m2= 0, T= 50 mK, r= 1.5,g1= 5κc,λ= 0.2κcandµ= 0.2κc. V. CONCLUSIONS We have theoretically investigated a scheme for the generation of the bipartite entanglement and Gaussian quantum steering in a double microwave cavity-magnon hybrid system where a two-mode squeezed microwave vacuum field is also transferred simultaneously into both cavities. We have obtained optimal parameter regimes for achieving the strong magnon-magnon entanglement and also explored the effectiveness of the scheme towards the mismatch of two cavity magnon couplings including the entanglement transfer efficiency. 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2023-07-19

We theoretically investigate a scheme to entangle two squeezed magnon modes in a double cavitymagnon system, where both cavities are driven by a two-mode squeezed vacuum microwave field. Each cavity contains an optical parametric amplifier as well as a macroscopic yttrium iron garnet (YIG) sphere placed near the maximum bias magnetic fields such that this leads to the excitation of the relevant magnon mode and its coupling with the corresponding cavity mode. We have obtained optimal parameter regimes for achieving the strong magnon-magnon entanglement and also studied the effectiveness of this scheme towards the mismatch of both the cavity-magnon couplings and decay parameters. We have also explored the entanglement transfer efficiency including Gaussian quantum steering in our proposed system

Enhanced bipartite entanglement and Gaussian quantum steering of squeezed magnon modes

2307.09846v1

arXiv:1603.09201v1 [cond-mat.mes-hall] 30 Mar 2016Low-damping transmission of spin waves through YIG/Pt-bas ed layered structures for spin-orbit-torque applications Dmytro A. Bozhko,1,2Alexander A. Serga,1,a)Milan Agrawal,1Burkard Hillebrands,1and Mikhail P. Kostylev3 1)Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universit¨ at Kaiserslautern, 67663 Kaiserslautern, Germany 2)Graduate School Materials Science in Mainz, Kaiserslauter n 67663, Germany 3)School of Physics, M013, University of Western Australia, C rawley, WA 6009, Australia (Dated: 12 June 2018) We show that in YIG-Pt bi-layers, which are widely used in experiments on the spin transfer torque and spin Hall effects, the spin-wave amplitude significantly decreases in comp arison to a single YIG film due to the excitation of microwave eddy currents in a Pt coat. By introducing a novel excitation geometry, where the Pt layer faces the ground plane of a microstrip line structure, we supp ressed the excitation of the eddy currents in the Pt layer and, thus, achieved a large increase in the transmissio n of the Damon-Eshbach surface spin wave. At the same time, no visible influence of an external dc curren t applied to the Pt layer on the spin-wave amplitude in the YIG-Pt bi-layer was observed in our experiments with YIG films of micrometer thickness. Magnonics addresses the transfer and processing of information using spin waves (SW) and their quanta, magnons.1One of the main challenges for this field is the finite lifetime ofmagnonseven in alow-dampingmaterial such as single crystal yttrium-iron-garnet(YIG).2A pos- sible way to overcome this issue is the compensation of SW damping via transfer of a spin torque to a magnetic medium by a spin-polarized electron current generated in an adjacent non-magnetic metal layer with high spin- orbit interaction. The decrease in the damping of spin waves propagating in a YIG film covered with a current conducting Pt film has already been reported.3,4How- ever, the initial spin-wave attenuation in such a struc- ture was unacceptably high for practical applications (≃35dB in Ref.3). Therefore, it is an important task to determine the origin of the excessive spin-wave damping in YIG-Pt bi-layers and find a way to reduce this harm- ful effect. The latter will only be possible if this damp- ing does not correspond to the spin-pumping effect – the process of a back transfer of a spin torque from magnons exited in a magnetic medium to a non-magnetic layer. Here, we show that in a Pt covered YIG film the SW amplitude decreases in comparison to a single YIG film due to the resistive losses attributed to the microwave eddy currents induced in a thin Pt layer by a propagat- ing spin wave. By placing near the Pt layer of a highly conducting metal plate, we suppressed the excitation of the eddy currents in platinum and, thus, achieved a large increase (up to 1000 times) in the transmission of the Damon-Eshbach surface spin wave. The experiments were performed using 2mm wide and 15mm long SW waveguidespreparedfrom a 6.7 µm thick YIG film grown by liquid phase epitaxy on a 500 µm thick gallium-gadolinium garnet (GGG) substrate. A Pt layer of 10nm thickness was deposited on top of the YIG surface. Spin waves were emitted and received by a)Electronic mail: serga@physik.uni-kl.de FIG. 1. Sketch of the experimental setup. (a) The conven- tional SW excitation geometry – a YIG-Pt bi-layer is placed near the microstrip antennas. (b) The inverted SW excitatio n geometry – the YIG-Pt bi-layer is close to the ground plate of the microstrip structure. two short-circuited gold-wire antennas of 50 µm diame- ter placed 7.6mm apart from each other (see Fig. 1). Spin-wave propagation was studied in a frequency band from 6.2 to 6.5GHz using a vector network analyzer (Anritsu MS46322A). A relatively small level of the ap- plied microwave power of 0.5mW was chosen to pre- vent the development of non-linear effects for the prop- agating spin waves. An external bias magnetic field H0= 1600Oe was applied in the YIG film plane per- pendicularly to the SW propagation direction. Thus, the geometry of the Damon-Eshbach (DE) surface spin wave was implemented.2 The reference experiment was performed in a conven- tional excitation geometry (as in Ref. 3), when a YIG film lays directly under the antennas (see Fig. 1(a) and insets in Fig. 2). In the case of a bare YIG film – i.e. without any Pt coating (Fig. 2(a)), the DE spin wave, which propagates near the YIG surface touching the an- tennas (see the sketch of spin-wave distribution over the YIG film thickness in inset in Fig. 2(a)), possesses rel-2 FIG. 2. Conventional excitation geometry. A YIG film is positioned directly under the antennas. The localization o f a Damon-Eshbach wave depends on the direction of the bias magnetic field H0(see insets). Dotted/blue lines correspond to the bias magnetic field pointing into the drawing plane. Solid/red lines correspond to the bias magnetic field pointi ng out of the drawing plane. (a) Transmission ( S12) charac- teristic of the bare YIG waveguide. (b) Transmission ( S12) characteristic of the YIG waveguide covered with 10nm Pt layer. (c) Excitation efficiency by the input antenna ( S22) for the bare YIG waveguide. (d) Excitation efficiency by the input antenna ( S22) for the YIG waveguide covered with 10nm Pt layer. atively low minimal transmission losses of around 10dB at 6.42GHz. The DE wave localized at the opposite film surface shows one order of magnitude weaker transmis- sion. This well known effect relates to the nonreciprocal excitation of a DE wave, whose efficiency depends on the relative orientation of the bias magnetic field H0and the SWpropagationdirection.2,5,6In spite ofthe pronounced difference in the transmissioncharacteristics,the efficien- cies of SW excitation, which are inverselyproportionalto the microwave reflection from the input antenna S22, are identical for two opposite orientations of the bias mag- netic field (see Fig. 2(c)). That is because in both cases the input antenna radiates spin waves in two opposite directions but only one of the emitted waves reaches the output antenna. In the case of the Pt-covered YIG film, the excita- tion efficiency remains almost the same as for the bare YIG waveguide (compare Fig. 2(d) to Fig. 2(c)) but the transmissionofthe DE wavelocalizednearthe Ptlayeris strongly suppressed (see the dashed curve in Fig. 2(b)). This suppression can be associated either with eddy cur- rents excited in the high-resistive ( RPt= 218Ohm) Pt layer by stray fields of the propagating spin wave or with the spin-pumping effect at the YIG-Pt interface. FIG. 3. Inverted excitation geometry. The YIG waveguide is positioned directly on the copper ground plate. Dotted/blu e lines correspond to the bias magnetic field pointing into the drawing plane. Solid/red lines correspond to the bias mag- neticfieldpointingoutofthe drawingplane. (a)Transmissi on (S12) characteristics of the bare YIG waveguide. (b) Trans- mission ( S12) characteristics of the YIG waveguide covered with 10nm Pt layer. (c) Excitation efficiency of the input antenna ( S22) for the bare YIG waveguide. (d) Excitation efficiency of the input antenna ( S22) for the YIG waveguide covered with a 10nm Pt layer. Insets demonstrate schematic representations of the surface spin wave localizations for dif- ferent directions of a bias magnetic field. At the second stage of our studies, in order to dis- tinguish between these two damping mechanisms, the YIG film was facing the ground plate (see Fig. 1(b) and Fig. 3). It is expected that if the Pt film is placed in direct contact with the bulk copper plate, than the eddy currents will mostly flow in the low resistive ground plate. Hence, no additional Ohmic losses due to currents in Pt are expected. At the same time, the spin pumping effect, which can potentially contribute to the excessive SW damping, should remain unchanged. The SW excitation efficiencies shown in Fig. 3(c) and Fig. 3(d) for the bare and Pt-covered YIG waveguides are close to those obtained in the conventional excitation geometry (compare with Figs. 2(c) and 2(d)). At the same time, one can see from Figures 3(a) and 3(b) that the wave, which propagates close to the Pt layer, experiences now no excessive damping and shows the same transmission losses as the wave in the bare YIG film. This fact explicitly confirms our assumption about the significant contribution of the SW induced eddy currents to the SW damping in Pt-YIG bi-layers. It should be noted that in the proposed excitation geometry spin waves are excited by microwave currents flowing in the ground plate rather than through the3 FIG. 4. Inverted excitation geometry. The YIG waveguide is positioned near the copper ground plate. An additional 80µm thick dielectric polypropylen (PP) spacer layer is intro- duced to electrically decouple the Pt layer from ground. Dot - ted/blue lines correspond to the bias magnetic field pointin g into the drawing plane. Solid/red lines correspond to the bi as magnetic field pointing out of the drawing plane. (a) Trans- mission ( S12) characteristics of the bare YIG waveguide. (b) Transmission ( S12) characteristics of the YIG waveguide cov- eredwith 10nmPtlayer. (c)Excitation efficiencyoftheinput antenna ( S22) for the bare YIG waveguide. (d) Excitation ef- ficiency of the input antenna ( S22) for the YIG waveguide covered with a 10nm Pt layer. Insets demonstrate schematic representations of the surface spin wave localizations for dif- ferent directions of a bias magnetic field. microstrip antenna, which is separated from the YIG film by the thick GGG layer. It decreases the excitation efficiency of short-wavelength spin waves because the microwave currents are localized more weakly in the extended ground plate than in the narrow microstrip. However, the change in the electrodynamic boundary conditions due to placing a highly conducting layer near the surface of the YIG film7makes the DE dispersion relation steeper, i.e., increases the eigen-frequencies of the excited long-wavelength spin waves. Moreover, the related increase in the SW group velocity decreases the SW propagation time and consequently the SW transmission losses. As a result, the SW transmission frequency band remains rather wide. The described geometry enables the surface spin waves to propagate rather long distances through the Pt- covered YIG-film waveguides. However, no application of a dc electric current to such a structure is possible be- cause of electric contact between the Pt layer and the Cu ground plate. In order to settle the issue, a 80 µm thick polypropylene spacerwas placed between the sample and the ground plate (see insets in Figs. 4(a) and 4(b)). Inthis case, the minimal transmission losses (Figs. 4(a) and 4(b)) remain the same as in the previous measurements (see Figs. 2(a), 3(a) and 3(b)), but the transmission bandwidth turns to be significantly narrower. The rea- sons for this narrowing are obvious. A microwave mag- netic field induced by microwave currents flowing in the ground plate is becoming more uniform with increase in the distance to the ground plate. This leads to a reduc- tion of the excited SW wavenumbers and, alongside with the vanishing of the effect of metallization, to a decrease in the frequency bandwidth. However, the coupling be- tween the Pt layer and the ground plate appears to be high enough to shunt the Pt film and reduce, thus, the eddy currents related SW losses. At the final stage, we used the circuit shown in inset in Fig. 4(b) in order to check the ability to control the SW damping by a dc electric current applied to the Pt layer. To avoid spurious heating effects the experiment was performed in the pulsed regime: overlapping input microwave pulse (20ns) and dc current pulse (350ns) were applied with 1kHz repetition rate. It is remarkable that in the studied structure the best SW transmission corresponds to the case when the DE wave is localized at the YIG-Pt interface. In the absence of the externally applied dc electric current, such localization results in rather high dc voltages induced in the Pt layer by the combined action of the spin pumping and inverse spin Hall effects. However, no influence of both positive and negative electric currents Idc=±370mA on the SW am- plitude was measured in the entire SW frequency band. In conclusion, we have demonstrated that by using a novel spin-wave circuitry, where a Pt-covered YIG-film waveguide faces the ground plane of a microstrip line structure, it is possible to reduce the excitation of SW- related eddy currents in the Pt layer and achieve, thus, in a YIG-Pt bi-layer the same transmission losses for the Damon-Eshbach surface spin wave as in a bare YIG film. At the same time, no influence of a dc electric current applied to the Pt layer on the amplitude of SW pulses propagating in the relatively thick YIG waveguide of mi- crometer thickness was observed in our experiments. Financial support by EU-FET InSpin 612759 and by the Australian Research Council is gratefully acknowl- edged. 1A.V. Chumak, V.I. Vasyuchka, A.A. Serga, and B. Hillebrands , Nature Phys. 11, 453 (2015). 2A. A. Serga, A. V. Chumak, and B. Hillebrands, J. Phys. D: Appl . Phys.43, 264002 (2010). 3Z. Wang, Y. Sun, M. Wu, V. Tiberkevich, and A. Slavin, Phys. Rev. Lett. 107, 146602 (2011). 4E. Padr´ on-Hern´ andez, A. Azevedo, and S. M. Rezende, Appl. Phys. Lett. 99, 192511 (2011). 5T. Schneider, A. A. Serga, T. Neumann, B. Hillebrands, and M. Kostylev, Phys. Rev. B 77, 214411 (2008). 6V. E. Demidov, M.P. Kostylev, K. Rott, P. Krzysteczko, G. Rei ss, and S. O. Demokritov, Appl. Phys. Lett. 95, 112509 (2009). 7J. P. Parekh, K. W. Chang, and H. S. Tuan, Circuits, Systems and Signal Processing 4, 9 (1985).

2016-03-30

We show that in YIG-Pt bi-layers, which are widely used in experiments on the spin transfer torque and spin Hall effects, the spin-wave amplitude significantly decreases in comparison to a single YIG film due to the excitation of microwave eddy currents in a Pt coat. By introducing a novel excitation geometry, where the Pt layer faces the ground plane of a microstrip line structure, we suppressed the excitation of the eddy currents in the Pt layer and, thus, achieved a large increase in the transmission of the Damon-Eshbach surface spin wave. At the same time, no visible influence of an external dc current applied to the Pt layer on the spin-wave amplitude in the YIG-Pt bi-layer was observed in our experiments with YIG films of micrometer thickness.

Low-damping transmission of spin waves through YIG/Pt-based layered structures for spin-orbit-torque applications

1603.09201v1

Magnetic Resonance in Defect Spins mediated by Spin Waves Clara M uhlherr, V. O. Shkolnikov, and Guido Burkard Department of Physics, University of Konstanz, D-78464 Konstanz, Germany In search of two level quantum systems that implement a qubit, the nitrogen-vacancy (NV) center in diamond has been intensively studied for years. Despite favorable properties such as remarkable defect spin coherence times, the addressability of NV centers raises some technical issues. The coupling of a single NV center to an external driving eld is limited to short distances, since an ecient coupling requires the NV to be separated by only a few microns away from the source. As a way to overcome this problem, an enhancement of coherent coupling between NV centers and a microwave eld has recently been experimentally demonstrated using spin waves propagating in an adjacent yttrium iron garnet (YIG) lm1. In this paper we analyze the optically detected magnetic resonance spectra that arise when an NV center is placed on top of a YIG lm for a geometry similar to the one in the experiment. We analytically calculate the oscillating magnetic eld of the spin wave on top of the YIG surface to determine the coupling of spin waves to the NV center. We compare this coupling to the case when the spin waves are absent and the NV center is driven only with the antenna eld and show that the calculated coupling enhancement is dramatic and agrees well with the one obtained in the recent experiment. I. INTRODUCTION The negatively charged nitrogen vacancy (NV) cen- ter in diamond is an optically active point defect with a ground state spin triplet, lying deep in the bandgap of diamond2,3. Its bright optical transition and the exis- tence of intersystem crossing provides a good mechanism for initialization and read out of the spin state of the center4. The ground state of the NV center is sensitive to magnetic and electric elds, as well as to strain5{7, and thus can be used as a nanosensor to detect them8{10. This makes the NV center extremely interesting for metrol- ogy. Apart from that, the long spin coherence time of the defect makes it interesting for quantum information purposes. Its state can eciently be manipulated with oscillating magnetic elds, that cause transitions between the levels of the spin triplet. In most of the experiments this oscillating eld was generated by an antenna placed in the vicinity of the center, which raises the issue of ad- dressability for many NV centers when the antenna can no longer be placed close to each of them. Recently an ex- periment was reported in which the NV center was placed on top of a ferromagnetic material (YIG) that can host propagating spin waves. Using an antenna as a source of the spin waves in this material, one can couple distinct NV centers to it just like dialog partners are connected by a signal line. In this work we theoretically treat the coupling of the spin waves to the NV centers for a spe- cial geometry of the device described below. We provide an analytical expression for the spin wave eld and de- termine the coupling enhancement that it produces with respect to the eld of the antenna only, when spin waves are absent. II. SETUP We calculate the eect of spin wave excitation inside ferromagnetic thin lms as used for example in the ex-periment presented by Andrich et al.1. The vicinity of a spin wave hosting material to a quantum system driven by external microwaves provides an additional compo- nent of the driving eld. In that way, spin wave excita- tion inside a ferromagnetic material aects the coupling between the driving eld and the given quantum system. In the experimental realization by Andrich et al.1the quantum system consisted of defect spins inside a col- lection of nanodiamonds patterned on the surface of the magnetic material as depicted in Fig. 1( a). A single nan- odiamond, which is selected with the laser focus, hosts an ensemble of500 NV centers with an isotropic electron g-factor of g23. The nanodiamond is embedded in a polydimethylsiloxane (PDMS) lm, which is on the top of a layered structure of YIG and gadolinium gallium gar- net (GGG). Since YIG is a ferromagnetic material with ultra-low spin wave damping, it is perfect for the usage as the spin wave medium. An ac current owing through a microstrip line (MSL) grown on the surface of the YIG generates a microwave driving eld, penetrating through the dierent materials and denoted as the antenna eld in Fig. 1(a). Due to the vicinity of the NV centers to the YIG surface, the defect spins are not only sensitive to the antenna eld, but also to the stray eld originating from the magnetic YIG lm. For a theoretical description of the system we use the coordinate system and the dening parameters sketched in Fig. 1( b). Choosing the YIG lm to have a thickness dand to lie in the xy-plane, the MSL orientation can be set as the y-axis and the MSL posi- tion marks the origin of the coordinate system. Further, the MSL has a width wand the probed nanodiamond is located in the xz-plane atxNV. Based on realistic dimensions of experimental systems, some assumptions concerning the boundary conditions of the system are reasonable. In Ref. 1, the YIG lm is only 3 :08µm thick, which is very thin compared to its dimensions of about 10 mm along the x- andy-axis1. Thus, the lm can be as- sumed to be innite in the xy-plane. The GGG substrate and the PDMS layer are a few hundreds of µm thick, soarXiv:1810.11841v1 [cond-mat.mes-hall] 28 Oct 20182 Figure 1. ( a) Electron spin resonance in NV spins driven by spin waves demonstrated by Andrich et al.1. An array of nanodiamonds is patterned inside a PDMS lm layered on top of a YIG thin lm. From a distant MSL grown on the YIG the antenna eld propagates through the PDMS and couples to the NV centers. Inside the YIG the microwave excitation by the MSL leads to spin waves propagating in the plane. Due to the given dimensions, the theoretical treatment can be reduced to a two-dimensional coordinate system as shown in ( b), which is a cross section of the setup in ( a). The layered structure lies in thexy-plane and the center of the MSL of width wmarks the origin of the coordinate system and is oriented along the y-direction. that they are treated as innite in the z-direction and the only boundary conditions to be fullled are those at the YIG interfaces, where both surrounding layers in re- gions I and III are approximated as non-magnetic. This assumption is justied due to their magnetic permeabil- ities being isotropic and close to 11112. The MSL has a width ofw= 5µm, making its height of about 200 nm negligible. Overall, due to the spatial expansion of the system along the y-direction and the invariance of the system under y-translation, we assume that all elds are independent of yand the problem is treated in two di- mensions. III. DRIVEN SPIN WAVES In order to calculate the spin wave spectrum of a fer- romagnetic thin lm and the resulting eld amplitude, we start from Maxwell's equations (MEs). If there is an external magnetic H-eld, a magnetization eld Mis built up in the magnetic lm. In general, the external H-eld can be decomposed into a static part H0and a time-dependent component h(t) originating from the microwave antenna eld. Consequently, the magnetiza- tion inside the material depends on the driving frequency !and will also have a time-dependent component m(t). The relation between both time-varying components is given by the constitutive equation m=Xh, where ma- terial properties and the geometry of the system enter via the susceptibility tensor X. In case of a strong static bias eldH0along they-direction, the magnetic lm is tangentially magnetized and the static component of magnetization M0saturates. Under these conditions, X takes the form of the Polder susceptibility X=0 @0i 0 1 0 i01 A; (1)with the frequency-dependent entries =!!M=(!2 0!2) and=!0!M=(!2 0!2)13. The parameters !0= 0H0 and!M= 0MSaccount for the characteristics of the material in an external eld H0. Here, andMSde- note the gyromagnetic ratio and the saturation magneti- zation of the lm and 0denotes the vacuum permeabil- ity. Using the Polder tensor (1) and assuming the electric permittivity of the materials to be 1 for simplicity, the single components of the four MEs in two dimensions form a system of eight coupled dierential equations for the magnetic and electric eld components Hi(x;z) and Ei(x;z) (i=x;y;z ) in each region I-III in Fig. 1( b). Since the MSL is located right at the PDMS-YIG inter- face and is assumed to be innitesimally thin, the ow- ing current is non-zero only at the boundary between the two upper layers I and II, whereas inside the bulk regions there are no free currents j owing, and thereby, no ad- ditional source terms. Referring to that, the elds inside the bulk regions I-III are obtained by solving the system of homogeneous MEs with j= 0 separately and the MSL current is included by matching the boundary conditions atz= 0 andz=dafterwards. Performing a one dimensional Fourier transform of the x-coordinate yields a system of ordinary dierential equa- tions in the kxz-space, where only one equation actually has to be solved, @2 zHx(kx;z) +a2Hx(kx;z) = 0; (2) wherea2= ((1 +)22)k2 0=(1 +)k2 xandk0= !=c. The other non-zero eld components Hz(kx;z) andEy(kx;z) can be expressed in terms of the solution Hx(kx;z) of (2) and its derivative @zHx(kx;z), Hz(kx;z) =ik2 0Hx(kx;z)ikx@zHx(kx;z) k2x(1 +)k2 0; (3) Ey(kx;z) =! kx(iHx(kx;z) + (1 +)Hz(kx;z):(4) Since the PDMS as well as the GGG layer are assumed to be innite in positive and negative z-direction, there3 are no incoming waves in these regions, which could be caused by re ections at any surfaces. Thus, the ansatz HI x(kx;z) =C1eiaPDMSz; HII x(kx;z) =C2eiaYIGz+C3eiaYIGz; HIII x(kx;z) =C4eiaPDMSz;(5) is chosen, where the superscripts I-III refer to the regions andaicorresponds to ain (2) in the corresponding ma- teriali. In order to obtain the actual amplitude of the magnetic eld, the coecients C1-C4have to be derived, so there is need to include existing boundary conditions, which arise at the two interfaces. In the absence of any surface currents, the parallel component of the H-eld and the orthogonal component of the B-eld are continu- ous at an interface. But since this is only the case for the lower interface at z=d, the upper boundary condition for the parallel H-eld component has to be considered more carefully. The eld component Hxis not continu- ous atz= 0, where the step between both regions equals the current density at the boundary. Hence, the proper boundary condition at the I-II interface is HI x(kx;0)HII x(kx;0) =j(kx;y) (6) in thekxz-space. The current density function j(x;z) describes the total current I0 owing through the in-nitesimally thin MSL of width w, what can be ex- pressed as j(x;z) =I0=w#(w=2x)#(x+w=2)(z) and the corresponding Fourier transform is j(kx;z) = j0sin(kxw=2)=kxwithj0= (2=)1=2I0=w. Combining this discontinuity of the parallel H-eld at the upper in- terface with the known continuity condition at the lower interface and the continuity of the orthogonal B-eld at both interfaces provides four boundary conditions in to- tal. Inserting the ansatz (5) nally leads to a system of linear equations determining the coecients C1C4. As long as the interacting quantum system is located above the magnetic thin lm, we deal require the solu- tion for positive z-values, so that only the eld ampli- tude in region I is playing a role for any coupling pro- cesses. Hence, it is sucient to concentrate only on the coecientC1. The system of equations can be further simplied by introducing approximations relying on the realistic experimental values1. A typical saturation mag- netizationMSof the ferromagnetic material is of the or- der of 103G and is achieved in an external magnetic eld B0of102G. Hence, the assumption !2 0!2is jus- tied for microwave excitation frequencies !in the GHz range and susceptibility parameters andof the order of 101and 100. Further, the wave vectors of microwaves of aboutk0101m1in vacuum are much smaller than the experimental wave vectors kx105m1, i. e. k2 0k2 x. Under these conditions, the parameter ain (2) can be approximated independently from the material as aPDMSaGGGaYIGikx. Hence, the result for C1 only depends on kx, C1(kx)j0sin(kxw=2) kx!M(!0+!M+!)e2kxd(!0+!M!)(2!0+!M+ 2!) !2 M+e2kxd(4!2(2!0+!M)2): (7) The full solution for the magnetic eld Hin the region I is given by HI x HI z  1 i C1(kx)ekxz 1 i HI(kx;z)=p 2;(8) where the amplitude function HI(kx;z) is introduced. From equation (8) as a function of kxandz, the ex- cited spin wave modes inside the ferromagnetic lm, i. e. the spin wave resonance condition, can be derived by determining the zeroes of the denominator. This yields the dispersion relation !(kx) =1 2q (2!0!M)2!2 Me2kxd; (9) that matches exactly the so called Damon-Eshbach sur- face waves (DESW), which were calculated for ferromag- netic thin lms in the absence of any surface currents14.Hence, the solution of the modied system with non- zero current is peaked around the DESW modes, which is reasonable against the background of made approxima- tions. Assuming large wave numbers kx, i. e.kxk0, the current function j(kx;z)sin(kxw=2)=kxdecreases in amplitude, so that the result for the resonance con- dition is identical to the DESW modes within the given approximation. An important feature of the calculated spin wave modes is that for large kxthe frequency in (9) saturates to a value that depends on the external magnetic eld B0. Thus, at a xed magnetic eld, spin wave excita- tions are limited to a xed range of frequencies between the limiting cases kx= 0 andkx!1 , p !0(!0!M)!!0!M=2: (10) The lower bound at kx= 0 corresponds to the so called uniform precession mode, where all the spins inside the4 Frequency (GHz) Frequency (GHz) Frequency Figure 2. Magnetic eld amplitude depending on the microwave frequency !at xed external eld B0= 50 G. The eld amplitudes are normalized to the applied current per unit length I0=w. (a) The wide range plot shows the step behavior at the upper bound !max swof spin wave excitation. ( b) Magnication of the range around the step frequency indicated by the black rectangle in ( a), which resolves a sharp decrease in the amplitude at e!sw. (c) Further magnication shows a second amplitude dip at e! sw, below!max sw= 2:73 GHz. material precess in phase, so that there is an oscillating magnetization, but no spatial propagation. Even in a strong external magnetic eld of 200 G as used, for ex- ample, in the reported experiment1, the uniform preces- sion mode of the considered YIG lm oscillates at about 1:6 GHz, which is far too small to stimulate magnetic dipole transitions in the spin triplet of the NV center. In contrast, the upper limit !max sw=!0!M=2 is reached for the same magnetic eld at 3:1 GHz, which lies, for example, in the range of NV resonances, as will be im- portant later. After deriving the driven spin wave modes, we are in- terested in the real space solution of the magnetic eld, to nally model the interaction with an NV ensemble and to give a quantitative expectation of the coupling strength. Therefore, the magnetic eld HIin (8) has to be Fourier transformed back into real space using the explicit form of the coecient C1(kx) in (7). The eld can be written as HI(x;z) = 1 i HI(x;z)=p 2 (11)with the Fourier transform of the amplitude function HI(x;z) =1p 2Z1 1HI(kx;z)eikxxdkx: (12) In order to calculate this integral, the denominator in (7) is expanded to rst order in kxaround its zero k0 x, which corresponds to the resonant wave number dened by (9), k0 x=1 2dln !2 M 4!2(2!0+!M)2 : (13) The remaining integral can be analytically evaluated by assuming that before the current in the MSL was switched on there were no spin waves. This imposes the rule how to go around the pole in the integral above. Ap- plying the Sokhotski-Plemelj theorem in case of a real line integral and restricting ourselves to the far eld regime, where the condition x(kxk0 x)1holds, the amplitude functionHI(x;z) in real space is HI(x;z)ip 2I0 wsin(k0 xw=2) k0xd!M(!0+!M+!) 4!2(2!0+!M)2e2k0 xd+!0+!M! 2!0+!M2! ek0 xzeik0 xx; (14) which is one of our main results. Based on this com- plex eld amplitude, the magnetic eld above the ferro- magnetic lm is known explicitly. Note, however, that the calculated solution only holds for frequencies within the range (10). The assumptions, which were made in order to solve the integral above, are not valid outside this frequency range, where the existence of the solution (14) at frequencies above !max swis not given. Hence, the given solution (14) is only valid forp !0(!0!M) !!0!M=2 and is set to 0 otherwise, which corre- sponds to the absence of spin waves. Within this deni-tion range, the absolute value of the eld amplitude in (14) at xed magnetic eld depends on the excitation fre- quency as plotted in Fig. 2. The wide range plot in Fig. 2(a) shows the expected behavior. As discussed before, an upper bound at !max swcuts o the spin wave driving regime. Above this limit, the amplitude rapidly drops to a level at least six orders of magnitude smaller. Below this limit, the amplitude increases until the resonance frequency!swfor DESW modes in (9) is reached, which becomes apparent as a high peak just below the cut-o frequency. Although the increase of jHIjfrom low fre-5 Figure 3. (a) Possible orientation of NV centers inside the diamond structure. Depending on the alignment of the connection line between a substitutional nitrogen atom (green) and the neighboring vacancy (red), the NV-axis (red) can be oriented in the depicted four directions. ( b) The crystal structure of diamond leads to four dierent angles Bibetween the magnetic eld and the possible NV-axes ^ ci. (c) Resonance frequencies of the magnetic dipole transitions within the ground state triplet of an ensemble of NV centers embedded in single crystal diamond. Each of the four possible NV center orientations ^ cicontributes two resonances at !indicated by the dierent line styles. The resonances depend on the alignment of the magnetic eld. For the left plot in ( c) the magnetic eld is B= 50at an angle 'B= 30with respect to one of the crystal axes and on the right the chosen parameters are B= 15;'B= 50. quencies towards !max swin Fig. 2(a) seems to be smooth at the rst sight, the magnications in ( b) and (c) high- light a substructure of amplitude dips very close to the maximum. The occurring amplitude dips correspond to the minima of the amplitude function in (14) induced by the zeros of the factor sin( k0 xw=2)=k0 x. The position of the amplitude dip at e! swin (c) diers by only a few hundreds of Hz from the resonance maximum and, therefore, might not be resolved in usual resonance experiments. In con- trast, the second dip in ( b) is more incisive and occurs at a frequency a few MHz from the maximum. This second decrease in the eld amplitude should be experimentally resolvable as discussed later in section V. IV. MAGNETICALLY DRIVEN NV SPINS The spin wave propagation through the magnetic thin lm can increase the interaction between the external eld and a quantum system. As a consequence, the driv- ing of magnetic dipole transitions of NV centers in di- amond can be performed more eciently. In order to model the coupling strength and to simulate the result- ing transition spectrum, we have to distinguish whether a single NV center or an NV ensemble in single crystal diamond is probed. A. Single NV The ground state of the NV center is a spin triplet, which is split into the mS= 0 andmS=1 sublevels. This ground state energy splitting Dof 2:87 GHz3origi- nates from spin-spin interaction and the Hamiltonian of the spin triplet is ^H0=D^S2 z+gBB0
S; (15)where the second term accounts for the Zeeman split- ting with an external magnetic eld B0= (Bx;By;Bz). The spin vector S= (^Sx;^Sy;^Sz) contains the ( S= 1) spin operators ^Si, which are dened in the basis of the mS= 0;1;1 spin projections corresponding to the ^Sz eigenstatesfj0i;jiandj+ig. In cases where the mag- netic eld is not aligned with the NV-axis ^ c, the Hamil- tonian (15) can be written as ^H0=D^S2 z+gBB0(sinB^Sx+ cosB^Sz); (16) where the relative angle Bquanties the orientation of the external magnetic eld in the frame of the NV cen- ter. Since the Hamiltonian (16) depends explicitly on B, the according eigenenergies and, consequently, the frequencies ! Bof the dipole transitions j0i$ji vary for dierent orientations of the magnetic eld with re- spect to the NV-axis. Hence, the transition spectrum of a single NV center consists of two frequency branches ! B, whose behavior with respect to varying bias eld B0 is determined by the given value of B. B. NV ensemble in single crystal diamond In case of an NV ensemble every single of the numer- ous contained NV centers contributes two branches to the transition spectrum. If the single NV centers were ran- domly oriented, all branches would combine to a blurred spectrum. But due to the particular symmetry of the diamond crystal, the actual transition spectrum of an NV ensemble inside a single crystal consists of single branches. For a single crystal diamond, the lattice has a xed orientation in the lab frame, but inside the diamond structure the actual orientation of an NV center is not controllable during fabrication. Thus, the NV axis can6 be aligned along the four crystal directions shown in Fig. 3(a), which are at a tetrahedral angle of t= 109:5to each other. A single nanodiamond contains NV centers with orientations equally distributed over these four di- rections. Accordingly, for a xed B0-eld orientation, the magnetic eld is aligned at a dierent angle with respect to each of these four directions. Consequently, there are not numerous overlaying resonance branches, but each of the four possible NV-axis orientations gives rise to two resonances and in total there are eight resonances for an NV ensemble. In order to describe the expected magnetic resonances with a formula, the NV center orientation is expressed in terms of the relative angles Bibetween the pos- sible NV-axis and the magnetic eld as shown in the Fig. 3(b). Choosing one of the NV-axes, ^ c0, to de- ne thez-direction and one of the carbon atoms to lie on thex-axis, the normalized magnetic eld vector and the four NV-axes ^ ciare parameterized in spherical coordinates as ^B= (sinBcos'B;sinBcos'B;cosB) and ^ci= (sinicos'i;sinicos'i;cosi) with0= 0, 1;2;3=t,'0;1= 0 and'2;3= 2=3;4=3. The in- troduced coordinate system is thus xed to the crystal lattice. Based on these vectors, the angles Bibetween the single NV-axes and the magnetic eld in Fig. 3 are related to the corresponding scalar product as cosBi=^B
^ci= sinBsinicos('B'i)+cosBcosi: (17) Inserting the angle Biobtained from this expression into the Hamiltonian for non-aligned magnetic eld (16) and calculating the transition frequencies between the corre- sponding eigenenergies yields the spectra depending on the polar as well as the azimuth angle of the magnetic eld as plotted with respect to the strength of the applied magnetic eld in Fig. 3( c). A comparison of the two plots corresponding to dierent parameter sets ( B;'B) high- lights the sensitivity of the resonance spectrum of single crystal diamond hosting NV centers at various orienta- tions to rotations of the crystal in a magnetic eld. In that way, information about the spatial orientations can be extracted by exploiting the existing relation. From experimental data of the eight dierent magnetic resonance frequencies in a nanodiamond of known crystal orientation, the angles Band'Bcan be calculated or, in turn, from a known magnetic eld alignment, the crys- tal directions of a nanodiamond can be derived. Hence, the resonance experiment can be used to sense a probe magnetic eld resolving its direction on the one hand and to measure the crystal structure of a nanodiamond indi- rectly on the other hand.15,16 C. Spin dynamics In order to model the quantum mechanical coupling of an NV ensemble to the calculated spin wave eld (14), the corresponding Bloch equations are solved. Therefore,we use a ve level scheme similar to17, which includes the NV energy levels relevant for the processes in an ODMR experiment. The NV is pumped by a continuous laser, which excites the system from its ground state3A2triplet fj0i;j1igto the excited state3E tripletfje0i;je1ig. From the excited triplet states, the system partially de- cays optically back to the ground state manifold and the intensity of the emitted uorescent photons is detected in ODMR experiments. Without external microwave drive and in the presence of optical excitation the uorescence would just have a constant intensity and the system would be polarized in thems= 0. due to the inter-system crossing channel via the excited singlet1A1statejsi. The induced resonance microwave transitions within the ground state manifold pump the system out of ms= 0 state and in the pres- ence of optical excitation lead to a change of uorescence intensity. The intensity in fact drops because now the system spends more time in the jsistate, which is not optically active at the same photon frequency. The dynamics of this mechanism are described by the time-dependent ve-dimensional density matrix (t) of the system written in the basis of fj0i;j1i;je0i;je1i;jsig. Thereby, only the driven mS= 1 triplet sublevels are regarded, since we neglect transitions from the singlet to the spin state that is not driven by the microwave eld. When the transition into the mS=1 state is driven, we use the replacement j1i!j 1i. The density matrix is determined by solving the Master equation in Lindblad form, @t=i[;^H] +X >0 LLy 1 2Ly L1 2Ly L ; (18) where the operators Ldescribe a non-unitary time evo- lution due to dissipative interactions between the sys- tem and the environment, which is given by the pho- ton bath of the pumping laser and the emitted uo- rescence. For the special case of the ve level system, the Hamiltonian ^Hfor the closed system in (18) is de- composed into the static part and the interaction part, ^H=^H0+^HI. The static Hamiltonian ^H0is diagonal containing the eigenenergies "iof the ve states jiiwith i= 0;1;e0;e1;s. Writing the classical microwave eld as Bmw(t) =Bmwcos(!t), the interaction Hamiltonian is given by ^HI= R 2 ei!t+ei!t
(j1ih0j+j0ih1j); (19) with the Rabi frequency R= h0jBmw
Sj1idenot- ing the coupling strength due to magnetic dipole inter- action between the driving eld and the ground state triplet. Here, the spin vector S= 1=2can be repre- sented by the Pauli matrices = (x;y;z). The op- eratorsLin (18) represent the various dissipation pro- cesses as well as the continuous pumping, which excites the system from the ground state manifold to the excited state manifold at pumping rate p, where the mSquan-7 tum number is left unchanged, i. e. L0 p= 1=2 pje0ih0j andL1 p= 1=2 pje1ih1j. The inverse process, the direct decay back into the ground states, happens at rate 0 and the corresponding operators are L0 0= 1=2 0j0ihe0j andL1 0= 1=2 0j1ihe1j. For the inter-system crossing the coupling between the lower level je0iandjsiis much smaller than the coupling between the highest level je1i andjsi, therefore, the latter is neglected. Hence, the corresponding operator is Les= 1=2 esjsihe1j. The second inter-system crossing from jsito the ground state triplet j0iandj1iis modeled to have the same probabilities for ending up in the nal states j0iandj1iwith the operators L0 sg= (sg=2)1=2j0ihsjandL1 sg= (sg=2)1=2j1ihsj. Fur- ther, theT1-decay at rate 1= 1=T1from themS=1 andmS= 0 ground states into the equilibrium, which is assumed to lie at equally populated states is included. We also treat the transverse relaxation including de- phasing inside the ground state spin triplet at rate 2. The corresponding operators can be written in terms of spin operators as L1= ( 1=2)1=2(j0ih1j+j1ih0j) and L2= ( 2=2)1=2^z. Inserting these operators and the Hamiltonians ^H0and ^HIinto the Master equation (18) yields the time derivative of the density matrix in the form of 15 coupled rst order dierential equations with time-dependent coecients. Since some of these equa- tions are redundant, a system of only seven dierential equations actually has to be solved, where the rotating wave approximation is used. In the special case of con- tinuous pumping a single measurement of the ODMR contrast at xed driving frequency !is detected over a time long enough to allow the system to evolve into a dynamical equilibrium. Hence, the solution of interest is a steady where the time derivative in (18) can be set to zero. Finally, the matrix elements ijare obtained by solving the remaining system of linear equations. D. Optical detection The ODMR intensity Ican be calculated from the ob- tained solution for the density matrix . It is given by the number of emitted photons, when the excited states je0iandje1idecay optically to the ground state triplet. Due to the existence of the inter-system crossing channel, only a part of the population in je1icontributes to the in- tensity. In detail, only the fraction 0=(0+es) remains within the spin triplet channel. Treating the detected in- tensity to be proportional to these eective probabilities, it can be written as I/e0e0+0 0+ ese1e1: (20) For the steady state solutions, the normalized intensity depending on the detuning ="1!turns out to be Lorentzian, I() =2 2+e 2; (21)wheree describes the width of a resonance centered at = 0, which is simplied assuming the following. First, the decay rate from the excited state je1iinto the triplet ground state and the inter-system crossing channel are of the same order, i. e. es0. Second, the ground state triplet is not completely depopulated during the experiment, which requires a pumping rate much lower than the decay rate, p0, and theT1-decay being much slower than the leakage via the inter-system cross- ing channel, 1sg.17Under these conditions, the parametere takes the compact form e =s 2e 2
2+4e 2(1 + p=(4sg)) 1+ p=4 2 R; (22) with the eective transverse relaxation rate e 2= 2+ p+1=2. So far, we assumed degenerate mS=1 sub- levels. But since the NV ensemble in the considered setup is placed in a static bias magnetic eld, these sublevels are split by the corresponding Zeeman energy. Thus, if the NV ensemble is driven with a microwave eld at fre- quency!, each of the transitions j0i$j +iandj0i$ji will be excited with dierent detunings =!!. In addition, the resonance spectrum of a single nanodia- mond is a combination of the resonances originating from four dierent alignments of the NV-axes as discussed for an NV ensemble. To calculate the ODMR spectrum of an NV ensemble inside a nanodiamond with eight occurring resonances, we assume the eight expected resonances be- ing far o-resonant with each other. Hence, at i += 0 the detuning from the second resonance i is large enough and single transitions are excited separately. Therefore, the contributions I(i ) in (21) from each of the eight resonances are summed up to the intensity of ODMR uorescence, I(!) =X j=4X i=1(!i j!)2 (!i j!)2+ 2: (23) Due to the large magnetic eld amplitudes of the driving eld in the experiment, the line width e in (22) is dom- inated by the second term, which means that the lines can be assumed to be purely power broadened17. In the case of low pumping power, the condition p=4sgis fullled ande can be further approximated by e s 4 e 2 1+ p=4 R: (24) For a plot of the ODMR intensity I(!) in dependence of the bias eld strength, we assume realistic values for the relaxation times T1= 1=1andT2= 1=2of the NV center to lie in the range of ms18andµs19and a pumping rate p106s1, so that the line width is approximately 2 R. Based on the intensity (23) with the simplied line width (24) and using the spin wave eld amplitude (14)8 Figure 4. ( a) and (b): ODMR intensity Idepending on the magnetic eld and the driving frequency. The used parameters correspond to a nanodiamond at xNV= 5µm away from the antenna and a driving power of 0 :5µW. For the magnetic eld orientation the arbitrary values B= 14and'B= 7are chosen. In ( b), a zoom of the black boxed area in ( a) is shown, which highlights the substructure of the resonances right at the cut-o frequency !max sw. calculated in section III, a theoretical ODMR spectrum is plotted in Fig. 4. In the frequency regime above the cut-o of spin wave excitation, i. e. ! > !max sw, we use a microwave driving eld amplitude corresponding to a pure antenna eld Hantwithout spin wave excitation in- side the YIG lm. The parameters xNVandPentering the function for Hsw/ant are chosen to be equal to the experimental spec- trum in Ref. 1 (Supplementary), which are given by the distance between the nanodiamond and the antenna xNV5µm and a driving power of P= 0:5µW. The angle of the bias eld is chosen arbitrary to be B= 14 and'B= 7. The zoom of two resonances within the black boxed area in ( a) right at the maximal spin wave excitation frequency is given in Fig. 4( b). The more the driving frequency abroaches the cut-o frequency !max sw, the stronger the resonances are broadened, which results from the maximum of the amplitude function HIat the upper frequency limit. In addition, the high resolution in Fig. 4(b) also shows a sharp peak in the intensity ap- parent as a bright line parallel to the border between the driving regimes. The reason for this intensity increase is the substructure of the amplitude function in the spin wave driving regime and it corresponds to the rst dip in Fig. 2 at frequency e!sw. Between this peak and the cut- o frequency, the spin wave eld takes a maximal value right before it drops to zero. This becomes apparent by the dark line in Fig. 4( b) crossing the resonance branches from the bottem left to the top right and a similar dark line occurs in experimental ODMR spectra1. This con- sistency between the experiment and the theory plot sug- gests, that the amplitude function (14) describes the eld amplitude in the spin wave regime at a satisfying level of accuracy.V. ENHANCEMENT Due to the presence of spin waves inside the magnetic lm, the magnetic eld coupling to an NV ensemble is enhanced compared to the amplitude of a pure antenna eld. Moreover, the spin wave eld has the great ad- vantage of higher decay length, which becomes appar- ent when inspecting the explicit dependencies of the spin wave eld in (14). The lack of x-dependence of the eld amplitude indicates a very important dierence from a pure antenna driving eld. The microwave eld of the antenna decays as 1 =x, so that the coupling strongly de- pends on the position of the interacting NV center. In a system of defect spins in a quantum system addressed or read out with an oscillating magnetic eld, informa- tion can be transferred over a distance, which increases with the length over which the eld amplitude is con- served. Hence, the x-independent spin wave eld ampli- tude in (14) appears to lead to a constant coupling over a large distance. Realistically, the coupling decays due to the neglected exchange energy terms in the suscepti- bility tensor. Regarding the corresponding terms would give rise to a magnetic amplitude damping in (14) along thex-coordinate. But YIG has a particularly low damp- ing parameter and the excited spin waves propagate at lowk, thus justifying the made approximation. Along the z-direction, ~C1decays exponentially with the distance to the surface, i.e. the spin wave modes are conned to the surface of the thin lm. This guarantees a highly local- ized eld, so that the eect of spin wave enhancement is only present in the vicinity to the magnetic lm. Using the obtained result for the spin wave eld al- lows us to investigate the enhancement of the coupling strength Rwith respect to the case without spin wave excitation. The enhancement factor is dened as = sw R= ant R, where sw Rdenotes the Rabi frequency in the9 Figure 5. (a) Plot of the resonance frequencies of an NV center (black) depending on the bias eld. The NV-axis is oriented in thexy-plane at an angle of 78. The black dashed line represents the step between spin wave and antenna regime. The intersections of the driving frequency (blue/red dashed) with one of the branches indicates possible measurement settings. (b) Dependence of the eective enhancement eon thex-component of the coupling NV ensemble. The theoretical model corresponds to the red line and the data points represent the experimental results of Andrich et al. presented in Ref. 1. case of spin wave excitation and ant Ris the same quan- tity for the pure antenna driving without any back action originating from magnetic material. More specically we calculate the antenna eld without regarding the stray eld of the YIG, which is reasonable in the regime with- out spin wave excitation, since the thin lm is located in an approximately homogeneous magnetic eld and due to its small height the in uence on the antenna eld be- comes negligible. Both quantities, the spin wave and the antenna eld, depend on the angle between the respec- tive eld and the NV center, which is non-zero in the general case. In a coordinate system S0withz0-axis parallel to the NV-axis ^ cand the antenna eld lying in the x0z0- plane, the magnetic elds of antenna and spin waves are parametrized as Bant=Bant(sinant;0;cosant) andBsw=Bsw(sinswcos'sw;sinswsin'sw;cossw), whereant/sw denotes the angle with respect to the ^ c- axis and the spin wave eld is at the angle 'swwith respect to the x0z0-plane. The Rabi frequency scales with the o-diagonal terms in Bsw/ant
S, so that sw R/ Bswsinswexp(i'sw) and analogously for the antenna driving eld. Inserting this into the enhancement ratio yields =Hsw Hantsinsw sinantei'sw: (25) Since the phase factor ei'swdoes not aect the absolute value of this enhancement, it is omitted in the following. Furthermore, we want to adapt our result to the exper- imental method described in1, where the resonance be- tween the same levels is measured twice in the dierent driving regimes, the antenna driving and the spin wave driving regime. To measure both regimes at the samefrequency, a resonance has to lie once above the maximal excitation frequency of spin waves and once below, so that the dierent regimes can be tuned by the bias mag- netic eld. For the realization of such a measurement, the orientation of the bias eld has to be non-collinear to the NV-axis, whereby the states j0iandjibecome coupled and the energy splitting does not depend linearly on the magnetic eld. Thus, the resonance branches are tilted as plotted in Fig. 5( a). The plot shows the NV resonances in the case of the NV-axis being aligned in thexy-plane at an angle of B= 78with respect to the bias magnetic eld. In Fig. 5( a) only six resonances are visible, because the magnetic eld encloses the same angle with two of the possible crystal directions lead- ing to these resonances being two-fold degenerate. An excitation frequency of 2 :86 GHz is on resonance with a bias eldB0antin the antenna driving regime, which is marked by the blue dot in Fig. 5( a). A second resonance (red dot) occurs in the regime of spin wave excitation at B0sw. For an optimal resolution of the detection in both regimes, the antenna current, i. e. the driving power Pis varied. The antenna eld decays rapidly with the distance of the NV center from the antenna, so that a driving power of the order of mW is required to obtain a clear signal. If the same driving power was used in the spin wave driving regime, the resonance would be signicantly power broadened and hence, a lower power at several µW is chosen. Hence, the enhancement ratio has to be corrected by the impact of varying driving power. Both, the antenna eld amplitude Hantand the spin wave eld amplitude Hswdepend linearly on the driving current I0= (2P=Z)1=2, which is an eective ac current through a wire with impedance Z. Finally, the eective enhancement ratio is obtained by normalizing the elds with respect to the driving power Pant/sw ,10 e=r Pant PswHsw Hantsinsw sinant= 4p 2r Pant Pswsin(k0 xw=2) k0x !M(!0+!M+!) 4!2(2!0+!M)2e2k0 xd+!0+!M! 2!0+!M2! ek0 xz lnw 2x
2=w 2+x
2sinsw sinant:(26) With this expression, we are able to make a theoret- ical prediction for the enhancement for an NV center located at xNV. Since the far eld approximation was made, the condition xk1has to be fullled and for wave numbers of the excited spin waves1of the order of105m1, the expression (26) applies for distances above10µm. For example, the resonances of a nan- odiamond at xNV= 20 µm driven by a eld at 2 :862 GHz occur in the antenna regime at 15 G and in the spin wave driving regime at 145 G. Using these parameters, the enhancement factor of the coupling between the defect spin and the driving eld is e117, which is close to the experimentally measured enhancement e100 de- tected via Rabi experiments1. Moreover, when distance xin expression (26) is varied, the enhancement factor e behaves as plotted in Fig. 5( b), along with the experi- mental data of Andrich et al.1. The theoretical prediction matches the experimental behavior, although the exper- imental data are slightly shifted to lower enhancement values with respect to the predicted theory curve. Nev- ertheless, the linear x-dependence of enot only shows good agreement with the trend of experimental data, but also an achievable enhancement factor of more than 400 at distances above 70 m which underscores the great ad- vantage of spin wave excitation inside the magnetic lm. For low magnetic damping the spin waves do not decay as fast as the microwave eld leading to a large enhance- ment far away from the antenna. VI. CONCLUSION We have presented a theoretical model that provides a complete description of real experimental hybrid systems consisting of driven spin waves coupled to NV spins. All results are obtained analytically and especially the de- scription of spin wave modes follows a fundamental ap- proach from the basics of Maxwell's equations. In that way, we derive an analytical solution for the magnetic eld amplitude, which allows for a theoretical calcula- tion of the coupling to an NV ensemble. Therefore, we regard every possible spatial orientation of an NV center in a real nano-diamond, thus obtaining the full spectrum capturing all eight resonances. Our theoretical intensity plots show good agreement with existing experimental data1, what clearly highlights the suitability of the de- veloped model. The existence of the substructure and an upper frequency bound limiting the spin wave driv- ing regime are important features, which follow from the theoretical calculations. They provide an accurate but at the same time very applicable method of tuning the drive of spin rotations of an NV center, where the resonant cou-pling below the cut-o frequency in the spin wave driving regime are calculated to be much stronger with respect to the case of pure antenna driving. This dierence in the coupling strength is a powerful tool to speed up qubit operations in the eld of quantum computing. The de- rived enhancement factor eis strongly dependent on the distance between the source of the microwave eld and the NV ensemble. Accordingly, the magnetic eld at the NV spin can be enhanced by two orders of magni- tude and the predicted behavior for longer distances is in good agreement with recent experiments1indicating the strength of our model as well as the applicability of the related assumptions and approximations. The presented solutions are obtained using the far eld approximation, which holds for distances above 10µm. This regime is of particular interest for technical realizations, since it is a general problem to couple a sensitive quantum system to a microwave antenna across distances of more than a fewµm in view of the rapidly decaying antenna eld. In contrast, we show that the spin wave eld is very stable across the distance enabling a coupling to NV centers far away from the antenna by using a magnetic thin lm. If the enhancing eect over a long distance is of partic- ular interest the made approximations are reasonable and the model provides good results. But in case of a mod- ied system operating at a dierent conditions, possibly the limits of our theoretical model could be reached. If a nanodiamond is positioned very close to the antenna, there is need of a solution outside the far eld regime, which is applicable for short distances. An exemplary real system would be an array of nanodiamonds, where the antenna is not grown besides but in between the ar- ray, so that the NV centers are partly in the near eld and partly in the far eld regime. Strictly speaking, this would require a mid eld solution as well. Further unconsidered eects, which had to be taken into account in order to optimize the presented model, are additional boundary eects caused by the sourround- ing material being non-perfect magnetic vaccum or spin wave re ections in case of the lm having nite expan- sion in the xy-plane. A renement of the model con- sidering those aspects would lead to important insights with respect to new applications of a spin wave mediated coupling. Since the spectrum of the spin waves is mainly determined by the shape of the ferromagnetic sample, the driven spin wave modes could be adapted by shap- ing, for example, a ferromagnetic sphere or a disk. In this way, the technique of the experiment of Andrich et al.1 could be transferred to various magnetic resonance ex- periments by changing the driven spin wave modes to lie on resonance with other defect spins like dierent color centers in diamond or vacancies in silicon carbide. 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2018-10-28

In search of two level quantum systems that implement a qubit, the nitrogen-vacancy (NV) center in diamond has been intensively studied for years. Despite favorable properties such as remarkable defect spin coherence times, the addressability of NV centers raises some technical issues. The coupling of a single NV center to an external driving field is limited to short distances, since an efficient coupling requires the NV to be separated by only a few microns away from the source. As a way to overcome this problem, an enhancement of coherent coupling between NV centers and a microwave field has recently been experimentally demonstrated using spin waves propagating in an adjacent yttrium iron garnet (YIG) film [1]. In this paper we analyze the optically detected magnetic resonance spectra that arise when an NV center is placed on top of a YIG film for a geometry similar to the one in the experiment. We analytically calculate the oscillating magnetic field of the spin wave on top of the YIG surface to determine the coupling of spin waves to the NV center. We compare this coupling to the case when the spin waves are absent and the NV center is driven only with the antenna field and show that the calculated coupling enhancement is dramatic and agrees well with the one obtained in the recent experiment.

Magnetic Resonance in Defect Spins mediated by Spin Waves

1810.11841v1

arXiv:1509.04018v1 [cond-mat.mtrl-sci] 14 Sep 2015Spectral characteristics of time resolved magnonic spin Se ebeck effect S. R. Etesami, L. Chotorlishvili, and J. Berakdar Institut f¨ ur Physik, Martin-Luther-Universit¨ at Halle- Wittenberg, 06099 Halle, Germany (Dated: 15 July 2021) Spin Seebeck effect (SSE) holds promise for new spintronic devices w ith low-energy consumption. The un- derlying physics, essential for a further progress, is yet to be fu lly clarified. This study of the time resolved longitudinal SSE in the magnetic insulator yttrium iron garnet (YIG) c oncludes that a substantial contri- bution to the spin current stems from small wave-vector subther mal exchange magnons. Our finding is in line with the recent experiment by S. R. Boona and J. P. Heremans, P hys. Rev. B 90, 064421 (2014). Technically, the spin-current dynamics is treated based on the Lan dau-Lifshitz-Gilbert (LLG) equation also including magnons back-action on thermal bath, while the formation of the time dependent thermal gradient is described self-consistently via the heat equation coupled to the m agnetization dynamics. Introduction. The spin counterpart of the Seebeck effect, the spin-Seebeck effect (SSE) refers to the emer- gence of a spin current upon applying a thermal bias. Aside fundamental interest, SSE is of relevance for a variety of applications, including spin-dependent ther- moelectric devices1. Since its first observation2, SSE hasbeenstudied extensivelybothexperimentally3–15and theoretically16–23and for a variety of systems. In partic- ular, SSE in ferromagnetic (FM) insulators4hints on a magnonic origin of the spin current5. Magnonic SSE has some advantages with respect to charge-carrier-related spin current in that magnon spin current propagatesover a length scale of up to a millimeter24while conduction- based spin current is usually much smaller due to spin- dependent scattering. The spectral characteristics of the magnons contributing to the magnonic SSE are still un- der debate. In a recent spatially resolved experiment6on the magnetic insulator yttrium iron garnet (YIG), the measured magnon temperature Tmwas related to the short wavelength exchange part of the magnon spectrum ω/parenleftbig/vectork/parenrightbig . An important observation is that a non-vanishing spin current emerges even for equal magnon and phonon temperatures Tm=Tph. Note that the standard narra- tive attributes the emergence of the magnonic spin cur- rent to the difference between magnon and phonon tem- peratures. A theoretical explanation to the observed fact was given in terms of the long-wavelength dipolar part of the magnon spectrum25. If long-wavelength dipolar magnons are weakly coupled to the phonons their life time is largerand the magnon temperature deviates from the phonon temperature. Thus, dipolar magnons might contribute to SSE. This explanation though comprehen- sible, does not exclude the contribution of the long- wavelength exchange subthermal magnons in the SSE. As was shown in a recent theoretical paper17constitu- tive issuein the formationofthe magnonicSSE is not the differencebetweenmagnonandphonontemperaturesbut rather the nonuniform magnon temperature profile lead- ing to a nonzero exchange spin torque and the magnon accumulation effect that drives the magnonic SSE. In the present paper we study the contribution of the long- wavelength (small wave-vector /vectork) exchange subthermalmagnons to the magnonic SSE. We find that contrary to the short-wavelength exchange thermal magnons, the long-wavelength exchange subthermal magnons do con- tribute substantially to the SSE. Our predictions are consistent with the recent experimental report by S. R. Boona and J. P. Heremans8. Magnon modes at thermal energies in YIG were found not responsible for the spin Seebeck effect. Subthermal magnons, i.e., those at ener- gies below about 30 ±10 [K], were found important for the spin transport in YIG at all temperatures. To assess the partial contribution to SSE of magnons with differ- ent frequencies ω/parenleftbig/vectork/parenrightbig we analyze the time resolved SSE adopting a low-pass filter, as done experimentally9. In time resolved SSE experiments the laser modulation fre- quency is the relevant control parameter. If it is smaller compared to the cutoff frequency of the filter then the filter does not affect the spin current, otherwise the low- pass filter cuts the spin current (external cutoff). Any cutoff observed for modulation frequencies smaller than cutoff frequency of the filter is thus intrinsic. In our sim- ulation we implemented this approach, which aside from mimicking the experimental situation bears some advan- tages against a discrete Fourier transform to obtain the spectral dependence of the spin current, as detailed in the supplementary materials26. We note, that while our study is limited to thermally induced magnonic transport in FM insulators, it can be in principle extended to include contributions from thermally activated carriers in metals or semiconductors. The method would rely however on further reasonable inputs such as the details of the spin torque current and the rescaled exchange interaction parameters. Theoretical background . In a FM insulator, the low-lying excitations are spin waves describable by Landau-Lifshitz-Gilbert (LLG) equation (for a compar- ison an implementation based on the Landau-Lifshitz- Miyasaki-Seki scheme has also been performed with full details and results being included in the Supplementary2 Materials26) ∂ ∂t/vectorM(/vector r,t) =−γ/vectorM(/vector r,t)×/bracketleftbigg H0ˆz+2A M2 S∇2/vectorM(/vector r,t)/bracketrightbigg +α MS/vectorM(/vector r,t)×∂ ∂t/vectorM(/vector r,t),(1) where/vectorM(/vector r,t) stands for the magnetization vector, γ is the gyromagnetic ratio, H0is the external mag- netic field, Ais the exchange stiffness, MSis the saturation magnetization and αrefers to the Gilbert damping constant. The LLG equation in the linear limit has the following solution Mx(/vector r,t) +iMy(/vector r,t)∝ exp/parenleftBig i/vectork·/vector r+iωkt/parenrightBig exp(−αωkt). The spin wave disper- sion relation ωk=γ/parenleftBig H0+2A MSk2/parenrightBig and the damping which depends on the wave vector αωk,5indicates that short wavevector /vectork(long wavelength)exchangemagnons are less damped. Thus magnons have different relaxation times. Assuming that the magnon-phonon scattering is the main source of damping, the magnon-phonon relax- ation (thermalization) time reads τk mp=/bracketleftbigg αγ/parenleftbigg H0+2A MSk2/parenrightbigg/bracketrightbigg−1 ,k= 2nπ/d,n= 0,±1,··· (2) wheredis the length of the FM insulator. Let us as- sume that the left edge of the FM insulator is heated up periodically with the modulation frequency ωmod(see Fig. 1). A time periodic thermal bias is meant to mimic qualitatively the action of laser pulses (in real exper- iment the laser electromagnetic energy is absorbed by the sample only partly, however. More details are found in the Supplementary Materials26). The inherent ther- mal loses relevant to the experiment are treated self- consistently by adopting an additional source term in the heat equation (see Eq. (SM-3) in the Supplemen- tary materials26). Thus, the temperature profile imple- mented in the LLG equation in our case is calculated self-consistently. For unraveling the role of magnons with different frequencies we follow the idea of using a low-pass filter, i.e., a filter with an extrinsic cutoff fre- quencyωcwhich detruncates the spin current if the mod- ulation frequency of the laser pulses exceeds the extrin- sic cutoff frequency of the filter ωmod> ωc. The ratio between ”true” I0and measured I/parenleftbig ωmod/parenrightbig spin current readsI/parenleftbig ωmod/parenrightbig =I0//radicalBig 1+/parenleftbig ωmod/ωc/parenrightbig2. Ifωmod< ωc, the measured spin current is not altered by the filter. The decay in the spin current in this case is ascribed to the intrinsic cutoff frequencies, which in turn are re- latedtothemagnon-phonon-relaxation-time(seeEq.(2)) Ωk mp= 2π/τk mp. In this way different internal cutoff fre- quencies can be observed. Model and simulations. We model a ferromagnetic insulator via a chain of FM cells arranged along the ˆ x axis (Fig. 1). The total energy density of the system of FIG. 1. Schematic of the contribution of the spin waves with different wave vectors kito the spin current in SSE. The temperature of the left edge of the system is varied in time: T0(t) =T00S(ωmodt), where S(ωmodt) is a rectangular pulse with a modulation frequency ωmodand levels 0 and 1. Tnand Mnrepresent the temperature and magnetization in each in- dividual cell, respectively, and are calculated self-cons istently via the heat and LLG equations (Eqs. (4) and (6)). Ncells reads: e=−H0N/summationdisplay n=1Mz n−2A a2M2 SN/summationdisplay n=1/vectorMn·/vectorMn+1,(3) whereais the size of the cell, /vectorMnis the magnetiza- tion vector of nthcubic cell and Ais the exchange stiff- ness. WeuseEq.(3)tomodelYttrium-iron-garnet(YIG) whichhasbeenemployedextensivelyinSSEexperiments. The effective magnetic field acting on the nthcell reads /vectorHeff n(t) =−∂e ∂/vectorMn=H0ˆz+2A a2M2 S/parenleftBig /vectorMn+1+/vectorMn−1/parenrightBig . A Gaussian-white noise /vector ηn(t) contribution to the effective magnetic field (with a correlation function ∝angb∇acketleftηni(t)ηmj(t+ ∆t)∝angb∇acket∇ight=2αKBTn(t) γMSa3δnmδijδ(∆t)) accounts for thermal fluctuations. Here, ∝angb∇acketleft···∝angb∇acket∇ightmeans average over different realization of the noise, nandmare cell num- bers and iandjarethe Cartesiancomponents. The time and site-dependent temperature Tn(t) obeys the follow- ing heat equation d dtTn(t) =κ ρCTn+1(t)−2Tn(t)+Tn+1(t) a2,(4) with the initial and the boundary conditions Tn(t= 0) = 0; n= 0,···N+1 T0(t) =T00S(ωmodt),TN+1(t) = 0.(5) κisthe phononicthermalconductivity, ρisthemassden- sity,Cis the phonon heat capacity, T00is the tempera- ture applied on the left edge and Sis a seriesof rectangu- lar laser pulses with the modulation frequency ωmod(see Fig. 1). For solving the heat equation we implemented a Forward-Time Central-Space (FTCS) scheme27. The hierarchy of the relaxation times for phonons τph=/bracketleftBig κ ρC/parenleftbigπ Na/parenrightbig2/bracketrightBig−1 ≈10[ns] (the number and size of cells N= 50,a= 10[nm] and phonon thermal conductivity κ= 6[W.m−1.K−1]) and magnons τmp=1 2αω0≈103[ns]3 0 10 20 30 40 50012345 Time/LBracket1ns/RBracket1Spincurrent1011/MultiΠly/LBracket1/HBars/Minus1/RBracket1Ωc/LBracket1GHz/RBracket1 /Infinity 2Π/MultiΠly10/Minus0.0 2Π/MultiΠly10/Minus0.6 2Π/MultiΠly10/Minus1.0 2Π/MultiΠly10/Minus1.4 FIG. 2. Spin current at the middle of a chain of 50 FM cells versus time and for different extrinsic cutoff frequencies ( ωc). We choose a= 10[nm], H0= 0.057[T],T00= 10[K], α= 0.1. The system is heated up periodically with the modulation frequencies of ωmod= 2π×10−1.0[GHz] and the spin current is statistically averaged over 1000 realization of the nois e. For the blue curve no cutoff frequency is implemented on the spin current ( ωc=∞) but for the red, green, orange and black curves the cutoff frequencies are ωc= 2π×100.0[GHz],ωc= 2π×10−0.6[GHz],ωc= 2π×10−1.0[GHz] and ωc= 2π× 10−1.4[GHz], respectively. (ferromagnetic resonance frequency and phenomenologi- cal damping constant ω0= 10 [Ghz], α≈10−4) allows an adiabatic decoupling which amounts, to a first order, to plug the obtained phonon temperature profile directly in the LLG equation and study so the magnetization dy- namics self-consistently. The magnetization dynamics is governed by a set of coupled LLG equations ∂ ∂t/vectorMn(t) = −γ 1+α2/vectorMn(t)×/bracketleftbigg /vectorHeff n(t)+α MS/vectorMn(t)×/vectorHeff n(t)/bracketrightbigg . (6) For the numerical integration of the coupled stochastic differential equations we utilize the Heun’s method28,29. The spin current tensor is calculated using the following TABLE I. Magnon-phonon relaxation times and the cor- responding frequencies according to Eq. (2) for N= 50, a= 10[nm], H0= 0.057[T] and α= 0.1. n 012345 |k|= 2πn/Na[108m−1]0.000.130.250.380.500.63 τk mp[ns] 1.0000.3030.0980.0460.0260.017 Ωk mp/2π[GHz] 1.03.310.221.637.758.4Ωc/LBracket1Hz/RBracket1/Slash12Π 6.3/MultiΠly107 1.0/MultiΠly108 1.6/MultiΠly108 2.5/MultiΠly108 4.0/MultiΠly108 6.3/MultiΠly108 1.0/MultiΠly109 1.6/MultiΠly109 2.5/MultiΠly109 4.0/MultiΠly109 6.3/MultiΠly109 1.0/MultiΠly1010 1.6/MultiΠly1010 2.5/MultiΠly1010 4.0/MultiΠly1010 /VertEllipsis2 1.0/MultiΠly1012 1071081091010101110120.60.650.70.750.80.850.90.951 Ωmod/LBracket1Hz/RBracket1/Slash12ΠNormalized spincurrent FIG. 3. Normalized spin current (Iωmod I2π×107[Hz]) at the middle of a chain of 50 FM cells versus the modulation frequency for different extrinsic cutoff frequencies ( ωc) with the parameters a= 10[nm], H0= 0.057[T],T00= 10[K], α= 0.1. The spin current is statistically averaged over 1000 noise realizat ions. Forωc<2π×109[Hz] the cascades follow the extrinsic cutoff frequencies which are characteristic of Low-Pass filter. Ho w- ever, for ωc≥2π×109[Hz] the cascades occur earlier than the corresponding extrinsic cutoff which are a sign of inhere nt intrinsic cutoff in the system. The arrows show the magnon- phonon frequencies (Ωk mp/2πin TABLE I) for different wave vectors evaluated theoretically (Eq. (2)) and coinciding w ith the appearance of intrinsic cutoff frequencies (cascades) i n the curves. formula: Iα n=−2Aa M2 S/summationtextn m=1∝angb∇acketleftMβ m(Mγ m−1+Mγ m+1)∝angb∇acket∇ightεαβγ whereα=x,y,zdefines the spin components of the tensor, while nstands for the cite number, εαβγis the Levi-Civita antisymmetric tensor and ∝angb∇acketleft···∝angb∇acket∇ightmeans aver- aging over the different realization of the noise5,17,19,22. In our model, because of the particular geometry of the system (1D chain aligned along ˆ xaxis) the only non-zero element of the spin current tensor is Iz n17. For the out- put signal we implemented a recursive low-pass filter27 Iout(t) =ωc∆t 1+ωc∆tIin(t) +1 1+ωc∆tIout(t−∆t). Here Iin andIoutare the spin currents before and after filtering procedure and ωcis the extrinsic cutoff frequency of the filter (see Supplementary Materials26). Results and discussion. Fig. 2 shows the spin See- beck current as a function of time for different extrinsic cutoff frequencies ωc. To each cutoff frequency a cer- tain color is attributed. The filter with the extrinsic cut- off frequency ωccuts the spin current if the modulation frequency of the laser pulses ωmodexceeds the extrinsic cutoffωmod> ωc. Thus, the larger the extrinsic cutoff of4 the filter ωc, the larger is the spin current. This is what we see in Fig. 2. For small extrinsic cutoff frequencies (see Fig. 3) ωc<2π×109[Hz] the spin current is detrun- cated extrinsically at modulation frequencies ωmod=ωc smaller than the first intrinsic cutoff frequency of the system Ωk=0 mp= 2π/τk=0 mp. Therefore, no inherent cutoff is observed in this case. However, for an elevated extrinsic cutoffωcwe observe a cascade of the inherent intrinsic cutoff at the frequencies Ωk mp= 2π/τk mp< ωc. All these intrinsic cutoff frequencies are in the subthermal regime of the magnon spectrum k < kmax≈108[m−1]6,8(TA- BLE I). To be confident while heating up the system the thermal magnons are also activated, the corresponding dispersion relation to our parameters is shown in Fig. 4. As can be seen, magnons with a broad range of frequen- cies, beyond subthermal regime are also activated. The subthermal regime of the spectrum is shown with a small green frame6,8. Γ/LParen1H0/Plus2A MS2/LParen11/MinusCos/LParen1ka/RParen1/RParen1 a2/RParen1 /Minus3/Minus2/Minus1 0 1 2 30123456 k/LBracket1108m/Minus1/RBracket1Ω/LBracket11012Hz/RBracket1 Subthermal regime FIG. 4. Spin-wave dispersion relation: The yellow back- ground shows the absolute value of the discrete fourier tran s- formation of mx+imybased on micromagnetic simulations33 for achain of 50FM cells underalinear temperature gradient . T1= 0,TN= 10 [K], a= 10[nm], H0= 0.057[T]. For small k, the dispersion relation reduces to ωk=γ/parenleftBig H0+2A MSk2/parenrightBig . The small green frame shows the subthermal regime of the spectrum. To ensure that our findings are not an artefact of a particular choice of parameters/model we performed the calculations for various temperature gradients, different applied external magnetic fields, and varying lengths of the chain (Fig. 5). In all these cases the intrinsic cut- off frequencies follow the corresponding magnon-phonon frequencies (Eq. (2)). Furthermore, the use of a white noise for a swift time-dependent heating might be ques- tioned. Therefore, we implemented the Landau-Lifshitz- Miyasaki-Seki scheme31to account for the back-action of the magnon subsystem to the surrounding phonon bath (see Supplementary Materials26) and arrived basi- callyatthesameconclusionthatsmall-wavevectors /vectorkex- change subthermal magnons contribute substantially totheformationofthermallyactivatedspin current. Hence, our finings are expected to be of some generalities for magnon-driven SSE and associated devices, for the em- ployed schemes are quite ubiquitous, had proven to be reliable for finite temperature spin dynamics, and the employed system parameters are generic. Acknowledgements. We thank K. Zakeri Lori, Y.-J. Chen and A. Sukhov for valuable discussions. Ωc/LBracket1Hz/RBracket1/Slash12Π 6.3/MultiΠly107 1.0/MultiΠly108 1.6/MultiΠly108 2.5/MultiΠly108 4.0/MultiΠly108 6.3/MultiΠly108 1.0/MultiΠly109 /VertEllipsis2 1.0/MultiΠly1012 1071081091010101110120.80.850.90.9511.05Normalized spincurrent/LParen1a/RParen1 N/Equal50 a/Equal10/LBracket1nm/RBracket1 H0/Equal0.057/LBracket1T/RBracket1 T00/Equal40/LBracket1K/RBracket1 R/Equal2000 Ωc/LBracket1Hz/RBracket1/Slash12Π 6.3/MultiΠly107 1.0/MultiΠly108 1.6/MultiΠly108 2.5/MultiΠly108 4.0/MultiΠly108 6.3/MultiΠly108 1.0/MultiΠly109 /VertEllipsis2 1.0/MultiΠly1012 1071081091010101110120.80.850.90.9511.05Normalized spincurrent/LParen1b/RParen1 N/Equal50 a/Equal10/LBracket1nm/RBracket1 H0/Equal0.57/LBracket1T/RBracket1 T00/Equal10/LBracket1K/RBracket1 R/Equal1000 Ωc/LBracket1Hz/RBracket1/Slash12Π 6.3/MultiΠly107 1.0/MultiΠly108 1.6/MultiΠly108 2.5/MultiΠly108 4.0/MultiΠly108 6.3/MultiΠly108 1.0/MultiΠly109 /VertEllipsis2 1.0/MultiΠly1012 1071081091010101110120.80.850.90.9511.05 Ωmod/LBracket1Hz/RBracket1/Slash12ΠNormalized spincurrent/LParen1c/RParen1 N/Equal100 a/Equal10/LBracket1nm/RBracket1 H0/Equal0.057/LBracket1T/RBracket1 T00/Equal20/LBracket1K/RBracket1 R/Equal1500 FIG. 5. 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2015-09-14

Spin Seebeck effect (SSE) holds promise for new spintronic devices with low-energy consumption. The underlying physics, essential for a further progress, is yet to be fully clarified. This study of the time resolved longitudinal SSE in the magnetic insulator yttrium iron garnet (YIG) concludes that a substantial contribution to the spin current stems from small wave-vector subthermal exchange magnons. Our finding is in line with the recent experiment by S. R. Boona and J. P. Heremans, Phys. Rev. B 90, 064421 (2014). Technically, the spin-current dynamics is treated based on the Landau-Lifshitz-Gilbert (LLG) equation also including magnons back-action on thermal bath, while the formation of the time dependent thermal gradient is described self-consistently via the heat equation coupled to the magnetization dynamics

Spectral characteristics of time resolved magnonic spin Seebeck effect

1509.04018v1

1 Crystal orientation dependent spin pumping in Bi 0.1Y2.9Fe5O12/Pt interface Ganesh Gurjar1,2,*, Vinay Sharma3,4, Avirup De5, Sunil Nair5, S. Patnaik1,*, Bijoy K. Kuanr4,* 1School of Physical Sciences, Jawaharlal Nehru University, New Delhi, INDIA 110067 2Department of Physics, Ramjas College, University of Delhi, New Delhi, INDIA 1100 07 3Departme nt of Physics, Morgan State University, Baltimore, MD, USA 21251 4Special Centre for Nanosciences, Jawaharlal Nehru Univ ersity, New Delhi, INDIA 110067 5Department of Physics, Indian Institute of Science Education and Research, Pune, INDIA 411007 Abstract Ferromagnetic resonance (FMR) based spin pumping is a versatile tool to quantify the spin mixing conductance and spin to charge conversion (S2CC) efficiency of ferromagnet/normal metal (FM/NM) heterostructure. The spin mixing conductance of FM/NM interfac e can also be tuned by the crystal orientation symmetry of epitaxial FM. In this work, we study the S2CC in epitaxial Bismuth substituted Yttrium Iron Garnet (Bi0.1Y2.9Fe5O12) thin films Bi-YIG ( 100 nm ) interface d with heavy metal platinum ( Pt (8 nm )) depo sited by pulsed laser deposition process on differen t crystal orientation Gd3Ga5O12 (GGG) substrates i.e. [100] and [111] . The crystal structure and surface roughness characterized by X -Ray diffraction and atomic force microscopy measurements establish epitaxial Bi -YIG[100], Bi -YIG[111] orientations and atomically flat surfaces respectively . The S2CC quantification has been realized by two complimentary techniques , (i) FMR -based spin pumping and inverse spin Hall effect (ISHE) at GHz frequency and (ii) temperature dependent spin Seebeck measurements. FMR -ISHE results demonstrate that the [111] oriented Bi-YIG/Pt sample shows significantly higher values of spin mixing conductance ((2.31±0.23) 1018 m-2) and spin Hall angle (0.01±0.001) as compared to the [10 0] oriented Bi - YIG/Pt . A longitudinal spin Seebeck measurement reveals that the [111] oriented sample has higher spin Seebeck coefficient (106.40±10 nV mm-1 K-1). Th is anisotropic nature of spin mixing 2 conductance and spin Seebeck coefficient in [111] and [100] orientation has been discussed using the magnetic environment elongation along the surface normal or parallel to the growth direction. Our results aid in understanding the role of crystal orientation symmetry in S2CC based spintronics devices. Keyw ords: Bi-YIG thin films, lattice mismatch, Pulsed Laser Deposition, ferromagnetic resonance, Gilbert damping, inhomogeneous broadening *Corresponding authors: ganeshgurjar4991@gmail.com, spatnaik@mail.jnu.ac.in , bijoykuanr@mail.jnu.ac.in 1. Introduction Spin pumping is defined as the transfer of electron spins from a ferromagnet (FM) to a normal metal (NM) in an FM/NM junction under magnetic precession [1–3]. In general, a time varying magnetization pumps a pure spin current into a NM contact and this idea of spin pumping has been realized by numerous experimental techniques in which excitation sources are varied from GHz to THz time scales [4–6]. This is also observed with temperature gradien t dependent spin Seebeck effect [7–9]. The efficiency of spin pumping relates to spin angular momentum transfer [10] and spin mixing conductance [11,12] . The nature of this efficiency is strongly affected by interface characteristics of NM in contact with either ferromagnetic metal (FM) [13] or ferrimagnetic insulator (FMI) [14]. A great deal of theoretical and experimental research has been focused on how to actuate, detect, and regulate the magnetization and spin currents in these systems [15]. One of the most important magnetic materials for spin pum ping is Bismuth substituted Yttrium Iron Garnet ( Bi-YIG, BixY3-xFe5O12). This is primarily due to its long spin -wave propagation length and low Gilbert damping [16–22]. Several experim ental studies on spin 3 pumping have been reported using the Bi-YIG/Pt system [8,23 –26] because of high spin orbit coupling (SOC) in Pt. The recent theoretical and experimental works on this subject predict that the spin pumping efficiency strongly depends on the interface cut and orientation of NM and FM structures [27–31]. This anisotropy has been explained on the basis of local magnetic moment of magnetic ions and generation of crystal field symmetry due to broken rotational symmetry of magnetic atoms [27]. Motivated by the se results, we investigate the role of crystal field symmetry in Bi-YIG/Pt multilayers. Here we discuss the tuning of spin pumping by controlling the interface cut dependent growth of 100 nm Bi-YIG film on Gd 3Ga5O12 (GGG) substrates. Our Bi-YIG/Pt het erostructures have been deposited using pulse d laser deposition (PLD) technique where GGG [100] and GGG [111] substrates are used to change the interface cut orientation. We utilize the broadband ferromagnetic resonance (FMR) technique to realize the spin pumpi ng based on the enhancement of Gilbert damping and spin to charge conversion (S2CC) using inverse spin Hall effect (ISHE) . The spin mixing conductanc e and spin Hall angle vary significantly with [111] and [100] orientations . To further corroborate our find ings, we have used the complementary technique based on spin Seebeck effect where the variation of spin Seebeck voltage due to interface cut orientation has been observed. 2. Experimental Technique The Bi-YIG thin film s of thickness 100 nm were g rown on [100] and [111] -oriented GGG substrates using PLD technique. Bare grown samples, GGG [100]/Bi-YIG and GGG [111]/Bi-YIG were labeled as Bi -YIG [100] and Bi -YIG [111] respectively . After that, a platinum thin film of 8 4 nm thickness with deposition ra te of 0.8Å/sec was successfully deposited in -situ on bare Bi -YIG using PLD wi th optimized growth parameters. Platinum deposited samples, GGG [100]/Bi-YIG/Pt, and GGG [111]/Bi-YIG/Pt were labeled as Bi -YIG[100]/Pt and Bi -YIG[111]/Pt, respectively . The size of grown samples is 2 mm 5 mm. Before deposition, GGG substrates were appropriately cleaned us ing acetone and isopropanol. All thin films were deposited at a base vacuum of 2×10-7 mbar. We used a 248 nm KrF excimer laser with a 10 Hz pulse rate to ablate the material from the target. During deposition, Oxygen pressure, target to substrate distance, and substrate temperature were all maintained at 0.15 mbar, 5.0 cm, and 825 oC, respectively. The growth rate of deposited Bi-YIG films was 6 nm/min. The as -grown films were in -situ annealed for 2 hours at 825 oC in the presence of oxygen (0.15 mbar). After deposition of Bi -YIG film the ta rget to substrate distance was changed to 3.0 cm and substrate temperature 100 oC , for deposition of the Pt film, Thickness was measured using atomic force microscopy (AFM) (WITec GmbH, Germany). FMR -based spin pumping measurements have been carried out in a vector network a nalyzer spectrometer, where samples are placed in a flip -chip arrangement on a microstrip line with DC ma gnetic field applied perpendicular to the high -frequency magnetic field (h RF) onto the film plane. The dc voltage generated due to ISHE was measured with the help of Keithley 2182 Nanovoltmet er. For spin Seebeck effect measurements, standard longitudinal s pin Seebeck effect (LSSE) geometry was used with a temperature gradient formed between the Pt layer and the GGG substrate, and the voltage (perpendicular to an applied magnetic field and temperature gradient) was measured using a Keithley 2182 Nanovoltmete r. 5 3. Results and Discussion 3.1 Structural, surface morphology and static magnetization study Bi-YIG crystallizes in cubic structure with the Ia -3d space group. Figure 1 (a) -(d) shows the XRD pattern of Bi -YIG and Bi -YIG/Pt bilayers on [100] and [111] cut GGG substrates. XRD patterns shown in Fig. 1 (a) and 1 (b) indicat e the single -crystalline growth of Bi -YIG and Bi -YIG/Pt thin films . Figures 1 (c) and 1 (d) show the evidence for successful growth of Pt layer on top of bare Bi-YIG films of [100] and [111] orientations. The growth of Pt film is nano -crystalline in nature. The lattice constant, lattice mismatch (with respect to substrate), and lattice volume obtained from XRD data are listed in Table 1. The cubic structure lattice constant 𝑎 is calculated using the formula 𝑎=𝜆√ℎ2+𝑘2+𝑙2 2𝑠𝑖𝑛𝑠𝑖𝑛 𝜃 (1) Where , 𝜆 is the wavelength of Cu -Kα radiation, 𝜃 is the diffraction angle, and [h, k, l] are the Miller indic es of the corresponding XRD peaks. The lattice mismatch (𝛥𝑎 𝑎) is calculated using the equation 𝛥𝑎 𝑎=(𝑎𝑓𝑖𝑙𝑚 − 𝑎𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 ) 𝑎𝑓𝑖𝑙𝑚×100 (2) Here, 𝑎𝑓𝑖𝑙𝑚 and 𝑎𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 indicate the latt ice constants of the film and substrate, respectively. The calculated value of lat tice constant are tabulated in T able 1 which are consiste nt with prior findings [16,21,32,33] . The l attice constant slightly increases in the case of [111] as compared to [100] because the distribution of Bi3+ at the dodecahedral site is dep endent on t he substrate 6 orientation [21,34,35] which leads to a comparatively larger lattice mismatch (or strain). Observed values of lattice mismatch are close to what has been reported earlier [36,37]. The lattice mismatch of the film plays an important role , a smaller lattice mismatch value can reduce the damping constant of the film [38,39] . Furthermore, t he surface roughness plays an important role in magnetization dynamics due to the generation of two magnon scattering in films with higher roughness. Figure 1 (e) -(h) shows room temperature AFM images with root mean square (RMS) roughness. We have observed RMS roughness around 0.35 nm or less for all grown Bi -YIG films which is comparable to previously reported YIG films [18,40] . We have observed that there RMS roughness remains unchanged for all grown films. Moreover, rough ness would be more affected by changes in growth factor s than by substrate orientation [18,21] . VSM magnetization measurements were performed at 300 K with an applied magnetic field parallel to the film plane (in -plane). The GGG substrate's paramagnetic co ntribution was properly subtracted . Magnetization plots of Bi -YIG thin films for [100] and [111] orientations are shown in Figure 1(i). The schematic of an applied magnetic field dire ction parallel to the film plane is shown in the inset of Fig. 1(i). Measu red saturation magnetization ( 0MS) values are 165.60 ± 20.10 mT Gauss for Bi -YIG [100] and 196.20 ± 19.10 mT for Bi -YIG [111]. Saturation magnetization error bars are related to s ample volume unc ertainty. The observed value of 0MS of as grown [100] and [111]-oriented Bi -YIG films are consistent with previous reports [33,41 –43]. Bismuth doping generally affects the magnetic and mechanical propertie s of YIG , However, the dependence of saturation magnetization on interface cut is minimal because the magnetization of YIG is induced via a super -exchange interaction at the d and a site (in Wycoff notation) between non -equivalent Fe3+ ions. Along [111], there is more contribution of Bi3+ ions co mpared to [100] orientation [21], 7 and Bismuth l ocated at the dodecahedral site dis tort the tetrahedral and octahedral Fe3+ ions[22]. Figure 1 (j) shows the crystal orientation dependent atomic model of YIG with Yttrium (Y) (red), Fe ‘d site’ (green) and Fe ‘a site’ (blue) atomic position. The inte rface cut dependent magnetic environment is highly anisotropic in [111] and [100] orientation whi ch is clearly depicted in Fig . 1(j). Pt interface with these different magnetic environments may ch ange its spin conversion efficiency [44] which will be discussed in next sections . Table 1: Lattice consta nt and roughness obtained from XRD and AFM. S. No. Sample Lattice constant (Å) Lattice Mismatch (%) Lattice volume (Å3) Roughness (nm) 1. Bi-YIG [100] 12.461 0.53 1935.15 0.36 2. Bi-YIG [111] 12.472 0.64 1939.80 0.33 3. Bi-YIG/Pt [100] 12.402 0.39 1907.81 0.29 4. Bi-YIG/Pt [111] 12.455 0.80 1932.31 0.34 3.2 Ferromagnetic resonance (FMR) study FMR measurements are performed at room temperature . Figure 2 (a) and (b) show the FM R absorption spectra of [100] and [111] -oriented thin films, respectively. When Pt is deposited on top of Bi -YIG, the observed FMR absorption spectra shift right in case of [100] and left in case of 8 [111] orientation with respect to bare Bi -YIG film. This shows that Pt deposition moderately affects the effective saturation magnetization in Bi -YIG[111] due t o its [111] texture . The FMR data at 10 GHz with its Lorentzian fit is shown in the inset of Fig. 2 (a) and 2 (b), and the schematic depicts the directi on of the applied DC magnetic field in the film plane. Inset in Fig. 2 (a) and (b) shows the enhancement of FMR linewidth in Bi -YIG/Pt samples as compared to bare Bi -YIG samples. The corresponding FMR linewidth enhancement is ~0.24 mT in [100] and ~ 0.50 mT in [111] oriented films. Th is enhancement in linewidth results from spin pumping . The coupling that transfers angular momentum from Bi -YIG to the metal contributes to the damping due to precess ion of Bi-YIG magnetization that leads to increasing linew idth. To calculate the increment in Gilbert damping due to Pt deposition, we have measured the FMR data at different resonance frequencies ( 𝑓= 2 GHz -12 GHz) with dc magnetic field applied parallel to film plane. The Lorentzian fitting was used to fit the FMR data and calculate the FMR linewidth (∆H) and resonance magnetic field (H rf) or different frequencies. From the fitting of Kittel's in -plane equation provided by Eq. 3[45], the gyromagnetic ratio (γ) and effective magnetization field ( 0𝑀𝑒𝑓𝑓) were calculated. 𝑓=𝛾 2𝜋0√(𝐻𝑟)(𝐻𝑟+𝑀𝑒𝑓𝑓) (3), Where , 0𝑀𝑒𝑓𝑓=0(𝑀𝑠−𝐻𝑎𝑛𝑖) is the effective field with anisotropy field 𝐻𝑎𝑛𝑖=2𝐾1 𝑀𝑠. Similarly, Gilbert damping parameter (α) and inho mogeneous broadening (∆H 0) linewidth were calculated from the fitting of the Landau –Lifshitz –Gilbert equation (LLG) provided by equation 4[45], 𝛥𝐻(𝑓)=𝛥𝐻 0+4𝜋𝛼 𝛾𝑓 (4) Figures 2 (c) and 2 (d) show Kittel and LLG fitted graphs for the [111] orientation, respectively, with an inset showing Kittel and LLG fitted graphs for the [100] orientation. 9 Table 2 lists the calculated parameters obtained from the FMR study. The measured Gilbert damping (α) is consistent with thin films used in spin -wave propagation studies [21,24,43] . For both orientations [100] and [111], the value of α increases when Pt is deposited on top of Bi -YIG films (Table 2). The value of α increases significantly in (111) orientation . However, in the case of pure YIG it is reported that Gilbert damping in [100] is more as compared to [111][28]. In our case higher α obtained in [111] may be ascribed qualitatively to the presence of Bi3+ ions, which additional induce spin -orbit coupling (SOC) [16,46,47] that causes local di stortion of Fe3+ so as to affect the magnetic properties as compared with pure YIG[22]. The higher lattice mismatch (strain) in (111) could also be the reason for increased electron scattering leading to higher damping [48]. Figure 1 (j) elucidates how more Fe atoms (i.e., more magnetic environment ) and also more yttrium sites are available for the distribution of Bi3+ ions along [111] orientation . This is the cause of higher lattice mismatch in Bi -YIG [111]. These findings account for the higher Gilbert da mping and 0𝑀𝑒𝑓𝑓 values in the [111] orientation. In conclusion, Bi -YIG/Pt with an orientation of [100] has the lowest damping factor. We note that t hese are the essential desirable factors used in spintronic devices that need longer propagation lengths for spin waves. Table 2: Damping and linewidth parameters obtained from FMR S. No. Sample α (10-4) ΔH 0 (Oe) 0Meff (Oe) 1. Bi-YIG [100] (2.97±0.11) 25.99±0.21 2124.89±8.80 2. Bi-YIG [111] (3.62±0.17) 26.69±0.33 2241.70±12.77 3. Bi-YIG/Pt [100] (3.74±0.14) 27.33±0.29 2030.12±40.19 10 4. Bi-YIG/Pt [111] (6.11±0.47) 25.19±0.92 2582.26±75.13 3.3 Inverse spin Hall effect (ISHE) study using FMR based spin pumping and spin Seebeck effect The induced crystal field anisotropy due to different interfaces strongly affects the tuning of spin mixing conductance because the magnetic Fe3+ ions can result in different spin currents according to its placement in [111] and [100] orientation. It is argued that the crystal field is elongated along the surface norma l in [111] orientation and parallel to the surface in [100] orientation [27]. This anisotropic nature of the crystal field gives the highest spin mixing conductance in [111] direction and considerably lower in [100] orientation. At Bi -YIG/Pt interfaces wher e the spin pumping or the transfer of spin angular momentum occurs, the spin mixing conductance ( g) plays a significant role in transfer of spin angular momentum. Spin pumping - induced damping is measured from the enhancement in the Gilbert damping con stant (e.g. due to Pt layer on top of Bi -YIG film ( 𝛼𝐵𝑖−𝑌𝐼𝐺 /𝑃𝑡−𝛼𝐵𝑖−𝑌𝐼𝐺)). This is required for calculating the spin mixing conductance at the Bi -YIG/Pt interface. Table 2 shows the Gilbert damping values of bare Bi-YIG and Pt deposited Bi-YIG samples. The value of g as (0.73±0.07) 1018 m-2 for [100] and (2.31±0.23) 1018 m-2 for [111] oriented samples are calculated using equation 5 [14]. The observed spin mixing conductance values are consistent with previous reports [14,24,49,50] and shows that it depends on crystal orientation. A larger spin mixing conductance in [111] oriented samples occur that indicates that [111] grown samples have a greater spin -injection efficiency than [100] oriented samples. This is a consequence of [111] oriented sample having a greater Gilbert damping constant enhancement owing to the Pt layer. 11 g=µ0𝑀𝑠𝑡𝐹 𝑔𝐵(𝛼𝐵𝑖−𝑌𝐼𝐺 /𝑃𝑡−𝛼𝐵𝑖−𝑌𝐼𝐺) (5) Where µ0𝑀𝑠 and 𝑡𝐹 are the magnetization and thickness of the Bi -YIG thin film, 𝐵is the Bohr magneton, 𝑔=𝛾ħ 𝐵 is the g -factor of Bi -YIG[22] and 𝛼𝐵𝑖−𝑌𝐼𝐺 /𝑃𝑡−𝛼𝐵𝑖−𝑌𝐼𝐺 is the enhancement in the Gilbert damping constant due to Pt layer on top of Bi -YIG film. ISHE measurements are carried out at room temperature using FMR technique. In this scheme, a dc magnetic field (H) is applied in the film plane and perpendicular to the sample length. ISHE voltage (V ISHE) is measured along the length of the Pt layer. A resonant rf field ( hrf), is also superimposed to cause the uniform precession of the Bi -YIG magnetization. When excited precession occurs at the inter face, angular momentum is transferred to the conduction electrons in Pt, resulting in a pure spin current (J S) in Pt. As a result of ISHE, the spin current travels across the Pt perpendicular to the plane and is transformed into charge current ( JC). As a consequence of this, a voltage difference develops between the two ends of the Pt layer that is measured using a Keithley nanovoltmeter. Figure 3 (a) shows the measured dc voltage (V dc) at frequency 10 GHz, and figure 3 (b) shows its Lorentzian fit ting[45] with symmetric and anti -symmetric contributions. Figure 3 (c) shows the symmetric contribution of the voltage (V ISHE) extracted from the measured dc voltage (Vdc). The i nset shows the schematic setup for FMR spin pumping for measurement of ISHE voltage. The ISHE voltages [14] given by equation 6 depends on several material parameters 𝑉𝐼𝑆𝐻𝐸 =−𝑒𝑆𝐻 𝜎𝑁𝑡𝑁+𝜎𝐹𝑡𝐹𝜆𝑆𝐷𝑡𝑎𝑛ℎ (𝑡𝑁 2𝜆𝑆𝐷) 𝑔𝑓𝐿𝑃 (𝛾ℎ𝑟𝑓 2𝛼𝜔)2 (6) Where e is the electron charge, 𝑆𝐻 is the spin Hall angle, 𝜎𝑁(𝜎𝐹) and 𝑡𝑁(𝑡𝐹) denotes the conductivity and thickness of the NM (FM) thin -film respectively, 𝜆𝑆𝐷=7.3 𝑛𝑚 is the spin 12 diffusion length in Pt[16], 𝑔 denotes interfacial spin mixing conductance, 𝜔=2𝜋𝑓 is the resonance frequency, L is the length of the sample, and ℎ𝑟𝑓=0.17 𝑂𝑒 in our FMR cavity at power = +15 dBm. The ellipticity of the magnetization precession gives rise to the factor 𝑃[13,14] . 𝑃=2𝜔[𝛾µ0𝑀𝑠+√(𝛾µ0𝑀𝑠)2+4𝜔2] (𝛾µ0𝑀𝑠)2+4𝜔2 =1.21 (7) Values of 𝑔 and 𝑆𝐻 are calculated using equations (5) -(7). Calculated values of 𝑆𝐻 for [100] orientation is (0.73±0.07) 10-2 and for [111] orientation is (1.01±0.10) 10-2. Figure 3 (d) shows the comparative results of 𝑔 and 𝑆𝐻 for [100] and [111] oriented samples. Consequently, the spin current density 𝐽𝑠 can be calculated using [14], 𝐽𝑠=𝜎𝑁𝑡𝑁+𝜎𝐹𝑡𝐹 𝑆𝐻𝜆𝑆𝐷𝑡𝑎𝑛 ℎ (𝑡𝑁 2𝜆𝑆𝐷) 𝑉𝐼𝑆𝐻𝐸 𝐿 (8) 𝐽𝑠 is calculated to be 0.59×107 A-m-2 for [100] and 0.70×107 A-m-2 for [111] orientation. 𝑔, 𝑆𝐻 and 𝐽𝑠 was significant ly larger in the case of [111] orientation, indicating the enhanced spin orbit contribution and spin transparency in [111] oriented Bi -YIG/Pt bilayer structure, resulting in maximum V ISHE signal ~76.30 μV . For the [100] oriented Bi -YIG/Pt film it was foun d to be ~46.31 μV. Our obtained transport parameters are in agreement with the reported literature values [14,51 – 58] in which order of spin mixing conductance values varies from 1016 m-2 to 1018 m-2. Similar ly order of spin Hall angle [57] and spin current density are also in agreement with published data[27,29 –31]. It show s the effect of crystal orientation on spin mixing conductance . Jia et al. [31] observed no crystal orientation dependent g in the pure YIG/Ag system . Cahaya et a l.[27], reported that asymmetry of spin mixing conductance depends not only on the density of exposed moments and but also on local point symmetry. The pumped spin -current anisotropy is dependent on the quadrupole moment, which is dependent on the orbital occup ancy of interface magnetic atoms. For half -filled shells ions (Fe3+) quadrupole moment is zero. It is necessary to break 13 symmetry, such as via strain or at an interface, to obtain a finite quadrupole moment. From the theoretical explanation given by [27], we conclude that our crystal orientation dependent findings may be due to rot ational symmetry break ing in Bi -YIG thin films owing to substrate lattice mismatch. As discussed already, a l arger lattice mismatch in [111] is observed compared to [100]. Furthermor e, we have us ed Bi -doped YIG system, where Bi3+ affects the magnetic properties of YIG[22] and also distribution of Bi3+ ions among dodecahedral sites depends on substrate orientation [21,59 –61]. Figure 1 (j) shows the interface cut dependent magnetic environment in [100] and [111] orientation. The interaction of Pt with these varied magnetic environments may affect its spin conversion efficiency. Hence, the enhancement in spin pumping parameters depends on the orienta tion of crystal growth [27,29 –31], which supports our experim ental findings of ISHE results. 3.4 Spin Seebeck effect The integration of magnetic insulators with Pt promises to usher in several application s of spin current technology to thermoelectric devices. In this regard, we discuss spin Seebeck study on the s ame films that were studied for invers e Spin Hall Effect previously. We note that t he doping of Bi softens the crystal and increases the growth induced anisotropy [22]. The effective SOC for Fe3+ ions is increased by mixing 6p orbitals of Bi ions with O -2p orbitals. Energy transmission between spin and phonon systems is also enhanced by SOC. Thus during spin thermoelectric generation, maximum energy transfer mechanism for the heat simulation of magnetization precession will result in larger spin currents in Bi -YIG films [23]. The spin Seebeck effec t measurements were performed in longitudinal spin Seebeck effect (LSSE) geom etry. Figure 4 (a) shows the schematic setup for LSSE measurements. Figu re 4 (b), 4 (c), and 4 (d) show the 14 comparative results of longitudinal spin Seebeck voltage (V LSSE) as a f unction of temperature (20 K to 300 K), applied magnetic field (up to ±0.2 Tesla) and temperature gradient (upto 11 K) respectively. At room temperature, the o bserved spin Seebeck coefficient is 106.40±10 nV mm-1 K-1 for [111] and 89.41±0.3 nV mm-1 K-1 for [100] oriented sample . Because these voltages are obtained across various effective resistances, we represent these voltages in their normalized form, i.e. 𝑉𝐿𝑆𝑆𝐸 =𝑉𝑑𝑐/𝑅𝛥𝑇𝐿 , where 𝑉𝑑𝑐 is the observed transverse voltage, R and L are the corresponding resistance and distance between contact probes, 𝛥𝑇 is the temperature gradient across the sample . The Spin Seebeck effect study indicates the significantly higher values of spin Seebeck coefficient observed in the case of the [111]. Thes e findings from spin Seebeck measurements also provide credence to the higher ISHE voltage generated in the [111] orientation that was seen in FMR spin pumping studies. According to these experimentally observed ISHE results, the spin mixing conductance of Bi-YIG/Pt bilayers depends on crystal cut and orientation [27]. We find that t he spin pumping efficiency of the [111] oriented sample is substantially greater than that of the [100] orientated sample. 4. Conclusion We have successfully grown the Bi -YIG(100 nm )/Pt(8 nm) bilayer using pulsed laser deposition technique on top of GGG substrates having [100] and [111] orientation s. AFM and XRD characterizations revealed that the deposited thin films are phase pure and have smooth surfaces. FMR based spin pumping re sults confirm that there are significantly higher values of spin mixing conductance (~ three times) and spin Hall angle in the case of the [111] orientated sample compared to [100] oriented sample . This indicates that significantly higher spin current can transfer to the Pt 15 from Bi -YIG [111]. Consequently, the [111] oriented sample has a higher spin pumping efficiency than the [100] oriented sample . From a longitudinal spin Seebeck measurement we note that the [111] orientated sample had a higher (~20 %) s pin Seebeck coefficient. In conclusion, t he results of inverse spin Hall effect and spin Seebeck experiments show that the fundamental parameters of spin pumping can be tuned effectively by substrate ’s orientation. Acknowledg ements This work is support ed by the MHRD -IMPRINT grant, DST (SERB, AMT, and PURSE -II) grant of Govt. of India. Ganesh Gurjar acknowledges CSIR, New Delhi for financial support. We acknowledge AIRF, JNU for access of PPMS facility. Conflict of interest statement The authors declare that they have no competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Data Availability statement The data that support the findings of this study are available from the corresponding author upon reasonable request. 16 References [1] Tserkovnyak Y, Brataas A and Bauer G E W 2002 Enhanced Gilbert Damping in T hin Ferromagnetic Films Phys. Rev. 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Lett. 110 206601 [56] Vlietstra N, Shan J, C astel V, Van Wees B J and Ben Youssef J 2013 Spin -Hall magnetoresistance in platinum on yttrium iron garnet: Dependence on platinum thickness and in -plane/out -of-plane magnetization Phys. Rev. B - Condens. Matter Mater. Phys. 87 184421 [57] Qiu Z, Ando K, Uchida K, Kajiwara Y, Takahashi R, Nakayama H, An T, Fujikawa Y and Saitoh E 2013 Spin mixing conductance at a well -controlled platinum/yttrium iron garnet interface Appl. Phys. Lett. 103 092404 [58] Li M, Zhang D, Jin L, Liu B, Zhong Z, Tang X, Meng H, Yang Q, Zhang L and Zhang H 2021 Interfacial chemical states and recoverable spin pumping in YIG/Pt Appl. Phys. Lett. 118 042406 [59] Callen H 1971 Growth -induced anisotropy by preferential site ordering in garnet crystals Appl. Phys. Lett. 18 311–3 [60] LeCraw R C, Luther L C and Gyorgy E M 1982 Growth -induced anisotropy and damping versus temperature in narrow linewidth, 1 -μm YIG(Bi, Ca, Si) bubble films J. Appl. Phys. 53 2481 –2 [61] Hansen P and Witter K 1985 Growth -induced uniaxial anisotropy of bism uth-substituted iron-garnet films J. Appl. Phys. 58 454–9 23 Figure Captions Figure 1: XRD patterns of GGG/Bi -YIG/Pt structure, Bi -YIG XRD pattern shown (a) with [100] and (b) [111] orientations, (c) and (d) shows the XRD pattern of Pt film with respec t to bare Bi - YIG films on [100] and [111] orientations, (e) -(h) AFM images of Bi -YIG and Bi -YIG/Pt films in [100] and [111] orientations, (i) VSM plot of Bi -YIG film in [100] and [111] orientations (one inset shows the schematic of applied magnetic fi eld direction in the film plane, second inset shows the schematic of sample grown structure) , (j) shows the crystal orientation dependent atomic model of YIG with Yttrium (Y) (red), Fe ‘d site’ (green) and Fe ‘a site’ (blue) atomic position. Figure 2: FMR absorption spectra of Bi -YIG and Bi -YIG/Pt films with (a) [100] and (b) [111] orientation. Inset shows FMR data at 10 GHz with its Lorentzian fit and enhancement in FMR linewidth in Pt deposited sample. The schematic shows the direction of the applied DC magnetic field in the film plane. (c) and (d) show frequency -dependent FMR field (H r) data fitted with Kittel equation and frequency -dependent FMR linewidth (ΔH) data fitted with LLG equation [111] orientation, respectively, with an inset showing Kittel a nd LLG fitted graphs for the [100] orientation. Figure 3: (a) show the measured dc voltage (V dc) at frequency 10 GHz, (b) show its Lorentzian fitting with symmetric and anti -symmetric contributions, (c) symmetric contribution of the voltage (VISHE) extrac ted from the measured dc voltage (V dc) (schematic setup of ISHE measurements), (d) shows the comparative results of spin mixing conductance ( 𝑔) and spin Hall angle ( 𝑆𝐻) for [100] and [111] oriented samples. 24 Figure 4: (a) Shows the schematic setup of spin Seebeck measurements in LSSE geometry, comparative results of longitudinal spin Seebeck voltage (V LSSE) as a function of (a) amb ient temperature, (b) applied magnetic field, (c) temperature gradient, for [100] and [111] oriented samples. 25 Figure 1: 26 Figure 2: 27 Figure 3: 28 Figure 4:

2023-01-16

Ferromagnetic resonance (FMR) based spin pumping is a versatile tool to quantify the spin mixing conductance and spin to charge conversion (S2CC) efficiency of ferromagnet/normal metal (FM/NM) heterostructure. The spin mixing conductance of FM/NM interface can also be tuned by the crystal orientation symmetry of epitaxial FM. In this work, we study the S2CC in epitaxial Bismuth substituted Yttrium Iron Garnet (Bi0.1Y2.9Fe5O12) thin films Bi-YIG (100 nm) interfaced with heavy metal platinum (Pt (8 nm)) deposited by pulsed laser deposition process on different crystal orientation Gd3Ga5O12 (GGG) substrates i.e. [100] and [111]. The crystal structure and surface roughness characterized by X-Ray diffraction and atomic force microscopy measurements establish epitaxial Bi-YIG[100], Bi-YIG[111] orientations and atomically flat surfaces respectively. The S2CC quantification has been realized by two complimentary techniques, (i) FMR-based spin pumping and inverse spin Hall effect (ISHE) at GHz frequency and (ii) temperature dependent spin Seebeck measurements. FMR-ISHE results demonstrate that the [111] oriented Bi-YIG/Pt sample shows significantly higher values of spin mixing conductance ((2.31+-0.23)x10^18 m^-2) and spin Hall angle (0.01+-0.001) as compared to the [100] oriented Bi-YIG/Pt. A longitudinal spin Seebeck measurement reveals that the [111] oriented sample has higher spin Seebeck coefficient (106.40+-10 nV mm-1 K-1). This anisotropic nature of spin mixing conductance and spin Seebeck coefficient in [111] and [100] orientation has been discussed using the magnetic environment elongation along the surface normal or parallel to the growth direction. Our results aid in understanding the role of crystal orientation symmetry in S2CC based spintronics devices.

Crystal orientation dependent spin pumping in Bi0.1Y2.9Fe5O12/Pt interface

2301.06477v1

Anomalous spin Hall angle of a metallic ferromagnet determined by a multiterminal spin injection/detection device T. Wimmer,1, 2,a)B. Coester,3S. Gepr ags,1R. Gross,1, 2, 4, 5S. T. B. Goennenwein,6H. Huebl,1, 2, 4, 5and M. Althammer1, 2,b) 1)Walther-Meiner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany 2)Physik-Department, Technische Universit at M unchen, 85748 Garching, Germany 3)School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Singapore 4)Nanosystems Initiative Munich (NIM), Schellingstrae 4, 80799 M unchen, Germany 5)Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, D-80799 M unchen, Germany 6)Institut f ur Festk orper- und Materialphysik and W urzburg-Dresden Cluster of Excellence ct.qmat, Technische Universit at Dresden, 01062 Dresden, Germany (Dated: July 5, 2021) We report on the determination of the anomalous spin Hall angle in the ferromagnetic metal alloy cobalt-iron (Co25Fe75, CoFe). This is accomplished by measuring the spin injection/detection eciency in a multiterminal device with nanowires of platinum (Pt) and CoFe deposited onto the magnetic insulator yttrium iron garnet (Y3Fe5O12, YIG). Applying a spin-resistor model to our multiterminal spin transport data, we determine the magnon conductivity in YIG, the spin conductance at the YIG/CoFe interface and nally the anomalous spin Hall angle of CoFe as a function of its spin diusion length in a single device. Our experiments clearly reveal a negative anomalous spin Hall angle of the ferromagnetic metal CoFe, but a vanishing ordinary spin Hall angle. This is in contrast to the results reported in Refs. 1 and 2 for the ferromagnetic metals Co and permalloy. The spin Hall eect (SHE) is at the origin of a plethora of transport eects relevant for spintronics ap- plications1{9. While the charge to spin current conversion eciency is conveniently expressed in terms of the phe- nomenological spin Hall angle  SH, its microscopic origin is the spin-orbit interaction causing spin-selective scatter- ing of charge carriers7,10. Many ferromagnetic metals ex- hibit a strong spin-orbit coupling, which manifests itself in various electrical transport eects, among them the anomalous Hall eect (AHE)11. The AHE hinges on the same physical principles as the SHE3,7. While the trans- verse charge current arising in the AHE has been studied for more than a century, the pure spin current part has only very recently received broad attention1,2,12,13. Recent developments in magnetotransport experi- ments with incoherent magnons (the quantized excita- tions of the magnetization)14{17oer a suitable plat- form for the investigation of the SHE and the anoma- lous spin Hall eect (ASHE) in ferromagnets1,2,18{20. In these experiments, a spin current is injected into an adjacent magnetic insulator via the SHE. More speci- vally, a DC charge current in a metallic electrode gener- ates a spin accumulation at the interface between the metal and the magnet, which in turn induces a non- equilibrium magnon accumulation in the magnet. The non-equilibrium magnons diuse in the magnet, and are detected in a second, electrically separate metallic elec- a)tobias.wimmer@wmi.badw.de b)matthias.althammer@wmi.badw.de Pt2 H φyz xPt1 CoFe YIG Pt3d Pt d CoFeM YIGMCoFe d CoFe+ -+ - + - + -Vdet VdetVdetIqFigure 1. Schematic depiction of the device, the electrical connection scheme and the coordinate system. A charge cur- rentIqis fed through the Pt2 electrode, resulting in a spin current injection into YIG via the SHE. The lateral diu- sion of the magnon spin current is electrically detected at the Pt electrodes ('Pt1' and 'Pt3') and the ferromagnetic metal electrode ('CoFe') as the detector voltage Vdet. The center-to- center distances between each of the Pt electrodes is constant, such thatdPt= 2dCoFe. trode as a voltage signal via the inverse SHE. In a recent work, Das et al. reported spin injection and detection in YIG via the AHE11using Py electrodes2. They found a magnetic eld enhanced injection/detection eciency of the permalloy (Py) electrodes due to the gradual increase of the eective anomalous spin Hall angle  ASHin Py. In this Letter, we report on the determination of ASHof the ferromagnetic metal alloy Co 25Fe75(CoFe)21 via all-electrical magnon transport measurements in thearXiv:1905.00663v1 [cond-mat.mes-hall] 2 May 20192 Pt1 Pt2 Pt3CoFe IcVdet Pt1 Pt2 Pt3CoFe IcVdet Pt1 Pt2 Pt3CoFe IcVdet(a) (b) (c) Figure 2. Resistance Rdetmeasured using dierent detector electrodes. (a) The signal at the Pt1 detector is taken as a reference measurement, with which we can characterize the magnon transport in the YIG layer. (b) The Pt3 detector signal is somewhat smaller than the Pt1 signal owing to the nite absorption of the magnon current by the CoFe elec- trode in between Pt2 and Pt3. (c) The Rdetassociated with the detector voltage recorded at the CoFe electrode shows a sign reversal, indicating that the anomalous spin Hall angle in CoFe is negative. magnetic insulator YIG. For this purpose, we utilize a multiterminal structure with four metallic electrodes { one made of CoFe and three made of Pt { deposited onto a YIG thin lm (see Fig. 1). Our device consists of a 1 µm thick, commercially avail- abe YIG lm grown on a GGG (Gd 3Ga5O12) substrate via liquid phase epitaxy. Both the Pt and the CoFe elec- trodes were deposited by DC sputtering and patterned via electron beam lithography and lift-o9. The CoFe electrode was additionally capped with a 2 :5 nm thick Al layer to prevent oxidation. In a further step, Al leads and bondpads were deposited to connect the device elec- trically. Each electrode has a width of w= 500 nm and a thickness of tPt=tCoFe = 7 nm. The lengths of the strips arelPt1=lPt3= 148 µm for the outer electrodes and lPt2=lCoFe = 162 µm for the inner ones. As indicated in Fig. 1, the center-to-center distances between the metal strips aredPt= 1:6µm anddCoFe = 0:8µm (cf. Fig. 1). For the injection of magnons, we apply a charge current Iq= 0:5 mA to the Pt2 electrode (the injector) and de- tect the magnon transport signal as the detector voltage Vdetat the Pt1, Pt3 and CoFe electrodes (see Fig. 1). In order to distinguish between electrically (via the SHE) and thermally (via Joule heating) injected magnons, we utilize the current reversal method15,16. Here, we focus on the magnon transport via the electrical SHE-induced spin current injection. All measurements are conducted in a superconducting magnet cryostat at a constant tem- perature of T= 280 K22. In order to compare between dierent detector signals, we dene a normalized signal amplitude as Rdet= (Vdet=Iq)
(Ainj=Adet), which ac- counts for the dierent interface areas Ainj(Adet) of the injector (detectors)16. To characterize the magnon transport in our device, we measure Rdetas a function of the magnetic eld ori-entation'for various in-plane eld magnitudes 0H. Corresponding data are shown in Fig. 2 (a)-(c) for three dierent external magnetic elds. Firstly, panel (a) shows the reference measurement using Pt1 as a detector. In accordance with Refs.14,15,23, we observe a sin2(')- dependence of Rdetwith reduced amplitudes for increas- ing external magnetic eld strengths. Secondly, panel (b) showsRdetrecorded across Pt3. Since the separation of the Pt1 and Pt3 strip to the injector strip Pt2 are the same (cf. 1), one would expect the same signal magni- tude. However, the Rdetmodulation recorded across the Pt3 strip is signicantly smaller, which we attribute to a partial absorption of the magnon spin current in the CoFe electrode located in between the Pt2 and Pt3 electrodes. Finally, panel (c) shows Rdetmeasured at the CoFe elec- trode. Interestingly, the polarity of the detected voltage is inverted. Since all strips were contacted with identical polarity in the experiments (see Fig. 1), we conclude that the anomalous spin Hall angle CoFe ASH in CoFe is negative compared to the positive spin Hall angle Pt SHin Pt10,24. This is in agreement with the negative spin Hall angles reported for both Co and Fe12. Unlike the magnetic eld suppression observed for the Pt detector strips, we nd a signicant enhancement of Rdetfor increasing magnetic elds up to 0H= 2 T for the CoFe detector. We at- tribute this to the eld-induced increase of CoFe ASH, quali- tatively similar to the results reported in Ref.2. For larger magnetic elds ( 0H= 7 T), however, we observe a sup- pression of the magnon transport signal. Since the CoFe magnetization MCoFe saturates around 0H= 2 T21, we attribute this eld suppression to the YIG magnon sys- tem in analogy to the situation observed for the Pt de- tectors14. Interestingly, we observe a distinct asymmetry in the magnitudes of the signal for strong magnetic elds 0H > 2 T, which is discussed in more detail in the Sup- plementary Information (SI)25. In Fig. 3 (a), we plot the anisotropic magnetoresistance (AMR) of the CoFe electrode by measuring its longitudi- nal resistance Rlongas a function of the magnetic eld strength (the sweep direction is indicated by arrows). The blue (green) colored lines correspond to the eld direction pointing perpendicular (parallel) to the strip length, while dark (light) colored lines correspond to the up (down) sweep of the magnetic eld strength (trace and retrace, respectively). Obviously, we observe a clear AMR with a maximum (minimum) in resistance for par- allel (perpendicular) eld alignment with respect to the strip length for a coercive eld of approximately 18 mT. Additionally, we nd a second peak at a characteristic eld of roughly 11 mT. This feature corresponds well to the switching eld observed for the longitudinal resis- tance of the Pt2 electrode, which is shown in Fig. 3 (b) (switching eld indicated by gray dashed lines). There- fore, the peaks in the resistance of the CoFe strip around 11 mT can be attributed to a spin Hall magnetoresistance (SMR)6contribution related to the magnetization rever- sal in the YIG lm. Most importantly, these magnetore- sistance measurements show that exchange coupling be-3 tween the two ferromagnetic layers is not relevant, since no exchange bias eect can be observed (which would lead to a shift of the hysteresis curves along the mag- netic eld axis). Figure 3 (c), (d) show the detector sig- nalsRdetas a function of the magnetic eld strength measured at the Pt1 and CoFe electrodes, respectively. For the reference detector (Pt1), we nd a continuous suppression of the magnon transport signal Rdetwith in- creasing magnetic eld strength when the eld is oriented perpendicular to the strips (blue data points in Fig. 3 (c)). For a parallel alignment of the eld and the strips (green data points), the signal vanishes. This response is quantitatively consistent with the eld-orientation de- pendent data (Fig. 2). Figure 3 (d) shows the magnetic eld dependence of Rdetwhen the CoFe electrode is used as the detector. Here, Rdetis zero for0H= 0 for both eld orientations. When the eld is oriented prependic- ular to the strips, Rdetrapidly increases and reaches its maximum at around 0H2 T. Since the injection and detection eciency of the magnons is maximized when the magnetizations MYIGandMCoFe are aligned per- pendicular to the electrodes2, the maximum of Rdetis expected when the magnetization MCoFe is fully satu- rated perpendicularly to the strips overcoming the shape anisotropy at around 2 T2. For larger magnetic elds, we again observe a eld-induced suppression of the signal, which was already discussed for the orientation depen- dent measurements in Fig. 2 and originates from the eld dependence of the magnon transport in YIG. Following Ref.2, the contributions of  SHand  ASHcan be sepa- rated by identifying the magnon transport signal at the switching eld and the saturation eld of the CoFe de- tector, respectively. It is, however, evident that the CoFe detector signal in Fig. 3 (d) becomes zero for small mag- netic elds, suggesting that there is no contribution from a pure SHE in CoFe, in contrast to the results reported for Py2. We note, however, that this observation strongly depends on whether or not the CoFe is in a multidomain state, since a net magnetization MCoFe perpendicular to the electrode could be counterbalanced by a positive SHE contribution. Clearly, we can verify again the asymmetry feature when the CoFe electrode is used as a detector25. Utilizing our multiterminal magnon transport device, we are able to extract the anomalous spin Hall angle CoFe ASH. To this end, we model the spin transport in our device by employing the spin-resistor circuit model pro- posed in Ref.26. This approach is valid as long as the distancedbetween the considered electrodes is smaller than the characteristic magnon diusion length min our YIG lm. We verify that dPt= 2dCoFe<  m 6µm for a comparable YIG lm (see SI). The equiva- lent spin-resistor circuit diagram for the Pt2-Pt1 con- tact pair and the three-terminal Pt2-CoFe-Pt3 contacts are shown in Fig. 4 (a) and (b), respectively. Here, the individual resistors are described by three dier- ent resistances: rstly, Rs i=ii=[liwtanh(ti=i)] is the spin resistance of electrode i(withi= Pt1, CoFe, Pt3) withithe spin diusion length and ithe elec- Pt1 Pt2 Pt3CoFe Ic (a) (c) (b) (d)VdetVdetFigure 3. Longitudinal resistance Rlongand magnon trans- port signal Rdetmeasured as a function of the magnetic eld strength for eld directions pointing perpendicular (blue) and parallel (green) to the strip length. Dark and light colored lines correspond to up- and down-sweep curves, respectively. (a)Rlongmeasured on the CoFe electrode, showing the AMR with a switching eld of 0H=18 mT. Additionally, a second switching at lower elds 0H=11 mT (indicated by gray dashed lines) is observed, which corresponds to the Rlong change measured on the Pt2 electrode (SMR) in (b). Here, the green curves correspond to the left vertical axis, while the blue lines refer to the right axis. (c), (d) show Rdetas a func- tion of the magnetic eld strength measured at the Pt1 and CoFe detector, respectively. trical resistivity. Furthermore, ti,liandwdenote the thickness, length and width of electrode i. Secondly, Rs int;i= 1=(giliw) is the interface spin resistance, with gi the interface spin conductance of electrode iand lastly Rs YIG=dCoFe=(mlPt2tYIG) is the YIG spin resistance for a distance of dCoFe withmthe magnon conductivity andtYIGthe thickness of the YIG lm. For both circuits shown in Fig. 4 (a) and (b), the "spin battery" of the net- work is characterized by the injected spin chemical po- tentialPt2 s;inj= 2Pt SHIcPt[RPt2=lPt2] tanh (tPt=(2Pt))27 at the YIG/Pt2 interface. Here, Ptis the spin diu- sion length of Pt, Pt SHis the spin Hall angle of Pt and RPt2is the electrical resistance of the Pt2 electrode. The spin chemical potential "drop" across each detector iis given via the measured detector voltages Vi detasi s;det= 2ti=(i (A)SHli) 1 + [cosh(ti=i)1]1
Vi det26. For each detectori, we can then calculate the spin transfer e- ciency as i s=i s;det Pt2 s;inj: (1) Applying Kirchho's laws to the spin-resistor network shown in Fig. 4 (a), we obtain the spin transfer eciency4 Rint, CoFes RPt2sRint, Pt2sRYIGsRYIGsRCoFes Rint, Pt3sRPt3s/uni03BCs, detPt3/uni03BCs, detCoFe /uni03BCs, injPt2RPt1sRint, Pt1sRYIGsRYIGsRint, Pt2sRPt2s/uni03BCs, detPt1 /uni03BCs, injPt2(a) (b) (c) (d) Pt1 Pt2 Pt3CoFe Ic Pt1 Pt2 Pt3CoFe Ic 10 mT0.1 T0.3 T0.5 T7 T Figure 4. Equivalent spin-resistor network for the Pt2-Pt1 contact pair (a) and the Pt2-CoFe-Pt3 contact congura- tion (b). (c) Experimentally determined absolute value of the anomalous spin Hall angle CoFe ASH as a function of the spin diusion length CoFe for various external magnetic elds. Here, the spin conductance of the YIG/CoFe interface was set to a constant value gCoFe = 41010S=m. (d) Anomalous spin Hall angle of CoFe as a function of the applied magnetic elds, assuming a spin diusion length CoFe = 6 nm. of the Pt1 detector as Pt1 s=Rs Pt1 Rs Pt2+Rs Pt1+Rs int;Pt2+Rs int;Pt1+ 2Rs YIG;(2) while analyzing the circuit shown in Fig. 4 (b), we nd Pt3 s=Rs Pt3 Rs tot(1 +); CoFe s =Rs CoFe Rs tot(1 +);(3a) (3b) for the Pt3 and CoFe detectors. Here, = [Rs int;CoFe + Rs CoFe]=[Rs YIG+Rs int;Pt3+Rs Pt3] andRs totis the total re- sistance of the spin-resistor network of Fig. 4 (b). On the basis of this model, we now calculate m,gCoFe of the YIG/CoFe interface and nally CoFe ASHof CoFe. We obtainmby equating Eqs. (1) and (2) for i= Pt125. Note, that the spin conductance gPtfor the YIG/Pt in- terfaces was independently determined via longitudinalSMR measurements25. We extract mfor the dierent magnetic eld values and nd m= 3:2104S=m for an external magnetic eld of 0H= 0:1 T. In a next step, we extract gCoFe. Since our experiment does not al- low to determine the spin diusion length CoFe of CoFe, we determine gCoFe as a function of CoFe from Eqs. (1) and (3a) for i= Pt3. Substituting the values extracted formfor each of the magnetic elds measured, we nd thatgCoFe only varies by 0:05 % when changing CoFe from 0 nm to 10 nm. Moreover, we nd gCoFe to vary between approximately 2 1010S=m for0H= 7 T and 41010S=m for0H= 0:5 T. Since the spin conduc- tance is not expected to depend on the applied magnetic eld, we adopt a constant value of gCoFe = 41010S=m in the following25. Then, we can extract CoFe ASH as a func- tion ofCoFefrom Eqs. (1) and (3b) for i= CoFe. The re- sult is shown in Fig.4 (c) for dierent magnetic elds. Ob- viously, CoFe ASH saturates as a function of CoFe at around 7 nm, which corresponds to the CoFe electrode thick- nesstCoFe. This is reasonable, since (experimentally) we do not expect any change of CoFe ASH forCoFe> tCoFe. Finally, we estimate the eld dependence of CoFe ASH by as- sumingCoFe = 6 nm28. Note, that the value of CoFe only aects the quantitative values for CoFe ASH, but the qualitative eld dependence remains the same. Plotting CoFe ASH as a function of magnetic eld in Fig. 4 (d), we nd that CoFe ASH rapidly increases with increasing mag- netic eld (as MCoFe saturates) and reaches its maximum value for about 2 T-3 T at 5 %. For permalloy, a spin Hall angle of 2 % was reported12. Clearly, the eld de- pendence of CoFe ASH in our experiment is determined by the magnetization MCoFe aligning perpendicularly to the CoFe strip length (i.e. along the magnetic hard axis). As detailed in the SI, however, the application of a Stoner- Wohlfarth model with uniaxial shape anisotropy29does not reproduce the observed eld dependence well, sug- gesting that the CoFe electrode is in a multidomain state for small magnetic elds. In conclusion, we demonstrated the determination of the anomalous spin Hall angle CoFe ASHof the ferromagnetic metal Co 25Fe75employing a multiterminal spin injec- tion/detection device. Using both paramagnetic Pt and ferromagnetic CoFe electrodes on the ferrimagnetic insu- lator YIG, we were able to determine the magnon con- ductivity of YIG, the spin conductance of the YIG/CoFe interface and nally the anomalous spin Hall angle of CoFe on a single device. We based our analysis on a spin-resistor model26and found that the pure SHE con- tribution in CoFe is negligible, which is in contrast to the nite SHE contribution reported for Py2. The anomalous spin Hall angle of CoFe was found to increase strongly by saturating MCoFe with an applied magnetic eld and shows a saturation value of 5 % for magnetic elds of 0H&2 T. 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2019-05-02

We report on the determination of the anomalous spin Hall angle in the ferromagnetic metal alloy cobalt-iron (Co$_{25}$Fe$_{75}$, CoFe). This is accomplished by measuring the spin injection/detection efficiency in a multiterminal device with nanowires of platinum (Pt) and CoFe deposited onto the magnetic insulator yttrium iron garnet (Y$_3$Fe$_5$O$_{12}$, YIG). Applying a spin-resistor model to our multiterminal spin transport data, we determine the magnon conductivity in YIG, the spin conductance at the YIG/CoFe interface and finally the anomalous spin Hall angle of CoFe as a function of its spin diffusion length in a single device. Our experiments clearly reveal a negative anomalous spin Hall angle of the ferromagnetic metal CoFe, but a vanishing ordinary spin Hall angle. This is in contrast to the results reported for the ferromagnetic metals Co and permalloy.

Anomalous spin Hall angle of a metallic ferromagnet determined by a multiterminal spin injection/detection device

1905.00663v1

Increased low-temperature damping in yttrium iron garnet thin lms C. L. Jermain,S. V. Aradhya, N. D. Reynolds, and R. A. Buhrman Cornell University, Ithaca, New York 14853, USA J. T. Brangham, M. R. Page, P. C. Hammel, and F. Y. Yang Ohio State University, Columbus, Ohio 43210, USA D. C. Ralph Cornell University, Ithaca, New York 14853, USA and Kavli Institute at Cornell for Nanoscale Science, Ithaca, New York, 14853, USA (Dated: December 7, 2016) We report measurements of the frequency and temperature dependence of ferro- magnetic resonance (FMR) for a 15-nm-thick yttrium iron garnet (YIG) lm grown by o-axis sputtering. Although the FMR linewidth is narrow at room tempera- ture (corresponding to a damping coecient = (9.00.2)104), comparable to previous results for high-quality YIG lms of similar thickness, the linewidth in- creases strongly at low temperatures, by a factor of almost 30. This increase cannot be explained as due to two-magnon scattering from defects at the sample interfaces. We argue that the increased low-temperature linewidth is due to impurity relaxation mechanisms that have been investigated previously in bulk YIG samples. We suggest that the low-temperature linewidth is a useful gure of merit to guide the optimiza- tion of thin-lm growth protocols because it is a particularly sensitive indicator of impurities.arXiv:1612.01954v1 [cond-mat.mes-hall] 6 Dec 20162 Yttrium iron garnet (Y 3Fe5O12, YIG) thin lms are of considerable interest for appli- cations in spintronics and magnonics, since YIG can have one of the lowest damping co- ecients of any magnetic material at room temperature.1High-quality lms of YIG and related garnets with thicknesses on the 10's of nm scale and below can be grown by pulsed- laser deposition (PLD),2{9o-axis sputtering,10{14and molecular-beam epitaxy (MBE).15 Ferromagnetic resonance (FMR) measurements at room temperature for lms grown by all of these techniques show that the FMR linewidth increases with decreasing lm thickness, and the frequency dependence of the linewidth has a nonlinear functional form for very thin lms.2,15,16This behavior has been attributed to two-magnon scattering at the lm interfaces that becomes increasingly dominant as the lm thickness decreases.17,18 We are interested in extending the use of ultra-thin YIG lms to cryogenic temperatures, for example so that we can use scannning SQUID microscopy19to study the manipulation of YIG devices by spin-orbit torques.20,21In the course of this work we have found that even apparently high-quality YIG lms, which possess small FMR linewidths at room tempera- ture, can have linewidths that increase dramatically with decreasing temperature. The 15 nm YIG lm featured in this paper has a linewidth that increases by a factor of 28 as the temperature is lowered from room temperature to 25 K. The linewidth of this thin YIG lm also shows an increasingly nonlinear frequency dependence as the temperature is lowered. We argue that these strong temperature dependencies cannot be explained by two-magnon scattering from the YIG interfaces. Instead, we suggest that the increased linewidth at low temperature is due to magnetic damping associated with impurity mechanisms that have been studied previously in bulk YIG samples.1,22{24 We grow our YIG lms by o-axis sputtering10,11,14on a (111)-oriented gadolinium gallium garnet (GGG, Gd 3Ga5O12) substrate (see details in supplementary material). We measure the FMR response using a broadband coplanar waveguide with simultaneous eld and power modulation.15The waveguide is installed in a continuous- ow He cryostat for temperature- dependent studies. Figure 1(a) and (b) show room temperature FMR results at 3 and 13 GHz respectively, for a 15 nm lm as deposited ( i.e., without post-annealing). The resonances correspond well to derivatives of individual Lorentzians, to which we t to extract the linewidth and resonance eld. In Fig. 1(c) we plot the Lorentzian full-width at half maximum (FWHM) linewidth
Hversus frequency. The slope of this curve corresponds to an eective Gilbert damping parameter = (9.00.2)104. This agrees well with3 previous measurements of a 14.0 nm YIG lm grown by o-axis sputtering,12which had a damping parameter = (11.60.7)104, and is within the range of measurements on PLD lms of similar thickness.2 Figure 2 shows how the in-plane FMR spectra of the same YIG lm vary as a function of temperature. With decreasing temperature the data show a very large increase in the linewidth
H, a shift in the resonance eld, and a reduction in the amplitude of the signal, visible in the normalized curves as a reduction of the signal-to-noise ratio. The reduction in signal amplitude is consistent with the linewidth increase, given that the amplitude is expected25to scale with (
H)2. Below roughly 37 K, the resonances become so broad that they are no longer distinguishable using the coplanar waveguide system. This strong temperature dependence is similar to results reported by Shigematsu et al. ,26but it is not universal in ultra-thin YIG lms: e.g., Haidar et al. have observed in YIG lms grown by PLD a damping coecient that decreased by approximately a factor of two upon decreasing Tfrom room temperature to 8 K.16 By analyzing similar FMR resonances obtained at dierent values of microwave frequency, we can extract both the frequency and temperature dependencies of
H(Fig. 3). The frequency dependence at room temperature has an approximately linear dependence, similar to previous studies of high-quality YIG thin lms in this thickness range.2,11,27As a function of decreasing temperature not only does the overall magnitude of the linewidth grow by a large factor, but at the same time there are strong deviations from linearity in the frequency dependence. These nonlinearities are qualitatively similar to what one might expect from two-magnon scattering from defects at the interfaces of the YIG lm, but as we will argue below this mechanism cannot explain the very strong variations with temperature. We will instead argue that these changes can be accounted for by impurity relaxation within the YIG lm. We can obtain greater sensitivity in the FMR experiments, and thereby extend our study to temperatures lower than 37 K, by performing measurements in an X-band cavity. This comes at the cost of operating at xed frequency (9.4 GHz). We perform background subtrac- tion using in- and out-of-plane measurements in the cavity, as described in the supplementary material. Figure 4 shows the Tdependence of the FMR linewidth in these cavity measure- ments, with a comparison to the broadband coplanar waveguide results. (The waveguide values are interpolated from measurements at 9 and 10 GHz.) We nd excellent quantitative4 agreement between the two types of measurements. The cavity measurements reveal that
Hhas a maximum near 25 K, with a clear decrease at lower temperatures. The maximum linewidth is 28 times larger than the room temperature result at this frequency. In order to evaluate possible mechanisms for these very strong changes in linewidth with temperature, we must rst characterize how the magnetic anisotropy in the YIG lm varies with temperature. We do this based on the measured FMR resonance elds, tting to the Kittel equation for a magnetic thin lm with an in-plane magnetic eld25 f=j j 2q Hk r(Hk r+ 4M e): (1) Here is the gyromagnetic ratio, Hk ris the in-plane resonance eld for a given xed frequency f, and 4M eparameterizes the shape anisotropy and any additional contributions to the perpendicular magnetic anisotropy. We obtain good ts (see Fig. 5(a)) with no additional in-plane anisotropy contribution. In Eq. (1) we do not include a renormalization shift in the resonance frequency that can result from two-magnon scattering because this is small on the scale important to our analysis.17,18We also neglect a small shift in resonance eld that can arise from a static dipole interaction between the YIG and the paramagnetic GGG substrate28,29because this is also small, less than a 1% shift for temperatures above 15 K (see the supplementary material). The values of 4 M ewe obtain from the ts to Eq. (1) at dierent temperatures are shown in Fig. 5(b). We nd that 4 M eis signicantly larger than the simple shape anisotropy generated by the YIG saturation magnetization, 4 M s (determined from vibrating sample magnetometry (VSM) measurements presented in the supplementary material), indicating the presence of a positive uniaxial anisotropy, Hs= 4M e4M s, favoring an in-plane magnetization. We have conrmed the value of Hs and the form of its temperature dependence using FMR measurements with an out-of-plane magnetic eld. Figure 5(a) shows the frequency dependence of the resonance position with an out-of-plane eld at room temperature, and Fig. 5(b) shows the extracted value of 4 M e as a function of temperature from both waveguide and cavity FMR measurements. The large value of Hsis greater than expected from surface anisotropy15or magneto-crystalline anisotropy25of cubic YIG alone, so we tentatively ascribe the result to a growth-induced anisotropy, such as caused by tetragonal distortion. This is consistent with predictions and observations in YIG lms grown by PLD,4where the anisotropy is highly dependent on the growth conditions. The temperature dependence of Hsthat we obtain is qualitatively5 consistent with the spin uctuation model,30,31which predicts Hs(T)/[Ms(T)]2. Given this characterization of Hs(T), we can now evaluate whether two-magnon scattering from surface defects, a mechanism that is expected to be active for ultra-thin YIG lms at room temperature,2,15is capable of explaining the large increase in the linewidth
Hthat we observe at low temperature. This eect causes a linewidth that is nonlinear with frequency f, following the form32
H2M= (T) sin1sp !2+ (!0=2)2!0=2p !2+ (!0=2)2+!0=2; (2) where!= 2fand!0= 4M e(T). The temperature dependence in this equation is dominated by the scattering coecient ( T), whose expected temperature dependence17is (T)/[Hs(T)]2. Given our determination of Hs(T) above (using MefromHk rts as a worst-case scenario), the temperature dependence expected from the two-magnon scattering mechanism is illustrated by the red line in Fig. 4. This mechanism can explain at most a factor of 4 increase in the linewidth as the temperature is reduced from 300 to 0 K, far less than the factor of 28 that we observe. It also is incapable of explaining the peak in
Hwe measure near 25 K. Similar conclusions follow if one assumes30,31thatHs(T)/[Ms(T)]2, together with our VSM measurements of Ms(T). An alternative mechanism that can account for a much stronger temperature dependence for
His impurity relaxation, for example due to rare earth or Fe2+impurities in the YIG lm. Researchers in the 1960's produced a rich body of literature which shows that the linewidth in bulk YIG samples can increase dramatically at low temperatures when impurity relaxation is active.1,22{24,33The frequency and temperature dependence of
H in our samples can be explained well using a model of slowly-relaxing impurities.1,24,34The contribution to the linewidth from this mechanism is expected to have the form
HSR=A(T)! 1 + (!)2; (3) whereA(T) is a frequency-independent prefactor and is a temperature-dependent time constant. The lines in Fig. 3 are ts assuming that the linewidths are governed by this functional form plus a linear-in-frequency temperature-independent background contribution equal to the room-temperature dependence (Fig. 1(c)). The t parameters are shown in the supplemental material. The maximum near 25 K in the temperature dependence of
H6 (Fig. 4) is very similar to previous measurements in bulk YIG,1and corresponds within the slowly-relaxing impurity model to the condition !1. In conclusion, even when a YIG lm has a narrow linewidth at room temperaure indi- cating an apparently high-quality lm, the linewidth can still increase dramatically at low temperature, by well over an order of magnitude. This is generally undesirable. For exam- ple, this will make manipulation of YIG lms by anti-damping spin-transfer torques much less ecient at low temperature, and may block it entirely for practical purposes. Based on measurements of the temperature and frequency dependence of the eect, we suggest that the increased low-temperature linewidth is due to slowly relaxing impurities, perhaps rare earth or Fe2+impurities introduced during growth.1Given the high degree of sensitivity of the low-temperature linewidth to these impurities, we suggest that the low- Tlinewidth can serve as a useful gure of merit for optimizing growth protocols for ultra-thin YIG lms. SUPPLEMENTARY MATERIAL See the supplementary material for detailed information on the YIG growth by o-axis sputtering, saturation magnetization and substrate susceptibility, broadband and cavity measurement systems, resonance eld shift from substrate dipolar elds, thickness depen- dence of the linewidths, and tting parameters extracted from the frequency dependence of the linewidths. ACKNOWLEDGMENTS We acknowledge G. E. Rowlands and S. Shi for their help in building a low-temperature cryostat, M. S. Weathers for her help with X-ray re ectivity measurements, and B. Dzikovski for his help with the low-temperature cavity measurements preformed at ACERT (National Biomedical Center for Advanced ESR Technology). This research was supported by the National Science Foundation (DMR-1406333 and DMR-1507274), NSF/MRSEC program through the Center for Emergent Materials (DMR-1420451), the NSF/MRSEC program through the Cornell Center for Materials Research (DMR-1120296), the NIH/NIGMS pro- gram through ACERT (P41GM103521), the US Department of Energy (DOE) (DE-FG02- 03ER46054), and the US Department of Defense - Intelligence Advanced Research Projects7 Activity (IARPA) through the U.S. Army Research Oce (W911NF-14-C-0089). The con- tent of the paper does not necessarily re ect the position or the policy of the Government, and no ocial endorsement should be inferred. This work made use of the Cornell Center for Materials Research shared facilities. clj72@cornell.edu 1M. Sparks, Ferromagnetic-Relaxation Theory (McGraw-Hill, New York, 1964) p. 226. 2O. d'Allivy Kelly, A. Anane, R. Bernard, J. Ben Youssef, C. Hahn, A. H. Molpeceres, C. Car- retero, E. Jacquet, C. Deranlot, P. Bortolotti, R. Lebourgeois, J.-C. Mage, G. de Loubens, O. Klein, V. Cros, and A. Fert, Appl. Phys. Lett. 103, 082408 (2013). 3P. C. Dorsey, S. E. Bushnell, R. G. Seed, and C. Vittoria, J. Appl. Phys. 74, 1242 (1993). 4S. A. Manuilov, S. I. Khartsev, and A. M. Grishin, J. Appl. Phys. 106, 123917 (2009). 5S. A. Manuilov and A. M. Grishin, J. Appl. Phys. 108, 013902 (2010). 6B. Heinrich, C. Burrowes, E. Montoya, B. Kardasz, E. Girt, Y.-Y. Song, Y. Sun, and M. Wu, Phys. Rev. Lett. 107, 066604 (2011). 7M. C. Onbasli, A. Kehlberger, D. H. Kim, G. Jakob, M. Kl aui, A. V. Chumak, B. Hillebrands, and C. A. Ross, APL Mater. 2, 106102 (2014). 8B. M. Howe, S. Emori, Hyung-Min Jeon, T. M. Oxholm, J. G. Jones, K. Mahalingam, Yan Zhuang, N. X. Sun, and G. J. Brown, IEEE Magn. Lett. 6, 3500504 (2015). 9C. Tang, M. Aldosary, Z. Jiang, B. Madon, K. Chan, J. E. Garay, and J. Shi, Appl. Phys. Lett. 108, 102403 (2016). 10H. L. Wang, C. H. Du, Y. Pu, R. Adur, P. C. Hammel, and F. Y. Yang, Phys. Rev. B 88, 100406 (2013). 11H. L. Wang, C. H. Du, Y. Pu, R. Adur, P. C. Hammel, and F. Y. Yang, Phys. Rev. Lett. 112, 197201 (2014). 12T. Liu, H. Chang, V. Vlaminck, Y. Sun, M. Kabatek, A. Homann, L. Deng, and M. Wu, J. Appl. Phys. 115, 17A501 (2014). 13H. Chang, P. Li, W. Zhang, T. Liu, A. Homann, L. Deng, and M. Wu, IEEE Magn. Lett. 5, 1 (2014).8 14J. T. Brangham, K.-Y. Meng, A. S. Yang, J. C. Gallagher, B. D. Esser, S. P. White, S. Yu, D. W. McComb, P. C. Hammel, and F. Yang, Phys. Rev. B 94, 054418 (2016). 15C. L. Jermain, H. Paik, S. V. Aradhya, R. A. Buhrman, D. G. Schlom, and D. C. Ralph, Appl. Phys. Lett. 109, 192408 (2016). 16M. Haidar, M. Ranjbar, M. Balinsky, R. K. Dumas, S. Khartsev, and J. Akerman, J. Appl. Phys. 117, 17D119 (2015). 17R. Arias and D. L. Mills, Phys. Rev. B 60, 7395 (1999). 18D. L. Mills and S. M. Rezende, Top. Appl. Phys. 87, 27 (2003). 19J. R. Kirtley, L. Paulius, A. J. Rosenberg, J. C. Palmstrom, C. M. Holland, E. M. Spanton, D. Schiessl, C. L. Jermain, J. Gibbons, Y. K. K. Fung, M. E. Huber, D. C. Ralph, M. B. Ketchen, G. W. Gibson, and K. A. Moler, Rev. Sci. Inst. 87, 093702 (2016). 20I. M. Miron, K. Garello, G. Gaudin, P.-J. Zermatten, M. V. Costache, S. Auret, S. Bandiera, B. Rodmacq, A. Schuhl, and P. Gambardella, Nature 476, 189 (2011). 21L. Liu, C.-F. C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, Science 336, 555 (2012). 22J. F. Dillon and J. W. Nielsen, Phys. Rev. Lett. 3, 30 (1959). 23E. G. Spencer, R. C. LeCraw, and R. C. Linares, Phys. Rev. 123, 1937 (1961). 24P. E. Seiden, Phys. Rev. 133, A728 (1964). 25D. Stancil and A. Prabhakar, Spin Waves (Springer US, Boston, MA, 2009). 26E. Shigematsu, Y. Ando, R. Ohshima, S. Dushenko, Y. Higuchi, T. Shinjo, H. J. Von Bardeleben, and M. Shiraishi, Appl. Phys. Express 9(2016), 10.7567/APEX.9.053002. 27Y. Sun, Y.-Y. Song, H. Chang, M. Kabatek, M. Jantz, W. Schneider, M. Wu, H. Schultheiss, and A. Homann, Appl. Phys. Lett. 101, 152405 (2012). 28M. Mary sko, Czechoslov. J. Phys. 39, 116 (1989). 29M. Mary sko, J. Magn. Magn. Mater. 101, 159 (1991). 30H. Callen and E. Callen, J. Phys. Chem. Solids 27, 1271 (1966). 31D. P. Pappas, J. Vac. Sci. Technol. B 14, 3203 (1996). 32K. Lenz, H. Wende, W. Kuch, K. Baberschke, K. Nagy, and A. J anossy, Phys. Rev. B 73, 144424 (2006). 33C. Kittel, J. Appl. Phys. 31, S11 (1960). 34A. M. Clogston, Bell Syst. Tech. 34, 739 (1955).9 FIGURES10 (a) (b) (c) FIG. 1. Normalized ferromagnetic resonance spectra at (a) 3 GHz and (b) 13 GHz for a 15 nm YIG lm at room temperature, with an in-plane applied magnetic eld. (c) The frequency dependence of the linewidth corresponds to an eective Gilbert damping constant = (9.00.2)104.11 FIG. 2. Normalized ferromagnetic resonance spectra at 3 GHz with an in-plane applied magnetic eld for the YIG lm at dierent temperatures. Dierent normalization factors are used for data at dierent temperatures; the actual amplitude of the resonances decreases strongly with decreasing temperature, as re ected in the decreasing signal-to-noise ratio. With decreasing temperature, we observe a large increase in the resonance linewidth.12 FIG. 3. Linewidths (Lorentzian FWHM) from the in-plane FMR spectra measured at dierent temperatures. Solid lines are ts to the sum of the frequency dependence expected from a slowly- relaxing impurity mechanism in addition to the room temperature linear behavior. The dashed line for 37 K is a guide to the eye.13 FIG. 4. FMR linewidth at 9.4 GHz as measured by two techniques: (open black triangles) cavity measurements and (blue squares) coplanar waveguide measurements. We observe a peak near 25 K, where
His 28 times larger than at room temperature. The solid red line indicates temperature dependence expected from two-magnon scattering; this dependence is too weak to explain the variation in
H.14 (a) (b) FIG. 5. FMR resonance eld as a function of frequency for (squares) an out-of-plane applied mag- netic eld at room temperature and (circles) and in-plane applied elds at various temperatures. Solid lines are ts to the Kittel equation. (b) Temperature dependence of the eective magnetiza- tion, determined from the Kittel ts for (black circles) in-plane and (open triangles) out-of-plane applied magnetic elds. The open triangles below 50 K are from cavity measurements. The red line is 4times the saturation magnetization, from a t to VSM measurements (see the supplementary material). The eective magnetization re ected in the magnetic anisotropy is signicantly greater than the saturation magnetization.

2016-12-06

We report measurements of the frequency and temperature dependence of ferromagnetic resonance (FMR) for a 15-nm-thick yttrium iron garnet (YIG) film grown by off-axis sputtering. Although the FMR linewidth is narrow at room temperature (corresponding to a damping coefficient $\alpha$ = (9.0 $\pm$ 0.2) $\times 10^{-4}$), comparable to previous results for high-quality YIG films of similar thickness, the linewidth increases strongly at low temperatures, by a factor of almost 30. This increase cannot be explained as due to two-magnon scattering from defects at the sample interfaces. We argue that the increased low-temperature linewidth is due to impurity relaxation mechanisms that have been investigated previously in bulk YIG samples. We suggest that the low-temperature linewidth is a useful figure of merit to guide the optimization of thin-film growth protocols because it is a particularly sensitive indicator of impurities.

Increased low-temperature damping in yttrium iron garnet thin films

1612.01954v1

Bistability of Cavity Magnon Polaritons Yi-Pu Wang,1Guo-Qiang Zhang,1Dengke Zhang,1,Tie-Fu Li,2, 1,yC.-M. Hu,3and J. Q. You1,z 1Quantum Physics and Quantum Information Division, Beijing Computational Science Research Center, Beijing 100193, China 2Institute of Microelectronics, Tsinghua National Laboratory of Information Science and Technology, Tsinghua University, Beijing 100084, China 3Department of Physics and Astronomy, University of Manitoba, Winnipeg R3T 2N2, Canada (Dated: October 8, 2018) We report the first observation of the magnon-polariton bistability in a cavity magnonics system consisting of cavity photons strongly interacting with the magnons in a small yttrium iron garnet (YIG) sphere. The bistable behaviors are emerged as sharp frequency switchings of the cavity magnon-polaritons (CMPs) and related to the transition between states with large and small number of polaritons. In our experiment, we align, respectively, the [100] and [110] crystallographic axes of the YIG sphere parallel to the static magnetic field and find very dierent bistable behaviors (e.g., clockwise and counter-clockwise hysteresis loops) in these two cases. The experimental results are well fitted and explained as being due to the Kerr nonlinearity with either positive or negative coe cient. Moreover, when the magnetic field is tuned away from the anticrossing point of CMPs, we observe simultaneous bistability of both magnons and cavity photons by applying a drive field on the lower branch. PACS numbers: 75.30.Ds, 71.36. +c, 42.65.Pc Both quantum information processing [1] and future quan- tum internet [2] inevitably need e cient quantum informa- tion transfers among di erent physical systems. Hybrid quan- tum systems may provide hopeful solutions to these prob- lems [3, 4]. Recently, cavity magnonics has attracted much interest (see, e.g., [5–12]), which involves cavity photons strongly or ultrastrongly [13] interacting with collective spin excitations in a millimeter-scale yttrium iron garnet (YIG) crystal. This hybrid system gives rise to new quasiparticles called cavity magnon-polaritons (CMPs) [14, 15]. Benefiting from low damping rates of both magnons and cavity photons, this hybrid system is also expected to become a building block of the future quantum information network. Now, a versatile quantum information processing platform based on the coher- ent couplings among magnons, cavity microwave photons [5– 7], optical photons [16–19], phonons [20] and superconduct- ing qubits [21, 22] is being established, with the strong cou- pling between magnons and cavity photons being the core of the hybrid quantum system. For the cavity magnonics system, in addition to the hy- bridization between magnons and cavity photons, nonlinear eect can also play an important role. Originating from the magnetocrystalline anisotropy in the YIG [23, 24], this non- linearity is related to the Kerr e ect of magnons and the nonlinearity-induced frequency shift has been demonstrated in the dispersive regime at cryogenic temperature [25]. In this Letter, we observe the CMP bistability in a cavity magnonics system. In a wide range of parameters, the frequency of the CMPs is found to jump up or down sharply at the switching points where the CMPs transition from one state to another. To clearly demonstrate the nonlinear e ect, we implement the experiment by aligning the [100] and [110] crystallographic axes of the YIG sphere parallel to the externally applied static magnetic field, respectively. The measured results show very dierent features in these two cases, such as the blue- or red-shift of the frequency of the CMPs and the emergence of the clockwise or counter-clockwise hysteresis loop related to the CMP bistability. Also, we theoretically fit the experimental results well and explain the very di erent phenomena of the CMPs as being due to the Kerr nonlinearity with either a pos- itive ornegative coecient. To our knowledge, this work is the first convincing obser- vation of the bistability in CMPs. As an important nonlin- ear phenomenon, bistability is not only of fundamental in- terest in studying dissipative quantum systems [26, 27], but also has potential applications in switches [28, 29] and mem- ories [30, 31]. Our cavity magnonics system o ers a new platform to explore these applications. Also, the employed Kittle-mode magnons have distinct merits, such as the tun- ability with magnetic field and the maintenance of good quan- tum coherence even at room temperature [7, 10]. These ad- vantages may bring new possibilities in exploring nonlinear properties of the system. Indeed, we also observe the simul- taneous bistability of both magnons and cavity photons at a very o -resonance point by applying a drive field on the lower branch, where the optical bistability is achieved via the mag- netic bistability. This observation shows that the CMPs can serve as a bridge /transducer between optical and magnetic bistabilities, which paves a new way for using one e ect to induce and control the other. The experimental setup for our hybrid system is schemat- ically shown in Fig. 1(a). We use a three-dimensional (3D) rectangular cavity made of oxygen-free copper with inner di- mensions 44 :022:06:0 mm3and have a small YIG sphere of diameter 1 mm glued on an inner wall of the cavity at a magnetic-field antinode of the cavity mode TE 102. The cavity has three ports, with ports 1 and 2 as the input and output ports for measuring transmission spectrum and also with a specially designed port (i.e., port 3) in the vicinity of the YIG sphere for conveniently loading a microwave drive field via a loop an-arXiv:1707.06509v2 [quant-ph] 6 Feb 20182 FIG. 1. (a) Schematic diagram of the experimental setup. The 3D cavity with a small YIG sphere embedded is placed in the static magnetic field B0generated by an electromagnet. Ports 1 and 2 of the cavity, connected to a vector network analyzer (VNA), are used for transmission spectroscopy and port 3 connected to a mi- crowave (MW) source is for driving the YIG sphere. At the bottom, the magnetic-field distribution of the TE 102mode is magnified for clarity. The small YIG sphere is placed beside a circular-loop an- tenna and located at the magnetic-field antinode of the TE 102mode. (b) Transmission spectrum of the CMPs measured versus the magnet coil current (i.e., the static magnetic field) and the frequency of the probe field. Two vertical dashed lines indicate, respectively, the res- onance and the very o -resonance points at which we show the CMP bistability. tenna. Here we focus on the Kittel mode which is a spatially uniform mode of the ferromagnetic spin waves [32]. To re- duce the disturbance from other magnetostatic modes [33, 34], the small YIG sphere is placed at the uniform field of the an- tenna. The whole cavity with the YIG sphere embedded is placed in a static magnetic field B0created by a high-precision tunable electromagnet at room temperature. This bias mag- netic field, the magnetic component of the microwave drive field, and the magnetic field of the TE 102mode are nearly per- pendicular to one another at the site of the small YIG sphere. When the frequency of the Kittel-mode magnons is tuned in resonance with the microwave photons of the cavity mode TE102, anticrossing of energy levels occurs owing to the strong coupling between magnons and cavity photons, which gives rise to two branches of CMPs [see Fig. 1(b)]. The magnon- photon coupling strength is found to be gm=2=41 MHz from the energy splitting at the resonance (anticrossing) point. The fitted cavity-mode linewidth is =2(1+2+3+ int)=2=3:8 MHz, where i(i=1;2;3) is the decay rate of the cavity due to the ith port and intis the intrinsic loss of the cavity. Also, the Kittel-mode linewidth is found to be m=2=17:5 MHz. It is clear that the system is in the strong- coupling regime because gm>; m. Here we demonstrate the nonlinear e ect in the cavity magnonics system by driving the YIG sphere with a mi- crowave field via the loop antenna. We first focus on the case with the magnons in resonance with the cavity photons, where a CMP is the maximal superposition of a magnon and 051015200 51015200 5010015020025030035005101520051015200 5101520- 60-40-2002 0406005101520/s948LP/2π = -14.1 MHz Scan forward Scan backward- - - - Theory(a)/s948 LP/2π = -12.1 MHzPolariton frequency shift ΔLP/2π (MHz)/s948 LP/2π = -10.1 MHzD rive power Pd (mW)(b)P d = 25 dBm Scan forward Scan backward- - - - TheoryP d = 23 dBmP d = 21 dBmF requency detuning δLP/2π (MHz)FIG. 2. (a)
LP=2versus PdwhenLP=2=14:1,12:1, and 10:1 MHz. (b)
LP=2versusLP=2when Pd=25, 23, and 21 dBm. The black circle (blue triangle) dots are the forward (backward)-scanning results. The red dashed curves are theoreti- cal results obtained using Eq. (2), with LP=2=( m+)=4= 10:65 MHz. In (a), the characteristic constant c=(2)3is fitted to be 3:15, 3:55, and 3:82 MHz3/mW for curves from top to bottom. In (b), c=(2)3is fitted to be 1 :85, 2:52, and 3:22 MHz3/mW for curves from top to bottom. The [100] crystallographic axis of the YIG sphere is aligned parallel to the static magnetic field B0and the magnons are in resonance with the cavity mode TE 102. a cavity photon. Also, the YIG sphere is aligned to have its [100] crystallographic axis parallel to the static magnetic field B0. In Fig. 2(a), we measure the frequency shift
LP of the lower -branch CMPs versus the drive power Pdfor dif- ferent values of the drive-field frequency detuning (see [35] for the measurement method). Here the angular frequency !LPof the lower -branch CMPs is tuned to be at the anticross- ing point Ain Fig. 1(b) and the drive-field frequency detuning LP!LP!dis relative to the lower -branch CMPs, where !dis the angular frequency of the drive field. At the reso- nance point with !m=!c(where!mis the frequency of the Kittel-mode magnons and !cis the frequency of the cavity mode TE 102), a hysteresis loop is clearly seen at LP<0, revealing the emergence of the CMP bistability in the cavity magnonics system. This hysteresis loop is counter-clockwise when considering the increasing and decreasing directions of the drive power. Moreover, its area reduces when decreasing the frequency detuning jLPj. In Fig. 2(b), we measure the fre- quency shift
LPof the same lower -branch CMPs versus the frequency detuning LPfor di erent values of Pd. When the increasing and decreasing directions of LPare considered, a counter-clockwise hysteresis loop is also clearly shown, and its area decreases when reducing Pd. For a small YIG sphere driven by a microwave field with frequency!d, when its [100] crystallographic axis is aligned parallel to the static magnetic field, the cavity magnonics sys- tem has the Hamiltonian (setting ~=1) [25] H=!caya+!mbyb+Kbybbyb+gm(ayb+aby)3 + d(byei!dt+bei!dt); (1) where ay(a) is the creation (annihilation) operator of the cavity photons at frequency !c,by(b) is the creation (annihilation) operator of the Kittel-mode magnons at frequency !m, and dis the drive-field strength. As shown in [25], the Kerr term Kbybbyb, with a positive coe cient K=0Kan 2=(M2Vm), is intrinsically due to the magnetocrystalline anisotropy in the YIG material. Here 0is the magnetic permeability of free space, Kanis the first-order anisotropy constant, =gB=~is the gyromagnetic ratio (with gbeing the g-factor and Bthe Bohr magneton), Mis the saturation magnetization, and Vm is the volume of the YIG sphere. Note that the Kerr e ect is strengthened when reducing Vm. We consider the case with j!LP!djj!UP!dj, where !UPis angular frequency of the upper -branch CMPs. In such a case, the drive field applied to the YIG sphere generates po- laritons in the lower branch much more than the polaritons in the upper branch. Using a quantum Langevin approach, we obtain a cubic equation for the CMP frequency shift
LP[35]  (
LP+LP)2+ LP 22
LPcPd=0; (2) where LPis the damping rate of the lower -branch CMP and cis a coe cient characterizing the coupling strength between the drive field and the lower -branch CMPs. This cubic equa- tion provides a steady-state solution for the frequency shift of thelower -branch CMPs as a function of both the drive-field frequency detuning LPand the drive power Pd. Under ap- propriate conditions, it has three solutions, two of them stable and the additional one unstable. This corresponds to the bista- bility of the system. In Fig. 2, we also show the theoretical results (dashed curves) obtained using Eq. (2), which fit the experimental results very well. This verifies the experimental observation of the CMP bistability in our cavity magnonics system. The two stable solutions of
LPin Eq. (2) correspond totwostates of the system with large andsmall number of po- laritons in the lower branch [35]. Thus, in Fig. 2, each sharp switching of the frequency shift
LPis related to the transition between these two states. In the resonance case with !m=!c, we further imple- ment experiment by aligning the [110] crystallographic axis of the YIG sphere parallel to the static magnetic field B0. In Fig. 3(a), we measure the frequency shift
LPof the lower - branch CMPs versus the drive power Pdfor di erent values of the drive-field frequency detuning LPrelative to the lower - branch CMPs. In sharp contrast to Fig. 2(a), bistability of the lower -branch CMPs is now observed at LP>0. The area of the hysteresis loop also decreases when reducing LP, but the hysteresis loop becomes clockwise. Also, we present in Fig. 3(b) the frequency shift
LPversus the frequency detun- ingLPfor di erent values of Pd. The hysteresis loop is also counter-clockwise, similar to that in Fig. 2(b). As shown in [35], when the YIG sphere is aligned with its [110] crystallographic axis parallel to the static magnetic field B0, the Hamiltonian of the cavity magnonics system takes the -20-15-10-50- 20-15-10-500 50100150200250300350-20-15-10-50-30-20-100- 30-20-100- 60-40-2002 04060-30-20-100/s948LP/2π = 17.2 MHz Scan forward Scan backward- - - - Theory(a)/s948 LP/2π = 13.2 MHzPolariton frequency shift ΔLP/2π (MHz)/s948 LP/2π = 9.2 MHzD rive power Pd (mW)Pd = 25 dBm Scan forward Scan backward- - - - Theory(b)P d = 23 dBmF requency detuning /s948LP/2π (MHz)Pd = 21 dBmFIG. 3. (a)
LP=2versus PdwhenLP=2=17:2, 13:2, and 9:2 MHz. (b)
LP=2versusLP=2when Pd=25, 23, and 21 dBm. The black circle (blue triangle) dots are the forward (backward)- scanning results. The red dashed curves are theoretical results ob- tained using Eq. (2), with LP=2=10:65 MHz. In (a), c=(2)3is fitted to be4,3:5, and3:1 MHz3/mW for curves from top to bot- tom. In (b), c=(2)3is fitted to be3:01,3:46, and3:6 MHz3/mW for curves from top to bottom. The [110] crystallographic axis of the YIG sphere is aligned parallel to B0and the magnons are in reso- nance with TE 102. same form as in Eq. (1), but the coe cient of the Kerr term becomes negative, K=130Kan 2=(16M2Vm). The corre- sponding theoretical results (dashed curves) obtained using Eq. (2) are shown in Fig. 3, which are also in good agree- ment with the experimental results. Moreover, Figs. 2 and 3 show that the CMPs have blue(red)-shift in frequency when the [100] ([110]) crystallographic axis of the YIG sphere is aligned along the direction of the static magnetic field. These are due to the positive and negative Kerr coe cients K’s in the two di erent cases [35]. Finally, we demonstrate the Kerr e ect of magnons at an o-resonance point much away from !m=!c, where the magnet coil current is 4 :8 A [see the left dashed line in Fig. 1(b)]. At this point, the frequency of the lower -branch CMPs is about 9 :91 GHz and the frequency of the upper - branch CMPs is about 10 :08 GHz, as indicated by points B andCin Fig. 1(b), respectively. Also, the [100] crystallo- graphic axis of the YIG sphere is aligned parallel to the static magnetic field B0. In Fig. 4(a), we present the frequency shift
LPof the lower -branch CMPs versus the drive power Pdfor dierent values of the drive-field frequency detuning LPrela- tive to the lower -branch CMPs. Moreover, the frequency shift
LPversus the frequency detuning LPis shown in Fig. 4(b) for di erent values of Pd. As in Fig. 2, we also see counter- clockwise hysteresis loops. Because the magnon is a domi- nating component of the lower -branch CMP at this very o - resonance point, nearly the bistability of magnons is actually observed here, directly owing to the magnon Kerr e ect in the YIG sphere. The theoretical results (dashed curves) obtained4 020400 20400 2040040800 40800 40800 120 120 501001502002503003500120240 24- 100- 500 5 01 00024/s948LP/2π = -31.8 MHz Scan forward Scan backward- - - - Theory(a)/s948 LP/2π = -27.8 MHzPolariton frequency shift ΔLP/2π (MHz)/s948 LP/2π = -23.8 MHz(b) Scan forward Scan backward- - - - TheoryPd = 25 dBmP d = 23 dBmP d = 21 dBm Scan forward Scan backward- - - - Theory(c)/s948LP/2π = -27.8 MHzPolariton frequency shift ΔUP/2π (MHz)/s948 LP/2π = -23.8 MHz/s948LP/2π = -31.8 MHzD rive power Pd (mW)Pd = 25 dBm(d) Scan forward Scan backward- - - - TheoryP d = 23 dBmP d = 21 dBmF requency detuning /s948LP/2π (MHz) FIG. 4. (a)
LP=2versus PdwhenLP=2=31:8,27:8, and 23:8 MHz. (b)
LP=2versusLP=2when Pd=25, 23, and 21 dBm. The black circle (green triangle) dots are the forward (backward)-scanning results. The red dashed curves are theoreti- cal results obtained using Eq. (2), with LP=2=16:8 MHz. In (a),c=(2)3is fitted to be 25 :2, 25:0, and 25:3 MHz3/mW for curves from top to bottom. In (b), c=(2)3is fitted to be 17 :0, 19:0, and 22:7 MHz3/mW for curves from top to bottom. (c) and (d)
UP=2 measured under the same conditions as in (a) and (b), respectively. The black circle (oragne triangle) dots are the forward (backward)- scanning results. The blue dashed curves (except for the parts around small dips) are also obtained using Eq. (2), with the fitted ratio 
UP=
LPfor curves from top to bottom being 0 :060, 0:062, and 0:061 in (c) and 0 :062, 0:063, and 0:061 in (d). The [100] crystal- lographic axis of the YIG sphere is aligned parallel to B0and the magnons are far o resonance with TE 102. In addition, the small dips in, e.g., (c) and (d) can be fitted by introducing a magnetostatic mode with the negative Kerr coe cient [35]. using Eq. (2) also agree well with the experimental results. Under the conditions same as in Figs. 4(a) and 4(b), we fur- ther show the frequency shift
UPof the upper -branch CMPs versus the drive power Pd[Fig. 4(c)], as well as the frequency shift
UPof the upper -branch CMPs versus the drive-field frequency detuning LPrelative to the lower -branch CMPs [Fig. 4(d)]. Here, as in Figs. 4(a) and 4(b), the drive field is still applied on the lower branch. Now the cavity photon is a dominating component of the upper -branch CMP at this very o -resonance point, so it is nearly the bistability of cavity photons that is observed in Figs. 4(c) and 4(d). As one knows, both optical [40, 41] and magnetic [42] bistabilities are inter- esting phenomena of nonlinear systems. Here we observe the simultaneous bistability of both magnons and cavity photonsat a very o -resonance point by applying the drive field only on the lower branch, where the optical bistability is achieved via the magnetic bistability. Also, the experimental results in Figs. 4(c) and 4(d) fit well with the theoretical results (dashed curves), where the fitted ratio 
UP=
LPis close to the theoretical value 0 :065 [35]. In addition, some small dips are observed in, e.g., Figs. 4(c) and 4(d). By fitting with these dips, we attribute them to other polaritons stemming from the coupling between cavity photons and a magnetostatic mode with the negative Kerr coe cient [35]. In conclusion, we have demonstrated the bistable behaviors of the CMPs using a hybrid system consisting of microwave cavity photons strongly interacting with the magnons in a YIG sphere. We find that the switching points where the CMPs transition from one state to another depend on the drive power and the drive-field frequency detuning. When implementing the experiment, we align, respectively, the [100] and [110] crystallographic axes of the YIG sphere parallel to the static magnetic field and find very di erent bistable behaviors in these two cases. We theoretically fit the experimental results well and put these di erent bistable behaviors down to the Kerr nonlinearity with either a positive or negative coe cient. The cavity magnonics system possesses the merits of tun- ability and compatibility with other quantum systems, in which the magnons can be tuned by an external magnetic field and the magnon-photon coupling can be tuned by moving the YIG sphere inside the cavity. With regard to the bistabil- ity of the CMPs, the area of the hysteresis loop can be eas- ily controlled by a lower drive power and a tunable drive- field frequency. This may bring potential applications in re- alizing low-energy switching devices. Also, we observe si- multaneous bistability of both magnons and cavity photons by tuning the magnetic field. It paves a new path to con- trol the conversion between magnetic and optical bistabili- ties. Moreover, our study provides possible routes to ex- plore other nonlinear e ects of the system such as the creation of a frequency comb for frequency conversion [43] and the chaos of CMPs. With the available CMP bistability, the cavity magnonics system can also provide a new platform to under- stand the dissipative phase transition and the related critical phenomena [26, 44, 45]. These open up di erent directions for future studies. This work was supported by the National Key Re- search and Development Program of China (Grant No. 2016YFA0301200), the NSFC (grant No. 11774022), the MOST 973 Program of China (Grant No. 2014CB848700), and the NSAF (Grant No. U1330201 and No. U1530401). C.M.H. was supported by the NSFC (Grant No. 11429401). Y .-P. W and G.-Q. Z contribute equally to this work. Present address: Department of Engineering, University of Cambridge, Cambridge CB3 0FA, United Kingdom ylitf@tsinghua.edu.cn5 zjqyou@csrc.ac.cn [1] M. Wallquist, K. Hammerer, P. Rabl, M. 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2017-07-20

We report the first observation of the magnon-polariton bistability in a cavity magnonics system consisting of cavity photons strongly interacting with the magnons in a small yttrium iron garnet (YIG) sphere. The bistable behaviors are emerged as sharp frequency switchings of the cavity magnon-polaritons (CMPs) and related to the transition between states with large and small number of polaritons. In our experiment, we align, respectively, the [100] and [110] crystallographic axes of the YIG sphere parallel to the static magnetic field and find very different bistable behaviors (e.g., clockwise and counter-clockwise hysteresis loops) in these two cases. The experimental results are well fitted and explained as being due to the Kerr nonlinearity with either positive or negative coefficient. Moreover, when the magnetic field is tuned away from the anticrossing point of CMPs, we observe simultaneous bistability of both magnons and cavity photons by applying a drive field on the lower branch.

Bistability of Cavity Magnon Polaritons

1707.06509v2

Proximity magnetoresistance in graphene induced by magnetic insulators D. A. Solis,1A. Hallal,1X. Waintal,2and M. Chshiev1 1Univ. Grenoble Alpes, CEA, CNRS, Grenoble INP*, IRIG-SPINTEC, 38000 Grenoble, France 2Univ. Grenoble Alpes, CEA, IRIG-PHELIQS, 38000 Grenoble, France We demonstrate the existence of Giant proximity magnetoresistance (PMR) eect in a graphene spin valve where spin polarization is induced by a nearby magnetic insulator. PMR calculations were performed for yttrium iron garnet (YIG), cobalt ferrite (CFO), and two europium chalcogenides EuO and EuS. We nd a signicant PMR (up to 100%) values dened as a relative change of graphene conductance with respect to parallel and antiparallel alignment of two proximity induced magnetic regions within graphene. Namely, for high Curie temperature (Tc) CFO and YIG insulators which are particularly important for applications, we obtain 22% and 77% at room temperature, respec- tively. For low Tc chalcogenides, EuO and EuS, the PMR is 100% in both cases. Furthermore, the PMR is robust with respect to system dimensions and edge type termination. Our ndings show that it is possible to induce spin polarized currents in graphene with no direct injection through magnetic materials. I. INTRODUCTION Graphene is a two-dimensional (2D) material1,2that has attracted a lot of interest in view of its unique physical properties and applications potential in di- verse elds such as electronics, spintronics and quantum computing3{5. Due to its weak spin orbit coupling6{15 graphene possesses a long spin relaxation time and lengths even at room temperature16. While these charac- teristics oer an optimal platform for spin manipulation, it remains however a challenge to achieve robust spin po- larization eciently at room temperature. Several methods have been proposed in order to introduce ferromagnetic order on graphene, among which functionalization with adatoms17, addition of defects18,19, and by means of proximity eect via an ad- jacent ferromagnet20{25. The latter approach attracted a lot of interest using magnetic insulators (MI) as a sub- strate to induce exchange splitting in graphene. When a material is placed on top of a magnetic insulator, it can acquire proximity induced spin polarization and ex- change splitting20resulting from the hybridization be- tweenpzorbitals with those of the neighboring magnetic insulator. For practical purposes, the implementation in spintronic devices of this kind of materials could lead to lower power consumption since no current injection across adjacent ferromagnet (FM) is required as in case of traditional spin injection techniques. Experimentally, the existence of proximity exchange splitting via mag- netic insulator in graphene have been demonstrated with exchange elds up to 100 T using the coupling between graphene and EuS23. For yittrium irog garnet/graphene (YIG/Gr) based system, using non-local spin transport measurements, Leutenantsmeyer et al.24demonstrated exchange eld strength of 0.2 T. Another possibility of inducing exchange splitting in graphene using FM metal, by separating them by alternative 2D material such as hexa-boron nitride (hBN), was also proposed theoreti- cally21. Recent studies have suggested the creation ofgraphene-based devices where EuO-graphene junction can act as a spin lter and spin valve simultaneously by gating the system26. It was also demonstrated27that a double EuO barrier on top of a graphene strip can exhibit negative dierential resistance making this system a spin selective diode. However, the drawback of using EuO is its low Curie temperature and the predicted strong electron doping20. It was proposed therefore using high Curie temperature materials such as YIG or cobalt fer- rite (CFO)28. Indeed, a large change in the resistance of a graphene-based spintronic device has been reported recently where the heavy doping induced by YIG could be treated by gating29. In this Letter we demonstrate the existence of Proxim- ity Magnetoresistance (PMR) eect in graphene for four dierent magnetic insulators (MI), YIG, CFO, europium oxide (EuO) and europium sulde (EuS). Using ab ini- tio parameters reported in Ref. [ 28], we show that for YIG and CFO based lateral graphene-based devices with armchair edges, PMR values could reach 77% and 22% at room temperature (RT), respectively. With chalco- genides, EuS and EuO, PMR values can reach 100% at 16 K and 70 K, respectively. In addition, we demon- strate the robustness of this eect with respect to sys- tem dimensions and edge type termination. Further- more, our calculations with spin-orbit coupling (SOC) included does not signicantly aect the PMR. These ndings will stimulate experimental investigations of the proposed phenomenon PMR and development of other proximity eect based spintronic devices. II. METHODOLOGY In order to calculate conductances and PMR, we em- ployed the tight-binding approach with scattering matrix formalism conveniently implemented within the KWANT package30. The system modeled is shown in Fig. 1 and comprises two identical proximity induced magnetic re- gions of width Wand length Lresulting from insula- tors with magnetizations M1andM2, separated by aarXiv:1906.04469v1 [cond-mat.mes-hall] 11 Jun 20192 FIG. 1. (color online) Lateral spintronic device comprising two magnetic insulators on top of a graphene sheet. The magnetic graphene regions have a length L, widthWand are separated by a distance d.distancedof nonmagnetic region of graphene sheet with armchair edges. Both magnetic graphene regions are sep- arated from the leads L1andL2by a small pure graphene region. In order to take into account the magnetism aris- ing in graphene from the proximity eects induced by the MI's, in the Hamiltonian are used the parameters obtained for dierent MI's in Ref. [ 28]. It is important to note that the magnetic regions do not aect the linear dispersion of graphene bands, except breaking the valley and electron-hole symmetry resulting in spin-dependent band splitting and doping. The discretized Hamiltonian for the magnetic graphene regions can be expressed as: H=X iX ltlcy (i+l)1ci0+h:c:+X i01X =0[+ (1)
]cy i[~ m:~ ]ci0+X i1X =0[ED+ (1)
s]cy ici (1) wherecy i(cy i) creates (annihilates) an electron of type = 0 for A sites and = 1 for B sites on the unit celliwith spin="(#) for up (down) electrons. ~ m and~ respectively represent a unit vector that points in the direction of the magnetization and the vector of Pauli matrices, so that ~ m:~  =mxx+myy+mzz. The anisotropic hopping tlconnects unit cells ito their nearest neighbor cells i+l. Parameters ,
,
sare dened via exchange spin-splittings e(h) of the elec- trons (holes) and spin-dependent band gaps
dened in Ref. [ 28]. EDindicates the Dirac cone position with respect to the Fermi level. The Hamiltonian for the whole device is obtained by making aforementioned parameters spatially dependent. To obtain hopping parameters of Hamiltonian (1), we tted tight-binding bands to those obtained from rst principles calculations in Ref. [ 28]. The results of the tting procedure in case of graphene magnetized by YIG, CFO, EuS and EuO are shown in Fig. 2(a), (b), (c) and (d), respectively. The corresponding hopping parameters are given in Table I. As one can see, the graphene bands obtained with tight-binding Hamiltonian given by Eq. 1 are in good agreement with those obtained using Density Functional Theory (DFT) conrming suitability of our model for transport calculations. Of note, due to the presence of supercial tension at the interface between CFO and graphene, hopping parameters in this case are anisotropic as they depend on direction to the nearest neighbor as specied in the inset of Fig. 2(b). The conductance for parallel and antiparallel congu- rations of magnetizations M1andM2in the linear re- sponse regime is then obtained according to: GP(AP)=e hX Z T P(AP)@f @E dE; (2)TABLE I. hopping parameters used in equation 1 for each magnetic insulator considered. Material Hopping di- rectionspin up (eV) spin down (eV) YIG t 3.6 3.8 CFOt1 1.38 1.44 t2 1:41ei0:011:48ei0:01 t3 1:36ei0:021:44ei0:02 EuS t 4.5 4.8 EuO t 4.9 4.3 whereT P(AP)indicates spin-dependent transmission probability for parallel(antiparallel) magnetizations con- gurations and f= 1=(e(E)=kBT+ 1) represents the Fermi-Dirac distribution with andTbeing electro- chemical potential (Fermi level) and temperature, respec- tively. It is important to note that temperature smearing has been taken into account using the Curie temperature of each MI. The PMR amplitude has been dened according to fol- lowing expression: PMR =GPGAP GP+GAP 100%; (3) In order to determine the impact of the system dimen- sions on the PMR, several calculations were carried out for dierent lengths, widths and separations of the mag- netic regions. Furthermore, we checked the robustness of PMR on edge type termination by calculating the PMR for systems with zigzag, armchair and rough edges. The latter were created by removing atoms and bounds ran- domly and deleting the dangling atoms at the new edges.3 FIG. 2. (color online) Band structure obtained using tight-binding Hamiltonian dened by Eq. (1) (solid lines) tted to the band structure from DFT spin majority (green open circles) and spin minority (black lled circles) data for the cases with (a) YIG, (b) CFO, (c) EuS and (d) EuO from Ref. [ 28]. The inset in (b) shows the anisotropic hoppings reported in Table I . III. RESULTS In Fig. 3 we present the PMR curves for lateral device structures based on YIG, CFO, EuS and EuO on top of a graphene sheet with armchair edges. Taking into account Curie temperatures for these materials, the curves were smeared out using 16 K (70 K) for EuS (EuO), and 300 K for YIG and CFO cases. For system with YIG we found a maximum PMR value of 77% while for CFO the value obtained was 22%. In case of chacolgenides EuS and EuO used, the maximum PMR values reach 100%. Among the materials studied, YIG represents the most suitable candidate for lateral spintronic applications due to both high Curie temperature and considerably large PMR value. In order to elucidate the underlying physics behind these PMR results, let us analyze details of the con- FIG. 3. (color online) Proximity magnetoresistance dened by Eq. 3 as a function of energy in respect to the Fermi level for YIG (blue circles), CFO(red squares), EuS(black dia- monds) and EuO(green triangles) using temperature smeared conductances at T=300 K, 300 K, 16 K and 70 K, respec- tively. System dimensions are L= 49:2 nm,W= 39:6 nm andd= 1:5 nm.ductance behaviour. In Fig. 4(a)-(b) we reproduce the graphene bands in proximity of YIG and corresponding transmission probabilities resolved in spin for P and AP congurations at T= 0 K for a system with dimensions L= 49:2 nm,W= 39:6 nm andd= 1:5 nm. One can see that for energies between -0.88 eV and -0.78 eV there is no majority spin states present and the only contri- bution to transmission T# Pis from minority spin channel (Fig. 4(b), red solid line). In other words, the situation within this energy range is half-metallic giving rise to maximum PMR values of 100% using \pessimistic" def- inition given by Eq. (3). The similar situation is for en- ergy ranges between -0.72 eV and -0.75 eV but this time the only contribution T" Pis from majority spin channel (Fig. 4(b), red dashed line). One should point out here that the conduction prole here is due combining both magnetic and nonmagnetic regions into one scattering region. The conductance of a pure graphene nanoribbon sheet represents quantized steps due to transverse con- nement with no conductivity at zero energy depending on its edges. Inducing magnetism within graphene sheet leads to symmetry breaking with the shift of exchange splitted gaps in the vicinity of Dirac cone region below the Fermi level. This leads to characteristic conductance prole with two minima at around -0.8 eV and 0 eV (not shown here) due to the Dirac cone regions of magne- tized and the pure graphene. The corresponding con- ductances for the parallel ( GP) and for the antiparallel (GAP) magnetic congurations at T= 300 K are shown in Fig 4(c). Interestingly, even at room temperature the PMR for YIG based structure preserves a very high value of about 77% as already pointed above, a behavior that is very encouraging for future experiments on PMR. As a guide to the eye with dashed lines we highlight the en- ergy value where the PMR has a maximum in Fig. 4. Since the edges may strongly in uence the aforemen- tioned properties of the system, we next explore the robustness of PMR against dierent edge types of the graphene channel of the proposed device. It is well known that electric eld can trigger half-metallicity in zigzag nanoribbons due to the antiferromagnetic inter- action of the edges31. On the other hand, graphene4 FIG. 4. (color online) (a) Band structure reproduced using the DFT parameters from Ref. [ 28] for graphene in proximity of YIG. (b) Transmission probabilities for majority (dash lines) and minority (solid) spin channel for parallel (red) and antiparallel (blue) magnetization congurations at T= 0 K for a system with dimensions L= 49:2 nm,W= 39:6 nm andd= 1:5 nm. (c) Resulting conductance for parallel (red circles) and antiparallel (blue squares) magnetization congurations at 300 K. (d) PMR for device with armchair (blue circles), rough (red squares) and zigzag (black triangles) edge termination of graphene. PMR proles as a function of (e) L, (f)Wand (g)d. (h) Dependence of PMR for the energy outlined by dashed line in (e), (f) and (g) as a function of L(black circles), W(red squares) and d(blue triangles). The green square highlights the region where PMR becomes independent of system dimensions. nanoribbons with armchair edges can display insulating or metallic behaviour depending on graphene nanoribbon (GNR) width32,33. Armchair and zigzag edges are par- ticular cases and the most symmetric edge directions in graphene. But one can cut GNR at intermediate angu- lar direction between these two limiting cases giving rise to an intermediate direction characterized by a chirality angle34. Graphene band structure is highly dependent on. When the angle is increased, the length of the edge states localized at the Fermi level decrease and eventu- ally disappear in the limiting case when = 30, i. e. when acquires armchair edge. In the laboratory condi- tions, graphene sheets are nite and have imperfections that in uence their transport properties. For defects at the edges, it has been demonstrated that rough edges can diminish the conductance of a graphene nanoribbon as was shown in Ref. [ 35] or may exhibit a nonzero spin conductance as reported in Ref. [ 36]. In order to demonstrate the robustness of PMR with respect to the edge type, we thus performed calculations with the same system setup (Fig. 1) but this time for various edge terminations. The resulting PMR behavior for the cases with armchair, rough edges and zigzag are shown in Fig. 4(d). The former have been modeled by creating extended vacancies distributed randomly. It is clear that the maximum PMR value does not present a signicant variation maintaining for all cases PMR val- ues around 75%. With this results in hand we can claim that the PMR is indeed robust with respect to edge ter-mination type. As a next step, we checked the dependence of the PMR on dierent system dimensions, i.e. the length of the magnetic region L, system width Wand the separation between the magnetic regions d. The corresponding de- pendences are presented respectively in Fig. 4(e),(f) and (g). One can see that for all energy ranges the PMR ratio has a tendency to increase as a function of Lapproaching limiting value of 77% at energies around -0.81 eV indi- cated by a dashed line Fig. 4(e). As for dependence of the PMR as a function of GNR width W, clear oscilla- tions due to quantum well states formation are present with a tendency to vanish as system widens (Fig. 4(f)). On a contrary, the PMR shows almost constant behav- ior as a function of separation between the magnets d (Fig. 4(g)) due to the fact that transport is in ballistic regime. For convenience, we summarize all these depen- dencies in Fig. 4(h) at energy -0.81 eV as a function of L,Wandd. One can clearly see that the PMR saturates as system dimensions are increased. At the same time, it shows the oscillations in the PMR for small Was well as the invariance of the PMR with respect to d. For large dimensions highlighted by the green box in Fig. 4(h), we can claim that the PMR is indeed robust, and the maxi- mum PMR value would be eventually limited only by the magnitude of the spin diusion length in the system. Finally, we consider the impact of spin-orbit coupling on the PMR. Despite weak SOC within graphene, the proximity of adjacent materials can induce the interfacial5 FIG. 5. (color online) PMR dependencies for three values of Rashba spin-orbit interaction parameter Rdened by Eq. (4) for YIG-based system with armchair edges and of dimensions L= 49:2 nm,W= 39:6 nm andd= 1:5 nm. The dashed line is a guide to the eye that shows the maximum value when R= 0 eV. Rashba SOC7. Rashba type SOC is included into our tight-binding approach adding the following term: HSO=iRX i0X lcy (i+l)1[x 0dx ly 0dy l]ci00+h:c: (4) where the vector ~dl= (dx l;dy l) connects the two nearest neighbours, Rindicates the SOC strength. The values ofRare generally lie in the range between 1-10 meV (see, for instance, in Ref. [ 37]). Keeping in mind this in- formation, we present in in Fig. 5 the PMR dependences for three values of spin-orbit interaction. One can see that increasing the strength of SOC Rlower the PMR. This behavior is expected and could be attributed to thefact that spin-orbit interaction mixes the spin channels. These dependencies allows us to conclude that PMR is quite robust also against SOC and even in the worst sce- nario remains of the order of 50 % (cf. black triangles and blue circles in Fig. 5). IV. CONCLUSIONS In this paper we introduced the proximity induced magnetoresistance phenomenon in graphene based lateral system comprising regions with proximity induced mag- netism by four dierent magnetic insulators. For YIG and CFO based devices we found PMR ratios of 77% and 22% at room temperature, respectively. For chalcogenide based systems, i.e. with EuS and EuO, we found PMR values of 100% for both at 16 K and 70 K, respectively. Very importantly, it is demonstrated that the PMR is robust with respect to system dimensions and edge type termination. Furthermore, the PMR survives in case of the presence of SOC decreasing only by about a half even in the case of considerably big SOC strength values. We hope this work will encourage further experimental re- search and will be useful for the development of novel generation of spintronic devices based on generation and exploring spin currents without passing charge currents across ferromagnets. V. ACKNOWLEDGMENTS We thank J. Fabian and S. Roche for fruitful discus- sions. This project has received funding from the Euro- pean Unions Horizon 2020 research and innovation pro- gramme under grant agreements No. 696656 and 785219 (Graphene Flagship). X.W. acknowledge support by ANR Gransport. 1K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. 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2019-06-11

We demonstrate the existence of Giant proximity magnetoresistance (PMR) effect in a graphene spin valve where spin polarization is induced by a nearby magnetic insulator. PMR calculations were performed for yttrium iron garnet (YIG), cobalt ferrite (CFO), and two europium chalcogenides EuO and EuS. We find a significant PMR (up to 100%) values defined as a relative change of graphene conductance with respect to parallel and antiparallel alignment of two proximity induced magnetic regions within graphene. Namely, for high Curie temperature (Tc) CFO and YIG insulators which are particularly important for applications, we obtain 22% and 77% at room temperature, respectively. For low Tc chalcogenides, EuO and EuS, the PMR is 100% in both cases. Furthermore, the PMR is robust with respect to system dimensions and edge type termination. Our findings show that it is possible to induce spin polarized currents in graphene with no direct injection through magnetic materials.

Proximity magnetoresistance in graphene induced by magnetic insulators

1906.04469v1

Giant magnon spin conductivity approaching the two- dimensional transport regime in ultrathin yttrium iron gar- net films X-Y . Wei1,*, O. Alves Santos1, C.H. Sumba Lusero1,†, G. E. W. Bauer1,2, J. Ben Youssef3, and B. J. van Wees1 1Physics of Nanodevices, Zernike Institute for Advanced Materials, University of Groningen, Ni- jenborgh 4, 9747 AG Groningen, The Netherlands 2WPI-AIMR &Institute for Materials Research &CSRN, Tohoku University, Sendai 980-8577, Japan 3Lab-STICC, CNRS- UMR 6285, Université de Bretagne Occidentale, 6 Avenue Le Gorgeu, 29238 Brest Cedex 3, France *e-mail: x.wei@rug.nl †Current address: Leibniz Institute for Solid State and Materials Research, IFW, 01069 Dresden, Germany Conductivities are key material parameters that govern various types of transport (elec- tronic charge, spin, heat etc.) driven by thermodynamic forces. Magnons, the elementary ex- citations of the magnetic order, flow under the gradient of a magnon chemical potential1–3in proportion to a magnon (spin) conductivity σm. The magnetic insulator yttrium iron garnet (YIG) is the material of choice for efficient magnon spin transport. Here we report an unex- pected giant σmin record-thin YIG films with thicknesses down to 3.7 nm when the number 1arXiv:2112.15165v3 [cond-mat.mes-hall] 12 Jan 2022of occupied two-dimensional (2D) subbands is reduced from a large number to a few, which corresponds to a transition from 3D to 2D magnon transport. We extract a 2D spin con- ductivity (≈1S)at room temperature, comparable to the (electronic) spin conductivity of the high-mobility two-dimensional electron gas in GaAs quantum wells at millikelvin tem- peratures4. Such high conductivities offer unique opportunities to develop low-dissipation magnon-based spintronic devices. The spin current density in metals is the difference of the up- and down-spin charge current densities measured in A/m2,which is driven by a gradient of the spin chemical potential (often called spin accumulation) ∇µs. The spin conductivity σ, defined asjs=j↑−j↓=σs∇µs/e, can be expressed in electrical units as S/m. In a magnetic insulator where charge currents are absent, each magnon carries angular momentum /planckover2pi1,which is equivalent to the spin current in metals carried by a pair of spin-up ( +/planckover2pi1/2) and spin-down (−/planckover2pi1/2) electrons that flow in opposite directions. A magnon current jmcan be defined as its number current times electron charge e. In magnetic insulator-based spintronic devices, magnon spin currents are injected, detected, and modulated by microwave striplines or electric contacts made from a heavy metal for charge-spin conversion5–9. The corresponding spin conductivity, magnon conductivity σm, is the current density divided by the gradient of the magnon chemical potential. The unit of the magnon conductivity in jm= σm∇µm/e, whereµmis the magnon chemical potential, is then the same as that of electrons in a metal1. The value of σm= 4×105S/m in a 210 nm thick YIG film at room temperature6 corresponds to the electronic conductivity of bad metals. 2The high magnetic and acoustic quality of magnetic insulators make them the ideal material for all-magnon logical circuits and magnon-based quantum information10, 11. An example of recent progress in magnon-based computing is an integrated magnonic half-adder based on 350 nm wide wave guides make from 85-nm-thick YIG films12. However, these devices operate with coherent magnons (∼GHz ) excited by narrow microwave striplines which can not be integrated into an all- electrical circuit. Therefore, it is attractive to inject magnons electrically13, but those are mainly thermal (∼Thz) and scatter much stronger at phonons. Also, scalability to smaller structure sizes, essential for future high-performance processing units, requires micro and nanofabrication in all dimensions. The first step is the growth of films of a few or even a single unit cell. Previously, magnon transport was reported in transistor structures on films down to about 10 nm, which shows that ultrathin films can maintain high quality and display intriguing non-linear magnon effects14, 15. However, the scattering by surface roughness is expected to be increase in even thinner films16. This could be an obstacle for magnon spin transport in ultrathin YIG films that hinders observation of a transition from three dimensional magnons to two dimensional magnon gas when the thermal wavelength λthermal = 2π/radicalbig /planckover2pi1γD/(kBT)(∼3 nm at room temperature) approaches the thickness of the films tYIG, where /planckover2pi1is the reduced Planck constant, γis the gyromagnetic ratio, Dis spin wave stiffness and kBis Boltzmann constant. Here we report measurements of the magnon conductivity of YIG films with thicknesses down to 3.7 nm. Much to our surprise, the magnon transport turns out to be strongly enhanced in the ultrathin regime. We report a drastical increase in magnon conductivity of up to σm= 1.6×108S/m at room temperature that even exceeds the electronic spin conductivity of high- 3purity copper. This increase is intimately connected to the small number of occupied subbands and apparent domination by the lowest subband in our films. These results can importantly boost the performance of magnon-based information technology10, 17. We employ a non-local configuration6(Figure 1a) of two Pt thin film strips with length L at a distance don top of YIG films grown on gallium gadolinium garnet (GGG) by liquid phase epitaxy. An electric charge current Ithrough the injector generates a transverse spin current due to the spin Hall effect18, resulting in a spin accumulation µsin Pt at the interface to YIG. The injector-conversion coefficient ηinj=µs/(eI)depends on the properties and dimensions of the Pt strip as explained in the Section I of the supplementary information (SI Section I). The effective interface spin conductance results from the exchange interaction across the interface and produces a magnon chemical potential µmon the YIG side of the interface that acts as a magnon source , whereµm≈µssince the interface spin resistance can be ignored (see SI Section III). The detector electrode is a magnon drain that absorbs magnons and converts them into a spin current jsentering the Pt detector electrode. The inverse spin Hall effect generates a detector voltage Vnlwith detector- conversion coefficient ηdet=Vnl/jdet s. By reciprocity, ηinj=ηdetwhen injector and detector contacts have the same properties (see SI Section I for details). Since the signal scales with L, a normalized non-local resistance can be defined as Rnl=Vnl/(IL). The magnon conductance follows from the measured non-local resistance Gm=1 ηinjηdetVnl I=RnlL ηinjηdet. (1) The magnon conductivity σmas a function of the thickness tYIGof the YIG films in Figure 1 then 4follows from the magnon spin conductance σm=Gmd tYIGL. (2) For the films with tYIGmuch smaller than the magnon relaxation length λmas well as the lateral device dimension, µmcan be considered constant in the zdirection . Therefore, we use following equation to describe magnon diffusion6, 19 Rnl=σmtYIGηinjηdet λmcschd λm→ σmtYIGηinjηdet d 2σmtYIGηinjηdet λmexp(−d λm)ford/lessmuchλm d/greatermuchλm. (3) When the spacing dis smaller than λm, it is the Ohmic regime in which the magnons are con- served,Rnl(Gm)∼d−1.Otherwise, the signal decays exponentially as a function of distance due to magnon relaxation. We measure Rnlat room temperature as a function of an external in-plane magnetic field Hexwith|Hex|= 50 mT, which we rotate in the plane (Figure 1a). We modulate the AC current Iby a low frequency (18 Hz) and detect the first/second harmonic signal Vnl(ω)/Vnl(2ω)by lock- in amplifiers (see Methods). Vnl(2ω)depends on the spin Seebeck generation and diffusion of magnons under an inhomogeneous temperature profile, which renders interpretation difficult20, 21 (see SI Section V). Therefore, we focus on Vnl(ω)that follows the formula R1ω nl(α) =R1ω nlcos2α+R1ω 0, (4) whereR1ω 0is an offset resistance (see Methods) and αis the angle of Hexwith thex-axis. In Figure. 2, the the angle-dependent measurements in various thickness YIG films show that R1ω nl becomes four times larger when the film is over fifty times thinner from 210 nm to 3.7 nm. We also 5observe a strongly increased non-local signal in ultrathin films in Figure 3 as a function of contact separation for a wide range of tYIGincluding results on ultrathin YIG films for 400 nm wide Pt strips and for thicker films tYIG≥210nm6, 22. Figure 4a emphasizes the dramatic enhancement of R1ω nlfor the thinnest films down to tYIG= 3.7nm and fixed d= 2.5µm, which can be attributed to thetYIGdependence of σmbecauseλm>2.5µm for all thicknesses (see SI Section IV for details). R1ω nlincreases with decreasing thickness and saturates for both the thinnest and thickest films. A finite-element model1can simulate the depth ( z) dependence of µmwhentYIG>λm(see SI Section I for details). This leads to a limiting σm→3×104S/m in Figure 4b for thicker films, which represents the bulk value. The simulated Rnlvalues ford= 2.5µm in Figure 4c have been fitted toR1ω nlin Figure 4a by conductivities that are strongly enhanced in the regime tYIG< λm. FortYIG= 3.7nm, the magnon conductivity σm= 1.6×108S/m is four orders of magnitude larger compared to the bulk value, exceeding the electronic conductivity of pure metals such as copper with σe= 6×107S/m23. The observed saturation at tYIG→0appears to reflect an increased role of surface roughness scattering that we do not model explicitly. A magnon conductivity that diverges for tYIG→0likeσm∼σ2D mt−1 YIGsimply suggests two- dimensional transport. In Figure 4c, it shows that σ2D msaturates for tYIG<10nm, i.e. higher 2D subbands do not contribute significantly even though they are still populated (see below). Extrapo- lation to zero thickness leads to σ2D m≈1S. This value at room temperature is comparable to that of the high-mobility two-dimensional electron gas at millikelvin temperatures, which is σ2D e≈1.4S in GaAs quantum wells4. 6The magnons propagate in the plane with wave vector kand form perpendicular standing spin waves (PSSW) in zdirection labeled by an integer n. The exchange interaction scales like ∼k2and dominates the magnon dispersion εnkat thermal energies ( ≈kBT) with small magne- todipolar corrections. A magnon with energy εnk=/planckover2pi1γD/parenleftbig k2+ (nπ/t YIG)2/parenrightbig contributes to the conduction proportional to its thermal occupation Nnk= 1/{exp [εnk/(kBT)]−1}. For YIG γ/2π= 28 GHz/T and the spin wave stiffness24D= 5×10−17Tm2. The highest occupied subbandndefined as n= int/parenleftBigg tYIG π/radicalBigg kBT /planckover2pi1γD/parenrightBigg (5) atεn0<kBTas a function of thickness25, where int(x) is the greatest integer no more than x. For tYIG= 3.7nm, only three approximately 2D subbands are occupied at room temperature. The simplest model for the magnon conductivity in ν(=2, 3) dimensions follows from the Boltzmann equation with a constant relaxation time τ σ(ν) m=e2τ(ν) kBT/integraldisplaydk (2π)ν/parenleftbigg∂εk /planckover2pi1∂kz/parenrightbigg2eεk/(kBT) (eεk/(kBT)−1)2(6) whereεk=/planckover2pi1γDk2. Magnetic freeze-out experiments show that the contributions from the low- frequency magnons ( ∼GHz ) is significant even at room temperature, presumably reflecting low mobilities of thermal exchange magnons26–28. This can be represented by a high momentum cut- offK∞∼1/nmat magnon frequencies ε∞//planckover2pi1∼THz . In the high temperature limit kBT/greatermuchεk the conductivities do not depend on γD: σ(3) m=2e2kBTτ(3) 3/planckover2pi12π2K∞, (7) σ(2) m=e2kBTτ(2) π/planckover2pi12logK∞ K0, (8) 7whereK0is a low momentum cutoff by the magnon gap of ε0//planckover2pi1∼GHz . By equating these equations with the experimental results σ(3) m≈3×104S/m and the present σ(2) m≈1S and using the scattering times as adjustable parameters, we arrive at τ(3)≈40fs andτ(2)≈0.1ns. The short scattering time in three dimensions can be explained by highly efficient magnon-phonon scattering at room temperature1. While the high-momentum cut-off plays an important role in 3 dimensions, the near independence of σ(2) memphasizes the importance of the near band-gap excitations for transport in two dimensions. Coherent magnons excited at GHz frequencies can propagate over cm’s in spite of their small group velocity because they scatter only weakly at phonons29. Their contribution has a much larger effect on transport in ultrathin films than in the bulk, which is consistent with the magnetic field and temperature dependence reported in the SI. The estimated scattering time of τ(2)= 0.1ns may be limited by the film roughness scattering. The precise mechanism can be elucidated only by more extensive experimental and theoretical studies of the temperature and field dependence. While magnon-based devices do not suffer from Joule heating, magnon transport is not dissipationless6, 30even for transport on length scales shorter than the magnon relaxation length where magnons are conserved. The observed giant magnon conductivity is therefore excellent news, implying low dissipation from magnon-phonon scattering even at room temperature. Ultra- thin films can therefore be driven with relative ease into the non-linear regime in e.g. magnon spin transistors14, 15, facilitating electrically-induced magnon Bose-Einstein condensation and magnon spin superfluidity31–33. The robustness of the transport of the magnetic order for thin films of close to the monolayer thickness should allow magnon transport in nanostructures such as constrictions, 8wires and dots with feature sizes of a few nanometer without loss of magnetic functionality. Methods Fabrication The YIG films are grown on Gd 3Ga5O12(GGG) substrates by liquid-phase epitaxy (LPE) at the Université de Bretagne Occidentale in Brest, France, with thicknesses from 3.7 nm to 53 nm The effective magnetization Meff(Hk−4πM s) and the magnetic relaxation (intrinsic damping parameter αand extrinsic inhomogenous linewidth ∆Hin) are determined by broadband ferromagnetic resonance (FMR) in the frequency range 2-40 GHz (see SI Section IV). The device patterns are written by three e-beam lithography steps, each followed by a standard deposition and lift-off procedure. The first step produces a Ti/Au marker pattern, used to align the subse- quent steps. The second step defines the platinum injector and detector strips, as deposited by dc sputtering in an Ar+ plasma at an argon pressure with thickness ˜8nm for all devices. The third step defines 5/75 nm Ti/Au leads and bonding pads, deposited by e-beam evaporation. Devices have an injector/detector length L= 30/25µm and the strip widths Ware 400 nm for series A and 100 nm for series B. The experimental results in main text are obtained from series A. The distance-dependent non-local resistances for series B can be found in SI Section III. Measurements All measurements were carried out by means of three SR830 lock-in amplifiers using excitation frequency of 18 Hz. The lock-in amplifiers are set up to measure the first and sec- ond harmonic responses of the sample. Current was sent to the sample using a custom built current source, galvanically isolated from the rest of the measurement equipment. V oltage measurements 9were made using a custom-built pre-amplifier (gain 103) and amplified further using the lock-in systems. The typical excitation currents applied to the samples are 200 µA (RMS) for series A and 20µm for series B. The in-plane coercive field of the YIG Bcis below 10 mT for all YIG samples, and we apply an external field to orient the magnetization using a physical property measurement system (PPMS). The samples are mounted on a rotatable sample holder with stepper motor. All experimental data in the main text have been collected at 300 K (room temperature) at an applied magnetic field of 50 mT. Simulations Our finite-element model implements the magnon diffusion equation in insulators in order to simulate transport of electrically injected magnons. We carried out the simulations by COMSOL MULTIPHYSICS (version 5.4) software package with technical details in the SI Section I. Acknowledgements We acknowledge the helpful discussion with J. Shan and T. Yu. We acknowledge the technical support from J. G. Holstein, H. de Vries, H. Adema, T. Schouten and A. Joshua. This work is part of the research programme "Skyrmionics" with project number 170, which is financed by the Dutch Research Council (NWO). The support by NanoLab NL and the Spinoza Prize awarded in 2016 to B. J. van Wees by NWO is also gratefully acknowledged. G.B. was supported by JSPS Kakenhi Grant 19H00645. 10Author contributions B.J.v.W. and X.W. conceived the experiments. X.W. designed and carried out the experiments, with help from O.A.S. J.B.Y . supplied the YIG samples used in the fabrication of devices. X.W., O.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 O.A.S., G.E.W.B. and B.J.v.W. All authors commented on the manuscript. References 1. Cornelissen, L. J., Peters, K. J. H., Bauer, G. E. W., Duine, R. A. & van Wees, B. J. Magnon spin transport driven by the magnon chemical potential in a magnetic insulator. Phys. Rev. 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Influence of yttrium iron garnet thickness and heater opacity on the nonlocal transport of electrically and thermally excited magnons. Phys. Rev. B 94, 174437 (2016). 1323. Laughton, M. A. & Say, M. G. Electrical engineer’s reference book (Elsevier, 2013). 24. Klingler, S. et al. Measurements of the exchange stiffness of YIG films using broadband fer- romagnetic resonance techniques. Journal of Physics D: Applied Physics 48, 015001 (2014). 25. Stamps, R. & Camley, R. Solid State Physics . No. V olume 65 in Solid State Physics (Elsevier Science, 2014). 26. Kikkawa, T. et al. Critical suppression of spin Seebeck effect by magnetic fields. Phys. Rev. B92, 064413 (2015). 27. Jin, H., Boona, S. R., Yang, Z., Myers, R. C. & Heremans, J. P. Effect of the magnon dispersion on the longitudinal spin Seebeck effect in yttrium iron garnets. Phys. Rev. B 92, 054436 (2015). 28. Jamison, J. S. et al. Long lifetime of thermally excited magnons in bulk yttrium iron garnet. Phys. Rev. B 100, 134402 (2019). 29. Streib, S., Vidal-Silva, N., Shen, K. & Bauer, G. E. W. Magnon-phonon interactions in mag- netic insulators. Phys. Rev. B 99, 184442 (2019). 30. Man, H. et al. Direct observation of magnon-phonon coupling in yttrium iron garnet. Phys. Rev. B 96, 100406 (2017). 31. Bender, S. A., Duine, R. A. & Tserkovnyak, Y . Electronic Pumping of Quasiequilibrium Bose-Einstein-Condensed Magnons. Phys. Rev. Lett. 108, 246601 (2012). 32. Demokritov, S. O. et al. Bose–Einstein condensation of quasi-equilibrium magnons at room temperature under pumping. Nature 443, 430–433 (2006). 1433. Divinskiy, B. et al. Evidence for spin current driven Bose-Einstein condensation of magnons. Nature Communications 12, 6541 (2021). 15Figure 1: Device layout : a)Schematic representation of the experimental geometry. Two Pt strips deposited on top of YIG serve as magnon detector and injector via the direct and inverse spin hall effect. A low-frequency ac current with rms value of Iacthrough the left Pt strip injects magnons. The center-to-center distance of the injector and the detector is dand the length of the injector/detector is L. A spin accumulation µsis formed at Pt|YIG interface due to the SHE when a charge current passes through the injector and excites magnon non-equilibrium underneath the injector. The diffusive magnons are absorbed at the drain, which induce a spin current density js. Using a lock-in technique, the first harmonic voltage is measured simultaneously by the right Pt strip, i.e. a magnon detector. αis the angle of external magnetic field Hex.b) SEM image of the geometry. The parallel vertical lines are the platinum injector and detector, and they are contacted by gold leads. Current and voltage connections are indicated schematically. The scale bar represents 2 µm. 16Figure 2: Dependence of non-local resistance on magnetization direction on ultrathin YIG films, for short center-to-center distances between the injector and the detector. The offset R1ω 0in Eq.4 has been subtracted. This shows R1ω nlincreases with decreasing thickness. Comparing with the reference from Cornelissen et al.6,R1ω nlsignificantly increases in ultrathin films. 17Figure 3: Non-local resistance as a function of injector-detector distance of the samples of series A andtYIGvarying from 3.7 nm to 50000 nm. The width of injector/detector is 400 nm. The results fortYIG≥210 nm are adopted from Cornelissen et al.6and Shan et al.22. 18Figure 4: a)The non-local resistance R1ω nlatd= 2.5µm as a function of tYIG. The results for tYIG≥210nm are adopted from Cornelissen et al.6and Shan et al.22.b)Thickness dependence of the magnon conductivity σmobtained by the best fit for different distances with statistical er- ror bars. c)Thickness dependence of the 2-dimension spin conductance σ(2) mand the non-local resistanceRnlfrom the simulation, values based on the best fit for the magnon conductivity. The saturation at tYIG→0indicates that the film approaches the two-dimensional regime in ultrathin limit. The obtained error bar here means the range of the best fitting results for the non-local resistance we can get from the FEM simulations (see SI Section II). 19Giant magnon spin conductivity approaching the two-dimensional transport regime in ultrathin yttrium iron garnet films X-Y. Wei, O. Alves Santos, C.H. Sumba Lusero, G. E. W. Bauer, J. Ben Youssef, and B. J. van Wees I. FINITE-ELEMENT MODELLING We measure magnon transport non-locally by monitoring the voltage in a Pt detector as a function of current a Pt injector. The electrical spin accumulation µs(expressed in energy), which is generated by a current Iin the injector at the YIG interface reads [S1] µs=2eIθPtls σetwtanht 2ls, (S1) wheree,t,w,θPt,lsandσsare the electronic charge, the Pt film thickness, width, spin Hall angle, spin relaxation length, and conductivity of the Pt contact. With parameters in Table I the charge current to spin accumulation conversion coefficient ηinj=µs/slash.left(eI)=0.05 Ω whenw=400 nm and 0 .21 Ω whenw=100 nm. NM1 Injector NM2 Detector Interface layer YIG 𝝁𝒔 𝒋𝒛𝒙𝒔xz dw w 𝐻𝑒𝑥 𝑀𝑡 𝑡𝑖𝑛𝑡 𝑡𝑌𝐼𝐺a. 𝜇𝑚𝜇Vb. znm xnm FIG. S1. a)The sample configuration and its dimensions in the model simulations. Eq. S1 and Eq. S4 describe the spin accumulation µsat the interface, used as input of the model, and the voltage build up in the detector, respectively. b)The magnon chemical potential profile for a 100 nm thick YIG film with a charge current I=20µA and 100 nm wide injector/detector contacts and parameters in Table. I. Magnon absorption by the detector electrode causes the dark region close to the detector. We calculate the diffusive magnon spin transport in YIG films numerically by finite-element method (FEM) [S2] in the configuration of Figure I a) using the COMSOL MULTIPHYSICS (version 5.4) software package. Since the length of the strips is much longer than the electrode separation, we use a 2D model, where the parameters depend on x and z only. Ohm’s Law for magnon current density jmin YIG is jm=−σm∇µm/slash.lefte. (S2) where ∇=ˆx∂x+ˆz∂zandσmis the magnon spin conductivity. The local magnon current is proportional to ∇µm. The diffusion equation ∇2µm=µm l2m, (S3) governs the magnon chemical potential µm,wherelmis the magnon relaxation length. We use the zero-current boundary condition (∇⋅n)µm=0 for the bottom as well as top surface that is not covered by Pt, where nis the surface normal.arXiv:2112.15165v3 [cond-mat.mes-hall] 12 Jan 20222 TABLE I. Parameters used to obtain the results in Figure I b) and Figure S2 b) and c). Parameter Symbol Value Magnon spin conductivity σm 5×105S/m (for 210 nm thickness YIG [S2]) Magnon spin relaxation length lm 5×10−6m Effective spin mixing conductance Geff s 2×1012S/m2 Pt spin Hall angle θPt 0.11 Pt conductivity σe 2×106S/m Pt spin relaxation length ls 1.5×10−9m The spin accumulation in the detector generates an magnon transport driving force proportional to ∂µs/slash.left∂zthat leads to a measurable voltage by the integral over the cross-section A=wt Vnl=θPt 2A/integral.disp A∂µs ∂zdA. (S4) The non-local resistance Rnl=Vnl/slash.left(IL)(in unit of Ω/m) can be compared with the experimental results. The detector efficiency is the same as that of the injector, ηdet=ηinj. The conversion of spin accumulation in Pt into magnons in YIG is governed by an effective spin conductance Geff s, which is a certain fraction, ≈0.06, of the spin mixing conductance [S3, S4]. It can be modelled by a hypothetical spacer between Pt and YIG with conductivity σint s=Geff stint, wheretint=1 nm is the spacer layer thickness. Figure I b) shows the magnon chemical potential profile µmin a 100 nm thick YIG under a charge current of I=20µA. Injector and detector with w=100 nm/t=8 nm have a center-to-center distance of 300 nm. Table I lists the parameters used to produce Figure Ib). The Pt detector strip acts as a spin sink, absorbing magnons and producing a dark YIG region at the bottom of the detector in Fig. Ib). II. YIG FILM THICKNESS DEPENDENCE We measured the non-local resistance in many samples in order to obtain the YIG film thickness and contact- distance dependence. Figure S2 a) shows the experimental results for the non-local resistance for 400 nm wide injector/detector, with center to center distance d=2.5µm, and for different YIG thicknesses tYIG. We distinguish two regimes: when tYIG>lm,R1ω nlsaturates at a small value that does not depend on tYIGanymore. On the other hand,R1ω nlincreases with decreasing YIG thickness when tYIG<lm, wherelmis the magnon relaxation length. Figure S2 b) shows that the non-local resistance calculated with a constant magnon spin conductivity decreases when the YIG film become thinner . Figure S2 c) shows the results for selected das a function of tYIGthat in contrast to experiments, show a completely different thickness dependence. 024681012141618200.010.11101001000R10nl (R/m) Distance (/m)YIG thickness (nm) 5.9 1077 15.8 2321 40 5000 75 10772 210 23208 500 50000 0.0010.010.11101000100200300400500600R11nl (3/m) YIG thickness (nm)Distance: 0.5 3m 0.8 m 1.5 µm 3.0 µm 12.0 µmb.a.c.Experimental dataSimulationSimulation FIG. S2. a)YIG thickness dependence of the measured non-local signal R1ω nlfor a injector/detector distance of d=2.5µm.b) Finite element model results with fixed magnon conductivity σm=5×105S/m obtained previously for tYIG=210 nm [S2]. c) R1ω nlthickness dependency for different distances, for a fixed value of magnon spin conductivity. This shows an opposite trend in comparison with a). In Figure S2 c), R1ω nlsaturates above tYIG>lmbecause the magnon current distribution does not change anymore when increasing tYIG. WhentYIG<lm, on the other hand, the non-local resistance vanishes when tYIG=0, clearly is3 opposite to the experimental trend. We have to conclude that σmdramatically increases the thin-film limit. We now use σmas a single adjustable parameters that fits the experimental results. The numerical simulations are summarized in Figure S3 a) to j). The values of spin relaxation length used in the simulations, shown in Table II, were obtained from the experimental measurements at large distances, where the exponential decay is dominant[S5, S6]. 0.010.11101001000 R1w nl (W/m) Data 8.0E3 S/m 1.4E4 S/m 2.5E4 S/m 4.5E4 S/m 8.0E4 S/mYIG thickness = 2700 nm 0 2 4 6 8 10 12 14 16 18 20 0.010.11101001000 R1w nl (W/m)Distance ( mm) Data 7.0E7 S/m 1.2E8 S/m 2.2E8 S/m 3.9E8 S/m 7.0E8 S/mYIG thickness = 5.9 nm 0.010.11101001000YIG thickness = 53 nmR1w nl (W/m) Data 2.0E6 S/m 3.6E6 S/m 6.3E6 S/m 1.1E7 S/m 2.0E7 S/m 0.010.11101001000YIG thickness = 210 nm R1w nl (W/m) Data 1.0E5 S/m 1.8E5 S/m 3.2E5 S/m 5.6E5 S/m 1.0E6 S/m 0.010.11101001000R1w nl (W/m) Data 1.0E4 S/m 1.8E4 S/m 3.2E4 S/m 5.6E4 S/m 1.0E5 S/mYIG thickness = 1500 nm 0 2 4 6 8 10 12 14 16 18 200.010.11101001000 R1w nl (W/m) Data 5.0E3 S/m 7.9E3 S/m 1.6E4 S/m 3.2E4 S/m 5.0E4 S/m Distance ( mm)YIG thickness = 50000 nm 0 2 4 6 8 10 12 14 16 18 200.010.11101001000R1w nl (W/m) Distance ( mm) Data 7.0E3 S/m 1.1E4 S/m 2.2E4 S/m 4.4E4 S/m 7.0E4 S/mYIG thickness = 12000 nm 0 2 4 6 8 10 12 14 16 18 20 0.010.11101001000Distance ( mm)R1w nl (W/m) Data 5.6E7 S/m 9.9E7 S/m 1.8E8 S/m 3.1E8 S/m 5.6E8 S/mYIG thickness = 3.7 nm 0.010.11101001000 R1w nl (W/m) Data 8.0E6 S/m 1.4E7 S/m 2.5E7 S/m 4.5E7 S/m 8.0E7 S/mYIG thickness = 15.8 nm 0.010.11101001000 Data 7.0E7 S/m 1.2E8 S/m 2.2E8 S/m 3.9E8 S/m 7.0E8 S/mR1w nl (W/m)YIG thickness = 7.9 nma. b. c. d. e. f. g. h. i. j. FIG. S3. Comparison between experimental data (black circles) and the two-dimensional diffusive model obtained for different values of the magnon spin conductivity σm, (lines brown to green). a)toj)present the distance dependence of the non-local resistance for different YIG thickness, from 3.7 nm up to 50 µm, indicated on the top of each figure. Based on that comparison we obtain the value of the magnon spin conductivity presented in Figure 5 of the main text. Each black circle in Figure S3 is an independent measurement and susceptible to variations in the individual sample parameters. It is clear from Figure S3 that the experiments can be well fitted by a σmthat depends on tYIG.4 TABLE II. Values of the magnon spin relaxation length adopted in the simulations. YIG thickness [nm] Spin relaxation length [ µm] 5.9 / 7.9 / 15.8 3.5 53 6.0 210 9.2 1500 4.0 2700 3.5 12000 / 50000 4.0 III. PT /divides.alt0YIG INTERFACE SPIN RESISTANCE The low interface magnon resistance resulting from the large effective spin mixing conductance makes the Pt strips act as a good spin source/sink. Figure S4 shows the calculated values of non-local resistance for wide range of effective spin conductance, Geff s. A significant change in the non-local resistance occurs only when we suppress Geff sto below 2×1012S/slash.leftm2. This value is lower than reported by Kohno et. al., 8 .8×1012S/slash.leftm2[S7] (after local annealing). In Figure S4, a higher Geff sdoes not significantly change the calculated non-local resistance, which represents that the giant increase of the non-local resistance we observed is not due to the increase of Geff s. Figure S5 shows an equivalent circuit model omitting magnon relaxation. The spin current passing through the circuit is dominated by the spin conductance of YIG since Rs int<Rs YIG. Therefore, Rs intcan be disregarded and the magnon transport measured by Rnlis determined by the magnon spin conductivity of YIG rather than the YIG /divides.alt0Pt interface conductance. 0 2 4 6 8 10 12 14 16 18 2011010010007.9 nm YIG 1.0E8 S/mNonlocal resistance ( W/m) Distance ( mm)Gseff(S/m2): 2.0E11 4.3E11 9.3E11 2.0E12 4.3E12 9.3E12 2.0E13 0 2 4 6 8 10 12 14 16 18 20101001000Nonlocal resistance ( W/m) Distance ( mm)Gseff(S/m2): 1.0E11 2.0E11 4.6E11 1.0E12 2.0E12 4.6E12 1.0E13210 nm YIG 4.5E5 S/ma) b) FIG. S4. Calculated non-local resistance for a range of values of the effective spin conductance Geff sof a 400 nm wide injec- tor/detector. a)tYIG=210 nm, and b)tYIG=7.9 nm YIG. In both cases, a significant change in the magnon transport only occurs forGeff s<2×1012S/slash.leftm2. FIG. S5. This equivalent circuit model includes the relevant electronic and magnonic spin resistances and it is valid for short distance (within the magnon relaxation length). µsis the spin accumulation induced by the spin Hall effect. Rs Ptis the spin resistance of the Pt strip (injector/detector). Rs intis the interface spin resistance of Pt /divides.alt0YIG.jsis the electronic spin current injected into the detector, and Rs YIGis the spin resistance of YIG, which is the parameter of interest obtained from the two-dimensional model. In an additional experiments we placed a third Pt strip between the injector and detector contacts. By absorbing magnons, the middle strip suppresses the non-local signal by a factor of 5 when d=2µm andtYIG=7.9 nm, and by a factor of 2 for d=4µm andtYIG=53 nm. From this, we obtain a best fit with Geff s≈2.5×1012S/slash.leftm2fortYIG=7.9 nm,5 andGeff s≈1.0×1012S/slash.leftm2fortYIG=53 nm. We therefore adopted Geff s=2×1012S/slash.leftm2as the standard value for the effective spin conductance in the other simulations. We confirm the high Geff sby comparing results for different widths of the injector/detector contacts. The non-local resistance from the Series B devices in Figure S6 with width w=100 nm, is roughly 10 times higher than that of Series A devices with w=400 nm, which is mainly due to the higher values of the injector/detector conversion efficiency, Eq. S1. The simulations show that when Geff sis large, magnons are injected/absorbed predominantly in/by only part of the interfaces, i.e., the magnon chemical potential in the YIG film covered by the Pt strip decays according to an absorption length Labsorb=/radical.alt1 σmtYIG/slash.leftGeffsas sketched in Figure S7 b) in terms of the interface chemical potential µmand spin current, /uni20D7Js, in the Pt detector. Figure S7 a) shows the calculated resistance for Geff s=2×1012S/m2, σm=1×108S/m,tYIG=3.7 nm and therefore Labsorb=430 nm that agrees with observations. This confirms our value forGeff seven for the thinnest films. FIG. S6. Non-local resistance as a function of injector-detector separation distance for series B devices. The thickness of YIG films 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 nm thickness YIG at same center-to-center distance. The lines are the simulation results based on a 2D-FEM model. The magnon 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 YIG. 𝜇𝑚𝐽𝑠 100 150 200 250 300 350 40005001000150020002500300035004000450050005500Nonlocal resistance ( W/m) Injector/detector width (nm) tYIG = 3.7 nm d = 1 ma. b. magnons FIG. S7. a)Simulation for the non-local resistance with different injector/detector width on 3.7 nm thickness YIG. The center-to-center distance between the injector and the detector is 1 µm. The simulations in Figure S7 are in agreement with the experimental results in Figure S6. b)Schematic illustration of the decay of the magnon chemical potential decay and spin current over the Pt width.6 IV. MAGNETIC PROPERTIES OF THE YIG FILMS Table. III summarizes the magnetic properties of the YIG films are determined by broadband ferromagnetic resonance in the range of 2-40 GHz. The magnon spin relaxation lengths obtained by fitting Eq. 3 in the main text, are shown in Figure. S8. TABLE III. Magnetic properties of the YIG films by FMR characterization. YIG thickness Gilbert damping Effective field Inhomogenous linewidth tYIG(nm) α(10−4)Heff=Hk−4πMs(Oe) ∆ Hinh(Oe) 3.7 5 ±2.1 1720 17 5.6 not available 1900 not available 5.9 10 ±2.4 1950 43 7.9 6 .3±4.6 1930 53 15.8 0 .6±0.5 1960 7.9 53 1 .0±0.2 1810 1 FIG. S8. Magnon spin relaxation length and Gilbert damping parameter as a function of YIG film thickness. The thickness of YIG films ranges from 3.7 nm to 50000 nm (shown on a logarithmic scale). The results from 210 nm to 50000 nm thickness samples are adopted from Cornelissen et al.[S5] and Shan et al.[S6]. The Gilbert damping parameter of the YIG films is also presented here. Note that for ultrathin YIG (around 10 nm), the magnon spin relaxation lengths are similar to each other but the Gilbert damping parameters are quite different.7 V. THE SECOND HARMONIC NON-LOCAL RESISTANCE The second harmonic signal in the non-local resistance is a measure of the thermal spin generation in the magnetic film by the spin Seebeck effect. Figures. S9 a) shows results for this R2ω nl(α) as a function of the direction of in-plane magnetic field. R2ω nlcan be extracted by R2ω nl(α)=1/slash.left2R2ω nlcosα+R2ω 0, (S5) whereαis the in-plane angle of Hexwith thex-axis.R2ω 0is an offset signal, possibly by an unintended conventional Seebeck voltage in the detector[S8]. We plot the observed amplitude R2ω nlas a function of contact separation in Figures. S9 b) for tYIG=5.9 nm, 7.9 nm, 15.9 nm (series A with 400 nm wide injector/detector strips). R2ω nlin ultrathin YIG is significantly reduced compared to that in thicker YIG films[S5, S6, S9], because the generation of thermal magnons by SSE due to the vertical temperature gradient become less effective. Figures. S9 b) show neither a simple Ohmic or exponential decay as a function of d, which indicates that the generation and transport of magnons is complex and requires detailed modelling of the temperature profile. Just like the magnon conductivity, the spin Seebeck coefficient should also depend on tYIG. A reliable extraction of both parameters as a function of thickness from the second harmonic signals appears impossible at this this time. FIG. S9. The second harmonic non-local signals of ultrathin YIG films. a)The angle-dependence for tYIG=7.9 nm and center-to-center distance of injector and detector d=1.5µm.b)Amplitude of the non-local signals as a function of dfor the samples from series A. VI. MAGNETIC FIELD DEPENDENCE OF THE NON-LOCAL SIGNAL IN ULTRATHIN FILMS We measured the non-local transport in the series A devices with tYIG=5.9 nm (Figure. S10) and tYIG=7.9 nm (Figure. S11) as a function of the strength of a magnetic field along x(in-plane, normal to the contacts). We use Eq. 3 from the main text to extract the field-dependence of the magnon conductivity and relaxation length, as shown in Figure. S12 and Figure. S13, where we use ηcs=ηsc=0.05 Ω from Section I. While the magnon spin relaxation length only slightly decreases, the increasing field suppresses the magnon spin conductivity stronger for the thinner samples. This is in line with the dominance of the lowest magnon subband in the thinnest samples, which is more susceptible to magnetic freeze-out of the thermal magnon population.8 FIG. S10. The non-local resistance R1ω nlas a function of magnetic field and contact distance of the A series sample with tYIG=5.9 nm. FIG. S11. The non-local resistance as function of magnetic field for tYIG=7.9 nm.9 FIG. S12.λmandσmas a function of field in a film with tYIG=5.9 nm. a)The magnon relaxation length λmdecreases slightly with increasing magnetic field. b)The magnetic field strongly suppresses the magnon conductivity σm.. FIG. S13. a)λmandb)σm, as a function of field in a film with tYIG=7.9 nm. VII. TEMPERATURE DEPENDENCE OF THE NON-LOCAL SIGNAL IN ULTRATHIN FILMS We measured the temperature dependence of the non-local signal on series A samples with tYIG=3.7 nm (Figure. S14), 5 .9 nm (Figure. S15) and 7 .9 nm (Figure. S16) and a magnetic field of 50 mT. R1ω nlfortYIG=3.7 nm at low temperatures is more than five times larger for d=1.3µm than for d=2µm. Since the decrease of Geff sdue to the temperature is equivalent for both devices. The large change of the ratio of R1ω nlford=1.3µm andd=2µm suggests thatR1ω nlis dominated by the magnon spin conductivity of YIG and not the Pt /divides.alt0YIG interface spin conductance, which supports the conclusions from Section III. Results of the fit by Eq. 3 from the main text are shown in Figure. S17 and Figure. S18. The magnon relaxation length only slightly changes with temperature, in contrast to that σmdecreases strongly with temperature. In thick YIG films,Rnl(T)decreases slowly at high temperatures and faster at low temperatures [S10, S11], but the decrease at low temperatures is much more pronounced for ultrathin thickness YIG. Also this result is consistent with the dominant role of the lowest magnon subband in thermal transport in the thinnest films, since the two-dimensional magnon gas is more susceptible to the freeze out of carriers at low temperatures.10 FIG. S14. The non-local resistance R1ω nlas a function of temperature for different contact distances on a YIG film with tYIG=3.7 nm. FIG. S15. Non-local resistance as a function of temperature for tYIG=5.9 nm.11 FIG. S16. Non-local resistance as a function of temperature for tYIG=7.9 nm. FIG. S17. a)λmandb)σmas a function of temperature for tYIG=5.9 nm.12 FIG. S18. a)λmandb)σmas a function of temperature for tYIG=7.9 nm. VIII. HIGHEST OCCUPIED EXCHANGE MODE OF YIG AT ROOM TEMPERATURE Figure. S19 shows the highest occupied exchange perpendicular standing spin waves (PSSW) 2D subbands nfor different thickness YIG films at room temperature. The number of the occupied subbands is n+1. It is reduced from over 104to a few when the thickness of the films is down to 3.7 nm. FIG. S19. Thickness dependency of the number of occupied PSSW modes defined as the highest occupied subband nat 300 K. [S1] Sinova, J., Valenzuela, S. O., Wunderlich, J., Back, C. H. & Jungwirth, T. Spin Hall effects. Rev. Mod. Phys. 87, 1213–1260 (2015). [S2] Cornelissen, L. J., Peters, K. J. H., Bauer, G. E. W., Duine, R. A. & van Wees, B. J. Magnon spin transport driven by the magnon chemical potential in a magnetic insulator. Phys. Rev. B 94, 014412 (2016). [S3] Weiler, M. et al. Experimental Test of the Spin Mixing Interface Conductivity Concept. Phys. Rev. Lett. 111, 176601 (2013).13 [S4] Qiu, Z. et al. Spin mixing conductance at a well-controlled platinum/yttrium iron garnet interface. Applied Physics Letters 103, 092404 (2013). [S5] Cornelissen, L. J., Liu, J., Duine, R. A., Ben Youssef, J. & van Wees, B. J. Long-distance transport of magnon spin information in a magnetic insulator at room temperature. Nature Physics 11, 1022–1026 (2015). [S6] Shan, J. et al. Influence of yttrium iron garnet thickness and heater opacity on the nonlocal transport of electrically and thermally excited magnons. Phys. Rev. B 94, 174437 (2016). [S7] Kohno, R. et al. Enhancement of YIG /divides.alt0Pt spin conductance by local Joule annealing. Applied Physics Letters 118, 032404 (2021). [S8] Sierra, J. F. et al. Thermoelectric spin voltage in graphene. Nature Nanotechnology 13, 107–111 (2018). [S9] Shan, J. et al. Criteria for accurate determination of the magnon relaxation length from the nonlocal spin Seebeck effect. Phys. Rev. B 96, 184427 (2017). [S10] Cornelissen, L. J., Shan, J. & van Wees, B. J. Temperature dependence of the magnon spin diffusion length and magnon spin conductivity in the magnetic insulator yttrium iron garnet. Phys. Rev. B 94, 180402 (2016). [S11] Gomez-Perez, J. M., V´ elez, S., Hueso, L. E. & Casanova, F. Differences in the magnon diffusion length for electrically and thermally driven magnon currents in Y 3Fe5O12.Phys. Rev. B 101, 184420 (2020).

2021-12-30

Conductivities are key material parameters that govern various types of transport (electronic charge, spin, heat etc.) driven by thermodynamic forces. Magnons, the elementary excitations of the magnetic order, flow under the gradient of a magnon chemical potential in proportion to a magnon (spin) conductivity $\sigma_{m}$. The magnetic insulator yttrium iron garnet (YIG) is the material of choice for efficient magnon spin transport. Here we report an unexpected giant $\sigma_{m}$ in record-thin YIG films with thicknesses down to 3.7 nm when the number of occupied two-dimensional (2D) subbands is reduced from a large number to a few, which corresponds to a transition from 3D to 2D magnon transport. We extract a 2D spin conductivity ($\approx1$ S) at room temperature, comparable to the (electronic) spin conductivity of the high-mobility two-dimensional electron gas in GaAs quantum wells at millikelvin temperatures. Such high conductivities offer unique opportunities to develop low-dissipation magnon-based spintronic devices.

Giant magnon spin conductivity approaching the two-dimensional transport regime in ultrathin yttrium iron garnet films

2112.15165v3

arXiv:1405.1929v1 [cond-mat.mtrl-sci] 8 May 2014Enhancement of Spin Pumping in Y3Fe5O12/Pt/Ni81Fe19Trilayer Film Daichi Hirobe,1,∗Ryo Iguchi,1Kazuya Ando,1,2,3Yuki Shiomi,1and Eiji Saitoh1,4,5,6 1Institute for Materials Research, Tohoku University, Send ai 980-8577, Japan 2Department of Applied Physics and Physico-Informatics, Keio University, Yokohama 223-8522, Japan 3PRESTO, Japan Science and Technology Agency, Saitama 332-0 012, Japan 4WPI Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 5CREST, Japan Science and Technology Agency, Chiyoda, Tokyo 102-0075, Japan 6The Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan (Dated: September 12, 2021) Abstract We study spin pumping in a Y 3Fe5O12(YIG)/Pt/Ni 81Fe19(Py) trilayer film by means of the inverse spin Hall effect (ISHE). When the ferromagnets are not excited simultaneously by a mi- crowave, ISHE-induced voltage is of the opposite sign at eac h ferromagnetic resonance (FMR). The opposite sign is consistent with spin pumping of bilayer film s. On the other hand, the voltage is of the same sign at each FMR when both the ferromagnets are excit ed simultaneously. Futhermore, the voltage greatly increases in magnitude. The observed vo ltage is unconventional; neither its sign nor magnitude can be expected from spin pumping of bilay er films. Control experiments show that the unconventional voltage is dominantly induced by sp in pumping at the Py/Pt interface. Interaction between YIG and Py layers is a possible origin of the unconventional voltage. 1I. INTRODUCTION Spintronics has attracted attention because it is expected to rea lize new magnetic mem- ories and computing devices by use of electronic spins.1A necessity of spintronics is to understand the nature of a spin current, a flow of spin angular mom entum. The concept of a spin current stems from spin-polarized transport in ferromagne tic metals. Ferromagnetic metals conduct a spin-dependent electric current, a charge curr ent accompanied by a spin current. This current induces interesting phenomena such as gian t magnetoresistance2and current-induced magnetization reversal3–5. The complete separation into spin and charge currents leads to the concept of a purespin current, a flow of electronic spins with nocharge currents. Spin pumping6–8is a versatile method for generating a pure spin current. This method uses magn etization precession in a ferromagnet to inject a spin current into an adjacent metal. Ferromagnet (F)/normal metal (NM) bilayer films have offered the mselves as the bench- mark for the validity of spin pumping. Experimental and theoretical studies have con- centrated on this structure and contributed to a solid understan ding of spin pumping. Spin-current injection enhances magnetization relaxation in F and b roadens ferromagnetic- resonance(FMR)linewidth. TheinjectioniselectricallydetectableinN Maswell; spin-orbit interaction in NM can convert an injected spin current into a charge current. This conver- sion is known as the inverse spin Hall effect9–14(ISHE) and induces an electric field EISHE that satisfies the relation10 EISHE∝Js×σ. (1) Here,Jsis the spatial direction of a spin current and σis the spin-polarization vector. Spin pumping in F/NM/F trilayer films will be of further interest since ne w collective excitation can be induced by magnetic couplings between the ferrom agnets. In fact, such excitation was verified by FMR measurements.7,16This paper presents an experimental investigation on spin pumping in a Y 3Fe5O12(YIG)/Pt/Ni81Fe19(Py) trilayer film by means of ISHE. DC voltage is found to be enhanced when both the ferroma gnets are excited simultaneously. The enhanced voltage is unconventional in the sens e that it cannot be explained byspin-pumping theoryofF/NMbilayer films. Control expe riments demonstrates that the unconventional voltage comes from spin pumping at the Py /Pt interface. We discuss the origin of the unconventional voltage and narrow down c ontributing factors on 2an experimental basis. II. METHOD Figure 1(a) shows a schematic illustration of a YIG /Pt/Py trilayer film. The YIG (Pt) layer was 1 mm wide, 3 mm long and 50 nm (5 nm) thick. The Py layer was 1 m m wide, 2 mm long and 10 nm thick. We grew the YIG layer on a Gd 3Ga5O12(GGG) substrate by a metal organic decomposition method15; we then sputtered the Pt layer on the YIG layer and finally deposited the Py layer by the electron-beam evaporation in a high vacuum. We attached two electrodes to the ends of the Pt layer to measure DC voltage. The film was placed at the center of a TE 011microwave cavity at which a microwave magnetic field (electric field) maximizes (minimizes) itself. A frequency of the microwave was 9.44 GHz. We applied a static magnetic field perpendicular to the mic rowave magnetic field. Figure 1(b) defines the out-of-plane magnetic-field angle θHand the magnetization angleθM,YIG(θM,Py) of YIG (Py). Spin pumping injects a spin current into the Pt layer, leaving the spin-polarization vector σparallel to the precession axis on temporal average [see Fig. 1(c)].10ISHE in the Pt layer converts the injected spin current into DC volta ge. DC voltage and an FMR spectrum were measured at room temperatu re with changing static-field magnitude or direction. III. RESULTS AND DISCUSSION Figure 2(a) shows spectra of the microwave absorption signal at θH=±90 deg. (i.e., with in-plane fields). We see FMR of Py (YIG) near µ0H= 120 (300) mT. Correlating with FMR, DC voltage appears as shown in Fig. 2(b). Remember the dir ection of relevant vectors: Js∝ba∇dblˆy,σ∝ba∇dblˆz, andEISHE∝ba∇dblˆx[see Fig. 1(c)]. Here, ˆx,ˆy, andˆzrepresent unit vectors in each direction. The voltage spectra are consistent with Eq. (1). At θH= 90 deg., voltage is negative at FMR of Py while it is positive at FMR of YIG. Th is opposite polarity is due to the opposite directions of spin-current injection ( Js∝ba∇dbl ±ˆy,σ∝ba∇dblˆz, and thus EISHE∝ba∇dbl ±ˆx). Moreover, when the field direction is reversed, the voltage sign r everses at each FMR. This reversal is due to the opposite directions of spin polarizat ion (Js∝ba∇dblˆy,σ∝ba∇dbl ±ˆz, and thus EISHE∝ba∇dbl ±ˆx). These results all support that the voltage is dominated by ISHE. 3The replacement of Pt with Cu reveals irrelevance of galvanomagnet ic effects such as the anomalous Hall effect and the planar Hall effect18. The replacement greatly reduces voltage. As a result, the voltage with Cu cannot reproduce the voltage with P t, even considering a conductance difference between Pt and Cu. This reduction is consis tent with the fact that ISHE is negligibly small in Cu owing to weak spin-orbit interaction. Figure 2(c) shows the θHdependence of a resonance field HFMR. The result is consistent with previous studies of bilayer films.17In terms of HFMR, we observed little difference between the trilayer film and its corresponding YIG/Pt and Py/Pt bila yer films. Different saturation magnetization between YIG and Py results in the differen tθHdependence of HFMRvia a demagnetization field along the y-axis [see Fig. 1(b)].17This enables us to control magnetization dynamics in each layer. At θH=±90 deg, resonance fields are different between YIG and Py. Thus, while one magnetic layer is on res onance with large precessional amplitude, theother isoff resonance withsmall prece ssional amplitude. Around θH=±20 deg., on the other hand, resonance fields are nearly identical: ma gnetization precesses simultaneously in the two ferromagnets. Figure 3(a) shows the magnetic-field dependence of voltage at var ious values of θH. We first confine ourselves to voltage at FMR of Py. Voltage monotonica lly decreases in mag- nitude as θHapproaches zero and changes its sign across θH= 0 deg.. This behavior is consistent with Eq. (1) because EISHE∝Js×σ∝sinθM,Pyholds [see Fig. 1(b)]. To fit the voltage spectra near FMR of Py, we used a sum of symmetric and asy mmetric Lorentzian functions10 V(H) =VSYM(∆H/2)2 (H−HFMR)2+(∆H/2)2+VASYM2∆H/2(H−HFMR) (H−HFMR)2+(∆H/2)2.(2) Here, ∆His the full width at half maximum and VSYM(VASYM) is the magnitude of a symmetric (asymmetric) component of voltage. VSYMcorresponds to ISHE and VASYMto the anomalous Hall effect in this study. Fig. 3(b) bottom shows that theθHdependence of VSYMisroughlyrepresented byasinecurveandrevealsnodifferencebet weentheYIG/Pt/Py trilayer film and a Py/Pt bilayer film. We next focus on voltage at FMR of YIG for θH>0 deg. Fig. 3(a) shows that as θH decreases from90deg., voltagemagnitude decreases andalmost v anishes at 30 deg.. Notable is that voltage of the opposite sign reappears at 20 deg.. This rever sal of sign cannot be 4explained by spin pumping in a YIG/Pt bilayer film. Application of Eq. (1) t o the bilayer film yields voltage of a constant sign for θH>0 deg. because EISHE∝Js×σ∝sinθM,YIG holds. An unconventional feature is not limited to the sign of voltage : the upper inset in Fig. 3(a) clearly shows enhancement of voltage near θH= 18 deg, where the resonance fields of YIG and Py are equal. Anomalies in sign and magnitude are summ arized in Fig. 3(b) and (c). At this stage, we cannot conclude that the unconve ntional voltage originates in spin pumping at the YIG/Pt interface. We also need to consider spin pumping at the Py/Pt interface; YIG and Py can affect mutual dynamics, or spin pu mping via magnetic interaction. We note that the observed unconventional voltage cannot be att ributed to heating effects due to microwave absorption. The microwave absorption at FMR hea ts the magnetic layer and may cause a heat current flowing from the magnetic layer to the Pt layer. Eventually DC voltage is induced by thermoelectric effects such as the ordinary Nernst effect, the anomalous Nernst effect and the longitudinal spin Seebeck effect19(LSSE). The DC voltage is proportional to H×∇T(ordinary Nernst effect) and M×∇T(anomalous Nernst effect and LSSE), where Tdenotes temperature. The magnitude of ∇Tis proportional to the microwave absorption at FMR IFMR; thus the DC voltage is proportional to |H|IFMRsinθH (ordinary Nernst effect) and |M|IFMRsinθM(anomalous Nernst effect and LSSE). Since IFMRdecreases as θHapproaches zero, the heating effects cannot reproduce the obs ervedθH dependence of voltage. Voltage anomalies appearing when HFMR,YIG=HFMR,Pysuggest that magnetization dy- namics, or spin pumping is affected by a magnetic coupling between the ferromagnets. One of the possibilities is a dynamic exchange coupling16, a coupling via spin currents mediated by conduction electrons in an intermediate NM layer. This coupling per sists on the spin- diffusion length scale in the NM layer and may exist in the YIG/Pt/Py trila yer film, because Pt has a spin diffusion length of around 10 nm. To examine a dynamic exc hange coupling, we prepared a YIG/SiO 2(20 nm thick)/Pt (5 nm thick)/Py (10 nm thick) quadrilayer film. Although a spin current is interrupted across the YIG/Pt interfac e, voltage anomalies per- sist near θH=±20 deg.. Moreover, the quadrilayer film reveals the similar θHdependence ofVSYMat FMR of YIG as observed in the trilayer film [see Fig. 4(a)]. The persis tence and the angular dependence both suggest that a dynamic exchange co upling is irrelevant to the trilayer film. 5The voltage anomalies in the YIG/SiO 2/Pt/Py quadrilayer film suggest that unconven- tional voltage originates in spin pumping at the Py/Pt interface. To r einforce this argument, we prepared a YIG/Pt (5 nm thick)/SiO 2(20 nm thick)/Py (10 nm thick) quadrilayer film, where the SiO 2layer suppresses a dynamic exchange coupling by interrupting a spin cur- rent across the Py/Pt interface. The unconventional voltage is f ound to disappear [see Fig. 4(b)]. This result further proves that unconventional voltage or iginates in spin pumping at the Py/Pt interface. We need to explain at least the following four features of the unconv entional voltage: (feature 1) the unconventional voltage persists even when the f erromagnets are separated by upto about20 nm; (feature 2)thevoltage originatesinspin pumping atthePy/Pt interface; (feature 3) the voltage appears when both the ferromagnets ar e excited simultaneously; (feature 4) the voltage takes extrema at θHwhere YIG and Py are equal in HFMR. The long-range nature (feature 1) suggests that a dipolar couplin g may be relevant to the unconventional voltage. As shown below, however, this coupling alo ne is unlikely to yield our results consistently. Magnetization dynamics in the YIG layer yie lds a time-dependent dipole field20,21. A sum of the time-dependent dipole field and a microwave field act on magnetization in the Py layer. The increased driving field may enhance the magnetization dynamics in the Py layer. Note that YIG and Py are greatly different in volume, the saturation magnetization, and the damping constant. These differ ences may enhance spin pumping of Py while having little influence on that of YIG. On the other h and, a static dipole coupling should lead to a resonance-field shift in the YIG/Pt/Py trilayer film. This shift will become prominent especially when YIG and Py are excited simu ltaneously.22At least we did not observe resonance-field shifts greater than an er ror range of 10 mT. It is unreasonable to infer that a dynamic dipole field enhances magnetiza tion dynamics while a static counterpart has little influence on resonance fields. The unc onventional voltage will need contributing factors other than a dipolar coupling. IV. CONCLUSION We investigated spin pumping in a YIG/Pt/Py trilayer film by means of IS HE. Un- conventional voltage appears when both the ferromagnets are e xcited simultaneously by a microwave. The voltage is attributed to spin pumping at the Py/Pt int erface. The voltage 6persists when a dynamic exchange coupling is suppressed. Several features are pointed out to characterize the unconventional voltage. The voltage persist s with the ferromagnets sep- arated by up to 20 nm (long-range feature). The voltage magnitud e increases when YIG and Py are equal in HFMR(dynamic feature). Considering these features, a time-depende nt dipolar coupling was proposed to explain the origin of the unconventio nal voltage. However, our experiments show that a static dipolar coupling does not shift re sonance fields. This result appears inconsistent with the proposed mechanism. The orig in of the unconventional voltage requires further elucidation. ACKNOWLEDGEMENTS This work was supported by CREST-JST ”Creaction of Nanosystem s with Novel Func- tions through Process Integration”, a Grant-in-Aid for Scientific Research (A) (24244051) from MEXT, Japan, The Murata Science Foundation, The Funding Pr ogram for Next Gen- eration World-Leading Researchers, The Asahi Glass Foundation, The Noguchi Institute, and The Mitsubishi Foundation. 7V Hmicrowave Py Pt YIG GGG Py YIGPt θM, Py θM, YIGθH H σPy Js,Py EISHE,Py EISHE Py Pt YIGMPy MYIGy x zy z σYIGJs,YIG EISHE,YIG σσYIGYIGJISHE ,YIG ,YIG MPy MYIG(a) (b) (c) VV+V 㸫V FIG. 1: (a) A schematic illustration of the YIG/Pt/Py trilay er film.His a static magnetic field. (b) Definitions of the magnetic-field angle θHand the magnetization angle θM.His a static magnetic field and Mis magnetization. (c) A schematic illustration of spin pump ing at the Py/Pt (left) and YIG/Pt interfaces (right). Mdenotes magnetization; Jsthe spatial direction of a spin current; σthe polarization vector; EISHEan ISHE-induced electric field. 8θH = -90 deg. θH = 90 deg.VH VH(a) (c) 1200 1000 800 600 400 200 0 0 MM YIG, YIG/Pt/Py YIG, YIG/Pt Py, YIG/Pt/Py Py, Py/Pt θH (deg.)(mT) FMRHµ0 90 -90 -30-20-1001020 V (µV) 350 300 250 200 150 100 µ0H (mT)dI/dH (a.u.) -2-1012 V (µV) 310 300 290 280 µ0H (mT) 90° -90° = -90°, YIG/Pt/Py = 90°, YIG/Pt/Py = 90°, YIG/Cu/PyθH θH θH 00 T)µ0HFMR, Py µ0HFMR, YIG (b) FIG. 2: (a) The in-plane field Hdependence of the microwave abosrption signal d I/dH.HFMRis a resonance field. (b) The in-plane field Hdependence of DC voltage Vfor the YIG/Pt/Py trilayer film (red and blue circles) and the YIG/Cu/Py trilayer film (bl ack circles). The inset shows the Hdependence of Vnear FMR of YIG. (c) The field-angle θHdependence of the resonance field HFMR. The filled circles represent data for the YIG/Pt/Py trilaye r film; the open triangles for the YIG/Pt and Py/Pt bilayer films. Data are blue for YIG and red fo r Py. 9V䢢䢢(µV) 1200 1000800 600 400 200 µ䢲H䢢(mT)V䢢䢢(µV) -200-1000100200 µ䢲(H-H䢢FMR, YIG䢢) (mT) 24° 22° 20° 19° 18° 16° 14° 20 ( µV) -100 -50 0 50 100 µ䢲(H-H FMR, YIG 䢢) (mT) 䢢= 19° 䢢20䢢(µV)䢢20䢢(µV)θΗ = 90 ° θΗ = 70 ° θΗ 䢢= 50° θΗ = 30 ° θΗ = 20 ° θΗ 䢢= 10° θΗ = 0° θΗ = -10 ° θΗ 䢢= -20 ° θΗ 䢢= -30 ° θΗ 䢢= -50 ° θΗ = -70 ° θΗ = -90 ° θHV䢢䢢(µV)(a) (d)θH θH θH θH θH θH θH θH θH θH θH θH θHFMR of Py FMR of YIG -1.001.0 -50 0 50800 400 0µ0H FMR (mT) -4-2024 VSYM/|VSYM, -90 deg.| YIG/Pt/Py at FMR of YIG YIG/Pt at FMR of Py YIG/Pt/Py Py/PtțH FMR = H FMR, Py - H FMR, YIG VSYM/|VSYM, -90 deg.| θH(deg.)ț(b) (c) FIG. 3: (a) The field Hdependence of DC voltage Vfor the YIG/Pt/Py trilayer film at magnetic- field angles θHbetween −90 and 90 deg. The blue (red) dashed line represents the reson ance field of YIG (Py). Orange (black) circles indicate unconventiona l (conventional) voltage at FMR of YIG. The upper inset shows the Hdependence of Vnear FMR of YIG. The lower inset shows experimental data (open circles) and fitting (dashed lines) atθH= 19 deg. Red and blue dashed lines are a symmetric Lorenzian function and the green dashe d line is an asymmetric Lorenzian function. (b) The magnetic-field angle θHdependence of the resonance-field difference δHFMR. δHFMRis defined as δHFMR=HFMR,Py−HFMR,YIG. Orange circles indicate where YIG and Py are equal in HFMR. (c) The magnetic-field angle θHdependence of the symmetric component of DC voltage VSYMat FMR of YIG. VSYM/|VSYM,−90deg|is the normalized DC voltage. Circles representdatafortheYIG/Pt/Pytrilayer film; triangles fo rtheYIG/Ptbilayer film. Orangecircles indicate where YIG and Py are equal in HFMR. (d) The magnetic-field angle θHdependence of the symmetric component of DC voltage VSYMat FMR of Py. VSYM/|VSYM,−90deg|is the normalized DC voltage. Circles represent data for the YIG/Pt/Py trilay er film; triangles for the YIG/Pt bilayer film. 10-0.4-0.200.20.4 V (µV) -20 -10 0 10 20 µ0(H-H FMR, YIG ) (mT)-10-50510 V (µV) at FMR of YIG at FMR of Py VSYM/VSYM, MAX VSYM/VSYM, MAX (a) (b) -1-0.5 00.51 -50 0 50-1-0.5 00.511.5 YIG/Pt/Py (reference) YIG/SiO 2/Pt/Py YIG/Pt/SiO 2/Py YIG/Pt/SiO 2/Py YIG/SiO 2/Pt/Py YIG/Pt/Py (reference) YIG/SiO 2/Pt/Py YIG /Pt/Py YIG/Pt/ /Py SiO 2 Js YIG /Pt/Py YIG/Pt/ /Py SiO 2 YIG/SiO 2/Pt/Py+ +- - θH(deg.)Py Pt YIGSiO 2Py Py Pt YIG YIGPt SiO 2Js JsJs = H FMR, Py H FMR, YIG FIG. 4: (a) The magnetic-field angle θHdependence of the symmetric component of DC voltage VSYMat FMR of YIG. The black circles represent data for the YIG/Si O2/Pt/Py quadrilayer film; the dark brown circles for the YIG/Pt/SiO 2/Py quadrilayer films; the sky blue circles for the YIG/Pt/Py trilayer film. The inset shows the magnetic field Hdependence of VSYMnear FMR of YIG.θHis such that YIG and Py are equal in the resonance field HFMR. The red (blue) arrow stresses a positive (negative) sign of VSYM. (b) The magnetic-field angle θHdependence of the symmetric component of DC voltage VSYMat FMR of Py. The black circles represent data for the YIG/SiO 2/Pt/Py quadrilayer film; the dark brown circles for the YIG/P t/SiO2/Py quadrilayer films; the light pink circles for the YIG/Pt/Py trilayer film. 11∗Electronic address: daichi.kinken@imr.tohoku.ac.jp 1Zutic, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). 2A. Fert, Rev. Mod. Phys. 80, 1517 (2008). 3J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). 4L. Berger, Phys. Rev. B 54, 9353 (1996). 5S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J . Schoelkopf, R. A. Buhrman, and D. C. Ralph, Nature 425, 380 (2003). 6S. Mizukami, Y. Ando, and T. Miyazaki, Phys. Rev. B 66, 104413 (2002). 7Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin , Rev. Mod. Phys. 77, 1375 (2005). 8A. Azevedo, L. H. Vilela Le˜ ao, R. L. Rodriguez-Suarez, A. B. Oliveira, and S. M. Rezende, J. Appl. Phys. 97, 10C715 (2005). 9T. Kimura, Y. Otani, T. Sato, S. Takahashi, and S. Maekawa, Ph ys. Rev. Lett. 98, 156601 (2007). 10E. Saitoh, M. Ueda, H.Miyajima, and G. Tatara, Appl. Phys. Le tt.88, 182509 (2006). 11S. O. Valenzuela and M. Tinkham, Nature 442, 176 (2006). 12T. Seki, Y. Hasegawa, S. Mitani, S. Takahashi, H. Imamura, S. Maekawa, J. Nitta, and K. Takanashi, Nat. Mater. 7, 125 (2008). 13S. Takahashi and S. Maekawa, Phys. Rev. Lett. 88, 116601 (2002). 14K. Ando, Y. Kajiwara, S. Takahashi, S. Maekawa, K. Takemoto, M. Takatsu, and E. Saitoh, Phys. Rev. B 78, 014413 (2008). 15T. Ishibashi, A. Mizusawa, M. Nagai, S. Shimizu, K. Sato, N. T ogashi, T. Mogi, M. Houchido, H. Sato, and K. Kuriyama, J. Appl. Phys. 97, 013516 (2005). 16B. Heinrich, Y. Tserkovnyak, G. Woltersdorf, A. Brataas, R. Urban, and G. E. W. Bauer, Phys. Rev. Lett. 90, 187601 (2003). 17K. Ando, S. Takahashi, J. Ieda, Y. Kajiwara, H. Nakayama, T. Y oshino, K. Harii, Y, Fujikawa, M. Matsuo, S. Maekawa, and E. Saitoh, J. Appl. Phys. 109, 103913 (2011). 18L. Chen, F. Matsukura, and H. Ohno, Nat. Commun. 4, 2055 (2013) 19K. Uchida, H. Adachi, T. Ota, H. Nakayama, S. Maekawa, and E. S aitoh, Appl. Phys. Lett. 97, 12172505 (2010). 20O. Dmytriiev, T. Meitzler, E. Bankowski, A. Slavin, and V. Ti berkevich, J. Phys. Condens. Matter22, 136001 (2010). 21S. Schafer, N. Pachauri, C. K. A. Mewes, T. Mewes, C. Kaiser, Q . Leng, and M. Pakala, Appl. Phys. Lett. 100, 032402 (2012). 22B. Heinrich, Z. Celinski, J. F. Cochran, W. B. Muir, J. Rudd, Q . M. Zhong, A. S. Arrott, K. Myrtle, and J. Kirschner, Phys. Rev. Lett. 64, 673 (1990) 13

2014-05-08

We study spin pumping in a $\mathrm{Y_3Fe_5O_{12}(YIG)/Pt/Ni_{81}Fe_{19}(Py)}$ trilayer film by means of the inverse spin Hall effect (ISHE). When the ferromagnets are not excited simultaneously by a microwave, ISHE-induced voltage is of the opposite sign at each ferromagnetic resonance (FMR). The opposite sign is consistent with spin pumping of bilayer films. On the other hand, the voltage is of the same sign at each FMR when both the ferromagnets are excited simultaneously. Futhermore, the voltage greatly increases in magnitude. The observed voltage is unconventional; neither its sign nor magnitude can be expected from spin pumping of bilayer films. Control experiments show that the unconventional voltage is dominantly induced by spin pumping at the Py/Pt interface. Interaction between YIG and Py layers is a possible origin of the unconventional voltage.

Enhancement of Spin Pumping in $\mathrm{Y_3Fe_5O_{12}/Pt/Ni_{81}Fe_{19}}$ Trilayer Film

1405.1929v1

Amplitude and Phase Noise of Magnons, UC Riverside (2019) Amplitude and Phase Noise of Magnons Sergey Rumyantsev1,2, Michael Balinskiy1, Fariborz Kargar1, Alexander Khitun1 and Alexander A. Balandin1 1Nano -Device Laboratory (NDL) , Department of Electrical and Computer Engineering, University of California, Riverside, California, USA 2Center for Terahertz Research and Applications (CENTERA), Institute of High Press ure Physics, Polish Academy of Sciences, Warsaw, Poland Abstract — The low -frequency amplitude and phase noise spectra of magnetization waves , i.e. magnons , was measured in the y ttrium iron garnet (YIG) waveguide s. This type of noise , which originates from the fluctuations of the physical properties of the YIG crystal s, has to be taken into account in the design of YIG -based RF generators and magnonic devices for data processing, sensing and imaging applications. It was found that the amplitude noise level of magnons depends strongly on the power level, increas ing sharply at the on -set of nonlinear dissipation. The noise spectra of both the amplitude and phase noise have the Lorentzian shape with the characteristic frequencies below 100 Hz. Keywords — magnon, magnetization wave, phase noise, amplitude noise, ran dom telegraph signal noise, RTS I. INTRODUCTION The majority of devices for in formation processing and sensing applications are based on the charge transfer in different media, e.g. semiconductors, metals, or vacuum . Recently , a completely different approach – termed magnonics – received significant attention . It is based on manipulation of the spin currents carried by the magnetization waves – magnons – in electrical insulators [1-7]. Spin current s in insulators avoid Ohmic losses and, therefore, Joule heatin g. A number of new devices based on magnon propagation have already been proposed and demonstrated for data processing, sensing and imaging applications [8-10]. The operation frequency of magnonic devices ranges from the low GHz to THz frequenc ies. The key material for th ese devices is yttrium iron garnet (YIG). Despite the strong interest to magnonic devices, their low -frequency noise characteristics remained largely unexplored [11]. YIG has also been used for filters and resonators, operating at frequenc ies up to 26 GHz [12-15]. Noise properties of RF generators based on YIG spheres and delay lines were studied in details [16-20]. The noise of a generator depends on many factors , and the generator scheme may include several noise sources. The YIG crystal is only one of them. However, to the best of our knowledge , the noise properties YIG material have not been rigorously addressed yet. We have recently reported on the low -frequency amplitude noise in YIG waveguides [ 11]. The low -frequency noise is a ubiqui tous phenomenon, present in all kinds of electronic materials and devices [ 21-33]. The low-frequency noise in magnonic devices is also an important metric, which deserves much more attention. In this paper , we report the results of the measurements of the low -frequency noise of magnons in a YIG waveguide, focusing on the phase noise. II. EXPERIMENTAL DETAILS The YIG -film of 9.6 μm thickness and 1.5 mm × 13. 5 mm in dimensions was grown on the g adolinium gallium garnet (GGG, Gd 3Ga5O12) substrate by the liquid phase epitaxy. The Ti/Au antenna e for spin wave (SW) e xcitation and detection were fabricated on the sur face of YIG -film waveguide (see inset in Figure 1) . The devices were placed in a magnetic field created by the permanent neodymium magnet. Depending on the orientation of the magnetic field , the spin waveguide structure supports either the magneto -static s urface spin waves (MSSW s) or backward volume magneto -static spin wave s (BVMS Ws). The M SSW can propagate either on the top surface of the YIG waveguide (surface waves) or at the interface between the YIG waveguide and GGG substrate (interface waves) . The st rength of the magnetic field corresponded to the f erromagnetic resonance (FMR) frequency of about 5 GHz. Antenna 1 or 3 were used to excite spin waves and antenna 2 was used as a receiver (see Fig ure 1). In order to confirm the generation and propagation of magnon current through the electrically insulating waveguide we measured the S -parameters of the waveguide as a function of frequency and magnetic field. The measured S – parameters were compared with known dispersion laws for BVMSW and MSSW. Good agre ement with the theory confirmed the type of propagating magnons, and allowed for tuning the fp-H space parameters for the magnon noise studies. These data also confirmed that the signal is not a result of direct electromagnetic coupling between antennas. Amplitude and Phase Noise of Magnons, UC Riverside (2019) Propagating in the waveguide, magnon current acquires variations in the amplitude and phase due to the fluctuations of the physical p roperties of the YIG thin film. In order to measure these fluctuations, the commercial Schottky diode (33330B Keysight Techn ologies Inc. ) was connected to the antenna 2 (see Fig ure 1). The DC detected signal from the diode was amplified by the low noise amplifier and analyzed by the FFT spectrum analyzer. III. RESULTS AND DISCUSSIO NS Fourier transform of the signal from the diode y ields the spectrum of the amplitude fluctuations . The amplitude noise of the magnons propagating along the interface between YIG waveguide and GGG substrate was the highest, and the lowest was the noise of the volume magnons. The high noise level of the interface magnons was explained by the YIG/GGG interface roughness, resulting in stronger fluctuations of the material parameters that govern magnon current propagation. The dependence of noise on power for interface and surfaces magnons reveals one or more maxima. The positions of the noise peaks correspond to the change of the slope of S 21 dependence on the power. The spectra of the amplitude fluctuations had the shape of the Lorentzian, SV~1/(1+f2/fc2) with the characteristic corner frequency fc<100- 1000H z. In the time domain, the magnon noise revealed itself as a random telegraph signal (RTS) noise. Very small changes in the input power of ~0.1dB led to the significant changes in the RTS noise and it spectrum. RTS noise is well known in electronic devices and charge density materials and devices [ 34-38]. Observation of RTS noise in the large magnon waveguides can be explained by the individual discrete macro events which contribute to both the noise and magnon dissipation process es [11]. Fluctuations of the speed of the magnon wave lead to the fluctuations of the phase of the wave. In order to measure the phase noise, the delay line itself with the three antennas , as shown in Fig ure 1, was used as a phase detector. For this purpose , the output of the exter nal generator was split and fed to antennas 1 and 3, correspondingly (see Fig ure 1). The line for the antenna 3 included an attenuator and a phase shifter. Using the attenuator , the power on the output antenna 2 was adjusted to be equal when powered separa tely in each of two configurations. The r esulting power on antenna 3 was equal to ~2 dBm. These two signals are merged at the plane, which corresponds to the antenna 3 (see Fig ure 1). Due to the interference, when both antennas 1 and 3 are powered, the power on the output is a function of the phase difference of these two signals. This phase difference is defined by the distance between antennas 2 and 3 and by the external phase shifter. Fluctuations of the magnon wave speed contribute to the phase differen ce, and are converted to the voltage fluctuations detected at the antenna 2. The s ymbols in Fig ure 2 show the DC voltage on the detector, which is proportional to the output power at the antenna 2 as a function of the phase shift adjusted by the external p hase shifter. The s olid line is a fit with A×cos2(Ψ/2) function ( A is a fitting parameter). The derivative, R, of this function is a conversion coefficient , which determines the spectral noise density of the voltage fluctuations: Sv=SΨ×R2 (SΨ is the spectral noise density of the phase fluctuations). Fig.1. Schematic view of the YIG waveguide with three antennae . 0 1 2 3 40.05.0x10-41.0x10-31.5x10-32.0x10-3Detector output voltage, V Ψ, radAcos2(Ψ/2) Fig. 2. DC voltage on the detector as a function of the phase shift adjusted by the phase shifter. The s ymbols show experimental data while the solid line is a fit with A ×cos2(Ψ/2) function. Figure 3 shows the spectral noise density S v at f=10 Hz as a function of the phase difference. The b lue symbols and line show the measured spectral nose density at f=10 Hz; the red symbols and dashed line show the level of the backgro und noise. As one can see , the noise is minimal at the phase differences Ψ≈0 and Ψ≈π. The noise at its maximum, i.e., at Ψ≈π/2 is more that an order of magnitude higher. The conversion coefficient, R , has its maximum at this phase. Therefore, we can conclu de that this amplitude noise is predominately due to the conversion of the phase noise. Figure 4 shows the calculated phase noise S Ψ = S v/R2. As seen, within the measurement accuracy, the phase noise is independent on Ψ, as expected. Although RTS noise was not found in the phase fluctuations, the noise spectra of the pha se noise within the frequency range 10 Hz < f < 1 kHz also had the form of the Lorentzian with the characteristic frequency within 10 Hz – 100 Hz (see Figure 5 ). With the excitation power of ~2 dBm on one of the antennas , the phase noise at the frequency of 10 Hz was measured to be around - 68 dB/Hz. Amplitude and Phase Noise of Magnons, UC Riverside (2019) We attributed the measured phase noise to the magnetization wave phase velocity fluctuations. The velocity fluctuations can be estimated as S v/V2=SΨ/Ψint2, where Ψint is the phase difference gained by the spin wave between the antennas, which can be measured by the vector network analyzer. Ψint was measured by method of substitution used “unwrapped phase” format of VNA. Total phase margin at operation frequency was first measured with our device, Ψ DUT, and then with electrically short SMA connector which substituted our device, Ψ conn. Interested value Ψ int was calculated as difference Ψ DUT - Ψconn. We found Ψint=93 rad, which yield s Sv/V2≈ 2×1011 Hz-1 at the frequency of the analysis f=10 Hz. This value of the velocity fluctuations can be used to estimate the phase fluctuations in the waveguides of an arbitrary length. 0 2 410-1610-1510-1410-13Noise S v, V2/Hz Ψ, radbackground Fig. 3. Spectral noise density Sv at f=10 Hz as a function of the phase difference. The d ashed line and symbols show the background noise. 0 1 2 3 4-120-110-100-90-80-70-60-50-40Phase noise dB/Hz Ψ, degree Fig. 4. Phase noise at f=10 Hz as a function of the phase shift. 100101102103-100-90-80-70-60phase noise, dB/Hz Frequency f, Hz Fig. 5. Phase noise spectrum of magnons in YIG wavegu ide. IV. CONCLUSIONS In conclusion, the low -frequency noise of magnons, propagating as magneto -static surface spin waves , was measured in YIG waveguides at the frequencies f<1 kHz. The noise spectra had the Lorentzian shape w ith the characteristic frequencies below 100 Hz. At these frequencies , the noise of magnons set s the limit for data processing, sensing and imaging applications . It can also contribute to the phase noise of RF devices based on YIG crystal s. ACKNOWLEDGMENT S The work at UC Riverside was sup ported as part of the Spins and Heat in Nanoscale Electronic Systems (SHINES), and Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences (BES) under Award No. SC0012670. S. 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Rumyantsev, M. S. Shur, and A. A. Balandin, Selective chemical vapor sensing with few -layer MoS 2 thin- film transistors: Comparison with graphene devices, Appl. Phys. Lett., vol. 106, p. 023115, 2015. [28] S. L. Rumyantsev ,C. Jiang, R. Samnakay,M. S. Shur, A. A. Balandin, “1/f Noise Characteristics of MoS 2 Thin -Film Transistors: Comparison of Single and Multilayer Structures,” IEEE Electron Device Lett., vol. 36, pp.517 -519, 2015. [29] M. A. Stolyarov, G. Liu, S. L. Rumyantsev, M. Shur, and A. A. Balandin, “Suppression of 1/f noise in near -ballistic h -BN-graphene -h- BN heterostructure field -effect transistors,” Appl. Phys. Lett., vol. 107, p 023106, 2015. [30] G. Liu, S.L. Rumyantsev, C. Jiang, M.S. Shur, and A.A. Balandin, “Selective Gas Sensing with h-BN Capped MoS 2 Heterostructure Thin Film Transistors,” IEEE El. D ev. Lett., vol.36, pp.1202 -1204, 2015 . [31] G. Liu, S. Rumyantsev, M. A. Bloodgood, T. T. Salgue ro, M.S Shur, and A. A. Balandin, “ Low-Frequency Electronic Noise in Quasi -1D TaSe 3 van der Waals Nanowires,” Nano Lett. vol.17, pp.377−383 , 2017. [32] K. Geremew, S. Rumyantsev, M. A. Bloodgood, T. T. Salguero and A. A. Balandin, “ Unique features of the g eneration –recombination noise in quasi -one-dimensional van der Waals nanoribbons, ” Nanoscale, vol. 10, pp.19749 –19756 , 2018. [33] R. Salgado, A. Mohammadzadeh, F. Kargar, A. Geremew, Chun -Yu Huang, M. A. Bloodgood, S. Rumyantsev, T. T. Salguero, and A. A. Balandin, “Low-frequency noise spectroscopy of charge -density -wave phase transitions in vertical quasi- 2D 1T -TaS 2 devices, ” Appl . Phys . Express vol.12, p. 037001 , 2019 . [34] Z. Çelik -Butler, P. Vasina, and N. V. Amarasinghe, “A method for locating the position of oxide traps responsible for random telegrap h signals in submicron MOSFET’s ,” IEEE Trans. Electron Devices vol. 47, pp. 646 –648, 2000. [35] M. J. Kirton, and M. J. Uren, “Noise in solid -state microstructures: a new per spective on individual defects, interface states and l ow- frequency (1/f) noise,” Adv. Phys , vol. 38, pp. 376–468, 1989. [36] I. Bloom, A. C. Marley, and M. B. Weissman, “Nonequilibrium dynamics of discrete fluctuators in charge -density waves in NbSe 3,” Phys. Rev. Lett. , vol. 71, pp.4385 –4388 , 1993. [37] I. Bloom, A. C. Marley, and M. B. Weissman, “Discrete fluctuators and broadband noise in the charge -density wave in NbSe 3,” Phys. Rev. B , vol. 50, pp. 5081– 5088 , 1994. [38] G. Liu, S. Rumyantsev , M. A. Bloo dgood , T. T. Salguero , and A. A. Balandin, “Low-Frequency Current Fluctuations and Sliding of the Charge Density Waves in Two -Dimensional Materials, ” Nano Letters , vol. 18 , pp. 3630 -3636 , 2018.

2019-08-30

The low-frequency amplitude and phase noise spectra of magnetization waves, i.e. magnons, was measured in the yttrium iron garnet (YIG) waveguides. This type of noise, which originates from the fluctuations of the physical properties of the YIG crystals, has to be taken into account in the design of YIG-based RF generators and magnonic devices for data processing, sensing and imaging applications. It was found that the amplitude noise level of magnons depends strongly on the power level, increasing sharply at the on-set of nonlinear dissipation. The noise spectra of both the amplitude and phase noise have the Lorentzian shape with the characteristic frequencies below 100 Hz.

Amplitude and Phase Noise of Magnons

1909.00085v1

arXiv:2007.15299v1 [quant-ph] 30 Jul 2020Coherent multi-mode conversion from microwave to optical w ave via a magnon-cavity hybrid system Yong Sup Ihn, Su-Yong Lee, Dongkyu Kim, Sin Hyuk Yim, and Zaeill Kim∗ Quantum Physics Technology Directorate, Agency for Defens e Development, Daejeon 34186, Korea (Dated: July 31, 2020) Coherent conversion from microwave to optical wave opens ne w research avenues towards long distant quantum network covering quantum communication, c omputing, and sensing out of the laboratory. Especially multi-mode enabled system is essen tial for practical applications. Here we experimentally demonstrate coherent multi-mode conversi on from the microwave to optical wave via collective spin excitation in a single crystal yttrium iron garnet (YIG, Y 3Fe5O12) which is strongly coupled to a microwave cavity mode in a three-dimensional re ctangular cavity. Expanding collective spin excitation mode of our magnon-cavity hybrid system fro m Kittel to multi magnetostatic modes, we verify that the size of YIG sphere predominantly plays a cr ucial role for the microwave-to-optical multi-mode conversion efficiency at resonant conditions. We also find that the coupling strength between multi magnetostatic modes and a cavity mode is manip ulated by the position of a YIG inside the cavity. It is expected to be valuable for designin g a magnon-hybrid system that can be used for coherent conversion between microwave and optical photons. I. INTRODUCTION Strong coupling induced by resonant light-matter in- teraction can give rise to coherent information trans- fer between distinct physical systems in quantum and classical information processing1–3. The coherent trans- fer of quantum state is a key role in realizing large- scale quantum optical networks and long distance quan- tum sensing and imaging4–9since it allows quantum in- formation to be exchanged between different systems that operate at different energy scales. A platform for transferring multi-mode states will be attractive to the practical application of quantum-enhance metrology and communication10,11. After the first demonstration of op- tical frequency conversion12, the photon frequency con- version has been implemented with crystals in optical domain, and with superconducting circuits in the mi- crowave domain13,14. Since it has great potential in realizing large-scale quantum optical networks with su- perconducting qubits, recently, the microwave to optical field conversionhas been intensivelyattracted and exper- imentally demonstrated by using intermediate systems, such as optomechanical systems15–20, electro-optical systems21, atomic ensembles22–25, and magnons26. So far, the maximum microwave-to-optical (MO) conver- sion efficiency has been demonstrated in a nanomechan- ical resonator system employing a nano-membrane that is combined with an optical cavity while it is coupled to a superconducting microwave resonator. Its conver- sion efficiency reached 47 % at low temperature16,17. A ferromagnetic material, an yttrium iron garnet (YIG), in a microwave cavity offers strong interaction between magnon and microwave cavity modes at both low and room temperatures since YIG has a Curie temperature of 560K and a net spin density of 2 .1×1022µBcm−3(µB; Bohr magneton) that is few orders of magnitude higher than a net spin density of 1016−1018µBcm−3in para- magnetic materials. High Verdet constant with sharp linewidth ofelectronspinresonancein microwavedomainalso makes YIG noticeable in Faraday effect27,28. Re- cently, YIG based materials have been studied on a novel concept for ultrafast magneto-optic polarization modula- tion with frequencies up to THz orders29,30. Longer spin excitationtime thanparamagneticspinsystemisanother advantage of YIG and its hybrid system31,32. In this hy- brid system, the MO conversion is achieved through the Faraday effect and Purcell effect. The magnetization os- cillation induced and amplified by the Purcell effect of a microwave cavity mode creates the sidebands to the incidental optical wave, resulting in coherent conversion between microwave and optical wave. So far, the MO conversion via YIG-cavity system has been focused only for the Kittel mode. In this paper, we report coherent multi-mode conver- sion from the microwave to optical wave fields, which is based on a hybrid system consisting of a sphere of YIG and a three-dimensional rectangular microwave cavity. We experimentally demonstrate and verify that the size of YIG is a dominant factor of the coherent multi-mode conversionefficiency. Theconversionefficiencyistheoret- ically derived by using the interaction Hamiltonian with the ferromagnetic-resonance (FRM or Kittel mode, KM) andahighermagnetostaticmode(MSM) andexperimen- tally characterized by normal-mode splitting, coupling strengths of the ferromagnetic-resonance, and magneto- static modes. For the near-uniform microwave cavity field, all the spins in the YIG sphere precess in phase that is called Kittel mode, and therefore the whole YIG sphere can be treated as a giant spin33,34. As the YIG sphere size increases, the microwave cavity field can no longer be treated as a uniform field, leading to the higher order magnon modes which are observed35–37. By posi- tioning a YIG off the uniform microwave cavity field re- gion, we can manipulate the coupling strength between multi magnetostatic modes and a cavity. As a result, it is shown that KM and MSM manifested in each YIG sample are successfully transferred through the conver- sion process from microwave field to optical wave field.2 D E FIG. 1. (a) Schematic of hybrid system for coherent cover- sion from microwave to light, where an YIG sphere is inserted into a rectangular microwave cavity. (b) Magnon-cavity hy- brid model. In the hybrid system, magnon mode ˆ smand a microwave cavity mode ˆ aare strongly coupled with a coupling strength gm. Here,κiandγmpresent internal cavity loss and intrinsic loss for magnon modes, respectively. A microwave field mode ˆ aiis coupled to a microwave cavity mode at a rate κe, while a traveling optical wave field mode ˆbiis coupled to the magnon mode at a rate δm. Through this process, mi- crowave field is converted to the traveling optical wave field with a conversion efficiency ηt. II. THEORETICAL CONVERSION MODEL Fig. 1(a) shows the main part of our hybrid system, a YIG sphere and a 3D rectangular microwave cavity. Due to the magnetic and optical properties, a highly polished YIG sphere can serve as an excellent magnon resonator at microwave frequencies. A 3D rectangular microwave cavity intrinsically maintains a low damping rate com- patible with one of magnon mode at room temperature and enhances the coupling rate between a magnon mode and a cavity mode through the Purcell effect. Then, the linearly polarized light travels through a YIG perpen- dicular to the magnetization direction along the static magnetic field. Finally the Faraday effect creates the op- tical sidebands, or polarization oscillations of the light induced by the magnetization oscillations26,38,39. Fig. 1(b) describes the schematic diagram for coherent con-version from microwave photons to optical photons. The total Hamiltonian( ˆHt) describing the conversion process including the KM and a higher MSM can be given by ˆHt=ˆHc+ˆHs+ˆHo, ˆHc=−i√2κe/bracketleftBig ˆa†ˆai(t)−ˆa† i(t)ˆa/bracketrightBig , ˆHs=ωcˆa†ˆa+ωKˆs† KˆsK+ωMˆs† MˆsM+gK/parenleftBig ˆa†ˆsK+ˆaˆs† K/parenrightBig +gM/parenleftBig ˆa†ˆsM+ˆaˆs† M/parenrightBig , ˆHo=−i/radicalbig 2δK/parenleftBig ˆsK+ ˆs† K/parenrightBig/bracketleftBig ˆbi(t)eiΩ0t−ˆb† i(t)e−iΩ0t/bracketrightBig −i/radicalbig 2δM/parenleftBig ˆsM+ ˆs† M/parenrightBig/bracketleftBig ˆbi(t)eiΩ0t−ˆb† i(t)e−iΩ0t/bracketrightBig , (1) where/planckover2pi1= 1. The subscripts KandMstand for the KM and a higher MSM, respectively. The MO coversion proceeds with the following steps in the Hamiltonians: ˆHc→ˆHs→ˆHo.ˆHcdescribes the interaction Hamil- tonian between an itinerant microwave mode ˆ aiand the microwave cavity mode ˆ awith external coupling rate κe, which results from the rotating-wave approximation. ˆHs including the system Hamiltonian describes the interac- tion Hamiltonian between cavity and magnon modes. gK andgMrepresent the magnon-microwave photon cou- pling strengths for KM and MSM, respectively. Here, we notethat gKandgMincludetheoverlappingcoefficient ξ, which is related to the space variation effect between the magnetic field of the cavity mode and magnon modes32. ˆa†(ˆa) is the creation (annihilation) operator for the mi- crowavephotonattheangularfrequency ωc. ˆs† K(ˆsK)and ˆs† M(ˆsM) represent the collective spin excitations for KM and MSM at angular frequency ωKandωM, respectively (see Appendix A). The number of spins in a YIG sphere can contribute to both KM and MSM. ˆHodescribes the interaction Hamiltonian between magnon modes ˆ sand a traveling optical photon mode ˆbiwith angular frequency Ω0.δKandδMrefer to the optical photon-magnon cou- pling rate for KM and MSM, respectively. Since it in- cludes both Stokes and anti-Stokes processes that are in- volved in the MO conversion, we leaves the Hamiltonian ˆHowithout the rotating-wave approximation. The MO conversion indicates that the itinerant microwave pho- ton mode ˆ aiis converted to the traveling optical photon modeˆbo. In our experiment, the strongly coupled magnons and cavitymicrowavephotonmodecanbedeterminedbynor- mal mode splittings in transmission spectra which are measured from the input and output channels (see Fig. 2(c)). According to the input-output relation40, equa- tions of motions for a cavity mode and magnon modes can be obtained from quantum Langevin equation (see Appendix A). As a result, the transmission for multi- magnon modes can be given by S21(ω) =−i2κe ω−ωc+iκt−/summationtext mg2mχm,(2)3 where χm(ω) = [ω−ωm+iγm]−1. (3) Here,κt=κe+κiis the total loss rate which includes both external coupling rate κeand internal loss of the cavityκi. For the conversion process from the microwave to op- tical wave, the conversion coefficients with amplification factorβmfor the KM and a MSM are obtained as (see Appendix A) S31,K(ω) =−2/radicalbig βKδKκegKχKχc/parenleftbig 1+g2 MχMχcTM/parenrightbig 1−g2 KχKχc(1+g2 MχMχcTM), (4) and S31,M(ω) =−2/radicalbig βMδMκegMχMχc/parenleftbig 1+g2 KχKχcTK/parenrightbig 1−g2 MχMχc(1+g2 KχKχcTK), (5) where χc(ω) = [ω−ωc+iκt]−1, Tm(ω) =/bracketleftbig 1−g2 mχmχc/bracketrightbig−1.(6) Therefore, the MO conversion efficiency including both KM and a MSM modes can be given by ηt(ω) =|S31,K(ω)|2+|S31,M(ω)|2.(7) Here, at the resonant condition ω−ωc=ω−ωK= ω−ωM= 0, the two-mode conversion efficiency can be represented in terms of cooperativities for CK=g2 K κtγK andCM=g2 M κtγM, ηt=4κe (1+CK+CM)2/bracketleftbigg δKβKC2 K g2 K+δMβMC2 M g2 M/bracketrightbigg .(8) In this work, the two-mode conversion efficiency given in Eq.(8) is well matched to the experimental results. If we expand the interaction Hamiltonian to possess higher order terms, the multi-mode MO conversion efficiency can be obtained by ηt=4κe (1+/summationtext mCm)2/summationdisplay m/bracketleftbigg δmβmC2 m g2m/bracketrightbigg ,(9) wheremis a mode index for each MSM. Since so far the MO conversion for multi-modes has not been reported in magnon-cavity system, our theoretical result can be applied to multi-mode conversionbasedon ferromagnetic material-hybrid systems. III. EXPERIMENTS AND RESULTS As a ferromagnetic material, we use commercial YIG spheres of diameter 0.45, 0.75, and 1 mm from Fer- risphere and Microsphere. A 3D rectangular cavity D E;+)URJGTGU F  [\ ][\ ]PP PP PP FIG. 2. (a) Numerical simulation of the magnetic field distri - bution of the fundamental mode TE 101inside the microwave cavity with the volume of 20 ×20×4 mm−3. An YIG sphere made by Ferrisphere Inc.41is mounted at the field maximum of the fundamental mode. (b) Transmission power (left y- axis) and phase (right y-axis) without a YIG sphere through the cavity as a function of the microwave frequency. ωc/2π= 10.632 GHz, κi/2π= 0.6 MHz, κe/2π= 2.1 MHz. Experi- mental data (black solid-circles and solid-diamonds), and the- oretical results (redand bluecurves). (c) Experimental se t-up for the microwave to optical light conversion. To examine th e hybrid system, the transmission data are taken by a vector network analyzer. For the conversion measurement, a 1550- nm cw laser with z-polarization is injected into a YIG sphere whose beam waist size is about 120 µm. The polarization of the light is oscillated by the magnetization oscillations, which is induced by the microwave driving field fed into the channel 1. Finally, the polarization oscillations of light is detec ted by a fast photodiode and amplified by a microwave amplifier with 30 dB amplification along the channel 3. HWP refers to the half-waveplate. is made of oxygen-free copper with the volume Vcof 20×20×4 mm3and the fundamental mode TE 101is used for magnetic-dipole coupling. Fig. 2(a) shows the microwavemagnetic field distribution of the fundamental mode TE 101at the resonant frequency ωc/2πof 10.598 GHz, simulated by COMSOL Multiphysics/circleR. The YIG4 D F E G H I% 7 % 7% 7 % 7% 7 % 7% 7 % 7% 7 % 7% 7 % 7% 7 % 7% 7 % 7 FIG. 3. (a) Measured microwave transmission spectrum, |S21(ω)|2of the 0.45-mm YIG-cavity hybrid system as a function of the microwave frequency and the static magnetic field. The ho rizontal and diagonal dashed lines (yellow) show the freque ncy of the fundamental mode TE 101and the Kittel mode frequency, respectively. The white-das hed line describes the dispersion of the resonance frequency obtained by diagonalizing ˆHsas given in Eq.(1). (b) Cross sections of |S21(ω)|2at static magnetic fields 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 data (solid dots). The individual data sets are vertically o ffset for clarity. (c) The phase S21(ω) with theoretical hand-fits at the static magnetic fields. (d) Measured MO conversion spect rum,|S31,K(ω)|2of the same system. Here, the MO conversion spectrum is obtained from the raw data which is amplified by a m icrowave amplifier (30 dB) and detected by a fast photodiode. (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 lines 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 static magnetic fields. The individual data sets are vertica lly offset for clarity. spheremounted onthe aluminarodalongthe crystalaxis /angbracketleft110/angbracketrightis placednearthe maximumofthe magneticfieldin order to get the largest coupling strength and the unifor- mity ofthe field as shownin Fig. 2(a). Fig. 2(b) presents measured transmission magnitude and phase without a YIGspherethroughthecavity. Asaresult, thefrequency of TE 101mode (ωc/2π) is 10.632 GHz, and the exter- nal cavity loss rate ( κe/2π) and the internal cavity loss (κi/2π) are 2.1 MHz and 0.6 MHz, respectively. In order to manipulate the magnetic field, a set of neodymium-iron-boron magnets applies a static mag- netic field of around 380 mT to the YIG sphere. The magnetic components of the microwave field perpendic- ular to the bias field induce the spin flip, and excite the magnon mode in YIG. The magnetic circuit consists of a set of permanent magnets and a pair of Helmholtz coils with 800 turns of wires for each. The cavity is placed at the center of a pair of Helmholtz coils, so a static mag- netic field along the z-axis is applied to the crystal axis /angbracketleft100/angbracketrightof the YIG sphere across the cavity. Helmholtz coils driven by a bipolar current supplier combine with the permanent magnetsand tunes the resonancefrequen- cies of magnon modes. The magnetic field measured bya flux gate sensor (3MTS) provides the field-to-current conversion rate of dB/dI= 70 Gauss /A. Fig. 2(c) shows the experimental set-up for the microwave to light con- version. We use temperature controlled butterfly diode laser to deliver 1550-nm cw input power of 5 mW before the YIG and get the transmission of 80 %. We also use some of optics and microwave components such as po- larizer and HWP to define the linear polarization, lens to focus the laser into the YIG, fast photo detector to receive the transmitted laser with optical side band, low noise microwave amplifiers with 30 dB amplification and isolators to increase the signal, and a vector network an- alyzer by probing the transmission through the hybrid system. Fig. 3(a) shows the measured microwave transmis- sion spectrum, |S21(ω)|2, of the hybrid system with the YIG diameter of 0.45 mm as a function of the frequency and the static magnetic field. A normal-mode splitting is clearly observed, and the avoided crossing manifests strong coupling between the Kittel mode and the mi- crowave cavity mode. As the magnetic field is swept, the Kittel mode approaches the cavity mode up to the degeneracy point. The horizontal dashed line shows the5 fundamental mode frequency of the cavity and the di- agonal dashed line presents the Kittel mode frequency, f11=ω11/2π=µ0γHo/2π(see Appendix B). White- dashed lines are the dispersion curves of the resonance frequency, ω±=ωc+ωK 2±1 2/radicalbig 4g2 K+(ωc−ωK)2, which are obtained from the diagonalization of the interaction Hamiltonian ˆHsin Eq.(1) without the last term. In or- der to quantify the coupling strength and the damping rate of ferromagnetic resonace frequency, the experimen- tal data at some of magnetic fields are hand-fitted into the theoretical transmission coefficient S21(ω) given in Eq.(2) (see Fig. 3(b)). As a result, the total cavity linewidth ( κt/2π), the coupling strength ( gK/2π), and the Kittel mode linewidth ( γK/2π) are determined as 2.7, 28.6, and 2.3 MHz, respectively. Here, the coupling strength gKcan be represented by gK=gB√ 2Ns= ξγ 2/radicalBig /planckover2pi1ωcµ0 Vc√ 2Ns, whereγis the electron gyromagnetic ratio,ωcis the cavity resonance frequency, Vcis the vol- ume of the cavity of 20 ×20×4 mm3, ands= 5/2 is the spin number in YIG26.gBis the coupling strength of a single Bohr magneton to the cavity which can be calcu- lated as 0.325 Hz for TE 101mode in our system. If we assumethatallthespinsintheYIGsphereareprecessing in phase, the coupling strength gKof the Kittel mode to the cavity mode is proportional to the square root of the numberofnetspins NK42. Inthiscase,almostallofspins contribute to the KM. The coupling strength gM/2πfor anMSM islessthan 1.0MHz, sothisterm canbe ignored here. The coefficient ξ≤1 indicates the spatial overlap and polarization matching conditions between the mi- crowave and the magnon modes32. From the extracted value of gK, we can deduce the number of net spins of NK= 1.51×1017. The fact which gKis much largerthan κtandγK, indicates that the system is in the strong coupling regime even at the room temperature. With the experimental parameters, we obtain a cooperativity ofCK=g2 K/κtγK= 132, indicating how well collective spins in the YIG sphere couple to the microwave cavity mode43,44. Fig. 3(c) shows cross-sectional experimental dataandtheoreticalcurvesforthe phasesof S21(ω). The phase values of S21(ω) range from −π/2 toπ/2 that are two times less than the phase values of S31(ω) as shown in Fig. 3(f). In Ref. [26], the phase of a reflection spec- trumS11(ω) shows the same feature as that of S31(ω) except the scale factor of 2. In this work, since all ex- perimental data are based on the S21(ω) transmission spectra, the phase of reflection spectrum S11(ω) is given in Appendix C. Here, Fig. 7 shows the similar feature to the phase of S31(ω). Fig. 3(c) and 3(f) also show that the phase of S21(ω) is clearly converted to the phase of S31(ω). Therefore, the conveyance of the phase clearly exhibits the coherent conversionfrom microwaveto light. Fig. 3(d) shows the measured power of the MO con- versioncoefficient, |S31,K(ω)|2, ofthehybridsystemwith the same YIG. The conversion spectrum |S31,K(ω)|2is almost similar to |S21(ω)|2, which implies that the mi- crowave field is coherently converted to the optical wavefield. Fig. 3(e) shows the cross sectional experimental data at some of magnetic fields in Fig. 3(d) that are hand-fitted into the theoretical transmission coefficient |S31,K(ω)|2given in Eq.(7). As a result, gK/2πand γK/2πare 28.5 MHz and 2.4 MHz, respectively that are quite close to the result of |S21(ω)|2. In orderto evaluate the MO conversion efficiency, we first estimate the opti- cal photon-magnon coupling rate δKwhich is given by δK=G2 Kl2 16VmnKP0 /planckover2pi1Ω026. Withl= 0.45 mm being the length of the YIG sample, nK= 3.16×1027m−3andVm= 4 3π×0.2253mm3being the spin density and the spatial volume of the magnetostatic mode, V= 3.8 radians/cm at 1.55µm49that result in GK= 4V/nK= 4.81×10−25 m2, andP0//planckover2pi1Ω0= 1.17×1017Hz forP0= 15 mW, we have δK/2π= 0.0036 Hz. Therefore, under the near resonant conditions at ω−ωc=ω−ωK= 0, the total conversion efficiency ηt=|S31,K|2can be approximated in terms of the coupling strength gK, ηt=/parenleftbigg2√δKκeCK gK(1+CK)/parenrightbigg2 . (10) With all obtained parameters for the YIG sphere with 0.45 mm diameter, we attain ηt= 8.45×10−11. This low conversion efficiency results from the small light-magnon coupling rate. Actually, the maximum conversion effi- ciency is occured at particular detunings from the cavity resonance and the Kittel mode frequency26. However, in this experiment, we are interested in the multi-mode conversion efficiency at the degenerate point. In orderto examinethe YIG sizedependence ofsystem parameters,wealsomeasuredthetransmissionspectraof YIG spheres with diameters of 0.75 and 1 mm, as shown in Fig. 4. Fig. 4(a) and 4(e) show transmission magni- tude,|S21(ω)|2, measured for YIG diameter of 0.75 and 1.0 mm, respectively. As the size of the YIG sphere in- creases, the larger number of spins in a bigger sphere can contribute the interaction with the microwave cavity mode, that makes the normal-mode splitting wider. As a result, we obtain the coupling strengths of gK/2π= 67.3 and 91 MHz for 0.75 and 1.0-mm YIG spheres so that the cooperativity CKfor the Kittel mode reaches up to about 3.5 ×103as shown in Table I. In addition to larger coupling strengths, another avoided level crossing feature is observed because of the strong coupling cor- responding to the nonuniform MSMs which can be also coupled to the cavity mode. The coupling strengths of MSM are gM/2π= 4 and 12 MHz, and the decay rates areγM/2π= 1.1 and 0.9 MHz for YIG spheres with 0.75 and 1.0 mm diameter, respectively. Based on the fitting parameters, 2D spectra of |S21(ω)|2for each case are simulated in Fig. 4(b) and (f). Here, the red-dashed line describes the nonuniform MSM which is identified by magnetostatic theory35,37. In general, the relation be- tween MSM frequencies and the external magnetic field is linear for i−|j|= 0 or 1 as mentioned in Appendix B. When the YIG sphere is subjected to an oscillating mag- netic field at ωijand a strong coupling regime is reached6 D E F H I JG K FIG. 4. (a) Measured |S21(ω)|2of the 0.75-mm YIG-cavity hybrid system as a function of the m icrowave frequency and the static magnetic field. The horizontal and diagonal dashed li nes show the frequency of the fundamental mode TE 101and the Kittel mode frequency, respectively. The white-dashed lin e describes the dispersion of the resonance frequency obtai ned by diagonalizing ˆHsas given in Eq.(1). (b) The simulated spectrum of |S21(ω)|2based on Eq.(2). For FMR or the Kittel mode, gK/2πandγK/2πare 67.3 and 1.1 MHz, respectively, and for MSM, gM/2πandγM/2πare 4 and 1.5 MHz, respectively. The red-dashed line refers to the (2,0) mode. (c) Measured MO conversion spectrum, ηtof the 0.75-mm YIG-cavity hybrid system. The measured spectrum is based on the raw data which i s amplified by a microwave amplifier (30 dB) and detected by a fast photodiode. (d) The simulated spectrum of ηtof the 0.75-mm YIG-cavity hybrid system based on Eq.(7). (e) Measured |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 or the Kittel mode, gK/2πandγK/2πare 91 and 0.95 MHz, respectively, and for MSM, gM/2πandγM/2πare 12 and 0.9 MHz, 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 hybrid system. The measured spectrum is based on the raw data which is amplified by a microwave amplifier (30 dB) and detected by a fast photodiode. (h) The simulated spectrum of ηtof the 1.0-mm YIG-cavity hybrid system based on Eq.(7). atHo, avoidedlevelcrossingsappearatthe regionswhere theresonancefrequenciesoftwosubsystemsarematched, that make it possible to distinguish a MSM with iandj associated to an avoided level crossing45–48. In our case, additional avoided level crossing is placed at the (2,0) mode which can be given by37 ω20=µ0γMs/radicalBigg/parenleftbiggHo Ms−1 3/parenrightbigg/parenleftbiggHo Ms+7 15/parenrightbigg .(11) Fig. 4(c) and (g) show the MO conversion spectra, ηt, measured for the YIG diameter of 0.75 and 1.0 mm, re- spectively. These MO conversion spectra present raw data which are amplified and detected by using a mi- crowave amplifier and a fast photodiode. One can find the same avoided level crossing features, as shown in Fig. 4(a) and 4(e), which clearly demonstrates the coherent conversion from microwave to optical photons. Based on the fitting parameters, 2D spectra of ηtfor each case are simulated in Fig. 4(d) and 4(h). As a result, when we take into account both KM and MSM contributions, the total conversion efficiency ηtare 5.12×10−12and 3.46×10−12for 0.75 and 1.0 mm YIG spheres, respec- tively. We summarize system parameters for each YIGsphere in Table I that are obtained from the two-mode MO conversion process. To examine the size dependence of a YIG sphere for MOconversionefficiency, wefirstevaluatethevolumede- pendence ofparametersusedforMLconversionefficiency at resonant conditions as shown in Fig. 5. According to the Ref. [30], it demonstrated that the coupling strength gKof the Kittel mode is proportional to the square root of the volume (or the number of spins) of YIG spheres. gKis linear-fitted to f(x) = 130 .97x, where xis the square root of volume V1/2. For the higher MSM, gM is not proportional to the linear function, but rather the quardratic function which is f(x) = 22.19x2(Fig. 5(a)). Thismight be due to the factthat the spin excitationsin- duced by non-uniform field do not linearly contribute to a higher mode. According to the relation of Cm=g2 m κtγm, the cooperativity CKis fitted to f(x) = 6569 .43x2and forCM,f(x) = 228.79x4is used. Inaddition, δKandδMalsohavethedependenceofthe number of spins. δKis fitted to f(x) = 0.693+0.625/x andδMis fitted to f(x) = 0.139/x2as shown in Fig. 5(b). By using these fitting values of system parameters, we obtain the theoretical fit curve for the MO conversion7 TABLE I. System parameters for two-mode MO conversion Parameter 0.45-mm dia. 0.75-mm dia. 1.0-mm dia. gK[MHz] 2 π×28.6 2 π×67.3 2 π×91.0 γK[MHz] 2 π×2.3 2 π×1.1 2 π×0.95 gM[MHz] <2π×1.0 2 π×4.0 2 π×12.0 γM[MHz] >2π×2.0 2 π×1.5 2 π×0.9 CK 132 1373 3487 CM 0.19 3.6 64 NK 1.51×10178.36×10171.53×1018 NM <1.84×10142.95×10152.66×1016 Vm[m3] 4.77 ×10−112.21×10−105.24×10−10 nK[m−3] 3.16 ×10273.79×10272.92×1027 nM[m−3]<3.87×10241.34×10255.07×1025 GK[m2] 4.81 ×10−254.02×10−255.21×10−25 GM[m2]> 3.93×10−221.14×10−222.99×10−23 δK[mHz] 2 π×3.61 2 π×1.80 2 π×1.75 δM[Hz] 2 π×2.95 2 π×0.512 2 π×0.101 ηt 8.45×10−115.12×10−123.46×10−12 Subscripts KandMdenote the Kittel mode and MSM, re- spectively. D E PP PP PPPPPPPP [[ /ScriptT /ScriptT FIG. 5. (a) Extracted values (solid circles) and fit curves for coupling strengths and cooperativities for the KM and MSM as a function of the square root of the YIG volume. (b) Extracted values (solid circles) and fit curves for optic al photon-magnon coupling rates for the KM and MSM and the total MO conversion efficiency as a function of the square root of the YIG volume.efficiency based on Eq.(8) as presented in Fig 5(b). As a YIG size increases, the total MO conversion efficiency at the resonant condition decreases since the increments of coupling strength and cooperativity lead to the drop in the MO conversion efficiency as given by Eq.(8). In our system, the conversion efficiency at the resonant condi- tion is limitted to 10−11order. This mainly comes from the small coupling rate δKandδMbetween the opti- cal photons and magnons although it depends on the experimental conditions such as the quality of the sam- ple and proper alignment. Therefore, we need to im- prove the coupling rate δmto enhance coherent quantum conversionefficiency between microwaveand optical pho- tons. There are several suggestions as mentioned in ref. [26]. One possible way was to use the optical whispering gallery modes (WGMs) of an YIG sphere50,51. No one hasachievedasignificantimprovement,however,suppos- edly due to the small overlap between the Kittel mode and WGMs. Other suggestions are to utilize a magnetic material with a large Verdet constant such as CrBr 3and iron garnet based on rare-earth atoms52–55. IV. DISCUSSION We clearly observe not only the YIG size dependence of the MO conversion but also the coupling strength be- tween the multi-magnetostatic mode and a cavity. But note that the multi-mode MO conversionfeatures arenot prominent compared to the single-mode MO conversion since the most of spins are involved in the KM mode that makes gMandCMmuch lower than the values of gKandCK. In order to make the dominant contribution of spins to higher MSM, we carefully position a 1.0 mm- YIG sphere off the uniform microwave mode region, so that a non-uniform MSM also apprearsat the degenerate point as shown in Fig. 6(a). In this configuration, the anti-crossingsduetothehigherMSMbecomelargersince the number of spins contributing to the higher mode in- creases. Fig. 6(b) shows the simultion result of |S21(ω)|2 based on Eq.(2). As a result, gK/2πandgM/2πare 83.4 and 25 MHz, respectively, which are few orders larger than decay rates of γK/2π=1.1 and γM/2π=0.5 MHz, that indicates the strong couplings between the cavity mode and the KM and MSM. Fig. 6(c) presents the 2D spectrum of ηt. Based on the measured data, ηtis sim- ulated by using Eq.(7) as shown in Fig. 6(d). We find out that the theoretical model is well matched with the experimental results. Here, we ignore higher modes in the tail of the spectrum because their contributions are very small in the MO conversion efficiency. The total multi-mode MO conversion efficiency is 1.02 ×10−11at the resonant condition. To date, adjustable MO conver- sion for multi-modes has not been reported in magnon- cavity system.8 D E F G FIG. 6. (a) Measured |S21(ω)|2through the 1.0-mm YIG-cavity hybrid system while YIG posit ion adjusted to enhance the MSMs. (b) The simulated spectrum of |S21(ω)|2from Eq.(2). (c) Measured ηtof the same system. The measured spectrum is based on the raw data which is amplified by a microwave ampli fier (30 dB) and detected by a fast photodiode. (d)The simulated spectrum of ηtfrom Eq.(7). V. CONCLUSION We have experimentally demonstrated coherent multi- mode conversion from microwave to optical fields via a YIG sphere in a rectangular microwave cavity. A large number of spins in ferromagnetic materials easily couple the collective excitation to cavity photons, that makes it possible to hybridize the microwave photon modes and magnetostatic modes. A traveling optical field is coupled to a microwavefield through this hybrid system. We first observed YIG size dependence of conversion efficiency by measuring the normal-mode splitting between the mag- netostatic modes and the microwavecavitymodes, where the coupling strength is in the order of magnitude larger than the decay rates. Based on our multi-mode MO con- version model, we analyzed all the system parameters with experimental data, confirming that the theoretical model is consistent with the experimental results. The total multi-mode conversion efficiency at the resonant condition reaches 1.02 ×10−11for 1.0 mm-YIG sphere. We also evaluate the multi-mode MO conversion effi- ciency by manipulating position of the YIG sphere in- side the cavity. These sharp and adjustable multi-mode conversion shows the possibility of coherent conversionof multi-mode quantum states while keeping coherence time. This work will also provide optimal design condi- tions of a cavity magnon-microwave photon system that can be used for coherent conversion between microwave and optical photons. ACKNOWLEDGMENTS This work was supported by a grant to Quantum Fre- quency Conversion Project funded by Defense Acquisi- tion Program Administration and Agency for Defense Development. We would like to thank prof. Changsuk Noh for his valuable comment on the quantum input- output relation and also thank Jinwon Yoo, prof. Won- min Son, and prof. Suyeon Cho for productive discussion on the hybrid system.9 Appendix A: The interaction Hamiltonian for the multi-mode microwave-to-optical wave conversion The Hamiltonian for the magnon-cavity system can be written as Hs=ωcˆa†ˆa+/summationdisplay m=K,M/bracketleftBig gµBBm zˆSm z+gm(ˆaˆSm ++ˆa†ˆSm −)/bracketrightBig , (A1) whereωcis the angular frequency of the cavity mode TE101, ˆa†(ˆa) is the microwave photon creation (anihila- tion) operator, and m=K,Mdenotes the Kittel mode (KM) and magneto static modes (MSM), respectively. g is the electron g-factor, µBis the Bohr magneton, and Bm zis the effective magnetic field affected by the magnon modes of the YIG sphere. The exchange interaction be- tween electron spins can be ignored because of the long- wavelength descrete modes of spins in the YIG sphere. Since the frequency of the corresponding magnon mode is different from each other, the Hamiltonian for each magnon mode can be written seperately. Here, ˆSmis the collective spin operator for magnon modes which is given by ( ˆSm x,ˆSm y,ˆSm z). These collective spin operators can be expressed in terms of the bosonic operators ˆ s† m, ˆsmby using the Holstein-Primakoff transformation56,57: ˆSm +=ˆSm x+iˆSm y= ˆs† m/radicalBig 2Sm−ˆs† mˆsm,ˆSm −=ˆSm x−iˆSm y= (/radicalBig 2Sm−ˆs† mˆsm)ˆsm, andˆSm z= ˆs† mˆsm−2Sm, whereSm is the total spin number of the corresponding magnon mode. For the low-lying excitations/angbracketleftbig ˆs† mˆsm/angbracketrightbig ≪2Sm, the Hamiltonian Hscan be obtained as Hs=ωcˆa†ˆa+/summationdisplay m=K,M/bracketleftbig ωmˆs† mˆsm+gm(ˆaˆs† m+ˆa†ˆsm)/bracketrightbig , (A2) whereωm=gµBBm zis the angular frequency of the corresponding magnon mode and gm=gB√ 2Sm= ξγ 2/radicalBig /planckover2pi1ωcµ0 Vc√ 2Sm. Here,γis the electron gyromagnetic ratio,ωcis the cavity resonance frequency, Vcis the vol- ume of the cavity, gBis the coupling strength of a single Bohr magneton to the cavity for TE 101mode, and ξis the spatial overlapping coefficient which is relevant to the spatial variation effect. In the Kittel mode, since the magnetic dipiole coupling between the spins engenders a uniform demagnetization field parallel to the magnetiza- tion in a sphere, the demagnetizing field plays no role in the magnetization dynamics for the Kittel mode. For the non-uniform profile for MSM, the variation in space plays a crucial role not only in the frequency calculation but also in the coupling with the exciting field as well as the light. Therefore, the interaction Hamiltonian of the multi- mode MO conversion can be given by Eq.(1) which con- sists of the magnon, microwave photon, optical photon, and their interactions. According to the input-output relation40, equations of motions for a cavity mode and magnon modes can be obtained from quantum Langevinequation. For the cavity mode ˆ a, ˙ˆa(t) =−i[ˆa,ˆHs]−κtˆa(t)−√ 2κeˆai(t) =−iωcˆa(t)−i(gKˆsK(t)+gMˆsM(t))−κtˆa(t) −√2κeˆai(t), (A3) where the total loss rate κt=κe+κiincludes both exter- nal coupling and internal losses of the cavity. The cavity mode ˆacan be given by ˆa(t) =χc(ω)/bracketleftbig gKˆsK(t)+gMˆsM(t)−i√ 2κeˆai(t)/bracketrightbig ,(A4) where χc(ω) = [ω−ωc+iκt]−1. (A5) Sincemagnonsdonotcoupledirectlytothecavity,noad- ditional input and output magnons are involved. There- fore, the equation of motion of ˆ sKcan be given by ˙ˆsK(t) =−i[ˆsK,ˆHs]−γKˆsK(t) =−iωKˆsK(t)−igKˆa(t)−γKˆsK(t) ˆsK(t) =χK(ω)gKˆa(t),(A6) where χK(ω) = [ω−ωK+iγK]−1. (A7) In the same manner, ˆ sMhas the similar form which is ˆsM(t) =χM(ω)gMˆa(t), (A8) where χM(ω) = [ω−ωM+iγM]−1. (A9) Substituting Eq. (A6) and Eq. (A8) into Eq. (A4) and applying the Fourier transform of the cavity mode ˆ a, we can derive ˆa(ω) =−i√ 2κeT−1ˆai(ω), (A10) where T=ω−ωc+iκt−(g2 KχK+g2 MχM).(A11) In our experiment, we obtain the transmission spectrum which can be determined by measuring the output port 2 from the input port 1. For no input in port 2 and the same external coupling rate at both ports, the boundary condition becomes ˆ ao,2(ω) =√2κeˆa(ω) that results in the transmission, S21=ˆao,2 ˆai,1=−i2κeT−1. (A12) For the transmission for multi modes, Eq. (A12) can be extended to T=ω−ωc+iκt−/summationtext mg2 mχm.10 In the conversion process from microwave to optical wave, we can obtain the equation of motions for magnon modes which are given by ˙ˆsK(t) =−i[ˆsK,ˆHs]−γKˆsK(t) −/radicalbig 2δK/parenleftBig ˆbi(t)eiΩ0t−ˆb† i(t)e−iΩ0t/parenrightBig ˙ˆsM(t) =−i[ˆsM,ˆHs]−γMˆsM(t) −/radicalbig 2δM/parenleftBig ˆbi(t)eiΩ0t−ˆb† i(t)e−iΩ0t/parenrightBig .(A13) As a result, magnon modes ˆ sKand ˆsMare written as ˆsK(t) =χK/bracketleftBig gKˆa(t)−i/radicalbig 2δKˆbi(t)/bracketrightBig ˆsM(t) =χM/bracketleftBig gMˆa(t)−i/radicalbig 2δMˆbi(t)/bracketrightBig .(A14) After substituting Eq. (A4) into Eq. (A14) and applying the Fourier transform, we can obtain magnon modes for KM and MSM which are given by ˆsK(ω) =gKgMχKχcTKˆsM(ω)−i√ 2κegKχKχcTKˆai(ω) −i/radicalbig 2δKχKTKˆbi(ω) ˆsM(ω) =gKgMχMχcTMˆsK(ω)−i√2κegMχMχcTMˆai(ω) −i/radicalBig 2δ′ MχMTMˆbi(ω), (A15) where TK(ω) =/bracketleftbig 1−g2 KχKχc/bracketrightbig−1 TM(ω) =/bracketleftbig 1−g2 MχMχc/bracketrightbig−1.(A16) If we substitute ˆ sK(ˆsM) into ˆsM(ˆsK) in Eq. (A15), we can obtain the MO conversion coefficients for KM and MSM. For the KM, by consideringthe Stokes (Ω = Ω 0− ω)andanti-Stokes(Ω = Ω 0+ω)processesandthebound- ary conditions ˆb† o(Ω0−ω) =ˆb† i(Ω0−ω)+√2δKˆcK(ω) and ˆbo(Ω0+ω) =ˆbi(Ω0+ω)+√2δKˆcK(ω)26, the conversion coefficient for the KM is obtained as S31,K(ω) =√βK 2i/parenleftBigg/angbracketleftBiggˆb† o(Ω0−ω) ˆai(ω)/angbracketrightBigg +/angbracketleftBiggˆbo(Ω0+ω) ˆai(ω)/angbracketrightBigg/parenrightBigg =−2/radicalbig βKδKκegKχKχc/parenleftbig 1+g2 MχMχcTM/parenrightbig 1−g2 KχKχc(1+g2 MχMχcTM), (A17) whereβKis the amplification factor. Here, we point out that, if we take into account the MO conversion of only the KM ( gM= 0), Eq. (A17) becomes the same result as the single-mode MO conversion coefficient in Ref. [26]. In the same manner, we can induce the MO conversion coefficient for a MSM by using similar boundary condi- tions and amplification factor of βMthat is given by S31,M(ω) =−2/radicalbig βMδMκegMχMχc/parenleftbig 1+g2 KχKχcTK/parenrightbig 1−g2 MχMχc(1+g2 KχKχcTK). (A18)Therefore,theconversionefficiencyforthetwo-modeMO conversion for the KM and a MSM can be obtained as ηt(ω) =|S31,K(ω)|2+|S31,M(ω)|2.(A19) Appendix B: Magnetostatic modes in a ferromagnetic sphere Magnons are spin excitations describing small pertur- bations to the magnetization of a ferromagnetic system. A small oscillating magnetic field in the plane perpendic- ular to the bias field can lead the alignment of spins to deviate slightly from the bias direction. The bias field exerts a torque on misaligned spins, and then the spins begin precessing around it. L.R. Walker first considered the relationshipbetweenthe resonancefrequencyand the internal static field of a ferromagnetic spheroid35,36. He assumed that the microwave magnetic fields in spheroids satisfy the magnetostatic approximations. The allowed resonant frequencies of MSMs in a sphere inserted in a microwave cavity can be derived from the character- istic equation in terms of associated Legendre function Pj i(ξ0)37. i+1+ξ0Pj i′(ξ0) Pj i(ξ0)±jχ2= 0, (B1) whereξ2 0= 1 +1 χ1,χ1=γ2MsHi γ2H2 i−f2,χ2=γMsf γ2H2 i−f2, Hi=H0−Ms 3, andPj i′(ξ0) =dPj i(ξ0) dξ0. Here, Hiand Hoareinternalandexternalmagneticfields, respectively. µ0Ms= 0.178 T (at 298 K)58,59is the saturation magne- tization, µ0is the vacuum permeability,γ 2π= 28 GHz/T is the gyromagnetic ratio, and fis the frequency. iand jare mode indices that i≥1 is an integer and jis also an integer obeying −i≤j≤i. For a single mode solution, it is labelled with ( i,j). For MSMs with i− |j|= 0 or 1, the relations between the resonant frequencies and the external magnetic field can be given by37,60 ωij µ0=γHo+/parenleftbiggj 2j+1−1 3/parenrightbigg γMs(i=j),(B2) ωij µ0=γHo+/parenleftbiggj 2j+3−1 3/parenrightbigg γMs(i=j+1).(B3) Here, the (1 ,1) FMR mode, known as the Kittel mode, is the lowest mode in which all spins precess in phase which gives a frequency given by ω11=µ0γHo. Appendix C: Microwave reflection spectrum Figure 7(a) and (b) show the measured reflection spec- trum|S11(ω)|2and the phase S11(ω) of the hybrid sys- tem with the 0.45 mm-dia YIG as a function of the mi- crowavefrequency. From the boundary conditon ˆ ao(ω) =11 D E FIG. 7. (a) Reflection coefficient |S11(ω)|2of the 0.45 mm-dia YIG-cavity hybrid system as a function of t he microwave frequency. (b) The phase S11(ω) of the 0.45 mm-dia YIG-cavity hybrid system as a function of the microwave frequency. 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2020-07-30

Coherent conversion from microwave to optical wave opens new research avenues towards long distant quantum network covering quantum communication, computing, and sensing out of the laboratory. Especially multi-mode enabled system is essential for practical applications. Here we experimentally demonstrate coherent multi-mode conversion from the microwave to optical wave via collective spin excitation in a single crystal yttrium iron garnet (YIG, Y3Fe5O12) which is strongly coupled to a microwave cavity mode in a three-dimensional rectangular cavity. Expanding collective spin excitation mode of our magnon-cavity hybrid system from Kittel to multi magnetostatic modes, we verify that the size of YIG sphere predominantly plays a crucial role for the microwave-to-optical multi-mode conversion efficiency at resonant conditions. We also find that the coupling strength between multi magnetostatic modes and a cavity mode is manipulated by the position of a YIG inside the cavity. It is expected to be valuable for designing a magnon hybrid system that can be used for coherent conversion between microwave and optical photons.

Coherent multi-mode conversion from microwave to optical wave via a magnon-cavity hybrid system

2007.15299v1

arXiv:1806.04764v1 [cond-mat.mes-hall] 12 Jun 2018Resonant spin wave excitation in magnetoplasmonic bilayer s by short laser pulses Stanislav Kolodny, Dmitry Yudin, and Ivan Iorsh ITMO University, Saint Petersburg 197101, Russia (Dated: June 14, 2018) In magnetically ordered solids a static magnetic field can be generated by virtue of the transverse magneto-optical Kerr effect (TMOKE). Moreover, the latter w as shown to be dramatically enhanced due to the optical excitation of surface plasmons in nanostr uctures with relatively small optical losses. In this paper we suggest a new method of resonant opti cal excitations in a prototypical bilayer composed of noble metal (Au) with grating and a ferro magnet thin film of yttrium iron garnet (YIG) via frequency comb. Based on magnetization dyn amics simulations we show that for the frequency comb with the parameters, chosen in resonant w ith spin-wave excitations of YIG, TMOKE is drastically enhanced, hinting towards possible te chnological applications in the optical control of spintronics systems. I. INTRODUCTION Ideas and concepts developed in photonics have been recently applied in related fields, in particular in physics of collinear magnets. The rapidly advancing area of magnonics represents photonics with spin waves, or, col- lective propagating magnetization excitations in mag- netic materials. Thanks to their unique linear and non- linear properties [1–5] spin waves can be successfully em- ployed in next generation spintronics devices and quan- tum computing. In the meantime, further implication in technology requires excitation of spin waves at short length- and time-scales. This can be achieved by the optical excitation of magnetic system with femtosecond laser pulses [6–12]. Despite a variety of ways of opti- cal generation of spin waves, the nonthermal magneto- optical processes such as inverse Faraday and inverse Kerr are of particular importance, since they are charac- terized by high time resolution and do not require heat- ing of the sample. The essence of the effect is that the external optical electric field Einduces magnetic field h∝E∗×Evia the nonlinear processes such as e.g. stimulated Raman scattering [13]. Thus induced mag- netic field drives spin waves in a magnetic structure. Re- cently, it was anticipated that in inhomogeneous media the expression E∗×Emay be non-zero even for the lin- earlypolarizedwave. Inplanargeometry,thiseffecttakes placeforTMpolarizationonly, andtheinduced magnetic field is perpendicular to the plane of incidence, realizing thus the inverse transverse magneto-optical Kerr effect (TMOKE) [14]. Among structures which allow inverse TMOKE are magnetoplasmonic systems [15], since they facilitate electric field localization and an effective chiral- ityσ∝ |E∗×E|/|E|2of surface plasmon polariton can approach a unity. It is clear that in continuous wave regime the induced magneticfield histime-independent, whereasunderfem- tosecond laser pulses this field follows the intensity pro- file. Therefore, a much higher efficiency of spin wave generation can be achieved in the case of resonant exci- tation, when the intensity profile is periodic with a fre- quency close to that of ferromagnetic resonance, which is typically of the order of 1 GHz. Recently, it has beendemonstratedexperimentally[16]thatusingclockedlaser excitation can substantially enhance the efficiency of the spin wave generation at the bottom of magnon band, followed by their diffusion. However, for the direct im- plication in data processing resonant excitation of spin waves with fixed both frequency and wavevector (and group velocity, as a result) is of vital importance. In this paper we propose a combination of a magne- toplasmonic structure and clocked laser excitation for the resonant excitation of spin waves. We first design the structure and numerically model the distribution of electromagnetic field under the clocked laser excitation. Using the data we evaluate the induced magnetic field and study magnetization dynamics based on numerical solution of Landau-Lifshitz-Gilbert (LLG) equation [17]. We provide an estimate for a realistic structure, where a resonant excitation of spin waves can be achieved with efficiency enhancement of at least two orders of magni- tude as compared to that in the free magnetic film under pulsed excitation. II. MODEL SYSTEM Among various magnetic materials of collinear order- ing yttrium iron garnet (YIG) is characterized by the best magneto-elecctrical and magneto-optical properties, which stand out this material for microwave applications [18]. YIG belongs to the class of magnetically ordered structures with cubic symmetry [19, 20] and rather low loss [21], and is successfully used in a broad range of spintronics applications. We therefore consider a hybrid nanostructure consisting of a thin film of YIG and Au discrete grating on the top (geometry of the system is schematicallydepicted inFig.1aundershortlaserpulses. To study magnetization dynamics induced by laser pulses we are to numerically solve LLG equation ∂M ∂t=−γM×Heff+αM×∂M ∂t,(1) whereγis the gyromagnetic ratio of an electron ( γ= 28 GHz/T). In general LLG equation describes precession of the magnetization, M, around the effective magnetic2 a) b) xyzkH0,M0 FIG. 1. (a) A schematic representation of the bilayer under c onsideration: a thin film of ferromagnetic YIG with the magne ti- zationM0aligned along the zaxis is covered by a layer of gold with grating. The system is p laced to the external magnetic fieldH0directed normal to the interface; and is irradiated with the laser pulses of intensity Iand duration t, so that the induced magnetic field hin YIG is perpendicular to the plain of incidence ( xOz). The latter is associated with surface plas- mon polaritons emerging right at the interface between two m etals. (b) The magnon dispersion relation in YIG obtained by numerical solution of Landau-Lifshitz-Gilbert equation. field,−γHeff=∂H/∂M, created by delectrons of a fer- romagnet, whereas the relaxation rate to the direction of the field is associated with the Gilbert damping param- eter,α. We assume the thin layer of YIG is initially po- larized along the zaxisM0=M0ez, as shown in Fig. 1a, and is placed to the external field H0. Being irradiated with laserpulses results in the emergenceofthe magnetic fieldhdirected perpendicular to the plain of incidence via TMOKE mechanism. The latter gives rise for the magnetization direction to slightly deviate from collinear ordering, and thus to spin waves excitation. To get spin wave spectrum we linearize Eq. (1) with respect to in- plane components of the magnetization m= (mx,my), i.e., we plug M=M0ez+mxex+myeyinto (1) on con- dition that |mx|,|my| ≪M0. We derive after straight forward algebra, ∂m ∂t=−γM0×Heff+α M0/parenleftbigg M0×∂m ∂t/parenrightbigg .(2) The properties of YIG can be well approximated using the microscopic Hamiltonian which includes Heisenberg exchangeinteraction, magnetocrystalline anisotropy, and Zeeman coupling. Thus, the effective field can be repre- sented as follows: −γHeff=h+H0+Hk+αik∂2m ∂xi∂xk−ˆNm,(3) where we assume the summation over repeated indexes; ˆNis the tensor of magnetocrystalline anisotropy, ˆNm= Nijmj. In Eq. (3) the exchange tensor αik=Aδikis reduced to a scalar owing to the cubic lattice symmetryof YIG with A=3·10−16m2, whileHkcorresponds to the effective anisotropy magnetic field [17]. Typically, the effective anisotropy field Hkis material-specific and equals 4.6 kA/m for YIG. In the following, we assume M0andHkare parallel to H0(see Fig.1a), thus being normal to Au–YIG interface. We put H0= 1 T, and set M0to be equal to the saturation magnetization of YIG, i.e.,Ms= 140 kA/m. It is therefore clear that with no hpresent in the system the vector product M0×Heff is zero, producing no magnetization precession. III. RESULTS AND DISCUSSION We put the Gilbert damping parameter α=3.2·10−4, although it can be decreased as shown in Ref. [22]. Fourier transforming the equation of motion (2) to m(ω,k) =G(ω,k)h(ω,k) makes it possible to evaluate the spin wave spectrum from G−1(ω,k) = 0. Thus ob- tained the magnon dispersion relation for the YIG thin film in the presence of his shown in Fig. 1b. One can clearly observe that magnons are positioned in 4-7 GHz range for k-vector ranging between 1 and 100 µm−1), and are characterized by a very narrow dispersion line (about 40 MHz) because of low Gilbert damping. There- fore, the direct optical excitation of magnons is hardly to be achieved. To bind optical waves (in the form of plasmons, for example) with spin waves we propose to use a gold grat- ing that amplifies the electric field and are to determine the field distribution due to strong field localization via3 6 5.5 44.5 3.5 3 0 10 20 30 40 |k|, 1f, GHza) c)b) d) 01|Ez/hmax| 01|Ex/hmax| 01|hy/hmax| 01|hy/hmax| FIG. 2. Distribution of electric field components Ex(a) andEz(b) inside the magnetoplasmonic bilayer, normalized by the maximum value of electric field. The induced magnetic field h∝E∗×Eis strongly localized at the Au-YIG interface (c), normalized by the maximum value of the field. Dispersion of h, created by the train of femtosecond optical pulses with repetition rate 4.5 GHz (d). Noteworthy, in these calculati ons we impose the periodic boundary conditions along xandyaxes. excitation of plasmonic resonances [23, 24]. To be more specific, we suppose: the thickness of Au layer dis 100 nm, periodicity ais100nm, the width wand theheight h of array’s element are 50 nm. The latter allows us to ex- cite plasmons on the Au-YIG interface with the k-vector to be 107m−1.To estimate the distribution of magnetic field inside the hybrid nanostructure we impose the pe- riodic boundaries in xandydirections. We further pro- ceed with eigenmode simulation in CST Microwave Stu- dio: the electric field distribution clearly reveals that ex- cited field of plasmons possesses two phase-shifted com- ponentsoftheelectricfield(see2a-b)insidetheYIGfilm. Thus, it induces time-independent magnetic polarization via the TMOKE mechanism h∝E∗×E[14]. The time-independent magnetic field hserves as a sourceofspin wavesinaccordancewithEq.(3). Tointro- duce time-dependence we propose to excite plasmons in the nanostructure by a train of Gaussian pulses with the repetition rate chosen in resonance with the frequency of magnons in the YIG thin film. We assume the dura-tion of each pulse is 40 ps that is shorter in comparison with the repetition rate. As we discussed earlier, the Au grating allows us to generate electromagnetic field with a fixed value of the wavevector k= 107m−1, thus, in full accordance with the magnon dispersion of YIG shown in Fig. 1a the resonance frequency of the spin waves in the magnetoplasmonic bilayer under consideration is 4.5 GHz. To be more realistic we put the finite size of hy- brid nanostructure limited by 1 µm. Such a value of repetition rate is difficult to achieve using only common methodsofGaussianpulsestraingeneration. Meanwhile, using a frequency comb technique facilitates overcoming this problembygeneratingthe Gaussianpulseswith high repetition rate up to THz range [25, 26]. The dispersion ofinducedmagneticfieldconsistsofadiscretesetofspots as shown in Fig.2d, the latter happens due to the peri- odicity of the structure and an excitation properties of frequency combs. Therefore, by solving the equation of motion with thus obtained hwe can get the spin waves dispersion m(ω,k) as well as to recover their time and4 |m|, x10-3a.u. 08 FIG. 3. Magnon dispersion relation in hybrid Au-YIG nanostr ucture. White dashed curve represents the dispersion line o f spin waves in YIG, crossing points of dashed lines represents the dispersion of h. space dependence by performing inverse discrete Fourier transformation. The resulting dispersion is depicted in Fig. 3. As it was expected there is only one spot on co- incidence of dispersion diagrams of excitation magnetic field and spin waves in YIG (hot spots of magnetic field diagram are represented by crossing of straight dashed pink lines and dispersion line of magnons in YIG is rep- resented by white dashed curve). The dependence of 2 4 6 8 10 Repetition frequency, GHz00.0020.0040.0060.0080.01|m|, a.u. -0.01 -0.005 0 0.005 0.01-0.01-0.00500.0050.01 a) b) mx ,
  m y FIG. 4. (a) Spin wave magnitude in the YIG thin film vs. repetition rate of femstosecond laser pulses. (b) Precessi on of/vector maround equilibrium point in time for different excitation frequency: brown curve – resonant excitation frequency 4.5 Ghz, blue curve – 4.48GHz, yellow curve – 4.52 GHz. |mon the repetition rate of femtosecond laser pulses for fixed values of kin range 1-10 GHz for maximum value of excitation magnetic field h/hmaxis presented in Fig. 4a, which clearly manifests the formation of a reso- nance mode. The resonance mode appears at 4.5 GHz.Also there are additional equidistant peaks are placed on lower frequencies. The main and additional peaks are very narrow because of low damping of spin waves in YIG which is described by Gilbert damping parameter as it was mentioned before. In addition, we study hodo- graph of the magnetization at different frequencies: In particular, results for 50 Gaussian pulses are shown in Fig. 4b. IV. CONCLUSIONS In the current studies we considered a typical magne- toplasmonic structure, that represents the bilayer of a thin film YIG covered by a layer of gold with grating and explicitly showedthat TMOKEemergingin such a struc- turemaybedramaticallyenhancedusingfrequencycomb technique. In fact, for the repetition rate chosen to be in resonant with the ferromagnetic resonanceof the mag- netic layer the direct numerical solution to LLG equation reveals a desired behavior. We believe that our results will trigger experimental activity in rapidly advancing area of research right in the border between plasmonics and spintronics. ACKNOWLEDGEMENTS We acknowledge the support from the Russian Science Foundation under the Project No. 17-12-01359.5 [1] S. O. Demokritov and A. N. Slavin, Magnonics: From fundamentals to applications (Springer Science & Busi- ness Media, 2012). [2] A. Khitun, M. Bao, and K. L. Wang, Magnonic logic circuits, J. Phys. D: Appl. Phys. 43, 264005 (2010). [3] A. Haldar, C. Tian, and A. O. Adeyeye, Isotropic trans- mission of magnon spin information without a magnetic field, Sci. Adv. 3, 1700638 (2017). [4] T. Fischer, M. Kewenig, D. A. Bozhko, A. A. Serga, I. I. Syvorotka, F. Ciubotaru, C. Adelmann, B. Hille- brands, and A. V. Chumak, Experimental prototype of a spin-wave majority gate , Appl. Phys. Lett. 110, 152401 (2017). [5] Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Ya- mazaki, K. Usami, and Y. Nakamura, Coherent coupling between a ferromagnetic magnon and a superconducting qubit, Science 349, 405 (2015). [6] M. van Kampen, C. Jozsa, J. T. Kohlhepp, P. LeClair, L. Lagae, W. J. M. de Jonge, and B. Koopmans, All-optical probe of coherent spin waves , Phys. Rev.Lett. 88, 227201 (2002). [7] A. V. Kimel, A. Kirilyuk, P. A. Usachev, R. V. Pisarev, A. M. Balbashov, and T. Rasing, Ultrafast non-thermal control of magnetization by instantaneous photomagnetic pulses, Nature 435, 655 (2005). [8] F. Hansteen, A. Kimel, A. Kirilyuk, and T. Rasing, Non- thermal ultrafast optical control of the magnetization in garnet films , Phys. Rev. B 73, 014421 (2006). [9] C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk, A. Tsukamoto, A. Itoh, and T. Rasing, All-optical magnetic recording with circularly polarized light , Phys. Rev. Lett. 99, 047601 (2007). [10] J.-Y. Bigot, M. Vomir, and E. Beaurepaire, Coherent ul- trafast magnetism induced by femtosecond laser pulses , Nature Phys. 5, 515 (2009). [11] T. Satoh, Y. Terui, R. Moriya, B. A. Ivanov, K. Ando, E. Saitoh, T. Shimura, and K. Kuroda, Directional control of spin-wave emission by spatially shaped light , Nature Photon. 6, 662 (2012). [12] Y. Au, M. Dvornik, T. Davison, E. Ahmad, and P. S. Keatley, Direct excitation of propagating spin waves by focused ultrashort optical pulses , Phys. Rev. Lett. 110, 097201 (2013). [13] D. Popova, A. Bringer, and S. Bl¨ ugel, Theoretical inves- tigation of the inverse Faraday effect via a stimulated Ra- man scattering process , Phys. 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2018-06-12

In magnetically ordered solids a static magnetic field can be generated by virtue of the transverse magneto-optical Kerr effect (TMOKE). Moreover, the latter was shown to be dramatically enhanced due to the optical excitation of surface plasmons in nanostructures with relatively small optical losses. In this paper we suggest a new method of resonant optical excitations in a prototypical bilayer composed of noble metal (Au) with grating and a ferromagnet thin film of yttrium iron garnet (YIG) via frequency comb. Based on magnetization dynamics simulations we show that for the frequency comb with the parameters, chosen in resonant with spin-wave excitations of YIG, TMOKE is drastically enhanced, hinting towards possible technological applications in the optical control of spintronics systems.

Resonant spin wave excitation in magnetoplasmonic bilayers by short laser pulses

1806.04764v1

1 Spin Seebeck effect through antiferromagnetic NiO Arati Prakash1, Jack Brangham1, Fengyuan Yang1, Joseph P. Heremans2,1,3 1The Ohio State University Dep artment of Physics, Columbus, Ohio 43210 2The Ohio State University Dep artment of Mechanical Engineering, Columbus, Ohio 43210 3The Ohio State University Dep artment of Materials Science and Engineering, Columbus, Ohio 43210 Abstract We report temperature -dependent spin-Seebeck measurements on Pt/YIG bilayers and Pt/NiO/YIG trilayers, where YIG ( Yttrium iron garnet , Y3Fe5O12) is an insulating ferrimagnet and NiO is an antiferromagnet at low temperature s. The thickness of the NiO layer is varied from 0 to 10 nm. In the Pt/YIG bilayers , the temperature gradient applied to the YIG stimulates dynamic spin injection into the Pt, which generates an inverse spin Hall voltage in the Pt. The presence of a NiO layer damp ens the spin injection exponent ially with a decay length of 20.6 nm at 180 K . The decay length increases with temperature and shows a maximum of 5.5 0.8 nm at 360 K. The temperature dependence of the amplitude of the spin-Seebeck signal without NiO shows a broad maximum of 6.50.5 V/K at 20 K. In the presence of NiO, the maximum shifts sharply to higher temperatures, likely correlated to the increase in decay length. This implies that NiO is mo st transparent to magnon propagation near the paramagnet -antiferromagnet transition . We do not see the enhancement in spin current driven into Pt reported in other papers when 1 -2 nm NiO layers are sandwiched between Pt and YIG. Introduction In the spin -Seebeck effect (SSE, longitudinal configuration) , 1 a heat current stimulates spin propagation across an interface between a ferro magnetic (FM) material and a normal metal (NM) , where it is converted into a transverse electric field by the inverse spin Hall effect 2 (ISHE) .2 The most commonly stu died materia l system is the Pt/YIG system. Recently,3,4,5 SSE signals have been reported in systems where the magnetic material is an antiferromagnet (AFM) . In principle,6 in the absence of a net spin polarization, an AFM cannot produce a spin -Seebeck signal. A net spin polarization can b e produced by bringing the AFM above the spin -flop transition by an applied magnetic field , resulting in recent experimental reports of SSE signals in Cr2O33 and MnF 2,4 as well as a theory for it.7 The effect on MnF 2 is particularly large, reaching over 40 μV/K at high field and low temperature (T).4 However, an uncertainty remains about the exact values of the temperature gradient applied to the AFM, and thus the absolute val ues of the spin-Seebeck coefficients . The effect on Cr 2O3 was much smaller3 and more in line with values hitherto observed on the Pt/ YIG system. We report here on SSE signal s measured on a thin AFM layer grown on a FM, the Pt/NiO/YIG trilayer structure, which was re cently reported to exhibit antiferromagnetic magnon transport .8 In that study, microwave -driven ferromagnetic resonance (FMR) in the YIG FM layer was used to produce a pumped spin current t hat flowed through the NiO AFM layer and reached the Pt layer , where it produced an ISHE electric field. Monitoring the amplitude of this field as a function of NiO layers thickness (varied from 0 to 10 0 nm) gave information about the propagation of a dynamic net magnetic moment through the AFM. The results showed t hat magnon ic transport through NiO layers thicker than 4 nm was attenuated exponentially with a decay length of about 10 1.2 nm at room temperature . However, the insertion of a 2 nm thin NiO layer between YIG and Pt enhanced the spin currents driven into Pt. A subsequent theory9 ascribes the origin of this effect to diffusion of thermal antif erromagnetic magnons in the NiO. If that is so, one would not expect to see the enhancement in the SSE measurements, and this is why the present experimental st udy was undertaken. We note that SSE results on the 3 Pt/NiO/YIG system were recently published5 after we started this study, in which an enhancement is reported . In contrast, we do not see the enhancement and we discuss the possible sources for that difference. In previous studies,10 the temperature dependence of the SSE in the YIG/Pt system without any NiO shows a broad maximum, attributed to the properties of the magnon dispersion. We report the same behavior here. This maximum was explained10 by the fact that t hermally driven magnons have a spectral distribution that covers almost the entire Bri llouin zone at room temperature, unlike FMR -driven magnons at ~10 GHz near the zo ne center . The magnon dispersion11 of YIG includes several optical -like branches that have practically no group velocity, and are not expected to contribute much to the SSE signal, but it also includes quasi - acoustic branches. The non -monoton ic temperature dependence then arises as follows. At the temperature s below the maximum , the SSE signal is limited by the thermal magnon population ; it increases with increasing temperature because higher energy magnons get excited , and in larger numbers. I n this temperature range, most magnons have rather similar group velocity (only the magnons with energies near about 10 GHz have near -zero group velocity). At temperatures a bove the maximum, the effects of the magnon inelastic mean free path12 and diffusio n length13 become more important, and the SSE now decreases with increasing temperature. It was further concluded from the magnetic -field dependence of the SSE measurements10,14 at high field that the magnon modes at energies below 40 K contribute the most to the SSE effect in the Pt/YIG system. The present study includes the temperature dependence of the SSE in YIG/Pt, which shows the same behavior as in Ref. [ 10], and to this we add temperature -dependent data of the SSE through the AFM, which was not included in the FMR study.8 4 Experiment YIG films were grown epitaxially on single crystal Gd3Ga5O12 (GGG) <111> substrates by ultrahigh vacuum off -axis sputtering.15,16,17 The YIG was grown at a substrate temperature of 750C while being rotated at 10 /s for optimal sample uniformity at a growth rate of 0.51 nm/min. The thickness of YIG films grown on GGG substrates was 250 nm, as it has been shown18 that the intensity of the SSE signal in YIG films grown on GGG increases with increasing YIG thickness, but saturates for thicknesses of about 250 nm. NiO and Pt layers were then deposited at room temperature o n these YIG films also by off -axis sputtering. A series of Pt/NiO/YIG samples with length 10 mm and width 5 mm were de posited with 6 nm thick Pt and 0-10 nm thick NiO. In all cases , the Pt deposit ion was done in-situ with the NiO growth. A summary of the samples studied is given in Table I. Sample TM (K) Pt(6nm) /YIG (250nm) 20 Pt(6nm) /NiO(1nm)/YIG 120 Pt(6nm) /NiO(2nm)/YIG 220 Pt(6nm)/ /NiO(5nm)/YIG 260 Pt(6nm)/ /NiO(10nm)/YIG 320 Table I p roperties of the samples studied. The SSE measurements wer e conducted as in Ref. [ 10], except for the way the temperature gradient was calculated . High temperature measurements were conducted in a liquid nitrogen flow cryostat from 80 to 420 K. Samples were mounted using Ap iezon H -grease ( T > 5 300 K) or N -grease ( T < 300 K) on electrically insulating cubic boron nitride (c-BN) heat- spreading pads with high in-plane thermal conductivity. Three r esistive heaters (120 Ohm) were applied to the c -BN pads and connected electrically in series , ensuring uniform heating . To switch the SSE signal, we swept the field between 50 mT. Low temperature measurements from 2 to 300 K were conducted on the Pt/YIG sample in a Physical Property Measurement System (PPMS) by Quantum Design . All experimental parameters were controlled as much as possible to be identical between the two systems. The voltage across the Pt layer s was measured with a Keithl ey 2182 nano voltmeter . Electric fields were calculated from raw traces (an example is shown in Fig. 1 a for a Pt (6nm) /YIG (250nm)/GGG bilayer ) of the voltage across the Pt strip, Vy, by averaging the values at -50 mT and + 50 mT, then dividing by the length Ly of the Pt layer to give the induced ISHE field ( y y yVEL in V/m ). Unlike in Ref [ 10], temperature gradients T are calculated from the measured heater power output Q per unit cross -sectional area of YIG (in units of W/ m2) and the measured values for bulk thermal conductivity of the YIG12 ( YIG ) at each temperature : YIGQT . This procedure assumes that there is negligible loss of heat between the heater and the heat sink, so that all the heat goes though the sample as a uniform flux. Heat spreaders assured uniformity and an adiabatic sam ple mount minimized heat losses. The high thermal conduct ance of the sample, due to the high thermal conductivity of GGG and the favorable geometry, combined with the extensive use of radiation shielding and of 25 m copper or manganin voltage and heater wires, ensure that heat losses are minimal. The procedure also assumes that the thermal conductivity of the 250 nm YIG film is the same as that of bulk YIG. In the absence of reliable data on the temperature dependence of the thermal conductivity of 250 nm -tick YIG films, this 6 hypothesis is not as easily justified. The error may affect the absolute values reported for the spin-Seebeck coefficient Sy, but the relative effect of adding NiO layers between the YIG and the Pt is much less sensitive to this hypothesis , since the thickness of the YIG layer is held constant . The values of Sy obtained following this procedure in different cryostats and in different runs are superimposed on each other in Fig. 1(b). The spread in the data, of the order of 30% at worst, gives a quantitative estimate of the accuracy with which we can make sample to sample comparisons, as we need to do to study the effect of NiO thickness. We developed this process, instead of using the metho d in Ref [ 10], because it is not sensitive to thermal contact resistances between films and substrate or between the different layers, while direct thermometry is. To test that, Quantum Design thermometry was mounted on the hot- and cold -side c -BN pads , but the resulting measurements were affected by thermal contact resistances between sample and heat sink, and proved to be less reproducible than the procedure used here. The values for the spin - Seebeck coefficient y yEST (in units of nV/K) , are reported as the ratios of electric field to temperature gradient, with the geometrical factors of the samples divided out. The procedure is consistent with our previous measurements ,10 enabling a quantitative comparison of all results. Results SSE signals are observed on all samples at room temperature . We begin with the cas e of the Pt/YIG sample (Table 1). We point out that the observed spin -Seebeck coefficients (Fig. 1 b) in the Pt (6nm) /YIG(250 nm) bilayer on GGG are one to two order s of magnitude larger than our own previously re ported signals for Pt on YIG single crystal ,10 but an important reason is simply the difference in the method used to calculate T. The maximum SSE signal is 6.50.5 V/K near 20 K. These results demonstrate the critical role of the Pt/YIG interface : the pristine interface between Pt and epitaxial YIG film allows the conduction electrons in the Pt directly 7 interact with the localized YIG magnetization in the absence of interfacial defects, which is highly desired for spin transfer acr oss the interface. In contrast , for our previous10 YIG bulk crystal with a polished surface (even for epi -ready YIG single crystals) , it is expected that there is defect layer at the YIG surface . Given that our previous FMR spin pumping study19 shows that a nonmagnetic insulating barrier of only 1 nm thickness at the interface can reduce the I SHE signal by a factor of ~250, it is reaso nable to expect that any defect layer will strongly affect the results. Figure 2 shows t he SSE of the Pt/NiO/YIG trilayer systems as a function of temperature. The SSE signal is hard to disce rn from the noise fl oor of 50 nV at some lo w-temperature cutoff point, varying from 100 to 180 K depending on sample thickness: Fig. 2 reports only data above this cut -off temperature for each sample . The temperature dependence of SSE shows a maximum at a temperature TM that decreases monotonically with the NiO t hickness, and is summarized in Table 1. TM is not necessarily the N éel temperature TN of NiO films , because two competing mechanisms are at work. On the one hand, it is known that TN in free NiO films is lower than that of bulk NiO and has some systematic variation with film thickness .20 On the other hand, it was shown21 by neutron diffraction on CoO/Fe 3O4 films in the same thickness range as those studied here that the ordering temperature of the CoO films is enhanced for small CoO thickness es. There is clearly an interaction between the Ni spins in NiO and the Fe spins in YIG by an exchange coupling or biasing effect. The positive slope of dSy/dT below TM indicates that the propagation of magnons though the NiO film improves as the thermal fluctuation in the AFM - ordered NiO is increased with increasing temperature. The negative slope of dSy/dT above TM in the Pt/NiO/YIG trilayers simply mirrors the negative slope behavior of the Pt/YIG system in the absence of NiO . 8 Figure 3a shows how the SSE signal decay s exponentially (i.e. 0exp( / )yS S th L where th is the NiO thickness, S0 is a prefactor, and L is an attenuation length) with increasing NiO thickness at 420, 360, 300, 240 and 180 K. We fit the attenuation length L to the data and report it as a function of temperature in Fig. 3b. The decay continues to be observed down to 100 K, but the number of samples on which the SSE is measurable has decreased. This behavior and the attenuation length at 300 K is within about 40% of that observed8 with FMR driven spin flu x. At first sight this is surprising, if we consider that the modes excited in FMR -driven spin flux are near the Brillo uin zone center and have a much larger spatial extent than the high energy modes exited in thermal spin fluxes. Recall that in FMR spin pumping of Pt/NiO/YIG trilayers, the excitation is uniform coherent precession of the YIG magne tization (or, zero -k magnons), while for thermally -driven SSE measurements of the same structure, the excitation is thermal magnons that have finite k and are diffusive. Therefore the ratio between the NiO film thickness and the magnon wavelength would be expected to be very different for FMR spin pumping and SSE ; thus, the decay length would be expected to follow th e same argument as well. The similar behaviors between the two experiments indicates that not all thermal magnons participate in the SSE, but only those at small k-values . This same conclusion was arrived at during the interpretation of the freeze -out of the SSE effect at high magnetic field.10,14 The attenuation length decreases with decreasing temperature below 300 K, and seems to have a broad maximum near 360 K, although the decrease above 360 K is within experimental error bars . This hints that two separate mechanisms are at work here, one related to thermal fluctuations of the magnons, and one to the coupling between Ni spins in NiO and Fe spins in the underlying YIG . Unlike what was observed with FMR -pumped magnons ,8 the SSE measurement of Pt/NiO/YIG for the 2 nm NiO reported here does not show an enhancement. This also contrast s 9 with the recently reported SSE measurements on similar structures using polycrystalline YIG substrates ,5 where an enhancement is observed similar to our earlier report of FMR spin pumping . We believe these two SSE measurements do not contradict each other because of the difference in measurement techniques and sample characteristics . In this work w e used 250 -nm YIG epitaxial films and in Ref. 5 , polished polycrystalline substrates were used . We also note that i n Ref. 5, the maximum magnitude of Sy for Pt(3 nm)/NiO(2 nm)/YIG(bulk) is about 1.1 V/K at ~260 K, while in Fig. 2 of this work, the maximum Sy for Pt(6 nm)/NiO(2 nm)/YIG(250 nm)/GGG is 0.35 V/K at 200 – 300 K. Considering that the resistance of a 3 nm Pt layer is typically a few times higher than that of a 6 nm Pt layer, these results of the same measurements on two similar samples are consistent. We believe that the AF M magnons and spin fluctuations in NiO are likely responsible for the spin current enhancement through NiO in Ref. 5 , similar to our earlier FMR spin pumping results .8 The absence of enhancement in this work is likely due to the differences in YIG samples (epitaxial films vs. polycrystalline bulk) and interfacial characteristics, as well as fact that our Pt/YIG sample alrea dy exhibit a large SSE signal. In summary, we report the transmission of thermal ly-driven magnons excited in the YIG films through a NiO layer in its antiferromagetic and its para magnetic state, with an attenuation length ranging between 2 and 5.5 nm at temperatures from 180 to 420 K . From this we conclude that the thermally -driven magnons that give rise to the SSE must be limited to the low -energy part of the magnon spectrum, consistent with previous work on the high -field suppression of the SSE. Acknow ledgement This work was primarily supported by the Army Research Office ( ARO ) MURI W911NF - 14-1-0016 and the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences, 10 under Grants No. DE -SC0001304. It is partially supported by the Center for Emergent Materials, an NSF MRSEC under grant DMR -1420451 . Figure captions Figure 1 . (a) A raw trace of the spin -Seebeck voltage Vy versus applied field at 300 K for an applied heater power of 0.495 W and (b) spin S eebeck Coefficient as a function of temperature for a Pt(6 nm)/YIG(250 nm) bilayer deposited on GGG <111>. In (b), t he results of several different runs in different instruments are superimposed on each other, illustrating the reproducibility of the results, which is of the order of 30%. A maximum Spin Seebeck Coefficient of ~6.50.5 V/K is observed on the YIG film w ithin a broad peak around 20 K. Figure 2. Temperature dependence of the SSE signal on the Pt/ YIG bilayer is compared to SSE signals from the YIG/NiO/Pt trilayers , with the NiO film thickness as noted . A decrease in SSE signal is observed with increas ing NiO film thickness. A maximum at a temperature TM is observe d for each of the YIG/NiO/Pt trilayers . TM increases with increasing NiO thickness. Figure 3. (a) Magnitude of the SSE signal at 420, 360, 300, 240, and 180 K plotted as a function of NiO thickness. The experimental values are the data points. The lines through them are fitted exponential decay functions, with a characteristic attenuation length scale given as a function of temperature in (b). 11 Figure 1 12 Figure 2 13 Figure 3 14 References: 1 K. Uchida, H. Adachi, T. Ota, H. Nakayama, S. Maekawa, and E. Saitoh, Appl. Phys. Lett. 97, 172505 (2010). 2 S. R. Boona, R. C. Myers and J. P. Heremans , Spin Caloritronics, Energy Environ. Sci. 7, 885- 910 (2014) . 3 S. Seki, T. Ideue, M. Kubota, Y. Kozuka, R. Takagi, M. Nakamura, Y. Kaneko, M. Kawasaki, and Y. Tokura, Phys. Rev. Lett. 115, 266601 (2015) . 4 S. M. Wu, W . Zhang, K. C. Amit KC, P . Borisov, J . E. Pearson, J. S . Jiang, D . Lederman, A . Hoffmann, and A . Bhattacharya, Phys. Rev. Lett. 116, 097204 (2016) . 5 W. W. Lin, K. Chen , S. F. Zhang , and C. -L. Chien, Phys. Rev. Lett. 116, 186601 (2016). 6 Y. Ohnuma, H. Adachi, E. Saitoh, and S. Maekawa, Phys. Rev. B 87, 014423 (2013). 7 S. M. Rezende, R. L. Rodríguez -Suárez and A. Azevedo, Phys. Rev. B 93, 014425 (2016) . 8 H. L. Wang, C. H. Du, P. C. Hammel, and F. Y. Yang, Phys. Rev. Lett. 113, 097202 (2014). 9 S. M. Rezende, R. L. Rodríguez -Suárez and A. Azevedo, Phys. Rev. B 93, 054412 (2016) . 10 H. Y. Jin, S. R. Boona, Z. H. Yang, R. C. Myers, and J . P. Heremans, Phys. Rev. B 92, 054436 (2015) . 11 J. S. Plant, J. Phys. C 16, 7037 (1983) . 12 S. R. Boona and J . P. Heremans, Phys. Rev. B 90, 064421 (2014) . 13 B. L. Giles, Z. H. Yang, J. S. Jamison, and R . C. Myers, Phys. Rev. B 92, 224415 (2015) . 14 T. Kikkawa, K. Uchida, S. Daimon, Z. Qiu, Y. Shiomi, and E. Saitoh, Phys. Rev. B 92, 064413 (2015). 15 A. J. Hauser, J. M. Lucy, H. L. Wang, J. R. Soliz, A. Holcomb, P. Morris, P. M. Woodward, and F. Y. Yang, Appl. Phys. Lett. 102, 032403 (2013) . 15 16 C. H. Du, R. Adur, H. L. Wang, A. J. Hauser, F. Y. Yang, and P. C. Hammel, Phys. Rev. Lett. 110, 147204 (2013). 17 H. L. Wang, C. H. Du, Y. Pu, R. Adur, P. C. Hammel, and F. Y. Yang, Phys. Rev. B 88, 100406(R) (2013). 18 A. Kehlberger, U . Ritzmann, D . Hinzke, E .-J. Guo, J . Cramer, G . Jakob, M . C. Onbasli, D . H. Kim, C . A. Ross, M . B. Jungfleisch, B . Hillebrands, U . Nowak, and M . Kläui, Phys. Rev. Lett. 115, 096602 (2015) . 19 C. H. Du, H. L. Wang, Y. Pu, T. L. Meyer, P. M. Woodward, F. Y. Yang, and P. C. Hammel, Phys. Rev. Lett. 111, 247202 (2013) ; H. L. Wang, C. H. Du, Y. Pu, R. Adur, P. C. Hammel, and F. Y. Yang, Phys. Rev. Lett. 112, 197201 (2014); C. H. Du, H. L. Wang, F. Y. Yang, and P. C. Hammel, Phys. Rev. B 90, 140407(R) (2014). 20 J. Nogues and I. K. Schuller, J. Magn. Magn. Mater. 192, 203 (1999) ; A. Baruth and S. Adenwalla, Phys. Rev. B 78, 174407 (2008) . 21 P. J. van der Zaag, Y. Ijiri, J. A. Borchers, L. F. Feiner, R. M. Wolf, J. M. Gaines, R. W. Erwin, and M. A. Verheijen, Phys. Rev. Lett. 84, 6102 (2000) .

2016-04-29

We report temperature-dependent spin-Seebeck measurements on Pt/YIG bilayers and Pt/NiO/YIG trilayers, where YIG (Yttrium iron garnet, Y$_3$Fe$_5$O$_{12}$) is an insulating ferrimagnet and NiO is an antiferromagnet at low temperatures. The thickness of the NiO layer is varied from 0 to 10 nm. In the Pt/YIG bilayers, the temperature gradient applied to the YIG stimulates dynamic spin injection into the Pt, which generates an inverse spin Hall voltage in the Pt. The presence of a NiO layer dampens the spin injection exponentially with a decay length of $2 \pm 0.6$ nm at 180 K. The decay length increases with temperature and shows a maximum of $5.5 \pm 0.8$ nm at 360 K. The temperature dependence of the amplitude of the spin-Seebeck signal without NiO shows a broad maximum of $6.5 \pm 0.5$ $\mu$V/K at 20 K. In the presence of NiO, the maximum shifts sharply to higher temperatures, likely correlated to the increase in decay length. This implies that NiO is most transparent to magnon propagation near the paramagnet-antiferromagnet transition. We do not see the enhancement in spin current driven into Pt reported in other papers when 1-2 nm NiO layers are sandwiched between Pt and YIG.

Spin Seebeck effect through antiferromagnetic NiO

1604.08659v2

1 Single-shot imaging of ultrafast all-optical magnet ization dynamics with a spatio-temporal resolution T. Zalewski and A. Stupakiewicz Faculty of Physics, University of Bialystok, 15-245 Bialystok, Poland Abstract. We present a laboratory system for single-shot magneto-opti cal (MO) imaging of ultrafast magnetization dynamics with MO Faraday’s rotation sensitivity of 4 mdeg/ µm. We create a stack of MO images repeatedly employing a single pair of a pump and defocused prob e pulses to induce and visualize MO changes in the sample. Both laser beams are independently wavelength-tunabl e allowing for a flexible, resonant adjustable two-color pump and probe scheme. To increase the MO contrast the pr obe beam is spatially filtered through a 50 µm aperture. We performed the all-optical switching experime nt in Co-doped yttrium iron garnet films (YIG:Co) to demonstrate the capability of the presented metho d. We determine the spatial-temporal distribution of the effective field of photo-induced anisotropy driving th e all-optical switching of the magnetization in YIG:Co film without an external magnetic field. Moreover, using this imaging method, we tracked the process of the laser- induced magnetization precession. I. INTRODUCTION The development of femtosecond lasers and time-resolved methods provide d the unprecedented capability for the all-optical coherent control of spins on ultrashort ti me scales 1. From an application perspective, the main challenge for obtaining an electronic-competiti ve solution is the integration of photonics and spintronic devices with triggering of magnetic order by a n ultrafast all-optical laser stimulus 2. The modern magneto-optical methods using pump-probe geometry are attra ctive tools for research in ultrafast magnetism, offering insight into the time-re solved dynamics of the light-induced processes on ultrashort timescales. The exceptionally sharp time resolution down to femtoseconds offers access to information inaccessible in the slow, static, or quasi-st atic regime experiments. At the same time, the spatial resolution of the magneto-optical microscopy 3 allows imaging of features as small as couple hundred nanometers, up to the light diffraction limit. The visualization of the ultrafast dynamics of photo-induced processes i n physics is critical for understanding the main interactions. The main advantage of such a techn ique is compounding both the spatial, and temporal behavior of the phenomena at the same time, givi ng three-dimensional sets of information. Image, as a 2D representation, gives briefly qualitati ve information, which by appropriate scaling of intensity level can be quantitive. Imaging methods, whi ch are based on pump and probe techniques are well-known for almost 40 years where the first process es were recorded on photographic films.4 Later, a huge amount of different variations of this general ide a were used very widely, mainly in chemistry and biology.5,6 However, the pump-and probe methods are inapplicable for all phenomena. Nonrepetable or difficult to reproduce ultrafast processes, such as irrev ersible chemical reactions, laser- induced damage, or scattering on a living tissue require so-called single-shot imaging. In this context, the “single-shot” term refers to capturing the whole event without repeating it. The whole measurement sequence is performed in real-time with only one laser pulse.7 In contradiction, it has to be noticed that in this article the discussed method is a repeatable pump and probe technique and the term single-shot refers to the magnetization switching by a single laser pulse. Recently, different fundamental mechanisms which are responsible for the ul trafast laser-induced magnetization dynamics have been revealed. In the case of the heat load in magnetic materials, they can be separated into thermal and non-thermal. The thermal effect has been observ ed in metals base on the ultrafast demagnetization during heating close to Curie point.8–10 The non-thermal effects are caused by the inverse Faraday effect 11 and the photo-magnetic effect.12 ,13 Since the discovery of these effects, rapid development, and increased interest in the possibility of magnetiz ation switching by laser pulses in a large class of materials occurred,14 and time-resolved imaging was a prominent tool for examining the spatio-temporal behavior of the magnetization. Two main approaches for time-resolved magneto-optical imaging can be noted. It can be realized via scanning a focused bea m through the sample 15–18 or via direct imaging on a CCD camera 19,20 which is also realized in this work. The method is widely known and it was used for examining plenty of different materials such a s GdFeCo metallic alloys,18,20–22 the 2 rare-earth orthoferrites DyFeO 323 and HoFeO 3,24 Co/Pt ferromagnetic multilayers,25 garnets.16,26 Moreover, the understanding of ultrafast magnetization dynamics was achi eved through the development of time-resolved imaging methods using femtosecond sources of X-ray radiation 27 and free-electron lasers.28 Recently, it was discovered that without any external magneti c field reversible photo-magnetic switching in Co-doped yttrium iron garnet films (YIG:Co) can be obtained only by a single laser pulse.29 Such switching mechanism is based on the light-induced effectiv e field of photo- induced magnetic anisotropy, which lifetime is about 20 ps, and it is sufficient to trigger angle precession of the magnetization. The photo-magnetic effect is the most intri guing due to the non-dissipative mechanism of all-optical switching of magnetization with the l owest heat load and fastest time switching. No long-timescale recovery from excessive heat load give s a high repetition rate limit of 20 GHz for a cold photomagnetic recording.30 Ultimately, imaging with only one ultrashort probe pulse from a lase r source is marked with a large number of technical difficulties. These are among others: nonuniformity of the illuminating light, temporal stability of the laser pulse train, separation between pump and probe signal, impact of the diffraction, and mechanical stability. Therefore, to sufficiently inve stigate the dynamic of such a fast process one has to develop a reliable and repeatable imaging system . For YIG:Co films such measurement has been also presented in 29 . However, obtained image contrast in performed measurements was marked as insufficient, concealing the desired i nformation in a spatially averaged, blurry picture. Detailed spatial analysis and determination of the spat ial distribution of the effective field of the photo-induced anisotropy were not performed. Here, we developed a spatially improved time-resolved system for s ingle-shot imaging of ultrafast laser- induced spin precession and switching. We used a two-color indepen dently tuned laser pump and probe pulses with duration <40 fs at a wide spectral range of 290 −2570 nm. Moreover, the two separate mechanical delay lines for both pump and probe beams allowed to ensure full flexibility and independence in settings desired time shift resolution <8 fs. We ap plied spatial filtering to the unfocused probe laser beam which allows to cleanse the beam of imperfections ge nerated by defects on the sample and the effect of the interference. We demonstrated time-resolved la ser-induced dynamics of magnetization precession and permanent switching with a high sensit ivity of magneto-optical rotation in YIG:Co thin films. Moreover, we determined the spatial distributi on of the field of photo-induced anisotropy driving the magnetization switching. This paper is organized as follows. In Sec. II, we describe the e xperimental details of the set-up for single-shot time-resolved imaging, the magnetic structure of the sa mple, and the image processing. In Sec. III, we demonstrate the imaging of all-optical photo-magnetic s witching and precession in YIG:Co film with temporal and spatial resolutions. The conclusions are in Sec. IV. II. EXPERIMENTAL DETAILS The set-up used for the magnetization dynamics imaging is presen ted in FIG. 1. We employed the time- resolved technique of magneto-optical polarized microscopy using exc itation and detection with a single femtosecond pump and probe pulses, respectively. A femtosecond laser s ystem is one-box integrated Ti:Sapphire amplifier (Coherent Astrella). Its output pulse energy is 8 mJ with the 1 kHz repetition rate and pulse duration not longer than 35 fs on 800 nm central wavelength. Next, the laser beam is divide d into two branches the probe and the pump beam. The magneto-optical imagi ng in ferrimagnetic dielectrics requires a two-color scheme between pump and probe due t o the spectrally-selective excitation of magnetic order and optimization of magneto-optical rotation for a probing. Two optical parametric amplifiers (Light conversion TOPAS-Prime) and optica l frequency mixers (Light conversion NirUVis) were applied to each branch with input pulse energy of 3.5 m J. The wavelength of both pump and probe pulses can be independently tuned in the 290 −2570 nm range. Afterward, laser pulses are temporarily shifted by ∆t defined by the mec hanical time-delay line up to 4.4 ns. The probe beam passes through a pinhole PH with a high damage thres hold and 50 µm diameter. Next, the beam goes through the polarizer P as a half-wave plat e and the adjustable lens with a focus distance of 10 mm. It stays on sample S slightly out of focus to illuminate a suffici ently large area. 3 FIG. 1. Magneto-optical setup scheme for the two colors ti me-resolved single-shot imaging. BS – beam splitter, OPAs – optical parametric amplifiers, PH – pinhole, SH1- mechanical sh utter, FM – flip mirror, L – adjustable lens, S – sampl e, C – coil, F- color filter, A – analyzer, P – polarizers, DG – delay gene rator. To retrieve magnetic contrast from images the princ iple of the magneto-optical polarizing microscope has been used. The light propagating through the sa mple is affected by the magneto-optical Faraday effect. It induces rotation of the plane of polarizat ion proportional to the magnetization component projected on the direction of the light propagation .3 In used Faraday (transmission) polarizing microscope geometry, linearly polarized probe pulse propagates through the sample and it is collected by the objective O. Afterward, the pulse passes thr ough the analyzer A. The mutual rotation between polariser and analyzer defines magneto-optical contra st. The CCD camera (Princeton Instruments ProEm 1024) working in the full-frame mode is used to record and digitize 16-bit images with resolution 1024×1024 pixels. To suppressed thermal noise and u nwanted charge accumulation, the built-in continuous cleaning of the array procedure was used between each frame. It removes any charge from the array until a trigger pulse is detected and sto ps as soon as the frame collection begins. The spot size of the probe laser beam localized in the sample surf ace was 350 µm in diameter which is significantly greater than the pump beam spot size of 130 µm in diameter. For adjusting the probe spot size focu sing lens L ( f = 100 mm) was used. It makes the beam divergent aft er its focal point resizing the illuminated area. To eliminate residual pump light the spectral bandwidth filter F was placed before the CCD camera . The pump pulse train is suppressed by the synchroni zed mechanical shutter SH1. It can release a single pulse to excite the sample. For safety reasons addit ional mechanical shutter, not marked in the figure was added to the probe branch. The recorded magneti c domain can be removed by an external in-plane magnetic field H produced by coil C. Moreover, for static imaging and magnetic domain st ructure observation, the sample also was illuminated by the white LED light let in by a flip mirror FM in normal incidence. To obtain high-quality images of magnetic domains the magneto-optical contr ast was optimized by setting the analyzer to respect of polarizer axis in the position which allo ws visualizing magnetic domains structure. It was done on a long time scale with an LED light source polarized in the same plane as the probe pulse. Next , the intensity of the illuminating probe laser pulse s, its uniformity, and angle of incidence were bala nced to minimalize the total impact of the imaging aberra tions, diffraction on surface defects, and created interface fringes concealing the image. A. Setup synchronization To automatize and ensure repeatability and stability of the experiment, the following measurement procedure which consists of four integral steps was applied. Firstly, the software sets the mechanical 4 delay line (Newport DL325) to the desired distance, which defines a time delay between the probe and pump pulses. The minimum incremental motion with gu aranteed bidirectional repeatability of ±0.15 µm corresponds to about 8 fs resolution, which is shor ter than a laser pulse duration. Secondly, the softw are automatically sends a predefined 100 ms duration vo ltage pulse to the erasing coil using a power suppl y (Kepco BOP 72-6 DL). It creates the magnetic field wit h an amplitude of H=100 Oe used to restore the initial state of the sample's magnetization. Next, w hile the pump beam is still suppressed by the mechanical shutter, only the probe beam illuminates the sample. The image recorded in this step acts a s the background image. Ensuring a new background ima ge for every measurement sequence helps to prevent the impact of long-time laser beam instabili ties. Finally, the mechanical shutter opens, and bot h beams – pump and probe illuminate the sample during the single shot of 35 fs creating the time-resolved image. The whole sequence is consistently repeated for every delay value ∆t. The electrical signal which acts as a trigger is de fined by the laser’s system regenerative amplifier Astrella. Its rising edge is synchronized with the op tical pulse generation (see Fig. 2a). To perform a single-shot imaging experiment in the above sequen ce, one has to be able to pick only one laser pulse from the pulse train on demand. Plenty of different solutions, mainly dependent on the laser repetition rate, can be applied to solve this issue. The conce ptually most simplistic approach is to use the mechanical shutter. Such devices have a large apertu re size which, does not affect the beam parameters. However, due to the large inertia of these devices they can be applied only for relatively small (tens of Hz) repetition rates. For higher repetition rates , electro-optic or acousto-optic modulators have to b e used. Shutter based on this type of device ensures perfect timing, but they introduce significant loss es in light transmission. Moreover in our case, the pu lse selection from both pump and probe pulse trains has to be independent. Significantly longer open-close time cycle (<35 ms) o f used mechanical shutter (Thorlabs SHB05T) and several millisecond CCD sensor data acquisition spe ed makes it necessary to decrease the laser repetit ion rate. Here, as presented in Fig. 2a, the laser repetition rate was reduced by the amplif ier’s built-in Pockels cell divider by 40 times, from 1 kHz to 25 Hz. Moreov er, to provide a proper pulses synchronization the digital time delay generator DG (Stanford Research Sy stems DG645) was applied. It is capable to pick not more than one trigger pulse per second, definin g the base system operation frequency at 1 Hz (Fig. 2b). This gives a sufficiently long time window for other components and ensures a single pulse selectivity. Because the opening and closing cycle of the mechanical shutter takes about 10 ms the fir st pulse is always suppressed. Thus, DG always chooses second, subsequent pulse (Fig. 2c). The introduced time shift from the triggering pulse to the sequent optical pulse ensures enough time for opening the mechanical shutter (Fig. 2d). The CCD ca mera is triggered right after the mechanical shutter. Its exposure time is set to 1 ms, but it is not essential in this type of experiment. The came ra requires additional dozen milliseconds for data acqu isition. FIG. 2. The timing synchronization scheme. a) The pump and probe pulses are formed from an electronically triggered amplifier'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 synchronizes the shutter and CCD camera with subsequen t one d) The mechanical shutter open-close time e) The CCD camera exposure and time required for data required. The delay be tween pump and probe is unnoticeable in the presented scale. 5 We note that our femtosecond laser system is also capable of worki ng in a single-shot or burst regime. The built-in Pockels cell can be steered directly, deterministi cally creating the required pulse train. However, every such event disturbs the amplifier’s cavity thermal equilibrium. We observed that in long term, it negatively affects the generated pulse train stabili ty. Using external modulators makes it possible to stabilize the laser cavity. Even for the decreased operation fre quency, the pulse train energy fluctuates less than in burst or single-shot regimes. The whole system is unifie d and controlled by the PC using the LabView software. It ensures flexibility and gives the possibility of integrating the system with various other components such as coils, step motors, modulators, or heaters, hence des igning many variants of experiments. B. Magnetic states in a garnet film The investigated sample is a 7.5 µm-thick film of Co-doped yttrium iron garnet 31 . The sample was grown by a liquid phase epitaxy method on a gadolinium gallium garnet substr ate with 4° miscut. The material is optically transparent in NIR 32 with a relatively low static Faraday rotation of about 0.4 ° at a wavelength of 650 nm 33 . The garnet has a cubic lattice with two antiferromagnetically coupled spin sublattices of Fe 3+ in both tetrahedral and octahedral sites. In both of the sublattic es, cobalt dopants replace the Fe 3+ with Co 2+ and Co 3+ ions.34 The addition of cobalt ions introduces strong magnetocrystalline anisotropy and large Gilbert damping α = 0.2.35 The negative cubic magnetic anisotropy gives 4 easy magnetization axes to be close to the cube diagonals of [111]-type directions (see Fig. 3a) The miscut was implemented to partially break the degenera cy between the magnetization states at room temperature and with no applied field and makes it e asier to magneto-optically distinguish the domain structure 29 . The orientations of easy magnetization axes are close to the cube diagonals [111] −type directions. Here, we focus on the simplest case switching between large domains M- and M+, state [11-1] and [1-11] respectively. FIG. 3. (a) Easy magnetization axes in YIG:Co film (b ) and image of remanent magnetic domain structure in the YIG:Co obtained at zero magnetic fields. The sample was illum inated by the LED with improved contrast by image proce ssing. Two magnetic states are visible: large domain (1) M- along [11-1], and small domain (5) along [111] directions. By applying the 100 Oe external field with 1 s duration in the direc tion along [110] the small domain pattern can be remo ved. On certain sample localization, only monodomain (single magnetic phase “ 1”) ROI can be selected. The size of the image is 380x 380 µm 2. 6 C. Imaging and data processing Imaging of the sample illuminated by the probe ultrashort laser pulse is not sufficiently efficient to directly visualize the magnetic domains with low magneto-optical contrast in the YIG:Co. To qualitatively improve the images a spatial filter was introduc ed into the optical setup. Spatial filter pinholes are useful components for maintaining high beam quality in hi gh-energy pulsed laser systems 36 . Such a filter was used to reduce the high-frequency noise in the pro be pulses beam profiles (see Fig. 4a). Also, the spatial shift of illumination connected with the delay line movement is minimized. To couple and decouple the beam through the pinhole two lenses with f = 40 mm we re used. The imaging beam is not a plane wave, therefore the light profile is not uniform. However, the presence of the magnetic domain structure can be easily distinguished on the images and on the profile of the cross-section (see Fig. 4b). FIG. 4. The magneto-optical images in YIG:Co film wh ich is illuminated with a single probe pulse. The dist ance profiles correspond to the yellow cross-section line marked in t he images. a) initial image was taken without a pinho le. Nonuniformity of the incident wavefront creates high-frequency noise concealing the magnetic structure character. b) ima ge obtained using a pinhole. As in FIG. 3 revealed domain structure corre sponds to the large (1) M- and small (5) M+ magnetic p hases schematically marked as stripes on the profile. Images a re represented in a normalized 8-bit grayscale. The i mages size is 380×380 µm2. To determine the character of pump-induced changes from the recorded 2D ima ges, one has to separate the contribution of the magneto-optical Faraday effect from pure optical effects. Any local light intensity changes such as probe beam instabilities or noise caused by the pump absorption in the s ample have an impact on the recorded image. Here, the visible difference comes from the rotation of the polarization plane of the probe beam, which passed through the sample. While propagating throug h the sample, the probe beam interacts with the magnetic domain structure. Different o rientation of the magnetization component of magnetic domains results in the different Faraday rota tion of the polarization plane, which is detected on the camera. The position of the analyzer defines the magneto-optic c ontrast. 7 FIG. 5. a) The single-shot imaging of YIG:Co at monod omain state with magnetization (1) M- along the [11- 1] direction before excitation, illuminated only by a single probe pulse wi th λ = 650 nm. b) The image recorded after a single p ump pulse with λ = 1300 nm for ∆t= 185 ps. The magnetization of the switched domain was along [1-11] direction and corresponds to a state (8). c) The differential image (B-A). The pump pola rization was E|| [100]. The slightly elongated shape of the switched do main is related to both the beam shape and the miscut angle of the sample and schematically marked (red dashed lin e) on a profile. The images of the domain structure can be visualized directly using only a single probe pulse. It is also valid to the dynamic situation within pump and probe pulses. However, for con trast improvement of these images and to suppress the impact of the probe’s wavefront nonuniform ity differential images were created. These images were obtained by subtracting image B recorded after single pump pulse irradiation and background initial image A (see Fig. 5). The most distinct advantage of imaging is that it reveals the spatial information of the domain, concerning its shape, structure, and size. The time-resolved addition to imaging allows determining the time scale of pump-induced spatial changes. Therefore, creating repe atable series of images, as was previously proposed in the measurement sequence, is an easy way to re trieve the information concerning the time and localization of pump-induced changes. Such an image stack contains information about the relative time delay between images, which is retrieved from the delay line d istance. 8 Spatial information can be obtained directly from a stack of image s, but several factors need to be considered. Firstly, to compare images obtained with non-constant illumi nation conditions, one has to uniformize their intensity levels. In order not to spoil the informat ion concerning the single image intensity levels, the contrast-enhancing procedure should be applied to all images in the same manner. Therefore, the histogram of the whole stack, instead of individual image s histograms, was used to rescale the intensity levels of each image. Here, float 32-bit representat ions of differential, images were rescaled to 0-256 range. Secondly, one has to be sure that the background intensity lev el is flat. For every image, it has to be at the same, uniform level. We note that without i t, differential images were spatially incomparable. In this case, we used additional post-processing algorithms.37 II. RESULTS We used a setup for single-shot time-resolved magneto-optical imagi ng to determine the photo-magnetic switching in YIG:Co film. The experimental data of photo-magnetic switching retrieved from the differential image stack were represented in a three-dimensional manner, highlighting the mutual temporal and spatial behavior of the change of magnetization orientat ion as shown in FIG. 6. The magneto-optical differential images for selected ∆t are shown on t he top panel in FIG. 6. The contrast in these images is due to the magneto-optical Faraday effect, wh ich is proportional to the perpendicular magnetization component. Change of intensity within the laser pump spot on t hese images allows tracking the evolution of the photo-magnetic switching. To visual ize the spatio-temporal redistribution of magnetization the 3D map was created from the intensity profiles of 150 differential images which were recorded using a single probe pulse for different ∆t. The intensi ty profile for every image, proportional to the time-resolved Faraday rotation, was used data for the 3D map. To improve the signal- to-noise ratio, instead of only one profile cross-section line, we chose the averaged profile from a rectangular selection of 10 µm width as shown by yellow on the image in Fig. 6. The color code on the map corresponds to the out-of-plane magnetization component. In this map, we obs erve the red area of photo-magnetic switching in YIG:Co between M- and M+ states which correspond to magnetization states (1) and (8) in FIG. 3a. FIG. 6. Three-dimensional representation of magnetiz ation dynamics under single pump pulse with a fluence o f 75 mJ/cm 2 in YIG:Co film. A negative time corresponds to the probe pulse illuminating the sample before the pump pulse. The characteristic blink in overlap position defines precisely ∆t=0 ps time. The top panel shows the differential imag es of magnetic domain structure obtained at various ∆t. The color on the diagram corresponds to the angle of time-resolved Faraday rotation which is proportional to the change of the out-of-plane magn etization component. The deep blue color is an initi al magnetization state (1) M-. The green color corresponds to the magnetizat ion aligned in-plane. The red color shows the area sw itched to state (8) M+. 9 The mechanism of presented photo-magnetic switching in YIG:Co film is triggered via the precession of the net magnetization.29 In this case, the pump light optically excites one of the possible and the most effective electronic transitions for Co ions being in the tetrahedra l sites.38 Such excitation is responsible for the change of magnetic anisotropy in a garnet. For an initial stat e, before pumping the intrinsic magnetic anisotropy is predominantly cubic with a small uniaxia l contribution. The strength of the effective field of photo-induced anisotropy is comparable to the intrin sic one.13 It can lower the energy barrier allowing the magnetization to switch to the second state. Be cause of the large damping, the lifetime of the photo-induced anisotropy is low and it decays at about 60 ps at room tempera ture which corresponds to the quarter of the laser-induced precession. The size of a switched domain can be determined by the spot of the pump beam. Here, it is given by the averaged laser optical power and total pump spot size on the sample, de termined by the system focusing. However, because the incident beam has a Gaussian shape, locall y the light intensity differs. It is highest on the beam axis, and it decreases away from the axis. Therefore, by selecting and ana lyzing a selected region of interest (ROI) one can distinguish the amplitude of photo-induced f ield-related temporal changes in FIG. 7a. Improved sensitivity of the set-up allowed to vis ualize and distinguish four characteristic regions with different behavior of the magnetizati on vector. The mean value from the selected ROIs is subjected to further analysis. Chosen ROIs are marked on the image with appropriate colors. All of them are circles with the same size of about 10 µm diameter and are shifted from the image center by given ∆ d. Choosing a larger region instead of only individual pixels can reduce noise impact. For the ∆ d > 60 µm ROI corresponds to the region which is unaffected by the pump puls e so it acts as a background. No change of magnetization out of plane component appears here. FIG. 7 a) The temporal behavior of the out-of-plane magnetization component for the different characte ristic regions of interest. b) The time-resolved precession around the state M- in the long-time domain for ∆ d = 40 µm. The signal obtained by the single- shot imaging is compared to the regular pump-probe dyn amic with the same experimental condition of the lase r pulses. 10 The background is used as the image stack reference. It has to be fl at to ensure that all images taken in the measurement series are comparable. For smaller ∆ d = 40 µm the incident pump intensity is still below the switching threshold. However, its intensity is high enough t o trigger the precessional movement of magnetization. With ∆ d = 20 µm and lower the incident pump intensity is high enough to obtain the switching of magnetization from state M- to state M+. After it, the out-of-plane component of magnetization is reversed. Presented behavior applies also to the averaged situat ion – ROI including whole switched spot. In the center of the spot ∆ d = 0 µm the local pump intensity is highest. It triggers the magnetization switching from one state to another. Previously, in reference 29 due to the insufficient magneto-optical contrast, this effect was hidden. Here, because of the greater imaging sensitivity, we can perform spatial analysis in detail. Thus, for a sufficiently la rge field of photo-induced anisotropy switching from the state (1) to (8) occurs. Next, after about 60 ps, when the magnetization was switched, we observed damped precession around the direction of the switched magneti zation state. The amplitude of this precession corresponds to the FMR mode was small and the magne tization remains at the switched state. This effect was predicted 38 but it was not observed experimentally. We compared precession for ∆ d = 40 µm measured through presented single-shot imaging with the signal obtained through highly sensitive pump-probe measurement (see Fig 7b). By that, we determined the time-resolved Faraday rotation sensitivity of our method as 30 mde g for the 7.5 µm garnets which can be scaled to about 4 mdeg/µm. The data presented as the pump-probe signal was obtained within the same setup but with a typical detection pump-probe scheme.39 The laser system worked with a repetition rate of 1 kHz. The probe beam was modulated by a chopper to 500 Hz and it was focused on the sample to 50 µm in diameter. Next, the probe was directed to the ha lf-wave plate and the Wollaston prism which separates it into two linearly polarized beams with orthogonal polarization. These beams are directed to the two separate branches of an auto-balanced photodiode , which monitors the change of their intensities corresponding to the Faraday rotation. The frequency modula tion alternating the pump and probe signal was applied to use lock-in amplifier (Zurich Instruments MFLI) based detection. Moreover, to improve the signal-to-noise ratio boxcar integrator (Stanford Re search Sr 250) was applied. III. CONCLUSIONS We developed and implemented an automated setup for the high-contrast time-resolved single-shot imaging of magnetization dynamic. By using two OPAs we obtained independent temporal and spectral tunability of two beams. The possibility of selecting the central w avelength from the wide spectrum for the pump beam is a key for inducing different resonant transitions responsi ble for the magnetization switching, especially in dielectrics. The spectral tuning of the probe beam is necessary to find the wavelength assuring an optimal balance between high magneto-optic al contrast and low absorption in the sample. Using two delay lines made it possible to set a pump d elay with an unchanging probe delay and vice versa. By applying the pinhole into the probe beam we limit ed the interference noise improving the imaging quality. We observed the switching in single-domain structure and spatially analyzed its dynamics. Moreover, right next to the switching with the high sensi tivity of 4 mdeg/ µm, we distinguished regions in which precession of the magnetization appears. The propos ed imaging method is characterized by full flexibility and very high sensitivity. It may be further use for examining the spatio-temporal behavior of more sophisticated, multi-state magneti c structures, as well as even finding completely new switching mechanisms in different materials. Acknowledgments. The authors thank D. Afanasiev and A.V. Kimel for the fruitful dis cussion. This work has been funded by the Foundation for Polish Science POIR.04.04.00-00-413C/17-00. REFERENCES 1 A. Kirilyuk, A. V. Kimel, and T. Rasing, Rev. Mod. Phys. 82 , 2731 (2010). 2 D. Sander, S.O. Valenzuela, D. Makarov, C.H. Marrows, E.E. 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2021-10-09

We present a laboratory system for single-shot magneto-optical (MO) imaging of ultrafast magnetization dynamics with high-sensitivity of MO rotation. We create a stack of MO images repeatedly employing a single pair of a pump and defocused probe pulses to induce and visualize MO changes in the sample. Both laser beams are independently wavelength-tunable allowing for a flexible, resonant adjustable two-color pump and probe scheme. To increase the MO contrast the probe beam is spatially filtered. We performed the all-optical switching experiment in Co-doped yttrium iron garnet films (YIG:Co) to demonstrate the capability of the presented method. We determine the spatial-temporal distribution of the effective field of photo-induced anisotropy driving the all-optical switching of the magnetization in YIG:Co film without an external magnetic field. Moreover, using this imaging method, we tracked the process of the laser-induced magnetization precession.

Single-shot imaging of ultrafast all-optical magnetization dynamics with a spatio-temporal resolution

2110.04506v1

1 The effect of the magnon dispersion on the longitudi nal spin Seebeck effect in yttrium iron garnets (YIG) Hyungyu Jin1, Stephen R. Boona1, Zihao Yang2, Roberto C. Myers2,3,4, and Joseph P. Heremans1,3,4 1. Department of Mechanical and Aerospace Engine ering, The Ohio State University, Columbus, OH, 43210 2. Department of Electrical and Computer Engineer ing, The Ohio State University, Columbus, OH, 43210 3. Department of Materials Science and Engineerin g, The Ohio State University, Columbus, OH, 43210 4. Department of Physics, The Ohio State University, Columbus, OH, 43210 ABSTRACT We study the temperature dependence of the long itudinal spin-Seebeck effect (LSSE) in a yttrium iron garnet Y 3Fe5O12 (YIG) / Pt system for samples of different thicknesses. In this system, the thermal spin torque is magnon-driven. The LSSE signal peak s at a specific temperature that depends on the YIG sample thickness. We also observe freeze-out of the LSSE signal at high magnetic fields, which we attribute to the opening of an energy gap in the ma gnon dispersion. We observe partial freeze-out of the LSSE signal even at room temperature, where kBT is much larger than the gap. This suggests that a subset of the magnon population with an energy below kBTC (TC ∼ 40 K) contribute disp roportionately to the LSSE; at temperatures below TC, we label these magnons subthermal magnons . The T-dependence of the LSSE at temperatures below the maximum is interprete d in terms of a new empirical model that ascribes most of the temperature dependence to that of the thermally driven magnon flux. 2 INTRODUCTION The spin Seebeck effect (SSE), first reported in 20081, is the experimental manifestation of thermal spin transfer torque. The SSE2,3 can be understood as the proce ss by which a spin current is generated by a temperature gradient ∇T in one material, where it is carried by either magnons or spin- polarized electrons, and this spin current then cro sses the interface into an adjacent normal metal (NM). Due to strong spin-orbit interactions in the NM, this injected spin current generates an electric field EISHE via the inverse spin-Hall effect (ISHE). Since this electric field is related to ∇T, it is possible in the regime of linear relations to define a spin-Seebeck coefficient SSSE= EISHE / ∇T. The effect manifests itself in two geometries, the tran sverse spin-Seebeck effect (TSSE)1,4,5,6,7,8 and the longitudinal spin- Seebeck effect (LSSE)9,10,11, in which the direction of the heat fl ux and of the spin flux are collinear and normal to the direction of the magneti zation, respectively. In the latter geometry, a spin Peltier effect has been observed12 to be the Onsager reciprocal of the LSSE. The potential issue of the admixture of a Nernst effect into the LSSE signal13 has also been addressed.14,15,16 The LSSE is well understood17 at this point; however, it is only applied to ferromagnetic insulators such as yttrium iron garnet Y 3Fe5O12 (YIG), since the geometry of the measurem ent means that LSSE signals are indistinguishable from classical Nernst signals if ferromagnetic conductors are used. The general theory for L SSE is given in Ref [17], and the effect in the YIG/Pt system is attributed to magnon transport.18 , 19 , 20 , 21 To explain this schematically, we can say that b ecause there is a magnon contribution κM to the thermal conductivity,22,23 a magnon heat flux jQM = -κM ∇T exists in the presence of a temper ature gradient, and thus there must also exist a flux of magnons, i.e ., a spin or magnetization flux jM. In the simplest possible approximation, by treating the magnons as an ideal gas and ignoring th e mode and frequency dependence of effects, one could even write that jM = jQM (gL μB)/(k BT) = - κM ∇T (gL μB)/(k BT). Here, gL is the Landé factor and μB 3 the Bohr magneton. This thermally dr iven magnetization flux then cro sses the interface through a process governed by a characteristic paramete r known as the spin-mixing conductance g↑↓, which is the ratio that relates the spin current generated in a material to the energy of the driving spin injection process.24 The directional spin polarization v ector is set by the magnetization M along the applied field H (as given in Fig. 1(b)). This overall spin polariz ation gives rise in the Pt metal to an inverse spin-H all field given by EISHE = D ISHE (jQM × M), which is measured as a voltage VISHE = | EISHE| L, where where DISHE is the , and L the length of the Pt strip. The dependence of LSSE on YIG thickness has been studied in thin films,25 along with the behavior at low18 and high26 temperatures and at high fields.16 Particularly interesting is that near the Curie temperature, a specific temperature dependence has been observed that does no t reflect that of the magnetization. Here, we present temperature-depende nt LSSE data at cryogenic temperatures on both a bulk single crystal and a 4 μm thick film, and we show that they are different. This difference is interpreted in terms of the magnon thermal mean free path. We also extend the low temperature measurements to high magnetic fields, where we see that the LSSE signal is decreased at high field, which we interpret in terms of the magnon gap openi ng due to Zeeman splitting of the magnon dispersion. This suppression is also observed at 300 K, where th e thermal energy is much larger than the magnon energy gap we open in our experiments. The fact that th ere still is a significant suppression of the LSSE at these temperatures and fields is taken as proof that low energy magnon modes appear to constitute a disproportionately large perc entage of the LSSE signal at all temper atures. This leads us to divide the magnon population into two categories: thermal magnons , which we define as those magnons at all energies up to T × kB, where T is the temperature of the experiment, and subthermal magnons, which we define as those at energies below kBTC. We define the cutoff temperature to be TC ∼ 40 K on the basis of 4 the amount of freeze-out measured at room temper ature, and note that the experimental magnon dispersion27 at energies of kBTC starts deviating significantly from a simple quadratic function of the wavevector. Obviously, the distinction between subthermal and thermal magnons only holds at T>TC, and that below TC both populations are the same. The concept of subthermal magnons is inspired by the same concept used for phonons in the context of electron-phonon drag in semiconductors.28 In phonon drag, the only phonons that can interact with the electrons are thos e with propagation vector limited to 2 kF, where kF is the wavevector of the electrons at the Fermi surface. In semiconductors where kF is much smaller than the size of the Brillouin zone, and at high temperatures where the phonons of energy kBT have a much larger k-vector, this cutoff means that only those “subthermal” phonons with smaller k- vectors can participate in drag. Through our LSSE data we present here, we offer evidence that subthermal magnons in YIG interact more strongly w ith the electrons in Pt, similarly to phonon drag, such that LSSE is driven not equally by all modes of the magnon population at th ermal equilibrium at any given temperature, but instead is biased toward low-energy subthermal magnons. Since this manuscript was written, a manuscript has appeared on arXiv29 which, for all intents and purposes, agrees very well with the da ta and conclusions we put forth here. In addition, we propose here an empirical model for the temperat ure dependence of the LSSE at low temperature and in thin films. The model takes into account the ch ange of the magnon dispersion, from quadratic at the zone edge to kinear (“pseudo-acoutic ” as defined by Plant27) at higher energies; this mode l explains the fact that the observed temperature dependence in thin films be low 100 K does not actually follow a power law of T, but varies slowly from a T1 law at the lowest temperatures, wher e the dispersion of the thermal magnons is mostly quadratic, to a T2 law at higher temperatures. 5 EXPERIMENTAL Two different samples (A and B) were measured in this study. Sample A consists of a 4 μm thick single-crystalline YIG film grown on a 0.5 mm thick gadolinium ga llium garnet (GGG) substrate with a 10 nm thick Pt layer deposited on top of the YIG. Sample B is a 0.5 mm thick single-crystalline YIG slab purchased from Princeton Scientific, Inc, upon which a 10 nm thick Pt layer was also deposited on one polished wide surface of the YIG slab using e-beam evaporation after careful cleaning. Dimensions ( Lx × Ly × 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 mm (sample B). For LSSE measurements, both samples were mounted in the same way as follows. First, two voltage leads were attached on two edges of the sample separated by Ly. Fine copper wires with 0.001 in. diameter were used for the voltage leads to minimi ze heat losses through the leads. Point contacts between the voltage leads and the sample were made using silver epoxy, and an effort was made to minimize the size of the contacts. The sample was sandwiched between tw o rectangular cubic boron nitride (c-BN) pads to promote uniform heat flux across the sample, and Apiezon N grease was used between the sample and c-BN pads to provi de uniform thermal contacts. Three 120 Ω resistive heaters connected in series were attached on the top of the upper c-BN pad by silver epoxy. A Cernox thermometer was attached to each c-BN pad usi ng Lakeshore VGE-7031 varnish. The whole sample block was then pushed onto the samp le platform wherein the Apiezon N grease was used as a thermal contact. The sample block and the platform were wr apped together by an insulated wire in order to mechanically secure the heterostructure and provide a solid thermal contact between them. Fig. 1(a) shows a picture of sample A mount ed in the way described above. 6 The LSSE measurements were performed usi ng a liquid helium cryostat for sample A and a Physical Properties Measurement Syst em (PPMS) for sample B. Sample B was also measured in the liquid helium cryostat for a cross-check, and the data obtained from the two different instruments showed good agreement in terms of the temperature dependence of LSSE signal. We observed that the magnitude of the signal can be slightly different for different measurement sets, possibly because of an error in the measured 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 small variability of the magnitude. The measurement configuration is shown in Fig. 1(b). First, the Cernox thermometers were calibrated as a function of temperature from room temperature down to 2 K. Resistances of the thermometers were measured at di fferent temperatures using an AC bridge, and those values were interpolated to extr act actual temperatures during LSSE measurements. LSSE measurements were made at different temperatures, and ~1h st abilization time was given after changing the base temperature to ensure thermal equilibrium between the sample and the system. After a temperature gradient zT∇ was applied across the sample, additional 15 ~ 30 min wait time was used. The transverse voltage Vy was measured using a Keithley 2182A nanovoltmeter while the magnetic field Hx was continuously swept between a same Hx value in both polarities and in both directions (+ Hx → -Hx and - Hx → +Hx). For low field measurement, Hx was swept between ± 800 Oe at 4 Oe/s, and for high field measurement, between ± 70 kOe at 30 Oe/s. The temperature difference zTΔ was separately measured from the attached thermometers using the same heater power as that used for the Vy measurement. The sample chamber was kept under a high vacuum (<10-6 torr) and covered by the gold-plated radiation shield to minimize convective and ra diative heat losses, respectively. 7 RESULTS AND DISCUSSION Raw traces of the measured transverse voltage Vy as a function of applied field Hx for both samples are shown in Fig. 2. Around room temperatur e, both samples show clear LSSE signals under an applied temperature gradient ∇Tz, wherein Vy switches sign with the magnetiz ation in the hysteresis loops (Fig 2(a), (d)). The ma gnitude of the switch, ∆Vy, is defined as the differe nce between the saturation voltages.7 The signals become noisier at low temperatures (Fig. 2(b), (e)) due to the smaller magnitude of both the voltage signals and the temperature difference s across the samples. The raw traces of sample B exhibit an additional loop at low Hx values, and the size of this loop is almost independent of temperature. While some of the previous studies reported a si milar repeatable trend in their LSSE signals,9,30 no explicit description has been gi ven thus far to explain these obs ervations. Based on the lack of temperature dependence of the loop si ze, we may tentatively attribute the additional loop to the presence of magnetic domains in the bulk sample . Fig. 2(c) and 2(f) show that ∆Vy varies linearly with ∆Tz for both samples, so that a spin Seebeck coefficient 2 yy z z yy zSE T L VL T≡∇ = Δ Δ // can be defined. The same LSSE measurement was extended up to Hx = ± 70 kOe for sample B. At Tavg = 296 K, a gradual reduction of Vy / ∆Tz is observed as Hx increases (Fig. 3(a)). This behavior around room temperature is similar to what has been reported by Kikkawa et al.16 also on Pt/YIG. The authors attributed the decrease of the LSSE signals to the suppression of magnon excitation caused by the magnon gap opening under an applied H. At low energies (B Tk ω=h < 30 K), the magnon dispersion relation for YIG can be expressed reasonably well using th e classic Heisenberg ferromagnet model for spin waves,31 which is given as32 22 kB L gBD a k ωμ=+h (1), 8 where h is Planck’s constant, ωk is the angular frequency of th e magnon mode with wave vector k, μB is the Bohr magneton, gL is the Landé factor ( gL = 2.046 for YIG33), D is a spin-wave stiffness parameter, and a is the size of the unit cell. Applying a non-zero external H creates a forbidden zone in the dispersion (Eq. (1)) as well as in the density of states of magnons due to the Zeeman effect,31 thus freezing out magnons with energy BLgB ωμ<h . Since thermally excited magnons are believed to induce the SSE in ferromagnetic insulators18,19,20,21,25, their suppression by an external H is expected to reduce the measured ISHE voltage. Fig. 3(b) shows that the suppression of Vy / ∆Tz in applied H is more pronounced at Tavg = 10 K. If we define Vy,max as the maximum value of the ISHE voltage obtained at low H, the Vy / ∆Tz at Hx = ± 70 kOe ( Vy,70kOe / ∆Tz) is less than 50 % of Vy,max / ∆Tz. Surprisingly, ab out 20% suppression is observed at all te mperatures up to Tavg = 296 K. We compared the magnitude of Vy,max / ∆Tz in these high H measurements (Fig. 3(c) and (d)) with that in the independently measured low H results on sample B (Fig. 2(d)-(f)), and found that they agree well with each other. The significant suppression of the LSSE signal at high H at Tavg = 10 K hints at which magnons are mainly responsible for the LSSE in Pt/YIG, which is further revealed by inve stigating the temperature dependence of the suppression effect. We can define ∆Sy = Sy,max – Sy,70kOe , where 2 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 dependence of ∆Sy / Sy,max derived from the high H measurements (Fig. 3) on sample B. Above Tavg = 30 K, the ∆Sy / Sy,max is almost constant at ~ 0.2 (inset in Fig. 4(a)), indicating that about 20% of the LSSE signal is suppressed at Hx = ± 70 kOe. The ∆Sy / Sy,max increases gradually below 30 K and then shows a drastic increase below 15 K reachi ng ~ 0.9 at about 7 K, suggesting th at the LSSE almost disappears at this temperature and field. For YIG, at Hx = ± 70 kOe, a gap of / 9.6 KBL BgH kμ = opens in the dispersion relation given by Eq. (1) as shown in Fig. 4(b) . Therefore, it is like ly that most magnons are 9 suppressed below 8 K in Hx = ± 70 kOe and so is the LSSE, which is consistent with the experimental result in Fig. 4(a). But, as had already been pointed out in Ref [23], magnon suppression cannot be complete in Hx = ± 70 kOe fields at temperatures above 10 K. Some amount of field suppression of the LSSE is measured at all temperatures, while not in specific heat or thermal conductivity.23 Here, we verified experimentally on another piece of Sample A that the thermal conductivity of YIG is independent of magnetic field at 300 K to within less than 1%, th e experimental accuracy of that measurement. We thus conclude that low-energy ma gnons (i.e. those of energy below ~ kB × TC) must play a more prominent role in producing the LSSE. We estimate the value of the critical temperature TC ∼ 40 K using Maxwell-Boltzmann statistics by setting ,max 0.2 / exp( / ) 1yy B L B CSS g B k T μ =Δ = − , but note that there is considerable uncertainty in the pr ecise energy scale of the relevant modes, since we expect all magnon modes to contribute to thermal spin transport and L SSE, just some to contribut e more than others. In particular, the inflexion point observed in the low- temperature behavior of ,max/yySSΔ at 15 K in Fig. 4(a) seems to single this temperature out as well. Similar to the case of phonons,28,34 we call these low-energy (E < kB TC), long-wavelength magnons subthermal magnons to distinguish them from the thermal magnons with relatively higher energies and short wa velengths that populate the magnon spectrum in thermal equilibrium at T > TC . We now discuss the temperature and thickness de pendence of the LSSE in Pt/YIG. Fig. 5 shows the temperature dependence of Sy for sample A and sample B between 10 K and 300 K. The Sy of sample A has a maximum at 180 K. The Sy of sample B, meanwhile, has a maximum at 70 K and a ~ T 1.19 (dashed line) dependence at low temperatur e. The longitudinal thermal conductivity κxxx was measured independently, but is not shown because th e data reproduce thos e of Slack & Oliver35 quite exactly. The temperature dependence of the resistivity of the Pt film on sample B was measured to change from 0.35 10 μΩ m below 5 K to 0.52 μΩ m at room temperature. The te mperature dependence of both these properties is quite different from that of the observed LSSE signal, which suggests that there is likely no significant correlation between them. In the absence of information about the temperature dependence of the spin Hall angle in Pt and of the spin mixing c onductance at the YIG/Pt inte rface, we concentrate on the temperature-dependence of the thermally driven magnon flux. Therefore, we assume that the temperature depe ndence of the LSSE signal arises primarily from that of the thermally driven magnon flux that reach es the interface. This in terpretation, supported by the results of Ref. [26], allows us to neglect the T-dependence of the ISHE, the spin-Hall angle in the Pt, and the spin-mixing conductance across the interface. In general, since the magnetization flux at the YIG/Pt interface is in YIG driven by a temperatur e gradient, we can describe this flux by ## T ML B L Bj gj g L Tμμ== ∇ ( 2 ) , where j# is the magnon number flux, and L#T is the effective kinetic coefficient that relates this number flux to the accelerating force, the temperature gradient . The number flux is also related to the magnon heat flux jQ, which is itself related to the magnon thermal conductivity κM by Fourier’s law QMj T κ=− ∇ . Using the Boltzman diffusion equation in the relaxation time ( τ) approximation, both kinetic coefficients can be written for each magnon mode as an in tegral over the magnon frequencies (or energy E):36 2 #T 0 2 0()() 1()()3MAX MAXE G E MGfLv E E d ET fEv E E dETτ κτ∂⎛⎞= ⎜⎟∂⎝⎠ ∂⎛⎞= ⎜⎟∂⎝⎠∫ ∫D D (3). 11 Here D(E) is the magnon density of states and f is the Bose-Einstein statisti cal distribution function, so that ()21x xfxe TT e∂=∂ − where B xEkT≡ is the reduced energy. At very low temperature where the magnon dispersion is purely quadratic27 and given by Eq. (1), the group velocity and density of states in the absence of magnetic field become: ()21 / 2 1/2 3/2 222 1()4GvD a E EE Daπ= =h D ( 4 ) . An experimental estimate of th e magnon thermal mean free path Ml is given by Ref. [23] for T > 2 K. In the ballistic regime where the ma gnon mean free path is limited by sample size, lM is temperature- independent and () /M G Evτ =l . Consequently, we can derive the kinetic coefficient of the magnon number flux as well as the thermal conductivity: () ()22 #T 2 22 0 3 3 2 2 22 031 2 1 1 3 1x MB x x MB Mxkx eLT d xDa e kx eTd xDa eπ κπ∞ ∞= − = −∫ ∫l h l h ( 5 ) . One recognizes in Eq. (5) the familiar magnon thermal conductivity T2 scaling law. In that case, Eq. (5) predicts that the thermally dr iven magnon flux, and thus the LSSE signal, should scale as T1 (recall jM = - κM ∇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. First, because the measurements are taken on bulk samples, the ballistic regime is not reached until T < 2K and there is a dependence of lM on T. The second reason is the non-parabolicity of the magnon 12 dispersions. Eq. (1) already breaks down at energies below 10 K: in deed Ref [27] shows that the temperature dependence of the saturation ma gnetization curves departs from the Bloch T3/2law already below 10K. At higher energies (> 8 me V) and temperatures, the magnon dispersion becomes linear27, and the thermal conductivity in the ballisti c regime is expected to follow a T3 law, with L#T ∝ T2. In terms of group velocity, it increases with energy when the dispersion is quadratic, but saturates at a maximum value vGS when the dispersion becomes linear. These feat ures can be captured by fitting the measured “pseudo-acoustic” magnon disp ersion curves in YIG27 (and ignoring the pseudo-opt ical modes) with a phenomenological model for the magnon dispersion writ ten by analogy with the electron dispersion in narrow gap semiconductors such as PbTe37 as: 22 0() ( 1 )EE ED a kEγ ≡+ = ( 6 ) , where E0 parametrizes the saturation value vGS of the group velocity vG, as shown below.38 The group velocity and DOS are given by: ()2 3/2 22 02 ' ' 1()4 '1 2GDav E Da E d dE Eγ γ γγ π γγ= = ≡= +h D (7). The value of E0 is derived from vGS by taking the limit 2 0lim ( )GS E GDEavv→∞ ==h. Ref [27] reports for D = 46 K × kB, vGS = 1.8 × 104 m s-1 along the (0,0,l) axes, and E0 = 270 K × kB. 13 The integrals in Eq. (3) can be fully evaluated and expre ssed in terms of Ml: () ()#T 2 2 0 2 2 2 01()3 1 1()3 1x MGxB x MM GxBeLE v E d EkT e eEvE d EkT eκ∞ ∞= − = −∫ ∫l lD D (8). In the ballistic regime with a constant Ml: () ()22 #T 2 22 0 0 33 2 2 22 0 03112 1 113 1x MB B x x MB B Mxkk T x eLT x d xDa E e kk T x eTx d xDa E eπ κπ∞ ∞⎛⎞=+ ⎜⎟ − ⎝⎠ ⎛⎞=+ ⎜⎟ − ⎝⎠∫ ∫l h l h (9). From Eqs (5) and (9), a temperature-dependen ce of the LSSE signal intermediate between T2 and T1 is expected, which is shown in Fig. 5 and appears to desc ribe the results on the thin-film sample (sample A) below the maximum quite well. We can now discuss the difference in the temperature dependence of Sy between sample A and sample B in Fig. 5 by considering the length scal e of subthermal magnons in terms of the sample thickness. Based on the suppression of LSSE at high fields, we can st ate that most of the LSSE signal observed at low temperatures comes from magnons at energies below kBTC. We can then extrapolate the mean free path data of Ref [23] in to the range of 10 K to 40 K, and we make the reasonable assumption that these subthermal magnons will have Mlof a few μm or above within this temperature range. Since Sample A is only 4 μm thick, we expect the subthermal magnons in this sample to be in the ballistic regime with a constant Ml = 4 μm, and therefore Eq. (9) to hold at low temperatures. This assumption of 14 ballistic magnon transport in sample A is expected to break down once the temperature becomes high enough that Ml becomes limited by the increased prevalence of magnon-phonon interactions.. In contrast to this situation, the subthermal magnons in the bulk sample B are expected to propagate diffusively at all temperatures due to the much larger YIG thickness (500 μm), since the magnon mean free path observed at 2K in Ref. [23] was only 100 μm. As in sample A, however, the magnon mean free path in sample B is also a f unction of temperature w ith a negative exponent,23 which results in a slower T dependence of the LSSE in sample A, as observed in Fig. 5. We attribute this difference to the fact that there is likely no purely ballistic magnon tr ansport in sample B, and thus inelastic magnon scattering starts to affect the subthermal magnons starting from even the lowest temperatures examined in this study. Furthermore, we can explain the shift in peak temperature between sample A and sample B by considering that magnons in sample B (the bulk crysta l) are sensitive to any mechanism that scatters magnons ov er a length scale shorter than 4 μm, whereas those in sample A (the 4 μm film) are not. In other words, bounda ry scattering persists as more important than diffuse inelastic magnon-phonon or magnon-magnon39 scattering for magnons up to TC ∼ 40 K in sample A, whereas the inelastic processes are already significant above 45 K in sample B. As a result, the maximum in Sy in sample B appears at 70 K, a much lo wer temperature than in sample A ( Tavg = 180 K). It is noted that the Sy of sample A is larger than that of sample B above 80 K and becomes smaller below 80 K because of the faster T dependence. Kehlberger et al.25 demonstrated that the LSSE signal increases as the thickness of YIG incr eases up to 100 nm and st arts to saturate for larger thicknesses at room temperature, while Rezende et al.18 observed no difference in LSSE between 8 μm thick and 1 mm thick YIG samples under the same ∇Tz. Both data are consistent with each other in the sense that 8 μm is likely to already be larger than the mean free path of subthermal magnons at room temperature (~ 100 15 nm), which can be estimated from the data in Ref. [2 3]. The same argument does not apply to our result, however, as the magnitude of Sy is quite different in sample A and sa mple B at room temperature. Indeed, we find that the magnitude of Sy can be a quite complex problem b ecause the magnitude of the voltage induced by the LSSE is sensitive to the Pt/YIG interface condition.40,41,42 Furthermore, because of the different origins of sample A and sample B in our st udy, we cannot exclude the pos sibility that the Pt/YIG interface condition varies between th e two samples, which may at least partially explain the observed difference in the absolute magnitude of Sy. CONCLUSIONS In summary, we present magnetic field and temper ature dependent data of the longitudinal Spin- Seebeck effect on the YIG/Pt system on a 4 μm thick film and on a bulk YIG. From the freeze-out of the LSSE signal in high-field data at different temper atures below 300 K, we conclude that subthermal magnons (defined as those with en ergies below a critical energy kBTC with TC of the order of 40 K) must play a more important role in the LSSE than most magnons present at thermal equilibrium at T > TC. We also attribute the temperature-depe ndence of the LSSE predominantly to that of the thermally-driven magnon flux, for which we develop a new model that takes into account the departure of the magnon dispersion relations from purely para bolic to a relation that more closely resembles that measured by neutron diffraction. Finally, we argue that it is this non-parabolic dispersion that is the reason behind the observed decrease in LSSE signal with decreasing te mperature below the maximum, which deviates from the expected T1 law for parabolic magnons when thei r mean free path is constant. 16 ACKNOWLEDGMENTS The work is supported as part of the ARO MU RI under award number W911N F-14-1-0016, US AFOSR MURI under award number FA9550-10-1-0533 (HJ), NSF grants CBET-1133589 and DMR 1420451, and NSF MRSEC (SB). The thin-film sample was kindl y supplied by Professors E. Saitoh and K. Uchida, Tohoku University, Sendai, Japan, whom we also acknowledge for useful conversations. 17 Figure Captions FIG. 1. (Color online) Experime ntal setup for LSSE measurement. (a) Experimental setup on a liquid helium cryostat used for both sample A and B. (b) Schematic illustration of the measurement geometry for 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. FIG. 2. (Color online) Lo w field LSSE measurement. (a) – (c) Transverse voltage V y as a function of magnetic field Hx (a,b), and ∆Vy versus applied temperature difference ∆Tz (c) for sample A. (d) – (f) Vy as a function of Hx (d,e), and ∆Vy versus ∆Tz (f) for sample B. FIG. 3. (Color online) High field L SSE measurement on sample B. (a), (b) Vy / ∆Tz as a function of Hx measured up to Hx = ± 70 kOe (“high field”) at Tavg = 296 K (a) and at Tavg = 10 K (b). (c), (d) Magnification of low field ( Hx = ± 3 kOe) results at Tavg = 296 K (c) and at Tavg = 10 K (d). FIG. 4. (Color online) (a) T dependence of ∆Sy / Sy,max for sample B. Here, ∆Sy / Sy,max ≡ (Sy,max – Sy,70kOe) / Sy,max with Sy,max and Sy,70kOe defined as in the text. ∆Sy / Sy,max = 1 indicates the fully suppressed Sy at H = ± 70 kOe (i.e. Sy,70kOe = 0). The inset shows the same data extended to 300 K in log scale. Tavg denotes the average sample temperature. (b) Schematic di agram of magnon dispersion of YIG below 0.6 THz (30 K) per Ref [27]. The red arrow indicates opening of a magnon gap equivalent to ~ 9 K when H = ± 70 kOe is applied. The red dashed lines mark the te mperature (~ 15 K) below which significant increase in ∆Sy / Sy,max starts to occur. 18 FIG. 5. (Color online) T dependence of the spin Seebeck coefficient Sy for the 4 μm film sample A (green diamond) and the bulk sample B (orange circle). Th e dashed lines represent a power law for the bulk sample, and values calculated from the model (Eq. 9) in the text for the thin-film sample, renormalized to go through the data. 19 Figure 1 (one column) H∇T Pt GGG substratez y x LxLzLy YIG heater thermometerYIGc-BNa b20 Figure 2 (two columns) -400 -200 0 200 400 Hx (Oe)-3-113Vy (μV) ΔTz = 2.5KTavg = 302.5KΔVy -400 -200 0 200 400 Hx (Oe)-0.3-0.10.10.3Vy (μV) ΔTz = 0.9KTavg = 36K -800 -400 0 400 800 Hx (Oe)-0.8-0.400.40.8Vy (μV) ΔTz = 1.1KTavg = 296.4KΔVy -800 -400 0 400 800 Hx (Oe)-150-5050150Vy (nV) ΔTz = 0.36KTavg = 13.2K01234 ΔTz (K)0246ΔVy (μV)Tavg = 302 ± 1K 00 . 8 1 . 6 ΔTz (K)00.8ΔVy (μV)41 ± 1K 0 0.4 0.8 1.2 ΔTz (K)00.40.81.21.6ΔVy (μV) Tavg = 296.4 K00 . 4 0 . 8 ΔTz (K)00.81.6ΔVy (μV) 33 ± 0.1Kac d fb e21 Figure 3 (one column) -40 0 40 Hx (kOe)-0.800.8Vy / ΔTz (μV K-1) Tavg = 296 K -2 0 2 Hx (kOe)-0.800.8Vy / ΔTz (μV K-1)-40 0 40 Hx (kOe)-0.400.4Vy / ΔTz (μV K-1) Tavg = 10 K -2 0 2 Hx (kOe)-0.400.4Vy / ΔTz (μV K-1)a bc dHigh field High field Low field Low field a cb d22 Figure 4 (one column) a b Excitation frequency (THz) T(K)k 300.6 00 15~ 9 K-π/2 +π/20 1 02 03 04 0 Tavg (K)00.51ΔSy / Sy,max 1 10 100 1000 Tavg (K)00.51ΔSy / Sy,max23 Figure 5 (one column) 24 References 1 K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, E. Saitoh, Nature 455, 778 (2008). 2 S. R. Boona, R. C. Myers, and J. P. Heremans, Energy Environ. Sci. 7, 885 (2014). 3 G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Nature Mater. 11, 391 (2012). 4 C. M. Jaworski, J. Yang, S. Mack, D. D. Awschalom, J. P. Heremans, and R. C. Myers, Nature Mater. 9, 898 (2010). 5 K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takaha shi, J. Ieda, T. Ota, Y. Kajiwara, H. Umezawa, H. Kawai, G.E.W. Bauer, S. Maekawa, and E. Saitoh, Nature Mater. 9, 894 (2010). 6 C. M. Jaworski, R. C. Myers, E. Johnston-Halperin, and J. P. Heremans, Nature 487, 210 (2012). 7 H. Jin, Z. Yang, R. C. Myers, and J. P. Heremans, Solid State Commun. 198, 40 (2014). 8 S. Bosu, Y. Sakuraba, K. Uchida, K. Saito , T. Ota, E. Saitoh, and K. Takanashi, Phys. Rev. B 83, 224401 (2011). 9 K. Uchida, H. Adachi, T. Ota, H. Nakayama, S. Maekawa, and E. Saitoh, Appl. Phys. Lett. 97, 172505 (2010). 10 R. Ramos, T. 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2015-04-03

We study the temperature dependence of the longitudinal spin-Seebeck effect (LSSE) in a yttrium iron garnet Y3Fe5O12 (YIG) / Pt system for samples of different thicknesses. In this system, the thermal spin torque is magnon-driven. The LSSE signal peaks at a specific temperature that depends on the YIG sample thickness. We also observe freeze-out of the LSSE signal at high magnetic fields, which we attribute to the opening of an energy gap in the magnon dispersion. We observe partial freeze-out of the LSSE signal even at room temperature, where kBT is much larger than the gap. This suggests that a subset of the magnon population with an energy below kB x TC (TC about 40 K) contribute disproportionately to the LSSE; at temperatures below TC, we label these magnons subthermal magnons. The T-dependence of the LSSE at temperatures below the maximum is interpreted in terms of a new empirical model that ascribes most of the temperature dependence to that of the thermally driven magnon flux.

The effect of the magnon dispersion on the longitudinal spin Seebeck effect in yttrium iron garnets (YIG)

1504.00895v1

arXiv:1608.01813v2 [cond-mat.other] 28 Aug 2016On supercurrents in Bose-Einstein magnon condensates in a Y IG ferrimagnet Dmytro A. Bozhko,1,2Alexander A. Serga,1Peter Clausen,1Vitaliy I. Vasyuchka,1 Gennadii A. Melkov,3Anna Pomyalov,4Victor S. L’vov,4and Burkard Hillebrands1,∗ 1Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universit¨ at Kaiserslautern, 67663 Kaisersla utern, Germany 2Graduate School Materials Science in Mainz, 67663 Kaisersl autern, Germany 3Faculty of Radiophysics, Electronics and Computer Systems , Taras Shevchenko National University of Kyiv, Kyiv 01601, U kraine 4Department of Chemical Physics, Weizmann Institute of Scie nce, Rehovot 76100, Israel Recently E. Sonin commented[1] on our preprint “Supercurre nt in a room temperature Bose- Einstein magnon condensate” [2, 3], arguing that our “claim of detection of spin supercurrent is premature and has not been sufficiently supported by presente d experimental results and their theo- retical interpretation.” We consider the appearance of thi s Comment as a sign of significant interest into the problem of supercurrents in Bose-Einstein magnon c ondensates. Here, we explicitly address E. Sonin’s comments and show that our interpretation of our e xperimental results as a detection of a magnon supercurrent is fully supported not only by the expe rimental results themselves, but also by independent theoretical analysis [4]. Recently E. Sonin submitted a short Comment on our preprint “ Supercurrent in a room temperature Bose- Einstein magnon condensate ”[2, 3], to arXiv[1], which we have addressed in a previous Note[5]. Here we dis- cuss the four main statements of Ref.[1] in more details. Statement I. “Observation of spin current in Yttrium- Iron-Garnet (YIG) is hardly possible” . This statement is based on two criteria for the existence of a steady supercurrent, formulated in Ref.[6] in terms directly applicable for easy-plane (anti)ferromagnets, where the minimum of the magnon frequency ω(k) corresponds to k= 0 and BEC is noth- ing else but the homogeneous precession of the magne- tization. The situation in tangentially magnetized YIG films is quite different: the dispersion relation ω(k) has two minima at non-zero wavevector values ±k0(with k0≃5·104cm−1) and the BEC spate is a magnon state with a wavelength of λ≃10−4cm, which at least ten times smaller than the hot-spot radius. In this case we have no problem with the anisotropy in the complex planeoftheBECphase(seee.g. Ref.[4])andthemagnon phase variation across the hot spot exceeds 2 πat least ten times. Statement II. “The authors did not consider a more common scenario of spin diffusion, which could probably successfully compete with their present interpretation. I t is worth noting that spin diffusion would be more effective in the condensed state than in a non-condensed gas of magnons” . We present below evidence that spin diffusion cannot be observed in our experiments and theoretical argu- ments that the spin diffusion is much smaller than the spin supercurrent. Experimental arguments. We explained in page 4 of our preprint, that “the laser heating process decreases locally the saturation magnetization via thermal excita-tion of high-energy magnons with terahertz frequencies, but practically does not increase the local population of low-energy magnons. As a result, the number of thermal magnons in the low-energy spectral region remains neg- ligibly small in comparison with the number of magnons originating from the parametric pumping process and cannot visibly affect the BEC dynamics”. Moreover, we observed a critical value of the bottom-magnon populations Ncr(see horizontal dashed line in Fig. 5) that separates two regions with different spin dynamics; in the upper region the only reasonable explanation for the observed spin dynamics is the existance of a supercurrent of magnons within the BEC, because a diffusive spin current is not a threshold effect, and, thus, cannot lead to the appearance of Ncr. Thus, we conclude: The above described experimental facts exclude the possibility of a significant contribution of the spin diffusion current to the observed spin dynamics . Theoretical arguments. To support the above conclu- sion theoretically, we could have chosen to present an analytical estimation of the relative roles of the magnon supercurrent and a normal diffusion current. Such an estimate has been discussed in a recent PRL paper by C.Sun, T. Nattermann and V.Pokrovsky [4] with the following result: “Thus, we expect that, in realistic cir- cumstances, the spin superfluid current (in YIG films) equal to 1022÷1023cm−2s−1is larger than the normal current by (3÷5)decimal orders ”. Due to experimen- tal limitations regarding the apparatus calibration, we cannot give the absolute value of the spin supercurrent, but we believe that for the purpose of our discussion the relative estimate, obtained in Ref.[4], is sufficient. Statement III. ”The authors investigated not station- ary, but time dependent spin currents.” Indeed, our experiments are dealing with time- dependent decay processes with a characteristic decay of time about (3 ÷4)·10−7s. This time is at least 10 times2 longer than the characteristic time of the four-magnon interaction, responsible for the creation of the magnon BEC state. Therefore we believe that our BEC can be considered as being quasi-stationary, at least in first- order approximation. However we agree with E. Sonin that an estimate of the effect of the non-stationarity is important in a consistent theory of the magnon BEC phenomenon. We hope to address this problem during ongoing research. Statement IV. ”The agreement (in our Ref.[2, 3]) looks very good indeed, probably due to using a fitting parame- ter.” We believe that such a good agreement stems from the fact, that our model reflects well the basic physics of the observed phenomena. If there exists an alternative quan- titative theoretical interpretation of the experimental re- sults, we would be happy to study it to compare, which model describes our experimental observations better. Conclusion. We consider E. Sonin’s Comment[1] as a sign of a wide interest to the subject and thank him for the constructive discussions. We believe that in this Note we clarifiedthe differences in the physicalsituations discussed in Ref.[1, 2, 5]. We are confident that our ex- perimental results confirm the existence of a magnon su- percurrent in a room-temperature magnon Bose-Einstein condensate. ∗hilleb@physik.uni-kl.de [1] E.B. Sonin, Comment on supercurrent in a room temperature Bose-Einstein magnon condensate, arXiv:1607.04720. [2] D. A. Bozhko, A. A. Serga, P. Clausen, V. I. Vasyuchka,F. Heussner, G. A. Melkov, A. Pomyalov, V. S. L’vov, and B. Hillebrands, Supercurrent in a room temperature Bose-Einstein magnon condensate, arXiv:1503.00482. [3] D. A. Bozhko, A. A. Serga, P. Clausen, V. I. Vasyuchka, F. Heussner, G. A. Melkov, A. Pomyalov, V. S. L’vov, and B. Hillebrands, Supercurrent in a room temperature Bose-Einstein magnon condensate, Nature Physics, 2016, DOI: 10.1038/NPHYS3838. [4] C. Sun, T. Nattermann, and V. Pokrovsky, Unconven- tional superfluidity in yttrium iron garnet films, Phys. Rev. Lett. 116, 257205 (2016). [5] D. A. Bozhko, A. A. Serga, P. Clausen, V. I. Vasyuchka, F. Heussner, G. A. Melkov, A. Pomyalov, V. S. L’vov, and B. Hillebrands, On supercurrents in Bose-Einstein magnon condensates in YIG ferrimagnet, arXiv:1608.01813v1 [6] E. B. Sonin, Adv. Phys. 59, 181 (2010). [7] R. E. Troncoso and ´A. S. N´ uez, Josephson effects in a Bose-Einstein condensate of magnons. Ann. Phys. 346, 182–194 (2014). [8] K. Nakata, K. A. van Hoogdalem, P. Simon, and D. Loss, Josephson and persistent spin currents in Bose-Einstein condensates ofmagnons. Phys.Rev.B 90, 144419 (2014). [9] K. Nakata, P. Simon, and D. Loss, Magnon transport through microwave pumping. Phys. Rev. B 92, 014422 (2015). [10] P. Nowik-Boltyk, O. Dzyapko, V.E. Demidov, N.G. Berloff, and S. O. Demokritov, Spatially non-uniform ground state and quantized vortices in a two-component Bose-Einstein condensate of magnons. Sci. Rep. 2, 482 (2012). [11] S. Autti et al.Self-trapping of magnon Bose-Einstein condensates in the ground state and on excited levels: from harmonic to boxconfinement. Phys. Rev. Lett. 108, 145303 (2012). [12] F. Li, W.M. Saslow, andV.L.Pokrovsky, Phase diagram for magnon condensate in yttrium iron garnet film. Sci. Rep.3, 1372 (2013).

2016-08-05

Recently E. Sonin commented [1] on our preprint "Supercurrent in a room temperature Bose-Einstein magnon condensate" [2,3], arguing that our "claim of detection of spin supercurrent is premature and has not been sufficiently supported by presented experimental results and their theoretical interpretation." We consider the appearance of this Comment as a sign of significant interest into the problem of supercurrents in Bose-Einstein magnon condensates. Here, we explicitly address E. Sonin's comments and show that our interpretation of our experimental results as a detection of a magnon supercurrent is fully supported not only by the experimental results themselves, but also by independent theoretical analysis [4].

On supercurrents in Bose-Einstein magnon condensates in YIG ferrimagnet

1608.01813v2

arXiv:1610.05760v1 [cond-mat.mtrl-sci] 18 Oct 2016Spin transport in antiferromagnetic NiO and magnetoresist ance in Y3Fe5O12/NiO/Pt structures Yu-Ming Hung,1Christian Hahn,1Houchen Chang,2Mingzhong Wu,2Hendrik Ohldag,3and Andrew D. Kent1 1)Department of Physics, New York University, New York, New Yo rk 10003 2)Department of Physics, Colorado State University, Fort Col lins, Colorado 80523 3)Stanford Synchrotron Radiation Lightsource, Menlo Park, C alifornia 94025 (Dated: 11 August 2018) We have studied spin transport and magnetoresistance in yttrium ir on garnet (YIG)/NiO/Pt trilayers with varied NiO thickness. To characterize the spin transport through NiO we excite ferromagnetic resonance in YIG with a microwave frequency magnetic field and detect the volta ge associated with the inverse spin- Hall effect (ISHE) in the Pt layer. The ISHE signal is found to decay e xponentially with the NiO thickness with a characteristic decay length of 3.9 nm. This is contrasted with t he magnetoresistance in these same structures. The symmetry of the magnetoresistive response is c onsistent with spin-Hall magnetoresistance (SMR). However, in contrast to the ISHE response, as the NiO thic kness increases the SMR signal goes towards zero abruptly at a NiO thickness of ≃4 nm, highlighting the different length scales associated with the spin-transport in NiO and SMR in such trilayers. I. INTRODUCTION Antiferromagnets have attracted a great deal of at- tention in spintronics because of their unique proper- ties, such as a low magnetic susceptibility and ter- ahertz spin dynamics.1,2In metallic antiferromagnet- based spintronics, electrical switching of the antiferro- magnetic domains in CuMnAs was sucesfully performed3 and spin pumping studies have demonstrated spin in- jection into IrMn.4,5In oxide spintronics, ferromag- net (FM)/antiferromagnet (AFM)/heavy metal trilayer structures have been used to study spin-transport in in- sulating antiferromagnets.6–13Experimental techniques include microwave-field-induced magnetization preces- sion in the FM to generate a spin-current in the AFM and the inverse spin Hall effect (ISHE) in the heavy metal to convert the spin-current transmitted through the AFM into a voltage.6–8,14Such experimental studies have shown that NiO can be an efficient spin-conductor. A theoretical model has been proposed that explains these experimental results by spin-currents conducted by evanescent spin-waves in NiO.15,16In similar structures, YIG/NiO/Pt, the spin-Hall magnetoresistance (SMR) has been measured as a function of temperature. Re- sults suggest that NiO can suppress the magnetic prox- imity effect measured in a 3 nm thick Pt layer and were further interpreted to indicate spin-transport between Pt and YIG through NiO layers.14 These two types of experiments have not been con- ducted on the same samples as a function of the NiO thickness. Here, we investigate spin transport through NiO layers with varied thickness at room temperature by exciting ferromagnetic resonance (FMR) in YIG and detecting the voltage signal across the Pt film associated with the ISHE. The SMR as a function of the NiO thick- ness was measured in the same samples. A comparisonof these results shows that the ISHE signal decreasesmono- tonically as a function of NiO thickness, while the SMR vanishes abruptly at a thickness of about 4 nm.II. EXPERIMENTAL METHODS YIG(20 nm)/NiO/Pt(5 nm) trilayers were deposited on gadolinium gallium garnet (Gd 3Ga5O12, GGG) sub- strates. YIG with a /angbracketleft111/angbracketrightorientation was deposited by magnetron sputtering.17,18The YIG film was subse- quently exposed to ambient conditions. Ar+ion clean- ing with a discharge voltage of 800 V was used prior to depositing NiO by radio frequency (rf) magnetron sput- tering. A 5 nm thick Pt layer was deposited in-situ by electron beam evaporation. Samples with 0 (i.e. without NiO), 2, 4, 6, and 10 nm thick NiO layers were prepared and studied. For FMR studies, we mount the sample on a Cu stripline with a width of 500 µm, similar to the experi- mentalgeometryin Ref.19. We applyamicrowavesignal at 3.85 GHz and measure the inverse spin Hall voltage VISHEacross the Pt film as a function of the applied field usingalock-intechnique. Themicrowavesignalisturned on and off at a few kHz and the lock-in response at this frequency is measured. Figure 1(a) shows a schematic of the sample and applied field geometry, while Fig. 1(b) shows a schematic of the magnetoresistance experiments. Magnetoresistance measurements of YIG/NiO/Pt trilay- ers were performed in a 4-wire configuration, i.e. with separate voltage and current contacts in a line. We ap- ply an external magnetic field of 0.2 T to align the mag- netization of YIG with the field. The samples are then rotated around three different axes to obtain the magne- toresistance as a function of angle. All the experiments presented in this paper were conducted at room temper- ature. III. RESULTS First we present measurements of the ISHE voltage in YIG(20 nm)/Pt(5 nm), thin films that do not have a NiO layer. Fig. 2(a) shows the ISHE voltage as a2 (a) (b) YIG NiO Pt V YIG NiO Pt FIG. 1. Schematics of (a) inverse spin Hall voltage measure- ment by microwave-field-induced magnetization precession in YIG(20 nm)/NiO(t nm)/Pt(5 nm) and (b) Pt resistance mea- surement as a function of angle of applied field ( α,β, andγ) with respect to three different axes of the same sample. function of the applied field. A peak in the ISHE voltage signalisseenattheresonancefieldoftheYIG.(TheFMR absorptiondataisnotshown.) Thereisalsoaminorpeak which is most likely due to inhomogneity in our extend film samples. As previously reported in Ref. 19–21, the measured ISHE voltage VISHEis an odd function of the magneticfield whichisasignaturethat the effect isbased on spin pumping into Pt and not a thermoelectric signal. In YIG/NiO/Pt trilayers the VISHEas a function of NiO thickness is shown in Fig. 2(b). We find slightly different resonance fields on different YIG samples, which may be associated with small variation in their magnetization. We checked that the shift in resonance field is symmetric in both field directions indicating that it is not caused by exchange bias introduced in the YIG by NiO. We extract the peak values of VISHEand plot it as a function of NiO thickness in Fig. 4. The VISHEsignal in YIG/NiO/Pt indicates that exciting FMR in YIG is able to produce spin-currents in NiO and spin-injection into Pt. We now discuss the magnetoresistance measurements of YIG/Pt and YIG/NiO/Pt. Magnetoresistance as a function of angle of applied field with respect to three different axes of the sample were measured. In these experiments, the applied field µ0H= 0.2 T is fixed. Fig- ure3 showsthe magnetoresistanceofYIG(20 nm)/NiO(2 nm)/Pt(5 nm) as a function of angle α(rotation in the x-y plane) in Fig. 3(a), β(rotation in the x-z plane) in Fig. 3(b), and γ(rotation in the y-z plane) in Fig. 3(c). The insets in Fig. 3 show the experimental geometry and definitions of angles, α,β, andγ. We measured the magnetoresistance with a resolution of∼10−6in all three geometries. If we assume SMR and anisotropic magnetoresistance (AMR) as possible contri- butions to the measured magnetoresistance, the angle dependent measurements in Fig. 3(a), 3(b), and 3(c)69 71 73 75 77 -4 -3 -2 -1 0 µ0H (mT) V ( µV) 0 nm 2 nm 4 nm 6 nm 10 nm (b) t -76 -73 -70 -4 -2 024 µ0H (mT) V ( µV) 70 73 76 µ0H (mT) (a) FIG. 2. Inverse spin Hall voltage VISHEmeasured as a func- tion of external field in (a) YIG(20 nm)/Pt(5 nm) and (b) YIG(20 nm)/NiO(t nm)/Pt(5 nm) with a 3.85 GHz rf exci- tation. should represent contributions of SMR and AMR, SMR only, and AMR only, respectively. The signal observed in Fig. 3(c) is of the order of the measurement noise. Given the measurement noise is ∼10−6, the peak value of the signal is at least a factor of 10 smaller than the SMR signal. This is consistent with the results reported in Ref. 14, where it was concluded based on no AMR response that there are no induced magnetic moments in Pt with a NiO interlayer. However, we also find neg- ligible signal when rotating in the y-z plane to sweep the angle γfor the YIG(20 nm)/Pt(5 nm) sample, which does not have the NiO interlayer. The AMR signal in- dicative of magnetic proximity effect found in Ref. 14 was measured in a sample with 3 nm Pt thickness as opposed to the thicker films used here and in Refs. 19, 22, where also no AMR was reported. If magnetic prox- imity effect is present in our samples, its contribution to the magnetoresistance is negligible compared to the contribution of SMR. Therefore, the resistance varia- tions in Figs. 3(a) and 3(b) can be attributed to the SMR, which describes how the Pt resistance reflects the itinerant electron-spin interactions at the NiO/Pt inter- face. The similar magnitude (∆ RMax/R0≃4×10−5) in Figs. 3(a) and 3(b) also suggest we only have a SMR signal. We observe a periodic response with a period of 180 degrees in Figs. 3(a)(b) which can be described withR−R0= ∆R= ∆RMaxsin2θ.θis the angle be- tween the magnetization Min YIG and the spin polar- ization from the spin Hall effect in Pt. We compute the equilibrium magnetization direction as a function of the applied field direction, αandβ, based on a macrospin model considering the demagnetization field and exter- nal field. The results are α=θin Fig. 3(a) and β=θ+ arcsin(sin(2 θ)Ms/2H) in Fig. 3(b). The sharp3 012345 012345ΔR/R 0 ( ×10 -5 ) -180 -90 0 90 180 012345 α,β,γ (deg.) (a) (c) /g2185 /g2868 (b) /g2868 /g2185 /g2868 /g2185 FIG. 3. Magnetoresistance of YIG(20 nm)/NiO(2 nm)/Pt(5 nm) as a function of the field angle rotated in the (a) x-y plane an angle α, (b) x-z plane an angle β, and (c) y-z plane an angle γ. The dashed line is the expected SMR response based on a macrospin model. peaks in the SMR signal at = +/-90 degree seen in Fig. 3(b) are due to the difference between the angles of the applied field ( β) and the YIG magnetization ( θ). The magnetization of YIG was determined by ferromagnetic resonance characterization to be µ0Ms= 0.176 T. As seen in the formofdashedlines in Fig. 3, plots of∆ R/R0 versus field angle ( α,β, andγ) fit the experimental data well. The results in Fig. 3 show that the direction of YIG magnetization can affect the resistance measured in Pt. This suggests that the YIG magnetization rotates the spins in the NiO which changes the spin-scattering and accumulation at the NiO/Pt interface and thus the re- sistance of the Pt film. We repeat the same experiment for YIG/Pt and YIG/NiO/Pt with t = 4 and 6 nm to extract the maximum value ∆ RMax/R0in Fig. 3(b) and plot it as a function of NiO thickness in Fig. 4. We now compare the two experiments as a func- tion of NiO thickness. We show the ISHE voltage of YIG/NiO/Pt as a function of NiO thickness and plot it using black circles on the left hand axis in Fig. 4. We fit theVISHEdata points with Ae−t/λand obtain a decay length of 3.9 nm. The blue squares plotted with refer- ence to the right axis in Fig. 4 represent the SMR signal of YIG/NiO/Pt as a function of NiO thickness. We see fromthe comparisoninFig. 4howboththeISHEvoltage and SMR decrease monotonically with NiO thickness. In contrast to the ISHE signal, the SMR shows an abrupt decrease at a thickness of 4 nm. This points to a dif- ference in how the critical length scales are determined in the two effects. In both cases spin flip-scattering of electrons in Pt on the NiO interface is involved. How- ever,theexponentialdecreaseassociatedwithdiffusionof0246810 012345 NiO thickness (nm) Max V( µV) 012345 ΔR/R 0(×10 -5 )VISHE VISHE =Ae (-t/λ) SMR FIG. 4. Black circles are the peak values of the ISHE voltage extracted from Fig. 2(b), while blue squares are the maxi- mum values of SMR ∆ RMax/R0extracted from Fig. 3(b) for different NiO thicknesses. Black dashed line is a fit to A e−t/λ. Blue dashed line is only a guide to the eyes. magnons in NiO15,16is not reproduced in the SMR data. On the contrary the insertion of a 2 nm NiO layer barely reduces the SMR amplitude. We can thus speculate that the source of the SMR, collinearity or non-collinearity between magnetic moments at the Pt/NiO interface and the electron spins, is abrubtly decoupled from the YIG magnetization above a certain NiO thickness. IV. CONCLUSION Inconclusion,wehavereportedISHEvoltageandSMR measurements in YIG/NiO/Pt layered structures. From the dependence on the NiO thickness, we found a mono- tonic decreaseof the VISHEand SMR signal with the NiO thickness. The sharp decrease of the SMR signal at t = 4 nm and the exponential decrease of the VISHEsignal suggests that the length scales of the phenomena are dis- tinct. The abrupt decrease of the SMR signal may reflect the blocking of the NiO moments but this remains to be tested. ACKNOWLEDGMENTS The growth of the YIG films at Colorado State Uni- versity was supported by the SHINES, an Energy Fron- tier Research Center funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Award SC0012670. The research at NYU was sponsored bytheInstituteforNanoelectronicsDiscoveryandExplo- ration (INDEX), a funded center of Nanoelectronics Re- search Initiative (NRI), a Semiconductor Research Cor- poration (SRC) program sponsored by NERC and NIST. 1T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich, Nature Nanotechnology 11, 231 (2016). 2V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono, and Y. Tserkovnyak, arXiv:1606.04284 (2016).4 3P. Wadley, B. Howells, J. elezn, C. Andrews, V. Hills, R. P. Campion, V. Novk, K. Olejnk, F. Maccherozzi, S. S. Dhesi, S. Y . Martin, T. Wagner, J. Wunderlich, F. Freimuth, Y. Mokrousov , J. Kune, J. S. Chauhan, M. J. Grzybowski, A. W. Rush- forth, K. W. Edmonds, B. L. Gallagher, and T. Jungwirth, Science351, 587 (2016). 4L. Frangou, S. Oyarzn, S. Auffret, L. Vila, S. Gambarelli, and V. Baltz, Physical Review Letters 116, 077203 (2016). 5W. Zhang, M. B. Jungfleisch, W. Jiang, J. E. Pear- son, A. Hoffmann, F. Freimuth, and Y. Mokrousov, Physical Review Letters 113, 196602 (2014). 6C. Hahn, G. d. Loubens, V. V. Naletov, J. B. Youssef, O. Klein, and M. Viret, Europhysics Letters 108, 57005 (2014). 7H. Wang, C. Du, P. C. Hammel, and F. Yang, Physical Review Letters 113, 097202 (2014). 8Z. Qiu, J. Li, D. Hou, E. Arenholz, A. T. N’Diaye, A. Tan, K.-i. Uchida, K. Sato, S. Okamoto, Y. Tserkovnyak, Z. Q. Qiu, and E. Saitoh, Nature Communication 7, 12670 (2016). 9T. Moriyama, S. Takei, M. Nagata, Y. Yoshimura, N. Mat- suzaki, T. Terashima, Y. Tserkovnyak, and T. Ono, Applied Physics Letters 106, 162406 (2015). 10K. Chen, W. Lin, C. L. Chien, and S. Zhang, Physical Review B 94, 054413 (2016). 11R. Cheng, D. Xiao, and A. Brataas, Physical Review Letters 116, 207603 (2016). 12R. Cheng, J. Xiao, Q. Niu, and A. Brataas, Physical Review Letters 113, 057601 (2014).13K. M. D. Hals, Y. Tserkovnyak, and A. Brataas, Physical Review Letters 106, 107206 (2011). 14T. Shang, Q. F. Zhan, H. L. Yang, Z. H. Zuo, Y. L. Xie, L. P. Liu, S. L. Zhang, Y. Zhang, H. H. Li, B. M. Wang, Y. H. Wu, S. Zhang, and R.-W. Li, Applied Physics Letters 109, 032410 (2016). 15R. Khymyn, I. Lisenkov, V. S. Tiberkevich, A. N. Slavin, and B. A. Ivanov, Physical Review B 93, 224421 (2016). 16S. M. Rezende, R. L. Rodrguez-Surez, and A. Azevedo, Physical Review B 93, 054412 (2016). 17H. Chang, P. Li, W. Zhang, T. Liu, A. Hoffmann, L. Deng, and M. Wu, IEEE Magnetics Letters 5, 1 (2014). 18E. L. Jakubisova, S. Visnovsky, H. Chang, and M. Wu, Applied Physics Letters 108, 082403 (2016). 19C. Hahn, G. de Loubens, O. Klein, M. Viret, V. V. Naletov, and J. Ben Youssef, Physical Review B 87, 174417 (2013). 20O. Mosendz, V. Vlaminck, J. E. Pearson, F. Y. Fradin, G. E. W. Bauer, S. D. Bader, and A. Hoffmann, Physical Review B 82, 214403 (2010). 21Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi, S. Maekawa, and E. Saitoh, Nature 464, 262 (2010). 22H. Nakayama, M. Althammer, Y. T. Chen, K. Uchida, Y. Kaji- wara, D. Kikuchi, T. Ohtani, S. Geprgs, M. Opel, S. Takahashi , R. Gross, G. E. W. Bauer, S. T. B. Goennenwein, and E. Saitoh, Physical Review Letters 110, 206601 (2013).

2016-10-18

We have studied spin transport and magnetoresistance in yttrium iron garnet (YIG)/NiO/Pt trilayers with varied NiO thickness. To characterize the spin transport through NiO we excite ferromagnetic resonance in YIG with a microwave frequency magnetic field and detect the voltage associated with the inverse spin-Hall effect (ISHE) in the Pt layer. The ISHE signal is found to decay exponentially with the NiO thickness with a characteristic decay length of 3.9 nm. This is contrasted with the magnetoresistance in these same structures. The symmetry of the magnetoresistive response is consistent with spin-Hall magnetoresistance (SMR). However, in contrast to the ISHE response, as the NiO thickness increases the SMR signal goes towards zero abruptly at a NiO thickness of $\simeq$ 4 nm, highlighting the different length scales associated with the spin-transport in NiO and SMR in such trilayers.

Spin transport in antiferromagnetic NiO and magnetoresistance in Y$_3$Fe$_5$O$_{12}$/NiO/Pt structures

1610.05760v1

Axion search with a quantum-limited ferromagnetic haloscope N. Crescini,1, 2,D. Alesini,3C. Braggio,2, 4G. Carugno,2, 4D. D'Agostino,5 D. Di Gioacchino,3P. Falferi,6U. Gambardella,5C. Gatti,3G. Iannone,5 C. Ligi,3A. Lombardi,1A. Ortolan,1R. Pengo,1G. Ruoso,1,yand L. Taarello7 (QUAX Collaboration) 1INFN-Laboratori Nazionali di Legnaro, Viale dell'Universit a 2, 35020 Legnaro (PD), Italy 2Dipartimento di Fisica e Astronomia, Via Marzolo 8, 35131 Padova, Italy 3INFN-Laboratori Nazionali di Frascati, Via Enrico Fermi 40, 00044 Roma, Italy 4INFN-Sezione di Padova, Via Marzolo 8, 35131 Padova, Italy 5INFN-Sezione di Napoli, Via Cinthia, 80126 Napoli, Italy and Dipartimento di Fisica, Via Giovanni Paolo II 132, 84084 Fisciano (SA), Italy 6IFN-CNR, Fondazione Bruno Kessler, and INFN-TIFPA, Via alla Cascata 56, 38123 Povo (TN), Italy 7INFN-Sezione di Padova, Via Marzolo 8, , 35131 Padova, Italy (Dated: January 27, 2020) A ferromagnetic axion haloscope searches for Dark Matter in the form of axions by exploiting their interaction with electronic spins. It is composed of an axion-to-electromagnetic eld transducer coupled to a sensitive rf detector. The former is a photon-magnon hybrid system, and the latter is based on a quantum-limited Josephson parametric amplier. The hybrid system consists of ten 2.1 mm diameter YIG spheres coupled to a single microwave cavity mode by means of a static magnetic eld. Our setup is the most sensitive rf spin-magnetometer ever realized. The minimum detectable eld is 5 :51019T with 9 h integration time, corresponding to a limit on the axion- electron coupling constant gaee1:71011at 95% CL. The scientic run of our haloscope resulted in the best limit on DM-axions to electron coupling constant in a frequency span of about 120 MHz, corresponding to the axion mass range 42 :4-43:1eV. This is also the rst apparatus to perform an axion mass scanning by changing the static magnetic eld. The axion is a beyond the Standard Model (BSM) hy- pothetical particle, rst introduced in the seventies as a consequence of the strong CP problem of quantum chro- modynamics (QCD) [1{3]. Present experimental eorts are directed towards \invisible" axions, described by the KSVZ [4, 5] and DFSZ [6, 7] models, which are extremely light and weakly coupled to the Standard Model parti- cles. Axions can be produced in the early Universe by dierent mechanisms [8{11], and may be the main con- stituents of galactic Dark Matter (DM) halos. Astro- physical and cosmological constraints [12, 13], as well as lattice QCD calculations of the DM density [14, 15], pro- vide a preferred axion mass window around tens of eV. Non-baryonic DM is where cosmology meets particle physics, and axions are among the most interesting and challenging BSM particles to detect. Their experimental search can be carried out with Earth-based instruments immersed in the Milky Way's halo, which are therefore called \haloscopes" [16]. Nowadays, haloscopes rely on the inverse Primako eect to detect axion-induced ex- cess photons inside a microwave cavity in a static mag- netic eld. Primako haloscopes allowed to exclude ax- ions with masses mabetween 1.91 and 3.69 eV [17{19], and, together with helioscopes [20], are the only exper- iments which reached the QCD-axion parameter space. The last years saw a ourishing of new ideas to search for axions and axion-like-particles (ALPs) [21{33]. Among these, the QUAX experiment [34, 35] searches for DM ax- ions through their coupling with the spin of the electron.This experiment aims to implement the idea of Ref. [36] as follows. The axion-electron interaction is described by the La- grangian Lae=gaee 2me@a e  5 e
; (1) wheregaeeis the axion-electron interaction constant, ais the axion eld, eandmeare the electron wavefunction and mass, and and 5are Dirac matrices. This vertex describes an axion-induced ip of an electron spin, which then decays back to the ground state emitting a photon. Sinceva, the relative speed between Earth and the DM halo, is small, we may use the non-relativistic limit of Euler-Lagrange equations and recast the interaction term Lae'2B
gaee 2e ra2B
Ba: (2) Here2Bandeare the spin and charge of the elec- tron,Bis the Bohr magneton, and Bais dened as the axion eective magnetic eld. As ra/va[36], the non-zero value of varesults in Ba6= 0. If accounting for the whole DM, the numeric axion den- sity isna'81012(42eV=ma) cm3. Forva'103c, wherecis the speed of light, the de Broglie wave- length and coherence time of the galactic axion eld are ra= 25 (42eV=ma) m, andra= 85 (ma=42eV)s [34, 35]. The eective eld frequency is proportional to the axion mass, !a=2= 10 (ma=42eV) GHz, while itsarXiv:2001.08940v1 [hep-ex] 24 Jan 20202 amplitude depends on the properties of the DM halo and of the axion model, Ba=gaee 2er na~ macmava'41023ma 42eV T;(3) where~is the reduced Planck constant. These features allows for the driving of a coherent interaction between Baand the homogeneous magnetization of a macroscopic sample. The sample is immersed in a static magnetic eldB0to couple the axion eld to the Kittel mode of uniform precession of the magnetization. The interaction yields a conversion rate of axions to magnons which can be measured by searching for oscillations in the sample's magnetization. Due to the angle between B0andBa, the resulting signal undergoes a full daily modulation [37]. The maximum axion-deposited power is related to Eq. (3) and to the characteristics of the receiver, namely number of spinsNsand system relaxation time s Pa= eBNs!aB2 as; (4) where eis the electron gyromagnetic ratio. To detect this signal we devised a suitable receiver. As it measures the magnetization of a sample, it is cong- ured as a spin-magnetometer used as an axion haloscope. The device consists of an axion eld transducer and of an rf detection chain. At high frequencies and in free eld, the electron spin resonance linewidth is dominated by radiation damping, which limits s[38{40]. To avoid this issue, the mate- rial is placed in a microwave cavity. If the frequency of the Kittel mode !m= eB0is close to the cavity mode frequency !c, the two resonances hybridize and the single mode splits into two, following an anticrossing curve [41, 42]. The B0-dependent hybrid modes frequen- cies are!1and!2and the cavity-material coupling is gcm= min(!2!1). Ifgcmis larger than the hybrid mode linewidths 1;2'( c+ m)=2, where cand m are the cavity and material dissipations, the system is in the strong coupling regime. To increase Pa,Nsand smust be large, so a suitable sample has a high spin density and a narrow linewidth. The best material iden- tied so far is Yttrium Iron Garnet (YIG), with roughly 21022spins=cm3and 1 MHz linewidth [43]. In the apparatus that we operated at the Labora- tori Nazionali di Legnaro of INFN, the TM110 mode of a cylindrical copper cavity is coupled to ten 2.1 mm- diameter spheres of YIG. The spherical shape is needed to avoid geometrical demagnetization. We devised an on- site grinding and polishing procedure to obtain narrow linewidth spherical samples starting from large single- crystals of YIG. The spheres are placed on the axis of the cavity, where the rf magnetic eld is uniform. Several room temperature tests were performed to de- sign the YIG holder: a 4 mm inner diameter fused silica pipe, containing 10 stacked PTFE cups, each one largeenough to host a free rotating YIG. Free rotation permits the spheres' easy axis self-alignment to the external mag- netic eld, while a separation of 3 mm prevents sphere- sphere interaction. The pipe is lled with 1 bar of helium and anchored to the cavity for thermalization. The cavity and pipe are placed inside the internal vacuum chamber (IVC) of a dilution refrigerator, with a base temperature around 90 mK. Outside the IVC, in a liquid helium bath, a superconducting magnet provides the static eld with an inhomogeneity below 7 ppm over all the spheres. FIG. 1: Measured (left) and modeled (right) transmis- sion functions of the HS. The right plot is the function fcdmn(!;! m), based on the second quantization of coupled harmonic oscillators, while the left one is a SO-to-Readout (see Fig. 2) transmission measurement with the JPA o, per- formed at 90 mK. Color scales are in arbitrary units (brighter colors corresponds to higher amplitudes). The dashed line in the left plot identies the hybrid mode frequencies !1, where we performed measurements. The resulting hybrid system (HS) has been studied by collecting a B0vs frequency transmission plot, re- ported in Fig. 1 (left). The measured plot is not a usual anticrossing curve. In our system the cavity frequency !c=2= 10:7 GHz and the expected coupling is of the order of 600 MHz, thus !2gets close and couples to a higher order mode of the cavity. This hybrid mode fur- ther splits into others, making the two oscillators de- scription unsuitable. Other disturbances are related to residual sphere-sphere interaction and to non-identical spheres. To model the HS, we write an hamiltonian based on two cavity modes, candd, and two magnetic modes, mandn Hcdmn =0 BB@!ci c 20gcmgcn 0!di d 2gdmgdn gcmgdm!mi m 2gmn gcngdngmn!ni n 21 CCA; (5) whereg,!and indicate their couplings, resonant fre- quencies and dissipations, respectively. Fig. 1 (right) shows the function fcdmn(!;!m) = det !I4Hcdmn
, whose maxima identify the resonance frequencies of the3 HS. By comparing the two plots of Fig. 1, one can see that the model appropriately describes the system, allowing us to extract the linewidths, frequencies and couplings of the modes through a t. The typical measured values are 1'1:9 MHz and gcm'638 MHz, yielding s'84 ns andNs'1:01021spins, respectively. Remarkably, the mode!1is not altered by other modes, thus we will use it to search for axion-induced signals. For a xed B0the linewidth of the hybrid mode is the haloscope sensitive band. By changing B0, we can perform a frequency scan along the dashed line of Fig. 1. Magnetmw-cavityYIGD1 D2-20SO -20 -10 -30Aux A1A2Readout-20 -30Pump JPAsp i90 mK4 K TT Calibration FIG. 2: Schematics of the apparatus. The cavity is reported in orange, the ten YIG spheres are in black, and the blue shaded region is permeated by a uniform magnetic eld. The cryogenic and room temperature HEMT ampliers are A1 and A2, respectively, and the JPA ports are the signal (s), idler (i) and pump (p). Superconducting cables are brown, the red- circledTs are the thermometers, SO is a source oscillator, and attenuators are shown with their reduction factor in dB. As inset, we show the calibration of the system gain and noise temperature, obtained by injecting signals in the SO line. The power injected in the HS is given in terms of an eective temperature proportional to Acal. The errors are within the symbol dimension. See text for further details. The electronic schematics, shown in Fig. 2, consists in four rf lines used to characterize, calibrate and operate the haloscope. The HS output power is collected by a dipole antenna (D1), connected to a manipulator by a thin steel wire and a system of pulleys to change its cou- pling. The source oscillator (SO) line is connected to aweakly coupled antenna (D2) and used to inject signals into the HS, the Pump line goes to a Josephson paramet- ric amplier (JPA), the Readout line amplies the power collected by D1, and Aux is an auxiliary line. The Read- out line is connected to an heterodyne as described in [35], where an ADC samples the down-converted power which is then stored for analysis. The JPA is a quan- tum limited amplier, with resonance frequency of about 10 GHz resulting in a noise temperature of 0.5 K. Its gain is close to 20 dB in a band of order 10 MHz, and its work- ing frequency can be tuned thanks to a small supercon- ducting coil [44]. Excluding some mode crossings, hybrid mode and JPA frequencies overlap between 10.2 GHz and 10.4 GHz, and allow us to scan the corresponding axion mass range. The procedure to calibrate all the lines of the setup is: (i) the transmittivity of the Aux-Readout path KARis measured by decoupling D1 or by detuning !1; (ii) for the Aux-SO and SO-Readout paths, KASandKSRare obtained by critically coupling D1 to the mode !1. The transmittivity of the SO line is KSO'p KSRKAS=KAR. If a signal of power Ainis injected in the SO line, the fraction of this power getting into the HS results Acal=AinKSO. SinceAcalis a calibrated signal, it can be used to measure gain and noise temperature of the Readout line. From this measurement we obtain a sys- tem noise temperature Tn= 1:0 K, and a gain of 70.4 dB from D1 to Readout (see Fig. 2). In our setup, the cou- pling of D1 can be varied of 8 dB, thus we estimate a calibration uncertainty of 16%. We measured the JPA gain, the HEMTs noise temperature, and the cavity tem- perature to get the noise budget detailed in Tab. I. The 0.12 K dierence may be due to unaccounted losses, or non-precise temperature control. Source Estimated Quantum noise 0.50 K Thermal noise 0.12 K HEMTs noise 0.25 K Expected total 0.87 K Measured total Tn0.99 K TABLE I: Noise budget of the apparatus. The measured noise is compatible with the estimated one. To double check the accuracy of the result, we measure the thermal noise of the HS. The noise dierence for !1 on and o the JPA resonance (dark blue and light blue) gives the noise added by the hybrid mode (orange curve), as shown in Fig. 3. The excess noise is compatible with a temperature of the HS 10 mK higher than the one of the nearest load, which is realistic. Similar results are obtained by changing the D1 antenna coupling for a xed B0. The axion search consisted in fty-six runs, each one4 FIG. 3: Thermal noise of the HS. The blue curves are the power measured at the Readout with !1in the JPA band- width (dark blue) and out of it (light blue). The dierence between the two is the HS noise (reported in orange). with xedB0. For every run a transmission measurement of the hybrid system is used to set !1, to critically couple D1 to it, and to measure 1. The frequency stability of !1resulted well below the linewidth within an interval of several hours, allowing long integration times. Data are stored with the ADC over a 2 MHz band around !1 for subsequent analysis. We FFT the data with a 100 Hz resolution bandwidth to identify and remove biased bins and disturbances in the down-converted spectra. To esti- mate the sensitivity to the axion eld, we rebin the FFTs with a resolution RBW '5 kHz, which at this frequency gives the best SNR for the axionic signal [36]. The spec- tra are tted to a degree ve polynomial to extract the residuals, whose standard deviation is the sensitivity of the apparatus. We veried that the analysis procedure excludes unwanted bins while preserving the signal and SNR by adding a fake axion signal to real data. Our data were collected in July 2019 in a total run time of 74 h. The average run length was 1 h, and each one was performed during the maximum of the daily- modulated axionic signal. The measured uctuations are compatible with the estimated noise in every run, and we detected no statistically signicant signal consis- tent with the DM axion eld. The minimum measured uctuation is P= 5:11024W, for the longest in- tegration time t= 9 h, where the Dicke prediction is kBTnp RBW=t= 4:81024W. In terms of rf mag- netic eld, this result corresponds to B= 5:51019T, which, to our knowledge, is a record one for an rf spin- magnetometer. The absence of fast rf bursts in the data is veried by using a 1 ms time resolution waterfall spec- trogram. Even if the minimum eld detectable by the haloscope is much larger than Ba, these measurements can still be a probe for ALPs, which may also constitute the totality of DM [46]. The 95% CL upper limit on the axion electroncoupling constant is gaee<e mavas kac2P 2B enaNss'1:71011:(6) The transduction coecient of the axionic signal kacwas calculated with a model similar to the one of Eq. (5) [47]. It essentially depends on !1and, in our bandwidth, re- sults 0:5<kac<1:0. The overall exclusion plot obtained with the ferromagnetic haloscope is given in Fig. 4. All the experimental parameters used to extract the limits from Eq. (6) are measured within every run, making the measurement highly self-consistent. These results improved the best previous limits [35] by roughly a factor 30 in gaeeand 50 in bandwidth. The improvement over the previous prototype is due to an in- creased material volume, to an almost quantum-limited noise temperature, and to longer integration times. No axion-mass scan was performed by previous experiments of this kind, and we now demonstrate that it is feasi- ble to tune a hybrid resonance over hundreds of MHz to search for axion-deposited power. Our prototype scanned a range of axion masses of about 0.7 eV with a eld vari- ation of 7 mT, drastically simplifying the tuning of the haloscope. In conclusion, we designed and developed a quantum- limited rf spin-magnetometer used as an axion haloscope. The instrument implements an axion-to-rf transducer, i. e. an hybrid system which embeds one of the largest quantity of magnetic material to date, and a detection electronics based on a quantum-limited JPA. The oper- ation of this instrument led to an axion search over a span of 0:7eV around 42 :7eV, with a maximum sen- sitivity togaeeof 1:71011. This, to our knowledge, is the best reported limit on the coupling of DM axions to electrons, and corresponds to a 1- eld sensitivity of 5:51019T, which is a record one. No showstoppers have been found so far, and hence a further upscale of the system can be foreseen. A superconducting cavity with a higher quality factor was already developed and tested [48]. It was not employed in this work since the YIG linewidth does not match the superconducting cav- ity one, and the improvement on the setup would have been negligible. With this prototype we reached the rf sensitivity limit of linear ampliers [49]. To further im- prove the present setup one needs to rely on bolometers or single photon/magnon counters [50]. Such devices are currently being studied by a number of groups, as they nd important applications in the eld of quantum infor- mation [51{54]. We are grateful to E. Berto, A. Benato and M. Rebes- chini, who did the mechanical work, F. Calaon and M. Tessaro who helped with the electronics and cryogenics, and to F. Stivanello for the chemical treatments. We thank G. Galet and L. Castellani for the development of the magnet power supply, and M. Zago who realized the5 FIG. 4: Exclusion plot at 95% CL on the axion-electron coupling obtained with the present prototype (excluded region reported in blue and error in light blue), and overview of other searches for the axion-electron interaction. The other results are from [35] (orange) and [45] (green), while the DFSZ axion line is at about gaee'1015. The inset is a detailed view of the reported result. technical drawings of the system. We deeply acknowledge the Cryogenic Service of the Laboratori Nazionali di Leg- naro, for providing us large quantities of liquid helium on demand. Electronic address: nicolo.crescini@phd.unipd.it yElectronic address: ruoso@lnl.infn.it [1] R. D. Peccei and Helen R. Quinn. CP conservation in the presence of pseudoparticles. Phys. Rev. Lett. , 38:1440{ 1443, Jun 1977. [2] Steven Weinberg. A new light boson? Phys. Rev. Lett. , 40(4):223, 1978. [3] Frank Wilczek. Problem of strong p and t invariance in the presence of instantons. Phys. Rev. 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2020-01-24

A ferromagnetic axion haloscope searches for Dark Matter in the form of axions by exploiting their interaction with electronic spins. It is composed of an axion-to-electromagnetic field transducer coupled to a sensitive rf detector. The former is a photon-magnon hybrid system, and the latter is based on a quantum-limited Josephson parametric amplifier. The hybrid system consists of ten 2.1 mm diameter YIG spheres coupled to a single microwave cavity mode by means of a static magnetic field. Our setup is the most sensitive rf spin-magnetometer ever realized. The minimum detectable field is $5.5\times10^{-19}\,$T with 9 h integration time, corresponding to a limit on the axion-electron coupling constant $g_{aee}\le1.7\times10^{-11}$ at 95% CL. The scientific run of our haloscope resulted in the best limit on DM-axions to electron coupling constant in a frequency span of about 120 MHz, corresponding to the axion mass range $42.4$-$43.1\,\mu$eV. This is also the first apparatus to perform an axion mass scanning by changing the static magnetic field.

Axion search with a quantum-limited ferromagnetic haloscope

2001.08940v1

Optical cooling of magnons Sanchar Sharma,1Yaroslav M. Blanter,1and Gerrit E. W. Bauer2, 1 1Kavli Institute of NanoScience, Delft University of Technology, 2628 CJ Delft, The Netherlands 2Institute for Materials Research & WPI-AIMR & CSRN, Tohoku University, Sendai 980-8577, Japan (Dated: April 10, 2018) Inelastic scattering of light by spin waves generates an energy ow between the light and magne- tization elds, a process that can be enhanced and controlled by concentrating the light in magneto- optical resonators. Here, we model the cooling of a sphere made of a magnetic insulator, such as yttrium iron garnet (YIG), using a monochromatic laser source. When the magnon lifetimes are much larger than the optical ones, we can treat the latter as a Markovian bath for magnons. The steady-state magnons are canonically distributed with a temperature that is controlled by the light intensity. We predict that such a cooling process can signicantly reduce the temperature of the magnetic order within current technology. A great achievement of modern physics is the Doppler cooling of trapped atoms by optical lasers [1, 2] down to temperatures of micro-Kelvin [3]. Subsequently, even macroscopic mechanical objects, such as membranes and cantilevers, have been cooled to their quantum me- chanical ground state [4{8] by blue shifting the stim- ulated emission using an optical cavity [4, 5]. `Cav- ity optomechanics' is a vibrant eld that achieved suc- cessful Heisenberg uncertainty-limited mechanical mea- surements, the generation of entangled light-mechanical states, and ultra-sensitive gravitational wave detection [8]. An optical cryocooler based on solid state samples [9] can be superior due to its compactness and lack of mov- ing components [10]. Optical cooling has been demon- strated for glass [9, 11] and envisioned for semiconductors [10, 12, 13]. An analogous cooling of a magnet would generate in- teresting opportunities. Magnetization couples to mi- crowaves [14{18], electric currents [18{20], mechanical motion [17, 21{23], and, indeed, light [24]. Spin waves are the elementary excitations of the ferromagnetic or- der, which are quantized as bosonic magnons. Similar to phonons, magnons may be considered non-interacting up to relatively high temperatures and are Planck- distributed at thermal equilibrium. However, there are important dierences as well: Magnons have mass and chirality [25, 26], both of which are tunable by an ex- ternal static magnetic eld. Their long wavelength dis- persion in thin lms is highly anisotropic, with minima in certain directions that can collect the Bose-Einstein condensate of magnons [27{29]. Magnons can be used as quantum transducers between microwaves and optical light [30] or between superconducting and ying qubits [31]. Motivated by the potential of a ferromagnet as a ver- satile quantum interface at low temperatures, we discuss here the potential of optical cooling of magnons. The magnon-photon interaction gives rise to inelastic Bril- louin light scattering (BLS) [32], which is a well estab- lished tool to study magnon dispersion and dynamics [24, 33, 34]. Recently, several groups carried out BLSexperiments on spheres made of ferrimagnetic insulator yttrium iron garnet (YIG) [35{40], which has a very high magnetic quality factor 105
[41{43] and sup- ports ferromagnetic-like magnons with long coherence times (s) [31, 44, 45]. YIG spheres are commercially available for microwave applications, but are also good infrared light cavities due to their large refractive index and high optical quality [46, 47], making them good op- tomagnonic resonators [35{39, 48{51]. Via proximity op- tical bers or prisms, external laser light can eciently excite `whispering gallery modes' (WGMs), i.e. the op- tical modes circulating in extremal orbits of dielectric spheroids [52, 53]. The BLS experiments on YIG spheres discovered a large asymmetry in the red- (Stokes) and blue-shifted (anti-Stokes) sidebands [36{39] due to selective resonant enhancement of the scattering cross section [38, 50, 51]. The asymmetry can be controlled by the polarization and wave vector of the light. When more photons are scattered into the blue than the red-shifted sidebands, light eectively extracts energy from the magnons. Op- tomagnonic scattering is enhanced for a `triple resonance condition' [38, 54{57] by tuning both the input and the scattered photon frequency to the optical resonances of the cavity. In contrast, optomechanical cooling [4, 5, 8] requires detuning the input laser from a cavity resonance with correspondingly reduced scattering and cooling rate. In this manuscript, we predict that modern technology and materials can signicantly reduce the temperature of the magnetic order, showing the potential to manipulate magnons using light. We derive below rate equations for photons and magnons to estimate the steady-state magnon number that can be reached as a function of material and device parameters. We consider a spherical magnetic insulator with high index of refraction that is transparent at the input light frequency (Fig. 1) and magnetization perpen- dicular to the WGM orbits that are excited by proximity coupling to an external laser. We single out two groups of magnon modes that couple preferentially to the WGMs [50]. The small angular momentum (including the Kit-arXiv:1804.02683v1 [cond-mat.mes-hall] 8 Apr 20182 Ain AoutWinMLWRMSWT HappInput Scattered Magnon FIG. 1. Optomagnonic cooling setup: A ferromagnetic sphere in contact with an optical waveguide. A magnetic eld Happ (into the paper) is applied to saturate the magnetization. In- put light with amplitude Ainis evanescently coupled to a WGMWin. We focus on anti-Stokes scattering by two types of magnons that are characterized by their angular momen- tum [50]. A small angular momentum magnon MSmaintains the direction of WGMs, converting WintoWT.Wincan be re ected into WRby absorbing a large angular momentum magnonML. Theoretically, both the cases can be treated in the same formalism. tel) magnons, MSin Fig. 1, and large angular momen- tum magnons, the chiral Damon-Eshbach (DE) modes ML. The theory presented below is valid for both types of magnons. We can understand the basic physics by the minimal model sketched in Fig. 2. We focus on a single incident WGMWpwith index pand frequency !p. It is occupied by [8] np=4Kp (p+Kp)2Pin ~!p(1) photons, with pbeing the intrinsic linewidth, Kpthe leakage rate into the proximity coupler, and Pinthe input light power. An optically active magnon M[with either small or large angular momentum] is annihilated Wp+ M!Wcor created Wp!Wh+Mby BLS, where WcandWhare blue and red-shifted sideband WGMs, respectively. We rst derive a simple semi-classical rate equation for the non-equilibrium steady-state magnon number, n(sc) m [the superscript distinguishes the estimate from nmas more rigorously derived below]. The thermal bath ab- sorbs and injects magnons at rates mn(sc) m(nth+ 1) and mnth n(sc) m+ 1 respectively, where mis the inversemagnon lifetime, nth= exp~!m kBT 11 (2) its equilibrium thermal occupation, !mthe magnon fre- quency and Tthe ambient temperature. The optical cooling rate is R0 cnpn(sc) m, whereR0 cis the anti-Stokes scattering rate of one Wp-photon by one M-magnon and we assumed that there are no photons in Wc. The lat- ter is justied because of small optomagnonic couplings compared to WGM dissipation rates, 20:11 GHz [36{38] while R0 cnpn(sc) mis at mostm21 MHz. In the absence of dissipation, Fermi's golden rule gives R0 c= 2jgcj2(!p+!m!c), where ~gcis the opto- magnonic coupling and f!p;!c;!mgare the frequencies offWp;Wc;Mg;respectively. When Wchas a nite life- time, the-function is broadened into a Lorentzian, giv- ing R0 c=jgcj2(c+Kc) (!p+!m!c)2+ (c+Kc)2=4; (3) wherecis its intrinsic linewidth, and Kcis its leak- age rate into the proximity coupler. Similarly, the opti- cal heating rate is R0 hnp n(sc) m+ 1 ;whereR0 his given by Eq. (3) with gc;!c;c!gh;!h;hand!m!!m. In deriving R0 c;h, we ignore the magnon linewidth since mc;h[38, 50]. In the steady state the cooling and heating rates are equal, leading to the estimate n(sc) m=mnth+R0 hnp m+ (R0cR0 h)np: (4) This agrees with the result from the more precise theory discussed below, thus capturing the essential processes correctly (a posteriori). However, the rate equation can- not access noise properties beyond the magnon number that are important for thermodynamic applications. Fur- ther, it does not dierentiate between a coherent preces- sion of the magnetization and the thermal magnon cloud with the same number of magnons. In order to model the cooling process more rigorously, we proceed from a model Hamiltonian for a system with three photon and one magnon mode. In the Hamiltonian ^HS=^H0+^Hom[50] ^H0=~!p^ay p^ap+~!c^ay c^ac+~!h^ay h^ah+~!m^my^m;(5) and ^axand ^mare the annihilation operators for photon Wxwithx2fp;c;hgand magnon M. The optomagnonic coupling in the rotating wave approximation reads [50] ^Hom=~gc^ap^ay c^m+~gh^ap^ay h^my+ h:c:; (6) wheregcandghare the scattering amplitudes and h :c:is the Hermitian conjugate.3 Wpκp Wc κcWh κhMκm Kp Kh KcR0 c R0 h FIG. 2. Light-induced cooling of a magnon, M. A proxim- ity ber or prism is coupled to the WGMs Wxwith a cou- pling constant Kx, excitingWpwhile collecting the scattered WcandWh. The photons are inelastically scattered by the magnonWp+M!WcandWp!Wh+Mat single particle ratesR0 candR0 hrespectively, derived in the text. All modes are coupled to their respective thermal baths by leakage rates x. Whencis much larger than the corresponding scatter- ing rate, the bath associated with Wccan become an ecient channel for dissipation of the magnons in M. In the rotating frame of the \envelope" operators ^Wx(t)4= ^ax(t)ei!xtand ^M(t)4= ^m(t)ei!mtthe (Heisen- berg) equation of motion for ^Mbecomes [4, 58] _^M=igh^Wp^Wy heihtig c^Wy p^Wceictm 2^Mpm^bm; (7) whereh=!h+!m!pandc=!c!m!pare the detunings from the scattering resonances. ^bm(t) is the stochastic magnetic eld generated by the interaction of Mwith phonons [59] and/or other magnons [60], whose precise form depends on the microscopic details of the interaction [61]. We assume that the correlators of ^bmobey the uctuation-dissipation theorem for thermal equilibrium [62, 63]. When mkBT=~, which is satised for m 21 MHz [36{38] and T50K, the (narrow band ltered) noise is eectively white and generates a canon- ical Gibbs distribution of the magnons in steady state [58]. Their statistics areD ^bm(t)E = 0,D ^by m(t0)^bm(t)E = nth(tt0) andD ^bm(t0)^by m(t)E = (nth+ 1)(tt0), where nthis dened in Eq. (2). For weak scattering relative to the input power, we can ignore any back-action on ^Wpsuch that its dynamics is governed only by the proximity coupling. When Wpis ina coherent state,D ^Wp(t)E =pnpandD ^Wy p(t0)^Wp(t)E = np, wherenpis given by Eq. (1). The photons in Wcare generated by ^Homand dissi- pated into their thermal bath, with Heisenberg equation of motion [4, 58] d^Wc dt=igc^Wp^Meictc+Kc 2^Wcpc^bcp Kc^Ac; (8) where ^bcand ^Acare noise operators. The physical ori- gins of ^bcand nite lifetime 1 care scattering by im- purities, surface roughness, and lattice vibrations. Kcis the leakage rate of Wcinto the proximity coupler and ^Ac is the vacuum noise from the latter into Wc. The noise sources are white for suciently small c.D ^Xc(t)E = 0,D ^Xy c(t0)^Xc(t)E = 0 andD ^Xc(t0)^Xy c(t)E =(tt0) where X2f^bc;^Acg, because the thermal occupation of photons at infrared and visible frequencies is negligibly small at room temperature e~!c=(kBT)0. The solution to Eq. (8) is ^Wc(t) =^Wc;th(t) +^Wc;om(t). The thermal contribution is, ^Wc;th=Zt 0e(c+Kc)(t)=2h pc^bc()p Kc^Ac()i d (9) where the origin of time is arbitrary. For t;t0! 1 , we get the equilibrium statisticsD ^Wy c;th(t0)^Wc;th(t)E = 0 and D ^Wc;th(t0)^Wy c;th(t)E = exp (c+Kc)jtt0j 2 ;(10) independent of the optomagnonic coupling. ^Wc;omcan be simplied by the adiabaticity of the magnetization dynamics that follows from mc. When ^Mis treated as a slowly varying constant ^Wc;om(t)igc^M(t)Zt 0e(c+Kc)(t)=2^Wp()eicd: (11) ^Wh(t) is obtained by the substitution c!hand ^M! ^Myin Eqs. (9-11). We can now rewrite Eq. (7) as d^M dt=m 2^M+pm^bm +^Oc+^Oh: (12) with cooling and heating operators that re ect the light scattering processes in Fig. 2: ^Oc=^Nc+i^c^M; (13) ^Oh=^Ny h+i^y h^M: (14) Focusing on the cooling process, we distinguish the self-energy, ^c=ijgcj2Zt 0e(ic+(c+Kc)=2)(t)^Wy p(t)^Wp()d;(15)4 from the noise operator, ^Nc(t) =ig c^Wy p(t)^Wc;th(t)eict: (16) In the weak-coupling regime we may adopt a mean-eld approximation by replacing ^cby its average, D ^cE =!c+ic 24=jgcj2np ci(c+Kc)=2; (17) where !cis the (reactive) shift of the magnon resonance and cthe optical contribution to the magnon linewidth. The noise ^Nccan be interpreted as the vacuum uc- tuations of Wcentering the magnon subsystem via the optomagnonic interaction. ^Nchas a very short corre- lation time(c+Kc)1[see Eq. (10)] compared to magnon dynamics 1 m, and thus can be treated as a white noise source withD ^Nc(t)E = 0,D ^Ny c(t)^Nc(t0)E = 0 , andD ^Nc(t0)^Ny c(t)E Vc(tt0). By integrating over time and using the correlation functions of ^Wpand ^Wc;th Vc=4jgcj2np(c+Kc) 42c+ (c+Kc)2= c; (18) dened in Eq. (17).  c=mat resonance c= 0 is the cooperativity between the magnons and Wc-photons due to the coupling mediated by Wp-photons. Analogous results hold for ^Oh, with substitutions c! hin Eqs. (15)-(18). We arrive at d^M dti(!c+ !h)^Mtot 2^Mptot^btot;(19) wheretot=m+ chandptot^btot=pm^bm ^Nc+^Ny h. The uctuations of the total noise follow from Eq. (18) D ^by tot(t0)^btot(t)E nm(tt0); (20) D ^btot(t0)^by tot(t)E (nm+ 1)(tt0); (21) where nm=mnth+ h m+ ch: (22) Eq. (19) is equivalent to the equation of motion for magnons in equilibrium after the substitutions !m! !m+ !c+ !h,m!tot, andnth!nm. It implies that the magnons in the non-equilibrium steady state are still canonically distributed with density matrix ^ne= exp~!m^nm kBTne Tr exp~!m^nm kBTne1 (23) where the number operator ^ nm= ^my^mand the non- equilibrium magnon temperature Tneis implicitly dened by Eq. (22) and nm= exp~!m kBTne 11 : (24)We getD ^MxE =D ^MyE = 0, which implies that light scattering does not induce a coherent magnon precession, in contrast to a resonant ac magnetic eld. nmis the av- erage number of magnons that can be larger or smaller than the equilibrium value nth. The result is consistent withn(sc) m[see Eq. (4)] because  c;h=R0 c;hnpas expected from Fermi's golden rule. The canonical distribution im- plies that the steady-state magnon entropy is maximized for the given number of magnons, nm. When hc> m;i.e. when heating by the laser overcomes the intrinsic magnon damping, the system be- comes unstable, leading to runaway magnon generation and self-oscillations [49, 64, 65]. The instability is reg- ularized by magnon-magnon scattering, not included in our theory. Here we focus on the cooling scenario in which  hc [50]. Magnon cooling can be monitored by the inten- sity of the blue-shifted sideband. Using the input-output formalism [58, 66] the scattered light amplitude in the rotating frame is ^Aout(t) =p Kc^Wc(t): (25) It can be converted into the output power by Pout= ~!cD ^Ay out(t)^Aout(t)E , which is independent of time in steady state. With impedance matching, p;c=Kp;c, and at the triple resonance, c= 0, Pout Pin=jgcj2 cpmnth m+ 2jgcj2np=c/1 1 +Pin=Ps;(26) dening the saturation power Ps4=~!ppcm 2jgcj2: (27) To leading order Pout/Pin[37, 50], but saturates when the magnon number becomes small, which is an exper- imental evidence for magnon cooling. Psis the input power that halves the number of magnons. For a YIG sphere with parameters !c!p= 2 300 THz (free space wavelength 1 m), an optical Q-factor !p=(2p) =!c=(2c) = 106, [37], magnon linewidth m= 21 MHz, and optomagnonic coupling gc= 210 Hz [50], we get Ps= 140 W. Trying to match this withPinis not useful since laser-induced lattice heating [10] will overwhelm the cooling eect. However, Pscan be signicantly reduced by large magnon-photon coupling. Doping YIG with bismuth can increase gctenfold [47], bringingPsdown to1 W. The spatial overlap between the magnons and photons [50] can be engineered in ellip- soidal or nanostructured magnets [67] which can increase gcfurther by an order of magnitude, giving Ps10mW. For an ambient temperature T= 1 K and magnon fre- quency!m= 210 GHz, the thermal magnon number nth= 1:62. ForPin=fPs=20; Ps;5Psgthe steady-state5 magnon numbers are nm=f1:55;0:81;0:27gand tem- peraturesTne=f0:96;0:60;0:31gK respectively. At an optimisticPs= 10mW, the above input power corre- sponds tonp=f3106;5107;3108gintra-cavity pho- tons respectively. Cooling is experimentally observable for relatively small powers Pin<Ps=20, which should be achievable by optimising the magnon-photon coupling. In summary, we estimate the cooling power due to BLS of light by magnons in an optomagnonic cavity. Due to the large mismatch of optical and magnonic time scales, the photon degree of freedom can be eliminated by renormalizing the magnon frequency and damping, cf. Eq. (19), and a light-controlled eective temperature Eq. (22). Current technology and materials are close to achieving signicant cooling of magnons, envisioning the possibility of light-controlled magnon manipulation. We thank J. Haigh, M. Elyasi, K. Satoh, K. Usami, and A. Gloppe for helpful inputs and discussions. This work is nancially supported by the Nederlandse Organ- isatie voor Wetenschappelijk Onderzoek (NWO) as well as JSPS KAKENHI Grant Nos. 26103006. 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2018-04-08

Inelastic scattering of light by spin waves generates an energy flow between the light and magnetization fields, a process that can be enhanced and controlled by concentrating the light in magneto-optical resonators. Here, we model the cooling of a sphere made of a magnetic insulator, such as yttrium iron garnet (YIG), using a monochromatic laser source. When the magnon lifetimes are much larger than the optical ones, we can treat the latter as a Markovian bath for magnons. The steady-state magnons are canonically distributed with a temperature that is controlled by the light intensity. We predict that such a cooling process can significantly reduce the temperature of the magnetic order within current technology.

Optical cooling of magnons

1804.02683v1

Single-Nitrogen-vacancy-center quantum memory for a superconducting ux qubit mediated by a ferromagnet Yen-Yu Lai,1, 2Guin-Dar Lin,1, 2Jason Twamley,3and Hsi-Sheng Goan1, 2, 1Department of Physics and Center for Theoretical Physics, National Taiwan University, Taipei 10617, Taiwan 2Center for Quantum Science and Engineering, National Taiwan University, Taipei 10617, Taiwan 3Centre for Engineered Quantum Systems, Department of Physics and Astronomy, Macquarie University, NSW 2109, Australia (Dated: May 1, 2018) We propose a quantum memory scheme to transfer and store the quantum state of a supercon- ducting ux qubit (FQ) into the electron spin of a single nitrogen-vacancy (NV) center in diamond via yttrium iron garnet (YIG), a ferromagnet. Unlike an ensemble of NV centers, the YIG mod- erator can enhance the eective FQ-NV-center coupling strength without introducing additional appreciable decoherence. We derive the eective interaction between the FQ and the NV center by tracing out the degrees of freedom of the collective mode of the YIG spins. We demonstrate the transfer, storage, and retrieval procedures, taking into account the eects of spontaneous decay and pure dephasing. Using realistic experimental parameters for the FQ, NV center and YIG, we nd that a combined transfer, storage, and retrieval delity higher than 0.9, with a long storage time of 10 ms, can be achieved. This hybrid system not only acts as a promising quantum memory, but also provides an example of enhanced coupling between various systems through collective degrees of freedom. PACS numbers: 03.65.Yz, 42.50.Dv, 03.67.-a, 03.65.Ta I. INTRODUCTION Superconducting qubits and related circuit-QED de- vices [1, 2] with excellent scalability, parametric tunabil- ity, and strong coupling with external elds are proving to be a powerful platform for quantum information pro- cessing. However, they suer from decoherence due to inevitable interactions with their surrounding environ- ments. In a complex quantum protocol, superconducting qubits may experience frequent idles times when they are not involved in active quantum gates. During this idle time, to prevent the decoherence of their quantum in- formation, one can transfer their quantum state to an adjacent quantum memory for better protection. A hybrid system that takes advantage of the fast oper- ation of superconducting qubits and long coherence times of a suitable quantum memory may yield good coherence preservation if the state transfer between them is quick enough, i.e., faster than the decoherence time of either system. The spin of a nitrogen-vacancy (NV) center in diamond, which has a relatively long coherence time even at room temperature [3], can be a candidate for such a quantum memory. This low decoherence rate also means that the NV center normally only couples weakly to a superconducting qubit. Such a weak coupling leads to a slow state transfer and coherence loss can be signicant. Ensembles of NV centers [4{8] may make the coupling stronger, but at the added cost of increased decoherence caused by internal spin-spin interactions, degrading the delity of the quantum memory. goan@phys.ntu.edu.twIn this paper, we propose a scheme to transfer quantum states faithfully between a superconducting ux qubit and a single-NV-center spin via the ferromagnetic ma- terial yttrium iron garnet (YIG) [9, 10]. YIG has been proposed as a mediator for classical magnetic elds to enhance the sensitivity of a NV magnetometer to achieve single nuclear spin detection [11]. In addition, the large number of spins in YIG with strong exchange interac- tion leads to collective-excitation modes with narrow linewidths at low temperature [12, 13]. These collective modes are known as quasiparticles or magnons [14], and have been shown to be capable of coupling to dierent kinds of quantum systems, such as superconducting mi- crowave cavity modes [13, 15, 16]. Magnons in YIG have also been proposed as a mediator of coherent coupling between two distant spins (e.g., two spatially distant NV- center spins) [17]. A CNOT gate between two single-NV- center spins separated by a distance of about 1 m with operation times of the order of a few tens of nanoseconds has been demonstrated [17]. This shows that a relatively strong coherent coupling between a single-NV-center spin and YIG magnons is feasible. On the other hand, a ux qubit (FQ) can display strong coherent coupling to an ensemble of NV centers exhibiting a collective cou- pling of70 MHz [8]. However the spin density of YIG (4:21021cm3) [13] is almost three orders of magni- tude larger than typical NV ensembles ( 51018cm3) [8]. This suggests that the coupling between a ux qubit and a small YIG sample may be similar or even stronger than between a ux qubit and a NV ensemble. In this paper, we show that we can achieve a substantially large eective coupling between a single-NV-center spin and a FQ by using the magnons in a small nearby YIG sample as a mediator without appreciably sacricing the transferarXiv:1804.11231v1 [quant-ph] 30 Apr 20182 and storage delity of the quantum state. When the size of the YIG is small enough, the Kit- tel mode (KM) of the YIG sample [13, 15] is gapped from the higher-energy modes and thus plays an impor- tant role in a low-temperature and low-excitation regime. In our scheme, we nd that the eective coupling and the spatial separation between the FQ and NV required to attain these coupling strengths via the YIG can be signicantly enhanced. The coupling attained using our proposal is of the order of several tenths of MHz, while the spatial separation can be increased to a few tenths ofm. This represents an enhancement in the coupling strength of 3{7 times over the direct FQ-NV coupling. More interestingly, it also represents a substantial en- hancement in spatial separation required between the FQ and NV. For comparison, a direct coupling scheme [18] nds a coupling strength of 100 kHz, but requires a mi- nuscule spatial separation of 20 nm. To achieve larger direct coupling strength, strengths comparable to those found using our scheme would require even tinier spatial separations, which may be physically unrealistic. In con- trast, in our proposal we are able to expand the spatial separation to a few tenths of a m scale, which is 10{20 times larger than the separation required to attain simi- lar coupling strengths via direct FQ-NV coupling. Thus our scheme can provide signicant couplings over a sep- aration, which is technically far easier to engineer. The quantum state transfer time with the coupling strength found in our scheme is considerably smaller than the de- coherence time of the FQ so that fast and faithful transfer can be realized without suering signicant decoherence. The paper is arranged as follows. In Sec. II, we derive the eective Hamiltonian and coupling strength between a FQ and a NV-center spin from a FQ-YIG-NV-center hybrid system. In Sec. III, we introduce a protocol for the transfer and storage of the quantum state. After that, simulations of the protocol are presented and discussed in Sec. IV, taking major decoherence eects into consider- ation. Finally, a short conclusion is given in Sec. V. All the details of derivations of equations and calculations are presented in Appendices A and B. II. MODEL The hybrid system in our proposal is schematically il- lustrated in Fig. 1 and contains three parts: the FQ, YIG, and a single-NV center. The noninteracting Hamiltonian describing the individual systems [with ( ~= 1)] can be written as Hs=HF+HY+HN; (1)where HF=1 2!F(z) F; (2) HY=JX hr;r0iSr
Sr0+ eBX rS(z) r; (3) HN=
ZS S(z) N2 eBS(z) N: (4) Here, the FQ is regarded as a typical two-level system de- scribed by the Hamiltonian HFin Eq. (2), with !Fthe transition frequency of the FQ and (z)the Pauli matrix (for details, see Appendix A). The Hamiltonian of the YIG in an external eld along the zaxis is given by HY in Eq. (3), where Srare the operators of the spin located at positionrin the YIG. The parameter Jis the exchange coupling between the spins inside the YIG. We consider the application of B=BL+B, an external magnetic eld along the zaxis and which is felt by the YIG and the NV center (see Fig. 1 and 2). Here, BLis a local magnetic eld generated by a micromagnet [19] without disturbing the FQ (for details, see Sec. IV), and Bis the tuneable magnetic eld whose value is set below the critical eld of the FQ. The tuneable dc magnetic eld could be gener- ated by a coil. The ground triplet states of the NV center is described by the Hamiltonian HNin Eq. (4), where S(z) Nis thezcomponent of the spin-1 operators of the single-NV center,
ZS= 2:87 GHz is the zero-eld split- ting of the ground triplets, and e=1:761011rad s1 T1is the gyromagnetic ratio of electron spin. To pro- ceed further, let us simplify the Hamiltonians a little bit. There are two single-photon transitions between j0Niand j1Niin the ground triplet states of the NV-center spin, and their energy gaps are !(1)=
ZS eB. We use, for instance, the NV-center spin states j0Niandj1Nias our storage qubit basis state. In the state transfer stage, this transition frequency is tuned to be resonant with the FQ transition frequency, while the j0Nitoj1Nitransi- tion is largely detuned (see Fig. 2). Further, as shown later, the eective coupling between the FQ and the NV- center storage qubit can be switched on and o (or very small) by varying the external Beld. Here, for the sake of deriving the eective Hamiltonian, we rst treat the NV-center spin as a two-level storage qubit by ignoring the far-detuned transition. We will take into account the eect of the existence of the far-detuned NV j1Nistate when we run numerical simulations for the dynamics of quantum state transfer and storage processes. We then transform the YIG Hamiltonian from a Heisenberg model to a magnon form with the Holstein-Primako transfor- mation [17, 20, 21] and harmonic approximation. As a result, the NV and YIG Hamiltonians can be rewritten as [17, 21] HN'1 2!N(z) N; (5) HY'X k!kay kak; (6)3 Figure 1. Schematic illustration of the proposed quantum memory. The FQ is regarded as a two-level system depending on the sign (direction) of its persistent current Ip, and the KM in YIG couples to both FQ and the single-NV-center spin with strengthsgFYandgY N, respectively. The external magnetic eld felt by the YIG and the NV center is B=BL+B (see Fig. 2), where BLis a local magnetic eld generated by a micromagnet without disturbing the FQ, and Bis a tuneable magnetic eld, generated by, e.g., a coil. where!N=!(1), and!k=sJa2k2+ eBare the fre- quencies of the NV storage qubit and magnon mode k, respectively, ay k(ak) is the creation (annihilation) opera- tor of magnon mode k,sis the maximum eigenvalue of the spin operator S(z) r, andais the lattice constant of the YIG. For a small-sized YIG sample, the boundary conditions at the surface are of great importance and the magnon modes become gapped. The FQ and NV interact indirectly via YIG. The cou- pling Hamiltonian thus has two parts: the FQ-YIG cou- pling and the YIG-NV coupling. The current in the loop of the FQ generates a magnetic eld which interacts with the spins in the YIG. The NV electron spin also cou- ples to these spins by dipole-dipole interaction. Under the rotating-wave approximation (RWA), the coupling Hamiltonian in terms of the YIG collective-excitation modes (derived in detail in Appendix A) reads Hc=HFY+HYN; (7) HFY'X k gFY(k)(+) Fak+H:C: ; (8) HYN'X k gYN(k)ay k() N+H:C: ; (9) where gFY(k) =0 2 eIpr 2s NNX rFeik
a rF; (10) gYN(k) =0 4 2 e~r 2s NNX rN3cos2rN1 r3 N eik
a; (11) are the coupling strength of the FQ and NV with magnon modekin the YIG, respectively. Here, 0is the vacuum permeability, Ipis the persistent current of the FQ, rF(rN) is the distance between a spin in the YIG and the FQ (the NV spin), Nis the number of the spins in the YIG, andrNis the angle between the vector connecting the NV and the spin in YIG, and the direction of the external magnetic eld. Under the condition that the Heisenberg interaction inside the YIG is much greater than the coupling between a qubit and any single spin in the YIG, the qubit then eectively interacts with the collective mode of all the spins in the YIG. The more spins in the YIG following this condition, the stronger the coupling. If one chooses a YIG sphere with a submicrometer di- ameter, then the energy levels of the YIG magnon modes are gapped, largely due to its small size. We consider only the simplest mode of the YIG, i.e., the KM [13, 15], with frequency !Kwhich is far from both the frequen- cies of the FQ and NV-center storage qubit. Thus the KM is in a virtual coupling regime with the FQ and NV- center storage qubit. To account for the overall eect, we use the Schrieer-Wol transformation (SWT) to de- rive the eective Hamiltonian up to the second order in the coupling strengths with the YIG by averaging out the far-o-resonance degrees of freedoms of the YIG. We then obtain (with detailed derivation shown in Appendix B) an eective Hamiltonian between the FQ and the NV- center qubit as He'1 2!F;e(z) F+1 2!N;e(z) N +gFN;e (+) F() N+H:C: ; (12) where !F;e=!F+F; (13) !N;e=!N+N; (14) are the eective frequencies of the FQ and the NV-center storage qubit with frequency shifts F=g2 FY(!K) !F!Kand N=g2 Y N(!K) (!N+Y N)!Kinduced by the KM of the YIG, re- spectively, and gFN;e(!K) =1 2gFY(!K)gYN(!K) 1 !F!K+1 (!N+YN)!K (15) is the eective coupling strength between the FQ and the NV-center qubit. Here, !Kdenotes the frequency of the KM. Note that to have a substantially large eective coupling, the detuning ( !F!K) and detuning ( !N+ YN!K) appearing in the denominator of Eq. (15) should not be too large. On the other hand, to keep the KM in a regime of virtual coupling with the FQ and NV-center storage qubit, they should not be too small. By varying the external magnetic eld, we can control the values of ( !F!K) and (!N!K). In particular, the change in !K, due to the variation of the magnetic4 Figure 2. Energy-level diagram of the FQ (left) and the ground triplet states of the NV-center spin (right). The NV- center spin states j1Niare degenerate (dashed line) and have a zero-eld splitting
ZSand an energy shift Ninduced by the indirect coupling scheme via YIG relative to state j0Ni (see red arrow). By tuning the magnetic eld on the NV- center spin to the value of B=Bres, one can control the tran- sition betweenj0Niandj1Nito be resonant with the FQ (see blue arrow) or to be o-resonant at B=Bo. The tran- sition betweenj0Niandj1Niis always set to be o-resonant with the FQ, i.e., is set to be a disconnected channel. eld, is opposite to the change of the energy dierence betweenj0Niandj1Ni[in contrast to the same energy change betweenj0Niandj1Niresulting in no change in (!(+1)!K)]. This makesj0Niandj1Nia better choice of storage qubit basis states. III. QUANTUM MEMORY Next we will use the derived eective Hamiltonian to investigate the dynamics and the delity of the proposed quantum memory scheme. There are two stages that we need to consider: state transfer stage and state storage stage. To better assess and calculate the delity of our scheme, we take all the lowest triplet states of the NV spin into account. In this case, the FQ couples to two transitions in these triplets separately and the eective Hamiltonian, given by Eq. (12), becomes He=1 2!F;e(z) F+X j=1!N;(j);e(B)jjNihjNj +g(+1)h (+) FS() N;(+1)+H:C:i +g(1)h (+) FS() N;(1)+H:C:i ; (16) Here in the spin-1 Hilbert space of the NV center, the eective NV spin frequencies are !N;(1);e(B) =
ZS eB+N;(1), where
ZSis the zero-eld splitting and N;(1)is the frequency shift. The eective couplingstrengths between the FQ and the NV spin transitions are denoted as g(1), corresponding to Eq. (15) with !N!!N;(1);e(B). The subscripts ( 1) in the ex- pression (and in the following), stand for the transitions betweenj0Niandj1Ni, respectively. The operators S() N;(1)with superscriptdenote the raising and lower- ing operators, respectively. Now we move to the interaction picture through the unitary transformation U= exp(itH0;e), where H0;e=1 2!F;e(z) F+P j=1!N;(j);e(Bres)jjNihjNjis the rst two terms of the eective Hamiltonian given by Eq. (16) with magnetic eld B=Bres, whereBresis the magnetic eld strength applied to the NV-center spin when the transition between j0Niandj1Nimatches the energy gap of the FQ (see Fig. 2). Then the eective interaction Hamiltonian Hintbecomes Hint=HN;int+HFN;int; (17) HN;int=X j=1B;(j)jjNihjNj; (18) HFN;int=g(1)h (+) FS() N;(1)+H:C:i ; +g(+1)h (+) FS() N;(+1)e2it eBres+H:C:i ;(19) and B;(1)= e(BBres): (20) WhenB;1is tuned to zero, i.e., B=Bres, theg(1)cou- pling terms start to transfer the quantum state from the FQ to the NV-center spin and the fast oscillating com- ponents in the g(+1)terms can be eectively neglected. We initially prepare the NV in the ground state, j N(0)i=j0Ni. Suppose that the FQ is in a general state characterized by angles and. Then the joint state is j (0)i= cosj1Fi+eisinj0Fi
j0Ni = cosj1F;0Ni+eisinj0F;0Ni: (21) After a transfer time t==(2g(1)), the target state in the interaction picture becomes j (t)i=icosj0F;1Ni+eisinj0F;0Ni =j0Fi icosj1Ni+eisinj0Ni
:(22) Once the state has been transferred to the NV-center spin, we turn o the coupling eectively by enlarging the mismatch of the frequencies between the FQ and the NV-center storage qubit. The quantum state can thus be stored for better coherence with dephasing time char- acterized by the NV-center spin's T2time. To retrieve the state from the NV-center storage qubit to the FQ, we tune to the NV-FQ resonance again. After a time t1==(2g(1)), the original state is restored in the FQ degrees of freedom j (tf)i=eiscosj1F;0Ni+eisinj0F;0Ni;(23)5 with an additional phase sthat comes from the coherent evolution during the storage time t2and since this is known it can be corrected. IV. RESULTS AND DISCUSSION We present the numerical results together with discus- sions to verify our scheme here. Before proceeding with our numerical calculations for the delity performance, we rst describe the system parameters used. It is as- sumed that the YIG is a sphere of radius about 45 nm and contains about 106spins. A local magnetic eld BLis generated by placing a micromagnet [19] of size 0:20:20:2m3with a uniform perpendicular magne- tization of about a hundred Gauss at a vertical distance 25{50 nm from the YIG. With this local magnetic eld, the frequency of the KM can reach GHz levels; further- more, by varying an external magnetic eld the frequency dierence (!F!K) can achieve a typical value of about 170 MHz. Since the direction of BLis parallel to the plane of the FQ, the FQ is insensitive to BL. Further- more, because the transverse distance from the micro- magnet boundary at which BLdrops to 0, is smaller than 0.1m [19], ifrFis considerably larger than the sum of this transverse distance and the distance from the YIG to the left boundary of the micromagnet (see Fig. 1), we can realize local magnetic eld control for the YIG without disturbing the FQ. Therefore, by choosing a rel- atively large value of rF0.25m and using a FQ with Ip=500 nA and a diamond with a single NV spin at a distancerN60 nm from the YIG, we can estimate the eective coupling strength gFN;eto be about 350{700 kHz according to Eqs. (A12), (A14), and (B14). The fre- quency shifts of both the FQ and NV-center spin due to the YIG coupling [see Eqs. (B8) and (B9)] are at about hundreds kHz, and are thus rather small and negligible in comparison with their own frequencies in the GHz range. Following the exposition of the system parameters, we now continue to show the numerical results and we plot the dynamics of the transfer process in Fig. 3. We choose the case where the eective coupling strength is 700 kHz and where the initial state is1=2 =p 1=2j1F;0Ni+p 1=2j0F;0Nisince this state is, in a more realistic situa- tion considered later, in uenced the most by the dephas- ing eect. During the transfer process (whose duration is 0.36s),j1F;0Niis transferred to j0F;1Ni, while j0F;0Niis left unchanged. The population of the other states remains zero, except that the j0F;1Nistate has a small probability ( 107) as shown in Fig. 3(b). This small probability is due to the detuning between the tran- sition frequency from j0Nitoj1Niand the frequency of the FQ, and one can reduce this probability further by making the detuning larger. To simulate the state transfer and storage processes in a more realistic setting, we use the master equation [22], which takes into account both spontaneous decay Figure 3. (a) Dynamics of the probabilities of the basis states of the quantum memory during a state transfer process for the initial state1=2 =p 1=2j1F;0Ni+p 1=2j0F;0Ni, and cou- pling strength 700 kHz. During the transfer process, j1F;0Ni is transferred toj0F;1Ni, whilej0F;0Niis unchanged. (b) Due to the detuning between the j0Nitoj1Nitransition and the FQ transition frequency, state j0F;1Nihas a negligible probability (107) during the process. and pure dephasing of the FQ and the NV-center spin, d dt=i[He;] + (s) F 2L[() F] + (p) F 2L[j1Fih1Fj] +X j=1 (s) N;(j) 2L[S() N;(j)] + (p) N;(j) 2L[jjNihjNj];(24) whereL[O] = 2OOyOyOOyO, is the Lindblad superoperator, and (s) qand (p) qare the spontaneous de- cay and the dephasing rates, respectively, of the species q. Note that they are related to the relaxation time T1and decoherence time T2asT1=1= (s)andT2=2=( (s)+2 (p)), respectively [22]. It has been reported recently that both the intrinsic T 1andT 2of FQ's could be about 10 s at 33 mK [23] and the value of the intrinsic T 2of an NV-center spin could be about 90 s at room temperature [24]. Fur- thermore, it has been shown that by applying dynamical decoupling pulse sequences, T1of ensembles of NV spins could be more than 10 sec and T2could be about 0.6 sec at 77 K [25]. In the simulations, the relevant decoherence times in the state transfer stage are the intrinsic T 1and T 2times of the FQ and the NV-center spin. In the state storage stage, however, the NV-center spin is eectively decoupled from the YIG and FQ due to the large detun- ing and thus dynamical decoupling pulse sequences can be applied to protect the NV from decoherence to main- tain the transferred state. Consequently, we can use the T1andT2values of the NV-center spin measured using dynamical decoupling [25] to estimate the delity in the state storage stage. Furthermore, the decoherence times6 of a NV center solely due to a coupling to a ferromagnet YIG have been estimated in Ref. [17], and theses times depend sensitively on the ratio of the magnon excitation gap to the YIG temperature. It has been shown that for a magnon gap of 100 eV and a temperature of 0.1 K, these times are typically much larger than the (intrinsic) decoherence times of the NV [17]. In other words, the induced decoherence solely due to coupling to the YIG for temperatures smaller than the magnon excitation gap is nondetrimental [17]. In our scheme, all of the compo- nents (the FQ, YIG, and NV) of the qubit and quantum memory are at the same low temperature as that of the FQ. Moreover, because of the small size of the YIG and the applied magnetic eld, this ratio of the magnon ex- citation gap to the temperature in our scheme is even bigger than that used for estimation in Ref. [17]. As a result, the eect of the induced decoherence solely due to coupling to the YIG will be neglected in our simulations. The eect of the linewidth of the YIG nanosphere can also be neglected. The linewidths of the KM of a single YIG sphere with submillimeter size have been measured to be about 1{2 MHz [12, 13]. Although linewidth mea- surement on a single YIG nanosphere is, to our knowl- edge, not available so far, a close case of YIG nanodisks with thickness about 20 nm and diameter ranging from 300{700 nm has been reported in Ref. [26]. There, the linewidth of the uniform mode, i.e., the lowest-energy ferromagnetic resonance mode of a nanodisk, was mea- sured to be about 7 MHz for a nanodisk with diame- ter 700 nm at frequency 8.2 GHz. However, since this measurement was performed at room temperature and the nanodisks with dierent diameters are arranged in a row with 3 m spacing (i.e., not completely a single-disk measurement), one may expect that the linewidth could be narrower if the measurements were performed for a actual single nanodisk at a low temperature of tens of mK and at the frequency down to the value of 2 GHz as in our proposal. Furthermore, the linewidths of the nanodisks do not change much with the diameter [27], at least within the range investigated in Ref. [26]. One may thus expect that the linewidth of a YIG nanosphere without surface defects at low temperature could be sim- ilar or at most at a few MHz level, which is still much smaller than the detuning ( 170 MHz) between the YIG nanosphere and other quantum systems in our proposal. Consequently, the eect of the linewidth of the KM of the YIG nanosphere does not appreciably aect the virtual excitation or virtual coupling picture in our proposal and thus is neglected in the subsequent calculations after the degrees of freedoms of the YIG are traced out. We then take the values of the decoherence and relax- ation times at higher temperatures [24, 25] to make a conservative evaluation of the performance of our quan- tum memory scheme through the delity of the state F=p h jj i; (25) wherej iis the target state and is the actual system density matrix. We can transform the Hamiltonian to therotating frame to obtain Hintas in Eqs. (17){(19). Since during the storage stage the system is tuned to be o- resonant, i.e., B;(1)g(1), the total system approx- imately undergoes free evolution during this stage. The delities of the quantum state memory for initial states j1iand1=2 are shown in Table I, in which results that make use of more conservative values for T 2= 20s for the NV center are also presented. The initial states j1iand1=2 are chosen because they are in uenced the most by the spontaneous decay and dephasing eect, respectively. We have also simulated for dierent ini- tial states ofj0i=j0F;0Ni,1=3 =p 2=3j1F;0Ni+p 1=3j0F;0Ni,1=4 =p 3=4j1F;0Ni+p 1=4j0F;0Ni, and1=5 =p 4=5j1F;0Ni+p 1=5j0F;0Ni, and the re- sult shows thatj1ihas the worst delity. This is be- cause during the transfer stage, the main factor caus- ing indelity is the decoherence of the FQ, and T 1and T 2of the FQ is about the same in our case so that the spontaneous decay rate is larger than the dephasing rate. Furthermore, in the transfer stage, switches take place betweenj1i=j1F;0Niandj0F;1Ni, while the state j0i=j0F;0Niis unchanged. When the portion of j1i in a general initial state decays into j0F;0Ni, the state transfer process of that portion will stop and will cause indelity. This results in the initial state j1ibeing the worst possible case for the parameters we used. Never- theless, if the eective coupling is stronger through the use of a YIG moderator containing more spins or if the FQ possesses a longer coherence time [28], the delity can be appreciably enhanced. In our scheme, the state transfer interaction can be eectively turned on and o depending on whether or not the NV storage qubit is resonant with the FQ. We thus can attempt to consider engineering a near-perfect step function of the external magnetic eld Bfrom 0 G (o) to 80 G (on). We choose the maximum mag- netic eld strength such that it is still lower than the critical eld of the FQ (the critical eld is 100 G for FQ made of aluminum; could be higher if made of other superconductor)[29]. However, due to technical limits on the charging and discharging times of circuits, the ramp- ing of the magnetic eld cannot be instantaneous and we take this rise time into account. We assume a vari- ation of the magnetic eld over 200 G in 10 ns, similar to what has been reported in experiments [30]. In our simulation, Bis switched from 0 to 80 G with either a linear or exponential ramping over a duration of 4 and 10 ns (see Fig. 4). The results shown in Table II indi- cate that the linear ramping has slightly lower delity than the exponential ramping. This is due to the fact that the linear ramping makes the system stay in the near-resonance regime longer and thus subject to FQ de- coherence longer. Shorter rise times of the magnetic eld can also improve the delity or, alternatively, one can ne tune the transfer time to correct the rise-time and fall-time eects.7 Table I. Fidelity at dierent steps for the initial states of j1i=j1F;0Niand1=2 =p 1=2j1F;0Ni+p 1=2j0F;0Niwith the storage time 10 ms. Initial state g(eff) FN T 2 F(FQ!NV) F(Storage) F(NV!FQ) 1=2700 kHz90s 0.9689 0.9598 0.9318 20s 0.9627 0.9548 0.9218 350 kHz90s 0.9421 0.9363 0.8880 20s 0.9307 0.9270 0.8709 j1i700 kHz90s 0.9317 0.9284 0.8653 20s 0.9268 0.9239 0.8562 350 kHz90s 0.8695 0.8668 0.7537 20s 0.8608 0.8581 0.7386 Table II. Fidelity F=p h tjj tiafter the transfer process with dierent lengths and types of the rise times for the initial states ofj1i=j1F;0Niand1=2 =p 1=2j1F;0Ni+p 1=2j0F;0Ni. State g(e) FN T 2 rise-time func. F(4 ns) F(10 ns) 1=2700 kHz90sexponential 0.9677 0.9647 linear 0.9672 0.9639 20sexponential 0.9613 0.9581 linear 0.9608 0.9573 350 kHz90sexponential 0.9412 0.9395 linear 0.9410 0.9392 20sexponential 0.9296 0.9278 linear 0.9295 0.9275 j1i700 kHz90sexponential 0.9294 0.9241 linear 0.9288 0.9228 20sexponential 0.9246 0.9193 linear 0.9240 0.9180 350 kHz90sexponential 0.8678 0.8648 linear 0.8677 0.8646 20sexponential 0.8591 0.8561 linear 0.8590 0.8559 Figure 4. Temporal variation of the magnetic eld with linear or exponential ramping from o-resonance (0 G) to resonance (80 G) in 4 ns at the beginning, and using an inverse ramping at the end of the state transfer stage. Only the rise-time and fall-time regimes are shown and the storage stage, during which the magnetic eld is xed, is not shown.V. CONCLUSION We have demonstrated how to couple a superconduct- ing FQ with a single electron spin of a NV-center spin via a collective KM in a ferromagnetic material, YIG. This scheme enhances the eective coupling between the FQ and the NV-center spin, allowing the single-NV-center spin to interact with the FQ at a longer spatial distance. This provides greater exibility for the design of hybrid quantum systems. We have proposed a protocol for quan- tum state memory and presented a quantitative analysis of the state transfer, taking into consideration the possi- ble decay channels and imperfect technical issues. This YIG architecture can be used not only as a quantum memory but also as a quantum transducer that couples a single-NV-center spin with other kinds of qubits or with a magnetic eld. ACKNOWLEDGMENTS G.D.L. and H.S.G. acknowledge support from the the Ministry of Science and Technology of Taiwan un-8 der Grants No. MOST 105-2112-M-002-015-MY3 and No. MOST 106-2112-M-002-013-MY3, from the Na- tional Taiwan University under Grant No. NTU-CCP- 106R891703, and from the thematic group program of the National Center for Theoretical Sciences, Taiwan. J.T. acknowledges support from the Center of Excellence in Engineered Quantum Systems. Appendix A: Coupling with magnons Here we describe how the magnons in YIG couple to the FQ or the single-NV spin, and give a detailed deriva- tion of their coupling strengths. The Hamiltonian of a FQ can be written as HF=1 2!F(z) F+1 2(x) F;where!F is the energy of the tunnel splitting, = 2Ip(0=2) is the energy bias, Ipis the persistent current of the FQ,  0 is the uxon, and  is the external ux. If  is tuned to the optimal point with  =  0=2 so that= 0, then one hasHF=1 2!F(z) F. As discussed in the main text, the FQ-YIG coupling comes from the Zeeman-like interac- tion of the spins in the YIG experienced in the magnetic eldBF(r) produced by the persistent current of the FQ (see Fig. 1). Since the persistent current carried by the side wire of the FQ loop near the YIG is along the z axis, the magnetic eld BF(r) =0 2r
Ip(x) Fgenerated by this persistent current is in the xaxis of the YIG [18]. Then the coupling Hamiltonian reads HFY=X r eBF(rF)
Sr =X r0 eIp 2rF (x) FS(x) r; (A1) whererFis the distance between a spin in the YIG and the FQ. The YIG-NV coupling Hamiltonian through the dipole-dipole interaction is written as HYN=0 4 2 eX rN3 (Sr
^rN) (N
^rN)(Sr
N) 2r3 N: (A2) Due to the external magnetic eld Bapplied in the z direction, which makes the frequencies !Nand eBmuch larger than the dipole-dipole coupling strength, we can apply the secular approximation to rewrite Eqs. (A2) as HYN=0 8 2 e~X r3cos2rN1 r3 Nh 3S(z) r(z) NSr
Ni ; (A3) whererNis the distance between a spin in the YIG and the NV spin, and rNis the angle between the vector, which connects the NV and the spin in YIG, and the direction of the external magnetic eld. We can rewritethis Hamiltonian as HYN'H0 YN+H(z) YN, where H0 YN=X rN0 2 e~ 43cos2rN1 r3 N S(+) r() N+S() r(+) N ; (A4) H(z) YN=X rN0 2 e~ 43cos2rN1 r3 N S(z) r(z) N: (A5) Magnons are low-energy spin-wave excitations, which are used to describe the collective behavior of the spins in YIG. Since the system of the quantum memory is at a temperature much lower than the Curie temperature of YIG,Tc=559 K, and the YIG is in an o-resonant cou- pling regime to the FQ and NV-center spin, one expects the excitation number to be very small. In this low- temperature and low-excitation regime, it is convenient to use the Holstein-Primako transformation [17, 20], S(z) r=s+ay rars; (A6) S() r=p 2sr 1nr 2sarp 2sar; (A7) S(+) r= S() ry ; (A8) to transform the spin operators in the YIG into bosonic operators, where each operator is associated with a par- ticle coordinate. Using the creation and annihilation op- erators of the magnon modes in the wave-vector repre- sentation, ay k=1p NX reik
ray r; (A9) ak=1p NX reik
rar; (A10) one can then rewrite the Hamiltonian of YIG as Eq. (6). Since the coupling strength in HFYis far smaller than!Fand!Y, it is valid to use the RWA and Eqs. (A6){(A10) to rewrite the FQ-YIG coupling Hamil- tonian, given by Eq. (A1), as H0 FY=X r0 eIp 2rF (+) FS() r+H:C: X kh gFY(k)(+) Fak+H:C:i ; (A11) with the coupling strength gFY(k) =0 2 eIpr 2s NNX rFeik
a rF: (A12) Similarly, the YIG-NV coupling reads H0 YN'X kh gYN(k)ay k() N+H:C:i (A13)9 where gYN(k) =0 4 2 e~r 2s NNX rN3cos2rN1 r3 N eik
a; (A14) is the coupling strength between the YIG and NV-center spin. Using Eq. (A6), one can also rewrite Eq. (A5) as H(z) YN'YN(z) N; (A15) whereYN=P rN0 4 2 e~3cos2rN1 r3 N sis the induced energy shift to the NV-center storage qubit due to the coupling with the YIG. Appendix B: Derivation of the eective Hamiltonian by the Schrier-Wol transformation Here we describe here the procedure to derive the eec- tive Hamiltonian between the the FQ and the NV-center spin. Following Schrier and Wol's approach [31{33], we can make a canonical transformation eon our orig- inal Hamiltonian H=H0+Hc, whereH0=Hs+H(z) YN withHsdened in Eq. (1), H(z) YNdened in Eq. (A15), andHcdened in Eq. (7). The Hamiltonian after the transformation reads ~H=eHe =H+ [;H] +1 2![;[;H]] +


: (B1) By choosing proper operator satisfying [H0;] =Hc, one has ~H'H0+1 2[;Hc]: (B2) In our case, = lim !0 i1 0Hc(t)etdt ; (B3) and ~H'H0lim !0i 21 0[Hc(t);Hc]etdt; (B4)whereHc(t) =eiH0tHceiH0t. Since the size of the YIG is small, the KM of YIG [13, 15] is gapped from the higher-energy modes. Thus, in a low-temperature and virtual-excitation regime of our scheme, we consider only the KM of the YIG to mediate the eective coupling strength between the FQ and the NV-center spin. To obtain the eective Hamiltonian between the FQ and the NV-center spin without considering the detailed dy- namics of the YIG, we trace out the degrees of freedoms of the YIG , i.e., He=h~HiY, withD ay KaKE Y=nK being the mean occupation number of the KM (similar to a mean-eld approximation). This is a good approx- imation as the YIG is in a low-temperature and low- excitation regime. Since Hc=H0 FY+H0 YN, we can rewrite Eq. (B4) by categorizing the terms of the com- mutators after a trace over the YIG degrees of freedom into two types, HeH0(Hs+Hc) (B5) The rst type in Eq. (B5) reads Hs= lim !0i 21 0h[H0 FY(t);H0 FY] + [H0 YN(t);H0 YN]iYetdt (B6) =1 2F(z) F+1 2N(z) N; (B7) where F=g2 FY(!K)1 !F!K ; (B8) N=g2 YN(!K)1 (!F+YN)!K ; (B9) with!Kdenoting the frequency of the KM, are the en- ergy shifts of the qubits of the FQ and NV systems, re- spectively. We demonstrate how to derive Eq. (B7) from Eq. (B6) by calculating the term containing Fexplicitly, and then the other term containing Ncan be obtained in a similar way. The commutator in the rst term of Eq. (B6) considering only the KM is10 h[H0 FY(t);H0 FY]iY=g2 FYD [(+) F(t)aK(t);() Fay K] + [() F(t)ay K(t);(+) FaK]E Y =g2 FYD ei(!F!K)th () F(+) F+(z) F ay KaK+ 1i +ei(!F!K)th () F(+) F(z) F ay KaK+ 1iE Y =g2 FYn ei(!F!K)th () F(+) F+(z) F(nK+ 1)i +ei(!F!K)th () F(+) F(z) F(nK+ 1)io : (B10) Then, integrating it over time as in Eq. (B6), one obtains lim !0i 21 0h[H0 FY(t);H0 FY]iYetdt=i 2g2 FYlim !01 0n ei(!F!K)th () F(+) F+(z) F(nK+ 1)i +ei(!F!K)th () F(+) F(z) F(nK+ 1)io etdt =g2 FYlim !01 !F!K+i1 2(2nK+ 1)(z) F+1 2IF =g2 FY1 !F!K1 2(2nK+ 1)(z) F+1 2IF '1 2g2 FY1 !F!K (z) F; (B11) where the constant energy term containing the identity operator IFcan be ignored, and nK!0 since the system is operated in a virtual-excitation regime. Equation (B11) is the rst term of Eq. (B7). Similarly, the second term of Eq. (B6) can be calculated and yields the term containing Nin Eq. (B7). The other type Hcin Eq. (B5) represents the eective coupling between the FQ and NV-center spin: Hc= lim !0i 21 0h[H0 FY(t);H0 YN] + [H0 YN(t);H0 FY]iYetdt (B12) =gFN;e (+) F() N+H:C: ; (B13) with gFN;e=1 2gFY(K)gYN(K)1 !F!K+1 (!N+YN)!K : (B14) We now show how to obtain Eq. (B13) from Eq. (B12). Following the same approach as in Eqs. (B10) and (B11), the rst term of the commutator in Eq. (B12) reads lim !0i 21 0h[H0 FY(t);H0 YN]iYetdt= lim !0i 21 0gFYgYND [(+) F(t)aK(t);ay K() N] + [() F(t)ay K(t);aK(+) N]E Yetdt =i 2gFYgYNlim !01 0h ei(!F!K)t (+) F() N +ei(!F!K)t () F(+) Ni etdt =gFYgYNlim !01 21 !F!K+i (+) F() N+1 21 !F!Ki () F(+) N =1 2gFYgYN1 !F!K (+) F() N+() F(+) N : (B15) The second term of the commutator in Eq. (B12) can be evaluated in a similar way and combining with Eq. (B15) give the results of Eqs. (B13) and (B14). Combining all these results, one arrives at the eective Hamiltonian of Eq. (12). [1] J. Clarke and F. K. Wilhelm, Nature (London) 453, 1031 (2008).[2] Z.-L. Xiang, S. Ashhab, J. Q. You, and F. Nori, Rev.11 Mod. Phys. 85, 623 (2013). [3] M. W. Doherty, N. B. Manson, P. Delaney, F. Jelezko, J. Wrachtrup, and L. C. Hollenberg, Physics Reports 528, 1 (2013). [4] J. Twamley and S. D. Barrett, Phys. Rev. B 81, 241202 (2010). [5] Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng, A. Dr eau, J.-F. Roch, A. Aueves, F. Jelezko, J. 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2018-04-30

We propose a quantum memory scheme to transfer and store the quantum state of a superconducting flux qubit (FQ) into the electron spin of a single nitrogen-vacancy (NV) center in diamond via yttrium iron garnet (YIG), a ferromagnet. Unlike an ensemble of NV centers, the YIG moderator can enhance the effective FQ-NV-center coupling strength without introducing additional appreciable decoherence. We derive the effective interaction between the FQ and the NV center by tracing out the degrees of freedom of the collective mode of the YIG spins. We demonstrate the transfer, storage, and retrieval procedures, taking into account the effects of spontaneous decay and pure dephasing. Using realistic experimental parameters for the FQ, NV center and YIG, we find that a combined transfer, storage, and retrieval fidelity higher than 0.9, with a long storage time of 10 ms, can be achieved. This hybrid system not only acts as a promising quantum memory, but also provides an example of enhanced coupling between various systems through collective degrees of freedom.

Single-Nitrogen-vacancy-center quantum memory for a superconducting flux qubit mediated by a ferromagnet

1804.11231v1

Spin-wave dispersion measurement by variable-gap propagating spin-wave spectroscopy Marek Va natka,1,Krzysztof Szulc,2Ond rej Wojewoda,1Carsten Dubs,3Andrii Chumak,4 Maciej Krawczyk,2Oleksandr V. Dobrovolskiy,4Jaros law W. K los,2and Michal Urb anek1, 5,y 1CEITEC BUT, Brno University of Technology, 612 00 Brno, Czech Republic 2ISQI, Faculty of Physics, Adam Mickiewicz University, Pozna n, Poland 3INNOVENT e.V. Technologieentwicklung, 07745 Jena, Germany 4Faculty of Physics, University of Vienna, A-1090 Wien, Austria 5Institute of Physical Engineering, Brno University of Technology, 616 69 Brno, Czech Republic (Dated: July 21, 2021) Magnonics is seen nowadays as a candidate technology for energy-ecient data processing in classical and quantum systems. Pronounced nonlinearity, anisotropy of dispersion relations and phase degree of freedom of spin waves require advanced methodology for probing spin waves at room as well as at mK temperatures. Yet, the use of the established optical techniques like Brillouin light scattering (BLS) or magneto optical Kerr eect (MOKE) at ultra-low temperatures is forbiddingly complicated. By contrast, microwave spectroscopy can be used at all temperatures but is usually lacking spatial and wavenumber resolution. Here, we develop a variable-gap propagating spin-wave spectroscopy (VG-PSWS) method for the deduction of the dispersion relation of spin waves in wide frequency and wavenumber range. The method is based on the phase-resolved analysis of the spin- wave transmission between two antennas with variable spacing, in conjunction with theoretical data treatment. We validate the method for the in-plane magnetized CoFeB and YIG thin lms in k?B andkkBgeometries by deducing the full set of material and spin-wave parameters, including spin-wave dispersion, hybridization of the fundamental mode with the higher-order perpendicular standing spin-wave modes and surface spin pinning. The compatibility of microwaves with low temperatures makes this approach attractive for cryogenic magnonics at the nanoscale. I. INTRODUCTION Properties of magnetic materials are of high interest due to several application concepts regarding, e.g., mem- ories, sensors, microwave devices, or logic devices [1]. In the emerging eld of magnonics, which utilizes spin- waves for data transport and processing, the essential system characteristic is the spin-wave dispersion relation. It provides a connection between the k-space and fre- quency space, and it also dictates other properties like the group velocity vgand decay length . In thin lms (approx. below 30 nm), only the fundamental mode is observed when the experimentally accessible frequency range is limited to few GHz. In contrast, the spin-wave dispersion of thicker lms can be rather complex as mul- tiple perpendicularly standing modes may appear in the spectrum, exhibiting frequency crossing and hybridiza- tion [2]. The spin-wave dispersion measurement is typi- cally done by k-resolved [2, 3] or phase-resolved [4, 5] Bril- louin Light Scattering (BLS). Current interest in quan- tum computing, quantum magnonics [6] and in super- conductor/ferromagnet hybrid systems[7, 8], rises the need for material characterization at ultra-low tempera- tures, where optical access is typically extremely compli- cated. All electrical measurements are usually preferable in these applications. marek.vanatka@live.com ymichal.urbanek@ceitec.vutbr.czThe spin-wave dispersion measurement is also possible using the propagating spin-wave spectroscopy (PSWS). The PSWS is a technique which uses a vector network analyzer (VNA) connected to a pair of microwave an- tennas (e.g., striplines or coplanar waveguides) by mi- crowave probes [9, 10]. The two antennas (i.e., spin- wave transmitter and receiver) have a gap between them over which the spin-waves propagate, as shown in Fig. 1(a). Transmitting antenna is powered by the VNA's microwave source and the receiving antenna serves as an induction pick-up detected by the VNA's second port. The antenna type determines the excitation properties and can be adjusted to the experiment. Main three types of antennas used in experiments are striplines (rectangu- lar wires), U-shaped ground-signal (GS) antennas, and coplanar waveguide (CPW) antennas. Schematic geom- etry of all three antenna types is shown in Fig. 1 (b). Striplines provide a continuous spectrum where the max- imum excited k-vector is limited by the stripline width. Ciubotaru et al. [11] showed scalability of the antennas, where 125 nm wide striplines provided a wide continu- ousk-vector band. Good alternatives are GS antennas, e.g. [9, 12, 13], or coplanar waveguides (CPW), e.g. [14{ 17], both providing a ltering capability for the k-vector spectrum allowing only specic ranges to exist. PSWS can be used on both nanostructured materials (stripes) [11{15, 18, 19] as well as layers [17, 20]. It was previously shown that the PSWS signal can be negligibly dierent for continuous layers and wide stripes [17]. PSWS signals can also be modeled [15, 17, 21]. In previous reports, the spin-wave dispersion was ex-arXiv:2107.09363v1 [cond-mat.mes-hall] 20 Jul 20212 tracted from PSWS spectra measured on yttrium iron garnet (YIG) using the CPW excitation. As the CPWs excitation spectrum exhibits distinct peaks in k-space, it allows extracting one point in spin-wave dispersion for each peak. The central k-vector of each peak is then as- signed to a frequency from either the envelope of the S21 sweep [18, 22] or by tting the S21spectrum [17]. This approach is limited to only several extracted points, and it is not easily transferable to metallic materials because of the low signal amplitude (compared to YIG) caused by large damping, making it impossible to use more than two peaks from the CPW antenna's excitation spectrum. Here, we show that the spin-wave dispersion measure- ment using VNA is possible with a high level of details determined by the VNA frequency step. II. MEASUREMENT SETUP AND SAMPLE PREPARATION Our setup uses Rohde & Schwarz ZVA50 VNA and GGB industries microwave probes to establish a con- nection to the two antennas lithographically fabricated on top of CoFeB and YIG thin lms. The lithogra- phy process consisted of e-beam patterning using PMMA resist, e-beam evaporation of Ti 5 nm/Cu 85 nm/Au 10 nm multilayer, and lift-o. The CoFeB lms (nom- inal thicknesses 30 nm and 100 nm) were magnetron- sputtered from Co 40Fe40B20(at. %) target on (100) GaAs substrate with 5 nm Ta buer layer. The (111) YIG lms were grown by liquid phase epitaxy on top of a 500 m, thick (111) gadolinium gallium garnet substrate [23, 24]. The samples were placed in a gap of a rotatable electro- magnet allowing to apply an in-plane magnetic eld up to 400 mT in an arbitrary direction with respect to the spin-wave propagation direction. The VNA was set us- ing a calibration substrate supplied with the microwave probes. A power sweep was performed before measur- ing each type of the sample-antenna combination to nd a suitable power level that avoids nonlinear phenomena [25] and maintains a sucient signal-to-noise ratio. The VNA controls and analyses the electric signals of transmitting and receiving antennas both in the domain of amplitude and phase, therefore, it can measure the phase acquired by the spectral components of spin wave while it propagates between antennas. Our analysis is based on the S21transmission parameter. The trans- mitted spin-wave signal is modied by a nonmagnetic background Sbackground 21 that is always present in the ex- periment due to direct electromagnetic crosstalk between the antennas. This background is constant for dierent values of static magnetic eld, and therefore it is possible to evaluate it as the median over all measured magnetic elds. The subtracted signal
S21is then calculated as:
S21=S21Sbackground 21 . Fig. 1 (c,d) shows the
S21signal, measured at 0 dBm power output for a 30 nm CoFeB thin lm, over the gap 1.8m in the k?Bgeometry (magnetostatic surfacewaves), and with 500 nm wide striplines used as excita- tion and detection antennas. This geometry is known to be nonreciprocal with an exponential distribution of the dynamic magnetization along the layer's thickness due to the surface localization of the mode [26, 27]. The higher signal amplitude in the + Bpart of the spectrum is caused by both the stronger excitation and by the larger induc- tion pick up from spin-waves propagating at the nearer surface. To achieve the best result, we can focus on + B part of the spectrum (or alternatively on the Bpart and use the reverse transmission parameter S21). Fig. gap kPort1Port2GND GNDGND GNDtransming an tenna receiving an tennavariable(a) GND GNDGND GNDGND GNDGND GNDGNDGND VNA port 1 VNA port 2 VNA port 2VNA port 1VNA port 1 VNA port 2striplines: GS antennas: CPW antennas(b) antenna types k k k Re( ) Im( ) FIG. 1. (a) Schematics of the PSWS experiment using a pair of stripline antennas. (b) Schematic geometry of the three most commonly used antenna types. The outcomes of the PSWS measurement: (c) real and (d) imaginary part of
S21(B;f) for 30 nm thick CoFeB layer 30 nm in k?B geometry and the gap width of 1 :8m. The measurement shows the nonreciprocity of spin-wave propagation for the k? Bgeometry which is re ected by the larger signal in + Belds than inBelds. The plots of (e) show real and imaginary parts of
S21(f), and (f) the unwrapped phase of
S21(f), at xed eld B= 20 mT.3 1(d) shows the plot of real and imaginary parts of
S21 measured at 20 mT. The corresponding phase, which was unwrapped, is shown in Fig. 1(e). The phase rises from the ferromagnetic resonance (FMR) frequency up until it reaches the antenna's excitation limits. Beyond this point, the signal loses its coherency due to insucient signal-to-noise ratio, and therefore the phase stops evolv- ing. The slope of the phase depends on the gap size { it changes more rapidly for wider gaps. In the next step, we repeat the measurement on multi- ple instances of identical antenna structures (with vary- ing gap width) prepared on the same 30 nm CoFeB thin lm sample. The phases measured over 11 gap widths are shown in Fig. 2(a). The phases are on the same level before the frequency reaches FMR, and then they start to rise. The lowest phase corresponds to the smallest gap width and the uppermost to the largest measured gap width. Then we project the measured phases into a phase { gap width plot [selected frequencies are plotted in Fig. 2(b)]. In this projection, the phase shows a lin- ear dependence (for a coherent plane wave) that can be tted; the slope equals to the k-vector at the given fre- quency. Now we can plot the extracted k-vectors against their frequencies, showing the resulting dispersion rela- tion in Fig. 2(c). To conrm the result, we remeasured the same sample using phase-resolved BLS, and we found a very good agreement. The comparison of the dispersion relations measured by both techniques (VNA { red, BLS { blue) is shown in Fig. 2(c), and the comparison of the measured phase evolution at the frequency of 11.5 GHz is shown in Fig. 2(d). III. MEASURED SPIN-WAVE DISPERSION RELATIONS Fig. 3(a) shows dispersion relations of the same 30 nm CoFeB thin lm measured in k?Bgeometry in mag- netic elds ranging from 20 mT to 380 mT with a step of 60 mT. The sample was also measured in kkBgeom- etry, but the measured signal was insucient to recon- struct the dispersion relation. For all measured elds, it was possible to evaluate the dispersion for k-vectors ranging from 0 up to 8 rad/ µm. The upper limit is given by the excitation eciency [15] of the used antenna (see blue lines in Fig. 3). By tting the measured dispersions using the Kalinikos-Slavin model [28], we were able to obtain material parameters of the measured thin lms (see black lines in Fig. 3 for the model ts and the gure caption for the tting results). In addition to the 30 nm CoFeB thin lm, we also mea- sured 100 nm thick CoFeB [Fig. 3(b)] and 100 nm thick YIG [Fig. 3(c)] lms to further explore the possibili- ties of the presented technique. The measurements were performed using dierent antenna types (stripline, GS, CPW) to see their in uence on the quality of the obtained dispersions, and they were also evaluated for multiple magnetic elds in k?Band in kkBgeometries. The 5 10 15 f (GHz)04 :8 :12 :Arg( "S21) (rad) 7GHz 8GHz 9GHz 10GHz 11GHz 12GHz 13GHz 14GHz 15GHzB=20mT 2 4 gap size (µm)04 :8 :Arg( "S21) (rad) 7GHz8GHz9GHz10GHz11GHz12GHz13GHz14GHz15GHz 0 5 k (rad/µm)51015f (GHz) VNA BLS 1 2 3 gap size (µm)02 :4 :Phase (rad)f=11.5GHz VNA BLS(a) (b) (c) (d)FIG. 2. Extraction of the dispersion relation from the out- comes of PSWS experiment on 30 nm CoFeB thin lm at B= 20 mT in k?Bgeometry. (a) The unwrapped
S21 phases measured over gap widths ranging from 0 :9m (the lowest line) to 2 :9m (the steepest line) with the step of 200 nm. (b) The representative ts of the phase at selected frequencies versus gap size where the slope of the t yields the desiredk-vector at that frequency. (c) The dispersion relation extracted for all frequencies within the range with sucient PSWS signal (red line). The dispersion obtained by phase- resolved BLS is plotted in the blue points for comparison. (d) The compared outcomes of the phase measurements for VNA (red circles) and phase-resolved BLS (blue squares) at single frequencyf= 11:5 GHz. 100 nm thick CoFeB lm was measured in the elds rang- ing from 20 to 380 mT with a step of 60 mT at a power of 0 dBm. In Fig. 3(b) we show the dispersion measured using 500 nm wide stripline antenna for k?Bgeometry and with 500 nm coplanar waveguide (signal and ground line widths, as well as signal-to-ground gap, were 500 nm) ink?Bgeometry. We were able to obtain a dispersion inkkBgeometry also using a 500 nm wide stripline antenna but with substantially worse quality due to its lower excitation eciency. The 100 nm thick YIG lm was measured in k?BandkkBgeometries in the elds ranging from 20 to 200 mT with a step of 20 mT at a power of30 dBm. In this case, it was possible to obtain a dispersion in both geometries using all types of antennas. Fig. 3(c) shows data acquired by GS antennas with gap widths from 1.0 to 3.4 m with 400 nm step. As in the case of the coplanar waveguide, the dimensions of the GS antennas were 500 nm (signal and ground line widths and signal-to-ground gap width). The dispersions are plotted for magnetic elds from 20 to 200 mT with a step of 20 mT and the power output was set to 30 dBm. In the dispersion measured on 100 nm thick CoFeB lm, we further identied visible hybridizations at the crossings' positions between the n= 0 fundamental spin- wave mode and n= 1;2;3 higher-order perpendicularly4 0 5 10 k (rad/µm)5101520253035f (GHz) Jexc (normalized) -10 -5 0 5 10 k (rad/µm)10203040f (GHz) Jexc (normalized) -10 -5 0 5 10 k (rad/µm)2468f (GHz) Jexc (normalized)(a) (b) (c) 20 mT 20 mT20 mT380 mT 260 mT 140 mT 380 mT 260 mT 140 mT140 mT 100 mT180 mT 60 mTP=0 dBm P=0 dBm P=-30 dBm FIG. 3. The dispersion relations measured using the PSWS technique (red points) compared to the analytic dispersions from Kalinikos-Slavin model (black lines) for (a) 30 nm thick CoFeB (b) 100 nm thick CoFeB, (c) 100 nm thick YIG layers. For clarity, the data for k?Bare plotted in the domain of positive wave numbers kand the kkBdata are presented with negative k. The samples were excited by stripline, CPW, and GS antennas with all lateral wire widths dimensions of 500 nm. The blue lines show calculated (and normalized) wave number excitation spectra of the antennas, calculated form the spatial distributions of microwave eld (the solid, dashed and dotted lines correspond to the spectra of stripline, GS, and CPW antenna, respectively). For Kalinikos-Slavin model we tted the following values of parameters: (a) Ms= 1230 kA/m (0Ms= 1:54 T), =2= 30:5 GHz/T,t= 29:0 nm; (b)Ms= 1310 kA/m ( 0Ms= 1:64 T), =2= 30:1 GHz/T; (c) Ms= 133 kA/m (0Ms= 0:166 T), =2= 28:3 GHz/T, where tstands for thickness of the layer. In tting procedure, we xed the exchange stiness to Aex= 15 pJ/m3for CoFeB lms and Aex= 3:6 pJ/m3for the YIG lm. standing spin-wave modes (see Fig. 4). The hybridiza- tions are present in the measured dispersion relation as gap openings, with a portion of the measured data evolv- ing towards smaller k-vectors [see Fig. 4(d) for the most prominent example]. This part of the dispersion has no physical meaning and is only a result of the data process- ing described earlier in the text. The hybridizations are also directly visible as dips in the magnitude Abs(
S21) of the transmission spectra [see Fig. 4(b)]. IV. NUMERICAL MODELING To investigate numerically the hybridization between the fundamental mode and perpendicularly standing modes, we solve Landau-Lifshitz equation (LLE) using the nite-element method (FEM) in the frequency do- main. The LLE was linearized and implemented in FEM solver (COMSOL Multiphysics) as a set of dierential equations for in-plane and the out-of-plane components of magnetization mkandm?: i!mk=j j0[(H02Aex 0Ms
)m?+Ms@x?'];(1) i!m?=j j0[(H02Aex 0Ms
)mk+Ms@xk'];(2) together with the equation for the scalar magnetostatic potential'derived from Maxwell equations in magneto- static approximation [29, 30]:
'@xkmk@x?m?= 0; (3) wherexkandx?denote the directions of the correspond- ing dynamic components of magnetization.In contrast to the analytical model of Kalinikos and Slavin, our numerical calculations truly reproduce the spin dispersion relation in the crossover dipolar-exchange regime [2], including the hybridizations between the fun- damental spin-wave mode and perpendicularly standing spin-wave modes. In our model, we also consider the presence of surface anisotropy Ks. The anisotropy is ex- pected to be dierent on the bottom and the top faces of the magnetic layer interfaced with dierent materi- als. The surface anisotropy and its asymmetry (between the top and bottom face) is responsible for the spin-wave pinning and the strength of the hybridization between fundamental and perpendicularly standing modes. The pinning is implemented in the boundary conditions [31] @x?mk= 0; (4) Aex@x?m?Ksm?= 0: (5) The numerical model (as well as the analytical model by Kalinikos and Slavin) is characterized by a few ma- terial parameters. We xed the value of exchange sti- ness toAex= 14 pJ/m2and layer's thickness to nom- inal valued= 100 nm. The values of saturation mag- netization Ms= 1275 kA/m and gyromagnetic ratio j j=2= 30:8 GHz/T were selected to t the ferromag- netic resonance frequency (i.e., the frequency of funda- mental mode at k= 0) and the slope of the dispersion for fundamental mode. The surface anisotropy Ksde- termines the spin-wave pinning and therefore is impor- tant both for the quantization (and the frequencies) of the perpendicularly standing modes and the strength of the hybridization between and perpendicularly standing modes. To obtain the proper values of the frequencies5 of perpendicularly standing modes, we did not need to reduce the thickness below the nominal value 100 nm but we had to introduce the non-zero Ksinstead, which seems to be a more realistic approach. The experimental data show that the hybridizations of fundamental mode with the perpendicularly standing mode are observed both for the perpendicularly standing modes quantized with even (n= 2) and odd number of nodes ( n= 1;3) across the layer. The eect for n= 1;3 is hardly visible in disper- sion relation [Fig. 4(c,e)] but is quite distinctive in the measurement of the transmission amplitude [Fig. 4(b)]. These results indicate that the asymmetric pinning (and dierent values of surface anisotropy on both faces of the CoFeB layer) must be considered to obtain less symmet- ric proles of spin-wave modes across the thickness of the layer, which in turn, gives the non-zero cross-sections be- tween fundamental mode and perpendicularly standing modes of odd numbers of nodes. There is always some degree of freedom for choosing the values of Kson both faces, therefore, we decided to consider the simplest case whereKs= 0 and 1600 J/m2at the top (interface with vacuum) and bottom (interface with GaAs). With these values, we succeeded in the induction of the hybridiza- tion of fundamental mode ( n= 0) with the rst ( n= 1) and third (n= 3) perpendicularly standing modes, char- acterized by the width comparable to the widths of the deeps of the transmission
S12[orange stripes in Fig. 4(a,b)]. It is worth noting that in the case of symmetric pinning (we took Ks= 700 J/m2on both faces), we do not observe the hybridization with the rst ( n= 1) and third (n= 3) perpendicularly standing modes [see blue dashed lines in Fig. 4(c,e)]. V. ADDITIONAL DATA ANALYSIS AND DISCUSSION From the measured dispersion relations, it is possi- ble to extract also other parameters important for spin- wave propagation. The group velocity vgcan be cal- culated as the numerical derivative of the dispersion vg= 2df=dkand the lifetime can be obtained as the numerical derivative of eld-dependent dispersion [29]: = (!@! @!B)1= (42f @f @B)1, where!B= Band is a damping constant. The decay length can be then calculated by multiply- ing the group velocity and the lifetime =vg. In Fig. 5(a), we plot the decay length obtained by the above- described procedure (orange line) and compare it with the decay length obtained using the more traditional ap- proach, i.e., by tting the exponential decay of the mag- nitude of
S21signal (blue line). Here, the decay length was obtained by tting [see Fig. 5(b) for representa- tive ts] the formula Abs(
S21) =Ieg=in logarithmic form ln[Abs(
S21)] = ln(I)1 g, wheregis the gap distance, and Iis a free parameter proportional to sig- nal strength. In Fig. 5(c), we plot eld-dependent life- time evaluated using the decay length obtained from the 0 0.5 k (rad/µm)21222324f (GHz) 11.5 22.5 k (rad/µm)242526f (GHz) 5 6 k (rad/µm)282930f (GHz)-3 Abs( ) (x10)0 2 4 6 8 k (rad/µm)202530f (GHz) 0 5(a) (e) (d) (c)(b) B=260 mTn=1n=2n=3 n=0 01-detail 02-detail 03-detailexperiment mod. asym. pinning mod. sym. pinningFIG. 4. (a) Measured dispersion relation (red points) for 100 nm thick CoFeB layer at 260 nm exhibiting hybridized spin-wave modes, visible mainly at the n= 02 crossing. The dashed line represents dispersion simulation with asymmet- rical surface pinning conditions with parameters KS= 0 and 1600 J/m2on top and bottom face of the layer, re- spectively. (b) Corresponding magnitude of the
S21trans- mission parameter exhibiting dips at the crossings' positions ([Abs(
S21)] data shown for the gap width of 1.8 m). (c-e) details of the 01, 02, and 03 crossings, respectively. Dashed blue lines correspond to the case of symmetric pinning ( KS= 700J/m2on both faces). derivative of eld-dependent dispersion ( @!=@!B, orange circles) and from tting the exponential decay of the mag- nitude of
S21signal (blue diamonds). Both approaches give roughly the same results for both the decay length and the lifetime, but the decay lengths obtained using the @!=@!Bapproach agree better with the analytical model [see Fig. 5(a), green line]. The lower quality of the data obtained from tting the exponential decay of the mag- nitude of
S21signal is caused by the limitation of our experimental arrangement. Due to the nite length of the excitation antenna, spin-wave caustics can form [32]. An example of such caustics measured by BLS microscopy is shown in Fig. 5(e). The phase in the caustic beam is spa- tially incoherent [33] and the focusing eect [34] causes modulation of the spin-wave intensity along the propa- gation direction. The associated adverse eects can be avoided by measuring at propagation distances that are suciently small with respect to the stripline length. In our experiments, we used the stripline length of 10 m, and the maximum measured propagation distance was 2.9m. For phase-resolved measurement such as evalu- ation of dispersions, the short propagation distance was sucient. On the other hand, the change in signal at 2.9m propagation distance was not sucient to obtain a fully reliable t of the exponential decay of the
S21 magnitude and at longer propagation distances the mea- surements were distorted by caustics. We also analyzed how the number of measured propa-6 gation distances (gaps) aected the reliability of the ob- tained dispersion. We took the data measured for 11 gap widths in total and then used dierent combinations of a reduced number of gap widths (starting from 2 up to 10) to evaluate the dispersion. The dispersion obtained from the reduced number of gap widths was then compared to the analytical t Kalinikos-Slavin model [28] of the dis- persion obtained from tting of the complete set of 11 measured points. As shown in Fig. 5(d), the mean fre- quency dierence from the reference dispersion can rise to the maximum of 400 MHz when using a combination of just two gap widths. This maximum quickly decreases to 200 MHz for ve gap widths and then stays around 100 MHz for combinations of six and more gaps. As can be seen from the presented data, the variable- gap approach allows the reconstruction of the spin-wave dispersion relation, including detailed features that might be used for analysis of the spin-wave system. In order to obtain the highest possible data quality, we need to fabri- cate multiple pairs of antennas where the gap width and the step in the gap width (an increase of the gap between two pairs of antennas) must be optimized for each exper- iment. Note that this approach does not require precise knowledge of the excitation spot location (phase origin) because it does not aect the t's slope. It is only essen- tial to know the relative dierences between the propa- gation distances, i.e., the gap-width step, which can be easily measured (and precisely fabricated using e-beam lithography). The antenna design is sample-specic and needs to be tailored to full experimental requirements. The an- tenna's type and geometry must be able to excite the expectedk-vector range with sucient eciency. The gap widths need to be in the optimum range con- sidering the caustics formation and the spin-wave decay length for the given material and geometry. For example, for CoFeB 30 nm in kkBgeometry at k= 5 rad/ m, we calculated decay lengths of 0.03 m and 0.45 m for magnetic elds of 20 and 200 mT, respectively. Here, the quality of the measured data, even for the smallest fabri- cated gap distance of 1 m, was not sucient to evaluate dispersion; thus, this geometry is not presented in Fig. 3(a). Before tting the data as plotted in Fig. 2(b), the phase needs to be correctly unwrapped. In the case of stripline antennas with a continuous excitation spectrum, it is possible to achieve correct unwrap even when the phases for neighboring gap widths at the same frequency are higher than rad by unwrapping the phase in the fre- quency spectrum [Fig. 2(a)]. It is because the frequency step size of the VNA is usually small and the phase shift of the neighboring points is always smaller than rad. On the other hand, the safest approach is to unwrap the phases when plotted against the gap size [Fig. 2(b)]. In this case, the phase change of neighboring points must be smaller than rad. This is necessary for antenna types with discrete excitation spectra (i.e., CPW, GS, ladders [42] or meanders [43]). meas. error error rangef=14 GHz f=16 GHz f=18 GHzB=100 mT k=2 r ad/µm B BLS intensity (counts)VNA signalGND GNDstripline 1 stripline 2(e)FIG. 5. The determination of decay length and lifetime from VG-PSWS measurement of 30 nm thick CoFeB lm (a- c,e) and the impact of the number of gaps on the quality of the result (d). (a) Blue line shows the spin-wave decay lengths obtained directly from exponential ts, orange line shows the decay length calculated by multiplication of the life time and the group velocity and the green line is calcu- lated from the Kalinikos-Slavin model. (b) Representative exponential ts of the
S21magnitude which were used to obtain the spin-wave propagation length. (c) Spin-wave life time extracted from the derivative of eld-dependent disper- sion using= 0:075 (orange circles), calculated by dividing the propagation length (from
S21magnitude) by the numer- ically calculated group velocity (blue diamonds) and calcu- lated from the Kalinikos-Slavin model (green line). (d) Mean frequency dierence of the dispersion obtained from the re- duced number of gap widths compared to the analytical t of the dispersion obtained from the complete set of 11 measured points. Blue points represent dierent combinations of gaps and the red area represents the dierence spread. (e) Spin- wave caustics caused by the nite length of the antenna, mea- sured on 100 nm CoFeB lm by BLS with exciation frequency of 11 GHz in the external magnetic eld of 26 mT. Caustics can destroy the phase coherence of propagating spin-waves. Note that all presented data were collected at propagation distances (gap widths) between 0.9 m to 2.9 m to avoid the negative eects of the caustics formation.7 TABLE I. Comparison of experimental techniques used for measurement of spin-wave dispersion relations. Lateral Resolution Resolution Phase max k Experimental resolution in k(rad/ m) inf(Hz) extraction (rad/ m) geometries in (nm) (MS, BV, FV) VG-PSWS [this work] - <0:1* 1 yes 10*** MS, BV, (FV possible) PSWS [17, 21] - 2*** 1 no 10*** MS, BV, FV MOKE [35, 36] 500 <0:1* 1
108** yes 10 BV, MS Conventional BLS [2, 37] 10,000 0.5 1
108no 23.6 BV, MS Phase-resolved BLS [4, 38, 39] 250 <0:1* 1
108** yes 7 BV, MS STXM [40, 41] 20 <0:1* 1
108** yes 30 BV, MS MS - magnetostatic surface spin waves, BV - backward volume spin waves, FV - forward volume spin waves * Thek-resolution can be tailored to the magnetic system under study by an appropriate design of the experiment geometry. **The resolution in fin these techniques is limited by the acquisition time. *** Excitation limited. The step in the gap width denes the maximum k- vector for which the dispersion can be measured. The phase change of the spin wave along the step distance has to be smaller than rad except for certain cases discussed later. On the other hand, small k-vectors may have a phase change that is too small over a short step in the gap width. If we require an accurate tting of smallk-vectors, the step in the gap width should be large enough to provide sucient phase change (we suggest at least 1 rad) while respecting the decay length. VI. METHOD COMPARISON In Table 1, we compare the variable gap propagat- ing spin-wave spectroscopy with other experimental tech- niques used to obtain spin-wave dispersion relations. To get detailed dispersion, one needs to have high resolu- tion in both the k-vector and the frequency. Optical and X-ray techniques have low frequency resolution, which is typically limited to hundreds of MHz. In the case of conventional BLS, the resolution is determined directly by the Fabry-Perot interferometer [44]. The optical tech- niques using microwave excitation, i.e., Magneto-Optical Kerr Eect microscopy (MOKE), Phase-resolved BLS and Scanning Transmission X-ray Microscopy (STXM), are limited by slow signal acquisition. It does not allow capturing the required span of frequencies with a high resolution in a reasonable time. The propagating spin- wave spectroscopy technique, which uses known positions of the excitation peaks of the CPW antenna, can cap- ture the data with a high resolution in frequency. How- ever, thek-resolution is limited with the nite widths of peaks in the excitation spectra of the used CPW anten- nas which in turn aect also the frequency resolution. Big advantage of the VG-PSWS technique is that it does not require direct optical access to the sample, mak- ing the method very suitable for, e.g., experiments at ultra-low temperatures. In addition, compared to other techniques for spin-wave dispersion relation measurement VG-PSWS can achieve high resolution in frequency andk-vector at the same time. This combination allows cap- turing smooth dispersion curves over the span of all ac- cessiblek-vectors and fast acquisition times allow repeat- ing the experiment in multiple magnetic eld strengths and orientations. The full set of eld-dependent disper- sion curves measured for both k?Band in kkBge- ometries present a robust 4-dimensional ( f,k,B, angle) dataset that can be further evaluated, and all essential material and spin-wave parameters can be extracted from it. A disadvantage of this method is the need for a set of antennas with multiple gap widths on top of the sample. The other techniques can obtain the dispersion either without any antenna (conventional BLS), with one exci- tation antenna (MOKE, Phase-resolved BLS, STXM) or with a pair of excitation and detection antennas (PSWS). However, the need for multiple antennas may be over- come in the future by using freestanding positionable an- tennas [45]. VII. CONCLUSION In conclusion, we presented a new method of extrac- tion of high-quality spin-wave dispersion relation from propagating spin-wave spectroscopy (PSWS) measure- ments performed over several propagation distances. We demonstrated this technique on CoFeB and YIG thin lms measured in k?BandkkBgeometries. The re- sults on CoFeB thin lm were veried by phase-resolved BLS measurement showing good agreement. When compared with the phase-resolved BLS, the VNA-based method provides more frequency measurement points in a shorter acquisition time. Fine detail measurement ca- pability was demonstrated on the measurement and anal- ysis of hybridized modes acquired on 100 nm thick CoFeB thin lm, revealing asymmetric surface pinning and the values of pinning parameters on both interfaces of the magnetic layer. The all electric nature of this method makes it very suitable for characterization of cryogenic and quantum magnonics systems and materials.8 ACKNOWLEDGMENTS The work was supported by MEYS CR (project CZ.02.2.69/0.0/0.0/19 073/0016948). 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2021-07-20

Magnonics is seen nowadays as a candidate technology for energy-efficient data processing in classical and quantum systems. Pronounced nonlinearity, anisotropy of dispersion relations and phase degree of freedom of spin waves require advanced methodology for probing spin waves at room as well as at mK temperatures. Yet, the use of the established optical techniques like Brillouin light scattering (BLS) or magneto optical Kerr effect (MOKE) at ultra-low temperatures is forbiddingly complicated. By contrast, microwave spectroscopy can be used at all temperatures but is usually lacking spatial and wavenumber resolution. Here, we develop a variable-gap propagating spin-wave spectroscopy (VG-PSWS) method for the deduction of the dispersion relation of spin waves in wide frequency and wavenumber range. The method is based on the phase-resolved analysis of the spin-wave transmission between two antennas with variable spacing, in conjunction with theoretical data treatment. We validate the method for the in-plane magnetized CoFeB and YIG thin films in $k\perp B$ and $k\parallel B$ geometries by deducing the full set of material and spin-wave parameters, including spin-wave dispersion, hybridization of the fundamental mode with the higher-order perpendicular standing spin-wave modes and surface spin pinning. The compatibility of microwaves with low temperatures makes this approach attractive for cryogenic magnonics at the nanoscale.

Spin-wave dispersion measurement by variable-gap propagating spin-wave spectroscopy

2107.09363v1

1 An Edge-Coupled Magnetostatic Bandpass Filter Connor Devitt, Graduate Student Member, IEEE , Renyuan Wang, Member, IEEE , Sudhanshu Tiwari, Member, IEEE , Sunil A. Bhave, Senior Member, IEEE Abstract —This paper reports on the design, fabrication, and characterization of an edge-coupled magnetostatic forward vol- ume wave bandpass filter. Using micromachining techniques, the filter is fabricated from a yttrium iron garnet (YIG) film grown on a gadolinium gallium garnet (GGG) substrate with inductive transducers. By adjusting an out-of-plane magnetic field, we demonstrate linear center frequency tuning for a 4th- order filter from 4.5 GHz to 10.1 GHz while retaining a fractional bandwidth of 0.3%, an insertion loss of 6.94 dB, and a -35dB rejection. We characterize the filter nonlinearity in the passband and stopband with IIP3 measurements of -4.85 dBm and 25.84 dBm, respectively. When integrated with a tunable magnetic field, this device is an octave tunable narrowband channel-select filter. Index Terms —Micromachining, magnetostatic wave (MSW), yttrium iron garnet (YIG), tunable bandpass filter, edge-coupled I. I NTRODUCTION COUPLED micro-electromechanical resonators with high quality factors (Q-factor) and miniaturized footprints have been an attractive technology for integration in wireless communication systems as narrowband channel-select filters. Wang et al [1] has demonstrated high-order micromechan- ical bandpass filters using one-dimensional (1D) arrays of mechanically-coupled resonators. However, for higher-order filters, long mechanical coupling beams become impractical due to size constraints [2] while sensitivity due to fabrica- tion variation causes increased insertion loss and passband distortion [3]. The high sensitivity of weak electrostatic or mechanical edge-coupled resonators due to structural asym- metries has been leveraged for sensitive parametric mass sensing applications in [2], but prohibits their use in filters. 2D microresonator arrays have shown some success by utilizing weak coupling in one dimension to achieve pass band shape and strong coupling in the other dimension to reduce effects of fabrication variation, but suffer from high insertion loss Manuscript received on XX XX, 2023; revised on XX XX, 2024; accepted on XX XX, 2024. This research was developed with funding from the Air Force Research Laboratory (AFRL) and the Defense Advanced Research Projects Agency (DARPA). The views, opinions and/or findings expressed are those of the authors and should not be interpreted as representing the official views or policies of the Department of Defense or the U.S. Government. This manuscript is approved for public release; distribution A: distribution unlimited. ( Corresponding authors: Connor Devitt, Renyuan Wang ) R.W. invented the device concept, and developed baseline model and device design. C.D. performed simulations on fabrication variations and total filter loss, chip fabrication and characterization, as well as data analysis. Manuscript was prepared by C.D. with inputs from R.W., S.T., and S.A.B. Connor Devitt (e-mail: devitt@purdue.edu), Sudhanshu Tiwari (e-mail: tiwari40@purdue.edu), and Sunil A. Bhave (e-mail: bhave@purdue.edu) are with the OxideMEMS Lab, Elmore Family School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907 USA. Renyuan Wang is with FAST Labs, BAE Systems, Inc., Nashua, NH 03060 USA (e-mail: renyuan.wang@baesystems.com). (a) (b) Fig. 1. (a)Chip microphotograph of multiple MSW bandpass filters and 1- port resonators fabricated on a YIG on GGG chip using YIG micromachining technology [6]. A 4-pole filter (on the left) and a 2-pole filter (on the right) are highlighted in red. A 1-port resonator is highlighted in orange. (b)Rendering of a 4-pole bandpass filter featuring four YIG resonators with gold electrodes conformally deposited over the etched YIG. Magnetic bias is oriented out-of- plane along the z-axis. [3]. Coupled resonator arrays utilizing magnetostatic waves (MSW) have the potential to overcome the weak coupling and extremely narrow bandwidths achieved by electrostatically coupled micromechanical resonators [4], [5] while introducing a degree of tunability. The magnetostatic wave resonance can be tuned more than an octave in frequency using a static magnetic field ensuring that the filter size does not scale to sub-micrometer dimensions at high-frequencies. Yttrium iron garnet (YIG) is the most widely used material for MSW devices due to its low Gilbert damping ( α= 2.8×10−4for a 100 nm film [7]) and experimentally demonstrated Q-factors exceeding 3000 [8], [9]. In state of the art YIG sphere filters [10]–arXiv:2312.10583v1 [physics.app-ph] 17 Dec 20232 [12], polished YIG resonators are attached to a thermally conductive rod and manually aligned to non-planar inductive loops acting as transducers. The assembled sphere and loop structures are then coupled through transmission lines similar to the coupling beams in [1] to synthesize a filter. Planar YIG resonators can magnetically couple if they are fabricated in close proximity (Fig. 1b), analogous to the electrically coupled mechanical resonators in [2], [13] or coupled electromagnetic resonators. Magnetically coupled YIG filters can be fabricated at scale using micromachining techniques on films grown on a gadolinium gallium garnet (GGG) substrate, allowing for miniaturization and eliminating the need for polishing and meticulous manual alignment. II. B ANDPASS FILTER DESIGN The bandpass filter shown in Fig. 1bconsists of a number of closely-spaced rectangular YIG magnetostatic forward volume wave (MSFVW) resonators with shorted 300 nm -thick gold in- ductive transducers conformally deposited over the outermost resonators. With an out-of-plane DC magnetic bias, the RF magnetic field from the transducers excite MSFVW modes in the YIG mesa. Forward volume waves are a family of highly dispersive modes in a thin film whose lowest order mode is described by the dispersion relation [14]: ω2=ω0 ω0+ωm 1−1−e−kmnt kmnt , (1) where ωm=µ0γmMs,ω0=µ0γmHeff DC,tis the film thickness, γmis the gyromagnetic ratio, µ0is the permeability of free space, kmnis the wave vector, Msis the saturation magnetization, and Heff DCis the effective DC magnetic field. Considering the limits as kmn→0andkmn→ ∞ in the dispersion relation, ωis restricted within the range [14]: ω0≤ω≤p ω0(ω0+ωm), (2) denoted as the spin wave manifold. When the planar dimen- sions of the thin YIG film are bounded, the magnetostatic waves reflects off the edges forming a standing waves with wave vectors approximately given by [15]–[19]: kmn=rπm l2 +πn w2 , m, n = 1,2,3. . . (3) where landware the length and width of the cavity respec- tively. These MSFVW resonances can be further understood through an analogy to Lamb waves in a piezoelectric plate [20]–[23]. An oscillating electric field perturbs the polarization of the piezoelectric film generating a stress field and exciting an acoustic wave. In the ferrimagnetic film, an oscillating magnetic field, conversely, perturbs the static magnetization leading to a precession of spins. Both the piezoelectric and magnetostatic cavities support a discrete number modes whose wave vectors depend on the cavity dimensions. Nonlinear dispersion relates the wave vectors to the resonant frequencies for both MSFVW and Lamb waves. For MSFVW, this leads to irregularly spaced modes which all reside in the spin wave manifold.Unique to MSW, the applied out-of-plane magnetic field shifts the MSFVW dispersion relation in frequency where the tuning rate for the fundamental mode is given by ∂ω ∂Heff DC=µ0γm2ω0+ωm 1−1−e−kmnt kmnt 2r ω0h ω0+ωm 1−1−e−kmnt kmnti,(4) which simplifies to µ0γm= 2.8 MHz /Oe(for YIG) when kmnt≪1. The magnetostatic scalar potential, ψ, decays exponentially outside the YIG mesa [14], [24] so the MSFVW resonance in one YIG mesa may couple to adjacent mesas if there is sufficient overlap in their scalar potentials [25]. Consequently, the spacing of adjacent resonators and their ver- tical sidewall profile are critical to control the inter-resonator coupling strength. For the 4-pole filter in Fig. 1b, the resonator spacings are s1= 10 µm,s2= 15 µm, and s3= 10 µm. Each has length l= 500 µm. The outermost resonators have a width w1,4= 70 µmwhile the inner resonators are slightly narrower atw2,3= 67 µmwhich was found to marginally improve insertion loss and higher order width mode suppression based on finite element simulation. Since each resonator length is much shorter than the elec- tromagnetic wavelength over the tuning range, the 4-pole filter can be modeled with the lumped element circuit show in Fig.2. A Butterworth-Van Dyke circuit [26], [27] is typically used to model an acoustic resonance where the mechanical mode is described by a series R-L-C tank circuit and the transducers introduce a shunt plate capacitance. In the case of magnetic resonators, the transducer introduces a parasitic series inductance and the MSFVW is modeled using a parallel R-L-C tank circuit instead. L0andR0represent the parasitic inductance and resistivity of the gold electrodes. Rm,Cm, and Lmrepresent the MSFVW resonance of each YIG mesa. Mnm is the inter-resonator coupling between adjacent YIG mesas while MIOrepresents input/output inductive coupling of the gold electrodes setting the out-of-band rejection level. Similar to a mechanical coupling coefficient, an effective coupling from the electrical to magnetostatic domain can be defined M12 M23 M34MIOL0 R0 Rm1 Lm1 Cm1 Rm2 Lm2 Cm2 Rm3 Lm3 Cm3 Rm4 Lm4 Cm4 L0 R0 Fig. 2. Lumped circuit model of an edge-coupled 4-pole MSW bandpass filter with electrically short transducers.3 Fig. 3. Measured frequency response of the 1-port resonator highlighted in Fig.1aat3962 Oe showing a Q= 2206 andk2 eff= 1.53% . by (5) where fpandfsrepresents the magnetic resonance and anti-resonance respectively. k2 eff=π 2fp fs cotπ 2fp fs (5) k2 eff is a function of the resonator design determined by the ratio of LmtoL0. Similar to an acoustic filter, k2 eff sets a bound on the maximum achievable filter bandwidth and impacts the passband ripple. The representative 1-port resonator (highlighted in Fig. 1a) exhibits a measured effective coupling and quality-factor of k2 eff= 1.53% andQ= 2206 at 3962 Oe with frequency response shown in Fig. 3. From the measured resonator impedance response, R0,L0,Rm,Cm, andLmcan all be extracted through separate fittings near the magnetostatic resonance and outside the spin wave manifold where the resonator behaves as an inductor. Using the same resonator parameters, the measured filter response excluding spurious modes can be fit to the lumped model in Fig. 2 by tuning MIO,M12, and M23under the assumption that each resonator has the same resonant frequency and the filter is symmetric. Fig. 8bshows the frequency response of the lumped model fitted to the measured S21 for a 4-pole filter along with the extracted model parameters. III. F ABRICATION The fabrication process for the MSW bandpass filter is outlined in Fig 4where steps (a)-(c) are adapted from [6]. A thick photoresist mask (SPR220-7.0) is patterned onto a 3µmYIG film grown via liquid phase epitaxially (LPE) on a 500µmGGG substrate. The YIG film is etched through at a rate of 36 nm /min using an optimized ion milling recipe for vertical sidewalls with sufficient intermittent cooling to prevent burning of the photoresist. Optimized lithography for the thick SPR photoresist is crucial to the filter’s final performance since the inter-resonator coupling factors, Mij, are sensitive to the physical separation of the etched YIG. With this process, we are able to fully etch resonator spacings as narrow as 3µm setting the maximum achievable Mij≤1.37% based on finite element simulation. After etching, the resist is removed and Fig. 4. Fabrication process of the bandpass filter: (a)3µmliquid phase epitaxy (LPE) YIG film grown on 500µmGGG substrate. (b)7.8µmthick photoresist (SPR220-7.0) patterned on YIG film as an etch mask. (c)3µm ion mill etch of YIG film at a rate of 36 nm /min.(d)Photoresist mask is removed and etched YIG is soaked in phosphoric acid at 80◦Cfor20 min .(e) Bi-layer photoresist (SPR220-7.0 and LOR 3B) mask for a liftoff is patterned onto the etched sample. (f)10 nm Ti and 300 nm Au is deposited using glancing angle e-beam evaporation followed by a liftoff process. Fig. 5. SEM of the MSFVW bandpass filter showing vertically etched YIG resonators with an inter-resonator spacing of 10.5µmand conformal gold electrodes over the edges of etched YIG. the sample is soaked in phosphoric acid at 80◦Cto remove redeposited material. For the gold electrodes, a SPR220-7.0 photoresist mask with a liftoff resist (LOR 3B) bi-layer is patterned for a liftoff process. Using a glancing angle e-beam evaporation, 300 nm of gold and a 10 nm titanium adhesion layer is conformally deposited over the etched YIG resonators. Finally, the sample is soaked in Remover PG overnight to complete the liftoff process. Fig. 5shows an SEM of a fabricated filter illustrating the vertically etched YIG and conformal gold electrodes. Due to the combination of bi-layer liftoff and angled metal deposition, the gold electrodes are larger than designed and show a tapered thickness.4 Fig. 6. Measured 4-pole MSW bandpass filter frequency response at different out-of-plane magnetic biases from 3205 Oe to5303 Oe . IV. E XPERIMENTAL RESULTS Filter s-parameters are measured using an Agilent PNA-L N5230A network analyzer with a pair of ground-signal (GS) probes from 3205 Oe to5303 Oe corresponding to a center frequency tuning over 5.6 GHz as shown in Fig. 6. A single- pole electromagnet powered by a constant current source provides the required out-of-plane magnetic bias (pictured in Fig.7) and a single-axis Gauss meter is used to map the source current to the applied field. Prior to each measurement, the device under test is aligned to the center of the electromagnet’s pole to ensure field uniformity. Fig. 8shows the frequency response near the passband for a 2-pole and 4-pole filter at 3864 Oe . Outside of the spin wave manifold, no magnetostatic waves may propagate so the filter behaves as two coupled inductors. Consequently, the out-of-band rejection is governed by the inductive coupling strength between the input and output electrodes. For the 2-pole filter with an electrode spacing of 67µm, the rejection is −25 dB while for the 4- pole filer with a spacing of 229µm, the rejection is −35 dB . At3864 Oe , the 2-pole filter shows an insertion loss (IL) of −3.55 dB and a 3 dB bandwidth of 57.0 MHz while the 4-pole filter has an IL of 6.94 dB and a 3 dB bandwidth of 17.0 MHz . From Fig. 8aand8b, higher-order magnetostatic spurious modes are visible to the right of the passband with a frequency separation of 29 MHz −40 MHz . As described in [15], the current distribution along the transducer length can excite Fig. 7. Experimental setup showing the fabricated filter chip resting on the pole of an electromagnet, two GS probes connected to one device under test, and an optical microscope used for probe landing and device alignment. (a) (b) Fig. 8. Frequency responses for the (a)2-pole and (b)4-pole filters high- lighted in Fig. 1near the passband at 3864 Oe .(b)also shows a comparison of the fitted lumped element model in Fig. 2with measured S21. The fitted circuit assumes all four resonators are identical and excludes the spurious pass bands caused by higher order MSFVW modes. either even or odd ordered modes. Based on the resonator dimensions and electrically short transducer, the frequency spacing of these spurs agree well with odd ordered length modes. Fig. 9shows the linear center frequency tuning at a rate of 2.7 MHz /Oeand the 3 dB -bandwidth over the applied bias for the 4-pole filter. With a well-calibrated bias field, the extrapolated center frequency tuning line should intersect −ωmat0 Oe . However, the Gauss meter used for the field Fig. 9. Measured 3 dB bandwidth and center frequency of the 4-pole MSFVW filter showing a tuning rate of 2.7 MHz /Oe.5 calibration is thicker than the fabricated chip, so the reported bias is underestimated by approximately 219 Oe . The filter’s tuning was also measured using a neodymium permanent magnet mounted on a 3-axis stage to precisely calibrate the field accounting for any thickness difference between the chip and sensor. In this setup, the extrapolated 0 Oe intersection is atωm=µ0γm·1751 Oe which agrees well the saturation magnetization of LPE YIG. Considering the total loss ( 1− |S11|2− |S21|2) for the 2- pole filter biased at 3652 Oe and measured far away from all magnetostatic resonances, an average of 43% of the input power is dissipated in the thin 300 nm gold transducers, radiated, or absorbed by the YIG on GGG substrate. Based on finite element simulations, the loss is primarily attributed to the resistance of the gold transducers. A second sample was fabricated with 3µmelectroplated gold to reduce resistive losses. Fig. 10compares the measured insertion loss and total loss for the same 2-pole filter with different gold thicknesses. As expected, the mean out-of-band loss shows significant improvement from 43% to only 12%. The average loss within the 3dB bandwidth exhibits a slight improvement dependent (a) (b) Fig. 10. (a)Measured S21 and (b)total loss for 2-pole filters with 300 nm and3µmthick gold transducers biased at 3652 Oe and3660 Oe respectively. Frequency is plotted relative to the center frequency to account for the slight difference in bias strength. Fig. 11. Two-tone IIP3 measurement in the passband of a 4-pole filter at 3652 Oe bias TABLE I UPPER INPUT TONE FREQUENCIES FOR IIP3 MEASUREMENTS Bias Field Stopband Low Passband Stopband High 3652 Oe 5.465 GHz 5.799 GHz 6.055 GHz TABLE II SUMMARY OF 4-POLE FILTER IIP3 Bias Field Stopband Low Passband Stopband High 3652 Oe ≥37.95 dBm −4.85 dBm 25.84 dBm on bias, ranging from 0.5% to14.5%. The insertion loss improvement reflects the change in mean in-band loss with a maximum improvement from 4.43 dB to2.92 dB around 3660 Oe . The linearity of the MSW bandpass filter is evaluated by measuring the input referred 3rdorder intercept point (IIP3) in the passband as well as the stopband both below and above the passband at a bias of 3652 Oe . The nonlinearity measurements are performed using two Keysight E8257D signal generators with a frequency separation of ∆f= 15 MHz . The higher of the two tone frequencies for each region are listed in Table I. A wideband power divider combines the two tones while an Agilent PXA spectrum analyzer measures the resultant spectrum. A two-stage calibration is performed to remove all cable and system losses at every tone frequency and input power level. Far away from the passband, the filter is expected to be linear and no intermodulation products were observed. Based on the maximum output power of the signal generators and noise floor of the spectrum analyzer, the lower stopband IIP3 is estimated to be greater than 37.95 dBm at3652 Oe . The passband shows the greatest nonlinearity with an IIP3 of −4.75 dBm at3652 Oe as shown in Fig. 11. The measured IIP3 in each frequency region is summarized in Table II. V. C ONCLUSION In this paper, we have demonstrated a novel edge-coupled highly-tunable magnetostatic bandpass filter using state-of- the-art micromachining fabrication techniques. The designed 2-pole and 4-pole filters have been tuned over an octave6 TABLE III PERFORMANCE COMPARISON WITH OTHER TUNABLE BANDPASS FILTERS Reference Frequency Tuning (GHz)Insertion Loss (dB)Bandwidth (MHz)Rejection (dB) This work (2-pole) 4.5-10.1 <6 29-39 >25 This work (4-pole) 4.5-10.1 <11 11-17 >35 YIG Sphere [10] 2-8 <5 20 >50 Magnetostatic Sur- face Wave [28]3.4-11.1 <5.1 18-25 >25 RF MEMS Tun- able Filter [29]6.5-10 <5.6 306-539 >50 Evanescent-Mode Cavity [30], [31]3-6.2 <4 15-25 >50 from 4.5 GHz to10.1 GHz showing a consistent passband shape with performance comparable to other state-of-the-art frequency tunable bandpass filters as summarized in Table III. We have also characterized the linearity of the filter in three distinct frequency regions. Our micromachining process enables precise control over the YIG mesa shape and spacings to synthesize miniaturized MSW channel-select filters analo- gous to electromagnetic cavity filter design. DATA AVAILABILITY The code and data used to produce the plots within this work will be released on the repository Zenodo upon publication. ACKNOWLEDGMENTS Chip fabrication was performed at the Birck Nanotechnol- ogy Center at Purdue. Resonator and filter measurements and characterization were performed at Seng-Liang Wang Hall at Purdue. The Purdue authors would like to thank Dave Lubelski for assistance with the glancing angle metal deposition and Yiyang Feng for discussions on the fabrication recipes. REFERENCES [1] K. Wang and C.-C. Nguyen, “High-order medium frequency microme- chanical electronic filters,” Journal of Microelectromechanical systems , vol. 8, no. 4, pp. 534–556, 1999. [2] P. Thiruvenkatanathan, J. Yan, J. Woodhouse et al. , “Enhancing para- metric sensitivity in electrically coupled mems resonators,” Journal of Microelectromechanical Systems , vol. 18, no. 5, pp. 1077–1086, 2009. [3] D. Weinstein, S. A. Bhave, M. 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Available: https://www.microlambdawireless.com/uploads/ pdfs/MLFD%20Series%20Dual-Two.pdf [11] Teledyne Technologies, “A new approach to yig- based band-reject filters – an introduction.” [Online]. Available: https://www.teledynedefenseelectronics.com/rf&microwave/ Documents/22-10%20Oct%20-%20Teledyne%20RF&M%20Super% 20Band-Reject%20Notch%20Filters%20AN%20v1.02.pdf [12] Micro Lambda Wireless Inc., “Technology description yig tuned filters – an introduction.” [Online]. Available: https://www.microlambdawireless. com/resources/ytfdefinitions2.pdf [13] P. Thiruvenkatanathan, J. Yan, J. Woodhouse et al. , “Ultrasensitive mode-localized mass sensor with electrically tunable parametric sen- sitivity,” Applied Physics Letters , vol. 96, no. 8, p. 081913, 2010. [14] D. D. Stancil, Theory of magnetostatic waves . Springer Science & Business Media, 2012. [15] W. S. Ishak and K.-W. Chang, “Tunable microwave resonators using magnetostatic wave in yig films,” IEEE transactions on microwave theory and techniques , vol. 34, no. 12, pp. 1383–1393, 1986. [16] W. S. Ishak, C. Kok-Wai, W. E. Kunz et al. , “Tunable microwave resonators and oscillators using magnetostatic waves,” IEEE transactions on ultrasonics, ferroelectrics, and frequency control , vol. 35, no. 3, pp. 396–405, 1988. [17] S. Hanna and S. Zeroug, “Single and coupled msw resonators for microwave channelizers,” IEEE Transactions on Magnetics , vol. 24, no. 6, pp. 2808–2810, 1988. [18] R. Marcelli, M. Rossi, and P. De Gasperis, “Coupled magnetostatic volume wave straight edge resonators for multipole microwave filtering,” IEEE Transactions on Magnetics , vol. 31, no. 6, pp. 3476–3478, 1995. [19] R. Marcelli, M. Rossi, P. De Gasperis et al. , “Magnetostatic wave single and multiple stage resonators,” IEEE Transactions on Magnetics , vol. 32, no. 5, pp. 4156–4161, 1996. [20] R. Wang, S. A. Bhave, and K. Bhattacharjee, “Design and fabrication of s 0lamb-wave thin-film lithium niobate micromechanical resonators,” Journal of Microelectromechanical Systems , vol. 24, no. 2, pp. 300–308. [Online]. Available: http://ieeexplore.ieee.org/document/7005450/ [21] K.-y. Hashimoto, RF bulk acoustic wave filters for communications . Artech House, 2009. [22] G. Esteves, T. R. Young, Z. Tang et al. , “Al0.68sc0.32n lamb wave resonators with electromechanical coupling coefficients near 10.28%,” Applied Physics Letters , vol. 118, no. 17, p. 171902. [Online]. Available: https://pubs.aip.org/apl/article/118/17/ 171902/236115/Al0-68Sc0-32N-Lamb-wave-resonators-with [23] G. Giribaldi, L. Colombo, and M. Rinaldi, “6–20 GHz 30% ScAlN lateral field-excited cross-sectional lam ´e mode resonators for future mobile RF front ends,” IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control , vol. 70, no. 10, pp. 1201–1212. [Online]. Available: https://ieeexplore.ieee.org/document/10243055/ [24] Y . 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Ding et al. , “Frequency tunable magnetostatic wave filters with zero static power magnetic biasing circuitry.” [Online]. Available: http://arxiv.org/abs/2308.009077 [29] K. Entesari and G. Rebeiz, “A differential 4-bit 6.5-10-GHz RF MEMS tunable filter,” IEEE Transactions on Microwave Theory and Techniques , vol. 53, no. 3, pp. 1103–1110. [Online]. Available: http://ieeexplore.ieee.org/document/1406317/ [30] H. Joshi, “Multi band RF bandpass filter design,” ISBN: 978-1- 124-15604-0 Publication Title: ProQuest Dissertations and Theses. [Online]. Available: https://www.proquest.com/dissertations-theses/multi-band-rf-bandpass-filter-design/docview/748222813/se-2 [31] H. Joshi, H. H. Sigmarsson, D. Peroulis et al. , “Highly loaded evanescent cavities for widely tunable high-q filters,” in 2007 IEEE/MTT-S International Microwave Symposium . IEEE, pp. 2133– 2136, ISSN: 0149-645X. [Online]. Available: http://ieeexplore.ieee.org/ document/4264292/

2023-12-17

This paper reports on the design, fabrication, and characterization of an edge-coupled magnetostatic forward volume wave bandpass filter. Using micromachining techniques, the filter is fabricated from a yttrium iron garnet (YIG) film grown on a gadolinium gallium garnet (GGG) substrate with inductive transducers. By adjusting an out-of-plane magnetic field, we demonstrate linear center frequency tuning for a $4^{\text{th}}$-order filter from 4.5 GHz to 10.1 GHz while retaining a fractional bandwidth of 0.3%, an insertion loss of 6.94 dB, and a -35dB rejection. We characterize the filter nonlinearity in the passband and stopband with IIP3 measurements of -4.85 dBm and 25.84 dBm, respectively. When integrated with a tunable magnetic field, this device is an octave tunable narrowband channel-select filter.

An Edge-Coupled Magnetostatic Bandpass Filter

2312.10583v1

Optical manipulation of a magnon-photon hybrid system C. Braggio,1,G. Carugno,1M. Guarise,1A. Ortolan,2,yand G. Ruoso2 1Dip. di Fisica e Astronomia and INFN, Sez di Padova, Via F. Marzolo 8, I-35131 Padova, Italy 2INFN, Laboratori Nazionali di Legnaro, Viale dell'Universit a 2, I-35020 Legnaro, Italy We demonstrate an all-optical method for manipulating the magnetization in a 1{mm YIG (yttrium-iron-garnet) sphere placed in a 0:17 T uniform magnetic eld. An harmonic of the frequency comb delivered by a multi-GHz infrared laser source is tuned to the Larmor frequency of the YIG sphere to drive magnetization oscillations, which in turn give rise to a radiation eld used to thoroughly investigate the phenomenon. The radiation damping issue that occurs at high frequency and in the presence of highly magnetizated materials, has been overcome by exploiting magnon-photon strong coupling regime in microwave cavities. Our ndings demonstrate an eective technique for ultrafast control of the magnetization vector in optomagnetic materials via polariza- tion rotation and intensity modulation of an incident laser beam. We eventually get a second-order susceptibility value of 107cm2/MW for single crystal YIG. Introduction. |Nonthermal control of spins by short laser pulses is one of the preferable means to achieve ul- trafast control of the magnetization in magnetic materi- als [see 1, and references therein], representing a break- through in potential applications ranging from high den- sity magnetic data storage [2], spintronics [3], to quantum information processing [4, 5]. One of the most interesting opto-magnetic mechanisms that allows for coherent con- trol of the magnetization in materials is the Inverse Fara- day (IF) eect, a Raman-like coherent scattering process that entails the generation of a magnetic excitation (i.e. magnon) in a medium undergoing the action of high{ intensity optical pulses. As it does not require absorption and takes place on a femtosecond time scale, it stands out as a promising mechanism to control the magnetiza- tion at high speeds. This principle has been successfully applied in dysprosium orthoferrite (DyFeO 3) with iso- lated femtosecond laser pulses, that act as magnetic eld pulses of a 0.3 T amplitude [6]. By way of the IF eect, vector control of magnetization in another antiferromag- netic crystal was also demonstrated by varying the delay between pairs of polarization-twisted ultrashort optical pulses [7]. In this work we introduce a new approach in opto{magnetism based on multi-gigahertz repetition rate lasers with optical carrier f0[8]. The power spectrum of such mode-locked laser sources, as detected by ultrafast photodiodes, is a frequency comb that consists of several harmonicsnfr, wherefris the repetition rate and nis a small number. Their gaussian envelope is determined by the optical pulse temporal prole [9]. For example, our 4.6 GHz passively mode-locked oscillator delivers 10 ps-duration pulses that give rise to a frequency comb up to 100 GHz, and the rst three harmonics have approximately the same amplitude. In principle, any harmonic of the comb can coherently drive the magnetization in the steady state through the process described in the present work, provided it is tuned to electron spin resonances (ESR) of the magnetized material. We study the spin dynamics in a hybridized system which consists of two strongly coupled oscillators, i.e. amicrowave cavity mode and a magnetostatic mode re- lated to ferromagnetic resonance (FMR) with uniform precession [10] of a single crystal yttrium-iron garnet Y3Fe5O12sphere. Magnetic garnets [11, 12] represent the ideal materials for such investigations for several rea- sons, including the possibility to realize large magneto- optical eects due to their strong spin-orbit coupling and intrinsically low magnetic damping [12{14]. We succeed to optically drive the precession of the spins electro-dynamically coupled to the cavity photons with the rst harmonic of the train of pulses at frtuned to one of the hybrid system's resonant frequencies. The process gives rise to a microwave eld that is measured with a loop antenna critically coupled to the cavity mode. In this way, we have identied a new observable for the spin precession to explore opto-magnetic phenom- ena. So far, experiments in this eld were performed with pump-probe apparatus based on femtosecond lasers [1, 7, 12, 15]. As it is well known, at very high values of frequency and magnetization, the energy radiated from oscillating magnetization through magnetic dipole radiation can be an issue for the dynamic control of the magnetization. For instance, in a polarized 1{mm YIG sphere with lin- ear susceptibility 30, the onset of radiation damping occurs at10 GHz [16]. However, radiation damping mechanism can be conveniently suppressed in the mi- crowave cavity-QED and strong coupling regimes [16, 17], as we detail in the following for hybridized systems. Moreover, under remarkable conditions of hybridization, we get the control of relevant experimental parameters such as the number of spins, rf absorbed power, and the involved relaxation times. System hybridization is how- ever not essential to observe the phenomenon described in the present work. In fact, we succeed in controlling the magnetization also in free space, but under experimental conditions that do not allow for accurate modeling. Hybridized system characterization. | Strong interac- tion between light (i.e. photons stored in a cavity) and magnetized materials has been accomplished in several experiments that are paving the way toward the develop-arXiv:1609.08147v1 [cond-mat.mes-hall] 24 Sep 20162 ment of quantum information technologies [18{20]. Hy- bridization is commonly investigated by measuring the microwave{cavity transmission spectrum as a function of the static magnetic eld, as summarized in Fig. 1 for our experimental setup, even though, very recently, some au- thors have reported electric detection via spin pumping [21]. A YIG sphere made by Ferrisphere Inc. with a ra- (a)(b) zx FIG. 1. Hybridizing magnons and microwave photons. (a) Cavity transmission spectrum measured as a function of the static magnetic eld at room temperature. (b) Simulated magnetic{eld distribution of the TE 102cavity mode. A static magnetic eld B extis applied normal to the xzplane, and the microwave magnetic eld at the YIG sphere (in black and not to scale in the representation) position is perpendic- ular to the static magnetic eld. The color map represents the amplitude of the cavity magnetic eld normalized to its maximum value. dius of 1 mm is mounted at the center of a 3D rectangular microwave cavity with dimensions 98 42:512:6 mm3. The cavity made of oxygen free copper has the TE 102 mode frequency !c=2'4:67 GHz, and its internal cav- ity lossint. This cavity has two ports characterized by the coupling coecients 1and2to the considered cav- ity mode. The sphere is glued to an alumina (aluminum{ oxide) rod that identies the crystal axis [110], perpen- dicular to the static magnetic eld Bext(yaxis) and par- allel to the TE 102microwave magnetic eld lines lying on thexzplane. Due to the strong coupling between the cavity mode and FMR mode an avoided crossing occurs when their resonant frequencies match. As derived in the input{output theory context [17, 20], when the static magnetic eld is tuned to drive the magnons in resonance with the cavity mode TE 102, the measured transmission coecient can be written as S21(!) =p12 i(!!c)c 2+jgmj2 i(!!FMR) m=2;(1) where!FMR and mare the frequency and linewidth of the FMR mode, c=2= (1+2+int)=2is the total cavity linewidth, and gmis the coupling strength of the FMR mode to the cavity mode. The latter parameter is proportional to the square root of the number of precess- ing spinsNs, i.e.gm=g0pNs, whereg0= ep 0~!c=Vcis the coupling strength of a single spin to the cavity mode, with e= 228 GHz/T electron gyromagnetic ratio,0permeability of vacuum and Vcis cavity volume. As discussed in the seminal paper of Bloembergen and Pound [16], the poles at the anti{crossing point !FMR = !c!0are given by p=i !0p jgmj2[(c m)=4]2 1 2kc 2+ m 2 ;(2) and their imaginary and real parts represent the frequen- cies !=!0p jgmj2[(c m)=4]2 and the linewidths = 1=2(kc+ m) of the hybridized modes, respectively. From eq. 2 hybridization clearly oc- curs only ifjgmj2[(c+ m)=4]2>0 and, as a con- sequence, hybridized modes have the same decay time, independent of the sample or cavity volume and cou- pling strengths, i.e.  = (2=c+ 2=2)1, where c= 2=cand2are the loaded cavity decay time and the spin{spin relaxation time, respectively. In the absence of hybridization, the term under square root in eq. 2 is neg- ative and thus the poles have the same frequency !0, with two relaxation times cand= (1=r+ 1=2)1, that correspond to the damping of the cavity mode and magnetization mode in the presence of radiation damping r=c=2=jgmj2. In our experimental apparatus for B ext'0:17 T we achieve a strong coupling regime with gm=2= 57 MHz, thus the involved precessing spins are Ns1020. Along with the mode frequencies f+= 4:7247 GHz and f= 4:6677 GHz, the t of the measured S21coecient to eq. 1 gives the mode decay times  '65 ns of the hybridized system, compatible with the value of 2provided by the manufacturer and the measured c. Photoinduced magnetization. | Once the hybrid sys- tem has been characterized, the experimental appara- tus illustrated in Fig. 2 is used to investigate the opto{ magnetic phenomenon. The 7.2 ps{duration, 1.55 m{ wavelength laser pulses are obtained at the idler output of an optical parametric oscillator (OPO), synchronously pumped by the second harmonic of a MOPA (master os- cillator power amplier) laser system that has been de- scribed elsewhere [23]. It is important to note that we are exploiting a non absorptive mechanism as the opti- cal wavelength is within the YIG transparency window (1:55m). The beam waist at the YIG position is 1:28 mm, and the average intensity of the incident pulses is 2:4 MW/cm2, obtained within <1s{duration macro{ pulses. In Fig. 3 we demonstrate the all{optical coherent con- trol of the magnon{photon mode at f= 4:67 GHz by employing a train of laser pulses with repetition rate fr tuned tof. The rise and decay time of the microwave pulse registered at the oscilloscope agrees with the mode decay time  = 65 ns we get through the S 12measure- ments within experimental errors. The duration of the3 MOPA LASERSH OPODMHS CGFF532 nm1064 nm1064 nm809 nm1554 nmMWM N S LCC B[...]500 nsUPD/2 YIG oscilloscope xyA FIG. 2. Schematic representation of the experimental ar- rangement. The 1064 nm{wavelength macro{pulse delivered by a MOPA laser is frequency{doubled (SH) to synchronously pump an optical parametric oscillator (OPO). The laser rep- etition rate, macro{pulse uniformity and energy are moni- tored at an InGaAs ultrafast photodiode (UPD), a coaxial waveguide device WM [22] and bolometer B respectively. The 809 nm OPO output beam intensity prole is adjusted at a digital laser camera LC. To make sure that only emission at 1550 nm impinges on the YIG sphere, several optical lters are inserted in the beam path. CGF is a 610 nm longpass colored glass lter, F transmits >1500 nm and M is a 1064 nm, high re ectivity dielectric mirror. HS (harmonic separator) is a di- electric mirror that transmits 1064 nm wavelength and has a high re ectivity for 532 nm whereas DM is a 1000 nm{cuto wavelength dichroic mirror. The microwave eld generated during the magnetization precession is detected by means of an antenna critically coupled to the TE 102mode and con- nected through a short transmission line to a 39 dB{gain am- plication stage A. The amplied signal is nally registered at a 20 GHz sampling oscilloscope. optical excitation is set to a value of te'0:5s>al- lowing us to control the system in its steady state. This diers from previous studies in opto{magnetism which were focused on the transient optical control of the mag- netization via single femtosecond laser pulses [see 1, and references therein]. Moreover, the YIG magnetization precesses in phase with the laser pulses, as demonstrated by juxtaposition in Fig. 3 (c) of the signal generated in the microwave cavity and the output of the laser macro{pulse monitor WM, i.e. a coaxial waveguide hosting a nonlin- ear crystal in which microwaves are generated through optical rectication [22]. Another important signature of the coherent precession of the magnetization is also shown in Fig. 3 (d), where the amplitude of the Fourier transform of the microwave signal is plotted for dierent values of the laser repetition rate fr. The data are tted to a lorentzian curve that takes into account the convolu- tion between the optical excitation and the prole of the hybridized mode at 4 :6711 GHz. As shown in Fig. 3 (b), the spectral component f+is also excited but with a much smaller strength. These results, combined with the -0.1-0.0500.050.1 -0.04-0.0200.020.04 0 200 400 600 800cavity - hybrid system macropulse monitor VC (V)VM (V) time (ns)(a) -140-120-100-80-60-40-20 4.554.64.654.74.754.8A (a.u.)frequency (GHz)(b) 390390.5391391.5392-0.1-0.0500.050.1 -0.04-0.0200.020.04 time (ns)VC (V)VM (V)(c) 0246810 4.644.654.664.674.684.69A (a. u.)frequency (GHz)(d)FIG. 3. (Color online) Optically-driven spin precession in the time and frequency domain. (a) Oscilloscope traces displaying both the amplied signal V Cdetected in the microwave cav- ity hosting the YIG sphere (blue) and the output V Mof the laser macro{pulse monitor (red). (b) Fourier transform am- plitude spectrum of the microwave signals displayed in (a). The logarithmic scale is used for the vertical axis. (c) 2 ns{ duration zoom out of (a) showing the magnetization precess- ing synchronously with the laser pulses. (d) Tuning the laser repetition rate to the hybridized frequency f. 00.20.40.60.811.2 0π/2π3π/22πAmplitude (a.u.)θ (rad)(a) 020406080 00.511.522.53Vrms (mV)I (MW/cm2)(b) FIG. 4. (a) Amplitude of the microwave signal in the cavity as a function of the laser polarization angle. (b) Magnetization dependence on the laser intensity. assessment of stationary precession of the macroscopic magnetization, unambiguously show that each macro{ pulse acts as an eective microwave eld on the ensemble of strongly correlated spins of the FMR mode. Discussion. | To conrm the nonthermal origin of the laser{induced magnetization precession and denitely at- tribute the observed opto{magnetic phenomenon to the IF eect, we investigated the dependence of the mi- crowave signal amplitude on the laser polarization [1] and the results are reported in Fig. 4 (a). Owing to the strong anisotropy of the magnetic susceptibility of YIG, the time{dependent magnetization vector Mis not ex- pected to be parallel to the wave vector kas is the case for IF in isotropic medium and circularly polarized light. Ac-4 tually the observed magnetization precession is induced by a second{order process conveniently described by a third{rank, axial time{odd tensor (2) ijk, provided that k is orthogonal to the [1 1 0] crystal direction d[7]. As the reference system axes xandycoincide with kandddi- rections, the photoinduced magnetization lies in the yz plane and reads Mi=Z d!(2) ijkE? j(!)Ek(!); (3) wherei= 2;3, andE2(!) =E(!) cos'andE3(!) = E(!) sin'; hereE(!) is the Fourier transform of the laser electric eld and 'is the polarization angle of the incident light with respect to the yaxis. Due to well{ known symmetries of second{order susceptibility [1], the non{vanishing components of ijkare233=222= 332=323(!) and equation 3 gives the components Mz=Z d!(!)jE(!)j2cos 2' (4) My=Z d!(!)jE(!)j2sin 2': (5) Around the hybridized mode frequencies !, the real and imaginary part of the complex susceptibility ( !) can be approximated by absorption ( !)00and dispersion ( !)0 components of magnetization [24]. In particular, at work- ing frequency !we have only absorption with no disper- sion, hence the susceptivity ( !) =  0!=2 becomes real, and does not aect the magnetization direction. Thus the fulllment of resonant condition also allow us to simplify the geometric description of the photoinduced magnetization vector. Indeed, to explain the 4{fold pe- riodicity of the plot displayed in Fig. 4 (a) we only need to realize that the cavity selects the Mz/cos 2'compo- nent via its geometric projection on TE 102mode (i.e. the z direction as shown in Fig. 1), and that the critically coupled antenna cannot distinguish between parallel and antiparallel orientation of Mz. Therefore the detected magnetization signal must be proportional to jcos 2'j, as conrmed by the t to the data in Fig. 4 (a). Figure 4 (b) shows instead the linearity of the measured spin oscilla- tions amplitude as a function of the pump laser intensity, in agreement with eq. 3 as well. The strength of the eective microwave eld Bethat drives theMzprecession can be estimated thanks to the peculiar dynamics of hybridization. In general, the ab- sorbed power in stationary conditions by a magnetized sample [24] is given by Pa=Vs B
dM dt ; whereh
idenotes the time average over one period. Moreover, at resonance and for a critically coupled in- ductive loop, the measured power in the microwave cav- ity isPa=2. In our experimental conditions, the absorbedpower by the YIG crystal at the frequency ! Pa=Vs0!2 B2 e 0(6) is written in terms of quantities that are measured or t- ted to the data, so that the second{order susceptivity can be readily estimated through  0=Pa0=(Vs!2 B2 e), whereBerepresents the laser induced eective mag- netic eld. Due to 1 =fdependence of the power spec- trum generated by downconversion of the picosecond fre- quency comb, the infrared optical eld average amplitude Bl=p 0I=c= 10 mT, at fo'190 THz optical fre- quency, is suppressed to Be= 2:5105BI= 0.25T atf4:7 GHz. With Pa= 3 nW estimated from the plots reported in Fig. 3, we eventually get  0107 cm2/MW. In summary, our experimental and theoretical ap- proach provides a purely optical, exible technique to manipulate the magnetization vector in YIG via polar- ization rotation and intensity modulation of the incident laser beam. Remarkably, the maximum control speed of this process is only limited by the bandwidth of currently available electro-optic devices. Unlike the ingenious op- tical method described in reference [7], here the mode- locked pulses impinging on the magnetized material allow for operation of the system in the steady state, opening a path on the ultrafast laser control of hybridized magnon- photon systems. It is worth mentioning that commer- cially available compact ultrafast oscillators with 200 pJ- energy output pulses [25] may foster applications of the present method in the opto{magnetism eld. The authors thank D. Budker and V. S. Zapasskii for carefully reading the manuscript and for useful discus- sions. C. B. and M. G. acknowledge partial nancial sup- port of the University of Padova under Progetto di Ate- neo (reference grant number CPDA135499/13). Techni- cal support by E. Berto is gratefully acknowledged. Electronic address: caterina.braggio@unipd.it yElectronic address: antonello.ortolan@lnl.infn.it [1] A. Kirilyuk, A. V. Kimel, and T. Rasing, Rev. Mod. Phys. 82, 2731 (2010). [2] C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk, A. Tsukamoto, A. Itoh, and T. Rasing, Phys. Rev. Lett. 99, 047601 (2007). [3] T. Li, A. Patz, L. Mouchliadis, J. Yan, T. A. Lograsso, I. E. Perakis, and J. Wang, Nature 496, 69 (2013). [4] Z.-L. Xiang, S. Ashhab, J. Q. You, and F. Nori, Rev. Mod. Phys. 85, 623 (2013). [5] D. Zhang, X.-M. Wang, T.-F. Li, X.-Q. Luo, W. Wu, F. Nori, and J. You, Npj Quantum Information 1, 15014 (2015). [6] A. V. Kimel, A. Kirilyuk, P. A. Usachev, R. V. Pis- arev, A. M. Balbashov, and T. Rasing, Nature 435, 655 (2005). [7] N. Kanda, T. Higuchi, H. Shimizu, K. Konishi, K. Yosh-5 ioka, and M. Kuwata-Gonokami, Nat. Commun. 2, 362 (2011). [8] U. Keller, Nature 424, 831 (2003). [9] S. T. Cundi and J. Ye, Rev. Mod. Phys. 75, 325 (2003). [10] C. Kittel, Introduction to solid state physics , 8th ed., edited by S. Vonsovskii (John Wiley & Sons, 2005). [11] F. Hansteen, A. Kimel, A. Kirilyuk, and T. Rasing, Phys. Rev. Lett. 95, 047402 (2005). [12] T. Satoh, Y. Terui, R. Moriya, B. A. Ivanov, K. Ando, E. Saitoh, T. Shimura, and K. Kuroda, Nat. Photon. 6, 662 (2012). [13] A. A. Serga, A. V. Chumak, and B. Hillebrands, J. Phys. D: Applied Physics 43, 264002 (2010). [14] 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, Nature 464, 262 (2010). [15] D. Bossini, S. Dal Conte, Y. Hashimoto, A. Secchi, R. V. Pisarev, T. Rasing, G. Cerullo, and A. V. Kimel, Nat Commun 7(2016). [16] N. Bloembergen and R. V. Pound, Phys. Rev. 95, 8 (1954). [17] M. O. Scully and M. S. Zubairy, Quantum Optics (Cam-bridge University Press, 1997). [18] H. Huebl, C. W. Zollitsch, J. Lotze, F. Hocke, M. Greifen- stein, A. Marx, R. Gross, and S. T. B. Goennenwein, Phys. Rev. Lett. 111, 127003 (2013). [19] X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, Phys. Rev. Lett. 113, 156401 (2014). [20] Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Us- ami, and Y. Nakamura, Phys. Rev. Lett. 113, 083603 (2014). [21] L. Bai, M. Harder, Y. P. Chen, X. Fan, J. Q. Xiao, and C. M. Hu, Phys. Rev. Lett. 114, 227201 (2015). [22] C. Braggio and A. F. Borghesani, Rev. Sci. Instrum. 85, 023105 (2014). [23] A. Agnesi, C. Braggio, L. Carr a, F. Pirzio, S. Lodo, G. Messineo, D. Scarpa, A. Tomaselli, G. Reali, and C. Vacchi, Opt. Express 16, 15811 (2008). [24] F. Fiorillo, Characterization and measurement of mag- netic materials (Elsevier, 2004). [25] L. Krainer, R. Paschotta, S. Lecomte, M. Moser, K. Weingarten, and U. Keller, IEEE J. Quant. Electron. 38, 1331 (2002).

2016-09-24

We demonstrate an all-optical method for manipulating the magnetization in a 1-mm YIG (yttrium-iron-garnet) sphere placed in a $\sim0.17\,$T uniform magnetic field. An harmonic of the frequency comb delivered by a multi-GHz infrared laser source is tuned to the Larmor frequency of the YIG sphere to drive magnetization oscillations, which in turn give rise to a radiation field used to thoroughly investigate the phenomenon. The radiation damping issue that occurs at high frequency and in the presence of highly magnetizated materials, has been overcome by exploiting magnon-photon strong coupling regime in microwave cavities. Our findings demonstrate an effective technique for ultrafast control of the magnetization vector in optomagnetic materials via polarization rotation and intensity modulation of an incident laser beam. We eventually get a second-order susceptibility value of $\sim10^{-7}$ cm$^2$/MW for single crystal YIG.

Optical manipulation of a magnon-photon hybrid system

1609.08147v1

(Accepted by Nature Scientic Reports: Jan. 29, 2013) Phase Diagram for Magnon Condensate in Yttrium Iron Garnet lm Fuxiang Li1, W.M. Saslow1, and V.L. Pokrovsky1;2 1Department of Physics, Texas A&M University, College Station, Texas77843-4242 and 2Landau Institute for Theoretical Physics, Chernogolovka, Moscow District,142432, Russia (Dated: May 26, 2022) Recently, magnons, which are quasiparticles describing the collective motion of spins, were found to undergo Bose-Einstein condensation (BEC) at room temperature in lms of Yttrium Iron Garnet (YIG). Unlike other quasiparticle BEC systems, this system has a spectrum with two degenerate minima, which makes it possible for the system to have two condensates in momentum space. Recent Brillouin Light scattering studies for a microwave-pumped YIG lm of thickness d= 5m and eldH= 1 kOe nd a low-contrast interference pattern at the characteristic wavevector Qof the magnon energy minimum. In this report, we show that this modulation pattern can be quantitatively explained as due to non-symmetric but coherent Bose-Einstein condensation of magnons into the two energy minima. Our theory predicts a transition from a high-contrast symmetric phase to a low-contrast non-symmetric phase on varying the dandH, and a new type of collective oscillations. PACS numbers: 75.10.-b, 75.60, 75.70, 75.85 Bose-Einstein condensation (BEC), one of the most intriguing macroscopic quantum phenomena, has been observed in equilibrium systems of Bose atoms, like4He [1, 2],87Rb [3] and23Na [4]. Recent experiments have extended the concept of BEC to non-equilibrium systems consisting of photons [5] and of quasiparticles, such as excitons [6], polari- tons [7{9] and magnons [10, 11]. Among these, BEC of magnons in lms of Yttrium Iron Garnet (YIG), discovered by the group of Demokritov [11{ 17], is distinguished from other quasiparticle BEC systems by its room temperature transition and two-dimensional anisotropic properties. In par- ticular, the spin-wave energy spectrum of a YIG lm shows two energetically degenerate minima. Therefore it is possible that the system may have two condensates in momentum space [18]. An ex- periment by Nowik-Boltyk et al. [17] indeed shows a low-contrast spatial modulation pattern, indicat- ing that there is interference between the two con- densates. Current theories [19{24] do not describe the appearance of coherence or the distribution of the two condensates. This report points out that a complete descrip- tion of BEC in microwave-pumped YIG lms must account for the 4th order interactions, includ- ing previously neglected magnon-non-conserving terms originating in the dipolar interactions. The theory explains not only the appearance of coher- ence but also quantitatively explains the low con-trast of the experimentally observed interference pattern. Moreover, the theory predicts that, on increasing the lm thickness dfrom a small value, there is a transition from a high-contrast symmet- ric phase S for d<dc, with equal numbers of con- densed magnons lling the two minimum states, to a low-contrast coherent non-symmetric phase NS ford > dc, with dierent numbers of condensed magnons lling the two minimum states. In com- paratively thin lms ( d<0:2m) the same transi- tion can be driven by an external magnetic eld H. Ford > d, wheredis another critical thickness (d> dc), the sum of phases of the two conden- sates changes from to 0; ford=dthe system is in a completely non-symmetric phase with only one condensate, for which there is no interference. In the experiment of Ref.[17] the lm thickness d exceededd. We suggest that the phase transitions may be identied by measuring the contrast of the spatial interference pattern for various dandH. We also predict a new type of collective magnetic oscillation in this system and discuss the possibil- ity of domain walls in non-symmetric phases. Results Phase Diagram.| We consider a YIG lm of thicknessdwith in-plane magnetic eld H(see in- set of Fig. 1). The 4-th order interaction of con-arXiv:1302.6128v1 [cond-mat.quant-gas] 25 Feb 20132 densate amplitudes reads [25{27]: ^V4=A[cy Qcy QcQcQ+cy Qcy QcQcQ] +2Bcy Qcy QcQcQ +C[cy QcQcQcQ+cy QcQcQcQ+h:c:]:(1) HerecQandcy Qare the annihilation and cre- ation operators for magnons in the two conden- sates located at the two energy minima (0 ;Q) in the 2-D momentum space (see. Fig.1). The coe- FIG. 1. The magnon spectrum in the kzdirection for d= 5m andH= 1 kOe. The inset is a schematic diagram of YIG lm. cients in Eq.(1) are: A=~!M 4SN[(13)FQ22(1F2Q)] DQ2 2SN[142]; B=~!M 2SN[(12)(1F2Q)(13)FQ] +DQ2 SN[122]; C=~!M 8SN[(3120 33+ 32)FQ +16 33(1F2Q)] +DQ2 SN3; (2) with1=u4+ 4u2v2+v4,2= 2u2v2and3= 3uv(u2+v2). Here,uandvare the coecients of Bogoliubov transformation (see the Methods sec- tion for details). S= 14:3 is the eective spin, Nis the total number of spins in the lm, Mis the mag- netization and ~!M= 4M with gyromagneticratio = 1:2105eV=kOe.Dis proportional to the exchange constant and Fk= (1ekd)=kd. Similar results for the coecients AandBwere obtained in Ref.[19]. The coecient C, which vi- olates magnon number conservation, has not been considered previously. Below we show that Cis the only source of coherence between the two conden- sates. The three coecients A,BandC, whose values as functions of Hare shown in Fig.2 for two typical values of d, determine the distribu- tion of condensed magnons in the two degenerate minima. Ref.[19] assumed a symmetric phase with condensed magnons in both minima having equal amplitudes and equal phases. Later, Ref.[20] as- sumed lling of only one minimum. More recently Ref.[24] considered Josephson-like oscillations by starting from two condensates with equal numbers of magnons but dierent phases. Our theory pre- dicts coherent condensates and the ratio of their amplitudes with no additional assumptions. In terms of condensate numbers NQand phases, the condensate amplitudes are cQ=p NQei. Substituting them into eq.(1) we nd: V4=A(N2 Q+N2 Q) + 2BNQNQ +2Ccos (N3 2 QN1 2 Q+N1 2 QN3 2 Q):(3) Clearly the dipole energy depends on the total phase  = ++. To minimize this energy, forC > 0 we have  = and forC < 0 we have  = 0. Fig.2 shows that the sign of Cchanges for dierentdandH, which indicates a transition of  between 0 and . For both C > 0 andC < 0 the dipole energy establishes a coherence between the two condensate amplitudes. In contrast to a Josephson-like interaction, the sum rather than the dierence of the two condensate phases is xed. Since the total number of condensed magnons Nc=NQ+NQis uniquely determined by the pumping (see Methods), the energy has only a sin- gle free variable, the so far unspecied dierence =NQNQ. In terms of Ncandthe conden- sate energy eq.(3) is: V4=1 2h (A+B)N2 c(BA)2 2jCjNcp N2c2i : (4) The ground state of the condensates depends on the criterion parameter
, dened as
AB+jCj: (5)3 FIG. 2. Interaction coecients A,BandC(in units of mK/N, with Nthe total number of spins in the lm) as a function of magnetic eld Hfor lm thickness (a) d= 1:0m and (b)d= 0:1m. When
>0,= 0 minimizes the energy, so the two minima have equal numbers of condensed magnons. This is the symmetric phase, with NQ=NQ. When
<0, the minimum shifts to2 N2c= 1C2 (BA)2. This is the non-symmetric phase (S), with NQ6=NQ. The transition from symmetric to non-symmetric phase (NS) at
= 0 is of the second order. There is no metastable state of these phases. At C= 0 one nds =Nc, which corresponds to a completely non-symmetric phase with only one condensate. The ground state of the non-symmetric phase is doubly-degenerate, corresponding to the two possible signs for . Fig.3 shows that for a lm thickness of about 0 :05m, the symmetric phase is energy favorable up to H= 1:2 T. Ford= 0:08m, on increasing Hto about 0:6 kOe, there is a transition from symmet- ric to non-symmetric phase. For the larger thick- nessesd= 0:1m andd= 1m, the ground state is non-symmetric for H > 0:3 kOe. Fig.4 gives the phase diagram in ( d;H) space. It has three dierent regions, separated by two critical transition lines, dc(H) andd(H), corre- sponding to
= 0 and C= 0, respectively. For d= 0:130:16the system possesses re-entrant behavior (NS  = , to NS  = 0, to NS  = ) asHincreases. As shown below, measurement of the contrast, or modulation depth [17], of the spa- tial interference pattern permits identication of the dierent condensate phases. FIG. 3. Transition criterion
from non-symmetric to symmetric phase,
(in units of mK/N), as a function of magnetic eld Hfor dierent thicknesses d. FIG. 4. The phase diagram for dierent values of thick- nessdand magnetic eld H. Zero Sound.| In two-condensate states the relative phase =+is a Goldstone mode. Its oscillation, coupled with the oscillation of the number density n=nQnQrepresents a new type of collective excitation, which we call zero sound (as in Landau's Fermi liquid, this mode is driven by the self-consistent eld rather than collisions). Solving a properly modied Gross- Pitaevskii equation (see Methods), we nd its spec- trum. In the symmetric phase its dispersion rela- tion is: !=r ~2k4 4m2+Nc
k2 m: (6)4 The magnon eective mass is of the order of the electron mass. The density of condensed magnons nc=Nc=Vis about 1018cm3and
10 mK=N. The sound speed for small kin this case isv0s=p Nc
=m, which is about 100 m =s. Near the transition point
= 0, the velocity of this zero sound goes to zero. For the non-symmetric case, the spectrum is: !=r ~2k4 4m2+Nc(BA)(1)k2 m;(7) where(BA)2 C2. In the experiment of Ref. [17], 104andBA= 8:4 mK=N, from which we estimate a sound speed of 3 103m=s. The dispersions of zero sound for symmetric and non- symmetric cases are shown in Fig.5. Note that the range of applicability of the linear approxima- tion decreases signicantly for small C, where one of the condensate densities is small and the phase uctuations grow. FIG. 5. Dispersion of zero sound as a function of wave vector in the direction of external magnetic eld for symmetric and non-symmetric cases, respectively. For the non-symmetric case, we choose H= 1 kOe and d= 5m. Domain Wall.| Since the ground state of the non-symmetric phase is doubly degenerate, it can consist of domains with dierent signs of  separated by domain walls. The width wof such a domain wall is of the order ofq ~2 2mNcj
j. For the data of experiment [17] we estimate that w10 m, and a domain wall energy per unit area of about 109J/m2.Discussion The ground state wave function ( z) gener- ally is a superposition of two condensate ampli- tudes (z) = (cQeiQz+cQeiQz)=p V, where cQ=p NQeiandVis the volume of the lm. The spatial structure of ( z) can be mea- sured by Brillouin Light Scattering (BLS), with intensity proportional to the condensate density j j2=nQ+nQ+2pnQnQcos(2Qz++). In their recent experiment, Nowik-Boltyk el al [17] observed the interference pattern associated with the ground state. They found that the con- trast of this periodic spatial modulation is far be- low 100%; of the order 3%. The present theory can quantitatively explain this result. In their ex- periment, Ref.[17] employ d= 5:1m andH= 1 kOe. Then eq.(2) for A,BandCgivesA=0:168 mK=N,B= 8:218 mK=NandC=0:203 mK=N, so
<0. This corresponds to the non-symmetric phase, where for spontaneous symmetry-breaking with=NQNQ>0 the ratio of the num- bers of magnons in the two condensates isNQ NQ C2 4(BA)2. The contrast is =j j2 maxj j2 min j j2max+j j2 min. Since CBandNQNQ, we have2q NQ NQ jCj jBAj. For the above values of A,BandC, is of order 2 :4%, in good agreement with experi- ment. The smallness of C(andA) relative to B is associated with the large parameter d=l, where l=q D  Mis an intrinsic length scale of the sys- tem andl106m. In terms of this parameter, jCj B(l d)2=3. In the experiment of [17] the contrast reaches a saturation value at a comparatively small pumping power, corresponding to the appearance of BEC. This agrees with our expression for , which de- pends only on lm thickness dand magnetic eld H. By varying dandH, the contrast can be changed. Specically, in the symmetric phase, = 1; in the non-symmetric phase,  < 1; and in the completely non-symmetric case with only one condensate ( d=d),= 0. Therefore, mea- surement of the contrast for dierent values of d andHcan give complete information about the phase diagram of the system, for comparison with the present theory. Fig.6 plots C,
andas functions of the lm thicknessdat xed magnetic eld H= 1 kOe. For5 smalldthe system is in the high-contrast sym- metric state. On increasing dabovedc= 0:07 m, the sign of
changes, corresponding to the transition from the symmetric to the low-contrast non-symmetric phase. As dfurther increases, to d= 0:17m,Cchanges sign, and the total phase  changes from to 0. Only at this point ddoes the zero-contrast phase (with only one condensate) appear. Correspondingly, a characteristic cusp in the contrast appears near d. FIG. 6. (a) Phase transition criterion
and inter- action coecient Cas functions of thickness dfor xed magnetic eld H= 1 kOe. (b) The contrast =j j2 maxj j2 min j j2max+j j2 minas a function of thickness dfor H= 1 kOe. S and NS denote symmetric and non- symmetric phase, respectively. To conclude, we have calculated the 4-th order magnon-magnon interactions in the condensate of a lm of YIG, including magnon-non-conserving terms that are responsible for the coherence of two condensates. sFor uciently thin YIG lms (d < 0:1) we predict a phase transition from symmetric to non-symmetric phase when the magnetic eld exceeds the modest value of 0:2 kOe. We also predict that within the non-symmetric phase there is a thickness d(H) where the modulation in the observed interference pattern should totally disappear. Methods Magnon Spectrum.| In a YIG lm with an in-plane external magnetic eld H, the magnon dispersion has been studied extensively [28{30]. At low energies, YIG can be described as a Heisenbergferromagnet with large eective-spin S= 14:3 [19, 24] on a cubic lattice. The Hamiltonian consists of three parts: H=JX hi;jiSi
Sj+HD HX iSz i;(8) the nearest neighbor exchange energy, the dipolar interaction and the Zeeman energy. We take yto be perpendicular to the lm and the magnetic eld to be in the plane along z. It is convenient to char- acterize the exchange interaction by the constant D= 2JSa2= 0:24 eV A2. The dipolar interaction can be calculated using the method indicated in Refs.[20, 30]. The competition between the dipo- lar interaction and exchange interaction leads to a magnon spectrum !kwith minima located at the two points in 2D wave-vector space given by k= (0;Q) (i.e. along z), with an energy gap
0. For lm thickness d= 5m and magnetic eld H= 1 kOe, we nd that Q= 7:5104cm1and
0= 2:7 GHz. In the experiment of [17] Qwas found to be about 3 :5104cm1, i.e. about half the predicted value. The reason for this may be associated with a rather shallow energy minimum as a function of wavevector. In such a situation small corrections to our approximate formula can have a large eect on the value of Q. The lowest band of the magnon spectrum can be calculated using the Holstein-Primako transformation [32], which expresses the spin operator Sin terms of boson operators aanday. To second order in aanday, the Hamiltonian eq.(8) is: H0=X kh Akay kak+1 2Bkakak+1 2B kay kay ki ;(9) with Ak= H0+Dk2+ 2M(1Fk) sin2+ 2MFk Bk= 2M(1Fk) sin2 2MFk (10) whereFk(1ekd)=kdandMis the magneti- zation of the material (4 M= 1:76 kG). Here,  is the angle between the 2D wave vector kand the magnetic eld direction ( z).H0of eq.(9) is diagonalized by the Bogoliubov transformation ak=ukck+vkcy kwithuk= (Ak+~!k 2~!k)1=2and vk=sgn(Bk)(Ak~!k 2~!k)1=2, leading to the magnon spectrum: ~!k= (A2 kjBkj2)1=2: (11)6 Fig.1 gives the magnon spectrum along kzfor typ- ical values of thickness dand magnetic eld H. Number of condensed magnons Nc= NQ+NQ.|Experimentally, the spin lattice relaxation time is of order 1 s, whereas the magnon-magnon thermalization time is of order 100 ns; the magnons are long-lived enough to equi- librate before decaying, thus making BEC possible [11]. After the thermalization time the pumped magnons go to a quasi-equilibrium state with a non-zero chemical potential . The number of pumped magnons Np=N(T;)N(T;0), where N(T;) =VP k1 e(!k)=T1, is determined by the pumping power and the magnon lifetime. is a monotonically increasing function of Npbut can- not exceed the energy gap
0. Therefore, on fur- ther pumping =
0and some of the pumped magnons fall into the condensate. The equation Npc=N(T;
0)N(T;0) thus denes the crit- ical line for condensation. Since
0Tand NpN(T;0) this equation can be satised at a rather high temperature. The total number of condensed particles is [11, 31] Nc=NpN(T;
0)+N(T;0) =N(T;)N(T;
0): (12) In exactly 2D systems BEC formally does not ex- ist since in the continuum approximation the sum inN(T;) diverges. However, for strong enough pumping the chemical potential approaches ex- ponentially close to the energy gap:
0
0exp(Np=N0), whereN0=VTm= ~2. For Np=N0>ln(T=
0) all pumped magnons occupy only one or two states Q. Eq.(12) determines only the total number of par- ticles in the condensate. The distribution of the condensate particles between the two minima re- mains undetermined in the quadratic approxima- tion. To resolve this issue we have shown that it is necessary to consider the fourth order terms in the Holstein-Primako expansion of the exchange and dipolar energy. Observe that terms of third order occur in this expansion of the dipolar interaction, but since the total momentum must be zero, such terms vanish for the condensate momenta (0 ;Q). Zero Sound.| We now provide details about calculating the zero sound spectrum. We consider small deviations from the static symmetric solution nQ=nQ=nc=2,+== 0, so that nQ=nc=2 +nwithn+=n=n=2 and+===2. Then E=Z dr~2 2m(jr +j2+jr j2)) +AV(j +j4+j j4+ 2BVj +j2j j2 +CV( + +  +  )(j +j2+j j2) ; On linearizing, the energy reads: E=Z dr~2 4mncjrnj2+~2nc 4mjrj2+
V 2n2 : Using the commutation relation [ ;n ] =i, and the equation of motion i~_= [;H ], we obtain: ~@ @t=~2 2mncr2n+
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2013-02-25

Recently, magnons, which are quasiparticles describing the collective motion of spins, were found to undergo Bose-Einstein condensation (BEC) at room temperature in films of Yttrium Iron Garnet (YIG). Unlike other quasiparticle BEC systems, this system has a spectrum with two degenerate minima, which makes it possible for the system to have two condensates in momentum space. Recent Brillouin Light scattering studies for a microwave-pumped YIG film of thickness d=5 $\mu$m and field H=1 kOe find a low-contrast interference pattern at the characteristic wavevector $Q$ of the magnon energy minimum. In this report, we show that this modulation pattern can be quantitatively explained as due to non-symmetric but coherent Bose-Einstein condensation of magnons into the two energy minima. Our theory predicts a transition from a high-contrast symmetric phase to a low-contrast non-symmetric phase on varying the $d$ and $H$, and a new type of collective oscillations.

Phase Diagram for Magnon Condensate in Yttrium Iron Garnet Film

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